Datasets:

aluncstokes

/

alg-geom.temp

like 0

9610

alg-geom/9610014

en

https://arxiv.org/abs/alg-geom/9610014

alg-geom/9610014

Hans Boden

H. U. Boden and K. Yokogawa

Moduli Spaces of Parabolic Higgs Bundles and Parabolic K(D) Pairs over Smooth Curves: I

25 pages, figure corrected

Internat. J. Math. 7 (1996) 573-598

10.1142/S0129167X96000311

This paper concerns the moduli spaces of rank two parabolic Higgs bundles and parabolic K(D) pairs over a smooth curve. Precisely which parabolic bundles occur in stable K(D), pairs and stable Higgs bundles is determined. Using Morse theory, the moduli space of parabolic Higgs bundles is shown to be a non-compact, connected, simply connected manifold, and a computation of its Poincar\'e polynomial is given.

alg-geom

\section{Introduction} Let $C$ be a compact curve. The correspondence between unitary representations of $\pi_1(C)$ and semistable bundles over $C$ of degree zero \cite{narasimhan-seshadri} was extended to non-compact curves $C_0$ by Mehta and Seshadri in \cite{mehta-seshadri}. If $C_0$ has compactification $C,$ they prove that semistable parabolic bundles over $C$ of parabolic degree zero correspond to unitary representations of $\pi_1(C_0)$ with fixed holonomy around $p \in C \setminus C_0.$ {Generalizing in a different direction, Hitchin and Donaldson \cite{hitchin, donaldson3} proved that representations of $\pi_1(C)$ correspond to semistable Higgs bundles over $C$ of degree zero.}\footnote{The non-abelian Hodge theorem, a further generalization of this, holds for arbitrary compact K\"ahler manifolds \cite{donaldson1, donaldson2, uhlenbeck-yau, corlette,simp1,simp4}.} A Higgs bundle includes the additional information of a Higgs field, which is a holomorphic map $\Phi : E \rightarrow E \otimes K,$ where $K$ denotes the canonical bundle. In the case of a parabolic bundle, the Higgs field is permitted to have poles of order one at the compactification points. Requiring these residues to be either {\it parabolic} or {\it nilpotent}, one obtains two moduli spaces: ${\cal P}_\aa,$ the moduli space of parabolic $K(D)$ pairs, and ${\cal N}_\aa,$ the moduli space of parabolic Higgs bundles. The subscript $\alpha$ refers to a particular choice of weights. In \cite{yoko1}, ${\cal P}_\aa$ is constructed using Geometric Invariant Theory and is proved to be a normal, quasi-projective variety. In \cite{konno}, ${\cal N}_\aa$ is constructed as a hyperk\"ahler quotient using gauge theory. Simpson's factorization theorem states that for $X$ a projective algebraic variety, any $SL(2, {\Bbb C})$ representation of $\pi_1(X)$ with Zariski dense image is either rigid or factors through an algebraic map from $X$ to an orbicurve \cite{simp3}. Because orbicurve representations can be interpreted as stable parabolic Higgs bundles \cite{scheinost}, it is important to understand these moduli spaces, which is the subject of our study here. Given a rank two parabolic bundle, we first establish algebraic conditions for the existence of a field making it stable as either a $K(D)$ pair or a Higgs bundle. One could use this to describe both moduli spaces, which we do for one particular case, but this approach appears too complicated to work in general. For that reason, we shift gears and study the topological properties of the moduli space of parabolic Higgs bundles, using the approach of Hitchin \cite{hitchin}. There is a circle action on ${\cal N}_\aa$ preserving its complex and symplectic structure and the associated moment map is a Morse function in the sense of Bott. We prove that ${\cal N}_\aa$ is a {non-compact}\footnote{The exception to this case is studied in \S 2.3}, connected, simply connected manifold and compute its Betti numbers, which turn out to be independent of the weights $\aa.$ This is surprising because it is not true for non-Higgs bundles: the Betti numbers of the moduli space ${\cal M}_\aa$ of parabolic bundles do depend in an essential way on $\aa$ (cf. \cite{bh}). In the sequel, we plan to extend these results to higher rank bundles. The paper is organized as follows. In \S 2.1 we define parabolic bundles with auxiliary fields and introduce the three moduli spaces ${\cal M}_\alpha, \, {\cal N}_\alpha,$ and ${\cal P}_\alpha.$ Tensor products, duals, and the Serre duality theorem for parabolic bundles are given in \S 2.2. In \S 3.1, we establish the algebraic conditions mentioned above, and in \S 3.2, we use these to characterize ${\cal P}_\alpha$ and ${\cal N}_\alpha$ in the case of ${\Bbb P}^1$ with three parabolic points. Turning our attention to ${\cal N}_\alpha$ in \S 4, we describe its construction in \S 4.1 as a hyperk\"ahler quotient, following \cite{konno}. In \S 4.2 we define the Morse function on ${\cal N}_\alpha^0$ and then prove our main results about the topology of ${\cal N}_\alpha^0$ in \S\S 4.3 and 4.4. Both authors are grateful to the Max-Planck-Institut f\"ur Mathematik for providing a stimulating intellectual environment as well as financial support. Warm thanks also to the VBAC Research Group of Europroj for travel funding, and to O. Garcia-Prada, L. G\"ottsche, N. Hitchin, Y. Hu, and D. Huybrechts for their advice. After submitting this paper, we learned that Nasatyr and Steer have obtained similar results studying orbifold Higgs bundles \cite{nasatyr-steer}. \section{Definitions and Preliminary Results} \subsection{Three moduli spaces} Let $X$ be a smooth curve of genus $g$ with $n$ marked points in the reduced divisor $D=p_1+\cdots + p_n$ and $E$ a holomorphic bundle over $X.$ \begin{defn} A parabolic structure on $E$ consists of weighted flags \begin{eqnarray*} &E_{p}=F_1(p) \supset \cdots \supset F_{s_p}(p) \supset 0&\\ &0\leq \alpha_1(p) < \cdots <\alpha_{s_p}(p) < 1& \end{eqnarray*} over each $p \in D.$ A holomorphic map $\phi:E^1 \longrightarrow E^2$ between parabolic bundles is called parabolic if $\alpha^1_i(p) > \alpha^2_j(p)$ implies $\phi(F^1_i(p)) \subset F^2_{j+1}(p)$ for all $p \in D.$ We call $\phi$ strongly parabolic if $\alpha^1_i(p) \geq \alpha^2_j(p)$ implies $\phi(F^1_i(p)) \subset F^2_{j+1}(p)$ for all $p \in D.$ \label{defn:pbundle} \end{defn} We use $E_*$ to denote the bundle together with a parabolic structure. Also, we use $\operatorname{ParHom}(E^1_*,E^2_*)$ and $\operatorname{ParHom}(E^1_*,\widehat{E}^2_*)$ to denote the sets of parabolic and strongly parabolic morphisms from $E^1$ to $E^2,$ respectively. (The decorative notation will become clear in \S 2.2.) If $\alpha^1_i(p) \neq \alpha^2_j(p)$ for all $i,j$ and $p \in D,$ then a parabolic morphism is automatically strongly parabolic. On the other hand, using the notation $\operatorname{ParEnd}(E_*) = \operatorname{ParHom}(E_*,E_*)$ and $\operatorname{ParEnd}^{\wedge}(E_*) = \operatorname{ParHom}(E_*,\widehat{E}_*),$ then strongly parabolic endomorphisms are nilpotent with respect to the flag data at each $p \in D.$ Let $K$ denote the canonical bundle of $X$ and give $E\otimes K(D)$ the obvious parabolic structure. \begin{defn} A parabolic $K(D)$ pair is a pair $(E, \Phi)$ consisting of a parabolic bundle $E$ and a parabolic map $\Phi:E \rightarrow E \otimes K(D).$ Such a pair is called a parabolic Higgs bundle if, in addition, $\Phi$ is a strongly parabolic morphism. \end{defn} Viewing $\alpha$ as a vector-valued function on $D,$ we use it as an index to indicate the parabolic structure on $E_*.$ Let $m_i(p) =\dim(F_i(p))-\dim(F_{i+1}(p)),$ the multiplicity of $\alpha_i(p),$ and $f_p = \frac{1}{2}(r^2 - \sum_{i=1}^{s_p} (m_i(p))^2),$ the dimension of the associated flag variety. Define the parabolic degree and slope of $E_*$ by \begin{eqnarray*} \operatorname{pardeg} E_* &=& \deg E + \sum_{p \in D} \sum_{i=1}^{s_p} m_i(p) \alpha_i(p),\\ \mu (E_*) &=& {{\operatorname{pardeg} E_*}\over{\operatorname{rank} E}}. \end{eqnarray*} If $L$ is a subbundle of $E,$ then $L$ inherits a parabolic structure from $E$ by pullback. We call the bundle $E_*$ {\it stable} ({\it semistable}) if, for every proper subbundle $L$ of $E,$ we have $\mu (L_*) < \mu (E_*)$ (respectively $\mu (L_*) \leq \mu (E_*)$). Likewise, we will call a parabolic $K(D)$ pair $(E_*,\Phi)$ {\it stable} (or {\it semistable}) if the same inequalities hold on those proper subbundles $L$ of $E$ which are, in addition, $\Phi$-invariant. Denote by ${\cal M}_\alpha$ the moduli space of $\alpha$-semistable parabolic bundles, by ${\cal N}_\alpha$ the moduli space of $\alpha$-semistable parabolic Higgs bundles, and by ${\cal P}_\alpha$ the moduli space of $\alpha$-semistable parabolic $K(D)$ pairs. By \cite{mehta-seshadri}, ${\cal M}_\alpha$ is a normal, projective variety of dimension $$\dim {\cal M}_\alpha = (g-1)r^2+1+\sum_{p \in D} f_p .$$ (If $g=0,$ this holds only when ${\cal M}_\alpha \neq \emptyset.$) Further, in \cite{yoko1,yoko2}, ${\cal P}_\alpha$ is shown to be a normal, quasi-projective variety of dimension $$\dim {\cal P}_\alpha = (2g-2+n)r^2+1$$ which contains ${\cal N}_\alpha$ as a closed subvariety of ${\cal P}_\alpha$ of dimension $$\dim {\cal N}_\alpha = 2(g-1)r^2+2+2\sum_{p \in D} f_p.$$ For generic $\alpha,$ a bundle (or pair) is $\alpha$-semistable $\Leftrightarrow$ it is $\alpha$-stable. In these cases, the moduli spaces ${\cal M}_\alpha, {\cal N}_\alpha$ and ${\cal P}_\alpha$ are smooth and can be described topologically as certain quotients of the gauge group ${\cal G}^{\Bbb C}=\operatorname{ParAut}(E_*).$ The same is true for ${\cal M}^0_\alpha, {\cal N}^0_\alpha$ and ${\cal P}^0_\alpha,$ the moduli spaces with fixed determinant and trace-free $\Phi.$ In this way, it is shown in \cite{konno} that ${\cal N}^0_\alpha$ is, for generic $\alpha,$ a smooth, hyperk\"{a}hler manifold of complex dimension $$\dim {\cal N}^0_\alpha = 2(g-1)(r^2-1)+2\sum_{p \in D} f_p .$$ \subsection{Parabolic sheaves and Serre duality} Some of the material in this section is a summary of results in \cite{yoko2} Suppose now that $E$ is a locally free sheaf on $X$ and $D=p_1+\cdots + p_n$ is a reduced divisor. \begin{defn} A parabolic structure on $E$ consists of a weighted filtration of the form \begin{eqnarray*} &E=E_0 = E_{\alpha_1} \supset \cdots \supset E_{\alpha_l} \supset E_{\alpha_{l+1}} = E(-D),&\\ &0=\alpha_0\leq \alpha_1 < \cdots < \alpha_l < \alpha_{l+1}=1.& \end{eqnarray*} We can define $E_x$ for $x \in [0,1]$ by setting $E_x = E_{\alpha_i}$ if $\alpha_{i-1} < x \leq \alpha_i,$ and then extend to $x \in {\Bbb R}$ by setting $E_{x+1} =E_{x}(-D).$ We call the resulting filtered sheaf $E_*$ a parabolic sheaf. We define the coparabolic sheaf $\widehat{E}_*,$ by $$\widehat{E}_x = \left\{\begin{array} {ll} E_x & \hbox{ if } x \neq \alpha_i\\ E_{\alpha_{i+1}} & \hbox{ if } x = \alpha_i. \end{array} \right.$$ A morphism of parabolic sheaves $\phi:E^1_*\rightarrow E^2_*$ is a called parabolic if $\phi(E^1_x) \subseteq E^2_x$ and strongly parabolic if $\phi(E^1_x) \subseteq \widehat{E}^2_x$ for all $x \in {\Bbb R}$. \end{defn} \begin{figure}[b] \begin{picture}(-80,100)(250,0) \put(-20,50){$E_*$} \put(0,10){\line(1,0){160}} \put(20,0){\line(0,1){90}} \put(120,0){\line(0,1){90}} \put(22,0){$0$} \put(122,0){$1$} \put(30,85){$E$} \put(0,80){\line(1,0){38}} \put(40,80){\circle*{4}} \put(40,0){$\aa_1$} \multiput(40,10)(0,4){12}{\line(0,1){2}} \multiput(40,78)(0,-4){4}{\line(0,-1){2}}\put(60,65){$E_{\aa_2}$} \put(42,60){\line(1,0){28}} \put(40,60){\circle{4}} \put(70,60){\circle*{4}} \put(70,0){$\aa_2$} \multiput(70,10)(0,4){7}{\line(0,1){2}} \multiput(70,58)(0,-4){4}{\line(0,-1){2}}\put(72,40){\line(1,0){28}} \put(90,45){$E_{\aa_3}$} \put(70,40){\circle{4}} \put(100,40){\circle*{4}} \put(100,0){$\aa_3$} \multiput(100,10)(0,4){5}{\line(0,1){2}} \multiput(100,38)(0,-4){2}{\line(0,-1){2}} \put(102,30){\line(1,0){38}} \put(100,30){\circle{4}} \put(140,30){\circle*{4}} \put(125,35){$E(-D)$} \put(140,0){$1+\aa_1$} \multiput(140,10)(0,4){2}{\line(0,1){2}} \multiput(140,28)(0,-4){2}{\line(0,-1){2}} \put(142,20){\line(1,0){18}} \put(140,20){\circle{4}} \put(180,50){$\Rightarrow$} \put(200,50){$\widehat{E}_*$} \put(220,10){\line(1,0){160}} \put(240,0){\line(0,1){90}} \put(340,0){\line(0,1){90}} \put(242,0){$0$} \put(342,0){$1$} \put(250,85){$E$} \put(220,80){\line(1,0){38}} \put(260,80){\circle{4}} \put(260,0){$\aa_1$} \multiput(260,10)(0,4){17}{\line(0,1){2}} \put(280,65){$E_{\aa_2}$} \put(260,60){\line(1,0){28}} \put(260,60){\circle*{4}} \put(290,60){\circle{4}} \put(290,0){$\aa_2$} \multiput(290,10)(0,4){12}{\line(0,1){2}} \put(290,40){\line(1,0){28}} \put(310,45){$E_{\aa_3}$} \put(290,40){\circle*{4}} \put(320,40){\circle{4}} \put(320,0){$\aa_3$} \multiput(320,10)(0,4){7}{\line(0,1){2}} \put(320,30){\line(1,0){38}} \put(320,30){\circle*{4}} \put(360,30){\circle{4}} \put(345,35){$E(-D)$} \put(360,0){$1+\aa_1$} \multiput(360,10)(0,4){5}{\line(0,1){2}} \put(360,20){\line(1,0){18}} \put(360,20){\circle*{4}} \end{picture} \caption{The simple relationship between $E_*$ and $\widehat{E}_*.$} \end{figure} We shall denote by $\operatorname{{\frak P}{\frak a}{\frak r}{\frak H}{\frak o}{\frak m}}(E^1_*, E^2_*)$ and $\operatorname{{\frak P}{\frak a}{\frak r}{\frak H}{\frak o}{\frak m}}(E^1_*, \widehat{E}^2_*)$ the sheaves of parabolic and strongly parabolic morphisms, and by $\operatorname{ParHom}(E^1_*,E^2_*)$ and $\operatorname{ParHom}(E^1_*, \widehat{E}^2_*)$ their global sections. We now show that there is an equivalence of the categories of parabolic bundles on $X$ and parabolic sheaves on $X.$ Given a parabolic bundle $E$ with flags and weights as in Definition \ref{defn:pbundle}, we define the filtered sheaf $E_*$ following Simpson \cite{simp5}. For $p \in D$ and $\alpha_{i-1}(p) < x \leq\alpha_i(p)$, set \begin{eqnarray*} E_x^p &=& \ker(E\rightarrow E_{p}/F_i(p)),\\ E_x &=& \bigcap_{p \in D} E^p_x. \end{eqnarray*} Now extend to all $x$ by $E_{x+1} = E_x(-D).$ Conversely, given a parabolic sheaf $E_*,$ the quotient $E/E_1$ is a skyscraper sheaf with support on $D$ and, for each $p \in D,$ we get weighted flags in $E_p$ by intersecting with the filtration at $p.$ To be precise, let $\alpha_1(p), \ldots, \alpha_{s_p}(p)$ be the subset of weights such that \begin{equation} \alpha_{i-1}(p) < x \leq \alpha_{i}(p) \Leftrightarrow (E_x/E_1)_p = (E_{\alpha_{i}(p)}/E_1)_p. \label{eqn:1} \end{equation} Setting $F_i(p) = (E_{\alpha_{i}(p)}/E_1)_p,$ we obtain a parabolic bundle in the sense of Definition \ref{defn:pbundle}. Suppose now $E^1$ and $E^2$ are parabolic bundles and $\phi\in\operatorname{ParHom}(E^1,E^2).$ We want to show that $\phi$ induces a morphism of the parabolic sheaves. So, suppose $\alpha^1_{i-1}(p) < x \leq\alpha^1_i(p)$ and $\alpha^2_{j-1}(p) < x \leq\alpha^2_j(p).$ Since $\alpha^1_i(p) > \alpha^2_{j-1}(p),\; \phi(F^1_i(p)) \subset F^2_j(p)$ and we see that $\phi$ maps $\ker(E^1\rightarrow E^1_{p}/F^1_i(p))$ to $\ker(E^2\rightarrow E^2_{p}/F^2_i(p))$ for all $p \in D,$ from which it follows that $\phi$ induces a map $\phi:E^1_x \rightarrow E^2_x.$ Suppose conversely that $E^1_*$ and $E^2_*$ are parabolic sheaves, $\phi \in \operatorname{ParHom}(E^1_*,E^2_*)$ and $\alpha^1_i(p) > \alpha^2_j(p).$ Set $x=\alpha^1_i(p)$ and $y=\alpha^2_{j+1}(p)$ for notational convenience. Then $\phi(E^1_x) \subset E^2_x.$ Since $x > \alpha^2_j(p),$ it follows from (\ref{eqn:1}) that $(E^2_x/E^2_1)_p \subset (E^2_{y}/E^2_1)_p$ and hence $\phi(F^1_i(p)) \subset F^2_{j+1}(p).$ It is not hard to see the same correspondence for strongly parabolic morphisms. Thus, we have an equivalence of the categories of parabolic bundles and parabolic sheaves. We use the definitions interchangeably and denote by $E_*$ a parabolic bundle or sheaf, reserving $E = E_0$ for the underlying holomorphic bundle. \medskip For the convenience of readers, we briefly summarize the results in \cite{yoko2} dealing with exact sequences and tensor products of parabolic sheaves. This is necessary for the statement of Serre duality for parabolic bundles, which is a tool we use throughout the paper. The category of parabolic sheaves ${\cal P}$ is not abelian, but is contained in an abelian category $\widetilde{{\cal P}}$ as a full subcategory. Objects in $\widetilde{{\cal P}}$ are also written by $E_*$ and a morphism $f:E_*^1\rightarrow E_*^2$ is a family of morphisms $f_x:E_x^1\rightarrow E_x^2.$ A coparabolic sheaf $\widehat{E}_*$ is realized in $\widetilde{{\cal P}}$. The set $\operatorname{ParHom}(E^1_*, \widehat{E}^2_*)$ is just the set of morphisms in $\widetilde{{\cal P}}.$ In $\widetilde{{\cal P}},$ a sequence \begin{equation} 0 \longrightarrow L_* \longrightarrow E_* \longrightarrow M_* \longrightarrow 0 \label{eqn:a} \end{equation} is exact if and only if the induced sequence at $x$ is exact for all $x\in {\Bbb R}.$ \medskip \noindent {\it Remark. \,} If the sequence (\ref{eqn:a}) is exact, then so is the sequence obtained by tensoring (\ref{eqn:a}) with any parabolic bundle (cf. Proposition 3.3 of \cite{yoko2}) and $$\operatorname{pardeg} E_* = \operatorname{pardeg} L_* +\operatorname{pardeg} M_*.$$ We can define dual parabolic sheaves $E_*^\vee$, parabolic tensor products $L_*\otimes M_*$, Hom-parabolic sheaves $\operatorname{{\frak P}{\frak a}{\frak r}{\frak H}{\frak o}{\frak m}}(L_*, M_*)_*$, and cohomology groups $\operatorname{Ext}^i(L_*,M_*)$. Clearly, $$\operatorname{pardeg} (L_*\otimes M_*) = \operatorname{rank}(M)\operatorname{pardeg} L_* + \operatorname{rank}(L)\operatorname{pardeg} M_*.$$ In addition, we have \begin{eqnarray*} \operatorname{Ext}^0(L_*,M_*) &=& H^0(L_*^\vee \otimes M_*) = H^0(\operatorname{{\frak P}{\frak a}{\frak r}{\frak H}{\frak o}{\frak m}}(L_*, M_*)) = \operatorname{ParHom}(L_*, M_*),\\ \operatorname{Ext}^1(L_*,M_*) &=& H^1(L_*^\vee \otimes M_*) = H^1(\operatorname{{\frak P}{\frak a}{\frak r}{\frak H}{\frak o}{\frak m}}(L_*,M_*)). \end{eqnarray*} We can identify $\operatorname{Ext}^1(M_*,L_*)$ with the set of equivalence classes of exact sequences of type (\ref{eqn:a}). The Serre duality theorem is generalized as follows (see Proposition 3.7 of \cite{yoko2}). \begin{prop}\label{serre-dual-thm} For parabolic sheaves $L_*$ and $M_*$, there is a natural isomorphism $$ \theta^i:H^i(L_*^\vee \otimes M_* \otimes K(D)) \stackrel{\simeq}{\longrightarrow} H^{1-i}(M_*^\vee \otimes \widehat{L}_*)^\vee.$$ \end{prop} Given $E_*$ and $\beta \in {\Bbb R}^n,$ define $E_*[\beta]_*,$ the parabolic sheaf $E_*$ shifted by $\beta,$ by $$E_*[\beta]_x = \bigcap_{i} E^{p_i}_{x+\beta_i}.$$ {\it Example. \;} {\it The Picard group of parabolic line bundles.}\\ A holomorphic bundle $E$ is regarded as a parabolic bundle with the trivial parabolic structure $E_p\supset 0, \aa_1(p)=0$ at each $p \in D.$ We call this the special structure on $E$. Note that every parabolic line bundle $L_*$ is gotten by shifting the special structure on the underlying bundle $L,$ i.e., there is a unique $\beta \in [0,1)^n$ with $L_*=L[\beta]_*$ Viewing ${\cal O}_X$ as a parabolic bundle with the special structure, then it is not difficult to verify that \begin{equation}\label{eqn:b} E_*[\beta]_* = E_* \otimes {\cal O}_X[\beta]_* \end{equation} Let $e_i$ denote the standard basis vector in ${\Bbb R}^n.$ From (\ref{eqn:b}) we have \begin{eqnarray*} E^1_*[\beta^1]_* \otimes E^2_*[\beta^2]_* &=& E^1_*\otimes E^2_*[\beta^1 + \beta^2]_*, \\ E_*[\beta]_*^\vee &=& E^\vee_*[-\beta]_*, \\ E_*[e_i]_* &=& E_*\otimes{\cal O}_X(-p_i). \end{eqnarray*} These three formulas determine the Picard group of parabolic line bundles on $X$. \medskip \noindent {\it Remark. \,} For any parabolic line bundle $L_*$, the stability (or semistability) of $E_*\otimes L_*$ is equivalent to that of $E_*.$ Similarly, the stability (or semistability) of $(E_*\otimes L_*,\Phi\otimes 1)$ is equivalent to that of $(E_*, \Phi).$ \medskip In particular, apply this to the case of a rank two parabolic bundle $E_*$ with full flags at each $p_i$ and weights $0 \leq \alpha_1(p_i) < \alpha_2(p_i) < 1.$ Using equation (\ref{eqn:b}) with $\beta_i =\frac{1}{2}(\aa_1(p_i)+\aa_2(p_i)-1),$ notice that $E_*[\beta]_*$ has weights $0 < a_1(p_i) < 1-a_1(p_i) < 1$ at $p_i,$ where $a_1(p_i) = \frac{1}{2}(\aa_1(p_i)-\aa_2(p_i)+1).$ \section{An Algebraic Description of the Moduli Spaces in Rank Two} \subsection{Criteria for the existence of stabilizing fields} In this section, we suppose that $E_*$ is a parabolic bundle of rank two with the weights $\aa_i \leq 1 - \aa_i$ at $p_i$ and that $n \geq 1.$ Consider the following existence questions: \begin{enumerate} \item[(I)] Does there exist $\Phi : E_* \rightarrow E_* \otimes K(D)$ with $(E_*, \Phi)$ stable? \item[(II)] Does there exist $\Phi : E_* \rightarrow \widehat{E}_* \otimes K(D)$ with $(E_*, \Phi)$ stable? \end{enumerate} Such $\Phi$ are called stabilizing fields. Of course, if $E_*$ is itself stable, then any $\Phi$ (e.g., $\Phi =0$) gives us an affirmative answer. The other possibilities are if $E_*$ is unstable (meaning not semistable) or if $E_*$ is strictly semistable. In either case, by choosing $L_*$ a line subbundle of maximal parabolic degree, we get a short exact sequence \begin{equation} \label{eq:ext1} 0 \longrightarrow L_* \stackrel{i}\longrightarrow E_* \stackrel{p}\longrightarrow M_* \longrightarrow 0 \end{equation} with $\mu(L_*) \geq \mu(E_*).$ Let $\xi \in H^1(M^\vee_* \otimes L_*)$ be the extension class representing (\ref{eq:ext1}). If $E_*$ is unstable, then $\mu(L_*) > \mu (E_*)$ and (\ref{eq:ext1}) is the Harder-Narasimhan filtration of $E_*$ and is canonical. If $E_*$ is strictly semistable, then $\mu(L_*) = \mu(E_*)$ and (\ref{eq:ext1}) is the Jordan-H\"{o}lder filtration of $E_*$ and is not, in general, canonical. For example, if $E_*$ is strictly semistable, then the subbundle $L_*$ is canonically determined if and only if the extension $\xi$ is nontrivial. In the following proposition, the assumption $g \geq 2$ is not essential and after the proof, we treat the case $g \leq 1.$ \begin{prop}\label{prop:list-st} If $g \geq 2$ and $E_*$ is not stable, then \begin{enumerate} \item[(i)] $(E_*, \Phi)$ is a stable parabolic $K(D)$ pair for some $\Phi \Leftrightarrow h^1(M^\vee_* \otimes \widehat{L}_*) \geq 1;$ \item[(ii)] $(E_*, \Phi)$ is a stable parabolic Higgs bundle for some $\Phi \Leftrightarrow h^1(M^\vee_* \otimes L_*) > 1$ or $h^1(M^\vee_* \otimes L_*) = 1$ and $\xi = 0.$ \end{enumerate} \end{prop} \begin{pf} Notice first of all that if such a $\Phi$ exists, then we can assume it is trace-free. Now consider the short exact sequences of the sheaves of parabolic and strongly parabolic bundle endomorphisms \begin{eqnarray} \label{eq:ses1} &0 \rightarrow E_*^{\vee} \otimes L_* \otimes K(D) \stackrel{\iota}{\longrightarrow} E_*^{\vee} \otimes_0 E_* \otimes K(D) \stackrel{\pi}{\longrightarrow} L_*^{\vee} \otimes M_* \otimes K(D) \rightarrow 0,& \\ \label{eq:ses2} &0 \rightarrow E_*^{\vee} \otimes \widehat{L}_* \otimes K(D) \stackrel{\hat{\iota}}{\longrightarrow} E_*^{\vee} \otimes_0 \widehat{E}_* \otimes K(D) \stackrel{\hat{\pi}}{\longrightarrow} L_*^{\vee} \otimes \widehat{M}_* \otimes K(D) \rightarrow 0,& \end{eqnarray} where $\pi, \hat{\pi}$ are the natural surjections, $\iota, \hat{\iota}$ are the natural isomorphisms to the kernels of $\pi, \hat{\pi}$ and $$E^\vee_* \otimes_0 E_* = \operatorname{{\frak P}{\frak a}{\frak r}{\frak E}{\frak n}{\frak d}}_0(E_*)$$ denotes the sheaf of trace-free endomorphisms of $E_*.$ Notice that $H^0(E_*^{\vee} \otimes L_* \otimes K(D))$ and $H^0(E_*^{\vee} \otimes \widehat{L}_* \otimes K(D))$ are the relevant subspaces of fields $\Phi$ for which $L_*$ is a $\Phi$-invariant subbundle. If $(E_*, \Phi)$ is stable, then $L_*$ is not $\Phi$-invariant, and $\pi_*(\Phi) \neq 0$ (similarly for $\hat{\pi}_*(\Phi)$). This proves one implication of the following claim. \begin{clm} \label{claim:1} Suppose that either $E_*$ is unstable or $\xi \neq 0,$ then \begin{enumerate} \item[(i)] for $\Phi \in H^0(E_*^{\vee} \otimes_0 E_* \otimes K(D)), (E_*, \Phi)$ is stable $\Leftrightarrow 0 \neq \pi_*(\Phi);$ \item[(ii)] for $\Phi \in H^0(E_*^{\vee} \otimes_0 \widehat{E}_* \otimes K(D)), (E_*, \Phi)$ is stable $\Leftrightarrow 0 \neq \hat{\pi}_*(\Phi).$ \end{enumerate} \end{clm} \begin{pf} To prove ($\Leftarrow$), we just show that $L_*$ is the unique parabolic subbundle of $E_*$ with $\mu(L_*) \geq \mu(E_*).$ Suppose $L'_*$ is another such subbundle. If $E_*$ is unstable, then $\mu(E_*) > \mu(M_*)$ and the projection $L'_* \rightarrow M_*$ is the zero map, which shows $L'_* = L_*$. On the other hand, if $L'_* \rightarrow M_*$ is not the zero map, then it is an isomorphism and defines a splitting of (\ref{eq:ext1}), hence $\xi =0.$ \end{pf} Now consider the coboundary maps in the cohomology sequences of (\ref{eq:ses1}) and (\ref{eq:ses2}) \begin{eqnarray*} &H^0(L_*^{\vee} \otimes M_*\otimes K(D)) \stackrel{\delta}{\longrightarrow} H^1(E_*^{\vee}\otimes L_*\otimes K(D)),& \\ &H^0(L_*^{\vee} \otimes \widehat{M}_*\otimes K(D)) \stackrel{\hat{\delta}}{\longrightarrow} H^1(E_*^{\vee}\otimes \widehat{L}_*\otimes K(D)).& \end{eqnarray*} Here $\delta$ is the zero map since by Serre duality $$h^1(E_*^{\vee}\otimes L_*\otimes K(D)) = h^0 (L_*^{\vee} \otimes \widehat{E}_*) = h^0(L_*^{\vee} \otimes \widehat{L}_*) + h^0(L_*^{\vee} \otimes \widehat{M}_*)=0 .$$ A diagram chase shows that the dual map of $\hat{\delta},$ $\hat{\delta}^\vee:H^0(L_*^\vee\otimes E_*)\rightarrow H^1(M_*^\vee\otimes L_*),$ maps $i$ to $\xi.$ Hence, $\hat{\delta}$ is the zero map if and only if $\xi =0.$ If $\xi \neq 0,$ then its image is one dimensional because $$h^1(E_*^{\vee}\otimes \widehat{L}_* \otimes K(D)) = h^0 (L_*^{\vee} \otimes E_*) = \begin{cases} 1 & \text{if $L_* \neq M_*$ or $\xi \neq 0,$} \\ 2 &\text{if $L_* = M_*$ and $\xi = 0.$} \end{cases}$$ In the cases covered by the claim, the proposition follows by another application of Serre duality $$h^0(L_*^{\vee}\otimes M_*\otimes K(D)) = h^1 (M_*^{\vee} \otimes \widehat{L}_*),$$ $$h^0(L_*^{\vee}\otimes \widehat{M}_* \otimes K(D)) = h^1 (M_*^{\vee} \otimes L_*).$$ The remaining cases follow by replacing the claim by the lemma below, which we note is the only step of the argument where we use the assumption $g \geq 2.$ \end{pf} \begin{lem} \label{lem:criterion} If $g \geq 2$ and $E_*$ is not stable, then \begin{enumerate} \item[(i)] $(E_*, \Phi)$ is a stable parabolic $K(D)$ pair for some $\Phi \Leftrightarrow \ker \delta \neq 0;$ \item[(ii)] $(E_*, \Phi)$ is a stable parabolic Higgs bundle for some $\Phi \Leftrightarrow \ker \hat{\delta} \neq 0.$ \end{enumerate} \end{lem} \begin{pf} Since the lemma is a consequence of the claim, when it applies, we can assume that $E_*$ is strictly semistable and $\xi =0.$ Furthermore, we only need to show ($\Leftarrow$). We introduce some notation. Define the intersection numbers $e_i$ and $\hat{e}_i$ by \begin{eqnarray*} e_i &=& \begin{cases} \dim L_{p_i} \cap F_2(p_i) & \text{if $F_2(p_i) \neq 0$,} \\ 1 & \text{if $F_2(p_i) =0,$} \end{cases}\\ \hat{e}_i &=& \dim L_{p_i} \cap F_2(p_i). \end{eqnarray*} If $\beta_i = \hat{e}_i + (-1)^{\hat{e}_i} \alpha_i$ and $\gamma_i= 1-\beta_i$ are the weights of $L_{p_i}$ and $M_{p_i},$ respectively, then $$\hat{e}_i = \begin{cases} 0 & \text{if $\beta_i \leq \gamma_i$,}\\ 1 & \text{if $\beta_i > \gamma_i$,} \end{cases} \quad \hbox{ and }\quad e_i = \begin{cases} 0 & \text{if $\beta_i < \gamma_i$,}\\ 1 & \text{if $\beta_i \geq \gamma_i$.} \end{cases}$$ Set $|e| = \sum e_i$ and $|\hat{e}| = \sum \hat{e}_i$ and notice that $e_i > \beta_i - \gamma_i$ and $\hat{e}_i \geq \beta_i - \gamma_i,$ with equality only when $\hat{e}_i = 0$ and $\beta_i = \gamma_i.$ If $\ker \delta \neq 0$ or $\ker \hat{\delta} \neq 0,$ then for generic $\Phi,$ $L_*$ is not $\Phi$-invariant. Suppose $L'_*$ ($\neq L_*$) is a line subbundle with $\mu(L'_*) \geq \mu(E_*).$ Semistability of $E_*$ implies $\mu(L'_*) = \mu(E_*).$ Then the restriction of $p$ to $L'_*,$ written $p_{L'} \, : \, L'_* \longrightarrow M_*,$ is an isomorphism since otherwise, $p_{L'} =0$ and $L'_* = L_*$. Such subbundles are identified with sections of $p$ and are parameterized by $H^0(M^\vee_* \otimes L_*).$ The relevant subspaces of $\Phi$ leaving $L'_*$ invariant are $H^0(E^\vee_* \otimes M_* \otimes K(D))$ and $H^0(E^\vee_* \otimes \widehat{M}_* \otimes K(D)).$ Thus, (i) will follow once we prove the inequality \begin{equation} \label{eq:ineq1} h^0(M^\vee_* \otimes L_*) + h^0(E^\vee_* \otimes M_* \otimes K(D))< h^0(E^\vee_* \otimes_0 E_* \otimes K(D)), \end{equation} which is equivalent to $h^0(M^\vee_* \otimes L_*) < h^0(M^\vee_* \otimes L_* \otimes K(D)).$ Likewise, (ii) will follow from \begin{equation} \label{eq:ineq2} h^0(M^\vee_* \otimes L_*) + h^0(E^\vee_* \otimes \widehat{M}_* \otimes K(D))< h^0(E^\vee_* \otimes \widehat{E}_* \otimes K(D)), \end{equation} which is equivalent to $h^0(M^\vee_* \otimes L_*) < h^0(M^\vee_* \otimes \widehat{L}_* \otimes K(D)).$ Since $\mu(M^\vee_* \otimes L_*) = 0,$ $$h^0(M^\vee_* \otimes L_*) = \begin{cases} 0 & \text{if $M_* \neq L_*,$} \\ 1 & \text{if $M_* = L_*.$} \end{cases}$$ On the other hand, because $h^1(M^\vee_* \otimes L_* \otimes K(D)) = h^0( L_*^\vee \otimes \widehat{M}_*) =0,$ it follows that \begin{eqnarray*} h^0(M^\vee_* \otimes L_* \otimes K(D)) &=& \deg (M^\vee \otimes L \otimes K({\textstyle{\sum}^n_{i=1}} e_i p_i)) + \chi(X)\\ &=& \deg L - \deg M +|e| + g-1. \end{eqnarray*} Notice that $\deg L - \deg M +|e| > \mu(L_*) - \mu(M_*) =0,$ hence (\ref{eq:ineq1}) holds provided \begin{equation} \label{eq:cond1} \deg L - \deg M +|e| \geq 2-g \hbox{ with equality } \Leftrightarrow L_* \neq M_*. \end{equation} This proves part (i) of the lemma when $g \geq 2.$ As for part (ii), notice that $$h^1(M^\vee_* \otimes \widehat{L}_* \otimes K(D)) = h^0( L_*^\vee \otimes M_*) = \begin{cases} 0 & \text{if $M_* \neq L_*,$} \\ 1 & \text{if $M_* = L_*,$} \end{cases}$$ and so (\ref{eq:ineq2}) follows as long as $\chi(M^\vee_* \otimes \widehat{L}_* \otimes K(D)) > 0.$ We have \begin{eqnarray*} \chi (M^\vee_* \otimes \widehat{L}_* \otimes K(D)) &=& \deg(M^\vee \otimes L \otimes K({\textstyle{\sum}^n_{i=1}} \hat{e}_i p_i)) + \chi(X)\\ &=& \deg L - \deg M +|\hat{e}| + g-1. \end{eqnarray*} Hence (\ref{eq:ineq2}) holds provided \begin{equation} \label{eq:cond2} \deg L - \deg M +|\hat{e}| \geq 2-g. \end{equation} But $\deg L - \deg M +|\hat{e}| \geq \mu(L_*) - \mu(M_*) = 0$ (with equality implying that $\beta_i = \gamma_i$ for all $i$). This proves part (ii) of the lemma when $g \geq 2.$ \end{pf} One can deduce the following corollary using Riemann-Roch. \begin{cor} If $g \geq 3,$ then for every semistable $E_*$, there exists a Higgs field $\Phi$ making $(E_*, \Phi)$ a stable parabolic Higgs bundle. \end{cor} We now explain how to extend these results to lower genus. Clearly, the proposition holds for $g \leq 1$ whenever $E_*$ is unstable or $\xi \neq 0$ by virtue of the claim. So assume that $E_*$ is semistable and $\xi = 0.$ The only place where we make essential use of the assumption $g \geq 2$ is in the proof of Lemma \ref{lem:criterion}. In particular, we observe from (\ref{eq:cond1}) and (\ref{eq:cond2}) that the inequalities (\ref{eq:ineq1}) and (\ref{eq:ineq2}) fail (respectively) if \begin{enumerate} \item[(i)] $0 < \deg L - \deg M +|e| \leq 2-g$ with equality $ \Leftrightarrow L_* = M_*,$ \item[(ii)] $0 \leq \deg L - \deg M +|\hat{e}| \leq 1-g.$ \end{enumerate} Thus, the only counterexamples to Lemma \ref{lem:criterion} for $g \leq 1$ are given by the semistable, split bundles $E_*$ satisfying (i) and (ii) along with the additional requirements (${\hbox{\rm i}}'$) $\ker \delta \neq 0$ and (${\hbox{\rm ii}}'$) $\ker \hat{\delta} \neq 0.$ First, we list these counterexamples to Lemma \ref{lem:criterion}, then we show that the bundles satisfying (i) and (ii) never give rise to any stable parabolic $K(D)$ pairs or stable parabolic Higgs bundles, respectively. If $E_*$ is semistable and split and satisfies (i) and (${\hbox{\rm i}}'$), i.e., if $h^1(M^\vee_* \otimes \widehat{L}_*) \geq 1,$ then there are but two possibilities: \begin{enumerate} \item[(i--a)] $(g,n) \in \{(0,2), (1,1)\},\, E_*= L_* \oplus M_*$ and $L_* = M_*,$ \item[(i--b)] $g=0,\, E_*= L_* \oplus M_*, \, \mu(L_*) = \mu(M_*), \, \deg L - \deg M + |e| = 1.$ \end{enumerate} Now if $E_*$ is semistable and split and satisfies (ii) and (${\hbox{\rm ii}}''$), i.e., if $h^1(M^\vee_* \otimes L_*) \geq 1,$ then again, we have only two possibilities: \begin{enumerate} \item[(ii--a)] $g=0,\, E_*= L_* \oplus M_*,\, \mu(L_*) = \mu(M_*),$ and $ 0 \leq \deg L - \deg M +|\hat{e}| \leq 1,$ \item[(ii--b)] $g=1,\, E_*= L_* \oplus M_*,\, L_* = M_*.$ \end{enumerate} We now show that if $E_*$ satisfies (i), then $(E_*, \Phi)$ is not stable for any $\Phi \in H^0(E^\vee_* \otimes_0 E_* \otimes K(D))$ and if $E_*$ satisfies (ii), then $(E_*, \Phi)$ is not stable for any $\Phi \in H^0(E^\vee_* \otimes_0 \widehat{E}_* \otimes K(D)).$ For example, suppose that $\deg L - \deg M +|e| = 2-g$ in (i), so that $L_* = M_*.$ Then either $g=0$ and $n=2$ or $g=1=n.$ In either case, $$H^0(E^\vee_* \otimes_0 E_* \otimes K(D)) = H^0(K(D))^{\oplus 3} = H^0({\cal O}_X)^{\oplus 3}.$$ Thus, any $\Phi$ is a constant matrix, one of whose eigenspaces determines a $\Phi$-invariant subbundle violating the condition for stability. Otherwise, if $\deg L - \deg M +|e| = 1-g$ in (i), then $h^0(M^\vee \otimes L_* \otimes K(D)) = 0$ so that $M_*$ is $\Phi$-invariant for all $\Phi.$ As for (ii), suppose first of all that $g=0$ and $\deg L - \deg M +|\hat{e}| \leq 1.$ Then $h^0(M^\vee_* \otimes \widehat{L}_* \otimes K(D)) =0$ and $M_*$ is $\Phi$-invariant for all $\Phi.$ Now if $g=1$ and $\deg L - \deg M +|\hat{e}| =0,$ then either $L_* \neq M_*$ and $M_*$ is $\Phi$-invariant for all $\Phi$ or $L_* = M_*$ and $H^0(E^\vee_* \otimes_0 \widehat{E}_* \otimes K(D)) = H^0({\cal O}_X)^{\oplus 3},$ in which case every $\Phi$ is a constant matrix. This proves the following proposition. \begin{prop} \label{prop:crit} If $E_*$ is not stable and $g \leq 1,$ then \begin{enumerate} \item[(i)] $(E_*, \Phi)$ is a stable parabolic $K(D)$ pair for some $\Phi \Leftrightarrow \, E_*$ is not one of the bundles occurring in \rom(i--a\rom) or \rom(i--b\rom) and $h^1(M^\vee_* \otimes \widehat{L}_*) \geq 1;$ \item[(ii)] $(E_*, \Phi)$ is a stable parabolic Higgs bundle for some $\Phi \Leftrightarrow \, E_*$ is not one of the bundles occurring in \rom(ii--a\rom) or \rom(ii--b\rom) and either $h^1(M^\vee_* \otimes L_*) > 1$ or $h^1(M^\vee_* \otimes L_*) = 1$ and $\xi = 0.$ \end{enumerate} \end{prop} We could ask questions (I) and (II) replacing stability with semistability. Of course, if $E_*$ itself is semistable, then so is $(E_*, \Phi)$ for any $\Phi.$ So we can assume that $E_*$ is unstable and apply the claim to determine precisely which $\Phi$ make $(E_*, \Phi)$ stable. One last comment is that if $(E_*, \Phi)$ is strictly semistable, then $E_*$ must also be strictly semistable. The converse, however, is false. \subsection{Example: Rank 2 parabolic bundles over ${\Bbb P}^1$ with 3 parabolic points} In this section, we describe the moduli spaces ${\cal M}_\alpha, {\cal N}_\alpha$ and ${\cal P}_\alpha$ of rank two bundles over $X={\Bbb P}^1$ with parabolic points in the reduced divisor $D=p_1+p_2+p_3.$ This case seems trivial as it turns out that ${\cal N}_\alpha$ is always just one point and that ${\cal P}_\alpha$ is always just the affine space ${\Bbb C}^5$. However, our complete description of this case sheds light on the general phenomenon that the moduli spaces ${\cal N}_\alpha$ and ${\cal P}_\alpha$ do not change when the weights are permitted to vary (even when ${\cal M}_\alpha$ becomes empty!). This trivial case is a prototype for such behavior. The simplest nontrivial cases are $X={\Bbb P}^1$ with 4 parabolic points and $X=C,$ an elliptic curve, with one parabolic point. In either case, ${\cal M}_\alpha,$ if nonempty, is ${\Bbb P}^1,$ and ${\cal N}_\alpha$ is a connected nonsingular noncompact surface containing the cotangent bundle of ${\Bbb P}^1$. There is a proper map from ${\cal N}_\alpha$ to ${\Bbb C}$ called the Hitchin map whose fibers over nonzero points $t \in {\Bbb C}$ are elliptic curves and whose fiber over $0$ is a union of five rational curves arranged in a $\widetilde{D}_4$ configuration. This case will be treated in the second part of this paper. We suppose that $\mu(E_*)=0$ and that the weights at $p_i$ are $\alpha_i$ and $1-\alpha_i$ for some $\alpha \in W = \{ (\alpha_1,\alpha_2,\alpha_3) \mid 0 < \alpha_i < \frac{1}{2} \}.$ Note that this is equivalent to saying that $\det E_* = {\cal O}_X$ (as parabolic bundles) and $E_*$ has full flags at each $p_i.$ For $e=(e_1,e_2,e_3),$ where $e_i \in \{0,1\},$ we use $\beta(\alpha,e)$ (or simply $\beta$) to denote the weights $\beta_i = e_i + (-1)^{e_i} \alpha_i.$ Let $$I = \{ (0,0,0), (0,1,1),(1,0,1),(1,1,0) \}.$$ Inside $W$ there are four hyperplanes $$H_e = \{ \alpha \mid \beta(\alpha,e) = 1 + \frac{|e|}{2} \}$$ for $e \in I$ whose complement $W \setminus \bigcup_{e \in I} H_e$ consists of five chambers: $C_e = \{ \alpha \mid \beta(\alpha,e) > 1 + \frac{|e|}{2} \}$ for $e \in I$ and $C_0 = \{ \alpha \mid \beta(\alpha,e) < 1 + \frac{|e|}{2} \hbox{ for all } e\in I\}.$ The following is an immediate consequence of the criteria established in the previous section. \begin{lem} If $(E_*, \Phi)$ is a semistable $K(D)$ pair, then the bundle $E_*$ is described as an extension \begin{equation} \label{eqn:ext2} 0 \longrightarrow L_* \longrightarrow E_* \longrightarrow L_*^\vee \longrightarrow 0 \end{equation} where $L_*$ satisfies $h^1(L_*^{\otimes 2})=1.$ \end{lem} \begin{pf} If $E_*$ is not stable, then by Proposition \ref{prop:crit}, we see that $h^1(L_*^{\otimes 2}) \geq 1.$ Since $\mu(L_*) \geq 0,$ we see that $\mu(L_*^{\otimes 2}) \geq 0,$ and because there are only three weights, this implies $\deg(L_*^{\otimes 2})_0 \geq -2.$ Thus $h^1(L_*^{\otimes 2}) = 1.$ If $E_*$ is stable, then by Grothendieck's Theorem, $E = {\cal O}(-1) \oplus {\cal O}(-2).$ Let $L_*$ be ${\cal O}(-1)$ with weights inherited as a subbundle of $E_*.$ Notice that $h^1(L_*^{\otimes 2}) \leq 1.$ But by stability of $E_*,$ the extension (\ref{eqn:ext2}) must be nontrivial, so $h^1(L_*^{\otimes 2}) \geq 1.$ \end{pf} We now determine all possible line subbundles $L_*$ with $h^1(L_*^{\otimes 2})=1.$ For fixed $\aa \in W,$ there are four possible line subbundles $L_*$ with $h^1(L_*^{\otimes 2})=1,$ namely $${L^e_*} = {\cal O}_X(-1-\frac{|e|}{2})[-\beta(\alpha,e)]$$ for $e \in I.$ We denote by $G^e_*$ the nontrivial extension gotten from (\ref{eqn:ext2}) with $L_*=L^e_*$. Notice that $G^e_*$ is unique up to isomorphism because $h^1({L^e_*}^{\otimes 2})=1.$ Let $F^e_*=L^e_*\oplus {L^e_*}^\vee.$ It is not hard to see that $G^e_*$ and $G^{e'}_*$ are isomorphic for $e, e' \in I.$ Set $G_* = G^e_*.$ This, together with the previous lemma, shows that if $(E_*, \Phi)$ is semistable, then $E_*$ is one of the five bundles in the set $\{ G_*, F^e_* \}.$ Recall that two bundles $E_*$ and $E'_*$ are called S-equivalent (written $E_* \sim_S E'_*$) if their associated graded bundles are isomorphic, i.e., if $\operatorname{gr} E_* \simeq \operatorname{gr} E'_*.$ We use $E_*$ to denote the isomorphism class of a bundle and $[E_*]$ for its S-equivalence class. \begin{prop} \begin{enumerate} \item If $\aa \in C_0,$ then ${\cal M}_\aa = \{{G_*}\}.$ \item If $\aa \in C_e, e \in I,$ then ${\cal M}_\aa = \emptyset.$ \item If $\aa \in H_e,$ then ${\cal M}_\aa = \{ [F^e_*] \}$ and ${G_*} \sim_S F^e_*$ are the two distinct isomorphism classes of semistable bundles. \end{enumerate} \end{prop} \begin{pf} From the above considerations, if $E_*$ is semistable, then $E_* =G_*$ or $F^e_*.$ But $G_*$ is stable if and only if $\aa \in C_0,$ and $F^e_*$ is never stable. On the other hand, if $\aa \in H_e,$ then $G_*$ and $F^e_*$ are clearly strictly semistable with associated graded bundle $L^e_* \oplus (L^e_*)^\vee.$ \end{pf} The next lemma shows which auxiliary fields can arise for these five bundles. \begin{lem} For any $\aa \in W,$ we have \begin{enumerate} \item[(i)] $G_*$ is simple, $h^0(G^\vee_* \otimes G_* \otimes K(D)) = 5,$ and $h^0(G^\vee_* \otimes \widehat{G}_* \otimes K(D))=0,$ \item[(ii)] $\operatorname{Aut} F^e_* = {\Bbb C}^*\! \times {\Bbb C}^*, \, h^0({F^e_*}^\vee \otimes F^e_* \otimes K(D)) = 5,$ and $h^0({F^e_*}^\vee \otimes \widehat{F}^e_* \otimes K(D))=1.$ \end{enumerate} \end{lem} \begin{pf} For $\aa \in C_0, \, G_*$ is stable, and therefore simple. But this property is independent of the weights, and it follows that for any $\aa \in W,$ \begin{eqnarray*} 1 &=& h^0(G_*^\vee\otimes G_*) = h^1(G_*^\vee\otimes \widehat{G}_* \otimes K(D)), \\ 0 &=& h^0(G_*^\vee\otimes \widehat{G}_*) = h^1(G_*^\vee\otimes{G}_* \otimes K(D)). \end{eqnarray*} Direct computation shows $\deg (G^\vee_* \otimes G_*)_0 = -3$ and $\deg (G^\vee_* \otimes \widehat{G}_*)_0 = -9,$ and part (i) follows using $K(D) = {\cal O}(1)$ and Riemann-Roch. As for (ii), since $({L^e_*}^{\otimes 2})_0 = {\cal O}(-2)$ and $({L^e_*}^{\otimes -2})_0 = {\cal O}(-1),$ every automorphism of $F^e_*$ is diagonal and $\operatorname{Aut} {F^e_*}= {\Bbb C}^*\! \times {\Bbb C}^*.$ Also, $h^0({L^e_*}^{\otimes 2}\otimes K(D))= 0,$ so every $\Phi\in H^0({F^e_*}^\vee\otimes F^e_*\otimes K(D))$ has the form $$ \Phi= \begin{pmatrix} a_1 & 0 \\ \phi & a_2 \end{pmatrix} $$ with $\phi\in H^0(L_*^{\otimes -2}\otimes K(D))={\Bbb C}$ and $a_i \in H^0(K(D))={\Bbb C}^2.$ Morevoer, $H^0({F^e_*}^\vee\otimes \widehat{F}^e_*\otimes K(D)) = H^0(L_*^{\otimes -2}\otimes K(D)),$ which completes the proof of part (ii) \end{pf} We can identify the action of $\operatorname{Aut}(F^e_*)$ on $H^0({F^e_*}^\vee\otimes F^e_*\otimes K(D)),$ it is given by conjugation $$(z_1,z_2) \cdot \Phi = \begin{pmatrix} z_1 & 0 \\0& z_2 \end{pmatrix} \begin{pmatrix} a_1 & 0 \\ \phi & a_2 \end{pmatrix} \begin{pmatrix} z_1^{-1} & 0 \\0& z_2^{-1} \end{pmatrix} = \begin{pmatrix} a_1 & 0 \\z_1^{-1}z_2 \phi & a_2 \end{pmatrix}.$$ Suppose that $\aa\in C_e$ and set $V=\operatorname{Ext}^1({L^e_*}^\vee,L^e_*)={\Bbb C}$. Let ${\cal E}_*$ be the universal parabolic bundle on $V \times X$ which, when restricted to $\{\xi\} \times X,$ is the bundle $G^\xi_*$ in (\ref{eqn:ext2}) with $L_* = L^e_*$ and extension class $\xi.$ For $\xi \neq 0, \, G^\xi_* \simeq G_*$ and obviously $G^0_* = F^e_*.$ Let $p_X$ and $p_V$ denote the two projection maps from $V \times X$ and define ${\cal L}_*$ to be the pullback bundle $p_X^* L^e_*.$ Consider the direct image sheaves of ${\cal E}^\vee_* \otimes {\cal E}_* \otimes K(D)$ and ${\cal L}_*^{\otimes -2} \otimes K(D)$ under $p_V,$ which, by the previous lemma, are locally free sheaves over $V$ whose associated vector bundles, $M$ and $N,$ are trivial with ranks 5 and 1, respectively. Notice that $N$ is canonically isomorphic to $V \times H^0({L^e_*}^{\otimes -2} \otimes K(D)).$ This is key to following construction. The canonical map $\tilde{\pi} : {\cal E}^\vee_* \otimes {\cal E}_* \otimes K(D) \rightarrow {\cal L}_*^{\otimes -2} \otimes K(D)$ of the previous section induces $ \tilde{\pi}_* : M \rightarrow N$ which is surjective, because the restriction of $\tilde{\pi}_*$ to a fiber above $\xi$ can be identified with $\pi^\xi_* : H^0({G^\xi_*}^\vee\otimes G^\xi_* \otimes K(D))\rightarrow H^0({L^e_*}^{\otimes -2}\otimes K(D)),$ whose cokernel is $H^1({G^\xi_*}^\vee \otimes L^e_* \otimes K(D)) = 0.$ Fix some $0 \neq \phi_0 \in H^0({L^e_*}^{\otimes -2}\otimes K(D))$ and set $Y = \tilde{\pi}_*^{-1}( \{\phi_0\} \times V) \simeq {\Bbb C}^4 \times V.$ \begin{prop} \begin{enumerate} \item If $\aa\in C_0,$ then ${\cal P}_\aa \simeq H^0(G_*^\vee\otimes G_* \otimes K(D)) = {\Bbb C}^5$ and ${\cal P}_\aa^0 \simeq {\Bbb C}^3$. \item If $\aa\in C_e,$ then ${\cal P}_\aa\simeq Y\simeq {\Bbb C}^5$ and ${\cal P}_\aa^0 = {\Bbb C}^3.$ \item If $\aa\in H_e,$ then ${\cal P}_\aa \simeq H^0(G_*^\vee\otimes {G_*} \otimes K(D)) = {\Bbb C}^5$ and its strictly semistable part can be identified with a hyperplane. \end{enumerate} \end{prop} \noindent {\it Remark.\,} In the course of the proof, we will determine the isomorphism classes of semistable parabolic $K(D)$ pairs. This differs from the above only for strictly semistable bundles, because the S-equivalence class of a stable bundle is precisely its isomorphism class. For $\aa \in H_e,$ we will find that there are three distinct components of isomorphism classes of strictly semistable bundles, each is just a copy of ${\Bbb C}^4.$ \begin{pf} Part (1) follows from the fact that $h^0({L^e_*}^{\otimes 2}\otimes K(D))=0,$ hence ${L^e_*}^\vee$ is a $\Phi$-invariant subbundle of $F^e_*$ for any $\Phi.$ Thus, if $(E_*, \Phi)$ is stable and $\aa \in C_0,$ then $E_* \simeq G_*.$ For part (2), if $(\xi,\Phi) \in Y,$ then the associated $K(D)$ pair $(G^\xi_*, \Phi)$ is stable by Claim \ref{claim:1} since $\pi^\xi_* (\Phi) = \phi_0 \neq 0$. This gives a map $\eta:Y\rightarrow {\cal P}_\aa,$ which we claim is a bijection. To see this, write $Y = Y' \cup Y'',$ where $Y' = Y|_{V \setminus 0}$ and $Y'' = Y|_0,$ and ${\cal P}_\aa = {\cal P}_\aa' \cup {\cal P}_\aa'',$ where ${\cal P}_\aa'$ and ${\cal P}_\aa''$ consist of the $K(D)$ pairs $(E_*, \Phi)$ with underlying bundle $E_*$ isomorphic to $G_*$ and $F^e_*,$ respectively. The restriction of $M$ to $V\setminus\{0\}$ is naturally isomorphic to $(V\setminus\{0\})\times H^0(G_*^\vee\otimes G_*\otimes K(D)).$ For $(\xi,\Phi)\in M|_{V\setminus\{0\}}$ and $t\in{\Bbb C}^*,$ $$\pi^{t \xi}_*(\Phi)=t^{-1}\pi^\xi_*(\Phi).$$ It follows from this formula that $\eta$ induces a bijection between $Y'$ and ${\cal P}_\aa'$ Using the description of the action of $\operatorname{Aut} F^e_*$ on $H^0({F^e_*}^\vee \otimes F^e_* \otimes K(D))$ following the proof of the previous lemma, every $(F^e_*, \Phi) \in {\cal P}_\aa''$ is isomorphic to $(F^e_*, \Phi_0)$ where $$ \Phi_0 = \begin{pmatrix} a_1 & 0 \\ \phi_0 & a_2 \end{pmatrix}. $$ Hence, $\eta$ gives a bijection between $Y''$ and ${\cal P}_\aa''.$ To prove (3), notice that we have a map $\eta:H^0(G_*^\vee\otimes {G_*}\otimes K(D))\rightarrow {\cal P}_\aa.$ Since ${\cal P}_\aa$ is normal, it is enough to show that this map is bijective. Now by Claim \ref{claim:1}, we see that ${\cal P}_\aa^s\simeq H^0(G_*^\vee\otimes {G_*}\otimes K(D))\setminus\operatorname{Ker}\pi_*.$ The strictly semistable bundles are pairs of the form $(F^e_*,\Phi)$ for any $\Phi,$ and $({G_*},\Phi)$ with $\Phi\in\operatorname{Ker}\pi_*.$ If $\Phi\in\operatorname{Ker}\pi_*$, the subbundle $L^e_*$ is $\Phi$-invariant and we get the extension of parabolic $K(D)$ pairs \begin{equation}\label{eq:ext} 0 \longrightarrow (L^e_*,\phi) \longrightarrow (G_*,\Phi) \longrightarrow ({L^e_*}^\vee,\psi) \longrightarrow 0. \end{equation} Thus $\operatorname{gr} (G_*, \Phi) = (L^e_*,\phi) \oplus ({L^e_*}^\vee,\psi)$ for $\Phi \in\operatorname{Ker}\pi_*.$ Consider now the map $$\lambda:\operatorname{Ker}\pi_* \longrightarrow H^0({L^e_*}^\vee\otimes L^e_*\otimes K(D))\oplus H^0(L^e_*\otimes {L^e_*}^\vee\otimes K(D))$$ defined by $\Phi\mapsto(\phi,\psi)$. For $\phi=\psi=0$, then the extension (\ref{eq:ext}) induces the zero map ${L^e_*}^\vee \rightarrow L^e_*\otimes K(D)$ (because $H^0({L^e_*}^{\otimes 2} \otimes K(D))=0$) and it follows that $\Phi=0.$ So $\lambda$ is injective. But the domain and range of $\lambda$ are both 4-dimensional, and so $\lambda$ is an isomorphism. Clearly $\operatorname{gr}(F^e_*,\Phi) = (L^e_*,\phi)\oplus({L^e_*}^\vee,\psi),$ and it follows that $\lambda$ gives a bijection between $\operatorname{Ker}\pi_*$ and ${\cal P}_\aa^{sss}.$ \end{pf} Choosing some $0 \neq \Phi_0 \in H^0({F^e_*}^\vee \otimes \widehat{F^e_*} \otimes K(D)) = {\Bbb C}$ and using the action of $\operatorname{Aut}(F^e_*),$ it is easy to verify that $(F^e_*, \Phi)$ is isomorphic to $(F^e_*, \Phi_0)$ for all $\Phi \neq 0.$ The proof of the last proposition is left as an entertaining exercise in applying the above lemmas. \begin{prop} \begin{enumerate} \item If $\aa \in C_0,$ then ${\cal N}_\aa = \{({G_*},0)\}.$ \item If $\aa \in C_e, e \in I,$ then ${\cal N}_\aa = \{(F^e_*,\Phi_0)\}.$ \item If $\aa \in H_e,$ then ${\cal N}_\aa = \{ [F^e_*,0] \}$ and $(G_*,0) \sim_S (F^e_*,0) \sim_S (F^e_*,\Phi_0)$ are the three distinct isomorphism classes of semistable Higgs bundles. \end{enumerate} \end{prop} \section{A Topological Description of ${\cal N}^0_\alpha$ in Rank Two} \subsection{The function spaces of Biquard and construction of Konno} We begin with a brief overview of the gauge theoretical description of ${\cal N}_\alpha$ following \cite{konno}. It is convenient to think of the parabolic bundle separate from its holomorphic structure, so we use $E_*$ to denote the underlying topological parabolic bundle (weights $\alpha$) and $\overline{\partial}_E$ its holomorphic structure. By tensoring with an appropriate line bundle, we can always assume that $\mu(E_*) = 0.$ We shall also restrict our attention to generic weights, i.e., weights $\aa$ for which $\alpha$-stability and $\alpha$-semistability coincide. Let ${\cal C}$ denote the affine space of all holomorphic structures on $E,$ and ${\cal G}_{\Bbb C}$ the group of smooth bundle automorphisms of $E$ preserving the flag structure. Introduce a metric $\kappa$ adapted to $E$ ($\kappa$ is unitary and smooth on $E|_{X \setminus D}$, but singular at $p \in D$ in a prescribed way, see Definition 2.3 \cite{biquard}), and let ${\cal A}$ denote the affine space of $\kappa$-unitary connections. Define ${\cal G}$ to be the subgroup of ${\cal G}_{\Bbb C}$ consisting of $\kappa$-unitary gauge transformations. Letting ${\cal C}_{ss}$ and ${\cal A}_{\hbox{\scriptsize \it flat}}$ be the subspaces of $\alpha$-semistable holomorphic structures and the flat connections, respectively, Biquard proved that $${\cal M}_\alpha \stackrel{\hbox{\scriptsize def}}{=} {\cal C}_{ss} / {\cal G}_{\Bbb C} \cong {\cal A}_{\hbox{\scriptsize \it flat}}/{\cal G}$$ by introducing the norms $\| \; \|_{D^p_k},$ defining the weighted Sobolev spaces ${\cal C}^p$ and ${\cal A}^p$ of $D^p_1$ holomorphic structures and $D^p_1 \, \kappa$-unitary connections, and taking quotients by the groups ${\cal G}_{\Bbb C}^p$ and ${\cal G}^p$ of $D^p_2$ gauge transformations for a certain $p>1$ \cite{biquard}. The same approach works for parabolic Higgs moduli, at least for generic weights, as was shown by Konno. The arguments in \cite{konno} are given for moduli with fixed determinant, but remain equally valid without this condition. We set \begin{eqnarray*} {\cal H} &=& \{ (\overline{\partial}_E, \Phi) \in {\cal C} \times \Omega^{1,0}(\operatorname{End} E) \mid \overline{\partial}_E \Phi = 0 \hbox{ on } X \setminus D \hbox{ and at each } p \in D, \\ & & \Phi \hbox{ has a simple pole with nilpotent residue with respect to the flag} \}. \end{eqnarray*} Note that ${\cal H}$ (this is denoted by ${\cal D}$ in \cite{konno}) is just the differential geometric definition of the space of parabolic Higgs bundle structures on $E_*,$ for example, the nilpotency condition implies that $\Phi$ is strongly parabolic. For $A \in {\cal A},$ we use $d_A$ for its covariant derivative, $F_A$ for its curvature, and $d_A''$ for the $(0,1)$ component of $d_A,$ so $d_A'' \in {\cal C}.$ Define ${\cal E} = {\cal A} \times \Omega^{0,1}(\operatorname{End} E)$ and ${\cal E}^p$ as its completion with respect to the norms $\| \; \|_{D^p_1},$ and set $${\cal E}_{\hbox{\scriptsize \it flat}} = \{ (d_A, \Phi) \in {\cal E}^p \mid d_A'' \Phi = 0, F_A + [\Phi, \Phi^*] = 0 \}.$$ (This last space is denoted ${\cal D}^p_{H \! E}$ by Konno.) Using the usual definition of stability on ${\cal H},$ Theorem 1.6 of \cite{konno} shows that for some $p>1,$ $${\cal N}_\alpha \stackrel{\hbox{\scriptsize def}}{=} {\cal H}_{ss} / {\cal G}_{\Bbb C} \cong {\cal E}_{\hbox{\scriptsize \it flat}}/{\cal G}^p.$$ The advantage of the second quotient is that it endows ${\cal N}_\alpha$ with a natural hyperk\"ahler structure, namely by viewing it as a hyperk\"ahler quotient of ${\cal E}^p$ (in the sense of \cite{HKLR}), whose hyperk\"ahler structure is given by the metric $$g((\xi,\phi),(\xi,\phi)) = 2 i \int_X \operatorname{Tr}(\xi^* \xi + \phi \phi^*),$$ which is K\"ahler with respect to each of three complex structures $$I(\xi,\phi) = (i \xi, i \phi), \quad J(\xi,\phi) = (i \phi^*,- i \xi^*), \quad K(\xi,\phi) = (- \phi^*, \xi^*).$$ \subsection{The Morse function for the moduli space of parabolic Higgs bundles} Assume that $E_*$ is a rank two parabolic bundle with generic weights $\alpha_i$ and $1-\alpha_i$ at $p_i$ and that $\mu_\alpha(E_*)=0.$ Write $\alpha=(\alpha_1,\ldots,\alpha_n).$ We will always assume $n\ge 1.$ We consider the moduli with fixed determinant and trace-free Higgs fields, requiring the following minor modifications in the definitions of the previous section: \begin{itemize} \item[(i)] the induced connection $d_\Lambda$ or holomorphic structure $\overline{\partial}_\Lambda$ on $\Lambda^2 E$ be fixed; \item[(ii)] the Higgs field be trace-free, i.e. $\Phi \in \Omega^{1,0}(\operatorname{End}_0 E).$ \end{itemize} We denote the corresponding spaces by ${\cal A}^0, {\cal C}^0, {\cal E}^0,$ and ${\cal H}^0.$ As in \cite{hitchin}, we consider the circle action defined on ${\cal E}^0$ by $e^{i\theta} \cdot (d_A,\Phi) = (d_A, e^{i\theta}\Phi).$ This action preserves the subspace ${\cal E}^0_{\hbox{\scriptsize \it flat}}$ and commutes with the action of the gauge group ${\cal G}^p,$ thus it descends to give a circle action $\rho$ on ${\cal N}^0_\alpha.$ This action commutes with the complex structure defined by $I$ and preserves the symplectic form $\omega_1(X,Y)=g(IX,Y),$ so the associated moment map $\mu_\rho(d_A, \Phi) = \frac{1}{4 \pi}\| \Phi \|_{D^p_1}^2,$ renormalized for convenience, is a Bott-Morse function and can be used to determine the Betti numbers of ${\cal N}^0_\alpha.$ We introduce some notation which will be used throughout the rest of this section. For any line subbundle $L_*$ of $E_*,$ let $e_i(L) = \dim L_{p_i} \cap F_2(p_i) \in \{0,1\}.$ The weight inherited by $L_*$ is then $\beta_i(L) = e_i +(-1)^{e_i} \alpha_i.$ We will often suppress the dependence on $L$ and simply write $e = (e_1, \ldots, e_n)$ and $\beta=(\beta_1, \ldots, \beta_n).$ We will also write $\beta(\alpha,e)$ when we want to emphasize the functional dependence of $\beta$ on $\alpha$ and $e.$ We also use $|e| = \sum_{i=1}^n e_i.$ \begin{thm} \label{thm:morse} \begin{itemize} \item[(a)] The map $\mu_\rho :{\cal N}^0_\alpha \longrightarrow {\Bbb R}$ is a proper Morse function. \item[(b)] Whenever nonempty, ${\cal M}^0_\alpha$ is the unique critical submanifold corresponding to the minimum value $\mu_\rho =0.$ The other critical submanifolds are given by ${\cal M}_{d,e}$ for an integer $d$ and $e \in {\Bbb Z}_2^n$ satisfying \begin{equation} \label{3a} -\sum_{i=1}^n \beta_i(\alpha,e) < d \leq g-1 -|e|/2. \end{equation} Along ${\cal M}_{d,e}, \, \mu_\rho$ takes the value $d + \sum_{i=1}^n \beta_i.$ \item[(c)] The critical submanifold ${\cal M}_{d,e}$ is $\widetilde{S}^{h_{d,e}}X,$ the $2^{2g}$ cover of the symmetric product $S^{h_{d,e}}X$ under the map $x \mapsto 2x$ on $J_X.$ Here, $h_{d,e} = 2g -2 -2d - |e|.$ \item[(d)] The Morse index of ${\cal M}_{d,e}$ is given by $\lambda_{d,e} = 2(n+2d +g-1 +|e|).$ \end{itemize} \end{thm} \noindent {\it Remark. \,} If $g=0,$ there are always $\alpha$ with ${\cal M}^0_\alpha = \emptyset$ (but ${\cal N}^0_\alpha \neq \emptyset$). For these $\alpha,$ the minimum value is achieved along some ${\cal M}_{d,e},$ which we identify in the next section. \begin{pf} Properness of $\mu_\rho$ follows from the global compactness result for parabolic bundles of Biquard (Theorem 2.14 in \cite{biquard}). This proves (a). All the other statements rely on the following correspondence between the circle action and the moment map given in \cite{frankel}. \begin{itemize} \item[(1)] Critical submanifolds are connected components of the fixed point set of $\rho.$ \item[(2)] The Morse index of a critical submanifold equals the dimension of the negative weight space of the infinitesimal circle action on its normal bundle. \end{itemize} Suppose that $(d_A,\Phi)$ is a fixed point of the circle action upstairs in ${\cal E}_{\hbox{\scriptsize \it flat}}.$ Then $\Phi = 0$ and this shows that one component of the fixed point set in ${\cal N}^0_\alpha$ consists of ${\cal M}^0_\alpha,$ the moduli of stable parabolic bundles with fixed determinant. The other fixed points arise from when $e^{i \theta}\cdot (d_A, \Phi)$ is gauge equivalent to $(d_A,\Phi),$ i.e., when there is a one parameter family $g_\theta \in {\cal G}^p$ such that \begin{eqnarray*} g^{-1}_\theta \Phi g_\theta &=& e^{i\theta}\Phi,\\ g^{-1}_\theta d_A g_\theta &=& d_A. \end{eqnarray*} By the first equation, $g_\theta$ is not central, and by the second, we see that $d_A$ is reducible and consequently the holomorphic parabolic bundle splits according to the eigenvalues of $g_\theta.$ Write $E_* = L_* \oplus M_*$ as a direct sum of parabolic bundles. We assume (wlog) that $\mu_\alpha(L_*) > 0 > \mu_\alpha(M_*).$ Let $d=\deg L$ and $e=(e_1,\ldots,e_n)$ where $e_i = \dim L_{p_i} \cap F_2(p_i).$ Then $L$ inherits the weight $\beta_i = e_i + (-1)^{e_i} \alpha_i$ at $p_i$ as a parabolic subbundle of $E_*$ and \begin{equation} 0 < \mu_\alpha(L_*) = d + \sum_{i=1}^n \beta_i. \label{3b} \end{equation} Since $g_\theta$ is diagonal with respect to this decomposition, $\Phi$ is either upper or lower diagonal, which means either $L$ or $M$ is $\Phi$-invariant. But $\alpha$-stability of the pair $(E_*,\Phi)$ implies that $$\Phi=\left(\begin{array}{cc} 0 & 0 \\ \phi&0 \end{array} \right),$$ where $0 \neq \phi \in \operatorname{ParHom}(L_*, \widehat{M}_* \otimes K(D)).$ Thus $$0 \neq H^0(L_{*}^{\vee} \otimes\widehat{M}_{*} \otimes K(D)) \\ =H^0(L^{\vee}\otimes M\otimes K({\textstyle{\sum}^n_{i=1}} (1-e_i)p_i)).$$ Let $|e| = \sum_{i=1}^n e_i,$ then a necessary condition is that \begin{equation} 0\leq \deg(L^{\vee}\otimes {M}\otimes K({\textstyle{\sum}^n_{i=1}} (1-e_i)p_i)) = 2(g-1) -2d - |e|. \label{3c} \end{equation} Now (\ref{3a}) follows from (\ref{3b}) and (\ref{3c}). We can use the defining equations for ${\cal E}^0_{\hbox{\scriptsize \it flat}}$ to determine the associated critical values. Take $(E_*, \Phi)$ as above, then $$0 = F_A + [\Phi, \Phi^*] = \left( \begin{array}{cc} F_L - \phi \phi^* & 0 \\ 0 & F_M + \phi^* \! \phi \end{array} \right).$$ Using the Chern-Weil formula for parabolic bundles (Proposition 2.9 of \cite{biquard}), we get $$\mu_\rho(d_A,\Phi) = \frac{1}{4\pi} \| \Phi \|^2 = \frac{i}{2\pi} \int_X \operatorname{Tr}(\Phi \Phi^*) = \frac{i}{2\pi} \int_X \phi \phi^* = \frac{i}{2\pi} \int_X F_L = \operatorname{pardeg} (L_*).$$ This completes the proof of (b). Given $E_* = L_* \oplus M_*$ and $\Phi$ as above, then the zero set of $\phi$ is a nonnegative divisor of degree $$h_{d,e} = \deg (L^{\vee}\otimes {M}\otimes K(\textstyle{\sum^n_{i=1}} (1-e_i)p_i)) =2g-2-2d-|e|$$ on $X,$ which is just an element of ${S}^{h_{d,e}}X.$ Conversely, given a nonnegative divisor of degree $h_{d,e},$ then we obtain a line bundle $U$ of degree $2d+n$ along with a section of $U^\vee \otimes K(\textstyle{\sum^n_{i=1}} (1-e_i)p_i))$ vanishing on that divisor. There are $2^{2g}$ choices of $L$ so that $U = L^{\otimes 2} \otimes \Lambda^2 E,$ and each choice gives a stable parabolic Higgs bundle $(E_*, \Phi).$ The line subbundle $L_*$ is canonically determined from $E_*,$ but $\Phi$ is only determined up to multiplication by a nonzero constant. However, it is easy to see that $(E_*, \Phi)$ is gauge equivalent to $(E_*, \lambda \Phi)$ for $\lambda \neq 0,$ and (c) now follows. We now calculate the index $\lambda_{d,e}$ of the critical submanifold ${\cal M}_{d,e},$ which is given by the negative weight space of the infinitesimal action of $\rho,$ or equivalently, of the gauge transformation $g_\theta.$ Letting $H^0(\operatorname{ParEnd}_0(E)) \cdot \Phi$ be the subspace of Higgs fields of the form $[\Psi, \Phi]$ for $\Psi \in H^0(\operatorname{ParEnd}_0(E)),$ then the subspace $$W = H^0(\operatorname{ParEnd}^\wedge_0(E)\otimes K(D)) / H^0(\operatorname{ParEnd}_0(E)) \cdot \Phi$$ is Lagrangian with respect to the complex symplectic form $$\omega((\xi_1,\phi_1),(\xi_2,\phi_2)) = \int_X \operatorname{Tr} (\phi_2 \xi_1 - \phi_1 \xi_2).$$ So once we determine the weights on $W,$ the weights on the dual space $W^*$ are given by $1-\nu$ for some weight $\nu$ on $W$ (since $\rho(\theta)^* \omega = e^{i \theta} \omega$). With respect to the decomposition $E_* = L_* \oplus M_*,$ we have $$g_\theta = \left( \begin{array}{cc} e^{-i \theta/2} &0 \\ 0 & e^{i\theta/2} \end{array} \right)$$ with weights $(0,1,-1)$ on $$\operatorname{ParEnd}^{\wedge}_0(E_*) = \operatorname{ParHom}(L_*, \widehat{L}_*) \oplus \operatorname{ParHom}(L_*, \widehat{M}_*) \oplus \operatorname{ParHom}(M_*, \widehat{L}_*).$$ Further, there are no negative weights on $H^0(\operatorname{ParEnd}_0(E)) \cdot \Phi$ and the weights on $W^*$ are $(1,0,2),$ so we get \begin{eqnarray*} \lambda_{d,e} &=& 2 h^0(M_*^{\vee}\otimes\widehat{L}_*\otimes K(D)) =2(n+2d+g-1+|e|). \end{eqnarray*} This completes the proof of (d). \end{pf} \subsection{The topology of ${\cal N}^0_\alpha$} Using the results of the previous section, we deduce the following theorem. \begin{thm} \label{thm:main} \begin{itemize} \item[(a)] If $g>0$ or $g=0$ and $n>3,$ then ${\cal N}^0_\alpha$ is noncompact. \item[(b)] The Betti numbers of ${\cal N}^0_\alpha$ depend only on the quasi-parabolic structure of $E_*.$ \item[(c)] If $g > 0$ or $g=0$ and $n \geq 3,$ then ${\cal N}^0_\alpha$ is connected and simply connected. \end{itemize} \end{thm} \begin{pf} Notice that, whenever $\dim{\cal N}^0_\alpha > 0,$ then for all $(d,e), \; \lambda_{d,e} < \dim{\cal N}^0_\alpha.$ Thus, the Morse function $\mu_\rho$ has no maximum value and (a) follows. The only case where $\dim{\cal N}^0_\alpha = 0$ is, of course, $g=0$ and $n=3.$ We first recall Theorem 3.1 of \cite{bh}. Let $W= \{\alpha \mid 0 < \alpha_i < \frac{1}{2} \}$ be the weight space and for any $(d,e),$ define the hyperplane $H_{d,e} = \{ \alpha \mid d+\beta(\alpha,e) = 0\}.$ The set $W \setminus \cup_{d,e} H_{d,e}$ consists of the generic weights, i.e., those for which stability and semistability coincide. Suppose $\delta \in H_{d,e},$ then stratifying ${\cal M}^0_{\delta}$ by the Jordan-H\"{o}lder type of the underlying parabolic bundle, we see that $${\cal M}^0_{\delta} = ({\cal M}^0_{\delta} \setminus \Sigma_{\delta}) \cup \Sigma_{\delta},$$ where $\Sigma_{\delta}$ consists of strictly semistable bundles, i.e., semistable bundles $E_*$ with $\operatorname{gr} E_* = L_* \oplus M_*$ for two parabolic line bundles of parabolic degree zero. Suppose that $\alpha$ and $\alpha'$ are generic weights on either side of $H_{d,e}$ and that $\operatorname{pardeg}_\alpha (L_*) < 0.$ If both ${\cal M}^0_{\alpha}$ and ${\cal M}^0_{\alpha'}$ are nonempty, then Theorem 3.1 of \cite{bh} states that there are canonical, projective maps $$\begin{array}{rcl}{\cal M}^0_{\alpha}& & {\cal M}^0_{\alpha'}\\ \phi \!\!\!\! & \searrow \; \swarrow & \!\!\! \phi'\\& {\cal M}^0_{\delta} \end{array}$$ which are isomorphisms on ${\cal M}^0_{\delta} \setminus \Sigma_{\delta}$ and are ${\Bbb P}^{a}$ and ${\Bbb P}^{a'}$ bundles along $\Sigma_{\delta},$ where $a = h^1(M_*^\vee \otimes L_*)-1$ and $a' = h^1(L_*^\vee \otimes M_*)-1.$ In particular, since $\Sigma_{\delta} = J_X,$ Corollary 3.2 of \cite{bh} gives $$P_t({\cal M}^0_{\alpha}) - P_t({\cal M}^0_{\alpha'}) = (P_t({\Bbb P}^{a})-P_t({\Bbb P}^{a'})) P_t(J_X).$$ To prove (b), we must show that $P_t({\cal N}^0_{\alpha}) = P_t({\cal N}^0_{\alpha'})$ for weights on either side of a hyperplane $H_{d,e}.$ Note that $d = \deg L$ and $e = e(L),$ and set $\hat{d}=-n-d$ and $\hat{e}_i = 1-e_i.$ Since $$d+\beta(\alpha,e) = \operatorname{pardeg}_\alpha(L) < 0 < \operatorname{pardeg}_{\alpha'}(L) = d + \beta(\alpha',e),$$ and $\hat{d} +\beta(\alpha', \hat{e}) < 0 < \hat{d}+\beta(\alpha,\hat{e}),$ it follows that the indexing sets of $(d,e)$ satisfying (\ref{3a}) for ${\cal N}^0_\alpha$ and ${\cal N}^0_{\alpha'}$ are identical except for $(d,e)$ and $(\hat{d},\hat{e})$ listed above; the pair $(d,e)$ satisfies (\ref{3a}) for $\alpha$ but not for $\alpha'$ and vice versa for $(\hat{d},\hat{e}).$ Thus, we claim \begin{eqnarray*} 0 &=& P_t({\cal M}^0_{\alpha}) - P_t({\cal M}^0_{\alpha'}) + t^{\lambda_{d,e}} P_t({\cal M}_{d,e})- t^{\lambda_{\hat{d},\hat{e}}} P_t({\cal M}_{\hat{d},\hat{e}}), \end{eqnarray*} which, setting $\Delta = t^{\lambda_{\hat{d},\hat{e}}} P_t({\cal M}_{\hat{d},\hat{e}}) - t^{\lambda_{d,e}} P_t({\cal M}_{d,e})$ is equivalent to \begin{equation} \label{3k} \Delta = \frac{( t^{2a'+2} -t^{2a+2}) (1+t)^{2g}}{1-t^2}. \end{equation} First, we compute $$h_{d,e} = 2g-2-2d -|e|, \quad \lambda_{d,e} = 2(n+2d+g-1+|e|), $$ $$h_{\hat{d},\hat{e}} = 2g-2+n+2d+|e|, \quad \lambda_{\hat{d},\hat{e}} = 2(g-1-2d-|e|).$$ Next, notice that if $h> 2g-2,$ then $P_t(\widetilde{S}^h(X))= P_t(S^h(X))$ (see p. 98 of \cite{hitchin}). But both $h_{d,e}$ and $h_{\hat{d},\hat{e}}$ are greater than $2g-2,$ which we see as follows. Since $\frac{e_i}{2} \leq \beta_i(\alpha,e) \leq \frac{1+e_i}{2},$ we have $\frac{|e|}{2} \leq \sum_{i=1}^n \beta_i(\alpha,e) \leq \frac{n+|e|}{2}.$ It now follows that $ 2d + |e| < 2d + 2\beta(\alpha,e) < 0$ and $2d+n+|e| > 2 d + 2 \sum_{i=1}^n \beta(\alpha',e) > 0.$ Now use the result of \cite{macdonald} to interpret $P_t(S^h X)$ as the coefficient of $x^h$ in $$\frac{(1+xt)^{2g}}{(1-x)(1-xt^2)},$$ and compute in terms of residues to see \begin{eqnarray*} \Delta&=& t^{\lambda_{\hat{d},\hat{e}}}P_t(S^{h_{\hat{d},\hat{e}} }X) - t^{\lambda_{d,e}}P_t(S^{h_{d,e}}X) \\ &=& \operatornamewithlimits{Res}_{x=0} \left( \frac{t^{\lambda_{\hat{d},\hat{e}}}}{x^{h_{\hat{d},\hat{e}}+1}} - \frac{t^{\lambda_{d,e}}}{x^{h_{d,e}+1}} \right)\left( \frac{(1+xt)^{2g}}{(1-x)(1-xt^2)}\right). \end{eqnarray*} This last function is analytic at $x=\infty$ and has a removable singularity at $x=1/t^2,$ thus \begin{eqnarray*} \Delta &=& -\operatornamewithlimits{Res}_{x=1} \left( \frac{t^{\lambda_{\hat{d},\hat{e}}}}{x^{h_{\hat{d},\hat{e}}+1}} - \frac{t^{\lambda_{d,e}}}{x^{h_{d,e}+1}} \right)\left( \frac{(1+xt)^{2g}}{(1-x)(1-xt^2)}\right)\\ &=& \frac{(t^{\lambda_{\hat{d},\hat{e}}} - t^{\lambda_{d,e}})(1+t)^{2g}}{1-t^2}. \end{eqnarray*} But we can compute directly that $2a'+2 = \lambda_{\hat{d},\hat{e}}$ and that $2a+2 = \lambda_{d,e}$ and (\ref{3k}) follows. This proves (b) in case both ${\cal M}^0_\alpha$ and ${\cal M}^0_{\alpha'}$ are nonempty. In case one of the moduli is empty, we use the following lemma (see the remark). To prove (c), we use the fact that ${\cal M}^0_\alpha$ is connected and simply-connected, which follows for $g=0$ from \cite{bauer} and for $g \geq 1$ from \cite{boden1}. Since $\lambda_{d,e}$ is always even, (c) will follow if $\lambda_{d,e} >0$ for all $(d,e).$ This is true if ${\cal M}^0_\alpha \neq \emptyset.$ However, if $g=0$ we must be careful since there are weights $\alpha$ with ${\cal M}_\alpha = \emptyset.$ In that case, we must show that there is a unique pair $(d,e)$ with $\lambda_{d,e} = 0,$ and also that ${\cal M}_{d,e}$ is connected and simply connected. This is the content of the following lemma. \end{pf} \begin{lem} \begin{enumerate} \item[(i)] If $g \geq 1,$ then $\lambda_{d,e} > 0$ for every $(d,e)$ satisfying \rom(\ref{3a}\rom). \item[(ii)] If $g = 0$ and $n \geq 3,$ then there is at most one pair $(d,e)$ satisfying \rom(\ref{3a}\rom) with $\lambda_{d,e} = 0.$ Such a pair $(d,e)$ exists if and only if ${\cal M}_\alpha = \emptyset,$ and in that case, ${\cal M}_{d,e} = {\Bbb P}^{n-3}.$ Here, $ {\cal M} = {\cal M}^0$ since $g=0.$ \end{enumerate} \end{lem} {\it Remark. \,} We now explain why this lemma proves part (b) of the Proposition when one of the moduli is empty. Suppose ${\cal M}_{\alpha}=\emptyset,$ then it follows that the moment map $\mu_\rho$ is positive with minimum value $d+\sum_{i=1}^n \beta(\aa,e)$ for the pair $(d,e)$ identified in part (ii) of the lemma. Since $(d,e)$ does not satisfy (\ref{3a}) for $\alpha',$ $H_{d,e}$ is the relevant hyperplane. This identifies the birth and death strata as ${\cal M}_{\aa'}$ and ${\cal M}_{d,e},$ and thus all the other strata for $\aa$ and $\aa'$ are identical. The rest follows from the fact that ${\cal M}_{\alpha'} = {\Bbb P}^{n-3},$ first proved by Bauer \cite{bauer}. \begin{pf} Suppose that $\lambda_{d,e} =0$ for a pair $(d,e)$ satisfying (\ref{3a}). We first show that $g=0.$ Recall that $\beta_i(\alpha,e) = e_i +(-1)^{e_i} \alpha_i.$ Using the fact that $0 = \lambda_{d,e} = n+2d+g+|e|-1,$ the condition (\ref{3a}) and the inequality $\beta_i(\alpha,e) < \frac{e_i +1}{2},$ we see that \begin{equation} \label{3d} \frac{n+|e|+g-1}{2} < \sum_{i=1}^n \beta_i(\alpha, e) < \frac{n+|e|}{2}. \end{equation} This is only possible if $g=0,$ which we now assume. Setting $\gamma_i = 1- \beta_i = (1-e_i)(1-\alpha_i)+e_i \alpha_i,$ then equation (\ref{3d}) is equivalent to $$\frac{n-|e|}{2} < \sum_{i=1}^n \gamma_i < \frac{n-|e|+1}{2}.$$ Writing $\gamma_i = \frac{1-e_i}{2} + (1-e_i)(\frac{1}{2}-\alpha_i) + e_i \alpha_i,$ we get immediately \begin{equation} \label{3e} 0< \sum_{i=1}^n (1-e_i)(\frac{1}{2}-\alpha_i) + e_i \alpha_i < \frac{1}{2}. \end{equation} The advantage of the (\ref{3e}) is that each summand is positive. We now prove uniqueness of the pair $(d,e).$ If $\lambda_{d',e'} = 0$ for $(d',e') \neq (d,e),$ then it follows that $|e|-|e'| = 2(d'-d)$ is even, which implies that $e_i \neq e'_i$ for at least two $i,$ which we assume (wlog) to include $i=1,2.$ Now $(\alpha,e)$ and $(\alpha, e')$ both satisfy the inequality (\ref{3e}). Add them together and notice that since $e_1 \neq e'_1$ and $e_2 \neq e'_2,$ the sum of the left hand sides is at least $ \alpha_1 + (1/2-\alpha_1) + \alpha_2 + (1/2-\alpha_2) = 1,$ which violates the (summed) inequality and therefore gives a contradiction. It follows from $\lambda_{d,e}=0$ and $g=0$ that $n+|e|-1$ is even and $h_{d,e} = n-3.$ Thus ${\cal M}_{d,e} = S^h X =S^h {\Bbb P}^1= {\Bbb P}^{n-3}.$ The rest of the lemma follows from the the inequality (\ref{3d}), together with the following proposition, which we have chosen to state as it is of independent interest. \end{pf} \begin{prop} \label{prop:null-ch} If $g=0,$ then the moduli space ${\cal M}_\alpha \neq \emptyset \Leftrightarrow$ \begin{equation}\label{3f} \sum_{i=1}^n e_i + (-1)^{e_i}\alpha_i < \frac{n+|e|-1}{2}. \end{equation} for every $e=(e_1,\ldots,e_n), \; e_i \in \{0,1\},$ with $n-|e|+1$ even. \end{prop} \noindent {\it Remark. \,} For $n=3,$ ${\cal M}_\alpha$ is either empty or a point. In this case, the proposition can be verified directly by comparing the inequalities (\ref{3f}) to the well-known fusion rules (or the quantum Clebsch-Gordan conditions): $${\cal M}_\alpha \neq \emptyset \Leftrightarrow |\alpha_1 - \alpha_2| \leq \alpha_3 \leq \min(\alpha_1 + \alpha_2, 1-\alpha_1-\alpha_2).$$ \begin{pf} Like the proof of part (b) of the theorem, we shall use the techniques of \cite{bh}. Recall the weight space $W = \{ \alpha \mid 0 \leq \alpha_i \leq 1/2 \}$ and the hyperplanes $H_{d,e} = \{ \alpha \mid d+\beta(\alpha,e)=0 \}$ defined earlier. We call connected components of $W \setminus \cup_{d,e} H_{d,e}$ {\it chambers}. A chamber $C$ is called {\it null} if the associated moduli space ${\cal M}_\alpha$ is empty in genus 0 for every $\alpha \in C.$ The proposition follows once we show that every null chamber is given by $C_{d,e} = \{ \alpha \mid d+\beta(\alpha,e)>0 \},$ where $2d=1-n-|e|.$ Associated to the configuration of hyperplanes in $W$ is a graph with one vertex for each chamber and an edge between two vertices whenever the two chambers are separated by a hyperplane. We shall see that in terms of this graph, null chambers have valency one. The (unique) hyperplane separating a null chamber from the rest of $W$ is called a {\it vanishing wall.} If $\delta \in H_{d,e},$ a vanishing wall, and $\alpha, \alpha'$ are nearby weights on either side of $H_{d,e},$ then the proof of Proposition 5.1 of \cite{bh} shows that ${\cal M}_\delta = \Sigma_\delta$ and, assuming that ${\cal M}_{\alpha'} = \emptyset,$ the map $\phi$ is a fibration with fiber ${\Bbb P}^{a},$ where $a = h^1(M_*^\vee \otimes L_*)-1.$ Moreover, $h^1(L_*^\vee \otimes M_*)=0$ and this last equation in fact characterizes vanishing walls. We claim that every vanishing hyperplane is given by $H_{d,e}$ for $2d=1-n-|e|.$ For if $d=\deg L$ and $e=e(L),$ then direct computation shows that $h^1(L_*^\vee \otimes M_*) =2d+n+|e|-1.$ On the other hand, if $n+|e|-1$ is even and $d = \frac{1-n-|e|}{2},$ then $H_{d,e}$ is a vanishing hyperplane. Along $H_{d,e},$ the relevant line bundles of parabolic degree 0 are given by $L_* = {\cal O}_X (\frac{-n-|e|+1}{2})[-\beta]_*$ and $M_* = {\cal O}_X (\frac{-n+|e|-1}{2})[-\gamma]_*,$ where $\delta \in H_{d,e}, \beta= \beta(\delta,e)$ and $\gamma_i = 1- \beta_i.$ Since $h^1(L_*^\vee \otimes M_*)=0$ and $h^1(M_*^\vee \otimes L_*)=n-2,$ it follows that the null chamber is defined by $C_{d,e}=\{ \alpha \mid \beta(\alpha,e) > \frac{n+|e|-1}{2} \}.$ To verify that this is indeed a chamber, we prove that no other hyperplane cuts through $C_{d,e}.$ This will also show that null chambers have valency one in the graph associated to the configuration of hyperplanes. Suppose to the contrary that $\alpha \in H_{d',e'} \cap C_{d,e}.$ Then we have $\sum (-1)^{e_i} \alpha_i > \frac{n-|e|-1}{2}$ and $\sum (-1)^{e'_i} \alpha_i = -|e'| -d' =k \in {\Bbb Z}.$ If $e_i = e_i' = 0,$ then $((-1)^{e_i} + (-1)^{e_i'}) \alpha_i < 1$ and in all other cases, $((-1)^{e_i} + (-1)^{e_i'}) \alpha_i \leq 0.$ Using a similar property for $e''= 1 -e',$ we see \begin{eqnarray*} \frac{n-|e|-1}{2} + k &<& \sum_{i=1}^n((-1)^{e_i} + (-1)^{e_i'}) \alpha_i < \sum_{e_i = e'_i=0} 1,\\ \frac{n-|e|-1}{2} -k &<& \sum_{i=1}^n((-1)^{e_i} + (-1)^{e_i''}) \alpha_i < \sum_{e_i = e''_i=0} 1. \end{eqnarray*} These are strict inequalities of integers, so after adding one to the left hand sides and summing the two inequalities (which are no longer strict), we see $n-|e|+1 \leq \sum_{e_i=0} 1 = n-|e|,$ a contradiction. \end{pf} \subsection{The Betti numbers of the moduli space of parabolic Higgs bundles} The results of the previous section show that the Betti numbers of ${\cal N}^0_\alpha$ depend only on the genus $g$ and number $n$ of parabolic points. In this section, we give a formula for the Poincar\'e polynomial of ${\cal N}^0_\alpha.$ Such a general calculation is not possible for $P_t({\cal M}^0_\alpha)$ without first specifying $\alpha,$ so take $\alpha=(\frac{1}{3},\ldots, \frac{1}{3^n}).$ Using Proposition \ref{prop:null-ch} (taking $e=(0,1,\ldots,1)$) it is clear that $\alpha$ lies in a null chamber. We could calculate $P_t({\cal M}^0_\alpha)$ using the Atiyah-Bott procedure for parabolic bundles as in \cite{boden1}, but there is an easier method which exploits the fact that $\alpha$ lies in a null chamber. First of all, using the results of \S 6.4 in \cite{boden1}, we get $$P_t({\cal M}^0_\alpha) = \frac{(1+t^2)^{n-1}(1+t^3)^{2g}}{(1-t^2)^2}- \frac{(1+t)^{2g}}{(1-t^2)} \sum_{\lambda,e}t^{2 d_{\lambda,e}}.$$ Note that $d_{\lambda,e}$ depends on $g$ ($d_{\lambda,e} =d_{\lambda,e}(g=0)+g$), but the indexing set $\{ \lambda,e\}$ is independent of $g$. Since ${\cal M}^0_\alpha(g=0) = \emptyset,$ this determines the sum and we see that $$P_t({\cal M}^0_\alpha) = \frac{(1+t^2)^{n-1}\left( (1+t^3)^{2g}-t^{2g}(1+t)^{2g}}{(1-t^2)^2\right)}.$$ It follows from Theorem \ref{thm:morse} that $$P_t({\cal N}^0_\alpha) = P_t({\cal M}^0_\alpha) + \sum_{d,e} t^{\lambda_{d,e}} P_t({\cal M}_{d,e}),$$ where the sum is taken over $(d,e)$ satisfying (\ref{3a}), which, for our choice of $\alpha,$ is simply $e_1 -|e| \leq d \leq [g-1 -\frac{|e|}{2}]$, where $[x]$ is the greatest integer less than $x.$ Setting $j=2d+n+|e|-1,$ then $j$ satisfies: $$n+2e_1 -|e|-1 \leq j \leq 2g+n-3 \quad \hbox{ and } \quad j-n-|e|+1 \hbox{ is even.}$$ Also $\lambda_{d,e} = 2(g+j)$ and $h_{d,e} = 2g+n-j-3.$ Fixing $e_1$ and $|e|,$ for each $d,$ there are $\left({{n-1}\atop{|e|-e_1}}\right)$ strata given by the choice of $e.$ Thus, for each $j,$ there are $q_j = \sum_{i=0}^{j} \left({{n-1}\atop{i}}\right)$ strata (note that $q_j = 2^{n-1}$ for $j \geq n-1$) and we see \begin{eqnarray*} \sum_{d,e} t^{\lambda_{d,e}} P_t({\cal M}_{d,e}) \!\!&=& \!\! \sum_{|e|=0}^n \left({{n-1}\atop{|e|-e_1}}\right) \sum_{d=e_1 - |e|}^{[g-1-|e|/2]} t^{\lambda_{d,e}} P_t(\widetilde{S}^{h_{d,e}}X) \\ \!\! &=& \!\! \sum_{j=0}^{2g+n-3} q_j t^{2(g+j)} P_t(\widetilde{S}^{2g+n-j-3}X)\\ \!\! &=& \!\! \sum_{j=0}^{n-2} q_j t^{2(g+j)} P_t(\widetilde{S}^{2g+n-j-3}X) + \sum_{j=0}^{2g-2} 2^{n-1}t^{2(g+n+j-1)} P_t(\widetilde{S}^{2g-j-2}X). \end{eqnarray*} We refer to the last two sums by $\widetilde{S}_1$ and $\widetilde{S}_2.$ Using the Binomial Theorem and the general formula (p. 98 of \cite{hitchin}) $\P_t(\widetilde{S}^{h}X) = (2^{2g}-1)\left({{2g-2}\atop{h}}\right)t^h +\P_t({S^{h}X}),$ we see that \begin{eqnarray*} \widetilde{S}_1 &=& \sum_{j=0}^{n-2} q_j t^{2(g+j)} P_t({S}^{2g+n-j-3}X) = S_1,\\ \widetilde{S}_2 &=& \sum_{j=0}^{2g-2} 2^{n-1}t^{2(g+n+j-1)} P_t({S}^{2g-j-2}X) + \sum_{j=0}^{2g-2} 2^{n-1} (2^{2g}-1) \left({{2g-2}\atop{j}} \right) t^{4g+2n+j-4}\\ &=& S_2 + 2^{n-1} (2^{2g}-1) t^{2(2g+n-2)}(1+t)^{2g-2}, \end{eqnarray*} where $S_1$ and $S_2$ are the sums obtained by removing the tildes from the summands of $\widetilde{S}_1$ and $\widetilde{S}_2.$ According to a result of \cite{macdonald}, $\P_t({S^{h}X})$ is the coefficient of $x^h$ in $$\frac{(1+xt)^{2g}}{(1-x)(1-xt^2)}.$$ This allows us to evaluate $S_i$ as follows: \begin{eqnarray*} S_1 &=& \operatornamewithlimits{Res}_{x=0} \left(\sum_{j=0}^{n-2}\frac{q_j t^{2(g+j)} (1+xt)^{2g}}{x^{2g+n-j-2}(1-x)(1-xt^2)}\right),\\ S_2 &=& \operatornamewithlimits{Res}_{x=0} \left( \frac{2^{n-1}t^{2(g+n-1)}(1+xt)^{2g}}{x^{2g-1}(1-x)(1-xt^2)^2}\right). \end{eqnarray*} But each of these rational functions is analytic at $x = \infty,$ so we can use the Cauchy Residue Formula to evaluate instead at the poles $x=1$ and $x=1/t^2.$ Letting $Q_n(t) = \sum_{k=0}^{n-2} q_k t^{2k}$ and noticing that $Q_n(1) = \sum_{k=0}^{n-2} q_k = 2^{n-2}(n-1),$ we get \begin{eqnarray*} S_1 &=& \left(Q_n(t) t^{2g} - 2^{n-2}(n-1) t^{2(2g+n-2)}\right) \frac{(1+t)^{2g}}{(1-t^2)},\\ S_2 &=& 2^{n-1}\left(t^{2(g+n-1)} + t^{4g+2n-3} \left( (2g-1)t - 2g \right) \right) \frac{(1+t)^{2g}}{(1-t^2)^2}. \end{eqnarray*} But since $Q_n(t)(1-t^2) + 2^{n-1}t^{2(n-1)} = (1+t^2)^{n-1},$ it follows that \begin{eqnarray*} P_t({\cal N}^0_\alpha) &=& P_t({\cal M}^0_\alpha) + \widetilde{S}_1 + \widetilde{S}_2 \\ &=& P_t({\cal M}^0_\alpha) + S_1 + S_2 + 2^{n-1}(2^{2g}-1) t^{2(2g+n-2)}(1+t)^{2g-2}\\\\ &=& \frac{(1+t^3)^{2g}(1+t^2)^{n-1} + 2^{n-1}t^{2n+4g-3}(1+t)^{2g}[(2g-1)t - 2g]}{(1-t^2)^2}\\ & & -\frac{2^{n-2}(n-1)t^{2n+4g-4}(1+t)^{2g}}{1-t^2} + 2^{n-1}(2^{2g}-1) t^{4g+2n-4}(1+t)^{2g-2}. \end{eqnarray*} Evaluating this at $t=-1$ shows that the Euler characteristic of ${\cal N}^0_\alpha$ is given by $$\chi({\cal N}^0_\alpha) = \begin{cases} (n-1)(n-2)2^{n-4} & \text{if } g=0, \\ 3\cdot2^n & \text{if } g=1, \\ 0 & \text{if } g \geq 2 \end{cases}$$ Theorem \ref{thm:main} would lead one to believe that the diffeomorphism type of ${\cal N}_\alpha^0$ depends only on the quasi-parabolic structure. We conjecture this is true in general. Subsequent to the writing of this paper, this conjecture was proved by H. Nakajima in rank two \cite{nakajima}.

9610

alg-geom/9610011

en

https://arxiv.org/abs/alg-geom/9610011

alg-geom/9610011

Bas Edixhoven

Bas Edixhoven

Special points on the product of two modular curves

hardcopy available at request to [email protected] LaTeX

We prove, assuming the Generalized Riemann Hypothesis for imaginary quadratic fields, that irreducible curves in the product of two modular curves that contain infinitely many complex multiplication points are either a Hecke correspondence or a fibre for one of the two projections. This gives evidence for a conjecture of Oort that says that irreducible components of the Zariski closure of a set of CM points in a Shimura variety are sub Shimura varieties.

alg-geom

\section{\@startsection {section}{1}{\z@}{-3.5ex plus -1ex minus -.2ex}{2.3ex plus .2ex}{\large\bf}} \def\subsection{\@startsection{subsection}{2}{\z@}{-3.25ex plus -1ex minus -.2ex}{1.5ex plus .2ex}{\normalsize\bf}} \newenvironment{eqn}{\refstepcounter{subsection} $$}{\leqno{\rm(\thesubsection)}$$\global\@ignoretrue} \newenvironment{subeqn}{\refstepcounter{subsubsection} $$}{\leqno{\rm(\thesubsubsection)}$$\global\@ignoretrue} \newenvironment{subeqarray}{\renewcommand{\theequation}{\thesubsubsection} \let\c@equation\c@subsubsection\let\cl@equation\cl@subsubsection \begin{eqnarray}}{\end{eqnarray}} \def\@rmkcounter#1{\noexpand\arabic{#1}} \def\@rmkcountersep{.} \def\@beginremark#1#2{\trivlist \item[\hskip \labelsep{\bf #2\ #1.}]} \def\@opargbeginremark#1#2#3{\trivlist \item[\hskip \labelsep{\bf #2\ #1\ (#3).}]} \def\@endremarkwithsquare{~\hspace{\fill}~$\Box$\endtrivlist} \makeatother \newenvironment{prf}[1]{\trivlist \item[\hskip \labelsep{\it #1.\hspace*{.3em}}]}{~\hspace{\fill}~$\Box$\endtrivlist} \newenvironment{proof}{\begin{prf}{\bf Proof}}{\end{prf}} \let\tempcirc=\mathord{\raise0.25ex\hbox{$\scriptscriptstyle\tempcirc$}} \def\mathord{\raise0.25ex\hbox{$\scriptscriptstyle\tempcirc$}}{\mathord{\raise0.25ex\hbox{$\scriptscriptstyle\tempcirc$}}} \def\nobreak\hskip.1111em\mathpunct{}\nonscript\mkern-\thinmuskip{:{\nobreak\hskip.1111em\mathpunct{}\nonscript\mkern-\thinmuskip{:} \hskip .3333emplus.0555em\relax} \setlength{\textheight}{220mm} \setlength{\textwidth}{160mm} \setlength{\evensidemargin}{0in} \setlength{\oddsidemargin}{0in} \setlength{\topmargin}{-1cm} \newcommand{{\bf Z}}{{\bf Z}} \newcommand{{\bf Q}}{{\bf Q}} \newcommand{{\bf R}}{{\bf R}} \newcommand{{\bf C}}{{\bf C}} \newcommand{{\bf F}}{{\bf F}} \newcommand{{\bf H}}{{\bf H}} \newcommand{{\bf P}}{{\bf P}} \newcommand{{\rm SL}}{{\rm SL}} \newcommand{{\rm PGL}}{{\rm PGL}} \newcommand{{\rm GL}}{{\rm GL}} \newcommand{\overline{\QQ}}{\overline{{\bf Q}}} \newcommand{{\rm End}}{{\rm End}} \newcommand{{\rm Lie}}{{\rm Lie}} \newcommand{{\rm Gal}}{{\rm Gal}} \newcommand{{\rm Pic}}{{\rm Pic}} \newcommand{{\rm discr}}{{\rm discr}} \newcommand{{\rm Spec}}{{\rm Spec}} \newcommand{{\rm H}}{{\rm H}} \newcommand{{\rm et}}{{\rm et}} \newcommand{\varepsilon}{\varepsilon} \newcommand{{\rm Li}}{{\rm Li}} \newcommand{{\rm pr}}{{\rm pr}} \newcommand{{\rm SO}}{{\rm SO}} \newcommand{\overline{H}}{\overline{H}} \newtheorem{theorem}[subsection]{Theorem.} \newtheorem{proposition}[subsection]{Proposition.} \newtheorem{lemma}[subsection]{Lemma.} \newtheorem{corollary}[subsection]{Corollary.} \newtheorem{tabel}[subsection]{Table.} \newtheorem{conjecture}[subsection]{Conjecture.} \newtheorem{sublemma}[subsubsection]{Lemma.} \newtheorem{subtheorem}[subsubsection]{Theorem.} \newremark{remark}[subsection]{Remark} \newremark{subremark}[subsubsection]{Remark} \newremark{notation}[subsection]{Notation} \newremark{example}[subsection]{Example} \newremark{definition}[subsection]{Definition} \renewcommand{\baselinestretch}{1.2} \begin{document} \title{Special points on the product of two modular curves.} \author{Bas Edixhoven\footnote{partially supported by the Institut Universitaire de France}} \maketitle \section{Introduction.}\label{section1} It is well known that the $j$-invariant establishes a bijection between ${\bf C}$ and the set of isomorphism classes of elliptic curves over~${\bf C}$, see for example \cite{Silverman1}. The endomorphism ring of an elliptic curve $E$ over ${\bf C}$ is either ${\bf Z}$ or an order in an imaginary quadratic extension of ${\bf Q}$; in the second case $E$ is said to be a CM elliptic curve (CM meaning complex multiplication). A complex number $x$ is said to be CM if the corresponding elliptic curve over ${\bf C}$ is~CM. A point $(x_1,x_2)$ in ${\bf C}^2$ is defined to be CM if both $x_1$ and $x_2$ are~CM. The aim of this article is to determine all irreducible algebraic curves $C$ in ${\bf C}^2$ containing infinitely many CM points. In other words, we want to determine all irreducible polynomials $f$ in ${\bf C}[x_1,x_2]$ that vanish at infinitely many CM points. The motivation for doing this comes from a conjecture of Frans Oort (see \cite[Chapter~IV, \S1]{Moonen1} for a precise statement), saying roughly that the irreducible components of the Zariski closure of any set of CM points in any Shimura variety are sub Shimura varieties. For the irreducible components of dimension zero this is trivially true. For those of dimension one Oort's conjecture was in fact stated earlier by Yves Andr\'e as a problem in \cite[Chapter~X, \S1]{Andre2}. We view ${\bf C}^2$ as the Shimura variety which is the moduli space of pairs of elliptic curves. Then the irreducible sub Shimura varieties of dimension one are the following: ${\bf C}\times\{x_2\}$ with $x_2$ a CM point, $\{x_1\}\times{\bf C}$ with $x_1$ a CM point, or the image in ${\bf C}^2$, under the usual map, of the modular curve $Y_0(n)$ for some integer $n\geq1$. Recall that, for $n\geq1$, $Y_0(n)$ is the modular curve classifying elliptic curves with a cyclic subgroup of order $n$, or, equivalently, cyclic isogenies of degree $n$ between elliptic curves. The usual map from $Y_0(n)$ to ${\bf C}^2$ sends an isogeny to its source and target, i.e., $\phi\colon E_1\to E_2$ is sent to $(j(E_1),j(E_2))$. We will prove the following result, giving evidence for the conjecture just mentioned. \begin{theorem}\label{thm1.1} Assume the generalized Riemann hypothesis for imaginary quadratic fields. Let $C$ be an irreducible algebraic curve in\/ ${\bf C}^2$ containing infinitely many CM points and such that neither of its projections to\/ ${\bf C}$ is constant. Then $C$ is the image of\/ $Y_0(n)$ for some $n\geq1$. \end{theorem} \begin{remark}\label{rmk1.2} In the proof of Theorem~\ref{thm1.1} we will see that the state of the art in analytic number theory is such that the Riemann hypothesis is ``almost not needed'' (see Remark~\ref{rmk5.4}). It is clear that Theorem~\ref{thm1.1} implies similar statements for curves contained in the product of two modular curves. In particular, if one assumes GRH, Oort's conjecture is true for curves contained in the product of two modular curves. \end{remark} \begin{remark}\label{rmk1.3} Ben Moonen has proved Oort's conjecture for the sets of CM points in moduli spaces of abelian varieties such that there exists a prime number $p$ at which all the CM points are canonical in the sense that they have an ordinary reduction of which they are the Serre-Tate canonical lift (see \cite[Chapter~IV, \S1]{Moonen1}). Yves Andr\'e has proved the conclusion of Theorem~\ref{thm1.1} with the Riemann hypothesis replaced by the assumption that the Zariski closure of $C$ in ${\bf P}^1\times{\bf P}^1$ meets $\{\infty\}\times{\bf C}$ only in points $(\infty,x_2)$ with $x_2$ a CM point (see \cite{Andre1}). In the case where $C$ meets the union of $\{\infty\}\times{\bf C}$ and ${\bf C}\times\{\infty\}$ only in $\infty\times\infty$ he has a very simple proof. \end{remark} The idea of the proof of Theorem~\ref{thm1.1} is the following. We use the Galois action on the set of CM $j$-invariants to show that for all but finitely many CM points $(x_1,x_2)$ on $C$ the CM fields of $x_1$ and $x_2$ coincide. Then we consider intersections of $C$ with its images under certain Hecke operators. The Riemann hypothesis implies that $C$ is actually contained in some of these images. To finish, we consider an irreducible component $X$ of the inverse image of $C$ in ${\bf H}\times{\bf H}$, the product of the complex upper half plane by itself, and show that the stabilizer of $X$ in ${\rm SL}_2({\bf R})\times{\rm SL}_2({\bf R})$ is of the kind it should be. \section{Some facts about CM elliptic curves.}\label{section2} Before we start with the proof of Theorem~\ref{thm1.1}, we need to recall some facts about CM elliptic curves. These facts can be found for example in \cite[Appendix~C, \S11]{Silverman1}. First of all, CM elliptic curves are defined over~$\overline{\QQ}$. Let $K$ be an imaginary quadratic extension of ${\bf Q}$, with a given embedding in~$\overline{\QQ}$. Let $O_K\subset K$ be the ring of integers. Every subring $A$ of $O_K$ of finite index is of the form $O_{K,f}:={\bf Z}+fO_K$ for a unique integer $f\geq1$. For $f\geq1$ let $S_{K,f}$ be the set of isomorphism classes of pairs $(E,\alpha)$, with $E$ an elliptic curve over $\overline{\QQ}$ and $\alpha\colon O_{K,f}\to{\rm End}(E)$ an isomorphism of rings inducing the given embedding of $K$ into $\overline{\QQ}$ via the action on ${\rm Lie}(E)$. The group $G_K:={\rm Gal}(\overline{\QQ}/K)$ acts on $S_{K,f}$. But also the Picard group ${\rm Pic}(O_{K,f})$ acts on $S_{K,f}$ by the following formula: \begin{eqn}\label{eqn2.1} (E,[L]) \mapsto E\otimes_{O_{K,f}}L, \end{eqn} where $L$ is an invertible $O_{K,f}$-module, $[L]$ its equivalence class and $E\otimes_{O_{K,f}}L$ the cokernel of the map $p\colon E^2\to E^2$ if $p\colon O_{K,f}^2\to O_{K,f}^2$ has cokernel $L$ (view $p$ as a matrix with coefficients in $O_{K,f}$). If we choose an embedding of $\overline{\QQ}$ in ${\bf C}$ and write $E({\bf C})$ as ${\bf C}$ modulo a lattice $\Lambda$, then $(E\otimes_{O_{K,f}}L)({\bf C})$ is the quotient of ${\bf C}\otimes_{O_{K,f}}L$ by $\Lambda\otimes_{O_{K,f}}L$. The actions by $G_K$ and ${\rm Pic}(O_{K,f})$ on $S_{K,f}$ commute. \begin{proposition}\label{prop2.2} The set $S_{K,f}$ is a ${\rm Pic}(O_{K,f})$-torsor, i.e., the action of\/ ${\rm Pic}(O_{K,f})$ is free and has exactly one orbit. \end{proposition} \begin{proof} (Sketch.) For every $(E,\alpha)$ and $\Lambda$ as above, ${\rm End}_{O_{K,f}}(\Lambda)=O_{K,f}$. Moreover, $O_{K,f}$ is of the form ${\bf Z}[x]/(g)$. It follows that $\Lambda$ is an invertible $O_{K,f}$-module. \end{proof} It follows that $G_K$ acts on $S_{K,f}$ via a morphism $G_K\to{\rm Pic}(O_{K,f})$. This morphism is surjective and unramified outside~$f$. The Frobenius element at a maximal ideal $m$ not containing $f$ is the element $[m]^{-1}$ of ${\rm Pic}(O_{K,f})$ (all this can be seen from deformation theory, using the theorem of Serre-Tate, or from class field theory). Let $H_{K,f}$ be the Galois extension of $K$ corresponding to this quotient ${\rm Pic}(O_{K,f})$ of~$G_K$. We remark that we have $H_{K,f}=K(j(E))$ for all $(E,\alpha)$ in~$S_{K,f}$. \section{The two CM fields are almost always equal.}\label{section3} Let $C_{\bf C}\subset{\bf C}^2$ be as in Theorem~\ref{thm1.1} (i.e., it is irreducible, it contains infinitely many CM points and its two projections to ${\bf C}$ are not constant). Since all CM points have coordinates in $\overline{\QQ}$, $C_{\bf C}$ is defined over $\overline{\QQ}$, in the sense that it is the locus of zeros of an irreducible polynomial, call it $f$, with coefficients in~$\overline{\QQ}$. It will be convenient for us to work with a curve defined over ${\bf Q}$, hence we let $C$ be the union of the finitely many conjugates of~$C_{\bf C}$. Then $C$ is defined by the product $F$ of the Galois conjugates of $f$, if we take $f$ such that it has a non-zero coefficient in~${\bf Q}$. Let $d_1$ and $d_2$ be the degrees of $F$ with respect to the second and first variable. Then $d_i$ is the degree of the $i$th projection from $C$ to~${\bf C}$. For $x$ in ${\bf C}$ we will denote the endomorphism ring of the corresponding elliptic curve by~${\rm End}(x)$. For a CM point $x$ in ${\bf C}$ we will call ${\bf Q}\otimes{\rm End}(x)$ the CM field of~$x$. Note that the isogeny class of a CM elliptic curve over $\overline{\QQ}$ consists of all elliptic curves with the same CM field. We want to prove that $C$ is the image in ${\bf C}^2$ of some~$Y_0(n)$. Our first step in this direction is the following proposition. \begin{proposition}\label{prop3.1} Let $C$ be as above. For all but finitely many CM points $(x_1,x_2)$ in $C$ the CM fields of $x_1$ and $x_2$ coincide. \end{proposition} \begin{proof} Suppose that $(x_1,x_2)$ is a CM point in $C(\overline{\QQ})$ such that the two CM fields $K_1$ and $K_2$ are different. Since $C$ is defined over ${\bf Q}$, ${\bf Q}(x_1,x_2)$ has degree at most $d_2$ over ${\bf Q}(x_1)$ and degree at most $d_1$ over~${\bf Q}(x_2)$. Let $L$ be the field generated by $K_1$ and $K_2$, and $M$ the intersection of $L(x_1)$ and~$L(x_2)$. Let us write ${\rm End}(x_i)=O_{K_i,f_i}$ for $i=1$ and~$2$. The field $L(x_i)$ is an abelian Galois extension of $L$, of degree at least $|{\rm Pic}(O_{K_i,f_i})|/2$. The degrees of $L(x_1,x_2)$ over $L(x_2)$ and $L(x_1)$ are equal to those of $L(x_1)$ and $L(x_2)$ over $M$, respectively. This gives us: \begin{eqn}\label{eqn3.2} |{\rm Pic}(O_{K_i,f_i})| \leq 2d_i[M:L]. \end{eqn} We will now work to get a suitable upper bound for~$[M:L]$. The group ${\rm Gal}(L(x_1,x_2)/{\bf Q})$ is an extension of ${\rm Gal}(L/{\bf Q})$ by the abelian group ${\rm Gal}(L(x_1,x_2)/L)$. Hence the action of ${\rm Gal}(L(x_1,x_2)/{\bf Q})$ on ${\rm Gal}(L(x_1,x_2)/L)$ by conjugation factors through an action of ${\rm Gal}(L/{\bf Q})$. In the same way, ${\rm Gal}(L/{\bf Q})$ acts on the two groups ${\rm Gal}(L(x_i)/L)$, which we view as subgroups of ${\rm Gal}(K_i(x_i)/K_i)$. Now ${\rm Gal}(L/{\bf Q})$ is equal to ${\rm Gal}(K_1/{\bf Q})\times{\rm Gal}(K_2/{\bf Q})$, hence equal to ${\bf Z}/2{\bf Z}\times{\bf Z}/2{\bf Z}$. The action of ${\rm Gal}(L/{\bf Q})$ on ${\rm Gal}(L(x_i)/L)$ factors through ${\rm Gal}(K_i/{\bf Q})$ and as such coincides with the restriction of the action of ${\rm Gal}(K_i/{\bf Q})$ on ${\rm Gal}(K_i(x_i)/K_i)={\rm Pic}(O_{K_i,f_i})$. \begin{lemma}\label{lemma3.3} Let $K$ be a quadratic imaginary field and $f\geq1$. Then the non-trivial element $\sigma$ of\/ ${\rm Gal}(K/{\bf Q})$ acts as $-1$ on\/ ${\rm Pic}(O_{K,f})$. \end{lemma} \begin{proof} The endomorphism $\sigma+1$ of ${\rm Pic}(O_{K,f})$ factors through the norm map from ${\rm Pic}(O_{K,f})$ to~${\rm Pic}({\bf Z})$. \end{proof} Now note that ${\rm Gal}(M/L)$ is a quotient of both ${\rm Gal}(L(x_i)/L)$, so the action of ${\rm Gal}(L/{\bf Q})$ on it is by the non-trivial character given by the first projection, but also by the second projection. This implies that ${\rm Gal}(M/L)$ is killed by multiplication by two. \begin{lemma}\label{lemma3.4} Let $K$ be an imaginary quadratic field and $f\geq1$. Then the dimension of the ${\bf F}_2$-vector space ${\rm Pic}(O_{K,f})\otimes{\bf F}_2$ is at most the number of odd primes dividing the discriminant ${\rm discr}(O_{K,f})$ of $O_{K,f}$ plus ten. \end{lemma} \begin{proof} (Sketch.) The exact bound we give does not matter so much, so we just give some indications. First one notes that there is an exact sequence: \begin{subeqn}\label{eqn3.4.1} (K\otimes{\bf Q}_2)^* \to {\rm Pic}(O_{K,f}) \to {\rm Pic}(O_{K,f}[1/2]) \to 0 . \end{subeqn} Let $S:={\rm Spec}(O_{K,f}[1/2])$ and $T:={\rm Spec}({\bf Z}[1/2])$. The Kummer sequence gives a surjection from ${\rm H}^1(S_{\rm et},{\bf F}_2)$ onto the $2$-torsion subgroup of ${\rm Pic}(S)$, which has the same dimension as ${\rm Pic}(S)\otimes{\bf F}_2$. One deals with ${\rm H}^1(S_{\rm et},{\bf F}_2)$ by projecting to~$T_{\rm et}$. \end{proof} Since ${\rm Gal}(M/L)$ is killed by 2 and a quotient of a subgroup of ${\rm Pic}(O_{K_i,f_i})$, we have: \begin{eqn}\label{eqn3.5} \log_2 [M:L] \leq |\{2\neq p | {\rm discr}(O_{K_i,f_i})\}| + 10, \qquad i\in\{1,2\}. \end{eqn} On the other hand, we have Siegel's theorem (see \cite{Oesterle1}), stating that: \begin{eqn}\label{eqn3.6} \log |{\rm Pic}(O_{K_i,f_i})| = (1/2 + {\rm o}(1))\log |{\rm discr}(O_{K_i,f_i})|, \qquad (|{\rm discr}(O_{K_i,f_i})|\to\infty). \end{eqn} Combining equations (\ref{eqn3.5}) and (\ref{eqn3.6}) shows that $|{\rm Pic}(O_{K_i,f_i})|/[M:L]$ tends to infinity as the discrminiant of $O_{K_i,f_i}$ tends to infinity. But then equation (\ref{eqn3.2}) can hold for only finitely many $(x_1,x_2)$. This ends the proof of Proposition~\ref{prop3.1}. \end{proof} \begin{remark}\label{rmk3.7} The proof of Proposition~\ref{prop3.1} shows actually more: the function on the set of CM points on $C$ that sends $(x_1,x_2)$ to $f_1/f_2$ takes only finitely many values. Using this, one can reduce the proof of Theorem~\ref{thm1.1} to the case where there are infinitely many CM points $(x_1,x_2)$ on $C$ with ${\rm End}(x_1)={\rm End}(x_2)$ (one replaces $C$ by its image under a suitable Hecke correspondence). As we do not know how to exploit this, we do not go into further detail. \end{remark} \begin{remark}\label{rmk3.8} Proposition~\ref{prop3.1} was also proved by Yves Andr\'e in \cite{Andre1}, and also by Ching-Li Chai (not published). \end{remark} \section{Intersecting $C$ with something.} \label{section4} We continue the proof of Theorem~\ref{thm1.1}. So we let $C$ be as before. At this point we already know that we have infinitely many CM points $(x_1,x_2)$ on $C$ for which $x_1$ and $x_2$ are isogeneous because they have the same CM field. We have to prove that there is an integer $n\geq1$ such that for infinitely many $(x_1,x_2)$ there exists an isogeny of degree $n$ between $x_1$ and~$x_2$. A direct approach for this is the following. Consider a CM point $(x_1,x_2)$ such that $x_1$ and $x_2$ have the same CM field, say $K$, and an isogeny from $x_1$ to $x_2$ of minimal degree, say~$n$. One can get an upper bound for $n$ in terms of the discriminants of the~${\rm End}(x_i)$. By Remark~\ref{rmk3.7}, one can assume that ${\rm End}(x_1)={\rm End}(x_2)=O_{K,f}$ and get an upper bound for $n$ from Minkowski's theorem on ideals of small norm representing elements of the class group; the bound is a constant times $|{\rm discr}(O_{K,f})|^{1/2}$. Then one considers the intersection of $C$ with~$Y_0(n)$. The degrees of both projections from $Y_0(n)$ to ${\bf C}$ are equal to $\psi(n)$, where $\psi(n)=n\prod_{p|n}(1+1/p)$. The Picard group of ${\bf P}^1\times{\bf P}^1$ (over a field, say ${\bf Q}$) is isomorphic to ${\bf Z}\times{\bf Z}$, the isomorphism sending an effective divisor to the degrees of its two projections to~${\bf P}^1$. The intersection form is the following: $(a,b)\cdot(c,d)=ad+bc$. Hence the intersection number of the Zariski closures in ${\bf P}^1\times{\bf P}^1$ of $C$ and $Y_0(n)$ is $\psi(n)(d_1+d_2)$. Since both curves we intersect are defined over ${\bf Q}$, the intersection contains all Galois conjugates of $(x_1,x_2)$, of which there are~$|{\rm Pic}(O_{K,f})|$. So if $|{\rm Pic}(O_{K,f})|$ exceeds $\psi(n)(d_1+d_2)$, the proof is finished, since then the intersection is not proper. Unfortunately, equation (\ref{eqn3.6}) does not imply such an inequality. Nevertheless, the idea of intersecting $C$ with something is a good one. Natural ``somethings'' to take are images of $C$ itself under Hecke correspondences. Again, we consider a CM point $(x_1,x_2)$ on $C$ such that the CM fields of $x_1$ and $x_2$ coincide. Let $K$, $f_1$ and $f_2$ be defined by: ${\rm End}(x_i)=O_{K,f_i}$. Let $f$ be the least common multiple of $f_1$ and~$f_2$. One easily checks that the field generated by $H_{K,f_1}$ and $H_{K,f_2}$ is $H_{K,f}$, hence the orbit of $(x_1,x_2)$ under the action of $G_K$ is a ${\rm Gal}(H_{K,f}/K)$-torsor. Recall from \S\ref{section2} that we can identify ${\rm Gal}(H_{K,f}/K)$ with~${\rm Pic}(O_{K,f})$. For $\sigma$ in ${\rm Gal}(H_{K,f}/K)$ corresponding to the class $[I]$ of an invertible ideal $I$ of $O_{K,f}$, there are isogenies from $x_1$ to $\sigma(x_1)$ and from $x_2$ to $\sigma(x_2)$ whose kernels are isomorphic, as $O_{K,f}$-modules, to~$O_{K,f}/I$. Hence if we take $I$ such that $O_{K,f}/I$ is a cyclic group of some order $n$, then $\sigma(x_i)$ is in $T_n(x_i)$ for $i$ equals 1 and~2, where $T_n$ is the correspondence on ${\bf C}$ that sends an elliptic curve to the sum (as divisors) of its quotients by its cyclic subgroups of order~$n$. (Let us note that this $T_n$ is not the same as the correspondence on ${\bf C}$ that is usually called $T_n$ if $n$ is not square free, since the usual one involves a sum over all subgroups of order~$n$.) Let $T_n\times T_n$ be the correspondence on ${\bf C}\times{\bf C}$ that is the product of $T_n$ on each factor: it sends a pair $(E_1,E_2)$ of elliptic curves to the sum of the $(E_1/G_1,E_2/G_2)$, where $G_i$ is a cyclic subgroup of order $n$ in~$E_i$. Then $(x_1,x_2)$ is in the intersection of $C$ and $(T_n\times T_n)C$, because $x_i$ is in $T_n(\sigma(x_i))$ and $(\sigma(x_1),\sigma(x_2))$ is in~$C$. Since both $C$ and $(T_n\times T_n)C$ are defined over ${\bf Q}$, their intersection contains all Galois conjugates of~$(x_1,x_2)$. Hence the intersection has at least $|{\rm Pic}(O_{K,f})|$ elements. Let us now calculate the degrees of the projections of $(T_n\times T_n)C$ to~${\bf C}$. By definition, $(T_n\times T_n)C$ consists of the $(x,y)$ such that there exist $u$ and $v$ in ${\bf C}$ with $(u,v)$ in $C$, and cyclic isogenies of degree $n$ from $u$ to $x$ and from $v$ to~$y$. Let $x$ be in~${\bf C}$. Then there are $\psi(n)$ $u$'s with $x\in T_n(u)$. For each such a $u$ there are $d_1$ $v$'s with $(u,v)$ on $C$. For each such a $v$ there are $\psi(n)$ $y$'s in~$T_n(v)$. This shows that the degree of the first projection of $(T_n\times T_n)C$ is $\psi(n)^2d_1$. Of course, for the second projection one has the analogous result. So, for the intersection number of $C$ and $(T_n\times T_n)C$ we find $2d_1d_2\psi(n)^2$. We conclude that if $|{\rm Pic}(O_{K,f})|$ is bigger than $2d_1d_2\psi(n)^2$, then $C$ is contained in $(T_n\times T_n)C$. The next thing to do is to see if there do exist ideals $I$ with the required properties. Let $x_1$, $x_2$, $K$ and $f$ be as above. Let $p$ be a prime number that splits in $O_{K,f}$, i.e., such that $O_{K,f}\otimes{\bf F}_p$ is isomorphic to ${\bf F}_p\times{\bf F}_p$. For $I$ we take one of the two maximal ideals containing~$p$. As explained above, we have the following implication: \begin{eqn}\label{eqn4.1} 2d_1d_2(p+1)^2 < |{\rm Pic}(O_{K,f})| \quad\mbox{\rm implies} \quad C\subset (T_p\times T_p)C . \end{eqn} Equation~(\ref{eqn3.6}) tells us that $|{\rm Pic}(O_{K,f})|=|{\rm discr}(O_{K,f})|^{1/2+{\rm o}(1)}$. So we want $p$ to be at most something as $|{\rm discr}(O_{K,f})|^{1/4}$. More precisely: \begin{proposition}\label{prop4.2} Suppose that there exists $\varepsilon>0$ such that, when $K$ ranges through all imaginary quadratic fields and $f$ through all positive integers, the number of primes $p<|{\rm discr}(O_{K,f})|^{1/4-\varepsilon}$ that are split in $O_{K,f}$ tends to infinity as $|{\rm discr}(O_{K,f})|$ tends to infinity. Then there are infinitely many primes $p$ such that $C$ is contained in $(T_p\times T_p)C$. \end{proposition} \begin{proof} Because we have infinitely many CM points $(x_1,x_2)$ on $C$, we know that the discriminants $|{\rm discr}(O_{K,f})|$ associated to them as above tend to infinity. The implication (\ref{eqn4.1}) and equation (\ref{eqn3.6}) give us the infinitely many required primes. \end{proof} \section{Existence of small split primes.} \label{section5} The aim of this section is to prove the hypothesis in Proposition~\ref{prop4.2}. It turns out that this is no problem at all if one assumes GRH for imaginary quadratic fields and uses the resulting effective Chebotarev theorem of Lagarias, Montgomery and Odlyzko as stated in~\cite{Serre1}. For $K$ an imaginary quadratic field, $f$ a positive integer and $x\geq2$ a real number, let $\pi_{K,f}(x)$ be the number of primes $p\leq x$ that are split in~$O_{K,f}$, let $d_K:=|{\rm discr}(O_K)|$ and let $d_{K,f}:=|{\rm discr}(O_{K,f})|$. Note that $d_{K,f}=f^2d_K$. As usual, let ${\rm Li}(x):=\int_2^xdt/\log(t)$. Theorem~4 of \cite{Serre1} and the second remark following it say that, for $x$ sufficiently big and for all $K$ as above for which GRH holds, one has: \begin{eqn}\label{eqn5.1} \left|\pi_{K,1}(x)-{1\over2}{\rm Li}(x)\right| \leq {1\over6}x^{1/2}\left(\log(d_K) + 2\log(x)\right) . \end{eqn} Since the number of primes dividing $f$ is at most $\log_2(f)$, equation (\ref{eqn5.1}) implies: \begin{eqn}\label{eqn5.2} \pi_{K,f}(x) \geq {x\over2\log(x)} \left({\rm Li}(x){\log(x)\over x} - {\log(x)\over 3x^{1/2}} \left(\log(d_K)+2\log(x)\right) - {2\log(x)\log(f)\over x\log(2)}\right). \end{eqn} If $x$ tends to infinity, ${\rm Li}(x)\log(x)/x$ tends to 1 and $\log(x)^2/x^{1/2}$ tends to~0. One checks easily that for $x$ sufficiently big (i.e., bigger than some absolute constant), and bigger than $\log(d_{K,f})^2(\log(\log(d_{K,f}))^2$, one has $\log(x)\log(d_K)/3x^{1/2}<c<1$, with $c$ independent of $K$ and~$f$. Under the same conditions, $\log(x)\log(f)/x$ tends to zero if $x$ tends to infinity. This means that we have proved the following proposition. \begin{proposition}\label{prop5.3} Let $C$ be as before (i.e., as in the beginning of~\S\ref{section3}). Assume GRH for all imaginary quadratic fields. Then there exist infinitely many primes $p$ such that $C$ is contained in $(T_p\times T_p)C$.~\hfill$\Box$ \end{proposition} \begin{remark}\label{rmk5.4} Of course, the question remains whether one can prove the hypothesis of Proposition~\ref{prop4.2} without assuming GRH. Etienne Fouvry tells me the following. He shows that for $r>0$ and all $n$, the set of $d_{K,f}$ such that the number of primes $p<d_{K,f}^r$ that are split in $O_{K,f}$ is at most $n$, has density zero (i.e., the number of such $d_{K,f}<x$ is ${\rm o}(x)$ for $x\to\infty$). Moreover, he says that the exponent $1/4$ is critical, in the sense that one can prove that for all $\varepsilon>0$, the number of primes $p<d_{K,f}^{1/4+\varepsilon}$ that are split in $O_{K,f}$ tends to infinity as $d_{K,f}$ tends to infinity. To prove this, he uses a result of Linnik and Vinogradov in \cite{LinnikVinogradov}, see also \cite{Friedlander}. The central point in \cite{LinnikVinogradov} is an upper bound for short character sums by Burgess, in which the exponent $1/4+\varepsilon$ appears. This $1/4$ has not moved in the last 30 years. \end{remark} \section{Some topological arguments.}\label{section6} In this section we finish the proof of Theorem~\ref{thm1.1} by combining Proposition~\ref{prop5.3} with the following theorem, which gives yet another characterization of modular curves. \begin{theorem}\label{thm6.1} Let $C$ in ${\bf C}^2$ be an irreducible algebraic curve. Let $d_1$ and $d_2$ be the degrees of its two projections to~${\bf C}$. Suppose that $d_1$ and $d_2$ are both non-zero, and that we have $C\subset (T_n\times T_n)C$ for some square free integer $n>1$ that is composed of primes $p\geq \max\{5,d_1\}$. Then $C$ is the image of\/ $Y_0(m)$ in\/ ${\bf C}^2$ for some $m\geq1$. \end{theorem} Let us first show that this theorem and Proposition~\ref{prop5.3} imply Theorem~\ref{thm1.1}. So let $C_{\bf C}$ and $C$ be as in the beginning of~\S\ref{section3}. Recall that $C$ is the union of the finitely many Galois conjugates of the irreducible component $C_{\bf C}$ of it. We know that there are infinitely many primes $p$ such that $C$ is contained in $(T_p\times T_p)C$. For such a prime $p$, let $T_{C,p}$ denote the correspondence on $C$ induced by $T_p\times T_p$. By this we mean the following. The correspondence $T_p\times T_p$ on ${\bf C}^2$ is given by the map from $Y_0(p)\times Y_0(p)$ to ${\bf C}^2\times{\bf C}^2$ that sends a point $(\phi,\psi)$ to $(s(\phi),s(\psi),t(\phi),t(\psi))$, where $s$ and $t$ stand for source and target, respectively. Take the inverse image of $C\times C$ in $Y_0(p)\times Y_0(p)$, and delete its zero-dimensional part; that, together with its two maps to $C$, is~$T_{C,p}$. We have to show that a suitable product $T_{C,p_1}\cdots T_{C,p_r}$ with $r\geq1$ and the $p_i$ distinct induces a non-trivial correspondence from $C_{\bf C}$ to itself, because then we can apply Theorem~\ref{thm6.1} to $C_{\bf C}$ with $n=p_1\cdots p_r$. Let $S$ be the finite set of irreducible components of~$C$. Then each $T_{C,p}$ induces a correspondence $T_{S,p}$ on $S$ that is surjective in the sense that both maps from $T_{S,p}$ to $S$ are surjective. Moreover, the Galois group $G_{\bf Q}$ acts transitively on $S$, and all $T_{S,p}$ are compatible with this action. Let $x_0$ in $S$ correspond to~$C_{\bf C}$. If there is some $T_{S,p}$ such that $x_0$ is in $T_{S,p}x_0$, we can take $n=p$. So suppose that for all $T_{S,p}$ we have $x_0\not\in T_{S,p}x_0$. Then we have for all $T_{S,p}$ and all $x$ that $x\not\in T_{S,p}x$. One now easily sees that there are $p_1,\ldots,p_r$ distinct with $1\leq r\leq|S|$ and $x_0\in T_{S,p_1}\cdots T_{S,p_r}x_0$. \begin{proof} (Of Theorem~\ref{thm6.1}.) We take an integer $n$ as in the theorem we are proving. Let $T_{C,n}$ be the correspondence on $C$ induced by $T_n\times T_n$, in the sense explained above. (In fact, for everything that follows we could also replace $T_{C,n}$ by one of its irreducible components, but it is useful to see how to exploit all of~it.) We view $T_{C,n}$ as a subset of $C\times C$. The image of $T_{C,n}$ under the map $({\rm pr}_1,{\rm pr}_1)$ from $C\times C$ to ${\bf C}\times{\bf C}$ is the image $T_n$ of $Y_0(n)$ in ${\bf C}\times{\bf C}$. Consider the commutative diagram: \begin{eqn}\label{eqn6.2} \renewcommand{\arraystretch}{1.5} \begin{array}{ccc} C & \to & {\bf C} \\ \uparrow & & \uparrow \\ T_{C,n} & \to & T_n \end{array} \end{eqn} in which the vertical maps are induced by the projections from $C\times C$ and ${\bf C}\times{\bf C}$ on the first factor. \begin{lemma}\label{lemma6.3} The map from $T_{C,n}$ to the fibred product $C\times_{\bf C} T_n$ induced by (\ref{eqn6.2}) is surjective. \end{lemma} \begin{proof} By construction, all four maps in (\ref{eqn6.2}) are finite as morphisms of (possibly reducible) algebraic curves. Therefore, the map from $T_{C,n}$ to $C\times_{\bf C} T_n$ is also a finite morphism of algebraic curves. Hence to show that it is surjective, it suffices to show that $C\times_{\bf C} T_n$ is irreducible, or, equivalently, that the tensor product of the function fields of $C$ and $Y_0(n)$ over ${\bf C}(j)$ is a field. For this, it is enough to prove that the tensor product with $Y_0(n)$ replaced by $Y(n)$ is a field ($Y(n)$ is the modular curve parametrizing elliptic curves with a symplectic basis of their $n$-torsion). The function field of $Y(n)$ is Galois over ${\bf C}(j)$ with Galois group ${\rm SL}_2({\bf Z}/n{\bf Z})/\{\pm1\}$. The group ${\rm SL}_2({\bf Z}/n{\bf Z})$ is isomorphic to the product of the ${\rm SL}_2({\bf F}_{p_i})$, $1\leq i\leq r$; one checks easily that it has no non-trivial subgroup of index at most~$d_1$. This means that the function fields of $C$ and $Y(n)$ are linearly disjoint. \end{proof} For reasons to become clear soon, we now first prove the following lemma. \begin{lemma}\label{lemma6.4} The orbits in $C$ of $T_{C,n}$ are not discrete for the strong topology. \end{lemma} \begin{proof} The morphism ${\rm pr}_1$ from $C$ to ${\bf C}$ is proper, hence the image of a closed subset of $C$ is closed in~${\bf C}$. In particular, the image of the closure of any subset of $C$ is the closure of its image. Hence it is enough to see that the images in ${\bf C}$ of the orbits of $T_{C,n}$ are not discrete. Let $x$ be in $C$, and let $y$ be its image in~${\bf C}$. Lemma~\ref{lemma6.3} implies that ${\rm pr}_1 T_{C,n}x=T_ny$, hence we just have to show that the orbits in ${\bf C}$ of $T_n$ are not discrete. For this we view ${\bf C}$ as the quotient of the complex upper half plane ${\bf H}$ by the group ${\rm SL}_2({\bf Z})$ via the map $\pi\colon\tau\mapsto j({\bf C}/({\bf Z}+{\bf Z}\tau))$. Let $x$ be in ${\bf C}$, and choose $\tau$ in~$\pi^{-1}x$. Then for all $a$ and $b$ in ${\bf Z}$, $\pi(\tau+a)$ and $\pi(n^b\tau)$ are in the orbit of $x$ under~$T_n$. By composing these operations, we see that $\pi(n^b\tau+a)$ and $\pi(\tau+n^{-b}a)$ are in the orbit of~$x$. Taking $a$ non-zero and $b$ big shows that the orbit is not discrete. \end{proof} We view ${\bf C}\times{\bf C}$ as the quotient of ${\bf H}\times{\bf H}$ by the group $\Gamma:={\rm SL}_2({\bf Z})\times{\rm SL}_2({\bf Z})$, via the map: \begin{eqn}\label{eqn6.5} \pi\colon {\bf H}\times{\bf H} \to {\bf C}\times{\bf C}, \quad (\tau_1,\tau_2) \mapsto (j({\bf C}/({\bf Z}+{\bf Z}\tau_1)),j({\bf C}/({\bf Z}+{\bf Z}\tau_2))). \end{eqn} Let $X$ be an irreducible component of the analytic subvariety $\pi^{-1}C$ of ${\bf H}\times{\bf H}$. The group $G:={\rm SL}_2({\bf R})\times{\rm SL}_2({\bf R})$ acts transitively on~${\bf H}\times{\bf H}$. We will study its subgroup $G_X$, the stabilizer of~$X$. What we have to prove is that $G_X$ is the graph of an inner automorphism of ${\rm SL}_2({\bf R})$; this automorphism then tells us for which $m$ our curve $C$ is the image of~$Y_0(m)$. The decisive step in the proof of this is to see that $G_X$ is not discrete (if $C$ is an arbitrary curve in ${\bf C}^2$, then $G_X$ is typically discrete). \begin{lemma}\label{lemma6.6} The group $G_X$ is an analytic subgroup of $G$. \end{lemma} \begin{proof} The action of $G$ on ${\bf H}\times{\bf H}$ is algebraic (it is given by fractional linear transformations). The subgroup $G_X$ consists of exactly those elements $g$ in $G$ that satisfy, for all $x$ in $X$, the two conditions $gx\in X$ and $g^{-1}x\in X$. All these conditions are analytic. \end{proof} \begin{lemma}\label{lemma6.7} The kernels of the two projections from $G_X$ to ${\rm SL}_2({\bf R})$ are discrete. \end{lemma} \begin{proof} This kernel $K$, say for the second projection, is the same as the stabilizer of $X$ in the subgroup ${\rm SL}_2({\bf R})\times\{1\}$ of~$G$. For all $\tau$ in ${\bf H}$, it stabilizes $X_\tau:=X\cap({\bf H}\times\{\tau\})$, which is discrete since $d_2>0$; hence the connected component $K^o$ of $K$ stabilizes every element of $X_\tau$. Now the stabilizer in ${\rm SL}_2({\bf R})$ of the element $i$ of ${\bf H}$ is~${\rm SO}_2({\bf R})$. Because $d_1>0$, $K^o$ is contained in all conjugates of ${\rm SO}_2({\bf R})$, which is~$\{\pm1\}$. \end{proof} \begin{lemma}\label{lemma6.8} The image in\/ ${\rm SL}_2({\bf Z})$ of\/ $\Gamma_X$, the stabilizer of\/ $X$ in\/ $\Gamma$, under the $i$th projection, has index at most~$d_i$. \end{lemma} \begin{proof} We do the proof for $i=2$. We factor the map $\pi\colon{\bf H}\times{\bf H}\to{\bf C}\times{\bf C}$ as follows: \begin{subeqn}\label{eqn6.8.1} {\bf H}\times{\bf H} \to {\bf C}\times{\bf H} \to {\bf C}\times{\bf C} . \end{subeqn} Let $Y$ be the image of $X$ in ${\bf C}\times{\bf H}$. Then $Y$ is an irreducible component of the inverse image $Z$ of $C$ in~${\bf C}\times{\bf H}$. The map from $X$ to $C$ is the quotient for the action of $\Gamma_X$, hence the map from $Y$ to $C$ is the quotient for the action of~${\rm pr}_2\Gamma_X$. It follows that ${\rm pr}_2\Gamma_X$ is the stabilizer in ${\rm SL}_2({\bf Z})$ of $Y$ in $Z$, so the set ${\rm SL}_2({\bf Z})/{\rm pr}_2\Gamma_X$ is the set of irreducible components of~$Z$. But $Z$ is also the fibred product of ${\rm pr}_2\colon C\to{\bf C}$ and ${\bf H}\to{\bf C}$, which implies that $Z$ has at most $d_2$ irreducible components. \end{proof} Lemmas~\ref{lemma6.6}, \ref{lemma6.7} and \ref{lemma6.8} are in fact valid for any curve $C$ in ${\bf C}^2$ for which $d_1$ and $d_2$ are non-zero. The next one crucially exploits that $C\subset (T_n\times T_n)C$. \begin{lemma}\label{lemma6.9} The topological group $G_X$ is not discrete. \end{lemma} \begin{proof} The subgroup $G_X$ of $G$ is analytic, hence closed. It contains~$\Gamma_X$. The inclusion $C\subset (T_n\times T_n)C$ implies that it contains some less trivial elements as well. The correspondence $T_n$ on ${\bf C}$ can be described as follows. Take $z$ in~${\bf C}$; take its inverse image in~${\bf H}$; apply the map $\tau\mapsto n\tau=({n\atop0}{0\atop1})\tau$ to it and take its image in ${\bf C}$; that is~$T_nz$. Another way to say this is: take representatives in ${\rm GL}_2({\bf Q})$ (there are $\psi(n)$ of them) $t_i$ for the quotient set ${\rm SL}_2({\bf Z})({n\atop0}{0\atop1}){\rm SL}_2({\bf Z})/{\rm SL}_2({\bf Z})$; then for $z$ in ${\bf C}$ and $\tau$ in ${\bf H}$ mapping to it, $T_nz$ is the image of the sum of the~$t_i\tau$. It follows that for each $(i,j)$ such that $(t_i,t_j)X$ is contained in $\pi^{-1}C$ we get an element $g_{i,j}$ in $G_X$ of the form $$ g_{i,j} = \gamma_{i,j,1}\cdot \left(n^{-1/2}\left({n\atop0}{0\atop1}\right), n^{-1/2}\left({n\atop0}{0\atop1}\right)\right) \cdot\gamma_{i,j,2}, $$ with $\gamma_{i,j,1}$ and $\gamma_{i,j,2}$ in~$\Gamma$. For $c$ in $C$ and $x$ in $X$ mapping to $c$, $T_{C,n}c$ is the image of the sum of the~$g_{i,j}x$. Let $H$ be the subgroup of $G_X$ generated by $\Gamma_X$ and these elements~$g_{i,j}$. We will prove that $H$ is not discrete. Let $\overline{H}$ be the closure of~$H$. We take an element $x$ in~$X$. The map from $G$ to ${\bf H}\times{\bf H}$ sending $g$ to $gx$ is proper, because the stabilizers of elements of ${\bf H}\times{\bf H}$ are compact. Hence $\overline{H} x$ is also the closure of~$Hx$. The subset $Hx$ of $X$ is discrete if and only if its image in $C$ is discrete, since $H$ contains $\Gamma_X$ and the map $X\to C$ is the quotient for the action of~$\Gamma_X$. By construction, the image of $Hx$ in $C$ is the orbit of $x$ for $T_{C,n}$, which, by Lemma~\ref{lemma6.4}, is not discrete. This proves that $G_X$ is not discrete. \end{proof} We can now quickly finish the proof of Theorem~\ref{thm6.1}. Consider the Lie algebra ${\rm Lie}(G_X)$, which by Lemma~\ref{lemma6.9} is non-zero. Lemma~\ref{lemma6.7} tells us that the two projections ${\rm pr}_i{\rm Lie}(G_X)$ are non-zero. But ${\rm pr}_i{\rm Lie}(G_X)$ is normalized by ${\rm pr}_i\Gamma_X$, which is Zariski dense in ${\rm SL}_2({\bf R})$ by Lemma~\ref{lemma6.8}. Since ${\rm Lie}({\rm SL}_2({\bf R}))$ is simple, it follows that ${\rm pr}_i{\rm Lie}(G_X)$ is equal to ${\rm Lie}({\rm SL}_2({\bf R}))$ for both~$i$. So, since ${\rm SL}_2({\bf R})$ is connected, $G_X$ projects surjectively on both factors ${\rm SL}_2({\bf R})$ of~$G$. Now we apply what is called Goursat's lemma. The kernel of ${\rm pr}_1\colon G_X\to{\rm SL}_2({\bf R})$ is a normal subgroup of ${\rm SL}_2({\bf R})$, viewed as ${\rm SL}_2({\bf R})\times\{1\}$. Since it is discrete and contains $\{1,-1\}$, it is $\{1,-1\}$. The same holds for the other projection, and $G_X$ is the inverse image in $G$ of the graph of an analytic automorphism, $\sigma$ say, of~${\rm SL}_2({\bf R})/\{\pm1\}$. Every such automorphism is inner. Since the ${\rm pr}_i\Gamma_X$ have finite index in ${\rm SL}_2({\bf Z})$, it follows that $\sigma$ is induced from an inner automorphism of the algebraic group~${\rm SL}_{2,{\bf Q}}$. The algebraic group of automorphisms of ${\rm SL}_{2,{\bf Q}}$ is~${\rm PGL}_{2,{\bf Q}}$. Since the map ${\rm GL}_2({\bf Q})\to{\rm PGL}_2({\bf Q})$ is surjective (for example by Hilbert~90), $\sigma$ is given by conjugation by some element $g$ in~${\rm GL}_2({\bf Q})$. So $G_X$ is the set $\{(h,\pm ghg^{-1})\,|\, h\in {\rm SL}_2({\bf R})\}$. Let $x$ be an element of $X$, and write it as $x=(\tau,h\tau)$ with $\tau$ in ${\bf H}$ and $h$ in~${\rm SL}_2({\bf R})$. Since $G_Xx$ is in $X$, which is of dimension two, the stabilizer of $x$ in $G_X$ has dimension at least one. Let $H$ be the stabilizer of $\tau$ in the connected component of identity $G_X^o$, for the action of $G_X^o$ on the first factor ${\bf H}$; then the stabilizer of $h\tau$ for the action on the second factor is the conjugate $g^{-1}hHh^{-1}g$ of~$H$. Since $H$ is of dimension one and connected (it is isomorphic to ${\rm SO}_2({\bf R})$) we must have $H=g^{-1}hHh^{-1}g$, i.e., $g^{-1}h$ normalizes~$H$. Since the normalizer of ${\rm SO}_2({\bf R})$ in ${\rm SL}_2({\bf R})$ is just ${\rm SO}_2({\bf R})$ itself, this means that $g^{-1}h$ is in $H$, or, equivalently, that $h\tau=g\tau$. This means that $X=\{(\tau,g\tau)\,|\,\tau\in{\bf H}\}$. We may replace $g$ by multiples $ag$ of it, with $a$ a non-zero rational number. So we can and do suppose that $g{\bf Z}^2$ is contained in ${\bf Z}^2$ and that ${\bf Z}^2/g{\bf Z}^2$ is cyclic, say of order~$m$. It is now clear that $C$ is~$Y_0(m)$. \end{proof} \section{Some remarks.}\label{section7} \begin{remark}\label{rmk7.1} Our proof of Theorem~\ref{thm1.1} shows in fact that, assuming GRH, for each pair $(d_1,d_2)$ of positive integers there exists an effectively computable number $B(d_1,d_2)$, such that on every irreducible curve $C$ in ${\bf C}^2$ of bi-degree $(d_1,d_2)$ that is defined over ${\bf Q}$ and not a modular curve there are at most $B(d_1,d_2)$ CM points. (Note that under GRH, the statement that $|{\rm Pic}(O_K)|/|{\rm Pic}(O_K)[2]|\to\infty$ is effective.) \end{remark} \begin{remark}\label{rmk7.2} It is not true that all irreducible curves $C$ in ${\bf C}^2$ with $C\subset(T_n\times T_n)C$ for some $n>1$ are the image of some~$Y_0(m)$. Here we construct some examples. Let $n>1$. Let $w_n$ be the Atkin-Lehner involution of $Y_0(n)$: it sends an isogeny to its dual. The correspondence $T_n$ on ${\bf C}$ has the following description. For $z$ in ${\bf C}$, take its inverse image in $Y_0(n)$, take the image of that under $w_n$ and then the image in~${\bf C}$. It follows that for an irreducible curve $C$ in ${\bf C}^2$ such that at least one of the irreducible components of its inverse image in $Y_0(n)\times Y_0(n)$ is stable under the involution $(w_n,w_n)$ we have $C\subset (T_n\times T_n)C$. Let $Z$ be the quotient of $Y_0(n)\times Y_0(n)$ by that involution. Bertini's theorem, see for example \cite[Theorem~6.3]{Jouanolou}, gives the existence of whole families of curves in $Z$ with irreducible inverse image in $Y_0(n)\times Y_0(n)$. Take $C$ to be the image in ${\bf C}^2$ of such an inverse image. \end{remark} \begin{remark}\label{rmk7.3} The condition that $n$ be square free in Theorem~\ref{thm6.1} should not be necessary; it is due to the laziness of the author. \end{remark} \begin{remark}\label{rmk7.4} It is very tempting to try to generalize the methods of this article to the general case of Oort's conjecture. \end{remark} \vspace{2\baselineskip}\noindent {\bf Acknowledgements.} I would like to thank Rutger Noot for interesting discussions on this subject that motivated me enough to work on it, and for his remarks on previous versions of this article. I thank Johan de Jong for his interest and the reference to~\cite{Serre1}. I am very grateful to Etienne Fouvry for a letter in which he explains in detail the results mentioned in Remark~\ref{rmk5.4}. Tim Dokshitzer pointed out a gap in a previous version of this article. I want to thank Fabrice Rouiller for helping me installing the necessary software on the computer with which this article is written. Finally, I am very grateful for an invitation to the Centre for Research in Mathematics at the Institut d'Estudis Catalans in Barcelona, where I could compile my somewhat chaotic and incomplete notes into this article.

9610

alg-geom/9610023

en

https://arxiv.org/abs/alg-geom/9610023

alg-geom/9610023

Fernando Torres

Rainer Fuhrmann, Arnaldo Garcia, and Fernando Torres

On maximal curves

LaTex2e, 17 pages; this article is an improved version of the paper alg-geom/9603013 (by Fuhrmann and Torres)

We study arithmetical and geometrical properties of maximal curves, that is, curves defined over the finite field F_{q^2} whose number of F_{q^2}-rational points reaches the Hasse-Weil upper bound. Under a hypothesis on non-gaps at a rational point, we prove that maximal curves are F_{q^2}-isomorphic to y^q + y = x^m, for some $m \in Z^+$. As a consequence we show that a maximal curve of genus g=(q-1)^2/4 is F_{q^2}-isomorphic to the curve y^q + y = x^{(q+1)/2}.

alg-geom

\section{Introduction} The interest on curves over finite fields was renewed after Goppa \cite{Go} showed their applications to Coding Theory. One of the main features of linear codes arising from curves is the fact that one can state a lower bound for their minimum distance. This lower bound is meaningful only if the curve has many rational points. The subject of this paper is the study of {\it maximal curves}. Let $X$ be a projective, geometrically irreducible and non-singular algebraic curve defined over the finite field $\mathbb F_\ell$ with $\ell$ elements. A celebrated theorem of Weil states that: \[ \#\,X(\mathbb F_\ell) \le \ell+1+2g\sqrt\ell, \] where $X(\mathbb F_\ell)$ denotes the set of $\mathbb F_\ell$-rational points of $X$ and $g$ is the genus of the curve. This bound was proved for elliptic curves by Hasse. The curve $X$ is called \emph{maximal over} $\mathbb F_\ell$ (in this case, $\ell$ must be a square; say $\ell = q^2$) if it attains the Hasse-Weil upper bound; that is, \[ \#\,X(\mathbb F_{q^2}) = q^2 + 1 + 2gq. \] Ihara \cite{Ih} shows that the genus of a maximal curve over $\mathbb F_{q^2}$ satisfies: \[ g \le (q-1)q/2. \] R\"uck and Stichtenoth \cite{R-Sti} show that the Hermitian curve (that is, the curve given by $y^q+y = x^{q+1}$) is the unique (up to $\mathbb F_{q^2}$-isomorphisms) maximal curve over $\mathbb F_{q^2}$ having genus $g = (q-1)q/2$. It is also known that the genus of maximal curves over $\mathbb F_{q^2}$ satisfies (see \cite{F-T} and the remark after Theorem 1.4 here): \[ g \le (q-1)^2/4 \qquad\text{or}\qquad g = (q-1)q/2\,. \] The Hermitian curve is a particular case of the following maximal curves over $\mathbb F_{q^2}$\,: \[ y^q + y = x^m, \text{ with $m$ being a divisor of $(q+1)$}. \] Note that the genus of the above curve is given by $g = (q-1)(m-1)/2$. In Section 1 we derive properties of maximal curves. The main tools being the application to the linear system ${\mathcal D} = |(q+1)P_0|$, $P_0$ a rational point, of St\"ohr-Voloch's approach \cite{S-V} to the Hasse-Weil bound via Weierstrass Point Theory; and Castelnuovo's genus bound for curves in projective spaces: \cite{C}, \cite[p. 116]{ACGH}, \cite[Corollary 2.8]{Ra}. A key result here is the fact that for any point $P$ of the curve, the divisor $qP+{\rm Fr}_{X}(P)$ is linearly equivalent to ${\mathcal D}$ (Corollary \ref{c1.2}). This is a consequence of the particular fashion of the characteristic polynomial $h(t)$ of the Frobenius endomorphism of the Jacobian of the curve, that is, $h(t)$ is a power of a linear polynomial. This property also affects the geometry of the curve. More precisely, we show that maximal curves over $\mathbb F_{q^2}$ of genus $g \ge q-1$ are non-classical curves for the canonical morphism (Proposition \ref{p1.7}). In some other cases one can deduce the non-classicality (for the canonical morphism) of the curve from the knowledge of $h(t)$. We will see this for the Deligne-Lusztig curve associated to the Suzuki group and to the Ree group (Proposition \ref{p?}). The non-classicality of the curve corresponding to the Suzuki group was already proved in \cite{G-Sti}. Our proof is different. It seems that the curve corresponding to the Ree group provides a new example of a non-classical curve. In Section 2, we characterize the curves \[ y^q + y = x^m,\,\,m \text{ being a divisor of $(q+1)$}, \] among the maximal curves over $\mathbb F_{q^2}$\,. This characterization being in terms of non-gaps at a rational point (Theorem \ref{t2.3}). Finally in Section 3, applying the results of Section 2, we show that \[ y^q + y = x^{(q+1)/2}\,, \text{ with $q$ odd}, \] is the unique (up to $\mathbb F_{q^2}$-isomorphisms) maximal curve over $\mathbb F_{q^2}$ with $g = (q-1)^2/4$. \section{Maximal curves} Throughout this paper we use the following notation: \begin{itemize} \item By a curve we mean a projective, geometrically irreducible, non-singular algebraic curve defined over a finite field. \item Let $k$ denote the finite field with $q^2$ elements, where $q$ is a power of a prime $p$. Let ${\bar k}$ denote its algebraic closure. \item The symbol $X(k)$ (resp. ${k}(X)$) stands for the set of $k$-rational points (resp. for the field of $k$-rational functions) of a curve $X$ defined over $k$. \item If $x\in {k}(X)$, then ${\rm div}(x)$ (resp. ${\rm div}_\infty(x)$) denotes the divisor (resp. the pole divisor) of the function $x$. \item Let $P$ be a point of a curve. Then $v_P$ (resp. $H(P)$) stands for the valuation (resp. for the Weierstrass non-gap semigroup) associated to $P$. We denote by $m_i(P)$ the $i$th non-gap at $P$. \item Let $D$ be a divisor on $X$ and $P\in X$. We denote by ${\rm deg}(D)$ the degree of $D$, by ${\rm Supp}(D)$ the support of $D$, and by $v_P(D)$ the coefficient of $P$ in $D$. If $D$ is a $k$-divisor, we set $$ L(D):= \{f\in {k}(X) \mid {\rm div}(f)+D \succeq 0\}, \quad\text{and}\quad \ell(D):={\rm dim}_k\, L(D). $$ \item The symbol ``$\sim$" denotes linear equivalence of divisors. \item The symbol $g^r_d$ stands for a linear system of projective dimension $r$ and degree $d$. \end{itemize} We first review some facts from Weierstrass Point Theory (see \cite{Sch} and \cite{S-V}). \noindent \textbf{Weierstrass points.}\label{wp} Let $X$ be a curve of genus $g$, and ${\mathcal D}=g^r_d$ be a base-point-free $k$-linear system on $X$. Then associated to a point $P\in X$ we have the Hermitian $P$-invariants $j_0(P)=0<j_1(P)<\ldots<j_r(P)\le d$ of ${\mathcal D}$ (also called the $({\mathcal D},P)$-orders). This sequence is the same for all but finitely many points. These finitely many points $P$, where exceptional $({\mathcal D},P)$-orders occur, are called the ${\mathcal D}$-Weierstrass points of $X$. The Weierstrass points of the curve are those exceptional points obtained from the canonical linear system. A curve is called {\it non-classical} if the generic order sequence (for the canonical linear system) is different from $\{0,1,\ldots,g-1\}$. Associated to the linear system ${\mathcal D}$ there exists a divisor $R$ supporting exactly the ${\mathcal D}$-Weierstrass points. Let $\epsilon_0<\epsilon_1<\ldots<\epsilon_r$ denote the $({\mathcal D},Q)$-orders for a generic point $Q\in X$. Then we have \begin{equation}\label{ineq1} \epsilon_i \le j_i(P),\quad \text{for each}\quad i=0,1,2,\dots,r \text{ and for any point $P$,} \end{equation} and also that \begin{equation}\label{degR} {\rm deg}(R)= (\epsilon_1+\ldots+\epsilon_r)(2g-2)+(r+1)d. \end{equation} Associated to ${\mathcal D}$ we also have a divisor $S$ whose support contains the set $X(k)$ of $k$-rational points on $X$. Its degree is given by \begin{equation*} {\rm deg}(S)=(\nu_1+\ldots+\nu_{r-1})(2g-2)+(q^2+r)d, \end{equation*} where the $\nu_i's$ form a subsequence of the $\epsilon_i's$. More precisely, there exists an integer $I$ with $0<I\le r$ such that $\nu_i=\epsilon_i$ for $i<I$, and $\nu_i=\epsilon_{i+1}$ otherwise. Moreover, for $P\in X(k)$, \begin{equation} v_P(S)\ge \sum_{i=1}^{r}(j_i(P)-\nu_{i-1}),\ \text{and}\quad \nu_i\le j_{i+1}(P)-j_1(P),\ \text{for each $i=1,2,\dots,r$}. \end{equation} \noindent \textbf{Maximal curves.} We study some arithmetical and geometrical properties of maximal curves. To begin with we recall the following basic result concerning Jacobians. Let $X$ be a curve, ${\rm Fr}_{\mathcal J}$ the Frobenius endomorphism (relative to the base field) of the Jacobian ${\mathcal J}$ of $X$, and $h(t)$ the characteristic polynomial of ${\rm Fr}_{\mathcal J}$. Let $ h(t)=\prod_{i=1}^{T}h^{r_i}_i(t)$ be the factorization over $\mathbb Z[t]$ of $h(t)$. Then \begin{equation}\label{car} \prod_{i=1}^{T}h_i({\rm Fr}_{\mathcal J})=0\qquad \mbox{on}\ \ {\mathcal J}. \end{equation} This follows from the semisimplicity of ${\rm Fr}_{\mathcal J}$ and the fact that the representation of endomorphisms of ${\rm Fr}_{\mathcal J}$ on the Tate module is faithful (cf. \cite[Thm. 2]{Ta}, \cite[VI, \S3]{L}). In the case of a maximal curve over $k = \mathbb F_{q^2}$, $h(t)=(t+q)^{2g}$. Therefore from (\ref{car}) we obtain the following result, which is contained in the proof of \cite[Lemma 1]{R-Sti}. \begin{lemma}\label{l1.1} The Frobenius map ${\rm Fr}_{\mathcal J}$ (relative to $k$) of the Jacobian ${\mathcal J}$ of a maximal curve over $k$ acts as multiplication by $(-q)$ on ${\mathcal J}$. \end{lemma} Let $X$ be a maximal curve over $k$. Fix $P_0\in X(k)$, and consider the map $f=f^{P_0}: X\to {\mathcal J}$ given by $P\to [P-P_0]$. We have \begin{equation}\label{car1} f\circ {\rm Fr} = {\rm Fr}_{\mathcal J}\circ f, \end{equation} where Fr denotes the Frobenius morphism on $X$ relative to $k$. Hence, from (\ref{car1}) and Lemma \ref{l1.1}, we get: \begin{corollary}\label{c1.2} For a maximal curve $X$ over $k$, it holds $$ {\rm Fr }(P)+qP \sim (q+1)P_0, \,\,\text{ for all points $P$ on $X$.} $$ \end{corollary} It follows then immediately that \begin{corollary}(\cite[Lemma 1]{R-Sti})\label{c1.3} Let $X$ be a maximal curve over $k$, $P_0, P_1 \in X(k)$. Then $(q+1)P_1\sim (q+1)P_0$. \end{corollary} Consider now the linear system ${\mathcal D} = g^{n+1}_{q+1}:= |(q+1)P_0|$. Corollary \ref{c1.3} says that ${\mathcal D}$ is a $k$-invariant of the curve. In particular, its dimension $n+1$ is independent of the choice of $P_0\in X(k)$. Moreover from Corollary \ref{c1.3} we have that $q+1\in H(P_0)$; i.e., $(q+1)$ is a non-gap at a rational point, and hence ${\mathcal D}$ is base-point-free. From now on the letter ${\mathcal D}$ will always denote the linear system $|(q+1)P_0|$, \,\,$P_0$ a rational point, $(n+1)$ being its projective dimension, $R$ will always mean the divisor supporting exactly the ${\mathcal D}$-Weierstrass points, and Fr will always stand for the Frobenius morphism on $X$ relative to $k$. \begin{theorem}\label{t1.4} For a maximal curve $X$ over $k$, the ${\mathcal D}$-orders satisfy (notations being as above): \begin{enumerate} \item[(i)] $\epsilon_{n+1}=\nu_n=q$. \item[(ii)] $j_{n+1}(P)=q+1$ if $P\in X(k)$, and $j_{n+1}(P)=q$ otherwise; in particular, all rational points over $k$ are ${\mathcal D}$-Weierstrass points of $X$. \item[(iii)] $j_1(P)=1$ for all points $P\in X$; in particular, $\epsilon_1=1$. \item[(iv)] If $n\ge2$, then $\nu_1 = \epsilon_1 = 1$. \end{enumerate} \end{theorem} \begin{proof} Statement (iii), for $P\in X(k)$, follows from (i), (ii) and the second inequality in (1.3). From Corollary \ref{c1.2} it follows the assertion (ii) and $\epsilon_{n+1}=q$. Furthermore, it also follows that $j_1(P)=1$ for $P\not\in X(k)$: in fact, let $P'\in X$ be such that ${\rm Fr }(P')=P$; then $P+qP'= {\rm Fr }(P')+ qP'\sim (q+1)P_0$. Now we are going to prove that $\nu_n=\epsilon_{n+1}$. Let $P\in X\setminus \{P_0\}$. Corollary \ref{c1.2} says that $\pi({\rm Fr }(P))$ belongs to the osculating hyperplane at $P$, where $\pi$ stands for the morphism associated to ${\mathcal D}$. This morphism $\pi$ can be defined by a base $\{f_0,f_1,\ldots,f_{n+1}\}$ of\linebreak $L((q+1)P_0)$. Let $x$ be a separating variable of $k(X)\mid k$. Then by \cite[Prop. 1.4(c), Corollary 1.3]{S-V} the rational function below is identically zero $$ w:= {\rm det} \begin{pmatrix} f_0\circ{\rm Fr} & \ldots & f_{n+1}\circ{\rm Fr}\\ D^{\epsilon_0}_x f_0 & \ldots & D^{\epsilon_0}_x f_{n+1}\\ \vdots & \vdots & \vdots\\ D^{\epsilon_n}_x f_0 & \ldots & D^{\epsilon_n}_x f_{n+1} \end{pmatrix}\,\,, $$ since it satisfies $w(P)=0$ for a generic point $P$. Let $I$ be the smallest integer such that the row $(f_0\circ{\rm Fr},\ldots,f_{n+1}\circ{\rm Fr})$ is a linear combination of the vectors $(D^{\epsilon_i}_x f_0,\ldots,D^{\epsilon_i}_x f_{n+1})$ with $i=0,\ldots,I$. Then according to \cite[Prop. 2.1]{S-V} we have $$ \{\nu_0<\ldots<\nu_n\}=\{\epsilon_0<\ldots<\epsilon_{I-1} <\epsilon_{I+1}<\ldots<\epsilon_{n+1}\}. $$ That $\epsilon_1=1$ follows from statement (iii). Suppose that $\nu_1 > 1$. Since $j_1(P)=1$ for all points of $X$, it follows from the proof of \cite[Thm. 1]{H-V} that \[ \#\,X(k) = (q+1)(q^2-1) - (2g-2). \] From the maximality of $X$, we then conclude $2g = (q-1)\cdot q$. \noindent On the other hand, $\pi$ is a birational morphism as follows from \cite[Prop. 1]{Sti-X} (see also Proposition \ref{p1.5}(iv) here). Then Castelnuovo's genus bound for curves in projective spaces applied to the morphism $\pi$ reads: \begin{equation}\label{eq1.4} 2g \le M\cdot(q-n+e) \le \begin{cases} (2q-n)^2/4n\,, &\quad \text{ if $n$ is even}\\ ((2q-n)^2-1)/4n\,, &\quad \text{ if $n$ is odd,} \end{cases} \end{equation} where $M$ is the integer part of $q/n$ and $e = q-M\cdot n$. We then conclude that $n=1$ and this finishes the proof of the theorem. \end{proof} \begin{remark} For a maximal curve $X$ with $n=1$, we have $\nu_1 = \epsilon_2 = q > 1$. Then the proof above shows that $2g = (q-1)\cdot q$. It then follows from \cite{R-Sti} that the curve $X$ is $k$-isomorphic to the Hermitian curve given by $y^q+y = x^{q+1}$. Also, if $n \ge 2$ then from Castelnuovo's formula (\ref{eq1.4}) we get $g \le (q-1)^2/4$. This is the main result of \cite{F-T}. \end{remark} The next proposition gives information on ${\mathcal D}$-orders and non-gaps at points of $X$. \begin{proposition}\label{p1.5} Let $X$ be a maximal curve over $k$ (notations being as before). Then: \begin{enumerate} \item[(i)] For each point $P$ on $X$, we have $\ell(qP) = n+1$; i.e., we have the following behaviour for the non-gaps at $P$ \[ 0 < m_1(P) <\cdots< m_n(P) \le q < m_{n+1}(P). \] \item[(ii)] If $P$ is not rational over $k$, the numbers below are ${\mathcal D}$-orders at the point $P$ \[ 0 \le q-m_n(P) < \cdots < q-m_1(P) < q. \] \item[(iii)] If $P$ is rational over $k$, the numbers below are exactly the $({\mathcal D},P)$-orders \[ 0 < q+1-m_n(P) <\cdots< q+1 - m_1(P) < q+1. \] In particular, if $j$ is a ${\mathcal D}$-order at a rational point $P$ then $q+1-j$ is a non-gap at $P$. \item[(iv)] If $P \in X(\mathbb F_{q^4})\backslash X(k)$, then $q-1$ is a non-gap at $P$. If $P\notin X(\mathbb F_{q^4})$, then $q$ is a non-gap at $P$. If $P \in X(k)$, then $q$ and $q+1$ are non-gaps at $P$. \item[(v)] Let $P$ be a non-Weierstrass point of $X$ (for the canonical morphism) and suppose that $n\ge2$, then we have for the non-gaps at $P$ that $m_{n-1}(P) = q-1$ and $m_n(P)=q$. \end{enumerate} \end{proposition} \begin{proof} Assertion (i) follows from Corollary 1.2. Let $m(P)$ be a non-gap at a point $P$ of $X$ with $m(P) \le q$, then by definition there exists a positive divisor $E$ disjoint from $P$ with \[ E \sim m(P)\cdot P. \] Summing up to both sides of the equivalence above the divisor $(q-m(P))\cdot P+{\rm Fr }(P)$, we get \[ E + (q-m(P))\cdot P + {\rm Fr }(P) \sim qP + {\rm Fr }(P) \sim (q+1)P_0\,. \] This proves assertions (ii) and (iii). To prove assertion (iv) we just apply (as in \cite[IV, Ex. 2.6]{Har}) the Frobenius morphism to the equivalence in Corollary \ref{c1.2}, getting \[ {\rm Fr}^2(P) + (q-1){\rm Fr }P \sim qP. \] The fact that $q$ and $q+1$ are non-gaps at any rational point follows from assertion (iii) taking $j=0$ and $j=1$. \noindent Now we are going to prove the last assertion (v). From assertion (iv) we know already \[ m_n(P) = q \quad{\rm and}\quad m_{n-1}(P) \le q-1. \] Suppose that $m_{n-1}(P) < (q-1)$. It then follows from Theorem 1.4 and the assertion (ii) above that the generic order sequence for the linear system ${\mathcal D}$ is as given below: \[ \epsilon_0=0 < \epsilon_1=1 < \epsilon_2 = q-m_{n-1}(P) <\cdots< \epsilon_n = q-m_1(P) < \epsilon_{n+1} = q. \] On the other hand, we have that Equation (1.1) implies \[ m_i(Q) \le m_i(P), \text{ for each $i$ and each $Q\in X$}. \] Thus at a rational point $Q \in X$, it follows from assertion (iii) that: \[ v_Q(R) \ge \sum_{i=1}^{n+1}(j_i(Q)-\epsilon_i) = 1 + \sum_{i=1}^{n-1}(m_i(P)- m_i(Q)+1) \ge n. \] From the maximality of $X$, Equation (1.2) and \cite[Thm. 1]{Ho}, we conclude that \[ n(q^2+2gq+1) \le {\rm deg }R \le (n+2)\epsilon_{n+1}(g-1) + (n+2)(q+1). \] Using that $\epsilon_{n+1} = q$, we finally have $nq^2 + qg(n-2) \le 2$. This contradicts the assumption that $n \ge 2$. \end{proof} \begin{example}\label{e1.6} By Theorem \ref{t1.4} we have that all rational points of the curve are ${\mathcal D}$-Weierstrass points. However, these sets may be different from each other as the following example shows: Let $X$ be the hyperelliptic curve defined by $x^2+y^5=1$ over $\mathbb F_{81}$. The curve $X$ is maximal because it is covered by the Hermitian curve $x^{10}+y^{10}=1$ (see \cite[Example VI.4.3]{Sti}). It has genus 2 and at a generic point $P$, we have $m_7(P)=9$. Hence we have ${\mathcal D}=|10P_0|=g^8_{10}$. All the canonical Weierstrass points are trivially rational points, and since $\#X(\mathbb F_{81})=118 > \#\,\{\text{Weierstrass points}\} = 6$, we have two possibilities for the $({\mathcal D},P)$-orders at rational points, namely: (a) If $P$ is a rational non-Weierstrass point; then its orders are $0,1,2,3,4,5,6,7,10$. (b) If $P$ is a Weierstrass point; then its orders are $0,1,2,3,4,5,6,8,10$. These computations follow from Proposition 1.5(iii). From the ${\mathcal D}$-orders in (a) above, we conclude that the generic order sequence for ${\mathcal D}$ is $0,1,2,3,4,5,6,7,9$. Hence, ${\rm deg}(R)=164$ and $v_P(R)=1$ (resp. $v_P(R)=2$) if $P$ satisfies (a) (resp. (b)) above. Since ${\rm deg }R -112\times1-6\times2=40>0$, we then conclude that there exist non-rational ${\mathcal D}$-Weierstrass points. The order sequence at such points must be $0,1,2,3,4,5,6,8,9$ and so there exist 40 non-rational ${\mathcal D}$-Weierstrass points, namely the fixed points of $\sigma\circ{\rm Fr}$, where $\sigma$ denotes the hyperelliptic involution. \end{example} By Proposition \ref{p1.5}(v) we have that $q-n$ is a lower bound for the genus of a maximal curve over $\mathbb F_{q^2}$. We are going to show that classical (for the canonical morphism) maximal curves attain such a bound. \begin{proposition}\label{p1.7} Let $X$ be a maximal curve over $k = \mathbb F_{q^2}$ and let $g \ge 2$ be its genus. Then \begin{enumerate} \item[(i)] If $g > q-n$ (with $n+1=\dim {\mathcal D}$ as before), then $X$ is non-classical for the canonical morphism. In particular, this holds for $g \ge q-1$. \item[(ii)] If $X$ is hyperelliptic and the characteristic is two, then $X$ has just one Weierstrass point for the canonical morphism. \end{enumerate} \end{proposition} \begin{proof} (i) If $X$ is classical, then at a generic point $P$ of the curve $X$ we have \[ m_i(P) = g+i, \quad \forall\,i \in \mathbb N. \] On the other hand, from Proposition 1.5(iv) we have $m_n(P) = q$. We then conclude that $g+n=q$. Now if $g\ge q-1$ and $X$ classical, then $n=1$. Therefore from the remark after Theorem 1.4 we would have $2g=q(q-1)$ and so $g=1$, a contradiction. (ii) Since $X$ is hyperelliptic, the Weierstrass points are the fixed points of the hyperelliptic involution. Let $P, Q$ be Weierstrass points of $X$ (they exist because the genus is bigger than one). From $2P\sim 2Q$ and $2\mid q$, we get $qP\sim qQ$. Therefore by Corollary \ref{c1.2}, $$ qQ+\text{ Fr}(Q)\sim qP+\text{ Fr}(P)\sim qQ+\text{ Fr}(P), $$ and so ${\rm Fr}(P)\sim {\rm Fr}(Q)$. This implies ${\rm Fr}(P)={\rm Fr}(Q)$, and consequently $P=Q$. \end{proof} \begin{remark} Hyperelliptic maximal curves are examples of classical curves for the canonical morphism. It would be interesting to investigate the maximal curves that are both non-hyperelliptic and classical for the canonical morphism. Examples of such curves are the one of genus 3 over $\mathbb F_{25}$ listed by Serre in \cite[\S4]{Se}, and the generalizations of Serre's example obtained by Ibukiyama \cite[Thm. 1]{I}. Another question is whether or not the condition $g=q-n$ characterizes classical (for the canonical morphism) maximal curves. \end{remark} Now we present two non-classical (for the canonical morphism) maximal curves over $\mathbb F_{q^2}$ of genus $g<q-1$. These are the so-called Deligne-Lusztig curves associated to the Suzuki group and to the Ree group. \begin{proposition}\label{p?} (I) Let $s\in \mathbb N,\ r:= 2^{2s+1},\ r_0:= 2^s$, and consider the curve $X$ over $\mathbb F_r$ defined by $$ y^r-y=x^{r_0}(x^r-x). $$ Then \begin{enumerate} \item[(i)] (\cite{H}, \cite{H-Sti}, \cite{Se}) The genus of $X$ is $g=r_0(r-1)$ and this curve is maximal over $\mathbb F_{r^4}$. \item[(ii)] (\cite{G-Sti}) The curve $X$ is non-classical for the canonical morphism. \end{enumerate} (II) Let $s\in N,\ r:=3^{2s+1},\ r_0:=3^s$, and consider the curve $X$ over $\mathbb F_r$ defined by $$ y^r-y=x^{r_0}(x^r-x),\qquad z^r-z=x^{2r_0}(x^r-x). $$ Then \begin{enumerate} \item[(i)] (\cite{H}, \cite{P}, \cite{Se}) The genus of $X$ is $g=3r_0(r-1)(r+r_0+1)$ and this curve is maximal over $\mathbb F_{r^6}$. \item[(ii)] The curve $X$ is non-classical for the canonical morphism. \end{enumerate} \end{proposition} \begin{proof} We first set some notations. We write ${\rm Fr}$ for the Frobenius morphism on $X$ relative to $\mathbb F_{r}$ and $h_i(t)$ for the characteristic polynomial of the Frobenius endomorphism (relative to $\mathbb F_{r^i}$) of the Jacobian of $X$. (I) From \cite[Prop. 4.3]{H}, \cite{H-Sti}, \cite{Se} we know that $g=r_0(r-1)$ and that $h_1(t)=(t^2+2r_0t+r)^g$. If $a_1$ and $a_2$ denote the roots of $h_1(t)$, then we have $a_1+a_2=-2r_0$ and $a_1a_2=r$. It then follows easily that $(a_1a_2)^4=r^4$ and $a_1^4+a_2^4=-2r^2$, and hence that $h_4(t)=(t+r^2)^{2g}$. This shows the maximality over $\mathbb F_{r^4}$. \noindent Now by (\ref{car}) we have ${\rm Fr}_{\mathcal J}^2+2r_0{\rm Fr}_{\mathcal J}+r=0$ on ${\mathcal J}$ and then by (\ref{car1}) we obtain $$ {\rm Fr}^2(P)+2r_0{\rm Fr}(P)+rP\sim (1+2r_0+r)P_0, $$ for all $P\in X$, $P_0\in X(\mathbb F_r)$. Now applying ${\rm Fr}$ to the equivalence above we get $$ {\rm Fr}^3(P)+(2r_0-1){\rm Fr}^2(P)+(r-2r_0){\rm Fr}(P)\sim rP. $$ Hence we conclude that $r\in H(P)$ at a general point $P\in X$, and since $g\ge r$, that $X$ is non-classical for the canonical morphism. (II) From \cite[Prop. 5.3]{H}, \cite{P}, \cite{Se} we already know the formula for the genus and that $h_1(t)=(t^2+r)^a(t^2+3r_0t+r)^b$ with $a, b\in \mathbb N$ and $a+b=g$. Let $a_1, a_2$ denote the roots of some factor of $h_1(t)$. Then in either case we get $(a_1a_2)^6=r^6$ and $a_1^6+a_2^6=-2r^3$ and hence that $h_6(t)=(t+r^3)^{2g}$. This shows the maximality of $X$ over $\mathbb F_{r^6}$. Finally as in the proof of item (I) we conclude that $r^2\in H(P)$ for a generic $P\in X$. Since $g\ge r^2$ the assertion follows. \end{proof} To finish this section on maximal curves, we study some properties involving the morphism $\pi: X\to \mathbb P^{n+1}$ associated to the linear system ${\mathcal D}=|(q+1)P_0|$. \begin{proposition}\label{p1.8} The following statements are equivalent: \begin{enumerate} \item[(i)] The morphism $\pi$ is a closed embedding, i.e. $X$ is $k$-isomorphic to $\pi(X)$. \item[(ii)] For all $P \in X(\mathbb F_{q^4})$, we have that $\pi(P)\in \mathbb P^{n+1}(k) \Leftrightarrow P \in X(k)$. \item[(iii)] For all $P \in X(\mathbb F_{q^4})$, we have that $q$ is a non-gap at $P$. \end{enumerate} \end{proposition} \begin{proof} Let $P\in X$. Since $j_1(P)=1$ (cf. Theorem \ref{t1.4}(iii)), we have that $\pi(X)$ is non-singular at all branches centered at $\pi(P)$. Thus $\pi$ is an embedding if and only if $\pi$ is injective. \begin{claim*} We have $\pi^{-1}(\pi(P))\subseteq \{P,{\rm Fr}(P)\}$ and if $\pi$ is not injective at the point $P$, then $P\in X(\mathbb F_{q^4})\setminus X(k)$ and $\pi(P)\in \mathbb P^{n+1}(k)$. \end{claim*} From Corollary \ref{c1.2} it follows that $\pi^{-1}(\pi(P))\subseteq \{P,{\rm Fr}(P)\}$. Now if $\pi$ is not injective at $P$, then $P\not\in X(k)$ and, since $P\in \pi^{-1}(\pi({\rm Fr}(P)))\subseteq \{{\rm Fr}(P), {\rm Fr}^2(P)\}$, we have ${\rm Fr}^2(P)=P$, i.e. $P\in X(\mathbb F_{q^4})\setminus X(k)$. Furthermore we have $\pi(P)=\pi({\rm Fr}(P))={\rm Fr}(\pi(P))$, i.e. $\pi(P)\in \mathbb P^{n+1}(k)$. This proves the claim above. From this claim the equivalence (i) $\Leftrightarrow$ (ii) follows immediately. As to the implication (i) $\Rightarrow$ (iii), we know that $\dim |{\rm Fr}(P)+qP-P-{\rm Fr}(P)|=\dim |{\rm Fr}(P)+qP| - 2$ (Corollary \ref{c1.2} and \cite[Prop. 3.1(b)]{Har}), i.e. $\ell((q-1)P)=n$ and so $q\in H(P)$, by Proposition 1.5(i). Finally we want to conclude from (iii) that $\pi$ is an embedding. According to the above claim it is sufficient to show that $\pi^{-1}(\pi(P))=\{P\}$, for $P \in X(\mathbb F_{q^4})\setminus X(k)$. Let then $P\in X(\mathbb F_{q^4})$. Since we have $q\in H(P)$, there is a divisor $D\in |qP|$ with $P\notin {\rm Supp}(D)$. In particular, $$ {\rm Fr}(P)+D\sim {\rm Fr}(P)+qP\sim (q+1)P_0, $$ and then $\pi^{-1}(\pi({\rm Fr}(P)))\subseteq {\rm Supp}({\rm Fr}(P)+D)$. So if $\pi^{-1}(\pi(P))=\{P,{\rm Fr}(P)\}$, then we would have that $P\in {\rm Supp}(D)$, a contradiction. This means altogether that $\pi$ is injective and so indeed a closed embedding. \end{proof} \begin{remark} Condition (iii) above is satisfied whenever $q\ge 2g$, and in most of the well known examples of maximal curves the morphism $\pi$ is always an embedding. Then a natural question is whether or not $\pi$ is an embedding for an arbitrary maximal curve. We conjecture that this property is a necessary condition for a maximal curve being covered by the Hermitian curve. \end{remark} \begin{proposition}\label{p1.9} Suppose that $\pi: X\rightarrow \mathbb P^{n+1}$ is a closed embedding. Let $P_0\in X(k)$ and assume furthermore that there exist $r, s \in H(P_0)$ such that all non-gaps at $P_0$ less than or equal to $q+1$ are generated by $r$ and $s$. Then the semigroup $H(P_0)$ is generated by $r$ and $s$. In particular, the genus of $X$ is equal to $(r-1)(s-1)/2$. \end{proposition} \begin{proof} Let $x, y\in k(X)$ with ${\rm div}_\infty(x)=sP_0$ and ${\rm div}_\infty(y)=rP_0$. Since we have that $q, q+1 \in H(P_0)$, then the numbers $r$ and $s$ are coprime. Let $\pi_2: X\rightarrow \mathbb P^2$, be given by $P\mapsto (1:x(P):y(P))$. Then the curves $X$ and $\pi_2(X)$ are birational and the image $\pi_2(X)$ is a plane curve given by an equation of the type below: $$ x^r+\beta y^s+\sum_{is+jr<rs} \alpha_{ij}x^iy^j=0, $$ where $\beta,\alpha_{ij}\in k$ and $\beta\neq 0$. We are going to prove that $\pi_2(P)$ is a non-singular point of the curve $\pi_2(X)$ for all $P\neq P_0$. From this it follows that $g=(r-1)(s-1)/2$ and also that $H(P_0)=\langle r,s\rangle$ (see \cite[Ch. 7]{Ful}, \cite{To}). Let $1,f_1,\ldots,f_{n+1}$ be a basis of $L((q+1)P_0)$, where $n+1:=\dim |(q+1)P_0|$. Then there exist polynomials $F_i(T_1,T_2)\in k[T_1,T_2]$ for $i=1,\ldots,n+1$, such that $$ f_i=F_i(x,y), \qquad \mbox{for}\qquad i=1,\ldots, n+1. $$ The existence of these polynomials follows from the hypothesis on the non-gaps at $P_0$ less than or equal to $(q+1)$. \noindent Consider the maps $\pi |(X\setminus\{P_0\}): X\setminus\{P_0\}\rightarrow \mathbb A^{n+1}$ given by $P\mapsto (f_1(P),\ldots,f_{n+1}(P))$; $\pi_2 |(X\setminus\{P_0\}): X\setminus\{P_0\}\rightarrow \mathbb A^2$, $P\mapsto (x(P),y(P))$; and $\phi : \mathbb A^2\rightarrow \mathbb A^{n+1}$, given by $(p_1,p_2)\mapsto (F_1(p_1,p_2),\ldots,F_{n+1}(p_1,p_2))$. Then the following diagram is commutative $$ \Atriangle[X\setminus\{P_0\}`{\mathbb A}^2`{\mathbb A}^r;\pi_2`\pi`\phi]. $$ Thus we have for a point $P$ of $X\setminus\{P_0\}$ and the corresponding local rings assigned to $\pi(P), \pi_2(P)$ the commutative diagram $$ \Atriangle[O_{\pi(X),\pi(P)}`O_{\pi_2(X),\pi_2(P)}`O_{X,P};f`c`h]\ , $$ where $h$ is injective since $k(X)=k(x,y)$, and $c$ is an isomorphism by assumption. Thus $\pi_2{X}$ is non-singular at $\pi_2{P}$. \end{proof} \section{Certain maximal curves} The curves we have in mind in this section are the ones given by (see \cite{G-V} and \cite{Sch}): \[ y^q + y = x^m,\,\, \text{ with $m$ being a divisor of $(q+1)$}. \] These are maximal curves (with $2g=(m-1)(q-1)$) since they are covered by the Hermitian curve. If $P_0$ is the unique point at infinity of this curve, then the semigroup of non-gaps at $P_0$ is generated by $m$ and $q$ and we have: \[ m\cdot n = q+1, \quad\text{where}\quad (n+2) = \ell((q+1)P_0). \qquad (**) \] The goal of this section is to give a proof that the above condition $(**)$ on non-gaps at a rational point $P_0$ characterizes the curves $y^q+y = x^m$ among the maximal curves over the finite field $k$. \begin{proposition}\label{p2.1} Let $X$ be a maximal curve of genus $g$. Suppose that there exists a rational point $P_0\in X(k)$ such that $n\cdot m=q+1$, with $m$ being a non-gap at $P_0$\,. Then, we have $2g=(q-1)(m-1)$. Also, there are at most two types of $({\mathcal D},P)$-orders at rational points $P \in X(k)$: {\bf Type 1.} The ${\mathcal D}$-orders at $P$ are\,\, $0,1,2,3,\dots,n,q+1$. In this case we have $v_P(R)=1$. {\bf Type 2.} The ${\mathcal D}$-orders at $P$ are\,\, $0,1,m,2m,\dots,(n-1)m, q+1$. In this case we have $w_2 := v_P(R) = n((n-1)m-n-1)/2\,\, + 2$. \noindent Moreover, the set of ${\mathcal D}$-Weierstrass points of $X$ coincides with the set of its $k$-rational points, and the order sequence for ${\mathcal D}$ is $0,1,2,\dots,n,q$. \end{proposition} \begin{proof} The morphism $\pi$ can be defined by $(1:y:\ldots:y^{n-1}:x:y^n)$, where $x, y \in k(X)$ are functions such that $$ {\rm div}_\infty(x)=qP_0\qquad {\rm and}\qquad {\rm div}_\infty(y)=mP_0. $$ The set of ${\mathcal D}$-orders at $P_0$ is of Type 2, as follows from Proposition 1.5(iii). \noindent Let $P\in X\setminus \{P_0\}$. From the proof of \cite[Thm. 1.1]{S-V} and letting $z=y-y(P)$, we have \begin{equation}\label{vals} v_P(z),\ldots,nv_P(z) \end{equation} are $({\mathcal D},P)$-orders. Thus, considering a non-ramified point for $y:X\to \mathbb P^1$, we conclude that the order sequence of the linear system ${\mathcal D}$ is given by $$ \epsilon_i = i\qquad {\rm for}\ \ i=1,\ldots, n, \quad {\rm and}\ \ \epsilon_{n+1}=q. $$ If $P$ is a rational point, by Theorem 1.4, we know that 1 and $(q+1)$ are $({\mathcal D},P)$-orders. We consider two cases: \begin{enumerate} \item[(1)] $v_P(z) = 1$: This implies that the point $P$ is of Type 1. \item[(2)] $v_P(z) > 1$: From assertion (2.2) above, it follows $n\cdot v_P(z) = q+1$ and hence $v_P(z)=m$. Then, we have that the point $P$ is of Type 2. \end{enumerate} If $P$ is not a rational point, by Theorem 1.4, we have that $j_{n+1}(P)=q$. If $v_P(z) > 1$ and using assertion (\ref{vals}), we get \[ n\cdot v_P(z) = q = n\cdot m-1. \] Hence $n=1$ and the $({\mathcal D},P)$-orders are $0,1,q$. This shows that $P$ is not a ${\mathcal D}$-Weierstrass point. If $v_P(z)=1$, again from assertion (\ref{vals}), we have that \[ 0,1,2,\ldots,n,\,\,q \] are the ${\mathcal D}$-orders at the point $P$; i.e., $P$ is not a ${\mathcal D}$-Weierstrass point. This shows the equality of the two sets below: \[ \{{\mathcal D}-\text{Weierstrass points of } X\} = \{k-\text{rational points of } X\}. \] The assertions on $v_P(R)$ follow from \cite[Thm. 1.5]{S-V}. Let $T_1$ (resp. $T_2$) denote the number of rational points $P\in X(k)$ whose $({\mathcal D},P)$-orders are of Type 1 (resp. Type 2). Thus we have from the equality in (1.2) $$ {\rm deg}(R)= (n(n+1)/2+q)(2g-2)+(n+2)(q+1)= T_1 + w_2 T_2. $$ Riemann-Hurwitz applied to $y:X\to \mathbb P^1({\bar k})$ gives $$ 2g-2=-2m + (m-1)T_2\,. $$ Since $T_1+T_2 = \#X(k)= q^2+2gq+1$, and using the two equations above, we conclude after tedious computations that $2g=(m-1)(q-1)$. This finishes the proof of the proposition. \end{proof} Now we are going to prove that maximal curves as in Proposition 2.1 are isomorphic to $y^q+y = x^m$. To begin with we first generalize \cite[Lemma 5]{R-Sti}. \begin{lemma}\label{l2.2} Notations and hypotheses as above. Then, the extension $k(X)\mid k(y)$ is a Galois cyclic extension of degree $m$. \end{lemma} \begin{proof} From the proof of Proposition 2.1 we see that the extension $k(X)\mid k(y)$ is ramified exactly at the rational points of Type 2 and that $T_2 = (q+1)$. Moreover, this extension is totally ramified at those points. Viewing the function $y$ on $X$ as a morphism of degree $m$ \[ y\colon X \to \mathbb P^1(\overline k), \] we consider the elements of the finite field $k$ belonging to the set below \[ V_1 = \{y(P) \mid P \in X(k) \quad \text{is of Type 1}\}. \] Since $T_2=(q+1)$ and the rational points of Type 2 are totally ramified for the morphism $y$, we must have $\#\,V_1 \le (q^2-q)$. Now, above each element of the set $V_1$ there are at most $m$ rational points of the curve $X$, those points being necessarily of Type 1, and hence: \[ \#\,X(k) = q^2+1+2gq = T_1+T_2 \le m\cdot (q^2-q)+(q+1). \] From the genus formula in Proposition 2.1, we then conclude that $\#\,V_1 = (q^2-q)$ and also that above each element of the set $V_1$ there are exactly $m$ rational points of the curve. Now the proof continues as in the proof of \cite[Lemma 5]{R-Sti}. We will repeat their argument here for completeness. Let $\widetilde F$ be the Galois closure of the extension $k(X)\mid k(y)$. The field $k$ is still algebraically closed in $\widetilde F$ since the elements of the set $V_1$ split completely in $k(X)\mid k(y)$. Moreover the extension $\widetilde F\mid k(X)$ is unramified, as follows from Abhyankar's lemma \cite[ch.III.8]{Sti}. Hence, \[ 2\tilde g-2 = [\widetilde F\colon F](2g-2), \] where $\tilde g$ denotes the genus of the field $\widetilde F$. The $(q^2-q)$ elements of the set $V_1$ split completely in $\widetilde F$ and then they give rise to $(q^2-q)m [\widetilde F\colon F]$ rational points of $\widetilde F$ over $k$. Then, from the Hasse-Weil bound, we conclude \[ (q^2-q)m[\widetilde F\colon F] \le q^2+2q+(2\tilde g-2)q = q^2+2q+[\widetilde F\colon F](2g-2)q. \] Substituting $2g = (m-1)(q-1)$ in the inequality above, we finally get: \[ [\widetilde F\colon F] \le \frac{q+2}{q+1} \text{ and hence } \widetilde F=F. \] Note that the extension is cyclic since there exist rational points (those of Type 2) that are totally ramified for the morphism $y$. \end{proof} \begin{theorem}\label{t2.3} Let $X$ be a maximal curve of genus $g$ such that there exists a rational point $P_0 \in X(k)$ with $m\cdot n = (q+1)$, where $m$ is a non-gap at $P_0$\,. Then the curve $X$ is $k$-isomorphic to the curve given by the equation: \[ y_1^q + y_1 = x_1^m\,. \] \end{theorem} \begin{proof} We know that $k(X)\mid k(y)$ is a Galois cyclic extension of degree $m$ and moreover that the functions $1,y,y^2,\ldots,y^{n-1}$ and $x$ form a basis for $L(qP_0)$. Let $\sigma$ be a generator of the Galois group of $k(X)\mid k(y)$. Since $P_0$ is totally ramified, then $\sigma(P_0)=P_0$ and hence $\sigma(L(qP_0)) = L(qP_0)$. Note that the functions $1,y,y^2,\ldots,y^{n-1}$ form a basis for the subspace $L((n-1)m\,P_0)$ and that $\sigma$ acts as the identity on this subspace. Since $m$ and $q$ are relatively prime, we can diagonalize $\sigma$ on $L(q\,P_0)$. Take then a function $v \in L(q\,P_0)$, $v \notin L((n-1)m\,P_0)$, satisfying $\sigma(v) = \lambda v$, with $\lambda$ a primitive $m$-th root of 1. \noindent Then denoting by $N$ the norm of $k(X)\mid k(y)$, we get $N(v) = (-1)^{m+1}\cdot v^m$. \noindent Hence $v^m \in k(y)$ and since it has poles only at $P_0$\,, we must have $v^m = f(y) \in k[y]$. Since $\text{div}_\infty(v) = \text{div}_\infty(x) =q\,P_0$\,, we see that $\text{deg }\,f(y)=q$. Now from the fact that there are exactly $(q+1)$ totally ramified points of $k(X) \mid k(y)$ and that all of them are rational, we conclude that $f(y) \in k[y]$ is separable and has all its roots in $k$. After a $k$-rational change of coordinates, we may assume that $f(0)=0$. Then, we get the following description for the set $V_1$\,: $V_1 = \{\alpha \in k \mid f(\alpha) \ne 0\}$. Knowing that all points of $X$ above $V_1$ are rational points over $k$ and from the equation $v^m = f(y)$, we get: \[ f^n(y) \equiv f^{nq}(y)\,\,\,\,\text{mod }(y^{q^2}-y). \qquad\qquad\qquad(*) \] \noindent{\bf Claim}. $f(y) = a_1\,y+a_q\,y^q$, with $a_1,a_q \in k^*$. We set $f(y) = \sum\limits_{i=1}^q a_i\,y^i$ and $f^n(y) = \sum\limits_{i=n}^{nq} b_i\,y^i$. Clearly, the fact that $a_1,a_q \in k^*$ follows from the fact that $f(y)$ is separable of degree $q$. Suppose that the set $I$ below is non-empty \[ I = \{2 \le i \le q-1 \mid a_i \ne 0\}, \] and then define \[ t = \min\,I \quad\text{and}\quad j = \max\,I. \] Clearly, we have $b_{(n-1)q+j} = n\cdot a_q^{n-1}\cdot a_j \ne 0$. Since the unique solution for $i$ in the congruence\,\,\, $i\,q \equiv (n-1)q+j \text{ mod }(q^2-1)$ , \,\,\,$i$ being smaller than $q^2$, is the one given by $i = (n-1) + j\,q$, it follows from ($*$) above that $b_{(n-1)q+j} = b_{(n-1)+j\,q}^q \ne 0$. \noindent It now follows that $\text{deg }(f^n(y)) = n\,q \ge (n-1)+j\,q$ and hence we get that $n-j \ge 1$ if $n \ge 2$. Note that if $n=1$, then we get $j \le 1$ and the proof of the claim is complete in this case. From now on we then assume $n\ge2$. We then conclude that $t \le j \le (n-1)$. Note that then $(n-1)+t < q$. Clearly, we also have $b_{n-1+t} = n\cdot q_1^{n-1}, a_t\ne 0$. Since the unique solution for $i$ in the congruence\,\,\,$i\,q \equiv n-1+t$\,\,$\text{mod }(q^2-1)$, \,\,\, $i$ being smaller than $q^2$, is the one given by $i = (n-1+t)q$, it follows from ($*$) above that $b_{n-1+t} = b_{(n-1+t)q}^q \ne 0$. \noindent As before, it now follows that $n\,q \ge (n-1+t)q$, and hence $t \le 1$. This gives the desired contradiction and hence the set $I$ is empty, thereby proving the claim. Now we are in a position to finish the proof of the theorem. Denoting \[ f(k) = \{f(\alpha) \mid \alpha \in k\} \quad\text{and}\quad H = \{\beta^m \mid \beta \in k\}, \] we have that $\mathbb F_q^*$ is a subgroup of $H\backslash\{0\}$ of index equal to $n$. Moreover, using the fact that above $V_1$ there are only rational points, we have: \[ f(k) \subseteq H = \underset{\ell=0}{\overset{n- 1}{\bigcup}}\,\xi^{\ell\cdot m}\,\mathbb F_q\,, \] where $\xi$ denotes a primitive element of the field $k$; i.e., $\xi$ is a generator for the multiplicative cyclic group $k^*$. Since $f(k)$ is a $\mathbb F_q$-linear subspace of $k$ as follows from the above claim, we conclude that its dimension is one and hence that $f(k) =\xi^{r\cdot m}\,\mathbb F_q$\,, for some $r$. Finally, putting $x_1 = \xi^{-r}\,v$ and $y_1 = \epsilon\,y$, where $\epsilon$ is the unique element of $k^*$ satisfying \[ \text{Tr }(\epsilon\,\alpha) = \xi^{-r\cdot m}\,f(\alpha), \quad \forall\, \alpha \in k, \] we conclude the proof of the theorem (Tr being the trace operator in $k(X) \mid k(y)$). \end{proof} \begin{remark} Notations being as above. Suppose that $m\cdot n \le q+1$ ($m$ being a non-gap at some rational point $P_0$ of $X$). Then, we have $q+1 \ge m\cdot n \ge m_n(P_0) = q$, where the last equality follows from Proposition 1.5(iv). In case that $m\cdot n=q$, we conjecture that $2g = (m-1)q$ and the curve is $\mathbb F_{q^2}$-isomorphic to a curve given by \[ F(y) = x^{q+1}, \] where $F(y)$ is a $\mathbb F_p$-linear polynomial of degree $m$. We have not been able to prove this possible result yet. We notice that if one could show that the morphism $\pi: X\to \mathbb P^{n+1}$ is a closed embedding, then by Proposition \ref{p1.9} we would have the claimed formula for $g$. Finally we also notice that $(m_1(P)-1)q/2$, $P\in X(k)$, is an upper bound for the genus of maximal curves. This follows from \cite[Thm. 1(b)]{Le}. \end{remark} \begin{example}\label{e2.4} There exist maximal curves that do not satisfy the hypothesis of Theorem \ref{t2.3}. We give two such examples below: (i) Let $X$ be the maximal curve over $\mathbb F_{25}$ and genus $g=3$ listed by Serre in \cite[\S4]{Se}. Let $m,5,6$ be the first three non-gaps at $P\in X(\mathbb F_{25})$. Here we have $6P_0=g^3_6$. We claim that $m=4$ (and so $nm>q+1$). Indeed, if $m=3$ by Proposition 2.1 we would have $g=4$. This example also shows a maximal curve where all the rational points are non-Weierstrass points: in fact, since $5 = {\rm char}(k)>2g-2$ the curve is classical. (ii) Let $X$ be a maximal curve over $\mathbb F_{q^2}$ of genus $g$. Suppose that $q\ge 2g+2$ (e.g. the maximal curves in Proposition \ref{p?} here, \cite[Thm. 3.12, Thm. 3.16]{G-Vl}, \cite[Thm. 1]{I}). Then $X$ does not satisfy the hypothesis of Theorem 2.3. In fact, for $P_0\in X(k)$ we have $m_{g+i}(P_0)=2g+i$ and then $n=q-g$. Therefore $m_1(P_0)n\ge 2n\ge q+2$, the last inequality following from $q \ge 2g+2$. \end{example} \section{Maximal curves of genus $(q-1)^2/4$.} As an interesting application of the preceding section we prove: \begin{theorem}\label{t3.1} Let $X$ be a maximal curve over $\mathbb F_{q^2}$ of genus $g = (q-1)^2/4$\,. Then the curve $X$ is $\mathbb F_{q^2}$-isomorphic to the one given by \[ y^q + y = x^{q+1/2}\,. \] \end{theorem} \begin{proof} From Equation (\ref{eq1.4}) which is Castelnuovo's genus bound applied to the linear system $|(q+1)P_0|$, we have $n=2$ (see remark before Proposition 1.5). From Theorem 2.3, it suffices to prove the existence of a rational point $P$ over $k$ with $m_1(P) = (q+1)/2$. This is clearly true for $q=3$ (since $g=1$ in this case) and hence we can assume $q \ge 5$. We prove firstly some lemmas: \begin{lemma}\label{l3.2} Let $P$ be a rational point over $k$ of the curve $X$ (hypothesis being as in Theorem 3.1). Then we have that\,\, $\ell(2(q+1)P) = 9$ and that either $m_1(P) =(q-1)$ or $m_1(P) = (q+1)/2$. Moreover, the divisor $(2g-2)P$ is a canonical divisor. \end{lemma} \begin{proof} Let $m_i = m_i(P)$ be the $i$-th non-gap at the rational point $P$. We have the following list of non-gaps at $P$: \[ 0 < m_1 < m_2 = q < m_3 = q+1 \le 2m_1 < m_1+m_2 < m_1+m_3 \le 2m_2 < m_2+m_3 < 2m_3\,. \] The inequality $2m_1 \ge (q+1)$ follows from the fact that $n=2$ and $q$ odd. Clearly, \begin{align*} &m_3 = 2m_1 \quad\text{if and only if}\quad m_1 = (q+1)/2; \quad\text{and}\\ &m_1+m_3 = 2m_2 \quad\text{if and only if}\quad m_1 = (q-1). \end{align*} Since $q \ge 5$, one cannot have both equalities above simultaneously. From the above list of non-gaps at $P$, it then follows that $\ell((2q+2)P) \ge 9$. Moreover, after showing that $\ell((2q+2)P) = 9$, it also follows that either $m_1 = (q+1)/2$ or $m_1 = (q-1)$. Let $\pi_2\colon X \to \mathbb P^{r+1}$ be the morphism associated to the linear system $|(2q+2)P|$; we already know that $r \ge 7$ and we have to show that $r=7$. Castelnuovo's bound for the morphism $\pi_2$ gives \[ 2g = \frac{(q-1)^2}{2} \le M\cdot(d-1-(r-e)),\qquad\qquad (*) \] where $d = 2q+2$, $M = \bigg[\dfrac{d-1}{r}\bigg]$ and $d-1 = M\cdot r+e$. Since $(r-e) \ge 1$ we have $d-1-(r-e) \le 2q$, and hence \[ (q-1)^2 \le 4qM \quad\text{and then}\quad q^2-q \le 4qM, \] since the right hand side above is a multiple of $q$. For $r \ge 9$, we now see that \[ q-1 \le 4M \le 4\cdot \frac{2q+1}{9}\,, \quad\text{and then}\quad q \le 13. \] The cases $q \le 13$ are discarded by direct computations in Equation $(*)$ above, and hence we have $r \le 8$. Now we use again Equation $(*)$ to discard also the possibility $r=8$. Since $q$ is odd, we have \begin{align*} 2q+1 \equiv 3(\text{mod}\,8) \quad &\text{or}\quad 2q+1 \equiv 7(\text{mod}\,8).\\ \intertext{It then follows} \begin{cases} M = (q-1)/4\\ \text{and } e=3 \end{cases} \qquad &\text{or}\qquad \begin{cases} M = (q-3)/4\\ \text{and } e=7. \end{cases} \end{align*} Substituting these two possibilities above in Equation $(*)$, one finally gets the desired contradiction; i.e., one gets \[ (q-1)^2 \le (q-1)(q-2) \quad \text{or}\quad (q-1)^2 \le (q-3)\cdot q.\qquad \] Now we prove the last assertion of the lemma. One can easily check that both semigroups $H_1$ and $H_2$ below are symmetric, with exactly $g=(q-1)^2/4$ gaps: \[ H_1 = \langle(q-1),q,q+1\rangle \quad\text{and}\quad H_2 = \big\langle \frac{q+1}{2},q\big\rangle. \] At a rational point $P$ on $X$ the Weierstrass semigroup $H(P)$ must then be equal to $H_1$ or $H_2$\,. Hence the semigroup $H(P)$ is necessarily symmetric and the last assertion follows. \end{proof} \begin{lemma}\label{l3.3} Let ${\mathcal D} = |(q+1)P|$ with $P$ being a rational point of $X$ (hypothesis as in Theorem \ref{t3.1}). Then at any non-rational point $Q$ of $X$, the $({\mathcal D},Q)$-orders are $0,1,2,q$. In particular the order sequence for ${\mathcal D}$ is $0,1,2,q$, and the set of ${\mathcal D}$-Weierstrass points is exactly the set of rational points. \end{lemma} \begin{proof} Let $0,1,j,q$ be the $({\mathcal D},Q)$-orders. Consider the following set $S$: \[ S = \{0,1,2,j,j+1,2j,q,q+1,q+j,2q\}. \] The set $S$ consists of $(2{\mathcal D},Q)$-orders, and hence from Lemma \ref{l3.2} we must have $\#\,S \le 9$. This eliminates the possibilities \[ 3 \le j \le (q-1)/2 \quad\text{and}\quad \frac{q+3}{2} \le j \le q-2, \] and it then follows that $j \in \{2,(q+1)/2,q-1\}$. From Lemma \ref{l3.2} we know that \[ (2g-2)P = \frac{(q+1)(q-3)}{2} P \quad\text{is canonical.} \] Then the following set $S(j)$ consists of orders at $Q$ for the canonical morphism \[ S(j) = \{a+bj+cq \mid a,b,c\in\mathbb N \text{ with } a+b+c \le \frac{q-3}{2}\}. \] One can check that $\#\,S(j) = (q-1)^2/4$ if the value of $j$ belongs to $\{2,(q+1)/2, q-1\}$, and hence that $S(j)$ consists of all canonical orders at the point $Q$. Then the set $H(j)$ below is necessarily a semigroup: \[ H(j) = \mathbb N \setminus (1+S(j)). \] This semigroup property on $H(j)$ is only satisfied for the value $j=2$, as one checks quite easily, and this finishes the proof of this lemma. \end{proof} Now we turn back to the proof of Theorem 3.1. Suppose that $m_1(P) = (q-1)$ at all rational points $P$ on the curve. It then follows from Proposition 1.5(iii) that the $({\mathcal D},P)$-orders are $0,1,2,q+1$ and hence $v_P(R) = 1$, where $R$ is the divisor supporting the ${\mathcal D}$-Weierstrass points. On the other hand, we have \[ \text{deg }R - \#\,X(k) = 3(2g-2) - (q-3)(q+1) = \frac 12 (q+1)(q-3). \] Since $q \ge 5$ and $v_P(R)=1$ for $P$ rational, we would then conclude the existence of non-rational points that are ${\mathcal D}$-Weierstrass points. This contradicts Lemma \ref{l3.3} and hence, from Lemma \ref{l3.2}, we finally conclude the existence of a rational point $P$ satisfying \[ m_1(P) = (q+1)/2. \] \end{proof} We can explore further the idea of the above proofs to obtain a partial analogue of the main result of \cite{F-T}, namely \begin{scholium} Let $X$ be a maximal curve over $k$ whose genus $g$ satisfies $$ (q^2-3q+2)/4<g\le (q-1)^2/4. $$ If $q$ is odd, neither $q$ is a power of 3 nor $q\not\equiv 3\pmod{4}$, then $g=(q-1)^2/4$. \end{scholium} Notice that Example 2.4(i) shows that the hypothesis on $g$ above is sharp. This Scholium is the first step toward a characterization of a maximal curve whose genus is $\frac{q-1}{2}(\frac{q+1}{t}-1)$ with $t\ge 3$.

9311

alg-geom/9311002

en

https://arxiv.org/abs/alg-geom/9311002

alg-geom/9311002

Rick Miranda

Ciro Ciliberto, Angelo Lopez, and Rick Miranda

Projective Degenerations of K3 Surfaces, Gaussian Maps, and Fano Threefolds

24 pages, AMS-LaTeX 1.1

10.1007/BF01232682

In this article we exhibit certain projective degenerations of smooth $K3$ surfaces of degree $2g-2$ in $\Bbb P^g$ (whose Picard group is generated by the hyperplane class), to a union of two rational normal scrolls, and also to a union of planes. As a consequence we prove that the general hyperplane section of such $K3$ surfaces has a corank one Gaussian map, if $g=11$ or $g\geq 13$. We also prove that the general such hyperplane section lies on a unique $K3$ surface, up to projectivities. Finally we present a new approach to the classification of prime Fano threefolds of index one, which does not rely on the existence of a line.

alg-geom

\section*{Introduction} Let $C$ be a stable curve of genus $g$. There is a natural map $\phi_C$, called the Gaussian map (or Wahl map) associated to $C$, \[ \phi_C: \bigwedge^2 H^0(C,\omega_C) \to H^0(C,\Omega^1_C \otimes \omega^{\otimes 2}_C) \] which is defined, in local coordinates, by sending $fdz\wedge gdz$ to $(f'g-fg'){(dz)}^3$. (Note that the target is just $H^0(C,\omega^{\otimes 3}_C)$ if $C$ is smooth.) J. Wahl \cite{wahl} has shown that if $C$ is the smooth hyperplane section of a $K3$ surface of degree $2g-2$ in $\Bbb P^g$, then the Gaussian map cannot be surjective; see also \cite{beauville-merindol}. More generally, F. Zak has shown that if the canonical curve $C$ is $k$-extendable, then the corank of the Gaussian map for $C$ is at least $k$ (see \cite{bertram-ein-lazarsfeld}). Moreover, the corank of the Gaussian map for $C$ governs the dimension of the tangent space to the Hilbert scheme at points representing cones over hyperplane sections of varieties to which $C$ extends. Thus the study of the Gaussian map for $C$ leads naturally to information concerning varieties whose curve section is the canonical image of $C$, namely $K3$ surfaces, Fano threefolds, etc., and their Hilbert schemes. In this paper we are concerned with the following main questions: What is the corank of the Gaussian map for the general curve $C$ which {\em is} the hyperplane section of a $K3$ surface? What consequences can we draw for Fano threefolds from the knowledge of this corank? Concerning the first question we prove that, if $g = 11$ or $g \geq 13$, then the general canonical curve which is the hyperplane section of a $K3$ surface, and which generates the surface's Picard group, has a Gaussian map with corank one (Theorem \ref{corank1_theorem}). In genus $12$, we prove that the general such curve has a Gaussian map with corank two (Proposition \ref{g=12corank2}). For genus $g=10$ and $g \geq 12$, the general curve has a surjective Gaussian map (see \cite{ciliberto-harris-miranda}). For $g \leq 9$ and $g=11$ the corank of the Gaussian map for the general curve has been studied in \cite{ciliberto-miranda1}. In genus $10$, F. Cukierman and D. Ulmer have shown that the corank of the Gaussian map is four for the general hyperplane section of a $K3$ surface. These results can be applied to the Hilbert scheme for $K3$ surfaces and Fano threefolds. Our main results concerning $K3$ surfaces are that, if $g = 11$ or $g \geq 13$, then a general hyperplane section of a $K3$ surface (whose Picard group is generated by the hyperplane class) lies on only one such surface, up to projectivities. This generalizes a result of S. Mukai \cite{mukai}. In low genera we give a new proof of a result of S. Mori and S. Mukai (see \cite{mori-mukai}) to the effect that the general curve of genus at most $9$ or equal to $11$ lies on a $K3$ surfaces. Turning to the Fanos, these techniques give a new approach to the classification of prime Fano threefolds which avoids completely the question of the existence of lines and the method of double projection. First we recover the sharp genus bound for prime Fanos, namely, that such Fanos exist only if $g \leq 10$ or $g = 12$. Then we prove that for these genera, the Hilbert scheme of prime Fanos is reduced and irreducible, and a dense open subset of it represents those prime Fanos which have been classically exhibited by Fano (in genus up to $10$) and by Iskovskih (in genus $12$). Returning to the question of the corank of the Gaussian map for hyperplane sections of $K3$ surfaces, previous partial results in this direction were obtained by C. Voisin \cite{voisin}; her results indicate that the corank of the Gaussian map is at most three for a general curve on a $K3$ surface of high enough genus. Our method for the proof of the corank one theorem is a degeneration technique, similar in spirit to that of \cite{ciliberto-harris-miranda} in that the degenerate curves are suitable graph curves. In that work, the only difficulty is to produce graph curves with surjective Gaussian map; by the general theory graph curves are limits of smooth curves, and by the semicontinuity one obtains the surjectivity for the general smooth curve. In this work the degeneration is quite a bit more tricky to construct. Firstly, we must produce graph curves with corank one Gaussian map, and then we must show that these particular stable curves are limits of hyperplane sections of $K3$ surfaces (with Picard group generated by the hyperplane section). We are therefore led to degenerating not only the curves but also the $K3$ surfaces themselves, to suitable configurations of unions of planes, whose hyperplane sections are then the desired graph curves. Thus the technical part of our proof is in producing the appropriate projective degenerations of $K3$ surfaces. We found it convenient to make the degeneration in two steps. We first show that the general union of two rational normal scrolls (each of degree $g-1$ in $\Bbb P^g$), meeting transversally along a smooth anti-canonical elliptic curve, can be smoothed to a $K3$ surface which has Picard group generated by its hyperplane class. This is described in Section \ref{defs_of_scrolls}, and uses standard deformation-theoretic arguments. We consider this an embedded version of an abstract Type II degeneration of $K3$ surfaces, as described in \cite{kulikov}; see also \cite{BGOD}. This degeneration, which to our knowledge has not been used or systematically constructed previously, is interesting in its own right in that it provides a purely algebraic construction of the component of the Hilbert scheme of such $K3$ surfaces. It is known by transcendental methods that this component is unique, but it would be interesting to provide an algebraic proof. Secondly we show that the union of the two scrolls can be degenerated to a suitable union of planes (whose dual complex is a decomposition of the sphere with a cubic graph as its $1$-skeleton). The particular unions of planes are described in Section \ref{union_of_planes}, and enjoy the property that the general hyperplane section are graph curves (a union of lines) having a corank one Gaussian map. The degeneration is constructed in Section \ref{degs_of_scrolls_to_planes} by projective methods; it is driven by a suitable degeneration of the elliptic double curve and certain linear systems on this curve which induce the scrolls. We consider this an embedded version of an abstract Type III degeneration of $K3$ surfaces, as described in \cite{kulikov}. Such degenerations are also of independent interest; in general it is not known whether a given configuration of planes can be smoothed. In particular we do not know whether any configuration of planes, whose dual complex is a decomposition of the sphere with a cubic graph as its $1$-skeleton, is a degeneration of a $K3$ surface. Our work suggests that this is the case under certain extra hypotheses (see the end of Section \ref{examples}), but we only produce a series of examples in this article, one for each genus. In Section \ref{Gaussmaps} we use the degenerations constructed in the previous sections to draw the conclusions about the Gaussian maps described above. In this section we also make the applications to $K3$ surfaces and Fano threefolds described above. The authors would like to thank the University of Rome II, the C.N.R., and Colorado State University for making the completion of this project possible. \section{Deformations of the Union of Two Scrolls to Smooth $K3$ Surfaces} \label{defs_of_scrolls} \subsection{Smooth Elliptic Curves, $g^1_2$'s, and Rational Normal Scrolls} \label{ENCandRNS} Before proceeding to the main part of the section, we need to recall some basic properties of elliptic normal curves and rational normal scrolls. Let $E$ be a smooth elliptic normal curve of degree $g+1$ in $\Bbb P^g$. Let $L$ be a $g^1_2$ on $E$. The union of the secants to the members of $L$ forms a rational normal scroll $R_L$, which is a surface of degree $g-1$ in $\Bbb P^g$. Moreover $E$ is an anti-canonical divisor in the scroll $R_L$. Conversely, every rational normal scroll containing a smooth elliptic normal curve $E$ can be constructed in this manner. Moreover, the general scroll is obtained in this way. In our situation we are considering the union of two scrolls which meet along an elliptic normal curve $E$. Given the above, we want to consider both of these scrolls as being determined by (different) $g^1_2$'s on $E$. The following Lemma shows that the simultaneous construction of the two scrolls is well-behaved. \begin{lemma} \label{2scrolls} Let $g$ be at least $3$, and let $E$ be an elliptic normal curve of degree $g+1$ in $\Bbb P^g$. Let $L_1$ and $L_2$ be two distinct $g^1_2$'s on $E$. Let $R_1$ and $R_2$ be the corresponding scrolls. Then $R_1$ and $R_2$ meet only along $E$, and meet transversally along $E$. \end{lemma} \begin{pf} The theorem is obvious for $g=3$, so we may assume $g \geq 4$. First we remark that $E$ has no $4$-secant $2$-planes; indeed, given any three points of $E$, the system of hyperplanes through the $3$ points cuts out on $E$ a complete linear series of degree at least $2$, and therefore has no base points. Now suppose that there is an intersection point $p$ of $R_1$ and $R_2$ outside $E$. There is at least one line from each of the scrolls through $p$, and these are different since the $g^1_2$'s are different; these two lines would span a $2$-plane, which is $4$-secant to $E$. A similar argument shows that the intersection is transverse; one replaces $p$ by the infinitely near point of tangency to $E$ at a supposed point of non-transversality. \end{pf} \subsection{The Deformation Theory for the Union of Two Scrolls} In this section we will prove the following theorem. \begin{theorem} \label{smoothing_scrolls} Fix $g \geq 3$, and let $R_1$ and $R_2$ be two smooth rational normal scrolls each of degree $g-1$ in $\Bbb P^g$. Assume that the two scrolls meet transversally along a smooth elliptic normal curve $E$ of degree $g+1$, which is anticanonical in each scroll. Then the union $R = R_1 \cup R_2$ is a flat limit of a family of smooth $K3$ surfaces of degree $2g-2$ in $\Bbb P^g$, which is represented by a reduced component ${\cal H}_g$ of the Hilbert scheme of dimension $g^2+2g+19$. \end{theorem} We note that the existence of an anticanonical smooth elliptic curve in each of the two scrolls almost implies they are general. Indeed, in even genus they are both forced to be isomorphic to $\Bbb F_1$, and in odd genus they are isomorphic either to $\Bbb F_0 (\cong \Bbb P^1\times\Bbb P^1$) or to $\Bbb F_2$. For the proof of the Theorem we need to introduce the $T^1$ sheaf for the union $R$ of the two scrolls. Recall that $T^1$ is defined by the sequence \[ 0 \to T_R \to T_{\Bbb P^g}|_R \to N_R \to T^1 \to 0, \] and for a variety with normal crossings, as $R$ is, this sheaf $T^1$ is locally free of rank one on the singular locus. By \cite{friedman}, we have the computation \begin{equation} \label{T1formula} T^1 \cong N_{E/R_1} \otimes N_{E/R_2} \end{equation} where $E$ is the double curve. In our case each sheaf $N_{E/R_i}$ is locally free rank one, of degree $8$; therefore $T^1$ has degree $16$. For our purposes we need the following. \begin{lemma} \label{F=T1} The sheaf $\;T^1$ is isomorphic to the cokernel of the inclusion $N_X \to N_R|_X$, where $X$ is either $R_1$ or $R_2$. \end{lemma} \begin{pf} A local computation shows that the cokernel $F$ in question is locally free of rank one on the double curve $E$. Consider the diagram \[ \begin{array}{ccccccc} T_{\Bbb P^g}|_R & \to & N_R & \to & T^1 & \to & 0 \\ & & & \searrow & & &\\ \downarrow & & & & N_R|_X & &\\ & & & \nearrow & & \searrow & \\ T_{\Bbb P^g}|_X & \to & N_X & & & & 0 \end{array} \] where the horizontal and diagonal rows are exact. The commutativity of the pentagon shows that $T^1$ surjects onto $F$. Since both are locally free rank one, this surjection must be an isomorphism. \end{pf} Theorem \ref{smoothing_scrolls} is, in turn, a consequence of the following. \begin{lemma} \label{2scroll_lemma} With the notation of Theorem \ref{smoothing_scrolls}, one has \begin{itemize} \item[(a)] $H^1(N_R) = H^2(N_R) = 0$, and \item[(b)] $\dim H^0(N_R) = g^2 + 2g + 19$. \end{itemize} \end{lemma} \begin{pf} Because of the transversality, the normal sheaf $N_R$ is locally free. We begin with the proof of (a). We have the exact sequence \[ 0 \to N_R|_{R_1}(-E) \to N_R \to N_R|_{R_2} \to 0, \] so that it suffices to show that if we set $X = R_1$ or $R_2$, then \begin{equation} \label{star} H^j(N_R|_X) = H^j(N_R|_X(-E)) = 0 \text{ for } j = 1,2. \end{equation} {}From the Lemma \ref{F=T1}, to prove (\ref{star}), it therefore suffices to prove \begin{equation} \label{(a)} H^j(N_X) = H^j(N_X(-E)) = 0 \text{ for } j = 1,2, \text{ and } \end{equation} \begin{equation} \label{(b)} H^j(T^1) = H^j(T^1\otimes {\cal O}_X(-E)) = 0 \text{ for } j = 1,2. \end{equation} \noindent {\em Proof of \ref{(a)}:} {}From the Euler sequence \[ 0 \to {\cal O}_X \to {{\cal O}_X(1)}^{g+1} \to T_{\Bbb P^g}|_X \to 0, \] it follows that $H^1(T_{\Bbb P^g}|_X) = H^2(T_{\Bbb P^g}|_X) = 0$. Note also that $H^2(T_X) = 0$, since \[ {H^2(T_X)}^* \cong H^0(\Omega^1_X \otimes \omega_X) = H^0(\Omega^1_X(-E)) \subseteq H^0(\Omega^1_X) = 0. \] Then, from the normal bundle sequence \[ 0 \to T_X \to T_{\Bbb P^g}|_X \to N_X \to 0 \] we see that $H^j(N_X) = 0$ for $j = 1,2$. By twisting the Euler sequence by ${\cal O}_X(-E)$, and remarking that \[ H^2({\cal O}_X(1)\otimes {\cal O}_X(-E)) \cong H^2(\omega_X(1)) \cong {H^0({\cal O}_X(-1))}^* = 0, \] we see that $H^2(T_{\Bbb P^g}(-E)|_X) = 0$; By twisting the normal bundle sequence by ${\cal O}_X(-E)$, we deduce that $H^2(N_X(-E))=0$. Dualizing the normal bundle sequence and taking cohomology, we see that \[ \begin{array}{ccccccc} H^0(\Omega^1_X) & \to & H^1(N^*_X) & \to & H^1(\Omega^1_{\Bbb P^g}|_X) & \stackrel{\phi}{\to} & H^1(\Omega^1_X). \\ \| & & & & \| & & \| \\ 0 & & & & {\Bbb C} & & {\Bbb C}^2 \end{array} \] Note that the map $\phi$ is injective, since its image is the span of the class of the hyperplane section of $X$ in $H^{1,1}(X)$. Therefore \[ H^1(N_X(-E)) \cong {H^1(N^*_X)}^* = \ker \phi = 0, \] proving (\ref{(a)}). \noindent {\em Proof of \ref{(b)}:} Of course since $T^1$ is supported on $E$, the $H^2$'s vanish. Moreover since $\deg(T^1) = 16$, $H^1(T^1) = 0$. Finally $\deg(T^1\otimes {\cal O}_X(-E)) = 8$, so that $H^1(T^1\otimes {\cal O}_X(-E))= 0$. This finishes the proof of (\ref{(b)}), and therefore of statement (a) of the Lemma. We now turn to the proof of statement (b) of the Lemma. We have that \begin{eqnarray*} \dim H^0(N_R) & = & \chi(N_R) \\ &=& \chi(N_R|_X) + \chi(N_R|_X(-E)) \\ &=& \chi(N_X) + \chi(T^1) + \chi(N_X(-E)) + \chi(T^1\otimes {\cal O}_X(-E)) \\ &=& g^2 +2g + 18 + \chi(N_X(-E)) \\ &=& g^2 + 2g + 18 + \chi(T_{\Bbb P^g}|_X(-E)) + \chi(T_X(-E)) \\ &=& g^2 + 2g + 18 + (g+1)\chi({\cal O}_X(1)\otimes {\cal O}_X(-E)) - \chi({\cal O}_X(-E)) + \chi(T_X(-E)). \end{eqnarray*} Riemann-Roch on the surface $X$ computes $\chi({\cal O}_X(1)\otimes {\cal O}_X(-E)) = 0$ and $\chi({\cal O}_X(-E)) = 1$, whereas $\chi(T_X(-E)) = \chi(\omega_X\otimes T_X) = \chi(\Omega^1_X) = 2$. This finishes the proof of statement (b), and the proof of the Lemma. \end{pf} Note that we have as a by-product of this analysis the following. \begin{corollary} \label{H0NtoH0T1} The natural map from $H^0(N_R)$ to $H^0(T^1)$ is surjective. \end{corollary} \begin{pf} First note that since $H^1(N_X) = 0$ (for $X = R_1$ or $R_2$), we have a surjection from $H^0(N_R|_X)$ to $H^0(T^1)$, by Lemma \ref{F=T1}. Therefore it suffices to show that $H^0(N_R)$ surjects onto $H^0(N_R|_X)$, which is a consequence of (\ref{star}). \end{pf} \begin{pf*}{Proof of Theorem \ref{smoothing_scrolls}} Because $H^1(N_R) = 0$, $R$ is represented by a smooth point $h$ in the Hilbert scheme. Therefore $R$ belongs to a single reduced component ${\cal H}_g$ of the Hilbert scheme of dimension $\dim H^0(N_R) = g^2 + 2g + 19$, by Lemma \ref{2scroll_lemma}. By Corollary \ref{H0NtoH0T1}, a general tangent vector to ${\cal H}_g$ at $h$ represents a first-order embedded deformation of $R$ which smooths the double curve. Therefore the general point in ${\cal H}_g$ represents a smooth irreducible surface $S$. Since $H^1({\cal O}_R) = H^1({\cal O}_R(1)) = 0$ ($R$ is Cohen-Macaulay), the same is true for $S$. Since the hyperplane section of $R$ is a stable canonical curve of genus $g$, the general hyperplane section of $S$ is also a smooth canonical curve. This shows that $S$ is regular, and has trivial canonical bundle. Therefore $S$ is a smooth $K3$ surface. \end{pf*} \subsection{The Picard Group of the General Smoothing} In this subsection we want to identify the component ${\cal H}_g$ of the Hilbert scheme which we have discovered as the component containing the union $R$ of the two scrolls. We will prove in this section that the general surface represented by a point in ${\cal H}_g$ has its Picard group generated by the hyperplane class. \begin{definition} A {\em prime} $K3$ surface of genus $g$ is a smooth $K3$ surface $S$ of degree $2g-2$ in $\Bbb P^g$ such that $\operatorname{Pic}(S)$ is generated by the hyperplane class. \end{definition} \begin{theorem} \label{Pic=Z} With the assumptions of Theorem \ref{smoothing_scrolls}, the general point of the component ${\cal H}_g$ of the Hilbert scheme containing the union $R$ of the two scrolls represents a smooth prime $K3$ surface of genus $g$. \end{theorem} \begin{pf} The argument is essentially the same as that given in \cite{griffiths-harris} and we will only outline the main steps. We first note that we may assume that the two $g^1_2$'s on $E$ (giving rise to the two scrolls $R_1$ and $R_2$ of $R$) and the hyperplane class on the double curve $E$ are linearly independent in $\operatorname{Pic}(E)$. Now consider a general flat family $S_t$ degenerating to $R$, parametrized by $t$ in a disc $\Delta$; the total space of this family in $\Bbb P^g \times \Delta$ is a singular threefold, with ordinary double points at the $16$ points $\{p_j\}$ corresponding to the zeroes of the section of $T^1$ determined by the first-order embedded deformation. Suppose now that we have a line bundle $W_t$ on the general fiber. We will assume for simplicity that $W_t$ is rationally determined, namely that there is a line bundle on the total space restricting to $W_t$ on the general fiber. If this is not the case, one can make a further base change and achieve this, at the cost of complicating somewhat the singularities of the total space. This is easily handled, and goes exactly like the argument in \cite[Appendix C]{griffiths-harris} Now make the total space $\cal S$ for the family smooth, by making a small resolution of each of the $16$ double points. This we can do in such a way that the central fiber $S_0$ is the union of the scroll $R_1$ and the $16$-fold blowup $\tilde{R}_2$ of $R_2$; each of the $16$ exceptional curves $E_i$ for the double point resolutions become $(-1)$-curves in $\tilde{R}_2$. The two surfaces $R_1$ and $\tilde{R}_2$ still meet transversally along $E$. We have two natural bundles on the total space $\cal S$, namely the hyperplane bundle $H$ and the bundle $M = {\cal O}_{\cal S}(\tilde{R}_2)$. We claim that $\operatorname{Pic}(S_0)$ is freely generated by the restriction $H_0$ and $M_0$ of these two bundles. If this is true, then we may finish the proof as follows. Take the line bundle $W$ on the total space which restricts to $W_t$ on the general fiber. By twisting $W$ appropriately with multiples of $H$ and $M$, we may assume that $W$ restricts to $H_0$ on the central fiber. Because the general fiber is a $K3$ surface, the dimension of $H^0(W_t)$ is numerically determined, and so is constant over the entire family; in particular, $h^0(W_t) = h^0(H_0) = g+1$. Hence if $\pi$ is the map from $\cal S$ to $\Delta$, we have that $\pi_*W$ is a trivial bundle of rank $g+1$ on $\Delta$. Moreover $W_t$ has no base points, since the limit $H_0$ has none. Therefore a general section of $\pi_*W$ provide us with a flat family of smooth curves in $\Bbb P^g$ of degree $2g-2$ degenerating to a general section of $H_0$ on the central fiber, which is a general hyperplane section of the union $R$. This limit is a canonical curve in $\Bbb P^g$, hence so is the general member. Therefore the general member of the linear system for $W_t$ is a canonical curve, which is a hyperplane section; hence $W_t$ is the hyperplane bundle. Alternatively, the degeneration argument shows that $W_t \cdot H_t = 2g-2$; if $W_t$ is not isomorphic to $H_t$, then the complete linear series cut out by $W_t$ on $H_t$ has degree $2g-2$ and dimension at least $g$. This is forbidden by Riemann-Roch, showing that $W_t \cong H_t$. To finish we must prove the claim that $\operatorname{Pic}(S_0)$ is freely generated by $H_0$ and $M_0$. Now $\operatorname{Pic}(S_0) \cong \operatorname{Pic}(R_1) \times_{\operatorname{Pic}(E)} \operatorname{Pic}(\tilde{R}_2)$, that is, a line bundle on $S_0$ is determined by line bundles on each component agreeing along the double curve. Let $H_i$ be the hyperplane class for $R_i$ (and $\tilde{R}_2$ if $i=2$), and let $F_i$ be the class of the fiber in $R_i$. Note that $F_i$ restricted to $E$ gives the corresponding $g^1_2$. We have $\operatorname{Pic}(R_1)$ generated by $H_1$ and $F_1$, and $\operatorname{Pic}(\tilde{R}_2)$ generated by $H_2$, $F_2$, and the $16$ exceptional classes $E_j$. Suppose that we have the two classes $a_1H_1 + b_1F_1$ on $R_1$ and $ a_2 H_2 + b_2 F_2 + \sum_j c_j E_j$ on $\tilde{R}_2$ which agree on the double curve $E$. Then \[ a_1 H_E + b_1 g^1_{2,1} = a_2 H_E + b_2 g^1_{2,2} + \sum_j c_j p_j \] must hold in $\operatorname{Pic}(E)$. By a standard monodromy argument (see \cite[Sublemma on page 36]{griffiths-harris}), all of the coefficients $c_j$ must be equal, say to $c$. Note that $\sum_j p_j \equiv 4H_E - (g-3)g^1_{2,1} - (g-3)g^1_{2,2}$, by (\ref{T1formula}). As noted above, since we are able to choose the $g^1_2$'s so that they, along with $H$, are independent, we see by equating coefficients, we have $a_1-a_2 = 4c$, $b_1 = (3-g)c$, and $b_2 = (g-3)c$. This proves that $\operatorname{Pic}(S_0)$ depends freely on the two parameters $c$ and $a_2$. The bundle $H_0$ is that given by $c=0$ and $a_2 = 1$; the bundle $M_0$ is given by $c=1$ and $a_2 = -2$. \end{pf} A priori there are many components to the Hilbert scheme of prime $K3$ surfaces of genus $g$, and what the above theorem shows is that our component ${\cal H}_g$ is one of them. However, the transcendental theory for $K3$ surfaces assures us that in fact there is only one component for the Hilbert scheme of prime $K3$ surfaces of genus $g$, which we will call the prime component. (All other components of the Hilbert scheme of $K3$ surfaces of degree $2g-2$ in $\Bbb P^g$ have the Picard group of the general member generated by a proper submultiple of the hyperplane class.) Thus we know that our component ${\cal H}_g$ is the (unique) prime component; but except where explicitly noted, we will not use this fact. \section{The Union of Planes and the Graph Curves} \label{union_of_planes} In this section we will describe the union of planes to which we will degenerate $K3$ surfaces in the component ${\cal H}_g$. The general hyperplane section of the union of planes is a union of lines, and both the union of planes and the union of lines are conveniently described by a graph. Indeed, these curves have been studied previously in several papers, (see \cite{bayer-eisenbud}, \cite{ciliberto-harris-miranda}, and \cite{ciliberto-miranda1}) and have been referred to as {\em graph curves}. \subsection{Graph Curves and Unions of Planes} First let us briefly recall the graph curve construction. A graph $G$ is {\em simple} if it has no loops or multiple edges; it is {\em trivalent} if each vertex lies on exactly three edges. Given a simple trivalent connected graph $G$, one forms an abstract stable curve $C_G$ by taking a smooth rational curve for each vertex of $G$, and connecting them transversally at one point as the edges of $G$ indicate. Since $G$ is trivalent, there is an integer $g$ such that $G$ has $2g-2$ vertices and $3g-3$ edges. This number $g$ is also the dimension of $H^1$ of the graph, and is the arithmetic genus of the graph curve $C_G$. A connected graph $G$ is {\em $n$-edge-connected} if the removal of less than $n$ edges never disconnects $G$. (Hence $1$-connected simply means connected; $2$-connected means that there are no ``disconnecting'' edges of $G$.) Recall that if $G$ is $3$-edge-connected, then the canonical map for the graph curve $C_G$ embeds $C_G$ into $\Bbb P^{g-1}$, \cite{bayer-eisenbud}; the components of $C_G$ go to straight lines. Now suppose that the trivalent graph $G$ is embedded in the plane, giving not only vertices and edges, but also faces. Using Euler's formula, one sees that there are $g+1$ faces in this decomposition of the plane. It is elementary to see that if $G$ is simple, trivalent, and $2$-edge-connected, then for each vertex, the three adjacent faces are different; we assume this from now on. Given the $2$-edge-connected planar graph $G$, we form a union of $2$-planes in $\Bbb P^g$ as follows. Take $g+1$ independent points in $\Bbb P^g$, and give a $1$-$1$ correspondence between these points and the faces of the decomposition. Each vertex of $G$ lies on three faces, which are then associated to three independent points in $\Bbb P^g$, generating a $2$-plane. The union of these $2$-planes over all vertices of $G$ will be denoted by $S_G$, although it depends not only on the graph $G$ but also on the planar embedding. We want to have the situation that the line intersections of the planes of $S_G$ are exactly encoded by the edges of $G$. This is the case if and only if $G$ is $3$-edge-connected (see \cite{ciliberto-miranda2}). I.e., if $G$ is $3$-edge connected, (as we will assume from now on), then two planes of $S_G$ intersect along a line if and only if the corresponding vertices of $G$ are joined by an edge. Note that a general hyperplane section of $S_G$ is a canonically embedded $C_G$. Moreover, arithmetically, the surface $S_G$ is a $K3$ surface; namely, the Hilbert function is that of a smooth $K3$ surface embedded in $\Bbb P^g$. This is seen by remarking that $S_G$ is arithmetically Cohen-Macaulay \cite[Lemma 2.12]{ciliberto-miranda1}, and has a stable canonical curve as its hyperplane section. \subsection{Examples of Planar Graphs} Now we will present examples of $3$-edge-connected trivalent planar graphs, which give us via the preceding constructions, both unions of planes and graph curves, embedded in projective space. We give one example for each genus $g$ at least $7$, and we denote that graph by $G_g$; the corresponding graph curve will be denoted by $C_g$, and the union of planes by $S_g$. \medskip \noindent {\bf The Odd Genus Case.} Suppose that $g = 2n+1$ is odd. Form a trivalent planar graph $G_{2n+1} = G_g$ with $2g-2 = 4n$ vertices $\{v_1,\dots,v_{4n}\}$. There are $3g-3 = 6n$ edges of $G_{2n+1}$, which can be described as follows. The vertices $v_1,\dots,v_n$ are arranged in a cycle, giving $n$ edges; similarly, the vertices $v_{n+1},\dots,v_{3n}$ lie in a cycle, giving $2n$ additional edges; finally, the vertices $v_{3n+1},\dots,v_{4n}$ lie in a cycle, giving $n$ more edges. All of these $4n$ edges are distinct. The edge $v_i$, for $1 \leq i \leq n$, is connected to the vertex $v_{n+2i-1}$; this adds $n$ more edges to the graph. Similarly, the vertex $v_i$, for $3n+1 \leq i \leq 4n$, is connected to the vertex $v_{2i-5n}$, giving the final $n$ edges to the graph $G_{2n+1}$. This graph $G_{2n+1}$ is planar, as can be seen from the following construction. Begin with $n$ points in the plane forming an $n$-gon; these will be the vertices $\{v_1,\dots,v_n\}$. Attach $n$ pentagons to the boundary of this $n$-gon, each meeting the original $n$-gon in one edge, and the neighboring pentagon on either side in one edge. Finally attach $n$ additional pentagons, each meeting two of the previous set of pentagons, and again meeting the neighboring pentagon on either side in one edge. We refer the reader to Figure 1. \medskip \noindent {\bf The Even Genus Case.} The graph in the case that $g$ is even is obtained from the graph $G_{g-1}$ constructed above by adding two vertices and one edge. One of the two additional vertices is added on the edge joining vertex $1$ to vertex $n$; the other is added on the opposite side of the central $n$-gon, on the edge joining vertex $[(n+1)/2]$ to vertex $[(n+3)/2]$. The added edge joins these two additional vertices. This graph is also planar; the construction is similar to the odd genus case. One begins instead with two polygons, one with $[(n+4)/2]$ sides, and one with $[(n+5)/2]$ sides, joined at one side. The union of the two form an $(n+2)$-gon with one interior edge. Attach $n$ pentagons to the boundary of the $(n+2)$-gon, with exactly two of the pentagons having one of their edges covering two of the edges of the $(n+2)$-gon, which the interior edge meets. Finally attach $n$ additional pentagons, as in the odd genus case. We refer the reader to Figure 1. Note that in this construction $g = 2n+2$. Also note that the numbering of the vertices is slightly different than in the odd genus case. In the next section we will describe a deformation of the surface $S_g$ to a union of two rational normal scrolls. Each of the scrolls degenerates to exactly half of the $2g-2$ planes. We will now describe the two sets of $g-1$ planes to which the scrolls individually degenerate. Recall that the vertices of the graph $G_g$ correspond to the $2$-planes of $S_g$. Therefore we may describe the desired subset of planes by indicating which of the vertices correspond to that subset. We will refer to one subset of $g-1$ vertices as the $\cal A$ subset, and the other as the $\cal B$ subset; we will refer to the corresponding union of planes by $S_A$ and $S_B$ respectively. We again have two cases, depending on the parity of the genus $g$. \medskip \noindent {\bf The Odd Genus Decomposition.} Recall here that $g = 2n+1$, so that each subset $\cal A$ and $\cal B$ consist of $2n$ vertices. We set: \[ \cal A = \{ v_1, v_{n+1},\dots,v_{3n-2}, v_{4n-1} \}, \text{ and } \] \[ \cal B = \{ v_2,\dots,v_n,v_{3n-1},v_{3n},v_{4n},v_{3n+1},\dots,v_{4n-2} \}. \] \medskip \noindent {\bf The Even Genus Decomposition.} Recall here that $g = 2n+2$, so that each subset $\cal A$ and $\cal B$ consist of $2n+1$ vertices. We set: \[ \cal A = \{ v_{n+1}, v_1,v_{n+3},v_{n+4},\dots,v_{3n},v_{4n+1} \},\text{ and } \] \[ \cal B = \{ v_2,v_3,\dots,v_{[(n+1)/2]},v_{n+2},v_{[(n+3)/2]},\dots, v_n, v_{3n+1},v_{3n+2},v_{4n+2},v_{3n+3},v_{3n+4},\dots,v_{4n} \}. \] The reader may consult Figure 2 for a representation of these decompositions. The order chosen above for the subsets $\cal A$ and $\cal B$ may seem a bit strange; however they are such that in that order, the vertices for both subsets form a path in the graph $G_g$. Since the vertices form a path, the corresponding sets of planes $S_A$ and $S_B$ are arranged also in a path, with each plane meeting two others along lines, except the first and last. This configuration is again Cohen-Macaulay, and the hyperplane section is a degenerate form of a rational normal curve, namely is a union of $g-1$ straight lines in $\Bbb P^{g-1}$. Hence each of the configurations $S_A$ and $S_B$ of planes is numerically a rational normal scroll, in the sense that it has the same Hilbert function. Of course, the union $S_A \cup S_B$ is the total configuration $S_g$. The two configurations intersect along a cycle of lines; the number of lines is $g+1$. We have drawn on Figure 2 a dotted path separating the two subsets of vertices $\cal A$ and $\cal B$; this path cuts the graph in exactly those edges which correspond to the lines in which the configurations $S_A$ and $S_B$ meet. Since the path meets each face exactly once, this set of $g+1$ lines forms a cycle through the coordinate points of $\Bbb P^g$ which were chosen to correspond to the faces of the planar decomposition. In particular, this intersection is a cycle of lines which is maximally embedded, and forms a degenerate elliptic normal curve in $\Bbb P^g$ of degree $g+1$. It is useful to have in mind the way in which the separate configurations $S_A$ and $S_B$ of planes are put together. This we show in Figure 3, for the odd genus case. Note that the numbering of the planes is of course the numbering of the corresponding vertices. For later use, an alternative description of these unions of planes will be needed. The intersection of the two chains of planes, as noted above, forms a cycle of $g+1$ lines. Number these (modulo $g+1$) in order around the cycle, and denote them therefore by $\ell_1,\ell_2,\dots,\ell_{g+1}$. Also number the intersection points of these lines in a standard way, so that they are denoted by $p_j$, where $p_j = \ell_j \cap \ell_{j+1}$. Now note that each plane in the configurations is the span of either two intersecting lines $\ell_j$ and $\ell_{j+1}$ (this is the case for the planes at the ends of the chains) or is the span of a line $\ell_j$ and a point $p_k$. Thus to describe the configurations one can simply give these spanning sets. This we now do for the configuration $S_A$ in the odd genus case, assuming that the lines are numbered as indicated in Figure 3. \begin{center} \begin{tabular}{||l|l||} \multicolumn{2}{c}{Table One} \\ \hline Plane & Span of \\ \hline $1$ & $\ell_1$, $\ell_{2n+2}$ \\ \hline $n+1$ & $p_1$, $\ell_{2n+1}$ \\ \hline $n+2i$, $1 \leq i \leq n-2$ & $p_i$, $\ell_{2n+1-i}$ \\ \hline $n+2i+1$, $1 \leq i \leq n-2$ & $p_{2n-i}$, $\ell_{i+1}$ \\ \hline $3n-2$ & $p_{n+2}$, $\ell_n$ \\ \hline $4n-1$ & $\ell_{n+1}$, $\ell_{n+2}$ \\ \hline \end{tabular} \end{center} \section{Degenerations of the Union of Two Scrolls to the Union of Planes} \label{degs_of_scrolls_to_planes} In this section we will describe a deformation of the union of planes $S_g$, which were constructed in the previous section, to a union of two rational normal scrolls. As mentioned above, each of the configurations $S_A$ and $S_B$ will be smoothed to a scroll; care must be taken that this is possible in such a way that the intersection curve also deforms correctly. Let us denote the two scrolls by $R_A$ and $R_B$; $R_A$ degenerates to $S_A$ and $R_B$ to $S_B$. The intersection of the scrolls will be a smooth elliptic normal curve which is an anti-canonical divisor in each scroll; this intersection curve will degenerate to the cycle of lines which is the intersection of the $S_A$ and $S_B$ configuration of planes. Our first step is to reduce the analysis of the degeneration to these intersection curves. The construction of scrolls as the secants to a $g^1_2$ on an elliptic normal curve as described in Section \ref{ENCandRNS} reduces our degeneration construction to the construction of an appropriate degeneration of the elliptic normal curve, along with the degeneration of the two $g^1_2$'s. \subsection{Cycle Degenerations of Elliptic Normal Curves} \label{cycle_degenerations} Let us next discuss the degeneration of an elliptic normal curve which we will use. For every integer $k \geq 2$, there is an abstract degeneration of smooth elliptic curves to a cycle of $k$ rational curves, namely the degeneration denoted by $I_k$ in Kodaira's original construction (see \cite{kodaira}). The total space of this degeneration is a smooth surface $T_k$, mapping to the disc $\Delta$, and the general fiber of the map is an elliptic curve. The central fiber is a cycle of $k$ $\Bbb P^1$'s, each of self-intersection $-2$. This degeneration is unique in the sense that any two such degenerations are isomorphic in a neighborhood of the central fiber. Moreover, one can arrange the isomorphism to identify one component of the central fiber of one with any component of the central fiber of the other, and also to make the identifications of the two cycles in any of the two orderings for the cycle. Suppose that $k \geq g+1$. Choose any $g+1$ of the $k$ components of the central fiber of the degeneration $T_k$. In addition, choose $g+1$ sections of the map to $\Delta$, each meeting one of the chosen $g+1$ components. These sections give a divisor of degree $g+1$ in each fiber of the map to $\Delta$, and the corresponding linear system embeds the fibers into $\Bbb P^g$ as projectively normal curves, giving a map, defined over $\Delta$, from $T_k$ to $\Bbb P^g \times \Delta$. The smooth fibers go to elliptic normal curves of degree $g+1$ in $\Bbb P^g$. The central fiber goes to a cycle of $g+1$ lines which span $\Bbb P^g$. In particular, each of the $g+1$ components which are met by the $g+1$ sections map to one of the lines in the cycle; the $k-g-1$ components not meeting any of the sections are contracted (to rational double points of type $A$). \subsection{The Degeneration of the $g^1_2$'s} For the degeneration of the $g^1_2$'s, we need to view a $g^1_2$ on an elliptic curve not as a linear system but as a $2$-$1$ branched cover of $\Bbb P^1$. With this point of view, a degeneration of a $g^1_2$ becomes a double covering of an appropriate degeneration of $\Bbb P^1$'s. This idea is essentially the notion of ``admissible coverings'', in a very special case. Firstly we present the relevant construction of such degenerations of $\Bbb P^1$ and the double covering. Start with a trivial family $\Bbb P^1 \times \Delta \to \Delta$. Blow up in the central fiber $m-1$ times, creating an abstract degeneration $P_m$ of $\Bbb P^1$ to a chain of $m$ $\Bbb P^1$'s; the ends of the chain are both $(-1)$-curves, and the interior members of the chain are all $(-2)$-curves. In order to have a double covering of $P_m$ whose general fiber is elliptic, the branch locus must meet the general fiber in $4$ points. Choose a smooth $4$-section $B$ which meets the central fiber only in the two end components, and meets each of these twice. Let $D_m(B)$ be the double covering of the surface $P_m$; it is a smooth surface, mapping to $\Delta$, and is a degeneration of elliptic curves. There are two cases for the behavior of the branch curve $B$ at the two ends of the chain: $B$ could either meet the end component at two points transversally, or $B$ could be tangent to the end component. Moreover either of these cases can happen at either end of the chain. This gives rise to three cases: \begin{itemize} \item[(i):] {\em The branch curve $B$ meets both end components transversally at two points each.}\\ In this case the double covering $D_m(B)$ is a degeneration of elliptic curves isomorphic to $T_{2m-2}$. Each of the interior $m-2$ components of $P_m$ splits into disjoint curves, and the two end components are doubly covered by a single component. \item[(ii):] {\em The branch curve $B$ meets one component transversally at two points, and is tangent to the other component at one point.}\\ In this case the double covering $D_m(B)$ is a degeneration of elliptic curves isomorphic to $T_{2m-1}$. Each of the interior $m-2$ components of $P_m$ again split, and the end component which $B$ meets transversally is also doubly covered by a single component. However the other end component to which $B$ is tangent also splits, into two components meeting transversally at the point lying over the point of tangency. \item[(iii):] {\em The branch curve $B$ is tangent to both end components.}\\ In this case the double covering $D_m(B)$ is a degeneration of elliptic curves isomorphic to $T_{2m}$. Each of the interior $m-2$ components of $P_m$ again split into disjoint curves, while the two end components to which $B$ is tangent split into two curves which meet transversally as above. \end{itemize} The double covering $D_m(B) \to P_m$ induces an involution on each fiber of $D_m(B)$ over $\Delta$. On the central fiber, this involution becomes a correspondence between the components. In case (i), the components which cover the end components carry a self-involution; the other components split into two chains (of length $m-2$), and the involution on the fiber makes a correspondence between the two chains of components. In case (ii), only the single component of $D_m(B)$ covering the component of $P_m$ which meets the branch curve $B$ in two points carries a self-involution; the remaining $2m-2$ components are paired. Finally, in case (iii), the $2m$ components are all paired, in two chains of $m$ components, starting and ending with the components covering the end components of $P_m$. These three situations lead us to define the following notion. \begin{definition} \label{allowable_double_correspondence} Let $\cal C$ be a cycle of smooth rational curves of length $k$. An {\em allowable double correspondence} on $\cal C$ is one of the following data: \begin{itemize} \item[(i):] If $k$ is even, with $k = 2m-2$, one gives a component of $\cal C$, its ``opposite'' component on the cycle, and a correspondence between the remaining two chains of components which associate components with the same distance from the two chosen components. In particular, if one numbers the components $C_i$ around the cycle so that the chosen components are $C_1$ and $C_m$, then in the two chains, the components $C_i$ and $C_{2m-i}$ are associated. \item[(ii):] If $k$ is odd, with $k = 2m-1$, one gives a component of $\cal C$, and a correspondence between the remaining components which associates components with the same distance from the chosen component. In particular, if one numbers the components $C_i$ around the cycle so that the chosen component is $C_1$, then the components $C_i$ and $C_{2m-i+1}$ are associated. \item[(iii):]If $k$ is even, with $k = 2m$, one gives a partition of the components of $\cal C$ into two chains of length $m$, and a correspondence between these chains, which associate components with the same distance from one of the two vertices of the cycle in which the two chains intersect. In particular, if one numbers the components $C_i$ around the cycle so that the first component on one of the two chains is $C_1$, then the components $C_i$ and $C_{2m-i+1}$ are associated. \end{itemize} \end{definition} We remark that most of this data is actually determined by a single choice. In case (i), if one chooses one of the components of the cycle $\cal C$, then the opposite component and the correspondence between the remaining components are all determined. Similarly in case (ii), it suffices to choose a single component, and in case (iii), a single vertex. It is obvious that for the degenerations of the $g^1_2$'s which we constructed above, each case in the construction (that is, each case for the branch locus $B$), gives the corresponding allowable double correspondence on the central fiber. Moreover, we have a converse to this construction. \begin{proposition} \label{adcprop} Let $X \to \Delta$ be an $I_k$ degeneration of elliptic curves, that is, $X$ is a smooth surface mapping to the disc $\Delta$ with general fiber an elliptic curve and with central fiber a cycle of $k$ $\Bbb P^1$'s, each having self-intersection $-2$. Suppose that on the central fiber there is given an allowable double correspondence. Then $X$ is isomorphic (in a neighborhood of the central fiber) to $D_m(B)$ for the appropriate $m$ and $B$, in such a way that the allowable double correspondences coincide. \end{proposition} \begin{pf} By the uniqueness of the $I_k$ degenerations, we know that $X \to \Delta$ is isomorphic to $D_m(B) \to \Delta$ for the appropriate $m$ and $B$. Moreover, as we mentioned above, given isomorphic $I_k$ degenerations, one can arrange the isomorphism to make any two components correspond. By adjusting the components which are associated via the isomorphism, we may clearly make the two allowable double correspondences to coincide. \end{pf} \subsection{Admissible Projective Embeddings of Degenerations of $g^1_2$'s} In this subsection we will combine the ideas in the previous subsections and explain how, in certain cases, with a degeneration of $g^1_2$'s and a suitable projective embedding, one can see the degeneration of the induced rational normal scrolls to a union of planes. The complete situation is the following. Suppose we are given an $I_k$ degeneration of elliptic curves $X \to \Delta$, abstractly isomorphic to the $T_k$ degeneration, with $k \geq g+1$. In addition, suppose that exactly $g+1$ of the components of the central fiber have been chosen, which induces a cycle degeneration of $X$ into $\Bbb P^g \times \Delta$ as described at the end of subsection \ref{cycle_degenerations}: the $g+1$ chosen components survive in the limit to form a cycle of lines. Finally we assume that an allowable double correspondence has been given on the central fiber of $X$. By Proposition \ref{adcprop}, we have a degeneration of $g^1_2$'s on $X$, which gives us on the general fiber $X_t$ of the map to $\Delta$ a $g^1_2$ determining a rational normal scroll $R_t$ in $\Bbb P^g \times \{t\}$. This is a flat family of surfaces over the punctured disc, and we want to understand the limit of these scrolls in $\Bbb P^g$. While this question is interesting in general, in our application we need only to understand some particular cases for the relationship between the choice of $g+1$ surviving components and the allowable double correspondence. These we describe next. \begin{definition} \label{compat} A choice of $g+1$ surviving components is {\em compatible} with an allowable double correspondence on the central fiber of a $T_k$ degeneration of elliptic curves if and only if the following conditions are satisfied: \begin{itemize} \item[(a)] In case (i) or (ii), no component which carries the self-involution can be one of the $g+1$ surviving components, that is, these end components must be contracted in the map to projective space. \item[(b)] In the two chains of components which are paired in the allowable double correspondence, the first component to survive (on either side) is paired with a surviving component on the other chain. \item[(c)] Except for these two pairs of surviving components, no other surviving component is paired with a surviving component. \end{itemize} \end{definition} This definition is exactly that which makes the analysis of the limit of the scrolls rather transparent. \begin{proposition} \label{planes} Suppose that a set of $g+1$ surviving components is chosen on the degeneration $T_k \to \Delta$, which is compatible with an allowable double correspondence. Then the flat limit of the rational normal scrolls $R_t$ is the union of $g-1$ $2$-planes, which are spanned by the following. \begin{itemize} \item[($\alpha$)] For each of the two pairs of surviving components as in (b) above, these components are mapped to lines which meet at one point; therefore they span a $2$-plane, which is part of the limit. \item[($\beta$)] For each of the other $g-3$ surviving components, which map to a line, the paired component is contracted to a point not on that line; this point and line span a $2$-plane, which is part of the limit. \end{itemize} \end{proposition} \begin{pf} Since the scroll is determined as the union of secants to the general $g^1_2$, the limit of the scrolls will contain the secants to the limit $g^1_2$. Recall that in our situation the degeneration of the $g^1_2$'s is obtained by the double covering of the central fiber, as in the example $D_m(B) \to P_m$. By definition, the secants in the limit are made between points of the cycle on $D_m(B)$ which map to the same point in the double covering. Therefore, in the limit, the secants are made between points of the paired components in the allowable double correspondence. When neither of a pair survives, we have no $2$-dimensional contribution to the limit. When both survive, as in case ($\alpha$), we clearly obtain the plane spanned by the components. When exactly one of the pair survives, as in case ($\beta$), we again have the plane spanned by the surviving component and by the point to which the contracted component is mapped. The union of these obvious $g-1$ $2$-planes contained in the limit is a reducible surface composed of a chain of $2$-planes, and is arithmetically a rational normal scroll, that is, it has the same Hilbert function as a rational normal scroll. This proves that this union of planes is exactly the flat limit of the scrolls $R_t$. \end{pf} \subsection{Examples} \label{examples} In this subsection we will exhibit the degeneration of the union of two scrolls in $\Bbb P^g$ to the union of $2g-2$ planes, in the configuration of the planar graphs given in the examples of Section \ref{union_of_planes}. Our method will be to give a $T_k$ degeneration of elliptic curves, together with a choice of $g+1$ surviving components, and two different allowable double correspondences on the central fiber, which are both compatible with the choice of surviving components. By Proposition \ref{planes}, this induces two different degenerations of scrolls through the general elliptic curve, both of which degenerate to planes in a well-controlled manner. We will show that the limit is exactly the union of the planes $S_A \cup S_B$ which we desire. Since the $g^1_2$'s in the limit are different, the $g^1_2$'s on the general fiber must be different, and hence the two scrolls $R^{(1)}_t$ and $R^{(2)}_t$ in the general fiber will meet transversally along the general elliptic curve by Lemma \ref{2scrolls}. The union of the planes $S_A \cup S_B$ is indeed the flat limit of the two scrolls $R^{(1)}_t$ and $R^{(2)}_t$. In fact, both $S_A \cup S_B$ and $R^{(1)}_t \cup R^{(2)}_t$ have the same Hilbert function, namely that of a smooth $K3$ surface of degree $2g-2$ in $\Bbb P^g$. \begin{theorem} \label{2scroll_limit} For each $g \geq 7$, the union of $2g-2$ $2$-planes $S_g$ described in Section \ref{union_of_planes} is a flat limit of a union of two rational normal scrolls, meeting transversally along a smooth elliptic curve, which is anti-canonical in each scroll. \end{theorem} As noted above, to prove the Theorem it is sufficient to exhibit the appropriate allowable double correspondences, which are compatible with a choice of $g+1$ surviving components on some $T_k$ degeneration of elliptic curves. The construction naturally breaks up into the odd and even genus cases, and we will treat these separately. \medskip \noindent {\bf The Odd Genus Case.} Recall that in this case we set $g = 2n+1$; now also set $k = 4g-8 = 8n-4$. This is the length of the cycle before the embedding into projective space. Number the components around the cycle from $1$ to $k=8n-4$, consecutively. As noted above we must give the $g+1 = 2n$ surviving components, as well as two different allowable double correspondences, which will give us the $\cal A$ and $\cal B$ configurations of planes in the limit. Let $\cal G$ be the set of indices of the surviving components. Then in this case: \[ \cal G = \{n-2,n+3,n+5,n+7,\dots,3n+1,5n-4,5n+1,5n+3,5n+5,\dots,7n-1\}. \] Now we give the two allowable double correspondences, which we also denote by $\cal A$ and $\cal B$. The numbering has been chosen so that $\cal A$ is the pairing of component $j$ with component $8n-3-j$: \[ \cal A: j \leftrightarrow 8n-3-j; \] this is the numbering used to describe case (iii) of the allowable double correspondence definition, in fact. The other allowable double correspondence $\cal B$ is also of type (iii), and is the pairing of component $j$ with component $2n+1-j \mod 8n-4$: \[ \cal B: j \leftrightarrow 2n+1-j \; \mod 8n-4. \] Let us check that $\cal G$ is compatible with $\cal A$; we will leave the analogous proof for $\cal B$ to the reader. Firstly, both $\cal A$ and $\cal B$ are of type (iii), so condition (a) of Definition \ref{compat} is automatic. Secondly, the first surviving components on the two chains of $\cal A$ at one end are indexed $n-2$ and $7n-1$, and these are paired; similarly, at the other end, the first surviving components are indexed $3n+1$ and $5n-4$, and these are paired. Thus condition (b) is satisfied. Finally, the indices of the interior components which survive on one side are all of the form $n + \text{ odd number }$, and these are paired with components whose indices are of the form $7n - \text{ even number }$; no such component survives in $\cal G$. Thus condition (c) of Definition \ref{compat} is also satisfied, and we have shown that $\cal G$ is compatible with $\cal A$. If we renumber the surviving components from $1$ to $g+1 = 2n+2$, simultaneously numbering the vertices in the standard way, then the planes of the limit of the $\cal A$ $g^1_2$, according to Proposition \ref{planes} clearly form the configuration $S_A$ described in Section \ref{union_of_planes} (in particular, in Table One) for this genus. The same is true for the $\cal B$ $g^1_2$: it has as the limit the configuration $S_B$. This completes the proof of Theorem \ref{2scroll_limit} in the odd genus case. \medskip \noindent {\bf The Even Genus Case.} Recall that in this case we set $g = 2n+2$; now also set $k = 8n + 6[(n-1)/2] - 11$. This is the length of the cycle before the embedding into projective space. Number the components consecutively around the cycle from $0$ to $k-1$. We must again give the $g+1 = 2n+3$ surviving components, as well as two different allowable double correspondences, which will give us the $\cal A$ and $\cal B$ configurations of planes in the limit. Let $\cal G$ be the set of indices of the surviving components. Then in this case: \begin{eqnarray*} \cal G &=& \{ [(n-2)/2],[(3n-7)/2], [(3n+1)/2],[(3n+5)/2],[(3n+9)/2],\dots,[(7n-7)/2], \\ & & 3[(3n-3)/2],2n+3[(3n-3)/2]-5, \\ & & 5[(3n-3)/2],5[(3n-3)/2]+2,5[(3n-3)/2]+4,\dots, \\ & & 8n+4[(n-1)/2]-[(n-2)/2]-11,8n+6[(n-1)/2]-[(n-2)/2]-11 \} \end{eqnarray*} Now we give the two allowable double correspondences, which we also denote by $\cal A$ and $\cal B$. The numbering has been chosen so that $\cal A$ is the pairing of component $j$ with component $k-j$: \[ \cal A: j \leftrightarrow k-j \; \mod k; \] note that $0$ is self-paired, and is the only such index, so that $\cal A$ is of type (ii) in the allowable double correspondence definition. The other allowable double correspondence $\cal B$ is also of type (ii), and is the pairing of component $j$ with component $2[(3n-3)/2]-j \mod k$: \[ \cal B: j \leftrightarrow 2[(3n-3)/2]-j \; \mod k. \] We leave to the reader in this even genus case to check that $\cal G$ is compatible with $\cal A$ and $\cal B$, and that the limit configuration of planes is exactly the union $S_A \cup S_B$ for this genus. This completes the proof of Theorem \ref{2scroll_limit}. As a consequence of Theorem \ref{2scroll_limit}, Theorem \ref{smoothing_scrolls}, and Theorem \ref{Pic=Z}, by combining the deformations of the planes to the two scrolls, and then the deformation of the two scrolls to the $K3$ surfaces, we have the following. \begin{corollary} \label{main_theorem} For each $g \geq 7$, the union of planes $S_g$ described in Section \ref{union_of_planes} is a flat limit of smooth prime $K3$ surfaces of genus $g$, and in particular is represented by a point of ${\cal H}_g$. Therefore, the graph curve $C_g$, which is the general hyperplane section of the union of planes $S_g$, is a flat limit of hyperplane sections of such smooth prime $K3$ surfaces of genus $g$. \end{corollary} These examples suggest that, in general, if the union of the $2g-2$ planes determined by a planar graph can be decomposed into two sets of $g-1$ planes, each of which form a chain of planes, then the union can be smoothed to a union of two scrolls. \subsection{Remarks for Low Values of the Genus} \label{low_genus_remarks} Actually, for low genus, it is not difficult to degenerate smooth $K3$ surfaces to unions of planes, whose dual graph is planar. For $g=3$, for example, one has the quartic surface degenerating to the union of the four coordinate planes in $\Bbb P^3$. Also for $g=4,5$, the general smooth $K3$ surface is a complete intersection, and the degeneration to the union of planes is straightforward. Up to genus $10$ we have also the following construction. There are smooth $K3$ surfaces in this range, obtained by intersecting a cone over a Del Pezzo surface with a general quadric. If one allows the Del Pezzo surface to degenerate to a cone over a cycle of lines, and the quadric to degenerate to a union of hyperplanes, then the smooth surface degenerates to a union of planes, whose dual graph is a ``prism'', that is, consists of two cycles of vertices joined in series. (In genus $10$ the smooth $K3$ surfaces are not prime, but they are prime for genus up to $9$.) Since the analysis is trivial in genus less than $6$, we now present another alternate degeneration in genus $7$ and $8$, taken from the planar graphs introduced in \cite{ciliberto-miranda1}. These graphs give graph curves with the minimal possible corank of the Gaussian map, which is neither the case for the prism graphs, nor for the graphs $G_7$ and $G_8$, and therefore it is interesting to note that the corresponding unions of planes are limits of $K$ surfaces. For completeness we set $\tilde{G}_6$, $\tilde{G}_7$, and $\tilde{G}_8$ to be the planar graphs depicted in \cite[page 429]{ciliberto-miranda1}. The graph $\tilde{G}_6$ is a prism over a pentagon, and can therefore be smoothed as described above. For the union of planes $\tilde{G}_7$, we return to the construction via the degeneration of the elliptic curve and the two $g^1_2$'s. Set $k=18$, and consider the $T_k$ degeneration of elliptic curves, numbered in order around the cycle from $1$ to $18$. The set $\cal G$ of surviving components is \[ \cal G = \{1,5,7,10,11,13,15,16\}. \] The allowable double correspondence giving the $\cal A$ set of planes is of type (iii), and pairs component $j$ with component $17-j\mod 18$. The allowable double correspondence giving the $\cal A$ set of planes is of type (i), with the components $3$ and $12$ being self-paired, and otherwise pairs component $j$ with component $6-j\mod 18$. For the union of planes $\tilde{G}_8$, we set $k=23$, and consider the $T_k$ degeneration of elliptic curves, numbered in order around the cycle from $1$ to $23$. The set $\cal G$ of surviving components is \[ \cal G = \{1,6,8,9,12,14,16,19,20\}. \] The allowable double correspondence giving the $\cal A$ set of planes is of type (ii), and pairs component $j$ with component $21-j\mod 23$; component $22$ is self-paired. The allowable double correspondence giving the $\cal B$ set of planes is also of type (ii), with component $15$ being self-paired, and otherwise pairs component $j$ with component $7-j\mod 23$. The reader may check that these are compatible allowable double correspondences, and give as the limit of the two scrolls, the union of planes described by the graphs $\tilde{G}_j$, for $j=7,8$. Indeed, the reader may reconstruct these graphs now from the data given above, by working out the union of planes. Finally we remark that the graph given in \cite{ciliberto-miranda1} for genus $9$ is the same as the $G_9$ graph of this paper. \section{Applications to Gaussian Maps, $K3$ Surfaces, and Fano Threefolds} \label{Gaussmaps} \subsection{The Corank One Theorem for $g=11$ and $g \geq 13$} \label{corank_one_section} Let $C$ be a stable curve. There is a natural map \[ \phi:\bigwedge^2H^0(\omega_C) \to H^0(\Omega^1_C\otimes\omega^{\otimes 2}_C) \] defined for a smooth curve in local coordinates by $\phi(fdz\wedge gdz) = (f'g-fg'){(dz)}^3$. Note that the corank of the Gaussian map is semi-continuous in moduli. Indeed, the question is local on the compactified moduli space $\overline{{\cal M}}_g$, and so one may work over the Kuranishi family for the stable curve. In this case, by constancy of dimensions and standard base change theorems, the domain and range of the Gaussian map fit together to form a vector bundle over the base, and the map is a map of bundles. The semi-continuity follows. Wahl \cite{wahl} has shown that a canonical curve which is the hyperplane section of a $K3$ surface cannot have a surjective Gaussian map (see also \cite{beauville-merindol}). Using the degeneration techniques developed in this article, we can make this theorem more precise. Let ${\cal H}_g$ be the component of the Hilbert scheme of $K3$ surfaces of degree $2g-2$ in $\Bbb P^g$ which degenerate to the union of two scrolls, as described in Theorem \ref{smoothing_scrolls}. \begin{theorem} \label{corank1_theorem} Suppose that $g=11$ or $g\geq 13$. Let $S$ be a $K3$ surface of degree $2g-2$ in $\Bbb P^g$ represented by a general point in ${\cal H_g}$. Then the general hyperplane section $C$ of $S$ has a Gaussian map with corank exactly one. \end{theorem} \begin{pf} First we remark that since the corank of the Gaussian map is semi-continuous, and it cannot be surjective by Wahl's result, it suffices to prove the theorem for a single stable curve $C$ which is the limit of general hyperplane sections of such $K3$ surfaces. For this purpose we use the graph curves $C_g$, which are the hyperplane sections of the union of planes $S_g$ introduced in Section \ref{union_of_planes}. These are limits of such $K3$ surfaces, as we have shown in Corollary \ref{main_theorem}. The Gaussian maps for such graph curves have been the subject of several papers by the authors. In particular, in \cite{miranda}, it is shown that each of these graph curves, in genus $11$ or genus at least $13$, have a corank one Gaussian map. This is also proved in \cite{ciliberto-franchetta1}, where a more systematic study is made and more general results are obtained. In any case this computation completes the proof of the Theorem. \end{pf} L. Ein has communicated to us an alternate approach to the corank one theorem. His approach would first prove the surjectivity of the relevant Gaussian map defined on the $K3$ surface itself; this would imply that every smooth hyperplane section would have a corank one Gaussian map, not simply the general one. His technique requires a decomposition of the hyperplane bundle which seems difficult to realize in low genera. However Ein's approach gives sharp results for hyperplane sections of $K3$ surfaces which are re-embedded via Veronese maps. We intend to treat this in a forthcoming paper. \subsection{The Cases of Genus Twelve and Ten} \label{g=12case} In genus $12$, the general curve has a surjective Gaussian map (see \cite{ciliberto-harris-miranda}). For the graph curve $C_{12}$ constructed in this article, and which is a limit of $K3$ sections by Corollary \ref{main_theorem}, the Gaussian map has corank two, by the computations presented in \cite{ciliberto-franchetta2}. Therefore by semi-continuity, the general $K3$ section in genus $12$ has corank either one or two. In fact it is two. We require a lemma. \begin{lemma} \label{h0N(-k)} Let $C$ be a general hyperplane section of a general prime $K3$ surface of genus $g$, with $g \geq 6$. If $k \geq 3$, then $H^0(N_C(-k)) = 0$. If $g \geq 7$, then $H^0(N_C(-2)) = 0$; if $g = 6$, then $\dim H^0(N_C(-2)) \leq 1$. When $k=1$, we have $\dim H^0(N_C(-1)) = g + \gamma_C$, where $\gamma_C$ is the corank of the Gaussian map for $C$. \end{lemma} \begin{pf} For $k \geq 3$, $H^0(N_C(-k)) = 0$ since the ideal of $C$ is generated by quadrics. For $k=2$, a computation on the graph curves introduced in this paper (namely those associated to the graphs $\tilde{G}_g$ for $g=7,8$ and the graphs $G_g$ for $g \geq 9$) shows that $H^0(N_C(-2))=0$ for $g \geq 7$. The computation is exactly that of \cite[Theorem 3.1]{ciliberto-miranda1}; in fact, this space is zero for any graph curve associated to a graph which is not a prism. The genus $6$ case was also computed in \cite[Theorem 3.1]{ciliberto-miranda1}, and the above statement follows from semi-continuity. The final statement concerning $H^0(N_C(-1))$ is \cite[Proposition 1.2(a)]{ciliberto-miranda2}. \end{pf} \begin{proposition} \label{g=12corank2} The general hyperplane section of a $K3$ surface of degree $22$ in $\Bbb P^{12}$ represented by a general point in ${\cal H}_{12}$ has corank two Gaussian map. \end{proposition} \begin{pf} We must only show that the general such hyperplane section cannot have corank one. By \cite{mukai}, the generic prime $K3$ surface of genus $12$ is the hyperplane section of a smooth Fano $3$-fold. (This is true in lower genera also; we discuss this in general below.) Therefore the general $K3$ hyperplane section $C$ is $2$-extendable, i.e., is a codimension two linear section of a smooth variety. If the corank of the Gaussian map for $C$ is one, then by Lemma \ref{h0N(-k)} we have $H^0(N_C(-2)) = 0$ and $\dim H^1(N_C(-1)) = g + 1$. These are the hypothesis of a theorem of Zak/L'vovsky (see \cite{lvovsky} or \cite{bertram-ein-lazarsfeld}), which concludes that $C$ cannot be $2$-extendable. Since this is a contradiction, we see that the corank of the Gaussian map must be two. \end{pf} By \cite{cukierman-ulmer}, the general hyperplane section of a prime $K3$ surface of genus $10$ has a corank $4$ Gaussian map. Using the techniques of this paper, it might be possible to prove, by exhibiting an example of such a curve, that the corank of the Gaussian map for the general prime $K3$ section of genus $10$ was at most four. We have been unable to construct an appropriate planar graph curve of genus $10$ with corank $4$ Gaussian map. We leave this as an open problem: to find such a graph curve. The cases of lower genera, in particular $g \leq 9$ and $g=11$, are discussed rather fully in \cite{ciliberto-miranda1}. Note that the above statements rely on the transcendental theory for $K3$ surfaces, which is used to insure that our component ${\cal H}_g$ is the prime component. \subsection{The Family of $K3$ Sections} \label{families} Let ${\cal H}_g$ be the component of the Hilbert scheme of $K3$ surfaces of degree $2g-2$ in $\Bbb P^g$ which degenerate to the union of two scrolls, as described in Theorem \ref{smoothing_scrolls}. In this and the next section we will invoke the transcendental theory and refer to this component ${\cal H}_g$ as the Hilbert scheme for prime $K3$ surfaces of genus $g$. Let ${\cal F}_g$ be the ``flag Hilbert scheme'' parametrizing pairs $(S,C)$, where $S$ is represented by a point of ${\cal H}_g$ and $C$ is a stable hyperplane section of $S$. Finally let ${\cal C}_g$ be the Hilbert scheme of degenerate stable canonical curves of genus $g$ in $\Bbb P^g$, which of course all live in some hyperplane. Note that \begin{eqnarray*} \dim {\cal H}_g &=& g^2 + 2g + 19, \\ \dim {\cal C}_g &=& g^2 + 4g - 4, \text{ and }\\ \dim {\cal F}_g &=& g^2 + 3g + 19. \end{eqnarray*} We have natural maps $p:{\cal F}_g \to {\cal C}_g$, and $q:{\cal F}_g \to {\cal H}_g$. In this subsection we will study some properties of the map $p$. We will fix the genus $g$ to be at least $6$; the lower genera are those of complete intersections, and all of the analysis is trivial. First let us compute the dimension of the general fiber of $p$. By standard deformation theory (see \cite{kleppe} or \cite{kleppe2}), the tangent space to the fiber at a point $(S,C)$ is isomorphic to $H^0(N_S \otimes I_{C/S})$, where $N_S$ is the normal bundle to $S$ and $I_{C/S}$ is the ideal sheaf of $C$ in $S$. We apply this remark to the pair $(X,C)$, where $C$ is a general point in the image of $p$, (i.e., $C$ is a general prime $K3$ surface section), and $X$ is a general cone over $C$. Since any projectively Cohen-Macaulay surface flatly degenerates to a cone over its hyperplane section, $X$ is indeed represented by a point in ${\cal H}_g$ and the pair $(X,C)$ by a point $(x,c)$ in ${\cal F}_g$. For the cone $X$, $H^0(N_X) \cong \oplus_{k\geq 0} H^0(N_C(-k))$; hence the tangent space to the fiber of $p$ at the point $(x,c)$ is isomorphic to $\oplus_{k\geq 1} H^0(N_C(-k))$. Hence, using Lemma \ref{h0N(-k)}, the tangent space to the fiber of $p$ at $(x,c)$ has dimension $g+\gamma_C$ for $g \geq 7$, and is at most $17$ for genus $6$. Now the discussion breaks into several cases. First assume that $6 \leq g \leq 9$, or $g = 11$. In this case we have an upper bound for the corank of the Gaussian map, by the graph curve computations made in \cite{ciliberto-miranda1}; this gives, by the above Lemma, and semi-continuity, an upper bound for the dimension of the general fiber of $p$. The answer (see \cite[Theorem 3.3]{ciliberto-miranda1} is that the fiber dimension is at most $23-g$. Hence in this range, we have that $p$ is surjective; in particular, the general fiber dimension is exactly $23-g$, and the inequality above in the genus $6$ case is an equality. The surjectivity of $p$ is also proved in \cite{mori-mukai}, in a different manner. We notice that, even ignoring the transcendental theory, one has that ${\cal H}_g$ is the only component of the Hilbert scheme of $K3$ surfaces which surjects onto $\cal C_g$, as proved in \cite{ciliberto-miranda1}. Next assume that $g \geq 13$. For this range, we know that the corank of the Gaussian map is one for the general curve $C$ in the image of $p$, by Theorem \ref{corank1_theorem}. Hence the tangent space to the fiber of $p$ at the cone point $(x,c)$ is $g+1$, and hence by semi-continuity, the tangent space to the fiber of $p$ at a general point is at most $g+1$. On the other hand, the fiber clearly has dimension at least $g+1$, by using projective transformations fixing the hyperplane section $C$. Therefore the general fiber has dimension $g+1$. Finally let us take up the cases of genus $12$ and $10$. In genus $12$, by the above Lemma and Proposition \ref{g=12corank2}, the dimension of the general fiber of $p$ is at most $14$. In fact it is exactly $14$, since the generic prime $K3$ surface of genus $12$ is the hyperplane section of a smooth Fano threefold, as described in \cite{mukai}. In genus $10$ the same remarks apply: using the results of \cite{cukierman-ulmer} the dimension of the general fiber of $p$ is again $14$. These computations are sufficient to show the following. \begin{proposition} \label{fiber_irred} Fix $g \geq 3$. Then the general fiber of the map $p$ is irreducible. \end{proposition} \begin{pf} Suppose that there are two components in the fiber. Then the point $(x,c)$, where $c$ is a general point in the image of $p$ and $x$ represents a cone over the curve represented by $c$, belongs to both components, and hence is a singular point of the fiber. However, we have shown above that the tangent space to the cone point is exactly the dimension of the fiber. Thus the fiber must be irreducible. \end{pf} The above Proposition was proved for low genera in \cite[Theorem 5.3]{ciliberto-miranda1}. Let $F_g$ be the moduli space for prime $K3$ surfaces of genus $g$. Let $I_g$ be the moduli space of pairs $(S,C)$, where $(S,{\cal O}_S(C)) \in F_g$, and $C$ is a stable curve on $S$. We have a natural projection $\psi:I_g \to F_g$, and a natural map $\pi: I_g \to {\cal M}_g$, where ${\cal M}_g$ is the moduli space for curves of genus $g$. We note that $\dim F_g = 19$, and $\dim I_g = g+19$. \begin{theorem} \label{moduli_map} For $3 \leq g \leq 9$ and for $g=11$, the map $\pi$ is dominant, and the general fiber is irreducible. For $g=10$, the codimension of the image of $\pi$ is one, and for $g=12$, the codimension of the image of $\pi$ is two. For $g \geq 13$, the map $\pi$ is birational onto its image. Moreover, for $g = 11$ and $g \geq 13$, the general canonical curve which is the hyperplane section of a prime $K3$ surface lies on a unique one, up to projective transformations. \end{theorem} The proof is an easy consequence of the previous Proposition, since the fibers of the map $\pi$ at the moduli space level are obtained by taking the fibers of the map $p$ at the Hilbert scheme level and dividing by the action of the projective group. \begin{corollary} \label{codim1conelocus} Let ${\cal H}_g$ be the Hilbert scheme of prime $K3$ surfaces of genus $g$. Let ${\cal X}_g$ be the locus in ${\cal H}_g$ representing cones over hyperplane sections of the surfaces in ${\cal H}_g$. Then the codimension of ${\cal X}_g$ in ${\cal H}_g$ is $\gamma_C + \sum_{k\geq 2}h^0(N_C(-k))$ where $C$ is a general curve in the image of $p$. In particular, ${\cal X}_g$ has codimension one in ${\cal H}_g$ if and only if $g=11$ or $g \geq 13$. \end{corollary} \begin{pf} Let $\delta$ be the dimension of the image of $p$. By the above discussion, we have that the general fiber of $p$ has dimension $g+\gamma_C + \sum_{k\geq 2}h^0(N_C(-k))$ where $C$ is a general curve in the image of $p$. Since the map $q$ is surjective with $g$-dimensional fibers, we see that ${\cal H}_g$ has dimension $\delta + \gamma_C + \sum_{k\geq 2}h^0(N_C(-k))$. On the other hand, it is elementary that ${\cal X}_g$ has dimension $\delta$. This proves the general statement, and the final statement follows from Theorem \ref{corank1_theorem}. \end{pf} We remark that if one considers a general point of ${\cal X}_g$, representing a cone $X$ with vertex $v$, and if we are in the corank one case of $g=11$ or $g\geq 13$, then the only embedded deformations of $X$ are either to smooth $K3$ surfaces (represented by points of ${\cal H}_g$) or to other cones. Therefore this singularity cannot be ``partially smoothed'': any deformation either is a total smoothing, or is topologically trivial. In conclusion, let us remark that the results here lead naturally to the following questions. Suppose that $g \geq 13$, and consider the moduli space ${\cal M}_g$ for curves of genus $g$. Let ${\cal N}_g$ be the locus representing curves with non-surjective Gaussian maps. Let ${\cal K}_g$ be the locus representing hyperplane sections of smooth prime $K3$ surfaces of genus $g$. Is ${\cal K}_g$ a component of ${\cal N}_g$? What are the components of ${\cal N}_g$ when it is not irreducible? The first question has a positive answer in genus $10$, by \cite{cukierman-ulmer}, and in this case ${\cal N}_g$ is irreducible. Finally, a related question is: does every stable curve with corank one Gaussian map lie on a numerical $K3$ surface? We know of no counterexamples to this; the other examples of curves with corank one Gaussian map are planar graph curves, or hypersurface sections of cones over canonical curves, all of which lie on numerical $K3$'s. \subsection{Fano Threefolds} \label{fanos} We note that Theorem \ref{moduli_map} extends the result of Mori and Mukai (see \cite{mori-mukai} and \cite{mukai}) which only proves that $\pi$ is generically finite if $g \geq 13$. Our more precise statement can be applied to the existence and irreducibility of families of Fano threefolds as we now explain. A {\em prime} Fano threefold of genus $g$ is a smooth projective threefold of degree $2g-2$ in $\Bbb P^{g+1}$ whose hyperplane class is anticanonical, and generates the Picard group. Note that by standard Noether-Lefschetz-type arguments, the general hyperplane section of a prime Fano threefold of genus $g$ is a prime $K3$ surface of genus $g$ (see e.g. \cite{moishezon}). Therefore the general curve section of a prime Fano threefold is a canonical curve which is at least $2$-extendable; hence the corank of the Gaussian map for these curve sections must be at least two, by Zak's Theorem (see \cite{bertram-ein-lazarsfeld}). Let ${\cal V}_g$ be the Hilbert scheme of prime Fano threefolds of genus $g$. Next we remark that the general hyperplane section $S$ of a prime Fano $V$ represented by a general point in {\em any} component of ${\cal V}_g$ describes a general prime $K3$ surface. In other words, the second projection from any component of the Hilbert scheme of pairs $(V,S)$ to ${\cal H}_g$ is surjective. This follows by considering the two maps $H^1(T_V) \to H^1(T_V|_S)$ and $H^1(T_S) \to H^1(T_V|_S)$. The second map is surjective: the next term in the cohomology sequence is $H^1(N_{S/V})$, and since $N_{S/V} \cong {\cal O}_S(1)$, this $H^1$ vanishes. The first map has corank at most one, since the next term in the cohomology sequence is $H^2(T_V(-1))$, which is Serre dual to $H^1(\Omega_V^1)$, and therefore has dimension one since $V$ is prime. On the other hand this first map cannot be surjective; if it were, every deformation of $S$ to first order would come from deforming $V$, and since there are non-algebraic deformations of $S$, this is not possible. Hence first map has corank exactly one and the second map is surjective, which, by standard deformation theory arguments, proves the above statement: a general hyperplane section of a general Fano (in any component of ${\cal V}_g$) is a general prime $K3$ surface. The following theorem now follows from the above remarks and Theorem \ref{corank1_theorem}. \begin{theorem} \label{nofanos} The Hilbert scheme ${\cal V}_g$ is empty for $g=11$ and $g \geq 13$. \end{theorem} The above theorem was first proved by V.A. Iskovskih in \cite{iskovskih2}, and reproved in the above spirit by S. Mukai in \cite{mukai}. Now let us turn to the question of irreducibility of the Hilbert scheme ${\cal V}_g$. We assume from now on that $g\geq 6$, since the lower genera are complete interesections and the theory is trivial. Now any prime Fano of genus $g$ can be flatly degenerated to a cone over its general hyperplane section. Moreover we know that ${\cal H}_g$ is the only component of the Hilbert scheme whose general point represents a prime $K3$ surface of genus $g$. Hence, by the same argument as was used in the proof of Proposition \ref{fiber_irred}, it will suffice to show that the cone over a $K3$ surface represented by a general point in ${\cal H}_g$ is represented by a smooth point of ${\cal V}_g$. Let $X$ be such a cone over a $K3$ surface $S$ with curve section $C$. The tangent space to the Hilbert scheme containing $X$ has dimension $h^0(N_X) = \sum_{k\geq 0}h^0(N_S(-k)) = h^0(N_S)+h^0(N_S(-1))+h^0(N_S(-2))$ since $S$ is cut out by quadrics. Now $h^0(N_S)$ is the dimension of ${\cal H}_g$, which is $g^2+2g+19$. The space $H^0(N_S(-1))$ is the tangent space to the fiber of the map $p:{\cal F}_g \to {\cal C}_g$ introduced in Section \ref{families}, and by the results there we deduce that this space has dimension $h^0(N_C(-1))+h^0(N_C(-2))$. Finally $h^0(N_S(-2))$ is, by semi-continuity upon deforming to a cone over $C$, at most $h^0(N_C(-2))$. Therefore, by Lemma \ref{h0N(-k)}, we have the following formula: \begin{equation} \label{H0NX} h^0(N_X) \leq \begin{cases} g^2 + 3g + 19 + \gamma_C & \text{ if }g\geq 7,\text{ and }\\ g^2 + 3g + 19 + \gamma_C+ 2 & \text{ if }g=6. \end{cases} \end{equation}. By the results of Fano and Iskovskih, for $6 \leq g \leq 10$ and $g=12$, (see \cite{iskovskih2} and \cite{mukai}), the Hilbert scheme ${\cal V}_g$ is non-empty; indeed, there are examples for these genera which provide families with the number of moduli and parameters shown in Table Two; also shown there is the corank $\gamma_C$ of the Gaussian map for the general curve section $C$. \begin{center} \begin{tabular}{||c|c|c|c||} \multicolumn{4}{c}{Table Two} \\ \hline genus $g$ & number of moduli & number of parameters & $\gamma_C$ \\ \hline 6 & 22 & 85 & 10\\ \hline 7 & 18 & 98 & 9 \\ \hline 8 & 15 & 114 & 7 \\ \hline 9 & 12 & 132 & 5 \\ \hline 10 & 10 & 153 & 4 \\ \hline 12 & 6 & 201 & 2\\ \hline \end{tabular} \end{center} The number of moduli for each of the examples are computed easily using projective arguments. The number of parameters for the family in $\Bbb P^{g+1}$ which these examples fill out is obtained simply by adding the number of projective transformations of $\Bbb P^{g+1}$, which is of course $g^2 + 4g +3$. The final column giving the value of the corank of the Gaussian map is taken from \cite[Theorem 3.2]{ciliberto-miranda1} and Section \ref{g=12case}. Now comparing the number of parameters for these families with the tangent space to the Hilbert scheme at the general cone, we see that the upper bound for the tangent space dimension given in (\ref{H0NX}) coincides for each of these genera with the number of parameters. Therefore we can conclude the following. \begin{theorem} \label{Vgirreducible} For $6 \leq g \leq 10$ and $g=12$, the point of ${\cal V}_g$ represented by a cone over a general prime $K3$ surface $S$ in ${\cal H}_g$ is a smooth point of ${\cal V}_g$. Moreover, ${\cal V}_g$ is irreducible, and the examples of Fano and Iskovskih fill out all of ${\cal V}_g$. Finally, the fiber of the second projection from pairs $(V,S)$ to ${\cal H}_g$ is irreducible; in genus $12$, this implies that the general prime $K3$ surface lies on a unique prime Fano, up to projective transformations. \end{theorem} Note that the above theorem can be viewed as a classification theorem for prime Fano threefolds. This approach to the theory of Fano threefolds avoids completely the necessity of proving the existence of lines and the method of double projection. It is clear that using this approach the entire theory and classification of Fano threefolds can be re-formulated. For the non-prime Fanos, a similar analysis can be made; this involves a careful study of the corank of the Gaussian map and the cohomology of the normal bundle for curves on general non-prime $K3$ surfaces. This we will return to in a later paper, as mentioned at the end of Section \ref{corank_one_section}.

9311

alg-geom/9311005

en

https://arxiv.org/abs/alg-geom/9311005

alg-geom/9311005

Charles Walter

Charles Walter

Irreducibility of Moduli Spaces of Vector Bundles on Birationally Ruled Surfaces

7 pages, LATeX 2.09

Let $S$ be a birationally ruled surface. We show that the moduli schemes $M_S(r,c_1,c_2)$ of semistable sheaves on $S$ of rank $r$ and Chern classes $c_1$ and $c_2$ are irreducible for all $(r,c_1,c_2)$ provided the polarization of $S$ used satisfies a simple numerical condition. This is accomplished by proving that the stacks of prioritary sheaves on $S$ of fixed rank and Chern classes are smooth and irreducible.

alg-geom

\section{\@startsection{section}{1}{\z@}{-3.25ex plus -1ex minus -.2ex}{1.5ex plus .2ex}{\large\bf}} \def\subsection{\@startsection {subsection}{2}{\z@}{3.25ex plus 1ex minus .2ex}{-0.5em}{\normalsize\sl}} \def\subsubsection{\@startsection {subsubsection}{3}{\z@}{3.25ex plus 1ex minus .2ex}{-0.5em}{\normalsize\sl}} \def\paragraph{\@startsection {paragraph}{3}{\z@}{2ex plus 0.6ex minus .2ex}{-0.5em}{\normalsize\sl}} \def\subparagraph{\@startsection {subparagraph}{3}{\parindent}{2ex plus 0.6ex minus .2ex}{-1pt}{\normalsize\sl}} \catcode`\@=12 \begin{document} \maketitle \begin{abstract} \noindent Let $S$ be a birationally ruled surface. We show that the moduli schemes $M_S(r,c_1,c_2)$ of semistable sheaves on $S$ of rank $r$ and Chern classes $c_1$ and $c_2$ are irreducible for all $(r,c_1,c_2)$ provided the polarization of $S$ used satisfies a simple numerical condition. This is accomplished by proving that the stacks of prioritary sheaves on $S$ of fixed rank and Chern classes are smooth and irreducible.\bigskip\ \end{abstract} One important recent result in the theory of vector bundles on algebraic surfaces is the theorem of Gieseker and Li that for any smooth projective surface $S$ and any ample divisor $H$ on $S$, the moduli scheme $% M_{S,H}(2,c_1,c_2)$ of S-equivalence classes of $H$-semistable torsion-free sheaves of rank $2$, determinant $c_1\in {\rm Pic}(S)$, and second Chern class $c_2$ is irreducible if $c_2\gg 0$. If $S$ is a surface of general type, the condition $c_2\gg 0$ is necessary because of an example of Gieseker with small $c_2$ where the moduli space is reducible. In contrast it has been known for quite some time that the moduli schemes $M_{{\bf P}% ^2,H}(r,c_1,c_2)$ is irreducible for all $(r,c_1,c_2)$ for which there exist semistable sheaves on the projective plane, and the same result is also known for ${\bf P}^1\times {\bf P}^1$. In this paper we extend this strong irreducibility result from ${\bf P}^2$ and ${\bf P}^1\times {\bf P}^1$ to all smooth projective surfaces of negative Kodaira dimension. To simplify our exposition, we will omit ${\bf P}% ^2$ although it can be handled by the same method. (Indeed our method is based on a method of Ellingsrud and Str\o mme which was developed for ${\bf P% }^2$.) So our surface $S$ possesses\ a morphism $\pi {:}~S\rightarrow C$ onto a smooth curve with connected fibers and with general fiber isomorphic to ${\bf P}^1$. We fix such a $\pi $. (Such a $\pi $ is unique if $% q(S)=g(C)>0$ or if $S={\bf P}({\cal O}_{{\bf P}^1}\oplus {\cal O}_{{\bf P}% ^1}(e))$ with $e>0$, but there can be many, even infinitely many, possible $% \pi $ for certain rational surfaces.) For $p\in C$, let $f_p=\pi ^{-1}(p)$. These $f_p$ are all numerically equivalent, and we write $f\in {\rm NS}(X)$ for the numerical class of these $f_p$. We prove \begin{theorem} \label{semistable}Let $\pi {:}~S\rightarrow C$ be a birationally ruled surface and $f\in {\rm NS}(S)$ the numerical class of a fiber of $\pi $. Let $H$ be an ample divisor on $S$ such that $H\cdot (K_S+f)<0$. Suppose $r\geq 2 $, $c_1\in {\rm NS}(S)$, and $c_2\in {\bf Z}$ are given. If the moduli scheme $M_{S,H}(r,c_1,c_2)$ of S-equivalence classes of $H$-semistable torsion-free sheaves of rank $r$ and Chern classes $c_1$ and $c_2$ is non-empty, then it is irreducible and normal. In addition, the open subscheme $M_{S,H}^s(r,c_1,c_2)$ parametrizing stable sheaves is smooth. \end{theorem} Our methods also show that the general $H$-semistable torsion-free sheaf in any of the $M_{S,H}(r,c_1,c_2)$ is locally free and that there is a dominant, generically finite map from an open subscheme of ${\rm Jac}% (C)\times {\rm Jac}(C)\times {\bf P}^m$ to $M_{S,H}^s(r,c_1,c_2)$ with $% m=2rc_2-(r-1)c_1+(r^2-2)g-r^2+1$ where $g$ is the genus of $C$. So if $S$ is a rational surface, then $M_{S,H}^s(r,c_1,c_2)$ is unirational. Ample divisors $H$ satisfying the hypothesis $H\cdot (K_S+f)<0$ exist on any birationally ruled surface because of Lemma \ref{H} below. On certain surfaces there may exist a divisor $D$ of degree $1$ on $C$ such that the divisor class $-K_S-\pi ^{*}(D)$ is effective. In that case all ample divisors $H$ satisfy $H\cdot (K_S+f)<0$, and the theorem holds for all possible polarizations. Examples of such surfaces include Del Pezzo surfaces, rational ruled surfaces, and ruled surfaces of the form ${\bf P}(% {\rm O}_C\oplus {\rm O}_C(D_0))$ with $D_0$ a divisor of degree at least $% 2g-1$ on the smooth projective curve $C$ of genus $g$. Our method of proof begins by adapting a definition from \cite{HL}. If $\pi {% :}~S\rightarrow C$ is a birationally ruled surface, then we will say that a coherent sheaf ${\cal E}$ on $S$ is {\em prioritary} (with respect to $\pi $% ) if it is torsion-free and if ${\rm Ext}^2({\cal E},{\cal E}(-f_p))=0\,$ for all $p\in C$. By the semicontinuity theorem the prioritary sheaves in any locally noetherian flat family of coherent sheaves on $S$ form an open subfamily. Hence the prioritary sheaves on $S$ are parametrized by an open substack of the stack of coherent sheaves on $S$. For a given $r\geq 1$, $c_1\in {\rm NS}(S)$, and $c_2\in {\bf Z}$, we will write ${\rm Coh}_S(r,c_1,c_2)$ for the stack of coherent sheaves of rank $r$ and Chern classes $c_1$ and $c_2$ (modulo numerical equivalence), and ${\rm % TF}_S(r,c_1,c_2)$ and ${\rm \Pr ior}_S(r,c_1,c_2)$ for the open substacks of, respectively, torsion-free and prioritary sheaves. We will derive Theorem \ref{semistable} from: \begin{proposition} \label{stack}Let $\pi {:}~S\rightarrow C$ be a birationally ruled surface. Suppose $r\geq 2$, $c_1\in {\rm NS}(S)$, and $c_2\in {\bf Z}$ are given. Then the stack ${\rm \Pr ior}_S(r,c_1,c_2)$ of prioritary sheaves on $S$ of rank $r$ and Chern classes $c_1$ and $c_2$ is smooth and irreducible. \end{proposition} The Proposition is proven in two steps. First we prove it for geometrically ruled surfaces. Then we prove that for $S\rightarrow S_1$ the blowup of a point, if the proposition holds for $S_1$ then it holds for $S$. Our method is based on the version of the method of Ellingsrud and Str\o mme (\cite{E}, \cite{ES}, \cite{HS}) as presented in \cite{LP} \S 9. The reader unfamiliar with algebraic stacks may wish to consult \cite{LMB}. We use stacks because in that context there exist natural universal families of coherent (or torsion-free or prioritary) sheaves. Alternative universal families which stay within the category of schemes would be certain standard open subschemes of Quot schemes. But these depend on the choice of a polarization ${\cal O}_S(1)$, of the Hilbert polynomial $P$, of twists $m\gg 0$ and of a vector space $H_m$ of dimension $P(m)$. One then deals with the scheme ${\rm Quot}_{S,{\cal O}_S(1)}^0(P,m)$ parametrizing all quotients $% \gamma {:}~H_m\otimes {\cal O}_S(-m)\TeXButton{-->>}{\twoheadrightarrow}% {\cal F}$ such that $H^i({\cal F}(m))=0$ for all $i\geq 1$ and such that the induced map $H_m\rightarrow H^0({\cal F}(m))$ is an isomorphism. One would prefer not to work in a context where one constantly has to refer to all these choices, particularly since no single set of choices will work when one deals with unlimited families. But strictly speaking, these Quot schemes are not that far from our point of view because the verification in \cite {LMB} (4.14.2) that the coherent sheaves on $S$ are parametrized by an algebraic stack ${\rm Coh}_S$ is done essentially by gluing together all the ${\rm Quot}_{S,{\cal O}_S(1)}^0(P,m)$ in the smooth Grothendieck topology. This paper was written in the context of the group on vector bundles on surfaces of Europroj and as a direct result of the Catania congress where the problem was mentioned by J.\ Le Potier. The author would also like to thank A.\ Hirschowitz for some pertinent comments. \section{Proof of the Theorem} We begin with two lemmas about coherent sheaves on ${\bf P}^1$ and the restriction of torsion-free sheaves on surfaces to curves in the surface. These lemmas are well known although they have usually been stated in terms of complete families or of versal deformation spaces instead of stacks. We state them without proof. \begin{lemma} \label{P1}Let $r\geq 2$ and $0\leq d<r$ be integers. Let ${\rm Coh}_{{\bf P}% ^1}(r,-d)$ be the stack of coherent sheaves of rank $r$ and degree $-d$ on $% {\bf P}^1$. (i)\quad If $d>0$, then sheaves not isomorphic to ${\cal O}_{{\bf P}% ^1}^{r-d}\oplus {\cal O}_{{\bf P}^1}(-1)^d$ form a closed substack of ${\rm % Coh}_{{\bf P}^1}(r,-d)$ of codimension at least $2$. (ii)\quad If $d=0$, then sheaves not isomorphic to ${\cal O}_{{\bf P}^1}^r$ form a closed substack of ${\rm Coh}_{{\bf P}^1}(r,0)$ of codimension $1$. Sheaves isomorphic neither to ${\cal O}_{{\bf P}^1}^r$ nor to ${\cal O}_{% {\bf P}^1}(1)\oplus {\cal O}_{{\bf P}^1}^{r-2}\oplus {\cal O}_{{\bf P}^1}(-1) $ form a closed substack of ${\rm Coh}_{{\bf P}^1}(r,0)$ of codimension at least $2$. \end{lemma} \begin{lemma} \label{restriction}Let $D$ be an effective Cartier divisor on a projective surface $S$. If ${\cal E}$ is a torsion-free sheaf on $S$ such that ${\rm Ext% }^2({\cal E},{\cal E}(-D))=0$, then the restriction map ${\rm TF}% _S(r,c_1,c_2)\rightarrow {\rm Coh}_D(r,c_1\cdot D)$ is smooth (and therefore open) in a neighborhood of $[{\cal E}]$. \end{lemma} We also need two lemmas for reduction steps in the proof of Proposition \ref {stack}. \begin{lemma} \label{Beil}Let $\pi {:}~S\rightarrow C$ be a geometrically ruled surface with a section $\sigma \subset S$. If ${\cal E}$ is a coherent sheaf on $S$ such that $\pi _{*}({\cal E}(-\sigma ))=R^1\pi _{*}({\cal E})=0$, then there is an exact sequence% $$ 0\rightarrow \pi ^{*}(\pi _{*}({\cal E}))\rightarrow {\cal E}\rightarrow \pi ^{*}(R^1\pi _{*}({\cal E}(-\sigma )))\otimes \Omega _{S/C}(\sigma )\rightarrow 0. $$ \end{lemma} \TeXButton{Proof}{\paragraph{Proof. }}This is a special case of a relative version of Beilinson's spectral sequence, but for lack of a precise reference we give the proof in full. Let $Y:=S\times _CS$. Then the diagonal $\Delta $ of $Y$ has Beilinson's resolution (cf.\ \cite{B})% $$ 0\rightarrow \Omega _{S/C}(\sigma )\TeXButton{boxtimes}{\boxtimes}{\cal O}% _S(-\sigma )\rightarrow {\cal O}_Y\rightarrow {\cal O}_\Delta \rightarrow 0. $$ Applying $R^i{\rm pr}_{1*}(-\otimes {\rm pr}_2^{*}({\cal E}))$ to this exact sequence gives a long exact sequence which is equivalent to the one asserted by the lemma because one always has $R^i{\rm pr}_{1*}({\cal F} \TeXButton{boxtimes}{\boxtimes}{\cal G})\cong {\cal F}\otimes \pi ^{*}(R^i\pi _{*}({\cal G}))$ if ${\cal F}$ is locally free and ${\cal G}$ coherent on $S$ because of the projection formula and \cite{H}, Chapter III, Proposition 9.3. \TeXButton{qed}{\hfill $\Box$ \medskip} \begin{lemma} \label{blowup}Let $S_1$ be a smooth surface $\alpha {:}~S\rightarrow S_1$ the blowup of a point $x$ of $S_1$. Let $E$ be the exceptional divisor in $S$% . Suppose that ${\cal E}$ is a coherent sheaf of rank $r$ on $S$ such that $% {\cal E}{\mid }_E\cong {\cal O}_E^{r-d}\oplus {\cal O}_E(-1)^d$ for some $d$% . Then $\alpha _{*}({\cal E})$ is locally free in a neighborhood of $x$, and there are exact sequences% \begin{eqnarray*} & 0\rightarrow \alpha ^{*}(\alpha _{*}({\cal E}))\rightarrow {\cal E}% \rightarrow {\cal O}_E(-1)^d\rightarrow 0, & \\ & 0\rightarrow {\cal E}(-E)\rightarrow \alpha ^{*}(\alpha _{*}({\cal E}% ))\rightarrow {\cal O}_E^{r-d}\rightarrow 0. & \end{eqnarray*} Moreover, for any divisor $D$ on $S_1$ we have ${\rm Ext}^2({\cal E},{\cal E}% (\alpha ^{*}(D)))\cong {\rm Ext}^2(\alpha _{*}({\cal E}),\alpha {\cal _{*}(E}% )(D))$. \end{lemma} \TeXButton{Proof}{\paragraph{Proof. }}Let ${\cal F}$ be the kernel of the composition $% {\cal E}\TeXButton{-->>}{\twoheadrightarrow}{\cal E}{\mid }_E\TeXButton{-->>} {\twoheadrightarrow}{\cal O}_E(-1)^d$. By general properties of elementary transforms the exact sequence $0\rightarrow {\cal O}_E^{r-d}\rightarrow {\cal E}{\mid }_E\rightarrow {\cal O}_E(-1)^d\rightarrow 0$ transforms into $% 0\rightarrow {\cal O}_E^d\rightarrow {\cal F}{\mid }_E\rightarrow {\cal O}% _E^{r-d}\rightarrow 0$. So ${\cal F}$ is trivial along $E$, and ${\cal F}% \cong \alpha ^{*}(\alpha _{*}({\cal F}))$. Applying $\alpha _{*}$ to the exact sequence $0\rightarrow {\cal F}\rightarrow {\cal E}\rightarrow {\cal O}% _E(-1)^d\rightarrow 0$, we see that $\alpha _{*}({\cal F})\cong \alpha _{*}(% {\cal E})$. Hence ${\cal F}\cong \alpha ^{*}(\alpha _{*}({\cal E}))$. The exact sequences asserted by the lemma are now the standard exact sequences of an elementary transform. By adjunction and the formula $K_S=\alpha ^{*}(K_{S_1})+E$ we see that% $$ {\rm Hom}(\alpha _{*}({\cal E}),\alpha {\cal _{*}(E})(-D+K_{S_1}))\cong {\rm % Hom}({\cal F},{\cal E}(-\alpha ^{*}(D)+K_S-E)) $$ or by Serre duality that ${\rm Ext}^2(\alpha _{*}({\cal E}),\alpha {\cal % _{*}(E})(D))\cong {\rm Ext}^2({\cal E},{\cal F}(\alpha ^{*}(D)+E))$. If we now apply the functor ${\rm Ext}^2({\cal E}(-\alpha ^{*}(D)),-)$ to the exact sequence $0\rightarrow {\cal E}\rightarrow {\cal F}(E)\rightarrow {\cal O}_E(-1)^{r-d}\rightarrow 0$ and note that $$ {\rm Ext}^i({\cal E}(-\alpha ^{*}(D)),{\cal O}_E(-1))\cong H^i(E,({\cal E}{% \mid }_E)^{\lor }(-1))=0 $$ for $i=1,2$, then we see that ${\rm Ext}^2({\cal E},{\cal F}(\alpha ^{*}(D)+E))\cong {\rm Ext}^2({\cal E},{\cal E}(\alpha ^{*}(D)))$. This completes the proof of the lemma. \TeXButton{qed}{\hfill $\Box$ \medskip}\ We now begin the proof of Proposition \ref{stack}. The smoothness of ${\rm % \Pr ior}_S(r,c_1,c_2)$ follows from ${\rm Ext}^2({\cal E},{\cal E})=0$ because this is the obstruction space for deformations of ${\cal E}$. So we concentrate on irreducibility. We begin with the special case of geometrically ruled surfaces. \paragraph{Proof of Proposition \ref{stack} when $\pi {:}~S\rightarrow C$ is a geometrically ruled surface.} We follow the method of Ellingsrud and Str\o mme as presented in \cite{LP} \S 9. We fix a section $\sigma \subset $$S$. Replacing ${\cal E}$ by an appropriate twist ${\cal E}(n\sigma )$ if necessary we may assume that $% d:=-c_1\cdot f$ satisfies $0\leq d<r$. The proof now divides briefly into two cases $d>0$ and $d=0$. If $d>$$0$, then by Lemmas \ref{P1} and \ref{restriction}, those ${\cal E}$ such that ${\cal E}{\mid }_{f_p}\TeXButton{ncong}{\ncong}{\cal O}% _{f_p}^{r-d}\oplus {\cal O}_{f_p}(-1)^d$ for some $p\in C$ are parametrized by a closed substack of ${\rm \Pr ior}_S(r,c_1,c_2)$ of codimension at least $1$. So we may restrict ourselves to the dense open substack ${\rm \Pr ior}% ^0 $ where ${\cal E}{\mid }_{f_p}\cong {\cal O}_{f_p}^{r-d}\oplus {\cal O}% _{f_p}(-1)^d$ for all $p\in C$. If $d=0$, then by an analogous argument, we may restrict ourselves to a dense open substack ${\rm \Pr ior}^0$ where ${\cal E}{\mid }_{f_p}\cong {\cal O}_{f_p}^r$ for all $p\in C$ except for a finite number of $p$ where $% {\cal E}{\mid }_{f_p}\cong {\cal O}_{f_p}(1)\oplus {\cal O}% _{f_p}^{r-2}\oplus {\cal O}_{f_p}(-1)$. In either case, we set ${\cal K}:=\pi _{*}({\cal E)}$. Since $R^1\pi _{*}(% {\cal E})=0$, the Leray-Serre spectral sequence implies that $\chi ({\cal E}% )=\chi ({\cal K})$. We may now calculate by Riemann-Roch that ${\cal K}$ is a vector bundle on $C$ of rank $r-d$ and degree $k:=\chi ({\cal E}% )+(r-d)(g-1)$ where $g$ is the genus of $C$. Let ${\cal L}=R^1\pi _{*}({\cal E}(-\sigma ))$. Then ${\cal L}$ is a sheaf on $C$ of rank $d$ and degree $l:=-\chi ({\cal E}(-\sigma ))+d(g-1)=-\chi (% {\cal E})+(c_1\cdot \sigma )-(r-d)(g-1)$. The sheaf ${\cal L}$ is locally free if $d>0$. By Lemma \ref{Beil} there is an exact sequence% $$ 0\rightarrow \pi ^{*}({\cal K})\rightarrow {\cal E}\rightarrow \pi ^{*}(% {\cal L})\otimes \Omega _{S/C}(\sigma )\rightarrow 0. $$ Now using the notations ${\rm Ext}_{+}$ and ${\rm Ext}_{-}$ of \cite{DLP}, p.\ 200, we have% $$ {\rm Ext}_{+}^i({\cal E},{\cal E})=H^i(\pi ^{*}({\cal K}^{\vee }\otimes {\cal L})\otimes \Omega _{S/C}(\sigma ))=0 $$ for all $i$. Hence ${\rm Ext}^i({\cal E},{\cal E})\cong {\rm Ext}_{-}^i(% {\cal E},{\cal E})$ for all $i$. Thus the infinitesimal deformations of $% {\cal E}$ are the same as the infinitesimal deformations of the filtered sheaf $0\subset \pi ^{*}({\cal K})\subset {\cal E}$. Furthermore, since $% {\rm Ext}^2(\pi ^{*}({\cal L})\otimes \Omega _{S/C}(\sigma ),\pi ^{*}({\cal K% }))=0$, we have a surjection% $$ {\rm Ext}_{-}^1({\cal E},{\cal E})\rightarrow {\rm Ext}_{{\cal O}_C}^1({\cal % K},{\cal K})\oplus {\rm Ext}_{{\cal O}_C}^1({\cal L},{\cal L})\rightarrow 0. $$ Hence a general infinitesimal deformation of ${\cal E}$ induces general infinitesimal deformations of ${\cal K}$ and ${\cal L}$. Since none of these deformations are obstructed, the morphism ${\rm \Pr ior}^0\rightarrow {\rm % Coh}_C(r-d,k)\times {\rm Coh}_C(d,l)$ defined by $[{\cal E}]\mapsto ([{\cal K% }],[{\cal L}])$ is dominant. The fibers of this morphism are irreducible since they are stack quotients of an open subscheme of the affine space $% {\rm Ext}^1(\pi ^{*}({\cal L})\otimes \Omega _{S/C}(\sigma ),\pi ^{*}({\cal K% }))$.The target of the morphism is irreducible since stacks of coherent sheaves of a fixed rank and degree on a smooth connected curve $C$ are irreducible. So ${\rm \Pr ior}^0$ and hence also ${\rm \Pr ior}_S(r,c_1,c_2)$ are irreducible. This completes of the proof of Proposition \ref{stack} when $\pi {:}~S\rightarrow C$ is a geometrically ruled surface. \TeXButton{qed} {\hfill $\Box$ \medskip} \paragraph{Proof of Proposition \ref{stack} in general.} We go by induction on the Picard number $\rho (S):={\rm rk}_{{\bf Z}}({\rm NS% }(S))$. The initial value is $\rho (S)=2$ which is the case of geometrically ruled surfaces which we just proved. So we may assume that $\rho (S)\geq 3$. Then $\pi {:}~S\rightarrow C$ is birationally but not geometrically ruled. So some fiber of $\pi $ contains an irreducible component $E\cong {\bf P}^1$ such that $E^2=-1$. We let $\alpha {:}~S\rightarrow S_1$ be the contraction of $E$, and we let $\beta {:}~S_1\rightarrow C$ be the morphism such that $% \pi $ factors as $\pi =\beta \alpha $. Since $\rho (S_1)=\rho (S)-1$, we may assume by induction that Proposition \ref{stack} holds for $\beta {:}% {}~S_1\rightarrow C$. Let $d=-c_1\cdot E$. By replacing ${\cal E}$ with an appropriate twist $% {\cal E}(nE)$ we may assume that $0\leq d<r$. Because $f_{\pi (E)}-E$ is effective, the condition ${\rm Ext}^2({\cal E},{\cal E}(-f_{\pi (E)}))=0$ implies ${\rm Ext}^2({\cal E},{\cal E}(-E))=0$. So by Lemmas \ref{P1} and \ref{restriction} the substack ${\rm \Pr ior}^1\subset {\rm \Pr ior}% _S(r,c_1,c_2)$ parametrizing prioritary sheaves ${\cal E}$ such that ${\cal E% }{\mid }_E\cong {\cal O}_E^{r-d}\oplus {\cal O}_E(-1)^d$ is open and dense. By Lemma \ref{blowup}, the application $[{\cal E}]\rightarrow [\alpha _{*}(% {\cal E})]$ defines a morphism ${\rm \Pr ior}^1\rightarrow {\rm \Pr ior}% _{S_1}(r,\alpha _{*}(c_1),c_2+\frac 12d(d-1))$. Moreover, the morphism realizes ${\rm \Pr ior}^1$ as a $d(r-d)$-dimensional Grassmannian bundle over the dense open substack of ${\rm \Pr ior}_{S_1}(r,\alpha _{*}(c_1),c_2+\frac 12d(d-1))$ of sheaves which are locally free at the center of the blowup. Since ${\rm \Pr ior}_{S_1}(r,\alpha _{*}(c_1),c_2+\frac 12d(d-1))$ is irreducible by the inductive hypothesis, we see that ${\rm \Pr ior}^1$ and hence also ${\rm \Pr ior}_S(r,c_1,c_2)$ are irreducible. This completes the proof of Proposition \ref{stack}. \TeXButton{qed}{\hfill $\Box$ \medskip} \paragraph{Proof of Theorem \ref{semistable}.} First we show that if $H\cdot (K_S+f)<0$, then any $H$-semistable sheaf $% {\cal E}$ is prioritary. But if ${\cal E}$ is $H$-semistable, then any nonzero torsion-free quotient ${\cal Q}$ of ${\cal E}$ would have $H$-slope satisfying $\mu _H({\cal Q})\geq \mu _H({\cal E})$, while any nonzero subsheaf ${\cal S}$ of ${\cal E}$ would have $H$-slope satisfying $\mu _H(% {\cal S})\leq \mu _H({\cal E})$. So if ${\cal E}$ were not prioritary, there would exist a $p\in C$ such that ${\rm Ext}^2({\cal E},{\cal E}(-f_p))\neq 0$% . There would then exist a nonzero $\phi \in {\rm Hom}({\cal E},{\cal E}% (K_S+f_p))\cong {\rm Ext}^2({\cal E},{\cal E}(-f_p))^{*}$. The image of $% \phi $ would then satisfy% $$ \mu _H({\cal E})\leq \mu _H({\rm im}(\phi ))\leq \mu _H({\cal E}% (K_S+f_p))=\mu _H({\cal E})+H\cdot (K_S+f), $$ contradicting $H\cdot (K_S+f)<0$. Thus the semistable sheaves on $S$ of rank $r$ and Chern classes $c_1$ and $% c_2$ are parametrized by an open substack $H{\rm -SS}\subset {\rm \Pr ior}% _S(r,c_1,c_2)$. This last stack is smooth and irreducible by Proposition \ref {stack}. So if there exist $H$-semistable sheaves on $S$ with that rank and those Chern classes, then $H{\rm -SS}$ is a smooth and irreducible stack. We now show that this implies that the moduli scheme $M_{S,H}(r,c_1,c_2)$ is normal and that the open subscheme $M_{S,H}^s(r,c_1,c_2)$ parametrizing $H$% -stable sheaves is smooth. We write ${\cal O}_S(1):={\cal O}_S(H)$. Since $H% {\rm -SS}$ is a limited family, there exists an integer $m\gg 0$ such that all $H$-semistable sheaves ${\cal E}$ in $H{\rm -SS}$ satisfy $H^i({\cal E}% (m))=0$ for $i\geq 1$ and have ${\cal E}(m)$ generated by global sections. Let $H_m$ be a vector space of dimension $h^0{\cal (E}(m))$ and let $Q^{ss}:=% {\rm Quot}_{S,H}^{ss}(m;r,c_1,c_2)$ denote the Hilbert-Grothendieck scheme parametrizing all quotients $\gamma {:}~H_m\otimes {\cal O}_S(-m) \TeXButton{-->>}{\twoheadrightarrow}{\cal F}$ such that ${\cal F}$ is $H$% -semistable of rank $r$ and Chern classes $c_1$and $c_2$ with $H^i({\cal F}% (m))=0$ for all $i\geq 1$ and such that the induced map $H_m\rightarrow H^0(% {\cal F}(m))$ is an isomorphism. Then according to the construction of \cite {LMB} (4.14.2), $H{\rm -SS}$ may be identified with the quotient stack$\ \left[ Q^{ss}/{\rm GL}(H_m)\right] $. Hence $Q^{ss}$ is a smooth and irreducible scheme. But $M_{S,H}(r,c_1,c_2)$ is the GIT quotient scheme $% Q^{ss}//({\rm SL}(H_m),{\cal L}_N)$ where ${\cal L}_N$ is Simpson's polarization of $Q^{ss}$ defined by ${\cal L}_N:=\det ({\rm pr}_{1*}({\cal U}% \otimes {\rm pr}_2^{*}({\cal O}_S(N))))$ where ${\cal U}$ is the universal sheaf on $Q^{ss}\times S$, the ${\rm pr}_i$ are the two projections, and $% N\gg m$ (cf.\ \cite{S} \S 1). Thus $M_{S,H}(r,c_1,c_2)$ is the GIT quotient of a smooth and irreducible variety. But such quotients, when nonempty, are always normal and irreducible varieties, and the points of the quotient corresponding to stable points are smooth. \TeXButton{qed}{\hfill $\Box$ \medskip} \section{Existence of Ample Divisors} We show that on any birationally ruled surface there exists an ample divisor satisfying the hypothesis of Theorem \ref{semistable}. \begin{lemma} \label{H}Let $S$ be a birationally ruled surface, $\pi {:}~S\rightarrow C$ a birational ruling, and $f\in {\rm NS}(X)$ the class of a fiber of $\pi $. Then there exists an ample divisor $H$ on $S$ such that $H\cdot (K_S+f)<0$. \end{lemma} \TeXButton{Proof}{\paragraph{Proof. }}First we consider the case of $S$ a geometrically ruled surface. Let $\sigma $ be a section of minimal self-intersection $-e$. According to \cite{H} Chapter V, Corollary 2.11, Propositions 2.20 and 2.21 and Exercise 2.14, we see that $K_S\equiv -2\sigma +(2g-2-e)f$ and that an $% H=\sigma +bf$ is ample if $b$ is sufficiently large. Since $H\cdot (K_S+f)=2g-1+e-2b$, by picking $b$ large enough we get an ample $H$ such that $H\cdot (K_S+f)<0$. Now we consider the case where $\pi {:}~S\rightarrow C$ is birationally but not geometrically ruled. Then some fiber of $\pi $ contains an exceptional divisor of the first kind $E$. Let $\alpha :S\rightarrow S_1$ be the contraction of $E$. By induction on the Picard number we may assume there exists an ample divisor $H_1$ on $S_1$ such that $H_1\cdot (K_{S_1}+f)<0$. {}From \cite{H} Chapter V, Proposition 3.3 and Exercise 3.3, we see that $% K_S=\alpha ^{*}(K_{S_1})+E$ and that $H:=2\alpha ^{*}(H_1)-E$ is ample. Then $H\cdot (K_S+f)=2(H_1\cdot (K_{S_1}+f))-E^2<0$. \TeXButton{qed} {\hfill $\Box$ \medskip}

9311

alg-geom/9311004

en

https://arxiv.org/abs/alg-geom/9311004

alg-geom/9311004

J. Winkelmann

On discrete Zariski-dense subgroups of algebraic groups

17 pages, plain TeX

We investigate for which linear-algebraic groups (over the complex numbers or any local field) there exists subgroups which are dense in the Zariski topology, but discrete in the Hausdorff topology. For instance, such subgroups exist for every non-solvable complex group.

alg-geom

\section{References} \let\c\Cedille \def\(##1\){} \frenchspacing \setbox0=\hbox{99.\enspace}\refindent=\wd0\relax \def\item##1##2:{\goodbreak \hangindent\refindent\hangafter=1\noindent\hbox to\refindent{\hss\refcite{##1}\enspace}\petcap \ignorespaces##2\rm: }} \def\refcite#1{\citeerrortrue\dorefcite#1[]\endcite\ifciteerror\errmessage{Citation error}\fi.} \def\dorefcite#1[#2]#3\endcite{\def\next{#2}% \ifx\next\empty\makecite{#1}{}\else\makecite{#2}{}\fi}% \def\AMS#1{\relax} \AMS{14G20,14L10,20G25,22E40} \def\otimes{\otimes} \def\smallmatrix#1{\dosmallmatrix#1}% \def\dosmallmatrix#1&#2\cr#3&#4\cr{\bigl({#1\atop#3 }{#2\atop#4}\bigr)}% \def\Rightarrow{\Rightarrow} \def\mathop{{\cal L}{\sl ie}}{\mathop{{\cal L}{\sl ie}}} \relax \defcitations{B,BS,BT,G,Mc,M,OP,R,Ro,S,T,Z} \titel{On discrete Zariski-dense subgroups of algebraic groups} \section{Introduction} Let $k$ be a local field, \ie a field equipped with an absolute value inducing a non-discrete locally compact topology. (It is well-known that any local field $k$ is isomorphic to one of the following: $\s C$, $\s R$, a finite extension of a field $\s Q_p$ of p-adic numbers or $\s F_q((t))$, where $\s F_q$ is a finite field.) Then every $k$-variety admits two topologies: the Zariski-topology and a Hausdorff topology induced by the absolute value on $k$. We are now interested in subgroups of $k$-groups for which these two topologies are as different as possible, \ie groups which are discrete in the Hausdorff topology, but dense in the Zariski-topology. A main motivation for our investigation was the following. For complex linear-algebraic groups a discrete cocompact subgroup is necessarily Zariski-dense. There are known obstructions to the existence of discrete cocompact subgroups, namely only unimodular groups may admit discrete cocompact subgroups. Thus one may ask whether these obstructions actually prohibit only discrete cocompact subgroups or prohibit {\sl all} Zariski-dense discrete subgroups. It turns out that there are many groups (in particular all non-solvable complex parabolic groups) which do admit discrete Zariski-dense subgroups although they do not admit any discrete cocompact subgroup. On the other hand there do exist linear-algebraic groups without any discrete Zariski-dense subgroup at all. The goal of this article is to give criteria as precise as possible for the existence and non-existence of discrete Zariski-dense subgroups. For non-solvable groups the main result is: \proclaim@{\bf}{Theorem Let $G$ be a Zariski-connected non-solvable $k$-group, defined over a local field $k$. Let $R$ denote the radical. For $char(k)=0$ the following statements are equivalent. \item{(i)} $G/R(k)$ is non-compact in the Hausdorff topology. \item{(ii)} $G(k)/R(k)$ is non-compact in the Hausdorff topology. \item{(iii)} $G/R$ is $k$-isotropic, \ie contains a positive-dimensional $k$-split torus. \item{(iv)} $G(k)$ is not amenable. \item{(v)} There exists a discrete Zariski-dense subgroup in $G(k)$. \endproclaim For $k=\s C$ this boils down to the following \proclaim@{\bf}{Theorem 1' Let $G$ be a non-solvable connected complex linear-algebraic group. Then $G$ admits a subgroup $\Gamma$ which is discrete in the Hausdorff topology and dense in the Zariski-topology. \endproclaim Since parabolic complex groups never contain lattices (they are never unimodular), this implies that there exist complex groups which admit discrete Zariski-dense subgroups although they do not admit lattices. For $char(k)>0$ we have partial results, in particular the implications $ (v)\Rightarrow (iv)\Rightarrow (ii)$ of Theorem 1 still hold. Concerning solvable groups we obtain quite different results for the various local fields. In the complex case, \ie $k=\s C$ we have a number of partial results. \proclaim@{\bf}{Theorem \def\par\noindent\sl{\par\noindent\sl} {\par\noindent\sl Reductive groups.} Let $G$ be a connected reductive solvable group, \ie $(\s C^*)^n$. Then for each $1\le k\le n $ there exists a Zariski-dense discrete subgroup $\Gamma$ in $G$ with $\Gamma\simeq(\s Z^k,+)$. (This is well-known). {\par\noindent\sl Unipotent groups.} A unipotent complex group $G$ admits a Zariski-dense discrete subgroup if and only if $G$ admits a real-algebraic subgroup $G_0$ which may be defined over $\s Q$ and is not contained in any proper complex subgroup of $G$. {\par\noindent\sl Borel groups.} A Borel group in a simple complex group never admits a Zariski-dense discrete subgroup. In contrast, Borel groups in $SL_2(\s C)\times SL_2(\s C)$ contain Zariski-dense discrete subgroups. {\par\noindent\sl Metabelian groups.} \item{(i)} Assume that $G=(\s C^*)\mathbin{\vrule height 4.5pt depth 0.1pt\kern-1.8pt\times}_\rho(\s C^k,+)$. Then unimodularity (\ie $\rho(\s C^*)\subset SL_k(\s C)$) is a necessary but in general not sufficient condition for the existence of a Zariski-dense discrete subgroup. \item{(ii)} A Borel group $B$ in $SL_2(\s C)\times SL_2(\s C)$ is an example for a semidirect product $B\simeq(\s C^*)^2\mathbin{\vrule height 4.5pt depth 0.1pt\kern-1.8pt\times}(\s C^2,+)$ which is not unimodular but nevertheless admits a discrete Zariski-dense subgroup. \item{(iii)} (Otte-Potters): Let $T$ be a maximal torus in $S=SL_k(\s C)$ and $G=T\mathbin{\vrule height 4.5pt depth 0.1pt\kern-1.8pt\times}_\rho(\s C^k,+)$ with $\rho$ given by the usual $S$-action on $\s C^k$. Then $G$ admits a Zariski-dense discrete cocompact subgroup. \endproclaim For $k=\s R$ we obtain the following \proclaim@{\bf}{Theorem Let $G$ be a connected solvable $\s R$-group. Then the following conditions are necessary, but in general not sufficient for the existence of discrete Zariski-dense subgroups in $G(\s R)$. \item{(i)} The commutator group $\cal D(G)$ may be defined over $\s Q$. \item{(ii)} The group is unimodular, \ie $Ad(G)\subset SL(\a g)$. If $G$ is unipotent, then condition $(i)$ is sufficient for the existence of discrete Zariski-dense subgroups (Condition $(ii)$ is automatically fulfilled, if $G$ is nilpotent). \endproclaim Finally, for non-archimedean fields in characteristic zero we are able to provide a complete description. \proclaim@{\bf}{Theorem Let $G$ be a Zariski-connected solvable $k$-group, $k$ a non-archi\-me\-dean local field with $char(k)=0$. Let $U$ denote the unipotent radical and $T$ a maximal $k$-split torus. Then there exists a discrete Zariski-dense subgroup of $G(k)$ if and only if the following two conditions are fulfilled: \item{(i)} $G$ is commutative. \item{(ii)} $\dim T\ge\max\{1,\dim U\}$. \endproclaim We basically use the notation of \cite B and \cite M. In particular a $k$-group $G$ is a linear-algebraic group defined over a field $k$ and $G(k)$ denotes the group of $k$-rational points. For a local field $k$, $G(k)$ carries an induced Hausdorff topology. Topological terms refer to this Hausdorff topology. Topological notions concerning the Zariski topology are preceded by the prefix {\sl "Zariski-"}. For $k=\s C$ the notions {\sl connected} and {\sl Zariski-connected} coincide. \section{Arithmetic Groups} Let $k$ be a local field and $G$ a simple group defined over $k$. Then there exists an {\sl arithmetic} subgroup $\Gamma\subset G(k)$, which is a lattice (\ie discrete with finite covolume) by the Borel-Harish-Chandra-Behr-Harder reduction theorem. Due to the Borel-Wang Density theorem $\Gamma$ is Zariski-dense unless $G$ is $k$-anisotropic. (For a local field $k$ a reductive group $G$ is $k$-anisotropic if and only if $G(k)$ is compact in the Hausdorff topology \cite{M,2.3.6. p.54}). Together this yields the following \proclaim@{\bf}{Theorem A Let $G$ be a Zariski-connected semisimple $k$-group without $k$-anisotropic factors, $k$ a local field. Then $G(k)$ contains a discrete Zariski-dense subgroup which furthermore is a lattice. \endproclaim In particular every connected complex semisimple group and every connected real semisimple group without compact factor admit a discrete Zariski-dense subgroup. \section{Preparations} We need the following theorem of Tits on the existence of free subgroups in linear groups. \proclaim@{\bf}{Theorem B (Tits) Let $G$ be a semisimple linear-algebraic group defined over a local field $k$, $\Lambda$ a Zariski-dense subgroup of $G(k)$. Then $\Lambda$ contains a free subgroup $H$ with infinitely many generators $f_i$, such that any two of these $f_i$ generate a Zariski-dense subgroup of $G$. \endproclaim \proclaim@{\bf}{Corollary}{}{\it} Let $G$ be a linear-algebraic group defined over a local field $k$, $\Lambda$ a subgroup. Then either $\Lambda$ contains a non-commutative free subgroup or it contains a solvable subgroup of finite index. \endproclaim For $char(k)=0$ the theorem is Th.3 in \cite T. For $char(k)>0$ it follows from Th. 4 in \cite T, because in a local field of positive characteristic only finitely many elements are algebraic over the prime field. (This follows from the result that such a field is isomorphic to $\s F_q((t))$ for some finite field $\s F_q$.) Note that such a statement does not hold for arbitrary (\ie not local) fields. For instance, let $k$ be an algebraic closure of a finite field $\s F_p$. Then $G=SL_2(k)$ is a group which contains neither a solvable subgroup of finite index nor a free subgroup. In fact any finitely generated subgroup of $G$ is finite. See \cite T for details. We will need a result in order to control the image of $k$-rational points under a $k$-morphism. \proclaim@{\bf}{Theorem C (Borel-Serre \cite{BS,p.153}) Let $k$ be a local field of characteristic zero, $G$, $H$ $k$-groups, $\rho:G\to H$ a surjective $k$-group homomorphism. Then $\rho(G(k))$ is a subgroup of finite index in $H(k)$. \endproclaim Unfortunately such a statement does not hold in positive characteristic. \proclaim@{\bf}{Example}{}{\rm} 1 Let $k=\s F_p((t))$, $G_m$ the multiplicative group of the field (\ie $G_m(k)=(k^*,\cdot)$) and $\rho:G_m\to G_m$ the group morphism given by $\rho(x)=x^p$. Then $\rho(G_m(k))=\{\sum_k a_kt^{kp}\}$ and the natural group homomorphism $\pi:G_m(k)\to G_m(k)/\rho(G_m(k))=Q$ maps $S=\{1+t+\sum_{k>0} a_kt^{kp}\}$ injectively into $Q$, hence $\rho(G_m(k))$ has infinite index in $\rho(G_m)(k)$. \endproclaim \section{Non-amenable Groups} \proclaim@{\bf}{Theorem Let $k$ be a local field of characteristic zero, $G$ be a Zariski-connected $k$-group. Assume that $G/R(k)$ is not compact, \ie not $k$-anisotropic. Then $G(k)$ admits a Zariski-dense discrete subgroup. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} By assumption there exists a surjective $k$-group morphism $\rho$ from $G$ to a simple $k$-isotropic group $S$. $S(k)$ admits a Zariski-dense discrete subgroup $\Lambda$. $I=\rho(G(k))$ is a subgroup of finite index in $S(k)$. Thus $\Lambda_1=\Lambda\cap I$ is of finite index in $\Lambda$, hence Zariski-dense and discrete in $S(k)$. Now $\Lambda_1$ contains a countable infinite subset $F=\{a_i:i\in\s N_0\}$ such that the elements of $F$ are free generators of a discrete subgroup $\Lambda_0\subset\Lambda_1$ and any two elements of $F$ generate a Zariski-dense subgroup of $S$. Choose $b_0,b_1\in G(k)$ such that $\rho(b_i)=a_i$ ($i=1,2$) and let $H$ denote the Zariski-closure of the subgroup generated by $b_0$ and $b_1$. Now $\rho$ maps $H$ surjectively on $S$, since $a_0$ and $a_1$ generate a Zariski-dense subgroup of $S$. Hence $\rho(H(k))$ is a subgroup of finite index in $S(k)$. Choose $c_i\in H(k)$, $n_i\in\s N$ ($i\ge 2$), such that $\rho(c_i)=a_i^{n_i}$. Let $\Sigma=\{s_i:i\in\s N, i\ge 2\}$ be a countable Zariski-dense subset of $A(k)$ where $A$ denotes the kernel of $\rho:G\to S$. Finally choose $b_i\in G(k)$ for $i\ge 2$ such that $b_i=c_i\cdot s_i$ and let $\Gamma$ denote the subgroup of $G(k)$ generated by the elements $b_i$ ($i\ge 0$). The map $\rho$ maps $\Gamma$ injectively into $\Lambda$, hence $\Gamma$ is discrete. Futhermore the construction implies that $\Gamma$ is Zariski-dense in $G$. \qed In the above proof we used the assumption $char(k)=0$ at two points: First we used that for a perfect field $k$ the radical of a $k$-group is defined over $k$. (A local field $k$ is perfect if and only if $char(k)=0$.) Second we used that for a surjective $k$-group homomorphism $\rho:G\to H$ the image of the $k$-rational points $\rho(G(k))$ has finite index in $H(k)$. By strengthening the assumptions one may circumvent these problems and obtain the following result. \proclaim@{\bf}{Proposition Let $k$ be a local field of arbitrary characteristic, $S$ a $k$-isotropic simple $k$-group, $H$ a $k$-group and $G$ a $k$-group which is $k$-isomorphic to a semi-direct product $S\mathbin{\vrule height 4.5pt depth 0.1pt\kern-1.8pt\times} H$. Then $G(k)$ admits a Zariski-dense discrete subgroup. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} The assumptions imply the existence of a $k$-group homomorphism $\rho$ from $G$ to $S$ which maps $G(k)$ surjectively on $S(k)$. Thus the proof for $char(k)=0$ can be carried over with the following modification. Using the semi-direct product structure we may choose $b_0$, $b_1$ inside the factor isomorphic to $S$. Then $H$ contains the factor isomorphic to $S$ and consequently $H(k)$ maps surjectively on $S(k)$ by $\rho$. \qed \medskip \section{Amenable Groups} \proclaim@{\bf}{Definition}{}{\rm} A locally compact topological group is called {\sl amenable} if for every continuous action on a compact metrizable topological space there exists an invariant probability measure. \endproclaim For any $k$-group defined over a local field $k$ the group $G(k)$ of $k$-rational points is a locally compact topological group in the Hausdorff topology. Thus we may apply the theory of amenable groups. We summarize some important properties of amenable groups. (see \cite G \cite Z for proofs and details). \item{i)} Compact groups are amenable, \item{ii)} Solvable groups are amenable, \item{iii)} Free discrete groups (with more than one generator) are not amenable, \item{iv)} If there is an exact sequence of topological groups $0\to A \to B \to C\to 0$, then $B$ is amenable iff $A$ and $C$ are amenable. \item{v)} Closed subgroups of amenable groups are amenable. Using these facts we can determine completely (for $char(k)=0$) when $G(k)$ is amenable. \proclaim@{\bf}{Lemma}{\the\lemno\global\advance\lemno by 1 }{\it} Let $k$ be a local field, $G$ a $k$-group, $R$ the radical of $G$. For $char(k)=0$ the follwing properties are equivalent: \item{1)} $G/R$ is $k$-anisotropic, \ie $G/R$ does not contain any positive-dimensional $k$-split torus; \item{2)} $G/R(k)$ is compact; \item{3)} $G(k)/R(k)$ is compact; \item{4)} $G(k)$ is amenable. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} For $1)\!\iff\! 2)$ see \cite M. The equivalence $2)\!\iff\! 3)$ follows from Theorem C, because $G(k)/R(k)$ is closed in $G/R(k)$ (\cite {BT,3.18.}). The implication $3)\Rightarrow 4)$ is a direct consequence of the above listed properties on amenable groups (i), ii) \& iv)). Finally, if $G/R(k)$ is not compact, then there exists a discrete subgroup of $G(k)$ containing a free subgroup (Th. A \& B). But an amenable group cannot contain a closed discrete non-commutative free subgroup. Hence $4)$ implies $2)$. \qed There is a similar result for arbitrary connected real Lie groups, see \cite{Z, 4.1.9 on p.62}. In positive characteristic the radical $R$ is not necessarily defined over $k$, hence $1)$ and $2)$ do not make sense. Even if $R$ is defined over $k$, it is not clear whether $G(k)/R(k)$ has finite index in $G/R(k)$. Therefore it is not clear whether $2)$ and $3)$ are equivalent. But at least the implication $3)\Rightarrow 4)$ is true in any characteristic. \proclaim@{\bf}{Proposition Let $G$ be a Zariski-connected $k$-group defined over a local field $k$. Assume that $G(k)$ is amenable and that there exists a Zariski-dense discrete subgroup $\Gamma$ of $G(k)$. Then $G$ is solvable. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} Amenability of $G$ implies amenability of $\Gamma$. By the result of Tits either $\Gamma$ contains a free subgroup or a solvable subgroup of finite index. Since the former is ruled out by amenability of $\Gamma$, $\Gamma$ contains a solvable subgroup $\Gamma_0$ of finite index which is still Zariski-dense in $G$. Hence $G$ is solvable. \qed \proclaim@{\bf}{Corollary}{}{\it} Let $G$ be a Zariski-connected $k$-group defined over a local field $k$, $R$ its radical. Assume that $G/R(k)$ is compact, but $G\ne R$. Then $G(k)$ does not contain any Zariski-dense discrete subgroup. \endproclaim \section{An obstruction for solvable groups} We will now derive an obstruction to the existence of discrete Zariski-dense subgroups in solvable groups. For $A,B\subset G$ let $[A,B]$ denote the subgroup generated by the commutators $aba^{-1}b^{-1}$. \proclaim@{\bf}{Lemma}{\the\lemno\global\advance\lemno by 1 }{\it} Let $A,B$ algebraic subgroups in $G$ and $\Gamma$ a subgroup of $G$ such that $A\cap\Gamma$ and $B\cap\Gamma$ are Zariski-dense in $A$ resp. $B$. Then $[A,B]\cap\Gamma$ is Zariski-dense in $[A,B]$. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} Let $C$ denote the set of commutators $aba^{-1}b^{-1}$ and $C^n=C\cdot\ldots\cdot C$. For $n$ sufficiently large the natural morphism $(A\times B)^n\to C^n\to [A,B]$ is dominant and therefore maps the Zariski-dense subset $\((A\cap\Gamma)\times(B\cap\Gamma)\)^n$ onto a Zariski-dense subset $\Lambda$ in $[A,B]$. \qed \def{\cal C}{{\cal C}} \proclaim@{\bf}{Definition}{}{\rm} For a group $G$ let ${\cal C}(G)$ denote the smallest collection of subgroups of $G$ such that $G\in{\cal C}(G)$ and $[A,B]\in{\cal C}(G)$ for all $A,B\in{\cal C}(G)$. \endproclaim The collected ${\cal C}(G)$ contains in particular all subgroups of the derived and the central series. Every $H\in{\cal C}(G)$ is a normal subgroup and for a $k$-group $G$ every $H\in{\cal C}(G)$ is defined over $k$ (\cite{B,p.58}). For a Zariski-connected group $G$ every $H\in{\cal C}(G)$ is again Zariski-connected. We will use this notation to deduce an obstruction to the existence of discrete Zariski-dense subgroups. \proclaim@{\bf}{Lemma}{\the\lemno\global\advance\lemno by 1 }{\it} Let $H$ be a Zariski-connected one-dimensional $k$-group, $k$ a local field, $\Gamma\subset H(k)$ an infinite discrete subgroup. Let $A$ denote the group of all $k$-group automorphisms of $H$ stabilizing $\Gamma$. Then $A$ is finite. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} If $H$ is $\bar k$-isomorphic to the multiplicative group $G_m$, then the group of {\sl all} automorphisms of $H$ is finite (because $z\to z^{-1}$ is the only non-trivial automorphism of $G_m$). Hence we may assume that $H\simeq G_a$. This isomorphism is given over some finite extension field $k'$ of $k$. (\cite{B,Th.10.9 \& Remark}). We may replace $k$ by $k'$, \ie we may assume that $H\simeq G_a$ as $k$-groups. Now any $k$-group automorphism of $H$ is given by $\mu_\lambda: z\mapsto\lambda z$ for some $\lambda\in k^*$. Since $k$ is locally compact and $\Gamma$ discrete, there exists an element $\gamma_0\in\Gamma$ such that $\abs{\gamma_0}\le\abs g$ for all $g\in\Gamma\setminus\{0\}$. This easily implies that $\abs\lambda=1$ for all $\mu_\lambda\in A$. Now the quotient map from $A$ to the $A$-orbit $A(\gamma_0)$ is injective and $A(\gamma_0)\subset\{x:\abs{x}=\abs{\gamma_0}\}$. Since $\{x:\abs{x}=c\}$ is compact for all $c$, it follows that $A$ is finite. \qed \proclaim@{\bf}{Proposition Let $k$ be a local field, $G$ a Zariski-connected $k$-group, $H\in{\cal C}(G)$. Assume that $H$ is one-dimensional and not central. Then $G$ does not admit any discrete Zariski-dense subgroup. \endproclaim \proclaim@{\bf}{Remark}{}{\rm} (1) Though not stated explicitly, the assumptions of this proposition imply that $G$ is solvable. To see this, note that for each $H\in{\cal C}(G)$ there exists a subgroup $I$ in the derived series of $G$ such that $I\subset H$. It follows that $G$ is solvable as soon as there exists a solvable subgroup $H\in{\cal C}(G)$. (2) As we will see below, solvable non-commutative p-adic groups never admit Zariski-dense discrete subgroups. Hence this proposition is interesting only for $k=\s R$, $k=\s C$ and $char(k)>0$. \endremark \proclaim@{\it}{Proof}{}{\ninepoint\rm} of Proposition 3 Assume that there exists a discrete Zariski-dense subgroup $\Gamma$ of $G(k)$. Thanks to Lemma 2 it is clear that $A\cap\Gamma$ must be Zariski-dense in $A$ for all $A\in{\cal C}(G)$. Hence $H\cap\Gamma$ is infinite. Now $\Gamma$ acts by conjugation on $H$, stabilizing $H\cap\Gamma$. With the help of the preceding lemma this implies that $\Gamma$ contains a subgroup $\Gamma_0$ of finite index such that $\Gamma_0$ centralizes $H$. Since $H$ is {\sl not} central in $G$, this contradicts the assumption that $\Gamma$ is Zariski-dense in $G$. \qed We will now apply this to Borel groups. \proclaim@{\bf}{Corollary}{}{\it} Let $S$ be a Zariski-connected simple $k$-group and assume that there exists a Borel subgroup $G$ defined over $k$. Then $G(k)$ does not admit any discrete Zariski-dense subgroup. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} Let $H$ be the one-dimensional unipotent subgroup of $G$ corresponding to the maximal root. It is easy to check that $H$ is not central and $H\in{\cal C}$. \qed As we will see below, simplicity is an essential condition for this result. We will demonstrate that a Borel group $B$ of the semisimple complex linear-algebraic group $SL_2(\s C)\times SL_2(\s C)$ {\sl does} admit discrete Zariski-dense subgroups. \bigskip \vfill \eject \section{Unipotent Groups in characteristic zero} \proclaim@{\bf}{Lemma}{\the\lemno\global\advance\lemno by 1 }{\it} Let $U$ be a unipotent $k$-group, $char(k)=0$. Then every (non-trivial) element of $U$ is of infinite order and $U(k)$ contains a Zariski-dense subgroup generated by $r=dim(U)$ elements. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} Recall that $U$ is $k$-split, \ie admits a sequence of $k$-subgroups $G=G_0\supset\ldots\supset G_s=\{e\}$ with $\dim G_i/G_{i+1}=1$. \cite{B, Cor. 15.5 (ii) on p.205}. Furthermore for $char(k)=0$ every one-dimensional unipotent $k$-group is $k$-isomorphic to $G_a$ (\cite{B,Th.10.9 \& Remark below}). Using these facts the assertions of the lemma follows easily by induction on $dim(U)$. \qed \proclaim@{\bf}{Lemma}{\the\lemno\global\advance\lemno by 1 }{\it} Let $U$ be a commutative unipotent $k$-group, $char(k)=0$, $g\in U\setminus\{e\}$ and $\s Z(g)\subset U$ the subgroup generated by $g$. Then the Zariski-closure of $\s Z(g)$ is one-dimensional. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} This follows immediately from the fact that $U$ is $\bar k$-isomorphic to a vector group $G_a^n$ (\cite{S,p.171}). \qed By induction one obtains the following consequence. \proclaim@{\bf}{Corollary}{}{\it} Let $U$ be a commutative unipotent $k$-group, $char(k)=0$, $\Gamma$ a Zariski-dense subgroup. Then $\Gamma$ can not be generated by less then $dim(U)$ elements. \endproclaim Combining these two lemmata we obtain \proclaim@{\bf}{Proposition Let $U$ be a commutative unipotent $k$-group, $char(k)=0$. Then $U$ is $k$-isomorphic to the vector group $(G_a)^r$. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} Let $r=dim(U)$ and $g_1,\ldots,g_r\in U(k)$ generators of a Zariski-dense subgroup. Let $\s Z(g_i)$ denote the subgroups generated by each $g_i$ and $H_i$ the Zariski-closure of the $\s Z(g_i)$. Since $g_i\in U(k)$, the $H_i$ are defined over $k$ (\cite{B,AG 14.4}). Furthermore the $H_i$ are one-dimensional and $k$-isomorphic to $G_a$. Since $U$ is commutative, the $k$-subgroups $H_i$ induce a $k$-group homomorphism $\rho$ from $H=\Pi H_i\simeq(G_a)^r$ to $U$. Now $\rho$ maps a Zariski-dense subgroup of $H(k)$ onto the Zariski-dense subgroup $\Gamma\subset U(k)$ generated by the $g_i$ and is therefore dominant and defined over $k$ (\cite{BT,1.4.}). A dominant $k$-group morphism between $k$-groups of the same dimension is surjective with finite kernel. But $char(k)=0$ implies that $(G_a)^r$ has no non-trivial finite subgroup. Thus $\rho$ is bijective. Again using $char(k)=0$ it follows that $\rho$ is an isomorphism. Finally, the inverse $\rho^{-1}$ maps the Zariski-dense set $\Gamma \subset U(k)$ into $H(k)$ and is therefore likewise defined over $k$. \qed Such a statement does not hold in positive characteristic, consider \eg Witt groups \cite{S,VII} or Examples 3 and 4 in Section 9. \proclaim@{\bf}{Corollary}{}{\it} 1 Let $U$ be a commutative unipotent $k$-group, $k$ an archimedean local field (\ie $\s R$ or $\s C$). Then $U$ admits a Zariski-dense discrete subgroup. \endproclaim \proclaim@{\bf}{Corollary}{}{\it} 2 Let $U$ be a commutative unipotent $k$-group, $k$ a non-archimedean local field of characteristic zero, $\Gamma$ a finitely generated subgroup of $U(k)$. Then $\Gamma$ is relatively compact in $U(k)$. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} This follows from the ultrametric condition via $U(k)\simeq G_a(k)^r$. \qed \proclaim@{\bf}{Corollary}{}{\it} 3 Let $G$ be a commutative unipotent $k$-group, $k$ non-archimedean local field of characteristic zero. Then $G(k)$ contains no discrete subgroups except $\{e\}$. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} Using induction on $dim(G)$ it is easy to prove that for $char(k)=0$ a unipotent $k$-group cannot contain a non-trivial finite subgroup. Therefore the preceding corollary implies that $\{e\}$ is the only finitely generated discrete subgroup of $G(k)$. Finally note that any discrete group must contain a finitely generated subgroup. \qed \proclaim@{\bf}{Corollary}{}{\it} 4 Let $G$ be a unipotent $k$-group, $k$ a non-archimedean local field of characteristic zero. Then $G(k)$ contains no discrete subgroups except $\{e\}$. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} Assume the contrary. Since every element in $G(k)$ is of infinite order, such a discrete subgroup would contain a subgroup $\Gamma$ isomorphic to $\s Z$. But then the Zariski-closure of $\Gamma$ would be a commutative unipotent $k$-group, thus contradicting the preceding corollary. \qed \proclaim@{\bf}{Proposition Let $G$ be a Zariski-connected solvable $k$-group for a non-archimedean local field $k$ with $char(k)=0$. Then any discrete subgroup $\Gamma\subset G(k)$ is commutative. In particular $G(k)$ cannot admit any discrete Zariski-dense subgroup unless $G$ is commutative. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} By standard results on solvable groups the commutator group $\cal D(G)$ of $G$ is unipotent. Thus for any discrete subgroup $\Gamma\subset G(k)$ we have $\Gamma\cap\cal D(G)(k)=\{e\}$, hence $\Gamma$ is commutative. This implies that the Zariski-closure of $\Gamma$ is commutative, too. \qed Now we turn to unipotent groups defined over archimedean fields. Here the question of the existence of discrete Zariski-dense subgroups has been settled by the following result of Malcev. \proclaim@{\bf}{Theorem D (Malcev, \cite{Mc}, see also \cite R) Let $G$ be a unipotent $k$-group, $k=\s R$. Then $G(k)$ admits a Zariski-dense discrete subgroup if and only if $G$ may be defined over $\s Q$. Each such discrete Zariski-dense subgroup is cocompact. \endproclaim In Malcev's article the condition that $G$ may be defined over $\s Q$ is replaced by the property that the Lie algebra may be defined over $\s Q$. However, for unipotent groups in characteristic zero the exponential map gives an isomorphism (as $k$-varieties) of the group and its Lie algebra. Using the Campbell-Hausdorff formula it follows that $G$ may be defined over $\s Q$ if and only if the Lie algebra can be defined over $\s Q$. Malcevs result immediately implies the following criterion for complex unipotent groups. \proclaim@{\bf}{Corollary}{}{\it} Let $G$ be a unipotent $k$-group, $k=\s C$. By "restriction of scalars" $G(\s C)$ is isomorphic (as topological group) to $\tilde G(\s R)$ for a unipotent group $\tilde G$ defined over $\s R$. Fix a continuous group isomorphism $\phi:\tilde G(\s R)\to G(\s C)$. $G(\s C)$ admits a Zariski-dense discrete subgroup if and only if $\tilde G$ admits a unipotent subgroup $H$ defined over $\s Q$ such that $\phi(H(\s R))$ is Zariski-dense in $G$. \endproclaim For the convenience of the reader we reformulate this in the language of Lie groups. \proclaim@{\bf}{Corollary}{}{\it} Let $G$ be a unipotent complex Lie group. Then $G$ admits a discrete subgroup which is dense in the algebraic Zariski-topology if and only if $G$ contains a real Lie subgroup $H$ such that \item{1)} The structure constants for $\mathop{{\cal L}{\sl ie}}(H)$ are rational numbers for a suitable base, \item{2)} $\mathop{{\cal L}{\sl ie}}(G)$ is the smallest complex vectorsubspace of $\mathop{{\cal L}{\sl ie}}(G)$ containing $\mathop{{\cal L}{\sl ie}}(H)$. \endproclaim \section{Real solvable groups} We will now use our results on real unipotent groups in order to deduce a statement about arbitrary real solvable groups. \proclaim@{\bf}{Proposition Let $G$ be a Zariski-connected solvable $k$-group, $k=\s R$. The following conditions are necessary for the existence of a Zariski-dense discrete subgroup. \item{(i)} The commutator group $\cal D(G)$ is defined over $\s Q$. \item{(ii)} $G$ is unimodular. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} Let $\Gamma$ be a discrete Zariski-dense subgroup in $G(\s R)$. Then the commutator group $\cal D(\Gamma)$ is Zariski-dense in $\cal D(G)$. Hence property $(i)$ is necessary. Moreover this implies that $\cal D(\Gamma)$ is cocompact in $\cal D(G)(\s R)$. This ensures unimodularity of $G$. \qed These conditions are not sufficient. For instance, let $$G(\s R)=\left\{\pmatrix{ \lambda^{-4} & w &&& \cr &1&&& \cr &&\lambda^2 & x & z \cr &&&\lambda & y \cr &&&& 1 \cr}:\lambda\in\s R^*;x,y,z\in\s R\right\}$$ $G$ fulfills all the conditions of the theorem, but likewise fulfills the obstruction criterion deduced in Proposition 3: $G''$ is one-dimensional, but not central. \section{Unipotent groups in positive characteristics} For local fields in positive characteristics we have only fragmentary results. A local field in positive characteristics is isomorphic to $\s F_q((t))$ for some $q=p^n$. Such a field contains infinite discrete subrings, \eg the ring generated by the elements $t^k$ for $k\le 0$. Hence it is easy to give examples of unipotent $k$-groups which do admit discrete Zariski-dense subgroups. However these discrete Zariski-dense subgroups are never finitely generated: \proclaim@{\bf}{Lemma}{\the\lemno\global\advance\lemno by 1 }{\it} Let $k$ be a field of positive characteristic, $U$ a Zariski-connected unipotent $k$-group and $\Gamma$ a finitely generated subgroup of $U(k)$. Then $\Gamma$ is finite. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} There is no loss in generality, if we assume $k$ to be algebraically closed. Then the lemma follows easily by induction, since any one-dimensional unipotent $k$-group is isomorphic to $G_a$. (In addition, one has to use the fact that for a finitely generated {\sl nilpotent} group every subgroup is again finitely generated \cite{R,Th.2.7.}). \qed This is in strong contrast to the situation in characteristic zero. For a local field $k$ with $char(k)=0$, a discrete subgroup of a solvable $k$-group is always finitely generated (This follows from \cite{R, Prop.3.8}). There also exist unipotent commutative $k$-groups in positive characteristic which do not contain any Zariski-dense discrete subgroup. \proclaim@{\bf}{Example}{}{\rm} 3 Let $k=\s F_p((t))$ and $G$ the one-dimensional unipotent $k$-group defined by $G=\{(x,y)\in G_a\times G_a:x^p-x=ty^p\}$.% \footnote*{This group has been studied by M. Rosenlicht \cite{Ro,p.46} for a different purpose.} Then $G(k)$ is compact and therefore does not admit any infinite discrete subgroup. To check compactness, let $x=\sum_i a_it^i$, $y=\sum_i b_it^i$. An explicit calculation shows that $(x,y)\in G$ if and only if $a_k^p=a_{kp}$ and $-a_{kp+1}=b_k^p$ for all $k\in\s Z$ and $a_k=0$ for all $k$ with $k \hbox{ \sl mod }p\not\in\{0,1\}$. This implies $a_k=0=b_k$ for all $k<0$. Hence $G(k)$ is a closed subgroup of $\cal O\times\cal O$, where $\cal O=\{x:\abs x\le 1\}$ denotes the (compact) additive group of local integers. Thus $G(k)$ is compact. \endproclaim Even for non-compact unipotent groups it is possible that there exist no discrete Zariski-dense subgroup. \proclaim@{\bf}{Example}{}{\rm} 4 Let $k=\s F_p((t))$. Let $U=G_a\times W_2$ where $W_2$ is the two-dimensional Witt group, \ie $W_2$ is $\s A^2$ as variety with the group multiplication given by $(x,y)\cdot(z,w)=(x+z,y+w+F(x,z))$ for $F(x,z)={1\over p}\left(x^p+y^p-(x+y)^p\right)$. (See \cite{S,VII.2} for more about Witt groups). As a variety $U=\s A^3$. Now let $H=\{(x,y,z)\in U:x^p-x=ty\}$. As a $k$-group $H$ is an extension of $G_a$ by the group $G$ studied in the previous example. Now $H(k)$ is non-compact, because it contains $G_a(k)$. Nevertheless there are no discrete Zariski-dense subgroups in $H(k)$. To see this, consider the group morphism $\phi:x\mapsto x^p$, where $x^p$ denotes the $p$-th power with respect to group multiplication in $H$. By standard results on Witt groups $\phi$ is a dominant morphism from $H$ to $A=\{(0,0,z)\}\subset U$ with $A\subset \ker\phi$. Now $H(k)/A(k)$ is compact, hence $\phi(H(k))$ is compact. Since $\phi(\Gamma)\subset\Gamma$, it follows that $\phi(\Gamma)$ is finite for every discrete subgroup $\Gamma\subset H(k)$. But $\phi(\Gamma)$ must be Zariski-dense in $\phi(H)=A$ for every Zariski-dense subgroup $\Gamma\subset H$. \endproclaim \section{General commutative groups} We will now deal with commutative groups which are not necessarily unipotent. The following decomposition theorem is a centerpiece for this investigation. \proclaim@{\bf}{Theorem D Let $G$ be a Zariski-connected commutative $k$-group, $char(k)=0$. Then $G$ admits a decomposition $G=G_u\times G_c\cdot G_i$ with $G_u$ unipotent, $G_c$ $k$-anisotropic torus and $G_i$ a $k$-split torus (\ie $G_i$ is $k$-isomorphic to $G_m(k)^r$). $G_c\cap G_i$ is finite and $G_u\cap(G_c\cdot G_i)=\{e\}$. All the groups $G_u$, $G_c$ and $G_i$ are defined over $k$. Moreover this decomposition is functorial, \ie any group morphism $\rho:G\to H$ between commutative $k$-groups will map $G_u$, $G_i$ and $G_c$ into $H_u$, $H_i$ resp. $H_c$. \endproclaim The same statement holds for positive characteristic except that $G_u$ is only $k$-closed and not necessarily defined over $k$. \proclaim@{\it}{Proof}{}{\ninepoint\rm} See \cite{B,p.121ff and p.137ff}. \qed \proclaim@{\bf}{Proposition Let $G$ be a Zariski-connected commutative $k$-group, where $k$ is a non-archi\-me\-dean local field with $char(k)=0$. Then $G(k)$ admits a discrete Zariski-dense subgroup if and only if \item{(1)} $G\ne G_c$ and \item{(2)} $\dim(G_i)\ge\dim(G_u)$. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} $G_c(k)$ is the maximal Zariski-connected compact subgroup of $G(k)$, hence condition (i) is clearly necessary. Now let $\Gamma$ be a finitely generated discrete subgroup of $G(k)$. Let $a_i$ denote the generators and $\pi_u:G\to G_u$ the natural projection. By Cor.2 to Prop.\ 4 the image of $\Gamma$ in $G_u$ under the natural projection is relatively-compact. Thus the projection of $\Gamma$ to $G_c\times G_u$ has a relatively compact image. It follows that the natural projection $\pi_i:G\to G_i$ has finite kernel if restricted to $\Gamma$. Now there is a proper continuous group homomorphism $G_i(k)\to\s Z^r$ with $r=\dim(G_i)$ induced by the logarithm of the absolute value on the local field $k$. Hence there is a group homomorphism $\Gamma\to\s Z^r$ with finite kernel. It follows that any discrete subgroup $\Lambda$ of $G(k)$ is finitely generated with $rank(\Lambda)\le\dim(G_i)$. Due to the corollary to Lemma 5 a finitely generated Zariski-dense subgroup $\Lambda_u$ of $G_u$ must fulfill $rank(\Lambda_u)\ge\dim(G_u)=n$. Therefore condition $(ii)$ is necessary for the existence of discrete Zariski-dense subgroups. On the other hand by the functoriality of the decomposition a subgroup $\Gamma$ is Zariski-dense in $G$ if and only if all the projections to $G_c$, $G_i$ and $G_u$ are Zariski-dense. Now $G_c(k)$ admits an element generating a Zariski-dense subgroup (\cite {B,Prop.8.8 \& Remark below}) and $G_u(k)$ admits a Zariski-dense subgroup generated by $dim(G_u)=n$ elements (Lemma 4). For any $x\in k^*$ with $\abs{x}\ne 1$ we obtain a Zariski-dense discrete subgroup $\Gamma$ of $G_i\simeq(k^*,\cdot)^m)$ by $\Gamma=\{(x^{n_1},\ldots,x^{n_m}):n_i\in\s Z\}$. Thus $dim(G_i)\ge\max(dim(G_i),1)$ is sufficient for the existence of discrete Zariski-dense subgroups in $G(k)$. \qed \proclaim@{\bf}{Remark}{}{\rm} For local fields of positive characteristic the "if"-part of the proposition still holds. However, the "only if"-part breaks down, \eg $G_a(k)$ admits a Zariski-dense discrete subgroup. \endremark The following is well-known and easy to prove. \proclaim@{\bf}{Lemma}{\the\lemno\global\advance\lemno by 1 }{\it} Let $G$ be a Zariski-connected commutative $k$-group for $k=\s R$ or $k=\s C$. For $k=\s R$ the group $G(k)$ admits a discrete cocompact Zariski-dense subgroup if and only if $G(k)$ is non-compact. For $k=\s C$ the group $G(k)$ is never compact and always admits a discrete Zariski-dense discrete cocompact subgroup. \endproclaim \section{Metabelian groups} We start with an auxiliary remark. \proclaim@{\bf}{Lemma}{\the\lemno\global\advance\lemno by 1 }{\it} Let $\rho:\s C^*\to GL(V)$ be a rational representation on a complex vector space $V$. Assume that all the weights are non-zero and let $\lambda\in\s C^*$. Then there exists a number $n\in\s N$ such that either all or none of the eigenvalues of $\rho(\lambda)$ are real. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} The eigenvalues of $\rho(\lambda)$ are $\lambda^{k_1},\ldots\lambda^{k_n}$ for some $k_1,\ldots k_n\in\s Z\setminus\{0\}$. If $\lambda^{k_i}\in\s R$ for an $i$, then $\rho(\lambda^{k_i})$ and $\rho(\lambda^{-k_i})$ have only real eigenvalues. \qed We use this to derive a necessary conition for the existence of Zariski-dense discrete subgroups in certain metabelian groups. \proclaim@{\bf}{Proposition Let $\rho:\s C^*\to GL(V)$ be a rational representation for which all the weights are distinct and non-zero. Let $G=\s C^*\mathbin{\vrule height 4.5pt depth 0.1pt\kern-1.8pt\times}_\rho V$ be the induced semidirect product. Assume that $G$ admits a discrete Zariski-dense subgroup $\Gamma$. Then $G$ is unimodular, \ie $\rho(\s C^*)\subset SL(V)$. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} Consider the projection $\tau:G\to\s C^*\simeq G/G'$. Now $\tau(\Gamma)$ is Zariski-dense in $\s C^*$. We claim that $\tau(\Gamma)$ is not relatively compact in $\s C^*$. Assume the contrary. Then $S=\overline{\tau(\Gamma)}$ is compact. Thus the action of $\Gamma$ by conjugation on $V$ factors through a compact group action on $V$. Hence the orbits of $\Gamma$ acting by conjugation on $\Gamma\cap V$ are relatively compact (and discrete), hence finite. Since $\Lambda=\Gamma\cap V$ is a finitely generated group, it follows that $\Gamma$ contains a subgroup of finite index which centralizes $\Lambda$. This is impossible, because $\Lambda$ is Zariski-dense in $V=[G,G]$ and $V$ is not central in $G$. Thus $\tau(\Gamma)\subset\s C^*$ is not relatively compact and in particular contains an element which is not of finite order, \ie not a root of unity. Let $\lambda$ be such an element. By the preceding lemma we may assume either all or none of the eigenvalues of $\rho(\lambda)$ are real. Now let $V=\oplus_\omega V_\omega$ be a decomposition of $V$ into eigenspaces of $\rho(\lambda)$. By assumption they are one-dimensional. Let $\pi_\omega:V\to V_\omega$ denote the respective projections. Since $\Gamma\cap V$ is Zariski-dense in $V$, it contains an element $v$ such that $\pi_\omega(v)\ne 0$ for all $\omega$. Let $\Sigma$ denote the smallest $\rho(\lambda)$-invariant subgroup of $V$ containing $v$. Then $W=\Sigma\otimes_{\s Z}\s R$ is a real subvectorspace of $V$, which is again invariant. It follows that $W=\oplus_\omega(W\cap V_\omega)$. If the eigenvalues are complex, then $W=V$. Since $\Sigma$ is a lattice in $W$, in this case $\Sigma$ is a lattice in $V$. If all eigenvaluies are real, then $W$ is a totally real subvectorspace with $V=W\oplus iW$. Thus in this case $\Sigma\oplus i\Sigma$ is a lattice in $V$. In any case $V$ admits a lattice, which is stable under $\rho(\lambda)$. Therefore $\rho(\lambda)\in SL(V)$. Since the group generated by $\lambda$ is Zariski-dense in $\s C^*$, it follows that $\rho(\s C^*)\subset SL(V)$. Thus $G$ is unimodular. \qed However, even in this special case unimodularity is not sufficient for the existence of a discrete Zariski-dense subgroup. \proclaim@{\bf}{Example}{}{\rm} 5 Let $G=\s C^*\mathbin{\vrule height 4.5pt depth 0.1pt\kern-1.8pt\times}_\rho\s C^3$ with the weights of $\rho$ given by $(2,-1,-1)$, \ie $\rho(\lambda)(x_1,x_2,x_3)=(\lambda^2x_1, \lambda^{-1}x_2,\lambda^{-1}x_3)$. Let $V=\s C^3=V_2\oplus V_{-1}$ where $V_\alpha$ denotes the weight space for $\alpha$. Assume that $G$ admits a Zariski-dense discrete subgroup $\Gamma$. Then $\Gamma'$ is Zariski-dense in $G'=V$. Hence $\Gamma$ contains an element $\gamma$ which is contained in $V$ but neither in $V_{2}$ nor in $V_{-1}$. Then $\gamma=\gamma_2+\gamma_{-1}$ with $0\ne\gamma_\alpha\in V_\alpha$. The elements $\gamma_\alpha$ span a two-dimensional subvectorspace $W$ of $V$. Consider the natural projection $\tau:G\to G/G'\simeq\s C^*$. By the considerations in the proof of the above proposition there exists an element $\delta\in\Gamma$ such that $\abs{\tau(\delta)}>1$ and $\tau(\delta)\in\s R$ or $(\tau\delta)^2\not\in\s R$ Let $\Lambda$ denote the subgroup of $\Gamma$ generated by $\gamma$ and $\delta$. Then by the same reasoning as in the above proof $\Lambda'$ or $\Lambda'+i\Lambda'$ must be lattice in $W$ which is stable under $\rho(\delta)$. But this is impossible, because $\abs{\det\left(\rho(\delta)|_W\right)}>1$. \endproclaim \section{A number-theoretical construction} A series of metabelian linear-algebraic groups over $\s C$ with discrete cocompact subgroups may be constructed by the following number-theoretic approach which generalizes a construction of Otte and Potters \cite{OP}. \footnote*{Otte and Potters studied the special case where $K$ is a totally real numberfield. For this case our theorem is equivalent to \cite{OP,3.2.}.} \proclaim@{\bf}{Theorem Let $K$ be a number field, $\cal O$ the ring of algebraic integers, $\cal O^*$ the multiplicative group of units, $r_1$ the number of real imbeddings, $r_2$ the number of pairs of conjugate complex embeddings and $r=r_1+r_2-1$. Then there is a semidirect product $G=T\mathbin{\vrule height 4.5pt depth 0.1pt\kern-1.8pt\times}_\rho V$ of a torus $T=(\s C^*)^m$ with and a vector group $V=\s C^{r+1}$ with $m\in\{r,r+1\}$ and $\{e\}=\ker\rho:T\to GL(V)$, such that $G$ contains a Zariski-dense discrete subgroup $\Gamma$ isomorphic to $\cal O^*\mathbin{\vrule height 4.5pt depth 0.1pt\kern-1.8pt\times}\cal O$. For $r_1=r_2=1$, we obtain $m=2$ and $G$ is isomorphic to a Borel group in $SL_2(\s C)\times SL_2(\s C)$. For totally real $K$ there exists a discrete cocompact subgroup $\Gamma_0$ isomorphic to $\cal O^*\mathbin{\vrule height 4.5pt depth 0.1pt\kern-1.8pt\times}(\cal O\times\cal O)$. Furthermore for totally real $K$, $\rho(T)$ is a maximal torus in $SL(V)$. \endproclaim \proclaim@{\it}{Proof}{}{\ninepoint\rm} Let $\zeta_1,\ldots,z_{r_1}$ and $\xi_1,\bar\xi_1,\ldots \xi_{r_2},\bar\xi_{r_2}$ denote the real resp. pairs of conjugate complex imbeddings of $K$ in $\s C$. Then $\phi=(\zeta_1,\ldots,\zeta_{r_1},\xi_1,\ldots,\xi_{r_2})$ embed $K$ into the complex vector space $V=\s C^{r+1}$ such that $W=K\otimes_{\s Q}\s R$ is embedded into $V$ as a real subvectorspace with $\phi(W)+i\phi(W)=V$. Furthermore $\phi(\cal O)$ is a cocompact lattice in $\phi(W)$ and therefore a discrete Zariski-dense subgroup of $V$. The action of $K^*$ on $K$ by multiplication induces an action on $V$. For each $x\in K^*$ this action is a diagonalizable endomorphism of $V$ with eigen-values $\zeta_1(x),\ldots,\zeta_{r_1}(x),\xi_1(x),\ldots \xi_{r_2}(x)$. It follows that $K^*$ acts on $V$ as a subgroup of a maximal torus $T$ of $GL_r(\s C)$. Thus we obtain an injective group homomorphism $\tau:K^*\to T$. Now $\tau(\cal O^*)$ stabilizes the discrete Zariski-dense subgroup $\cal O$ of $V$. It is easy to see that this implies that $\tau(\cal O^*)$ is discrete in $T$. Hence we obtain a discrete group $\Gamma=\tau(\cal O^*)\mathbin{\vrule height 4.5pt depth 0.1pt\kern-1.8pt\times}\phi(\cal O)$ which is Zariski-dense in its Zariski-closure $G=\bar\Gamma$. Clearly $V\subset G\subset T\mathbin{\vrule height 4.5pt depth 0.1pt\kern-1.8pt\times} V$ and $G'=[G,G]=V$. By the theorem of Dirichlet $\cal O^*$ is isomorphic to a direct product of a finite abelian group $A$ and $\s Z^r$. Hence $\mathop{{\fam=-1 dim}_{\s C}}(G/V)\ge r$. On the other hand $\mathop{{\fam=-1 dim}_{\s C}}(G/V)\le r+1$, since $\mathop{{\fam=-1 dim}_{\s C}}(T)=\mathop{{\fam=-1 dim}_{\s C}}(V)=r+1$. Now let us consider the special case $r_1=r_2=1$. Then for $\alpha\in\cal O^*$ we obtain $\abs{\det(\tau(\alpha))}=\abs{\zeta_1(\alpha)}\abs{\xi_1(\alpha)}$. Now $\abs{N_{K,\s Q}(\alpha)}=1$ implies that either $G$ is not unimodular or $\abs{\xi_1(\alpha)}=1=\abs{\zeta_1(\alpha)}$. However, the second alternative would imply that $\tau(\cal O^*)$ is relatively compact in $GL(V)$, which is impossible (see the proof of Prop. 9). Hence $G$ is not unimodular. By Prop. 9 it follows that $\tau(G)\subset GL(V)$ can not be one-dimensional. Hence $\tau(G)$ must be a maximal torus in $GL(V)=GL_2(\s C)$, which implies that $G$ is a Borel group in $S=SL_2(\s C)\times SL_2(\s C)$. Finally let us consider the special where $K$ is totally real, \ie $r_2=0$ (This is the case studied in \cite{OP}). Then $\phi(W)$ is totally real in $V$ and we obtain a cocompact lattice in $V$ by $\Lambda=\phi(\cal O)+i\phi(\cal O)$. Furthermore in this case $\det(\tau(x))= =N_{K,\s Q}(x)$ for all $x\in K^*$. Since the norm $N_{K,\s Q}(x)$ equals $1$ or $-1$ for all units $x\in\cal O^*$, it follows that $\tau(\cal O^*)$ admits a subgroup $\Delta$ of index $2$ or $1$ which is contained in $SL(V)$. Now $\Delta\mathbin{\vrule height 4.5pt depth 0.1pt\kern-1.8pt\times}\Lambda$ is a discrete cocompact subgroup in $T_0\mathbin{\vrule height 4.5pt depth 0.1pt\kern-1.8pt\times} V$ where $T_0=T\cap SL(V)$. \qed \medskip\sectno=-100 \section{References \item{[B]} Borel, A.: Linear algebraic groups. Second enlarged edition. Springer 1991. \item{BS} Borel, A.; Serre, J.P.: Th\'eor\`emes de finitude en cohomologie galoisienne. \sl Comm. Math. Helv. \bf 39\rm, 111--164 (1964) \item{BT} Borel, A.; Tits, J.: Homomorphismes "abstraits" de groupes alg\'ebriques simples. Ann. Math. \bf 97\rm, 499-571 (1973) \item{G} Greenleaf, F.: Invariant means on topological groups. Van Nostrand, New York 1969 \item{[Mc]} Malcev, A.: On a class of homogeneous spaces. \sl Izvestiya Akad. Nauk SSSR Ser. Math. \bf 13 \rm (1949)/ \sl AMS Transl. no. \bf 39 \rm (1951) \item{M} Margulis, G.A.: Discrete Subgroups of Semisimple Lie Groups. Springer Berlin Heidelberg New York 1989 \item{OP} Otte, M.; Potters, J.: Beispiele homogener Mannigfaltigkeiten. \sl Manu. math. \bf 10\rm, 117--127 (1973) \item{[R]} Raghunathan, M.S.: Discrete subgroups of Lie groups. \sl\rm Erg. Math. Grenzgeb. \bf 68 \rm, Springer (1972) \* \item{Ro} Rosenlicht, M.: Some rationality questions on algebraic groups. \sl Annali di Math. \bf 43\rm, 25-50 (1957) \item{S} Serre, J.P.: Algebraic Groups and Class Fields. Springer 1988. \item{T} Tits, J.: Free subgroups in linear groups. \sl J. Algebra \bf 20\rm, 250--270 (1972) \item{Z} Zimmer, R.J.: Ergodic Theory and Semisimple Groups. Birkh\"auser 1984 \endarticle \endinput

9412

alg-geom/9412006

en

https://arxiv.org/abs/alg-geom/9412006

alg-geom/9412006

Luca Barbieri-Viale

Luca Barbieri-Viale

On the Deligne--Beilinson cohomology sheaves

12 pages, LaTeX 2.09

Annals of K-Theory Vol. 1, Issue 1 (2016) 3-17

10.2140/akt.2016.1.3

We are showing that the Deligne--Beilinson cohomology sheaves ${\cal H}^{q+1}({\bf Z}(q)_{\cal D})$ are torsion free by assuming Kato's conjectures hold true for function fields. This result is `effective' for $q=2$; in this case, by dealing with `arithmetic properties' of the presheaves of mixed Hodge structures defined by singular cohomology, we are able to give a cohomological characterization of the Albanese kernel for surfaces with $p_g=0$.

alg-geom

\section*{Introduction} For $X$ a compact complex algebraic manifold the Deligne cohomology $H^*(X,\Z(\cdot)_{{\cal D}})$ is defined by taking the hypercohomology of the truncated De Rham complex augmented over $\Z$. The very extension of such a cohomology theory to arbitrary algebraic complex varieties is usually called Deligne--Beilinson cohomology ({\it e.g.\/}\ see \cite{GI} for definitions and properties or \cite{HV} for details). The associated Zariski sheaves ${\cal H}^*(\Z(\cdot)_{{\cal D}})$ have groups of global sections which are birational invariants of smooth complete varieties (see \cite{BV1}, \cite{BV2}). The motivation for this paper is to investigate these invariants. We can show that the Deligne--Beilinson cohomology sheaves ${\cal H}^{q+1}(\Z(q)_{{\cal D}})$ are {\it torsion free\,} by assuming Kato's conjectures hold true for function fields (see \S 2). In particular ${\cal H}^{3}(\Z(2)_{{\cal D}})$ is actually torsion free thanks to Merkur'ev--Suslin's Theorem on $K_2$. Thus these invariants {\it vanish\,} for unirational varieties. If only $H^2(X,{\cal O}_X)=0$ we then can show (see \S 3) that the group of global sections of ${\cal H}^3(\Z(2)_{{\cal D}})$ is {\it exactly\,} the kernel of the cycle map $CH^2(X)\to H^4(X,\Z(2)_{{\cal D}})$ in Deligne cohomology {\it i.e.\/}\ the kernel of the Abel--Jacobi map. This fact generalizes the result of H.Esnault for $0$-cycles ({\it cf.\/}\ \cite[Theorem~2.5]{H}), in the case of codimension $2$ cycles, to $X$ of arbitrary dimension and it is obtained by a different proof. Concerning the discrete part ${\cal F}^{2,2}_{\mbox{\scriptsize{$\Z$}}}$ of the Deligne--Beilinson cohomology sheaf ${\cal H}^2(\Z(2)_{{\cal D}})$ (as defined in {\it op.\/ cit.\/}\ or \S 2) we can describe, for any $X$ proper and smooth, the torsion of $H^1(X,{\cal F}^{2,2}_{\mbox{\scriptsize{$\Z$}}})$ in terms of `trascendental cycles' and $H^3(X,\Z)_{tors}$, and we can see that there are not non-zero global sections of it (see \S 3). For surfaces with $p_g=0$ we are able to compute the group of global sections of ${\cal H}^3(\Z(2)_{{\cal D}})$ --- Bloch's conjecture is that $H^0(X,{\cal H}^3(\Z(2)_{{\cal D}}))=0$ --- by means of some short exact sequences (see \S 4) involving the discrete part ${\cal F}^{2,2}_{\mbox{\scriptsize{$\Z$}}}$ and the Hodge filtration. In order to do that we are firstly arguing (see \S 1) with arithmetic resolutions of the Zariski sheaves associated with the presheaves of mixed Hodge structures defined by singular cohomology: the Hodge and weight filtrations do have corresponding coniveau spectral sequences, the $E_2$ terms of which are given by the cohomology groups of the Zariski sheaves associated to such filtrations.\\ I would like to thank B.Kahn for friendly useful conversations on some of the matters contained herein. \section*{Notations} Throughout this note $X$ is a complex algebraic variety. We will denote by $H^*(X,A)$ (resp. $H_*(X,A)$) the singular cohomology (resp. Borel--Moore homology) of the associated analytic space $\mbox{$X_{an}$}$ with coefficients in $A$ where $A$ would be $\Z,\Z/n, \C$ and $\C^*$; we let $H^*(X)$ (resp. $H_*(X)$) be the corresponding mixed Hodge structure (see \cite{D}). We will denote by $W_iH^*(X)$ (resp. $W_{-i}H_*(X)$) the $\mbox{$\bf Q$}$-vector spaces given by the weight filtration and by $F^iH^*(X)$ (resp. $F^{-i}H_*(X)$) the real vector spaces given by the Hodge filtration. For the ring $\Z$ of integers we will denote by $\Z(r)$ the Tate twist in Hodge theory and by $H^*(X,\Z(r)_{{\cal D}})$ the Deligne--Beilinson cohomology groups (see \cite{HV}, \cite{GI}). The Tate twist induces the twist $A\otimes\Z(r)$ in the coefficients which we will denote $A(r)$ for short. We will denote by ${\cal H}^*(A(r))$ and ${\cal H}^*(\Z(r)_{{\cal D}})$ the Zariski sheaves on a given $X$ associated to singular cohomology and Deligne--Beilinson cohomology respectively. \section{Arithmetic resolutions in mixed Hodge theory} Let $Z\hookrightarrow X$ be a closed subscheme of the complex algebraic variety $X$. According with Deligne \cite[8.2.2 and 8.3.8]{D} the relative cohomology groups $H^*_Z(X,\Z)$ (= $H^*(X {\rm mod} X-Z,\Z)$ in {\it op.\/ cit.\/}\ ) carry out a mixed Hodge structure fitting into long exact sequences \B{equation}\label{loc} \cdots \to H_Z^j(X)\to H_T^j(X)\to H_{T-Z}^j(X-Z)\to H_Z^{j+1}(X)\to \cdots \E{equation} for any pair $Z\subset T$ of closed subschemes of $X$. As it has been remarked in \cite{JA} the assignation $$Z\subseteq X \leadsto (H_Z^*(X),H_*(Z))$$ yields a Poincar\'e duality theory with supports (see \cite{BO} and furthermore we have that the above theory is appropriate for albegraic cycles in the sense of \cite{BV2}) with values in the abelian tensor category of mixed Hodge structures. In particular, by considering the presheaf of vector spaces $$U\leadsto F^iH^j(U)$$ (resp. $U\leadsto W_iH^j(U)$) and sheafifying it on a fixed variety $X$, we obtain Zariski sheaves ${\cal F}^i{\cal H}^j$ (resp. ${\cal W}_i{\cal H}^j$) filtering the sheaves ${\cal H}^j(\C)$. We then have: \B{prop}\label{arifilt} Let $X$ be smooth. The `arithmetic resolution' $$0\to{\cal H}^q(\C)\to \coprod_{x\in X^0}^{} i_x H^{q}(x) \to \coprod_{x\in X^1}^{} i_x H^{q-1}(x) \to \cdots\to \coprod_{x\in X^q}^{} i_x\C\to 0$$ is a bifiltered quasi-isomorphism $$({\cal H}^q(\C),{\cal F},{\cal W})\by{\simeq}(\coprod_{x\in X^{\odot}}^{} i_x H^{q-\odot}(x),\coprod_{x\in X^{\odot}}^{} i_x F, \coprod_{x\in X^{\odot}}^{} i_x W)$$ yielding flasque resolutions $$0\to gr^i_{{\cal F}}gr_j^{{\cal W}}{\cal H}^q(\C)\to\coprod_{x\in X^{0}}^{} i_x gr^i_{F}gr_j^{W}H^{q}(x)\to\cdots\to \coprod_{x\in X^{q}}^{} i_x gr^{i-q}_{F}gr_{j-2q}^{W}H^{0}(x)\to 0$$ \E{prop} \B{proof} Because of \cite[Theor.1.2.10 and 2.3.5]{D} the functors $F^n$, $W_n$ and $gr^n_F$ (any $n\in\Z$) from the category of mixed Hodge structures to that of real vector spaces are exact; $gr_n^W$ is exact as a functor from mixed Hodge structures to pure $\mbox{$\bf Q$}$-Hodge structures. So the claimed results are obtained via the `locally homologically effaceable' property (see \cite[Claim p. 191]{BO}) by construction of the arithmetic resolution (granted by \cite[Theor.4.2]{BO}). For example: by applying $F^i$ to the long exact sequences (\ref{loc}), taking direct limits over pairs $Z\subset T$ filtered by codimension and sheafifying, we do obtain a flasque resolution of length $q$ $$0\to {\cal F}^i{\cal H}^q\to \coprod_{x\in X^0}^{} i_x F^iH^{q}(x) \to \coprod_{x\in X^1}^{} i_x F^{i-1}H^{q-1}(x) \to \cdots$$ where $$F^*H^{*}(x)\mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, \limdir{\pp{ U \mbox{\ open\ } \subset \overline{\{ x\} }}} F^*H^*(U)$$ By this method we obtain as well a resolution of ${\cal W}_j$ $$0\to {\cal W}_j{\cal H}^q\to \coprod_{x\in X^0}^{} i_x W_jH^{q}(x) \to \coprod_{x\in X^1}^{} i_x W_{j-2}H^{q-1}(x) \to \cdots$$ These resolutions grant us of the claimed bifiltered quasi-isomorphism. (Note: for $X$ of dimension $d$, the fundamental class $\eta_X$ belongs to $W_{-2d}H_{2d}(X)\cap F^{-d}H_{2d}(X)$ so that `local purity' yields the shift by two for the weight filtration and the shift by one for the Hodge filtration). In the same way we obtain resolutions of $gr^i_{{\cal F}}$, $gr_j^{{\cal W}}$ and $gr^i_{{\cal F}}gr_j^{{\cal W}}$. \E{proof} We may consider the twisted Poincar\'e duality theory $(F^nH^{*},F^{-m}H_*)$ where the integers $n$ and $m$ play the role of twisting and indeed we have $$F^{d-n}H^{2d-k}_Z(X)\cong F^{-n}H_k(Z)$$ for $X$ smooth of dimension $d$. Via the arithmetic resolution of ${\cal F}^i{\cal H}^q$ we then have the following: \B{cor} Let assume $X$ smooth and let $i$ be a fixed integer. We then have a `coniveau spectral sequence' \B{equation}\label{conifilt} E^{p,q}_2 =H^p(X,{\cal F}^i{\cal H}^q) \mbox{$\Rightarrow$} F^iH^{p+q}(X) \E{equation} where $H^p(X,{\cal F}^i{\cal H}^q)=0$ if $q<\mbox{min}(i,p)$. \E{cor} \B{rmk} Concerning the Zariski sheaves $gr^i_{{\cal F}}{\cal H}^q$ and ${\cal H}^q/{\cal F}^i$ we indeed obtain corresponding coniveau spectral sequences as above. \E{rmk} Because of the maps of `Poincar\'e duality theories' $F^iH^*(-)\to H^*(-,\C)$ we do have as well, maps of coniveau spectral sequences; on the $E_2$-terms the map $$H^p(X,{\cal F}^i{\cal H}^q)\to H^p(X,{\cal H}^q(\C))$$ is given by taking Zariski cohomology of ${\cal F}^i{\cal H}^q \hookrightarrow {\cal H}^q(\C)$. For example: if $i<p$ we clearly have (by comparing the arithmetic resolutions): $H^p(X,{\cal F}^i{\cal H}^p)\cong H^p(X,{\cal H}^p(\C))$ and $$H^p(X,{\cal H}^p(\C))\cong NS^p(X)\otimes\C$$ by \cite[7.6]{BO} where $NS^p(X)$ is the group of cycles of codimension $p$ modulo algebraic equivalence. For $i=p$ we still have: \B{thm}\label{Nero} Let $X$ be smooth. Then $$H^p(X,{\cal F}^p{\cal H}^p)\cong NS^p(X)\otimes\C$$ \E{thm} \B{proof} Because of the Proposition~\ref{arifilt} $$H^p(X,{\cal F}^p{\cal H}^p)\cong {\rm coker} (\coprod_{x\in X^{p-1}}^{}F^1H^{1}(x) \to \coprod_{x\in X^p}^{} \C)$$ whence the canonical map $H^p(X,{\cal F}^p{\cal H}^p)\to NS^p(X)\otimes\C$ is surjective. To show the injectivity, via the arithmetic resolution we see that $$H^{2p-1}_{Z^{p-1}}(X,\C)\cong\mbox{ker}(\coprod_{x\in X^{p-1}}^{}H^{1}(x) \to \coprod_{x\in X^p}^{} \C)$$ where $H^{*}_{Z^{i}}$ denotes the direct limit of the cohomology groups with support on closed subsets of codimension $\geq i$; indeed this formula is obtained by taking the direct limit of (\ref{loc}) over pairs $Z\subset T$ of codimension $\geq p$ and $\geq p-1$ respectively, since $H^{2p-1}_{Z^{p}}=0$ and $$H^{2p}_{Z^{p}}(X,\C) = \coprod_{x\in X^p}^{} \C$$ Furthermore $$F^pH^{2p-1}_{Z^{p-1}}\cong\mbox{ker}(\coprod_{x\in X^{p-1}}^{}F^1H^{1}(x) \to \coprod_{x\in X^p}^{} \C)$$ and $$H^{2p-1}_{Z^{p-1}}/F^p\cong \coprod_{x\in X^{p-1}}^{}gr_F^0H^{1}(x)$$ since the arithmetic resolution of ${\cal H}^p/{\cal F}^p$ has lenght $p-1$. Thus we have that $$\mbox{image}(\coprod_{x\in X^{p-1}}^{}F^1H^{1}(x) \to \coprod_{x\in X^p}^{} \C)= \mbox{image}(\coprod_{x\in X^{p-1}}^{}H^{1}(x) \to \coprod_{x\in X^p}^{} \C)$$ \E{proof} \B{rmk} Note that by considering the sheaf ${\cal H}^q(\C)$(=${\cal F}^0{\cal H}^q$) on $X$ filtered by the subsheaves ${\cal F}^i{\cal H}^q$ we have as usual ({\it cf.\/}\ \cite[1.4.5]{D}) a spectral sequence $${}_{{\cal F}}E_1^{r,s}=H^{r+s}(X,gr^s_{{\cal F}}{\cal H}^q)\mbox{$\Rightarrow$} H^{r+s}(X,{\cal H}^q(\C))$$ with induced `aboutissement' filtration $$F^iH^p(X,{\cal H}^q) \mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, {\rm im}\, (H^p(X,{\cal F}^i{\cal H}^q)\to H^p(X,{\cal H}^q(\C)))$$ By the above Theorem we can see that $F^iH^p(X,{\cal H}^p)$ is uninteresting, since it gives the all N\'eron--Severi group if $i\leq p$ and vanishes otherwise. \E{rmk} \B{rmk} As an immediate consequence of this Theorem, via the coniveau spectral sequence (\ref{conifilt}), we see the well known fact that the cycle map $c\ell^p:NS^p(X)\otimes\C\to H^{2p}(X,\C)$ has its image contained in $F^pH^{2p}(X)$. \E{rmk} For any $X$ smooth and proper, we have that $F^2H^2(X)=H^0(X,{\cal F}^2{\cal H}^2)=H^0(X,\Omega^2_X)$ and \B{equation}\label{genus} H^0(X,{\cal H}^2/{\cal F}^2)\cong H^0(X,{\cal H}^2(\C))/H^0(X,\Omega^2_X)\cong \frac{H^2(X,\C)}{H^0(X,\Omega^2_X)\oplus NS(X)\otimes\C} \E{equation} where $H^0(X,{\cal H}^2(\C))\cong H^0(X,{\cal H}^2(\Z))\otimes\C$ and $H^0(X,{\cal H}^2(\Z)) = {\rm im}\, (H^2(X,\Z)\to H^2(X,{\cal O}_X))$ is the lattice of `trascendental cycles'. The formula (\ref{genus}) can be obtained, for example, by the exact sequence (given by the coniveau spectral sequence since ${\cal F}^2{\cal H}^1 =0$) $$0\to H^1(X,{\cal H}^1(\C))\to H^2(X)/F^2\to H^0(X,{\cal H}^2/{\cal F}^2)\to 0$$ because of $H^1(X,{\cal H}^1(\C))=NS(X)\otimes\C$. \section{Deligne--Beilinson cohomology sheaves} Let $X$ be smooth over $\C$. Let consider the Zariski sheaf ${\cal H}^*(\Z(r)_{{\cal D}})$ associated to the presheaf of Deligne--Beilinson cohomology groups $U\leadsto H^*(U,\Z(r)_{{\cal D}})$ on $X$. We have canonical long exact sequences of sheaves on $X$ \B{equation}\label{modf} \cdots\to {\cal H}^q(\Z(r))\to {\cal H}^q(\C)/{\cal F}^r \to {\cal H}^{q+1}(\Z(r)_{{\cal D}}) \to {\cal H}^{q+1}(\Z(r))\to\cdots \E{equation} \B{equation}\label{plusf} \cdots\to {\cal H}^{q}(\Z(r)_{{\cal D}}) \to {\cal H}^{q}(\Z(r))\oplus {\cal F}^r{\cal H}^q\to {\cal H}^q(\C)\to {\cal H}^{q+1}(\Z(r)_{{\cal D}})\to\cdots \E{equation} \B{equation}\label{star}\cdots\to {\cal F}^r{\cal H}^{q}\to{\cal H}^q(\C^*(r))\to {\cal H}^{q+1}(\Z(r)_{{\cal D}}) \to {\cal F}^r{\cal H}^{q+1}\to\cdots \E{equation} obtained by sheafifying the usual long exact sequences coming with Deligne--Beilinson cohomology (see \cite[Cor.2.10]{HV}). For example, if $r=0$ then ${\cal H}^q(\C)/{\cal F}^0=0$ and (\ref{modf}) yields the isomorphism ${\cal H}^{*}(\Z(0)_{{\cal D}}) \cong {\cal H}^{*}(\Z)$; (\ref{plusf}) splits in trivial short exact sequences and (\ref{star}) give us the following short exact sequence \B{equation}\label{universal} 0\to {\cal H}^q(\Z)\to{\cal H}^q(\C)\to{\cal H}^q(\C^*)\to 0 \E{equation} whenever the sheaves ${\cal H}^q(\Z)$ and ${\cal H}^{q+1}(\Z)$ are torsion free which is the case if $q\leq 2$ ({\it cf.\/}\ \cite[\S 3-4]{BV1} and \cite[p.1240]{BlS}). Torsion freeness of ${\cal H}^q(\Z)$ for all $q\geq 4$ is a conjectural property (see \cite[\S 7]{BV1}). In order to show that ${\cal H}^{q+1}(\Z)$ is torsion free it sufficies to see that ${\cal H}^{q}(\Z)\to{\cal H}^{q}(\Z/n)$ is an epimorphism for any $n\in\Z$; via the canonical map ${\cal O}^*_X\to{\cal H}^1(\Z)$ and cup--product we obtain a map $({\cal O}^*_X)^{\otimes q}\to{\cal H}^q(\Z)$. The composition of $$({\cal O}^*_X)^{\otimes q}\to{\cal H}^q(\Z)\to {\cal H}^{q}(\Z/n)$$ can be obtained as well as ({\it cf.\/}\ \cite[p.1240]{BlS}) the composition of $$({\cal O}^*_X)^{\otimes q}\by{sym} {\cal K}_q^M \to {\cal H}^q(\Z/n)$$ where by definition of Milnor's $K$-theory sheaf the symbol map $sym$ is an epimorphism thus we are left to show that the Galois symbol ${\cal K}_q^M \to {\cal H}^q(\Z/n)$ is an epimorphism (for the sake of exposition we are tacitly fixing an $n$-th root of unity yielding a non-canonical isomorphism ${\cal H}^{q}_{\acute{e}t}(\mu_n^{\otimes q})\cong {\cal H}^q(\Z/n)$); this last map can be obtained by mapping the Gersten complex for Milnor's $K$-theory to the Bloch--Ogus arithmetic resolution of the sheaf ${\cal H}^q(\Z/n)$ {\it i.e.\/}\ there is a commutative diagram $$\B{array}{ccc} {\cal K}_q^M &\to {\displaystyle \coprod_{\eta\in X^0}^{} i_{\eta} K^M_q(k(\eta))} &\to {\displaystyle\coprod_{x\in X^1}^{} i_x K_{q-1}^M(k(x))}\\ \downarrow & \downarrow & \downarrow\\{\cal H}^q(\Z/n) & \hookrightarrow {\displaystyle\coprod_{\eta\in X^0}^{} i_x H^{q}(\eta)}& \to {\displaystyle\coprod_{x\in X^1}^{} i_x H^{q-1}(x)} \E{array}$$ where $H^*(\mbox{point})$ is the Galois cohomology of $k(\mbox{point})$. Indeed O.Gabber announced the (universal) exactness of the above complex of Milnor's $K$-groups. Thus: by assuming Kato's conjecture {\it i.e.\/}\ $K^M_*(k(\mbox{point}))/n\cong H^*(\mbox{point})$, a diagram chase yields the desired projection ${\cal K}_q^M \to {\cal H}^q(\Z/n)$. (My thanks to B.Kahn for having directed my attention to Gabber's result.) Let consider the `discrete part' of the Deligne--Beilinson cohomology sheaves which is by definition ({\it cf.\/}\ \cite[\S 1]{H}) $${\cal F}^{r,q}_{\mbox{\scriptsize{$\Z$}}}\mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, \mbox{\ image\ }({\cal H}^q(\Z(r)_{{\cal D}}) \to {\cal H}^q(\Z(r)))$$ or, equivalently by (\ref{modf}), the integral part of ${\cal F}^r{\cal H}^q$. We may define the `trascendental part' of the Deligne--Beilinson cohomology sheaves as follows $${\cal T}^{r,q}_{{\cal D}}\mbox{\,$\stackrel{\pp{\rm def}}{=}$}\, \mbox{\ kernel\ }({\cal H}^q(\Z(r)_{{\cal D}}) \to {\cal H}^q(\Z(r)))$$ In particular, if $r=q$ we then have ({\it cf.\/}\ \cite[(1.3)$\alpha )$]{H}) the short exact sequence (by (\ref{star}) or (\ref{modf}) taking account of (\ref{universal})) \B{equation} \label{disc} 0\to {\cal H}^{q-1}(\C^*(q))\to {\cal H}^q(\Z(q)_{{\cal D}})\to {\cal F}^{q,q}_{\mbox{\scriptsize{$\Z$}}}\to 0 \E{equation} and moreover we have the following commutative diagram with exact rows and columns \B{equation}\B{array}{ccccccccc}\label{dia} &&0&&0&&0&&\\ &&\uparrow&&\uparrow&&\uparrow&&\\ 0&\to& {\cal H}^q(\Z(q))/{\cal F}^{q,q}_{\mbox{\scriptsize{$\Z$}}}&\to &{\cal H}^q(\C)/{\cal F}^q&\to & {\cal T}^{q,q+1}_{{\cal D}}&\to &0\\ &&\uparrow&&\uparrow&&\uparrow&&\\ 0&\to&{\cal H}^q(\Z(q))&\to&{\cal H}^q(\C)&\to&{\cal H}^q(\C^*(q)) &\to&0\\ &&\uparrow&&\uparrow&&\uparrow&&\\ 0&\to&{\cal F}^{q,q}_{\mbox{\scriptsize{$\Z$}}}&\to&{\cal F}^q&\to& {\cal F}^q/{\cal F}^{q,q}_{\mbox{\scriptsize{$\Z$}}}&\to&0\\ &&\uparrow&&\uparrow&&\uparrow&&\\ &&0&&0&&0&& \E{array}\E{equation} where the middle row is given by (\ref{universal}), the top one by (\ref{modf}) and the right-most column is obtained by (\ref{star}).\B{lemma} We have a short exact sequence of sheaves: \B{equation}\label{unidel} 0\to{\cal H}^q(\Z(r)_{{\cal D}})/n\to {\cal H}^q(\Z/n(r))\to {\cal H}^{q+1}((\Z(r)_{{\cal D}})_{n-tors}\to 0 \E{equation} for all $q, r\geq 0$ and $n\in\Z$.\E{lemma}\B{proof} The sequence (\ref{unidel}) is obtained from the long exact sequence (\ref{modf}) as follows. Since the sheaf ${\cal H}^{q+1}(\Z(r))$ is torsion free we do have that $${\cal T}^{r,q+1}_{{\cal D},n-tors}= {\cal H}^{q+1}((\Z(r)_{{\cal D}})_{n-tors}$$ Since the sheaf ${\cal H}^q(\C)/{\cal F}^r$ is uniquely divisible we have that $${\cal H}^{q+1}((\Z(r)_{{\cal D}})_{n-tors}= ({\cal H}^q(\Z(q))/{\cal F}^{r,q}_{\mbox{\scriptsize{$\Z$}}})\otimes \Z/n$$ because of the following short exact sequence $$0\to {\cal H}^q(\Z(r))/{\cal F}^{r,q}_{\mbox{\scriptsize{$\Z$}}}\to {\cal H}^q(\C)/{\cal F}^r\to {\cal T}^{r,q+1}_{{\cal D}}\to 0$$ Thus we get a short exact sequence $$0\to {\cal F}^{r,q}_{\mbox{\scriptsize{$\Z$}}}/n \to {\cal H}^q(\Z/n(r))\to {\cal H}^{q+1}((\Z(r)_{{\cal D}})_{n-tors}\to 0$$ by tensoring with $\Z/n$ the canonical one induced by the subsheaf ${\cal F}^{r,q}_{\mbox{\scriptsize{$\Z$}}}\hookrightarrow {\cal H}^q(\Z(r))$. Since ${\cal T}^{r,q}_{{\cal D}}$ is divisible we are done.\E{proof} By considering the Bloch--Beilinson regulators $${\cal K}_q^M\to {\cal H}^q(\Z(q)_{{\cal D}})$$ (simply obtained by the fact that ${\cal K}_1^M={\cal O}_X^*\cong {\cal H}^1(\Z(1)_{{\cal D}})$ and cup--product) we have that the composition of $${\cal K}_q^M\to {\cal H}^q(\Z(q)_{{\cal D}})\to {\cal H}^q(\Z(q))\to {\cal H}^q(\Z/n(q))$$ is the Galois symbol ({\it cf.\/}\ \cite[\S 0 p.375]{HM}). We thus have that the composition of $${\cal K}_q^M/n\to{\cal H}^q(\Z(q)_{{\cal D}})/n\hookrightarrow{\cal H}^q(\Z/n(q))$$ is an epimorphism if Kato's conjectures hold. Therefore, by comparing with (\ref{unidel}), we have a proof of the following: \B{thm}\label{Kato} Let assume that Kato's conjectures (for function fields) hold true. On a smooth $X$ we then have: \B{description}\item[{\it i)}] the sheaf ${\cal H}^q(\Z)$ is torsion free; \item[{\it ii)}] the sheaf ${\cal H}^{q+1}(\Z(q)_{{\cal D}})$ is torsion free; \item[{\it iii)}] there is a canonical isomorphism ${\cal H}^q(\Z(q)_{{\cal D}})\otimes \Z/n\cong {\cal H}^q(\Z/n(q))$ \E{description} for any $q\geq 0$. \E{thm} Note that the Theorem of Merkur'ev--Suslin (= Kato's conjecture for $K^M_2$) ensures the previous results when $q=2$. Moreover, by a standard argument ({\it cf.\/}\ \cite[\S 2]{BV1} and \cite{BV2}) we have the following: \B{cor} With the assumptions in the Theorem above, let suppose that $X$ is moreover unirational and complete. Then $$H^0(X,{\cal H}^{q+1}(\Z(q)_{{\cal D}}))=0$$ \E{cor} \section{Coniveau versus Hodge filtrations} We recall (see \cite{GI}) the existence of arithmetic resolutions of the sheaves ${\cal H}^{*}(\Z(\cdot)_{{\cal D}})$ thus the coniveau spectral sequence \B{equation}\label{conideli} {}_{{\cal D}}E^{p,q}_2 =H^p(X,{\cal H}^q(\Z(\cdot)_{{\cal D}})) \mbox{$\Rightarrow$} H^{p+q}(X,\Z(\cdot)_{{\cal D}}) \E{equation} and the formula (see \cite{GI}) $H^p(X,{\cal H}^p(\Z(p)_{{\cal D}}))\cong CH^p(X)$. By the spectral sequence (\ref{conideli}) we have a long exact sequence \B{equation}\label{delseq} 0\to H^1(X,{\cal H}^2(\Z(2)_{{\cal D}}))\to H^3(X,\Z(2)_{{\cal D}})\by{\rho} H^0(X,{\cal H}^3(\Z(2)_{{\cal D}}))\by{\delta}CH^2(X) \E{equation} The mapping $\delta$ is just a differential between ${}_{{\cal D}}E_2$-terms of the coniveau spectral sequence (\ref{conideli}); we still have $$\mbox{image\, }\delta = \ker (CH^2(X)\by{c\ell}H^4(X,\Z(2)_{{\cal D}}))$$ \B{prop}\label{alb} Let $X$ be proper and smooth. Then $$H^0(X,{\cal F}^{2,2}_{\mbox{\scriptsize{$\Z$}}})=0$$ the group $H^1(X,{\cal F}^{2,2}_{\mbox{\scriptsize{$\Z$}}})$ is infinitely divisible and $$H^1(X,{\cal F}^{2,2}_{\mbox{\scriptsize{$\Z$}}})_{tors}\cong H^0(X,{\cal H}^2(\mbox{$\bf Q$}/\Z(2)))$$ If $H^2(X,{\cal O}_X)=0$ then $$H^0(X,{\cal H}^3(\Z(2)_{{\cal D}})) \cong \ker (CH^2(X)\by{c\ell}H^4(X,\Z(2)_{{\cal D}}))$$ {\it i.e.\/}\ $\rho =0$ in (\ref{delseq}), and $H^1(X,{\cal F}^{2,2}_{\mbox{\scriptsize{$\Z$}}})\cong H^3(X,\Z)_{tors}$. \E{prop} \B{proof} By the canonical map of `Poincar\'e duality theories' $$H^{\sharp -1}(-,\C^*(\cdot))\to H^{\sharp}(-,\Z(\cdot)_{{\cal D}})$$ we do obtain a map of coniveau spectral sequences and the following commutative diagram \B{equation}\label{cocco}\B{array}{ccc} 0\to H^1(X,{\cal H}^2(\Z(2)_{{\cal D}}))&\to H^3(X,\Z(2)_{{\cal D}})&\by{\rho} H^0(X,{\cal H}^3(\Z(2)_{{\cal D}}))\\ \uparrow &\uparrow &\uparrow\\ 0\to NS(X)\otimes\C^*(2)&\to H^2(X,\C^*(2))&\to H^0(X,{\cal H}^2(\C^*(2)))\to 0 \E{array} \E{equation} where $H^1(X,{\cal H}^1(\C^*(2)))\cong NS(X)\otimes \C^*(2)$ and the left-most map is induced by the short exact sequence of sheaves (\ref{disc}) whence $H^0(X,{\cal F}^{2,2}_{\mbox{\scriptsize{$\Z$}}})$ and $H^1(X,{\cal F}^{2,2}_{\mbox{\scriptsize{$\Z$}}})$ are respectively the kernel and the cokernel of $NS(X)\otimes\C^*(2)\to H^1(X,{\cal H}^2(\Z(2)_{{\cal D}})))$ in fact: the cokernel is computed by the vanishing of $H^2(X,{\cal H}^1(\C^*(2)))$ and the kernel is obtained because of $H^0(X,{\cal H}^1(\C^*(2)))\cong H^1(X,\C^*(2))$, $H^0(X,{\cal H}^2(\Z(2)_{{\cal D}}))\cong H^2(X,\Z(2)_{{\cal D}})$ (note that $\C^*(2)\cong{\cal H}^1(\Z(2)_{{\cal D}})$ is flasque) and $H^1(X,\C^*(2))\cong H^2(X,\Z(2)_{{\cal D}})$ since $X$ is proper {\it i.e.\/}\ $F^2H^2\hookrightarrow H^2(X,\C^*(2))$. Furthermore we have that $H^3(X,\Z(2)_{{\cal D}})\cong H^2(X,\C^*(2))/F^2H^2$ and $H^0(X,{\cal F}^{2,2}_{\mbox{\scriptsize{$\Z$}}})$ vanishes because $F^2H^2\cap NS(X)\otimes\C^*(2)=0$. Since $H^0(X,{\cal H}^3(\Z(2)_{{\cal D}}))$ is torsion free (by the Theorem~\ref{Kato}) then ${\rm im}\,\rho$ is torsion free and infinitely divisible indeed, therefore $$H^1(X,{\cal H}^2(\Z(2)_{{\cal D}}))\otimes \mbox{$\bf Q$}/\Z = H^1(X,{\cal F}^{2,2}_{\mbox{\scriptsize{$\Z$}}})\otimes \mbox{$\bf Q$}/\Z =0$$ and by taking the torsion subgroups in the diagram (\ref{cocco}) we obtain the assertion about the torsion of $H^1(X,{\cal F}^{2,2}_{\mbox{\scriptsize{$\Z$}}})$. In order to show the second part of the statement, since $NS(X)\cong H^2(X,\Z)$, we then have (by the bottom row of the diagram (\ref{cocco}) above) that $H^0(X,{\cal H}^2(\C^*(2)))\cong H^3(X,\Z)_{tors}$ and its image in $H^0(X,{\cal H}^3(\Z(2)_{{\cal D}}))$ is equal to the image of $\rho$, whence the image of $\rho$ is zero since it is torsion free. Since $F^2H^2=0$ by diagram chase we obtain the last claim. \E{proof} \B{rmk} The group $H^0(X,{\cal H}^2(\mbox{$\bf Q$}/\Z))$ is effectively the extension of $H^0(X,{\cal H}^2(\Z))\otimes \mbox{$\bf Q$}/\Z$ by $H^3(X,\Z)_{tors}$ because $H^0(X,{\cal H}^3(\Z))$ is torsion free. \E{rmk} In order to detect elements in the misterious group of global sections $H^0(X,{\cal H}^3(\Z(2)_{{\cal D}}))$ we dispose of the image of $H^0(X,{\cal H}^2/{\cal F}^2)$, see (\ref{genus}), which is the same ({\it cf.\/}\ the diagram (\ref{dia})) as the image of $H^0(X,{\cal H}^2(\Z))\otimes \C/\mbox{$\bf Q$}(2)=H^0(X,{\cal H}^2(\C^*(2)))\otimes \mbox{$\bf Q$}$. Unfortunately these images cannot be the entire group, in general. Indeed we have that whenever the map $$H^0(X,{\cal H}^2(\C^*(2)))\to H^0(X,{\cal H}^3(\Z(2)_{{\cal D}}))$$ is surjective then $\rho$ is surjective in (\ref{delseq}) (because of (\ref{cocco})) whence the cycle map is injective which is not the case in general (indeed for any surface with $p_g\neq 0$ the cycle map is not injective by Mumford \cite{MU}). \section{Surfaces with $p_g=0$} In the following we let $X$ denote a complex algebraic surface which is smooth and complete. Let $A_0(X)$ be the subgroup of cycles of degree zero. Let $\phi: A_0(X)\to J^2(X)$ be induced by the canonical mapping to the Albanese variety. It is well known (see \cite[Theor.2 and Cor.]{GI}) that $c\ell\mid_{A_0(X)} =\phi$, where $c\ell$ is the cycle map in Deligne cohomology. We have: \B{equation}\label{cacca} H^1(X,{\cal H}^3(\Z(2)_{{\cal D}}))=0 \E{equation} Indeed: the sheaf ${\cal H}^4(\Z(2)_{{\cal D}})$ vanishes on a surface (as it is easy to see via the exact sequence (\ref{modf})) and by the spectral sequence (\ref{conideli}) we have that $H^1(X,{\cal H}^3(\Z(2)_{{\cal D}}))$ is the cokernel of the cycle map $CH^2(X)\by{c\ell}H^4(X,\Z(2)_{{\cal D}})$ but coker$(c\ell)=$coker$(A_0(X)\to J^2(X))=0$. Finally we have that $H^2(X,{\cal H}^3(\Z(2)_{{\cal D}}))\cong H^5(X,\Z(2)_{{\cal D}})=0$. Thus the only possibly non-zero terms in the spectral sequence (\ref{conideli}) are: those giving the exact sequence (\ref{delseq}), $H^0(X,{\cal H}^2(\Z(2)_{{\cal D}}))\cong H^2(X,\Z(2)_{{\cal D}})$ and $H^0(X,{\cal H}^1(\Z(2)_{{\cal D}}))=\C^*$. We know by Mumford (see \cite{MU}) that $A_0(X)\cong J^2(X)$ {\it i.e.\/}\ $\delta=0$ in (\ref{delseq}), implies $p_g=0$ therefore $H^0(X,{\cal H}^3(\Z(2)_{{\cal D}}))=0$ (by Proposition~\ref{alb}). Conversely we have the following:\\[5pt] {\bf Bloch's conjecture\ \ } {\it If $p_g=0$ then $A_0(X)\cong J^2(X)$.}\\[4pt] In order to test this conjecture we are reduced to compute the uniquely divisible group $H^0(X,{\cal H}^3(\Z(2)_{{\cal D}}))$ ({\it cf.\/}\ \cite{H}, \cite{GI}). We can show the following: \B{prop}\label{compute} Let $X$ be a smooth complete surface with $p_g=0$. We then have the following canonical short exact sequences \B{equation}\label{a} 0\to H^0(X,{\cal H}^3(\Z(2)_{{\cal D}}))\to H^1(X,{\cal F}^2/{\cal F}^{2,2}_{\mbox{\scriptsize{$\Z$}}})\to H^3(X,\C^*(2))\to 0 \E{equation} \B{equation}\label{b} 0\to H^0(X,{\cal H}^3(\Z(2)_{{\cal D}}))\to H^1(X,{\cal H}^2(\Z(2))/{\cal F}^{2,2}_{\mbox{\scriptsize{$\Z$}}}))\to H^3/F^2\to 0 \E{equation} where \B{equation}\label{c} 0\to F^2H^3\to H^1(X,{\cal F}^2/{\cal F}^{2,2}_{\mbox{\scriptsize{$\Z$}}})\to A_0(X)\to 0 \E{equation} \B{equation}\label{d} 0\to H^3(X,\Z)/tors\to H^1(X,{\cal H}^2(\Z(2))/{\cal F}^{2,2}_{\mbox{\scriptsize{$\Z$}}})\to A_0(X)\to 0 \E{equation} \E{prop} \B{proof} All these exact sequences are obtained by considering the exact diagram of cohomology groups associated with the diagram of sheaves (\ref{dia}) (where ${\cal T}^{2,3}_{{\cal D}} = {\cal H}^3(\Z(2)_{{\cal D}})$ on a surface) taking account of Theorem~\ref{Kato}, Theorem~\ref{Nero}, Proposition~\ref{alb} and the coniveau spectral sequence (\ref{conifilt}). For example, the sequence (\ref{a}) is obtained by taking the long exact sequence of cohomology groups associated with the right-most column of (\ref{dia}), the fact that $H^1(X,{\cal H}^2(\C^*(2)))\cong H^3(X,\C^*(2))$ on a surface and the formula (\ref{cacca}). For (\ref{b}) one has to use the top row of (\ref{dia}), the formulas (\ref{genus}) and (\ref{cacca}), and the fact that $H^3/F^2\cong H^1(X,{\cal H}^2/{\cal F}^2)$. The left-most column of (\ref{dia}) yields (\ref{d}) because of $H^2(X,{\cal F}^{2,2}_{\mbox{\scriptsize{$\Z$}}})\cong H^2(X,{\cal H}^2(\Z(2)_{{\cal D}}))\cong CH^2(X)$ by (\ref{disc}) ({\it cf.\/}\ \cite[1.3]{H}) and the map of sheaves ${\cal H}^2(\Z(2)_{{\cal D}})\to {\cal H}^2(\Z(2))$ induces the degree map on $H^2$. For (\ref{c}) one has to argue with the commutative square in the left bottom corner of (\ref{dia}) and the isomorphim $H^2(X,{\cal F}^2)\cong H^2(X,{\cal H}^2(\C))\cong \C$: remember that $H^1(X,{\cal F}^{2,2}_{\mbox{\scriptsize{$\Z$}}})=H^3(X,\Z)_{tors}$ whence it goes to zero in $F^2H^3\cong H^1(X,{\cal F}^2{\cal H}^2)$. \E{proof} \B{rmk} Because of this Proposition, Bloch's conjecture is equivalent to showing that the canonical injections of sheaves ${\cal F}^2/{\cal F}^{2,2}_{\mbox{\scriptsize{$\Z$}}}\hookrightarrow {\cal H}^2(\C^*(2))$ or ${\cal H}^2(\Z(2))/{\cal F}^{2,2}_{\mbox{\scriptsize{$\Z$}}}\hookrightarrow {\cal H}^2/{\cal F}^2$ remain injections on $H^1$. It would be very nice to know of any reasonable description of the Zariski cohomology classes of these subsheaves. \E{rmk}

9412

alg-geom/9412001

en

https://arxiv.org/abs/alg-geom/9412001

alg-geom/9412001

Wayne Raskind

Wayne M. Raskind

Higher $l$-adic Abel-Jacobi mappings and filtrations on Chow groups

33 pages, Latex. To appear in Duke Math. Journal

We define a filtration on the Chow groups of a smooth projective variety X over a field k by using the cycle map into continuous l-adic etale cohomology. The main theorem says that if k is a function field in one variable over a finite field, this filtration for zero cycles is of length at most one modulo the kernel of the cycle map.

alg-geom

\section*{Introduction} Let $X$ be a smooth projective variety over a field $k$ and denote by $CH^n(X)$ the group of codimension $n$ cycles modulo rational equivalence on $X$. The structure of this group is largely unknown when $n\geq 2$, especially when $k$ is a ``large'' field such as the complex numbers. Beginning with the work of Bloch [Bl3] and Beauville [Be1,2], the Chow ring of an abelian variety was found to have some structure arising from a filtration defined using Pontryagin products or the Fourier transform. In recent years, several people (Beilinson, Bloch, Murre, Nori, S. Saito, H. Saito; see [B1], [Bl4], [Mu], [N], [S1], [SH]) have made conjectures asserting the existence of certain filtrations on the Chow groups of any smooth projective variety. These conjectures would imply that the Chow groups have much more structure than has been hitherto uncovered. In this paper we define and study a filtration arising from the cycle map into $l$-adic \'etale cohomology. It is somewhat arithmetic in nature because it is necessary to first define it over a field which is finitely generated over its prime subfield. For varieties over larger fields, it may be defined by passing to the limit over finitely generated subfields. Others have studied this filtration, in particular Beauville (unpublished) and Jannsen (see [J4]). With their permission, I have included some of their results in this paper. Each of the filtrations defined so far has its advantages and disadvantages. The conjectural filtration of Beilinson arising from a spectral sequence relating extensions of mixed motives and motivic cohomology would imply many outstanding and deep conjectures about algebraic cycles. We will not repeat this story in detail here since it is treated very clearly in the paper of Jannsen [J4]. The conjectural filtration of Murre is known to exist in some cases, and it would also imply some of these conjectures, but to establish its existence in general appears to require the resolution of some very difficult conjectures. The filtrations of H. Saito and S. Saito are defined in a very explicit way, but it seems difficult to do specific calculations. The filtration in this paper has the advantage that it may be ``computed'' in some cases, and the powerful tools of arithmetic algebraic geometry may be brought to bear on it. However, it has two basic disadvantages. First, it is not at all obvious that it is independent of $l$. Second, suppose $X$ is defined over a field $k$ which is finitely generated over its prime subfield and $l$ is a prime number different from the characteristic of $k$. Jannsen [J1] has defined a cycle map into continuous \'etale cohomology (see \S 1): $$c_n: CH^n(X)\otimes \mbox{\bf Q}\to H^{2n}(X,\mbox{\bf Q}_l(n)).$$ The following is a more general version of a conjecture first made by Soul\'e over finite fields and by Jannsen and the author over number fields:\\ \noindent{\bf Conjecture:} $c_n$ is injective.\\ This is known when $n$=1 (a consequence of the finite generation of $CH^1(X)$), but except for special classes of varieties, we have little information for $n\geq 2$.\\ The Hochschild-Serre spectral sequence: $$H^r(k,H^q({\overline X},\mbox{\bf Q}_l(n)))\Longrightarrow H^{r+s}(X,\mbox{\bf Q}_l(n))$$ \noindent defines a filtration on $H^{2n}(X,\mbox{\bf Q}_l(n))$, which we pull back to $CH^n(X)\otimes \mbox{\bf Q}$ via the cycle map $c_n$. We call this the {\it $l$-adic filtration} on $CH^n(X)\otimes \mbox{\bf Q}$. By definition, the kernel of the cycle map is contained in all the steps of the filtration. For this reason, until we have more information on this kernel, we can only get results about the filtration modulo the kernel. We can define another filtration ${\cal F}^{\bullet}CH^n(X)\otimes \mbox{\bf Q}$ on the image of the cycle map by taking the intersection with the filtration obtained from the Hochschild-Serre spectral sequence. It is really this filtration about which we can say something.\\ In section 2 of this paper we refine the cycle map $c_n$ to get {\it higher $l$-adic Abel-Jacobi maps} from certain parts of the Chow group to Galois cohomology groups. For example, suppose that $n$=dim $X$ and let $A_0(X)$ denote the group of zero-cycles of degree zero modulo rational equivalence. Let $Alb(X)$ be the group of $k$-points of the Albanese variety of $X$ and $T(X)$ the kernel of the Albanese map: $$A_0(X)\to Alb(X).$$ \noindent Then we define a map: $$d_n^2: T(X)\otimes \mbox{\bf Q}\to H^2(k,H^{2n-2}(\overline X,\mbox{\bf Q}_l(n))).$$ \noindent This map generalizes the $l$-adic Abel-Jacobi map of Bloch [Bl1], which in this case is essentially the map obtained from Kummer theory on $Alb(X)$ (see \S 2 for more details). \\ For the results of this paper we will need to assume that if $X$ is a smooth projective variety over a local field $k$ with ring of integers ${\cal O}$, then $X$ has a regular proper model over $\cal O$. Assuming this, the main result of this paper is:\\ \noindent {\bf Theorem 0.1}: Let $k$ be a function field in one variable over a finite field of characteristic $p\neq l$ and $X$ a smooth projective variety of dimension $n$ over $k$. Then the map $d_n^2$ is zero.\\ \noindent Since a global field is of cohomological dimension 2 for $\mbox{\bf Q}_l$-cohomology, we have:\\ \noindent {\bf Corollary}: For $X$ as above of dimension $n$, we have ${\cal F}^iCH^n(X)\otimes \mbox{\bf Q}=0$ for all $i\geq 2$.\\ The following remarks may help clarify the meaning of these results. First of all, Beilinson and Bloch have made the conjecture that if $k$ is a global field and $X$ is a smooth projective variety over $k$ then the Albanese kernel $T(X)$ is a torsion group. Thus the theorem provides ``evidence'' for this conjecture for $k$ a global field of positive characteristic by showing that the cycle classes of elements of $T(X)$ in $H^{2n}(X,\mbox{\bf Q}_l(n))$ are trivial. This should be true for $k$ a number field as well, but we can only prove it when $H^{2n-2}({\overline X},\mbox{\bf Q}_l(n))$ is generated by the cohomology classes of algebraic cycles. In this case, Beauville showed me an easy proof which works over any field.\\ One of the key technical tools in this paper is the theory of continuous $l$-adic \'etale cohomology and the Hochschild-Serre spectral sequence relating the cohomology of a variety $X$ over a field $k$ with the cohomology over a separable closure ${\overline k}$ (see \S 1). Such spectral sequences do not exist in general for the usual $l$-adic cohomology defined by taking the inverse limit of cohomology with finite coefficients. Also, we rely on results of Jannsen on the Galois cohomology of global fields with coefficients in $l$-adic cohomology groups of algebraic varieties over $\overline k$ [J2]. Another key ingredient is Thomason's purity theorem for $\mbox{\bf Q}_l$-cohomology of schemes of finite type over the ring of integers in a local field of residue characteristic different from $l$ [T]. This enables us to construct cycle maps over rings of integers of local fields.\\ The $l$-adic Abel-Jacobi maps $d_n^i$ were discovered independently (and earlier) by Beauville and by Jannsen, who has also given a specific description of the map $d_n^2$ in terms of extension groups [J3]. \\ At least for $X$ smooth over a field, $CH^*(X)\otimes \mbox{\bf Q}\cong K_0(X)\otimes\mbox{\bf Q},$ and so the $l$-adic filtration may also be viewed as a filtration on $K_0(X)\otimes \mbox{\bf Q}$. There are Chern class maps: $$K_{j}(X)\otimes \mbox{\bf Q}\to \bigoplus_{i,j}H^{2i-j}(X,\mbox{\bf Q}_l(i))$$ \noindent which one can hope are injective over an absolutely finitely generated field. These can be used to define similar filtrations on all $K$-groups. We hope to develop this line of investigation further in another paper. \\ I would like to thank U. Jannsen for explaining some of his unpublished results to me and allowing me to include some of them in this paper. J.-L. Colliot-Th\'el\`ene listened to and helped me with several points in this paper, especially Proposition 3.2. A. Beauville informed me that one can easily prove that the $l$-adic filtration is of length at most $n$ on $CH^n$ modulo the kernel of the cycle map. He also showed me a much simpler proof than the one I had originally proposed of the triviality of the higher Abel-Jacobi map in the case where the cohomology of ${\overline X}$ is generated by the classes of algebraic cycles. I am grateful to him for allowing me to include these here. The referee made several helpful comments which have improved the exposition. Finally, I would like to thank K. Paranjape and S. Saito for useful discussions.\\ During the long period of thinking about and writing this paper, I enjoyed the hospitality of the Max-Planck Institut f\"ur Mathematik in Bonn, Universit\"at zu K\"oln, Universit\"at M\"unster, Universit\'e de Paris-Sud and the CRM of the Universitat Aut\`onoma de Barcelona. During my one-week visits to the aforementioned German universities, I was supported by the Deutsche Forschungsgemeinschaft. \section{Notation and Preliminaries} \parindent=0cm We denote by $X$ a smooth, projective, geometrically connected variety over a field $k$ and $l$ a prime number different from $\mbox{char.} k$. Let ${\overline k}$ be a separable closure of $k$ and $G=Gal({\overline k}/k), {\overline X}=X\times_k\overline k$. If $k$ is a global field and $S$ is a finite set of places of $k$, $G_S$ denotes the Galois group of a maximal extension of $k$ which is unramified outside $S$. For ${\cal X}$ a noetherian scheme, we denote by $CH^n({\cal X})$ the Chow group of codimension $n$ cycles modulo rational equivalence. For $X$ proper and geometrically connected over a field, $CH_0(X)$ is the group of zero-cycles modulo rational equivalence and $A_0(X)$ the group of zero-cycles of degree zero modulo rational equivalence. The group of $k$-points of the Albanese variety of $X$ will be denoted by $Alb(X)$ and $\alpha : A_0(X)\to Alb(X)$ denotes the Albanese map. We denote the kernel of this map by $T(X)$. If $M$ is an abelian group upon which $G$ acts continuously, we denote the continuous Galois cohomology groups of $G$ with values in $M$ by $H^*(k,M)$ (see \S 1 for a definition of these groups). The subgroup of $M$ consisting of elements killed by $l^m$ is denoted by $M_{l^m}$ and $$T_lM={\lim_{\stackrel{\leftarrow}{m}}}M_{l^m}.$$ We set $V_lM=T_lM\otimes_{\mbox{\bf Z}_l}\mbox{\bf Q}_l.$ As usual, we denote by $\mbox{\bf Z}_l(1)$ the $l$-adic sheaf of $l$-primary roots of unity. For $n>0, \mbox{\bf Z}_l(n)$ denotes $\mbox{\bf Z}_l(1)^{\otimes n}$; for $n<0, \mbox{\bf Z}_l(n)=Hom(\mbox{\bf Z}_l(-n),\mbox{\bf Z}_l)$. We use similar notation for $\mbox{\bf Q}_l$-sheaves.\\ Let $X$ be a smooth variety over a field $k$ and $l$ a prime number different from the characteristic of $k$. Jannsen has defined a cycle map [J1]: $$CH^n(X)\otimes \mbox{\bf Q} \to H^{2n}(X,\mbox{\bf Q}_l(n)),$$ \noindent where the group on the right is the continuous $l$-adic cohomology group with values in the $l$-adic sheaf $\mbox{\bf Q}_l(n)$. We briefly recall the definition of these groups. Let $({\cal F}_m)$ be a projective system of sheaves of $\mbox{\bf Z}/l^m$-modules on $X$. The functor which sends $({\cal F}_m)$ to $$\lim_{\stackrel{\leftarrow}{m}} H^0(X,({\cal F}_m))$$ is left exact, and we define $H^i(X,({\cal F}_m))$ to be the $i$-th right derived functor. When the inverse system is $\mbox{\bf Z}/l^m(j)$, we denote these groups by $H^i(X,\mbox{\bf Z}_l(j))$. Continuous \'etale cohomology with $\mbox{\bf Q}_l$-coefficients is defined by tensoring the $\mbox{\bf Z}_l$-cohomology with $\mbox{\bf Q}_l$. We shall need the following properties of this cohomology which can be found in Jannsen's paper [J1].\\ \noindent (i) There is an exact sequence: $$0\to {\lim_{\stackrel{\leftarrow}{m}}} ^1H^{i-1}(X,({\cal F}_m))\to H^i(X,({\cal F}_m))\to \lim_{\stackrel{\leftarrow}{m}} H^i(X,{\cal F}_m)\to 0.$$ In particular, if $H^i(X,\mbox{\bf Z}/l^m(j))$ is finite for all $i$ and $m$ then $$H^i(X,\mbox{\bf Z}_l(j))\to \lim_{\stackrel{\leftarrow}{m}} H^i(X,\mbox{\bf Z}/l^m(j))$$ is an isomorphism. This is true if, for example, $k$ is separably closed, finite or local. This is not true in general if $k$ is a number field.\\ \noindent (ii) Let $\overline k$ be a separable closure of $k$, $G=Gal(\overline k/k)$ and $\overline X=X\times _k\overline k$. Then there is a Hochschild-Serre spectral sequence: $$E_2^{p,q}=H^p(G,H^q(\overline X,\mbox{\bf Z}_l(j)))\Longrightarrow H^{p+q}(X,\mbox{\bf Z}_l(j)).$$ \noindent {\bf Theorem 1.1} (Jannsen): Let $X$ be a smooth projective variety over a field $k$. Then the Hochschild-Serre spectral sequence with $\mbox{\bf Q}_l$-coefficients degenerates at $E_2$.\\ Since this result has not been published, let us indicate the ingredients in the proof. The idea is to use Deligne's criterion for a spectral sequence to degenerate [D1]. In order to use this in this situation, one needs a good derived category of $\mbox{\bf Q}_l$-sheaves. One cannot use Deligne's definition given in ([D2], 1.1.2d) because the cohomology groups with finite coefficients need not be finite. Jannsen has given a good definition of such a derived category in the general case. Once this is done one applies the Hard Lefschetz Theorem to satisfy Deligne's criterion. Ekedahl has published another definition of the derived category of $\mbox{\bf Q}_l$-sheaves [E], and this should also suffice to prove the degeneration of the Hochschild-Serre spectral sequence. \section{The $l$-adic filtration} {}From the Hochschild-Serre spectral sequence for continuous cohomology: $$E_2^{p,q}=H^p(k,H^q(\overline X,\mbox{\bf Q}_l(n)))\Longrightarrow H^{p+q}(X,\mbox{\bf Q}_l(n)),$$ we get a filtration on $H^{2n}(X,\mbox{\bf Q}_l(n))$: $$H^{2n}(X,\mbox{\bf Q}_l(n))=F^0\supset F^1\supset . . .\supset F^{2n}$$ and $F^i/F^{i+1}=E_\infty ^{i,2n-i}$.\\ {\bf Definition 2.1}: Let $X$ be a smooth, projective, geometrically connected variety over a field $k$ which is finitely generated over its prime subfield. Then we define: $$F^iCH^n(X)\otimes \mbox{\bf Q}=c_n^{-1}[F^iH^{2n}(X,\mbox{\bf Q}_l(n))],$$ where $c_n$ is the cycle map (see \S 1).\\ If $k$ is not finitely generated, we define: $$F^iCH^n(X)\otimes \mbox{\bf Q}=\lim_{\stackrel{\rightarrow}{L}}F^iCH^n(X_L)\otimes \mbox{\bf Q},$$ where $L$ runs over all finitely generated fields contained in $k$ for which $X$ has a model over $L$.\\ Finally, set $${\cal F}^iCH^n(X)\otimes \mbox{\bf Q}=\mbox{Image}\,c_n \cap F^{i}H^{2n}(X,\mbox{\bf Q}_l(n)).$$ Of course, {\it a priori} ${\cal F}^{\bullet}CH^n(X)\otimes\mbox{\bf Q}$ is only a filtration on the image of the cycle map.\\ Let $X$ be defined over a field $L$ which is finitely generated over the prime subfield. By the definition of the filtration on $CH^n(X)\otimes \mbox{\bf Q}$ and the degeneration of the Hochschild-Serre spectral sequence (Theorem 1.1), we get maps: $$d_n^i:\, F^iCH^n(X)\otimes \mbox{\bf Q}\to H^i(k,H^{2n- i}({\overline X},\mbox{\bf Q}_l(n)))$$ which we call the {\it higher $l$-adic Abel-Jacobi mappings}. For $X$ over arbitrary $k$, we define the higher Abel-Jacobi map as the limit of the maps defined above for $X_L$, where $L$ runs over finitely generated subfields of $k$ over which $X$ has a model. \\ When $i$=1, these are the $l$-adic Abel-Jacobi mappings defined by Bloch [Bl1]. When $n=\mbox{dim}X$, this is the Albanese map followed by the map given by Kummer theory for $Alb(X)$, as we now explain.\\ Let $A_0(X)$ be the subgroup of $CH^n(X)$ consisting of cycles of degree zero and let $Alb(X)$ denote the group of $k$-points of the Albanese variety of $X$. Consider the Albanese map $$\alpha: A_0(X)\to Alb(X)$$ and its kernel $T(X)$. Then we have a commutative diagram with exact rows: $$\matrix{0 &\to & F^2&\to & F^1&\to& H^1(k,H^{2n- 1}(X,\mbox{\bf Q}(n)))&\to & 0\cr &&f\uparrow&&\uparrow&&\uparrow\cr 0&\to& T(X)\otimes \mbox{\bf Q}&\to& A_0(X)\otimes \mbox{\bf Q}&\to &Alb(X)\otimes \mbox{\bf Q}&\to &0}.$$ Here the right vertical map is given as follows. By Poincar\'e duality and the Weil pairing, we have an isomorphism: $$H^{2n-1}(\overline X,\mbox{\bf Q}_l(n))\cong V_l(Alb(\overline X)),$$ where $V_l$ denotes the Tate $\mbox{\bf Q}_l$-vector space of $Alb(\overline X)$. Then the right vertical map is defined by doing Kummer theory modulo $l^m$ on $Alb(X)$, passing to the inverse limit over $m$ and tensoring with $\mbox{\bf Q}_l$. The commutativity of the right square is proved in the appendix to this paper. This induces the map $f$. Composing $f$ with the map obtained from the spectral sequence: $$F^2\to E_\infty ^{2,2n-2},$$ we get a map $$d_n^2:\, T(X)\otimes \mbox{\bf Q}\to H^2(k,H^{2n-2}(\overline X,\mbox{\bf Q}_l(n))).$$ {\bf Remarks 2.1.1}: (i) It would be interesting to find a complex analytic analogue of the maps $d_n^i$. Of course, $d_n^1$ may be defined in the complex category as the Abel-Jacobi map of Griffiths. The target group of our $d_n^i$ may be reinterpreted as $Ext_{\mbox{\bf Q}_l}^i(\mbox{\bf Q}_l,H^{2n-i}({\overline X},\mbox{\bf Q}_l(n)))$ in the category of continuous Galois modules. The complex analogue of this is $Ext_{MHS}^i(\mbox{\bf Q},H^{2n-i}(X,\mbox{\bf Q}))$, where $MHS$ denotes the category of mixed Hodge structures. For $i=1$, this works fine, as it is well-known that $Ext_{MHS}^1(\mbox{\bf Z},H^{2n-1}(X,\mbox{\bf Z}))$ for $X/{\bf C}$ is the intermediate Jacobian of Griffiths. Unfortunately, all the higher $Ext$-groups are zero. Several people have suggested taking extensions in a suitable category of variations of Hodge structure.\\ (ii) The filtration $F_B$ of S. Saito (see [Sa3], Definition (1-4)) is contained in the $l$-adic filtration. It is not clear to me whether this is the case for the filtration $F_{BM}$ defined in (loc. cit. Definition (1-5)). For much more on the relations between various filtrations, see the paper of Jannsen [J4].\\ (iii) The $l$-adic Abel-Jacobi maps defined above have a conjectural mixed motivic interpretation. Recall that there are motivic cohomology groups with $\mbox{\bf Q}$-coefficients $H_{\cal M}^i(X,\mbox{\bf Q}(j))$ with $$H_{\cal M}^{2n}(X,\mbox{\bf Q}(n))=CH^n(X)\otimes \mbox{\bf Q}.$$ These may be defined by: $$H_{\cal M}^i(X,\mbox{\bf Q}(n))=gr_{\gamma}^{n}K_{2n-i}(X)\otimes \mbox{\bf Q},$$ where $\gamma$ denotes the gamma filtration in algebraic $K$-theory (here it is crucial that we are dealing with {\bf Q}-coefficients because the integral theory is much less developed). Beilinson [B1] has made the conjecture that these groups have a canonical filtration coming from a spectral sequence: $$Ext^p_{\cal M\cal M}(\mbox{\bf Q},H^q(X,\mbox{\bf Q}(j)))\Longrightarrow H_{\cal M}^{p+q}(X,\mbox{\bf Q}(j)).$$ Here the subscript $\cal M\cal M$ denotes the conjectural category of mixed motives on $X$ and the $Ext$ is taken in this category. The maps given above are in some sense ``$l$-adic realizations'' of the corresponding maps obtained from this conjectural spectral sequence. This point of view is very useful for learning what to expect. In fact, if Beilinson's conjecture on the existence of filtrations on Chow groups ([B1], [J4]) is true, then the filtration defined in this paper must agree with his (see [J4], Lemma 2.7).\\ We now prove some basic results on these filtrations.\\ {\bf Proposition 2.2} (Beauville): For $X$ as above, we have $$F^{n+j}CH^n(X)\otimes \mbox{\bf Q}=F^{n+1}CH^n(X)\otimes \mbox{\bf Q}$$ for all $j\geq 1.$ \\ {\bf Proof}: It suffices to show that the higher Abel-Jacobi maps $d_n^i$ are zero for $i>n$. To prove this, let $d$ be the dimension of $X$ and let $Y$ be a smooth hyperplane section of $X$. Then the diagram: $$\matrix{F^{n+j}CH^n(X)\otimes \mbox{\bf Q}&\to&H^{n+j}(k,H^{n- j}({\overline X},\mbox{\bf Q}_l(n)))\cr \downarrow&&\downarrow\cr F^{n+j} CH^{d+j}(X)\otimes \mbox{\bf Q}&\to&H^{n+j}(k,H^{2d- n+j}({\overline X},\mbox{\bf Q}_l(d+j)))}$$ is commutative. Here the left vertical arrow is given by intersecting cycles on $X$ with the class of the $(d+j-n)$-fold intersection of $Y$ with itself, and the right vertical arrow is given by cupping $(d+j-n)$-times with the class of $Y$ in $H^2(X,\mbox{\bf Q}_l(1))$. This commutativity is easily seen from the compatibility of the Hochschild-Serre spectral sequence and the cycle map with intersection products ([J1], Lemma 6.14). Now since $j\geq 1$, the bottom left group is zero. But the right vertical arrow is an isomorphism by the Hard Lefschetz Theorem, and hence the image of the top horizontal arrow is trivial. This proves the propostion.\\ Recall that the cohomology group $H^i({\overline X},\mbox{\bf Q}_l)$ is said to be {\it algebraic} if it is generated by Tate twists of cohomology classes of algebraic cycles. Thus if $i$ is odd, this means that the group is zero, and if $i=2j$ is even then the cycle map: $$CH^j({\overline X})\otimes \mbox{\bf Q}_l\to H^{2j}({\overline X},\mbox{\bf Q}_l(j))$$ is surjective (we keep the Tate twists because we will need them later). A basic conjecture of Bloch ([Bl2], Conjecture 0.4 in the case of surfaces) predicts that if $X$ is defined over $\bf C$ and the cohomology of $X$ is supported on codimension 1 subschemes, then the Albanese map: $$\alpha:\, A_0(X)\to Alb(X)$$ is an isomorphism. We now give an argument of Beauville to show how a weaker form of this conjecture is implied by the conjecture on injectivity of the cycle map for varieties over finitely generated fields (see the introduction to this paper).\\ {\bf Proposition 2.3} (Beauville): Let $X$ be a smooth projective variety of dimension $d$ over a field $k$. Let $i$ be a positive integer and assume that the cohomology groups $H^{2n-2i}({\overline X},\mbox{\bf Q}_l(i))$ and $H^{2d-2n+2i}({\overline X},\mbox{\bf Q}_l(d-n+i))$ are generated by algebraic cycles. Then the map: $$F^{2i}CH^n(X)\otimes\mbox{\bf Q}\to H^{2i}(k,H^{2n-2i}({\overline X},\mbox{\bf Q}_l(n)))$$ is zero. \\ {\bf Proof}: First note that for any finite extension $L/k$, the natural map: $$H^{2i}(k,H^{2n-2i}({\overline X},\mbox{\bf Q}_l(n)))\to H^{2i}(L,H^{2n-2i}({\overline X},\mbox{\bf Q}_l(n)))$$ is injective because its kernel is killed by $[L:k]$. By the algebraicity assumption, there is a finite extension $L/k$ over which $G=Gal({\overline k}/L)$ acts trivially on $H^{2n-2i}({\overline X},\mbox{\bf Q}_l(n-i))$ and on $H^{2d-2n+2i}({\overline X},\mbox{\bf Q}_l(d-n+i))$. Thus we may and do assume that $Gal({\overline k}/k)$ acts trivially on these groups. Let $Y_1,...Y_{\nu}$ be codimension $d-n+i$ cycles on $X$ whose cohomology classes $\xi_1,...,\xi_\nu$ form a basis for $H^{2d-2n+2i}({\overline X},\mbox{\bf Q}_l(d-n+i))$. Then the following diagram is commutative: $$\matrix{F^{2i}CH^n(X)\otimes \mbox{\bf Q}&\to&H^{2i}(k,H^{2n-2i}({\overline X},\mbox{\bf Q}_l(n)))\cr \downarrow&&\downarrow\cr [F^{2i}CH^{d+i}(X)\otimes\mbox{\bf Q}]^{\nu}&\to&[H^{2i}(k,H^{2d}({\overline X},\mbox{\bf Q}_l(d+i) ))]^{\nu}}.$$ Here the left vertical arrow is given by intersecting with the class of $Y_j$ on the $j$-th factor. The right vertical arrow is given by cupping with the class of $\xi_j$ on the $j$-th factor; it is an isomorphism by the assumption that the cohomology is algebraic and Galois acts trivially on $H^{2n-2i}({\overline X},\mbox{\bf Q}_l(n-i))$. The commutativity is easily established by the compatibility of the Hochschild-Serre spectral sequence and the cycle map with intersection product. Now the bottom left group is zero if $i\geq 1$. Hence the top horizontal map is zero. This completes the proof of the proposition. \\ {\bf Corollary 2.4}: Let $X$ be a smooth projective variety of dimension $d$ over an algebraically closed field $k$. assume that for $d\leq i\leq 2d-2$, the cohomology group $H^i({\overline X},\mbox{\bf Q}_l)$ is algebraic. Let $K$ be a finitely generated field of definition of $X$ and assume that the cycle map: $$c_d:\, CH^d(X_M)\otimes \mbox{\bf Q}\to H^{2d}(X_M,\mbox{\bf Q}_l(d))$$ is injective over any finitely generated field $M$ containing $K$. Then the Albanese map: $$\alpha:\: A_0(X)\to Alb(X)$$ is an isomorphism. In particular, if $k={\bf C}$ and $X$ is a surface with $H^0(X,\Omega^2_X)=0$, then injectivity of the cycle map $c_2$ over all finitely generated fields containing $K$ implies Bloch's conjecture.\\ {\bf Proof}: Let $a$ be in the kernel of $\alpha$. Then $a$ is defined over some finitely generated field $M$. The assumption on injectivity of the cycle map implies that the maps: $$F^iCH^d(X_M)\otimes \mbox{\bf Q}\to H^i(M,H^{2d-i}({\overline X},\mbox{\bf Q}_l(d)))$$ are injective. If $i\geq 2$ then the assumption that the cohomology is algebraic and Proposition 2.3 imply that the map is zero. When $i=1$, the map may be identified with $\alpha$ over $M$ followed by Kummer theory for $Alb(X_M)$ (see the appendix to this paper). Hence $\alpha$ is injective up to torsion. But by Roitman's theorem [Ro], [M2], the kernel of $\alpha$ has no torsion, so it is zero. The last statement follows immediately from the exponential sequence, Serre duality and the comparison theorem between singular and \'etale cohomology.\\ {\bf Remark 2.4.1}: This result should be true under the weaker hypothesis that for each $d\leq i\leq 2d-2$, the cohomology group $H^i({\overline X},\mbox{\bf Q}_l)$ is supported on codimension one subschemes. This is Bloch's conjecture in the general case.\\ {\bf Proposition 2.5}: Let $X$ be a smooth, projective variety over a field $k$ which is finitely generated over its prime subfield and $l$ a prime number different from the characteristic of $k$. Then the image of the cycle map: $$c_n: CH^n(X)\otimes \mbox{\bf Q}_l\to H^{2n}(X,\mbox{\bf Q}_l(n))$$ is a finite dimensional $\mbox{\bf Q}_l$-vector space.\\ {\bf Proof}: There exist an open subset $U$ of $\mbox{Spec}\mbox{\bf Z}[1/l]$, a regular ring $A$ of finite type and smooth over $U$ with fraction field $k$, and a smooth proper scheme ${\cal X}$ over $A$ with generic fibre $X$. There is a cycle map: $$CH^n({\cal X})\otimes \mbox{\bf Q}_l\to H^{2n}({\cal X},\mbox{\bf Q}_l(n))$$ (see e.g. ([S2], Lemma 5.3) or Proposition 2.9 below). Now $H^{2n}({\cal X},\mbox{\bf Q}_l(n))$ is a finite dimensional $\mbox{\bf Q}_l$ vector space (see e.g. [CT-R], Proof of Th\'eor\`eme 1.1, (d)) and since the map $$CH^n({\cal X})\otimes \mbox{\bf Q}_l\to CH^n(X)\otimes\mbox{\bf Q}_l$$ is surjective, we get the proposition.\\ {\bf Corollary 2.6}: With hypotheses as in Proposition 2.5, for any nonnegative integer $i$, the image of the higher $l$-adic Abel-Jacobi map: $$F^iCH^n(X)\otimes\mbox{\bf Q}_l\to H^i(k,H^{2n-i}({\overline X},\mbox{\bf Q}_l(n)))$$ is a finite dimensional $\mbox{\bf Q}_l$ vector space.\\ Proposition 2.5 and Corollary 2.6 are not very refined results since their proofs only use the existence of a smooth proper model of $X$ over an open subset of a model of $k$ over an open subset of Spec$\mbox{\bf Z}$. In order to get finer results, we need to assume the existence of a regular proper model ${\cal Y}$ of $k$ over $\mbox{\bf Z}$ together with a regular proper model of $X$ over ${\cal Y}$. In the rest of this section we assume we have such models. Our philosophy is that the image of the higher Abel-Jacobi map $d_n^i$ lies in the {\it unramified part} of $H^i(k,H^{2n-i}({\overline X},\mbox{\bf Q}_l(n))),$ in the following sense.\\ {\bf Definition 2.7}: Let $k$ be a field which is finitely generated over its prime subfield, $l$ a prime number which is invertible in $k$ and ${\cal Y}$ a regular proper model of $k$ over $\mbox{\bf Z}$. Let $F$ be an $l$-adic sheaf on ${\cal Y}$. We denote by $H^i_{nr}(k,F)^{(l)}$ the $\mbox{\bf Q}_l$-vector space: $$\mbox{Ker}[H^i(k,F)\to \prod_{y\in {\cal Y}^{1\prime}}H^{i+1}_y(A_y,F)].$$ Here ${\cal Y}^{1\prime}$ is the set of codimension one points of ${\cal Y}$ which do not lie above $l$ and $A_y$ is the valuation ring of $y$. By abuse of notation, we also denote by $F$ the sheaves on $k$ and $A_y$ induced by $F$. If $F$ is a locally constant sheaf and $l$ is invertible on ${\cal Y}$, it follows easily from purity that this is the usual definition of unramified cohomology (see [CT-O], for example).\\ Now let $X$ be a smooth projective variety over $k$. Assume that we have a regular proper scheme ${\cal Y}$ over $\mbox{\bf Z}[1/l]$ with function field $k$ and a regular proper scheme ${\cal X}$ together with a morphism: $$f:\,{\cal X}\to {\cal Y}$$ whose generic fibre is $X/k$. We consider the $l$-adic sheaf $F=R^{2n-i}f_*\mbox{\bf Q}_l(n)$, which is locally constant on the smooth locus of $f$, and we can define the unramified cohomology of $k$ with values in this sheaf. Then we have:\\ {\bf Conjecture 2.8}: With notation as above, the image of the higher Abel-Jacobi map $d_n^i$ is contained in $H^i_{nr}(k,F)^{(l)}$. \\ {\bf Remark 2.8.1}: In the definition of unramified cohomology and the conjecture, we have ignored points of ${\cal Y}$ lying above $l$, not because we don't think these are important, but rather because we are unable to formulate an intelligent conjecture on the image of the Abel-Jacobi map at these places. What is needed is a higher dimensional analogue of the finite part, exponential part and geometric part of Bloch-Kato [BK]. This will involve $l$-adic cohomology over a complete discretely valued field with nonperfect residue field of characteristic $l$, and this is not very well developed at the moment.\\ Another way to formulate Conjecture 2.8 is that the filtration on the Chow groups defined in 2.1 should extend to ${\cal X}$. We define this filtration using the following result:\\ {\bf Proposition 2.9}: Let $R$ be a ring which is of type (i), (ii) or (iii) below:\\ (i) of finite type over $\mbox{\bf Z}$\\ (ii) a localization of a ring as in (i)\\ (iii) the henselization of a ring as in (ii)\\ Let ${\cal X}$ be a regular scheme of finite type over $R$ and assume that $l$ is invertible on ${\cal X}$. Then there is a cycle map: $$c_n: CH^n({\cal X})\to H^{2n}({\cal X},\mbox{\bf Q}_l(n)).$$ {\bf Proof}: Thomason [T] has proved a purity theorem for $\mbox{\bf Q}_l$-cohomology of such schemes, and then the existence of a cycle map may be proven by copying the proof in e.g. ([M1], Chapter VI, \S 6).\\ {\bf Definition 2.10}: Let $f:\,{\cal X}\to {\cal Y}$ be a proper morphism of schemes of finite type over a ring $R$ as in 2.9 and $l$ a prime number invertible on ${\cal Y}$. Then we have the Leray spectral sequence for continuous cohomology ([J1], 3.10): $$H^p({\cal Y},R^qf_*\mbox{\bf Q}_l(n))\Longrightarrow H^{p+q}({\cal X},\mbox{\bf Q}_l(n)).$$ This defines a filtration on $H^{2n}({\cal X},\mbox{\bf Q}_l(n))$ which we can pull back to\newline $CH^n({\cal X})\otimes \mbox{\bf Q}$ via $c_n$. We denote the resulting filtration $F^{\bullet}CH^n({\cal X})\otimes \mbox{\bf Q}$. Of course, this filtration may depend on ${\cal Y}$, but this will be fixed in what follows.\\ The following is a properness conjecture for the filtration on a regular proper scheme over a discrete valuation ring.\\ {\bf Conjecture 2.11}: Let $A$ be a discrete valuation ring as in 2.9(ii) or (iii) with fraction field $k$ and $l$ a prime number invertible on $A$. Let ${\cal X}$ be a regular proper scheme of finite type over $A$ with generic fibre $X$. Then the natural map: $$F^iCH^n({\cal X})\otimes \mbox{\bf Q}\to F^iCH^n(X)\otimes \mbox{\bf Q}$$ is surjective.\\ {\bf Proposition 2.12}: With notation as in 2.11, assume that $A$ is Henselian and ${\cal X}$ is smooth over $A$. Then Conjecture 2.11 is true for ${\cal X}$.\\ {\bf Proof}: We need a lemma.\\ {\bf Lemma 2.13}: Let $A$ be a Henselian discrete valuation ring with fraction field $k$ and $(F_m)$ a projective system of locally constant $l$-primary sheaves on $A$. Then the natural map on continuous cohomology groups: $$H^i(A,(F_m))\to H^i(k,(F_m))$$ is injective.\\ {\bf Proof}: Let $s$ be the closed point of $S=\mbox{Spec}A$. For each $m$ we have an exact sequence: $$(*)...\to H^{i-1}(k,F_m)\stackrel{f}{\to} H^{i}_{s}(A,F_m)\to H^i(A,F_m)\to H^i(k,F_m)\to ...$$ Now the second group in this sequence is isomorphic to $H^{i-2}(s,F_m(-1))$ by purity for locally constant sheaves on $A$. The map $f$ is split surjective with a splitting being given by the map ($\pi$ a uniformizing parameter of $k$): $$H^{i-2}(s,F_m(-1))\cong H^{i-2}(A,F_m(-1))\stackrel{\cup\pi}{\to}H^{i-1}(k,F_m).$$ Hence the map obtained from $f$ above by passing to the inverse limit over $m$ is surjective as well. A similar statement holds for the case of the map $${\lim_{\stackrel{\leftarrow}{m}}}^1H^{i-1}(k,F_m)\to {\lim_{\stackrel{\leftarrow}{m}}}^1H^{i-2}(s,F_m(-1)).$$ Then by the property (i) of continuous cohomology discussed in \S 1 above, the map: $$H^{i-1}(k,(F_m))\to H^{i-2}(s,(F_m))$$ is surjective. Now there is a localization sequence like (*) above for continuous cohomology with support and a purity theorem for the cohomology with support in $s$ ([J1], 3.8 and same argument as in Theorem 3.17 of that paper). We conclude that the map $$H^i(A,(F_m))\to H^i(k,(F_m))$$ is injective. This completes the proof of the lemma.\\ Returning to the proof of the proposition, since ${\cal X}$ is smooth over $A$, the sheaves $F_{i,m}:=R^if_*\mbox{\bf Z}/l^m(n)$ are locally constant on $A$ (smooth and proper base change theorem), so the maps $$H^j(A,(F_{i,m}))\to H^j(k,(F_{i,m}))$$ are injective. Then the Leray spectral sequence $$H^p(A,R^qf_*(F_m))\Longrightarrow H^{p+q}({\cal X},(F_m))$$ degenerates at $E_2$ because it does so for $X/k$ (Theorem 1.1). Now an easy induction starting from the fact that the map $$CH^n({\cal X})\to CH^n(X)$$ is surjective proves the proposition.\\ {\bf Remarks 2.14}: (i) When ${\cal X}$ is not smooth over $A$, it seems much more difficult to prove properness of the filtration. In fact, it is not even known if the map $$F^1CH^n({\cal X})\otimes \mbox{\bf Q}\to F^1CH^n(X)\otimes \mbox{\bf Q}$$ is surjective. In Proposition 3.2 below we prove this when $n=\mbox{dim}X$, $A$ is Henselian and $k$ with finite or separably closed residue field.\\ (ii) It should be possible to prove Proposition 2.12 for any discrete valuation ring $A$ using results of Gillet [Gi]. His method may be used to show that if $F$ is a locally constant sheaf on $A$ with finite stalks, then the natural map: $$H^i(A,F)\to H^i(k,F)$$ is injective. We need this for continuous cohomology. In some sense, our result for Henselian rings is not what we really want because the fraction field $k$ is usually not finitely generated over the prime field. However, as will be shown in \S 3, in practice the Henselian case is very helpful.\\ \section{Reductions in the proof of the Main Theorem} In this section we show how to reduce Theorem 0.1 to a Hasse principle for Galois cohomology which has been proved in some cases by Jannsen. In order to do this, we need some local considerations and, unfortunately, a hypothesis:\\ {\bf Hypothesis 3.1}: Let $A$ be a discrete valuation ring with fraction field $k$ and $X$ a smooth projective variety over $k$. \\ (Resolution of singularities) $X$ has a regular proper model ${\cal X}$ over $A$. \\ Hypothesis 3.1 will be needed in order to have a cycle map for ${\cal X}$ (see Proposition 2.9) and for Proposition 3.2 below.\\ The following result may be found in ([B1], Lemma 2.2.6(b)). We give some of the details of the proof outlined there as well as the slight generalization we require.\\ {\bf Proposition 3.2}: Let $A$ be a Henselian discrete valuation ring with fraction field $k$ and residue field ${\bf F}$ which is of characteristic different from $l$. Assume that ${\bf F}$ is either finite or separably closed. Let $X$ be a smooth, projective, geometrically connected variety of dimension $n$ over $k$ and $\xi$ an element of $CH^n(X)\otimes \mbox{\bf Q}$ of degree 0. Then there exists a finite extension $L/k$ with valuation ring $B$, a regular proper model ${\cal X}$ of $X_L$ over $B$ and a cycle in $CH^n({\cal X})\otimes \mbox{\bf Q}_l$ which restricts to $\xi_L$ in $CH^n(X_L)\otimes\mbox{\bf Q}_l$ and whose cohomology class in $H^{2n}({\cal X},\mbox{\bf Q}_l(n))$ is zero.\\ {\bf Proof}: If $\xi$ is a zero-cycle of degree zero on $X$, its cohomology class is trivial in $H^{2n}({\overline X},\mbox{\bf Q}_l(n)).$ We claim that its cohomology class is zero in $H^{2n}(X,\mbox{\bf Q}_l(n))$. To see this, note that by Lemma 3.3 below, the group $A_0(X)$ is an extension of a torsion group by a group which is divisible prime to the residue charactersitic of $k$. Now the group $H^{2n}(X,\mbox{\bf Z}_l(n))$ contains no nontrivial $l$-divisible subgroup ([J1], Cor. 4.9). Thus the restriction of the integral cycle map to cycles of degree zero: $$A_0(X)\to H^{2n}(X,\mbox{\bf Z}_l(n))$$ has torsion image, so the class of $\xi$ in $H^{2n}(X,\mbox{\bf Q}_l(n))$ is trivial, as claimed.\\ Assume for the moment that $X$ is a curve. Let ${\cal X}$ be a regular proper model of $X$ over $A$ with special fibre $Y$. There is a commutative diagram with exact top row: $$\matrix{CH^0(Y)\otimes\mbox{\bf Q}_l&\to&CH^1({\cal X})\otimes\mbox{\bf Q}_l&\to&CH^1(X)\otimes\mbox{\bf Q}_l&\to&0\cr &&\downarrow&&\downarrow\cr &&H^2({\cal X},\mbox{\bf Q}_l(1))&\to &H^2(X,\mbox{\bf Q}_l(1))}.$$ The vertical maps are cycle maps and the top sequence is the exact sequence for the Chow groups of a scheme, an open subscheme and its complement. By Lemma 3.4 below, we can identify $H^2({\cal X},\mbox{\bf Q}_l(1))$ with\linebreak $(CH^0(Y_{{\overline {\bf F}}})\otimes \mbox{\bf Q}_l)^{*\,Gal({\overline {\bf F}}/{\bf F})}$, and the cycle map $$CH^1({\cal X})\otimes \mbox{\bf Q}\to H^2({\cal X},\mbox{\bf Q}_l(1))$$ constructed in Lemma 2.9 is given by intersecting with components of the geometric special fibre. Here * denotes $\mbox{\bf Q}_l$-linear dual. Now the intersection pairing on $CH^0(Y)/[Y]$ in ${\cal X}$ is nondegenerate, where $[Y]$ is the class of the whole special fibre ([L], Ch. 12, \S 4), and it induces an isomorphism: $$CH^0(Y)/[Y]\otimes \mbox{\bf Q}_l\to [CH^0(Y_{{\overline {\bf F}}})^0\otimes \mbox{\bf Q}_l]^{*\,Gal({\overline {\bf F}}/{\bf F})},$$ where the superscript 0 means the kernel of the degree map. If $\xi$ is a zero-cycle of degree zero on $X$ and $\tilde{\xi}$ is any lifting of $\xi$ to a codimension 1 cycle on ${\cal X}$, then the class of $\tilde{\xi}$ in $CH^0(Y)$ is of total degree zero, where total degree means the sum of the degrees on each irreducible component of $Y$. Hence we can modify $\tilde{\xi}$ by an element of $CH^0(Y)\otimes \mbox{\bf Q}_l$ so that its class in\linebreak $[CH^0(Y_{\overline{{\bf F}}})\otimes \mbox{\bf Q}_l]^*$ is trivial. This proves the lemma in the case of a curve. For the higher dimensional case, let $\xi$ be an element of $CH^n(X)\otimes \mbox{\bf Q}$ of degree zero. Then passing to a finite extension of $k$ if necessary (if $k$ is not perfect), we may find a smooth, projective, geometrically connected curve $C$ passing through $\xi$ ([C], [AK]). Let $B$ be the integral closure of $A$ in $L$ and let ${\cal C}$ be a regular proper model of $C$ over $B$. Then by blowing up, the rational map ${\cal C}\to {\cal X}$ may be resolved to get a morphism: $$f: \tilde{{\cal C}}\to {\cal X},$$ with $\tilde{{\cal C}}$ regular. Let $Z, Y$ be the special fibres of $\tilde{{\cal C}}$ and ${\cal X}$, respectively. Then the following diagram commutes: $$\matrix{CH^1(\tilde{{\cal C}})\otimes\mbox{\bf Q}_l&\to& CH^n({\cal X})\otimes\mbox{\bf Q}_l\cr \downarrow&&\downarrow\cr [CH^0(Z_{\overline{{\bf F}}})\otimes \mbox{\bf Q}_l]^*&\to&[CH^0(Y_{\overline{{\bf F}}})\otimes \mbox{\bf Q}_l]^*}.$$ Here the top horizontal map is the push-forward map and the bottom horizontal map is defined as the dual of the pullback map on irreducible components. By the first part of the proof, we can find a cycle $\tilde{\xi}$ in $CH^1(\tilde{{\cal C}})\otimes \mbox{\bf Q}_l$ whose image in $CH^1(C)\otimes\mbox{\bf Q}_l$ is equal to $\xi$ and whose class is trivial in $CH^0(Z)\otimes \mbox{\bf Q}_l$. Pushing $\tilde{\xi}$ to $CH^n({\cal X})\otimes \mbox{\bf Q}_l$, we get our desired cycle. This completes the proof of the proposition. \\ We now prove the two lemmas which were used in the proof of Proposition 3.2.\\ {\bf Lemma 3.3}: Let $k$ be a Henselian discrete valuation field. Assume that the residue field ${\bf F}$ is finite or separably closed. Let $X$ be a smooth projective variety over $k$. Then the group $A_0(X)$ is an extension of a torsion group by a group which is divisible prime to the characteristic of ${\bf F}$.\\ {\bf Proof}: As in the latter part of the proof of Proposition 3.2, given any zero-cycle on $X$, after passing to a finite extension of $k$ if necessary, we may find a smooth, projective, geometrically connected curve going through its support. By an easy norm argument, we see that it suffices to prove the result for the case of curves. This is implied by the following fact applied to the Jacobian of the curve:\\ {\bf Fact}: Let $k$ be as in the hypothesis of Lemma 3.3 and $A$ an abelian variety over $k$. Then the group $A(k)$ is an extension of a finite group by a group which is divisible prime to the characteristic of ${\bf F}$. \\ This is well-known if the residue field is finite. To see it when the residue field is separably closed, let ${\cal A}$ be the N\'eron model of $A$ over the valuation ring $R$ of $k$ with special fibre $\widetilde{{\cal A}}$ (see [BLR], Ch. 1, \S 1.3, Cor. 2 for the existence of ${\cal A}$). Let $A_0(k)$ be the subgroup of $A(k)$ consisting of points which specialize to elements in the connected component of identity $\widetilde{{\cal A}}_0$ of $\widetilde{{\cal A}}$. Then the quotient $A(k)/A_0(k)$ is finite. The group $\widetilde{{\cal A}}_0({\bf F})$ is the group of points of a connected algebraic group over the separably closed field ${\bf F}$, and hence is divisible prime to the characteristic of ${\bf F}$. Let $A_1(k)$ be the kernel of the surjective reduction map: $$A_0(k)\to \widetilde{{\cal A}}_0({\bf F}).$$ Then $A_1(k)$ is a pro-$p$ group , where $p$ is the characteristic of ${\bf F}$. Hence it is uniquely divisible prime to $p$. Putting everything together, we get the fact.\\ {\bf Lemma 3.4}: Let $A$ be a Henselian local ring with residue field ${\bf F}$ which is finite or separably closed. Let ${\cal X}$ a regular proper flat integral scheme of dimension $n$ over $A$ with special fibre $Y$. Let $l$ be a prime number different from the characteristic of ${\bf F}$. Then $$(\dagger)\:\: H^{2n}({\cal X},\mbox{\bf Q}_l(n))\cong [CH^0(Y_{{\overline {\bf F}}})\otimes \mbox{\bf Q}_l]^{*\,Gal({\overline {\bf F}}/{\bf F})},$$ where * denotes $\mbox{\bf Q}_l$-dual. The map $$CH^n({\cal X})\to [CH^0(Y_{{\overline {\bf F}}})\otimes \mbox{\bf Q}_l]^*$$ obtained by composing this isomorphism with the cycle map in Proposition 2.9 is given by intersecting with components of the geometric special fibre.\\ {\bf Proof}: This is known, but I could not find a good reference. By the proper base change theorem, we have: $$H^{2n}({\cal X},\mbox{\bf Q}_l(n))=H^{2n}(Y,\mbox{\bf Q}_l(n)).$$ This last group may be easily calculated as follows. Since the \'etale cohomology with $\mbox{\bf Q}_l$-coefficients of $Y$ and $Y_{red}$ is the same, we may assume that $Y$ is reduced. Let $Y_{sing}$ be the singular locus of $Y$. Then there is an exact sequence for cohomology with compact support: $$...\to H_c^{2n-1}(Y_{sing},\mbox{\bf Q}_l(n))\to H_c^{2n}(Y- Y_{sing},\mbox{\bf Q}_l(n))\to H_c^{2n}(Y,\mbox{\bf Q}_l(n))\to H_c^{2n}(Y_{sing},\mbox{\bf Q}_l(n)).$$ First assume that ${\bf F}$ is separably closed. Then since $Y_{sing}$ is a proper closed subset of $Y$, the groups on the ends are zero. By Poincar\'e duality for $Y-Y_{sing}$, we have: $$H_c^{2n}(Y-Y_{sing},\mbox{\bf Q}_l(n))\cong H^0(Y- Y_{sing},\mbox{\bf Q}_l)^*,$$ and this last group is easily seen to be isomorphic to the $\mbox{\bf Q}_l$-dual of $CH^0(Y)\otimes \mbox{\bf Q}_l.$ If ${\bf F}$ is finite, then we have an exact sequence: $$0\to H^1({\bf F},H^{2n-1}(Y_{{\overline {\bf F}}},\mbox{\bf Q}_l(n)))\to H^{2n}(Y,\mbox{\bf Q}_l(n))\to H^{2n}(Y_{{\overline {\bf F}}},\mbox{\bf Q}_l(n))^{Gal({\overline{\bf F}}/{\bf F})}\to 0.$$ By Deligne's theorem ([D2], Theorem 3.3.1), the weights of the geometric Frobenius acting on $H^{2n-1}(Y_{{\overline{\bf F}}},\mbox{\bf Q}_l(n))$ are $\leq -1$ and hence are nonzero. This implies that the group on the left is zero. Using what we have just proved for separably closed residue fields, we get the identification ($\dagger$). We leave the rest to the reader.\\ {\bf Remarks 3.5}: (i) If the field $k$ is perfect, then there is no need to pass to a finite extension in Proposition 3.2.\\ (ii) It is expected that, at least after tensoring with $\mbox{\bf Q}$, a cycle of arbitrary codimension which is homologically equivalent to zero on the generic fibre $X$ can be lifted to one which is homologically equivalent to zero on any regular proper model ${\cal X}$. \\ {\bf Proposition 3.6}: Let $k$ be a function field in one variable over a finite field, $X$ a smooth, projective variety of dimension $n$ over $k$ and $l$ a prime number different from char. $k$. Put $$M=H^{2n-2}(\overline X,\mbox{\bf Q}(n)).$$ Let $S$ be a finite set of places of $k$ including the bad reduction places of $X$. Then if $z$ is a zero-cycle in the Albanese kernel $T(X)\otimes \mbox{\bf Q}$, $d_n^2(z)$ is contained in the image of $$Ker[H^2(G_S,M)\to \bigoplus_{v\in S}^{} H^2(k_v,M)]$$ in $H^2(k,M)$, where $d_n^2$ is the map defined in Section 2.\\ {\bf Proof}: Let $U=Spec \: {\cal O}_{k,S}$. Then $X$ extends to a smooth, proper scheme ${\cal X}_U$ over $U$ and we can define a cycle map: $$CH^n({\cal X}_U)\to H^{2n}({\cal X}_U,\mbox{\bf Q}_l(n)).$$ Consider the Leray spectral sequence for the morphism $f:\,{\cal X}_U\to U$: $$H^p(U,R^qf_*\mbox{\bf Q}_l(n)))\Longrightarrow H^{p+q}({\cal X}_U,\mbox{\bf Q}_l(n)).$$ Here we denote by $R^qf_*\mbox{\bf Q}_l(n))$ the $l$-adic sheaf obtained from taking the higher direct images with finite coefficients. Given $z$ we can extend it to a cycle on ${\cal X}_U$ of codimension $n$ which is of degree zero on each fibre. This gives a cohomology class $\cal Z$ in $$F_1=Ker[H^{2n}({\cal X}_U,\mbox{\bf Q}_l(n))\to H^0(U,R^{2n}f_*\mbox{\bf Q}_l(n))].$$ From the spectral sequence there is a map: $$F_1\to H^1(U,R^{2n-1}f_*\mbox{\bf Q}_l(n)).$$ Claim: $\cal Z$ is zero in $H^1(U,R^{2n-1}f_*\mbox{\bf Q}_l(n)))$. To prove the claim, we note that since $z$ is in the Albanese kernel, its image in $H^1(k,H^{2n-1}(\overline X,\mbox{\bf Q}_l(n)))$ is zero. Then our claim follows from the following lemma:\\ {\bf Lemma 3.7}: The map: $$H^1(U,R^{2n-1}f_*\mbox{\bf Q}_l(n)))\to H^1(k,H^{2n-1}(\overline X,\mbox{\bf Q}_l(n)))$$ is injective.\\ {\bf Proof}: By the smooth and proper base change theorem, the $l$-adic sheaf ${\cal F}=R^{2n-1}f_*\mbox{\bf Q}_l(n)$ is locally constant on $U$. Hence $H^1_v(U,{\cal F})$=0 for any closed point $v$ of $U$ and this gives the injectivity of the map.\\ From the spectral sequence and the vanishing of the cohomology class $\cal Z$ in $H^1(U,R^{2n-1}f_*\mbox{\bf Q}_l(n))$, we see that this class comes from $H^2(U,R^{2n-2}f_*\mbox{\bf Q}_l(n))=H^2(G_S,H^{2n-2}(\overline X,\mbox{\bf Q}_l(n)))$. Our task is to show that this cohomology class vanishes in $H^2(k_v,H^{2n-2}(\overline X,\mbox{\bf Q}_l(n)))$ for all $v$ in $S$.\\ Let ${\cal X}_v$ be a regular proper model of $X_v$ over ${\cal O}_v$. By Proposition 3.2, after a finite extension of $L/k_v$, we can extend our original cycle $z$ on $X_v$ to an element of $CH^n({\cal X}_v)\otimes \mbox{\bf Q}_l$ in such a way that its cohomology class lies in $$H^{2n}({\cal X}_v,\mbox{\bf Q}_l(n))^0=Ker[H^{2n}({\cal X}_v,\mbox{\bf Q}_l(n)\to H^0({\cal O}_v,R^{2n}f_*\mbox{\bf Q}_l(n))].$$ Since the natural map: $$H^2(k_v,M)\to H^2(L,M)$$ is injective for any $\mbox{\bf Q}_l$-vector space $M$, it suffices to prove that our cohomology class vanishes in $H^2(L,M)$. {\bf Lemma 3.8}: $$H^{2n}({\cal X}_v,\mbox{\bf Q}_l(n))^0=0.$$ This will prove the proposition.\\ {\bf Proof}: From the Leray spectral sequence for $f:\,{\cal X}_v\to \mbox{Spec}\,{\cal O} _v$, we get the exact sequence: $$0\to H^2({\cal O}_v,R^{2n-2}f_*\mbox{\bf Q}_l(n)))\to H^{2n}({\cal X}_v,\mbox{\bf Q}_l(n))^0\to H^1({\cal O}_v,R^{2n-1}f_*\mbox{\bf Q}_l(n)))\to 0.$$ We claim that all three terms of this exact sequence are zero. The group on the left is zero because ${\cal O}_v$ has cohomological dimension one for torsion modules, hence for $l$-adic sheaves. As for the group on the right, two applications of the proper base change theorem give that it is isomorphic to $H^1({\bf F},H^{2n-1}(\overline Y,\mbox{\bf Q}_l(n)))$, where $\overline Y$ is the geometric special fibre of ${\cal X}_v$. By Deligne's theorem ([D2], 3.3.1), the geometric Frobenius acts on $H^{2n-1}(\overline Y,\mbox{\bf Q}_l(n))$ with weights which are $\leq -1$, hence nonzero. This shows that the group on the right vanishes and proves the lemma.\\ \section{A Hasse Principle and Completion of the Proof} In this section we prove a ``Hasse'' principle in the function field case. In the case where Frobenius acts semi-simply on $H^{2n-2}({\overline X},\mbox{\bf Q}_l(n))$, this is proven in Jannsen's paper ([J2], Theorem 4 and Remark 7). The more general result given here was shown to me by Jannsen, and I am grateful to him for allowing me to give his proof in this paper. To avoid copying his earlier proof, we assume some familiarity with his argument.\\ {\bf Theorem 4.1} (Jannsen): Let $k$ be a function field in one variable over a finite field ${\bf F}$ of characteristic $p$ and $l$ a prime number different from $p$. Let $V$ be a $\mbox{\bf Q}_l$-linear Galois representation of $G=Gal(k_{sep}/k)$ which is pure of weight -2, and let $S$ be a finite nonempty set of places of $k$ including those for which the restriction to the decomposition group $G_v$ is ramified. Then the natural map: $$H^2(G_S,V)\to \bigoplus_{v\in S}H^2(G_v,V)$$ \noindent is injective.\\ \noindent {\bf Proof}: Let $Y$ be a smooth, projective model of $k$ and let $U=Y-S, \overline U=U\times_{{\bf F}}{\overline {{\bf F}}}$ and $j:\,U\to Y$ the inclusion. Then by a theorem of Deligne ([D2], 3.4.1(iii)), $V$ is a semi-simple $\pi_1(\overline U)$-representation. Then we can write $V=V_1\bigoplus V_2$, where $\pi_1(\overline U)$ acts trivially on $V_2$ and $V_1^{\pi_1(\overline U)}=V_{1\:{\pi_1(\overline U)}}=0$. By Jannsen's proof in [J2], the result holds for $V_1$, so we prove it for $V_2$. Let $F$ be the smooth $\mbox{\bf Q}_l$-sheaf corresponding to $V_2$. Then we have the exact sequence of cohomology groups: $$(*)\,...H^2_S(Y,j_*F)\stackrel{g}{\to}H^2(Y,j_*F)\to H^2(U,F)\stackrel{f}{\to} H^3_S(Y,j_*F)\to ...$$ By excision and the fact that $H^2({\cal O}_v,j_*F)=H^2({\bf F},(j_*F)_{{\bf F}})=0$, we have: $$H^2(k_v,F)=H^3_v({\cal O}_v,j_*F)$$ for all $v\in S$, and the map $f$ is the localization map. Thus an element of the kernel of the localization map comes from $H^2(Y,j_*F).$ It will then suffice to show that the map $g$ in the exact sequence (*) above is surjective. Let $\Gamma =Gal({\overline {\bf F}}/{\bf F})$ and consider the exact sequence of $\Gamma$-modules: $$0\to H^1(\Gamma,H^1({\overline Y},j_*F))\to H^2(Y,j_*F)\to H^2({\overline Y},j_*F)^{\Gamma}\to 0.$$ By the fundamental theorem of Deligne ([D2], 3.2.3), the $\Gamma$-module $H^1({\overline Y},j_*F)$ is pure of weight -1, hence the left group is zero. Consider the following diagram: $$\matrix{H^2_S(Y,j_*F)&\to&H^2_{\overline S}({\overline Y},j_*F)^{\Gamma}\cr \downarrow&&\downarrow\cr H^2(Y,j_*F)&\to&H^2({\overline Y},j_*F))^{\Gamma}}.$$ We have just shown that the bottom horizontal map is an isomorphism and our goal is to show that the left vertical arrow is surjective. Since the top horizontal map is surjective, it will be enough to prove the surjectivity of the right vertical map. Let $\tilde{{\cal O}_v}$ denote the strict henselization of ${\cal O}_v$ and $\tilde{k_v}$ its fraction field. Then we have: $$ H^1(\tilde{k_v},F)\cong H^2_v(\tilde{{\cal O}_v},j_*F).$$ Now the pro-$l$-part of the absolute Galois group of $\tilde{k}$ is isomorphic to $\mbox{\bf Z}_l(1)$ as a $\Gamma$-module. Then by Pontryagin duality and the fact that $\pi_1({\overline U})$ is assumed to act trivially on $V_2$, we have $$H^1(\tilde{k_v},F)^{\Gamma}=(\hat{V_2}(1)_{\Gamma})^*,$$ where * denotes $\mbox{\bf Q}_l$-linear dual and $\hat{V_2}$ denotes the dual representation. But by Poincar\'e duality, we have: $$H^2({\overline Y},j_*F)^{\Gamma}\cong (\hat{V_2}(1)_{\Gamma})^*,$$ and since $S$ is nonempty, we obviously have injectivity of the dual map: $$(\hat{V_2}(1)_{\Gamma})\to [\bigoplus_{v\in S}\hat{V_2}(1)_{\Gamma}]^.$$ This completes the proof of the Theorem. Now we can prove the main theorem of this paper:\\ {\bf Theorem 4.2}: Let $X$ be a smooth, projective, geometrically connected variety of dimension $n$ over a function field $k$ in one variable over a finite field of characteristic $p$, and $l$ a prime number different from $p$. Assume Hypothesis 3.1 above. Then the map $d_n^2$ of section 1 is zero. \\ {\bf Proof}: Combine Proposition 3.6 and Theorem 4.1, taking $V=H^{2n-2}({\overline X},\mbox{\bf Q}_l(n))$.\\ {\bf Theorem 4.3}: Let $X$ be a smooth, projective, geometrically connected surface over a function field in one variable over a finite field of characteristic $p\geq 5$. Then the map $d_2^2$ is zero.\\ {\bf Proof}: Abhyankar [A] has proved resolution of singularities for 3-folds over an algebraically closed field of characteristic greater than or equal to 5. Given $X$, we can pass to a finite extension over which it has a smooth proper model. Then the kernel of $d_2^2$ will be killed by the degree of this extension, hence it is zero. \newpage \section*{Appendix} In this appendix we prove a technical lemma which shows that the Abel-Jacobi map as defined via the Hochschild-Serre spectral sequence agrees with the map defined by Kummer theory on $Alb(X)$. This fact is probably known to several people, but I have been unable to find a reference.\\ {\bf Lemma}: Let $X$ be a smooth projective geometrically connected variety of dimension $n$ over a field $k$, $l$ a prime number different from $\mbox{char.}k$. Consider the map $d_n^1$ defined in \S 1 above and the map: $$f_n^1: A_0(X)\to H^1(k,H^{2n-1}({\overline X},\mbox{\bf Q}_l(n)))$$ defined as the composite of the Albanese map $$\alpha: A_0(X)\to Alb(X)$$ and the map $$Alb(X)\to H^1(k,H^{2n-1}({\overline X},\mbox{\bf Q}_l(n)))$$ obtained from Kummer theory on $Alb(X)$. Then $d_n^1=f_n^1$.\\ {\bf Proof}: First assume that $X$ is a curve. The group $A_0(X)$ is generated by elements of the form $\mbox{cor}_{L/k}(P-Q)$, where $L$ is a finite extension of $k, P$ and $Q$ are rational points of $X_L$ and $$\mbox{cor}:\,A_0(X_L)\to A_0(X)$$ denotes corestriction. Then it suffices to prove that for any such $L/k$ and any two $L$-rational points $P,Q$ of $X$, the two maps above give the same element in $H^1(L,H^{1}({\overline X},\mbox{\bf Z}_l(1)))$. We suppress $L$ in what follows. Let $z=P-Q$, let $Z$ be its support and $U=X-Z$. Taking cohomology with support in ${\overline Z}$, we easily get the exact sequence: $$(*):\, 0\to H^1({\overline X},\mbox{\bf Z}/l^m(1))\to H^1({\overline U},\mbox{\bf Z}/l^m(1))\to \mbox{\bf Z}/l^m\to 0,$$ where the last group is generated by the class of $z$. Jannsen has shown ([J5], Lemma 9.4) that the extension given by (*) gives the element of $H^1(k,H^1({\overline X},\mbox{\bf Z}/l^m(1))$ which is associated to $P-Q$ via the map $d_1^1$. Now consider the Kummer sequence for $Pic^0({\overline X})$: $$0\to H^1({\overline X},\mbox{\bf Z}/l^m(1))\to Pic^0({\overline X})\stackrel{l^m}{\to}Pic^0({\overline X})\to 0,$$ where the group on the left is identified with the $l^m$-torsion of $Pic^0({\overline X})$. As is well-known, the boundary map: $$Pic^0({\overline X})^G\to H^1(k,H^1({\overline X},\mbox{\bf Z}/l^m(1)))$$ in the $G$-cohomology of this sequence may be may be obtained by pulling back the Kummer sequence by the $G$-map $$\mbox{\bf Z}\to Pic^0({\overline X})$$ corresponding to an element $y$ of $Pic^0({\overline X})^G$. One easily sees that the pullback extension is: $$(**):\: 0\to H^1({\overline X},\mbox{\bf Z}/l^m(1))\to D\to \mbox{\bf Z}\to 0,$$ where $D=\{x\in Pic^0({\overline X}): l^mx=ny\, \mbox{for some}\, n\in \mbox{\bf Z}\}$. We may define a $G$-map $$D/l^m\to H^1({\overline U},\mbox{\bf Z}/l^m(1))$$ as follows: Let $x\in D$, let ${\cal L}$ be the isomorphism class of line bundle associated to $x$ and ${\cal M}$ the isomorphism class of line bundle associated to $y$. Choose an isomorphism $${\cal L}^{\otimes l^m}\to {\cal M}^{\otimes n}.$$ Such an isomorphism is unique modulo $l^m$, since if we had two such, then we would get an isomorphism from ${\cal O}_{{\overline X}}$ to itself. Since ${\overline X}$ is proper, this is given by an element of ${\overline k}^*$, and hence must be the identity modulo $l^m$. But this means that the two isomorphisms were the same modulo $l^m$. Since the class of $y$ is supported on ${\overline Z}$, the restriction of this isomorphism to ${\overline U}$ gives a trivialization: $${\cal L}^{\otimes l^m}\cong {\cal O}_{\overline U},$$ and this data defines in a canonical way an element of $H^1({\overline U},\mbox{\bf Z}/l^m(1))$. It is easy to verify that via this map, the two extensions (*) and (**) give the same element of $H^1(k,H^1({\overline X},\mbox{\bf Z}/l^m(1)))$. Note that for this argument, we have used $\mbox{\bf Z}_l$-coefficients. In the case of $X$ of arbitrary dimension, Poincar\'e duality and the Weil pairing give an identification: $$H^{2n-1}({\overline X},\mbox{\bf Q}_l(n))=V_l(Alb({\overline X})).$$ Now given a zero-cycle $z$, if $k$ is perfect, we can find a smooth, projective curve $C$ going through it ([AK], [C]). By functoriality properties of the maps in question, we can easily reduce to the case of $C$ which we have done above. If $k$ is not perfect, such a curve $C$ may be found after an inseparable extension of $k$. Then an easy restriction-corestriction argument shows that the maps are the same. This completes the proof of the lemma. \newpage \section*{References} [A] Abhyankar, S.: {\bf Resolution of Singularities of Embedded Algebraic Surfaces}, Academic Press 1966\\ [AK] Altman, A. and Kleiman, S. {\em Bertini theorems for hypersurface sections containing a subscheme}, Comm. in Algebra {\bf 7} (1979), 775-790\\ [Be1] Beauville, A. {\em Quelques remarques sur la transformation de Fourier dans l'anneau de Chow d'une vari\'et\'e ab\'elienne}, in {\bf Algebraic Geometry: Tokyo/Kyoto 1982}, Lecture Notes in Mathematics 1016, Springer 1983\\ [Be2] Beauville, A. {\em Sur l'anneau de Chow d'une vari\'et\'e ab\'elienne}, Math. Ann. {\bf 273} (1986), 647-651\\ [B1] Beilinson, A. {\em Height pairing on algebraic cycles}, in {\bf Current Trends in Arithmetical Algebraic Geometry}, K. Ribet editor, Contemporary Mathematics Volume 67, American Mathematical Society 1987\\ [B2] Beilinson, A. {\em Absolute Hodge cohomology}, in {\bf Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory}, S. Bloch, R.K. Dennis, E.M. Friedlander and M. Stein editors, Contemporary Mathematics Volume 55, American Mathematical Society 1986\\ [Bl1] Bloch, S. {\em Algebraic cycles and values of $L$- functions}, J. f\"ur die Reine und Ang. Math. {\bf 350} (1984), 94-108\\ [Bl2] Bloch, S. {\em $K_2$ of Artinian $\mbox{\bf Q}$-algebras with application to algebraic cycles}, Communications in Algebra {\bf 3} (1975), 405-428\\ [Bl3] Bloch, S. {\em Some elementary theorems about algebraic cycles on abelian varieties}, Inventiones Math. {\bf 37} (1976), 215-228\\ [Bl4] Bloch, S. {\em Lectures on Algebraic Cycles}, Duke Univ. Press, 1980\\ [BK] Bloch, S. and Kato, K. {\em L-functions and Tamagawa numbers of motives}, in {\bf Grothendieck Festschrift}, Volume I, P. Cartier editor, Progress in Mathematics, Birkh\"auser 1990\\ [BLR] Bosch, S., L\"ukebohmert, W. and Raynaud, M. {\bf N\'eron Models}, Springer 1990\\ [CT-O] Colliot-Th\'el\`ene, J.-L. and Ojanguren, M. {\em Vari\'et\'es unirationnelles nonrationnelles: au d\'ela de l'exemple d'Artin-Mumford}, Inventiones Math. {\bf 97} (1989), 141-158\\ [CT-R] Colliot-Th\'el\`ene, J.-L. and Raskind, W. {\em Groupe de Chow de codimension deux des vari\'et\'es d\'efinies sur un corps de nombres: un th\'eor\`eme de finitude pour la torsion}, Inventiones Math. {\bf 105} (1991), 221-245\\ [C] Coray, D. {\em Two remarks on the Bertini theorems}, unpublished typescript, 1980\\ [D1] Deligne, P. {\em Th\'eor\`eme de Lefschetz difficile et crit\`eres de d\'eg\'en\'eresence de suites spectrales} Publ. Math. I.H.E.S. {\bf 35} (1968), 107-126\\ [D2] Deligne, P. {\em La Conjecture de Weil II}, Publ. Math. I.H.E.S. {\bf 52} (1980), 137-252\\ [E] Ekedahl, T. {\em On the adic formalism}, in {\bf Grothendieck Festschrift}, Volume II, P. Cartier editor, Progress in Mathematics, Birkh\"auser 1990\\ [Gi] Gillet, H. {\em Gersten's conjecture for the K-theory with torsion coefficients of a discrete valuation ring}, J. of Algebra {\bf 103} (1986), 377-380\\ [J1] Jannsen, U. {\em Continuous \'etale cohomology}, Math. Annalen {\bf 280} (1988), 207-245\\ [J2] Jannsen, U. {\em On the $l$-adic cohomology of varieties over number fields and its Galois cohomology}, in {\bf Galois Groups over Q}, Y. Ihara, K. Ribet and J.-P. Serre editors, Springer 1989\\ [J3] Jannsen, U. {\em letter to B. Gross}, 1989\\ [J4] Jannsen, U. {\em Motivic sheaves and filtrations on Chow groups}, in {\bf Motives: Seattle 1991}, U. Jannsen, S. Kleiman editors\\ [J5] Jannsen, U. {\bf Mixed Motives and Algebraic K-Theory}, Lecture Notes in Mathematics 1400, Springer 1990\\ [L] Lang, S. {\bf Diophantine Geometry}, Springer 1983\\ [M1] Milne J.S. {\bf Etale Cohomology}, Princeton Mathematical Series Volume 33, Princeton University Press 1980\\ [M2] Milne, J.S. {\em Zero-cycles on algebraic varieties in non-zero characteristic: Rojtman's theorem}, Composition Math. {\bf 47} (1982), 271-287\\ [Mu] Murre, J. {\em On a conjectural filtration on the Chow groups of an algebraic variety}, preprint 1992\\ [N] Nori, M. {\em Algebraic cycles and Hodge-theoretic connectivity}, Inventiones Math. {\bf 105} (1993) 349-373\\ [Ro] Roitman, A.A. {\em The torsion of the group of 0-cycles modulo rational equivalence}, Annals of Math. {\bf 111} (1980), 553-569\\ [S1] Saito, S. {\em Motives and filtrations on Chow groups}, preprint 1993\\ [S2] Saito, S. {\em On the cycle map for torsion algebraic cycles of codimension two}, Inventiones Math. {\bf 106} (1991), 443-460\\ [S3] Saito, S. {\em Remarks on algebraic cycles}, preprint 1993\\ [SH] Saito, H. {\em Generalization of Abel's Theorem and some finiteness properties of zero-cycles on surfaces}, preprint\\ [T] Thomason, R. {\em Absolute Cohomological Purity}, Bull. Math. Soc. France {\bf 112} (1984), 397-406\\ \end{document}

9412

alg-geom/9412017

en

https://arxiv.org/abs/alg-geom/9412017

alg-geom/9412017

V. Batyrev

Victor V. Batyrev and Lev A. Borisov

On Calabi-Yau Complete Intersections in Toric Varieties

27 pages, Latex

We investigate Hodge-theoretic properties of Calabi-Yau complete intersections $V$ of $r$ semi-ample divisors in $d$-dimensional toric Fano varieties having at most Gorenstein singularities. Our main purpose is to show that the combinatorial duality proposed by second author agrees with the duality for Hodge numbers predicted by mirror symmetry. It is expected that the complete verification of mirror symmetry predictions for singular Calabi-Yau varieties $V$ of arbitrary dimension demands considerations of so called {\em string-theoretic Hodge numbers} $h^{p,q}_{\rm st}(V)$. We restrict ourselves to the string-theoretic Hodge numbers $h^{0,q}_{\rm st}(V)$ and $h^{1,q}_{\rm st}(V)$ $(0 \leq q \leq d-r) which coincide with the usual Hodge numbers $h^{0,q}(\widehat{V})$ and $h^{1,q}(\widehat{V})$ of a $MPCP$-desingularization $\widehat{V}$ of $V$.

alg-geom

\section{Introduction} It was conjectured in \cite{bat.dual} that the polar duality for reflexive polyhedra induces the mirror involution for Calabi-Yau hypersurfaces in Gorenstein toric Fano varieties. The second author has proposed a more general duality which conjecturally induces the mirror involution also for Calabi-Yau {\em complete intersections} in Gorenstein toric Fano varieties. The verfication of predictions for Gromov-Witten invariants of Calabi-Yau complete intersections in ordinary and weighted projective spaces \cite{libgober,klemm-theisen} as well as in some toric varieties \cite{batyrev-straten} (see also \cite{berglund-hubsch,ell-str1,ell-str2,hosono}) can be considered as partial confirmations of this generalized mirror construction. If two {\em smooth} $n$-dimensional Calabi-Yau manifolds $V$ and $W$ form a mirror pair, then their Hodge numbers must satisfy the relation \begin{equation} h^{p,q}(V) = h^{n-p,q}(W), \label{h.dual} \end{equation} for all $0 \leq p,q \leq n$. However, the combinatorial involutions which were constructed in \cite{bat.dual,borisov} relate families of {\em singular} Calabi-Yau varieties. If $V$ is a Calabi-Yau complete intersection of semi-ample divisors in a Gorenstein toric Fano variety ${\bf P}$, then there exists always a partial desingularization $\pi \,: \, \widehat{V} \rightarrow V$ ($MPCP$-desingularization of $V$) such that: \begin{itemize} \item $\widehat{V}$ is again a Calabi-Yau complete intersection of semi-ample divisors in a projective toric variety $\widehat{\bf P}$; \item $\widehat{V}$ and $\widehat{\bf P}$ have only Gorenstein terminal abelian quotient singularities. \end{itemize} If $\widehat{V}$ is smooth (this is always the case for $n \leq 3$), we can use $\widehat{V}$ instead of $V$ for verification of the duality (\ref{h.dual}). In general, we have to change the cohomology theory and to consider the so called {\em string-theoretic Hodge numbers} $h_{\rm st}^{p,q}(V)$ for singular $V$. For mirror pair of {\em singular} $n$-dimensional Calabi-Yau varieties $V$ and $W$, we must have the duality for the string-theoretic Hodge numbers: \begin{equation} h^{p,q}_{\rm st} (V) = h^{n-p,q}_{\rm st}(W), \label{h.dual1} \end{equation} for all $0 \leq p,q \leq n$. The main properties of the string-theoretic Hodge numbers were considered in \cite{batyrev.dais}. These numbers satisfy the Poincar\'e duality and we have $h^{p,q}_{\rm st}(V) = h^{p,q}_{\rm st}(\widehat{V})$ for all $0 \leq p,q \leq n$. Moreover, one has the following properties: \begin{prop} Let $h^{p,q}(\widehat{V})$ denote the usual $(p,q)$-Hodge number of $\widehat{V}$. Then {\rm (i)} $h^{p,q}_{\rm st}(V) = h^{p,q}(\widehat{V})$ for all $p =0,1$ and $0 \leq q \leq n$; {\rm (ii)} $h^{p,q}_{\rm st}(V) = h^{p,q}(\widehat{V})$ for all $0 \leq p,q \leq n$ if $\widehat{V}$ is smooth. \label{string} \end{prop} The main purpose of this paper is to verify the duality (\ref{h.dual1}) for string-theoretic $(0,q)$ and $(1,q)$-Hodge numbers of Calabi-Yau complete intersections $V$ and its mirror partner $W$ predicted by the construction in \cite{borisov}. According to \ref{string}(i), it is sufficient to check the analgous duality for the usual Hodge numbers of the corresponding $MPCP$-desingularizations $\widehat{V}$ and $\widehat{W}$. \bigskip In section 2 we remind necessary facts from the theory of toric varieties. Section 3 is devoted to basic properties of complete intersections in toric varieties. In section 4 we explain the relation between Calabi-Yau complete intersection in Gorenstein toric Fano varieties and nef-partitions of reflexive polyhedra. In section 5 we prove the duality (\ref{h.dual1}) for $(0,q)$-Hodge numbers and give explicit formulas for them. It turned out to be not so easy to derive general formulas for $(1,q)$-Hodge numbers and to prove the duality (\ref{h.dual1}) for $(1,q)$-Hodge numbers in full generality. In sections 6 and 7 we give explicit formulas and prove the duality only for the alternative sum of $(1,q)$-Hodge numbers; i.e., for the Euler characteristics of the sheaves of $1$-forms on $\widehat{V}$ and $\widehat{W}$. In section 8, we derive explicit formulas for $(1,q)$-Hodge numbers of Calabi-Yau complete intersections of {\em ample} divisors. Finally, in section 9 we establish the duality (\ref{h.dual1}) for all $(1,q)$-Hodge numbers of Calabi-Yau complete intersections in projective spaces and their mirror partners. \section{Basic notations and statements} Let $M$ and $N = {\rm Hom}(M, {\bf Z})$ be dual free abelian groups of rank $d$, $M_{\bf R}$ and $N_{\bf R}$ the real scalar extensions of $M$ and $N$, $\langle *, * \rangle\; : \; M_{\bf R} \times N_{\bf R} \rightarrow {\bf R}$ the canonical pairing. We consider $M$ (resp. $N$) as the maximal lattice in $M_{\bf R}$ (resp. in $N_{\bf R}$). We denote by ${\bf T}$ the {\em affine algebraic torus} over ${\bf C}$: \[ {\bf T} : = {\rm Spec}\, {\bf C} \lbrack M \rbrack \cong {\bf C} \lbrack X_1^{\pm 1}, \ldots, X_d^{\pm 1} \rbrack. \] By a {\em lattice polyhedron} in $M_{\bf R}$ (resp. in $N_{\bf R}$) we always mean a convex polyhedron of dimension $\leq d$ whose vertices belong to $M$ (resp. in $N$). The {\em relative interior} of $\Delta$ is the set of interior points of $\Delta$ which is considered as a subset of the minimal ${\bf R}$-linear affine subspace containing $\Delta$. For any lattice polyhedron $\Delta$, we denote by $l^*(\Delta)$ the number of lattice points in the relative interior of $\Delta$. We set $b(\Delta) = (-1)^{{\rm dim}\, \Delta}l^*(\Delta)$ and denote by $l(\Delta)$ the number of lattice points in $\Delta$. A lattice polyhedron $\Delta$ defines the {\em projective toric variety} over ${\bf C}$: \[ {\bf P}_{\Delta} = {\rm Proj}\, S_{\Delta}, \] where $S_{\Delta}$ is the monomial subalgebra in the polynomial ring \[ {\bf C} \lbrack X_0, X_1^{\pm 1}, \ldots, X_d^{\pm 1} \rbrack \] spanned as ${\bf C}$-linear space by monomials $X_0^k X_1^{m_1} \cdots X_d^{m_d}$ such that the corresponding lattice point $(m_1, \ldots, m_d) \in M$ belongs to $k\Delta$. We remark that the dimension of ${\bf P}_{\Delta}$ equals ${\rm dim}\, \Delta$. If ${\rm dim}\, \Delta = d$, then ${\bf P}_{\Delta}$ can be considered as a projective compactification of ${\bf T}$. In the latter case, the irreducible components ${\bf D}_1, \ldots, {\bf D}_n$ of ${\bf P}_{\Delta} \setminus {\bf T}$ one-to-one correspond to $(d-1)$-dimensional faces $\Theta_1, \ldots, \Theta_n$ of $\Delta$. We denote by ${\bf e}_1, \ldots, {\bf e}_n$ the primitive lattice points in $N$ which define the linear equations for the affine hyperplanes containing $\Theta_1, \ldots, \Theta_n$ (in other words, ${\bf e}_1, \ldots, {\bf e}_n$ are primitive integral interior normal vectors to faces $\Theta_1, \ldots, \Theta_n$). The is another well-known definition of toric varieties ${\bf P}_{\Delta}$ via the normal fan $\Sigma = \{ \sigma_B \}$ consisting of all rational polyhedral cones \[ \sigma_B = {\bf R}_{\geq 0} {\bf e}_{i_1} + \cdots + {\bf R}_{\geq 0} {\bf e}_{i_s} \subset N_{\bf R} \] corresponding to those subsets $B = \{ i_1, \ldots, i_s \} \subset \{1, \ldots, n\}$ for which the intersection $\Theta_{i_1} \cap \cdots \cap \Theta_{i_s}$ is not empty. In this situation, we also use the notation ${\bf P}_{\Sigma}$ for toric varieties associated with $\Sigma$. It is known that every invertible sheaf ${\cal L}$ on any toric variety ${\bf P}$ admits a ${\bf T}$-linearization. By this reason, we shall consider in this paper only ${\bf T}$-linearized invertible sheaves on toric varieties. A ${\bf T}$-linearization of ${\cal L}$ induces the $M$-grading of the cohomology spaces \[ H^i({\bf P}, {\cal L}) = \bigoplus_{m \in M} H^i({\bf P}, {\cal L})(m). \] The convex hull $\Delta({\cal L})$ of all lattice points $m \in M$ for which $H^0({\bf P}, {\cal L})(m) \neq 0$ will be called the {\em supporting polyhedron for global sections of} ${\cal L}$. Recall the following well-known statement \cite{danilov,oda}: \begin{theo} There is one-to-one correspondence \[ {\cal L} \cong {\cal O}_{\bf P}(a_1 {\bf D}_1 + \cdots + a_n {\bf D}_n) \leftrightarrow \varphi,\;\; \; a_i = \varphi({\bf e}_i), \; i = 1, \ldots, k, \] between ${\bf T}$-linearized invertible sheaves ${\cal L}$ on ${\bf P} = {\bf P}_{\Sigma}$ and continious functions $\varphi \,: \, N_{\bf R} \rightarrow {\bf R}$ which are integral $($i.e., $\varphi(N) \subset {\bf Z})$ and ${\bf R}$-linear on every cone $\sigma \in \Sigma$. Moreover, the supporting polyhedron for the global sections of ${\cal L} = {\cal L}(\varphi)$ equals \[ \Delta({\cal L}) = \{ x \in M_{\bf R} | \langle x, y \rangle \geq - \varphi(y)\;\; \mbox{\rm for all $y \in N_{\bf R}$} \}. \] \label{1-1} \end{theo} By {\em semi-ample invertible sheaf} ${\cal L}$ on a projective toric variety ${\bf P}_{\Delta}$ we always mean an invertible sheaf ${\cal L}$ generated by global sections. We have the following \cite{danilov,oda}: \begin{prop} ${\cal L}$ is semi-ample if and only if the corresponding $\Sigma$-piecewise linear $\varphi$ is upper convex. Moreover, any semi-ample invertible sheaf ${\cal L}$ $($together with a ${\bf T}$-linearization$)$ on a toric variety ${\bf P}$ is uniquely determined by its supporting polyhedron $\Delta({\cal L})$: \[ {\cal L} \cong {\cal O}_{\bf P}(a_1 {\bf D}_1 + \cdots + a_n {\bf D}_n), \;\; \mbox{\rm where}\;\; a_i = - \min_{x \in \Delta({\cal L})} \langle x, {\bf e}_i \rangle. \] \label{semi-ample} \end{prop} \begin{dfn} {\rm Let $\Delta$ and $\Delta'$ be two lattice polyhedra. Then we call a polyhedron $\Delta'$ a {\em Minkowski summand} of $\Delta$ if there exist a positive integer $\mu$ and a lattice polyhedron $\Delta''$ such that $\mu \Delta = \Delta' + \Delta''$. } \end{dfn} Using \ref{semi-ample1}, one easily obtains: \begin{prop} A lattice polyhedron $\Delta'$ is the supporting polyhedron for global sections of a ${\bf T}$-linearized semi-ample invertible sheaf on ${\bf P}_{\Delta}$ if and only if $\Delta'$ is a Minkowski summand of $\Delta$. \label{semi-ample1} \end{prop} In the sequel, we shall use many times the following statement: \begin{theo} Let $D$ be a nef-Cartier divisor $($or, equivalently, ${\cal O}_{\bf P}(D)$ is a semi-ample invertible sheaf$)$ on a projective toric variety ${\bf P} = {\bf P}_{\Delta}$ $\Delta'$ the lattice polyhedron supporting the global sections ${\cal O}_{\bf P}(D)$. Then \[ H^i({\bf P}, {\cal O}_{\bf P}(-D)) = 0, \mbox{ if $i \neq {\rm dim}\, \Delta'$ } \] \[ H^i({\bf P}, {\cal O}_{\bf P}(-D)) = l^*(\Delta'), \mbox{ if $i = {\rm dim}\, \Delta'$ }. \] In particular the Euler characteristic $\chi({\cal O}_{\bf P}(-D))$ equals $b(\Delta')$. \label{cohomology} \end{theo} \noindent {\em Proof.} Let $k = {\rm dim}\, \Delta$. Then the invertible sheaf ${\cal O}_{\bf P}(D)$ defines the canonical morphism \[ \pi_D \; :\; {\bf P} \rightarrow {\bf V} = {\rm Proj} \oplus_{n \geq 0} H^0({\bf P}, {\cal O}_{\bf P}(nD)), \] where ${\bf V} \cong {\bf P}_{\Delta}$ is a $k$-dimensional projective toric variety, and ${\cal O}_{\bf P}(D) \cong \pi^*_D{\cal O}_{\bf V}(1)$. Therefore, \[ H^i({\bf P}, {\cal O}_{\bf P}(-D)) \cong H^i({\bf V}, {\cal O}_{\bf V}(-1)). \] Hence $H^i({\bf P}, {\cal O}_{\bf P}(-D)) = 0$ for $i > k$. On the other hand, $H^i({\bf V},{\cal O}_{\bf V}(-1)) = 0$ for $i < k$, and $H^k({\bf V},{\cal O}_{\bf V}(-1)) = l^*(\Delta)$ (see \cite{danilov,khov77}). \hfill $\Box$ \bigskip A complex $d$-dimensional algebraic variety $W$ is said to have only {\em toroidal singularities} if the $m$-adic completion of the local ring $(R, m)$ corresponding to any point $p \in W$ is isomorphic to the $m_{\sigma}$-completion of a semi-group ring $(S_{\sigma}, m_{\sigma})$, where $S_{\sigma} = {\bf C} \lbrack \sigma \cap M \rbrack$ for some $d$-dimensional rational convex polyhedral cone $\sigma \subset M_{\bf R}$ with vertex at $0 \in M$, and the maximal ideal $m_{\sigma} \subset S_{\sigma}$ is generated by all non-constant monomials. We formulate without proof the following technical statement which is a generalization of the classical Bertini's theorem: \begin{theo} Let $W$ be a complex projective algebraic variety with only toroidal singularities, ${\cal L}$ a semi-ample invertible sheaf on $W$, $D \subset W$ the set of zeros of a generic global section of ${\cal L}$. Then $D$ again has only toroidal singularities. In particular, $D$ is irreducible if ${\rm dim}\, D > 0$. \label{bertini} \end{theo} \section{Complete intersections} Let $\Delta_1, \ldots, \Delta_r$ be lattice polyhedra in $M_{\bf R}$. In this section ${\bf P}$ denotes the $d$-dimensional toric variety ${\bf P}_{\Delta}$, where $\Delta = \Delta_1 + \cdots + \Delta_r$ (without loss of generality, we assume ${\rm dim}\, \Delta = {\rm dim}\, M_{\bf R} = d$). By \ref{semi-ample1}, the lattice polyhedron $\Delta_i$ is the support polyhedron for global sections of some semi-ample invertible sheaf ${\cal L}_i$ on ${\bf P}$ $(i =1, \ldots, r )$. We identify $H^0({\bf P}, {\cal L}_i)$ with the space of all Laurent polynomials $f_i \in {\bf C}\lbrack M \rbrack$ having $\Delta_i$ as the Newton polyhedron. \begin{dfn} {\rm Lattice polyhedra $\Delta_1, \ldots, \Delta_r$ are called $k$-dependent if there exist $n > 0$ and $n$-element subset \[ \{ \Delta_{i_1}, \ldots, \Delta_{i_{n}} \} \subset \{ \Delta_1, \ldots, \Delta_r \}, \] such that \[ {\rm dim}\, (\Delta_{i_1} + \cdots + \Delta_{i_{n}}) < n+k-1. \] Lattice polyhedra which are not $k$-dependent will be called $k$-independent.} \end{dfn} \begin{rem} {\rm It follows immediately from definition that the $k$-independence of lattice polyhedra $\Delta_1, \ldots, \Delta_r$ implies the $l$-independence for $1 \leq l \leq k$. } \end{rem} \begin{theo} Let $Z_f \subset {\bf T}$ be a complete intersection of $r$ affine hypersurfaces $Z_{f_1}, \ldots, Z_{f_r}$ defined by a general system of the equations $f_1 = \cdots = f_r = 0$ where $f_i$ is a general Laurent polynomial with the Newton polyhedra $\Delta_i$ $(i =1, \ldots,r)$. Denote by $Z_i$ the closure of $Z_{f_i}$ in ${\bf P}$ $(i =1, \ldots, r)$. Let \[ V = Z_1 \cap \cdots \cap Z_r. \] Then the following statements hold: {\rm (i)} $V$ is non-empty if and only if $\Delta_1, \ldots, \Delta_r$ are $1$-independent; {\rm (ii)} if $\Delta_1, \ldots, \Delta_r$ are $2$-independent, then $V$ is irreducible; {\rm (iii)} if $\Delta_1, \ldots, \Delta_r$ are $k$-independent $(k \geq 3)$, then $h^1({\cal O}_{V}) = \cdots = h^{k-2}({\cal O}_{V}) = 0$. \label{c} \end{theo} \noindent {\em Proof. } Denote by ${\cal K}^*$ the Koszul complex \[ {\cal O}_{\bf P}(-Z_1 - \cdots - Z_r) \rightarrow \cdots \rightarrow \sum_{i < j} {\cal O}_{\bf P}(-Z_i -Z_j) \rightarrow \sum_i {\cal O}_{\bf P}(-Z_i) \rightarrow {\cal O}_{\bf P}. \] There are two spectral sequences $'E$ and $''E$ converging to the hypercohomology ${\bf H}^*({\bf P}, {\cal K}^*)$ (cf. \cite{griffiths}): \[ 'E_2^{p,q} = H^p({\bf P}, {\cal H}^q({\cal K}^*)), \] \[ ''E_2^{p,q} = H^q({\bf P}, H^p({\cal K}^*)). \] Since ${\cal K}^*$ is which an acyclic resolution of ${\cal O}_{V}$, we have ${\cal H}^q({\cal K}^*)=0$ for $q \neq r$, and ${\cal H}^r({\cal K}^*))={\cal O}_{V}$, the first spectral sequence degenerates and we get the isomorphisms \[ {\bf H}^{r+p}({\bf P}, {\cal K}^*) \cong H^{p}({\bf P}, {\cal O}_{V}) \cong H^{p}({V}, {\cal O}_{V} ). \] On the other hand, the second spectral sequence does not degenerate in general. The term ${''E}_2^{p,q}$ is the cohomology of the bicomplex \[ {''E}^{p,q}_1 = \bigoplus_{\{i_1, \ldots, i_{r-p}\}} H^q( {\bf P}, {\cal O}_{\bf P}(-Z_{i_1} - \cdots - Z_{i_{r-p}})). \] We can compute ${''E}^{p,q}_1$ using Proposition \ref{cohomology}. The statements (i)-(iii) will follow from the consideration of ${''E}$. (i) It is clear that $V$ is empty if and only if $Z_f$ is empty. Assume that $\Delta_1, \ldots, \Delta_r$ are $1$-dependent. Then there exists $n \geq 1$ such that $d' = {\rm dim}\, (\Delta_{i_1} + \cdots + \Delta_{i_{n}}) < n$. This means that we can choose the coordinates $X_1, \ldots, X_d$ on ${\bf T}$ in such a way that the polynomials $f_{i_1}, \ldots, f_{i_n}$ depends only on some $d'$ of them. Therefore, we obtain an overdetermined system $f_{i_1} = \cdots = f_{i_n} = 0$; i.e., $V$ is empty. Assume now that $V$ is empty, i.e., ${\cal K}^*$ is acyclic. If some $\Delta_i$ is $0$-dimensional, then $\Delta_1, \ldots, \Delta_r$ are $1$-dependent, and everything is proved. Otherwise, one has the non-zero cycle in ${''E}^{r,0}_1 \cong {{''E}}^{r,0}_2 \cong H^0({\bf P}, {\cal O}_{\bf P}) \cong {\bf C}$ which must be killed by some next non-zero differential \[ d_l\; :\; {''E}^{r-l,l-1}_{l} \rightarrow E^{r,0}_l\; (l \geq 2). \] Therefore ${''E}^{r-l,l-1}_{l} \neq 0$ for some $2 \leq l \leq r$. This implies that there exists an $l$-element subset $\{ i_1, \ldots, i_l \} \subset \{ 1, \ldots, r \}$ such that \[ H^{l-1}( {\bf P}, {\cal O}_{\bf P}(-Z_{i_1} - \cdots - Z_{i_l})) \neq 0.\] Applying \ref{cohomology}, we see that there exists an $l$-element subset \[ \{ \Delta_{i_1}, \ldots, \Delta_{i_l} \} \subset \{ \Delta_1, \ldots, \Delta_r \} \] such that \[ {\rm dim}\, (\Delta_{i_1} + \cdots + \Delta_{i_{l}}) = l-1, \] i.e., $\Delta_1, \ldots, \Delta_r$ are $1$-dependent. (ii) Assume that $\Delta_1, \ldots, \Delta_r$ are $2$-independent. By \ref{cohomology}, one has \[ {''E}^{r,1}={''E}^{r-s,s-1}_1={''E}^{r-s,s}_1 = 0\; \mbox{ for $1 \leq s \leq r$}. \] Hence \[ {''E}^{r,1}_l = {''E}^{r-s,s-1}_l = {''E}^{r-s,s}_l = 0\; \mbox{ for $1 \leq s \leq r$, $l \geq 1$}. \] So ${\bf H}^r({\bf P}, {\cal K}^*) \cong {\bf C} \cong H^0({V}, {\cal O}_{V})$. Therefore ${V}$ is connected. By \ref{bertini}, ${V}$ is irreducible. (iii) Assume that $\Delta_1, \ldots, \Delta_r$ are $k$-independent $(k \geq 3)$. By the same arguments using \ref{cohomology}, we obtain ${\bf H}^r({\bf P}, {\cal K}^*) \cong {\bf C}$, and ${\bf H}^{r+1}({\bf P}, {\cal K}^*) = \cdots = {\bf H}^{r+k -2}({\bf P}, {\cal K}^*) = 0$. This implies $h^1({\cal O}_{V}) = \cdots = h^{k-2}({\cal O}_{V}) = 0$. \hfill $\Box$ \medskip \begin{rem} {\rm Khovanski\^i announced (\cite{khov} p.41) that the statement \ref{c}(i) was first discovered and proved by D. Bernshtein using properties of mixed volumes in the following equivalent form:} { The affine variety $Z_f$ is empty if and only if there exists an $l$-dimensional affine subspace of $M_{\bf R}$ containing affine translates of some $l+1$ polyhedra from $\{ \Delta_1, \ldots, \Delta_r \}$.} \end{rem} \begin{coro} Assume that all lattice polyhedra $\Delta_1, \ldots, \Delta_r$ have positive dimension, $l^*(\Delta_1 + \cdots + \Delta_r ) = 1$, $d = {\rm dim}\, (\Delta_1 + \cdots + \Delta_r)$, and for any proper subset $\{ k_1, \ldots, k_s \} \subset \{ 1, \ldots, r\}$ one has $l^*(\Delta_{k_1} + \cdots + \Delta_{k_s}) = 0$. Then the following statements hold: {\rm (i)} ${V}$ is empty if and only if $d = r -1$; {\rm (ii)} ${V}$ consists of $2$ distinct points if and only if $d = r$; {\rm (iii)} ${V}$ is a smooth irreducible curve of genus $1$ if and only if $d = r+1$. {\rm (iv)} ${V}$ is a nonempty irreducible variety of dimension $d-r\geq 2$ having the property $h^{1}({\cal O}_{V}) = \cdots = h^{d-r-1}({\cal O}_{V}) = 0$ and $h^{0}({\cal O}_{V}) = h^{d-r}({\cal O}_{V})= 1$ if and only if $d \geq r+2$. \label{main.cor} \end{coro} \noindent {\em Proof. } It follows from Proposition \ref{cohomology} and our assumptions that ${''E}_1^{r,0}$ and ${''E}_1^{0,d}$ have dimension $1$ and all remaining spaces ${''E}_1^{p,q}$ are zero. (i) If $r = d+1$, then $Z_f$ is empty by dimension arguments. On the other hand, if $Z_f$ is empty, then $''E^{p,q}_l$ becomes acyclic for $ l \gg 0$. Note that the only nontrivial one-dimensional spaces ${''E}_1^{r,0}$ and ${''E}_1^{0,d}$ can kill each other only via the non-zero differential \[ d_l\; :\; {''E}_l^{0,d} \rightarrow {''E}_l^{r,0} \] where $r = d+1$ and $l = r$, i.e., $Z_f$ is empty if and only if $r = d+1$. Assume $r > d+1$. Then \[{\bf C} \cong {''E}_1^{0,d} \cong {''E}_{\infty}^{0,d} \cong {\bf H}^d({\bf P}, {\cal K}^*).\] On the other hand, the isomorphism ${\bf H}^{r+p}({\bf P}, {\cal K}^*) \cong H^{p}(C, {\cal O}_{V})$ implies ${\bf H}^{i}({\bf P}, {\cal K}^*) = 0$ for $i < r$. Contradiction. (ii) If $r = d$, then \[ {\bf C}^2 \cong {''E}_1^{0,d} \oplus {''E}_1^{r,0} \cong {''E}_{\infty}^{0,d} \oplus {''E}_{\infty}^{r,0} \cong H^0({V}, {\cal O}_{V}). \] Since ${V}$ is nonempty, one has ${\rm dim}\, {V} =0$, i.e., ${V}$ consists of $2$ distinct points. (iii)-(iv) For $r \leq d-1$, we have isomorphisms \[ {\bf C} \cong {''E}_1^{r,0} \cong H^{0}({V}, {\cal O}_{V}), \] \[ {\bf C} \cong {''E}_1^{0,d} \cong H^{d-r}({V}, {\cal O}_{V}), \] and \[ 0 \cong {''E}_1^{p,q} \cong H^{d-r}({V}, {\cal O}_{V})\; \mbox{ if $p + q \neq r,d$}. \] This proves (iii)-(iv). \hfill $\Box$ \section{Calabi-Yau varieties and nef-partitions} \begin{dfn} {\rm A lattice polyhedron $\Delta \subset M_{\bf R}$ is called {\em reflexive} if ${\bf P}_{\Delta}$ has only Gorenstein singularities and ${\cal O}_{\bf P}(1)$ is isomorphic to the anticanonical sheaf which is considered together with its natural ${\bf T}$-linearization. In this case, we call ${\bf P}_{\Delta}$ a {\em Gorenstein toric Fano variety}. } \end{dfn} \begin{rem} {\rm Since the ${\bf T}$-linearized anticanonical sheaf on ${\bf P}_{\Delta}$ is isomorphic to ${\cal O}_{{\bf P}_{\Delta}} ({\bf D}_1 + \cdots + {\bf D}_n)$, it follows from the above definition that any reflexive polyhedron $\Delta$ has $0 \in M$ as the unique interior lattice point. Moreover, \[ \Delta = \{ x \in M_{\bf R} | \langle x, {\bf e}_i \rangle \geq -1, \; i =1, \ldots, n \}, \] where ${\bf e}_1, \ldots, {\bf e}_n$ are the primi\-tive integ\-ral inte\-rior nor\-mal vec\-tors to codimension-$1$ faces $\Theta_1, \ldots, \Theta_n$ of $\Delta$. These properties of reflexive polyhedra were used in another their definition \cite{bat.dual}.} \end{rem} \begin{theo} {\rm \cite{bat.dual}} Let $\Delta$ be a reflexive polyhedron as above. Then the convex hull $\Delta^* \subset N_{\bf R}$ of the lattice vectors ${\bf e}_1, \ldots, {\bf e}_n$ is again a reflexive polyhedron. Moreover, $(\Delta^*)^* = \Delta$. \end{theo} \begin{dfn} {\rm The lattice polyhedron $\Delta^*$ is called {\em dual} to $\Delta$. } \end{dfn} Using the adjunction formula, \ref{semi-ample} and \ref{semi-ample1}, we immediately obtain: \begin{prop} Assume that a $d$-dimensional Gorenstein toric Fano variety ${\bf P}_{\Delta}$ contains a $(d-r)$-dimensional complete intersection $V$ $(r<d)$ of $r$ semi-ample Cartier divisors $Z_1, \ldots, Z_r$ such that the canonical $($or, equivalently, dualizing$)$ sheaf of $V$ is trivial. Then there exist lattice polyhedra $\Delta_1, \ldots, \Delta_r$ such that \[ \Delta = \Delta_1 + \cdots + \Delta_r. \] \end{prop} \begin{dfn} {\rm Let $\Sigma \subset N_{\bf R}$ be the normal fan defining a Gorenstein toric Fano variety ${\bf P}_{\Delta}$, $\varphi\, : \, N_{\bf R} \rightarrow {\bf R}$ the integral upper convex $\Sigma$-piecewise linear function corresponding to the ${\bf T}$-linearized anticanonical sheaf (by \ref{1-1}, $\varphi({\bf e}_i) = 1$, $i = 1, \ldots, n$), $\Delta = \Delta_1 + \cdots + \Delta_r$ a decomposition of $\Delta$ into a Minkowski sum of $r$ lattice polyhedra $\Delta_j$ $(j =1, \ldots, r)$, $\varphi = \varphi_1 + \cdots + \varphi_r$ the induced decomposition of $\varphi$ into the sum of integral upper convex $\Sigma$-piecewise linear functions $\varphi_j$ (see \ref{semi-ample}). Then $\Pi(\Delta) = \{ \Delta_1, \ldots, \Delta_r \}$ is called a {\em nef-partition of} $\Delta$ if $\varphi_j({\bf e}_i) \in \{ 0,1 \}$ for $i = 1 , \ldots, n$, $j =1, \ldots, r$. } \label{nef} \end{dfn} \begin{rem} {\rm There are reflexive polyhedra $\Delta$ which admit decompositions into Minkowski sum of two lattice polyhedra, but do not admit any nef-partition. For example, let $\Delta \subset {\bf R}^2$ be the convex hull of $4$ lattice points: $(1,0)$, $(-1,0)$, $(0,1)$ and $(0,-1)$. Then $\Delta$ is a $2$-dimensional reflexive polyhedron which admits a decomposition into Minkowski sum of two $1$-dimensional polyhedra $\Delta_1$ and $\Delta_2$, where $\Delta_1$ is the convex hull of $(-1,0)$ and $(0,-1)$, and $\Delta_2$ is the convex hull of $(0,0)$ and $(1,1)$. However, $\Delta$ does not admit any nef-partition.} \end{rem} \begin{rem} {\rm The notation in \ref{nef} are a little bit different from definitions and notations in \cite{batyrev-borisov,borisov}, but they are essentially equivalent.} \end{rem} \begin{dfn} {\rm Let $\Pi(\Delta) = \{ \Delta_1, \ldots, \Delta_r \}$ be a nef-partition of a reflexive polyhedron $\Delta$. We define the lattice polyhedron $\nabla_j \subset N_{\bf R}$ $( j=1, \ldots, r)$ as the convex hull of $0 \in N$ and all lattice vectors ${\bf e}_i \in \{ {\bf e}_1, \ldots, {\bf e}_n \}$ such that $\varphi_j({\bf e}_i) = 1$. } \end{dfn} \begin{theo} {\rm \cite{borisov} } The Minkowski sum $\nabla = \nabla_1 + \cdots + \nabla_r$ is a reflexive polyhedron. Moreover, $\Pi(\nabla) = \{ \nabla_1, \ldots, \nabla_r \}$ is a nef-partition of $\nabla$, and one has also \[ \nabla^* = {\rm Conv} \{ \Delta_1, \ldots, \Delta_r \}, \] \[ \Delta^* = {\rm Conv} \{ \nabla_1, \ldots, \nabla_r \}. \] \end{theo} \begin{dfn} {\rm The nef-partition $\Pi(\nabla)$ is called the {\em dual nef-partition}. } \end{dfn} \begin{exam} {\rm Let $\Delta_j \subset M_{\bf R}^{(j)}$ is a reflexive polyhedron $( j =1, \ldots, r)$, $\nabla_j : = \Delta^*_j \subset N_{\bf R}^{(j)}$ the corresponding dual reflexive polyhedron $(j =1, \ldots, r)$. We set $M_{\bf R} := M_{\bf R}^{(1)} \oplus \cdots \oplus M_{\bf R}^{(r)}$, $N_{\bf R} := N_{\bf R}^{(1)} \oplus \cdots \oplus N_{\bf R}^{(r)}$. Then $\Pi(\Delta) = \{ \Delta_1, \ldots, \Delta_r \}$ is a nef-partition of the reflexive polyhedron $\Delta = \Delta_1 + \cdots + \Delta_r \subset M_{\bf R}$, and $\Pi(\nabla) = \{ \nabla_1, \ldots, \nabla_r \}$ is the dual nef-partition of the reflexive polyhedron $\nabla = \nabla_1 + \cdots + \nabla_r \subset N_{\bf R}$.} \label{ex-split} \end{exam} By \cite{bat.dual}, a maximal projective triangulation ${\cal T}$ of $\Delta^* \subset N_{\bf R}$ defines a $MPCP$-desingularization $\pi_{\cal T} \, : \, \widehat{\bf P}_{\Delta} \rightarrow {\bf P}_{\Delta}$. The ${\bf T}$-invariant divisors on $\widehat{\bf P}_{\Delta}$ one-to-one correspond to lattice points on the boundary $\partial \Delta^*$. \begin{dfn} {\rm Denote by ${\cal V}(\Delta^*)$ the set $\partial \Delta^* \cap N$. If $v$ is a lattice point in ${\cal V}(\Delta^*)$, then the corresponding toric divisor on $\widehat{\bf P}_{\Delta}$ will be denoted by ${\bf D}(v)$. } \end{dfn} Since the anticanonical sheaf on $\widehat{\bf P}_{\Delta}$ is semi-ample, $\Delta_1, \ldots, \Delta_r$ are supporting polyhedra for global sections of some semi-ample invertible sheaves $\widehat{\cal L}_1, \ldots,\widehat{\cal L}_r$ on $\widehat{\bf P}_{\Delta}$. \begin{dfn} {\rm Let $\Pi(\Delta)$ be a nef-partition, $Z_f \subset {\bf T}$ a complete intersection of $r$ general affine hypersurfaces $Z_{f_1}, \ldots, Z_{f_r}$ defined by a general system of the polynomial equations $f_1 = \cdots = f_r = 0$ where $f_i$ is a general Laurent polynomial with the Newton polyhedra $\Delta_i$ $(i =1, \ldots,r)$. Denote by $\widehat{Z}_i$ the closure of $Z_{f_i}$ in $\widehat{\bf P}_{\Delta}$ $(i =1, \ldots, r)$. Define $V$ (resp. $\widehat{V}$) as the closure of $Z_f$ in ${\bf P}_{\Delta}$ (resp. in $\widehat{\bf P}_{\Delta}$; i.e., $\widehat{V} = \widehat{Z}_1 \cap \cdots \cap \widehat{Z}_r$). If $\Pi(\nabla)$ is the dual nef-partition, then the corresponding general complete intersection in ${\bf P}_{\nabla}$ (resp. in $\widehat{\bf P}_{\nabla}$) will be denoted by $W$ (resp. $\widehat{W}$). } \label{notation} \end{dfn} By the adjunction formula and \ref{bertini}, one immediately obtains: \begin{prop} In the above notations, assume that $V$ is nonempty and irreducible. Then $\widehat{V}$ is an irreducible $(d-r)$-dimensional projective algebraic variety with at most Gorenstein terminal abelian quotient singularities and trivial canonical class. In particular, $\widehat{V}$ is smooth if $d-r \leq 3$. \end{prop} In the next section we prove the following: \begin{theo} Let $\Pi(\Delta)$ be a nef-partition, $\Pi(\nabla)$ the dual nef-partition, $V$ $($resp. $W)$ corresponding to $\Pi(\Delta)$ $($resp. to $\Pi(\nabla))$ general complete intersections in ${\bf P}_{\Delta}$ $($ resp. in ${\bf P}_{\nabla})$. Then the following statements hold: {\rm (i)} $V$ is nonempty if and only if $W$ is nonempty; {\rm (ii)} $V$ is irreducible if and only if $W$ is irreducible; {\rm (iii)} $h^{i}({\cal O}_V) = h^{i}({\cal O}_W)$ for $0 \leq i \leq d-r$. \label{0-hodge} \end{theo} \begin{coro} Let $\Pi(\Delta)$ be a nef-partition, $\Pi(\nabla)$ the dual nef-partition, $\widehat{V}$ $($resp. $\widehat{W})$ corresponding to $\Pi(\Delta)$ $($resp. $\Pi(\nabla))$ general complete intersections in $\widehat{\bf P}_{\Delta}$ $($resp. in $\widehat{\bf P}_{\nabla})$. Then $\widehat{V}$ is an irreducible $(d-r)$-dimensional projective algebraic variety with at most Gorenstein terminal abelian quotient singularities and trivial canonical class if and only if $\widehat{W}$ has the same properties. \end{coro} \medskip \section{Semi-simplicity principle for nef-partitions} \noindent Let $\Delta$ be a $d$-dimensional reflexive polyhedron in $M_{\bf R}$, $\Pi(\Delta) = \{ \Delta_1, \ldots, \Delta_r \}$ a nef-partition of $\Delta$. \begin{dfn} {\rm We say that $\Pi(\Delta)$ {\em splits into a direct sum} \[ \Pi(\Delta) = \Pi(\Delta^{(1)}) \oplus \cdots \oplus \Pi(\Delta^{(k)}) \] if there exist convex lattice polyhedra $\Delta^{(1)}, \ldots, \Delta^{(k)} \subset \Delta$ satisfying the conditions (i) $ d = {\rm dim}\,\Delta^{(1)} + \cdots + {\rm dim}\, \Delta^{(k)} $ and \[ \Delta = \Delta^{(1)} + \cdots + \Delta^{(k)};\] (ii) for $1 \leq i \leq k$, the lattice point $0$ is cointained in the relative interior of $\Delta^{(i)}$, $\Delta^{(i)}$ is reflexive, and \[ \Pi(\Delta^{(i)}) = \{ \Delta_j \subset \Delta \mid \Delta_j \subset \Delta^{(i)} \} \] is a nef-partition of $\Delta^{(i)}$.} \end{dfn} \begin{dfn} {\rm Assume that $\Pi(\Delta)$ {splits into a direct sum} \[ \Pi(\Delta) = \Pi(\Delta^{(1)}) \oplus \cdots \oplus \Pi(\Delta^{(k)}). \] Denote by $M^{(i)}_{\bf R}$ the minimal ${\bf R}$-linear subspace of $M_{\bf R}$ containing $\Delta^{(i)}$ $(i =1, \ldots, k)$. We set also $M^{(i)} = M \cap M^{(i)}_{\bf R}$ $(i =1, \ldots, k)$. We say that $\Pi(\Delta)$ {splits into the direct sum} {\em over {\bf Z}} if \[ M = M^{(1)} \oplus \cdots \oplus M^{(k)}. \]} \end{dfn} It is easy to see the following: \begin{prop} Assume that a nef-partition $\Pi(\Delta)$ splits over ${\bf Z}$ into a direct sum $\Pi(\Delta^{(1)}) \oplus \cdots \oplus \Pi(\Delta^{(k)})$. Then the Gorenstein toric Fano variety ${\bf P}_{\Delta}$ is the product of the Gorenstein toric Fano varieties ${\bf P}_{\Delta^{(i)}}$ $(i = 1, \ldots, k)$. Moreover, \[ V = V^{(1)} \times \cdots \times V^{(k)}, \] \[ \widehat{V} = \widehat{V}^{(1)} \times \cdots \times \widehat{V}^{(k)}, \] where $V^{(i)}$ $($resp. $\widehat{V}^{(i)})$ is the Calabi-Yau complete intersections defined by the nef-partition $\Pi(\Delta^{(i)})$ in ${\bf P}_{\Delta^{(i)}}$ $($resp. in $\widehat{\bf P}_{\Delta^{(i)}})$, $i =1, \ldots, k$. \label{product} \end{prop} \begin{rem} {\rm In \ref{ex-split} we gave a simplest example of a nef-partition $\Pi(\Delta)$ which splits into a direct sum $\{\Delta_1\} \oplus \cdots \oplus \{\Delta_r\}$ over ${\bf Z}$. In general situation, if $\Pi(\Delta)$ {splits into a direct sum} $\Pi(\Delta) = \Pi(\Delta^{(1)}) \oplus \cdots \oplus \Pi(\Delta^{(k)})$, then $M^{(1)} \oplus \cdots \oplus M^{(k)}$ is a sublattice of finite index in $M$, but not necessarily the lattice $M$ itself.} \end{rem} \begin{exam} {\rm Let $\Delta = \Delta_1 + \Delta_2 \subset {\bf R}^4$ where \[ \Delta_1 = {\rm Conv}\{ (1,0,0,0), (0,1,0,0),(-1,0,0,0), (0,-1,0,0)\}; \] \[ \Delta_2 = {\rm Conv}\{ (0,0,1,0), (0,0,0,1),(0,0,-1,0), (0,0,0,-1)\}.\] We define the lattice $M \subset {\bf R}^4$ as \[ M = (\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}) + {\bf Z}^4. \] Then $\Delta$ splits into direct sum $\Pi(\Delta) = \{\Delta_1\} \oplus \{ \Delta_2 \}$. But it is not a splitting over ${\bf Z}$, because $M_1 \oplus M_2$ is a sublattice of index $2$ in $M$.} \end{exam} \begin{dfn} {\rm We call a nef-partition $\Pi(\Delta)$ {\em reducible} if there exists a subset \[\{ k_1, \ldots, k_s \} \subset \{1, \ldots, r \}\; ( 0< s < r) \] such that $\Delta_{k_1} + \ldots + \Delta_{k_s}$ contains $0$ in its relative interior. Nef-partitions which are not reducible are called {\em irreducible}. } \end{dfn} \begin{rem} {\rm (i) Notice that $\Delta_{k_1} + \ldots + \Delta_{k_s}$ contains $0$ in its relative interior if and only if \[ l^*(\Delta_{k_1} + \ldots + \Delta_{k_s}) = 1. \] (ii) It is clear that an irreducible nef-partition has no nontrivial splitting into a direct sum.} \label{nul} \end{rem} \begin{theo} Any nef-partition $\Pi(\Delta)$ has a unique decomposition into direct sum of irreducible nef-partitions. \label{decomposition} \end{theo} \noindent {\em Proof.} Assume that $\Pi(\Delta)$ is reducible. Choose lattice polyhedra $\Delta_{k_1}, \ldots, \Delta_{k_s}$ such that $\Delta' = \Delta_{k_1} + \ldots + \Delta_{k_s}$ contains $0$ in its relative interior. Denote \[ \Lambda' = {\rm Conv}(\Delta_{k_1}, \ldots, \Delta_{k_s}). \] It is clear that $\Lambda'$ also contains $0$ in its relative interior and has the same dimension as $\Delta_{k_1} + \ldots + \Delta_{k_s}$. Denote by $M'_{\bf R}$ the minimal linear subspace in $M_{\bf R}$ containing $\Delta'$. Denote by $\psi_1, \ldots, \psi_r$ the piecewise linear functions on $M_{\bf R}$ which determine the dual nef-partition $\Pi(\nabla)$ (see \ref{nef}). We set $\psi' = \psi_{k_1} + \cdots + \psi_{k_s}$, $\psi'' = \psi_1 + \cdots + \psi_r - \psi' = \psi_{j_1} + \cdots + \psi_{j_{r-s}}$. Then \[ \Lambda' = \{ x \in M_{\bf R}' \mid \psi'(x) \leq 1 \}. \] Since ${\psi}'$ is a non-negative integral convex piecewise linear function having value $1$ at the relative boundary $\partial \Lambda' \subset M_{\bf R}'$, the polyhedron $\Lambda'$ is reflexive. Since $\psi'$ is convex and non-negative \[ C' = \{ x \in M_{\bf R} \mid \psi'(x) = 0 \} \] is a convex subset of $M_{\bf R}$ containing all non-negative linear combinations of vertices of $\Delta_{j_1}, \ldots, \Delta_{j_{r-s}}$. One has $M_{\bf R}' \cap C' = 0$. Assume that $0$ is not contained in the relative interior of $C'$. Then, by separateness theorem for convex sets, there exists a non-zero element $y' \in N_{\bf R}$ such that $\langle M_{\bf R}', y' \rangle = 0$, and $\langle C' , y' \rangle \geq 0$. Therefore $\langle v, y' \rangle \geq 0$ for all vertices $v$ of $\Delta_1, \ldots, \Delta_r$. Contradiction. Thus $0$ is contained in the relative interior of $C'$. Since ${\bf R}_{\geq 0} C' \subset C'$, $C'$ is a linear subspace and $M_{\bf R} = M_{\bf R}' \oplus C'$. We put $M_{\bf R}'' = C'$. Define \[ \Lambda'' = \{ x \in M_{\bf R}'' \mid \psi''(x) \leq 1 \}. \] Then $\Lambda''= {\rm Conv}(\Delta_{j_1}, \ldots, \Delta_{j_{r-s}})$, and $\Delta'':= \Delta_{j_1} \oplus \cdots \oplus \Delta_{j_{r-s}})$ contains $0$ in its relative interior. Therefore $\Lambda''$ is also reflexive. Thus we obtain $\nabla^* = {\rm Conv}(\Lambda', \Lambda'')$, ${\rm dim}\, \Lambda' + {\rm dim}\, \Lambda'' = d$, and \[ \Pi(\Delta') = \{ \Delta_{k_1}, \ldots, \Delta_{k_s} \}, \] \[ \Pi(\Delta'') = \{ \Delta_{j_1}, \ldots, \Delta_{j_{r-s}} \} \] are nef-partitions of the reflexive polyhedra $\Delta'$ and $\Delta''$ corresponding to restricitions of $\psi_{k_1}, \ldots, \psi_{k_s}$ (resp. of $\psi_{j_1}, \ldots, \psi_{j_{r-s}}$) on $M'_{\bf R}$ (resp. on $M_{\bf R}''$). Now the statement of Theorem \ref{decomposition} follows by induction. \hfill $\Box$ \noindent {\bf Proof of Theorem \ref{0-hodge}.} Note that $V$ is nonempty if and only if $h^0(V, {\cal O}_V) \neq 0$. By \ref{bertini}, $V$ is irreducible if and only if $h^0(V, {\cal O}_V) = 1$. Therefore, (i) and (ii) follow from (iii). By \ref{main.cor} and \ref{nul}(i), the spectral sequence $''E^{p,q}$ which computes the cohomology of ${\cal O}_V$ degenerates at $''E_2^{p,q}$ for all irreducible nef-partitions. By \ref{product}, we obtain the degeneration of $''E^{p,q}$ at $''E_2^{p,q}$ for all nef-partitions $\Pi(\Delta)$ which split over ${\bf Z}$ into a sum of irreducible nef-partitions. According to Theorem \ref{decomposition}, in general situation we have a finite covering \[ \pi \; : \; {\bf P}_{\Delta} \rightarrow {\bf P}_{\Delta^{(1)}} \times \cdots \times {\bf P}_{\Delta^{(k)}} \] where the degree of $\pi$ equals the index of $M^{(1)} \oplus \cdots \oplus M^{(k)}$ in $M$, and $\Pi(\Delta^{(1)})$, $\ldots,$ $\Pi(\Delta^{(k)})$ are irreducible nef-partitions. Let \[ \tilde{\pi}\; ;\; V \rightarrow \tilde{V} := V^{(1)} \times \cdots \times V^{(k)} \] the finite covering induced by $\pi$. Then we obtain the canonical homomorphism of the spectral sequences: \[ \tilde{\pi}^*\; : \; ''\tilde{E}^{p,q} \rightarrow ''E^{p,q}. \] By \ref{nul}(i), we have the canonical isomorphism \[ ''\tilde{E}^{p,q}_1 \cong \rightarrow ''E^{p,q}_1. \] Since $''\tilde{E}^{p,q}$ degenerates at $''\tilde{E}^{p,q}_1$ (see the above arguments), $''E^{p,q}$ also degenerates at $''E^{p,q}_2$. Therefore, $h^i({\cal O}_V) = h^i({\cal O}_{\tilde{V}})$ $(i =1, \ldots, d-r)$. Analogously, $h^i({\cal O}_W) = h^i({\cal O}_{\tilde{W}})$ $(i =1, \ldots, d-r)$. Thus, we have reduced (iii) to already known case. \hfill $\Box$ \bigskip \begin{coro} Let $I = \{ 1, \ldots , r \}$. Denote by $|J|$ the cardinality of a subset $J \subset I$. Define \[ E(\Delta, t) = \sum_{i =0}^{d-r} h^i({\cal O}_V) t^i. \] Then \[ E(\Delta, t) = \sum_{J \subset I} l^*(\sum_{j \in J} \Delta_j) t^{{\rm dim}\left( \sum_{j \in J} \Delta_j \right) - |J|}. \] \end{coro} By Serre duality, and using the natural isomorphisms $H^{i}(V, {\cal O}_V) \cong H^i(\widehat{V}, {\cal O}_{\widehat{V}})$ ($i =1, \ldots, d-r)$, we also obtain: \begin{coro} Assume that $\widehat{V}$ is nonempty and irreducible. Then \[ h^i(\widehat{V}, {\cal O}_{\widehat{V}}) = h^{d-r-i}(\widehat{W}, {\cal O}_{\widehat{W}}). \] \label{o-dual} \end{coro} \section{$\chi(\Omega^1)$ for complete intersections} Now we want to calculate the Euler characteristic of $\Omega_{\widehat{{V}}}^1$ of a Calabi-Yau complete intersection $\widehat{{V}} = \widehat{Z}_1 \cap \cdots \cap \widehat{Z}_r$ in a $MPCP$-desingularization $\widehat{\bf P} : = \widehat{\bf P}_{\Delta}$ of a Gorenstein toric Fano variety ${\bf P} : = {\bf P}_{\Delta}$. \bigskip One has the standard exact sequence for a complete intersection: \[ 0 \rightarrow {\cal O}_{\widehat{V}}(-\widehat{Z}_1) \oplus \cdots \oplus {\cal O}_{\widehat{V}}(-\widehat{Z}_r) \rightarrow \Omega^1_{\widehat{\bf P}} \mid_{\widehat{V}} \rightarrow \Omega^1_{\widehat{V}} \rightarrow 0. \] On the other hand, for toric varieties with only quotient singularities there exists the exact sequence \cite{danilov,oda}: \[ 0 \rightarrow \Omega^1_{\widehat{\bf P}} \rightarrow {\cal O}_{\widehat{\bf P}}^d \rightarrow \bigoplus_{v \in {\cal V}(\Delta^*)} {\cal O}_{{\bf D}(v)} \rightarrow 0. \] By transversality of $\widehat{Z}_1, \ldots , \widehat{Z}_r$ to all strata on $\widehat{\bf P}$ we can restrict the last exact sequence on $\widehat{V}$ without obtaining additional Tor-sheaves: \[ 0 \rightarrow \Omega^1_{\widehat{\bf P}} \mid_{\widehat{V}} \rightarrow {\cal O}_{\widehat{V}}^d \rightarrow \bigoplus_{v \in {\cal V}(\Delta^*)} {\cal O}_{{\bf D}(v) \cap \widehat{V}} \rightarrow 0. \] Consequently, we obtain: \begin{prop} \[ \chi(\Omega_{\widehat{V}}^1) = \chi({\cal O}_{\widehat{V}}^d) - \sum_{j =1}^r \chi({\cal O}_{\widehat{V}}(-\widehat{Z}_j)) - \sum_{v \in {\cal V}(\Delta^*)} \chi({\cal O}_{{\bf D}(v) \cap \widehat{V}}). \] \label{lemm1} \end{prop} In order to compute $\chi({\cal O}_{\widehat{V}}(-\widehat{Z}_j))$, we consider the Koszul resolution \[ 0 \rightarrow {\cal O}_{\widehat{\bf P}}(-\widehat{Z}_1 - \cdots - \widehat{Z}_r) \rightarrow \cdots \] \[ \cdots \rightarrow \sum_{j < k} {\cal O}_{\widehat{\bf P}}( -\widehat{Z}_j - \widehat{Z}_k ) \rightarrow \sum_{j} {\cal O}_{\widehat{\bf P}}( -\widehat{Z}_j) \rightarrow {\cal O}_{\widehat{\bf P}} \rightarrow {\cal O}_{\widehat{V}} \rightarrow 0.\] Tensoring it by ${\cal O}_{\widehat{\bf P}}(-\widehat{Z}_i)$ and using \ref{cohomology}, we get \begin{prop} \[ \chi({\cal O}_{\widehat{V}}(-\widehat{Z}_i)) = - \sum_j b(\Delta_i + \Delta_j) + \sum_{j < k} b(\Delta_i + \Delta_j + \Delta_k) - \cdots \] \[ \cdots + (-1)^r b(\Delta_i + \Delta_1 + \cdots + \Delta_r). \] \label{lemm2} \end{prop} For the computation of $\chi({\cal O}_{{\bf D}(v) \cap \widehat{V}})$ we need the following: \begin{prop} Let $v$ an arbitrary lattice point in ${\cal V}(\Delta^*)$, $\Gamma(v)$ the minimal face of $\Delta^*$ containing $v$. Then $\Gamma(v)$ is a face of a polyhedron $\nabla_i$ for some $i$ $( 1 \leq i \leq r)$. \end{prop} \noindent {\em Proof. } If $v$ is a vertex of $\Delta^*$, then the statement is evident, because $\Delta^* = {\rm Conv}\{ \nabla_1, \ldots, \nabla_r \}$. Assume that $\Gamma(v)$ is a convex hull of $k > 1$ faces of polyhedra $\nabla_i$. Let $v_1, \ldots, v_m$ be vertices of $\Gamma(v)$. By assumption, $m \geq k$. Without loss of generality, we assume that $v_1 \in \nabla_1, \ldots, v_k \in \nabla_k$. Since $v$ belongs to relative interior of $\Gamma(v)$, there exist positive numbers $\lambda_1, \ldots, \lambda_m$ such that $\lambda_1 + \cdots + \lambda_m =1$ and $\lambda_1 v_1 + \cdots + \lambda_m v_m = v$. Choose an arbitrary vertex $w$ of the dual face $\Gamma^*(v) \subset \Delta$. Since $\Delta = \Delta_1 + \cdots + \Delta_r$, there exist $w_i \in \Delta_i$ such that $w = w_1 + \cdots + w_r$. It follows from definition of dual nef-partitions (cf. \cite{borisov}) that \[ \langle w_i, v_i \rangle \geq -1, \;\; 1 \leq i \leq k \] and \[ \langle w_i, v_j \rangle \geq 0,\; \; i \neq j,\; 1 \leq i \leq k, \; 1 \leq i \leq k. \] On the other hand, \[\langle w, v \rangle = -1 = \sum_{i=1}^m \sum_{j=1}^r \lambda_i \langle w_j, v_i \rangle. \] Since $\lambda_i > 0$, the last equality is possible only if all vertices $v_1, \ldots, v_m$ belong to {\em the same} polyhedron $\nabla_i$ for some $i$ $(1 \leq i \leq r)$. \hfill $\Box$ \begin{coro} \[ l(\Delta^*) = l(\nabla_1) + \cdots + l(\nabla_r) - r + 1. \] \end{coro} \bigskip \begin{dfn} {\rm Denote by $\Delta_j(v)$ the face of $\Delta_j$ which defines the Newton polyhedron for the equation of $\widehat{Z}_j$ in the toric divisor ${\bf D}(v) \subset \widehat{\bf P}_{\Delta}$. } \end{dfn} It is easy to prove the following: \begin{prop} Assume that the face $\Gamma(v) \subset \Delta^*$ is also a face of $\nabla_i$. Let $\Gamma^*(v)$ be the dual face of $\Delta$. Then \[ \Delta_i(v) = \{ x \in \Delta_i \mid \langle x, v \rangle = -1 \}, \] \[ \Delta_j(v) = \{ x \in \Delta_j \mid \langle x, v \rangle = 0\; \mbox{ if $ j \neq i $} \}, \] and \[ \Delta_1(v) + \cdots + \Delta_r(v) = \Gamma^*(v). \] \label{delta-v} \end{prop} \medskip Tensoring the Koszul resolution of ${\cal O}_{\widehat{V}}$ by ${\cal O}_{{\bf D}(v)}$, we obtain: \begin{prop} \[ \chi({\cal O}_{{\bf D}(v) \cap C}) = - \sum_j b(\Delta_j(v)) + \sum_{j < k} b(\Delta_j(v) + \Delta_k(v) ) - \cdots \] \[ \cdots + (-1)^r b(\Delta_1(v) + \cdots + \Delta_r(v)). \] \label{lemm3} \end{prop} Let $\nabla_i^0 = \nabla \cap {\cal V}(\Delta^*)$ $(i =1, \ldots, r)$, and $I := \{ 1, \ldots, r \}$. If we rewrite \[ \sum_{v \in {\cal V}(\Delta^*)} \chi({\cal O}_{{\bf D}(v) \cap C}) = \sum_{i =1}^r \sum_{v \in \nabla_i^0} \sum_{J \subset I} (-1)^{|J| + 1} b(\sum_{j \in J} \Delta_j(v)) = \] \[ = \sum_{i =1}^r \sum_{v \in \nabla_i^0}\sum_{i \not\in J \subset I} (-1)^{|J| + 1} b(\sum_{j \in J} \Delta_j(v)) + \sum_{i =1}^r \sum_{v \in \nabla_i^0}\sum_{i \in J \subset I} (-1)^{|J| + 1} b(\sum_{j \in J} \Delta_j(v)), \] then \ref{lemm1} and \ref{lemm2} imply: \begin{theo} \[ \chi(\Omega_{\widehat{V}}^1) = \chi({\cal O}_{\widehat{V}}^d) + \sum_{i =1}^r \sum_{J \subset I} (-1)^{|J| + 1} b(\Delta_i + \sum_{j \in J} \Delta_j ) + \] \[ + \sum_{i =1}^r \sum_{v \in \nabla_i^0}\sum_{i \not\in J \subset I} (-1)^{|J| + 1} b(\sum_{j \in J} \Delta_j(v)) + \] \[ + \sum_{i =1}^r \sum_{v \in \nabla_i^0}\sum_{i \in J \subset I} (-1)^{|J| + 1} b(\sum_{j \in J} \Delta_j(v)). \] \label{chi-formula} \end{theo} \section{Mirror duality for $\chi(\Omega^1)$} We prove that the involution $\Pi(\Delta) \leftrightarrow \Pi(\nabla)$ chahge the sign of $\chi(\Omega^1)$ by $(-1)^{d-r}$ (as it would follow from the expected duality $h^{1,q}(\widehat{V}) = h^{1,d-r-q}(\widehat{W})$, $0 \leq q \leq d-r$): \begin{theo} \[ \chi(\Omega_{\widehat{V}}^1) = (-1)^{d-r} \chi(\Omega_{\widehat{W}}^1). \] \label{chi-dual} \end{theo} For the proof we need some preliminary statements. \begin{prop} Let $i \in J \subset I$, $\Delta_i^0 = \Delta_i \cap {\cal V}(\nabla^*)$. Then a nonzero lattice point $w$ belongs to the relative interior of the polyhedron $\Lambda = \Delta_i + \sum_{j \in J} \Delta_j$ if and only if $w\in \Delta_i^0$ and the zero point $0 \in N$ belongs to the relative interior of $\sum_{j \not\in J} \nabla_j(w)$. Moreover, if this happens, then \[ {\rm dim}\,\left( \Delta_i + \sum_{j \in J} \Delta_j \right) + {\rm dim}\, \left( \sum_{j \not\in J} \nabla_j(w) \right) = d. \] \label{step1} \end{prop} \noindent {\em Proof.} First of all, let's check that the interior lattice point $w$ must belong to $\Delta_i.$ This means checking $\langle w, \nabla_k \rangle \geq 0$ for $k \neq i$ and $\langle w, \nabla_i \rangle \geq -1$, which follows easily from the fact that $w$ is a lattice point that belongs to $(1-\epsilon)\Lambda$ for some small positive $\epsilon.$ The polyhedron $\Lambda - w$ is defined by the inequalities \[ \langle x, v \rangle \geq -2 - \langle w, v \rangle, \; v \in \nabla_i, \] \[ \langle x, v \rangle \geq -1 - \langle w, v \rangle, \; v \in \nabla_j,\; j \in J,\; j \neq i, \] \[ \langle x, v \rangle \geq 0 - \langle w, v \rangle, \; v \in \nabla_j,\; j \not\in J. \] Since $\langle w, v \rangle \geq -1$ for $v \in \nabla_i$, and $\langle w, v \rangle \geq 0$ for $v \not\in \nabla_i$, only the inequalities $\langle x, v \rangle \geq 0$ for $v \in \nabla_j$, $\langle w , v \rangle = 0$ $(j \not\in J)$ give rise to nontrivial restrictions for the intersection of a small neighbourhood of $0 \in M_{\bf R}$ with $\Lambda - w$. Therefore, it remains to consider the halfspaces defined by the inequalities $\langle x, v \rangle \geq 0$ where $ v \in \nabla_j(w)$ $(j \not\in J)$. Denote by $L_w$ the convex cone in $M_{\bf R}$ defined by the inequalities \[ \langle x, v \rangle \geq 0, \;\; \mbox{ for all $v \in \nabla_j(w)$ and $j \not\in J$}. \] Then $0$ lies in the relative interior of $\Lambda - w$ if and only if $L_w$ is a linear subspace of $M_{\bf R}$. On the other hand, the cone \[ C_w = \sum_{j \not\in J} {\bf R}_{\geq 0} \nabla_j(w) \] is dual to $L_w$. Moreover, $C_w$ is a linear subspace in $N_{\bf R}$ if and only if $0$ is contained in the relative interior of $\sum_{j \not\in J} \nabla_j(w)$. It remains to note that a convex cone is a linear subspaces if and only if the dual cone is a linear subspace. In the latter case, ${\rm dim}\, L_w + {\rm dim}\, C_w = d$, i.e., ${\rm dim}\,\left(\Delta_i + \sum_{j \in J} \Delta_j \right) + {\rm dim}\, \left( \sum_{j \not\in J} \nabla_j(w) \right) = d$. \hfill $\Box$ \begin{coro} \[ \sum_{i =1}^r \sum_{J \subset I} (-1)^{|J| + 1} b(\Delta_i + \sum_{j \in J} \Delta_j ) = (-1)^{d-r} \left( \sum_{i =1}^r \sum_{w \in \Delta_i^0} \sum_{i \not\in J \subset I} (-1)^{|J| + 1} b(\sum_{j \in J} \nabla_j(w)) \right). \] \label{term2-3} \end{coro} \noindent {\em Proof.} Denote by $(-1)^{{\rm dim}\, \Theta}b^0(\Theta)$ the number of {\em nonzero} lattice points in the relative interior of a lattice polyhedron $\Theta$. Let $i \in J \subset I$, $J' = J \setminus \{i \}$. Note that $0$ is in the relative interior of $\Delta_i + \sum_{j \in J'} \Delta_j$ if and only if $0$ is in the relative interior of $\Delta_i + \sum_{j \in J} \Delta_j$. Since $|J| = |J'| +1$ and \[ {\rm dim}\,\left( \Delta_i + \sum_{j \in J'} \Delta_j \right) = {\rm dim}\, \left( \Delta_i + \sum_{j \in J} \Delta_j \right), \] we can have \[\sum_{i =1}^r \sum_{J \subset I} (-1)^{|J| + 1} b(\Delta_i + \sum_{j \in J} \Delta_j ) = \sum_{i =1}^r \sum_{i \in J \subset I} (-1)^{|J| + 1} b^0(\Delta_i + \sum_{j \in J} \Delta_j ). \] It remains to apply Proposition \ref{step1} and the property $|J| + |I \setminus J| = r$. \hfill $\Box$ \begin{prop} Let $i \in J \subset I$, and $v \in \nabla_i^0$ is a lattice point. Then a lattice point $w$ belongs to the relative interior of the polyhedron $\Lambda = \sum_{j \in J} \Delta_j(v)$ if and only $w \in \Delta_i^0$ and the lattice point $v$ belongs to the relative interior of $\nabla_i(w) + \sum_{j \not\in J} \nabla_j(w)$. Moreover, if this happens, then \[ {\rm dim}\, \left( \sum_{j \in J} \Delta_j(v) \right) + {\rm dim}\, \left( \nabla_i(w) + \sum_{j \not\in J} \nabla_j(w) \right) = d -1. \] \label{step2} \end{prop} \noindent {\em Proof.} We only need to prove the implication in one direction and the formula for the dimensions. By \ref{delta-v}, $\langle \Delta_j(v), v \rangle = 0$ for $j \in J$, $j \neq i$ and $\langle \Delta_i(v), v \rangle = -1$. Therefore, $\langle w, v \rangle = -1$, in particular, $w \neq 0$. Now we would like to prove that $w \in \Delta_i.$ Because of $\Lambda \subset \Delta$, this amounts to checking $\langle w, \nabla_k \rangle \geq 0$ for $k \neq i.$ Suppose there exists a vertex $v'$ of $\nabla_k$ such that $\langle w, v' \rangle = -1.$ Because $w$ lies in the relative interior of $\Lambda$, we get $\langle \Lambda, v' \rangle = -1.$ However, this is impossible, because $\Delta_j(v)$ contain zero if $j\neq i$, which leads to $\Delta_i(v) \subseteq \Lambda $. The polyhedron $\Lambda - w$ is defined by the conditions \[ \langle x, v' \rangle \geq -1 - \langle w, v' \rangle, \; v' \in \nabla_j,\; j \in J, \] \[ \langle x, v' \rangle \geq 0 - \langle w, v' \rangle, \; v' \in \nabla_j,\; j \not\in J, \] \[ \langle x, v \rangle = -1 - \langle w, v \rangle = 0. \] If we are interested in the neighbourhood of $0 \in M$, we are left with the inequalities $\langle x, v' \rangle \geq 0$, where $v'$ is either a vertex of $\nabla_j(w)$ $(j \not\in J)$, or $v'$ is the vertex of $\nabla_i(w)$, or $v'=-v$. The zero point is in the interior of $\Lambda - w$, hence the convex cone $L_w$ defined by these inequalities is a linear subspace. As a result, the dual to $L_w$ cone \[ C_w = {\bf R}_{\geq 0}\nabla_i(w)-{\bf R}_{\geq 0}v + \sum_{j \not\in J} {\bf R}_{\geq 0} \nabla_j(w). \] is also a linear subspace. Now we use \ref{delta-v} and the equality $\langle w, v \rangle = -1$ to conclude that $\{ y \in C_w | \langle w, y \rangle = 0\} $ is a linear subspace of dimension one less which equals \[{\bf R}_{\geq 0}\left(\nabla_i(w)-v\right) + \sum_{j \not\in J} {\bf R}_{\geq 0} \nabla_j(w). \] Because $\left(\nabla_i(w)-v\right)$ and $\nabla_j(w)$ contain zero, this implies that their sum has the same dimension and contains zero in its interior, which proves the first statement of the proposition. The formula for the dimensions follows from the duality of $L_w$ and $C_w.$ \hfill $\Box$ \begin{coro} \[ \sum_{i =1}^r \sum_{v \in \nabla_i^0} \sum_{i \in J \subset I} (-1)^{|J| + 1} b(\sum_{j \in J} \Delta_j(v)) = \] \[ = (-1)^{d-r} \left( \sum_{i =1}^r \sum_{w \in \Delta_i^0} \sum_{i \in J' \subset I} (-1)^{|J'| + 1} b(\sum_{j \in J'} \nabla_j(w)) \right). \] \label{term4} \end{coro} \noindent {\em Proof.} We set $J' = \{I \setminus J\} \cup \{i\}$. It remains to apply Proposition \ref{step2} and the property $|J| + |J'| = r +1$. \hfill $\Box$ \bigskip \noindent {\bf Proof of Theorem \ref{chi-dual}}: It remains to combine the statements \ref{o-dual}, \ref{term2-3} and \ref{term4} with the formula in \ref{chi-formula}. \hfill $\Box$ \bigskip \section{Complete intersections of ample divisors} Our next purpose is to give explicit formulas for $(*,1)$-Hodge numbers for $MPCP$-resolution $\widehat{V}$ of a Calabi-Yau complete intersection of ample divisors. Notice that in this case nef-partition is always irreducible. First, we glue together the following two exact sequences \[ 0 \rightarrow {\cal O}_{\widehat{V}}(-\widehat{Z}_1) \oplus \cdots \oplus {\cal O}_{\widehat{V}}(-\widehat{Z}_r) \rightarrow \Omega^1_{\widehat{\bf P}} \mid_{\widehat{V}} \rightarrow \Omega^1_{\widehat{V}} \rightarrow 0, \] \[ 0 \rightarrow \Omega^1_{\widehat{\bf P}} \mid_{\widehat{V}} \rightarrow {\cal O}_{\widehat{V}}^d \rightarrow \bigoplus_{v \in {\cal V}(\Delta^*)} {\cal O}_{{\bf D}(v) \cap {\widehat{V}}} \rightarrow 0 \] and obtain the complex \[ {\cal Q}^*\;: \; 0 \rightarrow {\cal O}_{\widehat{V}}(-\widehat{Z}_1) \oplus \cdots \oplus {\cal O}_{\widehat{V}}(-\widehat{Z}_r) \rightarrow {\cal O}_{\widehat{V}}^d \rightarrow \bigoplus_{v \in {\cal V}(\Delta^*)} {\cal O}_{ {\bf D}(v) \cap \widehat{V}} \rightarrow 0 \] whose cohomology ${\cal H}^i$ can be nontrivial only if $i =1$, and \[ {\cal H}^1 ({\cal Q}^*) \cong \Omega^1_{\widehat{V}}, \] i.e., the hypercohomology of ${\cal Q}^*$ coincides with the cohomology of $\Omega^1_{\widehat{V}}[1]$. \begin{prop} Assume that $\widehat{Z}_1, \ldots, \widehat{Z}_r$ are nef- and big-divisors; i.e., $\Delta_i$ is a Minkowski summand of $\Delta$ and ${\rm dim}\, \Delta_i = {\rm dim}\, \Delta = d$ $(i = 1, \ldots, r)$. Then \[ H^k({\cal O}_{\widehat{V}}(-\widehat{Z}_i)) = 0 \; \mbox{ for $i \neq d-r$ } \] and \[ h^{d-r}({\cal O}_{\widehat{V}}(-\widehat{Z}_i)) = \sum_{J \subset I} (-1)^{r - \mid J \mid } l^*(\Delta_i + \sum_{j \in J} \Delta_j). \] \label{coho1} \end{prop} \noindent {\em Proof. } Consider the Koszul resolution of ${\cal O}_{\widehat{V}}(-\widehat{Z}_i)$. Then the corresponding second spectral sequence degenerates in ${''}E_2$, because our assumptions on $\widehat{Z}_1, \ldots, \widehat{Z}_r$ imply ${''}E_1^{p,q} = 0$ for $q \neq d$. The latter immediately gives all statements. \hfill $\Box$ \begin{coro} Let ${\widehat{V}}$ be a complete intersection such that $d-r \geq 3$. Then, under assumptions in {\rm \ref{coho1}}, the second spectral sequence corresponding to the complex ${\cal Q}^*$ degenerates in ${''}E_2^*$, and one obtains the following relations \[ h^{d-r-1}(\Omega_{\widehat{V}}^1) = \sum_{i =1}^r h^{d-r}({\cal O}_{\widehat{V}}(-\widehat{Z}_i)) - d - \] \[ - \sum_{v \in {\cal V}(\Delta^*)} \left( h^{d-r-1}( {\cal O}_{{\bf D}(v) \cap {\widehat{V}}}) - h^{d-r-2}( {\cal O}_{{\bf D}(v) \cap {\widehat{V}}}) \right), \] \[ h^k(\Omega_{\widehat{V}}^1) = \sum_{v \in {\cal V}(\Delta^*)} h^{k-2}( {\cal O}_{{\bf D}(v) \cap {\widehat{V}}})\; \; \mbox{ for $2 \leq k \leq d-r-2$,} \] \[ h^1 (\Omega_{\widehat{V}}^1) = \sum_{v \in {\cal V}(\Delta^*)} h^{0}( {\cal O}_{{\bf D}(v) \cap {\widehat{V}}}) -d, \] \[ h^0(\Omega^1_{\widehat{V}}) = h^{d-r}(\Omega^1_{\widehat{V}}) =0 .\] \end{coro} \begin{prop} Assume that $\widehat{Z}_1, \ldots, \widehat{Z}_r$ are proper pullbacks of ample divisors on ${\bf P}_{\Delta}$; i.e., $\Delta_i$ and $\Delta$ are Minkowski summand of each other $( i =1, \ldots, r)$. Choose an element of $v \in {\cal V}(\Delta^*)$. Let $s$ be dimension of the minimal face $\Theta \subset \Delta^*$ containing $v$. Then the faces $\Delta_1(v), \ldots, \Delta_r(v)$ depend only on $\Theta$ $($we denote these faces by $\Theta^*_1, \ldots, \Theta^*_r$ $)$, and the following statements hold: {\rm (i)} If $d-r-s-1 >0$, then \[ h^{d-r-s-1}({\cal O}_{{\bf D}(v) \cap {\widehat{V}}}) = \sum_{J \subset I} (-1)^{r - \mid J \mid } l^*( \sum_{j \in J} \Theta_j^*), \] \[ h^0({\cal O}_{{\bf D}(v) \cap {\widehat{V}}}) =1, \] and \[ h^i({\cal O}_{{\bf D}(v) \cap {\widehat{V}}}) = 0\; \; \mbox{ for all $i \neq 0, d-r-s-1$}. \] {\rm (ii)} If $d-r-s-1 =0$, then \[ h^{0}({\cal O}_{{\bf D}(v) \cap {\widehat{V}}}) = 1 + \sum_{J \subset I} (-1)^{r - \mid J \mid } l^*( \sum_{j \in J} \Theta_j^*), \] and \[ h^i({\cal O}_{{\bf D}(v) \cap {\widehat{V}}}) = 0\; \; \mbox{ for all $i \neq 0$}. \] {\rm (iii)} If $d-r-s-1 <0$, then ${\bf D}(v) \cap {\widehat{V}}$ is empty. \label{coho2} \end{prop} \noindent {\em Proof. } Since $\widehat{Z}_1, \ldots, \widehat{Z}_r$ are proper pulbacks of ample divisors, the polyhedron $\Delta^*$ is combinatorially dual to each of $r+1$ polyhedra $\Delta$, $\Delta_1, \ldots, \Delta_r$. By this combinatorial duality, the faces $\Delta_i(v) \subset \Delta_i$ are dual to the face $\Theta$ and their arbitary sums have the same dimension $d-s-1$. In the sequel, we denote $\Delta_i(v)$ simply by $\Theta^*_i$. Consider the Koszul resolution of ${\cal O}_{{\bf D}(v) \cap {\widehat{V}}}$. Then the corresponding second spectral sequence degenerates in ${''}E_2$, because ${''}E_1^{p,q} = 0$ for $q \neq d-s-1, 0$, ${''}E_1^{0,p} = 0$ for $p \neq r$, and ${''}E_1^{0,r} \cong {\bf C}$. Now the statements (i)-(iii) are obvious. \hfill $\Box$ \begin{coro} Let ${\widehat{V}}$ be a complete intersection such that $d-r \geq 3$. Then, under assumptions in {\rm \ref{coho2}}, one obtains \[ h^{d-r-1}(\Omega_{\widehat{V}}^1) = \sum_{i =1}^r \left( \sum_{J \subset I} (-1)^{r - \mid J \mid } l^*(\Delta_i + \sum_{j \in J} \Delta_j) \right) - d - \] \[ - \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\, \Theta = 0} \\ {\scriptstyle \Theta \subset \Delta^*} \end{array}} \left(\sum_{J \subset I} (-1)^{r - \mid J \mid } l^*( \sum_{j \in J} \Theta_j^*) \right) + \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\, \Theta = 1} \\ {\scriptstyle \Theta \subset \Delta^*} \end{array}} l^*(\Theta) \cdot \left(\sum_{J \subset I} (-1)^{r - \mid J \mid } l^*( \sum_{j \in J} \Theta_j^*) \right), \] \[ h^k(\Omega_{\widehat{V}}^1) = \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\, \Theta = d-r-k -1} \\ {\scriptstyle \Theta \subset \Delta^*} \end{array}} l^*(\Theta) \cdot \left(\sum_{J \subset I} (-1)^{r - \mid J \mid } l^*( \sum_{j \in J} \Theta_j^*) \right)\; \mbox{ for $2 \leq k \leq d-r-2$,} \] \[ h^1 (\Omega_{\widehat{V}}^1) = {\rm Card}\{ \mbox{ {\rm lattice points in faces of dimension} $\leq d-r-1$} \} - d + \] \[ + \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\, \Theta = d-r-1} \\ {\scriptstyle \Theta \subset \Delta^*} \end{array}} l^*(\Theta) \cdot \left( \sum_{J \subset I} (-1)^{r - \mid J \mid } l^*( \sum_{j \in J} \Theta_j^*) \right), \] \[ h^0(\Omega^1_{\widehat{V}}) = h^{d-r}(\Omega^1_{\widehat{V}}) =0. \] \label{formulas} \end{coro} \begin{coro} Assume that $r =1$ and $d \geq 4$. Then the Hodge numbers $h^{p,1}(\widehat{V})$ have the following values \[ h^{0,1}(\widehat{V}) = h^{d-1,1}(\widehat{V}) = 0, \] \[ h^{1,1}(\widehat{V}) = l(\Delta^*) - d -1 - \left(\sum_{\begin{array}{c} {\scriptstyle \Theta^* \subset \Delta^*} \\ {\scriptstyle {\rm codim}\, \Theta^* =1} \end{array}} l^*(\Theta^*)\right) + \left(\sum_{\begin{array}{c} {\scriptstyle \Theta^* \subset \Delta^*} \\ {\scriptstyle {\rm codim}\, \Theta^* =2} \end{array}} l^*(\Theta^*)\cdot l^*(\Theta) \right), \] \[ h^{d-2, 1}(\widehat{V}) = l(\Delta) - d -1 - \left(\sum_{\begin{array}{c} {\scriptstyle \Theta^* \subset \Delta^*} \\ {\scriptstyle {\rm codim}\, \Theta^* =d}\end{array}} l^*(\Theta) \right)+ \left(\sum_{\begin{array}{c} {\scriptstyle \Theta^* \subset \Delta^*} \\ {\scriptstyle {\rm codim}\, \Theta^* =d-1}\end{array}} l^*(\Theta^*)\cdot l^*(\Theta) \right), \] \[ h^{p,1}(\widehat{V}) = \sum_{\begin{array}{c} {\scriptstyle \Theta^* \subset \Delta^*} \\ {\scriptstyle {\rm codim}\, \Theta^* =p+1}\end{array} }l^*(\Theta^*)\cdot l^*(\Theta) \\;\; \mbox{ for $2 \leq p \leq d-3$ }. \] \end{coro} \begin{prop} Let ${\widehat{V}}$ be a complete intersection of ample divisors $\widehat{Z}_1, \ldots, \widehat{Z}_r$ such that $d-r \geq 3$. Then one obtains \[ h^{d-r-1}(\Omega_{\widehat{V}}^1) = \sum_{i =1}^r \left( \sum_{J \subset I} (-1)^{r - \mid J \mid } l^*(\Delta_i + \sum_{j \in J} \Delta_j) \right) - d - \] \[ - \sum_{\begin{array}{c} {\scriptstyle {\rm dim}\, \Theta = 0} \\ {\scriptstyle \Theta \subset \Delta^*} \end{array}} \left(\sum_{J \subset I} (-1)^{r - \mid J \mid } l^*( \sum_{j \in J} \Theta_j^*) \right), \] \[ h^k(\Omega_{\widehat{V}}^1) = 0 \; \mbox{ for $k \neq 1,d-r-1$ ,} \] \[ h^1 (\Omega_{\widehat{V}}^1) = {\rm Card}\{ \mbox{ {\rm lattice points in faces of dimension} $\leq d-r-1$} \} - d. \] \label{formulas1} \end{prop} \noindent {\em Proof.} If $\widehat{Z}_1, \ldots, \widehat{Z}_r$ are ample, then ${\bf P}_{\Delta}$ has at most terminal singularities; i.e., $l^*(\Theta) = 0$ for all faces of $\Delta^*$ of positive dimension. This immediately implies the formulas. \hfill $\Box$ \begin{exam} {\sl Complete intersection $V_{d_1,d_2}$ of two hypersurfaces in ${\bf P}^5$.} {\rm For two cases below, we have \[ \sum_{ \begin{array}{c} {\scriptstyle {\rm codim}\, \Xi =1} \\ {\scriptstyle \Xi \subset \Delta} \end{array}} l^*(\Xi) = 30. \] \bigskip {\bf Case 1.} $d_1 = d_2 = 3$. Then $l(\Delta_1) = l(\Delta_2) = 56$, $l^*(2\Delta_1) = l^*(2\Delta_2) = 1$. Therefore, \[ h^{2,1}(V_{3,3}) = (112 - 7) - (30 + 1+1) + 0 = 73. \] {\bf Case 2.} $d_1 = 2$, $d_2 = 4$. Then $l(\Delta_1) = 21$, $l(\Delta_2) = 126$, $l^*(2\Delta_1) =0$, $l^*(2\Delta_2) = 21$. Therefore, \[ h^{2,1}(V_{2,4}) = (147 - 7) - (30 +21+0) + 0 = 89. \] } \end{exam} \begin{exam} {\sl Complete intersection $V_{2,2,3}$ in ${\bf P}^6$.} \[ h^{2,1}(V_{2,2,3}) = (28 + 28 + 84 - 9) - (7\cdot 6 + 1 +1 + 7 + 7 ) + 0 = 73. \] \end{exam} \begin{exam} {\sl Complete intersection $V_{2,2,2,2}$ in ${\bf P}^7$.} \[ h^{2,1}(V_{2,2,2,2}) = (4 \cdot 36 -11) - (8\cdot 7 + 3 \cdot 4 ) + 0 = 65. \] \end{exam} \section{Complete intersections in ${\bf P}^d$} Consider the case when a nef-partition $\Pi(\Delta) = \{\Delta_1, \ldots, \Delta_r\} $ defines a Calabi-Yau complete intersection $V = V_{d_1,\ldots, d_r}$ in projective space ${\bf P}^d$. This means that the polyhedra $\Delta_i = d_i \Lambda $ $(i = 1, \ldots, r)$ are $d_i$-multiples of a regular $d$-dimensional simplex $\Lambda$ and $d_1 + \cdots + d_r = d$. Our purpose is to compute the $(*,1)$-Hodge numbers of the mirror Calabi-Yau complete intersection $\widehat{W} = \widehat{Y}_1 \cap \cdots \cap \widehat{Y}_r$ defined by the dual nef-partition $\Pi(\nabla) = \{ \nabla_1, \ldots, \nabla_r \}$. In this case, $\nabla_i$ $(i = 1, \ldots, r)$ is a regular $d_i$-dimensional simplex. If for some $i$ we have $d_i = 1$, then the Calabi-Yau complete intersection in ${\bf P}^d$ reduces to a Calabi-Yau complete intersection in ${\bf P}^{d-1}$. So we can assume that $d_i \geq 2$ for all $i =1, \ldots r$. \begin{prop} In the above situation, one has \[ h^k({\cal O}( - \widehat{Y}_i - \sum_{j \in J} \widehat{Y}_j)) = 0\;\; \mbox{ for all $k$ } \] unless $J =I$, or $J \cup \{i \} = I$. Moreover \[ h^k({\cal O}( - \widehat{Y}_i - \sum_{j \in I} \widehat{Y}_j)) = 0\;\; \mbox{ for $k \neq d$ }, \] \[ h^d({\cal O}( - \widehat{Y}_i - \sum_{j \in I} \widehat{Y}_j)) = l(\nabla_i); \] and \[ h^k({\cal O}( - \sum_{j \in I} \widehat{Y}_j)) = 0\;\; \mbox{ for $k \neq d$ }, \] \[ h^d({\cal O}( - \sum_{j \in I} \widehat{Y}_j)) = 1. \] \end{prop} \noindent {\em Proof.} Note that the polyhedra $\nabla_i$, $\nabla_j$ ($ j \in J$) are regular simplices spanning {\em linearly independent} subspaces unless $i \in J$, $J =I$, or $J \cup \{i \} = I$. Therefore, $\nabla_i + \sum_{j \in J} \nabla_j$ is a regular simplex having no lattice points in its relative interior unless $i \in J$, $J =I$, or $J \cup \{i \} = I$. If $i \in J$, then $\nabla_i + \sum_{j \in J} \nabla_j$ has no lattice points in its relative interior since $l^*(2 \nabla_i) = 0$ for all $i \in I$ ($d_i = {\rm dim}\, \nabla_l \geq 2$). Hence the first statement follows from \ref{cohomology}. If $J = I$, or $j \cup \{ i \} = I$, then ${\rm dim}\,( \nabla_i + \sum_{j \in J} \nabla_j) = d$. By \ref{cohomology}, $h^k({\cal O}( - \widehat{Y}_i - \sum_{j \in I} \widehat{Y}_j)) = 0$ for $k \neq d$. The remaining statements follow from $l^*(\nabla_i + \nabla_1 + \cdots + \nabla_r) = l(\nabla_i)$ and $l^*(\nabla_1 + \cdots + \nabla_r) = 1$. \hfill $\Box$ \begin{coro} \[ h^k({\cal O}_{\widehat{W}}(-\widehat{Y}_i)) = 0 \; \mbox{ for $k \neq d-r$ } \] and \[ h^{d-r}({\cal O}_{\widehat{W}}(-\widehat{Y}_i)) = l(\nabla_i) -1. \] \end{coro} \noindent {\em Proof.} Tensoring the Koszul resolution ${\cal K}^*$ of ${\cal O}_{\widehat{W}}$ by ${\cal O}(- \widehat{Y}_i)$, we obtain the degenerated spectral sequence $``E^{p,q}$ from which one immediately obtains the statement. \hfill $\Box$ \begin{prop} For any vertex $w$ of $\nabla^*$ and any subset $J \subset I$, one has \[ l^*(\sum_{j \in J} \nabla_j(w)) = 0, \] unless ${\rm dim}\, \sum_{j \in J} \nabla_j(w) = 0$. \end{prop} \noindent {\em Proof.} We consider two cases: {\bf Case 1.} $J \neq I$. We know that $\sum_{j \in J} \nabla_j(w)$ is always a face of $\sum_{j \in J} \nabla_j$. On the other hand, the linear subspaces spanned by $\nabla_j$ are linearly independent. Since all $\nabla_i$ are regular simplices, there is no face of $\sum_{j \in J} \nabla_i$ of positive dimension containing a lattice point in its relative interior. {\bf Case 2.} $ J = I$. If there exists $ i \in I$ such that ${\rm dim}\, \nabla_i(w) = 0$, then setting $J' = J \setminus \{i \}$ we reduce all to Case 1. The polyhedron $\nabla$ has $d +2$ lattice points, but only those lattice points $v \in \nabla$ which satisfy the condition $\langle w,v \rangle \in \{0,-1\}$ might appear in $\nabla_j(w)$. Hence there exists a vertex of $\nabla$ which does not belong to any of $\nabla_i(w)$. Therefore linear subspaces spanned by $\nabla_i(w)$ are linearly independent. We again obtain the same statement, since all $\nabla_i(w)$ are regular simplices. \hfill $\Box$ Using the Koszul resolution, we obtain \begin{coro} \[ h^{k}({\cal O}_{{\bf D}(w) \cap \widehat{W}}) = 0\; \; \mbox{ for $k > 0$ } \] \end{coro} Therefore the second spectral sequence of the $3$-term complex ${\cal Q}^*$ degenerates in ${''}E_2$ and we have \begin{prop} Assume that $d - r \geq 3$. Then \[ h^k (\Omega^1_{\widehat{W}}) = 0 \; \mbox{ for $2 \leq d-r-2$ }, \] and \[ h^{d-r-1}(\Omega^1_{\widehat{W}}) = - d + \sum_{i=1}^r (l(\nabla_i) -1) = 1 = h^{1} (\Omega^1_V). \] \end{prop} Using the duality for the Euler characteristic from Theorem \ref{chi-dual} \[ - h^{1} (\Omega^1_{\widehat{W}}) + (-1)^{d-r-1} h^{d-r-1} (\Omega^1_{\widehat{W}}) = (-1)^{d-r} \left( - h^{1} (\Omega^1_V) + (-1)^{d-r-1} h^{d-r-1} (\Omega^1_{V}) \right), \] we obtain \[ h^{1} (\Omega^1_{\widehat{W}}) = h^{d-r-1}(\Omega^1_{V}). \] Therefore, we get the complete duality for $(1,q)$-Hodge numbers: \begin{theo} Let $V$ be a Calabi-Yau complete intersection of $r$ hypersurfaces in ${\bf P}^d$ and $d -r \geq 3$, $\widehat{W}$ a $MPCP$-desingularization of the Calabi-Yau complete intersection $W \subset {\bf P}_{\nabla}$ Then \[ h^{q} (\Omega^1_{\widehat{W}}) = h^{d-r-q}(\Omega^1_{V})\;\;\; \mbox{ for $0 \leq q \leq d-r$}. \] \end{theo}

9412

alg-geom/9412014

en

https://arxiv.org/abs/alg-geom/9412014

alg-geom/9412014

Ugo Bruzzo and Antony Maciocia

Hilbert schemes of points on some K3 surfaces and Gieseker stable bundles

8 pages, AMSLaTeX, no figures

By using a Fourier-Mukai transform for sheaves on K3 surfaces we show that for a wide class of K3 surfaces $X$ the punctual Hilbert schemes $\Hilb^n(X)$ can be identified, for all $n\geq 1$, with moduli spaces of Gieseker stable vector bundles on $X$ of rank $1+2n$. We also introduce a new Fourier-Mukai type transform for such surfaces.

alg-geom

\section*{Introduction} Let $(X,H)$ be a polarized K3 surface over ${\Bbb C}$ which also carries a divisor $\ell$ such that \begin{equation} H^2=2\,,\qquad\ell^2=-12\,,\qquad H\cdot\ell=0.\label{req}\end{equation} One must also include a technical condition which can be expressed in the form $H^0({\scrpt O}(\ell+2H))=0$. This will hold generically. There is an 18-dimensional family of such K3 surfaces which are called `reflexive' in \cite{BBHM}; these include generic Kummer surfaces. For any sheaf $\cal E$ on $X$ we denote its Mukai vector by $v(\cal E)$. Recall that $v({\cal E})=(\operatorname{rk}\cal E,c_1(\cal E),s(\cal E))$, where $$s(\cal E)=\operatorname{rk}\cal E+\o{ch}_2(\cal E)\,.$$ For such surfaces it can be shown that the moduli space ${\widehat X}=\cal M(2,\ell,-3)$ of Gieseker stable sheaves $\cal E$ on $X$ with Mukai vector $v(\cal E)=(2,\ell,-3)$ is isomorphic to $X$, and is formed by locally-free $\mu$-stable sheaves. The natural polarization on $\hat X$ is $\hat H$ which one can show is given by $2\ell+5H$ if we identify $\hat X$ with $X$. There is a divisor $\hat\ell$ on $\hat X$ which plays the role of $\ell$ and one can show that it is given by $-5\ell-12H$ if we identify $\hat X$ with $X$. In this paper we show the following result. \begin{thm} For any $n\geqslant 1$, the Hilbert scheme $\operatorname{Hilb}^n(X)$ of length-$n$ zero-di\-men\-sion\-al subschemes of $X$ is isomorphic, as an algebraic variety, to the moduli space ${\cal M}_n={\cal M}(1+2n,-n\hat\ell,1-3n)$ of Gieseker stable bundles on $X$ with respect to $\hat H$.\label{th:main} \end{thm} Notice that the theorem implies that all Gieseker stable sheaves are locally-free. It will also follow that they are never $\mu$-stable with respect to $\hat H$. In particular, the underlying smooth bundles do not carry anti-self-dual connections with respect to the K\"ahler metric associated to $\hat H$. The isomorphism $\operatorname{Hilb}^n(X)\simeq\cal M_n$ will be established by using a Fourier-Mukai (FM) transform of K3 surfaces (see \cite{BBH}). This FM transform preserves the natural complex symplectic structures of the moduli spaces \cite{Mac} and so $\operatorname{Hilb}^n(X)$ and $\cal M_n$ are isomorphic as complex hyperk\"ahler manifolds as well. Our results should be compared with those given in \cite{Z}, where for a general polarized K3 surface $(X,H)$ a birational map $\cal M(2,0,-1-n^2H^2)\to \operatorname{Hilb}^{2n^2H^2+3}(X)$ is constructed. Further birational identifications can be found in \cite{BC}. A similar version of Theorem 1 can be found in \cite{Mac2} for the case of an abelian surface. Our theorem is consistent with the results of \cite{GH} which show that moduli spaces of semi-stable torsion-free sheaves have the same Hodge numbers as Hilbert schemes of points of the K3 surface. \section{Fourier-Mukai Transforms for K3 Surfaces} Let us recall some definitions and properties related to the FM transform on $X$ introduced in \cite{Muk3}. Let $\cal Q$ denote the universal sheaf on $X\times{\widehat X}$ normalized by ${\bold R}\hat\pi_*({\cal Q})\cong{\scrpt O}_{{\widehat X}}[-1]$. This gives rise to a Fourier-Mukai transform $${\bold R}\Phi({\cal E})={\bold R}\hat\pi_*(\pi^*{\cal E}\tensor{\cal Q}).$$ In \cite{BBH} this is shown to give an equivalence of derived categories between the derived category $D(X)$ of complexes of coherent sheaves on $X$ and $D({\widehat X})$. The inverse is given by $${\bold R}\hat\Phi({\cal E})={\bold R}\pi_*(\hat\pi^*{\cal E}\tensor{\cal Q}^*)$$ up to a shift of complexes. \begin{dfn} We say that a sheaf $\cal E$ on $X$ is IT$_k$ if $H^j(X,\cal E\otimes\cal Q_\xi)=0$ for $j\neq k$, where $\cal Q_\xi=\cal Q\vert_{X\times\{\xi\}}$ for $\xi\in{\widehat X}$. We say that $\cal E$ is WIT$_k$ if $R^j\hat\pi_\ast(\pi^\ast\cal E\otimes\cal Q)=0$ for $j\neq k$, where $\pi$ and $\hat\pi$ are the projections of $X\times{\widehat X}$ onto the two factors. Obviously, any IT$_k$ sheaf is WIT$_k$. \end{dfn} For any WIT$_k$ sheaf $\cal E$ on $X$ we denote its FM transform as the sheaf on ${\widehat X}$ $$\widehat{\cal E}=R^k\Phi({\cal E}).$$ Note that $\cal E$ is IT$_k$ if and only if it is WIT$_k$ and its FM transform is locally free. The main properties of this FM transform are summarized as follows. \begin{prop} Let $\cal E$ be a sheaf on $X$. Then the Chern character of the transform of $\cal E$ is given by \begin{align*} \operatorname{ch}_0&=-\operatorname{ch}_0({\cal E})+c_1({\cal E})\cdot\ell+2\operatorname{ch}_2({\cal E})\\ \operatorname{ch}_1&=-c_1-(c_1\cdot H+5\operatorname{ch}_2({\cal E}))\hat\ell +(c_1\cdot\ell-2c_1\cdot H)\hat H\\ \operatorname{ch}_2&=-5\operatorname{ch}_2({\cal E})-2c_1\cdot\ell, \end{align*} where $c_1=c_1({\cal E})$. {}From this it follows that $\deg{\bold R}\Phi{\cal E}=-\deg\cal E$ and $\chi({\bold R}\Phi{\cal E})=-\chi({\cal F})$. \label{topinv}\end{prop} \begin{prop} {\rm (Invertibility)} Let $\cal E$ be a WIT$_k$ sheaf on $X$. Then its FM transform $\widehat{\cal E}=R^k\hat\pi_\ast(\pi^\ast\cal E\otimes\cal Q)$ is a WIT$_{2-k}$ sheaf on ${\widehat X}$, whose inverse FM transform $R^{2-k}\pi_\ast(\hat\pi^\ast\widehat{\cal E}\otimes{\cal Q}^\ast)$ is isomorphic to $\cal F$. \end{prop} \section{The isomorphism between the Hilbert scheme and the moduli space} We give a definition that extends to K3 surfaces the notion of homogeneous bundle on an abelian variety (cf.\ \cite{Muk1}). \begin{dfn} A coherent sheaf $\cal F$ on ${\widehat X}$ is {\em quasi-homogeneous} if it has a filtration by sheaves of the type $\widehat{{\scrpt O}_Y}$, where the $Y$'s are zero-dimensional subschemes of $X$, so that the associated grading is of the form $\oplus_k\cal Q_{p_k}$, with $p_k\in X$. \end{dfn} Let $W$ be a zero-dimensional subscheme of $X$; the structure sheaf ${\scrpt O}_W$ is IT$_0$, and its FM transform $\widehat{\scrpt O}_W$ is a quasi-homogeneous locally free sheaf. $\widehat{{\scrpt O}}_W$ is $\mu$-semistable, due to the following result. \begin{prop} Let $p\in X$. Any nontrivial extension \begin{equation} 0 \longrightarrow \cal Q_p \longrightarrow \cal F \longrightarrow \cal Q_p \longrightarrow 0 \label{ext} \end{equation} is $\mu$-semistable. Any destabilizing $\mu$-semistable subsheaf of $\cal F$ is isomorphic to $\cal Q_p$. \label{prop1}\end{prop} \begin{pf} The first part is standard. For the second part one observes that any torsion-free destabilising sheaf ${\cal Q}'$ of ${\cal F}$ which is $\mu$-semistable must have Chern character $(2,\ell,-5)$ because both ${\cal Q}'$ and ${\cal F}/{\cal Q}'$ must satisfy the Bogomolov inequality. Then it must also be locally-free by the Bogomolov inequality applied to ${\cal Q}'{}^{**}$. \end{pf} \begin{lemma} The ideal sheaf $\cal I_W$ is IT$_1$. \end{lemma} \begin{pf} Let $\xi\in{\widehat X}$. Then $H^0(X,\cal I_W\otimes\cal Q_\xi)\hookrightarrow H^0(X,\cal Q_\xi)=0$ because $\cal Q_\xi$ is $\mu$-stable, and $$H^2(X,\cal I_W\otimes\cal Q_\xi)^\ast\simeq \o{Ext}^0(\cal I_W\otimes\cal Q_\xi,{\scrpt O}_X)\simeq \o{Hom}(\cal I_W,\cal Q_\xi^\ast)=0$$ since $\cal Q_\xi$ is locally free. \end{pf} So by applying the FM transform to the sequence $$ 0 \longrightarrow \cal I_W \longrightarrow {\scrpt O}_X \longrightarrow {\scrpt O}_W \longrightarrow 0$$ we get \begin{equation} 0 \longrightarrow \widehat{\scrpt O}_W \longrightarrow \widehat{\cal I}_W \longrightarrow {\scrpt O}_{\widehat X} \longrightarrow 0.\label{due} \end{equation} \begin{prop} The FM transform $\widehat{\cal I}_W$ is Gieseker stable. \end{prop} \begin{pf} Since $\o{ch}(\widehat{\cal I}_W)=(1,0,-n)$, by the formulas in Proposition \ref{topinv} we obtain $$\operatorname{rk}\widehat{\cal I}_W=1+2n,\quad\operatorname{ch}_2\widehat{\cal I}_W=-5n,\quad\tilde p(\widehat{\cal I}_W)=\frac{\chi(\widehat{\cal I}_W)}{\operatorname{rk}\widehat{\cal I}_W}= \frac{2-n}{1+2n}>-\frac12.$$ Let $\cal A$ be a destabilizing subsheaf of $\widehat{\cal I}_W$, that we may assume to be Gieseker stable with a torsion-free quotient. Then we have $\tilde p(\cal A)\geqslant\tilde p(\widehat{\cal I}_W)>-\frac12$. Let $f$ denote the composite ${\cal A}\to\widehat{\cal I}_W\to{\scrpt O}_{{\widehat X}}$. There are two cases:\\ Case (i) $f=0$. Then there is a map $\cal A\to\widehat{\scrpt O}_W$. Let $g_k\colon\cal A\to\cal Q_{p_k}$ be the composition of this map with the canonical projection onto $\cal Q_{p_k}$. Since $\tilde p(\cal A)>-\frac12=\tilde p(\cal Q_{p_k})$ and both sheaves are Gieseker stable, we obtain $g_k=0$ for all $k$, which is absurd.\\ Case (ii) $f\ne 0$. We divide this into two further cases: $\operatorname{rk}\cal A=1$ and $\operatorname{rk}\cal A>1$. If $\operatorname{rk}\cal A=1$ we have $\cal A^\ast\simeq{\scrpt O}_{{\widehat X}}$; hence the sequence (\ref{due}) splits, which contradicts the inversion theorem $\widehat{\widehat{\cal I}_W}\simeq\cal I_W$. If $\operatorname{rk}\cal A>1$ we consider the exact sequences $$ 0 \longrightarrow \cal K_1 \longrightarrow \cal A \lRa{h} \cal B \longrightarrow 0\quad\mbox{and}\quad 0 \longrightarrow \cal B \longrightarrow {\scrpt O}_{{\widehat X}} \longrightarrow \cal K_2 \longrightarrow 0,$$ where $\operatorname{rk}{\cal K}_2=0,1$. If $\operatorname{rk}{\cal K}_2=1$ then $\cal B=0$, i.e.\ $f=0$ which is absurd, so that $\operatorname{rk}{\cal K}_2=0$, and $\cal B$ has rank one. We have an exact commuting diagram \begin{equation} \begin{CD} @. 0 @. 0 @. 0 @.\\ @. @AAA @AAA @AAA @.\\ 0 @>>> {\cal K}_3 @>>> {\cal K}_4 @>>> {\cal K}_2 @>>> 0 \\ @. @AAA @AAA @AAA @.\\ 0 @>>> \widehat{\scrpt O}_W @>>> \widehat{\cal I}_W @>>> {\scrpt O}_{{\widehat X}} @>>> 0 \\ @. @AgAA @AAA @AAh'A @.\\ 0 @>>> {\cal K}_1 @>>> {\cal A} @>h>> \cal B @>>> 0\\ @. @AAA @AAA @AAA @.\\ @. 0 @. 0 @. 0 @. \end{CD}\label{diagtwo} \end{equation} with $\mu({\cal K}_1)=0$, $0<\operatorname{rk}{\cal K}_1<2n$ and $f=h'\raise2pt\hbox{$\scriptscriptstyle\circ$} h$. If $n=1$ then $\widehat{\scrpt O}_W$ is $\mu$-stable, but this is a contradiction. For $n>1$, we may assume that $\cal K_1$ is $\mu$-semistable so that it is a direct summand of $\widehat{\scrpt O}_W$. Then $\cal K_3$ is locally free and $\operatorname{rk}\cal K_1\geqslant 2$. Moreover, $\mu(\cal B)\leqslant 0$ because $\cal B$ injects into ${\scrpt O}_{{\widehat X}}$, and $\mu(\cal B)\geqslant 0$ because $\mu(\cal K_1)\leqslant 0$. Then $\mu(\cal B)= \mu(\cal K_1)=0$. Since $\cal K_3$ is locally free the support of $\cal K_2$ is not zero-dimensional. So $\mu(\cal B)= 0$ implies $\cal K_2=0$ and $\cal K_3\simeq\cal K_4$. Finally, we consider the middle column in (\ref{diagtwo}). The sheaf $\cal A$ has rank greater than 2, and is Gieseker stable, so that it is IT$_1$. But $\widehat{\cal I}_W$ is WIT$_1$ while $\cal K_4$ is WIT$_2$. Then ${\cal A}\simeq\widehat{\cal I}_W$, but this is a contradiction. \end{pf} Note that $\widehat{\cal I}_W$ is never $\mu$-stable because (\ref{due}) destabilizes it. Let $\cal M_n$ be the moduli space $\cal M(1+2n,-n\hat\ell,1-3n)$ of Gieseker stable sheaves on ${\widehat X}$. The previous construction yields a map $\operatorname{Hilb}^n(X)\to\cal M_n$. This map is algebraic because the Fourier-Mukai transform is functorial and so preserves the Zariski tangent spaces (see \cite{Mac}). Another way to see this is to observe that the Fourier-Mukai transforms give a natural isomorphism of moduli functors and so give rise to an isomorphism of (coarse or fine) moduli schemes.\footnote{We would like to thank Daniel Hern\'andez Ruip\'erez for this observation.} We shall now show that the Fourier-Mukai transform is a surjection up to isomorphism. \begin{lemma} Any element $\cal F\in\cal M_n$ is WIT$_1$. \end{lemma} \begin{pf} Since $\tilde p(\cal F)>-\frac12$ and $\tilde p(\cal Q_p)=-\frac12$ there is no map $\cal F\to\cal Q_p$. This means that $H^2({\widehat X},\cal F\otimes\cal Q^\ast_p)=0$. We consider now nonzero morphisms $\cal Q_p\to\cal F$. Any such map is injective; otherwise it would factorize through a rank-one torsion-free sheaf $\cal B$ with $\mu(\cal B)>0$ (because $\cal Q_p$ is $\mu$-stable) and $\mu(\cal B)\leqslant 0$ (because $\cal F$ is $\mu$-semistable), which is impossible. Then $\cal Q_p$ is a locally free element of a Jordan-Holder filtration of $\cal F$. Since any such filtration has only a finite number of terms, and the associated grading $\o{gr}(\cal F)^{\ast\ast}$ is unique, there is only a finite number of $p$'s giving rise to nontrivial morphisms, i.e.\ $\o{Hom}(\cal Q_p,\cal F)\simeq H^0(X,\cal F\otimes\cal Q_p^\ast)$ does not vanish only for a finite set of points $p$. This suffices to prove that $\cal F$ is WIT$_1$ due to Proposition 2.26 of \cite{Muk3}. \end{pf} \begin{prop} The FM transform $\widehat{\cal F}$ of $\cal F$ is torsion-free. \end{prop} \begin{pf} Let $\cal T$ be the torsion subsheaf of $\widehat{\cal F}$, so that one has an exact sequence \begin{equation} 0 \longrightarrow \cal T \longrightarrow \widehat{\cal F} \longrightarrow \cal G \longrightarrow 0.\label{tor} \end{equation} Since $\cal T$ is supported at most by a divisor, and $\widehat{\cal F}$ is WIT$_1$, the sheaf $\cal T$ is WIT$_1$ as well. Moreover $\deg(\cal T)\geqslant 0$. If $\deg\cal T=0$ then $\cal T$ is IT$_0$, i.e.\ $\cal T=0$. Hence, we assume $\deg(\cal T)> 0$. The rank-one sheaf $\cal G$ is torsion-free and, by imbedding it into its double dual, we see that it is IT$_1$. Then, applying ${\bold R}\hat\Phi$ to (\ref{tor}), we get $$ 0 \longrightarrow \widehat{\cal T} \longrightarrow \cal F \longrightarrow \widehat{\cal G} \longrightarrow 0.$$ Since $\cal F$ is $\mu$-semistable we see that $$\deg\cal T=\deg \widehat{\cal T} \leqslant 0,$$ which is a contradiction. \end{pf} Now the Chern character of $\widehat{\cal F}$ is $(1,0,-n)$, so that it is the ideal sheaf of a zero-dimensional subscheme of $X$ of length $n$. We have therefore shown that the Fourier-Mukai transform surjects as a map $\operatorname{Hilb}^nX\to{\cal M}_n$. The inversion theorem for ${\bold R}\Phi$ therefore implies that the transform gives an isomorphism of smooth varieties. This establishes Theorem \ref{th:main}. \section{Another Fourier-Mukai Transform} We shall now show that ${\cal M}_1$ gives rise to another FM transform. \begin{lemma} $\dim H^1({\cal Q})=1$. \end{lemma} \begin{pf} This follows immediately from the degeneration of the Leray spectral sequence applied to $\hat\pi$ and the fact that ${\bold R}\hat\pi_*({\cal Q})={\scrpt O}_{{\widehat X}}[-1]$. \end{pf} This lemma shows that there is a unique extension \begin{equation} 0\longrightarrow{\cal Q}\longrightarrow{\cal E}\longrightarrow{\scrpt O}_{X\cross{\widehat X}}\lra0.\label{psiE} \end{equation} Both of the restrictions of ${\cal E}$ to the factors of $X\cross{\widehat X}$ give families of Gieseker stable bundles. Then the general theory of FM transforms (see \cite{Mac2}) implies that ${\cal E}$ gives rise to an FM transform which we denote by ${\bold R}\Psi$. \begin{prop}The FM transform ${\bold R}\Psi$ and its inverse ${\bold R}\hat\Psi$ satisfy the following: \begin{enumerate} \item ${\bold R}\Psi{\scrpt O}_X\cong{\scrpt O}_{{\widehat X}}[-2]$, \item $\operatorname{ch}({\bold R}\Psi{\cal F})=\chi({\cal F})\operatorname{ch}({\scrpt O}_{{\widehat X}}) +\operatorname{ch}({\bold R}\Phi{\cal F})$, where ${\cal F}$ is any sheaf on $X$ and \item ${\bold R}\hat\Psi{\cal I}_p\cong{\cal Q}^*_p[-1]$, for $p\in{\widehat X}$. \end{enumerate} \end{prop} \begin{pf} Apply ${\bold R}\hat\pi_*$ to the short exact sequence \ref{psiE} to obtain a long exact sequence. Note that ${\bold R}\hat\pi_*{\scrpt O}_{X\cross{\widehat X}}$ is concentrated in the 0th and 2nd positions and ${\bold R}\Phi{\cal Q}={\scrpt O}_{{\widehat X}}[-1]$. Then part (1) follows immediately. For the second part just twist \ref{psiE} by $\pi^*{\cal F}$ and apply ${\bold R}\hat\pi_*$. Then the formula follows from the facts that the Chern character is additive with respect to triangles in $D({\widehat X})$ and $${\bold R}\hat\pi_*(\pi^*{\cal E})={\bold R}\Gamma(X,{\cal E})\tensor{\scrpt O}_{{\widehat X}}$$ by the projection formula, where $\Gamma$ denotes the sections functor. By (1) we have ${\bold R}\hat\Psi{\scrpt O}_{{\widehat X}}={\scrpt O}_X$. We also have ${\bold R}\hat\Psi{\scrpt O}_p\cong{\cal E}^*_p$ from the definition of ${\bold R}\hat\Psi$. Then when we apply ${\bold R}\hat\Psi$ to the structure sequence of $p$ we obtain the short exact sequence $$0\longrightarrow {\scrpt O}_X\longrightarrow{\cal E}^*_p\longrightarrow R^1\hat\Psi{\cal I}_p\lra0.$$ This is just the dual of (\ref{due}). This completes the proof. \end{pf} Note that ${\cal I}_W$ does not satisfy WIT with respect to ${\bold R}\Psi$ but ${\cal I}_{\hat W}$ does satisfy WIT$_1$ with respect to ${\bold R}\hat\Psi$ and the transform is just the dual of the corresponding quasi-homogeneous bundle. \section{Concluding remarks} Since both the moduli spaces and the punctual Hilbert schemes have complex symplectic structures which are given by the cup product on $\operatorname{Ext}^1(E,E)$ the FM transforms will preserve the symplectic structures and so are complex symplectic isomorphisms. It follows immediately that the spaces are hyperk\"ahler isometric as well. Theorem \ref{th:main} has several immediate consequences which we can state in the following theorem. \begin{thm} Let $X$ be a reflexive K3 surface. \begin{enumerate} \item The moduli space ${\cal M}_n$ of Gieseker stable sheaves on $X$ is connected and projective. All points of ${\cal M}_n$ are locally-free. \item ${\cal M}_n$ contains no $\mu$-stable sheaves and so the moduli space of irreducible U$(2n+1)$-instantons, with fixed determinant ${\scrpt O}(\hat\ell)^{-n}$ and second Chern character $-5n$, is empty. \item The moduli space of all instantons with this type is isomorphic to the $n^{th}$ symmetric product $S^nX$. \end{enumerate} \end{thm} For the last part one uses the fact that any $\mu$-semistable sheaf of the given Chern character admits a surjection to ${\scrpt O}_X$ and so fits into a sequence of the form (\ref{due}).

9412

alg-geom/9412009

en

https://arxiv.org/abs/alg-geom/9412009

alg-geom/9412009

Michael J. Falk

Michael Falk, Hiroaki Terao

$\beta$-nbc bases for cohomology of local systems on hyperplane complements

minor additions to introduction and references, 14 pages. AMSLaTeX v1.2

We study cohomology with coefficients in a rank one local system on the complement of an arrangement of hyperplanes $\A$. The cohomology plays an important role for the theory of generalized hypergeometric functions. We combine several known results to construct explicit bases of logarithmic forms for the only non-vanishing cohomology group, under some nonresonance conditions on the local system, for any arrangement $\A$. The bases are determined by a linear ordering of the hyperplanes, and are indexed by certain ``no-broken-circuits" bases of $\A$. The basic forms depend on the local system, but any two bases constructed in this way are related by a matrix of integer constants which depend only on the linear orders and not on the local system. In certain special cases we show the existence of bases of monomial logarithmic forms.

alg-geom

\section{Special cases} In this section we assume that none of the $\lambda(X)$ is a nonnegative integer for dense $X \in L({\mathcal A}_{\infty}) - \{{\bf P}^{\ell} \}$. Recall \[ \omega_{B} = \omega_{i_{1} }\cdots \omega_{i_{r}} \] for any ordered base $B = (H_{i_{1} }, \ldots, H_{i_{r}})$. Since $\omega_{B}$ is a ``monomial'' while $\Xi(B)$ is a sum of several $\omega_{B}$, it might be desirable that the set $\{\left[\omega_{B}\right] ~|~ B\in \beta{\rm {\bf nbc}}({\mathcal A})\}$ form a basis for $H^{r}(M, {\mathcal L}_{\lambda})$. This will not hold in general, even for arrangements of rank two. In this section, we will prove that these monomial forms give a basis in two special situations: \begin{itemize} \item when the linear order on ${\mathcal A}$ is unmixed (Def. \ref{unmixeddef}), \item when the linear order on ${\mathcal A}$ is admissible (Def. \ref{admissibledef}) and $r=2.$ \end{itemize} \noindent The first case contains arrangements in general position, arrangements in general position to infinity, and normal arrangements among others. \begin{definition} \label{unmixeddef} {For each maximal $X\in L$, let \[ {\rm {\bf nbc}}_{X} := \{B \in{\rm {\bf nbc}} ~|~ \cap B = X\}. \] We say that $X$ is {\em unmixed} if either ${\rm {\bf nbc}}_{X}\subseteq \beta{\rm {\bf nbc}}$ or ${\rm {\bf nbc}}_{X}\cap \beta{\rm {\bf nbc}} = \emptyset.$ A linear order on ${\mathcal A}$ is {\em unmixed} if any maximal $X\in L$ is unmixed.} \end{definition} \begin{theorem} \label{unmixedtheorem} If the linear order on ${\mathcal A}$ is unmixed, then the set $\{\left[\omega_{B}\right] ~|~ B\in \beta{\rm {\bf nbc}}({\mathcal A})\}$ is a basis for $H^{r}(M, {\mathcal L}_{\lambda})$. \end{theorem} \begin{proof} Let $X\in L$ be an arbitrary maximal element with ${\rm {\bf nbc}}_{X} \subseteq \beta{\rm {\bf nbc}}.$ Then each $\Xi(B), ~B\in {\rm {\bf nbc}}_{X},$ belongs to $$ \sum_{B'\in {\rm {\bf nbc}}_{X}} \mathbb C \omega_{B'} \subseteq \sum_{B'\in \beta{\rm {\bf nbc}}} \mathbb C \omega_{B'}.$$ The result follows from Theorem \ref{main} and a dimension argument. \end{proof} \begin{example} \label{} {An arrangement ${\mathcal A}$ is said to be {\em in general position} if (i) $n = |{\mathcal A}| \geq \ell + 1$, (ii) $\mbox{\rm codim} (H_{i_{1}}\cap \cdots \cap H_{i_{k}}) = k$ whenever $1\leq k\leq \ell,$ and (iii) $H_{i_{1}}\cap \cdots \cap H_{i_{k}} = \emptyset$ whenever $k > \ell$. Let $X\in L$ be a maximal element. Then ${\rm {\bf nbc}}_{X} $ is a singleton. Thus any linear order on ${\mathcal A}$ is unmixed. We have \[ \beta{\rm {\bf nbc}}({\mathcal A}) = \{(H_{i_{1}}, \ldots, H_{i_{\ell}}) ~|~ 1 < i_{1} < \cdots < i_{\ell} \leq n \}, ~~\mbox{and}\] \[ \Xi(B) = \lambda_{i_{1}}\cdots \lambda_{i_{\ell}} \omega_{B}, \] where $B=(H_{i_{1}}, \ldots, H_{i_{\ell}}) \in\beta{\rm {\bf nbc}}({\mathcal A})$. This basis coincides with the basis constructed in \cite[p.292]{Aom1}.} \end{example} \begin{example} \label{} {An arrangement ${\mathcal A}$ is said to be {\em normal} \cite[1.4]{Var1} if $|{\mathcal A}_{X}| = \mbox{\rm codim}(X)$ for all $X\in L({\mathcal A})$. (It is equivalent to say that $\bigcup {\mathcal A}$ is a normal crossing divisor in $V$.) Arrangements in general position are normal. Let $X\in L$ be a maximal element. Then ${\rm {\bf nbc}}_{X} $ is again a singleton. Thus any linear order on ${\mathcal A}$ is unmixed. We have \[ \Xi(B) = \lambda_{i_{1}}\cdots \lambda_{i_{r}} \omega_{B}, \] where $B=(H_{i_{1}}, \ldots, H_{i_{r}}) \in\beta{\rm {\bf nbc}}({\mathcal A})$.} \end{example} \begin{example} \label{} {An arrangement ${\mathcal A}$ is said to be {\em in general position to infinity} \cite[6.2]{Yuz1} if $H_{i_{1}}\cap \cdots \cap H_{i_{k}} \neq \emptyset$ whenever $1 \leq k \leq \ell$. This implies that there are no parallels among the elements of $L$, so that the hyperplane at infinity is generic relative to ${\mathcal A}$. In particular, general position arrangements are in general position to infinity. Let $X\in L$ be a maximal element. Note that ${\mathcal A}_{X}$ is a central generic arrangement \cite[5.22]{OrT1}. We have ${\rm {\bf nbc}}_{X}\cap \beta{\rm {\bf nbc}} = \emptyset$ if and only if $X\subseteq H_{1}$. Otherwise ${\rm {\bf nbc}}_{X}\subseteq \beta{\rm {\bf nbc}}.$ Thus any linear order on ${\mathcal A}$ is unmixed. We have \[ \beta{\rm {\bf nbc}}({\mathcal A}) = \{(H_{i_{1}}, \ldots, H_{i_{r}}) ~|~ 1 < i_{1} < \cdots < i_{r} \leq n, i_{1} = \min {\mathcal A}_{X}, X = H_{i_{1}\cdots i_{r}}\}. \] } \end{example} \begin{example} \label{} {Let $H_{1} \in {\mathcal A}$ be generic, that is, $H_{1} $ transversely intersects $Y$ unless $Y\in L({\mathcal A} - \{H_{1} \})$ is maximal in $L({\mathcal A} - \{H_{1} \})$. Let $X\in L$ be a maximal element. Then ${\rm {\bf nbc}}_{X}\cap \beta{\rm {\bf nbc}} = \emptyset$ if and only if $X\subseteq H_{1}$. Otherwise ${\rm {\bf nbc}}_{X}\subseteq \beta{\rm {\bf nbc}}.$ Thus any linear order on ${\mathcal A}$ in which $H_{1}$ is the first hyperplane is unmixed. In this case, we have \[ \beta{\rm {\bf nbc}}({\mathcal A}) = \{(H_{i_{1}}, \ldots, H_{i_{r}})\in {\rm {\bf nbc}}({\mathcal A}) ~|~ 1 < i_{1} < \cdots < i_{r} \leq n \}. \] } \end{example} \begin{remark} {Suppose that ${\mathcal A}$ is complexified real and $r=\ell$. Then the number of bounded chambers of the real form of ${\mathcal A}$ is equal to $|\beta{\rm {\bf nbc}}|$. In this case Varchenko \cite[6.2]{Var1} associated a differential $\ell$--form $\eta_{\Delta}\in A^{\ell}({\mathcal A})$ (which is not necessarily a monomial) to each bounded chamber $\Delta$. Recall the definition of $U_{\lambda}$ from section 1 and define $\omega_{\Delta} = U_{\lambda} \eta_{\Delta}.$ The form $\omega_{\Delta} $ is called the hypergeometric form associated to $\Delta.$ The hypergeometric integrals give a determinant $\det \left[ \int_{\Gamma} \omega_{\Delta}\right].$ The main results in \cite{Var1} are beautiful formulas for the determinant when ${\mathcal A}$ is in general position (Thm.1.1), normal (Thm.1.4), or in general position to infinity (Thm.6.1). In particular, these determinants are nonzero, so that in these cases the set $\{\eta_{\Delta}\}$ gives a basis for $H^{\ell}(M, {\mathcal L}_{\lambda}).$ The relationship between $\{\eta_{\Delta}\}$ and $\{\Xi(B)\}$ is intriguing. They are not the same in general but coincide when ${\mathcal A}$ is normal or in general position to infinity.} \label{Varch} \end{remark} \noindent Finally we specialize to the case $r=2$. \begin{definition} \label{admissibledef} {The linear order on ${\mathcal A}$ is called {\em admissible} if there exists an integer $\nu$ such that $H_{i}$ and $H_{1} $ are parallel if and only if $1\leq i < \nu.$} \end{definition} If the linear order on ${\mathcal A}$ is admissible, then it is not difficult to see \[ \beta{\rm {\bf nbc}} ({\mathcal A}) = \{ (H_{i_{1} }, H_{i_{2} }) \in {\rm {\bf nbc}} ~|~ 1 < i_{1} < i_{2} \neq \nu\}. \] \begin{proposition} \label{admissibletheorem} Suppose ${\mathcal A}$ is an arrangement of rank 2 with admissible linear order. Then the set $\{ \left[\omega_{B} \right]~|~B\in \beta{\rm {\bf nbc}}\}$ gives a basis for $H^2 (M, {\mathcal L}_{\lambda}).$ \end{proposition} \begin{proof} Since $\dim H^{2}(M, {\mathcal L}_{\lambda}) = |\beta{\rm {\bf nbc}}|,$ it suffices to show that the set $\{ \left[\omega_{B} \right] ~|~B\in\beta{\rm {\bf nbc}}\}$ spans $H^{2}(M, {\mathcal L}_{\lambda})$. Define \[ N := \sum_{B\in\beta{\rm {\bf nbc}}} \mathbb C \omega_{B} + d_{\lambda} (A^{1}). \] We want to show $N = A^{2}. $ By Theorem \ref{main}, it is enough to show that $\Xi(B) \in N$ for all $B\in\beta{\rm {\bf nbc}}$. Let $B\in\beta{\rm {\bf nbc}}$ and $X = \cap B$. If $X$ is unmixed, then \[ \Xi(B) \in \sum_{B'\in{\rm {\bf nbc}}_{X}} \mathbb C \omega_{B'} \subseteq \sum_{B'\in\beta{\rm {\bf nbc}}} \mathbb C \omega_{B'} \subseteq N. \] Suppose that $X$ is mixed. Then $B = (H_{i}, H_{j}) \in \beta{\rm {\bf nbc}}$ with $1 < i < \nu < j$ and $X = H_{i} \cap H_{\nu} = H_{i} \cap H_{j}$. Note that $(H_{i}, H_{p}) \in\beta{\rm {\bf nbc}}$ for all $p > \nu.$ Thus $\omega_{p}\omega_{q} = \omega_{i}\omega_{q} - \omega_{i}\omega_{p} \in N $ if $\nu \not\in \{p, q\}$ and $H_{p} \cap H_{q} = X$. Therefore we have the following congruence relations modulo $N$: \begin{eqnarray*} \Xi(B) &=& \omega_{\lambda}(X) \wedge \lambda_{j}\omega_{j} \equiv \lambda_{\nu}\omega_{\nu} \wedge \lambda_{j} \omega_{j}\\ &=& \lambda_{\nu}\lambda_{j} (\omega_{ij} - \omega_{i \nu}) \equiv - \lambda_{\nu}\lambda_{j} \omega_{i \nu} \equiv \lambda_{j} d_{\lambda}(\omega_{i}) \equiv 0. \end{eqnarray*} This proves $\Xi(B)\in N.$ \end{proof} \noindent Proposition \ref{admissibletheorem} was independently proved by M. Kita \cite{Kit1}. He gives a direct proof which doesn't use Theorem \ref{main}. If the linear order is not admissible, the set $\{ \left[\omega_{B} \right] ~|~B\in\beta{\rm {\bf nbc}}\}$ does not give a basis for $H^{2} (M, {\mathcal L}_{\lambda})$ in general unless we impose additional unnatural genericity conditions on $\lambda$. \end{section} \begin{ack} We thank M.~Kita, P.~Orlik, A.~Varchenko and G.~Ziegler for helpful discussions. The second author is also grateful to H.~Kanarek for a helpful and inspiring conversation describing his work. \end{ack}

9412

alg-geom/9412002

en

https://arxiv.org/abs/alg-geom/9412002

alg-geom/9412002

E. Looijenga

Eduard Looijenga

Cellular decompositions of compactified moduli spaces of pointed curves

28 pages, amstex2.1, 5 figures available from author

To a closed connected oriented surface $S$ of genus $g$ and a nonempty finite subset $P$ of $S$ is associated a simplicial complex (the arc complex) that plays a basic r\^ ole in understanding the mapping class group of the pair $(S,P)$. It is known that this arc complex contains in a natural way the product of the Teichm\"uller space of $(S,P)$ with an open simplex. In this paper we give an interpretation for the whole arc complex and prove that it is a quotient of a Deligne--Mumford extension of this Teichm\"uller space and a closed simplex. We also describe a modification of the arc complex in the spirit of Deligne--Mumford.

alg-geom

\section{\global\advance\headnumber by1\global\labelnumber=0{{\the\headnumber}.\ }} \define\label{(\global\advance\labelnumber by1 \the\headnumber .\the\labelnumber )\enspace} \define\fig{\global\advance\fignumber by1{Fig.\ \the\fignumber} } \NoBlackBoxes \topmatter \title Cellular decompositions of compactified moduli spaces of pointed curves \endtitle \rightheadtext{Compactified moduli spaces} \author Eduard Looijenga \endauthor \address Faculteit Wiskunde en Informatica, Rijksuniversiteit Utrecht, PO Box 80.010, 3508 TA Utrecht, The Netherlands \endaddress \email looijenga\@math.ruu.nl \endemail \keywords mapping clas group, Teichm\"uller space, ribbon graph \endkeywords \abstract To a closed connected oriented surface $S$ of genus $g$ and a nonempty finite subset $P$ of $S$ is associated a simplicial complex (the arc complex) that plays a basic r\^ ole in understanding the mapping class group of the pair $(S,P)$. It is known that this arc complex contains in a natural way the product of the Teichm\"uller space of $(S,P)$ with an open simplex. In this paper we give an interpretation for the whole arc complex and prove that it is a quotient of a Deligne--Mumford extension of this Teichm\"uller space and a closed simplex. We also describe a modification of the arc complex in the spirit of Deligne--Mumford. \endabstract \endtopmatter \head Introduction \endhead Given a closed connected oriented differentiable surface $S$ of genus $g$ and a finite nonempty subset $P$ of $S$, then the mapping class group $\G (S,P)$ of this pair is the group of isotopy classes of sense preserving diffeomorphisms of $S$ that fix $S$ pointwise. Harer proved in a series of papers some remarkable properties of the cohomology of the $\G (S,P)$ (see \cite{\harerb} for an overview). In this work a central r\^ole is played by various simplicial complexes with an action of an appropriate mapping class group that have in common the property that stabilizers of simplices look like simpler mapping class groups. The complex depends on the context, but in all cases it can for a suitable pair $(S,P)$ be identified with a subcomplex of the {\it arc complex} $A(S,P)$. That complex is defined as follows: the vertices of $A(S,P)$ are ambient isotopy classes relative $P$ of embedded unoriented nontrivial loops and arcs in $S$ that connect two (possibly identical) points of $P$ and avoid all other points of $P$ (where a loop is considered trivial if it bounds an open disk in $S-P$) and finitely many such vertices span a simplex if we can respresent them by loops and arcs which do not meet outside $P$. We note that there is a piecewise linear map $\lambda$ from $A(S,P)$ to the simplex $\De _P$ spanned by $P$ characterized by the property that it sends a vertex represented by an arc (resp.\ a loop) to the barycenter of the $1$-simplex of $\Delta _P$ spanned by its end points (resp.\ the vertex of $\Delta _P$ representing the base point). An important property of this complex is that its interior can be identified with the product of the Teichm\"uller space $\T (S,P)$ of the pair $(S,P)$ (i.e., the space of isotopy classes relative $P$ of conformal structures on $S$) and the open simplex $\Delta _P^{\circ}$. We may therefore regard $A(S,P)$ as an extension of $\T (S,P)\times \Delta _P ^{\circ}$. In the applications alluded to there was no apparent need to know what this extension actually represents, and that may have been the reason that question received little attention. (An exception is the paper by Bowditch and Epstein \cite{\bowep} about which we shall say more below.) The situation changed with Kontsevich's work on a conjecture of Witten\ \cite{\konts}, where it became essential to interpret the part of $A(S,P)$ of lying over $\Delta _P^{\circ}$. In this article Kontsevich states the answer but omits a proof. The present paper grew out the desire to supply one and one of our main results now interprets all of $A(S,P)$ in terms of the Deligne--Mumford compactification of the moduli space $\Mod _g^P:=\G (S,P)\backslash\T (S,P)$. For a precise statement we refer to theorem \refer{8.6}. Suffice here to say that for every nonempty subset $Q$ of $P$ we describe a quotient space $K_Q\Mod _g^P$ of the Deligne--Mumford compactification of $\Mod _g^P$ and for every inclusion $Q\subset Q'$ a quotient map $K_Q\Mod _g^P\to K_{Q'}\Mod _g^P$ such that the geometric realization of the associated simplicial space over $\Delta _P$ can be identified with the orbit space $\G (S,P)\backslash A(S,P)$. In particular, $\G (S,P)\backslash A(S,P)$ is a quotient of the product of the Deligne--Mumford compactification and $\Delta _P$. We suspect that the compactifications $K_Q\Mod _g^P$ and the maps between them can be constructed in the category of projective varieties and morphisms so that $\G (S,P)\backslash A(S,P)$ becomes the geometric realization of a simplicial object in this category. We state the relevant conjectures in \refer{3.3} An intermediate result of our proof is a combinatorial description \refer{11.5} of (a thickened version of) the Deligne--Mumford compactification. More precisely, we equivariantly blow up $A(S,P)$ in a certain manner over its boundary (in the PL-category) to get a cell complex of which the orbit space naturally maps to $\overline{\Mod}{}_g^P\times\Delta _P$ with fibers products of simplices (or finite quotients thereof). This description may be helpful in determining which of the cohomology classes that Kontsevich introduced in $\Mod _g^P$ extend to $\overline{\Mod}{}_g^P$. A paper by Milgram--Penner \cite{\milpen} alludes to a combinatorial construction of the Deligne--Mumford compactification (for the case that $P$ is a singleton), but it is not clear to us whether what these authors have in mind coincides with our construction. The article by Epstein and Bowditch mentioned above came to our attention after this paper was essentially completed. It also gives an interpretation of the arc complex, but in this it differs from ours in two respects. First, it takes the hyperbolic point of view (which gives rise to a different embedding of thickened Teichm\"uller space in the arc complex) and second, our description is solely in terms of the Deligne--Mumford compactification. (For these reasons it is not clear to us whether it could take care of Kontsevich's assertion.) We adopted their term {\it arc complex} and we adapted our notation a little in order to avoid too blatant clashes with theirs. \smallskip The plan of the paper is as follows. The first seven sections are intended to have to some extent the characteristics of a review paper and were written with a nonexpert reader in mind. Yet they do contain results that we have not found in the literature. In the first section we collect facts about the Teichm\"uller spaces. The next two sections deal with certain extensions of them: we describe a boundary for Teichm\"uller spaces in the spirit of Harvey based on the Deligne--Mumford compactification and we introduce the quotients of the Deligne--Mumford compactification alluded to above. In section 4 we discuss some properties of the complex $A(S,P)$. The next two sections we introduce metrized ribbon graphs and explicate the relationship between this notion and the complex $A(S,P)$. In section 7 we invoke the fundamental results of Strebel, culminating in theorem \refer{7.5}. The subsequent sections are of more technical nature. In section 8 we describe the geometric objects that are parametrized by the points of $A(S,P)$. Our first main theorem \refer{8.6} is also stated there, but its proof is postponed to the last section. The remainder of the paper is mostly concerned with the combinatorial versions of notions related to the Deligne--Mumford compactification. In section 9 we introduce stable ribbon graphs of which we claim that it is the combinatorial analog of the notion of a stable curve. This is justified in section 10, where we show that a metrized stable ribbon graph can be obtained as the limit of a one-parameter family of ordinary metrized ribbon graphs. In the final section 11 we construct the modification $A(S,P)$ mentioned above and prove our second main theorem \refer{11.5}, namely that this modification is essentially a thickened Deligne--Mumford extension of $\T (S,P)$. Once this has been established, the proof of our first main theorem is easily completed. \remark{Acknowledgements} I thank K.\ Strebel for help with the proof of \refer{7.5}, S.\ Wolpert and J.\ Koll\'ar for correspondence regarding \refer{3.3} and A.J.\ de Jong for the observation mentioned in \refer{3.2}. \endremark \smallskip Throughout this paper $S$ stands for a compact connected oriented differentiable surface, $g$ for its genus, and $P$ for a finite nonempty subset of $S$. Therefore we often suppress $(S,P)$ in the notation and write $\G$, $A$, $\dots$. We assume that $S-P$ has negative Euler characteristic, which amounts to the requirement that if $g=0$, then $|P|\ge 3$. \head \section Teichm\"uller spaces \endhead \label If $T$ is an oriented $2$-dimensional vector space, then a conformal structure on $T$ determines an action of the circle group $U(1)$ on $T$ and in this way $T$ acquires the structure of a $1$-dimensional complex vector space. Clearly, the converse also holds. Thus, to give the oriented surface $S$ a conformal structure is equivalent to give its tangent bundle the structure of a complex line bundle. Such a structure comes from a (unique) complex-analytic structure on $S$, so that $S$ becomes a Riemann surface. By the uniformization theorem, its universal cover will be isomorphic to the upper half plane. A conformal structure on $S$ is given by a section of a fiber bundle whose fibre is the open convex subset in the vector space of quadratic forms on $\R ^2$ defined by the positive ones. The $C^{\infty}$-topology on this space defines a topology on the set $\conf (S)$ of conformal structures on $S$. (It also has a compatible structure of a convex set, so that $\conf (S)$ is contractible.) Let $\text{Diffeo} ^+(S\!,\!P)$ denote the group of sense preserving diffeomorphisms which leave $P$ pointwise fixed, and let $\text{Diffeo} ^0(S,P)$ denote its identity component. Its ``group of connected components'', $$ \G :=\text{Diffeo} ^+(S,P)/\text{Diffeo} ^0(S,P), $$ is the {\it mapping class group} of $(S,P)$. In this definition we may replace diffeomorphism by homotopy equivalence (relative $P$) or all natural choices in between such as PL-homeomorphism, quasiconformal homeomorphism or plain homeomorphism: we still get the same group. Clearly, $\text{Diffeo}^+(S,P)$ acts on the space of conformal structures on $S$. The orbit space with respect to its identity component: $$ \T :=\text{Diffeo} ^0(S,P)\backslash \conf (S) $$ is called the {\it Teichm\"uller space} of $(S,P)$. It comes with a natural action of $\G$. If we substitute for $\conf (S)$ the bigger space of conformal structures inducing the quasiconformal structure underlying the given differentiable structure and replace $\text{Diffeo}$ by the group of quasiconformal homeomorphisms of $S$ , then the result is the same. For many purposes this is actually the most useful characterization. The Fenchel-Nielsen parametrization shows that $\T $ is homeomorphic to an open disk. There is even a natural $\G$-invariant complex-analytic manifold structure on $\T $; if $t\in\T $ is represented by a Riemann surface $C$ which underlies $S$, the tangent space at $t$ is identified with $H^1(C,\theta _C(-P))$, where $\theta _C$ is the sheaf of holomorphic vector fields on $C$. The action of $\G$ on $\T $ is properly discontinous and $\G$ has a subgroup of finite index acting freely (for instance, the kernel of the representation of $\G$ on $H_1(C;\Z /3)$). This implies that the orbit space $$ \Mod _g^P:=\G \backslash\T $$ is in a natural way a normal analytic space with only quotient singularities. \medskip\label We can give $\T $ an interpretation as a moduli space: let us first define an {\it $P$-pointed Riemann surface} $(C,x)$ as a Riemann surface $C$ together with an injection $x:P\hookrightarrow C$ such that the automorphism group of the pair $(C ,x)$ is finite. Say that such an $P$-pointed Riemann surface $(C,x)$ is {\it $(S,P)$-marked} if we are given an sense preserving quasiconformal homeomorphism (henceforth abbreviated as $\qc$-homeomorphism) $f:S\to C$ which extends $x$, with the understanding that two such homeomorphisms define the same marking if they are $\qc$-isotopic relative $P$. Clearly, these markings are permuted in a simply-transitive manner by the mapping class group $\G $. An isomorphism of marked $P$-pointed Riemann surfaces $(C,x,f)$, $(C',x,f')$ is given by an sense preserving $\qc$-homeomorphism $h:C\to C'$ with $hx=x'$ such that $hf$ is $\qc$-isotopic to $f'$ modulo $P$. Now $\T (S,P)$ can be thought of as the space of isomorphism classes of $(S,P)$-marked Riemann surfaces. So $\Mod _g^P:=\G\backslash\T (S,P)$ can be identified with the set of isomorphism classes of $P$-pointed compact Riemann surfaces of genus $g$. It is a coarse moduli space which has a natural structure of a quasi-projective variety. Knudsen, Deligne and Mumford showed that there is a distinguished projective completion $\Mod _g^P\subset\overline{\Mod}{}_g^P$ by the coarse moduli space of stable $P$-pointed complex curves of genus $g$. (A {\it stable $P$-pointed complex curve} consists of a complete complex curve $C$ with only simple crossings and an injection $x$ of $P$ into the nonsingular part of $C$ such that $\Aut (C,x)$ is finite.) It is called the {\it Deligne--Mumford compactification}. \medskip\label Let $G=\G /\G _1$ be a finite factor group of $\G $ and put $$ \Mod _g^P[G]:=\G _1\backslash\T . $$ Then we have a ramified $G$-covering $\pi _G: \Mod _g^P[G]\to \Mod _g^P$. The rational cohomology of $\Mod _g^P$ is mapped by $\pi _G^*$ isomorphically onto the $G$-invariants of the rational cohomology of $\Mod _g^P[G]$. If $\G _1$ acts without fixed point on $\T $, then $\T $ can be regarded as a universal covering space of $\Mod _g^P[G]$, and as $\T $ is contractible, this implies that $\Mod _g^P[G]$ is a classifying space for $\G _1$. So the group cohomology of $\G _1$ is the singular cohomology of $\Mod _g^P[G]$. We get the same statement for $\G $ vis-\`a-vis $\Mod _g^P$, except that we must use rational coefficients: $$ H^{\bullet}(\Mod _g^P;\Q )=H^{\bullet}(\G ;\Q ). $$ This equality represents a gate between algebraic geometry (the left hand side) and combinatorial group theory (the right hand side). \head \section A boundary for Teichm\"uller space \endhead We shall give $\T$ a (noncompact) boundary with corners. This is an analogue of the Borel--Serre compactification for arithmetic groups and first appeared in a paper by W.J.\ Harvey \cite{\harvey }. \medskip We first recall that given a smooth manifold $M$ and a closed submanifold $N\subset M$ with orientable normal bundle, one has defined the oriented blowing-up $$ \pi :\Bl _N(M)\to M. $$ This is a manifold with boundary $\pi ^{-1}N$. The map is an isomorphism over $M-N$, whereas $\pi ^{-1}N\to N$ can be identified with the sphere bundle associated to the normal bundle (or more intrinsically, with the bundle of rays in that bundle) with its obvious projection onto $N$. Notice that in case the normal bundle has the structure of a complex line bundle, $\pi ^{-1}N\to N$ has the structure of a $U(1)$-bundle. This construction generalizes in a straightforward manner to the case where $N$ is a union of submanifolds with oriented normal bundles that intersect multi-transversally; in that case $\Bl _N(M)$ is a manifold with corners and the fibres of $\pi$ are products of spheres. \medskip Now let $(C,x)$ be a pointed stable curve of genus $g$. Let $\tilde C\to C$ be its normalization, denote by $\Sigma\subset\tilde C$ the pre-image of $C_{\sing}$ and consider the composite map $$ f : \Bl _{\Sigma}(\tilde C)\to\tilde C\to C. $$ For every $p\in C_{\sing}$, $f^{-1}(p)$ consists of two principal $U(1)$ homogeneous spaces. If we choose for every such $p$ an anti-isomorphism of these homegeneous spaces and glue accordingly, then we get an oriented surface over $C$, $S\to C$, of genus $g$ such that the pre-image of every singular point is a circle. We shall interpret the conformal structure on $f^{-1}C_{\reg}$ as a degenerate conformal structure on $S$. The choice of the anti-isomorphism over $p$ is the same thing as the choice of an anti-isomorphism between $T_pC'$ and $T_pC''$, where $C'$ and $C''$ are the local branches of $C$ at $p$, given up to a positive real scalar. But this amounts to choosing a ray in the complex line $T_pC'\otimes T_pC''$. If we denote that space of rays by $R_pC$, then our choices are effectively parametrized by $\prod _{p\in C_{\sing}} R_pC$; this is a principal homogeneous space of the torus $U(1)^{C_{\sing}}$ that we abbreviate by $R(Z)$. It is well-kmown that the complex lines $T_pC'\otimes T_pC''$ have an interpretation in terms of the deformation theory of $C$. Let us recall that there is a universal deformation $$ \bigr( (\Cal{C}, C)\to (B,O)\, ;\, x_{\Cal{C}}:(B,O)\times P\to \Cal{C}\bigl) $$ of $(C,x)$ with as base smooth complex-analytic germ $(B,O)$. Its universal character implies that the whole situation comes with with an action of the finite group $\Aut (C,x)$. The $\Aut (C,x)$-orbit space of the base can be identified with the germ of $\overline{\Mod}_g^P$ at the point defined by $(C,x)$. Each singular point $p$ of $C$ determines a smooth divisor $(D_p,O)$ in $(B,O)$ which parametrizes the deformations of $C$ that do not smooth the singularity $p$. The fiber over $O$ of the normal bundle of $D_p$, $T_OB/T_OD_p$, is canonically isomorphic to $T_pC'\otimes T_pC''$. The divisors $D_p$, $p\in C_{\sing}$, intersect with normal crossings so that their union $D$ defines an oriented blowing-up: $$ \pi : \Bl _D(B,O)\to (B,O). $$ The central fiber $\pi ^{-1}O$ is canonically identified with $R(C)$. So over it we have a canonical family of surfaces of genus $g$. It is easily seen that this true over all of $\Bl _D(B,O)$, so that we get a family of oriented genus $g$ surfaces $$ \Cal{S}\to \Bl _D(B,O). $$ This family is $P$-pointed. \smallskip Let $\hat B\to \Bl _D(B,O)$ be a universal cover. Since $\Bl _D(B,O)$ has the torus $R(C)$ as a deformation retract, the covering group is naturally isomorphic to the fundamental group of $U(1)^{C_{\sing}}$, i.e., to the free abelian group generated by $C_{\sing}$. It is known that the fundamental group of $U(1)^{C_{\sing}}$ maps injectively to the mapping class group of a fiber. So the covering transformations permute these markings freely. It also follows that $\hat B$ is contractible. If $\hat{\Cal{S}}\to\hat B$ is the pull-back of our family of surfaces, then is possible to mark the fibers simultaneously by means of trivialization $\hat{\Cal{S}}\to S$ relative the given pointing. This defines a map from $\hat{B}-\partial\hat B$ to $\T$. That map is a homeomorphism onto an open subset of $\T$. Now glue $\hat B$ to $\T$ by means of this map. This clearly endows $\T$ with a partial boundary with corners. This can be done over any neighborhood of the Deligne--Mumford compactification $\overline{\Mod}{}_g^P$ and the essential uniquess of this construction ensures that the result is a manifold with corners $\hat\T$ whose interior is $\T$. By construction, $\hat\T$ comes with a $\G$-action that extends the given one on $\T$. The construction also shows that $\G$ acts properly discontinuously on $\hat\T$ and that there is a natural proper map $\G\backslash\hat\T\to \overline{\Mod}_g^P$ whose fibres are finite quotients of real tori. There is also a universal family of genus $g$ surfaces over $\hat\T$. As a set, $\hat\T$ has the following moduli interpretation. Let us define a {\it stable conformal structure} on $S$ as being given by a closed one-dimensional submanifold $L\subset S-P$ and a conformal structure on $S-L$ having the property that contraction of every connected component of $L$ yields a stable $P$-pointed curve. The set of stable conformal structures is acted on by $\text{Diffeo} ^+(S,P)$ and the quotient by $\text{Diffeo} ^0(S,P)$ can be identified with $\hat\T$. The following proposition is well-known and tells us when a sequence in $\hat\T $ converges. \proclaim{\label Proposition} Let $L\subset S-P$ be a compact one-dimensional submanifold such that every connected component of $S-(P\cup L)$ has negative Euler characteristic. Let $(J_n)_{n=1}^{\infty}$ be a family of conformal structures on $S$ with the property that $(J_n|S-P)_n$ converges uniformly on compact subsets to a stable conformal structure $J_{\infty}$ on $S$. If $t_{\infty}$ denotes the corresponding element of $\hat \T$ and $t_n\in\T$ the image of $J_n$, then $(t_n)_n$ converges to $t_{\infty}$. \endproclaim \label In this paper, the space $\hat\T$ will play an auxiliary r\^ole; we will be more concerned with a quotient $\overline{\T }$ that is a kind of Stein factorization of the projection $\hat\T\to \overline{\Mod}{}_g^P$: $\overline{\T }$ is obtained by collapsing every connected component of a fiber of the latter map to a point. As these connected components are affine spaces (and hence noncompact in general), the result will not locally compact. Notice that $\G$ still acts on $\overline{\T }$, and that the orbit space $\G\backslash\overline{\T }$ can be identified with $\overline{\Mod}{}_g^P$. So $\overline{\T }\to \overline{\Mod}{}_g^P$ is a Galois covering with infinite ramification. \head \section Quotients of Deligne--Mumford compactifications \endhead We introduce certain quotients of $\Mod _g^P$ that are obtained by identifying points of the boundary of its Deligne--Mumford compactification and that arise naturally in a combinatorial setting. One such quotient plays a prominent r\^ole in Kontsevich's proof of a conjecture of Witten \cite{\konts}. Let us fix a nonempty subset $Q$ of $P$. If $(C,x)$ is an $P$-pointed stable curve, then the irreducible components of $C$ which contain a point of $Q$ make up a (not necessarily stable) $Q$-pointed curve $(C_Q,x|Q)$. The pairs $(C,x)$ for which every singular point of $C$ lies on $C_Q$ define a Zariski open subset $U_Q$ of $\overline{\Mod}{}_g^P$. We define an equivalence relation $R_Q$ on $U_Q$ as follows: two $P$-pointed stable curves $(C,x)$ and $(C',x')$ representing points of $U_Q$ are declared to be $R_Q$-equivalent if there exists an sense preserving homeomorphism $h:C\to C'$ such that $hx =x'$ and $h$ restricts to an analytic isomorphism of $C_Q$ onto $C'_Q$ as $Q$-pointed curves. We denote its quotient space by $K_Q{\Mod }_g^P$. The equivalence relation $R_Q$ has a natural extension $\overline{R}_Q$ to $\overline{\Mod}{}_g^P$ which is characterized by the property that if we keep both $C_Q$ and the singular points of $C$ on $C_Q$ fixed, but allow $C$ to acquire singularities outside $C_Q$, then we stay in the same equivalence class. So $K_Q{\Mod }_g^P$ may be regarded as a quotient of $\overline{\Mod}{}_g^P$. \proclaim{\label Lemma} The space $K_Q{\Mod }_g^P$ is compact Hausdorff. It contains ${\Mod }_g^P$ as an open-dense subset. \endproclaim \demo{Proof} The last assertion of the lemma is easy and is stated for the sake of record only. The first statement is a little ambiguous since it is not clear whether we give $K_Q{\Mod }_g^P$ the topology as a quotient of $U_Q$ or of $\overline{\Mod }_g^P$. A priori, the former could be finer than the latter, but we will show that they are the same. Now $\overline{\Mod}{}_g^P$ is compact and hence so is every quotient of it. It is therefore enough for us to verify that $K_Q{\Mod }_g^P$ is Hausdorff as a quotient of $\overline{\Mod}{}_g^P$. This will be a consequence of the following property of the compactification $\overline{\Mod}{}_g^P$. \smallskip Let $[(C_n,x_n)]_{n=1}^{\infty}$ be a sequence in $U_Q$ converging to $[(C,x)]$ and suppose that all the terms of this sequence have the same topological type. Then the intersection of $C_{n,Q}$ with the union of the other irreducible components of $C_n$ is a finite subset $Z_n$ of the smooth part of $C_{n,Q}$ of constant cardinality. Let $Z$ be a fixed finite set of this cardinality and choose for every $n$ a bijection $z_n:Z\cong Z_n$. Then $(C_{n,Q},x_n|Q\sqcup z_n)$ is a $(Q\sqcup Z)$-pointed curve, which is easily seen to be stable. If $h$ denotes the arithmetic genus of $C_{n,Q}$, then after passing to a subsequence, $[(C_{n,Q},x_n|Q\sqcup z_n)]_n$ will converge in $\overline{\Mod}{}_h^{Q\sqcup Z}$ to some $[(C^*,y\sqcup z)]$. The property alluded to is that $(C^*_Q, y)=(C_Q,x|Q)$. \smallskip To complete the proof, let $[(C_n,x_n)]_{n=1}^{\infty}$ and $[(C'_n,x'_n)]_{n=1}^{\infty}$ be sequences in $\overline{\Mod}{}_g^P$ converging to $[(C,x)]$ and $[(C',x')]$ repectively such that terms with the same index are $R_Q$-equivalent. We must show that $[(C,x)]$ and $[(C',x')]$ are $\overline{R}_Q$-equivalent. But this is immediate from the above mentioned property. \enddemo \medskip\label Here is a simple, but perhaps instructive example. Let $C$ be a smooth connected projective curve of genus $g\ge 2$. Then $C\times C$ parametrizes a subvariety of $\overline{\Mod}{} _g^{\{ 0,1\}}$. A point of the diagonal, $(p,p)\in C\times C$, represents the union of $C$ and $P^1(\C )$ with $p\in C$ identified with $\infty\in P^1(\C )$ and $i=0,1$ mapping to $i\in P^1(\C )$. Taking the image in $K_{\{ 0\}}{\Mod } _g^{\{ 0,1\}}$ means that we disregard the irreducible component $C$ and retain $P^1(\C )$ with its three points. So the composite map $C\times C\to K_Q{\Mod } _g^P$ contracts the diagonal. As A.J.\ de Jong pointed out to me, this contraction can be obtained algebraically as the normalization of the image of the difference map from $C\times C$ to the Jacobian of $C$. The contraction can also be realized by the line bundle on $C\times C$ that is the pull-back of the canonical sheaf under the projection $(p_0,p_1)\in C^2\to p_0\in C$ twisted by the diagonal (a positive tensor power of that bundle is without base points). \medskip\label Notice that the $\overline{R}_Q$ gets coarser as $Q$ gets smaller. In particular, for $Q\subset Q'$, there is a natural quotient mapping $K_{Q'}{\Mod }_g^P\to K_Q{\Mod }_g^P$. In this connection we venture the following \proclaim{Conjecture 1} The quotients $K_Q\Mod _g^P$ have the structure of a normal projective variety and such that the quotient map $\overline{\Mod}{}_g^P\to K_Q\Mod _g^P$ is a morphism. \endproclaim If the conjecture holds, then the natural maps $K_{Q'}{\Mod }_g^P\to K_Q{\Mod }_g^P$, $Q\subset Q'$, are morphisms, too. We actually expect the corresponding quotient to arise as the image under a certain linear system without base points. A special (but basic) case is when $P=Q$ is a singleton: \proclaim{Conjecture 2} The relatively dualizing sheaf of the universal stable curve of genus $g\ge 2$, $\overline{\Mod}{}_g^1\to\overline{\Mod}{}_g$, is semiample, i.e., a positive tensor power of it has no base points. \endproclaim S.\ Wolpert \cite{\wolpert} has shown that the natural metric on this relatively dualizing sheaf has nonnegative curvature and that this curvature is nonzero in directions transversal to the $R_1$-equivalence classes. Using this one can show that under the assumption of conjecture 2, a a positive power of the relatively dualizing sheaf defines a morphism of which the fibers are the $R_1$-equivalence classes. So conjecture 2 implies conjecture 1 for the case when $P=Q$ is a singleton. \medskip\label These extensions have Teichm\"uller counterparts: for every nonempty $Q\subset P$ we have a $\G $-equivariant quotient $K_Q\T $ of $\overline{\T }$ which contains $\T $ and for $Q\subset Q'$ a quotient mapping $K_{Q'}\T\to K_Q\T$. It is useful to have a moduli interpretation for these compactifications. We first remind the reader that one calls a complex-analytic space {\it weakly normal} if every continuous complex function on an open subset which is analytic outside a divisor is analytic. For curves this means that every singular point with $k$ branches is like the union of the coordinate-axes of $\C ^k$ at the origin. We make two definitions: A {\it $Q$-minimal $P$-pointed curve of genus $g$} consists of a connected weakly normal curve $C$, a map $x:P\to C$, and a function $\epsilon : C\to\Z _{\ge 0}$ with finite support (the {\it genus defect function}) such that \roster \item $x|Q$ is injective and its image is contained in $C_{\reg}\setminus x(P-Q)$ and meets every connected component of that space. \item The automorphism group of the triple $(C,x,\epsilon )$ is finite (equivalently: every connected component of $C_{\reg}\setminus (x(P)\cup\supp\epsilon )$ has negative Euler characteristic). \item $g=g(\hat C)+\sum_{z\in C} (\epsilon (z)+r(C,z)-1)$, where $\hat C$ is the normalization of $C$ and $r(C,z)$ is the number of branches of $(C,z)$ \endroster The above conditions imply the existence of a continuous map $f:S\to C$ that extends $x$ such that the pre-image of a point $z\in C$ is connected submanifold with boundary of $S$ of genus $\epsilon (z)$ with $r(C,z)$ boundary components if $\epsilon (z)+r(C,z)>1$ and a singleton else. If we are given such a map up to isotopy relative $P$, then we say that the $Q$-minimal $P$-pointed curve is {\it marked} by $(S,P)$. There is an obvious notion of isomorphism: two $Q$-minimal $P$-pointed curves $(C,x,\epsilon )$, $(C',x',\epsilon ')$ are declared isomorphic if there exists an isomorphism $h: C\to C'$ such that $x'h=x$ and $\epsilon 'h=\epsilon$. In the marked context we of course also require that $h$ respects the markings. \proclaim{\label Lemma} The isomorphism classes of (marked) $Q$-minimal $P$-pointed curves of genus $g$ are in bijective correspondence with the points of $K_Q\Mod _g^P$ ($K_Q\T$). \endproclaim \demo{Proof} We content ourselves with indicating how a $Q$-minimal curve $(C,x,\epsilon)$ determines an element of $K_Q\Mod _g^P$. Extend $x$ to a continuous map $f: S\to C$ as above. Let $L$ be the boundary of $f^{-1}(C_{\sing}\cup\supp\epsilon )$. Now collapse to a point every component of $L$ as well as every component of $f^{-1}(C_{\sing}\cup\supp\epsilon )$ that is homeomorphic to a cylinder and does not intersect $P$. Then $(\bar S,\pi x)$ is a stable $P$-pointed pseudosurface. The map $f$ factors through a map $\bar f:\bar S\to C$ and the irreducible components of $\bar S$ that are not contracted receive in this way a weakly normal complex structure. Extend this to a weakly normal complex structure (compatible with the given orientation) on $\bar S$. Then we get a stable $P$-pointed curve $C$. Its image in $K_Q{\Mod }_g^P$ only depends on $(C,x,\epsilon)$. \enddemo We can form the simplicial scheme $K_{\bullet}{\Mod }_g^P$. Its geometric realization is a quotient of $\overline{\Mod}{}_g^P$ such that the quotient map followed by the structure map $|K_{\bullet}{\Mod }_g^P |\to\De _P$ is the projection. We shall show that $|K_{\bullet}\Mod _g^P|$ is homeomorphic (over $\Delta _P$) to the semisimplicial complex $\G \backslash A$ that was defined in the introduction. We look at this complex in more detail in the next section. \head \section The arc complex \endhead \medskip\label We consider embedded unoriented loops and arcs $\alpha$ in $S$ which connect two (possibly identical) points of $P$ and avoid all other points of $P$. In case of a loop we also require that it be nontrivial in the sense that it does not bound an embedded disk in $S-P$. Let $\cA $ denote the set of isotopy relative $P$ of these arcs and loops. We endow this set with the structure of an abstract simplicial complex by stipulating that an $(l+1)$-element subset of $\cA $ defines an $l$-simplex if it is representable by arcs and loops that do not meet outside $P$. We denote the geometric realization of this complex by $A$. There is a piecewise linear map $\lambda$ from $A$ to the simplex $\De _P$ spanned by $P$ characterized by the property that it sends a vertex $\la\alpha\ra\in \cA $ to the barycenter of the end points of $\alpha$. So if $Q$ is a nonempty subset of $P$ and $\De _Q\subset\De _P$ the corresponding face, then $\lambda^{-1}\De _Q$ is a subcomplex of $A$ of which the $0$-simplices may be interpreted as the isotopy classes of embedded arcs and loops in $S-(P-Q)$ with end points in $Q$. \smallskip We say that the simplex $\langle\alpha _0,\dots ,\alpha _l\rangle$ is {\it proper} if its star is finite, that is, if it is contained in a finitely many simplices. This comes down to requiring that each connected component of $S-\cup _{\lambda}\alpha _{\lambda}$ is an embedded open disk which contains at most one point of $P$. The improper simplices make up a subcomplex $A_{\infty}\subset A$. We shall denote its complement $A-A_{\infty}$ by $A^{\circ}$. It is clear that $A$ has an action of $\G $ which preserves both $A_{\infty}$ and $\lambda$. \proclaim{\label Lemma} The group $\Gamma$ has only a finite number of orbits in the set of simplices of $\cA $. The dimension of a proper simplex is at least $2g-2+|P|$ and the dimension of every fiber of $\lambda$ is $6g-6+2|P|$. \endproclaim \demo{Proof} The first assertion is a consequence of the fact that up to homeomorphism there are only finitely many compact surfaces with an Euler characteristic bounded from below (the details are left to the reader). Let $a=\la\alpha _0,\dots,\alpha _l\ra $ be an $l$-simplex of $A$ and let $Q\subset P$ the set of points of $P$ that are end point of some $\alpha _{\lambda}$. This means that $l$ maps the relative interior of $a$ in the relative interior of $\De _Q$. If $a$ is a proper simplex, then the formula for the Euler characteristic gives $$ 2-2g=|Q|-(l+1)+d, $$ where $d$ is the number of connected components of $S-\cup _{\lambda}\alpha _{\lambda}$. Since every connected component contains at most one point of $P-Q$, we have $d\ge |P|-|Q|$. It follows that $l\ge 2g-2+|P|$. If $a$ is maximal in the pre-image of $\De _Q$, then every connected component of $S-\cup _{\lambda}\alpha _{\lambda}$ either is an open disk that contains precisely one point of $P-Q$ and is bounded by a single member of $a$ or contains no point of $P-Q$ and is bounded by three members of $a$. A straightforward computation shows that then $d={2\over 3}(l+1+|P|-|Q|)$. Substituting this in the formula for the Euler characteristic gives $l=6g-7+3|Q| +2(|P|-|Q|)=6g-6+2|P|+\dim \De _Q$. \enddemo \remark{Example} We take for $S$ the torus $\R ^2/\Z ^2$ and for $P$ the origin. An element of $\cA$ is uniquely represented by a circle which is also a subgroup of $S$. Such a subgroup is the image of a line in $\R ^2$ through the origin and another point of $\Z ^2$. In this way we obtain an identification of $\cA$ with the rational projective line $\bold{P}^1(\Q)$. The two circles defined by the relatively prime pairs of integers $(x_0,x_1)$ and $(y_0,y_1)$ define a $1$-simplex iff they do not intersect outside the origin. This is the case iff $x_0y_1-x_1y_0\not=\pm 1$, or equivalently, iff $x=(x_0,x_1)$ and $y=(y_0,y_1)$ make up a basis of $\Z ^2$. Then this $1$-simplex is adjacent to exactly two $2$-simplices, namely those defined by $\{ x,y,x+y\}$ and $\{ x,y,x-y\}$. A simplex is proper iff it is of dimension $>0$. The geometric realization of $A$ can be pictured in the upper half plane (with the vertex at $\infty$ missing) as a hyperbolic tesselation associated to a subgroup of the modular group of index two. \endremark \midspace{50mm}\caption{\fig The arc complex of a once-pointed torus} Let $b\cA $ denote the barycentric subdivision of $\cA $. So a vertex of $b\cA $ is the barycenter of a simplex $a$ of $\cA $ and a $k$-simplex of $b\Cal{A}$ is spanned by the barycenters of a strictly increasing chain $a_0<a_1<\cdots <a_k$ of simplices of $b\Cal{A}$. Let $\Cal{A}_{\pr}$ denote the full subcomplex of $b\Cal{A}$ whose vertices are the barycenters of proper simplices. Clearly, its geometric realization $A_{\pr}$ can be viewed as a subset of $A^{\circ}$. In the previous example we have drawn $A_{\pr}$ with dotted lines. \proclaim{\label Proposition} The fibres of $\lambda |A_{\pr}$ have dimension $4g-4+|P|$ and there is a natural $\G $-equivariant deformation retraction of $A^{\circ}$ resp. $A-A_{\pr}$ onto $A_{\pr}$ resp. $A_{\infty }$ which preserves the pre-image of every relatively open face of $\De _P$ under $\lambda$. \endproclaim \demo{Proof} A $k$-simplex of $\Cal{A}_{\pr}$ is represented by a chain $a_0<a_1<\cdots <a_k$ of simplices of $\Cal{A}$ with $a_0$ proper. According to the previous lemma $\dim a_0\ge 2g-2+|P|$ and $\dim a_k\le 6g-6+2|P|+\dim\De _Q$, where $Q\subset P$ is the smallest subset of $P$ such that $\lambda$ maps $a_k$ in $\De _Q$. So $k\le (6g-6+2|P|+\dim \De _Q)-(2g-2+|P|)=4g-4+|P|+\dim \De _Q$. It is easily verified that this value is attained. The proof of the remaining assertions is a standard argument in the theory of simplicial complexes, but let us give it nevertheless, say for $A_{\pr}\subset A^{\circ}$. If $x\in A^{\circ}=bA-bA_{\infty}$, then we can write $x=\sum _{i=0}^k x_ia_i$ with $a_0<a_1<\cdots <a_k$, $x_i>0$, and $a_k$ proper. Let $r$ be the first index such that $a_r$ is proper. Then $$ x':=\sum _{i=r}^k (\sum _{j=r}^k x_j)^{-1}x_ia_i\in A_{\pr} $$ and $x(t):=(1-t)x+tx'$ defines a deformation retraction of $A^{\circ}$ onto $A_{\pr}$. \enddemo Our goal is to construct a $\G $-equivariant homeomorphism of $A$ onto $|K_{\bullet}\T |$ which commutes with the given projections onto $\De _P$. For this we first need to discuss ribbon graphs. \head \section Ribbon graphs \endhead \label A {\it ribbon graph} is a nonempty finite graph in which we allow loops and multiple bonds, but not isolated points (in other words, a semi-simplicial complex of pure dimension $1$), such that for every vertex we are given a cyclic order of its outgoing edges. A finite graph embedded in an oriented surface acquires naturally such a structure. Conversely, a ribbon graph can be embedded in an oriented surface of which it is a deformation retract. For instance, \midspace{50mm}\caption{\fig Ambient surface of a ribbon graph} \noindent This surface can be compactified by adding a finite number of points so that the result is a surface. This compactification can be obtained in a purely combinatorial way as follows. Let $G$ be a ribbon graph. Denote by $X(G)$ its set of oriented edges (so that each edge determines two distinct elements of $X(G)$). Reversal of orientation defines a fixed point free involution $\sigma _1$ in $X(G)$. For $e\in X(G )$, let $v$ be its vertex of origin, and denote by $\sigma _0(e)\in X(G )$ the outgoing edge of $v$ that succedes $e$ relative the given cyclic order. This defines a permutation $\sigma _0$ of $X(G )$. We define the permutation $\sigma _{\infty}$ by the equality $\sigma _{\infty}\sigma _1\sigma _0=1$. \midspace{50mm}\caption{\fig The operations $\sigma _i$} Denote the orbit space of $\sigma _i$ in $X(G )$ by $X_i(G )$. For $i=0$ resp. $i=1$ it can be identified with the set of vertices resp. of (unoriented) edges of $G$; the elements of $X_{\infty}(G )$ are called {\it boundary cycles}. So $G $ can be reconstructed from $X(G )$ equipped with the permutations $\sigma _0$ and $\sigma _1$. (Indeed, any nonempty finite set equipped with a fixed point free involution and another permutation determines a ribbon graph.) \medskip\label Let $K$ be the two-simplex with vertices $v_0,\bar v_0,v_{\infty}$ with the orientation given by this order. The midpoint of the face $\la v_0, \bar v_0\ra$ is denoted $v_1$. Denote by a ``bar'' the involution of $K$ which interchanges $v_0$ and $\bar v_0$ and leaves $v_{\infty}$ (hence also $v_1$) fixed. We define a semi-simplicial complex $S(G )$ as a quotient of $K\times X(G )$ by identifying the oriented $1$-simplices $\la v_0,\bar v_0\ra \times \{e \}$ with $\la\bar v_0,v_0\ra\times\{ \sigma _1e\}$ and $\la v_0,v_{\infty}\ra \times\{ e\}$ with $\la \bar v_0,v_{\infty}\ra \times\{ \sigma_0 e\}$. Since the the disjoint union of the $X_0(G)$ and $X_{\infty}(G )$ appears here as the set of $0$-simplices, we will often regard these two as subsets of $S(G)$. In what follows a special r\^ole is played by the $0$-simplices that either belong to $X_{\infty}(G)$ or are a vertex of $G$ of valency $\le 2$. We shall call such points {\it distinguished}. We shall write $K_e$ for the image of $K\times \{ e\}$ and we call it the {\it tile} defined by $e$. The full subcomplex spanned by $X_0(G )$ can be identified with $G$, see the picture below. \midspace{50mm}\caption{\fig Combinatorial construction of the ambient surface} It is not difficult to see that the geometric realization of $S(G )$ is a compact surface. The given orientation of $K$ determines one of $S(G )$ and this orientation is compatible with the ribbon graph structure of $X(G )$. The surface has a piecewise linear structure and hence a quasiconformal structure. The image of $\la v_1,v_{\infty}\ra\times X(G )$ is the barycentric subdivision of another ribbon graph, called the dual of $G$, and denoted by $G ^*$. (It is essentially obtained by passing from $(X(G );\sigma _0,\sigma _1)$ to $(X(G );\sigma _{\infty},\sigma _1)$ and using a natural identification of $S(G ^*)$ with $S(G )$.) Observe that the edges of $G ^*$ are indexed by the edges of $G$. {\it Remark.} The permutations $\sigma _0,\sigma _1,\sigma _{\infty}$ associated to a ribbon graph $\G$ arise as monodromies in the following manner. Let $S_0$ be the topological sphere obtained from $K$ by identifying points on its boundary according to the involution ``bar'' and denote the image of $v_z$ by $z\in S_0$ ($z=0,1,\infty$). It is clear that there is a natural finite quotient map $S(G )\to S_0$. This map is ramified covering which branch locus $\{ 0,1,\infty\}$. The restriction to $K^{\circ}\hookrightarrow S_0$ is naturally identified with $K^{\circ}\times X(G )$ and the monodromy of $S(G )\to S_0$ around $z\in \{0,1,\infty\}$ is given by the permutation $\sigma _z$ acting on the second factor. \head \section Metrized ribbon graphs \endhead \medskip\label A {\it metric} on a ribbon graph is $G $ simply a map from its edges to $\R _{>0}$. If this map has in addition the property that the total length of the graph is $1$, then we call it a {\it unital metric}. A {\it conformal structure} on $G $ is a metric on every connected component of $G $, given up to a factor of proportionality. This is of course equivalent to be given a unital metric on every connected component of $G $. We denote the space of conformal structures on $G $ by $\conf (G )$. So for connected $G $, $\conf (G )$ may be identified with the open simplex spanned by the set of edges of $G $. \medskip\label Let $r:K\to [0,1]$ be the barycentric coordinate which is $1$ in $v_{\infty}\in K$ and $0$ in $v_0$ and $\bar v_0$ and identify $K-\{\infty \}$ with $\la v_0,\bar v_0\ra \times\R _{\ge 0}$, where the first component is an obvious projection and the second is given by $-\log r$. Suppose that we are given a ribbon graph $G $ with metric $l:X_1(G )\to\R _{>0}$. This determines a complete piecewise Euclidean metric on $S(G ) - X_{\infty }(G )$ as follows: give $(K-\{\infty \})\times \{e\}$ the metric which under its identification with $\la v_0,\bar v_0\rangle \times\R _{\ge 0}$ corresponds to the translation invariant product metric for which $\la v_0,\bar v_0\ra $ has length $l(e)$ and the second component has the standard metric. This descends to a metric on $S(G ) - X_{\infty }(G )$. The complement of the vertex set of $S(G )$ has a unique smooth structure for which this metric is Riemannian on that set. It is easy to check that its underlying conformal structure extends across the vertices, so that now $S(G )$ acquires a conformal structure. We denote the Riemann surface thus obtained by $C(G ,l)$. This Riemann surface comes with a meromomorphic quadratic differential $q_l$ whose absolute value gives the metric: if we identify the interior of the tile $K_e$ as a metric space in the obvious way with the Euclidean rectangle $\{ z\in \C: \Im (z)>0, |\Re (z)|< {1\over 2}l(e)\}$, then this is a complex-analytic chart and the quadratic differential is given by $dz\otimes dz$. One finds that $q_l$ has a pole of order two at each point of $X_{\infty}(G )$ and a zero at of order $k-2$ at each $k$-valent vertex of $G $ (so a pole of order one at a univalent vertex). This implies that successive outgoing oriented edges at a $k$-valent vertex make an angle of $2\pi/k$. There are no other singularities of $q_l$. Observe that as a piecewise-linear complex valued quadratic differential on $S(G )$, $q_l$ embodies all the extra structure: the smooth structure, the metric and (hence) the complex-analytic structure. Notice that the conformal structure on $S(G )$ only depends on the conformal structure on $G $ subordinate to $l$. Hence we can always assume that $l$ is unital on every connected component of $G $. If $v$ is a bivalent vertex of $G $, then ``forgetting'' that vertex yields a metrized ribbon graph of which the associated Riemann surface can be identified with $C(G ,l)$. \smallskip In case of a $l$ is a metric on a partial ribbon graph we do essentially the same construction where the metric on the incomplete edges should be thought of as having the value $\infty$. So if $e$ is an oriented edge without end point, then $K^{\circ}_e$ and $K^{\circ}_{\sigma _1(e)}$ are Euclidean quadrant (isomorphic to $\R _{\ge 0}^2$) such that $e$ corresponds to the positive $x$-axis in the former and to the positive $y$-axis in the latter. Again on verifies that the underlying conformal structure extends across the vertices so that we find a Riemann surface $C(G ,l)$. This time the quadratic differential $q_l$ may have higher order poles at the points of $X_{\infty}(G )$. In fact, the pole order at $\beta\in X_{\infty}(G )$ will be $2$ plus the number of incomplete (unoriented) edges that occur in $beta$. \medskip\label An {\it $P$-pointed ribbon graph} is an ribbon graph $G $ together with an injection $x: P\hookrightarrow X_{\infty}(G)\sqcup X_0(G )$ whose image contains all the distinguished points. Notice that in that case every connected component of $S(G)-x(P)$ has negative Euler characteristic: this is because $S(G)-X_{\infty}(G)$ admits $G$ as a deformation retract and every connected graph which is contractible (resp.\ a homotopy circle) has at least two (resp.\ one) vertices of valency at most $2$. \medskip Let $(G,x)$ be an $P$-pointed ribbon graph. If $s$ is an edge of $G $ which is neither isolated nor a loop, then collapsing that edge yields a ribbon graph $G /s$. It inherits an $P$-pointing iff not both of its vertices are in the image of $P$. The corresponding surface $S(G /s)$ is obtained as a quotient of $S(G )$ by collapsing the two tiles defined by $s$ according to the level sets of $r$. We call this an {\it edge collapse}. If $s$ is a non-isolated loop, and for some orientation $e$ of $s$, $e$ is by itself a boundary cycle, then it is still true that $G /s$ is a ribbon graph. In this case, $G /s$ inherits a $P$-pointing ifand only if the vertex of $s$ is not in the image of $P$. The surface $S(G /s)$ is then obtained by collapsing $K_e$ to a point (a {\it total collapse}) and by applying an edge collapse to the opposite tile $K_{\sigma _1e}$. In either case the quotient map $S(G)\to S(G/s)$ has in its homotopy class relative $P$ a unique isotopy class relative $P$ of $\qc$-homeomorphisms. \smallskip We can apply these two procedures successively to a collection $Z$ of edges of $G $ if and only if every connected component of the corresponding subgraph $G _Z\subset G$ is \roster \item either a tree with at most one marked vertex or \item a homotopy circle without marked vertices which contains an entire boundary cycle of $G $. \endroster We then say that $Z$ is {\it negligible}. So if $Z$ is negligible and $G /G _Z$ is the semi-simplicial complex obtained by collapsing every connected component of $G _Z$ to a point, then $G /G _Z$ has still the structure of a ribbon graph pointed by $P$ and the corresponding surface $S(G/G _Z)$ can be obtained by means of a succession of edge collapses and contractions of the tiles labeled by the oriented edges in $Z$. The quotient map $S(G)\to S(G/G _Z)$ determines an isotopy class relative $P$ of sense preserving $\qc$-homeomorphisms $S(G )\to S(G /G _Z)$. \smallskip An {\it almost-metric} on $G $ is a function $l: X_1(G )\to \R _{\ge 0}$ whose zero set $Z$ is negligible. It is clear that $l$ then factorizes over a metrized ribbon graph $G /G _Z$ with metric (still denoted) $l$ and we define $C(G ,l)$ simply as $C(G /G _Z, l)$. We have a corresponding notion of an {\it almost-conformal structure}. Denote the space of unital almost-conformal structures on $(G,x)$ by $\aconf (G ,x)$. It is clear that for a negligible $Z\subset X_1(G )$, we have a natural embedding of $\aconf (G /G _Z,x)$ in $\aconf (G,x)$. \medskip\label We now assume that $G $ is a connected ribbon graph. Over $\conf (G )$ lives a ``tautological'' topologically trivial family of metrized graphs and a corresponding family of Riemann surfaces. We extend the latter as a family of pseudosurfaces; in section 8 we give each of its fibers the structure of a weakly normal curve. The family appears as a factor of the projection $S(G)\times a(G)\to a(G)$ and is defined as follows. Any edge $s$ of $G $ determines by definition a vertex of $a(G)$. The codimension-one face opposite this vertex is identified with $a(G/s)$ and for each orientation $e$ of $s$, we apply an edge collapse to $K_e\times a(G/s)$ relative its projection onto $a(G/s)$. Likewise, every boundary cycle $\beta$ of $G$ determines a face $a(G/G_{\beta})$ of $a(G)$ and we perform a total collapse on the tiles $K_e\times a(G/G_{\beta})$ relative $a(G/G_{\beta})$ with $e\in\beta$. The result is a semisimplicial space $\CC (G)$ that comes with a projection $\pi _G:\CC (G)\to a(G)$. Over $l\in \conf (G )$ the fiber is the surface $S(G)$; it has a conformal structure which makes it canonically isomorphic to $C(G,l)$. That last fact is still true in case $l\in \aconf (G)$. The fiber $\CC (G)_l$ over an arbitrary $l\in a(G)$ is gotten as follows. Let $Z\subset X_1(G )$ be the zero set of $l$ and let $S(G )_Z$ be the quotient of $S(G )$ obtained by performing for every oriented edge $e$ of $Z$ a contraction or an edge collapse on $K_e$, depending on whether or not the boundary cycle of $G$ generated by $e$ is contained in $G_Z$. Then $\CC (G)_l$ can be identified with $S(G)_Z$. We will see in section 8 that $S(G)_Z$ is a pseudosurface and that $\CC (G)_l$ has a natural conformal structure on its smooth part given by quadratic differential. (This conformal structure determines a unique complex-analytic structure such that $\CC (G)_l$ is weakly normal.) \medskip\label We conclude this discussion with a few remarks. Every element of $X_0(G)\sqcup X_{\infty}(G)$ determines a section of $\CC (G)\to a(G)$. Those that are indexed by $P$ are disjoint over $\aconf (G)$. One can show that the complement of the sections defined by the elements of $X_0(G)\sqcup X_{\infty}(G)$ has a natural smooth structure. (To see this, use an atlas naturally indexed by the elements of $X_1(G)\sqcup X_{\infty}(G)$.) The conformal structures along the the fibers vary differentiably on this open subset. \head \section Moduli spaces \endhead \medskip\label We say that a ribbon graph $G $ is {\it $(S,P)$-marked} (or briefly, {\it marked}) if we are given a given isotopy class relative $P$ of sense preserving $\qc$-homeomorphisms $f: S\cong S(G)$ such that $f|P$ defines a $P$-pointing of $G $: $f$ maps $P$ to $X_{\infty}(G)\sqcup X_0(G)$ and its image contains the distinguished points. It is clear that $G$ permutes the markings. \medskip\label We claim that a marked ribbon graph is the same thing as a proper simplex of $\Cal{A}$. Let $f :S\cong S(G)$ be a marking. Regard the dual ribbon graph $G ^*$ as lying on $S(G )$. Then the pre-image of every edge of $G^*$ under $f$ connects two points of $P$ and therefore the collection of these determines a simplex $a(G,f)$ of $\Cal{A}$. A connected component of $S-G ^*$ is given by a vertex of $G $; it contains one or no point of $P$ depending on whether this vertex is marked by $P$. If the vertex is unmarked it has valency $k\ge 3$ and the connected component is $k$-gon. So distinct edges of $G^*$ yield distinct vertices of $a(G,f)$ and $a(G,f)$ is a proper simplex. We also notice that the space of unital metrics $\conf (G )$ may be identified with the relative interior of $a(G,f)$; we shall therefore denote that relative interior by $\conf (G,f)$. Conversely, if $a=\la\alpha _0,\dots ,\alpha _l\ra$ is a proper simplex of $A$, then the union of the $\alpha _i$'s define a ribbon graph $G _a$ on $S$ with vertex set contained in $P$. It is easily seen that the inclusion $G _a\subset S$ extends to a $\qc$-homeomorphism $S(G_a)\to S$ such that $X_{\infty}(G_a)$ is mapped in $P$. If we identify $S(G_a ^*)$ with $S(G ,S)$, then we see that $G_a$ has in a natural way the structure of a marked ribbon graph. We remark that $\conf (G,f)$ has maximal dimension iff all vertices of $G$ are trivalent (so that $P$ maps bijectively onto the set boundary cycles of $G$). \proclaim{\label Lemma} Let $a$ be a proper simplex of $A$ as above with associated marked ribbon graph $(G ,f)$. Let $Z\subset X_1(G )$ be a set of edges of $G $ and let $a(G /G _Z)$ be the codimension $|Z|$ face of $a$ opposite the face defined by $Z$. Then $Z$ is negligible if and only if $a(G /G _Z)$ is proper and in that case $S(G /G _Z)$ inherits an marking (denoted $f/Z$). \endproclaim \demo{Proof} It is enough to show this in case $Z$ has only one element and this we leave to the reader. \enddemo So given a marking $f$, then the space of unital almost-metrics $\aconf (G ,f|P)$ may be identified with $|a(G ,f)|\cap A^{\circ}$. We denote the latter by $\aconf (G ,f)$. The restriction of $\lambda :A\to\De _P$ to $\aconf (G ,f)$ has the following simple description: for $p\in P$ the corresponding barycentric coordinate $\lambda _p$ is in case $f(p)$ corresponds to a boundary cycle, half the length of that cycle and it is zero otherwise. \smallskip Remember that every proper simplex of $A$ is of the form $a(G,f)$ and that over such a simplex we have defined in section 6 the family $\CC (G )\to a(G ,f)$. As each inclusion of proper simplices is canonically covered by an inclusion of the corresponding families, this gives us a global family $\pi :\CC \to A$. This family comes with sections labeled by $P$. Summing up: \proclaim{\label Proposition} The set of points of $A^{\circ}$ is naturally interpreted as the set isomorphism classes of marked ribbon graphs endowed with a unital metric. It is obtained from the spaces $\aconf (G ,f)$ by identifying $\aconf (G /G _Z,f/Z)$ with its image in $\aconf (G ,f)$ for every negligible $Z\subset X_1(G )$. Moreover, $A$ supports a family $\pi :\CC \to A$ of weakly normal curves with sections indexed by $P$. Over $A^{\circ}$ these sections are disjoint, the family is locally trivial with fiber $S$ and each fiber comes with a complex structure which varies continuously with the base point. \endproclaim In the next section we shall discuss the fibers over $A_{\infty}$.\par The family $\pi $ restricted to $A^{\circ}$ defines a classifying map $\Phi : A^{\circ}\to\T $. This map is continuous and clearly $\G$-equivariant. The following theorem is a rather direct consequence of the work of Strebel. \proclaim{\label Theorem} The map $$ \Psi ^{\circ}:=(\Phi,\lambda) :A^{\circ}\to\T \times\De _P $$ is a homeomorphism. \endproclaim The observation that Strebel's work leads to theorems of this type is due to Thurston, Mumford and Harer \cite{\harera}. (We did not come across this version, though.) For the proof we must discuss Jenkins-Strebel differentials first. Let $R$ be a Riemann surface. If $q$ is a meromorphic quadratic differential on $R$, then at each point $p$ of $R$ where $q$ has neither a zero nor a pole the tangent vectors at $p$ on which $q$ takes a real value $\ge 0$ form a real line in $T_zC$. This defines a foliation on $R$ minus the singular set of $q$. If the union of the closed leaves of this foliation is dense in $R$, then $q$ is called a {\it Jenkins-Strebel differential}. Suppose $q$ is such a differential. Then a local consideration shows that $q$ has no poles of order $>2$ and that the double residue at a pole of order $2$ is a negative real number. The form $q$ determines a Riemann metric $|q|$ on the complement of the singular set of $q$. This metric is locally like $|dz|^2$ and hence flat. The union $K$ of the non-closed leaves and the singular points of $q$ of order $\ge -1$ is closed in $R$. It is an embedded graph with a singularity of order $k$ being a vertex of valency $k+2$; it is called the {\it critical graph} of $q$. Each connected component of the complement of $K$ is either a flat annulus (metrically a flat cylinder) or a disk containing a unique pole of order two (metrically outside this pole a flat semi-infinite cylinder) or a copy of $\C -\{ 0\}$. Suppose that $R$ is the complement of a finite subset of a compact Riemann surface $C$. Then $q$ is also a Jenkins-Strebel differential on $C$ and the closure $\overline{K}$ of $K$ in $C$ is an embedded graph. (When $C$ has genus zero it may happen that this closure becomes a closed orbit on $C$, so $\overline{K}$ may depend on $R$. It can be shown however, that this is the only such case.) Clearly, $\overline{K}$ has the structure of a ribbon graph. Notice that $q$ defines a metric on it. \proclaim{\label Theorem} {\rm(Strebel)} Let $(C,x)$ be a compact connected $P$-pointed Riemann surface such that is not the two-pointed Riemann sphere and let $\lambda\in\Delta _P$. Then there exists a Jenkins--Strebel differential $q$ on $C$ with the property that the union of the closed leaves of $q$ form semi-infinite cylinders around the points of $x(p)$ with $\lambda (p)\not= 0$ (of circumference $\lambda (p)$) and the points $x(p)$ with $\lambda (p)=0$ lie on the critical graph of $q$. Moreover, such a $q$ is unique. \endproclaim \demo{Proof} Denote by $Q\subset P$ denote the zero set of $\lambda$ and put $Q':=P-Q$. If $|Q'|\ge 2$, then the asserted properties follow from Theorem $23.5$ of \cite{\strebel} applied to the Riemann surface $C-x(Q)$ with circumferences given by $p\in Q'\mapsto \lambda (p)$. (The fact that $q$ will have at the points of $Q$ order $\ge -1$ follows from the discussion above.) In case $Q'$ is a singleton $\{ p\}$, then Theorem $23.2$ of \cite{\strebel} implies that there is Jenkins--Strebel differential on $C-x(Q)$ for which all the closed leaves belong to the cylinder about $p$. This differential is unique up to a positive real scalar factor and hence the theorem follows in this case, too. \enddemo We shall refer to $\lambda$ as a {\it circumference function} of $(C,x)$, the name being suggested by the above theorem. So such a function determines a metrized ribbon graph $(G _{\lambda},l_{\lambda})$ in $C$ (denoted by $\overline{K}$ in the discussion above). Notice that if $\lambda (p)=0$, then $x(p)$ is a univalent vertex or an interior point of an edge of $G _{\lambda}$; if $\lambda (p)\not= 0$, then $x(p)$ defines a boundary cycle of $G _{\lambda}$. Moreover, all univalent vertices and boundary cycles of $G _{\lambda}$ are thus obtained. In other words, $G _{\lambda}$ is in a natural manner an $P$-pointed ribbon graph. The associated $P$-pointed curve $C(G _{\lambda},l_{\lambda})$ is canonically isomorphic to $(C,x)$: this is clear on the complement of the union of $x(P)$ and the vertex set of $G _{\lambda}$. Hence it is true everywhere. \demo{Proof of \refer{7.5}} The above discussion shows that $\Psi ^{\circ}$ has a unique inverse, in other words, that it is bijective. Since $\Psi ^{\circ}$ is continuous and has locally compact domain and range, it must be a homeomorphism. \enddemo \proclaim{\label Corollary} {\rm (Harer \cite{\harera})} For nonempty $P$, the moduli space $\Mod _g^P$ has the homotopy type of a finite semi-simplicial complex of dimension $\le 4g-4+|P|$. In particular, $\Mod _g^P$ has no homology or cohomology in dimension $>4g-4+|P|$. \endproclaim \demo{Proof} Choose $p\in P$ and regard $p$ as a vertex of $\De _P$. Then $\T $ is by \refer{7.5} equivariantly homeomorphic to $\lambda ^{-1}(p)\cap A^{\circ}$. Now apply \refer{4.3}. \enddemo \head \section Minimal models \endhead In this section we introduce a combinatorial analogue of a $Q$-minimal $P$-pointed curve. Here $(G ,x)$ is a connected marked ribbon graph. \medskip\label We say that a set $Z$ of edges of $G$ is {\it semistable} if no component of $G _Z$ is the set of edges of a negligible subset and every univalent vertex of $G_Z$ is in the image of $x$. Then every component of $G_Z$ which is contractible contains at least two vertices in $x(P)$. A component which is a homotopy circle without a vertex in $x(P)$ is necessarily a topological circle which is not a boundary cycle of $G$. It is clear that every subset $Z\subset X_1(G)$ has a maximal semistable subset $Z^{\sst}$. Notice that $Z^{\sst}-Z$ is a negligible subset of $X_1(G)$ so that if we put $G':=G/G_{Z^{\sst}-Z}$, then $S(G')$ is $\qc$-homeomorphic relative $P$ to $S(G)$. We sometimes regard $G_{Z^{\sst}}$ as a graph on $S(G')$, so that with this convention $G/G_Z=G'/G'_{Z^{\sst}}$. \medskip\label Let be given a proper subset $Z$ of $X_1(G)$. We can associate to $Z$ two ribbon graphs: one with edge set $Z$ and another with edge set $X_1(G)-Z$. In the first case we give $G _Z$ an induced structure of ribbon graph by telling how the corresponding operator $\sigma _0$ acts on $X(G _Z)$: it sends $e\in X(G _Z)$ to the first term of the sequence $(\sigma _0^k(e))_{k\ge 1}$ which is in $X(G _Z)$. The second case is in a sense dual to the first: we define a ribbon graph $G/G _Z$ with $X_1(G )-Z$ as its set of edges and the corresponding operator $\sigma _{\infty}$ sends $e\in X(G )-X(G _Z)$ to the first term of the sequence $(\sigma _{\infty}^k(e))_{k\ge 1}$ which is not in $X(G _Z)$. This ribbon graph naturally maps onto a subgraph of $G$, but this map need not be injective as it may identify distinct vertices of $G/G_Z$. A vertex of $G/G_Z$ that is in the image of an oriented edge in $Z^{\sst}$ will be called {\it exceptional}. Any such vertex corresponds to a boundary cycle of $G_{Z^{\sst}}$ that is not a boundary cycle of $G$ (and vice versa), reason for us to call such boundary cycles {\it exceptional} also. \proclaim{\label Lemma} There is a natural identification mapping of $S(G/G_Z)\to S(G)_Z$. This map identifies two distinct points if and only if both are exceptional vertices of $S(G/G _Z)$ that come from a boundary cycle of the same component of $G _{Z^{\sst}}$. In particular, $S(G)_Z$ is a pseudosurface whose combinatorial normalization is $S(G/G_Z)$. Moreover, every distinguished point of $G/G_Z$ comes from a distinguished point of $G$ or is exceptional. \endproclaim \demo{Proof} Straightforward. \enddemo In this situation we have a genus defect function $\epsilon : S(G)_Z\to \Z _{\ge 0}$ which assigns to the image of an exceptional vertex the genus of the corresponding component of $S(G_{Z^{\sst}})$ and is zero else. \medskip\label Choose an $l\in a(G)$. In \refer{6.4} we constructed a map $\pi _{G }:\CC (G )\to a(G )$ and we noticed that that the fiber over $l$, $\CC (G )_l$, can be identified with $S(G )_Z$, where $Z$ is the zero set of $l$. Since $l$ determines a unital metric on $G/G_Z$, we have a Riemann surface $C(G/G_Z,l)$ with underlying space $S(G/G_Z)$. We use the previous lemma to give $\CC (G)_l$ the unique complex-analytic structure for which $\CC (G )_l$ is weakly normal and $C(G/G_Z,l)\to C(G)_l$ is its normalization. \proclaim{\label Proposition} Let $Q$ be the set of $p\in P$ that map to a boundary cycle of $G$ of positive length. Then $(Q,\epsilon ,P\to S(G )\to \CC (G )_l)$ give $\CC (G )_l$ the structure of a $Q$-minimal $P$-pointed curve. \endproclaim \demo{Proof} We verify the defining properties of \refer{3.4}. The property for $p\in P$ to belong to $Q$ is equivalent to $x(p)\in X_{\infty}(G /G _Z)$. The first property now follows. For the second we must show that $S(G /G _Z)-X_{\infty}(G /G _Z)-\{\text{exceptional vertices}\}$ has negative Euler characteristic. But this follows from the fact that this is (by \refer{8.3}) just the complement of the set of distinguished points on $G/G_Z$. The verification of the third property is left to the reader. \enddemo Suppose we are given a marking $f$ of $G$ that extends the pointing by $x$. This determines a marking of $\CC (G )_l$ by $(S,P)$. In view of the moduli interpretation \refer{3.5}, the structure present on $\CC (G )_l$ determines a point of $K_Q\T$. By letting $l$ vary over the elements of $a(G,f)$, we thus obtain a map $a(G,f)\to |K_{\bullet}\T |$ commuting with the given maps of domain and range to $\Delta _P$. For a negligible edge $s$ of $G$ the restriction of this map to $a(G /s,f/s)$ coincides with the one defined for that simplex. This results in an $\G$-equivariant map $\Psi :A\to |K_{\bullet}\T |$. We can now state our first main result. It gives an analytic interpretation of $A$: \proclaim{\label Theorem} The map $\Psi : A\to |K_{\bullet}\T |$ is a $\G$-equivariant continuous bijection that commutes with the given maps to $\De _P$. \endproclaim The main difficulty is to show that $\Psi$ is continuous. We postpone the proof to a point where we have treated the combinatorial version of the Deligne--Mumford compactification. The reader may wonder whether $\Psi $ is a homeomorphism. The answer is that it is not, as is illustated by the case $g=1$, $P$ a singleton: then $|K_{\bullet}\T |$ is the union of the upper half plane and $P^1(\Q )$. Near $\infty$ it has the horocyclic topology but the topology it receives from its triangulation is much finer: a subset of the upper half plane is the complement of a neighborhood of $\infty$ if and only if its intersection with any vertical strip of bounded width is bounded. \head \section Stable ribbon graphs \endhead Here we introduce the ribbon graph analogue of a stable $P$-pointed curve. That our definition is the natural one may not be immediately obvious, but that this is indeed the case will become apparent in the discussion following the definition and in section 10. \medskip\label Suppose we are given a ribbon graph $G$ and an injection $x:P\to X_0(G)\sqcup X_{\infty}(G)$. We no longer assume that $x(P)$ contains the set of distinguished points of $S(G)$, but instead we suppose given a subset $\Sigma\subset X_0(G)\sqcup X_{\infty}(G)$ which contains both $x(P)$ and the distinguished points of $G$ and an involution $\iota$ on the complement $\Sigma - x(P)$. We define inductively the {\it order} of a connected component of $G$ as follows: a connected component is of order zero if it contains a point of $x(P)\cap X_{\infty}(G)$; a connected component has order $\le k+1$ if it contains a distinguished point $p$ such that $\iota (p)$ lies on a component of order $\le k$. We say that $(G,x,\iota )$ is a {\it stable $P$-pointed ribbon graph} if \roster \item every component has an order and \item for every $p\in X_{\infty}(G)$ on a component of order $k>0$, $\iota (p)$ is on a component of order $k-1$. \endroster (So in the situation (2) we must have $\iota (p)\in X_0(G)$.) \medskip\label A stable $P$-pointed ribbon graph $(G,x,\iota )$ determines a stable $P$-pointed pseudosurface $(S(G,\iota ),x)$: it is obtained from the surface $S(G)$ by identifying the points (of $\Sigma -x(P)$) according to the involution $\iota $. If this surface is connected, then it has a {\it genus $g$} characterized by the condition that $2-2g$ is the Euler characteristic of the smooth part of $S(G,\iota )$. We have seen that a conformal structure $l$ on $G$ determines a conformal structure on $S(G)$ so that we have a compact Riemann surface $C(G ,l)$. This in turn, determines a weakly normal complex-analytic structure on $S(G,\iota )$. With that structure, $(S(G,\iota ),x)$ becomes a stable $P$-pointed curve $(C(G,\iota ,l),x)$. This curve has additional structure: to every point $p\in x(P)\cup S(G,\iota )_{\sing}$ is assigned a nonnegative number $\lambda (p)$, namely half the length of the corresponding boundary cycle (with respect to the componentwise unital metric defining the conformal structure) in case the point comes from $X_{\infty}(G)$ and zero else. Notice that $\lambda (p)=0$ if $x(p)$ lies on a single irreducible component of $S(G ,\iota )$ or if $p\in P$ and $x(p)\in X_0(G)$, and that the sum of the values of $\lambda$ on each irreducible component is $1$. This suggests to extend the notion of a {\it circumference function} to the case of a stable connected $P$-pointed pseudosurface $(S',x)$ as as a function $\lambda :x(P)\cup S' _{sing}\to\R _{>0}$ which possesses these properties. So the space of circumference functions on $(S',x)$ is a product of simplices (with a factor for each irreducible component). \medskip\label Just as for smooth $P$-pointed curves, the datum of a cicumference function $\lambda$ on a stable $P$-pointed curve $(C,x)$ permits us to go in the opposite direction: apply Strebel's theorem \refer{7.6} componentwise to the normalisation $(\hat C,\lambda )$. This determines a Jenkins-Strebel differential $q$ on $\hat C$ with the properties mentioned there. In particular, we have a critical graph $(G ,l)$ in $\hat C$ which contains the zeroes of $\lambda$. Moreover, each $p\in\supp (\lambda)$ determines (and is determined by) a boundary cycle of $G$ and the length of that boundary cycle is $\lambda (p)$. The associated Riemann surface $C(G ,l)$ is naturally isomorphic to $\hat C$. \medskip\label Let now $(G,x)$ be a $P$-pointed ribbon graph. We describe how a proper subset of $X_1(G)$ (or rather, strictly decreasing sequences of such) define stable $P$-pointed ribbon graphs. First two definitions. Let $Z$ be a semistable set of edges of $G$. Recall that then every component of $G_Z$ that is a homotopy circle without a vertex in $x(P)$ is necessarily a topological circle (and is not a boundary cycle of $G$). If this does not happen, i.e., if every component of $G_Z$ that is a topological circle contains a vertex in the image of $x$, then we say that $Z$ is {\it stable}. It is clear that every subset $Z\subset X_1(G)$ has a maximal semistable subset $Z^{\st}$; it is a union of components of $Z^{\sst}$. Forgetting the bivalent vertices of $G_{Z^{\st}}$ that are in $x(P)$ yields a ribbon graph with the same underlying topological space as $G_{Z^{\st}}$; we denote this ribbon graph by $\bar G_{Z^{\st}}$ and its set of edges by $\bar Z^{\st}$. It is clear that the set of distinguished points of $S(G_{Z^{\st}})$ coincides with $X_{\infty}(G_{\bar Z^{\st}})$. A metric on $G_{Z^{\st}}$ determines one on $\bar G_{Z^{\st}}$. \medskip\label Let $Z$ be a proper subset of $X_1(G)$ and put $G(Z):=G/G_Z\sqcup G_{Z^{\st}}$. It is clear that the pointing $x$ determines an injection $\tilde x$ of $P$ in the set of $0$-simplices of $G(Z)$. The proof of the following lemma is easy and left to the reader \proclaim{\label Lemma} The set of distinguished points of $G(Z)$ that are not in the image of $\tilde x$ comes with a natural involution $\iota$ so that $G(Z)$, $\tilde x$ and $\iota$ define a stable $P$-pointed ribbon graph. The associated $P$-pointed stable pseudosurface $S(G;Z)$ is obtained from $S(G/G_Z)$ and $G_{Z^{\st}}$ by identifying each exceptional vertex of $S(G/G_Z)$ with the corresponding exceptional element of $X_{\infty}(G_{Z^{\st}})$ and then contracting every irreducible component that corresponds to a component of $G_{Z^{\sst}}-G_{Z^{\st}}$. A conformal structure on $\tilde G$ determines one on $S(\tilde G,\iota)$ and turns the latter into a stable $P$-pointed curve. \endproclaim \label We may of course repeat this construction for a set of edges of $G_{\bar Z^{\st}}$. In order to be able to state this we introduce the following notions. A {\it permissible sequence} for $(G ,x)$ is a sequence $Z _{\bullet}=(X_1(G ) \! =\! Z_0,Z_1, Z_2,\dots ,Z_k)$ such that $Z _{\kappa}\subset \bar Z_{\kappa -1}^{\st}$ and $G_{Z _{\kappa}}$ does not contain a connected component of $\bar G_{Z_{\kappa -1}^{\st}}$. A {\it stable metric} relative such a sequence is given by a conformal structure on every difference $\bar G _{Z_{\kappa }^{\st}}-G _{Z_{\kappa +1}}$. So this may be given by a sequence of functions $l_{\kappa}:Z_{\kappa}^{\st}\to\R {\ge 0} $ such that $l_{\kappa}$ has zero set $Z_{\kappa +1}$ ($\kappa =0,1,\dots $). (So $l_{\bullet}$ determines $Z _{\bullet}$.) The previous discussion generalizes in a straightforward way to: \proclaim{\label Proposition} Let $Z _{\bullet}$ be a permissible sequence for $(G ,x)$. Then the disjoint union of the ribbon graphs $G_{\bar Z^{\st}_{\kappa}}/G_{Z_{\kappa +1}}$ ($\kappa =0,1,\dots )$ is in a natural way a stable $P$-pointed ribbon graph $(G(Z_{\bullet}),\tilde x,\iota )$. A stable metric $l_{\bullet}$ relative $Z _{\bullet}$ defines a conformal structure on $S(G,Z_{\bullet})$ and turns it into a stable $P$-pointed curve $C(G, l_{\bullet})$. \endproclaim \head \section Stable limits \endhead In this section we fix a connected $P$-pointed ribbon graph $(G,x)$. We explain how the stable pseudosurface associated to a permissible sequence for $G$ arises as a limit of Riemann surfaces $C(G,l(t))$. \medskip\label We shall use a blowing up construction in the PL-category. The basic construction starts out from a collection $\beta$ of oriented edges of $G$ that defines an oriented circular subgraph $G _{\beta}$ of $G$. Let $U_{\beta}$ be the union of the relatively open simplices that have a point of $G _{\beta}$ in their closure; this is a regular neighborhood of $G _{\beta}$ PL-homeomorphic to an open cylinder. Notice that $U_{\beta}-G _{\beta}$ has two connected components, one which contains the interiors of the tiles associated to the elements of $\beta$; we denote that component $U_{\beta}^+$ and the other by $U_{\beta}^-$. By means of the barycentric coordinates of the simplices in $U^+_{\beta}$ we have defined a piecewise-linear function $U^+_{\beta}\to [0,1)$ which measures the distance to $G _{\beta}$. Let $\phi _{\beta}: U_{\beta}\to [0,1)$ be its extension by zero on $U_{\beta}$; this is a continuous PL-function. Let $(U_{\beta}\times \R _{\ge 0})^{\,\widetilde{}}$ be the closure of the graph of the function $$ (u,t)\in (U_{\beta}-G _{\beta})\times\R _{>0}\mapsto [-\log (1-\phi _{\beta}(u):t]\in P^1(\R ) $$ in $U_{\beta}\times\R _{\ge 0}\times P^1(\R )$. The projection $(U_{\beta}\times \R _{\ge 0})^{\,\widetilde{}}\to U_{\beta}\times\R _{\ge 0}$ is clearly a PL-homeomorphism over the complement of $G _{\beta}\times\{ 0\}$ whereas the pre-image of $G _{\beta}\times\{ 0\}$ is $G _{\beta}\times\{ 0\}\times [0,\infty ]$. The strict transform of $U_{\beta}^+\times\{ 0\}$ resp.\ $U_{\beta}^-\times\{ 0\}$ meets $G _{\beta}\times\{ 0\}\times [0,\infty ]$ in $G _{\beta}\times\{ 0\}\times \{\infty \}$ resp.\ $G _{\beta}\times\{ 0\}\times \{ 0\}$. So the total transform of $U_{\beta}\times\{ 0\}$ is a kind of thickening of $U_{\beta}$ (see the figure below). \midspace{50mm}\caption{\fig Blowing up of an oriented cycle} In particular, this total transform is PL-homeomorphic to $U_{\beta}$; indeed, the projection $(U_{\beta}\times \R _{\ge 0})^{\,\widetilde{}}\to \R _{\ge 0}$ is trivial. We glue $(U_{\beta}\times\R _{\ge 0})^{\,\widetilde{}}\to \R _{\ge 0}$ to $(S(G )\times \R _{\ge 0})-(G _{\beta}\times\{ 0\})$ via their common open subset $U_{\beta}\times\R _{\ge 0} -G _{\beta}\times\{ 0\}$ and obtain a modification $(S(G )\times\R _{\ge 0})_{\beta}^{\,\widetilde{}}\to S(G )\times\R _{\ge 0}$. For $e\in\beta$, the tile $K_e\times\{ 0\}$ lifts PL-homeomorphically to $(S(G )\times\R _{\ge 0})_{\beta}^{\,\widetilde{}}$. We apply an edge collapse to all these lifted copies and denote the result $(S(G )\times\R _{\ge 0})_{\beta}^{\,\widehat{}}$. The pre-image of $S(G)\times\{ 0\}$ is denoted by $S(G;\beta )$. It is a pseudosurface that is PL-homeomorphic to the the space obtained from $S(G)$ by contracting $G_{\beta}$. It comes with an injection of $P$ in its regular part. \medskip\label We now fix a proper subset $Z$ of $X_1(G )$ and show how $S(G ;Z)$ is obtained as a one-parameter degeneration of $S(G )$. First we assume that $Z$ is stable. We carry out the previous construction for each boundary cycle of $Z$. It is easily seen that these can be performed independently so that we have defined a modification $$ (S(G )\times\R _{\ge 0})^{\,\widehat{}}_Z\to S(G )\times\R _{\ge 0}. $$ The crucial remark is that this projection $(S(G )\times \R _{\ge 0})^{\,\widehat{}}_Z\to \R _{\ge 0}$ is trivial over $\R _{>0}$ with fiber $S(G )$ whereas the fiber over $0$ is canonically isomorphic to $S(G;Z)$. In case $Z$ is not stable, we first apply the preceding procedure to $Z^{\st}$ and next we collapse the strict transforms of the tiles indexed by the oriented members of $Z-Z^{\st}$ (a total collapse or an edge collapse, depending on whether the boundary cycle generated by the corresponding oriented edge is in $G_Z$ or not). The order of these operations can be reversed; in particular, we can first pass to $G':=G/G _{Z-Z^{\sst}}$ and the image $Z'$ in $X_1(G')$ (so that $Z'$ is now semistable), then perform edge collapses on the tiles indexed by the oriented members of $Z'-Z'{}^{\st}$ (these make up a union of circular components of $G_{Z'}$) and finally apply the preceding construction with $Z'{}^{\st}$. Then the fiber over $0$ can be identified with $S(G;Z)$ as before. \smallskip We already observed that conformal structures $l_0$ on $G/G_Z$ and $l_1$ on $G_Z$ determine a conformal structure on $S(G;Z)$, turning it into a stable $P$-pointed curve $C(G,(l_0,l_1))$ whose normalization is the disjoint union of the Riemann surfaces $C(G/G_Z,l_0)$ and $C(G_{Z^{\st}},l_1)$. We may obtain such conformal structures by means of a degeneration of a family of metrics on $S(G)$. To be concrete, let $l$ be a metric on $G$ and let for $t>0$, $l(t)$ be the metric on $G$ which takes on an edge $s$ the value $tl(s)$ if $s\in Z$ and remains $l(s)$ if not. We give the fiber of $(S(G )\times \R _{\ge 0})^{\,\widehat{}}_Z\to \R _{\ge 0}$ over $t\in \R _{>0}$ (which is just $S(G)$) the corresponding metric structure (denoted $m_t$). The regular part of the fiber over $0$ is given the metric structure $m_0$ defined by the restrictions $l_0$ resp.\ $l_1$ of $l$ to $X_1(G )-Z$ resp.\ $Z$. This is in general not a continuous family of metrics, but for the underlying conformal structures the situation is different. To see this, let $\phi _Z:S(G)\to \R _{\ge 0}$ be the piecewise-linear function that takes the value $0$ on every vertex in $G_Z$, $U_Z\subset S(G)$ the set where $\phi _Z<1$ and put $f_Z:=-\log (1-\phi _Z): U_Z\to\R _{\ge 0}$. It is clear from our definition of $m_t$ that the set $f_Z<a$ with metric $m_t$ is conformally equivalent to subset $f_Z<t^{-1}a$ with metric $m_1$. In fact, we have \proclaim{\label Lemma} Suppose that the pointing $x$ of $G$ has been extended to a marking by $(S,P)$. Then the map $\R _{\ge 0}\to \overline{\T }$, which assigns to $t>0$ resp.\ $t=0$ the isomorphism class of $C(G,l(t))$ resp.\ $C(G, (l_0,l_1))$ is continuous. \endproclaim \demo{Proof} There is no loss of generality in assuming that $G_Z$ has no negligible components. The continuity on $\R _{>0}$ is clear. To prove continuity at $0$ we wish to invoke \refer{2.1}. This requires that we trivialize our family locally. At the points of $S(G;Z)$ outside the exceptional set this is no problem and it is clear that relative a suitable trivialisation the complex structures converges uniformly on compact subsets. At the points of $S(G;Z)$ outside the strict transform we trivialize as follows. Choose a piecewise-linear retraction $r_Z:U_Z\to G_Z$ so that $(r_Z,f_Z)$ defines a PL-homeomorphism $h$ of $U_Z-G_Z$ onto $\tilde G_Z\times\R_{>0}$, where $\tilde G_Z$ is the disjoint union of the boundary cycles of $G_Z$. Let $k$ denote its inverse and for $t>0$, let $k_t(p,s)=k(p,st)$. Then $$ (p,s,t)\in \tilde G_Z\times\R _{> 0}\times\R _{>0}\mapsto (k_t(p,s),t) $$ extends to a PL-homeomorphism of $\tilde G_Z\times \R _{> 0}\times\R _{\ge 0}$ onto an open subset of $(S(G)\times \R _{\ge 0})^{\,\widehat{}}_Z$ so that for $t=0$ we get a PL-homeomorphism $k_0$ of $\tilde G_Z\times \R _{> 0}$ onto the complement of the union of the strict transform of $S(G)$ and $G_Z$ in $S(G;Z)$. We must show that the conformal structure $J_t$, $t\ge 0$ on $\tilde G_Z\times (0,1)$ defined by pull-back of the given conformal structure on $C(G,l(t))$ under $k_t$ depends continuously on $t$. This is proved using explicit coordinates. We leave that to the reader. \enddemo The preceding can be iterated in an obvious way and yields: \proclaim{\label Proposition} If $Z_{\bullet}$ is a permissible sequence, then there is defined an iterated modification: $$ (S(G)\times \R _{\ge 0})^{\,\widehat{}}_{Z_{\bullet}}\to \R _{\ge 0}. $$ This fibration is canonically trivialized (relative $x$) over $\R _{>0}$ with fiber $S(G)$, whereas the fiber over $0$ is canonically homeomorphic to $S(G;Z_{\bullet})$. Suppose that the pointing $x$ of $G$ has been extended to a marking by $(S,P)$. Given a metric $l$ on $G$, let $l(t)$ be the metric on $G$ that on $G_{Z_{\kappa}}-G_{Z_{\kappa +1}}$ is equal to $t^{\kappa}l$ ($t>0$) and let $l_{\bullet}$ be the stable metric relative $Z _{\bullet}$ that is defined by the restrictions of $l$. Then the map $\R_{\ge 0}\to \overline{\T }$ which assigns to $t\in\R_{>0}$ resp.\ $t=0$ the isomorphism type of $C(G,l(t))$ resp.\ $S(G)\cong C(G,l_{\bullet})$ is continuous. \endproclaim \head \section Deligne--Mumford modification of the arc complex \endhead Let $(G,x)$ be a connected $P$-pointed ribbon graph. Recall that we have defined the family $\pi :\CC (G )\to a(G)$ which over the interior $\conf (G )$ of $a(G)$ is trivialized with fiber $S(G)$. We are going to modify this family over the locus where this family is not locally trivial. This will also modify the base and the result will be a family parametrizing stable pointed pseudosurfaces with stable conformal structures. \medskip Let $\Cal{Z}(G )$ denote the collection of stable subsets $Z\subset X_1(G )$ with $G_Z$ connected. For $l\in \conf (G )$ and $Z\in \Cal{Z}(G )$, we let $\pi _Z(l)$ denote the unital metric on $G _{\bar Z}$ which is proportional to $l|G _Z$. Let $\hat a(G )$ be the closure of the graph of the map $l\in \conf (G )\mapsto (\pi _Z(l)\in \conf (G _Z))_Z$ in $a(G )\times \prod _{Z\in \Cal{Z}(G )} a(G _{\bar Z})$. \proclaim{\label Proposition} There is a natural bijection between the points of $\hat a(G )$ and the set of stable conformal structures on $G$. \endproclaim \demo{Proof} Let $(l^{(n)})_{n=1}^{\infty}$ be a sequence in $\conf (G )$. By passing to a subsequence, we may assume that for every $Z\in\ZZ (G)$, the sequence $(\pi _Z(l^{(n)}))_n$ converges (to $l_Z$, say). Write $l_0$ for $l_{X_1(G)}$, let $Z(l_0)$ be the zero set of $l_0$ and put $Z_1:=Z(l_0)^{\st}$. Notice that $\Cal{Z}(G_{Z_1})$ is just a subset of $\Cal{Z}(G_Z)$. So for each $Z\in \Cal{Z}(G_{Z_1})$ we have a function $l_{Z}:Z\to [0,1]$ whose sum is $1$. Applying this to the connected components of $G_{Z_1}$ yields a function $l_1:Z_1\to [0,1]$ that on each connected component of $G_{Z_1}$ sums up to $1$. We proceed with induction: if $l_{\kappa}: Z_{\kappa}\to [0,1]$ has been constructed, then let $Z(l_{\kappa})$ be the zero set of $l_{\kappa}$. We put $Z_{\kappa +1}:=Z(l_{\kappa})^{\st}$ and define $l_{\kappa +1}: Z_{\kappa +1}\to [0,1]$ by letting it on each connected component $G_{Z}$ of $G_{Z_{\kappa +1}}$ be equal to $l_{Z}$. Then $Z_{\bullet}$ is a permissible sequence for $(G ,x)$ by construction. It comes naturally with a unital stable metric $l_{\bullet}$ relative this sequence. This stable metric determines every $l_Z$: for $Z\in\Cal{Z}(G)$, let $\kappa$ be such that $G_Z\subset G_{Z _{\kappa}}$ and $G_Z\not\subset G_{Z _{\kappa +1}}$. Then $G_Z$ is contained in a connected component $G_{Z'}$ of $G_{Z_{\kappa}}$. Since $Z\not\subset Z _{\kappa +1}$, $l_{Z'}|Z$ (and hence $l_{\kappa }|Z$) is not identically zero. It then follows that $l_Z$ is the unital metric proportional to $l_{\kappa }|Z$. On the other hand, \refer{10.4} shows that every stable metric thus arises. \enddemo \medskip\label If $Z$ is a negligible set of edges of $G$, then $\hat a(G /G _Z)$ can be identified with the subset of $\hat a(G )$ parametrizing stable metrics $l_{\bullet}$ of which each term vanishes on $Z$. Hence if we endow the ribbon graphs with markings, then the closed cells $\hat a(G ,f)$ can be glued together to yield a modification $$ \hat A\to A. $$ It is clear that $\hat A$ comes with a decomposition into cells. Such a cell admits a description in terms of arc complexes as follows: it is of the form $\sigma _0\times\sigma _1\times\dots$, where each $\sigma _{\kappa}$ is a cell (a product of simplices) of the arc complex associated to a (not necessarily connected) pointed surface $(S_{\kappa},P_{\kappa})$. These pointed surfaces (and hence these cells) are defined inductively: $(S_0,P_0):=(S,P)$ and $\sigma _0$ is an arbitrary simplex of $A$. For $\kappa\ge 1$, let $\bar S'_{\kappa}$ be the pseudosurface obtained from $S_{\kappa -1}$ by contracting the arcs that make up $\sigma _{\kappa -1}$, $S'_{\kappa}$ its normalisation, and let $P'_{\kappa}\subset S'_{\kappa}$ the pre-image of the image of $P_{\kappa -1}$. Let $(S_{\kappa},P_{\kappa})$ be obtained from $(S'_{\kappa},P'_{\kappa})$ by discarding all components that are one- or two-pointed spheres. The connected components of $S_{\kappa}$ label the factors of $\sigma _{\kappa}$ so that each factor is made up of arcs in that component. We require that these arcs connect only points of $P_{\kappa}$ that map to singular points of $\bar S'_{\kappa -1}$. Under the projection $\hat A\to A$ this cell maps to $\sigma _0$. It is possible to give a complete description of the incidence relations between these cells, but we omit this. \medskip\label We shall define a family of surfaces $\hat\CC (G )$ over $\hat a(G )$. Let $Z_{\bullet}$ be a permissible sequence for $G$ of {\it connected stable} subsets, which we here regard as a strictly decreasing sequence of connected stable subsets of $X_1(G)$, and consider the map $$ I_{Z_{\bullet}}: S(G)\times\conf (G)\to \prod _{\kappa \ge 1} (S(G)\times \R _{>0}),\quad (u,l)\mapsto (u,l(Z_{\kappa})/l(Z_{\kappa -1}))_{\kappa}. $$ The closure of its graph in $S(G)\times a(G)\times \prod _{\kappa \ge 1}(S(G)\times \R _{\ge 0})^{\,\widehat{}}_{Z_{\kappa}}$ is denoted by $(S(G)\times a(G))^{\,\widehat{}}_{Z_{\bullet}}$. Similarly, we denote the closure of the graph of $$ \conf (G)\to\prod _{\kappa \ge 1}\R _{>0},\quad l\mapsto (l(Z_{\kappa})/l(Z_{\kappa -1}))_{\kappa} $$ in $a(G)\times\prod _{\kappa \ge 1}\R _{\ge 0}$ by $a(G)^{\,\widehat{}}_{Z_{\bullet}}$. Since the functions $l(Z_{\kappa})/l(Z_{\kappa -1})$ extend continuously to $\hat a(G)$, this is a quotient of $\hat a(G)$. We have a projection $$ (S(G)\times a(G))^{\,\widehat{}}_{Z_{\bullet}}\to a(G)^{\,\widehat{}}_{Z_{\bullet}}. $$ Any fiber over a point of $a(G)^{\,\widehat{}}_{Z_{\bullet}}$ that has all its coordinates in $\prod _{\kappa \ge 1}\R _{\ge 0}$ equal to zero is isomorphic to $S(G;Z_{\bullet})$. \smallskip We do this for all such sequences simultaneously. To be precise, let $\ZZZ (G)$ be the collection of strictly decreasing sequences of connected stable subsets of $X_1(G)$, and consider the map $$ I=(I_{Z_{\bullet}}):S(G)\times\conf (G)\to \prod _{Z_{\bullet}} \prod _{\kappa \ge 1} (S(G)\times \R _{> 0}). $$ The closure of its graph in $$ \hat a(G)\times \prod _{Z_{\bullet}} \prod _{\kappa \ge 1} (S(G)\times \R _{\ge 0})^{\,\widehat{}}_{Z_{\kappa}} $$ is denoted $\hat\CC (G)$ and the projection of $\hat\CC (G)$ onto $\hat a(G)$ by $\hat \pi _G$. The preceding discussion shows: \proclaim{\label Proposition} If $l_{\bullet}$ is a stable metric with associated permissible sequence $Z_{\bullet}$, then the fibre $\hat \pi _{G } ^{-1}(l_{\bullet})$ is naturally homeomorphic to $S(G;Z_{\bullet})$. \endproclaim We endow the fiber $\hat \pi _{G }^{-1}(l_{\bullet})$ with the conformal structure prescribed by the stable metric $l_{\bullet}$ so that $\hat\pi _G$ defines a family of stable $P$-pointed stable curves. For marked ribbon graphs this construction is compatible in the sense that if $Z\subset X_1(G )$ is negligible, then $\hat \pi _{G /G _Z}:\hat\CC (G /G _Z)\to \hat a(G /G _Z)$ can be identified with the restriction of $\hat \pi _{G }$ over $\hat a(G /G _Z)$. We may therefore glue these maps to each other to get a family $\hat \pi :\hat\CC \to \hat A$ of stable $P$-pointed curves. Each fiber of $\hat \pi $ maps to a fiber of $\pi $, so that we have a commutative diagram $$ \CD \hat\CC @>>> \CC \\ @V\hat \pi VV @V\pi VV\\ \hat A @>>> A \endCD $$ of spaces with $\G$-action. We have also have a classifying map that extends $\Phi$: $$ \hat\Phi :\hat A\to\overline{\T } . $$ It is clearly $\G$-equivariant. Our second main result reads as follows: \proclaim{\label Theorem} The map $\hat\Phi:\hat A\to \overline{\T }$ is a $\G $-equivariant continuous surjection. The pre-image of the class of a marked stable $P$-pointed curve $(C,[f])$ under $\hat\Phi$ can be identified with the space of circumference functions \refer{9.2} of $(C,x)$. In particular, $\hat\Phi$ drops to a continuous surjection of $\G \backslash\hat A $ onto the Deligne--Mumford compactification $\overline{\Mod}{}^P_g$. \endproclaim \demo{Proof} Let $(C,[f])$ be as in the theorem. The construction described in \refer{9.3} produces for every circumference function of $(C,x)$ a marked ribbon graph $(G ,f)$ plus a stable metric $l_{\bullet}$ on $G$ which reconstructs $(C,[f])$ for us. This defines an element of $\hat a(G ,f)$ and one verifies that its image in $\hat A$ is unique. It remains to show that $\hat\Phi$ is continuous. It is enough to prove that its restriction to every closed cell $\hat a(G ,f)$ is. Since $\hat a(G ,f)$ is second countable and $\overline{\T }$ is Hausdorff, we only need to verify that the image of a converging sequence $(l_{\bullet}^{(n)})_n$ in $\hat a(G ,f)$ under $\hat\Phi$ has a limit point. Then after passing to a subsequence we may assume that $(l_{\bullet}^{(n)})_n$ is in the relative interior of a single cell, say of $\hat a(G ,f)$. The desired property then follows from \refer{2.1} as in the proof of \refer{10.3}. \enddemo We can now finish the proof of our first main result, too. \medskip \demo{Proof of \refer{8.6}} The map $\hat\Phi$ and the projection $\hat A\to A\to\Delta _P$ together define a map from $\hat A$ to $\overline{\T }\times\Delta _P$. If we compose the latter with the quotient map $\overline{\T }\times\Delta _P\to |K_{\bullet}\T |$ we get a map $\hat\Psi :\hat A\to |K_{\bullet}\T |$. The theorem above implies that the fibers of $\hat\Psi$ and the fibers of $\hat A\to A$ coincide. The induced bijection $A\to |K_{\bullet}\T |$ is just $\Psi$. Since $A$ has the quotient topology for the projection $\hat A\to A$, it follows that $\Psi$ is continuous. \enddemo \Refs \ref\no 1 \by B.H.\ Bowditch \&\ D.B.A.\ Epstein \paper Natural triangulations associated to a surface \jour Topology \vol 27 \pages 91-117 \yr 1988 \endref \ref\no 2 \by P.\ Deligne \&\ D.\ Mumford \paper The irreducibility of the space of curves of given genus \jour Inst\. Hautes \'Etudes Sci\. Publ\. Math\. \vol 36 \yr 1969 \pages 75--109 \endref \ref\no 3 \by J.L.\ Harer \paper The virtual cohomological dimension of the mapping class group \jour Invent. Math. \vol 84 \yr 1986 \pages 157--176 \endref \ref\no 4 \by J.L.\ Harer \paper The cohomology of the moduli space of the space of curves \pages 138--221 \inbook Theory of Moduli \ed E.\ Sernesi \bookinfo Lecture Notes in Math. \vol 1337 \publ Springer \publaddr Berlin and New York \yr 1988 \endref \ref\no 5 \by W.J.\ Harvey \paper Boundary structure of the modular group \inbook Riemann surfaces and related topics \bookinfo Annals of Math. Studies \eds I.\ Kra and B.\ Maskit \publ Princeton UP \yr 1981 \pages 245--251 \endref \ref\no 6 \by M.\ Kontsevich \paper Intersection theory on the moduli space of curves and the matrix Airy function \jour Comm. Math. Phys. \vol 147 \yr 1992 \pages 1--23 \endref \ref\no 7 \by J.\ Milgram \&\ R.C.\ Penner \paper Riemann's moduli space and the symmetric group \inbook Mapping class groups and moduli spaces of Riemann surfaces \eds C.--F.\ B\"odigheimer \&\ R.\ Hain \bookinfo Contemp.\ Math. \vol 150 \pages 247--290 \yr 1993 \endref \ref\no 8 \by R.C.\ Penner \paper Perturbative series and the moduli space of punctured surfaces \jour J.\ Diff.\ Geom. \vol 27 \yr 1988 \pages 35--53 \endref \ref\no 9 \by K.\ Strebel \book Quadratic differentials \bookinfo Ergebnisse der Math.\ u.\ ihrer Grenzgebiete, 3.\ Folge \vol 5 \publ Springer \publaddr Berlin and New York \yr 1984 \endref \ref\no 10 \by S.\ Wolpert \paper The hyperbolic metric and the geometry of the universal curve \jour J.\ Diff.\ Geom. \vol 31 \pages 417--472 \yr 1990 \endref \endRefs \enddocument

9412

alg-geom/9412007

en

https://arxiv.org/abs/alg-geom/9412007

alg-geom/9412007

Dan Edidin

D. Edidin, W. Graham

Characteristic classes and quadric bundles

Duke Mathematical Journal, to appear, 29pages LaTeX

We construct Euler and Stiefel-Whitney classes of vector bundles with quadratic form by analyzing the intersection theory of the associated quadric bundles. We also compute the Chow rings of quadric and isotropic flag bundles. Along the way, we prove a conjecture of Fulton on top Chern classes of maximal isotropic sub-bundles of an even rank quadratic vector bundle.

alg-geom

\section{Introduction} \label{s.intro} In this paper we construct Stiefel-Whitney and Euler classes in Chow cohomology for algebraic vector bundles with nondegenerate quadratic form. These classes are not in the algebra generated by the Chern classes of such bundles and are new characteristic classes in algebraic geometry. On complex varieties, they correspond to classes with the same name pulled back from the cohomology of the classifying space $BSO(N,{\Bbb C})$. The classes we construct are the only new characteristic classes in algebraic geometry coming from the classical groups (\cite{T2}, \cite{E-G-T}). We begin by using the geometry of quadric bundles to study Chern classes of maximal isotropic subbundles. If $V\rightarrow X$ is a vector bundle with quadratic form, and if $E$ and $F$ are maximal isotropic subbundles of $V$ then we prove (Theorem \ref{t.chern}) that $c_{i}(E)$ and $c_{i}(F)$ are equal mod 2. Moreover, if the rank of $V$ is $2n$, then $c_{n}(E) = \pm c_{n}(F)$, proving a conjecture of Fulton. We define Stiefel-Whitney and Euler classes as Chow cohomology classes which pull back to Chern classes of maximal isotropic subbundles of the pullback bundle. Using the above theorem we show (Theorem \ref{char.exist}) that these classes exist and are unique, even though $V$ need not have a maximal isotropic subbundle. These constructions also make it possible to give ``Schubert" presentations, depending on a fixed maximal isotropic flag, of the Chow rings of quadric and isotropic flag bundles. The presentation for the flag bundle is closely related to Fulton's work on Schubert varieties in isotropic flag bundles (\cite{F1}, \cite{F2}). An interesting aspect of this work is that it emphasizes the difference between principal $SO(N)$ bundles in the Zariski and \'etale topologies. As is well known (\cite{Sem-Chev}), the group $SO(N)$, unlike $SL(N)$, is not {\em special}; i.e. not all principal bundles which are locally trivial in the \'etale topology are locally trivial in the Zariski topology. This fact manifests itself as follows. The Stiefel-Whitney classes in $A^*(X;{\Bbb Z}/2{\Bbb Z})$ we construct are for bundles which are locally trivial in the Zariski topology. In the \'etale topology it is not always possible to construct these classes, as there are bundles for which the corresponding topological Stiefel-Whitney classes are not algebraic \cite{T1}. Similarly, we construct an integral Euler class for Zariski locally trivial $SO(2n)$ bundles, but for general $SO(2n)$ bundles only a power of 2 times the Euler class is integral \cite{T1}. The paper is organized as follows. Section \ref{s.intro} is the introduction. Section \ref{s.prelim} is largely a collection of standard and/or easy facts about vector bundles with quadratic form, quadric bundles and isotropic flag bundles which we use and which have not been adequately presented in the literature. The remainder of the paper is new. In Section \ref{s.chern} we prove the theorem on Chern classes of maximal isotropic subbundles, including Fulton's conjecture. Section \ref{s:Euler} contains the construction of Stiefel-Whitney and Euler classes for bundles with quadratic form which are locally trivial in the Zariski topology. In Section \ref{s:topology} we show that for smooth complex varieties the classes we construct map to the classes with the same names in singular cohomology. Section \ref{s.halfchow} deals with bundles whose structure group reduces to $SO(2n)$ but which are not locally trivial. In particular, we show that there is a characteristic class $y_n \in A^nX$ such that $\frac{1}{2^{n-1}}y_n$ is an Euler class. We also give a natural (with ${\Bbb Z}[\frac{1}{2}]$ coefficients) presentation of the Chow ring of the quadric bundle as an algebra with two generators and two relations, such that the Euler and Chern classes of $V$ are the coefficients in the relations. This presentation is the $SO(2n)$ analogue of the presentation of $A^*{\Bbb P}(V)$ over $A^*X$, where the Chern classes of V are the coefficients in the sole relation. Viewing the isotropic flag variety $Fl(V)$ as a tower of quadrics bundles, we also describe $A^*(Fl(V);{\Bbb Z}[\frac{1}{2}])$. Finally, in Section \ref{s.zchow} we give a ``Schubert'' presenations for $A^*Q$ and $A^*(Fl(V))$, which depend on a fixed maximal isotropic subbundle $F \subset V$. {\bf Acknowledgements.} We would like to thank William Fulton for his generous help, in particular for informing us of his conjecture and suggesting that the Euler class was related to top Chern classes of maximal isotropic subbundles. Some of the facts presented in Section 2 are based on his lectures at the University of Chicago. We are also grateful to Burt Totaro for his example showing that the Euler class need not be integral for bundles which are not locally trivial in the Zariski topology. \section{Preliminaries} \label{s.prelim} Throughout this paper, unless otherwise noted, all schemes are assumed to be of finite type over an arbitrary field. \medskip {\bf Operational Chow groups} If $X$ is a scheme, then $A_*X$ denotes the Chow homology groups defined in \cite[Chapter 1]{Fulton}, and $A^*X$ is the operational Chow cohomology ring of \cite[Chapter 17]{Fulton}. An element $c \in A^pX$ is a collection of homomorphisms $A_*X' \rightarrow A_{*-p} X'$ for all $X' \rightarrow X$ which is compatible with proper pushforward, flat pull-back, and interstions. By construction, if $X \stackrel{f} \rightarrow Y$ is any map there is always a ring homomorphism $f^*:A^*Y \rightarrow A^*X$. If $X$ is regular, then $A^*X$ is just the usual Chow groups with multiplication given by intersection product. Now if $R$ is any coefficient ring, we can also define $A^*(X;R)$ as maps from $A_*X' \otimes R \rightarrow A_*X' \otimes R$ for all $X' \rightarrow X$ satisfying the same conditions as above. There is always a homomorphism $A^*X \otimes R \rightarrow A^*(X;R)$. If $X$ admits a resolution of singularities then standard arguments (cf. \cite{Kimura}) show that $X$ is an isomorphism. \medskip {\bf Quadratic Forms} A quadratic form $q$ on a vector space $V$ will always mean a {\em hyperbolic} quadratic form: in other words, we assume that $V$ (of dimension $2n$ or $2n+1$) has an isotropic subspace of dimension $n$ (if the ground field is algebraically closed of characteristic not equal to 2, this is automatically satisfied). We can choose a basis $v_{1}, v_{2}, \ldots$ of $V$ such that the quadratic norm $q(v,v)$ of a vector $v = \sum x_{i}v_{i}$ is given by $$q(v,v) = x_{1}x_{2n} + \ldots + x_{n}x_{n+1} \mbox{ (if dim $V = 2n$)}$$ $$q(v,v) = x_{1}x_{2n+1} + \ldots + x_{n}x_{n+2} + a x_{n+1}^2 \mbox{ (if dim $V = 2n+1$)}$$ where $a$ is a nonzero scalar. The reason our form must be hyperbolic is that otherwise there may be no nonzero isotropic vectors in $V$. For example, there are no nonzero isotropic vectors for ${\Bbb R}^2$ with quadratic form $x_{1}^{2} + x_{2}^{2}$. While the theorems we prove are for schemes of arbitrary fields, we will abuse notation and refer to the group that preserves a quadratic form on a $N$ dimensional vector space as $O(N)$, even though this notation is only correct when the field is algebraically closed. Let $\mbox{Iso}(V)$ be the variety of maximal isotropic subspaces of $V$. The groups $SO(N)$ and $O(N)$ act on $\mbox{Iso}(V)$. The $O(N)$ action is always transitive, but the $SO(N)$ action is only transitive if $N$ is odd. If $N$ is even, there are two $SO(N)$ orbits and two maximal isotropic subspaces $E$ and $F$ of $V$ lie in the same $SO(N)$ orbit if and only if $\mbox{dim }E \cap F \equiv n(\mbox{mod }2)$. We will say that $E$ and $F$ are in the same family if they are in the same $SO(N)$ orbit, and in opposite families otherwise. Let $Fl(V)$ denote the flag variety of complete isotropic flags of a certain length in $V$. This length is $n$ if dim $V=2n+1$ and $n-1$ if dim $V=2n$. If dim $V=2n$ we let $Fl_n(V)$ denote the variety of isotropic flags of length $n$. A point in one of these varieties corresponds to a flag $E_1 \subset E_2 \subset \ldots $ of isotropic subspaces of $V$ (where $\mbox{dim }E_i=i$). In the even dimensional case, every length $n-1$ isotropic flag can be completed to exactly two isotropic length $n$ flags, and the space $Fl_n(V)$ is a disconnected double cover of $V$. Again, the groups $SO(N)$ and $O(N)$ act. The $SO(N)$ action on $Fl(V)$ is transitive, but there are two $SO(2n)$ orbits on $Fl_n(V)$; two length $n$ isotropic flags $E.$ and $F.$ are in the same $SO(N)$ orbit if and only if $\mbox{dim }E_{n} \cap F_{n} \equiv n(\mbox{mod }2)$. Let $Q \subset {\Bbb P}(V)$\footnote{Throught this paper ${\Bbb P}(V)$ will denote the scheme of lines in $V$.} denote the quadric of isotropic vectors. If $E$ and $F$ are maximal isotropic subspaces then ${\Bbb P}(E)$ and ${\Bbb P}(F)$ are subvarieties of $Q$ called rulings. ${\Bbb P}(E)$ and ${\Bbb P}(F)$ represent the same class in $A_{n-1}(Q)$ if and only if $E$ and $F$ lie in the same $SO(N)$ orbit. Thus, if dim $V=2n$, there are two families of rulings giving rise to two Chow classes $e$ and $f$, and, letting $h$ denote the hyperplane section, we have $h^{n-1} = e+f$. If dim $V=2n+1$ there is only one family and $h^{n}=2e$. \begin{lemma} \label{quad} $(a)$ If $N=2n$ is even, then a basis for $A_*(Q)$ is $1,h,h^2, \ldots, h^{n-1},e,he,\ldots,h^{n-1}e$. $(b)$ If $N=2n+1$ is odd, then a basis for $A_*(Q)$ is $1,h,h^2, \ldots, h^{n-1},e,he,\ldots,h^{n-1}e$. $\triangle$ \end{lemma} Remark: Because a quadric has a decomposition into affine cells, $A_*(U \times Q) = A_*(U) \otimes A_*(Q)$. This fact will be essential when we compute the Chow rings of quadric bundles. \medskip Proof: (a) In any characteristic the hyperbolic quadric has an affine cellular decompostion with 1 cell in every dimension, except 2 cells in dimension $n-1$. Hence $A_i(Q) = {\Bbb Z}$ for $i \neq n-1$, and $A_{n-1}(Q)={\Bbb Z} \oplus {\Bbb Z}$. Since $e$ is a class of a linear space, $h^{n-1}e= 1$. Hence, for $i \leq n-1$, $h^i$ is a generator of $A_{n-i}(Q)$. Likewise, the $h^ie$ are also generators of $A_{i}(Q)$ for $i \leq n-1$. This proves (a). The proof of (b) is similar. $\triangle$ \medskip In the sequel we use the following relations in the Chow ring of an even-dimensional quadric. \begin{lemma} \label{rels} Let $Q \subset {\Bbb P}^{2n-1}$ be an even-dimensional quadric. Then $e^2 = f^2=0$ and $ef = 1$ if $n$ is even, and $e^2 = f^2=1$ and $ef = 0$ if $n$ is odd. \end{lemma} Proof: First note that $e$ and $f$ both have degree 1, so $h^{n-1}e=h^{n-1}f = 1$. Since $h^{n-1} = e+f$, it follows that $e^2 = f^2$. Also, since $h^{2n-2} = (e+f)^2=2$, it follows either that $e^2=f^2=1$ and $ef=0$, or, $e^2=f^2=0$ and $ef=1$. The linear system spanned by $e$ is an $SO(2n)$ orbit of isotropic $n$-planes. One such $n-$plane is given by $x_1=x_2 = \ldots =x_n=0$. When $n$ is even, this plane can be taken to the plane $x_{n+1} = x_{n+2} = \ldots = x_{2n}=0$ via an element of $SO(2n)$ (an even permutation). Projectivizing, we see that there are two disjoint elements in the linear system spanned by $e$. Hence $e^2=0$. When $n$ is odd, the same argument shows that there are elements in the linear system corresponding to $e$ that are disjoint from elements of $f$. Thus $ef=0$. $\triangle$ \medskip The constructions above also work for vector bundles. We adopt the convention that a quadratic form $q$ on a vector bundle $V \rightarrow X$ means a section of $S^{2}(V^*)$ which restricts to a nondegenerate hyperbolic quadratic form on each fiber. The constructions of $Q$, $Fl(V)$, and $Fl_n(V)$ all carry over to the bundle case. A maximal isotropic subbundle is defined to be an isotropic subbundle of rank $n$ (where the rank of $V$ is $2n$ or $2n+1$). If the rank of $V$ is even and the structure group reduces to $SO(2n)$ (this is automatically satisfied in the Zariski locally trivial case -- see below) then the pullback of $V$ to $Fl(V)$ has two tautological maximal isotropic subbundles. If the rank is odd then there is only one tautological maximal isotropic subbundle over $Fl(V)$. These facts will be used extensively in the sequel. The description of $SO(2n)$ orbits of maximal isotropic subspaces yields the following fact about maximal isotropic subbundles. \begin{prop} \label{p.const} If $E$ and $F$ are maximal isotropic subbundles of $V \rightarrow X$, where the rank of $V$ is even, then $\mbox{{\em dim }} E_x \cap F_x$ is constant {\em mod} $2$ on each component of $X$. $\triangle$ \end{prop} We say the pair $(V,q)$ is Zariski locally trivial (or simply locally trivial) if there exists a Zariski open covering $\{U_i\}$ of $X$, such that $V_{|U_i} \simeq X \times {\Bbb A}^{N}$ and the under the isomorphisms the quadratic norm on a fiber over $u \in U_i$ is given by: $$\mbox{(rank $V=2n$) } q(v,v) = v_1v_{2n} + \ldots + v_nv_{n+1}$$ $$\mbox{(rank $V=2n+1$) } q(v,v) = v_1v_{2n+1} + \ldots + v_{n}v_{n+2} + f(u)v_{n+1}^2.$$ Here $v=(v_1, \ldots , v_N)$ is a vector in the fiber over $u$, and $f$ is a nowhere vanishing function on $U_i$. (Swan \cite[Cor 1.2]{Swan} has shown that any $(V,q)$ is locally trivial in the \'etale topology; i.e., there is an \'etale cover $\{U_i \stackrel{\pi_i} \rightarrow X \}$ such that $(\pi_i^*V,\pi_i^*q)$ is as above). \medskip {\bf Remarks:} Not all bundles with quadratic form are locally trivial in the Zariski topology, as the following example shows. {\bf Example:} Let $X={\Bbb C}^*$ be the $1$ dimensional torus. Let $V = X \times {\Bbb C}^2$ be a trivial rank two vector bundle. Define a quadratic form on $V$ by the rule that over a point $\lambda\in {\Bbb C}^*$ the quadratic norm is $v_1^2 + \lambda v_2^2$. The pair $(V,q)$ is locally trivial in the \'etale topology but not in the Zariski topology. Maximal isotropic subbundles need not exist, even if $V$ is Zariski locally trivial; we give a simple example in Section \ref{s.halfchow}. However, if $V$ has a maximal isotropic subbundle, then the pair $(V,q)$ is locally trivial. If the rank of $V$ is odd, then it is not always possible to choose trivializations so that the functions $f(u)$ above are identically $1$, even if $V$ has a maximal isotropic subbundle. For instance, if $V$ is a self-dual line bundle, then the trivializations can be so chosen if and only if $V$ is the trivial line bundle. If the pair $(V,q)$ is locally trivial in the Zariski topology, then so is the associated quadric bundle. This assertion is obvious if the rank is even. In the odd rank case, it suffices to show that if $V \rightarrow U$ is a trivial vector bundle with quadratic form as above, then the quadric bundle $Q$ on $U$ is isomorphic to $U \times Q_{pt}$, where $Q_{pt}$ is the quadric over a point. The desired isomorphism maps $u \times (v_{1},\ldots,v_{2n+1})$ to $(f(u)v_{1},\ldots,f(u)v_{n},v_{n+1},\ldots, v_{2n+1})$. Note that if $(V,q)$ is Zariski locally trivial of rank $2n$ then the structure group of $V$ reduces to $SO(2n)$. The reason is that the principal bundle is also Zariski locally trivial. This bundle is a priori an $O(2n)$ bundle, but any Zariski locally trivial bundle on a connected space has the same number of components as the fiber. Thus (assuming $X$ connected) the $O(2n)$ principal bundle has two components, implying that the structure group reduces. If the rank is $2n+1$ we cannot conclude that the structure group reduces to $SO(2n+1)$, because the associated $O(2n+1)$ principal bundle need not be Zariski locally trivial (since the function $f(u)$ need not be identically 1). The following lemma well be needed in the sequel. \begin{lemma} \label{thom.bundle} Let $f:Y \rightarrow X$ be a bundle with complete fibers which is locally trivial in the Zariski topology. Then $f^*$ is injective for Chow cohomology with any coefficients. \end{lemma} Proof: Because $Y$ is a locally trivial in Zariski topology, $f:Y \rightarrow X$ is a proper Chow envelope in the sense of \cite[Chapter 18]{Fulton}. The lemma follows from \cite[Lemma 2.1]{Kimura}. $\triangle$ \medskip {\bf Remark on characteristic 2.} The results of this paper hold in characteristic 2 (and if we assume that the constant $a$ in the odd rank quadratic form is a square in the field of definition). The reason is that the integral quadrics $x_1x_{2n} + \ldots x_{n-1}x_{n+1}$ or $x_1x_{2n+1} + \ldots x_{n}x_{n+2} + x_{n+1}^2$ are smooth over $Spec\;{\Bbb Z}$. Likewise we can construct an isotropic flag variety which is smooth over $Spec\;{\Bbb Z}$. By specializing from characteristic 0, we can prove all the necessary intersection theoretic properties of the quadric and flag varieties defined over ${\Bbb F}_2$. Since our constructions depend only on intersection theory, the results follow. For consistency of notation we take $SO(N)$ in characteristic 2 to mean the connected component of the identity in $O(N)$. Note that in characteristic 2, even if the field is algebraically closed not all forms are hyperbolic. For example the quadratic form $x_1^2 + x_2^2$ over $\overline{{\Bbb F}}_2$ is not hyperbolic. \section{Chern Classes of Isotropic Subbundles} \label{s.chern} The purpose of this section is to prove the following theorem. Part (c) is Fulton's original conjecture. \begin{thm} \label{t.chern} Let $V$ be a vector bundle on a connected scheme $X$, of rank $N$ equal to $2n$ or $2n+1$, equipped with a non-degenerate quadratic form, and let $d=N-n$. Let $E$ and $F$ be maximal isotropic subbundles of $V$. $(a)$ There exist classes $c_{i}$ and $d_{i}$ in $A^*(X)$ such that $2c_{i} = c_i(E)+ c_i(F)$ and $2d_{i} = c_i(V/E) +c_i(V/F)$. These classes are functorial for maps $X^\prime \rightarrow X$. $(b)$ If $E$ and $F$ are in the same (resp. opposite) families, then $c_d(V/E) = c_d(V/F)$ (resp. $-c_d(V/F)$). $(c)$ If the rank of $V$ is $2n$, then $c_n(E)=c_n(F)$ if $E$ and $F$ are in the same family, and $c_n(E)=-c_n(F)$ if $E$ and $F$ are in opposite families. \end{thm} Proof: Note that (c) follows immediately from (b), because in the even rank case, we may identify $V/E$ and $V/F$ with $E^*$ and $F^*$ respectively. If $Y \rightarrow X$, then $c_i(V/E) \cap [Y]$ and $c_i(V/F) \cap [Y]$ are both supported in $A_*(Y)$. Replacing $X$ with $Y$ it suffices to check the relations after capping with $[X]$. In the remainder of the proof, we will use the notations $c_i(V/E)$, etc.,to mean $c_i(V/E) \cap [X] \in A_*(X)$. Let $Q\rightarrow X$ be the quadric bundle associated to the quadratic form on $V$. Let $i:Q \hookrightarrow {\Bbb P}(V)$ be the inclusion, and let $\pi:{\Bbb P}(V) \rightarrow X$ and $\rho:Q \rightarrow X$ be the projections. Because $E$ and $F$ are isotropic subbundles, ${\Bbb P}(E)$ and ${\Bbb P}(F)$ are subvarieties of $Q$. Now we will examine $[{\Bbb P}(E)]$ and $[{\Bbb P}(F)]$ in the Chow groups of $Q$. Set $\gamma = [{\Bbb P}(E)]$, and let $h$ be the pullback to $A_*(Q)$ of the hyperplane class $H$ of ${\Bbb P}(V)\rightarrow X$. Since ${\Bbb P}(E) \subset Q$ is a regular embedding we can define a bivariant class $\gamma \in A^*(Q \rightarrow X)$ by the formula $\alpha \rightarrow \gamma \cap \rho^*\alpha \in A_*(Q)$. \begin{lemma} \label{gen} Every $\beta \in A_*(Q)$ can be written as $$\sum_{i=0}^{i=n-1} h^i \rho^*(\alpha_i) + \sum_{i=0}^{i=n-1} h^i \gamma \cap \rho^*(\beta_i).$$ \end{lemma} Proof of Lemma \ref{gen}: By Lemma \ref{quad}, the classes listed generate the Chow groups of the fibers. Because $V$ has a maximal isotropic subbundle, $Q \rightarrow X$ is locally trivial in the Zariski topology. The conclusion then follows by Noetherian induction as in \cite[Proposition 1.9]{Fulton}. $\triangle$ \medskip We now finish the proof of the theorem. First suppose that the $E$ and $F$ are in the same family (this is automatically satisfied in the odd rank case). Then $[{\Bbb P}(E)]$ and $[{\Bbb P}(F)]$ agree on each fiber of $Q$. Thus in $A_*(Q)$, $$[{\Bbb P}(E)] -[{\Bbb P}(F)]= \sum_{i > 0} h^{d-1-i} \cap \rho^*(\alpha_{i}),$$ for some classes $\alpha_{i}$ in $A_*(X)$. But $$i_*(h^j \cap \rho^* \alpha) = c_1({\cal O}(Q)) \cap H^j \cap \pi^*(\alpha).$$ Since the quadric bundle corresponds to a section of ${\cal O}_{{\Bbb P}(V)}(2)$, $c_1({\cal O}(Q))=2H$. Thus $$i_*([{\Bbb P}(E)] -[{\Bbb P}(F)])= \sum_{i>0} 2H^i \cap \pi^*(\alpha_{d-i}).$$ On the other hand, \cite[Example 3.2.17]{Fulton} implies that $$i_*([{\Bbb P}(E)] - [{\Bbb P}(F)])= \sum_{i \geq 0} H^i \cap \pi^* (c_{d-i}(V/E) - c_{d-i}(V/F)).$$Comparing these expressions, we see that $c_d(V/E) = c_d(V/F)$, which proves part (b). Moreover, defining $d_{i} = \alpha_{i} + c_i(V/F)$, we see that $2d_{i} = c_i(V/E) +c_i(V/F)$ as desired. In the even case, to get the classes $c_i$, we simply define $c_i = (-1)^i d_{i}$. In the odd case, we no longer have $V/E$ and $V/F$ isomorphic to $E^*$ and $F^*$. However, an easy computation (using the fact that $c_1(V) = c_1(E^{\perp}/E) = c_1(F^{\perp}/F)$) shows that $$ c(E^*) + c(F^*) = \frac{c(V/E) + c(V/F)}{1+c_1(V)}. $$ Therefore, if we define the $c_i$ by the formula $$ \sum_{i=0}^{n} (-1)^i c_i = \frac{\sum_{i=0}^{n} d_i}{1+c_1(V)}. $$ we get $2c_{i} = c_i(E) +c_i(F)$, as desired. The classes $c_i$ and $d_i$ are obviously functorial for maps $X^\prime \rightarrow X$. This proves the theorem in the case where $E$ and $F$ are in the same family. In the case where $E$ and $F$ are in opposite families we compare $[{\Bbb P}(E)]$ and $h^{n-1} - [{\Bbb P}(F)]$ to obtain the result. $\triangle$ \section{Construction of Euler and Stiefel-Whitney Classes} \label{s:Euler} Let $V \rightarrow X$ be a vector bundle of rank $N=2n$ or $N=2n+1$ with a nondegenerate quadratic form $q:V \otimes V \rightarrow {\cal O}$. \begin{defn} (a) A {\em Stiefel-Whitney class} for $V \rightarrow X$ is a class $w_{2i} \in A^i(X;{\Bbb Z}/2{\Bbb Z})$ such that for any $f:Y \rightarrow X$ and any maximal isotropic subbundle $E \subset f^*V$ we have $f^*w_{2i} \equiv c_i(E)(\mbox{mod } 2)$ (b) If the rank of $V$ equals $2n$, then an {\em Euler class} for $V \rightarrow X$ is a class $x \in A^n(X)$ such that for any $f:Y \rightarrow X$, and any maximal isotropic subbundle $E \subset f^*V$ we have $f^*x = \pm c_n(E)$. \end{defn} The purpose of this section is to prove the existence of these classes when the pair $(V,q)$ is locally trivial in the Zariski topology. For the remainder of this section we will therefore assume that $(V,q)$ is Zariski locally trivial. We may repeat this assumption for emphasis. {\bf Remark.} Note that if $E \subset V$ is a maximal isotropic subbundle, then $c_n(E)$ is an Euler class for $V$. In Section \ref{s:topology} we will explain the connnection between our defintions and those given in topology. The next proposition states some basic properties of Stiefel-Whitney and Euler classes. \begin{prop} \label{p.basic} $(1)$ Pullbacks of Stiefel-Whitney (resp. Euler) classes are Stiefel-Whitney (resp. Euler) classes for the pullback bundle. $(2)$ Stiefel-Whitney classes are unique. $(3)$ An Euler class is unique up to sign; i.e., if $x,y$ are both Euler classes for a bundle $V_n$, then $x = \pm y$. $(4)$ If $x$ is an Euler class of $V_n$, then $x^2 = (-1)^{n}c_{2n}(V_n)$. \end{prop} Proof: Property (1) follows from the definition. Next consider the map $f:Fl(V) \rightarrow X$. The bundle $f^*V$ has a maximal isotropic subbundle $E$. Suppose $w_{2i}$ and $w_{2i}^\prime$ are Stiefel-Whitney classes in degree $A^i(X)$. By definition, $$f^*w_{2i}=f^*w_{2i}^\prime =c_i(E)(\mbox{mod 2}).$$ Since $Fl(V) \rightarrow X$ is locally trivial, $f^*$ is injective (Lemma \ref{thom.bundle}), so $w_{2i} = w_{2i}^\prime$ in $A^*(X;{\Bbb Z}/2{\Bbb Z})$, proving $(2)$. Likewise, $f^*x$ and $f^*y$ are equal to $\pm c_n(V^\prime_n)$, so $x=\pm y$ as Chow cohomology classes. This proves (3). Finally, note that, via the quadratic form, we can identify $V_n^{\prime *} = f^*V_n/V_n^\prime$. Thus, $$c_{2n}(f^*V_n) = c_n(V_n^\prime) c_n(V_n^{\prime *}) = (-1)^nf^*(x^2).$$ Since $f^*$ is injective, (4) follows. $\triangle$ \begin{thm} \label{char.exist} Let $V \rightarrow X$ be a vector bundle with a nondegenerate quadratic form $q$. If the pair $(V,q)$ is locally trivial in the Zariski topology then $V$ has Stiefel-Whitney classes $w_{2i} \in A^i(X;{\Bbb Z}/2{\Bbb Z})$. If the rank of $V$ is $2n$, then $V$ has two Euler classes $x_n$ and $-x_n$ in $A^n(X)$. \end{thm} {\bf Remark 1.} Proposition \ref{p.basic} and Theorem \ref{char.exist} together show that Stiefel-Whitney and Euler classes are characterstic classes for vector bundles with quadratic form which are locally trivial in the Zariski topology. \medskip {\bf Remark 2.} In topology the sign of the Euler class is determined by the choice of an orientation of the bundle, or in other words, a reduction of structure group to $SO(N)$. The same holds in the algebraic case. This is equivalent to choosing a maximal isotropic subbundle of the pullback of $V$ to the flag bundle. \medskip Proof: Consider the Cartesian diagram $$\begin{array}{ccc} Fl(V) \times_{X} Fl(V) & \stackrel{g'}\rightarrow & Fl(V) \\ \downarrow\scriptsize{f'} & & \scriptsize{f}\downarrow\\ Fl(V) & \stackrel{g}\rightarrow & X \end{array}$$ Let $E \subset f^*V$ be one of the two tautological maximal isotropic subbundles. (The choice of $E$ is equivalent to choosing an orientation.) To show that $w_{2i}$ exists we must show that $c_i(E)(\mbox{mod 2})$ is the pullback (mod 2) of a class $A^*X$. Let $F \subset g^*V$ be a maximal isotropic subbundle. Then $c_i(F) (\mbox{mod 2})$ is a Stiefel-Whitney class for $g^*V$. By Theorem \ref{t.chern}, $$f^{\prime*}c_i(F) \equiv c_{i}(g^{\prime *}E) \equiv g^{\prime *} c_{i}(E)(\mbox{mod 2}).$$ Now $Fl(V) \rightarrow X$ is a proper Chow envelope, so by \cite[Theorem 2.3]{Kimura} $c_i(E)$ is a pullback (mod 2). To complete the proof we must show that for any $q:X^\prime \rightarrow X$, and any maximal isotropic subbundle $F \subset q^*V$, we have $q^*w_{2i} \equiv c_i(F) (\mbox{mod 2})$. To prove this consider the Cartesian diagram $$\begin{array}{ccc} Fl(q^*V) & \stackrel{q'}\rightarrow & Fl(V) \\ \downarrow\scriptsize{f'} & & \scriptsize{f}\downarrow\\ X' & \stackrel{g}\rightarrow & X \end{array}$$ Chasing the diagram, and applying Theorem \ref{t.chern}, it follows that $f^{\prime *}q^*(x_n) = \pm f^{\prime *}(c_n(F))$. Since $f^*$ is injective mod 2, the existence of Stiefel-Whitney classes follows. If $V$ has rank $2n$, then we can prove the existence of Euler classes in $A^n(X)$ by the method above, using the fact that $c_n(E)=\pm c_n(F)$ for any two maximal isotropic subbundles. $\triangle$ {\bf Remark.} If the bundle and quadratic form are not locally trivial in the Zariski topology it is still possible to construct (using our techniques) classes $\pm y_n \in A^n(X)$ (Theorem \ref{euler.haexist}) such that $\pm \frac{y_n}{2^{n-1}}$ are Euler classes. On the other hand, the calculations of B. Totaro \cite{T1} show that Stiefel-Whitney classes do not exist and that the Euler class is not integral for all quadratic vector bundles which are not locally trivial. In particular, he shows that on a variety approximating the classifying space $BSO(4)$ there is a tautological quadratic vector bundle where the topological Stiefel-Whitney and Euler classes are not represented by complex manifolds. \section{Connections with topology} \label{s:topology} In this section we explain the connection between the definitions of Euler and Stiefel-Whitney classes in the algebraic case and the usual definitions of these classes in topology. One consequence of this connection is a proof of a topological analogue of Theorem \ref{t.chern}. Throughout this section, unless otherwise stated, we work in the setting of topology rather than algebraic geometry. We begin by recalling a few facts about characteristic classes and classifying spaces. The space $BO(N,{\Bbb C})$ is the classifying space for rank $N$ complex vector bundles with nondegenerate quadratic form, and $BO(N,{\Bbb R})$ is the classifying space for rank $N$ real vector bundles. Let ${\cal V} \rightarrow BO(N,{\Bbb C})$ and ${\cal W} \rightarrow BO(N,{\Bbb R})$ be the universal vector bundles. The bundle ${\cal V}$ (resp. ${\cal W}$) has a nondegenerate (resp. positive definite real) quadratic form. Complexifying the quadratic form on ${\cal W}$ gives a nondegenerate quadratic form on ${\cal W} \otimes {\Bbb C}$. Because topologically the groups $O(N,{\Bbb C})$ and $O(N,{\Bbb R})$ are homotopy equivalent, we can identify the spaces $BO(N,{\Bbb C})$ and $BO(N,{\Bbb R})$. Under this identification, ${\cal V} = {\cal W} \otimes {\Bbb C}$, as bundles with quadratic form. Because this is the universal case, we have the following lemma. \begin{lemma} \label{l:class} Let $V \rightarrow X$ be a complex vector bundle with nondegenerate quadratic form. Then there exists a real vector bundle $W \rightarrow X$ with positive definite quadratic form, such that $V \simeq W \otimes {\Bbb C}$ as vector bundles with quadratic form. Moreover, the classifying maps $X \rightarrow BO(N,{\Bbb C})$ of $V$ and $W$ coincide. $\triangle$ \end{lemma} The cohomology ring $H^{*}(BO(N,{\Bbb C}); {\Bbb Z} / 2 {\Bbb Z})$ is isomorphic to the polynomial ring \\ ${\Bbb Z} / 2 {\Bbb Z} [w_{1}, \ldots, w_{N}]$. The Stiefel-Whitney classes of a complex vector bundle $V$ with nondegenerate quadratic form (resp. real vector bundle $W$) over $X$ are defined to be the pullbacks of these classes via the classifying map to $H^{*}(X; {\Bbb Z} / 2 {\Bbb Z})$. If the structure group of $V$ (resp. $W$) reduces to $SO(N,{\Bbb C})$ (resp. $SO(N,{\Bbb R})$), then the classifying map $X \rightarrow BO(N,{\Bbb C})$ lifts (in exactly two ways, assuming $X$ connected) to a map $X \rightarrow BSO(N,{\Bbb C})$. If $N=2n$, we single out a universal class $x_{n} \in H^{n}(BSO(2n,{\Bbb C}); {\Bbb Z})$; the Euler classes of $V$ (resp. $W$) are defined to be the pullbacks of $x_{n}$ to $H^{n}(X; {\Bbb Z})$ via the two lifts. These pullback classes differ only by sign. If $V$ and $W$ are as in Lemma \ref{l:class}, then because the classifying maps of $V$ and $W$ coincide, their Stiefel-Whitney and Euler classes coincide as well. \begin{prop} \label{p:topiso} Let $V \rightarrow X$ be a complex vector bundle with nondegenerate quadratic form and let $E$ be a maximal isotropic subbundle of $V$. Then the even Stiefel-Whitney classes of $V$ are the mod $2$ reductions of the Chern classes of $E$. Moreover, if the rank of $V$ is $2n$, then the structure group of $V$ reduces to $SO(2n,{\Bbb C})$, the Euler classes of $V$ are $\pm c_{n}(E)$, and the odd Stiefel-Whitney classes of $V$ vanish. \end{prop} Proof: Choose $W$ as in Lemma \ref{l:class}. Let $V_{{\Bbb R}}$ and $E_{{\Bbb R}}$ denote $V$ and $E$ viewed as real vector bundles. We have maps of real bundles $$ E_{{\Bbb R}} \hookrightarrow V_{{\Bbb R}} \simeq W \oplus iW \rightarrow W . $$ The composition $E_{{\Bbb R}} \rightarrow W$ is injective, since $E$ is isotropic and the quadratic form on $W$ is positive definite. Assume now that the rank of $V$ is $2n$. Then the ranks of $E_{{\Bbb R}}$ and $W$ are equal, so $E_{{\Bbb R}} \simeq W$. Since $W$ is isomorphic to the realification of a complex vector bundle, it is orientable as a real vector bundle, which means that the structure group of $W$ reduces to $SO(2n,{\Bbb R})$. This implies that the structure group of $V$ reduces to $SO(2n,{\Bbb C})$. The Stiefel-Whitney and Euler classes of $V$ equal those of $W$, which in turn equal those of $E_{{\Bbb R}}$. The assertions of the proposition then follow from \cite[Problem 14-B and Definition, p. 158]{Milnor-Stasheff}. This proves the proposition in the even rank case. In the odd rank case, we can write $W \simeq E_{{\Bbb R}} \oplus L$, where $L$ is a real line bundle. The proposition then follows from the Whitney product formula and \cite[Problem 14-B]{Milnor-Stasheff}. $\triangle$ \medskip The preceding proposition shows that the Euler and Stiefel-Whitney classes we have defined in Chow cohomology bear the same relation to the Chern classes in Chow cohomology as the Euler and Stiefel-Whitney classes in topology do to the Chern classes in topology, and hence justifies our use of these names. In the topological setting, the properties of Stiefel-Whitney and Euler classes given in Proposition \ref{p:topiso} yield the following corollary, which is a topological analogue of Theorem \ref{t.chern}. (In the algebraic case, the logic is reversed: we use Theorem \ref{t.chern} to prove the existence of Stiefel-Whitney and Euler classes.) \begin{cor} \label{c:weakeuler} Let $V \rightarrow X$ be a complex vector bundle with nondegenerate quadratic form, of rank $N$ equal to $2n$ or $2n+1$, and suppose that $E$ and $F$ are maximal isotropic subbundles of $V$. Then the mod $2$ reductions of $c(E)$ and $c(F)$ are equal. Moreover, if $N=2n$, then $c_{n}(E) = \pm c_{n}(F)$. $\triangle$ \end{cor} For smooth complex varieties the existence of a natural map from Chow cohomology to singular cohomology yields the following proposition. \begin{prop} \label{p:correspond} Let $X$ be a smooth complex variety and let $V \rightarrow X$ be a vector bundle with quadratic form, locally trivial in the Zariski topology. Then, under the natural map $co:A^i(X;R) \rightarrow H^{2i}(X;R)$ (for $R={\Bbb Z}/2{\Bbb Z}$ or $R={\Bbb Z}$) the algebraic Stiefel-Whitney and Euler classes map to the corresponding topological classes. \end{prop} Proof: Both the algebraic and topological Stiefel-Whitney classes (resp. Euler classes) pull back to Chern classes of bundles on $Fl(V)$. Since $co$ maps algebraic Chern classes to topological Chern classes (\cite[Chapter 19]{Fulton}), the images of our algebraic classes pull back on $Fl(V)$ to the pullbacks of the corresponding topological classes. The proposition then follows from the naturality of $co$ and the following lemma. \begin{lemma} \label{thom.top} If $f:Y \rightarrow X$ is a proper Chow envelope of smooth complex varieties, then the pullback $f^*:H^*(X;R) \rightarrow H^*(Y;R)$ is injective for any coefficient group $R$. \end{lemma} Proof of Lemma \ref{thom.top}: Consider the maps $cl:A_i(Y) \rightarrow H^{BM}_{2i}(Y)$ and $cl:A_i(X) \rightarrow H^{BM}_{2i}(X)$ from Chow groups to Borel-Moore homology defined in \cite[Chapter 19]{Fulton}. Let $s=[\tilde{X}] \in A_*(Y)$, where $\tilde{X} \subset Y$ is a subvariety mapping birationally to $X$. Then $f_*(s)=[X]$. Since the class map is compatible with proper pushforward, $$f_*(cl(s))=cl([X]) = \mu_X,$$ where $\mu_X$ is the fundamental homology class of $X$. If $x \in H^*(X)$, then by naturality of cap product, $$f_*(f^*x \cap cl(s)) = x \cap f_*(cl(s)) = x \cap \mu_X.$$ On the other hand, $\_ \cap \mu_X$ is the Poincare duality pairing (\cite[Chapter 19]{Fulton}). Thus, if $f^*x = 0 \in H^*(Y;R)$, then the Poincare image of $x$ in $H_*^{BM}(X;R)$ is zero. However, $X$ is smooth, so the pairing is perfect, and thus $x=0$. Therefore $f^*$ is injective. $\triangle$ \section{Chow rings of quadric and flag bundles with half integer coefficients} \label{s.halfchow} In this section we study even rank bundles with quadratic form. We will not assume that bundle and its quadratic form are locally trivial in the Zariski topology. Instead, we assume, throughout this section, that the structure group reduces to $SO(2n)$. We show (Theorem \ref{euler.haexist}) that every such bundle $V \rightarrow X$ has characteristic classes $\pm y_{n} \in A^{n}X$, such that $\frac{y_{n}}{2^{n-1}}$ is an Euler class in $A^*(X; {\Bbb Z}[\frac{1}{2}] )$. We use these Euler classes to compute the Chow rings tensored with ${\Bbb Z}[\frac{1}{2}]$ of the associated quadric and isotropic flag bundles. There is no discussion of the odd rank case because (over ${\Bbb Z}[\frac{1}{2}]$) the Chow groups of a quadric bundle are generated by powers of the hyperplane section. \paragraph{Construction of the isotropic flag bundle as a tower of quadrics} Let $V=V_n$ be a vector bundle of rank $2n$ with a non-degenerate quadratic form. Let $Q=Q_{n-1} \rightarrow X$ be the corresponding quadric bundle. We have the following diagram: $$\begin{array}{ccc} Q & \stackrel{i}\rightarrow& {\Bbb P}(V)\\ & \searrow \scriptsize{\rho} & \downarrow \scriptsize{\pi}\\ & & X \end{array}$$ Let $V_{n}$ also denote the pullback bundle on ${\Bbb P}(V)$, $S_{n}$ the tautological subbundle, and $S_{n}^{\perp}$ the orthogonal complement of $S_{n}$ in $V_{n}$. Although $S_{n}$ is not a subbundle of $S_{n}^{\perp}$, the pullback $i^{*}S_{n}$ is a subbundle of $i^{*}S_{n}^{\perp}$. Set $V_{n-1} = i^*S_n^\perp/i^*S_n$; this is a rank $2n-2$ vector bundle on $Q_{n-1}$ with nondegenerate quadratic form. Let $Q_{n-2} \rightarrow Q_{n-1} $ be the corresponding quadric bundle. In the same way, construct $V_{n-2}$, a rank $2n-4$ vector bundle on $Q_{n-2}$. Continuing, we get a tower $$\begin{array}{cccccccccc} Q_0 & \rightarrow & {\Bbb P}(V_1) & & & & & & &\\ & & \downarrow & & & & & &\\ & & Q_1 & \rightarrow & {\Bbb P}(V_2) & & & & &\\ & & & & \downarrow & & & & &\\ & & & & & \ldots & \rightarrow & {\Bbb P}(V_{n-1}) & &\\ & & & & & & & \downarrow & &\\ & & & & & & & Q_{n-1} & \rightarrow & {\Bbb P}(V_n)={\Bbb P}(V)\\ & & & & & & & & & \downarrow\\ & & & & & & & & & X \end{array}$$ (There is an analogous tower in the odd rank case, but it is not essential to our discussion.) Over $Q_{1}$ there is then a rank $2$ vector bundle $V_{1}$ with nondegenerate quadratic form. The corresponding quadric bundle $Q_{0}$ is a double cover of $Q_{1}$. \begin{lemma} \label{l:structuregroup} With notation as above, if the structure group of $V_{n} \rightarrow X$ reduces to $SO(2n)$, then the structure group of $V_{n-1} \rightarrow Q_{n-1}$ reduces to $SO(2n-2)$. Moreover, if $V_n$ with its quadratic form is locally trivial in Zariski topology, then $V_{n-1}$ (with its quadratic form) is as well. $\triangle$ \end{lemma} The proof is straightforward, and we omit it. Since we are assuming that the structure group of $V_{n} \rightarrow X$ reduces to $SO(2n)$, the above lemma implies that the structure group of $V_{1} \rightarrow Q_{1}$ reduces to $SO(2)$. \begin{lemma} \label{l.rank2} Let $V_1 \rightarrow X$ be a rank $2$ vector bundle with nondegenerate quadratic form whose structure group reduces to $SO(2)$. Then $V$ splits into a direct sum $V_1^\prime \oplus V_1^{\prime \prime}$ of isotropic line bundles. \end{lemma} Proof: If $W$ is a vector space of rank 2 with a non-degenerate quadratic form, then $W$ has exactly 2 isotropic lines. These lines intersect in the origin, and each is invariant under $SO(2)$. Thus, if the transition functions of the bundle $V_1$ lie in $SO(2)$, we can construct two isotropic sub-linebundles of $V_1$ whose direct sum is $V$. $\triangle$ \medskip To avoid cumbersome notation, we will temporarily write $V_i$, $S_i$, etc. for the pullbacks of these bundles to $Q_1$. The preceding lemma implies that the bundle $V_1 \rightarrow Q_1$ splits into a direct sum $V_1^\prime \oplus V_1^{\prime \prime}$. This implies that each of the bundles $V_2, V_3, \ldots V_n$ has two isotropic subbundles when pulled back to $Q_1$. They can be defined as follows. Assume by induction that $V_{i-1}^\prime$ and $V_{i-1}^{\prime\prime}$, maximal isotropic subbundles of $V_i$ are defined. By definition we have a surjective map $$S_i^\perp \rightarrow S_i^\perp/S_i = V_{i-1}.$$ Define $V_i^\prime$ (resp. $V_i^{\prime \prime}$) to be the preimage of $V^\prime_{i-1}$ (resp $V_{i-1}^{\prime \prime}$). Then $V_i^\prime$ and $V_i^{\prime \prime}$ are maximal isotropic subbundles of $V_i$. Note that by construction, \begin{equation} \label{eqn} c_i(V_i^\prime) = h_i c_{i-1}(V^\prime_{i-1}) \end{equation} where $h_i = c_1(S_i^*)$ is the hyperplane class on ${\Bbb P}(V_i)$. The space $Q_1$ can be identified with the bundle $Fl(V)$ of isotropic flags in $V$ of length $n-1$. On $Fl(V)$ the bundle $V$ has a tautological flag of subbundles, which we will denote $E_1 \subset E_2 \subset \ldots \subset E_{n-1}$. In this notation, the bundles $V_i$ (pulled back to $Fl_{n-1}$) are the quotients $E_{n-i}^\perp/E_{n-i}$ (taking $E_0 = 0$). The bundle $V$ has two maximal isotropic subbundles $V_n^\prime$ and $V_n^{\prime \prime}$. Using the fact that the bundles $V_1^\prime$ and $V_1^{\prime \prime}$ are dual, direct calculation, via Equation \ref{eqn}, shows that $$c_n(V_n^\prime) = -c_n(V_n^{\prime \prime}).$$ \begin{prop} \label{pushpull}Let $f:Fl(V_n) \rightarrow X$ be the projection. Then there is a (canonical) class $s \in A^*(Fl(V_n))$ such that for any $Y \rightarrow X$ and any $x \in A^*(X)$, $f_*(s \cdot f^*x \cap [Fl(V_n) \times_{X}Y ] ) = 2^{n-1}x \cap [Y]$. Hence $f^*$ is injective with ${\Bbb Z}[\frac{1}{2}]$ coefficients. \end{prop} Proof: We identify $Fl(V_n) = Q_1$. If $\rho_i:Q_{i-1} \rightarrow Q_{i}$ is a quadric in our tower, $x_{i} \in A^*(Q_i)$, and $Y \rightarrow Q_{i}$, then $\rho_{i*}(h_i^{2i-2} \cdot \rho_i^* x \cap [Q_{i-1} \times_{X}Y] ) = 2x \cap [Y]$. Thus $s = h_2^2 h_3^4 \ldots h_{n}^{2n-2}$ (pulled back to $Fl(V_n))$ is the desired class. $\triangle$ The following is an immediate consequence of Proposition \ref{pushpull}. \begin{thm} \label{euler.haexist} Let $V$ be a vector bundle of rank $2n$ with a non-degenerate quadratic form (not necessarily locally trivial in the Zariski topology) whose structure group reduces to $SO(2n)$. Then there are characteristic classes $\pm y_n \in A^n(X)$ such that $\pm\frac{y_n}{2^{n-1}}$ are Euler classes in $A^n(X; {\Bbb Z}[\frac{1}{2}] )$. \end{thm} {\bf Remark:} Theorem \ref{euler.haexist} strengthens a result of Vistoli \cite{Vistoli} who showed (implicitly) the existence of an Euler class in $A^*X \otimes {\Bbb Q}$ for arbitrary principal $SO(2n)$ bundles. \medskip Proof: The existence of the class $s$ from Proposition \ref{pushpull} implies that if $c$ is in the kernel of the map $A^{*} Fl(V) \stackrel{p_1^* - p_2^*} \rightarrow Fl(V) \times_X Fl(V)$ then the class $d$ in $A^*X$ defined by the formula $$d \cap [Y] = f_*(s \cdot c \cap [Fl(V_n) \times_{X}Y ])$$ satisfies $f^*d = 2^{n-1} c$. This is an analogue of \cite[Theorem 2.3]{Kimura} used above, and is proved in the same way. Setting $c=c_n(E)$ where $E$ is one of the two tautological maximal isotropic subbundles of $f^*V$, produces a class $y_n \in A^{*} X$ such that $f^*y_n = 2^{n-1}c_n(E)$. The classes $\pm y_n$ are natural with respect to pullbacks, because the class $s$ of Proposition \ref{pushpull} is natural. $\triangle$ \paragraph{Computation of Chow groups} We now describe $A^*(Q; {\Bbb Z}[\frac{1}{2}])$ as an algebra over $A^*(X ;{\Bbb Z}[\frac{1}{2}])$. Iterating over the quadric tower used above, we are also able to compute $A^*(Fl(V);{\Bbb Z}[\frac{1}{2}])$ as an algebra over $A^*(X; {\Bbb Z}[\frac{1}{2}] )$. Before we state the theorems, we need some further properties of Euler classes. \begin{prop} \label{euler.recur} Let $x_{n-1}$ be an Euler class for the associated bundle $V_{n-1} \rightarrow Q_{n-1}$, and let $h$ be the pullback of the hyperplane section on ${\Bbb P}(V)={\Bbb P}(V_n)$. Then $hx_{n-1} = \rho^*x_n$, where $x_n \in A^n(X;{\Bbb Z}[\frac{1}{2}])$ is an Euler class for $V$. If the bundle is locally trivial, then the identity holds integrally. \end{prop} Proof: Let $f:Fl(V) \stackrel{g} \rightarrow Q \stackrel{\rho} \rightarrow X$ be the projection. Then $f^*x_n = c_n(V^\prime_n)$ where $V_n^\prime$ is a maximal isotropic subbundle of $f^*V$. On the other hand, $g^*x_{n-1} = c_{n-1}(V_{n-1}^\prime)$ where $V_{n-1}^\prime$ is a maximal isotropic subbundle of $f^*V_{n-1}$. By Equation \ref{eqn}, $c_n(V_{n}^\prime) =hc_{n-1} (V_{n-1}^\prime)$, so the relation holds on $Fl(V)$. Since $f^*$ is injective up to 2-torsion (or in the locally trivial case -- integrally injective) the relation follows. $\triangle$ \medskip The following theorem is the key connection between Euler classes and Chow groups of quadric bundles. We use the same notation as Proposition \ref{euler.recur} above. \begin{thm} \label{euler.fiber} On a fiber of $Q_{n-1} \rightarrow X$ the Euler class $x_{n-1}$ restricts to $\pm (e_{n-1} -f_{n-1})$, where $e_{n-1}$ and $f_{n-1}$ are the two ruling classes on the fiber. \end{thm} Proof: Since we are working on a fiber, we may assume $X$ is a point, and $V=V_n$ is a vector space. Consider the tower above. The space $Q_0$ is the flag variety of length $n$ isotropic flags in $V$. It is a disconnected double cover of $Q_1 = Fl(V)$. Any element $g \in O(2n)$ induces an automorphism of this tower. At each step denote the induced maps $Q_i \rightarrow Q_i$ and ${\Bbb P}(V_i) \rightarrow {\Bbb P}(V_i)$ by $g_i$. By construction, the $g_i$'s are compatible for different values of $i$. If $g$ is not in the identity component of $O(2n)$, then $g_0:Q_0 \rightarrow Q_0$ acts by exchanging the two sheets of $Q_0$. \begin{lemma} \label{euler.auto} Let $g$ be an element of $O(2n)$ not in the identity component. Then $g_1^*(c_i(V_i^\prime) = -c_i(V_i^\prime)$. \end{lemma} Proof of Lemma \ref{euler.auto}: It is clear that the hyperplane class in any of the projective bundles in the tower is invariant under pullback by the automorphism induced by $g$. Since $c_i(V_i^\prime) = h_ic_{i-1}(V_{i-1}^\prime)$, it suffices to prove the proposition when $i=1$. Now $Q_0$ is the union of two components, ${\Bbb P}(V_1^\prime)$ and ${\Bbb P}(V_1^{\prime\prime})$, and $i_{0_*}([{\Bbb P}(V_1^{\prime\prime})] - [{\Bbb P}(V_1^{\prime})]) = \pi_1^*(2c_1 (V_1^\prime)).$ Since $g$ exchanges the two sheets of $Q_0$, $$g_0^*([{\Bbb P}(V_1^\prime)] -[{\Bbb P}(V_1^{\prime\prime})]) = -([{\Bbb P}(V_1^\prime)] - [{\Bbb P}(V_1^{\prime\prime})]).$$ Thus, $$g_0^*(\pi_1^*(2c_1(V_1^\prime)))= \pi_1^*(g_1^*(2c_1(V_1^\prime))) =-\pi_1^*(2c_1(V_1^\prime)).$$ Since $\pi_1^*$ is injective, and we are working with coefficients in ${\Bbb Z}[\frac{1}{2}]$, the lemma follows. $\triangle$ \medskip Since $e_{n-1}$ and $f_{n-1}$ generate $A_{n-1}(Q_{n-1})$ (Lemma \ref{quad}), we can write $x_{n-1} = ae_{n-1} + bf_{n-1}$, where $a$ and $b$ are in ${\Bbb Z}[\frac{1}{2}]$. Thus, $i_*(x_{n-1}) =(a+b)h_n$. Let $g$ be an element of $O(2n)$ not in the identity component. Then in the notation of the preceding proposition, $g_{n-1}^*x_{n-1}= -x_{n-1}$. Thus $a+b =0$. Hence we can write $x_{n-1} = \lambda(e_{n-1} - f_{n-1})$ for some $\lambda\in {\Bbb Z}[\frac{1}{2}]$. Since $x_{n-1}$ is an Euler class, $x_{n-1}^2=(-1)^nc_{2n-2}(V_{n-1})$ (Proposition \ref{p.basic}). By direct calculation, $$c(V_{n-1}) = \frac{c(V_n)}{1-h_{n}^2}.$$ But $c(V_n) = 1$ since $V_n$ is trivial. Thus, $$ x_{n-1}^2= (-1)^{n} c_{2n-2}(V_{n-1}) = (-1)^{n} h_n^{2n-2}=(-1)^{n} 2.$$ On the other hand, $x_{n-1}^2=\lambda^2(e_n-f_n)^2$, so by Lemma \ref{rels}, $x_{n-1}^2 = \lambda^2 \cdot (-1)^n\cdot 2$. Therefore, $\lambda = \pm 1$ as desired. $\triangle$ \begin{thm} \label{q.algebra} Let $X$ be a scheme and let $V\rightarrow X$ be a vector bundle of rank $2n$ whose structure group reduces to $SO(2n)$ with respect to a non-degenerate quadratic form on $V$. Let $Q \subset {\Bbb P}(V)$ be the associated quadric bundle. Then $A^*(Q;{\Bbb Z}[\frac{1}{2}])= A^*(X;{\Bbb Z}[\frac{1}{2}])[h,x_{n-1}]/I $, where $I$ is the ideal generated by the relations $$hx_{n-1} = \rho^*(x_n)$$ $$x_{n-1}^2 = (-1)^{n-1} c_{2n-2}(V_{n-1}) = (-1)^{n-1}(h^{2n-2} + h^{2n-4}c_{2}(V) + \ldots + h^{2}c_{2n-4}(V) + c_{2n-2}(V)).$$ \end{thm} {\bf Remark:} The relation $$h^{2n} + h^{2n-2}c_{2}(V)+ \ldots + c_{2n}(V) = 0 ,$$ inherited from $A^*({\Bbb P}(V))/A^*(X)$, can be easily derived from the two relations above. If $V \rightarrow X$ has rank $2n+1$ then $$A^*Q=A^*X[h]/h^{2n+1} + h^{2n-1}c_2(V) + \ldots + c_{2n}(V).$$ {\bf Example:} Theorem \ref{q.algebra} can be used to show that not all rank vector bundles of rank $N$ whose structure group reduces to $SO(N)$ have maximal isotropic subbundles. For an example when $N$ is even, let $Q_2 \subset {\Bbb P}^5$ be a smooth quadric of rank 3. On $Q_2$ we can define the bundle $V_2=S^\perp/S$, where $S={\cal O}(1)$, and $S^\perp$ is the orthogonal bundle with respect to the quadratic form on $Q$. Direct calculation shows that $V_2$ is locally trivial in the Zariski topology, so the bundle has rank 4 and structure group reduces to $SO(4)$. We claim, however, that $V_2$ has no isotropic subbundle of rank 2. Suppose to the contrary, that $E \subset V_2$ was such a bundle. Then $c_2(E) = x_2$, while $c_1(E) = ah$ for some constant $a$, since $h$ generates $A^1(Q)$. On the other hand, $$c(E)c(E^*)=c(V_2)=1+h^2+h^4.$$ Thus, $$(1 +ah+x_2)(1-ah+x_2)=1+h^2+h^4.$$ In particular, $a^2h^2+2x_2=h^2$, which is impossible since $h^2$ and $x_2$ are independent in $A^2(Q)$. Therefore, no such bundle $E$ exists. For an example of an odd rank bundle, let $V$ be the rank 3 bundle $S^\perp/S$ on the quadric hypersurface in ${\Bbb P}^4$. A similar calculation (using only the hyperplane class) shows that $V$ has no isotropic line subbundles. \medskip Now let $f:Fl(V) \rightarrow X$ be the flag bundle over $X$. Let $E_1 \subset E_2 \subset \ldots \subset E_{n-1} \subset E_{n} \subset f^*V$ be one of the two tautological isotropic flags. Via the quadratic form, we can extend it to a complete flag $E_1 \subset \ldots \subset E_n \ldots \subset E_{2n} =f^*V$. Set $h_i=c_1(E_{n-i+1}/E_{n-i})$ (where $E_0=0$). \begin{thm} \label{fl.algebra} Let $V$ be as in the statement of Theorem \ref{q.algebra}. Then $$A^*(Fl(V); {\Bbb Z}[\frac{1}{2}] ) = A^*(X; {\Bbb Z}[\frac{1}{2}])[h_1, \ldots, h_n]/I $$ where $I$ is the ideal generated the relations $$\prod_{i=1}^{n} (1-h_i^2) = f^*c(V)$$ $$h_1\;h_2 \ldots h_n=f^*x_n$$ \end{thm} {\bf Remark:} The choice of notation $h_i$ in the above theorem is consistent with its use previously, since for $i>1$, $c_1(E_{n-i+1}/E_{n-i})$ is the pullback of the hyperplane class of the projective bundle ${\Bbb P}(V_i)$.\\ Proofs of Theorems \ref{q.algebra} and \ref{fl.algebra}: We give the proof assuming $X$ is smooth to avoid using the language of bivariant intersection theory (\cite[Chapter 17]{Fulton}). A reader familiar with this theory can easily extend the proof to singular schemes. Theorem \ref{fl.algebra} follows from Theorem \ref{q.algebra} by iterating over the tower of quadrics constructed previously. Thus it suffices to prove Theorem \ref{q.algebra}. First note that $A^*X[h,x_n]/I$ has a basis of monomials $h^ix_n^\alpha$, where $0 \leq i \leq n-1$, and $\alpha \in \{0,1\}$. Since $x_{n-1}$ restricts on a fiber to the difference of two rulings, these classes restrict to a basis for the Chow group tensored with ${\Bbb Z}[\frac{1}{2}]$ of a fiber (Lemma \ref{quad}). At this point we would like to apply Noetherian induction and conclude that $h^ix_n^\alpha$ is a basis for the Chow groups of $Q$ over $A^*(X)$. Unfortunately, the bundle $Q \stackrel{\rho} \rightarrow X$ need not be locally trivial in the Zariski topology. However, $Q \times_X Fl(V)$ is Zariski locally trivial over $Fl(V)$ since the pullback of $V$ has a maximal isotropic subbundle. Thus, by Noetherian induction, our classes pull back to a basis for $A^*(Q\times_X Fl(V); {\Bbb Z}[\frac{1}{2}] )$ over $A^*(Fl(V))$. On the other hand, the pullback $f^*:A^*(X; {\Bbb Z}[\frac{1}{2}] ) \rightarrow A^*(Fl(V); {\Bbb Z}[\frac{1}{2}] )$ has a section (see Proposition \ref{pushpull}). Pushing forward via the section, we conclude that $\{h^ix_n^\alpha\}$ is a basis for $A^*(Q; {\Bbb Z}[\frac{1}{2}] )$. To complete the proof, we must check the relations. The relation $hx_{n-1}= \rho^*(x_n)$ is Proposition \ref{euler.recur}. Since $x_{n-1}$ is an Euler class for the bundle $V_{n-1}$, $x_{n-1}^2 = (-1)^{n-1}c_{2n-2}(V_{n-1})$ (Proposition \ref{p.basic}). Since $c_{2n-2}(V_{n-1}) = \left\{ \frac{c(V)}{1-h^2} \right\}_{2n-2}$, the second relation follows. $\triangle$ \medskip {\bf Remark:} In general $A^*(Fl(V))$ need not be free over $A^*(X)$. If it were, then one could define an integral Euler class for bundles whose structure group reduces to $SO(2n)$, but were not locally trivial in the Zariski topology. However, we noted before that B. Totaro (\cite{T1}) has shown that this can not be true. {\bf Remark:} If the stucture group $V \rightarrow X$ does not reduce to $SO(2n)$, then the flag variety $f:Fl_n(V) \rightarrow X$ of length $n$ isotropic flags is connected. If $E \subset f^*V$ is a tautological rank $n$ isotropic subbundle, then arguments similar to those used in Lemma \ref{euler.auto} shows that $c_n(E) = 0 \in A^*(Fl_n(V); {\Bbb Z}[\frac{1}{2}] )$. Thus the Euler class is also 2-torsion, a fact consistent with topology, since non-orientable bundles only have Thom classes with ${\Bbb Z}/2{\Bbb Z}$ coefficients. This calculation also shows that $c_{2n}(V)$ is 2-torsion as well. \medskip \section{Chow groups of quadric and isotropic flag bundles with integer coefficients} \label{s.zchow} Let $V \rightarrow X$ be a vector bundle with a quadratic form, and let $F \subset V$ be a fixed maximal isotropic subbundle. The purpose of this section is to compute the Chow rings of the associated quadric and flag bundles. The presentation will depend on the particular subbundle $F$. \paragraph{Chow rings of quadric bundles} \begin{thm} \label{zch.quad} Let $V$ be a vector bundle of rank $N$ with with a nondegenerate quadratic form on a scheme $X$. Let $Q \stackrel{\rho} \rightarrow X$ be the associated quadric bundle and $h$ the hyperplane class on $Q$. Assume that $F \subset V$ is a maximal isotropic subbundle and set $\gamma = [{\Bbb P}(F)] \subset Q$. Then $$A^*(Q) = A^*X[h,\gamma]/I$$ where $I$ is the ideal generated by the relations $$2h\gamma= h^{n} -c_1(F)h^{n-1} + \ldots + (-1)^nc_n(F)$$ $$\gamma^2= (-1)^{n-1}(c_{n-1}(F) + c_{n-3}(F)h^2 + \ldots )\gamma$$ if $N=2n$, and the relations $$2h\gamma = h^{n+1} + c_1(V/F)h^{n} + \ldots c_{n+1}(V/F)$$ $$\gamma^2 = (c_n(V/F) +c_{n-2}(V/F)h^2 + \ldots)\gamma$$ if $N=2n+1$. \end{thm} Proof: We give the proof only in the even case, as the odd case is analogous. As in the proof of Theorem \ref{euler.haexist} we assume $X$ is smooth. A basis for $A^*X[h,\gamma]/I$ as an $A^*X$ module is the monomials $1,h,h^2, \ldots h^{n-1},\gamma,h\gamma, \ldots , h^{n-1}\gamma$. These classes restrict to a basis of the Chow groups of the fiber, so they form a basis for $A^*(Fl(V))$ as a $A^*(X)$ module (All bundles are locally trivial in Zariski topology, so we can apply Noetherian induction). To prove the theorem we must check the relations. Let $i:Q \hookrightarrow {\Bbb P}(V)$ be the inclusion. From the proof of Theorem \ref{t.chern}, we know that $i_*\gamma = H^n -c_1(F)H^{n-1} + \ldots + (-1)^n c_n(F)$, where $H$ is the hyperplane section on ${\Bbb P}(V)$. Hence the right hand side of the first relation is $i^*i_*\gamma$. On the other hand, the normal bundle to $Q$ in ${\Bbb P}(V)$ is $2h$. Thus, by the self intersection formula, $i^*i_*\gamma= 2h\gamma$. This proves the first relation. The second relation follows from the self intersection formula $\gamma^2=j_*(c_{n-1}(N_{{\Bbb P}(F)}Q))\gamma$ where $j_*:{\Bbb P}(F) \hookrightarrow Q$ is the inclusion. The normal bundle is $$N_{{\Bbb P}(F)}Q=N_{{\Bbb P}(F)}{\Bbb P}(V)/N_{Q}{\Bbb P}(V) = \frac{V/F \otimes S^*}{(S^*)^{\otimes 2}}.$$ Tensoring the righthand side with $S \otimes S^*$, and identifying $V/F$ with $F^*$ we can identify the normal bundle as $\frac{F^*}{S^*}\otimes S^*$. The formula then follows by direct calculation. $\triangle$ \paragraph{Splitting bundles with quadratic form} Let $V= L_1 \oplus L_2 \oplus \ldots \oplus L_N$ be a direct sum of line bundles of the form $L_1 \oplus L_2 \oplus \ldots \oplus L_n \oplus L_n^* \oplus \ldots \oplus L_1^*$ if $N=2n$, and $L_1 \oplus \ldots \oplus L_n \oplus M \oplus L_n^* \oplus \ldots \oplus L_1$, if $N=2n+1$, where $M$ is a self-dual line bundle. Define a quadratic form $q_{std}$ on $V$ by the rule that $q_{std}$ restricted to $L_i \oplus L_{N-i+1}$ is the canonical pairing, and $L_i$ is orthogonal to $L_j$ otherwise. We will call $q_{std}$ the standard quadratic form on $V$. Suppose that $q$ is a quadratic form on $V$ such that $q$ restricted to $L_i \oplus L_{N-i+1}$ is the canonical pairing and such that (with respect to $q$), $L_i$ is orthogonal to $L_1, \ldots , L_{N-i}$. \begin{lemma} Let $V$, $q$ and $q_{std}$ be as above. Then, (assuming that the base field has characteristic not equal to 2), there exists a vector bundle automorphism $g$ of $V$ such that $g$ is the identity on $L_1 \oplus \ldots L_n$ and such that $g^*q_{std} = q$. $\triangle$\end{lemma} The proof of this lemma is an exercise in ``bundle-ized'' linear algebra which we omit. {\bf Remark:} Fulton \cite{F1} shows that there is a deformation of $(V,q)$ to $(V,q_{std})$ that induces an isomorphism of the Chow rings of the associated isotropic flag bundles. As a result, Theorems \ref{ch.dn} and \ref{ch.bn} are still valid in characteristic 2. The following quadratic splitting principle (in characteristic not equal to 2) is an application of the lemma above. \begin{prop} \label{q.split} Let $V \rightarrow X$ be a vector bundle of rank $N=2n$ or $N=2n+1$ with a nondegenerate quadratic form. Let $E_1 \subset \ldots E_n$ be a maximal isotropic flag of subbundles of $V$. Set $L_i=E_i/E_{i-1}$ and if $N$ is odd, set $M=E_n^\perp/E_n$. Then there exists $f:Y \rightarrow X$ with $f^*:A^*X \rightarrow A^*Y$ injective, and an isomorphism of $f^*V$ with $L_1 \oplus \ldots L_n \oplus L_n^* \oplus \ldots \oplus L_1^*$ or $L_1 \oplus \ldots L_n \oplus M \oplus L_n^* \ldots \oplus L_1^*$, as vector bundles with quadratic form ( where $f^*V$ is given the quadratic form inherited from $V$, and the direct sum of line bundles is given the standard form described above). Furthermore, the subbundle $E_i$ corresponds to the sum $L_1 \oplus \ldots \oplus L_i$. \end{prop} Proof: Extend $E_1 \subset \ldots \subset E_n$ to a complete flag $E_1 \subset \ldots \subset E_N = V$ by setting $E_{n+i}=E_{n-i}^\perp$. Then $L_{n+i} =E_{n+i}/E_{n+i-1} \simeq L_{n-i+1}^*$. Then the complete splitting principle as stated in \cite{F1} (cf. \cite[Theorem 8.3]{Gillet}) gives $f:Y \rightarrow X$ with $f^*$ injective and and isomorphism of $f^*V$ with $L_1 \oplus \ldots L_n \oplus L_n^* \oplus L_1^*$ or $L_1 \oplus \ldots L_n \oplus M \oplus L_n^* \ldots \oplus L_1^*$, as vector bundles. Let $q$ be the quadratic form on $L_1 \oplus \ldots \oplus L_N$ pulled back by isomorphism from the quadratic form on $f^*V$. By construction, $q$ restricts to the dual pairing on $L_i \oplus L_{N-i+1}$. Moreover, since we began with a maximal isotropic flag, each $L_i$ is orthogonal to $L_1, \ldots L_{N-i}$. By the above lemma, there exists a vector bundle automorphism of $L_1 \oplus\ldots \oplus L_N$ taking $q$ to $q_{std}$. This proves the proposition. $\triangle$ \paragraph{Presentation of the Chow rings of flag bundles} Let $V$ be a vector bundle with a nondegenerate quadratic form on a scheme $X$. We assume the existence of a maximal isotropic flag $F_{\cdot}= F_1 \subset F_2 \ldots \subset F_n \subset V$. To compute the Chow ring we need only assume the existence of a maximal isotropic subbundle $F$. However, assuming the existence of the full flag $F_\cdot$ gives a presentation consistent with the Schubert variety ideas of \cite{F1}. Let $\pi:Fl(V) \rightarrow X$ be the bundle of maximal isotropic flags in $V$. (Recall that if the rank $V$ is $2n$, then the flag variety has two isomorphic connected components. In this case the notation $Fl(V)$ will refer to one of these components, which may be identified with $Fl_{n-1}(V)$.) Set $y_i = -c_1(\pi^*F_i/\pi^*F_{i-1})$. The following two theorems (depending on whether the rank of $V$ is even or odd) describe $A^*Fl(V)$ as an algebra over $A^*X$. \begin{thm} \label{ch.dn} ({\em $D_n$ case}): If $V$ has rank $2n$ then $A^*Fl(V) =A^*X[x_1,x_2, \ldots , x_n,c_1, \ldots c_{n-1}]/I$ where $I$ is the ideal generated by the following relations (for $1 \leq i \leq n$): $$e_i(-x_1^2,\ldots ,-x_n^2) = \pi^*c_{2i}(V)$$ $$(-1)^pc_p^2 + (-1)^{p-1}2c_{p-1}c_{p+1} + \ldots - 2c_{2p-1}c_{1} +2c_{p} = c_{2p} - c_{2p-1}e_1(x_1,\ldots x_n) + \ldots + (-1)^pc_p e_p(x_1, \ldots , x_n)$$ $$2c_i = e_i(x_1, \ldots , x_n) + e_i(y_1, \ldots , y_n)$$ ($c_k=0$ for $k \geq n$). \end{thm} Recall that if $a_1, \ldots a_n$ is a set of variables then $e_i(a_1, \ldots , a_n)$ denotes the $i$-th elementary symmetric polynomial in those variables. \begin{thm} \label{ch.bn} ({\em $B_n$ case}): If $V$ has rank $2n+1$, then $A^*Fl(V) = A^*X[x_1, \ldots , x_n,c_1, \ldots , c_n]/I$ where $I$ is the ideal generated by the following relations (for $1 \leq i \leq n$) $$e_i(-x_1^2, \ldots , -x_n^2) = \pi^*c_{2i}(V)$$ $$le_i(-x_1^2, \ldots , -x_n^2) = \pi^*c_{2i+1}(V)$$ $$2c_i = e_i(x_1+l, \ldots , x_n +l) + e_i(y_1+l,y_2+l, \ldots , y_n +l)$$ \begin{eqnarray*} (-1)^pc_p^2 & + & (-1)^{p-1}2c_{p-1}c_{p+1} + \ldots - 2c_{2p-1}c_{1} +2c_{p}\\ &=& c_{2p} - c_{2p-1}e_1(x_1+l,\ldots x_n+l) + \ldots + (-1)^pc_p e_p(x_1+l, \ldots , x_n+l) \end{eqnarray*} where $l = \pi^*c_1(V)$ \end{thm} Proof of Theorem \ref{ch.dn}. If $V$ has rank $2n$, let $E_1 \subset E_2 \ldots \subset E_n$ be a tautological flag on $\pi^*V$ such that $E_n$ and $\pi^*F_n$ are in opposite ruling families. Set $x_i = c_i(E_i/E_{i-1})$. Since $\pi^*V/E_n = E^*_n$, $e_i(-x_1^2, \ldots -x_n^2)= c_{2i}(\pi^*V)$ proving the first relation. Let $\rho:Q \rightarrow Fl(V)$ be the quadric bundle associated to $\pi^*V$. Define classes $c_i \in A^*(Fl(V))$ by the relation $$[{\Bbb P}(E_n)] + [{\Bbb P}(\pi^*F_n)] = h^{n-1} - h^{n-2}\rho^*c_1 + h^{n-3}\rho^*c_2 - \ldots + (-1)^{n-1}\rho^*c_{n-1}$$ Since $\rho_* (h^n[{\Bbb P}(E_n)]\rho^*c_i) = c_i$, the $c_i$'s are uniquely determined. Furthermore, by the proof of Theorem \ref{t.chern}, $2c_i = c_i(E_n) + c_i(F_n) =e_i(x_1, \ldots, x_n) + e_i(y_1, \ldots, y_n)$. Note that $A^*X[x_1, \ldots x_n,c_1, \ldots c_{n-1}]/I$ has a basis of monomials of the form $$x_1^{a_1}x_2^{a_2}\cdots x_n^{a_n} c_1^{\alpha_1}c_2^{\alpha_2} \cdots c_{n-1}^{\alpha_{n-1}}$$ where $0 \leq a_i \leq n-i$ and $\alpha_i \in \{0,1\}$. On the other hand, Fulton \cite{F1} (generalizing a result stated in \cite{Marlin}) has shown that these monomials form a basis for $A^*(Fl(V))$ as an $A^*X$ module. Therefore, to prove the theorem it suffices to check that the second relation holds. By Proposition \ref{q.split} there is a map $f:Y \rightarrow Fl(V)$ such that $f^*$ is injective and $\pi^*V$ pulls back to $L_1 \oplus L_2 \ldots L_n \oplus L_n^* \oplus \ldots \oplus L_1$ with the standard quadratic form. To check a relation, it suffices to check it for the totally split bundle on $Y$. Furthermore, there is always a map $g:Y^\prime \rightarrow Y$ with $g^*$ injective such that $Y^\prime$ is quasi-projective (this is Chow's Lemma combined with Nagata's embedding theorem, see \cite[Section 18.3]{Fulton}). We may therefore assume that $Y$ is quasi-projective. In particular there are maps $j_i:Y \rightarrow {\Bbb P}^{k_i}$ such that $L_i = j_i^*M_i$ for some line bundle $M_i$ on ${\Bbb P}^{k_i}$. Setting $Z={\Bbb P}^{k_1} \times {\Bbb P}^{k_2} \times \ldots \times {\Bbb P}^{k_n}$, there is a map $j:Y \rightarrow Z$ such $L_1 \oplus L_2 \oplus \ldots L_n \oplus L_n^* \oplus \ldots \oplus L_n^*$ is the pullback of direct sum of line bundles and their duals on $Z$. To prove the relation, it suffices to check it in $A^*Z$. However, since $A^*Z$ is torsion free, we need only check that the relation holds up to multiplication by a constant. Since $2c_p = e_p(x_1, \ldots , x_n) + e_p(y_1, \ldots y_n)$, multiplying both sides of the potential relation by 4 we reduce to showing that the degree $2p$ term in $$(c(E_n) + c(\pi^*F_n))(c(E_n^*) + c(\pi^*F^*_n))$$ equals the degree $2p$ term in $$c(E_n)(c(E^*_n) + c(\pi^*F_n)) + c(E_n^*)(c(E_n) + c(\pi^*F_n)).$$ This equality is immediate since $$c(E_n)c(E_n^*) = c(\pi^*F_n)c(\pi^*F_n^*) = c(\pi^*V).$$ This proves Theorem \ref{ch.dn}. $\triangle$ Proof of Theorem \ref{ch.bn}. If $V$ has rank $2n+1$, let $E_1 \subset E_2 \subset \ldots \subset E_n$ be the tautological isotropic flag on $Fl(V)$. Again set $x_i = c_1(E_i/E_{i-1})$. Let $L=E_{n+1}/E_{n}$. Then $l = c_1(\pi^*V)$, and $c(V)=c(E)c(E^*)(1+l)$. This shows that the first two relations hold. Again let $\rho:Q \rightarrow Fl(V)$ be the quadric bundle associated to $\pi^*(V) \otimes L$. Define the $c_i$ by the equation $$[{\Bbb P}(E_n \otimes L) + {\Bbb P}(\pi^*F_n \otimes L)] = h^n -h^{n-1}\rho^*c_1 +h^{n-1}\rho^*c_2- \ldots + (-1)^n \rho^*c_n.$$ The proof of Theorem \ref{t.chern} also shows that $2c_i = c_i(E_n \otimes L) + c_i(\pi^*F_n \otimes L)$, thereby proving the last relation. Fulton \cite{F1} has also shown that the appropriate monomials generate $A^*(Fl(V))$ as an $A^*(X)$ module, so that proving the theorem reduces to checking the quadratic relation on the $c_i$'s. The proof of the quadratic relation in the $B_n$ case is more or less the same as that given above. There is one twist. The totally split pullback of $\pi^*V$ is $L_1 \oplus L_2 \oplus \ldots L_n \oplus M \oplus L_n^* \oplus \ldots \oplus L_1^*$, where $M$ is self dual. However, $\pi^*(V)\otimes M$ totally splits as $M_1 \oplus M_2 \ldots \oplus M_n \oplus {\cal O} \oplus M_n^* \oplus \ldots \oplus M_1^*$ (where $M_i = L_i \otimes M$). We can then reduce to checking the relation on a product of projective spaces. The relation on the $c_p$'s then follows from the fact that $$c(E_n \otimes M)c(E^*_n \otimes M) = c(\pi^*F_n \otimes M) c(\pi^*F_n^* \otimes M).$$ This proves Theorem \ref{ch.bn}. $\triangle$

9412

alg-geom/9412020

en

https://arxiv.org/abs/alg-geom/9412020

alg-geom/9412020

R.Scognamillo

An elementary approach to the abelianization of the Hitchin system for arbitrary reductive groups

25 pages, LaTex. In the revised version, the most relevant changes are in the proofs contained in section 3. The major ones concern the proof of theorem 3.2 (theorem 3.1 in the revised version)

We consider the moduli space of stable principal G-bundles over a compact Riemann surface C of genus >1, with G a reductive algebraic group. We explicitly construct a map F from the generic fibre of the Hitchin map to a generalized Prym variety associated to a suitable Galois covering of C. The map F has finite fibres. In case G=PGl(2) one can check that the generic fibre of F is a principal homogeneous space with respect to a product of 2d-2 copies of Z/2Z where d is the degree of the canonical bundle over C. However in case the Dynkin diagram of G does not contain components of type $B_{n}$ n>0, or when the commutator subgroup (G,G) is simply connected the map F is injective.

alg-geom

\section*{Introduction} We consider here the moduli space $\cal M$ of stable principal $G$-bundles over a compact Riemann surface $C$, with $G$ an algebraic complex group. We denote by $K$ the canonical bundle over $C$. In \cite{hi} N.Hitchin defined an analytic map $\cal H$ from the cotangent bundle $T^{*}\cal M$ to the "characteristic space" $\cal K$ by associating to each $G$-bundle $P$ and section $s\in H^{0}(C,adP\otimes K)$ the spectral invariants of $s$. Hitchin showed for $G=Gl(n),SO(n),Sp(n)$ that the generic fibre of $\cal H$ is an open set in an abelian variety ${\cal A}$. In fact, he considers in each case a non-singular spectral curve $S$ covering $C$: when $G=Gl(n)$, ${\cal A}$ is identified with the Jacobian $J(S)$ ; in the other cases, there is a naturally defined involution on $S$ and $\cal A$ is the associated Prym variety. More recently, Faltings extended these results and described an abelianization procedure for the moduli space of Higgs $G$-bundles, with $G$ any reductive group (see \cite{fa} ). If $T\! \subset \! G$ is a fixed maximal torus with Weyl group $W$, one may construct for each given generic element $\phi\in{\cal K}$ a ramified covering \mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } of $C$ having $\mbox{$\mid \! W\! \mid $} $ sheets. The combined action of $W$ on \mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } and on the group of one parameter subgroups of $T$ induces an action on the space of all principal $T$-bundles $\tau $ over \mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } and we may consider the subvariety $\widehat{\cal P}$ of those $\tau $ which are $W$-invariant in this sense. The connected component ${\cal P}_{0}$ of $\widehat{\cal P}$ which contains the trivial $T$-bundle is an abelian variety. In \cite{fa} it is shown that the generic fibre of the Hitchin map is a principal homogeneous space with respect to a group (namely the first \'{e}tale cohomology group of $C$ with coefficients in a suitably defined group scheme) which is isogenous to $\widehat{\cal P}$. In the present paper, by means of mostly elementary techniques, we explicitly construct a map ${\cal F}$ from each connected component ${\cal H}^{-1}(\phi )_{c}$ of ${\cal H}^{-1}(\phi )$ to ${\cal P}_{0}$ and show that $\cal F$ has finite fibres. We use the classical theory of representations of finite groups to compute $dim\; {\cal P}_{0}=dim\; {\cal M}$ and conclude that the image under ${\cal F}$ of ${\cal H}^{-1}(\phi )$ contains a Zariski open set in ${\cal P}_{0}$. In case $G=PGl(2)$ one can check directly that the generic fibre of ${\cal F}_{c}: {\cal H}^{-1}(\phi )_{c}\rightarrow {\cal P}_{0}$ is a principal homogeneous space with respect to a product of $(2\cdot deg\; K-2)$ copies of $\ze /2\ze$. However in case the Dynkin diagram of $G$ does not contain components of type $B_{l}$, $l\geq 1$ or when the commutator subgroup $(G,G)$ is simply connected the map ${\cal F}_{c}$ is injective. Such results were announced in our previous paper \cite{sc} , in which we showed that ${\cal P}_{0}$ is isogenous to a "spectral" Prym-Tjurin variety $\mbox{$P_{\lambda}$}$ for each given dominant weight $\lambda $. Results concerning the description of the Hitchin fibre in terms of generalized Prym varieties were also announced in R.Donagi, {\em Spectral covers}, preprint, alg-geom/9505009 (1995). \section{The Hitchin map for any reductive group} \label{sec-pre} We denote by $C$ a compact Riemann surface of genus g$\; \geq 2$ and by $G$ a reductive algebraic group over the field of complex numbers. We also write $\mbox{\bf{g}} $ as the Lie algebra of $G$. The moduli space of stable principal $G$-bundles over $C$ is a quasi-projective complex variety $\cal M$ with $\; dim{\cal M}$=(g$-1)dimG+dimZ(G)$ , $Z(G)$ being the center of $G$. Note here that semistability for a principal $G$-bundle $P$ corresponds to semistability for the holomorphic vector bundle $adP$ associated to the adjoint representation \mbox{$Ad:G\rightarrow $gl$(\mbox{\bf{g}} )\; $} (\cite{ab} , \cite{ra} ). We denote by $K$ the canonical line bundle over $C$. By deformation theory and Serre duality, a point in the cotangent bundle $T^{*}\cal M$ of $\cal M$ is a pair $(P,s)$ with $P$ a stable principal $G$-bundle over $C$ and $s$ a section of the vector bundle $adP\otimes K$. The ring of polynomials on $\mbox{\bf{g}} $ which are invariant with respect to the adjoint action is freely generated by homogeneous polynomials $h_{1},\ldots ,h_{k}$. Each $h_{i}$ induces a map ${\cal H}_{i}:adP\otimes K\rightarrow K^{d_{i}}$ where $d_{i}= deg\; h_{i}$, and the Hitchin map \[ {\cal H}\; :\; T^{*}{\cal M}\longrightarrow {\cal K}={\displaystyle {\oplus_{i=1}^{k}}}H^{0}(C,K^{d_{i}})\] takes $(P,s)$ to the element in $\cal K$ whose $i$-th component is the composition of ${\cal H}_{i}$ with $s$ ( \cite{hi} ). It is a remarkable fact that the dimension of $\cal K$ is equal to the dimension of $\cal M$. Moreover the map $\cal H$ is surjective. This fact can be deduced from the existence of very stable $G$-bundles (see \cite{lau}, \cite{br} ,\cite{kp} Lemma 1.4). We fix once and for all a maximal torus $T\! \subset \! G$ with associated root system $R=R(G,T)$ and Weyl group \mbox{$W\! =\! N_{G}(T)/T$}. We also fix a subset $R^{+}\! \subset \! R$ (or equivalently a Borel subgroup $B\! \supset \! T$). If $\mbox{\bf{t}} $ denotes the Lie algebra of $T$, the differential of each root $\alpha \in R$ induces a map \( d\alpha :\mbox{\bf{t}} \otimes K\rightarrow K\) and the homogeneous $W$-invariant polynomials $\sigma_{1},\ldots ,\sigma_{k}$ on $\mbox{\bf{t}} $ obtained by restriction of $h_{1},\ldots ,h_{k}$ define a Galois covering \[ \underline{\sigma} =(\sigma_{1},\ldots ,\sigma_{k}):\mbox{\bf{t}} \otimes K\longrightarrow {\textstyle \oplus_{i=1}^{k}}K^{d_{i}}\] whose discriminant $\Xi$ is given by the zeroes of the $W$-invariant function $\prod _{\alpha\in R}d\alpha$ . For generic $\phi \in {\cal K}\! =H^{0}(C,{\textstyle \oplus_{i}}K^{d_{i}})$, we consider the curve $\widetilde{C} :=\phi ^{*}(\mbox{\bf{t}} \otimes K)$. This is a ramified covering of $C$ having $\; m =\mid \! W\! \mid $ sheets, whose branch locus $Ram$ satisfies by construction \begin{equation} \label{eq:ram} {\cal O}(Ram)\cong K^{\mid R\mid }\equiv K^{(dimG-rankG)}\; .\end{equation} If we indicate by $\iota: \mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } \rightarrow \mbox{\bf{t}} \otimes K$ the natural inclusion map, we have by definition, for each $w\in W$, \begin{equation} \label{eq:ad} \iota(w\; \eta)\! =\! Ad(n_{w})\; \iota (\eta) \end{equation} where $n_{w}\! \in N_{G}(T)$ is any representative of $w$. Note also that, if $\pi:\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } \rightarrow C$ denotes the projection map, $d\alpha \circ \iota $ is a holomorphic section of $\pi^{*}K$. \[ \begin{array}{rccc} & \widetilde{C} & \stackrel{\iota}{\longrightarrow } & \mbox{\bf{t}} \otimes K \\ {\scriptstyle \pi} & \downarrow & & \downarrow \\ & C & \stackrel{\phi}{\longrightarrow } & {\textstyle \oplus _{i}}K^{d_{i}} \end{array} \] As a consequence of our genericity hypothesis, \mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } has the following properties: \\ a) it is smooth and irreducible. \\ b) each ramification point $p\in \pi^{-1}(Ram)$ has index 1; i.e. is a simple zero for the section $\prod_{\alpha\in R^{+}}(d\alpha \circ \iota ):\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } \rightarrow \pi^{*}K^{\mid R\mid /2}$. \\[.2cm] This may be checked as follows. Let us denote by $\pi_{i}:K^{d_{i}}\rightarrow C$, $i=1,\ldots ,k$ and $q:\mbox{\bf{t}} \otimes K\rightarrow C$ the projections. Moreover for every $i=1,\ldots ,k$ let us denote by $\gamma _{i}:K^{d_{i}}\rightarrow\pi_{i}^{*}K^{d_{i}}$ the tautological section. For each $i$ we consider those sections of $q^{*}K^{d_{i}}$ that have the form $s=c\cdot\sigma_{i}^{*}\gamma _{i}+q^{*}a_{i}$ for some $c\in\mbox{\bf{C}} $ and $a_{i}\in H^{0}(C,K^{d_{i}})$. As $c$ varies in $\mbox{\bf{C}} $ and $a_{i}$ in $H^{0}(C,K^{d_{i}})$ the zero divisor of $s$ forms a linear system $\delta_{i}$ of divisors in $\mbox{\bf{t}} \otimes K$ that has no base points since the linear system $\mid K^{d_{i}}\mid $ on $C$ has no base points. For $\phi =(a_{1},\ldots ,a_{k})\in {\cal K}$, the curve $\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } $ is defined by the equations $\sigma_{i}^{*}\gamma _{i}=q^{*}a_{i}$, $i=1,\ldots , k$. One immediately checks that the map \[ \begin{array}{lcc} K^{d_{i}} & \longrightarrow & \mbox{\bf{P}} } \newcommand{\qu}{\mbox{\bf{Q} } ^{dim\; H^{0}(C,K^{d_{i}})}\\ x & \longmapsto & [\gamma _{i}(x),\pi_{i}^{*}a_{i,1}(x),\ldots ,\pi_{i}^{*}a_{i,m_{i}}(x)] \end{array} \] where the $a_{i,j}$'s form a basis of $H^{0}(C,K^{d_{i}})$ has image of dimension 2 and that $\sigma_{1}:\mbox{\bf{t}} \otimes K\rightarrow K^{d_{1}}$ is dominant. By Bertini's theorem (see \cite{jou}, theorem 6.3) the divisor $X_{1}\in\delta_{1}$ of the section $\sigma_{1}^{*}(\gamma _{1}-\pi_{1}^{*}a_{1})=\sigma_{1}^{*}\gamma _{1}-q^{*}a_{1}$ with $a_{i}$ generic in $H^{0}(C,K^{d_{i}})$ is smooth and irreducible. If $k\geq 2$, we next consider the linear system on $X_{1}$ given by the restriction of $\delta_{2}$. Since the polynomial $\sigma_{2}$ is algebraically independent from $\sigma_{1}$ the map $\sigma_{2}\mid_{X_{1}}:X_{1}\rightarrow K^{d_{2}}$ is dominant. We use the same argument as above and from Bertini's theorem we obtain that the divisor $X_{2}\subset X_{1}$ of the section $\sigma_{2}^{*}\gamma _{2}-q^{*}a_{2}\mid_{X_{1}}$ with generic $a_{2}$ is smooth and irreducible. We can repeat the same argument for the linear system $\delta_{i}\mid_{X_{i-1}}$ for every $i\leq k$ (since the map $\sigma_{i}\mid_{X_{i-1}}:X_{i-1}\rightarrow K^{d_{i}}$ is dominant) and thus prove a) . As for the statement b) one may consider the restriction of the linear systems above both to the discriminant locus $\Xi$ and to the locus ${\cal Z}\subset\Xi$ where $\prod _{\alpha\in R^{+}}d\alpha$ vanishes with multiplicity $\geq 2$ (${\cal Z}=Sing\; \Xi$). Again from Bertini's theorem one obtains that $\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } $ does not intersect $\cal Z$ and intersects $\Xi\setminus {\cal Z}$ transversely. \\[.2cm] \mbox{{\em Remark} } 1.1. For each $\alpha \in R^{+}$, let $s_{\alpha }\in W$ denote the corresponding reflection. As a consequence of condition b) above we may consider the ramification locus in \mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } as a disjoint union: ${\cal D}=\coprod_{\alpha\in R^{+}}{\cal D}_{\alpha}$, with ${\cal D}_{\alpha}=\{ \mbox{zeroes of } \; \; d\alpha \circ\iota \} =\{ \eta\in \mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } \mid s_{\alpha}\; \eta\; =\eta\}$ . By our previous considerations ${\cal D}_{\alpha }$ belongs to the linear system $\mid \! \pi^{*}K\! \mid $. In case $G$ is simple and simply laced, i.e. $W$ acts transitively on the set of roots $R$, we may write for each $y\in Ram$ \[ \pi^{-1}(y)=\! {\displaystyle {\coprod _{\alpha \in R^{+}}}}{\cal D}_{\alpha }^{y} \label{eq:div} \] where $\; {\cal D}_{\alpha }^{y}:={\cal D}_{\alpha }\cap \pi^{-1}(y)$ is nonempty for every $\alpha \in R^{+}$. If $G$ is not simply laced and has connected Dynkin diagram, $R$ is the union of two $W$-orbits $R_{1},\; R_{2}$, each one consisting of roots having the same length. Then we have \\ \begin{eqnarray} \label{eq:orb} \pi^{-1}(y)=\! \! {\displaystyle {\! \! \coprod _{\alpha \in R_{1}\cap R^{+}}}}\! {\cal D}_{\alpha }^{y} & \mbox{\makebox[1.5cm]{or} } & \pi^{-1}(y)=\! \! {\displaystyle {\! \! \coprod _{\alpha \in R_{2}\cap R^{+}}}}\! {\cal D}_{\alpha }^{y} \end{eqnarray} depending on whether $y$ corresponds to a short or a long root. More generally, if the Dynkin diagram of $G$ has more than one connected component, we have as many different "kinds" of fibers \[ \pi^{-1}(y)=\! \! {\displaystyle {\! \! \coprod _{\alpha \in R_{j}\cap R^{+}}}}\! {\cal D}_{\alpha }^{y}\] as are the $W$-orbits $R_{j}\! \subset \! R$. Since for each $\alpha \in R^{+}$ we have $\mid \! {\cal D}_{\alpha }\! \mid =\mbox{$\mid \! W\! \mid $} \cdot deg\; K$ and each fibre over a branch point consists of $\mbox{$\mid \! W\! \mid $} /2$ points, the number of fibres which correspond to the same orbit $R_{j}$ is equal to \begin{eqnarray} n_{j}& = & \mid R_{j}^{+}\mid \cdot \mbox{$\mid \! W\! \mid $} \cdot deg\; K/\frac{1}{2} \mbox{$\mid \! W\! \mid $} \nonumber \\ \label{eq:nj} & = & \mid R_{j}\mid \cdot \; deg\; K\; . \end{eqnarray} \vspace{.3cm} Let now $X(T)$ be the group of characters on $T$ and consider the group $H^{1}(\widetilde{C},T)$ of isomorphism classes of holomorphic principal $T$-bundles over $\widetilde{C}$. Each pair $(\tau ,\mu )$ with $\tau $ a principal $T$-bundle, $\mu \in X(T)$ defines a line bundle $\tau _{\mu }\equiv \tau \times _{\mu }\! {\bf C}$ and this way $H^{1}(\widetilde{C},T)$ is identified with \[ Pic(\widetilde{C})\otimes X(T)^{*},\] $X(T)^{*}\! \equiv \! Hom(X(T),\ze )$ being the dual group. For the same reason, the group of isomorphism classes of topologically trivial principal $T$-bundles is a tensor product \[ J(\widetilde{C})\otimes X(T)^{*}\] (here, as usual, $J(\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } )$ denotes the group of divisors with zero degree modulo linear equivalence ). Now, the action of $W$ on the sheets of $\widetilde{C}$ induces an action on $J(\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } )$. On the other hand, $W$ acts by conjugation on $T$, hence on $X(T)^{*}$. If $\tau \! =D_{1}\otimes \chi_{1}+\cdots +D_{l}\otimes \chi_{l}$ is a principal $T$-bundle over $\widetilde{C}$ and $w\in W$ an element of the Weyl group, we set \[ ^{w}\tau =w\; D_{1}\otimes \; ^{w}\chi _{1}+\cdots +w\; D_{l}\otimes \; ^{w}\chi_{l}\; .\] \begin{defin} The generalized Prym variety $\; {\cal P}=[J(\widetilde{C})\otimes X(T)^{*}]^{W}$ consists of those isomorphism classes of topologically trivial $T$-bundles $\tau $ which satisfy $\; ^{w}\tau \equiv \tau \; $ for each $w\in W$. \end{defin} Note that $\cal P$ is an algebraic group whose connected component of the identity ${\cal P}_{0}$ is an abelian variety. \section{Computing the dimension of $\cal P$} The following can be deduced from the above mentioned Faltings' result describing the generic Hitchin fibre as isogenous to $\widehat{\cal P}=[Pic(\widetilde{C})\otimes X(T)^{*}]^{W}$ (\cite{fa}, theorem III.2) and the fact (due to G.Laumon and proved in \cite{fa}, theorem II.5) that all Hitchin fibers have the same dimension: \begin{prop} \label{it:prd} The dimension of $\cal P$ is equal to the dimension of $\cal M$. \end{prop} In this section we want to give a direct proof of such statement. If we set ${\cal S}\equiv \! X(T)\otimes_{\ze }\mbox{\bf{C}} $ and denote by $H^{1}$ the first cohomology $W$-representation $H^{1}(\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } ,\mbox{\bf{C}} )$, by Doulbault theorem we have \[ dim{\cal P}=\frac{1}{2} dim[H^{1}\otimes {\cal S}^{*}]^{W}=\frac{1}{2} dim\; Hom_{W}({\cal S},H^{1}).\] We will compute \( M\equiv dim\; Hom_{W}({\cal S},H^{1})\) by use of the classical theory of representations of finite groups and associated characters (for more details about this subject, see for example \cite{se} ). For any $W$-representation $V$ considered here, we denote by $\chi_{V}:W\rightarrow \mbox{\bf{C}} $ its character (for $\rho :W\rightarrow Gl(V)$ the homomorphism defining the representation, we have by definition $\chi_{V}(w)=trace(\rho(w))\; \forall w\in W$). By the theory of characters of finite groups we have \begin{equation} \label{eq:car} M=<\chi_{{\cal S}},\chi_{H^{1}}>\end{equation} where $\scp $ is the usual scalar product between characters. If $N$ is the number of connected components of the Dynkin diagram $\Pi$ of $G$ and $h=dimZ(G)$ we have a decomposition \[ {\cal S}=\underbrace{{\cal B}\oplus \cdots \oplus {\cal B}}_{h} \oplus {\cal S}_{1}\oplus \cdots \oplus {\cal S}_{N}\] where $\cal B$ is the 1-dimensional trivial representation and ${\cal S}_{i}$ the irreducible reflection representation corresponding to the $i$-th component of $\Pi $, $i=1,...,N$. Then we may rewrite (\ref{eq:car}) as \begin{equation} \label{eq:dim} M=h<\chi_{{\cal B}},\chi_{H^{1}}>+\sum_{i=1}^{N}<\chi_{{\cal S}_{i}},\chi_{H^{1}}>. \end{equation} We observe that $W$ acts trivially on the cohomology groups $H^{0}(\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } ,\mbox{\bf{C}} )\cong H^{2}(\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } ,\mbox{\bf{C}} )\cong \mbox{\bf{C}} $. Hence the Lefschetz character $\chi_{L}\equiv \chi_{H^{0}}-\chi_{H^{1}}+\chi_{H^{2}}$ satisfies $\chi_{L}=2\chi_{{\cal B}}-\chi_{H^{1}}$ and we have \begin{eqnarray} \label{eq:lef} <\chi_{{\cal B}},\chi_{H^{1}}> & = & 2-<\chi_{{\cal B}},\chi_{L}> \\ \label{eq:lefbis} <\chi_{{\cal S}_{i}},\chi_{H^{1}}> & = & -<\chi_{{\cal S}_{i}},\chi_{L}>.\end{eqnarray} On the other hand, it is well known (Hopf trace formula, see e.g.\cite{cr} ) that the Lefschetz character satisfies \[ \chi_{L}=\chi_{\widetilde{C}^{0}}-\chi_{\widetilde{C}^{1}}+\chi _{\widetilde{C}^{2}}\] $\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } ^{n}$ being the free $\mbox{\bf{C}} $-module generated by the $n$-cells of some cellular decomposition of $\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } $ ($\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } ^{n}\cong H_{n}(K^{n},K^{n-1};\mbox{\bf{C}} )$, with $K^{j}$ the $j$-th skeleton of $\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } $, $j=n,n-1$). We choose one finite triangulation $\Delta$ of $C$ whose set of vertices contains all branch points. We denote by $C^{n}$ the free module generated by the $n$-cells of $\Delta$ for $n=1,2$ , and by $C_{0}^{0}$ and $D_{j}$ the free modules whose generators are respectively all vertices not lying in the branch locus $Ram$ and all branch points corresponding to the same $W$-orbit $R_{j}\! \subset \! R$ (see {\em Remark} 1.1.). Let $N'$ be the number of $W$-orbits in $R$, and for each $j=1,\ldots N'$ let us fix one positive root $\alpha _{j}\! \in \! R_{j}^{+}$ and set $H_{j}=\{ 1,s_{\alpha _{j}}\} \! \subset \! W$. We denote by $Ind^{W}_{H_{j}}(B_{j})$ the $W$-representation induced by the 1-dimensional trivial representation $B_{j}$ of $H_{j}$ (by definition ,$Ind^{W}_{H_{j}}(B_{j})=\oplus_{[w]\in W/H_{j}}\mbox{\bf{C}} v_{[w]}$ with $W$ acting by $u\circ v_{[w]}=v_{[uw]}$). We have the following isomorphisms of $W$-modules: \begin{eqnarray*} \mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } ^{2} & \cong & \mbox{\bf{C}} [W]\otimes C^{2} \\ \mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } ^{1} & \cong & \mbox{\bf{C}} [W]\otimes C^{1} \\ \mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } ^{0} & \cong & \mbox{\bf{C}} [W]\otimes C^{0}_{0}\oplus \bigoplus_{j=1}^{N'}Ind^{W}_{H_{j}}(B_{j})\otimes D_{j} \\ & \equiv & \mbox{\bf{C}} [W]\otimes C^{0}_{0}\oplus \bigoplus_{j=1}^{N'}(Ind^{W}_{H_{j}}(B_{j}))^{n_{j}}\end{eqnarray*} where $\mbox{\bf{C}} [W]$ denotes as usual the regular representation and the $n_{j}$'s are defined as in (\ref{eq:nj}) . \\ By Frobenius reciprocity formula we have \[ <\chi_{\cal B},\chi_{Ind^{W}_{H_{j}}(B_{j})}>=<\chi_{B_{j}},\chi_{B_{j}}>=1\; ;\] and since from the general theory each irreducible $W$-representation occurs as a subrepresentation of $\mbox{\bf{C}} [W]$ as many times as is its dimension, we obtain \begin{equation} \label{eq:ada} <\chi_{{\cal B}},\chi_{L}>=rk\; C^{2}-rk\; C^{1}+rk\; C^{0}_{0}+\mid Ram\mid =(2-2g).\end{equation} Analogously, we have \[ <\chi_{{\cal S}_{i}},\chi_{L}>=(rk\; C^{2}-rk\; C^{1}+rk\; C^{0}_{0})dim{\cal S}_{i}+\sum_{j=1}^{N'}n_{j}<\chi_{B_{j}},\chi _{res_{j}{\cal S}_{i}}>\] where $res_{j}{\cal S}_{i}$ denotes the representation obtained by restriction to $H_{j}$.\\ Now, given some positive root $\alpha\in R^{+}$, the corresponding reflection $s_{\alpha}\in W$ acts trivially on ${\cal S}_{i}$ whenever $\alpha\! \notin \! {\cal S}_{i}$ , otherwise it acts trivially on one subspace of codimension 1 in ${\cal S}_{i}$. Thus we get \begin{eqnarray} <\chi_{{\cal S}_{i}},\chi_{L}> & = & (rk\; C^{2}-rk\; C^{1}+rk\; C^{0}_{0})dim{\cal S}_{i}+\sum_{R_{j}\subset {\cal S}_{i}} n_{j}(dim{\cal S}_{i}-1)+\nonumber \\ & + & \sum_{R_{j}\not\subset {\cal S}_{i}} n_{j}\cdot dim{\cal S}_{i}\nonumber \\ \label{eq:adb} & = & (2-2g)\; dim{\cal S}_{i}-\sum_{R_{j}\subset {\cal S}_{i}} n_{j}\; . \end{eqnarray} By substituting (\ref{eq:ada}) and (\ref{eq:adb}) respectively in (\ref{eq:lef}) and (\ref{eq:lefbis}) and then (\ref{eq:lef}) and (\ref{eq:lefbis}) in (\ref{eq:dim}) , we finally obtain \begin{eqnarray*} M & = & 2h+(2g-2)(h+\sum_{i=1}^{N}dim{\cal S}_{i})+\sum_{j=1}^{N'} n_{j} \\ & = & 2h+(2g-2)dim\; T+\mid \! Ram\! \mid .\end{eqnarray*} Since $dim\; T+\mid \! R\! \mid =dim\; G$, by (\ref{eq:ram}) we get \[ dim{\cal P}\equiv \frac{1}{2} M=(g-1)dim\; G+h. \] \section{The main results} \label{sec-res} In this section we will define a map $\cal F$ from each component of the generic Hitchin fibre to the abelian variety ${\cal P}_{0}$ and study its properties. We first show how one can associate to each given pair $(P,s)\in {\cal H}^{-1}(\phi)$ a $T$-bundle ${\cal T}={\cal T}(P,s)$ which satisfies $^{w}\! {\cal T} \cong {\cal T} \; \forall w\in W$. For $\phi\in {\cal K}$ generic, let then $P$ be a principal $G$-bundle and $s\in H^{0}(C,adP\otimes K)$ such that $(P,s)\in {\cal H}^{-1}(\phi)$. We first consider the restriction $P_{0}$ of $P$ to the open set $C_{0}$. Since for every $\xi\in C_{0}$, $s(\xi)\in \mbox{\bf{g}} $ is regular semisimple (for an analysis of the regular elements in $\mbox{\bf{g}} $ , see for example \cite{ko}), we have a morphism of vector bundles \[ [s,\hspace{.2cm} ]:adP_{0}\longrightarrow adP_{0}\otimes K\] whose kernel $\cal N$ is a bundle of Cartan subalgebras in $\mbox{\bf{g}} $. We thus have a section \[ \gamma :C_{0}\rightarrow P/N_{G}(T)\equiv P\times_{G} G/N_{G}(T)\] locally defined by $\gamma(\xi)=\nu(\xi)N_{G}(T)$ where $\nu(\xi)\in G$ satisfies $Ad\; \nu(\xi)\mbox{\bf{t}} ={\cal N}_{\xi}\equiv c_{\mbox{\bf{g}} }(s(\xi))$. If we pull back $P_{0}$ over $\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} }_{0}$ we actually have a section \begin{equation}\label{eq:sect} \varphi :\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} }_{0}\rightarrow \pi^{*}P_{0}/T \end{equation} locally defined by $\varphi(\eta)=\mu(\eta) T$ where $\mu(\eta)\in G$ satisfies \begin{equation}\label{eq:diag} Ad\;\mu(\eta) (\iota(\eta))=s(\pi(\eta)).\end{equation} Thus over $\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} }_{0}$ the bundle $\pi^{*}P$ has a reduction of its structure group to $T$. Moreover, from (\ref{eq:ad}) we have for each $w\in W$ \begin{equation} \label{eq:nw} \varphi(w\eta)=\mu(\eta) n_{w}^{-1}T\end{equation} which implies that such $T$-reduction $\tau_{0}=\varphi^{*}(\pi^{*}P_{0})$ is $W$-invariant with respect to the action previously defined. Now if we consider a Borel subgroup $B\subset G$ containing $T$, the inclusion map $T\hookrightarrow B$ and $\varphi$ define a section $:\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} }_{0}\rightarrow\pi^{*}P\times_{G} G/B$. Since $G/B$ is a complete variety, by the valuative criterion of properness this section can be extended to the whole curve $\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } $ and we thus obtain (uniquely up to isomorphisms) a $B$-reduction $P_{B}$ of the $G$-bundle $\pi^{*}P$ such that $P_{B}\mid_{\scriptsize \mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} }_{0}} $ is the $B$-extension of $\tau_{0}$.\\[.1cm] If $(\; ,\; )$ denotes a $W$-invariant scalar product on $X(T)_{\ze}\otimes {\bf{R} }$ and $\beta \in R$, we define as usual the one parameter subgroup $\beta '\in Hom(X(T),\ze)$ by \begin{equation} \label{eq:'} \beta '(\lambda)=<\lambda ,\beta >\equiv \frac{2(\lambda ,\beta )}{(\beta ,\beta )}\; \; \forall \lambda\in X(T).\end{equation} We want to prove the following: \begin{th} \label{it:thw} Let $\tau_{B} =\tau(P,s)$ be the $T$-bundle associated to $P_{B}$ via the natural projection $B\rightarrow T$. Let us fix one theta characteristic $\frac{1}{2} K$ and consider the $T$-bundle $K_{\rho}=\frac{1}{2}\pi^{*}K\otimes \sum_{\beta \in R^{+}} \beta '$, where $R^{+}\subset R$ is the subset of positive roots that corresponds to $B$. Then ${\cal T}(P,s):=\tau_{B} +K_{\rho}$ is $W$-invariant. \end{th} The proof will be organized in a few lemmas. We first observe that since $W$ is generated by the simple reflections it suffices to show \begin{equation} \label{eq:tbun} ^{s_{\alpha }}\tau_{B} \cong \tau_{B} +\pi^{*}K\otimes \alpha ' \end{equation} for every simple root $\alpha $. In fact we have ${\displaystyle \sum_{\beta \in R^{+}} s_{\alpha }(\beta ')=\sum _{\stackrel{\beta \in R^{+}}{\beta \neq \alpha}} \beta ' -\alpha '} $, so, if relation (\ref{eq:tbun}) holds, one has $^{s_{\alpha }}(\tau_{B} +K_{\rho})\cong \tau_{B} +K_{\rho}$. In terms of line bundles associated to characters on $T$, relation (\ref{eq:tbun}) can be rewritten as \begin{equation} \label{eq:fon} (^{s_{\alpha }}\tau_{B} -\tau_{B})\times_{\lambda }\mbox{\bf{C}} \cong \mbox{$<\lambda,\alpha>$}\pi^{*}K\; \; \; \forall \lambda \in X(T). \end{equation} Given a simple root $\alpha $, let us denote by $s_{\alpha}(B)$ the Borel subgroup $n_{\alpha }Bn_{\alpha }^{-1}$, where $n_{\alpha}\in N_{G}(T)$ represents $s_{\alpha}$. One analogously obtains another $T$-bundle $\tau_{s_{\alpha} (B)}$ such that $\tau_{s_{\alpha} (B)}\mid_{\scriptsize\widetilde{C}_{0}}\cong\tau_{0}$ from the completion of $\tau_{0}$ to an $s_{\alpha}(B)$-reduction $P_{s_{\alpha}(B)}$. The first lemma treats the relationship between $\tau_{B}$ and $\tau_{s_{\alpha} (B)}$. \begin{lem}\label{it:borels} We have $\tau_{s_{\alpha}(B)}\cong\; ^{s_{\alpha}}\tau_{B}$. \end{lem} \mbox{{\em Proof.} } We consider an open covering $\{ V_{h}\} _{h\in H}$ of $C$ over which $P$ and the canonical bundle $K$ can be trivialized and with the property that each $V_{h}$ contains at most one branch point. We choose a \v{C}ech covering ${\cal U}=\{ U_{h}\}_{h\in H}$ of \mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } to be given by all open sets $U_{h}=\pi^{-1}(V_{h})$ (by definition each $U_{h}$ is stable with respect to the action of $W$). For $h\in H$ we choose frames $e_{1}^{h},\ldots ,e_{q}^{h}$ for the vector bundle $adP\otimes K$ over $V_{h}\subset C$, $q$ being equal to the dimension of $\mbox{\bf{g}} $. With respect to this choice the section $s:C\rightarrow ad P\otimes K$ is locally given by "coordinates" $s_{h}:V_{h}\rightarrow \mbox{\bf{g}} $ satisfying \begin{equation} \label{eq:shl} s_{h}=Ad\; g_{hl}\cdot k_{hl}s_{l}\; \; \mbox{ for } V_{h}\cap V_{l}\neq \emptyset , \end{equation} $g_{hl}$ and $k_{hl}$ being transition functions for $P$, $K$ respectively. Let $\iota_{h}:U_{h}\rightarrow \mbox{\bf{t}} $ be coordinates for $\iota:\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} }\rightarrow\mbox{\bf{t}} \otimes K$. We define $J\subset H$ to be the subset of those indices $j$ such that $V_{j}$ contains a branch point and set $I=H\setminus J$. For each $h\in H$ we fix maps $\mu_{h}:U_{h}\rightarrow G$ such that for each $i\in I$ $\mu_{i}$ satisfies \begin{equation}\label{eq:mui} Ad\;\mu_{i}(\eta) (\iota_{i}(\eta))=s_{i}(\pi(\eta))\end{equation} (compare with (\ref{eq:diag}) ) and the 0-chain $\{ \mu_{h}(\eta) B\}_{h\in H}$ defines the section $\widehat{\varphi}_{B}:\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} }\rightarrow\pi^{*}P/B$ completing $\varphi$ in (\ref{eq:sect}). By definition, the $B$-bundle $P_{B}$ is represented by the cocycle $\{ b_{hl}\} \in{\cal Z}^{1}({\cal U},B)$ where $b_{hl}(\eta) \equiv \mu_{h} (\eta) ^{-1}g_{hl}(\pi(\eta) )\mu_{l} (\eta)$. Define $\{ b'_{hl}\} \in{\cal Z}^{1}({\cal U},s_{\alpha}(B))$ by $b'_{hl}(\eta)=n_{\alpha} b_{hl}(s_{\alpha}\eta)n_{\alpha }^{-1}\; \forall\eta\in U_{h}\cap U_{l}$. We have $b'_{hl}(\eta) \equiv n_{\alpha}\mu_{h} (s_{\alpha}\eta) ^{-1}g_{hl}(\pi(\eta) )\mu_{l} \setan_{\alpha}^{-1}$, hence $\{ b'_{hl}\}$ represents an $s_{\alpha} (B)$-reduction of $\pi^{*}P$. On the other hand, from (\ref{eq:nw}) we have $\{ \mu_{i} \setan_{\alpha}^{-1}T\}_{i\in I}=\{ \mu_{i} (\eta) T\}_{i\in I}$ hence $\{ b'_{hl}\} $ represents $P_{s_{\alpha}(B)}$. Now, if we denote by $p:B\rightarrow T,\; p':s_{\alpha}(B)\rightarrow T$ the natural projections we have $p'\circ b'_{hl}(\eta) =n_{\alpha} (p\circ b_{hl}(s_{\alpha}\eta))n_{\alpha }^{-1}$ (since every Borel subgroup is a semidirect product of its maximal torus and its maximal unipotent subgroup). Since $\{ n_{\alpha} (p\circ b_{hl}(s_{\alpha}\eta))n_{\alpha }^{-1}\} $ are by definition transition functions for $^{s_{\alpha}}\tau_{B}$, we thus have an isomorphism $\tau_{s_{\alpha}(B)}\cong\; ^{s_{\alpha}}\!\tau_{B}$.\hspace{.2cm} $\Box$\\ We keep the notations of the proof of lemma \ref{it:borels}. For each positive root $\beta\in R^{+}$, we shall denote by $\beta_{h}:U_{h}\rightarrow \mbox{\bf{C}} $ the coordinates of the section of $\pi^{*}K$ over \mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } given by the composition $d\beta \circ \iota$ (see \S \ref{sec-pre}). Our next step consists in finding suitable transition functions $b_{ji}$ for $P_{B}$ on intersections $U_{i}\cap U_{j}$ with $j\in J$. Indeed, we will find suitable maps $\mu_{j}:U_{j}\rightarrow G$ with $j\in J$ defining the completed section $\widehat{\varphi}_{B}$. We fix nilpotent generators $\{ X_{\gamma}\}_{\gamma \in R^{+}}$ in the Lie algebra $\mbox{\bf{b}} $ of $B$ with $ad\; t(X_{\gamma} )=\gamma (t)X_{\gamma}$ $\forall t\in \mbox{\bf{t}} $, $\; \forall\gamma \in R^{+}$. In general, the completion $\hat{\varphi}_{B}:\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} }\rightarrow \pi^{*}P/B$ of our $\varphi$ above is locally given by holomorphic maps $f_{j}:U_{j}\rightarrow G$ with $j\in J$ such that \begin{equation} \label{eq:ff} Ad\; f_{j} (\eta) ^{-1}s_{j}(\pi(\eta))=\iota_{j} (\eta) +\sum_{\gamma\in R^{+}} a_{\gamma}(\eta) X_{\gamma}\; . \end{equation} By \mbox{{\em Remark} } 1.1 , for $j\in J$ the set $U_{j}$ is a union of open sets $\bigcup_{\beta\in R(j)\cap R^{+}}U_{j,\beta}$ where $R(j)$ is some $W$-orbit of roots depending on $j$ and each $U_{j,\beta}$ contains only those ramification points that are zeroes for $\beta_{j}$ . \begin{lem} \label{it:bor} There exists a holomorphic map $\mu_{j} :U_{j}\rightarrow G$ satisfying for each $\beta \in R(j)\cap R^{+}$ and $\eta\in U_{j,\beta }$ \begin{equation} \label{eq:jb} Ad\; \mu_{j} (\eta) ^{-1}s_{j}(\pi(\eta))=\iota_{j} (\eta) +X_{\beta} \; . \end{equation} \end{lem} \mbox{{\em Proof.} } We construct $\mu_{j}$ separately on each open set $U_{j,\beta }$. By our genericity hypothesis we may assume for a ramification point $p\in U_{j,\beta }$ \begin{equation} \label{eq:fff} Ad\; f_{j} (p) ^{-1}s_{j}(\pi(p) )=\iota_{j} (p) + X_{\beta} \end{equation} with $\beta_{j}(p) \equiv d\beta(\iota_{j}(p) )=0$.\\ Let $\alpha$ be the root with minimal height in $R^{+}\setminus\{ \beta\} $ such that $a_{\alpha}(\eta) $ in (\ref{eq:ff}) is not identically zero. The map $c_{j}(\eta) =exp\frac{a_{\alpha}(\eta) }{\alpha_{j}(\eta) }X_{\alpha} :U_{j,\beta }\rightarrow G$ is holomorphic on each fixed connected component of $U_{j,\beta}$ and by evaluating $Ad\; c_{j}(\eta) $ on the right-hand side of (\ref{eq:ff}) we get \[ Ad\; c_{j}(\eta) (\iota_{j} (\eta) + \sum_{\gamma\in R^{+}} a_{\gamma}(\eta) X_{\gamma} ) =\iota_{j} (\eta) + a'_{\beta}(\eta) X_{\beta} + \sum _{\stackrel{\gamma\in R^{+}\setminus \{ \beta \} }{\gamma >\alpha}} a_{\gamma}(\eta) X_{\gamma} \; .\] By an induction argument we can then assume \begin{equation} Ad\; f_{j} (\eta) ^{-1}s_{j}(\pi(\eta) )=\iota_{j} (\eta) +a_{\beta}(\eta) X_{\beta} \end{equation} where $a_{\beta }(p)=1$ (since we may multiply $f_{j}$ by a suitable constant in $T$). Consider now the map $d_{j}(\eta) =exp\frac{a_{\beta}(\eta) -1}{\beta_{j}(\eta) }X_{\beta} :U_{j}\rightarrow G$. Since $p$ is a simple zero for $\beta_{j}(\eta) $, $d_{j}(\eta) $ is holomorphic on each chosen connected component of $U_{j,\beta}$. We have \[ Ad\; d_{j}(\eta) (\iota_{j} (\eta) +a_{\beta}(\eta) X_{\beta} )=\iota_{j} (\eta) + X_{\beta} \] and the claim of our lemma is proved. \makebox[1.2cm][r]{$\Box$} \\ For each $j\in J$, define $u_{j} :U_{j \rightarrow B$ by $u_{j}(\eta) =exp\; \frac{X_{\beta} }{\beta _{j}(\eta) }$ whenever $\eta\in U_{j,\beta }$. We have \begin{equation} \label{eq:uj} Ad\; u_{j}(\eta)^{-1}\iota_{j}(\eta)=\iota_{j}(\eta)+X_{\beta}\; .\end{equation} We may represent the completed section $\widehat{\varphi}_{B}$ by $\{ \mu_{h}(\eta) B\}$ where the $\mu_{i} $'s are as in (\ref{eq:mui}) for every $i\in I$ and the $\mu_{j} $'s satisfy (\ref{eq:jb}) for every $j\in J$. By substituting (\ref{eq:mui}) and (\ref{eq:jb}) in (\ref{eq:shl}) and replacing $\iota_{j}(\eta)+X_{\beta} $ with $Ad\; u_{j}(\eta)^{-1}\iota_{j}(\eta) $ we obtain transition functions on each nonempty intersection $U_{j}\cap U_{i}$ \begin{equation} \label{eq:gp} b_{ji}(\eta) \equiv \mu_{j} (\eta) ^{-1}g_{ji}(\pi(\eta) )\mu_{i} (\eta) =u_{j}^{-1}(\eta) t_{ji}(\eta) \end{equation} where $t_{ji}(\eta) :U_{i}\cap U_{j}\rightarrow T$ is holomorphic (as $u_{j}$ is holomorphic on $U_{i}\cap U_{j}$ ). Since each element in $B$ can be written uniquely as a product of a unipotent element by an element in $T$ we have $t_{ji}=p\circ b_{ji}$. We now compare $P_{B}$ with $P_{s_{\alpha}(B)}$. By definition we only need to compare them around the ramification points. As set of nilpotent generators in the Lie algebra of $s_{\alpha}(B)$ we may choose $\{ X_{\beta}\} _{\beta \in R^{+}\setminus\{\alpha \} }\cup\{ Ad\; n_{\alpha} (X_{\alpha})\}$. Thus from lemma \ref{it:bor} we may define a section $\hat{\varphi}_{s_{\alpha}(B)}:\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} }\rightarrow \pi^{*}P/s_{\alpha}(B)$ completing $\varphi$ by \begin{eqnarray*} \hat{\varphi}_{s_{\alpha}(B)}(\eta) & = & \mu_{j}\detas_{\alpha}(B)\; \makebox{ for }\; \eta\in U_{j}\setminus U_{j,\alpha } \\ \hat{\varphi}_{s_{\alpha}(B)}(\eta) & = & \mu_{j}\setan_{\alpha}^{-1}s_{\alpha}(B)\; \makebox{ for }\; \eta\in U_{j,\alpha } \end{eqnarray*} where the $G$-valued maps $\mu_{j}$ satisfy (\ref{eq:jb}). From this we see that $P_{s_{\alpha}(B)}$ and $P_{B}$ are isomorphic on $\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} }\setminus {\cal D}_{\alpha }$ and that on all intersection sets $U_{j,\alpha }\cap U_{i}$ with $j\in J$ we have transition functions for $P_{s_{\alpha}(B)}$ of the form \begin{equation}\label{eq:b'} b'_{ji}(\eta) =n_{\alpha}\mu_{j}(s_{\alpha}\eta) ^{-1}\mu_{j}(\eta) b_{ji}(\eta) . \end{equation} If we apply lemma \ref{it:bor} to the set $s_{\alpha}(R^{+})$ of positive roots corresponding to $s_{\alpha}(B)$ we obtain on $U_{j,\alpha }\cap U_{i}$ a factorization $b'_{ji}(\eta) ={u'_{j}}^{-1}(\eta) t'_{ji}(\eta) $ with $u'_{j}(\eta) =exp\; \frac{Ad\; n_{\alpha}(X_{\alpha} ) }{-\alpha _{j}(\eta) }=n_{\alpha} {u_{j}}^{-1}\detan_{\alpha}^{-1}$ and $t'_{ji}(\eta) =p'\circ b'_{ji}(\eta) $ (compare with (\ref{eq:gp}) ). Let us denote by I the identity element in $G$. From (\ref{eq:b'}) and lemma \ref{it:borels} a meromorphic section of $\; ^{s_{\alpha}}\tau_{B} -\tau_{B}$ is given by a 0-cochain $\{t_{h} \}_{h\in H}\in{\cal C}^{0}({\cal U},T)$ where \begin{eqnarray} t_{h}(\eta) & = & \mbox{I~} \mbox{ whenever } h\in I \mbox{ or } h\in J \mbox{ and } \eta\notin U_{j,\alpha } \label{eq:sezi}\\ t_{j}(\eta) & = & n_{\alpha} u_{j}(\eta)^{-1}\mu_{j}(s_{\alpha}\eta) ^{-1}\mu_{j}(\eta) u_{j}(\eta)^{-1} \; \; \forall\eta\in U_{j,\alpha },\; j\in J. \label{eq:sez} \end{eqnarray} By (\ref{eq:jb}) on each $U_{j,\alpha }$ the map $h_{j}(\eta) =\mu_{j}(s_{\alpha}\eta) ^{-1}\mu_{j}(\eta)$ satisfies \begin{equation}\label{eq:Fj} Ad\; h_{j}(\eta) (\iota_{j}(\eta) +X_{\alpha} )=\iota_{j}(s_{\alpha}\eta) +X_{\alpha} =Ad\; n_{\alpha}(\iota_{j}(\eta) )+X_{\alpha} . \end{equation} Choose $X_{-\alpha} \in \mbox{\bf{g}} $ so that $X_{\alpha} ,X_{-\alpha} ,h_{\alpha }:=[X_{\alpha} ,X_{-\alpha} ]\in \mbox{\bf{t}} $ generate a Lie subalgebra ${\bf h}_{\alpha }\subset \! \mbox{\bf{g}} $ with ${\bf h}_{\alpha }\cong sl(2)$ and $d\alpha (h_{\alpha })=2$. Define \[ F_{j}(\eta) =exp(\alpha _{j}(\eta) X_{-\alpha} )\; \; \forall \eta\in U_{j,\alpha }.\] Since $F_{j}(\eta)$ satisfies $Ad\; F_{j}(\eta) (\iota_{j}(\eta) +X_{\alpha} )=Ad\; n_{\alpha}(\iota_{j}(\eta) )+X_{\alpha} $, by (\ref{eq:Fj}) we have on $U_{j,\alpha }$ \begin{equation} \label{eq:c} \mu_{j} (s_{\alpha}\eta) ^{-1}\mu_{j} (\eta) =F_{j}(\eta) \cdot L_{j}(\eta) \end{equation} where for each $\eta\in U_{j,\alpha }$, $L_{j}(\eta) \in B$ lies in the centralizer of $\iota_{j}(\eta) +X_{\alpha} \in \mbox{\bf{b}} $. Note that for $q$ any ramification point in $U_{j,\alpha }$ we have by definition \begin{equation} \label{eq:uno} L_{j}(q)=\mbox{I} .\end{equation} In particular the map $L_{j}$ is holomorphic. Since when $\eta\in U_{j,\alpha }$ is not a ramification point $~\iota_{j}(\eta) + X_{\alpha}$ is regular semisimple and by (\ref{eq:uj}) one has $c_{\mbox{\bf{g}} }(\iota_{j}(\eta) + X_{\alpha})=Ad\; u_{j}(\eta)^{-1}\mbox{\bf{t}} $ , the holomorphic $T$-valued map $l_{j}(\eta) =p\circ L_{j}(\eta)$ has the form \begin{equation} \label{eq:lj} l_{j}(\eta) =u_{j}(\eta) L_{j}(\eta) u_{j}(\eta) ^{-1}.\end{equation} Relation (\ref{eq:sez}) becomes \begin{equation}\label{eq:tjz} t_{j}(\eta) =z_{j}(\eta)\cdot l_{j}(\eta) \end{equation} where the map $z_{j}(\eta)\equiv n_{\alpha} u_{j}(\eta)^{-1} F_{j}(\eta) u_{j}(\eta) ^{-1}$ has values in $T$ and is holomorphic everywhere in $U_{j,\alpha }$ but on the ramification points. The connected subgroup $H_{\alpha }\subset \! G$ generated by $exp(X_{\alpha} ),exp(X_{-\alpha} ),exp(h_{\alpha })$ is isomorphic to a copy of $Sl(2)$ or $PGl(2)$ in $G$ and one can compute $z_{j}(\eta) $ directly in terms of two by two matrices. In the $Sl(2)$ case, denoting by $\varrho$ the isomorphism $:H_{\alpha }\rightarrow Sl(2)$, one has for some $c\in\mbox{\bf{C}} ^{*}$ \begin{eqnarray} \nonumber \varrho (z_{j}(\eta) ) & = & \mp\left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right) \left( \begin{array}{cc} 1 & -c/\alpha _{j}(\eta) \\ 0 & 1 \end{array}\right) \left( \begin{array}{cc} 1 & 0 \\ \alpha _{j}(\eta) /c & 1 \end{array}\right) \left( \begin{array}{cc} 1 & -c/\alpha _{j}(\eta) \\ 0 & 1 \end{array}\right) \\ \label{eq:cru} & = & k\cdot diag(\alpha _{j}(\eta) ,\alpha _{j}(\eta) ^{-1}) \end{eqnarray} where $k\in T$ is a constant and $\alpha _{j}(\eta) $ are the coordinates of the section $d\alpha \circ\iota$, according to our previous notations. As for $H_{\alpha }\stackrel{\varrho}{\cong}PGl(2)$ one gets \begin{equation} \label{eq:crup} \varrho (z_{j}(\eta) )= \overline{k\cdot diag(\alpha _{j}(\eta) ,\alpha _{j}(\eta) ^{-1})} \end{equation} where the bar indicates the image under the factor map $:Gl(2)\rightarrowPGl(2)$. Let now $T_{\alpha }\subset \! T$ be the identity component of the subgroup $Ker(\alpha )=\{ t\in T\; \mid \; \alpha (t)=1\} $. The centralizer $Z_{\alpha }$ in $G$ of $T_{\alpha }$ is a reductive group of semisimple rank 1 having Lie algebra ${\bf z}=\mbox{\bf{t}} \oplus \mbox{\bf{C}} X_{\alpha} \oplus \mbox{\bf{C}} X_{-\alpha} $, and it is known that such a group is a product $T'\times H$ , $T'$ being a torus and H being a copy of $Sl(2)$, $PGl(2)$ or $Gl(2)$. The case $H=Sl(2)$ is characterized by the group of characters $X(T)$ being an orthogonal direct sum $\ze \chi_{1} \oplus X'$, with $\chi_{1}=\sqrt{\alpha }$. If we compose any $\lambda \in X'$ with the 0-chain $\{ t_{h}\} _{h\in H}$ defined by (\ref{eq:sezi}) and (\ref{eq:sez}) we obtain a nowhere vanishing holomorphic section of the line bundle $(^{s_{\alpha} }\tau_{B}-\tau_{B})\times _{\lambda }\mbox{\bf{C}} $. If instead we compose $\chi_{1}$ to $\{ t_{h}\} _{h\in H}$, by (\ref{eq:tjz}) and (\ref{eq:cru}) we get an holomorphic section for $(^{s_{\alpha} }\tau_{B}-\tau_{B})\times _{\chi_{1}}\mbox{\bf{C}} $ having simple zeroes exactly on the locus ${\cal D_{\alpha }}$. Thus relation (\ref{eq:fon}) is satisfied (see \mbox{{\em Remark} } 1.1). The case $H=PGl(2)$ is characterized by $X(T)$ being an orthogonal direct sum $\ze \alpha \oplus X'$. For $\lambda \in X'$, we get the same result as for the $Sl(2)$ case. For $\lambda =\alpha $ we find instead an holomorphic section for $(^{s_{\alpha} }\tau_{B}-\tau_{B})\times _{\lambda }\mbox{\bf{C}} $ having zeroes of multiplicity two on ${\cal D_{\alpha }}$ . This proves (\ref{eq:fon}). In case $H=Gl(2)$, we have an orthogonal direct sum $X(T)=X'\oplus \ze \chi_{1}\oplus \ze \chi_{2}$ with $\alpha =\chi_{1}\cdot \chi_{2}^{-1}$. Composing $\lambda \in X'$ gives us again $^{s_{\alpha} }\tau_{B}\times _{\lambda }\mbox{\bf{C}} \cong \tau_{B}\times _{\lambda }\mbox{\bf{C}} $ as in the previous cases. If we compose $\chi_{1}$ we obtain an holomorphic section of $(^{s_{\alpha} }\tau_{B}-\tau_{B})\times _{\chi_{1}}\mbox{\bf{C}} $ having simple zeroes exactly on ${\cal D_{\alpha }}$. If we compose $\chi_{2}$ we obtain a meromorphic section of $(^{s_{\alpha} }\tau_{B}-\tau_{B})\times _{\chi_{2}}\mbox{\bf{C}} $ having simple poles exactly on ${\cal D_{\alpha }}$. Thus relation (\ref{eq:fon}) holds also in this case and theorem \ref{it:thw} is proved. We thus have a map \[ \begin{array}{cccc} {\cal T}: & {\cal H}^{-1}(\phi) & \rightarrow & \widehat{{\cal P}} \equiv [Pic(\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } )\otimes X(T)^{*}]^{W} \nonumber \\ & (P,s) & \longmapsto & \tau (P,s)+K_{\rho}\; \; . \end{array} \] Note that from (\ref{eq:tbun}) and lemma \ref{it:borels} \hspace{.1cm} $\cal T$ does not depend on the choice of the Borel subgroup $B\supset T$ (or of the subset of positive roots in $R$). \begin{defin} Let ${\cal H}^{-1}(\phi)_{c}$ be some connected component of ${\cal H}^{-1}(\phi)$. For a fixed point $(P',s')\in {\cal H}^{-1}(\phi)_{c}$ we define ${\cal F}_{c}:{\cal H}^{-1}(\phi)_{c}\rightarrow {\cal P}_{0}$ by \[ {\cal F}_{c}(P,s)={\cal T}(P,s)-{\cal T}(P',s')\equiv {\tau(P,s)}-{\tau(P',s')}\; . \] \end{defin} Such definition does not depend on our previous choice of the theta characteristic $\frac{1}{2} K$. We now want to study the fibers of ${\cal F}_{c}$. First we make the following \\ {\em Remark} \ref{sec-res}.1 \hspace{.2cm} For $i\in I$, the maps $\mu_{i} (\eta)$ in (\ref{eq:mui}) are defined up to multiplication to the right by some holomorphic map $m_{i}:U_{i}\rightarrow T$. As for $j\in J$, any other holomorphic map $\mu_{j} '(\eta)$ satisfying (\ref{eq:jb}) has the form $\mu_{j} '(\eta) =\mu_{j} (\eta) M_{j}(\eta) $ where for every $\alpha \in R(j)\cap R^{+}$, $M_{j}(\eta) :U_{j,\alpha }\rightarrow B$ is holomorphic and such that $M_{j}(\eta) \in c_{G}(\iota_{j}(\eta) +X_{\alpha} )$. If we replace $\mu_{j}$ and $\mu_{i}$ with the new maps $\mu_{j} '(\eta) $ and $\mu_{i} '(\eta) =\mu_{i} (\eta) m_{i}(\eta) $, we obtain from $(P,s)$ and $B$ an equivalent cocycle $\{ m_{h}^{-1}t_{hi} m_{i}\} $ representing $\tau_{B}$. Since for every $j\in J$ and $q\in U_{j}\cap {\cal D}_{\alpha }$ $~\iota_{j}(q)+X_{\alpha} \in \mbox{\bf{b}} $ is regular, we have $c_{G}(\iota_{j}(q)+X_{\alpha} )=T_{\alpha }{\cal U}_{\alpha }$, where $T_{\alpha }$ is the identity component of $Ker(\alpha :T\rightarrow\mbox{\bf{C}} ^{*})$ and ${\cal U}_{\alpha }$ is the unipotent 1-dimensional subgroup corresponding to the root $\alpha $. Hence the $T$-valued map $m_{j}(\eta) :=p\circ M_{j}(\eta) \equiv u_{j}(\eta) M_{j}(\eta) u_{j}(\eta) ^{-1}$ satisfies for every $\alpha \in R(j)\cap R^{+}$ \begin{equation} \label{eq:norm} \alpha (m_{j}(q))=1\; \mbox{ $\forall q\in U_{j}\cap {\cal D}_{\alpha }$ .} \end{equation} \begin{lem} \label{it:l2} Let $(P,s),(Q,v)$ be pairs in ${\cal H}^{-1}(\phi)$ such that $\tau(P,s)$ and $\tau(Q,v)$ are isomorphic. Let $\{ t_{hl}\}$ and $\{ \tilde{t}_{hl}\}$ with $h,l\in H$ be cocycles representing $\tau(P,s)$ and $\tau(Q,v)$ respectively and suppose \begin{equation} \tilde{t}_{hl}=m_{h}^{-1}t_{hl}m_{l} \label{eq:t1p} \end{equation} where the maps $m_{h}:U_{h}\rightarrow T$ are holomorphic and satisfy condition {\em (\ref{eq:norm})} for every $j\in J$ and $\alpha \in R(j)\cap R^{+}$. Then $Q$ is isomorphic to $P$ and $v=s$. \end{lem} \mbox{{\em Proof.} } For what concerns $P$ and the construction of $\tau (P,s)$ we keep the notations used in the proof of theorem \ref{it:thw} . In particular we still consider a C\v{e}ch covering ${\cal U}=\{ U_{h}\}_{h\in H}$ of $\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } $ consisting of $W$-invariant open sets as it was first defined in the proof of lemma \ref{it:borels}. For each nonempty intersection $U_{h}\cap U_{l}$ we have transition functions for the $B$-reduction $Q_{B}$ of $\pi^{*}Q$ having the form: \begin{eqnarray}\label{eq:gpt} \tilde{b}_{ji}(\eta) & = & \tilde{\mu}_{j}(\eta)^{-1}\tilde{g}_{ji}(\pi(\eta) )\tilde{\mu}_{i}(\eta) =u_{j}(\eta)^{-1}\tilde{t}_{ji}(\eta) \; \; \forall j\in J,i\in I \\ \tilde{b}_{hi}(\eta) & = & \tilde{\mu}_{h}(\eta)^{-1}\tilde{g}_{hi}(\pi(\eta) )\tilde{\mu}_{i}(\eta) =\tilde{t}_{hi}(\eta) \; \; \forall i,h\in I \end{eqnarray} where $\{ \tilde{g}_{hl}\}_{h,l\in H}$ are transition functions for the $G$-bundle $Q$ and $\tilde{\mu}_{i}$, $\tilde{\mu}_{j}$ are defined analogously as $\mu_{i}$ and $\mu_{j}$ in (\ref{eq:gp}). For $j\in J$, define $M_{j}:U_{j}\rightarrow B$ by \begin{equation} \label{eq:mj} M_{j}:=u_{j}^{-1}m_{j}u_{j} \; \; \; \mbox{ (see \mbox{{\em Remark} } \ref{sec-res}.1 ).} \end{equation} The hypothesis of the lemma provide that $M_{j}$ is holomorphic on $U_{j,\alpha }$ for each $\alpha \in R(j)\cap R^{+}$ and we have $M_{j}(\eta) \in c_{G}(\iota_{j}(\eta) +X_{\alpha} )\; \forall \eta \in U_{j,\alpha }$ by definition of $u_{j}$. Define the holomorphic maps \begin{eqnarray*} \Gamma_{i} & = & \mu_{i}\; m_{i}\; \tilde{\mu}_{i}^{-1}\; \; \forall i\in I \mbox{ and} \\ \Gamma_{j} & = & \mu_{j} M_{j}\tilde{\mu}_{j}^{-1}\; \; \forall j\in J. \end{eqnarray*} From (\ref{eq:gpt}), (\ref{eq:gp}) and (\ref{eq:t1p}) we obtain the equivalence condition between cocycles on $\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} }$: \[ \tilde{g}_{hl}(\pi(\eta) )=\Gamma_{h}(\eta) ^{-1}g_{hl}(\pi(\eta) )\Gamma_{l}(\eta) \; \; \; \forall\eta\in U_{h}\cap U_{l}\; \forall h,l\in H.\] The claim of the lemma is then proved provided we show that the maps $\Gamma_{l}$ are invariant with respect to the action of $W$ on the sheets of $\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} }$. In fact if we indicate by $\{ v_{h}\} _{h\in H}$ the coordinates of $v$ so that $v_{h}=Ad\tilde{g}_{hl}\cdot k_{hl}v_{l}$, by our definition of the maps $\tilde{\mu}_{l}$, $\tilde{\mu}_{h}$ we have:\[ Ad\; \Gamma_{l}\; v_{l}=s_{l}\; \; \forall l\in H.\] Since $W$ is generated by the simple reflections, it suffices to show $\Gamma_{l}(s_{\alpha}\eta) =\Gamma_{l}(\eta)$ for every simple reflection $s_{\alpha}$. From (\ref{eq:nw}) we have for each $i\in I$ \begin{equation}\label{eq:mis} {\mu}_{i}(s_{\alpha}\eta) ^{-1}{\mu}_{i} (\eta) =n_{\alpha} {l}_{i}(\eta) \end{equation} for suitable holomorphic maps $l_{i}:U_{i}\rightarrow T$. By evaluating the transition functions $t_{hi}=\mu_{h}^{-1}g_{hi}\mu_{i}$ with $h,i\in I$ on $s_{\alpha} \eta$ and replacing $\mu_{i}(s_{\alpha}\eta)$ with ${\mu}_{i} (\eta) {l}_{i}(\eta)^{-1}n_{\alpha}^{-1}$ and $\mu_{h}(s_{\alpha}\eta)$ with ${\mu}_{h} (\eta) {l}_{h}(\eta)^{-1}n_{\alpha}^{-1}$ we obtain \begin{equation} \label{eq:t} {t}_{hi}(s_{\alpha}\eta) =n_{\alpha} {l}_{h}(\eta) {t}_{hi}(\eta) {l}_{i}(\eta)^{-1} n_{\alpha}^{-1}.\end{equation} Analogously, if we define $\tilde{l}_{i}:U_{i}\rightarrow T$ by \begin{equation}\label{eq:mtis} \tilde{\mu}_{i}(s_{\alpha}\eta)^{-1}\tilde{\mu}_{i}(\eta)=n_{\alpha}\tilde{l}_{i}(\eta), \end{equation} we have \begin{equation} \label{eq:td} \tilde{t}_{hi}(s_{\alpha}\eta) =n_{\alpha} \tilde{l}_{h}(\eta) \tilde{t}_{hi}(\eta) \tilde{l}_{i}(\eta)^{-1} n_{\alpha}^{-1}.\end{equation} By replacing $\tilde{t}_{hi}$ with $m_{h}^{-1}t_{hi}m_{i}$ in both sides of (\ref{eq:td}) and substituting (\ref{eq:t}) in the left-hand side, we obtain an equality both sides of which contain only factors with values in $T$. We cancel $t_{hi}(\eta)$ and obtain \[ m_{h}(\eta)\cdot n_{\alpha}^{-1} m_{h}(s_{\alpha}\eta)^{-1} n_{\alpha}\cdot\tilde{l}_{h}(\eta)^{-1}\cdot l_{h}(\eta) =m_{i}(\eta)\cdot n_{\alpha}^{-1} m_{i}(s_{\alpha}\eta)^{-1} n_{\alpha}\cdot\tilde{l}_{i}(\eta)^{-1}\cdot l_{i}(\eta) \] for every $\eta\in U_{h}\cap U_{i}$, $i,h\in I$. We can repeat the same calculation on intersection sets $U_{i}\cap U_{j}$ with $j\in J$ and $i\in I$. What we need is the analog for $j\in J$ of the relations (\ref{eq:mis}) and (\ref{eq:mtis}). On each open set $U_{j,\alpha }$ the map $\mu_{j}(\eta)$ is related with $\mu_{j}(s_{\alpha}\eta)$ via the identity (\ref{eq:c}). If for each $\beta \in R^{+}\setminus\{ \alpha \}$ we define $n_{\alpha \beta }\in N(T)$ to be the representative of $s_{\alpha}$ satisfying $Ad\; n_{\alpha ,\beta }(X_{\beta} )=X_{s_{\alpha}(\beta )}$, by construction of the maps $\mu_{j}$ in lemma (\ref{it:bor}) we have for $\eta\in U_{j,\beta }$ \begin{equation}\label{eq:mjs} {\mu}_{j}(s_{\alpha}\eta) ^{-1}{\mu}_{j} (\eta) = n_{\alpha ,\beta }{L}_{j}(\eta) \end{equation} where $L_{j}(\eta)$ is a suitable element in the centralizer of $\iota_{j}(\eta) +X_{\beta} $. We analogously define $\tilde{L}_{j}:U_{j}\rightarrow B$ $\forall j\in J$ by \begin{eqnarray} \label{eq:ctd} \tilde{\mu}_{j}(s_{\alpha}\eta) ^{-1}\tilde{\mu}_{j} (\eta) & = & F_{j}(\eta) \tilde{L}_{j}(\eta) \; \; \mbox{ for $\eta\in U_{j,\alpha }$} \\ \label{eq:mtjs} \tilde{\mu}_{j}(s_{\alpha}\eta) ^{-1}\tilde{\mu}_{j} (\eta) & = & n_{\alpha ,\beta }\tilde{L}_{j}(\eta) \; \; \mbox{ for $\eta\in U_{j,\beta }$ with $\beta \neq\alpha $ } \end{eqnarray} and set for each $\eta\in U_{j}$ \begin{eqnarray} \label{eq:plj} {l}_{j}(\eta) &:= & p\circ L_{j}(\eta) =u_{j}(\eta) {L}_{j}(\eta) u_{j}(\eta) ^{-1}\\ \label{eq:pltj} \tilde{l}_{j}(\eta) & := & p\circ \tilde{L}_{j}(\eta) =u_{j}(\eta) \tilde{L}_{j}(\eta) u_{j}(\eta) ^{-1}.\end{eqnarray} One uses (\ref{eq:c}), (\ref{eq:ctd}) and the fact that the map $z_{j}(\eta)=n_{\alpha} u_{j}^{-1}(\eta) F_{j}(\eta) u_{j}^{-1}(\eta)$ (see (\ref{eq:tjz}) ) is holomorphic $T$-valued outside the ramification points (hence it commutes with any other map with values in $T$), to obtain by the same procedure described above for all pairs of indices $h,i\in I$ \[ m_{j}(\eta)\cdot n_{\alpha}^{-1} m_{j}(s_{\alpha}\eta)^{-1} n_{\alpha}\cdot\tilde{l}_{j}(\eta)^{-1}\cdot l_{j}(\eta) =m_{i}(\eta)\cdot n_{\alpha}^{-1} m_{i}(s_{\alpha}\eta)^{-1} n_{\alpha}\cdot\tilde{l}_{i}(\eta)^{-1}\cdot l_{i}(\eta) \] for each $\eta\in U_{j,\alpha }\cap U_{i}$. One uses (\ref{eq:mjs}) and (\ref{eq:mtjs}) to prove the same identity for all $\eta\in U_{j,\beta }\cap U_{i}$ with $\beta \neq\alpha $. In conclusion, the maps \\ $m_{h}(\eta)\cdot n_{\alpha}^{-1} m_{h}(s_{\alpha}\eta)^{-1} n_{\alpha}\cdot \tilde{l}_{h}(\eta)^{-1}\cdot l_{h}(\eta):U_{h}\rightarrow T$ with $h\in H$ are the restriction to $U_{h}$ of a global holomorphic map on $\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} }$, hence are equal to some constant ${\bf c}$. We compute such map on one ramification point $q\in U_{j,\alpha }$ . Since we have $l_{j}(q)=\tilde{l}_{j}(q)=\mbox{I} $ (compare with (\ref{eq:uno}) ) and $\alpha (m_{j} (q))=1$ by hypothesis, we obtain ${\bf c}=$I , i.e. \begin{equation} \label{eq:lm} m_{h}(s_{\alpha}\eta) = n_{\alpha} m_{h}(\eta) \cdot l_{h}(\eta)\cdot \tilde{l}_{h}(\eta) ^{-1} n_{\alpha}^{-1}\; \; \; \; \forall h\in H. \end{equation} By use of (\ref{eq:mis}), (\ref{eq:mtis}) and this last identity we find $\Gamma_{i}(s_{\alpha}\eta) =\Gamma_{i}(\eta)$ for each $\eta\in U_{i}$, $i\in I$. As for $j\in J$, if $\eta$ is in $U_{j,\alpha }$ we have by (\ref{eq:c}) and (\ref{eq:ctd}), by the definition of $M_{j}$, $l_{j}$ and $\tilde{l}_{j}$ and by (\ref{eq:lm}) \begin{eqnarray*} \Gamma_{j}(s_{\alpha}\eta) & = & \mu_{i}(\eta) u_{j}(\eta)^{-1} l_{j}(\eta)^{-1}z_{j}(\eta)^{-1}m_{j}(\eta) l_{j}(\eta)\tilde{l}_{j}(\eta)^{-1} z_{j}(\eta)\tilde{l}_{j}(\eta) u_{j}(\eta)\tilde{\mu}_{j}(\eta)^{-1}=\\ & = & \Gamma_{j}(\eta) . \end{eqnarray*} If $\eta$ is in $U_{j,\beta }$, one proves \( \Gamma_{j}(s_{\alpha}\eta) =\Gamma_{j}(\eta) \) by using (\ref{eq:mjs}), (\ref{eq:mtjs}), (\ref{eq:lm}) and the identity (following from the above definition of $n_{\alpha ,\beta }$) $n_{\alpha ,\beta }\; u_{j}(s_{\alpha}\eta)\; n_{\alpha ,\beta }^{-1}=u_{j}(\eta)$. \makebox[2cm][r]{$\Box$} \begin{lem} \label{it:l3} Let $(P,s),(Q,v)$ be pairs in ${\cal H}^{-1}(\phi)$ such that $\tau(P,s)$ and $\tau(Q,v)$ are isomorphic. Let $\{ t_{hl}\}$ and $\{ \tilde{t}_{hl}\}$ with $h,l\in H$ be cocycles representing $\tau(P,s)$ and $\tau(Q,v)$ respectively and write \begin{equation} \tilde{t}_{hl}=m_{h}^{-1}t_{hl}m_{l} \end{equation} for suitable holomorphic maps $m_{h}:U_{h}\rightarrow T$ with $h\in H$. Up to multiplying each $m_{h}$ by one and the same suitably chosen element in $T$, the following holds: \\ {\em (i)} for each positive root $\alpha \in R^{+}$ and $q\in U_{j}\cap {\cal D}_{\alpha }$ we have $\alpha (m_{j}(q))=\mp 1$. {\em (ii)} if for $\alpha \in R^{+}$ there exists some character $\lambda \in X(T)$ such that \begin{equation} \label{eq:sc} <\lambda ,\alpha >=1\; , \end{equation} we have $\alpha (m_{j}(q))=1$ $\forall q\in U_{j}\cap {\cal D}_{\alpha }$. \end{lem} \mbox{{\em Proof.} } Choose one ramification point $q_{\alpha }\in {\cal D}_{\alpha }$ for each $\alpha \in \Delta $, $q_{\alpha }\in U_{j(\alpha )}$ for suitable $j(\alpha )\in J$. Up to multiplying the maps $\{ m_{h}\}_{h\in H}$ by a suitable element in $T$ we may assume \begin{equation} \label{eq:ga} \alpha (m_{j(\alpha )}(q_{\alpha }))=1\; \; \forall \alpha \in \Delta. \end{equation} We keep the same notation as before. We consider the maps $\{ l_{h}\} $ and $\{ \tilde{l}_{h}\} $, $h\in H$ as in (\ref{eq:mis}), (\ref{eq:mtis}), (\ref{eq:plj}) and (\ref{eq:pltj}) and let $\alpha $ be some simple root. From the proof of lemma (\ref{it:l2}) one has that the maps $m_{h}(\eta)\cdot n_{\alpha}^{-1} m_{h}(s_{\alpha}\eta)^{-1} n_{\alpha}\cdot \tilde{l}_{h}(\eta)^{-1}\cdot l_{h}(\eta):U_{h}\rightarrow T$ are the restriction of a global holomorphic map on $\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} }$. Computing such map on $q_{\alpha }$ gives us by (\ref{eq:ga}) and the fact that we have $l_{j}(q)=\tilde{l}_{j}(q)=\mbox{I} \; \; \forall q\in {\cal D}_{\alpha }\cap U_{j} $ \begin{equation} \label{eq:s} m_{j}(q)\cdot n_{\alpha}^{-1}m_{j}(s_{\alpha} q)^{-1}n_{\alpha} \cdot \tilde{l}_{j}(q)^{-1}\cdot l_{j}(q)=\mbox{I} \; \; \forall q\in {\cal D}\cap U_{j}, j\in J \end{equation} and \[ m_{j}(q)=\; \; n_{\alpha}^{-1}m_{j}(s_{\alpha} q)n_{\alpha}\; \; \forall q\in {\cal D}_{\alpha }\cap U_{j},\; j\in J. \] By evaluating $\alpha :T\rightarrow\mbox{\bf{C}} ^{*}$ on both sides of this last identity we obtain \[ \alpha ^{2}(m_{j}(q))=1.\] If moreover $\alpha $ satisfies condition (\ref{eq:sc}), evaluating $\lambda $ on both sides of the same identity gives \( \lambda (m_{j}(q))=\lambda (m_{j}(q))\cdot\alpha ^{-1}(m_{j}(q))\), or \[ \alpha (m_{j}(q))=1.\] The claim of the theorem is thus proved for every simple root. Consider now $q\in {\cal D}_{\beta }$ with $\beta \in R^{+}\setminus\Delta$. Note that for $q\in U_{j}$, from the definition of $l_{j}$ and $\tilde{l}_{j}$ and the fact that $L_{j}(q)$ and $\tilde{L}_{j}(q)$ belong to the centralizer in $G$ of $\iota_{j}(q)+X_{\beta} $ we have \begin{equation}\label{eq:bl} \beta (l_{j}(q))=\beta (\tilde{l}_{j}(q))=1 \end{equation} (compare with (\ref{eq:norm}) in {\em Remark} 3.1). By evaluating $\beta :T\rightarrow\mbox{\bf{C}} ^{*}$ on both sides of (\ref{eq:s}) as $\alpha $ runs over all simple roots we obtain $\beta (m_{j}(q))=\beta (n_{\alpha}^{-1}m_{j}(s_{\alpha} q)n_{\alpha} )\; \; $ $\forall\alpha \in\Delta$, hence \[ \beta (m_{j}(q))=\beta (n_{w}^{-1}m_{j}(wq)n_{w})\; \; \forall w\in W.\] On the other hand, we know that there exist $\alpha \in\Delta$ and $u\in W$ with $u(\alpha )=\beta $. We thus have \[ \beta (m_{j}(q))=\beta (n_{u}m_{j}(u^{-1}q)n_{u}^{-1})= \alpha (m_{j}(u^{-1}q))=\mp 1.\mbox{\makebox[2cm][r]{$\Box$} } \] \begin{th} \label{it:cor1} Suppose $G$ has one of the following properties: \\ a) the commutator group $(G,G)$ is simply connected ;\\ b) the Dynkin diagram of $G$ has no component of type $B_{l},\; l\geq 1$.\\ Then the map ${\cal T}:{\cal H}^{-1}(\phi)\rightarrow\widehat{{\cal P}}$ is injective. \end{th} \mbox{{\em Proof.} } In case $(G,G)$ is simply connected the fundamental weights are elements in $X(T)$; in particular condition (\ref{eq:sc}) in lemma \ref{it:l3} is satisfied for every root $\alpha \in R^{+}$ and our claim follows from lemma \ref{it:l2}. As for the case $G$ satisfies condition $b)$, we see from the Dynkin diagram of all simple groups of type different from $B_{l}$, $l\geq 1$ and $G_{2}$ that for every $\alpha \in R^{+}$ there exists another root $\beta $ with $<\beta ,\alpha >=1$. On the other hand the type $G_{2}$ is simply connected. \makebox[2cm][r]{$\Box$} \begin{th} \label{it:cor2} Let $a\geq 1$ be the cardinality of the subset $A\subset R^{+}$ of those roots which do not satisfy condition {\em (\ref{eq:sc})} in lemma {\em \ref{it:l3}}. If $d$ denotes the degree of $\pi^{*}K$, the fibre of $\cal T$ consists of at most $2^{a(d-1)}$ points. \end{th} \mbox{{\em Proof.} } Let $(P,s)\in {\cal H}^{-1}(\phi)$, $\tau(P,s)$ be as in theorem \ref{it:thw} and suppose there exists a pair $(Q,v)\in {\cal H}^{-1}(\phi)$ such that $\tau (Q,v)\cong\tau (P,s)$. Let $\{ t_{hl}\}_{h,l\in H}$ and $\{ \tilde{t}_{hl}\}_{h,l\in H}$ be cocycles representing $\tau(P,s)$ and $\tau(Q,v)$ respectively and write \( \tilde{t}_{hl}=m_{h}^{-1}t_{hl}m_{l} \) for suitable holomorphic maps $m_{h}:U_{h}\rightarrow T$ with $h\in H$. From the proof of lemma \ref{it:l3} we can assume that for $a$ chosen ramification points $q\in {\cal D}_{\beta }$, one for each $\beta \in A$, and every other ramification point $q\in {\cal D}_{\beta }$ with $\beta \notin A$, condition $\beta (m_{j}(q))=1$ (for suitable $j\in J$) holds. If $(Q,v)$ is distinct from $(P,s)$, by lemmas \ref{it:l2} and \ref{it:l3} there exists some $\alpha \in A$ and some $p_{\alpha }\in U_{j}\cap {\cal D}_{\alpha }$ (with suitable $j\in J$) such that condition \begin{equation} \label{eq:-1} \alpha (m_{j}(p_{\alpha }))=-1 \end{equation} is satisfied. Moreover, two pairs for which relation (\ref{eq:-1}) holds for exactly the same set of ramification points coincide by \mbox{{\em Remark} } \ref{sec-res}.1. \makebox[2cm][r]{$\Box$} \\[.1cm] From theorems \ref{it:cor1} and \ref{it:cor2} and from proposition \ref{it:prd} we obtain the following \begin{cor} The image under ${\cal F}$ of the generic Hitchin fibre ${\cal H}^{-1}(\phi)$ contains a Zariski open set in ${\cal P}_{0}$. \end{cor} \subsection{The $PGl(2)$ case.} Let $\phi\in H^{0}(C,K^{2})$ be generic. Let $P$ be a $PGl(2)$-bundle over $C$ and $s\in H^{0}(C,adP\otimes K)$ such that ${\cal H}(P,s)=\phi$. We indicate by $pr:Gl(2)\rightarrow PGl(2)=Gl(2)/\mbox{\bf{C}} ^{*}$ the factor map and as maximal torus $T\subset PGl(2)$ we choose the one obtained by restricting $pr$ to the maximal torus $\widetilde{T}\subset Gl(2)$ given by all diagonal matrices. We also set $\mbox{\bf{t}} =Lie\; T,\; \tilde{\mbox{\bf{t}} }=Lie\; \widetilde{T}$. In this setting, $\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } =\phi^{*}(\mbox{\bf{t}} \otimes K)$ is a ramified double covering of $C$ whose ramification divisor $\cal D$ satisfies by definition ${\cal O}({\cal D})\cong \pi^{*}K$. Let $\{ V_{h}\}_{h\in H}$ and $\{ U_{h}\}_{h\in H}$ be open coverings of $C$ and $\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} }$ defined as before. If $\{ g_{hl}:V_{h}\cap V_{l}\rightarrow PGl(2) \} _{h,l\in H}$, are transition functions for $P$, it is known that there exists some rank 2 vector bundle F, hence some principal $Gl(2)$-bundle $\widetilde{P}$, with transition functions $\tilde{g}_{hl}$ satisfying \begin{equation} \label{eq:pr} pr\circ \tilde{g}_{hl}=g_{hl}\; \; \forall h,l\in H .\end{equation} Moreover, any other rank 2 vector bundle $F'$ has the same property if and only if $F'\cong F\otimes L$ for some line bundle $L\in Pic(C)$. Note also that this implies $deg\; F\equiv deg\; F'\; \; (mod\; \; 2)$ (since $deg(F\otimes L)=deg\; F\cdot deg\; L^{2}$). For the sake of simplicity for any $F$ satisfying relation (\ref{eq:pr}) we write $P=pr(F)$. For $\widetilde{P}$ as above, we clearly have an isomorphism $ad\; \widetilde{P}\otimes K\cong (ad\; P\otimes K)\oplus K$ and given some fixed generic section $x:C\rightarrow K$ we may define $\tilde{s}\in H^{0}(ad\; \widetilde{P}\otimes K)$ by $\tilde{s}=s\oplus x$. We set $\tilde{\phi}={\cal H}_{Gl(2)}(\widetilde{P},\tilde{s})\in H^{0}(C,K\oplus K^{2})$ ( the subscript indicating that we are in the $Gl(2)$ setting ) and observe that the covering $\tilde{\phi}^{*}(\tilde{\mbox{\bf{t}} }\otimes K)$ of $C$ coincides with $\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} }$. Then it is clear from the argument above that we have a surjective map \[ "pr":{\cal H}_{Gl(2)}^{-1}(\tilde{\phi})\rightarrow {\cal H}_{PGl(2)}^{-1}(\phi)\; .\] This also shows that ${\cal H}_{PGl(2)}^{-1}(\phi)$ has two components ${\cal H}_{PGl(2)}^{-1}(\phi)_{0}$, ${\cal H}_{PGl(2)}^{-1}(\phi)_{1}$: namely $(Q,v)\in {\cal H}_{PGl(2)}^{-1}(\phi)$ is contained in ${\cal H}_{PGl(2)}^{-1}(\phi)_{0}$ or ${\cal H}_{PGl(2)}^{-1}(\phi)_{1}$ depending on the parity of the degree of those $F$ which satisfy $pr(F)=Q$. We now look at our construction in the $Gl(2)$ case. If we indicate by $\chi_{1}$ and $\chi_{2}$ the coordinate functions on $\widetilde{T}$ and set $\tilde{\alpha }=\chi_{1}\cdot \chi_{2}^{-1}$, $\sigma=s_{\tilde{\alpha }}$, we have by definition \[ {\cal P}_{Gl(2)}=\{ Q\otimes \chi '_{1}\oplus \sigma ^{*}Q\otimes \chi '_{2}\mid Q\in J(\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} })\} \equiv J(\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } )\] (the one parameter subgroups $\chi_{i}'$ being defined by $\chi_{i}(\chi '_{j})=(\chi_{i},\chi_{j}),\; j=1,2$) and \[ \widehat{{\cal P}}_{Gl(2)}=Pic(\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } ).\] The map ${\cal T}:{\cal H}_{Gl(2)}^{-1}(\tilde{\phi})\rightarrow Pic(\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } )$ is injective (see theorem \ref{it:cor1}), dominant and by Hitchin's theory (see \cite{hi} ) it preserves the parity of the degrees. By the argument above the generic fibre of the map $"pr"$ is a principal homogeneous space with respect to \( \Lambda=\{ M\in Pic(\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} })\; \mid M=\pi^{*}L,\; \; L\in Pic(C)\}. \) In this setting the map $\pi^{*}:Pic(C)\rightarrow Pic(\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} })$ is injective (since $\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} }\rightarrow C$ is a ramified covering: see e.g \cite{mum} ), hence $\Lambda$ coincides with $Pic(C)$. Since $Pic(\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} })^{\mbox{\em \scriptsize even}}/Pic(C)$ and $Pic(\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} })^{\mbox{\em \scriptsize odd}}/Pic(C)$ are both principal homogeneous spaces with respect to the connected group $J(\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} })/J(C)$, it follows that the components ${\cal H}_{PGl(2)}^{-1}(\phi)_{0}$, ${\cal H}_{PGl(2)}^{-1}(\phi)_{1}$ are connected. Now, let $\chi '$ be the one parameter subgroup in $T\subsetPGl(2)$ given by composing $pr$ with $\chi '_{1}$ (we have $X(T)^{*}=\ze \chi '$). By definition, we have $\widehat{{\cal P}} _{PGl(2)}={\cal P}_{PGl(2)}=\{ Q\otimes \chi '\; \mid Q\in J(\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} }),\sigma^{*}Q\cong Q^{-1}\} $ and, since $\pi^{*}:J(C)\rightarrow J(\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} })$ is injective, this is just the Prym variety $P(\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } ,\sigma)\subset J(\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} })$. From theorem \ref{it:thw} the $\widetilde{T}$-bundle $\tilde{\tau}=\tau(\widetilde{P},\tilde{s})$ has transition functions $t_{hl}:U_{h}\cap U_{l}\rightarrow \widetilde{T}$ of the form \[ t_{hl}(\eta) =diag(\; q_{hl}(\eta) ,\sigma^{*}q_{hl}(\eta) \cdot k_{hl}(\pi(\eta))\; \; ).\] One can easily check that the maps \[ pr\circ t_{hl}(\eta) = q_{hl}(\eta) \cdot \sigma^{*}q_{hl}(\eta) ^{-1}\cdot k_{hl}(\pi(\eta))^{-1}:U_{h}\cap U_{l}\rightarrow \mbox{\bf{C}} ^{*}\] are transition functions for $\tau=\tau(P,s)$. In other words, if we use the additive notation, we have ${\cal T}_{PGl(2)}(P,s)=(1-\sigma^{*})\circ {\cal T}_{Gl(2)}(\widetilde{P},\tilde{s})$. Moreover, if $\widetilde{P}'$ is another $Gl(2)$-bundle inducing via the factor map $pr$ the same $PGl(2)$-bundle $P$, we have that $\tau(\widetilde{P}',\tilde{s})$ has transition functions $t_{hr}(\eta)\cdot l_{hr}(\pi(\eta))$, where $\{ l_{hr}:V_{h}\cap V_{r}\rightarrow \mbox{\bf{C}} ^{*}\} _{h,r\in H}$ define some line bundle $L$ over $C$. We thus have the following commutative diagram: \[ \begin{array}{rcccl} & Pic(\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} }) & \stackrel{(1-\sigma^{*})}{\longrightarrow } & P(\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } ,\sigma ) & \\ \mbox{\scriptsize ${\cal T}_{Gl(2)}$} & \uparrow & & \uparrow \! \! & \mbox{\scriptsize ${\cal T}_{PGl(2)}$} \\ & {\cal H}_{Gl(2) }^{-1}(\tilde{\phi}) & \stackrel{"pr"}{\longrightarrow } & {\cal H}_{PGl(2)}^{-1}(\phi)_{0}\coprod {\cal H}_{PGl(2)}^{-1}(\phi)_{1} & \end{array} \] If we set $\Lambda '=\{ N\in Pic(\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} })\; \mid N=\sigma^{*}N\} $, we see that all sufficiently general fibres of the dominant map ${\cal T}_{PGl(2)}$ are principal homogeneous spaces with respect to $\Lambda '/\Lambda$. It is known (see \cite{mum} ) that $\Lambda '/\Lambda $ is isomorphic to $(\ze /2\ze )^{(d-1)}$, $d$ being the number of ramification points of $\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} }$ or, in this setting, the degree of $\pi^{*}K$ . Note here that the number of $\ze /2\ze$ factors reaches its maximum with respect to the estimate given in theorem \ref{it:cor2} . Since each component ${\cal H}_{PGl(2)}^{-1}(\phi)_{c}$ , $c=0,1$, is connected, we have that the generic fibre of ${\cal F}_{c}:{\cal H}_{PGl(2)}^{-1}(\phi)_{c} \rightarrow P(\mbox{$\widetilde{C} $} } \newcommand{\ze}{{\bf{Z} } ,\sigma)$ consists of $2^{(d-2)}$ points.\\[.5cm] {\em Acknowledgments.}\\ I wish to express my big debt to my advisor Corrado De Concini for sharing his ideas on the subject and my gratefulness to Vassil Kanev for discussions of crucial importance concerning the algebro-geometric aspects of the problem.\\ Also, it is a pleasure for me to thank one of the referees for his interesting remarks and his contribution in improving the paper.

9412

alg-geom/9412022

en

https://arxiv.org/abs/alg-geom/9412022

alg-geom/9412022

Amnon Neeman

Amnon Neeman

Grothendieck duality via homotopy theory

31 pages, AMS-LaTeX. To appear in Jour. AMS.

Grothendieck proved that if $f:X\longrightarrow Y$ is a proper morphism of nice schemes, then $Rf_*$ has a right adjoint, which is given as tensor product with the relative canonical bundle. The original proof was by patching local data. Deligne proved the existence of the adjoint by a global argument, and Verdier showed that this global adjoint may be computed locally. In this article we show that the existence of the adjoint is an immediate consequence of Brown's representability theorem. It follows almost as immediately, by ``smashing'' arguments, that the adjoint is given by tensor product with a dualising complex. Verdier's base change theorem is an immediate consequence.

alg-geom

\section{Introduction} \label{S0} Let $f: X \rightarrow Y$ be a proper morphism of schemes. Then, under mild hypotheses on $f, \ X$ and $Y$, Grothendieck proved that there is a natural isomorphism $$ R f_* {\cal RH}om^{}_X(x,f^! y) \simeq {\cal RH}om^{}_Y (Rf_*x,y) $$ of objects in the derived category. We should perhaps briefly remind the reader what this means. Let $D^+(qc/X)$ be the derived category of bounded--below chain complexes of quasi-coherent sheaves on $X$. Let $Rf_*: D^+(qc/X) \rightarrow D^+(qc/Y)$ be the right derived functor of $f_*$. Then the statement above asserts that $Rf_*$ has a right adjoint, denoted $f^!$, and that $f^!$ behaves well with respect to pullbacks by open immersions. Grothendieck's original proof of the existence of $f^!$ was constructive. It amounted to a local computation. Since derived categories are basically unsuited for local computations, the argument turns out to be quite unpleasant; see \cite{H}. There is an abstract way to prove the existence of $f^!$ in the literature. It is due to Deligne~\cite{D}. The approach was developed and elaborated by Verdier~\cite{V}, where it is shown that almost everything in \cite{H} can be obtained directly from Deligne's result. All these results assumed the scheme $X$ Noetherian. Lipman recently developed a deep theory for removing the Noetherian hypotheses. The reader is referred to \cite{L1} and \cite{L2}. Unfortunately, none of the approaches generalizes well to $D$--modules. Given a morphism $f: X \rightarrow Y$, one can define a morphism $Rf_+$ on the corresponding derived categories of complexes of $D$--modules. It turns out that $R f_+$ has a right adjoint. But the existence of the right adjoint has until now always been proved by factoring $f$ suitably, defining certain trace maps which $\grave{a}$ {\em priori} depend on the factorization, and finally proving that they are independent of factorizations. What we propose to do here is show that all of the results are direct consequences of a very simple, general statement about triangulated functors on triangulated categories. \bigskip \nin {\bf Definition~\ref{compactly generated}} {\em Let ${\cal S}$ be a triangulated category. Suppose that all small coproducts exist in ${\cal S}$. Suppose there exists a set $S$ of objects of ${\cal S}$ such that \begin{itemize} \item[(1)] For every $s \in S$, Hom$(s,-)$ commutes with coproducts. \item[(2)] If $y$ is an object of ${\cal S}$ and Hom$(s,y) = 0$ for all $s \in S$, then $y = 0$. \end{itemize} \nin Such a triangulated category ${\cal S}$ is called {\em compactly generated.}}\bigskip \nin The Brown Representability Theorem, in one version, states: \bigskip \noindent{\bf Theorem~\ref{adjoints}}. {\em Let ${\cal S}$ be a compactly generated triangulated category, ${\cal T}$ any triangulated category. Let $F: {\cal S} \rightarrow {\cal T}$ be a triangulated functor. Suppose $F$ respects coproducts; that is, the natural map $$ \coprod_{\lambda \in \Lambda} F(s_\lambda) \rightarrow F\left(\coprod_{\lambda \in \Lambda} s_\lambda\right) $$ is an isomorphism. Note that although we are not assuming that ${\cal T}$ has coproducts, we are assuming that the object on the right is a coproduct of the objects on the left. Then there exists a right adjoint for $F$, namely a functor $G: {\cal T} \rightarrow {\cal S}$, for which there is a natural isomorphism $$ Hom_{\cal S}\left(x,Gy\right)=Hom_{\cal T}\left(Fx,y\right). $$ } \bigskip \nin We will see that Grothendieck duality is an immediate consequence. Even the non--noetherian statements come very cheaply. Sections~\ref{S1} and \ref{S1.5} are introductory. They give the definitions and elementary properties of compactly generated categories. They also discuss why the categories one naturally wants to consider are examples. In particular, the derived category of quasi-coherent sheaves on $X$, and the derived category of quasi-coherent sheaves of $D$-modules. There is, however, a technical point. In the context of Brown representability, it is essential to have triangulated categories with direct sums. Thus we must work with {\em unbounded} derived categories. Sections~\ref{S1} and \ref{S1.5} also discuss why the natural functors one might consider, for instance $R f_*$, respect coproducts. It should definitely be noted that nothing in Section~\ref{S1} is new. Except for the terminology, the results can certainly all be found in SGA6. There is also an excellent exposition of them in Thomason's~\cite{TT}. But there are two reasons for giving a complete and self--contained exposition of these facts. One is to keep this article self--contained. But more importantly, both \cite{SGA6} and \cite{TT} are pre--Bousfield. This needs to be made precise. As the reader will easily observe by studying the dates, \cite{TT} came long after Bousfield's \cite{B1}, \cite{B2} and \cite{B3}. But as far as the author knows, it was not until \cite{BN} that it was realised that Bousfield's techniques applied to these problems. And \cite{BN} is more recent than \cite{TT}. Ever since \cite{BN}, the present author has delighted in taking every opportunity to point out that Bousfield's techniques make all the old results much clearer, more general and easier to prove. Section~\ref{S1} is to be taken in this vein. In fact, perhaps the entire article is little more than a manifestation of the above. As for Section~\ref{S1.5}, it redoes the classical theory from the new perspective offered by Thomason's localisation theorem. Classically, the proofs that $D(qc/X)$ is compactly generated relied on finding enough line bundles on $X$. It follows from Thomason's localisation theorem that one does not need line bundles. This is explained in Section~\ref{S1.5}. Again, the observation is not new, it was first made by Thomason in \cite{TT}. Section~\ref{S1.5} is the only section in this article which assumes familiarity either with Thomason's article on the subject, or with \cite{STT}. There is, for the reader's benefit, a summary of the needed results in Theorem~\ref{Thomason's localisation}. A reader willing to assume that all his/her schemes have ample line bundles can skip Section~\ref{S1.5}, except for the last page and a half which discuss homotopy colimits. There is a technical point which perhaps deserves mention. In the literature, one frequently considers some other triangulated categories. For instance, the category $D(qc/X)$ of complexes of quasi--coherent sheaves on $X$ may be replaced by $D^{}_{qc}(X)$, the category of complexes of sheaves of ${\cal O}^{}_X$ modules on $X$, with quasi--coherent cohomology. There is a natural functor $$ F:D(qc/X)\longrightarrow D^{}_{qc}(X). $$ It turns out that this functor is an equivalence of categories if $X$ is quasi--compact and separated; see \cite{BN}, Corollary 5.5. A similar statement holds for the derived categories of $D$--modules; I leave it to the reader to state and prove the analogue. We will make no explicit use of this fact. It is only mentioned to comfort the experts who might be concerned about such things. A similar statement also is valid for maps. Given a morphism $f:X \la Y$, there are induced maps $Rf_*:D(qc/X)\la D(qc/Y)$ and $Rf_*:D(X)\la D(Y)$. It is comforting to know that they agree; there is a commutative diagram $$ \begin{array}{ccc} D(qc/X) & \la & D(qc/Y) \\ \downarrow& &\downarrow\\ D(X) & \la & D(Y). \end{array} $$ It is possible to give a proof based on homotopy theoretic ideas as in \cite{BN}, but for a written proof we refer the reader to \cite{L1}, Proposition~3.9.2. Sections \ref{S2} and \ref{S3} contain the proof of the Brown Representability Theorem, Theorem~\ref{adjoints}. The proof itself is very short and simple, and although not very different from Brown's proof in \cite{B2}, we include it for the reader's convenience. The last two sections, Sections \ref{S4} and \ref{S5}, treat the same problems that are addressed by Verdier in \cite{V}. What is different here is (1) that we work with unbounded derived categories, and (2) that the argument is entirely based on the behaviour of coproducts. In Section~\ref{S4} we ask when does the right adjoint $f^!$ of $R f_*: D(qc/X) \rightarrow D(qc/Y)$ respect coproducts? It turns out that one can give a simple, satisfactory criterion, and what makes the question ``natural'' is that $f^!$ respects coproducts precisely when $$ f^!(y) \simeq Lf^*(y) \otimes f^! {\cal O}_Y. $$ In other words, $f^!$ respects coproducts when there is a dualizing complex $f^!{\cal O}_Y$, and $f^!$ is given as tensor product with $f^!{\cal O}_Y$. Note that we will always write $\otimes$ for the left derived functor $^L\otimes$. The final question we address is when the functor $f^!$ localizes well; that is, when it gives an isomorphism in $D(qc/Y)$, as in Grothendieck's original theorem. Although our treatment here is not complete, we give a useful sufficient criterion. The reader should note that Theorem 2 in Verdier~\cite{V}, page 394 seems better than the result we get here; but this is partly a delusion. Verdier's theorem is about $D^+_{qc}(X)$ whereas Proposition~\ref{the vanishing} deals with $D^{}_{qc}(X)$. The difference between bounded and unbounded derived categories is crucial here. Without boundedness hypotheses Theorem 2 in Verdier~\cite{V} fails. Another way to say this is that certain functors that come up in the proof commute with coproducts in $D^+_{qc}(X)$ but not with coproducts in $D^{}_{qc}(X)$. See the proof of Proposition~\ref{the vanishing, bounded} for a proof of Verdier's result based on coproducts (in the non--noetherian case too). See Example~\ref{counterexample} for a counterexample showing that without some conditions, one cannot expect Verdier's theorem to hold in the unbounded derived category. It should emphasised that here also, the topological techniques work without any hypothesis that the schemes be noetherian. We recover, therefore, Lipman's results. The point of this article is that the Grothendieck duality is very easy to prove by homotopy theoretic techniques. It is an immediate consequence of Brown representability. But the reader is not assumed familiar with homotopy theory; hence the first few sections, in which we attempt to give something like a self--contained treatment. The reader is not assumed familiar with much of the literature on the subject, especially not with previous articles by the present author. What is assumed is some familiarity with derived categories. Since we will be working almost entirely with unbounded derived categories, the reader should be familiar with the work of Spaltenstein \cite{S} on extending $Rf_*$ and tensor products to unbounded complexes. There is a brief account of Spaltenstein's results in Sections 1 and 2 of \cite{BN}; this account is recommended because it uses the notations and terminology of the topologists, which we will also follow here. Finally, I wish to thank Morava for directing me to the problem. I read {\em Residues and Duality} many years ago, before I learned any topology. But Morava kept suggesting that there is more there than meets the eye. There is a topological analogue one would like to understand. On rereading {\em Residues and Duality}, it became clear to me that even the algebraic geometry could be understood better. I also want to thank Alonso, Jeremias, Kuhn and Lipman for helpful conversations and useful comments. Lipman was especially helpful, reading earlier versions of this manuscript, pointing out gaps and making many suggestions. The referee also pointed out helpful simplifications and corrections. The $D$-module problem of directly constructing the right adjoint of $R f_+$ was posed by Nick Katz 9 years ago. By a happy accident, I am now able to solve it. \section{Preliminaries, approached classically} \label{S1} Let ${\cal T}$ be a triangulated category. There are several hypotheses one likes to place on ${\cal T}$ in order to work with it. \dfn{sums exist} The category ${\cal T}$ is said to {\em contain small coproducts} if, for any small set $\Lambda$ and any collection $\{t_\lambda, \: \lambda \in \Lambda\}$ of objects $t_\lambda \in \mbox{Ob}({\cal T})$ indexed by $\Lambda$, the categorical coproduct $$ \displaystyle\coprod_{\lambda \in \Lambda} t_\lambda $$ exists in ${\cal T}$.\edfn \rmk{R1.comm} It turns out that one can prove two things: \begin{description} \sthm{R1.1.1} The suspension functor commutes with coproducts; that is the natural map $$ \coprod_{\lambda \in \Lambda} \Sigma t_\lambda \la \Sigma\left\{\coprod_{\lambda \in \Lambda} t_\lambda\right\} $$ is an isomorphism.\esthm \sthm{R1.1.2} The coproduct of any set of triangles is a triangle. \esthm \end{description} The proof of this may be found in the yet--to--be--completed \cite{NV}. If the reader is unhappy with this, just modify Definition~\ref{sums exist} so that a category $\ct$ containing small coproducts is assumed to satisfy \ref{R1.1.1} and \ref{R1.1.2}. \ermk \exm{sums in $D(qc/X)$} Let $X$ be a scheme. Let $D(qc/X)$ be the derived category of chain complexes of quasi-coherent sheaves over $X$. Suppose $\{ x_\lambda |\: \lambda \in \Lambda \}$ is a set of objects of $D(qc/X)$. That is, each $x_\lambda$ is a chain complex $$ \rightarrow x_\lambda^{n-1} \stackrel{\partial}{\rightarrow} x_\lambda^n \stackrel{\partial}{\rightarrow} x_\lambda^{n+1} \rightarrow . $$ Then it is nearly trivial that the chain complex $$ \longrightarrow \displaystyle\coprod_{\lambda \in \Lambda} x_\lambda^{n-1} \stackrel{\coprod\partial}{\longrightarrow} \displaystyle\coprod_{\lambda \in \Lambda} x_\lambda^n \stackrel{\coprod\partial}{\longrightarrow} \displaystyle\coprod_{\lambda \in \Lambda} x_\lambda^{n+1} \longrightarrow $$ is the coproduct of the $x_\lambda$ in the category $D(qc/X)$. It is also nearly trivial that coproducts of triangles are triangles. For a proof, see \cite{BN}, Corollary 1.7.\eexm \lem{$Rf_*$ respects sums} Let $X$ be a quasi-compact, separated scheme and $Y$ be a scheme. Let $f: \: X \rightarrow Y$ be a separated morphism. Let $R f_*: \: D(qc/X) \rightarrow D(qc/Y)$ be the direct image functor. Then the natural map $$ \displaystyle\coprod_{\lambda \in \Lambda} R f_* x_\lambda \rightarrow R f_*\left\{\displaystyle\coprod_{\lambda \in \Lambda} \: x_\lambda\right\} $$ is an isomorphism; that is, $Rf_*$ respects coproducts.\elem \noindent{\bf Proof.} The question being local in $Y$, we may assume $Y$ affine. Since $X$ is quasi-compact, it may be covered by finitely many open affines: $X = \displaystyle\bigcup^n_{i=1} U_i$, with $U_i$ affine. We will prove the lemma by induction on the number $n$ of open affines. If $n = 1$, then $X = U_1$ is affine. Thus the map $$ \mbox{Spec}(S) = X \rightarrow Y = \mbox{Spec}(R) $$ corresponds to a ring homomorphism $R \rightarrow S$. The category $D(qc/X)$ is just $D(S)$, the derived category of chain complexes of $S$--modules. The functor $Rf_*: D(S) \rightarrow D(R)$ just takes a chain complex of $S$--modules and views it as a chain complex of $R$--modules. This clearly preserves coproducts. If $n > 1,$ let $U = U_1, \ V = \displaystyle\bigcup^{n}_{i=2} U_i$. Then $U \cap V = \displaystyle\bigcup^{n}_{i=2} (U_1 \cap U_i)$, and $U_1 \cap U_i$ is affine because $X$ is separated. Thus both $V$ and $U \cap V$ are unions of $n-1$ affines. By induction, the theorem holds for the maps $f_U: U \rightarrow Y, \ f_V:V \rightarrow Y$ and $f_{U \cap V}: U \cap V \rightarrow Y$. Let $i_U: U \hookrightarrow X, \ i_V: V \hookrightarrow X$, and $i_{U \cap V}: U \cap V \hookrightarrow X$ be the open immersions. Then any object $x$ of $D(qc/X)$ admits a triangle \begin{picture}(400,100) \put(150,10) {$R(i^{}_{U \cap V})_* i_{U \cap V}^* (x)$} \put(180,70) {$\longrightarrow$} \put(120,70) {$x$} \put(220,70) {$R(i^{}_U)_* i^*_U x \oplus R(i^{}_V)_* i^*_V(x)$} \put(125,34) {$\scriptstyle{(1)}$} \put(135,40) {$\nwarrow$} \put(230,40) {$\swarrow$} \end{picture} \nin Thus we deduce a triangle \begin{picture}(400,100) \put(140,10) {$Rf_* R(i^{}_{U \cap V})_* i_{U \cap V}^*(x)$} \put(180,70) {$\longrightarrow$} \put(110,70) {$Rf_*(x)$} \put(230,70) {$Rf_*R(i^{}_U)_*i^*_U(x) \oplus Rf_* R(i^{}_V)_*i^*_V(x)$} \put(125,34) {$\scriptstyle{(1)}$} \put(135,40) {$\nwarrow$} \put(230,40) {$\swarrow$} \end{picture} \noindent But now \begin{eqnarray*} Rf_* R(i^{}_U)_* & = & R(f^{}_U)_* \\ Rf_* R(i^{}_V)_* & = & R(f^{}_V)_* \\ \mbox{and} \rule{1cm}{0cm} Rf_* R(i^{}_{U \cap V}) & = & R(f^{}_{U \cap V})_* \rule{1cm}{0cm} \end{eqnarray*} all commute with coproducts by the induction hypothesis. The functors $i^*_U, \ i^*_V$, and $i^*_{U \cap V}$ commute with coproducts because they have right adjoints. Therefore, in the morphism of triangles $$ \begin{array}{ccccc} \displaystyle\coprod_{\lambda \in \Lambda} Rf_*(x_\lambda) & \rightarrow & \displaystyle\coprod_{\lambda \in \Lambda} [R(f^{}_U)_* i^*_U(x_\lambda) \oplus R(f^{}_V)_* i^*_V(x_\lambda)] & \rightarrow & \displaystyle\coprod_{\lambda \in \Lambda} R(f^{}_{U \cap V})_* i^*_{U \cap V} (x_\lambda) \\ \alpha \downarrow && \beta \downarrow && \gamma \downarrow \\ Rf_*\left(\displaystyle\coprod_{\lambda \in \Lambda} x_\lambda\right) & \rightarrow & R(f^{}_U)_* i^*_U \left( \displaystyle\coprod_{\lambda \in \Lambda} x_\lambda\right) \oplus R(f^{}_V)_* i^*_V \left(\displaystyle\coprod_{\lambda \in \Lambda} x_\lambda\right) & \rightarrow & R(f^{}_{U \cap V})_* i^*_{U \cap V} \left(\displaystyle\coprod_{\lambda \in \Lambda} x_\lambda\right) \end{array} $$ the maps $\beta$ and $\gamma$ are isomorphisms; hence so is $\alpha$. \hfill $\Box$ \bigskip \cor{sums} If in Lemma~\ref{$Rf_*$ respects sums} we take $Y$ to be the scheme Spec$({\Bbb Z})$, then $Rf_*$ is just the derived functor of the global section functor. We deduce that if $X$ is a quasi-compact, separated scheme then the functors $H^i$ $$ H^i(X,-): D(qc/X) \rightarrow \{\mbox{abelian groups}\} $$ respect coproducts.\ecor \dfn{compact objects} An object $c$ of ${\cal T}$ is called {\em compact} if, for any coproduct of objects of ${\cal T}$ $$ \mbox{Hom}_{\cal T} \left(c, \displaystyle\coprod_{\lambda \in \Lambda} t_\lambda \right) = \displaystyle\coprod_{\lambda \in \Lambda} \mbox{Hom}_{\cal T} (c, t_\lambda). $$ \edfn \medskip \noindent{\bf Observation.} The suspension of a compact object is compact. \dfn{compactly generated} The triangulated category ${\cal T}$ is called {\em compactly generated} if ${\cal T}$ contains small coproducts, and there exists a small set $T$ of compact objects of ${\cal T}$, such that $$ \mbox{Hom} (T,x) = 0 \Rightarrow x = 0. $$ In other words, if $x$ is an object of ${\cal T}$, and for every $t \in T$, Hom$(t,x) = 0$, then $x$ must be the zero object.\edfn \dfn{generating set} If ${\cal T}$ is a compactly generated triangulated category, then a set $T$ of compact objects of ${\cal T}$ is called a {\em generating set} if \begin{itemize} \item[(1)] Hom$(T,x) = 0 \Rightarrow x = 0;$ \item[(2)] $T$ is closed under suspension; $T = \Sigma T$. \end{itemize}\edfn \rmk{2 is easy} Let $T$ be any set of objects in ${\cal T}$ as in Definition~\ref{compactly generated}. Then $$ \displaystyle\bigcup_{i \in {\Bbb Z}} \Sigma^i T $$ is a generating set as in Definition~\ref{generating set}. (1) holds because it holds for $T$; and clearly the set $$ \displaystyle\bigcup_{i \in {\Bbb Z}} \Sigma^iT $$ is stable under suspension. It needs only be remarked that any suspension of a compact object is compact, and hence the set consists only of compact objects of ${\cal T}$.\ermk \exm{quasi-projective variety} Let $X$ be a quasi-compact, separated scheme. Let ${\cal T} = D(qc/X)$ be the category of chain complexes of quasi-coherent sheaves on $X$. By Example~\ref{sums in $D(qc/X)$} we know that ${\cal T}$ contains small coproducts. Now let ${\cal L}$ be any line bundle on $X$. View ${\cal L}$ as an object of $D(qc/X)$; it is the complex of sheaves which is just ${\cal L}$ in degree 0. Then $$ \mbox{Hom}\left({\cal L}, \displaystyle\coprod_{\lambda \in \Lambda} t_\lambda\right) = H^0 \left({\cal L}^{-1} \otimes \displaystyle\coprod_{\lambda \in \Lambda} t_\lambda\right). $$ Now tensor product respect coproducts, since it has a right adjoint. The functor $H^0$ respects coproducts by Corollary~\ref{sums}. It follows that ${\cal L}$ is a compact object of ${\cal T} = D(qc/X)$. Now suppose ${\cal L}$ is an ample line bundle. For any $m \in {\Bbb Z}, \ {\cal L}^m$ is compact. For any $n \in {\Bbb Z}, \ \Sigma^n {\cal L}^m$ is also compact. Let $$ T = \{\Sigma^n {\cal L}^m| \: m,n \in {\Bbb Z}\}. $$ I assert that $T$ is a generating set for ${\cal T}$. Suppose $x \neq 0$ is an object of ${\cal T}$. Then it has some non-trivial sheaf cohomology; for some $n, \ {\cal H}^{-n}(x) \neq 0$. But ${\cal H}^{-n}(x)$ is a quasi-coherent sheaf on $X$, and because ${\cal L}$ is ample, ${\cal L}^t \otimes {\cal H}^{-n}(x)$ has non-zero global sections for some $t >> 0$. If $x$ is the complex $$ \rightarrow x^{-n-1} \stackrel{\partial}{\rightarrow} x^{-n} \stackrel{\partial} {\rightarrow} x^{-n+1} \rightarrow, $$ then there is a surjective map of quasi-coherent sheaves on $X$ $$ \mbox{ker}(x^{-n} \rightarrow x^{-n+1}) \rightarrow {\cal H}^{-n}(x), $$ and it follows that, choosing $t$ large enough, we can find a class $$ s \in H^0({\cal L}^t \otimes x^{-n}) $$ which maps to 0 in $H^0({\cal L}^t \otimes x^{-n+1})$, and whose image in $H^0({\cal H}^{-n}({\cal L}^t \otimes x))$ is non-zero. This immediately gives us a non-zero map $$ \Sigma^n {\cal L}^{-t} \rightarrow x. $$ If all such maps vanish, so must $x$.\eexm \exm{families of line bundles} In Example~\ref{quasi-projective variety} we can replace a single ample line bundle by a family; if $X$ admits a family of line bundles $\{{\cal L}_\alpha | \: \alpha \in A\}$ which is jointly ample, then $$ T = \{\Sigma^n {\cal L}^m_\alpha | \: m,n \in {\Bbb Z}, \: \alpha \in A\} $$ will do for a generating set of compact objects. See \cite{SGA6}, page 171.\eexm \exm{$D$-modules} Let $X$ be a smooth quasi-compact variety of finite type over a field $k$ of characteristic zero. Let ${\cal T} = D\left(\frac{qc\: D\mbox{-modules}}{X}\right)$ be the derived category of chain complexes of quasi-coherent $D$-modules over $X$. Once again, it is trivial that ${\cal T}$ contains small coproducts. Because $X$ is smooth, one knows by a trick of Kleiman that there is an ample family of line bundles; cover $X$ by open affines $X = \displaystyle \bigcup_{i\in I} U_i$ and let ${\cal L}_i$ be the line bundle ${\cal O}(D_i)$, where $D_i$ is the divisor $X-U_i$. Then the ${\cal L}_i$'s form an ample family. Now observe $$ \mbox{Hom}_{D\left(\frac{qc\: D\mbox{-modules}}{X}\right)} \left({\cal D}^{}_X \otimes_{{\cal O}^{}_X} {\cal L}_i^m, x\right) = \mbox{Hom}_{D\left(qc/X\right)} \left({\cal L}_i^m,x\right). $$ \nin From this it follows that $\Sigma^n\left({\cal D}^{}_X \otimes_{{\cal O}^{}_X} {\cal L}_i^m\right)$ are compact for all $i, m$ and $n$, and the set $$ T = \left\{\Sigma^n \left({\cal D}^{}_X \otimes_{{\cal O}^{}_X} {\cal L}_i^m\right)\left| \: i\in I,\quad m,n \in {\Bbb Z}\right.\right\} $$ is a generating set.\eexm \exm{other compacts in $D(qc/X)$} Let $X$ be a quasi-compact, separated scheme. In Example~\ref{quasi-projective variety} we proved that any line bundle ${\cal L}$ on $X$, viewed as an object of $D\left(qc/X\right)$, is compact. Given ample families of line bundles on $X$, we used this to construct a generating set. Let us now observe that if $c \in D\left(qc/X\right)$ is any perfect complex on $X$, it is compact. Recall that a complex $c$ is perfect if, locally on $X$, it is isomorphic to a bounded complex of finitely generated, projective ${\cal O}^{}_X$-modules.\eexm \noindent{\bf Proof.} Let $\displaystyle\coprod_{\lambda \in \Lambda} \: x_\lambda$ be a coproduct in $D\left(qc/X\right)$. Let $$ {\cal RH}om\left(c, \displaystyle\coprod_{\lambda \in \Lambda} x_\lambda\right) $$ be the sheaf ${\cal RH}om$; it is an object of $D\left(qc/X\right)$. I assert that the natural map in $D\left(qc/X\right)$ $$ \phi_c: \: \coprod_{\lambda \in \Lambda} {\cal RH}om (c, x_\lambda) \rightarrow {\cal RH}om\left(c, \coprod_{\lambda \in \Lambda} x_\lambda\right) $$ is an isomorphism whenever $c$ is a perfect complex. The problem is local in $X$; we may therefore assume that $X$ is affine, and $c$ is a bounded complex of finitely generated projective ${\cal O}_X$-modules. If $c$ is $\Sigma^m{\cal O}_X$, then $\phi_c$ is clearly an isomorphism. If $c$ is $\Sigma^m{\cal O}^n_X$, a finite direct sum of $\Sigma^m {\cal O}_X$'s, then $\phi_c$ is also an isomorphism. If $c = c' \oplus c''$ and $\phi_c$ is an isomorphism, so are $\phi_{c'}$ and $\phi_{c''}$. Hence $\phi_c$ is an isomorphism whenever $c$ is a suspension of a finitely generated projective module. Now if we have a triangle \begin{picture}(400,100) \put(182,10) {$c''$} \put(180,70) {$\longrightarrow$} \put(135,70) {$c$} \put(230,70) {$c'$} \put(145,34) {$\scriptstyle{(1)}$} \put(150,40) {$\nwarrow$} \put(210,40) {$\swarrow$} \end{picture} \noindent and $\phi_{\Sigma^mc'}$ and $\phi_{\Sigma^mc''}$ are isomorphisms for all $m \in {\Bbb Z}$, then it follows from the 5-lemma that $\phi_{\Sigma^mc}$ is an isomorphism for all $m \in {\Bbb Z}$. Therefore the full subcategory of $c$'s such that $\phi_{\Sigma^mc}$ is an isomorphism for all $m \in {\Bbb Z}$ is triangulated and contains the finitely generated projective ${\cal O}_X$-modules. Hence it contains finite complexes of finitely generated projectives. Thus we have proved that for $c$ a perfect complex, $$ \phi_c : \ \displaystyle\coprod_{\lambda \in \Lambda} {\cal RH}om (c,x_\lambda) \rightarrow {\cal RH}om\left(c, \displaystyle\coprod_{\lambda \in \Lambda} x_\lambda\right) $$ is an isomorphism. But \begin{eqnarray*} \mbox{Hom}\left(c, \displaystyle\coprod_{\lambda \in \Lambda} x_\lambda\right) & = & H^0\left[{\cal RH}om\left(c, \coprod_{\lambda \in \Lambda} x_\lambda \right)\right] \\ & = & H^0\left[\coprod_{\lambda \in \Lambda}{\cal RH}om\left(c, x_\lambda \right)\right] \\ & = & \coprod_{\lambda \in \Lambda}H^0\left[{\cal RH}om\left(c, x_\lambda \right)\right] \\ & = & \coprod_{\lambda \in \Lambda}\mbox{Hom}\left(c,x_\lambda\right), \end{eqnarray*} where the third equality is by Corollary~\ref{sums}, which assures us that $H^0$ respects coproducts. \hfill $\Box$ \bigskip \exm{other compacts in $D(qc D-modules/X)$} Let $X$ be a quasi-compact, separated smooth scheme of finite type over a field $k$ of characteristic 0. Then by Example~\ref{$D$-modules} we know that objects of the form $\Sigma^n\{{\cal D}_X \otimes_{{\cal O}_{X}} {\cal L}\}$ are compact in ${\cal T} = D\left(\frac{qc\: D\mbox{-modules}}{X}\right)$. But much as in Example~\ref{other compacts in $D(qc/X)$}, it can be shown that any bounded complex with coherent cohomology is compact. Note that because ${\cal D}_X$ is locally of finite projective dimension, any coherent sheaf can locally be replaced by a finite chain complex of finitely generated projectives. We leave the details to the reader.\eexm \section{The approach using Thomason's localisation theorem} \label{S1.5} Let $\cal T$ be a triangulated category. In this section, and for the remainder of the article, we adopt the notation that ${\cal T}^c$ stands for the full subcategory of compact objects in $\cal T$. Let $X$ be a quasi--compact, separated scheme. In Section~\ref{S1} we saw how to prove that the category $D(qc/X)$ is compactly generated, provided $X$ has an ample family of line bundles; see Example~\ref{families of line bundles}. On the other hand, we also know that if $X$ is arbitrary (that is, quasi--compact and separated but not necessarily possessing any line bundles), every perfect complex on $X$ is compact. In this section we will use Thomason's localisation theorem to prove that $D(qc/X)$ is compactly generated, even without line bundles. First, we quote the theorem to which we will appeal \thm{Thomason's localisation} Let $\cal S$ be a compactly generated triangulated category. Let $R$ be a set of compact objects of $\cal S$ closed under suspension. Let $\cal R$ be the smallest full subcategory of $\cal S$ containing $R$ and closed with respect to coproducts and triangles. Let $\cal T$ be the Verdier quotient ${\cal S}/{\cal R}$. Then we know: \sthm{Thom1} The category $\cal R$ is compactly generated, with $R$ as a generating set.\esthm \sthm{Thom1.5} If $R$ happens to be a generating set for all of $\cal S$, then ${\cal R}={\cal S}$.\esthm \sthm{Thom2} If $R\subset\cal R$ is closed under the formation of triangles and direct summands, then it is all of ${\cal R}^c$. In any case, ${\cal R}^c={\cal R}\cap{\cal S}^c$. \esthm \sthm{Thom3} Suppose $t$ is a compact object of $\cal T$. Then there is an object $t'\in{\cal T}^c$ and an object $s\in {\cal S}^c$ and an isomorphism in $\cal T$ $$ s\simeq t\oplus t'. $$ Thus, $t$ might not be the isomorphic in $\cal T$ to a compact object in $\cal S$, but it is the direct summand of an object isomorphic in $\cal T$ to a compact object of $\cal S$. Furthermore, $t'$ may be chosen to be $\Sigma t$, or any other object whose sum with $t$ is zero in $K_0$.\esthm \sthm{Thom4} Given an object $s\in {\cal S}^c$, an object $s'\in\cal S$, and a morphism in $\cal T$ $s\la s'$, then there is a diagram in $\cal S$ $$ \begin{array}{ccccccccc} & & \tilde s & & \\ &\swarrow& &\searrow& \\ s & & & & s' \end{array} $$ where $\tilde s$ is compact, in the triangle $r\la \tilde s\la s\la \Sigma r$, the object $r$ is in ${\cal R}^c$, and when we reduce the diagram to $\cal T$, the composite of the map $\tilde s\la s'$ with the inverse of $\tilde s\la s$ is the given map $s\la s'$.\esthm \ethm \rmk{The proofs} For triangulated categories $D(qc/X)$ where $X$ is a scheme, the theorem is due to Thomason \cite{TT}. In the generality in which the theorem is stated, it may be found in \cite{STT}. Note that in \cite{STT} we assume not only that $\cal S$ is compactly generated, but that the generating set may be taken to be ${\cal S}^c$; in particular, we assume ${\cal S}^c$ to be small. The reader will note that this is inessential to any of the arguments in \cite{STT}. The only point where the proof uses the hypothesis on the smallness of ${\cal S}^c$ is in showing \ref{Thom3}, and that comes at the very end. So in any case we know the other properties. And in the proof of \ref{Thom3}, it suffices to know that $\cal S$ is compactly generated, and that by \ref{Thom1.5} it then follows that if $S$ is a generating set, the category ${\cal R}\subset{\cal S}$ which is the smallest category containing $S$ and closed with respect to triangles and coproducts is all of $\cal S$. With this comment, we now give the references for the proofs. \ref{Thom1.5} really follows from the proof in \cite{STT}. In the notation there, if $j^*:{\cal S}\la{\cal T}$ is the natural functor and $j_*:{\cal T}\la{\cal S}$ its right adjoint, then the objects $j_*j^*x$ are characterised by the fact that $Hom(R,j_*j^*x)=0$. Since $R$ generates $\cal S$, it follows that for every $x$, $j_*j^*x=0$. But the identity on $j^*x$ factors as $$ j^*x\la j^*j_*j^*x\la j^*x $$ and the middle object is zero. Hence $j^*x=0$. This is true for every $x$, which means that the category $\cal T$ is zero, and hence ${\cal R}={\cal S}$. For the remaining statements, \ref{Thom2} is Lemma~2.2 of \cite{STT}. \ref{Thom3} goes as follows. The existence of $t'$ is Lemma~2.6 of \cite{STT}, but as noted above, the proof needs to be modified slightly to account for the fact that we are only assuming $\cal S$ compactly generated. The fact that $t'$ may be taken to be $\Sigma t$, or even anything else which is isomorphic to $-t$ in $K_0$, may be found in the appendix to \cite{STT}. \ref{Thom4} is Lemma~2.5 of \cite{STT}. This leaves us with \ref{Thom1}, which is trivial, so let us give the proof. Suppose $r$ is an object of $\cal R$ such that $Hom(R,r)=0$. Consider the full subcategory $^\perp r\subset\cal R$, defined by $$ ^\perp r=\left\{x\in {\cal R}|Hom(\Sigma^n x,r)=0 \hbox{\rm\ for all\ } n\in{\Bbb Z} \right\}. $$ Clearly, $^\perp r$ is triangulated and closed under coproducts, and contains $R$. Hence it must be all of $\cal R$, in particular $^\perp r$ contains $r\in\cal R$. Thus $Hom(r,r)=0$, hence $r=0$. \ermk \cor{all the compacts are perf} Let $X$ be a quasi--compact, separated scheme, and suppose we know that $D(qc/X)$ is compactly generated, and that the generating set consists of some perfect complexes. Then the category of all perfect complexes on $X$ is nothing other than $D(qc/X)^c$.\ecor \rmk{for now} For now, we only know that $D(qc/X)$ is compactly generated when there is an ample family of line bundles. So for now Corollary~\ref{all the compacts are perf} only applies in that case. But this will change by the end of the section.\ermk \nin {\bf Proof of Corollary~\ref{all the compacts are perf}.}\ \ Let ${\cal S}=D(qc/X)$, and let $R$ be the set of perfect complexes in Theorem~\ref{Thomason's localisation}. By hypothesis, $R$ generates $\cal S$, and hence by \ref{Thom1.5}, ${\cal R}={\cal S}=D(qc/X)$, and $\cal T$ is trivial. But $R$ is closed with respect to direct summands and triangles. Closure with respect to triangles is clear. Closure with respect to direct summands asserts that a direct summand of a perfect complex is perfect. This is local, so we may assume $X$ affine. For affine $X$ this is Proposition~3.4 in \cite{BN}. By \ref{Thom2} we therefore know that $R={\cal R}^c$; every compact object in $D(qc/X)$ is a perfect complex.\qqed Now we come to the existence of compacts on a general $X$. \pro{compacts exist in general} Let $X$ be a quasi--compact, separated scheme. Then the category $D(qc/X)$ is compactly generated.\epro \nin It might be useful to state a lemma that clearly implies Proposition~\ref{compacts exist in general}, and which we will prove. \lem{extending maps, special case} Let $X$ be a quasi--compact, separated scheme. Let $U\subset X$ be a quasi--compact, open subscheme. Let $x$ be an arbitrary object of $D(qc/X)$, and let $u$ be a perfect complex in $D(qc/U)$. Suppose that we are given a map in $D(qc/U)$ of the form $u\la x$. Then there exists a perfect complex $u'$ in $D(qc/U)$ so that the map $$ u\oplus u'\stackrel{\pi_1}\la u\la x $$ lifts to $D(qc/X)$. There exists a perfect complex $\tilde u$ on $X$, restricting to $u\oplus u'$ on $U$, and a map $\tilde u\la x$, defined on $X$, which restricts on $U$ to the given map $u\oplus u'\stackrel{\pi_1}\la u\la x$.\elem \nin {\bf Proof that Lemma~\ref{extending maps, special case} implies Proposition~\ref{compacts exist in general}.}\ \ Take $U$ to be an open affine. Then $U$ admits an ample family of line bundles; the trivial bundle is ample. Thus we already know that $U$ is compactly generated, and that the compact objects are the perfect complexes. If $x$ is any object of $D(qc/X)$ and the restriction of $x$ to $U$ is non-zero, there is a perfect complex $u$ on $U$ and a non--zero map $u\la x$ on $U$. By Lemma~\ref{extending maps, special case} this map can be extended to a non--zero map $\tilde u\la x$ on all of $X$, where $\tilde u$ is perfect. Thus, unless the restriction of $x$ to every open affine $U\subset X$ is zero, there is a non--zero map from a perfect complex to $x$. But if the restriction of $x$ to every open affine $U\subset X$ vanishes, then $x$ vanishes.\qqed \nin {\bf Proof of Lemma~\ref{extending maps, special case}.}\ \ Suppose first that $X$ is affine. Then $X$ has an ample line bundle; after all, the trivial bundle is ample. We therefore already know that $D(qc/X)$ is compactly generated. Furthermore, the compacts are precisely the perfect complexes. Now let $D_{X-U}(qc/X)\subset D(qc/X)$ be the full subcategory whose objects are complexes supported on $X-U$. That is, $$ D_{X-U}(qc/X)=\left\{x\in D(qc/X)|\hbox{the restriction of $x$ to $U$ is acyclic.}\right\} $$ Lemma 6.1 of \cite{BN} shows that even $D_{X-U}(qc/X)$ is compactly generated, in fact, generated by the suspensions of one compact object in $D(qc/X)$. In Theorem~\ref{Thomason's localisation}, let $\cal S$ be the category $D(qc/X)$, and let $R$ be a generating set for $D_{X-U}(qc/X)$. This makes ${\cal R}=D_{X-U}(qc/X)$. Now $\cal T$ is easily identified with $D(qc/U)$. Thomason localisation theorem applies, and we discover first that by \ref{Thom3} the complex $u\oplus\Sigma u$ may be lifted to a perfect complex $\tilde u$ in $D(qc/X)$, and then by \ref{Thom4} that the map $u\oplus \Sigma u\la x$ can be lifted to a map $\tilde u\la x$ on all of $X$, possibly after changing the choice of $\tilde u$ lifting $u\oplus \Sigma u$. Suppose next that $X=U\cup W$, where $W$ is affine. We know by the above that the restriction of the map $u\la x$ can be extended from $U\cap W$ to all of $W$. Precisely, there is a perfect complex $\tilde u$ on $W$ and a map $\tilde u\la x$ of complexes on $W$, so that the restriction to $U\cap W$ is isomorphic to the map $u\oplus\Sigma u\la x$. Thus, if $j^{}_W:W\la X$, $j^{}_U:U\la X$ and $j^{}_{U\cap W}:U\cap W\la X$ are the open immersions, we have an isomorphism on $U\cap W$ of $\{j^{}_{U\cap W}\}^*\{u\oplus\Sigma u\}$ with $\{j^{}_{U\cap W}\}^*\tilde u$. We are given the maps $\beta$ and $\gamma$, which we can complete to a morphism of triangles of complexes on $U\cup W=X$ $$ \begin{array}{ccccc} \hat u & \rightarrow & \{j^{}_U\}_*\{u\oplus\Sigma u\}\oplus\{j^{}_W\}_*\tilde u & \rightarrow & \{j^{}_{U\cap W}\}_*\{j^{}_{U\cap W}\}^*\{u\oplus\Sigma u\}\\ \alpha \downarrow && \beta \downarrow && \gamma \downarrow \\ x & \rightarrow & \{j^{}_U\}_*\{j^{}_{U}\}^*x\oplus\{j^{}_W\}_*\{j^{}_{W}\}^*x & \rightarrow & \{j^{}_{U\cap W}\}_*\{j^{}_{U\cap W}\}^*x \end{array} $$ and it is easy to check that $\hat u$ is perfect and the map $\hat u\la x$, defined on all of $X$, is just a lifting of $u\oplus\Sigma u\la x$ from $U$ to all of $X$. Since $X$ is quasi--compact, it can be covered by finitely many open affines, and in finitely many steps of extending from $U$ to $U\cup W$, with $W$ affine, we extend to all of $X$.\qqed Theorem~\ref{Thomason's localisation} can also be used to construct compactly generated categories. The point being that given a compactly generated category $\cal S$ and a set of compact objects $R$ in it, the categories $\cal R$ and $\cal T$ are compactly generated. \exm{holo} Let $X$ be a smooth, quasi-compact, separated scheme of finite type over a field $k$ of characteristic 0. Let ${\cal S} =D\left(\frac{qc\: D\mbox{-modules}}{X}\right) $ be the derived category of chain complexes of quasi-coherent $D$-modules on $X$. Let $R \subset {\cal S}$ be the set $$ R = \left\{x \in \mbox{Ob}({\cal S})\left| \begin{array}{l} {\cal H}^i(x) = 0 \mbox{ for all but finitely many } i, \mbox{ and} \\ {\cal H}^i(x) \mbox{ is holonomic for all } i. \end{array} \right.\right\} $$ Because holonomic modules are coherent, it follows from Example~\ref{other compacts in $D(qc D-modules/X)$} that $R$ consists of compact objects of ${\cal T}$. Let ${\cal R}$ be as above. Then ${\cal R}$ is a compactly generated triangulated category, with $R$ for a generating set. We will call this ${\cal R}$ by the name $D(\mbox{holo}/X)$.\eexm The key tool one uses is the homotopy colimit. Let ${\cal T}$ be a triangulated category containing small coproducts. Let $$ X_0 \rightarrow X_1 \rightarrow X_2 \rightarrow \cdots $$ be a sequence of objects and morphisms in ${\cal T}$. Then $\begin{array}[t]{c} \mbox{hocolim} \\ \vspace{-.75cm} \\ \longrightarrow \end{array} X_i$ is by definition the third edge of the triangle $$ \begin{array}{rcccc} \displaystyle\coprod_i X_i & & \begin{array}{c} 1{\rm -shift} \\ \vspace{-.75cm} \\ \longrightarrow \end{array} && \displaystyle\coprod_i X_i \\ & {\scriptstyle{(1)}} \nwarrow && \swarrow & \\ && \begin{array}{c} {\rm hocolim} \\ \vspace{-.75cm} \\ \longrightarrow \end{array} X_i && \end{array} $$ \bigskip \lem{maps to colimits} Suppose $c$ is a compact object of ${\cal T}$, and suppose $$ X_0 \rightarrow X_1 \rightarrow X_2 \rightarrow \cdots $$ is a sequence of objects and morphisms in ${\cal T}$. Suppose ${\cal T}$ admits small coproducts. Then $$ \mbox{Hom}\left(c, \begin{array}[t]{c} {\rm hocolim} \\ \vspace{-.75cm} \\ \longrightarrow \end{array} X_i \right) = \displaystyle\lim_{\rightarrow} \mbox{ Hom}(c,X_i). $$\elem \noindent{\bf Proof.} Consider the triangle $$ \begin{array}{rcccc} \displaystyle\coprod_i X_i & & \begin{array}{c} 1{\rm -shift} \\ \vspace{-.75cm} \\ \longrightarrow \end{array} && \displaystyle\coprod_i X_i \\ & {\scriptstyle{(1)}} \nwarrow && \swarrow & \\ && \begin{array}{c} {\rm hocolim} \\ \vspace{-.75cm} \\ \longrightarrow \end{array} X_i && \end{array} $$ Applying the homological functor Hom$(c, -)$ we get a long exact sequence. In particular $$ \begin{array}{ccccc} \mbox{Hom}\left(c, \displaystyle\coprod_i X_i\right) & \stackrel{\gamma}{\longrightarrow} & \mbox{Hom}\left(c, \mbox{hocolim} X_i\right) & \longrightarrow & \mbox{Hom}\left(c, \displaystyle\coprod_i \Sigma X_i\right) \\ &&&& \rule{0cm}{.01cm} \\ &&&& \downarrow 1\mbox{-shift} \\ &&&& \rule{0cm}{.01cm} \\ &&&& \mbox{Hom}\left(c, \displaystyle\coprod_i \Sigma X_i\right) \end{array} $$ is exact. But $c$ is compact, and hence in the following commutative square the columns are isomorphisms $$ \begin{array}{ccc} \displaystyle\coprod_i \mbox{Hom}\left(c, \Sigma X_i\right) & \stackrel{1\mbox{-shift}}{\longrightarrow} & \displaystyle\coprod_i \mbox{Hom}\left(c, \Sigma X_i\right) \\ && \rule{0cm}{.01cm} \\ |\downarrow \wr && |\downarrow \wr \\ && \rule{0cm}{.01cm} \\ \mbox{Hom}\left(c, \displaystyle\coprod_i\Sigma X_i\right) & \stackrel{1\mbox{-shift}}{\longrightarrow} & \mbox{Hom}\left(c, \displaystyle\coprod_i\Sigma X_i\right). \end{array} $$ But the top row is clearly injective. Hence the bottom row is injective, and we deduce that $\gamma$ is surjective. We now have a commutative diagram $$ \begin{array}{ccccccc} \displaystyle\coprod_i\mbox{Hom}\left(c, X_i\right) & \stackrel{1\mbox{-shift}}{\longrightarrow} & \displaystyle\coprod_i \mbox{Hom}\left(c, X_i\right) &&&& \\ &&&& \rule{0cm}{.01cm} &&\\ |\downarrow \wr && |\downarrow \wr && &&\\ &&&& \rule{0cm}{.01cm} &&\\ \mbox{Hom}\left(c, \displaystyle\coprod_iX_i\right) & \stackrel{1\mbox{-shift}}{\longrightarrow} & \mbox{Hom}\left(c, \displaystyle\coprod_i X_i\right) & \stackrel{\gamma}{\longrightarrow} & \mbox{Hom}\left(c, \begin{array}{c} \mbox{hocolim} \\ \vspace{-.75cm} \\ \longrightarrow \end{array} X_i \right) &\longrightarrow&0\\ \end{array} $$ where the bottom row is exact. The top row identifies Hom$\left(c, \begin{array}{c} \mbox{hocolim} \\ \vspace{-.75cm} \\ \longrightarrow \end{array} X_i \right)$ as $\displaystyle\lim_\rightarrow $ Hom$(c,X_i)$. \hfill $\Box$ \section{Brown Representability} \label{S2} In this section, we will prove: \bigskip \thm{Brown representability} Let ${\cal T}$ be a compactly generated triangulated category. Let $H: {\cal T}^{op} \rightarrow Ab$ be a homological functor. That is, $H$ is contravariant and takes triangles to long exact sequences. Suppose the natural map $$ H\left(\displaystyle\coprod_{\lambda \in \Lambda} t_\lambda\right) \rightarrow \displaystyle\prod_{\lambda \in \Lambda} H(t_\lambda) $$ is an isomorphism for all small coproducts in ${\cal T}$. Then $H$ is representable.\ethm \noindent{\bf Proof.} Let $T$ be a generating set for ${\cal T}$. Let $U_0$ be defined as $$ U_0 = \displaystyle\bigcup_{t \in T} H(t). $$ Thus elements of $U_0$ can be thought of as pairs $(\alpha,t)$ with $\alpha \in H(t)$. Put $$ X_0 = \displaystyle\coprod_{(\alpha,t) \in U_{0}} t. $$ Then $$ H(X_0) = \displaystyle\prod_{(\alpha,t) \in U_{0}} H(t), $$ and there is an obvious element in $H(X_0)$, namely the element which is $\alpha \in H(t)$ for $(\alpha,t) \in U_0$. Call this element $\alpha_0 \in H(X_0)$. The construction is such that if $t \rightarrow X_0$ is the inclusion of $t$ into $X_0 = \displaystyle\coprod_{(\alpha,t) \in U_{0}} t$ corresponding to $(\alpha, t) \in U_0$, then the induced map $H(X_0) \rightarrow H(t)$ takes $\alpha_0 \in H(X_0)$ to $\alpha \in H(t)$. To give an object $X_0$ and an element $\alpha_0 \in H(X_0)$ is by Yoneda's lemma the same as giving a natural transformation $$ \phi_0: \mbox{ Hom}(-, X_0) \rightarrow H, $$ and what we have seen is precisely that $$ \phi_0(t): \mbox{ Hom}(t,X_0) \rightarrow H(t) $$ is surjective, for all $t \in T$. Suppose that for some $i \geq 0$ we have defined an object $X_i$ of ${\cal T}$, and a natural transformation $$ \mbox{Hom}(-,X_i) \rightarrow H. $$ Define $U_{i+1}$ by $$ U_{i+1} = \displaystyle\bigcup_{t \in T} \ker\left\{ \mbox{Hom}(t,X_i) \rightarrow H(t)\right\}. $$ An element of $U_{i+1}$ can be thought of as a pair $(f,t)$ where $t \in T$ and $f: t \rightarrow X_i$ is a morphism. Put $$ K_{i+1} = \displaystyle\coprod_{(f,t) \in U_{i+1}} t, $$ and let $K_{i+1} \rightarrow X_i$ be the map which is $f$ on the factor $t$ corresponding to the pair $(f,t)$. Let $X_{i+1}$ be given by the triangle $$ \begin{array}{rcccc} K_{i+1} && \longrightarrow && X_i \\ & {\scriptstyle{(1)}} \nwarrow && \swarrow & \\ && X_{i+1} && \end{array} $$ We have a map Hom$(-,X_i) \rightarrow H$, which by Yoneda's lemma corresponds to an element $\alpha_i \in H(X_i)$. Under the map \begin{eqnarray*} H(X_i) \rightarrow H(K_{i+1}) & = & H\left(\displaystyle\coprod_{(f,t) \in U_{i+1}} t\right) \\ & = & \displaystyle\prod_{(f,t) \in U_{i+1}} H(t) \end{eqnarray*} the element $\alpha_i \in H(X_i)$ maps to zero; the $f: t \rightarrow X_i$ were chosen so that the induced map Hom$(t,X_i) \rightarrow H(t)$ vanishes. But $H$ is a homological functor; the exact sequence $$ H(X_{i+1}) \stackrel{k}{\rightarrow} H(X_i) \stackrel{j}{\rightarrow} H(K_{i+1}) $$ coupled with the fact that $j(\alpha_i) =0$, guarantees that there exists $\alpha_{i+1} \in H(X_{i+1})$ with $k(\alpha_{i+1}) = \alpha_i$. Choose such an $\alpha_{i+1}$. There is a corresponding natural transformation $$ \mbox{Hom}(-,X_{i+1}) \rightarrow H $$ rendering commutative the triangle \begin{picture}(400,100) \put(110,10) {$\mbox{Hom}(-,X_{i+1})$} \put(200,10) {$\longrightarrow$} \put(260,10) {$H$} \put(180,70) {$\mbox{Hom}(-,X_i)$} \put(170,40) {$\swarrow$} \put(240,40) {$\searrow$} \end{picture} \nin Let $X = \begin{array}{c} \mbox{hocolim} \\ \vspace{-.75cm} \\ \longrightarrow \end{array} X_i$. I assert: \begin{itemize} \item[(1)] There is a natural transformation Hom$(-,X) \rightarrow H$ rendering commutative the triangles \begin{picture}(400,100) \put(120,10) {$\mbox{Hom}(-,X)$} \put(200,10) {$\longrightarrow$} \put(260,10) {$H$} \put(180,70) {$\mbox{Hom}(-,X_i)$} \put(170,40) {$\swarrow$} \put(240,40) {$\searrow$} \end{picture} \noindent for every $i$. \item[(2)] The natural transformation Hom$(-,X) \rightarrow H$ is an isomorphism. \end{itemize} \noindent{\bf Proof of (1).} consider the triangle $$ \begin{array}{rcccc} \displaystyle\coprod_i X_i && \stackrel{1\mbox{-shift}}{\longrightarrow} && \displaystyle\coprod_i X_i \\ & {\scriptstyle{(1)}} \nwarrow && \swarrow & \\ && \stackrel{\mbox{hocolim}}{\longrightarrow} X_i = X && \end{array} $$ Applying the cohomological functor $H$, we get an exact sequence $$ \begin{array}{ccccc} H(X) & \longrightarrow & H\left(\displaystyle\coprod_i X_i \right) & \stackrel{1\mbox{-shift}}{\longrightarrow} & H\left(\displaystyle\coprod_i X_i \right) \\ &&&& \rule{0cm}{.01cm} \\ && \| && \| \\ &&&& \rule{0cm}{.01cm} \\ && \displaystyle\prod_i H(X_i) & \stackrel{1\mbox{-shift}}{\longrightarrow} & \displaystyle\prod_i H(X_i). \end{array} $$ The element $\displaystyle\prod_i \alpha_i \in \displaystyle\prod_i H(X_i)$ is in the kernel of (1-shift), and hence there is an $\alpha \in H(X)$ mapping to it. By Yoneda, $\alpha$ corresponds to a natural transformation $$ \mbox{Hom}(-,X) \rightarrow H, $$ and the fact that $\alpha$ maps to $\prod \alpha_i \in H(\coprod X_i)$ means that the diagram \begin{picture}(400,100) \put(120,10) {$\mbox{Hom}(-,X)$} \put(200,10) {$\longrightarrow$} \put(260,10) {$H$} \put(180,70) {$\mbox{Hom}(-,X_i)$} \put(170,40) {$\swarrow$} \put(240,40) {$\searrow$} \end{picture} \noindent commutes for all $i$. \bigskip \noindent{\bf Proof of (2).} It remains to show that $$ \phi: \mbox{ Hom}(-,X) \rightarrow H $$ constructed above is an isomorphism. Let us begin with objects $t \in T$. We will show that, for all $t \in T$, $$ \phi(t): \mbox{ Hom}(t,X) \rightarrow H(t) $$ is an isomorphism. Observe the commutative diagram $$ \begin{array}{ccccc} && \mbox{Hom}(t,X_0) && \\ & \swarrow && \searrow & \\ \mbox{Hom}(t,X) && \longrightarrow && H(t) \end{array} $$ Since we know that Hom$(t,X_0) \rightarrow H(t)$ is surjective, it follows that $$ \mbox{Hom}(t,X) \rightarrow H(t) $$ is surjective. It remains only to prove it injective. Let $f \in \mbox{ Hom}(t,X)$ with $\phi(t)(f) = 0$. Now $f \in \mbox{ Hom} (t,X) = \mbox{ Hom}\left(t, \begin{array}{c} \mbox{hocolim} \\ \vspace{-.75cm} \\ \longrightarrow \end{array} X_i\right)$. But as $t \in T$ is compact, we have by Lemma~\ref{maps to colimits} that $$ \mbox{Hom}\left(t,\begin{array}{c} \mbox{hocolim} \\ \vspace{-.75cm} \\ \longrightarrow \end{array} X_i\right) = \displaystyle\lim_\rightarrow \mbox{ Hom} (t,X_i). $$ In other words, there exists $f_i: t \rightarrow X_i$ so that the composite $$ t \stackrel{f_i}{\longrightarrow} X_i \longrightarrow X $$ is $f$. But the diagram \begin{picture}(400,100) \put(120,10) {$\mbox{Hom}(t,X)$} \put(200,10) {$\longrightarrow$} \put(205,1) {$k$} \put(260,10) {$H(t)$} \put(180,70) {$\mbox{Hom}(t,X_i)$} \put(170,40) {$j \swarrow$} \put(240,40) {$\searrow$} \end{picture} \noindent commutes, and $j(f_i) = f$ while $k(f) = 0$. It follows that $f_i \in \ker \{\mbox{Hom}(t,X_i) \rightarrow H(t)\}$, that is, $(f_i,t) \in U_{i+1}$. This means that $f_i: t \rightarrow X_i$ factors through the map $h$ in the triangle \begin{picture}(400,100) \put(90,15) {$\displaystyle\left\{\coprod_{(f_{i},t) \in U_{i+1}}t\right\} = K_{i+1}$} \put(220,15) {$\longleftarrow$} \put(225,4) {$\scriptstyle{(1)}$} \put(260,15) {$X_{i+1}$} \put(210,70) {$X_i$} \put(170,44) {$h \nearrow$} \put(240,44) {$\searrow g$} \end{picture} \noindent and hence $g \circ f_i = 0$. But the map $$ X_i \stackrel{g}{\longrightarrow} X_{i+1} \stackrel{\bar{g}}{\longrightarrow} X $$ satisfies \begin{eqnarray*} f & = & \{ \bar{g} \circ g\} \circ f_i \\ & = & \bar{g} \circ \{g \circ f_i\} \\ & = & 0. \end{eqnarray*} Thus $\phi(t): \mbox{ Hom}(t,X) \rightarrow H(t)$ is an isomorphism whenever $t \in T$. Let ${\cal S} \subset {\cal T}$ be the full subcategory of objects $y \in {\cal T}$ such that, for all $n\in {\Bbb Z}$, the map $\phi(\Sigma^n y): \mbox{ Hom}(\Sigma^n y,X) \rightarrow H(\Sigma^n y)$ is an isomorphism. Then the category ${\cal S}$ contains $T$, and is closed with respect to the formation of coproducts and triangles. To finish our proof of Theorem~\ref{Brown representability}, we need the lemma \lem{$T$ generates} Let ${\cal S} \subset {\cal T}$ be a full, triangulated subcategory containing $T$ and closed under the formation of ${\cal T}$-coproducts of its objects. Then ${\cal S} = {\cal T}$.\elem \noindent{\bf Proof.} Let ${\cal S}$ be the smallest subcategory of ${\cal T}$ which is full, triangulated, closed with respect to ${\cal T}$-coproducts of its objects, and contains $T$. Let $Z$ be an object of ${\cal T}$. Let $H = \mbox{Hom}_{\cal T}(-,Z)$ be viewed as a homological functor on ${\cal S}$. Then ${\cal S}$ is compactly generated, with a generating set $T$. We can therefore apply what we have proved so far about Brown representability to the functor $H$ on ${\cal S}$; there is an object $X$ of ${\cal S}$, a natural transformation of functors on ${\cal S}$ $$ \phi: \mbox{Hom}_{\cal S} (-,X) \rightarrow \mbox{Hom}_{\cal T}(-,Z), $$ and this natural transformation is an isomorphism on a full, triangulated subcategory of ${\cal S}$ containing $T$ and closed with respect to ${\cal S}$-coproducts of its objects. But ${\cal S}$ is minimal with this property; hence $$ \mbox{Hom}_{\cal S} (-,X) \rightarrow \mbox{Hom}_{\cal T} (-,Z) $$ is an isomorphism of functors on ${\cal S}$. By Yoneda's lemma, this means there is a morphism in ${\cal T}$ $$ X \rightarrow Z $$ so that, for every object, $s$ of ${\cal S}$, $$ \mbox{Hom}(s,X) \rightarrow \mbox{Hom}(s,Z) $$ is an isomorphism. Complete $X \rightarrow Z$ to a triangle in ${\cal T}$ $$ \begin{array}{rcccc} X && \longrightarrow && Z \\ & {\scriptstyle{(1)}} \nwarrow & & \swarrow & \\ && Y && \end{array} $$ One easily deduces that, for every object $s$ of $\cal S$, $\mbox{Hom}(s,Y) = 0$. But $T \subset {\cal S}$, and hence for every object $t \in T, \ \mbox{Hom}(t,Y) = 0$. But because $T$ generates, $Y = 0$, and hence $X \rightarrow Z$ is an isomorphism. Thus $Z$ is an object of ${\cal S}$, and since $Z \in {\cal T}$ was arbitrary, ${\cal S} = {\cal T}$. \hfill $\Box$ \section{The adjoint functor theorem and examples} \label{S3} \thm{adjoints} Let ${\cal S}$ be a compactly generated triangulated category, ${\cal T}$ any triangulated category. Let $F: {\cal S} \rightarrow {\cal T}$ be a triangulated functor. Suppose $F$ respects coproducts; the natural maps $$ F(s_\lambda) \rightarrow F\left(\displaystyle\coprod_{\lambda \in \Lambda} s_\lambda\right) $$ make $F\left(\displaystyle\coprod_{\lambda \in \Lambda} s_\lambda\right)$ a coproduct of ${\cal T}$. Then $F$ has a right adjoint $G: {\cal T} \rightarrow {\cal S}$.\ethm \noindent{\bf Proof.} Let $t$ be an object of ${\cal T}$, and consider the functor on ${\cal S}$ $$ s \mapsto \mbox{ Hom}_{\cal T} (F(s),t). $$ This functor is homological and takes coproducts to products; we have \begin{eqnarray*} \mbox{Hom}_{\cal T}\left(F\left(\displaystyle\coprod_{\lambda \in \Lambda} s_\lambda\right),t\right) & = & \mbox{Hom}_{\cal T}\left(\coprod_{\lambda \in \Lambda} F(s_\lambda),t\right) \\ & = & \prod_{\lambda \in \Lambda} \mbox{Hom}_{\cal T}(F(s_\lambda),t). \end{eqnarray*} Hence, by Theorem~\ref{Brown representability}, this functor is representable; there is a $G(t) \in {\cal S}$ with $$ \mbox{Hom}_{\cal T}(F(s),t) = \mbox{Hom}_{\cal S}(s,G(t)), $$ and by standard arguments, $G$ extends to a functor, right adjoint to $F$. \hfill $\Box$ \exm{$qc$ adjoints} Let $f: X \rightarrow Y$ be a separated morphism of quasi-compact separated schemes. Then $$ Rf_*: D(qc/X) \rightarrow D(qc/Y) $$ has a right adjoint $$ f^!:D(qc/Y) \rightarrow D(qc/X). $$\eexm \noindent{\bf Proof.} We need only show that $Rf_*$ is triangulated and respects coproducts; the fact that it is triangulated is obvious, the fact that it respects coproducts is Lemma~\ref{$Rf_*$ respects sums}. \hfill $\Box$ \exm{$D$-module adjoints} Let $f: X \rightarrow Y$ be a separated morphism of smooth, quasi-compact, separated schemes of finite type over a field $k$ of characteristic 0. Then $$ Rf_+: D\left(\frac{qc \: D\mbox{-modules}}{X}\right) \rightarrow D\left(\frac{qc \: D\mbox{-modules}}{Y}\right) $$ has a left adjoint.\eexm \noindent{\bf Proof.} Since $Rf_+$ is clearly a triangulated functor, the real point is to prove that it respects coproducts. But $Rf_+$ is given as $$ Rf_+(x) \stackrel{\rm def}{=} Rf_*\left({\cal D}^{}_{Y \leftarrow X} \otimes_{{\cal D}^{}_{X}}x\right) $$ and tensor product trivially respects coproducts, while $Rf_*$ does by Lemma~\ref{$Rf_*$ respects sums}. \hfill $\Box$ \bigskip \exm{holomorphic adjoints} With $f: X \rightarrow Y$ as in Example~\ref{$D$-module adjoints}, let $D(\mbox{holo}/X)$ and $D(\mbox{holo}/Y)$ be as in Example~\ref{holo}; that is, $D(\mbox{holo}/X)$ is the smallest full, triangulated category of $D\left(\frac{qc \: D\mbox{-modules}}{X}\right)$ closed with respect to coproducts and containing the bounded complexes with holonomic cohomology. It is well known that $$ Rf_+: D\left(\frac{qc \: D\mbox{-modules}}{X}\right) \rightarrow D\left(\frac{qc \: D\mbox{-modules}}{Y}\right) $$ takes complexes with holonomic cohomology to complexes with holonomic cohomology. Since $Rf_+$ is triangulated and respects coproducts, it takes $D(\mbox{holo}/X)$ to $D(\mbox{holo}/Y)$. It induces a functor, which we also denote $Rf_+$, $$ Rf_+: D(\mbox{holo}/X) \rightarrow D(\mbox{holo}/Y). $$ This functor has a left adjoint.\eexm \noindent{\bf Proof.} By Example~\ref{holo}, $D(\mbox{holo}/X)$ is a compactly generated triangulated category. The functor $Rf_+$ is the restriction to $D(\mbox{holo}/X)$ of a functor on $D\left(\frac{qc \: D\mbox{-modules}}{X}\right)$ respecting coproducts; hence it respects coproducts. By Theorem~\ref{Brown representability}, the adjoint exists. \hfill $\Box$ \section{Commuting with coproducts} \label{S4} In Section~\ref{S3}, we constructed an adjoint to a functor $F: {\cal S} \rightarrow {\cal T}$. Precisely, if $F: {\cal S} \rightarrow {\cal T}$ is a triangulated functor, one sometimes has a right adjoint $G: {\cal T} \rightarrow {\cal S}$. Being a right adjoint, $G$ certainly respects products. It turns out to be interesting to know when $G$ respects coproducts. \thm{respecting coproducts} Let ${\cal S}$ be a compactly generated triangulated category, and let ${\cal T}$ be any triangulated category. Let $F: {\cal S} \rightarrow {\cal T}$ be a triangulated functor respecting coproducts, and let $G: {\cal T} \rightarrow {\cal S}$ be its right adjoint, which exists by Theorem~\ref{adjoints}. Let $S$ be a generating set for ${\cal S}$. Then $G: {\cal T} \rightarrow {\cal S}$ respects coproducts if and only if for every $s \in S, \ F(s)$ is a compact object of ${\cal T}$.\ethm \noindent{\bf Proof.} $\Rightarrow$ Suppose $G$ preserves coproducts. Let $s \in S$ be some object. Then $$ \begin{array}{cccl} \mbox{Hom}_{\cal T} \left(F(s),\displaystyle\coprod_{\lambda \in \Lambda} x_\lambda\right) & = & \mbox{Hom}_{\cal S}\left(s,G\left( \displaystyle\coprod_{\lambda \in \Lambda} x_\lambda\right)\right)& \\*[3pt] & = & \mbox{Hom}_{\cal S}\left(s,\displaystyle\coprod_{\lambda \in \Lambda} G\left(x_\lambda\right)\right)&\quad\hbox{because G commutes} \\*[-10pt] &&&\quad\hbox{with coproducts}\\*[10pt] & = & \displaystyle\coprod_{\lambda \in \Lambda}\mbox{Hom}_{\cal S}\left(s, G\left(x_\lambda\right)\right)&\quad\hbox{because $s$ is compact} \\*[20pt] & = & \displaystyle\coprod_{\lambda \in \Lambda} \mbox{Hom}_{\cal T} (F(s),x_\lambda),& \end{array} $$ and this proves that $F(s)$ is compact in ${\cal T}$. \medskip \nin $\Leftarrow$ Suppose that, for all $s \in S, \ F(s)$ is compact. Let $\displaystyle\coprod_{\lambda \in \Lambda} x_\lambda$ be a coproduct in ${\cal T}$. Then for any $s \in S$, $$ \begin{array}{cccc} \mbox{Hom}\left(s,G\left(\displaystyle\coprod_{\lambda \in \Lambda} x_\lambda\right)\right) & = & \mbox{Hom}\left(F(s),\displaystyle\coprod_{\lambda \in \Lambda} x_\lambda\right) &\\ & = & \displaystyle\coprod_{\lambda \in \Lambda} \mbox{Hom} (F(s),x_\lambda)&\quad\hbox{because $F(s)$ is compact} \\*[20pt] & = & \displaystyle\coprod_{\lambda \in \Lambda} \mbox{Hom}(s,G(x_\lambda))& \\ & = & \mbox{Hom}\left(s,\displaystyle\coprod_{\lambda \in \Lambda} G(x_\lambda)\right)&\quad\hbox{because $s$ is compact}. \end{array} $$ In other words, the natural map $$ \displaystyle\coprod_{\lambda \in \Lambda} G(x_\lambda) \rightarrow G\left(\displaystyle\coprod_{\lambda \in \Lambda} x_\lambda\right) $$ induces a natural transformation $$ \phi: \mbox{Hom}_{\cal S} \left(-\:, \displaystyle\coprod_{\lambda \in \Lambda} G(x_\lambda)\right) \rightarrow \mbox{Hom}_{\cal S}\left(-\:,G\left( \displaystyle\coprod_{\lambda \in \Lambda} x_\lambda\right)\right), $$ and $\phi(s)$ is an isomorphism for all $s \in S$. But then in the triangle $$ \displaystyle\coprod_{\lambda \in \Lambda} G(x_\lambda) \rightarrow G\left(\displaystyle\coprod_{\lambda \in \Lambda} x_\lambda\right)\la Z\la \Sigma\left\{\displaystyle\coprod_{\lambda \in \Lambda} G(x_\lambda)\right\} $$ the object $Z$ must satisfy $Hom({\cal S}, Z)=0$. But as $\cal S$ generates, $Z=0$ and $$ \displaystyle\coprod_{\lambda \in \Lambda} G(x_\lambda) \rightarrow G\left(\displaystyle\coprod_{\lambda \in \Lambda} x_\lambda\right) $$ is an isomoprphism.\hfill $\Box$ \bigskip \exm{example} Let $f: X \rightarrow Y$ be a pseudo--coherent proper morphism of separated, quasi--compact schemes. Suppose that $f$ is of finite Tor-dimension. Then $Rf_*: D(qc/X) \rightarrow D(qc/Y)$ takes perfect complexes to perfect complexes, by \cite{Ki}. It follows that it takes a set of generators of $D(qc/X)$ to a set of compact objects of $D(qc/Y)$. Hence $f^!$ commutes with coproducts. What makes this interesting is Theorem~\ref{$f^!$ commutes with sums} which follows. But in the theorem we appeal to the projection formula, and since I do not know a reference which proves it in the generality we want, the following is the sketch of a proof.\eexm \pro{projection formula} Let $f:X\la Y$ be a morphism of separated, quasi--compact schemes. Let $D(X)$ be the derived category of all ${\cal O}_X$--modules, $D(Y)$ the derived category of all ${\cal O}_Y$--modules. Let $y$ be an object of $D(Y)$, $x$ an object of $D(X)$. Then there is a natural map, in $D(Y)$, $$ y{^L\otimes}Rf_*x \la Rf_*\left\{Lf^*y{^L\otimes} x\right\}. $$ If $y$ is in $D(qc/Y)\subset D(Y)$ and $x$ is in $D(qc/X)\subset D(X)$, this natural map is an isomorphism.\epro \noindent {\bf Proof.}\ \ The existence and naturality of the map really comes from the definition of $Rf_*$, $Lf^*$ and the derived tensor product. To define $Rf_*$ of an object in $D(X)$, one expresses the derived category $D(X)$ as a quotient of the the homotopy category $K(X)$ by the acyclic complexes $E(X)$, and notes that the subcategory $L(X)$ of Bousfield local objects in $K(X)$ with respect to $E(X)$ maps isomorphically to $D(X)$. Here, $L(X)$ consists of the special complexes of injectives of Spaltenstein's \cite{S}. $Rf_*$ is just $f_*$ on $L(X)$. The tensor product, and $Lf^*$, depend for their construction on the fact that $D(X)$ is also isomorphic to the subcategory of Bousfield colocal objects, denoted here $\tilde L(X)$. Concretely, these are complexes of objects $j_!{\cal O}_U$, where $j:U\la X$ is the inclusion of an open affine, and $j_!$ is extension by zero. Replacing $y$ by a colocal object on $Y$ and $x$ by a local object on $X$, the tensor products become the natural ones and we have an isomorphism $$ y\otimes f_*x \simeq f_*\left\{f^*y\otimes x\right\}. $$ The left hand side identifies immediately with $y{^L\otimes}Rf_*x$, by definition of the derived tensor product and $Rf_*$. On the right, the part in bracket identifies with $Lf^*y{^L\otimes} x$, again by definition. Hence a natural isomorphism $$ y{^L\otimes}Rf_*x \simeq f_*\left\{Lf^*y{^L\otimes} x\right\}. $$ The problem is that, in general, $Lf^*y{^L\otimes} x$ is not Bousfield local (=a complex of injectives), and hence $f_*$ of it is not the same as $Rf_*$. But for any complex $x$, there is a natural map $f_*x\la Rf_*x$, and this gives the natural map $$ y{^L\otimes}Rf_*x \la Rf_*\left\{Lf^*y{^L\otimes} x\right\}. $$ It remains to show that the restriction of this map to the subcategories of complexes of quasi--coherents is an isomorphism. Fix $x\in D(qc/X)$; we have a natural transformation of functors in $y\in D(qc/Y)$. First, the problem is now local in $Y$ and we may therefore assume $Y$ affine. Secondly, on the category $D(qc/X)$, $Rf_*$ respects coproducts by Lemma~\ref{$Rf_*$ respects sums}. Tensor product and $Lf^*$ always respect sums, so the map is a natural transformation of two functors in $y$ respecting coproducts. For each $y\in D(qc/Y)$, let $\phi(y)$ be the map $$ y{^L\otimes}Rf_*x \la Rf_*\left\{Lf^*y{^L\otimes} x\right\}. $$ Let ${\cal R}\subset D(qc/Y)$ be the full subcategory of all $y$'s such that $\phi(\Sigma^n y)$ is an isomorphism for all $n$. The category $\cal R$ is closed with respect to triangles and coproducts. It clearly contains ${\cal O}_Y$. Since $Y$ is affine, ${\cal O}_Y$ is ample and generates $D(qc/Y)$ by Example~\ref{quasi-projective variety}. By Lemma~\ref{$T$ generates}, it follows that $\cal R$ is all of $D(qc/Y)$.\qqed \thm{$f^!$ commutes with sums} Let $f: X \rightarrow Y$ be a morphism of schemes. Suppose $Rf_*$ has a right adjoint $f^!$ which commutes with coproducts. Suppose $Y$ is quasi-compact and separated. Then there is a natural isomorphism of functors $D(qc/Y) \rightarrow D(qc/X)$, which on objects gives $$ f^!(y) \simeq (Lf^*y) \otimes_{{\cal O}_{X}} (f^!{\cal O}_Y). $$ Conversely, if $f^!$ is naturally isomorphic to the functor on the right, it respects coproducts.\ethm \noindent{\bf Proof.} $\Leftarrow$ Suppose we have a natural isomorphism of functors in $y$ $$ f^!(y) \simeq (Lf^*y) \otimes_{{\cal O}_{Y}}(f^!{\cal O}_Y). $$ Since $Lf^*$ has a right adjoint, it respects coproducts. Tensor products respect coproducts, so we deduce that $f^!$ respects coproducts. \medskip \nin $\Rightarrow$ We will show that there is a natural map $$ (Lf^*y) \otimes_{{\cal O}_{X}}(f^!{\cal O}_Y)\la f^!(y) $$ and that this map is an isomorphism whenever $y$ is compact. Then, if $f^!$ respects coproducts, it will easily follow that this natural map is an isomorphism for all $y$. Let us prove a slightly more general fact. We will actually show that, for any $y'$ in $D(qc/Y)$, there is a natural map $$ (Lf^*y) \otimes_{{\cal O}_{X}}(f^!y')\la f^!(y\otimes_{{\cal O}_Y}y') $$ which is an isomorphism if $y$ is compact. The case $y'={\cal O}_Y$ is then the above. In any case, there is a natural map $$ \mu: Rf_* f^! y' \rightarrow y', $$ the counit of adjunction. For every $y \in D(qc/Y)$, $$ Rf_*\left[(Lf^*y) \otimes_{{\cal O}_{X}} f^!y'\right] = y \otimes_{{\cal O}_{Y}} Rf_* f^! y' $$ by the projection formula. Hence there is a natural map $$ \begin{array}{rcc} Rf_*\left[Lf^*(y) \otimes_{{\cal O}_{X}} f^!y'\right] & = & y \otimes_{{\cal O}_{Y}} Rf_*f^!y' \\*[10pt] && \hspace*{1cm} \downarrow 1_y \otimes \mu \\*[10pt] && y \otimes_{{\cal O}_{Y}} y'. \end{array} $$ By adjunction, we have a map $$ Lf^*(y) \otimes_{{\cal O}_{X}} f^!y'\la f^!\left(y\otimes_{{\cal O}_{Y}}y'\right) $$ This is our natural map. We need to show that for compact $y$ it is an isomorphism. Let $y$ be compact, and let $x$ be an arbitrary complex in $D(qc/X)$. It suffices to show that the natural map above induces an isomorphism after applying $Hom(x,-)$. Let us therefore reflect what $Hom(x,-)$ does. To begin with, put $\hat y={\cal RH}{\rm om}_{{\cal O}_Y}(y,{\cal O}_Y)$. Then $Lf^*\hat y={\cal RH}{\rm om}_{{\cal O}_X}(Lf^*y,{\cal O}_X)$, and since $y$ and $Lf^*y$ are perfect complexes, there are natural isomorphisms $$ Hom_X(-\otimes Lf^*\hat y\,,\,-)=Hom_X(-\,,\,Lf^*y\otimes-) $$ and $$ Hom_Y(-\otimes\hat y\,,\,-)=Hom_Y(-\,,\,y\otimes-). $$ Now, the map $$ Lf^*(y) \otimes_{{\cal O}_{X}} f^!y'\la f^!\left(y\otimes_{{\cal O}_{Y}}y'\right) $$ induces a map after applying $Hom(x,-)$. By definition, this takes a map $\gamma\in Hom\left(x\,,\,Lf^*(y) \otimes_{{\cal O}_{X}} f^!y'\right)$ to a map $x\la f^!(y\otimes_{{\cal O}_{Y}} y')$. By the adjunction, this map corresponds to a map $\gamma':Rf_*x\la y\otimes_{{\cal O}_{Y}} y'$. But we know what $\gamma'$ is; it is the composite of $Rf_*\gamma$ with the natural counit $$ \begin{array}{rcc} Rf_*\left[Lf^*(y) \otimes_{{\cal O}_{X}} f^!y'\right] & = & y \otimes_{{\cal O}_{Y}} Rf_*f^!y' \\*[10pt] && \hspace*{1cm} \downarrow 1_y \otimes \mu \\*[10pt] && y \otimes_{{\cal O}_{Y}} y'. \end{array} $$ We need to show this correspondence to be an isomorphism. But \begin{eqnarray*} \mbox{Hom}_{{\cal O}_{X}} \left(x\,,\,Lf^*y \otimes_{{\cal O}_{X}} f^!y'\right) & = & \mbox{Hom}_{{\cal O}_{X}}\left(x \otimes Lf^*\hat y\,,\, f^!y'\right) \\ & = & \mbox{Hom}_{{\cal O}_{Y}}\left(Rf_* \{x \otimes Lf^*\hat y\}\,,\,y'\right) \\ & = & \mbox{Hom}_{{\cal O}_{Y}}\left( Rf_*(x)\otimes \hat y\,,\,y'\right) \\ & = & \mbox{Hom}_{{\cal O}_Y} \left(Rf_*(x)\,,\,y \otimes_{{\cal O}_{Y}} y'\right), \end{eqnarray*} where the third equality is the projection formula. This isomorphism is easily identified with $Hom(x,-)$ applied to the map $$ \phi:Lf^*(y) \otimes_{{\cal O}_{X}} f^!y'\la f^!\left(y\otimes_{{\cal O}_{Y}}y'\right). $$ We therefore know that $\phi$ is an isomorphism if $y$ is compact. Now assume that $f^!$ respects coproducts. For fixed $y'$, the functor $$ y \mapsto (Lf^*y) \otimes_{{\cal O}_{X}} f^!y' $$ is a triangulated functor in $y$; $Lf^*$ is, as is tensor product. The functor $f^!$ is the adjoint of a triangulated functor, hence triangulated; see \cite{SH}, Lemma 3.9. Thus $\phi$ is a natural transformation of triangulated functors, both of which respect coproducts. Let ${\cal S} \subset D(qc/Y)$ be the full subcategory $$ {\cal S} = \left\{y \in {\cal O}b\left[D(qc/Y)\right]| \: \phi(\Sigma^ny) \mbox{ is an isomorphism for all } n \in {\Bbb Z}\right\}. $$ Then ${\cal S}$ contains the generating set of compact objects, is triangulated and closed with respect to $D(qc/Y)$-coproducts. Thus ${\cal S} = D(qc/Y)$. Hence, $\phi$ is a natural isomorphism. The theorem is the special case $y'={\cal O}_Y$ of the above. \hfill $\Box$ \bigskip \rmk{traditional} In the traditional literature on the subject, $f^!{\cal O}_Y$ is called the {\em dualizing complex}, and plays a key role in the theory.\ermk \section{A sheaf version} \label{S5} The traditional way to state Grothendieck's duality theorem comes in a sheaf version. Let $f: X \rightarrow Y$ be a proper morphism of noetherian, separated schemes. One would like to deduce that $f^!$, the adjoint we have for $Rf_*$, gives an isomorphism in the category of sheaf homomorphisms $$ {\cal RH}om(Rf_*x,y) \simeq Rf_* {\cal RH}om(x,f^!y). $$ Note that the counit $u: Rf_*f^!y \rightarrow y$ defines in any case a well-defined natural transformation $$ \phi: Rf_* {\cal RH}om(x,f^!y) \rightarrow {\cal RH}om(Rf_*x,y), $$ and the only question is whether it is an isomorphism. Once we apply the functor $H^0(Y,-)$, it becomes an isomorphism; this is because $f^!$ is adjoint to $Rf_*$. To say that the map is an isomorphism of sheaves is to say that the derived functor of $\Gamma(U,-)$ gives an isomorphism $R\Gamma(U,\phi)$ for every open set $U \subset Y$. Concretely, it says that if we take the commutative diagram $$ \begin{array}{ccc} f^{-1}U & \stackrel{j'}{\hookrightarrow} & X \\ f' \downarrow && \downarrow f \\ U & \stackrel{j}{\hookrightarrow} & Y, \end{array} $$ then $(f')^!j^*$ and $(j')^*f^!$ are naturally isomorphic. It is easy to see that $j^*Rf_* = Rf'_*(j')^*$. Taking right adjoints, we deduce \lem{$^*$ OK} If the diagram $$ \begin{array}{ccc} f^{-1}U & \stackrel{j'}{\hookrightarrow} & X \\ f' \downarrow && \downarrow f \\ U & \stackrel{j}{\hookrightarrow} & Y, \end{array} $$ is given by pulling back an open immersion $j:U\la Y$, then there is a natural isomorphism $$f^!R{j_*} = R{j'_*}(f')^!$$.\qqed\elem Therefore $$ (j')^*\{Rj'_*(f')^!\}j^* = (j')^*\{f^!R{j_*}\}j^*. $$ Hence $(f')^!j^* = (j')^*f^!R{j_*}j^*$ because $(j')^*R{j'_*}$ is the identity functor. Thus we must convince ourselves that the natural unit of adjunction $1 \rightarrow R{j_*}j^*$ induces an isomorphism $$ (j')^*f^! \rightarrow (j')^*f^!R{j_*}j^*. $$ Let $Z=Y-U$ be the (closed) complement of $U\subset Y$. There is a triangle of functors $$ \begin{array}{c} R\Gamma_Z \rightarrow 1 \rightarrow Rj_*j^* \\ \longleftarrow \\ (1) \end{array} $$ where $R\Gamma_Z$ is Grothendieck's local cohomology functor. Sometimes $R\Gamma_Z$ is denoted $i_*i^!$, because it is also the counit of an adjunction. But since the $i^!$ and $i_*$ are not the type of map we have been considering here, the notation might lead to confusion. Note that $Z\subset Y$ is a Zariski closed subset, but we have {\em not} given it a scheme structure. Hence the $i_*$ and $i^!$ of this article make no sense. It suffices to show that $$ (j')^* f^! R\Gamma_Z $$ is the zero functor. It is enough to show this for open sets $U \subset X$ which form a basis for the topology. This is what we will do. \bigskip \pro{the vanishing} Let $f: X \rightarrow Y$ be a morphism of schemes such that $Rf_*: D(qc/X) \rightarrow D(qc/Y)$ has a right adjoint $f^!$. Suppose $f^!$ respects coproducts. Suppose $U \subset Y$ is an open subset, $Z = Y-U$ the complement. Suppose $U$ and $Y$ are quasi--compact and separated. Then, in the notation above, the composite $$ (j')^* f^! R\Gamma_Z = 0. $$\epro \noindent{\bf Proof.}\ \ We wish to show that $(j')^* f^!$ vanishes on any object of the form $R\Gamma_Z (y)$. This means concretely that if $y$ is a complex which is acyclic away from $Z$, then $f^!y$ must be shown acyclic off $f^{-1}Z$. But by Theorem~\ref{$f^!$ commutes with sums}, $$ f^!y=Lf^*y\otimes f^!{\cal O}_Y. $$ Clearly, if $y$ is supported on $Z$, then $Lf^*y$ is supported on $f^{-1}Z$, and hence so is its tensor product with $f^!{\cal O}_Y$.\qqed From the work of Verdier for the Noetherian case, Lipman in general, we know that for the bounded--below derived category more is true. Let us state their theorem, then prove it by coproduct techniques. \pro{the vanishing, bounded} Let $f: X \rightarrow Y$ be a pseudo--coherent, proper morphism of quasi--compact, separated schemes. Then $Rf_*: D^+(qc/X) \rightarrow D^+(qc/Y)$ has a right adjoint $f^!$. Furthermore, in the notation above, the composite $$ (j')^* f^! R\Gamma_Z = 0. $$\epro \noindent{\bf Proof.}\ \ Let us begin by showing that the right adjoint of $f^!: D(qc/Y) \rightarrow D(qc/X)$ takes bounded--below complexes to bounded--below complexes, and is therefore also a right adjoint to $Rf_*: D^+(qc/X) \rightarrow D^+(qc/Y)$. Since $X$ is quasi-compact, it may be covered by finitely many open affines, say $n$ of them. But since $f$ is separated, the open affines may be used to compute $Rf_*$, via the \v Cech complex. Since the \v Cech complex has only $(n+1)$ terms, it follows that if $x\in D^-(qc/X)$ vanishes above dimension $l$, then $Rf_*x$ vanishes above dimension $l+n$. In the notation of {\it t}--structure truncations, if $x\in {D(qc/X)}^{\le l}$, then $Rf_*x\in {D(qc/Y)}^{\le l+n}$. Pick any $x\in {D(qc/X)}^{\le l}$, and $y\in {D(qc/Y)}^{\ge l+n+1}$. Then $$ Hom(x,f^!y)=Hom(Rf_*x,y)=0 $$ since $Rf_*x\in {D(qc/Y)}^{\le l+n}$ and $y\in {D(qc/Y)}^{\ge l+n+1}$. But $x\in {D(qc/X)}^{\le l}$ was arbitrary; thus $f^!y\in {D(qc/X)}^{\ge l+1}$. In other words, $y\in {D(qc/Y)}^{\ge l}$ implies $f^!y\in {D(qc/X)}^{\ge l-n}$. In particular, if $y\in D^+(qc/Y)$, then $f^!y\in D^+(qc/X)$. It remains to show the vanishing of $ (j')^* f^! R\Gamma_Z. $ We need to show that if $y$ is an object of $D^+(qc/Y)$, which is acyclic off $Z$, then $f^!y$ is acyclic off $f^{-1}Z$. Let us first make an observation. Suppose $y$ is arbitrary, vanishing off $Z$. Suppose $Y=V_1\cup V_2$ expresses $Y$ as a union of two open sets $V_1$ and $V_2$. There is then a triangle $$ y\la {j^{}_{V_1}}_*{j^{}_{V_1}}^*y\oplus {j^{}_{V_2}}_*{j^{}_{V_2}}^*y \la {j^{}_{V_1\cap V_2}}_*{j^{}_{V_1\cap V_2}}^*y\la\Sigma y $$ where $j^{}_{W}:W\hookrightarrow Y$ is the inclusion of the open set $W$ in $Y$. From the triangle, it clearly suffices to show the vanishing of $(j')^* f^!$ on the complexes ${j^{}_{W}}_*{j^{}_{W}}^*y$ where $W$ is any of $V_1$, $V_2$ or $V_1\cap V_2$. Suppose $X$ can be covered by $n$ affines. If $V_1$ is affine and $V_2$ the union of $n-1$ affines, then each of $V_1$, $V_2$ and $V_1\cap V_2$ can be covered by at most $n-1$ affines. By induction we therefore easily show that it suffices to prove the vanishing of $(j')^* f^!$ on ${j^{}_{W}}_*{j^{}_{W}}^*y$ where $W\subset X$ is affine; in other words, we need to show that $f^!{j^{}_{W}}_*{j^{}_{W}}^*y$ vanishes off $f^{-1}Z$. Even better, we may replace $y$ by $z={j^{}_{W}}^*y$. It will suffice to show that, given an open affine $W\subset Y$ and $z\in D^+(qc/W)$ whose support is in $Z\cap W$, then $f^!{j^{}_{W}}_*z$ is supported on $f^{-1}Z$. Now consider the pullback square $$ \begin{array}{ccc} f^{-1}W & \stackrel{j_{f^{-1}W}}{\hookrightarrow} & X \\ f' \downarrow && \downarrow f \\ W & \stackrel{j_W}{\hookrightarrow} & Y. \end{array} $$ By Lemma~\ref{$^*$ OK} there is a natural isomorphism ${\{j_{f^{-1}W}\}}_*{\{f'\}}^!z=f^!{\{j_W\}}_*z$. We need to show that this complex (either of the two isomorphic versions) is acyclic off $f^{-1}Z$. From the description as ${\{j_{f^{-1}W}\}}_*{\{f'\}}^!z$, it clearly suffices to show that ${\{f'\}}^!z$ is supported on $f^{-1}\{Z\cap W\}$. In other words, we are reduced to studying the problem for the map $f:f^{-1}W\la W$. Thus we may assume $Y$ affine. Next it is clear that if $Z=\cap Z_i$, it is enough to prove the statement for each $Z_i$. We may therefore assume that $Z$ is a ``divisor'' in $W$. That is, there exists a global function $\gamma\in\Gamma(Y,{\cal O}_Y)$, so that $Z$ is the divisor defined by $\gamma$. On $Y$ we have perfect complexes, namely the desuspensions of the mapping cones of the maps $$ \gamma^k:{\cal O}_Y\la {\cal O}_Y. $$ Call these perfect complexes $b_k$. Then $b_k$ fits in a triangle $$ b_k\la {\cal O}_Y\stackrel{\gamma^k}\la {\cal O}_Y \la \Sigma b_k. $$ There is a map $b_k\la b_{k+1}$, given by completing the following commutative square $$ \begin{array}{ccc} {\cal O}_Y & \stackrel{\gamma^k}\la & {\cal O}_Y \\ 1\downarrow & & \downarrow\gamma \\ {\cal O}_Y & \stackrel{\gamma^{k-1}}\la & {\cal O}_Y \end{array} $$ to a map of triangles $$ \begin{array}{ccccccc} b_k&\la &{\cal O}_Y & \stackrel{\gamma^k}\la & {\cal O}_Y &\la &\Sigma b_k\\ \downarrow& &1\downarrow & & \downarrow\gamma & &\downarrow \\ b_{k+1}&\la &{\cal O}_Y & \stackrel{\gamma^{k+1}}\la & {\cal O}_Y &\la &\Sigma b_{k+1} \end{array} $$ There is furthermore a map $b_k\la {\cal O}_Y$, which is a map from the direct system. Applying the functor $(-)\otimes z$ to this direct system, we get a direct system with a map $$ b_k\otimes z\la {\cal O}_Y\otimes z=z. $$ This induces a (non--canonical) map on homotopy colimits $$ \hbox{hocolim}\left(b_k\otimes z\right)\la z $$ and since the cohomology of $z$ is supported on $Z$, that is annihilated by some power of $\gamma$, it is easy to show that the map is a cohomology isomorphism, hence an isomorphism in the derived category. Note that in the construction above, $b_k\otimes z$ can be obtained by tensoring the triangle $$ b_k\la {\cal O}_Y\stackrel{\gamma^k}\la {\cal O}_Y \la \Sigma b_k $$ with the object $z$. There is a triangle $$ b_k\otimes z\la z\stackrel{\gamma^k}\la z \la \Sigma b_k\otimes z. $$ We know that $z\in D^+(qc/Y)$. Suppose $z\in D(qc/Y)^{\ge l}$. Then from the triangle, $b_k\otimes z$ also lies in $D(qc/Y)^{\ge l}$. Because $z$ is the homotopy colimit of $b_k\otimes z$, there is a triangle on $Y$ $$ \bigoplus_{k}\left[ b_k\otimes z\right]\la \bigoplus_{k}\left[ b_k\otimes z\right]\la z\la \Sigma\bigoplus_{k}\left[ b_k\otimes z\right] $$ which expresses $z$ as the homotopy colimit. Applying $f^!$, we have a triangle $$ f^!\left\{\bigoplus_{k}\left[ b_k\otimes z\right]\right\}\la f^!\left\{\bigoplus_{k}\left[ b_k\otimes z\right]\right\}\la f^!z\la f^!\left\{\Sigma\bigoplus_{k}\left[ b_k\otimes z\right]\right\} $$ and to show that $f^!z$ is supported on $f^{-1}Z$, it suffices to show that the other two terms in the triangle are. Now note that in any case we there is a natural isomorphism $$ f^!\left[ b_k\otimes z\right]= Lf^* b_k\otimes f^!z $$ because $b_k$ is compact, and by the proof Theorem~\ref{$f^!$ commutes with sums}. Now $Lf^* b_k$ is supported on $f^{-1}Z$, hence its tensor product with $f^!{j^{}_W}_*z$ also is. It follows that $f^!\left[ b_k\otimes z\right]$ is supported on $f^{-1}Z$. We need to show that $\displaystyle f^!\left\{\bigoplus_{k}\left[ b_k\otimes z\right]\right\}$ is supported on $f^{-1}Z$. It will clearly suffice to show that the natural map $$ \bigoplus_{k}f^!\left[ b_k\otimes z\right]\la f^!\left\{\bigoplus_{k}\left[b_k\otimes z\right]\right\} $$ is an isomorphism. This we will now do. Now let $x$ be an arbitrary perfect complex on $X$. Since we are only assuming that the map $f:X\la Y$ is proper and of finite type, we do not know that $Rf_*x$ is perfect. However, we do know, by \cite{Ki}, that locally it can be resolved by finite dimensional vector bundles to arbitrary length. That is, $Y$ can be covered by open sets $V$, and for each $V$ there is a triangle $$ q\la p\la Rf_*x\la\Sigma q $$ where $p$ is perfect and $q\in D(qc/V)^{\le l-1}$. Since $Y$ is affine, this can even be done globally. Such a triangle exists on all of $Y$. We deduce \begin{eqnarray*} \hbox{Hom}\left(x,f^!\left\{\bigoplus_{k}\left[b_k\otimes z\right] \right\}\right) &=&\hbox{Hom} \left(Rf_*x,\bigoplus_{k}\left[b_k\otimes z\right]\right) \\ &=&\hbox{Hom} \left(p,\bigoplus_{k}\left[b_k\otimes z\right]\right) \\ &=&\bigoplus_{k}\hbox{Hom} \left(p,\left[b_k\otimes z\right]\right) \\ &=&\bigoplus_{k}\hbox{Hom} \left(Rf_*x,\left[b_k\otimes z\right]\right) \\ &=&\bigoplus_{k}\hbox{Hom} \left(x,f^!\left[b_k\otimes z\right]\right)\\ &=&\hbox{Hom} \left(x,\bigoplus_{k}f^!\left[b_k\otimes z\right]\right) \end{eqnarray*} But we have a map, defined on $Y$, $$ \bigoplus_{k}f^!\left[b_k\otimes z\right]\la f^!\left\{\bigoplus_{k}\left[b_k\otimes z\right]\right\} $$ and we have just shown that if we apply the functor $Hom(x,-)$ to this map where $x\in D(qc/X)$ is compact, we get an isomorphism. But then the map $$ \hbox{Hom}\left(x, \bigoplus_{k}f^!\left[b_k\otimes z\right] \right)\la \hbox{Hom}\left(x, f^!\left\{\bigoplus_{k}\left[b_k\otimes z\right]\right\} \right) $$ is an isomorphism for all $x$ in the subcategory generated by the compacts in $D(qc/X)$; that is for any $x\in D(qc/X)$. Thus the map $$ \bigoplus_{k}f^!\left[b_k\otimes z\right]\la f^!\left\{\bigoplus_{k}\left[b_k\otimes z\right]\right\} $$ is an isomorphism in $D(qc/X)$.\qqed \rmk{Verdier} The results of Verdier apply not only to open immersions, but to arbitrary flat map. In other words, he proves, in Theorem~2 of \cite{V}, the following. Suppose we have a cartesian square of noetherian schemes $$ \begin{array}{ccc} X'&\stackrel {g'}\longrightarrow& X\\ {\scriptstyle f'}\!\downarrow\,\quad & &{\scriptstyle f}\!\!\downarrow\,\,\,\, \\ Y'&\stackrel {g}\longrightarrow& Y \end{array} $$ \nin with $f$ and $f'$ proper, $g$ and $g'$ flat. Then there is a natural isomorphism $$ \{g'\}^*f^!=\{f'\}^!g^*. $$ What interests us here is not so much the best statement possible, but the relation with preserving coproducts. To illustrate this, we will give the following counterexample.\ermk \exm{counterexample} Let $R$ be a noetherian ring (eg $\Bbb Z$), and let $S=R[\epsilon]/(\epsilon^2)$. There is a homomorphism $S\la R$ sending $\epsilon$ to 0. This gives a map of schemes from $X=Spec(R)$ to $Y=Spec(S)$. This map is certainly proper, and of finite type. Let us denote this map $f:X\la Y$. For affine maps, $f^!$ is easy to describe. For any $S$--module $N$ and $R$--module $M$, we can view $M$ as an $S$--module via the homomorphism $S\la R$. This is the functor $f_*:\{R\!-\!\hbox{mod}\}\la \{S\!-\!\hbox{mod}\}$. There is a canonical ismorphism $$ Hom_S\left(M,N\right)=Hom_R\left(M,Hom_S(R,N)\right). $$ This allows us to view $f^!$ as the derived $Hom$ $$ f^!N=RHom_S(R,N) $$ for any $N\in D(qc/Y)=D(S)$. One way to get the derived functor of $Hom$ is to take projective resolutions in the first variable. There is an obvious projective resolution for $R$ as an $S$--module; the chain complex $$ \cdots\stackrel\epsilon\la S\stackrel\epsilon\la S\la R\la 0 $$ is exact. Let $N$ be $R$, viewed as an $S$--module. Then $$ f^!R= RHom_S(R,R)=\prod_{i=0}^\infty \Sigma^{-i}R $$ is the complex which is $R$ in every positive dimension, with differential zero. Now consider the complex $$ N=\prod_{k=0}^\infty\Sigma^{k}R $$ that is, the complex with zero differential which is $R$ in every negative dimension, viewed as a complex of $S$--modules. Because $f^!$ is a right adjoint, it respects products. Thus, \begin{eqnarray*} f^!N&=&f^!\left\{\prod_{k=0}^\infty\Sigma^{k}R\right\}\\ &=& \prod_{k=0}^\infty f^!\left\{\Sigma^{k}R\right\}\\ &=& \prod_{k=0}^\infty\prod_{i=0}^\infty \Sigma^{k-i}R \end{eqnarray*} Choose an element $\gamma\in R\subset S=R[\epsilon]/(\epsilon^2)$. If we restrict to the open subset where $\gamma$ is inverted, then $f^!N$ restricts to $$ {\{j'\}}^*f^!N=\left\{f^!N\right\}\otimes_R R\left[\frac1\gamma\right]. $$ On the other hand, if we first restrict to the open subset, note that we are in exactly the same situation; $S\left[\frac1\gamma\right] =R\left[\frac1\gamma\right][\epsilon]/(\epsilon^2)$. The complex $N$ restricts to $$ j^*N=\prod_{k=0}^\infty \Sigma^{k}R\left[\frac1\gamma\right]. $$ If this is not clear, note that because the $R$'s are placed in different dimensions, the coproduct agrees with the product. The natural map $$ \bigoplus_{k=0}^\infty \Sigma^{k}R\la \prod_{k=0}^\infty \Sigma^{k}R=N $$ is a homology isomorphism. But $j^*$, being a left adjoint, respects coproducts; hence $$ j^*\left(\bigoplus_{k=0}^\infty \Sigma^{k}R\right)= \bigoplus_{k=0}^\infty j^*\Sigma^{k}R =\bigoplus_{k=0}^\infty \Sigma^{k}R\left[\frac1\gamma\right] $$ and, once again, the natural map $$ \bigoplus_{k=0}^\infty \Sigma^{k}R\left[\frac1\gamma\right]\la \prod_{k=0}^\infty \Sigma^{k}R\left[\frac1\gamma\right] $$ is a homology isomorphism. We can therefore use our last computation, replacing $R$ by $R\left[\frac1\gamma\right]$, to deduce that $$ {\{f'\}}^!j^*N=\prod_{k=0}^\infty\prod_{i=0}^\infty \Sigma^{k-i}R\left[\frac1\gamma\right] $$ In other words, we have just computed everything. It remains to check whether the natural map is an isomorphism ${\{j'\}}^*f^!N\la {\{f'\}}^!j^*N$. The map is just $$ \left\{\prod_{k=0}^\infty\prod_{i=0}^\infty \Sigma^{k-i}R\right\} \otimes_R R\left[\frac1\gamma\right]\la \prod_{k=0}^\infty\prod_{i=0}^\infty \Sigma^{k-i}R\left[\frac1\gamma\right] $$ If we just look at the induced map on $H^0$, the product is over all $k=i$. The map on $H^0$ is therefore $$ \left\{\prod_{i=0}^\infty R\right\}\otimes_R R\left[\frac1\gamma\right]\la \prod_{i=0}^\infty R\left[\frac1\gamma\right] $$ This map clearly is not an isomorphism in general. If $\gamma$ is neither nilpotent nor invertible in $R$, then the element $$ \prod_{i=0}^\infty \frac1{\gamma^i} $$ is a well--defined member of the right hand side, but not the image of something on the left.\eexm

9308

alg-geom/9308004

en

https://arxiv.org/abs/alg-geom/9308004

alg-geom/9308004

Robert Friedman

Robert Friedman

Vector bundles and $SO(3)$ invariants for elliptic surfaces III: The case of odd fiber degree

68 pages, AMS-TeX

This paper, the last in a series of three, studies vector bundles on an elliptic surface whose determinant has odd intersection number with a general fiber and uses this study to calculate certain coefficients of Donaldson polynomials.

alg-geom

\section{Introduction.} Let $S$ be a simply connected elliptic surface with at most two multiple fibers, of multiplicities $m_1$ and $m_2$, where one or both of the $m_i$ are allowed to be 1. In this paper, the last of a series of three, we shall study stable rank two vector bundles $V$ on $S$ such that $\det V \cdot f$ is odd, where $f$ is a general fiber of $S$. Thus necessarily the multiplicities $m_1$ and $m_2$ are odd as well. Bundles $V$ such that $\det (V|f)$ has even degree for a general fiber $f$ have been studied extensively [3], [4, Part II], and as we shall see the case of odd fiber degree is fundamentally different. Thus we shall have to develop the analysis of the relevant vector bundles from scratch, and the results in this paper are for the most part independent of those in [3] and [4]. Our goal in this paper is to give a description of the moduli space of stable rank two bundles with odd fiber degree and then to use this information to calculate certain Donaldson polynomials. Before stating our main result, recall that, for an elliptic surface $S$, $J^d(S)$ denotes the elliptic surface whose general fiber is the set of line bundles of degree $d$ on the general fiber of $S$. We shall prove the following two theorems: \theorem{1} Let $\frak M_t$ be the moduli space of stable rank two bundles $V$ on $S$ \rom(with respect to a suitable ample line bundle\rom) with $\det V \cdot f = 2e+1$ and $4c_2(V) - c_1^2(V) - 3\chi (\scrO _S) = 2t$. Then $\frak M_t$ is smooth and irreducible and is birational to $\Sym ^tJ^{e+1}(S)$. \endstatement \medskip \theorem{2} Let $\gamma _t$ be the Donaldson polynomial of degree $2t$ corresponding to the choice of moduli space $\frak M_t$. Let $\kappa \in H_2(S;\Zee)$ be the primitive element such that $m_1m_1\kappa = f$. Then, for all $\Sigma \in H_2(S)$, \roster \item"{(i)}" $\gamma _0 = 1$. \item"{(ii)}" $\gamma _1(\Sigma, \Sigma) = (\Sigma ^2) + ((m_1^2m_2^2)(p_g(S) + 1)- m_1^2-m_2^2) (\Sigma \cdot \kappa)^2$. \item"{(iii)}" $\gamma _2(S)(\Sigma, \Sigma ,\Sigma, \Sigma ) = 3(\Sigma ^2)^2 + 6C_1(\Sigma ^2 )(\Sigma \cdot \kappa)^2 + (3C_1^2 - 2C_2)(\Sigma \cdot \kappa)^4$, where $$\align C_1&= (m_1^2m_2^2)(p_g(S) + 1)-m_1^2-m_2^2;\\ C_2 &= (m_1^4m_2^4) (p_g(S) + 1) -m_1^4-m_2^4. \endalign$$ \endroster \endstatement \medskip Let us outline the basic ideas behind the proof of Theorem 1. Standard arguments show that, for a suitable choice of an ample line bundle $L$ on $S$, a rank two vector bundle $V$ with $c_1(V)\cdot f= 2e+1$ is $L$-stable if and only if its restriction to a general fiber $f$ is stable. A pleasant consequence of the assumption of odd fiber degree is that there is a unique stable bundle of a given determinant of odd degree on each smooth fiber $f$. Using this, it is easy to show that there exists a rank two vector bundle $V_0$ whose restriction to {\it every\/} fiber $f$ is stable, and that $V_0$ is unique up to twisting by a line bundle. The bundle $V_0$ is the progenitor of all stable bundles on $S$, in the sense that every stable rank two vector bundle is obtained from $V_0$ by making elementary modifications along fibers. Generically, this involves choosing $t$ smooth fibers $f_i$ and line bundles $\lambda _i$ of degree $e+1$ on $f_i$. These choices define the birational isomorphism from the moduli space to $\Sym ^tJ^{e+1}(S)$. Given the above analysis of stable bundles, the main problem in computing Donaldson polynomials is to fit together all of the various possible descriptions of stable bundles into a universal family whose Chern classes can be calculated. This is easier said than done! Even in the case where $S$ has a section, the construction of the universal bundle for the four-dimensional moduli space, which just involves well-known techniques of extensions and elementary modifications, is already quite involved. We shall therefore proceed differently, and try to describe the moduli spaces and Chern classes involved up to contributions which only depend on the analytic type of a neighborhood of the multiple fibers. But we shall not try to analyze these contributions explicitly. Instead we shall repeatedly use the fact that an elliptic surface with $p_g=0$ and just one multiple fiber is a rational surface, and thus its Donaldson polynomials are the same as those for an elliptic surface with $p_g=0$ and with a section, or equivalently no multiple fibers. Thus if we know these, we can try to interpolate this knowledge into the general case. We shall use this idea twice. The first application will be to calculate the invariant $\gamma _1$. Here the moduli space is $J^{e+1}(S)$ and a lengthy calculation with the Grothendieck-Riemann-Roch theorem identifies the divisor corresponding to the $\mu$-map up to a rational multiple of the fiber, which depends only on the multiplicities. Appealing to the knowledge of the invariant for a rational surface enables us to determine this multiple. Of course, it is likely that the exact multiple could also be determined by a direct calculation. In order to calculate the polynomial $\gamma _2$, we shall use a variant of this idea. In this case, the divisor corresponding to the $\mu$-map is essentially known from the corresponding calculation in the case of $\gamma _1$. However what changes is the moduli space itself: the presence of multiple fibers means that the birational map from the moduli space to $\operatorname{Hilb}^2J^{e+1}(S)$ is not a morphism, and the actual moduli space differs from $\operatorname{Hilb}^2J^{e+1}(S)$ in codimension two. Thus while the divisors are known, their top self-intersection is not. Again using the rational elliptic surfaces, we are able to determine the discrepancy between the self-intersection of the $\mu$-divisors in $\operatorname{Hilb}^2J^{e+1}(S)$ and in the actual moduli space. The methods used here are in a certain sense the analogue in algebraic geometry of gluing techniques for ASD connections. Although the actual arguments are rather involved, the main point to emphasize here is that the coefficients of the Donaldson polynomial are quite formally determined by the knowledge of the polynomial for a rational surface. It is natural to wonder if the techniques in this paper can be pushed further to determine $\gamma _t$ for all $t$. I believe that this should be possible, although one necessary and so far missing ingredient in this approach is the knowledge of the multiplication table for divisors in $\operatorname{Hilb}^2 J^{e+1}(S)$. Here is a rapid description of the contents of the paper. In Section 1 we describe some general results on rank two vector bundles on an elliptic curve. In Section 2 these results are extended to cover the case of an irreducible nodal curve of arithmetic genus one. In Section 3 we give the classification of stable bundles on an elliptic surface $S$ and prove Theorem 1. In Section 4 we specialize to the case of a surface with a section. Our purpose here is twofold: First, we would like to show how many of the results of the preceding section take a very concrete form in this case. Secondly, we shall make a model for a piece of the four-dimensional moduli space which we shall need to use later. In Section 5 we calculate the two-dimensional invariant $\gamma _1$ in case $S$ has a section. This calculation has already been done by a different method in Section 4 and will be redone in full generality. However it seemed worthwhile to do this special case in order to make the general calculation more transparent. The next three sections are devoted to calculating $\gamma _1$ in general. The outline of the argument is given in Section 6. We construct a coherent sheaf which is an approximation to the universal bundle over the moduli space, over a branched cover $T$ of $S$. We determine its Chern classes via a lengthy calculation using the Grothendieck-Riemann-Roch theorem, which is given in Section 7. The necessary correction terms are identified via the results in Section 8. In Sections 9 and 10 we deal with the invariant $\gamma _2$. Once again the outline of the calculation is given first and the technical details are postponed to Section 10. The paper concludes with an appendix which collects some general results about elementary modifications. \section{Notation, conventions, and preliminaries.} All spaces are over $\Cee$, all sheaves are coherent sheaves in the classical topology unless otherwise specified. We do not distinguish between a vector bundle and its locally free sheaf of sections. Given s subvariety $Y$ of a compact complex manifold $X$, we denote the associated cohomology class by $[Y]$. If $V$ is a rank two vector bundle on a complex manifold or smooth scheme $X$, we shall frequently need to consider the first Pontrjagin class of $\operatorname{ad}V$, which is $c_1^2(V) - 4c_2(V)$. We will denote this expression by $p_1(\operatorname{ad}V)$. We shall occasionally and incorrectly use the shorthand $p_1(\operatorname{ad}V)$, for an arbitrary coherent sheaf $V$, to denote $c_1^2(V) - 4c_2(V)$. Given a vector bundle $V$, we shall need to know how $p_1(\operatorname{ad}V)$ changes under elementary modifications. Recall that an elementary modification is defined as follows. Let $X$ be a smooth scheme and let $D$ be an effective divisor on $X$, not necessarily smooth, with $i\: D\to X$ the inclusion. Let $L$ be a line bundle on $D$. Then $i_*L$ is a coherent sheaf on $X$, which we shall frequently just denote by $L$. Suppose that $V_0$ is a rank two vector bundle on $X$ and that $V_0 \to i_*L$ is a surjective homomorphism. Let $V$ be the kernel of the map $V \to i_*L$. Then $V$ is again a rank two vector bundle on $X$ (and in particular it is locally free). We call $V$ an {\sl elementary modification\/} of $V_0$. The change in $p_1$ is given as follows: \lemma{0.1} Let $X$ be a smooth scheme and let $D$ be an effective divisor on $X$, not necessarily smooth. Let $L$ be a line bundle on $D$ and $V_0$ a rank two vector bundle, and suppose that there is an exact sequence $$0 \to V \to V_0 \to i_*L \to 0,$$ where $i\: D \to X$ is the inclusion. Then $$p_1(\operatorname{ad}V) - p_1(\operatorname{ad}V_0) = 2c_1(V_0)\cdot [D] + [D]^2 -4 i_*c_1(L).$$ \endstatement \proof The proof follows easily from standard formulas for $c_1(V)$ and $c_2(V)$, cf\. [7] or [5]. \endproof Next we will recall some properties of the scheme $\operatorname{Hilb}^2S$, where $S$ is an algebraic surface. In general, we denote by $\operatorname{Hilb}^nS$ the smooth projective scheme parametrizing $0$-dimensional subschemes of $S$ of length $n$. There is a universal codimension two subscheme $\Cal Z \subset S \times \operatorname{Hilb}^nS$. We may describe the case $n=2$ quite explicitly. Let $\tilde H$ be the blowup of $S\times S$ along the diagonal $\Bbb D$ and let $\tilde \Bbb D$ be the exceptional divisor. There is an involution $\iota$ of $\tilde H$ whose fixed set is $\tilde \Bbb D$. We claim that the quotient $\tilde H /\iota$ is naturally $\operatorname{Hilb}^2S$. Indeed, if $\tilde\Bbb D_{12}$ and $\tilde\Bbb D_{13}$ are the proper transforms in $S\times \tilde H$ of the subsets $$\Bbb D_{1j} = \{\, p \in S\times S\times S \mid \pi _1(p) = \pi _j(p)\,\},$$ then $\tilde \Cal Z = \tilde \Bbb D_{12} + \tilde \Bbb D_{13}$ is a codimension two subscheme of $S\times \tilde H$ which is easily seen to be a local complete intersection. Thus it defines a flat family of subschemes of $S$ and so a morphism $\pi \:\tilde H \to \operatorname{Hilb}^2S$. It is easy to see that the induced morphism $\tilde H/\iota \to \operatorname{Hilb}^2S$ is an isomorphism. The projection $\Cal Z \to \operatorname{Hilb}^2S$ is a double cover which identifies $\Cal Z$ with $\tilde H$. Given $\alpha \in H_2(S)$, we can define the element $D_\alpha \in H^2(\operatorname{Hilb}^2S)$ by taking slant product with $[\Cal Z] \in H^4(S\times \operatorname{Hilb}^2S)$. If for example $\alpha = [C]$ where $C$ is an irreducible curve on $S$, then $D_\alpha$ is represented by the effective divisor consisting of pairs $\{x, y\}$ of points of $S$ such that either $x$ or $y$ lies on $C$. The inverse image $\pi ^*D_\alpha\in H^2(\tilde H)$ is the pullback of the class $1\otimes \alpha + \alpha \otimes 1\in H^2(S\times S)$. There is also the class in $ H^2(\operatorname{Hilb}^2S)$ represented by the divisor $E$ of subschemes of $S$ whose support is a single point. Since $\pi$ is branched over $E$, the class $[E]$ is divisible by $2$ and $\pi ^*[E]= 2\tilde \Bbb D$. Using this it is easy to check that the map $\alpha \mapsto D_\alpha$ defines an injection $H_2(S) \to H^2(\operatorname{Hilb}^2S)$ and that $H^2(\operatorname{Hilb}^2S) = H_2(S) \oplus \Zee\cdot [E/2]$. Finally the multiplication table in $H^2(\operatorname{Hilb}^2S)$ can be determined from the fact that $\tilde H$ is the blowup of $S\times S$ along the diagonal and that the normal bundle of the diagonal in $S\times S$ is the tangent bundle of $S$: we have $$\gather D_\alpha ^4 = 3(\alpha ^2)^2; \qquad D_\alpha ^3\cdot E = 0; \qquad D_\alpha ^2\cdot E^2 = -8(\alpha ^2); \\ D_\alpha \cdot E^3 = -8(c_1(S)\cdot \alpha); \qquad E^4 = 8(c_2(S) - c_1(S)^2). \endgather$$ Finally we need to say a few words about calculating Donaldson polynomials. Let $M$ be a closed oriented simply connected 4-manifold with a generic Riemannian metric $g$, and let $P$ be a principal $SO(3)$-bundle over $M$ with invariants $w_2(P)=w$ and $p_1(P) = p$. There is a Donaldson polynomial $\gamma _{w,p}(S)$ defiend via the moduli space of $g$-ASD connections on $P$, together with a choice of orientation for this space. If $b_2^+(M)>1$, then this polynomial is independent of $g$, whereas if $b_2^+(M)=1$ then it only depends on a certain chamber in the positive cone of $H^2(M; \Ar)$. If $M=S$ is a complex surface, $\Delta$ is a holomorphic line bundle such that $w= c_1(\Delta )\mod 2$ and $g$ is a Hodge metric corresponding to an ample line bundle $L$, there is a diffeomorphism of real analytic spaces from the moduli space of $g$-ASD connections on $P$ to the moduli space of $L$-stable rank two vector bundles $V$ on $S$ with $c_1(V) = \Delta$ and $c_2(V) = (\Delta ^2-p)/4$. We denote this moduli space for the moment by $\frak M$. We shall always choose the orientation of the moduli space of $g$-ASD connections which agrees with the natural complex orientation of $\frak M$. If $\frak M$ is smooth, compact, and of real dimension $2d$ and there is a universal bundle $\Cal V$ over $S\times \frak M$, then slant product with $-p_1( \ad \Cal V)/4$ defines a homomorphism $\mu$ from $H_2(S)$ to $H^2(\frak M)$. In general we can define the holomorphic vector bundle $\ad \Cal V$ even when the universal bundle $\Cal V$ does not exist. To see this, note that there is always a universal $\Pee ^1$-bundle $\pi \: \Pee (\Cal V)\to S\times \frak M$, and taking $\pi _*$ of the relative tangent bundle gives $\ad \Cal V$. Thus given a class $\Sigma \in H_2(S)$, we can evaluate $\mu (\Sigma)^d$ on the fundamental class of $\frak M$ and this gives the value $\gamma _{w,p}(\Sigma, \dots, \Sigma)$. For the applications in this paper, since the moduli spaces always have the correct dimension and in particular are empty if $-p-3\chi(\scrO_S)<0$, the moduli spaces of complex dimension zero and two are compact. For the four-dimensional moduli space, we can calculate $\gamma _{w,p}$ by choosing an appropriate compactification of $\frak M$. For the purposes of gauge theory, there is the Uhlenbeck compactification. For the purposes of algebraic geometry, there is the Gieseker compactification $\overline{\frak M}$. Following O'Grady [11], the divisors $\mu (\Sigma)$ extend naturally to divisors $\nu(\Sigma)$ on $\overline{\frak M}$, which we shall continue to denote by $\mu(\Sigma)$. If there is a universal sheaf $\Cal V$ on the Gieseker compactification, then the $\mu$-map is again defined by taking slant product with $-p_1(\ad \Cal V)/4$. In general, for holomorphic curves $\Sigma$ (which would suffice for the applications in this paper) we can use determinant line bundles on the moduli functor. For a general $\Sigma \in H_2(S)$, we can define $\mu(\Sigma)$ for the moduli spaces that arise in this paper (where there are no strictly semistable sheaves) as follows: there exists a universal coherent sheaf $\Cal E$ over $S\times U$, where $U$ is the open subset of an appropriate Quot scheme corresponding to stable torsion free sheaves with the appropriate Chern classes. Thus we can define an element of $H^2(U)$ by taking slant product with $p_1(\ad \Cal E)$. As $\overline{\frak M}$ is a quotient of $U$ by a free action of $PGL(N)$ for some $N$, $H^2(\overline{\frak M})\cong H^2(U)$, and this defines $\mu(\Sigma)$ in general. We can now evaluate $\mu (\Sigma)^d$ on the fundamental class of $\overline{\frak M}$. By recent results of Li [9] and Morgan [10] the result is again $\gamma _{w,p}$. Strictly speaking, their results are stated with certain extra assumptions. However, the cases we will need in this paper involve the following situation: all moduli spaces are smooth of the expected dimension and there are no strictly semistable torsion free sheaves. Under these assumptions, the proofs in e.g\. [10] go over essentially unchanged. \section{1. Review of results on vector bundles over elliptic curves.} We recall the following well-known result of Atiyah [1]: \theorem{1.1} Let $V$ be a rank two vector bundle over a smooth curve $C$ of genus $1$. Then exactly one of the following holds: \roster \item"{\rm (i)}" $V$ is a direct sum of line bundles; \item"{\rm (ii)}" $V$ is of the form $\Cal E \otimes L$, where $L$ is a line bundle on $C$ and $\Cal E$ is the \rom(unique\rom) extension of $\scrO _C$ by $\scrO _C$ which does not split into the direct sum $\scrO _C\oplus \scrO _C$; \item"{\rm (iii)}" $V$ is of the form $\Cal F_p\otimes L$, where $L$ is a line bundle on $C$, $p \in C$, and $\Cal F_p$ is the unique nonsplit extension of the form $$0 \to \scrO _C \to \Cal F_p \to \scrO _C(p) \to 0.\qed $$ \endroster \endstatement We shall not prove (1.1) but shall instead prove the analogous statement in the slightly more complicated case of a singular curve in Section 2. \corollary{1.2} Let $V$ be a stable rank two bundle over a smooth curve $C$ of genus $1$. Then $\deg V$ is odd, say $\deg V = 2e +1$. Moreover, for every line bundle $\lambda$ of degree $e+1$ we have $\dim \operatorname{Hom}(V, \lambda) = 1$, $H^1(V\spcheck\otimes \lambda ) = 0$, and there is an exact sequence $$0 \to \mu \to V \to \lambda \to 0,$$ where $\mu$ is a line bundle of degree $e$ on $C$, uniquely determined by the isomorphism $$\mu \otimes \lambda = \det V,$$ and the surjection $V \to \lambda$ is unique mod scalars. \endstatement \proof Clearly, if $V$ is stable we must be in case (iii) of the theorem. Conversely, suppose that $V$ is as in (iii). We shall show that $V$ is stable. It suffices to show that $\Cal F_p$ is stable. Let $M$ be a line bundle on $C$ of degree at least $\det \Cal F_p /2 = 1/2$ such that there is a nonzero map $M \to \Cal F_p$. Clearly $\deg M \leq 1$ and $\deg M = 1$ if and only if $M = \scrO _C(p)$. Since $\deg M \geq 1/2$, $\deg M = 1$ and $M = \scrO _C(p)$. But then $\Cal F_p$ is the split extension, contradicting the definition of $\Cal F_p$. Thus $\Cal F_p$ is stable. Now let $V$ be a stable bundle of degree $2e+1$, so that there exists a line bundle $L$ of degree $e$ on $C$ with $V = \Cal F_p \otimes L$. Then, if $\lambda$ is a line bundle of degree $e+1$, we have an exact sequence $$0 \to \operatorname{Hom} (L\otimes \scrO_C(p), \lambda) \to \operatorname{Hom} (V, \lambda) \to \operatorname{Hom}(L, \lambda) \to H^1(\lambda \otimes L^{-1}\otimes \scrO_C(-p)).$$ If $\lambda = L\otimes \scrO_C(p)$, then by assumption there exists a surjection $V\to \lambda$. If $\varphi _1$ and $\varphi _2$ are two nonzero maps from $V$ to $\lambda$, then for every $p\in C$ there is a scalar $c$ such that $\varphi _1 -c\varphi _2$ vanishes at $p$, and thus defines a map $V\to \lambda \otimes \scrO_C(-p)$. By stability this map must be zero, so that $\varphi _1 =c\varphi _2$. Thus the surjection is unique mod scalars. If $\lambda \neq L\otimes \scrO_C(p)$, then $\lambda\otimes L^{-1}\otimes \scrO_C(-p)$ is a line bundle of degree zero on $C$ which is not trivial. Hence $H^1(\lambda \otimes L^{-1}\otimes \scrO_C(-p))=0$, and $\operatorname{Hom}(V, \lambda) \cong \operatorname{Hom}(L, \lambda) $. Moreover $\operatorname{Hom}(L, \lambda) = H^0(L^{-1}\otimes \lambda)$ has dimension one since $\deg (L^{-1}\otimes \lambda) = 1$. Thus there is a nontrivial map $V\to \lambda$, which is unique mod scalars. If it is not surjective, there is a factorization $V\to \lambda \otimes \scrO_C(-q) \subset \lambda$, and this contradicts the stability of $V$. Lastly we see that $H^1(V\spcheck\otimes \lambda) \cong H^1(L^{-1}\otimes \lambda)$, and this last group is zero since $\deg (L^{-1}\otimes \lambda) = 1$. \endproof We can generalize the last statement of (1.2) as follows. \lemma{1.3} Let $C$ be a smooth curve of genus one. \roster \item"{(i)}" Let $V$ be a stable rank two vector bundle over $C$ and suppose that $\deg V = 2e +1$. Let $d \geq e+1$, and let $\lambda$ be a line bundle on $V$ of degree $d$. Then $\dim \operatorname{Hom}(V, \lambda) = 2d-2e-1$, and there exists a surjection from $V$ to $\lambda$. Conversely, with $V$ as above, let $\lambda$ be a line bundle such that there exists a nonzero map from $V$ to $\lambda$. Then $\deg \lambda \geq e+1$. \item"{(ii)}" Suppose that $V=L_1\oplus L_2$ is a direct sum of line bundles $L_i$ with $\deg V = 2e+1$ and $\deg L_1 \leq e<\deg L_2$. Let $\lambda$ be a line bundle on $C$ with $d=\deg \lambda > \deg L_2$. Then $\dim \operatorname{Hom}(V, \lambda) = 2d-2e-1$, and there exists a surjection from $V$ to $\lambda$. Conversely, if $\lambda$ is a line bundle and there exists a surjection from $L_1\oplus L_2$ to $\lambda$, then either $\deg \lambda > \deg L_2$ or $\lambda = L_2$ or $\lambda = L_1$. If $\lambda = L_2$, then $\dim \operatorname{Hom}(V, \lambda) = 2d-2e$, where $d=\deg L_2= \deg \lambda$. \endroster \endstatement \proof We shall just prove (i), as the proof of (ii) is simpler. Let $\lambda$ be a line bundle on $C$ of degree $d\geq e+1$. We may assume that $\deg \lambda >e+1$, the case $\deg \lambda = e+1$ having been dealt with in (1.2). There is an exact sequence $$0 \to L_1 \to V \to L_2 \to 0,$$ where $\deg L_1 = e$ and $\deg L_2 = e+1$. Thus there is an exact sequence $$0 \to H^0(L_2^{-1}\otimes \lambda) \to \operatorname{Hom}(V, \lambda) \to H^0(L_1^{-1}\otimes \lambda)\to H^1(L_2^{-1}\otimes \lambda).$$ We have $\deg (L_1^{-1}\otimes \lambda )= d-e>0$ and $\deg (L_2^{-1}\otimes \lambda )= d-e-1>0$. Thus $H^1(L_2^{-1}\otimes \lambda) =0$, $\dim H^0(L_1^{-1}\otimes \lambda) = d-e$, and $\dim H^0(L_2^{-1}\otimes \lambda) = d-e-1$. So $\dim \operatorname{Hom}(V, \lambda) = 2d-2e-1$. To see the last statement, let $Y$ be the set of elements $\varphi$ of $\operatorname{Hom}(V, \lambda)$ such that $\varphi$ is not surjective. Then $Y$ is the union over $x\in C$ of the spaces $\operatorname{Hom}(V, \lambda\otimes \scrO _C(-x))$, each of which has dimension at most $2d-2e -3$. So the dimension of $Y$ is at most $2d-2e -2$. Thus $\operatorname{Hom}(V, \lambda)-Y$ is nonempty, and every $\varphi \in \operatorname{Hom}(V, \lambda)-Y$ is a surjection. The final statement of (i) is then an immediate consequence of the stability of $V$. \endproof For future use let us also record the following lemmas: \lemma{1.4} Let $C$ be a smooth curve of genus one and let $\xi$ be a line bundle on $C$ of degree zero such that $\xi ^{\otimes 2} \neq 0$. Let $V$ be a stable rank two vector bundle on $C$. Then $\Hom (V, V\otimes \xi) = 0$. \endstatement \proof Since $\deg V = \deg (V\otimes \xi)$ and both are stable, a nonzero map between them must be an isomorphism, by standard results on stable bundles. However $\det (V\otimes \xi) = \det V \otimes \xi ^{\otimes 2} \neq \det V$, and so the bundles cannot be isomorphic. Thus there is no nonzero map from $V$ to $V\otimes \xi$. \endproof \corollary{1.5} Let $\bold F\subset X$ be a scheme-theoretic multiple fiber of odd multiplicity $m$ of an elliptic surface, and let $F$ be the reduction of $\bold F$. Let $\bold V$ be a rank two vector bundle on $\bold F$ whose restriction $V$ to $F$ is stable. Then $\dim _\Cee\Hom (\bold V, \bold V)= 1$ and every nonzero map from $\bold V$ to itself is an isomorphism. \endstatement \proof Let $\xi$ be the normal bundle of $F$ in $X$. Thus $\xi$ has order $m$. For $a>0$, let $aF$ denote the subscheme of $X$ defined by by the ideal sheaf $\scrO_X(-aF)$. Thus $\bold F=mF$ and there is in general an exact sequence $$0 \to \xi ^{-a} \to \scrO_{(a+1)F} \to \scrO_{aF} \to 0.$$ Tensor the above exact sequence by $Hom (\bold V, \bold V) = \bold V\spcheck \otimes \bold V$ and take global sections. This gives an exact sequence $$0 \to \Hom (V, V\otimes \xi ^{-a}) \to \Hom (\bold V|(a+1)F, \bold V|(a+1)F) \to \Hom (\bold V|aF, \bold V|aF).$$ For $a=1$ we have $\dim _\Cee\Hom (\bold V|F, \bold V|F)= \dim _\Cee\Hom (V,V) = 1$. For $1\leq a \leq m-1$, $\xi ^{-a}$ is a nontrivial line bundle of odd order. Thus by (1.4) $\Hom (V, V\otimes \xi ^{-a}) = 0$. It follows that the map $\Hom (\bold V|(a+1)F, \bold V|(a+1)F) \to \Hom (\bold V|aF, \bold V|aF)$ is an injection, so that by induction $\dim _\Cee \Hom (\bold V|(a+1)F, \bold V|(a+1)F)\leq 1$. On the other hand multiplication by an element of $H^0(\scrO_{(a+1)F})=\Cee$ defines a nonzero element of $\Hom (\bold V|(a+1)F, \bold V|(a+1)F)$. Thus $\dim _\Cee \Hom (\bold V|(a+1)F, \bold V|(a+1)F) = 1$ for all $a\leq m-1$, and in particular $\dim _\Cee \Hom (\bold V|mF, \bold V|mF)=1$. \endproof \section{2. The case of a singular curve.} Our goal in this section will be to show that the statements of the previous section hold for rank two vector bundles on singular nodal curves $C$. Let $C$ be an irreducible curve of arithmetic genus one, which has one node $p$ as a singularity. Locally analytically, then, $\hat {\Cal O}_{C,p} \cong \Cee [[x,y]]/(xy)$. Let $a\: \tilde C \to C$ be the normalization map, and let $p_1$ and $p_2$ be the preimages of the singular point on $\tilde C$. We begin by giving a preliminary discussion concerning torsion free sheaves on $C$. \definition{Definition 2.1} A {\sl torsion free rank one sheaf\/} on $C$ is a coherent sheaf which has rank one at the generic point of $C$ and has no local sections vanishing on an open set. It is well known that every torsion free rank one sheaf on $C$ is either a line bundle or of the form $a_*L$, where $L$ is a line bundle on $\tilde C$. For example, the maximal ideal sheaf of the singular point $p$ of $C$ is $a_*L$, where $L= \scrO_{\tilde C}(-p_1-p_2)$, where $p_1$ and $p_2$ are the preimages of $p$ in $\tilde C$. Here the line bundle $L$ has degree $-2$ on $\tilde C$. We define the {\sl degree} of a torsion free rank one sheaf $F$ on $C$ by $\deg F = \chi (F)$. By the Riemann-Roch theorem on $C$, $\deg F$ is the usual degree in case $F$ is a line bundle, whereas for $F= a_*L$, an easy calculation shows that $\chi (F) = \deg L +1$. (Note that, in case $p_a(C)$ is arbitrary, we would have to correct by a term $p_a(C) -1$, which is zero in our case, to get the usual answer for a line bundle.) It is easy to check that, if $F$ is a rank one torsion free sheaf on $C$ and $L$ is a line bundle, then $\deg (F\otimes L) = \deg F + \deg L$. \enddefinition \medskip \lemma{2.2} If $F_1$ and $F_2$ are torsion free rank one sheaves on $C$, then so is $Hom (F_1, F_2)$, and $$\deg Hom (F_1, F_2) = \cases \deg F_2 - \deg F_1, &\text{if one of the $F_i$ is a line bundle}\\ \deg F_2 - \deg F_1 + 1,&\text{if neither $F_1$ nor $F_2$ is a line bundle}. \endcases$$ Finally if $\deg F_2>\deg F_1$ and neither is a line bundle, then the natural map from $\Hom (F_1, F_2)\otimes \scrO_{C,p}$ to $\Hom _{\scrO_{C,p}}(\frak m_p, \frak m_p)$ is surjective, where $\frak m_p$ is the maximal ideal of $\scrO_{C,p}$. \endstatement \proof The proof is clear if $F_1$ is a line bundle. Thus we may assume that $F_1$ is of the form $a_*L$ for a line bundle $L$ on $\tilde C$. First assume that $F_2$ is a line bundle. An easy calculation shows that $Hom (F_1, F_2) = a_*(L^{-1} \otimes \scrO_{\tilde C}(-p_1-p_2))\otimes F_2$. This is just the local calculation $Hom _{\scrO _{C,p}}(a_*\widetilde{\scrO _{C,p}}, \scrO _{C,p}) = \frak m_p$, where $\frak m_p$ is the maximal ideal of $\scrO _{C,p}$ and the isomorphism is canonical. Thus $Hom (F_1, F_2)$ is again a torsion free rank one sheaf and $$\deg Hom (F_1, F_2) = -\deg L + 1-2 +\deg F_2 = \deg F_2 - \deg F_1.$$ Now assume that $F_1 = a_*L_1$ and $F_2 = a_*L_2$. Again using a local calculation $Hom _{\scrO _{C,p}}(a_*\widetilde{\scrO _{C,p}},a_*\widetilde{\scrO _{C,p}}) = a_*\widetilde{\scrO _{C,p}}$, where the isomorphism is also canonical, it is easy to check that $Hom (a_*L_1, a_*L_2) = a_*(L_1^{-1}\otimes L_2)$, and so $Hom (F_1, F_2)$ is again a torsion free rank one sheaf. Moreover $$\deg Hom (F_1, F_2) = \deg L_2 -\deg L_1 +1= \deg F_2-\deg F_1 + 1.$$ To see the final statement, again writing $F_1 = a_*L_1$ and $F_2 = a_*L_2$, we have $Hom (a_*L_1, a_*L_2) = a_*(L_1^{-1}\otimes L_2)$. Moreover the global sections of $L_1^{-1}\otimes L_2$ separate the points $p_1$ and $p_2$. It is then easy to see that the map $\Hom (F_1, F_2)\otimes \scrO_{C,p} \to \Hom _{\scrO_{C,p}}(\frak m_p, \frak m_p) = \widetilde{\scrO _{C,p}}$ is surjective. \endproof \lemma{2.3} Let $F$ be a torsion free rank one sheaf on $C$. If $\deg F >0$, or if $\deg F = 0$ and $F$ is not trivial, then $h^0(F) = \deg F$ and $h^1(F) =0$. If $\deg F < 0$, or if $\deg F = 0$ and $F\neq \scrO _C$, then $h^0(F) = 0$ and $h^1(F) = \deg F$. \endstatement \proof If $\deg F \geq 0$ and $F$ is not $\scrO _C$, then the claim that $h^0(F) = \deg F$ is clear if $F$ is a line bundle and follows from $h^0(a_*L) = \deg L +1$ in case $L$ is a line bundle of degree at least $-1$ on $\tilde C \cong \Pee ^1$. In this case, since by definition $\deg F = \chi (F) = h^0(F)$, we must have $h^1(F) = 0$. The proof of the second statement is similar. \endproof Next let us consider extensions of torsion free sheaves. The maximal ideal $\frak m_p$ has the following local resolution, where we set $R = \scrO _{C,p}$: $$\dots \to R \oplus R \to R \oplus R \to \frak m_p \to 0,$$ where the maps $R \oplus R \to R \oplus R$ alternate between $(\alpha, \beta) \mapsto (x\alpha, y\beta)$ and $(\alpha, \beta) \mapsto (y\alpha, x\beta)$. A calculation shows that $\operatorname{Ext} ^1_R(\frak m_p, \frak m_p)$ has length two. More intrinsically it is isomorphic to $\scrO _{\tilde C}(-p_1-p_2)/\scrO _{\tilde C}(-2p_1-2p_2)$. Thus as an $R$-module, $\operatorname{Ext} ^1_R(\frak m_p, \frak m_p) \cong \tilde R/\tilde{\frak m}_p$, where $\tilde R$ is the normalization of $R$ and $\tilde{\frak m}_p = \frak m_p\tilde R$. We can describe the $\tilde R$-action on $\operatorname{Ext} ^1_R(\frak m_p, \frak m_p)$ more invariantly as follows: multiplication by $r \in \tilde R$ gives an endomorphism $\frak m_p\to \frak m_p$, and hence an action of $\tilde R$ on $\operatorname{Ext} ^1_R (\frak m_p, \frak m_p)$. We leave to the reader the straightforward verification that this action is the same as the action on $\operatorname{Ext} ^1_R(\frak m_p, \frak m_p)$ implicit in the isomorphism $\operatorname{Ext} ^1_R(\frak m_p, \frak m_p) \cong \tilde R/\tilde{\frak m}_p$ given above. There is an induced action of the invertible elements ${\tilde R}^*$ on $(\operatorname{Ext} ^1_R(\frak m_p, \frak m_p) -0)/\Cee ^* = \Pee ^1$. Since $R^*$ acts trivially, this induces an action of ${\tilde R}^*/R^* \cong \Cee ^*$ on $(\operatorname{Ext} ^1_R(\frak m_p, \frak m_p) -0)/\Cee ^*$. It is easy to see that there are three orbits of this action: an open orbit isomorphic to $\Cee ^*$ and two closed orbits which are points in $\Pee ^1$, corresponding to the case of an element $e \in \operatorname{Ext} ^1_R(\frak m_p, \frak m_p)$ such that $\tilde R\cdot e \neq \operatorname{Ext} ^1_R(\frak m_p, \frak m_p)$. Given an element $e\in \operatorname{Ext} ^1_R(\frak m_p, \frak m_p)$, denote the corresponding extension of $\frak m_p$ by $\frak m_p$ by $M_e$. Note that two extensions $M_e$ and $M_{e'}$ such that $e$ and $e'$ lie in the same $\tilde R^*$-orbit are abstractly isomorphic as $R$-modules, via a diagram of the form $$\CD 0 @>>> \frak m_p @>>> M_{re} @>>> \frak m_p @>>> 0 \\ @. @V=VV @VVV @V{\times r}VV @. \\ 0 @>>> \frak m_p @>>> M_e @>>> \frak m_p @>>> 0, \endCD$$ where $r\in \tilde R^*$ is such that $re = e'$. \lemma{2.4} $M_e$ is locally free if and only if the image of $e$ in $$\Bigl(\operatorname{Ext} ^1_R(\frak m_p, \frak m_p) -0\Bigr)\big/\Cee ^*$$ is not a closed orbit of $\tilde R$. \endstatement \proof Consider the long exact Ext sequence $$\operatorname{Hom}_R(\frak m_p, \frak m_p) \to \operatorname{Ext} ^1_R (\frak m_p, \frak m_p) \to \operatorname{Ext} ^1_R(M_e, \frak m_p).$$ We see that $\operatorname{Ext} ^1_R(M_e, \frak m_p)$ contains as a submodule $\operatorname{Ext} ^1_R(\frak m_p, \frak m_p)/\tilde R\cdot e$. Thus if $\tilde R\cdot e \neq \operatorname{Ext} ^1_R(\frak m_p, \frak m_p)$, then $\operatorname{Ext} ^1_R(M_e, \frak m_p) \neq 0$ and so $M_e$ is not locally free. Conversely suppose that the image of $e$ does not lie in one of the closed orbits. Since every two extensions in the same orbit are abstractly isomorphic, it will suffice to exhibit one locally free extension of $\frak m_p$ by $\frak m_p$. However we have the obvious surjection $R\oplus R \to \frak m_p$ given above, and its kernel is easily seen to be isomorphic to $\frak m_p$ again. \endproof We leave as an exercise for the reader the description of the extensions corresponding to the closed orbits. Let us also note that, using the resolution above, a short computation shows that $\operatorname{Ext}^1_R(\frak m_p, R) = 0$. Thus there is no locally free $R$-module $M$ which sits in an exact sequence $$0 \to R \to M \to \frak m_p \to 0.$$ Globally, we have the following: \lemma{2.5} Let $n$ be a positive integer and let $\delta$ be a line bundle of degree one on $C$. \roster \item"{(i)}" There is a unique rank two vector bundle $V_{n, \delta}$ on $C$ such that $\det V_{n, \delta} = \delta$ and such that there is an exact sequence $$0 \to F \to V_{n, \delta} \to F' \to 0,$$ where $F$ and $F'$ are torsion free rank one sheaves of degrees $n$ and $1-n$ respectively, and $F$ and $F'$ are not locally free. \item"{(ii)}" Let $G$ be a torsion free rank one subsheaf of $V_{n, \delta}$. Then either $\deg G \leq -n$ or $G$ is contained in $F$. \item"{(iii)}" The vector bundle $V_{n, \delta}$ is indecomposable for all $n$ and $\delta$ and $V_{n, \delta} \cong V_{n', \delta'}$ if and only if $n=n'$ and $\delta = \delta '$. \endroster \endstatement \proof To see (i), let $F$ and $F'$ be the unique torsion free rank one sheaves of degrees $n$ and $1-n$ respectively which are not locally free. Let us evaluate $\operatorname{Ext}^1(F', F)$. From the local to global Ext spectral sequence, there is an exact sequence $$0 \to H^1(Hom (F', F)) \to \operatorname{Ext}^1(F', F) \to H^0(Ext ^1(F', F)) \to 0.$$ Now $\chi (Hom (F', F)) = \deg Hom (F', F) = h^0(Hom (F', F))$, since $$\deg Hom (F', F) = 2n>0$$ by (2.2) and (2.3). So $H^1(Hom (F', F)) = 0$. Thus $\operatorname{Ext}^1(F', F) \cong H^0(Ext ^1(F', F))$. Moreover, the set of all locally free extensions is naturally a principal homogeneous space over $H^0(\scrO_{\tilde C}^*/\scrO _C^*) = \tilde R^*/R^*$. On the other hand, from the exact sequence $$0 \to \scrO _C^* \to \scrO_{\tilde C}^* \to \scrO_{\tilde C}^*/\scrO _C^* \to 0,$$ we have a natural isomorphism $\operatorname{Pic}^0C \cong H^0(\scrO_{\tilde C}^* /\scrO _C^*)=\tilde R^*/R^*$. Let $\partial\: \tilde R^*/R^* \to \operatorname{Pic}^0C$ be the coboundary map; it is an isomorphism. Given $e \in \operatorname{Ext}^1(F', F) \cong H^0(Ext ^1 (F', F))$, let $V_e$ be the extension corresponding to $e$. A straightforward exercise in the definitions shows that, for $r\in \tilde R$, $$\det V_{r\cdot e} = \partial (r) \otimes \det V_e.$$ From this it is clear that there is a unique extension $V_{n, \delta}$ with determinant $\delta$. Next we prove (ii). Let $G$ be a torsion free rank one subsheaf, possibly a line bundle, of $V_{n, \delta}$ such that $\deg G >-n$. We have an exact sequence $$0 \to \operatorname{Hom}(G,F) \to \operatorname{Hom}(G, V_{n, \delta}) \to \operatorname{Hom}(G,F').$$ Moreover $\operatorname{Hom}(G,F') = H^0(Hom(G,F'))$. First suppose that either $\deg G > 1-n$ or that $G$ is locally free. The torsion free sheaf $Hom(G,F')$ has degree either $1-n-\deg G$ or $2-n - \deg G$, depending on whether $G$ is or is not locally free. In any case it has degree $\leq 0$ and is not locally free, so that $H^0(Hom(G,F')) =0$, by (2.2) and (2.3). So every such $G$ is contained in $F$. In the remaining case where $\deg G =1-n$ and $G$ is not locally free, then $G=F'$. Since $\operatorname{Hom}(F',F') \cong k^*$, every nonzero homomorphism from $F'$ to itself is an isomorphism. Thus the exact sequence defining $V_{n, \delta}$ would be split, contrary to assumption. Hence this last case is impossible. To see (iii), let $G$ be a torsion free rank one subsheaf of degree at least $1-n$ such that $V_{n, \delta}/G$ is torsion free. Then by (ii) $G=F$. This clearly implies that $V_{n, \delta} \cong V_{n', \delta'}$ if and only if $n=n'$ and $\delta = \delta '$ and that $V_{n, \delta}$ is indecomposable. \endproof \theorem{2.6} Let $C$ be an irreducible curve of arithmetic genus one, which has one node as a singularity. Let $V$ be a rank two vector bundle on $C$ and suppose that $\deg \det V = 2e+1$. Then $V$ is one of the following: \roster \item"{(i)}" A direct sum of line bundles; \item"{(ii)}" $L\otimes \Cal F_x$, where $L$ is a line bundle of degree $e$, $x\in C$ is a smooth point, and $\Cal F_x$ is the unique nontrivial extension $$0 \to \scrO _C \to \Cal F_x\to \scrO _C(x)\to 0;$$ \item"{(iii)}" $L\otimes V_{n, \delta}$, where $L$ is a line bundle of degree $e$ and $V_{n, \delta}$ is the rank two vector bundle described in \rom{(2.5)}. In this case, the subsheaf $L\otimes F$, where $F$ is the subsheaf in the definition of $V_{n, \delta}$, is the maximal destabilizing subsheaf. \endroster \endstatement \proof Clearly we may assume that $\deg V = 1$. By the Riemann-Roch theorem, $h^0(V) \neq 0$. Thus there is a map $\scrO _C \to V$. If this map is the inclusion of a subbundle, then $V$ is given as an extension $$0 \to \scrO _C \to V \to \scrO _C(x)\to 0$$ for some smooth point $x\in C$. Either this extension splits, in which case we are in case (i), or it does not in which case we are in case (ii). Now suppose that the map $\scrO_C \to V$ vanishes at some point. There is a largest rank one subsheaf $F$ of $V$ containing the image of $\scrO_C$, and $\deg F = n>0$. The quotient $V/F = F'$ is torsion free. If $F$ is a line bundle, then so is $F'$, since locally $\operatorname{Ext}^1_R(\frak m_p, R) = 0$. In this case $(F')^{-1}\otimes F$ has degree $2n-1 >0$, so that the extension splits and $V$ is the direct sum of $F$ and $F'$. Hence we are in case (i). Otherwise $F$ and $F'$ are not locally free. It follows that $V= V_{n, \delta}$ for $\delta = \det V$, and we are in case (iii). The last statement in (iii) then follows from the last paragraph of the proof of (2.5). \endproof Finally let us show that a statement analogous to (1.3) continues to hold for the case of a singular curve. \lemma{2.7} Let $C$ be an irreducible nodal curve of arithmetic genus one. \roster \item"{(i)}" Let $V$ be a stable rank two vector bundle over $C$ and suppose that $\deg V = 2e +1$. Let $d \geq e+1$, and let $\lambda$ be a torsion free rank one on $V$ of degree $d$. Then $\dim \operatorname{Hom}(V, \lambda) = 2d-2e-1$, and there exists a surjection from $V$ to $\lambda$. Moreover, if $\lambda$ is a line bundle on $C$ such that there exists a nonzero map from $V$ to $\lambda$, then $\deg \lambda \geq e+1$. Finally, if $d=e+1$, then $H^1(V\spcheck \otimes \lambda) = 0$. \item"{(ii)}" Suppose that $V=L_1\oplus L_2$ is a direct sum of line bundles $L_i$ with $\deg V = 2e+1$ and $\deg L_1 \leq e<\deg L_2$. Let $\lambda$ be a rank one torsion free sheaf on $C$ with $d=\deg \lambda > \deg L_2$. Then $\dim \operatorname{Hom}(V, \lambda) = 2d-2e-1$, and there exists a surjection from $V$ to $\lambda$. Moreover, if $\lambda$ is a rank one torsion free sheaf on $C$ such that there exists a surjection from $V$ to $\lambda$, then either $d=\deg \lambda > \deg L_2$ or $\lambda = L_2$ or $\lambda = L_1$ and $\dim \operatorname{Hom}(V, \lambda) = 2d-2e$. \item"{(iii)}" Suppose that $V = L\otimes V_{n, \delta}$ for some $n$, where $L$ is a line bundle of degree $e$ and that $L_2$ is the subsheaf $L\otimes F$ of $V$ of degree $e+n$ corresponding to the subsheaf $F$ of $V_{n, \delta}$ in the definition of $V_{n, \delta}$ and that $L_1$ is the quotient $V/L_2$. Let $\lambda$ be a rank one torsion free sheaf on $C$ with $d=\deg \lambda > \deg L_2 = e+n$. Then $\dim \operatorname{Hom}(V, \lambda) = 2d-2e-1$. Moreover, if there exists a surjection from $V$ to $\lambda$ then either $\deg \lambda > e+n$ or $\lambda = L_1$ and $\dim \operatorname{Hom}(V, \lambda) = 1$. \endroster \endstatement \proof The proof of (i) and (ii) follows the same lines as the proof of (1.3), with minor modifications, given Lemmas 2.2 and 2.3. Let us prove (iii) in the case where $\lambda$ is not locally free (the proof in the other case is slightly simpler). By definition there is an exact sequence $$0 \to L_2 \to V \to L_1 \to 0,$$ where $L_1$ and $L_2$ are not locally free and $\deg L_2 = e+n$, $\deg L_1 = 1-n+e$. There is a long exact sequence $$0 \to \operatorname{Hom}(L_1, \lambda ) \to \operatorname{Hom}(V, \lambda) \to \operatorname{Hom}(L_2, \lambda ) \to \operatorname{Ext}^1(L_1, \lambda).$$ Moreover, by the long exact sequence for $\operatorname{Ext}$, we have an exact sequence $$0 \to H^1((Hom (L_1, \lambda)) \to \operatorname{Ext}^1(L_1, \lambda) \to H^0(Ext ^1(L_1, \lambda)).$$ Since $\deg Hom (L_1, \lambda) = d-e+n +1 \leq 0$ and $Hom (L_1, \lambda)$ is not locally free, $H^1((Hom (L_1, \lambda)) = 0$ by (2.3). Moreover $H^0(Ext ^1(L_1, \lambda)) = \Cee ^2$. We claim that the composite map $\operatorname{Hom}(L_2, \lambda ) \to \operatorname{Ext}^1(L_1, \lambda) \to H^0(Ext ^1(L_1, \lambda))$ is surjective. Since $\deg L_2>\deg \mu$, the map $\operatorname{Hom}(L_2, \lambda)\to \operatorname{Hom}_R(\frak m_p, \frak m_p)\cong \tilde R$ is onto the quotient $\tilde R/\tilde \frak m_p$ by the last statement in (2.2). Thus the image of the map $\operatorname{Hom}(L_2, \lambda ) \to H^0(Ext ^1(L_1, \lambda))$ contains the orbit $\tilde R\cdot \xi \subseteq \operatorname{Ext}^1_R(\frak m_p,\frak m_p)$, where $\xi$ is the extension class. Since $V$ is locally free, this orbit is all of $\operatorname{Ext}^1_R(\frak m_p,\frak m_p)$ by the proof of (2.4), and so the map $\operatorname{Hom}(L_2, \lambda ) \to H^0(Ext ^1(L_1, \lambda))$ is onto. It follows that $$\align \dim \operatorname{Hom}(V, \lambda) &= \operatorname{Hom}(L_1, \lambda) + \operatorname{Hom}(L_2, \lambda) -2\\ &= d-(e+n) +1 + d-(1-n+e) + 1 - 2 = 2d -2e-1. \endalign$$ Let us finally consider the case when there is a surjection from $V$ to $\lambda$. Let the degree of $\lambda$ be $d+e$. Thus there is a surjection from $V_{n, \delta}$ to $\lambda \otimes L^{-1}$, which is of degree $d$. Let $G$ be the kernel of the map $V_{n, \delta} \to \lambda \otimes L^{-1}$. Then $\deg G = 1-d$. By (2.5)(ii), either $1-d \leq -n$ or $G\subseteq F$. Thus either $d>e$ or $\lambda = L_1$. In the last case, there is a unique surjection from $V$ to $L_1$ mod scalars, by the proof of (2.5)(iii). \endproof \section{3. A Zariski open subset of the moduli space.} Let $\pi \:S \to \Pee ^1$ be an algebraic elliptic surface of geometric genus $p_g(S)=p_g$. We shall always assume that the only singular fibers of $\pi$ are either irreducible nodal curves or multiple fibers with smooth reduction. Denote the multiple fibers by $F_1$ and $F_2$ and suppose that the multiplicity of $F_i$ is $m_i$. We shall assume that the multiple fibers lie over points where the $j$-invariant of $S$ is unramified. We denote by $J(S)$ the associated Jacobian elliptic surface or basic elliptic surface. For an integer $n$, $J^n(S)$ denotes the relative Picard scheme of line bundles on the fibers of degree $n$ (see for example Section 2 in Part I of [4]). Hence $J(S) = J^0(S)$ and $S = J^1(S)$. If $n$ is relatively prime to $m_1m_2$, then $J^n(S)$ again has two multiple fibers of multiplicities $m_1$ and $m_2$. We always have $p_g(J^n(S)) = p_g$. If $\Delta$ is a divisor on $S$, we let $f\cdot \Delta$ denote the {\sl fiber degree\/}, i.e\. the degree of the line bundle $\Delta$ on a smooth fiber $f$. Let $\operatorname{Pic}^{\text {v}}S$ denote the set of {\sl vertical} divisor classes, i.e\. the set of divisor classes spanned by the class of a fiber and the classes of the reductions of the multiple fibers. With our assumptions $\operatorname{Pic}^{\text {v}}S \cong \Zee \cdot \kappa$, where $m_1m_2\kappa =f$ (see also [6] Chapter 2 Corollary 2.9). Clearly $\operatorname{Pic}^{\text {v}}S$ is the kernel of the natural map from $\operatorname{Pic} S$ to the group of line bundles on the generic fiber. In general let $\eta = \Spec k(\Pee ^1)$ be the generic point of $\Pee ^1$ and let $\bar \eta = \Spec \overline{k(\Pee ^1)}$, where $\overline{k(\Pee ^1)}$ is the algebraic closure of $k(\Pee ^1)$. Let $S_\eta$ be the restriction of $S$ to $\eta$ and $S_{\bar\eta}$ be the pullback of $S_\eta$ to $\bar\eta$. Define $V_\eta$ to be the restriction of $V$ to $S_\eta$ and similarly for $V_{\bar\eta}$. \definition{Definition 3.1} An ample line bundle $L$ on $S$ is {\sl $(\Delta, c)$-suitable\/} if for all divisors $D$ on $S$ such that $-D^2 + D\cdot \Delta \leq c$, either $f\cdot(2D - \Delta) = 0$ or $$\operatorname{sign} f\cdot (2D - \Delta) = \operatorname{sign}L\cdot (2D - \Delta).$$ Given the pair $(\Delta, c)$, we set $w = \Delta \bmod 2 \in H^2(S; \Zee/2\Zee)$ and let $p = \Delta ^2 - 4c$. Thus $(\Delta, c)$ and $(\Delta ',c')$ correspond to the same values of $w$ and $p$ if and only if $\Delta ' = \Delta + 2F$ for some divisor class $F$ and $c' = c + \Delta \cdot F + F^2$. An easy calculation shows that the property of being $(\Delta, c)$-suitable therefore only depends on the pair $(w,p)$, and we will also say that $L$ is $(w, p)$-suitable. \enddefinition \medskip We have the following, which is Lemma 3.3 in Part I of [4]: \lemma{3.2} For all pairs $(\Delta, c)$, $(\Delta, c)$-suitable ample line bundles exist. \qed \endstatement \medskip \definition{Definition 3.3} Let $\Delta$ be a divisor on $S$ and $c$ an integer. Fix a $(\Delta, c)$-suitable line bundle $L$. We denote by $\frak M(\Delta, c)$ the moduli space of equivalence classes of $L$-stable rank two vector bundles $V$ on $S$ with $c_1(V) = \Delta$ and $c_2(V) = c$. Here $V_1$ and $V_2$ are {\sl equivalent\/} if there exists a line bundle $\scrO _S(D)$ such that $V_1$ is isomorphic to $V_2 \otimes \scrO _S(D)$. In particular, since $\det V_1 = \det V_2$, the divisor $2D$ is linearly equivalent to zero, and in fact $V_1$ and $V_2$ must be isomorphic since there is no 2-torsion in $\Pic S$. As the notation suggests and as we shall shortly show, the scheme $\frak M(\Delta, c)$ does not depend on the choice of the $(\Delta, c)$-suitable line bundle $L$. Given a divisor $\Delta$ on $S$ and an integer $c$, we let $w = \Delta \mod 2$ and $p = \Delta ^2 - 4c$. The moduli space $\frak M(\Delta, c)$ only depends on $w$ and $p$ and we shall also denote it by $\frak M(w,p)$. \enddefinition Now fix an odd integer $2e+1$. We shall consider rank two vector bundles $V$ such that the line bundle $\det V$ has fiber degree $2e+1$. However, it will be convenient not to fix the determinant of $V$. In this section we shall show that the moduli space $\frak M(w,p)$ is smooth and irreducible, and shall describe a Zariski open and dense subset of it explicitly. The basic idea is to show first that there is a largest integer $p_0$ such that $\frak M(w,p_0)$ in nonempty and that there is a unique element in $\frak M(w,p_0)$, corresponding to the bundle $V_0$. For all other $p<p_0$, the bundles in $\frak M(w,p)$ are obtained by elementary modifications of $V_0$ along fibers. Let us begin by recalling the following result (Corollary 4.4 in Part I of [4]): \theorem{3.4} Let $V$ be a rank two bundle with $\det V = \Delta$ and $c_2(V) = c$. Suppose that $\det V$ has fiber degree $2e+1$. Let $L$ be a $(\Delta, c)$-suitable ample line bundle, and suppose that $V$ is $L$-stable. Then there exists a Zariski open subset $U$ of $\Pee ^1$ such that, if $f$ is a fiber of $\pi$ corresponding to a point of $U$, then $f$ is smooth and $V|f$ is stable. Conversely, if there exists a smooth fiber $f$ such that $V|f$ is stable, then $V$ is $L$-stable for every $(\Delta,c)$-suitable ample line bundle $L$. \qed\endstatement \medskip Next we show that there exist bundles satisfying the hypotheses of Theorem 3.4: \lemma{3.5} Let $\delta$ be a line bundle on the generic fiber $S_\eta$ of odd degree $2e+1$. Then there exists a rank two vector bundle $V$ such that the restriction of $\det V$ to $S_\eta$ is $\delta$ and such that there exists a smooth fiber $f$ for which the restriction $V|f$ is stable. \endstatement \proof Let $\Delta _0$ be a line bundle on $S$ restricting to $\delta$ on $S_\eta$. Fix a smooth fiber $f$. By (1.1) there exists a stable bundle $E$ on $f$ with determinant equal to $\Delta _0|f$. Let $H$ be a line bundle on $S$ such that $\deg (H|f) \geq e+1$. Then by (1.3) there is a surjection from $E$ to $H|f$, and thus $E$ is given as an extension $$0 \to (H^{-1}\otimes \Delta _0)|f) \to E \to (H|f) \to 0.$$ This extension corresponds to a class in $H^1(f; (H^{\otimes -2}\otimes \Delta _0)|f)$. We would like to lift this exact sequence to an exact sequence on $S$. Of course, we can replace $\Delta _0$ by $\Delta _0 + Nf$ for an integer $N$ and get the same restriction to $f$. It suffices to show that, for some $N$, the map $$H^1(S; H^{\otimes -2}\otimes \Delta _0 \otimes \scrO _S(Nf)) \to H^1(f; (H|f)^{\otimes -2} \otimes \Delta _0)$$ is surjective. The cokernel of this map is contained in $$H^2(S;H^{-2}\otimes \Delta _0 \otimes \scrO_S((N-1)f)) = H^0(S; H^2\otimes \Delta _0^{-1} \otimes \scrO _S((-N+1)f\otimes K_S)^*.$$ Clearly $H^0(S; H^2\otimes \Delta _0^{-1} \otimes \scrO _S((-N+1)f\otimes K_S) =0$ if $N\gg 0$, and thus there is an extension on $S$ inducing $E$. \endproof \noindent {\bf Note.} We could also have proved (3.5) by descent theory. \medskip Before we state the next lemma, recall that a stable vector bundle $V$ on $S$ is {\sl good} if $H^2(S; \operatorname{ad}V) =0$. This means that $V$ is a smooth point of the moduli space, which has dimension $-p_1(\operatorname{ad}V) - 3\chi (\scrO _S)$. Thus the content of the next lemma is that the moduli space is always smooth of the expected dimension. \lemma{3.6} Let $V$ be a rank two bundle on $S$ such that the restriction of $V$ to the generic fiber is stable. Then $V$ is good. \endstatement \proof By Serre duality, $H^2(S; \operatorname{ad}V) =0$ if and only if $H^0(\operatorname{ad}V\otimes K_S) = 0$. A section $\varphi$ of $H^0(\operatorname{ad}V\otimes K_S)$ gives a trace free endomorphism of $V_{\bar \eta}$ (since $K_S$ has trivial restriction to the generic fiber). But $V_{\bar \eta}$ is simple, so that $\varphi$ has trivial restriction to the generic fiber. Hence $\varphi = 0$. \endproof \lemma{3.7} Let $V_1$ and $V_2$ be rank two bundles on $S$ whose restrictions to the generic fibers are stable and have the same determinant \rom(as a line bundle on $S_\eta$\rom). Then there exists a divisor $D$ on $S$, lying in $\operatorname{Pic}^{\text{v}}S$, and an inclusion $V_1\otimes \scrO _S(D) \subseteq V_2$. Moreover for an appropriate choice of $D$ we have an exact sequence $$0\to V_1\otimes \scrO _S(D) \to V_2 \to Q \to 0,$$ where $Q$ is supported on fibers or reductions of fibers and the map $V_1\otimes \scrO _S(D) \to V_2$ does not vanish on any fiber. \endstatement \proof By assumption $V_1$ and $V_2$ have isomorphic restrictions to $S_\eta$. An isomorphism between these extends to give a map $V_1\otimes \scrO_S(D_1) \to V_2 \otimes \scrO_S(D_2)$, where $D_i$ have trivial restriction to the generic fiber. Twisting gives a map $\varphi \:V_1 \otimes \scrO_S(D') \to V_2$, where $D'$ has trivial restriction to the generic fiber. By construction $\varphi$ is an isomorphism on the generic fiber, so $\varphi$ is an inclusion. The determinant $\det \varphi$ is a nonzero section of $\det V_1^{-1} \otimes \scrO _S(-2 D')\otimes \det V_2$, which restricts trivially to the generic fiber. Thus $\det V_1^{-1} \otimes \scrO _S(-2 D')\otimes \det V_2 = \scrO_S(\sum _in_iF_i + nf)$, where the $F_i$ are the multiple fibers, $f$ is a general fiber and $n_i$, $n$ are $\geq 0$. Here $Q$ has support whose reduction is the sum of the $F_i$ for which $n_i \neq 0$ plus some smooth fibers. If $\varphi$ vanishes identically on a fiber or fiber component $F$, then it factors: $$V_1 \otimes \scrO _S(D')\subset V_1 \otimes \scrO _S(D'+F) \to V_2.$$ So after enlarging $D'$ to a new divisor $D$ we can assume that this doesn't happen. Thus $D$ is as desired. \endproof \corollary{3.8} Let $V_1$ and $V_2$ be two rank two bundles on $S$ with the following property: for every curve $F$ which is a reduced fiber or the reduction of a multiple fiber, the restriction of $V_i$ to $F$ is stable. Then there exists a divisor $D\in \operatorname{Pic}^{\text{v}}S$ such that $V_2 = V_1\otimes \scrO _S(D)$. \endstatement \proof Find a nonzero map $\varphi \: V_1\otimes \scrO _S(D) \to V_2$ which does not vanish on $F$ for every $F$ the reduction of a fiber, via (3.7). For all $F$, $V_1\otimes \scrO _S(D)|F$ and $ V_2|F$ are stable bundles of the same degree and $\varphi|F$ is a nonzero map between them. Thus $\varphi|F$ is an isomorphism for all $F$ and so $\varphi $ is an isomorphism as well. \endproof \corollary{3.9} Suppose that $V_0$ is a rank two vector bundle satisfying the hypotheses of \rom{(3.8)}: the restriction $V|F$ is stable for every reduction $F$ of a fiber component. Let $\Delta = \det V_0$ and $c = c_2(V_0)$. Then $\frak M (\Delta, c)$ consists of a single reduced point corresponding to the bundle $V_0$. Thus necessarily $p_1(\operatorname{ad}V_0) = -3\chi (\scrO _S)$. \endstatement \proof If $V'$ is another such, $V' = V_0 \otimes \scrO _S(D)$, and so $V'$ and $V_0$ are equivalent. By (3.6) $V_0$ is good. Thus $\frak M (\Delta, c)$ is a single reduced point. Moreover the dimension of $\frak M (\Delta, c)$ is $-p_1(\operatorname{ad}V_0) - 3\chi (\scrO _S) = 0$, and so $p_1(\operatorname{ad}V_0) = -3\chi (\scrO _S)$. \endproof Next we establish the existence of such a $V_0$. Before we do so let us pause to record the following lemma. \lemma{3.10} Let $$0 \to V_1 \to V_2 \to Q\to 0$$ be an exact sequence of coherent sheaves on $S$, where $V_1$ and $V_2$ are rank two vector bundles and $Q=i_*M$ where $i\: F\to S$ is the inclusion of a reduced fiber or the reduction of a multiple fiber, and $M$ is a torsion free rank one sheaf on $F$. Then: \roster \item"{(i)}" We have the following formula for $p_1(\operatorname{ad}V_2)$: $$\align p_1(\operatorname{ad}V_2) &= p_1(\operatorname{ad}V_1) + 4\Bigl(\deg M - \frac{\deg (V_2|F) }2\Bigr)\\ &= p_1(\operatorname{ad}V_1) + 4\Bigl(\deg M - \frac{\deg (V_1|F) }2\Bigr). \endalign$$ \item"{(ii)}" If we define $Q'$ by the exact sequence $$0 \to V_2\spcheck \to V_1\spcheck \to Q' \to 0,$$ then $V_i\spcheck \cong V_i \otimes (\det V_i)^{-1}$ is a twist of $V_i$ and $Q' = Ext ^1(Q, \Cal O_X)$ is of the form $i_*M'$, where $M'$ is a torsion free rank one sheaf on $F$ with $\deg M' = -\deg M$. Finally $M$ is locally free if and only if $M'$ is locally free. \endroster \endstatement \noindent {\it Proof.} The first equality in (i) follows from (0.1) if $M$ is locally free, with minor modifications in general. To see the second, since $\det V_2 = \det V_1\otimes \Cal O_S(F)$ and $F^2=0$, we have $$\deg (V_1|F) = \deg (\det V_1|F) = \deg (\det V_2|F) = \deg (V_2|F).$$ To prove (ii), note that, after trivializing the bundles $V_i$ in a Zariski open set $U$, the map $V_1\to V_2$ is given by a $2\times 2$ matrix $A$ with coefficients in $\Cal O_U$, and so the dual map corresponds to the matrix ${}^tA$. A local calculation shows that $Q'=i_*M'$, where $M'$ is a torsion free rank one sheaf on $F$, where $F$ is locally defined by $\det A$, and that $M'$ is locally free if and only if $M$ is locally free. To calculate $\deg M'$, use the formula in (i) for $\deg M'$, noting that $p_1(\operatorname{ad} V_i\spcheck) = p_1(\operatorname{ad} V_i)$ and that $\deg (V_1\spcheck|F) = - \deg (V_1|F)$. Putting this together gives $$\align 4\deg M' &=p_1(\operatorname{ad}V_1) -p_1(\operatorname{ad}V_2) -2\deg (V_1|F)\\ &= -4\deg M. \qed \endalign$$ \medskip Using the above, we shall show the following, which together with (3.9) proves (i) of Theorem 2 of the introduction. \proposition{3.11} Given a line bundle $\delta$ on $S_\eta$ of odd degree, there exists a rank two bundle $V_0$ on $S$ such that the restriction of $\det V_0$ to $S_\eta$ is $\delta$ and such that the restriction $V|F$ is stable for every reduction $F$ of a fiber component. The rank two bundle $V_0$ is unique up to equivalence: if $V_1$ is any other bundle with this property, then there exists a line bundle $\scrO _S(D)$ such that $V_1 \cong V_0 \otimes \scrO _S(D)$. Moreover $p_1(\operatorname{ad} V_0) \geq p_1(\operatorname{ad}V)$ for every rank two bundle $V$ such that the restriction of $\det V$ to $S_\eta$ is $\delta$ and such that there exists a smooth fiber $f$ for which the restriction $V|f$ is stable, with equality if and only if $V = V_0 \otimes \scrO _S(D)$. \endstatement \proof Begin with $V$ such that $\det V|S_\eta=\delta$ and such that there exists a smooth fiber $f$ for which the restriction $V|f$ is stable. Such $V$ exist by (3.5). If there exists an $F$ such that $V|F$ is not stable, then there is a torsion free quotient $Q$ of $V|F$ such that $\deg Q < (\deg (V|F))/2$. Define $V'$ by the exact sequence $$0 \to V' \to V \to Q \to 0,$$ where we abusively denote by $Q$ the sheaf $i_*Q$, where $i$ is the inclusion of $F$ in $S$. Using (i) of (3.10), $$p_1(\operatorname{ad}V) = p_1(\operatorname{ad}V') + 4\Bigl(\deg Q - \frac{\deg (V|F) }2\Bigr).$$ Thus $p_1(\operatorname{ad}V') > p_1(\operatorname{ad} V)$. If $V'$ satisfies the conclusions of (3.11), we are done. Otherwise repeat this process. At each stage $p_1$ strictly increases. But $p_1$ is bounded from above, either from Bogomolov's inequality or using the fact that the dimension of the moduli space is always $-p_1 -3\chi (\scrO _S) \geq 0$, by (3.6). Hence this process terminates and gives a $V_0$ as desired. By (3.8) $V_0$ is unique up to twisting by a line bundle, and the final statement is clear from the method of proof. \endproof Next we shall interpret the proof of (3.11) as saying that every stable bundle $V$ is obtained from $V_0$ by an appropriate sequence of elementary modifications. \definition{Definition 3.12} Let $V$ be a rank two vector bundle on $S$ whose restriction to the generic fiber is stable. Let $F$ be a fiber on $S$ and $Q$ be a torsion free rank one sheaf on $F$, viewed as a sheaf on $S$. A surjection $V \to Q$ is {\sl allowable\/} if $$2\deg Q > \deg (V|F).$$ Thus if $\deg (V|F) = 2e+1$, then $\deg Q \geq e+1$. If $W$ is defined as an elementary modification $$0\to W \to V \to Q \to 0,$$ then we shall say that the elementary modification $W$ is {\sl allowable\/} if the surjection $V \to Q$ is allowable. It then follows from (3.10) that, if $W$ is an allowable elementary modification of $V$, then $p_1(\operatorname{ad}W) < p_1(\operatorname{ad}V)$. \enddefinition Let $Q$ be a rank one torsion free sheaf on a fiber $F$, viewed also as a sheaf on $S$, and let $d = \deg Q$. It is an easy consequence of (1.3) and (2.7) that if $V \to Q$ is allowable and $\deg (V|f) = 2e+1$, then $d>e$ and either $\dim \operatorname{Hom}(V,Q) = 2d-2e-1$ or $\dim \operatorname{Hom}(V,Q) = 2d-2e$ and $Q$ is a uniquely specified rank one torsion free sheaf on $F$. With this said, we have the following: \proposition{3.13} Let $\delta$ be a line bundle on $S_\eta$ and let $V$ be a stable rank two bundle on $S$ such that the restriction of $\det V$ to $S_\eta$ is $\delta$. Then there is a sequence $V_0$, $V_1$, \dots, $V_n=V$ such that $V_{i+1}$ is an allowable elementary modification of $V_i$ for $i=1, \dots , n-1$. Moreover $2n \leq p_1(\ad V_0)- p_1(\ad V)$. Finally if $V$ is obtained from $V_0$ from a sequence of allowable elementary modifications then $\dim \Hom (V, V_0) =1$. \endstatement \proof The construction given in the proof of (3.11) is the following: Begin with $V$. If $V\neq V_0$, then there is a fiber $F$, a rank one torsion free sheaf $Q$ on $F$, and an elementary modification $$0 \to V' \to V \to Q \to 0,$$ where $\deg Q < \deg (V|F)/2$. If $V' \neq V_0$, we repeat the process. So, noting that $V \cong V\spcheck\otimes \det V$ and using the notation of (3.10)(ii) it will suffice to show that the dual elementary modification $$0\to V\spcheck\otimes \det V \to (V')\spcheck\otimes \det V \to Q'\otimes \det V \to 0$$ is allowable, since then we have obtained $V$ as an allowable elementary modification of $(V')\spcheck\otimes \det V $. But we have $\deg Q' = -\deg Q$ by (3.10)(ii), and so $$\align \deg ( Q'\otimes \det V) &= -\deg Q + \deg (V|F)\\ &> \frac{\deg (V|F)}{2}. \endalign$$ Thus the surjection $(V')\spcheck\otimes \det V \to Q'\otimes \det V $ is allowable. The statement about the number of elementary modifications follows since an allowable elementary modification always decreases $p_1$ by a quantity whose absolute value is at least 2. Finally let us show that $\dim \Hom (V, V_0) =1$. Since $\dim \Hom (V_0, V_0) =1$, it is enough by induction on the number of elementary modifications to show the following: suppose given an exact sequence $$0 \to V_2 \to V_1 \to Q \to 0,$$ where $\deg Q > \deg (V_i|F)/2$. Then $\Hom (V_1, V_0) \to \Hom (V_2, V_0)$ is an isomorphism. For simplicity we shall just give the argument in case $Q$ is locally free on $F$. In any case $\Hom(Q, V_0)=0$ since $Q$ is a torsion sheaf and the cokernel of the map is $$\Ext ^1(Q, V_0) = H^0(Ext ^1(Q, V_0)) = H^0(Q\spcheck \otimes (V_0|F)) = \Hom (Q, V_0|F).$$ Since $V_0|F$ is stable and $\deg Q > \deg (V_i|F)/2= \deg (V_0|F)/2$, this last group is zero. \endproof Putting all this together, we shall describe a Zariski open subset of the moduli space. Let us first observe that the moduli space $\frak M(\Delta, c)$ is always good and of dimension $$4c - \Delta ^2 -3\chi (\scrO _S) = -p - 3\chi (\scrO _S).$$ By the canonical bundle formula for an elliptic surface, $$K_S= \scrO _S((p_g-1)f + (m_1-1)F_1+(m_2-1)F_2),$$ where $F_i$ are the reductions of the multiple fibers. As $m_1$ and $m_2$ are odd, $$K_S\cdot \Delta \equiv p_g -1\mod 2.$$ By the Wu formula, $\Delta ^2 \equiv p_g -1\mod 2$ as well. Hence $$4c - \Delta ^2 -3\chi (\scrO _S) \equiv 0\mod 2,$$ and the dimension of the moduli space is always an even integer $2t$. Now suppose that $\delta$ is a line bundle on the generic fiber $S_\eta$ of odd degree. Then there exists a divisor $\Delta$ on $S$ which restricts to $\delta$ and $\Delta$ is determined up to a multiple of $\kappa$. Mod 2, the only possibilities are $\Delta$ and $\Delta +\kappa$. Note that $(\Delta +\kappa)^2 = \Delta ^2 + 2(\Delta \cdot \kappa) \equiv \Delta ^2 + 2\mod 4$. Thus if we also fix $\Delta ^2\mod 4$, there is a unique choice of $w = \Delta \mod 2$. Fix an integer $t\geq 0$ and let $-p = 2t + 3\chi (\scrO _S)$. There is then a unique class $w\in H^2(S; \Zee/2\Zee)$ with $w^2 \equiv p \mod 4$ such that $w$ is the mod two reduction of a divisor $\Delta$ which restricts to $\delta$ on $S_\eta$. Given $\delta$ and $t$, we shall denote the corresponding moduli space by $\frak M_t$. The following theorem is a more precise version of Theorem 1 of the Introduction: \theorem{3.14} In the above notation, $\frak M_t$ is nonempty, smooth and irreducible, and is birational to $\operatorname{Sym}^tJ^{e+1}(S)$. More precisely, there exists a Zariski open and dense subset $U$ of $\frak M_t$ which is isomorphic to the open subset of $\operatorname{Sym}^tJ^{e+1}(S)$ consisting of $t$ line bundles $\lambda_1, \dots , \lambda _t$ of degree $e+1$ lying on smooth \rom(and reduced\rom) fibers of $\pi$ such that the images $\pi (\lambda _i)$ are distinct points of $\Pee ^1$, where we continue to denote by $\pi$ the projection from $J^{e+1}(S)$ to $\Pee ^1$. \endstatement \proof Let us describe the set $U$. Given the line bundle $\delta$ on $S_\eta$, let $\deg \delta = 2e+1$. If $f$ is a smooth fiber and $\lambda$ is a line bundle of degree $e+1$ on $f$, the restriction $V_0|f$ sits in an exact sequence $$0 \to \mu \to V_0|f \to \lambda \to 0,$$ where $\mu \otimes \lambda = \delta$. Once we have fixed $\lambda$, the surjection $V_0|f \to \lambda$ is unique mod scalars. Now fix $t$ distinct smooth fibers $f_1, \dots, f_t$ and line bundles $\lambda _i$ of degree $e+1$ on $f_i$. Let $Q_i$ be the sheaf $\lambda _i$ viewed as a sheaf on $S$ and let $Q = \bigoplus _iQ_i$. We shall consider the set of vector bundles $V$ described by an exact sequence $$0 \to V \to V_0 \to Q \to 0.$$ The set of all such $V$ is clearly parametrized by the open subset $U$ of $\operatorname{Sym}^tJ^{e+1}(S)$ consisting of $t$ line bundles $\lambda_1, \dots , \lambda _t$ lying on smooth (reduced) fibers of $\pi$ such that the images $\pi (\lambda _i)$ are distinct points of $\Pee ^1$. For such a bundle $V$, we also have $$p_1(\operatorname{ad}V) = p_1(\operatorname{ad}V_0) -2t.$$ We shall first construct a family of bundles parametrized by $U$ (more precisely, we shall construct ``universal" bundles over the product of $S$ with a finite cover of $U$), thereby giving a morphism from $U$ to $\frak M_t$ which is easily seen to be an open immersion. Finally we shall show that $U$ is in fact dense in $\frak M_t$. \medskip \noindent {\bf Step I.} Let $U$ be the open subset of $\operatorname{Sym}^tJ^{e+1}(S)$ described above, and let $\tilde U$ be defined as follows: $$\tilde U = \{\,(\lambda_1, \dots, \lambda _t)\in \bigl(J^{e+1}(S)\bigr)^t\: \{\lambda_1, \dots, \lambda _t\} \in U\,\}.$$ We shall try to construct a universal bundle $\Cal V$ over $S\times \tilde U$ as follows. Let $\Cal Z \subset S \times U$ be defined by $$\Cal Z = \{\, (p, \{\lambda_1, \dots , \lambda_t\}) \in S \times U : \text{ for some $i$, $\pi (p) = \pi (\lambda_i)$}\,\}.$$ Thus given a point $u = \{\lambda_1, \dots , \lambda_t\} \in U$, $$(S\times \{u\}) \cap \Cal Z = \coprod _{i=1}^t (f_i\times \{u\}),$$ where $f_i$ is the fiber corresponding to $\lambda_i$. Clearly $\Cal Z$ is a smooth divisor in $S\times U$. Analogously, we have the pulled back divisor $\tilde {\Cal Z} \subset S\times \tilde U$. In fact, $\tilde {\Cal Z}$ breaks up into a disjoint union of divisors $\tilde {\Cal Z}_i$, where for example $$\tilde {\Cal Z}_1 = \bigl(S\times _{\Pee ^1}J^{e+1}(S)\bigr)\times J^{e+1}(S)^{t-1},$$ and the other $\tilde {\Cal Z}_i$ are defined by taking the fiber product over $\Pee ^1$ of $S$ with the $i^{\text {th}}$ factor of $J^{e+1}(S)^t$. Thus each $\tilde {\Cal Z}_i$ fibers over $\tilde U$ and the fiber is an elliptic curve. Let $\rho _i\: \tilde {\Cal Z}_i\to S\times _{\Pee ^1}J^{e+1}(S)$ be the projection. Over $S\times _{\Pee ^1}J^{e+1}(S)$, there is a relative Poincar\'e bundle $\Cal P_{e+1}$. Actually, $\Cal P_{e+1}$ really just exists locally around sufficiently small neighborhoods of smooth nonmultiple fibers of $J^{e+1}(S)$, or in irreducible \'etale neighborhoods $\psi\:\Cal U_0\to J^{e+1}(S)$ of smooth nonmultiple fibers, but we will write out all the arguments as if there were a global bundle. We shall return to this point in Section 7. So we should really replace $\tilde U$ by $\tilde U_0$ defined by $$\tilde U_0 = \{\,(x_1, \dots, x_t) \in \Cal U_0^t: (\psi(x_1), \dots, \psi(x_t))\in U\,\}.$$ We can define the divisors $\tilde\Cal Z_i$ on $S\times \tilde U_0$ as well. Thus we have $\rho _i^*\Cal P_{e+1}$, which is a line bundle on $\tilde {\Cal Z}_i$. By extension, we can view $\rho _i^*\Cal P_{e+1}$ as a coherent sheaf on $S\times \tilde U_0$. \lemma{3.15} For every $i$, there is a line bundle $\Cal L_i$ on $\tilde U_0$ with the following property: There is a surjection $$\pi _1^*V_0\to \bigoplus _{i=1}^t\Bigl(\rho _i^*\Cal P_{e+1}\otimes \pi _2^*\Cal L_i \Bigr),$$ and the surjection is unique up to multiplying by the pullback of a nowhere vanishing function on $\tilde U_0$. \endstatement \proof We have $$\align \operatorname{Hom}(\pi _1^*V_0 , \bigoplus _{i=1}^t\rho _i^*\Cal P_{e+1}) &= H^0(\bigl(\pi _1^*V_0\bigr)\spcheck\otimes \Bigl[\bigoplus _{i=1}^t\rho _i^*\Cal P_{e+1})\Bigr] \\ &= H^0\Bigl(\tilde U_0; \bigoplus _{i=1}^tR^0\pi _2{}_*\Bigl(\bigl(\pi _1^*V_0\bigr)\spcheck\otimes \rho _i^*\Cal P_{e+1}\Bigr)\Bigr). \endalign$$ By base change and (1.2), the sheaf $R^0\pi _2{}_*\Bigl(\bigl(\pi _1^*V_0\bigr)\spcheck\otimes \rho _i^*\Cal P_{e+1} \Bigr)$ is a line bundle on $\tilde U_0$, which we denote by $\Cal L_i^{-1}$. Choosing a nowhere vanishing section of $\scrO _{\tilde U_0}$ gives an element of $$\align \operatorname{Hom}(\pi_1^*V_0, \rho _i^*\Cal P_{e+1}\otimes \pi _2^*\Cal L_i)&= H^0\Bigl(\tilde U_0; R^0\pi _2{}_*\Bigl(\bigl(\pi _1^*V_0\bigr)\spcheck\otimes \rho _i^*\Cal P_{e+1}\otimes \pi _2^*\Cal L_i)\Bigr)\Bigr)\\ &= H^0(\tilde U_0;\Cal L_i^{-1} \otimes \Cal L_i) = H^0(\tilde U_0; \scrO _{\tilde U_0}). \endalign$$ Since the $\tilde Z_i$ are disjoint, we can make such a choice for each $i$ to obtain the desired surjection. \endproof \noindent {\bf Note.} We shall essentially calculate $\Cal L_i$ in Section 7. \medskip Making a choice of a surjection from $\pi _1^*V_0$ to $\bigoplus _{i=1}^t\Bigl(\rho _i^*\Cal P_{e+1}\otimes \pi _2^*\Cal L_i \Bigr)$ gives a rank two vector bundle $\Cal V$ over $S\times \tilde U_0$ defined by the exact sequence $$0 \to \Cal V \to \pi _1^*V_0\to \bigoplus _{i=1}^t\Bigl(\rho _i^*\Cal P_{e+1}\otimes \pi _2^*\Cal L _i\Bigr)\to 0.$$ Thus there is a morphism $\tilde U _0\to \frak M_t$. It is easy to see that this morphism descends to a morphism of schemes $U\to \frak M_t$ whose image is the set of bundles described at the beginning of the proof of (3.14). Clearly the morphism $U\to \frak M_t$ is injective. By Zariski's Main Theorem it is an open immersion. This concludes the proof of Step I. \medskip \noindent {\bf Step II.} Now we must show that the open set $U$ constructed above is Zariski dense. To do so, we shall make a standard moduli count which essentially shows that the closed subset $\frak M_t-U$ may be parametrized by a scheme of dimension strictly smaller than $\dim \frak M_t =2t$. Consider the set of all allowable elementary modifications of a fixed vector bundle $V'$ with $\deg (V'|F) = 2e+1$. Thus there is a reduced fiber or the reduction of a multiple fiber, say $F$, and a rank one torsion free sheaf $Q$ on $F$ with $\deg Q =d\geq e+1$. By (1.3) and (2.7), there is a surjection from $V'$ to $Q$, and the set of all such has dimension $2d-2e-1$ or $2d-2e$. Let $V$ be the kernel of such a surjection. By (3.10), $$p_1(\operatorname{ad}V') = p_1(\operatorname{ad}V) + 4d-4e-2.$$ Thus the number of moduli of all $V$ is $$\align -p_1(\operatorname{ad}V)-3\chi (\scrO _S) &= -p_1(\operatorname{ad} V') - 3\chi (\scrO _S)+4d-4e-2. \endalign $$ On the other hand, for $d$ and $V'$ fixed, the above construction depends on $2d-2e$ parameters. If $F$ is generic, there is one parameter to choose $F$. Next, either $\dim \operatorname{Hom}(V', Q) = 2d-2e -1$ or $2d-2e$, and in this last case $Q$ is fixed. Taking the homomorphisms mod scalars the number of moduli is either $2d-2e -2$ or $2d-2e-1$. In the first case the choice of $Q$ is one more parameter, but not in the second case. Thus we always get $2d-2e -1$ parameters for the choice of the sheaf $Q$ and the surjection $V' \to Q$. Adding in the choice of $F$ gives $2d-2e$ moduli. For the above construction to account for a Zariski open subset of the moduli space, we clearly must have $V'$ a general point of its moduli space, $F$ a general fiber, and $2d-2e\geq 4d-4e-2$. It follows that $d\leq e+1$, and hence since $d>e$ that $d=e+1$. Arguing by induction, we may assume that $V$ is obtained from $V_0$ by performing successive elementary modifications along distinct fibers $F_i$ which are smooth and nonmultiple and with respect to line bundles $\mu _i$ on $F_i$ of degree exactly $e+1$. In this case $V$ is in the open set $U$ described above. \endproof \noindent{\bf Notation 3.16.} Given a line bundle $\delta$ on $S_\eta$ and a nonnegative integer $t$, we let $\frak M_t$ be the moduli space defined prior to (3.14) of equivalence classes of stable bundles $V$ with $-p_1(\ad V) = 2t + 3\chi (\scrO_S)$, such that $w_2(V)$ is the mod two reduction of a divisor $\Delta$ with $\Delta |S_\eta = \delta$. Thus $\frak M_t$ depends only on $\delta$ and $t$. Let $\overline{\frak M}_t$ denote the Gieseker compactification of $\frak M_t$. \section{4. The case where $S$ has a section.} In this section, we shall assume that there is a section $\sigma$ on $S$, so that $m_1 = m_2 = 1$. In this case, $\sigma ^2 = -(1+p_g(S))$. Our goal is to give a very explicit description of the set of stable bundles on $S$ such that $\det V$ has the same restriction to the generic fiber as $\sigma$. Thus $\det V = \sigma +nf$ for some integer $n$. We begin with a lemma on various cohomology groups which will be used often. \lemma{4.1} Let $S$ be an elliptic surface with a section $\sigma$. Let $p_g = p_g(S)$. \roster \item"{(i)}" For all integers $a$, $h^0(-\sigma + af) = 0$. \item"{(ii)}" For all integers $a$, $$h^1(-\sigma + (p_g+1-a)f) = \cases 0, & a>0\\ -a+1, & a\leq 0. \endcases$$ \item"{(iii)}" For all integers $a$, $$h^2(-\sigma + (p_g+1-a)f) = \cases a-1, & a\geq 2\\ 0, & a\leq 1. \endcases$$ \endroster \endstatement \proof Clearly $h^0(-\sigma + af) = 0$ for all integers $a$. Likewise $R^0\pi _* \scrO_S(-\sigma + af) = 0$ for all $a$. In addition $R^2\pi _* \scrO_S(-\sigma + af) = 0$ for all $a$ since $\pi$ has relative dimension one. Thus, from the Leray spectral sequence, we see that $$\gather H^1(\scrO_S(-\sigma + (p_g+1-a)f)) = H^0(R^1\pi _* \scrO_S(-\sigma + (p_g+1-a)f))\\ H^2(\scrO_S(-\sigma + (p_g+1-a)f)) = H^1(R^1\pi _* \scrO_S(-\sigma + (p_g+1-a)f)). \endgather$$ Thus we must determine the sheaf $R^1\pi _* \scrO_S(-\sigma + (p_g+1-a)f)$ on $\Pee ^1$. Now $R^1\pi _* \scrO_S(-\sigma + (p_g+1-a)f) = R^1\pi _* \scrO_S(-\sigma )\otimes \scrO _{\Pee ^1}(p_g+1-a)$. To calculate $R^1\pi _* \scrO_S(-\sigma )$, we use the exact sequence $$0 \to \scrO_S(-\sigma) \to \scrO _S \to \scrO _{\sigma} \to 0.$$ Taking the long exact sequence for $R^i\pi _*$ gives $R^1\pi _*\scrO_S(-\sigma) \cong R^1\pi _* \scrO _S$, and, by e.g\. [6] Chapter 1 (3.18), $R^1\pi _* \scrO _S \cong \scrO _{\Pee ^1}(-p_g-1)$. So $R^1\pi _* \scrO_S(-\sigma + (p_g+1-a)f) \cong \scrO _{\Pee ^1}(-a)$, and (ii) and (iii) follow from the usual calculations for $\Pee ^1$. \endproof Next we shall determe the unique stable vector vector bundle $V_0$ (up to equivalence) which satisfies $-p_1(\ad V_0) =3\chi (\scrO_S) $. \proposition{4.2} Let $S$ be a nodal elliptic surface with a section $\sigma$. \roster \item"{(i)}" If $p_g(S)$ is odd, set $k=(1+p_g(S))/2$. Then there is a unique nonsplit extension $$0 \to \scrO _S(kf) \to V_0 \to \scrO _S(\sigma - kf)\to 0,$$ and $\det V = \sigma$, $-p_1(\ad V_0) = 3\chi (\scrO _S)$, and the restriction of $V_0$ to every fiber is stable. \item"{(ii)}" If $p_g(S)$ is even, set $k = p_g(S)/2$. Then there is a unique nonsplit extension $$0 \to \scrO _S(kf) \to V_0 \to \scrO _S(\sigma - (k+1)f)\to 0,$$ and $\det V = \sigma -f$, $-p_1(\ad V_0) = 3\chi (\scrO _S)$, and the restriction of $V_0$ to every fiber is stable. \endroster \endstatement \proof We shall just consider the case where $p_g$ is odd; the other case is identical. First note that $H^1(S; \scrO _S(-\sigma +2kf)) = H^1(-\sigma + (p_g+1)f)$ has dimension one, by (4.1)(ii). Thus there is a unique nonsplit extension up to isomorphism. Clearly $\det V_0 = \sigma$ and $-p_1(\ad V_0) = 4k -\sigma ^2 = 3(1+p_g)$. Finally we claim that the restriction of $V_0$ to every fiber is stable. It suffices to show that the restriction of $V_0$ to every fiber $f$ is the nontrivial extension of $\scrO_f(p)$ by $\scrO_f$, where $p$ is the point $\sigma \cdot f$. Thus we must consider the restriction map $$H^1(S; \scrO _S(-\sigma +2kf) \to H^1(f; \scrO _S(-\sigma +2kf)|f).$$ Its kernel is $H^1(S; \scrO _S(-\sigma +(2k-1)f))= H^1(S; \scrO _S(-\sigma +p_gf))$. Again by (4.1)(ii) this group is zero, so that $H^1(S; \scrO _S(-\sigma +2kf) \to H^1(f; \scrO _S(-\sigma +2kf)|f)$ is an injection and hence an isomorphism since both spaces have dimension one. It follows that $V_0|f$ is stable for every $f$ and is thus the unique bundle up to equivalence satisfying the hypotheses of (3.8). \endproof The bundle $V_0$ (with a slightly different normalization) has been described independently by Kametani and Sato [8]. Let us now consider the case where $V$ is a stable bundle with $-p_1(\ad V) - 3\chi (\scrO _S) = 2t\geq 0$. \proposition{4.3} With $S$ as above, let $V$ be a stable rank two vector bundle over $S$ such that $\det V=\sigma +nf$ for some $n$ and $-p_1(\ad V) - 3\chi (\scrO _S) = 2t$. \roster \item"{(i)}" If $p_g$ is odd and we set $k= (1+p_g)/2$, then, after twisting by a line bundle of the form $\scrO_S(af)$, there exist an integer $s$, $0\leq s \leq t$, and an exact sequence $$0 \to \scrO _S((k-s)f) \to V \to \scrO _S(\sigma +(-k+s-t)f)\otimes I_Z\to 0.$$ Here $Z$ is a codimension two local complete intersection subscheme of length $s$. Moreover the inclusion of $\scrO _S((k-s)f)$ into $V$ is canonically given by the map $\pi ^*\pi _*V \to V$. If $\varphi\:\scrO_S(af) \to V$ is a sub-line bundle, then $\varphi$ factors through the inclusion $\scrO _S((k-s)f) \to V$. \item"{(ii)}" If $p_g$ is even and we set $k= p_g/2$, then, after twisting by a line bundle of the form $\scrO_S(af)$, there exist an integer $s$, $0\leq s \leq t$, and an exact sequence $$0 \to \scrO _S((k-s)f) \to V \to \scrO _S(\sigma +(-k-1+s-t)f)\otimes I_Z\to 0.$$ Here $Z$ is again a codimension two local complete intersection subscheme of length $s$. Finally the inclusion of $\scrO _S((k-s)f)$ into $V$ is canonically given by the map $\pi ^*\pi _*V \to V$, and every nonzero map $\scrO_S(af) \to V$ factors through $\scrO _S((k-s)f)$. \endroster \endstatement \proof We shall just write down the argument in case $p_g$ is odd. By (3.13), possibly after twisting, $V$ is obtained from $V_0$ by a sequence of $r\leq t$ allowable elementary modifications. In particular $V$ may be identified with a subsheaf of $V_0$, and $\det V = \sigma -rf$. There is the map from $\scrO_S(kf)$ to $V_0$, and clearly the image of the subsheaf $\scrO_S((k-r)f)$ lies in $V$. Of course, the map $\scrO_S((k-r)f) \to V$ may vanish along a divisor, but this divisor must necessarily be a union of at most $r$ fibers. Thus there is an integer $u$ with $0\leq u\leq r$ and an exact sequence for $V$ of the form $$0 \to \scrO_S((k-r+u)f)\to V \to \scrO _S(\sigma +(-k-u)f)\otimes I_Z \to 0.$$ Using the condition that $-p_1(\ad V) - 3(p_g+1) = 2t$ gives $$4\ell(Z) +4(k-r+u) + (1+p_g) -2r -3(1+p_g)= 2t.$$ Solving, we get $$-r +2u+2\ell(Z) = t.$$ Let $s = \ell(Z)$. Twisting the exact sequence by $\scrO_S(bf)$, where $b= u+\ell(Z) -t$, gives a new exact sequence (where we rename $V$ by $V\otimes \scrO_S(bf)$) $$0 \to \scrO_S((k-s)f)\to V \to \scrO_S(\sigma + (-k+s-t)f)\otimes I_Z \to 0.$$ Clearly $s=\ell(Z) \geq 0$ and since $2s = t+r-2u$ with $u\geq 0$, $r\leq t$, we have $s\leq t$. This gives the desired expression of $V$ as an extension. Since the restriction of this extension to the generic fiber is not split, the map $$R^0\pi _* (\scrO_S(\sigma + (-k+s-t)f)\otimes I_Z) \to R^1\pi _* \scrO_S((k-s)f)$$ is injective. Thus $\pi _*V = \pi _*\scrO_S((k-s)f) = \scrO_{\Pee ^1}((k-s))$ and the map $\pi ^*\pi _*V \to V$ is just the inclusion $\scrO_S((k-s)f) \to V$. Finally if $\scrO_S(af) \to V$ is nonzero then $\pi _*\scrO_S(af) \to \pi _*V = \pi _*\scrO_S((k-s)f)$ is nonzero as well, and the last assertion of the proposition is then clear. \endproof There is an analogue of (4.3) for Gieseker semistable torsion free sheaves: \proposition{4.3$'$} With $S$ and $k$ as above, suppose that $V$ is a rank two torsion free sheaf with $c_1(V) = \Delta = \sigma +nf$ for some $n$ and $c_2(V) =c$ such that $V$ is Gieseker semistable with respect to a $(\Delta, c)$-suitable line bundle. Suppose that $-p_1(\operatorname{ad}V)-3\chi (\scrO_S) = 2t$. Then the restriction of $V$ to a general fiber of $S$ is stable, and after twisting by $\scrO_S(af)$ for some $a$ there are zero-dimensional subschemes $Z_1$ and $Z_2$ of $S$, not necessarily local complete intersections, an integer $s$ with $0\leq s\leq t$, and an exact sequence $$0 \to \scrO _S((k-s)f)\otimes I_{Z_1} \to V \to \scrO _S(\sigma +(-k+s-t)f) \otimes I_{Z_2}\to 0,$$ if $p_g=2k-1$ is odd, and $$0 \to \scrO _S((k-s)f)\otimes I_{Z_1} \to V \to \scrO _S(\sigma +(-k-1+s-t)f) \otimes I_{Z_2}\to 0$$ if $p_g=2k$ is even. Moreover $\ell(Z_1) +\ell (Z_2) = s$. \endstatement \proof The double dual $V\spcheck{}\spcheck$ of $V$ is a semistable rank two vector bundle. Thus it is stable and fits into an exact sequence as in (i) or (ii) of (4.3). Thus (4.3$'$) follows from manipulations along the lines of the proof of (4.3). \endproof Next let us consider when an extension as in (4.3) can be unstable. For simplicity we shall just write out the case where $p_g$ is odd. \proposition{4.4} Suppose that $p_g=2k-1$ is odd and that $V$ is an extension of the form $$0 \to \scrO _S((k-s)f) \to V \to \scrO _S(\sigma +(-k+s-t)f)\otimes I_Z\to 0,$$ where $\ell (Z)=s$. Let $s_0$ be the smallest integer such that $h^0(\scrO_S(s_0f)\otimes I_Z)\neq 0$. Thus $0\leq s_0\leq s$, and $s_0 = 0$ if and only if $s=0$. If $V$ is unstable, then the maximal destabilizing subbundle is equal to $\scrO_S(\sigma -af)$, where $$t+k-(s-s_0) \leq a \leq t+k.$$ Thus if $s=s_0$ the only possibility is $\scrO_S(\sigma -(t+k)f)$. \endstatement \proof The maximal destabilizing subbundle has a torsion free quotient. Clearly, it restricts to $\sigma$ on the generic fiber, and thus must be of the form $\scrO_S(\sigma -af)$ for some integer $a$. Using the exact sequence $$0 \to \scrO_S(\sigma -af) \to V \to \scrO_S((a-t)f)\otimes I_{Z'} \to 0,$$ where $Z'$ is a codimension two subscheme, and the fact that $$\align c_2(V) &= k-s+s = k\\ &= a-t+\ell (Z'), \endalign$$ we see that $a\leq t+k$. On the other hand, there is a nonzero map from $\scrO_S(\sigma -af)$ to $\scrO _S(\sigma +(-k+s-t)f)\otimes I_Z$ and thus a nonzero section of $\scrO _S((-k+s-t+a)f)\otimes I_Z$. Thus $$-k+s-t+a \geq s_0,$$ or in other words $a\geq t+k-(s-s_0)$. \endproof \corollary{4.5} With assumptions as above, suppose that $Z=\emptyset$, so that $V$ is an extension $$0 \to \scrO _S(kf) \to V \to \scrO_S(\sigma - (k+t)f) \to 0.$$ Then $V$ is stable if and only if it is not the split extension. In this case we can identify the set of all nonsplit extensions with $\Sym ^t\sigma$, and an extension $V$ corresponding to $\{p_1, \dots, p_t\}\in \Sym ^t\sigma$ has unstable restriction to a fiber $f$ if and only if $p_i\in f$ for some $i$. \endstatement \proof As we are in the case $s=0$ of (4.4), if $V$ is unstable then the destabilizing line bundle is $\scrO_S(\sigma - (k+t)f)$, which splits the exact sequence. Conversely, if the sequence is not split, then $V$ is stable. The set of nonsplit extensions of $\scrO_S(\sigma - (k+t)f)$ by $\scrO _S(kf)$ is parametrized by $\Pee H^1(\scrO_S(-\sigma +(2k+t)f))$. By (4.1) $H^1(\scrO_S(-\sigma +(2k+t)f)) \cong H^0(R^1\pi _*\scrO_S(-\sigma +(2k+t)f)) = H^0(\Pee ^1; \scrO_{\Pee^1}(t))$. Moreover $\Pee H^0(\Pee ^1; \scrO_{\Pee^1}(t)) = \Sym ^t\sigma$ by associating to a section the set of points where it vanishes. This says that the extension $V$ restricts to the split extension on a fiber $f$ exactly when the corresponding section of $\scrO_{\Pee^1}(t)$ vanishes at the point of $\Pee ^1$ under $f$. \endproof Next we analyze the generic case where $\ell (Z)=t$. \proposition{4.6} Suppose that $p_g=2k-1$ is odd and that $V$ is an extension of the form $$0 \to \scrO _S((k-t)f) \to V \to \scrO _S(\sigma -kf)\otimes I_Z\to 0,$$ where $\ell (Z)=t>0$. \roster \item"{(i)}" A locally free extension $V$ as above exists if and only if $Z$ has the Cayley-Bacharach property with respect to $|\sigma +(t-2)f|$. \item"{(ii)}" Suppose that $s_0=t$ or $t-1$ in the notation of \rom{(4.4)}, and that $\Supp Z\cap \sigma = \emptyset$. Then $\dim \Ext ^1(\scrO _S(\sigma -kf) \otimes I_Z,\scrO _S((k-t)f))=1$. A locally free extension exists in this case if $s_0=t$. \item"{(iii)}" Suppose that $Z$ consists of $t$ points lying in distinct fibers, exactly one of which lies on $\sigma$. Then $\dim \Ext ^1(\scrO _S(\sigma -kf) \otimes I_Z,\scrO _S((k-t)f))=1$. A locally free extension exists in this case if and only if $t=1$. \item"{(iv)}" If $s_0 \leq t-1$, for example if $Z$ contains two distinct points lying on the same fiber, then $V$ is unstable. \item"{(v)}" If $s_0 =t$, then $V$ is stable if no point of $Z$ lies on $\sigma$. Likewise if $t=1$ and $Z\subset \sigma$, then $V$ is not stable. \endroster \endstatement \proof The long exact sequence for $\Ext$ gives $$\gather H^1(-\sigma +2k-t)f) \to \Ext ^1(\scrO _S(\sigma -kf) \otimes I_Z,\scrO _S((k-t)f))\to \\ \to H^0(\scrO_Z) \to H^2(-\sigma +2k-t)f). \endgather$$ By (4.1)(ii) $H^1(-\sigma +2k-t)f) =0$. The map $H^0(\scrO_Z) \to H^2(-\sigma +2k-t)f)$ is dual to the map $H^0(\scrO_S(\sigma +(t-2)f))\to H^0(\scrO_Z)$ defined by restriction. Thus (i) follows by definition. As for (ii), since $\Supp Z\cap \sigma = \emptyset$ and $H^0(\scrO_S(\sigma +(t-2)f)) = H^0(\scrO_S((t-2)f))$ under the natural inclusion, clearly $H^0(\scrO_S(\sigma +(t-2)f)\otimes I_Z) = H^0(\scrO_S((t-2)f)\otimes I_Z)$. By assumption $H^0(\scrO_S((t-2)f)\otimes I_Z)=0$, so that the map $H^0(\scrO_S (\sigma +(t-2)f))\to H^0(\scrO_Z)$ is an inclusion. But $h^0(\scrO_S (\sigma +(t-2)f))=t-1$ and $h^0(\scrO_Z) = t$. Thus the cokernel has dimension one. It is clear that if $s_0 = t$ and $Z$ is reduced, then it has the Cayley-Bacharach property with respect to $|\sigma +(t-2)f|$. A more involved argument left to the reader handles the nonreduced case. Thus a locally free extension exists. This proves (ii), and the proof of (iii) is similar. To see (iv), note that if $s_0 \leq t-1$, then there is a section of $\scrO_S((t-1)f)\otimes I_Z$. Consider the exact sequence $$0 \to \scrO_S(-\sigma +(2k-1)f) \to Hom (\scrO_S(\sigma - (k+t-1)f), V) \to \scrO_S((t-1)f)\otimes I_Z \to 0.$$ Since $H^1(\scrO_S(-\sigma +(2k-1)f))=0$ by (4.1), the section of $\scrO_S((t-1)f)\otimes I_Z $ lifts to define a nonzero homomorphism from $\scrO_S(\sigma - (k+t-1)f)$ to $V$. Thus $V$ is unstable. Finally we must prove (v). The bundle $V$ is stable if and only if its restriction to a general fiber $f$ is stable. Let $f$ be a fiber not meeting $\Supp Z$. Then there is a natural map $\Ext ^1(\scrO _S(\sigma -kf) \otimes I_Z,\scrO _S((k-t)f))\to \Ext^1(\scrO_f(p), \scrO_f) = H^1(\scrO_f(-p))$. This fits into an exact sequence $$\gather H^0(\scrO_f(-p))\to\Ext ^1(\scrO _S(\sigma -kf) \otimes I_Z,\scrO _S((k-t-1)f))\to \\ \to \Ext ^1(\scrO _S(\sigma -kf) \otimes I_Z,\scrO _S((k-t)f))\to H^1(\scrO_f(-p)). \endgather$$ Since $H^0(\scrO_f(-p)) =0$ and $h^1(\scrO_f(-p)= \dim \Ext ^1(\scrO _S(\sigma -kf) \otimes I_Z,\scrO _S((k-t)f))=1$, by (ii) and (iii), it will suffice to show that $\dim \Ext ^1(\scrO _S (\sigma -kf) \otimes I_Z,\scrO _S((k-t-1)f))\geq1$ if $\Supp Z \cap \sigma \neq \emptyset$. Now since $H^1(\scrO_S(-\sigma +(2k-t-1)f))=0$ by (4.1), $\Ext ^1(\scrO _S (\sigma -kf) \otimes I_Z,\scrO _S((k-t-1)f))$ is dual to the cokernel of the restriction map $H^0(\scrO_S(\sigma +(t-1)f))\to H^0(\scrO_Z)$. Since $s_0 =t$, by definition $h^0(\scrO_S((t-1)f)\otimes I_Z)=0$. Thus if $\Supp Z \cap \sigma =\emptyset$, then $H^0(\scrO_S(\sigma +(t-1)f))$ and $H^0(\scrO_S((t-1)f))$ have the same image in $H^0(\scrO_Z)$ and $H^0(\scrO_S((t-1)f))\to H^0(\scrO_Z)$ is injective. As both $H^0(\scrO_S((t-1)f))$ and $H^0(\scrO_Z)$ have dimension $t$, the map between them is an isomorphism and the cokernel is zero. It follows that $V$ restricts to a stable bundle on $f$. Likewise, if $t=1$ and $ Z\subset \sigma$, then clearly the map $H^0(\scrO_S(\sigma +(t-1)f))\to H^0(\scrO_Z)$ cannot be surjective, and so the cokernel is nonzero. Thus $V$ restricts on $f$ to an unstable bundle for almost every fiber $f$, so that $V$ is unstable. \endproof Let us give another proof for (4.6)(v). Using (4.4) we know that the maximal destabilizing line bundle, if it exists, must necessarily be of the form $\scrO_S(\sigma -(t+k)f)$. There is an exact sequence $$0 \to \scrO_S(-\sigma +2kf) \to Hom (\scrO_S(\sigma -(t+k)f), V) \to \scrO_S(tf)\otimes I_Z \to 0,$$ and $V$ is unstable if and only if the nonzero section of $\scrO_S(tf)\otimes I_Z$ lifts to a homomorphism from $\scrO_S(\sigma -(t+k)f)$ to $V$. The nonzero section of $\scrO_S(tf)\otimes I_Z$ defines an exact sequence $$0 \to \scrO_S \to \scrO_S(tf)\otimes I_Z \to Q\to 0.$$ Here if $Z$ consists of points $z_i$ on distinct fibers $f_i$, then $Q= \bigoplus _i\scrO_{f_i}(-z_i)$. The coboundary map from $H^0(\scrO_S(tf)\otimes I_Z)$ to $H^1(\scrO_S(-\sigma +2kf))$ is given by taking cup product of the nonzero section with the extension class $\xi$ in $\Ext^1(\scrO_S(tf)\otimes I_Z,\scrO_S(-\sigma +2kf))$ corresponding to $V$. It is easy to see by the naturality of the pairing that this is the same as taking the image of $\xi$ in $\Ext ^1(\scrO_S, \scrO_S(-\sigma +2kf))= H^1(\scrO_S(-\sigma +2kf))$ using the above exact sequence. Taking the long exact Ext sequence and using the fact that $H^0(\scrO_S(-\sigma +2kf)) =0$, there is an exact sequence $$\gather 0\to \Ext ^1(Q,\scrO_S(-\sigma +2kf))\to \Ext ^1(\scrO_S(tf)\otimes I_Z,\scrO_S(-\sigma +2kf)) \to \\ \to H^1(\scrO_S(-\sigma +2kf)). \endgather$$ Since $\dim \Ext ^1(\scrO_S(tf)\otimes I_Z,\scrO_S(-\sigma +2kf))=1$, we see that $\xi \mapsto 0$ if and only if $\Ext ^1(Q,\scrO_S(-\sigma +2kf)) \neq 0$. So we shall show that $\Ext ^1 (Q,\scrO_S(-\sigma +2kf)) = 0$ if and only if the support of $Z$ does not meet $\sigma$. First consider the case where $Z$ consists of points $z_i$ on distinct fibers $f_i$. Then $Q= \bigoplus _i\scrO_{f_i}(-z_i)$, and standard arguments (cf\. [6] Chapter 7 Lemma 1.27) show that $\Ext ^1(\scrO_{f_i}(-z_i), \scrO_S(-\sigma +2kf)) = H^0(\scrO_{f_i}(z_i-p_i))$, where $p_i = f_i\cap \sigma$. This group is then zero unless $z_i = p_i$. We shall briefly outline the argument in the case where $\Supp Z$ is a single point $z$ supported on a fiber $f$ (the proof in the general case is then just a matter of notation). In this case $Q= \bigl(\scrO_S(tf)\otimes I_Z\bigr)/\scrO_S \cong I_Z/I_{tf}$, where $I_{tf}$ is the ideal of the nonreduced subscheme $tf$. Moreover the assumption that $s_0=t$ implies that $t$ is the smallest integer $s$ such that $x^s\in I_Z$, where $x$ is a local defining function for the fiber $f$. Our goal now is again to prove that $\Ext ^1(Q, \scrO_S(-\sigma +2kf)) = 0$. Now the sheaf $Q$ has a filtration by subsheaves whose successive quotients are $$Q_n=I_Z\cap I_{nf}/I_Z\cap I_{(n+1)f}\cong \bigl(I_Z\cap I_{nf}+I_{(n+1)f}\bigr)/I_{(n+1)f},$$ for $0\leq n\leq t-1$. It is easy to see that each such quotient is a torsion free rank one $\scrO_f$-module contained in $I_{nf}/I_{(n+1)f}\cong \scrO_f$. Thus it is a line bundle on $f$ of strictly negative degree, necessarily of the form $\scrO_f(-a_nz)$, unless $(I_Z\cap I_{nf}+I_{(n+1)f})/I_{(n+1)f} = I_{nf}/I_{(n+1)f}$, or in other words $I_Z\cap I_{nf}+I_{(n+1)f} = I_{nf}$. In this case, in the local ring of $z$ we would have $x^n = h + gx^{n+1}$, where $x$ is a local defining function for $f$ and $h\in I_Z$. But then $h= x^n(1-gx)$, so that $x^n\in I_Z$, contradicting the fact that $x^t$ is the smallest power of $x$ which lies in $I_Z$. Hence $Q_n \cong \scrO_f(-a_nz)$ with $a_n\geq 1$. A standard argument with Chern classes shows that $$c_2(Q) = t[z] = \sum _{n=0}^{t-1}c_2(Q_n)=-\sum _{n=0}^{t-1} \deg Q_n,$$ where $c_2(Q)$, $c_2(Q_n)$ are taken in the sense of sheaves on $S$ and $\deg Q_n$ is in the sense of line bundles on $f$. Thus $\deg Q_n = -1$ for all $n$ and $Q_n = \scrO _f(-z)$. It follows that $\Ext ^1(Q_n, \scrO_S(-\sigma +2nf))= H^0(\scrO_f(z-p))$ where $p=\sigma \cap f$. This group is zero if $z\neq p$ and is nonzero otherwise. Thus $\Ext ^1(Q, \scrO_S(-\sigma +2nf)) = 0$ if $z\neq p$ and $\Ext ^1(Q, \scrO_S(-\sigma +2nf)) \neq 0$ if $z=p$. \medskip We shall now reverse the above constructions and try to find a universal bundle in the case where the dimension of the moduli space is 2 or 4. For simplicity we shall just consider the case where $p_g$ is odd. \medskip \noindent {\bf The two-dimensional invariant.} \medskip Let $\frak M_1$ denote the moduli space of equivalence classes of stable rank two bundles $V$ for which $-p_1(\ad V) -3\chi (\scrO_S) =2$. Thus $\frak M_1$ is compact. Since $p_g$ is odd, we may fix the determinant of $V$ to be $\sigma -f$. Our goal is to show the following: \theorem{4.7} $\frak M_1 \cong S$. Moreover there is a universal bundle $\Cal V$ over $S\times S$, and $$p_1(\ad \Cal V)/ [\Sigma] = (2(\sigma \cdot \Sigma) -2p_g(f\cdot \Sigma))f -4(f\cdot \Sigma)\sigma -4\Sigma.$$ Thus, as $-4\mu (\Sigma) = p_1(\ad \Cal V)/ [\Sigma]$, we have $$\mu (\Sigma)^2 = (\Sigma )^2 + (p_g-1)(f\cdot \Sigma)^2.$$ \endstatement \proof It follows from (4.3), (4.5), and (4.6)(v) that if $V$ is stable with $-p_1(\ad V) -3\chi (\scrO_S) =2$ and $c_1(V)=\sigma -f$, then either there is an exact sequence $$0 \to \scrO _S((k-1)f) \to V \to \scrO _S(\sigma -kf)\otimes \frak m_q \to 0$$ with $\frak m_q$ the maximal ideal of a point $q\notin \sigma$ or there is an nonsplit exact sequence $$0 \to \scrO _S(kf) \to V \to \scrO _S(\sigma + (-k-1)f) \to 0. $$ In this case the set of all nonsplit extensions is isomorphic to $\sigma$. Thus the moduli space $\frak M_1$ is made up of $S-\sigma$, together with a copy of $\sigma$. To glue these two pieces, we shall construct a universal bundle over $S\times S$ by taking extensions and then making an elementary modification. To this end, let $\Bbb D$ be the diagonal in $S\times S$. Consider the extension $\Cal W$ over $S\times S$ defined as follows: $$0 \to \pi _1^*\scrO_S((k-1)f) \otimes \pi _2^*\Cal L \to \Cal W \to \pi _1^*\scrO _S(\sigma -kf)\otimes I_{\Bbb D} \to 0.$$ Here, using the relative Ext sheaves and standard exact sequences we should take $$\align \Cal L^{-1}&= Ext ^1_{\pi _2}(\pi _1^*\scrO _S(\sigma -kf)\otimes I_{\Bbb D}, \pi _1^*\scrO_S((k-1)f)) \\ &= \pi _2{}_*(\det N_{\Bbb D}\otimes \pi _1^*(\scrO _S(-\sigma + (2k-1)f))\\ &= \pi _2{}_*(\scrO _{\Bbb D}(-(p_g-1)f) \otimes \pi _1^*(\scrO _S(-\sigma + p_gf) \\ &= \scrO _S(-\sigma + f). \endalign$$ With this choice of $\Cal L$, we find that $\operatorname{Ext}^1(\pi _1^*\scrO _S(\sigma -kf)\otimes I_{\Bbb D}, \pi _1^*\scrO_S((k-1)f) \otimes \pi _2^*\Cal L) \cong H^0(\scrO _{\Bbb D})$ and that the unique nontrivial extension is indeed locally free. This defines $\Cal W$, and an easy computation gives $$\align c_1(\Cal W) &= \pi _1^*(\sigma -f) + \pi _2^*c_1(\Cal L);\\ p_1(\ad \Cal W) &= 2\pi _1^*(-\sigma + (2k-1)f)\cdot \pi _2^*(\sigma - f) -4[\Bbb D] + \cdots, \endalign$$ where the omitted terms do not affect slant product. The restriction $W$ of $\Cal W$ to the slice $S\times \{q\}$ is the unique nontrivial extension of $\scrO _S(\sigma -kf)\otimes \frak m_q$ by $\scrO _S((k-1)f)$. By (4.6)(v) $W$ is stable if and only if $q$ does not lie on $\sigma$. To remedy this problem, we shall make an elementary modification along $S\times \sigma$. Note that, if $W$ is given as an extension $$0 \to \scrO _S((k-1)f) \to W \to \scrO _S(\sigma -kf)\otimes \frak m_q \to 0,$$ where $q\in \sigma$, then the maximal destabilizing sub-line bundle of $W$ must be $\scrO _S(\sigma + (-k-1)f)$ by (4.4) and thus there is an exact sequence $$0 \to \scrO _S(\sigma + (-k-1)f) \to W \to \scrO _S(kf) \to 0.$$ It follows that $\pi _2{}_*Hom(\Cal W|(S\times \sigma), \pi _1^*\scrO _S(kf))$ is a line bundle $\Cal M$ and the natural map $$\Cal W \to i_*(\pi _1^*\scrO _S(kf) \otimes \pi _2^*\Cal M)$$ is surjective. Thus we can define $\Cal V$ by taking the associated elementary modification. By construction there is an exact sequence $$0 \to \Cal V \to \Cal W \to i_*(\pi _1^*\scrO _S(kf) \otimes \pi _2^*\Cal M) \to 0.$$ Moreover for each $q\in \sigma$ there is an exact sequence $$0 \to \scrO _S(kf) \to \Cal V|S\times \{q\} \to \scrO _S(\sigma +(-k-1)f) \to 0.$$ Thus by (4.5) $\Cal V|S\times \{q\}$ is stable provided that this extension does not split. We state this fact explicitly as a lemma, whose proof will be deferred until later: \lemma{4.8} In the above notation, the extension for $\Cal V|S\times \{q\}$ does not split. \endstatement \medskip Assuming the lemma, the restriction of $\Cal V$ to each slice is stable and thus $\Cal V$ defines a morphism from $S$ to the moduli space $\frak M_1$. It is clear that this morphism is a bijection between two smooth surfaces and is thus an isomorphism. Moreover, by (0.1), $$p_1(\ad \Cal V) = p_1(\ad \Cal W) + 2c_1(\Cal W)\cdot [S\times \sigma] + [S\times \sigma]^2-4i_*c_1(\pi _1^*\scrO _S(kf) \otimes \pi _2^*\Cal M).$$ Plugging in for $c_1(\Cal W)$ and $p_1(\ad \Cal W)$ gives $$p_1(\ad \Cal V) = 2\pi _1^*(-\sigma + (2k-1)f)\cdot \pi _2^*(\sigma - f) - 4[\Bbb D] +2\pi _1^*(\sigma -f)\cdot \pi _2^*\sigma -4\pi _1^*(kf) \cdot \pi _2^*\sigma + \cdots,$$ where as usual the omitted terms do not affect slant product. Thus collecting terms and taking slant product gives $$-4\mu (\Sigma) = 2(\sigma\cdot \Sigma)f -2p_g(f\cdot \Sigma)f-4(f\cdot \Sigma) \sigma - 4\Sigma,$$ as claimed in the statement of Theorem 4.4. This concludes the proof of Theorem 4.7. \endproof \demo{Proof of \rom{(4.8)}} We shall use the criterion (A.4) of the Appendix and the discussion following it to see that the extension does not split. Given $q\in \sigma$, let $W= \Cal W|S\times \{q\}$. We need to show: \roster \item"{(i)}" $ \Hom (\scrO_S(\sigma +(-k-1)f), \scrO_S(kf))=0$. \item"{(ii)}" The map (coming from the usual long exact sequences) $$\gather R^0\pi _2{}_*\bigl(\pi _1^*\scrO_S(\sigma -kf) \otimes I_{\Bbb D} \otimes \pi _1^* \scrO _S(-\sigma +(k+1)f)\bigr) = R^0\pi _2{}_*\bigl(\pi _1^*\scrO_S(f) \otimes I_{\Bbb D}\bigr) \to \\ \to R^1\pi _2{}_*\pi _1^*(\scrO _S((k-1)f) \otimes \scrO _S(-\sigma +(k+1)f)) = R^1\pi _2{}_*\pi _1^*(\scrO _S(-\sigma + 2kf)) \endgather$$ vanishes simply along $\sigma$. \item"{(iii)}" $H^1(\scrO_S(f)\otimes \frak m_q)$ is independent of $q$, and is nonzero only if $p_g=0$. Moreover $H^2(\scrO_S(-\sigma +2kf))=0$. \item"{(iv)}" At each point of $\sigma$, the map $H^1(\scrO _S(-\sigma + 2kf))\to H^1(\scrO _S(kf)\otimes \scrO _S(-\sigma +(k+1)f)) = H^1(-\sigma + (2k+1)f)$ induced by the map $H^1(\scrO _S(-\sigma + 2kf))\to H^1(W\otimes \scrO _S(-\sigma +(k+1)f)$ followed by the natural map $H^1(W\otimes \scrO _S(-\sigma +(k+1)f)\to H^1(\scrO _S(kf)\otimes \scrO _S(-\sigma +(k+1)f)$ is injective. \endroster The statement (i) is clear. To prove (ii), we shall calculate $R^1\pi _2{}_*(\Cal W \otimes \scrO _S(\sigma +(-k-1)f))$ by an argument similar to the second proof of (4.6)(v). By base change $R^0\pi _2{}_*(\pi _1^*\scrO _S(f) \otimes I_{\Bbb D})=\Cal L_1$ is a line bundle on $S$. From the definition of $\Cal W$ and $\Cal L$ the sheaf $Ext ^1_{\pi _2}(\pi _1^*\scrO _S(f)\otimes I_{\Bbb D} , \pi _1^*\scrO _S(-\sigma +2kf))\otimes \Cal L$ is the trivial line bundle. A global section induces the map $\Cal L_1 \to R^1\pi _2{}_*\pi _1^*\scrO _S(\sigma +2kf))\otimes \Cal L$. The cokernel of this map is a subsheaf of $R^1\pi _2{}_*(\Cal W \otimes \scrO _S(-\sigma +(k+1)f))$. To determine where the map vanishes, use the exact sequence $$0 \to \pi _2^*\Cal L_1 \to \pi _1^*\scrO _S(f)\otimes I_{\Bbb D}\to \Cal P \to 0.$$ Here the map $\pi _2^*\Cal L_1 \to \pi _1^*\scrO _S(f)\otimes I_{\Bbb D}$ is the natural one and a calculation in local coordinates shows that it vanishes simply along $D = S\times _{\Pee ^1}S\subset S\times S$. It follows that, up to a line bundle pulled back from the second factor $\Cal P = \scrO_D(-\Bbb D)$. Thus $\Cal P$ is up to sign a Poincar\'e bundle. Now $\Ext ^2(\scrO _S(f)\otimes \frak m_q,\scrO _S(-\sigma +2kf))=0$ since $H^2(\scrO _S (-\sigma +(2k-1)f))=0$. Thus $Ext ^2_{\pi _2}(\pi _1^*\scrO _S(f)\otimes I_{\Bbb D} , \pi _1^*\scrO _S(-\sigma +2kf))=0$ and there is an exact sequence $$\gather Ext _{\pi _2}^1(\Cal P, \pi _1^*\scrO _S(-\sigma +2kf))\to Ext ^1_{\pi _2}(\pi _1^*\scrO _S(f)\otimes I_{\Bbb D} , \pi _1^*\scrO _S(-\sigma +2kf)) \to \\ \to R^1\pi _2{}_*\pi _1^*\scrO _S(\sigma +2kf))\to Ext _{\pi _2}^2(\Cal P, \pi _1^*\scrO _S(-\sigma +2kf))\to 0. \endgather$$ It follows from the naturality of the pairings involved that the image of the map $$Ext ^1_{\pi _2}(\pi _1^*\scrO _S(f)\otimes I_{\Bbb D} , \pi _1^*\scrO _S(-\sigma +2kf)) \to R^1\pi _2{}_*\pi _1^*\scrO _S (\sigma +2kf))$$ is, up to a twist by the line bundle $\Cal L$, the image of $\Cal L_1$. Note that the restriction of $\Cal P\spcheck \otimes \pi _1^*\scrO _S(-\sigma +2kf)$ to $\pi _2^{-1}(q)\subset D$, where $q$ is any point except the singular point on a singular fiber, is $\scrO _f(p-q)$, where $f$ is the fiber containing $q$ and $p = f\cap \sigma$. Ignoring the possible double points of $D$, we have by standard arguments $$\gather Ext ^1_{\pi _2}(\pi _1^*\scrO _S(f) \otimes I_{\Bbb D} , \pi _1^*\scrO _S(-\sigma +2kf))=\pi _2{}_*(\Cal P\spcheck \otimes \pi _1^*\scrO _S(-\sigma +2kf))\\ Ext ^2_{\pi _2}(\pi _1^*\scrO _S(f) \otimes I_{\Bbb D} , \pi _1^*\scrO _S(-\sigma +2kf))= R^1\pi _2{}_*(\Cal P\spcheck \otimes \pi _1^*\scrO _S(-\sigma +2kf)) \endgather$$ (where $\Cal P\spcheck$ means that the dual is taken as a line bundle on $D$). Thus $\pi _2{}_*(\Cal P\spcheck \otimes \pi _1^*\scrO _S(-\sigma +2kf)) =0$ and $R^1\pi _2{}_*(\Cal P\spcheck \otimes \pi _1^*\scrO _S(-\sigma +2kf))$ is supported on $\sigma$. To calculate its length, we have (again ignoring the double points of $D$ which will not cause trouble) an exact sequence $$0 \to \scrO_D(\Bbb D -\pi _1^*\sigma +\pi _1^*(2kf)) \to \scrO_D(\Bbb D + \pi _1^*(2kf)) \to \scrO_D(\Bbb D + \pi_1^*(2kf))|\pi _1^*\sigma \cap D\to 0.$$ Now $\pi _1^*\sigma \cap D \cong S$ via $\pi _2$ and under this isomorphism $\scrO_D(\Bbb D + \pi_1^*(2kf))|\pi _1^*\sigma \cap D \cong \scrO_S(\sigma +2kf)$. The map $$R^0\pi _2{}_*\scrO_D(\pi_1^*(2kf)) \to R^0\pi _2{}_*\scrO_D(\Bbb D+\pi_1^*(2kf))$$ is an isomorphism, since the induced map on $H^0$'s for the restriction to each fiber of $\pi _2$ is an isomorphism. Using the exact sequence $$0\to \scrO_D(\pi_1^*(-\sigma + 2kf)) \to \scrO_D(\pi_1^*(2kf)) \to \scrO_S(2kf) \to 0,$$ it follows that the image of $R^0\pi _2{}_*\scrO_D(\pi_1^*(2kf)) = R^0\pi _2{}_*\scrO_D(\Bbb D+\pi_1^*(2kf))$ in $$R^0\pi _2{}_*\scrO_D(\Bbb D + \pi_1^*(2kf))|\pi _1^*\sigma \cap D= \scrO_S(\sigma +2kf)$$ is just the image of $\scrO_S(2kf)$ in $\scrO_S(\sigma +2kf)$. Thus this map vanishes simply along $\sigma$, and its cokernel, which is $$R^1\pi _2{}_*\scrO_D(\Bbb D -\pi _1^*\sigma +\pi _1^*(2kf)) = R^1\pi _2{}_* (\Cal P\spcheck \otimes \pi _1^*\scrO _S(-\sigma +2kf)),$$ is a line bundle on $\sigma$. It follows that the map of line bundles $$Ext ^1_{\pi _2}(\pi _1^*\scrO _S(f)\otimes I_{\Bbb D} , \pi _1^*\scrO _S(-\sigma +2kf))\otimes \Cal L \to R^1\pi _2{}_*\pi _1^*\scrO _S(\sigma +2kf))\otimes \Cal L$$ vanishes simply along $\sigma$, so that we are in the situation of (A.4): the cokernel contributes torsion of length one. To see that the above is exactly the torsion in $R^1\pi _2{}_*(\Cal W\otimes \scrO_S(-\sigma +(k+1)f))$ follows from (iii), as in the discussion after (A.4). To see (iii), use the exact sequence $$0\to \scrO_S(f)\otimes \frak m_q \to \scrO_S(f) \to \Cee _q \to 0.$$ The long exact cohomology sequence shows that $H^1(\scrO_S(f)\otimes \frak m_q) \cong H^1(\scrO_S(f))$. It is easy to see that this last group is zero if $p_g >0$ and has dimension one if $p_g=0$ (and in any case its dimension is obviously independent of $q$). Finally $H^2(\scrO_S(-\sigma +2kf))=0$ by (4.1). Thus we have identified the torsion in $R^1\pi _2{}_*(\Cal W\otimes \scrO_S(-\sigma +(k+1)f))$, compatibly with base change. We finally need to check that the induced map $$H^1(\scrO_S(-\sigma +2kf)) \to H^1(\scrO_S(-\sigma +(2k+1)f))$$ is injective. But this map is induced from the composition of the map of sheaves $\scrO_S(-\sigma +2kf) \to W\otimes \scrO_S(-\sigma +(k+1)f)$ together with the map $W\otimes \scrO_S(-\sigma +(k+1)f) \to \scrO_S(-\sigma +(2k+1)f)$. This composition is then a nonzero map from $\scrO_S(-\sigma +2kf)$ to $\scrO_S(-\sigma +(2k+1)f)$ and so fits into an exact sequence $$0 \to \scrO_S(-\sigma +2kf) \to \scrO_S(-\sigma +(2k+1)f) \to \scrO_f(-p) \to 0.$$ Since $H^0(\scrO_f(-p))=0$, the map the map $H^1(\scrO_S(-\sigma +2kf)) \to H^1(\scrO_S(-\sigma +(2k+1)f))$ is injective. Thus the extension class for $\Cal V|S\times \{q\}$ is nonzero, and we are done. \endproof \medskip \noindent {\bf The four-dimensional invariant.} \medskip We again assume that $p_g$ is odd and list the possible types of extensions for a stable bundle. The generic case (Type 1) is where there exists a codimension two subscheme $Z$ with $\ell(Z)=2$ and an exact sequence $$0 \to \scrO _S((k-2)f) \to V \to \scrO _S(\sigma -kf) \otimes I_Z \to 0.\tag Type 1$$ Other possibilities (Types 2 and 3 respectively) are $$\gather 0 \to \scrO _S((k-1)f) \to V \to \scrO _S(\sigma +(-k-1)f) \otimes \frak m_q \to 0;\tag Type 2\\ 0 \to \scrO _S(kf) \to V \to \scrO _S(\sigma +(-k-2)f) \to 0. \tag Type 3 \endgather$$ Here $\frak m_q$ is the maximal ideal of a point $q$. Finally there is also the case where $V$ is not locally free. In this case the double dual of $V$ fits into an extension $$0 \to \scrO _S((k-1)f) \to V\spcheck{}\spcheck \to \scrO _S(\sigma +(-k-1)f) \to 0$$ which must be nonsplit if $V$ is to be stable, in which case $V\spcheck{}\spcheck$ is just a twist of $V_0$. One possibility is that $V$ is given as the unique non-locally free extension of $\scrO _S(\sigma +(-k-1)f) \otimes \frak m_q$ by $\scrO _S((k-1)f)$ as in the second exact sequence above. The remaining possibility (Type 4) is that $V$ is given as an extension $$0 \to \scrO _S((k-1)f)\otimes \frak m_q \to V \to \scrO _S(\sigma +(-k-1)f) \to 0.\tag Type 4$$ For a fixed $q$, the set of all such extensions is parametrized by a $\Pee ^1$, one point of which correspond to a $V$ such that $V\spcheck{}\spcheck$ is unstable. Our goal here is to give a very brief sketch of the following, where we use the notation of the introduction for divisors on $\operatorname{Hilb}^2S$: \theorem{4.9} The moduli space $\frak M_2$ of dimension $4$ is isomorphic to $\operatorname{Hilb}^2S$, and for all $\Sigma \in H_2(S)$, $$\mu (\Sigma) = D_{\alpha _2} -\bigl((f\cdot \Sigma)/2\bigr)E,$$ where, setting $\alpha _1 = \mu _1(\Sigma)$ to be the class computed by the $\mu$-map for the two-dimensional invariant, $$\align \alpha _2 &= \Sigma +\bigl(-(\sigma \cdot \Sigma) + (p_g+1)(f\cdot \Sigma)/2\bigr)f + (f\cdot \Sigma)\sigma\\ &= \alpha _1 + \bigl((f\cdot \Sigma)/2\bigr)f. \endalign$$ \endstatement Thus an easy calculation using the multiplication table for $\operatorname{Hilb}^2S$ gives the following: \corollary{4.10} In the above notation, $$\mu (\Sigma )^4 = 3(\Sigma ^2)^2 + 6(p_g-1)(\Sigma ^2)(f\cdot \Sigma)^2 + \bigl[3(p_g+1)(p_g-1)-8(p_g-1)\bigr](f\cdot \Sigma)^4.$$ \endstatement We shall not give a complete proof of (4.9) here, but shall outline the argument and prove some statements which will be used later. In Sections 9 and 10, we shall prove a more general statement which will imply (4.9). We begin as before by analyzing the generic case, Type 1. Let $Z$ be a codimension two subscheme of $S$ with $\ell (Z) =2$. Let $D_\sigma$ be the effective divisor of $\operatorname{Hilb}^2S$ which is the closure of the locus of pairs $\{z_1, z_2\}$ where $z_1\in \sigma$. Then arguing as in the proof of (4.6)(i)--(iii), we see that $$\dim \Ext ^1(\scrO _S(\sigma -kf)\otimes I_Z, \scrO _S((k-2)f) = \cases 1, & \text{if $Z \notin \Sym ^2\sigma \subset \operatorname{Hilb}^2S$;}\\ 2, &\text{otherwise.} \endcases$$ In case $Z\notin D_\sigma$, the unique extension class mod scalars corresponds to a locally free extension. If $Z\in D_\sigma - \Sym ^2\sigma$, then the unique nontrivial extension is not locally free. If $Z\in \Sym ^2\sigma$, then there exist locally free extensions. Next we must analyze when a locally free extension is stable. Let $\Cal D$ be the irreducible divisor in $\operatorname{Hilb}^2S$ corresponding to the divisor $S\times _{\Pee ^1}S \subset S\times S$. Equivalently $$\Cal D=\{\, Z\in \operatorname{Hilb}^2S\mid h^0(\scrO_S(f)\otimes I_Z) = 1\,\}.$$ The divisor $\Cal D$ is smooth, although $S\times _{\Pee ^1}S$ is singular at the finitely many pairs of points $(x,x)$, where $x$ is a double point of a singular fiber. One way to see this is as follows. The divisor $S\times _{\Pee ^1}S$ has ordinary threefold double points at the singularities. Moreover it contains the diagonal $\Bbb D\subset S\times S$, which is smooth and passes through the double points. It is well-known (and easy to check) that the blowup of a threefold double point $(xy-zw)$ along a subvariety of the form $(x-z, y-w)$ gives a small resolution of the singularity. Thus the proper transform of $S\times _{\Pee ^1}S$ in the blowup of $S\times S$ along $\Bbb D$ is smooth, and $\Cal D$ is the quotient of this proper transform by an involution whose fixed point set is smooth (it is $\Bbb D$). Thus $\Bbb D$ is smooth. \lemma{4.11} Let $V$ be a vector bundle given by an extension $$0 \to \scrO _S((k-2)f) \to V \to \scrO _S(\sigma -kf) \otimes I_Z \to 0,$$ where $\ell (Z)=2$. Then $V$ is not stable if and only if either $Z\in \Sym ^2\sigma$ or $Z\in \Cal D$. If $Z\in \Cal D$, then the maximal destabilizing sub-line bundle is $\scrO_S(\sigma +(-k-1)f)$ and there is an exact sequence $$0 \to \scrO_S(\sigma +(-k-1)f) \to V \to \scrO_S((k-1)f)\otimes \frak m_q \to 0.$$ Here $q=z_1+z_2-p$, where $f$ is the unique fiber containing $Z=\{z_1, z_2\}$, $p=\sigma \cap f$, and the addition is with respect to the group law on $f$ \rom(if $f$ is singular and $\Supp Z$ meets the singular point then $q$ is the singular point as well\rom). If $Z\in \Sym ^2\sigma-\Cal D$, then the maximal destabilizing sub-line bundle is $\scrO_S(\sigma -(k+2)f)$. \endstatement \proof If $Z\notin D_\sigma$, then we have seen in (4.6)(v) that $V$ is unstable if and only if $Z\in \Cal D$, and in this case the destabilizing sub-line bundle must be $\scrO_S(\sigma +(-k-1)f)$ by (4.4). The quotient is torsion free and by a Chern class calculation it must be $\scrO_S((k-1)f)\otimes \frak m_q$ for some point $q$. To identify the point $q$, let us assume for simplicity that $\Supp Z$ does not meet the singular point of a singular fiber, we can restrict the two exact sequences for $V$ to the fiber $f$ containing $Z$. From these we see that there are surjective maps $V|f \to \scrO _f(p-z_1-z_2)$ and $V|f \to \scrO _f(-q)$. Since $\deg V|f= 1$, it splits and the unique summand of negative degree is thus $\scrO _f(p-z_1-z_2) \cong \scrO _f(-q)$. It follows that $q=z_1+z_2-p$. The case where $\Supp Z$ contains the singular point of a singular fiber is similar. If $Z\in D_\sigma$, then since $V$ is locally free $Z\in \Sym ^2\sigma$. Arguments as in (4.6) then show that $V$ is unstable. If moreover $Z\notin \Cal D$, then by (4.4) the maximal destabilizing sub-line bundle is $\scrO_S(\sigma -(k+2)f)$. \endproof Our next task will be to construct a universal sheaf $\Cal V$ over $\operatorname{Hilb}^2S-D_\sigma$. We begin by finding a sheaf $\Cal W$ as follows: let $\Cal Z\subset S\times \operatorname{Hilb}^2S$ be the universal subscheme, and consider the relative extension sheaf $Ext ^1_{\pi _2}(\pi _1^*\scrO _S(\sigma -kf)\otimes I_{\Cal Z}, \pi _1^*\scrO_S((k-2)f)$. Since $H^1(\scrO_S(-\sigma +(2k-2)f)) = 0$, there is an exact sequence $$\gather 0 \to Ext ^1_{\pi _2}(\pi _1^*\scrO _S(\sigma -kf)\otimes I_{\Cal Z}, \pi _1^*\scrO_S((k-2)f) \to\\ \to R^0\pi _2{}_*Ext ^1(\pi _1^*\scrO _S(\sigma -kf)\otimes I_{\Cal Z}, \pi _1^*\scrO_S((k-2)f). \endgather$$ Over the complement of $\Sym ^2\sigma$, $Ext ^1_{\pi _2}(\pi _1^*\scrO _S(\sigma -kf)\otimes I_{\Cal Z}, \pi _1^*\scrO_S((k-2)f)$ is a line bundle on $\operatorname{Hilb}^2S - \Sym ^2\sigma$ which we denote by $\Cal L^{-1}$ and thus there is a coherent sheaf $\Cal W$ defined by $$0 \to \pi _1^*\scrO_S((k-2)f) \otimes \Cal L \to \Cal W \to \pi _1^*\scrO _S(\sigma -kf)\otimes I_{\Cal Z} \to 0.$$ However, if $Z\in \Cal D$, then $\Cal W|S\times \{Z\}$ is not stable, and if $Z\in D_\sigma$ then $\Cal W|S\times \{Z\}$ is neither locally free nor stable. We shall first study $\Cal W|S\times (\operatorname{Hilb}^2S -D_\sigma)$, and shall denote this for simplicity again by $\Cal W$. There is a unique point $q= z_1+z_2-p$ such that $\Cal W|S\times \{Z\}$ maps surjectively to $\scrO_S((k-1)f)\otimes \frak m_q$, and so we expect to be able to make an elementary transformation along $\Cal D$. Indeed, since $\dim \Hom (\scrO_S(\sigma +(-k-1)f), \Cal W|S\times \{Z\})=1$ for all $Z\in \Cal D$, there are line bundles $\Cal L_1$ and $\Cal L_2$ on $\Cal D$ and an exact sequence $$0 \to \pi _1^*\scrO_S(\sigma +(-k-1)f)\otimes \pi _2^*\Cal L_1 \to \Cal W|S\times \Cal D \to \pi _1^*\scrO_S((k-1)f)\otimes \pi _2^*\Cal L_2 \otimes I_{\Cal Y} \to 0,$$ where $\Cal Y$ is the set $$\{\,(q, z_1, z_2)\in S\times _{\Pee ^1}\Cal D\mid q=z_1+z_2-p\,\}.$$ It is easy to check from the definition that $\Cal Y$ is smooth and that the map $\Cal Y \to \Cal D$ is an isomorphism. Thus we may define $\Cal V$ by the exact sequence $$0 \to \Cal V \to \Cal W \to i_*\pi _1^*\scrO_S((k-1)f)\otimes \pi _2^*\Cal L_2 \otimes I_{\Cal Y} \to 0,$$ where $i$ is the inclusion of $S\times \Cal D$ in $S\times (\operatorname{Hilb}^2S -D_\sigma)$. We then have the following: \proposition{4.12} The sheaf $\Cal V$ is a reflexive sheaf, flat over $\operatorname{Hilb}^2S -D_\sigma$. The restriction of $\Cal V$ to each slice $S\times \{Z\}$ is a stable torsion free sheaf, which is locally free if and only if $Z\notin \Cal D$. \endstatement \proof By (A.2) of the Appendix, $\Cal V$ is reflexive and flat over $\operatorname{Hilb}^2S -D_\sigma$. For each $Z\in \Cal D$, if $V_Z$ is the restriction of $\Cal V$ to the slice $S\times \{Z\}$, there is an exact sequence $$0 \to \scrO_S((k-1)f)\otimes \frak m_q \to V_Z \to \scrO_S(\sigma +(-k-1)f)\to 0,$$ by (A.2) again. If $Z\notin \Cal D$ then $V_Z =\Cal V|S\times \{Z\}$ is locally free and stable. Thus we need only check that the double dual of $V_Z$, for $Z\in \Cal D$, is the unique nonsplit extension of $\scrO_S(\sigma +(-k-1)f)$ by $\scrO_S((k-1)f)$, which will imply that $V_Z\spcheck{}\spcheck$ is up to a twist the stable bundle $V_0$. To verify that the double dual of $V_Z$ is a nonsplit extension amounts to the following: the extension class corresponding to $V_Z$ lives in $\Ext ^1( \scrO_S(\sigma +(-k-1)f), \scrO_S((k-1)f)\otimes \frak m_q) = H^1(\scrO_S(-\sigma +2kf)\otimes \frak m_q)$, and we must show that its image in $H^1(\scrO_S(-\sigma +2kf)$ is nonzero. To do this we shall use the result (A.4) of the Appendix. Let $M= \scrO _S(\sigma -(k+1)f)$ and $L=\scrO_S((k-1)f)$. Clearly $\Hom (M,L) = 0$. By the definition of $\Cal W$ there is an exact sequence $$0 \to \pi _1^*\scrO _S(-\sigma +(2k-1)f)\otimes \pi _2^*\Cal L \to \Cal W \otimes \pi _1^*M^{-1} \to \pi _1^*\scrO _S(f)\otimes I_{\Cal Z} \to 0.$$ By (4.1) $R^1\pi _2{}_*\pi _1^*\scrO _S(-\sigma +(2k-1)f) = R^2\pi _2{}_*\pi _1^*\scrO _S(-\sigma +(2k-1)f) = 0$. Thus $R^1\pi _2{}_*\Cal W \otimes \pi _1^*M^{-1} \cong R^1\pi _2{}_*\bigl(\pi _1^*\scrO _S(f)\otimes I_{\Cal Z}\bigr)$. To analyze $R^1\pi _2{}_*\bigl(\pi _1^*\scrO _S(f)\otimes I_{\Cal Z}\bigr)$, use the exact sequence $$0 \to \pi _1^*\scrO _S(f)\otimes I_{\Cal Z} \to \pi _1^*\scrO _S(f) \to \scrO_{\Cal Z}\otimes \pi _1^*\scrO _S(f) \to 0.$$ It is easy to check that $R^1\pi _2{}_*\pi _1^*\scrO _S(f) =0$ if $p_g>0$, and is a line bundle if $p_g=0$. Clearly $R^1\pi _2{}_*(\scrO _{\Cal Z} \otimes \pi _1^*\scrO_S(f)) = 0$. Thus the torsion in $R^1\pi _2{}_*\bigl(\pi _1^*\scrO _S(f)\otimes I_{\Cal Z}\bigr)$ is the cokernel of the map between two rank two vector bundles on $\operatorname{Hilb}^2S$ $$R^0\pi _2{}_*\pi _1^*\scrO _S(f) \to R^0\pi _2{}_*\bigl(\scrO_{\Cal Z}\otimes \pi _1^*\scrO _S(f)\bigr).$$ Sine $\Cal D$ is a smooth divisor, by using elementary divisors for the vector bundle map we can describe this cokernel by describing what it looks like at the generic point. It is a simple exercise in local coordinates to identify the determinant of the vector bundle map with a local equation for $\Cal D$ at the generic point. Thus the torsion in $R^1\pi _2{}_*\Cal W \otimes \pi _1^*M^{-1}$ is a line bundle on $\Cal D$, which is identified with the torsion in $R^1\pi _2{}_*\bigl(\pi _1^*\scrO _S(f)\otimes I_{\Cal Z}\bigr)$. Similar statements hold via standard base change results if we restrict to a first order neighborhood of $\Cal D$, where torsion is to be interpreted in the sense of (A.4)(ii) of the appendix. Next, let $Z= \{z_1, z_2\}\in \Cal D$ and let $W$ be the extension corresponding to the restriction of $\Cal W$ to the slice $S\times \{Z\}$, we must identify the corresponding extension class, i.e\. the image of the one-dimensional vector space $H^1(\scrO_S(f)\otimes I_Z)$ in $H^1( M^{-1} \otimes L\otimes \frak m_q)$ and its further image in $H^1( M^{-1} \otimes L)$. Using the two exact sequences $$\gather 0\to \scrO _S \to W \otimes M^{-1} \to M^{-1}\otimes L\otimes \frak m_q\to 0;\\ 0\to\scrO_S(-\sigma + (2k-1)f) \to W \otimes M^{-1}\to \scrO_S(f)\otimes I_Z \to 0, \endgather$$ we see that the composite map $\scrO _S \to \scrO_S(f)\otimes I_Z$ is nonzero and gives the nontrivial section. Now the quotient of $\scrO_S(f)\otimes I_Z$ by $\scrO_S$ is $\scrO_f(-z_1-z_2)$. Thus there is an induced map $M^{-1}\otimes L\otimes \frak m_q \to \scrO_f(-z_1-z_2)$ which must factor through the natural map $M^{-1}\otimes L\otimes \frak m_q = \scrO_S(-\sigma + 2kf)\otimes \frak m_q \to \scrO _f(-p-q)$. (Here as usual $p =\sigma \cap f$.) As the induced map $\scrO _f(-p-q) \to \scrO_f(-z_1-z_2)$ is nonzero, it is an isomorphism, and we recover the fact that $q= z_1+z_2 -p$. Using the commutativity of $$\CD 0 @>>> \scrO_S(f)\otimes I_Z @>>> \scrO_S(f) @>>> \scrO_Z @>>> 0\\ @.@VVV @VVV @| @.\\ 0 @>>> \scrO_f(-z_1-z_2) @>>> \scrO_f @>>>\scrO_Z @>>> 0, \endCD$$ we also see that the image of $H^1(\scrO_S(f)\otimes I_Z)$ in $H^1(\scrO_f(-z_1-z_2)$ is the same as the image of $H^0(\scrO_Z)$ in $H^1(\scrO_f(-z_1-z_2))$. There is a commutative diagram $$\CD @. H^1(\scrO_S(-\sigma +2kf)\otimes \frak m_q) @>>> H^1(\scrO_S(-\sigma +2kf))\\ @. @VVV @VVV\\ H^1(\scrO_f(-z_1-z_2)) @>{\cong}>> H^1(\scrO _f(-p-q)) @>>> H^1(\scrO _f(-p)). \endCD$$ Moreover the map $H^1(\scrO_S(-\sigma +2kf)) \to H^1(\scrO _f(-p))$ is an isomorphism. So the problem is the following one: does the image of $H^0(\scrO_Z)$ in $H^1(\scrO_f(-z_1-z_2))$ map to zero in $H^1(\scrO _f(-p))$? The image of $H^0(\scrO_Z)$ in $H^1(\scrO_f(-z_1-z_2))$ is dual to the image of $H^0(\scrO _f)$ in $H^0(\scrO_f(z_1+z_2))$, giving a section vanishing at $z_1$ and $z_2$. On the other hand the kernel of the map $H^1(\scrO _f(-p-q)) \to H^1(\scrO _f(-p))$ is dual to the image of the map $H^0(\scrO_f(p)) \to H^0(\scrO_f(p+q))$, and the corresponding section of $\scrO_f(p+q)$ vanishes at $p$ and $q$. So the only way that this can equal the image of $H^0(\scrO_Z)$ is for $z_1$ or $z_2$ to equal $p$, i.e\. $Z\in D_\sigma$. Conversely, if $Z\notin D_\sigma$, then the image of the extension class in $H^1( M^{-1} \otimes L)$ is not zero. Thus the double dual of the restriction of $\Cal V$ to $S\times \{Z\}$ is a nonsplit extension and so it is stable. \endproof This is as far as we shall go in this section in calculating the four-dimensional invariant. But let us sketch here how to obtain the full formula in (4.8). We will prove a more general statement in Section 10, where we will use (4.12). First, to deal with the fact that $\dim \Ext ^1(\scrO _S(\sigma -kf)\otimes I_Z, \scrO _S((k-2)f)$ jumps along $\Sym ^2\sigma$, blow up $\Sym ^2\sigma$ inside $\operatorname{Hilb}^2S$. Let the exceptional divisor be $G$. After blowing up, we can asume that the extension is not locally trivial along $G$. There is thus a universal extension of torsion free sheaves $\tilde \Cal W$ over $S\times \operatorname{Bl}_{\Sym ^2\sigma} \operatorname{Hilb}^2S$. Now make an elementary modification along $\Cal D$, replacing unstable Type 1 extensions with $Z\in \Cal D - \Sym ^2\sigma $ with stable Type 4 extensions. Next make an elementary modification along $D_\sigma$, replacing unstable Type 1 extensions with $Z\in D_\sigma$ with Type 2 extensions; this also fixes some of the unstable Type 4 extensions. Finally make an elementary modification along $G$ to replace the remaining unstable extensions with Type 3 extensions. At this point every member of the family is a stable torsion free sheaf, and the induced morphism to $\overline{\frak M}_2$ blows $G$ back down again to $\Sym ^2\sigma$. The morphism $\operatorname{Hilb}^2S \to \overline{\frak M}_2$ is then an isomorphism. Keeping track of the Chern classes gives the formula in Theorem 4.8. Finally, we state a general conjecture: \medskip \noindent{\bf Conjecture 4.13.} If $S$ has a section, then the map of (3.14) extends to an isomorphism $\operatorname{Hilb}^tS \to \overline{\frak M}_t$. \medskip If the conjecture is true, then the method of test surfaces used in the proof of Lemma 9.2 can be used to show that the $\mu$-map is given by the following formula (where we use the notation of the Introduction for divisors in $\operatorname{Hilb}^tS$ as well): $$\mu (\Sigma) = D_{\alpha _t} - \bigl((f\cdot \Sigma)/2\bigr)E,$$ where $$\align \alpha _t &= \Sigma + \bigl((-(\sigma \cdot \Sigma) +(p_g-1+t)(f\cdot \Sigma))/2\bigr)f +(f\cdot \Sigma)\sigma \\ &= \alpha _1 + (t-1)\bigl((f\cdot \Sigma)/2\bigr)f. \endalign$$ \section{5. Calculation of the invariant for dimension two and no multiple fibers.} Our goal in this and the following three sections will be a complete calculation of the Donaldson polynomial invariant $\gamma_{w,p}$ in case $-p -3\chi (\scrO_S) = 2$. In this case, the moduli space is compact of real dimension four and complex dimension two, and may be identified with the algebraic surface $J^{e+1}(S)$. We shall begin with the case where $S$ has a section $\sigma$ and $e=-2$. We have already described how to calculate the invariant in this case in the last section. However, we shall give another method for doing so here, since it will serve to explain the construction in the general case. In fact, we shall reprove (in a slightly different guise) the formula in (4.7): $$-4\mu (\Sigma ) = (2(\sigma \cdot \Sigma) -2p_g(f\cdot \Sigma))f -4(f\cdot \Sigma)\sigma -4\Sigma.$$ To describe the $\mu$-map, we begin by describing a universal bundle over $S$. Recall that every bundle $V$ with $-p_1(\ad V) -3\chi (\scrO_S) = 2$ is obtained from the fixed bundle $V_0$ by a single allowable elementary modification. For convenience we will look at the case where $e=-2$. Thus we shall normalize $V_0$ to have $\det V_0\cdot f = -3$ and $$-p_1(\ad V_0) -p = c_1(V_0)^2 - 4c_2(V_0) = 3(1+p_g(S)).$$ As $V_0$ is well-defined up to twisting, so that we can assume that $c_1(V_0) = -3\sigma$, if $p_g(S)$ is odd, and $c_1(V_0) = -3\sigma +f$ if $p_g(S)$ is even. (Here we could use the explicit description of $V_0$ from the preceding section, or use the congruence $p\equiv 1+p_g\mod 4$ to see that these choices always give $c_1(V_0)^2 \equiv p\mod 4$.) We shall just consider the argument in case $p_g$ is odd. Setting $c= c_2(V_0)$, we have $$4c - (-3\sigma )^2 = 3(1+p_g)$$ and thus $$c = -\frac32(1+p_g).$$ If $V$ is stable, with $-p_1(\ad V) =3(1+p_g) +2$, then there is an exact sequence $$0 \to V \to V_0 \to Q \to 0,$$ where $Q$ is a rank one torsion free sheaf on a fiber $f$ with $\deg Q = -1$ and $\det V_0 \cdot f = -3$, and conversely every such $V$ is stable. We need to parametrize such sheaves $Q$ as a family over $S\times S$, where the first factor should be viewed as the surface and the second as the moduli space. To do so, let $\pi _1$ and $\pi _2$ be the projections of $S\times S$ to the first and second factors, let $\Bbb D$ denote the diagonal inside $S\times S$ and let $D= S\times _{\Pee^1}S$ be the fiber product. Thus $D$ is a Cartier divisor, which is not however smooth at the images of pairs of double points. At such a point $D$ has the local equation $xy = zw$, and thus $D$ has an ordinary double point in dimension three. The diagonal $\Bbb D$ is of course contained as a hypersurface in $D$, but this hypersurface fails to be Cartier at the singular points of $D$. Let $\Cal P = I_{\Bbb D}/I_D$. In local analytic coordinates, $\Cal P$ looks like $$(x-z, y-w)R/(xy-zw)R$$ near the double point, where $R = \Cee\{x,y,z,w\}$. We claim that the sheaf $\Cal P$ is flat over $S$ (the second factor). Indeed there is an exact sequence $$0 \to I_{\Bbb D}/I_D \to \scrO _D \to \scrO_{\Bbb D} \to 0.$$ Moreover $\scrO_{\Bbb D}$ is obviously flat over $S$ and $\scrO _D$ is flat over $S$ since $D$ is a local complete intersection inside $S\times S$. Thus $\Cal P$ is flat over $S$ also. Given $q \in S$ denote $\Cal P|\pi _2^{-1}(q)$ by $\Cal P_q$, where we shall identify $\Cal P_q$ with the corresponding torsion sheaf on $S$. If $q$ is not a singular point of a nodal fiber, then $\Cal P_q = \scrO_f(-q)$, where $f$ is the fiber containing $q$ and we have identified $\scrO_f(-q)$ with its direct image on $S$ under the inclusion. If $q$ is the singular point of a singular fiber, then in local analytic coordinates $\Cal P_q$ is given by $$(x-z, y-w)R/(xy-zw, z,w)R \cong (x,y)\Cee\{x,y\}/(xy)\Cee\{x,y\}.$$ Thus globally $\Cal P_q$ is the maximal ideal of $q$, in other words it is the unique torsion free rank one sheaf of degree $-1$ on the singular fiber which is not locally free. Fix as above $V_0$ to be a stable rank two vector bundle on $S$ of fiber degree $-3$ such that the restriction of $V_0$ to every fiber is stable. Thus as we have seen in (1.2) and (2.7)(i), $\dim \Hom (V_0, \Cal P_q) = h^0(V_0\spcheck \otimes \Cal P_q) = 1$ and $h^1 (V_0\spcheck \otimes \Cal P_q) = 0$. It follows via flat base change as in the proof of (3.15) that $\pi _2{}_*((\pi _1^*V_0)\spcheck \otimes \Cal P)$ is a line bundle on $S$. We let $\Cal L$ denote the dual line bundle. Thus $$\align \Hom (\pi _1^*V_0, \Cal P \otimes \pi _2^*\Cal L) &= H^0(S\times S;(\pi _1^*V_0) \spcheck \otimes \Cal P \otimes \pi _2^*\Cal L) \\ &= H^0(S; \pi _2{}_*((\pi _1^*V_0)\spcheck \otimes \Cal P) \otimes \Cal L)\\ &= H^0(S; \Cal L^{-1}\otimes \Cal L) = H^0(S; \scrO_S). \endalign$$ Thus there is a nonzero map $\pi _1^*V_0 \to \Cal P \otimes \pi _2^*\Cal L$, essentially unique, and its restriction to each fiber $\pi _2^{-1}(q)$ is also nonzero. We may then define a universal bundle by the exact sequence $$0 \to \Cal V \to \pi _1^*V_0 \to \Cal P \otimes \pi _2^*\Cal L \to 0.$$ \lemma{5.1} The sheaf $\Cal V$ is locally free and its restriction to each slice $S\times \{q\}$ is a stable rank two vector bundle $V_q$ with $-p_1(\ad V_q) - 3\chi (\scrO_S) = 2$. The resulting morphism $S\to \frak M_1$ is an isomorphism. \endstatement \proof There is an exact sequence $$0 \to V_q \to V_0 \to \Cal P_q \to 0.$$ Thus $V_q$ is locally free for all $q$ and so is $\Cal V$. By construction $V_q$ has stable restriction to every fiber except the one containing $q$. Thus $V_q$ is stable. The statement about $p_1(\ad V_q)$ is clear. Finally, examining the description of (3.13), we see that the map $S\to \frak M_1$ is a bijection. Since $\frak M_1$ is smooth, the map is therefore an isomorphism. \endproof We now turn to calculating the Chern classes of $\Cal V$. By (0.1) $$p_1(\ad\Cal V) - p_1(\ad \pi ^*V_0) = 2c_1(V_0)\cdot D + [D]^2 -4i_*c_1(\Cal P \otimes \pi _2^*\Cal L),$$ where $i\:D \to S\times S$ is the inclusion. Here the sheaf $\Cal P \otimes \pi _2^*\Cal L$ fails to be a line bundle exactly at the singular points of $D$, which does not affect the Chern classes $c_1$ and $c_2$. Thus we can simply define $i_*c_1(\Cal P \otimes \pi _2^*\Cal L)$ to be the unique extension of the class $i_*c_1(\Cal P \otimes \pi _2^*\Cal L|D_{\text{reg}})$. Next we claim: \lemma{5.2} In $H^2(S\times S)$, we have $[D] = f\otimes 1 + 1\otimes f$. \endstatement \proof Let $C$ be a Riemann surface embedded in $S$, and consider $([C]\otimes [x])\cup [D]$, where $x$ is a point of $S$. This is the same as $\#\bigl((C\times \{x\})\cap D\bigr)$, where the points are counted with signs. Clearly this intersection is the same as $\#(C\cap f)$. A similar argument holds for $([x]\otimes [C])\cap [D]$. Thus $[D]$ and $f\otimes 1 + 1\otimes f$ define the same element of $H^2(S\times S)$. \endproof It follows that, up to a term not affecting slant product, $$p_1(\ad\Cal V) - p_1(\ad \pi ^*V_0) = -6\sigma \otimes f + 2f\otimes f -4i_* c_1(\Cal P \otimes \pi _2^*\Cal L).$$ Next we must calculate the most interesting term in the expression for $p_1(\ad \Cal V)$ above, the term $c_1(\Cal P \otimes \pi _2^*\Cal L)$, viewed as a coherent sheaf on $D$. As far as $c_1$ is concerned, we can ignore the singularities of $D$. Thus $$\align c_1(\Cal P \otimes \pi _2^*\Cal L|D_{\text{reg}}) &= c_1(I_{\Bbb D}/I_D|D_{\text{reg}}) + \pi _2^*c_1(\Cal L)\\ &= -[\Bbb D] + \pi _2^* c_1(\Cal L). \endalign$$ Here $[\Bbb D]$ is viewed as a divisor on $D_{\text{reg}}$. However the unique extension of $i_*[\Bbb D]$ to an element of $H^4(S\times S)$ is clearly again $[\Bbb D]$, where we now view $\Bbb D$ as a codimension two cycle on $S\times S$. Now let $\alpha = c_1(\Cal L^{-1}) \in H^2(S)$. Then $$\align i_*\pi _2^*c_1(\Cal L^{-1}) &= i_*i^*(1\otimes \alpha)\\ &= i_*i^*(1)\cup (1\otimes \alpha) = [D]\cup (1\otimes \alpha)\\ &= f\otimes \alpha + 1\otimes [f\cdot \alpha]. \endalign$$ Thus up to a term which does not affect slant product, $i_*\pi _2^*c_1(\Cal L^{-1}) = f\otimes \alpha$. To calculate this term, we shall use the following lemma: \lemma{5.3} $\alpha = c_1(\Cal L^{-1}) = -3\sigma -\frac52(p_g+1)f$. \endstatement \proof We shall apply the Grothendieck-Riemann-Roch theorem to calculate the Chern classes of $$\Cal L^{-1} = \pi _2{}_*((\pi _1^*V_0)\spcheck\otimes \Cal P).$$ We have $$\ch((\pi _2)_!((\pi _1^*V_0)\spcheck\otimes \Cal P)\Todd S = \pi _2{}_* \Bigl[\ch ((\pi _1^*V_0)\spcheck\otimes \Cal P))\cdot \Todd (S\times S)\Bigl].$$ Now $H^1(V_0 \spcheck \otimes Q) = 0$ for all $Q$ a torsion free rank one sheaf on a fiber $f$, so that $(\pi _2)_!((\pi _1^*V_0)\spcheck\otimes \Cal P) = \pi _2{}_*((\pi _1^*V_0)\spcheck\otimes \Cal P) = \Cal L^{-1}$ and the left hand side above is just $c_1(\Cal L^{-1})\Todd S$. Now we can also multiply by $(\Todd S)^{-1}$ to get $$\align c_1(\Cal L^{-1}) &= \pi _2{}_*\Bigl[\ch ((\pi _1^*V_0)\spcheck\otimes \Cal P))\cdot \Todd (S\times S)\Bigl]\cdot (\Todd S)^{-1}\\ &= \pi _2{}_*\Bigl[\ch ((\pi _1^*V_0)\spcheck\otimes \Cal P))\cdot \Todd (S\times S)\cdot \pi _2^*(\Todd S)^{-1}\Bigl]\\ &= \pi _2{}_*\Bigl[\ch ((\pi _1^*V_0)\spcheck\otimes \Cal P))\cdot \pi _1^*\Todd S\Bigl]\\ &=\pi _2{}_*\Bigl[\ch (\pi _1^*V_0)\spcheck\cdot \pi _1^*\Todd S\cdot \ch(\Cal P)\Bigl], \endalign$$ using the multiplicativity of the Todd class. Moreover $$\ch (V_0\spcheck) = 2-c_1(V_0) + \frac{c_1(V_0)^2 - 2c_2(V_0}2 = 2 + 3\sigma -6(1+p_g)[\pt]$$ and $$\Todd S = 1 - \frac{(p_g-1)}2f + (p_g+1)[\pt].$$ So $$\pi _1^*\ch (V_0\spcheck)\cdot \pi _1^*\Todd S = 2 + 3\sigma \otimes 1-(p_g-1)f \otimes 1 + N[\pt]\otimes 1,$$ where $$N= \frac{(3\sigma)^2 -2c}2 +2(p_g+1) -\frac32(p_g-1) = \frac{-5p_g +1}2,$$ using the fact that $c = -\frac32(1+p_g)$. Next we compute $\ch\Cal P= \ch (I_\Bbb D/I_D) = \ch I_\Bbb D - \ch I_D$. Now $I_D = \scrO_{S\times S}(-D)$, so that $\ch I_D = 1 -[D] + [D]^2/ 2 -\cdots$. As for $\ch I_\Bbb D$, we have $\ch I_\Bbb D = 1- \ch \scrO _\Bbb D$. Applying the Grothendieck-Riemann-Roch formula to the inclusion $j\: \Bbb D \to S\times S$ gives $\ch \scrO _\Bbb D = j_*((\Todd N_{\Bbb D/S\times S})^{-1})$, where $N_{\Bbb D/S\times S}$ is the normal bundle of $\Bbb D$ in $S\times S$, and so is equal to the tangent bundle $T_S$ on $\Bbb D$. Thus $$\align \ch\scrO _\Bbb D &= j_*\Bigl((1-\frac{(p_g-1)}2f +(1+p_g)[\pt])^{-1}\Bigr)\\ &=j_*(1+\frac{(p_g-1)}2f -(1+p_g)[\pt])\\ &= [\Bbb D] + \frac{(p_g-1)}2j_*f- (1+p_g)j_*[\pt]. \endalign$$ Collecting up the terms through degree 3 (which are the only ones which will contribute) gives $$\ch\Cal P = [D] - \frac{[D]^2}2 - [\Bbb D] - \frac{p_g-1}2j_*f+\cdots.$$ Putting this together, we see that $\alpha$ is the degree one term in $$\pi _2{}_*\Bigl[(2 + 3\sigma \otimes 1-(p_g-1)f\otimes 1 + N[\pt]\otimes 1)\cdot ([D] - \frac{[D]^2}2 - [\Bbb D] - \frac{p_g-1}2j_*f)\Bigr].$$ Recalling that $D=f\otimes 1 +1\otimes f$ and that $[D]^2/2 = f\otimes f$, we must apply $\pi _2{}_*$ to $$\pi _2{}_*((p_g-1)j_*f -3(\sigma \otimes 1)\cdot [\Bbb D] + (p_g-1)(f\otimes 1)\cdot [\Bbb D] -3(\sigma \otimes 1)\cdot (f\otimes f) +N[\pt] \otimes f).$$ The result is then $$-(p_g-1)f -3f -3\sigma +(p_g-1)f +Nf=-3\sigma + (N-3)f,$$ as claimed. \endproof The above lemma thus implies that $$-4i_*c_1(\Cal P \otimes \pi _2^*\Cal L)= 4[\Bbb D] -12f\otimes \sigma -10(p_g+1)f\otimes f.$$ Putting this together gives (neglecting all terms which do not affect slant product) $$p_1(\ad \Cal V) = -6(\sigma \otimes f) + (-10p_g-8)f\otimes f -12f\otimes \sigma + 4[\Bbb D] +\cdots.$$ We may finally summarize our calculations as follows: \lemma{5.4} In the above notation, $$-4\mu (\Sigma) = \Bigl[-6(\sigma\cdot\Sigma) +(-10p_g-8)(f\cdot \Sigma)\Bigr]f -12(f\cdot \Sigma)\sigma + 4\Sigma.$$ Thus $\mu (\Sigma )^2 = (\Sigma)^2 + (p_g-1)(f\cdot \Sigma)^2$. \qed \endstatement \medskip At first glance, this formula looks quite different from the previous formula $$-4\mu (\Sigma ) = (2(\sigma \cdot \Sigma) -2p_g(f\cdot \Sigma))f -4(f\cdot \Sigma)\sigma -4\Sigma.$$ However, the surface $S$ (viewed as the moduli space) has an involution $\iota$, coming from taking $x\mapsto -x$ on each fiber using $\sigma$ as the identity section. This involution corresponds to viewing $S$ as the double cover of a rational ruled surface as in [6] Chapter 1. Since $S$ has only nodal singular fibers, it follows that on $H^2(S)$, $\iota$ fixes $\sigma$ and $f$ and is equal to $-\Id$ on the orthogonal complement $\{f, \sigma\}^\perp$. It is then an easy exercise to see that for a general $\Sigma$ we have $$\iota ^*(\Sigma) = -\Sigma + 2\Bigl[(\sigma \cdot \Sigma)+(p_g+1)(f\cdot\Sigma)\Bigr]f + 2(f\cdot \Sigma)\sigma.$$ Applying $\iota$ then exchanges the above two expressions for $\mu (\Sigma)$. Clearly this discrepancy arose as follows. In the general scheme for identifying the moduli space implicit in (3.14) and (4.7) we used not $\Cal P$ but its dual. However it was technically slightly simpler not to make this choice in the Riemann-Roch calculation above. Thus the identifications of the moduli space differ by $-\Id$. \section{6. The case of multiple fibers.} Having done the rather tedious calculation in the preceding section in case $S$ has a section, we must now move on to deal with the case where $S$ has multiple fibers. Fortunately, it will turn out that much of the calculation in this case exactly follows the pattern of the previous calculation. Before getting into the nitty-gritty, let us fix notation. Let $\pi\:S\to \Pee ^1$ be a nodal surface with at most two multiple fibers of odd multiplicity. Fix a divisor on the generic fiber $S_\eta$ of odd degree $2e+1$. Let $V_0$ be a rank two vector bundle on $S$ with $c_1(V_0) = \Delta$ and $c_2(V_0) = c$, whose restriction to the reduction of every fiber is stable. Thus $4c - \Delta ^2 = 3(p_g+1)$ and so $$2c = \frac{\Delta ^2 +3(p_g+1)}2.$$ We would like to construct a universal bundle using $J^{e+1}(S)$. Unfortunately, this is not in general possible, and we shall instead use a finite cover. Thus we fix an elliptic surface $T$ together with a map $T\to S$, such that $T$ has a section. We may further assume that $T$ is obtained as follows: choose a smooth multisection $C$ of $\pi$, for example a general hyperplane section of $S$ in some projective embedding. For $C$ sufficiently general, we may assume that $C$ meets the multiple fibers transversally and that the map $C\to \Pee ^1$ is not branched at any points corresponding to singular nonmultiple fibers of $\pi$. Then set $T$ to be the normalization of $S\times _{\Pee ^1}C$. It follows that the only singular fibers of $T$ lie over singular nonmultiple fibers of $S$, and that $T$ has a section $\sigma$. If $d$ is the degree of $C\to \Pee ^1$, then at the point of $\Pee ^1$ lying under the multiple fiber $F_i$ of multiplicity $m_i$, $C\to \Pee ^1$ is branched to order $m_i$ at exactly $d/m_i$ points. Let $\varphi\:T\to S$ be the natural map and $\rho\:T\to C$ be the elliptic fibration, so that we have a commutative diagram $$\CD T@>{\varphi}>>S\\ @V{\rho}VV @VV{\pi}V \\ C@>>> \Pee ^1. \endCD$$ Now we can state the main result of this section: \theorem{6.1} There exists a vector bundle $\tilde \Cal V$ over $S\times T$ with the following properties: \roster \item"{(i)}" The restriction of $\tilde \Cal V$ to each slice $S\times \{p\}$ is a stable rank two vector bundle $V$ with $\det V = \Delta -f$ and $-p_1(\ad V) -3\chi(\scrO _S) =2$. \item"{(ii)}" The morphism $T\to \frak M_1$ induced by $\tilde \Cal V$ has degree $d$. \item"{(iii)}" If $\tilde \mu\: H_2(S) \to H^2(T)$ is the map induced by slant product with the class $-p_1(\ad \tilde \Cal V)/4$, then, setting $\delta =[\Delta]$, $$\align -4\tilde \mu (\Sigma) &= \Bigl[\delta ^2 -(1+p_g) -4(e+2)^2(1+p_g) +2 + c(e,m_1) + c(e,m_2)\Bigr] (f\cdot \Sigma)df \\ &-4(e+2)(\varphi ^*\delta \cdot \sigma)(f\cdot \Sigma)f - 4(e+2)(\varphi ^* \Sigma \cdot \sigma)f +2d(\delta \cdot \Sigma)f\\ & +4(f\cdot \Sigma)\varphi ^*\delta-8(e+2)(f\cdot \Sigma)\sigma +4\varphi ^*\Sigma, \endalign$$ where $c(e,m_i)$ depends only on $m_i$ and $e$ and on an analytic neighborhood of the multiple fiber, and not on $S$ or $p_g$, and where $c(e,1) =0$. \endroster \endstatement \medskip We shall defer the proof of Theorem 6.1 to the next two sections. The constant $c(e, m_i)$ in fact might depend {\it a priori\/} on the particular choice of the multiple fiber. However, as we shall see from Theorem 6.3, the choice of the fiber and of $e$ does not matter. Let us begin with a calculation of $\mu (\Sigma)^2$: \lemma{6.2} With notation as in \rom{(6.1)}, we have $$16\tilde \mu (\Sigma)^2 = 16d(\Sigma)^2 +16d(p_g-1-c(e,m_1) -c(e,m_2))(f\cdot \Sigma)^2.$$ Thus as $\tilde \mu (\Sigma)^2 = d\mu (\Sigma)^2$, we have $$\align \mu (\Sigma)^2 &=(\Sigma)^2 +(p_g-1-c(e,m_1) -c(e,m_2))(f\cdot \Sigma)^2\\ &=(m_1m_2)^2 (p_g-1-c(e,m_1) -c(e,m_2))(\kappa\cdot \Sigma)^2, \endalign$$ where $\kappa$ is the primitive class such that $m_1m_2\kappa = f$. \endstatement \proof This is a tedious calculation. \endproof \theorem{6.3} With notation as in the statement of \rom{(6.1)}, we have $$c(e,m_i) = -1 + \frac{1}{m_i^2}.$$ \endstatement \proof By symmetry it suffices to consider $i=1$. Choose a general nodal rational elliptic surface $S_0$ with a single multiple fiber of multiplicity $m_1$. We can assume that an analytic neighborhood of the multiple fiber in $S_0$ is analytically isomorphic to a neighborhood of $F_1$ in $S$, which is possible since we assumed that the multiple fibers did not lie over branch points of the $j$-function of $S$. Since $m_1|2e+1$, there exists a divisor $\Delta$ on $S_0$ with $\Delta \cdot f = 2e+1$. Thus we may use $S_0$ to calculate $c(e, m_1)$. Now setting $p_g =0$ and $m_2=1$ in the formula of (6.2) gives the coefficient of $(\kappa \cdot \Sigma)^2$ in the Donaldson polynomial: it is $(m_1)^2(-1 -c(e,m_1))$. On the other hand, $S_0$ is orientation-preserving diffeomorphic to a rational elliptic surface $S_1$ with a section, by a diffeomorphism $\psi$ which carries $\kappa$ to the class of a fiber. Using (3.5) of Part I of [4], this diffeomorphism must then carry a $(w,p)$-suitable chamber for $S_1$ to a $(\psi ^*w,p)$-suitable chamber for $S_0$. The Donaldson polynomial for $S_1$ and a $(w,p)$-suitable chamber is then sent under $\psi ^*$ to the $\pm$ the Donaldson polynomial for $S_0$ and a $(\psi ^*w,p)$-suitable chamber. Normalizing the orientations so that the leading coefficients agree (these are both $(\Sigma ^2)$), the coefficients of $(\kappa \cdot \Sigma)^2$ must agree also. We have already calculated the coefficient of $(\kappa \cdot \Sigma)^2$ for $S_1$ (by two different methods): it is $-1$. Thus $$(m_1)^2(-1 -c(e,m_1)) = -1.$$ Hence $c(e,m_1) = -1 + 1/m_1^2$, as claimed. \endproof Thus we get the formula for $\mu (\Sigma)^2$ stated in (ii) of Theorem 2 of the Introduction: \corollary{6.4} The two-dimensional Donaldson polynomial is given by the formula $$\align \mu (\Sigma)^2 &= (\Sigma)^2 +(m_1m_2)^2 (p_g-1 + 1-\frac{1}{m_1^2} + 1-\frac{1}{m_2^2}) (\kappa\cdot \Sigma)^2\\ &= (\Sigma)^2 +\bigl[(m_1m_2)^2 (p_g+1) - m_1^2 -m_2^2\bigr](\kappa\cdot \Sigma)^2. \qed \endalign$$ \endstatement For future reference we note the following lemma: \lemma{6.5} If $f$ denotes the general fiber of $\frak M_1= J^{e+1}(S) \to \Pee^1$, then $$\mu (\Sigma)\cdot f = 2(f\cdot \Sigma).$$ \endstatement \noindent {\it Proof.} It suffices to calculate $\tilde \mu (\Sigma )\cdot f$, where $\tilde \mu$ is as defined in (6.1)(iii) and here $f$ denotes a general fiber on $T$. But using the formula in (6.1)(iii) gives $$\tilde \mu (\Sigma )\cdot f = -(f\cdot \Sigma)(2e+1) +2(e+2) (f\cdot \Sigma) - (f\cdot \Sigma) = 2(f\cdot \Sigma). \qed$$ \section{7. Proof of Theorem 6.1: a Riemann-Roch calculation.} We return to the notation of the preceding section. Our goal in this section will be to approximate the universal bundle by a coherent sheaf which is essentially an elementary modification of $\pi _1^*V_0$, where $V_0$ is as described at the beginning of the preceding section and $\pi _i$ denotes the $i^{\text{th}}$ projection now on $S\times T$. We have the map $\varphi \: T\to S$ of elliptic surfaces covering the map $\rho \:C\to \Pee ^1$ of the base curves. Let $\Gamma$ be the graph of $\varphi$ in $S\times T$ and let $H$ be the graph of the composition $\psi\: T@>{\rho}>>C@>{\sigma}>>T@>{\varphi}>>S$, where we view $\sigma$ temporarily not as a curve in $T$ but rather as a morphism. Let $D = S\times _{\Pee^1}T\subset S\times T$ and let $\tilde D$ be the normalization of $D$. The singularities of $D$ are of two types. The first type consists of points $(p,q)$ where $\varphi(q) = p$ and $p$ and $q$ are the singular points on a nodal fiber. At such points $D$ has an ordinary double point as in the case where $S$ has a section. The second type of singularity is along a multiple fiber $F_i$. At a point of $\Pee ^1$ lying under $F_i$, the map $C\to \Pee ^1$ is branched to order $m_i$. Thus, in local analytic coordinates $x,y,z,w$ on $S\times T$ the divisor $D$ has the local equation $x^{m_i} = z^{m_i}$. If $R$ is the local ring of $D$ at such a point and $\tilde R$ is its normalization, then the inclusion $R\subseteq \tilde R$ is given by $$\Cee\{x,y,z,w\}/(x^{m_i} - z^{m_i}) \hookrightarrow \bigoplus _k\Cee\{x,y,w\},$$ where the map from $R$ to the $k^{\text{th}}$ factor in the direct sum is given by setting $z = \zeta ^kx$ for $\zeta = e^{2\pi \sqrt{-1}/m_i}$. Let $\tilde F_i = \varphi ^{-1}(F_i)$ and let $E_i$ be a component of $\tilde F_i$. There is thus an induced map $\nu _i \: E_i \to F_i$ which is \'etale of degree $m_i$. We also have maps $D\to T$ and $\tilde D\to D$. Clearly $D$ and $\tilde D$ are flat over $T$ (note that $\tilde D$ is smooth away from the images of pairs of double points). The calculations above for $R$ and $\tilde R$ show that the scheme-theoretic fiber of $D$ at a point $q\in E_i$ is $F_i$ as a multiple fiber and that $i_*\scrO _{\tilde D}$ restricted to this fiber is $\nu _i{}_*\scrO _{E_i}$. Since a section cannot pass through a singular point of a fiber, the graph $H$ avoids the double point singularities of $D$. Denote also by $H$ the pullback of $H$ to $\tilde D$. Then $H$ is a Cartier divisor on $\tilde D$. Define $$\Cal P = i_*\scrO _{\tilde D}(-\Gamma + (e+2)H).$$ This notation does not define $\Cal P$ near the double points of $D$, but as $H$ does not pass through the double points and $\tilde D=D$ in a neighborhood of the double points we can just glue $\Cal P$ to $I_\Gamma/I_D$ at the double points. Equivalently we could just take the push-forward of the restriction of $i_*\scrO _{\tilde D}(-\Gamma + (e+2)H)$ to $D_{\text{reg}}$. Finally we shall let $\pi _1$ and $\pi _2$ denote the first and second projections on $S\times T$. \lemma{7.1} The sheaf $\pi _2{}_*((\pi _1^*V_0)\spcheck \otimes \Cal P)$ is a line bundle on $T$, whose dual is denoted $\Cal L$. Moreover $$R^1\pi _2{}_*((\pi _1^*V_0)\spcheck \otimes \Cal P) = 0.$$ \endstatement \proof Letting $h\: \tilde D \to T$ and $j\: \tilde D \to S\times S@>{\pi _1}>> S$ be the natural maps, it is clear that $$\pi _2{}_*((\pi _1^*V_0)\spcheck \otimes \Cal P) = h_*((j^*V_0)\spcheck \otimes \scrO _{\tilde D}(-\Gamma + (e+2)H)).$$ So we must check that the restriction of $(j^*V_0)\spcheck \otimes \scrO _{\tilde D}(-\Gamma + (e+2)H)$ to each fiber of $h$ has $h^0 =1$ and $h^1=0$. The only new case is the case corresponding to a multiple fiber. In this case the restriction to the fiber is $(\nu _i^*V_0)\spcheck \otimes L$, where $L$ is a line bundle of degree $e+1$ on $E_i$. The degree of $\nu _i^*V_0$ is $m_i(\deg V/m_i) = 2e+1$ and $\nu _i^*V_0$ is stable since it is the pullback of the stable bundle $V_0|F_i$. Thus by (1.2), $H^0(E_i;(\nu _i^*V_0)\spcheck \otimes L)$ has dimension one and $H^1(E_i;(\nu _i^*V_0)\spcheck \otimes L) = 0$. \endproof Thus arguing as in the case of a section there is a unique nonzero map (mod scalars) $$\pi _1^*V_0 \to \Cal P \otimes \pi _2^*\Cal L.$$ Unfortunately, if there are multiple fibers this map is no longer surjective. We shall return to this point in the next section. Our remaining goal in this section is to calculate $\Cal L$: \lemma{7.2} With $\Cal L^{-1}= \pi _2{}_*((\pi _1^*V_0)\spcheck \otimes \Cal P)$ and $\delta =[\Delta]$, we have $$c_1(\Cal L^{-1}) =\Big[\frac{\delta ^2}4-\frac{1+p_g}4 -(e+2)^2(1+p_g) \Big]df -(e+2)(\varphi ^*\delta \cdot \sigma)f + \varphi ^*\delta -2(e+2)\sigma.$$ \endstatement \proof As before we shall apply the Grothendieck-Riemann-Roch theorem to find $c_1(\pi _2{}_*((\pi _1^*V_0)\spcheck \otimes \Cal P))$: it is the degree one term in $$\pi _2{}_*(\pi _1^*\ch V_0\spcheck \cdot\pi _1^*\Todd S \cdot \ch i_*\scrO _{\tilde D}(-\Gamma + (e+2)H)).$$ We have $$\ch V_0\spcheck = 2-\delta +\fracwithdelims(){\delta ^2-2c}2[\pt],$$ where $\delta =[\Delta]$, and $$\Todd S = 1 + \frac{r}2f + (1+p_g)[\pt],$$ where $$-r = (p_g+1) - \frac{1}{m_1} - \frac{1}{m_2}.$$ Thus the product of the first two terms above is $\pi _1^*(2-\delta + rf +M[\pt])$, where $$M= \frac{\delta ^2-2c}2 + 2(1+p_g) -\frac{r}2(2e+1).$$ Since we have $$\delta ^2-2c = \frac{\delta ^2 -4c}2 + \frac{\delta ^2}2 = -\frac32(1+p_g) + \frac{\delta ^2}2,$$ we can rewrite this as $$M= \frac{\delta ^2}4 + \frac54(1+p_g) -\frac{r}2(2e+1).$$ Next we must calculate $\ch i_*\scrO _{\tilde D}(-\Gamma + (e+2)H))$. Again using the Grothendieck-Riemann-Roch theorem, and setting $G= -\Gamma + (e+2)H$ for notational simplicity, we have $$\ch i_*\scrO _{\tilde D}(G) = i_* \bigl[\ch \scrO _{\tilde D}(G)\cdot(\Todd N_i)^{-1}\bigr],$$ where $N_i$ is the normal bundle to the immersion $i$. Now $\ch \scrO _{\tilde D}(G) = 1 + G + G^2/2 + \cdots$. As for $N_i$, locally at the multiple fiber $F_i$ $D$ is the union of $m_i$ sheets, and so $$N_i = \scrO _{\tilde D}(D - (m_1-1) B_1 - (m_2-1)B_2),$$ where $B_i = F_i \times \tilde F_i$. It follows that $$(\Todd N_i)^{-1} = 1-\frac{D - (m_1-1) B_1 - (m_2-1)B_2}2 +\cdots$$ and so $$\align \ch i_*\scrO _{\tilde D}(G) = D + G &-i_*\fracwithdelims(){D - (m_1-1) B_1 - (m_2-1)B_2}2 + i_*\fracwithdelims(){G^2}2 \\ &-i_*\fracwithdelims(){G\cdot (D - (m_1-1) B_1 - (m_2-1)B_2)}2 +\cdots . \endalign$$ So we must take the degree three term in the product of the above expression with $\pi _1^*(2-\delta + rf +M[\pt])$ and then apply $\pi _2{}_*$. First, a calculation along the lines of (5.2) shows that $$[D] = f\otimes 1 + d(1\otimes f),$$ where $f$ denotes either the class of a fiber in $S$ or $T$, depending on the context. The degree three term above is then a sum of three terms: $T_1+T_2+T_3$, where $$\align T_1 &= M([\pt]\otimes 1)\cdot D \\ T_2 &= -G\cdot (\delta \otimes 1) + G\cdot (rf\otimes 1) \\ &-\frac12i_*( D - (m_1-1) B_1 - (m_2-1)B_2)\cdot (-\delta \otimes 1 + rf\otimes 1)\\ T_3 &= i_*(G^2 -G\cdot i^*D + (m_1-1)(G\cdot B_1) + (m_2-1)(G\cdot B_2)). \endalign$$ Let us now apply $\pi _2{}_*$ to these terms. First $$\pi _2{}_* T_1= \pi _2{}_*(Md)[\pt]\otimes f= (Md)f.$$ To calculate $\pi _2{}_*T_2$, first note the following, whose proof is an easy verification: \lemma{7.3} For every $\alpha \in H^2(S)$, $$\align \pi _2{}_*(\Gamma \cdot\alpha \otimes 1) &= \varphi ^*\alpha;\\ \pi _2{}_*(H\cdot \alpha \otimes 1) &= (\varphi ^*\alpha \cdot \sigma)f. \qed \endalign$$ \endstatement \medskip So the terms involving $G$ in $\pi _2{}_*T_2$ give $$\align -(e+2)(\varphi ^*\delta \cdot \sigma)f &+ \varphi ^*\delta + (e+2)r(\varphi ^*f \cdot \sigma)f- r\varphi ^*f\\ =-(e+2)(\varphi ^*\delta \cdot \sigma)f &+ \varphi ^*\delta + (e+1)rdf, \endalign$$ where we have used $\varphi ^*f = df$. To handle the terms involving $B_i$, note that $i_*[B_i] = m_i[F_i\times \tilde F_i]$. Also $[F_i] = (1/m_i)f$ and $\tilde F_i$ consists of $d/m_i$ copies of $f$ (the fiber on $T$) so that $$[F_i\times \tilde F_i] = \fracwithdelims(){d}{m_i^2}f\otimes f; \qquad i_*[B_i] = \frac{d}{m_i}f\otimes f.$$ Also $i_*D = D^2 = 2d(f\otimes f)$. Thus $$-\frac12 i_*(D - (m_1-1) B_1 - (m_2-1)B_2) = -\frac{d}2\left(\frac1{m_1} + \frac1{m_2}\right)(f\otimes f).$$ The product of this term with $f\otimes 1$ is zero, and we are left with the product with $-\delta \otimes 1$, which contributes $$ \frac{d(2e+1)}2\left(\frac1{m_1} + \frac1{m_2}\right)f.$$ Combining these, we see that $$\pi _2{}_*T_2 =-(e+2)(\varphi ^*\delta \cdot \sigma)f + \varphi ^*\delta + (e+1)rdf+ \frac{d(2e+1)}2\left(\frac1{m_1} + \frac1{m_2}\right)f.$$ We turn now to the term $\pi _2{}_*T_3$. We have $G^2 = (e+2)^2H^2 -2(e+2)H \cdot \Gamma + \Gamma ^2$. To calculate $\pi _2{}_*$ applied to these terms, we shall use the following lemma: \lemma{7.4} \roster \item"{(i)}" $\pi _2{}_*i_*H^2 = \pi _2{}_*i_*\Gamma ^2 = -d(1+p_g)f$. \item"{(ii)}" $\pi _2{}_*i_*H\cdot \Gamma = \sigma$. \endroster \endstatement \proof To see (i), note that we have an exact sequence $$0 \to N_{\Gamma /\tilde D} \to N_{\Gamma /S\times T} \to N_i \to 0.$$ Also $(\Gamma ^2)_{\tilde D} = \phi_*c_1(N_{\Gamma /\tilde D})$, where $\phi\: \Gamma \to \tilde D$ is the inclusion. Now $$\align c_1(N_{\Gamma /\tilde D}) &= c_1(N_{\Gamma /S\times T}) - c_1(N_i)\\ &= c_1(\pi _1^*T_S|\Gamma) -(D-(m_1-1)B_1-(m_2-1)B_2)|\Gamma\\ &= \varphi ^*(rf) - ((f\otimes 1 +d(1\otimes f)-(m_1-1)(B_1\cdot \Gamma)-(m_2-1)(B_2\cdot\Gamma))\\ &= \Bigl[-d\left(p_g+1-\frac1{m_1} -\frac1{m_2}\right)-\left(2d -\frac{d(m_1-1)} {m_1} - \frac{d(m_2-1)}{m_2}\right)\Big]f\\ &= -d(p_g+1)f. \endalign$$ Thus $\pi _2{}_*i_*\Gamma ^2 = -d(1+p_g)f$. A similar calculation handles $\pi _2{}_*i_*H^2$. The proof of (ii) is an easy calculation. \endproof Thus $$\pi _2{}_*G^2 = -d(p_g+1)((e+2)^2 +1)f - 2(e+2)\sigma.$$ The remaining term is $-\pi _2{}_*(G\cdot(D-(m_1-1)B_1-(m_2-1)B_2))$. We have seen in the course of the proof of Lemma 7.4 that $$\align \pi _2{}_*\Gamma \cdot (D-(m_1-1)B_1-(m_2-1)B_2) &= \pi _2{}_*H\cdot (D-(m_1-1)B_1-(m_2-1)B_2)\\ &= \left(\frac1{m_1} + \frac1{m_2}\right)df. \endalign$$ Thus $$\pi _2{}_*G\cdot (D-(m_1-1)B_1-(m_2-1)B_2)= d(e+1)\left(\frac1{m_1} + \frac1{m_2}\right)f.$$ In all then, $$\pi _2{}_*T_3 = d\Big[ -(p_g+1)((e+2)^2 +1)- (e+1)\left(\frac1{m_1} + \frac1{m_2}\right)\Big]f - 2(e+2)\sigma.$$ Combining terms, we have $$c_1(\Cal L^{-1}) =\Big[\frac{\delta ^2}4-\frac{1+p_g}4 -(e+2)^2(1+p_g) \Big]df -(e+2)(\varphi ^*\delta \cdot \sigma)f + \varphi ^*\delta -2(e+2)\sigma,$$ as claimed. This concludes the proof of Lemma 7.2. \endproof \section{8. Proof of Theorem 6.1: Conclusion.} We keep the notation of the two previous sections. We begin by constructing a ``universal bundle" $\tilde \Cal V$ over $S\times T$. Begin with the morphism $\pi _1^*V_0 \to \Cal P \otimes \pi _2^*\Cal L$ defined in the previous section, and let $\tilde \Cal V$ be the kernel. By construction $\tilde \Cal V$ is locally free away from $(F_1\times \tilde F_1)\amalg (F_2\times \tilde F_2)$. There is an exact sequence: $$ 0 \to \tilde \Cal V \to \pi _1^*V_0 \to \Cal P \otimes \pi _2^*\Cal L \to \Cal Q_1\oplus \Cal Q_2 \to 0 .$$ where $\Cal Q_i$ is supported on $F_i\times \tilde F_i$. Now $\tilde F_i$ is a disjoint union of $d/m_i$ fibers of $T$. Let $\Cal Q = \Cal Q_1 \oplus \Cal Q _2$ and let $c$ denote the total Chern polynomial. Then $$c(\tilde \Cal V) = \pi _1^*c(V_0) \cdot c(\Cal P \otimes \pi _2^*\Cal L)^{-1} \cdot c(\Cal Q).$$ Thus if we let $\pi _1^*c(V_0) \cdot c(\Cal P \otimes \pi _2^*\Cal L)^{-1} = 1 + x_1 + x_2+ \cdots$, then $$\align c_2(\tilde \Cal V) &= x _2 + c_2(\Cal Q);\\ c_1(\tilde \Cal V)^2 -4c_2(\tilde \Cal V) &= x_1 ^2 -4x _2 -4 c_2(\Cal Q). \endalign$$ Now we claim that Theorem 6.1 is a consequence of the following two results: \theorem{8.1} There exist integers $q(e, m_i)$ such that $$c_2(\Cal Q_i)= dq(e, m_i)[F_i\times f].$$ Here the integer $q(e, m_i)$ depends only on an analytic neighborhood of $F_i$ and $e$ but not on $S$ or $p_g(S)$. \endstatement \medskip \theorem{8.2} The coherent sheaf $\tilde \Cal V$ is locally free. \endstatement \medskip \demo{Proof that \rom{(8.1)} and \rom{(8.2)} imply Theorem \rom{(6.1)}} Let us consider the restriction of $\tilde \Cal V$ to a slice $S\times \{q\}$. In all cases this restriction is a vector bundle $V$ whose restriction to every smooth fiber $f$ of $S$ not equal to the fiber containing $\varphi(q)$ is $V_0|f$. Thus the restriction of $V$ to such a fiber $f$ is stable, and so $V$ is stable by (3.4). Now if $\varphi(q)$ does not lie on a multiple fiber, there is an exact sequence $$0 \to V \to V_0 \to Q \to 0,$$ where $Q$ is the direct image of the line bundle on $f$ corresponding to the divisor $(e+2)\psi(q) - \varphi(q)$, which has degree $e+1$. Thus $c_1(V) = \Delta -f$ and $p_1(\ad V) = p_1(\ad V_0) -2$. This establishes (i) of Theorem 6.1. Note also that the map $q\mapsto (e+2)\psi(q) - \varphi(q)$ defines a rational map from $T$ to $J^{e+1}(S)$ (which in fact is a morphism) and the map $T \to \frak M_1$ factors through the map $T\to J^{e+1}(S)$, compatibly with the identification of a dense open subset of $J^{e+1}(S)$ with a dense open subset of $\frak M_1$ given in (3.14). Next let us calculate the degree of the induced morphism $T\to \frak M_1$. Fix a general smooth fiber $f$ of $S$, a line bundle $L$ on $f$ of degree $e+1$ and a vector bundle $V$ which is uniquely specified by an exact sequence $$0 \to V \to V_0 \to i_*L \to 0,$$ where $i\: f \to S$ is the inclusion. We shall count the preimage of $V$ in $T$. If $f$ is general, then $T \to S$ is unbranched over $f$ and the preimage of $f$ consists of $d$ distinct fibers $f_1, \dots, f_d$. Moreover $\varphi$ restricts to an isomorphism from $f_i$ to $f$ for each $i$. The image of $f_i$ under $\psi$ is a single point $p_i \in f$ corresponding to the point $\sigma \cap f_i$. Now clearly there is a unique point $q_i \in f_i$ such that $$L = \scrO _f((e+2)p_i - \varphi (q_i)).$$ Thus the preimage of $V$ consists of $d$ distinct points, and so the map $T\to \frak M_1$ has degree $d$. Lastly we must calculate $p_1(\ad \tilde \Cal V)$. We begin by calculating $\pi _1^*c(V_0) \cdot c(\Cal P \otimes \pi _2^*\Cal L)^{-1}$. Here $\pi _1^*c(V_0) = 1 + \pi _1^*\delta + \pi _1^*c[\pt]$. As for the term $c(\Cal P \otimes \pi _2^*\Cal L)$, we clearly have $c_1(\Cal P \otimes \pi _2^* \Cal L) = D = (f\otimes 1) + d(1\otimes f)$. On the other hand, with the notation of Section 7 we may apply the Grothendieck-Riemann-Roch theorem to the immersion $i\: \tilde D \to S\times T$ to obtain $$\ch (\Cal P \otimes \pi _2^*\Cal L) = i_*\bigl[\ch \scrO _{\tilde D}((e+2)H-\Gamma)(\Todd N_i)^{-1}\cdot\pi _2^*\ch \Cal L)\bigr] .$$ A calculation similar to those in Section 7 shows that this is equal to $$i_*\Bigl[1 + (e+2)H - \Gamma - \pi _2^*\alpha - \frac{D-(m_1-1)B_1 - (m_2-1)B_2}2+\cdots \Bigr],$$ where $\alpha = c_1(\Cal L^{-1})$ has been calculated in Lemma 7.2, and further manipulation gives $$-2c_2(\Cal P\otimes \pi _2^*\Cal L) = -2[D]^2 + 2\Bigl[(e+2)H - \Gamma - \pi _2^*\alpha \cdot [D] -d\left(1-\frac1{m_1} + 1-\frac1{m_2}\right)(f\otimes f)\Bigr].$$ Recalling that $\pi _1^*c(V_0) \cdot c(\Cal P \otimes \pi _2^*\Cal L)^{-1} = 1 + x_1 + x_2+ \cdots$, we have $$(1+ x_1 + x_2+\cdots )(1+ [D] + c_2(\Cal P \otimes \pi _2^*\Cal L)) = 1 + \pi _1^*\delta + \pi _1^*c[\pt].$$ Thus $x_1 = \pi _1^*\delta - [D]$ and $$x_2 = \pi _1^*c[\pt] - \pi _1^*\delta \cdot [D] +[D]^2 -c_2 (\Cal P \otimes \pi _2^*\Cal L).$$ A calculation then shows that $$\align x_1^2 -4x_2 &= \pi _1^* p_1(\ad V_0)+2 \pi _1^*\delta \cdot [D] +[D]^2 -4(e+2)[H] + 4[\Gamma] + 4\pi _2^*\alpha \cdot [D] \\ &+ 4\left(1-\frac1{m_1} + 1-\frac1{m_2}\right)d(f\otimes f). \endalign$$ There are correction terms $b(m_i) = 1-1/m_i$ depending on the multiple fibers. Now $$\align p_1(\ad \tilde \Cal V) &=x_1^2-4x_2 -4c_2(\Cal Q)\\ &= \pi _1^* p_1(\ad V_0)+2 \pi _1^*\delta \cdot [D] +[D]^2 -4(e+2)[H] + 4[\Gamma] + 4\pi _2^*\alpha \cdot [D] \\ &+ 4(b(m_1) - q(e, m_1)/m_1 + b(m_2) - q(e, m_2)/m_2 )d(f\otimes f), \endalign$$ where the terms $b(m_i)$, $q(e, m_i)$ depend only on an analytic neighborhood of the multiple fiber and are both 0 if $m_i =1$. Let $c(e, m_i) = 4(b(m_i) - q(e, m_i)/m_i)$. Taking slant product of this expression with $[\Sigma]$, using the fact that $[\Gamma ] \backslash [\Sigma] = \varphi ^*\Sigma$ and $[H] \backslash [\Sigma] = (\varphi ^*\Sigma\cdot \sigma)f$, and plugging in the expression for $\alpha$ given by Lemma 7.2 gives the final formula in Theorem 6.1(iii). \endproof \demo{Proof of \rom{(8.1)}} Choose an analytic neighborhood $X$ of $F_i$. We may assume that $X$ fibers over the unit disk in $\Cee$. Then $\varphi ^{-1}(X)$ consists of $d/m_i$ copies of $\tilde X$, which is the normalization of the pullback of $X$ by the map from the disk to itself defined by $z = w^{m_i}$. Restrict $\varphi$ and $V_0$ to this local situation, and let $D$ now denote the fiber product inside $X\times \tilde X$ and $\tilde D$ its normalization. We can similarly define the codimension two subsets $\Gamma$ and $H$. Let us examine the dependence of the terms $V_0$ and $\scrO _{\tilde D}((e+2)H-\Gamma)$ on the various choices. First, suppose that $V_0$ and $V_0'$ are two different choices of a bundle over $X$ whose determinants have fiber degree $2e+1$ and whose restrictions to the reduction of every fiber are stable. Then $\det V_0 \otimes (\det V_0')^{-1}$ has fiber degree zero. On the other hand, from the exponential sheaf sequence $$H^1(X; \scrO _X) \to \Pic X \to H^2(X; \Zee)$$ and the identification $H^2(X; \Zee) \cong H^2(F_i;\Zee ) \cong \Zee$, it follows that the group of line bundles of fiber degree zero is divisible. Thus there is a line bundle $L$ on $X$ such that $\det V_0' = \det (V_0\otimes L)$. The proof of Corollary 3.8 shows that $V_0'$ and $V_0\otimes L$ differ by twisting by a line bundle pulled back from the disk, which is necessarily trivial. Thus $V_0' \cong V_0\otimes L$. The remaining choice was the choice of a section $\sigma$ of $\tilde X$. Given two such choices $\sigma _1$ and $\sigma_2$, we have two divisors $H_1$ and $H_2$ on $\tilde D$, and two line bundles $\scrO _{\tilde D}((e+2)H_1-\Gamma)$ and $\scrO _{\tilde D}((e+2)H_2-\Gamma)$. Their difference is the line bundle $\scrO _{\tilde D}((e+2)(H_1-H_2)$. The restriction of $\scrO _{\tilde D}((e+2)H_i-\Gamma)$ to each fiber $f$ of the map $\tilde D \to \tilde X$ over $q\in \tilde X$ is the line bundle $\scrO _f((e+2)p_i - q)$, where $p_i = \sigma _i \cap f$ and we can identify the fiber over $q$ with the fiber on $\tilde X$ containing $q$ via $\varphi$. Let $\Psi \: \tilde X \to \tilde X$ be the inverse of the map given by translation by the divisor of fiber degree zero $(e+2)(\sigma _1-\sigma _2) - c_1(\tilde L)$, where $\tilde L$ is the pullback to $\tilde X$ of $L$. Thus $\Psi ^{-1}(q) = q + (e+2)(p_1-p_2)-\lambda$, where $q\in f$ and $\lambda$ is the line bundle $L|f$. Now $\operatorname{Id}\times \Psi$ acts on $X \times \tilde X$, preserving the divisor $D$ and acting as well on the normalization $\tilde D$. Clearly the line bundles $\scrO _{\tilde D}((e+2)H_2-\Gamma)\otimes \pi _1^*L$ and $(\operatorname{Id}\times \Psi)^*\scrO _{\tilde D}((e+2)H_1-\Gamma)$ have isomorphic restrictions to each fiber of the map $\tilde D \to \tilde X$. Thus they differ by the pullback of a line bundle $L'$ on $\tilde X$. Thus we have an isomorphism $$ (\Id \times \Psi )^*(\pi _1^*V_0\spcheck\otimes i_*\scrO _{\tilde D}((e+2)H_1-\Gamma)) \cong (\pi _1^*V_0)\spcheck \otimes i_*\scrO _{\tilde D} ((e+2)H_2-\Gamma)\otimes \pi _1^*L\otimes \pi _2^*L' $$ and a similar isomorphism when we apply $R^0\pi _2{}_*$. Lastly every map $\pi ^*V_0 \to i_*\scrO _{\tilde D}((e+2)H_1-\Gamma)$ which corresponds to an everywhere generating section of the line bundle $\pi _2{}_*Hom (\pi _1^*V_0, i_*\scrO _{\tilde D}((e+2)H_1-\Gamma)$ under the natural map $$\pi _2^*\pi _2{}_*Hom (\pi _1^*V_0, i_*\scrO _{\tilde D}((e+2)H_1-\Gamma) \to Hom (\pi _1^*V_0, i_*\scrO _{\tilde D}((e+2)H_1-\Gamma)$$ is determined up to multiplication by a nowhere vanishing function on $\tilde X$. It now follows that, up to twisting by the pullback of the line bundle $L'$ on $\tilde X$, we may identify the map $\pi _1^*V_0'\to i_*\scrO _{\tilde D}((e+2)H_2-\Gamma)$, up to a nowhere vanishing function on $\tilde X$ and up to twisting by the pullback of a line bundle on $\tilde X$, with the pullback under $(\operatorname{Id}\times \Psi)^*$ of the corresponding map from $\pi _1^*V_0$ to $i_*\scrO _{\tilde D}((e+2)H_1-\Gamma)$. In particular the cokernels of these maps, viewed as sheaves supported on $F_i \times E_i$, have the same length. But the lengths of the cokernels are exactly what is needed to calculate $c_2(\Cal Q_i)$, in the notation of the beginning of this section. Thus we have established (8.1). \endproof \noindent {\bf Remark.} We could easily show directly by a slight modification of the proof above that the integers $q(e, m_i)$ defined above are independent of $e$. \medskip \demo{Proof of \rom{(8.2)}} We begin with the following (see also (A.2)(i)): \lemma{8.3} The sheaf $\tilde \Cal V$ is reflexive. \endstatement \proof Since $\tilde \Cal V$ is a subsheaf of the locally free sheaf $\pi _1^*V_0$, it is torsion free. Thus it will suffice to show that every section $\tau$ of $\tilde \Cal V$ defined on an open set of the form $W-Z$, where $W$ is an open subset of $S\times T$ and $Z$ is a closed subvariety of $W$ of codimension at least two, extends to a section of $\tilde \Cal V$ over $W$. Now locally (after possibly shrinking $W$) $\tilde \Cal V$ is given by an exact sequence $$0 \to \tilde \Cal V|W \to \scrO _W^2 \to i_*\scrO _{\tilde D}|W .$$ Now viewing the section $\tau$ as a section of $\scrO _W^2$ over $W-Z$, it extends as a section of $\scrO _W^2$ by Hartogs' theorem. Let $\tilde \tau$ be the unique extension. Then the image of $\tilde \tau$ in $i_*\scrO _{\tilde D}|W$ vanishes on $D-Z$, which is nonempty. Clearly then it is zero. Thus the extension $\tilde \tau$ defines a section of $\tilde \Cal V$ extending $\tau$, so that $\tilde \Cal V$ is reflexive. \endproof Returning to the proof of (8.2), let $U = S\times T -(F_1\times \tilde F_1) - (F_2 \times \tilde F_2)$. By Lemma 8.3, $\tilde \Cal V$ is a reflexive sheaf which is locally free on $U$. We claim that $\tilde \Cal V$ is everywhere locally free. The problem is local around each point $(x,y)$ of $F_i\times \tilde F_i$. Since $\tilde \Cal V$ is reflexive, it will suffice to show the following: each point $y$ of $\tilde F_i$ has a neighborhood $\Cal N$ such that $\tilde \Cal V|(S\times \Cal N)\cap U$ has an extension to a locally free sheaf over $S\times \Cal N$. Let $T_0 = T- \tilde F_1-\tilde F_2$. Clearly $T_0$ is the inverse image of $J^{e+1}(S)-F_1-F_2$ under the natural morphism from $T$ to $J^{e+1}(S)$. The restriction of $\Cal V_U$ to $S\times T_0$ is a bundle over $S\times T_0$ in the sense of schemes since it is the restriction of a coherent sheaf over $S\times T$. Thus it induces a morphism of schemes from $T_0$ to $\frak M_1$. If we denote the points of $\frak M_1$ corresponding to multiple fibers by $F_1$ and $F_2$ again, then it is easy to see that the map of (3.14) extends to an embedding $J^{e+1}(S)-F_1-F_2 \to \frak M_1-(F_1\cup F_2)$. Thus the map $T_0 \to \frak M_1-(F_1\cup F_2)$ is proper. This map extends to a rational map from $T$ to $\frak M_1$. After blowing up $T$, there is a morphism from the blowup $\tilde T$ to $\frak M_1$. The image of $\tilde T-T_0$ must clearly lie inside the two elliptic curves in $\frak M_1$ corresponding to elementary modifications along $F_1$ or $F_2$. Since there are no nonconstant maps from $\Pee ^1$ to an elliptic curve, every exceptional curve on $\tilde T$ is mapped to a point, and the map $T_0 \to \frak M_1$ extends to a morphism $\Phi \:T\to \frak M_1$. Clearly the morphism $\Phi \:T\to \frak M_1$ identifies $\frak M_1$ with $J^{e+1}(S)$. Given $y\in \tilde F_i$, choose a neighborhood $N_0$ of $\Phi(y)$ in $\frak M_1$ such that there exists a universal bundle over $S\times N_0$, and let $\Cal N$ be the component of $\Phi ^{-1}(N_0)$ containing $y$. Thus there is a universal vector bundle $\Cal W$ over $S\times \Cal N$. By construction $\Cal W|S\times (\Cal N -\tilde F_i)$ and $\tilde\Cal V|S\times (\Cal N -\tilde F_i) $ have isomorphic restrictions to every slice $S\times \{z\}$. Thus $\pi _2{}_*Hom (\Cal W, \tilde \Cal V)$ is a torsion free rank one sheaf on $\Cal N$, which is thus an ideal sheaf on $\Cal N$ if $\Cal N$ is small enough. We may assume that $\pi _2{}_*Hom (\Cal W, \tilde \Cal V)|\Cal N-\{y\}$ is just the structure sheaf. Choosing an everywhere generating section of $\pi _2{}_*Hom (\Cal W, \tilde \Cal V)|\Cal N-\{y\}$ gives a homomorphism $\Cal W|S\times (\Cal N-\{y\}) \to \tilde \Cal V|S\times (\Cal N-\{y\})$. This homomorphism is an isomorphism over $S\times (\Cal N -\tilde F_i)$ and is nonzero at a general point of $S\times ((\Cal N -\{y\})\cap \tilde F_i)$. As both $\Cal W$ and $\tilde \Cal V$ are vector bundles away from $F_i\times (\Cal N \cap \tilde F_i)$ whose restrictions to every smooth fiber of $S$ in every slice are stable, it follows that $\Cal W|S\times (\Cal N-\{y\}) \to \tilde \Cal V|S\times (\Cal N-\{y\})$ is an isomorphism in codimension one. Since both sheaves are reflexive, they are isomorphic. Finally $\Cal W$ and $\tilde \Cal V$ are two reflexive sheaves which are isomorphic on the complement of the codimension two set $S\times \{y\}\subset S\times \Cal N$, so they are isomorphic. Thus $\tilde \Cal V$ is locally free. \endproof \section{9. The four-dimensional invariant.} Our goal in this section will be to calculate the four-dimensional invariant. What follows is an outline of the calculation. Let $\overline{\frak M}_2$ denote the moduli space of Gieseker stable torsion free sheaves on $S$ of dimension four. As we have seen, $\overline{\frak M}_2$ is smooth and irreducible and birational to $\operatorname{Hilb}^2J^{e+1}(S)$. In fact, we shall begin by establishing a more precise statement. Let $Y_i\subset \operatorname{Hilb}^2J^{e+1}(S)$ be the subset of codimension two consisting of subschemes of $J^{e+1}(S)$ whose support has reduction contained in the multiple fiber $F_i$ on $J^{e+1}(S)$. Clearly $Y_i$ has two components: one component is just $\operatorname{Sym}^2F_i$, the closure of the locus of two distinct points lying on $F_i$, and the other is a $\Pee ^1$-bundle over $F_i$ corresponding to nonreduced subschemes whose support is a point on $F_i$. There is a similar subscheme $Y_i'$ of $\overline{\frak M}_2$, consisting of torsion free sheaves $V$ on $S$ such that either $V$ is not locally free and the unique point where $V$ is not locally free lies on $F_i$ or $V$ is a bundle obtained from $V_0$ up to equivalence by taking two elementary modifications along line bundles on $F_i$. We claim: \lemma{9.1} The isomorphism defined in \rom{(3.14)} from a Zariski open subset of $\Sym ^2J^{e+1}(S)$ to an open subset of $\frak M_2$ extends to an isomorphism $\operatorname{Hilb}^2J^{e+1}(S) - Y_1-Y_2 \to \overline{\frak M}_2 - Y_1'-Y_2'$. \endstatement \medskip Let us remark that, in case there are multiple fibers, the birational map above does not extend to a morphism. This follows from the identification of the function $d(e, m_i)$ below, and can also be seen directly as follows. The moduli space $\overline{\frak M}_2$ contains the set of nonlocally free sheaves, which is a smooth $\Pee^1$-bundle over $S$. The corresponding subset of $\operatorname{Hilb}^2J^{e+1}(S)$ is the image of the blowup of $J^{e+1}(S)\times _{\Pee ^1}J^{e+1}(S)$ along the diagonal (which is not a Cartier divisor) under the involution. It is easy to see that this image is not normal along the image of $F_i\times F_i$ if $m_i>1$. There is an isomorphism $H^2(\operatorname{Hilb}^2J^{e+1}(S) - Y_1-Y_2 ) \cong H^2(\overline{\frak M}_2 - Y_1'-Y_2')$, so that by restriction we can view $\mu (\Sigma)$ as an element of $H^2(\operatorname{Hilb}^2J^{e+1}(S) - Y_1-Y_2 ) \cong H^2(\operatorname{Hilb}^2J^{e+1}(S))$. Denote this element of $H^2(\operatorname{Hilb}^2J^{e+1}(S))$ by $\mu '(\Sigma)$. In fact, it is easy to identify this element: let $\alpha _1 = \mu _1(\Sigma ) \in J^{e+1}(S)$ be given by the $\mu$-map for the two-dimensional invariant, and set $$\alpha _2 = \alpha _1 + \frac{(f\cdot \Sigma)}2f.$$ Then we have the following formula: \lemma{9.2} $$\mu '(\Sigma) = D_{\alpha _2} -\frac{(f\cdot \Sigma)}2E.$$ \endstatement Now $\alpha _1^2$ is just the value of the two-dimensional invariant, which we shall write as $(\Sigma ^2) + C_1(\kappa\cdot \Sigma)^2$, where $C_1 = m_1^2m_2^2(p_g+1) -m_1^2-m_2^2$. Thus $$\align \alpha _2^2 &= \alpha _1 ^2 + (f\cdot \Sigma)(\alpha _1\cdot f) \\ &= \alpha _1 ^2 + 2(f\cdot \Sigma)^2 \\ &= (\Sigma ^2) + (C_1+2m_1^2m_2^2)(\kappa\cdot \Sigma)^2, \endalign$$ where we have used Lemma 6.5 to conclude that $\alpha _1\cdot f = 2(f\cdot \Sigma)$. Thus a routine calculation with the multiplication table in $\operatorname{Hilb}^2J^{e+1}(S)$ gives: \lemma{9.3} $$\align \mu '(\Sigma)^4 =& 3(\Sigma ^2)^2 + 6C_1(\Sigma ^2)(\kappa \cdot \Sigma)^2 + \\ &+ \Bigl[ 3C_1^2-(2(p_g+1) +12)m_1^4m_2^4 + 8(m_1^3m_2^4 + m_1^4m_2^3)\Bigr] (\kappa \cdot \Sigma)^4.\qed \endalign$$ \endstatement Of course, this is a calculation on $\operatorname{Hilb}^2J^{e+1}(S)$, not on $\overline{\frak M}_2$. To get an answer on $\overline{\frak M}_2$, we shall argue that the above formula must be corrected by terms which only depend on the multiplicities of the multiple fibers and not on $p_g$. \lemma{9.4} There exist a function $d(e, m_i)$, depending only on $e$ and an analytic neighborhood of the multiple fiber $F_i$ in $S$, with the following properties: \roster \item"{(i)}" $d(e, 1)=0$. \item"{(ii)}" $\mu (\Sigma )^4 - \mu '(\Sigma )^4 = m_1^4m_2^4(d(e,m_1) +d(e, m_2))(\kappa \cdot \Sigma )^4.$ \endroster \endstatement \medskip We can now complete the proof of (iii) of Theorem 2 in the Introduction. It follows from (9.3) and (9.4) that the coefficient of $(\kappa \cdot \Sigma)^4$ in $\mu (\Sigma )^4$ is given by $$3C_1^2-(2(p_g+1) +12)m_1^4m_2^4 + 8(m_1^3m_2^4 + m_1^4m_2^3) + m_1^4m_2^4(d(e,m_1) +d(e, m_2)).$$ To calculate $d(e, m_i)$, take as before $S$ to be a rational surface with a multiple fiber of multipicity $m_1$. In this case, arguing as in the proof of (6.3), the coefficient of $(\kappa \cdot \Sigma )^4$ is the same as the coefficient of $(\kappa \cdot \Sigma )^4$ for the rational surface with no multiple fibers. To calculate this coefficient, we apply (9.4) and (9.3) with $m_1=m_2 =1$ and $p_g=0$, to see that $\mu (\Sigma )^4 = \mu' (\Sigma )^4$ and thus that the coefficient of $(\kappa \cdot \Sigma )^4$ is $3-14 +16 =5$. Now taking $p_g =0$ and $m_1$ arbitrary and $m_2=1$ in the above formulas gives $C_1=-1$ and $$5 = 3(-1)^2 -14m_1^4 +8(m_1^3+m_1^4) +m_1^4d(e, m_1).$$ Thus $m_1^4d(e, m_1) = 2-8m_1^3 +6m_1^4$, or $$d(e, m_1) = \frac{2}{m_1^4}-\frac{8}{m_1} +6.$$ Plugging this into the expression above for the coefficient for $(\kappa \cdot \Sigma)^4$ in the general case gives $$\align 3C_1^2-(2(p_g+1)& +12)m_1^4m_2^4 +12m_1^4m_2^4 +2m_1^4 + 2m_2^4\\ = 3C_1^2-&2((p_g+1)m_1^4m_2^4 -m_1^4 - m_2^4). \endalign$$ We may write this answer more neatly as $3C_1^2-2C_2$, where $$\align C_1 &= m_1^2m_2^2(p_g+1)-m_1^2-m_2^2;\\ C_2 &= m_1^4m_2^4(p_g+1)-m_1^4-m_2^4.\qed \endalign$$ \section{10. Proof of Lemmas 9.1, 9.2, and 9.4.} In this section we shall give a proof of the remaining results from the previous section. \demo{Proof of Lemma \rom{9.1}} The lemma asserts the existence of an isomorphism from $\operatorname{Hilb}^2J^{e+1}(S) - Y_1-Y_2$ to $\overline{\frak M}_2 - Y_1'-Y_2'$ extending the isomorphism given in (3.14). The isomorphism of (3.14) is defined on the open set $U$ of $\operatorname{Hilb}^2J^{e+1}(S)$ consisting of pairs of points $\{z_1, z_2\}$ such that $z_1$ and $z_2$ lie in distinct fibers, neither of which is singular or multiple. We must show that the map extends over the set of pairs $\{z_1, z_2\}$, where $z_1$ and $z_2$ lie in distinct fibers, one or both of which may be singular or multiple, as well as over the set of pairs $Z$ where either $Z$ is nonreduced but the support of $Z$ does not lie in a multiple fiber or where $Z =\{z_1, z_2\}$ with $z_1$ and $z_2$ lying in the same nonmultiple fiber. Let us first consider the case where $z_1$ and $z_2$ lie in distinct fibers. As in Section 7, choose an elliptic surface $T \to C$ with a section such that $C$ is a finite cover of $\Pee ^1$, generically branched except below the multiple fibers and $T$ is the normalization of $S\times _{\Pee ^1}C$. Let $\varphi \: T\to S$ be the natural map. There is also the map $\varphi _{e+1}\: T\to J^{e+1}(S)$ defined by $\Cal P$, i.e\. if $q\in T$, $f$ is the fiber containing $q$ and $p = f\cap \sigma$, then $\varphi _{e+1}(q) = \scrO_f((e+2)p-q)$. We have constructed a universal bundle $\tilde \Cal V \to S\times T$ in Section 7 for the choices of $w$ and $p$ corresponding to the two-dimensional invariant. Let $\tilde U\subset T\times T$ be the open set of pairs of points $(y_1, y_2)$ such that $\varphi (y_1)$ and $\varphi (y_2)$ lie in different fibers. Let $\tilde \Cal V_1$ be the pullback of $\tilde \Cal V$ to $S\times \tilde U$ via the natural projection of $S\times \tilde U\subset S\times T\times T$ onto the first and second factors. We also have the coherent sheaf $\Cal P$ on $S\times T$ defined at the beginning of Section 7. Let $\Cal P'$ be the pullback of $\Cal P$ to $S\times \tilde U$ defined by the projection of $S\times T\times T$ to the first and third factor. Thus given a point $(y_1, y_2) \in \tilde U$, the restriction of $\tilde \Cal V_1$ to the slice through $(y_1, y_2)$ is an elementary modification of $V_0$ along the fiber containing $\varphi (y_1)$ and the restriction of $\Cal P'$ to the slice through $(y_1, y_2)$ is the direct image of a line bundle of degree $e+1$ on the fiber through $\varphi (y_2)$. Thus, leting $\pi _2$ denote the projection $S\times \tilde U \to \tilde U$, $\pi _2{}_*\bigl(\tilde \Cal V_1\spcheck \otimes \Cal P'\bigr)$ is a line bundle on $\tilde U$, whose inverse we denote by $\Cal L'$. Define $\tilde \Cal V_2$ as the kernel of the natural map $\tilde \Cal V_1 \to \Cal P' \otimes \Cal L'$. The proof of (8.2) shows that $\tilde \Cal V_2$ is a vector bundle whose restriction to each slice $S\times \{(y_1, y_2)\}$ is stable. The induced map $\tilde U \to \frak M_2$ then descends to a map from the open subset of $\operatorname{Hilb}^2J^{e+1}(S)$ consisting of points lying in distinct fibers to $\frak M_2$. (In fact, the proof shows that this morphism extends to a morphism defined on the complement of the divisor $E$ of nonreduced points together with the proper transforms of $\Sym ^2F_1$ and $\Sym ^2F_2$.) Next we must extend the morphism over the points of $\operatorname{Hilb}^2J^{e+1}(S)$ corresponding to points lying in the same nonmultiple fiber and nonreduced points whose support does not lie in a multiple fiber. In order to do so, we will need the model for elliptic surfaces with a section constructed in Section 4. Let $Z$ be a point of $\operatorname{Hilb}^2J^{e+1}(S)$ such that $\Supp Z$ lies in a single nonmultiple fiber $f$, and let $X$ be a small neighborhood of $f$ mapping properly to a disk inside $\Pee ^1$. Thus there is a biholomorphic map from $X$ to a neighborhood of the corresponding fiber in the Jacobian surface $J(S)$, and we may further assume that the image of $\Supp Z$ does not meet the identity section $\sigma$ under this map. Now the results of Section 4, after tensoring by $\scrO_X(e\sigma)$, give a rank two vector bundle $V_0'$ over $X$ whose restriction to every fiber is stable of degree $2e+1$ and a rank two reflexive sheaf $\Cal V_0$ over $X\times (\operatorname{Hilb}^2X-D_\sigma)$, flat over $H = \operatorname{Hilb}^2X-D_\sigma$, whose restriction to each slice is an elementary modification of $V_0'$. Let $V_0$ denote as usual the bundle on $S$ whose restriction to every fiber is stable. Then as in the proof of (8.1) there is a line bundle $L$ on $X$ such that $V_0|X \cong V_0'\otimes L$. The sheaf $\Cal V_0 \otimes \pi _1^*L$ has the following property. Let $\Cal B \subset X\times H$ be the set $$\Cal B =\{\,(x, z_1, z_2)\mid \pi (x) = \pi (z_i) \text{\, for some $i$}\,\}.$$ Let $p$ be a point of $H$ and $\Bbb U$ a small neighborhood of $p$, which we can identify with a neighborhood of $Z\in \operatorname{Hilb}^2J^{e+1}(S)$. We can assume that $\Bbb U$ is a polydisk. There is a proper map $\Pi \: (X\times \Bbb U) - \Cal B \to (D_0 \times \Bbb U)-\Cal B'$ induced by $\pi \: X\to D_0$, where $D_0$ is the disk which is the base curve of $X$ and $$\Cal B' = \{\,(t, z_1, z_2)\mid t = \pi (z_i) \text{\, for some $i$}\,\}.$$ By construction the restrictions of $\Cal V_0 \otimes \pi _1^*L$ and $\pi _1^*V_0$ to each fiber of $\Pi$ are isomorphic stable bundles on the fiber, which is reduced (possibly nodal). Thus $R^0\Pi _*Hom (\Cal V_0 \otimes \pi _1^*L, \pi _1^*V_0)$ is a line bundle $\Cal F$ on $(D_0 \times \Bbb U)-\Cal B'$. Both $\Cal V_0 \otimes \pi _1^*L$ and $\pi _1^*V_0$ extend to coherent sheaves on $X\times \Bbb U$. Therefore $R^0\Pi _*Hom (\Cal V_0 \otimes \pi _1^*L, \pi _1^*V_0) =\Cal F$ extends to a coherent sheaf on $D_0 \times \Bbb U$, which we shall continue to denote by $\Cal F$. Replacing $\Cal F$ by its double dual if necessary, we can assume that it is reflexive, and thus since its rank is one that it is a line bundle. Since by assumption every line bundle on $D_0\times \Bbb U$ is trivial, $R^0\Pi _* Hom (\Cal V_0 \otimes \pi _1^*L, \pi _1^*V_0)$ is a trivial line bundle on $(D_0 \times \Bbb U)-\Cal B'$, and we can thus choose an everywhere generating section. This section corresponds to a homomorphism from $\Cal V_0 \otimes L$ to $\pi _1^*V_0$ over $(X\times \Bbb U) - \Cal B$ which is an isomorphism on every fiber. It follows that we can glue $\Cal V_0 \otimes L$ to $\pi _1^*V_0$ over $(X\times \Bbb U) - \Cal B$. Since $\{X\times \Bbb U, (S\times \Bbb U)-\Cal B\}$ is an open cover of $S\times \Bbb U$ whose intersection is $(X\times \Bbb U) - \Cal B$, we have constructed an coherent sheaf on $S\times \Bbb U$, flat over $\Bbb U$. In this way we have extended the morphism from $\tilde U \cap \Bbb U$ over all of $\Bbb U$. So the morphism $U\to \overline{\frak M}_2$ extends over all the points $Z\in \operatorname{Hilb}^2J^{e+1}(S)$ such that $A\notin Y_1\cup Y_2$. Clearly its image is exactly $\overline{\frak M}_2 -Y_1'-Y_2'$. \endproof \demo{Proof of Lemma \rom{9.2}} We shall show that the divisor $\mu '(\Sigma)$ which is the natural extension of the restriction of $\mu (\Sigma)$ to $\operatorname{Hilb}^2J^{e+1}(S) - Y_1-Y_2$ to a divisor on $\operatorname{Hilb}^2J^{e+1}(S)$ is equal to $D_{\alpha _2} -\bigl((f\cdot \Sigma)/2\bigr)E$. Recall that $H^2(\operatorname{Hilb}^2J^{e+1}(S)) \cong H^2(J^{e+1}(S))\oplus \Zee \cdot [E/2]$. Also, given a point $y\in J^{e+1}(S)$, there is an induced morphism $\tau _y\: \operatorname{Bl}_yJ^{e+1}(S) \to \operatorname{Hilb}^2J^{e+1}(S)$ defined on $J^{e+1}(S) -\{y\}$ by $\tau _y(x) = \{x,y\}$. If $E_y$ is the exceptional divisor on $\operatorname{Bl}_yJ^{e+1}(S)$, then it is easy to see that $\tau _y^*D_\alpha = \alpha$ for all $\alpha \in H^2(J^{e+1}(S))$ (where we have identified $H_2(J^{e+1}(S))$ and $H^2(J^{e+1}(S))$ and identified $H^2(J^{e+1}(S))$ with a subspace of $H^2(\operatorname{Bl}_yJ^{e+1}(S))$). Also $\tau _y^*[E] = 2[E_y]$, which can easily be checked by going up to the double cover of $\operatorname{Hilb}^2J^{e+1}(S)$ which is the blowup of $J^{e+1}(S) \times J^{e+1}(S)$ along the diagonal. Similarly, suppose that $\varphi\:T\to S$ is a finite cover as usual and consider the morphism $\varphi _{e+1}\: T\to J^{e+1}(S)$ defined by $\Cal P$, i.e\. if $q\in T$, $f$ is the fiber containing $q$ and $p = f\cap \sigma$, then $\varphi _{e+1}(q) = \scrO_f((e+2)p-q)$. Suppose that $y$ is a general point of $J^{e+1}(S)$ (and so does not lie on a multiple or singular fiber) and let $\varphi _{e+1}^{-1}(y) = \{y_1, \dots , y_d\}$. Then there is an induced map $\tau \: T-\{y_1, \dots, y_d\}\to \operatorname{Hilb}^2J^{e+1}(S)$, and clearly we have $\tau ^*D_\alpha = \varphi ^*\alpha$. In particular the map $\tau ^*$ is injective on the subspace $H_2(J^{e+1}(S))$, and we can determine $\mu '(\Sigma)$ provided that we know $\tau ^*\mu '(\Sigma)$ and $\mu '(\Sigma)|E_y$. Note finally that the image of $\tau$ and $E_y$ are contained in $\operatorname{Hilb}^2J^{e+1}(S) - Y_1-Y_2$, so that we can calculate the $\mu$-map by finding a universal family of coherent sheaves on $S\times (T-\{y_1, \dots, y_d\})$ and over $S\times E_y$. To find such a family, begin with the bundle $\tilde \Cal V$ over $S\times T$. We know that $\tilde \mu (\Sigma) = \varphi _{e+1}^*\alpha _1$, where $\tilde \mu$ is the natural $\mu$-map defined on $T$ and $\alpha _1 = \mu (\Sigma)$ is the $\mu$-map for the two-dimensional invariant. Fix a general fiber $f$ of $S$ and a point $y\in J^{e+1}(S)$ corresponding to a line bundle $\lambda$ of degree $e+1$ on $f$. Let $f_1, \dots, f_d$ be the fibers on $T$ lying above $f$ and $y_1, \dots, y _d$ the points of $T$ corresponding to $\lambda$. We shall perform an elementary modification along the divisor $f\times T$ with respect to the line bundle $\pi _1^*\lambda$. This will run into trouble along $y_1, \dots, y_d$, so that we will restrict to $S\times (T-\{y_1, \dots, y_d\})$. The upshot will be a family of stable torsion free sheaves on $S\times (T-\{y_1, \dots, y_d\})$ such that the induced morphism $T-\{y_1, \dots, y_d\} \to \overline {\frak M}_2$ is $\tau$. First let us calculate $\Hom (\tilde \Cal V|S\times (T-\{y_1, \dots, y_d\}), \pi _1^*\lambda)$. If $V_t$ is the restriction of $\tilde \Cal V$ to the slice $S\times \{t\}$, then $V_t$ is an elementary modification of $V_0$ either at a fiber different from $\lambda$ or along $f$ with respect to a line bundle $\lambda '$ of degree equal to $\deg \lambda$ but with $\lambda '\neq \lambda$. It follows that the map $\Hom (V_0, \lambda) \to \Hom (V_t, \lambda)$ defined by the inclusion $V_t\subset V_0$ is a map between two one-dimensional spaces by (1.3)(i), and its kernel is $H^0((\lambda ')^{-1}\otimes \lambda)=0$. Thus $\Hom (V_0, \lambda) \cong \Hom (V_t, \lambda)$ and the induced map $R^0\pi _2{}_*\pi _1^*(V_0\spcheck \otimes \lambda) \to R^0\pi _2{}_*\bigl(\tilde \Cal V|S\times (T-\{y_1, \dots, y_d\})\spcheck\otimes \pi _1^*\lambda\bigr)$ is an isomorphism. As $R^0\pi _2{}_*\pi _1^*(V_0\spcheck \otimes \lambda)$ is the trivial line bundle, there is a unique homomorphism mod scalars from $\tilde \Cal V|S\times (T-\{y_1, \dots, y_d\})$ to $\pi _1^*\lambda$ and its restriction to each slice is the corresponding nonzero homomorphism on the slice. Let $\tilde \Cal V_2$ be the kernel, so that there is an exact sequence $$0 \to \tilde \Cal V_2 \to \tilde \Cal V|S\times (T-\{y_1, \dots, y_d\})\to \pi _1^*\lambda. $$ Note that the right arrow fails to be surjective over the slice $S\times \{t\}$ only if $\varphi (t) \in f$, and in this case it vanishes at one point. Thus by (A.5) $\tilde \Cal V_2$ is reflexive and flat over $T-\{y_1, \dots, y_d\}$, and is a family of torsion free sheaves parametrized by $T-\{y_1, \dots, y_d\}$. The restriction of $\tilde \Cal V_2$ to a general fiber in every slice is stable, and thus $\tilde \Cal V_2$ is a flat family of stable torsion free sheaves. Clearly the corresponding morphism to $\overline {\frak M}_2$ is $\tau$. Next we claim that $$p_1(\ad \tilde \Cal V_2) = p_1(\ad \tilde \Cal V) -2d(f\otimes f) +\cdots,$$ where the omitted terms do not affect slant product. Indeed the defining map $\tilde \Cal V|S\times (T-\{y_1, \dots, y_d\})\to \pi _1^*\lambda$ is surjective in codimension two, so that in calculating $p_1(\ad \tilde \Cal V_2)$ we can in fact apply the formula (0.1) as if the map were surjective. Now (0.1) gives $$p_1(\ad \tilde \Cal V_2) = p_1(\ad \tilde \Cal V) +2c_1(\tilde \Cal V) \cdot (f\otimes 1) - 4i_*(\lambda \otimes 1 )\cdot (f\otimes 1).$$ Using $c_1(\tilde \Cal V) = \pi _1^*c_1(V_0) -\bigl[(f\otimes 1) +d(1\otimes f)\bigr]$ and plugging in gives the claimed formula for $p_1(\ad \tilde \Cal V_2)$. Thus $$\align -(p_1(\ad \tilde \Cal V_2)\backslash \Sigma)/4 &= -(p_1(\ad \tilde \Cal V)\backslash \Sigma)/4 +d(f\cdot \Sigma)f/2\\ &= \varphi _{e+1}^*(\alpha _1) + \bigl((f\cdot \Sigma)/2\bigr)\varphi _{e+1}^*(f)\\ &= \varphi _{e+1}^*(\alpha _2), \endalign$$ and the pullback of $\mu '(\Sigma)$ to $T-\{y_1, \dots, y_d\}$ under $\tau$ is just $\varphi _{e+1}^*(\alpha _2)$. It follows that $\mu '(\Sigma) = D_{\alpha _2} + aE$ for some rational number $a$. To determine the coefficient of $E$ in $\mu '(\Sigma)$, fix a general fiber $f$ of $S$ and a line bundle $\lambda$ of degree $e+1$ on $f$, which corresponds to a point $y\in J^{e+1}(S)$. The set of points of $\operatorname{Hilb}^2J^{e+1}(S)$ whose support is $\{y\}$ is a curve $E_y\cong \Pee ^1$. We shall construct a universal sheaf $\Cal V_2$ over $S\times E_y$ and show that $-(p_1(\ad \Cal V_2)\backslash \Sigma)/4 = (f\cdot \Sigma)$. Begin with $V$ which is obtained from $V_0$ by a single elementary modification along $\lambda$. Thus $V|f = \lambda \oplus \mu$ with $\deg \lambda = e+1$ and $\deg \mu =e$. By (1.3)(ii) $\dim \Hom(V, \lambda) = 2$ and there is a unique nonzero homomorphism from $V$ to $\lambda$ which is not surjective, indeed which vanishes exactly at the point corresponding to the degree one line bundle $\lambda \otimes \mu^{-1}$. Identify $\Pee(\Hom(V, \lambda))$ with $\Pee ^1$ (and with $E_y$). There is a general construction [5] of a universal homomorphism $\Phi\: \pi _1^*V\otimes \pi _2^*\scrO_{\Pee ^1}(-1) \to \pi _1^*\lambda$. Thus we can define $\Cal V_2$ to be its kernel: $$0 \to \Cal V_2 \to \pi _1^*V\otimes \pi _2^*\scrO_{\Pee ^1}(-1) \to \pi _1^*\lambda.$$ By (A.5) in the appendix, $\Cal V_2$ is reflexive and flat over $\Pee ^1$, and is a family of torsion free sheaves, which are locally free except for the point of $\Pee ^1$ corresponding to the non-surjective homomorphism. The restriction of $\Cal V_2$ to a general fiber in every slice is stable, so that the restriction of $\Cal V_2$ to each slice is a stable torsion free sheaf. The induced map to $\overline{\frak M}_2$ is easily seen to be one-to-one with image $E_y$. We may again calculate $p_1(\ad \Cal V_2)$ by the formula of (0.1), noting that $c_1( \pi _1^*V\otimes \pi _2^*\scrO_{\Pee ^1}(-1) = \pi _1^*c_1(V) + 2 \pi _2^*c_1(\scrO_{\Pee ^1}(-1))$: $$p_1(\ad \Cal V_2)= \pi _1^*p_1(\ad V)+ 2(\pi _1^*c_1(V) + 2\pi _2^*c_1(\scrO_{\Pee ^1}(-1)))\cdot (f\otimes 1)-4i_*\pi _1^*c_1(\lambda).$$ The only term which matters for slant product is the term $4\pi _2^*c_1(\scrO_{\Pee ^1}(-1))\cdot (f\otimes 1)$. Thus $$\mu '(\Sigma)\cdot E_y = -(1/4)4(-1)(f\cdot \Sigma) = (f\cdot \Sigma).$$ Bearing in mind that $E\cdot E_y = -2$, it follows that the coefficient of $E$ in $\mu '(\Sigma)$ is $-(f\cdot \Sigma)/2$. So putting this all together gives the final answer for $\mu '(\Sigma)$ in (9.2). \endproof \demo{Proof of \rom{(9.4)}} The basic idea of the proof is similar to the idea of the proof of (9.1). Fix an analytic neighborhood $X$ of the multiple fiber $F_i$ as usual. Let $X_e$ be the corresponding subset of $J^{e+1}(S)$. Then $\Bbb X = \operatorname{Hilb}^2X_e$ may be identified with an analytic open subset of $\operatorname{Hilb}^2J^{e+1}(S)$ which is a neighborhood of $Y_i$. Under the birational map $\operatorname{Hilb}^2J^{e+1}(S) \dasharrow \overline {\frak M}_2$, the open set $\Bbb X$ corresponds birationally to an open set $\Bbb X'$ which is a neighborhood of $Y_i'$. Moreover $\Bbb X - Y_i \cong \Bbb X' - Y_i'$. Now let $S_0$ be another nodal elliptic surface containing a multiple fiber of multiplicity $m_i$ and let $X_0$ be an analytic neighborhood of the multiple fiber. Let $\Delta _0$ be a divisor on $S_0$ of fiber degree $2e+1$ and let $V_0'$ be a rank two vector bundle whose restriction to every fiber is stable. We suppose that $X_0$ is biholomorphic to $X$ and identify them. We may then define $\Bbb X_0$ and $\Bbb X_0'$ analogously. There are also closed subsets of $\Bbb X_0$ and $\Bbb X_0$ corresponding to $Y_i$ and $Y_i'$, which we shall again denote by $Y_i$ and $Y_i'$. Of course $\Bbb X_0 \cong \Bbb X$ under the identification $X_0 \cong X$. The main claim is then the following: \claim{} There is a biholomorphic map $\Bbb X_0' \cong \Bbb X'$ which is compatible with the isomorphisms $$\Bbb X_0' - Y_i' \cong \Bbb X _0 -Y_i \cong \Bbb X-Y_i \cong \Bbb X' -Y_i'.$$ \endstatement \proof For emphasis, we will write $\overline{\frak M}_2(S)$ for the moduli space for $S$, and similarly $\overline{\frak M}_2(S_0)$ for the moduli space for $S_0$. We shall glue $\Bbb X_0'$ to $\overline{\frak M}_2(S)-Y_i'$ along $\Bbb X-Y_i$, and show that the result maps to $\overline{\frak M}_2(S)$, compatibly with the inclusion $\overline{\frak M}_2(S)-Y_i' \subseteq \overline{\frak M}_2(S)$. This will define a proper morphism from $\Bbb X_0'$ to $\Bbb X'$ of degree one which is an isomorphism in codimension one, and thus is an isomorphism by Zariski's Main Theorem. We must show that the inclusion $\overline{\frak M}_2(S)-Y_i' \subseteq \overline{\frak M}_2(S)$ extends to a morphism from $\Bbb X_0'$ to $\overline{\frak M}_2(S)$. It suffices to do so locally around each point of $\Bbb X_0'$. Given an arbitrary point $p\in \Bbb X_0'$, let $\Bbb U\subset \Bbb X_0'$ be an open neighborhood of $p$ which is biholomorphic to a polydisk, so that $\Pic \Bbb U = 0$, and such that there exists a universal sheaf $\Cal V_{\Bbb U}$ over $S_0\times \Bbb U$. Denote again the restriction of $\Cal V_{\Bbb U}$ to $X_0 \times \Bbb U = X\times \Bbb U$ by $\Cal V_{\Bbb U}$. Letting as usual $V_0$ denote the rank two bundle on $S$ whose restriction to every fiber is stable, we have seen that there is a line bundle $L$ on $X$ such that $V_0'\otimes L\cong V_0$. Now view $\Bbb U - Y_i'$ as an open subset of $\Bbb X-Y_i\subset \operatorname{Hilb} ^2X$. As in the proof of (9.1) we have the locus $\Cal B \subset X\times \Bbb U$ which is the closure of the set $$\{\,(x, z_1, z_2)\mid \pi (x) = \pi (z_i) \text{\, for some $i$}\,\}.$$ The set $\Cal B$ is a closed analytic subset both of $X\times \Bbb U$ and of $S\times \Bbb U$. The two sets $(S\times \Bbb U)-\Cal B$ and $X\times \Bbb U$ cover $S\times \Bbb U$ and their intersection is $(X\times \Bbb U)-\Cal B$. We shall show that there is an isomorphism of the restriction of $\Cal V_{\Bbb U}\otimes \pi _1^*L$ to $(X\times \Bbb U)-\Cal B$ with $\pi _1^*V_0$. Let $\Pi \: (X\times \Bbb U)-\Cal B \to (D_0\times \Bbb U)-\Cal B'$ be the projection, where $D_0$ is the base of $X$ and, as in the proof of (9.1), $$\Cal B' = \{\,(t, z_1, z_2)\mid t = \pi (z_i) \text{\, for some $i$}\,\}.$$ By construction, the restriction of $\Cal V_{\Bbb U}\otimes \pi _1^*L$ to the reduction of every fiber of $\Pi$ is stable, and hence isomorphic to the restriction of $\pi _1^*V_0$ to the fiber. Consider $R^0\Pi _*Hom (\Cal V_{\Bbb U}\otimes \pi _1^*L, \pi _1^*V_0)$. By base change and (1.5) for the case of a multiple fiber, this is a line bundle on $(D_0\times \Bbb U)-\Cal B'$. On the other hand, both $\Cal V_{\Bbb U}\otimes \pi _1^*L$ and $\pi _1^*V_0$ extend to coherent sheaves on $X\times \Bbb U$, so that $R^0\Pi _*Hom (\Cal V_{\Bbb U}\otimes \pi _1^*L, \pi _1^*V_0)$ also extends to a coherent sheaf on $D_0 \times \Bbb U$. Arguing as in the proof of (9.1), $R^0\Pi _*Hom (\Cal V_{\Bbb U}\otimes \pi _1^*L, \pi _1^*V_0)|(D_0\times \Bbb U)-\Cal B'$ is a trivial line bundle and we may choose a section of $Hom (\Cal V_{\Bbb U}\otimes \pi _1^*L, \pi _1^*V_0)$ which generates the fiber at every point. This section then defines an isomorphism from $\Cal V_{\Bbb U}\otimes \pi _1^*L$ to $\pi _1^*V_0)$ over $(X\times \Bbb U)-\Cal B$. Thus we may define a coherent sheaf over $S\times \Bbb U$, flat over $\Bbb U$, which by construction is a family of stable torsion free sheaves on $S$. This sheaf defines a morphism from $\Bbb U$ to $\overline{\frak M}_2(S)$ which is the desired extension. Doing this for a neighborhood of every point of $\Bbb X_0'$ defines the extension over all of $\Bbb X_0'$. \endproof We return to the proof of (9.4). The proof will now follow from standard algebraic topology. We have the moduli spaces $\operatorname{Hilb}^2J^{e+1}(S)$ and $\overline{\frak M}_2(S)$ for $S$ and corresponding moduli spaces $\operatorname{Hilb}^2J^{e+1}(S_0)$ and $\overline{\frak M}_2(S_0)$ for $S_0$. There are also the divisors $\mu'(\Sigma)$ on $\operatorname{Hilb}^2J^{e+1}(S)$ and $\mu (\Sigma)$ on $\overline{\frak M}_2(S)$, as well as the corresponding divisors $\mu _0'(\Sigma_0)$ and $\mu _0(\Sigma _0)$ for $S_0$. Here $\Sigma \in H_2(S)$ and $\Sigma _0 \in H_2(S_0)$. Finally we have the open sets $\Bbb X =\Bbb X_0$ and $\Bbb X' \cong \Bbb X_0'$. \claim{} If $\Sigma \cdot f = \Sigma _0\cdot f$, then $\mu'(\Sigma )|H^2(\Bbb X) = \mu '(\Sigma _0)|H^2(\Bbb X)$. \endstatement \proof We have $H^2(\Bbb X) \cong H^2(Y_i)$ by restriction. Here $Y_i$ is the total transform of $\Sym ^2F_i$ in $\operatorname{Hilb}^2X$ and consists of two components. One of these is the proper transform of $\Sym ^2F_i$ and the other is the $\Pee ^1$-bundle over $F_i$ consisting of nonreduced length two subschemes whose support lies in $F_i$. Clearly, if $D_\alpha + aE$ is a divisor in $H^2(\operatorname{Hilb}^2J^{e+1}(S))$, then the restriction of $D_\alpha + aE$ to $Y_i$ depends only on $\alpha \cdot f$ and $a$. By Lemma 9.2, $\mu'(\Sigma ) = D_{\alpha _2} -\bigl((f\cdot \Sigma)/2\bigr)E$, where $\alpha _2 \cdot f = \alpha _1\cdot f = 2(f\cdot \Sigma)$. A similar statement holds for $\mu '_0(\Sigma _0)$. Thus $\mu'(\Sigma )|H^2(\Bbb X) = \mu '(\Sigma _0)|H^2(\Bbb X)$. \endproof To compare $\mu '(\Sigma)^4$ with $\mu (\Sigma )^4$, shrink $\Bbb X$ slightly so that it is a manifold with boundary $\partial$. Thus $\Bbb X'$ can also be shrunk slightly so that its boundary is $\partial$. Form the closed oriented 8-manifold $\Bbb Y$ which is $\Bbb X$ glued to $-\Bbb X'$ along $\partial$. Doing the same construction with $\Bbb X_0$ and $\Bbb X_0'$ gives an 8-manifold $\Bbb Y_0$ diffeomorphic to $\Bbb Y$. Given $\Sigma \in H_2(S)$, the divisor $\mu '(\Sigma)$ induces a class $\xi(i) \in H^2(\Bbb Y)$, and likewise $\Sigma _0 \in H_2(S_0)$ induces a class $\xi_0(i) \in H^2(\Bbb Y_0)$. It follows from the above claim that if $\Sigma \cdot f = \Sigma _0\cdot f$, then the classes $\xi(i)$ and $\xi_0(i)$ agree under the natural identification of $\Bbb Y$ with $\Bbb Y_0$. Next we claim \claim{} $\mu '(\Sigma)^4-\mu (\Sigma )^4 = \xi(1)^4+\xi (2)^4$. \endstatement \proof After passing to a multiple, we can assume that $\mu '(\Sigma)$ is represented by a submanifold $\bold M$ of $\operatorname{Hilb}^2J^{e+1}(S)$. Perturbing slightly, we can find four submanifolds $\bold M_1, \dots, \bold M_4$ of $\operatorname{Hilb}^2J^{e+1}(S)$ whose signed intersection is $\mu '(\Sigma)^4$. The closure of the image of $\bold M_i$ in $\Bbb X'$ is the image of a blowup of $\bold M_i$. Restricting $\bold M_i$ to $\Bbb X$ and glue it to its closure in image in $\Bbb X'$ gives a stratified subspace $M_i$ of $\Bbb Y$ representing $\xi(i)$. Taking small general deformations of $\bold M_i$ gives stratified subspaces $M_1', M_2', M_3', M_4'$ meeting transversally in finitely many points whose signed intersection number calculates $\xi(i) ^4$. After a small perturbation, we may glue $M_i'\cap \Bbb X_0$ to $\bold M_i| \operatorname{Hilb}^2J^{e+1}(S)-\Bbb X$ and use these to calculate $\mu (\Sigma )^4$. Clearly the discrepancy between $\mu '(\Sigma )^4$ and $\mu (\Sigma )^4$ is counted by $\xi (1)^4+\xi (2)^4$. \endproof Now $\xi (i)^4$ depends only on $(f\cdot \Sigma)$, $e$, and an analytic neighborhood $X$ of the multiple fiber and is homogeneous of degree four in $\Sigma$. Thus we can write $\xi (i)^4 = d(e, m_i)(f\cdot \Sigma )^4$ for some rational number $d(e,m_i)$ depending only on the analytic neighborhood $X$, where $d(e,1)=0$. Using the previous claim, we see that $$\mu '(\Sigma)^4-\mu (\Sigma )^4 = m_1^4m_2^4(d(e, m_1) + d(e, m_2))(\kappa \cdot \Sigma )^4,$$ as claimed in Lemma 9.4. \endproof \section{Appendix: Elementary modifications.} In this appendix, we consider the following problem (and its generalizations): let $X$ be a smooth projective scheme or compact complex manifold, let $T$ be smooth and let $D$ be a smooth divisor on $T$. Suppose that $\Cal W$ is a rank two vector bundle over $X\times T$, and that $L$ is a line bundle on $X$. Let $i\: X\times D \to X\times T$ be the inclusion, and suppose that there is a surjection $\Cal W \to i_*\pi _1^*L$ defining $\Cal V$ as an elementary modification: $$0 \to \Cal V \to \Cal W \to i_*\pi _1^*L \to 0.$$ For $t\in T$, let $W_t = \Cal W|X\times \{t\}$ and $V_t= \Cal V|X\times \{t\}$. If $0$ is a reference point of $D$, then there are two extensions $$\align 0\to M \to &W_0 \to L \to 0; \\ 0 \to L \to &V_0 \to M \to 0. \endalign$$ In particular the second exact sequence defines an extension class $\xi \in H^1(M^{-1}\otimes L)$. We want a formula for $\xi$ and in particular we want to know some conditions which guarantee that $\xi \neq 0$. \proposition{A.1} Let $\theta$ be the Kodaira-Spencer map for the family $\Cal W$ of vector bundles over $X$, so that $\theta$ is a map from the tangent space of $T$ at $0$ to $H^1(Hom(W_0, W_0))$. Let $\partial/\partial t$ be a normal vector to $D$ at $0$. Then the image of $\theta(\partial/\partial t)$ in $H^1(M^{-1}\otimes L)$ under the natural map $H^1(Hom(W_0, W_0)) \to H^1(Hom(M,L)) =H^1(M^{-1}\otimes L)$ is independent mod scalars of the choice of $\partial/\partial t$ and is the extension class corresponding to $V_0$. \endstatement \proof Since $W_0$ is given as an extension, there is an open cover $\{U_i\}$ of $X$ and transition functions for $W_0$ with respect to the cover $\{U_i\}$ of the form $$\bar{A}_{ij} = \pmatrix \lambda _{ij} & *\\ 0 &\mu _{ij} \endpmatrix.$$ Letting $t$ be a local equation for $D$ near $0$, we can then choose transition functions for $\Cal W$ of the form $A_{ij} = \bar{A}_{ij} + tB_{ij}$. With these choices of trivialization, a basis of local sections for $\Cal V$ on $U_i\times T$ is of the form $\{e_1, te_2\}$. Thus the transition functions for $\Cal V$ are given by $$ \pmatrix 1& 0\\0 &t^{-1} \endpmatrix \cdot \bigl(\bar{A}_{ij} + tB_{ij}\bigr) \cdot \pmatrix 1& 0\\0 &t \endpmatrix.$$ If $B_{ij} = \pmatrix a&b\\c&d\endpmatrix$, then a calculation shows that the transition functions are equal to $$\pmatrix \lambda _{ij} & t* \\ 0 & \mu _{ij} \endpmatrix + \pmatrix ta & t^2b\\ c & td \endpmatrix= \pmatrix \lambda _{ij} & 0 \\ c & \mu _{ij} \endpmatrix + tB'_{ij}.$$ Here $c$ is a matrix coefficient which naturally corresponds to the image of $B_{ij}$ in $Hom (M,L)$. The proposition is just the intrinsic formulation of this local calculation. \endproof \noindent {\bf Note.} The proof shows that, if the extension does split, then we can repeat the process, viewing $V_0$ again as extension of $L$ by $M$. Either this procedure will eventually terminate, creating a nonsplit extension at the generic point of $D$, or $\Cal W$ was globally an extension in a neighborhood of $D$. \medskip Let us give another proof for (A.1) in intrinsic terms which, although less explicit, will generalize. There are canonical identifications $H^1(Hom(W_0, W_0)) = \Ext ^1(W_0, W_0)$ and $H^1(Hom(M,L))=\Ext ^1(M,L)$. For simplicity assume that $\dim T = 1$. Note that if we restrict the defining exact sequence for $\Cal V$ to $X\times C$, where $C$ is a smooth curve in $T$ transverse to $D$, then the sequence remains exact (since $\operatorname{Tor}^R _1(R/tR, R/sR) =0$ if $t$ and $s$ are relatively prime elements of the regular local ring $R$). Thus we can always restrict to the case where $\dim T=1$. Now $\Spec \Cee [t]/(t^2)$ is a subscheme of $T$, and we can restrict $\Cal W$ to $\Spec \Cee [t]/(t^2)$ to get a bundle $\Cal W_\varepsilon$. The bundle $\Cal W_\varepsilon$ is naturally an extension $$0 \to W_0 \to \Cal W_\varepsilon \to W_0 \to 0,$$ and the associated class in $\Ext ^1(W_0, W_0)$ is the Kodaira-Spencer class. The natural map $\Ext ^1(W_0, W_0) \to \Ext ^1(M,L)$ is defined on the level of extensions as follows: given an extension $\Cal W_\varepsilon$ of $W_0$ by $W_0$, let $\Cal E$ be the preimage of $M$ in $\Cal W_\varepsilon$, so that there is an exact sequence $$0 \to W_0 \to \Cal E \to M \to 0.$$ Given the map $W_0 \to \Cal E$, the quotient $\Cal F = \bigl(\Cal E \oplus L\bigr)/W_0$, where $W_0$ maps diagonally into each summand, surjects onto $M$ by taking the composition of the projection to $\Cal E$ with the given map $\Cal E \to M$. The kernel is naturally $L$. Thus $\Cal F$ is an extension of $M$ by $L$, and it is easy to see that $\Cal F$ corresponds to the image of the extension class for $\Cal W_\varepsilon$ under the natural map. Finally note that, since $W_0 \to L$ is surjective, there is a natural identification of $\Cal F= \bigl(\Cal E \oplus L\bigr)/W_0$ with $\Cal E/M$ where we take the image of $M$ under the map $M\to W_0 \to \Cal E$. On the other hand, restricting the defining exact sequence for $\Cal V$ to $\Spec \Cee [t]/(t^2)$ gives a new exact sequence $$0 \to L \to \Cal V_\varepsilon \to \Cal W_\varepsilon \to L \to 0.$$ If we set $\Cal E$ to be the image of $\Cal V_\varepsilon$ in $\Cal W_\varepsilon$, then it is clear that $\Cal E$ is the inverse image of $M\subset W_0$ under the natural map. Now there is an isomorphism $\Cal V_\varepsilon/L \cong \Cal E$, and it is easy to see that this isomorphism identifies $V_0$ with $\Cal E/M$ under the natural maps, compatibly with the extensions. Thus the extension of $M$ by $L$ defined by $V_0$ has an extension class equal to the image of the extension class of $\Cal W_\varepsilon$ in $\Ext ^1(M,L)$ under the natural map. With this said, here is the promised generalization of (A.1): \proposition{A.2} With notation at the beginning of this section, let $\Cal W$ be a rank two reflexive sheaf over $X\times T$, flat over $T$, let $D$ be a reduced divisor on $T$, not necessarily smooth and let $i\: D \to T$ be the inclusion. Suppose that $L$ is a line bundle on $X$ and that $\Cal Z$ is a codimension two subscheme of $X\times D$, flat over $D$. Suppose further that $\Cal W \to i_*\pi _1^*L\otimes I_{\Cal Z}$ is a surjection, and let $\Cal V$ be its kernel: $$0 \to \Cal V \to \Cal W \to i_*\pi _1^*L\otimes I_{\Cal Z} \to 0.$$ Then: \roster \item"{(i)}" $\Cal V$ is reflexive and flat over $T$. \item"{(ii)}" For each $t\in D$, there are exact sequences $$\align 0 \to M\otimes I_{Z'} \to &W_t \to L\otimes I_Z \to 0;\\ 0\to L\otimes I_Z \to &V_t \to M\otimes I_{Z'} \to 0, \endalign$$ where $Z$ is the subscheme of $X$ defined by $\Cal Z$ for the slice $X\times \{t\}$ and ${Z'}$ is a subscheme of $X$ of codimension at least two. \item"{(iii)}" If $D$ is smooth, then the extension class corresponding to $V_t$ in $\Ext ^1(M\otimes I_W, L\otimes I_Z)$ is defined by the image of the normal vector to $D$ at $t$ under the composition of the Kodaira-Spencer map from the tangent space of $T$ at $t$ to $\Ext ^1(W_t, W_t)$, followed by the natural map $\Ext ^1(W_t, W_t) \to \Ext ^1(M\otimes I_{Z'}, L\otimes I_Z)$. \endroster \endstatement \proof First note that $\Cal V$ is a subsheaf of $\Cal W$ and is therefore torsion free. Given an open set $U$ of $X\times T$ and a closed subscheme $Y$ of $U$ of codimension at least two, let $s$ be a section of $\Cal V$ defined on $U-Y$. Then $s$ extends to a section $\tilde s$ of $\Cal W$ over $U$ since $\Cal W$ is reflexive. Moreover the image of $\tilde s$ in $H^0(U\cap D; L\otimes I_{\Cal Z})$ vanishes in codimension one and thus everywhere. Thus $\tilde s$ defines a section of $\Cal V$ over $U$, and so $\Cal V$ is reflexive. That it is flat over $T$ follows from the next lemma: \lemma{A.3} Let $R$ be a ring and $t$ an element of $R$ which is not a zero divisor. Let $I$ be an $R/tR$-module which is flat over $R/tR$. For an $R$-module $N$, let $N_t$ be the kernel of multiplication by $t$ on $N$. \roster \item"{(i)}" For all $R$-modules $N$, $\operatorname{Tor}_1^R(I,N) = I\otimes _{R/tR}N_t$, and $\operatorname{Tor}_i^R(I,N) = 0$ for all $i>1$. \item"{(ii)}" Suppose that there is an exact sequence of $R$-modules $$0 \to M_2 \to M_1 \to I \to 0,$$ where $M_1$ is flat over $R$. Then $M_2$ is flat over $R$ as well. \endroster \endstatement \proof The statement (i) is easy if $I=R/tR$, by taking the free resolution $$0\to R @>{\times t}>>R \to R/tR \to 0.$$ Thus it holds more generally if $I$ is a free $R/tR$-module. In general, start with a free resolution $F^\bullet$ of $I$. By standard homological algebra (see e.g\. [EGA III 6.3.2]) there is a spectral sequence with $E_1$ term $\operatorname{Tor}_p^R(F^q, N)$ which converges to $\operatorname{Tor}_{p+q}^R(I,N)$. The only nonzero rows correspond to $p=0,1$ and the row for $p=1$ is the complex $F^\bullet \otimes _{R/tR}N_t$. Since $I$ is flat, this complex is exact except in dimension zero and is a resolution of $I\otimes _{R/tR}N_t$. Since $F^q\otimes _RN = F^q\otimes _{R/tR}(N\otimes _RR/tR)$, the flatness of $I$ over $R/tR$ implies that the row for $p=0$ is exact except in dimension zero. Thus $\operatorname{Tor}_1^R(I,N) = I\otimes _{R/tR}N_t$. The second statement now follows since, for every $R$-module $N$, the long exact sequence for $\operatorname{Tor}$ defines an isomorphism, for all $i\geq 1$, from $\operatorname{Tor}_i^R(M_2, N)$ to $\operatorname{Tor}_{i+1}^R(I, N)=0$. \endproof Returning to (A.2), let us prove (ii). There is a surjection $W_t \to L\otimes I_Z$ and the kernel of this surjection is a rank one torsion free sheaf on $X$, which is thus of the form $M\otimes I_{Z'}$ for some subscheme ${Z'}$ of $X$ of codimension at least two. Now there is an exact sequence $$Tor_1^{\scrO_{X\times T}}(i_*\pi _1^*L\otimes I_{\Cal Z}, \scrO _{X\times \{t\}}) \to V_t \to W_t \to L\otimes I_Z \to 0.$$ In the $Tor_1$ term, the first sheaf is an $\scrO_{X\times D}$-module, flat over $D$, and the second is an $\scrO_D$-module. Using (i) of (A.3) identifies $Tor_1^{\scrO_{X\times T}}(i_*\pi _1^*L\otimes I_{\Cal Z},\scrO _{X\times \{t\}})$ with $L\otimes I_Z$. Thus we obtain the exact sequence for $V_t$. Finally, the identification of the extension class in (iii) is formally identical to the second proof of (A.1) given above and will not be repeated. \endproof Next we shall give some criteria for when the extension is nonsplit. The simplest case is when the Kodaira-Spencer map is an isomorphism at $0$. In this case we can check whether or not the extension is split by looking at the map $\Ext ^1(W_t, W_t) \to \Ext^1(M\otimes I_{Z'}, L\otimes I_Z)$. Thus the problem is essentially cohomological. A similar application concerns the case where $T$ is the blowup of a universal family along the locus where the sheaves are extensions of $L\otimes I_Z$ by $M\otimes I_{Z'}$. In our applications, however, we shall need a more general situation and will have to analyze some first order information about the family $\Cal W$. For simplicity we shall assume that $\dim T=1$, with $t$ a coordinate. It is an easy consequence of (A.3)(i) that the general case can be reduced to this special case by taking a curve in $T$ transverse to $D$. \proposition{A.4} In the notation of \rom{(A.3)}, let $\Cal W_\varepsilon$ be the restriction of $\Cal W$ to $\Spec \Cee [t]/t^2$. Suppose that \roster \item"{(i)}" $\Hom (M\otimes I_{Z'}, L\otimes I_Z) =0$. \item"{(ii)}" The map from $\Ext ^1(M\otimes I_{Z'}, \Cal W_\varepsilon)/t\Ext ^1(M\otimes I_{Z'}, \Cal W_\varepsilon)$ to $\Ext ^1(M\otimes I_{Z'}, \Cal W_\varepsilon)$ induced by multiplication by $t$ has a one-dimensional kernel. \endroster Then we may identify the kernel with a line in $\Ext ^1(M\otimes I_{Z'}, W_0)$, and if the image of this line in $\Ext ^1(M\otimes I_{Z'}, L\otimes I_Z)$ is $\Cee \cdot \xi$ then the corresponding extension class is $\xi$. \endstatement \proof From the first assumption $\dim \Hom (W_0, W_0) =1$. Thus if $\theta$ is the Kodaira-Spencer class, there is an exact sequence $$0\to \Ext ^1(M\otimes I_{Z'}, W_0)/\Cee \cdot\theta \to \Ext ^1(M\otimes I_{Z'}, \Cal W_\varepsilon)\to \Ext ^1(M\otimes I_{Z'}, W_0).$$ Multiplication by $t$ induces the natural map $$\im(\Ext ^1(M\otimes I_{Z'}, \Cal W_\varepsilon))\subseteq \Ext ^1(M\otimes I_{Z'}, W_0) \to \Ext ^1(M\otimes I_{Z'}, W_0)/\Cee \cdot\theta.$$ If this map has a kernel then clearly $\theta \in \im(\Ext ^1(M\otimes I_{Z'}, \Cal W_\varepsilon))$ and the kernel is $\Cee \cdot \theta$. The image of the kernel in $\Ext ^1(M\otimes I_{Z'}, L\otimes I_Z)$ is then just the image of the Kodaira-Spencer class. \endproof Here is the typical way we will apply the above: suppose that $\Cal W$ is locally free and that $Z' = \emptyset$. Then $\Ext ^1(M\otimes I_{Z'}, \Cal W_\varepsilon) = R^1\pi _2{}_*(\Cal W_\varepsilon \otimes \pi _1^*M^{-1})$. Suppose in addition that $\Cal W$ is globally an extension: $$0 \to \pi _1^*\Cal L_1 \to \Cal W \to \pi _1^*\Cal L_2\otimes I_{\Cal Y} \to 0,$$ where $\Cal Y\subset X\times T$ is flat over $T$. Thus there is a map $$R^0\pi _2{}_*\bigl(\pi _1^*\Cal L_2\otimes I_{\Cal Y}\otimes \pi _1^*M^{-1}\bigr) \to R^1\pi _2{}_*\pi _1^*(\Cal L_1\otimes M^{-1})$$ whose cokernel sits inside $R^1\pi _2{}_*(\Cal W\otimes \pi _1^*M^{-1})$. A similar statement is true when we restrict to $\Spec \Cee [t]/(t^2)$. Now suppose that $\dim H^0(X;\Cal L_2\otimes M^{-1}\otimes I_{Y_t})$ is independent of $t$. Then the sheaves $R^0\pi _2{}_*\bigl(\pi _1^*\Cal L_2\otimes I_{\Cal Y}\otimes \pi _1^*M^{-1}\bigr)$ and $R^1\pi _2{}_*\pi _1^*(\Cal L_1\otimes M^{-1})$ are locally free and compatible with base change, by [EGA III, 7.8.3, 7.8.4, 7.7.5] so if we know that the map between them has a determinant which vanishes simply along $D$ then the same will be true for the restrictions to $\Spec \Cee [t]/(t^2)$. The image in $R^1\pi _2{}_*(\Cal W\otimes \pi _1^*M^{-1})$ is the direct image of a line bundle $\Cal K$ on $D$. Furthermore suppose that $\dim H^1(X;\Cal L_2\otimes M^{-1} \otimes I_{Y_t})$ is independent of $t$. Then $R^0\pi _2{}_*\bigl(\pi _1^*\Cal L_2\otimes I_{\Cal Y}\otimes \pi _1^*M^{-1}\bigr)$ is locally free and compatible with base change. If it is nonzero suppose further that $R^2\pi _2{}_*\pi _1^*(\Cal L_1\otimes M^{-1}) =0$. Thus the torsion part of $R^1\pi _2{}_*(\Cal W\otimes \pi _1^*M^{-1})$ is just $\Cal K$ and the restriction of $\Cal K$ to $\Spec \Cee [t]/(t^2)$ gives the kernel of multiplication by $t$ as in (ii). We can then take the map from the torsion part of $R^1\pi _2{}_*(\Cal W\otimes \pi _1^*M^{-1})_0$, namely the image of $H^1(\Cal L_1\otimes M^{-1})$, to $H^1(L\otimes I_Z\otimes M^{-1}) = \Ext ^1(M, L\otimes I_Z)$ and this image gives the extension class. \medskip We will also need to consider a slightly different situation. Suppose that $\Cal W$ is a rank two vector bundle on $X\times T$, $E$ is a smooth divisor on $X$ and $L$ is a line bundle on $E\times T$. Let $j\: E\times T \to X\times T$ be the inclusion and let $\Phi\:\Cal W \to j_*L$ be a morphism. We may think of $\Phi$ as a family of morphisms parametrized by $T$. In local coordinates $\Phi$ is given by two functions $f$, $g$ on $E\times T$, whose vanishing defines a subscheme $Y$ of $E\times T$. Away from the projection $\pi _2(Y)$ of $Y$ to $T$, $\Phi$ defines a family of elementary modifications which degenerates over $\pi _2(Y)$ at the points of $Y$. \proposition{A.5} Let $\Phi\:\Cal W \to j_*L$ be a morphism as above and suppose that the cokernel of $\Phi$ is supported on a nonempty codimension two subset $Y$ of $E\times T$, necessarily a local complete intersection. Suppose further that, for each $t\in T$, the codimension of $Y\cap (X\times \{t\})$ in $X\times \{t\}$ is at least two if $Y\cap (E\times \{t\})\neq \emptyset$. Let $\Cal V$ be the kernel of $\Phi$. Then $\Cal V$ is a reflexive sheaf, flat over $T$, and its restriction to each slice $X\times \{t\}$ is a torsion free sheaf on $X$. \endstatement \proof The proof of (A.2)(i) shows that $\Cal V$ is reflexive. As for the rest, the problem is local around a point of $Y$. Let $R$ be the local ring of $X\times T$ at a point $(x,t)$, $R'$ the local ring of $T$ at $t$, and $S$ the local ring of $X\times \{t\}$ at $(x,t)$. Let $u$ be the local equation for $E$ in $X\times T$. Then locally $\Phi$ corresponds to a map $R\oplus R \to R/uR$, necessarily given by elements $\bar f, \bar g \in R/uR$. Lift $\bar f$ and $\bar g$ to elements $f, g\in R$. Then $(u,f,g)R$ is the ideal of $Y$ in $R$, and $Y$ has codimension three in $X\times T$. Thus $u, f,g$ is a regular sequence, any two of the three are relatively prime, and necessarily $\dim R\geq 3$. The kernel $M$ of the map $R\oplus R \to R/uR$ given by $(a,b)\mapsto a\bar f + b\bar g$ is clearly generated by $(-g,f)$, $(u,0$, and $(0, u)$. These three elements define a surjection $R\oplus R\oplus R \to M$. The kernel of this surjection is easily calculated to be $R\cdot(u, g, -f)$. Thus there is an exact sequence $$0 \to R \to R\oplus R\oplus R \to M \to 0.$$ This sequence restricts to define $$S\to S\oplus S\oplus S \to M\otimes _RS \to 0.$$ Here the image of $S$ in $S\oplus S\oplus S$ is equal to $S\cdot (u, g, -f)$, where we denote the images of $u,f,g$ in $S$ by the same letter. By hypothesis, not all of $u$, $f$, $g$ vanish on $X\times \{t\}$ and so this map is injective. By the local criterion of flatness $M$ is flat over $R'$. Finally we must show that $M\otimes _RS$ is a torsion free $S$-module. By hypothesis $u, g, -f$ generate the ideal of a subscheme of $\Spec S$ of codimension at least two and thus $u$ does not divide both $f$ and $g$ in $S$. Given $h\in S$ with $h\neq 0$, suppose that $hm = 0$ for some $m\in M\otimes _RS$. Then there is $(a,b,c)\in S\oplus S\oplus S$ such that $h(a,b,c) = \alpha (u, g, -f)$. We claim that $u|a$. To see this, let $n$ be the largest integer such that $u^n|h$. Then $u^n|hb = \alpha g$ and likewise $u^n|\alpha f$. Since at least one of $f$, $g$ is prime to $u$, $u^n|\alpha$. But then $u^{n+1}|\alpha u = ha$, so that $u|a$. If $a=ua'$, then $\alpha = ha'$ and so $hb= ha'g$ and $b=a'g$. Likewise $c=a'(-f)$. Thus $(a,b,c) = a'(u, g, -f)$ and its image in $M\otimes _RS$ is zero. It follows that $M\otimes _RS$ is torsion free. \endproof Let us finally remark that we can calculate the class $p_1(\ad \Cal V)$, in the above notation, by applying the formula (0.1), since $\Phi$ is surjective in codimension two. \Refs \widestnumber\no{EGA} \ref \no 1\by M. Atiyah \paper Vector bundles over an elliptic curve \jour Proc. London Math. Soc. \vol 7\yr 1957 \pages 414--452\endref \ref \no 2\by S. K. Donaldson and P. B. Kronheimer \book The Geometry of Four-Manifolds \publ Clarendon \publaddr Oxford \yr 1990 \endref \ref \no 3\by R. Friedman \paper Rank two vector bundles over regular elliptic surfaces \jour Inventiones Math. \vol 96 \yr 1989 \pages 283--332 \endref \ref \no 4\bysame \paper Vector bundles and $SO(3)$-invariants for elliptic surfaces I, II \toappear \endref \ref \no 5 \bysame \book Stable Vector Bundles on Algebraic Varieties \bookinfo in preparation \endref \ref \no 6\by R. Friedman and J. W. Morgan \book Smooth 4-manifolds and Complex Surfaces \toappear \endref \ref \no 7\by W. Fulton \book Intersection Theory \publ Springer-Verlag \publaddr Berlin Heidelberg \yr 1984 \endref \ref \no 8 \by Y. Kametani and Y. Sato \paper $0$-dimensional moduli spaces of stable rank $2$ bundles and differentiable structures on regular elliptic surfaces \paperinfo preprint \endref \ref \no 9 \by J. Li \paper Algebraic geometric interpretation of Donaldson's polynomial invariants of algebraic surfaces \toappear \endref \ref \no 10 \by J. W. Morgan \paper Comparison of the Donaldson polynomial invariants with their algebro-geo\-metric analogues \toappear \endref \ref \no 11 \by K. O'Grady \paper Algebro-geometric analogues of Donaldson's polynomials \jour Inventiones Math. \vol 107\yr 1992 \pages 351--395 \endref \ref \no EGA \by A. Grothendieck and J. Dieudonn\'e \paper \'Etude cohomologique des faisceaux coh\'erents \jour Publ. Math. I.H.E.S. \vol 17 \yr 1963 \pages 137--223\endref \endRefs \enddocument \bye

9308

alg-geom/9308005

en

https://arxiv.org/abs/alg-geom/9308005

alg-geom/9308005

Richard Hain

Richard Hain

The existence of higher logarithms

25 pages, AMS-LaTeX

In this paper we prove the existence of all higher logarithms as multivalued and ordinary Deligne cohomology classes.

alg-geom

\section{Introduction}\label{intro} Denote by $G^p_q$ the Zariski open subset of the grassmannian of $q$-dimensional linear subspaces of ${\Bbb P}^{p+q}$ which are transverse to each coordinate hyperplane and each of their intersections. Intersecting elements of $G^p_q$ with each of the $p+q+1$ coordinate hyperplanes of ${\Bbb P}^{p+q}$ defines $p+q+1$ maps $$ A_i : G^p_q \to G^p_{q-1}, \qquad 0\le i \le p+q. $$ The spaces $G^p_q$ with $0 \le q \le p$ and the face maps $A_i$ form a truncated simplicial variety $G^p_\bullet$. In \cite[\S 12]{hain-macp} (see also \cite{b-mcp-s}) the $p$th higher logarithm is defined as a certain element of the ``multivalued Deligne cohomology'' of $G^p_\bullet$. In that paper the existence of only the first three higher logarithms was established. In this paper we establish the existence of all higher logarithms, but in a sense slightly weaker than that of \cite{hain-macp} --- we show that for each $p$, there is a Zariski open subset $U^p_\bullet$ of $G^p_\bullet$ on which the $p$th higher logarithm is defined as a multivalued (and ordinary) Deligne cohomology class. This should be sufficient to show that the $p$th Chern classes on the algebraic $K$-theory of a complex algebraic variety is represented by the $p$th higher logarithm (cf.\ \cite{goncharov:chern}, \cite{hain-yang}). The key new technical ingredient is the construction of a topology on the generic part of each Grassmannian which is coarser than the Zariski topology and where each open contains another which is both a $K(\pi,1)$ and a rational $K(\pi,1)$. Hanamura and MacPherson \cite{hanamura-macp_2} have a geometric construction of the part of each higher logarithms that lies in the multivalued de Rham complex of $G^p_\bullet$. The part of our higher logarithm that lies in the multivalued de~Rham complex of $G^p_\bullet$ is only defined generically, so their result is stronger than ours in this respect (cf. Remark \ref{hana-mac-const}), but our result is stronger than theirs in that we construct higher logarithms as both multivalued and ordinary Deligne cohomology classes. One part of the cocycle defining the $p$th higher logarithm is a multivalued function $L_p$ defined on the Zariski open subset $U^p_{p-1}$ of the self dual Grassmannian of $p$ planes in ${\Bbb C}^{2p}$. The cocycle condition implies that this multivalued function satisfies the canonical $2p+1$ term functional equation $$ \sum_{i=0}^{2p} (-1)^i A_i^\ast L_p = 0. $$ In the cases $p=1,2,3$, the function $L_p$ has a single valued cousin $D_p$ which also satisfies the functional equation $$ \sum_{i=0}^{2p} (-1)^i A_i^\ast D_i = 0. $$ The first function $D_1$ is simply $\log|\phantom{x}|$, the second is the Bloch-Wigner function, and the third, whose existence was established in \cite[\S 11]{hain-macp}, can be expressed in terms of Ramakrishnan's single valued cousin of the classical trilogarithm, as was proved by Goncharov \cite{goncharov:trilog}. The functional equation implies that $D_p$ ($p=1,2,3$) represents an element of $H^{2p-1}(GL_p({\Bbb C}),{\Bbb C}/{\Bbb R}(p))$. This class is known to be a non-zero rational multiple of the Borel element, the class used to define the Borel regulator (\cite{bloch}, \cite{dupont}, \cite{goncharov:trilog}, \cite{yang}, see also \cite{hain:poly}). The single valued cousins of the higher logarithms constructed in this paper are constructed in \cite{hain-yang} where it is shown that each represents a non-zero rational multiple of the Borel class. We now discuss the content of this paper in more detail. As in \cite{hain-macp}, the algebra of multivalued differential forms on an algebraic manifold will be denoted by ${\widetilde{\Omega}}^\bullet(X)$. We will usually denote the ring of multivalued functions ${\widetilde{\Omega}}^0(X)$ by ${\widetilde{\O}}(X)$. There is a weight filtration $W_\bullet$ on ${\widetilde{\Omega}}^\bullet(X)$ which gives it the structure of a filtered d.g.\ algebra. The category of complex algebraic manifolds and regular maps between them will be denoted by ${\cal A}$. Since the pullback of a multivalued function under a regular map $X\to Y$ is not well defined, it is necessary to refine the category ${\cal A}$ in order that ${\widetilde{\Omega}}^\bullet$ becomes a well defined functor. Such a refinement $\widetilde{\A}$ of ${\cal A}$ is defined in \cite[\S 2]{hain-macp}. The objects of $\widetilde{\A}$ are universal coverings $\widetilde{X}\to X$ of objects of ${\cal A}$, and the morphisms are commutative squares $$ \begin{array}{ccc} \widetilde{X} & \to & \widetilde{Y} \\ \downarrow & & \downarrow \\ X & \to & Y \end{array} $$ where the bottom arrow is a morphism of ${\cal A}$. The truncated simplicial variety $G^p_\bullet$ has a natural lift to a simplicial object of $\widetilde{\A}$, \cite[(5.4)]{hain-macp}. Denote the Deligne-Beilinson cohomology of a smooth (simplicial) variety $X$ by $H_{\D}^\bullet(X,{\Bbb Q}(p))$. In \cite[\S 12]{hain-macp} the multivalued Deligne cohomology of a simplicial object $X_\bullet$ of $\widetilde{\A}$ was defined. It will be denoted by $H_{\calM\D}^\bullet(X_\bullet,{\Bbb Q}(p))$. There are several equivalent ways to define rational $K(\pi,1)$ spaces, but for our purposes in the introduction, perhaps the most pertinent comment is that a Zariski open subset $U$ of a grassmannian is a rational $K(\pi,1)$ if and only if $$ H^\bullet(W_l{\widetilde{\Omega}}^\bullet(U)) = {\Bbb C} $$ for all $l\ge 0$. Using this, we show in Section \ref{pf_dunno} that if $X_\bullet$ is a simplicial object of $\widetilde{\A}$ and each $X_q$ is a rational $K(\pi,1)$, then there is a natural isomorphism $$ H_{\D}^\bullet(X_\bullet,{\Bbb Q}(p)) \approx H_{\calM\D}^\bullet(X_\bullet,{\Bbb Q}(p)). $$ This was stated without proof in \cite[(12.3)]{hain-macp}. The main idea of this paper is to exploit this fact by replacing $G^p_\bullet$ by a Zariski open subset $U^p_\bullet$ where each $U^p_q$ is a rational $K(\pi,1)$. Once one has done this and established that $U^p_\bullet$ lifts to a simplicial object of $\widetilde{\A}$, the proof of the existence of higher logarithms is relatively straightforward --- there is a natural $GL_p({\Bbb C})$ bundle over $U^p_\bullet$ whose $p$th Chern class is an element of $$ H_{\D}^{2p}(U^p_\bullet,{\Bbb Q}(p)) \approx H_{\calM\D}^{2p}(U^p_\bullet,{\Bbb Q}(p)). $$ The $p$th higher logarithm is a suitable rational multiple of this class.% \footnote{If one only wants the multivalued function, or the higher logarithm in the sense of \cite[(6.1)]{hain-macp}, then one can appeal directly to the analogue of \cite[(8.9)]{hain-macp} for $U^p_\bullet$ --- cf. (\ref{exist_1}).} To recapitulate, one of the main obstacles to proving the existence of the $p$th higher logarithm is to establish the existence of such a Zariski open subset $U^p_\bullet$ of $G^p_\bullet$ where each $U^p_q$ is a rational $K(\pi,1)$. If the Zariski topology had the property that each open set contains another open which is a rational $K(\pi,1)$, then one could easily find the sought after open subset $U^p_\bullet$ of $G^p_\bullet$. Unfortunately, this is not the case (cf. \cite[(9.7)]{hain:cycles}). For this reason we introduce a coarser topology on the $G^p_q$, called the {\it constructible topology}, which does enjoy this property. This is done in Section \ref{topology}. We conclude the introduction with a brief description of the constructible topology and the idea behind the proof of the existence of $U^p_\bullet$. The first point is that each $\xi \in G^p_q$ determines an ordered configuration of $p+q+1$ points in ${\Bbb P}^{p-1}$, no $p$ of which lie on a hyperplane; the configuration is well defined up to projective equivalence. To see how this works, note that the set of $(q+1)$-dimensional planes in ${\Bbb P}^{p+q}$ which contain $\xi$ comprise a projective space of dimension $p-1$. The $j$th point of the configuration is the point of this projective space which corresponds to the join of the $j$th standard basis vector with $\xi$. Each such configuration determines a configuration of hyperplanes in ${\Bbb P}^{p-1}$ --- the hyperplanes are those determined by the $(p-1)$-element subsets of the points. The configuration corresponding to an element of $G^3_2$ and the corresponding arrangement of hyperplanes in ${\Bbb P}^2$ are illustrated in Figure \ref{fiber}. A configuration of hyperplanes in ${\Bbb P}^{p-1}$ corresponds to a central configuration of hyperplanes in ${\Bbb C}^p$. Denote the central configuration in ${\Bbb C}^p$ which corresponds to $\xi \in G^p_q$ by ${\cal C}(\xi)$. One's natural instinct when trying to understand the topology of the $G^p_q$ is to use the face maps $A_i : G^p_q \to G^p_{q-1}$ to study them inductively. The ``standard mistake'' is to believe that all such face maps are fibrations. If they were, life would be easier, but less interesting. It is worthwhile to see how the face maps fail to be fibrations as it is relevant to the proof of the existence of $U^p_\bullet$. Observe that the fiber of the face map $A_i : G^p_q \to G^p_{q-1}$ over the point $\xi \in G^p_{q-1}$ is the complement of the arrangement ${\cal C}(\xi)$ in ${\Bbb C}^p$. The simplest example where a face map is not a fibration is provided by any of the face maps $A_i : G^3_3 \to G^3_2$. The projectivization of the generic fiber is the complement of an arrangement determined by six points in ${\Bbb P}^2$, no three of which lie on a line, and where no three of the lines they determine are concurrent, except at one of the points $x_0,\dots, x_5$. The complement of the arrangement on the right hand side of Figure \ref{fiber} is the projectivization of a special fiber of $A_6 : G^3_3 \to G^3_2$ as there is an exceptional triple point. Since the topology of the fiber of $A_6 : G^3_3 \to G^3_2$ is not constant, $A_6$ is not a fibration. \begin{figure} \begin{picture}(400,200) \put(10,0){\begin{picture}(200,200) \put(80,150){\circle*{6}} \put(88,147){$x_0$} \put(30,50){\circle*{6}} \put(10,57){$x_1$} \put(130,50){\circle*{6}} \put(120,32){$x_2$} \put(80,100){\circle*{6}} \put(65,77){$x_3$} \put(180,100){\circle*{6}} \put(172,110){$x_4$} \put(30,130){\circle*{6}} \put(15,114){$x_5$} \end{picture}} \put(250,0){\begin{picture}(200,200) \put(80,150){\circle*{6}} \put(88,147){$x_0$} \put(30,50){\circle*{6}} \put(10,57){$x_1$} \put(130,50){\circle*{6}} \put(120,32){$x_2$} \put(80,100){\circle*{6}} \put(65,77){$x_3$} \put(180,100){\circle*{6}} \put(172,110){$x_4$} \put(30,130){\circle*{6}} \put(15,114){$x_5$} \put(10,10){\line(1,2){90}} \put(80,-10){\line(0,1){200}} \put(150,10){\line(-1,2){90}} \put(10,30){\line(1,1){160}} \put(10,50){\line(1,0){180}} \put(170,10){\line(-1,1){160}} \put(200,100){\line(-1,0){190}} \put(200,90){\line(-2,1){190}} \put(201,107){\line(-3,-1){195}} \put(200,120){\line(-1,-1){130}} \put(10,122){\line(5,2){170}} \put(30,10){\line(0,1){180}} \put(10,146){\line(5,-4){170}} \put(10,142){\line(5,-3){180}} \put(200,96){\line(-5,1){190}} \end{picture}} \end{picture} \caption{}\label{fiber} \end{figure} The basic closed subsets of $G^p_q$ in the constructible topology are defined to be the closure of the set of points $\xi$ where the combinatorics of ${\cal C}(\xi)$ is fixed. For example, the closure of the set of points in $G^3_2$ where the lines $x_0x_2$, $x_1x_3$ and $x_4x_5$ intersect in a single point (as in Figure \ref{fiber}) is a closed subset of $G^3_2$ in the constructible topology. The combinatorial objects which parameterize the closed sets are called {\it templates}. Observe that $A_0 : G^3_3 \to G^3_2$ is a fibration over the constructible open subset of $G^3_2$ which consists of all $\xi$ for which the projectivization of ${\cal C}(\xi)$ contains no exceptional triple points. By passing to a constructible open subset of $G^3_3$, one can arrange for the generic fiber of $A_0$ to be the complement of an arrangement of fiber type, and by restricting $A_0$ to a smaller constructible open subset of $G^3_2$ we may assume that $A_0$ is a fibration whose fiber is the complement of an arrangement of fiber type. Since arrangements of fiber type are rational $K(\pi,1)$s, this provides, via (\ref{fibration}), the inductive step needed for finding the open subset $U^p_\bullet$ of $G^p_\bullet$ in which each $U^p_q$ is a rational $K(\pi,1)$. It is assumed that the reader is familiar with \cite{hain-macp}. \medskip \noindent{\sl Conventions.} In this paper, all simplicial objects are strict --- that is, they are functors from the category $\Delta$ of finite ordinals and {\it strictly} order preserving maps to, say, the category of algebraic varieties. As is standard, the finite set $\{0,1,\dots,n\}$ with its natural ordering will be denoted by $[n]$. Let $r$ and $s$ be positive integers with $r \le s$. Denote the full subcategory of $\Delta$ whose objects are the ordinals $[n]$ with $r\le n \le s$ by $\Delta[r,s]$. An $(r,s)$ {\it truncated} simplicial object of a category ${\cal C}$ is a contravariant functor from $\Delta[r,s]$ to ${\cal C}$. The word {\it simplicial} will be used generically to refer to both simplicial objects and truncated simplicial objects. By Deligne cohomology, we shall mean Beilinson's refined version of Deligne cohomology as defined in \cite{beilinson:ahc}. It can be expressed as an extension $$ 0 \to \Ext^1_{\cal H}({\Bbb Q},H^{k-1}(X,{\Bbb Q}(p))) \to H_{\D}^k(X,{\Bbb Q}(p)) \to \Hom_{\cal H}({\Bbb Q},H^k(X,{\Bbb Q}(p)))\to 0 $$ where ${\cal H}$ denotes the category of ${\Bbb Q}$ mixed Hodge structures. \medskip \noindent{\sl Acknowledgements.} I would like to thank Jun Yang for helpful discussions regarding material in Sections \ref{existence} and \ref{proof}. \medskip \section{Constructible Configurations and Templates} Fix a ground field ${\Bbb F}$. Denote the projective space ${\Bbb P}^m({\Bbb F})$ over ${\Bbb F}$ by ${\Bbb P}^m$. By a {\it configuration} of $n$ points in ${\Bbb P}^m$, we shall mean an element ${\vec{x}}$ of $\left({\Bbb P}^m\right)^n$. A {\it subconfiguration} of ${\vec{x}}$ is any element of $({\Bbb P}^m)^l$, $l\le n$, obtained by deleting some of the components of ${\vec{x}}$. A {\it linear configuration} in ${\Bbb P}^m$ is a finite collection of linear subspaces of ${\Bbb P}^m$. The {\it complete configuration} ${\widehat{\H}}$ associated to a linear configuration ${\cal H} = \{L_1,\ldots, L_l \}$ in ${\Bbb P}^m$ is the configuration consisting of the $L_j$ and all of their non-empty intersections. A linear configuration is {\it complete} if it equals its completion. The {\it join} of two linear subspaces $L_1$, $L_2$ of ${\Bbb P}^m$ is the smallest linear subspace of ${\Bbb P}^m$ which contains them both. It will be denoted by $L_1 \ast L_2$. \begin{definition}\label{derived} The set ${\cal D}({\vec{x}})$ of linear configurations in ${\Bbb P}^m$ {\it derived} from a particular configuration ${\vec{x}}$ of $n$ points in ${\Bbb P}^m$ is the unique set of linear configurations in ${\Bbb P}^m$ which satisfies the following properties: \begin{enumerate} \item the completion of the configuration consisting of all hyperplanes in ${\Bbb P}^m$ that are spanned by a subconfiguration of ${\vec{x}}$ is in ${\cal D}({\vec{x}})$; \item every ${\cal H} \in {\cal D}({\vec{x}})$ is complete; \item if ${\cal H} \in {\cal D}({\vec{x}})$, $L\in {\cal H}$ and ${\cal X}$ is a subconfiguration of ${\vec{x}}$ such that $L \ast \spn {\cal X}$ is a hyperplane, then the completion of ${\cal H} \cup \{L \ast \spn {\cal X}\}$ is also in ${\cal D}({\vec{x}})$. \end{enumerate} \end{definition} \begin{figure}[htp] \begin{picture}(400,200) \put(10,0){\begin{picture}(200,200) \put(80,150){\circle*{6}} \put(75,157){$x_0$} \put(30,50){\circle*{6}} \put(25,57){$x_1$} \put(130,50){\circle*{6}} \put(125,57){$x_2$} \put(80,100){\circle*{6}} \put(75,107){$x_3$} \put(180,100){\circle*{6}} \put(175,107){$x_4$} \end{picture}} \put(250,0){\begin{picture}(200,200) \put(80,150){\circle*{6}} \put(30,50){\circle*{6}} \put(130,50){\circle*{6}} \put(80,100){\circle*{6}} \put(180,100){\circle*{6}} \put(10,10){\line(1,2){90}} \put(80,-10){\line(0,1){200}} \put(150,10){\line(-1,2){90}} \put(10,30){\line(1,1){120}} \put(10,50){\line(1,0){180}} \put(170,10){\line(-1,1){140}} \put(200,100){\line(-1,0){180}} \put(200,90){\line(-2,1){150}} \put(201,107){\line(-3,-1){195}} \put(200,120){\line(-1,-1){130}} \end{picture}} \end{picture} \caption{}\label{points} \end{figure} \begin{example} Let ${\vec{x}}$ be the configuration $(x_0,x_1,x_2,x_3,x_4)$ of 5 points in ${\Bbb P}^2({\Bbb R})$ depicted in the left half of Figure \ref{points}. The right half of Figure \ref{points} depicts the configuration defined in (1) of the definition of ${\cal D}({\vec{x}})$. Every other linear configuration ${\cal H}$ in ${\cal D}({\vec{x}})$ contains this configuration. The first linear configuration depicted in Figure \ref{config} is in ${\cal D}({\vec{x}})$, the second is not. \end{example} \begin{figure} \begin{picture}(400,200) \put(10,0){\begin{picture}(200,200) \put(80,150){\circle*{6}} \put(30,50){\circle*{6}} \put(130,50){\circle*{6}} \put(80,100){\circle*{6}} \put(180,100){\circle*{6}} \put(10,10){\line(1,2){90}} \put(80,-10){\line(0,1){200}} \put(150,10){\line(-1,2){90}} \put(10,30){\line(1,1){120}} \put(10,50){\line(1,0){180}} \put(170,10){\line(-1,1){140}} \put(200,100){\line(-1,0){180}} \put(200,90){\line(-2,1){150}} \put(201,107){\line(-3,-1){195}} \put(200,120){\line(-1,-1){130}} \put(200,96){\line(-5,1){170}} \end{picture}} \put(250,0){\begin{picture}(200,200) \put(80,150){\circle*{6}} \put(30,50){\circle*{6}} \put(130,50){\circle*{6}} \put(80,100){\circle*{6}} \put(180,100){\circle*{6}} \put(10,10){\line(1,2){90}} \put(80,-10){\line(0,1){200}} \put(150,10){\line(-1,2){90}} \put(10,30){\line(1,1){120}} \put(10,50){\line(1,0){180}} \put(170,10){\line(-1,1){140}} \put(200,100){\line(-1,0){180}} \put(200,90){\line(-2,1){150}} \put(201,107){\line(-3,-1){195}} \put(200,120){\line(-1,-1){130}} \put (20,117){\line(1,0){190}} \end{picture}} \end{picture} \caption{}\label{config} \end{figure} The class of all order preserving functions $r : P \to {\Bbb N}$ from a partially ordered set into ${\Bbb N}$ forms a category ${\cal P}$. A morphism $\Phi$ from $r_1 : P_1 \to {\Bbb N}$ to $r_2: P_2 \to {\Bbb N}$ is an order preserving function $\phi : P_1 \to P_2$ such that the diagram $$ \begin{matrix} P_1 & \stackrel{\phi}{\longrightarrow} & P_2 \\ r_1 \searrow & & \swarrow r_2 \\ & {\Bbb N} & \end{matrix} $$ commutes. We shall denote the {\it $k$-dimensional elements} $r^{-1}(k)$ of $P$ by $P_k$. \begin{definition} A {\it template} is an isomorphism class of objects of the category ${\cal P}$. \end{definition} Templates are a generalization of matroids. Each linear configuration ${\cal H}$ in ${\Bbb P}^m$ is a partially ordered set---the linear subspaces are ordered by inclusion. Define a rank function $r: {\cal H} \to {\Bbb N}$ by defining $r(L) = \dim L$ for each $L \in {\cal H}$. In this way we associate a template to each linear configuration. The template of a linear configuration ${\cal H}$ derived from a configuration of points ${\vec{x}}$ in ${\Bbb P}^m$ has additional structure; namely, the marking of the points of ${\vec{x}}$. For this reason, we now define marked templates. Denote the set $\{0,1,\ldots,n\}$ by $[n]$. One can consider the class of triples $(P,r,\psi)$, where $P$ is a partially ordered set, $r:P \to {\Bbb N}$ is an order preserving function, and where $\psi : [n] \to P_0$ is a function. These form a category ${\widetilde{\cal P}}_n$; the morphisms are order preserving maps which preserve the rank functions $r$ and the markings $\psi$. \begin{definition} An {\it $n$-marked template} is an isomorphism class of objects of the category ${\widetilde{\cal P}}_n$. \end{definition} Marked templates are a generalization of oriented matroids. Each linear configuration derived from a configuration ${\vec{x}}$ of $n+1$ points in ${\Bbb P}^m$ determines an $n$-marked template. If ${\vec{x}} = (x_0,x_1,\ldots , x_n)$, then the marking $\psi : [n] \to {\cal H}_0$ is defined by $\psi(j) = x_j$. We will view ${\cal D}({\vec{x}})$ as a set of marked linear configurations. Denote the set of $n$-marked templates $$ \left\{ T({\cal H}) : {\cal H} \in {\cal D}({\vec{x}})\right\} $$ associated to a configuration ${\vec{x}}$ of $n+1$ points in ${\Bbb P}^m$ by ${\cal T}({\vec{x}})$. Taking ${\cal H}$ to $T({\cal H})$ defines a bijection $$ {\cal D}({\vec{x}}) \to {\cal T}({\vec{x}}). $$ We shall denote the element of ${\cal D}({\vec{x}})$ which corresponds to $T\in {\cal T}({\vec{x}})$ by ${\cal H}_T$. The group of projective equivalences $PGL_{m+1}$ acts on the set of linear configurations in ${\Bbb P}^m$. Observe that if two linear configurations are projectively equivalent, they determine the same template. Consequently, ${\cal T}({\vec{x}})$ depends only on the projective equivalence class of ${\vec{x}}$. \section{Hyperplane Arrangements of Fiber Type} We retain the notation of the previous section. We inductively define what it means for an arrangement of hyperplanes in ${\Bbb F}^n$ to be of {\it fiber type}. First, every arrangement of distinct points in ${\Bbb F}$ is of fiber type. An arrangement of hyperplanes ${\cal H}$ in ${\Bbb F}^n$ is of fiber type if there is a linear projection $\phi:{\Bbb F}^n\to {\Bbb F}^{n-1}$ and an arrangement of hyperplanes ${\cal A}$ in ${\Bbb F}^{n-1}$ of fiber type such that \begin{enumerate} \item[(a)] the arrangement $\phi^{-1}{\cal A}$ is a sub-arrangement of ${\cal H}$; \item[(b)] the image under $\phi$ of each element of ${\cal H} - \phi^{-1}{\cal A}$ is all of ${\Bbb F}^{n-1}$; \item[(c)] for each $u\in {\Bbb F}^{n-1}-\cup{\cal A}$, the number of points in the induced arrangement of points $\phi^{-1}(u)\cap {\cal H}$ of $\phi^{-1}(u)$ by ${\cal H}$ is independent of $u$. \end{enumerate} When ${\Bbb F}$ is ${\Bbb R}$ or ${\Bbb C}$, the conditions (a), (b) and (c) imply that the projection $\psi : {\Bbb C}^n - \bigcup{\cal H} \to {\Bbb C}^{n-1} - \bigcup {\cal A}$ is a topological fiber bundle. \begin{proposition}\label{fiber_type} If ${\vec{x}}$ is a configuration of $n$ points in ${\Bbb P}^m$, then, for each $T \in {\cal T}({\vec{x}})$, there is $T'\in {\cal T}({\vec{x}})$ such that ${\cal H}_T \subseteq {\cal H}_{T'}$ and such that ${\cal H}_{T'}$ is an arrangement of fiber type. \end{proposition} \begin{pf} We prove the result by induction on $m$. The result is trivially true when $m=1$. Now suppose that $m\ge 1$. The image of $(x_0,\ldots, x_{n-1})$ under the linear projection $$ \phi : {\Bbb P}^m - \{x_n\} \to {\Bbb P}(T_{x_n}{\Bbb P}^m) \approx {\Bbb P}^{m-1} $$ is an $(n-1)$-marked configuration ${\vec{z}}$ of points in ${\Bbb P}^{m-1}$. The linear subspaces $L\in {\cal H}_T$ which contain $x_0$ induce a linear arrangement $\overline{{\cal H}}_T$ in ${\Bbb P}(T_{x_n}{\Bbb P}^m)$. It is easy to check that $\overline{{\cal H}}_T \in {\cal D}({\vec{z}})$. By induction, there is a template $S\in {\cal T}({\vec{z}})$ such that the arrangement ${\cal H}_S$ in ${\Bbb P}(T_{x_n}{\Bbb P}^m)$ is of fiber type and contains $\overline{{\cal H}}_T$. The inverse image of ${\cal H}_S$ under $\phi$ is an arrangement $\widetilde{{\cal H}}_S$ of hyperplanes in ${\Bbb P}^m$ each of whose hyperplanes contains $x_n$. The completion of the linear arrangement $$ {\cal H} := \widetilde{{\cal H}}_S \cup {\cal H}_T $$ is an element of ${\cal D}({\vec{x}})$. The projection $\phi$ induces a linear projection $$ \psi : {\Bbb P}^m - \cup {\cal H} \to {\Bbb P}(T_{x_n}{\Bbb P}^m) - \cup {\cal H}_S $$ whose fibers are punctured lines. Adding to ${\cal H}$ the hyperplanes in ${\Bbb P}^m$ which are the join of $x_n$ with a codimension 2 stratum of ${\cal H}$ we obtain a linear arrangement ${\cal H}'$ in ${\Bbb P}^m$ such that the restriction of $\psi$ to ${\Bbb P}^m - \cup {\cal H}'$ is a linear map $$ {\Bbb P}^m - \cup {\cal H}' \to {\Bbb P}(T_{x_n}{\Bbb P}^m) - \cup {\cal H}_S $$ each of whose fibers is ${\Bbb P}^1$ minus the same number of points. That is, the arrangement ${\cal H}'$ is of fiber type. Let $T'\in {\cal T}({\vec{x}})$ be the template which corresponds to the completion of ${\cal H}'$. Then $\cup {\cal H}_{T'} = \cup {\cal H}'$, and so ${\cal H}_{T'}$ is an arrangement of fiber type which contains ${\cal H}_T$. \end{pf} \section{The Generic Grassmannians}\label{topology} As in the previous sections, ${\Bbb F}$ will denote a fixed ground field. First recall that the grassmannian $G(q,{\Bbb P}^{p+q})$ of $q$-dimensional subspaces of ${\Bbb P}^{p+q}$ can be viewed as the orbit space $$ \left\{ (v_0,v_1,\ldots,v_{p+q}) \in ({\Bbb F}^p)^{p+q+1} : v_0, \ldots, v_{p+q} \text{ span } {\Bbb F}^p\right\}/GL_p({\Bbb F}) $$ where $GL_p$ acts diagonally (cf. \cite[\S 5]{hain-macp}). The generic part $G^p_q$ of $G(q,{\Bbb P}^{p+q})$ is defined to be the set of those points in $G(q,{\Bbb P}^{p+q})$ which correspond to $(p+q+1)$-tuples of vectors $(v_0,\ldots,v_{p+q})$ in ${\Bbb F}^p$ where each $p$ of the vectors are linearly independent. The torus $$ ({\Bbb G}_m)^{p+q} \approx ({\Bbb G}_m)^{p+q+1}/\text{ diagonal} $$ acts on $G^p_q$ via the action $$ (\lambda_0,\ldots,\lambda_{p+q}) : (v_0,\ldots,v_{p+q}) \mapsto (\lambda_0 v_0,\ldots,\lambda_{p+q} v_{p+q}). $$ The quotient space is the variety $$ Y^p_q := \left\{(x_0,\ldots,x_{p+q}) \in ({\Bbb P}^{p-1})^{p+q+1} : \text{each $p$ of the points span ${\Bbb P}^{p-1}$} \right\}/PGL_p. $$ The morphism $\pi : G^p_q \to Y^p_q$ is a principal $({\Bbb G}_m)^{p+q}$-bundle with a section \cite[(5.9)]{hain-macp}. Consequently, $$ G^p_q \approx Y^p_q \times ({\Bbb G}_m)^{p+q}. $$ Denote the point of ${\Bbb P}(V)$ which corresponds to $v\in V-\{0\}$ by $[v]$. The point $v$ of $G^p_q$ corresponding to the orbit of $(v_0,v_1,\ldots,v_{p+q})$ determines the point $$ {\vec{x}}(v) = ([v_0],[v_1],\ldots,[v_{p+q}]) $$ of $Y^p_q$. We can therefore associate to each point of $G^p_q$ the set ${\cal T}({\vec{x}}(v))$ of $(p+q)$-marked templates. For each $(p+q)$-marked template $T$, define the subset $E^p_q(T)$ of $G^p_q$ to be the Zariski closure of $$ \left\{v \in G^p_q : T \in {\cal T}({\vec{x}}(v)) \right\}. $$ We will define two templates $T_1$ and $T_2$ to be {\it $(p,q)$-equivalent} if the subvarieties $E^p_q(T_1)$ and $E^p_q(T_2)$ of $G^p_q$ are equal. \begin{example} The two configurations in Figure \ref{templates} determine templates $T_1$ and $T_2$, respectively. Both $E^3_4(T_1)$ and $E^3_4(T_2)$ are proper subvarieties of $G^3_4$, and $E^3_4(T_2)$ is a proper subvariety of $E^3_4(T_1)$. \end{example} \begin{figure}[htp] \begin{picture}(440,220)(0,0) \put(20,0){\begin{picture}(220,220) \put(50,50){\circle*{6}} \put(50,150){\circle*{6}} \put(150,150){\circle*{6}} \put(150,50){\circle*{6}} \put(20,125){\circle*{6}} \put(180,125){\circle*{6}} \put(100,35){\circle*{6}} \put(100,165){\circle*{6}} \put(20,20){\line(1,1){160}} \put(0,125){\line(1,0){200}} \put(20,180){\line(1,-1){160}} \put(100,10){\line(0,1){180}} \end{picture}} \put(240,0){\begin{picture}(220,220) \put(50,50){\circle*{6}} \put(50,150){\circle*{6}} \put(150,150){\circle*{6}} \put(150,50){\circle*{6}} \put(20,100){\circle*{6}} \put(180,100){\circle*{6}} \put(100,35){\circle*{6}} \put(100,165){\circle*{6}} \put(20,20){\line(1,1){160}} \put(0,100){\line(1,0){200}} \put(20,180){\line(1,-1){160}} \put(100,10){\line(0,1){180}} \end{picture}} \end{picture} \caption{}\label{templates} \end{figure} Define $F^p_q(T) \subseteq Y^p_q$ to be the quotient of $E^p_q(T)$ by the torus action. Observe that: \begin{proposition}\label{prod} For each template $T$, the varieties $E^p_q(T)$ and $F^p_q(T) \times ({\Bbb G}_m)^{p+q}$ are isomorphic. \qed \end{proposition} \begin{definition} The {\it constructible topology} on $G^p_q$ is the topology whose closed sets are finite unions of the sets $E^p_q(T)$. The {\it constructible topology} on $Y^p_q$ is the topology whose closed sets are finite unions of the sets $F^p_q(T)$. The constructible topology on a subset of $G^p_q$ or $Y^p_q$ is the topology induced from the constructible topology on $G^p_q$ or $Y^p_q$. In particular, the sets $E^p_q(T)$ and $F^p_q(T)$ have constructible topologies. \end{definition} Evidently, the closed subsets of $G^p_q$ are precisely the inverse images of closed subsets of $Y^p_q$ under the projection $G^p_q \to Y^p_q$. Note that the constructible topology is coarser than the Zariski topology. \begin{proposition}\label{generic} For each $(p,q)$-marked template $T$, $$ \left\{v \in F^p_q(T) : T \in {\cal T}({\vec{x}}(v))\right\} $$ is a constructible open subset of $F^p_q(T)$. \qed \end{proposition} \section{Rational $K(\pi,1)$ Spaces}\label{rat_space} In this section we briefly review the definition and basic properties of rational $K(\pi,1)$ spaces. Relevant references include \cite{kohno}, \cite{falk}, \cite{hain:cycles} and \cite{hain-macp}. As motivation, recall that if a topological space $X$ is a $K(\pi,1)$, then there is a natural isomorphism $$ H^\bullet(X,{\Bbb M}) \approx H^\bullet(\pi_1(M),M) $$ where $M$ is a $\pi_1(M,\ast)$ module, and ${\Bbb M}$ denotes the corresponding local system over $X$. One can define the continuous cohomology of a group $\pi$ by $$ H_{cts}^\bullet(\pi;{\Bbb Q}) = \lim_{\stackrel{\to}{\Gamma}} H^\bullet(\Gamma,{\Bbb Q}) $$ where $\Gamma$ ranges over all finitely generated nilpotent quotients of $\pi$. There is an evident map $$ H_{cts}^\bullet(\pi,{\Bbb Q}) \to H^\bullet(\pi,{\Bbb Q}). $$ A topological space $X$ is defined to be a rational $K(\pi,1)$ if the composition $$ H_{cts}^\bullet(\pi_1(X),{\Bbb Q}) \to H^\bullet(\pi_1(X),{\Bbb Q}) \to H^\bullet(X,{\Bbb Q}) $$ is an isomorphism. Every nilmanifold is clearly both a $K(\pi,1)$ and a rational $K(\pi,1)$. In particular, the circle is both a $K(\pi,1)$ and a rational $K(\pi,1)$. The following results will be used in Section \ref{main_thm}. Proofs of them can be found in \cite[\S5]{hain:cycles}. \begin{theorem}\label{wedge} The one point union of two rational $K(\pi,1)$s is a rational $K(\pi,1)$. In particular, every Zariski open subset of ${\Bbb C}$ is both a $K(\pi,1)$ and a rational $K(\pi,1)$. \qed \end{theorem} \begin{theorem}\label{fibration} Suppose that $f:X \to Y$ is a fiber bundle with fiber $F$. If $Y$ and $F$ are rational $K(\pi,1)$s, and if the natural action of $\pi_1(Y,y)$ on each cohomology group of $F$ is unipotent, then $X$ is a rational $K(\pi,1)$. \qed \end{theorem} Since each Zariski open subset of ${\Bbb C}$ is both a $K(\pi,1)$ and a rational $K(\pi,1)$, and since the monodromy representations associated to a linear fibration is trivial \cite[(5.12)]{hain:cycles} we obtain the following result: \begin{corollary}\label{fiber-type} The complement of an arrangement of hyperplanes in ${\Bbb C}^n$ which is of fiber type is both a $K(\pi,1)$ and a rational $K(\pi,1)$. \qed \end{corollary} \section{The Main Theorem}\label{main_thm} In this section, we prove the following result. \begin{theorem}\label{main} Each constructible open subset of $G^p_q({\Bbb C})$ contains a constructible open subset which is both a $K(\pi,1)$ and a rational $K(\pi,1)$. \end{theorem} \begin{remark} It is easy to show that in the cases of $G^p_0$ and $G^p_1$, the constructible topology is trivial. That is, the only constructible open sets in these spaces are the the empty set and the whole space. Thus Theorem \ref{main} implies that $G^p_0({\Bbb C})$ and $G^p_1({\Bbb C})$ are $K(\pi,1)$s and rational $K(\pi,1)$s. This is clear in the case of $G^p_0$, and is proved directly in the case of $G^p_1$ in \cite[(8.6)]{hain-macp}. \end{remark} The proof of Theorem \ref{main} occupies the rest of this section. Since the classes of rational $K(\pi,1)$s and $K(\pi,1)$s are closed under products, and since each constructible open subset of $G^p_q$ is a product of the corresponding constructible open subset of $Y^p_q$ with $({\Bbb C}^\ast)^{p+q}$, we need only prove that each constructible open subset of $Y^p_q$ contains a constructible open set which is a both a $K(\pi,1)$ and a rational $K(\pi,1)$. Suppose that $0\le i \le {p+q}$. The {\it $i$th face map}, $$ A_i : Y^p_q \to Y^p_{q-1} $$ is defined by forgetting the $i$th point of a configuration of $p+q$ points in ${\Bbb P}^{p-1}$. The $i$th {\it dual face map} $$ B_i : Y^p_q \to Y^{p-1}_q $$ is obtained by projecting all but the $i$th point of a configuration of $p+q$ points in ${\Bbb P}^{p-1}$ onto a generic ${\Bbb P}^{p-2}$ using the $i$th point as the center of the projection. \begin{proposition}\label{cont} If $T$ is a $(p+q)$-marked template, then for each integer $i$ satisfying $0\le i \le p+q+1$, there is a $(p+q+1)$-marked template $A^iT$ (resp.\ $B^iT$) whose $(p,q+1)$-equivalence class (resp.\ $(p+1,q)$-equivalence class) depends only on the $(p,q)$-equivalence class of $T$. Moreover, $$ A_i^{-1}F^p_q(T) = F^p_{q+1}(A^iT),\quad A_i^{-1}E^p_q(T) = E^p_{q+1}(A^iT) $$ and $$ B_i^{-1}F^p_q(T) = F^{p+1}_{q}(B^iT),\quad B^iE^p_q(T) = E^{p+1}_{q}(B^iT). $$ In particular, the face maps and dual face maps are continuous with respect to the constructible topology. \end{proposition} \begin{pf} For simplicity of notation, we take $i=p+q+1$. Suppose that $T$ is a $(p+q)$-marked template. Represent it by the object $(P,r,\psi)$ of the category ${\widetilde{\cal P}}_{p+q}$. Denote the marked elements $\psi(0), \ldots,\psi(p+q)$ of $P_0$ by $p_0,\ldots,p_{p+q}$. For each subset $I$ of $\{0,\ldots,p+q\}$, denote the element of $P$ which is the least upper bound of $\{p_i: i\in I\}$ by $p_I$. Set $r_I = r(p_I)$. Define $A^i T$ and $B^i T$ both to be the isomorphism class of the completion of the marked ordered set obtained from $(P,r,\psi)$ by adding one extra element $p_{p+q+1}$ to $P_0$, and elements $p_I\ast p_{p+q+1}$ to $P_{1+r_I}$. This is the ``smallest'' template $T'$ for which $A_iT' = T$. For example, if $T$ is the 6-marked template associated to the configuration on the left hand side of Figure \ref{invim}, then $A^6T$ is the 7-marked template associated to the configuration on the right hand side of Figure \ref{invim}. \begin{figure} \begin{picture}(440,220)(0,0) \put(20,0){\begin{picture}(220,220) \put(50,50){\circle*{6}} \put(50,150){\circle*{6}} \put(150,150){\circle*{6}} \put(150,50){\circle*{6}} \put(100,40){\circle*{6}} \put(100,160){\circle*{6}} \put(20,20){\line(1,1){160}} \put(20,180){\line(1,-1){160}} \put(100,-10){\line(0,1){220}} \end{picture}} \put(280,0){\begin{picture}(220,220) \put(50,50){\circle*{6}} \put(50,150){\circle*{6}} \put(150,150){\circle*{6}} \put(150,50){\circle*{6}} \put(100,40){\circle*{6}} \put(100,160){\circle*{6}} \put(0,100){\circle*{6}} \put(-6,112){$x_6$} \put(20,20){\line(1,1){160}} \put(20,180){\line(1,-1){160}} \put(100,-10){\line(0,1){220}} \put(-20,80){\line(1,1){130}} \put(-20,88){\line(5,3){170}} \put(-21,93){\line(3,1){200}} \put(-20,120){\line(1,-1){130}} \put(-20,112){\line(5,-3){170}} \put(-21,107){\line(3,-1){200}} \end{picture}} \end{picture} \caption{}\label{invim} \end{figure} For $v\in Y^{p+1}_q$, it is clear that $T \in {\cal T}({\vec{x}}(A_iv))$ if and only if $A^i T \in {\cal T}({\vec{x}}(v))$. It follows that $$ A_i^{-1}F^p_q(T) = F^{p+1}_q(A^iT) $$ and that the $(p+1,q)$-equivalence class of $A^i T$ depends only on the $(p,q)$-equivalence class of $T$. The corresponding statement for $E^p_q(T)$ follows from (\ref{prod}). The statements with $A$ replaced by $B$ follow using the dual argument. \end{pf} The following definition is an analogue of (\ref{derived}) for templates. It is used only in the proof of the next result. \begin{definition}\label{generate} Suppose that $(P,r,\psi)$ is an object of the category ${\widetilde{\cal P}}_n$. Suppose that $f: [m] \to [n]$ is an order preserving injection. Define the subset $Q$ of $P$ {\it generated} by $f$ to be the smallest subset of $P$ which contains $\left\{\psi\circ f(j) : j\in [m]\right\}$ and is closed under the following operations: \begin{enumerate} \item if $S\subseteq \im f$, then the least upper bound of $S$ in $P$ is in $Q$; \item if $S\subseteq Q$, then the greatest lower bound of $S$ in $P$ is in $Q$; \item if $v\in Q$ and $S\subseteq \im f$, then the greatest lower bound of $S$ and $v$ in $P$ is in $Q$. \end{enumerate} \end{definition} \begin{proposition}\label{closed} If $T$ is a $(p+q)$-marked template, then for each integer $i$ satisfying $0\le i \le p+q$, there is a $(p+q-1)$-marked template $A_iT$ (resp.\ $B_iT$) whose $(p,q-1)$-equivalence class (resp.\ $(p-1,q)$-equivalence class) depends only on the $(p,q)$-equivalence class of $T$. Moreover, $A_iF^p_q(T)$ is a constructible open subset of $F^p_{q-1}(A_iT)$, $A_iE^p_q(T)$ is a constructible open subset of $E^p_{q-1}(A_iT)$, $B_iF^p_q(T)$ is a constructible open subset of $F^{p-1}_{q}(B_iT)$ and $B_iE^p_q(T)$ is a constructible open subset of $E^{p-1}_{q}(B_iT)$. \end{proposition} \begin{pf} Suppose that $T$ is a $(p+q)$-marked template. Let $(P,r,\psi)$ be an object of the category ${\widetilde{\cal P}}_{p+q}$ which represents $T$. Let $Q$ be the partially ordered subset of $P$ generated by the $i$th face map $d_i : [p+q] \to [p+q+1]$---that is, the unique order preserving injection which omits the value $i$. Let $(Q,r,\psi\circ d_i)$ be the object of ${\widetilde{\cal P}}_{p+q-1}$ where $r$ is the restriction of the rank function of $P$. Define $A_{p+q}T$ to be the $(p+q-1)$-marked template which is represented by $(Q,r,\psi)$. For example, if $T$ is the template associated to the configuration in the left hand side of Figure \ref{less}, then $A_2 T$ is the template corresponding to the configuration on the right hand side of Figure \ref{less} \begin{figure} \begin{picture}(400,200) \put(10,0){\begin{picture}(200,200) \put(80,150){\circle*{6}} \put(30,50){\circle*{6}} \put(130,50){\circle*{6}} \put(80,100){\circle*{6}} \put(180,100){\circle*{6}} \put(122,35){$x_2$} \put(10,10){\line(1,2){90}} \put(80,-10){\line(0,1){200}} \put(150,10){\line(-1,2){90}} \put(10,30){\line(1,1){120}} \put(10,50){\line(1,0){180}} \put(170,10){\line(-1,1){140}} \put(200,100){\line(-1,0){180}} \put(200,90){\line(-2,1){150}} \put(201,107){\line(-3,-1){195}} \put(200,120){\line(-1,-1){130}} \put(200,96){\line(-5,1){170}} \end{picture}} \put(250,0){\begin{picture}(200,200) \put(80,150){\circle*{6}} \put(30,50){\circle*{6}} \put(80,100){\circle*{6}} \put(180,100){\circle*{6}} \put(10,10){\line(1,2){90}} \put(80,-10){\line(0,1){200}} \put(10,30){\line(1,1){120}} \put(200,100){\line(-1,0){180}} \put(200,90){\line(-2,1){150}} \put(201,107){\line(-3,-1){195}} \end{picture}} \end{picture} \caption{}\label{less} \end{figure} It is clear that $A_iF^p_q(T) \subseteq F^p_{q-1}(A_i T)$. That $A_iF^p_q(T)$ is a constructible open subset of $F^p_{q-1}(A_0T)$ follows directly from (\ref{generic}). The corresponding statements for $E^p_q(T)$ follows from (\ref{prod}). The statements with $A$ replaced by $B$ follow using the dual argument. \end{pf} \begin{example} An example where $A_0F^p_q(T)$ is a proper subset of $F^p_{q-1}(A_0T)$ is given in Figure \ref{face}. If $T$ is the 13-marked template associated to the left hand figure, then the right hand configuration is an element of $F^3_8(A_0T)-A_0(F^3_9(T))$. \end{example} \begin{figure} \begin{picture}(400,200) \put(10,0){\begin{picture}(200,200) \put(20,35){\circle*{6}} \put(20,65){\circle*{6}} \put(80,35){\circle*{6}} \put(80,65){\circle*{6}} \put(35,120){\circle*{6}} \put(35,180){\circle*{6}} \put(65,120){\circle*{6}} \put(65,180){\circle*{6}} \put(165,35){\circle*{6}} \put(165,165){\circle*{6}} \put(150,65){\circle*{6}} \put(150,135){\circle*{6}} \put(100,100){\circle*{6}} \put(80,97){$x_0$} \put(10,10){\line(1,1){180}} \put(10,190){\line(1,-1){180}} \put(150,10){\line(0,1){180}} \put(10,30){\line(2,1){80}} \put(10,70){\line(2,-1){80}} \put(30,110){\line(1,2){40}} \put(30,190){\line(1,-2){40}} \end{picture}} \put(250,0){\begin{picture}(200,200) \put(20,35){\circle*{6}} \put(20,65){\circle*{6}} \put(80,35){\circle*{6}} \put(80,65){\circle*{6}} \put(35,120){\circle*{6}} \put(35,180){\circle*{6}} \put(65,120){\circle*{6}} \put(65,180){\circle*{6}} \put(165,35){\circle*{6}} \put(165,165){\circle*{6}} \put(100,65){\circle*{6}} \put(100,135){\circle*{6}} \put(10,10){\line(1,1){180}} \put(10,190){\line(1,-1){180}} \put(100,10){\line(0,1){180}} \put(10,30){\line(2,1){80}} \put(10,70){\line(2,-1){80}} \put(30,110){\line(1,2){40}} \put(30,190){\line(1,-2){40}} \end{picture}} \end{picture} \caption{}\label{face} \end{figure} \begin{corollary}\label{example} If $T$ is a $(p+q)$-marked template, then $$ F^p_q(T) \subseteq F^p_q(A^iA_i T)\quad\text{and} \quad E^p_q(T) \subseteq E^p_q(A^iA_i T) $$ for all integers $i$ satisfying $0\le i \le p+q$, and if $T$ is a $(p+q)$-marked template, then $$ F^p_q(T) \subseteq F^p_q(B^iB_i T)\quad\text{and} \quad E^p_q(T) \subseteq E^p_q(B^iB_i T) $$ for all integers $i$ such that $0\le i \le p+q+1$. \qed \end{corollary} We are now ready to prove Theorem \ref{main}. Throughout the remainder of this section, the ground field will be ${\Bbb C}$, unless explicitly stated to the contrary. Fix $p>0$. The proof is by induction on $q$. When $q = 0$, $Y^p_q$ is a point and the result is trivially true. Now suppose that $q>0$ and that the result is true for $Y^p_{q-1}$. Suppose that $U$ is a non-empty constructible open subset of $Y^p_q$. The idea behind the proof is to replace $U$ by a smaller constructible open set $L$ whose image under $A_0$ is a constructible open subset of $Y^p_{q-1}$ and such that the map $L \to A_0L$ is a fibration whose fibers are complements of arrangements of hyperplanes in ${\Bbb P}^{p-1}$ of fiber type and whose monodromy representations are trivial. Using the inductive hypothesis, one then finds a constructible open subset $L'$ of $A_0L$ which is both a $K(\pi,1)$ and a rational $K(\pi,1)$. It will then follow from (\ref{fibration}) that $A_0^{-1}(L')$ is the sought after constructible open subset of $U$. We now give the details. Our first task is to find a constructible open subset $W$ of $Y^p_{q-1}$ such that the restriction of $A_0$ to $U\cap A_0^{-1}W$ is a family of hyperplane complements where each relative hyperplane is proper over the base. There are $(p+q)$-marked templates $T_1,\ldots, T_l$ such that $$ U = Y^p_q - \bigcup_{j=1}^l F^p_q(T_j). $$ For each $j$, either $F^p_{q-1}(A_0T_j)$ is all of $Y^p_{q-1}$ or is a proper closed subvariety. We may suppose that $F^p_{q-1}(A_0T_j)$ is $Y^p_{q-1}$ when $j\le k$ and is a proper subvariety when $j > k$. When $j \le k$, set $$ C_j = Y^p_{q-1} - A_0F^p_q(T_j); $$ this is a constructible closed proper subset of $Y^p_{q-1}$ by (\ref{generic}). Set $$ W = Y^p_{q-1} - \left(\bigcup_{j\le k} C_j \cup \bigcup_{j>k} F^p_{q-1}(A_0 T_j)\right). $$ Then $W$ is a non-empty constructible open subset of $Y^p_{q-1}$ and the restriction of $A_0 : A_0^{-1}W \to W$ to $F^p_q(T_j)$ is proper and surjective when $j\le k$. Now $$ A_0^{-1}W \cap U = A_0^{-1}W - \bigcup_{j\le k} F^p_q(T_j). $$ is a constructible open subset of $U$. The fiber of $$ A_0 : A_0^{-1}W - \bigcup_{j\le k} F^p_q(T_j) \to W $$ over $(x_1,\ldots,x_{p+q})$ is the complement of an arrangement of hyperplanes in ${\Bbb P}^{p-1}$ which is derived from the configuration $(x_1,\ldots,x_{p+q})$ and where each relative hyperplane is proper over $W$. Our next task is to replace $W$ by a smaller constructible open set $O$ such that the restriction of $A_0$ to $U\cap A_0^{-1}O$ is a fiber bundle over $O$. We say that two linear configurations in ${\Bbb C}^m$ have the {\it same combinatorics} if their associated partially ordered sets are isomorphic, or equivalently, if they determine the same template. \begin{proposition}\label{bundle} There is a non-empty constructible open subset $O$ of $W$ such that the restriction of $$ A_0 : A_0^{-1}W - \bigcup_{j\le k} F^p_q(T_j) \to W $$ to $A_0^{-1}(O)$ has the property that each of its fibers is the complement of a linear configuration with the same combinatorics. Consequently, $$ A_0 : A^{-1}(O) \to O $$ is a fiber bundle where the action of $\pi_1(O,\ast)$ on the homology of the fibers is trivial. \end{proposition} \begin{pf} Let ${\Bbb F}$ be the function field of $Y^p_{q-1}$. Then $\bigcup_{j\le k} F^p_q(T_j)$ is a configuration of hyperplanes in ${\Bbb P}^{p-1}({\Bbb F})$. For generic $v\in W$, the combinatorics of the restriction of this configuration to the fiber of $A_0$ over $v$ has the same combinatorics as this configuration over the generic point of $Y^p_{q-1}$. The set of $v$ for which the combinatorics is different is a closed constructible subset $F$ of $Y^p_{q-1}$. The desired constructible open subset of $Y^p_{q-1}$ is then $O = W-F$. \end{pf} Next we further shrink both $U$ and $O$ to make the fibers of $A_0$ to $U\cap A_0^{-1}O$ complements of arrangements of hyperplanes of fiber type. \begin{proposition} There is a non-empty constructible open subset $O'$ of $O$ and a $(p+q)$-marked template $T$ such that $A_0F^p_q(T)$ contains $O'$ and such that $$ F^p_q(T) \supseteq \bigcup_{j\le k} F^p_q(T_j) $$ and the map $$ A_0^{-1}O' - F^p_q(T) \to O' $$ induced by $A_0$ is a fiber bundle all of whose fibers are complements of arrangements of hyperplanes of fiber type. \end{proposition} \begin{pf} As in the proof of the previous result, we shall denote the function field of $Y^p_{q-1}$ by ${\Bbb F}$. The points $x_1,\ldots,x_{p+q}$ are defined over ${\Bbb F}$, and therefore may be regarded as a configuration ${\vec{x}}({\Bbb F})$ of points in ${\Bbb P}^{p-1}({\Bbb F})$. The set $$ \bigcup_{j\le k} F^p_q(T_j), $$ is a configuration of hyperplanes defined over ${\Bbb F}$ and thus determines an element of ${\cal D}({\vec{x}}({\Bbb F}))$. Let $T'\in {\cal T}({\vec{x}}({\Bbb F}))$ be the corresponding template. By (\ref{fiber_type}), there is a template $T\in {\cal T}({\vec{x}}({\Bbb F}))$ such that $$ {\cal H}_{T} \supseteq \bigcup_{j\le k} F^p_q(T_j) $$ and such that ${\cal H}_{T}$ is an arrangement of fiber type. Since $T$ is defined over the generic point of $Y^p_{q-1}$, it follows from (\ref{closed}) that $A_0F^p_q(T)$ is a constructible open subset of $Y^p_{q-1}$. Moreover, the set of $v \in A_0F^p_q(T)$ for which the combinatorics of the restriction of ${\cal H}(T)$ to the fiber of $A_0$ over $v$ is given by $T$ is a constructible open subset of $A_0F^p_q(T)$. Let $O'$ be the intersection of this open set with $O$. \end{pf} By our inductive hypothesis, the constructible open set $O'$ of $Y^p_{q-1}$ contains a constructible open subset $L$ which is a $K(\pi,1)$ and a rational $K(\pi,1)$. Since $A_0^{-1}L$ is a non-empty constructible open subset of $Y^p_q$, $$ V = A_0^{-1}(L) - \left(F^p_q(T) \cup \bigcup_{j\le k} F^p_q(T_j)\right) $$ is also a non-empty constructible open subset of $U$. Further, the map $$ A_0 : V \to L $$ is a fibration each of whose fibers is the complement of an arrangement of hyperplanes in ${\Bbb P}^{p-1}$ of fiber type. It follows from (\ref{fiber-type}) that the fibers are $K(\pi,1)$s and rational $K(\pi,1)$s. Since the base is a $K(\pi,1)$ and a rational $K(\pi,1)$, and since the monodromy is trivial (\ref{bundle}), it follows from (\ref{fibration}) that $V$ is a $K(\pi,1)$ and a rational $K(\pi,1)$. This completes the proof of Theorem \ref{main}. \section{Existence and Uniqueness of Higher Logarithms} \label{existence} In this section, we first establish the existence and uniqueness of the $p$th higher logarithm in the sense of \cite[(6.1)]{hain-macp}, but with $G^p_\bullet$ replaced by a suitably chosen Zariski open subset $U^p_\bullet$. We then show how to construct the generalized $p$-logarithm, a multivalued Deligne cohomology class, in the sense of \cite[(12.4)]{hain-macp}, but with $G^p_\bullet$ replaced by $U^p_\bullet$. We shall use the notation and definitions of \cite{hain-macp}. We will say that a simplicial variety $U_\bullet$ is a {\it subvariety} of the simplicial variety $X_\bullet$ if each $U_q$ is a subvariety of $X_q$, and if the inclusion $U_\bullet \hookrightarrow X_\bullet$ is a morphism of simplicial varieties. We will say that $U_\bullet$ is an {\it open} (resp.\ {\it closed, dense, constructible}) subset of $G^p_\bullet$ if each $U_q$ is open (resp.\ { closed, dense,constructible}) in each $X_q$. There are analogous definitions with $G^p_\bullet$ replaced by $Y^p_\bullet$. \begin{proposition}\label{good_sub} For each positive integer $p$, each dense constructible open subset $V^p_\bullet$ of the truncated simplicial variety $G^p_\bullet$ contains a dense constructible open subset $U^p_\bullet$ where each $U^p_q$ is a rational $K(\pi,1)$. In particular, $G^p_\bullet$ contains a dense constructible open subset $U^p_\bullet$ where each $U^p_q$ is a $K(\pi,1)$ and a rational $K(\pi,1)$. \end{proposition} \begin{pf} The only dense constructible open subset of $G^p_0$ is $G^p_0$ itself. So $V^p_0=G^p_0$, and we must take $U^p_0=G^p_0$. Suppose that $m > 0$ and that $U^p_q$ has been constructed when $q< m$ such that each $U^p_q$ is dense in $G^p_q$, $U^p_q \subseteq V^p_q$, and such that $A_i(U^p_q) \subseteq U^p_{q-1}$ whenever $0 < q < m$. Now, $$ V^p_m \cap \bigcap_{i=0}^m A_i^{-1}U^p_{m-1} $$ is a non-empty constructible open subset of $G^p_m$. So, by (\ref{main}), it contains a non-empty, and therefore dense, constructible open subset $V^p_m$ of $G^p_m$. The result now follows by induction. \end{pf} In order to apply the multivalued de Rham complex functor, we will need to know that such a constructible open subset $U^p_\bullet$ of $G^p_\bullet$ can be lifted to a truncated simplicial object in the category $\widetilde{\A}$ defined in the introduction and in \cite[\S 2]{hain-macp}. \begin{theorem}\label{lift} Each constructible open subset $U^p_\bullet$ of $G^p_q$ can be lifted to a truncated simplicial object of the category $\widetilde{\A}$. \end{theorem} The lift is natural in the following sense: it comes with a lift $\tilde{\imath}$ of the inclusion $i:U^p_\bullet \hookrightarrow G^p_\bullet$ such that if $j: V^p_\bullet\hookrightarrow U^p_\bullet$ is an inclusion of constructible open subsets of $G^p_\bullet$, then $\tilde{\imath}\tilde{\jmath} = \widetilde{\imath\jmath}$. As the proof of this theorem is technical; it is given in a separate section, \S \ref{proof}. Next, we show how to construct the $p$th higher logarithm in the sense of \cite[(6.1)]{hain-macp} defined on some constructible dense open subset of $G^p_\bullet$. The following fact is is a direct consequence of \cite[(7.8)]{hain-macp} and \cite[(8.2)(i)]{hain-macp}. \begin{proposition}\label{acyclic} If the complex algebraic variety $X$ is a rational $K(\pi,1)$ with $q(X)=0$, then for all $l\ge 0$, the complex $W_l{\widetilde{\Omega}}^\bullet(X)$ is acyclic. \end{proposition} The existence of the higher logarithms is now an immediate consequence of (\ref{good_sub}), (\ref{lift}), (\ref{acyclic}) and \cite[(9.7)]{hain-macp}: \begin{theorem}\label{exist_1} For each integer $p\ge 1$, there is a dense constructible open subset $U^p_\bullet$ of the simplicial variety $G^p_\bullet$ which has a lift to the category $\widetilde{\A}$, and there is an element $Z_p$ of the double complex $W_{2p}{\widetilde{\Omega}}^\bullet (U^p_\bullet)$, unique up to a coboundary, whose coboundary is the ``volume form'' $$ {dx_1\over x_1} \wedge \ldots \wedge {dx_p\over x_p} \in \Omega^p(G^p_0). $$ \end{theorem} \begin{remark} With a little more care, one can arrange for each $U^p_q$ to be invariant under the action of the symmetric group $\Sigma_{p+q+1}$ on $G^p_q$ and for the symbol (as defined in \cite[p.~444]{hain-macp}) of each component of $Z_p$ to span a copy of the alternating representation. One should note, however, that it seems difficult to arrange for each $U^p_q$ to be a rational $K(\pi,1)$ and be preserved by the action of $\Sigma_{p+q+1}$. \end{remark} \begin{remark}\label{hana-mac-const} Hanamura and MacPherson \cite{hanamura-macp_2} give an explicit construction of all higher logarithms in the double complex $W_{2p}{\widetilde{\Omega}}(G^p_\bullet)$. In particular, they show that it is not necessary to pass to a Zariski open subset of $G^p_\bullet$ as we did. \end{remark} Next, we establish the existence of higher logarithms as Deligne cohomology classes. For this, we shall assume the reader is familiar with the definition of the multivalued Deligne cohomology functor $H_{\calM\D}^\bullet({\underline{\phantom{x}}},{\Bbb Q}(p))$ defined in \cite[\S 12]{hain-macp}. The key point here is the following result, a slightly stronger version of which was stated in \cite[(12.3)]{hain-macp}, and which we will prove in \S \ref{pf_dunno}. Recall from the introduction that $H_{\D}^\bullet$ denotes Beilinson's absolute Hodge cohomology. \begin{theorem}\label{dunno} Suppose that $X_\bullet$ is a truncated simplicial variety with a lift to $\widetilde{\A}$. If each $X_q$ is a rational $K(\pi,1)$, then for each integer $p$, there is a natural isomorphism $$ H_{\calM\D}^\bullet(X_\bullet,{\Bbb Q}(p)) \approx H_{\D}^\bullet(X_\bullet,{\Bbb Q}(p))). $$ \end{theorem} Granted this, the construction of the generalized $p$th higher logarithm as an element of $H_{\calM\D}^\bullet(U^p_\bullet,{\Bbb Q}(p))$ is relatively straightforward. \begin{theorem}\label{gen_log} If $U^p_\bullet$ is a dense subvariety of $G^p_\bullet$ where each $U^p_q$ is a rational $K(\pi,1)$, then there is an element of $$ H_{\calM\D}^{2p}(U^p_\bullet,{\Bbb Q}(p)) $$ whose restriction to $G^p_0$ is the volume form. \end{theorem} \begin{pf} Let $V^p_m$, be the subvariety of $$ \left\{(v_0,v_1,\ldots,v_{m}): v_j \in {\Bbb C}^p\right\} $$ which consists of those $(m+1))$-tuples of vectors where each set of $\min(m+1,p)$ of the vectors is linearly independent. When $m\ge p$, there is a natural projection $V^p_m\to G^p_{m-p}$ which is a principal $GL_p({\Bbb C})$-bundle. Define face maps $$ A_i : V^p_m \to V^p_{m-1} $$ by omitting the $i$th vector. Denote the corresponding simplicial variety by $V^p_\bullet$. Denote the truncated simplicial space which consists only of those $V^p_m$ with $p\le m \le 2p$ by $\widetilde{V}^p_\bullet$. There is a natural projection $\widetilde{V}^p_\bullet \to G^p_\bullet$ which is a principal $GL_p({\Bbb C})$-bundle. We would like to say that this bundle has a Chern class $$ c_p \in H_{\D}^{2p}(G^p_\bullet,{\Bbb Q}(p)). $$ Since the variety $G^p_\bullet$ is truncated (it has no simplices in dimensions $< p$), the existence of such a Chern class is not immediate. Our next task is to establish the existence of this class. We do this using the Borel construction. Let $E_\bullet$ be, say, the standard simplicial model for the universal bundle associated to $GL_p({\Bbb C})$. What is important for us is that $E_\bullet$ is a simplicial variety with the homotopy type of a point and on which $GL_p({\Bbb C})$ acts freely. Let $P_\bullet$ be the bisimplicial variety $V_\bullet \times E_\bullet$. It has the homotopy type of $V^p_\bullet$. Denote the quotient of $P_\bullet$ by the diagonal action of $GL_p({\Bbb C})$ by $B_\bullet$. Since $GL_p({\Bbb C})$ acts freely on $P_\bullet$, the quotient map $$ P_\bullet \to B_\bullet $$ is a principal $GL_p({\Bbb C})$ bundle. By \cite{beilinson}, this bundle has a Chern class \begin{equation}\label{chern} c_p \in H_{\D}^{2p}(B_\bullet,{\Bbb Q}(p)). \end{equation} Denote the truncated simplicial variety consisting of those $B_m$ with $p\le m \le 2p$ by $\widetilde{B}_\bullet$ and denote the restriction of the bundle $P_\bullet$ to $\widetilde{B}_\bullet$ by $\widetilde{P}_\bullet$. Then there is a commutative diagram $$ \begin{matrix} \widetilde{P}_\bullet & \to & \widetilde{V}^p_\bullet \\ \downarrow & & \downarrow \\ \widetilde{B}_\bullet & \to & G^p_\bullet\\ \end{matrix} $$ of principal $GL_p({\Bbb C})$ bundles obtained by collapsing out $E_\bullet$. Since the action of $GL_p({\Bbb C})$ on $\widetilde{V}^p$ is free, the bottom arrow is a homotopy equivalence of simplicial varieties, and therefore induces an isomorphism on Deligne cohomology. We can therefore restrict the Chern class (\ref{chern}) to $G^p_\bullet$ to obtain a class in $$ H_{\D}^{2p}(G^p_\bullet,{\Bbb Q}(p)). $$ It follows from (\ref{dunno}) that we can restrict this class to $U^p_\bullet$ to obtain a class $C_p$ in $$ H_{\calM\D}^{2p}(U^p_\bullet,{\Bbb Q}(p)) $$ provided that each $U^p_q$ is a rational $K(\pi,1)$. Finally, to prove Theorem \ref{gen_log}, we have to show that the restriction of $C_p$ to $U^p_0=G^p_0$ is a non-zero multiple of the volume form in $H^p(G^p_0)$. It is proved in \cite{hain-yang} that the restriction of $C_p$ to $G^p_0$ is $(p-1)! \mathrm{vol}$. It follows that $C_p/(p-1)!$ is a generalized $p$-logarithm. This completes the proof of Theorem \ref{gen_log}. \end{pf} \section{Proof of Theorem \ref{lift}} \label{proof} In the proof we shall need the following construction. Let $$ X^p_q = {\Bbb C}^{p+q+1} - \Delta, $$ where $\Delta$ denotes the fat diagonal---that is, the locus of points in ${\Bbb C}^{p+q+1}$ where the coordinates are not all distinct. Define $$ A_i : X^p_q \to X^p_{q-1} $$ by deleting the $i$th coordinate: $$ A_i: (t_0,\ldots,t_{p+q}) \mapsto (t_0,\ldots,\widehat{t_i},\ldots,t_{p+q}). $$ Denote the truncated simplicial variety consisting of those $X^p_q$ with $0\le q \le p$ by $X^p_\bullet$. We can define a morphism $\phi : X^p_\bullet \to G^p_\bullet$ by taking $(t_0,\ldots,t_{p+q})$ to the $GL_p({\Bbb C})$ orbit of the $(p+q+1)$-tuple of vectors $$ \begin{pmatrix} 1 \cr t_{0} \cr t_{0}^2 \cr \vdots \cr t_{0}^{p-1} \end{pmatrix} ,\quad \begin{pmatrix} 1 \cr t_{1} \cr t_{1}^2 \cr \vdots \cr t_{1}^{p-1} \end{pmatrix} ,\,\dots \quad \begin{pmatrix} 1 \cr t_{p+q} \cr t_{p+q}^2 \cr \vdots \cr t_{p+q}^{p-1} \end{pmatrix}. $$ This map is easily seen to be a well defined morphism of simplicial varieties. It induces a morphism $\overline{\phi} : X^p_\bullet \to Y^p_\bullet$. \begin{lemma}\label{dense} The image of $X^p_q$ in $G^p_q$ is dense in $G^p_q$ in the constructible topology. \end{lemma} \begin{pf} In view of (\ref{prod}), we need only prove that the image of $X^p_q$ in $Y^p_q$ is dense in $Y^p_q$ in the constructible topology. We do this by induction on $q$. Since $Y^p_0$ is a point, the result is trivially true when $q=0$. Suppose that $q>0$. Denote the constructible closure of the image of $X^p_q$ in $Y^p_q$ by $C^p_q$. By induction, $C^p_{q-1} = Y^p_{q-1}$. The intersection of $C^p_q$ with each fiber of $A_0 : Y^p_q \to Y^p_{q-1}$ is a constructible closed subset of the fiber. Note that the fiber of $A_0 : Y^p_q \to Y^p_{q-1}$ is the complement of a linear arrangement in ${\Bbb P}^{p-1}$ and that each of its constructible closed subsets is the intersection of the fiber with a finite union of linear subspaces of ${\Bbb P}^{p-1}$. Since the intersection of $C^p_q$ with the fiber is an open subset of a rational normal curve in ${\Bbb P}^{p-1}$, and since each rational normal curve is non-degenerate, it follows that the fiber of $A_0 : C^p_q \to Y^p_{q-1}$ equals the fiber of $A_0 : Y^p_q \to Y^p_{q-1}$. It follows that $C^p_q = Y^p_q$. \end{pf} Denote the topological analogue of the category $\widetilde{\A}$ by $\widetilde{\text{Top}}$. Observe that a simplicial object of ${\cal A}$ has a lift to the category $\widetilde{\A}$ if and only if it has a lift to the category $\widetilde{\text{Top}}$. \begin{proposition} Suppose that $Y_\bullet$ and $Z_\bullet$ are simplicial topological spaces where each $Y_n$ and $Z_n$ is path connected. If $f : Y_\bullet \to Z_\bullet$ is a morphism of simplicial spaces, and if $Y_\bullet$ has a lift to $\widetilde{\text{Top}}$, then both $Z_\bullet$ and $f$ have lifts to $\widetilde{\text{Top}}$. \end{proposition} \begin{pf} We use the equivalence of the category $\widetilde{\A}$ with the category ${\cal A}_\ast$ which is constructed in \cite[\S 2]{hain-macp}. We first construct a simplicial object of ${\cal A}_\ast$ which corresponds to the lift of $Y_\bullet$ to a simplicial object of $\widetilde{\A}$. Let $\widetilde{Y}_\bullet$ be the simplicial object of $\widetilde{\A}$ which is the lift of $Y_\bullet$. Choose a base point $y_n'$ of $\widetilde{Y}_n$ for each $n$, and let $y_n$ be its image in $Y_n$. Each strictly order preserving map $\phi : [m] \to [n]$ induces a morphism $A_\phi' : \widetilde{Y}_n \to \widetilde{Y}_m$ of $\widetilde{\A}$ which covers the face map $A_\phi : Y_n \to Y_m$. Since each $\widetilde{Y}_n$ is connected and simply connected, there is a unique homotopy class of paths in $\widetilde{Y}_m$ from $y_m'$ to $A_\phi'(y_n')$. Its image in $Y_m$ is a distinguished homotopy class of paths $\gamma_\phi$ in $Y_m$ from $y_m$ to $A_\phi(y_n)$. The pair $(A_\phi,\gamma_\phi)$ is a morphism $(Y_n,y_n) \to (Y_m,y_m)$ in the category ${\cal A}_\ast$ and the collection $(Y_n,y_n)$ of pointed spaces together with the maps $(A_\phi,\gamma_\phi)$ is a simplicial object of ${\cal A}_\ast$. We now use this to construct a lift of $X_\bullet$ to ${\cal A}_\ast$. Let $x_n = f(y_n)$. For each order preserving injection $\phi : [m] \to [n]$, let $\mu_\phi$ be the homotopy class $f\cdot \gamma_\phi$ of paths in $X_m$ from $x_m$ to $A_\phi(x_n)$. The collection of pointed spaces $(X_n,x_n)$ together with the pairs $(A_\phi,\mu_\phi)$ is easily seen to be a simplicial object of ${\cal A}_\ast$. Take $\widetilde{X}_n$ to be the standard model of the universal covering space of $(X_n,x_n)$ --- it consists of homotopy classes $\rho$ of paths that emanate from $x_n$. The face maps $A_\phi$ lift to face maps $A_\phi'$ by defining $A_\phi'(\rho)$ to be the homotopy class of paths $\mu_\phi\cdot \rho$ in $\widetilde{X}_m$. This is a simplicial object of $\widetilde{\A}$ which lifts $X_\bullet$. \end{pf} \begin{corollary}\label{top-lift} Suppose that $Y_\bullet$ and $Z_\bullet$ are simplicial topological spaces where each $Y_n$ and $Z_n$ is path connected. If $f : Y_\bullet \to Z_\bullet$ is a morphism of simplicial spaces, and if each simplex $Y_n$ of $Y_\bullet$ is simply connected, then $Z_\bullet$ has a canonical lift to $\widetilde{\text{Top}}$ such that $f$ is a morphism of $\widetilde{\text{Top}}$.\qed \end{corollary} The following result is needed in the proof of the theorem. \begin{lemma}\label{poly-lem} Suppose that $f \in {\Bbb R}[t_1,\dots,t_n]$. If $f\neq 0$, there is a real number $K > 1$ such that $f$ is bounded away from zero in the region $$ D_n(K) := \left\{(t_1,\dots,t_n) : t_1 \ge K, t_2 \ge Ke^{t_1}, \dots t_n \ge Ke^{t_{n-1}}\right\}. $$ \end{lemma} \begin{pf} The proof is by induction on the number of variables $n$. The result is trivially true when $n=1$. Now suppose that $n>1$ and that the result has been proved for polynomials with fewer than $n$ variables. Set $x = (t_1,\dots, t_{n-1})$ and $y = t_n$. We can write \begin{equation}\label{poly} f = a_d(x)y^d + a_{d-1}(x)y^{d-1} + \dots + a_1(x)y + a_0(x) \end{equation} where each $a_j(x) \in {\Bbb R}[t_1,\dots,t_{n-1}]$ and $a_d\neq 0$. If $d=0$, then we are in the previous case and the result holds by induction. So assume that $d>0$. By induction, there exist real constants $C>0$ and $K>1$ such that $ |a_d(x)| \ge C $ for all $x\in D_{n-1}(K)$. By a standard estimate, the roots $\theta(x)$ of the polynomial (\ref{poly}) satisfy $$ |\theta(x)| \le 1 + \max_{0 \le j < d} \left|{a_j(x) \over a_d(x)}\right| \le 1 + \max_{0 \le j < d} {|a_j(x)| \over C} \le \|x\|^l $$ for some positive integer $l$ and for each $x \in D_{n-1}(K)$. Observe that if $(t_1,\dots,t_{n-1}) \in D_{n-1}(K)$, then $$ 1 < K \le t_1 \le t_2 \le \dots \le t_{n-1} $$ so that $\|x\| \le t_{n-1}$ when $x\in D_{n-1}(K)$. It follows from the previous inequality that $$ 1 + |\theta(x)| < e^{t_{n-1}} \le y $$ provided that $t_{n-1}$ is sufficiently large, which can be arranged by increasing $K$ if necessary. Since $$ f(x,y) = \pm\, a_d(x) \prod_{j=1}^d (y-\theta_j(x)), $$ it follows that if $(x,y) \in D_n(K)$, then $|f(x,y)| \ge C$. \end{pf} \begin{pf*}{Proof of Theorem \ref{lift}} We first give a brief proof of (\ref{lift}) in the case when $U^p_\bullet = G^p_\bullet$. For each $q$, the subset $$ \Delta^p_q := \left\{(t_0,\ldots,t_{p+q}) : t_i \in {\Bbb R}, 0 < t_0 < t_1 < \cdots < t_{p+q}\right\} $$ of $X^p_q({\Bbb R})$ is contractible. Moreover, each of the face maps $A_i$ maps $\Delta^p_q$ into $\Delta^p_{q-1}$. It follows that we have morphisms $$ \Delta^p_\bullet \hookrightarrow X^p_\bullet \to G^p_\bullet $$ of truncated simplicial spaces. Since each $\Delta^p_q$ is contractible, $\Delta^p_\bullet$ has a unique lift to a simplicial object of the category $\widetilde{\text{Top}}$. If follows from (\ref{top-lift}) that both $X^p_\bullet$ and $G^p_\bullet$ have lifts to $\widetilde{\text{Top}}$, and therefore to $\widetilde{\A}$. The strategy in the general case is similar. We seek a simplicial space $D_\bullet$, each of whose simplices is contractible, which maps to $U^p_\bullet$. It follows from (\ref{dense}) that the pullback of $U^p_q$ to $X^p_q$ is a proper open subvariety$V^p_q$ of $X^p_q$. By standard arguments, there is a non zero polynomial $f_q\in {\Bbb R}[t_0,\dots, t_{p+q}]$ such that $$ X^p_q - f_q^{-1}(0) \subseteq V^p_q. $$ It follows from (\ref{poly-lem}) that there is a real number $K_q > 1$ such that $$ D^p_q(K_q):= \left\{(t_0,\dots,t_{p+q}) \in X^p_q({\Bbb R}) : t_0 \ge K_q, t_j \ge K_qe^{t_{j-1}} \text{ when } j\ge 1 \right\} \subset V^p_q. $$ Let $$ K = \max_{0\le q \le p} K_q. $$ Then $D^p_q(K) \subset V^p_q$. It is not difficult to show that $A_i(D^p_q(K)) \subseteq D^p_{q-1}$ for each $i$. It follows that the $D^p_q(K)$, with $0\le q \le p$, form a truncated simplicial space $D^p_\bullet(K)$ which maps to $V^p_\bullet$, and therefore to $U^p_\bullet$. It is not difficult to show that each $D^p_q(K)$ is contractible. It follows from (\ref{top-lift}) that $U^p_q$ lifts to a simplicial object of $\widetilde{\A}$. \end{pf*} \section{Proof of \ref{dunno}\label{pf_dunno}} We only give a detailed sketch of the proof. First we prove the result when $X_\bullet$ is replaced by a single space. Denote the Malcev Lie algebra associated to the pointed space $(Y,y)$ by ${\frak{p}}(Y,y)$. Now suppose that $Y$ is a complex algebraic manifold. Recall from \cite[p.~470]{hain-macp} that the multivalued Deligne cohomology of the object $(Y,y)$ of the category $\widetilde{\A}$ is defined to be the cohomology of the complex\footnote{Note that there is a typo in the definition of $MD(X,{\Bbb Q}(p))$ in \cite[p.~470]{hain-macp} --- one should quotient out by $F^p{\cal C}({\frak{g}},{\widetilde{\Omega}}^\bullet)$ as defined on op cit, p.~469 and not just by $F^p{\widetilde{\Omega}}^\bullet$.} \begin{multline*} MD(Y,{\Bbb Q}(p)) := \cone\Bigl( W_{2p} \Hom^{cts}_{\Bbb Q}(\Lambda^\bullet {\frak{p}}(Y,y),{\Bbb Q}) \oplus \\ W_{2p}\Hom^{cts}_{\Bbb C}(\Lambda^\bullet {\frak{p}}(Y,y),F^p{\widetilde{\Omega}}^\bullet(Y,y)) \to W_{2p}\Hom^{cts}_{\Bbb C}(\Lambda^\bullet {\frak{p}}(Y,y),{\widetilde{\Omega}}^\bullet(Y,y)) \Bigr)[-1]. \end{multline*} We shall need a ${\Bbb Q}$ analogue of $\Hom^{cts}_{\Bbb C}(\Lambda^\bullet {\frak{p}}(Y,y),{\widetilde{\O}}(Y))$. This will be constructed using continuous cohomology of certain path spaces. The space of paths in a topological space $Y$ which go from $y\in Y$ to $z\in Y$ will be denoted by $P_{y,z}Y$. The homology group $H_0(P_{y,z}Y,{\Bbb Q})$ has a natural topology which agrees with the filtration of $H_0(P_{y,y}Y,{\Bbb Q})\approx{\Bbb Q}\pi_1(Y,y)$ by powers of its augmentation ideal when $y=z$ (cf. \cite[\S 3]{hain-zucker}). Denote the continuous dual of $H_0(P_{y,z}Y,{\Bbb Q})$ by $H_{cts}^0(P_{y,z}Y,{\Bbb Q})$. These groups fit together to form a local system over $Y\times Y$ whose fiber over $(y,z)$ is $H_{cts}^0(P_{y,z}Y,{\Bbb Q})$. It is a direct limit of unipotent local systems over $Y\times Y$ and a direct limit of unipotent variations of mixed Hodge structure when $Y$ is a smooth algebraic variety \cite{hain-zucker}. There are two natural inclusions of $\Hom^{cts}_{\Bbb Q}(\Lambda^\bullet {\frak{p}}(Y,a),{\Bbb Q})$, ($a=y,z$), into $$ \Hom^{cts}_{\Bbb Q} (\Lambda^\bullet {\frak{p}}(Y,y) \otimes_{\Bbb Q} \Lambda^\bullet {\frak{p}}(Y,z),H_{cts}^0(P_{y,z}Y)). $$ They are induced by the two projections of $(Y,y)\times (Y,z)$ onto $(Y,a)$ and by the inclusion of the constants into $H_{cts}^0(P_{y,z}Y)$. We shall denote them by $\phi_1$ and $\phi_2$, respectively. \begin{proposition} If $H_1(Y,{\Bbb Q})$ is finite dimensional, then $\phi_1$ and $\phi_2$ are both quasi-isomorphisms. \end{proposition} \begin{pf} We prove the result for $\phi_2$, the other case being similar. By a standard spectral sequence argument, it suffices to show that $$ \Hom^{cts}_{\Bbb Q}(\Lambda^\bullet {\frak{p}}(Y,y), H_{cts}^0(P_{y,z}Y)) $$ is acyclic. We may, without loss of generality, take $y=z$. Because $H_1(Y)$ is finite dimensional, each graded quotient of the topology on $H_0(P_{y,y}Y,{\Bbb Q})$ is finite dimensional, and it follows that the dual of $H_{cts}^0(P_{y,y}Y),{\Bbb Q})$ is isomorphic to the completion of $H_0(P_{y,y}Y,{\Bbb Q})\approx {\Bbb Q}\pi_1(Y,y)$. This, in turn, is isomorphic to the completion of $U{\frak{p}}(Y,y)$. So there is a natural isomorphism $$ \Hom^{cts}_{\Bbb Q}(\Lambda^\bullet{\frak{p}}(Y,y),H_{cts}^0(P_{y,y}Y)) \approx \Hom^{cts}_{\Bbb Q}(\Lambda^\bullet{\frak{p}}(Y,y) \comptensor U{\frak{p}}(Y,y),{\Bbb Q}) $$ of chain complexes. This last complex is acyclic, as it is the continuous dual of an acyclic complex (cf.\ \cite[(3.9)]{hain:cycles}). \end{pf} The following result is a straightforward refinement of the previous result. \begin{proposition}\label{qism} If $Y$ is a smooth algebraic variety, then each of the complexes in the previous result is a a complex of mixed Hodge structures, and the two natural inclusions $\phi_1$ and $\phi_2$ of $\Hom^{cts}_{\Bbb Q}(\Lambda^\bullet {\frak{p}}(Y,a),{\Bbb Q})$ ($a=y,z$) into $$ \Hom^{cts}_{\Bbb Q}(\Lambda^\bullet {\frak{p}}(Y,y) \otimes_{\Bbb Q} \Lambda^\bullet {\frak{p}}(Y,z), H_{cts}^0(P_{y,z}Y)) $$ are quasi-isomorphisms in the category of complexes of mixed Hodge structures. \qed \end{proposition} Next, observe that each $F \in{\widetilde{\O}}(Y,y)$ induces a linear map $$ H_0(P_{y,z}Y) \to {\Bbb C} $$ by taking the path $\gamma$ to the difference $F(z) - F(y)$ where the branch of $F$ at $z$ is obtained by analytically continuing $F$ along $\gamma$. It follows from standard properties of iterated integrals that this map is continuous. Consequently, we obtain a linear map $$ {\widetilde{\O}}(Y,y) \to H_{cts}^0(P_{y,z}Y,{\Bbb C}). $$ It follows from \cite[\S 3]{hain-macp} and Chen's de Rham Theorem for the fundamental group that when $q(Y)=0$, this map is an isomorphism of $W_\bullet$ filtered vector spaces. This isomorphism is $\pi_1(Y,y)$-equivariant with respect to the standard actions of $\pi_1(Y,y)$ on ${\widetilde{\O}}(Y,y)$ and $H_{cts}^0(P_{y,z}Y,{\Bbb C})$. Recall that there is a natural homomorphism $$ \theta : \Hom^{cts}_{\Bbb C}(\Lambda^\bullet {\frak{p}}(Y,y), {\Bbb C}) \to \Omega^\bullet(Y) $$ of $W_\bullet$ filtered d.g.\ algebras \cite[(7.7)]{hain-macp}. Fix a point $z$ of $Y$. Consider the complex \begin{multline*} \cone\Bigl(W_{2p} \Hom^{cts}_{\Bbb Q}\left(\Lambda^\bullet {\frak{p}}(Y,y) \otimes_{\Bbb Q} \Lambda^\bullet {\frak{p}}(Y,z), H_{cts}^0(P_{y,z}Y,{\Bbb Q})\right) \oplus \\ W_{2p}\Hom^{cts}_{\Bbb C}(\Lambda^\bullet {\frak{p}}(Y,y),F^p{\widetilde{\Omega}}^\bullet(Y,y)) \to W_{2p}\Hom^{cts}_{\Bbb C}(\Lambda^\bullet {\frak{p}}(Y,y),{\widetilde{\Omega}}^\bullet(Y,y)) \Bigr)[-1]. \end{multline*} Here $$ \Hom^{cts}_{\Bbb Q}(\Lambda^\bullet {\frak{p}}(Y,y) \otimes_{\Bbb Q} \Lambda^\bullet {\frak{p}}(Y,y), H_{cts}^0(P_{y,z}Y,{\Bbb Q})) $$ is mapped into $$ W_{2p}\Hom^{cts}_{\Bbb C}(\Lambda^\bullet {\frak{p}}(Y,y),{\widetilde{\O}}(Y,y)\otimes \Omega^\bullet(Y)) $$ using the identification of $H_{cts}^0(P_{y,z}Y,{\Bbb C}))$ with ${\widetilde{\O}}(Y,y)$ and the map $\theta$ in the second factor. It is straightforward to check it is a chain map. Define a map from this complex to $MD(Y,{\Bbb Q}(p))$ by defining it to be $\phi_1$ on the first factor and the identity on the other two factors. Since $\phi_1$ is a $W_\bullet$ filtered quasi-isomorphism, this map is a quasi-isomorphism. Next, Define $MD'(X,{\Bbb Q}(p))$ to be the complex $$ \cone\Bigl(W_{2p}\Hom^{cts}_{\Bbb Q}(\Lambda^\bullet {\frak{p}}(Y,y),{\Bbb Q}) \oplus F^p W_{2p}\Omega^\bullet(Y)) \to W_{2p}\Omega^\bullet(Y)) \Bigr)[-1], $$ where the map $\Hom^{cts}_{\Bbb Q}(\Lambda^\bullet {\frak{p}}(Y,y),{\Bbb Q}) \to \Omega^\bullet(Y)$ is induced by $\theta$. It can be mapped to the previous complex using $\phi_2$ on the first factor and the obvious inclusions on the other two factors. Since $$ \Hom^{cts}_{\Bbb C}(\Lambda^\bullet {\frak{p}}(Y,y),{\widetilde{\O}}^\bullet(Y,y)) $$ is an acyclic complex of mixed Hodge structures, it follows that this map is also a quasi-isomorphism. That is, we can equally well compute $H_{\calM\D}^\bullet(Y,{\Bbb Q}(p))$ using the complex $MD'(Y,{\Bbb Q}(p))$. A map of $MD'(Y,{\Bbb Q}(p))$ into a standard complex that computes $H_{\D}^\bullet(Y,{\Bbb Q}(p))$ can now be constructed using the techniques of the proof of \cite[(11.7)]{carlson-hain}. Taking homology, we obtain a map $$ \psi : H_{\calM\D}^\bullet(Y,{\Bbb Q}(p)) \to H_{\D}^\bullet(Y,{\Bbb Q}(p)) $$ for all smooth varieties. \begin{proposition}\label{iso} If $Y$ is a rational $K(\pi,1)$, then $\psi$ is an isomorphism. \end{proposition} \begin{pf} The homology of the complex $\Hom^{cts}_{\Bbb Q}(\Lambda^\bullet {\frak{p}}(Y,y),{\Bbb Q})$ is the continuous cohomology $H_{cts}^\bullet({\frak{p}}(Y,y))$ of the Lie algebra ${\frak{p}}(Y,y)$. The natural map $$ H_{cts}^\bullet({\frak{p}}(Y,y)) \to H^\bullet(Y,{\Bbb Q}), $$ is a morphism of mixed Hodge structures \cite[(11.7)]{carlson-hain}. Since the multivalued Deligne cohomology is constructed as a cone, we have a long exact sequence $$ \cdots \to W_{2p}H_{cts}^{k-1}(Y,{\Bbb Q}) \oplus F^pW_{2p}\Omega^{k-1}(Y) \stackrel{\theta - i}{\longrightarrow} W_{2p}\Omega^{k-1} (Y) \to H_{\calM\D}^k(Y,{\Bbb Q}(p)) \to \cdots $$ where $i$ denotes the inclusion of $F^p\Omega^\bullet$ into $\Omega^\bullet$. The map $\psi$ induces a map from this long exact sequence into the standard long exact sequence $$ \cdots \to W_{2p}H^{k-1}(Y,{\Bbb Q}) \oplus F^pW_{2p}H^{k-1}(Y) \to W_{2p}H^{k-1}(Y,{\Bbb C}) \to H_{\D}^k(Y,{\Bbb Q}(p)) \to \cdots $$ When $Y$ is a rational $K(\pi,1)$, each of the maps $$ H_{cts}^\bullet(Y,{\Bbb Q}) \stackrel{\theta}{\to} \Omega^\bullet(Y) \to H^\bullet(Y,{\Bbb C}) $$ is a $(W_\bullet,F^\bullet)$ bifiltered quasi-isomorphism \cite[(8.2)(i),(iii)]{hain-macp}. The result now follows using the 5-lemma. \end{pf} One can take $y=z$ in each of the chain maps above. If one does this, the assignment of each of these complexes to an object of the category ${\cal A}_\ast$ defined in \cite[\S 2]{hain-macp} is a functor. Suppose that $X_\bullet$ is a simplicial object of the category $\widetilde{\A}$. Choose a base point $x_n$ of each $X_n$; $X_\bullet$ now determines a simplicial object of the category ${\cal A}_\ast$, and we may apply any of the functors above to $X_\bullet$ to obtain a double complex. Using standard arguments, we see that the total complex associated to the double complex $MD'(X_\bullet,{\Bbb Q}(p))$ computed the multivalued Deligne cohomology of $X_\bullet$ and that there is a map $$ \Psi : H_{\calM\D}^\bullet(X_\bullet,{\Bbb Q}(p)) \to H_{\D}^\bullet(X_\bullet,{\Bbb Q}(p)). $$ When each $X_n$ is a rational $K(\pi,1)$, it is not difficult to show, using an argument similar to the proof of (\ref{iso}) and the skeleton filtration, that $\Psi$ is an isomorphism. \section{Higher Logarithms and Extensions of Tate Variations} \label{conclusion} The second remark is that the higher logarithms we have constructed generically on $G^p_\bullet$ are related to extensions of (Tate) variations of mixed Hodge structures. Indeed, by \cite[(12.1)]{carlson-hain} and \cite[(8.6)]{hain:cycles}, if a space $X$ is a rational $K(\pi,1)$ with $q(X)=0$, then there are natural isomorphisms $$ H_{\D}^\bullet(X,{\Bbb Q}(p)) \approx \Ext^\bullet_{{\cal H}(X)}({\Bbb Q},{\Bbb Q}(p)) \approx \Ext^\bullet_{{\cal T}(X)}({\Bbb Q},{\Bbb Q}(p)) $$ where ${\cal H}(X)$ and ${\cal T}(X)$ denote the categories of unipotent variations of mixed Hodge structure over $X$ and Tate variations of mixed Hodge structure over $X$, respectively. Thus if $X_\bullet$ is a simplicial variety where each $X_n$ is a rational $K(\pi,1)$ with $q(X)=0$, then, in some sense, we may identify $H_{\calM\D}^\bullet(X_\bullet,{\Bbb Q}(p))$ with the ``hyper-ext" group of extensions of ${\Bbb Q}$ by ${\Bbb Q}(p)$ associated to $X_\bullet$ (cf. \cite[\S 10]{hain:cycles}).

9308

alg-geom/9308002

en

https://arxiv.org/abs/alg-geom/9308002

alg-geom/9308002

Elizondo Javier

Javier Elizondo

The Euler Series of Restricted Chow Varieties

15 pages, LaTex

Let X be an algebraic projective variety in {\bf P}^n. Denote by {\cal C}_{\lambda} the space of all effective cycles on X whose homology class is \lambda \in H_{2p} (X,{\bf Z}). It is easy to show that {\cal C}_{\lambda} is an algebraic projective variety. Let \chi ({\cal C}_{\lambda} be its Euler characteristic. Define the Euler series of X by E_{p} = \sum_{\lambda\in{C}} \, \chi({\cal C}_{\lambda} \lambda \in \, {\bf Z}[[C]] where {\bf Z}[[C]] is the full algebra over {\bf Z} of the monoid C of all homology classes of effective p-cyles on X. This algebra is the ring of function (with respect the convolution product) over C. Denote by {\bf Z}[C] the ring of functions with finite support on C. We say that an element of {\bf Z}[[C]] is rational if it is the quotient of two elements in {\bf Z}[C]. If a basis for homology is fixed we can associated to any rationa element a rational function and therefore compute the Euler characteristic of {\cal C}_{\lambda}. We prove that E_p is rational for any projective variety endowed with an algebraic torus action in such a way that there are finitely many irreducible invariant subvarieties. If it is smooth we also define the equivariant Euler series and proved it is rational, we relate both series and compute some classical examples. The projective space {\bf P}^n, the blow up of {\bf P}^n at a point, Hirzebruch surfaces, the product of {\bf P}^n with {\bf P}^m.

alg-geom

\section{The Euler Series} In \cite{la&yau-hosy} B.Lawson and S.S.Yau introduced a series that becomes a formal power series when a basis for homology is fixed. They proved it is a rational function for the cases of ${\bf P}^{n}$ and ${\bf P}^{n} \times {\bf P}^{m}.$ In this section we define a more general series and state the problem of its rationality in intrinsic terms for any algebraic projective variety. We follow an approach suggested by E.Bifet \cite{emili-cartas}. The definition of the monoid $C,$ rationality and lemma ~\ref{fibers} are taken from him. We start with some basic definitions and some of their properties. Throughout this article any algebraic projective variety $X$ comes with a fixed embedding $ X \stackrel{j}{\hookrightarrow} {\bf P}^{n}.$ An {\bf effective p-cycle} c on $X$ is a finite (formal) sum \mbox{$c=\sum n_{s} \, V_{s}$} where each $n_{s}$ is a non-negative integer and each $V_{s}$ is an irreducible \mbox{p-dimensional} subvariety of $X$. From now on, we shall use the term {\bf cycle} for {\bf effective cycle}. For any projective algebraic variety $ X \stackrel{j}{\hookrightarrow} {\bf P}^{n}$ we denote by ${\cal C}_{p,d} (X)$ the {\bf Chow variety} of $X$ of all cycles of dimension p and degree d in ${\bf P}^{n}$ with support on $X.$ By convention, we write ${\cal C}_{p,0} (X) \, = \, \{ \emptyset \}.$ Let $\lambda$ be an element in $H_{2p} (X,\bf {Z})$ and denote by ${\cal C}_{\lambda} (X)$ the space of all cycles on $X$ whose homology class is $\lambda.$ Note that ${\cal C}_{\lambda} (X)$ is contained in ${\cal C} _{p,d} (X),$ where $j_{\ast} \, \lambda \, = \, d[{\bf {P}}^{p}].$ \begin{lemma} \label{compacto} Let $\lambda$ be an element of $H_{2p}(X,{\bf Z})$, then ${\cal C}_{\lambda} (X)$ is a projective algebraic variety. \end{lemma} {\bf Proof:} Let us write ${\cal C} _{p,d} (X)= \cup _{i=1}^{M} {\cal C} _{p,d}^{i} (X)$, where ${\cal C} _{p,d}^{i} (X)$ are its irreducible components. Suppose ${\cal C}_{\lambda} (X) \cap {\cal C} _{p,d}^{i_{o}} (X) \neq \emptyset.$ Any two cycles in ${\cal C} _{p,d}^{i_{o}}$ are algebraically equivalent, hence they represent the same element in homology. Therefore ${\cal C} _{p,d}^{i_{o}} (X) \subset {\cal C}_{\lambda} (X)$ for some $\lambda.$ Consequently ${\cal C}_{\lambda} (X)= \cup_{j=1}^{l} {\cal C} _{p,d}^{i_{j}} (X),$ where ${\cal C}_{\lambda} (X) \cap {\cal C} _{p,d}^{i_{j}} (X) \neq \emptyset$ for $j \, = \, 1, \ldots ,l .$ \hfill \rule{.5em}{1em} For an effective $p$-cycle $c \,=\, \sum \, n_i \, V_i,$ we denote by $[c]$ its homology class in $H_{2p}\,(X,{\bf Z}).$ Now, let $C$ be {\bf the monoid of homology classes of effective p-cycles} in $H_{2p}\,(X,{\bf Z}),$ and let ${\bf Z}[[C]]$ be the {\bf set of all integer valued functions on $C$}. We shall write the elements of ${\bf Z}[[C]]$ as $$ \sum_{\lambda\in{C}} \;a_{\lambda}\, \lambda \:\:\:\:\:\mbox{where }\,\,a_{\lambda} \, \in \, {\bf Z}. $$ The following lemma allows us to prove that ${\bf Z}[[C]]$ is a ring, \begin{lemma} \label{fibers} Let $C$ be the monoid of homology classes of effective p-cycles on $X.$ \,Then $$ +: \,C \, \times \, C \longrightarrow \, C $$ has finite fibers. \end{lemma} {\bf Proof:} If $X \, = \, {\bf P}^{n}$ the result is obvious since $C$ is isomorphic to ${\bf N}.$ Let $$ j_{\ast} :\, H_{2p}\,(X,{\bf Z}) \, \longrightarrow \, H_{2p}\,({\bf P}^{n},{\bf Z}) $$ be the homomorphism induced by the embedding $ X \, \stackrel{j}{\hookrightarrow} \, {\bf P}^{n}. $\, Denote by $C^{\prime} \, \cong \, {\bf N}$ the monoid of homology classes of p-cycles on ${\bf P}^{n}.$ It follows from the proof of lemma ~\ref{compacto} that $$ j_{\ast}| _{\scriptstyle C} :\, C \, \longrightarrow \, C^{\prime} $$ has finite fibers. Finally, the lemma follows from the commutative diagram below $$ \begin{array}{ccccc} && {\scriptstyle +} && \\ & C\times C & \longrightarrow & C & \\ {j_{\ast}\times j_{\ast} \,|}_{\scriptstyle {C\times C} }& \downarrow && \downarrow & {j_{\ast}\,|}_{\scriptstyle C} \\ & C^{\prime}\times C^{\prime} & \to & C^{\prime} & \\ && {\scriptstyle +} && \end{array} $$ \hfill \rule{.5em}{1em} \\ Directly from this lemma we obtain: \begin{proposition} \label{ring} Let $C$ and $Z[[C]]$ defined as above. Then $Z[[C]]$ is a ring under the convolution product, i.e. $$ (f \cdot g) \,(\lambda ) \,= \, \sum_{\lambda={\mu_{1}}+{\mu_{2}}} \,f(\mu_{1}) g(\mu_{2}). $$ \end{proposition} We are ready for the following definition. \begin{definition} \label{ec} Let $X$ be a projective algebraic variety. The {\bf Euler series} of $X,$ in dimension p, is the element $$ E_{p} \, = \sum_{\lambda} \chi \, ({\cal C}_{\lambda}) \, \lambda \, \, \, \; \in \, \, \, \; {\bf {Z}}[[C]] \; \; $$ where ${\cal C}_{\lambda}(X)$ is the space of all effective cycles on $X$ with homology class $\lambda,$ and $\chi \, ({\cal C}_{\lambda}(X))$ is the Euler characteristic of ${\cal C}_{\lambda}$. \end{definition} By convention, if ${\cal C}_{\lambda}$ is the empty set then its Euler characteristic is zero. Let ${\bf Z}[C]$ be the monoid-ring of $C$ over ${\bf Z}.$ This ring consists of all elements of ${\bf Z}[[C]]$ with finite support. We arrive at the following definition, \begin{definition} \label{rational} We say that an element of ${\bf Z}[[C]]$ is rational if it is the quotient of two elements of ${\bf Z}[C].$ \end{definition} {\bf Remark:} Denote by $H$ the homology group $H_{2p}(X,{\bf Z})$ together with a fixed basis $\cal A.$ Consider $H$ as a multiplicative group in the standard way and suppose that $C$ is isomorphic to the monoid of the natural numbers. Then it is easy to see that ${\bf Z}[[C]]$ is isomorphic to the ring of formal power series in as many variables as the rank of $H.$ Therefore ${\bf Z}[C]$ is the ring of polynomials and the last definition just says when a formal power series is a rational function. We are interested in the following problem.\newline {\bf Problem:} \label{problem} \, When is the {\bf Euler series} rational in the sense of the last definition? For cycles of dimension zero on a projective algebraic variety, we have directly from the computation in \cite{mcd-sym} that $$ E_{0} \, = \, \frac{1}{{\left( 1-t \right)}^{\chi (X)}} $$ The present article, in particular, recovers the result for $X\,=\,{\bf P}^{n}$ which was worked out in \cite{la&yau-hosy}. \section{Varieties with a Torus Action} Throughout this section $X$ means a projective algebraic variety, on which an algebraic torus $T$ acts linearly having only a finite number of invariant irreducible subvarieties of dimension p. In particular, we will see that the result is true for any projective toric variety. Let us denote by $H$ the homology group $H_{2p}\,(X,{\bf Z}).$ The action of $T$ on $X$ induces an action on the Chow variety ${\cal C} _{p,d} (X).$ Let $\lambda$ be an element in $H$ and denote by ${\cal C}_{\lambda}^{T}$ the fixed point set of ${\cal C} _{\lambda} (X)$ under the action of $T.$ Then its Euler characteristic $\chi \left( {\cal C}_{\lambda}^{T} \right)$ is equal to the number of invariant subvarieties of $X$ with homology class $\lambda.$ We have \begin{equation} \label{lawson} E_{p}(\lambda) \, = \, \chi \left( {\cal C}_{\lambda} \right) \, = \, \chi \left( {\cal C}_{\lambda}^{T} \right) \end{equation} where the first equality is just the definition of $E_{p}$ and the last one is proved in \cite{la&yau-hosy}. The following theorem tells us that $E_{p}$ is rational. \begin{theorem} \label{main} Let $E_{p}$ be the Euler series of $X.$ Denote by $V_1, \ldots , V_N$ the p-dimensional invariant irreducible subvarieties of $X.$ Let $e_{[V_i]} \in {\bf Z}[C]$ be the characteristic function of the subset $\{ [V_i] \}$ of $C.$ Then \begin{equation} \label{ecp} E_{p}=\, \prod_{1\leq i\leq N}\left( \frac{1}{1-e_{[V_i]}}\right) \end{equation} \end{theorem} {\bf Proof:} For each $V_i$ define $f_i$ in ${\bf Z}[[C]]$ by \begin{equation} \label{function} f_i(\lambda)=\left\{ \begin{array}{ll} 1 & \mbox{if $\lambda=n\cdot [V_i]$, $n\geq 0$} \\ 0 & \mbox{otherwise.}\end{array} \right. \end{equation} It is easy to see from equations (\ref{lawson}) and (\ref{function}) that $E_{p}$ can be written as \begin{equation} \label{ecr} E_{p} \, = \, \prod_{1\leq i\leq N} f_i. \end{equation} The theorem follows because of the equality $$ 1=\left( 1-e_{[{V}_{i}]} \right) \cdot f_i $$ \hfill \rule{.5em}{1em} Observe that if we fix a basis for $H$ modulo torsion, then the elements of ${\bf Z}[C]$ can be identified with some Laurent polynomials. Under this identification any rational element of ${\bf Z}[[C]]$ is a rational function. The next lemma tells us that the result is true for any projective toric variety. \begin{lemma} \label{ciclos} Let $X$ be a projective (perhaps singular) toric variety. Then any irreducible subvariety V of $ X$ which is invariant under the torus action is the closure of an orbit. Therefore, any invariant cycle has the form $$ c \, = \, \sum \, n_{i} \, \overline{\cal O}_{i} $$ where each $n_{i}$ is a nonnegative integer and each $\overline{\cal O}_{i}$ is the closure of the orbit ${\cal O}_{i}$. \end{lemma} {\bf Proof: } The fan $\Delta$ associated to $X$ is finite because $X$ is compact. Hence there is a finite number of cones, therefore a finite number of orbits. Let V be an invariant irreducible subvariety of $X.$ We can express V as the closure of the union of orbits. Since there is a finite number of them we must have that $$ V \, = \, {\overline{\cal O}}_{1} \, \cup \, {\overline{\cal O}}_{2} \, \cup \cdots \cup \, {\overline{\cal O}}_{N} $$ where $\overline{\cal O}_{i}$ is the closure of an orbit. Finally since V is irreducible, there must be $i_{0}$ such that $V \, = \, {\overline{\cal O}}_{{i}_{0}}$. \hfill \rule{.5em}{1em} \section{Smooth Toric Varieties} In this section we give an equivariant version of the Euler series and find a relation between the equivariant and not equivariant Euler series. The use of equivariant cohomology allows us to analize the Euler series from a geometrical point of view. This approach might help to understand other cases. Throughout this section, unless otherwise stated, $X$ is a smooth projective toric variety, and we use cohomology instead of homology by applying Poincar\'e duality. Let $H$ and $H_{T}$ be the cohomology group $H^{2p}(X,{\bf Z})$ and the equivariant cohomology group $H_{T}^{2p}(X,{\bf Z})$ of $X,$ respectively. Denote by $\Delta$ the fan associated with $X.$ Let ${\cal C}_{\lambda}^{T}$ and ${\cal C}_{\lambda}$ be the spaces of all p-dimensional effective invariant cycles and p-dimensional effective cycles on $X$ with cohomology class $\lambda.$ It is proved in \cite{la&yau-hosy} that \begin{equation} \label{blaine} \chi \left( {\cal C}_{\lambda}^{T} \right) \, = \, \chi \left( {\cal C}_{\lambda} \right). \end{equation} The next lemma is crucial for the following results, \begin{lemma} \label{finito} Let $\lambda$ be an element in $H.$ Then ${\cal C}_{\lambda}^{T}$ is a finite set. \end{lemma} {\bf Proof:} By lemma ~\ref{ciclos} we know that any invariant effective cycle $c$ in ${\cal C}_{\lambda}^{T}$ has the form $c \, = \, \sum_{i=1}^{N} \, {\beta}_{i} \, {\overline{\cal O}}_{i} $ with $\beta_{i} \in {\bf N},$ we obtain that ${\cal C}_{\lambda}^{T} $ has a countable number of elements. We know that ${\cal C}_{\lambda}$ is a projective algebraic variety (see lemma ~\ref{compacto}), and since ${\cal C}_{\lambda}^{T}$ is Zariski closed in ${\cal C}_{\lambda},$ we have that ${\cal C}_{\lambda}^{T}$ is a finite set. \hfill \rule{.5em}{1em} Our next step is to define the equivariant Euler series for $X.$ \\ Let ${\overline{\cal O}}$ be an irreducible invariant cycle in a smooth toric variety (lemma ~\ref{ciclos}). Since ${\overline{\cal O}} \subset X$ is smooth, we have an equivariant \mbox{Thom-Gysin} sequence $$ \cdots \longrightarrow H_{T}^{i-2cod\,\overline{\cal O}} (\overline{\cal O}) \longrightarrow \, H_{T}^{i} \, (X) \longrightarrow \, H_{T}^{i} \,(X-\overline{\cal O}) \longrightarrow \, \cdots $$ and we define ${[\overline{\cal O}]}_{T}$ as the image of 1 under $$ H_{T}^{0} \, (\overline{\cal O}) \longrightarrow H_{T}^{2cod\,\overline{\cal O}} \, (X). $$ Let $\{ D_{1}, \ldots, D_{K} \}$ be the set of T-invariant divisors on $X.$ To each $D_{i}$ we associate the variable \, $t_{i}$ \, in the polynomial ring \,${\bf Z}[t_{1}, \ldots, t_{K}].$\, Let $\cal I$ be the ideal generated by the (square free) monomials $\{ t_{{i}_{1}} \cdots t_{{i}_{l}} | \, D_{{i}_{1}} + \cdots + D_{{i}_{l}} \not\in \, \Delta \}.$ It is proved in \cite{bcp-regular} that \begin{equation} \label{eqring} {\bf Z}[t_{1}, \ldots, t_{K}] / {\cal I} \, \cong \, H_{T}^{\ast} (X, \, {\bf Z}) \end{equation} The arguments given there also prove the following. \begin{proposition} \label{independent} For any T-orbit $\cal O$ in a smooth projective toric variety $X$, one has $$ [\overline{\cal O}]_{T} \, = \, \prod_{{\overline{\cal O}} \subset{D_{i}}} \, [D_{i}]_{T} . $$ Furthermore if $\cal O$ and ${\cal O}^{\prime}$ are distinct orbits, then $$ [{\overline{\cal O}}]_{T} \, \not= \, [\overline{{\cal O}^{\prime}}]_{T}. $$ \end{proposition} It is natural to define the cohomology class for any effective invariant cycle \ $V = \sum \, m_{i} \, {\overline{\cal O}}_{i}$ \ as \ ${[V]}_{T} = \sum_{i} \, m_{i} \, {[{\overline{\cal O}}_{i}]}_{T}.$ where ${\cal O}_{i} \not= \, {\cal O}_{j}$\ if $i\not= j$ In a similar form as we define $C,$ ${\bf Z}[[C]]$ and ${\bf Z}[C],$ we denote by $C_{T}$ the monoid of equivariant cohomology classes of invariant effective cycles of codimension p, by ${\bf Z}[[C_{T}]]$ the set of functions on $C_{T},$ and by ${\bf Z}[C_{T}]$ the set of functions with finite support on $C_{T}.$ Since $C_{T} \simeq {\bf N}^{N}$ where $N$ is the number of orbits of codimension p, we obtain that $$ +: \, C_{T} \times C_{T} \, \longrightarrow \, C_{T} $$ has finite fibers. Observe that if $\pi : H_{T} \rightarrow H$ denotes the standard surjection, we obtain from lemma ~\ref{finito} that $$ \pi : C_{T} \longrightarrow C $$ is onto with finite fibers. We arrive at the following definition: \begin{definition} \label{eec} Let $X$ be a smooth projective toric variety and let $H_{T} \, (X)$ be the equivariant cohomology of $X$. Let us denote by ${\cal C}^{T}_{\xi}$ the space of all invariant effective cycles on $X$ whose equivariant cohomology class is $\xi .$ The {\bf equivariant Euler series} of $X$ is the element $$ E_{p}^{T} = \sum_{\xi} \, \chi \left( {\cal C}^{T}_{\xi} \right)\, \xi \; \; \; \in \; \; {\bf Z}[[C_{T}]] $$ where the sum is over $C_{T}.$ \end{definition} Let us define the ring homomorphism $$ J: \, {\bf Z}[[C_{T}]] \, \longrightarrow \, {\bf Z}[[C]] $$ by $$ J(\xi) \, = \, \sum_{\lambda} \left( \sum_{\pi(\beta)=\lambda} \, a_{\beta} \right) \lambda $$ where $\xi \, = \, \sum_{\beta} \, a_{\beta}\, \beta.$ This is well defined since $\pi$ has finite fibers. \begin{theorem} \label{ec-eec} Let $X$ be a smooth projective toric variety. Denote by $E_{p}$, $E_{p}^{T}$ and $J$ the Euler series, the equivariant Euler series and the ring homomorphism defined above. Then $ J \left( E_{p}^{T} \right) \, = \, E_{p}.$ Furthermore, $$ E_{p}^{T} \,=\,\prod_{1\leq{i}\leq{N}} \, \left( \frac{1}{1-e_{[{\overline{\cal O}}_i]_{T}}} \right) $$ and therefore $$ E_{p} \,=\,\prod_{1\leq{i}\leq{N}} \, \left( \frac{1}{1-e_{[{\overline{\cal O}}_i]}} \right) $$ \end{theorem} {\bf Proof: } We define for each orbit ${\cal O}_{i}$ an element $ f_{i}^{T} \in {\bf Z}[[C_{T}]]$ by \begin{equation} \label{ec.e} f^T_i(\xi)=\left\{ \begin{array}{ll} 1 & \mbox{if $\xi=n\cdot [{\overline {\cal O}_i}]_T$, $n\geq 0$} \\ 0 & \mbox{otherwise.\mbox{ } } \end{array} \right. \end{equation} and denote by $e_{\xi}$ the characteristic function of $\{\xi\}.$ It follows from both definition (\ref{eec}) and equation (\ref{ec.e}) that $$ E_{p}^{T} \, = \, \prod_{1\leq{i}\leq{N}} \, f_{i}^{T} \, $$ and $$ \left( 1-e_{[{\overline {\cal O}_{i}}]_T} \right) \cdot f_{i}^{T} \, = \, 1. $$ Therefore the equivariant Euler series is rational and \begin{equation} \label{rationale} E_{p}^{T} \, = \,\prod_{1\leq{i}\leq{N}} \left( \frac{1}{1-e_{[{\overline{\cal O}_{i}}]_{T}}} \right). \end{equation} For each $V_i$ we defined (see equation \ref{function}) a function $f_i$ in ${\bf Z}[[C]]$ by $$ f_i(\lambda)=\left\{ \begin{array}{ll} 1 & \mbox{if $\lambda=n\cdot [V_i]$, $n\geq 0$} \\ 0 & \mbox{otherwise.}\end{array} \right. $$ And we know from theorem ~\ref{main} that $$ E_p \, = \, \prod_{i=1}^{N} \, f_i \mbox{ \, \, \, \, \, \, with \, \, \, \, \, \,} f_i \cdot \left( 1-e_{[V_{i}]} \right) \, = \, 1. $$ Now, the result follows since $\pi ([{\overline{\cal O}}_i]_{T}) \, = \, [{\overline{\cal O}}_{i}]$ and $J$ is a ring homomorphism satisfying $$ J \left( f_{i}^{T} \right) \,=\, f_{i} \mbox{ \, \, \, \, \, \, and \, \, \, \, \, \, } J \left( e_{\sigma}\right) \,=\, e_{\pi(\sigma)}. $$ \hfill \rule{.5em}{1em} \section{Some Examples} \noindent {\bf I) The projective space ${\bf P}^{n}$} Let \, $X = {\bf P}^{n}$ \, be the complex projective space of dimension n. Let $\{e_{1},\ldots,e_{n}\}$ be the standard basis for ${\bf R}^{n}$. Consider $A \, = \, \{e_{1},\ldots,e_{n+1}\}$ a set of generators of the fan $\Delta$ \, where $e_{n+1} = - \sum_{i=1}^{n} e_{i}$. We have the following equality $$ H^{\ast}(X, \, {\bf Z}) \, \cong \, {\bf Z} \, [t_{1}, \ldots, t_{n+1}] \, /I $$ where I is the ideal generated by $$ i) \; \; \; t_{1} \cdots t_{n+1} $$ and $$ ii) \; \; \; \sum_{j=1}^{n+1} \, e_{i}^{\ast} (e_{j}) \, t_{j} \; \;\; \mbox{\, \, for\, \, }i\,=\,1,\ldots,n \, , $$ where $e_{i}^{\ast} \, \in \, ({\bf R}^{n})^{\ast}$ is the element dual to $e_{i}$.\\ However $ii)$ says that $\; t_{i} \sim t_{j} \; $ for all. Therefore $$ H^{\ast} \, (X, \, {\bf Z}) \, = \, {\bf Z} \, [t] \, / t^{n+1}. $$ Consequently, any two cones of dimension p represent the same element in cohomology, and Theorem ~\ref{ec-eec} implies that $$ {\displaystyle \prod_{i=1}^{(_{\; \; \, p}^{n+1})} {\left( \frac{1}{1-t} \right)} \, = \, \left( \frac{1}{1-t} \right)^{(_{\; \; \; p}^{n+1})} \, = \, E_{p}}. $$ \noindent {\bf II) $P^{n} \times P^{m}$} Recall that $X( \Delta \times {\Delta}^{\prime}) \, \cong \, X(\Delta) \times X({\Delta}^{\prime})$. Using the same notation as in example I, we have that a set of generators of $\Delta \times \Delta '$ is given by $$ \{e_{1}, \, \ldots, \, e_{n}, e_{n+1}, \ldots, e_{n+m}, e_{n+m+1}, e_{n+m+2}\} $$ with $e_{n+m+1} \, = \, - \sum_{i=1}^{n} \, e_{i}$ and $e_{n+m+2} \, = \, - \sum_{i=n+1}^{n+m} \, e_{i}.$ and $\{e_{1}, \ldots, e_{n+m}\}$ is a basis for $P^{n} \times P^{m}$. Then $$ H^{\ast} \, (X, \, {\bf Z}) \, = \, {\bf Z} \, [t_{1}, \ldots , t_{n+m+2}] \, / \, I $$ where I is the ideal generated by $$ i)\ \ \{t_{1} \cdots t_{n} t_{n+m+1}, \; \; t_{n+1} \cdots t_{n+m}t_{n+m+2}, \; \; \prod_{i=1}^{n+m+2} t_{i} \} $$ and $$ ii)\ \ \sum_{j=1}^{n+m+2} \, e_{i}^{\ast} (e_{j}) \, t_{j}\ \; \; \; \; i\, = \, 1,\ldots, n+m \, . $$ {}From $ii)$ we obtain, \begin{eqnarray*} t_{i} \, \sim \, t_{n+m+1} & & \mbox{if \ \ $1 \leq i \leq n $} \\ t_{j} \, \sim \, t_{n+m+2} & & \mbox{if \ \ $n+1 \leq j \leq n+m $} \end{eqnarray*} The number of cones of dimension p is equal to $\sum_{k+l=p} \, (_{\; \; \; k}^{n+1}) \,(_{\; \; \; l}^{m+1})$. \\ Denote by ${\displaystyle t_{k,l} \, = \, t_{{i}_{1}} \cdots t_{{i}_{k}} t_{{j}_{1}} \cdots t_{{j}_{l}}}.$ Then $$ \prod_{k+l=p} {\left( \frac{1}{1-t_{k,l}} \right)}^{(_{\; \; \,k}^{n+1}) \, (_{\;\;\,l}^{m+1})} \, = \, E_{p} $$ where $t_{k,l} \, = \, t_{n+m+1}^{k} \, t_{n+m+2}^{l}$. \noindent {\bf III) Blow up of ${\bf P}^{n}$ at a point.} The fan $\tilde{\Delta}$ associated to the blow up $\tilde{{\bf P}^{n}}$ of the projective space at the fixed point given by the cone ${\bf R}^{+} e_{2} + \cdots + {\bf R}^{+} e_{n+1}$ is generated by $\{e_{1}, \, \ldots, \, e_{n+1}, e_{n+2}\}$ where $e_{n+2} \, = \, -e_{1}$. Denote by $D_{i}$ the 1-dimensional cone \, ${\bf R}^{+}\, e_{i}$ \, and by $s_{i}$ its class in cohomology where $$ H^{\ast} \, (X, \, {\bf Z}) \, = \, {\bf Z} \, [s_{1}, \ldots , s_{n+2}] \, / \, I $$ and I is the ideal generated by $$ i)\ \ \{s_{{i}_{1}} \cdots s_{{i}_{k}}\, | \, D_{{i}_{1}}+ \cdots + D_{{i}_{k}} \, \mbox{is not in } \tilde{\Delta} \} $$ and $$ ii)\ \ \sum_{j=1}^{n+2} \, e_{i}^{\ast} (e_{j}) \, s_{j}\ \; \; \; \; i\, = \, 1,\ldots, n \, . $$ However $ii)$ is equivalent to $$ii) \, \, \, s_{2}\sim \cdots \sim s_{3} \sim s_{n+1} \; \; \; \mbox{and} \; \; \; s_{1} \sim s_{n+1}+s_{n+2}$$ Note that a p-dimensional cone cannot contain both $D_{n+2}$ and ${D_{1}}$. The reason is that $D_{n+2}$ is generated by $- e_{1}$ and $D_{1}$ by $e_{1},$ but by definition, a cone does not contain a subspace of dimension greater than 0. We would like to find a basis for $H^{\ast}(\tilde{{\bf P}^{n}})$ and write any monomial of degree p in terms of it. Consider the monomial $ s_{{i}_{1}} \cdots s_{{i}_{p}}$. There are three possible situations: \noindent {\bf 1)} $s_{{i}_{j}}$ is different from both $s_{n+2}$ and $s_{n+1}$. In this situation we have from $ii)$ that $s_{{i}_{1}} \cdots s_{{i}_{p}} \, = \, s_{n+1}^{p}$. \noindent{\bf 2)} $s_{n+2}$ is equal to $s_{{i}_{j}}$ for some $j \, = \, 1,\ldots, p$. Then from $ii)$ we obtain that $s_{{i}_{1}} \cdots s_{{i}_{p}} \, = \, s_{n+1}^{p-1} \, s_{n+2}$. \noindent {\bf 3)} $s_{1}$ is equal to $s_{{i}_{j}}$ for some $j \, = \, 1,\ldots, p$. Then from $ii)$ we obtain $s_{{i}_{1}} \cdots s_{{i}_{p}} \, = \, (s_{n+1} + s_{n+2}) \, s_{n+1}^{p+1} \, = \, s_{n+1}^{p} + s_{n+2} \, s_{n+1}^{p-1}$ which is the sum of 1) and 2). \noindent We conclude that $s_{n+1}^{p}$ and $s_{n+2} s_{n+1}^{p-1}$ form a basis for $H^{2p}$ if $p < n$. If $p=n$ then $s_{n+1}^{p} \, = \, 0$ and the only generator is $s_{n+2} s_{n+1}^{p-1}$. Let us call $s_{n+1}$ by $t_{1}$ and $s_{n+2} s_{n+1}^{p-1}$ by $t_{2}$. The Euler series for $\tilde{{\bf P}^{n}}$ is: $$ {\displaystyle E_{p} \, = \, {\left(\frac{1} {1-t_{1}}\right)}^{(_{p}^{n})} \, {\left(\frac{1}{1-t_{1}t_{2}}\right)}^{(_{p-1}^{\, \, \, n})} \, {\left(\frac{1}{1-t_{2}}\right)}^{(_{p-1}^{\, \, \, n})}} \; \; \; \;{\mbox if }\; \; p < n \, . $$ $$ {\displaystyle E_{p} \, = \, {\left(\frac{1}{1-t_{2}}\right)}^{(_{\; \; \; p}^{n+2})}} \; \; \; \;{\mbox if }\; \; p\,=\,n \, . $$ \noindent {\bf IV) Hirzebruch surfaces} A set of generators for the fan $\Delta$ that represents the Hirzebruch surface $X(\Delta)$ is given by $\{e_{1}, \ldots , e_{4} \}$ with $\{e_{1}, e_{2}\}$ the standard basis for ${\bf R}^{2}$, and $e_{3} \, = \, -e_{1} + ae_{2}, \; \; a > 1$ and $e_{4} \, = \, -e_{2}$. With the same notation as in the last examples, we have $$ H^{\ast} (X(\Delta)) \, = \, {\bf Z}[t_{1}, \ldots, t_{4}] \, / \, I $$ where I is generated by $$ i) \; \; \; \{ t_{1}t_{3}, \, t_{2}t_{4} \} $$ and $$ ii) \; \; \; \{ t_{1} -t_{3}, \, t_{2}+ at_{3} -t_{4} \} $$ from $ii)$ we have the following conditions for the $t_{i}$'s in $H^{\ast}(X)$ \begin{equation} \label{coh} \; \; \; \; t_{1} \, \sim \, t_{3} \; \; \mbox{and }\; \; t_{2} \, \sim \, (t_{4}-at_{3}). \end{equation} A basis for $H^{\ast} (X)$ is given by $\{ \{0\}, t_{3}, t_{4}, t_{4}t_{1} \}$ \, (see \cite{da-tova},\cite{ful-tova}). The Euler series for each dimension is: \noindent {\bf 1)} Codimension 2: There are four orbits (four cones of dimension 2), and all of them are equivalent in homology. From Theorem ~\ref{ec-eec} we obtain $$ E_{2} \, = \, {\left(\frac{1}{1-t}\right)}^{4} $$ {\bf 2)} Codimension 1: Again, there are four orbits (four cones of dimension 1), and the relation among them, in homology, is given by ~\ref{coh}. From theorem ~\ref{ec-eec} we obtain $$ E_{1} \, = \, {\left(\frac{1}{1-t_{3}}\right)}^{2} \, \left(\frac{1}{1-t_{4}}\right) \, \left(\frac{1}{1-t_{3}^{-a} t_{4}}\right). $$ {\bf 3)} Codimension 0: The only orbit is the torus itself so $$ E_{0} \, = \, \left(\frac{1}{1-t}\right). $$

9308

alg-geom/9308003

en

https://arxiv.org/abs/alg-geom/9308003

alg-geom/9308003

Brussee Rogier

R. Brussee

Some remarks on the Kronheimer-Mrowka classes of algebraic surfaces

6 pages, Latex 2.09

Define the Donaldson series of a simply connected 4-manifold by q(X) = \sum_d q_d(X)/d! Recently Kronheimer and Mroka have announced the result that the Donaldson series of so called simple 4-manifolds can be written as q(X) = e^{Q/2}\sum_{i=1}^p a_i e^{K_i} where $Q$ is the intersection form and the $K_i \in H^2(X,\Z)$ are the {\it Kronheimer-Mrowka classes}. We prove that for simple simply connected algebraic surfaces the $K_i$ are algebraic classes and that they are closely related to the canonical class $K_X$. For simple simply connected minimal surfaces of general type we prove $K_i^2 \le K_X^2$ with equality if and only if $K_i = \pm K_X$. Remark: although no gauge theory is used in this paper it should have a cross reference with the as yet non existent e-print service for low dimensional topology.

alg-geom

\section{Introduction} Recently Kronheimer and Mrowka have announced a very interesting result, which sheds new light on the Donaldson polynomials \cite{KM-}. They find recurrence relations between the Donaldson polynomials, by finding relations between the polynomials and the minimal genus of a smooth real surface representing an homology class. To be more precise we need a definition. For a simply connected 4-manifold $X$ with odd $b_+ \ge 3$, we denote the ${\rm SU}(2)$ polynomials on $H_0(X) \oplus H_2(X)$ by $q_k(X)$. $X$ is called {\sl simple} if we have $$ q_k(X)(pt^2,-) = 4 q_{k-1}(X), \qquad d= 4k -\numfrac32(1+b_+). $$ For simple 4-manifolds it is convenient to label the polynomials by their degree on $H_2(X)$ i.e. we define $$ q_d(X) =\cases{q_k(X)|_{H_2(X)} &if $d = 4k -\numfrac32(1 + b_+)$, \\ q_k(X)(pt,-)|_{H_2(X)}&if $d = 4k - 2 -\numfrac32(1 +b_+)$, \\ 0 & otherwise} $$ The {\em Donaldson series} is then the formal power series $q(X) = \sum_d q_d(X) / d!\,$. \theorem KM. (Kronheimer, Mrowka) For every simple 4-manifold $X$ there exist a finite number of {Kronheimer-Mrowka classes} $K_1,\ldots, K_p\in H^2(X)$ and non zero rational numbers $a_1,\ldots a_p$ such that \itm{(i)} $q(X) = e^{Q / 2} \sum_{i = 1}^n a_i e^{K_i}$, \itm{(ii)} $K_i \equiv w_2(X) \pmod 2$, for all $i=1,\ldots,p$, \itm{(iii)} if $K_i \in \{K_1,\ldots, K_p\}$ then $-K_i\in \{K_1,\ldots K_p\}$, \itm{(iv)} for every homologically nontrivial connected real surface $\Sigma$ with $\Sigma^2 \ge 0$ and every Kronheimer-Mrowka class $K_i$ we have $$ 2g(\Sigma) - 2 \ge \Sigma^2 + K_i\cdot \Sigma. $$ Here $Q$ is the intersection form. The Kronheimer Mrowka classes will be abbreviated KM-classes and are called the basic classes in \cite{KM-}. Condition (ii) is reminiscent of the Wu formula, whereas condition (iv) is similar to the the adjunction formula for the genus of a smooth algebraic curve. This suggests that for complex surfaces, the KM- classes should be closely related to the canonical divisor $K_X$. Indeed in the examples and the conjectural expression for the Donaldson series of elliptic surfaces in the announcement, the KM- classes are of type $K_i = \alpha_i K_{\min} + \sum \beta_{ij} E_j$ where $K_{\min}$ is the canonical divisor of the minimal model, $E_1,\ldots E_l$ the $(-1)$-curves, $\alpha_i$ a rational number with $|\alpha_i| \le 1$ and $\beta_{ij} = \pm 1$. A formulation which is a little less obvious but which generalises better, as we will see, is $K_i = C-D$ where $C$, $D$ are divisors such that $K_X = C+D$ and such that a multiple is effective. Hence in these cases the canonical class is a KM- class which is extremal from an algebraic geometric point of view. To formulate the general relationship, we recall that the effective cone of a complex surface $\rmmath{NE}(X)$, is the positive rational cone in $\rmmath{NS}(X)_{\Bbb Q} \subset H^2(X,{\Bbb Q})$ generated by effective divisors. Let $\rmmath{\overline{NE}}(X)$ be its closure in the norm topology. \theorem main. For every KM- class $K_i$ on a simple simply connected algebraic surface $X$, there is a unique decomposition $K_X = C_i + D_i$ in ${\Bbb Z}$-divisors $C_i,D_i \in \rmmath{\overline{NE}}(X)$ such that $ K_i = C_i - D_i $. In particular $K_i = K_X$ if and only if there is a smooth hyperplane section $H$ such that $2g(H) -2 = H^2 +K_i\cdot H$. The results in \cite{Kronheimer:genus} seem to indicate that a KM- class and a hyperplane section $H$ as in the theorem exist, at least when there is an $\omega \in H^0(K)$ such that for sufficiently large $k$, $q_k(\omega + \bar\omega) \ne 0$. For minimal surfaces of general type this would imply the invariance of the canonical class up to sign under orientation preserving self-diffeomorphisms. \corollary gentype. Assume in addition that $X$ is minimal and of general type, then $K_i^2 \le K_X^2$ with equality if and only if $K_i =\pm K_X$. By the Lefschetz (1,1) theorem \cite[p. 163]{G&H}, the algebraicity of the KM- classes is equivalent to the $K_i$ being of type $(1,1)$, In fact this is what we will prove, using that the Donaldson polynomials are of pure Hodge type as in \cite{(-1)-curve}. Since we assume that $p_g >0$ this shows that the lattice spanned by the $K_i$ is a proper sublattice. Moreover since the $K_i$ are defined by the differentiable structure, they are contained in the fixed lattice of the variation of Hodge structures defined by a family of complex structures on $X$. In favourable circumstances this should force the KM- classes to be in $H^2(X,{\Bbb Z}) \cap [K_{\min}, E_1,\ldots, E_l]$. Another implication of the theorem is that the KM- classes are trivial on $H^{0,2}(X) \subset H^2(X,{\Bbb C})$. Thus we get the following corollary. \corollary forms. If $X$ is a simple and simply connected surface, then for all $\omega \in H^0(K_X)$ we have $$ q(\omega + \bar\omega) = q_0 e^{\int \omega \wedge \bar\omega} $$ where $q_0$ is the Donaldson polynomial of degree 0 It would be rather interesting to understand this formula from an algebraic geometric point of view, possibly clarifying the role of the simple\-ness condition. Combining the corollary with O'Grady's non vanishing result, we see that $q_0 \ne 0$ if $X$ is of general type, $p_g$ is odd and $|K_{\min}|$, the linear system of the canonical class of the minimal model, contains a reduced curve \cite[th. 2.4]{O'Grady}, \cite[appendix]{JunLi:Kodaira}, \cite[th. 1]{Morgan}. In a similar vein, since the Neron-Severi group has a non degenerate intersection form, the $K_i$ are determined by their intersection products with divisors. Thus we get \corollary alg. For a simple simply connected surface, the Donaldson series $q(X)$ is determined by $q(X)|_{\rmmath{NS}(X)}$. The corollary says that by knowing the algebraic part of all Donaldson polynomials we can reconstruct the transcendental part as well, i.e. in the simple case, the polynomials defined by Jun-Li \cite{JunLi:polynomials} contain as much information as the full polynomials. Moreover since $\rmmath{\overline{NE}} \cap K_X -\rmmath{\overline{NE}}$ is a bounded subset of $\rmmath{NS}(X,{\Bbb Q})$, and the $K_i$ are integral, at least in principle, we get an effective bound on the number of Kronheimer Mrowka classes, hence on the number of polynomials one has to compute in order to reconstruct the Donaldson series. \section{Proof of theorem \protect\ref{main} and corollary \protect\ref{gentype}} We have to prove that the KM- classes are of type $(1,1)$. Accepting this, the rest of the statement of theorem \ref{main} and corollary \ref{gentype} is a consequence of property (ii), (iii), and (iv) of the KM- classes. By property (ii) we can write $K_i = K_X - 2D$ for some ${\Bbb Z}$-divisor $D$. For every very ample line bundle ${\cal O}(H)$, we choose a smooth connected hyperplane section $H$. Then we get $$ 2g(H)-2 = H^2 + H\cdot K_X \ge H^2 + H\cdot K_i = H^2 +H\cdot K_X - 2H \cdot D $$ i.e. $D \cdot H \ge 0$. Thus by the duality of the closure of the effective cone $\rmmath{\overline{NE}}(X)$ and the nef cone, we conclude that $D \in \rmmath{\overline{NE}}(X)$ \cite[prop. 2.3]{Wilson:birational}. We also have $ C:= (K_X + K_i)/2 \in \rmmath{\overline{NE}}$ by property (iii). Rewriting we get $K_X = C+D$ and $K_i= C-D$ as claimed. Note that nothing is gained by applying the inequality to other smooth connected divisors $C$ with $C^2 \ge 0$, since such divisors are nef. Finally note that $H\cdot K_i = 2g(H)-2-H^2 = H\cdot K_X$ if and only if $D = 0$, since $\rmmath{\overline{NE}} \cap H^\perp = (0)$. This proves theorem \ref{main} up to algebraicity. Now assume temporarily that $X$ is minimal and of general type. Write $K_i^2 = K_X^2 + 4(D^2 -K_X \cdot D)$. Since $K_X$ is nef, we get $K_i^2 \le K_X^2$ if $D^2 \le 0$, with equality iff $D^2 = K\cdot D = 0$. The latter is equivalent to $D=0$ by the Hodge index theorem. Interchanging $K_i$ and $-K_i$ if necessary, we are left with the case $C^2>0$, $D^2>0$. Since $C,D \in \rmmath{\overline{NE}}$, we have $C\cdot D >0$ by the numerical connectedness of $K_X$ \cite[prop VII.6.1]{BPV} (strictly speaking we need that $C$ and $D$ are effective, but the proof of [loc. cit] carries over without change). Thus $K_X^2 > D^2$. Then again by the Hodge index theorem we get $(K\cdot D)^2 \ge K^2D^2 > (D^2)^2$, so $K_i^2 < K^2$. This proves the corollary. To prove that the KM- classes are of type (1,1), we need a slight generalization of \cite[prop. 3.1]{(-1)-curve}. For simplicity we restrict ourselves to the ${\rm SU}(2)$ case and a statement about Hodge types, but the proof can easily be modified to show that all ${\rm SO}(3)$ polynomials $q_{L,k}(X)$ with $L \in \rmmath{NS}(X)$ come form an algebraic cycle (cf. [loc. cit.]). Consider the Donaldson polynomial $q_d$ as an element of $S^d H^2(X)$ en\-dowed with its natural Hodge structure. \lemma pure. For every $d \ge 0$, the Donaldson polynomials $q_d$ are pure of type $(d,d)$. \proof. We temporarily index the polynomials on $H_2(X)$ by $k = c_2$. By \cite[prop. 3.1]{(-1)-curve} the lemma is true for $q_k$ with odd $k \gg 0$ and evaluated on 2 dimensional classes. Here sufficiently large means that the moduli space ${\cal M}_k(X)$ of stable bundles on $X$ for generic polarisation $H$ is generically smooth and reduced of the proper dimension, and that the lower moduli spaces ${\cal M}_{k'}(X)$ for $k'< k$ have sufficiently low dimension, so that we can apply Morgan's comparison result \cite{Morgan}. For the general case we use stabilization. By \cite[th. 2.1.1]{Morgan-O'Grady}, for every $k$ we can find an $l_0$, and $\epsilon_1,\ldots, \epsilon_{l}$ sufficiently small such that for a generic polarisation on the $l \ge l_0$-fold blow-up of type $H - \sum \epsilon_i E_i$, the moduli space ${\cal M}_{k+l}(\^X(x_1,\ldots, x_l))$ is generically smooth of the proper dimension and the lower moduli spaces ${\cal M}_{k'}(\^X)$ with $k' <k$, have sufficiently low dimension. Thus $q_{k+l}(\^X(x_1,\ldots x_l))$ is pure of type $(d+4l,d+4l)$ Now by the blow-up formula \cite[th. 4.3.1]{F&M}, we have $$ q_k(X) = (-{\numfrac12})^l q_{k+l}(\^X(x_1,\ldots x_l))(E_1^4,\ldots,E_l^4,-). $$ Clearly the exceptional divisors $E_i$ are pure of type $(1,1)$, hence $q_k(X)$ is pure of type $(d,d)$. Finally $$ q_k(X)(pt,-) = -{\numfrac12} q_{k+1}(\^X(x))(E^6,-), $$ by \cite[below cor. 4.3.2]{F&M} (without proof) or \cite{consum}, and so we can use the same argument as above. \endproof The rest of the proof is now straightforward. Let $$ C(X) = e^{-Q/2}q(X) = \sum a_i e^{K_i}. $$ Then $C(X) = \sum C_d(X)$, where $C_d(X)$ is a homogeneous polynomial of pure type $(d,d)$, since both $\exp(-Q/2)$ and $q(X)$ is a sum of homogeneous polynomials of pure type. Write $K_i = \alpha_i + \beta_i + \bar\alpha_i$, with $\alpha_i \in H^{0,2}$, $\beta_i \in H^{1,1}$. Let $z \in H^{1,1}$ be variable and denote by $\ddz{}{}$ the directional derivative. Then we have $$ \ddz n{C(X)} = \sum_{d=0}^\infty \ddz n{C_d} = \sum a_i \<\beta_i,z>^n e^{K_i}. $$ Clearly $\ddz n{C_d}$ has pure type $(d-n,d-n)$. Thus, upon restriction to $H^{0,2}(X)$ only the constant term contributes, and we get $$ \left.\ddz n{C(X)}\right|_{H^{0,2}} = \txt{constant} = \sum a_i \<\beta_i,z>^n e^{\bar\alpha_i}. $$ We conclude that $$ \sum_{\alpha_i \ne 0} a_i \<\beta_i,z>^n e^{\bar\alpha_i} =0. $$ Since $z$ and $n$ are arbitrary we find that the sum must be empty, i.e. all KM- classes $K_i$ are of type $(1,1)$. \endproof

9704

alg-geom/9704003

en

https://arxiv.org/abs/alg-geom/9704003

alg-geom/9704003

Alex Degtyarev

A. Degtyarev and V. Kharlamov

Empty real Enriques surfaces and Enriques-Einstein-Hitchin 4-manifolds

10 pages, AmS-TeX with amsppt (must be run twice)

The Arnoldfest (Toronto, ON, 1997), 131--140, Fields Inst. Commun., v24, Amer. Math. Soc., Providence, RI, 1999

We prove that the moduli space of empty real Enriques surfaces (and, thus, the moduli space of compact orientable 4-dimensional Einstein manifolds whose universal covering is a K3-surface and \pi_1(E) = Z/2 x Z/2) is connected. The proof is based on a systematic study of real elliptic pencils and gives explicit models of all empty real Enriques surfaces.

alg-geom

\chapter{\protect\chapter@toc}\let\ifnum\pageno<\z@\romannumeral-\pageno\else\number\pageno\fi\relax \def\protect{\noexpand\noexpand\noexpand}% \Protect@@~\Protect@@\@@@xref\Protect@@\pageref\Protect@@\nofrills \edef\@tempa{\@ifundefined#1{}{\write#1{#2}}}\expandafter}\@tempa\@esphack} \def\writeauxline#1#2#3{\@writeaux\@auxout {\string\@auxline{#1}{#2}{#3}{\ifnum\pageno<\z@\romannumeral-\pageno\else\number\pageno\fi}}} {\let\newwrite\relax \gdef\@openin#1{\make@letter\@input{\jobname.#1}\tau@mpcat} \gdef\@openout#1{\global\expandafter\newwrite\csname tf@-#1\endcsname \immediate\openout\csname tf@-#1\endcsname \jobname.#1\relax}} \def\auxlinedef#1{\expandafter\def\csname do@-#1\endcsname} \def\@auxline#1{\@ifundefined{\do@-#1}{\expandafter\eat@iii}% {\expandafter\expandafter\csname do@-#1\endcsname}} \def\begin@write#1{\bgroup\def\do##1{\catcode`##1=12 }\dospecials\do\~\do\@ \catcode`\{=\@ne\catcode`\}=\tw@\immediate\write\csname#1\endcsname} \def\end@writetoc#1#2#3{{\string\tocline{#1}{#2\string\page{#3}}}\egroup} \def\do@tocline#1{% \@ifundefined{\tf@-#1}{\expandafter\eat@iii} {\begin@write{tf@-#1}\expandafter\end@writetoc} } \auxlinedef{toc}{\do@tocline{toc}} \let\protect\empty \def\Protect#1{\@ifundefined{#1@P@}{\PROTECT#1}{}} \def\PROTECT#1{% \expandafter\let\csname\eat@bs#1@P@\endcsname#1% \edef#1{\noexpand\protect\@xname{#1@P@}}} \Protect\operatorname \Protect\operatornamewithlimits \Protect\qopname@ \Protect\qopnamewl@ \Protect\text \Protect\topsmash \Protect\botsmash \Protect\smash \Protect\widetilde \Protect\widehat \Protect\thetag \Protect\therosteritem \Protect\Cal \Protect\Bbb \Protect\bold \Protect\slanted \Protect\roman \Protect\italic \Protect\boldkey \Protect\boldsymbol \Protect\frak \Protect\goth \Protect\dots \Protect\cong \Protect\lbrace \let\{\lbrace \Protect\rbrace \let\}\rbrace \let\root@P@@\root \def\root@P@#1{\root@P@@#1\of} \def\root#1\of{\protect\root@P@{#1}} \let\@frills@\identity@ \let\@txtopt@\identyty@ \let\@numopt@\identity@ \def\ignorespaces\@txtopt@{\ignorespaces\@txtopt@} \def\@numopt@{\@numopt@} \newif\if@write\@writetrue \def\checkstar@#1{\DN@{\@writetrue \ifx\next*\DN@####1{\let\@numopt@\eat@\checkstar@@{#1}}% \else\DN@{\let\@numopt@\identity@#1}\fi\next@}\FN@\next@} \def\checkstar@@#1{\DN@{% \ifx\next*\DN@####1{\@writefalse#1}% \else\DN@{\@writetrue#1}\fi\next@}\FN@\next@} \def\checkfrills@#1{\DN@{% \ifx\next\nofrills\DN@####1{#1}\def\@frills@####1{####1\nofrills}% \else\DN@{#1}\let\@frills@\identity@\fi\next@}\FN@\next@} \def\checkbrack@#1{\DN@{% \ifx\next[\DN@[####1]{\def\@txtopt@########1{####1}% #1}% \else\DN@{\let\@txtopt@\identity@#1}\fi\next@}\FN@\next@} \def\check@therstyle#1#2{{\DN@{#1}\ifx\@txtopt@\identity@\else \DNii@##1\@therstyle{}\def\@therstyle{\DN@{#2}\nextii@}% \def\nextiii@##1##2\@therstyle{\expandafter\nextii@##1##2\@therstyle}% \expandafter\nextiii@\@txtopt@\@therstyle.\@therstyle\fi \expandafter}\next@} \newif\if@theorem \let\@therstyle\eat@ \def\@headtext@#1#2{{\disable@special\let\protect\noexpand \def\chapter{\protect\chapter@rh}\Protect@@\nofrills \edef\@tempa{\noexpand\@frills@\noexpand#1{#2}}\expandafter}\@tempa} \let\AmSrighthead@\checkfrills@{\@headtext@\AmSrighthead@} \def\checkfrills@{\@headtext@\AmSrighthead@}{\checkfrills@{\@headtext@\AmSrighthead@}} \let\AmSlefthead@\checkfrills@{\@headtext@\AmSlefthead@} \def\checkfrills@{\@headtext@\AmSlefthead@}{\checkfrills@{\@headtext@\AmSlefthead@}} \def\@head@@#1#2#3#4#5{\if@theorem\else \@frills@{\csname\expandafter\eat@iv\string#4\endcsname}\@name{#1font@}\fi \@name{#1\ST@LE}{\counter#3}{\ignorespaces#5\unskip}% \if@write\writeauxline{toc}{\eat@bs#1}{#2{\counter#3}{\ignorespaces#5}}\fi \if@theorem\else\expandafter#4\fi \ifx#4\endhead\ifx\@txtopt@\identity@\else \headmark{\@name{#1\ST@LE}{\counter#3}{\ignorespaces\@txtopt@{}}}\fi\fi \ignorespaces} \let\subsubheadfont@\empty \ifx\undefined\endhead\Invalid@\endhead\fi \def\@head@#1{\checkstar@{\checkfrills@{\checkbrack@{\@head@@#1}}}} \def\@thm@@#1#2#3{% \@frills@{\csname\expandafter\eat@iv\string#3\endcsname} {\@theoremtrue\check@therstyle{\@name{#1\ST@LE}}\ignorespaces\@txtopt@ {\counter#2}\@theoremfalse}% \expandafter\envir@stack\csname end\eat@bs#1\endcsname \@name{#1font@}\ignorespaces} \let\proclaimfont@\empty \def\@thm@#1{\checkstar@{\checkfrills@{\checkbrack@{\@thm@@#1}}}} \def\@capt@@#1#2#3#4#5\endcaption{\bgroup \edef\@tempb{\global\footmarkcount@\the\footmarkcount@ \global\@name{#2{\Bbb N}@M}\the\@name{#2{\Bbb N}@M}}% \def\shortcaption##1{\def\sh@rtt@xt####1{##1}}\let\sh@rtt@xt\identity@ \DN@##1##2##3{\false@\fi\iftrue}% \ifx\@frills@\identity@\else\let\notempty\next@\fi #4{\@tempb\@name{#1\ST@LE}{\counter#2}}\@name{#1font@}#5\endcaption \if@write\writeauxline{#3}{\eat@bs#1}{{} \@name{#1\ST@LE}{\counter#2}% \if\notempty{#5}.\enspace\fi\sh@rtt@xt{#5}}\fi\egroup} \def\@capt@#1{\checkstar@{\checkfrills@{\checkbrack@{\@capt@@#1}}}} \let\captiontextfont@\empty \def\newfont@def#1#2{\@ifundefined{#1font@} {\@xdefname{#1font@}{\@xname{.\expandafter\eat@iv\string#2font@}}}{}} \def\newhead@#1#2#3#4{{% \gdef#1{\@therstyle\@therstyle\@head@{#1#2#3#4}}\newfont@def#1#4% \@ifundefined{#1\ST@LE} {\expandafter\gdef\csname\eat@bs#1\ST@LE\endcsname{\headstyle}}{}% \@ifundefined{#2}{\gdef#2{\headtocstyle}}{}% \@@addto\moretocdefs@{\\#1#1#4}}} \outer\def\newhead#1{\checkbrack@{\expandafter\newhead@\expandafter #1\@txtopt@\headtocstyle}} \outer\def\newtheorem#1#2#3#4{{% \gdef#2{\@thm@{#2#3#4}}\newfont@def#2\endproclaim% \@xdefname{\end\eat@bs#2}{\noexpand\revert@envir \@xname{\end\eat@bs#2}\noexpand#4}% \expandafter\gdef\csname\eat@bs#2\ST@LE\endcsname{\proclaimstyle{#1}}}}% \outer\def\newcaption#1#2#3#4#5{{\let#2\relax \edef\@tempa{\gdef#2####1\csname end\eat@bs#2\endcsname}% \@tempa{\@capt@{#2#3{#4}#5}##1\endcaption}\newfont@def#2\endcaptiontext% \expandafter\gdef\csname\eat@bs#2\ST@LE\endcsname{\captionstyle{#1}}% \@@addto\moretocdefs@{\\#2#2\endcaption}\newtoc{#4}}} {\let\newtoks\relax \outer\gdef\newtoc#1{{% \expandafter\ifx\csname do@-#1\endcsname\relax \global\auxlinedef{#1}{\do@tocline{#1}}{}% \@@addto\tocsections@{\make@toc{#1}{}}\fi}}} \toks@\expandafter{\itembox@} \toks@@{{\let\therosteritem\identity@\let\rm\empty \edef\next@{\edef\noexpand\@lastmark{\therosteritem@}}\expandafter}\next@} \edef\itembox@{\the\toks@@\the\toks@} \def\firstitem@false{\let\iffirstitem@\iffalse \global\let\lastlabel\@lastlabel} \def\rosteritemref#1{\hbox{\therosteritem{\@@@xref{#1}}}} \def\local#1{\label@\@lastlabel{\lastlabel-i#1}} \def\loccit#1{\rosteritemref{\lastlabel-i#1}} \def\xRef@P@{\gdef\lastlabel} \def\xRef#1{\@xref{#1}\protect\xRef@P@{#1}} \let\Ref\xRef \def\iref@P@{\gdef\lastref} \def\itemref#1#2{\rosteritemref{#1-i#2}\protect\iref@P@{#1}} \def\iref#1{\@xref{#1}\itemref{#1}} \def\ditto#1{\rosteritemref{\lastref-i#1}} \def\tag\counter\equation{\tag\counter\equation} \def\eqref#1{\thetag{\@@@xref{#1}}} \def\tagform@#1{\ifmmode\hbox{\rm\else\protect\rom@P@{\fi (\ignorespaces#1\unskip)\iftrue}\else}\fi} \let\AmSfnote@\makefootnote@ \def\makefootnote@#1{{\let\footmarkform@\identity@ \edef\next@{\edef\noexpand\@lastmark{#1}}\expandafter}\next@ \AmSfnote@{#1}} \def\ifnum\insertpenalties>0\line{}\fi\vfill\supereject{\ifnum\insertpenalties>0\line{}\fi\vfill\supereject} \def\find@#1\in#2{\let\found@\false@ \DNii@{\ifx\next\@nil\let\next\eat@\else\let\next\nextiv@\fi\next}% \edef\nextiii@{#1}\def\nextiv@##1,{% \edef\next{##1}\ifx\nextiii@\next\let\found@\true@\fi\FN@\nextii@}% \expandafter\nextiv@#2,\@nil} \def\checkstar@\include@@#1{{\ifx\@auxout\@subaux\DN@{\errmessage {\labelmesg@ Only one level of \string\checkstar@\include@\space is supported}}% \else\edef\@tempb{#1}\@numopt@\ifnum\insertpenalties>0\line{}\fi\vfill\supereject \xdef\@include@{\relax\bgroup\ifx\@numopt@\identity@\ifnum\insertpenalties>0\line{}\fi\vfill\supereject \else\let\noexpand\@immediate\noexpand\empty\fi}% \DN@##1 {\if\notempty{##1}\edef\@tempb{##1}\DN@####1\eat@ {}\fi\next@}% \DNii@##1.{\edef\@tempa{##1}\DN@####1\eat@.{}\next@}% \expandafter\next@\@tempb\eat@{} \eat@{} % \expandafter\nextii@\@tempb.\eat@.% \let\next@\empty \if\expandafter\notempty\expandafter{\@tempa}% \edef\nextii@{\write\@mainaux{% \noexpand\string\noexpand\@input{\@tempa.aux}}}\nextii@ \@ifundefined\@includelist{\let\found@\true@} {\find@\@tempa\in\@includelist}% \if\found@\@ifundefined\@noincllist{\let\found@\false@} {\find@\@tempb\in\@noincllist}\else\let\found@\true@\fi \if\found@\@ifundefined{\cl@#1}{}{\DN@{\csname cl@#1\endcsname}}% \else\global\let\@auxout\@subaux \xdef\@auxname{\@tempa}\xdef\@inputname{\@tempb}% \@numopt@\immediate\openout\@auxout=\@tempa.aux \@numopt@\immediate\write\@auxout{\relax}% \DN@{\@input\@inputname \@include@\closeaux@@\egroup\make@auxmain}\fi\fi\fi \expandafter}\next@} \def\checkstar@\include@{\checkstar@\checkstar@\include@@} \def\includeonly#1{\edef\@includelist{#1}} \def\noinclude#1{\edef\@noincllist{#1}} \newwrite\@mainaux \newwrite\@subaux \def\make@auxmain{\global\let\@auxout\@mainaux \xdef\@auxname{\jobname}\xdef\@inputname{\jobname}} \begingroup \catcode`\(\the\catcode`\{\catcode`\{=12 \catcode`\)\the\catcode`\}\catcode`\}=12 \gdef\closeaux@@((% \def\\##1(\@Waux(\global##1=\the##1) \edef\@tempa(\@Waux(% \gdef\expandafter\string\csname cl@\@auxname\endcsname{)% \\\pageno\\\footmarkcount@\closeaux@\@Waux(})\@immediate\closeout\@auxout)% \expandafter)\@tempa) \endgroup \let\closeaux@\empty \let\@immediate\immediate \def\@Waux{\@immediate\write\@auxout} \def\readaux{% \W@{>>> \labelmesg@ Run this file twice to get x-references right} \global\everypar{}% {\def\\##1{\global\let##1\relax}% \def\/##1{\gdef##1{\preambule@cs##1}}% \disablepreambule@cs}% \make@auxmain\make@letter{\setboxz@h{\@input{\jobname.aux}}}\tau@mpcat \immediate\openout\@auxout=\jobname.aux \immediate\write\@auxout{\relax}\global\let\end\endmain@} \everypar{\global\everypar{}\readaux} {\toks@\expandafter{\topmatter} \global\edef\topmatter{\noexpand\readaux\the\toks@}} \let\@@end@@\end \let\theend\end \def\endmain@{\ifnum\insertpenalties>0\line{}\fi\vfill\supereject\@immediate\closeout\@auxout \make@letter\def\newlabel##1##2##3{}\@input{\jobname.aux}% \W@{>>> \labelmesg@ Run this file twice to get x-references right}% \@@end@@} \def\disablepreambule@cs{\\\disablepreambule@cs} \def\preambule@cs#1{\warning@{Preamble command \string#1\space ignored}} \def\proof{\checkfrills@{\checkbrack@{% \check@therstyle{\@frills@{\demo}{\ignorespaces\@txtopt@{Proof}}{}} {\ignorespaces\@txtopt@{}\envir@stack\endremark\envir@stack\enddemo}% \envir@stack\nofrillscheck{\frills@{\qed}\revert@envir\endproof\enddemo}\ignorespaces}}} \def\nofrillscheck{\frills@{\qed}\revert@envir\endproof\enddemo}{\nofrillscheck{\ignorespaces\@txtopt@@{\qed}\revert@envir\nofrillscheck{\frills@{\qed}\revert@envir\endproof\enddemo}\enddemo}} \let\AmSRefs\Refs \let\AmSref\@ref@}\ignorespaces}} \let\AmSrefstyle\refstyle \let\plaincite\cite \def\citei@#1,{\citeii@#1\eat@,} \def\citeii@#1\eat@{\@@@xref{#1}} \def\cite#1{\protect\plaincite{\citei@#1\eat@,\unskip}} \def\refstyle#1{\AmSrefstyle{#1}\uppercase{% \ifx#1A\relax \def\@ref@##1{\AmSref\xdef\@lastmark{##1}\key##1}% \else\ifx#1C\relax \def\@ref@{\AmSref\no\counter\refno}% \else\def\@ref@{\AmSref}\fi\fi}} \refstyle A \checkbrack@{\expandafter\newcounter@\@txtopt@{{}}}\refno\null \gdef\Refs{\checkstar@{\checkbrack@{\csname AmSRefs\endcsname \nofrills{\ignorespaces\@txtopt@{References}% \if@write\writeauxline{toc}{vartocline}{\ignorespaces\@txtopt@{References}}\fi}% \def\@ref@}\ignorespaces}}{\@ref@}\ignorespaces}}} \let\@ref@}\ignorespaces}}\xref \def\tocsections@{\make@toc{toc}{}} \let\moretocdefs@\empty \def\newtocline@#1#2#3{% \edef#1{\def\@xname{#2line}####1{\expandafter\noexpand \csname\expandafter\eat@iv\string#3\endcsname####1\noexpand#3}}% \@edefname{\no\eat@bs#1}{\let\@xname{#2line}\noexpand\eat@}% \@name{\no\eat@bs#1}} \def\maketoc#1#2{\Err@{\Invalid@@\string\maketoc}} \def\newtocline#1#2#3{\Err@{\Invalid@@\string\newtocline}} \def\make@toc#1#2{\penaltyandskip@{-200}\aboveheadskip \if\notempty{#2} \centerline{\headfont@\ignorespaces#2\unskip}\nobreak \vskip\belowheadskip \fi \@openin{#1}\@openout{#1}% \vskip\z@} \def\readaux\checkfrills@{\checkbrack@{\@contents@}}{\readaux\checkfrills@{\checkbrack@{\@contents@}}} \def\@contents@{\toc@{\ignorespaces\@txtopt@{Contents}}\envir@stack\endcontents% \def\nopagenumbers{\let\page\eat@}\let\newtocline\newtocline@ \def\tocline##1{\csname##1line\endcsname}% \edef\caption##1\endcaption{\expandafter\noexpand \csname head\endcsname##1\noexpand\endhead}% \ifmonograph@\def\vartoclineline{\Chapterline}% \else\def\vartoclineline##1{\sectionline{{} ##1}}\fi \let\\\newtocline@\moretocdefs@ \ifx\@frills@\identity@\def\\##1##2##3{##1}\moretocdefs@ \else\let\tocsections@\relax\fi \def\\{\unskip\space\ignorespaces}\let\maketoc\make@toc} \def\endcontents{\tocsections@\vskip-\lastskip\revert@envir\endcontents \endtoc} \@ifundefined\selectf@nt{\let\selectf@nt\identity@}{} \def\textonlyfont@#1#2{% \expandafter\def\csname\eat@bs#1@P@\endcsname{% \RIfM@\Err@{Use \string#1\space only in text} \else\edef\f@ntsh@pe{\string#1}\selectf@nt#2\fi}% \edef#1{\noexpand\protect\@xname{#1@P@}\empty}} \tenpoint \def\newshapeswitch#1#2{\gdef#1{\selectsh@pe#1#2}\PROTECT#1} \def\shapeswitch#1#2#3{\expandafter\gdef\csname\eat@bs#1\string#2\endcsname{#3}} \shapeswitch\rm\bf\bf \shapeswitch\rm\tt\tt \shapeswitch\rm\smc\smc \newshapeswitch\em\it \shapeswitch\em\it\rm \shapeswitch\em\sl\rm \def\selectsh@pe#1#2{\relax\@ifundefined{#1\f@ntsh@pe}{#2} {\csname\eat@bs#1\f@ntsh@pe\endcsname}} \def\@itcorr@{\leavevmode\skip@\lastskip\unskip\/% \ifdim\skip@=\z@\else\hskip\skip@\fi} \def\protect\rom@P@@P@#1{\@itcorr@{\selectsh@pe\rm\rm#1}} \def\protect\rom@P@{\protect\protect\rom@P@@P@} \def\protect\Rom@P@@P@#1{\@itcorr@{\rm#1}} \def\protect\Rom@P@{\protect\protect\Rom@P@@P@} {\catcode`\-=11 \gdef\wr@index#1{\@writeaux\tf@-idx{\string\indexentry{#1}{\ifnum\pageno<\z@\romannumeral-\pageno\else\number\pageno\fi}}} \gdef\wr@glossary#1{\@writeaux\tf@-glo{\string\glossaryentry{#1}{\ifnum\pageno<\z@\romannumeral-\pageno\else\number\pageno\fi}}}} \def\@itcorr@\bgroup\em\checkstar@\emph@{\@itcorr@\bgroup\em\checkstar@\@itcorr@\bgroup\em\checkstar@\emph@@} \let\index\wr@index \let\glossary\wr@glossary \def\@itcorr@\bgroup\em\checkstar@\emph@@#1{\@numopt@{\wr@index{#1}}\ignorespaces#1\unskip\egroup \DN@{\DN@{}\ifx\next.\else\ifx\next,\else\DN@{\/}\fi\fi\next@}\FN@\next@} \def\catcode`\"\active{\catcode`\"\active} {\catcode`\"\active\gdef"#1"{\@itcorr@\bgroup\em\checkstar@\emph@{#1}}} \def\@openout{idx}{\@openout{idx}} \def\@openout{glo}{\@openout{glo}} \def\endofpar#1{\ifmmode\ifinner\endofpar@{#1}\else\eqno{#1}\fi \else\leavevmode\endofpar@{#1}\fi} \def\endofpar@#1{\unskip\penalty\z@\null\hfil\hbox{#1}\hfilneg\penalty\@M} \@@addto\disablepreambule@cs{% \\\readaux \/\Monograph \/\@openout{idx} \/\@openout{glo} } \catcode`\@\labelloaded@ \def\headstyle#1#2{\@numopt@{#1.\enspace}#2} \def\headtocstyle#1#2{\@numopt@{#1.}\space #2} \def\specialsectionstyle#1#2{#2} \def\specialtocstyle#1#2{#2} \checkbrack@{\expandafter\newcounter@\@txtopt@{{}}}\section\null \checkbrack@{\expandafter\newcounter@\@txtopt@{{}}}\subsection\section \checkbrack@{\expandafter\newcounter@\@txtopt@{{}}}\subsubsection\subsection \newhead\specialsection[\specialtocstyle]\null\endspecialhead \newhead\section\section\endhead \newhead\subsection\subsection\endsubhead \newhead\subsubsection\subsubsection\endsubsubhead \def\firstappendix{\global\sectionno=0 % \counterstyle\section{\Alphnum\sectionno}% \global\let\firstappendix\empty} \def

9704

alg-geom/9704010

en

https://arxiv.org/abs/alg-geom/9704010

alg-geom/9704010

Christoph Lossen

Gert-Martin Greuel, Christoph Lossen and Eugenii Shustin

Plane curves of minimal degree with prescribed singularities

33 pages, LaTeX 2e, corrected some typos, simplified proofs of Lemmas 3.1, 4.1

10.1007/s002220050254

We prove that there exists a>0 such that for any integer d>2 and any topological types S_1,...,S_n of plane curve singularities, satisfying $\mu(S_1)+...+\mu(S_n) \leq ad^2$, there exists a reduced irreducible plane curve of degree d with exactly n singular points of types S_1,...,S_n, respectively. This estimate is optimal with respect to the exponent of d. In particular, we prove that for any topological type S there exists an irreducible polynomial of degree $d \leq 14\sqrt{\mu(S)}$ having a singular point of type S.

alg-geom

\section*{Introduction} Throughout the article we consider all objects to be defined over an algebraically closed field ${\mathbf K}$ of characteristic zero. In the paper we deal with the following classical problem: given an integer \mbox{$d\ge 3$} and types \mbox{$S_1,\dots,S_n$} of plane curve singularities, does there exist a reduced irreducible plane curve of degree $d$ with exactly $n$ singular points of types \mbox{$S_1,\dots,S_n$}, respectively? The complete answer is known for nodal curves \cite{Sev}: an irreducible curve of degree $d$, with $n$ nodes as its only singularities, exists if and only if $$0\le n\le \frac{(d-1)(d-2)}{2}\: .$$ For other singularities, even for ordinary cusps there is no complete answer. Namely, various restrictions are found (from Pl\"ucker formulae to inequalities by Varchenko \cite{Va} and Ivinskis \cite{HF,Ivi}), which read \begin{equation} \label{0.1} \sum_{i=1}^n \sigma(S_i) < \alpha_2 d^2+\alpha_1 d+\alpha_0, \;\;\;\alpha_2= \mbox{const}>0, \end{equation} with some positive invariants $\sigma$ of singular points which are at most quadratic in d. We want to give an {\bf asymptotically optimal} sufficient existence condition, that is a condition of type $$\sum_{i=1}^n\sigma(S_i)<\alpha d^2+o(d^2),\quad \alpha\leq\alpha_2,$$ providing (\ref{0.1}) is necessary. Note that an asymptotically exact condition, that is \mbox{$\alpha = \alpha_2$}, is hardly attainable. For example, there exist curves of degree \mbox{$d=2\cdot 3^k$}, \mbox{$(k=1,2,\dots)$} with $9(9^k-1)/8=9d^2/32+O(d)$ ordinary cusps \cite{H}. But here the number of conditions imposed by the cusps is \mbox{$d^2/16+O(d)$} more than the dimension of the space of curves of degree $d$, therefore one cannot expect that all intermediate quantities of cusps may be realized. The only previously known general sufficient condition for the existence of a curve with given singularities was (see \cite{Sh}) \begin{equation} \label{no} \sum_{i=1}^n(\mu(S_i)+4)(\mu(S_i)+5)\le\frac{(d+3)^2}{2}\ . \end{equation} It is not asymptotically optimal, because the left--hand side may be about $d^4$. The goal of this paper is: \begin{theorem} \label{0.2} For any integer \mbox{$d\ge 1$} and topological types \mbox{$S_1,\dots,S_n$} of plane curve singularities, satisfying \begin{equation} \label{0.3} \sum_{i=1}^n \mu(S_i) \le \frac{d^2}{392}\: , \end{equation} there exists a reduced irreducible plane projective curve of degree $d$ with exactly $n$ singular points of types \mbox{$S_1,\dots,S_n$}, respectively. \end{theorem} This estimate is asymptotically optimal, because always $$\sum_{i=1}^n \mu(S_i)<d^2\ .$$ The constant in (\ref{0.3}) is not the best possible. Our method could give a bigger constant, providing more tedious computations. For certain classes of singularities such as simple or ordinary, there are much better results (see, for instance, \cite{GLS}, section 3.3 and \cite{Sh}). The problem is of interest even for one individual singularity. Given a singularity $S$, what is the minimal degree of a reduced irreducible plane projective curve having this singularity at the origin? The classical upper bound is the determinacy bound \mbox{$\mu(S)+2$} \cite{To}, whereas a lower bound is \mbox{$\sqrt{\mu(S)}+1$} (coming from intersecting two generic polars and B\'ezout's Theorem). We claim \begin{theorem} \label{0.4} For any topological type $S$ of plane curve singularities there exists a reduced irreducible plane projective curve of degree \mbox{$\le 14\sqrt{\mu(S)}$} with singularity $S$ at the origin. \end{theorem} We should like to thank the Deutsche Forschungsgemeinschaft and the grant G 039-304.01/95 of the German--Israeli Foundation for financial support. \section{Strategy of the Proof} \setcounter{equation}{0} To emphasize what is new in our approach we describe shortly the main previously known constructions. The first one is to construct, somehow, a curve of the given degree, which is degenerate with respect to the required curve, and then to deform it in order to obtain the prescribed singularities. For example, Severi \cite{Sev} showed that singular points of a nodal curve, irreducible or not, can be smoothed out or preserved independently. Hence, taking the union of generic straight lines and smoothing out suitable intersection points, one obtains irreducible curves with any prescribed number of nodes, allowed by Pl\"ucker's formulae. Attempts to extend this construction on other singularities give curves with a number of singularities bounded from above by a linear function of the degree $d$ (see, for example, \cite{GrM} for curves with nodes, cusps and ordinary triple points), because of the very restrictive requirement of the independence of deformations of singular points. The second way consists of a construction especially adapted to the given degree and given collection of singularities. It may be based on a sequence of rational transformations of the plane applied to a more or less simple initial curve in order to get the required curve. Or it may consist in an invention of a polynomial defining the required curve. This can be illustrated by constructions of singular curves of small degrees as, for instance, in \cite{Wl1}, \cite{Wl2}, or by the construction of cuspidal curves as in \cite{H}, cited in the Introduction. Two main difficulties do not allow the appliance of this approach to a wide class of degrees and singularities: (1) for any new degree or singularity one has to invent a new construction, (2) even if one has constructed a curve with a lot of singularities, like in \cite{H}, it is hard to check that these singular points can be smoothed out independently and any intermediate numbers of singularities can be realized. Another idea, based on a modification of the Viro method of gluing polynomials (see the original method in \cite{Vi}) and on the independence of singular point deformations, was suggested in \cite{Sh}. This method, from the very beginning, requires a collection of ``base curves'' with given singularities (as, for instance, in Theorem \ref{0.4}), which originally provides only non--optimal results such as (\ref{no}) for arbitrary singularities. In our proof we use the previous constructions and introduce the following new element. With reduced germs of plane curves we associate a class ${{\mathcal G} {\mathcal S}}$ of irreducible zero--dimensional schemes, called below {\bf generalized singularity schemes}. Further we proceed in three main steps. {\it Step 1}. Given a topological type $S$ of plane curve singularities, we show that there exists a scheme \mbox{$X\in{{\mathcal G}{\mathcal S}}$} with \mbox{$\deg X\le a_1\mu(S)$}, \mbox{$a_1=\mbox{const}>0$} such that the relation $$h^1(\P^2,{\mathcal J}_{X/\P^2}(d))=0,$$ where \mbox{${\mathcal J}_{X/\P^2}\subset{{\mathcal O}}_{\P^2}$} is the ideal sheaf of $X$, suffices for the existence of a curve of degree $d$ with a singular point of type $S$ (see Lemmas \ref{4.1}, \ref{4.11} below). {\it Step 2}. For our purposes we have to provide the previous $h^1$--vanishing as \mbox{$d\le a_2\sqrt{\deg X}$}, \mbox{$a_2=\mbox{const}>0$}. To do this, we observe that in the first step, $X$ can be replaced by a generic scheme $X'$ in the same Hilbert scheme. Then we follow basically Hirschowitz \cite{HA}, who obtained, in an analogous manner, the $h^1$--vanishing for schemes of generic fat points in the plane. Namely, we fix a straight line $L$ and apply an inductive procedure described in Sections 3 and 4, which consists of a passage from $X$ and $d$ to the residue scheme $X:L$ (called below the {\bf reduction of} $X$) and $d-1$. Each time we have to verify that $X:L$ belongs to ${\mathcal G}{\mathcal S}$ (Proposition \ref{1.11}), and that $$a_3d\le\deg(X\cap L)\le d+1,\quad a_3=\mbox{const}>0.$$ The latter relation is achieved by means of two operations: {\bf specialization} of the scheme $X$ with respect to $L$ (Lemma \ref{1.14}), and {\bf extension} of $X$ (Definitions \ref{1.19}, \ref{1.21}, Lemma \ref{1.20}) when the specialization fails. {\it Step 3}. The final stage is a construction of curves with many singular points, done by means of a version of the Viro method (Section 6). Given topological singularities \mbox{$S_1,\dots,S_n$}, we find curves of degrees $$d_i\le a_4\sqrt{\mu(S_i)},\quad i=1,\dots ,n,\quad a_4=\mbox{const}>0,$$ each having a singular point of the corresponding type. Then we take a curve of degree $$d\le a_5\sqrt{d_1^2+...+d_n^2},\quad a_5=\mbox{const}>0,$$ with $n$ generic points of multiplicities \mbox{$d_1,\dots ,d_n$}, respectively, and deform these points in order to obtain the given singularities on a curve of any degree $$d\ge a_6\sqrt{\mu(S_1)+...+\mu(S_n)},\quad a_6=\mbox{const}>0.$$ \section{Singularity Schemes, Reductions and Extensions} \setcounter{equation}{0} Throughout this section, $S$ denotes a smooth surface, \mbox{$z \in S$}, and $C$ a reduced curve on $S$. Since our statements are local, we may assume that $C$, or the germ $(C,z)$, is given by a power series which, by abuse of notation, is also denoted by $C$ or by $(C,z)$. If \mbox{$z \not\in C$}, then $(C,z)$ denotes the empty germ or a unit of ${\mathcal O}_{S,z}$. Later, $z$ denotes also a finite set of points of $S$ and $(C,z)$ the corresponding multigerm. \begin{definition}\label{1.1} The {\bf multiplicity} of $C$ at $z$ is the non--negative integer \[ \mbox{$\mathrm{mt}$} \,(C,z) = \operatorname{max} \{n \in {\mathbf Z} \mid C \in \mathfrak{m}^n_z\}, \] where $\mathfrak{m}_z$ is the maximal ideal of ${\mathcal O}_{S,z}$, the analytic local ring of $S$ at $z$. If \mbox{$z \in C$}, we define, as usual, (cf.\ \cite{Z}, \cite{W}, \cite{Te}, \cite{BK}) the {\bf topological type} (or {\bf equisingularity type}) of the germ $(C,z)$ by the following discrete characteristic: the embedded resolution tree of $(C,z)$ and the multiplicities of the total transforms of $(C,z)$ at infinitely near points (including $z$). Two germs with the same topological types are called equivalent (notation $\sim$). \end{definition} \begin{definition}\label{1.2} $z$ is called an {\bf essential} point of $C$ if \mbox{$z \in C$}, and if the germ $(C,z)$ is not smooth. If \mbox{$z \in C$} and if \mbox{$q \not= z$} is infinitely near to $z$, we denote by $C_{(q)}$, respectively $\widehat{C}_{(q)}$, the corresponding strict, respectively total, transforms under the composition of blowing--ups \mbox{$\pi_{(q)} : S_{(q)} \longrightarrow S$} defining $q$. We call $q$ {\bf essential} if it is not a node (ordinary double point) of the union of $C_{(q)}$ with the reduced exceptional divisor. \end{definition} We shall introduce now the singularity scheme, respectively the generalized singularity scheme, of $(C,z)$, which are zero--dimensional subschemes of $S$ and which encode to a certain extent the topological type of $(C,z)$, respectively together with some higher order tangencies. For \mbox{$z \in C$} let \mbox{$T(C,z)$} denote the (infinite) complete embedded resolution tree of $(C,z)$ with vertices the points infinitely near to $z$. It is naturally oriented, inducing a partial ordering on its vertices such that \mbox{$z < q$} for all \mbox{$q \in T(C,z) \backslash \{z\}$}. If \mbox{$z \not\in C$} we define $T(C,z)$ to be the empty tree. Moreover, let \[ T^\ast (C,z) := \{q \in T(C,z) \mid q \mbox{ is essential}\} \] denote the tree of essential points of $(C,z)$, which is a finite subtree of $T (C,z)$. \begin{definition}\label{1.3} Let \mbox{$T^\ast \!\subset T(C,z)$} be a finite, connected subtree containing the essential tree $T^\ast(C,z)$. For any point \mbox{$q \in T^\ast$} and any \mbox{$f \in {\mathcal O}_{S, z}$} denote by $f_{(q)}$, respectively $\hat{f}_{(q)}$, the strict, respectively total, transform under the modification $\pi_{(q)}$ defining $q$. Put \mbox{$m_q := \mbox{$\mathrm{mt}$} \,(C_{(q)}, q)$}, \mbox{$\hat{m}_q := \mbox{$\mathrm{mt}$} \,(\widehat{C}_{(q)}, q)$} and define the ideal \[ J := J(C,T^\ast) := \{f \in {\mathcal O}_{S,z} \mid \mbox{$\mathrm{mt}$} \,(\hat{f}_{(q)}, q) \ge \hat{m}_q,\;\; q \in T^\ast\} \subset {\mathcal O}_{S,z} \] and the subscheme of $S$ defined by $J$, \[ X := X(C,T^\ast) = Z(J),\quad {\mathcal O}_{X,z} := {\mathcal O}_{S,z}/J, \] which is concentrated on $\{z\}$. $X$ is called a {\bf generalized singularity (scheme)} and the class of zero--dimensional subschemes of $S$, constructed in this way, is denoted by ${\mathcal G} {\mathcal S}$. The subclass of schemes \mbox{$X \in {\mathcal G} {\mathcal S}$} with \mbox{$T^\ast \!= T^\ast(C,z)$} is denoted by ${\mathcal S}$, \mbox{$X \in {\mathcal S}$} is called a {\bf singularity (scheme)}. \end{definition} \begin{examples} \begin{enumerate} \item Let $(C,z)$ be smooth. If \mbox{$T^\ast \!= \emptyset$}, we obtain \mbox{$J = {\mathcal O}_{S,z}$} and \mbox{$X = \emptyset$}. If \mbox{$T^\ast \!= \{ z \!=\! q_0,q_1,\dots, q_n\}$} and \mbox{$C =y$} with respect to local coordinates $(x,y)$ at $z$, then \mbox{$J = \langle y, x^{n+1}\rangle$}. \item If $(C,z)$ is an ordinary $r$--fold singularity ($r$ smooth branches with different tangents) and if \mbox{$T^\ast \!= \!\{z\}\; (= T^\ast(C,z))$} then \mbox{$J = \mathfrak{m}^r_z$}. \end{enumerate} \end{examples} The following lemma shows the relation of $X$ to equisingular deformations of $(C,z)$. Note that the germ $(C,z)$ defining $X$ is not uniquely determined by $X$, and that the tree $T^\ast$ is part of the data of $X$. We write $J_X$ and $T^\ast_X$ for the ideal $J$ and the tree $T^\ast$ belonging to $X$. By the following lemma, though $(C,z)$ is not uniquely determined by $X$, all topological invariants of $(C,z)$ can be associated uniquely to $X$. In particular, we define the {\bf multiplicity} $\mbox{$\mathrm{mt}$} \,X$, the {\bf Milnor number} $\mu(X)$ and the {$\mathbf \delta$}--{\bf invariant} $\delta(X)$ as those of $(C,z)$. \pagebreak[3] \begin{lemma}\label{1.4} Let \mbox{$X \in {\mathcal G} {\mathcal S}$} be a generalized singularity scheme. \begin{enumerate} \item[(i)] If $X$ is defined by $(C,z)$ then a generic element of $J_X$ is topologically equivalent to $(C,z)$. \item[(ii)] The set of base points of the ideal $J_X$ is equal to $T^\ast_X$, that is, the strict transforms of two generic elements in $J_X$ intersect exactly in $T^\ast_X$. \end{enumerate} \end{lemma} \begin{proof} Adding a generic element \mbox{$f\in J_X$} to the equation of \mbox{$(C,z)\subset (S,z)$} defines another generic element of $J_X$ having exactly the given multiplicities $\hat{m}_q$, \mbox{$q\in T^{\ast}$}, in particular, it is topologically equivalent to $(C,z)$, since $T^\ast(C,z) \subset T^\ast$. \\ Moreover, this shows that the strict transforms of two generic elements have the same multiplicity \mbox{$m_q\geq 1$} at \mbox{$q\in T^\ast_X$}. On the other hand, let \mbox{$q\in T(C,z)\backslash T^\ast_X$}, $Q$ be the corresponding branch of $(C,z)$ and $\bar{q}$ the predecessor of $q$ in \mbox{$T(C,z)$}. Then, slightly changing the tangent direction of the strict transform of $Q$ in $\bar{q}$, blowing down and composing with the other branches of $(C,z)$ defines an element of $J_X$ whose strict transform at $q$ does not contain $q$. \end{proof} The concepts developed so far generalize immediately to multigerms $(C,z)$, \mbox{$z= \{z_1, \dots, z_k\} \subset S$}. Then $T(C,z)$ and $T^\ast(C,z)$ are finite unions of trees. For \mbox{$T^\ast(C,z) \subset T^\ast \subset T(C,z)$} such that \mbox{$T^\ast \!\cap T(C,z_i)$} is a finite and connected subtree, we can define \mbox{$J(C,T^\ast) \subset {\mathcal O}_{S,z} = \prod\nolimits_{i=1}^k {\mathcal O}_{S,z_i}$} and \mbox{$X = X(C,T^\ast) = Z(J)$}, as before. $X$ is then a reducible subscheme of $S$, concentrated on \mbox{$z_1, \dots, z_k$}. Let $\mbox{$\mathrm{mt}$} \,X,\; \mu(X),\; \delta(X)$ denote the sum of the corresponding invariants at \mbox{$z_1, \dots, z_k$}. We need this generalization after blowing up. Let \mbox{$z \in C \subset S$} be a point and \mbox{$\widehat{S} \longrightarrow S$} be the blowing--up of $z$. We denote by $\widehat{C}$, respectively $C^\ast$, the total, respectively strict, transform of $C$, $E$ the reduced exceptional divisor and \mbox{$\hat{z} := E \cap C^\ast$}. \mbox{$(C^\ast\!, \hat{z}) \subset (\widehat{S}, \hat{z})$} is a (multi)germ. For any \mbox{$f \in {\mathcal O}_{S,z}$} satisfying \mbox{$\mbox{$\mathrm{mt}$} \,(f,z) \ge m := \mbox{$\mathrm{mt}$} \,(C,z)$} we may divide the total transform $\hat{f}$ by the $m$'th power of $E$ and we shall denote this multigerm at $\hat{z}$ by \mbox{$\hat{f} : mE$}. If \mbox{$m = \mbox{$\mathrm{mt}$} \,(f,z)$}, then \mbox{$\hat{f} :m E = f^\ast$} is the strict transform of $f$. Note that for \mbox{$q \in T(C,z)\backslash \{z\}$} \[ \mbox{$\mathrm{mt}$} \,(\hat{f}_{(q)}, q) = \mbox{$\mathrm{mt}$} \,((\hat{f} : mE)_{(q)}, q) + k(q)\cdot m, \] where \mbox{$k(q)\in {\mathbf N}$} is independent of $f$. This holds especially for \mbox{$f = C$}, hence we obtain for \mbox{$T^\ast(C,z) \subset T^\ast \subset T(C,z)$}: \begin{lemma}\label{1.5} With the above notations: \[ f \in J(C,T^\ast) \Leftrightarrow \left( \mbox{$\mathrm{mt}$} \,(f,z) \ge m \mbox{ and } \hat{f} : mE \in J(C^\ast, T^\ast \backslash\{z\})\right). \] \end{lemma} Let us denote by \mbox{$\mbox{$\mathrm{mt}$} \,(X,q) := m_q$}, \mbox{$q \in T^\ast_X$}, the {\bf multiplicity of $\bf X$ at $\bf q$} and by \mbox{deg$(X) := \dim_K({\mathcal O}_{S,z}/J_X)$} the {\bf degree of $\bf X$}. \begin{lemma}\label{1.6} For \mbox{$X \in {\mathcal G} {\mathcal S}$}, \mbox{$T^\ast \!= T^\ast_X$} and \mbox{$m_q = \mbox{$\mathrm{mt}$} \,(X,q)$} we have \[ \deg(X) \,=\, \sum_{q \in T^\ast} \frac{m_q(m_q + 1)}{2} \,=\, \delta(X) + \sum_{q \in T^\ast} m_q. \] \end{lemma} \begin{proof} The second equality follows from $$ \delta(C,z) = \sum_{q\in T^\ast} \frac{m_q(m_q-1)}{2}.$$ For the proof of the first one, cf.\ \cite{Casas-Alvero}, Proposition 6.1. \end{proof} \begin{lemma}\label{1.7} Let \mbox{$X \in {\mathcal S}$} be defined by a germ $(C,z)$ and let \mbox{$I^{es} \subset {\mathcal O}_{S,z}$} be the equisingularity ideal of $(C,z)$ in the sense of Wahl $($\cite{W}, cf.\ also \cite{DH}$)$, then \mbox{$I^{es} \supset J_X$}. \end{lemma} \begin{proof} Clear from the definitions. \end{proof} \begin{lemma}\label{1.8} Let \mbox{$Q_1, \dots, Q_r$} denote the branches of $(C,z)$ and \mbox{$J = J(C,T^\ast)$}. Then \[ f \in J \Leftrightarrow (f, Q_j) \ge 2\delta(Q_j) + \sum_{i\not= j} (Q_i, Q_j) + \sum_{q \in T^\ast \cap Q_j} \mbox{$\mathrm{mt}$} \,(Q_{j,(q)}, q), \] where \mbox{$T^\ast \!\cap Q_j = \{q \in T^\ast \!\mid q \in Q_{j,(q)}\}$} and $(f,g)$ denotes the intersection multiplicity. \end{lemma} \begin{proof} As shown before, the multiplicity of the strict transform at \mbox{$q\in T^\ast$} of a generic element \mbox{$g\in J$} fulfills \mbox{$\mbox{$\mathrm{mt}$} \,(g_{(q)},q)=\mbox{$\mathrm{mt}$} \,(X,q)=m_q$}. In particular, we obtain for each branch $Q_j$ of $(C,z)$ \begin{eqnarray*} \left(f,Q_j \right) \; \geq \; \left( g,Q_j \right) & \ge & \sum_{q\in T^\ast \cap Q_j} m_q \cdot \mbox{$\mathrm{mt}$} \,(Q_{j,(q)},q)\\ & = & \sum_{q\in T^\ast \cap Q_j} \sum_{i=1}^r \mbox{$\mathrm{mt}$} \,(Q_{i,(q)},q) \cdot \mbox{$\mathrm{mt}$} \,(Q_{j,(q)},q)\\ & = & 2\delta(Q_j) + \sum_{i\neq j} (Q_i,Q_j) + \sum_{q\in T^\ast \cap Q_j} \mbox{$\mathrm{mt}$} \,(Q_{j,(q)},q)\;=:\; \alpha_j\,. \end{eqnarray*} Hence, we have the inclusion \mbox{$ J \subset J_1 := \bigcap_{j=1}^r \left\{ f\in {\mathcal O}_{S,z} \mid (f,Q_j)\geq \alpha_j \right\}.$} We can consider both as ideals in ${\mathcal O}_{C,z}$ and have to show that \mbox{$\dim_K ({\mathcal O}_{C,z}/J_1) = \deg (X)$}. To do so, let $n$ denote the injection $$ n: {\mathcal O}_{C,z} \hookrightarrow \prod_{j=1}^r {\mathbf C}\{t_j\} =: \overline {{\mathcal O}} $$ induced by a parametrization of $(C,z)$ and consider the image \mbox{$n(J_1)\subset \overline{{\mathcal O}}$}. For an element \mbox{$f \in \overline{{\mathcal O}}$} the conditions on the intersection multiplicities \mbox{$(f,Q_j)$} read as \mbox{$f \in \prod_{j=1}^r t_j^{\alpha_j}\cdot {\mathbf C}\{t_j\}$}. Hence \begin{eqnarray*} \dim_K\left({\mathcal O}_{C,z}\big/J_1\right) & \geq & \dim_K\big(\overline{{\mathcal O}}\big/ \textstyle{\prod}_{j=1}^r \,t_j^{\alpha_j}\cdot {\mathbf C}\{t_j\}\big) - \dim_K \big( \overline{{\mathcal O}}\big/ {\mathcal O}_{C,z}\big)\\ & = & \sum_{j=1}^r \alpha_j - \delta(C,z)\; =\; \deg(X) .\qquad \qquad \qquad \qquad \;\;\qed \end{eqnarray*} \renewcommand{\qed}{}\end{proof} \begin{definition}\label{1.9} Two generalized singularities \mbox{$X_0, X_1 \in {\mathcal G} {\mathcal S}$}, centred at $z$, are called {\bf isomorphic}, $X_0 \cong X_1$, if they are isomorphic as subschemes of $S$. $X_0$ and $X_1$ are called {\bf equivalent}, \mbox{$X_0 \sim X_1$}, if there exist germs (respectively multigerms, if the $X_i$ are reducible) \mbox{$(C_0, z)$} defining $X_0$ and \mbox{$(C_1, z)$} defining $X_1$, and a $T^\ast \!$--equimultiple family over some (reduced) open connected subset $T$ of ${\mathbf A}^1$ having $(C_0,T^\ast_0)$ and $(C_1,T^\ast_1)$ as fibres. Here, by a {\bf $T^\ast \!$--equimultiple family} over a (reduced) algebraic $k$--scheme $T$ we denote a flat family with section $\sigma$, \arraycolsep0.1cm \renewcommand{\arraystretch}{1.1} $$ \begin{array}{rrl} \makebox[0.7cm]{${\mathcal C}$} & \hookrightarrow & S\times T\\ \makebox[0.3cm]{$\sigma \nwarrow$} & & \swarrow \\ & \makebox[0.1cm]{$T$ ,} \end{array} $$ of reduced plane curve singularities \mbox{$({\mathcal C}_t,\sigma(t))\subset S=S\times \{t \}$} which admits a simultaneous, embedded resolution, together with sections $\sigma_q$ through infinitely near points, defining a family ${\mathcal T}^\ast$ of trees $T^\ast_t$, \mbox{$T^\ast({\mathcal C}_t, \sigma(t)) \subset T^\ast_t \subset T({\mathcal C}_t,\sigma(t))$} such that the total transform of ${\mathcal C}$ is equimultiple along $\sigma_q$, \mbox{$\sigma_q(t)\in T^\ast_t$}. We denote such a family by {\bf $({\mathcal C},{\mathcal T}^\ast)$}. \end{definition} Since any \mbox{$X \in {\mathcal G}{\mathcal S}$} is defined by a generic element in \mbox{$J_X \subset {\mathcal O}_{S,z}$}, isomorphic schemes $X_0, X_1$ are defined by isomorphic germs which can be connected by a family of isomorphisms. Hence, \mbox{$X_0 \cong X_1$} implies \mbox{$X_0 \sim X_1$}. \begin{definition}\label{1.10} Let \mbox{$X \in {\mathcal G}{\mathcal S}$} with centre $z$ and \mbox{$L = (L,z)$} be a smooth (mul\-ti)germ. Define \mbox{$X \cap L$} to be the scheme--theoretic intersection. Set \begin{align*} T^\ast_X \cap L & := \{q \in T^\ast_X \mid q \in L_{(q)}\},\\ J_X : L & :=\{f \in {\mathcal O}_{S,z} \mid f L \in J_X\},\\ X : L & := Z(J_X :L). \end{align*} We call \mbox{$X : L$} the {\bf reduction of $\bf X$} by $L$. \end{definition} \begin{proposition}\label{1.11} Let \mbox{$X=X(C,T^\ast) \in {\mathcal G}{\mathcal S}$} and \mbox{$L \subset S$} be smooth at $z$, the centre of $X$. \begin{enumerate} \item[(i)] The reduction \mbox{$X :L$} is a generalized singularity centred at $z$ and its tree \mbox{$T^\ast_{X:L} = T^\ast\!:L$} is a subtree of $T^\ast$. \item[(ii)] \mbox{$\mbox{$\mathrm{mt}$} \,(X,q)-1 \le \mbox{$\mathrm{mt}$} \,(X:L,q) \le \mbox{$\mathrm{mt}$} \,(X,q)$} for all \mbox{$q \in T^\ast$}. \item[(iii)] \mbox{$\deg(X:L) = \deg(X) - \deg(X \cap L)$}, \mbox{$\;\deg(X \cap L) = \! \underset{q \in T^\ast \cap L}{\sum} \mbox{$\mathrm{mt}$} \,(X,q)$}. \item[(iv)] Let $L$ be a line in $\P^2$, then there exists an exact sequence of ideal sheaves on $\P^2$ \[ 0 \longrightarrow {\mathcal J}_{X:L/\P^2}(d\!-\!1) \stackrel{\cdot L}{\longrightarrow} {\mathcal J}_{X/\P^2}(d) \longrightarrow {\mathcal J}_{X \cap L/L}(d) \longrightarrow 0. \] \end{enumerate} \end{proposition} \begin{proof} (iv) is obvious and (iii) follows from (iv), respectively the fact that $$\deg (X\cap L) = \mbox{$\mathrm{mt}$} \,(C\cap L,z) = \!\sum_{q\in T^\ast \cap L} \!\mbox{$\mathrm{mt}$} \,(X,q).$$ (i) will be proved by induction on $\deg(X)$. Again, we may begin with \mbox{$\deg(X) = 0$}, which implies \mbox{$X \!= \emptyset$}, \mbox{$T^\ast \!= \emptyset$} and \mbox{$X : L = \emptyset$}, \mbox{$T^\ast \!: L = \emptyset$}. If \mbox{$\deg(X) > 0$} then \mbox{$z \in T^\ast$} and we consider the blowing--up \mbox{$\pi : \widehat{S} \longrightarrow S$} of $z$. By Lemma \ref{1.6} the strict transform $C^\ast$ of $C$ fulfills \mbox{$\deg(X(C^\ast \!, T^\ast\backslash\{z\})) < \deg(X)$}, hence by the induction assumption there exists a (multi)germ $D^{\ast}$ and a subtree \mbox{$T_{D^\ast} \subset T^\ast \backslash\{z\}$} such that \begin{gather*} J(C^{\ast}, T^\ast \backslash\{z\}) : L^\ast = J(D^{\ast}, T_{D^\ast})\\ T^\ast(D^{\ast}) \subset T_{D^\ast} \subset T(D^{\ast}), \end{gather*} moreover, we can choose $D^{\ast}$ generically in \mbox{$J(C^\ast, T^\ast \backslash\{z\}) : L^\ast$} (such that $\mbox{$\mathrm{mt}$} \,(D^\ast)$ and the intersection multiplicity $(D^\ast \!, E)$ are minimal). Blowing down $D^{\ast}$ we obtain a germ $D$ at $z$. Let \mbox{$m:=\mbox{$\mathrm{mt}$} \,(C,z)$}. \underline{Case 1}: $\mbox{$\mathrm{mt}$} \,(D,z) = m-a < m$. Define the germ $C^\prime$ at $z$ by \mbox{$C^\prime := D \cdot L_1 \cdot \ldots \cdot L_{a-1}$}, where \mbox{$L_1, \dots, L_{a-1}$} are smooth germs with generic tangent directions at $z$. Then \[ \begin{array}{lcl} f \in J(C,T^\ast) : L & \!\Leftrightarrow & \!\mbox{$\mathrm{mt}$} \,(f,z) \ge m\!-\!1,\;L^\ast \hat{f} \!:\! (m\!-\!1) E \in J(C^{\ast}, T^\ast \backslash\{z\})\\ & \!\Leftrightarrow & \!\mbox{$\mathrm{mt}$} \,(f,z) \ge m\!-\!1,\; \hat{f}\! :\! (m\!-\!1) E \in J(D^{\ast},T_{D^\ast})\\ & \!\Leftrightarrow & \!\mbox{$\mathrm{mt}$} \,(f,z) \ge m\!-\!1, \;\hat{f}\! :\! (m\!-\!1) E \in J(D^{\ast} \!\!\cdot L^\ast_1 \!\cdot \ldots \cdot L^\ast_{a-1}, T_{D^\ast})\\ & \!\Leftrightarrow & \!f \in J (C^\prime, T^{\prime\ast}) \end{array} \] with \mbox{$T^{\prime\ast} := T_{D^\ast} \cup \{z\}$}. \underline{Case 2}: \mbox{$\mbox{$\mathrm{mt}$} \,(D,z) = m$}. By the induction assumption, there exists a (multi)germ $\bar{D}^\ast$ and a subtree \mbox{$T_{\bar{D}^\ast} \subset T_{D^\ast} \subset T^\ast \backslash\{z\}$} such that \begin{gather*} J(D^{\ast}, T_{D^\ast}) : E = J(\bar{D}^\ast, T_{\bar{D}^\ast})\\ T^\ast(\bar{D}^\ast) \subset T_{\bar{D}^\ast} \subset T(\bar{D}^\ast). \end{gather*} We choose $\bar{D}^\ast$ generically in the ideal \mbox{$J(D^\ast, T_{D^\ast}) : E$}. Since \mbox{$D^{\ast}\! \in J(\bar{D}^\ast\!, T_{\bar{D}^\ast})$}, \mbox{$p:= (D^{\ast}\!, E)\! -\! (\bar{D}^{\ast}\!, E) \ge 0$}, and we define \mbox{$C^\prime :=\bar{D} \cdot L_1 \cdot \ldots \cdot L_p$}, where \mbox{$L_1, \dots, L_p$} denote generic smooth germs at $z$ and $\bar{D}$ is the blowing--down of $\bar{D}^\ast$. Again \[ f \in J(C,T^\ast) :L \;\Leftrightarrow \; \mbox{$\mathrm{mt}$} \,(f,z) \ge m\!-\!1, \;L^\ast \hat{f} : (m\!-\!1) E \in J (C^{\ast}, T^\ast \backslash \{z\}). \] Assuming \mbox{$\mbox{$\mathrm{mt}$} \,(f,z) = m\!-\!1$} and \mbox{$L^\ast \hat{f} :(m\!-\!1) E \in J(C^{\ast}, T^\ast \backslash\{z\})$} we would have \mbox{$f^\ast \!\in J(C^{\ast}\!, T^\ast \backslash\{z\}) : L^\ast = J(D^{\ast}\!,T_{D^\ast})$}, hence \mbox{$m\!-\!1 = (f^\ast \!, E) \ge (D^{\ast} \!, E) = m$}. Thus \[ \begin{array}[l]{lcl} f \in J(C,T^\ast) : L & \Leftrightarrow & \mbox{$\mathrm{mt}$} \,(f,z) \ge m, \;\: \hat{f} : mE\in J(D^{\ast}, T_{D^\ast}) :\!E = J(\bar{D}^\ast, T_{\bar{D}^\ast})\\ & \Leftrightarrow & \mbox{$\mathrm{mt}$} \,(f,z) \ge m,\;\: \hat{f} : mE \in J(\bar{D}^\ast \cdot L^\ast_1 \cdot \ldots \cdot L^\ast_p, T_{\bar{D}^\ast})\\ & \Leftrightarrow & f \in J(C^\prime,T^{\prime\ast}\!:= T_{\bar{D}^\ast} \cup \{z\}). \end{array} \] Note that in this case we have \mbox{$\mbox{$\mathrm{mt}$} \,(C^\prime,z)=\mbox{$\mathrm{mt}$} \,(C,z)$}, while in the first case we had \mbox{$\mbox{$\mathrm{mt}$} \,(C^\prime,z)=\mbox{$\mathrm{mt}$} \,(C,z)-1$}. This implies (ii). \end{proof} \begin{examples} \begin{enumerate} \itemsep0.1cm \item Let \mbox{$(C,z)$} be a node, \mbox{$C=y^2-x^2$} with respect to local coordinates \mbox{$(x,y)$} at $z$, \mbox{$T^\ast=T^\ast(C,z)=\{z\}$}, then for each $L$ the reduction \mbox{$X:L$} is the generalized singularity given by a smooth germ at $z$ and the tree \mbox{$T^\ast_{X:L}=T^\ast=\{z\}$}. \item In the case of an $A_{2k-1}$--singularity (\mbox{$k\geq 2$}) \mbox{$C=y^2-x^{2k}$} with the tree of essential points \mbox{$T^\ast=\{z=q_0,\dots,q_{k-1}\}$} we have \mbox{$J_X=\langle y^2,yx^k,x^{2k}\rangle $}. If \mbox{$L=y$}, then \mbox{$J_{X:L}=\langle y,x^k\rangle$} and \mbox{$X:L$} is the generalized singularity given by the smooth germ \mbox{$y-x^k$} and the tree \mbox{$T^\ast_{X:L}=\{z=q_0,\dots ,q_{k-1}\}$}. On the other hand, if $L=x$, then \mbox{$J_{X:L}=\langle y^2,yx^{k-1},x^{2k-1}\rangle$} and \mbox{$X:L$} is the generalized singularity scheme given by \mbox{$(y-x^k)(y+x^{k-1})$} and the tree \mbox{$T^\ast_{X:L}=T^\ast $}. \item For an $A_{2k}$--singularity \mbox{$C=y^2-x^{2k+1}$} with the tree of essential points \mbox{$T^\ast=\{z=q_0,\dots,q_{k+1}\}$} we have \mbox{$J_X=\langle y^2,yx^{k+1},x^{2k+1}\rangle $}. If \mbox{$L=y$}, then \mbox{$J_{X:L}=\langle y,x^{k+1} \rangle$} and \mbox{$X:L$} is the generalized singularity given by the smooth germ \mbox{$y-x^{k+1}$} and the tree \mbox{$T^\ast_{X:L}=\{z=q_0,\dots,q_k\}$}. On the other hand, if $L=x$, then \mbox{$J_{X:L}=\langle y^2,yx^k,x^{2k}\rangle $} and \mbox{$X:L$} is the singularity scheme given by an $A_{2k-1}$--singularity. \end{enumerate} \end{examples} \begin{definition}\label{1.12} Denote by \mbox{${\mathcal G}{\mathcal S}_1 \subset {\mathcal G}{\mathcal S}$} the subclass of such $X$ defined by germs $(C,z)$ with all branches smooth. \end{definition} \smallskip \begin{lemma}\label{1.13} The class ${\mathcal G}{\mathcal S}_1$ is closed with respect to the equivalence relation $\sim$ and with respect to reduction by $L$. \end{lemma} \begin{proof} The first statement is obvious, the second is a consequence of the proof of \ref{1.11}. \end{proof} \begin{lemma}\label{1.14} Let \mbox{$X = X(C,T^\ast) \in {\mathcal G}{\mathcal S}$} be non--empty and $L$ smooth at $z$. \begin{enumerate} \item[(i)] There exists a branch $Q$ of $(C,z)$ such that \mbox{$T^\ast \cap L \subset T^\ast \cap Q$}. \item[(ii)] If $Q$ is a non--singular branch of $(C,z)$ and if \mbox{$M \subset T^\ast$} is a connected subtree with \mbox{$T^\ast\!\cap L \subset M \subset T^\ast\!\cap Q$} then there exists \mbox{$X_1 = X(C_1, T^\ast_1) \in {\mathcal G}{\mathcal S}$}, \mbox{$X_1 \cong X$}, $Q_1$ a smooth branch of \mbox{$(C_1, z) \cong (C,z)$}, \mbox{$M \cong M_1 \subset Q_1 \cap T^\ast_1$} such that \mbox{$T^\ast_1 \cap L = M_1$}. \end{enumerate} \end{lemma} \begin{proof} (i) is obvious. For the proof of (ii), we may assume that \mbox{$z\in L\cap T^\ast$}. We choose coordinates $(x,y)$ at z such that \mbox{$L=y$} and the non--singular branch $Q$ is given by \mbox{$y-\sum_{i=1}^\infty \alpha_i x^i$}. Let \mbox{$N:=\#(M)$} and $f$ be the power series defining \mbox{$(C,z)$}. The germs $$ C_t\,=\,f\left(x\,,\,y+t\cdot \!\left(\textstyle{\sum\limits_{i=1}^{N-1}} \alpha_i x^i+\beta x^N\right)\right)\,, \;\;\beta \in K \mbox{ generic,}$$ define an equianalytic family such that \mbox{$(C_0,z)=(C,z)$} and \mbox{$(C_1,z)$} has a branch $Q_1$ given by \mbox{$y-\sum_{i\geq N} \tilde{\alpha}_i x^i$}. Especially for the corresponding trees \mbox{$T^\ast_1\cong T^\ast$}, respectively \mbox{$M_1\cong M$}, we have \mbox{$M_1 \subset Q_1 \cap T^\ast_1$} and, since $\beta$ was chosen generically, $$ \#(T^\ast_1 \cap L )= \#(T^\ast_1\cap L \cap Q_1) = N = \#(M_1), $$ hence \mbox{$T^\ast_1\cap L = M_1$}. \end{proof} \pagebreak[3] \begin{lemma}\label{1.15} Let \mbox{$X = X(C,T^\ast) \in {\mathcal G}{\mathcal S}$}, $Q$ be a smooth branch of $(C,z)$ and $L$ be smooth at $z$. \begin{enumerate} \item[(i)] If \mbox{$T^\ast \cap Q \subset T^\ast \cap L$} then \mbox{$\mbox{$\mathrm{mt}$} \, (X:L) = \mbox{$\mathrm{mt}$} \, X -1$}. \item[(ii)] If \mbox{$T^\ast \cap Q = T^\ast \cap L$} then \mbox{$X :L$} is defined by the germ $(C^\prime \!,z)$ and the tree \mbox{$T^\ast\!\cap T(C^\prime \!,z)$}, where $C^\prime$ is a factor of $C$ such that \mbox{$C = C^\prime Q$}. \end{enumerate} \end{lemma} \begin{proof} Let \mbox{$C=C^\prime\cdot Q$} at $z$. Then, obviously \mbox{$C^\prime \in J_X:L$}, hence the multiplicity of a generic element is at most \mbox{$\mbox{$\mathrm{mt}$} \,X-1$}. Thus, (i) follows from \ref{1.11} (ii).\\ If \mbox{$T^\ast \!\cap Q = T^\ast \!\cap L$}, then we have for each \mbox{$q\in T^\ast$} $$ \mbox{$\mathrm{mt}$} \,(\widehat{C}^\prime_{(q)},q)= \mbox{$\mathrm{mt}$} \,(\widehat{C}_{(q)},q)-\mbox{$\mathrm{mt}$} \,(\widehat{L}_{(q)},q),$$ which implies (ii). \end{proof} \begin{lemma}\label{1.16} Let \mbox{$X = X(C,T^\ast) \in {\mathcal G}{\mathcal S}$} and $(L,z)$ be a smooth germ. Then we have \mbox{$\deg((X:L) \cap L) \le \deg(X \cap L)$}. Moreover, \begin{enumerate} \item[(i)] if \mbox{$\deg((X:L) \cap L) = \deg (X \cap L)$} then \mbox{$\mbox{$\mathrm{mt}$} \,(X:L) = \mbox{$\mathrm{mt}$} \,X$}; \item[(ii)] if \mbox{$\deg((X:L) \cap L)) < \deg(X \cap L)$} then either \mbox{$\mbox{$\mathrm{mt}$} (X:L) < \mbox{$\mathrm{mt}$} \, X $} or the defining germ $(C',z)$ of \mbox{$X:L$} has a branch $Q'$ satisfying \mbox{$ T^\ast_{X:L} \cap Q' \subset T^\ast_{X:L} \cap L$}. In any case, \mbox{$\mbox{$\mathrm{mt}$} (X:L^2) \le \mbox{$\mathrm{mt}$} \, X -1$}. \end{enumerate} \end{lemma} \begin{proof} By Prop.\ \ref{1.11} (ii), \mbox{$\mbox{$\mathrm{mt}$} \,(X:L,q) \le \mbox{$\mathrm{mt}$} \,(X,q)$} for all \mbox{$q \in T^\ast$}, hence the inequality. Therefore, \mbox{$\deg((X:L) \cap L) = \deg(X \cap L)$} implies \mbox{$\mbox{$\mathrm{mt}$} \,(X:L,q) = \mbox{$\mathrm{mt}$} \,(X,q)$} for any \mbox{$q \in T^\ast\! \cap L$}, in particular for \mbox{$q = z$}. This implies (i).\\ Let \mbox{$\deg((X:L) \cap L) < \deg(X \cap L)$} and \mbox{$T^\ast\! \cap L=\{z\!=\!q_0,q_1,\dots,q_\ell\}$}. Recall the construction of \mbox{$X:L=X(C^\prime,T^{\prime\ast})$} in the proof of Proposition \ref{1.11} (i). In \mbox{Case 1} we had $$ \mbox{$\mathrm{mt}$} \,(X:L)=\mbox{$\mathrm{mt}$} \,(C^\prime \!,z)=\mbox{$\mathrm{mt}$} \,(C,z)-1=\mbox{$\mathrm{mt}$} \,X-1. $$ In Case 2, $C^\prime$ was given as $$ C^\prime=\bar{D}\cdot L_1 \cdot \ldots \cdot L_p \:\mbox{ with }\:p=(C^\ast \!,E)-(\bar{D}^\ast \!,E).$$ Assume that there is no branch $Q'$ of $C^\prime$ such that \mbox{$T^{\prime\ast} \!\cap Q' \subset T^{\prime\ast} \!\cap L$}, in particular, \mbox{$p=0$}. Then \mbox{$\bar{D}^\ast\!=C^{\prime\ast}$}, the strict transform of $C^\prime$, and $$\mbox{$\mathrm{mt}$} \,(C_{(q_0)})=(C^\ast \!,E)=(C^{\prime\ast} \!,E)= \mbox{$\mathrm{mt}$} \,(C^\prime_{(q_0)}).$$ On the other hand, the intersection multiplicity \mbox{$(C^{\prime\ast} \!,E)$} is just the sum of all \mbox{$\mbox{$\mathrm{mt}$} \,(C^\prime_{(q)})$} with \mbox{$q\in T(C^\prime)\cap E_{(q)}$}. By the above assumption all those $q$ are essential for $C^\prime$, which implies $$ \sum_{q\in T^\ast(C^\prime)\cap E_{(q)}} \mbox{$\mathrm{mt}$} \,(C^\prime_{(q)})= (C^{\prime\ast} \!,E) = (C^\ast \!,E)= \sum_{q\in T(C)\cap E_{(q)}} \mbox{$\mathrm{mt}$} \,(C_{(q)}).$$ Since \mbox{$T^\ast(C^\prime)\subset T^\ast(C)$}, it follows \mbox{$\mbox{$\mathrm{mt}$} \,(C^\prime_{(q)})=\mbox{$\mathrm{mt}$} \,(C_{(q)})$} for all \mbox{$q\in T^\ast(C^\prime)\cap E_{(q)}$}, especially \mbox{$\mbox{$\mathrm{mt}$} \,(C^\prime_{(q_1)})=\mbox{$\mathrm{mt}$} \,(C_{(q_1)})$}. By induction, we obtain \mbox{$\mbox{$\mathrm{mt}$} \,(C^\prime_{(q_i)})=\mbox{$\mathrm{mt}$} \,(C_{(q_i)})$} for each \mbox{$i\in \{0,\dots\!\,,\ell\}$}, which is impossible (cf.~Proposition \ref{1.11} (iii)). \end{proof} \begin{definition} Given a $T^\ast\!$--equimultiple family of plane curve singularities \mbox{$({\mathcal C},{\mathcal T}^\ast)$} over a reduced algebraic $K$--scheme $T$ (as defined in \ref{1.9}), we define $$ \mathfrak{J}({\mathcal C},{\mathcal T}^\ast):=\left\{ f\!\in\! {\mathcal O}_{S\times T,\sigma(T)} \,\big| \, \mbox{$\mathrm{mt}$} \,(\hat{f}_{\sigma_q(t)},\sigma_q(t))\geq \mbox{$\mathrm{mt}$} \,(\hat{{\mathcal C}}_{\sigma_q(t)},\sigma_q(t) ) \mbox{ for } q \!\in \!T^\ast_t,\, t\!\in \!T\right\} $$ and $${\mathcal X}({\mathcal C},{\mathcal T}^\ast):= {\mathcal O}_{S\times T,\sigma(T)}\big/\mathfrak{J} ({\mathcal C},{\mathcal T}^\ast).$$ A flat family ${\mathcal X}$ of fat points in \mbox{$S\times T$} is called a {\bf family of generalized singularity schemes}, if \mbox{${\mathcal X} ={\mathcal X}({\mathcal C},{\mathcal T}^\ast)$} for some $T^\ast\!$--equimultiple family \mbox{$({\mathcal C},{\mathcal T}^\ast)$}. \end{definition} Since we consider only reduced base spaces $T$, then flatness just means that the total length is constant, which holds for a family \mbox{${\mathcal X}({\mathcal C},{\mathcal T}^\ast)$} by Lemma \ref{1.6}. It is easily seen that the functor $$\underline{{\mathcal G}{\mathcal S}} \,: \:T \mapsto \{\mbox{families of generalized singularity schemes over }T \}$$ is representable by a locally closed subscheme $GS$ of the punctual Hilbert scheme of $S$. \begin{proposition}\label{1.17a} Let \mbox{$X \in {\mathcal G}{\mathcal S}$}, $L$ be smooth and \mbox{$Y = X:L$}. For almost all \mbox{$Y^\prime \sim Y$} satisfying \mbox{$\deg(Y^\prime \cap L) = \deg(Y \cap L)$} there exists a generalized singularity scheme \mbox{$X^\prime \sim X$} such that \mbox{$\deg(X^\prime \cap L) = \deg(X \cap L)$} and \mbox{$Y^\prime = X^\prime :L$}. \end{proposition} \begin{proof} \footnote{We should like to thank I.~Tyomkin for an idea leading to the present proof.} Let \mbox{${\mathcal X} ={\mathcal X}({\mathcal C},{\mathcal T}^\ast)$} be a family of generalized singularity schemes over the reduced base space $T$, \mbox{$t\in T$}. The construction of \mbox{$X:L$} given in the proof of \ref{1.11} shows that we can simultaneously reduce the fibres of ${\mathcal X}$ by $L$. Hence we have a natural transformation $$ \rho_L: \underline{{\mathcal G}{\mathcal S}}\longrightarrow \underline{{\mathcal G}{\mathcal S}}\:,\; {\mathcal X} \mapsto {\mathcal X} :L $$ inducing a morphism \mbox{$\rho_L : GS \longrightarrow GS$}. Notice that two generalized singularity schemes $X_1$,$X_2$ are equivalent if and only if they are in the same connected component of $GS$. Therefore, to prove the proposition, it is enough to show that the restriction \mbox{$ \rho_{L,X}: GS_{L,X} \longrightarrow GS_{L,Y}$} of $\rho_L$ to the connected component $GS_{L,X}$ of \mbox{$\{X^\prime \in GS \mid \deg(X^\prime \!\cap L)= \deg(X\cap L)\}$} containing $X$ is dominant. But this follows immediately from the fact that the dimension of the fibre \mbox{$ \rho_{L,X}^{-1}(Y)$} is just \mbox{$ \#(T^\ast_X)-\left(\#(T^\ast_Y)+\#(T^\ast_X \cap L) -\#(T^\ast_Y \cap L)\right) = \dim (GS_{L,X}) - \dim (GS_{L,Y}).$} \end{proof} In the following, we shall introduce the second basic operation on generalized singularities, the extension. For this, it is convenient to work with the field $K\{\{x\}\}$ of fractional power series \[ \sum^\infty_{i=0} \alpha_i x^{i/n}, \alpha_i \in K, n \in {\mathbf N}. \] Any germ $(C,z)$ of a reduced curve singularity may be given, with respect to suitable local coordinates $x,y$, as \[ C = \prod^m_{i=1} (y - \xi_i(x)),\;\; m = \mbox{$\mathrm{mt}$} \,(C,z),\;\; \xi_1, \dots, \xi_m \in K\{\{x\}\}. \] Moreover, if $(C,z)$ is irreducible and $t=x^{1/m}$, then $$C = \prod^m_{i=1} (y - \xi(\eta^it)),\;\; \xi \in K[[t]],$$ with $\eta$ a primitive $m$--th root of unity. We define the intersection multiplicity of two fractional power series $\xi_i, \xi_j\in K\{\{x\}\} $ to be \[ (\xi_i, \xi_j) := \max \,\{\rho \in {\mathbf Q} \mid x^\rho \mbox{ divides } \xi_i(x) - \xi_j(x)\}. \] \begin{lemma}\label{1.17} Let \mbox{$X = X(C,T^\ast(C,z)) \in {\mathcal S}$} be a singularity scheme and $(C,z)$ given as above. Then \[ \deg(X) = \sum_{1\le i < j \le m} (\xi_i, \xi_j) + \sum^m_{i=1} \underset{j \not=i}{\max} \,(\xi_i, \xi_j) + \frac{m-r}{2}, \] where \mbox{$m = \mbox{$\mathrm{mt}$} \,(C,z)$} and $r$ is the number of branches of $(C,z)$. \end{lemma} \begin{proof} It is well--known that the intersection multiplicity at $z$ of the polar $P_q(C)$ (\mbox{$q=(0\!:\!1\!:\!0)$}) given by the power series \[ \frac{\partial C}{\partial y} = \sum_{i=1}^m \prod_{j\neq i} (y-\xi_j(x)) \] and the curve C fulfills \mbox{$ \sum_{i\neq j} (\xi_i,\xi_j)=\mbox{$\mathrm{mt}$} \,(P_q(C)\cap C,z)=2\delta(C,z)+m-r$}.\\ Hence, by Lemma \ref{1.6}, it suffices to show \[ \sum_{q\in T^\ast(C)} m_q=\sum_{i=1}^m \underset{j\neq i}{\max} \,(\xi_i,\xi_j)+m-r. \] In the case of an irreducible germ $(C,z)$ it follows from the above description of the $\xi_i$ that the numbers $(\xi_i,\xi_j)$ do only depend on the characteristic terms of the Puiseux expansion, and the statement is an immediate consequence of the algorithm to compute the multiplicity sequence from the Puiseux pairs (cf.~\cite{BK}).\\ In the case of a reducible germ $(C,z)$, we have to investigate, additionally, the case of two branches \mbox{$Q_k=\prod_{i=1}^{m_k} (y-\xi^{(k)}(\eta_k^i t))$} \mbox{($k\!\in\!\{1,2\}$)} such that \mbox{$T(Q_1)\cap T(Q_2)$} contains a non--essential point $q$ of $T(Q_1)$ and for all branches \mbox{$Q\neq Q_1$} of \mbox{$(C,z)$} and all successors $\hat{q}$ of $q$ in \mbox{$T(Q_1)$} we have \mbox{$\hat{q}\not\in T(Q)$}. In this case, obviously, \mbox{$m_2=M m_1$}, \mbox{$M\in {\mathbf N}$}, and we can assume the maximum intersection multiplicity of the fractional power series \mbox{$\xi^{(1)}_i (x)=\xi^{(1)}(\eta^{Mi}x^{1/m_1})$} ($\eta $ a primitive $m_2$--th root of unity, \mbox{$i\in \{1,\dots,m_1\}$}) with any other fractional power series in the equation of $(C,z)$ to be realized by \mbox{$\xi^{(2)}_{Mi} (x)=\xi^{(2)}(\eta^{Mi}x^{1/m_2})$}. Then we have \begin{eqnarray*} \sum_{q\in T^\ast(C)} m_q(Q_1) & = & \frac{1}{M} \sum_{q\in T^\ast(C)} m_q(Q_1)m_q(Q_2) - \sum_{q\in T^\ast(C)} m_q(Q_1)(m_q(Q_1)-1) \\ &=& \frac{1}{M} \cdot \mbox{$\mathrm{mt}$} \,(Q_1\cap Q_2,z)-2\cdot \delta (Q_1)\\ & = & \sum_{i=1}^{m_1} \sum_{j=1}^{m_1} (\xi_i^{(1)},\xi_{Mj}^{(2)}) - \sum_{i\neq j} (\xi_i^{(1)},\xi_j^{(1)}) + m_1 -1, \end{eqnarray*} and the statement follows from the fact that for \mbox{$i\neq j$} the intersection multiplicities \mbox{$(\xi_i^{(1)},\xi_j^{(1)})$} and \mbox{$ (\xi_i^{(1)},\xi_{Mj}^{(2)})$} coincide. \end{proof} \begin{lemma}\label{1.18} Let \mbox{$X = X(C,T^\ast) \in {\mathcal G}{\mathcal S}$}, $L$ be smooth at $z$ and \mbox{$q \in T^\ast \!\cap L\backslash \{z\}$}. Let \[ C = \prod^n_{i=1} (y - \xi_i(x)) \prod^m_{i=n+1} (y-\xi_i(x)),\;\; m = \mbox{$\mathrm{mt}$} \,(C,z),\;\; \xi_i \in K\{\{x\}\}, \] be decomposed so that \mbox{$\xi_1, \dots, \xi_n$} are all fractional power series belonging to branches $Q$ of $(C,z)$ with $q \in Q_{(q)}$. Then there exists an integer \mbox{$k \ge 0$} such that \begin{align*} k < (\xi_i, \xi_j) & \mbox{ for } 1 \le i < j \le n,\\ k \ge(\xi_i, \xi_j) & \mbox{ for } 1 \le i \le n < j \le m. \end{align*} More precisely, if \mbox{$L=y$}, then \[ \xi_i(x) = \sum_{\rho \ge 0} \alpha^{(i)}_\rho x^\rho,\quad \alpha^{(i)}_\rho \in K,\; \rho \in {\mathbf Q} \] belongs to $Q$ with \mbox{$q \in Q_{(q)}$} if and only if \mbox{$\alpha^{(i)}_\rho \!= 0$} for \mbox{$\rho \le k$}. \end{lemma} \begin{proof} Let $L=y$ and \mbox{$T^\ast \!\cap L = \{z\!=\!q_0,q_1,\dots,q_\ell\}$}. Moreover, let the branch $Q$ be given by \[ Q=y^p+a_1(x)y^{p-1}+\dots +a_p(x)= \prod_{i=1}^p\left(y-\xi (\eta^i x^{1/p})\right), \] where \mbox{$\xi(t)=\sum_{j=0}^\infty \alpha_j t^{j} \in K[[t]]$}. To prove the lemma, it is sufficient to show that for each \mbox{$k\in \{1,\dots \!\,,\ell \}$} \[ q_k\in Q_{(q_k)} \;\;\Leftrightarrow \;\; \alpha_j=0 \,\mbox{ for each }\, j\leq k\!\cdot \! p. \] We proceed by induction on the length $k+1$ of the tree \mbox{$\{q_0,\dots, q_k\}$}. Obviously, \mbox{$z=q_0\in Q$} if and only if \mbox{$a_p(0)=0$}, that is, if and only if \mbox{$\alpha_0=0$}. Furthermore, the total transform of $Q$ at $q_1$ reads as \mbox{$\widehat{Q}_{(q_1)}=\prod_{i=1}^p (uv-\xi(\eta^i u^{1/p}))$}, hence \mbox{$q_1\in Q_{(q_1)}$} if and only if \mbox{$\xi(t)=\sum_{j=p+1}^\infty \alpha_j t^{j} $}. Then $Q_{(q_1)}$ has the equation \[ Q_{(q_1)} = \prod_{i=1}^p \left(v-\tilde{\xi}(\eta^iu^{1/p})\right) \] at $q_1$, where \mbox{$\tilde{\xi}(t)=\sum_{j=1}^\infty \alpha_{j+p} t^{j}$}, and we complete the proof by applying the induction hypothesis to $Q_{(q_1)}$ and the tree \mbox{$\{q_1,\dots , q_k\}\subset \left(T^\ast\backslash\{z\}\right)\cap L$}. \end{proof} \begin{definition}\label{1.19} Using the notations and hypotheses of Lemma \ref{1.18}, let \[ \xi_i(x) = \sum_{\rho > k} \alpha_\rho^{(i)} x^\rho,\; i = 1, \dots, n. \] Define a germ \mbox{$(C(q), z)$} by \[ C(q) := \prod^n_{i=1} (y - x \xi_i(x)) \prod^m_{i = n+1} (y - \xi_i(x)). \] Call $C(q)$ the {\bf extension of} $\bf C$ at $q$. \end{definition} \begin{lemma}\label{1.20} The tree $T^\ast(C(q))$ of essential points of $C(q)$ has the following structure: insert in $T^\ast(C)$ a new point $q^\prime$ between $q$ and its predecessor $\bar{q}$. Moreover, \mbox{$\mbox{$\mathrm{mt}$} \,(C(q)_{(p)},p) = \mbox{$\mathrm{mt}$} \,(C_{(p)},p)$} for all \mbox{$p \in T^\ast(C(q))\backslash \{q^\prime\} = T^\ast(C)$} and \[ \mbox{$\mathrm{mt}$} \,(C(q)_{(q^\prime)}, q^\prime) = \sum_{Q:\;q\in Q_{(q)}} \mbox{$\mathrm{mt}$} \,(Q_{(\bar{q})}, \bar{q}). \] \end{lemma} Any tree $T^\ast$ containing $T^\ast(C)$ becomes extended by this operation to a tree $T^\ast(q)$. We call $T^\ast(q)$ the {\bf extension of} $\bf T^\ast$ at $q$. \begin{proof} As in the proof of \ref{1.18}, let \mbox{$T^\ast\!\cap L=\{z\!=\!q_0,q_1,\dots , q_\ell\}$}. An easy consideration shows that for \mbox{$q=q_k$} the strict transform \mbox{$C(q)_{(q)}$} of the extension of $C$ at $q$ has the local equation \[ \prod_{i=1}^n \big(v-\frac{1}{u^{k-1}}\xi_i(u)\big) = \prod_{i=1}^n \big(v-\textstyle{\sum\limits_{\rho >0}} \alpha^{(i)}_{\rho +k} u^{\rho +1}\big) \] while \mbox{$C(q)_{(q_{k+1})}$} is given by \[ \prod_{i=1}^n \big(v-\frac{1}{u^{k}}\xi_i(u)\big) = \prod_{i=1}^n \big(v-\textstyle{\sum\limits_{\rho >0}} \alpha^{(i)}_{\rho +k} u^{\rho }\big) \] which corresponds to the equation of $C_{(q)}$ at $q$. Hence, the structure of \mbox{$T^\ast(C(q))$} can be described as in the lemma. Moreover, notice that for \mbox{$i\leq k$} $$ \mbox{$\mathrm{mt}$} \,(C(q)_{(q_i)},q_i)= n+ \!\sum_{Q:\;q\not\in Q_{(q)}} \!\mbox{$\mathrm{mt}$} \,(Q_{(q_i)},q_i),$$ which implies the statement about the multiplicities. \end{proof} \begin{definition}\label{1.21} Let \mbox{$X = X(C,T^\ast) \in {\mathcal G}{\mathcal S}$}, $L$ be a smooth germ at $z$ and $q \in T^\ast \!\cap L \backslash \{z\}$. We define the {\bf extension of} $\bf X$ at $q$ to be \[ X(q) := X(C(q),\, T^\ast(q)) \in {\mathcal G}{\mathcal S}. \] \end{definition} \begin{examples} \begin{enumerate} \itemsep0.1cm \item Let \mbox{$(C,z)$} be an ordinary cusp, \mbox{$C=y^2-x^3$} with respect to local coordinates \mbox{$(x,y)$} at $z$, \mbox{$T^\ast=T^\ast(C,z)=\{z=q_0,q_1,q_2\}$} with (strict) multiplicities \mbox{$m_z=2$}, \mbox{$m_{q_1}=m_{q_2}=1$}; we write \mbox{$T^\ast= \underset{2}{\ast}-\underset{1}{\ast}-\underset{1}{\ast}$}\,.\\ Moreover, let \mbox{$L=y$}, that is \mbox{$T^\ast \cap L=\{q_0,q_1\}$}. The extension $X(q_1)$ is given by \mbox{$y^2-x^5$} and the tree \mbox{$T^\ast(q_1)= \underset{2}{\ast}-\underset{2}{\ast}-\underset{1}{\ast}- \underset{1}{\ast}$}\,. \item Let \mbox{$X\in {\mathcal S}$} be given by \mbox{$C=(y^2-x^3)(y^2-x^5)$} and the tree of essential points $$T^\ast=\underset{4}{\ast}- \underset{3}{\ast} <\!\!\! \renewcommand{\arraystretch}{0.7} \begin{array}{l} \overset{1}{\ast}-\overset{1}{\ast}\\ \underset{1}{\ast} \end{array}. $$ If \mbox{$L=y$}, then \mbox{$T^\ast\cap L=\{z=q_0,q_1,q_2\}$} and the extension $X(q_1)$ is given by \mbox{$(y^2-x^5)(y^2-x^7)$} and the tree $$T^\ast(q_1)=\underset{4}{\ast}- \underset{4}{\ast}-\underset{3}{\ast} <\!\!\! \renewcommand{\arraystretch}{0.7} \begin{array}{l} \overset{1}{\ast}-\overset{1}{\ast}\\ \underset{1}{\ast} \end{array}, $$ whence the extension $X(q_2)$ is given by \mbox{$(y^2-x^3)(y^2-x^7)$} and $$T^\ast(q_2)=\underset{4}{\ast}-\underset{3}{\ast} <\!\!\! \renewcommand{\arraystretch}{0.7} \begin{array}{l} \overset{2}{\ast}-\overset{1}{\ast}-\overset{1}{\ast}\\ \underset{1}{\ast} \end{array}. $$ \end{enumerate} \end{examples} \begin{lemma}\label{1.22} With the notations of the preceding definition, assume that $$H^1(S,{\mathcal J}_{X(q)/\P^2}(d)) = 0.$$ Then there exists an \mbox{$X^\prime \in {\mathcal G}{\mathcal S}$}, \mbox{$X^\prime \sim X$} such that \mbox{$\deg(X^\prime \cap L) = \deg(X \cap L)$} and \mbox{$H^1(S,{\mathcal J}_{X^\prime/\P^2}(d)) = 0$}. \end{lemma} \begin{proof} Let \mbox{$(x,y)$} be coordinates in a neighbourhood of \mbox{$z=(0,0)$}, such that \mbox{$L=y$} and $(C,z)$ is given as in Lemma \ref{1.18}. For \mbox{$t\in {\mathbf A}^1$} define \[ C_t:= \underbrace{\prod_{i=1}^n\big(y-((1\!-\!t)x+t)\cdot\xi_i(x)\big)}_{\textstyle C^1_t} \:\cdot \underbrace{\prod_{i=n+1}^m\big(y-\xi_i(x)\big)}_{\textstyle C^2}. \] For \mbox{$t\neq 0$} there is an obvious isomorphism \mbox{$\varphi_t: (C^1_t,z)\stackrel{\cong}{\longrightarrow} (C^1_1,z)=(C^1\!,z)\subset (C,z)$} and for two branches $Q^1_t$ (resp.~$Q^2$) of \mbox{$(C^1_t,z)$} (resp.~\mbox{$(C^2,z)$} the intersection mul\-ti\-plicity at $z$ fulfills \mbox{$\mbox{$\mathrm{mt}$} \,(Q^1_t\cap Q^2,z)= \mbox{$\mathrm{mt}$} \,(Q^1_1\cap Q^2,z)$}, hence \mbox{$(C_t,z)\sim (C,z)$}. Moreover, for \mbox{$t\neq 0$} sufficiently small, $C_t$ has an ordinary $n$--fold point at $z_t=(-t/(1\!-\!t),0)$.\\ Define $T^\ast_t$ as the union of $\{z_t\}$ with the tree $T^{2\ast}$ corresponding to $C^2$ and the tree induced by $\varphi_t$ from $T^{1\ast}$ (corresponding to $C^1$). $T^\ast_t$ is well--defined since \mbox{$L\cap C_t\supset L\cap C$} for each \mbox{$t\in {\mathbf A}^1$}. Thus, we have defined a family ${\mathcal X}$ with fibres \mbox{$X_t=X(C_t,T^\ast_t)$} centred at the multigerm \mbox{$\{z,z_t\}$}. Obviously ${\mathcal X}$ is flat in \mbox{$t=0$} since for small \mbox{$t\neq 0$} (by \ref{1.6} and \ref{1.20}) $$\deg (X_t) = \deg (X)+\frac{n(n+1)}{2} = \deg (X(q))= \deg(X_0).$$ Hence, the family $\mathfrak{J}$ of ideals \mbox{$J_{X_t}=J(C_t,T^\ast_t)$} is flat in \mbox{$t=0$}, which implies, by semicontinuity, the vanishing of \mbox{$H^1(S,{\mathcal J}_{X_t/\P^2}(d))$} for small \mbox{$t\neq 0$}. \end{proof} \begin{remark}\label{1.25}{\rm The family $C_t$ of the above proof defines a deformation of the germ $(C(q),z)$ to \mbox{$(C_t,\{z, z_t\})$}, where \mbox{$(C_t,z) \sim (C,z)$} and \mbox{$(C_t,z_t)$} is an ordinary $n$--fold point, $n$ as in Lemma \ref{1.20}. In particular, $(C(q), z)$ is a degeneration of a germ which is topologically equivalent to $(C,z)$. } \end{remark} \section{$h^1$--vanishing criterion for zero--dimensional schemes of class ${\mathcal G}{\mathcal S}_1$ in the plane} \setcounter{equation}{0} \begin{lemma} \label{2.1} For any \mbox{$d\ge 1$} and \mbox{$X\in{\mathcal G}{\mathcal S}_1$} satisfying \begin{equation} \label{2.2} \deg X<(3-2\sqrt{2})(d-\mbox{$\mathrm{mt}$} \, X)^2 \end{equation} there is \mbox{$X'\sim X$} with \mbox{$h^1({\mathcal J}_{X'/\P^2}(d))=0$}. \end{lemma} \begin{proof} We shall prove the following statement. Let \mbox{$L\subset\P^2$} be a fixed straight line. There exist \mbox{$\alpha,\beta\ge 0$} such that for any integer \mbox{$d\ge 1$} and \mbox{$X\in{\mathcal G}{\mathcal S}_1$} satisfying \begin{align} \deg X & \,\le \; \beta(d-\mbox{$\mathrm{mt}$} \, X)^2 \label{2.3}\\ \deg(X\cap L) & \,\le \; d-\alpha \,\frac{\deg X}{d} \label{2.4} \end{align} there exists \mbox{$X'\sim X$} with \mbox{$\deg(X'\cap L)=\deg(X\cap L)$}, \mbox{$h^1({\mathcal J}_{X'/\P^2}(d))=0$}. Moreover, in Step 2, we show that for our approach the maximal possible value for $\beta$ is attained at \begin{equation} \label{2.10} \alpha=\sqrt{2}+1,\quad \beta=3-2\sqrt{2}\: . \end{equation} {\it Step 1}. Assume that $X$ is an ordinary singularity, that means \mbox{$T^\ast_X=\{z\}$}. Then the ideal of $X$ in \mbox{${\mathcal O}_{\P^2,z}$} is defined by the vanishing of the coefficients of all monomials lying under the diagonal \mbox{$\,[(0,\mbox{$\mathrm{mt}$} \, X),(\mbox{$\mathrm{mt}$} \, X,0)]\,$} in the Newton diagram. Since \mbox{$\mbox{$\mathrm{mt}$} \, X<d$} by (\ref{2.3}), these (linear) conditions are independent, hence \mbox{$h^1({\mathcal J}_{X/\P^2}(d))=0$}. So, further on, we can suppose that \mbox{$\deg(X) >0$} and that $X$ is not an ordinary singularity. We proceed by induction in $d$. For \mbox{$d\leq 2$} there is nothing to consider. In the induction step, we reduce $X$ by $L$ and have to show \renewcommand{\arraystretch}{1.7} \begin{equation} \label{2.7} \begin{array}{c} \deg(X:L)\;=\;\deg X-\deg(X\cap L)\;\le\;\beta(d-1-\mbox{$\mathrm{mt}$} \,(X:L))^2 \ ,\\ \deg((X:L)\cap L)\;\le\;\deg(X\cap L)\;\le\; d-1-\alpha\,\displaystyle{\frac{\deg(X:L)}{d-1}} \: . \end{array} \end{equation} Then, by the induction assumption \mbox{$h^1({\mathcal J}_{Y/\P^2}(d-1))=0$} for some \mbox{$Y\sim X:L$}, \mbox{$\deg(Y\cap L)=\deg((X:L)\cap L)$}. By Proposition \ref{1.17a} there exists \mbox{$X'\sim X$} with \mbox{$X':L=Y$} and \mbox{$\deg(X'\cap L)=\deg(X\cap L)$}. Since \mbox{$h^1({\mathcal J}_{X\cap L/L}(d))=0$}, because \mbox{$\deg(X'\cap L)\le d-\alpha\cdot\deg X/d<d+1$}, we obtain by Proposition \ref{1.11} (iv) the desired relation \mbox{$h^1({\mathcal J}_{X'/\P^2}(d))=0$}. {\it Step 2}. Assume that \begin{equation} \label{2.6} \deg(X\cap L)\:=\:d-\alpha \,\frac{\deg X}{d}\: . \end{equation} Due to \mbox{$\mbox{$\mathrm{mt}$} \,(X:L)\le\mbox{$\mathrm{mt}$} \, X$}, (\ref{2.3}) and (\ref{2.6}), the first inequality in (\ref{2.7}) will follow from $$\beta(d-\mbox{$\mathrm{mt}$} \, X)^2-d+\alpha \,\frac{\beta(d-\mbox{$\mathrm{mt}$} \, X)^2}{d} \;\le\;\beta(d-1-\mbox{$\mathrm{mt}$} \, X)^2,$$ which is equivalent to $$d^2(1-\alpha\beta-2\beta) +\alpha\beta(2d-\mbox{$\mathrm{mt}$} \, X)\cdot\mbox{$\mathrm{mt}$} \, X+2\beta d\cdot\mbox{$\mathrm{mt}$} \, X+\beta d\;\ge\; 0,$$ hence, due to \mbox{$\mbox{$\mathrm{mt}$} \, X\le d$}, it is enough to impose the condition \begin{equation} \label{2.8} 1\;\ge\;(\alpha+2)\beta\: . \end{equation} The second inequality in (\ref{2.7}) will follow from $$(\alpha+\alpha^2)\cdot\deg X \;\le\; (\alpha-1)d^2+d\: ,$$ which, by (\ref{2.3}), holds true as \begin{equation} \label{2.9} \alpha-1\;\ge\;\beta(\alpha+\alpha^2). \end{equation} We are interested in $\beta$ as large as possible. The inequality (\ref{2.9}) gives $$\beta\;\le\; \frac{\alpha-1}{\alpha+\alpha^2}\;\le\; 3-2\sqrt{2},$$ and the maximal value is attained at \mbox{$\alpha=\sqrt{2}+1$}. So, from now on we suppose (\ref{2.10}), especially the condition (\ref{2.8}) is satisfied. {\it Step 3}. Assume that $$\deg(X\cap L)\,<\,d-\alpha\,\frac{\deg X}{d}\ ,$$ $X$ is not an ordinary singularity, there exists a branch $Q$ of \mbox{$(C,z)$} such that \mbox{$T^\ast_X\cap L=T^\ast_X\cap Q$}, and there is no branch $Q'$ of $(C,z)$ with \mbox{$T^\ast_X\cap L\subsetneq T^\ast_X\cap Q'$}. In this case \mbox{$T^\ast_X\cap L$} consists of at least two points. Therefore \mbox{$\deg(X:L)<\deg X$} and \mbox{$\deg((X:L)\cap L) \le \deg(X\cap L)-2$}. Moreover, by (\ref{2.3}) and (\ref{2.8}), we have \mbox{$\,\alpha \deg X /d\leq d\!-\!1$}, hence $$ \deg((X:L)\cap L)\; \le \; (d-1)-\alpha\,\displaystyle{\frac{\deg(X:L)}{d-1}}\:. $$ On the other hand, by Lemma \ref{1.15}, \mbox{$\mbox{$\mathrm{mt}$} \,(X:L)=\mbox{$\mathrm{mt}$} \, X-1$}, thus $$\deg(X:L)\;< \;\deg X \;\le\; \beta(d-\mbox{$\mathrm{mt}$} \, X)^2 \;=\; \beta(d-1-\mbox{$\mathrm{mt}$} \,(X:L))^2.$$ {\it Step 4}. Assume that $$\deg(X\cap L)<d-\alpha\,\frac{\deg X}{d}\: ,$$ and there is a branch $Q$ of \mbox{$(C,z)$} such that: (1) \mbox{$T^\ast_X \cap Q$} consists of points \mbox{$z_1=z,\dots ,z_r$}, naturally ordered, (2) \mbox{$T^\ast_X \cap L$} consists of points \mbox{$z_1,\dots ,z_s$}, \mbox{$1\leq s<r$}, (3) the multiplicity \mbox{$m=\mbox{$\mathrm{mt}$} \,(C_{(z_{s+1})},z_{s+1})$} satisfies \begin{equation} \label{2.11} \deg(X\cap L)+m\:>\:d-\alpha\,\frac{\deg X}{d}\: . \end{equation} In this case, Lemma \ref{1.6} gives \begin{equation} \label{2.12} \deg X\,>\,\frac{(\mbox{$\mathrm{mt}$} \, X)^2+m^2}{2}\: . \end{equation} Since \mbox{$m\le\mbox{$\mathrm{mt}$} \, X$} and due to (\ref{2.11}), the first inequality in (\ref{2.7}) will follow from $$\beta(d-\mbox{$\mathrm{mt}$} \, X)^2-d+\alpha\,\frac{\beta(d-\mbox{$\mathrm{mt}$} \, X)^2}{d}+\mbox{$\mathrm{mt}$} \, X \:\le\: \beta(d-1-\mbox{$\mathrm{mt}$} \, X)^2,$$ or, equivalently, from \begin{equation} \label{2.13} (1-2\beta-\alpha\beta)\left(d-\mbox{$\mathrm{mt}$} \,X\right)+\beta + \alpha \beta \,\frac{\mbox{$\mathrm{mt}$}\, X}{d} \left(d- \mbox{$\mathrm{mt}$} \,X \right)\: \geq \:0\,, \end{equation} which holds true if (\ref{2.8}) is satisfied. Similarly, the second inequality in (\ref{2.7}) will follow from $$d-\alpha\,\frac{\deg X}{d} \;\le\; d-1- \alpha\,\frac{\deg X-d+\alpha\deg X/d+ m}{d-1},$$ which, by (\ref{2.3}), is satisfied, if \begin{equation} \label{2.15} (\alpha-1)d^2+d-\alpha md \ge (\alpha+\alpha^2) \,\beta\,(d-\mbox{$\mathrm{mt}$} \, X)^2 \ . \end{equation} The coefficient of $d^2$ is zero by (\ref{2.10}), hence, it is enough to show that $$2\beta(\alpha\!+\!\alpha^2)\,d\cdot\mbox{$\mathrm{mt}$} \, X- \beta(\alpha\!+\!\alpha^2)(\mbox{$\mathrm{mt}$} \, X)^2-\alpha md\;\ge\; 0\ ,$$ or, equivalently, $$m\,\le\,\beta(1+\alpha)\left(2\lambda-\frac{\lambda^2}{d}\right) \!\;=:\, \varphi(\lambda)\,, \qquad \lambda=\mbox{$\mathrm{mt}$} \, X.$$ Indeed, due to (\ref{2.12}) an (\ref{2.3}), we have \mbox{$\lambda\leq \frac{\sqrt{2\beta}}{1+\sqrt{2\beta}}\,d$} and \begin{equation} \label{2.16} m\,\le\,\psi(\lambda)\,:=\,\left\{ \begin{array}{cl} \lambda &\mbox{for }\; 0\le\lambda\le \frac{\sqrt{\beta}}{1+\sqrt{\beta}}\,d,\\ \sqrt{2\beta(d-\lambda)^2-\lambda^2} &\mbox{for }\; \frac{\sqrt{\beta}}{1+\sqrt{\beta}}\,d \le \lambda\le \frac{\sqrt{2\beta}}{1+\sqrt{2\beta}}\,d\,. \end{array} \right. \end{equation} Since \mbox{$\varphi(d\sqrt{\beta}/(\sqrt{\beta}\!+\!1))=\psi(d\sqrt{\beta}/ (\sqrt{\beta}\!+\!1))$} as (\ref{2.10}) holds, and $\varphi$ is increasing concavely in the segment \mbox{$[0\,,\,d\sqrt{2\beta}/(\sqrt{2\beta}\!+\!1)]$}, we obtain \mbox{$m\le\psi(\lambda)\le \varphi(\lambda)$}. {\it Step 5}. Assume that $$\deg(X\cap L)\:<\:d-\alpha\,\frac{\deg X}{d}\ ,$$ and that there is a (smooth) branch $Q$ of \mbox{$(C,z)$} such that: (1) \mbox{$T^\ast_X \cap Q$} consists of points \mbox{$z_1=z,\dots,z_r$}, naturally ordered, (2) \mbox{$T^\ast_X \cap L$} consists of points \mbox{$z_1,\dots,z_s$}, \mbox{$1\leq s<r$}, (3) the multiplicity \mbox{$m=\mbox{$\mathrm{mt}$} \,(C_{(z_{s+1})},z_{s+1})$} satisfies $$\deg(X\cap L)+m\:\le\: d-\alpha\,\frac{\deg X}{d}\ .$$ Then by Lemma \ref{1.14} we specialize the point $z_{s+1}$ on the line $L$ and consider the new scheme \mbox{$\widetilde{X}\sim X$} with \mbox{$\deg(\widetilde{X}\cap L)=\deg(X\cap L)+m$}. By the semi--continuity of cohomology, \mbox{$h^1({\mathcal J}_{\widetilde{X}/\P^2}(d))=0$} yields \mbox{$h^1({\mathcal J}_{X/\P^2}(d))=0$}. Thus, specializing points of $Q$ onto $L$ we come to one of the cases studied above. \end{proof} \pagebreak[3] \section{$H^1$--Vanishing Criterion for Zero--Dimensional Schemes of Class ${\mathcal G}{\mathcal S}$ in the plane} \setcounter{equation}{0} For a scheme \mbox{$X\in{\mathcal G}{\mathcal S}$} denote by \mbox{$\mbox{$\mathrm{mt}$}_s X$} the sum of the multiplicities of all singular branches of the underlying germ \mbox{$(C,z)$}. Note that $\mbox{$\mathrm{mt}$} \, X$, $\mbox{$\mathrm{mt}$}_s X$ are invariant with respect to the extension (cf.~(\ref{1.19})). \begin{lemma} \label{3.1} For any integer \mbox{$d\ge 1$} and any \mbox{$X\in{\mathcal G}{\mathcal S}$} satisfying \begin{equation} \label{3.2} \deg X\:\le\:\beta_0(d-\mbox{$\mathrm{mt}$} \, X-\mbox{$\mathrm{mt}$}_s X)^2\ , \end{equation} where \mbox{$\beta_0=(\alpha_0+8)^{-1}=0.10340\dots$} and \mbox{$\alpha_0=(31-3\sqrt{85})/2 =1.6706\dots$} is the positive root of the equation $$\left(\frac{\sqrt{4\alpha^3+\alpha^2-4\alpha}+\alpha-2}{2(1+\alpha+\alpha^2)} \right)^2=\frac{1}{\alpha+8}\ ,$$ there is \mbox{$X'\sim X$} with \mbox{$h^1({\mathcal J}_{X'/\P^2}(d))=0$}. \end{lemma} \begin{proof} As in the proof of Lemma \ref{2.1}, we shall obtain a more general statement. Let \mbox{$L\subset\P^2$} be a fixed straight line. There exist \mbox{$\alpha,\beta>0$} such that, for any integer \mbox{$d\ge 1$} and any \mbox{$X\in{\mathcal G}{\mathcal S}$}, satisfying $$\deg X\:\le\:\beta(d-\mbox{$\mathrm{mt}$} \, X-\mbox{$\mathrm{mt}$}_s X)^2,\quad \deg(X\cap L)\:\le \:d-\alpha\,\frac{\deg X}{d}\ ,$$ there exists \mbox{$X'\sim X$} with \mbox{$h^1({\mathcal J}_{X'/\P^2}(d))=0$}, \mbox{$\deg(X'\cap L)=\deg(X\cap L)$}. Finally we show that for $\alpha$, $\beta$ we can take the values $\alpha_0$ and $\beta_0$, respectively. {\it Step 1}. In the case \mbox{$X\in{\mathcal G}{\mathcal S}_1$} the proof of Lemma \ref{2.1} gives sufficient conditions on $\alpha$, $\beta$, namely (\ref{2.8}), (\ref{2.9}) in the Steps 2, 3, and the inequality (\ref{2.15}) in Step 4. Due to (\ref{2.16}), it is sufficient to check the inequality (\ref{2.15}) after removing the term $d$ and substituting \mbox{$d\sqrt{\beta}/(\sqrt{\beta}+1)$} for $\mbox{$\mathrm{mt}$} \, X$ and $m$, or, equivalently, to have \begin{equation} \label{3.3} \beta\:\le\:\left(\frac{\sqrt{4\alpha^3+\alpha^2-4\alpha}+\alpha-2}{ 2(1+\alpha+\alpha^2)}\right)^2\ . \end{equation} {\it Step 2}. Assume that \mbox{$X\in{\mathcal G}{\mathcal S} \backslash {\mathcal G}{\mathcal S}_1$}. Since \mbox{$\mbox{$\mathrm{mt}$}_s X\le\mbox{$\mathrm{mt}$} \, X$}, we can perform the inductive procedure described in the proof of Lemma \ref{2.1} under assumptions (\ref{2.8}), (\ref{2.9}), (\ref{3.3}), until the following situation occurs: \begin{itemize} \item[(1)] $X$ satisfies (\ref{3.2}); \item[(2)] let \mbox{$T^\ast_X\cap L=\{q_1,\dots ,q_N\}$}, such that for each branch $D$ going through \mbox{$q:=q_N$} $$ \mbox{$\mathrm{mt}$} \,(D_{(q_1)},q_1)=\ldots = \mbox{$\mathrm{mt}$} \,(D_{(q_{N-1})},q_{N-1})> \mbox{$\mathrm{mt}$} \,(D_{(q)},q)>0$$ (especially $D$ is not a smooth branch), and \begin{equation} \label{3.4} \deg(X\cap L)\:<\:d-\alpha\,\frac{\deg X}{d}-m_q\ , \end{equation} where $m_q$ is the multiplicity of $X$ at $q$. \end{itemize} Again, in the induction step, it is sufficient to show the two inequalities \renewcommand{\arraystretch}{1.7} \begin{equation} \label{3.4A} \begin{array}{c} \deg X-\deg(X\cap L)\;\le\;\beta(d-1-\mbox{$\mathrm{mt}$} \,(X:L)-\mbox{$\mathrm{mt}$}_s(X:L))^2 \ ,\\ \deg((X:L)\cap L)\;\le\;\deg(X\cap L)\;\le\; d-1-\alpha\,\displaystyle{\frac{\deg(X:L)}{d-1}} \: . \end{array} \end{equation} Consider the possible situations. {\it Step 3}. Under the hypotheses of the second step, assume that $$\deg(X\cap L)\;\ge\; d-2m'-\alpha\,\frac{\deg X}{d-m'}\ ,$$ where $m'$ is the sum of the multiplicities of all branches, going through $q$. Then the first inequality in (\ref{3.4A}) will follow from $$\alpha\,\frac{\beta(d\!-\!\mbox{$\mathrm{mt}$} \, X\!-\!\mbox{$\mathrm{mt}$}_s X)^2}{d\!-\!m'}\:\le\: (d\!-\!2m')+\beta-2\beta(d\!-\!\mbox{$\mathrm{mt}$} \, X\!-\!\mbox{$\mathrm{mt}$}_sX)\ .$$ Since \mbox{$m'\le\mbox{$\mathrm{mt}$}_s X \le \mbox{$\mathrm{mt}$} \, X$}, replacing the left--hand side by \mbox{$\alpha\beta(d-\mbox{$\mathrm{mt}$} \, X-\mbox{$\mathrm{mt}$}_sX)$}, and replacing the term \mbox{$d-2m'$} in the right--hand side by \mbox{$d-\mbox{$\mathrm{mt}$} \, X-\mbox{$\mathrm{mt}$}_sX$}, one obtains a stronger inequality, namely $$0\:\le\:\beta+(1-2\beta-\alpha\beta)(d-\mbox{$\mathrm{mt}$} \, X-\mbox{$\mathrm{mt}$}_sX)\ ,$$ which is an immediate consequence of (\ref{2.8}). The second inequality of (\ref{3.4A}) will follow from $$d-1-\alpha\,\frac{\deg X-\deg(X\cap L)}{d-1}\;\ge\;\deg(X\cap L)\ .$$ We replace $\deg X$ and $\deg(X\cap L)$ by the upper bounds (\ref{3.2}), (\ref{3.4}) and obtain $$(\alpha-1)d^2+d\;\ge\;\beta(\alpha+\alpha^2)(d-\mbox{$\mathrm{mt}$} \, X-\mbox{$\mathrm{mt}$}_sX)^2- m_q(d-1-\alpha)\ ,$$ which holds true by (\ref{2.9}). {\it Step 4}. Under the hypotheses of the second step, assume that \begin{equation} \label{3.5} \deg(X\cap L)\:<\:d-2m'-\alpha\,\frac{\deg X}{d-m'}\ . \end{equation} In this case we have to exert ourselves to obtain an analogue to the first inequality in (\ref{3.4A}). For that, we shall perform the following $m'$--step algorithm. Let \mbox{$1\le j\le m'$} and let \mbox{$X_{j-1}\in{\mathcal G}{\mathcal S}$} be defined by a germ \mbox{$(C_{j-1},z)$} and a tree $T^\ast_{j-1}$ (\mbox{$X_0=X,\ C_0=C,\ T^*_0=T^*$}), such that at the endpoint $q$ of \mbox{$T^\ast_{j-1}\cap L$} the strict transform of $C_{j-1}$ has the multiplicity $m_q^{(j-1)}$ (\mbox{$m_q^{(0)}=m_q$}), and $$\deg(X_{j-1}\cap L)\:<\:d-m'-\alpha\,\frac{\deg X}{d\!-\!m'}\ .$$ The $j$-th step of the algorithm appears as follows: introduce $$s_j:=\min\left\{l\ge 0\ \Big|\ \deg(X_{j-1}\cap L)+lm'_j\,\ge\, d-2m'-\alpha\,\frac{\deg X}{d\!-\!m'}\right\},$$ where $m'_j$ is the sum of the multiplicities of all branches of \mbox{$(C_{j-1},z)$} going through $q$ at the preceding point \mbox{$\overline{q} \in T^\ast_{j-1}\cap L$}. In particular, \mbox{$m'_1=m'$} and \mbox{$s_1\geq 1$}. Define $X'_{j-1}$ as the extension $$X'_{j-1}:=X_{j-1}\underbrace{(q)\dots(q)}_{s_j\ \mbox{\scriptsize times}}\ , \:\mbox{ and }\: X_j:=X'_{j-1}:L\ .$$ Note that in the previous formula, in the definition of $X_{j-1}$ and in the assumption of Step 2, we denote different points by $q$. But all these points appear in the extension operation introduced above, and the notation $q$ moves to new points of new schemes as was described in the assertion of Lemma \ref{1.20}. Due to Lemma \ref{1.22}, again it is enough to show \begin{align} \deg(X_{m'}\cap L) &\: \le \; d-m'-\alpha\,\frac{\deg X_{m'}}{d-m'}\ ,\label{3.8}\\ \deg X_{m'} & \:\le\; \beta(d-m'-\mbox{$\mathrm{mt}$} \, X_{m'}-\mbox{$\mathrm{mt}$}_sX_{m'})^2 \label{3.9} \end{align} to complete the induction step. \smallskip\noindent We define the set $$\Lambda\,:=\: \bigl\{\!\:j\in[1,m'\!-\!\!\:1]\:\big|\: m'_{j+1}<m'_{j} \!\:\bigr\} \:=\: \bigl\{\!\:j_1,j_2,\dots, j_\ell\!\:\bigr\}\,,$$ \mbox{$j_{k+1}> j_{k}$}, that is, by Proposition \ref{1.11} (ii), we have $$ m'_{j_k+1}\: =\: m'_{j_k+2}\:=\ldots =\: m'_{j_{k+1}}\:=\:m'\!-\!\!\:k $$ for any \mbox{$k=0,\dots,\ell$} (where \mbox{$j_0:=0$} and \mbox{$j_{\ell+1}:=m'$}). We set $$ N_k \,:=\, \sum_{i=j_{k}+1}^{j_{k+1}} s_{i}\,. $$ Note that if \mbox{$j\in \Lambda$}, that is, if \mbox{$m'_{j+1}=m'_{j}-1$}, then there are two possibilities: first, it might be that \mbox{$\mbox{$\mathrm{mt}$}\, X_j=\mbox{$\mathrm{mt}$} \,X_{j-1}\!\!\:-\!\!\:1\ $}. Secondly, if this is not the case, then Lemma \ref{1.16} (ii) gives at least the existence of a branch $Q'$ of the germ \mbox{$(C_j,z)$} with \mbox{$T^\ast_j\cap Q'\subset T^\ast_j\cap L$}, and Lemma \ref{1.15} implies that \mbox{$\mbox{$\mathrm{mt}$} \,X_{j+1}=\mbox{$\mathrm{mt}$}\, X_{j}\!\!\:-\!\!\:1$}. In any case, we have \mbox{$\mbox{$\mathrm{mt}$}_s X_j=\mbox{$\mathrm{mt}$}_s X_{j-1}\!\!\:-\!\!\:1\ .$} Hence, we can estimate $$ \widetilde{\ell}\::=\: (\mbox{$\mathrm{mt}$}\, X\!\!\:+\!\!\:\mbox{$\mathrm{mt}$}_s X) - (\mbox{$\mathrm{mt}$}\, X_{m'}\!\!\:+\!\!\: \mbox{$\mathrm{mt}$}_s X_{m'}) \:\left\{ \renewcommand{\arraystretch}{1.1} \begin{array}{ll} \geq 0 & \text { if $\ell = 0$}\,,\\ \geq \ell\!\!\;+\!\!\;1 & \text { if $\ell\neq 0$} \,, \end{array} \right. $$ To run an induction step, it is sufficient to show that \begin{equation} \label{3.14} \deg X_{m'} \,\leq \, \deg X\,, \qquad \deg X_{m'} \, \leq \, \beta (d\!\!\:-\!\!\:m'\!+\!\!\:\widetilde{\ell}\!\!\:-\!\!\:\mbox{$\mathrm{mt}$}\, X\!\!\:-\!\!\:\mbox{$\mathrm{mt}$}_s X)^2. \end{equation} By construction, we have \begin{eqnarray*} \deg X_{m'} & = & \deg X + \sum_{k=0}^{\ell} N_k \!\;\frac{(m'\!-\!\!\:k)(m'\!-\!\!\:k\!\!\:+\!\!\:1)}{2} \!\;- \sum_{j=0}^{m'-1} \deg(X'_{j}\cap L) \\ & = & \deg X + \frac{(m'\!-\!\!\:\ell\!\;)(m'\!-\!\!\:\ell\!\!\;+\!\!\:1)}{2} \cdot \sum_{k=0}^{\ell} N_k \!\:- \sum_{j=0}^{m'-1} \deg(X'_{j}\cap L) \\ && \phantom{\deg X } + \sum_{k=0}^{\ell-1} \bigl((m'\!-\!\!\:k)\cdot (N_0+\ldots + N_{k})\bigr) \,, \end{eqnarray*} and we can estimate \begin{equation} \label{partial sum} \deg(X_{j_{k+1}-1}\cap L) \: \geq \: (m' \!-\!\!\:k) \cdot (1+N_0+\ldots + N_{k}) \,, \end{equation} for any \mbox{$k=0,\dots,\ell\!\!\;-\!\!\:1$}. On the other hand, in the $j$-th step we have $$d\!\!\;-\!\!\;2m'\!+\!\!\;m'_{j}\!\!\;-\!\!\;\alpha\,\frac{\deg X}{d\!\!\:-\!\!\:m'} \:>\: \deg(X'_{j-1}\cap L) \: \geq \: d\!\!\;-\!\!\;2m'\!-\!\!\;\alpha\,\frac{\deg X}{d\!\!\:-\!\!\:m'} \,,$$ \mbox{$j=0,\dots,m'\!-\!\!\:1$}. In particular, this together with (\ref{partial sum}) implies that $$ (m' \!-\!\!\:\ell\!\:) \cdot \sum_{k=0}^{\ell} N_k \:\leq \: \deg(X_{m'-1}\cap L) -(m'\!-\!\!\: \ell) \: < \: d\!\!\;-\!\!\;2m'\!-\!\!\;\alpha\,\frac{\deg X}{d\!\!\:-\!\!\:m'} \,. $$ Hence, we can estimate \begin{eqnarray*} \deg X_{m'} & \leq & \deg X + \frac{m'\!-\!\!\:\ell\!\!\;+\!\!\:1}{2} \cdot (m'\!-\!\!\:\ell\!\;)\cdot \sum_{k=0}^{\ell} N_k \!\:- \sum_{\renewcommand{\arraystretch}{0.5} \begin{array}{c} \scriptstyle{ k=0 } \\ \scriptstyle{ k+1\not \in \Lambda} \end{array} }^{m'-1} \deg(X'_{k}\cap L)\\ & \leq & \deg X + \left(\frac{m'\!-\!\!\:\ell\!\!\;+\!\!\:1}{2} - m'\!\!\:+\!\!\;\ell\!\:\right) \cdot \left(d\!\!\;-\!\!\;2m'\!-\!\!\;\alpha\,\frac{\deg X}{d\!\!\:-\!\!\:m'} \right)\\ & \leq & \deg X - \frac{m'\!-\!\!\:1\!\!\:-\!\!\;\ell}{2} \cdot \left(d\!\!\;-\!\!\;2m'\!-\!\!\;\alpha\,\frac{\deg X}{d\!\!\:-\!\!\:m'} \right) \,, \end{eqnarray*} whence the first inequality in (\ref{3.14}). The second will follow from \begin{eqnarray*} \lefteqn{\beta(d\!\!\:-\!\!\:m'\!+\!\!\:\widetilde{\ell}\!\!\:-\!\!\:\mbox{$\mathrm{mt}$}\, X\!\!\:-\!\!\:\mbox{$\mathrm{mt}$}_s X)^2}\hspace{2cm}\\ &\geq & \beta(d\!\!\:-\!\!\:\mbox{$\mathrm{mt}$}\, X\!-\!\!\:\mbox{$\mathrm{mt}$}_s X)^2- \frac{m'\!-\!\!\:1\!\!\:-\!\!\;\ell}{2} \cdot \left(d\!\!\;-\!\!\;2m'\!-\!\!\;\alpha\,\frac{\deg X}{d\!\!\:-\!\!\:m'} \right) \end{eqnarray*} or, equivalently, \begin{eqnarray*} \lefteqn{ 2\beta(m'\!-\!\!\;\widetilde{\ell}\,) (d\!\!\:-\!\!\:\mbox{$\mathrm{mt}$}\, X\!-\!\!\:\mbox{$\mathrm{mt}$}_s X)}\hspace{2cm}\\ &\leq & \frac{m'\!-\!\!\:1\!-\ell}{2}\cdot \left(d-2m'\!\!\:- \alpha\beta\,\frac{(d\!\!\:-\!\!\:\mbox{$\mathrm{mt}$}\, X\!-\!\!\:\mbox{$\mathrm{mt}$}_s X)^2}{ d\!\!\:-\!\!\:m'}\right) +(m'\!-\!\!\;\widetilde{\ell}\,)^2. \end{eqnarray*} Since \mbox{$m'\le\mbox{$\mathrm{mt}$}_s X \le \mbox{$\mathrm{mt}$}\, X $}, this holds whenever \begin{equation} \label{3.23} 0 \: \leq \: \frac{1\!\!\:-\!\!\: \alpha\beta}{4\beta} \cdot (m'\!-\!\!\:1\!\!\:-\!\!\;\ell)(d\!\!\:-\!\!\:2m')- (m'\!-\!\!\;\widetilde{\ell}\,)(d\!\!\:-\!\!\:2m')\, . \end{equation} Note that for fixed $\ell$ the right-hand side takes its minimum for the minimal possible value of $\widetilde{\ell}$. Hence, it suffices to consider two cases: \smallskip \noindent {\it Case 4A.} \mbox{$\,\widetilde{\ell}=\ell=0\!\;$}. Then, since \mbox{$m'\!\geq 2$}, (\ref{3.23}) is implied by \begin{equation} \label{3.13} \frac{1}{\beta} \: \geq \: 8+\alpha\,. \end{equation} \smallskip \noindent {\it Case 4B.} \mbox{$\,\widetilde{\ell}=\ell+1\geq 2\!\;$}. Then (\ref{3.23}) holds if \mbox{$\frac{1}{\beta} \geq 4+\alpha$}, which is a consequence of (\ref{3.13}). {\it Step 5}. Finally we look for the maximal value of $\beta$, satisfying (\ref{2.8}), (\ref{2.9}), (\ref{3.3}), (\ref{3.13}) with any \mbox{$\alpha>1$}. It is easily shown that $\beta_0$ mentioned in the assertion of Lemma \ref{3.1} is this maximal value. {\it Step 6}. The induction base for \mbox{$d\le 6$} is trivial. Indeed, for \mbox{$X\in {\mathcal G}{\mathcal S}\backslash {\mathcal G}{\mathcal S}_1$} the right--hand side of (\ref{3.2}) does not exceed \mbox{$\beta_0(6-2-2)^2<1$}. \end{proof} \section{Existence of Curves with one Singular Point} \setcounter{equation}{0} We start with the following auxiliary statement: \begin{lemma} \label{4.1} Let a scheme \mbox{$Y\in{\mathcal G}{\mathcal S}\cap{\mathcal S}$} be defined by a germ \mbox{$(C,z)$} with branches \mbox{$Q_1,\dots,Q_p$} and the tree \mbox{$T^\ast=T^\ast(C)$}. Let \mbox{$f\in J_Y$} satisfy \begin{equation} \label{5.2} \mbox{$\mathrm{mt}$} \,(f,z)\,=\,\mbox{$\mathrm{mt}$} \, Y,\quad(f, Q_i)\,>\!\!\sum_{q\in T^*\!\cap Q_i}\!m_q\cdot \mbox{$\mathrm{mt}$} \,(Q_{i,(q)},q),\ \,i=1,\dots,p. \end{equation} Then the germ \mbox{$(f,z)$} also defines the singularity scheme $Y$, in particular \mbox{$(f,z)$} and \mbox{$(C,z)$} have the same topological type (which we denote by $Y$). \end{lemma} \begin{proof} {\it Step 1}. In the case \mbox{$Y\in{\mathcal G}{\mathcal S}_1\cap{\mathcal S}$} this easily can be shown by induction on $\deg Y$. The induction base with \mbox{$Y=\emptyset$} and $C$ being non--singular at $z$, is trivial. Hence, assume \mbox{$\deg Y>0$} and blow up the point $z$. If $w$ is an intersection point of the strict transform $C^\ast$ of $C$ with the exceptional divisor $E$, then, for any branch \mbox{$Q_{i,(w)}$} of $C^\ast$ centred at $w$, we have \begin{equation} \label{4.2} (f_{(w)},Q_{i,(w)})\:=\:(f\cdot Q_i)-\mbox{$\mathrm{mt}$} \,(f,z) \:>\! \sum_{q\in T^\ast\cap Q_i\backslash \{z\} }\!\!\!m_q\:\ge\: 0\, . \end{equation} Thus, \mbox{$w\in f_{(w)}$} and, if $C^\ast$ is non--singular at $w$, then \mbox{$\mbox{$\mathrm{mt}$} \,(f_{(w)},w)\ge 1=\mbox{$\mathrm{mt}$} \,(C^\ast\!,w)$}. If $C^\ast$ is singular at $w$, then by Lemma \ref{1.8}, \mbox{$f_{(w)}\in J_{Y^\ast_{(w)}}$}, where \mbox{$Y^\ast_{(w)}\in{\mathcal G}{\mathcal S}_1$} is defined by the germ \mbox{$(C^\ast\!,w)$} and its tree of essential points. Especially, again, \mbox{$\mbox{$\mathrm{mt}$} \,(f_{(w)},w)\ge\mbox{$\mathrm{mt}$} \,(C^\ast\!,w)$}. On the other hand, $$\mbox{$\mathrm{mt}$} \,(f,z)\:\ge\:\sum_{w\in C^\ast\cap E}\!\!\mbox{$\mathrm{mt}$} \,(f_{(w)},w)\:\ge\!\sum_{w\in C^\ast\cap E}\! \mbox{$\mathrm{mt}$} \,(C^\ast\!,w)\:=\:(C^\ast\!,E)\:=\:\mbox{$\mathrm{mt}$} \, Y,$$ which implies \mbox{$\mbox{$\mathrm{mt}$} \,(f_{(w)},w)=\mbox{$\mathrm{mt}$} \,(C^\ast\!,w)$} and, together with (\ref{4.2}) and the induction assumption, \mbox{$(f_{(w)},w)$} defines the same singularity scheme as \mbox{$(C^\ast\!,w)$}. Since, moreover, $C^\ast$ and $f_{(w)}$ are transversal to $E$ at $w$, \mbox{$(f,z)$} and \mbox{$(C,z)$} define the same singularity scheme $Y$. {\it Remark}. Let \mbox{$(C,z)$} be given in local coordinates $x$,$y$ at $z$ by $$ C(x,y)\:=\: \prod_{i=1}^m \left(y-\sum_{j=1}^\infty a_{ij}x^j\right) $$ and define the {\bf essential part} $\Gamma_{es}$ of the Newton diagram $\Gamma$ of \mbox{$C(x,y)$} at the origin as the union of \begin{itemize} \item[(i)] all the integral points \mbox{$(i,j)\in \Gamma$} with positive $i,\,j$, \item[(ii)] a point \mbox{$(n,0)\in\Gamma$}, if it is not an endpoint of an edge \mbox{$[(n,0),(n',1)]\subset\Gamma$}, \item[(iii)] a point \mbox{$(0,n)\in\Gamma$}, if it is not an endpoint of an edge \mbox{$[(0,n),(1,n')]\subset\Gamma$}. \end{itemize} We claim that \mbox{$f(x,y)$} has the same essential part $\Gamma_{es}$ of the Newton diagram at the origin. This is easily shown by induction on $\deg Y$, using the transformation \begin{equation} \label{4.3} (x,y)\mapsto(x,xy) \end{equation} as the blowing--up at $z$. {\it Step 2}. Now assume that $Y$ is arbitrary in ${\mathcal G}{\mathcal S}\cap {\mathcal S}$. We apply induction on the number $$N\,:=\,\sum_Q\sum_{q\in T^\ast\!\cap Q}\!\mbox{$\mathrm{mt}$} \,(Q_{(q)},q),$$ where $Q$ runs through all singular branches of $C$. The case \mbox{$N=0$} means just \mbox{$Y\in{\mathcal G}{\mathcal S}_1$}. If \mbox{$N>0$} then, again, we blow up the point $z$. Each intersection point of the strict transform $C^\ast$ of $C$ with the exceptional divisor $E$ corresponds to a straight line $W$ through $z$, tangent to $C$. Without restriction, we can suppose that in local coordinates $x,y$ at $z$ we have $W=y$ and $C(x,y)$ decomposes in local branches \mbox{$Q_1,\dots,Q_p$} with $$ Q_k=\prod_{i=1}^{s_k} \left( y-\xi_i^{(k)} (x)\right),\;\;\;\xi_i^{(k)} (x)=\sum_{j=1}^\infty a^{(k)}_{ij}x^{j/s_k},\;\;\; s_k=\mbox{$\mathrm{mt}$} \,(Q_k,z), $$ The (covering) transformation \begin{equation} \label{4.4} (x,y)\mapsto(x^M\!,y) \;\mbox{ with } \,M:=\prod_{k=1}^p s_k \end{equation} turns \mbox{$(C,z)$} into a germ \mbox{$(\widetilde{C},z)$} with multiplicity \mbox{$\mbox{$\mathrm{mt}$} \,(\widetilde{C},z)=\mbox{$\mathrm{mt}$} \,(C,z)=\mbox{$\mathrm{mt}$} \, Y$} and only non--singular branches $$Q_i^{(k)}\:=\: y-\sum_{j=1}^{\infty}a^{(k)}_{ij}x^{jM/s_k},\quad i=1,\dots,s_k,\; k=1,\dots,p\,.$$ Let \mbox{$\widetilde{Y}\in{\mathcal G}{\mathcal S}_1$} be defined by the germ \mbox{$(\widetilde{C},z)$} and the tree \mbox{$T^\ast(\widetilde{C})$}. We shall show that the transform \mbox{$\widetilde{\varphi}(x,y):= \varphi(x^M\!,y)$} of any element \mbox{$\varphi\in J_Y$} belongs to \mbox{$J_{\widetilde{Y}}$}. By Lemma \ref{1.8}, we have for any (fixed) \mbox{$i=1,\dots,s_k$}, $$ \left(\widetilde{\varphi}, Q_i^{(k)}\right) \: = \: \displaystyle\frac{M}{s_k}(\varphi, Q_k) \:\ge\: \displaystyle\frac{M}{s_k}\left(2\delta(Q_k)+\sum_{l\ne k}(Q_l, Q_k)+\!\!\sum_{q\in T^\ast\!\cap Q_k}\!\!\!\mbox{$\mathrm{mt}$} \,(Q_{k,(q)},q)\right)\, . $$ Hence, using the considerations in the proof of Lemma \ref{1.17}, \begin{align*} \left(\widetilde{\varphi}, Q_i^{(k)}\right) &\: \geq \; \left(\sum_{j\neq i} (Q_i^{(k)},Q_j^{(k)})-M+\frac{M}{s_i}\right) + \left( \sum_{l\neq k} \sum_{j=1}^{s_l} (Q_i^{(k)},Q_j^{(l)}) \right)\\ &\quad\quad\quad\quad + \displaystyle \left( \max_{(j,l)\neq (i,k)} (Q_i^{(k)}, Q_j^{(l)})+ M - \frac{M}{s_i} \right)\\ &\:=\; \displaystyle \sum_{(j,l)\ne(i,k)} (Q_i^{(k)}, Q_j^{(l)})\,+\,\# \,\{ q\in T^\ast(\widetilde{C})\cap Q_i^{(k)} \} \, , \end{align*} and Lemma \ref{1.8} implies \mbox{$\widetilde{\varphi}\in J_{\widetilde{Y}}$}. The germ \mbox{$\widetilde{f}(x,y)=f(x^M\!,y)$}, clearly, satisfies \mbox{$\mbox{$\mathrm{mt}$} \,(\widetilde{f},z)=\mbox{$\mathrm{mt}$} \, Y$}. Due to (\ref{5.2}), the previous computation with \mbox{$\widetilde{f}$} instead of $\widetilde{\varphi}$ gives $$(\widetilde{f},Q_i^{(k)})\;>\!\!\sum_{q\in T^\ast\!(\widetilde{C})\cap Q_i^{(k)}}\!\!\mbox{$\mathrm{mt}$} \,(\widetilde{C}_{(q)},q),\quad i=1,\dots,s_k,\; k=1,\dots,p\,.$$ Hence, by Step 1, \mbox{$(\widetilde{f},z)$} defines the same singularity scheme as \mbox{$(\widetilde{C},z)$}. Denote by $\Gamma$ the Newton diagram of \mbox{$C(x,y)$} at the origin. Evidently, the Newton diagram $\widetilde{\Gamma}$ of $\widetilde{C}$ and its essential part $\widetilde{\Gamma}_{es}$ are obtained from $\Gamma$, $\Gamma_{es}$ by the transformation \mbox{$(I,J)\mapsto(MI,J)$}. As established above, $\widetilde{f}$ has the same essential part $\widetilde{\Gamma}_{es}$ of the Newton diagram at the origin. Therefore, $\Gamma_{es}$ is the essential part of the Newton diagram of $f(x,y)$ at the origin. Let $\Gamma'$ be the part of $\Gamma$ corresponding to the branches of $C$ tangent to $y$, and $$(n_1,m_1),\dots,(n_l,m_l),\quad m_1>\dots>m_l=0,$$ be the vertices of $\Gamma'$. Applying the blowing--up (\ref{4.3}) at $z$, we easily obtain that the Newton diagram of \mbox{$C^\ast(x,y)$} at \mbox{$w=(0,0)$} has the vertices $$(0,m_1),\ (n_2\!-\!n_1,m_2),\dots,\ (n_{l-1}\!-\!n_1,m_{l-1}),\ (n_l\!-\!n_1,0),$$ and that \mbox{$f_{(w)}(x,y)$} has the Newton diagram with vertices $$(0,m_1),\ (n_2\!-\!n_1,m_2),\dots,\ (n_{l-1}\!-\!n_1,m_{l-1}),\ (r,0),$$ where $r$ may be different from \mbox{$n_l\!-\!n_1$} only in the case \mbox{$m_{l-1}=1$}. In particular, this means that \begin{equation} \label{4.9} \mbox{$\mathrm{mt}$} \,(f_{(w)},w)=\mbox{$\mathrm{mt}$} \,(C^\ast\!,w),\quad(f_{(w)}, E)=\mbox{$\mathrm{mt}$} \,(C^\ast\!\cap E,w)\, . \end{equation} On the other hand, for any branch \mbox{$Q_{i,(w)}$} of $C^\ast$ centred at $w$, we have $$(f_{(w)},Q_{i,(w)})\,=\,(f, Q_i)-\mbox{$\mathrm{mt}$} \,(f,z)\cdot\mbox{$\mathrm{mt}$} \, (Q_i,z)\,>\!\! \sum_{q\in T^\ast\!\cap Q_i\backslash \{z\}}\!\!m_q\cdot\mbox{$\mathrm{mt}$} \,(Q_{i,(q)},q)\ .$$ This, together with (\ref{4.9}) and the induction assumption, implies that \mbox{$(f_{(w)}\cdot E,w)$} defines the same singularity scheme as \mbox{$(C^\ast\!\cdot E,w)$}. Finally, blowing down all the germs $f_{(w)}$, \mbox{$w\in C^\ast\!\cap E$}, one obtains that \mbox{$(f,z)$} also defines the singularity scheme $Y$. \end{proof} \begin{definition} \label{4.10} Let $F$ be a curve of degree $d$ with an isolated singular point $z$. The germ \mbox{$(H_{\P^2}^{es,d},F)$} of the equisingular stratum \mbox{$H_{\P^2}^{es,d} \subset \P^N=\P(\Gamma({\mathcal O}_{\P^2}(d)))$}, \mbox{$N=d(d\!+\!3)/2$}, at $F$ (cf.~\cite{GrL}) is called {\bf T--smooth}, if it is smooth and, for any \mbox{$d'>d$}, it is a transversal intersection in \mbox{$\P(\Gamma({\mathcal O}_{\P^2}(d')))$} of \mbox{$(H_{\P^2}^{es,d'}\!,F)$} and \mbox{$\P(\Gamma({\mathcal O}_{\P^2}(d)))$}, included in \mbox{$\P(\Gamma({\mathcal O}_{\P^2}(d')))$} by \mbox{$C\mapsto CL^{d'-d}$}, where $L$ is a fixed generic straight line not passing through $z$. \end{definition} \begin{lemma} \label{4.11} For any scheme \mbox{$X\in{\mathcal G}{\mathcal S}\cap{\mathcal S}$}, and any positive integer $d$, satisfying \begin{equation} \label{4.12} \deg X+\mbox{$\mathrm{mt}$} \, X+1\:<\:\left\{ \begin{array}{cl} (3-2\sqrt{2})(d-\mbox{$\mathrm{mt}$} \, X-2)^2\,,&\mbox{if} \ X\in{\mathcal G}{\mathcal S}_1,\\ \beta_0(d-\mbox{$\mathrm{mt}$} \, X-\mbox{$\mathrm{mt}$}_sX-2)^2\,,&\mbox{if} \ X\in{\mathcal G}{\mathcal S}\backslash{\mathcal G}{\mathcal S}_1, \end{array} \right. \end{equation} $\beta_0$ as above in Lemma \ref{3.1}, there exists an irreducible curve $F$ of degree $d$ with a singular point $z$ of (topological) type $X$ as its only singularity such that the germ \mbox{$(H_{\P^2}^{es,d},F)$} is T--smooth. \end{lemma} \begin{proof} {\it Step 1}. Let \mbox{$(C,z)$} be a defining germ of the scheme $X$ and consider the germ \mbox{$(\widetilde{C},z)$}, \mbox{$\widetilde{C}=C\cdot L$}, where $L$ is a generic straight line through $z$. Note that $$T^\ast(\widetilde{C})=T^\ast(C),\quad\mbox{$\mathrm{mt}$} \,(\widetilde{C},z)=\mbox{$\mathrm{mt}$} \,(C,z)+1\ .$$ Now we introduce (1) the scheme $X'$, defined by the germ \mbox{$(C,z)$} and the tree $T^\ast_{X'}$, containing $T^\ast(C)$ and the first non--essential points of all local branches of \mbox{$(C,z)$}, and (2) the scheme $\widetilde{X}$, defined by the germ \mbox{$(\widetilde{C},z)$} and the tree \mbox{$T^\ast(C)$}. Clearly, \begin{equation} \label{4.13} \begin{split} &\mbox{$\mathrm{mt}$} \, X'=\,\mbox{$\mathrm{mt}$} \, X,\quad\mbox{$\mathrm{mt}$} \,\widetilde{X}=\,\mbox{$\mathrm{mt}$} \, X+1,\\ \deg X'\leq&\,\deg X+\mbox{$\mathrm{mt}$} \, X,\quad\deg\widetilde{X}\leq\, \deg X+\mbox{$\mathrm{mt}$} \, X+1,\\ \end{split} \end{equation} and in addition, for any local branch $Q$ of \mbox{$(C,z)$} and any elements \mbox{$f\in{\mathcal J}_{X'/\P^2,z}$}, \mbox{$g\in{\mathcal J}_{\widetilde X/\P^2,z}$}, \begin{equation} \label{4.14} \begin{split} &(f, Q)\:>\!\sum_{q\in T^\ast(C)\cap Q}\mbox{$\mathrm{mt}$} \,(C_{(q)},q),\\ (g, Q)\:\ge&\sum_{q\in T^\ast\!(C)\cap Q}\!\!\mbox{$\mathrm{mt}$} \,(\widetilde{C}_{(q)},q)\:>\! \sum_{q\in T^\ast\!(C)\cap Q}\!\!\mbox{$\mathrm{mt}$} \,(C_{(q)},q).\\ \end{split} \end{equation} By (\ref{4.12}), (\ref{4.13}) and the Lemmas \ref{2.1}, \ref{3.1}, we may assume that \begin{equation} \label{4.16a} h^1({\mathcal J}_{X'/\P^2}(d-1))=h^1({\mathcal J}_{\widetilde{X}/\P^2}(d-1))=0. \end{equation} Hence, due to \mbox{$X'\subsetneq\widetilde{X}$}, there exists a (generic) curve \begin{equation} \label{4.15} f\in H^0({\mathcal J}_{X'/\P^2}(d\!-\!1))\backslash H^0({\mathcal J}_{\widetilde{X} /\P^2}(d\!-\!1)), \end{equation} which, by (\ref{4.13}), (\ref{4.14}), satisfies the condition of Lemma \ref{4.1}. Thus, \mbox{$(f,z)$} defines the singularity scheme $X$. Replacing, if necessary, multiple components of $f$ (which do not go through $z$) by distinct components, we obtain a reduced curve $f$ of degree \mbox{$d-1$}. If $f$ is irreducible, then we are done. Otherwise, we shall use Bertini's Theorem to construct the desired irreducible curve $F$ of degree $d$. For that, let a straight line $L$ meet $f$ at \mbox{$d-1$} distinct non--singular points $w_1,\dots,w_{d-1}$. Obviously, \mbox{$h^1({\mathcal J}_{\{w_i\}/L}(1))=0$} for each \mbox{$i=1,\dots,d\!-\!1$} and, by (\ref{4.16a}) \mbox{$h^1({\mathcal J}_{X'/f}(d\!-\!1))=0$}. First, observe that this implies \begin{equation} \label{4.18} h^1({\mathcal J}_{X'\cup\{w_i\}/fL}(d))=0,\quad 1\le i\le d\!-\!1. \end{equation} Indeed, the first morphism \mbox{$\mbox{id}_1\otimes L+f \otimes \mbox{id}_2$} in the exact sequence $$0\longrightarrow{\mathcal O}_{f}(d-1)\oplus{\mathcal O}_{L}(1)\longrightarrow{\mathcal O}_{fL}(d) \longrightarrow {\mathcal O}_{f\cap L}\longrightarrow 0$$ maps the sheaf \mbox{${\mathcal J}_{X'/f}(d-1)\oplus{\mathcal J}_{\{w_i\}/L}(1)$} injectively to the sheaf \mbox{${\mathcal J}_{X'\cup\{w_i\}/fL}(d)$}. Now, consider the commutative diagram $$ \arraycolsep0.1cm \begin{array}{ccccccccc} \ &\ &\ &\ & 0 &\ &\ &\ &\ \\ \ &\ &\ &\ & \downarrow &\ &\ &\ &\ \\ \ &\ &\ &\ &{\mathcal O}_{\P^2} &\ &\ &\ &\ \\ \ &\ &\ &\ & \downarrow &\ &\ &\ &\ \\ 0 &\longrightarrow &{\mathcal J}_{X'\cup\{w_i\}/\P^2}(d) &\longrightarrow &{\mathcal O}_{\P^2}(d) &\longrightarrow &{\mathcal O}_{X'\cup\{w_i\}} &\longrightarrow & 0 \\ \ &\ &\ &\ & \downarrow &\ &\| &\ &\ \\ 0 &\longrightarrow &{\mathcal J}_{X'\cup\{w_i\}/fL}(d) &\longrightarrow &{\mathcal O}_{fL}(d) &\longrightarrow &{\mathcal O}_{X'\cup\{w_i\}} &\longrightarrow & 0 \\ \ &\ &\ &\ & \downarrow &\ &\ &\ &\ \\ \ &\ &\ &\ & 0 &\ &\ &\ &\ \end{array} $$ to deduce from (\ref{4.18}) the surjectivity of \mbox{$H^0({\mathcal O}_{\P^2}(d))\to H^0({\mathcal O}_{X'\cup\{w_i\}})$}, which is equivalent to $$ h^1({\mathcal J}_{X'\cup\{w_i\}/\P^2}(d))=0,\quad 1\le i\le d\!-\!1. $$ In particular, there exist curves $$\varphi_i\in H^0({\mathcal J}_{X'/\P^2}(d))\backslash H^0({\mathcal J}_{X'\cup\{w_i\}/ \P^2}(d)), \quad 1\le i\le d\!-\!1,$$ and, by Bertini's theorem, the generic member $\Phi$ of the linear family $$f\cdot L+\lambda_1\varphi_1+\dots+\lambda_{d-1}\varphi_{d-1}$$ is irreducible and smooth outside of the base points, whereas at $z$ it has a singularity of the (topological) type $X$. Assume that $\Phi$ has singular points \mbox{$w_1,\dots,w_p$} different from $z$. Using the preceding arguments, we show that there exist curves $$\Phi_j\in H^0({\mathcal J}_{X'/\P^2}(d))\backslash H^0({\mathcal J}_{X'\cup\{w_j\}/\P^2} (d)), \quad j=1,\dots,p.$$ Finally, by Bertini's theorem, a generic member $F$ of the linear family $$f\cdot L+\lambda_1\varphi_1+\dots+\lambda_{d-1}\varphi_{d-1} +\mu_1\Phi_1+\dots+\mu_p\Phi_p$$ has no singularities outside $z$. {\it Step 2}. Note that, by Lemma \ref{1.7}, \mbox{$h^1({\mathcal J}_{X'/\P^2}(d))=0$} implies \mbox{$h^1({\mathcal J}^{\mbox{\scriptsize es}}(d))=0$}, where \mbox{${\mathcal J}^{\mbox{\scriptsize es}}\subset{\mathcal O}_{\P^2}$} is the ideal sheaf of the zero--dimensional scheme given by the equisingularity ideal \mbox{$I^{\mbox{\scriptsize es}}\subset {\mathcal O}_{\P^2,z}$} of \mbox{$(F,z)$}. But the latter equality yields the required T--smoothness of the germ \mbox{$(H_{\P^2}^{es,d},F)$} (\cite{GrK,GrL}). \end{proof} \section{Proof of Theorems 1 and 2} \setcounter{equation}{0} \begin{definition} \label{5.1} Let \mbox{$X\in {\mathcal S}$} be a singularity scheme. By $s(X)$ we denote the minimal integer such that there exists an irreducible curve $F$ of degree $s(X)$ with a singular point $z$ of type $X$ as its only singularity, and such that the germ \mbox{$(H_{\P^2}^{es,d},F)$} of the equisingular stratum \mbox{$H_{\P^2}^{es,d} \subset \P^N=\P(\Gamma({\mathcal O}_{\P^2}(d)))$}, \mbox{$N=d(d\!+\!3)/2$}, at $F$ is T--smooth. \end{definition} \begin{lemma} \label{5.1a} Let \mbox{$X\in {\mathcal S}$} be a singularity scheme. Then $s(X)\leq \sigma(X)$, where $$\sigma(X):= \left\{ \renewcommand{\arraystretch}{1.7} \begin{array}{cl} \left[\sqrt{2} \,\mbox{$\mathrm{mt}$} \, X\right] +1, &\mbox{ if }\ X \mbox{ is ordinary,}\\ \displaystyle \left[(1\!+\!\sqrt{2})\sqrt{\deg X\!+\!\mbox{$\mathrm{mt}$} \, X\!+\!1}\right]\!+\!\mbox{$\mathrm{mt}$} \, X\!+\!3, &\mbox{ if } X\in{\mathcal G}{\mathcal S}_1,\\ \left[\sqrt{(\deg X\!+\!\mbox{$\mathrm{mt}$} \, X\!+\!1)\beta_0^{-1}}\right]\!+\!\mbox{$\mathrm{mt}$} \, X\!+\!\mbox{$\mathrm{mt}$}_sX\!+\!3, & \mbox{ if } X\not\in{\mathcal G}{\mathcal S}_1. \end{array} \right. $$ \end{lemma} \begin{proof} By Lemma \ref{4.11}, \mbox{$s(X)\le\sigma(X)$} for any non--ordinary singularity $X$. The case of an ordinary singularity was already treated in \cite{GLS} and the result follows from Lemma \ref{5.3} below. \end{proof} \begin{definition} \label{5.2a} Let \mbox{$\overline{z}=(z_1,\dots,z_n)$} be a tuple of distinct points in $\P^2$, and \mbox{$\overline{m}=(m_1,\dots,m_n)$} be a vector of positive integers, then we denote by \mbox{$X(\overline{z},\overline{m})$} the zero--dimensional scheme in $\P^2$ defined by the ideals \mbox{$\mathfrak{m}_{z_i}^{m_i}\subset{\mathcal O}_{\P^2,z_i}$}, \mbox{$i=1,\dots,n$}. \end{definition} \begin{lemma}[GLS, section 3.3] \label{5.3} Let \mbox{$z_1,\dots,z_n$} be distinct generic points in $\P^2$, and the positive integers \mbox{$d,m_1,\dots,m_n$} satisfy $$\sum_{i=1}^n\frac{m_i(m_i+1)}{2}\:<\:\frac{d^2+6d-1}{4}- \left[\frac{d}{2}\right]\, .$$ Then (1) there exists an irreducible curve $F_d$ of degree $d$ with ordinary singular points \mbox{$z_1,\dots,z_n$} of multiplicities \mbox{$m_1,\dots,m_n$}, respectively, as its only singularities; (2) for \mbox{$\overline{z}=(z_1,\dots,z_n)$}, \mbox{$\overline{m}=(m_1,\dots,m_n)$}, \begin{equation} \label{5.4} h^1({\mathcal J}_{X(\overline{z},\overline{m})/ \P^2}(d))=0\, . \end{equation} \end{lemma} \begin{lemma} \label{5.5} If a positive integer $d$ and the singularities \mbox{$S_1,\dots,S_n$} satisfy the inequality $$\sum_{i=1}^n \frac{(s(S_i)+1)(s(S_i)+2)}{2}\:<\: \frac{d^2+6d-1}{4}-\left[ \frac{d}{2}\right],$$ then there exists an irreducible curve $F$ of degree $d$ with exactly $n$ singular points of (topological) types \mbox{$S_1,\dots,S_n$}, respectively. \end{lemma} \begin{proof} Due to Lefschetz's principle (cf.~\cite{JL}, Theorem 1.13), we suppose, without loss of generality, \mbox{${\mathbf K}={\mathbf C}$}. Let $$G_i(x,y)=\sum_{0\le k+l\le s(S_i)}a^{(i)}_{kl} x^ky^l,\quad \deg G_i=s(S_i),\quad i=1,\dots,n,$$ be affine irreducible curves such that each of them has exactly one singular point at \mbox{$(0,0)$} of type \mbox{$S_1,\dots,S_n$}, respectively. On the other hand, let \mbox{$z_1,\dots,z_n$} be distinct generic points in $\P^2$, then Lemma \ref{5.3} implies the existence of an irreducible curve $G'$ of degree $d$ having ordinary singularities at \mbox{$z_1,\dots,z_n$} of multiplicities \mbox{$m_i=s(S_i)+1$}, \mbox{$i=1,\dots,n$}, as its only singularities. For any \mbox{$i=1,\dots,n$}, let us fix affine coordinates $x_i$, $y_i$ in a neighbourhood of $z_i$. The relation (\ref{5.4}) means that an affine neighbourhood $U$ of $G'$ in \mbox{$\P^N=\P(\Gamma({\mathcal O}_{\P^2}(d)))$} can be parametrized by the following independent parameters: to any \mbox{$\Phi\in U$} we assign \begin{itemize} \itemsep0.1cm \item[(1)] coefficients \mbox{$A^{(i)}_{kl}$}, \mbox{$0\le k+l\le s(S_i)$}, from the representation $$\Phi(x_i,y_i)=\sum_{0\le k+l\le d}\! A^{(i)}_{kl}x^k_iy^l_i$$ of $\Phi$ in the coordinates $x_i$, $y_i$, for any \mbox{$i=1,\dots,n$}, \item[(2)] some parameters $B_j$, \mbox{$1\le j\le r:=N-\sum_{i=1}^n (s(S_i)\!+\!1)(s(S_i)\!+\!2)/2$}. \end{itemize} First we deform $G'$ into a curve $G$ by addition of the leading form, that is, the part of degree $s(S_i)$, of \mbox{$G_i(x,y)$} as the $s(S_i)$--jet at $z_i$, \mbox{$i=1,\dots,n$}. Without loss of generality we may suppose that $G$ is irreducible and has no singularities outside \mbox{$\{z_1,\dots,z_n\}$}. Moreover, in the local coordinates $x_i$, $y_i$, $G$ is represented as $$G^{(i)}(x_i,y_i)=\sum_{d\ge k+l\ge s(S_i)}\!a^{(i)}_{kl}x_i^ky_i^l,$$ \mbox{$i=1,\dots,n$}, and corresponds to the parameter values \mbox{$A^{(i)}_{kl} = 0$}, if \mbox{$k+l<s(S_i)$}, \mbox{$A^{(i)}_{kl} = a^{(i)}_{kl}$}, if \mbox{$k+l=s(S_i)$}, and \mbox{$B_j=b_j$}, \mbox{$j=1,\dots,r$}. We shall look for the desired curve $F$ close to $G$, given by parameters $$ A^{(i)}_{kl}(\tau) = \left\{ \begin{array}{cr} \tau^{s(S_i)-k-l}a^{(i)}_{kl}(\tau) & \mbox{ if } \,k+l<s(S_i)\\ a^{(i)}_{kl}(\tau) & \mbox{ if } \,k+l=s(S_i) \end{array} \right. \; \mbox{ and } \; B_j=b_j(\tau)\,, $$ where \mbox{$a^{(i)}_{kl}(\tau)$}, \mbox{$b_j(\tau)$} are smooth functions in a neighbourhood of zero such that $$a^{(i)}_{kl}(0)\,=\,a^{(i)}_{kl},\quad b_j(0)\,=\,b_j,$$ for all \mbox{$i,j,k,l$}. Let us fix some \mbox{$1\leq i\leq n$}. In a neighbourhood of the point $z_i$ we have $$F(x_i,y_i)=\sum_{0\le k+l\le s(S_i)}\!\!\tau^{s(S_i)-k-l} a^{(i)}_{kl}(\tau)x_i^ky_i^l+\!\! \sum_{d\geq p+q>s(S_i)}\!a^{(i)}_{pq}(\tau)x_i^py_i^q\,,$$ where the coefficients \mbox{$a^{(i)}_{pq}(\tau)$}, \mbox{$p+q>s(S_i)$}, are affine functions in the parameters \mbox{$A^{(s)}_{kl}$}, \mbox{$s\neq i$}, and $B_j$, \mbox{$j=1,\dots,r$}. The transformation \mbox{$(x_i,y_i)\mapsto (\tau x_i,\tau y_i)$} turns $F$ for sufficiently small \mbox{$\tau \neq 0$} into a curve $$F_i(x_i,y_i)=\sum_{0\le k+l\le s(S_i)}\! a^{(i)}_{kl}(\tau) x_i^ky^l_i\,+\!\!\sum_{d\geq p+q>s(S_i)} \!\tau^{p+q-s(S_i)}a^{(i)}_{pq} (\tau) x_i^py_i^q$$ close to $G_i$ in \mbox{$\P^N=\P(\Gamma({\mathcal O}_{\P^2}(d)))$}. By the definition of $s(S_i)$, the germ of the equisingular stratum \mbox{$H_{\P^2}^{es,d}\subset \P^N$} at $G_i$ can be described by \mbox{$c(S_i)$} equations $$\varphi^{(i)}_u(A^{(i)}_{kl})=0,\quad u=1,\dots,c(S_i),$$ on the coefficients \mbox{$A^{(i)}_{kl}$}, \mbox{$0\le k+l\le d$}, of a curve $$H(x_i,y_i)=\sum_{0\le k+l\le d} \!A^{(i)}_{kl}x_i^ky_i^l,$$ such that there exists \mbox{$\Lambda_i\subset\{(k,l)\in{\mathbf Z}^2\,|\: k,l\ge 0, \ k+l\le s(S_i)\}$}, \mbox{$ \mbox{card}(\Lambda_i)=c(S_i)$}, with $$\mbox{det}\left(\frac{\partial\varphi^{(i)}_u}{\partial A^{(i)}_{kl}}\right)_{ \renewcommand{\arraystretch}{0.6} \begin{array}{c} \scriptstyle{u=1,\dots,c(S_i)}\\ \scriptstyle{(k,l)\in\Lambda_i} \end{array}} \ne 0$$ at the point \mbox{$A^{(i)}_{kl}=a^{(i)}_{kl}$}, \mbox{$0\le k\!+\!l\le s(S_i)$} and \mbox{$ A^{(i)}_{pq}=0$}, \mbox{$ p\!+\!q>s(S_i)$}. Thus, the condition on $F$ to have singular points of types \mbox{$S_1,\dots,S_n$} can be expressed as the system of equations \begin{equation} \label{5.6} \varphi^{(i)}_u \left( \{a^{(i)}_{kl}(\tau)\,|\:k\!+\!l\le s(S_i)\},\; \{ \tau^{p+q-s(S_i)}a^{(i)}_{pq}(\tau)\,|\: p\!+\!q > s(S_i)\} \right) =0\,, \end{equation} \mbox{$u=1,\dots,c(S_i)$}, \mbox{$i=1,\dots,n$}. From the above it follows immediately that at the point \mbox{$\tau=0$} the determinant $$\mbox{det} \left( \frac{\partial \varphi^{(i)}_u}{ \partial a^{(m)}_{kl}}\right)_{ \renewcommand{\arraystretch}{0.6} \begin{array}{c} \scriptstyle{u=1,\dots,c(S_i)}\\ \scriptstyle{(k,l)\in\Lambda_m}\\ \scriptstyle{m,i=1,\dots,n} \end{array}} =\prod_{i=1}^n \mbox{det}\left(\frac{\partial \varphi^{(i)}_u}{\partial a^{(i)}_{kl}}\right)_{ \renewcommand{\arraystretch}{0.6} \begin{array}{c} \scriptstyle{u=1,\dots,c(S_i)}\\ \scriptstyle{(k,l)\in\Lambda_i} \end{array}} $$ does not vanish, which implies the existence of an appropriate solution to (\ref{5.6}), and, hereby, the existence of a curve $F$ with $n$ singular points of types \mbox{$S_1,\dots,S_n$}. The only thing we should explain, is why $F$ has no other singular points. Let us consider $F$ as a polynomial function. First, note that $F$ is a small deformation of the function $G$, which has \mbox{$(d\!-\!1)^2-(s(S_1)\!-\!1)^2-\dots -(s(S_n)\!-\!1)^2$} critical points out of the zero level. Second, we deform each ordinary critical point $z_i$ of $G$ by means of the function $G_i$ which has \mbox{$(s(S_i)\!-\!1)^2-\mu(S_i)$} critical points out of the zero level. Hence, $F$ has at least \mbox{$(d\!-\!1)^2-\mu(S_1)-...-\mu(S_n)$} critical points out of the zero level, that means the $n$ constructed singular points are the only singular points of the curve $F$. \end{proof} Thus, to prove the Theorems 1 and 2 from the introduction, by Lemma \ref{5.1a}, it suffices to prove the following \begin{proposition} \label{5.8} For any singularity $X\in {\mathcal S}$, $$(\sigma(X)\!+\!1)(\sigma(X)\!+\!2)\le 196\,\mu(X)\, .$$ \end{proposition} \begin{proof} If $X$ is an ordinary singularity, then \mbox{$\mu(X)=(\mbox{$\mathrm{mt}$} \, X\!-\!1)^2$}; therefore $$(\sigma(X)\!+\!1)(\sigma(X)\!+\!2)\,\leq \,2(\mbox{$\mathrm{mt}$} \, X\!+\!1)(\mbox{$\mathrm{mt}$} \, X\!+\!3)\,\leq\, 30\,\mu(X).$$ For an arbitrary singularity $X\in {\mathcal S}$, defined by the germ \mbox{$(C,z)$}, we have $$\mu(X)\,=\,2\delta(X)-r+1\,=\,\sum_{q\in T^\ast\!(C)} m_q(m_q\!-\!1)-r+1,$$ where $r$ is the number of local branches. Note that the number of the essential points $q$ with \mbox{$m_q=1$} does not exceed \mbox{$\mbox{$\mathrm{mt}$} \, X<\sqrt{\mu(X)}+1$}. Hence, by Lemma \ref{1.6}, $$ 2(\deg X+\mbox{$\mathrm{mt}$} \, X+1) \: \leq \: \sum_{m_q>1} m_q(m_q\!+\!1) +4\mbox{$\mathrm{mt}$} \, X+2 \:< \: 3\left(\sqrt{\mu(X)}+\sqrt{2}\right)^2\,; $$ thus $$ \sigma(X) \: < \: \sqrt{ \frac{\deg X+\mbox{$\mathrm{mt}$} \, X+1}{\beta_0}}+2\mbox{$\mathrm{mt}$} \, X+3 \: < \: \left( \sqrt{\frac{3}{2\beta_0}}+2\right)\sqrt{\mu(X)}+\sqrt{\frac{3}{\beta_0}}+5\,, $$ and, finally, it follows that \mbox{$(\sigma(X)\!+\!1)(\sigma(X)\!+\!2)$} is smaller than $$ \left(\sqrt{\frac{3}{2\beta_0}} \!+\!2\right)^2\mu(X)+ \left(\sqrt{\frac{3}{\beta_0}}\!+\!6\right)\left(\sqrt{\frac{3}{\beta_0}} \!+\!7\right) +\left(2 \sqrt{\frac{3}{\beta_0}}\!+\!13 \right) \left(\sqrt{\frac{3}{2\beta_0}}\!+\!2\right)\sqrt{\mu(X)}, $$ which for \mbox{$\mu(X)\ge 2$} does not exceed \mbox{$196\mu(X)$}. \end{proof}

9704

alg-geom/9704002

en

https://arxiv.org/abs/alg-geom/9704002

alg-geom/9704002

Yi Hu and Wei-Ping Li

Birational Models of the Moduli Spaces of Stable Vector Bundles over Curves

To appear in Intern. Journal of Math., AMS-LaTeX

We give a method to construct stable vector bundles whose rank divides the degree over curves of genus bigger than one. The method complements the one given by Newstead. Finally, we make some systematic remarks and observations in connection with rationality of moduli spaces of stable vector bundles.

alg-geom

\section{Introduction} \label{sec:introduction} \begin{say} Let $X$ be a complete non-singular algebraic curve of genus $g \ge 2$ over an algebraically closed field $k$ of characteristic 0 (for simplicity we assume that $k$ is the field of complex numbers ${\Bbb C}$). The moduli space ${\cal M}(n, L)$ of semi-stable vector bundles of rank $n$ and determinant $L$ with degree ${\rm deg} L=d$ over $X$ is an irreducible projective variety of dimension $(n^2 -1)(g-1)$. The question whether the moduli space ${\cal M}(n, L)$ is rational is very subtle. The affirmative answer is known for many cases through the works of Narasimhan-Ramanan ([{\bf NR}]), Newstead ([{\bf N1, N2}]), and Tjurin ([{\bf T3}]). In proving rationality in [{\bf N1, N2}], Newstead conjectured a systematic way to construct (generic) stable vector bundles $F$ from (generic) stable vector bundles $F'$ of lower rank via extensions of the following type: $$\exact{{\cal O}_X^{\oplus r}}{F}{F'}$$ where ${\rm rank}(F)=n$ and ${\rm deg}F=d=n(g-1)+r$ for some $0<r<n$. He proved the conjecture in [{\bf N1, N2}] for some cases and Grzegorczyk completed the proof for all cases in her paper [{\bf G}]. In [{\bf N2}], this extension was used as an induction step to show that the following moduli spaces of stable vector bundles are rational. \end{say} \begin{thm} ({\rm [{\bf N2}]}) \label{newsteadrationalitythm} The moduli space ${\cal M}(n, L)$ is rational in the following cases: \begin{enumerate} \item $d=1$ mod $n$ or $d=-1$ mod $n$; \item $\gcd (n, d)=1$ and $g$ is a prime power; \item $\gcd(n, d)=1$ and the sum of the two smallest distinct prime factors of $g$ is greater than $n$. \end{enumerate} \end{thm} \begin{say} Clearly, from that $d=n(g-1)+r$ and $0<r<n$, one sees that the above method of constructing stable vector bundles leaves out the case when rank divides degree. {\sl It is one of the objectives of the current paper to find a method of constructing stable vector bundles in complement to that of} [{\bf N1, N2, G}]. \end{say} \begin{thm} \label{constructingstablebundles} Let $n$, $d$ be positive integers such that $d=ng$. Let $L={\cal O}_X(P_0)$ where $P_0$ is a special effective divisor defined in \ref{choosedivisor}. Then there exists a non-empty Zariski open subset of the moduli space ${\cal M}(n, L)$ consisting of vector bundles $V$ such that \begin{enumerate} \item $h^1(X, V)=0$. \item ${\cal O}_X^{\oplus n}$ is a sub-sheaf of $V$. \item there exists an exact sequence $$0\longrightarrow V^*\longrightarrow {\cal O}_X^{\oplus n}\mapright{\varphi} {\cal O}_P\longrightarrow 0\eqno(1)$$ where $P$ is a divisor $P=p_1+\ldots +p_d$ such that the map ${\varphi}$ in $Hom({\cal O}_X^{\oplus n}, {\cal O}_P)$ is stable with respect to the action of $Aut({\cal O}_X^{\oplus n}) \times Aut({\cal O}_P)$. \end{enumerate} \end{thm} \begin{rem} We point out that Tjurin ([{\bf T1, T2}]) studied these moduli spaces using the different method of matrix divisors. \end{rem} For the purpose of comparison, we mention a result of Grzegorczyk [{\bf G}] who proved the following conjecture of Newstead: \begin{thm} {\rm ([{\bf N1, G}]) } \label{gthm} Let, $n$, $d$, $r$, be natural numbers such that $d=n(g-1)+r$, $0<r<n$. Then there exists a non-empty Zariski open subset of the moduli space ${\cal M}(n, L)$ consisting of vector bundles $F$ such that \begin{enumerate} \item $h^1(F)=0$. \item ${\cal O}_X^{\oplus r}$ is a sub-bundle of $F$. \item the quotient bundle $F/{\cal O}_X^{\oplus }$ is stable. \end{enumerate} \end{thm} This theorem covers all moduli spaces except those where $n\, |\, d$. Our theorem above deals exactly the complement. \begin{say} The tool that we used to construct stable vector bundles is that of elementary transformations which were probably first studied by Maruyama ([{\bf M}]). We choose a special effective divisor $P_0$ (see \ref{choosedivisor}) whose degree equals $ng$. Take $L$ to be ${\cal O}_X(P_0)$. Let $U$ be a non-empty Zariski open subset of the linear system $|P_0|$ satisfying certain generic condition (see \ref{chooseU}). Let $P$ be an effective divisor in $U$. Now given a surjective map $\varphi\in Hom({\cal O}_X^{\oplus n}, {\cal O}_P)$ (which is called an elementary transformation), we get an exact sequence $$0\longrightarrow {W}\longrightarrow {{\cal O}^{\oplus n}_X}\mapright{\varphi} {\cal O}_P\longrightarrow 0.$$ Let $V=W^*$. Then we see that ${\rm rank} V=n,\quad {\rm det} V={\cal O}_X(P)={\cal O}_X(P_0)=L.$ Then Theorem \ref{constructingstablebundles} (see also \S\S 3--5) basically asserts that if the elementary transformation satisfies some generic conditions, then $V$ is stable and $h^1(V)=0$. Furthermore, the stable vector bundles $V$ obtained this way form a non-empty Zariski open subset of the moduli space and hence it leads to a birational model for the moduli space. \end{say} \begin{say} In the end of this paper, we shall systematically explore that the inductive method in ([{\bf N2}]) can actually be extended to prove rationality of a rather larger class of moduli spaces. This class consists of much more cases than those listed in Theorem \ref{newsteadrationalitythm}. For example, when $g=6$, $n=15, d=77$, the moduli space ${\cal M}(n, L)$ is rational. Also, the moduli spaces ${\cal M}(n, L)$ with $$n =11+7m, \quad {\rm deg}L=62+35m$$ are all rational over any genus $6$ curve for $m\ge 7$. But the set $\{ g=6, n=15, d=77 \}$ and $\{ g=6, n=11+7m, d= 62+35m \}$ satisfy none of the three conditions as listed in Theorem \ref{newsteadrationalitythm}. In general, the moduli space ${\cal M}(n, L)$ is rational if $n(g-1) < d={\rm deg}L < ng$ and the pair $(n; d)$ can be ``linked'' to $(1; g)$ by some successive arithmetic reductions of the following types: \begin{enumerate} \item $(n'; d')=(ng-d; d-k(ng-d))$ for some non-negative integer $k$ such that $n'(g-1) < d' \le n'g$; or \item $(n''; d'')=(d-n(g-1); n(2g-1)-d-k(d-n(g-1)))$ for some non-negative integer $k$ such that $n''(g-1) < d'' \le n''g$. \end{enumerate} This fact (Theorems \ref{newsteadpairthm} and \ref{ballicopairthem}), together with some other results, is systematically explored in \S 6. Also, we shall point out how our Theorem \ref{constructingstablebundles} is related to other moduli spaces by similar arithmetic reductions as above. \end{say} \begin{say} The paper is structured as follows. In \S 2, we recall the Abel-Jacobi theory and other technical results that will be used in the latter sections. \S 3 deals with elementary transformations and how to use them to construct stable vector bundles as stated in Theorem \ref{constructingstablebundles}. \S 4 studies the group action of $Aut({\cal O}_P)$ and $Aut({\cal O}_X^{\oplus n})$ on $Hom({\cal O}_X^{ \oplus n}, {\cal O}_P)$. \S 5 concerns birational models for the moduli spaces and completes the proof of our main result Theorem \ref{constructingstablebundles}. Finally, in \S 6, we make some systematic remarks and observations on rationality of moduli spaces. \end{say} \begin{say} We fix the following notations: \par\noindent $V$ is a locally free sheaf, by abuse of notation, we also call it a vector bundle. \par\noindent $V^*$ is the dual of $V$, i.e. $V^*={\cal H}om({\cal O}_X, V)$. \par\noindent $H^i(V)$ is the cohomology $H^i(X, V)$. \par\noindent $h^i(V)$ is the dimension of $H^i(V)$. \par\noindent $n$ is the rank of the vector bundle $V$. \par\noindent $d$ is the degree of the vector bundle $V$. \par\noindent ${\rm det} V=\wedge^n V$ is the determinant line bundle. \par\noindent $L$ is a line bundle with ${\rm deg} L=d$. \par\noindent $(M)^d= M\times\ldots\times M$ ($d$-times) \par\noindent (2.1), (2.2), ... , are the indices labeling formulas. \par\noindent 2.1, 2.2, ... , are the indices labeling theorems, remarks, paragraphs, and so on. \end{say} This note stems from our study of the rationality problem of moduli spaces when both of the authors were visiting Max-Planck-Institute f\"ur Mathematik in the summer of 1993. Due to the fact that there are few papers in the literature addressing the moduli spaces when the rank divides the degree, we decided to write up this note and make it available to interested readers. \medskip\noindent {\bf Acknowledgments:} The hospitality and financial support from the Max-Planck-Institute (summer, 1993) are gratefully acknowledged. We thank P. Newstead, Igor Dolgachev, A. Beauville for their attention on this work. W.L. thanks M. Reid and MRC of University of Warwick for providing a stimulating environment for research. Y.H. acknowledges the Centennial Fellowship award by the American Mathematical Society. \section{Preliminaries} \begin{defn} Let $X$ be a smooth projective curve over the field of complex numbers and $V$ a rank-$n$ algebraic vector bundle over $X$. $V$ is stable (semi-stable) if for any proper sub-vector bundle $F$ of $V$, $${{\rm deg }F\over {\rm rank }F}<\,\,(\le)\,\,{{\rm deg}V\over {\rm rank V}}. \eqno(2.1)$$ Since any torsion free coherent sheaf over a curve is locally free, we can get a slightly modified definition of stability ( or semi-stability): $V$ is stable ( or semi-stable) if for any proper sub-vector bundle $F$ of $V$ whose cokernel $V/F$ is a vector bundle, (2.1) holds. \end{defn} \begin{say} Let $L$ be a line bundle over $X$. Throughout this paper, ${\cal M}(n, \,L)$ represents the moduli space of rank-$n$ semi-stable vector bundles over $X$ with ${\rm det}V \buildrel\sim\over =L$. \end{say} \begin{say} \label{choosedivisor} We begin with a special divisor whose corresponding line bundle will be chosen as the fixed determinant of our semi-stable bundles. Let $\omega_1,\ldots, \omega_g$ be a basis of $H^0(K_X)$. By the Abel-Jacobi theory (see [{\bf GH}]), for general points $x_1,\ldots, x_g\in X$, the determinant $$ \left| \begin{array}{clcr} \omega_1(x_1) & \omega_1(x_2) & \ldots & \omega_1(x_g) \\ \vdots & \vdots & \ddots & \vdots \\ \omega_g(x_1)& \omega_g(x_2) & \ldots & \omega_g(x_g) \end{array} \right| $$ is not zero. We now choose, once and for all, $d=ng$ many distinct points $q_1, \ldots, q_{ng}$ such that the determinants $$\left| \begin{array}{cccc} \omega_1(q_{i+1})&\omega_1(q_{i+2})& \ldots & \omega_1(q_{i+g}) \\ \vdots&\vdots&\ddots&\vdots \\ \omega_g(q_{i+1})&\omega_g(q_{i+2})&\ldots&\omega_g(q_{i+g}) \end{array} \right| \neq 0 \eqno(2.2) $$ for $i=0, g, 2g, \ldots, (n-1)g$. Here $n$ is a positive integer bigger than one which will be taken later on as the rank of vector bundles. Let $P_0$ be the effective divisor $P_0=q_1+\ldots+q_{ng}$. The linear system $|P_0|$ is a projective space of dimension $(n-1)g$. This can be easily seen by Riemann-Roch. \end{say} \begin{say} For a technical reason as we shall see in \S 3, we need to consider the determinant $$\left| \begin{array}{cccc} b_{11}\omega_1(x_1)&b_{21}\omega_1(x_2)&\cdots&b_{d1}\omega_1(x_d)\\ b_{11}\omega_2(x_1)&b_{21}\omega_2(x_2)&\cdots&b_{d1}\omega_2(x_d) \\ \vdots&\vdots&\ddots&\vdots \\ b_{11}\omega_g(x_1)&b_{21}\omega_g(x_2)&\cdots&b_{d1}\omega_g(x_d) \\ b_{12}\omega_1(x_1)&b_{22}\omega_1(x_2)&\cdots&b_{d2}\omega_1(x_d) \\ \vdots&\vdots&\ddots&\vdots \\ b_{12}\omega_g(x_1)&b_{22}\omega_g(x_2)&\cdots&b_{d2}\omega_g(x_d) \\ \vdots&\vdots&\ddots&\vdots \\ b_{1n}\omega_1(x_1)&b_{2n}\omega_1(x_2)&\cdots&b_{dn}\omega_1(x_d) \\ \vdots&\vdots&\ddots&\vdots \\ b_{1n}\omega_g(x_1)&b_{2n}\omega_g(x_2)&\cdots&b_{dn}\omega_g(x_d) \end{array} \right|. \eqno (2.3)$$ Take $b_{ij}$'s as unknowns, the determinant at the point $(q_1,\cdots, q_d)\in (X)^d$ (chosen as in \ref{choosedivisor}) is a non-vanishing polynomial in variables $b_{ij}$. The way to see this is as follows. We expand the determinant of (2.3) and check the coefficient of $$b_{11}b_{21}\ldots b_{g1}b_{(g+1)2}\ldots b_{(2g)2}\ldots b_{(d-g)n}\ldots b_{dn}.$$ By the technique of minor expansions in the determinant theory, the coefficient is $$\hbox{$\prod\limits_{i=0}^{(n-1)g}$} \left| \begin{array}{cccc} \omega_1(q_{i+1})&\omega_1(q_{i+2})& \ldots&\omega_1(q_{i+g}) \\ \vdots&\vdots&\ddots&\vdots \\ \omega_g(q_{i+1})&\omega_g(q_{i+2})&\ldots&\omega_g(q_{i+g}) \end{array} \right| $$ which is not zero by the choices of $q_i$'s. Hence the determinant (2.3) as a polynomial in variables $b_{ij}$ is not identically zero. \end{say} \begin{say} \label{chooseU} Choose a collection of $b_{ij}$'s such that the determinant at $(q_1,\cdots, q_d)$ is not zero. With these $b_{ij}$'s fixed, the zero locus of the determinant (2.3) defines a divisor in $|P_0|$. So we can choose a non-empty Zariski open subset $U$ of $|P_0|$ such that every divisor $P=p_1+\dots+p_d$ in $U$ satisfies the property that $p_i$'s are distinct and that the determinant (2.3) with the chosen $b_{ij}$'s is non-zero at $P\in U$. \end{say} \begin{say} It is known that ${\cal M}(n, L) \cong {\cal M}(n, L')$ if ${\rm deg} L={\rm deg} L'$. This can be seen as follows. There exists a line bundle $\bar L\in Pic^0(X)$ such that $\bar L^{\otimes n}=L^*\otimes L'\in Pic^0(X)$. We have that ${\rm det} (V\otimes \bar L)={\rm det} V\otimes \bar L^{\otimes n}=L\otimes L^*\otimes L'=L'={\rm det}V'$. Hence there is a bijection $${\cal M}(n, L)\mapright{\otimes \bar L} {\cal M}(n,L').\eqno(2.4)$$ Since these two moduli spaces are coarse, the map (2.4) must be an isomorphism. Therefore, we have some freedom in choosing a line bundle $L$ in a way we like without affecting the isomorphic type of the moduli space ${\cal M}(n, L)$. In this paper, we choose $L={\cal O}_X(P_0)$. \end{say} \section{Stable vector bundles whose ranks divide their degrees} This section is entirely devoted to the case when the rank of the bundles divides the degree. We shall use elementary transformations to construct generic rank-$n$ stable vector bundles $V$ with degree $ng$ whose $H^1(V)$ is trivial. In the section that follows, we shall discuss the necessary group actions on the space of elementary transformations, which will lead to a birational model for the moduli space. \begin{defn} Let $W_1$ $W_2$ be two rank-$n$ vector bundles over an algebraic variety. Suppose that there exists an injective morphism from $W_1$ to $W_2$, then we can write a short exact sequence $$0\longrightarrow{W_1}\longrightarrow {W_2}\mapright{\varphi}{Q} \longrightarrow 0$$ where $Q$ is a torsion sheaf. In this case, we say that $W_1$ is an elementary transformation of $W_2$ with respect to the surjective map $W_2\mapright{\varphi} Q \longrightarrow 0$; or we may simply say that the map $\varphi$ is an elementary transformation. \end{defn} \begin{say} \label{elemtrans} Let $P=p_1+\ldots +p_d$ be an effective divisor over the curve $X$ such that $P\in U$ (see \ref{chooseU}). Consider an elementary transformation: $$0\longrightarrow {W}\longrightarrow{{\cal O}_X^{\oplus n}}\mapright{\varphi} {{\cal O}_P} \longrightarrow 0. $$ $W$ has to be a vector bundle. Let $V=W^*$ or $W=V^*$. Then the above exact sequence can be rewritten as $$0\longrightarrow {V^*}\longrightarrow {{\cal O}_X^{\oplus n}}\mapright{\varphi}{{\cal O}_P}\longrightarrow 0. \eqno(3.1)$$ A simple computation leads to $${\rm rank}V=n,\quad {\rm det}V={\cal O}_X(P)={\cal O}_X(P_0)=L,\quad\hbox{and } {\rm deg}V=d.$$ Such elementary transformations $\varphi$ are classified by surjective maps in $$Hom({\cal O}_X^{\oplus n}, {\cal O}_P)=\hbox{$\bigoplus\limits_{i=1}^d$} Hom ({\cal O}_X^{\oplus n},{\cal O}_{p_i})\buildrel\over= ({\Bbb C}^n)^d. \eqno (3.2)$$ \end{say} \begin{notationnum} A map $\varphi$ in (3.1) can be expressed, under the isomorphisms in (3.2), as a matrix $$\left( \begin{array}{c} \varphi_1 \\ \vdots \\ \varphi_d \end{array} \right) = \left( \begin{array}{ccc} a_{11}&\ldots&a_{1n} \\ \vdots&\ddots&\vdots \\ a_{d1}&\ldots&a_{dn} \end{array} \right) \eqno (3.3) $$ where $\varphi_i$ represents the vector $(a_{i1},\ldots, a_{in})$ in $Hom({\cal O}_X^{\oplus n}, {\cal O}_{p_i}) \cong {\Bbb C}^n$. \end{notationnum} \begin{rem} \label{norowzero} $\varphi$ is a surjection if and only if no rows $\varphi_i$ in (3.3) are zero vectors. \end{rem} \begin{say} Tensor the exact sequence (3.1) by $K_X$ and then take the long cohomological exact sequence, we get $$0\longrightarrow H^0(V^*\otimes K_X)\longrightarrow H^0({\cal O}_X^{\oplus n} \otimes K_X)\mapright {\varphi^0} H^0({\cal O}_P) $$ $$\longrightarrow H^1(V^*\otimes K_X) \longrightarrow H^1({\cal O}_X^{\oplus n}\otimes K_X)\longrightarrow 0. \eqno (3.4) $$ \end{say} We need to introduce two generic conditions on elementary transformations (3.1). We shall then show that if an elementary transformation satisfies these two conditions, then the vector bundle $V$ is stable and $h^1(V)=0$. \begin{condition} We define the following: \begin{enumerate} \item{(A)} An elementary transformation $\varphi$ in $Hom({\cal O}_X^{\oplus n},{\cal O}_P)$ is said to satisfy Condition A if the induced map $\varphi^0$ of $\varphi$ in (3.4) $$\varphi^0\colon H^0(K_X^{\oplus n})\mapright{} H^0({\cal O}_P)$$ is an isomorphism. \item{(B)} An elementary transformation $\varphi$ in $Hom({\cal O}_X^{\oplus n},{\cal O}_P)$ is said to satisfy Condition B if any $n$ many $\varphi_i$'s in (3.3) are linearly independent. \end{enumerate} \end{condition} \begin{rem} \label{conditionsab} \par \begin{enumerate} \item From the exact sequence (3.4), we see that if $\varphi$ satisfies Condition A, then $h^0(V^*\otimes K_X)=0$, or $h^1(V)=0$ by Serre duality. Converse is also true. That is, if $h^1(V)=0$, then $\varphi^0$ is injective. Because $h^0({\cal O}_X^{\oplus n} \otimes K_X) = h^0({\cal O}_P) = d$, it has to be an isomorphism. Hence Condition A is equivalent to $h^1(V)=0$. By Riemann-Roch, it is also equivalent to $h^0(V)=n$. \item Generic $d\times n$ matrix satisfies Condition B. Also this property is clearly invariant under the natural action of $Aut({\cal O}_X^{\oplus n})$ and $Aut ({\cal O}_P)$ on $Hom({\cal O}_X^{\oplus n}, {\cal O}_P)$. \end{enumerate} \end{rem} \begin{lem} Fix any divisor $P$ in $U$. Generic elementary transformation $\varphi$ in (3.1) satisfies Condition A. \end{lem} \smallskip\noindent{\it Proof. } First notice that $h^0(K_X^{\oplus n})=ng =d =h^0({\cal O}_P)$. Let $\omega_1,\ldots,\omega_g$ be a basis of $H^0(K_X)$ as chosen in \ref{choosedivisor}. Take $$\underbrace{(\omega_1, 0,\ldots,0)}_n,\ldots,(\omega_g,0\ldots,0),\ldots, (0,\ldots, 0,\omega_1),\ldots, (0,\ldots, 0, \omega_g)$$ as a basis of $H^0(K_X^{\oplus n})$. One checks that $$\varphi^0(\omega_1,0\ldots, 0)= (a_{11}\omega_1(p_1),a_{21}\omega_1(p_2), \ldots, a_{d1}\omega_1(p_d)).$$ \par By the similar calculation for other elements of the basis, we get a natural matrix representation of the map $\varphi^0$: $$\left( \begin{array}{cccc} a_{11}\omega_1(p_1)&a_{21}\omega_1(p_2)&\cdots&a_{d1}\omega_1(p_d) \\ a_{11}\omega_2(p_1)&a_{21}\omega_2(p_2)&\cdots&a_{d1}\omega_2(p_d) \\ \vdots&\vdots&\ddots&\vdots \\ a_{11}\omega_g(p_1)&a_{21}\omega_g(p_2)&\cdots&a_{d1}\omega_g(p_d) \\ a_{12}\omega_1(p_1)&a_{22}\omega_1(p_2)&\cdots&a_{d2}\omega_1(p_d) \\ \vdots&\vdots&\ddots&\vdots \\ a_{12}\omega_g(p_1)&a_{22}\omega_g(p_2)&\cdots&a_{d2}\omega_g(p_d) \\ \vdots&\vdots&\ddots&\vdots \\ a_{1n}\omega_1(p_1)&a_{2n}\omega_1(p_2)&\cdots&a_{dn}\omega_1(p_d) \\ \vdots&\vdots&\ddots&\vdots \\ a_{1n}\omega_g(p_1)&a_{2n}\omega_g(p_2)&\cdots&a_{dn}\omega_g(p_d) \end{array} \right) \eqno (3.5)$$ Hence $\varphi$ satisfies Condition A iff the determinant of (3.5) is not zero. Regarding the determinant of (3.5) as a polynomial in variables $a_{ij}$ with $\omega_i(p_j) $'s fixed, either the polynomial is identically zero or the zero locus of this polynomial is a divisor in the space $Hom({\cal O}_X^{\oplus n}, {\cal O}_P)$. However we know that if $P$ is in $U$, the determinant of the matrix (3.5) is not zero at $a_{ij}=b_{ij}$ (see \ref{chooseU}). Hence the set of the elementary transformations $\varphi$ satisfying Condition A is a non-empty open dense subset of the space $Hom({\cal O}_X^{\oplus n}, {\cal O}_P)$.\hfill\qed \begin{cor} For generic elementary transformations $\varphi\in Hom({\cal O}_X^{\oplus n}, {\cal O}_P)$, $$0\longrightarrow V^*\mapright{}{\cal O}_X^{\oplus n}\mapright{\varphi} {\cal O}_P\mapright{}0 \qquad \hbox{where $P\in U$},$$ $h^0(V)=n$ or equivalently $h^1(V)=0$. \end{cor} \begin{thm} \label{stable} Fix a divisor $P\in U$. If an elementary transformation $\varphi$ satisfies Condition A and B, then $V$ is stable and satisfies: \begin{enumerate} \item ${\rm deg} V =ng$ and ${\rm det}V={\cal O}_X(P)=L$; \item $h^1(V)=0 $. \end{enumerate} \end{thm} \smallskip\noindent{\it Proof. } Suppose $V$ is not stable. Then there exist rank-$f$ vector bundle $F$, rank-$f'$ vector bundle $F'$, and an exact sequence: $$\exact{F}{V}{F'}\eqno(3.6)$$ such that ${\rm deg}F\ge \displaystyle {f\cdot{\rm deg} V\over n} =fg$. Take the dual of (3.1), we get an exact sequence $$\exact{{\cal O}_X^{\oplus n}}{V}{{\cal O}_P}.\eqno(3.7)$$ Combining (3.6) and (3.7), we obtain a commutative diagram of exact sequences: $$\begin{array}{cccccccccccc} &&&&0 \\ &&&&\mapup{} \\ &&&&F' \\ &&&&\mapup{} \\ 0&\mapright{}&{\cal O}_X^{\oplus n}&\mapright{}&V&\mapright{}&{\cal O}_P& \mapright{}&0 \\ &&\mapup{}&&\mapup{}&&\mapup{} \\ 0&\mapright{}&E&\mapright{}&F&\mapright{}&{\cal O}_Q&\mapright{}&0 \\ &&\mapup{}&&\mapup{}&&\mapup{} \\ &&0&&0&&0 \end{array} \eqno (3.8) $$ where $E$ is a rank-$f$ vector bundle and $Q$ is a subset of $P$. \par Diagram (3.8) can be extended into another commutative diagram of exact sequences: $$\begin{array}{ccccccccccc} &&0&&0&&0 \\ &&\mapup{}&&\mapup{}&&\mapup{} \\ 0&\mapright{}&E'&\mapright{}&F'&\mapright{}&{\cal O}_R&\mapright{}&0 \\ &&\mapup{}&&\mapup{}&&\mapup{} \\ 0&\mapright{}&{\cal O}_X^{\oplus n}&\mapright{}&V&\mapright{}&{\cal O}_P&\mapright{}&0 \\ &&\mapup{}&&\mapup{}&&\mapup{} \\ 0&\mapright{}&E&\mapright{}&F&\mapright{}&{\cal O}_Q&\mapright{}&0 \\ &&\mapup{}&&\mapup{}&&\mapup{} \\ &&0&&0&&0 \end{array} \eqno (3.9)$$ where $R$ is a subset of $P$. >From Riemann-Roch, we get $h^0(F)={\rm deg}F+f(1-g)+h^1(F)\ge f$ and equality holds iff $h^1(F)=0$ and ${\rm deg} F=fg$. \par Take long cohomological exact sequence of the diagram (3.9), we get a commutative diagram of exact sequences: $$\begin{array}{ccccccccccccccccc} 0&\mapright{}&H^0(E')&\mapright{\alpha_1}&H^0(F')&\mapright{\alpha_2}& H^0({\cal O}_R) \\ &&\mapup{}&&\mapup{\phi_1}&&\mapup{\varphi_1} \\ 0&\mapright{}&H^0({\cal O}_X^{\oplus n})&\mapright{\beta_1}& H^0(V)&\mapright{\beta_2}&H^0({\cal O}_P) \\ &&\mapup{}&&\mapup{\phi_2}&&\mapup{\varphi_2} \\ 0&\mapright{}&H^0(E)&\mapright{\gamma_1}&H^0(F)&\mapright{\gamma_2} &H^0({\cal O}_Q) \\ &&\mapup{}&&\mapup{}&&\mapup{} \\ &&0&&0&&0. \end{array} \eqno (3.10)$$ Since the map $\beta_2$ is a zero map by Condition A, we get $\gamma_2$ is also a zero map, hence $h^0(E)=h^0(F)$. \par Assume that $h^0(F)>f$, then $h^0(E)=h^0(F)>f$. Consider the sub-sheaf $E_1$ of $E$ generated by global sections of $E$. Since $E$ is a sub-sheaf of ${\cal O}_X^{\oplus n}$, $H^0(E)$ is a subspace of $ H^0({\cal O}_X^{\oplus n})$. Hence the sub-sheaf $E_1$ is a trivial sheaf with rank equal to $h^0(E)>f$. This contradicts to the fact that ${\rm rank}E=f$. Thus $h^0(F)$ must equal $f$, therefore $h^1(F)=0$ and ${\rm deg} F=fg$. In this case, we get $${\cal O}^{\oplus f}\hookrightarrow E\hookrightarrow {\cal O}^{\oplus n}.$$ The first map forces ${\rm deg} E\ge 0$ and the second map forces ${\rm deg}E\le 0$. Hence ${\rm deg}E=0$ and the map ${\cal O}_X^{\oplus f} \longrightarrow E$ is an isomorphism. From the exact sequence $\exact{{\cal O}_X^{\oplus f}}{F}{{\cal O}_Q}$, we can see that the number of points in $Q$ is $\ell(Q)={\rm deg} F=fg$. Hence $\ell(R)=ng -fg =f'g$. \par Now we consider the dual diagram of (3.9): $$\begin{array}{ccccccccccccc} {}&&0&&0&&0 \\ &&\mapdown{}&&\mapdown {}&&\mapdown{} \\ 0&\mapleft{}&{\cal O}_R&\mapleft{}&E^{'*}&\mapleft{}&F^{'*}&\mapleft{}&0 \\ &&\mapdown{}&&\mapdown {}&&\mapdown{} \\ 0&\mapleft{}&{\cal O}_P&\mapleft{}&{\cal O}_X^{\oplus n}&\mapleft{}&V^*& \mapleft{}&0 \\ &&\mapdown{}&&\mapdown{}&&\mapdown{} \\ 0&\mapleft{}&{\cal O}_Q&\mapleft{}&{\cal O}^{\oplus f}_X&\mapleft{}&F^*& \mapleft{}&0 \\ &&\mapdown{}&&\mapdown{}&&\mapdown{} \\ &&0&&0&&0. \end{array} \eqno (3.11)$$ \par Since $h^1(V)=0$ (due to Condition A), $h^0(V^*\otimes K_X)=0$ by the Serre duality. Also $h^0(K_X)\ge 1$ because we have assumed that $g\ge 2$. Hence $h^0(V^*)=0$ and $h^0(F^{'*})=0$ as well. Since $h^1(F)=0$, by the similar argument, $h^0(F^*)=0$. Now we take the cohomology of the diagram (3.11), we get $$\begin{array}{ccccccccc} 0&&0 \\ \mapdown{}&&\mapdown{} \\ H^0({\cal O}_R)&\mapleft{\rho_2}&H^0(E^{'*})&\mapleft{}&0 \\ \mapdown{\pi_2}&&\mapdown{\sigma_2} \\ H^0({\cal O}_P)&\mapleft{\varphi_0}&H^0({\cal O}_X^{\oplus n})&\mapleft{}&0 \\ \mapdown{\pi_1}&&\mapdown{\sigma_1} \\ H^0({\cal O}_Q)&\mapleft{\rho_1}&H^0({\cal O}_X^{\oplus f})&\mapleft{}&0 \\ \mapdown{}&&\mapdown{} \\ 0&&0 \end{array}$$ where $\varphi_0$ is the induced map of $\varphi$ on the cohomology. (Notice that the map $\varphi_0$ here is different from the map $\varphi^0$ we defined in (3.4).) \par The surjectivity of $\sigma_1$ is due to the nature of the map ${\cal O}_X^{\oplus n}\longrightarrow {\cal O}_X^{\oplus f}$. Hence $h^0(E^{'*})=n-f=f'$. (In fact, we can show that $E^{'*}={\cal O}_X^{\oplus f'}$, but we don't need such a strong statement.) \par $\pi_1$ is a natural coordinate projection. By Condition B, $\pi_1|_{{\rm Im}\varphi_0 }$ is either a surjection if $\ell(Q)\le n$ or an injection if $\ell (Q)\ge n$. \par If $\ell(Q)\le n$, $\pi_1|_{{\rm Im}\varphi_0}$ is a surjection. Hence $\rho_1$ has to be a surjection. But $h^0({\cal O}_X^{\oplus f})=f<fg=h^0({\cal O}_Q)$, a contradiction. \par If $\ell(Q)\ge n$, $\pi_1|_{{\rm Im}\varphi_0}$ is an injection. ${\rm Im}(\pi_1\circ \varphi_0 )$ has rank $n$. But ${\rm Im}(\pi_1\circ \varphi_0)={\rm Im}(\rho_1\circ\sigma_1)$ and ${\rm Im}(\rho_1\circ\sigma_1)$ has rank $f<n$, a contradiction. \par Hence $V$ has to be stable. \hfill\qed \section{Group actions on $Hom({\cal O}_X^{\oplus n}, {\cal O}_P)$} In this section, we study the actions of $Aut ({\cal O}_X^{\oplus n})$ and $Aut ({\cal O}_P)$ on $Hom({\cal O}_X^{\oplus n}, {\cal O}_P)$. \begin{say} Recall that every surjective map $\varphi$ in $Hom({\cal O}_X^{\oplus n}, {\cal O}_P)$ gives an exact sequence $$0\longrightarrow {V^*}\mapright{\phi} {{\cal O}_X^{\oplus n}}\mapright{\varphi} {\cal O}_P\longrightarrow 0.$$ Hence for an elementary transformation $\varphi$, we get a vector bundle $V^*$, or equivalent $V$. The group $Aut({\cal O}_P)$ and $Aut({\cal O}_X^{\oplus n})$ act on $Hom({\cal O}_X^{\oplus n}, {\cal O}_P)$ as follows. Let $\rho\in Aut ({\cal O}_X^{\oplus n})$ and $\sigma\in Aut({\cal O}_P)$. The action of $\rho$ and $\sigma$ on $\varphi$ gives a new elementary transformation $\varphi'=\sigma\circ \varphi\circ \rho^{-1}$. It fits into the following exact sequence: $$0\longrightarrow V^* \buildrel{\rho\circ \phi}\over \longrightarrow {\cal O}_X^{\oplus n}\mapright {\varphi'}{\cal O}_P\longrightarrow 0.$$ This means that $\varphi'$ gives arise to the same bundle $V^*$, or equivalently, $V$. So we have: \end{say} \begin{lem} \label{orbitimpliesisom} Suppose two elementary transformations $\varphi, \varphi'\in Hom({\cal O}_X^{\oplus n}, {\cal O}_P)$ are in the same orbit under the actions of $Aut ({\cal O}_X^{\oplus n})$ and $Aut ({\cal O}_P)$. Then the corresponding vector bundles (which are kernels of them) are isomorphic. \end{lem} It is natural to ask whether the converse is also true. In general, it might well be that two elementary transformations in different group orbits give arise to two isomorphic vector bundles. However we have the following lemma: \begin{lem} \label{isomimpliesorbit} Let $V$ and $V'$ be obtained from two elementary transformations $\varphi$ and $\varphi' $ in $Hom({\cal O}_X^{\oplus n}, {\cal O}_P)$ and $Hom({\cal O}_X^{\oplus n}, {\cal O}_Q)$ respectively both of which satisfy Condition A ($P$ and $Q$ are not assumed to be the same). If $V$ and $V'$ are isomorphic, then $P=Q$ and $\varphi$ and $\varphi'$ are in the same group orbit under the actions of $Aut({\cal O}_X^{\oplus n})$ and $Aut ({\cal O}_P)$. \end{lem} \smallskip\noindent{\it Proof. } Let $f\colon V\mapright{} V'$ be the isomorphism. Consider the following exact sequences $$0\longrightarrow V^*\longrightarrow {\cal O}_X^{\oplus n}\mapright{\varphi} {\cal O}_P\mapright{} 0, $$ $$0\longrightarrow V^{'*}\longrightarrow {\cal O}_X^{\oplus n}\mapright{\varphi'} {\cal O}_Q\mapright{} 0. $$ Take the duals of these two exact sequences, we have $$0\mapright{}{\cal O}_X^{\oplus n}\mapright{\alpha}V\mapright{\beta} {\cal O}_P\mapright{}0,\eqno(4.1)$$ $$0\mapright{}{\cal O}_X^{\oplus n}\mapright{\alpha'}V'\mapright{\beta'} {\cal O}_Q\mapright{}0.\eqno(4.2)$$ Since $\varphi$ and $\varphi'$ satisfy Condition A, from Remark \ref{conditionsab}, we have that $h^0(V)=h^0(V')= n$. Take the cohomology of the two exact sequences (4.1) and (4.2), we get two isomorphisms $$H^0({\cal O}_X^{\oplus n})\mapright{\alpha^0} H^0(V),\qquad H^0({\cal O}_X^{\oplus n})\mapright{\alpha^{\prime 0}} H^0(V')$$ where $\alpha^0$ (or $\alpha^{'0}$) is the induced map of $\alpha$ (or $\alpha^{'}$) on cohomologies. Hence ${\rm Im}\alpha$ (or ${\rm Im}\alpha'$) is the sub-sheaf of $V$ (or $V'$) generated by $H^0(V)$ (or $H^0(V')$). Since $H^0(V)$ is mapped isomorphically to $H^0(V')$ by the map $f^0$ which is the induced map of the given isomorphism $f: V \rightarrow V' $, ${\rm Im}\alpha$ is mapped isomorphically to ${\rm Im}\alpha'$ by the map $f$ from $V$ to $V'$. Therefore, there is an induced morphism $\rho\in Aut({\cal O}_X^{\oplus n})$ making the following diagram commute: $$\begin{array}{cccccccccccc} 0 \mapright{}&{\cal O}_X^{\oplus n}&\mapright{\alpha}&V \\ &\mapdown{\rho}&&\mapdown{f} \\ 0\mapright{}&{\cal O}_X^{\oplus n}&\mapright{\alpha'}&V' \end{array} $$ which in turn induces a morphism $\sigma\colon {\cal O}_P\mapright{}{\cal O}_Q$. That is to say, there exists a commutative diagram: $$\begin{array}{cccccccccccc} 0\mapright{}&{\cal O}_X^{\oplus n}&\mapright{\alpha}&V& \mapright{\beta}&{\cal O}_P&\mapright{}0 \\ &\mapdown{\rho}&&\mapdown{f}&&\mapdown{\sigma} \\ 0\mapright{}&{\cal O}_X^{\oplus n}&\mapright{\alpha'}& V'&\mapright{\beta'}& {\cal O}_Q&\mapright{}0. \end{array} \eqno(4.3)$$ Notice that points in $P$ (or $Q$) are the points where the map $\alpha$ (or $\alpha'$) fails to be an isomorphism. Hence $Q$ must equal $P$ and $\sigma$ must be an automorphism of ${\cal O}_P$. \par Take the dual of (4.3), we get $$\begin{array}{cccccccccccc} 0\mapright{}&V^*&\mapright{}&{\cal O}_X^{\oplus n}& \mapright{\varphi}&{\cal O}_P&\mapright{}0 \\ &\mapup{f^*}&&\mapup{\rho^*}&&\mapup{\sigma^*} \\ 0\mapright{}&V^{'*}&\mapright{}&{\cal O}_X^{\oplus n}& \mapright{\varphi'}& {\cal O}_P&\mapright{}0 \end{array}$$ where $\rho^*\in Aut ({\cal O}_X^{\oplus n})$ and $\sigma^*\in Aut ({\cal O}_P)$. So $\varphi =\sigma^*\circ \varphi'\circ (\rho^*)^{-1}$. \par Hence $\varphi$ and $\varphi'$ are in the same group orbit. \hfill\qed \begin{say} Lemma \ref{orbitimpliesisom} and Lemma \ref{isomimpliesorbit} tell us that for generic elementary transformation $\varphi\in Hom({\cal O}_X^{\oplus n}, {\cal O}_P)$, its group orbit $Aut({\cal O}_P)\cdot \varphi\cdot Aut({\cal O}_X^{\oplus n})$ classifies stable vector bundles up to bundle isomorphisms. This leads us to the quotient space $$ Aut ({\cal O}_P) \backslash Hom({\cal O}_X^{\oplus n}, {\cal O}_P)/ Aut({\cal O}_X^{\oplus n}).$$ \end{say} \begin{say} We have the following identifications: $$Hom({\cal O}_X^{\oplus n},{\cal O}_P)\buildrel \sim\over= H^0({\cal O}_P\otimes {\cal O}_X^{\oplus n}) $$ $$\cong H^0({\cal O}_{p_1}^{\oplus n})\oplus \ldots\oplus H^0({\cal O}_{p_d}^{\oplus n}) \cong ({\Bbb C}^n)^d. \eqno(4.4)$$ Under the isomorphism in (4.4), the group actions become the natural ones: \begin{enumerate} \item $Aut ({\cal O}_X^{\oplus n})=GL(n)$ acts on the space $\bigoplus\limits_{i=1}^d H^0({\cal O}_{p_i}^{\oplus n})=({\Bbb C}^n)^d$ diagonally (on each component $H^0({\cal O}_{p_i}^{\oplus n})={\Bbb C}^n$ the action is the standard one); \item $Aut ({\cal O}_P^{\oplus n})= ({\Bbb C^*})^d$ acts on $Hom({\cal O}_X^{\oplus n}, {\cal O}_P)=({\Bbb C}^n)^d$ by component-wise scalar multiplications. \end{enumerate} In a more down-to-earth way, we may write an element $\varphi\in ({\Bbb C}^n)^d$ as $(\varphi_1, \ldots, \varphi_d)$ where each $\varphi_i\in {\Bbb C}^n$. Then for $g\in GL(n)$, $$g\cdot \varphi =(g\varphi_1,\ldots, g\varphi_d),$$ and for $c=(c_1,\ldots, c_d)\in ({\Bbb C^*})^d$, $$c\cdot \varphi=(c_1\varphi_1,\ldots, c_d\varphi_d).$$ \end{say} \begin{lem} \label{invariant} Condition A (or B) is invariant under the actions of $Aut({\cal O}_P)$ and $Aut({\cal O}_X^{\oplus n})$. That is, if $\varphi$ is an elementary transformation in $Hom({\cal O}_X^{\oplus n}, {\cal O}_P)$ satisfying Condition A (or B), then every element in the orbit $Aut({\cal O}_P)\cdot \varphi\cdot Aut({\cal O}_X^{\oplus n})$ also satisfies Condition A (or B). \end{lem} \smallskip\noindent{\it Proof. } From Remark \ref{conditionsab}, we know that Condition A is equivalent to $h^1(V)=0$. From Lemma \ref{orbitimpliesisom}, we know that $\varphi$ and $\sigma\circ\varphi\circ \rho^{-1}$ induce the same vector bundles $V$ for any $\sigma\in Aut({\cal O}_P)$ and $\rho\in Aut ({\cal O}_X^{\oplus n})$. Thus if $\varphi$ satisfies Condition A, so does $\sigma\circ\varphi\circ \rho^{-1}$. The invariance of Condition B under the actions of $Aut({\cal O}_P)$ and $Aut ({\cal O}_X^{\oplus n})$ is straightforward from the definition of Condition B. \hfill\qed \begin{notationnum} \begin{enumerate} \item Define $N$ to be the space of elementary transformations in $Hom({\cal O}_X^{\oplus n}, {\cal O}_P)$, i.e. $$N\buildrel\sim\over= ({\Bbb C}^n-\{{\bf 0}\})^d$$ where ${\bf 0}$ is the zero vector in ${\Bbb C}^n$. \item Define $\overline{N}$ to be the quotient space of $N$ by the action of $Aut({\cal O}_P)$, i.e. $$\overline{N}= ({\Bbb C}^*)^d\backslash ({\Bbb C}^n-\{{\bf 0}\})^d \buildrel\sim\over= ({\Bbb P}_{n-1})^d.$$ \item Define $\widehat{N}$ to be the (non-separated) quotient space of $N$ by the action of groups $Aut({\cal O}_P)$ and $Aut({\cal O}_X^{\oplus n})$, i.e. $$\widehat{N} \buildrel\sim\over=({\Bbb C}^*)^d\backslash ({\Bbb C}^n-\{{\bf 0}\})^d/GL(n) \buildrel\sim\over=({\Bbb P}_{n-1})^d/PGL(n)$$ where $PGL(n)$ acts on $({\Bbb P}_{n-1})^d$ diagonally. \item Define $N_A$ (or $N_B$) to be the subset of elementary transformations in $N$ satisfying Condition A (or B); define $\overline{N}_A$ (or $\overline{N}_B$) to be the quotient space of $N_A$ (or $N_B$) by the action of the group $Aut({\cal O}_P)$; and define $\widehat{N}_A$ (or $\widehat{N}_B$) to be the quotient space of $N_A$ (or $N_B$) by the action of the groups $Aut({\cal O}_P)$ and $Aut({\cal O}_X^{\oplus n})$. Clearly $\widehat{N}_A$ and $\widehat{N}_B$ are non-empty Zariski open subsets of $\widehat{N}$ and we shall see that $\widehat{N}_B$ is separated, quasi-projective, and rational (see \ref{git} below). \end{enumerate} \end{notationnum} \begin{say} \label{git} Now take an element $\varphi=(\varphi_1,\ldots,\varphi_d)$ in $N$. Consider its image $\overline\varphi=(\overline\varphi_1,\ldots,\overline\varphi_d)$ in $\overline N$. $PGL(n)$ acts on $\overline N=({\Bbb P}_{n-1})^d$. The geometric invariant theory of this standard $PGL(n)$ action on $\overline N=({\Bbb P}_{n-1})^d$ can be found in [{\bf MF}]. If $\varphi$ satisfies Condition B, then any $n$-many $\overline\varphi_{i_1},\ldots,\overline\varphi_{i_n}$ will be linearly independent. In the context of geometric invariant theory, such $\overline\varphi$'s are necessarily stable with respect to all linearizations. It follows from [{\bf MF}] that $\widehat{N}_B$ is separated, quasi-projective, and rational. For the convenience to the reader, we shall give a brief account of these standard results. Consider the diagonal action of $PGL(n+1)$ on $({\Bbb P}_n)^{m+1}$. Let $p_i$ be the projection of $({\Bbb P}_n)^{m+1}$ to the $i$-th factor. Let $L_i=p_i^*{\cal O}_{{\Bbb P}_n}(1)$. Then ${\cal L}=(L_1\otimes \ldots \otimes L_{m+1})^{n+1}$ admits a $PGL(n+1)$-linearization. Assume that $m\ge n+1$. The following is the Definition 3.7 / Proposition 3.4 in [{\bf MF}]. \begin{prop} The set of stable points in $({\Bbb P}_n)^{m+1}$ with respect to $\cal L$ is the open subset of $({\Bbb P}_n)^{m+1}$ whose geometric points $x=(x^{(0)}, \ldots, x^{(m)})$ are those points such that for every proper linear subspaces $L\subset {\Bbb P}_n$, $${\hbox{ number of points $x^{(i)}$ in $L$}\over m+1}<{ {\rm dim} L+1\over n+1.} \eqno(4.5)$$ \end{prop} It is an easy exercise to check the following: \begin{cor} Let $x=(x^{(0)}, \ldots, x^{(m)})$ be a point in $({\Bbb P}_n)^{m+1}$ such that any $(n+1)$-many $x^{(i)}$ are linearly independent in ${\Bbb P}_n$, then $x$ is a stable point with respect to $\cal L$. \end{cor} \begin{defn} \label{defnstable} We say an elementary transformation $\varphi$ in $Hom({\cal O}_X^{\oplus n}, {\cal O}_P)$ is stable if $\overline \varphi\in\overline N$ is stable with respect to the linearization $\cal L$. \end{defn} \end{say} Apply the geometric invariant theory to our situation, we get: \begin{cor} An elementary transformation $\varphi\in Hom({\cal O}_X^{ \oplus n}, {\cal O}_P)$ is stable if $\varphi$ satisfies Condition B. \end{cor} Now we may rewrite Theorem \ref{stable} as follows: \begin{prop} \label{re-stable} Fix a divisor $P$ in $U$. Let $\varphi$ be an elementary transformation in $Hom({\cal O}_X^{\oplus n}, {\cal O}_P)$. If $\varphi$ satisfies Condition A and B, then the vector bundle $V$ obtained from $\varphi$ is a stable vector bundle with ${\rm deg}V=d=ng$ and ${\rm rank} V=n$ such that \begin{enumerate} \item $h^1(V)=0$. \item ${\varphi}$ is stable (in the sense of Definition \ref{defnstable}). \item The set of such stable bundles is isomorphic to the set $\widehat{{ N}}_A\cap\widehat{{ N}}_B$. Furthermore, $\widehat{{ N}}_A\cap\widehat{{ N}}_B$ is quasi-projective, rational, and of dimension $d(n-1)-n^2+1$. \end{enumerate} \end{prop} \begin{say} Now we see that we can construct a stable vector bundle such that it satisfies (1), (2) and (3) of Theorem \ref{constructingstablebundles}. Next, we need to prove that such vector bundles are generic in the moduli space ${\cal M}(n, L)$. For this, we have to let $P$ move in $U$. Then we get a set $\cal B$ of stable vector bundles. $\cal B$ has dimension $${\rm dim} \widehat{{ N}}_A\cap\widehat{{ N}}_B+{\rm dim} U =(n-1)ng-n^2+1+(n-1)g=(n^2-1)(g-1)$$ which is the same as the dimension of the moduli space ${\cal M}(n, L)$. It is natural to ask if this space $\cal B$ is actually a non-empty Zariski open subset of the moduli space. Indeed, we shall show that this is the case in the next section. \end{say} \section{A birational model for ${\cal M}(n, L)$} In this section, using relative extension sheaf, we will construct a family of stable bundles which provides an injection to the moduli space ${\cal M}(n, L)$ and gives a non-empty Zariski open subset of ${\cal M}(n, L)$. In the previous sections, the method we used to construct stable vector bundles is the elementary transformations. Here we shall use extensions instead. The two approaches are related: one is the dual of the other. Below we will elaborate on this. \begin{say} Given an elementary transformation $\varphi\in Hom({\cal O}_X^{\oplus n}, {\cal O}_P)$, i.e., $$0\longrightarrow {V^*}\longrightarrow {{\cal O}_X^{\oplus n}}\mapright{\varphi}{{\cal O}_P}\longrightarrow 0\eqno(5.1)$$ Take the dual of the exact sequence (5.1), we get $$\exact{{\cal O}_X^{\oplus n}}{V}{{\cal O}_P}.\eqno(5.2)$$ Hence $V$ is an extension of ${\cal O}_P$ by ${\cal O}_X^{\oplus n}$. Such extensions are classified by the extension group $Ext^1({\cal O}_P, {\cal O}_X^{\oplus n})$. There is an isomorphism $$Ext^1({\cal O}_P, {\cal O}_X^{\oplus n})\buildrel\sim\over = H^0({\cal E}xt^1({\cal O}_P, {\cal O}_X^{\oplus n})) \buildrel\sim\over=H^0({\cal O}_P\otimes {\cal O}_X^{\oplus n})$$ $$\buildrel\sim\over=H^0( {\cal O}_X^{\oplus n}|_{p_1})\oplus \ldots \oplus H^0({\cal O}_X^{\oplus n}|_{p_d})\buildrel\sim\over = ({\Bbb C}^n)^d. $$ \end{say} The extension (5.2) in general does not give arise to a vector bundle in the middle. However, writing an extension class $e$ as $(e_1, \ldots, e_d) \in ({\Bbb C}^n)^d$, we have the following: \begin{lem} The extension $e$ gives a vector bundle $V$ in (5.2) if and only iff $e_i\ne 0$ for all $i$. \end{lem} \smallskip\noindent{\it Proof. } See Lemma 16 of [{\bf B1}].\hfill\qed \begin{rem} The above lemma is equivalent to Remark \ref{norowzero}. Actually there exists a dictionary between elementary transformations and their corresponding extensions. For example, under this dictionary, the groups $Aut({\cal O}_P)$ and $Aut ({\cal O}_X^{\oplus n})$ act on $Ext^1({\cal O}_P, {\cal O}_X^{\oplus n})$ in the same way as they act on $Hom({\cal O}_X^{\oplus n}, {\cal O}_P)$ after we identify $Ext^1({\cal O}_P, {\cal O}_X^{\oplus n})$ with $Hom({\cal O}_X^{\oplus n}, {\cal O}_P)$. We leave the existence of the dictionary to the reader. \end{rem} \begin{notationnum} \begin{enumerate} \item Define $Z'\subset X\times U$ to be the universal divisor $$Z'=\{(x,D)\in X\times U|x\in D\}.$$ \item Define $\pi_i$ to be the natural projection from $X\times U$ to the $i$-th factor. \item Define ${\cal E}$ to be the relative extension sheaf $${\cal E}={\cal E}xt^1_{\pi_2}({\cal O}_{Z'}, \pi^*_1{\cal O}_X^{\oplus n}).$$ The fiber of $\cal E$ over a point $P\in U$ is isomorphic to $Ext^1({\cal O}_P, {\cal O}_X^{\oplus n})$. \item The general linear group $GL(n)$ acts on ${\cal E}$ via acting on ${\cal O}_X^{\oplus n}$. Meanwhile, the group scheme ${\cal T} = (\pi'_2)_* {\cal O}_{Z'}^*$ also acts on ${\cal E}$ where $\pi'_2$ is the restriction of $\pi_2$ to $Z'$. This group scheme is a twisted torus. The fiber of ${\cal T}$ over a point $P \in U$ is $Aut({\cal O}_P^{\oplus n} ) \cong ({\Bbb C}^*)^d $. The fiber-wise action of ${\cal T}$ on ${\cal E}$ is just the action of $Aut({\cal O}_P)$ on $Ext^1({\cal O}_P, {\cal O}_X^{\oplus n})$. The actions of the group scheme ${\cal T}$ and the (global) group $GL(n)$ commute. Hence we get an action of ${\cal G} = {\cal T} \times GL(n)$ where the diagonal multiplicative group ${\Bbb C}^*$ appears in both ${\cal T}$ and $GL(n)$. \end{enumerate} \end{notationnum} \begin{lem} \label{fiber-wise} The natural morphism: $${\cal E}_{[P]}\longrightarrow Ext^1({\cal O}_P,{\cal O}_X^{\oplus n})\eqno(5.3)$$ is an isomorphism and ${\cal E}$ is a locally free sheaf over $U$. \end{lem} \smallskip\noindent{\it Proof. } Note that dim$Ext^1({\cal O}_P,{\cal O}_X^{\oplus n})=nd$ is a constant. By Satz 3 of [{\bf BPS}], $\cal E$ is a locally free sheaf and the natural morphism (5.3) is an isomorphism. \hfill\qed \begin{prop} Let ${\Bbb V}={\Bbb V}({\cal E}^*)$ be the vector bundle associated to the locally free sheaf $\cal E$ following Grothendieck's notation. Then over ${\Bbb V}$, there is a universal extension $$\exact{q_1^*{\cal O}_X^{\oplus n}\otimes q_2^*{\cal A}} {{\cal V}}{\gamma^*{\cal O}_{Z'}}\eqno(5.4)$$ where $q_i$ is the projection from $X\times {\Bbb V}$ to its $i$-th factor and $\gamma$ is the projection from $X\times {\Bbb V}$ to $X\times U$ and ${\cal A}$ is some line bundle on ${\Bbb V}$. Moreover, $\cal V$ is flat over $\Bbb V$. \end{prop} \smallskip\noindent{\it Proof. } Since $Ext^0({\cal O}_P,{\cal O}_X^{\oplus n})=0$ for all $P\in U$, ${\cal E}xt^0_{\pi_2}({\cal O}_{Z'}, \pi^*_1{\cal O}_X^{\oplus n})=0$. By Lemma \ref{fiber-wise}, ${\cal E}xt^1_{\pi_2}({\cal O}_{Z'}, \pi^*_1{\cal O}_X^{\oplus n})$ commutes with base change (according to Lange's terminology [{\bf L}]). Hence by Corollary 3.4 of [{\bf L}], we get the universal extension (5.4). Since ${\cal O}_{Z'}$ is flat over $U$, both $\gamma^*{\cal O}_{Z'}$ and $q_1^*{\cal O}_X^{\oplus n}\otimes q^*_2{\cal A}$ are flat over ${\Bbb V}$. Hence ${\cal V}$ is flat over $\Bbb V$. \hfill\qed \begin{rem} If we use $e$ to represent a point in ${\Bbb V}$ corresponding to an extension (5.2), then the restriction of (5.4) to $X\times e$ is just the extension (5.2) and ${\cal V}|_{X\times e}\buildrel\sim\over= V$. \end{rem} \begin{thm} \label{birationalmodel} The moduli space ${\cal M}(n, L)$ is birational to the quotient space of a non-empty Zariski open subset of ${\Bbb V}$ by the action of the group ${\cal G}={\cal T} \times GL(n)$. \end{thm} \smallskip\noindent{\it Proof. } Let ${\Bbb V}^0$ be the subset of ${\Bbb V}$ consisting of extensions (5.2) whose corresponding elementary transformations (via the dictionary) satisfy Conditions A and B. By Lemma \ref{invariant}, ${\Bbb V}^0$ is invariant under the action of ${\cal G}$. By Proposition \ref{re-stable} and semi-continuity using the flatness of $\cal V$, ${\Bbb V}^0$ is a non-empty Zariski open subset of ${\Bbb V}$. Restrict ${\cal V}$ to $X\times {\Bbb V}^0$, we get a family of stable bundles over $X\times {\Bbb V}^0$. Since the moduli space ${\cal M}(n, L)$ is coarse, the family induces a morphism: $$\Phi\colon {\Bbb V}^0\longrightarrow {\cal M}(n,L).\eqno(5.5)$$ Now by Lemma \ref{orbitimpliesisom} and Lemma \ref{isomimpliesorbit}, we see that points of ${\Bbb V}^0$ are in the same ${\cal G}$ orbit if and only if they correspond to isomorphic stable bundles. Thus by passing to the quotient, we get a natural induced map $$\overline\Phi \colon {\Bbb V}^0/({\cal T} \times GL(n)) \longrightarrow {\cal M}(n, L)$$ which is an injective morphism. By calculating dimensions, we get $${\rm dim}({\Bbb V}^0/{\cal G})={\rm dim}{\Bbb V}^0-{\rm dim} ({\cal T} \times GL(n)) + 1 ={\rm dim}{\cal M}(n, L).$$ (Notice that a ``$+1$'' modification appears in the first equality because the multiplicative group ${\Bbb C}^*$ appears in both ${\cal T}$ and $GL(n)$.) Hence the morphism $\overline\Phi$ is birational.\hfill\qed \begin{rem} It can be seen that the non-empty Zariski open subset ${\Bbb V}_0/{\cal G}$ of ${\cal M}(n, L)$ is a fibration over the rational variety $U$ whose typical fiber is also rational since it is birational to the geometric quotients of $({\Bbb P}_{n-1})^d$ by $PGL(n)$ (which are known to be rational) (see also [{\bf T1, T2}]). Unfortunately, this fibration seems not to be locally trivial in Zariski topology. The twisted torus ${\cal T}$ is responsible for the problem. \end{rem} \begin{say} {\sl proof of Theorem \ref{constructingstablebundles}:} Theorem \ref{constructingstablebundles} clearly follows from the combination of Proposition \ref{re-stable} and Theorem \ref{birationalmodel}. \end{say} \section{Some remarks on rationality} We shall provide a thorough and {\sl systematic} account of the inductive method in [{\bf N1}] and prove several theorems that are stronger than Theorem \ref{newsteadrationalitythm}. The essential ideas are, however, contained in [{\bf N1}]. Throughout, $(n; d)$ stands for a pair of integers; while $\gcd (n, d)$ stands for the greatest common factor of the two integers $n$ and $d$. \begin{say} Let $V$ be a stable vector bundle in ${\cal M}(n, L)$ and $(n; d)$ satisfies $$ n(g-1)<d<ng. \eqno(6.1)$$ By Riemann-Roch, $\chi(V)={\rm deg}V+n(1-g)=d-n(g-1).$ Set $r=\chi(V)$. Then $0<r<n$ by (6.1). In ([{\bf N1}]) and ([{\bf G}]), it was shown that for generic bundles $V$ in ${\cal M}(n, L)$, $V$ satisfies $h^1(V)=0$ and there exists an exact sequence $$\exact{{\cal O}_X^{\oplus r}}{V}{V'} \eqno(6.2)$$ such that $V'$ is stable. That is $V' \in {\cal M}(n', L)$ where $n'=n-r$. One can always find a non-negative integer $k$ such that $$ n'(g-1)< d'=d-kn' \le n'g. \eqno (6.3)$$ Then ${\cal M}(n', L) \cong {\cal M}(n', L')$ where $L'=L \otimes (M^*)^{\otimes n'}$ for some line bundle $M$ of degree $k$ and ${\rm deg}L' = d'$. This suggests that there exist a rational map from ${\cal M}(n, L)$ to ${\cal M}(n', L')$. \end{say} \begin{defn} Let $(n;d)$ be a pair of positive integers satisfying (6.1). Let $n'=ng-d$, $d'=d-kn'$ for some non-negative integer $k$ so that $n'(g-1)<d'\le n'g$. We call $(n';d')$ the reduction of $(n;d)$, we denote this process of reduction by $$(n;d)\longrightarrow (n';d').$$ \end{defn} \begin{say} Sometimes, it is necessary to apply the same procedure to the dual bundle $V^*$. In other words, instead of constructing $V$ by an extension (6.2), we may do it for $V^*$. Precisely, let again $(n;d)$ satisfy (6.1). Choose a line bundle $M$ with $\hbox{deg}M=2g-1$. Then $$n(g-1)<\hbox{deg}(V^*\otimes M)=-d+n(2g-1)<ng.$$ Hence $V^*\otimes M$ is a stable vector bundle in the moduli space ${\cal M}(n, L^*\otimes M^{\otimes n})$. The pair $(n; {\rm deg}(L^*\otimes M^{\otimes n}))$ satisfies (6.1). We then apply the reduction to $(n; n(2g-1)-d)$, and obtain $(n';d')$ where $$n'=d-n(g-1), \; d'=n(2g-1)-d-kn',\; \hbox{and} \;n'(g-1) < d' \le n'g $$ for some non-negative integer $k$. \end{say} \begin{defn} Let $(n;d)$ be a pair of integers satisfying the hypothesis (6.1). Let $n'=d-n(g-1)$ and $d'=n(2g-1)-d-kn'$ for some non-negative integer $k$ such that $n'(g-1)< d' \le n'g$. We call $(n'; d')$ the dual reduction of $(n; d)$. We denote this process also by $$(n;d)\longrightarrow (n';d'). $$ \end{defn} \begin{prop} Let $(n;d)$ be a pair satisfying (6.1). Let $(n'; d')$ be the reduction (or dual reduction) of $(n;d)$. Let ${\cal M}(n, L)$ and ${\cal M}(n', L')$ be the moduli spaces such that ${\rm deg} L=d$, ${\rm deg} L'=d'$ and $L=L'\otimes M^{\otimes n'}$ for some line bundle $M$ of non-negative degree $k$ with $h^0(M)\ne 0$. Then there exists a non-empty Zariski open subset ${\cal M}^0(n, L)\subset {\cal M}(n, L)$ and a morphism $\Phi: {\cal M}^0(n, L)\mapright{} {\cal M}(n', L')$ such that the image of $\Phi$ is a non-empty Zariski open subset of ${\cal M}(n', L')$. \end{prop} \smallskip\noindent{\it Proof. } We only prove the proposition when $(n;d) \rightarrow (n'; d')$ is a reduction. The same arguments work for a dual reduction equally well. Take ${\cal M}^0(n, L)$ to be the non-empty Zariski open subset of ${\cal M}(n, L)$ as in Theorem \ref{gthm}. Recall that a stable vector bundle $V$ in ${\cal M}^0(n, L)$ has $h^1(V)=0$ and sits in the exact sequence (6.2) where $r=n-n'$ and $V'\in {\cal M}(n', L)$. Let $F'=V'\otimes(M^*)^{\otimes n'}$. Then $F'$ is a stable bundle in $ {\cal M}(n', L')$. Define the map $$\Phi\colon {\cal M}^0(n, L) \mapright{} {\cal M}(n', L')$$ by setting $$\Phi(V)=F'$$ First of all, the map is well-defined. This amounts to saying that if $V_1$ and $V_2$ are two isomorphic stable bundles in ${\cal M}^0(n, L)$, i.e. there exists two exact sequences $$0\mapright{}{\cal O}_X^{\oplus r}\mapright{\alpha_1}V_1\mapright{\beta_1} V_1'\mapright{}0,$$ $$0\mapright{}{\cal O}_X^{\oplus r}\mapright{\alpha_2}V_2\mapright{\beta_2} V_2'\mapright{}0$$ where $h^0(V_1)=h^0(V_2)= r$ and $V'_1$ and $V_2'$ are stable, then $V_1'$ and $V_2'$ are isomorphic. In fact, since $h^0(V_1)=r$, ${\cal O}_X^{ \oplus r}$ is the sub-sheaf of $V_1$ generated by global sections of $H^0(V_1)$. Same is true for $V_2$. Hence if $f$ is an isomorphism from $V_1$ to $V_2$, $f$, restricted to ${\cal O}_X^{\oplus r}$, maps ${\cal O}_X^{\oplus r} \subset V_1$ to ${\cal O}_X^{\oplus r}\subset V_2$ isomorphically, hence $V'_1$ is isomorphic to $V'_2$. Next we need to show that $\Phi$ is a morphism. Consider the construction of ${\cal M}(n, L)$ via geometric invariant theory, we let $\cal Q$ be the Quot scheme, $\cal U$ be the universal quotient sheaf over $X\times {\cal Q}$ and ${\cal G}$ be the group such that $ {\cal M}(n, L)$ is the GIT quotient of $\cal Q$ by $\cal G$. Let $\Pi$ be the quotient map from ${\cal Q}$ to ${\cal M}(n, L)$. Let ${\cal Q}_0\subset \cal Q$ be the inverse image of $ {\cal M}^0(n, L)$ under the map $\Pi$. Since bundles in ${\cal M}^0(n, L)$ are stable bundles, the pre-image of $V\in {\cal M}^0(n, L)$ under the map $\Pi$ consists of bundles isomorphic to $V$. Clearly we can define a map $$\widetilde \Phi\colon {\cal Q}_0\mapright{} {\cal M}(n', L')$$ in the same way as we defined $\Phi$. Because ${\cal M}(n', L')$ is a coarse moduli space and since $\cal U$ is a universal quotient sheaf over $X\times {\cal Q}_0$, we conclude that $\widetilde\Phi$ is a morphism. Since ${\cal M}^0(n, L)$ is the geometric quotient of ${\cal Q}_0$ by ${\cal G}$ and any orbit (e.g., the fiber $\Pi^{-1}(V)$) is mapped to a single point $F'$ in ${\cal M}(n', L')$ under the map $\widetilde \Phi$, by the universality of GIT quotients, the morphism $\widetilde \Phi: {\cal Q}_0\mapright{} {\cal M}(n', L')$ induces a morphism on the quotient ${\cal M}^0(n, L) \mapright{} {\cal M}(n', L')$. Evidently, the induced morphism is just the map $\Phi$ defined earlier. Finally, we need to show that the image of $\Phi$ contains a non-empty Zariski open subset of ${\cal M}(n', L')$. Recall that the combination of Lemma 5 and Lemma 6 in [{\bf N1}] says that if $V'$ is stable, $h^1(V')=0$, ${\rm deg} V'=d$, $n(g-1)<d<ng$ and $r=d-n(g-1)$, then there exists a stable vector bundle $V$ sitting in the exact sequence (6.2) with $h^1(V)=0$. {}From a result in [{\bf G}], we know that there exists a non-empty Zariski open subset $ {\cal M}_0(n', L')$ of ${\cal M}(n', L')$ consisting of $F'$ with $h^1(F')=0$. Hence $h^0(F^{'*}\otimes K_X)=0$ by Serre duality. Since $h^0(M)\ne 0$, we must have $h^0(F^{'*}\otimes (M^*)^{\otimes n'} \otimes K_X)=0$. Recall that $V'=F'\otimes M^{\otimes n'}$. Hence $h^1(V')=h^1(F'\otimes M^{\otimes n'})=0$ by Serre duality. Now Lemma 5 and Lemma 6 in [{\bf N1}] imply that there exists a stable vector bundle $V$ in ${\cal M}^0(n, L)$ such that $\Phi(V)=F'$. So ${\rm Im}\Phi$ contains ${\cal M}_0(n', L')$. Therefore it contains a non-empty Zariski open subset of ${\cal M}(n', L')$. The case when $n'=1$, $d'=g$ needs some special remarks. According to our convention, $V'=L$ in this case. Since $ n\ge 2$, we have $${\rm deg} L =d\ge n(g-1)+1\ge 2g-2+1=2g-1.$$ Therefore $h^1(L)=h^0(L^*\otimes K_X)=0$ because ${\rm deg} (L^*\otimes K_X)\le -2g+1+2g-2= -1$. Hence Lemma 5 and Lemma 6 in [{\bf N1}] also apply. \hfill\qed \begin{say} \label{genericfibration} Now we need to analyze the structure of the rational map $$\Phi\colon {\cal M}(n, L) ---> {\cal M}(n', L').$$ Extensions (6.2) are classified by the extension group: $$Ext^1(V', {\cal O}_X^{\oplus r})=H^1(V^{'*})^{\oplus r}.$$ The group $Aut ({\cal O}_X^{\oplus r})=GL(r)$ acts on the extension group. Lemma 1 of [{\bf N1}] showed that the $GL(r)$-orbits correspond to the equivalent classes of stable vector bundles. Let $F'\in {\cal M}_0(n', L')$. Then $\Phi^{-1}(F')$ is a non-empty Zariski open subset of $Ext^1(V', {\cal O}_X^{\oplus r})/GL(r)$ which is known to be rational (see Lemma 2, [{\bf N1}]). The rational map $\Phi$ can be regarded as a fibration (over a non-empty Zariski open subset of ${\cal M}(n', L')$). \end{say} \begin{say} \label{localtriviality} Next, we need to know when the fibration $$\Phi\colon {\cal M}(n, L) ---> {\cal M}(n', L')$$ is locally trivial in Zariski topology over the range of $\Phi$. This admits the affirmative answer if the moduli space ${\cal M}(n', L')$ is fine, i.e. if ${\cal M}(n', L')$ admits a universal bundle (if and only if $\gcd(n', d')=1$). This is because the existence of the universal bundle will allow us to get the locally triviality by using the relative extension sheaf (cf. Lemma 3, [{\bf N1}]). If $\gcd(n', d') \ne 1$, we still have the map $\Phi$, but it seems very likely that the fibration is no longer locally trivial in Zariski topology. \end{say} The following is implicit in [{\bf N2}], which was used to obtain Theorem \ref{newsteadrationalitythm}. \begin{prop} \label{keyinduction} Let $(n;d) \rightarrow (n';d')$ be a reduction or a dual reduction. \begin{enumerate} \item If $(n';d')$ is coprime, then the fibration ${\cal M}(n, L) ---> {\cal M}(n', L')$ is locally trivial in Zariski topology. Furthermore, ${\cal M}(n, L)$ is birational to ${\Bbb C}^m\times {\cal M}(n,' L')$ where $m=(n^2-n^{'2})(g-1)$. \item If $(n';d')$ is coprime and ${\cal M}(n', L')$ is rational, then ${\cal M}(n, L)$ is rational. \end{enumerate} \end{prop} \begin{say} \label{reductionsequence} An interesting question is whether the above induction method works only for the pairs stated in Theorem \ref{newsteadrationalitythm}. A closer examination of reductions and dual reductions tells us that the method can actually be extended to work for a much larger class of moduli spaces, which we now describe. Given a pair $(n;d)$ satisfying (6.1), if $n>1$, we can apply a reduction or a dual reduction to $(n;d)$ to get another pair $(n';d')$. Three possibilities can occur: \begin{enumerate} \item $n'=1$. \item $n' \ge 2$ and $d'=n'g$. \item $n'\ge 2$ and $n'(g-1)<d-kn'< n'g$ for some integer $k\ge 0$. \end{enumerate} In the case that (3) occurs, we can continue reductions and dual reductions. Keep doing this, the process will eventually terminate and we get a sequence of reductions and dual reductions: $$(n;d)\longrightarrow (n_1; d_1)\longrightarrow \ldots \longrightarrow (n_t; d_t)\eqno(6.4)$$ such that \begin{enumerate} \item either $n_t=1, d_t=g$, \item or $d_t=n_tg$, $n_t\ge 2$. \end{enumerate} \end{say} \begin{defn} \label{newsteadpair} Let $(n; d)$ be a pair of positive integers satisfying (6.1). The pair is called a nice pair if either $n=1$ or after successive reductions and dual reductions (6.4), we get $n_t=1$. \end{defn} \begin{rem} It can happen that after a sequence of reductions and dual reductions, $(n;d)$ is reduced to $(n_t;d_t)$ with $n_t \ge 2$ and $d_t=n_tg$ while after another sequence of reductions and dual reductions, $(n;d)$ is reduced to $(1; g)$. According to our definition, this kind of pair is nice. Take the pair $(7; 8)$ when $g=2$ as an example. After two reductions we get $$(7; 8) \to (6;8) \to (4;8).$$ However, after a dual reduction, we get $$(7; 8) \to (1;2).$$ The latter will imply that ${\cal M} (7, L)$ with $\deg L = 8$ is rational (cf. Theorem \ref{newsteadpairthm} below), while the former won't. \end{rem} Below is a simple but useful observation. \begin{lem} \label{newsteadpairlem} Let $(n; d)$ be a pair satisfying (6.1). Let $(n'; d')$ be the reduction or dual reduction of $(n; d)$. If $\gcd (n', d')=1$, then $\gcd (n, d)=1$. \end{lem} \smallskip\noindent{\it Proof. } Assume $(n;d)\longrightarrow (n';d')$ is a reduction: $$(n'; d') = (ng - d; d- k(ng -d))\eqno(6.5)$$ for some non-negative integer $k$. \par If an integer $m$ divides $n$ and $d$, from (6.5), it is clear that $m$ divides $n'=ng-d$ and $d'=d-k(ng-d)$. Hence the lemma is proved in this case. \par Assume $(n;d)\longrightarrow (n';d')$ is a dual reduction. $$(n'; d')=(ng-n(2g-1)+d; n(2g-1)-d-k(ng-n(2g-1)+d))\eqno(6.6)$$ for some non-negative integer $k$. \par If an integer $m$ divides $n$ and $d$, from (6.6), it is clear that $m$ divides $n'$ and $d'$ as well. Hence the lemma is also proved in this case. \hfill\qed \begin{rem} \label{goodpairrem} \begin{enumerate} \item On any forwarded (dual) reduction path, $\gcd$ is a non-decreasing function. \item All nice pairs are coprime. \item The converse of Lemma \ref{newsteadpairlem} is not true which is exactly and the {\it only} place where the proposition of \S 3 in [{\bf N1}] fails. \end{enumerate} \end{rem} \begin{prop} \label{rationalityinductioncor} Let $(n;d)\mapright{}(n_1;d_1)\mapright{}\cdots\mapright{}(n_s; d_s)$ be a sequence of reductions and dual reductions. Suppose ${\rm gcd}(n_s, d_s)=1$, then ${\cal M}(n, L)$ is birational to ${\Bbb C}^x \times {\cal M}(n_s, L_s)$ where ${\deg}L_s = d_s$ and $x=(n^2-n^2_s)(g-1)$. \end{prop} \smallskip\noindent{\it Proof. } By assumption, we have a sequence of dominant rational maps: $${\cal M}(n, L) ---> {\cal M}(n_1, L_1) ---> \ldots ---> {\cal M}(n_s, L_s). $$ Since $(n_s, d_s)=1$, all the above moduli spaces are fine moduli spaces by Lemma \ref{newsteadpairlem}. Hence each of the above rational map is generically a locally trivial fibration in Zariski topology whose typical fiber is rational (cf. \ref{localtriviality} and Proposition \ref{keyinduction}). The proposition then follows immediately. \hfill\qed When $(n_s; d_s)=(1; g)$, we obtain: \begin{thm} \label{newsteadpairthm} If $(n; d)$ is a nice pair, then the moduli space ${\cal M}(n, L)$ with $d={\rm deg} L$ is rational. \end{thm} One checks that all pairs in Theorem \ref{newsteadrationalitythm} are nice pairs. The following gives an example of a nice pair not contained in Theorem \ref{newsteadrationalitythm}. \begin{exmp} Choose $g=6$, $n=15$, $d=77$. One checks that $(n;d)=(15;77)$ satisfies the inequality (3.1) but satisfies none of the conditions (1), (2), (3) in Theorem \ref{newsteadrationalitythm}. Apply reductions twice, we obtain $$(15;77) \rightarrow (13:77) \rightarrow (1;6).$$ Hence the moduli space ${\cal M}(15, L)$ with ${\rm deg}L=77$ is rational. \end{exmp} \begin{say} In view of Theorem \ref{newsteadpairthm}, it is desirable to find ways of constructing all nice pairs. The following two lemmas help to characterize arithmetically the iterated construction of all such pairs starting from $(1;g)$. \end{say} \begin{lem} \label{iterationlem1} Fix a genus $g \ge 2$. \begin{enumerate} \item $(n;d) \rightarrow (1; g)$ is a (one-step) reduction if and only if $d=ng-1$. Consequently, $\gcd(d, g)=1$. \item $(n;d) \rightarrow (1; g)$ is a (one-step) dual reduction if and only if $d=ng-n+1$. Consequently, $\gcd(d+n, g)=1$. \end{enumerate} \end{lem} \smallskip\noindent{\it Proof. } (1) If $d=ng-1$ for $n\ge 2$, it is easy to see that after one reduction we get $(1;g)$. Now suppose $(n;d)\rightarrow (1;g)$ is a reduction. From (6.5), we get $n=\displaystyle{g+(k+1)\over g}$ and $d=g+k$ for some integer $k\ge 0$. Hence $k+1$ must be $mg$ for some integer $m\ge 1$, $n=m+1\ge 2$ and $d=g+mg-1=ng-1$. It is clear that $\gcd(d, g)=1$. Similar arguments prove (2). \hfill\qed \begin{rem} Lemma \ref{iterationlem1} is equivalent to Theorem \ref{newsteadrationalitythm} (1). \end{rem} Next, we have \begin{lem} \label{iterationlem2} Fix a genus $g \ge 2$. Let $(n'; d')$ be a nice pair with either $\gcd(d', g)=1$ or $\gcd(d'+n', g)=1$. Then, \begin{enumerate} \item there exists a pair $(n;d)$ having $(n'; d')$ as a reduction if and only if $\gcd(n', g)=1$. In particular, $\gcd(d, g)=1$. \item there exists a pair $(n; d)$ having $(n'; d')$ as a dual reduction if and only if $\gcd(n', g)=1$. In particular $\gcd(d+n, g)=1$. \end{enumerate} \end{lem} \smallskip\noindent{\it Proof. } Consider the equations in (6.5): $$n'=ng-d,\qquad d'=d-kn'.$$ Add these equations together, we get an equation $$ng=d'+(k+1)n'.$$ Since $\gcd(d', g)=1$ (the case when $\gcd(d'+n', g)=1$ can be proved similarly), this equation has integral solutions for $n$ and $k$ iff $\gcd(n', g)=1$. And if such solutions exist, $d=ng-n'$, so $\gcd(d, g)=1$. The similar arguments prove (2). \hfill\qed \begin{cor} \label{iterationcor} If $(n; d)$ is a nice pair, then either $\gcd(d, g)=1$ or $\gcd (d+n, g)=1$. \end{cor} \begin{rem} In summary, we have the following. Start with any $(n;d)$. After successive reductions and dual reductions, we get $$(n;d) \rightarrow (n_1; d_1) \rightarrow \ldots \rightarrow (n_t;d_t).$$ Correspondingly, we have a sequence of dominant rational maps: $${\cal M}(n, L) ---> {\cal M}(n_1, L_1) ---> \ldots ---> {\cal M}(n_t, L_t) \eqno(6.7)$$ where ${\rm deg} L_i=d_i$ fro all $i$. The terminator is either $(1; g)$ or has that $n_t | d_t$ and $n_t\ge 2$. The first case is covered in Theorem \ref{newsteadpairthm}. For the latter case, the sequence (6.7) ends at the moduli space studied in the previous sections. In the sequence (6.7), it is possible that some fibrations are locally trivial in Zariski topology but the rest are not. The local triviality is known if the base moduli space of a fibration is fine. Otherwise, we believe that it is not. It seems that the rationality problem boils down to how well we know about the moduli spaces ${\cal M}(n_t, L_t)$ when $n_t | d_t$. \end{rem} \begin{say} The above procedure of proving rationality (for coprime pairs only) can be slightly generalized to include an even larger class of pairs (essentially due to an observation by Ballico in [{\bf Ba}]). The observation rests on the following easy lemma: \end{say} \begin{lem} {\rm (Transferring Lemma).} \label{transferringlemma} Suppose that we have the following projections: $$\begin{array}{clc} X && Y \\ \searrow && \swarrow \\ &Z& \\ \end{array}$$ such that \begin{enumerate} \item $X = {\Bbb C}^x \times Z$; \item $Y = {\Bbb C}^y \times Z$; \item $x \ge y$; \item $Y$ is rational. \end{enumerate} Then $X$ is rational. \end{lem} \smallskip\noindent{\it Proof. } $X = {\Bbb C}^x \times Z \cong {\Bbb C}^{x - y} \times ({\Bbb C}^y \times Z) = {\Bbb C}^{x - y} \times Y$. \hfill\qed \begin{say} Consider a diagram $$\begin{array}{ccccccccccccc} (n;d) &&(n_1;d_1) &&& \ldots && (n_{t-1}; d_{t-1}) &&(n_t; d_t) \\ \searrow && \swarrow \searrow &&& \ldots && \swarrow \searrow && \swarrow \\ &(\ell_1; k_1)&& (\ell_2; k_2) && \ldots && & (\ell_t; k_t) \end{array} \eqno (6.8) $$ where each pair in the diagram is a pair of positive integers satisfying (6.1) and each downward arrow represents successive reductions and dual reductions. For example, $(\ell_1; k_1)$ is obtained from $(n; d)$ and $(n_1; d_1)$ by some sequences of reductions and dual reductions, respectively. It is allowed that $(n_t; d_t)=(\ell_t; k_t)$. Correspondingly, we get a diagram of dominant rational maps: $$\begin{array}{ccccccccccccc} {\cal M}(n,L)&&{\cal M}(n_1, L_1)&& \ldots && {\cal M}(n_t, L_t) \\ \searrow && \swarrow \searrow && \ldots && \swarrow \\ &{\cal M}(\ell_1, K_1)&& {\cal M} ( (\ell_2; K_2) & \ldots & {\cal M}(\ell_t, K_t) \end{array} $$ where $L_i$ is a line bundle with deg$L_i=d_i$ and $K_i$ is a line bundle with deg$K_i=k_i$. \end{say} \begin{defn} \label{admissible} The diagram (6.8) is called admissible if $n \ge n_i$ and $\gcd (l_i, k_i) = 1$ for all $i$. \end{defn} \begin{thm} \label{ballicothm} Suppose the diagram (6.8) is admissible. Then ${\cal M}(n, L)$ is birational to ${\Bbb C}^{z_t}\times {\cal M}(n_t,L_t)$ where $z_t=(n^2-n^2_t)(g-1)\ge 0$ is the difference of the dimension of two moduli spaces. \end{thm} \smallskip\noindent{\it Proof. } We shall use Proposition \ref{rationalityinductioncor} and Lemma \ref{transferringlemma} repeatedly. Let $x= \dim {\cal M}(n, L)$, $x_i = \dim {\cal M}(n_i, L_i)$, and $y_i = \dim {\cal M}(l_i, K_i)$. For simplicity, we use the notation $X \cong^{bir} Y$ to indicate that $X$ and $Y$ are birational equivalence. Then we have $${\cal M}(n, L) \cong^{bir} {\Bbb C}^{x - y_1} \times {\cal M} (l_1, K_1) = {\Bbb C}^{x - x_1} \times {\Bbb C}^{x_1 - y_1} \times {\cal M} (l_1, K_1) $$ $$ \cong^{bir} {\Bbb C}^{x - x_1} \times {\cal M} (n_1, L_1) \cong^{bir} {\Bbb C}^{x - x_1} \times{\Bbb C}^{x_1 - y_2} \times {\cal M} (l_2, K_2) $$ $$ = {\Bbb C}^{x - y_2} \times {\cal M} (l_2, K_2) = {\Bbb C}^{x - x_2} \times{\Bbb C}^{x_2 - y_2} \times {\cal M} (l_2, K_2) $$ $$\cong^{bir} {\Bbb C}^{x - x_2} \times {\cal M} (n_2, L_2) \cong^{bir} \ldots \cong^{bir} {\Bbb C}^{x - x_t} \times {\cal M} (n_t, L_t).$$ The theorem then follows. \hfill\qed \begin{defn} \label{ballicopair} A pair $(n; d)$ is called a fine pair if there exists an admissible diagram (6.8) such that $(n_t; d_t)$ is a nice pair. Fine pairs are also necessarily coprime. \end{defn} \begin{thm} \label{ballicopairthem} If $(n; d)$ is a fine pair, then ${\cal M}(n, L)$ is rational. \end{thm} \smallskip\noindent{\it Proof. } It follows from Theorem \ref{ballicothm} and Theorem \ref{newsteadpairthm}. \hfill\qed \begin{exmp} Assume $g=6$. Then the pair $(7; 38)$ is not a nice pair by Corollary \ref{iterationcor} ($\gcd(38, 6)=2$ and $\gcd(38+7, 6)=3$). Notice that $\gcd(7, 38)=1$. One can check the following statements: \begin{enumerate} \item $(7; 38)$ is the (one-step) dual reduction of $(11+7m; 62+35m)$ for all integer $m\ge 0$. \item $m=7$ is the smallest number such that the pair $(11+7m; 62+35m)$ is a nice pair. (When $m=7$, the pair is $(60; 307)$ and we actually have $$(60; 307)\longrightarrow (53; 307)\longrightarrow (11; 65) \longrightarrow (1; 6)$$ where all $``\longrightarrow "$ are (one-step) reductions.) \item When $m$ is even, $(11+7m; 62+35m)$ is not a nice pair. (The dual reduction of the pair gives $(7;38)$ which is not a nice pair; a reduction gives $(4+7m; 62+35m)$ which is not a nice pair either since $2$ divides $\gcd(4+7m, 62+35m)$ when $m$ is even.) \end{enumerate} However $(11+7m; 62+35m)$ are fine pairs for all $m\ge 7$: $$\begin{array}{clc} (11+7m; 62+35m) && (60; 307) \\ \searrow && \swarrow \\ &(7; 38)& \\ \end{array}$$ Hence the corresponding moduli spaces ${\cal M}(11+7m, L)$ where ${\rm deg}L=62+35m$ are rational for all $m\ge 7$ by Theorem \ref{ballicopairthem}. The above provides infinitely many fine pairs that are not nice pairs. \end{exmp} \begin{rem} Finally, some remarks are in order. \begin{enumerate} \item There are coprime pairs that are not fine pairs. \item In [{\bf N2}], Newstead defined a good pair to be a coprime pair $(n; d)$ whose corresponding moduli space is rational. Nice pairs and fine pairs are good. The converse may not be true. To find all good pairs (that are not nice or fine) seems requiring methods different than the one explored in this paper. \item There is an algorithm to locate nice and fine pairs on the lattice cone in the $xy$-plane defined by inequalities: $$ x(g-1) < y \le xg$$ for any fixed $g \ge 2$. A reduction takes the form $$(x, y) \rightarrow (gx -y, y - k (gx-y))$$ for some non-negative integer $k$. A dual reduction takes the form $$(x, y) \rightarrow (y - (g-1) x, (2g-1)x -y - k (y-(g-1)x))$$ for some non-negative integer $k$. \item By using the techniques of variations of moduli spaces of parabolic bundles, H. Boden and K. Yokogawa were able to prove the rationality of certain moduli spaces. (see [{\bf B, BH}]. \end{enumerate} \end{rem} \noindent {\bf Reference} \vskip 1cm \noindent {[{\bf BPS}]} C. Banica, M. Putinar, G. Schumacher, Variation der globalen Ext in Deformationen kompakter komplexer R\"aume {\it Math. Ann.} {\bf 250} (1980), 135-155 \noindent {[{\bf Ba}]} E. Ballico, Stable rationality for the variety of vector bundles over an algebraic curve, {\it J. London Math. Soc.} {\it 2nd Series} {\bf 30} (1984), 21-26. \noindent [{\bf B}] H. Boden, Rationality of the moduli space of vector bundles over a smooth curve, preprint IHES/M/95/73. \noindent [{\bf BY}] H. Boden, K. Yokogawa, Rationality of moduli spaces of parabolic bundles, alg-geom/9610013 \noindent {[{\bf B1}]} J.E. Brosius, Rank-2 vector bundles on ruled surfaces, I, {\it Math. Ann.} {\bf 265} (1983), 155-168. \noindent {[{\bf B2}]} J.E. Brosius, Rank-2 vector bundles on ruled surfaces, II, {\it Math. Ann.} {\bf 266} (1983), 199-214. \noindent {[{\bf GH}]} P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley, New York (1978). \noindent {[{\bf G}]} I. Grzegorczyk, On Newstead's conjecture on vector bundles on algebraic curves, {\it Math. Ann.} {\bf 300} (1994), 521-541. \noindent {[{\bf L}]} H. Lange, Universal families of extensions, {\it Journal of Algebra} {\bf 83} (1983), 101-112. \noindent {[{\bf M}]} M. Maruyama, Elementary transformationations in the theory of algebraic vector bundles, {\it LNM} {\bf 961} (1982), 241-266. \noindent {[{\bf MF}]} D. Mumford and J. Fogarty, Geometric Invariant Theory, {\it A Series of Modern Surveys in Mathematics}, Springer-Verlag (1982). \noindent {[{\bf NR}]} M. S. Narasimhan and S. Ramanan: Moduli of vector bundles on a compact Riemann surface, {\it Ann. of Math.} {\bf 82} (1965), 540-567. \noindent {[{\bf N1}]} P. Newstead, Rationality of moduli spaces of stable bundles, {\it Math. Ann. } {\bf 215} (1975), 251-268. \noindent {[{\bf N2}]} P. Newstead, Rationality of moduli spaces of stable bundles, {\it Correction,} {\it Math. Ann. } {\bf 249} (1980), 281-282. \noindent {[{\bf R}]} S. Ramanan: The moduli spaces of vector bundles over an algebraic curve, {\it Math. Ann. } {\bf 200} (1973), 69-84. \noindent {[{\bf T1}]} A. Tjurin, Classification of vector bundles over an algebraic curve of arbitrary genus, {\it Amer. Math. Soc. Transl.} {\bf 63} (1967), 245-279. \noindent {[{\bf T2}]} A. Tjurin, Classification of $n$-dimensional vector bundles over an algebraic curve of arbitrary genus, {\it Amer. Math. Soc. Transl.} {\bf 73} (1968), 196-211. \noindent {[{\bf T3}]} A. Tjurin, On the classification of two-dimensional vector bundles over an algebraic curve of arbitrary genus, {\it Izv. AKa. Nauk SSSR Ser. Mat.} {\bf 28} (1964), 21-52. \vskip .5cm \noindent \noindent Y.H. Department of Mathematics, University of California, Berkeley, CA 94720, USA. hu@@math.berkeley.edu \vskip .5cm \noindent W.L. Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, HK. mawpli@@uxmail.ust.hk \end{document}

9704

alg-geom/9704023

en

https://arxiv.org/abs/alg-geom/9704023

alg-geom/9704023

Ugo Bruzzo

Claudio Bartocci, Ugo Bruzzo, Daniel Hernandez Ruiperez and Jose' M. Mu~noz Porras

Mirror symmetry on K3 surfaces via Fourier-Mukai transform

15 pages, AMS-LaTeX v1.2. Final version to appear in Commun. Math. Phys

Commun. Math. Phys. 195 (1998) 79-93.

10.1007/s002200050380

SISSA Ref. 61/97/fm/geo

We use a relative Fourier-Mukai transform on elliptic K3 surfaces $X$ to describe mirror symmetry. The action of this Fourier-Mukai transform on the cohomology ring of $X$ reproduces relative T-duality and provides an infinitesimal isometry of the moduli space of algebraic structures on $X$ which, in view of the triviality of the quantum cohomology of K3 surfaces, can be interpreted as mirror symmetry.

alg-geom

\section{Introduction} In a recent approach of Strominger, Yau and Zaslow \cite{SYZ}, the phenomenon of mirror symmetry on Calabi-Yau threefolds admitting a $T^3$ fibration is interpreted as T-duality on the $T^3$ fibres. According to this formulation one would like to {\it define} the mirror dual to a Calabi-Yau manifold (of any dimension) as a compactification of the moduli space of its special Lagrangian submanifolds (the $T^3$ tori in the above case) endowed with a suitable complex structure \cite{SYZ,Morr,Hit}. In two dimensions this means that one considers a K3 surface elliptically fibred over the projective line, $p \colon X \to {\Bbb{P}}^1$. A mirror dual to $X$ can be identified with the component $\M$ of the moduli space of simple sheaves on $X$ having Mukai vector $(0,\mu,0)\in H^\bullet(X,{\Bbb Z})$, where $\mu$ is the cohomology class defined by the fibres of $p$. The mirror map between the Hodge lattices of $X$ and $\M$ should be given by a suitable Fourier-Mukai transform \cite{Morr,eng,DM}. In this paper we show that a Fourier-Mukai transform on elliptically fibred K3 surfaces provides indeed a description of mirror symmetry. The Fourier-Mukai transform not only maps special Lagrangian 2-cycles to 0-cycles, as noticed by Morrison and others, but also reproduces the correct duality transformations on 4-cycles and on 2-cycles of genus 0. It turns out that the Fourier-Mukai transform does not define an automorphism of the cohomology ring of the K3 surface which swaps the directions corresponding to complex structures with the directions corresponding to complexified K\"ahler structures. In this sense our treatment is different from other approaches, cf.~e.g.~\cite{Asp,G,GW}. However, we are able to obtain an isometry between the tangent space to the deformations of complex structures on $X$ and the tangent space to the deformations of ``complexified K\"ahler structures'' on the mirror manifold. We also note that the map determined by the Fourier-Mukai transform has a correct action on the mass of the so-called BPS states. In order to describe this ``geometric mirror symmetry'' two modifications must be introduced in the construction we have above outlined. First, we regard the mirror dual to the elliptic K3 surface $X$ as its compactified relative Jacobian ${\widehat X}$ (this is actually isomorphic to $\M$); secondly, we define a Fourier-Mukai in a relative setting (cf.~\cite{Muk3} for a relative Fourier-Mukai transform for abelian schemes). Moreover, the relative transform we define, once restricted to the smooth fibres, reduces to the usual Fourier-Mukai transform for abelian varietes; in this way the reduction of mirror symmetry to relative T-duality in the spirit of \cite{SYZ} is achieved. It should be stressed that this analysis shows that the moduli space $\M$ is isomorphic to the original K3 surface $X$ as an algebraic variety, in accordance with the fact that, under this interpretation of mirror symmetry, a K3 surface is mirror to itself \cite{SYZ}. This fact, together with the the existence of an isometry between the above mentioned spaces of deformations, is consistent with the triviality of the quantum cohomology of a K3 surface (in particular, the Weil-Petersson metric on the moduli space of complexified K\"ahler structures bears no instantonic corrections). To go through some more detail, the Fourier-Mukai functor ${\bold T}}\def\bS{{\bold S} $ we define transforms a torsion-free rank-one zero-degree sheaf concentrated on an elliptic fibre of $X$ to a point of the compactified relative Jacobian ${\widehat X}$; accordingly, ${\bold T}}\def\bS{{\bold S} $ enjoys the T-duality property of relating $2$-cycles to $0$-cycles. Furthermore, ${\bold T}}\def\bS{{\bold S} $ induces an isometry $$\psi\colon H^{1,1}({\widehat X},{\Bbb C})/\o{Pic}({\widehat X})\otimes{\Bbb C}\to H^{1,1}( X,{\Bbb C})/ \o{Pic}( X) \otimes{\Bbb C}\,.$$ The quotient $H^{1,1}({\widehat X},{\Bbb C})/\o{Pic}({\widehat X})\otimes{\Bbb C}$ can be regarded as the tangent space at ${\widehat X}$ to the space of deformations of algebraic structures on ${\widehat X}$ which preserve the Picard lattice, and similarly, $H^{1,1}( X,{\Bbb C})/\o{Pic}( X) \otimes{\Bbb C}$ is to be identified with the tangent space to the space of deformations of K\"ahler structures on $X$ preserving the Picard lattice. With these identifications in mind, the isometry $\psi$ can be regarded as an ``infinitesimal'' mirror map. {From} a mathematical viewpoint the transform we define here provides another example of a Fourier-Mukai transform on K3 surfaces in addition to the one given in \cite{BBH}. The paper is organized as follows. In Section 2 we fix notations, define the relative Fourier-Mukai functor and prove its first properties. In Section 3 we prove that it is invertible and thus gives rise to an equivalence of derived categories. In Section 4 we study the action of the Fourier-Mukai transform on the cohomology ring of the K3 surface $X$. In Section 5 we discuss how the Fourier-Mukai transform can be regarded as a mirror duality for string theories compactified on an elliptic K3 surface. \par\vbox to4mm{}\par\section{The basic construction} Let $p\colon X\to {\Bbb{P}}^1$ be a minimal algebraic elliptically fibred K3 surface (all algebraic varieties will be over ${\Bbb C}$). The fibration $p$ has singular fibres; these has been classified by Kodaira \cite{Kod}. We assume that $p\colon X\to{\Bbb{P}}^1$ has a section $e\colon{\Bbb{P}}^1\hookrightarrow X$ and write $H=e({\Bbb{P}}^1)$. We shall denote by $X_t$ the fibre of $p$ over $t\in{\Bbb{P}}^1$, and by $i_t\colon X_t\hookrightarrow X$ the inclusion. \smallskip {\it A compactification of the relative Jacobian.} Let $\M$ be the moduli space of simple sheaves on $X$, of pure dimension 1 and Chern character $(0,\mu,0)$, where $\mu$ is the cohomology class of the fibres of $p$. Results of Simpson \cite{Simp} imply that $\M$ is a smooth projective surface (actually, a minimal K3 surface, cf.~\cite{Muk3}). One may define a morphism \begin{align} \gamma\colon X & \to \M \\ x & \mapsto (i_t)_\ast ({\frak m}_x\otimes {\cal O}_{X_t}(e(t))) \label{e:gamma}\end{align} where $X_t\ni x$, and ${\frak m}_x$ is the ideal sheaf of $x$ in $X_t$. Let ${\cal U}\subset{\Bbb{P}}^1$ be the open subset supporting the smooth fibres of $p$, and let ${J(\rest X,{\U})}$ be the relative Jacobian variety. The restriction of $\gamma$ to $\rest X,{{\cal U}}$ factors as $$\rest X,{{\cal U}} \iso {J(\rest X,{\U})}\hookrightarrow \M\,, $$ where the isomorphism $\rest X,{{\cal U}} \iso {J(\rest X,{\U})}$ is given by $x\mapsto {\cal O}_{X_t}(e(t)-x)={\frak m}_x\otimes{\cal O}_{X_t}(e(t))$ and ${J(\rest X,{\U})}\hookrightarrow \M$ associates with any zero-degree torsion-free sheaf $L_t$ over $X_t$ its direct image $(i_t)_\ast L_t$. Then $\gamma$ is birational, and $X\simeq\M$ since they both are smooth projective surfaces and $X$ is minimal. We want now to construct a suitable compactification of the relative Jacobian of $p\colon X\to {\Bbb{P}}^1$. We denote by ${\bold Pic}^-_{X/\cpuno}$ the functor which to any morphism $f\colon S\to{\Bbb{P}}^1$ of algebraic varieties associates the space of $S$-flat sheaves on $p_S\colon X\times_{\cpuno} S\to S$, whose restrictions to the fibres of $p_S$ are torsion-free, of rank one and degree zero. Two such sheaves ${\cal F}$, ${\cal F}'$ are considered to be equivalent if ${\cal F}'\simeq{\cal F}\otimes p_S^\ast{\cal L}$ for a line bundle ${\cal L}$ on $S$ (cf.~\cite{AK}). Due to the existence of the section $e$, ${\bold Pic}^-_{X/\cpuno}$ is a sheaf functor. \begin{prop} The functor ${\bold Pic}^-_{X/\cpuno}$ is represented by an algebraic variety $\hat p\colon{\widehat X} \to{\Bbb{P}}^1$, which is isomorphic to $X$. \end{prop} \begin{pf} If we denote by ${\bold h}_X$, ${\bold h}_{\M}$ the functors of points of $X$, $\M$ as schemes over ${\Bbb{P}}^1$, the isomorphism $\gamma\colon {\bold h}_X\iso {\bold h}_{\M}$ factors as $$ {\bold h}_X@>\varpi>>{\bold Pic}^-_{X/\cpuno} @>\alpha>> {\bold h}_{\M} $$ where $\varpi$ and $\alpha$ are defined (over the closed points) by $\varpi(x)= {\frak m}_x\otimes{\cal O}_{X_t}(e(t))$ and $\alpha(L_t)=(i_t)_\ast L_t$ for any zero-degree torsion-free sheaf $L_t$ over $X_t$. Both morphisms of functors are immersions and their composition is an isomorphism, so that they are isomorphisms as well. Then, ${\bold Pic}^-_{X/\cpuno}$ is represented by a fibred algebraic variety $\hat p\colon{\widehat X}\to{\Bbb{P}}^1$, and $\varpi\colon X\to{\widehat X}$ is an isomorphism. \end{pf} We denote by $\hat e\colon{\Bbb{P}}^1\to{\widehat X}$ the canonical section; one has $\hat e=\varpi\circ e$. Moreover, we denote by $\pi$, $\hat\pi$ the projections of the fibred product $X\times_{\cpuno} {\widehat X}$ onto the factors. \begin{remark} The Picard functor is also representable by an open dense subscheme $J$ of ${\widehat X}$, the relative Jacobian $J\to{\Bbb{P}}^1$ of $X\to{\Bbb{P}}^1$. If $X^s\subset X$ denotes the complement of the singular points of the fibres of $\pi$, then $\varpi$ gives an isomorphism $X^s\iso J$ of schemes over ${\Bbb{P}}^1$. One should notice that in general the Jacobian variety $J\to{\Bbb{P}}^1$ is different from $\o{Pic}^0(X/{\Bbb{P}}^1)\to{\Bbb{P}}^1$. This scheme can be obtained from $J\to{\Bbb{P}}^1$ by removing the images by $\varpi$ of those components of the singular fibres of $p\colon X\to{\Bbb{P}}^1$ that do not meet the image $H$ of the section. \qed\end{remark} The representability of ${\bold Pic}^-_{X/\cpuno}$ means there exists a coherent sheaf ${\cal P}$ on $X\times_{\cpuno}{\widehat X}$ flat over ${\widehat X}$, whose restrictions to the fibres of $\hat\pi$ are torsion-free, and of rank one and degree zero, such that \begin{equation} \o{Hom}\,_{{\Bbb{P}}^1}(S,{\widehat X}) \to {\bold Pic}^-_{X/\cpuno}(S)\,,\qquad\qquad \phi \mapsto [(1\times\phi)^\ast{\cal P}] \label{e:univ}\end{equation} is an isomorphism of functors. ${\cal P}$ is defined up to tensor product by the pullback of a line bundle on ${\widehat X}$, and is called the universal Poincar\'e sheaf. To normalize the Poincar\'e sheaf we notice that the isomorphism $\varpi\colon X\iso{\widehat X}$ is induced, according to the universal property (\ref{e:univ}), by the sheaf ${\cal I}_\Delta\otimes p_1^\ast{\cal O}_X(H)$ on $X\times_{\cpuno} X$, where $p_1$ is the projection onto the first factor and ${\cal I}_\Delta$ is the ideal sheaf of the diagonal $\delta\colon X\hookrightarrow X\times_{\cpuno} X$; this sheaf is flat over the second factor and has zero relative degree. Then \begin{equation} {\cal P}=(1\times\varpi^{-1})^\ast\left({\cal I}_\Delta\otimes p_1^\ast{\cal O}_X(H)\right) \otimes\hat \pi^\ast{\cal L} \label{e:poin} \end{equation} for a line bundle ${\cal L}$ on ${\widehat X}$. Restriction to $H\times_{\cpuno}{\widehat X}$ gives $\rest{{\cal P}},{H\times_{\cpuno}{\widehat X}}\simeq {\cal O}_{{\widehat X}}(-2)\otimes{\cal L}$, which is a line bundle. We can then normalize ${\cal P}$ by letting \begin{equation} \rest{\cal P},{H\times_{\cpuno} {\widehat X}}\simeq {\cal O}_{{\widehat X}}\,.\label{e:norm}\end{equation} We shall henceforth assume that ${\cal P}$ is normalized in this way. We shall denote by ${\cal P}_\xi$ the restriction $\rest{{\cal P}},{\hat\pi^{-1}(\xi)}$. As a consequence of (\ref{e:poin}), ${\cal P}$ is flat over $X$ as well. \begin{remark} Since the moduli space $\M$ is fine, on $X\times \M\simeq X\times {\widehat X}$ there is a universal sheaf ${\cal Q}$. This is the sheaf that gives rise to the morphism $\gamma$ (cf.~Eq.~(\ref{e:gamma})). One can show that ${\cal Q}$ is supported on $X\times_{\cpuno}{\widehat X}$, and its restriction to its support is isomorphic to ${\cal P}$ (up to tensoring by a pullback of a line bundle on ${\widehat X}$).\qed\end{remark} The dual ${\cal P}^\ast$ of the Poincar\'e bundle is a coherent sheaf on $X\times_{\cpuno}{\widehat X}$ whose restrictions to the fibres of $\hat\pi$ are torsion-free, rank one, and of degree zero. As ${\cal P}^\ast$ is flat over ${\widehat X}$ it defines a morphism $\iota\colon{\widehat X}\to {\widehat X}$. Since ${\cal E}xt\, ^1_{{\cal O}_{X_t}}({\cal P}_\xi,{\cal O}_{X_t})=0$ for every point $\xi\in{\widehat X}$ (here $t=\hat p(\xi)$) and ${\cal E}xt\,^1_{{\cal O}_{X\times_{\cpuno}{\widehat X}}}({\cal P},{\cal O}_{X\times_{\cpuno}{\widehat X}})=0$ by (\ref{e:poin}), the base change property for the local Ext's (\cite{AK}, Theorem 1.9) implies that $({\cal P}^\ast)_\xi\simeq ({\cal P}_\xi)^\ast$. Then, the morphism $\iota \colon{\widehat X}\to {\widehat X}$ maps any rank-one torsion-free and zero-degree coherent sheaf ${\cal F}$ on a fibre $X_t$ to its dual ${\cal F}^\ast$. By (\ref{e:univ}) one has $(1\times\iota)^\ast{\cal P}\simeq{\cal P}^\ast\otimes\hat\pi^\ast{\cal N}$ for some line bundle ${\cal N}$ on ${\widehat X}$, which turns out to be trivial by (\ref{e:norm}). Then \begin{equation} (1\times\iota)^\ast{\cal P}\simeq{\cal P}^\ast\,. \label{e:inv}\end{equation} The morphism $\iota\circ\iota\colon{\widehat X}\to{\widehat X}$ is the identity on the Jacobian ${J(\rest X,{\U})}\subset{\widehat X}$; by separateness $\iota\circ\iota=\o{Id}$, and (\ref{e:inv}) implies ${\cal P}\simeq{\cal P}^{\ast\ast}\,.$ Then, every coherent sheaf ${\cal F}$ on $X\times_{\cpuno} S$ flat over $S$ whose restrictions to the fibres of $X\times_{\cpuno} S \to S$ are torsion-free and of rank one and degree zero is reflexive, ${\cal F}\simeq{\cal F}^{\ast\ast}$. \begin{prop} The relative Jacobian of the fibration $\hat p\colon{\widehat X}\to {\Bbb{P}}^1$ admits a compactification which is isomorphic to $X$ as a fibred variety over ${\Bbb{P}}^1$, and the relevant universal Poincar\'e sheaf may be identified with ${\cal P}^\ast$. \end{prop} \begin{pf} By (\ref{e:poin}), the sheaf ${\cal P}^\ast$ is flat over $X$. Proceeding as above, one proves that $({\cal P}^\ast)_x\simeq ({\cal P}_x)^\ast$ for every point $x\in X$, which means that the restrictions of ${\cal P}^\ast$ to the fibres of $\pi$ are torsion-free sheaves of rank one and degree zero. So ${\cal P}^\ast$ defines a morphism $X\to \widehat{{\widehat X}}$ of schemes over ${\Bbb{P}}^1$. If ${\cal U}\subset{\Bbb{P}}^1$ denotes as above the open subset supporting the smooth fibres of $p$, this morphism restricts to an isomorphism $X_{\vert{\cal U}}\simeq J({\widehat X})_{\vert{\cal U}}$ of schemes over ${{\cal U}}$. Since ${\widehat X}$ is minimal and $X$ is smooth and has no $(-1)$-curves, $X\simeq\widehat{{\widehat X}}$. \end{pf} The roles of $X$ and ${\widehat X}$ are so completely interchangeable. \par\smallskip {\it The Fourier-Mukai functors.} For any morphism $f\colon S\to{\Bbb{P}}^1$ let us consider the diagram $$\begin{CD} (X\times_{\cpuno}{\widehat X})_S @>\hat\pi_S>> {\widehat X}_S \\ @V\pi_SVV @VV\hat p_SV \\ X_S @>p_S>> S \end{CD} $$ We shall systematically denote objects obtained by base change to $S$ by a subscript $S$. (Note that $(X\times_{\cpuno}{\widehat X})_S\simeq X_S\times_S{\widehat X}_S$). We define the Fourier-Mukai functors $\bS _S^i$, $i=0,1$ by associating with every sheaf ${\cal F}$ on $X_S$ flat over $S$ the sheaf on ${\widehat X}_S$ $$\bS_S^i({\cal F})=R^i\hat\pi_{S\ast}(\pi_S^\ast{\cal F}\otimes{\cal P}_S)\,.$$ The Fourier-Mukai functors mapping sheaves on $X$ to sheaves on ${\widehat X}$ will be denoted by $\bS^i$. \begin{defin} We say that a coherent sheaf ${\cal F}$ on $X_S$ flat over $S$ is WIT$_i$ if $\bS _S^j({\cal F})=0$ for $j\ne i$. We say that ${\cal F}$ is IT$_i$ if it is WIT$_i$ and $\bS _S^i({\cal F})$ is locally free. \end{defin} One should notice that, due to the presence of the fibred instead of the cartesian product, the WIT$_0$ and IT$_0$ conditions are not equivalent: for instance $\kappa(x)$ (the skyscraper sheaf concentrated at $x\in X$) is WIT$_0$ but not $IT_0$. Since the fibres of $\hat\pi_S$ are one-dimensional the first direct image functor commutes with base change. \begin{prop} Let ${\cal F}$ be a sheaf on $X_S$, flat over $S$. For every morphism $g\colon T\to S$ one has $g_{{\widehat X}}^\ast\bS_S^1({\cal F})\simeq \bS_T^1(g_X^\ast{\cal F})$, where $g_X\colon X_T\to X_S$, $g_{{\widehat X}}\colon{\widehat X}_T\to{\widehat X}_S$ are the morphisms induced by $g$. \qed\label{basechange}\end{prop} The zeroth direct image does not commute with base change; however, a weaker property holds. \begin{prop} Let ${\cal F}$ be a sheaf on $X_S$, flat over $S$. For every point $\xi\in {\widehat X}_S$, the natural base change morphism $$ \hat\pi_{S\ast}(\pi_S^\ast{\cal F}\otimes{\cal P}_S)\otimes\kappa(\xi)\to H^0(X_s,{\cal F}_s\otimes{\cal P}_\xi) $$ is injective (here $s=\hat p_S(\xi)$\rm). \label{basechange2} \end{prop} \begin{pf} Let ${\frak m}_\xi$ denote the ideal sheaf of $\xi\in {\widehat X}_S$. Since $\hat\pi_S$ is flat, $\hat\pi_S^\ast{\frak m}_\xi $ is the ideal sheaf of the fibre $\hat\pi_S^{-1}(\xi)\simeq X_s$ in $X_S\times_S{\widehat X}_S$. Let us write ${\cal N}=\pi_S^\ast{\cal F}\otimes{\cal P}_S$ and ${\cal N}_\xi=\rest{{\cal N}},{\hat\pi_S^{-1}(\xi)}$. Since ${\cal N}$ is flat over ${\widehat X}_S$ there is an exact sequence $$ 0@>>>\hat\pi_S^\ast{\frak m}_\xi \otimes{\cal N} @>>> {\cal N} @>>> j_\ast {\cal N}_\xi @>>> 0\,, $$ where $j\colon \hat\pi_S^{-1}(\xi)\simeq X_s\hookrightarrow X_S\times_S{\widehat X}_S$ is the natural immersion. By taking direct images we obtain $$ 0 @>>> \hat\pi_{S\ast} (\hat\pi_S^\ast{\frak m}_\xi \otimes{\cal N}) @>>> \hat\pi_{S\ast}({\cal N}) @>\eta>> \hat\pi_{S\ast}(j_\ast{\cal N}_\xi)=H^0(X_s,{\cal N}_\xi)\,. $$ By the projection formula, $\hat\pi_{S\ast} (\hat\pi_S^\ast{\frak m}_\xi \otimes{\cal N})\simeq {\frak m}_\xi\otimes \hat\pi_{S\ast}{\cal N}$, and then $\ker\eta\simeq {\frak m}_\xi\cdot{\cal N}$; this implies that the base change morphism ${\cal N}\otimes\kappa(\xi)\to H^0(X_s,{\cal N}_\xi)$ is injective.\end{pf} \smallskip {\it Fourier-Mukai transform of rank 1 sheaves.} A first manifestation of geometric mirror symmetry is the fact that the Fourier-Mukai transform of a torsion-free rank-one zero-degree coherent sheaf on a fibre $X_t$ is a skyscraper sheaf concentrated at a point of ${\widehat X}_t$. By Proposition \ref{basechange} the basic ingredients to compute the functors $\bS^\bullet$ are the Fourier-Mukai transforms $\bS_{{\widehat X}}^\bullet({\cal P})$ of the universal Poincar\'e sheaf ${\cal P}$ on $X_{{\widehat X}}=X\times_{\cpuno}{\widehat X}$. The relevant higher direct images of ${\cal P}$ and ${\cal P}^\ast$ are computed as follows. (For every algebraic variety $q\colon Y\to{\Bbb{P}}^1$ over ${\Bbb{P}}^1$ and every coherent sheaf ${\cal N}$ on $Y$ we denote by ${\cal N}(n)$ the sheaf ${\cal N}\otimes q^\ast{\cal O}_{{\Bbb{P}}^1}(n)$.) \begin{thm} $\phantom{xxxxx}$ \begin{list}{}{\itemsep=2pt} \item[1] $\bS_{{\widehat X}}^1({\cal P})\simeq \zeta_\ast{\cal O}_{{\widehat X}}(-2)$, where $\zeta\colon {\widehat X}\hookrightarrow {\widehat X}\times_{\cpuno}{\widehat X}$ is the graph of the morphism $\iota$. \item[2.] $\bS_{{\widehat X}}^0({\cal P})=0$. \item[3.] $\bS_{{\widehat X}}^1({\cal P}^\ast)\simeq \delta_\ast{\cal O}_{{\widehat X}}(-2),\quad \bS_{{\widehat X}}^0({\cal P}^\ast)=0$, where $\delta\colon{\widehat X}\hookrightarrow{\widehat X}\times_{\cpuno}{\widehat X}$ is the diagonal immersion. \item[4.] $R^1\hat\pi_\ast{\cal P}\simeq R^1\hat\pi_\ast{\cal P}^\ast \simeq \hat e_\ast{\cal O}_{{\Bbb{P}}^1}(-2)$, while the zeroth direct images vanish. \end{list} \label{pm}\end{thm} A result similar to the second formula can be found in \cite{Muk3} for the case of relative abelian schemes. To prove Theorem \ref{pm} we need some preliminary results. \begin{lemma} Let $Y$ be a proper connected curve of arithmetic genus 1 and ${\cal F}$ a torsion-free rank-one and zero-degree sheaf on $Y$. Then $H^1(Y,{\cal F})\neq 0$ if and only if ${\cal F}\simeq{\cal O}_Y$. \label{huno} \end{lemma} \begin{pf} One has $H^0(Y,{\cal F})\neq 0$ by Riemann-Roch and $H^0(Y,{\cal F}^\ast) \neq 0$ by duality. Let $\tau$ and $\sigma$ be nonzero sections of ${\cal F}$ and ${\cal F}^\ast$ respectively. Let $\rho$ be the composition $$\begin{CD} {\cal F} @>>> {\cal F}^{\ast\ast} @> \sigma^\ast>> {\cal O}_Y\,.\end{CD}$$ Since $\rho\circ\tau\ne 0$, the morphism $\rho\circ\tau$ consists in the multiplication by a nonzero constant, which may be set to 1. Then $\rho\circ\tau=\o{id}$, so that ${\cal F}\simeq {\cal O}_Y\oplus\M$, where $\M$ has rank zero; hence $\M=0$, and ${\cal F}\simeq{\cal O}_Y$. \end{pf} \begin{lemma} Let $\xi,\mu\in {\widehat X}_t$. \begin{list}{}{\itemsep=2pt} \item[1.] The sheaf ${\cal P}_\xi\otimes{\cal P}_\mu$ has torsion if and only if $\xi$ is a singular point of the fibre ${\widehat X}_t$ and $\mu=\xi$. \item[2.] The evaluation morphism ${\cal P}_\xi\otimes{\cal P}_\xi^\ast\to{\cal O}_{X_t}$ induces an isomorphism $H^1(X_t,{\cal P}_\xi\otimes{\cal P}_\xi^\ast)\simeq H^1(X_t,{\cal O}_{X_t})$. \item[3.] If $\mu\ne\iota(\xi)$ then $H^1(X_t,{\cal P}_\xi\otimes{\cal P}_\mu)=0$. \end{list} \label{eval} \end{lemma} \begin{pf} 1. We have ${\cal P}_\xi={\frak m}_x(e(t))$ for a point $x\in X_t$, and $\xi$ is singular in ${\widehat X}_t$ iff $x$ is singular in $X_t$. If $\xi$ or $\mu$ are not singular, then one of the sheaves ${\cal P}_\xi$, ${\cal P}_\mu$ is locally free, and ${\cal P}_\xi\otimes{\cal P}_\mu$ is torsion-free. Otherwise, ${\cal P}_\xi={\frak m}_x(e(t))$ and ${\cal P}_\mu={\frak m}_y(e(t))$ for singular points $x,y\in X_t$. If $\mu\ne\iota(\xi)$ then ${\frak m}_x\otimes{\frak m}_y$ is torsion-free. Finally, if $\mu=\iota(\xi)$ then ${\frak m}_x(2e(t))={\frak m}_y^\ast$, so that $x=y$ (because ${\frak m}_x$, ${\frak m}_y$ are not locally-free only at $x$, $y$, respectively). Thus $\mu=\xi$, and ${\frak m}_x\otimes{\frak m}^\ast_x$ has torsion at $x$. 2. The only nontrivial case is when ${\cal P}_\xi$ is not locally free. Let $x\in X_t$ be the singular point corresponding to $\xi$. We have an exact sequence $$ 0@>>>{\frak m}_x@>>>{\cal P}_\xi\otimes{\cal P}_\xi^\ast @>>>{\frak m}_x/{\frak m}_x^2 @>>>0\,, $$ which implies that $H^1(X_t,{\frak m}_x)\to H^1(X_t,{\cal P}_\xi\otimes{\cal P}_\xi^\ast)$ is an epimorphism. Since the composition $H^1(X_t,{\frak m}_x)\to H^1(X_t,{\cal P}_\xi\otimes{\cal P}_\xi^\ast) \to H^1(X_t,{\cal O}_{X_t})$ is an isomorphism, $H^1(X_t,{\cal P}_\xi\otimes{\cal P}_\xi^\ast)\simeq H^1(X_t,{\cal O}_{X_t})$ is an isomorphism as well. 3. Follows from 1 and Lemma \ref{huno}. \end{pf} In order to compute the Fourier-Mukai transform $\bS_{{\widehat X}}^\bullet({\cal P})$ of the Poincar\'e sheaf ${\cal P}$ on $X\times_{\cpuno} {\widehat X}$ we consider the diagram $$ \begin{CD} X\times_{\cpuno}{\widehat X}\times_{\cpuno}{\widehat X} @>\pi_{23}>> {\widehat X}\times_{\cpuno}{\widehat X} \\ @V\pi_{12}VV @VV\hat p_1V \\ X\times_{\cpuno}{\widehat X} @>\hat \pi >> {\widehat X} \end{CD} $$ The Poincar\'e sheaf on $X\times_{\cpuno}{\widehat X}\times_{\cpuno}{\widehat X}$ is ${\cal P}_{{\widehat X}}=\pi_{13}^\ast{\cal P}$ and the Fourier-Mukai transforms of ${\cal P}$ are $\bS_{{\widehat X}}^\bullet({\cal P})=R^\bullet\pi_{23\ast}(\pi_{12}^\ast{\cal P}\otimes \pi_{13}^\ast{\cal P})$. \smallskip{\it Proof of 1 of Theorem \ref{pm}.} We have ${\cal P}\otimes{\cal P}^\ast= (1\times\zeta)^\ast(\pi_{12}^\ast{\cal P}\otimes\pi_{13}^\ast{\cal P})$. The composition of the epimorphism $\pi_{12}^\ast{\cal P}\otimes\pi_{13}^\ast{\cal P}@>>>(1\times\zeta)_\ast ({\cal P}\otimes{\cal P}^\ast)$ with the evaluation morphism $(1\times\zeta)_\ast({\cal P}\otimes{\cal P}^\ast)@>>>(1\times\zeta)_\ast ({\cal O}_{X\times_{\cpuno}{\widehat X}})$ gives a morphism $$ \pi_{12}^\ast{\cal P}\otimes\pi_{13}^\ast{\cal P}@>>>(1\times\zeta)_\ast ({\cal O}_{X\times_{\cpuno}{\widehat X}})\,. $$ We have then a morphism $$ \bS_{{\widehat X}}^1({\cal P})=R^1\pi_{23\ast}(\pi_{12}^\ast{\cal P}\otimes\pi_{13}^\ast{\cal P}) @>>> R^1\pi_{23\ast} ((1\times\zeta)_\ast({\cal O}_{X\times_{\cpuno}{\widehat X}}))\simeq \zeta_\ast{\cal O}_{{\widehat X}}(-2)\,. $$ Since the first direct image functor commutes with base change, Lemma \ref{eval} implies that $\bS_{{\widehat X}}^1({\cal P})$ is supported on $\zeta({\widehat X})$. Moreover the fibre of the previous morphism at a point $\zeta(\xi)$, with $\xi\in {\widehat X}_t$, is $H^1(X_t,{\cal P}_\xi\otimes{\cal P}_\xi^\ast)@>>>H^1(X_t,{\cal O}_{X_t})$, which is an isomorphism by Lemma \ref{eval}.\qed\smallskip Let $f\colon S\to{\Bbb{P}}^1$ be a morphism and ${\cal F}$ a coherent sheaf on $X\times_{\cpuno} S$ flat over $S$ whose restrictions to the fibres of $p_S$ are torsion-free and have rank one and degree zero. Let $\phi\colon S\to{\widehat X}$ be the morphism determined by the universal property (\ref{e:univ}), so that $$ (1\times\phi)^\ast{\cal P}\simeq{\cal F}\otimes p_S^\ast{\cal L}\,, $$ for a line bundle ${\cal L}$ on $S$. Let $\Gamma\colon S\hookrightarrow {\widehat X}_S$ be the graph of the morphism $\iota\circ\phi\colon S\to{\widehat X}$. \begin{lemma} $\bS_S^1({\cal F})\otimes \hat p_S^\ast{\cal L}\simeq \Gamma_{\ast}{\cal O}_S(-2)\,,\quad \bS_S^0({\cal F})=0$. \label{SS}\end{lemma} \begin{pf} The formula for $\bS_S^1({\cal F})$ follows from Proposition \ref{basechange} and 1 of Theorem \ref{pm} after some standard computations. The second formula is proved as follows. From Proposition \ref{basechange2} we have the exact sequence $$ 0 @>>> \hat\pi_{S\ast}(\pi_S^\ast{\cal F}\otimes{\cal P}_S)\otimes\kappa(\xi) @>>> H^0(X_s,{\cal F}_s\otimes{\cal P}_\xi)$$ where $s=\hat p_S(\xi)$. If $\xi\notin\Gamma(S)$, $H^0(X_s,{\cal F}_s\otimes{\cal P}_\xi)=0$ by Lemma \ref{eval} and $\hat\pi_{S\ast}(\pi_S^\ast{\cal F}\otimes{\cal P}_S)\otimes\kappa(\xi)=0$ as well. If $\xi\in\Gamma(S)$ the first direct image $\bS_S^1({\cal F})$ is not locally-free at $\xi$ since it is concentrated on the image of $\Gamma$, and then the second arrow is not surjective; but $H^0(X_s,{\cal F}_s\otimes{\cal P}_\xi)$ is one-dimensional by Lemma \ref{eval}, so that $\hat\pi_{S\ast}(\pi_S^\ast{\cal F}\otimes{\cal P}_S)\otimes\kappa(\xi)=0$. \end{pf} \smallskip{\it End of proof of Theorem \ref{pm}.} 2 is proved by applying Lemma \ref{SS} with $S={\widehat X}$ and $\phi$ the identity, while to prove 3 one chooses $S={\widehat X}$ and $\phi=\iota$. Taking $S={\Bbb{P}}^1$ and $\phi=\hat e$ one proves the claims of 4 concerning the sheaf ${\cal P}$. To prove the claims for ${\cal P}^\ast$ one notices that Lemma \ref{SS} still applies after replacing ${\cal P}$ by ${\cal P}^\ast$.\qed\smallskip We can now compute the Fourier-Mukai transform of sheaves on $X$ corresponding to points in ${\widehat X}$. \begin{corol} Let ${\cal F}$ be a rank-one, zero-degree, torsion-free coherent sheaf on a fibre $X_t$. Then $$ \bS^0_t({\cal F})=0\,,\qquad \bS ^1_t({\cal F})=\kappa([{\cal F}^\ast])\,, $$ where $[{\cal F}^\ast]$ is the point of ${\widehat X}_t$ defined by ${\cal F}^\ast$. \qed\label{cor2}\end{corol} \par\vbox to4mm{}\par\section{Inversion of the Fourier-Mukai transform} The Fourier-Mukai functor defines a functor $D^-(X)\to D^-({\widehat X})$ given by $\bS(F)= R\hat\pi_\ast(\pi^\ast F{\overset L \otimes}{\cal P})$ (here $D^-(X)$ is the subcategory of the derived category of coherent ${\cal O}_X$-modules consisting of the complexes bounded from above). To state the invertibility properties of this functor in a neat way we define a modified functor ${\bold T}}\def\bS{{\bold S} \colon D^-(X)\to D^-({\widehat X})$ by ${\bold T}}\def\bS{{\bold S} (F)=\bS(F{\overset L\otimes}{\cal O}_X(1))$. A natural candidate for the inverse of ${\bold T}}\def\bS{{\bold S} $ is the functor $\widehat{{\bold T}}\def\bS{{\bold S}}\colon D^-({\widehat X})\to D^-(X)$ given by $$\widehat{{\bold T}}\def\bS{{\bold S}}(G)= \widehat{\bS} (G{\overset L\otimes}{\cal O}_{{\widehat X}}(1))\qquad \text{where}\qquad\widehat{\bS}(G') =R\pi_\ast(\hat\pi^\ast G'{\overset L\otimes}{\cal P}^\ast)\,.$$ Since the relative dualizing complexes of $\pi$ and $\hat\pi$ are both isomorphic to ${\cal O}_{X\times_{\cpuno}{\widehat X}}(2)[1]$, relative duality gives: \begin{prop} For every objects $F$ in $D^-(X)$ and $G$ in $D^-({\widehat X})$ one has functorial isomorphisms $$ \o{Hom}_{D^-({\widehat X})}(G,{\bold T}}\def\bS{{\bold S} (F))\simeq\o{Hom}_{D^-(X)}(\widehat{{\bold T}}\def\bS{{\bold S}}(G), F[-1]) $$ $$ \o{Hom}_{D^-(X)}(F,\widehat{{\bold T}}\def\bS{{\bold S}}(G)) \simeq\o{Hom}_{D^-({\widehat X})}({\bold T}}\def\bS{{\bold S} (F),G[-1])\,. $$ \qed\end{prop} \begin{thm} For every $G\in D^-({\widehat X})$, $F\in D^-(X)$ there are functorial isomorphisms $$ {\bold T}}\def\bS{{\bold S} (\widehat{{\bold T}}\def\bS{{\bold S}}(G))\simeq G[-1]\,,\quad \widehat{{\bold T}}\def\bS{{\bold S}}({\bold T}}\def\bS{{\bold S} (F))\simeq F[-1] $$ in the derived categories $D^-({\widehat X})$ and $D^-(X)$, respectively. \end{thm} \begin{pf} Let $\hat\pi_1$ and $\hat\pi_2$ be the projections onto the two factors of ${\widehat X}\times_{\cpuno}{\widehat X}$. Then $ {\bold T}}\def\bS{{\bold S} (\widehat{{\bold T}}\def\bS{{\bold S}}(G))=R \hat\pi_{2,\ast}(\hat\pi_1^\ast G{\overset L\otimes}\widetilde{{\cal P}}) \otimes{\cal O}_X(2)$ (see \cite{Muk1,Muk3,Muk4,BBH} for similar statements), with $$ \widetilde{{\cal P}}=R\pi_{23,\ast}(\pi_{12}^\ast{\cal P}^\ast\otimes \pi_{13}^\ast{\cal P})\,. $$ By Theorem \ref{pm} $\widetilde{{\cal P}}\simeq\delta_\ast({\cal O}_{{\widehat X}}(-2))[-1]$ in the derived category, and ${\bold T}}\def\bS{{\bold S} (\widehat{{\bold T}}\def\bS{{\bold S}}(G))\simeq G[-1]$. The second statement follows from the first by interchanging the roles of $X$ and ${\widehat X}$. \end{pf} So ${\bold T}}\def\bS{{\bold S} $ establishes an equivalence of triangulated categories. \begin{corol} Let ${\cal F}$ be a WIT$_i$ sheaf on $X$. Then its Fourier-Mukai transform ${\bold T}}\def\bS{{\bold S}^i(F)$ is a WIT$_{1-i}$ sheaf on ${\widehat X}$, whose Fourier-Mukai transform $$\widehat{{\bold T}}\def\bS{{\bold S}}^{1-i}({\bold T}}\def\bS{{\bold S}^i(F))=R^{1-i}\pi_\ast(\hat\pi^\ast {\bold T}}\def\bS{{\bold S}^i(F) \otimes{\cal P}^\ast(1))$$ is isomorphic to ${\cal F}$. \label{inv2} \qed\end{corol} We also have a property of preservation of the Hom groups, which is sometimes called ``Parseval theorem.'' \begin{prop} There are functorial isomorphisms $$ \o{Hom}_{D^-({\widehat X})} (G,\bar G) \simeq \o{Hom}_{D^-(X)}(\widehat{\bS}(G),\widehat \bS(\bar G))\simeq \o{Hom}_{D^-(X)} (\widehat{{\bold T}}\def\bS{{\bold S}}(G),\widehat {\bold T}}\def\bS{{\bold S}(\bar G)) $$ $$ \o{Hom}_{D^-(X)}(F,\bar F) \simeq \o{Hom}_{D^-({\widehat X})}(\bS(F),\bS(\bar F))\simeq \o{Hom}_{D^-({\widehat X})}({\bold T}}\def\bS{{\bold S}(F), {\bold T}}\def\bS{{\bold S}(\bar F)) $$ for $F$, $\bar F$ in $D^-(X)$ and $G$, $\bar G$ in $D^-({\widehat X})$. \qed\end{prop} \begin{corol} Let ${\cal F}$, ${\cal F}'$ be coherent sheaves on $X$. If ${\cal F}$ is WIT$_i$ and ${\cal F}'$ is WIT$_j$, we have $$ \o{Ext}^h({\cal F},{\cal F}')\simeq\o{Ext}^{h+i-j}(\bS^i({\cal F}),\bS^j({\cal F}')) \simeq\o{Ext}^{h+i-j} ({\bold T}}\def\bS{{\bold S}^i({\cal F}),{\bold T}}\def\bS{{\bold S}^j({\cal F}'))\,. $$ for $h=0,1$. In particular, if ${\cal F}$ is WIT$_i$ there is an isomorphism $\o{Ext}^h({\cal F},{\cal F})\simeq\o{Ext}^h({\bold T}}\def\bS{{\bold S}^i({\cal F}),{\bold T}}\def\bS{{\bold S}^i({\cal F}))$ for every $h$, so that ${\bold T}}\def\bS{{\bold S}^i({\cal F})$ is simple if ${\cal F}$ is. \qed\label{simple}\end{corol} \begin{remark} Moduli spaces of sheaves on holomorphic symplectic surfaces carry a holomorphic symplectic structure, which is given by the Yoneda pairing $\o{Ext}^1({\cal F},{\cal F})\otimes \o{Ext}^1({\cal F},{\cal F})\to\o{Ext}^2({\cal F},{\cal F})\simeq {\Bbb C}$ (cf.~\cite{Muk2}), where one identifies $\o{Ext}^1({\cal F},{\cal F})$ with the tangent space to the moduli space at the point corresponding to the sheaf ${\cal F}$. Whenever the Fourier-Mukai transform establishes a morphism between such moduli spaces, Corollary \ref{simple} implies that the morphism is symplectic. \qed\label{sympl}\end{remark} \par\vbox to4mm{}\par\section{Action on the cohomology ring} The cohomology ring $H^\bullet(X,{\Bbb Z})$ carries a bilinear pairing, usually called {\it Mukai pairing,} defined as $$(a,b,c)\cdot(a',b',c')=(b\cup b'-a\cup c'-a'\cup c)\,\backslash\,[X], $$ and the same is true for $H^\bullet({\widehat X},{\Bbb Z})$ (here $\,\backslash\,$ denotes the slant product). We define an isomorphism ${\bold f}\colon H^\bullet(X,{\Bbb Q})\to H^\bullet({\widehat X},{\Bbb Q})$ and want to show that in terms of ${\bold f} $ one can introduce an isometry between the tangent space to the moduli space of algebraic structures on ${\widehat X}$ and the space of deformations of the complexified K\"ahler structure on $X$, which can be regarded as a geometric realization of mirror symmetry. We define the map ${\bold f} $ basically as in \cite{Muk4}, but the properties of this map are slightly different, since we are working in a relative setting, and the relative dualizing sheaf is nontrivial. Also, we must take coefficients in ${\Bbb Q}$ because the relative Todd characters involved in the definition of the ${\bold f}$ map do not have integral square roots. \smallskip {\it The ${\bold f} $ map.} We now define the ${\bold f} $ map and describe its basic properties. We shall be concerned with varieties fibred over ${\Bbb{P}}^1$, $\phi_Y\colon Y\to{\Bbb{P}}^1$, with a section $\sigma_Y\colon {\Bbb{P}}^1 \hookrightarrow Y$. Since $\sigma_Y^\ast\circ\phi_Y^\ast=1$, there is a decomposition $$H^\bullet(Y,{\Bbb Q})\simeq \phi_Y^\ast H^\bullet({\Bbb{P}}^1,{\Bbb Q})\oplus H^\bullet_\phi(Y,{\Bbb Q}) $$ where $H^\bullet_\phi(Y,{\Bbb Q})=\ker\sigma_Y^\ast$. One has in particular $$ H^0_\phi(Y,{\Bbb Q})=0, \quad H^2 (Y,{\Bbb Q})={\Bbb Q}\mu_Y\oplus H^2_\phi(Y,{\Bbb Q}),\quad H^{2i}_\phi(Y,{\Bbb Q})= H^{2i}(Y,{\Bbb Q})\quad \text{for $i\ge2$}\,. $$ We define in $H^{\text{even}} (Y,{\Bbb Q})$ an involution $^\ast$ by letting \begin{equation}\begin{align*} \alpha^\ast =(-1)^i\alpha&\qquad \text{if}\quad \alpha\in H^{2i}_\phi(Y,{\Bbb Q}) \\ (\phi_Y^\ast\eta)^\ast =\phi_Y^\ast\eta&\qquad\text{if}\quad \eta\in H^{2i}({\Bbb{P}}^1,{\Bbb Q}) \,.\end{align*} \end{equation} Turning back to the case where $X$ is an elliptic K3 surface, satisfying all the properties we have so far stated, we define morphisms $${\bold f} \colon H^\bullet(X,{\Bbb Q})\to H^\bullet({\widehat X},{\Bbb Q}),\qquad {\bold f} '\colon H^\bullet({\widehat X},{\Bbb Q})\to H^\bullet(X,{\Bbb Q})$$ by letting $${\bold f} (\alpha)=\hat\pi_\ast(Z\,\pi^\ast\alpha),\qquad {\bold f} '(\beta)=\pi_\ast(Z^\ast\,\hat\pi^\ast\beta)\,.$$ where $$Z=\sqrt{\o{td}\hat\pi}\,\o{ch}({\cal P}\otimes\pi^\ast{\cal O}_X(1))\, \sqrt{\o{td}\pi}\,.$$ \begin{lemma} The maps ${\bold f} $, ${\bold f} '$ have the following properties: \begin{list}{}{\itemsep=2pt} \item[1.] ${\bold f} \circ {\bold f} '(\beta)=-\beta$; \item[2.] ${\bold f} $ and ${\bold f} '$ are $H^\bullet({\Bbb{P}}^1,{\Bbb Q})$-module isomorphisms; \item[3.] ${\bold f} (\mu)=-\hat w$, where $\hat w$ is the fundamental class of ${\widehat X}$; \item[4.] ${\bold f} (H)=1+\hat w$; \item[5.] ${\bold f} (1)=-\hat\mu-\Theta+\hat w$, where $\hat \mu$ is the divisor given by the fibres of $\hat p\colon {\widehat X}\to{\Bbb{P}}^1$, and $\Theta=\hat e({\Bbb{P}}^1)$. \item[6.] $\beta\cdot {\bold f} (\alpha)= -{\bold f}'(\beta)\cdot\alpha$ for $\alpha\in H^\bullet_p(X,{\Bbb Q})$, $\beta\in H^\bullet_{\hat p}({\widehat X},{\Bbb Q})$. \item[7.] ${\bold f} $ establishes an isometry between $H^\bullet_p(X,{\Bbb Q})$ and $H^\bullet_{\hat p}({\widehat X},{\Bbb Q})$. \end{list} \label{newlemma}\end{lemma} \begin{pf} Property 1 is proved as in \cite{Muk4}, p.~382, provided that suitable adaptations to the relative case are done. One also proves that ${\bold f} '\circ {\bold f} (\alpha)=-\alpha$, so that 2 follows. To prove 3, let ${\cal L}$ be a flat line bundle on a smooth fibre $X_t$ of $p$. One knows that $\o{ch} i_{t\ast}({\cal L})=i_{t\ast}(1)=\mu$ since the normal bundle to $X_t$ is trivial. By Corollary \ref{cor2} we have $$\bS ^0(i_{t\ast}{\cal L})=0,\qquad\bS ^1(i_{t\ast}{\cal L})= k([{\cal L}^\ast])$$ where $[{\cal L}]\in{\widehat X}_t$ is the isomorphism class of ${\cal L}$. By Riemann-Roch we get $-\hat w = {\bold f} (\mu)$. (This implies ${\bold f} '(\hat\mu)=w$; after swapping $X$ and ${\widehat X}$ we get ${\bold f} '(\hat\mu)=-w$ which implies ${\bold f} (w)=\hat\mu$.) To prove 4 we apply Riemann-Roch to $$\bS ^0({\cal O}_H)={\cal O}_{{\widehat X}},\qquad \bS ^1({\cal O}_H)=0\,. $$ 5 is now straightforward. Using these results one proves 6 as in \cite{Muk4}. 7 follows from 1 and 6. \end{pf} If one defines a modified, $H^\bullet({\Bbb{P}}^1,{\Bbb Q})$-valued Mukai pairing by letting $\alpha\cdot \alpha'= p_\ast(\alpha^\ast \cup\alpha')$ then the map ${\bold f} $ establishes an isometry $H^\bullet(X,{\Bbb Q})\iso H^\bullet({\widehat X},{\Bbb Q})$ as $H^\bullet({\Bbb{P}}^1,{\Bbb Q})$-modules. \begin{prop} For all $\alpha\in H^\bullet(X,{\Bbb Q})$, the $H^0({\widehat X},{\Bbb Q})$-component of ${\bold f} (\alpha)$ is $\mu\cdot \alpha$. As a consequence, ${\bold f} $ induces an isometry $\tilde {\bold f} \colon \mu^\perp/{\Bbb Q}\mu\to H^2({\widehat X},{\Bbb Q})$. \label{ftilde}\end{prop} \begin{pf} We already know that ${\bold f} (w)^0=0$ and ${\bold f} (1)^0=0$, so we may assume $\alpha\in H^2(X,{\Bbb Q})$. Then, ${\bold f} (\alpha)^0=\pi^\ast\alpha\,\backslash\,\mu=\alpha\cdot\mu$. Thus ${\bold f} (\alpha)^0=0$ for $\alpha\in\mu^\perp$. We now define $\bar {\bold f} \colon \mu^\perp\to H^2({\widehat X},{\Bbb Q})$ by taking $\bar {\bold f} (\alpha)$ as the $H^2$-component of ${\bold f} (\alpha)$. One has that $\bar {\bold f} (\alpha)=0$ if and only if ${\bold f} (\alpha)=s\hat w$ ($s\in{\Bbb Q}$), and then $\alpha=-s\hat\mu$, which proves that $\ker\bar {\bold f} ={\Bbb Q}\mu$, and $\bar {\bold f} $ induces an injective morphism $\tilde {\bold f} \colon \mu^\perp/{\Bbb Q}\mu\hookrightarrow H^2({\widehat X},{\Bbb Q})$. If $\beta\in H^2({\widehat X},{\Bbb Q})$, ${\bold f} '(\beta)\cdot \mu=0$, and $\beta=\tilde {\bold f} (-{\bold f} '(\beta))$, thus finishing the proof. \end{pf} \begin{remark} The cohomology lattice $H^\bullet(X,{\Bbb Z})$ contains a hyperbolic sublattice $U$ generated by $\mu$ and $H$, and the hyperbolic sublattice $V=H^0(X,{\Bbb Z})\oplus H^4(X,{\Bbb Z})$. {From} Proposition \ref{ftilde} we see that (after identifying $X$ and ${\widehat X}$) the map ${\bold f}$ swaps the lattices $U$ and $V$. \end{remark} \smallskip {\it Topological invariants of the Fourier-Mukai transform.} Let us assume at first that the Picard number of $X$ is two; then the Picard group of $X$ reduces to the hyperbolic lattice $U$ (this happens when $X$ has 24 singular fibres consisting in elliptic curves with a nodal singularity). It is then possible to compute the invariants of the Fourier-Mukai transform of a sheaf on $X$ by means of the Riemann-Roch formula, expressed in the form $$\o{ch}{\bold T}}\def\bS{{\bold S} ^\bullet({\cal F})=\frac1{\sqrt{\td \hat p}}{\bold f} ((\o{ch}{\cal F})\sqrt{\td p})\,.$$ In particular, let us assume that ${\cal F}$ is WIT$_i$, and set $\widehat {\cal F}={\bold T}}\def\bS{{\bold S} ^i({\cal F})$, and $$\o{ch}{\cal F}=r+a\,H+b\,\mu+c\,w\qquad\text{where}\quad r=\o{rk}{\cal F}\,.$$ We then have \begin{equation} (-1)^i\,\o{rk}\widehat{\cal F}=a,\qquad (-1)^i\,c_1(\widehat{\cal F})=-r\,\Theta+c\,\hat\mu,\qquad (-1)^i\, \o{ch}_2(\widehat{\cal F})=-b\,\hat w\,.\label{e:RR1} \end{equation} In the same way, if ${\cal E}$ is a WIT$_i$ sheaf on ${\widehat X}$, with $$\o{ch}{\cal E}=r+a\,\Theta+b\,\hat\mu+c\, \hat w$$ after setting $\widehat {\cal E} = \widehat{{\bold T}}\def\bS{{\bold S}}^i{\cal E}$ we have \begin{equation} (-1)^i\,\o{rk}\widehat{\cal E}=a, \qquad (-1)^i\,c_1(\widehat{\cal E})=-r\,H+c\,\mu,\qquad (-1)^i\, \o{ch}_2(\widehat{\cal E})=-b\,w\,.\label{e:RR2} \end{equation} One obtains similar formulae also in the case when the Picard group has higher rank; in the Appendix we treat the case when $X$ has also singular fibres of type $I_n$ (according to Kodaira's classification \cite{Kod}). \par\vbox to4mm{}\par\section{Fourier-Mukai functor as mirror symmetry} We would like now to examine some facts which pinpoint the relations between the relative Fourier-Mukai transform on elliptic K3 surfaces and mirror symmetry. \smallskip (a) The formulae (\ref{e:RR1}) and (\ref{e:RR2}) establish a morphism \begin{equation} \begin{align*} H^0(X,{\Bbb Z})\oplus \o{Pic}(X) \oplus H^4(X,{\Bbb Z}) & \to H^0({\widehat X},{\Bbb Z}) \oplus \o{Pic}({\widehat X}) \oplus H^4({\widehat X},{\Bbb Z}) \\ r + a\,H + b\,\mu + c\,w & \mapsto a - r\,\Theta + c\,\hat\mu -b\, \hat w \end{align*}\end{equation} together with its inverse. According to these formulae, the cycle corresponding to a 0-brane is mapped to a special Lagrangian 2-cycle of genus 1 (i.e.~to the cycle homologous to $\hat\mu$), and {\it vice versa,} while a 4-brane is mapped to a special Lagrangian 2-cycle of genus 0, and {\it vice versa.} So one recovers the transformation properties of D-branes under T-duality as known from string theory \cite{OOY}. One should notice that, according to Corollary \ref{cor2}, a fibre of $X$, regarded as supersymmetric 2-cycle, is mapped to 0-brane (point) lying in the same fibre, thus giving rise to a relative (fibrewise) T-duality. \smallskip (b) Mirror symmetry should consist in the identification of the moduli space of complex structures on an $n$-dimensional Calabi-Yau manifold $X$ with the moduli space of ``complexified K\"ahler structures'' on the mirror manifold ${\widehat X}$. The tangent spaces to the two moduli spaces are the cohomology groups $H^{n-1,1}(X,{\Bbb C})$ and $H^{1,1}({\widehat X},{\Bbb C})$, respectively. We want to show that when $X$ is an (algebraic) elliptic K3 surface the ${\bold f} $ map establishes an isometry between the subspaces of these tangent spaces which describe ``algebraic deformations,'' in a sense that we shall clarify hereunder. We denote by $\phi$ the complexification of $\tilde {\bold f} $ and by $\psi\colon H^2 ({\widehat X},{\Bbb C})\to H^\bullet(X,{\Bbb C})$ its inverse. \begin{prop} The map $\psi$ establishes an isometry $$\frac{H^{1,1} ({\widehat X},{\Bbb C})}{\o{Pic}({\widehat X})\otimes{\Bbb C}} \iso\frac{H^{1,1}(X,{\Bbb C})}{\o{Pic}(X)\otimes{\Bbb C}}$$ \end{prop} \begin{pf} Let $\Omega$, $\bar\Omega$ be generators of $H^{2,0}({\widehat X},{\Bbb C})$ and $H^{0,2}({\widehat X},{\Bbb C})$. Since ${\widehat X}$ is a moduli space of sheaves on $X$, by Remark \ref{sympl} the classes $\psi(\Omega)$ and $\psi(\bar\Omega)$ lie in $H^{2,0}(X,{\Bbb C})$ and $H^{0,2}(X,{\Bbb C})$, respectively. The result then follows from Lemma \ref{newlemma} and Proposition \ref{ftilde}. \end{pf} The space ${H^{1,1}({\widehat X},{\Bbb C})}/{\o{Pic}({\widehat X}) \otimes{\Bbb C}}$ may be naturally identified with the tangent space at ${\widehat X}$ to the space of deformations of algebraic structures on ${\widehat X}$ which preserve the Picard lattice. Analogously, the space ${H^{1,1}(X,{\Bbb C})}$ can be regarded as the space of deformations of the K\"ahler structure of $X$, and its quotient ${H^{1,1}(X,{\Bbb C})}/{\o{Pic}(X)\otimes{\Bbb C}}$ as the space of deformations of the K\"ahler structure which preserve the Picard lattice. The map $\psi$ can then be thought of as a mirror transformation in the algebraic setting. Since the Weil-Petersson metrics on both spaces are expressed in terms of the Mukai pairing, which is preserved by $\psi$, we see that $\psi$ establishes an isometry between the tangent spaces to the two moduli spaces, consistently with the fact that the quantum cohomology of a K3 surface is trivial. \smallskip (c) The mass of a BPS state, which is represented by a D-brane wrapped around a 2-cycle $\gamma$, is given by the expression \cite{GK} $$M=\frac{\left\vert\int_\gamma\Omega\right\vert} {\left(\int_X\Omega\wedge\bar\Omega\right)^{\frac12}}= \frac{\left\vert\gamma\cdot[\Omega]\right\vert}{\left([\Omega] \cdot[\bar\Omega] \right)^{\frac12}}$$ where $\Omega$ denotes a holomorphic 2-form on $X$, and $[\Omega]$ its cohomology class in $H^{2,0}(X,{\Bbb C})$. The map $\phi$ evidently preserves this quantity. \smallskip As a final remark, we would like to mention \cite{HO}, where the authors consider a Fourier-Mukai transform on the cartesian product $X\times{\widehat X}$ given by the ideal sheaf of the diagonal and use it to define a T-duality between $X$ and ${\widehat X}$. A Riemann-Roch computation is then advocated to support an interpretation of the duality of the baryonic phases in $N=2$ super Yang-Mills theory. Thus, the geometric setting and the physical implications of this construction are different from those of the present paper. \smallskip {\it Conclusions.} It should be stressed that in this picture, in accordance with \cite{SYZ,Morr}, and differently to other proposals that have been recently advocated (cf.~e.g.~\cite{Asp,GW,G}), the mirror dual to a given elliptic K3 surface $X$ is isomorphic to $X$. Of course this does not imply that the mirror map is to be trivial, and indeed the Fourier-Mukai transform seems to establish such a map, at least at cohomological level, and at the ``infinitesimal'' level as far as the moduli spaces of complex structures and the moduli space of complexified K\"ahler structures are concerned. It would be now of some interest to develop a similar construction in terms of a generalized Fourier-Mukai transform in higher dimensional cases, where the mirror dual is not expected to be isomorphic to the original variety; however, one may conjecture that the derived categories of the two varieties are equivalent. \medskip\noindent {\bf Acknowledgements.} We thank C.~G\'omez, C.-S.~Chu, and especially C.~Imbimbo for useful discussions. This research was partly supported by the Spanish DGES through the research project PB95-0928, by the Italian Ministry for Universities and Research, and by an Italian-Spanish cooperation project. The first author thanks the Tata Institute for Fundamental Research, Bombay, for the very warm hospitality and for providing support during the final stage of preparation of this paper. \par\vbox to4mm{}\par\section{Appendix} In order to be able to compute the topological invariants of the Fourier-Mukai transform of a sheaf on $X$ we need to describe the action of the ${\bold f} $ map on the generators of the Picard group. In this Appendix we assume that the elliptic K3 surface $X$ has singular fibres which are of type $I_n$, $n \ge 3$ or are elliptic nodal curves; every singular fibre of type $I_n$ is a reducible curve whose irreducible components are $n$ smooth rational curves which intersect pairwise. The section $e$ intersects only one irreducible component of each singular fibre. The Picard group $\pic X$ is generated by the divisors $\mu$ and $H$ and by $r$ divisors $\alpha_1,\dots,\alpha_r$ given by the irreducible components $C_i$ of the singular fibres of type $I_n$ which do not meet the section $e$. Since $\alpha_i\cdot\mu=0$ and $\alpha_i\cdot H = {\bold f} (\alpha_i)\cdot(1+\hat w)=0$ we have ${\bold f} (\alpha_i)=\beta_i\in H^2({\widehat X},{\Bbb Q})$. \begin{prop} The sheaf ${\cal O}_X(-C_i)$ is WIT$_1$, and ${\bold T}}\def\bS{{\bold S}^1 \left[{\cal O}_X(-C_i)\right]\simeq {\cal O}_{\Sigma_i}(-1)$, where $\Sigma_i$ is a section of ${\widehat X}$ whose associated cohomology class is $\Theta+\hat\mu+\beta_i$. \end{prop} \begin{pf} By base change for every $t\in{\Bbb{P}}^1$ one has \begin{equation} {\bold T}}\def\bS{{\bold S}^i\left[{\cal O}_X(-C_i)\right]\otimes{\cal O}_{{\widehat X}_t}\simeq {\bold T}}\def\bS{{\bold S}^i_t({\cal L}_t)\label{e:rf} \end{equation} where ${\cal L}_t=\rest{{\cal O}_X(-C_i)},{X_t}$ so that ${\cal O}_X(-C_i)$ is WIT$_1$. By Riemann-Roch one has \begin{equation}\o{ch}{\bold T}}\def\bS{{\bold S}^1\left[{\cal O}_X(-C_i)\right]=\Theta+\hat\mu+\beta_i \label{e:LRR}\end{equation} {From} equation (\ref{e:rf}) we see that ${\bold T}}\def\bS{{\bold S}^i \left[{\cal O}_X(-C_i)\right] \otimes{\cal O}_{{\widehat X}_t}$ is concentrated at the point in ${\widehat X}_t$ corresponding to the flat line bundle ${\cal L}_t^\ast$ on $X_t$, whence the first claim follows. The second is a consequence of formula (\ref{e:LRR}). \end{pf} \par\vbox to4mm{}\par

9704

alg-geom/9704016

en

https://arxiv.org/abs/alg-geom/9704016

alg-geom/9704016

Geir Ellingsrud

Geir Ellingsrud, Manfred Lehn

On the irreducibility of the punctual Quotient Scheme of a Surface

10 pages, Latex2e

We prove that the Quot-scheme of finite quotients of a vector bundle which are of a given length and supported in one point, is irreducible and of the expected dimension.

alg-geom

\section{Elementary Modifications}\label{modific} Let $X$ be a smooth projective surface and $x\in X$. If $N$ is a coherent ${\cal O}_X$-sheaf, $e(N_x)=\hom_X(N,k(x))$ denotes the dimension of the fibre $N(x)$, which by Nakayama's Lemma is the same as the minimal number of generators of the stalk $N_x$. If $T$ is a coherent sheaf with zero-dimensional support, we denote by $i(T_x)=\hom_X(k(x),T)$ the dimension of the socle of $T_x$, i.e.\ the submodule ${\rm Soc}(T_x)\subset T_x$ of all elements that are annihilated by the maximal ideal in ${\cal O}_{X,x}$. \begin{lemma}\label{comparisonofiande}--- Let $[q:E\to T]\in{\rm Quot}(E,\ell)$ be a closed point and let $N$ be the kernel of $q$. Then the socle dimension of $T$ and the number of generators of $N$ at $x$ are related as follows: $$e(N_x)=i(T_x)+r.$$ \end{lemma} {\em Proof}. Write $e(N_x)=r+i$ for some integer $i\geq 0$. Then there is a minimal free resolution $0\lra{\cal O}_{X,x}^i\stackrel{\alpha}{\lra}{\cal O}_{X,x}^{r+i}\lra N_x\lra0$, where all coefficients of the homomorphism $\alpha$ are contained in the maximal ideal of ${\cal O}_{X,x}$. We have ${\rm Hom}(k(x),T_x)\cong{\rm Ext}_X^1(k(x),N_x)$ and applying the functor ${\rm Hom}(k(x),\,.\,)$ one finds an exact sequence $$0\lra{\rm Ext}_X^1(k(x),N_x)\lra{\rm Ext}_X^2(k(x),{\cal O}_{X,x}^i) \stackrel{\alpha'}{\lra}{\rm Ext}_X^2(k(x),{\cal O}_{X,x}^{r+i}).$$ But as $\alpha$ has coefficients in the maximal ideal, the homomorphism $\alpha'$ is zero. Thus ${\rm Hom}(k(x),T)\cong{\rm Ext}_X^2(k(x),{\cal O}_{X,x}^i)\cong k(x)^i$. \hspace*{\fill}$\Box$ The main technique for proving the theorem will be induction on the length of $T$. Let $N$ be the kernel of a surjection $E\to T$, let $x\in X$ be a closed point, and let $\lambda:N\to k(x)$ be any surjection. Define a quotient $E\to T'$ by means of the following push-out diagram: $$\begin{array}{ccccccccc} &&0&&0\\ &&\uparrow&&\uparrow\\ 0&\lra&k(x)&\stackrel{\mu}{\lra}&T'&\lra&T&\lra&0\\ &&{\scriptstyle\lambda}\uparrow{\phantom{\scriptstyle\lambda}}&&\uparrow&& \|\\ 0&\lra&N&\lra&E&\lra&T&\lra&0\\ &&\uparrow&&\uparrow&&\\ &&N'&=&N'\\ &&\uparrow&&\uparrow\\ &&0&&0 \end{array} $$ In this way every element $\langle\lambda\rangle\in\IP(N(x))$ determines a quotient $E\to T'$ together with an element $\langle\mu\rangle\in \IP({\rm Soc}(T'_x)\makebox[0mm]{}^{{\scriptstyle\vee}})$. (Here $W\makebox[0mm]{}^{{\scriptstyle\vee}}:={\rm Hom}_k(W,k)$ denotes the vector space dual to $W$.) Conversely, if $E\to T'$ is given, any such $\langle\mu\rangle$ determines $E\to T$ and a point $\langle\lambda\rangle$. We will refer to this situation by saying that $T'$ is obtained from $T$ by an elementary modification. We need to compare the invariants for $T$ and $T'$: Obviously, $\ell(T')= \ell(T)+1$. Applying the functor ${\rm Hom}(k(x),\,.\,)$ to the upper row in the diagram we get an exact sequence $$0\longrightarrow k(x)\longrightarrow{\rm Soc}(T'_x)\to{\rm Soc}(T_x)\longrightarrow {\rm Ext}_X^1(k(x),k(x))\cong k(x)^2,$$ and therefore $|i(T_x)-i(T'_x)|\leq 1$. Moreover, we have $0\leq e(T'_x)-e(T_x)\leq 1$. Two cases deserve more attention: \begin{lemma}\label{caseeincreases}--- Consider the natural homomorphisms $g:N(x)\to E(x)$ and $f:{\rm Soc}(T'_x)\to T'\to T'(x)$. The following assertions are equivalent \begin{enumerate} \item $e(T')=e(T)+1$ \item $\langle\mu\rangle\not\in\IP({\rm ker}(f)\makebox[0mm]{}^{{\scriptstyle\vee}})$ \item $\langle\lambda\rangle\in\IP({\rm im} (g))$. \end{enumerate} Moreover, if these conditions are satisfied, then $T'\cong T\oplus k(x)$ and $i(T'_x)=i(T_x)+1$. \end{lemma} {\em Proof}. Clearly, $e(T')=e(T)+1$ if and only if $\mu(1)$ represents a non-trivial element in $T'(x)$ if and only if $\mu$ has a left inverse if and only if $\lambda$ factors through $E$.\hspace*{\fill}$\Box$ \begin{lemma}\label{caseiincreases}--- Still keeping the notations above, let $E\to T'_{\lambda}$ be the modification of $E\to T$ determined by the point $\langle\lambda\rangle\in\IP(N(x))$. Similarly, for $\langle\mu'\rangle\in \IP({\rm Soc}(T_x)\makebox[0mm]{}^{{\scriptstyle\vee}})$ let $T^-_{\mu'}=T/\mu'(k(x))$. If $i(T'_{\lambda,x})=i(T_x)+1$ for all $\langle\lambda\rangle\in\IP(N(x))$, then $i(T_x)=i(T^-_{\mu',x})-1$ for all $\langle\mu'\rangle\in\IP({\rm Soc}(T_x)\makebox[0mm]{}^{{\scriptstyle\vee}})$ as well. \end{lemma} {\em Proof}. Let $\Phi:{\rm Hom}_X(N,k(x))\to{\rm Hom}_k({\rm Ext}_X^1(k(x),N),{\rm Ext}_X^1 (k(x),k(x)))$ be the homomorphism which is adjoint to the natural pairing $${\rm Hom}_X(N,k(x))\otimes{\rm Ext}_X^1(k(x),N)\to {\rm Ext}_X^1(k(x),k(x)).$$ Identifying ${\rm Soc}(T_x)\cong{\rm Ext}_X^1(k(x),N)$, we see that $i(T'_{\lambda,x})=1+i(T_x)-{\rm rank}(\Phi(\lambda))$. The action of $\Phi(\lambda)$ on a socle element $\mu':k(x)\to T$ can be described by the following diagram of pull-backs and push-forwards $$\begin{array}{ccccccccc} 0&\to&N&\to&E&\to&T&\to&0\\ &&\|&&\uparrow&&\phantom{{\scriptstyle\mu}}\uparrow{\scriptstyle\mu'}\\ 0&\to&N&\to&N^-_{\mu'}&\to&k(x)&\to&0\\ &&{\scriptstyle\lambda} \downarrow\phantom{{\scriptstyle\lambda}}&&\downarrow&&\|\\ 0&\to&k(x)&\to&\xi&\to&k(x)&\to&0 \end{array}$$ The assumption that $i(T'_{\lambda,x})=1+i(T_x)$ for all $\lambda$, is equivalent to $\Phi=0$. This implies that for every $\mu'$ and every $\lambda$ the extension in the third row splits, which in turn means that every $\lambda$ factors through $N^-_{\mu'}$, i.e.\ that $N(x)$ embeds into $N^-_{\mu'}(x)$. Hence, for $T^-_{\mu'}=E/N^-_{\mu'}={\rm coker}(\mu)$ we get $i(T^-_{\mu',x})=e(N^-_{\mu',x})-r=e(N_x)+1-r=i(T_x)+1$.\hspace*{\fill}$\Box$ \section{The Global Case}\label{globalcasesection} Let $Y_\ell={\rm Quot}(E,\ell)\times X$, and consider the universal exact sequence of sheaves on $Y_\ell$: $$\ses{{\cal N}}{{\cal O}_{{\rm Quot}}\otimes E}{{\cal T}}.$$ The function $y=(s,x)\mapsto i({\cal T}_{s,x})$ is upper semi-continuous. Let $Y_{\ell,i}$ denote the locally closed subset $\{y=(s,x)\in Y_\ell|i({\cal T}_{s,x})=i\}$ with the reduced subscheme structure. \begin{proposition}\label{globprop}--- $Y_\ell$ is irreducible of dimension $(r+1)\ell+2$. For each $i\geq 0$ one has ${\rm codim}(Y_{\ell,i},Y_\ell)\geq 2i$, \end{proposition} Clearly, the first assertion of the theorem follows from this. {\em Proof}. The proposition will be proved by induction on $\ell$, the case $\ell=1$ being trivial: $Y_1=\IP(E)\times X$, the stratum $Y_{1,1}$ is the graph of the projection $\IP(E)\to X$ and $Y_{1,i}=\emptyset$ for $i\geq 2$. Hence suppose the proposition has been proved for some $\ell\geq 1$. We describe the `global' version of the elementary modification discussed above. Let $Z=\IP({\cal N})$ be the projectivization of the family ${\cal N}$ and let $\varphi=(\varphi_1,\varphi_2):Z\to Y_\ell={\rm Quot}(E,\ell)\times X$ denote the natural projection morphism. On $Z\times X$ there is canonical epimorphism $$\Lambda:(\varphi_1\times{\rm id}_X)^*{\cal N}\to({\rm id}_Z,\varphi_2)_*\varphi^*{\cal N}\to ({\rm id}_Z,\varphi_2)_*{\cal O}_{Z}(1)=:{\cal K}.$$ As before we define a family ${\cal T}'$ of quotients of length $\ell+1$ by means of $\Lambda$: $$\begin{array}{ccccccccc} 0&\lra&{\cal K}&\lra&{\cal T}'&\lra&(\varphi_1,{\rm id}_X)^*{\cal T}&\lra&0\\ &&\Lambda\Big\uparrow\phantom{\Lambda}&&\Big\uparrow&&\Big\|\\ 0&\lra&(\varphi_1,{\rm id}_X)^*{\cal N}&\lra&{\cal O}_Z\otimes E&\lra&(\varphi_1,{\rm id}_X)^*{\cal T}& \lra&0 \end{array}$$ Let $\psi_1:Z\to {\rm Quot}(E,\ell+1)$ be the classifying morphism for the family ${\cal T}'$, and define $\psi:=(\psi_1,\psi_2:=\varphi_2):Z\to Y_{\ell+1}$. The scheme $Z$ together with the morphisms $\varphi:Z\to Y_\ell$ and $\psi: Z\to Y_{\ell+1}$ allows us to relate the strata $Y_{\ell,i}$ and $Y_{\ell+1,j}$. Note that $\psi(Z)=\bigcup_{j\geq1}Y_{\ell+1,j}$. The fibre of $\varphi$ over a point $(s,x)\in Y_{\ell,i}$ is given by $\IP({\cal N}_s(x))\cong\IP^{r-1+i}$, since $\dim({\cal N}_s(x))=r+i(T_{s,x})=r+i$ by Lemma \ref{comparisonofiande}. Similarly, the fibre of $\psi$ over a point $(s',x)\in Y_{\ell+1,j}$ is given by $\IP({\rm Soc}({\cal T}'_{s',x})\makebox[0mm]{}^{{\scriptstyle\vee}})\cong\IP^{j-1}$. If $T'$ is obtained from $T$ by an elementary modification, then $|i(T')-i(T)| \leq 1$ as shown above. This can be stated in terms of $\varphi$ and $\psi$ as follows: For each $j\geq 1$ one has: $$\psi^{-1}(Y_{\ell+1,j})\subset\bigcup_{|i-j|\leq1}\varphi^{-1}(Y_{\ell,i}).$$ Using the induction hypothesis on the dimension of $Y_{\ell,i}$ and the computation of the fibre dimension of $\varphi$ and $\psi$, we get $$\dim(Y_{\ell+1,j})+(j-1)\leq\max_{|i-j|\leq 1}\{(r+1)\ell+2-2i+(r-1+i)\}$$ and $$\dim(Y_{\ell+1,j})\leq(r+1)(\ell+1)+2-2j-\min_{|i-j|\leq 1}\{i-j+1\}.$$ As $\min_{|i-j|\leq 1}\{i-j+1\}\geq0$, this proves the dimension estimates of the proposition. It suffices to show that $Z$ is irreducible. Then ${\rm Quot}(E,\ell+1)=\psi_1(Z)$ and $Y_{\ell+1}$ are irreducible as well. Since $X$ is a smooth surface, the epimorphism ${\cal O}_{{\rm Quot}}\otimes E\to {\cal T}$ can be completed to a finite resolution $$0\longrightarrow{\cal A}\longrightarrow{\cal B}\longrightarrow{\cal O}_{{\rm Quot}}\otimes E \longrightarrow {\cal T}\longrightarrow0$$ with locally free sheaves ${\cal A}$ and ${\cal B}$ on $Y_{\ell}$ of rank $n$ and $n+r$, respectively, for some positive integer $n$. It follows that $Z=\IP({\cal N})\subset \IP({\cal B})$ is the vanishing locus of the composite homomorphism $\varphi^*{\cal A}\to\varphi^*B\to{\cal O}_{\IP({\cal B})}(1)$. In particular, assuming by induction that $Y_\ell$ is irreducible, $Z$ is locally cut out from an irreducible variety of dimension $(r+1)\ell+2+(r+n-1)$ by $n$ equations. Hence every irreducible component of $Z$ has dimension at least $(r+1)(\ell+1)$. But the dimension estimates for the stratum $Y_{\ell,i}$ and the fibres of $\varphi$ over it yield: $$\dim(\varphi^{-1}(Y_{\ell,i}))\leq (r+1)\ell+2-2i+(r+i-1)=(r+1)(\ell+1)-i,$$ which is strictly less than the dimension of any possible component of $Z$, if $i\geq 1$. This implies that the irreducible variety $\varphi^{-1}(Y_{\ell,0})$ is dense in $Z$. Moreover, since the fibre of $\psi$ over $Y_{\ell+1,1}$ is zero-dimensional, $\dim(Y_{\ell+1})=\dim(Y_{\ell+1,1})+2=\dim(Z)+2$ has the predicted value.\hspace*{\fill}$\Box$ \section{The Local Case}\label{localcasesection} We now concentrate on quotients $E\to T$, where $T$ has support in a single fixed closed point $x\in X$. For those quotients the structure of $E$ is of no importance, and we may assume that $E\cong{\cal O}^r_X$. Let $Q^r_\ell$ denote the closed subset $$\Big\{[{\cal O}^r_X\to T]\in{\rm Quot}({\cal O}^r_X,\ell)|\,\,{\rm Supp}(T)=\{x\}\Big\}$$ with the reduced subscheme structure. We may consider $Q^r_\ell$ as a subscheme of ${Y_{\ell,1}}$ by sending $[q]$ to $([q],x)$. Then it is easy to see that $\varphi^{-1}(Q^r_\ell)=\psi^{-1}(Q^r_{\ell+1})$. Let this scheme be denoted by $Z'$. We will use a stratification of $Q^r_\ell$ both by the socle dimension $i$ and the number of generators $e$ of $T$ and denote the corresponding locally closed subset by $Q^{r,e}_{\ell,i}$. Moreover, let $Q^{r}_{\ell,i}=\bigcup_{e}Q^{r,e}_{\ell,i}$ and define $Q^{r,e}_\ell$ similarly. Of course, $Q^{r,e}_{\ell,i}$ is empty unless $1\leq i\leq \ell$ and $1\leq e \leq \min\{r,\ell\}$. To prove the second half of the theorem it suffices to show: \begin{proposition}\label{localcase}--- $Q^r_\ell$ is an irreducible variety of dimension $r\ell-1$. \end{proposition} \begin{lemma}\label{dimensionestimatesforstrata}--- $\dim(Q^{r,e}_{\ell,i}) \leq(r\ell-1)-(2(i-1)+\binom{e}{2}).$ \end{lemma} {\em Proof}. By induction on $\ell$: if $\ell=1$, then $Q^r_1\cong\IP^{r-1}$, and $Q^{r,e}_{1,i}=\emptyset$ if $e\geq 2$ or $i\geq 2$. Assume that the lemma has been proved for some $\ell\geq 1$. Let $[q':{\cal O}_X^r\to T']\in Q^{r,e}_{\ell+1,j}$ be a closed point. Suppose that the map $\mu:k(x)\to T'(x)$ represents a point in $\psi^{-1}([q'])=\IP({\rm Soc}(T'_x)\makebox[0mm]{}^{{\scriptstyle\vee}})$ and that $T_\mu={\rm coker}(\mu)$ is the corresponding modification. If $i=i(T_{\mu,x})$ and $\varepsilon=e(T_{\mu,x})$, then, according to Section \ref{modific}, the pair $(i,\varepsilon)$ can take the following values: \beeq{fourpossibilities} (i,\varepsilon)=(j-1,e-1),\,(j-1,e),\,(j,e)\,\mbox{ or }\, (j+1,e), \end{eqnarray} in other words: $$\psi^{-1}(Q^{r,e}_{\ell+1,j})\subset \varphi^{-1}(Q^{r,e-1}_{\ell,j-1})\cup\bigcup_{|i-j|\leq1} \varphi^{-1}(Q^{r,e}_{\ell,i}).$$ Subdivide $A=Q^{r,e}_{\ell,j}$ into four locally closed subsets $A_{i,\varepsilon}$ according to the generic value of $(i,\varepsilon)$ on the fibres of $\psi$. Then $$\dim(A_{i,\varepsilon})+(j-1)\leq \dim(Q^{r,\varepsilon}_{\ell,i})+d_{i,\varepsilon},$$ where $d_{i,\varepsilon}$ is the fibre dimension of the morphism $$\varphi:\psi^{-1}(A_{i,\varepsilon})\cap\varphi^{-1}(Q^{r,\varepsilon}_{\ell,i}) \lra Q^{r,\varepsilon}_{\ell,i}.$$ By the induction hypothesis we have bounds for $\dim(Q^{r,\varepsilon}_{\ell,i})$, and we can bound $d_{i,\varepsilon}$ in the four cases (\ref{fourpossibilities}) as follows: A) If $[q:{\cal O}_X^r\to T]\in Q^{r,e-1}_{\ell,j-1}$ is a closed point with $N={\rm ker}(q)$, then according to Lemma \ref{caseeincreases} \begin{eqnarray*} \varphi^{-1}([q])\cap\psi^{-1}(A_{e-1,j-1})&\cong&\IP({\rm im}(g:N(x)\to k(x)^r))\\ &\cong&\IP({\rm ker}(k(x)^r\to T(x))\cong \IP^{r-e}, \end{eqnarray*} since ${\rm im}(k(x)^r\to T(x))\cong k^{e-1}$. Hence $d_{j-1,e-1}=r-e$ and \begin{eqnarray*} \dim(A_{j-1,e-1})&\leq&\dim Q^{r,e-1}_{\ell,j-1}+(r-e)-(j-1)\\ &\leq&\left\{(r\ell-1)-2(j-2)-\binom{e-1}{2}\right\}+(r-e)-(j-1)\\ &=&\left\{(r(\ell+1)-1)-2(j-1)-\binom{e}{2}\right\}-(j-2). \end{eqnarray*} Note that this case only occurs for $j\geq2$, so that $(j-2)$ is always nonnegative. B) In the three remaining cases $$\varepsilon=e\mbox{ and } i=j-1,\, j, \mbox{ or } j+1$$ we begin with the rough estimate $d_{i,e}\leq r+i-1$ as in Section \ref{globalcasesection}. This yields: \begin{eqnarray} \label{rechnung2} \dim(A_{i,e})&\leq&\left\{(r\ell-1)-2(i-1)-\binom{e}{2}\right\}+(r+i-1)-(j-1)\\ \label{rechnung3} &=&\left\{(r(\ell+1)-1)-2(j-1)-\binom{e}{2}\right\}-(i-j). \end{eqnarray} Thus, if $i=j$ we get exactly the estimate asserted in the Lemma, if $i=j+1$ the estimate is better than what we need by 1, but if $i=j-1$, the estimate is not good enough and fails by 1. It is this latter case that we must further study: let $[q:{\cal O}^r_X\to T]$ be a point in $Q^{r,e}_{\ell,j-1}$ with $N={\rm ker}(q)$. By Lemma \ref{caseiincreases} there are two possibilities: \begin{itemize} \item[---]{\em Either} the fibre $\varphi^{-1}([q])\cap\psi^{-1}(A_{j-1,e})$ is a {\em proper} closed subset of $\IP(N(x))$ which improves the estimate for the dimension of the fibre $\varphi^{-1}([q])$ by 1, \item[---]{\em or} this fibre {\em equals} with $\IP(N(x))$, in which case we have $i(T^-)=i(T)+1$ for every modification $T^-={\rm coker}(\mu^-:k(x)\to T)$. But, as we just saw, calculation (\ref{rechnung3}), applied to the contribution of $Q^{r,e}_{\ell-1,j}$ to $Q^{r,e}_{\ell,j-1}$, shows that the dimension estimate for the locus of such points $[q]$ in $Q^{r,e}_{\ell,j-1}$ can be improved by 1 compared to the dimension estimate for $Q^{r,e}_{\ell,j-1}$ as stated in the lemma. \end{itemize} Hence in either case we can improve estimate (\ref{rechnung3}) by 1 and get $$\dim(A_{j-1,e})\leq(r(\ell+1)-1)-2(j-1)-\binom{e}{2}$$ as required. Thus, the lemma holds for $\ell+1$.\hspace*{\fill}$\Box$ \begin{lemma}\label{intheclosureofelessthanr}--- $\psi(\varphi^{-1}(Q^{r,e}_{\ell}))\subset\overline{Q^{r,e}_{\ell+1}}$. \end{lemma} {\em Proof}. Let $[q:{\cal O}^r_X\to T]\in Q^{r,e}_{\ell,i}$ be a closed point with $N={\rm ker}(q)$. Then $\varphi^{-1}([q])=\IP(N(x))\cong\IP^{r+i-1}$ and $\varphi^{-1}([q])\cap\psi^{-1}(Q^{r,e+1}_{\ell+1})\cong \IP({\rm im} (G))\cong \IP^{r-e-1}$. Since we always have $e\geq 1,i\geq 1$, a dense open part of $\varphi^{-1}([q])$ is mapped to $Q^{r,e}_{\ell+1}$.\hspace*{\fill}$\Box$ \begin{lemma}\label{inductionstepforr}--- If $r\geq 2$ and if $Q^{r-1}_\ell$ is irreducible of dimension $(r-1)\ell-1$, then $Q^{r,<r}_{\ell}:=\bigcup_{e<r}Q^{r,e}_{\ell}$ is an irreducible open subset of $Q^{r}_\ell$ of dimension $r\ell-1$. \end{lemma} {\em Proof}. Let $M$ be the variety of all $r\times(r-1)$ matrices over $k$ of maximal rank, and let $\ses{{\cal O}_M^{r-1}}{{\cal O}_M^r}{{\cal L}}$ be the corresponding tautological sequence of locally free sheaves on $M$. Consider the open subset $U\subset M\times Q^{r}_\ell$ of points $(A,[{\cal O}^{r}\to T])$ such that the composite homomorphism $${\cal O}^{r-1}\stackrel{A}{\lra}{\cal O}^{r}\lra T$$ is surjective. Clearly, the image of $U$ under the projection to $Q^{r}_\ell$ is $Q^{r,<r}_\ell$. On the other hand, the tautological epimorphism $${\cal O}_{U\times X}^{r-1}\to{\cal O}_{U\times X}^{r}\to({\cal O}_M\otimes{\cal T})|_{U\times X}$$ induces a classifying morphism $g':U\to Q^{r-1}_\ell$. The morphism $g=(pr_1,g'):U\to M\times Q^{r-1}_\ell$ is surjective. In fact, it is an affine fibre bundle with fibre $$g^{-1}(g(A,[{\cal O}^{r-1}\to T]))\cong{\rm Hom}_k({\cal L}(A),T)\cong {\mathbb A}_k^\ell.$$ Since $Q^{r-1}_\ell$ is irreducible of dimension $(r-1)\ell-1$ by assumption, $U$ is irreducible of dimension $r\ell-1+\dim(M)$, and $Q^{r,<r}_\ell$ is irreducible of dimension $r\ell-1$.\hspace*{\fill}$\Box$ {\em Proof of Proposition \ref{localcase}.} The irreducibility of $Q^{r}_{\ell}$ will be proved by induction over $r$ and $\ell$: the case $(\ell=1,r \mbox{ arbitrary })$ is trivial; whereas $(\ell\mbox{ arbitrary },r=1)$ is the case of the Hilbert scheme, for which there exist several proofs (\cite{Briancon}, \cite{EllingsrudStromme}). Assume therefore that $r\geq 2$ and that the proposition holds for $(\ell,r)$ and $(\ell+1,r-1)$. We will show that it holds for $(\ell+1,r)$ as well. Recall that $Z':=\varphi^{-1}(Q^{r}_\ell)=Q^r_{\ell}\times_{Y_{\ell}}Z$. Every irreducible component of $Z'$ has dimension greater than or equal to $\dim(Q^{r}_\ell)+r-1=r(\ell+1)-2$ (compare Section \cite{globalcasesection}). On the other hand, $\dim(\varphi^{-1}(Q^r_{\ell,i}))\leq r\ell-1-2(i-1)+(r+i-1)=r(\ell+1)-i$. Thus an irreducible components of $Z'$ is either the closure of $\varphi^{-1}(Q^r_{\ell,1})$ (of dimension $r(\ell+1)-1)$) or the closure of $\varphi^{-1}(W)$ for an irreducible component $W\subset Q^r_{\ell,2}$ of maximal possible dimension $r\ell-3$. But according to Lemma \ref{intheclosureofelessthanr} the image of $\varphi^{-1}(W)$ under $\psi$ will be contained in the closure of $Q^{r,<r}_{\ell+1}$, unless $W$ is contained in $Q^{r,r}_{\ell,2}$. But Lemma \ref{dimensionestimatesforstrata} says that $Q^{r,r}_{\ell,2}$ has codimension $\geq 2+\binom{r}{2}\geq 3$ if $r\geq 2$, and hence cannot contain $W$ for dimension reasons. Hence any irreducible component of $Z'$ is mapped by $\psi$ into the closure of $Q^{r,<r}_{\ell+1}$ which is irreducible by Lemma \ref{inductionstepforr} and the induction hypothesis. This finishes the proof of the proposition.\hspace*{\fill}$\Box$

9704

alg-geom/9704014

en

https://arxiv.org/abs/alg-geom/9704014

alg-geom/9704014

Tom C. Braden

Tom C. Braden and Robert D. MacPherson

Intersection Homology of Toric Varieties and a Conjecture of Kalai

16 pages, 1 figure AMSLaTeX

We prove that the intersection homology Poincare' polynomial P(X) of an affine toric variety X is bounded below by the product P(Y)P(X/Y), where Y is the closure of any orbit in X and X/Y is a slice transverse to the orbit. This proves a combinatorial conjecture of Kalai for rational polytopes.

alg-geom

\section{$g$-numbers of polytopes} Let $P \subset {\mathbb R}^d$ be a $d$-dimensional convex polytope, i.e.\ the convex hull of a finite collection of points affinely spanning ${\mathbb R}^d$. The set of faces of $P$, ordered by inclusion, forms a poset which we will denote by ${\cal F}(P)$. We include the empty face $\emptyset = \emptyset_P$ and $P$ itself as members of ${\cal F}(P)$. It is a graded poset, with the grading given by the dimension of faces. By convention we set $\dim \emptyset = -1$. Faces of $P$ of dimension $0$, $1$, and $d-1$ will be referred to as vertices, edges, and facets, respectively. Given a face $F$ of $P$, the poset ${\cal F}(F)$ is clearly isomorphic to the interval $[\emptyset, F] \subset {\cal F}(P)$. The interval $[F, P]$ is the face poset of the polytope $P/F$ defined in the introduction. Given the polytope $P$, there are associated polynomials (first introduced in \cite{S1}) $g(P) = \sum g_i(P)q^i$ and $h(P) = \sum h_i(P)q^i$, defined recursively as follows: \begin{itemize} \item $g(\emptyset) = 1$ \item $h(P) = \Sigma_{\emptyset \le F < P} (q-1)^{\dim P - \dim F - 1} g(F)$, and \item $g_0(P) = h_0(P)$, $g_i(P) = h_i(P) - h_{i-1}(P)$ for $0<i\le \dim P/2$, and $g_i(P) = 0$ for all other $i$. \end{itemize} The coefficients of these polynomials will be referred to as the $g$-numbers and $h$-numbers of $P$, respectively. For our purposes, the $g$-polynomials will be of primary interest; the $h$-polynomials will not play a role here. These numbers clearly depend only on the poset ${\cal F}(P)$. In fact, as Bayer and Billera \cite{BB} showed, they depend only on the flag numbers of $P$: given a sequence of integers $I = (i_1, \dots, i_n)$ with $0 \le i_1 < i_2 < \dots < i_n \le d$, an $I$-flag is an $n$-tuple $F_1 < F_2 < \dots < F_n$ of faces of $P$ with $\dim F_k = i_k$ for all $k$. The $I$-th {\em flag number\/} $f^{}_I(P)$ is the number of $I$-flags. Letting $P$ vary over all polytopes of a given dimension $d$, the numbers $g_i(P)$ and $h_i(P)$ can be expressed as a ${\mathbb Z}$-linear combination of the $f^{}_I(P)$. Conjecturally all the $g_i(P)$ should be nonnegative for all $P$. This is known to be true for $i = 1, 2$ \cite{K}. For higher values of $i$, it can be proved for rational polytopes using the interpretation of $g_i(P)$ as an intersection homology Betti number of an associated toric variety. \begin{prop} \label{gnneg} If $P$ is a rational polytope, then $g_i(P) \ge 0$ for all $i$. \end{prop} \section{Relative $g$-polynomials} The following proposition defines a relative version of the classical $g$-polynomials. \begin{prop}\label{grdef} There is a unique family of polynomials $g(P,F)$ associated to a polytope $P$ and a face $F$ of $P$, satisfying the following relation: for all $P, F$, we have \begin{equation}\label{grdefeq} \sum_{F \le E \le P} g(E, F)g(P/E) = g(P).\end{equation} \end{prop} \begin{proof} The equation \eqref{grdefeq} can be used inductively to compute $g(P, F)$, since the left hand side gives $g(P, F)\cdot 1$ plus terms involving $g(E, F)$ where $\dim E< \dim P$. The induction starts when $P = F$, which gives $g(F, F) = g(F)$. Uniqueness is clear. \end{proof} As an example, if $F$ is a facet of $P$, then $g(P, F) = g(P) - g(F)$. Just as before we will denote the coefficient of $q^i$ in $g(P, F)$ by $g_i(P,F)$. We have the following notion of relative flag numbers. Let $P$ be a $d$-polytope, and $F$ a face of dimension $e$. Given a sequence of integers $I = (i_1, \dots, i_n)$ with $-1 \le i_1 < i_2 < \dots < i_n \le d$ and a number $1 \le k \le n$ with $i_k \ge e$, define the relative flag number $f^{}_{I,k}(P, F)$ to be the number of $I$-flags $(F_1, \dots,F_n)$ with $F \le F_k$. Note that letting $k = n$ and $i_n = d$ gives the ordinary flag numbers of $P$ as a special case. Also note that the numbers $f^{}_{I, k}$ where $i_k = e$ give all products of the form $f^{}_J(F)f^{}_{J'}(P/F)$. \begin{prop} Fixing $\dim P$ and $\dim F$, the relative $g$-number \\ $g_i(P, F)$ is a ${\mathbb Z}$-linear combination of the $f_{I, k}(P, F)$. \end{prop} \begin{proof} Use induction on $\dim P/F$. If $P = F$, then we have $g(P,P) = g(P)$ and the result is just the corresponding result for the ordinary flag numbers. If $P \ne F$, the equation \eqref{grdefeq} gives \[ g(P, F) = g(P) - \sum_{e = \dim F}^{\dim P - 1} \sum_{\substack{ \dim E = e \\ F\le E < P}} g(E, F)g(P/E).\] For every $e$ the coefficients of the inner summation on the right hand side are ${\mathbb Z}$-linear combinations of the $f_{I,k}(P, F)$, using the inductive hypothesis. \end{proof} The following theorem is the main result of this paper. It will be a consequence of Theorem \ref{grint}. \begin{thm} \label{grnneg} If $P$ is a rational polytope and $F$ is any face, then \\ $g_i(P, F) \ge 0$ for all $i$. \end{thm} \begin{cor}[Kalai's conjecture] If $P$ is a rational polytope and $F$ is any face, then \[g(P) \ge g(F)g(P/F),\] where the inequality is taken coefficient by coefficient. \end{cor} \begin{proof} For any face $E$ of $P$ the polytope $P/E$ is rational, so we have $g(P) = g(F, F)g(P/F) \mathop{+}$ other nonnegative terms. \end{proof} \section{Some examples and formulas} This section contains further combinatorial results on the relative $g$-polynomials. They are not used in the remainder of the paper. First, we give an interpretation of $g_1(P, F)$ and $g_2(P, F)$ analogous to the ones Kalai gave for the usual $g_1$ and $g_2$ in \cite{K}. We begin by recalling those results from \cite{K}. Give a finite set of points $V \subset {\mathbb R}^d$ define the space $\mathop{{\mathcal A}{\it ff}}(V)$ of affine dependencies of $V$ to be \[\{\, a \in {\mathbb R}^V \mid \Sigma_{v\in V} a_v = 0,\, \Sigma_{v\in V} a_v\cdot v = 0\,\}.\] If $V_P$ is the set of vertices of a polytope $P \subset {\mathbb R}^d$, then $\mathop{{\mathcal A}{\it ff}}(V_P)$ is a vector space of dimension $g_1(P)$. To describe $g_2(P)$ we need the notion of stress on a framework. A framework $\Phi = (V, E)$ is a finite collection $V$ of points in ${\mathbb R}^d$ together with a finite collection $E$ of straight line segments (edges) joining them. Given a finite set $S$, we denote the standard basis elements of ${\mathbb R}^S$ by $1_s, s\in S$. The space of stresses ${\cal S}(\Phi)$ is the kernel of the linear map \[\alpha\colon {\mathbb R}^E \to {\mathbb R}^V \otimes {\mathbb R}^d,\] defined by \[\alpha(1_e) = 1_{v_1}\otimes (v_1 - v_2) + 1_{v_2}\otimes (v_2 - v_1),\] where $v_1$ and $v_2$ are the endpoints of the edge $e$. A stress can be described physically as an assignment of a contracting or expanding force to each edge, such that the total force resulting at each vertex is zero. To a polytope $P$ we can associate a framework $\Phi_P$ by taking as vertices the vertices of $P$, and as edges the edges of $P$ together with enough extra edges to triangulate all the $2$-faces of $P$. Then $g_2(P)$ is the dimension of ${\cal S}(\Phi_P)$. Given a polytope $P$ and a face $F$, define the closed union of faces $N(P,F)$ to be the union of all facets of $P$ containing $F$. Note that $N(P,\emptyset) = \partial P$, and $N(P, P) = \emptyset$. Let $V_N$ be the set of vertices of $P$ in $N(P, F)$, and define a framework $\Phi_N$ by taking all edges and vertices of $\Phi_P$ contained in $N(P, F)$. \begin{thm} We have \[g_1(P, F) = \dim_{\mathbb R} \mathop{{\mathcal A}{\it ff}}(V_P)/ \mathop{{\mathcal A}{\it ff}}(V_N), \text{and}\] \[g_2(P, F) = \dim_{\mathbb R} {\cal S}(\Phi_P)/{\cal S}(\Phi_N),\] using the obvious inclusions of $\mathop{{\mathcal A}{\it ff}}(V_N)$ in $\mathop{{\mathcal A}{\it ff}}(V_P)$ and ${\cal S}(\Phi_N)$ in ${\cal S}(\Phi_P)$. \end{thm} The proof for $g_1$ is an easy exercise; the proof for $g_2$ will appear in a forthcoming paper \cite{BM}. Next, we have a formula which shows that $g(P, F)$ can be decomposed in the same way $g(P)$ was in Proposition \ref{grdef}. Given two faces $E, F$ of a polytope $P$, let $E \vee F$ be the unique smallest face containing both $E$ and $F$. \begin{prop} For any polytope $P$ and faces $F' \le F$ of $P$, we have \[g(P, F) = \sum_{F'\le E} g(E, F')g(P/E, (E\vee F)/E).\] \end{prop} \begin{proof} As usual, we show that this formula for $g(P, F)$ satisfies the defining relation of Proposition \ref{grdef}. Fix $F' \le F$, and define $\hat{g}(P, F)$ to be the above sum. Then we have \begin{eqnarray*} \sum_{F\le D} \hat{g}(D,F)g(P/D) & = & \sum_{\substack{F'\le E \\ F\vee E\le D}} g(P/D)g(E, F')g(D/E, (E\vee F)/E)\\ &=& \sum_{F'\le E} g(E, F')g(P/E)\\ & =& g(P). \end{eqnarray*} Since the computation of $g(P, F)$ from Proposition \ref{grdef} only involves computation of $g(E, F)$ for other faces $E$ of $P$, this proves that $\hat{g}(P,F) = g(P, F)$, as required. \end{proof} Finally, we can carry out the inversion implicit in Proposition \ref{grdef} explicitly. First we need the notion of polar polytopes. Given a polytope $P \subset {\mathbb R}^d$, we can assume that the origin lies in the interior of $P$ by moving $P$ by an affine motion. The polar polytope $P^*$ is defined by \[P^* = \{\,x \in ({\mathbb R}^*)^d \mid \langle x, y \rangle \le 1 \ \text{for all}\ y \in P\,\}.\] The face poset ${\cal F}(P^*)$ is canonically the opposite poset to ${\cal F}(P)$. Define ${\bar g}(P) = g(P^*)$. \begin{prop} We have \begin{equation}\label{eq1} g(P, F) = \sum_{F\le F' \le P} (-1)^{\dim P - \dim F'} g(F'){\bar g}(P/F').\end{equation} \end{prop} \begin{proof} We use the following formula, due to Stanley \cite{St}: For any polytope $P \ne \emptyset$, we have \begin{equation}\label{eq2} \sum_{\emptyset \le F \le P} (-1)^{\dim F} {\bar g}(F)g(P/F) = 0. \end{equation} Now define $\hat{g}(P, F)$ to be the right hand side of \eqref{eq1}. We will show that the defining property \eqref{grdefeq} of Proposition \ref{grdef} holds. Pick a face $F$ of $P$. We have, using \eqref{eq2}, \[ \sum_{F \le E \le P} g(E, F)g(P/E) = \sum_{F \le F' \le E \le P} (-1)^{\dim E - \dim F'}g(F'){\bar g}(E/F')g(P/E) \] \begin{eqnarray*} & = & \sum_{F\le F' \le P} g(F') \sum_{F' \le E \le P} (-1)^{\dim E - \dim F'} {\bar g}(E/F')g(P/E) \\ & = & g(P), \end{eqnarray*} as required. \end{proof} \section*{Introduction to \S\S 4 - 6} Sections 4-6 of this paper concern the topology of algebraic varieties. They may be read independently from the combinatorics of \S 1-3. The principal result, Theorem 16 of \S 5, is this: Consider a subvariety $Y$ of a complex algebraic variety $X$. Suppose that there is a blowup $p\colon\widetilde{X}\to X$ such that $p^{-1}Y$ has a neighborhood in $\widetilde{X}$ that is homeomorphic to a line bundle over $p^{-1}Y$. Then the restriction of the intersection homology sheaf ${\mathbf {IC^{\textstyle \cdot}}}(X)$ of $X$ to $Y$ is a direct sum of shifted intersection homology sheaves. For our applications, we need a slight strengthening of this result. The neighborhood of $p^{-1}Y$ will only be a Seifert bundle $E\to B$, a generalization of a line bundle which allows fibers to be quotients by cyclic groups. These are treated in \S 4. In \S 6 we apply the principal result to the inclusion of toric varieties $Y_F\subset X_P$, where $Y_F \simeq X_{P/F}$ is the closure of the torus orbit corresponding to $F$. If $x$ is the unique torus-fixed point of $X$, then $g_i(P,F)$ measures the number of copies of the intersection homology sheaf ${\mathbf {IC^{\textstyle \cdot}}}(\{x\})$ that appear with shift $2i$ in the restriction of the intersection homology sheaf of $X_P$ to $Y_F$. \section{Seifert bundles} In this section we investigate maps of algebraic varieties $E \to B$ which are nearly line bundles, but which allow fibers to be quotients by cyclic groups. \begin{defn} A {\em Seifert bundle \/} is an affine map $\pi\colon E \to B$ of algebraic varieties, together with a section (which we will sometimes call the zero section) $s\colon B \to E$, and an algebraic ${\mathbb C}^*$-action on $E$, so that: \begin{itemize} \item giving $B$ the trivial ${\mathbb C}^*$-action, $\pi$ and $s$ are ${\mathbb C}^*$-equivariant, \item each fiber $\pi^{-1}(b)$ is a curve whose normalization is isomorphic to the complex line, on which ${\mathbb C}^*$ acts by multiplication by a character $x \mapsto x^{n_b}$, $n_b>0$. \end{itemize} \end{defn} \begin{lemma} If $(E, B, \pi, s)$ is a Seifert bundle, and $n_b$ is a constant on all of $B$, then $E$ is topologically a complex line bundle over $B$. \end{lemma} \begin{proof} It is enough to show this locally, so assume $B$ is an affine variety, and take $b\in B$. Then $E$ is also affine, and the ${\mathbb C}^*$ action on $E$ induces an action, and hence a nonnegative grading, on the coordinate ring $A(E)$. Take a polynomial $f$ which doesn't vanish on $\pi^{-1}(b)$; we can assume it is homogeneous. Shrinking $B$ if necessary, we can assume $f$ doesn't vanish on any fiber of $\pi$. Let $Y \subset E$ be the subvariety defined by the equation $(f = 1)$. We will show that $\pi|_Y\colon Y \to B$ is proper. The lemma follows from this claim; if $\pi|_Y$ is proper, then the natural bijection $Y/G \to B$, where $G$ is the group of $d$th roots of unity, $d = \deg f$, is proper and hence a homeomorphism. Then, since the $G$-action is free, $Y \to B$ is a covering map. For a small (topological) neighborhood $U$ of $b$ we can thus define a continuous section $\sigma\colon B\to Y$. The map $(b, t) \mapsto t\cdot \sigma(b)$ gives the required local trivialization of $E$. To show the claim, take a compact set $K \subset B$. Choosing a homogeneous system $(f_1, \dots, f_s)$ of generators for $A = A(E)$ over its zeroth graded piece $A_0$ defines an embedding \[E \subset B \times {\mathbb C}^n\] as a closed subvariety, and the ${\mathbb C}^*$ action on $E$ is given by a linear ${\mathbb C}^*$ action on ${\mathbb C}^n$. Let $r>0$ be the smallest character of ${\mathbb C}^*$ appearing in a diagonalization of this action; it is the smallest of the degrees of the $f_i$. Then if $S^{2n-1}$ is the set of elements of norm one in ${\mathbb C}^n$, the set \[\pi^{-1}(K) \cap (B \times S^{2n-1})\] is compact, and so the values $|f|$ takes on it are bounded away from zero, say by $\delta$. Thus $\pi^{-1}(K) \cap Y$ is a closed subset of $K\times N_{1/\delta^r}$, where $N_a \subset {\mathbb C}^n$ is the closed ball of vectors of norm $\le a$, and so is compact. \end{proof} \begin{cor} \label{Sbcor} Any Seifert bundle $E$ over $B$ maps to a (topological) line bundle $E'$ over $B$ by a finite map. \end{cor} \begin{proof} We can take the least common multiple \[n = \mathop{\rm lcm}\limits_{b \in B} n_b\] of the numbers $n_b$, since there are only finitely many distinct values of $n_b$. Setting $E' = E/G$ where $G$ is the group of $n$th roots of unity in ${\mathbb C}^*$ does the trick. \end{proof} Intuitively, the zero-section map $s$ for a Seifert bundle will be a ``${\mathbb Q}$-homology normally nonsingular inclusion". We have the following generalization of a result of \cite{GM} about normally nonsingular inclusions: \begin{prop}\label{Sbprop} Let $(E, B, \pi, s)$ be a Seifert bundle. Then there is an isomorphism \[s^*{\mathbf {IC^{\textstyle \cdot}}}(E) \cong {\mathbf {IC^{\textstyle \cdot}}}(B).\] \end{prop} We need a small lemma first. \begin{lemma}\label{fgqIC} Let $X$ be a pseudomanifold, acted on by a finite group $G$, and let $Y$ be a $G$-invariant subspace. Then there is an isomorphism \[IH_*(X/G, Y/G; {\mathbb Q}) \cong IH_*(X, Y; {\mathbb Q})^G\] between the intersection homology of the pair $(X/G, Y/G)$ and the $G$-stable part of the intersection homology of $(X, Y)$. \end{lemma} \begin{proof} Give $X$ a $G$-invariant triangulation. Then the intersection homology of $X$ can be expressed by means of simplicial chains of the barycentric subdivision, see \cite[Appendix]{MV}. Now the standard argument in \cite[p. 120]{Br} can be applied. \end{proof} \begin{proof}[Proof of Proposition \ref{Sbprop}] By Corollary \ref{Sbcor}, we can map $E$ to a line bundle $E'$ by a finite map. Let $s'$, $\pi'$ denote the section and projection maps for $E'$. Let $\mathbf{A} = s^*{\mathbf {IC^{\textstyle \cdot}}}(E)$, $\mathbf{A'} = (s')^*{\mathbf {IC^{\textstyle \cdot}}}(E')$. Because $E'$ is a line bundle over $B$, $\mathbf{A'}$ is isomorphic to ${\mathbf {IC^{\textstyle \cdot}}}(B)$. Let $U \subset B$ be a Zariski open subset where $n_b$ is constant. Then $E|_U = \pi^{-1}(U)$ is a line bundle over $B$, so $F|_U$ is a one-dimensional constant local system. We will show that for any point $p \in B$ there are isomorphisms \[j_p^*\mathbf{A} \cong j_p^*\mathbf{A'},\ j_p^!\mathbf{A} \cong j_p^!\mathbf{A'}\] between the stalks and costalks of $\mathbf{A}$ and $\mathbf{A'}$ (or more precisely isomorphisms between their cohomology groups), where $j_p$ is the inclusion. It follows that $F$ satisfies the perversity axioms defining the intersection homology sheaf from \cite{GM}. To show the claim, note that since the ${\mathbb C}^*$ action retracts both $E$ and $E'$ onto $B$, we have isomorphisms \[\mathbf{A} \cong R\pi_*{\mathbf {IC^{\textstyle \cdot}}}(E), \ \mathbf{A'} \cong R\pi'_*{\mathbf {IC^{\textstyle \cdot}}}(E').\] So we can describe the stalks and costalks of $F$ and $F'$ as follows. Let $N$ be a small neighborhood of $p$ in $B$, and let $L$ be its boundary. Then we have \begin{eqnarray*} {\mathbb H}^i j_p^*\mathbf{A} & = & IH_{n-i}(\pi^{-1}(N), \pi^{-1}(L); {\mathbb Q}),\\ {\mathbb H}^i j^!_p\mathbf{A} & = & IH_{n-i}(\pi^{-1}(N); {\mathbb Q}), \end{eqnarray*} where $n$ is the real dimension of $B$, and similarly for $\mathbf{A'}$. The claim now follows from Lemma \ref{fgqIC}, using the fact that the finite group $G$ is contained in ${\mathbb C}^*$ and hence acts trivially on the intersection homology groups above. \end{proof} \section{Seifert resolutions} \begin{defn} A {\em Seifert resolution\/} of an inclusion $Y \subset X$ of irreducible algebraic varieties is a variety $\widetilde{X}$ together with a proper, surjective map $p\colon \widetilde{X}\to X$, so that, if $\widetilde{Y} = p^{-1}(Y)$, then $p$ induces an isomorphism of $\widetilde{X} \setminus \widetilde{Y}$ with $X \setminus Y$, and the inclusion $\widetilde{Y} \subset \widetilde{X}$ is the zero section of a Seifert bundle. \end{defn} Now suppose $X$ is a connected normal algebraic variety with a nontrivial algebraic ${\mathbb C}^*$ action. Let $Y$ be an irreducible subvariety contained in the fixed point set of $X$. Let $U = \{\, x\in X \mid \overline{{\mathbb C}^*\cdot x} \cap Y \ne \emptyset\,\}$. We say that $Y$ is an {\em attractor \/} for the ${\mathbb C}^*$ action if for all points $x\in U$ the limit $\lim_{t\to 0} t\cdot x$ exists and lies in $Y$ and the only points $x \in U$ for which $\lim_{t \to \infty} t\cdot x$ lies in $Y$ are already in $Y$. \begin{thm} \label{srex} If $U$ is an open neighborhood of $Y$ and $Y$ is an attractor, then the pair $(Y, X)$ has a Seifert resolution. \end{thm} \begin{proof} We will show that $(Y, U)$ has a Seifert resolution; this will be enough. By \cite{Su}, every point $y \in Y$ has a ${\mathbb C}^*$-invariant affine neighborhood $U_y \subset U$. Let $A_y$ be its coordinate ring. The ${\mathbb C}^*$ action induces a grading on $A_y$ which is nonnegative because $Y$ is an attractor. Further, if $R_y$ is the coordinate ring of $Y_y = U_y \cap Y$, the natural quotient map $A_y \to R_y$ identifies $R_y$ with the zeroth graded piece of $A_y$. Thus there is a projection map $\rho_y\colon U_y \to Y_y$; these glue to give $\rho\colon U \to Y$. Furthermore, the varieties and maps $\mathop{\rm Proj}(A_y) \to Y_y$ glue to give a variety $\widetilde{Y}$ and a proper map $q\colon \widetilde{Y}\to Y$ (in other words, we let $\widetilde{Y} = (U \setminus Y)/{\mathbb C}^*$). We also have a map $k\colon U\setminus Y \to \widetilde{Y}$ satisfying $q\circ k = \rho|_{U\setminus Y}$. Define a morphism $U\setminus Y \to U\times_Y \widetilde{Y}$ by sending $x$ to $(x, k(x))$. Let $\widetilde{U}$ be the closure of the image of this map, and let $p\colon \widetilde{U}\to U$ and $\pi\colon \widetilde{U}\to \widetilde{Y}$ be the restrictions of the projections of $U\times_Y \widetilde{Y}$ on the first and second factor, respectively. $\widetilde{U}$ will be the required Seifert resolution. Note that $p^{-1}(Y) = Y \times_Y \widetilde{Y} \cong \widetilde{Y}$. The map $p$ is proper, because the projection \[U\times_Y \widetilde{Y} \to U \cong U \times_Y Y\] is proper. It is now easy to check that $\widetilde{U}$ is a Seifert bundle over $\widetilde{Y}$. \end{proof} \begin{defn} Call an object $\mathbf{A}$ in $D^b(X)$ {\em pure}\/ if it is a direct sum of shifted intersection homology sheaves \begin{equation} \label{pureeq}\bigoplus_\alpha {\mathbf {IC^{\textstyle \cdot}}}(Z_\alpha; {\cal L}_\alpha)[n_\alpha], \end{equation} where each $Z_\alpha$ is an irreducible subvariety of $X$, ${\cal L}_\alpha$ is a simple local system on a Zariski open subset $U_\alpha$ of the smooth locus of $Z_\alpha$, and $n_\alpha$ is an integer. \end{defn} \begin{lemma}\label{KSps}If $\mathbf{A, B}$ are objects in $D^b(X)$ and $\mathbf{A} \oplus \mathbf{B}$ is pure, then so is $\mathbf{A}$. \end{lemma} \begin{proof} Denote $\mathbf{A} \oplus \mathbf{B}$ by $\mathbf{C}$. Since $\mathbf{C}$ is pure, it is isomorphic to the direct sum \[\bigoplus_{i \in {\mathbb Z}} {}^p\!H^i(\mathbf{C})[-i]\] of its perverse homology sheaves. Each ${}^p\!H^i(\mathbf{C}) = {}^p\!H^i(\mathbf{A}) \oplus {}^p\!H^i(\mathbf{B})$ is a pure perverse sheaf, and since the category of perverse sheaves is abelian, ${}^p\!H^i(\mathbf{A})$ is pure. Then the composition \[\bigoplus {}^p\!H^i(\mathbf{A})[-i] \to \bigoplus {}^p\!H^i(\mathbf{C})[-i] \cong \mathbf{C} \to \mathbf{A}\] induces an isomorphism on all the perverse homology sheaves, and hence is an isomorphism (see \cite{BBD}). \end{proof} Also note that the decomposition \eqref{pureeq} of a pure object $\mathbf{A}$ is essentially unique: any other such decomposition will be the same up to a reordering of the terms and replacing the local system ${\cal L}_\alpha$ by another local system ${\cal L}'_\alpha$ on $U'_\alpha$, so that ${\cal L}_\alpha$ and ${\cal L}'_\alpha$ agree on $U_\alpha \cap U'_\alpha$. \begin{thm} \label{Srmain} If a pair of varieties $(Y, X)$ has a Seifert resolution, then the pullback $j^*{\mathbf {IC^{\textstyle \cdot}}}(X)$ of the intersection homology sheaf by the inclusion is a pure object in $D^b(Y)$. \end{thm} \begin{proof} Consider the fiber square \[\begin{CD} \widetilde{Y} @>{\tilde{\jmath}}>> \widetilde{X}\\ @VVqV @VVpV\\ Y @>j>> X\\ \end{CD}\] where $j, \tilde{\jmath}$ are the inclusions, and $q = p|_{\widetilde{Y}}$. Because $p$ and $q$ are proper we have \[Rq_*\tilde{\jmath}^*{\mathbf {IC^{\textstyle \cdot}}}(\widetilde{X}) \cong j^*Rp_*{\mathbf {IC^{\textstyle \cdot}}}(\widetilde{X}).\] The left hand side is $Rq_*{\mathbf {IC^{\textstyle \cdot}}}(\widetilde{Y})$ by Proposition \ref{Sbprop}, which is pure by the decomposition theorem of \cite{BBD}. The decomposition theorem also implies that $\mathbf{A}=Rp_*{\mathbf {IC^{\textstyle \cdot}}}(\widetilde{X})$ is pure, and because $\widetilde{X} \to X$ is an isomorphism on a Zariski dense subset, the intersection homology sheaf of $X$ must occur in $\mathbf{A}$ with zero shift. Thus the right hand side becomes \[j^*({\mathbf {IC^{\textstyle \cdot}}}(X)) \oplus j^*\mathbf{A}',\] where $\mathbf{A}'$ is pure. The result now follows from Lemma \ref{KSps}. \end{proof} \section{Toric varieties} We will only sketch the properties of toric varieties that we will need. For a more complete presentation, see \cite{F}. Throughout this section let $P$ be a $d$-dimensional rational polytope in ${\mathbb R}^d$. Define a toric variety $X_P$ as follows. Embed ${\mathbb R}^d$ into ${\mathbb R}^{d+1}$ by \[(x_1, \dots, x_d) \mapsto (x_1, \dots, x_d, 1),\] and let $\sigma = \sigma^{}_P$ be the cone over the image of $P$ with apex at the origin in ${\mathbb R}^{d+1}$. It is a rational polyhedral cone with respect to the standard lattice $N = {\mathbb Z}^{d+1}$. More generally, if $F$ is a face of $P$, let $\sigma^{}_F$ be the cone over the image of $F$; set $\sigma^{}_\emptyset = \{0\}$. Then define $X = X_P$ to be the affine toric variety $X_\sigma$ corresponding to $\sigma$. It is the variety $\mathop{\rm Spec} {\mathbb C}[M \cap \sigma^\vee]$, where \[\sigma^\vee = \{ x \in ({\mathbb R}^{d+1})^* \mid \langle x, y \rangle \ge 0 \quad\text{for all}\quad y\in \sigma\,\}\] is the dual cone to $\sigma$, $M$ is the dual lattice to $N$, and ${\mathbb C}[M \cap \sigma^\vee]$ if the semigroup algebra of $M \cap \sigma^\vee$. It is a $(d+1)$-dimensional normal affine algebraic variety, on which the torus $T = \mathop{\rm Hom}(M, {\mathbb C}^*)$ acts. Let $f_v\colon X_P \to {\mathbb C}$ be the regular function corresponding to the point $v \in M \cap \sigma^\vee$. The orbits of the action of $T$ on $X$ are parametrized by the faces of $P$. Let $F$ be any face of $P$, including the empty face, and let \[\sigma^\bot_F = \{\, x\in \sigma^\vee \mid \langle x, y\rangle = 0 \quad\text{for all}\quad y\in \sigma^{}_F\,\}\] be the face of $\sigma^\vee$ dual to $\sigma^{}_F$. Then the variety \[O_F := \{\,x\in X \mid f_v(x) \ne 0 \iff v\in M\cap \sigma^\bot_F\,\}\] is an orbit, isomorphic to the torus $({\mathbb C}^*)^{d - e}$, where $e = \dim F$. Furthermore, all $T$-orbits arise this way. Thus $X_P$ has a unique $T$-fixed point $\{p\} = O_P$. Given a face $F$, the union \[ U_F = \bigcup_{E\le F} O_{E}\] is a $T$-invariant open neighborhood of $O_F$. There is a non-canonical isomorphism $U_F \cong O_F \times X_F$ where $X_F$ is the affine toric variety corresponding to $F$, considered as a polytope in the affine space spanned by $F$, with the lattice given by its intersection with $N$. If $O'_E$ denotes the orbit of $X_F$ corresponding to a face $E \le F$, then $O_E$ sits in $U_F \cong O_F \times X_F$ as $O_F \times O'_E$. The union \[Y_F = \bigcup_{F\le E} O_{E}\] is the closure $\overline{O_F}$. It is isomorphic to the affine toric variety $X_{P/F}$. More precisely, $Y_F$ is the affine toric variety corresponding to the cone $\tau = \sigma/\sigma_F$, the image of $\sigma$ projected into ${\mathbb R}^{d+1}/\mathop{\rm span}\sigma^{}_F$, with the lattice given by the image of $N$. It is an easy exercise to show that $\tau$ is the cone over a polytope of the type $P/F$. The connection between toric varieties and $g$-numbers of polytopes is given by the following result. Proofs appear in \cite{DL,Fie}. \begin{prop} \label{BKM}The local intersection homology groups of $X_P$ are described as follows. Let $x$ be a point in $O_F$, and let $j_x$ be the inclusion. Then \[\dim {\mathbb H}^{2i}j^*_x{\mathbf {IC^{\textstyle \cdot}}}(X_P) = g_i(F),\] and ${\mathbb H}^kj_x^*{\mathbf {IC^{\textstyle \cdot}}}(X_P)$ vanishes for odd $k$. \end{prop} Now fix a face $F$ of $P$. \begin{lemma} There exists a ${\mathbb C}^*$ action coming from a one-dimensional subtorus of $T$ so that the fixed point set is $Y_F = \overline{O_F}$ and for any $x \in X_P$, \[\lim_{t\to 0} t\cdot x \in Y_F.\] \end{lemma} \begin{proof} Let $a \in N\cap \sigma$ be a lattice point in the relative interior of $\sigma^{}_F$. This defines a ${\mathbb C}^*$ action on $X_P$ by letting, for all $t\in {\mathbb C}^*$, $x\in X_P$, and $v \in M\cap \sigma^\vee$, \[f_v(t \cdot x) = t^{\langle a, v\rangle}f_v(x).\] The required property of this action is clear. \end{proof} Thus we can apply Theorem \ref{srex} to obtain a Seifert resolution $\widetilde{Y}$ of the pair $(Y_F, X_P)$. Although we will not need this, a description of $\widetilde{Y}$ is quite interesting. Let $\Delta(a)$ be the fan obtained by coning off all the faces of $\sigma$ to the one-dimensional cone $\tau$ containing $a$. Then $\widetilde{Y}$ is the toric variety $X_{\Delta(a)}$. So by Theorem \ref{Srmain}, if $j\colon Y_F \to X_P$ is the inclusion, the pullback $\mathbf{A} = j^*{\mathbf {IC^{\textstyle \cdot}}}(X_P)$ is a direct sum \[\bigoplus_\alpha {\mathbf {IC^{\textstyle \cdot}}}(Z_\alpha, {\cal L}_\alpha)[n_\alpha]\] of shifted simple intersection homology sheaves. \begin{lemma} All the terms in this decomposition are of the form \[{\mathbf {IC^{\textstyle \cdot}}}(Y_{E}, {\mathbb Q}_{O_{E}})[n]\] where $E \ge F$ and $-n$ is a nonnegative even integer. \end{lemma} \begin{proof} Since the sheaf ${\mathbf {IC^{\textstyle \cdot}}}(X_P)$ is invariant under the action of $T$, so is the pullback $\mathbf{A}$; it follows that all the varieties $Z_\alpha$ are $T$-invariant. Second, the isomorphism $U_F \cong O_F \times X_F$ implies that the homology sheaves of ${\mathbf {IC^{\textstyle \cdot}}}(X_P)$, and hence of $\mathbf{A}$, are constant on each orbit; thus no nonconstant local systems can occur. Finally, the assertion about the shifts follows from Proposition \ref{BKM}. \end{proof} Thus we can write \begin{equation} \label{Asplits} \mathbf{A} = \bigoplus_{E\ge F} \bigoplus_{i\ge 0}{\mathbf {IC^{\textstyle \cdot}}}(Y_{E}; {\mathbb Q})[-2i] \otimes V^i_{E}, \end{equation} for some finite dimensional ${\mathbb Q}$-vector spaces $V^i_{E}$. Now we come to the main result, which gives an interpretation of the combinatorially defined polynomials $g(P, F)$ for rational polytopes which implies nonnegativity, and hence Theorem \ref{grnneg}. Let $\{p\} = O_P$ be the unique $T$-fixed point of $X_P$. \begin{thm} \label{grint} The relative $g$-number $g_i(P,F)$ is given by \[g_i(P, F) = \dim^{}_{\mathbb Q} V^i_P.\] \end{thm} \begin{proof} Taking this for the moment as a definition of $g(P,F)$, we will show that the defining relation of Proposition \ref{grdef} holds. First we need to interpret the vector spaces $V^i_E$ for $F\le E \ne P$. There is a commutative diagram of inclusions \[\begin{CD} Y'_F @>j'>> X_E \\ @Vk'VV @VVkV \\ Y_F @>j>> X_P \end{CD}\] where $j'$ be the inclusion of $Y'_F = \overline{O'_F}$ in $X_E$, $k$ is the inclusion of $X_E$ in $U_E \cong O_E \times X_E$ as $\{x\} \times X_E$, and $k'$ is the restriction of $k$. Then $k$ is a normally nonsingular inclusion, so we have \[(j')^*k^*{\mathbf {IC^{\textstyle \cdot}}}(X_P) = (j')^*{\mathbf {IC^{\textstyle \cdot}}}(X_{E}) =\] \[\bigoplus_{F\le F'\le E} \bigoplus_{i\ge 0}{\mathbf {IC^{\textstyle \cdot}}}(Y'_{F'}; {\mathbb Q})[-2i] \otimes W^i_{F'}\] for some vector spaces $W^i_{F'}$. On the other hand, since $k'$ is a normally nonsingular inclusion, it is also equal to \[(k')^*\mathbf{A} = \bigoplus_{F\le F'\le E} \bigoplus_{i\ge 0}{\mathbf {IC^{\textstyle \cdot}}}(Y'_{F'}; {\mathbb Q})[-2i] \otimes V^i_{F'}.\] Comparing terms, we see that $W^i_{F'} \cong V^i_{F'}$, so we have \[\dim^{}_{\mathbb Q} V^i_E = g_i(E, F).\] The theorem now follows; the defining relation of Proposition \ref{grdef} expresses two different ways of writing the dimensions of the stalk intersection homology groups of $X_P$ at the fixed point $p$. One the one hand, they are given by the coefficients of $g(P)$, by Proposition \ref{BKM}. On the other hand they are given by \[\sum_{F \le E \le P} g(E, F)g(P/E),\] using \eqref{Asplits}. \end{proof}

9704

alg-geom/9704012

en

https://arxiv.org/abs/alg-geom/9704012

alg-geom/9704012

Mikhail Zaidenberg

K. Oguiso, M. Zaidenberg

On fundamental groups of elliptically connected surfaces

Latex

A compact complex manifold $X$ is called elliptically connected if any pair of points in $X$ can be connected by a chain of elliptic or rational curves. We prove that the fundamental group of an elliptically connected compact complex surface is almost abelian. This confirms a conjecture which states that the fundamental group of an elliptically connected K\"ahler manifold must be almost abelian.

alg-geom

\section*{Introduction} We use below the following \bigskip \noindent {\bf 0.1. Definition.} Let $X$ be a compact complex space. We say that $X$ is {\it elliptically} (resp. {\it torically}) {\it connected} if any two points $x',\,x'' \in X$ can be joined by a finite chain of (possibly, singular) rational or elliptic curves (resp. of holomorphic images of complex tori). Here we discuss the following \bigskip \noindent {\bf 0.2. Conjecture} [Z]. {\it Let $X$ be a compact K\"ahler manifold. If $X$ is torically connected, then the fundamental group $\pi_1 (X)$ is almost abelian (or, for a weaker form, almost nilpotent).} \bigskip A group $G$ is called {\it almost abelian}\footnote{or {\it virtually abelian}, or {\it abelian-by-finite}.} (resp. {\it almost nilpotent, almost solvable, etc.}) if it contains an abelian (resp. nilpotent, solvable, etc.) subgroup of finite index. Obviously, each of these properties is stable under finite extensions. \medskip More generally, we may ask whether a K\"ahler variety connected by means of chains of subvarieties with almost abelian (resp. almost nilpotent, almost solvable, etc.) fundamental groups has itself such a fundamental group. \smallskip It is known that {a rationally connected} (i.e. connected by means of chains of rational curves) compact K\"ahler manifold is simply connected [KMM, Cam1 (3.5), Cam2 (5.7), Cam3 ($2.4'$), Cam4 (5.2.3)] (see also [Se]). Moreover, it follows from [Cam1 (2.2), Cam2 (5.2), (5.4); Cam4 (5.2.4.1)] that \smallskip \noindent {\it a) if a compact K\"ahler manifold $X$ is connected by means of chains of holomorphic images of simply connected varieties, then $\pi_1 (X)$ is a finite group; \smallskip \noindent b) if $X$ as above can be covered by holomorphic images of complex tori passing through a point $x_0 \in X$, then the group $\pi_1 (X)$ is almost abelian.} \smallskip This gives a motivation for the above conjecture. Another kind of motivation is provided by the following function--theoretic consideration. We introduce the next \bigskip \noindent {\bf 0.3. Definition.} We say that a complex space $X$ is {\it sub--Liouville} if its universal covering space $U_X$ is {\it Liouville}, i.e. if any bounded holomorphic function on $U_X$ is constant. \bigskip The complex tori yield examples of sub--Liouville compact manifolds. By a theorem of Lin [Li], any quasi--compact complex variety $X$ with an almost nilpotent fundamental group $\pi_1 (X)$ is a sub--Liouville one. It is easily seen that any complex space with countable topology, connected by means of chains of sub--Liouville subspaces, is itself sub--Liouville [DZ, (2.3)]. In particular, any torically connected variety is sub--Liouville. Thus, the question arises whether such a variety should also satisfy the assumption of Lin's Theorem, which is just Conjecture 0.2 in its weaker form. \bigskip \noindent {\bf 0.4.} Note that even in its weaker form the conjecture fails for non--K\"ahlerian compact complex manifolds. An example (communicated by J. Winkelmann\footnote{we are thankful to J. Winkelmann for a kind permission to mention it here.}) is a complex 3-fold which is a quotient of $SL_2 ( I \!\!\!\! C)$ by a discrete cocompact subgroup (for details see Appendix below). \bigskip In this note we consider the simplest case of complex surfaces. We prove the following \bigskip \noindent {\bf 0.5. Theorem.} {\it Let $S$ be a smooth compact complex surface. If $S$ is torically connected, then the group $\pi_1 (X)$ is almost abelian.} \bigskip \noindent {\bf 0.6.} The above conjecture can be also formulated for non--compact K\"ahler manifolds, in particular, for smooth quasi--projective varieties. To this point, in Definition 0.1 one should consider, instead of chains of rational or elliptic curves (resp. compact complex tori), the chains of non--hyperbolic quasi--projective curves\footnote{i.e. those with abelian fundamental groups} (resp. products of compact tori and factors $( I \!\!\!\! C^* )^m,\,m \in \Bbb N$). L. Haddak\footnote{unpublished} has checked that Theorem 0.5 holds true for smooth quasi--projective surfaces. The proof is based on the Fujita classification results for open surfaces [Fu]. \bigskip \section*{Proof of Theorem 0.5} \bigskip \noindent In the proof we use the following two lemmas. \bigskip \noindent {\bf 1.1. Lemma} [Fu, Thm. 2.12; No, Lemma 1.5.C]. {\it Let $X$ and $Y$ be connected compact complex manifolds, and let $f:\,X \to Y$ be a dominant holomorphic mapping. Then $f_* \pi_1 (X) \subset \pi_1 (Y)$ is a subgroup of finite index. In particular, if the group $\pi_1 (X)$ is almost abelian (resp. almost nilpotent, almost solvable), then so is $\pi_1 (Y)$.} \bigskip \noindent {\bf 1.2. Lemma.} {\it Every elliptically connected smooth compact complex surface $S$ is projective.} \bigskip \noindent {\it Proof.} If the algebraic dimension $a(S)$ were zero, then $S$ would have only a finite number of irreducible curves [BPV, IV.6.2] and hence, it would not be elliptically connected. In the case when $a(S) = 1$, $S$ is not elliptically connected, either. Indeed, such an $S$ is an elliptic surface [BPV, VI.4.1], and any irreducible curve on it is contained in a fibre of the elliptic fibration $\pi\,:\,S \to B$, where $B$ is a smooth curve (because, if an irreducible curve $E \subset S$ were not contained in a fibre of $\pi$, then one would have $E\cdot F > 0$, where $F$ is a generic fibre of $\pi$, and hence $(E + nF)^2 > 0$ for $n$ large enough, which would imply that $S$ is projective [BPV, IV.5.2], a contradiction). Thus, $a(S) = 2$, and therefore, $S$ is projective (see e.g. [BPV, IV.5.7]). \hfill $\Box$ \bigskip \noindent {\bf 1.3.} {\it Proof of Theorem 0.5.} Let $S$ be a smooth compact complex surface. Suppose that $S$ is torically connected. Then either $S$ itself is dominated by a complex torus, and then, by Lemma 1.1.$c)$, the group $\pi_1 (S)$ is almost abelian, or $S$ is elliptically connected. Consider the latter case. Due to the bimeromorphic invariance of the fundamental group, we may assume $S$ being minimal. $S$ being elliptically connected, by Lemma 1.2 it is a projective surface with a rational or elliptic curve passing through each point of $S$. Certainly, the Kodaira dimension $k(S) \le 1$. According to the possible values of $k(S)$, consider the following cases. \medskip \noindent a) Let $k(S) = -\infty$. Then $S$ is either a rational surface or a non--rational ruled surface over a curve $E$. In the first case, $S$ is simply connected. In the second one, $E$ should be an elliptic curve. Indeed, since $S$ is elliptically connected, $E$ is dominated by a rational or elliptic curve $C \setminus S$, and therefore it is itself rational or elliptic. The surface $S$ being non--rational, $E$ must be elliptic. Thus, we have a relatively minimal ruling $\pi :\,S \to E$, which is a smooth fibre bundle with a fibre $ I \!\! P^1$. {}From the exact sequence $${\bf 1} = \pi_2 (E) \to {\bf 1} = \pi_1 ( I \!\! P^1 ) \to \pi_1 (S) \to \pi_1 (E) \to {\bf 1}$$ we obtain $\pi_1 (S) \cong \pi_1 (E) \cong Z \!\!\! Z^2$. \medskip \noindent b) Let $k(S) = 0$. By the Enriques--Kodaira classification (see e.g. [GH, p.590] or [BPV, Ch. VI]), there are the following four possibilities: \smallskip \noindent * $S$ is a K3--surface, and then $\pi_1 (S) = ${\bf 1}. \smallskip \noindent * $S$ is an Enriques surface, and then $\pi_1 (S) \cong Z \!\!\! Z / 2 Z \!\!\! Z$. \smallskip \noindent * $S$ is an abelian surface, and then $\pi_1 (S) \cong Z \!\!\! Z^4$. \smallskip \noindent * $S$ is a hyperelliptic surface, and then, being a finite non--abelian extension of $ Z \!\!\! Z^4$, the group $\pi_1 (S)$ is almost abelian. \smallskip Note that in the last two cases $S$ is dominated by a torus. \medskip \noindent c) Suppose further that $k(S) = 1$. Then $S$ is an elliptic surface [GH, p. 574]; let $\pi_S\, :\,S \to B$ be an elliptic fibration. Since $S$ is elliptically connected, the base $B$ is dominated by a rational or elliptic curve $C \subset S$. Hence, $B$ itself is rational or elliptic. Fix a dominant morphism $g :\,E \to C$ from a smooth elliptic curve $E$. Set $f = \pi_S \circ g \,:\,E \to B$, and consider the product $X = S \times_B E$. The elliptic fibration $\pi_X \,:\,X \to E$ obtained from $\pi_S\, :\,S \to B$ by the base change $f \,:\,E \to B$ has a regular section $\sigma \,:\, E \ni e \longmapsto (e, g(e)) \in X = E \times_B S$. Passing to a normalization and a minimal resolution of singularities $X' \to X$ we obtain a smooth surface $X'$ with an elliptic fibration $\pi_{X'}\, :\,X' \to E$ and a section $\sigma' : \,E \to X'$. Thus, $\pi_{X'}$ has no multiple fibre. Replacing $X'$ by a birationally equivalent model we may also assume this fibration to be relatively minimal. If it were no singular fibre, then $\pi_{X'}$ would be a smooth morphism, and so $\chi (X') = \chi(F) \chi (E) = 0$, where $F$ denotes the generic fibre of $\pi_{X'}$. The formula for the canonical class of a relatively minimal elliptic surface [GH, p.572] implies that $K_{X'} = \pi_{X'}^* (L)$, where $L$ is a line bundle over $E$. Hence, $c_2(X') = K_{X'}^2 = 0$, and by the Noether formula, $$\chi ({\cal O}_{X'}) = {c_1(X')^2 + c_2(X') \over 12} = {K_{X'}^2 + \chi (X') \over 12} = 0\,.$$ Thus, ${\rm deg}\, L = 2g(E) - 2 + \chi (O_{X'}) = 0$. Therefore, the line bundle $K_{X'}$ is trivial, and so $k(X') = 0$, in contradiction with our assumption (indeed, since $X'$ rationally dominates $S$ and $k(S) =1$, we have $k(X') \ge 1$). Hence, $\pi_{X'}\, :\,X' \to E$ is a minimal elliptic fibration with a singular fibre. By Proposition 2.1 in [FM, Ch. II], we have $\pi_1 (X') \cong \pi_1 (E) \cong Z \!\!\! Z^2$. Since $S$ is dominated by a surface birationally equivalent to $X'$, whose fundamental group is isomorphic to those of $X'$, by Lemma 1.1.$c)$, the group $\pi_1(S)$ is almost abelian. This completes the proof of Theorem 0.5. \hfill $\Box$ \bigskip \noindent {\bf 1.4.} {\it Remark.} For explicit examples of smooth elliptic surfaces $\pi_S\,:\,S \to I \!\! P^1$ with a section $\sigma\,:\, I \!\! P^1 \to S\,\,(\pi_S \circ \sigma = {\rm id}_{ I \!\! P^1})$ of Kodaira dimension $1$, one may consider a (crepant) resolution of a surface in the projective bundle $ I \!\! P({\cal O} \bigoplus {\cal O}(-2m) \bigoplus {\cal O}(-3m))$ over $ I \!\! P^1$, where $m \ge 3$, defined by a general Weierstrass equation (see e.g. [Ka, Mi]). In the same way, replacing $ I \!\! P^1$ by $ I \!\! P^n$ and taking $m \ge n + 2$, one can construct examples of elliptically connected smooth projective varieties $X$ of Kodaira dimension $k(X) = $ dim$\,X - 1$. \bigskip \section*{APPENDIX: Winkelmann's example} \bigskip We present here the example mentioned in (0.4) above, of an elliptically connected smooth compact non-K\"ahlerian 3-fold $X$ such that the group $\pi_1(X)$ contains a non--abelian free subgroup, and hence, is not even almost solvable. We are grateful to D. Akhiezer for the detailed exposition reproduced below. Let $\Gamma \subset SL_2( I \!\!\!\! C)$ be a discrete cocompact subgroup. Due to Selberg's Lemma, there exists a torsion free subgroup of $\Gamma$ of finite index. Replacing $\Gamma$ by this subgroup we may assume $\Gamma$ being torsion free. By the Borel Density Theorem (see [Ra, 5.16]), $\Gamma$ is Zariski dense in $SL_2( I \!\!\!\! C)$. Set $X = SL_2( I \!\!\!\! C)/\Gamma$. Thus, $X$ is a (non--K\"ahlerian) compact homogeneous 3-fold with the fundamental group $\pi_1(X) \cong \Gamma$. Suppose that $\Gamma$ has a solvable subgroup $\Gamma' \subset \Gamma$ of finite index. Then $\Gamma'$ being Zariski dense in $SL_2( I \!\!\!\! C)$, we would have that $SL_2( I \!\!\!\! C)$ is solvable, too. $SL_2( I \!\!\!\! C)$ being simple, $\Gamma$ cannot be almost solvable. By Tits' alternative [Ti], $\Gamma$ must contain a non--abelian free subgroup. Let $x \sim y$ mean that the points $x$ and $y$ in $X$ can be connected in $X$ by a chain of rational or elliptic curves. To show that $X$ is elliptically connected, it is enough to check this locally. That is to say, to show the existence of a neighborhood $U_0$ of the point $x_0 := \,${\bf e}$\,\cdot \Gamma$ in $X= SL_2( I \!\!\!\! C)/\Gamma$ such that $x \sim x_0$ for any point $x \in U_0$. Suppose we can find three one--dimensional algebraic tori (i.e. one--parametric subgroups isomorphic to $G_m \cong I \!\!\!\! C^* $) $A_0,\,A_1,\,A_2 \subset SL_2( I \!\!\!\! C)$ such that \smallskip \noindent (i) $ A_i/ (A_i \cap \Gamma)$ is compact, and therefore, the image $E_i$ of $A_i$ in $X$ is a smooth elliptic curve, $i = 0,\,1,\,2$; \smallskip \noindent (ii) the Lie subalgebras ${\goth a}_i \subset {\goth{sl}}_2( I \!\!\!\! C),\,\,i = 0,\,1,\,2$, span ${\goth{sl}}_2( I \!\!\!\! C)$. \smallskip Then, by (ii), any point $x$ in a small enough neighborhood $U_0$ of the point $x_0 \in X$ can be presented as $a_0 a_1 a_2 \cdot x_0$ with some $a_i \in A_i,\,\,i = 0,\,1,\,2$. Hence, by (i), $x$ and $x_0$ are joined in $X$ by the chain of elliptic curves $E_0,\,\,E_1' := a_0E_1,\,\,E_2' := a_0a_1E_2$. Indeed, we have $$x_0,\,a_0 \cdot x_0 \in A_0 \cdot x_0 = E_0\,,$$ $$a_0\cdot x_0,\,\,a_0a_1\cdot x_0 \in a_0A_1\cdot x_0 = E_1'\,,$$ $$a_0a_1 \cdot x_0,\,\,x = a_0a_1a_2\cdot x_0 \in a_0a_1A_2 \cdot x_0 = E_2' \,.$$ This proves that $X$ is elliptically connected. To find three tori $A_0,\,A_1,\,A_2$ in $SL_2( I \!\!\!\! C)$ with properties (i) and (ii) note that the Zariski dense torsion free subgroup $\Gamma \subset SL_2( I \!\!\!\! C)$ must contain at least one semisimple element ${\gamma} \neq \,${\bf e}. Indeed, there exists a Zariski open subset $\Omega \subset G$ such that all elements of $\Omega$ are semisimple. We may assume that {\bf e} is not in $\Omega$. Since $\Gamma$ is Zariski dense in $G$, $\Gamma$ can not be contained in $G \setminus \Omega$. Thus, there is a semisimple element $\gamma \ne$ {\bf e} in $\Gamma$. Let $A_0 \subset SL_2( I \!\!\!\! C)$ be a torus which contains ${\gamma}$, and let $v \in {\goth a}_0,\,\,v \neq 0$. In view of the Zariski density of $\Gamma$, the orbit of $v$ by the adjoint action of $\Gamma$ on ${\goth{sl}}_2( I \!\!\!\! C)$ generates ${\goth{sl}}_2( I \!\!\!\! C)$. Hence, we can find ${\gamma}_1,\,{\gamma}_2 \in \Gamma$ such that $v,\,\,$Ad$\,({\gamma}_1) \cdot v$ and Ad$\,{\gamma}_2 \cdot v$ form a basis of ${\goth{sl}}_2( I \!\!\!\! C)$. Then for $A_i$ we can take the torus ${\gamma}_iA_0{\gamma}_i^{-1}$ through ${\gamma}_i{\gamma}_0{\gamma}_i^{-1},\,\,i=1,\,2$. Clearly, (ii) is fulfilled and (i) follows from the fact that $\Gamma$ has no torsion. Finally, $X = G/\Gamma$ is non-K\"ahler. Indeed, by a theorem of Borel and Remmert (see [Ak, 3.9]), a complex compact homogeneous K\"ahler manifold is a product of a simply connected projective variety and a torus. Thus, it has an abelian fundamental group. Here $\Gamma$ is certainly non--abelian. \section*{References} {\footnotesize \noindent [Ak] D.N. Akhiezer. {\it Lie group actions in complex analysis}, Vieweg, Braunschweig/Wiesbaden, 1995 \noindent [BPV] W. Barth, C. Peters, A. Van de Ven. {\it Compact Complex Surfaces}, Springer, Berlin e.a. 1984 \noindent [Cam1] F. Campana. {\it On twistor spaces of the class $\cal C$}, J. Diff. Geom. 33 (1991), 541--549 \noindent [Cam2] F. Campana. {\it Remarques sur le rev\^etement universel des vari\'et\'es k\"ahl\'eriennes compactes}, Bull. Soc. math. France, 122 (1994), 255--284 \noindent [Cam3] F. Campana. {\it Fundamental group and positivity of cotangent bundles of compact K\"ahler manifolds}, J. Algebraic Geom. 4 (1995), 487--502 \noindent [Cam4] F. Campana. {\it Kodaira dimension and fundamental group of compact K\"ahler manifolds}, Dipart. di Matem. Univ. degli Studi di Trento, Lect. Notes Series 7, 1995 \noindent [DZ] G. Dethloff, M. Zaidenberg. {\it Plane curves with C--hyperbolic complements}, Pr\'epublication de l'Institut Fourier de Math\'ematiques, 299, Grenoble 1995, 44p. Duke E-print alg-geom/9501007 \noindent [FM] R. Friedman, J. W. Morgan. {\it Smooth four--manifolds and complex surfaces}. Berlin e.a.: Springer, 1994 \noindent [Fu] T. Fujita. {\it On the topology of non--complete algebraic surfaces}, J. Fac. Sci. Univ. Tokyo, Ser. 1A, 29 (1982), 503--566 \noindent [GH] Ph. Griffiths, J. Harris. {\it Principles of Algebraic Geometry.} NY: Wiley, 1978 \noindent [Ka] A. Kas. {\it Weierstrass normal forms and invariants of elliptic surfaces}, Trans. Amer. Math. Soc. 225 (1977), 259--266 \noindent [Ko] K. Kodaira. {\it Collected works}, Princeton Univ. Press, Princeton, New Jersey, 1975 \noindent [KMM] J. Kollar, Y. Miyaoka, S. Mori. {\it Rationally connected varieties}, J. Algebraic Geom. 1 (1992), 429--448 \noindent [Li] V. Ja. Lin. {\it Liouville coverings of complex spaces, and amenable groups}, Math. USSR Sbornik 60 (1988), 197--216 \noindent [Mi] R. Miranda. {\it The moduli of Weierstrass fibrations over $ I \!\! P^1$}, Math. Ann. 255 (1981), 379-394 \noindent [No] M.V. Nori. {\it Zariski's conjecture and related problems}, Ann. scient. Ec. Norm. Sup. 16 (1983), 305--344 \noindent [Ra] M. S. Raghunathan. {\sl Discrete subgroups of Lie groups}, Berlin e.a.: Springer, 1972 \noindent [Se] J.-P. Serre. {\sl On the fundamental group of a unirational variety}, J. London Math. Soc. 34 (1959), 481--484 \noindent [Ti] J. Tits. {\sl Free subgroups in linear groups}, J. Algebra 20 (1972), 250--270 \noindent [Z] M. Zaidenberg, Problems on open algebraic varieties. In: {\it Open problems on open varieties (Montreal 1994 problems), P. Russell (ed.)}, Pr\'epublication de l'Institut Fourier des Math\'ematiques 311, Grenoble 1995, 23p. E-print alg-geom/9506006 \bigskip \bigskip {\it Added in proofs.} Recently F. Campana has proved the above Conjecture 0.2 in its stronger form, and obtained interesting generalizations. \noindent Keiji Oguiso: \noindent Department of Mathematical Sciences, University of Tokyo, Komaba Megro, Tokyo, Japan \smallskip \noindent e-mail: [email protected] \bigskip \noindent Mikhail Zaidenberg: \noindent Universit\'{e} Grenoble I, Institut Fourier des Math\'ematiques, BP 74, 38402 St. Martin d'H\'eres--c\'edex, France \smallskip \noindent e-mail: [email protected]} \end{document}

9704

alg-geom/9704020

en

https://arxiv.org/abs/alg-geom/9704020

alg-geom/9704020

Elham Izadi

E. Izadi

Second order theta divisors on Pryms

AMS-Latex, 11 pages, the exposition has been modified, Proposition 5 has been modified

Van Geemen and van der Geer, Donagi, Beauville and Debarre proposed characterizations of the locus of jacobians which use the linear system of $2\Theta$-divisors. We give new evidence for these conjectures in the case of Prym varieties.

alg-geom

\section{Preliminaries}\label{prelim} Let $\pi : \widetilde{C} \rightarrow C$ be an \'etale double cover of a smooth non-hyperelliptic curve $C$ of genus $g$. Let $\alpha$ be the point of order $2$ in $Pic^0 C$ associated to the double cover $\pi$ so that we have $\pi_*{\cal O}_{\widetilde{C} }\cong{\cal O}_C\oplus\alpha$. Choose an element $\beta$ of $Pic^0 C$ such that $\beta^{\otimes 2} = \alpha$ and a theta-characteristic $\kappa$ on $C$ such that $h^0( C,\kappa )$ and $h^0(\widetilde{C} ,\pi^* (\kappa\otimes\beta ))$ are even. Symmetric principal polarizations on $J\widetilde{C} = Pic^0\widetilde{C}$ and $JC = Pic^0 C$ can be defined as the reduced divisors \[ \widetilde{\Theta} := \{ D \in Pic^0 \widetilde{C} : h^0(\widetilde{C}, D \otimes \pi^* (\kappa \otimes \beta) ) > 0 \} \] \[ \Theta := \{ D \in Pic^0 C : h^0(C, D \otimes \kappa ) > 0 \} \: . \] With these definitions, the inverse image of $\widetilde{\Theta}$ by the morphism $\pi^* : JC\rightarrow J\widetilde{C}$ is the divisor $\Theta_{\beta} +\Theta_{\beta^{-1}}$ where $\Theta_{\beta}$ is the translate of $\Theta$ by $\beta$. The Prym variety $( P,\Xi )$ of the double cover $\pi :\widetilde{C}\rightarrow C$ is defined by the reduced varieties \[ P := \{ E \in Pic^0 \widetilde{C} : Nm(E) \cong {\cal O}_C, h^0(\widetilde{C}, E \otimes \pi^*(\kappa \otimes \beta)) \equiv 0 \hbox{ mod }2 \} \: , \] \[ \Xi := \{ E \in P : h^0(\widetilde{C}, E \otimes \pi^* (\kappa \otimes \beta)) > 0 \} \] where $Nm : Pic \widetilde{C} \rightarrow Pic C$ is the Norm map (see \cite{mumford74} pages 331-333 and 340-342). As divisors we have $ 2\Xi = P.\widetilde{\Theta}$. Also define \[ P' := \{ E \in Pic^{2g-2} \widetilde{C} : Nm(E) \cong \omega_C, h^0( \widetilde{C}, E) \equiv 0 \hbox{ mod }2 \} \] and \[ \Xi' :=\{ E \in P' : h^0( \widetilde{C}, E ) > 0\}\; . \] For any $E \in P'$, put \[ \Xi_E := \{ D \in P : D \otimes E \in \Xi' \} \: . \] Since $Nm(E)\cong\omega_C$, we have $\omega_{\widetilde{C}} \otimes E^{-1} \cong\sigma^* E$. By the theorem of the square, the divisor $\Xi_E + \Xi_{\sigma^* E}$ is in the linear system $|2 \Xi |$ and, by Wirtinger Duality (see \cite{mumford74} pages 335-336), such divisors span $| 2 \Xi |$. Furthermore, if $E \in \Xi'$, then $\Xi_E + \Xi_{\sigma^* E}$ is in $| 2 \Xi |_0$ and such divisors span $| 2 \Xi |_0$ (by Wirtinger duality, the span of such divisors is the span of $\phi(\Xi)$ where $\phi : P\rightarrow |2\Xi |^*$ is the natural morphism, this span is a hyperplane in $|2\Xi |^*$ which can therefore be identified with $| 2\Xi |_0$ by Wirtinger duality, also see \cite{welters86} page $18$). We now explain how Proposition \ref{mainprop} follows from results of Welters and Debarre and how Proposition \ref{propinf00} follows from results of Debarre. In this paragraph only, suppose that $(P,\Xi)$ is a {\em general} Prym variety. Then $E$ is an element of the singular locus $Sing(\Xi')$ of $\Xi'$ if and only if $h^0(\widetilde{C}, E)\geq 4$ (\cite{welters872} page 168). Therefore, for every $E\in Sing(\Xi')$ and $(p,q )\in\widetilde{C}^2$, we have $h^0 (\widetilde{C}, E\otimes{\cal O}_{\widetilde{C}} (p +q -\sigma p -\sigma q)) > 0$. Hence $\Sigma\subset\Xi_E$ and $\Xi_E +\Xi_{\sigma^* E}$ is in $| 2\Xi |_{00}'$. Since $\Xi$ is symmetric, the tangent cones at $0$ to $\Xi_E$ and $\Xi_{\sigma^* E}$ are equal. It follows from these facts and the irreducibility of $Sing(\Xi')$ for $p\geq 6$ (see \cite{debarre89} Th\'eor\`eme 1.1 page 114) that $V_{00}'$ is set-theoretically contained in $\cap_{E\in Sing(\Xi')} \Xi_E$ and $V_{inf,00}'$ is set-theoretically contained in the intersection of the tangent cones at $0$ to $\Xi_E$ for $E$ a point of multiplicity $2$ on $\Xi'$. This latter can be rephrased as: $V_{inf,00}'$ is set-theoretically contained in the intersection of the tangent cones to $\Xi'$ at its points of multiplicity $2$. It is proved in \cite{welters872} (Theorem 2.6 page 169) that, for $p\geq 16$, the intersection $\cap_{E\in Sing(\Xi')} \Xi_E$ is the set-theoretical union of $\Sigma$ and, possibly, some curves and points. This proves Proposition \ref{mainprop}. By \cite{debarre89} Th\'eor\`eme 1.1 page 114, the tangent cones to $\Xi'$ at its points of multiplicity $2$ generate the space of quadrics containing $\chi C$ for $p\geq 6$ and, for $p\geq 8$ (see \cite{debarre89} Corollaire 2.3 page 129, also see \cite{lazarsfeld95} and \cite{langesernesi96}), the Prym-canonical curve $\chi C$ is cut out by quadrics. This proves Proposition \ref{propinf00}. \section{Pull-backs of divisors to $\widetilde{C}^2$} Let $\rho : \widetilde{C}^2 \rightarrow P\subset J\widetilde{C}$ be the morphism \[ \rho : (s,t) \longmapsto [s,t] := {\cal O}_{\widetilde{C}}(s+t-\sigma s-\sigma t) \] so that $\Sigma$ is the image of $\widetilde{C}^{2}$ by $\rho$. The morphism $\rho$ lifts to a finite morphism $\widetilde{\rho} :\widetilde{C}^2\hspace{3pt}\to \hspace{-19pt}{\rightarrow} \:\:\widetilde{\Sigma}\subset\widetilde{P}$. We have \begin{lemma} For any $E \in \Xi'$ such that $h^0(\widetilde{C},E) = 2$ and $|E|$ contains reduced divisors, the inverse image of $\Xi_E$ in $\widetilde{C}^2$ is the divisor \[ \rho^*\Xi_E = D_{\sigma^* E} +\Delta' \] where $D_E$ is the reduced divisor \[ D_E := \{ (p,q) : h^0( \widetilde{C}, E\otimes{\cal O}_{\widetilde{C}}(-p-q) ) > 0 \} \] and $\Delta'$ is the ``pseudo-diagonal'' of $\widetilde{C}^2$, i.e., the reduced curve $\Delta' := \{ (p, \sigma p) \in \widetilde{C}^2 \} = \rho^{-1}( 0 )$. \label{XiEC2} \end{lemma} {\em Proof :} We have the equality of sets \[ \rho^*\Xi_E = \{ (p,q) : E \otimes [p,q] \in \Xi' \} = \{ (p,q) : h^0(\widetilde{C}, E \otimes [p,q]) > 0 \} \:\: . \] First suppose that $| E |$ has no base points. By \cite{mumford71} (page 187, Step II and its proof), for any $p \in \widetilde{C}$, we have $h^0(\widetilde{C}, E \otimes {\cal O}_{\widetilde{C}}(p-\sigma p)) = h^0(\widetilde{C}, E) - 1$ if $\sigma p$ is not a base point of $|E|$ and $h^0(\widetilde{C}, E \otimes {\cal O}_{\widetilde{C}}(p-\sigma p)) = h^0(\widetilde{C}, E) + 1$ if $\sigma p$ is a base point of $|E|$. Since $|E|$ has no base points and $h^0(\widetilde{C}, E) = 2$, we have $h^0(\widetilde{C}, E \otimes {\cal O}_{\widetilde{C}}(p-\sigma p)) = 1$ for all $p\in\widetilde{C}$. Similarly, $h^0 (\widetilde{C}, E \otimes [p,q]) = h^0 (\widetilde{C}, E\otimes{\cal O}_{\widetilde{C}}(p-\sigma p)) - 1 =0$ if $\sigma q$ is not a base point of $| E\otimes{\cal O}_{\widetilde{C}}(p-\sigma p)|$ and $h^0(\widetilde{C}, E \otimes [p,q]) = h^0(\widetilde{C}, E\otimes{\cal O}_{\widetilde{C} }( p -\sigma p)) + 1 = 2$ if $\sigma q$ is a base point of $| E\otimes{\cal O}_{\widetilde{C} } (p -\sigma p)|$. Let $D_{\sigma p}$ be the unique divisor of $|E|$ containing $\sigma p$, then $D_{\sigma p} +p -\sigma p$ is the unique element of $| E \otimes{\cal O}_{\widetilde{C}} (p-\sigma p)|$. So, to obtain $h^0(\widetilde{C}, E \otimes [p,q]) > 0$, we need $\sigma q$ to be a base point of $|E \otimes{\cal O}_{\widetilde{C}} (p-\sigma p)|$, i.e., the point $\sigma q$ appears in $D_{\sigma p} +p - \sigma p$. Therefore, either $\sigma q = p$ or $h^0(\widetilde{C}, E \otimes {\cal O}_{\widetilde{C}}(-\sigma p - \sigma q)) > 0$. So the support of $\rho^* \Xi_E$ is $D_{\sigma^* E} \cup \Delta'$. An easy degree computation on the fibers of $\widetilde{C}^2$ over $\widetilde{C}$ by the two projections shows then that $\rho^* \Xi_E = D_{\sigma^* E} + \Delta'$ as divisors. If $| E |$ has base points, then its base points are distinct because $| E |$ contains reduced divisors. Then, by \cite{griffithsharris78} page 287, the cohomology class of $D_{\sigma^* E}$ is the same as in the case where $|E|$ has no base points. Hence, since $\rho^* \Xi_E = D_{\sigma^* E} + \Delta'$ as divisors in the case where $|E|$ has no base points, we see that $\rho^*\Xi_E - \Delta'$ and $D_{\sigma^* E}$ have the same cohomology class. Since the support of $D_{\sigma^* E}$ is clearly contained in the support of $\rho^*\Xi_E -\Delta'$ and these are both effective divisors, they are equal. \qed \begin{remark} Suppose that $E$ as above can be written as $E = \pi^* M\otimes {\cal O}_{\widetilde{C}}(B)$ where $B$ is the base divisor of $E$ and $M$ is an invertible sheaf on $C$ with $h^0( C, M) = 2$. Then $D_{\sigma^* E} =\Delta' + D'$ for some effective divisor $D'$. Therefore $\rho^*\Xi_E = D' + 2\Delta'$ which agrees with the fact that in such a case $E\in Sing (\Xi')$ so that $\Xi_E$ is singular at $0$ (see \cite{mumford74} pages 342-343). \end{remark} Let $\Delta \subset \widetilde{C}^2$ be the diagonal, let $p_i :\widetilde{C}^2\rightarrow\widetilde{C}$ be the projection onto the $i$-th factor and let $\omega_{\widetilde{C}^2 }$ be the canonical sheaf of $\widetilde{C}^2$. Then $\omega_{\widetilde{C}^2 }\cong p_1^*\omega_{\widetilde{C} }\otimes p_2^*\omega_{\widetilde{C} }$ and we have K\"unneth's isomorphism $H^0 (\widetilde{C}^2 ,\omega_{\widetilde{C}^2 } )\cong H^0 (\widetilde{C} ,\omega_{\widetilde{C} } )^{\otimes 2}$. Let $I_2(\widetilde{C}) \subset S^2H^0(\widetilde{C}, \omega_{\widetilde{C}} )\subset H^0(\widetilde{C} ,\omega_{\widetilde{C}})^{\otimes 2} = H^0(\widetilde{C}^2,\omega_{\widetilde{C}^2 })$ be the vector space of quadratic forms vanishing on the canonical image $\kappa \widetilde{C}$ of $\widetilde{C}$. Fix an embedding $H^0(\widetilde{C}^2 ,\omega_{\widetilde{C}^2 }\otimes{\cal O}_{\widetilde{C}^2 }( - 2 \Delta) )\subset H^0(\widetilde{C}^2,\omega_{\widetilde{C}^2 } )$ obtained by multiplication by a nonzero global section of ${\cal O}_{\widetilde{C}^2 }( 2 \Delta)$ (note that $h^0(\widetilde{C}^2,{\cal O}_{\widetilde{C}^2 }( 2 \Delta)) = 1$ because $\Delta$ has negative self-intersection, hence any two such embeddings differ by multiplication by a constant). Then it is easily seen that \[ I_2(\widetilde{C}) = S^2H^0(\widetilde{C}, \omega_{\widetilde{C}}) \cap H^0(\widetilde{C}^2 ,\omega_{\widetilde{C}^2 }\otimes{\cal O}_{\widetilde{C}^2 }( - 2 \Delta) )\subset H^0(\widetilde{C}^2,\omega_{\widetilde{C}^2 } ) \: . \] Similarly fix embeddings $H^0(\widetilde{C}^2 ,\omega_{\widetilde{C}^2 }\otimes{\cal O}_{\widetilde{C}^2 }( - 2 \Delta - 2\Delta') )\subset H^0(\widetilde{C}^2 ,\omega_{\widetilde{C}^2 }\otimes{\cal O}_{\widetilde{C}^2 }( - 2 \Delta) )\subset H^0(\widetilde{C}^2,\omega_{\widetilde{C}^2 }\otimes{\cal O}_{\widetilde{C}^2} (- 2 \Delta+2 \Delta' ) )$. For $E\in\Xi'$ such that $h^0(\widetilde{C}, E)=2$, the linear system $|E|$ has no base points and $E\not\cong\sigma^* E$, it is well-known (see, e.g., \cite{ACGH} page 261) that $q_E :=\cup_{D \in |E|} \langle D \rangle =\cup_{D \in |\sigma^*E|} \langle D\rangle$ is a quadric of rank $4$ whose rulings cut the divisors of $|E|$ and $|\sigma^*E|$ on $\widetilde{C}$. We need the following \begin{lemma}\label{Zsqs} \begin{enumerate} \item\label{restrictC2} We have \[ \rho^* {\cal O}_P (2 \Xi) \cong \omega_{\widetilde{C}^2 }\otimes{\cal O}_{\widetilde{C}^2} (- 2 \Delta+2 \Delta' ), \] \[ \rho^*\Gamma_0\subset I_2 (\widetilde{C} )\subset H^0( \widetilde{C}^2 ,\omega_{\widetilde{C}^2 }\otimes{\cal O}_{\widetilde{C}^2}( - 2\Delta ) )\subset H^0( \widetilde{C}^2, \omega_{\widetilde{C}^2 }\otimes{\cal O}_{\widetilde{C}^2}(- 2 \Delta+2 \Delta' ) )\; , \] and \[ \rho^*\Gamma_{ 00 }\subset I_2 (\widetilde{C} )\cap H^0( \widetilde{C}^2,\omega_{\widetilde{C}^2 }\otimes{\cal O}_{\widetilde{C}^2} ( - 2\Delta - 2 \Delta' ) )\subset H^0( \widetilde{C}^2 ,\omega_{\widetilde{C}^2 }\otimes{\cal O}_{\widetilde{C}^2} ( - 2\Delta + 2\Delta' ) ). \] \item\label{Zs} For any $s \in \Gamma_0$, let $q(s)$ be the quadric in $|\omega_{\widetilde{C} }|^*$ with equation $\rho^* s\in I_2 (\widetilde{C} )$. Then, in $\widetilde{C}^2$, the zero locus of $\rho^* s\in H^0 (\widetilde{C}^2 ,\omega_{\widetilde{C}^2 }\otimes{\cal O}_{\widetilde{C}^2}( - 2\Delta +2 \Delta' ) )$ is \[ Z(\rho^* s) = Z_{q(s)} + 2 \Delta' \] where $Z_{q(s)}$ is a divisor with support \[ \{ (p,q) \in \widetilde{C}^2 : \langle p+q \rangle \subset q(s) \}. \] If $s\in\Gamma_0$ is general, then $Z_{q(s)}$ is reduced. If $Z(s) = \Xi_E + \Xi_{\sigma^* E}$ for some $E\in\Xi'$ such that $h^0(\widetilde{C}, E) = 2$, then $q(s) = q_E :=\cup_{D\in | E |}\langle D\rangle$. \item \label{tangcone} If $s \in \Gamma_0 \setminus \Gamma_{00}$, then \[ q(s) \cap ({\Bbb P} T_0 P = | \omega_C \otimes \alpha |^*) \subset{\Bbb P} T_0 J\widetilde{C} = |\omega_{\widetilde{C}} |^* \] is the projectivized tangent cone $\tau_{Z(s)}$ to $Z(s)\subset P$ at $0$. \item\label{multDs} For any $s\in\Gamma_0$, the multiplicity of $\rho^*s$ at the generic point of $\Delta'$ is even $\geq 2$ and if $\rho^*s$ vanishes on $\Delta'$ with multiplicity $\geq 4$, then either $s \in \Gamma_{00}$ or $\tau_{Z(s)}$ contains the Prym-canonical curve $\chi C$. \end{enumerate} \end{lemma} {\em Proof : }\ref{restrictC2}. Let $E$ be an invertible sheaf of degree $2g-2$ on $\widetilde{C}$ such that $h^0(\widetilde{C}, E)=2$ and $|E|$ has no base points. Let $s_1$ and $s_2$ be two general sections of $E$. Then \[ s_1 \otimes s_2 - s_2 \otimes s_1 \in \Lambda^2 H^0(\widetilde{C}, E) \subset H^0(\widetilde{C},E)^{\otimes 2} \] and, as in the the proof of Lemma \ref{XiEC2}, it is easily seen that $Z(s_1 \otimes s_2 - s_2 \otimes s_1) = D_E +\Delta$. Now, supposing that $E \otimes \sigma^*E \cong \omega_{\widetilde{C}}$, from the natural map \[ \psi_E : H^0(\widetilde{C}, E) \otimes H^0(\widetilde{C}, \sigma^*E) \longrightarrow H^0(\widetilde{C}, \omega_{\widetilde{C}}) \] we obtain the map \[ \begin{array}{rcl} H^0(\widetilde{C}, E)^{\otimes 2} \otimes H^0(\widetilde{C}, \sigma^*E)^{\otimes 2} & \longrightarrow & H^0(\widetilde{C}, \omega_{\widetilde{C}})^{\otimes 2}\cong H^0(\widetilde{C}^2, \omega_{\widetilde{C}^2}) \\ t_1\otimes t_2\otimes\sigma^* u_1\otimes\sigma^* u_2 & \longmapsto & \psi_E (t_1\otimes\sigma^* u_1 )\otimes\psi_E (t_2\otimes\sigma^* u_2 ) \end{array} \] which induces the map \[ \phi_E : \Lambda^2 H^0(\widetilde{C}, E) \otimes \Lambda^2 H^0(\widetilde{C}, \sigma^*E) \longrightarrow S^2 H^0(\widetilde{C}, \omega_{\widetilde{C}}) \] Put \[ t := (s_1 \otimes s_2 - s_2 \otimes s_1) \otimes (\sigma^* s_1 \otimes \sigma^* s_2 - \sigma^* s_2\otimes \sigma^* s_1) \] then $Z(\phi_E(t))$ is in the linear system $|\omega_{\widetilde{C}^2 }|$ and is equal to $D_E + \Delta + D_{\sigma^* E} + \Delta$. Hence $D_E + D_{\sigma^* E}$ is in the linear system $|\omega_{\widetilde{C}^2 }\otimes{\cal O}_{\widetilde{C}^2 } ( - 2 \Delta ) |$. If $s\in\Gamma_0$ is such that $Z(s) =\Xi_E +\Xi_{\sigma^* E}$, then, by Lemma \ref{XiEC2}, we have $Z(\rho^* s) = D_E + D_{\sigma^* E} + 2\Delta'$. So $Z(\rho^* s) - 2\Delta' = Z(\phi_E(t)) -2\Delta$ and $\rho^*{\cal O}_P ( 2\Xi )\cong\omega_{\widetilde{C}^2 }\otimes{\cal O}_{\widetilde{C}^2 } ( - 2 \Delta + 2\Delta')$. Since $Z(\rho^* s) - 2\Delta' = Z(\phi_E(t)) -2\Delta$, the section $\rho^* s$ of $\rho^* {\cal O}_P (2 \Xi)\otimes{\cal O}_{\widetilde{C}^2} (- 2 \Delta' )\cong\omega_{\widetilde{C}^2 }\otimes{\cal O}_{\widetilde{C}^2} (- 2 \Delta )$ is a nonzero constant multiple of $\phi_E (t )\in I_2 (\widetilde{C} )\subset H^0 (\widetilde{C}^2 ,\omega_{\widetilde{C}^2 }\otimes{\cal O}_{\widetilde{C}^2 } ( - 2 \Delta ) )$. Since such $s$ generate $\Gamma_0$, this proves that $\rho^*\Gamma_0\subset I_2 (\widetilde{C} )$. The rest of part \ref{restrictC2} easily follows now. \ref{Zs}. Let $s$ and $E$ be as above and suppose furthermore that $E\not\cong\sigma^* E$. With the above notation, let $X_1, X_2, X_3, X_4$ be the images of, respectively, \[ s_1 \otimes \sigma^* s_1, s_2 \otimes \sigma^* s_1, s_1 \otimes \sigma^* s_2, s_2 \otimes \sigma^* s_2 \] by the map $\psi_E$. Then $\phi_E (t) = X_1X_4-X_2X_3$. By, e.g. \cite{ACGH} page 261, the polynomial $X_1X_4-X_2X_3$ is an equation for $q_E$. Therefore, since $\rho^* s$ is a constant nonzero multiple of $\phi_E (t)$, we have $q(s) = q_E$. By continuity, for all $s'$ such that $Z(s') =\Xi_M +\Xi_{\sigma^* M}$ for some $M\in\Xi'$ with $h^0 (\widetilde{C}, M) = 2$, we have $q(s') = q_M$. A line is in $q_E$ if and only if it is in a linear space of one of the two rulings. Therefore a secant $\langle p+q\rangle$ to $\kappa\widetilde{C}$ is in $q_E$ if and only if $h^0(\widetilde{C}, E\otimes{\cal O}_{\widetilde{C}} (-p-q)) > 0$ or $h^0(\widetilde{C},\sigma^* E\otimes{\cal O}_{\widetilde{C}} (-p-q)) > 0$. Hence \[ D_E + D_{\sigma^* E} = \{ (p,q) : \langle p+q \rangle\subset q_E \} = Z_{q_E} = Z_{q(s)} \] and $Z_{q(s)}$ is a reduced divisor in this case. Hence $Z_{q(s)}$ is reduced for general $s\in\Gamma_0$. Now the rest of part \ref{Zs} follows by linearity. \ref{tangcone}. When $E$ is a smooth point of $\Xi'$, the intersection $q_E\cap{\Bbb P} T_0 P = 2{\Bbb P} T_E\Xi'$ is the projectivized tangent cone at $0$ to $\Xi_E +\Xi_{\sigma^* E}$ (see \cite{mumford74} pages 342-343). Hence \ref{tangcone} also follows by linearity. \ref{multDs}. This immediately follows from the facts that $\rho^* s\in I_2 (\widetilde{C} )\subset S^2 H^0 (\widetilde{C} ,\omega_{\widetilde{C} })\subset H^0 (\widetilde{C}^2 ,\omega_{\widetilde{C}^2 })$ and that $\chi C$ is the tangent cone at $0$ to $\Sigma$. \qed \section{More divisors in $|2 \Xi |_{00}$}\label{newdiv} For any $M \in Pic^{g-1}C$, let $\widetilde{\Theta}_{\pi^* M}$ be the translate of $\widetilde{\Theta}$ by $\pi^* (\kappa\otimes\beta\otimes M^{-1})$, i.e., \[ \widetilde{\Theta}_{\pi^* M} :=\{ E\in Pic^{2g-2}\widetilde{C} : h^0(\widetilde{C}, E\otimes\pi^* M ) > 0\}\; . \] We have \begin{proposition}\label{newdivG0} The divisor $P.\widetilde{\Theta}_{\pi^* M}$ is in the linear system $|2 \Xi |$. It is in $| 2 \Xi |_0$ if $h^0 (C , M)$ is positive. \end{proposition} {\em Proof :} We first prove that all the divisors $P.\widetilde{\Theta}_{\pi^* M}$ are linearly equivalent as $M$ varies in $Pic^{g-1} C$. Let $\psi : JC\rightarrow Pic^0 P$ be the morphism of abelian varieties which sends $M\otimes\kappa^{ -1 }\otimes\beta^{ -1}$ to ${\cal O}_P (P.\widetilde{\Theta}_{\pi^* M} - P.\widetilde{\Theta} )\in Pic^0 P$. For $C$ general, the abelian variety $JC$ is simple, hence, since its dimension is not equal to the dimension of $Pic^0 P$, the morphism $\psi$ is trivial, i.e., its image is ${\cal O}_P$. By continuity, this is the case for all $C$. Therefore, since $P.\widetilde{\Theta} = 2\Xi$, all the divisors $P.\widetilde{\Theta}_{\pi^* M}$ are linearly equivalent to $2\Xi$. The second assertion is now immediate. \qed \begin{proposition}\label{newdivG00} The divisor $P.\widetilde{\Theta}_{\pi^* M}$ is an element of $| 2 \Xi |_{00}$ if $h^0(C, M) \geq 2$. \end{proposition} {\em Proof :} We prove the Proposition in the case where $h^0(\widetilde{C} ,\pi^* M) = 2$ and it will follow in all cases by semi-continuity. By, e.g. \cite{ACGH} page 261, the tangent cone to $\widetilde{\Theta}_{\pi^*M}$ at $0$ is the quadric $q_{\pi^* M} :=\cup_{\delta\in |\pi^* M|}\langle\delta\rangle$ in $|\omega_{\widetilde{C} }|^* ={\Bbb P} T_0 J\widetilde{C}$. Let $\pi$ also denote the projection ${\Bbb P} T_0 J\widetilde{C} = |\omega_{\widetilde{C} }|^*\rightarrow |\omega_{ C }|^* ={\Bbb P} T_0 JC$ with center $|\omega_C\otimes\alpha |^* ={\Bbb P} T_0 P$. Since $q_{\pi^* M} = \pi^* (q_M ) =\pi^* (\cup_{\delta\in | M|}\langle\delta\rangle)$, we see that $q_{\pi^* M}$ contains ${\Bbb P} T_0 P\subset{\Bbb P} T_0 J\widetilde{C}$ and the multiplicity of $P.\widetilde{\Theta}_{\pi^*M}$ at $0$ is at least $3$. Since this multiplicity is even, we have $P.\widetilde{\Theta}_{\pi^*M}\in |2\Xi |_{00}$. \qed \section{The base locus of $L |_{\widetilde{\Sigma}}$}\label{secproof} In this section we prove Proposition \ref{mainprop2} and Theorem \ref{maintheorem}. Recall that we have assumed $C$ to be non-hyperelliptic. We have \begin{proposition}\label{qpiD} If $M\in Pic^{g-1} C$ is such that $h^0 ( C , M) = h^0 (\widetilde{C},\pi^* M) = 2$ and $|M|$ and $|\omega_C\otimes M^{-1 }|$ contain reduced divisors, then \[ \rho^* (P.\widetilde{\Theta}_{\pi^* M}) = D_M + D_{\omega_C\otimes M^{-1}} + 4\Delta' \] where $D_M$ is the reduced divisor with support \[ \{ (p,q) : h^0 (C, M\otimes{\cal O}_C (-\pi p -\pi q)) > 0\}\; . \] \end{proposition} {\em Proof :} First note that $P.\widetilde{\Theta}_{\pi^* M}$ does not contain $\Sigma$. Indeed, for general points $p$ and $q$ in $\widetilde{C}$, we have $h^0 (\widetilde{C} ,\pi^* M\otimes{\cal O}_{\widetilde{C}} (p+q)) = 2$ and $h^0 (\widetilde{C} ,\pi^* M\otimes{\cal O}_{\widetilde{C}} (-\sigma p -\sigma q)) = 0$. Therefore $p$ and $q$ are base points for $| \pi^* M\otimes{\cal O}_{\widetilde{C}} (p+q) |$ and $h^0 (\widetilde{C} ,\pi^* M\otimes{\cal O}_{\widetilde{C}} (p+q -\sigma p -\sigma q)) = 0$. Now note that $\rho^* (P.\widetilde{\Theta}_{\pi^* M}) - 4\Delta'$ is effective because, by Proposition \ref{newdivG00}, the divisor $P.\widetilde{\Theta}_{\pi^* M}$ has multiplicity at least $4$ at the origin. Then, using Lemma \ref{Zsqs}, Proposition \ref{newdivG0} and \cite{griffithsharris78} page 287, one easily computes that $\rho^* (P.\widetilde{\Theta}_{\pi^* M}) - 4\Delta'$ and $D_M + D_{\omega_C\otimes M^{-1}}$ have the same cohomology class. It is therefore sufficient to prove that the support of $\rho^* (P.\widetilde{\Theta}_{\pi^* M}) - 4\Delta'$ contains the support of $D_M + D_{\omega_C\otimes M^{-1}}$. By definition the support of $\rho^* (P.\widetilde{\Theta}_{\pi^* M})$ is the set \[ \{ (p,q) : h^0 (\widetilde{C} ,\pi^* M\otimes{\cal O}_{\widetilde{C} } (p + q -\sigma p -\sigma q) > 0\} \] which, by Riemann-Roch, Serre Duality and the isomorphism $\omega_{\widetilde{C} }\cong\pi^*\omega_C$, is equal to the set \[ \{ (p,q) : h^0 (\widetilde{C} ,\pi^* (\omega_C\otimes M^{-1} )\otimes{\cal O}_{\widetilde{C} } (\sigma p +\sigma q -p-q) > 0\}\; . \] Therefore the support of $\rho^* (P.\widetilde{\Theta}_{\pi^* M})$ contains the union of $\Delta'$ and the sets \[ \{ (p,q) : h^0 (C, M\otimes{\cal O}_C ( -\pi p-\pi q) ) > 0 \} \] and \[ \{ (p,q) : h^0 (C,\omega_C\otimes M^{-1}\otimes{\cal O}_C ( -\pi p-\pi q) ) > 0 \}\; . \] \qed \vskip20pt The following implies Theorem \ref{maintheorem} \begin{proposition}\label{rhobase} The inverse image by $\widetilde{\rho}$ of the support of the base locus of $L |_{\widetilde{\Sigma}}$ is contained in the set of elements $(p,q)$ of $\widetilde{C}^2$ such that $\langle\pi p + \pi q \rangle$ is contained in the intersection of the quadrics containing the canonical curve $\kappa C$. In particular, if $C$ is not trigonal, then the base locus of $L$ does not intersect $\widetilde{\Sigma}$. \end{proposition} {\em Proof :} Let $W_{g-1}^1$ be the subvariety of $Pic^{g-1} C$ parametrizing invertible sheaves with $h^0(C,M) = 2$. By Proposition \ref{newdivG00} the base locus of $L |_{\widetilde{\Sigma}}$ is contained in $\widetilde{\Sigma}\cap\left(\cap_{M \in W_{g-1}^1}\epsilon^{-1}_* ( P .\widetilde{\Theta}_{\pi^* M} )\right)$ where $\epsilon^{-1}_*( P .\widetilde{\Theta}_{\pi^* M} ) =\epsilon^* (P .\widetilde{\Theta}_{\pi^* M}) - 4{\cal E}$. We have $\widetilde{\rho}^* (\epsilon^{-1}_* (P .\widetilde{\Theta}{\pi^* M})) =\rho^* (P .\widetilde{\Theta}_{\pi^* M}) - 4\Delta'$. It immediately follows from Proposition \ref{qpiD} that the inverse image $\widetilde{\rho}^*\left(\cap_{M\in W_{g-1}^1}\epsilon^{-1}_* ( P . \widetilde{\Theta}_{\pi^* M} )\right)$ is supported on the set of elements $(p,q)$ of $\widetilde{C}^2$ such that $\langle\pi p +\pi q\rangle$ is contained in $q_M$ for all $M\in W_{g-1}^1$. Since the quadrics of the form $q_M$ generate $| I_2(C) |$ (see \cite{green84} and \cite{smithvarley90}) and the base locus of $|I_2(C)|$ in the canonical space $|\omega_C |^*$ does not contain any secants to $\kappa C$ for $C$ non-trigonal (see \cite{ACGH} page 124), the proposition follows. \qed \begin{remark}\label{rembase} For $C$ trigonal, $(P,\Xi )$ is the jacobian of a curve, say $X$. Then, using the set-theoretical equalities $V_{00} = X-X$ and $V_{inf,00} = \kappa X$, one can show that the inverse image by $\widetilde{\rho}$ of the support of the base locus of $L |_{\widetilde{\Sigma} }$ is {\em equal} to the set of $(p,q)\in\widetilde{C}^2$ such that $\langle\pi p +\pi q\rangle$ is contained in the intersection of the quadrics containing the canonical curve $\kappa C$. \end{remark} Let \[ \tau_2 :\Gamma_0\longrightarrow H^0 ({\Bbb P} T_0 P,{\cal O}_{{\Bbb P} T_0 P} (2)) \] be the map which to $s\in\Gamma_0$ associates the quadric term of its Taylor expansion at $0$. Then $\tau_2$ is onto because its kernel is $\Gamma_{00}$ which has dimension $2^p - 1 -\frac{1}{2} p(p+1) = $ dim$(\Gamma_0) -h^0 ({\Bbb P} T_0 P,{\cal O}_{{\Bbb P} T_0 P} (2))$. We have \begin{lemma} For $s\in\Gamma_0$, \[ s\in\Gamma_{00}\Longleftrightarrow q(s)\supset{\Bbb P} T_0 P\; . \] \end{lemma} {\em Proof :} By Lemma \ref{Zsqs} part \ref{tangcone} and with the notation there, if $s\in\Gamma_0\setminus\Gamma_{ 00}$, then $\tau_{Z(s)} = q(s)\cap{\Bbb P} T_0 P$. So the projectivizations of the two maps $\tau_2$ and $s\mapsto (\rho^* s) |_{{\Bbb P} T_0 P}\in I_2 (\widetilde{C} ) |_{{\Bbb P} T_0 P}$ are equal. Hence, there exists $\lambda\in{\Bbb C}^*$ such that, for every $s\in\Gamma_0$, we have $\lambda\tau_2 (s) = (\rho^* s) |_{{\Bbb P} T_0 P}$. So \[ s\in\Gamma_{00}\Leftrightarrow\tau_2 (s) = 0\Leftrightarrow (\rho^* s) |_{{\Bbb P} T_0 P} = 0\Leftrightarrow q(s)\supset{\Bbb P} T_0 P\; . \] \qed \vskip20pt Let $I_2 (\widetilde{C} ,\alpha)$ be the subvector space of $I_2 (\widetilde{C})$ consisting of elements which vanish on ${\Bbb P} T_0 P$. By the above lemma and because all elements of $\Gamma_0$ are even, the map $\rho^*$ sends $\Gamma_{00}$ into the subspace $I_2 (\widetilde{C} ,\alpha)^+$ of $\sigma$-invariant elements of $I_2 (\widetilde{C} ,\alpha)$. We have \begin{lemma}\label{I2inv} The subspace $I_2 (\widetilde{C} ,\alpha)^+$ is equal to $I_2 (C)\stackrel{\pi^* }{\subset} I_2 (\widetilde{C})$. \end{lemma} {\em Proof :} The $\sigma$-invariant and $\sigma$-anti-invariant parts of $H^0 (\widetilde{C} ,\omega_{\widetilde{C}} )$ are, respectively, $H^0 (C ,\omega_{C} )$ and $H^0 (C ,\omega_{C}\otimes\alpha )$. Therefore in the decomposition \[ S^2 H^0 (\widetilde{C} ,\omega_{\widetilde{C}} ) = S^2 H^0 (C ,\omega_{C} )\oplus H^0 (C ,\omega_{C} )\otimes H^0 (C ,\omega_{C}\otimes\alpha )\oplus S^2 H^0 (C ,\omega_{C}\otimes\alpha ) \] the space $S^2 H^0 (C ,\omega_{C} )\oplus S^2 H^0 (C ,\omega_{C}\otimes\alpha )$ is the $\sigma$-invariant part of $S^2 H^0 (\widetilde{C} ,\omega_{\widetilde{C}} )$. So $S^2 H^0 (C ,\omega_{C} )$ is the subspace of $\sigma$-invariant elements of $S^2 H^0 (\widetilde{C} ,\omega_{\widetilde{C}} )$ which vanish on ${\Bbb P} T_0 P$. Therefore $I_2 (\widetilde{C} ,\alpha)^+$ is contained in $S^2 H^0 (C ,\omega_{C} )$ and $I_2 (\widetilde{C} ,\alpha)^+$ is the subspace of elements of $S^2 H^0 (C ,\omega_{C} )$ which vanish on $\kappa\widetilde{C}$. This is precisely $I_2(C)$. \qed \begin{corollary} The dimension of $\Gamma_{00}'$ is at least $2^p - 2 - p^2 +p$ and at most $2^p - 1 -\frac{1}{2} p(p+1) - 3 = 2^p - 4 -\frac{1}{2} p(p+1)$. The codimension of ${\cal Q}_{00}'$ in ${\cal Q}_{00}$ is at least $2$. \end{corollary} {\em Proof :} By Lemma \ref{I2inv}, we have $\Gamma_{00}' = Ker (\rho^* :\Gamma_{00}\rightarrow I_2 (C)\stackrel{\pi^* }{\subset} I_2 (\widetilde{C}))$. Hence the dimension of $\Gamma_{00}'$ is at least dim$(\Gamma_{00}) -$ dim$(I_2 (C)) = 2^p - 1 -\frac{1}{2} p(p+1) -\frac{1}{2} (g-2)(g-3) = 2^p - 1 -\frac{1}{2} p(p+1) -\frac{1}{2} (p-1)(p-2) = 2^p - 2 - p^2 +p $. To prove the upper bound for the dimension of $\Gamma_{00}'$, we prove that the dimension of $\rho^*\Gamma_{00}$ is at least $3$. For $C$ non-trigonal, this is an immediate consequence of the result of Proposition \ref{rhobase} which says that $\widetilde{\rho}^* L$ has no base points and the fact that $\widetilde{\rho}^* L$ has positive self-intersection number: this number is easily computed using Lemma \ref{XiEC2} and the calculation of the cohomology class of $D_E$ on page 287 of \cite{griffithsharris78}. Now suppose $C$ trigonal. The linear system $\widetilde{\rho}^* L$ contains the divisors $\widetilde{\rho}^* (P.\widetilde{\Theta}_{\pi^* M}) - 4\Delta'$ for $M\in W^1_{g-1}$. Since $g\geq 5$, the curve $C$ has a unique linear system of degree $3$ and dimension $1$ and we denote the associated invertible sheaf of degree $3$ by $M_0$. Choose three general effective divisors $N$, $N'$ and $N''$ of degree $g-4$ on $C$ and put $M = M_0\otimes{\cal O}_C( N)$, $M' = M_0\otimes{\cal O}_C( N')$ and $M'' = M_0\otimes{\cal O}_C( N'')$. Let $s$, $s'$ and $s''$ be sections whose divisors of zeros are respectively $\widetilde{\rho}^* (P.\widetilde{\Theta}_{\pi^* M}) - 4\Delta'$, $\widetilde{\rho}^* (P.\widetilde{\Theta}_{\pi^* M'}) - 4\Delta'$ and $\widetilde{\rho}^* (P.\widetilde{\Theta}_{\pi^* M''}) - 4\Delta'$. If a non-trivial linear combination of $s, s', s''$ is zero, then the coefficients of at least two of $s, s', s''$, say $s$ and $s'$, are non-zero. Then the restrictions of $s$ and $s'$ to $\widetilde{\rho}^* (P.\widetilde{\Theta}_{\pi^* M''}) - 4\Delta'$ have the same divisors. By Proposition \ref{qpiD}, we have $\widetilde{\rho}^* (P.\widetilde{\Theta}_{\pi^* M}) - 4\Delta' = D_{M} +D_{\omega_C\otimes {M}^{-1}}$. Hence $\widetilde{\rho}^* (P.\widetilde{\Theta}_{\pi^* M}) - 4\Delta' = D_{M_0} + p_1^* (\pi^* N) +p_2^* (\pi^* N) +D_{\omega_C\otimes {M}^{-1}}$. Similarly, $\widetilde{\rho}^* (P.\widetilde{\Theta}_{\pi^* M'}) - 4\Delta' = D_{M_0} + p_1^* (\pi^* N') +p_2^* (\pi^* N') +D_{\omega_C\otimes {M'}^{-1}}$ and $\widetilde{\rho}^* (P.\widetilde{\Theta}_{\pi^* M''}) - 4\Delta' = D_{M_0} + p_1^* (\pi^* N'') +p_2^* (\pi^* N'') +D_{\omega_C\otimes {M''}^{-1}}$. Let $p$ be a point of $\widetilde{C}$ such that $\pi p$ is a point of $N''$. Then $F_p := p_1^* (p)\cong\widetilde{C}$ is contained in $\widetilde{\rho}^* (P.\widetilde{\Theta}_{\pi^* M''}) - 4\Delta'$. Hence $(\widetilde{\rho}^* (P.\widetilde{\Theta}_{\pi^* M}) - 4\Delta' )|_{F_p} = (\widetilde{\rho}^* (P.\widetilde{\Theta}_{\pi^* M'}) - 4\Delta' )|_{F_p}$. Equivalently, $(p_2^* (\pi^* N) + D_{\omega_C\otimes {M}^{-1}} )|_{F_p} = (p_2^* (\pi^* N') + D_{\omega_C\otimes {M'}^{-1}} )|_{F_p}$. Since $N$, $N'$, $N''$ are general, there are unique divisors, say $G$ and $G'$ in, respectively, $|\omega_C\otimes {M}^{-1} |$ and $|\omega_C\otimes {M'}^{-1} |$ containing $\pi p$. Then, after identifying $F_p$ with $\widetilde{C}$, the previous equality of divisors becomes $\pi^* N + \pi^* G -p-\sigma p =\pi^* N' + \pi^* G' -p-\sigma p$. Since $N'$ is general and $G$ depends only on $N$ and $N''$, this is not possible. Hence there are no non-trivial linear relations between $s, s', s''$ and $\widetilde{\rho}^* L$ is at least a net. In the non-trigonal case, the codimension of ${\cal Q}_{00}'$ in ${\cal Q}_{00}$ is at least $2$ because the restriction of $\widetilde{\rho}^* L$ to $\Delta'$ has positive degree and no base points, therefore it is at least a pencil. In the trigonal case, use divisors as above to show that $(\widetilde{\rho}^* L)|_{\Delta'}$ contains at least two distinct divisors. \qed \begin{remark} In case $p=4$, the above gives a second proof of the result of \cite{I3} (page 148) saying that the dimension of $\Gamma_{00}'$ is $2$. \end{remark} \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}

9704

alg-geom/9704001

en

https://arxiv.org/abs/alg-geom/9704001

alg-geom/9704001

Maxim Braverman

Alexander Braverman and Maxim Braverman

Tempered currents and the cohomology of the remote fiber of a real polynomial map

LaTeX 2e

Let $p:R^n\to R$ be a polynomial map. Consider the complex $S'\Omega^*(\RR^n)$ of tempered currents on $R^n$ with the twisted differential $d_p=d-dp$ where $d$ is the usual exterior differential and $dp$ stands for the exterior multiplication by $dp$. Let $t\in R$ and let $F_t=p^{-1}(t)$. In this paper we prove that the reduced cohomology $\tilda H^k(F_t;C)$ of $F_t$ is isomorphic to $H^{k+1}(S'\Omega^*(\RR^n),d_p)$ in the case when $p$ is homogeneous and $t$ is any positive real number. We conjecture that this isomorphism holds for any polynomial $p$, for $t$ large enough (we call the $F_t$ for $t >> 0$ the remote fiber of $p$) and we prove this conjecture for polynomials that satisfy certain technical condition. The result is analogous to that of A. Dimca and M. Saito, who give a similar (algebraic) way to compute the reduced cohomology of the generic fiber of a complex polynomial.

alg-geom

\section{#2}\label{S:#1}% \ifShowLabels \TeXref{{S:#1}} \fi} \newcommand{\ssec}[2]{\subsection{#2}\label{SS:#1}% \ifShowLabels \TeXref{{SS:#1}} \fi} \newcommand{\refs}[1]{Section ~\ref{S:#1}} \newcommand{\refss}[1]{Section ~\ref{SS:#1}} \newcommand{\reft}[1]{Theorem ~\ref{T:#1}} \newcommand{\refl}[1]{Lemma ~\ref{L:#1}} \newcommand{\refp}[1]{Proposition ~\ref{P:#1}} \newcommand{\refc}[1]{Corollary ~\ref{C:#1}} \newcommand{\refd}[1]{Definition ~\ref{D:#1}} \newcommand{\refr}[1]{Remark ~\ref{R:#1}} \newcommand{\refe}[1]{\eqref{E:#1}} \newenvironment{thm}[1]% { \begin{Thm} \label{T:#1} \ifShowLabels \TeXref{T:#1} \fi }% { \end{Thm} } \renewcommand{\th}[1]{\begin{thm}{#1} \sl } \renewcommand{\eth}{\end{thm} } \newenvironment{lemma}[1]% { \begin{Lem} \label{L:#1} \ifShowLabels \TeXref{L:#1} \fi }% { \end{Lem} } \newcommand{\lem}[1]{\begin{lemma}{#1} \sl} \newcommand{\end{lemma}}{\end{lemma}} \newenvironment{propos}[1]% { \begin{Prop} \label{P:#1} \ifShowLabels \TeXref{P:#1} \fi }% { \end{Prop} } \newcommand{\prop}[1]{\begin{propos}{#1}\sl } \newcommand{\end{propos}}{\end{propos}} \newenvironment{corol}[1]% { \begin{Cor} \label{C:#1} \ifShowLabels \TeXref{C:#1} \fi }% { \end{Cor} } \newcommand{\cor}[1]{\begin{corol}{#1} \sl } \newcommand{\end{corol}}{\end{corol}} \newenvironment{defeni}[1]% { \begin{Def} \label{D:#1} \ifShowLabels \TeXref{D:#1} \fi }% { \end{Def} } \newcommand{\defe}[1]{\begin{defeni}{#1} \sl } \newcommand{\end{defeni}}{\end{defeni}} \newenvironment{remark}[1]% { \begin{Rem} \label{R:#1} \ifShowLabels \TeXref{R:#1} \fi }% { \end{Rem} } \newcommand{\rem}[1]{\begin{remark}{#1}} \newcommand{\end{remark}}{\end{remark}} \newcommand{\eq}[1]% { \ifShowLabels \TeXref{E:#1} \fi \begin{equation} \label{E:#1} } \newcommand{ \end{equation} }{ \end{equation} } \newcommand{ \begin{proof} }{ \begin{proof} } \newcommand{ \end{proof} }{ \end{proof} } \newcommand\alp{\alpha} \newcommand\Alp{\Alpha} \newcommand\bet{\beta} \newcommand\gam{\gamma} \newcommand\Gam{\Gamma} \newcommand\del{\delta} \newcommand\Del{\Delta} \newcommand\eps{\varepsilon} \newcommand\zet{\zeta} \newcommand\tet{\theta} \newcommand\Tet{\Theta} \newcommand\iot{\iota} \newcommand\kap{\kappa} \newcommand\lam{\lambda} \newcommand\Lam{\Lambda} \newcommand\sig{\sigma} \newcommand\Sig{\Sigma} \newcommand\vphi{\varphi} \newcommand\ome{\omega} \newcommand\Ome{\Omega} \newcommand\calA{{\mathcal{A}}} \newcommand\calB{{\mathcal{B}}} \newcommand\calC{{\mathcal{C}}} \newcommand\calD{{\mathcal{D}}} \newcommand\calE{{\mathcal{E}}} \newcommand\calF{{\mathcal{F}}} \newcommand\calG{{\mathcal{G}}} \newcommand\calH{{\mathcal{H}}} \newcommand\calI{{\mathcal{I}}} \newcommand\calJ{{\mathcal{J}}} \newcommand\calK{{\mathcal{K}}} \newcommand\calL{{\mathcal{L}}} \newcommand\calM{{\mathcal{M}}} \newcommand\calN{{\mathcal{N}}} \newcommand\calO{{\mathcal{O}}} \newcommand\calP{{\mathcal{P}}} \newcommand\calQ{{\mathcal{Q}}} \newcommand\calR{{\mathcal{R}}} \newcommand\calS{{\mathcal{S}}} \newcommand\calT{{\mathcal{T}}} \newcommand\calU{{\mathcal{U}}} \newcommand\calV{{\mathcal{V}}} \newcommand\calW{{\mathcal{W}}} \newcommand\calX{{\mathcal{X}}} \newcommand\calY{{\mathcal{Y}}} \newcommand\calZ{{\mathcal{Z}}} \newcommand\bfa{{\mathbf a}} \newcommand\bfA{{\mathbf A}} \newcommand\bfb{{\mathbf b}} \newcommand\bfB{{\mathbf B}} \newcommand\bfc{{\mathbf c}} \newcommand\bfC{{\mathbf C}} \newcommand\bfd{{\mathbf d}} \newcommand\bfD{{\mathbf D}} \newcommand\bfe{{\mathbf e}} \newcommand\bfE{{\mathbf E}} \newcommand\bff{{\mathbf f}} \newcommand\bfF{{\mathbf F}} \newcommand\bfg{{\mathbf g}} \newcommand\bfG{{\mathbf G}} \newcommand\bfh{{\mathbf h}} \newcommand\bfH{{\mathbf H}} \newcommand\bfi{{\mathbf i}} \newcommand\bfI{{\mathbf I}} \newcommand\bfj{{\mathbf j}} \newcommand\bfJ{{\mathbf J}} \newcommand\bfk{{\mathbf k}} \newcommand\bfK{{\mathbf K}} \newcommand\bfl{{\mathbf l}} \newcommand\bfL{{\mathbf L}} \newcommand\bfm{{\mathbf m}} \newcommand\bfM{{\mathbf M}} \newcommand\bfn{{\mathbf n}} \newcommand\bfN{{\mathbf N}} \newcommand\bfo{{\mathbf o}} \newcommand\bfO{{\mathbf O}} \newcommand\bfp{{\mathbf p}} \newcommand\bfP{{\mathbf P}} \newcommand\bfq{{\mathbf q}} \newcommand\bfQ{{\mathbf Q}} \newcommand\bfr{{\mathbf r}} \newcommand\bfR{{\mathbf R}} \newcommand\bfs{{\mathbf s}} \newcommand\bfS{{\mathbf S}} \newcommand\bft{{\mathbf t}} \newcommand\bfT{{\mathbf T}} \newcommand\bfu{{\mathbf u}} \newcommand\bfU{{\mathbf U}} \newcommand\bfv{{\mathbf v}} \newcommand\bfV{{\mathbf V}} \newcommand\bfw{{\mathbf w}} \newcommand\bfW{{\mathbf W}} \newcommand\bfx{{\mathbf x}} \newcommand\bfX{{\mathbf X}} \newcommand\bfy{{\mathbf y}} \newcommand\bfY{{\mathbf Y}} \newcommand\bfz{{\mathbf z}} \newcommand\bfZ{{\mathbf Z}} \newcommand\QQ{\mathbb{Q}} \newcommand\WW{\mathbb{W}} \newcommand\EE{\mathbb{E}} \newcommand\RR{\mathbb{R}} \newcommand\TT{\mathbb{T}} \newcommand\YY{\mathbb{Y}} \newcommand\UU{\mathbb{U}} \newcommand\II{\mathbb{I}} \newcommand\OO{\mathbb{O}} \newcommand\PP{\mathbb{P}} \renewcommand\AA{\mathbb{A}} \renewcommand\SS{\mathbb{S}} \newcommand\DD{\mathbb{D}} \newcommand\FF{\mathbb{F}} \newcommand\GG{\mathbb{G}} \newcommand\HH{\mathbb{H}} \newcommand\JJ{\mathbb{J}} \newcommand\KK{\mathbb{K}} \newcommand\LL{\mathbb{L}} \newcommand\ZZ{\mathbb{Z}} \newcommand\XX{\mathbb{X}} \newcommand\CC{\mathbb{C}} \newcommand\VV{\mathbb{V}} \newcommand\BB{\mathbb{B}} \newcommand\NN{\mathbb{N}} \newcommand\MM{\mathbb{M}} \newcommand\grA{{\mathfrak{A}}} \newcommand\gra{{\mathfrak{a}}} \newcommand\grB{{\mathfrak{B}}} \newcommand\grb{{\mathfrak{b}}} \newcommand\grC{{\mathfrak{C}}} \newcommand\grc{{\mathfrak{c}}} \newcommand\grD{{\mathfrak{D}}} \newcommand\grd{{\mathfrak{d}}} \newcommand\grE{{\mathfrak{E}}} \newcommand\gre{{\mathfrak{e}}} \newcommand\grF{{\mathfrak{F}}} \newcommand\grf{{\mathfrak{f}}} \newcommand\grG{{\mathfrak{G}}} \newcommand\grg{{\mathfrak{g}}} \newcommand\grH{{\mathfrak{H}}} \newcommand\grh{{\mathfrak{h}}} \newcommand\grI{{\mathfrak{I}}} \newcommand\gri{{\mathfrak{i}}} \newcommand\grJ{{\mathfrak{J}}} \newcommand\grj{{\mathfrak{j}}} \newcommand\grK{{\mathfrak{K}}} \newcommand\grk{{\mathfrak{k}}} \newcommand\grL{{\mathfrak{L}}} \newcommand\grl{{\mathfrak{l}}} \newcommand\grM{{\mathfrak{M}}} \newcommand\grm{{\mathfrak{m}}} \newcommand\grN{{\mathfrak{N}}} \newcommand\grn{{\mathfrak{n}}} \newcommand\grO{{\mathfrak{O}}} \newcommand\gro{{\mathfrak{o}}} \newcommand\grP{{\mathfrak{P}}} \newcommand\grp{{\mathfrak{p}}} \newcommand\grQ{{\mathfrak{Q}}} \newcommand\grq{{\mathfrak{q}}} \newcommand\grR{{\mathfrak{R}}} \newcommand\grr{{\mathfrak{r}}} \newcommand\grS{{\mathfrak{S}}} \newcommand\grs{{\mathfrak{s}}} \newcommand\grT{{\mathfrak{T}}} \newcommand\grt{{\mathfrak{t}}} \newcommand\grU{{\mathfrak{U}}} \newcommand\gru{{\mathfrak{u}}} \newcommand\grV{{\mathfrak{V}}} \newcommand\grv{{\mathfrak{v}}} \newcommand\grW{{\mathfrak{W}}} \newcommand\grw{{\mathfrak{w}}} \newcommand\grX{{\mathfrak{X}}} \newcommand\grx{{\mathfrak{x}}} \newcommand\grY{{\mathfrak{Y}}} \newcommand\gry{{\mathfrak{y}}} \newcommand\grZ{{\mathfrak{Z}}} \newcommand\grz{{\mathfrak{z}}} \newcommand\nek{,\ldots,} \newcommand\sdp{\times \hskip -0.3em {\raise 0.3ex \hbox{$\scriptscriptstyle |$}}} \newcommand\area{\operatorname{area}} \newcommand\Aug{\operatorname{Aug}} \newcommand\Aut{\operatorname{Aut}} \newcommand\Char{\operatorname{Char}} \newcommand\Cl{\operatorname{Cl}} \newcommand\cf{{\rm \,cf\,}} \newcommand\operatorname{Cone}{\operatorname{Cone}} \newcommand\cont{\operatorname{cont}} \newcommand\codim{\operatorname{codim}} \newcommand\conv{\operatorname{conv}} \newcommand\Conv{\operatorname{Conv}} \newcommand\const{\operatorname{const}} \newcommand\Const{\operatorname{Const}} \newcommand\Det{\operatorname{Det}} \newcommand\diag{\operatorname{diag}} \newcommand\diam{\operatorname{diam}} \newcommand\Diam{\operatorname{Diam}} \newcommand\dist{\operatorname{dist}} \newcommand\Dom{\operatorname{Dom}} \newcommand\dom{\operatorname{dom}} \newcommand\End{\operatorname{End\,}} \newcommand\Ext{\operatorname{Ext}} \newcommand\esssup{\operatorname{ess\ sup}} \newcommand\Ran{{\rm Ran}} \newcommand\RANK{\operatorname{rank}} \newcommand\Geo{\operatorname{Geo}} \newcommand\GL{\operatorname{GL}} \newcommand\Gr{\operatorname{Gr}} \newcommand\gl{\operatorname{gl}} \newcommand\grad{\operatorname {\rm grad}} \newcommand\Hom{\operatorname {Hom}} \newcommand\RHom{\operatorname {RHom}} \newcommand\im{\operatorname {im}} \newcommand\IM{\operatorname{Im}} \newcommand\Ind{\operatorname{Ind}} \newcommand\ind{\operatorname{ind}} \newcommand\Inf{\operatorname{Inf}} \newcommand\Int{\operatorname{Int}} \newcommand\Min{\operatorname{Min}} \newcommand\MOD{\operatorname{mod}} \newcommand{\operatorname{Mor}}{\operatorname{Mor}} \newcommand{\operatorname{Ob\,}}{\operatorname{Ob\,}} \newcommand\ord{\operatorname{ord}} \newcommand\Ka{\operatorname{Ka}} \newcommand\Ker{\operatorname{Ker}} \newcommand\PGL{{\rm PGL}} \newcommand\PGSp{{\rm PGSp}} \newcommand\Plt{\operatorname{Plt}} \newcommand\proj{\operatorname{proj}} \newcommand\Proj{\operatorname{Proj}} \newcommand\res{\rm res} \newcommand\rk{\operatorname{rk}} \newcommand\Range{\operatorname{Range}} \newcommand\RE{\operatorname{Re}} \newcommand\Res{\operatorname{Res}} \newcommand\rot{\operatorname{rot}} \newcommand\Max{\operatorname{Max}} \newcommand\Maximum{\operatorname{Maximum}} \newcommand\Minimum{\operatorname{Minimum}} \newcommand\Minimize{\operatorname{Minimize}} \newcommand\Prob{\operatorname{Prob}} \newcommand\sech{\rm sech} \newcommand\sgn{\operatorname{sgn}} \newcommand{\operatorname{sign}}{\operatorname{sign}} \newcommand\SL{{\rm SL}} \newcommand\Sbm{\operatorname{Sbm}} \newcommand\SO{{\rm SO}} \newcommand\SPAN{\operatorname{span}} \newcommand\spec{{\rm spec}} \newcommand\supess{\operatorname{sup\ ess}} \newcommand\supp{\operatorname{supp}} \newcommand\Supp{\operatorname{Supp}} \newcommand\Sup{\operatorname{Sup}} \newcommand\Sym{\operatorname{Sym}} \newcommand\tr{\operatorname{tr}} \newcommand\Tr{\operatorname{Tr}} \newcommand\Tor{\operatorname{Tor}} \newcommand\Var{\operatorname{Var}} \newcommand\Vol{\operatorname{Vol}} \newcommand\oa{{\overline{a}}} \newcommand\oA{{\overline{A}}} \newcommand\ob{{\overline{b}}} \newcommand\oB{{\overline{B}}} \newcommand\oc{{\overline{c}}} \newcommand\oC{{\overline{C}}} \newcommand\oD{{\overline{D}}} \newcommand\od{{\overline{d}}} \newcommand\oE{{\overline{E}}} \renewcommand\oe{{\overline{e}}} \newcommand\of{{\overline{f}}} \newcommand\oF{{\overline{F}}} \newcommand\og{{\overline{g}}} \newcommand\oG{{\overline{G}}} \newcommand\oh{{\overline{h}}} \newcommand\oH{{\overline{H}}} \newcommand\oI{{\overline{I}}} \newcommand\oj{{\overline{j}}} \newcommand\oJ{{\overline{J}}} \newcommand\ok{{\overline{k}}} \newcommand\oK{{\overline{K}}} \newcommand\oL{{\overline{L}}} \newcommand\om{{\overline{m}}} \newcommand\oM{{\overline{M}}} \newcommand\oN{{\overline{N}}} \newcommand\oO{{\overline{O}}} \newcommand\oo{{\overline{o}}} \newcommand\op{{\overline{p}}} \newcommand\oP{{\overline{P}}} \newcommand\oq{{\overline{q}}} \newcommand\oQ{{\overline{Q}}} \newcommand\OR{{\overline{r}}} \newcommand\oS{{\overline{S}}} \newcommand\os{{\overline{s}}} \newcommand\ot{{\overline{t}}} \newcommand\oT{{\overline{T}}} \newcommand\ou{{\overline{u}}} \newcommand\oU{{\overline{U}}} \newcommand\ov{{\overline{v}}} \newcommand\oV{{\overline{V}}} \newcommand\ow{{\overline{w}}} \newcommand\oW{{\overline{W}}} \newcommand\ox{{\overline{x}}} \newcommand\oX{{\overline{X}}} \newcommand\oy{{\overline{y}}} \newcommand\oY{{\overline{Y}}} \newcommand\oz{{\overline{z}}} \newcommand\oZ{{\overline{Z}}} \newcommand\oalp{{\overline{\alpha}}} \newcommand\obet{{\overline{\bet}}} \newcommand\odel{{\overline{\del}}} \newcommand\oDel{{\overline{\Del}}} \newcommand\ocup{{\overline{\cup}}} \newcommand\ovarphi{{\overline{\varphi}}} \newcommand\ochi{{\overline{\chi}}} \newcommand\oeps{{\overline{\eps}}} \newcommand\oeta{{\overline{\eta}}} \newcommand\ogam{{\overline{\gam}}} \newcommand\okap{{\overline{\kap}}} \newcommand\olam{{\overline{\lambda}}} \newcommand\oLam{{\overline{\Lambda}}} \newcommand\omu{{\overline{\mu}}} \newcommand\onu{{\overline{\nu}}} \newcommand\oOme{{\overline{\Ome}}} \newcommand\ophi{\overline{\phi}} \newcommand\oPhi{{\overline{\Phi}}} \newcommand\opi{{\overline{\pi}}} \newcommand\oPsi{{\overline{\Psi}}} \newcommand\opsi{{\overline{\psi}}} \newcommand\orho{{\overline{\rho}}} \newcommand\osig{{\overline{\sig}}} \newcommand\otau{{\overline{\tau}}} \newcommand\otet{{\overline{\theta}}} \newcommand\oxi{{\overline{\xi}}} \newcommand\oome{\overline{\ome}} \newcommand\opart{{\overline{\partial}}} \newcommand\ua{{\underline{a}}} \newcommand\ub{{\underline{b}}} \newcommand\uc{{\underline{c}}} \newcommand\uD{{\underline{D}}} \newcommand\uk{{\underline{k}}} \newcommand\ue{{\underline{e}}} \newcommand\uj{{\underline{j}}} \newcommand\ul{{\underline{l}}} \newcommand\uL{{\underline{L}}} \newcommand\uo{{\underline{o}}} \newcommand\uO{{\underline{O}}} \newcommand\uP{{\underline{P}}} \newcommand\uQ{{\underline{Q}}} \newcommand\um{{\underline{m}}} \newcommand\uM{{\underline{M}}} \newcommand\un{{\underline{n}}} \newcommand\us{{\underline{s}}} \newcommand\ut{{\underline{t}}} \newcommand\uu{{\underline{u}}} \newcommand\uv{{\underline{v}}} \newcommand\uV{{\underline{V}}} \newcommand\ux{{\underline{x}}} \newcommand\uX{{\underline{X}}} \newcommand\uy{{\underline{y}}} \newcommand\uz{{\underline{z}}} \newcommand\ualp{{\underline{\alp}}} \newcommand\ubet{{\underline{\bet}}} \newcommand\uchi{{\underline{\chi}}} \newcommand\udel{{\underline{\del}}} \newcommand\uell{{\underline{\ell}}} \newcommand\ueps{{\underline{\eps}}} \newcommand\ueta{{\underline{\eta}}} \newcommand\uGam{{\underline{\Gamma}}} \newcommand\unu{{\underline{\nu}}} \newcommand\uome{{\underline{\omega}}} \newcommand\utet{{\underline{\tet}}} \newcommand\ulam{{\underline{\lam}}} \newcommand\hata{{\widehat{a}}} \newcommand\hatA{{\widehat{A}}} \newcommand\hatb{{\widehat{b}}} \newcommand\hatc{{\widehat{c}}} \newcommand\hatC{{\widehat{C}}} \newcommand\hatB{{\widehat{B}}} \newcommand\hatD{{\widehat{D}}} \newcommand\hate{{\widehat{e}}} \newcommand\hatE{{\widehat{E}}} \newcommand\hatf{{\widehat{f}}} \newcommand\hatF{{\widehat{F}}} \newcommand\hatg{{\widehat{g}}} \newcommand\hatG{{\widehat{G}}} \newcommand\hath{{\widehat{h}}} \newcommand\hatH{{\widehat{H}}} \newcommand\hati{{\hat{i}}} \newcommand\hatI{{\hat{I}}} \newcommand\hatj{{\widehat{j}}} \newcommand\hatJ{{\widehat{J}}} \newcommand\hatk{{\widehat{k}}} \newcommand\hatK{{\widehat{K}}} \newcommand\hatL{{\widehat{L}}} \newcommand\hatm{{\widehat{m}}} \newcommand\hatM{{\widehat{M}}} \newcommand\hatn{{\widehat{n}}} \newcommand\hatN{{\widehat{N}}} \newcommand\hatp{{\widehat{p}}} \newcommand\hatP{{\widehat{P}}} \newcommand\hatr{{\widehat{r}}} \newcommand\hatR{{\widehat{R}}} \newcommand\hatq{{\widehat{q}}} \newcommand\hatQ{{\widehat{Q}}} \newcommand\hatT{{\widehat{T}}} \newcommand\hatu{{\widehat{u}}} \newcommand\hatU{{\widehat{U}}} \newcommand\hatV{{\widehat{V}}} \newcommand\hatv{{\widehat{v}}} \newcommand\hatw{{\widehat{w}}} \newcommand\hatW{{\widehat{W}}} \newcommand\hatx{{\widehat{x}}} \newcommand\hatX{{\widehat{X}}} \newcommand\haty{{\widehat{y}}} \newcommand\hatY{{\widehat{Y}}} \newcommand\hatZ{{\widehat{Z}}} \newcommand\hatz{{\widehat{z}}} \newcommand\hatalp{{\widehat{\alpha}}} \newcommand\hatdel{{\widehat{\delta}}} \newcommand\hatDel{{\widehat{\Delta}}} \newcommand\hatbet{{\widehat{\beta}}} \newcommand\hateps{{\hat{\eps}}} \newcommand\hatgam{{\widehat{\gamma}}} \newcommand\hatGam{{\widehat{\Gamma}}} \newcommand\hatlam{{\widehat{\lambda}}} \newcommand\hatmu{{\widehat{\mu}}} \newcommand\hatnu{{\widehat{\nu}}} \newcommand\hatOme{{\widehat{\Ome}}} \newcommand\hatphi{{\widehat{\phi}}} \newcommand\hatPhi{{\widehat{\Phi}}} \newcommand\hatpi{{\widehat{\pi}}} \newcommand\hatpsi{{\widehat{\psi}}} \newcommand\hatPsi{{\widehat{\Psi}}} \newcommand\hatrho{{\widehat{\rho}}} \newcommand\hatsig{{\widehat{\sig}}} \newcommand\hatSig{{\widehat{\Sig}}} \newcommand\hattau{{\widehat{\tau}}} \newcommand\hattet{{\widehat{\theta}}} \newcommand\hatvarphi{{\widehat{\varphi}}} \newcommand\hatZZ{{\widehat{\ZZ}}} \newcommand\tilA{{\widetilde{A}}} \newcommand\tila{{\widetilde{a}}} \newcommand\tilB{{\widetilde{B}}} \newcommand\tilb{{\widetilde{b}}} \newcommand\tilc{{\widetilde{c}}} \newcommand\tilC{{\widetilde{C}}} \newcommand\tild{{\widetilde{d}}} \newcommand\tilD{{\widetilde{D}}} \newcommand\tilE{{\widetilde{E}}} \newcommand\tilf{{\widetilde{f}}} \newcommand\tilF{{\widetilde{F}}} \newcommand\tilg{{\widetilde{g}}} \newcommand\tilG{{\widetilde{G}}} \newcommand\tilh{{\widetilde{h}}} \newcommand\tilk{{\widetilde{k}}} \newcommand\tilK{{\widetilde{K}}} \newcommand\tilj{{\widetilde{j}}} \newcommand\tilm{{\widetilde{m}}} \newcommand\tilM{{\widetilde{M}}} \newcommand\tilH{{\widetilde{H}}} \newcommand\tilL{{\widetilde{L}}} \newcommand\tilN{{\widetilde{N}}} \newcommand\tiln{{\widetilde{n}}} \newcommand\tilO{{\widetilde{O}}} \newcommand\tilP{{\widetilde{P}}} \newcommand\tilp{{\widetilde{p}}} \newcommand\tilq{{\widetilde{q}}} \newcommand\tilQ{{\widetilde{Q}}} \newcommand\tilR{{\widetilde{R}}} \newcommand\tilr{{\widetilde{r}}} \newcommand\tilS{{\widetilde{S}}} \newcommand\tils{{\widetilde{s}}} \newcommand\tilT{{\widetilde{T}}} \newcommand\tilt{{\widetilde{t}}} \newcommand\tilu{{\widetilde{u}}} \newcommand\tilU{{\widetilde{U}}} \newcommand\tilv{{\widetilde{v}}} \newcommand\tilV{{\widetilde{V}}} \newcommand\tilw{{\widetilde{w}}} \newcommand\tilW{{\widetilde{W}}} \newcommand\tilX{{\widetilde{X}}} \newcommand\tilx{{\widetilde{x}}} \newcommand\tily{{\widetilde{y}}} \newcommand\tilY{{\widetilde{Y}}} \newcommand\tilZ{{\widetilde{Z}}} \newcommand\tilz{{\widetilde{z}}} \newcommand\tilalp{{\widetilde{\alpha}}} \newcommand\tilbet{{\widetilde{\beta}}} \newcommand\tildel{{\widetilde{\delta}}} \newcommand\tilDel{{\widetilde{\Delta}}} \newcommand\tilchi{{\widetilde{\chi}}} \newcommand\tileta{{\widetilde{\eta}}} \newcommand\tilgam{{\widetilde{\gamma}}} \newcommand\tilGam{{\widetilde{\Gamma}}} \newcommand\tilome{{\widetilde{\ome}}} \newcommand\tillam{{\widetilde{\lam}}} \newcommand\tilmu{{\widetilde{\mu}}} \newcommand\tilphi{{\widetilde{\phi}}} \newcommand\tilpi{{\widetilde{\pi}}} \newcommand\tilpsi{{\widetilde{\psi}}} \renewcommand\tilome{{\widetilde{\ome}}} \newcommand\tilOme{{\widetilde{\Ome}}} \newcommand\tilPhi{{\widetilde{\Phi}}} \newcommand\tilQQ{{\widetilde{\QQ}}} \newcommand\tilrho{{\widetilde{\rho}}} \newcommand\tilsig{{\widetilde{\sig}}} \newcommand\tiltau{{\widetilde{\tau}}} \newcommand\tiltet{{\widetilde{\theta}}} \newcommand\tilvarphi{{\widetilde{\varphi}}} \newcommand\tilxi{{\widetilde{\xi}}} \newcommand\twolongrightarrow{\ \hbox{$\longrightarrow\hskip -17pt \longrightarrow$}\ } \renewcommand{\>}{\rangle} \newcommand{\<}{\langle} \setlength{\textwidth}{6.3in} \addtolength{\oddsidemargin}{-1.7cm} \addtolength{\evensidemargin}{-1.7cm} \def\calO{\calO} \def\Ome{\Ome} \def\calS'\w{\calS'\Ome} \def\RR^n{\RR^n} \def\partial{\partial} \def\bullet{\bullet} \def\n{\nabla} \def\operatorname{Cone}{\operatorname{Cone}} \begin{document} \title[The cohomology of the remote fiber of a real polynomial map]{Tempered currents and the cohomology of the remote fiber of a real polynomial map} \author{Alexander Braverman \and Maxim Braverman} \address{School of Mathematical Sciences\\ Tel-Aviv University\\ Ramat-Aviv 69978, Israel} \email{[email protected]} \address{Department of Mathematics\\ Ohio State University\\ Columbus, Ohio 43210} \email{[email protected]} \begin{abstract} Let $p:\RR^n\to\RR$ be a polynomial map. Consider the complex $\calS'\w^{\bullet}(\RR^n)$ of tempered currents on $\RR^n$ with the twisted differential $d_p=d-dp$ where $d$ is the usual exterior differential and $dp$ stands for the exterior multiplication by $dp$. Let $t\in\RR$ and let $F_t=p^{-1}(t)$. In this paper we prove that the reduced cohomology $\tilH^k(F_t;\CC)$ of $F_t$ is isomorphic to $H^{k+1}(\calS'\w^{\bullet}(\RR^n),d_p)$ in the case when $p$ is homogeneous and $t$ is any positive real number. We conjecture that this isomorphism holds for any polynomial $p$, for $t$ large enough (we call the $F_t$ for $t\gg 0$ {\it the remote fiber} of $p$) and we prove this conjecture for polynomials that satisfy certain technical condition (cf. \reft{main'}). The result is analogous to that of A.~Dimca and M.~Saito (\cite{DimSa}), who give a similar (algebraic) way to compute the reduced cohomology of the generic fiber of a complex polynomial. \end{abstract} \maketitle \sec{introd}{Introduction} \ssec{0.1}{The Dimca-Saito theorem}Let $p:\CC^n \to \CC$ be a complex polynomial. Let $F$ denote the {\em generic fiber} of $p$ (it is well-defined as a topological space). In \cite{DimSa}, A.~Dimca and M.~Saito have given the following algebraic way to compute the cohomology of $F$. Let $\Ome^{\bullet}$ denote the De Rham algebra of polynomial differential forms on $\CC^n$. Define a differential $d_p$ on $\Ome^{\bullet}$ by $$ d_p(\ome)=d\ome-dp\wedge \ome $$ \begin{Thm}[\textbf{Dimca-Saito}]\label{T:DS}\sl There exists an isomorphism $$ H^{k+1}(\Ome^{\bullet},d_p)\simeq \tilH^{k}(F,\CC) \quad\text{for }k=0,1\nek n-1 $$ where $\tilH^{\bullet}(F,\CC)$ denotes the reduced cohomology of $F$ with coefficients in $\CC$. \end{Thm} \ssec{0.2}{The main result} The main purpose of this paper is to describe certain real analogue of \reft{DS}. Namely, let now $p:\RR^n\to \RR$ be a real polynomial. Then (cf. \cite{Varch1}) the topological type of the fiber $F_t=p^{-1}(t)$ does not depend upon $t$ provided $t$ is large enough. We shall refer to $F_t$ as to {\it remote fiber} of $p$ and we shall be interested in the cohomology of $F_t$. Let $\calO:=\CC\, [x_1\nek x_n]$ denote the ring of complex polynomials on $\RR^n$ and let \/ $\calS'(\RR^n)$ \/ denote the $\calO$ module of tempered complex valued distributions on $\RR^n$. Let $\Ome^\bullet$ be the complex of global algebraic differential forms on $\RR^n$. Consider the space $$ \calS'\w^\bullet(\RR^n)=\calS'(\RR^n)\otimes_{\calO} \Ome^\bullet $$ of tempered currents and define a differential $d_p:\calS'\w^{\bullet}(\RR^n)\to \calS'\w^{\bullet+1}(\RR^n)$ on $\calS'\w^{\bullet}(\RR^n)$ by $$ d_p(\ome)=d\ome - dp\wedge \ome \qquad \text{for} \quad \ome\in\calS'\w^{\bullet}(\RR^n). $$ In this paper we discuss the following \begin{Conj}\label{T:mainC}\sl For any real polynomial $p:\RR^n\to \RR$ the following isomorphism holds: \eq{main} H^{k+1}(\calS'\w^{\bullet}(\RR^n),d_p)=\tilH^{k}(F_t;\CC), \qquad k=0,1,\dots,n-1 \end{equation} where $\tilH$ denotes reduced cohomology. \end{Conj} In particular, we prove the following \th{main} Assume that $p:\RR^n\to \RR$ is a homogeneous polynomial map of degree $m$, i.e. $p(sx)=s^mp(x)$ for any $x\in\RR^n, \ s\in \RR$. For any $t>0$, the isomorphism \refe{main} holds. \eth \rem{rem} {\it a)} \ By definition, the reduced cohomology of any topological space $X$ is the cohomology of the complex $$\begin{CD} 0\to \CC @>\eps>> H^0(X;\CC)@>0>>\cdots @>0>>H^k(X;\CC)\to\cdots \end{CD}$$ where $\eps=0$ if $X$ is empty and $\eps$ is the natural map $\CC=H^0(pt;\CC)\to H^0(X;\CC)$ coming from the projection $X\to pt$ otherwise. Therefore in the case when $X$ is empty one should have $\tilH^{-1}(X;\CC)=\CC$ (but $\tilH^{-1}(X;\CC)=0$ if $X$ contains at least one point). With this convention, \reft{main} remains true also for $k=-1$. {\it b) \ }Note that if instead of the complex $\calS'\w^{\bullet}(\RR^n)$ we considered the complex of all currents with the same differential $d_p$, then we would get a complex quasi-isomorphic to the usual complex of currents on $\RR^n$ with the ordinary exterior differential $d$ (since $d$ and $d_p$ are conjugate to one another by means of the function $e^p$). Therefore if we do not impose any growth conditions on our currents we will not get any interesting cohomology. \end{remark} \ssec{0.4}{Sketch of the proof} \ {\it Step 1. \ }Let $U_t=\{x\in\RR^n:\ p(x)>t\}$ (note that $U_t$ might be empty). Then, $U_t$ is diffeomorphic to the product $F_t\times (0,\infty)$, for any $t>0$. Using the long cohomological sequence of the pair $(\RR^n,U_t)$ one can easily see that $\tilH^{k-1}(F_t,\CC)=H^k(\RR^n,U_t;\CC)$. {\it Step 2. \ }Let $\calD'\Ome^{\bullet}(U_t)$ denote the complex of all currents on $U_t$. In \refss{S'} we define certain subcomplex $\calS'\w^{\bullet}(U_t)$ of $\calD'\Ome^{\bullet}(U_t)$ ({\it the complex of tempered currents on $U_t$}) and prove that its natural inclusion into $\calD'\Ome^{\bullet}(U_t)$ is a quasi-isomorphism. {\it Step 3. \ }Let $\calD'\Ome^{\bullet}(\RR^n)$ denote the complex of all currents on $\RR^n$. Let $\theta (s)$ be a smooth function on $\RR$, such that $\theta (s)=s$ for $s<1$ and $\theta (s)=0$ for $s>2$. Define $\tilp:\RR^n\to \RR$ by $\tilp(x)=\theta (p(x))$. Let $\calS'_p\Ome^{\bullet}(\RR^n)$ denote the space of all currents $\ome$ on $\RR^n$, such that $e^{\tilp}\ome\in \calS'\w^{\bullet}(\RR^n)$. Then we show in \refl{S'p} that $\calS'_p\Ome^{\bullet}(\RR^n)$ is a subcomplex of $\calD'\Ome^{\bullet}(\RR^n)$ and the natural embedding $\calS'_p\Ome^{\bullet}(\RR^n)\hookrightarrow\calD'\Ome^{\bullet}(\RR^n)$ is a quasi-isomorphism. Let now $\rho$ denote the natural map from $\calS'_p\Ome^{\bullet}(\RR^n)$ to $\calS'\w^{\bullet}(U_t)$ (restriction to $U_t$). It follows from step 2 and from the above statement that the complex $$ \operatorname{Cone}^\bullet(\rho)= \calS'_p\Ome^\bullet(\RR^n)\oplus \calS'\Ome^{\bullet-1}(U_t) $$ computes the relative cohomology $H^{\bullet}(\RR^n,U_t;\CC)$. {\it Step 4. \ }The map $\Phi_1:\ \ome\to e^{-p}\ome$ defines a morphism of complexes $\calS'\w^{\bullet}(\RR^n)\to \calS'_p\Ome^\bullet(\RR^n)$. Moreover, every element in the image of $\Phi_1$ is rapidly decreasing along the rays $R_x=\big\{ sx:\, s>0\big\}$, for any $x\in F_t$. This enables us to extend $\Phi_1$ to an explicit map $ \Phi:\ \calS'\w^{\bullet}(\RR^n)\to \operatorname{Cone}^{\bullet}(\rho) $. In order to do that we need the following notations. Let $\mu_s:\RR^n\to \RR^n$ denote the multiplication by $s$ and let $\mu_s^*:\calD'\Ome^\bullet(\RR^n)\to \calD'\Ome^\bullet(\RR^n)$ be the corresponding pull-back map. Consider the {\em Euler vector field} $ \calR=\sum_{i=1}^n x_i\frac \partial{\partial x_i} $ on $\RR^n$ and let $\iot_\calR, \ \calL_\calR$ denote the interior multiplication by $\calR$ and the Lie derivative along $\calR$. Then \eq{dmu'} \frac d{ds}\mu_s^*(\ome)= \mu_s^*\big(\calL_\calR\ome\big)s^{-1} \quad \text{for any} \quad\ome\in \calD'\Ome^\bullet(\RR^n). \end{equation} We define the map $ \Phi: \calS'\Ome^\bullet(\RR^n)\to \operatorname{Cone}^\bullet(\rho)= \calS'_p\Ome^\bullet(\RR^n)\oplus \calS'\Ome^{\bullet-1}(U_t) $ by the formula \eq{2.4'} \Phi:\ome\mapsto \big(\Phi_1\ome, \Phi_2\ome\big)= \left(e^{-p}\ome,\, -\int_1^\infty \mu_s^*(e^{-p}\iot_{\calR}\ome)\, \frac{ds}s\right). \end{equation} The integral in \refe{2.4'} converges since $e^{-p(sx)}$ decreases exponentially in $s$ as $s$ tends to infinity. One uses \refe{2.4'} to show that $\Phi$ commutes with differentials. Finally we prove by an explicit calculation that $\Phi$ is a quasi-isomorphism. Therefore $\calS'\w^{\bullet}(\RR^n)$ computes $H^{\bullet}(\RR^n,U_t;\CC)$, which is isomorphic to $\tilH^{\bullet-1}(F_t,\CC)$ by step 1. \ssec{general}{The general case} Let now $p:\RR^n\to \RR$ be an arbitrary polynomial. Set $v=\frac{\n p}{|\n p|^2}$. Then the Lie derivative of $p$ along $v$ is equal to 1. In the Appendix we show that the flow of $v$ is globally defined on $U_t$ if $t$ is large enough. We denote this flow by $g_s:U_t\to U_t$ and let $g_s^*: \calD'\Ome^\bullet(U_t)\to \calD'\Ome^\bullet(U_t)$ be the corresponding pull-back of currents. Then $g_s^*(p)=p+s$. In particular, we obtain a new proof of topological equivalence of the fibers $F_t$ with $t\gg 0$. Denote $$ \tilv \ = \ pv $$ and let $\iot_\tilv, \ \calL_\tilv$ denote the interior multiplication by $\tilv$ and the Lie derivative along $\tilv$. The flow $\tilg_s$ of $\tilv$ is defined on $U_t, t\gg 0$. The flow $\tilg_s$ and the vector field $\tilv$ are connected by the formula $$ \frac d{ds}\tilg_s^*(\ome) \ = \ \tilg_s^*\big(\calL_\tilv\ome\big) \quad \text{for any} \quad\ome\in \calD'\Ome^\bullet(\RR^n), $$ which is similar to \refe{dmu'} (if $p$ is a homogeneous polynomial of degree $m$ then $\mu_{s}= g_{m\ln s}$). One can easily check that $\tilg_s^*(p)=e^sp$. One can try to define a map $\Phi: \calS'\Ome^\bullet(\RR^n)\to \operatorname{Cone}^\bullet(\rho)$ by formula $$ \Phi:\ome\mapsto \big(\Phi_1\ome, \Phi_2\ome\big)= \left(e^{-p}\ome,\, -\int_1^\infty \tilg_s^*(e^{-p}\iot_{\tilv}\ome)\, ds\right), $$ similar to \refe{2.4'}. The only problem here is that we were not able to prove that the integral in the definition of $\Phi_2$ converges to a tempered current. However, if the map $\Phi_2:\calS'\Ome^\bullet(\RR^n)\to \calS'\Ome^{\bullet-1}(\RR^n)$ is well defined a verbatim repetition of the proof of \reft{main} gives the following \th{main'}Suppose that $p:\RR^n\to \RR$ is a real polynomial and $\tilg_s, s>0$ is a one-parameter semigroup of diffeomorphisms $U_t\to U_t$ such that $\tilg_s^*(p)=e^{ms}p$. Let $\tilv=\frac{d}{ds}_{|_{s=0}}g_s$. If for any tempered current $\ome$ the integral $$ \int_1^\infty \tilg_s^*(e^{-p}\iot_{\tilv}\ome)\, ds $$ converges to a tempered current, then the isomorphism \refe{main} holds. \eth \ssec{example}{Example} Consider the polynomial of two variables $p(x,y)=x^2-x-y$. Set $U_0=\big\{ (x,y)\in \RR^2:\, p(x,y)>0\big\}$ and define a one parameter semigroup $g_s$ of diffeomorphisms of $U_0$ by the formula $$ g_s(x,y) \ = \ \left( e^{s/2}x, e^{s}x-e^{s/2}x+e^{s}y \right). $$ Then $g_s^*p=e^sp$. Clearly, all other conditions of \reft{main'} are satisfied. Hence, the isomorphism \refe{main} holds for $p(x,y)$. \subsection*{Acknowledgments}It is a great pleasure for us to express our gratitude to J.~Bernstein and M.~Farber; the paper was considerably influenced by communications with them. It was M.~Farber who sugested to use the map \refe{2.4'} for the study of $H^\bullet(\Ome^\bullet,d_p)$. We are also thankful to N.~Zobin, M.~Zaidenberg and S.~Kaliman. The first author would like to thank Institute for Advance Study for hospitality. \sec{cur}{Complexes of Currents} In this section we review some facts about complexes of currents which will be used in the proof of \reft{main}. Let $p:\RR^n\to \RR$ be a homogeneous polynomial map of degree $m$, i.e. $p(sx)=s^mp(x)$. Let $U_t=\nolinebreak{\big\{x\in\RR^n:\, p(x)>t\big\}}$, where $t\in \RR$. \ssec{D'}{The complex of currents} By $\Ome_c^\bullet(U_t)$ we denote the De Rham complex of compactly supported complex valued $C^\infty$-forms on $U_t$. The cohomology of $\Ome_c^\bullet(U_t)$ is called the {\em compactly supported cohomology} of $U_t$. Recall that if $ 0\to C^0\overset{d}{\to} C^1\overset{d}{\to} \cdots\overset{d}{\to} C^n\to 0 $ is a complex of topological vector spaces then the dual complex to $(C^\bullet,d)$ is, by definition, the complex $$\begin{CD} 0\to (C^n)^*@>d^*>> (C^{n-1})^*@>d^*>>\cdots @>d^*>> (C^0)^*\to 0, \end{CD}$$ where $(C^i)^*$ denotes the topological dual of the space $C^i$ and $d^*$ denotes the adjoint operator of $d$. The complex of currents $\calD'\Ome^\bullet(U_t)$ on $U_t$ is the complex dual to $\Ome_c^\bullet(U_t)$. By the Poincar\'e duality for non-compact manifolds (cf. \cite{BottTu}), the cohomology of $\calD'\Ome^\bullet(U_t)$ is equal to the cohomology of $U_t$. Analogously, one defines the complex $\calD'\Ome^\bullet(\RR^n)$ of currents on $\RR^n$. Let $r:\calD'\Ome^\bullet(\RR^n)\to \calD'\Ome^\bullet(U_t)$ be the restriction. Recall that the cone $\operatorname{Cone}^\bullet(r)$ of $r$ is the complex $$ \operatorname{Cone}^\bullet(r)=\calD'\Ome^\bullet(\RR^n)\oplus \calD'\Ome^{\bullet-1}(U_t), \qquad d:(\ome,\alp)\mapsto (d\ome,\ome-d\alp). $$ The cohomology of $\operatorname{Cone}^\bullet(r)$ is equal to the relative cohomology $H^\bullet(\RR^n,V;\CC)$ of the pair $(\RR^n,V)$. \ssec{S'}{The complex of tempered currents} The space $\calS(\RR^n)$ of Schwartz (rapidly decreasing) functions on $\RR^n$ is the set of all $\phi\in C^\infty(\RR^n)$ such that for any linear differential operator $L:C^\infty(\RR^n)\to C^\infty(\RR^n)$ with polynomial coefficients \eq{1.1} \sup_{x\in\RR^n} |L\phi(x)| <\infty. \end{equation} The topology in $\calS(\RR^n)$ defined by the semi-norms in the left-hand side of \refe{1.1} makes $S(\RR^n)$ a Fr\'echet space. Recall that by $\Ome^\bullet$ we denote the De Rham complex of global algebraic differential forms on $\RR^n$. The {\em complex of Schwartz forms} on $\RR^n$ is the complex $$ \calS\Ome^\bullet(\RR^n)=\calS(\RR^n)\otimes_{\calO} \Ome^\bullet \qquad $$ with natural differential. By the {\em complex of Schwartz forms on $U_t$} we will understand the subcomplex of $\calS\Ome^\bullet(\RR^n)$ consisting of the forms $\ome$ such that there exists a real number $\eps=\eps(\ome)$ such that the support of $\ome$ lies in $U_{t+\eps}$. The complex $\calS'\Ome^\bullet(\RR^n)$ of {\em tempered currents} on $\RR^n$ is, by definition, the dual complex to $\calS\Ome^\bullet(\RR^n)$. Similarly, the complex $\calS'\Ome^\bullet(U_t)$ of {\it tempered currents on $U_t$} is, the dual complex to $\calS\Ome^\bullet(U_t)$. \lem{t1t2} For any $t_1> t_2 >0$ the natural map $i: \calS'\Ome^\bullet(U_{t_2})\to \calS'\Ome^\bullet(U_{t_1})$ is a homotopy equivalence of complexes. \end{lemma} \begin{proof} Let $\mu_s:\RR^n\to \RR^n$ denote the multiplication by $s$ and let $\mu_s^*:\calD'\Ome^\bullet(\RR^n)\to \calD'\Ome^\bullet(\RR^n)$ be the corresponding pull-back map. Clearly, $\mu_s^*$ preserves the space of tempered currents. Set $\tau=(t_1/t_2)^{1/m}$. Then $\mu_\tau(U_{t_2})=U_{t_1}$. In particular, we can consider $\mu^*_{\tau}$ as a map from $\calS\Ome^\bullet(U_{t_1})$ to $\calS\Ome^\bullet(U_{t_2})$. To prove the lemma we will show that $\mu^*_\tau$ is a homotopy inverse of $i$. Consider the {\em Euler vector field} $$ \calR=\sum_{i=1}^n x_i\frac \partial{\partial x_i} $$ on $\RR^n$ and let $\iot_\calR, \ \calL_\calR$ denote the interior multiplication by $\calR$ and the Lie derivative along $\calR$. Then \eq{dmu} \frac d{ds}\mu_s^*(\ome)= \mu_s^*\big(\calL_\calR\ome\big)s^{-1} \quad \text{for any} \quad\ome\in \calD'\Ome^\bullet(\RR^n). \end{equation} Note that if $\ome$ is a tempered current so are $\iot_\calR\ome$ and $\calL_\calR\ome$. For any current $\ome$, set $$ H\ome \ = \ \int^{\tau}_1 \mu^*_s(\iot_{\calR}\ome)\, \frac{ds}s. $$ The operators $\mu_s^*$ and $\iot_{\calR}$ preserve the space of tempered currents. Hence, so does $H$. Using \refe{dmu} and the Cartan homotopy formula \eq{Cartan} \calL_\calR=d\iot_\calR+ \iot_\calR d \end{equation} we obtain $$ (dH+Hd)\, \ome \ = \ \mu_\tau^*\ome \ - \ \ome, $$ for any current $\ome$. The lemma is proven. \end{proof} \lem{S'D'} For any $t> 0$ the embedding $\calS'\Ome^\bullet(U_t)\hookrightarrow \calD'\Ome^\bullet(U_t)$ is a homotopy equivalence of complexes. In particular, the cohomology of the complex $\calS'\Ome^\bullet(U_t)$ is equal to the cohomology $H^\bullet(U_t;\CC)$ of $U_t$. \end{lemma} \begin{proof} By \refl{t1t2} it is enough to show that the embedding $i:\calS'\Ome^\bullet(U_1)\hookrightarrow \calD'\Ome^\bullet(U_1)$ is a quasi-isomorphism. Let $h_s:U_1\to U_1, \ s>0$ denote the map defined by the formula $$ h_s:x\mapsto \frac{1+s}{1+s|x|}\cdot x. $$ Here $|x|$ denotes the norm of the vector $x\in \RR^n$. Let $h_s^*:\calD'\Ome^\bullet(U_1)\to \calD'\Ome^\bullet(U_1)$ denote the corresponding pull-back. Then (cf. \refe{dmu}) \eq{dh} \frac{d}{ds}h^*_s(\ome) \ = \ \frac{1-|x|}{(1+s)(1+s|x|)} \, h^*_s(\calL_\calR\ome), \end{equation} for any current $\ome$. Note also that $h_s^*$ preserves the space of tempered currents. Clearly, $h_0$ is the identity map. The image of $h_1:U_1\to U_1$ lies in the compact set $$ \big\{x\in \RR^n:\ |x|\le 2\big\}. $$ Hence, $h_1^*\ome$ is a tempered current for any $\ome\in\calD'\Ome^\bullet(\RR^n)$. Note also that $h_s^*$ preserves the space of tempered currents for any $s>0$. We will prove that the map $h_1^*: \calD'\Ome^\bullet(U_1)\to \calS'\Ome^\bullet(U_1)$ is a homotopy inverse of the embedding $i:\calS'\Ome^\bullet(U_1)\hookrightarrow \calD'\Ome^\bullet(U_1)$. For any $\ome\in \calD'\Ome^\bullet(U_1)$, set $$ H\ome \ = \ \int_0^1\, \frac{1-|x|}{(1+s)(1+s|x|)} \, h_s^*(\iot_\calR\ome)\, ds. $$ Here, $\iot_\calR$ denote the operator of interior multiplication by $\calR$. Using \refe{dh} and \refe{Cartan} we obtain \eq{homot'} (dH+Hd)\, \ome = h_1^*\, \ome -\ome, \qquad \ome\in \calD'\Ome^\bullet(U_1). \end{equation} Thus the map $i\circ h_1^*: \calD'\Ome^\bullet(U_1)\to \calD'\Ome^\bullet(U_1)$ homotopic to the identity map. Since the operators $h_s^*$ and $\iot_\calR$ preserve the space of tempered currents, so does $H$. Hence, \refe{homot'} implies that the map $h_1^*\circ i: \calS'\Ome^\bullet(U_1)\to \calS'\Ome^\bullet(U_1)$ is also homotopic to the identity map. \end{proof} \ssec{1.5.}{The complex $\calS'_p\Ome^\bullet(\RR^n)$} We will need the following twisted version of the complex of tempered currents on $\RR^n$. Fix a smooth function $\tet:\RR\to \RR$ such that $$ \tet(s)=\cases s \quad \text{if} \quad s<1,\\ 0 \quad \text{if} \quad s>2. \endcases $$ and define $\tilp(x)=\tet(p(x)), x\in \RR^n$. Note that the current $d\tilp\wedge \ome$ is tempered for any tempered current $\ome$. \lem{S'p}The space $\begin{CD} \calS'_p\Ome^\bullet(\RR^n)=\big\{ \ome \in \calD'\Ome^\bullet(\RR^n):\, e^{\tilp}\ome\in \calS'\Ome^\bullet(\RR^n) \big\} \end{CD}$ is a subcomplex of $\calD'\Ome^\bullet(\RR^n)$ and the embedding $\calS'_p\Ome^\bullet(\RR^n)\hookrightarrow \calD'\Ome^\bullet(\RR^n)$ is a quasi-isomorphism. \end{lemma} \begin{proof} Suppose $\ome\in \calS'_p\Ome^\bullet(\RR^n)$, i.e. $e^{\tilp}\ome\in \calS'\Ome^\bullet(\RR^n)$. Then $$ e^{\tilp} d\ome=d(e^{\tilp}\ome)-d\tilp\wedge e^{-\tilp}\ome \in \calS'\Ome^\bullet(\RR^n), $$ i.e. $d\ome\in \calS'_p\Ome^\bullet(\RR^n)$. Hence $\calS'_p\Ome^\bullet(\RR^n)$ is a subcomplex of $\calD'\Ome^\bullet(\RR^n)$. Clearly, the embedding $\calS'_p\Ome^\bullet(\RR^n)\hookrightarrow \calD'\Ome^\bullet(\RR^n)$ induces an isomorphism of 0-cohomology. To prove \refl{S'p} it remains to show that the $k$-th cohomology $H^k(\calS'_p\Ome^\bullet(\RR^n)), \ k>0$ of $\calS'_p\Ome^\bullet(\RR^n)$ vanishes. We will use the notation introduced in the proof of \refl{t1t2}. In particular, $\mu_s:\RR^n\to \RR^n$ is the multiplication by $s$ and $\calR$ is the Euler vector field on $\RR^n$. Let $\ome$ be a closed current, $d\ome=0$. Using \refe{dmu} and the Cartan homotopy formula $\calL_\calR=d\iot_\calR+\iot_\calR d$, we obtain $$ \ome-\mu_0^*(\ome) =d\, \int_0^1\mu_s^*\big(\iot_\calR\ome\big)\, \frac{ds}s. $$ (Note that the integral in the left hand side converges, because $\calR$ vanishes at 0). If $\ome$ is a $k$-current, $k>0$, then $\mu_0^*(\ome)=0$. Hence, to finish the proof we need only to show that \eq{temp} \int_0^1\mu_s^*\big(\iot_\calR\ome\big)\, \frac{ds}s\in \calS'_p\Ome^\bullet(\RR^n) \end{equation} for any $\ome\in \calS'_p\Ome^\bullet(\RR^n)$. Set $\bet =e^\tilp\ome\in \calS'\Ome^\bullet(\RR^n)$. Then \eq{new} e^\tilp\int_0^1\mu_s^*\big(\iot_\calR\ome\big)\, \frac{ds}s= \int_0^1 e^{\tilp(x)-\tilp(sx)} \mu_s^*\big(\iot_\calR\bet\big)\, \frac{ds}s. \end{equation} Since, for any $s\in [0,1]$, the function $\tilp(x)-\tilp(sx)$ is bounded from above, all the derivatives of the function $e^{\tilp(x)-\tilp(sx)}$ are bounded by polynomials. It follows that $s\mapsto e^{\tilp(x)-\tilp(sx)} \mu_s^*\big(\iot_\calR\bet\big)s^{-1}$ defines a continuous map $[0,1]\to \calS'\Ome^\bullet(\RR^n)$. Hence the current \refe{new} is tempered and \refe{temp} holds. \end{proof} \ssec{1.7}{} For any $\ome\in \calS'_p\Ome^\bullet(\RR^n)$, $t>0$ the restriction of $\ome$ on $U_t$ is a tempered current on $U_t$. Hence, the restriction map $\rho:\calS'_p\Ome^\bullet(\RR^n)\to \calS'\Ome^\bullet(U_t)$ is defined. \lem{Cone} The complexes $\operatorname{Cone}^\bullet(\rho)$ and $\operatorname{Cone}^\bullet(r)$ (cf. \refss{S'}) are quasi-~isomorphic. In particular, the cohomology of $\operatorname{Cone}^\bullet(\rho)$ equals the relative cohomology of the pair $(\RR^n,U_t)$. \end{lemma} \begin{proof} Let $i:\calS'_p\Ome^\bullet(\RR^n)\to \calD'\Ome^\bullet(\RR^n),\, j:\calS'\Ome^\bullet(U_t)\to \calD'\Ome^\bullet(U_t)$ denote the natural inclusions. Consider the commutative diagram \eq{1.5} \begin{CD} \calS'_p\Ome^\bullet(\RR^n) @>\rho>> \calS'\Ome^\bullet(U_t)\\ @ViVV @VVjV \\ \calD'\Ome^\bullet(\RR^n) @>r>> \calD'\Ome^\bullet(U_t) \end{CD} \end{equation} According to Lemmas \ref{L:S'D'} and \ref{L:S'p} the vertical arrows of this diagram are quasi-isomorphisms. Hence, \refe{1.5} induces a quasi-isomorphism between $\operatorname{Cone}^\bullet(\rho)$ and $\operatorname{Cone}^\bullet(r)$. \end{proof} In the proof of \refl{2.6} we will also need the following \lem{Cone'} Suppose $t_1>t_2>0$ and let $$ \rho_1:\calS'_p\Ome^\bullet(\RR^n)\to \calS'\Ome^\bullet(U_{t_1}), \qquad \rho_2:\calS'_p\Ome^\bullet(\RR^n)\to \calS'\Ome^\bullet(U_{t_2}) $$ denote the corresponding restrictions. The natural map $\operatorname{Cone}^\bullet(\rho_2)\to \operatorname{Cone}^\bullet(\rho_1)$ is a quasi-isomorphism. \end{lemma} \begin{proof} Consider the commutative diagram \eq{1.6} \begin{CD} \calS'_p\Ome^\bullet(\RR^n) @>\rho_2>> \calS'\Ome^\bullet(U_{t_2})\\ @| @VVV \\ \calS'_p\Ome^\bullet(\RR^n) @>\rho_1>> \calS'\Ome^\bullet(U_{t_1}) \end{CD} \end{equation} By \refl{t1t2}, the right vertical arrow of this diagram is a quasi-isomorphism. Hence, \refe{1.6} induces a quasi-isomorphism between $\operatorname{Cone}^\bullet(\rho_2)$ and $\operatorname{Cone}^\bullet(\rho_1)$. \end{proof} \sec{proof}{Proof of Theorem 0.2} \ssec{FtUt}{Cohomology of $F_t$ as relative cohomology} Fix $t> 0$ and set $$ F_t=p^{-1}(t); \qquad U_t= \big\{x\in\RR^n:\, p(x)>t\big\}. $$ Then $U_t$ is diffeomorphic to the the product $F_t\times (0,+\infty)$. In particular, $U_t$ has the same cohomology as $F_t$. Using the long exact sequence of the pair $(\RR^n,U_t)$, we obtain \eq{2.3} \tilH^k(F_t;\CC)=H^{k+1}(\RR^n,U_t;\CC), \qquad k=0,1,\dots, n-1, \end{equation} where $H^\bullet(\RR^n,U_t;\CC)$ denotes the relative cohomology of the pair $(\RR^n,U_t)$ and $\tilH^\bullet(F_t;\CC)$ denotes the reduced cohomology of $F_t$. From \refl{Cone} and \refe{2.3} we see that to prove \reft{main} it is enough to show that the complexes $(\calS'\Ome^\bullet(\RR^n),d_p)$ and $\operatorname{Cone}^\bullet(\rho)$ are quasi-isomorphic. \ssec{S'Cone}{A map from $\calS'\Ome^\bullet(\RR^n)$ to $\operatorname{Cone}^\bullet(\rho)$} Recall that by $\mu_s:\RR^n\to \RR^n$ we denote the multiplication by $s\in\RR$. Then $\mu_s(U_t)=U_{s^mt}$. In particular, if $s\ge 1$, then $\mu_s$ may be considered as a map from $U_t$ to itself. Let $\mu^*_s:\calD'\Ome^\bullet(U_t)\to \calD'\Ome^\bullet(U_t)$ denote the corresponding pull-back map. Then $\mu_s^*p(x)=p(\mu_sx)=s^mp(x)$. Recall also that $\iot_{\calR}$ denote the interior multiplication by the Euler vector field $\calR=\sum x_i\frac{\partial}{\partial x_i}$. Note that if $\ome$ is a tempered current on $U_t$ than so are $\iot_{\calR}\ome$ and $\mu_s^*\ome$. Recall from \refss{1.7} that $\rho: \calS'_p\Ome^\bullet(\RR^n)\to \calS'\Ome^\bullet(U_t)$ denotes the restriction. We define the map $$ \Phi: \calS'\Ome^\bullet(\RR^n)\to \operatorname{Cone}^\bullet(\rho)= \calS'_p\Ome^\bullet(\RR^n)\oplus \calS'\Ome^{\bullet-1}(U_t) $$ by the formula \eq{2.4} \Phi:\ome\mapsto \big(\Phi_1\ome, \Phi_2\ome\big)= \left(e^{-p}\ome,\, -\int_1^\infty \mu_s^*(e^{-p}\iot_{\calR}\ome)\, \frac{ds}s\right). \end{equation} The integral in \refe{2.4} converges since $e^{-p(sx)}$ decreases exponentially in $s$ as $s$ tends to infinity. It follows from \refe{dmu}, \refe{Cartan} that the map $\Phi: \calS'\Ome^\bullet(\RR^n)\to \operatorname{Cone}^\bullet(\rho)$ commutes with differentials, i.e. $$ \Phi_1d_p\ome \ = \ d\Phi_1\ome; \qquad \Phi_2 d_p\ome \ =\ \Phi_1\ome_{|_U} \ - \ d\Phi_2\ome. $$ \lem{2.5} The map $ H^\bullet\big( \calS'\Ome^\bullet(\RR^n),d_p\big)\to H^\bullet\big(\operatorname{Cone}^\bullet(\rho)\big) $ induced by $\Phi$ is injective. \end{lemma} \begin{proof} Suppose that $\ome$ is a tempered current on $\RR^n$ and that $\Phi\ome$ is a coboundary in $\operatorname{Cone}^\bullet(\rho)$. Then there exists $\alp\in \calS'_p\Ome^\bullet(\RR^n)$ and $\bet\in \calS'\Ome^{\bullet-1}(U_t)$ such that $$ e^{-p}\ome \ = \ d\alp; \qquad -\int_1^\infty \mu_s^*(e^{-p}\iot_{\calR}\ome)\, \frac{ds}s \ = \ \alp_{|_U} \ - \ d\bet. $$ Choose $j\in C^\infty(\RR)$ such that $j(s)=0$ if $s\le t+1$ and $j(s)=1$ if $s\ge t+2$ and set $\phi(x)=j(p(x)), \ (x\in\RR^n)$. Then the support of $\phi$ is contained in $U_t$. Hence $\phi\bet$ may be considered as a current on $\RR^n$. Since all the derivatives of $\phi$ are bounded by polynomials, $\phi\bet\in \calS'_p\Ome^\bullet(\RR^n)$. Define $\oalp=\alp-d(\phi\bet)$. Then $\ome= d_p(e^{p}\oalp)$. So to prove the lemma we only need to show that $e^p\oalp$ is a tempered current. Let $\psi(x)=j(p(x)-1)$. It is enough to prove that $e^p(1-\psi)\oalp$ and $e^p\psi\oalp$ are tempered currents. Since $(1-\psi)\oalp\in \calS'_p\Ome^\bullet(\RR^n)$ vanishes when $p(x)>t+2$, we see from the definition of $\calS'_p\Ome^\bullet(\RR^n)$ that $e^p(1-\psi)\oalp$ is a tempered current. On the support of $\psi$ the function $\phi$ is identically equal to 1. Hence $\psi\oalp=\psi(\alp-d\bet)$ and \begin{multline}\notag e^p\psi\oalp=e^p\psi(\alp-d\bet)= -\psi\int_1^\infty e^p \mu_s^*(e^{-p}\iot_{\calR}\ome)\, \frac{ds}s=\\ -\psi\int_1^\infty e^{p(x)-p(\mu_sx)} \mu_s^*(\iot_{\calR}\ome)\, \frac{ds}s \in\calS'\Ome^\bullet(\RR^n). \end{multline} \end{proof} \lem{2.6} The map $ H^\bullet\big( \calS'\Ome^\bullet(\RR^n),d_p\big)\to H^\bullet\big(\operatorname{Cone}^\bullet(\rho)\big) $ induced by $\Phi$ is surjective. \end{lemma} \begin{proof} Choose $\eps>0$ such that $t-\eps>0$ and set $U_{t-\eps}=\big\{x\in\RR^n:\, p(x)>t-\eps\big\}$. Let $\rho_\eps:\calS'_p\Ome^\bullet(\RR^n)\to \calS'\Ome^\bullet(U_{t-\eps})$ be the restriction. By \refl{Cone'}, any cohomology class $\xi$ of the complex $\operatorname{Cone}^\bullet(\rho)$ may be represented by a pair $(\alp, \bet_{|_U})$, where $\alp\in \calS'_p\Ome^\bullet(\RR^n)$ and $\bet\in \calS'\Ome^\bullet(U_{t-\eps})$. Fix $j\in C^\infty(\RR)$ such that $j(s)=0$ if $s\le t-\eps/2$ and $j(s)=1$ if $s\ge t$ and set $\phi(x)=j(p(x)), \ (x\in\RR^n)$. Then all the derivatives of $\phi$ are bounded by polynomials and the support of $\phi$ is contained in $U_{t-\eps}$. Hence, $\phi\bet$, considered as a current on $\RR^n$, belongs to the space $\calS'_p\Ome^\bullet(\RR^n)$. The cohomology class of the pair $ \big(\alp- d(\phi\bet),\, 0\big)\in \operatorname{Cone}^\bullet(\rho) $ equals $\xi$. Set $\ome=e^p(\alp-d(\phi\bet))$. Since the current $\alp-d(\phi\bet)$ vanishes on $U_t$, we see from the definition of the space $\calS'_p\Ome^\bullet(\RR^n)$ that $\ome$ is a tempered current. Clearly, $\Phi\ome=\big(\alp- d(\phi\bet),\, 0\big)$. Hence, $\xi$ belongs to the image of the map $H^\bullet\big( \calS'\Ome^\bullet(\RR^n),d_p\big)\to H^\bullet\big(\operatorname{Cone}^\bullet(\rho)\big)$ \end{proof} From Lemmas \ref{L:2.5} and \ref{L:2.6}, we see that the complexes $(\calS'\Ome^\bullet(\RR^n),d_p)$ and $\operatorname{Cone}^\bullet(\rho)$ are quasi-isomorphic. \reft{main} follows now from \refl{Cone} and \refe{2.3}.

9704

alg-geom/9704006

en

https://arxiv.org/abs/alg-geom/9704006

alg-geom/9704006

Carlos Simpson

Carlos Simpson

A closed model structure for $n$-categories, internal $Hom$, $n$-stacks and generalized Seifert-Van Kampen

Corrects an error in the proof of Theorem 5.1, by rewriting locally. Doesn't change the rest of the text, so numerous expositional problems are left untouched

We define a closed model category containing the $n$-nerves defined by Tamsamani, and admitting internal $Hom$. This allows us to construct the $n+1$-category $nCAT$ by taking the internal $Hom$ for fibrant objects. We prove a generalized Seifert-Van Kampen theorem for Tamsamani's Poincar\'e $n$-groupoid of a topological space. We give a still-speculative discussion of $n$-stacks, and similarly of comparison with other possible definitions of $n$-category.

alg-geom

\section*{A closed model structure for $n$-categories, internal $\underline{Hom}$, $n$-stacks and generalized Seifert-Van Kampen} Carlos Simpson\newline CNRS, UMR 5580, Universit\'e Paul Sabatier, 31062 Toulouse CEDEX, France. \bigskip \numero{Introduction} The purpose of this paper is to develop some additional techniques for the weak $n$-categories defined by Tamsamani in \cite{Tamsamani} (which he calls {\em $n$-nerves}). The goal is to be able to define the internal $Hom(A,B)$ for two $n$-nerves $A$ and $B$, which should itself be an $n$-nerve. This in turn is for defining the $n+1$-nerve $nCAT$ of all $n$-nerves conjectured in \cite{Tamsamani}, which we can do quite easily once we have an internal $Hom$. It is essentially clear {\em a priori} that we cannot just take an internal $Hom$ on all of the $n$-nerves of Tamsamani, and in fact some simple examples support this: any strict $n$-category may be considered in an obvious way as an $n$-nerve i.e. a presheaf of sets over $\Delta ^n$ satisfying certain properties, but the morphisms of the resulting presheaves are the same as the strict morphisms of the original strict $n$-categories; on the other hand one can see that these strict morphisms are not enough to reflect all of the ``right'' morphisms. \footnote{The simplest example which shows that the strict morphisms are not enough is where $G$ is a group and $V$ an abelian group and we set $A$ equal to the category with one object and group of automorphisms $G$, and $B$ equal to the strict $n$-category with only one $i$-morphism for $i<n$ and group $V$ of $n$-automorphisms of the unique $n-1$-morphism; then for $n=1$ the equivalence classes of strict morphisms from $A$ to $B$ are the elements of $H^1(G,V)$ so we would expect to get $H^n(G,V)$ in general, but for $n>1$ there are no nontrivial strict morphisms from $A$ to $B$. } Our strategy to get around this problem will be based on the idea of {\em closed model category} \cite{Quillen}. We will construct a closed model category containing the $n$-nerves of Tamsamani. Then we can simply take as the ``right'' $n$-nerve of morphisms, the internal $Hom(A,B)$ whenever $A$ and $B$ are {\em fibrant} objects in the closed model category (all objects will be cofibrant in our case). This strategy is standard practice for topologists. As usual, in order to define a closed model category we first have to enlarge the class of objects under consideration. Instead of $n$-nerves as defined by Tamsamani we look at $n$-pre-nerves (i.e. presheaves of sets over the cartesian product of $n$ copies of the standard simplicial category) which satisfy the constancy condition---C1 in Tamsamani's definition of $n$-nerve---and call these {\em $n$-precats} (this notion being in between the pre-nerves and nerves of \cite{Tamsamani}, we take a different notation). An $n$-precat may be interpreted as a presheaf on a certain quotient $\Theta^n$ of $\Delta ^n$, in particular we obtain a category $PC_n$ of objects closed under all limits, with internal $Hom$ etc. We follow the method of constructing a closed model category developed by Jardine-Joyal \cite{Jardine} \cite{Joyal} in the case of simplicial presheaves. The cofibrations are essentially just monomorphisms (however we cannot---and don't---require injectivity for top-degree morphisms, just as sets or categories with monomorphisms are not closed model categories \cite{Quillen}). The main problem is to define a notion of weak equivalence. Our key construction is the construction of an $n$-nerve $Cat(A)$ for any $n$-precat $A$, basically by throwing in freely all of the elements which are required by the definition of nerve \cite{Tamsamani} (although to make things simpler we use here a definition of nerve modified slightly to ``easy nerve''). Then we say that a morphism of $n$-precats $A\rightarrow B$ is a {\em weak equivalence} if $Cat(A)\rightarrow Cat(B)$ is an exterior equivalence of $n$-nerves in the sense of \cite{Tamsamani}. The fibrant morphisms are characterized in terms of cofibrations and weak equivalences by a lifting property, in the same way as in \cite{Jardine}. One new thing that we obtain in the process of doing this is the notion of pushout. The category of $n$-precats is closed under direct limits and in particular under pushouts. Applying the operation $Cat$ then gives an {\em $n$-categorical pushout}: if $A\rightarrow B$ and $A\rightarrow C$ are morphisms of $n$-nerves then the categorical pushout is $Cat(B\cup ^AC)$. The main lemma which we need to prove (Lemma \ref{pushout} below) is---again just as in \cite{Jardine}---the statement that a pushout by a trivial cofibration (i.e. a cofibration which is a weak equivalence) is again a trivial cofibration. After that the rest of the arguments needed to obtain the closed model structure are relatively standard following \cite{Jardine} when necessary. Once the closed model structure is established, we can go on to define internal $Hom $ and construct the $n+1$-nerve $nCAT$. Using these we can, in principal, define the notion of $n$-stack. Our discussion of $n$-stacks is still at a somewhat speculative stage in the present version of the paper, because there are several slightly different notions of a family of $n$-categories parametrized by a $1$-category ${\cal X}$ and ideally we would like to---but don't yet---know that they are all the same (as happens for $1$-stacks). The notion of categorical pushout which we developed as a technical tool actually has a geometric consequence: we obtain a generalized Seifert-Van Kampen theorem (Theorem \ref{svk} below) for the Poincar\'e $n$-groupoids $\Pi _n(X)$ of a space $X$ which were defined by Tamsamani (\cite{Tamsamani} \S 2.3 ff). If $X$ is covered by open sets $U$ and $V$ then $\Pi _n(X)$ is equivalent to the category-theoretic pushout of $\Pi _n(U)$ and $\Pi _n(V)$ along $\Pi _n(U\cap V)$. We define the {\em nonabelian cohomology} of $X$ with coefficients in a fibrant $n$-precat $A$ as $H(X,A):= Hom (\Pi _n(X), A)$. The generalized Seifert-Van Kampen theorem implies a Mayer-Vietoris statement for this nonabelian cohomology. There are many possible approaches to the notion of $n$-category and, without pretending to be exhaustive, I would like to point out some of the other possibilities here for comparison. \newline ---One of the pioneering works in the search for an algebraic approach to homotopy of spaces is the notion of $Cat ^n$-groups of Brown and Loday. This is what is now known as the ``cubical'' approach where the set of objects can itself have a structure for example of $n-1$-category, so it isn't quite the same as the approach we are looking for (commonly called the ``globular'' case). \newline ---Gordon, Powers and Street have intensively investigated the cases $n=3$ and $n=4$ \cite{Gordon-Power-Street}, following the path set out by Benabou for $2$-categories \cite{Benabou}. \newline ---In \cite{Grothendieck} A. Grothendieck doesn't seem to have hit upon any actual definition but gives a lot of nice intuition about $n$-categories. \newline ---On p. 41 of \cite{Grothendieck} starts a reproduction of a letter from Grothendieck to Breen dated July 1975, in which Grothendieck acknowledges having recieved a proposed definition of non-strict $n$-category from Breen, a definition which according to {\em loc. cit} ``...has certainly the merit of existing...''. It is not clear whether this proposed construction was ever worked out. \newline ---In \cite{Street}, R. Street proposes a definition of weak $n$-category as a simplicial set satisfying a certain variant of the Kan condition where one takes into account the directions of arrows. \newline ---Kapranov and Voevodsky in \cite{Kapranov-Voevodsky} construct, for a topological space $X$, a ``Poincar\'e $\infty$-groupoid'' which is a strictly associative $\infty$-groupoid but where the arrows are invertible only up to equivalence. This of course raises the question to know if strictly associative $n$-categories would be a sufficient class to yield the correct $n+1$-category $nCAT$. As pointed out in the footnote above, one wonders in particular whether there is a closed model structure to go along with these strict $n$-categories. \newline ---In his recent preprint \cite{Batanin} M. Batanin develops some ideas towards a definition of weak $\infty$-category based on operads. In the introduction he mentions a letter from Baez and Dolan to Street dating to November 29, 1995 which contains some ideas for a definition of weak $n$-category; and he states that Makkai, Hermida and Power have worked on the idea contained in this letter. \newline ---M. Rosellen told me in September 1996 that he was working on a version using the theory of operads (cf \cite{Adams} for example). Just as our current effort is based on Segal's delooping machine, there should probably be an $n$-category machine analogous to any of the other various delooping machines, and in fact the problems are almost identical: the basic problem of doing $n$-categories comes down to doing delooping while keeping track of the non-connected case and not requiring things to be invertible up to homotopy (cf the last section of \cite{Tamsamani} for some arguments relating $n$-categories and delooping machines). \newline ---J. Baez and J. Dolan have developed their theory originating in the letter refered to above, a definition of $n$-categories based on operads, in a preprint \cite{BaezDolanIII} of February 1997. In this preprint they discuss operads, give their definition of $n$-category and of certain morphisms of $n$-categories, and define the homotopy category of $n$-categories which they conjecture to be equivalent to the homotopy category for other definitions such as the category $Ho-n-Cat$ mentionned in \cite{Tamsamani}. The main problem which needs to be accomplished in any of these points of view is to obtain an $n+1$-category (hopefully within the same point of view) $nCAT$ parametrizing the $n$-categories of that point of view. This is the main thing we are doing here for Tamsamani's point of view. As far as I know, the present one is the first precise construction of the $n+1$-category $nCAT$. Once several such points of view are up and running, the comparison problem will be posed: to find an appropriate way to compare different points of view on $n$-categories and (one hopes) to say that the various points of view are equivalent and in particular that the various $n+1$-categories $nCAT$ are equivalent via these comparisons. It is not actually clear to me what type of general setup one should use for such a comparison theory, although the first thing to try would be to explore a theory of ``internal closed model category'', a closed model category with internal $Hom$: any reasonable point of view on $n$-categories should probably yield an internal closed model category $n--C$ (such as the $PC_n$ we obtain below) and furthermore $nCAT$ should be an object in $(n+1)--C$. Comparison between the theories might then be possible using a version of Quillen's adjoint functor approach \cite{Quillen}. We give an indication of how to start on comparison in \S 11 by sketching how to obtain a functor from any internal closed model category containing $Cat$, to the our closed model category of $n$-categories. Having a good theory of $n$-categories should open up the possibility to pursue any of the several programs such as that outlined by Grothendieck \cite{Grothendieck}, the generalization to $n$-stacks and $n$-gerbs of the work of Breen \cite{Breen}, or the program of Baez and Dolan in topological quantum field theory \cite{BaezDolan}. Once the theory of $n$-stacks is off the ground this will give an algebraic approach to the ``geometric $n$-stacks'' considered in \cite{geometricN}. We clarify the pretentions to rigor of the various sections of this paper. \S\S 2--7 are supposed to be a first version of something precise and correct (although at the time of this first version I haven't checked all of the details in a very thorough way). The same holds for \S 9 on Seifert-Van Kampen. On the other hand, the discussion of \S 8 on $n$-stacks is blatantly speculative; and the discussion of \S 10 on nonabelian cohomology is very incomplete. {\em Acknowledgements:} I would specially like to thank Z. Tamsamani and A. Hirschowitz. This work follows up the definition and original work on $n$-nerves done by Z. Tamsamani in his thesis \cite{Tamsamani}. Much of what is done in the present paper was suggested by discussions with Tamsamani. More recently in preparing some joint work with A. Hirschowitz on universal geometric $n$-stacks related to Brill-Noether, Hirschowitz asked repeatedly for an algebraic approach to $n$-stacks which would be more natural than the approach passing through presheaves of topological spaces or simplicial presheaves. The present work owes much to A. Hirschowitz's questions and suggestions, as well as to his perseverance in asking for an algebraic approach to $n$-stacks. I would also like to thank R. Brown for pointing out the importance of the notion of push-out and Seifert-Van Kampen, and G. Maltsiniotis and A. Brugui\`eres for helpful discussions. {\em Ce papier est dedi\'e \`a Nicole, Chlo\'e et L\'eo.} \numero{Preliminaries} Let $\Delta$ be the standard category of ordered finite sets. Let $\Theta ^n$ be the quotient of the cartesian product $\Delta ^n$ obtained by identifying all of the objects $(M, 0, M')$ for fixed $M = (m_1,\ldots , m_k)$ and variable $M'= (m'_1, \ldots , m'_{n-k-1})$. The object of $\Theta ^n$ corresponding to the class of $(M,0,M')$ with all $m_i >0$ will be denoted $M$. The object $(1, \ldots , 1)$ ($k$ times) will be denoted $1^k$. We permit concatenation in our notation for objects, thus $M, m$ denotes the object $(m_1, \ldots , m_k , m)$ (when this makes sense, that is when $k<n$). The class of $(0,\ldots , 0)$ will be denoted by $0$. We give the explicit construction of $\Theta ^n$. If $M=(m_1,\ldots , m_k)$ and $M' = (m'_1,\ldots , m'_l)$ then set $M,0$ equal to the concatenation of $M$ with $(0,\ldots , 0)$ in $\Delta ^n$ and similarly for $M',0$. We define an equivalence relation on morphisms $\varphi = (\varphi _1, \ldots , \varphi _n)$ from $M, 0$ to $M', 0$ by saying $\varphi \sim \varphi '$ whenever there exists $j$ such that $\varphi _i = \varphi '_i$ for $i\leq j$ and $\varphi _j: m_j \rightarrow m'_j$ factors through the object $0\in \Delta$ (which is the one-point set). This equivalence relation is compatible with composition so we obtain a category $\Theta ^n$ by taking as morphisms the quotient of the morphisms in $\Delta ^n$ by this equivalence relation. There is an obvious projection from $\Delta ^n$ to $\Theta ^n$. We assume familiarity with \cite{Tamsamani}. An $n$-precat is a presheaf of sets on $\Theta ^n$. This corresponds to an $n$-prenerve in Tamsamani's notation (i.e. presheaf of sets on $\Delta ^n$) which satisfies his axiom C1 in the definition of $n$-nerve. Let $PC_n$ denote the category of $n$-precats. An $n$-precat is an {\em $n$-category} (or {\em $n$-nerve} in the notation of Tamsamani \cite{Tamsamani} which is the sense which we will always assign to the terminology ``$n$-category'' below) if it satisfies certain additional conditions \cite{Tamsamani}. We give an easier version of these conditions which we call an {\em easy $n$-category}. We start with the notion of easy equivalence between two easy $n$-categories---this is not circular because the notion of easy $n$-category will only use the notion of easy equivalence for morphisms of $n-1$-categories. If $A$ and $B$ are easy $n$-categories then a morphism $f: A\rightarrow B$ (of $n$-precats, i.e. of presheaves on $\Theta ^n$) is an {\em easy equivalence} if for all $v\in B_{1^k}$ (called a $k$-morphism of $B$) and all $a,a' \in A_{1^{k-1}}$ with $s(v)=f(a)$ and $t(v)=f(a')$ and $s(a)=s(a')$ and $t(a)=t(a')$ (here $s$ and $t$ denote the morphisms ``source'' and ``target'' from $T_{1^{k}}$ to $T_{1^{k-1}}$ for any $n$-precat $T$), there exists $u\in A_{1^k}$ with $s(u)= a$ and $t(u)= a'$ and $f(u)=v$. A {\em marked easy equivalence} is the data of a morphism $f$ together with choices $u(a,a', v)$ in every situation as above. The reader is cautioned that we will still need Tamsamani's notion of equivalence (which he calls ``\'equivalence ext\'erieure'' \cite{Tamsamani} \S 1.3) for our closed model category structure below. The notion of easy equivalence is mainly just used when it is an ingredient in the notion of $n$-category. Before giving the definition of easy $n$-category we introduce the following notation. If $T$ is an $n$-precat then for any $M= (m_1, \ldots , m_k)$ we denote by $T_{M/}$ the $n-k$-precat obtained by restricting $T$ to the subcategory of objects of $\Theta^n$ of the form $(M,M')$ for variable $M'$. This differs from the notation of Tamsamani who called this just $T_M$; our notation with a slash is necessitated by the notation $M$ for objects of $\Theta ^n$. (Sorry about these slight notational changes but is is much easier for us to use $\Theta ^n$ for what will be done below). With these notations, an $n$-precat $A$ is an {\em easy $n$-category} if: \newline ---for each $m$, $A_{m/}$ is an easy $n-1$-category; and \newline ---the morphisms $$ A_{m/} \rightarrow A_{1/} \times _{A_0} \ldots \times _{A_0} A_{1/} $$ are easy equivalences. Note here that $A_0$ is set which is the fiber over the object $0\in \Theta ^n$ (which exits slightly from our notational convention; it is the class of objects $0, M'$ of $\Delta ^n$ but here there is no ``$M$'' to put into the notation so we put ``$0$'' instead). A {\em marked easy $n$-category} is an easy $n$-category provided with the addional data of markings for the $A_{m/}$ and markings for the easy equivalences going into the definition. These two conditions amount (recursively) to saying that we have markings for all of the morphisms of the form $$ A_{M,m/} \rightarrow A_{M,1/} \times _{A_M} \ldots \times _{A_M} A_{M,1/}. $$ The notion of marking as we have defined above actually makes sense for any $n$-precat, and an $n$-precat with a marking is automatically an easy $n$-category. For this reason, arbitrary inverse limits of marked easy $n$-categories (indexed by systems of morphisms which preserve the markings) are again marked easy $n$-categories. Suppose $A$ is an $n$-precat. We define the {\em marked easy $n$-category generated by $A$} denoted $Cat(A)$ by $$ Cat (A)= \lim _{\leftarrow , {\cal C}} T $$ where the limit is taken over the category ${\cal C}$ whose objects are triples $(T,\mu ,f)$ with $(T, \mu )$ a marked easy $n$-category ($\mu$ denotes the marking) and $f: A\rightarrow T$ is a morphism of $n$-precats. The morphisms of ${\cal C}$ are morphisms of $n$-precats (i.e. morphisms of presheaves on $\Theta ^n$) required to preserve $f$ and the marking $\mu$. By the principle given in the previous paragraph, this inverse limit is again a marked easy $n$-category. The construction $Cat(A)$ is the key to the rest of what we are going to say. The description of $Cat(A)$ given above is one of cutting it down to size. There is also a creative description. In order to explain this we first discuss certain push-outs of $n$-precats. An object of $\Theta ^n$ represents a presheaf (i.e. $n$-precat). If $M$ is an object we denote the $n$-precat represented by $M$ as $h(M)$. A morphism of $n$-precats $h(M)\rightarrow A$ is the same thing as an element of $A_M$. Note that direct limits exist in the category of $n$-precats (as in any category of presheaves). In particular push-outs exist. We construct the following standard $n$-precats. Let $M= (m_1, \ldots , m_l)$ with $l\leq n-1$, and let $m\geq 1$ (although by the remark below we could also restrict to $m\geq 2$). Let $-1 \leq k \leq n-l-1$. We will state the constructions by universal properties (although we give an explicit construction later). Note that these universal properties admit solutions because we work in the category of presheaves over a given category $\Theta ^n$ so the necessary limits exist. Define $\Sigma = \Sigma (M, [m], \langle k,k+1\rangle )$ to be the universal $n$-precat with $k$-morphisms $a,b$ in $\Sigma _{M,m/}$ (i.e. $a,b\in \Sigma _{M,m, 1^k}$) and a $k+1$-morphism $$ v= (v_1, \ldots , v_m) \in (\Sigma _{M,1/} \times _{\Sigma _{M,0}} \ldots \times _{\Sigma _{M,0}}\Sigma _{M,1/} )_{1^{k+1}} $$ such that $s(a)=s(b)$, $t(a)=t(b)$, and such that the images of $a$ and $b$ by the usual map to the product of $\Sigma _{M,1/}$ are $s(v)$ and $t(v)$ respectively. Note that $h=h(M,m, 1^{k+1})$ is the universal $n$-precat with a $k+1$-morphism $u$ in $\Sigma _{M,m/}$ (i.e. $u\in \Sigma _{M,m,1^{k+1}}$). Note that setting $a$ to $s(u)$, $b$ to $t(u)$ and $v$ to the image of $u$ by the usual map to the product, we obtain (by the universal property of $\Sigma$) a morphism $$ \varphi = \varphi (M, [m], \langle k,k+1\rangle ) : \Sigma (M, [m], \langle k,k+1\rangle ) \rightarrow h(M,m,1^{k+1}) $$ We will show below that $\varphi$ is a cofibration, \footnote{ The definition, from \S 3 below, is that a morphism $A\rightarrow B$ of $n$-precats is a {\em cofibration} if for every $M= (m_1, \ldots , m_k)$ with $k <n$, the morphism $A_M \rightarrow B_M$ is injective. at the same time giving an explicit construction of $\Sigma$ as a pushout of representable presheaves. Before doing that, we mention the modifications to the above definition necessary for the boundary cases $k=-1$ and $k=n-l-1$. For $k=-1$, $\Sigma (M, [m], \langle -1,0\rangle )$ is the universal $n$-precat with an object $$ v= (v_1, \ldots , v_m) \in \Sigma _{M,1} \times _{\Sigma _{M,0}} \ldots \times _{\Sigma _{M,0}}\Sigma _{M,1} . $$ As $h(M,m,0)$ is the universal $n$-precat with an object $u\in h_{M,m}$ we have an object $v$ as above for $h$ (the image of $u$ by the usual map) so we obtain $\Sigma \rightarrow h$. For $k=n-l-1$, $\Sigma (M, [m], \langle n-l-1,n-l\rangle )$ is the universal $n$-precat with $a,b\in \Sigma _{M,m, 1^{n-l-1}}$ such that $s(a)=s(b)$ and $t(a)=t(b)$ and such that $a$ and $b$ map to the same elements of $$ (\Sigma _{M,1/} \times _{\Sigma _{M,0}} \ldots \times _{\Sigma _{M,0}}\Sigma _{M,1/} )_{1^{n-l-1}} . $$ Note that $h=h(M, m, 1^{n-l})$ is normally speaking not defined because the length of the multiindex $(M,m,1^{n-l})$ is $n+1$. Thus we formally define this $h$ to be equal to $h(M,m, 1^{n-l-1})$ and take the elements $a=b$ equal to the canonical $n-l-1$-morphism in $h_{M,m/}$. This gives a morphism $\Sigma \rightarrow h$. We will now give an explicit construction of $\Sigma$ and use this to show that $\Sigma \rightarrow h$ is a cofibration. (The boundary cases will be left to the reader). In general the universal $n$-precat with a collection of elements with certain equalities required, is a quotient of the disjoint union of the representable $n$-precats corresponding to the elements we want, by identifying pairs of morphisms from the representable $n$-precats corresponding to the elements which need to be equal. We do this in several steps. First, the universal $n$-precat $\Upsilon = \Upsilon (M, [m], 1^k)$ with element $$ v= (v_1, \ldots , v_m) \in (\Upsilon _{M,1/} \times _{\Sigma _{M,0}} \ldots \times _{\Upsilon _{M,0}}\Upsilon _{M,1/} )_{1^{k}} $$ is constructed as the quotient of the disjoint union of $m$ copies of $h(M,1,1^k)$ making $m-1$ identifications over pairs of maps $$ h(M,1,1^k) \leftarrow h(M) \rightarrow h(M,1,1^k). $$ This is the same as taking the pushout of the diagram $$ h(M,1,1^k) \leftarrow h(M) \rightarrow h(M,1,1^k) \leftarrow \ldots \leftarrow h(M) \rightarrow h(M,1, 1^k). $$ Now $\Sigma (M, [m], \langle k,k+1\rangle )$ is the quotient of the disjoint union $$ h(M, m, 1^k)^a\sqcup h(M, m, 1^k)^b \sqcup \Upsilon (M, [m], 1^{k+1}) $$ by the following identifications (the superscripts $a$ and $b$ in the above notation are there to distinguish between the two components, which correspond respectively to choosing $a$ and $b$). There are two maps (dual to $s$ and $t$) $s^{\ast}, t^{\ast}: h(M,m,1^{k-1})\rightarrow h(M,m, 1^k)$, and we identify over the pairs of morphisms $$ h(M, m, 1^k)^a \stackrel{s^{\ast}}{\leftarrow} h(M,m, 1^{k-1})\stackrel{s^{\ast}}{\rightarrow} h(M, m, 1^k)^b $$ and $$ h(M, m, 1^k)^a \stackrel{t^{\ast}}{\leftarrow} h(M,m, 1^{k-1})\stackrel{t^{\ast}}{\rightarrow} h(M, m, 1^k)^b. $$ Then we also identify over the pairs of maps $$ h(M,m,1^k)^a\leftarrow \Upsilon (M, [m], 1^{k}) \stackrel{s^{\ast}}{\rightarrow}\Upsilon (M, [m], 1^{k+1}) $$ and $$ h(M,m,1^k)^b\leftarrow \Upsilon (M, [m], 1^{k}) \stackrel{t^{\ast}}{\rightarrow}\Upsilon (M, [m], 1^{k+1}) $$ where the left maps are induced by the collection of principal morphisms $1\rightarrow m$. The result of all these identifications is $\Sigma (M, [m], \langle k,k+1\rangle )$. This gives an explicit construction for those wary of just defining things by the universal property. We now show that the morphism $\varphi : \Sigma \rightarrow h$ is a cofibration in the sense of \S 3 below, i.e. injective on all levels except the top one. We will say that a diagram of $n$-precats of the form $$ A\stackrel{\displaystyle \rightarrow}{\rightarrow} B \rightarrow C $$ (where the two compositions are the same) is {\em semiexact} if the morphism from the coequalizer of the two arrows to $C$ is a cofibration in the sense of \S 3 below. Our above construction gives $\Sigma$ as the coequalizer of $$ \Upsilon ' \sqcup \Upsilon ' \sqcup h' \sqcup h' \stackrel{\displaystyle \rightarrow}{\rightarrow} \Upsilon \sqcup h^a \sqcup h^b $$ where $$ \Upsilon = \Upsilon (M, [m], 1^{k+1}), \;\;\; \Upsilon ' = \Upsilon (M, [m], 1^{k}), $$ $$ h^a (\mbox{resp.}\,\, h^b) := h(M,m,1^k),\;\;\; h' := h(M,m, 1^{k-1}). $$ We would like to prove that $$ \Upsilon ' \sqcup \Upsilon ' \sqcup h' \sqcup h' \stackrel{\displaystyle \rightarrow}{\rightarrow} \Upsilon \sqcup h^a \sqcup h^b \rightarrow h(M, m, 1^{k+1}) $$ is semiexact. To prove this it suffices (by a simple set-theoretic consideration) to show that $$ h' \sqcup h' \stackrel{\displaystyle \rightarrow}{\rightarrow} h^a \sqcup h^b \rightarrow h(M, m, 1^{k+1}) $$ $$ \Upsilon ' \stackrel{\displaystyle \rightarrow}{\rightarrow} \Upsilon \sqcup h^a \rightarrow h(M, m, 1^{k+1}) $$ and $$ \Upsilon ' \stackrel{\displaystyle \rightarrow}{\rightarrow} \Upsilon \sqcup h^b \rightarrow h(M, m, 1^{k+1}) $$ are semiexact. The first statement follows from the claim that for any $M= (m_1, \ldots , m_l)$, $$ h(M) \sqcup h(M)\stackrel{\displaystyle \rightarrow}{\rightarrow} h(M, 1) \sqcup h(M, 1) \rightarrow h(M,1,1) $$ is semi-exact. Let $P= (p_1, \ldots , p_n)$ be an element of $\Theta ^n$ (some of the $p_j$ may be zero). A morphism $P\rightarrow (M, 1)$ corresponds to a collection of morphisms $f_i:p_i \rightarrow m_i$ for $i\leq l$ and $f_{l+1}:p_{l+1} \rightarrow 1$, up to equivalence. The equivalence relation is obtained by saying that if one of the morphisms factors through $0$ then the subsequent ones don't matter. The first thing to note is that the two morphisms $(I, s), (I,t): h(M, 1)\rightarrow h(M, 1, 1)$ are injective, as follows directly from the above description. Now suppose that $f, g: P\rightarrow (M, 1)$ are two morphisms such that $(I,s)\circ f = (I, t)\circ g$. Since $s: 0\rightarrow 1$ composed with anything is different from $t: 0\rightarrow 1$ composed with the same, this means that one of the $f_i$ must factor through $0$ for $i\leq l+1$, and that $f_j=g_j$ for $j\leq i$. If it is the case for $i\leq l$ then $f$ and $g$ both come from the morphism $(f_1, \ldots , f_l): P \rightarrow M$, which is equivalent to $(g_1, \ldots , g_l)$, via either one of the morphisms $M\rightarrow (M, 1)$. If $i=l+1$ then $f_{l+1}=g_{l+1}$ factors through one of the two morphisms $0\rightarrow 1$, so $f$ and $g$ both come from the morphism $(f_1, \ldots , f_l)=(g_1, \ldots , g_l)$ via the morphism $M\rightarrow (M, 1)$ corresponding to the morphism $0\rightarrow 1$ occuring above. Thus $f$ and $g$ are equivalent in the coequalizer, giving the claim for this paragraph and thus the first of our semiexactness statements. For the next semiexactness statement, we first note that $\Upsilon \rightarrow h(M, m, 1^{k+1})$ is cofibrant. In fact we can describe $\Upsilon$ as a subsheaf of $h(M,m, 1^{k+1})$ as follows. For $P= (p_1 , \ldots , p_n)$ the morphisms from $P$ to $(M,m, 1^{k+1})$ are the sequences of morphisms $f= (f_1, \ldots , f_{l+k +2})$ with $f_i : p_i \rightarrow m_i$ (or taking values in $m$ or $1$ as appropriate depending on $i$). Such a morphism is contained in $\Upsilon _P$ if and only if the morphism $f_{l+1}: p_{l+1}\rightarrow m$ factors through one of the principal morphisms $1\rightarrow m$ (we leave to the reader to verify that $\Upsilon$ is equal to this subsheaf). Suppose $f\in \Upsilon _P$ and $g= (g_1, \ldots , g_{l+k+1})\in h^a_P$, projecting to the same element of $h(M,m,1^{k+1})_P$. Note that $g$ projects to the element $(g_1, \ldots , g_{l+k+1}, s)$ where $s: 0\rightarrow 1$ denotes the source map (or really its dual but for purposes of the present argument we omit the dual notation). In particular $f$ is equivalent to $(g_1, \ldots , g_{l+k+1}, s)$, which implies that (up to changing $f$ and $g$ in their equivalence classes) $g_{l+1}$ factors through one of the principal maps $1\rightarrow m$ and $f_{l+k+2}=s$. This exactly means that $f$ comes from $\Upsilon '_P \rightarrow \Upsilon _P$ and $g$ from $\Upsilon ' _P \rightarrow h^a_P$. Thus $f$ and $g$ are equivalent in the coequalizer, giving the second of our semiexactness statements. The proof of the third semiexactness statement is the same as that of the second (although $s$ above would be replaced by $t$). This completes the proof that the standard morphisms $\Sigma \rightarrow h$ are cofibrations (modulo the boundary cases which we have left to the reader). An $n$-precat $A$ is an easy $n$-category if and only if every morphism $\Sigma (M,[m], \langle k,k+1\rangle )\rightarrow A$ extends to a morphism $h(M, m, 1^{k+1})\rightarrow A$. A marked easy $n$-category is an $n$-precat $A$ together with choice of extension for every morphism $\Sigma (M,[m], \langle k,k+1\rangle )\rightarrow A$. Finally, we say that a {\em partially marked $n$-precat} is an $n$-precategory provided with a distinguished subset $\mu$ of the set of all morphisms of the form $f:\Sigma (M,[m], \langle k,k+1\rangle )\rightarrow A$, and for each such morphism, a chosen extension $f^{\mu}$ to $h(M, m, 1^{k+1})$. If $(A, \mu )$ is a partially marked $n$-precat, then we define a new partially marked $n$-precat $Raj(A, \mu )$ by taking the pushout via $\varphi (M, [m], \langle k,k+1\rangle )$ for all morphisms $$ \Sigma (M,[m], \langle k,k+1\rangle )\rightarrow A $$ which are not in the subset $\mu$ of marked ones. {\em Remark:} In the above notations, if $m=1$ then $$ \Sigma (M, [1], \langle k,k+1\rangle ) = \Upsilon (M, [1], 1^{k+1}) = h(M, 1 , 1^{k+1}) $$ so the pushout by $\varphi (M, [1], \langle k,k+1\rangle )$ is trivial and we can ignore these cases if we like in the previous notation (and also in the notion of marking). \begin{lemma} If $A$ is an $n$-precat then the marked easy $n$-category $Cat(A)$ is obtained by iterating infinitely many times (i.e. over the first countable ordinal) the operation $(A', \mu ') \mapsto Raj(A', \mu ')$, starting with $(A, \emptyset )$. \end{lemma} {\em Proof:} If $(B, \nu )$ is a marked easy $n$-category and $(A', \mu ') \rightarrow (B, \nu )$ is a morphism compatible with the partial marking of $A'$, then there is a unique extension to a morphism $Raj( A', \mu ' )\rightarrow B$ compatible with the partial marking of $Raj(A', \mu ')$. It follows that if we set $Cat '(A)$ equal to the result of the iteration described in the lemma, then there is a unique morphism $Cat '(A)\rightarrow B$ compatible with the partial marking of $Cat '(A)$ and extending the given morphism $A\rightarrow B$. But $Cat '(A)$ is fully marked. By the universal property of $Cat (A)$ this implies that $Cat (A)=Cat '(A)$. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} We will often have a need for the following construction. If $A$ is an $n$-precat then iterate (over the first countable ordinal) the operation $(A', \mu ') \mapsto Raj (A', \emptyset )$. Call this $BigCat(A)$. Another way to describe this consruction is that we throw in an infinite number of times the pushouts of all of the required diagrams (which is in some sense a more obvious way to obtain an $n$-category). There is an obvious morphism $Cat (A) \rightarrow BigCat (A)$. One of the advantages of the $BigCat$ construction is that $BigCat(A) \cong BigCat (BigCat(A))$ (although the natural maps are not this isomorphism). More generally, we will use the terminology ``reordering'' below to indicate that a sequence of pushouts can be done in any order (subject to the obvious condition that the things over which the pushouts are being done exist at the time they are done!), which yields isomorphisms such as $BigCat(A) \cong BigCat (BigCat(A))$. If $B\leftarrow A \rightarrow C$ is a diagram of $n$-categories, then we define the {\em category-theoretic pushout} to be $Cat (B\cup ^AC)$. It is again an $n$-category. We will also often use just the pushout of $n$-precats, i.e. the pushout of presheaves over $\Theta ^n$. \numero{The closed model category structure} We now come to the first main definition. A morphism $A\rightarrow B$ of $n$-precats (that is, a morphism of presheaves on $\Theta ^n$) is a {\em weak equivalence} if the induced morphism $Cat(A)\rightarrow Cat(B)$ is an exterior equivalence of $n$-categories in the sense of Tamsamani (\cite{Tamsamani} \S 1.3). Note in particular that we don't require it to be an easy equivalence---which would be too strong a condition. The second main definition is relatively easy: we would like to say that a morphism $A\rightarrow B$ of $n$-precats is a cofibration if it is a monomorphism of presheaves on $\Theta ^n$. However, this doesn't work out well at the top degree (for example, the category of sets with isomorphisms as weak equivalences and injections as cofibrations, is not a closed model category \cite{Quillen}). Thus we leave the top level alone and say that a morphism $A\rightarrow B$ of $n$-precats is a {\em cofibration} if for every $M= (m_1, \ldots , m_k)$ with $k <n$, the morphism $A_M \rightarrow B_M$ is injective. A cofibration which is a weak equivalence is called a {\em trivial cofibration}. The third definition which goes along automatically with these two is that a morphism $A\rightarrow B$ of $n$-precats is a {\em fibration} if it satisfies the lifting property for trivial cofibrations, that is if every time $U\hookrightarrow V$ is a trivial cofibration and $U\rightarrow A$ and $V\rightarrow B$ are morphisms inducing the same $U\rightarrow B$ then there exists a lifting to $V\rightarrow A$ compatible with the first two morphisms. We recall from \cite{Quillen} the definition of {\em closed model category}, as well as from \cite{QuillenAnnals} an equivalent set of axioms. \begin{theorem} \label{cmc} The category of $n$-precats with the weak equivalences, cofibrations and fibrations defined above, is a closed model category. \end{theorem} \subnumero{Some lemmas} The proof of Theorem \ref{cmc} is by induction on $n$. Thus we may assume that the theorem and all of the lemmas contained in the present section and \S\S 4-6 are true for $n'$-precats for all $n'<n$. In view of this, we state all (or most) of the lemmas before getting to the proofs. Our proof will be modelled on the proof of Jardine that simplicial presheaves on a site form a closed model category \cite{Jardine}. The main lemma that we need (which corresponds to the main lemma in \cite{Jardine}) is \begin{lemma} \label{pushout} Suppose $A\rightarrow B$ is a trivial cofibration and $A\rightarrow C$ is any morphism. Let $D = B\cup ^AC$ be the push-out of these two morphisms (the push-out of $n$-precats). Then the morphism $C\rightarrow D$ is a weak equivalence. \end{lemma} This lemma speaks of push-out of $n$-precats. Applying the construction $Cat$ we obtain a notion of push-out of $n$-categories: if $A\rightarrow B$ and $A\rightarrow C$ are morphisms of $n$-categories (i.e. morphisms of the corresponding $n$-precats) then define the {\em push-out $n$-category} to be $$ Cat ( B\cup ^AC ). $$ \begin{corollary} If $A\rightarrow B$ is an equivalence of $n$-categories then $$ C\rightarrow Cat ( B\cup ^AC ) $$ is an equivalence of $n$-categories. \end{corollary} We will come back to push-out below in the section on Siefert-Van Kampen. Going along with the previous lemma is something that we would like to know: \begin{lemma} \label{equiv} If an $n$-precat $A$ is an $n$-category in the sense of \cite{Tamsamani} then the morphism $A\rightarrow Cat(A)$ (resp. the morphism $A\rightarrow BigCat(A)$) is an equivalence of $n$-categories in the sense of \cite{Tamsamani}. \end{lemma} Another lemma which is an important technical point in the proof of everything is the following. An $n$-precat $A$ can be considered as a collection $\{ A_{m/}\}$ of $n-1$-precats (functor of $\Delta$ and the first element is a set). We obtain the collection $\{ Cat(A_{m/})\}$ which is a functor from $\Delta$ to the category of $n-1$-precats. Divide by the equivalence relation setting the $0$-th element to a constant $n-1$-precat, in this way we obtain a new $n$-precat denoted $Cat _{\geq 1} (A)$. \begin{lemma} \label{partialCat1} Suppose that $A$ and $B$ are $n$-precats and $f: A\rightarrow B$ is a morphism which induces an equivalence on the $n-1$-categories $Cat (A_{m/})\rightarrow Cat (B_{m/})$. Then $Cat(A)\rightarrow Cat(B)$ (resp. $BigCat(A)\rightarrow BigCat(B)$) is an equivalence. \end{lemma} \begin{corollary} \label{partialCat} The morphism $$ Cat (A) \rightarrow Cat (Cat _{\geq 1}(A)) $$ is an equivalence of $n$-categories. \end{corollary} {\em Proof:} The morphism $A \rightarrow Cat _{\geq 1}(A)$ satisfies the hypotheses of the previous lemma so the corollary follows from the lemma. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \begin{lemma} \label{coherence} For any $n$-precat $A$, the morphism $A\rightarrow Cat(A)$ (resp. $A\rightarrow BigCat(A)$) is a weak equivalence. \end{lemma} The closed model structure that we already have by induction for $n-1$-precats allows us to deduce some things about $n$-categories. Let $HC_{n-1}$ denote the localization of $PC_{n-1}$ by inverting the set weak equivalences, which is also (see \ref{htytype} below) the localization of $n-1$-categories by inverting the set of equivalences. We know from the closed model structure \cite{Quillen} that this is equivalent to the category of fibrant (and automatically cofibrant) objects where we take as morphisms, the homotopy classes of morphisms. We also know that a morphism in $PC_{n-1}$ is a weak equivalence if and only if it projects to an isomorphism in $HC_{n-1}$ (\cite{Quillen}, Proposition 1, p. 5.5). In particular by \ref{equiv} in degree $n-1$ we know that a morphism of $n-1$-categories is an equivalence if and only if it projects to an isomorphism in $HC_{n-1}$. Suppose $A$ is an $n$-category. For $x,y\in A_0$ we have an $n-1$-category $A_1(x,y)$ which we could denote by $Hom_A(x,y)$. Let $LHom_A(x,y)$ denote the image of this object in the localization $HC_{n-1}$. On the other hand, the truncation $T^{n-1}A$ is a $1$-category. We claim that for $x$ fixed, the mapping $y\mapsto LHom_A(x,y)$ is a functor from $T^{n-1}A$ to $HC_{n-1}$. Similarly we claim that for $y$ fixed the mapping $x\mapsto LHom_A(x,y)$ is a contravariant functor from $T^{n-1}A$ to $HC_{n-1}$. These claims give some meaning at least in a homotopic sense to the notion of ``composition with $f: y\rightarrow z$'' as a map $LHom _A(x,y)\rightarrow LHom_A(x,z)$. We prove the first of the two claims, the proof for the second one being identical. Note that these arguments are generalizations of what is mentionned in \cite{Tamsamani} Proposition 2.2.8 and the following remark. Suppose $f\in A_1(y,z)$. Then let $A_2(x,y,f)$ be the homotopy fiber of $A_2(x,y,z)\rightarrow A_1(y,z)$ over the object $f$ (this is calculated by replacing the above map by a fibrant map and taking the fiber). The condition that $A_2$ be equivalent to $A_1\times _{A_0}A_1$ implies that this homotopy fiber maps by an equivalence to $A_1(x,y)$. On the other hand it maps to $A_1(x,z)$ and this diagram gives a morphism $LHom _A(x,y)\rightarrow LHom _A(x,z)$ in the localized category $H_{n-1}$. We just have to check that this morphism is independent of the choice of $f$ in its equivalence class. For this we use Proposition \ref{intervalK} below (there is no circularity because we are discussing $n-1$-categories here). If $f$ is equivalent to $g$ as elements of the $n-1$-category $A_1(y,z)$ then let $K$ denote the $n-1$ category given by \ref{intervalK}; there is a morphism $K\rightarrow A_1(y,z)$ sending $0$ to $f$ and $1$ to $g$, and since $K$ is a contractible object (weakly equivalent to $\ast$) this proves that the homotopy fibers over $f$ or $g$ are equivalent to the homotopy fiber product with $K$; we have a single map from here to $A_1(x,z)$ so our two maps induced by $f$ and $g$ are homotopic. Associativity is given by a similar argument using $A_3$ which we omit. Once we have our functors $T^{n-1}A\rightarrow HC_{n-1}$ we obtain the following type of statement: suppose $f$ is an equivalence between $u$ and $x$, then composition with $f$ induces an equivalence $LHom_A(x,y)\cong LHom _A(u,y)$ (and similarly for composition in the second variable). \begin{lemma} \label{remark} If $$ A\stackrel{f}{\rightarrow }B \stackrel{g}{\rightarrow } C $$ is a pair of morphisms of $n$-categories such that any two of $f$, $g$ or $g\circ f$ are equivalences in the sense of \cite{Tamsamani} then the third is also an equivalence. If $f: A\rightarrow B$ and $g:B\rightarrow A$ are two morphisms of $n$-categories such that $fg$ is an equivalence and $gf$ is the identity then $f$ and $g$ are equivalences. \end{lemma} {\em Proof:} The fact that composition of equivalences is an equivalence is \cite{Tamsamani} Lemme 1.3.5. The statement concluding that $f$ is an equivalence if $g$ and $g\circ f$ are, is a direct consequence of Tamsamani's interpretation of equivalence in terms of truncation operations (\cite{Tamsamani} Proposition 1.3.1). For the conclusion for $g$, note first of all that on the level of truncations $T^nA\rightarrow T^nB \rightarrow T^nC$ the fact that $f$ and $gf$ are isomorphisms of sets implies that $g$ is an isomorphism of sets. This gives essential surjectivity. Now suppose $x,y$ are objects of $B$. Choose objects $u,v$ of $A$ and equivalences $f(u)\sim x$ and $f(v)\sim y$. Then composition with these equivalences induces an isomorphism in the localized category $HC_{n-1}$ between $LHom _B(x,y)$ and $LHom_B(f(u), f(v))$ (see the discussion preceeding this lemma). The image under $g$ of this isomorphism is the same as composition with the images of the equivalences, so we have a diagram $$ \begin{array}{ccc} LHom _B(x,y) & \rightarrow & LHom _C(g(x), g(y)) \\ \downarrow &&\downarrow \\ LHom _B(f(u), f(v)) & \rightarrow & LHom _C(gf(u), gf(v)) \end{array} $$ in the category $HC_{n-1}$. The horizontal arrows are the localizations of the arrows $g:B_1(x,y)\rightarrow C_1(g(x), g(y))$ etc., and the vertical arrows are composition with our chosen equivalences, isomorphisms in $HC_{n-1}$. On the other hand, the bottom arrow fits into a diagram $$ LHom _A(u,v) LHom _B(f(u), f(v)) \rightarrow LHom _C(gf(u), gf(v)) $$ where the first arrow induced by $f$ is an isomorphism, and the composed arrow induced by $gf$ is an isomorphism; thus the bottom arrow of the previous diagram is an isomorphism therefore the top arrow is an isomorphism in $H_{n-1}$. This implies that the morphism $B_1(x,y)\rightarrow C_1(g(x), g(y))$ is an equivalence of $n-1$-categories. This is what we needed to prove to complete the proof that $g$ is an equivalence. We turn now to the second paragraph of the lemma: suppose $f$ and $g$ are morphisms of $n$-categories such that $fg$ is an equivalence and $gf$ is the identity. The corresponding fact for sets shows that $T^nf$ and $T^ng$ are isomorphisms between the sets of equivalence classes of objects $T^nA$ and $T^nB$. Suppose $x,y\in A_0$. Note that $gf(x)=x$ and $gf(y)=y$. We obtain morphisms $$ A_1(x,y) \stackrel{f}{\rightarrow} B_1(fx,fy) \stackrel{g}{\rightarrow} A_1(gfx,gfy)=A_1(x,y) \stackrel{f}{\rightarrow} B_1(fgfx, fgfy) = B_1(fx,fy). $$ We have again that $gf$ is the identity on $A_1(x,y)$ and $fg$ is an equivalence on $B_1(fx,fy)$. Inductively by our statement for $n-1$-categories, the morphism $f: A_1(x,y)\rightarrow B_1(fx,fy)$ is an equivalence. This implies that $f: A\rightarrow B$ is an equivalence and hence that $g$ is an equivalence (by the first paragraph of the lemma). \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \begin{corollary} \label{htytype} The localized category of $n$-precats modulo weak equivalence is equivalent to the category $Ho-n-Cat$ of $n$-categories localized by equivalence defined in \cite{Tamsamani}. \end{corollary} {\em Proof:} The functor $Cat$ sends weak equivalences to equivalences (by Lemma \ref{equiv} together with Lemma \ref{remark}). Thus it induces a functor $c$ on localizations. Let $i$ be the functor induced on localizations by the inclusion of $n$-categories in $n$-precats. The natural transformation $A\rightarrow Cat(A)$ gives a natural isomorphism $1 \cong c\circ i$ of functors on the localization of $n$-categories. On the other hand, Lemma \ref{coherence} says that the same natural transformation induces a natural isomorphism $1\cong i\circ c$ of functors on the localization of $n$-precats. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} We will prove lemmas \ref{pushout}, \ref{equiv} and \ref{partialCat1} all at once in one big induction on $n$. Thus we may assume that they hold for $n' < n$. All lemmas from here until the end of the big induction presuppose that we know the inductive statement for $n'<n$. {\em Remark on the passage between $Cat$ and $BigCat$ in \ref{equiv} and \ref{partialCat1}:} The statements for $Cat$ and $BigCat$ are equivalent. Take Lemma \ref{equiv} for example. If $A\rightarrow Cat(A)$ is an equivalence for any $n$-category $A$ then $BigCat(A)$ can be constructed as the iteration over the first countable ordinal of the operation $A' \mapsto Cat (A')$ (and starting at $A$). The morphisms at each stage in the iteration are equivalences, so it follows that the morphism $A\rightarrow BigCat(A)$ is an equivalence. On the other hand, suppose we know that $A\rightarrow BigCat(A)$ is an equivalence for any $n$-category $A$. Then $Cat(A)\rightarrow BigCat(Cat(A))$ is an equivalence, but by reordering $BigCat(Cat(A))=BigCat(A)$. Thus the hypothesis also gives that $A\rightarrow BigCat(Cat(A))$ is an equivalence. Lemma \ref{remark} then implies that $A\rightarrow Cat(A)$ is an equivalence. We obtain the required statement concerning \ref{partialCat1} by using the fact that $Cat(A)\rightarrow BigCat(A)$ (resp. $Cat(B)\rightarrow BigCat(B)$) is an equivalence---note that our proof of \ref{partialCat1} comes after our proof of \ref{equiv} below---and applying \ref{remark}. \subnumero{A simplified point of view} We started to see, in the proof of Lemma \ref{remark}, a simplified or ``derived'' point of view on $n$-categories. We will expand on that a bit more here. When we use the statements of the above lemmas for $n-1$-categories, they may be considered as proved in view of our global induction. The homotopy or localized category $HC_{n-1}$ of $n-1$-precats modulo weak equivalence, also equal to the localization of Tamsamani's $(n-1)-Cat$ by equivalences, admits direct products. There is a functor $T^{n-1}: HC_{n-1} \rightarrow Sets$ related to the inclusion $i: Sets \subset HC_{n-1}$ by morphisms $iT^{n-1}(X)\rightarrow X$ and $T^{n-1}iS \cong S$ (the first is only well defined in the localized category). Thus if $A\times B\rightarrow C$ is a morphism in $HC_{n-1}$ then we obtain the map $A\times iT^nB \rightarrow C$. Note that $HC_{n-1}$ admits fibered products over objects of the form $i(S)$ for $S$ a set, since these are essentially just direct products. (However the homotopy category does not admit general fibered products nor, dually, does it admit pushouts.) We can define the notion of $HC_{n-1}$-category, as simply being a category in the category $HC_{n-1}$ such that the object object is a set. Applying the functor $T^{n-1}$ yields a category, and this category acts on the morphism objects of the previous one, using the above remark. If $A$ is an $n$-category then taking $A_0$ as set of objects and using the object $LHom_A(x,y)$ as morphism object in $HC_{n-1}$ we obtain an $HC_{n-1}$-category which we denote $HC_{n-1}(A)$. We can write $$ Hom _{HC_{n-1}(A)}(x,y) := LHom_A(x,y) $$ which in turn is, we recall, the image of $A_1(x,y)$ in the localization of the category of $n-1$-precats. This is what we used in the proof of Lemma \ref{remark} above. The truncation operation $T^{n-1}$ applied to $HC_{n-1}(A)$ gives the $1$-category $T^{n-1}A$. We obtain again the action of this category on the morphism objects in $HC_{n-1}(A)$. In the next section we will be interested in the notion of {\em $HC_{n-1}$-precategory}, a functor $F:\Delta \rightarrow HC_{n-1}$ sending $0$ to a set. An $HC_{n-1}$-precategory $F$ is an $HC_{n-1}$-category if and only if the usual morphisms $$ F_p\rightarrow F_1\times _{F_0} \ldots \times _{F_0}F_1 $$ are isomorphisms. If $A$ is an $n$-precat then let $HC_{n-1}(A)$ denote the $HC_{n-1}$-precategory which to $p\in \Delta$ associates the image of $A_{p/}$ in the localized category $HC_{n-1}$. Here is a small remark which is sometimes useful. \begin{lemma} \label{HCequivCat} Suppose $f:A\rightarrow B$ is a morphism of $n$-categories and suppose that $HC_{n-1}(A)\rightarrow HC_{n-1}(B)$ is an equivalence in $HC_{n-1}Cat$. Then $f$ is an equivalence. \end{lemma} \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} We can also make a similar statement for $HC_{n-1}$-precats under the condition of requiring an isomorphism on the set of objects. \begin{lemma} \label{HCequivPreCat} Suppose $f:A\rightarrow B$ is a morphism of $n$-precats and suppose $HC_{n-1}(A)\rightarrow HC_{n-1}(B)$ is an isomorphism of functors $\Delta \rightarrow HC_{n-1}Cat$. Then $f$ is a weak equivalence. \end{lemma} {\em Proof:} This is just a restatement of Lemma \ref{partialCat1} (in particular it is not available for use in degree $n$ until we have proved \ref{partialCat1} below). \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} It would have been nice to be able to have an operation on $HC_{n-1}$-precategories which, when applied to $HC_{n-1}(A)$ yields $HC_{n-1}(Cat(A))$. This doesn't seem to be possible (although I don't have a counterexample) because the construction we discuss in the next section relies heavily on pushouts but these don't exist in $HC_{n-1}$. If this had been possible we would have been able to formulate a notion of weak equivalence for $HC_{n-1}$-precats and in particular we would have been able to give a stronger formulation in the previous lemma. We end this discussion by pointing out that information is lost in passing from $A$ to $HC_{n-1}(A)$. (See the next paragraph for some counterexamples but I don't have counterexamples for all of the nonexistence statements which are made.) Let $HC_{n-1}Cat$ (resp. $HC_{n-1}PreCat$) denote the categories of $HC_{n-1}$-categories (resp. $HC_{n-1}$-precategories). The functors $n-Cat\rightarrow HC_{n-1}Cat$ and $n-Cat \rightarrow HC_n$ do not enter into a commutative triangle with a morphism between $HC_n$ and $HC_{n-1}Cat$ in either direction. The only thing we can say is that there is an obvious notion of equivalence between two $HC_{n-1}$-categories, and if we let $Ho-HC_{n-1}-Cat$ denote the category of $HC_{n-1}$-categories localized by inverting these equivalences, then there is a factorization $$ n-CAT \rightarrow HC_n \rightarrow Ho-HC_{n-1}-Cat $$ but the second arrow in the factorization is not an isomorphism. In particular, when we pass from $A$ to $HC_{n-1}(A)$ we lose information. Nonetheless, it may be helpful especially from an intuitive point of view to think of an $n$-category in terms of its associated object $HC_{n-1}(A)$ which is a category in the homotopy category of $n-1$-categories. The topological analogy of the above situation (which can be made precise using the Poincar\'e groupoid and realization constructions \cite{Tamsamani}---thus providing some counterexamples to support some of the the nonexistence statements made in the previous paragraph) is the following: if $X$ is a space then for each $x,y\in X$ we can take as $h(x,y)$ the space of paths from $x$ to $y$ viewed as an object in the homotopy category $Ho(Top)$. We obtain a category in $Ho(Top)$. If $X$ is connected it is a groupoid with one isomorphism class, thus essentially a group in $Ho(Top)$. This group is just the loop space based at any choice of point, viewed as a group in $Ho(Top)$. It is well known (\cite{Adams} \cite{Tanre}) that this object does not suffice to reconstitute the homotopy type of $X$, thus our functor from $Top$ to the category of groupoids in $Ho(Top)$ does not yield a factorization of the localization functor $Top\rightarrow Ho(Top)$. On the other hand, since there is no way to canonically choose a collection of basepoints for an object in $Ho(Top)$, there probably is not a factorization in the other direction either. \subnumero{Another simplified point of view} We now give another set of remarks relating the present approach to $n$-categories with the usual standard ideas. This is based on the following observation. The proof of the lemma is based on some ideas from the next section so the reader should look there before trying to follow the proof. We have put the lemma here for expository reasons. \begin{lemma} \label{fibrantpieces} If $A$ is a fibrant $n$-precat then the $A_{p/}$ are fibrant $n-1$-precats. \end{lemma} {\em Proof:} Fix objects $x_0, \ldots , x_p\in A_0$. We show that $A_{p/}(x_0, \ldots , x_p)$ is fibrant. Suppose $U\rightarrow V$ is a trivial cofibration of $n-1$-precats. Let $B$ (resp. $C$) be the $n$-precat with objects $0, \ldots , p$ and such that $B_{q/}(i_0, \ldots , i_q)$ (resp. $C_{q/}(i_0, \ldots , i_q)$) is the disjoint union of $U$ (resp. $V$) over all morphisms $f:q\rightarrow p$ such that $f(q)=i_q$. Then (as can be seen by the discussion of the next section) $B\rightarrow C$ is a trivial cofibration. A morphism $B\rightarrow A$ (resp. $C\rightarrow A$) is the same thing as a morphism $U\rightarrow A_{p/}(x_0,\ldots , x_p)$ (resp. $V\rightarrow A_{p/} (x_0,\ldots , x_p)$). It follows immediately that if $A$ is fibrant then $A_{p/}(x_0,\ldots , x_p)$ has the required lifting property to be fibrant. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} Now we can use the closed model category structure on $PC_{n-1}$ to analyze the collection of $A_{p/}$ when $A$ is fibrant. Recall that morphisms in the localized category between fibrant and cofibrant objects are represented by actual morphisms \cite{Quillen}. Thus the morphism $$ A_{2/} \rightarrow A_{1/} \times _{A_0} A_{1/} $$ which is an equivalence, can be inverted and then followed by the projection to the third edge of the triangle to give $$ A_{1/} \times _{A_0} A_{1/}\rightarrow A_{2/} \rightarrow A_{1/}. $$ We get a morphism ``composition'' $$ m:A_1(x,y)\times A_1(y,z) \rightarrow A_1(x,z) $$ which represents the composition $$ LHom_A(x,y)\times LHom_A(y,z)\rightarrow LHom_A(x,z) $$ of the previous ``simplified point of view''. Of course our composition morphism $m$ is not uniquely determined but depends on the choice of inversion of the original equivalence. In particular $m$ will not in general be associative. However $A_{3/}$ gives a homotopy in the sense of Quillen between $m(m(f,g),h)$ and $m(f, m(g,h))$. This can be turned into a homotopy in the sense of the $n-1$-categories of morphisms (an exercise left to the reader). \numero{Calculus of ``generators and relations''} For the proofs of \ref{equiv} and \ref{partialCat1} we need a close analysis of an operation which when iterated yields $BigCat$. This analysis will lead us to a point of view which generalizes the idea of generators and relations for an associative monoid. At the end we draw as a consequence one of the main special lemmas needed to treat the special case \ref{specialcase} in the proof of \ref{pushout}. The overall goal of this section is to investigate the operation $A\mapsto Cat(A)$ in the spirit of looking at the simplicial collection of $n-1$-precats $A_{p/}$ as a functor from $\Delta$ to our closed model category in degree $n-1$. We would like to understand the transformation which this functor undergoes when we apply the operation $Cat$ to $A$. We first describe a general type of operation which we often encounter. Suppose $A$ is an $n$-precat and suppose $A_{m/}\rightarrow B$ is a cofibration of $n-1$-precats provided with a morphism $\pi :B\rightarrow A_0 \times \ldots \times A_0$ making the composition $$ A_{m/}\rightarrow B\rightarrow A_0 \times \ldots \times A_0 $$ equal to the usual morphism (there are $m+1$ factors $A_0$ in the product). We can alternatively think of this as a collection of cofibrations $$ A_{m/} (x_0, \ldots , x_m) \rightarrow B(x_0,\ldots , x_m) $$ for all sequences of objects $x_i \in A_0$. Then we define the cofibration of $n$-precats $$ A\rightarrow {\cal I} (A; A_{m/}\rightarrow B) $$ as follows (the projection $\pi$ is part of the data even though it is not contained in the notation). For any $p$, ${\cal I} (A; A_{m/}\rightarrow B)_{p/}$ is the multiple pushout of $A_{p/}$ and $A_{m/}\rightarrow B$ over all morphisms $A_{m/} \rightarrow A_{p/}$ coming from morphisms $p\rightarrow m$ which do not factor through $0$. Functoriality is defined as follows: if $q\rightarrow p$ is a morphism then for any $f:p\rightarrow m$ such that the composition $q\rightarrow m$ doesn't factor through $0$, we define the morphism of functoriality on the part of the pushout corresponding to $f$ as the identity in the obvious way; on the other hand, if $f: p\rightarrow m$ is a morphism such that the composition $q\rightarrow m$ factors through $0\rightarrow m$ then we obtain (from the projection $\pi$) a morphism $B\rightarrow A_0$ extending the morphism $A_{m/}\rightarrow A_0$ and so that part of the pushout is sent into the image of $A_0$ in $A_{q/}$. We call $A\rightarrow {\cal I} (A; A_{m/}\rightarrow B)$ the {\em pushout of $A$ induced by $A_{m/}\rightarrow B$}. Using Lemma \ref{pushout} in degree $n-1$ we find that if $A_{m/}\rightarrow B$ is a trivial cofibration then the morphisms $$ A_{p/} \rightarrow {\cal I} (A; A_{m/}\rightarrow B)_{p/} $$ are trivial cofibrations. This operation occurs notably in the process of doing $Cat$ or $BigCat$ to $A$. Fix $m\geq 1$, $M=(m_1, \ldots , m_l)$, $m'$ and $k$. Let $$ \Sigma := \Sigma (m, M, [m'], \langle k,k+1 \rangle ), $$ and $$ \varphi := \varphi (m, M, [m'], \langle k,k+1 \rangle ):\Sigma \rightarrow h(m,M,m', 1^{k+1}). $$ Suppose again $a: \Sigma\rightarrow A$ is a morphism and let $C$ be the pushout $n$-precat of $A$ and $\varphi$ over $a$. In this case note that $\Sigma$ and $h(m,M,m', 1^{k+1})$ are pushouts of diagrams of objects entirely within the category $(m, \Theta ^{n-1})$ of objects of the form $(m,M')$. The restriction of $A$ to this category is just $A_{m/}$. Let $\psi : A_{m/}\rightarrow F$ be the pushout $n-1$-precat of $\varphi$ over $a$ considered in this way. Then $C = {\cal I} (A; A_{m/}\rightarrow F)$ is the pushout of $A$ induced by $\psi$ (note that $\psi$ admits a projection $\pi$ in an obvious way). The proof is that $h(m,M,m', 1^{k+1})$ has exactly the same description as a pushout of $\Sigma$. Suppose $A$ is an $n$-precat. Define a new $n$-precat $Fix(A)$ by iterating the above operation of pushout by all standard cofibrations $\varphi (m, M, [m'], \langle k,k+1 \rangle )$, over all possible values of $m$, $M$, $m'$ and $k$, and repeating this operation a countable number of times. By reordering, $Fix(A)$ may be seen as obtained from $A$ by a sequence of standard pushouts of the form $$ A' \rightarrow {\cal I} (A', A'_{m/} \rightarrow BigCat(A'_{m/})). $$ In particular it is clear that each $A_{p/}\rightarrow Fix(A)_{p/}$ is a trivial cofibration. On the other hand it is also clear that the $Fix(A)_{p/}$ are $n-1$-categories (they are obtained by iterating operations of the form, taking $BigCat$ then taking a bunch of pushouts then taking $BigCat$ and so on an infinite number of times---and such an iteration is automatically an easy $n-1$-category). In order to get to $Cat(A)$ or $BigCat(A)$ we need another type of operation which relates the different $A_{m/}$. Suppose $A$ is an $n$-precat, fix $m\geq 2$ and suppose that we have a diagram $$ A_{m/} \stackrel{f}{\rightarrow} Q \stackrel{g}{\rightarrow} A_{1/} \times _{A_0} \ldots \times _{A_0}A_{1/} $$ with the first arrow cofibrant. Then we define the pushout $A\rightarrow {\cal J} (A; f,g)$ as follows. ${\cal J} (A; f,g)_{p/}$ is the multiple pushout of $A_{m/}\rightarrow A_{p/}$ and $A_{m/} \rightarrow Q$ over all maps $p\rightarrow m$ not factoring through any of the principal maps $1\rightarrow m$. The morphisms of functoriality are defined in the same way as for the construction ${\cal I}$ using the map $g$ here. {\em Remark:} This pushout changes the object over $1\in \Delta$ because there are morphisms $1\rightarrow m$ (the faces other than the principal ones) which don't factor through the principal face maps. The remaining of our standard pushouts which are not covered by the operation ${\cal I}$ are covered by this operation ${\cal J}$. Fix some $m\geq 2$ and $k$. Write $$ \Sigma\;\;\; \mbox{for} \;\;\; \Sigma ([m], \langle k,k+1 \rangle ), $$ and $$ \varphi := \varphi ([m], \langle k,k+1 \rangle ):\Sigma \rightarrow h(m, 1^{k+1}). $$ We have a diagram of $n-1$-precats $$ \Sigma _{m/} \stackrel{f(m,k)}{\rightarrow} h(1^{k+1}) \stackrel{g(m,k)}{\rightarrow} \Sigma _{1/} \times _{\Sigma _0} \ldots \times _{\Sigma _0} \Sigma _{1/}, $$ and via this diagram $$ h(m, 1^{k+1}) = {\cal J} (\Sigma ; f(m,k),g(m,k)). $$ It follows that if $A$ is an $n$-precat and $\Sigma \rightarrow A$ is a morphism then the standard pushout $B$ of $A$ along $\varphi$ is of the form $B = {\cal J} (A; f,g)$ for appropriate maps $f$ and $g$ induced by the above ones. We need to have some information about decomposing and commuting the operations ${\cal I}$ and ${\cal J}$. Suppose $$ A_{m/} \stackrel{\varphi}{\rightarrow} P \stackrel{\pi}{\rightarrow} A_0 \times \ldots \times A_0 $$ is a morphism. Let $$ \eta :A_{1/}\rightarrow B $$ denote the multiple pushout of $A_{1/}$ by $\varphi$ over all of the principal morphisms $1\rightarrow m$ (with projection $\nu : B\rightarrow A_0\times A_0$). We obtain a factorization $$ A_{m/} \stackrel{\varphi}{\rightarrow} P \stackrel{\psi}{\rightarrow} B\times _{A_0} \ldots \times _{A_0} B $$ and we have $$ {\cal I} (A; \varphi , \pi )= {\cal J} ({\cal I} (A; \eta ,\nu ); \varphi , \psi ). $$ In this way we turn an operation of the form ${\cal I}$ for $m$ into an operation of the form ${\cal I}$ for $1$ followed by an operation of the form ${\cal J}$ for $m$. We define a type of operation combining operations of the form ${\cal J}$ for $m$ with operations of the form ${\cal I}$ for $1$. However, we would like to keep track of certain sub-$n-1$-precats of $A_{m/}$ and $A_{1/}$. So we say that an {\em $(m,1)$-painted $n$-precat} (or just {\em painted $n$-precat} in the current context where $m$ is fixed) is an $n$-precat $A$ together with cofibrations of $n-1$-precats $A^{\ast}_{m/} \rightarrow A_{m/}$ and $A^{\ast}_{1/} \rightarrow A_{1/}$. We require a lifting of the standard morphism to $$ A^{\ast}_{m/} \rightarrow A^{\ast}_{1/} \times _{A_0} \ldots \times _{A_0} A^{\ast}_{1/}. $$ Suppose $(A,A^{\ast}_{m/},A^{\ast}_{1/})$ is a painted $n$-precat, and suppose that we have morphisms $$ A^{\ast}_{1/}\stackrel{\eta}{\rightarrow} B\stackrel{\nu}{\rightarrow}A_0\times A_0 $$ and $$ A^{\ast}_{m/} \stackrel{\varphi}{\rightarrow} P \stackrel{\psi}{\rightarrow} B\times _{A_0} \ldots \times _{A_0} B $$ compatible with the previous lifting of the standard morphism to the painted parts. Let $\eta '$, $\nu '$, $\varphi '$ and $\psi '$ be obtained by taking the pushouts of the above with $A_{1/}$ or $A_{m/}$. Then we define a new painted $n$-precat $$ {\cal J} '(A; \eta , \nu ; \varphi , \psi ) := {\cal J} ({\cal I} (A; \eta ',\nu '); \varphi ', \psi '), $$ with painted parts $(P, B)$ replacing $(A_{m/}^{\ast}, A_{1/}^{\ast})$. This operation now behaves well under iteration: the composition of two such operations is again an operation of the same form. Furthermore, our operations ${\cal I}$ and ${\cal J}$ coming from standard trivial cofibrations can be interpreted as operations of the above type if the original $\Sigma \rightarrow A$ sends the arrows $(a,b, v_i)$ into the painted parts $A_{m/}^{\ast}, A_{1/}^{\ast}$. These operations are exactly designed to do two things: replacing the painted parts by their associated $n$-categories; and getting the standard map to being an equivalence. In particular, starting with $A^{\ast}_{1/} = A_{1/} $ and $A^{\ast}_{m/} = A_{m/}$, there is a sequence of operations coming from standard trivial cofibrations (concerning only $m$ and $1$) such that, when interpreted as operations on painted $n$-precats, combine into one big operation of the form ${\cal J}'$ where $\eta : A^{\ast}_{1/} \rightarrow B$ is a trivial cofibration to an $n-1$-category, and where the morphism $$ P \stackrel{\psi}{\rightarrow} B\times _{A_0} \ldots \times _{A_0} B $$ is an equivalence of $n$-categories. Going back to the original definition of the operation ${\cal J} '$ in terms of ${\cal J}$ and ${\cal I}$ we find that an appropriate sequence of trivial cofibrations can be reordered into an operation of the form $A\mapsto A' = {\cal I} (A, \eta , \nu )$ followed by ${\cal J} (A'; f,g)$ for $$ A'_{m/}\stackrel{f}{\leftarrow} {\cal G} [m](A) \stackrel{g}{\rightarrow} A'_{1/}\times _{A_0} \times \ldots \times _{A_0} A'_{1/} $$ where $g$ is a weak equivalence. Recall that the morphism $A\rightarrow A'$ coming before the operation ${\cal J}$ has the property that the $A_{p/}\rightarrow A'_{p/}$ are weak equivalences. Note also that we can assume that $A'_{1/}$ is an $n-1$-category, because it is equal to $B$---there are no morphisms $1\rightarrow 1$ other than the identity and those which factor through $0$---and $B$ can be chosen to be an $n-1$-category. (This paragraph is the conclusion we want; the discussion of painted $n$-precats was just a means to arrive here and will not be used any further below.) Let $Gen [m](A)$ denote the result of the previous operation, which we can thus write as $$ Gen [m](A)= {\cal J} (A'; A'_{m/}\stackrel{f}{\rightarrow} {\cal G} [m](A) \stackrel{g}{\rightarrow} A'_{1/}\times _{A_0} \times \ldots \times _{A_0} A'_{1/}). $$ The pushouts chosen as above may be assumed to contain, in particular, all of the standard pushouts of the second type for $m$. Put $Gen _1(A)=Fix(A)$ and $Gen _i (A):= Fix(Gen[i](Gen _{i-1}(A)))$ for $i\geq 2$. Let $Gen(A)$ be the inductive limit of the $Gen _i(A)$. Finally, iterate the operation $A'\mapsto Gen(A')$ a countable number of times. It is clear that, by reordering, this yields $BigCat(A)$, since on the one hand all of the necessary pushouts occur, whereas on the other hand only the standard pushouts are used. \subnumero{Proofs of \ref{equiv} and \ref{partialCat1}} The above description yields immediately the proofs of these two lemmas. {\em Proof of \ref{equiv}:} Suppose $A$ is an $n$-precat such that $A_{1/}$ is an $n-1$-category and such that $(\ast )$ for all $m$ the morphisms $$ A_{m/} \rightarrow A_{1/}\times _{A_0} \ldots \times _{A_0} A_{1/} $$ are weak equivalences of $n-1$-precats. Fix $m$ and apply the operation $Gen [m](A)$. Let $A'$ be the intermediate result of doing the preliminary operations ${\cal I}$. The morphism $$ f:A'_{m/} \rightarrow {\cal G} [m](A) $$ is a trivial cofibration of $n$-precats, using: \newline (1)\,\, the fact that ${\cal G} [m](A)\rightarrow A'_{1/}\times _{A_0} \ldots \times _{A_0} A'_{1/}$ is a weak equivalence; \newline (2)\,\, the fact that $A'_{1/}$ are $n$-categories equivalent to $A_{1/}$, and noting that direct products of $n$-categories (or fibered products over sets) preserve equivalences; and \newline (3)\,\, the hypothesis that $A_{m/}$ is weakly equivalent to the product of the $A_{1/}$, again coupled with the fact that $A'_{m/}$ is weakly equivalent to $A_{m/}$ because the operations ${\cal I}$ preserve the weak equivalence type of the $A_{p/}$. It now follows from the definition of the operation ${\cal J} (A' ; f,g)$ that the morphisms $$ A_{p/} \rightarrow A'_{p/} \rightarrow {\cal J} (A' ; f,g)_{p/} $$ are weak equivalences. Thus (under the hypothesis $(\ast )$ above) the morphism $A\rightarrow Gen [m](A)$ induces weak equivalences $A_{p/}\rightarrow Gen [m](A)_{p/}$. The same holds always for the operation $Fix$, and iterating these we obtain the conclusion (under hypothesis $(\ast )$) that $$ A_{p/}\rightarrow BigCat(A)_{p/} $$ are weak equivalences. In the hypotheses of \ref{equiv}, $A$ is an $n$-category, so $A_{1/}$ is an $n-1$-category and hypothesis $(\ast )$ is satisfied. It follows from above that the morphisms $$ A_{p/} \rightarrow BigCat(A)_{p/} $$ are weak equivalences of $n-1$-precats, but since both sides are $n-1$-categories this implies that they are equivalences of $n-1$-categories (using Lemma \ref{equiv} in degree $n-1$). Therefore $A\rightarrow BigCat(A)$ is an equivalence of $n$-categories, completing the proof of \ref{equiv}. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} {\em Proof of \ref{partialCat1}:} Suppose $A\rightarrow B$ is a morphism of $n$-precats which induces weak equivalences of $n-1$-precats $A_{m/} \rightarrow B_{m/}$ for all $m$. Replacing $A$ by $Fix(A)$ and $B$ by $Fix(B)$ conserves the hypothesis. We show that replacement of $A$ by $Gen [m](A)$ and $B$ by $Gen[m](B)$ conserves the hypothesis. Let $A'$ (resp. $B'$) denote the intermediate $n$-precats used in the definition of $Gen[m](A)$ (resp. $Gen[m](B)$). These are the results of applying operations ${\cal I}$ to $A$ and $B$, and we may assume as in the previous proof that $A'_{1/}$ and $B'_{1/}$ are $n-1$-categories. We do these operations in a canonical way so as to preserve morphisms $A'\rightarrow B'$ and $Gen[m](A)\rightarrow Gen[m](B)$. Note that $A_{1/}\rightarrow A'_{1/}$ is a weak equivalence and the same for $B$. Our hypothesis now implies that the morphism $A'_{1/}\rightarrow B'_{1/}$ is an equivalence of $n-1$-categories. Therefore $$ A'_{1/} \times _{A'_0} \ldots \times _{A'_0}A'_{1/} \rightarrow B'_{1/} \times _{B'_0} \ldots \times _{B'_0}B'_{1/} $$ is a weak equivalence. As before $$ Gen [m](A)= {\cal J} (A'; A'_{m/}\stackrel{f_A}{\rightarrow} {\cal G} [m](A) \stackrel{g_A}{\rightarrow} A'_{1/}\times _{A_0} \times \ldots \times _{A_0} A'_{1/}), $$ with $g_A$ being a weak equivalence. Similarly $$ Gen [m](B)= {\cal J} (B'; B'_{m/}\stackrel{f_B}{\rightarrow} {\cal G} [m](B) \stackrel{g_B}{\rightarrow} B'_{1/}\times _{B_0} \times \ldots \times _{B_0} B'_{1/}), $$ with $g_B$ a weak equivalence. It follows immediately that $$ {\cal G} [m] (A) \rightarrow {\cal G} [m](B) $$ is a weak equivalence and hence (using the descripition of ${\cal J}$ as well as the fact that weak equivalences on the components induce weak equivalences of pushouts, which we know by induction for $n-1$-precats) the morphism $Gen [m](A)\rightarrow Gen [m](B)$ induces weak equivalences $$ Gen [m](A)_{p/}\rightarrow Gen [m](B)_{p/}. $$ This shows that the operation $Gen[m]$ preserves the hypothesis of \ref{partialCat1}. Taking limits we get that $Gen$ and finally that $BigCat$ preserve the hypothesis: we get that for all $p$, $$ BigCat (A)_{p/} \rightarrow BigCat (B)_{p/} $$ is a weak equivalence. This implies that $BigCat(A)\rightarrow BigCat(B)$ is an equivalence of $n$-categories, finishing the proof of \ref{partialCat1}. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \subnumero{$1$-free ordered precats} Suppose $A$ is an $n$-precat. We say that $A$ is {\em $1$-free ordered} if there is a total order on the set $A_0$ of objects (which we suppose for simplicity to be finite) such that the following properties are satisfied: \newline (FO1)---for any sequence $x_0,\ldots , x_n$ which is out of order (i.e. some $x_i$ is strictly bigger than $x_{i+1}$), $A_{m/}(x_0,\ldots , x_m)=\emptyset$; \newline (FO2)---for any sequence $x_0, \ldots , x_n$ with $x_{i-1}\leq x_i$ the morphism $$ A_{m/}(x_0,\ldots , x_m)\rightarrow A_{1/}(x_0,x_m) $$ is a weak equivalence; and \newline (FO3)---for any stationary sequence $A_{m/}(x,\ldots , x)$ is weakly equivalent to $\ast$. Properties (FO1) and (FO2) properties are preserved under the operation $BigCat$. Indeed, the standard cofibrations $\Sigma \rightarrow h$ go between $n$-precats which satisfy these conditions, and these conditions are preserved by pushouts over diagrams of morphisms which are order-respecting (i.e. morphisms respecting $\leq$) between $1$-free ordered $n$-precats. Note that if $A$ satisfies (FO1) then any morphism $\Sigma \rightarrow A$ must respect the order. The condition (FO3) is preserved by pushouts of morphisms which are strictly order-preserving, and also by the operation $Fix$ (which is essentially the same as $Cat _{\leq 1}$). If $A$ is $1$-free ordered, using (FO1) and (FO3) we get that in order to obtain $Cat(A)$ it suffices to use the operation $Fix$ and pushouts for trivial cofibrations $\Sigma \rightarrow h(m, 1^{k+1})$ via morphisms $\Sigma \rightarrow A$ which are strictly order preserving. Thus $BigCat(A)$ again satisfies (FO3). Furthermore, note that these trivial cofibrations do not change the homotopy type for adjacent objects, so if $x,y$ are adjacent in the ordering then $$ A_{1/}(x,y)\rightarrow BigCat(A)_{1/}(x,y) $$ is a weak equivalence. We obtain the following conclusion: \begin{lemma} \label{freeness} Suppose $A$ is a $1$-free ordered $n$-precat with finite object set. For two objects $x,y\in A_0$ let $x=x_0, \ldots , x_m = y$ be the maximal strictly increasing ordered sequence going from $x$ to $y$. Let $A':= BicCat(A)$. Then the morphisms $$ A'_{1/}(x,y)\leftarrow A'_{m/}(x_0,\ldots , x_m) \rightarrow A_{1/}(x_0,x_1)\times _{A_0} \ldots \times _{A_0} A_{1/}(x_{m-1}, x_m) $$ are weak equivalences. In particular if $A\rightarrow B$ is a morphism of $1$-free ordered $n$-precats (with finite object sets) preserving the ordering, inducing an isomorphism on object sets and inducing equivalences $A_{1/} (x,y)\rightarrow B_{1/}(x,y)$ for all pairs of adjacent objects $(x,y)$ then $A\rightarrow B$ is a weak equivalence. \end{lemma} {\em Proof:} In the diagram $$ A'_{1/}(x,y)\leftarrow A'_{m/}(x_0,\ldots , x_m) \rightarrow A'_{1/}(x_0,x_1)\times _{A_0} \ldots \times _{A_0} A'_{1/}(x_{m-1}, x_m) $$ the left arrow is a weak equivalence by property (FO2) for $A'$ (we have shown that that property is preserved under passage from $A$ to $A'$); and the right arrow is a weak equivalence because $A'$ is an $n$-category. The product on the right may be replaced by that appearing in the statement of the lemma since $A_{1/}(x_{i-1},x_{i})\rightarrow A'_{1/}(x_{i-1},x_{i})$ is an equivalence because $x_{i-1}$ and $x_i$ are adjacent. This gives the first statement of the lemma. For the second statement, note that (using the same notation for objects of $A$ and $B$, and using the notations $A'$ and $B'$ for associated $n$-categories) the first statement implies that $A'_{1/} (x,y)\rightarrow B'_{1/}(x,y)$ is a weak equivalence for any pair of objects $x,y$. This implies that $A'\rightarrow B'$ is an equivalence of $n$-categories. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \subnumero{Characterization of weak equivalence} We close this section by mentioning a proposition which gives a sort of uniqueness for the notion of weak equivalence. \begin{proposition} \label{general} Suppose $F: PC_n \rightarrow PC_n$ is a functor with natural transformation $i_A: A\rightarrow F(A)$ such that: \newline (a)---for all $A$, $F(A)$ is an $n$-category; \newline (b)---if $A$ is an $n$-category then $i_A$ is an iso-equivalence of $n$-categories (recall that this means an equivalence inducing an isomorphism on sets of objects); and \newline (c)---for any $n$-precat $A$ the morphism $F(i_A): F(A) \rightarrow F(F(A))$ is an equivalence of $n$-categories. \newline Then for any $n$-precat $A$ the morphism $A\rightarrow F(A)$ is a weak equivalence. \end{proposition} {\em Proof:} Put $F'(A):= Cat (F(A))$. It is a marked easy $n$-category. We have a morphism $k_A := Cat (i_A):Cat(A)\rightarrow F'(A)$. Letting $j_A: A\rightarrow Cat(A)$ denote the inclusion and $i'_A$ the map $A\rightarrow F'(A)$, note that $k_Aj_A = i'_A$. The functor $F'$ again satisfies the properties (a), (b) and (c) above. For property (c) note that the map $F'(i'_A)$ is obtained by applying $Cat$ to the composed map $$ F(A) \stackrel{F(i_A)}{\rightarrow} F(F(A))\stackrel{F(j_{F(A)})}{\rightarrow} F(Cat(F(A))), $$ but the first map is an equivalence by hypothesis and the second is an equivalence because of the diagram $$ \begin{array}{ccc} F(A) & \rightarrow & F(F(A)) \\ \downarrow && \downarrow \\ Cat (F(A)) & \rightarrow & F(Cat (F(A))) \end{array} $$ where the top arrow is $i_{F(A)}$ which is an equivalence by (b) (but it is different from $F(i_A)$!), the left vertical arrow is the equivalence $i_{F(A)}$ (by (a)) and the bottom arrow is the equivalence $i_{Cat(F(A))}$ again an equivalence by (b). We have the following diagram: $$ \begin{array}{ccccc} Cat(A) & \rightarrow & Cat(Cat(A)) &&\\ \downarrow & & \downarrow &&\\ F' (A) & \rightarrow & F'(Cat(A)) & \rightarrow & F'(F'(A)) \end{array} . $$ The morphism on the top is $Cat(j_A)$, and the vertical morphisms are $k_A$ and $k_{Cat(A)}$ respectively. The morphisms on the bottom are $F'(j_A)$ and $F'(k_A)$ respectively. The diagram comes from naturality of $k$. The top arrow $Cat(j_A)$ and the middle vertical arrow $k_{Cat(A)}$ are equivalences, as is the composition along the bottom $F'(i'_A)$. On the other hand, all of the arrows are identities on the sets of objects. Thus, when morphisms are equivalences they are in fact iso-equivalences, and in particular equivalences on the level of the $n-1$-categories $(\cdot )_{p/}$. The closed model structure for $n-1$ implies that a morphism of $n-1$-categories is a weak equivalence (hence an equivalence) if and only if it projects to an isomorphism in the localized category. Look at the images of the above diagram in the localized category of $n-1$-precats after applying the operation $(\cdot )_{p/}$. The equivalences that we know show that the bottom left arrow goes to an arrow which has a left and right inverse. It follows that it goes to an invertible arrow, i.e. an isomorphism, in the localized category. Its left inverse, the left vertical arrow, must also go to an isomorphism. This implies that for each $p$, the map $Cat(A)_{p/} \rightarrow F'(A)_{p/}$ is an equivalence. By definition then $A\rightarrow F(A)$ is a weak equivalence. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \numero{Compatibility with products} The goal of the present section is to prove the following theorem. \begin{theorem} \label{ce} Suppose $A$ and $B$ are $n$-precats. Then the morphism $$ A\times B \rightarrow Cat(A)\times Cat(B) $$ is a weak equivalence. \end{theorem} Before getting to the proof, we give some corollaries. \begin{corollary} \label{ProdInterval} Suppose $B$ is an $n$-precat. Let $\overline{I}$ be the $1$-category with two isomorphic objects denoted $0$ and $1$, considered as an $n$-precat. Then the morphisms $$ Cat(B)\stackrel{i_0, i_1}{\rightarrow } Cat (B\times \overline{I}) \stackrel{p}{\rightarrow} Cat(B) $$ are equivalences of $n$-categories, where $i_0$ and $i_1$ come from the inclusions $0\rightarrow \overline{I}$ and $1\rightarrow \overline{I}$ and $p$ comes from the projection on the first factor. \end{corollary} {\em Proof:} Note that the morphism $\overline{I} \rightarrow Cat(\overline{I})$ is a weak equivalence by Lemma \ref{equiv}. Thus Theorem \ref{ce} says that $B\times \overline{I} \rightarrow Cat (B)\times \overline{I}$ is a weak equivalence. On the other hand, the morphism $Cat(B)\times \overline{I} \rightarrow Cat(B)$ is a weak equivalence, so $B\times \overline{I} \rightarrow Cat(B)$ is a weak equivalence. The morphism $B\rightarrow Cat(B)$ is of course a weak equivalence, so Lemma \ref{remark} implies that the two morphisms $$ B\stackrel{i_0, i_1}{\rightarrow } B\times \overline{I} \stackrel{p}{\rightarrow} B $$ are weak equivalences, which is the same statement as the corollary. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \begin{corollary} \label{forInternalHom} Suppose $A\rightarrow A'$ is a weak equivalence. Then for any $B$, $A\times B\rightarrow A'\times B$ is a weak equivalence. \end{corollary} {\em Proof:} By Theorem \ref{ce} we have that $$ A\times B \rightarrow Cat(A)\times Cat(B), \;\;\; A'\times B \rightarrow Cat(A')\times Cat(B) $$ are weak equivalences. By hypothesis, the map $Cat(A)\rightarrow Cat(A')$ is an equivalence of $n$-categories. It follows (from any of several characterizations of equivalences of $n$-categories, see for example \cite{Tamsamani} Proposition 1.3.1) that $Cat(A)\times Cat(B)\rightarrow Cat(A')\times Cat(B)$ is an equivalence, which gives the corollary. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \subnumero{Proof of Theorem \ref{ce}} We start by making some preliminary reductions. First we claim that it suffices to prove that if $B$ is any $n$-precat and $\Sigma = \Sigma (M,[m], \langle k , k+1 \rangle )$ and $h= h(M,m, 1^{k+1})$ then the morphism $\Sigma \times B \rightarrow h\times B$ is a weak equivalence. Suppose that we know this statement. Then, noting that the proof of \ref{pushout} below doesn't use Theorem \ref{ce} in degree $n$ in the case of a pushout by a trivial cofibration which is an isomorphism on objects (which is the case for $\Sigma \times B \rightarrow h\times B$) we obtain that for any $A$ and any morphism $\Sigma \rightarrow A$ the morphism $$ A\times B \rightarrow A\times B \cup ^{\Sigma \times B} h\times B $$ is a weak equivalence. The morphism $A\times B \rightarrow Cat(A)\times Cat(B)$ is obtained by iterating operations of this form (either on the variable $A$ or on the variable $B$ which works the same way). Therefore it would follow from the hypothesis of our claim that the morphism of \ref{ce} is a weak equivalence. We are now reduced to proving that $\Sigma \times B \rightarrow h\times B$ is a weak equivalence. In the previous notations if $M$ has length strictly greater than $0$ then the $\Sigma _{p/} \rightarrow h_{p/}$ are weak equivalences. Thus $$ (\Sigma \times B)_{p/} = \Sigma _{p/} \times B_{p/} \rightarrow h_{p/} \times B_{p/} = (h\times B)_{p/} $$ is a weak equivalence, and by Lemma \ref{partialCat1} it follows that $\Sigma \times B \rightarrow h\times B$ is a weak equivalence. Thus we are reduced to treating the case where $M$ has length zero, that is $$ \Sigma = \Sigma ([m], \langle k , k+1 \rangle )\;\;\; \mbox{and} \;\;\; h= h(m, 1^{k+1}). $$ Let $\Sigma ^{nu}=\Sigma ^{nu}([m], 1^{k+1}) $ denote the pushout of $m$ copies of $h(1, 1^{k+1})$ over the standard $h(0)$. We claim that it suffices to prove that $$ \Sigma ^{nu} \times B \rightarrow h\times B $$ is a weak equivalence. This claim is proved by induction on $k$. Note that $\Sigma$ is obtained from $\Sigma ^{nu}$ by a sequence of standard cofibrations over $\Sigma ' \rightarrow h'$ which are for smaller values of $k$. Assuming that we have treated all of the cases $\Sigma ^{nu}\times B\rightarrow h\times B$ and assuming our present claim for smaller values of $k$, we obtain that $\Sigma ^{nu}\times B \rightarrow \Sigma \times B$ is a trivial cofibration. It follows from Lemma \ref{remark} that $\Sigma \times B \rightarrow h\times B$ will be a trivial cofibration. We are now reduced to proving that the morphism $$ \Sigma ^{nu}([m], 1^k) \rightarrow h(m, 1^k) $$ induces a weak equivalence $\Sigma ^{nu}\times B \rightarrow h\times B$ (we have changed the indexing $k$ here from the previous paragraphs). Our next reduction is based on the following observation. Suppose we know this statement for $n$-precats $A$, $B$ and $C$ with cofibrations $A\rightarrow B$ and $A\rightarrow C$. Let $P:= B\cup ^AC$. The morphisms $\Sigma ^{nu}\times A \rightarrow h\times A$ (resp. $\Sigma ^{nu}\times B \rightarrow h\times B$, $\Sigma ^{nu}\times C \rightarrow h\times C$) are weak equivalences inducing isomorphisms on objects. As remarked above, the proof of \ref{pushout} below doesn't use Theorem \ref{ce} in degree $n$ when concerning pushouts by weak equivalences which are isomorphisms on objects. On the other hand, the morphism $\Sigma ^{nu}\times P\rightarrow h\times P$ may be obtained as a successive coproduct by these previous morphisms; thus it is a weak equivalence. We may apply this observation to infinite iterations of cofibrant pushouts. But note that any $n$-precat $B$ may be expressed as an iterated pushout of representable objects $h(M)$ over the boundaries $\partial h(M) \rightarrow h(M)$. The boundaries are in turn iterated pushouts over representable objects. >From the remark of the previous paragraph, it follows that we are reduced to proving that $\Sigma ^{nu}\times B\rightarrow h\times B$ is a weak equivalence when $B$ is a representable object. We will write $B=h(u , M)$, distinguishing the first variable from the rest. We next define the following operation. Suppose $C$ is an $n-1$-precat and suppose $D$ is a $1$-precat; then we define $D\oplus C$ to be the $n$-precat with $(D\oplus C)_0:= D_0$ and for any $p$, $(D\oplus C)_{p/}$ is the union for $f\in D_p$ of $C$ when $f$ is not totally degenerate and $\ast$ when $f$ is totally degenerate (we say that $f\in D_p$ is {\em totally degenerate} if it is in the image of the morphism $D_0 \rightarrow D_p$). The morphisms of functoriality for $p\rightarrow q$ are obtained by projecting $C$ to the final $n-1$-precat $\ast$ for elements $f\in D_q$ going to totally degenerate elements in $D_p$. This notation is useful because $h(u , M)= h(u ) \oplus h(M)$. Thus $h(m, 1^k) = h(m)\oplus h(1^k)$; and finally $$ \Sigma ^{nu}([m], 1^k)= \Sigma ^{nu} ([m]) \oplus h(1^k). $$ The operation $\oplus$ is compatible with pushouts: if $B\leftarrow A \rightarrow C$ is a diagram of $n-1$-precats then for any $1$-precat $X$, $$ X\oplus (B\cup ^AC) = (X\oplus B) \cup ^{X\oplus A} (X\oplus C). $$ On the other hand, if $C\rightarrow C'$ is a weak equivalence then $X\oplus C \rightarrow X\oplus C'$ is a weak equivalence (by applying \ref{partialCat1}). The object $h(M)$ is weak equivalent to a pushout of objects of the form $h(1^k)$, with the pushouts being over boundaries which are themselves weak equivalent to pushouts of objects of the same form. Since, as we have seen above, changing things by pushouts or by weak equivalences (which are isomorphisms on objects) in the second variable, preserves the statement in question. Thus it suffices to treat, in the previous notations, the case $M=1^j$. We have now boiled down to the basic case which needs to be treated: we must show that $$ \Sigma ^{nu}([m], 1^k) \times h(u, 1^j) \rightarrow h(m, 1^k)\times h(u,1^j) $$ is a weak equivalence. To do this, use the standard subdivision of the product of simplices $h(m)\times h(u)$ into a coproduct of simplices identified over their boundaries. \footnote{ This is basically the only place in the paper where we really use the fact that we have taken the category $\Delta$ and not some other category such as the semisimplicial category or a truncation of $\Delta$ using only objects $m$ for $m\leq m_0$. One can see for example that the statement \ref{ce} for $1$-precats is no longer true if we try to replace $\Delta$ by the semisimplicial category throwing out the degeneracy maps---this example comes down to saying that the product of two free monoids on two sets of generators is not the free monoid on the product of the sets of generators.} In this last part of the proof there was an error in version 1: on p. 31 the line claiming that ``$B^{(a,b)} = B^{<(a,b)} \cup ^{B^{\hat{x}}} B^x$'' was not true. Furthermore the notation of this part of the proof was relatively difficult to follow. Thus we rewrite things in the present version 2. This error and its correction were found during discussions with R. Pellissier, so I would like to thank him. The basic idea remains the same as what was said in version 1. The objects of $h(m, 1^k)\times h(u, 1^j)$ may be denoted by $(i,j)$ with $i=0,\ldots , m$ and $j=0,\ldots , u$. These should be arranged into a rectangle. The problem is to understand the composition as we go from $(0,0)$ to $(m,u)$. There are many different paths (i.e. sequences of points which are adjacent on the grid and where $i$ and $j$ are nondecreasing) and the goal is to say that the composition along the various different paths is the same. The reason is that when one changes the path by the smallest amount possible at a single square, that is to say changing ``up then over'' to ``over then up'', the composition doesn't change. This elementary step was done correctly in the original proof (see in v1 the statements that the morphisms $A^{\hat{x}}\rightarrow B^{\hat{x}}$ and $A^x\rightarrow B^x$ are weak equivalences). Then one has to put these elementary steps together to conclude the desired statement for the big rectangular grid. This requires an inductive argument along the lines of what was done in v1 but in v1 the induction wasn't organized correctly and was based on a mistaken claim as pointed out above. So let's rewrite things and hope for the best! Put $$ A:= \Sigma ^{nu}([m], 1^k) \times h(u, 1^j) $$ and $$ B:= h(m, 1^k)\times h(u,1^j). $$ Note that $A$ is a coproduct of things of the form $h(1,1^k) \times h(u, 1^j)$. We want to show that the morphism $A\rightarrow B$ is a weak equivalence. The precats $A$ and $B$ share the same set of objects which we denote by $Ob$, equal to the set of pairs $(i,j)$ with $0\leq i \leq m$ and $0 \leq j\leq u$. Suppose $S \subset Ob$ is a subset of objects. We denote by $A\{ S\} $ (resp. $B\{ S\}$) the full sub-precat of $A$ (resp. $B$) whose object-set is $S$. By ``full sub-precat'' we mean that for any sequence $x_0, \ldots , x_k$ in $S$, $$ A\{ S\} _{k/}(x_0,\ldots , x_k):= A _{k/}(x_0,\ldots , x_k) $$ and the same for $B\{ S\}$. We will use this for subsets $S$ of the form ``notched sub-rectangle plus a tail''. By a ``sub-recatangle'' we mean a subset of the form $$ S' = \{ (i,j) : \;\;\; 0 \leq i \leq m',\;\;\ 0\leq j \leq u' \} $$ and by a ``tail'' we mean a subset of the form $$ S'' = \{ (i_k, j_k) : \;\;\; 0 \leq k \leq r \} $$ where $i_k \leq i_{k+1} \leq i_k + 1$ and $j_k \leq j_{k+1} \leq j_k + 1$. A tail $S''$ that goes with a rectangle $S'$ as above, is assumed to have $i_0= m'$, $j_0 = u'$, $i_r = m$, $j_r=u$. In other words, the tail is a path going from the upper corner of $S'$ to the upper corner of $Ob$, and the path goes by steps of at most one in both the $i$ and the $j$ directions. Finally, a ``notched sub-rectangle'' is a subset of the form $$ S' = \{ (i,j) : \;\;\; (0 \leq i \leq m' \, \mbox{and} \, 0 \leq j \leq u'-1 ) \, \mbox{or} \, (v'\leq i \leq m' \, \mbox{and} j=u')\} . $$ We call $(m',u', v')$ the {\em parameters} of $S'$, and if necessary we denote $S'$ by $S'(m',u',v')$. Note that $u'\geq 1$ here, and $0\leq v' \leq m'$. A rectangle with $u'=0$ may be considered as part of a tail; thus, modulo the initial case where all of $S$ is a tail, which will be treated below, it is safe to assume $u' \geq 1$. If $v'=0$ then $S'$ is just the rectangle of size $m' \times u'$. If $v'=m'$ then $S'$ is a rectangle of size $m'\times (u'-1)$, plus the first segment of a tail going from $(m', u'-1)$ to $(m',u')$. Thus if $S''$ is a tail from $(m',u')$ to $(m,u)$, we get that $$ S'(m',u',0) \cup S'' $$ is a rectangle of size $m'\times u'$ plus a tail, whereas $$ S'(m',u',m') \cup S'' $$ is a rectangle of size $m'\times (u'-1)$ plus a tail. We prove by induction on $(m',u')$ that if $R$ is of the form $R=S'\cup S''$ for $S'$ a rectangle of size $m'\times u'$ and $S''$ a tail from $(m',u')$ to $(m,u)$, then $A\{ R\} \rightarrow B\{ R\}$ is a weak equivalence. We treat the case of $m',u'$ and suppose that it is known for all strictly smaller rectangles (i.e. for $(m'', u'')\neq (m',u')$ with $m'' \leq m'$ and $u''\leq u'$) and all tails. In the current part of the induction we assume that $u' \geq 1$. The case $u'=0$ (which is really the case of a tail $S''$ going all the way from $(0,0)$ to $(m,u)$) will be treated below. In particular, we know that $$ A \{ S'(m',u',m') \cup S'' \} \rightarrow B \{ S'(m',u',m') \cup S'' \} $$ is a weak equivalence (cf the above description of $S'(m',u',m')$). Now we treat the case where $S$ is a notched rectangle plus tail of the form $$ S = S'(m',u',v') \cup S'' $$ for $0\leq v' \leq m'$. For an $S$ of this notched form, we claim again that $A\{ S\} \rightarrow B\{ S\}$ is a weak equivalence. We prove this by descending induction on $v'$, the initial case $v'=m'$ being obtained above from the case of rectangles of size $m'\times (u'-1)$. Thus we may fix $v'< m'$ and assume that it is known for $$ \overline{S}= S'(m',u',v'+1) \cup S'', $$ in other words we may assume that $A\{ \overline{S}\} \rightarrow B\{ \overline{S}\}$ is a weak equivalence. We analyze how to go from $\overline{S}$ to $S$. Note that $S$ has exactly one object more than $\overline{S}$, the object $$ x := (v',u'). $$ Let $S^x$ denote the subset of objects $(i,j)\in S$ such that either $(i,j)\leq x$ or $(i,j) \geq x$. Here we define the order relation by $$ (i,j) \leq (k,l) \Leftrightarrow i\leq k \;\; \mbox{and}\;\; j \leq l. $$ With respect to this order relation, note that for a sequence of objects $(x_0, \ldots , x_p)$, we have that $B_{p/} (x_0,\ldots , x_p)$ is nonempty, only if $x_0\leq x_1 \leq \ldots \leq x_p$. (The same remark holds {\em a fortiori} for $A$ because of the map $A\rightarrow B$.) In particular, if $(x_0,\ldots , x_p)$ is a sequence of objects of $S$ such that $B_{p/} (x_0,\ldots , x_p)$ is nonempty and such that some $x_a=x$ then all of the $x_b$ are in $S^x$. Let $\overline{S}^x = \overline{S} \cup S^x$. We claim $(\ast )$ that $$ A\{ S\} = A\{ \overline{S}\} \cup ^{A\{ \overline{S}^x\} } A\{ S^x \} , $$ and similarly that $$ B\{ S\} = B\{ \overline{S}\} \cup ^{B\{ \overline{S}^x\} } B\{ S^x \} . $$ These are the statements that replace the faulty lines of the proof in version 1. To prove the claim, suppose $(x_0, \ldots , x_p)$ is a sequence of objects of $S$. It suffices to show that $$ B\{ S\}_{p/}(x_0,\ldots ,x_p) $$ is the pushout of $$ B\{ \overline{S}\}_{p/}(x_0,\ldots ,x_p) $$ and $$ B\{ S^x\}_{p/}(x_0,\ldots ,x_p) $$ over $$ B\{ \overline{S}^x\}_{p/}(x_0,\ldots ,x_p) $$ (the proof is the same for $A$, we give it for $B$ here). If none of the objects $x_a$ is equal to $x$, then either the sequence stays inside $\overline{S}^x$, in which case: $$ B\{ S\}_{p/}(x_0,\ldots ,x_p) = B\{ \overline{S}\}_{p/}(x_0,\ldots ,x_p)= B\{ \overline{S}^x\}_{p/}(x_0,\ldots ,x_p)= B\{ S^x\}_{p/}(x_0,\ldots ,x_p); $$ or else the sequence doesn't stay inside $\overline{S}^x$, in which case $$ B\{ S\}_{p/}(x_0,\ldots ,x_p) = B\{ \overline{S}\}_{p/}(x_0,\ldots ,x_p) $$ but $$ B\{ \overline{S}^x\}_{p/}(x_0,\ldots ,x_p)= B\{ S^x\}_{p/}(x_0,\ldots ,x_p)=\emptyset . $$ In both of these cases one obtains the required pushout formula. On the other hand, if some $x_a$ is equal to $x$, then either the sequence doesn't stay inside $S^x$ in which case all terms are empty (cf the above remark), or else it stays inside $S^x$ in which case $$ B\{ S\}_{p/}(x_0,\ldots ,x_p) = B\{ S^x\}_{p/}(x_0,\ldots ,x_p) $$ but $$ B\{ \overline{S}\}_{p/}(x_0,\ldots ,x_p)= B\{ \overline{S}^x\}_{p/}(x_0,\ldots ,x_p)= \emptyset . $$ Again one obtain the required pushout formula. This proves the claim $(\ast )$. We now note that both $S^x$ and $\overline{S}^x$ are of the form, a rectangle of size $v'\times (u'-1)$, plus a tail. The first step of the tail for $S^x$ goes from $(v',u'-1)$ to $x=(v',u')$. The next step goes to $(v'+1,u')$. The first step of the tail for $\overline{S}^x$ goes from $(v',u'-1)$ directly to $(v'+1, u')$. Both tails continue horizontally from $(v'+1,u')$ to $(m',u')$ and then continue as the tail $S''$ from there to $(m,u)$. By our induction hypothesis, we know that $$ A\{ S^x\} \rightarrow B\{ S^x\} $$ and $$ A\{ \overline{S}^x\} \rightarrow B\{ \overline{S}^x\} $$ are weak equivalences. Recall from above that we also know that $A\{ \overline{S}\} \rightarrow B\{ \overline{S}\}$ is a weak equivalence. These morphisms from full sub-precats of $A$ to full sub-precats of $B$ are all isomorphisms on sets of objects, so by the remark at the start of the proof of Theorem \ref{ce}, we can use Lemma \ref{pushout} for pushouts along these morphisms. A standard argument shows that the morphism $$ A\{ \overline{S}\} \cup ^{A\{ \overline{S}^x\} }A\{ S^x\} \rightarrow B\{ \overline{S}\} \cup ^{B\{ \overline{S}^x\} }B\{ S^x\} $$ is a weak equivalence. For clarity we now give this standard argument. First, $$ A\{ \overline{S}\} \rightarrow A\{ \overline{S}\} \cup ^{A\{ \overline{S}^x\} }B\{ \overline{S}^x\} $$ is a trivial cofibration (which again induces an isomorphism on sets of objects). One can verify that the morphism $$ A\{ \overline{S}\} \cup ^{A\{ \overline{S}^x\} }B\{ \overline{S}^x\} \rightarrow B\{ \overline{S}\} $$ is a cofibration. It is a weak equivalence by Lemma 3.8. Thus it is a trivial cofibration inducing an isomorphism on sets of objects. Similarly, the morphism $$ B\{ \overline{S}^x\} \cup ^{A\{ \overline{S}^x\} }A\{ S^x\} \rightarrow B\{ S^x\} $$ is a trivial cofibration inducing an isomorphism on sets of objects. Thus the pushout morphism (pushing out by these two morphisms at once) $$ (A\{ \overline{S}\} \cup ^{A\{ \overline{S}^x\} }B\{ \overline{S}^x\} ) \cup ^{B\{ \overline{S}^x\} } (B\{ \overline{S}^x\} \cup ^{A\{ \overline{S}^x\} }A\{ S^x\} ) $$ $$ \rightarrow B\{ \overline{S}\} \cup ^{B\{ \overline{S}^x\} }B\{ S^x\} $$ is a weak equivalence. Finally, note that $$ (A\{ \overline{S}\} \cup ^{A\{ \overline{S}^x\} }B\{ \overline{S}^x\} ) \cup ^{B\{ \overline{S}^x\} } (B\{ \overline{S}^x\} \cup ^{A\{ \overline{S}^x\} }A\{ S^x\} ) $$ $$ = (A\{ \overline{S}\} \cup ^{A\{ \overline{S}^x\} }A\{ S^x\} ) \cup ^{A\{ \overline{S}^x\} } B \{ \overline{S}^x\} $$ so the morphism from $A\{ \overline{S}\} \cup ^{A\{ \overline{S}^x\} }A\{ S^x\} $ to this latter, is also a weak equivalence. Putting these all together we have shown that the morphism $$ A\{ S\} = A\{ \overline{S}\} \cup ^{A\{ \overline{S}^x\} }A\{ S^x\} $$ $$ \rightarrow B\{ \overline{S}\} \cup ^{B\{ \overline{S}^x\} }B\{ S^x\} = B\{ S\} $$ is a weak equivalence. This completes the proof of the inductive step for the descending induction on $v'$, so we obtain the result for $v'=0$, in which case $S$ is a rectangle of size $m'\times u'$ plus a tail; in turn, this gives the inductive step for the induction on $(m',u')$. After all of this induction we are left having to treat the initial case $u'= 0$, where all of $S$ is a tail going from $(0,0)$ to $(m,u)$. This part of the proof is exactly the same as in version 1: what we call the ``tail'' here corresponds to the sequence which was denoted $x$ in version 1. If $S$ is a tail, then $A\{ S\} $ and $B\{ S\}$ are $1$-free ordered $n$-precats, so by Lemma \ref{freeness} it suffices to check that for two adjacent objects $x,y$ in the sequence corresponding to $S$, the morphisms $$ A_{1/}(x,y)\rightarrow B_{1/}(x,y) $$ are weak equivalences. In fact these morphisms are isomorphisms. If $x=(i,j)$ and $y=(i,j+1)$ then $$ A_{1/}(x,y) = B_{1/}(x,y) = h(1^j); $$ if $x=(i,j)$ and $y=(i+1,j)$ then $$ A_{1/}(x,y) = B_{1/}(x,y) = h(1^k); $$ and if $x=(i,j)$ and $y=(i+1,j+1)$ then $$ A_{1/}(x,y) = B_{1/}(x,y) = h(1^k)\times h(1^j). $$ Thus the criterion of \ref{freeness} implies that $A\{ S\}\rightarrow B\{ S\}$ is a weak equivalence. This completes the initial case of the induction. Combining this initial step with the inductive step that was carried out above, we obtain the result in the case where $S=Ob$ is the whole rectangle of size $m\times u$; in this case $A\{ S\} = A$ and $B\{ S\} = B$, so we have completed the proof that $$ A=\Sigma ^{nu}([m], 1^k) \times h(u, 1^j) \rightarrow h(m, 1^k)\times h(u,1^j)=B $$ is a weak equivalence. This completes the proof of Theorem \ref{ce}. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} The above proof went basically along the same lines as the proof of version 1, but here we met {\em all} possible tails going from $(0,0)$ to $(m,u)$ along the way in our induction, whereas in version 1 only some of the tails were met. This should have been taken as an indication that the proof in version 1 was incorrect. I would like again to thank R. Pellissier for an ongoing careful reading which turned up this problem. His reading has also turned up numerous other problems in the exposition or organisation of the argument (the reader has no doubt noticed!); however, I have chosen in the present version 2 to make only a minimalist correction of the above problem. \numero{Proofs of the remaining lemmas and Theorem \ref{cmc}} We can assume \ref{equiv}, \ref{partialCat1} and the corollary \ref{partialCat} for degree $n$ also. {\em Proof of \ref{coherence}:} We have to prove that the morphism $f:Cat (A) \rightarrow Cat(Cat(A))$ is an equivalence. Note that this is not the same morphism as the standard inclusion, rather it is the morphism induced by $A\rightarrow Cat (A)$. In particular, \ref{coherence} is not just an immediate corollary of \ref{equiv}. To obtain the proof, note that $Cat(A)$ is marked, so we have a morphism $$ r:Cat (Cat(A))\rightarrow Cat (A) $$ of marked easy $n$-categories, inducing the identity on the standard map $i:Cat (A) \rightarrow Cat (Cat (A))$. The morphism $r$ is an equivalence because the standard map $i$ is an equivalence by \ref{equiv}. On the other hand, the morphism of $n$-precats $A\rightarrow Cat (A)$ induces the morphism $f$ of marked easy $n$-categories. We obtain a morphism $r\circ f$ of marked easy $n$-categories $Cat (A)\rightarrow Cat (A)$ extending the standard map $A\rightarrow Cat (A)$. By the universal property of $Cat (A)$, $r\circ f$ is the identity. Thus (applying our usual Lemma \ref{remark}) the morphism $f$ is an equivalence. {\em Proof of \ref{pushout}:} We first treat the following special case of \ref{pushout}. Say that a morphism of $n$-categories $A\rightarrow B$ is an {\em iso-equivalence} if it is an equivalence and an isomorphism on objects. This is equivalent to the condition that for all $m$, the morphism $A_{m/}\rightarrow B_{m/}$ is an equivalence of $n-1$-categories. \begin{lemma} \label{isopushout} Suppose $A\rightarrow B$ is an iso-equivalence of $n$-categories and $A\rightarrow C$ is a morphism of $n$-categories. Then the morphism $$ C \rightarrow Cat (B\cup ^AC) $$ is an equivalence (in fact, even an iso-equivalence). \end{lemma} {\em Proof:} By our inductive hypotheses, the morphisms $$ C_{m/} \rightarrow Cat (B_{m/} \cup ^{A_{m/}}C_{m/}) $$ are equivalences of $n-1$-categories. Setting $D= B\cup ^AC$ we have $$ D_{m/}=B_{m/} \cup ^{A_{m/}}C_{m/} $$ and note that $D_0=C_0$. Hence the morphism of $n$-precats $$ C\rightarrow Cat _{\geq 1}(D) $$ is an equivalence on the level of each $C_{m/}$. But the condition of being an $n$-category depends only on the equivalence type of the $C_{m/}$, in particular $Cat _{\geq 1}(D)$ is an $n$-category (in this special case only---this is not a general principle!). Note in passing that the morphism $$ C \rightarrow Cat _{\geq 1}(D) $$ is an equivalence of $n$-categories. By Lemma \ref{equiv} in degree $n$, the morphism $$ Cat _{\geq 1}(D)\rightarrow Cat (Cat _{\geq 1}(D)) $$ is an equivalence. On the other hand, by Lemma \ref{partialCat} in degree $n$ the morphism $$ Cat (D) \rightarrow Cat (Cat _{\geq 1}(D)) $$ is an equivalence. By Lemma \ref{remark} at the start of the proof of all of the lemmas, this shows that $C\rightarrow Cat(D)$ is an equivalence. This proves Lemma \ref{isopushout}. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} The next lemma is the main special case which has to be treated by hand. This proof uses Corollary \ref{ProdInterval}, which in turn is where we use the full simplicial structure of $\Delta$. It seems likely (from some considerations in topological examples) that the following lemma would not be true if we looked only at functors from $\Delta_{\leq k}$ to $n$-precats. \begin{lemma} \label{specialcase} Suppose $\overline{I}$ is the category with two objects and exactly one isomorphism between them. Let $0$ denote one of the objects. Then for any $n$-category $A$ and object $c\in A_0$, if $0\rightarrow A$ denotes the corresponding morphism, the push-out morphism $$ A \rightarrow Cat (\overline{I} \cup ^0 A) $$ is an equivalence. \end{lemma} {\em Proof:} Let $\overline{I}^2$ denote $\overline{I}\times \overline{I}$. There is a morphism $h:\overline{I}^2\rightarrow \overline{I}\times \overline{I}$ equal to the identity on $\overline{I}\times \{ 0\}$ and on $\{ 0\} \times \overline{I}$, and sending $\overline{I}\times \{ 1\}$ to $(0, 1)$. Let $B:= \overline{I} \cup ^0A$. Then $$ B\times \overline{I} = \overline{I}^2 \cup ^{\{ 0\} \times \overline{I}}C\times \overline{I}. $$ Using $h$ on the first part of this pushout we obtain a map $$ f:B\times \overline{I}\rightarrow B\times \overline{I} $$ such that $f|_{B\times \{ 0\}}$ is the identity and $f|_{B\times \{ 1\}}$ is the projection $B\rightarrow A$ obtained from the projection $\overline{I}\rightarrow \{ 0\}$. By Corollary \ref{ProdInterval}, the morphisms $$ i_0:Cat (B)\times \{ 0 \} \rightarrow Cat (B \times \overline{I}) $$ and $$ i_1:Cat (B)\times \{ 1 \} \rightarrow Cat (B \times \overline{I}) $$ are equivalences of $n$-categories. Next, the morphism $f$ induces a morphism $$ g:Cat (B\times \overline{I})\rightarrow Cat (B) $$ such that the composition with $i_0$ is the identity $Cat(B)\rightarrow Cat(B)$ and the composition with $i_1$ is the factorization $Cat(B)\rightarrow Cat(A)\rightarrow Cat(B)$. Looking at $g\circ i_0$ we conclude that $g$ is an equivalence of $n$-categories. Therefore (since $i_1$ is an equivalence) the composition $g\circ i_1$ is an equivalence of $n$-categories. Now we have morphisms $Cat(A)\rightarrow Cat(B)$ and $Cat(B)\rightarrow Cat(A)$ such that the composition in one direction is the identity, and the composition in the other direction is an equivalence of $n$-categories. This implies that the two morphisms are equivalences of $n$-categories by Lemma \ref{remark}. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} In preparation for the next corollary, we discuss a sort of ``versal semi-interval'' $\overline{J}$. Ideally we would like to have an $n$-precat which is weakly equivalent to $\ast$, containing two objects $0$ and $1$ such that for any $n$-category (easy, perhaps) $A$ with two equivalent objects $a$ and $b$, there exists a morphism from our ``interval'' to $A$ taking $0$ to $a$ and $1$ to $b$. I didn't find an easy way to make this construction. The problem is somewhat analogous to the problem of finding a canonical inverse for a homotopy equivalence, solved in a certain topological context in \cite{flexible} but which seems quite complicated to put into action here in view of the fact that our $n$-category $A$ might not be fibrant (we don't yet have the closed model structure!). Thus we will be happy with a cruder version. Let $\overline{J}$ be the universal easy $n$-category with two objects $0$ and $1$ and a ``marked inner equivalence'' $u :0\rightarrow 1$. The quasi-inverse of $u$ will be denoted by $v$. The marking means a structure of choice of morphism whenever necessary for the definition of inner equivalence, as well as a choice of diagram (i.e. a partial marking in the sense defined at the start of the paper) whenever necessary for things to make sense. In practice this means that we start with objects $0$ and $1$, add the morphisms $u$ and $v$, add the diagrams over $2\in \Delta$ mapping to $(u,v)$ and $(v,u)$, and (letting $w$ and $y$ denote the compositions resulting from these diagrams) add (inductively by the same construction for $n-1$-categories) equivalences between $w$ and $e$ (resp. $y$ and $e$) where $e$ denote the identities. Let $\overline{L}\subset \overline{J}$ be the full-sub-$n$-category whose object set is $\{ 0\}$. The morphism $\overline{L}\rightarrow \overline{J}$ is automaticallly an equivalence since it is an isomorphism on morphism $n-1$-categories and is essentially surjective since by construction $1\in \overline{J}$ is equivalent to $0$. By the universal property of $\overline{J}$ we obtain a morphism $\overline{J}\rightarrow \overline{L}$ sending $u$ to $e$ and $v$ to $w$, sending our $2$-diagrams to degenerate diagrams in $\overline{L}$ and sending our homotopies to the corresponding homotopies in $\overline{L}$. The composition $$ \overline{L}\rightarrow \overline{J}\rightarrow \overline{L} $$ is the identity. On the other hand we have an obvious map $\overline{J}\rightarrow \overline{I}$, so we obtain a map $$ \overline{J}\rightarrow \overline{L}\times \overline{I}. $$ This map is compatible with the inclusions of $\overline{L}$ and hence is an equivalence of $n$-categories. It is also an isomorphism on objects so it is an iso-equivalence. \begin{corollary} \label{specialJ} Let $\overline{L}\subset \overline{J}$ be as above. Then for any $n$-category $A$ and morphism $\overline{L}\rightarrow A$, the push-out morphism $$ A \rightarrow Cat (\overline{J} \cup ^{\overline{L}} A) $$ is an equivalence. \end{corollary} {\em Proof:} Let $B= (\overline{L}\times \overline{I}) \cup ^{\overline{L}} A$ and $C= \overline{J} \cup ^{\overline{L}} A$ The morphism $\overline{J}\rightarrow (\overline{L}\times \overline{I})$ is an iso-equivalence so it satisfies the hypothesis of \ref{partialCat1}. By \ref{pushout} for $n-1$-precats, the morphism $C\rightarrow B$ also satisfies the hypothesis of \ref{partialCat1}. Therefore the morphism $Cat(C)\rightarrow Cat(B)$ is an equivalence. It suffices to show that $A\rightarrow Cat(B)$ is an equivalence. For this, note that $$ \overline{L} \cup ^0\overline{I} \rightarrow \overline{L}\times \overline{I} $$ is a weak equivalence by Lemma \ref{specialcase}. Similarly $$ A \rightarrow A \cup ^0\overline{I} $$ is a weak equivalence, and $$ B = (A\cup ^0\overline{I}) \cup ^{(\overline{L} \cup ^0\overline{I})} \overline{L}\times \overline{I} $$ so composing these two gives that $A\rightarrow B$ is a weak equivalence. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \begin{corollary} Suppose $A\rightarrow B$ is a cofibrant equivalence of $n$-categories such that the objects of $A$ form a subset of the objects of $B$ whose complement has one object. Then the push-out morphism $$ C \rightarrow Cat (B \cup ^A C) $$ is an equivalence. \end{corollary} {\em Proof:} We may replace $B$ by $Cat(B)$ since the morphism $B\rightarrow Cat(B)$ is a sequence of standard pushouts, so the corresponding morphism on pushouts of $C$ is also a sequence of standard pushouts so the conclusion for $Cat(B)$ implies the conclusion for $B$ (by Lemma \ref{remark}). Thus we may assume that $B$ is an easy $n$-category. Let $A'$ be the full sub-$n$-category of $B$ consisting of the objects of $A$. The pushout of $C$ from $A$ to $A'$ is a weak equivalence by Lemma \ref{isopushout}. Thus we may assume that $A=A'$. Let $b$ denote the single new object of $B$. It is equivalent to an object $a\in A$. By the universal property of $\overline{J}$ there is a morphism $\overline{J}\rightarrow B$ sending $0$ to $a$ and $1$ to $b$. Since $A$ is now a full sub-$n$-category of $B$, this morphism sends $\overline{L}$ to $A$. Let $E$ denote the push-out $$ E := Cat (A \cup ^{\overline{L}} \overline{J}). $$ By the prevoius corollary, $A\rightarrow E$ is an equivalence. Our morphism $\overline{J} \rightarrow B$ gives a morphism $$ E \rightarrow B $$ (use the marking of $B$ to go from $Cat(B)$ back to $B$) and this is an equivalence since $$ A\rightarrow B \rightarrow Cat(B) \;\;\; \mbox{and}\;\;\; A \rightarrow E $$ are equivalences. But the morphism $E\rightarrow B$ induces an isomorphism on objects. Now we have $$ C \cup ^A (A \cup ^{\overline{L}} \overline{J})= C \cup ^{\overline{L}}\overline{J} $$ so $$ C \rightarrow Cat (C \cup ^A (A \cup ^{\overline{L}} \overline{J})) $$ is an equivalence by the previous corollary. It is obvious from the construction of $Cat$ (resp. $BigCat$) via pushouts, together with the reordering of these pushouts, that $$ BigCat (C \cup ^A (A \cup ^{\overline{L}} \overline{J})) = BigCat (C \cup ^ACat (A \cup ^{\overline{L}} \overline{J})) =BigCat(C\cup ^AE). $$ Thus (since taking $BigCat$ is equivalent to taking $Cat$ by Lemma \ref{equiv} ---which we now know---and the reordering principle) $$ C\rightarrow Cat(C \cup ^AE) $$ is an equivalence. Now $$ Cat (C\cup ^AE) \rightarrow Cat (C \cup ^A B) $$ is an equivalence because $E_{p/}\rightarrow B_{p/}$ is an equivalence so by \ref{pushout} in degree $n-1$, $$ (C\cup ^AE )_{p/} \rightarrow (C \cup ^A B)_{p/} $$ is an equivalence and by Lemma \ref{partialCat1} we get the desired statement. Combining, we get that $$ C\rightarrow Cat (C \cup ^A B) $$ is an equivalence. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} {\em Proof of Lemma \ref{pushout}:} Suppose $A\rightarrow B$ is a cofibration of $n$-categories which is an equivalence. By applying the previous corollary inductively (adding one object at a time) we conclude that the push-out is an equivalence. Finally we treat the case where $A$, $B$ and $C$ are only $n$-precats rather than $n$-categories. If $\Sigma \rightarrow h$ is one of our standard pushout diagrams and if $\Sigma \rightarrow A$ is a morphism then $$ (B\cup ^{\Sigma} h) \cup ^{A\cup ^{\Sigma} h}(C\cup ^{\Sigma} h) = (B\cup ^AC)\cup ^{\Sigma}h. $$ This implies that $$ BigCat(B)\cup ^{BigCat(A)}BigCat(C) $$ is obtained by a collection of standard pushouts from $B\cup ^AC$, so in particular (by reordering) $$ BigCat(BigCat(B)\cup ^{BigCat(A)}BigCat(C))=BigCat(B\cup ^AC). $$ Now our hypothesis is that $BigCat(A)\rightarrow BigCat(B)$ is an equivalence (note also that it is a cofibration since $A\rightarrow B$ is a cofibration). By our proof of \ref{pushout} for the case of $n$-categories (and the equivalence between $Cat$ and $BigCat$ which we now know by \ref{equiv}) we conclude that $$ BigCat(C)\rightarrow BigCat(BigCat(B)\cup ^{BigCat(A)}BigCat(C)) $$ is an equivalence, which is to say that $$ BigCat(C)\rightarrow BigCat(B\cup ^AC) $$ is an equivalence. Thus $C\rightarrow B\cup ^AC$ is a weak equivalence. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} {\em Remark:} The semi-interval $\overline{J}$ we have constructed above is not contractible (i.e. equivalent to $\ast$). However for some purposes we would like to have such an object. We have the following fact (which is not used in the proof of Theorem \ref{cmc} in degree $n$---but which we put here for expository reasons): \begin{proposition} \label{intervalK} There is an $n$-category $K$ such that $K\rightarrow \ast$ is an equivalence, together with objects $0,1\in K$ such that if $A$ is an $n$-category and if $a,b$ are two equivalent objects of $A$ then there is a morphism $K\rightarrow A$ sending $0$ to $a$ and $1$ to $b$. \end{proposition} {\em Proof:} Since this proposition is not used in degree $n$ in the proof of Theorem \ref{cmc}, we can apply Theorem \ref{cmc}. We say that $K$ is {\em contractible} if the morphism $K\rightarrow \ast$ is an equivalence. In view of the versal property of $\overline{J}$, it suffices to construct a contractible $K$ with objects $0,1$ and a morphism $K\rightarrow \overline{J}$ sending $0$ to $0$ and $1$ to $1$. From the original discussion of $\overline{J}$ we have an equivalence $\overline{J}\rightarrow \overline{L}\times \overline{I}$. Using the closed model structure, factor the constant morphism as $$ \overline{I}\rightarrow M \rightarrow \overline{L}\times \overline{I} $$ into a composition of a trivial cofibration followed by a fibration. Note that $M$ is contractible. Set $$ K:=\overline{J}\times _{\overline{L}\times \overline{I}}M. $$ The morphism $\overline{J} \rightarrow \overline{L}\times \overline{I}$ is an isomorphism on objects, so for each $p$, $\overline{J}_{p/} \rightarrow (\overline{L}\times \overline{I})_{p/}$ is an equivalence of $n-1$-categories. Note also that $M _{p/}\rightarrow (\overline{L}\times \overline{I})_{p/}$ are fibrations. By the fact that weak equivalences are stable under fibrant pullbacks for $n-1$-categories (Theorem \ref{properness}), we have that $$ K_{p/}= \overline{J}_{p/}\times _{(\overline{L}\times \overline{I})_{p/}}M _{p/} \rightarrow M_{p/} $$ are weak equivalences, which in turn implies that $K\rightarrow M$ is a weak equivalence. In particular, $K$ is contractible. Since the morphism $$ M \rightarrow \overline{L}\times \overline{I} $$ is surjective on objects, there are objects $0,1\in K$ mapping to $0,1\in \overline{J}$. This completes the construction. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \begin{corollary} \label{FibImpliesCat} If $A$ is a fibrant $n$-precat then $A$ is automatically an easy $n$-category. \end{corollary} {\em Proof:} To show this it suffices to show that the morphisms $$ \varphi :\Sigma (M, [m], \langle k,k+1 \rangle ) \rightarrow h(M,[m], 1^{k+1}) $$ are trivial cofibrations. But $\varphi$ is a cofibration which is the first step in an addition of arbitrary pushouts of our standard morphisms $\varphi$, so by reordering of these pushouts the above inclusion extends to an isomorphism $$ BigCat (\Sigma (M, [m], \langle k,k+1 \rangle ) )\cong BigCat (h(M,[m], 1^{k+1})). $$ Since $Cat (B) \rightarrow BigCat(B)$ is an equivalence by Lemma \ref{equiv} applied to $Cat(B)$ plus reordering, this implies that the morphism $\varphi$ above is a trivial cofibration. By the definition of fibrant, $A$ must then satisfy the extension property to be an easy $n$-category. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \subnumero{The proof of Theorem \ref{cmc}} We follow the proof of Jardine-Joyal that simplicial presheaves form a closed model category, as described in \cite{Jardine}. The proof is based on the axioms CM1--CM5 of \cite{QuillenAnnals}. {\em Proof of CM1:} \,\, The category of $n$-precats is a category of presheaves so it is closed under finite (and even arbitrary) direct and inverse limits. {\em Proof of CM2:} \,\, Given composable morphisms $$ X\stackrel{f}{\rightarrow}Y\stackrel{g}{\rightarrow}Z, $$ if any two of $f$ or $g$ or $g\circ f$ are weak equivalences then the same two of $Cat(f)$, $Cat(g)$ or $Cat(g\circ f)$ are equivalences of $n$-categories in the sense of \cite{Tamsamani} and by Lemma \ref{remark} the third is also an equivalence; thus the third of our original morphisms is a weak equivalence. {\em Proof of CM3:} \,\, This axiom says that ``the classes of cofibrations, fibrations and weak equivalences are closed under retracts''. Jardine \cite{Jardine} doesn't actually discuss the retract condition other than to say that it is obvious in his case, and a look at Quillen yields only the conclusion that the diagram on p. 5.5 of \cite{Quillen} for the definition of retract is wrong (that diagram has no content related to the word ``retract'', it just says that one arrow is the composition of three others). Thus---since I am not sufficiently well acquainted with other possible references for this---we are reduced to speculation about what Quillen means by ``retract''. Luckily enough, this speculation comes out to be non-speculative in the end. We say that $f: A\rightarrow B$ is a {\em weak retract} of $g:X\rightarrow Y$ if there is a diagram $$ \begin{array}{ccccc} A&\stackrel{i}{\rightarrow}&X&\stackrel{r}{\rightarrow}&A\\ \downarrow &&\downarrow && \downarrow\\ B&\stackrel{j}{\rightarrow}&Y&\stackrel{s}{\rightarrow}&B \end{array} $$ such that $r\circ i=1_A$ and $s\circ j=1_B$. There is also another notion which we call {\em strong retract} obtained by using the same diagram but with the arrows going in the opposite direction on the bottom. It turns out that if $f$ is a strong retract of $g$ then $f$ is also a weak retract of $g$: the strong retract condition can be stated as the condition $j\circ f \circ r = g$ (along with the retract conditions $ri=1$ and $sj=1$). Applying $s$ on the left we obtain $fr=sg$ and applying $i$ on the right we obtain $jf=gi$, these two conditions giving the weak retract condition. Thus for our purposes, if we can show that the classes of maps in question are closed under weak retract, this implies that they are also closed under strong retract, and we don't actually care which of the two definitions was intended in \cite{Quillen}! We start out, then, with the condition that $f$ is a weak retract of $g$ using the notations of the diagram given above. If $g$ satisfies any lifting property then $f$ satisfies the same lifting property, using the retractions. This shows that if $g$ is a fibration then $f$ is a fibration. Furthermore, if $g$ is a cofibration then it is injective over any object $M=(m_1,\ldots , m_k)$ with $k<n$. It follows from the retractions that $f$ satisfies the same injectivity conditions (one has the same diagram of retractions on the values of all of the presheaves over the object $M$). Thus $f$ is a cofibration. Suppose $g$ is a weak equivalence, we would like to show that $f$ is a weak equivalence. Replacing the whole diagram by $Cat$ of the diagram, we may assume that $A$, $B$, $X$, and $Y$ are $n$-categories and $g$ is an equivalence. Suppose $x,y$ are objects of $A$. Then denoting by the same letters their images in $B$, $X$ and $Y$ we obtain morphisms of $n-1$-categories $f_1(x,y): A_{1/}(x,y)\rightarrow B_{1/}(x,y)$ and $g_1(x,y): X_{1/}(x,y)\rightarrow Y_{1/}(x,y)$ such that $g$ is a retract of $f$ in the category of arrows of $n-1$-precats. Furthermore $g_1(x,y)$ is an equivalence of $n-1$-categories. It follows by induction (since we may assume CM3 known for $n-1$-categories) that $f_1(x,y)$ is an equivalence of $n-1$-categories. In order to prove that $f$ is an equivalence we have to prove that it is essentially surjective. Suppose $w$ is an object of $B$. Then $i(w)$ is an object of $Y$ so by essential surjectivity of $g$ there is an object $u$ of $X$ with an equivalence $e:g(u)\cong i(w)$ (i.e. a pair of elements $e'\in Y_1(g(u),i(w))$ and $e'' \in Y_1(g(u),i(w))$ such that their compositions, which are well defined in the truncation $T^{n-1}Y_1( g(u),g(u))$ and $T^{n-1}Y_1( i(w),i(w))$ are the identities in these truncations). Applying the retractions $r$ and $s$ we obtain an element $r(u)\in A_0$ and an equivalence $s(e)$ between $fr(u)$ and $si(w)=w$. This proves essential surjectivity of $f$, completing the verification of CM3. {\em Proof of CM4:}\,\, The first part of CM4 is exactly Lemma \ref{pushout}. The second part follows from the first by the same trick as used by Jardine (\cite{Jardine} pp 64-65) and ascribed by him to Joyal \cite{Joyal}. {\em Proof of CM5(1):}\,\, For our situation, the cardinal $\alpha$ refered to in Jardine is the countable infinite one $\omega$. Suppose $A\rightarrow C$ is a trivial cofibration. We claim that if $B$ is an $\omega$-bounded subobject of $C$ (by this we mean a sub-presheaf over $\Theta^n$) then there is an $\omega$-bounded subobject $B_{\omega}\subset C$ as well as an $\omega$-bounded subobject $A_{\omega}\subset A\times _BB_{\omega}$ such that $B\subset B_{\omega} \subset C$ and such that $A_{\omega} \rightarrow B_{\omega}$ is a trivial cofibration. (Note that in our situation cofibrations are not necessarily injective morphisms of presheaves, so $A_{\omega}$ is not necessarily equal to $A\times _BB_{\omega}$ the latter of which could be uncountable). To prove the claim, note that for a given element in $B_M$ for some $M$, the statement that it is contained in an $n$-precat which is weakly equivalent to $A$ can in principal be written out explicitly involving only a countable number of elements of various $A_{M'}$ and $B_{M'}$. Iterate this operation starting with all of the elements of $B$ and repeatedly applying it to all of the new elements that are added. The iteration takes place a countable number of times, and each time we add on a countable union of countable objects. At the end we arrive at $A_{\omega} \subset B_{\omega}$ which is an $\omega$-bounded trivial cofibration. Using this claim, the rest of Jardine's arguments of (\cite{Jardine}, Lemmas 2.4 and 2.5) work and we obtain the statement that every morphism $f:X\rightarrow Y$ of $n$-precats can be factored as $f=p\circ i$ where $i$ is a trivial cofibration and $p$ is fibrant---\cite{Jardine} Lemma 2.5, which is CM5(1). Note that the only sentence in Jardine's argument which needs further verification is the fact that filtered colimits of trivial cofibrations are again trivial cofibrations; and this holds in our case too. {\em Proof of CM5(2):} We have to prove that any morphism $f$ may be factored as $f=q\circ j$ where $q$ is a fibrant weak equivalence and $j$ a cofibration. It suffices to construct a factorization $f=q\circ j$ with $j$ a cofibration and $q$ a weak equivalence, for then we can apply CM5(1) to factor $q$ as a product of a trivial cofibration and a fibration, the latter of which is automatically also a weak equivalence by CM2. Thus we now search for $f=q\circ j$ with $q$ a weak equivalence and $j$ a cofibration. The reader may wish to think about this in the case of $1$-categories to get an idea of what is happening and to see why this part is actually easy modulo some small details: we multiply the number of objects in each isomorphism class in the target category to have the morphism injective on the sets of objects. If $f:A\rightarrow B$ is a morphism of $n$-precats then we define a canonical factorization $A\rightarrow N(A,B)\rightarrow B$ in the following way. Let $L(A)$ denote the $1$-category (considered as an $n$-category) whose set of objects is equal to $A_0$ and which has exactly one morphism between any pair of objects. Note that $L(A)\rightarrow \ast$ is a weak equivalence. The tautological map $A_0 \rightarrow L(A)_0$ lifts to a unique map of $n$-precats $t:A\rightarrow L(A)$. Set $N(A,B):= L(A) \times B$ with the diagonal map $(t,f) :A\rightarrow N(A,B)$ and the second projection $p: N(A,B)\rightarrow B$. Note that $p$ is a weak equivalence (by an appropriate generalization of Corollary \ref{ProdInterval}) and $(t,f)$ is injective on objects. Now suppose by induction that we have constructed for every morphism $f':A'\rightarrow B'$ of $n-1$-precats a factorization $A'\rightarrow M(A',B')\rightarrow B'$ as a composition of a weak equivalence and a cofibration, functorial in $f'$. (To start the induction for $n=0$ we set $M(A',B'):= B'$ recalling that all morphisms are cofibrations in this case.) Suppose $f: A\rightarrow C$ is a morphism of $n$-precats such that $A_0 \hookrightarrow C_0$ is injective. Define a presheaf on $\Delta \times \Theta ^{n-1}$ denoted $P(A,C)$, with factorization $A\rightarrow P(A,C)\rightarrow C$ as follows. Put $$ P(A,C)_{p/} := M(A_{p/}, C_{p/}). $$ By functoriality this is a functor from $\Delta ^o$ to $n-1$-precats, and we have a factorization $$ A_{p/} \rightarrow P(A,C)_{p/} \rightarrow C_{p/}. $$ The second morphisms in the factorization are equivalences, and the first morphisms are cofibrations. The only problem is that $P(A,C)_{0/}$ is not a set: it is an $n$-category which is equivalent to the set $C_0$. For any $p$ there is a morphism $\psi _p:P(A,C)_{0/}\rightarrow P(A,C)_{p/}$ which, because it is a section of any one of the morphisms back to $0$, is a cofibration and in fact even injective in the top degree. If $p\rightarrow q$ is a morphism in $\Delta$ then $P(A,C)_{q/} \rightarrow P(A,C)_{p/}$ composed with $\psi _q$ is equal to $\psi _p$. Hence if we set $$ Q(A,C)_{p/}:= P(A,C)_{p/} \cup ^{P(A,C)_{0/}} C_0 $$ then $Q(A,C)_{p/}$ is functorial in $p\in \Delta$. Now $Q(A,C)_{0/}= C_0$ is a set rather than an $n-1$-precat so $Q(A,C)$ descends to a presheaf on $\Theta ^n$. We have a morphism $A\rightarrow Q(A,C)$ projected from the above morphism into $P(A,C)$. We also have a morphism $Q(A,C)\rightarrow C$ because the composed morphism $$ P(A,C)_{0/} \rightarrow P(A,C)_{p/} \rightarrow C_{p/} $$ factors through the unique morphism $C_0 \rightarrow C_{p/}$. The composition of these morphisms is the morphism $A\rightarrow C$. We next claim that the second morphism $Q(A,C)\rightarrow C$ is a weak equivalence. It suffices by Lemma \ref{partialCat1} to prove that $Q(A,C)_{p/}\rightarrow C_{p/}$ are weak equivalences, but the facts that $P(A,C)_{0/}\rightarrow C_0$ is a weak equivalence and $P(A,C)_{0/}\rightarrow P(A,C)_{p/}$ a cofibration imply (inductively using the closed model structure for $n-1$-precats) that $P(A,C)_{p/}\rightarrow Q(A,C)_{p/}$ is a weak equivalence. Now $P(A,C)_{p/} \rightarrow C_{p/}$ being a weak equivalence implies that $Q(A,C)_{p/}\rightarrow C_{p/}$ is a weak equivalence. This proves the claim. Finally we claim that $A\rightarrow Q(A,C)$ is a cofibration. It suffices to prove that the $A_{p/}\rightarrow Q(A,C)_{p/}$ are cofibrations. We know by the inductive hypothesis that $A_{p/} \rightarrow P(A,C)_{p/}$ are cofibrations. By the pushout definition of $Q(A,C)_{p/}$ and using the fact that $P(A,C)_{0/}$ is a sub-presheaf of $P(A,C)_{p/}$, it suffices to prove that the map $$ A_{p/} \times _{P(A,C)_{p/}} P(A,C)_{0/} \rightarrow C_0 $$ is cofibrant. In fact we show below that for any $M$ of length $<n-1$, $$ A_{p,M} \times _{P(A,C)_{p,M}} P(A,C)_{0,M} = A_0\subset A_{p,M} $$ which implies what we want, since we have assumed that $A_0\rightarrow C_0$ is injective. Note that the notation $P(A,C)_{0,M}$ means $(P(A,C)_{0/})_M$, and we don't have in this case that this is a constant $n-1$-category so the usual rule saying that $P(A,C)_{0,M}$ should be equal to $P(A,C)_{0,0}$ doesn't apply. Fix any one of the maps $e:p\rightarrow 0 \rightarrow p$. This gives a map $A_{p/} \rightarrow A_{p/}$ whose image is automatically $A_0$. This implies that the fixed subsheaf of the endomorphism $e$ is equal to $A_0$. The endomorphism acts compatibly on $P(A,C)_{p/}$ and the fixed point subsheaf there is $P(A,C)_{0/}$. For any $M$ of length $<n-1$ we have an inclusion $A_{p,M} \hookrightarrow P(A,C)_{p,M}$. This is compatible with the endomorphisms $e$ on both sides, so the intersection of $A_{p,M}$ with the fixed point set $P(A,C)_{0,M}\subset P(A,C)_{p,M}$ is the fixed point set $A_{0}\subset A_{p,M}$. This shows the statement of the previous paragraph. This completes the proof that $A\rightarrow Q(A,C)\rightarrow C$ is a factorization of the desired type, when $A_0 \rightarrow C_0$ is injective on objects. Note also that $Q(A,C)$ is functorial in the morphism $A\rightarrow C$. Suppose now that $A\rightarrow B$ is any morphism. We put $$ M(A, B):= Q(A, N(A,B)). $$ We have the factorization $A\rightarrow N(A,B) \rightarrow B$ with the first arrow injective on objects and the second arrow a weak equivalence, discussed at the start of the proof. The first arrow is then factored as $A\rightarrow M(A,B) \rightarrow N(A,B)$ with the first arrow a cofibration and the second arrow a weak equivalence. The factorization $A\rightarrow M(A,B)\rightarrow B$ therefore has the desired properties, and furthermore it is functorial in $A\rightarrow B$ (this is needed in order to continue with the induction on $n$). This completes the proof of CM5(2). \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} We refer to \cite{Quillen} for all of the consequences of Theorem \ref{cmc}. Recall also that a closed model category is said to be {\em proper} if it satisfies the following two axioms: \newline {\bf Pr}(1)\,\, If $A\rightarrow B$ is a weak equivalence and $A\rightarrow C$ a cofibration then $C\rightarrow B\cup ^AC$ is a weak equivalence; \newline {\bf Pr}(2)\,\, If $B\rightarrow A$ is a weak equivalence and $C\rightarrow A$ a fibration then $B\times _AC\rightarrow C$ is a weak equivalence. \begin{theorem} \label{properness} The closed model category $PC_n$ satisfies axiom {\bf Pr}(1); and it satisfies axiom {\bf Pr}(2) for equivalences $B\rightarrow A$ between $n$-categories; however it doesn't satisfy axiom {\bf Pr}(2) in general. \end{theorem} {\em Proof:} We will prove stability of weak equivalences under coproducts. Suppose $A\rightarrow B$ is a cofibration, and suppose $A\rightarrow C$ is a weak equivalence. We would like to show that $B\rightarrow B\cup ^AC$ is a weak equivalence. For this we use a version of the ``mapping cone''. Recall that $\overline{I}$ is the category with two isomorphic objects $0,1$ and no other morphisms. The morphism $B\times \{ 1\} \rightarrow B \times \overline{I}$ is a trivial cofibration, so $$ B\cup ^AC\rightarrow D:= (B\cup ^AC)\cup ^{B\times \{ 1\} } B\times \overline{I} $$ is a trivial cofibration. It follows that the projection $D\rightarrow B\cup ^AC$ deduced from $B\times \overline{I}\rightarrow B$ is a weak equivalence. Let $$ E:= (B\times \{ 0\} )\cup ^{A\times \{ 0\} }A \times \overline{I} $$ and note that $B\times \{ 0\} \rightarrow E$ is a weak equivalence (since it is pushout by the trivial cofibration $A\times \{ 0\} \rightarrow A\times \overline{I}$) hence $E\rightarrow B\times \overline{I}$ is a trivial cofibration. Thus the morphism $$ E \cup ^{A\times \{ 1\} } C \rightarrow B\times \overline{I} \cup ^{A\times \{ 1\} } C $$ is a trivial cofibration. But note that $B\times \overline{I} \cup ^{A\times \{ 1\} } C =D$ so $$ E \cup ^{A\times \{ 1\} } C \rightarrow D $$ is a weak equivalence. Finally, $$ E \cup ^{A\times \{ 1\} } C = B\times \{ 0\} \cup ^{A\times \{ 0\}} (A\times \overline{I} \cup ^{A\times \{ 1\} } C) $$ and the morphism $$ A\times \{ 0\}\rightarrow A\times \overline{I} \cup ^{A\times \{ 1\} } C $$ is a weak equivalence because it projects to $A\rightarrow C$ which is by hypothesis a weak equivalence. Therefore the map $$ B\times \{ 0\} \rightarrow E \cup ^{A\times \{ 1\} } C $$ is a weak equivalence, and from above $B\times \{ 0\} \rightarrow D$ is a weak equivalence. Following by the projection $D\rightarrow B\cup ^AC$ which we have seen to be a weak equivalence, gives the standard map $B\rightarrow B\cup ^AC$ which is therefore a weak equivalence. This proves the first half of properness. We now prove the second statement, proceeding as usual by induction on $n$. Factoring $B\rightarrow A$ into a cofibration followed by a fibration and treating the fibration, we can assume that the morphism is a cofibration (note that a fibration which is a weak equivalence, over an $n$-category, is again an $n$-category so the hypotheses are preserved). Let $A'\subset A$ be the full sub-$n$-category consisting of the objects which are in the image of $B_0$. Let $C':= A'\times _AC$. The morphism $B\rightarrow A'$ is an iso-equivalence so the $B_{p/}\rightarrow A'_{p/}$ are equivalences of $n-1$-categories. The morphisms $C'_{p/} \rightarrow A'_{p/}$ are fibrant, so by the inductive hypothesis $$ (C'\times _{A'} B)_{p/} = C'_{p/} \times _{A'_{p/}} B_{p/} \rightarrow C'_{p/} $$ are equivalences. This implies that $$ C\times _{A} B = C'\times _{A'} B \rightarrow C' $$ is an equivalence. Now $C'$ is a full sub-$n$-category of $C$ (meaning that for any objects $x_0,\ldots , x_m$ of $C'$ the morphism $C'_{m/}(x_0,\ldots , x_m)\rightarrow C_{m/}(x_0,\ldots , x_m)$ is an isomorphism), so to prove that $C'\rightarrow C$ is an equivalence it suffices to prove essential surjectivity. Suppose $x$ is an object of $C$. It projects to an object $y$ in $A$ which is equivalent to an object $y'$ coming from $A'$. By \ref{intervalK} there is a morphism $K\rightarrow A$ sending $0$ to $y$ and $1$ to $y'$. The object $x$ provides a lifting to $C$ over $\{ 0\}$, so by the condition that $C\rightarrow A$ is fibrant there is a lifting to $K\rightarrow C$ sending $0$ to $x$ and $1$ to an object lying over $y'$. In particular $1$ goes to an object of $C'$. This shows that $x$ is equivalent to an object of $C'$, the essential surjectivity we needed. We now sketch an example showing why axiom {\bf Pr}(2) can't be true in general. Let $A$ be the category $I^{(2)}$ with objects $0,1,2$ and one morphism $i\rightarrow j$ for $i\leq j$ ($i,j=0,1,2$). Let $B$ be the sub-$1$-precat obtained by removing the morphism $0\rightarrow 2$ (it is the pushout of two copies of $I$ over the object $1$). The morphism $B\rightarrow A$ is a weak equivalence. Let $C$ be a $1$-category with three objects $x_0, x_1, x_2$ and morphisms from $x_i$ to $x_j$ only when $i\leq j$. There is automatically a unique morphism $C\rightarrow A$ sending $x_i$ to $i$. One can see that this morphism is fibrant. We can choose $C$ so that the composition morphism $$ C_1(x_0,x_1)\times C_1(x_1,x_2)\cong C_2(x_0, x_1, x_2)\rightarrow C_1(x_0,x_2) $$ is not an isomorphism. Let $D:= Cat(B\times _AC)$. There is a unique morphism $D\rightarrow C$ extending the second projection morphism, and this morphism takes $D_1(x_0,x_1)$ (resp. $D_1(x_1,x_2)$) isomorphically to $C_1(x_0,x_1)$ (resp. $C_1(x_1,x_2)$). However, the composition morphism for $D$ is an isomorphism $$ D_1(x_0,x_1)\times D_1(x_1,x_2)\stackrel{cong}{\rightarrow} D_1(x_0,x_2). $$ Thus the morphism $D_1(x_0,x_2)\rightarrow C_1(x_0,x_2)$ is not an isomorphism; thus $D\rightarrow C$ is not an equivalence and $B\times _AC\rightarrow C$ is not a weak equivalence. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} As one last comment in this section we note the following potentially useful fact. \begin{lemma} \label{automatic} If $f:A\rightarrow A'$ is a fibrant morphism of $m$-precats then it is again fibrant when considered as a morphism of $n$-precats for $n\geq m$. \end{lemma} {\em Proof:} Suppose $m< n$. Define the {\em brutal truncation} denoted $\beta \tau _{\leq m}$ from $n$-precats to $m$-precats as follows. If $B$ is an $n$-precat then put $$ \beta\tau _{\leq n-1}(B)_M := B_M $$ for $M= (m_1, \ldots , m_k)$ with $k <m$ whereas for $M$ of length $m$ put $$ \beta\tau _{\leq n-1}(B)_M:=B_M /\langle B_{M, 1} \rangle $$ where $\langle B_{M, 1} \rangle$ denotes the equivalence relation on $B_M$ generated by the image of $B_{M,1}\rightarrow B_M\times B_M$. This should not be confused with the ``good'' truncation operation $T^{n-m}$ of \cite{Tamsamani}, as in general they will not be the same (however they are equal in the case of $n$-groupoids). We claim that brutal truncation is compatible with the operation $BigCat$, that is $$ BigCat (\beta \tau _{\leq m} B) = \beta \tau _{\leq m} (BigCat(B)). $$ To prove this claim, we note two things: \newline (1) \,\, that if $\Sigma \rightarrow h$ is one of our standard cofibrations for $n$-precats then $\beta \tau _{\leq m}\Sigma \rightarrow \beta \tau _{\leq m}h$ is a standard cofibration for $m$-precats; and \newline (2)\,\, that any map $\beta \tau _{\leq m}\Sigma \rightarrow \beta \tau _{\leq m}B$ comes from a map $\Sigma \rightarrow B$ or---in the top degree case---at least from a map $\Sigma \rightarrow BigCat(B)$. By reordering we find that the two sides of the above equation are the same, which gives the claim. Next we claim that brutal truncation preserves weak equivalences. From the previous claim it suffices to note that it preserves equivalences of $n$-categories, and this follows from the fact that brutal truncation of $1$-categories takes equivalences to isomorphisms of sets. Finally, it is immediate from the definitions that brutal truncation takes cofibrations of $n$-precats to cofibrations of $m$-precats (using of course the fact that there is no injectivity on the top degree morphisms for cofibrations of $m$-precats). Suppose $A$ is an $m$-precat, and let $Ind^n_m(A)$ denote $A$ considered as an $n$-precat (for this we simply set $Ind^n_m(A)_{M,M'}:= A_M$ for $M$ of degree $m$ and any $M'$ or for $M$ of degree $<m$ and empty $M'$). Then (speaking of absolute $Hom$ here rather than internal $Hom$ as in the next section) $$ Hom (\beta \tau _{\leq m} B, A) = Hom (B, Ind^n_m(A)), $$ in other words $\beta \tau _{\leq m}$ and $Ind^n_m$ are adjoint functors. We can now prove the lemma. If $f$ is a fibrant morphism of $m$-precats and $B\rightarrow C$ is a trivial cofibration of $n$-precats then $\beta\tau _{\leq m}B\rightarrow \beta\tau _{\leq m}C$ is a trivial cofibration of $m$-precats, so $f$ has the lifting property for this latter. By adjointness $Ind^n_m(f)$ has the lifting property for $B\rightarrow C$. Therefore $Ind ^n_m(f)$ is fibrant. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \numero{Internal $\underline{Hom}$ and $nCAT$} Recall the result of Corollary \ref{forInternalHom}: that direct product with any $n$-precat preserves weak equivalences. Direct product also preserves cofibrations, so it preserves trivial cofibrations. This property is not a standard property of any closed model category, it is one of the nice things about our present situation which allows us to obtain the right thing by looking at internal $Hom$ of $n$-precats. \begin{theorem} \label{hom} Suppose $A$ is an $n$-precat and $B$ is a fibrant $n$-precat. Then the internal $ \underline{Hom}(A,B)$ of presheaves over $\Theta ^n$ is a fibrant easy $n$-category. Furthermore if $B'\rightarrow B$ is a fibrant morphism then $\underline{Hom} (A, B')\rightarrow \underline{Hom} (A, B)$ is fibrant. Similarly if $A\rightarrow A'$ is a cofibration and if $B$ is fibrant then $\underline{Hom}(A', B)\rightarrow \underline{Hom}(A,B)$ is fibrant. \end{theorem} {\em Proof:} Note that it suffices to prove that $\underline{Hom} (A,B)$ is fibrant, for \ref{FibImpliesCat} then shows that it is an easy $n$-category. A morphism $S\rightarrow \underline{Hom} (A,B)$ is the same thing as a morphism $S\times A\rightarrow B$. Suppose $S\rightarrow T$ is a trivial cofibration. Then $S\times A \rightarrow T \times A$ is a trivial cofibration. It follows immediately from the definition of $B$ being fibrant that any map $S\times A \rightarrow B$ extends to a map $T \times A \rightarrow B$. Thus $\underline{Hom} (A, B)$ is fibrant. Similarly if $B'\rightarrow B$ is fibrant then any map $T\times A\rightarrow B$ with lifting $S\times A\rightarrow B'$ admits a compatible lifting $T\times A\rightarrow B$. Thus $\underline{Hom} (A,B')\rightarrow \underline{Hom}(A, B)$ is fibrant. Suppose $A\rightarrow A'$ is cofibrant, and $B$ fibrant. We show that $\underline{Hom} (A',B)\rightarrow \underline{Hom}(A, B)$ satisfies the lifting property to be fibrant. Say $S\rightarrow S'$ is a trivial cofibration, and suppose we have maps $S'\rightarrow \underline{Hom}(A, B)$ and lifting $S\rightarrow \underline{Hom}(A', B)$. These are by definition maps $S'\times A \rightarrow B$ and $S\times A'\rightarrow B$ which agree over $S\times A$. These give a morphism $$ f:S\times A' \cup ^{S\times A} S ' \times A \rightarrow B. $$ The morphism $$ g:S\times A' \cup ^{S\times A} S ' \times A \rightarrow S' \times A' $$ is a cofibration. Lemma \ref{pushout} applied to the trivial cofibration $S\times A\rightarrow S'\times A$ implies that the morphism $$ S\times A'\rightarrow S\times A' \cup ^{S\times A} S '\times A $$ is a weak equivalence. On the other hand the morphism $S\times A' \rightarrow S'\times A'$ is a weak equivalence by \ref{forInternalHom}, so by Lemma \ref{remark} the morphism $g$ is a weak equivalence. Thus the fact that $B$ is fibrant means that our morphism $f$ extends to a morphism $S'\times A' \rightarrow B$, and this gives exactly the desired lifting property for the last statement of the theorem. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \begin{lemma} \label{stability} Suppose $A\rightarrow A'$ is a weak equivalence, and $B$ fibrant. Then $\underline{Hom} (A', B)\rightarrow \underline{Hom} (A, B)$ is an equivalence of $n$-categories. If $B\rightarrow B'$ is an equivalence of fibrant $n$-precats then $\underline{Hom}(A,B)\rightarrow \underline{Hom}(A,B')$ is an equivalence. Suppose $A\rightarrow B$ and $A\rightarrow C$ are cofibrations. Then $$ \underline{Hom} (B\cup ^AC, D) = \underline{Hom} (B, D) \times _{\underline{Hom}(A,D)}\underline{Hom}(C,D). $$ \end{lemma} {\em Proof:} The last statement is of course immediate, because for any $S$ we have $(B\cup ^AC)\times S = (B\times S ) \cup ^{(A\times S)} (C\times S)$. We treat the other statements. Suppose first that $A\rightarrow A'$ is a trivial cofibration. Suppose that $S\rightarrow T$ is any cofibration. Suppose we have maps $T\rightarrow \underline{Hom}(A, B)$ lifting over $S$ to $S\rightarrow \underline{Hom}(A', B)$. We claim that the lifting extends to $T$; then the characterization of weak equivalences in (\cite{Quillen} \S 5, Definition 1, Property M6, part (c)) will imply that $\underline{Hom}(A', B)\rightarrow \underline{Hom}(A,B)$ is a weak equivalence. To prove the claim, note that our data correspond to a morphism $$ T\times A \cup ^{S\times A} S\times A' \rightarrow B. $$ The morphism $S\times A \rightarrow S\times A'$ is a trivial cofibration, so the morphism $$ T\times A \rightarrow T\times A \cup ^{S\times A} S\times A' $$ is a trivial cofibration, and since $T\times A \rightarrow T\times A'$ is a weak equivalence we get that the morphism $$ T\times A \cup ^{S\times A} S\times A' \rightarrow T\times A' $$ is a trivial cofibration. The fibrant property of $B$ implies that our map extends to a map $T\times A' \rightarrow B$, so we get the required lifting to $T\rightarrow \underline{Hom}(A', B)$. This implies that $\underline{Hom}(A',B)\rightarrow \underline{Hom}(A,B)$ is a fibrant weak equivalence. Next we treat the case of any weak equivalence $A\rightarrow A'$. Let $C$ be the $n$-precat pushout of $A\times 0\rightarrow A\times \overline{I}$ and $A\times 0 \rightarrow A' \times 0$. Since $0\rightarrow \overline{I}$ is a trivial cofibration, the various morphisms $$ A\hookrightarrow C, \;\; A' \hookrightarrow C, \;\; C \rightarrow A' $$ (the first sending $A$ to $A\times 1$, the second sending $A'$ to $A' \times 0$ and the third coming from the projection $A\times \overline{I} \rightarrow A'$) are all weak equivalences. We have a composable pair of morphisms $$ \underline{Hom} (A', B) \rightarrow \underline{Hom} (C, B) \rightarrow \underline{Hom} (A', B) $$ composing to the identity, and where the second arrow is an equivalence by the previous paragraph since $A'\rightarrow C$ is a trivial cofibration. Therefore the first arrow is an equivalence. Next, the morphism $\underline{Hom} (C, B)\rightarrow \underline{Hom} (A,B)$ obtained from the trivial cofibration $A\rightarrow C$ (going to $A\times 1$) is an equivalence, so the composed map $\underline{Hom} (A', B)\rightarrow \underline{Hom} (A,B)$ is an equivalence. This is the map induced by our original $A\rightarrow A'$. This completes the proof of the first part of the lemma. We now turn to the second part and treat first a fibrant weak equivalence $B\rightarrow B'$. Note first that such a morphism satisfies the lifting property for any cofibrations (this is the other half of CM4 which comes from Joyal's trick). We prove that $\underline{Hom}(A,B)\rightarrow \underline{Hom}(A,B')$ satisfies lifting for any cofibration (which as above implies that it is a fibrant weak equivalence). Suppose $S\rightarrow T$ is a cofibration. and $T\rightarrow \underline{Hom} (A,B')$ is a map with lifting over $S$ to a map $S\rightarrow \underline{Hom} (A,B)$. These correspond to maps $T\times A \rightarrow B'$ and lifting to $S\times A \rightarrow B$. The morphism $S\times A \rightarrow T\times A $ is a cofibration so by the lifting property of $B\rightarrow B'$ for any cofibration, there is a lifting to $T\times A\rightarrow B$ compatible with the given map on $S$. This establishes the necessary lifting property to conclude that $\underline{Hom}(A,B)\rightarrow \underline{Hom}(A,B')$ is a fibrant equivalence. Next suppose that $i:B\rightarrow B'$ is a trivial cofibration of fibrant $n$-precats. The lifting property for $B$ lets us choose a retraction $r: B'\rightarrow B$ such that $ri= 1_B$. Let $p:B' \cup ^BB' \rightarrow B'\rightarrow B'$ be the projection which induces the identity on both of the components $B'$. Note that $B'\rightarrow B' \cup _BB'$ is a trivial cofibration by \ref{pushout} so the projection $p$ is a weak equivalence (using \ref{remark}). Choose a factorization $$ B' \cup ^BB' \rightarrow P \rightarrow B' $$ with the first morphism cofibrant and the second morphism fibrant (whence $P$ fibrant itself); and both morphisms weak equivalences. Let $q: B' \cup ^BB'\rightarrow B'$ be the morphism inducing the retraction $r$ on the first copy of $B'$ and the identity on the second copy. Since $B'$ is fibrant this extends to a morphism we again denote $q: P\rightarrow B'$. The result of the previous paragraph implies that the morphism $$ p:\underline{Hom}(A, P)\rightarrow \underline{Hom}(A, B') $$ is an equivalence, which implies that either of the two morphisms $$ j_0, j_1:\underline{Hom}(A, B')\rightarrow \underline{Hom}(A, P) $$ (refering to the two inclusions $j_0,j_1: B' \rightarrow P$) are equivalences. Now we have that the composition $$ \underline{Hom}(A, B')\stackrel{j_1}{\rightarrow} \underline{Hom}(A, P) \stackrel{q}{\rightarrow} \underline{Hom}(A, B') $$ is the morphism induced by $qj_1= 1_{B'}$ thus it is the identity. The fact that the morphism induced by $j_1$ (the first of the above pair) is an equivalence implies that the morphism induced by $q$ (the second in the above sequence) is an equivalence. But since the morphism induced by $j_0$ is an equivalence, we get that the morphism induced by $qj_0=ir$ is an auto-equivalence of $\underline{Hom}(A,B')$. The morphism induced by $ri=1_B$ is of course the identity. The last part of Lemma \ref{remark} now implies that the morphism $$ i: \underline{Hom}(A,B)\rightarrow \underline{Hom}(A,B') $$ is an equivalence. This proves the statement in case of a trivial cofibration. Finally note that any equivalence of $n$-categories $B\rightarrow B'$ decomposes as a composition $B\rightarrow C \rightarrow B'$ where the first arrow is a trivial cofibration and the second a fibration and weak equivalence. Note that $C$ is fibrant since by hypothesis $B'$ is fibrant. Thus our two previous discussions apply to give that the two morphisms $$ \underline{Hom}(A,B)\rightarrow \underline{Hom}(A,C) \rightarrow \underline{Hom}(A,B') $$ are equivalences, their composition is therefore an equivalence. This completes the proof of the first paragraph of Lemma \ref{stability}. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} For any fibrant $n$-categories $A$, $B$ and $C$ we have composition morphisms $$ \underline{Hom} (A, B) \times \underline{Hom}(B,C) \rightarrow \underline{Hom}(A,C), $$ which are associative. Define a simplicial $n$-category $nCAT$ by setting $nCAT_0 $ equal to a set of representatives for {\em isomorphism} classes of fibrant $n$-categories, and by setting $$ nCAT_m(A_0, \ldots , A_m):= \underline{Hom} (A_0, A_1) \times \ldots \times \underline{Hom} (A_{m-1}, A_m), $$ with simplicial structure given by the above compositions. Since $nCAT_0$ is a set considered as $n$-precat, this simplicial $n$-precat (presheaf on $\Delta \times \Theta ^{n}$) descends to a presheaf on $\Theta ^{n+1}$, in other words it is an $n+1$-precat. The composition gives the necessary conditions in the first degree, and in higher degrees the fact that $\underline{Hom} (A,B)$ are $n$-categories completes what we need to know to conclude that $nCAT$ is an $n+1$-category. If $A$ and $B$ are $n$-categories but not necessarily fibrant then let $Fib(A)$ and $Fib(B)$ be their fibrant replacements (given by the above construction for Theorem \ref{cmc} for example). We call these the {\em fibrant envelopes}. The ``right'' $n$-category of morphisms from $A$ to $B$ is $\underline{Hom}(Fib(A), Fib(B))$. We will sometimes use the notation $$ HOM (A, B):= \underline{Hom}(Fib(A), Fib(B)). $$ We obtain an $n+1$-category equivalent to $nCAT$ by taking all $n$-categories as objects and taking the $HOM (A,B)$ as morphism $n$-categories. {\em Question:} Describe the fibrant envelope of the $n+1$-category $nCAT$. This would be important if one wants to consider weak morphisms $A\rightarrow nCAT$ as families of $n$-categories indexed by $A$ in a meaningul way. We have almost proved Conjecture (\cite{Tamsamani} between 1.3.6 and 1.3.7) on the existence of $nCAT$. We just have to check that the truncation of $nCAT$ down to a $1$-category is equivalent to the localization of the category of $n$-categories by the subcategory of morphisms which are equivalences. In Corollary \ref{htytype} above, we have seen that the localization in question is equal to the localization of $PC_n$ by the weak equivalences. We now know that $PC_n$ is a closed model category, and Quillen shows in this case that the $Hom$ in the localized category is equal to the set of {\em homotopy classes} of morphisms between fibrant and cofibrant objects. In our (second) definition above, we took $nCAT$ to be the category of fibrant (and automatically cofibrant) $n$-precats. The $Hom$ in the localized category is thus the set of homotopy classes of maps. On the other hand, the truncation of $nCAT$ down to a $1$-category is obtained by replacing the $\underline{Hom}(A,B)$ $n$-categories by their sets of equivalence classes of objects. Thus, to prove the conjecture we simply need to show that for $A$ and $B$ fibrant, the set of equivalence classes of objects in the $n-1$-category $\underline{Hom} (A,B)$ is equal to the set of homotopy classes of maps from $A$ to $B$. Note that the objects of $\underline{Hom} (A,B)$ are again just the maps from $A$ to $B$ so we are reduced to showing the following lemma. \begin{lemma} If $A$ and $B$ are fibrant $n$-precats then two morphisms $f,g: A\rightarrow B$ are homotopic in the sense of \cite{Quillen} if and only if the corresponding elements of the $n$-category $\underline{Hom}(A,B)$ are equivalent. \end{lemma} {\em Proof:} Suppose $f$ and $g$ are homotopic (\cite{Quillen} p. 0.2). Then there is an object $A'$ with morphisms $i,j: A\rightarrow A'$ each inducing a weak equivalence, with a projection $p:A'\rightarrow A$ such that the compositions $pi$ and $pj$ are the identity, and a morphism $h: A'\rightarrow B$ such that $hi=f$ and $hj=g$. We may assume that $A'$ is fibrant. Then we obtain pullback morphisms on the $\underline{Hom}$ $n$-categories and in particular, two morphisms $$ i^{\ast}, j^{\ast} :\underline{Hom} (A', B) \rightarrow \underline{Hom} (A, B) $$ and a morphism $$ p^{\ast}: \underline{Hom} (A, B)\rightarrow \underline{Hom} (A', B) $$ which are weak equivalences by Lemma \ref{stability}. These induce isomorphisms on the sets of equivalence classes which we denote $T^n\underline{Hom} (A,B)$ etc., so we have $$ T^ni^{\ast}, T^nj^{\ast} :T^n\underline{Hom} (A', B) \cong T^n \underline{Hom} (A, B) $$ and $$ T^np^{\ast}:T^n \underline{Hom} (A, B)\cong T^n \underline{Hom} (A', B). $$ Here, as before, we have that $T^ni^{\ast} \circ T^np^{\ast}$ and $T^n j^{\ast}\circ T^np^{\ast}$ are equal to the identity. This implies that $T^ni^{\ast} = T^nj^{\ast}$ and hence that their applications to the class of $h$ give the same equivalence class. The results are respectively the classes of $f$ and of $g$, hence $f$ is equivalent to $g$. Conversely suppose $f$ and $g$ are equivalent as objects in $\underline{Hom}(A,B)$. Then by Proposition \ref{intervalK} there is a contractible $K$ with $0,1\in K$ and a morphism $K\rightarrow \underline{Hom}(A,B)$ taking $0$ to $f$ and $1$ to $g$. This yields (by the universal property of the internal $\underline{Hom}$) a morphism $h:A\times K\rightarrow B$. This morphism together with the various others gives a homotopy from $f$ to $g$. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \begin{corollary} {\rm (\cite{Tamsamani} Conjecture 1.3.6-7)} The $n+1$-category $nCAT$ yields when truncated down to a $1$-category the localization $Ho-n-Cat$ of \cite{Tamsamani}. \end{corollary} \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \numero{$n$-stacks} We can give a preliminary discussion of the notion of $n$-stack, following the lines that are already well known for simplicial presheaves and even $n$-stacks of $n$-groupoids (approached via topological spaces in \cite{flexible}, discussed for $n=2,3$ in \cite{Breen}). Our present discussion will be incomplete, basically for the following reason: if ${\cal X}$ is a $1$-category, there are several natural types of objects which represent the idea of a family of $n$-categories indexed (contravariantly) by ${\cal X}$, and we would like to know that all of these notions are equivalent. The main possible versions are: \newline 1. \,\, A functor ${\cal X} ^o\rightarrow nCAT$, which if we take the second point of view on $nCAT$ presented above, is the same thing as a presheaf of fibrant $n$-categories over ${\cal X}$; \newline 2.\,\, A weak functor from ${\cal X} ^o$ to $nCAT$, in other words a functor from ${\cal X} ^o$ to $Fib(nCAT)$ or (what is basically the same thing) an element of $HOM({\cal X} ^o, nCAT)$ i.e. a morphism in $(n+1)CAT$; \newline 3.\,\, A ``fibered $n$-category over ${\cal X}$'', which would be a morphism of $n$-categories ${\cal F} \rightarrow {\cal X}$ (note that a $1$-category considered as an $n$-category is automatically fibrant by \ref{automatic}) satisfying some condition analogous to the definition of fibered $1$-category---I haven't written down this condition (note however that it is distinct from the condition that the morphism be fibrant in the sense we use in this paper). Here is what I currently know about the relationship between these points of view. From (1) one automatically gets (2) just by composing with the morphism $nCAT \rightarrow Fib(nCAT)$ to the fibrant envelope. From (2) one should be able to get (3) by pulling back a universal fibered $n+1$-category over $Fib(nCAT)$. To construct this universal object, first construct a universal $n+1$-category ${\cal U} \rightarrow nCAT$ (with fibers the $n$-categories being parametrized---in particular this morphism is relatively $n$-truncated) then replace the composed morphism ${\cal U} \rightarrow Fib(nCAT)$ by a fibrant morphism. Finally, from (3) one should be able to get (1) by applying the ``sections functor'': if ${\cal F} \rightarrow {\cal X}$ is a fibered $n$-category then define $\Gamma ({\cal X} , {\cal F} )$ to be the $n$-category fiber (calculated in the correct homotopic sense) over $1_{{\cal X}} \in HOM ({\cal X} , {\cal X})$ of $$ HOM ({\cal X} , {\cal F} )\rightarrow HOM ({\cal X} , {\cal X} ). $$ Require now that ${\cal F} \rightarrow {\cal X}$ be a fibrant morphism (if this doesn't come into the condition of being fibered already). Then $$ X\in {\cal X}\mapsto \Gamma ({\cal X} /X, {\cal F} \times _{{\cal X}} ({\cal X} /X )) $$ should be strictly functorial in the variable $X$ yielding a presheaf ${\cal X} ^o \rightarrow nCAT$ which is notion (1). The condition of being a fibered category should imply (as it does in the case $n=1$) that the morphism $$ \Gamma ({\cal X} /X, {\cal F} \times _{{\cal X}} ({\cal X} /X ))\rightarrow {\cal F} _X:= {\cal F} \times _{{\cal X}} \{ X\} $$ be an equivalence of $n$-categories (one might even try to take this condition as the definition of being fibered but I'm not sure if that would work). Finally we would like to show that doing these three constructions in a circle results in an essentially equivalent object. The previous paragraph is for the moment speculative, the main questions left open being the definition of ``fibered $n$-category'' and the construction of the universal family. However, for the rest of this section we will discuss the theory of $n$-stacks supposing that the above equivalences are known. Denote by $\int$ the operation going from (1) to (3). Suppose ${\cal X}$ is a site. There are a couple of different ways of approaching the notion of $n$-stack over ${\cal X}$. Our first definition will be modelled on what was done in \cite{flexible}. A fibered $n$-category ${\cal F} \rightarrow {\cal X}$ is an {\em $n$-stack} if for any $X\in {\cal X} /X$ and any sieve ${\cal B} \subset {\cal X} /X$ the morphism $$ \Gamma ({\cal X} /X, {\cal F} \times _{{\cal X}} ({\cal X} /X ))\rightarrow \Gamma ({\cal B} , {\cal F} \times _{{\cal X}} {\cal B} ) $$ is an equivalence of $n$-categories. If ${\cal F} \rightarrow {\cal X}$ is a fibered $n$-category then we (should be able to) construct the {\em associated $n$-stack} by iterating $n+2$ times the operation $$ L({\cal F} ):= \int \left( X\mapsto \lim _{\rightarrow , {\cal B} \subset {\cal X} /X} \Gamma ({\cal B} , {\cal F} \times _{{\cal X}} {\cal B} ) \right) . $$ This conjecture is based on the corresponding result for flexible sheaves in \cite{flexible}. The second main type of approach is to combine the theory of simplicial presheaves of Jardine-Joyal-Brown-Gersten (cf \cite{Jardine}) with the discussion in the present paper to obtain a closed model structure for the category of presheaves of $n$-precats over ${\cal X}$. In this case the fibrant condition would imply the condition of being an $n$-stack. To give the definitions (without proving that we get a closed model category) it suffices to define weak equivalence---the cofibrations being just the maps which over each object of ${\cal X}$ are cofibrations of $n$-precats, and the fibrations then being defined by the lifting property for trivial cofibrations. (As usual the main problem would then be to prove that pushout by a trivial cofibration is again a trivial cofibration---and for this we could probably just combine the proofs of Jardine/Joyal \cite{Jardine} and the present paper.) If $A\rightarrow B$ is a morphism of presheaves of $n$-precats over ${\cal X}$ then we obtain a morphism $Cat(A)\rightarrow Cat(B)$ of presheaves of easy $n$-categories (where $Cat(A)(X):= Cat(A(X))$). We will say that $A\rightarrow B$ is a weak equivalence if $Cat(A)\rightarrow Cat(B)$ is a weak equivalence of presheaves of $n$-categories, notion which we now define. Let $T$ denote Tamsamani's truncation operation \cite{Tamsamani} which is functorial so it extends to presheaves of $n$-categories. A morphism of presheaves of $n$-categories $A\rightarrow B$ is a {\em weak equivalence in top degree} if for every $n$-morphism of $B$ and lifting of its source and target to $n-1$-morphisms in $A$, there exists a unique lifting to an $n$-morphism in $A$. Now we say that a morphism $A\rightarrow B$ of presheaves of $n$-categories is a {\em weak equivalence} if for every $k$ the morphism $$ Sh(T^kA)\rightarrow Sh(T^kB) $$ is a weak equivalence in top degree, where $Sh$ denotes the stupid sheafification operation (i.e. sheafify each of the presheaves $A_M$). In this point of view, if $A$ is a presheaf of $n$-categories over ${\cal X}$ then we define the {\em associated $n$-stack} to be the fibrant object equivalent to $A$ in the previous presumed closed model category. \subnumero{$n$-categories as $n-1$-stacks} Heuristically we can define a structure of {\em site} on $\Delta$ where the coverings of an object $m$ are the collections of morphisms $\lambda _i \hookrightarrow m$ where $\lambda _i = \{ a_i , a_i + 1, \ldots , b_i\}$ such that $a_{i+1}=b_i$. If $A$ is an $n$-precat then the collection $\{ A_{p/}\}$ may be thought of as a presheaf of $n-1$-precats over $\Delta $. The condition to be an $n$-category is that this should be a presheaf of $n-1$-categories which should satisfy descent for coverings, i.e. it should be an $n-1$-stack of $n-1$-categories over this site. The construction $Cat$ is essentially just finding the $n-1$-stack associated to an $n-1$-prestack by an operation similar to that described in \cite{flexible}. The main problems above are caused by the fact that this site doesn't admit fiber products. It might be a good idea to replace this site by its associated topos, the category of categories, which would lead to the yoga: {\em that an $n$-category is an $n-1$-stack over the topos of categories.} It might be possible, by treating $n$-stacks at the same time as $n$-categories, to simplify the arguments of the present paper by recursively defining $n$-categories as $n-1$-stacks. I haven't thought about this any further. \numero{The generalized Seifert-Van Kampen theorem} Our closed model category structure allows us (with a tiny bit of extra work) to obtain the analogue of the Siefert-Van Kampen theorem for the Poincar\'e $n$-groupoid of a topological space $\Pi _n(X)$ defined by Tamsamani (\cite{Tamsamani}, \S 2.3). \begin{theorem} \label{svk} If $X$ is a space covered by open subsets $X = U\cup V$ then (setting $W:= U\cap V$) $\Pi _n (X)$ is equivalent to the category-theoretic pushout of the diagram $$ \Pi _n (U) \leftarrow \Pi _n (W) \rightarrow \Pi _n (V). $$ \end{theorem} In order to prove this theorem we recall Tamsamani's realization functor from $n$-precats to topological spaces (\cite{Tamsamani} \S 2.5). There is a covariant functor $R: \Delta ^n \rightarrow Top$ which associates to $M= (m_1, \ldots , m_n)$ the product $R^{m_1}\times \ldots \times R^{m_n}$ where $R^m$ denotes the usual topological $m$-simplex. If $A: (\Delta ^n)^o\rightarrow Sets$ is a presheaf of sets then Tamsamani defines the {\em realization of $A$} in the standard way combining $R$ and $A$. We denote this $\langle R, A\rangle$ because it is a sort of pairing of functors. If $A$ is an $n$-precat in our notations then pull it back to a presheaf on $\Delta ^n$ and apply the realization (we still denote this as $\langle R, A\rangle$). The functor $\langle R, \cdot \rangle$ obviously preserves pushouts. {\em Caution:} The realization functor does not preserve cofibrations. It takes injective morphisms of presheaves over $\Theta^n$ to cofibrations of spaces, but the cofibrations which are not injective in the top degree are taken to non-injective morphisms. Recall Tamsamani's Proposition 3.4.2(ii): \begin{proposition} If $A\rightarrow B$ is an equivalence of $n$-categories then $$ \langle R, A \rangle \rightarrow \langle R, B \rangle $$ is an equivalence of spaces. \end{proposition} {\em Proof:} The proof of \cite{Tamsamani} for $n$-groupoids using induction on $n$ also works for $n$-categories. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} Say that a morphism $X\rightarrow Y$ of topological spaces is an {\em $n$-weak equivalence} if it is an isomorphism on $\pi _0$ and for any choice of basepoint in $X$ it is an isomorphism on $\pi _i$ for $i\leq n$. This is equivalent to saying that it induces a weak equivalence on the Postnikov tower up to stage $n$. \begin{corollary} \label{equivtoequiv} The realization functor $\langle R, \cdot \rangle$ from $n$-precats to $Top$ takes weak equivalences to $n$-weak equivalences of topological spaces. \end{corollary} {\em Proof:} Realization takes our standard trivial cofibrations $\Sigma \rightarrow h$ to homotopy equivalences of topological spaces. This is essentially the content of the constructions of retractions in the proof of Theorem 2.3.5 (that $\Pi _n(X)$ is an $n$-nerve) of \cite{Tamsamani}. For all except the upper boundary cases, the standard trivial cofibrations are taken to cofibrations of topological spaces. Pushout by the injective standard trivial cofibrations becomes pushout by a trivial cofibration of spaces, whence a homotopy equivalence. In order to deal with the upper boundary cases we introduce the following notation: $$ \rangle R, A \langle _{q} $$ denotes the $q$-skeleton of the realization of $A$, that is the realization taken over all $M$ with $\sum m_i \leq q$. Then, if $q\leq n-1$ the functor $\rangle R, \cdot \langle _{q}$ takes cofibrations to cofibrations of spaces. Suppose $\varphi : \Sigma \rightarrow h$ is a standard trivial cofibration in one of the boundary cases. Using the notation of \S 2 we can write $\Sigma$ as the coequalizer of $$ h' \sqcup h' \sqcup \Upsilon ' \rightarrow h^a \sqcup h^a $$ (the component $\Upsilon$ which appears on the right in the general case disappears in the upper boundary case). The map $\Sigma \rightarrow h$ is given by the map $h^a \sqcup h^b\rightarrow h$ which in this case is two times the identity (because $h(M, m, 1^{k+1})$ which doesn't exist is replaced by $h:= h(M, m, 1^k)= h^a=h^b$). The cells in $\langle R,\Sigma \rangle $ of dimension $n$ are automatically of the form $h(1^l, 1, 1^k)$ for maps $1^l\rightarrow M$ and $1\rightarrow m$. There are two such which are identified whenever $1\rightarrow m$ is not one of the principal morphisms. The cells coming from principal $1\rightarrow m$ occur only once in the realization of $\Sigma$ already. It follows (since any non-principal $1\rightarrow m$ is a path which is homotopic to a concatenation of principal $1\rightarrow m$) that the $n$-cells which are identified are homotopic. Note that on the level of cells of dimension $<n$ the morphism $\Sigma \rightarrow h$ is an isomorphism. In particular, pushout via $\varphi$ over any $\Sigma \rightarrow A$ preserves the $n-1$-skeleton of the realization, and the $n$-cells which are identified are homotopic. In this boundary case the pushout by $\varphi$ is surjective, in particular it is surjective on $n+1$-cells. A surjective morphism of cell complexes which is an isomorphism on $n-1$-skeleta and which only identifies $n$-cells which are homotopic (relative the $n-1$-skeleton) is an $n$-weak equivalence. This completes the proof that pushout by any of our standard trivial cofibrations $\varphi$ induces an $n$-weak equivalence. It follows by construction of the operation $Cat$ that for any $n$-precat $A$ the morphism $$ \langle R, A \rangle \rightarrow \langle R, Cat(A) \rangle $$ is an $n$-weak equivalence of spaces. Now we can complete the proof: if $A\rightarrow B$ is a weak equivalence then by definition $Cat(A)\rightarrow Cat(B)$ is an equivalence of $n$-categories so in the diagram $$ \begin{array}{ccc} \langle R, A \rangle & \rightarrow & \langle R, B \rangle \\ \downarrow && \downarrow \\ \langle R, Cat(A) \rangle & \rightarrow & \langle R, Cat(B) \rangle \end{array} $$ the vertical arrows are $n$-weak equivalences from the previous argument and the bottom arrow is a weak equivalence by the proposition, so the top arrow is an $n$-weak equivalence of spaces. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \begin{lemma} \label{groupoid} If $A\rightarrow B$ and $A\rightarrow C$ are morphisms of $n$-groupoids with one being a cofibration, then the category-theoretic pushout $Cat(B\cup ^AC)$ is an $n$-groupoid. \end{lemma} {\em Proof:} We say that an $n$-precat is a {\em pre-groupoid} if its associated $n$-category is a groupoid. We prove that the pushout of pre-groupoids is again a pre-groupoid, and we proceed by induction on $n$ so we may assume this is known for $n-1$-pre-groupoids. Suppose now that $A$, $B$ and $C$ are $n$-groupoids with morphisms as in the statement of the lemma. Then the $A_{p/}$, $B_{p/}$ and $C_{p/}$ are $n-1$-groupoids, and $$ (B\cup ^AC)_{p/} = B_{p/}\cup ^{A_{p/}}C_{p/}. $$ In particular, $(B\cup ^AC)_{p/}$ are $n-1$-pre-groupoids. The process of going from this collection of $n-1$-precats to the collection corresponding to $Cat (B\cup ^AC)$ as described in \S 4, uses only iterated pushouts by the various $(B\cup ^AC)_{p/}$ in various combinations. Since we know by induction that pushouts of $n-1$-pre-groupoids are again $n-1$-pre-groupoids, it follows that $Cat(B\cup ^AC)_{p/}$ are $n-1$-groupoids. It suffices now to show that the truncation of $Cat(B\cup ^AC)$ down to a $1$-category is a groupoid. But this truncation is the same as the brutal truncation since we know that the $Cat(B\cup ^AC)_{p/}$ are $n-1$-groupoids. On the other hand, brutal truncation commutes with the operations $Cat$ and pushout, therefore the truncation of $Cat(B\cup ^AC)$ is the pushout of the truncations of $A$, $B$ and $C$ which are groupoids. Finally, the $1$-category pushout of groupoids is again a groupoid, so $Cat(B\cup ^AC)$ is a groupoid. To complete the proof it remains to be seen that the pushout of $n$-pregroupoids is a pregroupoid. Suppose $A$, $B$ and $C$ are $n$-pre-groupoids. Then by reordering $$ BigCat(B\cup ^AC) = BigCat(Cat(B) \cup ^{Cat(A)}Cat(C)). $$ Thus $BigCat(B\cup ^AC)$ is the category-theoretic pushout of $n$-groupoids so by the previous argument it is an $n$-groupoid. This shows that $B\cup ^AC$ is an $n$-pre-groupoid, completing the proof of the induction step. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} {\em Proof of Theorem \ref{svk}:} Note first of all that Tamsamani's proof that $\Pi_n(X)$ is an $n$-category (\cite{Tamsamani} Theorem 2.3.6) actually shows that it is an easy $n$-category. With the notations of the theorem, we have a diagram of easy $n$-categories $$ \begin{array}{ccc} \Pi _n(W) & \rightarrow & \Pi _n(U) \\ \downarrow && \downarrow \\ \Pi _n(V) & \rightarrow & \Pi _n(X). \end{array} $$ Let $A$ be the pushout $n$-precat of the upper and left arrows. We have a morphism $A\rightarrow \Pi _n(X)$, and hence (non-uniquely) $Cat(A)\rightarrow \Pi _n(X)$ since the latter is an easy $n$-category. The realization of $A$ is the pushout of the realizations of $\Pi _n(U)$ and $\Pi _n(V)$ over $\Pi _n(W)$. These last realizations are $n$-weak equivalent to $U$, $V$ and $W$ respectively (\cite{Tamsamani} 3.3.4), so the realization of $A$ is $n$-weak equivalent to the pushout of $U$ and $V$ over $W$, in other words to $X$. Thus the morphism $A\rightarrow \Pi _n(X)$ induces an $n$-weak equivalence on realizations. On the other hand we have seen above that $A\rightarrow Cat(A)$ induces an $n$-weak equivalence on realizations. Thus the morphism $Cat(A)\rightarrow \Pi _n(X)$ induces an $n$-weak equivalence on realizations. Lemma \ref{groupoid} implies that $Cat(A)$ is an $n$-groupoid. Applying the functor $\Pi _n$ again and using Proposition 3.4.4 of \cite{Tamsamani} we conclude that $Cat(A)\rightarrow \Pi _n(X)$ is an equivalence of $n$-groupoids. This proves the theorem. \hfill $/$\hspace*{-.1cm}$/$\hspace*{-.1cm}$/$\vspace{.1in} \numero{Nonabelian cohomology} If $A$ is a fibrant $n$-category and $X$ a topological space then define the {\em nonabelian cohomology of $X$ with coefficients in $A$} to be $H(X, A): = \underline{Hom} (\Pi _n(X), A)$. It is an $n$-category. This satisfies Mayer-Vietoris: if $U,V\subset X$ and $W=U\cap V$ then $$ m:H(X, A) \rightarrow H(U,A)\times _{H(W,A)}H(V,A) $$ is an equivalence of $n$-categories (where the fiber product is understood to be the homotopic fiber product obtained by replacing one of the morphisms with a fibrant morphism). To see this, note that if $A$ is fibrant then for any cofibration $B\rightarrow C$ the morphism $\underline{Hom}(C,A)\rightarrow \underline{Hom}(B,A)$ is fibrant. To prove this claim it suffices to remark that if $S\rightarrow T$ is a trivial cofibration then $$ S\times C \cup ^{S\times B} T\times B \rightarrow T \times C $$ is a trivial cofibration, now apply the universal property of the internal $\underline{Hom}$ to obtain the lifting property in question. In particular, note that $H(U,A)\rightarrow H(W,A)$ and $H(V,A)\rightarrow H(W,A)$ are fibrations (since open inclusions induce cofibrations of $\Pi _n$). Thus the above fiber product is the homotopic fiber product. We now prove that the Mayer-Vietoris map $m$ is an equivalence of $n$-categories. It is the same as the map $$ \underline{Hom} (\Pi _n(X), A)\rightarrow \underline{Hom} (\Pi _n(U)\cup ^{\Pi _n(W)}\Pi _n(V), A). $$ But we have seen in \ref{svk} that the morphism $$ \Pi _n(U)\cup ^{\Pi _n(W)}\Pi _n(V)\rightarrow \Pi _n(X) $$ is a trivial cofibration. Thus, to complete the proof it suffices to note (from \ref{stability}) that for any trivial cofibration $B\rightarrow C$ the morphism $\underline{Hom} (C,A)\rightarrow \underline{Hom} (B,A)$ is an equivalence of $n$-categories. If we take cohomology of a CW complex $X$ with coefficients in a fibrant groupoid $A$ then $H(X,A)$ is equivalent to $\Pi _n(\underline{Hom} _{Top}(X, \langle R,A\rangle )$. To prove this note that $\Pi _n$ is adjoint to the realization $\langle R, A \rangle$, which implies on the level of internal $\underline{Hom}$ that for any $n$-precat $B$ and space $U$, $$ \underline{Hom}(B, \Pi _n(U)) = \Pi _n (\underline{Hom}_{Top}(\langle R, B\rangle , U) $$ where $\underline{Hom}_{Top}$ denotes the compact-open mapping space. Corolary \ref{equivtoequiv} and the adjointness imply that for any space $U$, $\Pi _n(U)$ is fibrant. On the other hand, Tamsamani proves in \cite{Tamsamani} \S 3 that for any $n$-groupoid $A$ the morphism $$ A\rightarrow \Pi _n (\langle R, A \rangle ) $$ is an equivalence of $n$-groupoids. Thus if $A$ is a fibrant $n$-groupoid we have an equivalence $$ \underline{Hom}(\Pi _n(X), A) \rightarrow \underline{Hom}(\Pi _n(X), \Pi _n (\langle R, A \rangle )) = \Pi _n \underline{Hom}_{Top}(\langle R, \Pi _n(X) \rangle , \langle R, A \rangle ). $$ On the other hand, again from \cite{Tamsamani} \S 3 we know that $\langle R, A \rangle $ is $n$-truncated, and there is a space $W$ and diagram $$ X \leftarrow W \rightarrow \langle R, \Pi _n(X) \rangle $$ where the left morphism is a weak homotopy equivalence and the right morphism induces an isomorphism on homotopy groups in degrees $\leq n$. Thus, under the assumption that $X$ is a CW complex (which allows us to obtain weak equivalences when we apply $\underline{Hom}_{Top}$) we obtain a diagram of weak homotopy equivalences $$ \underline{Hom}_{Top}(X , \langle R, A \rangle ) \rightarrow \underline{Hom}_{Top}(W , \langle R, A \rangle ) \leftarrow \underline{Hom}_{Top}(\langle R, \Pi _n(X) \rangle , \langle R, A \rangle ). $$ Combining with the above we get a diagram of equivalences $$ \underline{Hom}(\Pi _n(X), A)\rightarrow \Pi _n \underline{Hom}_{Top}(W , \langle R, A \rangle ) \leftarrow \Pi _n \underline{Hom}_{Top}(X , \langle R, A \rangle ). $$ Thus the nonabelian cohomology with coefficients in an $n$-groupoid coincides with the approach using topological spaces. Of course even the nonabelian cohomology with coefficients in an $n$-category $A$ which isn't a groupoid doesn't really give a new homotopy invariant since all of the information is contained in the Poincar\'e $n$-groupoid $\Pi _n(X)$. However, it might give some interesting special cases to study. Once the theory of $n$-stacks gets off the ground, we should be able to interpret $H(X,A)$ as the $n$-category of global sections of the $n$-stack associated to the constant presheaf $U\mapsto A$ over the site $Site(X)$ of disjoint unions of open subsets of $X$. More generally if ${\cal X}$ is any site and $\underline{A}$ is a presheaf of $n$-categories over ${\cal X}$ (or a fibered $n$-category over ${\cal X}$) then the $n$-category of global sections of the $n$-stack associated to $\underline{A}$ is the {\em nonabelian cohomology} $H({\cal X}, \underline{A})$. We now treat the example mentionned in the footnote of the introduction. Suppose $G$ is a group and $U$ an abelian group, and let $A$ (resp. $B$) be the strict $1$-category with one object and group of automorphisms $G$ (resp. the strict $n$-category with one arrow in each degree $<n$ and group $U$ of automorphisms in degree $n$). Let $A'$ (resp. $B'$) be fibrant replacements for $A$ and $B$. We would like to show that the set $T^n\underline{Hom}(A',B')$ is equal to the group cohomology $H^n(G, U)$. For the moment, the only way I see to do this is to pass by topology using the Seifert-Van Kampen theorem. Let $X=K(G,1)$ and $Y= K(U,n)$. Then $\Pi _n(X)$ is equivalent to $A$ and $\Pi _n(Y)$ is equivalent to $B$. Similarly in the other direction $\langle R, B' \rangle $ is equivalent to $\langle R , B \rangle$ which in turn is equivalent to $Y$. By the above discussion $H(X, B')$ is equivalent to $\Pi _n(\underline{Hom} (X, Y))$. The truncation $T^nH(X,B')$ is thus equal to $\pi _0(\underline{Hom} (X,Y))$ which (as is well-known) is $H^n(X, U)= H^n(G, U)$. But by definition $H(X,B')=\underline{Hom} (\Pi _n(X), B')$ which is equivalent to $\underline{Hom} (A', B')$. This gives the desired statement. The above argument is clearly not ideal, since we are looking for a purely algebraic approach to these types of problems. It seems likely that the algebraic techniques of \cite{Quillen} with appropriate small additional lemmas would permit us to give a purely algebraic proof of the result of the previous paragraph. \numero{Comparison} As pointed out in the introduction, there are many different theories of weak $n$-categories in the process of becoming reality, and this will pose the problem of comparison. As an initial step we give a construction of functors modeled on Tamsamani's Poincar\'e $n$-groupoid construction. We denote our ``Poincar\'e $n$-category'' functors by $\Upsilon_n$ to avoid confusion with the $\Pi _n$ (specially on the fact that the $\Upsilon_n$ will not take images in $n$-groupoids). This section is only a sketch, with many details of proofs missing. In particular the following proposed set of axioms for internal model categories is a preliminary attempt only. Suppose ${\cal C}$ is a closed model category with the following additional properties: \newline (IM1)---${\cal C}$ admits an internal $\underline{Hom}$; \newline (IM2)---If $A$ and $B$ are fibrant and cofibrant objects then $\underline{Hom} (A,B)$ is fibrant; \newline (IM3)---If $A\rightarrow A'$ is a cofibration (resp. trivial cofibration) of fibrant and cofibrant objects, and if $B'\rightarrow B$ is a fibration (resp. trivial fibration) of fibrant and cofibrant objects, then $\underline{Hom}(A', B')\rightarrow \underline{Hom}(A,B)$ is a fibration (resp. a trivial fibration); \newline (IM4)---Internal $\underline{Hom}$ takes cofibrant pushout in the first variable (resp. fibrant fiber product in the second variable) to fiber product. We call ${\cal C}$ an {\em internal closed model category}. Suppose now that ${\cal C}$ is an internal closed model category with an inclusion $i:Cat\subset {\cal C}$ having the following properties: \newline (a)---$i(\emptyset )$ is the initial object and $i(\ast )$ is the final object of ${\cal C}$; (b)---$i$ is compatible with disjoint union; \newline (c)---$i$ takes values in the fibrant and cofibrant objects of ${\cal C}$; \newline (d)---$i$ takes the internal $\underline{Hom}$ in $Cat$ to the internal $\underline{Hom}$ of ${\cal C}$. \newline (e)---Let $I$ denote the category with two objects $0,1$ and one non-identity morphism from $0$ to $1$. Let $I^{(m)}$ denote the symmetric product of $m$ copies of $I$, which is the category with objects $0,\ldots , m$ and one morphism from $i$ to $j$ when $i\leq j$. Then we require that the morphism from the ${\cal C}$-pushout of the diagram $$ i(I) \leftarrow \ast \rightarrow i(I) \leftarrow \ast \ldots \ast \rightarrow i(i) $$ to $i(I^{(m)})$ be a cofibrant weak equivalence in ${\cal C}$. {\em Remark:} Our closed model categories $PC_n$ are internal, and (for $n\geq 1$) have functors $i: Cat\hookrightarrow PC_n$ satisfying properties (a)--(e) above. These seem to be reasonable properties to ask of any closed model category representing a theory of $n$-categories (or $\infty$-categories). However, some of the properties are of a rather technical nature so it is possible that some technically slightly different approach to comparison would be needed---the present section is just a first attempt. Suppose $({\cal C} , i)$ is an internal closed model category with inclusion $i$ having the above properties. Let ${\cal C} _f$ denote the subcategory of fibrant objects. Then for any $n$ we define a functor $\Upsilon _n: {\cal C} \rightarrow n-Cat \subset PC_n$, which we call the ``Poincar\'e $n$-category'' functor. These functors will have the property that they take weak equivalences in ${\cal C} _f$ to equivalences of $n$-categories, and will be compatible with direct products (hence with fiber products over sets). The definition is by induction. First of all, $\Upsilon _0(X)$ is defined bo be equal to the set of homotopy classes of maps $\ast\rightarrow X$. Then, supposing that we have defined $\Upsilon _{n-1}$ we define for any $X\in {\cal C} _f$ the simplicial object $U$ of ${\cal C}$ by: $U_0$ is the set (note that sets are categories so $i$ gives an inclusion of sets into ${\cal C}$) of morphisms $\ast\rightarrow X$; and for $x_0,\ldots , x_m\in U_0$, $U_m(x_0,\ldots , x_m)$ is the fiber of $$ \underline{Hom}(I^{(m)}, X)\rightarrow \underline{Hom}(\{ 0, \ldots , m\} , X) $$ over the point $(x_0, \ldots , x_m)$. Then $U_m$ is the disjoint union of the $U_m(x_0,\ldots , x_m)$ over all sequences of $x_i \in U_0$. Axiom IM4 and condition (e) imply that the usual morphism $$ U_m \rightarrow U_1\times _{U_0} \ldots \times _{U_0}U_1 $$ is a weak equivalence. With the above notation set $$ \Upsilon _n(X)_{m/}:= \Upsilon _{n-1}(U_m). $$ This simplicial $n-1$-category is an $n$-category since $\Upsilon _{n-1}$ is compatible with direct products and preserves weak equivalences. Note that $\Upsilon _n$ is obviously compatible with direct products. One has to prove that $\Upsilon _n$ preserves weak equivalences (we leave this out for now). \subnumero{Examples} Tamsamani's functor $\Pi _n$ is essentially an example of the functor functor $\Upsilon _n$ for ${\cal C} = Top$ and $i: Cat \rightarrow Top$ the realization functor. Our definition of $\Upsilon _n$ is a generalization of the definition of (\cite{Tamsamani} \S 3). For ${\cal C} = PC_{n'}$ we obtain the functor $\Upsilon _n$. If $n=n'$ it is essentially the identity; for $n< n'$ is is the truncation $T^{n'-n}$; and for $n>n'$ it is the induction $Ind ^{n'}_n$. Note however that the induction doesn't preserve pushouts, so $\Upsilon _n$ will not necessarily preserve pushouts in general (where by pushouts here we mean the replacement of pushouts by weak-equivalent objects of ${\cal C} _f$). For any given theory ${\cal C}$ of $n$-categories satisfying the above properties, one would like to check that the functor $\Upsilon_n$ is an equivalence of homotopy theories in the sense of \cite{Quillen} (or at least that it induces an isomorphism of localized categories). If $n$ is correctly chosen to correspond to the level of ${\cal C}$ then one would try to show that $\Upsilon _n$ preserves pushouts. There are examples of ${\cal C}$ which are not equivalent to $PC_n$, such as $Top$ or, for example, the category of ``Segal categories'', i.e. simplicial spaces whose first object is a set and which satisfy Segal's condition (cf \cite{Tamsamani} \S 3). Even if we look at Segal categories whose elements are $n-1$-truncated, the functor $\Upsilon _n$ will go into the $n$-precats $A$ whose $A_{p/}$ are $n-1$-groupoids, in particular $\Upsilon _n$ will not be essentially surjective. Similarly, one can imagine looking at a theory ${\cal C}$ of $n$-categories with extra structure. For example $Top$ is basically the theory of $n$-categories where the $i$-morphisms have essential inverses. Baez and Dolan propose another type of extra structure of ``adjoints'' rather than inverses, in relation to topological quantum field theory \cite{BaezDolan}. It is possible in this case that ${\cal C}$ would again be an internal closed model category and that we would have a functor $i$. The resulting functor $\Upsilon_n$ would then be essentially the functor of ``forgetting the extra structure'' and taking the underlying $n$-category. A more fundamental example of the above phenomenon will be the closed model category of $n$-stacks. This retracts onto that of $n$-categories: the inclusion being the constant stack functor and the morphism $\Upsilon _n$ being the global section functor. Of course in this situation we don't expect $\Upsilon _n$ to be an equivalence of theories. This example shows that more is needed than just the above axioms for ${\cal C}$ in order to prove that the composition in the other direction is the identity.

9409

alg-geom/9409003

en

https://arxiv.org/abs/alg-geom/9409003

alg-geom/9409003

Yi Hu

Yi Hu and Wei-Ping Li

Variation of the Gieseker and Uhlenbeck Compactifications

24 pages, AmsLaTex

In this article, we study the variation of the Gieseker and Uhlenbeck compactifications of the moduli spaces of Mumford-Takemoto stable vector bundles of rank 2 by changing polarizations. Some {\it canonical} rational morphisms among the Gieseker compactifications are proved to exist and their fibers are studied. As a consequence of studying the morphisms from the Gieseker compactifications to the Uhlebeck compactifications, we show that there is an everywhere-defined {\it canonical} algebraic map between two adjacent Uhlenbeck compactifications which restricts to the identity on some Zariski open subset.

alg-geom

\section{Introduction}\label{s:intro} Let $X$ be an algebraic surface with $p_g=0$ and $H$ an ample divisor over $X$. The moduli space ${\cal M}^\mu_H$ of the Mumford-Takemoto $H$-stable rank two vector bundles has turned out to be a key ingredient in the Donaldson theory of smooth topology of algebraic surfaces. In fact, Donaldson showed that the moduli space ${\cal N}_H$ of $SU(2)$-ASD connections on $X$ with respect to the Hodge metric induced by $H$ is homeomorphic to the moduli space ${\cal M}^\mu_H$ of Mumford-Takemoto $H$-stable rank two vector bundles. Hence the study of this moduli space is important for the application of the Donaldson theory. It is obvious that the moduli space ${\cal M}^\mu_H \cong {\cal N}_H$ depends on the polarization $H$. The effect on the moduli space of $H$-stable bundles when changing the polarization has been considered before by Donaldson \cite{D}, Friedman-Morgan \cite{FM}, Mong \cite{M} and Qin \cite{Q1}, among others. In particular, Qin \cite{Q1} gave a very systematic treatment. \par However, for many important applications, e.g., computing Donaldson's polynomials, just considering the open variety ${\cal M}^\mu_H$ is not sufficient. In fact, Donaldson polynomials are computed on the Uhlenbeck compactification of ${\cal N}_H \cong {\cal M}^\mu_H$. So instead of considering variation of moduli spaces of Mumford-Takemoto $H$-stable rank-two vector bundles ${\cal M}^\mu_H$ for different $H$, we take the step further to consider the variations of the Gieseker and Uhlenbeck compactifications of the moduli space ${\cal M}^\mu_H$ of $H$-stable bundles. \par Gieseker constructed the moduli space ${\cal M}_H$ of $H$-semi-stable torsion-free coherent sheaves and showed that it is a projective scheme. Since ${\cal M}_H$ contains the moduli space ${\cal M}^\mu_H$ as a Zariski open subset, ${\cal M}_H$ can be considered as a compactification of ${\cal M}^\mu_H$. According to the Uhlenbeck weak compactness theorem, ${\cal M}^\mu_H \cong {\cal N}_H$ also admits a Gauge theoretic compactification. This compactification is called the Uhlenbeck compactification, denoted by $\overline{{\cal N}_H}$. \par It appears that the works before only considered variation of ${\cal M}^\mu_H$ and set-theoretic comparison of the moduli spaces ${\cal M}^\mu_H$ by varying $H$ (cf. \cite{Q1}). However, in this paper, not only we take into account the variation of compactifications of ${\cal M}^\mu_H$ but also our consideration is on the level of morphisms. Namely, we address the existence of morphisms amongst the moduli spaces. In particular, we showed that there are {\it enough} canonical algebraic rational maps amongst the Gieseker compactifications and canonical everywhere-defined algebraic maps amongst the Uhlenbeck compactifications. Moreover, we gave some explicit description of these morphisms and maps (see Theorem 5.1 and \S 7). One of advantages of this consideration is that these maps carry considerable information which may allow one to trace the geometry and topology from one moduli space (and its compactifications) to another. \par The main result of this paper may be summarized as follows. Let $C_X$ be the K\"ahler cone of $X$ which is the closed convex cone in $Num(X) \otimes {\Bbb R}$ spanned by all ample divisors. There are certain natural wall and chamber structures in $C_X$ such that an ample divisor $H$ lies on a wall if and only if it possesses non-universal {\it strictly} MT $H$-semistable bundles. Let $C$ and $C'$ be two adjacent chambers with a common face $F = \overline{C} \cap \overline{C'}$. Pick up divisors $H$, $H'$, and $H_0$ in $C$, $C'$ and $F$, respectively. Then there are two canonical rational morphisms ${\varphi}$ and $\psi$ amongst the Gieseker compactifications which descend to two everywhere-defined algebraic maps $\overline{\varphi}$ and $ \overline{\psi}$ amongst the Uhlenbeck compactifications $$\begin{matrix}{\cal M}_H(c_2)&\buildrel{\varphi}\over{-->} & {\cal M}_{H_0}(c_2)& \buildrel{\psi}\over{<--} {\cal M}_{H'}(c_2)\\ \mapdown{\gamma_H}&&\mapdown{\gamma_{H_0}}&\mapdown{\gamma_{H'}}\\ \overline{{\cal N}_H}(c_2)&\mapright{\overline{\varphi}}&\overline{{\cal N}}_{H_0}(c_2) &\mapleft{\overline{\psi}} \overline{{\cal N}}_{H'}(c_2) \end{matrix}$$ such that the above diagram commutes (see Theorem 5.1 and Theorem 7.8 for more details). Here the morphisms $\gamma$ are the morphisms from the Gieseker compactifications to their corresponding Uhlenbeck compactifications as constructed by J. Li \cite{Li}. Although $\varphi$ and $\phi$ are just rational maps and hence are not surjective, $Im\varphi\cup Im\phi={\cal M}_{H_0}(c_2)$. \par Another interesting result in this paper is about Uhlenbeck compactifications. Uhlenbeck compactification $\overline {\cal N}_H(c_2)$ is, in general, a closed subset of $\coprod\limits_{j=0}^{c_2}\widetilde{\cal N}_H(j)\times Sym^{c_2-j}(X)$. It is unknown whether $\overline {\cal N}_H(c_2)=\coprod\limits_{j=0}^{c_2} \widetilde{\cal N}_H(j)\times Sym^{c_2-j}(X)$. When $p_g(X)=0$, we are able to give an affirmative answer. \par Some of our considerations are inspired by a recent paper of Dolgachev and the first author \cite{DH} where they treated the variational problem of geometric invariant theory quotients. However, we would like to point out that the variational problem of the Gieseker compactifications and the Uhlenbeck compactifications is considerably different from that of GIT! Notably, the differences include, amongst other:\par (1) in general, there are infinitely many moduli spaces that are distinct to each other in nature, while in the GIT case, the number of naturally distinct quotients is finite; \par (2) in general, there only exist rational maps among the Gieseker compactifications, while in the GIT case, morphisms among quotients are always defined everywhere. Quite surprisingly, the maps among the Uhlenbeck compactifications are defined everywhere. \par Another inspiration is Jun Li's paper on the relations between the Uhlenbeck compactification and the Gieseker compactification. Because a morphism from the Gieseker compactification to the Uhlenbeck compactification is constructed, using results of the variation of the Gieseker compactification, we can get some results on the variation of the Uhlenbeck compactification. J. Morgan \cite{Mo} also studied the map from Gieseker compactification to Uhlenbeck compactification. \par We mention that Friedman and Qin \cite{FQ} obtained stronger relations among the Gieseker compactifications and applied their results to good effect on computing the Donaldson's invariants. Also, after this work was completed, we learnt the work \cite{MW} and received a copy of it. However, neither of \cite{FQ} and \cite{MW} stresses on the Uhlenbeck compactifications. \par {\bf Acknowledgment}: We would like to thank the following people who helped us in various ways during the course of this work: Robert Friedman, Jun Li, Zhenbo Qin, and Yungang Ye. We also would like to thank Max-Planck-Institut f\"ur Mathematik for inviting both of us to visit the institute in the summer of 1993. \section{Background materials} Let $X$ be an algebraic surface. \begin{defn}\label{t:i1} Let $V$ be a rank-two torsion-free coherent sheaf over $X$. Let $H$ be an ample divisor on $X$ which will be called a stability polarization (or, a polarization for short). $V$ is said to be Gieseker $H$-stable ($H$-semi-stable) if for any rank one subsheaf $L$ of $V$, $$\chi(L(nH))<\,\,(\le )\,\, {1\over 2} \chi(V(nH))\qquad \hbox{for $n\gg0$.}$$ $V$ is strictly Gieseker $H$-semi-stable if in addition there exists $L\subset V$ such that $$\chi(L(nH))= {1\over 2} \chi(V(nH)) \qquad \hbox{for $n\gg0$}.$$ \end{defn} \par There is another notion of stability namely, the Mumford-Takemoto stability. \begin{defn}\label{t:i2} $V$ is said to be Mumford-Takemoto $H$-stable ($H$-semi-stable) if for any rank one subsheaf $L$ of $V$, $$c_1(L)\cdot H<\,\,(\le )\,\, {1\over 2} c_1(V)\cdot H.$$ $V$ is strictly Mumford-Takemoto $H$-semi-stable if in addition there exists rank one subsheaf $L\subset V$ such that $$c_1(L)\cdot H= {1\over 2} c_1(V)\cdot H.$$ \end{defn} \begin{rk}\label{r:i1} In this article, unless otherwise stated, when we say $V$ is $H$-stable ($H$-semi-stable), we shall mean Gieseker $H$-stable ($H$-semi-stable). We abbreviate Mumford-Takemoto $H$-stable ($H$-semi-stable) as M-T$H$-stable ($H$-semi-stable). \end{rk} \par Also, in this paper, the following convention will be adopted. $V$, $V'$, etc. represent rank two torsion free coherent sheaves and $L$, $L'$, $M$, $M'$, ect. represent rank one torsion free coherent sheaves. \par Suppose $V$ is strictly $H$-semi-stable. Then following Harder-Narishimhan filtration on semi-stable sheaves, we have that $V$ sits in an exact sequence $$\exact{L}{V}{L'}$$ with $$\chi(L(nH))={1\over 2}\chi(V(nH)).$$ This exact sequence needs not to be unique but $gr V=L\oplus L'$ is uniquely determined by $V$. We say that two strictly semi-stable bundles $V$ and $V'$ are $s$-equivalent if $grV=grV'$ (see \cite{Gi}). \par Throughout this paper, we use ${\cal M}_H(c_1, c_2)$, or ${\cal M}_H$ if the Chern classes are obvious from the context, to represent the moduli space of $H$-semi-stable sheaves $V$ over $X$ with $c_1(V)=c_1$ and $c_2(V)=c_2$. That is, ${\cal M}_H$ is the set of $H$-semi-stable sheaves modulo $s$-equivalence. Gieseker \cite{Gi} showed that ${\cal M}_H$ is a projective scheme. We use ${\cal M}^\mu_H(c_1, c_2)$ (or ${\cal M}^\mu_H$) to represent M-T $H$-stable vector bundles $V$ with $c_1(V)=c_1$ and $c_2(V)=c_2$. \section{Walls and Chambers}\label{s:chambers} \begin{defn} The K\"ahler cone $C_X$ of $X$ is the closed convex cone in ${\rm Num}(X)\otimes {\Bbb R}$ spanned by ample divisors. \end{defn} For the purpose of comparing moduli spaces for varying polarizations, we will introduce certain walls in the K\"ahler cone $C_X$. These walls arise naturally from semi-stability. Let $V$ be a rank 2 torsion-free coherent sheaf and $L$ be a subsheaf of rank 1. By Riemann-Roch formula, we have $$\chi(V(nH))=\chi(V)+n^2H^2-nH\cdot K_X+nH\cdot c_1(V),$$ $$\chi(L(nH))=\chi(L)+{n^2\over 2}H-{n\over 2}H\cdot K_X+nH\cdot c_1(L).$$ \par Hence $$2\chi(L(nH))-\chi(V(nH))=(2\chi(L)-\chi(V))+n(2c_1(L)-c_1(V))\cdot H.$$ Therefore we obtain the following: \par ($i$) $V$ is $H$-stable if and only if for any given subsheaf $L$ one of the following holds: \par {}~(1) $(2c_1(L)-c_1(V))\cdot H < 0$; \par {}~(2) $(2c_1(L)-c_1(V))\cdot H = 0$ but $2\chi(L)- \chi(V) <0$. \par ($ii$) Likewise, $V$ is strictly $H$-semi-stable if and only if for any given subsheaf $L$ (1) or (2) of the above holds except that for some subsheaves $L$, we have $(2c_1(L)-c_1(V))\cdot H = 0$ and $2\chi(L)- \chi(V) = 0$. \par In fact, in the above, we can always assume that the cokernel $V/L$ is torsion free. In particular, if $V$ is strictly $H$-semi-stable, $V$ sits in an exact sequence \begin{equation}\label{e:c1} \exact{L}{V}{L'} \end{equation} with $(2c_1(L) - c_1(V) )\cdot H = 0$ and $2\chi(L)=\chi(V)$. Clearly, that $V$ is M-T $H$-stable implies that $V$ is $H$-stable. The converse is not true, however. Notice that $V$ is strictly M-T $H$-semi-stable if and only if for some subsheaf $L$, $(2c_1(L)-c_1(V))\cdot H = 0$, while in the Gieseker case we need to require that $2\chi(L)- \chi(V) = 0$. So the Gieseker stability is finer than M-T stability. This is the main feature that distinguishes the variation problem of Gieseker's stability from that of M-T stability. \begin{defn}\label{t:c1} Let $\tau\in {\rm Num}(X)$ be of the form $2c_1(L) - c_1$ where $L$ is a rank 1 sheaf. Assume further that $-c\le \tau^2<0$ where $c$ is a fixed positive number. We define the hyperplane of type $\tau$ as $$W^{\tau}=\{\,h\in C_X |\,\tau\cdot h=0\}$$ $W$ is called a $c$-wall (or just a wall). \end{defn} \par Let ${\cal W}$ be the set of $c$-walls in $C_X$. It can be shown that for fixed $c$, the $c$-walls are locally finite. Following \cite{Q1}, we give the following definition. \begin{defn}\label{t:c2} A $c$-chamber (or just a chamber) $C$ in $C_X$ is a connected component of the complement, $C_X - \bigcup_{W \in {\cal W}} W$, of the union of the $c$-walls. A wall $W$ is called a wall of a chamber $C$ if $W \cap \overline{C}$ contains a non-empty open subset of $W$. In this case, The relative interior of $W \cap \overline{C}$ is an {\it open face} (or just {\it face}) of $C$. If $F$ is a face of $C$, then there is unique chamber $C' \ne C$ which also has $F$ as a face. In this case, the chamber $C$ and $C'$ lie on opposite sides of the wall containing the common face $F$. \end{defn} It is obvious that each chamber (and each of its faces) is a convex cone in $C_X$. In fact, it is a polyhedral cone if its closure is contained entirely (except for the origin) in the interior of $C_X$. \par Now we fix $c=4c_2-c_1^2$, once and for all. \par Suppose $C$ and $C'$ share the same face $F$ lying on a wall $W^{\tau}$. Then $\tau\cdot H$ is either positive for all $H\in C$ or negative for all $H\in C$. A similar conclusion holds for $H'\in C'$. Thus we may assume that $\tau \cdot C>0$ and $\tau\cdot C'<0$. For simplicity, we shall say that $C$ is the upper chamber and $C'$ is the lower chamber. In many places of this paper, we shall use $H$ ($H'$) to represent an ample divisor in the chamber $C$ ($C'$) respectively, $H_0$ to represent an ample divisor on the face $F$, and $\widetilde H$ to represent an arbitrary ample divisor. \par The following proposition partially justifies the definition of chambers. \par \begin{prop}\label{p:c1} Let $C$ be a chamber in the K\"ahler cone. Then ${\cal M}_H={\cal M}_{H_1}$ for any two $H, H_1 \in C$. \end{prop} \begin{pf} We shall prove this proposition by producing contradiction. Without loss of generality, assume that there exists $V\in {\cal M}_H\backslash {\cal M}_{H_1}$. Then there exists an exact sequence \equref{e:c1} such that either we have $2c_1(L)\cdot H_1 > c_1(V)\cdot H_1$ or we have $2c_1(L)\cdot H_1 = c_1(V)\cdot H_1$ but $2\chi(L)>\chi(V)$. That is, if setting $\tau=2c_1(L)-c_1(V)$, then $\tau\cdot H_1\ge 0$ and \begin{equation}\label{e:c2} \hbox{if }\tau\cdot H_1 = 0, \hbox{ then } 2\chi(L)>\chi(V). \end{equation} Since $V$ is $H$-semi-stable, we must have $\tau\cdot H\le 0$ and \begin{equation}\label{e:c3} \hbox{if }\tau\cdot H=0, \hbox{ then } 2\chi(L)\le \chi(V). \end{equation} Clearly \equref{e:c2} and \equref{e:c3} cannot hold simultaneously. Therefore $\tau$ cannot be numerically trivial. \par Now we choose $H_2=(-\tau\cdot H)H_1+(\tau\cdot H_1)H$. Obviously, $\tau \cdot H_2=0$. Because the chamber $C$ is a convex cone, $-\tau\cdot H\ge 0$ and $ \tau\cdot H_1 \ge 0$, we obtain that $H_2$ is also an ample line bundle in $C$. Since $\tau$ is not numerically trivial, by Hodge index theorem, $\tau^2<0$. \par On the other hand, if we calculate the Chern classes from the exact sequence \equref{e:c1}, we will get $$ c_2(V)=c_1(L)\cdot(c_1(V)-c_1(L))+c_2(L)+c_2(L') \ge c_1(L)\cdot (c_1(V)-c_1(L)).$$ After some simplifications, we get $$\tau^2\ge c_1(V)^2-4c_2(V)=-c.$$ So $\tau$ defines a $c$-wall. Hence $H_2$ is in the chamber $C$ as well as on the $c$-wall $W^\tau$, a contradiction. \end{pf} \section{Variation of ${\cal M}_H$ for different polarizations}\label{s:p} Let $C$ and $C'$ be two chambers with a common face $F \subset W^\tau$. In this section, we will compare the moduli spaces ${\cal M}_H$, ${\cal M}_{H'}$ and ${\cal M}_{H_0}$ where $H \in C, H' \in C'$, and $H_{0} \in F$. \begin{defn}\label{t:p1} $V$ is universally stable (semi-stable) if $V$ is stable (semi-stable) with respect to any polarization. \end{defn} In this section, we will investigate what kind of $H$-stable vector bundles are not $H_0$-semi-stable and so on. We will have a series of propositions of similar nature. \begin{prop}\label{p:p8} Let $V$ be a $H$-semi-stable sheaf of rank 2. Suppose that $V$ is not universally semi-stable. Then $V$ must be $H$-stable. In another word, every semi-sable sheaves in ${\cal M}_H$ is $H$-stable unless it is universally semi-stable. \end{prop} \begin{pf} Suppose $V$ is $H$-semi-stable, only two cases may happen: $V$ is M-T $H$-stable, or $V$ is strictly M-T $H$-semi-stable. If $V$ is M-T $H$-stable, then it must be $H$-stable. So assume that $V$ is strictly M-T $H$-semi-stable. Then there exists a subsheaf $L\subset V$ such that $V$ sits in the exact sequence \equref{e:c1} with $2c_1(L)\cdot H=c_1(V)\cdot H$. If $\tau = 2c_1(L)-c_1(V)$ is not numerically trivial, then by Hodge index theorem, $\tau^2<0$ and $\tau^2\ge c_1(V)^2-4c_2(V)=-c$. Hence $\tau$ defines a $c$-wall and $H$ lies on the wall. This contradicts to the assumption that $H\in C$. \par Hence we must have $2c_1(L)-c_1(V)$ is numerically trivial. Therefore $(2c_1(L)-c_1(V)) \cdot \tilde{H} =0$ for any ample divisor $\tilde{H}$, in particular, for the ample divisor $H$. Since $V$ is $H$-semi-stable, we must have $2\chi(L) \le \chi(V)$. From here, we will show that $V$ is $\tilde{H}$- semi-stable for any $\tilde{H}$. \par In fact, assume that $M$ is a subsheaf of $V$. If $M$ is a subsheaf of $L$, then $c_1(M)\cdot \tilde{H}\le c_1(L)\cdot \tilde{H}$ and $\chi(M)\le \chi(L)\le \displaystyle{1\over 2}\chi(V)$. Otherwise, $M$ admits an injection $M\hookrightarrow L'$. Either $c_1(L')-c_1(M)$ is effective, or $c_1(L')=c_1(M)$. In the first case, we obtain $c_1(M)\cdot \tilde{H}<c_1(L)\cdot \tilde{H} =\displaystyle{1\over 2}c_1(V)\cdot \widetilde H$. In the latter case, i.e., $c_1(L')=c_1(M)$, we take double dual of the exact sequence \equref{e:c1}, we get \begin{equation}\label{e:p0} 0\rightarrow L^{**}\rightarrow V^{**}\rightarrow L^{'**}I_Z\rightarrow 0, \end{equation} $$V^{**}\hookleftarrow M^{**}=L^{'**}.$$ Hence $\ell(Z)=0$ and the exact sequence \equref{e:p0} splits, i.e. $$V^{**}=L^{'**}\oplus L^{**}.$$ Therefore, the exact sequence \equref{e:c1} splits, i.e. $V=L \oplus L'$. Since $2c_1(L)\cdot H=c_1(V)\cdot H=2c_1(L')\cdot H$ and $V$ is $H$-semi-stable, we must have $2\chi(L)=\chi(V)=2\chi(L')$. Hence $2c_1(M)\cdot \tilde{H}=c_1(V)\cdot \tilde{H}$ and $2\chi(M)\le 2\chi(L')=\chi(V)$. This implies that $V$ is $\tilde{H}$-semi-stable. That is, $V$ is a universally semi-stable sheaf. But this contradicts to the assumption that $V$ is not universally semi-stable. \end{pf} \begin{rk}\label{r:p2} The argument in the proof above to show that the exact sequence \equref{e:c1} splits will be used (or referred) later on. \end{rk} \begin{cor}\label{p: c1} Suppose $2c_1(L)-c_1(V)$ is numerically trivial, then any non-splitting exact sequence \equref{e:c1} gives a universally semi-stable sheaf provided $2\chi(L)\le\chi(V)$. \end{cor} \begin{prop}\label{p:p2} Let $V$ be strictly $\tilde{H}$-semi-stable sitting in an exact sequence \equref{e:c1} with $2c_1(L)\cdot \tilde{H} =c_1(V)\cdot \tilde{H}$ and $2\chi(L)=\chi(V)$. \par ($i$) If the exact sequence \equref{e:c1} doesn't split, then the subsheaf $L$ satisfying $2c_1(L)\cdot \tilde{H} =c_1(V)\cdot \tilde{H}$ and $2\chi(L)=\chi(V)$ is unique. \par ($ii$) Any $V$ sitting in \equref{e:c1} satisfying $2c_1(L)\cdot \tilde{H} =c_1(V)\cdot \tilde{H}$ and $2\chi(L)=\chi(V)$ is strictly $\tilde{H}$-semi-stable. \end{prop} \begin{pf} We only need to show the uniqueness of $L$. \par Suppose otherwise, we have two exact sequences $$\exact{L}{V}{L'},$$ $$\exact{M}{V}{M'}$$ satisfying $2c_1(L)\cdot \tilde{H}=2c_1(M)\cdot \tilde{H}=c_1(V)\cdot \tilde{H}$, $2\chi(L)=2\chi(M)=\chi(V)$ and $M\ne L$. \par If $M$ is a subsheaf of $L$, since $\chi(L)=\chi(M)$ and $c_1(L)\cdot \tilde{H}=c_1(M)\cdot \tilde{H}$, then $L=M$, a contradiction. \par Hence $M$ admits an injection into $L'$. By the similar argument as mentioned in \rkref{r:p2}, the exact sequence splits: $V=L\oplus L'$ and $M=L'$. But we have assume that the sequence \equref{e:c1} does not split. \end{pf} {}From now on, we will be mainly concentrating on non-universally semi-stable sheaves. \begin{thm}\label{p:p3} Let $V$ be a non-universally semi-stable sheaf of rank 2. Assume $V$ is $H_0$-stable. Then one of the following holds. \par ($i$) If $V$ is M-T $H_0$-stable, then $V$ is M-T $H$-stable as well as M-T $H'$-stable. In particular, $V$ is $H$-stable as well as $H'$-stable. \par ($ii$) If $V$ is not M-T $H_0$-stable, then $V$ is either $H$-stable or $H'$-stable, but cannot be both. \end{thm} \begin{pf} Since $V$ is $H_0$-stable, then for subsheaf $L\subset V$ with torsion free cokernel, there exists an exact sequence \equref{e:c1} such that either $2c_1(L)\cdot H_0< c_1(V)\cdot H_0$ or $2c_1(L)\cdot H_0= c_1(V)\cdot H_0$ and $2\chi(L)<\chi(V)$. \par If $(2c_1(L)-c_1(V)\cdot H_0<0$, then $(2c_1(L)-c_1(V)\cdot H<0$ and $(2c_1(L)-c_1(V)\cdot H'<0$. Otherwise there would exist a $c$-wall separating $H_0$ with $H$ or $H_0$ with $H'$. \par If $2c_1(L)\cdot =c_1(V)\cdot H_0$, then $2\chi(L)<\chi(V)$. Since we assumed that $V$ is not universally stable, hence $2c_1(L)-c_1(V)$ is not numerically trivial. Therefore $2c_1(L)-c_1(V)$ defines the $c$-wall where $H_0$ lies. Hence $$\hbox{either}\qquad(2c_1(L)-c_1(V))\cdot H>0\qquad\hbox{ or }\qquad (2c_1(L)-c_1(V))\cdot H<0.$$ \par Since $V$ is $H_0$-stable, the exact sequence \equref{e:c1} doesn't split and subsheaf $L$ satisfying $2c_1(L)\cdot =c_1(V)\cdot H_0$ is unique. \par Assume $(2c_1(L)-c_1(V))\cdot H<0$. For any subsheaf $M$ of $V$, if $M$ is a subsheaf of $L$, we have $2c_1(M)\cdot H<c_1(V)\cdot H$. Otherwise, $M$ admits an injection into $L'$. \par If $c_1(L')-c_1(M)$ is an effective divisor, then $2c_1(M)\cdot H_0<2c_1(L')\cdot H_0=c_1(V)\cdot H_0$. Hence $(2c_1(M)-c_1(V))\cdot H_0<0$. Therefore $(2c_1(M)-c_1(V))\cdot H<0$. \par If $c_1(L')=c_1(M)$, then by the argument mentioned in \rkref{r:p2}, the exact sequence \equref{e:c1} splits. Since $V$ is $H_0$-stable, we get a contradiction. Hence $2c_1(M)\cdot H\cdot <c_1(V)\cdot H$. Therefore $V$ is M-T $H$-stable. \par Assume $(2c_1(L)-c_1(V))\cdot H>0$, then $(2c_1(L)-c_1(V))\cdot H'<0$, hence by the similar argument $V$ is M-T $H'$-stable. \par The proof of ($i$) and ($ii$) will follow easily. For example, for ($ii$), if $V$ is not M-T $H_0$-stable, there will exist subsheaf $L\subset V$ such that $2c_1(L)\cdot H_0=c_1(V)\cdot H_0$. Hence if $2c_1(L)\cdot H<c_1(V)\cdot H$, then $2c_1(L)\cdot H'> c_1(V)\cdot H'$. In other words, if $V$ is $H$-stable, then $V$ cannot be $H'$ stable and vice-versa. \end{pf} \begin{thm}\label{p:p4} Let $V$ be a sheaf of rank 2 which is not universally semi-stable. Assume that $V$ is strictly $H_0$-semi-stable and sits in the non-splitting exact sequence \equref{e:c1}. Then $V$ is either $H$-stable or $H'$-stable, but can not be both. If the exact sequence \equref{e:c1} splits, then $V$ is neither $H$-stable nor $H'$-stable. \end{thm} \begin{pf} Assume that $V$ is not $H$-stable nor $H'$-stable. Then there exist two exact sequences \begin{equation}\label{e:p1} \exact{N}{V}{N'} \end{equation} \begin{equation}\label{e:p2} \exact{M}{V}{M'} \end{equation} such that $2c_1(N)\cdot H\ge c_1(V)\cdot H$ and $2c_1(M)\cdot H'\ge c_1(V)\cdot H'$. Hence \begin{equation}\label{e:p3} 2c_1(N)\cdot H_0\ge c_1(V)\cdot H_0\hbox{ and }2c_1(M)\cdot H_0\ge c_1(V)\cdot H_0 \end{equation} \par Since $V$ is not universally semi-stable, it is easy to show that $2c_1(N)\cdot H>c_1(V)\cdot H$ and $2c_1(M)\cdot H'>c_1(V)\cdot H'$. Since $V$ is strictly $H_0$-semi-stable, we must have \begin{equation}\label{e:p4} 2c_1(N)\cdot H_0\le c_1(V)\cdot H_0\hbox{ and }2c_1(M)\cdot H_0\le c_1(V)\cdot H_0 \end{equation} Combining \equref{e:p3} and \equref{e:p4}, we must have $$2c_1(N)\cdot H_0=c_1(V)\cdot H_0=2c_1(M)\cdot H_0.$$ \par Hence \begin{equation} 2c_1(N)\cdot H'< c_1(V)\cdot H'\hbox{ and }2c_1(M)\cdot H< c_1(V)\cdot H \end{equation} Since $V$ is not universally stable, we have \begin{equation}\label{e:p5} 2c_1(L)\cdot H\ne c_1(V)\cdot H\hbox{ and }2c_1(L)\cdot H'\ne c_1(V)\cdot H' \end{equation} If $2c_1(L)\cdot H<c_1(V)\cdot H$, then $N$ cannot be a subsheaf of $L$, since otherwise, we would have $2c_1(L)\cdot H\ge 2c_1(N)\cdot H>c_1(V)\cdot H 2c_1(L)\cdot H$, a contradiction. Hence $N$ admits an injection to $L'$. Then by the argument mentioned in \rkref{r:p2}, the exact sequence \equref{e:c1} splits, a contradiction. \par If $2c_1(L)\cdot H>c_1(V)\cdot H$, then $L$ cannot be a subsheaf of $M$, since otherwise, we would have $c_1(V)\cdot H>2c_1(M)\cdot H\ge 2c_1(L)\cdot H>c_1(V)\cdot H$, a contradiction. Hence $L$ admits an injection to $M'$. Then by the argument mentioned in \rkref{r:p2}, the exact sequence \equref{e:c1} splits, a contradiction. \par Hence we proved that $V$ is either $H$-stable or $H'$-stable. It is easy to see that either $2c_1(L)\cdot H>c_1(V)\cdot H$ which implies that $V$ is not $H$-stable, or $2c_1(L)\cdot H<c_1(V)\cdot H$, equivalently, $2c_1(L)\cdot H'>c_1(V)\cdot H'$, which implies that $V$ is not $H'$-stable. \par If the exact sequence \equref{e:c1} splits, by the same argument as in the previous paragraph, $V$ is neither $H$-stable nor $H'$-stable. \end{pf} Next, we give a criterion for strictly $H_0$-semi-stable sheaves. \begin{prop}\label{p:p5} Assume that $V$ is not universally semi-stable. $V$ is strictly $H_0$-semi-stable iff $V$ sits in the exact sequence \equref{e:c1} where $c_1(L)\cdot H_0=c_1(L')\cdot H_0$ and $\chi(L)=\chi(L')$. \end{prop} \begin{pf} Easy. \end{pf} \begin{defn}\label{p:d2} Let $L$, $L'$ be two rank one torsion free coherent sheaves such that $c_1(L)+c_1(L')=c_1$, $c_1(L)\cdot c_1(L')+c_2(L)+c_2(L')=c_2$. If $c_1(L)-c_1(L')$ is numerically trivial, then the pair ($L$, $L'$) is called $U$-pair. Otherwise, it is called NU-pair. \end{defn} \begin{prop}\label{p:p6} Suppose ($L$, $L'$) is a NU-pair, $c_1(L)\cdot H_0 =c_1(L')\cdot H_0$ and $2c_1(L)\cdot H<c_1\cdot H$. Then \par ($i$) every non-splitting exact sequence \begin{equation}\label{e:p7} \exact{L}{V}{L'} \end{equation} gives an $H$-stable sheaf $V$. \par ($ii$) $V$ is $H_0$-stable if $\chi(L)<\chi(L')$; \par ($iii$) $V$ is strictly $H_0$-semi-stable if $\chi(L)=\chi(L')$; \par ($iv$) $V$ is $H_0$-unstable if $\chi(L)>\chi(L')$. \end{prop} \begin{pf} Consider a subsheaf $M\subset V$. If $M$ is a subsheaf of $L$, then $$2c_1(M)\cdot H\le 2c_1(L)\cdot H< c_1(V)\cdot H.$$ Otherwise, $M$ admits an injection into $L'$. Hence $$2c_1(M)\cdot H_0\le 2c_1(L')\cdot H_0=c_1(V)\cdot H_0.$$ \par If $2c_1(M)\cdot H_0=c_1(V)\cdot H_0=2c_1(L')\cdot H_0$, by the argument mentioned in \rkref{r:p2}, the exact sequence \equref{e:p7} splits, a contradiction. \par Hence we must have $2c_1(M)\cdot H_0<c_1(V)\cdot H_0$. Hence $2c_1(M)\cdot H<c_1(V)\cdot H$. \par The proves of (ii), (iii) and (iv) are quite straight forward. \end{pf} \begin{prop}\label{p:p7} Assume that $V$ is not universally semi-stable. \par ($i$) If $V$ is $H$-stable and strictly M-T $H_0$-semi-stable sitting in the exact sequence \equref{e:c1} with $c_1(L)\cdot H_0=c_1(L')\cdot H_0$, then any sheaf $V'$ sitting in the non-splitting exact sequence \begin{equation}\label{e:p6} \exact{L'}{V'}{L} \end{equation} is $H'$-stable. \par ($ii$) If in addition, $V$ is not $H_0$-semi-stable, then $V'$ is $H_0$-semi-stable. \end{prop} \begin{pf} Since $V$ is not universally semi-stable, the pair ($L$, $L'$) is an NU-pair. Because $V$ is $H$-stable and strictly M-T $H_0$-semi-stable, we obtain $2c_1(L)\cdot H < c_1\cdot H$ and $2c_1(L)\cdot H_0 = c_1 \cdot H_0$. Thus we get $2c_1(L) \cdot H' > c_1\cdot H'$, or equivalently, $2c_1(L') \cdot H'< c_1 \cdot H'$. Apply \propref{p:p6} to the pair ($L'$, $L$), we get the conclusion. \end{pf} \begin{prop}\label{p:p9} Fix the first Chern class $c_1$ and a $c$-wall $W^{\tau}$ where $c=4c_2-c_1^2$. For $c_2'\ge c_2$, $W^{\tau}$ is also a $c'$-wall where $c'=4c_2'-c_1^2$. Suppose that $\tau\cdot H_0=0$, $\tau\cdot H<0$ and $\tau+c_1=2c_1(L)$ for some line bundle $L$. Then for $c_2'\gg0$, there exists a $H$-stable sheaf $V$ with $c_1(V)=c_1$ and $c_2(V)=c_2'$ such that it is not $H_0$-semi-stable. \end{prop} \begin{pf} When $c_2(L')\gg 0$ , $Ext^1(L', L)\ne 0$. Hence there exists a non-splitting exact sequence $$\exact{L}{V}{L'}$$ where $L'$ is a rank one subsheaf such that $c_1(L')=c_1-c_1(L)$ and $c_2(L')=c_2'-c_2(L)-c_1(L)\cdot c_1(L')$. From \propref{p:p6}, $V$ is $H$-stable with $c_2(V)=c_2'$ and $c_1(V)=c_1$. \par $$2\chi(L)=c_1^2(L)-c_1(L)\cdot K_X+2\chi({\cal O}_X),$$ $$\chi(V)=\displaystyle{c_1^2-c_1\cdot K_X\over 2}-c_2'+2\chi({\cal O}_X).$$ If $c_2'\gg 0$, then $2\chi(L)>\chi(V)$. Since $(2c_1(L)-c_1(V))\cdot H_0=0$, hence $V$ is not $H_0$-semi-stable. \end{pf} \par In some propositions above, one may have noticed that we have used the term ``non-splitting exact sequence'' several times. Suppose we have an exact sequence \equref{e:c1} with $c_1(L)\cdot H_0=c_1(L')\cdot H_0$ and $\chi(L)=\chi(L')$, then $V$ is strictly $H_0$-semi-stable and is $s$-equivalent to $gr(V)=L\oplus L'$. However, if $(L, L')$ is a NU-pair and if $V=L\oplus L'$, then $V$ is neither $H$-stable nor $H'$-stable. Therefore if $Ext^1(L', L)=0$, then there would exist a class in ${\cal M}_{H_0}$ represented by $V=L\oplus L'$ such that $V$ is not s-equivalent to any $H_0$-semi-stable sheaf which is $H$-semi-stable, nor is it s-equivalent to any $H_0$-semi-stable sheaf which is $H'$-semi-stable. In the following, we are going to show that the situation above cannot happen. This fact guarantees (ii) of our main theorem in the next section. \par Let ${\cal F}$ and ${\cal F}'$ be two torsion free coherent sheaves. Define (see \cite{Q2}) $$\chi({\cal F}',{\cal F})=\sum^2_{i=0}{\rm dim}Ext^i({\cal F}', {\cal F}).$$ \begin{prop}\label{p:n1} $$\chi({\cal F}',{\cal F})=ch({\cal F}')^*\cdot ch({\cal F})\cdot td(X)_{H^4 (X;{\Bbb Z})}$$ where $*$ acts on $H^{2i}(X;{\Bbb Z})$ by $(-1)^i\cdot Id$. \end{prop} \begin{cor}\label{c:n1} Let $L$ and $L'$ be two torsion free rank one sheaves. Let $\tau=c_1(L)-c_1(L') $. Let $V$ sit in the exact sequence \equref{e:c1}. Then \begin{equation}\label{e: n1} \chi(L', L)={\tau^2\over 4} -{K_X\cdot \tau\over 2} -c_2(V)+{c_1^2(V)\over 4}+ \chi({\cal O}_X) \end{equation} \end{cor} \begin{pf} {}From the exact sequence \equref{e:c1}, we get $\tau = 2c_1(L)-c_1(V)$ and $c_2(L)+c_2(L')=c_2(V)-c_1(L)\cdot c_1(L')$. $$\begin{array}{ll} &\chi(L', L)=ch(L')^*\cdot ch(L)\cdot td(X)\\ =&(1-c_1(L')+\displaystyle {c_1(L')^2\over 2} -c_2(L'))\cdot (1+c_1(L)+ \displaystyle{ c_1(L)^2\over 2} -c_2(L)) \\ &~~\cdot (1-\displaystyle{ 1\over 2}K_X+\chi({\cal O}_X))_{H^4}\\ =&(1+c_1(L)-c_1(L')-c_1(L)\cdot c_1(L')+\displaystyle{ c_1(L)^2\over 2} + \displaystyle{ c_1(L')^2\over 2} -c_2(L)-c_2(L'))\\ &~~\cdot (1-\displaystyle{ 1\over 2}K_X+\chi({\cal O}_X))_{H^4}\\ =&\displaystyle{ (c_1(L)-c_1(L'))^2\over 2}- \displaystyle{ K_X\cdot (c_1(L)-c_1(L'))\over 2}-c_2(L)-c_2(L') +\chi({\cal O}_X)\\ =&\displaystyle{ \tau^2\over 2}-\displaystyle{ K_X\cdot \tau\over 2}-c_2(V)+ (c_1(V)-c_1(L))\cdot c_1(L)+\chi({\cal O}_X)\\ =&\displaystyle{ \tau^2\over 2}-\displaystyle{ K_X\cdot \tau\over 2} -c_2(V)+\displaystyle{ c_1(V)^2-\tau^2\over 4}+\chi({\cal O}_X)\\ =&\displaystyle{ \tau^2\over 4}-\displaystyle{ K_X\cdot \tau\over 2} -\displaystyle{ 4c_2(V)-c_1(V)^2-3\chi({\cal O}_X)\over 4} + \displaystyle{ \chi({\cal O}_X)\over 4}\\ =&\displaystyle{\tau^2\over 4} -\displaystyle{ K_X\cdot \tau\over 2} - \displaystyle{ d\over 4}+\displaystyle{ \chi({\cal O}_X)\over 4} \end{array}$$ where $d=4c_2(V)-c_1(V)^2-3\chi({\cal O}_X)$. \end{pf} \begin{prop}\label{p:n2} With assumption on $L$ and $L'$ as in \propref{c:n1}. In addition, assume that $\tau\cdot H_0=0$ and $d$, which is the virtual dimension of the moduli space, is non-negative, then $$-\chi(L, L')-\chi(L', L)>0.$$ \end{prop} \begin{pf}Since ($L$, $L'$) is a NU-pair, $\tau$ is not numerically trivial, hence $\tau^2<0$. {}From \corref{c:n1}, $$-\chi(L, L')-\chi(L', L)={d\over 2}+{-\tau^2-\chi({\cal O}_X)\over 2} ={d\over 2} +{-\tau^2-1+q\over 2}\ge {d\over 2}\ge 0.$$ Suppose $-\chi(L, L')-\chi(L', L)=0$, then $q=0, \tau^2=-1, 4c_2(V)-c_1^2(V)=3$. \par {}From the exact sequence \equref{e:c1}, we get $$-\tau^2+4c_2(L)+4c_2(L')=4c_2(V)-c_1^2(V)=3.$$ Hence $1+4(c_2(L)+c_2(L'))=3$, or $4(c_2(L)+c_2(L'))=2$, impossible. \end{pf} \begin{cor}\label{c:n2} ${\rm dim}Ext^1(L, L')+{\rm dim}Ext^1(L', L)>0$. And there either exists a non-splitting exact sequence $$\exact{L}{V}{L'},$$ or a non-splitting exact sequence $$\exact{L'}{V}{L}.$$ \end{cor} \begin{pf} Easy consequence of \propref{p:n2}. \end{pf} \begin{cor}\label{c:n3} Suppose ($L$, $L'$) is a NU-pair satisfying $c_1(L)\cdot H_0=c_1(L')\cdot H_0$, $\chi(L)=\chi(L')$, then there exists a non-splitting exact sequence $$\exact{L}{V}{L'}\qquad\hbox{or}\qquad \exact{L'}{V}{L}$$ such that $V$ is strictly $H_0$-semi-stable. \end{cor} \begin{pf} An easy consequence of \corref{c:n2} and \propref{p:p5}. \end{pf} \section{Canonical rational morphisms among the Gieseker compactifications}\label{s:g} In this section, we shall draw some conclusions on the variations of Gieseker compactifications following many discussions in the previous section. \par Again, we place ourselves in the following situation. Let $C$ and $C'$ be two chambers with a common face $F \subset W^\tau$. We assume that $C$ is the upper chamber and $C'$ is the lower chamber with respect to $\tau$. That is, $C \cdot \tau > 0$ and $C' \cdot \tau <0$. We will derive some canonical morphisms among the moduli spaces ${\cal M}_H$, ${\cal M}_{H'}$ and ${\cal M}_{H_0}$ where $H \in C, H' \in C'$, and $H_{0} \in F$. \begin{thm}\label{t:n1} There are two {\it canonical} rational algebraic maps $${\cal M}_H \buildrel{\varphi}\over{-->} {\cal M}_{H_0} \buildrel{\psi}\over{<--} {\cal M}_{H'}$$ with the following properties: \par ($i$) $\varphi$ and $\psi$ are isomorphisms over ${\cal M}^\mu_{H_0} \subset {\cal M}_{H_0}$. \par ($ii$) $Im\varphi \cup Im\psi={\cal M}_{H_0}$. \par ($iii$) If $V\in {\cal M}_{H_0}$ is universally semi-stable, then the inverse image $\varphi^{-1}(grV)$ or $\varphi^{-1}(V)$ consists of a single point. The same conclusion holds for $\psi$. \par($iv$) If $V\in {\cal M}_{H_0}$ is not universally semi-stable and is $H_0$-stable, then the inverse image $\varphi^{-1}(V)$ ($\psi^{-1}(V)$) consists of a single point or is an empty set; More precisely, \par~~~~($a$) If $V$ is MT $H_0$-stable, then $\varphi^{-1}(V)$ ($\psi^{-1}(V)$) consists of a single point; \par~~~~($b$) If $V$ is strictly MT $H_0$-semistable, then $V$ sits in the exact sequence \equref{e:c1}; \par~~~~~~~~($b1$) If $\chi(L)>\chi(L')$, $\varphi$ is not defined over ${\Bbb P}(Ext^1(L, L')) \subset {\cal M}_H$, but $\psi$ sends ${\Bbb P}(Ext^1(L', L)) \subset {\cal M}_{H'}$ injectively to ${\cal M}_{H_0}$; \par~~~~~~~~($b2$) If $\chi(L) < \chi(L')$, $\psi$ is not defined over ${\Bbb P}(Ext^1(L', L)) \subset {\cal M}_{H'}$, but $\varphi$ sends ${\Bbb P}(Ext^1(L, L')) \subset {\cal M}_H$ injectively to ${\cal M}_{H_0}$. \par ($v$) If $V\in {\cal M}_{H_0}$ is not universally semi-stable and is strictly $H_0$-semi-stable, then $V$ sits in the exact sequence \equref{e:c1} with $\chi(L)=\chi(L')$, the inverse image of $grV\in {\cal M}_{H_0}$ by $\varphi$ is ${\Bbb P}(Ext^1(L, L'))$, and the inverse image of $grV\in {\cal M}_{H_0}$ by $\psi$ is ${\Bbb P}(Ext^1(L', L))$. \end{thm} \begin{pf} First of all, for a given sheaf $V$ in ${\cal M}_H$, if $V$ is also a semi-stable sheaf with respect to $H_0$, then we can define a map which sends $V \in {\cal M}_H$ (or $grV$) to $V$ (or $grV$) as a point in ${\cal M}_{H_0}$. It is easy to see that this gives rise to a well-defined map $\varphi$ from a Zariski open subset of ${\cal M}_H$ to ${\cal M}_{H_0}$. Obviously, $\varphi$ is defined and restricts to the identity over ${\cal M}^\mu_{H_0} \subset {\cal M}^\mu_H \subset {\cal M}_H $. It remains to show the algebraicity of the map $\varphi$. The proof is a standard one. So we only brief it. Recall from the construction of the moduli space ${\cal M}_H$ (see \cite{Gi}), ${\cal M}_H$ is the quotient of ${\cal Q}_H^{ss}$ by the group $PGL(N)$ (we adopt the notations from \cite{Li}). By the universality of the quotient scheme, there is a universal quotient sheaf ${\cal F}$ over $X \times {\cal Q}_H^{ss}$ with the usual property. Now by the axiom of the coarse moduli, there is a rational map from ${\cal Q}_H^{ss}$ to ${\cal M}_{H_0}$. Clearly this map respects the group action (send an orbit of $PGL(N)$ to a point by Proposition 4.2), thus by passing to the quotient, we get a rational map from ${\cal M}_H$ to ${\cal M}_{H_0}$, and this map is by definition the map $\varphi$. Hence $\varphi$ is a morphism. \par The other map $\psi$ can be treated similarly. \par Property ($i$) and ($iii$) follows immediately from the above explanation. \par ($iv $) and ($v$) follow as consequences of \propref{p:p6} and \propref{p:p7}. To prove ($ii$), take a $H_0$-semi-stable sheaf $V$. \par If $V$ is universally semi-stable, the conclusion follows by definition. \par If $V$ is not universally semi-stable but is $H_0$-stable, then the conclusion follows from \propref{p:p3} \par If $V$ is not universally semi-stable and is strictly $H_0$-semi-stable, then the conclusion follows from \corref{c:n3} and \propref{p:p4}. \end{pf} \begin{prop}\label{v:p5} Let the situation be as in \thmref{t:n1}. If $c_2\gg0$, then the map $\varphi\colon {\cal M}_H(c_1, c_2) --> {\cal M}_{H_0}(c_1, c_2)$ and $\psi\colon {\cal M}_{H'}(c_1, c_2) --> {\cal M}_{H_0}(c_1, c_2)$ are genuine rational maps (in the sense that they can not be extended to everywhere). \end{prop} \begin{pf} It follows from \propref{p:p9}. \end{pf} \section{From Gieseker's compactification to Uhlenbeck's compactification}\label{s:u} In this section, we will study the Uhlenbeck compactification of moduli spaces using its relation with the Gieseker compactification. We will use a technique established by Jun Li \cite{Li} where he compared the Gieseker compactification and the Uhlenbeck compactification. We assume that $q=0$ and $c_1=0$ through out this section. Our analysis relies heavily on the results of Jun Li \cite{Li}. \par Following the notations in \cite{Li}, let $H$ be an ample divisor and $g$ the corresponding Hodge metric on $X$. We use $\widetilde {\cal N}_{H}(j)$ to represent the moduli space of ASD connections, with respect to the Riemannian metric $g$, on an $SU(2)$ principal bundle $P$ over $X$ with $c_2(P)=j$, and ${\cal N}_{H}(j)$ to represent the moduli space of irreducible ASD connections. ${\cal N}_{H}(j)$ is known by a Donaldson's theorem to be homeomorphic to the moduli space of Mumford-Takemoto $H$-stable vector bundles with $c_1=0$ and $c_2=j$. We adopt the notation $\overline{\cal N}_H(c_2)$ to represent the Uhlenbeck compactification. The virtual dimension of the moduli space ${\cal M}_H(c_2)$ or $\overline{\cal N}_H(c_2)$ is $d=4c_2-3$. \par Uhlenbeck compactification theorem tells us that $\overline{\cal N}_H(c_2)$ is a closed subset of $\coprod\limits_{j=0}^{c_2}\widetilde{\cal N}_H(j)\times Sym^{c_2-j}(X)$. However, we didn't know whether $\overline {\cal N}_H(c_2)$ is the total space. The main conclusion (\thmref{t:u2}) of this section will give a complete answer to this question. \par In what follows, we shall quote a useful theorem proved by J. Li (cf. Theorem 0.1, \cite{Li}). \begin{thm}\label{t:u1} {\rm \cite{Li}} There is a complex structure on $\overline{\cal N}_H(c_2)$ making it a reduced projective scheme. Furthermore, if we let ${\cal M}_H^{\mu}(c_2)$ be the open subset of ${\cal M}_H(c_2)$ consisting of locally free M-T $H$-stable sheaves and let $\overline{\cal M}_H^{\mu}(c_2)$ be the closure of ${\cal M}_H^{\mu}(c_2)$ in ${\cal M}_H(c_2)$ endowed with reduced scheme structure, then there is a morphism $$\gamma\colon \overline{\cal M}_H^{\mu}(c_2)\longrightarrow \overline{\cal N}_H(c_2)$$ extending the homeomorphism between the set of M-T $H$-stable rank two vector bundles and the set of gauge equivalent classes of irreducible ASD connections with fixed Chern classes. \end{thm} It is known that when $c_2$ is large enough, ${\cal M}_H(c_2)$ is irreducible (\cite{GL}) and thus $\overline{\cal M}_H^{\mu}(c_2) = {\cal M}_H(c_2)$ and $\gamma({\cal M}_{H}(c_2)) = \gamma(\overline{\cal M}_H^{\mu}(c_2))$. In \S 5 of \cite{Li}, a continuous map $$\overline \sigma: \gamma(\overline{\cal M}_H^{\mu}(c_2)) \rightarrow \coprod\limits^{c_2}_{j=0}\widetilde {\cal N}_{H}(j)\times Sym^{c_2-j}X$$ is defined and $\overline \sigma$ identifies $\gamma(\overline{\cal M}_H^{\mu}(c_2))$ isomorphically with the Uhlenbeck compactification $\overline{\cal N}_H(c_2) \subset \coprod\limits^{c_2}_{j=0}\widetilde {\cal N}_{H}(j)\times Sym^{c_2-j}X$. We will use $\widetilde \sigma$ to stand for the map from $\gamma({\cal M}_{H}(c_2))$ to $\coprod\limits^{c_2}_{j=0}\widetilde {\cal N}_{H}(j)\times Sym^{c_2-j}X$ defined in the proof of Theorem 5 in \cite{Li}. If ${\cal M}_{H}(c_2)$ is normal, then this map $\widetilde \sigma$ is simply the map $\overline \sigma$. Otherwise, $\widetilde \sigma$ is an extension of $\overline \sigma$. \par The definition of $\widetilde\sigma$ and $\overline\sigma$ are given in J. Li's paper \cite{Li}. It is recommended that the reader consult J. Li's paper to get familiar with these maps since in this and next section, we make use of these maps a lot. \begin{rk}\label{r:u3} In the rest of the paper, rather than directly working on the Uhlenbeck compactification $\overline{\cal N}_H(c_2)$, we will be working on $\gamma({\cal M}_{H}(c_2))$ instead. One should keep in mind that $\gamma({\cal M}_{H}(c_2))$ can be identified via $\overline \sigma$ with the Uhlenbeck compactification $\overline{\cal N}_H(c_2)$ when ${\cal M}_{H}(c_2)$ is irreducible (and this can be ensured by requiring $c_2$ to be large \cite{GL}). For small $c_2$, $\overline{\cal N}_H(c_2)$ is contained in $\gamma({\cal M}_{H}(c_2))$ via the identification with $\gamma(\overline{\cal M}_H^{\mu}(c_2))$. In this case, $\gamma({\cal M}_{H}(c_2))$ is slightly larger than ${\cal M}_{H}(c_2)$. \end{rk} \begin{notation}\label{n:u1} Let $Z$ be a zero-cycle. $red(Z)$ will be the reduced scheme with multiplicity counted at each point. \end{notation} We are going to prove the following proposition. \begin{prop}\label{p:u1} Assume that $H$ is an ample divisor away from $c$-walls. \par{($i.$)} If $c_2\ge 2$, then ${\rm Im}\widetilde \sigma =\coprod\limits^{c_2}_{j=0}{\cal N}_{H}(j)\times Sym^{c_2-j}X$. \par {($ii.$)} If $c_2=1$, then ${\rm Im}\widetilde \sigma = {\cal N}_{H}(1)$. In particular, $ {\cal N}_{H}(1)$ is compact. \end{prop} In order to prove the proposition, we divide into several lemmas. \begin{lem}\label{l:u1} Suppose $V\in {\cal M}_{H}(c_2)$ is strictly M-T $H$-semi-stable, then $V$ sits in the exact sequence \begin{equation}\label{e:u1} \exact{I_Z}{V}{I_{Z'}} \end{equation} for some zero-cycles $Z$ and $Z'$ such that $$\widetilde \sigma(\gamma(V))=({\cal O}_X\oplus {\cal O}_X,\, red(Z\cup Z')).$$ \end{lem} \begin{pf} Since $V$ is strictly M-T $H$-semi-stable, hence there exist torsion free coherent sheaves of rank one $L$ and $L'$ such that $V$ sits in the exact sequence $$\exact{L}{V}{L'}$$ with $c_1(L)\cdot H=0$. \par Since $H$ is away from $c$-walls, $c_1(L)$ is the trivial divisor by Hodge index theorem. Hence we have the exact sequence \equref{e:u1}. \par By the proof of Lemma 3.3. in \cite{Li}, we get $I_Z\oplus I_{Z'}\in \Gamma(\gamma(V))$. Therefore $$\widetilde \sigma(\gamma(V))=({\cal O}_X\oplus {\cal O}_X,\, red(Z\cup Z')) \in {\cal M}_{H}(0)\times Sym^{c_2}(X). $$ Note that ${\cal M}_{H}(0)$ consists of a single point represented by ${\cal O}_X\oplus {\cal O}_X$. \end{pf} \begin{rk}\label{r:u2} Due to the same reason, the universally semi-stable sheaves can only be sheaves sitting in the exact sequence \equref{e:u1}. \end{rk} \begin{lem}\label{l:u2} Assume that $c_2\ge 2$. For any point $({\cal O}_X\oplus {\cal O}_X, x)$ in ${\cal M}_{H}(0)\times Sym^{c_2}(X)$, choose a zero-cycle $Z$ of length $c_2$ such that $red(Z)=x$. Then there exists a non-splitting exact sequence \begin{equation}\label{e:u2} \exact{I_Z}{V}{{\cal O}_X} \end{equation} such that $V$ is $H$-semi-stable. \end{lem} \begin{pf} Let's calculate ${\rm dim} Ext^1({\cal O}_X, I_Z)=h^1(I_Z)$. $$\begin{array}{ll} h^1(I_Z)&=-\chi(I_Z)+h^0(I_Z)+h^2(I_Z)\\ & \ge -\chi(I_Z)=-(-c_2+1)\ge 1.\\ \end{array}$$ Hence there exists a nonsplitting exact sequence \equref{e:u2}. \par {\it Claim:} $V$ is $H$-semi-stable. \par In fact, let $M$ be a rank one subsheaf of $V$. Notice that $$2\chi(I_Z)=2(-\ell(Z)+1)=-2\ell(Z)+2=-2c_2+2<-c_2+2=\chi(V).$$ If $M$ is a subsheaf of $I_Z$, then $c_1(M)\cdot H\le 0$ and $2\chi(M)\le 2\chi(I_Z)<\chi(V)$. \par Otherwise $M$ is a subsheaf of ${\cal O}_X$. Hence $c_1(M)\cdot H\le 0$. If $c_1(M)\cdot H=0$, again $c_1(M)$ has to be the trivial divisor. Hence the exact consequence \equref{e:u2} splits, a contradiction. Therefore $c_1(M)\cdot H<0$. \par Thus, we proved the claim. \end{pf} \begin{cor}\label{c:u1} Assume that $H$ is an ample divisor away from $c$-walls. Then $${\rm Im}(\widetilde \sigma)\supset {\cal M}_{H}(0)\times Sym^{c_2}(X).$$ \end{cor} \begin{lem}\label{l:u3} Given any element $(A, x)\in \widetilde{\cal N}_{H}(j)\times Sym^{c_2-j}X$ where $j>0$, $A$ is an irreducible ASD corresponding to a M-T $H$-stable bundle $V_j$. In particular $${\cal N}_H(j)=\widetilde{\cal N}_H(j).$$ \par Choose a zero-cycle $Z$ of length $\ell(Z)=c_2-j$ such that $red(Z)=x$. Then the elementary transformation $V$ in $$\exact{V}{V_j}{{\cal O}_Z}$$ is an M-T $H$-stable bundle in ${\cal M}_{H}(c_2)$. \end{lem} \begin{pf} Any reducible ASD in ${\cal N}_{H}(j)$ takes form of $L\oplus L^{-1}$ where $L$ is a line bundle with $c_1(L)\cdot H=0$ and $c_1(L)^2=-j<0$. Since $H$ is away from walls, such $L$ doesn't exist. Hence $A$ corresponds to M-T $H$-stable bundle. The second statement is clear. \end{pf} \begin{cor}\label{c:u2} With assumption and notations as in \lemref{l:u3}. Then $\widetilde\sigma(V)=(A, x)$. \end{cor} \begin{pf} $V^{**}=V_j$. By definition of $\widetilde\sigma$, $\widetilde\sigma(V) =(A, x)$. \end{pf} Combination of \corref{c:u1} and \corref{c:u2} gives a proof of ($i$) of \propref{p:u1}. \begin{lem}\label{l:u4} Assume $c_2=1$. Then there is no $H$-semi-stable sheaf $V$ which is strictly M-T $H$-semi-stable. \end{lem} \begin{pf} Suppose that $V$ is strictly M-T $H$-semi-stable, by \lemref{l:u1}, $V$ sits in the exact sequence \equref{e:u1} with $\ell(Z)+\ell(Z')=c_2=1$. Since $V$ is also $H$-semi-stable, $2\chi(I_Z)=-2\ell(Z)+2\le \chi(V)=1$. Hence $\ell(Z)$ has to be one and $\ell(Z')=0$. However, $$\begin{array}{ll} &{\rm dim}Ext^1({\cal O}_X, I_Z)\\ =&h^1(I_Z)=h^0(I_Z)+h^2(I_Z)-\chi(I_Z)\\ =&h^2({\cal O}_X)-(-\ell(Z)+1)=p_g=0.\\ \end{array} $$ Hence the exact sequence (1) splits, i.e. $V={\cal O}_X\oplus I_Z$. Since $2\chi({\cal O}_X)=2>\chi(V)=1$, $V$ will not be $H$-semi-stable, a contradiction. \end{pf} This lemma proves ($ii$) of \propref{p:u1}. Hence we finished the proof of \propref{p:u1}. \par In the following, we consider the case where our polarization is on a face. \begin{lem}\label{l:u5} Suppose $H_0$ is an ample divisor on a face. Suppose $V\in {\cal M}_{H_0}(c_2)$ is strictly M-T $H_0$-semi-stable, then either $V$ sits in \equref{e:u1} and hence satisfies the conclusion of \lemref{l:u1}, or $V$ sits in the exact sequence \begin{equation}\label{e:u3} \exact{L\otimes I_Z}{V}{L^{-1}\otimes I_{Z'}} \end{equation} such that $c_1(L)\cdot H=0$ and $L$ is not the trivial line bundle. Moreover, $$\widetilde\sigma(\gamma(V))=(L\oplus L^{-1},\,red(Z\cup Z')).$$ \end{lem} \begin{pf} The same as the proof of \lemref{l:u1} \end{pf} For the first case in \lemref{l:u5}, \lemref{l:u1}, \lemref{l:u2} and \lemref{l:u3} still hold with some minor modifications. Let's prove the following lemma which deals with the latter case in \lemref{l:u5}. \begin{lem}\label{l:u6} Without loss of generality, let's assume $c_1(L)\cdot K_X\ge 0$. For any point $(L\oplus L^{-1}, x)$ in $\widetilde{\cal N}_{H_0}^j\times Sym^{c_2-j}(X)$ where $0<j=-c_1(L)^2$, choose a zero-cycle $Z$ of length $c_2-j$ such that $red(Z)=x$. Then there exists an $H_0$-semi-stable sheaf $V$ in the non-splitting exact sequence \begin{equation}\label{e:u4} \exact{L\otimes I_Z}{V}{L^{-1}}. \end{equation} \end{lem} \begin{pf} The proof is similar to that of \lemref{l:u2}. $$\begin{array}{ll} &{\rm dim}Ext^1(L^{-1}, L\otimes I_Z)=h^1( L^{\otimes 2}\otimes I_Z)\\ \ge&-\chi\bigl(L^{\otimes 2}\otimes I_Z\bigr) =-(\displaystyle{2c_1(L)\cdot (2c_1(L)-K_X)\over2}-\ell(Z)+1)\\ =&-2c_1^2(L)+c_1(L)\cdot K_X+\ell(Z)-1\\ =&c_2+c_1(L)\cdot K_X+j-1\ge c_2>0.\\ \end{array}$$ Hence there exists a non-splitting exact sequence \equref{e:u4}. Notice that $$\begin{array}{ll} &~~2\chi(L\otimes I_Z)=2(\displaystyle{c_1(L)^2-c_1(L)\cdot K_X\over 2}-\ell(Z)+1)\\ &=-j-c_1(L)\cdot K_X-2\ell(Z)+2\\ &=-c_2+2-c_1(L)\cdot K_X-\ell(Z)\\ &\le -c_2+2=\chi(V).\\ \end{array}$$ \par Let's prove that $V$ is $H_0$-semi-stable. \par Let $M$ be a rank one subsheaf of $V$. If $M$ is a subsheaf of $L\otimes I_Z$, then $$c_1(M)\cdot H_0\le c_1(L)\cdot H_0\qquad\hbox{and}\qquad 2\chi(M)\le 2\chi(L\otimes I_Z)\le \chi(V).$$ \par Otherwise, $M$ is a subsheaf of $L^{-1}$. Hence $c_1(L^{-1})-c_1(M)$ is effective or trivial and $c_1(M)\cdot H_0\le c_1(L^{-1})\cdot H_0=0$. If $c_1(M)\cdot H_0 =c_1(L^{-1})\cdot H_0$, then $c_1(L^{-1})-c_1(M)$ is the trivial divisor, hence the exact sequence \equref{e:u4} splits, a contradiction. Therefore, $c_1(M)\cdot H_0<c_1(L^{-1})\cdot H_0=0$. \par Hence $V$ is $H_0$-semi-stable. \end{pf} By the similar argument, we get the following proposition \begin{prop}\label{p:u2} Assume that $H_0$ is an ample divisor on a face. \par{($i.$)} If $c_2\ge 2$, then ${\rm Im}\widetilde \sigma =\coprod\limits^{c_2}_{j=0}\widetilde {\cal N}_{H_0}(j)\times Sym^{c_2-j}X$. \par {($ii.$)} If $c_2=1$, then ${\rm Im}\widetilde \sigma =\widetilde {\cal N}_{H_0}(1)$. In particular, $\widetilde {\cal N}_{H_0}(1)$ is compact. \end{prop} \begin{thm}\label{t:u2} Let $\widetilde H$ be an arbitrary polarization. If ${\cal M}_{\widetilde H}(c_2)$ is irreducible, then the Uhlenbeck compactification is the total space, i.e. $$\overline {\cal N}_{\widetilde H}(c_2)=\hbox{$\coprod\limits_{j=0}^{c_2}$} \widetilde {\cal N}_{\widetilde H}(j)\times Sym^{c_2-j}(X)$$ when $c_2\ge 2$; When $c_2 = 1$, we have $$\overline {\cal N}_{\widetilde H}(1)= \widetilde {\cal N}_{\widetilde H}(1).$$ \end{thm} \section{Canonical regular morphisms among the Uhlenbeck compactifications} In the following, we are going to study the variation of the Uhlenbeck compactifications. \par Let's recall some notations and results of J. Li \cite{Li}. Let ${\cal Q}_H$ be the Grothendieck's quotient scheme parameterizing all quotient sheaves $F$ of ${\cal O}_X^{\oplus N}$ with $\hbox{det}F=H^{\oplus 2n}$ and $c_2(F\otimes H^{-n})=c_2$. Let ${\cal Q}_H^{\mu}\subset {\cal Q}_H$ be the open set consisting of all M-T $H$-semi-stable quotient sheaves. Let ${\cal Q}^{ss}_H\subset {\cal Q}_H$ be the open set consisting of all $H$-semi-stable quotient sheaves. \begin{rk}\label{r:u4} J. Li constructed \cite{Li} a commutative diagram $$\begin{matrix}{\cal Q}^{ss}_H\subset {\cal Q}_H^{\mu} &\mapright{\gamma_{{\cal Q}_H}} &{\Bbb P}^k\\ \mapdown{\pi}&&\Vert\\ {\cal M}_H(c_2)&\mapright{\gamma_H}&{\Bbb P}^k \end{matrix}$$ for some projective space ${\Bbb P}^k$ such that (Lemma 3.2, \cite{Li}) $\gamma_{{\cal Q}_H}({\cal Q}^{\mu}_H\cap\overline{\cal Q}^{ss}_H)$ is identical to $\gamma_H({\cal M}_H(c_2))$ as sets where $\overline{\cal Q}^{ss}_H$ is the closure of ${\cal Q}^{ss}_H$ in ${\cal Q}_H$. \end{rk} Since we are comparing spaces depending on different stability polarizations, we will use subscripts to distinguish different maps for different spaces, for example, $\gamma_H$, $\overline\sigma_H$, ${\cal Q}_H$, etc. \par We also assume through out this section that the moduli spaces ${\cal M}_H(c_2)$ and ${\cal M}_{H_0}(c_2)$ are normal. For example, when $c_2$ is sufficiently large, ${\cal M}_H(c_2)$ and ${\cal M}_{H_0}(c_2)$ are both normal, and generic $H$-stable sheaves are $H_0$-stable, too. \par Since we assumed that ${\cal M}_H(c_2)$ and ${\cal M}_{H_0}(c_2)$ are both normal, the map $\widetilde\sigma$ defined and discussed in section \secref{s:u} becomes the map $\overline \sigma$ defined by J. Li. \par We are going to define a map $\overline\varphi$ from Uhlenbeck compactification of the moduli space $\overline{\cal N}_H(c_2)$ to $\overline{\cal N}_{H_0}(c_2)$. \begin{notation}\label{n:h1} We use $\ell_x(Q)$ to represent the length of the torsion sheaf $Q$ at the point $x$. \end{notation} \par Using Lemma 3.2 and Theorem 4 in [7], we can define a map $$\overline\varphi\colon \overline {\cal N}_H(c_2)\longrightarrow \overline {\cal N}_{H_0}(c_2)$$ as follows: \par Any element in $\overline {\cal N}_H(c_2)$ can be represented by $\overline\sigma_H(\gamma_H(V))$ for some $H$-semi-stable sheaf $V\in {\cal M}_H(c_2)$. We know that $V$ is either M-T $H_0$-stable or strictly M-T $H_0$-semi-stable. If $V$ is the former, we define $$\overline\varphi(\overline\sigma_H(\gamma_H(V)))=\overline\sigma_{H_0}(\gamma_{ H_0}(V)).$$ \par If $V$ is the latter, then $V$ sits in an exact sequence $$\exact{L\otimes I_Z}{V}{L^{-1}\otimes I_{Z'}}.$$ \par Then we define $$\overline\varphi(\overline\sigma_H(\gamma_H(V)))=(L\oplus L^{-1}, \hbox{$\sum$}(\ell_x(Z)x+\ell_x(Z')x)).$$ \begin{rk}\label{r:h2} This map can be regarded as the induced map from $\varphi$ between Gieseker compactifications. \end{rk} \begin{prop}\label{p:h1} The map $\overline\varphi$ is well-defined. \end{prop} \noindent {\it Proof}. Suppose an element in $\overline {\cal N}_H(c_2)$ can be represented by $\overline\sigma_H(\gamma_H(V))$ and $\overline\sigma_H(\gamma_H(V'))$ for some $H$-semi-stable sheaves $V$ and $V'$. Then $V$ and $V'$ sit in the exact sequences $$\exact{V}{V^{**}}{Q},$$ $$\exact{V}{V^{'**}}{Q'}$$ where $Q$ and $Q'$ are supported at zero-dimensional schemes. \par By the definition of $\overline\sigma_H$ and $\gamma_H$ (see \cite{Li}), $\overline\sigma_H(\gamma_H(V)) =(V^{**}, \hbox{$\sum$}\ell_x(Q)x)$ and $\overline\sigma_H(\gamma_H(V'))=(V^{'**}, \hbox{$\sum$}\ell_x(Q')x)$. Hence $V^{**}=V^{'**}$ and $\hbox{$\sum$}\ell_x(Q)x=\hbox{$\sum$}\ell_x(Q')x$. \par If $V$ is M-T $H_0$-stable, then $V^{**}=V^{'**}$ is M-T $H_0$-stable. Hence $V'$ is $H_0$-stable. Then $$\begin{array}{ll} &\overline\varphi(\overline\sigma_H(\gamma_H(V)) =\overline\sigma_{H_0}(\gamma_{H_0}(V))\\ =&(V^{**}, \hbox{$\sum$}\ell_x(Q)x) =(V^{'**}, \hbox{$\sum$}\ell_x(Q')x)\\ =&\overline\sigma_{H_0}(\gamma_{H_0}(V)) =\overline\varphi(\overline\sigma_H(\gamma_H(V')).\\ \end{array}$$ \par Otherwise, $V^{**}=V^{'**}$ is strictly M-T $H_0$-semi-stable and $V$ sits in the exact sequences \begin{equation}\label{e:h0} \exact{L\otimes I_{Z_1}}{V}{L^{-1}\otimes I_{Z_2}} \end{equation} and $$\exact{V}{V^{**}}{Q}.$$ \par By taking double dual of the exact sequence \equref{e:h0} , we get $$\exact{L}{V^{**}=V^{'**}}{L^{-1}\otimes I_Z}.$$ \par Hence we get $$\exact{L/L\otimes I_{Z_1}}{V^{**}/V=Q}{L^{-1}\otimes I_Z/L^{-1}\otimes I_{Z_2}}.$$ \par Hence $$\ell_x(Q)+\ell_x(Z)=\ell_x(Z_1)+\ell_x(Z_2).$$ \par Therefore $$\begin{array}{ll} &\overline\varphi(\overline\sigma_H(\gamma_H(V))\\ =&(L\oplus L^{-1}, \hbox{$\sum$}\ell_x(Z_1)x+\hbox{$\sum$}\ell_x(Z_2)x)\\ =&(L\oplus L^{-1}, \hbox{$\sum$}\ell_x(Q)x+\hbox{$\sum$}\ell_x(Z)x).\\ \end{array}$$ \par By the similar argument as above for $V'$, we get $$\begin{array}{ll} &\overline\varphi(\overline\sigma_H(\gamma_H(V'))\\ =&(L\oplus L^{-1}, \hbox{$\sum$}\ell_x(Q')x+\hbox{$\sum$}\ell_x(Z)x).\\ \end{array}$$ Hence $$\begin{array}{ll} &\overline\varphi(\overline\sigma_H(\gamma_H(V))\\ =&(L\oplus L^{-1}, \hbox{$\sum$}\ell_x(Q)x+\hbox{$\sum$}\ell_x(Z)x)\\ =&(L\oplus L^{-1}, \hbox{$\sum$}\ell_x(Q')x+\hbox{$\sum$}\ell_x(Z)x)\\ =&\overline\varphi(\overline\sigma_H(\gamma_H(V')).\\ \end{array}$$ $\Box$ \begin{rk}\label{r:h3} Since $\varphi$ is just a rational map, it might be expected that the induced map $\overline\varphi$ should also be only defined on a Zariski open subset. However, the following two observations may be useful in understanding the differences: \par (1) Uhlenbeck compactification losses track of Gieseker strictly semi-stability. It only respects M-T semi-stability ( see Lemma 3.3 in \cite{Li}). \par (2) When we regard the morphism defined by J. Li $$\overline\sigma_H\circ\gamma_H\colon{\cal M}_H(c_2)\mapright{}\overline{\cal N}_H(c_2)$$ as a blowing-down, then although $${\cal M}_H(c_2)\buildrel{\varphi}\over{-->}{\cal M}_{H_0}(c_2)$$ is a rational map, after blowing downs on ${\cal M}_H(c_2)$ and ${\cal M}_{H_0}(c_2)$ respectively, the induced map $\overline{\cal N}_H(c_2)\rightarrow \overline{\cal N}_{H_0}(c_2)$ becomes a well-defined map. Thus we have the following commutative diagram $$\begin{matrix}{\cal M}_H(c_2)&\buildrel{\varphi}\over{-->}&{\cal M}_{H_0}(c_2)\\ \mapdown{\gamma_H}&&\mapdown{\gamma_{H_0}}\\ \overline{{\cal N}_H}(c_2)&\mapright{\overline{\varphi}}&\overline{{\cal N}}_{H_0}(c_2). \end{matrix}$$ \end{rk} \par Next, we are going to show that the map $\overline\varphi$ is continuous in the classical complex topology. \par \begin{thm}\label{t:h1} The map $$\overline\varphi\colon \overline{\cal N}_H(c_2)\longrightarrow \overline{\cal N}_{H_0}(c_2)$$ is continuous in analytic topology. \end{thm} \begin{pf} The argument pretty much follows the argument in the proof of theorem 5 in \cite{Li}. \par Since $\overline{\cal N}_H$ and $\overline{\cal N}_{H_0}$ are both compact, it suffices to show that if $\lim s_n=s$ in $\overline{\cal N}_H$ and $\lim \overline\varphi (s_n)=t$ in $\overline{\cal N}_{H_0}$, then $\overline\varphi(s)=t$. \par Since $\gamma_H (\overline{\cal M}_H^\mu )= \overline{\cal N}_H$ is compact, ${\cal M}_H^{\mu}$ is dense in $ \overline{\cal M}_H^\mu$, and generic $H$-stable sheaves are also $H_0$-stable, it suffices to show the following statement: assume that $\{V_i\}$ is a sequence of $H$- and also $H_0$-stable locally free sheaves, $\lim V_i=V$ in ${\cal M}_H$, and $\lim\overline\varphi \circ \overline\sigma_H\circ \gamma_H(V_i)=t$ in $\overline{\cal N}_{H_0}$. Then $\overline\varphi(V)=t$. \par If $V$ is $H$-stable and $H_0$-semi-stable, clearly, the map $\overline\varphi $ in the neighborhood of $V$ is induced from $\varphi$. Since $\varphi$ is continuous, $\overline\varphi$ is continuous at $V$. \par Now suppose $V$ is $H$-stable and not $H_0$ semi-stable. It is clear that $V$ is strictly MT $H_0$-semi-stable. Since continuity is a local problem, we can consider things locally. In classical topology there exists an open subset $U$ of ${\cal M}_H$ containing $V$ and a universal sheaf ${\cal V}$ over $U\times X$ such that for any $u\in U$, ${\cal V}|_u$ represents $u$ in ${\cal M}_H$. Since every $H$-stable sheaf is $H_0$-semi-stable, by Cor 1.4 in \cite{Gi}, we know that there exists an integer $N$ such that $h^i({\cal V}|_u(NH_0))=0$ for $i\ge 1$ and $H^0({\cal V}|_u(NH_0))$ generates ${\cal V}|_u(NH_0)$. By base change theorem, we see that ${\cal V}(NH_0)|_{U'}$ is a quotient of ${\cal O}^{\oplus r}_{ U'\times X}$ where $U'$ is an open subset of $U$ containing $V$ and $r=h^0({\cal V}|_u(NH_0))$. \par By the universality of the quotient scheme ${\cal Q}_{H_0}$, we see that there exists an analytic morphism $$f\colon U'\longrightarrow {\cal Q}_{H_0}.$$ \par Without loss of generality, we may assume that $V_i\in U'$. Hence $$\lim_{i\to \infty}f(V_i)=f(V)\qquad \hbox{in ${\cal Q}_{H_0}$}.$$ \par Therefore $f(V)\in {\cal Q}^{\mu}_{H_0}\cap\overline{\cal Q}^{ss}_{H_0}$. \par Since $\overline\varphi\circ\overline\sigma_H\circ\gamma_H(V_i)= \overline\sigma_{H_0}\circ\gamma_{{\cal Q}_{H_0}}\circ f(V_i)$, we have $$\begin{array}{ll} &\lim\overline\varphi\circ\overline\sigma_H\circ\gamma_H(V_i)= \lim\overline\sigma_{H_0}\circ\gamma_{{\cal Q}_{H_0}}\circ f(V_i)\\ =&\overline\sigma_{H_0}\circ\gamma_{{\cal Q}_{H_0}}\circ f(V)=t= \overline\varphi\circ\overline\sigma_H\circ\gamma_H(V),\\ \end{array}$$ where the last equality comes from the definition of $\overline\varphi$. \end{pf} \par Using the following lemma, the above theorem implies immediately the algebraicity of the map $\overline\varphi\colon \overline{\cal N}_H(c_2)\longrightarrow \overline{\cal N}_{H_0}(c_2)$. \begin{lem} Let $X$ and $Y$ be two algebraic varieties. Let $U$ be a Zariski dense open subset of $X$ with an algebraic morphism $\varphi_U\colon U\longrightarrow Y$ which extends to $\varphi\colon X\longrightarrow Y$ continuously in analytic topology. Then $\varphi$ is an algebraic morphism. \end{lem} \begin{pf} Consider the graph of the maps $\varphi$ and $\varphi_U$: \begin{equation} \rm{graph}(\varphi)\subset X\times Y, \end{equation} \begin{equation} \rm{graph}(\varphi_U)\subset U\times X\subset X\times Y. \end{equation} Take the closure of $\rm{graph}(\varphi_U)$ inside $X\times Y$, call it $\overline{\rm{graph}(\varphi_U)}$. Since $\varphi$ is continuous, it is easy to see that $\overline{\rm{graph}(\varphi_U)}={\rm{graph}(\varphi)}$. \par In fact, for any element $(x, y)$ in $\overline{\rm{graph}(\varphi_U)}$, there exists a sequence $(x_n, y_n) \in{\rm{graph}(\varphi_U)}$ such that $(x_n, y_n)\rightarrow (x, y)$ as $n\rightarrow \infty$. Since $\varphi(x_n)=y_n$, \begin{equation} y=\lim_{n\rightarrow \infty}y_n=\lim_{n\rightarrow\infty}\varphi(x_n)= \varphi(\lim_{n\rightarrow\infty}x_n)=\varphi(x). \end{equation} Hence $\varphi\colon X\rightarrow Y$ can be regarded as a composition of \begin{equation} X\buildrel{\cong}\over\longrightarrow \rm{graph}\varphi = \overline{\rm{graph}(\varphi_U)} \hookrightarrow X\times Y \buildrel{proj}\over\longrightarrow Y. \end{equation} Therefore, $\varphi$ must be an analytic morphism. Since $X$ and $Y$ are all algebraic varieties, $\varphi$ must be an algebraic morphism. \end{pf} \begin{cor} The map $\overline\varphi\colon \overline{\cal N}_H(c_2)\longrightarrow \overline{\cal N}_{H_0}(c_2)$ is algebraic. \end{cor} \par In the rest of the section, we will study the inverse image of the map $\overline\varphi$. \begin{lem}\label{l:h1} Let $L$ and $L'$ be line bundles. Suppose $V$ sits in a non-splitting exact sequence $$\exact{L\otimes I_Z}{V}{L'\otimes I_{Z'}}$$ with $2c_1(L)\cdot H_0=c_1(V)\cdot H_0$. Then $V$ is strictly M-T $H_0$-semi-stable. \par If in addition, $2c_1(L)\cdot H<c_1(V)\cdot H$, then $V$ is M-T $H$-stable. \end{lem} \begin{pf} That $V$ is strictly M-T $H_0$-semi-stable is clear. \par Let $M$ be a rank one subsheaf of $V$. If $M$ is a subsheaf of $L\otimes I_Z$, then $$2c_1(M)\cdot H\le 2c_1(L)\cdot H<c_1(V)\cdot H.$$ Otherwise, $M$ is a subsheaf of $L'\otimes I_{Z'}$. Hence $$2c_1(M)\cdot H_0\le 2c_1(L')\cdot H_0=c_1(V)\cdot H_0.$$ \par If $2c_1(M)\cdot H_0<2c_1(L')\cdot H_0 = c_1(V)\cdot H_0$, then $2c_1(M)\cdot H<c_1(V)\cdot H$, since otherwise, $2c_1(M)-c_1(V)$ would define an $c$-wall between $H$ and $H_0$, a contradiction. \par If $2c_1(M)\cdot H_0= 2c_1(L')\cdot H_0$, then $L'=M$. Hence the exact sequence splits, a contradiction. \end{pf} Now we divide the study of inverse image into several cases. \par (i) Suppose $(A, x)=({\cal O}_X\oplus {\cal O}_X, x)\in {\cal N}_{H_0}(0)\times Sym^{c_2}(X)$ for $c_2\ge 2$. Then $\overline\varphi^{-1}((A, x))=(A, x)\in {\cal N}_{H}(0)\times Sym^{c_2}(X)$. Hence the inverse image of a point in the lowest stratum is just a single point. \par (ii) Suppose $(A, x)\in \coprod\limits_{j\ge 1}^{c_2}{\cal N}_{H_0}(j)\times Sym^{c_2-j}(X)$, then $A$ corresponds to an M-T $H_0$-stable vector bundles $V_j$. $V_j$ is also M-T $H$-stable. Then it is easy to see that $$\overline\varphi^{-1}((A, x))=(A,x)\in\coprod_{j\ge 1}^{c_2}{\cal N}_H(j) \times Sym^{c_2-j}(X).$$ Hence the inverse image of $\overline\varphi$ of a single point in $\coprod\limits_{j\ge 1}^{c_2}{\cal N}_H(j) \times Sym^{c_2-j}(X)$ is also just a single point. \par (iii) Suppose $(A, x)=(L\oplus L^{-1}, x)$ with $L\cdot H_0=0$ and $c_1(L)^2=-j$, i.e. $$A\in \widetilde {\cal N}_{H_0}(j)-{\cal N}_{H_0}(j).$$ Without loss of generality, we assume that $L\cdot H<0$. Assume that $(A, x)= \overline\varphi(\widetilde\sigma\gamma_H(V))$. By the way how the maps $\widetilde \sigma$ and $\overline \varphi$ are defined, it is easy to see that $V$ sits in the non-splitting exact sequence \begin{equation}\label{e:h2} \exact{L\otimes I_Z}{V}{L^{-1}\otimes I_{Z'}}. \end{equation} By \lemref{l:h1}, $V$ is M-T $H$-stable. \par The inverse image of $(A, x)$ is rather complicated due to the arbitrariness of $Z$ and $Z'$ in the exact sequence \equref{e:h2}. We can only give a rough description of the inverse image. \par Consider one extreme case where $V$ sits in a non-splitting exact sequence $$\exact{L\otimes I_Z}{V}{L^{-1}}.$$ We know that $V$ is $H$-stable. Clearly, $V^{**}$ sits in the exact sequence $$\exact{L}{V^{**}}{L^{-1}}.$$ Hence $$\overline\varphi^{-1}((L\oplus L^{-1}, x))\supset ({\Bbb P}(H^2(L^{\oplus 2}), x) \subset {\cal M}_H(j)\times Sym^{c_2-j}(X).$$ \par For other cases, it is easy to see that $\overline\varphi^{-1}((L\oplus L^{-1}, x))$ consists of all $$(V^{**}, x')\in {\cal M}_H(j')\times Sym^{c_2-j'}(X)$$ where $V$ sits in the exact sequence \equref{e:h2}, $c_2(V^{**})=j'$, $x'\subset x$, and we have the exact sequence $\exact{V}{V^{**}}{{\cal O}_{Z''}}$ where $red(Z'')=x'$. \par In another word, $\overline\varphi^{-1}((L\oplus L^{-1}, x))$ consists of $$(V_{j'}, x')\in {\cal M}_H(j')\times Sym^{c_2-j'}(X)$$ where $c_2(V_{j'})=j'$, $x'\subset x$, $V_j$ is locally free sheave sitting in the non-splitting exact sequence $$\exact{L}{V_{j'}}{L^{-1}I_{Z'}}$$ where $red(Z')=x-x'$. \par In general, the preimage of the map $\overline\varphi$ may contain points in $\overline {\cal N}_H(c_2)$ from every stratum ${\cal N}_H(j')$ for $j'\ge j$. The intersection $\overline\varphi^{-1}((A, x))\cap{\cal N}_H(j')$ of the preimage with each stratum may not be closed. But put all these strata together, $\overline\varphi^{-1}((A, x))$ will be closed. \par Summarize the above, we have \begin{thm} Let the situation be as in Theorem 5.1. Then we have the following commutative diagram $$\begin{matrix} {\cal M}_H(c_2)&\buildrel{\varphi}\over{-->}& {\cal M}_{H_0}(c_2)& \buildrel{\psi}\over{<--} &{\cal M}_{H'}(c_2)\\ \mapdown{\gamma_H}&&\mapdown{\gamma_{H_0}}&&\mapdown{\gamma_{H'}}\\ \overline{{\cal N}_H}(c_2)&\mapright{\overline{\varphi}}&\overline{{\cal N}}_{H_0}(c_2) &\mapleft{\overline{\psi}} &\overline{{\cal N}}_{H'}(c_2) \end{matrix}$$ such that \par ($i$) $\overline{\varphi}$ and $\overline{\psi}$ are induced from $\varphi$ and $\psi$; both are well-defined everywhere and are algebraic maps; \par ($ii$) $\overline{\varphi}$ and $\overline{\psi}$ are homeomorphisms over the Zariski open subset $\coprod\limits_{j = 0}^{c_2}{\cal N}_{H_0}(j) \times Sym^{c_2-j}(X)$; \par ($iii$) Let $(A, x) \in (\widetilde {\cal N}_{H_0}(j)-{\cal N}_{H_0}(j)) \times Sym^{c_2-j}(X)$. The the preimages of $\overline{\varphi}$ and $\overline{\psi}$ over $(A, x)$ are contained in $\coprod\limits_{j' \ge j} \widetilde {\cal N}_{H_0}(j') \times Sym^{c_2-j'}(X)$. \end{thm}

9409

alg-geom/9409004

en

https://arxiv.org/abs/alg-geom/9409004

alg-geom/9409004

Dr. Yakov Karpishpan

Yakov Karpishpan

Riemann reciprocity in higher dimensions

15 pages, LaTeX

The reciprocity law for abelian differentials of first and second kind is generalized to higher-dimensional varieties. It is shown that $H^1(V)$ of a polarized variety $V$ is encoded in the Laurent data along a curve germ in $V$, with the polarization form on $H^1(V)$ corresponding to the {\em one-dimensional} residue pairing. This associates an {\em extended abelian variety} to $V$; if $V$ is an abelian variety itself, our construction ``extends" it, even when $V$ is not a Jacobian.

alg-geom

\section{Introduction} The reciprocity law, or bilinear relation, for abelian differentials of the first and second kind is classically formulated as follows. Let $X$ be a compact Riemann surface of genus $g$. Suppose $\omega$ is a holomorphic one-form (an abelian differential of the first kind) and $\eta$ is a meromorphic one-form with exactly one pole, necessarily of order $\geq 2$, at some point $p\in X$ (an abelian differential of the second kind). Let $A_1,\ldots,A_g, B_1\ldots, B_g$ (resp. $A'_1,\ldots,A'_g, B'_1,\ldots, B'_g$) denote the periods of $\omega$ (resp. $\eta$) over a standard symplectic basis for $H_1(X,{\bf Z})$. Then \begin{equation} \label{classic} \sum_{j=1}^{g}A_jB'_j - A'_jB_j = 2\pi i \,\mbox{\rm res}_p\ (f\eta)\ , \end{equation} where $f$ is a holomorphic function defined near $p$, with $df=\omega$. The left-hand side of (\ref{classic}) can be more easily recognized for what it is when written as a matrix product $$ (A_1 \ldots B_g)\left( \begin{array}{cc} 0 & I_g\\ -I_g & 0 \end{array} \right) \left(\begin{array}{c} A'_1\\ \vdots\\ B'_g \end{array}\right)\ . $$ The forms $\omega$ and $\eta$ represent cohomology classes in $H^1(X,{\bf C})$ determined by their vectors of periods. The polarization form $Q$ is defined as the dual of the intersection form on $H_1(X,{\bf Z})$, and so the matrix product above equals $Q([\omega],[\eta])$. As to the right-hand side of (\ref{classic}), let $g$ be a meromorphic function defined near $p$ with $dg=\eta$. Writing $<f,g>$ for the residue pairing $\mbox{\rm res}_p (fdg)$, one has the following version of (\ref{classic}): \begin{equation} \label{modern} Q([\omega],[\eta])=2\pi i\,<f,g>\ . \end{equation} It is this statement that we wish to generalize in this paper for varieties of dimension greater than one. The pointed Riemann surface is replaced by a smooth complex projective variety $V$ with an irreducible ample divisor $D$ and a regular point $p$ on $D$ (we do not assume that $D$ is smooth). We also choose a smooth curve germ ${\cal X}$ on $V$ through $p$, transversal to $D$. The Chern class of $D$ defines a polarization form $Q$ on $H^1(V,{\bf C})$. The claim is that $Q$ is again expressed via the {\em one-dimensional} residue pairing. Specifically, let $\omega$ be a holomorphic one-form (a simple abelian differential of the first kind) on $V$, and let $\eta$ be a {\em closed} meromorphic one-form on $V$ with poles only along $D$ (a simple abelian differential of the second kind). Such forms again represent cohomology classes in $H^1(V,{\bf C})$ (Proposition \ref{V-D:V}), and in Theorem \ref{main} we establish a similar relation \begin{equation} \label{our} Q([\omega],[\eta])=(-1)^{n-1}2\pi i\,<f,g>\ , \end{equation} where now $f$ and $g$ are, respectively, holomorphic and meromorphic functions on ${\cal X}$, whose differentials are the one-forms $\omega$ and $\eta$ {\em pulled back to} ${\cal X}$: $$ df=\omega|_{{\cal X}}\ ,\hspace{1in}dg=\eta|_{{\cal X}}\ , $$ and $<f,g>=\mbox{\rm res}_pfdg$ denotes the residue pairing on ${\cal X}$ at $p$. Returning to the one-dimensional case, with additional notation and terminology a more complete statement is possible. Let $\H={\bf C}((z))$ (the field of formal Laurent power series), $\H_+={\bf C}[[z]]$, and $\H'={\bf C}((z))/{\bf C}$. The pairing $<f,g>=\mbox{\rm res}_{z=0}fdg$ on $\H$ is skew-symmetric; it is non-degenerate on $\H'$. Choosing a formal local parameter $u:\hat{{\cal O}}_{X,p}\stackrel{\cong}{\longrightarrow}\H_+$ on $X$ at $p$ defines maps $$ \Gamma(X,{\cal O}_X(*p))\longrightarrow \H'\ \ \ \mbox{and}\ \ \ \Gamma(X,\Omega^1_X(*p))\longrightarrow \H dz\ , $$ both of which will also be denoted $u$. Put $K_0=u(\Gamma(X,{\cal O}_X(*p)))$ and $\Omega=\{f\in\H'\,|\,df\in u(\Gamma(X,\Omega^1_X(*p)))\}$. Then $K_0\subset \Omega$ are each other's annihilators in $\H'$, i.e. $\Omega=K_0^{\perp}$ and $<\ ,\ >$ induces a perfect pairing on $K_0^{\perp}/K_0$. The reciprocity law now says that there is a symplectic isomorphism \begin{equation} \label{basic:iso} (H^1(X,{\bf C}),\ Q)\longrightarrow (K_0^{\perp}/K_0,\ 2\pi i\,<\ ,\ >)\ {}. \end{equation} Noting that $(H^1(X,{\bf C}),Q)$ is a polarized Hodge structure, we may go further and identify all of its components in terms of the Laurent data. Thus $H^{1,0}=F^1H^1(X,{\bf C})$ is simply $K_0^{\perp}\cap\H'_+$. As to the integral structure, put $K=\{f\in\H'\,|\,e^f\in u(\Gamma(X-\{p\},{\cal O}^*))\}$. Then $K_0\subset K\subset K_0^{\perp}$ and $\Lambda:=K/K_0$ is isomorphic to $H^1(X,{\bf Z})$. Finally, let $U\subset K_0^{\perp}/K_0$ be the image of $H^{0,1}$ under (\ref{basic:iso}) and, in turn, let $Z$ denote the preimage of $U$ under the projection $K_0^{\perp}\rightarrow K_0^{\perp}/K_0$. Then $Z$ is a maximal isotropic subspace of $\H'$. These results, most of which may already be found in \cite{SW}, led Arbarello and De Concini \cite{AD} to codify triples $(Z,K_0,\Lambda)$ with properties as above under the name of {\em extended abelian varieties}. They are also called {\em extended Hodge structures of weight one} in \cite{K}. Evidently, to each such there corresponds a unique polarized Hodge structure of weight one, although going back there are infinitely many choices (see (\ref{choices})). Thus one has a ``de-extension" map with infinite fibers \begin{equation} \label{de-extend} \left\{\mbox{extended abelian varieties}\right\}\longrightarrow \left\{\mbox{abelian varieties}\right\}\ . \end{equation} In this paper it is shown that all of the above generalizes to higher-dimen\-sion\-al varieties. We again meet the subspaces $K_0\subset\Omega$ of $\H'$ with $\Omega=K_0^{\perp}$, and a reciprocity law analogous to (\ref{basic:iso}) holds (Theorem \ref{main:two}): there is a symplectic isomorphism \begin{equation} \label{basic:iso:two} (H^1(V,{\bf C}),\ Q)\longrightarrow (K_0^{\perp}/K_0,\ (-1)^{n-1}2\pi i<\ ,\ >)\ . \end{equation} We also construct the remaining components of an extended abelian variety whose associated Hodge structure is that of $H^1(V,{\bf C})$ for any variety $V$ with a divisor $D$ as above (Theorem \ref{EHS}). In particular, this applies when $V$ is an abelian variety and $D$ is its theta divisor. Thus we obtain a method of ``inverting" the ``de-extension" map (\ref{de-extend}), or ``extending" a Hodge structure of weight one, even if the latter did not come from geometry. This may be useful in approaching the Schottky problem, which was, in fact, our original motivation. We leave with a question suggested by the present work: does this generalize for differential forms of higher degree, and is there a good notion of an extended Hodge structure of weight higher than one? \noindent{\bf Acknowledgements}. During work on this paper the author had helpful conversations with Enrico Arbarello and Roberto Silvotti. This project has also been influenced by the attempts of M. Sato and his co-workers at a higher-dimensional generalization of Krichever's theory, sketched in \cite{Sa}. \section{$H^1$ of a polarized variety} \label{H1} We will be studying a smooth complex projective variety $V$ with an irreducible ample divisor $D$, not necessarily smooth. By Grothendieck's Algebraic De Rham Theorem \cite{Gro}, \begin{equation} H^k(V-D,{\bf C})\cong\frac{\Gamma(\tilde{\Omega}_V^k(*D))} {d\Gamma(\Omega_V^{k-1}(*D))}\ , \end{equation} where the tilde in $\tilde{\Omega}_V^k(*D)$ denotes the subsheaf of $d$-closed forms in $\Omega_V^k(*D)$. \begin{Prop} \label{V-D:V}With the above assumptions, $$ H^1(V,{\bf C})\cong H^1(V-D,{\bf C})\cong \frac{\Gamma(\tilde{\Omega}_V^1(*D))} {d\Gamma({\cal O}_V(*D))}\ . $$ \end{Prop} \ \\ \noindent {\bf Proof.\ \ } First, assume $D$ is smooth. Then there is a sequence of sheaf complexes $0\rightarrow\Omega_V^{\bullet}\rightarrow \Omega_V^{\bullet}(\log D)\rightarrow\Omega_D^{\bullet-1}\rightarrow 0$ inducing $$ 0\longrightarrow H^1(V)\longrightarrow H^1(V-D)\stackrel{P.R.}{\longrightarrow} H^0(D)\stackrel{\gamma}{\longrightarrow} H^2(V)\longrightarrow\ldots $$ Here $P.R.$ stands for the Poincar\'e Residue and $\gamma$ denotes the Gysin map, which is obtained by applying Poincar\'e duality on the source and the target of the map $H_{2n-2}(D)\rightarrow H_{2n-2}(V)$, where $n=\dim_{{\bf C}}V$. The latter map is injective: the image of a suitable generator of $H_{2n-2}(D)(\cong{\bf C})$ is the fundamental class of $D$ in $H_{2n-2}(V)$, which cannot be zero by the irreducibility of $D$. Hence $\gamma$ is injective too, which proves the assertion. In the general case, when $D$ is not smooth, the above arguement does not work. Instead, consider the spectral sequence $$ E_2^{p,q}=H^p(V,\H^q\Omega_V^{\bullet}(*D))\Rightarrow H^*(V-D,{\bf C})\ {}. $$ It yields the very same exact sequence $$ 0\longrightarrow H^1(V)\longrightarrow H^1(V-D)\stackrel{R}{\longrightarrow} H^0(D)\stackrel{\nu}{\longrightarrow} H^2(V)\longrightarrow\ldots $$ where now $R$ is the cohomological residue map induced by the sheaf isomorphism $\H^1\Omega_V^{\bullet}(*D)\cong{\bf C}_D$, and $\nu$ is the map sending $1_D$ to (the Poincar\'e dual of) the fundamental class of $D$ in $H^2(V)$ (see \cite{GH}, p. 458). Again, the map $\nu$ must be injective, i.e. $H^1(V)\cong H^1(V-D)$ in all cases. \ $\displaystyle\Box$\\ \ \par We note that the isomorphism of the above Proposition makes the Hodge filtration on $H^1(V,{\bf C})$ obvious: $$ F^1H^1(V,{\bf C})=\Gamma(V,\Omega_V^1)\ . $$ There is also a corresponding isomorphism of quotients \begin{equation} \frac{\Gamma(V,\tilde{\Omega}^1_V(*D))} {\Gamma(V,\Omega^1_V) + d\Gamma(V,{\cal O}_V(*D))} \stackrel{\cong}{\longrightarrow}H^1(V,{\cal O}_V)\ . \end{equation} For future use, we will need to define it explicitly. Let ${\cal U}=\{U_{\alpha}\}$ be an affine open cover of $V$ or an acyclic refinement of such; thus $\check{H}^1({\cal U},{\cal O}_V)\cong H^1(V,{\cal O}_V)$. The \v{C}ech cochain $\{g_{\alpha\beta}\in\Gamma(U_{\alpha}\cap U_{\beta},{\cal O}_V)\}$ giving the image of $\eta\in\Gamma(V,\tilde{\Omega}^1_V(*D))$ in $H^1(V,{\cal O}_V)$ can be described as follows: $$ \{g_{\alpha\beta}\}=\check{\delta}\{\mu_{\alpha}\}\ , $$ where $\mu_{\alpha}$ is a meromorphic function on $U_{\alpha}$ such that $\eta|_{U_{\alpha}}-d\mu_{\alpha}$ is a holomorphic one-form on $U_{\alpha}$. In particular, when $U_{\alpha}$ is simply connected, we may assume that $\eta|_{U_{\alpha}}=d\mu_{\alpha}$, and if $\eta$ is already holomorphic over $U_{\alpha}$, then we may take $\mu_{\alpha}=0$. \section{Polarization and residues} \label{polarization} The Hodge structure on $H^1(V,{\bf C})$ is polarized by the divisor $D$. Concretely, the cup product \begin{equation} \label{QQ} Q:H^1(V,{\bf C})\stackrel{\smile}{\otimes} H^1(V,{\bf C}) \stackrel{\smile c_{[D]}^{n-1}}{\longrightarrow} H^{2n}(V,{\bf C}) \stackrel{\int}{\longrightarrow}{\bf C} \end{equation} gives an integrally-defined perfect pairing. Here $c_{[D]}\in H^2(V,{\bf Z})$ stands for the first Chern class of ${\cal O}_V(D)$, and $\int$ denotes the topological trace, i.e. the map $H^{2n}_{DR}(V,{\bf C})\stackrel{\sim}{\rightarrow}{\bf C}$ obtained by integrating $C^{\infty}$ complex-valued $2n$-forms over $V$. Thinking of $c_{[D]}$ as an element of $H^1(V,\Omega_V^1)$, we will generally prefer a more algebraic version \begin{equation} \label{Q} Q:H^0(V,\Omega_V^1)\stackrel{\smile}{\otimes} H^1(V,{\cal O}_V) \stackrel{\smile c_{[D]}^{n-1}}{\longrightarrow} H^n(V,\Omega_V^n)\cong H^{2n}(V,{\bf C}) \stackrel{\int}{\longrightarrow}{\bf C} \end{equation} \refstepcounter{Thm No notational distinction is made between the classes $[\omega]$, $[\eta]$ in $H^1(V,{\bf C})$ and in $H^0(V,\Omega_V^1)$ or $H^1(V,{\cal O}_V)$. Likewise, both pairings above are denoted $Q$. This should cause no confusion, since the pairings are compatible with the natural maps $H^0(V,\Omega_V^1)\hookrightarrow H^1(V,{\bf C})$ and $H^1(V,{\bf C})\,\longrightarrow\hspace{-12pt H^1(V,{\cal O}_V)$. Even more algebraically, $\int=(2\pi i)^n tr$, where $tr$ is Grothendieck's trace isomorphism (cf. \cite{D}, (2.2)). As the latter is the least explicit part in the definition of $Q$, let us elaborate on it. First recall from \cite{L} and \cite{GH} a construction of the local trace $$ H^n_{\{p\}}(V,\Omega_V^n) \stackrel{tr_p}{\longrightarrow}{\bf C}\ . $$ Let $U$ be a polycylindrical neighborhood of $p$ in $V$. Then $$ H^n_{\{p\}}(V,\Omega_V^n)\cong H^n_{\{p\}}(U,\Omega_U^n)\ , $$ by excision. And $U$ being Stein (and $n\geq 1$) implies $$ H^n_{\{p\}}(U,\Omega_U^n)\cong H^{n-1}(U-\{p\},\Omega_U^n)\ . $$ These isomorphisms are composed with the residue morphism $$ \mbox{\rm Res}: H^{n-1}(U-\{p\},\Omega_U^n)\stackrel{\cong}{\longrightarrow}{\bf C}\ . $$ To define $\mbox{\rm Res}$, let $t_1,\ldots,t_n$ be a coordinate system on $U$ centered at $p$. Thus $D_j=\{t_j=0\}$ are $n$ smooth hypersufaces in $U$ intersecting normally at $p$. We assume that $D_1=D\cap U$, although this will not be necessary until the next section. Put $U_j=U-D_j$, and let $D^+=D_1+\ldots +D_n$. Evidently, the $U_j$'s form an acyclic open cover of $U-\{p\}$. Then a class in $H^{n-1}(U-\{p\},\Omega_U^n)$ is represented by a holomorphic form on $U_1\cap\ldots \cap U_n=U-D^+$. In fact, we may assume that this form is a restricton to $U_1\cap\ldots \cap U_n$ of a meromorphic form $\psi\in\Gamma(U,\Omega_U^n(*D^+))$. The form $\psi$ can be expanded in a Laurent series: $$ \psi=\sum a_{k_1,\ldots, k_n}t^{k_1}\cdots t^{k_n}dt_1\wedge\ldots\wedge dt_n\ . $$ Finally, $$ \mbox{\rm Res} \,\psi\stackrel{\rm def}{=} a_{-1,\ldots,-1}\ . $$ It is a basic fact of residue theory that $\mbox{\rm Res} \,\psi$ is independent of the parameter system $(t_1,\ldots,t_n)$. We note one property of $$ \mbox{\rm Res}: \Gamma(U,\Omega_U^n(*D^+))\longrightarrow {\bf C} $$ for future reference. Assume $\psi\in\Gamma(U,\Omega_U(*D^+))$ has only {\em simple} poles along $D_2,\ldots,D_n$, i.e. can be written as $$ \psi=\frac{h\,dt_1\wedge\ldots\wedge dt_n}{t_1^m\,t_2\cdots t_n}\ , $$ with $h$ a holomorphic function. Then \begin{equation} \label{Res:res} \mbox{\rm Res}\,\psi = \mbox{\rm res}_0\, \frac{\tilde{h}\,dt_1}{t_1^m}\ , \end{equation} where $\tilde{h}(t_1)=h(t_1,0,\ldots,0)$, and $\mbox{\rm res}_0$ denotes the usual residue at $0$ of a meromorphic one-form in one variable. The reason for bringing in the local trace is the following commutative diagram \cite{L}: $$ \begin{array}{rcl} H^n_{\{p\}}(V,\Omega_V^n) & \stackrel{\cong}{\longrightarrow} & H^n(V,\Omega_V^n)\\ & & \\ tr_p\searrow\cong & & \cong\swarrow tr\\ & & \\ & {\bf C} & \end{array} $$ To tie in with the above description of $tr_p$, it remains to explain the isomorphism $$ H^n(V,\Omega_V^n)\longrightarrow H^n_{\{p\}}(V,\Omega_V^n) \longrightarrow H^{n-1}(U-\{p\},\Omega_U^n) $$ in \v{C}ech cohomology. Take an affine (and hence acyclic) open covering ${\cal U}=\{U_{\alpha}\}$ of $V$, so that $H^n(V,\Omega_V^n)\cong\check{H}^n({\cal U},\Omega_V^n)$. We may assume that the neighborhood $U$ of $p$ selected above is entirely contained in some $U_{\alpha}$. Then, throwing in $U_i=U-D_i$ and $U_0:=U$, we get an acyclic refinement $\bar{{\cal U}}$ of ${\cal U}$. Hence $\check{H}^n(\bar{{\cal U}},\Omega_V^n)$ is still isomorphic to $H^n(V,\Omega_V^n)$. Now, an $(n-1)$-cochain over $U-\{p\}$ with coefficients in $\Omega_U^n$ is a section of $\Omega_U^n$ over $U_1\cap\ldots\cap U_n$. Regarding it as a section of $\Omega_V^n$ over $U_0\cap U_1\cap\ldots\cap U_n$ defines an $n$-cochain with coefficients in $\Omega_V^n$. This is the cochain map underlying the isomorphism $$ H^{n-1}(U-\{p\},\Omega_U^n)\stackrel{\cong}{\longrightarrow} H^n(V,\Omega_V^n)\ . $$ To define its inverse, just read this backwards: take a cocycle in $\check{C}^n(\bar{{\cal U}},\Omega_V^n)$, isolate its component in $\Gamma(U_0\cap U_1\cap\ldots\cap U_n,\Omega_V^n)$, and view it as an element in $\Gamma(U_1\cap\ldots\cap U_n,\Omega_U^n)$. \section{The Laurent data} Let $p$ be a regular point on $D$, and ${\cal X}$ a germ of a smooth curve through $p$, normal to $D$, cut out by $t_2=\ldots=t_n=0$. $z:=t_1|_{{\cal X}}$ defines a coordinate on ${\cal X}$, identifying $\hat{{\cal O}}_{{\cal X},p}$ with ${\bf C}[[z]]$. We will use the notation $\H={\bf C}((z))$, $\H_+= {\bf C}[[z]]$, and $\H'=\H/{\bf C}$, and we will write $u$ for the Laurent expansion in terms of $z$ of the global meromorphic objects on $V$ {\em restricted to} ${\cal X}$. Thus we have two compositions, both denoted $u$: $$ \Gamma(V,{\cal O}_V(*D))\,\longrightarrow\hspace{-12pt {\cal O}_{{\cal X},p}(*p)\hookrightarrow\H\,\longrightarrow\hspace{-12pt \H' $$ and $$ \Gamma(V,\Omega_V(*D))\,\longrightarrow\hspace{-12pt \Omega^1_{{\cal X},p}(*p)\hookrightarrow\H dz\ . $$ Now we introduce two important subspaces of $\H'$ carrying information about $V$: $K_0:=u(\Gamma(V,{\cal O}_V(*D)))$ and $$ \Omega:=\{f\in\H'\,|\,df\in u(\Gamma(V,\tilde{\Omega}^1_V(*D)))\}\ {}. $$ It is obvious that $K_0\subset\Omega$ and that $u$ (followed by $d^{-1}:\H dz\stackrel{\sim}{\rightarrow}\H'$) induces a surjection \begin{equation} \label{basic:surj} H^1(V,{\bf C})\cong\frac{\Gamma(V,\tilde{\Omega}^1_V(*D))} {d\Gamma(V,{\cal O}_V(*D))}\,\longrightarrow\hspace{-12pt\frac{\Omega}{K_0}\ . \end{equation} We also have surjective morphisms $$ H^0(V,\Omega_V^1)=\Gamma(V,\tilde{\Omega}^1_V)\longrightarrow \Omega\cap\H'_+ $$ and $$ H^1(V,{\cal O}_V)\cong\frac{\Gamma(V,\tilde{\Omega}^1_V(*D))} {\Gamma(V,\Omega^1_V)+d\Gamma(V,{\cal O}_V(*D))}\longrightarrow \frac{\Omega}{(\Omega\cap\H'_+)+K_0}\ . $$ The space $\H$ carries a symplectic form $$ <f,g>=\mbox{\rm res}_0\,(fdg)\ , $$ which is non-degenerate on $\H'$. This form induces a symplectic pairing, also denoted $<\ ,\ >$: \begin{equation} \label{form:symp} (\Omega\cap\H'_+)\times \frac{\Omega}{\Omega\cap\H'_+}\longrightarrow{\bf C}\ . \end{equation} A comparison of this with the polarization form $Q$ (\ref{Q}) constitutes our main result. \section{The generalized reciprocity law} \begin{Thm} \label{main} Let $\omega\in\Gamma(V,\Omega_V^1)$ and $\eta\in\Gamma(V,\tilde{\Omega}_V^1(*D))$ represent classes in $H^0(V,\Omega_V^1)$ and $H^1(V,{\cal O}_V)\cong \Gamma(V,\tilde{\Omega}_V^1(*D))/ \Gamma(V,\Omega_V^1)+d\Gamma(V,{\cal O}_V^1(*D))$, respectively, and suppose $u(\omega)=df$ and $u(\eta)=dg$ for some $f\in\Omega\cap\H'_+$ and $g\in\Omega$. Then $$ Q([\omega],[\eta])=(-1)^{n-1}2\pi i\,<f,g>\ . $$ \end{Thm} \ \\ \noindent {\bf Proof.\ \ } We will work in the covering $\bar{{\cal U}}$ as above. The class $[\omega]$ is represented by the cochain$\{\omega_{\alpha}=\omega|_{U_{\alpha}}\}$, whereas $[\eta]=[\{g_{\alpha\beta}\in\Gamma(U_{\alpha}\cap U_{\beta},{\cal O}_V)\}]$ has been described as the end of Section \ref{H1}. We single out $$ g_{01}=\mu_1|_{U_0\cap U_1}-\mu_0|_{U_0\cap U_1}\ : $$ since $U_0=U$ is contractible, we may assume $d\mu_0=\eta|_{U}$, and since $\eta$ is holomorphic over $U_1$, we may take $\mu_1=0$. Thus $g_{01}=-\mu_0|_{U_0\cap U_1}$ is a holomorphic function on $U_0\cap U_1$, with $\eta|_{U_0\cap U_1}=-dg_{01}$. And $c_{[D]}\in H^1(V,\Omega_V^1)$ is represented by \begin{equation} \label{c} \left\{-\frac{1}{2\pi i} \frac{dh_{\alpha\beta}}{h_{\alpha\beta}}+d\ell\right\}\ , \end{equation} where $h_{\alpha\beta}=0$ is a local equation of $D$ in $U_{\alpha}\cap U_{\beta}$ and $\ell\in\Gamma(U_{\alpha}\cap U_{\beta},{\cal O}^*_V)$ extends across $D$ as an invertible holomorphic function, if $D$ meets the closure of $U_{\alpha}\cap U_{\beta}$. As explained in Section \ref{polarization}, $$ Q([\omega],[\eta])=(2\pi i)^n tr\,([\omega]\smile[\eta]\smile c_{[D]}^{n-1})\ , $$and the trace can be computed locally, by taking the residue of the component at $U_0\cap U_1\cap\ldots\cap U_n$ of a \v{C}ech cochain representing $[\omega]\smile[\eta]\smile c_{[D]}^{n-1}$ in $H^n(V,\Omega_V^n)$. Thus it suffices to take the wedge product of the restrictions to $U_0\cap U_1\cap\ldots\cap U_n$ of $\omega_0$, $g_{01}$, and of the \v{C}ech components of $c_{[D]}$ over $U_0\cap U_1$, $U_1\cap U_2$, \ldots,$U_{n-1}\cap U_n$. However, in the case of the cochain (\ref{c}) we have no information on the singularities these components have along the $D_j$'s, except the one over $U_1\cap U_2$, which is $-\frac{1}{2\pi i}\frac{dt_1}{t_1}+d\ell$. This is unsuitable for computing the residue. Thus we need other cochains representing $c_{[D]}$. Now, replacing a divisor by a linearly equivalent one does not affect the Chern class. And just as $v_1t_1^N\in{\cal O}_{V,p}$ is a germ of a global meromorphic function on $V$ for an appropriate $N$ and some $v_1\in\Gamma(U,{\cal O}^*_U)$, we shall assume temporarily that the same is true of the other coordinate functions $t_2,\ldots,t_n$. Then $v_{j1}(t_j/t_1)^N$ ($j=2,\ldots,n$) also come from meromorphic functions on $V$. Consequently, for each $j=2,\ldots,n$ there is a cochain representing $c_{[ND]}$ whose component on $U_{j-1}\cap U_j$ is $-\frac{N}{2\pi i}\frac{dt_j}{t_j}+d\ell_j$, with $\ell_j$ extending as an invertible holomorphic function on all of $U$. Dividing by $N$ gives new cochains representing $c_{[D]}$. Using these cochains, \begin{eqnarray} \nonumber \lefteqn{Q([\omega],[\eta])=}\\ \nonumber & = & (2\pi i)^n \mbox{\rm Res}\ \omega_0 g_{01}\wedge \left(-\frac{1}{2\pi i}\frac{dt_2}{t_2}+d\ell_2\right)\wedge\ldots\wedge \left(-\frac{1}{2\pi i}\frac{dt_n}{t_n}+d\ell_n\right)\\ & = & (-1)^{n-1}2\pi i\, \mbox{\rm Res}\ \omega_0 g_{01} \wedge\frac{dt_2}{t_2}\wedge\ldots\wedge\frac{dt_n}{t_n} \label{Q:Res} \end{eqnarray} The last reduction is possible because $d\ell_j$'s do not contribute to the residue, having no poles or zeroes along any component of $D^+$. At this point we recall that the residue is independent of the parameter system (see, e.g. \cite{L}). Thus we may return to the one in which $t_1$ is a local equation of $D$, while $t_2,\ldots,t_n$ are arbitrary. By (\ref{Res:res}) the multidimensional residue in (\ref{Q:Res}) reduces to the one-dimen\-sion\-al $$ (-1)^{n-1}2\pi i\, \mbox{\rm res}_0\,(g_{01}\omega_0|_{{\cal X}})= (-1)^{n-1}2\pi i\,<-g,f>\ , $$ and we end up with $$ Q([\omega],[\eta])=(-1)^{n-1}2\pi i\,<f,g>\ . $$ \ $\displaystyle\Box$\\ \ \par \begin{Cor} \label{zero} $<x,y>=0$ whenever $x\in\Omega\cap \H'_+$ and $y\in K_0$. Consequently, the pairing (\ref{form:symp}) induces $$ (\Omega\cap\H'_+)\times \frac{\Omega}{\Omega\cap\H'_++K_0}\longrightarrow{\bf C}\ , $$ also denoted $<\ ,\ >$, and the surjection $$ H^0(V,\Omega_V^1)\times H^1(V,{\cal O}_V)\longrightarrow (\Omega\cap\H'_+)\times \frac{\Omega}{\Omega\cap\H'_++K_0} $$ transforms the pairing $Q$ on the source into $(-1)^{n-1}2\pi i\,<\ ,\ >$ on the target. \end{Cor} \ $\displaystyle\Box$\\ \ \par \begin{Cor} \label{main:2} The surjection $$ H^1(V,{\bf C})\cong\frac{\Gamma(V,\tilde{\Omega}_V^1(*D))} {d\Gamma(V,{\cal O}_V(*D))}\longrightarrow\frac{\Omega}{K_0} $$ is a symplectic isomorphism. \end{Cor} \ \\ \noindent {\bf Proof.\ \ } The map in question induces a graded surjection \begin{eqnarray*} \lefteqn{H^0(V,\Omega_V^1)\oplus H^1(V,{\cal O}_V)\cong}\\ & \cong &\Gamma(V,\Omega_V^1)\oplus \frac{\Gamma(V,\tilde{\Omega}_V^1(*D))} {\Gamma(V,\Omega_V^1)+d\Gamma(V,{\cal O}_V(*D))}\\ & \longrightarrow & (\Omega\cap\H'_+)\oplus\frac{\Omega}{\Omega\cap\H'_++K_0} \end{eqnarray*} which transforms the polarization form $Q$ on the left into the residue pairing $(-1)^{n-1}2\pi i\,<\ ,\ >$ on the right. However, $Q$ is non-degenerate, i. e. for any non-zero $x\in H^0(V,\Omega_V^1)$ there exists $y\in H^1(V,{\cal O}_V)$ with $Q(x,y)\neq 0$, and similarly for any non-zero $y\in H^1(V,{\cal O}_V)$. Therefore, the image of any such $x$ or $y$ in $\Omega\cap\H'_+$ (resp. in $\Omega/\Omega\cap\H'_++K_0$) cannot be $0$. This shows that the graded map is an isomorphism --- in fact, a symplectic one, --- which implies the same for the original map. \ $\displaystyle\Box$\\ \ \par \section{Extended Hodge structure} In this last section we will show that the above constructions can be completed to a full {\em extended Hodge structure of weight one} (= {\em an extended abelian variety}). We already have $K_0\subset\H'$. Now we use the isomorphism $H^1(V,{\bf C})\cong\Omega/K_0$ to define $U$ as the image of $H^{0,1}\subset H^1(V,{\bf C})$ in $\Omega/K_0$. Put $Z=$ the preimage of $U$ in $\Omega$ under the projection $\Omega\rightarrow\Omega/K_0$. Evidently, $U$ is a complement of $\Omega\cap\H'_+$ in $\Omega/K_0$, and $Z$ is a complement of $\H'_+$ in $\H'$. And since $U$ projects isomorphically onto $\Omega/\Omega\cap\H'_++K_0$ under the projection $$ \frac{\Omega}{K_0}\longrightarrow\frac{\Omega}{\Omega\cap\H'_++K_ 0}\ , $$ $U$ is perfectly paired with $\Omega\cap\H'_+$ under the pairing induced by $<\ ,\ >$ on $\Omega/K_0$. From this we deduce that $Z$ is a maximal isotropic subspace of $\H'$. Furthermore, $\Omega\subseteq K_0^{\perp}$, i. e. $Z\subset K_0^{\perp}$. Thus $Z$ is sandwiched between $K_0$ and $K_0^{\perp}$: $$ K_0\subset Z \subset K_0^{\perp}\ . $$ \begin{Lemma} The subspaces $K_0$ and $\Omega$ of $\H'$ are annihilators of each other with respect to the symplectic form $<\ ,\ >$. \end{Lemma} \ \\ \noindent {\bf Proof.\ \ } By (\ref{zero}) $K_0\subseteq \Omega^{\perp}$. And by (\ref{main:2}) and by non-degeneracy of $Q$, ${<\ ,\ >}$ induces a nondegenerate pairing on $\Omega/K_0$. However, $<\ ,\ >$ also induces a nondegenerate pairing on $\Omega/\Omega^{\perp}$, which is a surjective image of $\Omega/K_0$. But any symplectic surjection of a vector space with a non-degenerate symplectic form must be an isomorphism, as the proof of (\ref{main:2}) shows. Hence $\Omega^{\perp}=K_0$. Now, $\Omega\subseteq K_0^{\perp}$, hence $K_0^{\perp\perp}\subseteq\Omega^{\perp}=K_0$. But $K_0\subseteq K_0^{\perp\perp}$; so $K_0=K_0^{\perp\perp}$. Then $<\ ,\ >$ is nondegenerate on $K_0^{\perp}/K_0^{\perp\perp}=K_0^{\perp}/K_0$. In particular, its maximal isotropic subspaces must be of dimension $\frac{1}{2}\dim\,(K_0^{\perp}/K_0)$. However, $Z/K_0$ is already a maximal isotropic subspace in $K_0^{\perp}/K_0$, and it is only of dimension $g=\frac{1}{2}\dim\,(\Omega/K_0)$. Therefore, $K_0^{\perp}=\Omega$. \ $\displaystyle\Box$\\ \ \par Combining this lemma with Corollary \ref{main:2} yields \begin{Thm} \label{main:two} There is a symplectic isomorphism $$ (H^1(V,{\bf C}),\ Q)\longrightarrow (K_0^{\perp}/K_0,\ (-1)^{n-1}2\pi i<\ ,\ >)\ . $$ \end{Thm} We now introduce the remaining components of the extended abelian variety. Define $\Lambda\subset\Omega/K_0=K_0^{\perp}/K_0$ as the image of the lattice $H^1(V,{\bf Z})$ under the isomorphism $H^1(V,{\bf C})\stackrel{\sim}{\rightarrow}K_0^{\perp}/K_0$, and let $K$ be the preimage of $\Lambda$ under the projection $K_0^{\perp}\rightarrow K_0^{\perp}/K_0$. With this notation we may summarize our results as follows. \begin{Thm} \label{EHS} The triple $(Z,K_0,\Lambda)$ associated to the pointed polarized variety $(V,D,p)$ is an extended abelian variety. \end{Thm} Indeed, the definition of an extended abelian variety in \cite{AD} calls for $Z$ to be a maximal isotropic subspace of $\H'$ with $\H'_+\oplus Z=\H'$ , $K_0$ must be a subspace of $Z$ and $\Lambda$ a lattice in $K_0^{\perp}/K_0$ such that $$ (\Lambda,K_0^{\perp}/K_0,2\pi i\,<\ ,\ >) $$ constitutes a polarized Hodge structure of weight one, with the Hodge decomposition induced by the direct sum decomposition of $\H'$: the $(1,0)$-component of $K_0^{\perp}/K_0$ is $K_0^{\perp}\cap\H'_+$, and the $(0,1)$-component is ${(K_0^{\perp}\cap Z)/K_0}$. These conditions have been established already. \refstepcounter{Thm In \cite{SW} and \cite{AD} the polarization form $Q$ on the first cohomology of a Riemann surface corresponds to $\frac{1}{2\pi i}<\ ,\ >$ instead of our $2\pi i\,<\ ,\ >$. The discrepancy is due to a different convention adopted in these papers: they identify $\Lambda$ with $H^1(X,2\pi i{\bf Z})$ rather than $H^1(X,{\bf Z})$, as we do. \refstepcounter{Thm \label{choices} The construction of an extended abelian variety associated to $V$ obviously depends on the choice of the ample divisor $D$ and the point $p$. It also depends on the coordinate system $(t_1,\ldots,t_n)$ at $p$, without which we would not be able to define the Laurent expansions and hence the map $u$. However, all such choices are equally good for our purposes. We end with one last observation. In the curve case we had $$ K=\{f\in\H'\mid e^{2\pi if}\in u(\Gamma(X-\{p\},{\cal O}_X^*))\}= u(\Gamma(X,{\cal O}_X(*p)/{\bf Z}))\ . $$ It turns out, $K$ admits a similar identification in the multidimensional situation. \begin{Lemma} $K=u(\Gamma(V,{\cal O}_V(*D)/{\bf Z}))$. \end{Lemma} \ \\ \noindent {\bf Proof.\ \ } Let us write $\tilde{K}$ for $u(\Gamma(V,{\cal O}_V(*D)/{\bf Z}))$. Our point of departure in identifying $K$ with $\tilde{K}$ is the exact sequence $$ 0\longrightarrow{\bf Z}_V\longrightarrow{\cal O}_V(*D)\longrightarrow{\cal O}_V( *D)/{\bf Z}\longrightarrow 0\ . $$ The corresponding cohomology sequence reads, in part, $$ H^0(V,{\cal O}_V(*D))\longrightarrow H^0(V,{\cal O}_V(*D)/{\bf Z})\longrightarrow H^1(V,{\bf Z})\longrightarrow H^1(V,{\cal O}_V(*D)) $$ The last term is isomorphic to $H^1(V-D,{\cal O}_V)=0$. Therefore, applying $u$ yields $$ 0\longrightarrow K_0\longrightarrow \tilde{K}\longrightarrow \tilde{K}/K_0\longrightarrow 0\ . $$ Thus $u$ identifies $H^1(V,{\bf Z})$ with $\tilde{K}/K_0$. It is easy to see that $\tilde{K}\subset\Omega$, and we conclude that $\tilde{K}=K$. \ $\displaystyle\Box$\\ \ \par \refstepcounter{Thm Connecting $\Gamma(V,{\cal O}_V(*D)/{\bf Z})$ with $\Gamma(V-D,{\cal O}_V^*)$ by means of the exponential sequence on $V-D$, we also get $$ K=\{f\in\H'\mid e^{2\pi i f}\in u(\Gamma(V-D,{\cal O}_V^*))\}\ . $$

9409

alg-geom/9409008

en

https://arxiv.org/abs/alg-geom/9409008

alg-geom/9409008

Yoshioka Kota

K\=ota Yoshioka

Chamber structure of polarizations and the moduli of stable sheaves on a ruled surface

12 pages, amslatex

In this paper we shall generalize the chamber structure of polarizations defined by Qin, and as an application we shall compute the Picard groups of moduli spaces of stable sheaves on a non-rational ruled surface.

alg-geom

\section{Introduction} Let $X$ be a smooth projective surface defined over $\Bbb C$ and $H$ an ample divisor on $X$. Let $M_H(r;c_1,c_2)$ be the moduli space of stable sheaves of rank $r$ whose Chern classes $(c_1,c_2) \in H^2(X,\Bbb Q)\times H^4(X,\Bbb Q)$ and $\overline{M}_H(r;c_1,c_2)$ the Gieseker-Maruyama compactification of $M_H(r;c_1,c_2)$. When $r=2$, these spaces are extensively studied by many authors. When $r \geq 3$, Drezet and Le-Potier [D1],[D-L] investigated the structure of moduli spaces on $\Bbb P^2$, and Rudakov [R] treated moduli spaces on $\Bbb P^1 \times \Bbb P^1$. In this paper, we shall consider moduli spaces of rank $r \geq 3$ on a ruled surface which is not rational. In particular, we shall compute the Picard group of $\overline{M}_H(r;c_1,c_2)$. Let $\pi:X \to C$ be the fibration, $f$ a fibre of $\pi$ and $C_0$ a minimal section of $\pi$ with $(C_0^2)=-e$. We assume that $e>2g-2$, where $g$ is the genus of $C$. Then $K_X$ is effective, and hence $(K_X,H)<0$ for any ample divisor $H$. In particular, $M_H(r;c_1,c_2)$ is smooth with the expected dimension $2r^2 \Delta-r^2(1-g)+1$. In section 2, we shall generalize the chamber structure of Qin [Q2]. As an application, we shall consider the difference of Betti numbers of moduli spaces on a ruled surface. Although we cannot generalize the method in [Y2, 0] directly, by using Qin's method we can generalize it to any rank case. In [Y2], we computed the number of $\mu$-semi-stable sheaves of rank 2 on a ruled surface defind over $\Bbb F_q$. So, in principle, we can compute the Betti numbers of $M_H(3;c_1,c_2)$ on $\Bbb P^2$. Combining chamber structure with another method, G\"{o}ttsche [G\"{o}] also considered the difference of Hodge numbers (and hence Betti numbers) of moduli spaces of rank 2. Matsuki and Wentworth [M-W] also generalized the chamber structure of polarizations. Combining another chamber structure, they showed that the rational map between two moduli spaces is factorized to a sequence of flips. In sections 4 and 5, we assume that $X$ is a ruled surface which is not rational. Then, in the same way as in [Q1], we can give a condition for the existence of stable sheaves. Since we had computed the Picard group $Pic(\overline{M}_H(r;c_1,c_2))$ in case of $(c_1,f)=0$ [Y3], we assume that $0<(c_1,f)<r$. In section 5, we shall compute the Picard group of $\overline{M}_H(r;c_1,c_2)$, which is a generalization of [Q1] to $r \geq 3$. The proof is the same as that in [D-N]. As is well known, it is difficult to treat the moduli spaces on rational ruled surfaces (cf. [D-L], [R]). However we can also check that $M_H(r;c_1,c_2)$ is emply or not in principle. I would like to thank Professor S. Mori for valuable suggestions. \section{chamber structure} \subsection{}Notation Let $X$ be a smooth projective surface defined over $\Bbb C$. Let $NS(X)$ be the Neron-Severi group of $X$ and $Num(X)=NS(X)/\text{torsion}$. Let $C(X) \subset Num(X) \otimes _{\Bbb Z}\Bbb R$ be the ample cone. We denote the moduli space of stable sheaves of rank $r$ with Chern classes $(c_1,c_2) \in H^2(X,\Bbb Q)\times H^4(X,\Bbb Q)$ by $M_H(r;c_1,c_2)$ and the Gieseker-Maruyama compactification of $M_H(r;c_1,c_2)$ by $\overline{M}_H(r;c_1,c_2)$. We denote the open subscheme of $M_H(r;c_1,c_2)$ consisting of $\mu$-stable sheaves by $M_H(r;c_1,c_2)^{\mu}$ and the open subscheme consisting of $\mu$-stable vector bundles by $M_H(r;c_1,c_2)^{\mu}_0$. For a torsion free sheaf $E$ on $X$, we set $\mu(E)=\frac{c_1(E)}{\operatorname{rk}(E)} \in H^2(X,\Bbb Q)$ and $\Delta(E)=\frac{1}{\operatorname{rk}(E)}(c_2(E)-\frac{\operatorname{rk}(E)-1}{2\operatorname{rk}(E)}c_1(E)^2) \in H^4(X,\Bbb Q)$. For a $x \in H^2(X,\Bbb Q)$, we set $P(x)=(x,x-K_X)/2+\chi(\cal O_X)$. For a scheme $S$, we denote the projection $X \times S \to S$ by $p_S$. \subsection{} In this section, we shall generalize the chamber structure of polarizations in [Q2]. For a torsion free sheaf $E$, we set $\gamma(E):=(\operatorname{rk}(E),\mu(E),\Delta(E)) \in H^0(X,\Bbb Q) \times H^2(X,\Bbb Q) \times H^4(X,\Bbb Q)$. For $\gamma \in \prod_{i=0}^2 H^{2i}(X, \Bbb Q)$, let $M_H^{\gamma}$ be the set of torsion free sheaves $E$ defined over $\Bbb C$ with $\gamma(E)=\gamma$ which is $\mu$-semi-stable with respect to $H$. \begin{lem}\label{lem:1} Let $E$ be a torsion free sheaf which is defined by an extension $ 0 \to F_1 \to E \to F_2 \to 0$. Then $\Delta(E)=\frac{\operatorname{rk}(F_1)}{\operatorname{rk}(E)}\Delta(F_1)+\frac{\operatorname{rk}(F_2)}{\operatorname{rk}(E)}\Delta(F_2)- \frac{\operatorname{rk}(F_1)\operatorname{rk}(F_2)}{2\operatorname{rk}(E)^2}((\mu(F_1)-\mu(F_2))^2)$. \end{lem} \begin{lem} Let $B$ be a subset of $C(X)$. Let $\cal F_B(\gamma)$ be the set of filtrations $F:0 \subset F_1 \subset F_2 \subset \cdots \subset F_{s-1} \subset F_s=E$ which satisfies (1) $\gamma(E)=\gamma$, (2) $\Delta_i=\Delta(F_i/F_{i-1}) \geq 0$ and (3) there is an element $H \in B$ with $(\mu(F_{i-1})-\mu(F_i),H)=0$ for $2 \leq i \leq s$. If $B$ is compact, then $S_B(\gamma)=\{(\gamma(F_1),\cdots,\gamma(F_s))| \text{$F_i$ is the $i$-th filter of $F \in \cal F_B(\gamma)$}\}$ is a finite set. \end{lem} \begin{pf} We denote $\mathrm{gr}_i(F):=F_i/F_{i-1}$ by $E_i$. By using Lemma \ref{lem:1} successively, we see that \begin{equation}\label {eq:2.1} \Delta(E)=\sum_{i=1}^s \frac{\operatorname{rk}(E_i)}{\operatorname{rk}(E)}\Delta(E_i)-\sum_{i=2}^s\frac{\operatorname{rk}(E_{i-1})}{2 \operatorname{rk}(E_i) \operatorname{rk}(E)}((\mu(F_{i-1})-\mu(F_i))^2). \end{equation} By the Hodge index theorem, we get $-((\mu(F_{i-1})-\mu(F_i))^2) \geq 0$ and $-((\mu(F_{i-1})-\mu(F_i))^2)=0$ if and only if $\mu(F_{i-1})-\mu(F_i)=0$. By [F-M, II, Lemma 1.4], the set of ${c_1(F_i)}$ is finite. Hence ${\Delta_i}$ is finite. Therefore $S_B(\gamma)$ is a finite set. \end{pf} \begin{rem} For a filtration $F:0 \subset F_1 \subset F_2 \subset \cdots \subset F_{s-1} \subset F_s=E$ which belongs to $\cal F_B(\gamma)$, $F':0 \subset F_i \subset F_s$ belongs to $\cal F_B(\gamma)$ for $1 \leq i \leq s-1$. In fact \eqref{eq:2.1} implies that $\Delta(F_i)\geq 0$ and $\Delta(F_s/F_i)\geq 0$. \end{rem} \begin{defn} For an element $F:0 \subset F_1 \subset F_2 \subset \cdots \subset F_{s-1} \subset F_s=E$ of $\cal F_{C(X)}(\gamma)$, we define a wall $W^F:=\cup_{i}\{ H \in C(X)|(\mu(F_s)-\mu(F_{i}),H)=0 \}$, where $i$ runs for $1 \leq i \leq s-1$ with $\mu(F_s)-\mu(F_i) \ne 0$. By the above lemma, $\cup_F W^F$ is locally finite. We shall call the connected component of $C(X) \setminus \cup_F W^F$ by chamber. \end{defn} \begin{lem}\label{lem:2} Let $H$ and $H'$ be ample divisors which belong to a chamber $\cal C$. Let $E$ be a $\mu$-semi-stable sheaf with respect to $H$. Then $E$ be also $\mu$-semi-stable with respect to $H'$, and hence we may denote $M_H^{\gamma}$ by $M_{\cal C}^{\gamma}$. \end{lem} \begin{pf} Assume that $E$ is not $\mu$-semi-stable with respect to $H'$. We shall construct a wall which separates $H'$ from $H$. There is a filtration $F$ of $E$ such that $(\mu(F_{i-1})-\mu(F_i),H')>0$, $2\leq i \leq s$ and $\Delta(\operatorname{gr}_i(E))\geq 0$, $1 \leq i \leq s$. In fact, let $F:0 \subset F_1 \subset F_2 \subset \cdots \subset F_{s-1} \subset F_s=E$ be the Harder-Narasimhan filtration of $E$ with respect to $H'$. Then, the Bogomolov-Gieseker inequality implies that $\Delta(F_i/F_{i-1}) \geq 0$. Let $H_t=H'+t(H-H'), 0 \leq t \leq 1$ be a line segment joining $H$ and $H'$. There is a $t_1 \in \Bbb Q$ such that $(\mu(E_i)-\mu(E_{i+1}),H_t)>0$ for $t<t_1$ and $(\mu(E_j)-\mu(E_{j+1}),H_{t_1})=0$ for some $j$. Let $\{F_1',F_2',\dots,F_{s(t_1)}' \}$ be a subset of $\{F_1,F_2,\dots,F_s \}$ such that $(\operatorname{rk}(F_i'),(F_i',H_{t_1})) \in \Bbb Q \times \Bbb Q$, $1 \leq i \leq s(t_1)$ are vertices of the convex hull of $\{(\operatorname{rk}(F_i),(F_i,H_{t_1})) \}_{i=1}^{s(t_1)}$. By using Lemma \ref{lem:1}, we see that $\Delta(F_i'/F_{i-1}') \geq 0$. Assume that $s(t_1) \ne s$. Applying this argument successively, we obtain a filtration $F'':0 \subset F_1'' \subset F_2'' \subset \cdots \subset F_u''=E$ such that $\Delta(F_i''/F_{i-1}'')\geq 0$ and $(\mu(F_i'')-\mu(F_{i+1}''),H_{t'})=0$ for some $t'$ with $0 < t' \leq 1$, moreover $\mu(F_i'')-\mu(F_{i+1}'')\ne 0$. This implies that $H_{t'}$ belongs to a wall, which is a contradiction. \end{pf} \begin{defn} Let $W$ be a wall and $\cal C$ a chamber such that $\overline{\cal C}$ intersects $W$. Let $H$ be an ample divisor belonging to $\overline{\cal C} \cap W$ and $H_1$ an ample divisor which belongs to $\cal C$. $V_{H,\cal C}^{\gamma}$ be the set of $\mu$-semi-stable sheaves with respect to $H$ such that $E$ is not $\mu$-semi-stable with respect to $H_1$ and $\gamma(E)=\gamma$. \end{defn} We shall investigate the set $V_{H,\cal C}^{\gamma}$. We set $H_t=H_1+t(H-H_1)$, and $B=\{H_t| 0 \leq t \leq 1 \}$. For an element $E$ of $V_{\cal C,H}^{\gamma}$, $S_B(\gamma)$ is a finite set. Hence $S=S_B(\gamma) \cup \operatornamewithlimits{\cup}\limits_{F \in \cal F_B(\gamma)}\cup_i S_B(\gamma(\operatorname{gr}_i(E)))$ is a finite set. Then there is a number $t'$ such that for all $t$ with $t' \leq t < 1$, $F$ is the Harder-Narasimhan filtration of $E$ with resrect to $H_t$ if and only if $F$ is that with respect to $H_{t'}$. In fact, let $W^G$ be a wall defined by a $(\gamma(G_1),\cdots,\gamma(G_s))\in S$ and $I=\{H_t|t' \leq t \leq 1\}$ an interval which is contained in $B \setminus \cup_{G}W^G$. Let $F:0 \subset F_1 \subset F_2 \subset \cdots \subset F_{s-1} \subset F_s=E$ be the Harder-Narasimhan filtration of $E$ with respect to $H_{t'}$. In the same way as in the proof of Lemma \ref{lem:2}, there is a subset $\{F_1',F_2',\dots,F_{s'}' \}$ of $\{F_1,F_2,\dots,F_s \}$ such that $F':0 \subset F_1' \subset F_2' \subset \cdots \subset F_{s'}'=E$ belongs to $\cal F_I(\gamma)$. By the choice of $t'$, we get $(\mu(F_i')-\mu(F_{i+1}'),H)=0$. Moreover since $S_I(\gamma(F'_i/F'_{i-1}))$ is a subset of $S$, $\{F_1',F_2',\dots,F_{s'}' \}$ must be $\{F_1,F_2,\dots,F_s \}$. Thus $F$ belongs to $\cal F_B(\gamma)$. If $\operatorname{gr}_i(E)$ is not $\mu$-semi-stable with respect to some $H_t$ with $t'<t \leq 1$, then $t'$ and $t$ are separated by a wall (Lemma 2.3), which is a contradiction. Thus $F$ is the Harder-Narasimhan filtration of $E$ for all $H_t$, $t' \leq t<1$. Therefore we get the following proposition. \begin{prop} Let $C$ be a 2-dimensional vector space such that $C \cap \cal C \ne \emptyset$ and $H \in C$. (1)There is an element $H_1 \in C$ and $V_{H,\cal C}^{\gamma}$ is the set of torsion free sheaves $E$ such that $E$ has the Harder-Narasimhan filtration $F$ with respect to $H_1$ which is also Harder-Narasimhan filtration with respect to $H_t, 0 \leq t<1$, and $F$ belongs to $\cal F_{\{H\}}(\gamma)$. (2) $M_H^{\gamma}=M_{\cal C}^{\gamma} \amalg V_{H,\cal C}^{\gamma}$. \end{prop} \section{Equivariant cohomology of $M_H^{\gamma}$ } \subsection{} Let $C$ be a smooth projective curve with genus $g$ and $\pi:X \to C$ a ruled surface. Let $C_0$ be a minimal section of $\pi$ with $(C_0^2)=-e$. We assume that $e>\chi=2g-2$. In this section, we shall define a cohomology of $M_H^{\gamma}$ and consider the effect of change of polarizations. For a scheme $S$, we denote the projection $S \times X \to S$ by $p_S$. Let $D=nH$, $n \gg 0$ be an ample divisor such that for an element $E \in M_H^{\gamma}$, $E(D)$ is generated by global sections and $H^j(X,E(D))=0$ $j>0$. Let $Q^{\gamma}$ be an open subscheme of $Quot_{\cal O_X(-D)^{\oplus N}/X/\Bbb C}$ such that for a quotient $\cal O_X(-D)^{\oplus N} \to E$, $E$ belongs to $M_H^{\gamma}$, $H^0(X, \cal O_X^{\oplus N}) \cong H^0(X, E(D))$ and $H^j(X, E(D))=0 ,j>0$. Since $(K_X,H)<0$, $Q^{\gamma}$ is smooth. Let $H^*_{GL(N)}(Q^{\gamma},\Bbb Q):=H^*(Q^{\gamma} \times_{GL(N)}E(GL(N)),\Bbb Q)$ be the equivariant cohomology of $Q^{\gamma}$, where $E(GL(N))$ is the universal $GL(N)$-bundle over the classifying space. \begin{lem} $H^*_{GL(N)}(Q^{\gamma},\Bbb Q)$ does not depend on the choice of $Q^{\gamma}$. We denote this cohomology by $\tilde H^*(M_H^{\gamma},\Bbb Q)$ and the Poincar\'{e} polynomial $\sum_i \dim \tilde{H}^i(M_H^{\gamma},\Bbb Q) z^i$ by $\tilde P (M_H^{\gamma},z)$. \end{lem} \begin{pf} Let $Q_i^{\gamma}$ $(i=1,2)$ be an open subscheme of $Quot_{\cal O_X(-D_i)^{\oplus N_i}/X/\Bbb C}$ which satisfies the above conditions. Let $q_i:\cal O_{Q_i^{\gamma} \times X}(-D_i)^{\oplus N_i} \to \cal U_i$ be the universal quotient on $Q_i^{\gamma} \times X$. From the construction, $p_{Q_1^{\gamma}*}\cal U_1(D_2)$ is a locally free sheaf on $Q_1^{\gamma}$. Let $\varphi: \Bbb V=\Bbb V(\cal Hom(\cal O_{Q_1^{\gamma}}^{\oplus N_2}, p_{Q_1^{\gamma}*}\cal U_1(D_2))^{\vee}) \to Q_1^{\gamma}$ be a vector bundle over $Q_1^{\gamma}$ and $h_1:\cal O_{\Bbb V}^{\oplus N_2} \to \varphi^*p_{Q_1^{\gamma}*}\cal U_1(D_2)$ the universal homomorphism. Let $\cal G$ be the open subscheme of $\Bbb V$ such that $h_1$ is an isomorphism. Then $\cal G$ is a principal $GL(N_2)$-bundle over $Q_1^{\gamma}$ and there is a surjection $\cal O_{\cal G \times X}(-D_2)^{\oplus N_2} \to \cal U_1$. For a $S$-valued point of $\cal G$, there is a flat family of quotients $\cal O_{S \times X}(-D_1)^{\oplus N_1} \to \cal E$ and an isomorphism $\cal O_S^{\oplus N_2} \cong p_{S*}\cal E(D_2)$. It defines a surjection $q:\cal O_{S \times X}(-D_2)^{\oplus N_2} \to \cal E$, and conversely for a surjection $q$, it defines an isomorphism $\cal O_S^{\oplus N_2} \cong p_{S*}\cal E(D_2)$. Thus we obtain the following. \begin{equation} \cal G(S)=\left.\left\{(\cal E,q_1,q_2)\left| \begin{aligned} & \text{$\cal E$ is a flat family of coherent sheaves which belong to $M_H^{\gamma}$,}\\ & \text{ and $q_i:\cal O_{S \times X}(-D_i)^{\oplus N_i} \to \cal E$ is a surjective homomorphism.} \end{aligned} \right. \right\}\right/\sim \end{equation} where $(\cal E,q_1,q_2) \sim (\cal E',q_1',q_2')$ if and only if there is an isomorphism $\psi:\cal E \to \cal E'$ with $q_i'=\psi \circ q_i$. Hence $\cal G$ can be regarded as a $GL(N_1)$-bundle over $Q_2^{\gamma}$. For simplicity, we denote $GL(N_i)$ by $G_i$. $\cal G$ has a natural $G_1\times G_2$-action and $\cal G \times_{G_1 \times G_2} EG_1 \cong Q_1^{\gamma} \times_{G_1} EG_1$. Therefore $\cal G \times_{G_1 \times G_2}EG_1 \times EG_2 \to Q_1^{\gamma} \times _{G_1}EG_1$ is a $EG_2$-bundle. Thus $H^*_{G_1\times G_2}(\cal G,\Bbb Q) \cong H^*_{G_1}(Q_1^{\gamma},\Bbb Q)$. In the same way $H^*_{G_1\times G_2}(\cal G,\Bbb Q) \cong H^*_{G_2}(Q_2^{\gamma},\Bbb Q)$. Hence $H^*_{G_1}(Q_1^{\gamma},\Bbb Q) \cong H^*_{G_2}(Q_2^{\gamma},\Bbb Q)$, which implies that $\tilde{H}^*(M_H^{\gamma},\Bbb Q)$ is well defined. \end{pf} \subsection{} Let $\cal C$ be a chamber and $W$ a wall with $\overline{\cal C} \cap W \ne \emptyset $. Let $H$ be an ample divisor on $W$ and $H' \in \cal C$ an ample divisor which is sufficiently close to $H$. For a sequence of $\gamma_i=(r_i,\mu_i,\Delta_i)$, $1 \leq i \leq s$ with $(\mu_i-\mu_{i+1},H)=0$ and $(\mu_i-\mu_{i+1},H')>0$, we set \begin{equation} V_{H,\cal C}^{\gamma_1,\cdots,\gamma_s}=\left\{E \left| \begin{aligned} & \text{$E$ is not $\mu$-semi-stable with respect to $H'$ and for the Harder-}\\ & \text{Narasimhan filtration $F:0 \subset F_1 \subset \cdots \subset F_s=E$, $\gamma(F_i/F_{i-1})= \gamma_i$.} \end{aligned} \right. \right \}. \end{equation} Let $Q^{\gamma_1,\cdots,\gamma_s}$ be the subscheme of $Q^{\gamma}$ such that for a quotient $\cal O_X(-D)^{\oplus N} \to E$, $E$ belongs to $V_{H,\cal C}^{\gamma_1,\cdots,\gamma_s}$, and $\Gamma_{H,\cal C}$ the set of sequence $(\gamma_1,\cdots,\gamma_s)$. By [D-L, 1], $Q^{\gamma_1,\cdots,\gamma_s}$ is a smooth locally closed subscheme of $Q^{\gamma}$. For the same $D$, let $Q^{\gamma_i}$ be an open subscheme of $Quot_{\cal O_X(-D)^{\oplus N_i}/X/ \Bbb C}$ a quotient $\cal O_X(-D)^{\oplus N_i} \to E$ is contained in $Q^{\gamma_i}$ if and only if $E$ belongs to $M_H^{\gamma_i}$. In the same way as in [Y2, Appendix] (cf. [A-B],[K]), we obtain the following theorem. \begin{thm}\label{thm:1} (1) \begin{align*} H_{GL(N)}^*(Q^{\gamma_1,\cdots,\gamma_s},\Bbb Q) & \cong \otimes_iH_{GL(N_i)}^*(Q^{\gamma_i},\Bbb Q)\\ & \cong \otimes_i\tilde H^*(M_{\cal C}^{\gamma_i},\Bbb Q). \end{align*} \newline (2) $d_{\gamma_1,\cdots,\gamma_s}:=\operatorname{codim} Q^{\gamma_1,\cdots,\gamma_s}= -\sum_{i<j}r_ir_j(P(\mu_j-\mu_i)-\Delta_i-\Delta_j)$. \newline (3)$$ \tilde P(M_H^{\gamma},z)=\tilde P(M_{\cal C}^{\gamma},z)+\sum_{(\gamma_1,\cdots,\gamma_s) \in \Gamma_{H,\cal C}}z^{2d_{\gamma_1,\cdots,\gamma_s}}\prod_{i=1}^s\tilde P(M_{\cal C}^{\gamma_i}, z).$$ \end{thm} \begin{pf} The proof is the same as that in [Y2, Appendix], so we shall give a sketch of the proof. Let $q_i:\cal O_{Q^{\gamma_i} \times X}(-D)^{\oplus N_i} \to \cal F_i$ be the universal quotient. We set $Z=\prod_{i=1}^sQ^{\gamma_i}$ and denote the $i$-th projection by $\varpi_i$. Then the quotient $\oplus_i q_i:\oplus_{i=1}^s \varpi_i^* \cal O_{Q^{\gamma_i} \times X}(-D)^{\oplus N_i} \to \oplus_{i=1}^s \varpi_i^* \cal F_i$ defines a morphism $Z \to Q^{\gamma}$, which is an immersion. We set $Y_1=Z$. We shall define a sequence of schemes $Y_s \to \cdots \to Y_2 \to Y_1$ and quotients $\oplus _{j=1}^i\cal O_{Y_j \times X}(-D)^{\oplus N_j} \to \cal F_{1,2,\cdots,i}$ $1 \leq i \leq s$ as follows. Let $\psi_2:Y_2 \to Y_1$ be the vector bundle defined by a locally free sheaf $\operatorname{Hom}_{p_{Y_1}}(\varpi_2^*(\ker q_2), \varpi_1^* \cal F_1)$. There is a family of quotients $q_{1,2}:\oplus_{i=1}^2 \cal O_{Y_2 \times X}(-D)^{\oplus N_i} \to \cal F_{1,2}$, which induces $q_1$ and $q_2$. For $\psi_i:Y_i \to Y_{i-1}$ and $\oplus _{j=1}^i\cal O_{Y_j \times X}(-D)^{\oplus N_j} \to \cal F_{1,2,\cdots,i}$, $\operatorname{Hom}_{p_{Y_i}}(\ker q_{i+1}, \cal F_{1,2,\cdots,i})$ is a locally free sheaf on $Y_i$. Let $q_{i+1}:Y_{i+1} \to Y_i$ be the associated vector bundle on $Y_i$. Then there is a quotient $\oplus _{j=1}^{i+1}\cal O_{Y_j \times X}(-D)^{\oplus N_j} \to \cal F_{1,2,\cdots,i+1}$. Let $P_{\gamma_1, \cdots ,\gamma_s}$ be the parabolic subgroup of $GL(N)$ which preserves the filtration $0 \subset \cal O_{Z \times X}(-D)^{\oplus N_1} \subset \oplus _{i=1}^2 \cal O_{Z \times X}(-D)^{\oplus N_i} \subset \cdots \subset \oplus_{i=1}^s \cal O_{Z \times X}(-D)^{\oplus N_i}$. Then $Q^{\gamma_1, \cdots ,\gamma_s} \cong GL(N) \times_{ P_{\gamma_1, \cdots ,\gamma_s}}Y_s$. The assertions follow from this (cf. [A-B, 7]). \end{pf} \begin{cor} Let $\cal C'$ be another chamber with $\overline{\cal C'} \cap W \ne \emptyset$. Then $$ \tilde P(M_{\cal C'}^{\gamma},z)=\tilde P(M_{\cal C}^{\gamma},z)+\sum_{(\gamma_1,\cdots,\gamma_s) \in \Gamma_{H,\cal C}}\left\{z^{2d_{\gamma_1,\cdots,\gamma_s}}\prod_{i=1}^s\tilde P(M_{\cal C}^{\gamma_i}, z) -z^{2d_{\gamma_s,\gamma_{s-1},\cdots,\gamma_1}}\prod_{i=1}^s\tilde P(M_{\cal C'}^{\gamma_i}, z) \right \}.$$ \end{cor} \begin{rem} In the same way, we denote the set of $\mu$-semi-stable sheaves defined over $\Bbb F_q$ by $M_{\cal C}^{\gamma}(\Bbb F_q)$. By using [D-R], we see that \begin{multline}\label{eq:7} \sum_{E \in M_{\cal C'}^{\gamma}(\Bbb F_q)}\frac{1}{\# \operatorname{Aut}(E)}= \sum_{E \in M_{\cal C}^{\gamma}(\Bbb F_q)}\frac{1}{\# \operatorname{Aut}(E)}\\ +\sum_{(\gamma_1,\cdots,\gamma_s) \in \Gamma_{H,\cal C}}\left\{q^{d_{\gamma_s,\gamma_{s-1},\cdots,\gamma_1}} \prod_{i=1}^s \sum_{E \in M_{\cal C}^{\gamma_i}(\Bbb F_q)}\frac{1}{\#\operatorname{Aut}(E)} -q^{d_{\gamma_1,\cdots,\gamma_s}}\prod_{i=1}^s \sum_{E \in M_{\cal C'}^{\gamma_i}(\Bbb F_q)}\frac{1}{\#\operatorname{Aut}(E)} \right \}. \end{multline} By using the Weil conjectures [De] and results of Kirwan [K], we can also obtain this corollary (cf. [Y2, Proposition 4.3]). By using \eqref{eq:7}, [Y2, Theorem 0.1] and analoguous argument to the proof of [Y1, Proposition 0.3], in principle, we can compute the Betti numbers of the moduli spaces of stable sheaves of rank 3 on $\Bbb P^2$ (in case of $c_1=0$, see [Y2, 4]). For example, we obtain the following. \begin{align*} P(M_H(3;1,2),z) &=1+z^2+z^4,\\ P(M_H(3;1,3),z) &=1+2z^2+5z^4+8z^6+10z^8+8z^{10}+5z^{12}+2z^{14}+z^{16}, \\ P(M_H(3;1,4),z) &=1+2z^2+6z^4+12z^6+24z^8+38z^{10}+54z^{12}+59z^{14}\\ &\qquad \qquad+54z^{16}+38z^{18}+24z^{20}+12z^{22}+6z^{24}+2z^{26}+z^{28}. \end{align*} \end{rem} \begin{rem} Let $X$ be a K3 or an Abelian surface and assume that Bogomolov-Gieseker inequality holds. Then \eqref{eq:7} also holds. By using induction on $r$, we see that $\sum\limits_{E \in M^{\gamma}_{\cal C}(\Bbb F_q)}\frac{1}{\#\operatorname{Aut}(E)}$ does not depend on $\cal C$ (cf. [G\"{o}]). \end{rem} \section{The existence of stable sheaves.} \subsection{} In this section, we assume that $X$ is not rational, that is, $g \geq 1$ and assume that $e>-\chi=2g-2$. We denote $C_0+xf$ by $H_x$. Let $W_x$ be a wall containing $H_x$ and let $\cal C_x^+$ (resp. $\cal C_x^-$) a chamber containing $H_x+\epsilon f$ (resp. $H_x-\epsilon{}f$) with $0< \epsilon \ll 1$. For $(\gamma{}_1,\cdots,\gamma{}_s) \in \Gamma{}_{H_x,\cal C_x^-}$, we shall prove that $d_{\gamma{}_1,\cdots,\gamma{}_s} \geq 2$. Since $(\mu{}_j-\mu{}_i)^2<0$ and $\Delta{}_i \geq 0$, it is enough to prove that $r_ir_j(\mu{}_j-\mu{}_i,K_X/2) \geq 1$ for $i <j$. We denote $r_ir_j(\mu{}_j-\mu{}_i)$ by $aC_0-bf$, and then $a$ and $b$ are positive integer. A simple calculation shows that $r_ir_j(\mu{}_j-\mu{}_i,K_X/2)=b+(g-1+e/2)a \geq 3/2$. Therefore $d_{\gamma{}_1,\cdots,\gamma{}_s} \geq 2$. In particular, if $M_{\cal C_x^+}^{\gamma{}}$ is not empty, then $M_{\cal C_x^-}^{\gamma{}}$ is not empty. {}From this we obtain the following proposition. \begin{prop}\label{prop:1} For a triplet $\gamma=(r,\mu,\Delta)$ with $0<(\mu,f)<1$, there exists a $\mu$-semi-stable sheaf $E$ of $\gamma(E)=\gamma$ with respect to $H_x$ if and only if $x \leq \frac{e}{2}+\frac{r^2}{r_1r_2} \Delta$. \end{prop} \begin{pf} Assume that $M_{H_x}^{\gamma}$ is not emty. Since $(K_X+f,H)<0$, the deformation theory implies that there is a $\mu$-semi-stable sheaf $E$ and an exact sequence \begin{equation} 0 \to F_1(C_0) \to E \to F_2 \to 0, \end{equation} where $F_1$ and $F_2$ are torsion free sheaves with $(\mu(F_1),f)=(\mu(F_2),f)=0$. We denote $\operatorname{rk}(F_i), \mu(F_i)$ and $\Delta(F_i)$ by $r_i,\mu_i$ and $\Delta_i$ respectively. $(\mu(F_1(C_0))-\mu(F_2),H)=(\mu_1-\mu_2+C_0,C_0+xf)=(\mu_1-\mu_2,C_0)-e+x \leq 0$. Thus $x \leq -(\mu_1-\mu_2,C_0)+e$. On the other hand, $\Delta(E)=\frac{r_1}{r} \Delta(F_1(C_0))+\frac{r_2}{r} \Delta_2 -\frac{r_1r_2}{2r^2}((\mu(F_1(C_0))-\mu(F_2))^2) \geq -\frac{r_1r_2}{2r^2}(-e+2(\mu_1-\mu_2,C_0))$. Thus $\Delta(E) \geq \frac{r_1r_2}{2r^2}(x-e/2)$, and hence $x \leq \frac{e}{2}+\frac{r^2}{r_1r_2} \Delta$. We shall next prove that the above condition is sufficient. Let $E$ be a vector bundle defined by the following exact sequence. $$ 0 \to F_1(C_0) \to E \to F_2 \to 0, $$ where $F_1$ (resp. $F_2$) is the pull-back of a semi-stable vector bundle of rank $r_1$ (resp. $r_2$) on $C$ with degree $d_1=r_1d+\frac{r_1^2-r_1}{2}e-c_2$ (resp. $d-d_1$). Then $E$ is $\mu$-semi-stable with respect to $H'=C_0+(\frac{e}{2}+\frac{r^2}{r_1r_2} \Delta(E))f$. For general $H_x$, the claim follows immediately. \end{pf} \begin{cor}\label{cor:1} If $H$ is not on a wall, then there is a $\mu$-stable sheaf. Moreover, $\operatorname{codim}(\overline{M}_H(r;c_1,c_2) \setminus M_H(r;c_1,c_2)^{\mu}) \geq 2$. \end{cor} \begin{pf} We may assume that $r \geq 4$. Let $F:0 \subset F_1 \subset F_2 \subset \cdots \subset F_s=E$ be a Jordan-H\"{o}lder filtration of a $\mu$-semi-stable sheaf $E$. We set $E_i=F_i/F_{i-1}$. Then $-\chi(E_i,E_j)=\operatorname{rk}(E_i)\operatorname{rk}(E_j)(g-1+\Delta(E_i)+\Delta(E_j))$. By Proposition \ref{prop:1}, we see that $\operatorname{rk}(E_i)\Delta(E_i) > e/4$. Hence $-\chi(E_i,E_j)>(\operatorname{rk}(E_i)+\operatorname{rk}(E_j))e/4\geq e \geq 1$. The claim follows from this, (cf. [D-L] and [Y3, Proposition 2.3]). \end{pf} \begin{cor}\label{cor:2} If $H$ is not on a wall, then $\overline{M}_H(r,c_1,c_2)$ is locally factorial. \end{cor} \begin{pf} This follows from the above corollary and [D2]. \end{pf} \begin{rem}\label{rem:2} For $(\gamma_1,\cdots ,\gamma_s) \in \Gamma_{H_x,\cal C_x^-}$, $d_{\gamma_1,\cdots ,\gamma_s} \geq 3$. To prove this assertion, we may assume that $s=2$. In the same way, we denote $r_1r_2(\mu_2-\mu_1)$ by $aC_0-bf$. Since $r_1r_2(\mu_2-\mu_1,K_X/2)=b+((g-1)+e/2)a$, the assertion holds unless $a=b=1$. Assume that $a=b=1$, and then $(\mu_1,f)$ or $(\mu_2,f)$ is not an integer. Hence we may assume that $(\mu_1,f)$ is not an integer. By Proposition \ref{prop:1}, $r_1 \Delta_1 \geq \frac{1}{2}(x-\frac{e}{2})$. Since $x=b/a+e=e+1$, $r_1 \Delta_1 >1/2$. Therefore, we get $d_{\gamma_1,\gamma_2} \geq 3$. \end{rem} \section{The Picard group of $\overline{M}_H(r;c_1,c_2)$} \subsection{} In this section, we shall compute the Picard group of $\overline{M}_{H_x}(r;c_1,c_2)$ under the assumption that $H_x$ does not lie on a wall. By Corollary \ref{cor:1} and \ref{cor:2}, $Pic(\overline{M}_{H_x}(r;c_1,c_2))=Pic(M_{H_x}(r;c_1,c_2)^{\mu})$. By using [Y1, Theorem 0.4], we see that $\operatorname{codim}(M_{H_x}(r;c_1,c_2)^{\mu} \setminus M_{H_x}(r;c_1,c_2)^{\mu}_0) \geq r-1$, and hence we shall compute $Pic(M_{H_x}(r;c_1,c_2)^{\mu}_0)$. For a $\mu$-stable vector bundle $E \in M_{H_x}(r;c_1,c_2)^{\mu}_0$, $E^{\vee}$ is $\mu$-stable. Hence $Pic(M_{H_x}(r;c_1,c_2)^{\mu}_0) \cong Pic(M_{H_x}(r;-c_1,c_2)^{\mu}_0)$. Therefore we may assume that $0<(r,f) \leq r/2$. For $c_1=r_1C_0+df$, we set $r_2=r-r_1,d_1=r_1d+\frac{r_1^2-r_1}{2}e-c_2$ and $d_2=d-d_1$. We shall first define a morphism $\overline{M}(r;c_1,c_2) \to J^{d_1} \times J^{d_2}$. Let $\cal Q$ be an open subscheme of quot-scheme $Quot_{V/X/\Bbb C}$ such that $\overline{M}_{H_x}(r;c_1,c_2) =\cal Q/PGL(N)$ and $V \otimes \cal O_{\cal Q \times X} \to \cal U$ the universal quotients. $\cal L=\det((1_{\cal Q} \times \pi)_! \cal U(-C_0))$ is a line bundle on $\cal Q \times C$. It defines a morphism $\lambda_{\cal Q}:\cal Q \to J^{d_1}$. It is easy to see that $\lambda_{\cal Q}$ is $PGL(N)$-invariant. Thus we get a morphism $\lambda:\overline{M}(r;c_1,c_2) \to J^{d_1}$. The line bundle $\det \cal U \otimes \cal O(-r_1C_0)\otimes \cal L^{\vee}$ defines a morphism $\nu: \overline{M}(r;c_1,c_2) \to J^{d_2}$. Therefore we obtain the required morphism $\lambda \times \nu:\overline{M}(r;c_1,c_2) \to J^{d_1} \times J^{d_2}$. For simplicity, we denote $M_H(r;c_1,c_2)_0^{\mu}$ by $M_0^{\mu}$. We set $$ M^0=\{E|E \in M_0^{\mu} \text{ and }E_{|\pi{}^{-1}(P)} \cong \cal O_{\pi{}^{-1}(P)}(1)^{\oplus r_1} \oplus \cal O_{\pi{}^{-1}(P)}^{\oplus r_2} \text{ for all $P \in C$} \}. $$ Assume that $r_1 \ne 1$. Since $e >\chi$, we get $(K_X+f,H_x)<0$. Then, by the deformation theory, we see that $\operatorname{codim}(M_0^{\mu} \setminus M^0)\geq 2$. Next we assume that $r_1=1$. For a fibre $l$, we set $$ Z_l=\{E|E \in M_0^{\mu},\;E_l \cong \cal O_l(1)^{\oplus 2} \oplus \cal O_l(-1) \oplus \cal O_l^{\oplus r_2-2} \}. $$ Then we see that $\operatorname{codim} Z_l=2$ unless $Z_l=\emptyset$. We set $Z=\cup_{P \in C}Z_{\pi{}^{-1}(P)}$. $Z$ is a locally closed subscheme of $M_0^{\mu}$. For a point $E$ of $Z$, there is an exact sequence $0 \to L(C_0) \to E \to F \to 0$ , where $L$ is a line bundle with $c_1(L)=(d_1+1)f$ and $F$ is a torsion free sheaf with $c_1(F)=(d_2-1)f$ and $c_2(F)=1$. We set $x_0=\frac{e}{2}+\frac{r^2}{r_1r_2} \Delta$ and $x_1=\frac{e}{2}+\frac{r^2}{r_1r_2} (\Delta-\frac{1}{r})$. If $x_1<x<x_0$, then $(H_x,\mu{}(L(C_0)))>(H_x,\mu{}(E))$, which is a contradiction. Thus $Z=\emptyset$. Assume that $x<x_1$. Then for some $L$ and $F$ such that $F$ is semi-stable, there is an exact sequence \begin{equation}\label{eq:2} 0 \to L(C_0) \to E \to F \to 0 \end{equation} such that $E$ is semi-stable (see the proof of Proposition \ref{prop:1}). Thus $Z$ is not empty. Therefore, to compute the Picard group of $\overline{M}_{H_x}(r;c_1,c_2)$, it is enough to consider $Pic(M^0 \cup (Z^0 \cap M_0^{\mu}) )$, where $Z^0$ is the subscheme of $M_H(r;c_1,c_2)$ consisting of stable sheaves which are defined by the exact sequence \eqref{eq:2}. \subsection{} We set $V_i=\cal O_X(-nH_x)^{\oplus N_i}$, $(i=1,2$). Let $Quot_{V_1/X/\Bbb C}^{\gamma_1}$ ( resp. $Quot_{V_2/X/\Bbb C}^{\gamma _2}$) be a quot-scheme parametrizing all quotients $V_1\to F_1$ (resp. $V_2 \to F_2$) such that $\gamma(F_1)=(r_1,C_0+\frac{d_1}{r_1}f,0)$ (resp. $\gamma(F_2)=(r_2,\frac{d_2}{r_2}f,0)$). Let $Q_i$ ($i=1,2$) be the open subscheme of $Quot_{V_i/X/\Bbb C}^{\gamma_i}$ consisting all quotients $V_i \to F_i$ which satisfy \begin{enumerate} \item $F_i$ is $\mu$-semi-stable with respect to $H_x$, \item $F_{i|\pi^{-1}(\eta)}$ is a semi-stable vector bundle, where $\eta$ is the generic point of $C$, \item $H^0(X,V_i(nH_x)) \cong H^0(X,F_i(nH_x))$, $H^j(X,F_i(nH_x))=0,j>0$. \end{enumerate} Let $V_i \otimes \cal O_{Q_i \times X} \to \cal F_i$ be the universal quotient, and $\cal K_i$ the universal subsheaf. If we choose a sufficiently large integer $n$ and a suitable $N_i$, then all $\mu$-semi-stable sheaves which satisfy (ii) are parametrized by $Q_i$. We set $\cal F_1'=\cal F(-C_0)$ and $\cal F_2'=\cal F_2$. Let $g_i:G_i= Gr(p_{Q_i*}(\cal F_i'),r_i-1) \to Q_i$ be the grassmannian bundle over $Q_i$ parametrizing rank $r_i-1$ subbundle of $p_{Q_i*}(\cal F_i')$ and $\cal U_i$ the universal subbundle of rank $r_i-1$. Since $p_{Q_i*}(\cal F_i')$ is $PGL(N_i)$-linearized, $G_i$ is $PGL(N_i)$-linearized. Let $G_i'=\{x \in G_i|\cal U_x \otimes \cal O_X \text{ is a subbundle of }(\cal F_i')_x \}$. Let $h_i:D_i=\Bbb P(\cal Hom (\cal O_{G'_i}^{(r_i-1)},\cal U)^{\vee})\to G'_i$ be a projective bundle and $\cal O_{D_i}(1)$ the tautological line bundle on $D_i.$ On $D_i$, there is a homomorphism $\delta:\cal O_{D_i}^{\oplus (r_i-1)} \to h_i^*\cal U \otimes \cal O_{D_i}(1)$. Let $D_i'=\{x \in D_i |\delta_x \text{ is an isomorphism} \}$ be an open set of $D_i$. Setting $\widetilde{\cal F_i'}=(g_i \circ h_i \times 1_X)^*\cal F_i'$ and $\widetilde{\cal U}=p^*_{D_i'} h_i^* \cal U$, there is an injective homomorphism on $D_i' \times X$: $\cal O_{D_i' \times X}^{\oplus (r_i-1)} \to \widetilde{\cal U} \otimes p_{D_i'}^*\cal O_{D_i'}(1) \to \widetilde{\cal F_i'} \otimes p_{D_i'}^*\cal O_{D_i'}(1)$. The quotient $\widetilde{\cal F_i'}\otimes p_{D_i'}^*\cal O_{D_i'}(1)/ \cal O^{\oplus (r_i-1)}_{D_i' \times X}$ is a flat family of line bundles of degree $d_i$. Thus we obtain an extension \begin{equation} 0 \to \cal O^{\oplus (r_i-1)}_{D_i' \times X} \to \widetilde{\cal F_i'}\otimes p_{D_i'}^*\cal O_{D_i'}(1) \to \det(\widetilde{\cal F_i'}\otimes p^*_{D_i'}\cal O_{D_i'}(1)) \to 0. \label{eq:1} \end{equation} We set $Q=Q_1 \times Q_2$, $D=D_1' \times D_2'$ and $I=PGL(N_1)\times PGL(N_2)$. Then, in the same way as in [D-N, 7.3.4], we obtain the following exact sequence: \begin{equation} 0 \to Pic^{I}(Q_1 \times Q_2) \to Pic^{I}(D_1 \times D_2) \to T \to 0, \label{eq:5} \end{equation} where $T$ is a finite abelian group with $\#T=\frac{(r_1-1)d_1}{n_1}\frac{(r_2-1)d_2}{n_2}$. Let $\cal P_i$ be a poincar\'{e} line bundle of degree $d_i$ on $J^{d_i}\times X$. Let $\cal V_i:=\operatorname{Ext}^1_{p_{J^{d_i}}}(\cal P_i,\cal O_{J^{d_i} \times X}^{\oplus (r_i-1)})$ be the relative extension sheaf on $J^{d_i}$. The base change theorem implies that $\cal V_i$ is locally free. Let $\mu_i:\Bbb P_i=\Bbb P(\cal V_i^{\vee}) \to J^{d_i}$ be the projection and $\cal O_{\Bbb P_i}(1)$ the tautological line bundle on $\Bbb P_i$. On $\Bbb P_i$, there is a universal family of extensions: \begin{equation} 0 \to \cal O_{\Bbb P_i \times X}^{\oplus (r_i-1)} \to \cal E_i \to \mu_i^* \cal P \otimes \cal O_{\Bbb P_i}(-1) \to 0. \label{eq:4} \end{equation} We set $\Bbb P_i^{ss}=\{y \in \Bbb P_i|\text{$(\cal E_i)_y$ is semi-stable}\}$. The extension \eqref{eq:1} gives a morphism $k':D_i' \to \Bbb P_i^{ss}$ such that the pull--back of \eqref{eq:4} is \eqref{eq:1}. Since $k'$ is $PGL(N_i)$-invariant and each fibre of $k'$ is an orbit of $PGL(N_i)$, [M-F-K, Proposition 0.2] implies that $\Bbb P_i^{ss}$ is a geometric quotient of $D_i'$ by $PGL(N_i)$. By [Y3, 4.2], $k'$ has a local section. Since the action of $PGL(N_i)$ is set-theoretically free, Z.M.T. implies that $D_i'$ is a Zariski locally trivial fibre bundle. In particular, $Pic^{I}(D)=Pic(\Bbb P_1^{ss}\times \Bbb P_2^{ss})$ ([SGA I, 8]). The base change theorem implies that $\cal V=\operatorname{Ext}^1_{p_Q}(\cal F_2,\cal F_1)$ is a locally free sheaf on $Q$. Let $\Bbb P_Q=\Bbb P(\cal V^{\vee}) \to Q$ be the projective bundle associated to $\cal V^{\vee}$ and $\cal O_{\Bbb P_Q}(1)$ the tautological line bundle. On $\Bbb P(\cal V^{\vee}) $, there is a universal extension \begin{equation} 0 \to \cal F_1 \to \cal E \to \cal F_2 \otimes \cal O_{\Bbb P_Q}(-1) \to 0. \label{eq:6} \end{equation} $I$ acts on $\Bbb P(\cal V^{\vee})$ and $\cal E$ is a $GL(N_1) \times PGL(N_2)$-linearized. Let $\Bbb P_Q^s$ be the open subscheme of $\Bbb P(\cal V^{\vee})$ parametrizing stable sheaves. Then there is a surjective morphism $\lambda:\Bbb P_Q^s \to M^0$. It is easy to see that $\lambda$ is $I$-invariant and each fibre is an orbit of this action. Thus $M^0$ is a geometric quotient of $\Bbb P_Q^s$ by $I$. Let $S \to M^0$ be a smooth and surjective morphism such that there is a universal family $\cal E$. Then there is an exact sequence $0 \to \cal G_1(C_0) \to \cal E \to \cal G_2 \to 0$, where $\cal G_1=\pi^*\pi_* \cal E(-C_0)$ and $\cal G_2=\cal E/\cal G_1(C_0)$. There is an open covering $\{U_i \}$ of $S$ such that $p_{U_i*} \cal G_1$ and $p_{U_i*} \cal G_2$ are free $\cal O_{U_i}$-module. Then it defines a morphism $U_i \to Q$ and hence we get $U_i \to \Bbb P_Q^s$. Therefore we get a local section of $\Bbb P_Q^s \times _{M^0}S \to S$. Since $S \to M^0$ is a smooth morphism, $\Bbb P_Q^s \times _{M^0}S$ is smooth. By using Z.M.T., we see that $\Bbb P_Q^s \times _{M^0}S \to Q$ is a Zariski locally trivial $I$-bundle. By the descent theory ([SGA I, 8]), $\Bbb P_Q^s \to M^0$ is a $I$-bundle. We shall prove that $\operatorname{codim}(\Bbb P_Q\setminus \Bbb P^s_Q) \geq 2$. In the same way as in the proof of Theorem\ref{thm:1}, we define $Y_2=\Bbb V(\operatorname{Hom}_{p_Q}(\cal K_2,\cal F_1)^{\vee})$. Then, there is a quotient $\cal O_{Y_2\times X}(-D)^{\oplus N} \to \cal F_{1,2}$. It defines a closed immersion $Y_2 \hookrightarrow \cal Q$. Let $Y_2^s$ be the open subscheme of $Y_2$ parametrizing all quotients which are stable with respect to $H_x$. Corollary \ref{cor:1} implies that $\operatorname{codim}(Y_2\setminus Y_2^s)\geq 2$. Note that each fibre of $Y_2 \to \Bbb V(\operatorname{Ext}_{p_Q}(\cal F_2,\cal F_1)^{\vee})$ is contained in an orbit of $P_{\gamma_1,\gamma_2}$, where $P_{\gamma_1,\gamma_2}$ is the parabolic subgroup of $GL(N)$ defined in the proof of Theorem \ref{thm:1}. {}From this, we obtain that $\operatorname{codim}(\Bbb P_Q \setminus \Bbb P^s_Q) \geq 2$. On $D_i$, there is a $GL(N_i)$-linearized line bundle such that the action of the center $\Bbb C^{\times}$ is multiplication by constants. In the same way, we obtain the following exact and commutative diagram: \begin{equation} \begin{CD} @[email protected]@.0@. \\ @.@VVV @VVV @VVV @. \\ 0 @>>> Pic^{I}(Q) @>>> Pic^{I}(\Bbb P_Q^s) @>>> \frac{n_1 n_2}{n} \Bbb Z @>>> 0 \\ @. @VVV @VVV @VVV @. \\ 0 @>>> Pic^{I}(D) @>>> Pic^{I}(\Bbb P_D^s) @>>> \Bbb Z @>>> 0 \\ @. @VVV @VVV @VVV @. \\ 0 @>>> T @>>> T' @>>> \Bbb Z/ \frac{n_1 n_2}{n}\Bbb Z @>>> 0 \\ @.@VVV @VVV @VVV @. \\ @.0 @[email protected]@. \end{CD} \end{equation} where $T'$ is a finite abelian group with $\#T'=\frac{(r_1-1)d_1(r_2-1)d_2}{n}$. \subsection{} Let $K(X)$ be the Grothendieck group of $X$. Let $K^0(X)$ be the subgroup of $K(X)$ which is generated by $\cal O_X-\cal O_X(-D)$ and $\cal O_C-\cal O_C(-D)$, $D,D' \in Pic^0(X)$. Then $K^0(X) \cong Pic^0(X)\oplus Alb(X)$. We shall represent the class in $K(X)$ of $\cal O_X,\cal O_X(-f), \cal O_X(-C_0)$ and $\cal O_X(-C_0-f)$ by $e_1,e_2,e_3$ and $e_4$ respectively. Then $K(X) \cong K^0(X) \oplus L$, where $L$ is the free $\Bbb Z$-module of rank 4 generated by $e_i$, $1 \leq i \leq 4$. Let $\varepsilon$ be the class in $K(X)$ of a torsion free sheaf of rank $r$ with Chern classes $c_1,c_2$ and let $K(r;c_1,c_2)$ be the kernel of a homomorphism $K(X) \to \Bbb Z:x \mapsto \chi(\varepsilon \otimes x)$. Let $\cal E$ be a family of stable sheaves of rank $r$ with Chern classes $c_1,c_2$ parametrized by a smooth scheme $S$. Then $\det(p_{S!}(\cal E \otimes x))$, $x \in K(r;c_1,c_2)$ defines a line bundle on $S$. Thus we obtain a homomorphism $\kappa{}_S:K(r;c_1,c_2) \to Pic(S)$. We can also define $\kappa{}:K(r;c_1,c_2) \to Pic(M_H(r;c_1,c_2))$, (see [Y3, 4.3]). $K(r;c_1,c_2)=K^0(X) \oplus K$ where $K=K(r;c_1,c_2) \cap L$. \begin{lem}\label{lem:6} If $r_1 \ne 1$, then $K(r;c_1,c_2) \to Pic(M_H(r;c_1,c_2))/Pic(J^{d_1} \times J^{d_2})$ is surjective. \end{lem} \begin{pf} We denote $Pic^I(\Bbb P_D)/Pic(J^{d_1} \times J^{d_2})$ by $N$. Since $Pic^I(D)/Pic(J^{d_1} \times J^{d_2}) \cong Pic(\Bbb P_1 \times \Bbb P_2)/Pic(J^{d_1} \times J^{d_2}) \cong \Bbb Z^{\oplus 2}$, we get $N \cong \Bbb Z^{\oplus 3}$. We shall prove that $\#(N/\operatorname{im}(\kappa_{\Bbb P_D}))=\#T'$. We denote the image of $\cal O_{\Bbb P_1}(1), \cal O_{\Bbb P_2}(1)$ and $\cal O_{\Bbb P_D}(1)$ to $N$ by $\nu_1,\nu_2$ and $\nu$ respectively. Let $\theta:Pic^I(\Bbb P_D) \to N$ be the quotient homomorphism. We shall define $A_i$ $(1 \leq i \leq 4)$ as follows: \begin{equation} \left\{ \begin{split} A_1 &:=\theta (\det p_{\Bbb P_D!}\cal E)=-((2\chi+2d_1-e)\nu_1+(\chi+d_2)\nu_2+(r_2\chi+d_2)\nu),\\ A_2 &:=\theta (\det p_{\Bbb P_D!}\cal E(-f))=-((2\chi+2d_1-2-e)\nu_1+(\chi+d_2-1)\nu_2+(r_2\chi+d_2-r_2)\nu),\\ A_3 &:=\theta (\det p_{\Bbb P_D!}\cal E(-C_0))=-(\chi+d_1)\nu_1,\\ A_4 &:=\theta (\det p_{\Bbb P_D!}\cal E(-C_0-f))=-(\chi+d_1-1)\nu_1. \end{split} \right. \end{equation} Let $\phi:L \to N $ be a homomorphism such that $\phi(e_i)=A_i$. Then a simple calculation shows that $\phi(K)=\operatorname{im}(\kappa_{\Bbb P_D})$. There is the following exact and commutative diagram \begin{equation} \begin{CD} @[email protected]@.0@. \\ @.@.@VVV @VVV @. \\ @.@. \ker \phi @= \ker \phi @. \\ @.@.@VVV @VVV @. \\ 0 @>>> K @>>> L @> \psi>> n\Bbb Z @>>> 0 \\ @. @| @VV{\phi}V @VVV @. \\ 0 @>>> K @>>> \phi(L) @>>> \phi(L)/K @>>> 0 \\ @.@.@VVV @VVV @. \\ @.@. 0 @.0@. \end{CD} \end{equation} It is easy to see that $\ker \phi$ is generated by $(\chi+d_1-1)e_3-(\chi+d_1)e_4$ and $N/\phi(L) \cong \Bbb Z/(r_2-1)d_2 \Bbb Z$. Hence $\psi(\ker \phi)=(r_1-1)d_1 \Bbb Z$. Therefore $\#N/K=\#(N/\phi(L))\# (\phi(L)/K)=\frac{(r_1-1)d_1(r_2-1)d_2}{n}$, and hence $\# N/K=\#T'$. Thus, we obtain our lemma. \end{pf} \begin{lem}\label{lem:14} The restriction of $\kappa:K(r;c_1,c_2) \to Pic(M(r;c_1,c_2))$ to $K^0(X)$ is injective and its image is $(\lambda \times \det)^*(Pic^0(J^{d_1} \times J^{d_2}))$. \end{lem} \begin{pf} If $D=\sum_ia_i \pi^*(R_i), a_i \in \Bbb Z$, then we see that $\kappa_{\Bbb P_D}(\cal O_X(D)-\cal O_X)= \otimes_i (\cal P_{1R_i}^{\otimes 2}\otimes \cal P_{2R_i})^{\otimes a_i}$ and $\kappa_{\Bbb P_D}(\cal O_{C_0}(D)-\cal O_{C_0}) =\otimes_i \cal P_{1R_i}^{\otimes a_i}$. The assertion follows immediately from this. \end{pf} We shall first consider the case of $g \geq 2$. \begin{thm} Assume that $g \geq 2$ and $H_x$ does not lie on a wall. (1) If $r_1 \ne 1$, then $Pic(\overline{M}_{H_x}(r;c_1,c_2)) \cong Pic(J^{d_1} \times J^{d_2}) \oplus \Bbb Z^{\oplus 3}$. (2) If $r_1=1$, then \begin{equation*} Pic(\overline{M}_{H_x}(r;c_1,c_2)) \cong \left\{ \begin{aligned} & Pic(J^{d_1} \times J^{d_2}) \oplus \Bbb Z^{\oplus 2}, \;x_1<x<x_0\\ & Pic(J^{d_1} \times J^{d_2}) \oplus \Bbb Z^{\oplus 3}, \;x<x_1. \end{aligned} \right. \end{equation*} (3) $Pic(\overline{M}_{H_x}(r;c_1,c_2))/Pic(J^{d_1}\times J^{d_2})$ is generated by the image of $\kappa$. \end{thm} \begin{pf} In the same way as in the proof of [Y3, Lemma 4.2], we see that $\operatorname{codim}(\Bbb P_i^{ss}) \geq 2$, ($i=1,2$) for a sufficiently large $d$. Then (1) follows immediately from Lemma \ref{lem:6} and Lemma \ref{lem:14}. We shall treat the case $r_1=1$. In the same way, we can define $Q_1,Q_2,D_1$ and $D_2$, where $D_1=Q_1$. Moreover we can define a projective bundle $\Bbb P_D$, where $D=D_1 \times D_2$. Then it is easy to see that $Pic^I(\Bbb P_D)/Pic(J^{d_1} \times J^{d_2}) \cong \Bbb Z^{\oplus 2}$ and $\# (Pic^I(\Bbb P_D)/Pic^I(\Bbb P_{Q_1 \times Q_2}))=\frac{(r_2-1)d_2}{n_2}$. We denote $Pic^I(\Bbb P_D)/Pic^I(\Bbb P_{Q_1 \times Q_2})$ by $N$. In the same way as in Lemma \ref{lem:6}, we denote the image of $\cal O_{\Bbb P_2}(1)$ and $\cal O_{\Bbb P_D}(1)$ to $N$ by $\nu_2$ and $\nu$ respectively. Let $\phi:L \to N$ be the homomorphism such that $\phi(e_1)=-((\chi+d_2)\nu_2+(r_2 \chi+d_2)\nu), \phi(e_2)=-((\chi+d_2-1)\nu_2+(r_2 \chi+d_2-r_2)\nu),\phi(e_3)=\phi(e_4)=0$. Then $\phi(K)=\operatorname{im}(\kappa_{\Bbb P_D})$. There is the following exact and commutative diagram: \begin{equation} \begin{CD} @[email protected]@.0@. \\ @.@VVV @VVV @VVV @. \\ 0 @>>> K \cap \ker \phi @>>> \ker \phi @>>> \psi(\ker \phi) @>>> 0\\ @.@VVV @VVV @VVV @. \\ 0 @>>> K @>>> L @> \psi>> \Bbb Z @>>> 0 \\ @. @VVV @VV{\phi}V @VVV @. \\ 0 @>>> \phi(K) @>>> \phi(L) @>>> \phi(L)/K @>>> 0 \\ @.@VVV @VVV @VVV @. \\ @. 0@. 0 @.0@. \end{CD} \end{equation} Since $\ker \phi=\Bbb Z e_3 \oplus \Bbb Z e_4$, we get $\Bbb Z/\psi(\ker \phi)=0$, and hence $\phi(K)=\phi(L)$. A simple calculation shows that $\# N/\phi(L)=\frac{(r_2-1)d_2}{n_2}$, therefore $K \to Pic(M^0)/Pic(J^{d_1} \times J^{d_2})$ is surjective. {}From this we get $Pic(M_{\cal C_{x_0}^-}(r;c_1,c_2))=Pic(J^{d_1} \times J^{d_2}) \oplus \Bbb Z^{\oplus 2}$, and hence we obtain the assertion for $x_1<x<x_0$. We shall next prove the claim for $x<x_1$. It is sufficient to compute $Pic(M_{\cal C_{x_1}^-}(r;c_1,c_2))$. We set $V_i=\cal O_X(-nH_{x_1})^{\oplus N_i}$, $(i=3,4$). Let $Quot_{V_3/X/\Bbb C}^{\gamma_3}$ ( resp. $Quot_{V_4/X/\Bbb C}^{\gamma _4}$) be a quot-scheme parametrizing all quotients $V_3\to F_3$ (resp. $V_4 \to F_4$) such that $\gamma(F_3)=(r_1,C_0+\frac{d_1+1}{r_1}f,0)$ (resp. $\gamma(F_4)=(r_2,\frac{d_2-1}{r_2}f,\frac{1}{r_2})$). Let $Q_i$ ($i=3,4$) be the open subscheme of $Quot_{V_i/X/\Bbb C}^{\gamma_i}$ consisting quotients $V_i \to F_i$ which satisfy \begin{enumerate} \item $F_i$ is $\mu$-semi-stable with respect to $H_x$, \item $F_{i|\pi^{-1}(\eta)}$ is a semi-stable vector bundle, \item $H^0(X,V_i(nH_{x_1})) \cong H^0(X,F_i(nH_{x_1}))$, $H^j(X,F_i(nH_{x_1}))=0,j>0$. \end{enumerate} Let $V_i \otimes \cal O_{Q_i \times X} \to \cal F_i$ be the universal quotient, and $\cal K_i$ the universal subsheaf. If we choose a sufficiently large integer $n$ and a suitable $N_i$, then all $\mu$-semi-stable sheaves which satisfy (ii) are parametrized by $Q_i$. We set $R=Q_3 \times Q_4$. The base change theorem implies that $\cal W=\operatorname{Ext}^1_{p_R}(\cal F_4,\cal F_3)$ is a locally free sheaf on $R$. Let $\Bbb P_R=\Bbb P(\cal W^{\vee}) \to R$ be the projective bundle associated to $\cal W^{\vee}$ and $\cal O_{\Bbb P_R}(1)$ the tautological line bundle. On $\Bbb P(\cal W^{\vee}) $, there is a universal extension \begin{equation} 0 \to \cal F_3 \to \cal E \to \cal F_4 \otimes \cal O_{\Bbb P_R}(-1) \to 0. \label{eq:3} \end{equation} $PGL(N_3)\times PGL(N_4)$ acts on $\Bbb P(\cal W^{\vee})$ and $\cal E$ is a $GL(N_3) \times PGL(N_4)$-linearized. Let $\Bbb P_R^s$ be the open subscheme of $\Bbb P(\cal W^{\vee})$ parametrizing stable sheaves. Then there is a surjective morphism $\lambda:\Bbb P_R^s \to Z^0 \subset M(r;c_1,c_2)$. It is easy to see that $\lambda$ is $PGL(N_3)\times PGL(N_4)$-invariant and each fibre is an orbit of this action. In the notation of {\bf 5.1}, we denote the pull--back of $Z^0$ to $\cal Q$ by $\cal Z^0$. [D-L, 1] implies that $\cal Z^0$ is smooth. Hence $Z^0$ is a geometric quotient of $\Bbb P_R^s$ by $PGL(N_3)\times PGL(N_4)$. On $\cal Z^0$, there is an exact sequence $ 0 \to \cal G_1(C_0) \to \cal U_{|\cal Z^0 \times X} \to \cal G_2 \to 0$, where $\cal G_1$ is a flat family of line bundles with $c_1=(d_1+1)f$ and $\cal G_2$ a flat family of torsion free sheaves of rank $r-1$ with Chern classes $((d_2-1)f,1)$. Then the normal bundle $\cal O_{\cal Z^0}(\cal Z^0)$ is isomorphic to $\operatorname{Ext}^1_{p_{\cal Z^0}}(\cal G_1(C_0),\cal G_2)$. It is easy to see that the pull--back of $\operatorname{Ext}^1_{p_{\cal Z^0}}(\cal G_1(C_0),\cal G_2)$ to $\cal Z^0 \times _{Z^0} \Bbb P_R^s$ is isomorphic to the pull--back of $\operatorname{Ext}^1_{p_{\Bbb P^s_R}}(\cal F_3,\cal F_4 \otimes \cal O_{\Bbb P_R}(-1))$ to $\cal Z^0 \times _{Z^0} \Bbb P_R^s$. Since $Pic(\Bbb P_R^s) \to Pic(\cal Z^0 \times _{Z^0} \Bbb P_R^s)$ is injective, the pull--back of $\cal O_{Z^0}(Z^0)$ to $\Bbb P_R^s$ is isomorphic to $\operatorname{Ext}^1_{p_{\Bbb P^s_R}}(\cal F_3,\cal F_4 \otimes \cal O_{\Bbb P_R}(-1))$. By virtue of Remark \ref{rem:2}, we get $\operatorname{codim}(\Bbb P_R \setminus \Bbb P_R^s) \geq 2$. Let $\lambda' \times \det:\overline{M}(r_2;(d_2-1)f,1) \to C \times J^{d_2-1}$ be the morphism defined in [Y3]. Let $J^{d_1+1}\times C \times J^{d_2-1} \to J^{d_1} \times J^{d_2}$ be the morphism sending $(L,P,L')$ to $(L \otimes \cal O_C(-P),L' \otimes \cal O_C(P))$. Then the composition $Z^0 \to \overline{M}(r_1;(d_1+1)f,0) \times \overline{M}(r_2;(d_2-1)f,1) \to J^{d_1+1}\times C \times J^{d_2-1} \to J^{d_1} \times J^{d_2}$ is the same as the restriction of $\lambda \times \nu$ to $Z^0$. It is easy to see that $K \cap \ker \phi=\Bbb Z((\chi+d_1-1)e_3-(\chi+d_1)e_4)$, and hence $L:=\kappa((\chi+d_1-1)e_3-(\chi+d_1)e_4))$ can be written as $\cal O(nZ^0)\otimes \cal L$, where $\cal L$ is the pull--back of a line bundle on $J^{d_1} \times J^{d_2}$. We shall prove that $n=-1$. Since $\det(\cal E(-C_0))=\det(\cal F_3(C_0))\otimes \det(\cal F_4 \otimes \cal O_{\Bbb P_R}(-1))$, the restriction of $\det(\cal E(-C_0))$ to a fibre of $\Bbb P_R \to R$ is $\cal O(1)$. From this, we see that $n=-1$. (3) follows from the proof of (1) and (2). \end{pf} \subsection{} We shall treat the case of $g=1$. We assume that $(r_1,d_1) \ne 1$ and $(r_2,d_2)=1$, and then there is an integers $r_2'$ and $d_2'$ such that $r_2d_2'-r_2'd_2=1$ and $0<r_2'<r_2$. We set $r_2''=r_2-r_2'$ and $d_2''=d_2-d_2'$. Let $W$ be the subset of $M^0$ whose element $E$ has the following filtration $F:0 \subset F_1 \subset F_2 \subset F_3=E$ such that \begin{enumerate} \item $\operatorname{rk}(F_1)=r_1$, $c_1(F_1(-C_0))=d_1f$ and $c_2(F_1(-C_0))=0$, \item $\operatorname{rk}(F_2/F_1)=r_2'$, $c_1(F_2/F_1)=d_2'f$ and $c_2(F_2/F_1)=0$, \item $\operatorname{rk}(F_3/F_2)=r_2''$, $c_1(F_3/F_2)=d_2''f$ and $c_2(F_3/F_2)=0$. \end{enumerate} If $\frac{r}{r_1}\left(\frac{d_2''}{r_2''}-\frac{d}{r} \right) < x < \frac{d_2}{r_2}-\frac{d}{r}$, then for a general element $E$ of $W$, the Harder-Narasimhan filtration is $F':0 \subset F_2 \subset F_3=E$. Let $Q_2'$ be an open subscheme of $Quot_{V_2/X/\Bbb C}$ whose point $y$ satisfies that $\cal F_y$ is semi-stable or the Harder-Narasimhan filtration of $\cal F_y$ is $0 \subset G_1 \subset \cal F_y$, where $G_1$ is a semi-stable vector bundle of rank $r_2'$ and degree $d_2'$. We shall replace $Q_2$ by $Q_2'$, and construct $\Bbb P_D$ and $\Bbb P_D^s$. Let $\widetilde{W}$ be the open subscheme of $\Bbb P_D \setminus \Bbb P_D^s$ whose point defines an element of $W$. Then, there is an exact sequence $ \Bbb Z \widetilde{W} \to Pic^I(\Bbb P_D) \to Pic^I(\Bbb P_D^s) \to 0.$ In the same way, we see that $\cal O_{\widetilde{W}}(\widetilde{W})$ is a primitive element of $Pic(\widetilde{W})$. Note that $Pic^I(\Bbb P_D)$ is isomorphic to $Pic(\overline{M}_{H_{x'}}(r,c_1,c_2))$, $x'<\frac{r}{r_1}(\frac{d_2''}{r_2''}-\frac{d}{r})$. Therefore $Pic^I(\Bbb P_D^s) \cong Pic(J^{d_1} \times J^{d_2}) \oplus \Bbb Z^{\oplus 3}/\Bbb Z \widetilde{W} \cong Pic(J^{d_1} \times J^{d_2})\oplus \Bbb Z^{\oplus 2}$. In the same way, we obtain the following theorem \begin{thm} If $g=1$, then $Pic(\overline{M}_H(r;c_1,c_2)) \cong Pic(J^{d_1} \times J^{d_2}) \oplus \Bbb Z^{\oplus a}$, $a=1,2$ or $3$. \end{thm}

9409

alg-geom/9409001

en

https://arxiv.org/abs/alg-geom/9409001

alg-geom/9409001

Paul Meurer

The Number of Rational Quartics on Calabi-Yau hypersurfaces in Weighted Projective Space P(2,1^4)

28 pages, AMSLaTeX 1.1

We compute the number of rational quartics on a general Calabi-Yau hypersurface in weighted projective space P(2,1^4). The result agrees with the prediction made by mirror symmetry.

alg-geom

\section{Introduction} \label{intro} In this note we will compute the number of rational quartics (see \secref{quartics}) on a general Calabi-Yau hypersurface in weighted projective space ${{\bold P}(2,1^4)}$. The number was asked for for the first time by Sheldon Katz \cite{katz:ratcur}, and David Morrison \cite{morr:picfux} computed it using mirror symmetry methods. Our result, which agrees with the mirror symmetry computation, is: \begin{thm} There are 6028452 rational quartics on a general Calabi-Yau hypersurface in weighted projective space ${{\bold P}(2,1^4)}$. \end{thm} The method used can be sketched as follows: We show that the irreducible component ${\tilde{\cal H}}_4$ of the Hilbert scheme of ${{\bold P}(2,1^4)}$ containing the rational quartics is smooth and can be embedded into the irreducible component ${\cal H}_4$ of the Hilbert scheme of $\bold P^4$ containing the elliptic quartic curves. There, it can be characterized as one component of the fixed point scheme of a natural involution. On the other hand, ${\cal H}_4$ is well-known and explicitly described by Dan Avritzer and Israel Vainsencher~\cite{avvai}. Together, this leads to an explicit description of ${\tilde{\cal H}}_4$. The number of rational quartics on a general Calabi-Yau hypersurface is given as the integral of the top Chern class of a certain vector bundle on ${\tilde{\cal H}}_4$. It will be computed by a formula of Bott's, which expresses the integral of a homogeneous polynomial in the Chern classes of a bundle on a smooth, compact variety with a $\bold C^*$-action in terms of data given by the induced linear actions on the fibers of the bundle and the tangent bundle in the (isolated) fixed points of the action. {\it Acknowledgement.} I would like to express my thanks to Stein Arild Str{\o}mme for many helpful conversations. \section{Relations between the Hilbert schemes of $X$ and $X/\Gamma$} In this section, we investigate relations between the Hilbert schemes of a projective scheme $X$ and its quotient $X/\Gamma = Y$ w.r.t.\ the action of a finite group $\Gamma$. To each irreducible component ${\cal H}$ of $\bold{Hilb}_{X/\Gamma}$, we will find a subscheme $Z$ of $\bold{Hilb}_X$ mapping birationally onto ${\cal H}$. The morphism $\phi: Z\to {\cal H}$ is generally no isomorphism, but in the situation interesting us primarily (i.e.\ ${\cal H} =$ irreducible component of $\bold{Hilb}_{{{\bold P}(2,1^4)}}$ containing the rational quartics), it actually is an isomorphism.\\ We assume all schemes to be defined over an algebraically closed field $k$ of characteristic 0. First, we state some nice simple properties of finite quotients $\rho: X\to X/\Gamma=Y$ for later reference. \begin{lem}\label{quotients} \begin{enumerate} \item $\rho_*$ and $(\rho_*(\cdot ))^\Gamma$ are exact functors from $\Gamma$-linearized quasicoherent ${\cal O}_X$-modules to quasicoherent ${\cal O}_Y$-modules. \item Let $V\subseteq X$ be $\Gamma$-invariant. Then we have $${\cal O}_{V/\Gamma} = (\rho_*{\cal O}_V)^\Gamma,\qquad {\cal I}_{V/\Gamma} = (\rho_*{\cal I}_V)^\Gamma.$$ \item $(\rho_*(\cdot ))^\Gamma$ commutes with cohomology, i.e. $$\operatorname{H}^i(X,\cal F)^\Gamma = \operatorname{H}^i(Y,(\rho_*\cal F)^\Gamma).$$ \end{enumerate} \end{lem} \begin{pf} (i) $\rho$ is a finite map.\\ (ii) follows immediately from the definition of $V/\Gamma$ and (i).\\ (iii) Note that the functors $\operatorname{H}^i(X,\cdot )^\Gamma$ and $\operatorname{H}^i(Y,(\rho_*(\cdot ))^\Gamma)$ are left-exact and equal for $i=0$. But the category of quasicoherent ${\cal O}_X$-modules has enough injectives, so they are equal for all $i$. \end{pf} \begin{lem}\label{flatquotient} Let $X$ be a quasiprojective scheme and $\Gamma$ a finite group acting on $X$. Let $$ \begin{array}{ccc} V& \hookrightarrow& S\times X\\ \downarrow& \swarrow& \\ S& & \end{array} $$ be a family of subschemes of $X$, flat over $S$ and invariant under the action of $\Gamma$. Then $V/\Gamma \subseteq S\times X/\Gamma$ is flat over $S$. \end{lem} \begin{pf} We can assume that $S$ and $X$ are affine, since flatness is a local property and $X\to Y$ is an affine map. Write $S = \operatorname{Spec} A$, $X = \operatorname{Spec} R_0$, $R=A\otimes R_0$. On every $R$-module $M$, we define an $R^\Gamma$-linear endomorphism $\Phi_M: M\to M$ by \begin{equation}\label{average} \Phi_M(x) := |\Gamma|^{-1}\sum_{\gamma\in\Gamma}\gamma(x). \end{equation} Then $M^\Gamma=\Phi_M(M)$, and $M$ splits as $M = M^\Gamma\oplus \ker\Phi_M$. In particular, we have ${\cal O}_V = \tilde M$ for some $A$-flat $R$-module $M$, so ${\cal O}_{V/\Gamma} = \tilde{M^\Gamma}$, and $M^\Gamma$ is $A$-flat as a direct summand of a flat $A$-module. \end{pf} For a finite group $\Gamma$ acting on a quasiprojective scheme $H$, we can define the {\it fixed point scheme} $H^\Gamma$ of the $\Gamma$-action as follows: If $H=\operatorname{Spec} R$ is affine, then $H^\Gamma$ is defined by the ideal generated by $\ker(\Phi_R)$. In general, cover $H$ by affine invariant open sets $U$ and let $(H^\Gamma)_{|U} := U^\Gamma$. $H^\Gamma$ can be characterized as the maximal subscheme of $H$ having $\Gamma$-invariant structure sheaf. Note that the group $\Gamma$ acts in a natural way on $\bold{Hilb}_X$: if $\lbrack C\rbrack\in\bold{Hilb}_X$ and $\gamma\in\Gamma$, the action is given by $\gamma(\lbrack C\rbrack) := \lbrack\gamma(C)\rbrack$.\\ Let now ${\cal H}_Y$ be an irreducible component of $\bold{Hilb}_Y$ with universal family ${\cal W}$. For any open set $U\subseteq{\cal H}_Y$, denote by ${\cal W}_U$ the restriction of ${\cal W}$ to $U$ and let $\rho_U := \rho\times id_U$. The lift of the family ${\cal W}_U$ to $X$ is given by $\rho_U^{-1}{\cal W}_U = {\cal W}_U\times_{Y\times U}X\times U$. \begin{thm}\label{birational} Suppose that there exists an open subset $U\subseteq{\cal H}_Y$ such that $\rho_U^{-1}{\cal W}_U$ is flat. Then there is a uniquely determined irreducible component $Z$ of $(\bold{Hilb}_X)^\Gamma$ mapping birationally to~${\cal H}_Y$ by the map $\lbrack C\rbrack\mapsto \lbrack C/\Gamma\rbrack$. \end{thm} \noindent{\it Remark.} The existence of such an open set is guaranteed for example if ${\cal H}_Y$ is reduced (cf. \cite{mumford:curves}, Lect.~8: Flattening stratifications). \begin{pf} $\rho_{{\cal H}_Y}^{-1}{\cal W}$ is in general not flat over ${\cal H}_Y$, but by assumption, there is an open subset $U\subseteq {\cal H}_Y$ such that $\rho_U^{-1}{\cal W}_U$ is flat. We can assume that $U = \operatorname{Spec} A$. By the universal property of the Hilbert functor, we thus get a morphism $\psi: U\to {\cal H}_X$, where ${\cal H}_X$ is some irreducible component of $\bold{Hilb}_X$. Moreover, $\psi$ factors through an irreducible component $Z$ of $({\cal H}_X)^\Gamma$. Let ${\cal V}_Z\subseteq X\times Z$ be the restriction of the universal family on ${\cal H}_X$ to $Z$. If we denote $id_X\times\psi$ by $\psi_X$ etc., it is immediate that $\rho_{U*}\psi_X^*\cal F=\psi_Y^*\rho_{Z*}\cal F$ for any coherent sheaf $\cal F$ on $X\times Z$. On the other hand, ${\cal O}_{{\cal W}_U} = (\rho_{U*}\rho_U^*{\cal O}_{{\cal W}_U})^\Gamma$; a simple computation shows then that ${\cal W}_U$ is the pullback of ${\cal V}_Z/\Gamma$ by $\psi$. By \lemref{flatquotient}, ${\cal V}_Z/\Gamma\subseteq Y\times Z$ is flat over $Z$, therefore we get a morphism $\phi$ from $Z$ to the same component ${\cal H}_Y$ such that ${\cal V}_Z/\Gamma$ is the pullback of the universal family ${\cal W}_Y$. Since $\rho_U^{-1}{\cal W}_U$ is flat over $U$, it is again easy to see that ${\cal V}_{\phi^{-1}U}$ is the pullback of $\rho_U^{-1}{\cal W}_U$ by $\phi$. We can summarize the situation in the following commutative diagram: $$ \begin{array}{ccccccccc} \rho_U^{-1}{\cal W}_U & \longrightarrow& {\cal V}_Z & \supseteq & {\cal V}_{\phi^{-1}U} & & \longrightarrow & & \rho_U^{-1}{\cal W}_U\\ \downarrow& &\downarrow& & & & & & \downarrow\\ {\cal W}_U&\longrightarrow &{\cal V}_Z/\Gamma& &\longrightarrow & & {\cal W} & \supseteq & {\cal W}_U\\ \downarrow& &\downarrow& & & &\downarrow& &\downarrow\\ U &\stackrel{\psi}{\longrightarrow} & Z & &\stackrel{\phi}{\longrightarrow} & & {\cal H}_Y & \supseteq & U \end{array} $$ Since all maps in the first and second rows are pullbacks by $\psi$ or $\phi$, it follows by the universal property of the Hilbert functor that $\phi\circ\psi=id_U$ and $\psi\circ\phi_{\phi^{-1}U}=id_{\phi^{-1}U}$. Therefore, $\phi: Z\to{\cal H}_Y$ is birational. \end{pf} {\it Example.} A simple example showing that the map $\phi:Z\to{\cal H}_Y$ dosen't need to be an isomorphism is the following: Take $X=\bold P^2$ and $\Gamma = \{id,\iota\}$, where $\Gamma$ acts on $\bold P^2$ by $\iota x_0=-x_0$, and $\iota x_i=x_i$ for $i=1,2$. Then $Y=X/\Gamma$ is the weighted projective space $\bold P(2,1,1)$ (cf. \defref{def:weighted_pr_sp}). Let ${\cal H}^1_{\bold P(2,1,1)}$ be the Hilbert scheme of subschemes of length 1 in $\bold P(2,1,1)$, which is isomorphic to $\bold P(2,1,1)$ itself, and singular in the point $(1,0,0)$. On the other hand, since the inverse image of a general point in $\bold P(2,1,1)$ is a pair of points in $\bold P^2$, $Z$ lies in the Hilbert scheme ${\cal H}^2_{\bold P^2}$ of subschemes of length 2 in $\bold P^2$, which is a $\bold P^2$-bundle over $\check{\bold P}^2$, hence smooth. By \corref{cor:smooth}, $Z$ is smooth too, hence $Z$ can't be isomorphic to $\bold P(2,1,1)$. In fact, $Z$ is the blow up of $\bold P(2,1,1)$ in the singular point.\\ The following proposition is certainly well known. Since we didn't find a reference for it, we give a proof. \begin{prop} Let $H$ be a smooth scheme and $\Gamma$ a finite group acting on $H$. \begin{enumerate} \item The fixed point scheme $H^\Gamma$ of $\Gamma$ is a smooth subscheme of $H$. \item The Zariski tangent space $\cal T_{H^\Gamma}(x)$ to $H^\Gamma$ in a point $x$ is equal to $\cal T_H(x)^\Gamma$. \end{enumerate} \end{prop} \begin{pf} (i) Let $p\in H^{\Gamma}$ be a closed point and $R = {\cal O}_{p,X}$; consider the induced action of $\Gamma$ on $R$. Then the ideal $\bold a_\Gamma$ of $H^{\Gamma}$ in $R$ is generated by $\ker\Phi_R$. Since $H$ is smooth, $R$ is a regular (noetherian) local ring. We will show that also $R/\bold a_\Gamma$ is regular, which proves the proposition. Let $\bold m\subseteq R$ be the maximal ideal. $\Gamma$ induces a linear action on the vector space $\bold m/\bold m^2 = V$. Let $\hat x_1,\ldots,\hat x_d$ be a base of the subspace of $V$ invariant under $\Gamma$, and complete it by $\hat y_1,\ldots,\hat y_{n-d}$ to a base of $V$, such that the $\hat y_j$ satisfy $\Phi_R(\hat y_j)=0$. This can be achieved by choosing arbitrary $y_j$ and setting $\hat y_j=y_j-\Phi_R(y_j)$. By Nakayama's lemma, the $\hat x_i$, $\hat y_j$ lift to generators $\tilde x_i$, $\tilde y_j$ of $\bold m$. Again we can construct new ring elements by averaging: Let $x_i := \Phi_R(\tilde x_i)$, $y_j := \tilde y_j-\Phi_R(\tilde y_j)$. The images of $x_i$, $y_j$ in $R/\bold m^2$ are $\hat x_i$, $\hat y_j$. We conclude that also $x_i$, $y_j$ generate $\bold m$. We will show that $\bold a := (y_1,\ldots,y_{n-d})$ is equal to $\bold a_\Gamma$; then $R/\bold a_\Gamma$ is a regular local ring, and $H^\Gamma$ is smooth. The ideal $\bold a$ is clearly contained in $\bold a_\Gamma = (\ker\Phi_R)$. On the other hand, suppose that $$\ker\Phi_R\subseteq \bold a + \bold m^r,$$ (this is trivially true for $r=1$), and let $b=\sum_i\alpha_ix_i+\sum_j\beta_jy_j \in\ker\Phi_R$ be an arbitrary element. We can write $$b = b-\Phi_R(b) = \sum (\alpha_i-\Phi_R(\alpha_i))\,x_i + \sum\beta_jy_j-\Phi_R(\beta_jy_j).$$ The first sum is contained in $\bold a+\bold m^{r+1}$ because $\alpha_i-\Phi_R(\alpha_i)$ lies in $\ker\Phi_R\subseteq\bold a +\bold m^r$. The summands $\beta_jy_j$ lie in $\bold a$. Write now \begin{eqnarray*} \Phi_R(\beta_jy_j) &= &\Phi_R\big(\Phi_R(\beta_j)\,y_j+(\beta_j-\Phi_R(\beta_j))\,y_j\big)\\ &=&\Phi_R((\beta_j-\Phi_R(\beta_j))\,y_j)\\ &=&|\Gamma|^{-1}\sum_{\gamma}\gamma(\beta_j-\Phi_R(\beta_j))\,\gamma(y_j). \end{eqnarray*} In the last sum, both factors of the summands lie in $\ker\Phi_R$, hence $\Phi_R(\beta_jy_j)$ lies in $\bold a +\bold m^{r+1}$, too, and together we have that $b\in\bold a +\bold m^{r+1}$, hence $$\ker\Phi_R\subseteq \bold a + \bold m^{r+1}.$$ Since $\bigcap_r\bold m^r=\emptyset$, we get by induction on $r$ that $$\ker\Phi_R\subseteq\bold a.$$ Hence $\bold a_\Gamma = \bold a$, and $H^\Gamma$ is smooth. (ii) is immediate. \end{pf} With the notation of \thmref{birational}, we have \begin{cor}\label{cor:smooth} $Z$ is smooth if ${\cal H}_X$ is smooth. \end{cor} \section{Rational quartics in ${{\bold P}(2,1^4)}$ and elliptic quartics in $\bold P^4$} \label{quartics} \begin{defn}\label{def:weighted_pr_sp} A {\em weighted projective $n$-space} $\bold P(k_0,\ldots, k_n)$ with positive integer weights $k_0,\ldots,k_n$ is defined as $\operatorname{Proj}$ of the graded ring $R := k\lbrack y_0,\ldots,y_n\rbrack$, the variables $y_0,\ldots,y_n$ having weights $k_0,\ldots,k_n$. We can assume that the $k_0,\ldots,k_n$ are coprimal. There is an isomorphism of graded rings $R\cong k\lbrack x_0^{k_0},\ldots,x_n^{k_n}\rbrack$, the $x_i$ having weight 1, and in the following we will work with the $x_i$ rather than the $y_i$. \end{defn} We will use the abbreviation ${\tilde{\bold P}}$ for an arbitrary weighted projective space. \smallskip Let $\Gamma_k$ be the group of $k$-th roots of unity, and let $\Gamma := \Gamma_{k_0}\times\ldots\times\Gamma_{k_n}$. $\Gamma$ acts on $\bold P^n= \operatorname{Proj} k(x_0,\ldots,x_n)$ in the obvious way, namely through the action of $\Gamma_{k_i}$ on the $i$-th projective coordinate by multiplication. ${\tilde{\bold P}}$ can then be described as the geometric quotient of $\bold P^n$ by this action of $\Gamma$: $$\bold P(k_0,\ldots,k_n) = \bold P^n/\Gamma.$$ We denote by $\rho$ the quotient map $\bold P^n\to{\tilde{\bold P}}$, which is a finite ramified covering map, induced by the inclusion $k\lbrack x_0^{k_0},\ldots,x_n^{k_n}\rbrack\subset k\lbrack x_0,\ldots,x_n\rbrack$. The ramification locus of $\rho$ is the union of the fixed point sets of the non-identity elements of $\Gamma$. The singular points of ${\tilde{\bold P}}$ are exactly the points $\stackrel{i}{(0,\ldots,0,1,0,\ldots,0)}$ such that $k_i>1$. We can define twisting sheaves in the usual way by setting ${\cal O}_{\tilde{\bold P}}(r) := R(r)\tilde{\ }$. In general, these sheaves need not to be invertible, but when ${\tilde{\bold P}} = \bold P(k,1^n)$, then ${\cal O}_{\tilde{\bold P}}(k)$ is a very ample line bundle and induces a Veronese type embedding of ${\tilde{\bold P}}$ into a big projective space $\bold P^N$: the image of ${\tilde{\bold P}}$ is the projective cone over the image $v_k(\bold P^{n-1})$ of the Veronese embedding of $\bold P^{n-1}$ by ${\cal O}_{\bold P^{n-1}}(k)$. There is a natural projection from the singular point onto $\bold P^{n-1}$ induced by the inclusion $k\lbrack x_1,\ldots,x_n\rbrack\subset k\lbrack x_0^k,x_1,\ldots,x_n\rbrack$. In $\bold P^N$, this projection is given by projecting the cone down to $v_k(\bold P^{n-1})$ from the point $(1,0,\ldots,0)$.\\ We will need the following \begin{lem} The quotient map $\rho:\bold P^n\to\bold P(k,1^n)$ is flat away from the singular point $(1,0,\ldots,0)$. \end{lem} \begin{pf} $\bold P^n$ can be covered by the $\Gamma$-invariant affine open sets $D_+(x_i)$, $i=0,\ldots,n$, and the union $\bigcup_{i=1}^n D_+(x_i)$ is equal to $\bold P^n-\{(1,0,\ldots,0)\}$. It suffices to show that all the maps $D_+(x_i)\to D_+(x_i)/\Gamma$ are flat for $i=1,\ldots,n$. Let $$R_i= k\Big\lbrack{x_0\over x_i},\ldots,{x_{i-1}\over x_i}, {x_{i+1}\over x_i},\ldots,{x_n\over x_i}\Big\rbrack.$$ Then the invariant ring is $$R_i^\Gamma= k\Big\lbrack\Big({x_0\over x_i}\Big)^{\!k},{x_1\over x_i},\ldots, {x_{i-1}\over x_i},{x_{i+1}\over x_i},\ldots,{x_n\over x_i}\Big\rbrack,$$ and we have $D_+(x_i)=\operatorname{Spec} R_i$, $D_+(x_i)/\Gamma= \operatorname{Spec} R_i^\Gamma$. Thus the ring $R_i$ is equal to $\bigoplus_{s=0}^{k-1}({x_0\over x_i})^s\cdot R_i^\Gamma$, hence a free $R_i^\Gamma$-module, and $D_+(x_i)$ is flat over $D_+(x_i)/\Gamma$. \end{pf} \begin{cor}\label{flatlift} Let $V\hookrightarrow S\times\bold P(k,1^n)$ be a flat family over $S$ such that no fiber of $V$ contains the singular point of $V$. Then $\rho_S^{-1}V\subseteq S\times\bold P^n$ is flat over $S$. \end{cor} \begin{pf} $V$ is a flat family in $\bold P(k,1^n)-\{(1,0,\ldots,0)\}$. The claim follows from the lemma and by transitivity and base change stability of flatness. \end{pf} There is no intrinsic notion of degree of a curve in a weighted projective space ${\tilde{\bold P}}$. In the case of a weighted projective space of type ${\tilde{\bold P}} = \bold P(k,1^n)$, however, we agree to measure the degree of curves (or, more generally, the Hilbert polynomial of subschemes) w.r.t.\ the embedding described above, i.e.\ $\deg C =\deg({\cal O}_{\tilde{\bold P}}(k)_{|C})$.\\ Let us now specialize to the case ${\tilde{\bold P}} = {{\bold P}(2,1^4)}$, the case we are primarily interested in. ${\tilde{\bold P}}=\operatorname{Proj}(k\lbrack x_0^2,x_1,\ldots,x_4\rbrack)$ is the quotient of $\bold P^4$ by the group $\Gamma=\{1,\iota\}\cong\bold Z_2$, the action of $\Gamma$ being given by $\iota x_0=-x_0$, $\iota x_i=x_i$ for $i=1,\ldots,4$. We will consider rational quartics, i.e.\ rational curves of degree (in the above sense) four in ${\tilde{\bold P}}$. Let $C\subseteq{\tilde{\bold P}}$ be a rational quartic and \begin{eqnarray}\label{parametrization} \bold P^1 & \to & C\nonumber \\ (s,t) & \mapsto & \big(f_0(s,t),\ldots,f_4(s,t)\big) \end{eqnarray} a parametrization of $C$. The image of $C$ under the embedding ${\tilde{\bold P}}\hookrightarrow\bold P^N$ induced from ${\cal O}_{\tilde{\bold P}}(2)$ is parametrized as $$(s,t)\quad\mapsto\quad (f_0,f_1^2,f_1f_2,\ldots,f_4^2)$$ and is a degree four curve by definition. Hence $f_0$ has degree four and $f_1,\cdots,f_4$ have degree two. Furthermore, we see that the projection of a rational quartic from the singular point to $\bold P^3$ is a conic in $\bold P^3$. We denote by ${\tilde{\cal H}}_4$ the irreducible component of $\bold{Hilb}_\WP$ containing the rational quartics. (To see that the rational quartics are really contained in {\it one} irreducible component, observe that the parameter space of rational quartic curves (without degenerations) is fibered over the space of conics in $\bold P^3$ (which is irreducible). Each fiber is irreducible, too, as one can seefrom \eqref{parametrization} by specifying a parametrization $(f_1,\ldots,f_4)$ of a conic and letting $f_0$ vary. Hence the space of rational quartics is irreducible, and ${\tilde{\cal H}}_4$ is the irreducible component of $\bold{Hilb}_\WP$ containing that space.) The universal curve on ${\tilde{\cal H}}_4$ will be denoted by ${\cal C}$. Let now $C$ be a general rational quartic in ${\tilde{\cal H}}_4$. The quotient map $\rho^{-1}C\to\rho^{-1}C/\Gamma=C$ exhibits $\rho^{-1}C$ as a double cover of $C$, and the ramification locus consists of the four points where $f_0=0$ in the parametrization \eqref{parametrization}. By Hurwitz's theorem, $\rho^{-1}C$ is an elliptic curve, and it is again a quartic because $\rho^{-1}C$ intersects the hyperplane $\{x_0=0\}$ in four points. Since $C$ doesn't contain the singular point of ${\tilde{\bold P}}$, there is an open subset $U\subseteq{\tilde{\cal H}}_4$ containing $\lbrack C\rbrack$ such that no fiber of ${\cal C}_U$ contains the singular point. By \corref{flatlift}, ${\cal C}_U$ lifts to a flat family $\rho_U^{-1}{\cal C}_U$ in $\bold P^4$. Denote by ${\cal H}_4$ the component of $\bold{Hilb}_{\bold P^4}$ parametrizing the smooth elliptic quartic curves in $\bold P^4$ and their degenerations. According to \thmref{birational}, there is an irreducible subscheme $Z_4$ of ${\cal H}_4$ mapping birationally to ${\tilde{\cal H}}_4$ by $\lbrack C'\rbrack\mapsto \lbrack\rho(C')\rbrack$. Since ${\cal H}_4$ is smooth (see below), it follows by \corref{cor:smooth} that $Z_4$ is smooth, too.\\ \noindent{\it Remark.} It is clear that the above considerations are valid as well for rational quartics in $\bold P(2,1^3)$ instead of $\bold P(2,1^4)$. Thus, if we denote by ${\cal H}_3$ the irreducible component of $\bold{Hilb}_{\bold P^3}$ parametrizing the elliptic quartic curves in $\bold P^3$, there is a smooth irreducible subscheme $Z_3$ of ${\cal H}_3$ mapping birationally to ${\tilde{\cal H}}_3$ by $\lbrack C'\rbrack\mapsto \lbrack\rho(C')\rbrack$.\\ In order to obtain the explicit description of $Z_4$ that we need for the calculations, and particularly to prove the claimed isomorphy of $Z_4$ and ${\tilde{\cal H}}_4$, we will have to look at the description of ${\cal H}_4$ given by Dan Avritzer and Israel Vainsen\-cher:\\ Let $G := \operatorname{Grass}_2(\operatorname{H}^0({\cal O}_{\bold P^3}(2)))$ be the Grassmannian of pencils of quadric surfaces in $\bold P^3$, and denote by $G'$ the image of the (well-defined) map \begin{eqnarray}\label{H3toG'} {\cal H}_3 &\to &\operatorname{Grass}_8(\operatorname{H}^0({\cal O}_{\bold P^3}(3))\\ \lbrack C \rbrack &\mapsto &\lbrack \operatorname{H}^0({\cal I}_C(3))\rbrack\ .\nonumber \end{eqnarray} On $G$, we have a canonical family of subschemes of $\bold P^3$: the fiber in a point $g\in G$ is the base locus of the pencil represented by $g$. In the same way, $G'$ gives rise to a family of subschemes of $\bold P^3$: the fiber in a point $g'$ is the base locus of the linear system of cubic surfaces represented by $g'$. Denote by $B$ the subscheme of $G$ where the family on $G$ is not flat, and denote by $D$ the subscheme of $G'$ where the family defined by $G'$ is not flat ($B$ consists of pencils with a fixed component, and $D$ is the scheme of planes in $\bold P^3$ with an embedded subscheme of length 2). Then we have: \goodbreak \begin{thm}\label{thm:elliptic} \begin{enumerate} \item (Avritzer, Vainsencher \cite{avvai}) ${\cal H}_3$ is isomorphic to a two-fold blow up of $G$. More precisely, $G'$ is isomorphic to the blow up of $G$ along $B$ and ${\cal H}_3$ is isomorphic to the blow up of $G'$ along $D$. The ideal of every (degenerated) curve in ${\cal H}_3$ is generated in degree $\leq 4$. In particular, ${\cal H}_3$ is smooth of dimension~16. \item ${\cal H}_4$ is fibered locally trivially over $\check{\bold P}^4$ in a natural way with fiber ${\cal H}_3$; i.e., the restriction of the universal curve over ${\cal H}_4$ to the fiber over a hyperplane $h\cong\bold P^3$ in $\check{\bold P}^4$ is the universal elliptic quartic curve in $h\cong\bold P^3$. \end{enumerate} \end{thm} \begin{pf} (ii) We have to show that all degenerations of elliptic quartic curves in $\bold P^4$ span exactly a $\bold P^3$. Then the fibration is given by projecting a point $\lbrack C \rbrack\in{\cal H}_4$ to the hyperplane it spans. It is clear that no degeneration can possibly span less than a $\bold P^3$, for then this degeneration would already be contained in ${\cal H}_3$. On the other hand, for a general elliptic curve, $\dim(\operatorname{H}^0(\cal I_C(1)))=1$; if a degeneration $C_0$ spanned all of $\bold P^4$, then $\dim(\operatorname{H}^0(\cal I_C(1)))$ would drop to 0 in $C=C_0$, in contradiction to upper semicontinuity of $\dim(\operatorname{H}^0(\cal I_C(1)))$. The fibration is locally trivial because $\operatorname{PGL}(4)$ acts transitively on $\bold P^4$ and this action lifts to an action on ${\cal H}_4$. \end{pf} As a corollary of this theorem we are now able to derive an analogous description of $Z_3$ and $Z_4$. Consider therefore the inclusion of grassmannians $$\tilde G:= \operatorname{Grass}_2\bigl(\operatorname{H}^0({\cal O}_{\bold P(2,1^3)}(2))\bigr) \hookrightarrow G = \operatorname{Grass}_2\bigl(\operatorname{H}^0({\cal O}_{\bold P^3}(2))\bigr)$$ induced by the natural inclusion \begin{equation}\label{inclusion} i:\operatorname{H}^0({\cal O}_{\bold P(2,1^3)}(2))\hookrightarrow\operatorname{H}^0({\cal O}_{\bold P^3}(2)). \end{equation} \begin{prop}\label{prop:hilbertscheme} $Z_3$ is smooth of dimension 10 and isomorphic to the proper transform of $\tilde G$ under the twofold blow up map $b:{\cal H}_3\to G$. \end{prop} \begin{pf} $\Gamma$ acts on $G$. If $Z'$ is an irreducible component of $G^\Gamma$ not contained in the blow up locus $B$, then the proper transform of $Z'$ is an irreducible component of ${\cal H}_3^\Gamma$. On the other hand, it is easy to see that $\tilde G\subseteq G$ {\it is} a component of $G^\Gamma$. Furthermore, curves in $\tilde G-B$ map to rational quartics; this proves that the proper transform of $\tilde G$ in ${\cal H}_3$ is the right component, i.e.\ equal to $Z_3$. \end{pf} We turn to the explicit description of $Z_3$ as a twofold blow up of $\tilde G$. A pencil in $\tilde B = \tilde G \cap B$ is a pencil generated by two quadratic polynomials $F_1,F_2$ in $\operatorname{H}^0({\cal O}_{{\tilde{\bold P}}^3}(2))$ with a common linear factor, thus we have $F_1 = f_1g$, $F_2 = f_2g$, and they must be independent of $x_0$. It follows that the scheme described by such a pencil projects to the union of a line and a point in $\bold P^2\cong \{x_0=0\}$ (or a degeneration thereof) under projection from the singular point. The image is described by the same equations $F_1 = f_1g$, $F_2 = f_2g$. We conclude that $\tilde B$ is isomorphic to $\check{\bold P}^2\times\bold P^2$ and has dimension 4. According to \cite{avvai}, a plane in $\bold P^3$ with an embedded subscheme of length 2 has ideal $$(x_1(x_1-x_4)x_3, x_2x_3, x_3^2) \qquad\hbox{ or }\qquad (x_1^2x_3, x_2x_3, x_3^2)$$ up to projective equivalence, depending on whether the subscheme has support in two points or one point. Consequently, ideals in $\tilde D$, the intersection of $D$ with the proper transform $\tilde G'$ of $\tilde G$, have, up to $\Gamma$-invariant projective equivalence, the form $$(x_3(ax_0^2+bx_1^2), x_3x_2, x_3^2)\ ,\qquad (a,b)\in \bold P^1.$$ Geometrically, $\tilde D$ parameterizes hyperplanes through $(1,0,0,0)$ with an embedded $\Gamma$-invariant subscheme of length 2. By projecting the hyperplane down to $\bold P^2\cong \{x_0=0\}$, we see that $\tilde D$ is fibered over $\check{\bold P}^2$, with fiber $F$ isomorphic to $\bold P^1\times\bold P^1$. If $F$ is the fiber over the point $\{x_3=0\}$, the isomorphism is given by \begin{eqnarray*} \bold P^1\times\bold P^1\!\quad &\to &\quad F\\ (a,b)\times(c,d) &\mapsto &(x_3(ax_0^2-b(cx_1+dx_2)^2), x_3(dx_1-cx_2),x_3^2)\ . \end{eqnarray*} So $\tilde D$, too, is smooth of dimension 4.\\ \noindent{\it Remark.} This is another proof of the smoothness of $Z_3$. \begin{prop}\label{prop:isomorphy} The map $\phi:Z_3\to{\cal H}_3$ is an isomorphism. \end{prop} \begin{pf} Since $\phi$ is birational and ${\tilde{\cal H}}_3$ is irreducible, it is enough to show that the tangent map $d\phi:\cal T_{Z_3}([C])\to\cal T_{{\tilde{\cal H}}_3}([C/\Gamma])$ is injective in every point. Consider a $\bold C^*$-action on $\bold P^3$ acting diagonally w.r.t.\ the coordinates $x_0,\ldots,x_3$ and having isolated fixed points. We can choose the action in such a way that the induced action on ${\cal H}_3$ has isolated fixed points, the fixed points being given by monomial ideals. Furthermore, the action leaves $Z_3$ invariant, hence descends to an action on ${\tilde{\cal H}}_3$ (more about torus actions in \secref{calc}). Suppose now that $[C]\in Z_3$ is a point where $d\phi$ is not injective. Then $d\phi$ is not injective in any point of $\bold C^*\!\cdot\! [C]$. Let $[C_0]\in\overline{\bold C^*\!\cdot\! [C]}$ be a fixed point in the closure. $d\phi$ cannot be injective in $[C_0]$ neither, because then it would be injective on a whole neighborhood of $[C_0]$ which would contain also points of $\bold C^*\!\cdot\! [C]$. Therefore it suffices to show that $d\phi$ is injective in all points of $Z_3$ defined by monomial ideals. A sufficient condition for this to happen is given by the following \begin{lem} Let $\lbrack C \rbrack\in Z_3$ be a point whose homogeneous ideal $I_C$ is generated by invariant monomials. Assume that there is no invariant monomial $m$ of degree 3 not contained in $I_C$ such that all monomials $x_1m,x_2m,x_3m$ lie in $I_C$. Then $d\phi$ is injective in $\lbrack C \rbrack$. \end{lem} We can check explicitly by looking at all types of monomial ideals, which we will determine in \secref{calc}, that the assumption of the lemma is always met. This is a boring but simple exercise, and we will only point out the reasoning for one case. Take for example the fixed curve $C$ with ideal $I_C= (x_1x_2,x_1x_2,x_0^2x_2)$, and let $m$ be a monomial of degree 3 not contained in $I_C$. Since $m$ is invariant, there are two possibilities: \begin{enumerate} \item $x_0^2$ divides $m$. Then we have $m=x_0^2x_i$, $i=1$ or $3$, and $x_im=x_0^2x_i^2$ is certainly not contained in $I_C$. \item $x_0^2$ does not divide $m$. We have either $x_1\not\:\mid m$, then $x_2m$ or $x_3m \not\in I_C$, because not both can be multiples of $x_0^2x_2$; or $x_1\mid m$, then $m=x_1^3$, and $x_1m\not\in I_C$. \end{enumerate} All other cases can be treated in a similar way. Thus $d\phi$ is injective everywhere, and the proposition is proved. It remains to prove the lemma. Let $[C]\in Z_3$ be a point with homogeneous ideal $I_C=(h_1,\ldots,h_r)$, the $h_i$ being $\Gamma$-invariant monomials of degree $d_i$ (i.e., they contain $x_0$ in even power). There is a natural injection \begin{equation} \cal T_{{\cal H}_3}([C])\hookrightarrow\operatorname{Hom}({\cal I}_C, {\cal O}_C). \end{equation} The resolution of ${\cal I}_C$ defined by $I_C$ gives rise to a diagram \begin{equation} \begin{array}{ccccc} \bigoplus_i{\cal O}_{\bold P^3}(-d_i)& \stackrel{I_C}{\longrightarrow}& {\cal I}_C &\rightarrow & 0\\ &\searrow\bar f &\downarrow f & & \\ & &{\cal O}_C & & \end{array} \end{equation} for every $f\in\operatorname{Hom}({\cal I}_C,{\cal O}_C)$, and hence to an injective map \begin{equation} \operatorname{Hom}({\cal I}_C,{\cal O}_C) \stackrel{j}{\hookrightarrow} \bigoplus_i\operatorname{Hom}({\cal O}_{\bold P^3}(-d_i),{\cal O}_C)\cong \bigoplus_i\operatorname{H}^0({\cal O}_C(d_i)) \end{equation} which sends $f$ to $\bar f$. Now let $I'_C = (h'_1,\ldots, h'_{r'})$ be a homogeneous ideal of $C$ generated by monomials $h'_i$ of {\it even} degree. In concrete terms, construct $I'_C$ from $I_C$ by retaining the generators of even degree and replacing each generator $h_i$ of odd degree by generators $x_0h_i,\ldots, x_3h_i$. It is clear that $I_C$ and $I'_C$ both define the same scheme $C$. Let \begin{equation}\label{I_C} \bigoplus_{i,\lambda}{\cal O}_{\bold P^3}(-d'_{i,\lambda}) \stackrel{I'_C}{\to}{\cal I}_C\to 0 \end{equation} be the corresponding surjection, where $d'_{i,\lambda} = d_i+1$ and $\lambda = 0,\ldots, 3$ if $d_i$ is odd and $d'_{i,\lambda} = d_i$ and $\lambda = -1$ (say) if $d_i$ is even. By applying the exact functor $(\rho_{X*}(\cdot ))^\Gamma$ to \eqref{I_C}, we get a surjective map $$\bigoplus_{i,\lambda}{\cal O}_{\tilde{\bold P}}(-d'_{i,\lambda}) \stackrel{(\Phi(h'_{i,\lambda}))_{i,\lambda}}{\longrightarrow} {\cal I}_{C/\Gamma}\to 0. $$ But when $d_i$ is odd, then $\Phi(h'_{i,0})=\Phi(x_0h_i)=0$, and we don't lose anything if we let the sum run only over indices $(i,\lambda)$ with $\lambda\ne 0$. The diagram \begin{equation} \begin{array}{ccccc} \bigoplus_{i,\lambda}{\cal O}_{\bold P^3}(-d'_{i,\lambda}) & \stackrel{I'_C}{\longrightarrow} &{\cal I}_C & \to & 0\\ \downarrow\mu & & \Arrowvert& & \\ \bigoplus_i{\cal O}_{\bold P^3}(-d_i)& \stackrel{I_C}{\longrightarrow}& {\cal I}_C &\rightarrow & 0 \end{array} \end{equation} induces a diagram \begin{equation} \begin{array}{ccc} \operatorname{Hom}({\cal I}_C,{\cal O}_C) & \stackrel{j}{\hookrightarrow} &\bigoplus_i\operatorname{H}^0({\cal O}_C(d_i)) \\ \Arrowvert & & \downarrow \kappa \\ \operatorname{Hom}({\cal I}_C,{\cal O}_C) & \stackrel{j'}{\hookrightarrow} &\bigoplus_{i,\lambda}\operatorname{H}^0({\cal O}_C(d'_{i,\lambda})). \end{array} \end{equation} The map $\kappa$ sends an element $\ (0,\ldots,\stackrel{i}{m},\ldots, 0)\ $ to $\ (0,\ldots,\stackrel{(i,-1)}{m},\ldots, 0)\ $ if $d_i$ is even and to $\ (0,\ldots,\stackrel{(i,0)}{x_0m},\ldots,\stackrel{(i,3)}{x_3m}, \ldots, 0)\ $ otherwise. The tangent space to $Z_3$ in $[C]$ is equal to $(\cal T_{{\cal H}_3}([C])^\Gamma$, and there is a commutative diagram \begin{equation} \begin{array}{ccc} \cal T_Z([C]) & \hookrightarrow &\operatorname{Hom}({\cal I}_C,{\cal O}_C)^\Gamma \\ \downarrow d\phi & & \downarrow \sigma \\ \cal T_{{\tilde{\cal H}}_3}([C/\Gamma]) & \hookrightarrow & \operatorname{Hom}((\rho_*{\cal I}_C)^\Gamma,(\rho_*{\cal O}_C)^\Gamma) \end{array} \end{equation} (recall that ${\cal I}_{C/\Gamma} = (\rho_*{\cal I}_C)^\Gamma$ etc.). $\sigma$ is induced from the functor $(\rho_{X*}(\cdot ))^\Gamma$. The last two diagrams fit together into a big diagram \begin{equation} \begin{array}{ccccc} \cal T_Z([C]) & \hookrightarrow &\operatorname{Hom}({\cal I}_C,{\cal O}_C)^\Gamma & \stackrel{j}{\hookrightarrow}&\bigoplus_i\operatorname{H}^0({\cal O}_C(d_i))^\Gamma \\ &&& \searrow & \downarrow\kappa\\ \big\downarrow d\phi &&\big\downarrow\sigma && \bigoplus_{i,\lambda}\operatorname{H}^0({\cal O}_C(d'_{i,\lambda}))^\Gamma\\ &&&&\downarrow\tau\\ \cal T_{{\tilde{\cal H}}_3}([C/\Gamma]) & \hookrightarrow & \operatorname{Hom}((\rho_*{\cal I}_C)^\Gamma,(\rho_*{\cal O}_C)^\Gamma)&\hookrightarrow & \bigoplus_{i,\lambda\atop\lambda\ne 0} \operatorname{H}^0({\cal O}_C^\Gamma(d'_{i,\lambda})).\\ \end{array} \end{equation} The map $\tau$ simply forgets the components with indices $(i,0)$ and is an isomorphism on the complement. Now, in order to prove that $d\phi$ is injective, it suffices to show that the composition $\tau\circ\kappa$ is injective. By computing resolutions of all the ideals ${\cal I}_C$ (in Macaulay, for example), we see that they all are at least 3-regular, which means that $\operatorname{H}^p({\cal I}_C(m-p)) = 0$ for all $p>0$, $m\geq 3$; thus $\operatorname{H}^1({\cal I}_C(d))=0$ for $d\geq 2$. Therefore $\operatorname{H}^0({\cal O}_C(d_i))^\Gamma$ is generated by the invariant monomials of $\operatorname{H}^0({\cal O}_{\bold P^3}(d_i))$ not contained in $\operatorname{H}^0({\cal I}_C(d_i))^\Gamma$. Thus it is enough to check that no such monomial is contained in $\ker(\tau\circ\kappa)$. For monomials of even degree this is clear, because $\tau\circ\kappa$ is the identity on the direct summands $\operatorname{H}^0({\cal O}_C(d_i))^\Gamma$ for $d_i$ even. An invariant monomial $m$ (which for short will stand for $(0,\ldots,m,\ldots,0)$) of odd degree $d_i$ ($d_i=3$ in our case for all ideals) is mapped to $[(0,\ldots,x_1m,x_2m,x_3m,\ldots,0)]$ by $\tau\circ\kappa$. Thus $\tau\circ\kappa(m)$ is zero exactly when $x_1m,x_2m,x_3m$ all lie in $\operatorname{H}^0({\cal I}_C(d_i+1))^\Gamma$ (or, what amounts to the same, in $I_C$). \end{pf} Now we are able to deduce the analogue to \thmref{thm:elliptic} (ii). \begin{prop}\label{fiberingofhilb} $Z_4$ is isomorphic to ${\tilde{\cal H}}_4$, and ${\tilde{\cal H}}_4$ is a locally trivial fibration over $\check{\bold P}^3$ with fibers isomorphic to ${\tilde{\cal H}}_3$. Thus ${\tilde{\cal H}}_4$ is smooth of dimension 13. \end{prop} \begin{pf} We can proceed as in the proof of \thmref{thm:elliptic} (ii). A smooth rational quartic $C$ in ${{\bold P}(2,1^4)}$ spans exactly a hyperplane $H\cong\bold P(2,1^3)$ (i.e.\ a hypersurface with a linear equation $l(x_1,\ldots,x_4)$; use the fact that the lift of $C$ to $\bold P^4$ is an invariant elliptic quartic). Thus, the projection of $C$ to $\bold P^3\subseteq\bold P(2,1^3)$ from the singular point spans a hyperplane in $\bold P^3$, i.e.\ is a point in $\check{\bold P}^3$. Again, no degeneration can possibly span less than a hyperplane because it then would already be contained in ${\tilde{\cal H}}_3$. But every degeneration in ${\tilde{\cal H}}_3$ spans the whole $\bold P(2,1^3)$. A semicontinuity argument shows again that no degeneration can span more than a hyperplane. Thus the map \begin{eqnarray*} {\tilde{\cal H}}_4 &\to &\check{\bold P}^3\\ \lbrack C\rbrack &\mapsto & [span(pr(C))] \end{eqnarray*} is welldefined, and clearly a locally trivial fibration. Consider now the map $$ Z_4\to{\tilde{\cal H}}_4\to\check{\bold P}^3 $$ which exhibits $Z_4$ as a locally trivial fibration over $\check{\bold P}^3$, with fiber $Z_3$. But $Z_3$ is isomorphic to ${\tilde{\cal H}}_3$, so $Z_4$ is isomorphic to ${\tilde{\cal H}}_4$. \end{pf} \section{Rational quartics on Calabi-Yau hypersurfaces} {}From now on, we work over the ground field $\bold C$. A hypersurface in ${\tilde{\bold P}} = {{\bold P}(2,1^4)}$ given by a polynomial of weighted degree 6 has trivial canonical bundle, i.e., is Calabi-Yau (the following is the only point where the Calabi-Yau comes into play). Consider the ideal sequence of the universal family $p:{\cal C}\to{\tilde{\cal H}}_4$ of rational quartics in ${\tilde{\bold P}}$: $$0\to\cal I_{\cal C}\to {\cal O}_{{\tilde{\cal H}}_4\times{\tilde{\bold P}}} \to{\cal O}_{\cal C}\to 0\ .$$ By twisting with ${\cal O}(6)$ and taking direct images under $p$, we get the sequence $$0\to p_*\cal I_{\cal C}(6)\to p_*{\cal O}_{{\tilde{\cal H}}_4\times{\tilde{\bold P}}}(6) \to p_*{\cal O}_{\cal C}(6)\to R^1p_*\cal I_{\cal C}(6)\ .$$ If we assume for a moment that $R^1p_*\cal I_{\cal C}(6)$ vanishes and that the (zeroth) direct images are locally free, this sequence reduces to $$0\to p_*\cal I_{\cal C}(6)\to \operatorname{H}^0({\cal O}_{{\tilde{\cal H}}_4\times{\tilde{\bold P}}}(6))_{{\tilde{\cal H}}_4} \stackrel{\rho}{\to} p_*{\cal O}_{\cal C}(6)\to 0\ .$$ Now take a section of $\operatorname{H}^0({\cal O}_{{\tilde{\cal H}}_4\times{\tilde{\bold P}}}(6))_{{\tilde{\cal H}}_4}$ which is induced from a generic section $F$ of ${\cal O}_{{\tilde{\cal H}}_4\times{\tilde{\bold P}}}(6)$ and so represents a Calabi-Yau hypersurface $X_F$. If $\lbrack C\rbrack\in{\tilde{\cal H}}_4$ is a given curve, then the induced section $\rho(F)$ of $p_*{\cal O}_{\cal C}(6)$ restricted to the fiber $\operatorname{H}^0(C, {\cal O}_C(6))$ over $\lbrack C\rbrack\in{\tilde{\cal H}}_4$ is equal to the restriction of $F$ to $C$. Hence, $\rho(F)$ vanishes exactly when $C$ is contained in $X_F$. Since the rank of $p_*{\cal O}_{\cal C}(6)$ equals the dimension of ${\tilde{\cal H}}_4$, Kleiman's Bertini theorem (cf.\ \cite{kleiman:bertini}, Remark 6) implies that the zero scheme of the section $\rho(F)$ is finite and nonsingular; hence the length of this scheme is equal to the number of rational quartics on a generic Calabi-Yau hypersurface in ${\tilde{\bold P}}$. This number is given by the integral \begin{equation}\label{integral} \int_{{\tilde{\cal H}}_4}c_{13}(p_*{\cal O}_{\cal C}(6))\ , \end{equation} $c_{13}$ being the top Chern class. It remains to prove the claimed facts about the direct image sheaves. \begin{prop} $p_*\cal I_{\cal C}(6)$ and $p_*{\cal O}_{\cal C}(6)$ are locally free sheaves, and $R^1p_*\cal I_{\cal C}(6)$ vanishes. \end{prop} \begin{pf} We show first that $\operatorname{H}^i(\cal I_C(6)) = 0$ for all curves $\lbrack C\rbrack\in{\tilde{\cal H}}_4$, $i\geq 1$, and that $\dim\operatorname{H}^0(\cal I_C(6))$ is constant on ${\tilde{\cal H}}_4$. Since $\dim\operatorname{H}^i({\tilde{\bold P}},{\cal I}_C(6))$ is an upper semicontinuous function on ${\tilde{\cal H}}_4$, we have to show the vanishing of $\operatorname{H}^i({\tilde{\bold P}},{\cal I}_C(6))$ only for all degenerations with monomial ideals, and the constantness of $\dim\operatorname{H}^0({\cal I}_C(6))$ for those and a generic curve. Namely, let $\lbrack C\rbrack\in{\tilde{\cal H}}_4$ and suppose $\dim\operatorname{H}^i(\cal I_C(6)) = d$. Take a one-dimensional torus action on ${\tilde{\bold P}}$ such that the monomial curves are the fixed points of the action. For all $t\in \bold C^*$, the schemes $C_t := tC$ are projectively equivalent, hence the cohomology groups have all the same dimension $d_t = d$. But the limit $C _0 = \lim_{t\to 0}C_t$ is a monomial curve, and by semicontinuity, $\dim\operatorname{H}^i({\cal I}_{C_0}(6)) \geq \dim\operatorname{H}^i({\cal I}_C(6)) $. As mentioned before, all the ideals of monomial degenerations of elliptic quartics $C'\subseteq\bold P^4$ are at least 3-regular, thus $\operatorname{H}^i(\bold P^4,\cal I_{C'}(6)) = 0$ for all $i>0$. By \lemref{quotients}, it follows that $\operatorname{H}^i({\tilde{\bold P}},{\cal I}_C(6)) = 0$ for all $i>0$ and all monomial degenerations of rational quartics. The constantness of $\dim\operatorname{H}^0({\cal I}_C(6))$ can also be verified by an explicit computation. Since ${\cal O}_{\cal C}(6)$ is flat over ${\tilde{\cal H}}_4$, $\cal I_{\cal C}(6)$ is a flat sheaf, too. By the previous result and cohomology and base change theorems (\cite{thehartshorne}, III 12.11, 12.9) we conclude that $R^1p_*{\cal I}_{\cal C}(6) = 0$ and that $p_*{\cal I}_{\cal C}(6)$ and $p_*{\cal O}_{\cal C}(6)$ are locally free. \end{pf} \section{TheCalculation} \label{calc} We will calculate the integral \eqref{integral} by Bott's formula, as follows (cf. \cite{bott:formula} and \cite{geirsas:twcub2}; these ideas are largely due to Geir Ellingsrud and Stein Arild Str{\o}mme): Suppose we are given a $\bold C^*$-action on ${\tilde{\bold P}}$ which induces a $\bold C^*$-action with isolated fixed points on ${\tilde{\cal H}}_4$. This action in turn induces an equivariant $\bold C^*$-action on the tangent bundle $\cal T_{{\tilde{\cal H}}_4}$ and on $p_*{\cal O}_{\cal C}(6)$. Therefore, in a fixed point $x = \lbrack C \rbrack$ of the action, the respective fibers $\cal T_{{\tilde{\cal H}}_4}(x)$ and $p_*{\cal O}_{\cal C}(6)\otimes\bold C(x)$ are $\bold C^*$-representations. As torus representations, they decompose into a direct sum of one-dimensional representations. Let $w_1(x),\ldots,w_r(x)$ resp.\ $\tau_1(x),\ldots,\tau_r(x)$ be the corresponding weights. \goodbreak Then Bott's formula says in our context: \begin{equation}\label{bott} \int_{{\tilde{\cal H}}_4} c_{13}(p_*{\cal O}_{\cal C}(6)) = \sum_{x\in{\tilde{\cal H}}_4^{\bold C^*}}{\tau_1(x)\cdot\ldots\cdot\tau_r(x)\over w_1(x)\cdot\ldots\cdot w_r(x)}. \end{equation} Let $T\subseteq \operatorname{GL}(5)$ be a maximal torus which acts diagonally on ${\tilde{\bold P}}$ w.r.t.\ the coordinates $x_0,\ldots,x_4$ of $\bold P^4$. There are characters $\lambda_0,\ldots,\lambda_4$ on $T$ such that for any $t\in T$, we have $t\cdot x_i = \lambda_i(t)\cdot x_i$, and these characters generate the representation ring of $T$, i.e., if $W$ is a finite representation of $T$, we can write {\it cum granum salis:} \goodbreak $$W = \sum a_{p_0\ldots p_4}\lambda_0^{p_0}\cdot\ldots\cdot\lambda_4^{p_4}\ .$$ The action of $T$ descends to an action on ${\tilde{\bold P}} = \bold P^4/\Gamma$. In the following, we will compute the torus representations of the induced $T$-action on ${\tilde{\cal H}}_4$ in the fibers of $p_*{\cal O}_{\cal C}(6)$ and $\cal T_{{\tilde{\cal H}}_4}$ in all fixed points. It is easy to see that a point $x\in{\tilde{\cal H}}_4$ is fixed exactly when the graded ideal of the corresponding curve is generated by monomials. Then we choose a one-parameter subgroup $\bold C^*\subseteq T$ with no non-trivial $\bold C^*$-weight in the tangent space of any fixed point. Such a one-parameter subgroup is given by a point $(w_0,\ldots,w_4)$ in the weight lattice $\operatorname{Hom}(\bold C^*,T)\cong\bold Z^5$; the corresponding characters on $\bold C^*$ are given by $\lambda_i(t)=t^{w_i}$. If $\lambda_0^{p_0}\cdot\ldots\cdot\lambda_4^{p_4}$ is the character of the $\bold C^*$-representation on an invariant one-dimensional subspace of the tangent space in a fixed point, the corresponding weight is given by \begin{equation}\label{weight} w=p_0w_0+\ldots+p_4w_4. \end{equation} All these weights are nonzero if the weight vector $(w_0,\ldots,w_4)$ is chosen to avoid simultaneously all the (finitely many) hyperplanes in the weight lattice defined by the linear forms \eqref{weight}. Such a choice is clearly possible. (In the concrete calculation of the integral \eqref{bott} by the {\sc Maple}-program listed in the appendix, we try randomly chosen weights; if none of the denominators in the summands of \eqref{bott} is zero --- which would result in a ``division by zero" error --- the choice is valid.) Our choice of the weights guarantees that the $\bold C^*$-action on ${\tilde{\cal H}}_4$ has isolated fixed points (in fact, the same fixed points as the action of $T$), hence we will be able to apply Bott's formula.\\ We will first calculate the fixed points and tangent space representations for ${\tilde{\cal H}}_3$ and afterwards use the fact that ${\tilde{\cal H}}_4$ is a locally trivial fibration over $\check{\bold P}^3$ with fiber~${\tilde{\cal H}}_3$. This is being done by calculating the fixed points of the $T$-action and the $T$-representation on the tangent spaces in each successive step of the blow up, i.e.\ on $\tilde G$, on $\tilde G'$, and finally on ${\tilde{\cal H}}_3$ (Note that there are induced $T$-actions on those spaces and that fixed points lie over fixed points). To compute the data of a blow up, say, of $X$ with center $B$, we have to get hold of the fiber $\cal N(x)$ of the normal bundle $\cal N=\cal N_{B/X}$ to the subvariety $B$ to blow up in each fixed point $x\in B$. We can achieve this by computing $T$-semiinvariant base vectors of $\cal N(x)$. Every such vector $\xi$ gives rise to a fixed point $x_\xi$ in the proper transform $B'$ of $B$: $x_\xi$ is the inverse image of $x$ in the proper transform of a curve tangent to $\xi$ in $x$. The tangent space at $x_\xi$ in the blow up $X' = Bl_B X$ is then given as \begin{equation}\label{tangentofblowup} \cal T_{X'}(x_\xi) \cong L_\xi\oplus\cal T_B(x)\oplus\cal T_{\bold P(\cal N^\vee(x))}(x_\xi)\ , \end{equation} where $L_\xi$ is the span of $\xi$ in the normal space $\cal N(x)$. The isomorphism is equivariant. (Recall that the exceptional divisor of the blow up with center $B$ is isomorphic to the projective bundle $\bold P(\cal N^\vee_{B/X})$, with normal space in $x_\xi$ isomorphic to $L_\xi$; cf.\ for instance \cite{fulton:intersect}, B.~6.) \\ Let us now look at the concrete calculations. We denote by $\tilde B'$ (resp.\ $\tilde D'$) the proper transform of $\tilde B$ in $\tilde G$ (resp.\ of $\tilde D$ in ${\tilde{\cal H}}_3$). \begin{prop} There are 126 fixed points in ${\tilde{\cal H}}_3$, and they are projectively equivalent to one of the following 25 types listed below. (To each fixed point type, we give the permutations of the variables which generate the remaining fixed points of that type.) \begin{equation*} \def\vrule height12pt depth0pt width0pt{\vrule height12pt depth0pt width0pt} \def\vrule height0pt depth6pt width0pt{\vrule height0pt depth6pt width0pt} \begin{tabular}{|c|c|c|c|} \hline\vrule height12pt depth0pt width0pt{\bf lies} & & & {\bf number of} \\ {\bf in}\vrule height0pt depth6pt width0pt & \raisebox{1.5ex}[0cm][0cm]{\bf fixed point type} & \raisebox{1.5ex}[0cm][0cm]{\bf permutations} & {\bf fixed points} \\ \hline & \vrule height12pt depth0pt width0pt $(x_0^2,x_1^2)$ & cyclic & 3\\ \raisebox{-1.5ex}[0cm][0cm]{$\tilde G-\tilde B$} & \vrule height12pt depth0pt width0pt $(x_0^2,x_1x_2)$ & permutations & 3\\ & \vrule height12pt depth0pt width0pt $(x_1^2,x_2^2)$ & of & 3\\ & \vrule height12pt depth0pt width0pt\vrule height0pt depth6pt width0pt $(x_1x_2,x_3^2)$ & $x_1,x_2,x_3$ & 3\\ \hline &\vrule height12pt depth0pt width0pt $(x_1x_2,x_1x_3,f),$ & $x_1 \leftrightarrow x_2$ & \raisebox{-1.5ex}[0cm][0cm]{6 $\times$ 4}\\ \raisebox{-1.6ex}[0cm][0cm]{$\tilde B'-\tilde D$} & \vrule height0pt depth6pt width0pt $f\in \{x_0^2x_2, x_0^2x_3, x_2^3, x_2^2x_3, x_2x_3^2, x_3^3\}$ & and $x_1 \leftrightarrow x_3$ &\\ & \vrule height12pt depth0pt width0pt $(x_1^2,x_1x_2,g),$ & all permutations & \raisebox{-1.5ex}[0cm][0cm]{3 $\times$ 6}\\ & \vrule height0pt depth6pt width0pt $g\in\{x_2^3, x_22x_3, x_2x_3^2, x_0^2x_2, x_1x_3^2, x_1x_0^2\}$ & of $x_1,x_2,x_3$ &\\ \hline & \vrule height12pt depth0pt width0pt $(x_1^2, x_1x_2, x_1x_3^2,f),$ & \raisebox{-3pt}[0cm][0cm]{all} & \raisebox{-1.5ex}[0cm][0cm]{6 $\times$ 6} \\ \raisebox{-1.5ex}[0cm][0cm]{$\tilde D'$} & \vrule height0pt depth6pt width0pt $f\in\{x_2x_3^3, x_2^2x_3^2, x_3^4, x_2^4, x_2^3x_3, x_0^2x_2^2\}$ & \raisebox{-2pt}[0cm][0cm]{permutations} &\\ & \vrule height12pt depth0pt width0pt $(x_1^2, x_1x_2, x_1x_0^2,g),$ & \raisebox{3pt}[0cm][0cm]{of} & \raisebox{-1.5ex}[0cm][0cm]{6 $\times$ 6} \\ & \vrule height0pt depth6pt width0pt $g\in \{x_0^2x_2x_3, x_2^2x_3^2, x_0^4, x_2^4, x_2^3x_3, x_0^2x_2^2\}$ & \raisebox{4pt}[0cm][0cm]{$x_1,x_2,x_3$} &\\ \hline \end{tabular} \end{equation*} The tangent space in a fixpoint $x$ in $\tilde G$ is given by $\cal T_{\tilde G}(x) = \sum \lambda_\gamma\lambda_\delta\lambda^{-1}_\alpha\lambda^{-1}_\beta,$ where $\alpha$, $\beta$ ($\alpha\leq\beta$) run over the pairs of indices of the monomials in $I_x$, whereas $\gamma$, $\delta$ ($\gamma\leq\delta$) run over all except these indices. The tangent space in each fixed point $x_\xi$ lying over a fixed point $x$ in $\tilde B$ resp.\ $\tilde D$ is calculated by formula \eqref{tangentofblowup}, where the tangent spaces and normal spaces to $\tilde B$ resp.\ $\tilde D$ are given by: \begin{equation*} \def\vrule height12pt depth0pt width0pt{\vrule height12pt depth0pt width0pt} \def\vrule height0pt depth6pt width0pt{\vrule height0pt depth6pt width0pt} \begin{tabular}{|c|rl|} \hline\vrule height12pt depth0pt width0pt\vrule height0pt depth6pt width0pt $I_{x_\xi}$ & {\bf tangent\!\!\!\!\!} & {\ \bf and normal spaces} \\ \hline\vrule height12pt depth0pt width0pt\vrule height0pt depth6pt width0pt & $\cal T_{\tilde B}(x)\quad =$ & $\lambda_1\lambda_2^{-1} + \lambda_1\lambda_3^{-1} + \lambda_2\lambda_1^{-1} + \lambda_3\lambda_1^{-1}$ \\ $(x_1x_2,x_1x_3,f)$ & $\cal N_{\tilde B/\tilde G}(x)\quad =$ & $\lambda_0^2\lambda_1^{-1}\lambda_2^{-1} + \lambda_0^2\lambda_1^{-1}\lambda_3^{-1} + \lambda_2\lambda_1^{-1} +{}$\\ \vrule height0pt depth6pt width0pt & &$\lambda_3\lambda_1^{-1} + \lambda_3^2\lambda_1^{-1}\lambda_2^{-1} + \lambda_2^2\lambda_1^{-1}\lambda_3^{-1} $ \\ \vrule height12pt depth0pt width0pt\vrule height0pt depth6pt width0pt & $\cal T_{\tilde B}(x)\quad =$ & $\lambda_3\lambda_1^{-1} + \lambda_3\lambda_2^{-1} + \lambda_2\lambda_1^{-1} + \lambda_3\lambda_1^{-1}$ \\ $(x_1^2,x_1x_2,g)$ & $\cal N_{\tilde B/\tilde G}(x)\quad =$ & $\lambda_0^2\lambda_1^{-1}\lambda_2^{-1} + \lambda_0^2\lambda_1^{-2} + \lambda_3^2\lambda_1^{-1}\lambda_2^{-1} +{}$\\ \vrule height0pt depth6pt width0pt& &$\lambda_3^2\lambda_1^{-2} + \lambda_2\lambda_3\lambda_1^{-2} + \lambda_2^2\lambda_1^{-2} $ \\ \hline \vrule height12pt depth0pt width0pt\vrule height0pt depth6pt width0pt & $\cal T_{\tilde D}(x)\quad =$ & $\lambda_3\lambda_1^{-1} + \lambda_2\lambda_1^{-1} + \lambda_3\lambda_2^{-1} + \lambda_0^2\lambda_3^{-2}$ \\ $(x_1^2, x_1x_2, x_1x_3^2,f)$ & $\cal N_{\tilde D/\tilde G'}(x)\quad =$ & $\lambda_3\lambda_1^{-1} + \lambda_3^2\lambda_1^{-1}\lambda_2^{-1} + \lambda_2^3\lambda_1^{-1}\lambda_3^{-2} +{}$\\ \vrule height0pt depth6pt width0pt & &$\lambda_2^2\lambda_1^{-1}\lambda_3^{-1} + \lambda_2\lambda_1^{-1} + \lambda_0^2\lambda_2\lambda_1^{-1}\lambda_3^{-2} $ \\ \vrule height12pt depth0pt width0pt\vrule height0pt depth6pt width0pt & $\cal T_{\tilde D}(x)\quad =$ & $\lambda_3\lambda_1^{-1} + \lambda_2\lambda_1^{-1} + \lambda_3\lambda_2^{-1} + \lambda_3^2\lambda_0^{-2}$ \\ $(x_1^2, x_1x_2, x_1x_0^2,g)$ & $\cal N_{\tilde D/\tilde G'}(x)\quad =$ & $\lambda_3\lambda_1^{-1} + \lambda_0^2\lambda_1^{-1}\lambda_2^{-1} + \lambda_2^3\lambda_1^{-1}\lambda_0^{-2} +{}$\\ \vrule height0pt depth6pt width0pt & & $\lambda_2^2\lambda_3\lambda_0^{-2}\lambda_1^{-1} + \lambda_2\lambda_1^{-1} + \lambda_2\lambda_3^2\lambda_0^{-2}\lambda_1^{-1} $ \\ \hline \end{tabular} \end{equation*} \end{prop} \begin{pf} The fixed points $x$ in the grassmannian $\tilde G$ are readily determined, their ideals have the form $I_x = (x_ix_j, x_kx_l)$ with the obvious restrictions on the indices. Let $V_2 := \operatorname{H}^0({\cal O}_{{\tilde{\bold P}}^3}(2))$, $V_{I_x} := \bold C\cdot x_ix_j\oplus\bold C\cdot x_kx_l$. The tangent space in a fixed point $x$ is given by $$\cal T_{\tilde G}(x) = \operatorname{Hom}(V_{I_x}, V_2/V_{I_x}) = \sum \lambda^{-1}_\alpha\lambda^{-1}_\beta\lambda_\gamma\lambda_\delta\ ,$$ the indices being as specified in the proposition.\\ {\it First blow up.} The subvariety $\tilde B\subseteq\tilde G$ to blow up consists of pencils with a fixed component, and the fixed points in $\tilde B$ are of type $(x_1x_2, x_1x_3)$ and $(x_1^2, x_1x_2)$. Consider the fixed point $x$ with ideal $I_x = (x_1x_2, x_1x_3)$. The tangent space to $\tilde G$ in $x$ is given by \begin{eqnarray}\label{tangentfistbl} \cal T_{\tilde G}(x) &= &\lambda_1^{-1}\lambda_2^{-1}\lambda_0^2 + \lambda_1^{-1}\lambda_3^{-1}\lambda_0^2 + \lambda_2^{-1}\lambda_1 + \lambda_3^{-1}\lambda_1 + 2\lambda_1^{-1}\lambda_2 +\\ &&2\lambda_1^{-1}\lambda_3 + \lambda_1^{-1}\lambda_2^{-1}\lambda_3^2 + \lambda_1^{-1}\lambda_3^{-1}\lambda_2^2\ .\nonumber \end{eqnarray} First, we will determine a semiinvariant basis for the fiber $\cal N_{\tilde B/\tilde G}(x)$ of the normal bundle of $\tilde B$ in $\tilde G$. Let $\xi\in \cal T_{\tilde G}(x)$ be a semiinvariant tangent vector in $x$, given as $\xi = {\xi_1\choose\xi_2}$ w.r.t.\ the basis $\{x_1x_2,x_1x_3\}$ of $V_{I_x}$ (i.e., $\xi(ax_1x_2,bx_1x_3) = (a\xi_1x_1x_2,b\xi_2x_1x_3)$). Since $\xi$ is semiinvariant, $\xi_1$ and $\xi_2$ are scalar multiples of a common Laurent monomial $\mu_\xi$ in $x_0,\ldots,x_3$ of degree 0. Furthermore, the torus representation on the subspace spanned by $\xi$ is obtained by formally substituting $\lambda_i$ for $x_i$, $i=0,\ldots,3$ in $\mu_\xi$. Clearly this monomial in the $\lambda_i$ has to be one of the summands in \eqref{tangentfistbl}. Now $I_\xi(t) = (x_1x_2+t\xi_1x_1x_2, x_1x_3+t\xi_2x_1x_3)$ is a curve through $x$ with tangent direction $\xi$ in $x$. We see that the semiinvariant tangent vectors $$\xi = {x_1x_2^{-1}\choose 0}, {0\choose x_1x_3^{-1}}, {x_2x_1^{-1}\choose x_2x_1^{-1}}, {x_3x_1^{-1}\choose x_3x_1^{-1}} $$ are tangent to $\tilde B$; the curves given by $I_\xi(t)$ are even contained in $\tilde B$. The vectors $$ {x_2x_1^{-1}\choose -x_2x_1^{-1}}, {x_3x_1^{-1}\choose -x_3x_1^{-1}}, {x_0^2x_1^{-1}x_2^{-1}\choose 0}, {x_3^2x_1^{-1}x_2^{-1}\choose 0}, {0 \choose x_0^2x_1^{-1}x_3^{-1}}, {0 \choose x_2^2x_1^{-1}x_3^{-1}} $$ complete the previous ones to a semiinvariant basis of $\cal T_{\tilde G}(x)$, thus they represent (modulo $\cal T_{\tilde B}(x)$) a semiinvariant basis of $\cal N_{\tilde B/\tilde G}(x)$. In order to compute the fixed points of $\tilde G'$ lying over $x$, we consider the curves $I_\xi(t)$ for $\xi$ in that basis. Each of them defines a flat family of curves over $\bold A^1-\{0\}$. We can extend this family in a unique way to a flat family over $\bold A^1$. That flat family induces a map $\bold A^1\to{\tilde{\cal H}}_3$ such that the image $x_\xi'$ of $0\in\bold A^1$ maps onto a fixed point $x_\xi$ of $\tilde G'$ under the blow up map ${\tilde{\cal H}}_3\to\tilde G'$. The ideal $I_{x_\xi}$ corresponding to $x_\xi$ is the subideal generated in degree 3 of the ideal corresponding to $x_\xi'$ (cf.\ \eqref{H3toG'}). To actually compute the ideal $I_{x_\xi}$, we use a flattening algorithm described by Bayer and Mumford (\cite{baymum:compute}, Ch.1): \\ \begin{prop} Let $I(t) = (m_1(t),\ldots,m_r(t))$ be the ideal of a family of schemes over $\bold A^1$ and suppose that all $m_i := m_i(0)$ are monomials. Let $I=(m_1,\ldots,m_r)$ be the ideal of the central fiber. Consider the following algorithm: \begin{enumerate} \item Take a minimal syzygy of two generators $m_i$, $m_j$ of the ideal $I$, i.e., a relation $$h_im_i+h_jm_j=0,$$ with $h_i$ and $h_j$ coprime monomials, which does not lift to a syzygy in $I_\xi(t)$. This means that we get a relation $$h_im_i(t)+h_jm_j(t) = t^sg$$ with $g\neq 0$ and $t \not\:\mid g$. Add the polynomial $g$ to the generators of $I(t)$ to get new ideals $I'(t)$ and $I'=I'(0)$. \item Repeat this process finitely often until all syzygies lift to syzygies in $I'(t)$. \end{enumerate} Then the resulting ideal $I'(t)$ is flat in $t=0$, and the ideal of the fiber is equal to~$I'$. \end{prop} In our case, all $g$'s are monomials and $s$ is equal to 1 in the first step. Furthermore, we are only interested in generators of degree 3, so we can stop the iteration when we have added all monomials of degree 3, and that is the case already after the first step, as can easily be seen. In concrete terms, the only syzygy $x_3\cdot(x_1x_2)-x_2\cdot(x_1x_3)=0$ lifts to $I_\xi(t)$ as \begin{eqnarray*} x_3\cdot(x_1x_2+t\xi_1x_1x_2)-x_2\cdot(x_1x_3+t\xi_2x_1x_3) &= &t\cdot(x_3\xi_1x_1x_2-x_2\xi_2x_1x_3)\\ &=: &t\cdot f_\xi\ . \end{eqnarray*} All together, the fixed points in the blow up $\tilde G'$ lying over $x$ are: $$I_\xi = (x_1x_2,x_1x_3,f_\xi), \quad f_\xi\in \{x_0^2x_2, x_0^2x_3, x_2^3, x_2^2x_3, x_2x_3^2, x_3^3\}\ ,$$ plus the ideals obtained by the permutations $x_1 \leftrightarrow x_2$ and $x_1 \leftrightarrow x_3$. The calculation for points $x$ of type $I_x = (x_1^2, x_1x_2)$ is exactly analogous.\\ {\it Second blow up.} In $\tilde G'$, we have to blow up the non-flat locus $\tilde D\subseteq \tilde G'$. The fixed points $x$ of $\tilde G'$ contained in $\tilde D$ are of type $I_x = (x_1^2, x_1x_2, x_1x_3^2)$ and $I_x = (x_1^2, x_1x_2, x_1x_0^2)$. The tangent space to $\tilde G'$ in the first fixed point is given by (cf.\ \eqref{tangentofblowup}): \begin{eqnarray}\label{tangentsecondbl} \cal T_{\tilde G}(x) &= &2\lambda_1^{-1}\lambda_3 + 2\lambda_1^{-1}\lambda_2 + \lambda_2^{-1}\lambda_3 + \lambda_3^{-2}\lambda_0^2 + \lambda_1^{-1}\lambda_2^{-1}\lambda_3^2 +\\ &&\lambda_1^{-1}\lambda_3^{-2}\lambda_2^3 + \lambda_1^{-1}\lambda_3^{-1}\lambda_2^2 + \lambda_1^{-1}\lambda_3^{-2}\lambda_2\lambda_0^2\ .\nonumber \end{eqnarray} We will determine (the torus representation of) the normal space $\cal N(x)$ to $\tilde D$ in $x$. A curve in $\tilde G'$ through $x$ tangent to $\xi\in\cal T_{\tilde G'}(x)$ is given modulo $t^2$ as $$I_\xi(t) = (x_1^2+t\xi_1x_1^2, x_1x_2+t\xi_2x_1x_2, x_1x_3^2+t\xi_3x_1x_3^2)\ ,$$ where $\xi = (\xi_1, \ldots,\xi_3)^t$, and if $\xi$ is semiinvariant, then the $\xi_i$ are scalar multiples of a common Laurent monomial $\mu_\xi$ of degree zero. Again, by lifting a syzygy relation from $I_x$ to $I_\xi(t)$ (this can be done modulo $t^2$, too), we calculate the monomial $f_\xi$ we have to add to $I_x$ in order to get the fixed point in the blow up ${\tilde{\cal H}}_3$ corresponding to $\xi$. Since the syzygies of all pairs of monomials in $I_x$ generate the whole syzygy module, we only need to consider such pairs. Suppose that lifting the syzygy of the pair $(m_1, m_2)$ results in the right monomial $f_\xi$; then $f_\xi$ is equal to $lcm(m_1,m_2)\cdot\mu_\xi$. This monomial is supposed to have degree four. The two syzygies which can possibly yield a monomial of degree four are those between $x_2$ and $x_1x_3^2$ and between $x_1x_2$ and $x_1x_3^2$. Consider the first pair. We have $f_\xi = x_1^2x_3^2\cdot\mu_\xi$. But from \eqref{tangentsecondbl} it is apparent that the Laurent monomials $\mu_\xi$ contain $x_1$ to the power -1 or higher, that means that $f_\xi$ contains $x_1$ as a factor. Hence the resulting ideal $I_\xi=(x_1^2,x_1x_2,x_1x_3^2,f_\xi)$ is contained in the ideal $(x_1)$ and certainly doesn't correspond to a flat degeneration. So we only have to consider the pair $x_1x_2$, $x_1x_3^2$. We claim that the monomials $\mu_\xi$ corresponding to a semiinvariant basis of $\cal N(x)$ are \begin{eqnarray}\label{mus} \{x_1^{-1}x_3,\ x_1^{-1}x_2,\ x_1^{-1}x_2^{-1}x_3^2,\ x_1^{-1}x_3^{-2}x_2^3, \ x_1^{-1}x_3^{-1}x_2^2,\ x_1^{-1}x_3^{-2}x_2x_0^2\}\ . \end{eqnarray} The two other possibilities for $\mu_\xi$, namely $x_2^{-1}x_3$ and $x_3^{-2}x_0^2$, are excluded because they again lead to ideals contained in the ideal $(x_1)$. On the other hand, $\cal N(x)$ has dimension 6; together this proves the claim. So, the fixed points in the blow up ${\tilde{\cal H}}_3$ lying over $I_x = (x_1^2, x_1x_2, x_1x_3^2)$ are $$I_\xi = (x_1^2, x_1x_2, x_1x_3^2,f_\xi), \quad f_\xi\in \{x_2x_3^3, x_2^2x_3^2, x_3^4, x_2^4, x_2^3x_3, x_0^2x_2^2\}\ .$$ The torus representation of $\cal N(x)$ is gotten by formally summing up the monomials in \eqref{mus} and substituting $\lambda$ for $x$. To get all fixed points, we have to add the fixed points generated by all permutations of the variables $x_1$, $x_2$ and $x_3$. The calculation for points $x$ of type $I_x = (x_1^2, x_1x_2, x_1x_0^2)$ is again analogous. Now we have determined all fixed points of the torus action on ${\tilde{\cal H}}_3$: they are the fixed points of all stages of the blow up process minus the fixed points lying in the blow up loci. \end{pf} {\it Fixed points and tangent spaces in ${\tilde{\cal H}}_4$.} A fixed curve in ${\tilde{\bold P}}$ spans a $T$-invariant hyperplane ($\cong \bold P(2,1^3)$) in ${\tilde{\bold P}}$, which is given by an equation $x_i = 0$, $i\in\{1,\ldots,4\}$. Thus, if for instance $i=4$, we get the ideals of all fixed curves lying in $\{x_4=0\}$ by adjoining the monomial $x_4$ to the ideals previously determined. The ideals of the remaining fixed points are obtained by cyclically permuting the variables $x_1,\ldots,x_4$ in these ideals. Thus, ${\tilde{\cal H}}_4$ containes 504 fixed points all together. According to \propref{fiberingofhilb}, ${\tilde{\cal H}}_4$ is fibered over $\check{\bold P}^3$ as \begin{eqnarray*} \pi: {\tilde{\cal H}}_4 &\to &\check{\bold P}^3 \\ \lbrack C \rbrack & \mapsto & \lbrack\hbox{ projection of } span(C) \hbox{ to } \bold P^3\subseteq{\tilde{\bold P}}\rbrack\ . \end{eqnarray*} The tangent space of ${\tilde{\cal H}}_4$ in a fixed point $x=\lbrack C \rbrack$ therefore decomposes equivariantly as a direct sum $$\cal T_{{\tilde{\cal H}}_3}(x)\oplus\cal T_{\check{\bold P}^3}(\pi(x)) .$$ Let $C$ be an invariant curve spanning the singular hyperplane $\{x_i=0\}$ and let $V_i = \bold C\cdot x_i$, $V = \bold C\cdot x_1\oplus\ldots\oplus\bold C\cdot x_4$. Then the torus representation of the tangent space to $\check{\bold P}^3$ is equal to $$\cal T_{\check{\bold P}^3}(\pi(x)) = \operatorname{Hom}(V_i, V/V_i) = \sum_{j\neq i}\lambda_j\lambda_i^{-1}\ $$ Finally, the torus representations on the fibers $\operatorname{H}^0({\cal O}_C(6))$ of $p_*{\cal O}_{\cal C}(6)$ are easily determined: $\operatorname{H}^0({\cal O}_C(6))$ is spanned by those monomials of degree 6 that don't lie in the ideal $\cal I_C$. \section{Appendix: Program Listing} \begin{verbatim} # Calculation of the number of rational quartics on a general # Calabi-Yau hypersurface in P(2,1,1,1,1). ### some setup and utility routines # is_invariant decides whether a monomial is invariant under # var -> -var, i.e. whether the variable var occurs in even power is_invariant := proc(mon, var) mon = subs(var=-var,mon) end: # invariant_subspace returns the invariant subspace of a vector space # (w.r.t. var -> -var) invariant_subspace := proc(J, var) local i, inv; inv := 0: for i from 1 to nops(J) do if is_invariant(op(i,J),var) then inv := inv + op(i,J) fi od: inv: end: # setuptorus(n,m) sets up the data of an (n+1)-dimensional torus # with characters x0,...xn. The representation ring is identified # with the Laurent series ring Z[xi,1/xi, i=0..n]. V[i] denotes the # canonical representation of H^0(o_P(2,1^n)(i)). setuptorus := proc(N,M) local genf,i,k; genf := expand(convert(series( 1/product('1-x.i*t',i=0..N),t,M+1),polynom)): for i from 0 to M do V[i] := invariant_subspace(sort(coeff(genf,t,i)),x0) od; i := 'i': dualsubstring := seq(x.i=x.i^(-1),i=0..N): variables := [seq(x.i,i=0..N), seq(x.i^(-1),i=0..N)]: end: dualrep := proc(C) sort(expand(subs(dualsubstring,C))) end: simpleweights := proc(p) local mon; expand(p); if p=0 then RETURN(0) fi; coeffs(",variables,'mon'); convert([mon],`+`); end: # twist(J,k) computes the representation of the degree k part of # the ideal J: twist := proc(J,k) local i,j,res,mon,deg; res := 0: for j from 1 to nops(J) do mon := op(j,J): if k >= degree(mon) then res := res + V[k-degree(mon)]*mon: fi: od: simpleweights(res): end: # If C* -> T is the one-parameter subgroup given by weights # w0,..,w4, the character xi restricts to t^(wi), i=0..4. # In order to calculate the enumerator in Bott's formula, we have # to calculate the product of all of them. prodwts := proc(f) local t,cof,mon,res,i; cof := [coeffs(f,variables,mon)]; mon := subs(x0=t^w0,x1=t^w1,x2=t^w2,x3=t^w3,[mon]); res := 1; for i from 1 to nops(mon) do res := res*subs(t=1,diff(mon[i],t))^cof[i] od; res end: ### here begins the actual calculation # First, we calculate all data for H_3. After that, we use the # fiber structure of H_4 with fiber H_3 to calculate the torus # action on H_4. setuptorus(3,6); tw := 6: # we are going to calculate o(6) fp := 0: # fp counts the number of fixpoints # ideals and tangent spaces of the fixed points in the grassmannian for i0 from 0 to 3 do for i1 from i0 to 3 do for i2 from i0 to 3 do for i3 from i2 to 3 do if i0 <> i2 and i0 <> i3 and i1 <> i2 and i1 <> i3 then idd := x.i0*x.i1 + x.i2*x.i3: # ideal of fixed point if is_invariant(idd,x0) then # we have to filter fp := fp+1: # out the invariant id[fp] := idd: # ideals T[fp] := expand((V[2]-id[fp]) # tangent space of *dualrep(id[fp])): # the grassmannian o[fp] := # torus representation of V[tw]-twist(id[fp],tw): # the fiber of p_*O_C(6) fi fi od od od od: # first blow up idd := x1*x2+x1*x3: # ideal of the first type TB := expand((x2+x3)/x1 + x1/x2 + x1/x3): # tangent space to the # subvariety B to blow up N := expand((V[2]-idd)*dualrep(idd)) - TB: # normal space to B for i from 1 to nops(N) do # normal direction to the TL := op(i,N): # exceptional divisor TPN := expand((N-TL)/TL): # tangent space to the fiber T_[i] := TB + TL + TPN: # total tangent space T_[i] := unapply(T_[i], x1,x2,x3): id_[i] := idd+x1*x2*x3*TL: # ideal of the fixed point o_[i] := V[tw]-twist(id_[i],tw): # torus representation of o_[i] := unapply(o_[i], x1,x2,x3): # the fiber of p_*O_C(6) id_[i] := unapply(id_[i], x1,x2,x3): # in a fixed point od: # permutations of the variables that generate the remaining # fixed points permut1 := [x1,x2,x3], [x2,x1,x3], [x3,x2,x1]: for j from 1 to 3 do # the remaining fixed points for i from 1 to nops(N) do fp := fp+1: T[fp] := T_[i](op(permut1[j])): o[fp] := o_[i](op(permut1[j])): id[fp] := id_[i](op(permut1[j])): od: od: idd := x1^2+x1*x2: # ideal of the second type TB := expand(x3*dualrep(x1+x2) + (x2+x3)*dualrep(x1)): N := expand((V[2]-idd)*dualrep(idd)) - TB: for i from 1 to nops(N) do TL := op(i,N): TPN := expand((N-TL)/TL): T_[i] := TB + TL + TPN: T_[i] := unapply(T_[i], x1,x2,x3): id_[i] := idd+x1^2*x2*TL: if divide(id_[i],x1) then o_[i] := 0 else # the ideals whose o_[i] := V[tw]-twist(id_[i],tw) fi: # generators have a o_[i] := unapply(o_[i], x1,x2,x3): # common factor are lying id_[i] := unapply(id_[i], x1,x2,x3): # in the next blow up od: # locus # permutations of the variables that generate the remaining # fixed points permut2 := [x1, x2, x3], [x2, x1, x3], [x1, x3, x2], [x3, x2, x1], [x3, x1, x2], [x2, x3, x1]: for j from 1 to 6 do for i from 1 to nops(N) do fp := fp+1: T[fp] := T_[i](op(permut2[j])): o[fp] := o_[i](op(permut2[j])): id[fp] := id_[i](op(permut2[j])): if o[fp] = 0 then fp := fp-1 fi: od: od: # second blow up idd := x1^2+x1*x2: TL_ := [x3^2/(x1*x2), x0^2/(x1*x2)]: for k from 1 to nops(TL_) do TD := expand(x3/x1 + x3/x2 + x2/x1 + ((x3^2+x0^2)/(x1*x2)-TL_[k])/TL_[k]): N := expand(x3/x1 + TL_[k] + (x2^2+x2*x3+x3^2+x0^2)/(x1^2*TL_[k])): for i from 1 to nops(N) do # normal direction to the exceptional divisor: TL := op(i,N): # tangent space to the fiber: TPN := expand((N-TL)/TL): # total tangent space: T_[i] := TD + TL + TPN: T_[i] := unapply(T_[i], x1,x2,x3): # ideal of the fixed point: id_[i] := idd+x1^2*x2*TL_[k]+x1^2*x2^2*TL_[k]*TL: # torus representation of the fiber of p_*O_C(6) # in a fixed point o_[i] := V[tw]-twist(id_[i],tw): o_[i] := unapply(o_[i], x1,x2,x3): id_[i] := unapply(id_[i], x1,x2,x3): od: for j from 1 to 6 do for i from 1 to nops(N) do fp := fp+1: T[fp] := T_[i](op(permut2[j])): o[fp] := o_[i](op(permut2[j])): id[fp] := id_[i](op(permut2[j])): if o[fp] = 0 then fp := fp-1 fi: od: od: od: lprint(`number of fixpoints =`, fp): # Euler characteristic # of H_3 = 126 # weight vector # the weights are experimentally chosen in such a way that the C* # representation has no trivial weight in any tangent space. That # ensures that the C* action has isolated fixed points. WG[1] := 4: WG[2] := 17: WG[3] := 55: WG[4] := 160: WG[0] := 267: for i from 1 to fp do # calculate the summands of Bott's num := prodwts(o[i]): # formula as a function of the den := prodwts(T[i]): # weights, still leaving out the fra[i] := unapply(num/den, # contribution of the base P^3 w0,w1,w2,w3): # of the fibration H_4 -> P^3 od: perm_h := [0,2,3,4], [0,3,4,1], # permutation of the hyperplanes [0,4,1,2], [0,1,2,3]: # in P(2,1^4) containing # the fixed curves result := 0: # summing up variable for k from 1 to 4 do h := perm_h[k]: TG := (WG[h[2]]-WG[k]) # tangent space of P^3 *(WG[h[3]]-WG[k])*(WG[h[4]]-WG[k]): for i from 1 to fp do # summing up over all fixed points result := result+ fra[i](WG[h[1]],WG[h[2]],WG[h[3]],WG[h[4]])/TG: od: od: lprint(`number of curves =`, result): # 6028452 \end{verbatim} \bibliographystyle{amsplain} \ifx\undefined\leavevmode\hbox to3em{\hrulefill}\, \newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,} \fi

9611

alg-geom/9611011

en

https://arxiv.org/abs/alg-geom/9611011

alg-geom/9611011

Tohsuke Urabe

Tohsuke Urabe (Department of Mathematics, Tokyo Metropolitan University, Japan)

Dynkin Graphs, Gabri\'{e}lov Graphs and Triangle Singularities

10 pages with six postscript figures. AMSLaTeX v 1.2.figures. Adobe PDF version is available also at my private homepage, http://urabe-lab.math.metro-u.ac.jp/PreprintListE.html

Fourteen kinds of triangle singularities with modality one in Arnold's classification list are discussed. We consider which kinds of combinations of rational double points can appear on small deformation fibers of the singularities. We show that possible combinations of rational double points can be described by a unique principle from the view point of Dynkin graphs.

alg-geom

\section{Review of results by Russian mathematicians} \label{review} In this article we assume that every variety is defined over the complex field $\mathbf{C}$. First I explain some results by Arnold and Gabri\'{e}lov briefly. In \cite{arnold;gauss} Arnold has introduced an invariant $m$ called \emph{modality} or \emph{modules number}, and has given a long classification list of hypersurface singularities. Modality $m$ is a non-negative integer. Though we find singularities of any dimension in Arnold's list, we consider singularities of dimension two in particular. His class of singularities with $m=0$ coincides with the class of rational double points. It is well known that each rational double point corresponds to a connected Dynkin graph of type $A$, $D$ or $E$ in the theory of Lie algebras. (Durfee~\cite{durfee;charac}.) The class with $m=1$ consists of three subclasses. ($\lambda$ is a parameter.) \begin{enumerate} \item Three simple elliptic singularities: $J_{10},\;X_9,\;P_8$ \item Cusp singularities $T_{p,\,q,\,r}$. $\left( {\frac{1}{p}+\frac{1}{q}+\frac{1}{r}<1} \right)$: $x^p+y^q+z^r+\lambda xyz=0$ $\left( {\lambda \ne 0} \right)$. \item fourteen triangle singularities (These fourteen are also called exceptional singularities.) $$\begin{array}{llllllll} E_{12}&Z_{11}&Q_{10}&\hspace{10 mm}&W_{12}&S_{11}&\hspace{10 mm}&U_{\,12}\\ E_{13}&Z_{12}&Q_{11}&\hspace{10 mm}&W_{13}&S_{12}\\ E_{14}&Z_{13}&Q_{12} \end{array}$$ $$\begin{array}{c} E_{12}\,:\;x^7+y^3+z^2+\lambda x^5y=0\hspace{5 mm} W_{12}\,:\;x^5+y^4+z^2+\lambda x^3y^2=0\\ U_{12}\,:\;x^4+y^3+z^3+\lambda x^2yz=0. \end{array}$$ \end{enumerate} (As for the other defining polynomials see Arnold~\cite{arnold;gauss}.) His list continues in the case $m\ge 2$, but we do not refer further. We go on to Gabri\'{e}lov's results. (Gabri\'{e}lov~\cite{gabrielov;unimodular}.) Let $f\left( {x,\,y,\,z} \right)=0$ be one of defining polynomials of fourteen hypersurface triangle singularities. It defines a singularity at the origin. We consider the \emph{Milnor fiber}, i.e., $$F=\left\{ {\,\left( {x,\,y,\,z} \right)\in \mathbf{C}^3\;\left| {\;\left| x \right|^2+\left| y \right|^2+\left| z \right|^2<\varepsilon ^2, \;f\left( {x,\,y,\,z} \right)=t\,} \right.} \right\}$$ where $\epsilon$ is a sufficiently small positive real number and $t$ is a non-zero complex number whose absolute value is sufficiently small compared with $\epsilon$. The pair $$ \left( H_2\left( F,\;\mathbf{Z} \right),\;the\ intersection\ form\right) $$ is called the \emph{Milnor lattice}, and $\mu =\mathrm{rank} \,H_2\left( F,\;\mathbf{Z} \right)$ is called the \emph{Milnor number} of the singularity. Gabri\'{e}lov has computed the Milnor lattice for fourteen hypersurface triangle singularities. According to him, there exists a basis $e_{\,1},\;e_2,\;\ldots ,\;e_\mu $ of $H_2\left( F,\;\mathbf{Z} \right)$ such that each $e_{\,i}$ is a vanishing cycle (thus in particular $e_{\,i}\cdot e_{\,i}=-2$) and the intersection form is represented by the dual graph below. \vspace*{2 mm} \epsfig{file=wall_fig1.eps} In the above graph the basis $e_{\,1},\;e_2,\;\ldots ,\;e_\mu $ has one-to-one correspondence with vertices. Edges indicate intersection numbers. Two vertices corresponding to $e_{\,i}$ and $e_j$ are not connected, if $e_{\,i}\cdot e_j=0$. They are connected by a single solid edge, if $e_{\,i}\cdot e_j=1$. They are connected by a double dotted edge, if $e_{\,i}\cdot e_j=-2$. Three integers $p_1,\;p_2,\;p_3$ are the numbers of vertices in the corresponding arm.\nopagebreak[4] They depend on the type of the triangle singularity. The corresponding triplets $\left( {p_{\,1},\;p_2,\;p_3} \right)$ to the above fourteen symbols are as follows: $$\begin{array}{llllllll} \left( {2,\;3,\;7} \right)&\left( {2,\;4,\;5} \right)& \left( {3,\;3,\;4} \right)&\hspace{10 mm}& \left( {2,\;5,\;5} \right)&\left( {3,\;4,\;4} \right)&\hspace{10 mm}& \left( {4,\;4,\;4} \right)\\ \left( {2,\;3,\;8} \right)&\left( {2,\;4,\;6} \right)& \left( {3,\;3,\;5} \right)&\hspace{10 mm}& \left( {2,\;5,\;6} \right)&\left( {3,\;4,\;5} \right)\\ \left( {2,\;3,\;9} \right)&\left( {2,\;4,\;7} \right)& \left( {3,\;3,\;6} \right) \end{array}$$ (Thus the above figure is the graph for $S_{12}$.) The main part of the above graph below is called the \emph{Gabri\'{e}lov graph}. \vspace*{2 mm} \epsfig{file=wall_fig2.eps} \vspace*{\fill} \pagebreak[4] The Gabri\'{e}lov graph defines a lattice $P^*$ with a basis $e_{\,1},\;e_2,\;\ldots ,\;e_{\mu -2}$ , if we apply the above mentioned rule. It is easy to check that $P^*$ has signature $\left( 1,\;\mu -3 \right)$, and $H_2\left( F,\;\mathbf{Z} \right)\cong P^*\oplus H$ as lattices. Here $H=\mathbf{Z}\,u+\mathbf{Z}\,v$ denotes the hyperbolic plane, i.e., a lattice of rank 2 with a basis $u$, $v$ satisfying $u\cdot u=v\cdot v=0$ and $u\cdot v=v\cdot u=1$, and $\oplus $ denotes the orthogonal direct sum. \nopagebreak[4] \section{Singularities on deformation fibers of triangle singularities} \label{triangle} A finite disjoint union of connected Dynkin graphs is also called a Dynkin graph. Let $T$ denote one of the above fourteen symbols of hypersurface triangle singularities. By $PC\left( T \right)$ we denote the set of Dynkin graphs $G$ with several components such that there exists a small deformation fiber $Y$ of a singularity of type $T$ satisfying the following conditions: \begin{enumerate} \item $Y$ has only rational double points as singularities. \item The combination of rational double points on $Y$ corresponds to graph $G$ exactly. \end{enumerate} Here, the type of each component of $G$ corresponds to the type of a rational double point on $Y$, and the number of components of each type corresponds to the number of rational double points of each type on $Y$. If $G$ has $a_k$ of components of type $A_k$ for each $k\ge 1$, $d_\ell $ of components of type $D_\ell $ for each $\ell \ge 4$ and $e_m$ of components of type $E_m$ for $m=6,\;7,\;8$, we identify $G$ with the formal sum $G=\sum {a_k\,A_k}+\sum {d_\ell \, D_\ell} +\sum {e_m\,E_m}$. Mr. F.-J. Bilitewski informed me that he had a complete listing of Dynkin graphs of $PC\left( T \right)$ for every $T$ of the above fourteen. \begin{thm} \label{main} Let $T$ be one of the above fourteen symbols of hypersurface triangle singularities. Let $G$ be a Dynkin graph with only components of type $A$, $D$ or $E$. The following conditions \textup{\textbf{(A)}} and \textup{\textbf{(B)}} are equivalent: \begin{description} \item[(A)] $G\in PC\left( T \right)$. \item[(B)] Either \textup{\textbf{(B-1)}} or \textup{\textbf{(B-2)}} holds. \begin{description} \item[(B-1)] $G$ can be made by an elementary transformation or a tie transformation from a Dynkin subgraph of the Gabri\'{e}lov graph of type $T$. \item[(B-2)] $G$ is one of the following exceptions: \end{description} \end{description} \vspace*{-5 mm} \[ \begin{array}{l} \begin{array}{rl} &\mbox{\hspace{5 mm}Exceptions}\\ T & =Z_{\,13}\,:\;A_7+A_4\\ T & =S_{\,11}\,:\;2A_4+A_{\,1}\\ T & =U_{\,12}\,:\;2D_4+A_2,\;A_6+A_4,\;A_5+A_4+A_{\,1},\;2A_4+A_1 \end{array} \\ \mbox{The other eleven triangle singularities: None} \end{array} \] \end{thm} An elementary transformation and a tie transformation in the above theorem are operations by which we can make a new Dynkin graph from a given Dynkin graph. \begin{defn} Elementary transformation: The following procedure is called an elementary transformation of a Dynkin graph: \begin{enumerate} \item Replace each connected component by the corresponding extended Dynkin graph. \item Choose in an arbitrary manner at least one vertex from each component (of the extended Dynkin graph) and then remove these vertices together with the edges issuing from them. \end{enumerate} \end{defn} An extended Dynkin graph is a graph obtained from a connected Dynkin graph by adding one vertex and one or two edges. (Bourbaki~\cite{bourbaki;lie}.) Below we show extended Dynkin graphs. Numbers in the figures below are the coefficients of the maximal root, which will appear in the definition of a tie transformation. We can get the corresponding Dynkin graph, if we erase one vertex with the attached number $1$ and edges issuing from it. \vspace*{5 mm} \epsfig{file=wall_fig3.eps} \vspace{\fill} \pagebreak[4] \begin{defn} Tie transformation: Assume that by applying the following procedure to a Dynkin graph $G$ we have obtained the Dynkin graph $\bar G$. Then, we call the following procedure a tie transformation of a Dynkin graph: \begin{enumerate} \item Replace each component of $G$ by the extended Dynkin graph of the same type. Attach the corresponding coefficient of the maximal root to each vertex of the resulting extended graph $\tilde G$. \item Choose, in an arbitrary manner, subsets $A$, $B$ of the set of vertices of the extended graph $\tilde G$ satisfying the following conditions: \begin{description} \item[$\left\langle a \right\rangle$] $A\cap B=\emptyset $ \item[$\left\langle b \right\rangle$] Choose arbitrarily a component $\tilde G''$ of $\tilde G$ and let $V$ be the set of vertices in $\tilde G''$. Let $\ell $ be the number of elements in $A\cap V$. Let $n_{\,1},\;n_2,\;\ldots ,\;n_\ell $ be the numbers attached to $A\cap V$. Also, let $N$ be the sum of the numbers attached to elements in $B\cap V$. (If $B\cap V=\emptyset $, $N=0$.) Then, the greatest common divisor of the $\ell +1$ numbers $N,\;n_{\,1},\;n_2,\;\ldots ,\;n_\ell$ is $1$. \end{description} \item Erase all attached integers. \item Remove vertices belonging to $A$ together with the edges issuing from them. \item Draw another new vertex called $\theta $. Connect $\theta $ and each vertex in $B$ by a single edge. \end{enumerate} \end{defn} \begin{rem} After following the above procedure 1--5, the resulting graph $\bar G$ is often not a Dynkin graph. We consider only the cases where the resulting graph $\bar G$ is a Dynkin graph, and then we call the above procedure a tie transformation. The number $\#\left( B \right)$ of elements in the set $B$ satisfies $0\le \#\left( B \right)\le 3$. $\ell =\#\left( {A\cap V} \right)\ge 1$. \end{rem} \begin{exmp} We consider the case $T=W_{\,13}$. The Gabri\'{e}lov graph in this case is the following, and it has a Dynkin subgraph of type $E_8+A_2$: \vspace*{3 mm} \epsfig{file=wall_fig4.eps} \vspace*{3 mm} First we apply a tie transformation to $E_8+A_2$. In the second step of the transformation we can choose subsets $A$ and $B$ as follows: \vspace*{4 mm} \epsfig{file=wall_fig5.eps} \pagebreak[4] For the component of type $E_8$, $\ell =1$, $n_{\,1}=4$, $N=1$ and thus $G.C.D.\left( {n_{\,1},\;N} \right)=1$. For the component $A_{\,2}$, $\ell =1$, $n_{\,1}=1$, $N=1$ and thus $G.C.D.\left( {n_{\,1},\;N} \right)=1$. One sees that the condition $\left\langle b \right\rangle $ is satisfied. As the result of the transformation one gets a graph of type $A_6+D_5$. By our theorem one can conclude $A_6+D_5\in PC\left( {W_{\,13}} \right)$. Second we apply an elementary transformation to $E_8+A_2$. \vspace*{4 mm} \epsfig{file=wall_fig6.eps} \vspace*{3 mm} As in the above figure we can get $E_6+2A_2$. Thus $E_6+2A_2\in PC\left( {W_{\,13}} \right)$. \end{exmp} \nopagebreak[4] \section{K3 surfaces and lattice theory} \label{k3} It is known that fourteen hypersurface triangle singularities have interesting property called the strange duality. (Pinkham~\cite{pinkham;duality}.) Let $T$ be one of the above fourteen symbols of hypersurface triangle singularities. Associated with $T$, we have another symbol $T^*$ also in the above fourteen symbols of hypersurface triangle singularities. This $T^*$ is called the \emph{dual} of $T$. The dual of the dual coincides with the original one, i.e., $(T^*)^* = T$. For the following six singularities the dual coincides with itself, i.e., $T^* = T$: $E_{\,12},\;Z_{\,12},\;Q_{\,12},\;W_{\,12},\;S_{\,12},\;U_{\,12}$. For the following four pairs the dual is another member of the pair: $\left\{ {\,E_{\,13},\;Z_{\,11}\,} \right\},\;\left\{ {\,E_{\,14},\; Q_{\,10}\,} \right\},\;\left\{ {\,Z_{\,13},\;Q_{\,11}\,} \right\},\; \left\{ {\,W_{\,13},\;S_{\,11}\,} \right\}$. Following Looijenga~\cite{looijenga;triangle}, we explain the relation between triangle singularities and K3 surfaces below. Let $T$ be one of the above fourteen symbols of hypersurface triangle singularities. Let $\Gamma^*$ be the Gabri\'{e}lov graph of the dual $T^*$. We can define a reducible curve $IF$ on a surface whose dual graph coincides with $\Gamma^*$. The curve $IF=IF\left( T \right)$ is called \emph{the curve at infinity} of type $T$. The irreducible components are all smooth rational curves $C$ with $C\cdot C=-2$ and have one-to-one correspondence with vertices of $\Gamma^*$. For two components $C$, $C'$ of $IF$ the intersection number $C\cdot C'$ is equal to one or zero, according as the corresponding vertices in $\Gamma^*$ are connected in $\Gamma^*$ or not. Let $G$ be a Dynkin graph with components of type $A$, $D$ or $E$ only. Assume that there exists a smooth K3 surface $Z$ satisfying the following conditions \textbf{(a)} and \textbf{(b)}: \begin{description} \item[(a)] $Z$ contains the curve at infinity $IF=IF\left( T \right)$ of type $T$ as a subvariety. \item[(b)] Let $E$ be the union of all smooth rational curves on $Z$ disjoint from $IF$. The dual graph of the components of $E$ coincides with graph $G$. \end{description} (Note that an irreducible curve $C$ on a K3 surface is a smooth rational curve if, and only if, $C\cdot C=-2$.) Contracting every connected component of $E$ to a rational double point and then removing the image of $IF$, we obtain an open variety $\tilde Y$. \begin{prop}[Looijenga~\cite{looijenga;triangle}] \label{looijenga} \begin{enumerate} \item Under the above assumption there exists a small deformation fiber $Y$ of a singularity of type $T$ homeomorphic to $\tilde Y$. \item Let $Y$ be a small deformation fiber of a singularity of type $T$. Assume that $Y$ has only rational double points as singularities, and the combination of rational double points on $Y$ corresponds to a Dynkin graph $G$. Then, there exists a K3 surface satisfying \textup{\textbf{(a)}} and \textup{\textbf{(b)}}, and the corresponding $\tilde Y$ is homeomorphic to $Y$. \end{enumerate} \end{prop} By the above proposition our study is reduced to the study of K3 surfaces containing the curve $IF=IF\left( T \right)$. K3 surfaces are complicated objects, but it is known that by the theory of periods we can reduce the study of K3 surfaces to the study of lattices. Below we explain several terminologies in the lattice theory. (Urabe~\cite{urabe;quardli}.) A free module over $\mathbf{Z}$ of finite rank equipped with an integral symmetric bilinear form $\left( {\;\;\;,\;\;} \right)$ is called a \emph{lattice}. Besides, if a free module $L$ over $\mathbf{Z}$ of finite rank has a symmetric bilinear form $\left( {\;\;\;,\;\;} \right)$ with values in rational numbers, then $L$ is called a \emph{quasi-lattice}. For simplicity we write $x^2=\left( {x,\;x} \right)$. Let $L$ be a quasi-lattice and $M$ be a submodule. The submodule $$\tilde M=\left\{ x\in L\, \left| {\,mx\in M\ } \mbox{for some non-zero integer} \ m\,\right. \right\}$$ is called the \emph{primitive hull} of $M$ in $L$. We say that $M$ is \emph{primitive}, if $M=\tilde M$, and an element $x\in L$ is primitive, if $M=\mathbf{Z}\,x$ is primitive. We say that an embedding $M\to L$ of quasi-lattices is a \emph{primitive embedding}, if the image is primitive. If $M$ is non-degenerate and primitive as a sub-quasi-lattice, we can define the canonical induced bilinear form on the quotient module $L/M$. Let $L$ be a quasi-lattice, and $FL$ be a submodule of $L$ such that the index $\#\left( L/ FL \right)$ is finite. Set \begin{eqnarray*} R=\left\{ \,\alpha \in FL\;\left| \;\alpha ^2=-2\, \right. \right\} \cup\left\{ \,\beta \in L\;\left| \;\beta ^2=-1\ or\ -2/ 3\; \right.\right\}\\ \hspace{7 mm}\cup \left\{ \,\gamma \in L\;\left| \;\gamma ^2=-1/ 2,\; 2\gamma \in FL\; \right. \right\} \end{eqnarray*} The set $R=R(L,\;FL)$ is called the \emph{root system} of $\left( {L,\;FL} \right)$, and every element $\alpha \in R$ is called a \emph{root}. If the pair $\left( {L,\;FL} \right)$ satisfies the following conditions \textbf{(R1)} and \textbf{(R2)}, then $\left( {L,\;FL} \right)$ is called a \emph{root module}: \begin{description} \item[(R1)] $2\left( {x,\;\alpha } \right)/\alpha ^2$ is an integer for every $x\in L$ and $\alpha \in R$. \end{description} Under \textbf{(R1)}, for every $\alpha \in R$ we can define an isomorphism $s_\alpha \,:\;L\to L$ preserving the bilinear form, by setting for $x\in L$ $s_\alpha \left( x \right)=x-2\left( {x,\;\alpha } \right)\alpha / \alpha ^2$. \begin{description} \item[(R2)]$s_\alpha \left( {FL} \right)=FL$ for every $\alpha \in R$. \end{description} Let $\left( {L,\;FL} \right)$ be a root module. If $L=FL$, we say that it is \emph{regular} and abbreviate $FL$. Let $M$ be a submodule of $L$. It is easy to check that the pair $\left( {M,\;FL\cap M} \right)$ is again a root module. Below we identify $M$ with the pair $\left( {M,\;FL\cap M} \right)$. If the root system of $M$ and the root system of $\tilde M$ coincide, then we say that $M$ is \emph{full}. An embedding $M\to L$ of quasi-lattices is a \emph{full embedding}, if the image is full. Let $G$ be a Dynkin graph with several components of type $A$, $D$ or $E$ only. We can define a lattice and its basis such that the corresponding dual graph coincides with $G$. This lattice is called the \emph{root lattic}e of type $G$ and is denoted by $Q(G)$. $Q(G)$ is a regular root module with a basis $\alpha _{\,1},\;\alpha _{\,2},\; \ldots ,\;\alpha _r$ with $\alpha _{\,i}^2=-2$ for every $i$. Let $\Lambda _N$ denote the even unimodular lattice with signature $\left( {N,\;16+N} \right)$ for $N\ge 0$. The isomorphism class of $\Lambda _N$ is unique if $N\ge 1$ and thus $\Lambda _N\cong \Lambda _{N-1}\oplus H$. For a K3 surface $Z$ the second cohomology group $H^2\left( {Z,\;\mathbf{Z}} \right)$ with the intersection form is a lattice isomorphic to $\Lambda _3$. Let $P=P\left( T \right)$ be the lattice whose dual graph is the Gabri\'{e}lov graph $\Gamma^*$ of the dual $T^*$. Assume that there exists a K3 surface $Z$ satisfying the above condition \textbf{(a)}. The classes of the components of $IF$ generate a primitive sublattice in $H^2\left( {Z,\;\mathbf{Z}} \right)$, which is isomorphic to $P$. \begin{prop} \label{embedding} \begin{enumerate} \item If $N\ge 1$, there is a primitive embedding $P\to \Lambda _N$. \item If $N\ge 2$, a primitive embedding $P\to \Lambda _N$ is unique up to automorphisms of $\Lambda _N$. \item If $N\ge 1$, for any embedding $P\to \Lambda _N$, the pair $\left( {\Lambda _N/\tilde P,\;F_N} \right)$ is a root module, where $F_N$ is the image of the orthogonal complement of $P$ in $\Lambda _N$ by the canonical surjective homomorphism $\Lambda _N\to \Lambda _N/ \tilde P$. \item For any primitive embedding $P=P\left( T \right)\to \Lambda _2$ the orthogonal complement $F_2$ of $P$ in $\Lambda _2$ has a basis whose dual graph coincides with the Gabri\'{e}lov graph of type $T$. \end{enumerate} \end{prop} With aid of Looijenga's results in \cite{looijenga;triangle} we can show the following: \begin{prop} \label{full} We fix a primitive embedding $P\to \Lambda _3$. There exists a K3 surface $Z$ satisfying the above conditions \textup{\textbf{(a)}} and \textup{\textbf{(b)}} if, and only if, there is a full embedding $Q\left( G \right)\to \Lambda _3/P$. \end{prop} \begin{cor} \label{criterion} $G\in PC\left( T \right)$ if, and only if, there is a full embedding $Q\left( G \right)\to \Lambda _3/P\left( T \right)$. \end{cor} By Proposition~\ref{full} our study has been reduced to the lattice theory. Next, we have to consider properties of the lattice $P=P\left( T \right)$ depending on $T$ closely. Let $T$ be one of fourteen symbols of hypersurface triangle singularities. \begin{prop} \label{property} We fix $N\ge 1$. \begin{enumerate} \item For any $T$ and for any embedding $P(t)\to \Lambda _N$ the quasi-lattice $\Lambda _N/\tilde P\left( T \right)$ does not contain an element $\beta$ with $\beta ^2=-1$. \item The root module $\left( {\Lambda _N/\tilde P\left( T \right),\;F_N} \right)$ contains a root $\gamma$ with $\gamma ^2=-1/2$ for some embedding $P\left( T \right)\to \Lambda _N$ if, and only if, $T=E_{\,13},\;Z_{\,12},\;Q_{\,11},\;W_{\,13}$ or $U_{\,12}$. It contains a root $\gamma$ with $\gamma ^2=-1/2$ for some primitive embedding $P\left( T \right)\to \Lambda _N$ if, and only if, $T=E_{\,13},Z_{\,12}$ or $\ Q_{\,11}$. \item The root module $\left( {\Lambda _N/\tilde P\left( T \right),\;F_N} \right)$ contains a root $\beta$ with $\beta ^2=-2/3$ for some embedding $P\left( T \right)\to \Lambda _N$ if, and only if, $T=E_{\,14},\;Z_{\,13}$ or $Q_{\,12}$. \end{enumerate} \end{prop} Consider the case where $(L,\;FL)$ is a root module such that the bilinear form on $L$ has signature $(1,\;\mathrm{rank}\, L-1)$. In this case we can apply the hyperbolic geometry, and we can give the generalization of the theory in the negative definite case such as the Weyl chamber and the Dynkin graph. The generalized Dynkin graph in this case is called the \emph{Coxeter-Vinberg graph}. (Vinberg~\cite{vinberg;group}.) We need consider the Coxeter-Vinberg graph of $\left( {\Lambda _{\,2}/P\left( T \right),\;F_2} \right)$. By Proposition~\ref{embedding}.4 we can expect that it is related to the Gabri\'{e}lov graph. We fix a primitive embedding $P\left( T \right)\to \Lambda _2$. \begin{prop} \label{vinberg-g} Let $\tilde \Gamma$ denote the Coxeter-Vinberg graph of $\left( {\Lambda _{\,2}/P\left( T \right),\;F_2} \right)$. \begin{enumerate} \item We can draw $\tilde \Gamma$ in finite steps if, and only if, $T\ne S_{\,11},\;S_{\,12}$. \item If $T\ne W_{\,12},\;W_{\,13},\;S_{\,11},\;S_{\,12},\;\ U_{\,12}$, every vertex in $\tilde \Gamma$ corresponds to a root. \item If $T=W_{\,12}$, every vertex in $\tilde \Gamma$ corresponds to either a root $\alpha$ with $\alpha ^2=-2$ or an element $\delta$ with $\delta ^2=-2/ 5$. \item If $T=W_{\,13}\ $ or $U_{\,12}$, every vertex in $\tilde \Gamma$ corresponds to either a root $\alpha$ with $\alpha ^2=-2$ or an element $\delta$ with $\delta ^2=-1/2$ and $2\delta \notin F_2$. \item If $T=E_{\,12},\;Z_{\,11},$ or $\ Q_{\,10}$, the Gabri\'{e}lov graph coincides with $\tilde \Gamma$. \item If $T=E_{\,13},\;E_{\,14},\;Z_{\,12},\;Z_{\,13},\;Q_{\,11},\;Q_{\,12}, \;W_{\,12},$ or $\ U_{\,12}$, the Gabri\'{e}lov graph is the subgraph of $\tilde \Gamma$ consisting of all vertices corresponding to a root $\alpha$ with $\alpha ^2=-2$. \item If $T=W_{\,13}$, the Gabri\'{e}lov graph is the maximal subgraph of $\tilde \Gamma$ such that every vertex corresponds to a root $\alpha$ with $\alpha ^2=-2$, and if $\alpha$, $\beta$ are roots corresponding to two vertices, then $\left( {\alpha ,\;\beta } \right)\ne -2$. \end{enumerate} \end{prop} We can explain main ideas in the verification of our Theorem~\ref{main} here. Let $\overline {PC}\left( T \right)$ denote the set of all Dynkin graphs made from a Dynkin subgraph of the Gabri\'{e}lov graph of type $T$ by an elementary transformation or a tie transformation. We assume that $G\in \overline {PC}\left( T \right)$ was made from a Dynkin subgraph $G'$ of the Gabri\'{e}lov graph. Besides, we fix a primitive embedding $P\to \Lambda _N$ for $N=2,\;3$. By Proposition~\ref{embedding}.4 there is a primitive embedding $Q\left( {G'} \right)\to F_2$. By the theory of elementary and tie transformations (Urabe~\cite{urabe;elem}, \cite{urabe;tie}.) we can conclude that there is a full embedding $Q\left( G \right)\to F_3\cong F_2\oplus H$ into the regular root module $F_3$. Assume $T\ne E_{\,13},\;Z_{\,12},\;Q_{\,11},\;E_{\,14},\;Z_{\,13}, \;Q_{\,12}$ here. By Proposition~\ref{property} the composition $Q\left( G \right)\to F_3\subset \Lambda _3/P$ defines a full embedding into the root module $\left( {\Lambda _3/P,\;F_3} \right)$ in these cases. By Corollary~\ref{criterion} we have $G\in PC\left( T \right)$. Thus $\overline {PC}\left( T \right)\subset PC\left( T \right)$. Next, we determine the difference $PC\left( T \right)-\overline {PC}\left( T \right)$. Let $r$ be the number of vertices of a graph $G$. It is easy to see that if $G\in PC\left( T \right)$, then $r\le \mu -2$. In case $T\ne S_{\,11},\;S_{\,12}$, using Proposition~\ref{vinberg-g} we can show that conditions $G\in PC\left( T \right)$ and $G\in \overline {PC}\left( T \right)$ are equivalent if $r\le \mu -5$. Thus we can assume $r=\mu -2,\;\mu -3$ or $\mu -4$. For triangle singularities the Milnor number $\mu$ is relatively small, and it is easy to check whether a Dynkin graph $G$ belongs to $PC\left( T \right)-\overline {PC}\left( T \right)$ case-by-case. To tell the truth, we could not succeed in finding any effective method except case-by-case checking. This is a weak point of our theory. I regret this fact and hope that somebody can improve it. If $T=S_{\,11}$ or $S_{\,12}$, the checking becomes more complicated since we have no Coxeter-Vinberg graph. Now, if $T\ne W_{\,12},\;W_{\,13},\;S_{\,11},\;S_{\,12},\;\ U_{\,12}$, then because of Proposition~\ref{vinberg-g}.2 we can formulate another theorem. (Urabe~\cite{urabe;triangle}.) In this another theorem we start from not a Gabri\'{e}lov graph but a Dynkin graph possibly with a component of type $BC_{\,1}$ or $G_2$, and the number of transformations is not one but two. There, no exception appears even in the case $Z_{\,13}$. (We can make $A_7+A_4$ from $E_7+G_2$ by two tie transformations.) For $T=E_{\,13},\;Z_{\,12},\;Q_{\,11},\;E_{\,14},\;Z_{\,13}$ or $Q_{\,12}$ our theorem in this article follows from this theorem in another formulation. Also for $T=W_{\,12},\;W_{\,13},\;S_{\,11},\;S_{\,12},\;U_{\,12}$ the theorem in another formulation is possible, but becomes very complicated, because Proposition~\ref{vinberg-g}.2 does not hold for them. It is not worth mentioning. Details of the verification will appear elsewhere. Now, it is very strange that our Theorem~\ref{main} has a few exceptions in a few cases. Perhaps this is because our theory has a missing part. \vspace*{3 mm} \noindent \textbf{Problem.} Find the missing part of our theory and give a simple characterization of the set $PC\left( T \right)$ without exceptions. \vspace*{3 mm} This problem may be very difficult, but I believe that there exists a solution.

9611

alg-geom/9611010

en

https://arxiv.org/abs/alg-geom/9611010

alg-geom/9611010

Serkan Hosten

Serkan Hosten

On the Complexity of Smooth Projective Toric Varieties

8 pages

In this paper we answer a question posed by V.V. Batyrev. The question asked if there exists a complete regular fan with more than quadratically many primitive collections. We construct a smooth projective toric variety associated to a complete regular fan $\Delta$ in R^d with $n$ generators where the number of primitive collections of $\Delta$ is at least exponential in $n-d$. We also exhibit the connection between the number of primitive collections of $\Delta$ and the facet complexity of the Gr\"obner fan of the associated integer program.

alg-geom

\section{Introduction} \noindent In this paper we give an affirmative answer to the following question posed by V.V.Batyrev \cite{Bat}: \vskip 0.3cm \noindent {\bf Question:} {\sl Does there exist a complete regular $d$-dimensional fan $\Delta$ with $n$ generators such that $\Delta$ has more than $(n-d-1)(n-d+2)\slash2$ primitive collections for $n-d > 1$? } \vskip 0.3cm \noindent In Section 2 we prove the following theorem which answers the above question. \begin{theorem} There exists a complete regular $d$-dimensional fan $\Delta$ with $n$ generators where the number of primitive collections of $\Delta$ is more than $2^{\frac{1}{2}(n-d)}$. \end{theorem} \vskip 0.3cm \noindent A fan $\Delta \subset {\bf R}^d$ that covers ${\bf R}^d$ is a complete fan. If we require the full-dimensional cones in $\Delta$ to be simplicial with integral generators which form a ${\bf Z}$-basis for ${\bf Z}^d$, then $\Delta$ is said to be {\it regular} (see below for formal definitions). If a fan $\Delta \subset {\bf R}^d$ is generated by the $1$-dimensional cones defined by the vectors in ${\cal A} = \{a_1, a_2, \ldots, a_n\} \subset {\bf Z}^d$, then the {\it primitive collections} of $\Delta$ are defined as follows: \begin{definition} A nonempty subset ${\cal P} = \{a_{i_1},a_{i_2},\ldots,a_{i_k}\}$ of ${\cal A}$ is called a {\bf primitive collection} if for each generator $a_{i_s} \in {\cal P}$ the elements ${\cal P} \setminus a_{i_s}$ generate a $(k-1)$-dimensional cone in $\Delta$, while ${\cal P}$ does not generate a $k$-dimensional cone in $\Delta$. \end{definition} \noindent The primitive collections of a complete regular fan $\Delta \subset {\bf R}^d$ with $n$ generators are studied in \cite{Bat} to classify $d$-dimensional smooth complete toric varieties with $n-d=3$. The question we study asks whether there exists a complete regular fan $\Delta$ where the number of primitive collections of $\Delta$ is at least {\it quadratic} in $n-d$. Theorem 1.1 constructs a complete regular fan with {\it exponentially} many primitive collections. The same theorem can be restated in the language of Gr\"obner bases of toric varieties: Theorem 3.1 shows that there exists a toric variety $X$ with a square-free initial ideal whose number of minimal generators is exponential in the codimension of $X$. In the last section we make a connection between two conjectures: one of them appears in the context of the complexity of complete regular fans (Conjecture 7.1 in \cite{Bat}) and the other one is about the complexity of {\it Gr\"obner fans} in the context of {\it integer programming} (Conjecture 6.1 in \cite{ST}). \noindent In this article we will use results from the theory of {\it coherent triangulations} of a vector configuration. In order to prove Theorem 1.1 we need the following definitions which connect coherent triangulations and complete regular projective fans: A complete fan $\Delta \in {\bf R}^d$ with $n$ generators ${\cal A} = \{a_1, a_2, \ldots, a_n\} \subset {\bf Z}^d$ is said to be {\it regular} if every $d$-dimensional cone $\sigma \in \Delta$ is simplicial and the generators $\{a_{i_1}, a_{i_2}, \ldots, a_{i_d}\}$ of $\sigma$ form a ${\bf Z}$-basis of ${\bf Z}^d$. \begin{definition} A complete regular fan $\Delta$ is said to be {\bf projective} if there exists a {\bf support function} $\phi : {\bf R}^d \rightarrow {\bf R}$ such that \begin{enumerate} \item $\phi$ is convex and $\phi({\bf Z}^d) \subset {\bf Z}$, \item $\phi$ is linear on each cone of $\Delta$ with $\phi\mid_{\sigma} \neq \phi\mid_{\tau}$ for distinct $d$-dimensional cones $\sigma$ and $\tau$. \end{enumerate} \end{definition} \noindent It is a well-known fact that if $V(\Delta)$ is the smooth complete $d$-dimensional toric variety that is associated with $\Delta$ then $V(\Delta)$ is projective if and only if $\Delta$ is projective \cite{Oda}. In this paper we will use an equivalent definition of a projective toric variety via {\it coherent} triangulations. \begin{definition} A triangulation $T$ of a vector configuration ${\cal A} = \{a_1, a_2, \ldots, a_n\} \in {\bf R}^d$ is a polyhedral complex consisting of simplical cones which cover $pos({\cal A}) = \{x \in {\bf R}^d\,: \, x = \sum_{i=1}^{n} \lambda_i a_i, \, \lambda_i \geq 0\}$. A triangulation $T$ of ${\cal A}$ is said to be {\bf coherent} if there exists a support function $\phi$ on $T$ as in Definition 1.3 (see \cite{BFS}, \cite{GKZ}). \end{definition} \section{Exponential lower bound} \noindent To give an exponential lower bound for the number of primitive collections of a complete regular fan we will use the example given in Proposition 6.7 of \cite{ST}. \begin{definition} Given a matrix $B \in {\bf Z}^{d \times n}$, the {\bf chamber complex} $\Gamma(B)$ of $B$ is the coarsest polyhedral complex that refines all triangulations of $B$ and covers $pos(B)$. \end{definition} \begin{proposition} \cite{BGS} Let $A$ be a Gale transform of $B \in {\bf Z}^{d \times n}$, i.e. let $A \in {\bf Z}^{(n-d) \times n}$ such that $$0 \longrightarrow {\bf R}^{d} \stackrel{B^T}{\longrightarrow} {\bf R}^{n} \stackrel{A}{\longrightarrow} {\bf R}^{(n-d)} \longrightarrow 0$$is exact. Then there is a bijection between the coherent triangulations of $A$ and the $d$-dimensional chambers of $\Gamma(B)$ given by $$T = \bigcup_{i=1}^{t} \sigma_i \Longleftrightarrow C = \bigcap_{i=1}^{t} \sigma^{\ast}_i$$ where $\sigma_i = pos(a_{i_1},a_{i_2},\ldots,a_{i_{n-d}})$ are the cones of the coherent triangulation $T$ and $\sigma^{\ast}_i = pos(\{b_j: 1 \leq j \leq n, \,j \neq i_1, i_2, \ldots, i_{n-d}\})$ are the cones of $B$ containing the chamber $C$. \end{proposition} \noindent Now we construct a complete regular projective fan which has exponentially many primitive collections. Let $B$ be the node-edge incidence matrix of the complete bipartite graph $K_{n,m}$ where $n=2k-1$ and $m=2k+1$. $B = \{e_i \times e'_j: \, 1 \leq i \leq n, 1 \leq j \leq m \}$ where $e_i \in {\bf R}^n$ and $e'_j \in {\bf R}^m$ are unit vectors. $B$ has rank $n+m-1$ and is unimodular, i.e. any subdeterminant of $B$ is $0$ or $\pm 1$ (\cite{Schr}, p.273). The cone $pos(B)$ consists of all non-negative vectors $(u_1,\ldots,u_n) \times (v_1,\ldots,v_m)$ such that $u_1+\cdots+u_n = v_1+\cdots+v_m$. Let $A \in {\bf Z}^{(n-1)(m-1) \times nm}$ be a Gale transform of $B$. By Proposition 2.2, for every chamber in the chamber complex $\Gamma(B)$ there exists a corresponding coherent triangulation of $A$. We consider a special chamber in $\Gamma(B)$. The one-dimensional cone generated by $(\frac{1}{n},\ldots,\frac{1}{n}) \times (\frac{1}{m},\ldots,\frac{1}{m})$ is in $pos(B)$ and we claim that it is in the interior of a full-dimensional chamber. The facets of full-dimensional chambers of $\Gamma(B)$ correspond to the cocircuits of the oriented matroid of $B$ (\cite{DHSS}, Lemma 2.7). In our situation, a cocircuit of $B$ corresponds to a cut $(C_+,C_-;D_+,D_-)$ in $K_{n,m}$ where $(C_+,C_-)$ is a partition of $\{1,\ldots,n\}$ and $(D_+,D_-)$ is a partition of $\{1,\ldots,m\}$. The corresponding hyperplane is defined by $$\sum_{i \in C_+} u_i - \sum_{i \in C_-} u_i - \sum_{j \in D_+} v_j + \sum_{j \in D_-} v_j \, = \, 0. \eqno (1)$$ Since $n$ and $m$ are relatively prime, $(\frac{1}{n},\ldots,\frac{1}{n}) \times (\frac{1}{m},\ldots,\frac{1}{m})$ cannot lie on any of these hyperplanes. So it must be in the interior of a full-dimensional chamber. This chamber is called the {\it central chamber} of $pos(B)$. Let $\Delta$ be the corresponding coherent triangulation of $A$. Since $B$ is unimodular, so is $A$, and therefore $\Delta$ consists of simplical cones whose generators form a ${\bf Z}$-basis for ${\bf Z}^{(n-1)(m-1)}$. Because there exists a strictly positive vector in $im(B^T) = ker(A)$, any vector in ${\bf R}^{(n-1)(m-1)}$ is in $pos(A)$. This shows that $\Delta$ is a complete regular fan. \noindent Now we show that every column of $A$ is a generator of $\Delta$. By Proposition 2.2 it is enough to show that for every column $b_{i,j} = e_i \times e'_j \in B$ there exists a cone $\tau$ which contains the central chamber but which does not have $b_{i,j}$ as a generator. Suppose $\tau$ is a cone that contains the central chamber and has $b_{i,j}$ as a generator. Since the natural action of the product of symmetric groups $S_n \times S_m$ on $B$ fixes the central chamber, for any $\pi \times \sigma \in S_n \times S_m$, $(\pi \times \sigma)(\tau)$ covers the central chamber as well. We can pick $\pi$ such that $\pi(i) = i$. If $b_{i,k}, \, k \neq j$ does not appear as a generator of $\tau$ we can choose $\sigma$ such that $\sigma(k) = j$, and we would be done. So suppose $b_{i,k}$ is a generator of $\tau$ for $k = 1,\ldots,m$. Since these vectors are linearly independent and $rank(B)=n+m-1$, there are exactly $n-1$ generators of $\tau$ which are not of the above form. But $\tau$ contains $(\frac{1}{n},\ldots,\frac{1}{n}) \times (\frac{1}{m},\ldots,\frac{1}{m})$ in its interior, so these remaining $n-1$ generators are of the form $b_{k,s_k}, \, k =1,\ldots,n, \, k \neq i$. Each of these generators must have the coefficient $\frac{1}{n}$ in the unique expression that expresses $(\frac{1}{n},\ldots,\frac{1}{n}) \times (\frac{1}{m},\ldots,\frac{1}{m})$ in terms of generators of $\tau$. But $\frac{1}{n} > \frac{1}{m}$, and this shows that $\Delta$ is a complete regular projective fan generated by the columns of $A$. \noindent In order to give an exponential lower bound on the primitive collections of $\Delta$ constructed above we establish a link between its {\it circuits} and its primitive collections. \begin{definition} Let ${\cal A}$ be a vector configuration in ${\bf Z}^d$. A collection of linearly dependent vectors $Z \subseteq {\cal A}$ is called a {\bf circuit} if any proper subset of $Z$ is linearly independent. \end{definition} \noindent We will call the circuits of the generators of a complete fan $\Delta$ the circuits of $\Delta$. If $Z$ is a circuit of $\Delta$, the unique (up to sign) dependence relation $ \sum_i \lambda_i z_i = 0$ partitions $Z$ into two subsets, namely $Z_+ = \{z_i \in Z: \lambda_i > 0\}$ and $Z_i = \{z_i \in Z:\lambda_i < 0\}$. In this case, there exist precisely two triangulations of $Z$: $\, t_+(Z) = \{Z \setminus z_i: z_i \in Z_+\}$ and $t_-(Z) = \{Z \setminus z_i: z_i \in Z_-\}$. Note that $relint(pos(Z_+)) \bigcap relint(pos(Z_-)) \neq \emptyset$ (see \cite{BLSWZ}). Given a triangulation $\Delta$ and a circuit $Z$ of $\Delta$ such that $t_+(Z)$ is a subcomplex of $\Delta$, one can get via a {\it bistellar flip} another triangulation $\Delta'$ such that $t_-(Z)$ is a subcomplex of $\Delta'$. For the details we refer to [GKZ, pp. 231-233]. The next lemma makes the connection between the circuits and primitive collections of $\Delta$. \begin{lemma} Let $\Delta \subset {\bf R}^d$ be a complete regular fan (i.e. a triangulation) and let $Z$ be a circuit such that $t_+(Z)$ is a subcomplex of $\Delta$. Then $Z_+$ is a primitive collection. Moreover, if $Z'$ is a different circuit where $t_+(Z')$ is a subcomplex of $\Delta$, then $Z_+ \neq Z'_+$. \end{lemma} \noindent {\sl Proof:} Clearly $Z_+$ does not generate a cone in $\Delta$. By definition of $t_+(Z)$, for all $z \in Z_+, \, pos(Z_+ \setminus z)$ is a face of $t_+(Z)$, and hence is a cone in $\Delta$. Each of these cones must be $(card(Z_+)-1)$-dimensional, since otherwise $Z$ cannot be a circuit. This shows $Z_+$ is a primitive collection. For the second statement, assume $Z_+ = Z'_+$. Since $t_+(Z) \neq t_+(Z')$, the respective subcomplexes $K$ and $K'$ of $\Delta$ on which the bistellar flips are supported are different as well. But $pos(Z_+) \bigcap relint(K) \neq \emptyset$ and $pos(Z'_+) \bigcap relint(K') \neq \emptyset$, which implies $relint(K) \bigcap relint(K') \neq \emptyset$. This cannot happen since $K$ and $K'$ are distinct subcomplexes of $\Delta$. This contradiction completes the proof. $\Box$ \vfill \eject \noindent For the main theorem we need the following result which provides the link between bistellar flips (and hence the primitive collections) of $\Delta$ and the corresponding chamber in the dual configuration. \begin{theorem}(\cite{GKZ}, p.233) Let ${\cal A} \subset {\bf Z}^d$ be a vector configuration and let ${\cal B}$ be a Gale transform of ${\cal A}$. If $\Delta$ and $\Delta'$ are two coherent triangulations of ${\cal A}$, then $\Delta$ and $\Delta'$ differ by a bistellar flip if and only if the corresponding chambers in $\Gamma({\cal B})$ share a facet. \end{theorem} \noindent {\sl Proof of Theorem 1.1:} Let $B$ be the node-incidence matrix of $K_{n,m}$ where $n=2k-1$ and $m=2k+1$ and let $A$ be a Gale transform of $B$. Let $\Delta$ be the coherent triangulation of $A$ that corresponds to the central chamber in $\Gamma(B)$. As we established before $\Delta$ is a complete regular projective fan generated by the columns of $A$. Lemma 2.4 and Theorem 2.5 imply that the number of primitive collections of $\Delta$ should be at least the number of facets of the central chamber in $\Gamma(B)$. The following proposition shows that there are at least exponentially many such facets. The proof of the proposition can be found in \cite{ST}, but we include its proof for completeness. \begin{proposition} The central chamber in $\Gamma(B)$ which correponds to $\Delta$ has at least $4^k$ facets. \end{proposition} \noindent {\sl Proof:} If $H$ is a hyperplane defined by the equation (1), we will call $(card(C_+),card(D_+))$ the {\it type} of $H$. Now starting at the point $(\frac{1}{n},\ldots,\frac{1}{n}) \times (\frac{1}{m},\ldots,\frac{1}{m})$ and moving in the direction of $(-1,\ldots,-1,n-1) \times (0,\ldots,0)$ to a generic point $(a,\ldots,a,a+1-na) \times (\frac{1}{m},\ldots,\frac{1}{m})$ we cross a facet of type $(r,s)$ with $n \in C_-$ whenever $$\, r \cdot a \,\, - \,\, (n-r-1) \cdot a - (a+1-na) \,\, - \,\, s / m \,\, + \,\, (m-s)/m \quad = \quad 2 \cdot r \cdot a \,-\, 2 \cdot s / m \quad = \quad 0. $$ From here we get $a = s / mr$ and since $a < 1 / n$ we like to find $r$ and $s$ which minimize the positive integer $m \cdot r - n \cdot s$. The unique solution is $r = k$ and $s = k + 1$ and since $S_n \times S_m$ acts transitively on the set of hyperplanes of type $(r,s)$, we conclude that every hyperplane of this type is a facet of the central chamber. When $k \geq 3$ there are $2k-1 \choose k$ $2k+1 \choose k+1$ $> 2^k \cdot 2^k = 4^k$ such facets. $\Box$ \section{Connections to integer programming} \noindent In this section we will first state Theorem 1.1 in terms of the {\it Gr\"obner basis} (\cite{AL}, \cite{CLO}) of an {\it integer program}. Subsequently we will relate two conjectures, one that appears in the context of smooth projective toric varieties and the other one in the context of integer programming. An integer program can be stated as follows: $$ \hbox{minimize} \,\, c \cdot x \,\,\,\, \hbox{subject to} \,\, A \cdot x \, = \, b, \,\, x \in {\bf N}^n$$ where $A \in {\bf Z}^{d \times n}$ with $rank(A) = d$, $b \in {\bf Z}^d$ and $c \in {\bf R}^n$. The reduced Gr\"obner basis of the {\it toric ideal} $I_A = \langle x^\alpha - x^\beta: \, \alpha, \beta \in {\bf N}^n, \, A \cdot \alpha = A \cdot \beta \rangle$ with respect to the term order induced by the cost vector $c$ provides a {\it test set} for solving this integer program (see \cite{AL}, \cite{CT}, \cite{ST}, \cite{Th} for details). \begin{theorem} Let $A$ be a Gale transform of the node-edge incidence matrix $B$ of $K_{n,m}$ with $n=2k-1$ and $m=2k+1$. Then $I_A$ is generated by $x_{i1} x_{i2} \cdots x_{im} - 1, \, i = 1,\ldots,n$ and $x_{1j} x_{2j} \cdots x_{nj} -~1, \, j = 1, \ldots,m$, and the reduced Gr\"obner basis of $I_A$ with respect to the degree lexicographic term order contains at least $4^k$ elements. \end{theorem} \noindent {\sl Proof:} The rows of $B$ constitute a ${\bf Z}$-basis for $ker(A) \bigcap {\bf Z}^{nm}$. The above binomials correspond to the rows of $B$ and the ideal they generate is contained in $I_A$. But since the sum of all the rows of $B$ is a strictly positive vector, these binomials generate $I_A$ (see Lemma 2.1 in \cite{SWZ}). The degree lexicographic term order $\succ_{deglex}$ can be represented by the cost vector $c = (1,1,\ldots,1)$ refined by the lexicographic order. Since $A$ is unimodular, the reduced Gr\"obner basis of $I_A$ with respect to $\succ_{deglex}$ consists of square-free binomials (\cite{St}, Corollary 8.9) and the initial term of each binomial corresponds to a minimal non-face (i.e. a primitive collection) of the coherent triangulation $\Delta$ induced by $c$ (\cite{St}, Theorem 8.3). As the vector $B\cdot c$ is in the central chamber of $\Gamma(B)$, Proposition 2.2 implies that the triangulation $\Delta$ induced by $c$ is the same as the complete regular fan we considered in the proof of Theorem 1.1. This finishes the proof. $\Box$ \begin{example}($3 \times 5$ Complete Bipartite Graph) \end{example} Let $B$ be the node-incidence matrix of $K_{3,5}$. If we associate with every column $b_{ij} = e_i \times e^{\prime}_j, \, \, i = 1,2,3, \,\, j = 1,2,3,4,5$, the variable $x_{ij}$, then $$I_A \, = \, \langle x_{11}x_{21}x_{31} - 1, x_{12}x_{22}x_{32} - 1, x_{13}x_{23}x_{33} - 1, x_{14}x_{24}x_{34} - 1, x_{15}x_{25}x_{35} - 1,$$ $$x_{11}x_{12}x_{13}x_{14}x_{15} - 1, x_{21}x_{22}x_{23}x_{24}x_{25} - 1, x_{31}x_{32}x_{33}x_{34}x_{35} - 1 \rangle.$$ There are 30 facets of the central chamber of $B$ and indeed the reduced Gr\"obner basis of $I_A$ with respect to $\succ_{deglex}$ consists of 50 binomials. \vskip 0.3 cm \noindent In relation to smooth complete projective varieties, the following conjecture is posed in \cite{Bat}. \begin{conjecture} (\cite{Bat}, Conjecture 7.1) For any $d$-dimensional smooth complete toric variety defined by a complete regular fan $\Delta$ with $n$ generators, there exists a constant $N(n-d)$ depending only on $n-d$ such that the number of primitive collections in $\Delta$ does not exceed $N(n-d)$. \end{conjecture} Another conjecture with a similar flavor is stated about the complexity of {\it Gr\"obner cones} in the setting of integer programming in \cite{ST} (Conjecture 6.1), and here we will give a connection between the two conjectures along the lines of the previous section. Given an integer program defined by a matrix $A$, two generic cost vectors $c$ and $c'$ are considered to be equivalent if the respective reduced Gr\"obner bases of $I_A$ are the same. The set of all such equivalent cost vectors associated to a fixed reduced Gr\"obner basis of the toric ideal of $A$ is an open polyhedral cone and the collection of the closures of all such cones and their faces constitute a fan called the {\it Gr\"obner fan} of $A$ (\cite{MR}, \cite{BM}, \cite{St}, \cite{ST}). \begin{conjecture} (\cite{ST}, Conjecture 6.1) There exists a function $\varphi$ such that, for every matrix $A \in {\bf Z}^{d \times n}$ of rank $d$, every cone of the Gr\"obner fan of $A$ has at most $\varphi(n-d)$ facets. \end{conjecture} This conjecture is true for $n-d \leq 2$. For the case $n-d = 3$, $\varphi(3) = 4$ under certain genericity assumptions on the matrix $A$ and this was proved by I.~B\'ar\'any and H.~Scarf in \cite{BS}. The following proposition points to a connection between $N$ and $\varphi$: \begin{proposition} (\cite{ST}, Corollary 3.18) Let $A \in {\bf Z}^{d \times n}$ with $rank(A)=d$ be a unimodular matrix and let $B$ be a Gale transform of $A$. Then the Gr\"obner fan of $A$ and $\Gamma(B)$ coincide. \end{proposition} In the light of this proposition one can formulate a specialized version of Conjecture 3.4. \begin{conjecture} There exists a function $\varphi'$ such that, for every unimodular matrix $A \in {\bf Z}^{d \times n}$ of rank $d$, every cone of $\Gamma(B)$ has at most $\varphi'(n-d)$ facets where $B$ is a Gale transform of $A$. \end{conjecture} \begin{theorem} If there exist $N$ and $\varphi'$ as above, then $N(n-d) \geq \varphi'(n-d)$ for all $n$ and $d$. \end{theorem} \noindent {\sl Proof:} Fix $n$ and $d$, and suppose that $A \in {\bf Z}^{d \times n}$ is a unimodular matrix with a coherent triangulation $\Delta$ such that the corresponding chamber in $\Gamma(B)$, where $B$ is a Gale transform of $A$, has $\varphi'(n-d)$ facets. If $\Delta$ uses all columns of $A$ as generators, then by the results of the previous section we would be done. Otherwise we can refine $\Delta$ into another complete regular fan by adding the missing generators. This will not destroy the primitive collections in $\Delta$ associated with the bistellar flips as in Lemma 2.4. Hence $N(n-d) \geq \varphi'(n-d)$. $\Box$ \vskip 0.5cm \noindent {\bf Acknowledgement} The author thanks Bernd Sturmfels for pointing out Batyrev's open problem.

9611

alg-geom/9611004

en

https://arxiv.org/abs/alg-geom/9611004

alg-geom/9611004

Alexandre Kabanov (Max-Planck-Institut and Michigan State Univ.)

The Second Cohomology with Symplectic Coefficients of the Moduli Space of Smooth Projective Curves

23 pages, latex2e with amslatex and xy-pic, to appear in Compositio Mathematica

Each finite dimensional irreducible rational representation V of the symplectic group Sp_{2g} determines a generically defined local system \V over the moduli space M_g of genus g smooth projective curves. We study H^2(M_g;\V) and the mixed Hodge structure on it. Specifically, we prove that if g>5, then the natural map IH^2(MS_g;\V)-->H^2(M_g;\V) is an isomorphism where MS_g is the Satake compactification of M_g. Using the work of Saito we conclude that the mixed Hodge structure on H^2(M_g;\V) is pure of weight 2+r if \V underlies a variation of Hodge structure of weight r. We also obtain estimates on the weight of the mixed Hodge structure on H^2(M_g;\V) for 2<g<6. Results of this article can be applied in the study of relations in the Torelli group T_g.

alg-geom

\section{Introduction} \label{sec:intro} The moduli space ${\mathcal M}_g$ of smooth projective curves of genus $g$ is a quasi-projective variety over ${\mathbb C}$. Its points correspond to isomorphism classes of smooth projective complex curves of genus $g$. It has only finite quotient singularities, and therefore behaves like a smooth variety. This space has several natural compactifications. In this article we will be interested in the so called Satake compactification $\widetilde{\mathcal M}_g$ of ${\mathcal M}_g$. The period map determines an inclusion of ${\mathcal M}_g$ into ${\mathcal A}_g$, the moduli space of principally polarized abelian varieties. The Satake compactification $\widetilde{\mathcal M}_g$ is the closure of ${\mathcal M}_g$ inside $\overline{\mathcal A}_g$, the Satake compactification of ${\mathcal A}_g$. It has quite complicated singularities at its boundary $\widetilde{\mathcal M}_g - {\mathcal M}_g$. Each representation of the algebraic group $\operatorname{Sp}_{2g}$ gives rise to an orbifold local system over ${\mathcal M}_g$. To explain this we introduce the mapping class group ${\Gamma}_g$. It is the group of connected components of the group of the orientation preserving diffeomorphisms of a compact orientable surface $S$ of genus $g$. The group ${\Gamma}_g$ is the orbifold fundamental group of ${\mathcal M}_g$, and representations of ${\Gamma}_g$ give rise to orbifold local systems over ${\mathcal M}_g$. There is a natural surjective map $$ {\Gamma}_g \rightarrow \operatorname{Aut} (H_1(S; {\mathbb Z}), \cap) $$ where $\cap$ is determined by the intersection pairing. The right hand group is isomorphic to $\operatorname{Sp}_{2g}({\mathbb Z})$. So each finite dimensional rational representation $V$ of an algebraic group $\operatorname{Sp}_{2g}$ gives rise to a {\it symplectic} orbifold local system ${\mathbb V}$ over ${\mathcal M}_g$. Since ${\mathbb V}$ is generically defined over $\widetilde{\mathcal M}_g$, one can consider the intersection cohomology groups $IH^\bullet (\widetilde{\mathcal M}_g; {\mathbb V})$. There is a natural restriction map $$ IH^\bullet (\widetilde{\mathcal M}_g; {\mathbb V}) \rightarrow H^\bullet ({\mathcal M}_g; {\mathbb V}). $$ The main result of this article is \begin{theoremvoid} (cf. Th.~\ref{thm:main}) The natural restriction map $$ IH^k (\widetilde{\mathcal M}_g; {\mathbb V}) \rightarrow H^k ({\mathcal M}_g; {\mathbb V}) $$ is an isomorphism when $k=1$ for all $g\ge 3$, and when $k=2$ for all $g\ge 6$. \end{theoremvoid} The group $H^1({\mathcal M}_g; {\mathbb V})$ is easily computed when $g\ge 3$ for all symplectic local systems ${\mathbb V}$ using Johnson's fundamental work \cite{johnson} (cf.~\cite{hain:torelli}). Let $X$ be an algebraic variety. From Saito's work \cite{saito:intro}, \cite{saito:mhm} we know that $H^\bullet(X;{\mathbb V})$ has natural mixed Hodge structure (MHS) if ${\mathbb V} \to X$ is an admissible polarized variation of Hodge structure, and $IH^\bullet(X;{\mathbb V})$ has natural mixed Hodge structure if ${\mathbb V}$ is a generically defined admissible polarized variation of Hodge structure over $X$. Further if $X$ is compact and ${\mathbb V}$ is pure of weight $r$, then $IH^\bullet(X;{\mathbb V})$ is pure of weight $k+r$. \begin{theoremvoid} (cf. Cor.~\ref{thm:purity}, Cor.~\ref{cor:semipurity}) If $g\ge 6$ and ${\mathbb V} \rightarrow {\mathcal M}_g$ is a variation of Hodge structure of weight $r$ whose underlying local system is symplectic, then the natural mixed Hodge structure on $H^2({\mathcal M}_g;{\mathbb V})$ is pure of weight $2+r$. If $3\le g <6$, then the weights of the mixed Hodge structure on $H^2({\mathcal M}_g; {\mathbb V})$ lie in $\{ 2+r, 3+r \}$. \end{theoremvoid} Each symplectic local system ${\mathbb V}$ associated to an irreducible representation $V$ of $\operatorname{Sp}_{2g}$ underlies a variation of Hodge structure over ${\mathcal M}_g$ which is unique up to Tate twist. It is convenient to fix the weight of the variation of Hodge structure ${\mathbb V}(\lambda)$ associated to a dominant integral weight $\lambda$. Fix fundamental weights $\lambda_1, \lambda_2, \dots, \lambda_g$ of $\operatorname{Sp}_{2g}$. If $\lambda= a_1\lambda_1 + a_2\lambda_2 +\cdots + a_g\lambda_g$, define $|\lambda|= a_1+ 2a_2+ \cdots +ga_g$. This is the smallest integer $r$ such that $V(\lambda) \subseteq H_1(S)^{\otimes r}$. (A good reference is \cite{fulton:harris}.) Then ${\mathbb V}(\lambda)$ can be realized uniquely as a variation of Hodge structure of weight $|\lambda|$. Harer proved in \cite{harer:stability} that the cohomology $H^k({\mathcal M}_g; {\mathbb Z})$ stabilizes when $g \ge 3k$, and Ivanov later improved the range of stability \cite{ivanov:stability}. \cite{ivanov:twist}. He showed that $H^k({\mathcal M}_g; {\mathbb Z})$ stabilizes when $g\ge 2k+2$. In \cite{ivanov:twist} Ivanov also proved that $H^k ({\mathcal M}_{g,1}; {\mathbb V}(\lambda))$ is independent of $g$ when $g\ge 2k+2+|\lambda|$. (The space ${\mathcal M}_{g,1}$ is the moduli space of curves with a marked non-zero tangent vector.) In \cite{looijenga:stable} Looijenga calculated the stable cohomology groups of ${\mathcal M}_g$ with symplectic coefficients as a module over stable cohomology groups of ${\mathcal M}_g$ with trivial coefficients. In particular, this implies that $H^k ({\mathcal M}_g; {\mathbb V}(\lambda))$ is independent of $g$ when $g\ge 2k+2+2|\lambda|$. Looijenga's result also provides very specific information about the MHS on $H^k ({\mathcal M}_g; {\mathbb V}(\lambda))$. Combined with computations of $H^k({\mathcal M}_g; {\mathbb Q})$ in low dimensions due to Harer \cite{harer:second}, \cite{harer:third}, \cite{harer:fourth}, it implies that $H^k ({\mathcal M}_g; {\mathbb V}(\lambda))$ is pure of weight $k+|\lambda|$ when $k\le 4$ and $g$ is in the stability range. In particular, $H^2({\mathcal M}_g; {\mathbb V}(\lambda))$ is pure of weight $2+|\lambda|$ when $g\ge 6+2|\lambda|$. Recently, Pikaart proved in \cite{pikaart} that the stable cohomology $H^k ({\mathcal M}_g; {\mathbb Q})$ is pure of weight $k$. Combined with Looijenga's computations, this shows that $H^k ({\mathcal M}_g; {\mathbb V}(\lambda))$ is pure of weight $k+|\lambda|$ whenever $g\ge 2k+2+2|\lambda|$. Unlike the stability range, our purity range is {\it independent} of $|\lambda|$. This is important for the following application which was the motivation for this article. The Torelli group $T_g$ is the kernel of the surjective homomorphism ${\Gamma}_g \to \operatorname{Sp}_{2g}({\mathbb Z})$. One can consider the Malcev Lie algebra ${\mathfrak t}_g$ associated to $T_g$. (For definitions see \cite{hain:completion}). This Lie algebra is an analogue of the Lie algebra associated to the pure braid group on $m$ strings, which is important in the study of Vassiliev invariants and conformal field theory. By a result of Johnson \cite{johnson}, $T_g$ is finitely generated when $g\ge 3$. Thus, ${\mathfrak t}_g$ is also finitely generated when $g\ge 3$. It is not known for any $g\ge 3$ whether $T_g$ is finitely presented or not. In \cite{hain:lietor} Hain gives an explicit presentation of ${\mathfrak t}_g$ for $g\ge 3$. More specifically, he proves that for each choice of $x_0 \in {\mathcal M}_g$ there is a canonical MHS on ${\mathfrak t}_g$ which is compatible with the bracket. Thus, $$ {\mathfrak t}_g \otimes {\mathbb C} \cong \prod_m \Gr{W}{-m} {\mathfrak t}_g \otimes {\mathbb C} $$ where $\Gr{W}{\bullet}$ are the graded quotients of the MHS associated to a choice of $x_0$. Hain proves that for all $g\ge 3$ $$ \Gr{W}{\bullet} {\mathfrak t}_g = \L (H_1 ({\mathfrak t}_g)) / (R_g) $$ where $\L$ stands for the free Lie algebra, and $R_g$ is a set of relations. According to a result of Johnson \cite{johnson} $H_1 ({\mathfrak t}_g)$ is isomorphic as an $\operatorname{Sp}_{2g}$-module to $V({\lambda}_3)$. Using the above theorem about the MHS on $H^2({\mathcal M}_g; {\mathbb V})$ Hain proves that the relations $R_g$ are quadratic when $g\ge 6$, and quadratic and possibly cubic when $g=3,4,5$. Moreover, he explicitly calculates all quadratic relations. This implies that ${\mathfrak t}_g$ is finitely presented for all $g\ge 3$. We shall outline the proof of the first theorem above. There are three main steps in the proof. The first step is to notice that if $g\ge 3$, then the boundary $\widetilde{\mathcal M}_g - {\mathcal M}_g$ of the Satake compactification has one irreducible component of codimension two, and all other irreducible components have codimension three. This immediately implies that $H^1({\mathcal M}_g; {\mathbb V}) \cong IH^1(\widetilde{\mathcal M}_g; {\mathbb V})$. The codimension two irreducible component of $\widetilde{\mathcal M}_g - {\mathcal M}_g$ has a Zariski open subset isomorphic to ${\mathcal M}_1 \times {\mathcal M}_{g-1}$. We denote it by $X$. (In the paper we work with a smooth Zariski open subset of $X$. However this is just a technical detail, and we do not want to draw an attention to it here.) Let $N^\ast$ be the link bundle of $X$ in $\widetilde{\mathcal M}_g$. We denote by $\pi$ the corresponding projection. Then there is an exact sequence $$ 0 \to IH^2(\widetilde{\mathcal M}_g; {\mathbb V}) \to H^2({\mathcal M}_g; {\mathbb V}) \to H^0(X; R^2 \pi_\ast {\mathbb V}), $$ and the last morphism factors through the edge homomorphism $$ \psi: H^2(N^\ast; {\mathbb V}) \to H^0(X; R^2 \pi_\ast {\mathbb V}) $$ of the Leray--Serre spectral sequence of $\pi$. Therefore it suffices to show that $\psi$ is the trivial homomorphism. The second step is to understand the link bundle $N^\ast$. Let $L$ be the pull-back under $pr_2: X \to {\mathcal M}_{g-1}$ of the unit relative tangent bundle over ${\mathcal M}_{g-1}$, and $\tilde{\pi}$ be the corresponding projection $L \to X$. We show that $L$ is a two-to-one unramified covering of $N^\ast$. (This is done in Section \ref{sec:link}.) Here we need to assume that $g\ge 4$. Denote by $\widetilde{\mathbb V}$ the pull-back of the local system ${\mathbb V}$ to $L$, and by $\tilde{\psi}$ the edge homomorphism $H^2(L; \widetilde{\mathbb V}) \to H^0(X; R^2 \tilde{\pi}_\ast \widetilde{\mathbb V})$ of the Leray--Serre spectral sequence of $\tilde{\pi}$. There is a commutative diagram $$ \begin{CD} H^2(L; \widetilde{\mathbb V}) @> \tilde{\psi} >> H^0(X; R^2 \tilde{\pi}_\ast \widetilde{\mathbb V}) \\ @AAA @AAA \\ H^2(N^\ast; {\mathbb V}) @> \psi >> H^0 (X; R^2 \pi_\ast {\mathbb V}) \\ \end{CD} $$ where both vertical maps are inclusions. This implies that $\psi$ is trivial, if $\tilde{\psi}$ is trivial. The third step is to show that $\tilde{\psi}$ is trivial. The local system $\widetilde{\mathbb V}$ extends to the stratum $X \cong {\mathcal M}_1 \times {\mathcal M}_{g-1}$, and splits over it according to the branching rule for the standard inclusion of $\operatorname{Sl}_2 \times \operatorname{Sp}_{2g-2}$ into $\operatorname{Sp}_{2g}$. The bundle map $\tilde{\pi}$ respects this splitting. Thus, it suffices to show that $\tilde{\psi}$ is trivial for each irreducible symplectic local system $\overline{\mathbb V}$ over $X$. We complete the computation using Schur's lemma and the fact, due to Harer \cite{harer:fourth}, that $H^2({\Gamma}_{g,1}; H^1(S))$ is trivial when $g\ge 4$. (One can also use a result from \cite[Sec.~7]{harer:third} that $H^2({\Gamma}_{g,1}; H^1(S))$ is trivial when $g\ge 9$.) \begin{acknowledgements} I would like to thank Richard Hain for his helpful suggestions and numerous discussions during my graduate studies at Duke University. John Harer helped to complete the last step in the proof of the main theorem. I would also like to thank Sloan Foundation for providing me with the Doctoral Dissertation Fellowship that allowed me to spend the Fall Semester of 1994 at the Institute for Advanced Study at Princeton. Conversations with Pierre Deligne, Alan Durfee, Mikhail Grinberg, Eduard Looijenga, William Pardon, Martin Pikaart, Chad Schoen, Mark Stern, and Steven Zucker were also very helpful. I would like to thank the referee who suggested many shortcuts in this paper. \end{acknowledgements} \section{Basic facts about the moduli space of curves} \label{sec:moduli} In this section we recall the definitions and basic properties of the moduli spaces of curves, and the corresponding mapping class groups. The moduli space ${\mathcal M}_{g,r}^s$ parameterizes the isomorphism classes of smooth complex projective curves of genus $g$ with $s$ marked points and $r$ marked non-zero holomorphic tangent vectors. The existence of such moduli spaces follows from geometric invariant theory. These moduli spaces are known to be normal quasi-projective varieties \cite[Th.~5.11, Th.~7.13]{mumford:git}. One can also construct ${\mathcal M}_{g,r}^s$ using {Teichm\"uller}\ theory. This approach allows us to establish the relation between the moduli spaces and the corresponding mapping class groups. Let $S$ denote a smooth compact orientable surface of genus $g$. Fix $s+r$ distinct points $p_1, \dots, p_{r+s}$ on $S$, and $r$ non-zero tangent vectors $v_1, \dots, v_r$ at points $p_1, \dots, p_r$ respectively. One can consider triples $$ (C, (q_1,\dots, q_{r+s}, w_1, \dots, w_r), [f]), $$ where $C$ is a smooth projective genus $g$ curve, $q_1, \dots, q_{r+s}$ are distinct points on $C$, $w_1, \dots, w_r$ are non-zero holomorphic tangent vectors at $q_1, \dots, q_r$ respectively, and $f: C \to S$ is an orientation preserving diffeomorphism such that $f(q_i)= p_i$ and $f_\ast (w_i)= v_i$ (we use the canonical identification of the holomorphic tangent space with the underlying real tangent space). We denote by $[f]$ the isotopy class of $f$ relative to $\{ q_1, \dots, q_{r+s}, w_1, \dots, w_r \}$. Two triples $$ (C_j, (q_1^j, \dots, q_{r+s}^j, w_1^j, \dots, w_r^j), [f_j]), \quad j=1,2, $$ are called {\it equivalent} if there exists a biholomorphism $h: C_1 \to C_2$ such that $h(q_i^1) =q_i^2$, $h_\ast(w_i^1) =w_i^2$, and $[f_2 \circ h]= [f_1]$ where the homotopy is required to preserve the marked points and tangent vectors. The space of equivalence classes ${\mathcal T}_{g,r}^s$ is called the {\it {Teichm\"uller}\ space} \cite{harer:review}, \cite[p.~26]{harer:third}. It is known that ${\mathcal T}_{g,r}^s$ is a contractible complex manifold of dimension $3g-3+s+2r$ when $2g-2+s+2r>0$. The {\it mapping class group} ${\Gamma}_{g,r}^s$ is defined to be $\operatorname{Diff}^+(S)/ \operatorname{Diff}^+_0(S)$, where $\operatorname{Diff}^+(S)$ is the group of orientation preserving diffeomorphisms of $S$, which leave the marked points $p_1, \dots, p_{r+s}$ and marked tangent vectors $v_1, \dots, v_r$ fixed, and $\operatorname{Diff}^+_0(S)$ is the connected component of the identity. If $g>0$, then the group ${\Gamma}_{g,r}^s$ is torsion free when either $r>0$, or $s>2g+2$. The group ${\Gamma}_{g,r}^s$ acts on ${\mathcal T}_{g,r}^s$ as follows. If $g\in {\Gamma}_{g,r}^s$, then $$ g (C, (q_1,\dots, w_r), [f])= (C, (q_1,\dots, w_r), [g\circ f]). $$ The quotient space ${\Gamma}_{g,r}^s \!\! \setminus \!\! {\mathcal T}^s_{g,r}$ is the moduli space ${\mathcal M}_{g,r}^s$ of curves with $s$ marked points and $r$ marked tangent vectors. The group ${\Gamma}_{g,r}^s$ acts on ${\mathcal T}_{g,r}^s$ by biholomorphisms, and this action is properly discontinuous and virtually free. It follows that ${\mathcal M}_{g,r}^s$ is a complex analytic variety with only finite quotient singularities. This analytic structure agrees with the one coming from geometric invariant theory. If ${\Gamma}_{g,r}^s$ is torsion free, then the action is free, and ${\mathcal M}_{g,r}^s$ is smooth. \begin{notation} We shall omit indices $r$ and $s$ from ${\mathcal T}_{g,r}^s$, ${\Gamma}_{g,r}^s$, and ${\mathcal M}_{g,r}^s$ when they are equal to zero. We shall use both ${\mathcal M}_1^1$ and ${\mathcal M}_1$ to denote the moduli space of elliptic curves. \end{notation} \begin{remark} One can also consider ${\mathcal M}^{[s]}_{g}$, the moduli space of genus $g$ curves with a marked set of cardinality $s$. It is the quotient of ${\mathcal M}^s_{g}$ by the natural action of the symmetric group on $s$ letters. This action permutes the marked points. \end{remark} The singular locus of ${\mathcal M}_g$ is contained in the locus of curves with non-trivial automorphisms. When $g\ge 3$, we denote by ${\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_g$ the locus of curves with only trivial automorphisms. This is a smooth Zariski open subset of ${\mathcal M}_g$ whose complement has codimension $g-2$. There are natural surjective morphisms between different moduli spaces which correspond to forgetting marked points and marked tangent vectors \cite{knudsen:II}. We will consider the morphisms ${\mathcal M}_g^1 \to {\mathcal M}_g$ and ${\mathcal M}_{g,1} \to {\mathcal M}_g^1$. The first morphism ${\mathcal M}_g^1 \to {\mathcal M}_g$ is called the ``universal curve''\cite[p.~218]{eisenbud:harris}. Its fiber over a point $[C]\in {\mathcal M}_g$ is $C/\! \operatorname{Aut} C$. On the level of the mapping class groups there is a corresponding short exact sequence \cite{birman:braids} $$ 1 \to \pi_1(S) \to {\Gamma}_g^1 \to {\Gamma}_g \to 1. $$ The morphism ${\mathcal M}_{g,1} \to {\mathcal M}_g^1$ ``forgets'' the tangent vector, but remembers its base point. When $g\ge 2$ it is the frame bundle of the relative holomorphic tangent bundle to the universal curve. On the level of the mapping class groups there is a corresponding short exact sequence \cite{birman:braids} $$ 1 \to {\mathbb Z} \to {\Gamma}_{g,1} \to {\Gamma}_g^1 \to 1. $$ The composition of the two morphisms discussed above is the morphism ${\mathcal M}_{g,1} \to {\mathcal M}_g$ obtained by forgetting the tangent vector. If $C$ is a curve without non-trivial automorphisms, then the fiber over $[C] \in {\mathcal M}_g$ is isomorphic to $T^u C$, the frame bundle of the holomorphic tangent bundle of the curve $C$. The corresponding homomorphism of the mapping class groups is ${\Gamma}_{g,1} \to {\Gamma}_g$. One can also consider finite index level subgroups ${\Gamma}_{g,r}^s [l]$ of ${\Gamma}_{g,r}^s$ for each integer $l$. The level $l$ subgroup is defined to be the subgroup of ${\Gamma}_{g,r}^s$ which acts trivially on $H_1(S; {\mathbb Z}/ l{\mathbb Z})$. Consequently, one has a short exact sequence $$ 1 \to {\Gamma}_{g,r}^s [l] \to {\Gamma}_{g,r}^s \to \operatorname{Sp}_{2g}({\mathbb Z}/ l{\mathbb Z}) \to 1. $$ The quotient ${\Gamma}_{g,r}^s [l]\setminus {\mathcal T}_{g,r}^s$ is isomorphic to ${\mathcal M}_{g,r}^s [l]$, the moduli space of smooth projective curves with a level $l$ structure which is defined in Section \ref{sec:comp}. It is well-known that for all $g\ge 1$ and $l\ge 3$, the group ${\Gamma}_{g,r}^s [l]$ acts freely on ${\mathcal T}_{g,r}^s$. Thus for each $l\ge 3$ the moduli space ${\mathcal M}_{g,r}^s [l]$ is a smooth finite cover of ${\mathcal M}_{g,r}^s$. When ${\mathcal M}_{g,r}^s$ is different from ${\mathcal M}_1$ and ${\mathcal M}_2$ each representation of ${\Gamma}_{g,r}^s$ determines an orbifold local system over ${\mathcal M}_{g,r}^s$. When ${\mathcal M}_g$ is either ${\mathcal M}_1$ or ${\mathcal M}_2$ we consider only such representations of ${\Gamma}_g$ that for each $[C] \in {\mathcal M}_g$ represented by a curve with only two automorphisms, the stabilizer of $(C,[f])\in {\mathcal T}_g$ acts trivially on the representation space. These representations give rise to orbifold local systems over ${\mathcal M}_1$ and ${\mathcal M}_2$. Let $V$ be a representation of ${\Gamma}_{g.r}^s$ on a rational vector space, and let ${\mathbb V}$ be the associated orbifold local system over ${\mathcal M}_{g,r}^s$. The contractibility of the {Teichm\"uller}\ space implies that for all $g\ge 1$: $$ H^\bullet ({\Gamma}_{g,r}^s; V) \cong H^\bullet ({\mathcal M}_{g,r}^s; {\mathbb V}) \cong H^\bullet ({\mathcal M}_{g,r}^s [l]; {\mathbb V}[l])^{\operatorname{Sp}_{2g} ({\mathbb Z}/ l{\mathbb Z})}. $$ \section{Compactifications of the moduli space of curves} \label{sec:comp} In this section we recall some basic properties of the Satake compactification and the Deligne--Mumford compactification of the moduli spaces of curves. We start with the Deligne--Mumford compactification of ${\mathcal M}_g^s$. A {\it stable curve} is a reduced connected curve which has only nodes as singularities, and a finite automorphism group \cite{deligne:mumford}. The {\it Deligne--Mumford compactification} $\overline{\mathcal M}_g^s$ of ${\mathcal M}_g^s$ is the moduli space of stable projective curves. It is a normal projective variety in which ${\mathcal M}_g^s$ is a Zariski open subset \cite{deligne:mumford}, \cite[Th.~5.1]{mumford:stable}. The singularities of $\overline{\mathcal M}_g^s$ are contained in the locus of stable curves with non-trivial automorphisms \cite[p.~218]{eisenbud:harris}. We will describe the boundary $\overline{\mathcal M}_g^s - {\mathcal M}_g^s$ in the case when $s=0$. The boundary $\overline{\mathcal M}_g-{\mathcal M}_g$ is the union of irreducible divisors $$ \bigcup_{i=0}^{[{g/ 2}]} \Delta_i, $$ where each divisor $\Delta_i$ has the following property. When $i=0$ there is birational morphism $\overline{\mathcal M}_{g-1}^{[2]} \to \Delta_0$; when $1\le i < g-i$ there is birational morphism $\overline{\mathcal M}_i^1 \times \overline{\mathcal M}_{g-i}^1 \to \Delta_i$; and when $i=g-i$ there is a birational morphism from the ${\mathbb Z}/2{\mathbb Z}$-quotient of $\overline{\mathcal M}_i^1 \times \overline{\mathcal M}_i^1$ to $\Delta_i$. \begin{definition} (cf. \cite[Def.~10.5]{popp}) A {\it level $l$ structure} on a stable curve $C$ is a symplectic monomorphism $H^1(C; {\mathbb Z}/l{\mathbb Z}) \to ({\mathbb Z}/l{\mathbb Z})^{2g}$, where $({\mathbb Z}/l{\mathbb Z})^{2g}$ has the standard symplectic structure. \end{definition} Note that a level $l$ structure on a smooth curve $C$ is just a choice of a symplectic basis for $H^1(C;{\mathbb Z}/l{\mathbb Z})$, or, equivalently, for $H_1(C;{\mathbb Z}/l{\mathbb Z})$ because the symplectic form determines the canonical identification between homology and cohomology. The same is true for a singular stable curve $C$ whose dual graph is a tree. From now on we assume that $l\ge 3$. Denote by ${\mathcal M}_g [l]$ the moduli space of smooth curves with a level $l$ structure. It is isomorphic to the quotient of ${\mathcal T}_g$ by the action of ${\Gamma}_g [l]$ (cf. Sec.~\ref{sec:moduli}). The moduli space ${\mathcal M}_g [l]$ is a smooth quasi-projective variety, and the forgetful morphism ${\mathcal M}_g[l] \to {\mathcal M}_g$ is a Galois covering \cite[Prop.~5.8]{deligne:mumford}, \cite[Th.~1.8]{oort:steenbrink}. When $g\ge 2$ and $l\ge 3$ there exists the moduli space of stable curves with a level $l$ structure $\overline{\mathcal M}_g [l]$, which is a compactification of ${\mathcal M}_g [l]$ \cite[p.~106]{deligne:mumford}, \cite[Bem.~1]{mostafa}, \cite[Rem.~2.3.7]{jong:pikaart}. This is a projective variety according to \cite[Th.~4, III.8]{mumford:red}, and there is a finite morphism $\overline{\mathcal M}_g [l] \to \overline{\mathcal M}_g$ determined by forgetting a level $l$ structure. In \cite{mostafa} Mostafa proves that $\overline{\mathcal M}_g [l]$ is not smooth, at least when $g\ge 3$. However, in this article we are interested in particular strata of the boundary of $\overline{\mathcal M}_g [l]$. The irreducible component $\Delta_1$ of the boundary of $\overline{\mathcal M}_g$ contains a Zariski open subset isomorphic to ${\mathcal M}_1^1 \times {\mathcal M}_{g-1}^1$. Consider the inverse image of this subset under the finite morphism above. It is a finite disjoint union of locally closed subvarieties of codimension one each of which is isomorphic to ${\mathcal M}_1^1 [l] \times {\mathcal M}_{g-1}^1 [l]$. According to \cite[Lem.~1]{mostafa}, \cite[p.~240]{looijenga:prym} all points of this inverse image are smooth points of $\overline{\mathcal M}_g [l]$. To introduce the Satake compactification $\widetilde{\mathcal M}_g$ of ${\mathcal M}_g$ we use the space ${\mathcal A}_g$, the moduli space of principally polarized abelian varieties of dimension $g$. It is the quotient of the Siegel upper-half space by the action of $\operatorname{Sp}_{2g}({\mathbb Z})$. The space ${\mathcal A}_g$ is a quasi-projective variety \cite[Th.~7.10]{mumford:git}. Among other compactifications, it admits the Satake compactification $\overline{\mathcal A}_g$ which is a projective variety \cite{satake}. The moduli space ${\mathcal M}_g$ is isomorphic to the image of the period map ${\mathcal M}_g \to {\mathcal A}_g$ which is a locally closed subvariety of ${\mathcal A}_g$ \cite[Cor.~3.2]{oort:steenbrink}. The closure $\widetilde{\mathcal M}_g$ of ${\mathcal M}_g$ in the Satake compactification $\overline{\mathcal A}_g$ of ${\mathcal A}_g$ is called the {\it Satake compactification of} ${\mathcal M}_g$ (cf. \cite{baily:1962}). There exists a birational morphism $\alpha: \overline{\mathcal M}_g \to \widetilde{\mathcal M}_g$ which is the identity on ${\mathcal M}_g$, and sends the point $[C]$ corresponding to a stable curve $C$ to the polarized Jacobian of its normalization \cite[p.~211]{knudsen:III}. The image of the boundary $\overline{\mathcal M}_g - {\mathcal M}_g$ under $\alpha$ is the boundary $\widetilde{\mathcal M}_g - {\mathcal M}_g$ of the Satake compactification. It follows that when $g\ge 3$ the boundary $\widetilde{\mathcal M}_g - {\mathcal M}_g$ has $[g/2]$ irreducible components each of which except one has codimension three in $\widetilde{\mathcal M}_g$. The irreducible component $\Phi_1$ which is the image of $\Delta_1 \subset \overline{\mathcal M}_g$ has codimension two. It contains a Zariski open subset isomorphic to ${\mathcal M}_1 \times {\mathcal M}_{g-1}$. One can also construct the {\it Satake compactification} $\widetilde{\mathcal M}_g [l]$ of ${\mathcal M}_g [l]$. Denote by ${\mathcal A}_g [l]$ the moduli space of principally polarized abelian varieties with a level $l$ structure. A point in ${\mathcal A}_g [l]$ is represented by an abelian variety $A$ of dimension $g$ and a symplectic basis of $H_1 (A; {\mathbb Z}/l{\mathbb Z})$. It is a quasi-projective variety \cite[Th.~7.9]{mumford:git}, \cite[Th.~1.8]{oort:steenbrink} which is smooth when $l\ge 3$. The space ${\mathcal A}_g [l]$ has the Satake compactification $\overline{\mathcal A}_g [l]$ which is a normal projective variety \cite[p.~124]{popp}, \cite{satake}. If $g=1,2$, then ${\mathcal M}_g [l]$ is isomorphic to a Zariski open subset of ${\mathcal A}_g [l]$, and we define the Satake compactification $\overline{\mathcal A}_g [l]$ of ${\mathcal A}_g [l]$ to be the Satake compactification of ${\mathcal M}_g [l]$. If $g\ge 3$, then the morphism ${\mathcal M}_g [l] \to {\mathcal A}_g [l]$ is not injective. In this case we define $\widetilde{\mathcal M}_g [l]$ to be the normalization of $\widetilde{\mathcal M}_g$ with respect to ${\mathcal M}_g [l]$. It follows from this definition that $\widetilde{\mathcal M}_g [l]$ is a projective variety \cite[III.8, Th.~4]{mumford:red}, and that the morphism ${\mathcal M}_g [l] \to {\mathcal M}_g$ extends to a finite morphism $\widetilde{\mathcal M}_g [l] \to \widetilde{\mathcal M}_g$. One can also show that there is a birational morphism $\alpha^l:\overline{\mathcal M}_g [l] \to \widetilde{\mathcal M}_g [l]$ with connected fibers which is the identity on ${\mathcal M}_g [l]$, and fits into the commutative diagram $$ \begin{CD} \overline{\mathcal M}_g [l] @>\alpha^l >> \widetilde{\mathcal M}_g [l] \\ @VVV @VVV \\ \overline{\mathcal M}_g @>\alpha >> \widetilde{\mathcal M}_g. \\ \end{CD} $$ The boundary $\widetilde{\mathcal M}_g [l] - {\mathcal M}_g [l]$ is the union of irreducible components each of which has codimension either two, or three in $\widetilde{\mathcal M}_g [l]$. The image of each component $\Phi_1^\beta$ of codimension two under the morphism $\widetilde{\mathcal M}_g [l] \to \widetilde{\mathcal M}_g$ is the codimension two component $\Phi_1$ of $\widetilde{\mathcal M}_g - {\mathcal M}_g$. One can show that each $\Phi_1^\beta$ contains a Zariski open subset $Z_\beta$ such that these subsets do not intersect each other, and each of them is isomorphic to a smooth Zariski open subset of ${\mathcal M}_1 [l] \times {\mathcal M}_{g-1} [l]$. \section{Codimension two stratum of the Satake compactification} \label{sec:link} In this section we analyze the link of the codimension two boundary stratum $\Phi_1$ inside the Satake compactification of the moduli space $\widetilde{\mathcal M}_g$. More precisely, we study the local links of the points in a smooth Zariski open subset of $\Phi_1$, and we show that $\widetilde{\mathcal M}_g$ is equi-singular along this Zariski open subset. We will need this in Section \ref{sec:main}. For the rest of this section we assume that $g\ge 4$. Recall that $\Phi_1$ contains a Zariski open subset $X$ isomorphic to ${\mathcal M}_1 \times {\mathcal M}_{g-1}$. We identify it with ${\mathcal M}_1 \times {\mathcal M}_{g-1}$. Then a point in $X$ is represented by a pair of isomorphism classes of curves $([C_1],[C_2])$. Let ${X}^\circ$ be a Zariski open subset of $X$ defined as follows. Recall that in Section \ref{sec:moduli} we defined ${\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_g$ to be the locus of curves with only trivial automorphisms when $g\ge 3$. We define ${\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_1$ to be the locus of elliptic curves with exactly two automorphisms. Then ${X}^\circ$ is the subset of $X$ corresponding to ${\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_1 \times {\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_{g-1}$. In this section we study the link of ${X}^\circ$ in ${\mathcal M}_g \cup {X}^\circ \subset \widetilde{\mathcal M}_g$. Let $N$ be a regular neighborhood of ${X}^\circ$ in ${\mathcal M}_g \cup {X}^\circ$. The complement $N^\ast= N- {X}^\circ$ is a deleted regular neighborhood of ${X}^\circ$. Recall that ${\mathcal M}_{g-1,1} \to {\mathcal M}_{g-1}$ is a surjective morphism defined by forgetting the holomorphic tangent vector. Let $L_2$ be the inverse image of ${\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_{g-1}$ in ${\mathcal M}_{g-1,1}$, and $\tilde{\pi}_2: L_2 \to {\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_{g-1}$ be the corresponding map. The fiber of $\tilde{\pi}_2$ over $[C_2] \in {\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_{g-1}$ is $T^u C_2$, the punctured holomorphic tangent bundle. Denote by $L$ the product ${\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_1 \times L_2$, and by $\tilde{\pi}$ the pull-back of $\tilde{\pi}_2$ to ${X}^\circ$: $$ \begin{CD} L= {\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_1 \times L_2 @> pr_2 >> L_2 \\ @V \tilde{\pi} VV @V \tilde{\pi}_2 VV \\ {X}^\circ= {\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_1 \times {\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_{g-1} @> pr_2 >> {\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_{g-1}. \\ \end{CD} $$ \begin{lemma} \label{lem:link} The bundle $\tilde{\pi}: L= {\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_1 \times L_2 \to {X}^\circ$ is a two-to-one unramified cover of the punctured regular neighborhood $N^\ast$. The corresponding fix point free action of ${\mathbb Z}/2{\mathbb Z}$ on $L$ sends a vector $v$ to $-v$. \end{lemma} \begin{proof} The morphism ${\mathcal M}_{g-1,1} \to {\mathcal M}_{g-1}$ factors as $$ {\mathcal M}_{g-1,1} \to {\mathcal M}_{g-1}^1 \to {\mathcal M}_{g-1}. $$ Denote by $Y_2$ the inverse image of ${\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_{g-1}$ under the second morphism. Then the commutative diagram above factors as $$ \begin{CD} L= {\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_1 \times L_2 @> pr_2 >> L_2 \\ @V \pi^c VV @V \pi^c_2 VV \\ Y= {\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_1 \times Y_2 @> pr_2 >> Y_2 \\ @V \bar{\pi} VV @V \bar{\pi}_2 VV \\ {X}^\circ= {\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_1 \times {\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_{g-1} @> pr_2 >> {\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_{g-1}, \\ \end{CD} $$ where $\pi_2^c$ (resp. $\bar{\pi}_2$) is the restriction of ${\mathcal M}_{g-1,1} \to {\mathcal M}_{g-1}^1$ (resp. ${\mathcal M}_{g-1}^1 \to {\mathcal M}_{g-1}$) to $L_2$ (resp. $Y_2$), and $\pi^c$ (resp. $\bar{\pi}$) is its pull-back along $pr_2$. At the same time $Y= {\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_1 \times Y_2$ is isomorphic to a smooth Zariski open subset of the boundary component $\Delta_1$ in the Deligne--Mumford compactification. We identify $Y$ with this Zariski open subset. Then the morphism $\bar{\pi}: Y \to {X}^\circ$ is the restriction of the morphism $\alpha: \overline{\mathcal M}_g \to \widetilde{\mathcal M}_g$ to $Y$. The morphism $\alpha$ is the identity when restricted to ${\mathcal M}_g$. Therefore a deleted regular neighborhood $N^\ast$ of ${X}^\circ$ in ${\mathcal M}_g \cup {X}^\circ$ and a deleted regular neighborhood of the divisor $Y$ in ${\mathcal M}_g \cup Y \subset \overline{\mathcal M}_g$ can be chosen to be the same. The deleted neighborhood of $Y$ is homeomorphic to the punctured normal bundle of $Y$ in ${\mathcal M}_g \cup Y$. Note that the only non-trivial automorphism of a pair $(C_1,x_1),(C_2,x_2)$ representing a point in $Y$ is induced by the elliptic involution of $(C_1,x_1)$. It follows that ${\mathbb Z}/2{\mathbb Z}$ acts on the space of versal deformations of the stable curve $(C_1,x_1),(C_2,x_2)$, and this action fixes the divisor that is the locus of the singular curves \cite[Chap.~13, Lem.~(1.6)]{acgh:II}. Therefore the fiber of the normal bundle of $Y \subset \Delta_1$ at the point $[(C_1,x_1),(C_2,x_2)]$ is isomorphic to the ${\mathbb Z}/2{\mathbb Z}$ quotient of $T_{x_1}C_1 \otimes T_{x_2}C_2$, where the generator of ${\mathbb Z}/2$ acts as $-id$. Thus $N^\ast$ is the ${\mathbb Z}/2{\mathbb Z}$ quotient of the ${\mathbb C}^\ast$-bundle $L'$ over $Y$ whose fiber at $[(C_1,x_1),(C_2,x_2)]$ is $T_{x_1}C_1 \otimes T_{x_2}C_2 - \{ 0 \}$. It is well-known that the moduli space of elliptic curves ${\mathcal M}_1$ is isomorphic to ${\mathbb C}$. It contains two distinguished points that correspond to the two elliptic curves with exceptional automorphisms. It follows that the space ${\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_1$ is isomorphic to ${\mathbb C}- \{ 2\ \text{points} \}$. All line bundles over this space are trivial. Therefore the bundle $L'$ is the pull-back of the punctured relative tangent bundle of the morphism $Y_2 \to {\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_{g-1}$. The punctured relative tangent bundle of the morphism $Y_2 \to {\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_{g-1}$ is $\pi^c_2: L_2 \to Y_2$. Hence, one has a commutative diagram $$ \begin{CD} L' @>>> L_2 \\ @VVV @V \pi^c_2 VV \\ Y={\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_1 \times Y_2 @> pr_2 >> Y_2, \\ \end{CD} $$ where $L'$ is the pull-back of $L_2$. We conclude that the bundles $L'$ and $L$ are isomorphic, and $N^\ast$ is the ${\mathbb Z}/2{\mathbb Z}$ quotient of $L$, where ${\mathbb Z}/2{\mathbb Z}$ action sends a vector in a fiber of $\pi^c$ to its opposite. \end{proof} It follows from the lemma above that $\widetilde{\mathcal M}_g$ is equi-singular along ${X}^\circ$. We expressed $N^\ast$ as a bundle ${X}^\circ$ whose fiber over the point $([C_1,C_2])$ is equal to $T^u C_2$, the frame bundle of the holomorphic tangent bundle of $C_2$. \section{Main theorem} \label{sec:main} In this section we prove the main theorem of this article. The proof consists of a sequence of lemmas and propositions. We assume that the reader is familiar with intersection cohomology, and suggest the references \cite{bbd}, \cite{borel:ic}, \cite{gmII}. \begin{notation} For the rest of the paper we omit $R^\bullet$ from the notation for the derived functors. For example, if $f:X\to Y$ is a continuous map between topological spaces, then $f_\ast = R^\bullet f_\ast$. \end{notation} As we mentioned before each representation of the mapping class group ${\Gamma}_g$, at least when $g\ge 3$, determines an orbifold local system over ${\mathcal M}_g$. In this section we consider only the {\it symplectic} local systems, that is local systems arising from finite dimensional rational representations of the algebraic group $\operatorname{Sp}_{2g}$. We fix a symplectic representation $V$ of ${\Gamma}_g$, and denote the corresponding orbifold local system by ${\mathbb V}$. \begin{theorem} \label{thm:main} The natural map $IH^k(\widetilde{\mathcal M}_g;{\mathbb V}) \to H^k({\mathcal M}_g;{\mathbb V})$ induced by the inclusion is an isomorphism, when \begin{center} $\begin{array}{l@{,\quad}l} k=0 & g\ge 1;\\ k=1 & g\ge 3; \\ k=2 & g\ge 6. \end{array}$ \end{center} \end{theorem} The first statement is trivial and included only for the sake of completeness. The statement concerning the first cohomology is also rather simple. Indeed, in Section \ref{sec:comp} we saw that if $g\ge 3$, then the boundary $\widetilde{\mathcal M}_g - {\mathcal M}_g$ of the Satake compactification has codimension two in $\widetilde{\mathcal M}_g$. This, and the properties of intersection cohomology immediately imply the statement of the theorem for $k=1$. The non-trivial part of this theorem concerns the second cohomology. \begin{remark} If $g\ge 3$, then the map $IH^1(\widetilde{\mathcal M}_g;{\mathbb V}) \to H^1({\mathcal M}_g;{\mathbb V})$ is an isomorphism for an arbitrary orbifold local system ${\mathbb V}$ determined by a representation of ${\Gamma}_g$ on a rational vector space. This can be easily seen from the above argument. \end{remark} Combining this with the computations of $H^1({\mathcal M}_g;{\mathbb V})$ in \cite{hain:torelli}, \cite{johnson} one gets the following corollary. \begin{corollary} If $g\ge 3$ and ${\mathbb V}(\lambda)$ is a generically defined local system corresponding to the representation of $\operatorname{Sp}_{2g}$ with the highest weight $\lambda$, then $$ IH^1 (\widetilde{\mathcal M}_g; {\mathbb V}(\lambda)) \cong \left \{ \begin{array}{ll} {\mathbb Q} & \quad {\mbox{when}} \quad \lambda= \lambda_3 ; \\ 0 & \quad {\mbox{otherwise}}. \qquad \qquad \qed \end{array} \right. $$ \end{corollary} The rest of this section is devoted to the proof of the isomorphism in second cohomology. We assume that $g\ge 4$. Recall that we denote by $\Phi_1$ the codimension two irreducible component of the boundary of $\widetilde{\mathcal M}_g$, and by ${X}^\circ$ its Zariski open subset isomorphic to ${\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_1 \times {\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_{g-1}$. \begin{notation} We denote by ${\mathcal S}^\bullet$ the intersection cohomology sheaf ${\mathcal I \mathcal C}^\bullet ({\mathbb V})$ on $\widetilde{\mathcal M}_g$ corresponding to the local system ${\mathbb V}$. The following diagram defines the notation for the inclusions: $$ {\mathcal M}_g \stackrel{i}{\hookrightarrow} {\mathcal M}_g \cup {X}^\circ \stackrel{j}{\hookleftarrow} {X}^\circ. $$ \end{notation} First, we use again that the boundary of $\widetilde{\mathcal M}_g$ has only one irreducible component of codimension two, namely $\Phi_1$, and all other irreducible components have codimension three. This and the properties of intersection cohomology imply that the restriction $$ IH^2(\widetilde{\mathcal M}_g; {\mathbb V}) \to IH^2({\mathcal M}_g \cup {X}^\circ; {\mathbb V}) $$ is an isomorphism, and there is an exact sequence \begin{multline*} 0 \to IH^2({\mathcal M}_g \cup {X}^\circ; {\mathbb V}) \to H^2({\mathcal M}_g; {\mathbb V}) \stackrel{\phi}{\to} \\ H^3 ({X}^\circ; j^! {\mathcal S}^\bullet) \cong H^0 ({X}^\circ; {\mathcal H}^3 j^! {\mathcal S}^\bullet). \end{multline*} Therefore to prove the theorem it suffices to show that $\phi$ from the exact sequence above is the zero morphism. The distinguished triangle \ecouple{j^! {\mathcal S}^\bullet} {} {j^* {\mathcal S}^\bullet} {} {j^* i_* {\mathbb V}} {[1]} implies that ${\mathcal H}^3 j^! {\mathcal S}^\bullet \cong {\mathcal H}^2 j^* i_* {\mathbb V}$. Then the morphism $\phi$ composed with this isomorphism can be factored as $$ H^2({\mathcal M}_g; {\mathbb V}) \to H^2({X}^\circ; j^* i_* {\mathbb V}) \stackrel{\psi}{\to} H^0({X}^\circ; {\mathcal H}^2 j^* i_* {\mathbb V}). $$ The sheaf $j^* i_* {\mathbb V}$ is called the {\it local link cohomology functor} \cite[p.~57]{durfee:saito}. It expresses the cohomology of $N^\ast$, the link of ${X}^\circ$ in ${\mathcal M}_g \cup {X}^\circ$. We denote by $\pi$ the corresponding projection $N^\ast \to {X}^\circ$. Then the morphism $\psi$ from the exact sequence above can be written as $$ \psi: H^2(N^\ast; {\mathbb V}) \to H^0({X}^\circ; {\mathcal H}^2 \pi_\ast {\mathbb V}). $$ One can easily check that $\psi$ is the edge homomorphism associated to the Leray--Serre spectral sequence determined by $\pi$. In order to prove the theorem it is enough to show that $\psi$ is the trivial homomorphism when $g\ge 6$, and the rest of this section deals with the proof of this fact. First we want to understand the behavior of the local system ${\mathbb V}$ over $N^\ast$. We start with the following lemma. \begin{lemma} \label{lem:split} The orbifold local system ${\mathbb V}$ over $N^\ast$ splits into a direct sum of symplectic orbifold local systems determined by rational representations of $\operatorname{Sl}_2 \times \operatorname{Sp}_{2g-2}$. \end{lemma} \begin{proof} Recall that a symplectic orbifold local system is determined by a representation of ${\Gamma}_g$ which is the pull-back of an algebraic representation $V$ of $\operatorname{Sp}_{2g}$. Choose a level $l\ge 3$. The inverse image of ${X}^\circ \subset \widetilde{\mathcal M}_g$ in $\widetilde{\mathcal M}_g [l]$ has several connected components. Let $N_l^\ast$ be a deleted regular neighborhood of one of them. Then one has a commutative diagram $$ \begin{CD} N_l^\ast @>>> {\mathcal M}_g [l] @>>> {\mathcal A}_g [l] \\ @VVV @VVV @VVV \\ N^\ast @>>> {\mathcal M}_g @>>> {\mathcal A}_g. \\ \end{CD} $$ (Recall that $A_g$ stands for the moduli space of principally polarized abelian varieties.) Denote by ${\mathbb V}_l$ the pull-back of ${\mathbb V}$ to ${\mathcal M}_g [l]$, and by ${\mathbb V}_l'$ the local system over $A_g [l]$ determined by $V$. Both ${\mathbb V}_l$ and ${\mathbb V}_l'$ are genuine local systems, and ${\mathbb V}_l$ is the pull-back of ${\mathbb V}_l'$ under ${\mathcal M}_g [l] \to {\mathcal A}_g [l]$. The product ${\mathcal A}_1 \times {\mathcal A}_{g-1}$ is canonically embedded in ${\mathcal A}_g$. Its inverse image under ${\mathcal A}_g [l] \to {\mathcal A}_g$ consists of several connected component, each of which is isomorphic to ${\mathcal A}_1 [l]\times {\mathcal A}_{g-1} [l]$. The image of $N_l^\ast$ in ${\mathcal A}_g [l]$ is contained in a tubular neighborhood of one of these connected components. The local system ${\mathbb V}_l'$, restricted to this connected component, splits according to the branching law of the inclusion $\operatorname{Sl}_2 \times \operatorname{Sp}_{g-2} \hookrightarrow \operatorname{Sp}_{2g}$. It follows that the local system ${\mathbb V}_l$ splits over $N_l^\ast$ according to the same branching law. In addition, ${\mathbb V}_l$ is constant on the fibers of the composite $$ N_l^\ast \longrightarrow N^\ast \stackrel{\pi}{\longrightarrow} {X}^\circ. $$ Thus the splitting of ${\mathbb V}_l$ over $N_l^\ast$ descends to the splitting of ${\mathbb V}$ over $N^\ast$. \end{proof} Our aim is to show that the morphism $\psi$ is trivial. Therefore without loss of generality we can assume that ${\mathbb V}$ is a local system over $N^\ast$ determined by an irreducible algebraic representation of $\operatorname{Sl}_2 \times \operatorname{Sp}_{2g-2}$ with highest weight $(\mu, \nu)$. Note that $\mu$ is just a non-negative integer. We consider two cases. First, assume that $\mu$ is odd. The morphism $N_l^\ast \to N^\ast$ is a Galois covering with the Galois group $\operatorname{Sl}_2 ({\mathbb Z}/l{\mathbb Z}) \times \operatorname{Sp}_{2g-2} ({\mathbb Z}/l{\mathbb Z})$. The element $(-id, id)$ of this group leaves the fibers of $N_l^\ast \to N^\ast \to {X}^\circ$ fixed because it corresponds to the involution of the elliptic curve, and acts as $-id$ on the local system ${\mathbb V}_l$. It follows that ${\mathcal H}^2 \pi_\ast {\mathbb V}$ is the trivial local system, and we have nothing more to prove. Next, assume that $\mu$ is even. Then $(-id, id)$ acts trivially on the local system ${\mathbb V}_l$, and therefore the local system ${\mathbb V}$ extends to ${X}^\circ$. This means that ${\mathbb V}$ is the restriction to $N^\ast$ of a local system defined on the whole regular neighborhood $N$ of ${X}^\circ$. Denote the restriction of this local system to ${X}^\circ$ by $\overline{\mathbb V}$. Then $\overline{\mathbb V}$ is isomorphic to ${\mathbb W}_1 (\mu) \boxtimes {\mathbb W}_2 (\nu)$, the symplectic local system over ${\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_1 \times {\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_{g-1}$ determined by the highest weight $(\mu, \nu)$. The local system ${\mathbb V}$ is the pull-back of $\overline{\mathbb V}$ under $\pi: N^\ast \to {X}^\circ$. Lemma \ref{lem:link} says that $N^\ast$ is the ${\mathbb Z}/2{\mathbb Z}$ quotient of the bundle $L$ defined in Section \ref{sec:link}, and $\pi$ is induced by the projection $\tilde{\pi}: L \to {X}^\circ$. We denote by $\widetilde{\mathbb V}$ be the pull-back of ${\mathbb V}$ to $L$. Then $$ \widetilde{\mathbb V} =\tilde{\pi}^\ast \overline{\mathbb V} \cong\tilde{\pi}^\ast ({\mathbb W}_1(\mu) \boxtimes {\mathbb W}_2(\nu)). $$ Let $B_\bullet$ be the Leray--Serre spectral sequence determined by $\pi$, and $A_\bullet$ be the Leray--Serre spectral sequence determined by $\tilde{\pi}$. Let $\tilde{\psi}$ be the edge homomorphism $H^2(L; \widetilde{\mathbb V}) \to H^0 ({X}^\circ;{\mathcal H}^2 \tilde{\pi}_\ast\widetilde{\mathbb V})$ associated $A_\bullet$. The two-fold covering map $L \to N^\ast$ induces the map of the spectral sequences $B_\bullet \to A_\bullet$. One has for each $q$ \cite[p.~85]{bredon:sheaf} $$ {\mathcal H}^q \pi_\ast {\mathbb V} = ({\mathcal H}^q \tilde{\pi}_\ast \widetilde{\mathbb V})^{{\mathbb Z}/2{\mathbb Z}}. $$ It follows that the induced map $B_2^{0,q} \to A_2^{0,q}$ is an inclusion of global ${\mathbb Z}/2{\mathbb Z}$ invariants. The homomorphism $H^2(N^\ast; {\mathbb V}) \to H^2(L; \widetilde{\mathbb V})$ is also an inclusion of ${\mathbb Z}/2{\mathbb Z}$ invariants, and one has a commutative diagram $$ \begin{CD} H^2(L; \widetilde{\mathbb V}) @>\tilde{\psi} >> H^0 ({X}^\circ;{\mathcal H}^2 \tilde{\pi}_\ast\widetilde{\mathbb V})\\ @AAA @AAA \\ H^2(N^\ast; {\mathbb V}) @>\psi >> H^0({X}^\circ;{\mathcal H}^2 \pi_\ast{\mathbb V}), \\ \end{CD} $$ where both vertical maps are inclusions. It follows that if $\tilde{\psi}$ is trivial, then $\psi$ is trivial. We shall show that $\tilde{\psi}$ is trivial. Note that $\tilde{\pi}_\ast \widetilde{\mathbb V}$ is quasi-isomorphic to $\tilde{\pi}_\ast {\mathbb Q} \otimes \overline{\mathbb V}$. It follows that $ {\mathcal H}^2 \tilde{\pi}_\ast \widetilde{\mathbb V}$ is isomorphic to ${\mathcal H}^2 \tilde{\pi}_\ast {\mathbb Q} \otimes \overline{\mathbb V}$. \begin{lemma} The local system ${\mathcal H}^2 \tilde{\pi}_\ast{\mathbb Q}$ over ${X}^\circ$ is isomorphic to ${\mathbb W}_1(0) \boxtimes {\mathbb W}_2(\nu_1)$. \end{lemma} \begin{proof} The bundle $\tilde{\pi}$ is the pull-back of the bundle $\tilde{\pi}_2: L_2 \to {\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_{g-1}$ to ${X}^\circ$ (see Lem.~\ref{lem:link}). It follows that the local system ${\mathcal H}^2 \tilde{\pi}_\ast {\mathbb Q}$ is the exterior tensor product of the constant local system ${\mathbb W}_1(0)$ over ${\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_1$ and ${\mathcal H}^2 \tilde{\pi}_{2\ast} {\mathbb Q}$. Recall that $\tilde{\pi}_2$ factors as $$ L_2 \stackrel{\pi^c_2}{\longrightarrow} Y_2 \stackrel{\bar{\pi}_2}{\longrightarrow} {\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_{g-1}, $$ where $\bar{\pi}_2$ is the restriction of the universal curve to $Y_2$, and $\pi_2^c$ is a punctured relative tangent bundle to $\bar{\pi}_2$. Therefore we have a Gysin long exact sequence of local systems \begin{equation} \label{gysin} \cdots \to {\mathcal H}^0 \bar{\pi}_{2\ast} {\mathbb Q} \stackrel{e}{\longrightarrow} {\mathcal H}^2 \bar{\pi}_{2\ast} {\mathbb Q} \to {\mathcal H}^2 \tilde{\pi}_{2\ast} {\mathbb Q} \to {\mathcal H}^1 \bar{\pi}_{2\ast} {\mathbb Q} \to 0 \end{equation} where $e$ is the multiplication by the Euler class. The Euler class is non-zero, because the genus of $C_2$ is greater than one. It follows that $e$ is an isomorphism on rational cohomology. Thus we conclude that ${\mathcal H}^2 \tilde{\pi}_{2\ast} {\mathbb Q}$ is isomorphic to ${\mathcal H}^1 \bar{\pi}_{2\ast} {\mathbb Q}$. The local system ${\mathcal H}^1 \bar{\pi}_{2\ast} {\mathbb Q}$ is isomorphic to ${\mathbb W}_2(\nu_1)$, the local system corresponding to the standard representation of $\operatorname{Sp}_{2g-2}$. It follows that $$ {\mathcal H}^2 \pi^\ast{\mathbb Q} \cong {\mathbb W}_1(0)\boxtimes {\mathbb W}_2(\nu_1). \quad \qed $$ \renewcommand{\qed}{} \end{proof} The lemma above shows that the edge homomorphism $\tilde{\psi}$ is of the form \begin{equation*} H^2(L;\tilde{\pi}^\ast({\mathbb W}_1(\mu)\boxtimes{\mathbb W}_2(\nu))) \to H^0({X}^\circ;{\mathbb W}_1(\mu)\boxtimes ({\mathbb W}_2(\nu_1)\otimes{\mathbb W}_2(\nu))). \end{equation*} \begin{lemma} The space $H^0({X}^\circ;{\mathbb W}_1(\mu)\boxtimes ({\mathbb W}_2(\nu_1)\otimes{\mathbb W}_2(\nu)))$ is isomorphic to ${\mathbb Q}$ if $\mu=0$ and $\nu=\nu_1$, and zero otherwise. \end{lemma} \begin{proof} Applying the K\"unneth formula one gets \begin{align*} H^0({X}^\circ;{\mathbb W}_1(\mu)\boxtimes ({\mathbb W}_2(\nu_1)&\otimes{\mathbb W}_2(\nu))) \cong \\ H^0({\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_1;{\mathbb W}_1(\mu))&\otimes H^0({\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_{g-1};{\mathbb W}_2(\nu_1)\otimes{\mathbb W}_2(\nu)). \end{align*} The zero cohomology of a space with coefficients in a local system is equal to the space of global invariants of the local system. An irreducible symplectic local system has no global invariants unless it is constant. This implies that $H^0({\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_1; {\mathbb W}_1(0)) \cong {\mathbb Q}$, and $H^0({\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_1; {\mathbb W}_1(\mu))=0$ if $\mu\ne 0$. Similarly, $H^0({\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_{g-1};{\mathbb W}_2(\nu_1)\otimes{\mathbb W}_2(\nu))$ is equal to zero, unless the local system ${\mathbb W}_2(\nu_1)\otimes{\mathbb W}_2(\nu)$ contains a constant local system as a direct summand. This occurs if and only if the tensor product $W_2(\nu_1)\otimes W_2(\nu)$ of irreducible representations of $\operatorname{Sp}_{2g-2}({\mathbb Q})$ contains a copy of the trivial representation. It is known that all irreducible representations of the symplectic group are self-dual. Therefore the trivial part of that representation is equal to \begin{align*} (W_2(\nu_1)\otimes W_2(\nu))^{\operatorname{Sp}_{2g-2}({\mathbb Q})} & = W_2(\nu_1)\otimes_{\operatorname{Sp}_{2g-2}({\mathbb Q})}W_2(\nu) \\ & \cong \operatorname{Hom}_{\operatorname{Sp}_{2g-2}({\mathbb Q})}({\mathbb W}_2(\nu_1);{\mathbb W}_2(\nu)). \end{align*} By Schur's lemma the latter term is isomorphic to ${\mathbb Q}$, if $\nu=\nu_1$, and $0$ otherwise. \end{proof} It follows that $\tilde{\psi}$ is trivial unless $\overline{\mathbb V} \cong {\mathbb W}_1(0) \boxtimes {\mathbb W}_2(\nu_1)$. In the remaining part of this section we study this case. To simplify the notation we denote ${\mathbb W}_1(0) \boxtimes {\mathbb W}_2(\nu_1)$ by ${\mathbb W}_2(\nu_1)$. \begin{lemma} \label{lem:g6} If $g\ge 6$, then the homomorphism $$ \tilde{\psi}: H^2(L;\tilde{\pi}^\ast{\mathbb W}_2(\nu_1)) \to H^0({X}^\circ; {\mathbb W}_2(\nu_1)^{\otimes 2}) $$ is the zero map. \end{lemma} \begin{proof} Note that $\tilde{\psi}$ factors through the $A_\infty^{0,2}$ term of the spectral sequence $$ A_2^{p,q} = H^p({X}^\circ; {\mathcal H}^q \tilde{\pi}_\ast {\mathbb Q} \otimes {\mathbb W}_2(\nu_1)) \Rightarrow H^{p+q} (L; \tilde{\pi}^\ast {\mathbb W}_2(\nu_1)). $$ Therefore, it suffices to prove that $A_\infty^{0,2}=0$. The morphism $\tilde{\pi}_2: L_2 \to {\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_{g-1}$ gives rise to a Leray--Serre spectral sequence \begin{equation*} C_2^{p,q}=H^p({\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_{g-1};{\mathcal H}^q\tilde{\pi}_{2\ast} {\mathbb Q} \otimes {\mathbb W}_2(\nu_1)) \Rightarrow H^{p+q}(L_2;\tilde{\pi}_{2}^\ast {\mathbb W}_2(\nu_1)). \end{equation*} The bundle $\tilde{\pi}: L \to {X}^\circ$ is the pull-back of $L_2$, and the local system ${\mathbb W}_2(\nu_1)$ over ${X}^\circ$ is also the pull-back from the second factor. It follows that the morphism of spectral sequences $C_\bullet \to A_\bullet$ induced by the projection $pr_2: {X}^\circ \to {\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_{g-1}$ is an inclusion of a direct summand. (Here we mean that for each $(r,p,q)$ the term $C_r^{p,q}$ is a direct summand of $A_r^{p,q}$, and all differentials $d_r$ respect this splitting.) Note that $A_2^{0,2} \cong C_2^{0,2}$. Indeed, \begin{align*} A_2^{0,2}=H^0({X}^\circ; {\mathbb W}_2(\nu_1)^{\otimes 2}) &\cong H^0({\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_1;{\mathbb Q})\otimes H^0({\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_{g-1}; {\mathbb W}_2(\nu_1)^{\otimes 2})\\ &\cong H^0({\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_{g-1};{\mathbb W}_2(\nu_1)^{\otimes 2})=C_2^{0,2} \end{align*} because ${\mathcal H}^2 \tilde{\pi}_{2\ast} {\mathbb Q} \cong {\mathbb W}_2(\nu_1)$ according to exact sequence \eqref{gysin}. It follows that $A_{\infty}^{0,2} \cong C_{\infty}^{0,2}$. The final step is to show that $C_\infty^{0,2}=0$. There is a surjective homomorphism $$ H^2(L_2; \tilde{\pi}_2^\ast{\mathbb W}_2(\nu_1)) \to C_\infty^{0,2}, $$ associated to the spectral sequence $C_\bullet$. Therefore it suffices to show that $$ H^2(L_2; \tilde{\pi}_2^\ast {\mathbb W}_2(\nu_1))=0. $$ The complement of the Zariski open subset ${\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_{g-1}$ of ${\mathcal M}_{g-1}$ has complex codimension $g-3$ (cf. Section \ref{sec:moduli}). It follows that $L_2$ also has complex codimension $g-3$ in ${\mathcal M}_{g-1,1}$. Thus $$ H^2(L_2; \tilde{\pi}_2^\ast {\mathbb W}_2(\nu_1)) \cong H^2({\mathcal M}_{g-1,1}; {\mathbb W}_2(\nu_1)) $$ when $g-3\ge 3$. The mapping class group of ${\mathcal M}_{g-1,1}$ is ${\Gamma}_{g-1,1}$, and their rational cohomology are the same. In particular, $$ H^2({\mathcal M}_{g-1,1}; {\mathbb W}_2(\nu_1)) \cong H^2({\Gamma}_{g-1,1}; W_2(\nu_1)) = H^2({\Gamma}_{g-1,1}; H^1(S;{\mathbb Q})) $$ where $S$ is a reference surface of genus $g-1$. In Lemma \ref{lem:h3} below (based on a result of Harer) we show that $H^2({\Gamma}_{g,1}; H^1(S;{\mathbb Q}))$ when $g\ge 5$. This implies that $C_\infty^{0,2}=0$, and therefore both homomorphisms $\tilde{\psi}$ and $\psi$ are zero homomorphisms when $g\ge 6$. \end{proof} \begin{lemma} \label{lem:h3} If $g\ge 5$, then $H^2({\Gamma}_{g,1}; H^1(S;{\mathbb Q}))=0$. \end{lemma} \begin{proof} All cohomology groups are considered with rational coefficients. The homomorphism ${\Gamma}_{g,1}^1 \to {\Gamma}_{g,1}$ is defined by forgetting a fixed point, and therefore is surjective. We can choose a fixed point in a neighborhood of the base point of a fixed tangent vector. This determines a splitting of the homomorphism above. As the associated spectral sequence has two rows, the existence of splitting implies that the spectral sequence degenerates at $E_2$. Hence, $H^2({\Gamma}_{g,1}; H^1(S))$ is a direct summand of $H^3({\Gamma}_{g,1}^1)$ \cite[Sec.~7]{harer:third}. Thus it suffices to prove that $H^3({\Gamma}_{g,1}^1)=0$. There is a short exact sequence of groups $$ 1 \to {\mathbb Z} \to {\Gamma}_{g,2} \to {\Gamma}_{g,1}^1 \to 1. $$ It determines a Gysin long exact sequence $$ \dots \to H^1({\Gamma}_{g,1}^1) \to H^3({\Gamma}_{g,1}^1) \to H^3({\Gamma}_{g,2}) \to \dots $$ We know that the last term is trivial according to Theorem 3.1 from \cite{harer:fourth}. The first term is trivial by \cite[Prop.~5.2]{hain:torelli}. It follows that the middle term $H^3({\Gamma}_{g,1}^1)$ is also zero. \end{proof} \begin{remark} In Theorem 3.1 from \cite{harer:fourth} Harer gives an explicit description of a basis of $H_3({\Gamma}_{4,2}) \cong {\mathbb Q}$. Using this one can deduce that $H_3({\Gamma}_{4,1}^1)$ is trivial, and therefore that $H^2({\Gamma}_{4,1}; H^1(S;{\mathbb Q}))$ is trivial. \end{remark} Recall that each irreducible symplectic local system over ${\mathcal M}_g$ is determined by its highest weight ${\lambda}$. If ${\lambda}_1, \dots, {\lambda}_g$ is a set of fundamental weights of $\operatorname{Sp}_{2g}$, then ${\lambda}$ is uniquely expressed as $\sum_{i=1}^g a_i {\lambda}_i$ for some non-negative integers $a_i$. We defined $|{\lambda}|$ to be $\sum_{i=1}^g i a_i$. \begin{definition} We say that an irreducible symplectic local system over ${\mathcal M}_g$ determined by the highest weight ${\lambda}$ is {\it even} if $|{\lambda}|$ is even, and it is {\it odd} if $|{\lambda}|$ is odd. \end{definition} The following corollary is a consequence of the proof of the main theorem. \begin{corollary} \label{cor:even:main} If ${\mathbb V}$ is an even local system, then the natural map $$ IH^2(\widetilde{\mathcal M}_g;{\mathbb V}) \to H^2({\mathcal M}_g;{\mathbb V}) $$ is an isomorphism when $g\ge 4$. \end{corollary} \begin{proof} The estimate $g\ge 6$, rather than $g\ge 4$, appears in the proof of Lemma \ref{lem:g6}. This lemma deals with the case when a symplectic local system ${\mathbb V}$ restricted to $N^\ast$ has a direct summand isomorphic to the irreducible local system $\pi^\ast ({\mathbb W}_1(0) \boxtimes {\mathbb W}_2(\nu_1))$. Note that ${\mathbb V}$ contains such direct summand if and only if the corresponding algebraic representation $V$ of $\operatorname{Sp}_{2g}$ restricted to the subgroup $\operatorname{Sl}_2 \times \operatorname{Sp}_{2g-2}$ contains a copy of $W_1(0) \boxtimes W_2(\nu_1)$. The branching rule of $\operatorname{Sp}_{2g}$ over $\operatorname{Sl}_2 \times \operatorname{Sp}_{2g-2}$ respects even and odd components. Therefore if $V$ is even, then its restriction cannot contain $W_1(0) \boxtimes W_2(\nu_1)$. This implies that in this case $\tilde{\psi}$ is trivial for all $g\ge 4$. \end{proof} \section{Mixed Hodge theory} \label{sec:mhs} In this section we consider the mixed Hodge structure on $H^2({\mathcal M}_g; {\mathbb V})$ where ${\mathbb V}$ is an irreducible symplectic local system. We prove that the mixed Hodge structure on $H^2({\mathcal M}_g;{\mathbb V})$ is pure when $g\ge 6$. We also prove that if $g= 3,4,5$, then the mixed Hodge structure on $H^2({\mathcal M}_g;{\mathbb V})$ has at most two weights. In this section we assume that $g\ge 3$. We use results of the theory of mixed Hodge modules developed by M.~Saito. For definitions and results we refer the reader to \cite{saito:intro}, \cite{saito:mhm}. In this paper we use only the formal properties of mixed Hodge modules. \begin{notation} Let $H=(H_{\mathbb Q},H_{\mathbb C},W_\bullet,F^\bullet)$ be a rational mixed Hodge structure where ${\mathbb W}_\bullet$ denotes the weight filtration, and $F^\bullet$ denotes the Hodge filtration. Denote the graded quotient $W_kH_{\mathbb Q} /W_{k-1}H_{\mathbb Q}$ by $\Gr{W}{k}H$. We shall say that an integer $m$ is a {\it weight} of a mixed Hodge structure $H$ if $\Gr{W}{m}H\ne 0$. We use abbreviations: MHS for mixed Hodge structure, and MHM for mixed Hodge module. \end{notation} In \cite{deligne:hodge} Deligne proved that the rational cohomology of every quasi-projective variety possesses a natural MHS. In \cite{saito:mhm} Saito proved that the cohomology and intersection cohomology of an algebraic variety with coefficients in an admissible variation of MHS carry MHSs. The definition of an admissible variation of MHS is given for curves in \cite{steenbrink:zucker}, and in general in \cite{kashiwara} (also see \cite[2.1]{saito:intro}). There is a strong belief that when both MHSs of Deligne and Saito exist they are the same. Let ${\mathbb V}$ be an irreducible symplectic local system over ${\mathcal M}_g$ determined by highest weight ${\lambda}$. This is clear that the restriction of the local system ${\mathbb V}$ to ${\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_g$ underlies a polarized variation of Hodge structure of geometric origin. Therefore the restriction of ${\mathbb V}$ to ${\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_g$ is an admissible variation of Hodge structure. The local system ${\mathbb V}$ is irreducible, therefore the corresponding variation of Hodge structure is unique up to Tate twist \cite[Prop.~8.1]{hain:torelli}. We fix ${\mathbb V}$ as a variation of Hodge structure by decreeing its weight to be $|\lambda|$. According to the theory of MHMs both $IH^q(\widetilde{\mathcal M}_g;{\mathbb V})$ and $H^q ({\mathcal M}_g; {\mathbb V})$ carry natural MHSs \cite[pp.~146-147]{saito:intro}. The MHS on $H^q ({\mathcal M}_g; {\mathbb V})$ can be defined using either the smooth covers ${\mathcal M}_g [l]$ for $l\ge 3$, or the isomorphism $H^q ({\mathcal M}_g; {\mathbb V}) \cong IH^q ({\mathcal M}_g; {\mathbb V})$ where in the second term we consider the restriction of ${\mathbb V}$ to ${\hspace{-.39cm}{\phantom{\mathcal M}}^{\circ_g$. This is easy to check that all these ways lead to the same MHS. \begin{theorem} \label{thm:purity} If $g\ge 6$, or if $g\ge 4$ and ${\mathbb V}$ is an even local system, then the mixed Hodge structure on $H^2({\mathcal M}_g;{\mathbb V}(\lambda))$ is pure of weight $2+|\lambda|$. \end{theorem} \begin{proof} The theory of MHM implies that the restriction \begin{equation*} IH^2(\widetilde{\mathcal M}_g;{\mathbb V}) \to H^2({\mathcal M}_g;{\mathbb V}) \end{equation*} is a morphism of MHSs, and according to Theorem \ref{thm:main} and Corollary \ref{cor:even:main} this is an isomorphism. The space $\widetilde{\mathcal M}_g$ is a projective variety. It follows that the MHS on $IH^2(\widetilde{\mathcal M}_g;{\mathbb V})$ is pure of weight $2+|\lambda|$ \cite[pp.~221-222]{saito:mhm}. Thus the MHS on $H^2({\mathcal M}_g;{\mathbb V})$ is also pure of weight $2+|\lambda|$. \end{proof} In the rest of this section we deal with the MHS on $H^2({\mathcal M}_g [l];{\mathbb V})$ where ${\mathcal M}_g [l]$ is the moduli space of curves with a level $l$ structure, and ${\mathbb V}$ is a symplectic local system ${\mathbb V}({\lambda})$ which underlies a variation of Hodge structure of weight $|{\lambda}|$. We assume that $l\ge 3$, and therefore ${\mathcal M}_g [l]$ is smooth and ${\mathbb V}$ is a genuine (not only orbifold) local system. There exists a natural MHS on $H^q({\mathcal M}_g [l]; {\mathbb V})$ for each $q\ge 0$. \begin{theorem} \label{thm:semipurity} If $l\ge 3$ and $g \ge 3$, then $\Gr{W}{k}H^2({\mathcal M}_g[l];{\mathbb V})=0$ for $k > 3+|\lambda|$ and $k<2+|\lambda|$. \end{theorem} \begin{proof} In the beginning we recall some facts from Section \ref{sec:comp}. The moduli space ${\mathcal M}_g [l]$ has the Satake compactification $\widetilde{\mathcal M}_g [l]$ which is a projective variety. The boundary $\widetilde{\mathcal M}_g [l] - {\mathcal M}_g [l]$ has codimension two in $\widetilde{\mathcal M}_g [l]$, and each codimension two irreducible component $\Phi_1^\beta$ has a Zariski open subset $Z_\beta$ such that the subsets $Z_\beta$ do not intersect each other, and each of them is isomorphic to a smooth Zariski open subset of ${\mathcal M}_1 [l] \times {\mathcal M}_{g-1} [l]$. \begin{notation} We denote by ${\mathcal S}^\bullet$ the intersection cohomology sheaf ${\mathcal I \mathcal C}^\bullet({\mathbb V})$ on $\widetilde{\mathcal M}_g [l]$. The following diagrams defines the notation for the inclusions: $$ {\mathcal M}_g [l] \stackrel{i}{\hookrightarrow} {\mathcal M}_g [l] \cup (\cup_\beta Z_\beta) \stackrel{j}{\hookleftarrow} \cup_\beta Z_\beta, $$ and we denote by $j_\beta$ the restriction of $j$ to $Z_\beta$. This notation is similar to that in Section \ref{sec:main}. \end{notation} It follows that one has an exact sequence $$ 0 \to IH^2(\widetilde{\mathcal M}_g[l];{\mathbb V}) \to H^2({\mathcal M}_g[l];{\mathbb V}) \to H^3(\cup_\beta Z_{\beta}; j^! {\mathcal S}^\bullet) $$ in the category of MHSs. Taking graded quotients with respect to weight filtration is an exact functor. Therefore for every $k$ there is an exact sequence $$ 0 \to \Gr{W}{k}IH^2(\widetilde{\mathcal M}_g[l];{\mathbb V}) \to \Gr{W}{k}H^2({\mathcal M}_g[l];{\mathbb V}) \to \Gr{W}{k}H^3(\cup_\beta Z_{\beta}; j^! {\mathcal S}^\bullet). $$ Since the space $\widetilde{\mathcal M}_g[l]$ is a projective variety, and ${\mathbb V}$ is a polarized variation of Hodge structure of geometric origin of weight $|{\lambda}|$, the intersection cohomology $IH^2(\widetilde{\mathcal M}_g[l];{\mathbb V})$ has a pure MHS of weight $2+|{\lambda}|$. To prove the theorem we will show that $\Gr{W}{k}H^3(\cup_\beta Z_{\beta}; j^! {\mathcal S}^\bullet)= 0$ unless $k=3+ |{\lambda}|$. As the sets $Z_\beta$ are disjoint it suffices to show that each $H^3(Z_\beta; j_\beta^! {\mathcal S}^\bullet)$ has a pure MHS of weight $3+|{\lambda}|$. From now on we fix an arbitrary index $\beta$, and omit $\beta$ from the notation for $Z_\beta$ and $j_\beta$. The sheaf $j^! {\mathcal S}^\bullet$ is constructible, and $\widetilde{\mathcal M}_g [l]$ is equi-singular along $Z$. Therefore ${\mathcal H}^3 j^! {\mathcal S}^\bullet$ is a local system over $Z$. The standard argument implies that there is an isomorphism of MHSs \begin{equation*} H^3(Z; j^! {\mathcal S}^\bullet) \cong H^0 (Z; {\mathcal H}^3 j^! {\mathcal S}^\bullet), \end{equation*} and there is an isomorphism of MHMs \begin{equation} \label{sheaves} {\mathcal H}^3 j^! {\mathcal S}^\bullet \cong {\mathcal H}^2 j^\ast i_\ast {\mathbb V}. \end{equation} We will show that these MHMs are pure of weight $3+|{\lambda}|$. Recall that $j^\ast i_\ast {\mathbb V}$ expresses cohomology of the link of $Z$ in ${\mathcal M}_g [l] \cup Z$. The inverse image of $Z$ under the birational morphism $\alpha^l: \overline{\mathcal M}_g [l] \to \widetilde{\mathcal M}_g [l]$ is a smooth locally closed divisor. We denote it by $Y$. Then the link of $Z$ in ${\mathcal M}_g [l] \cup Z$ is the same as the link of $Y$ in ${\mathcal M}_g [l] \cup Y$. We use this to find the weights on ${\mathcal H}^2 j^\ast i_\ast {\mathbb V}$. The following commutative diagram introduces the notation: $$ \begin{CD} {\mathcal M}_g[l] @>\kappa >> {\mathcal M}_g[l] \cup Y @<\mu << Y \\ @V=VV @V\alpha^l VV @V\bar{\pi} VV \\ {\mathcal M}_g[l] @>i >> {\mathcal M}_g[l] \cup Z @<j << Z. \\ \end{CD} $$ The local link cohomology functor of $Y$ is $\mu^\ast \kappa_\ast{\mathbb V}$. Therefore one expects that $j^\ast i_{\ast}{\mathbb V} \simeq \bar{\pi}_\ast \mu^\ast \kappa_\ast{\mathbb V}$. (The sign $\simeq$ denotes an isomorphism in the derived category of MHMs.) Indeed, both $\alpha^l$ and $\bar{\pi}$ are proper maps, therefore $\alpha^l_\ast= \alpha^l_!$ and $\bar{\pi}_\ast= \bar{\pi}_!$. It follows that for an arbitrary sheaf ${\mathcal F}^\bullet$ on ${\mathcal M}_g [l] \cup Y_\beta$ one has that $j^\ast\alpha^l_\ast {\mathcal F}^\bullet \simeq \bar{\pi}_\ast \mu^\ast {\mathcal F}^\bullet$ \cite[Prop.~10.7]{borel:ic}. Therefore \begin{equation} \label{links} j^\ast i_{\ast}{\mathbb V}\simeq j^\ast\alpha^l_\ast \kappa_\ast{\mathbb V} \simeq \bar{\pi}_\ast \mu^\ast \kappa_\ast{\mathbb V}. \end{equation} Thus ${\mathcal H}^2 j^\ast i_{\ast} {\mathbb V} \cong {\mathcal H}^2 \bar{\pi}_\ast \mu^\ast \kappa_\ast{\mathbb V}$ is an isomorphism of MHMs. The variation of Hodge structure ${\mathbb V}$ on ${\mathcal M}_g [l]$ extends to a variation of Hodge structure on ${\mathcal M}_g [l] \cup Y_\beta$ because ${\mathbb V}$ is pulled back from ${\mathcal A}_g[l]$. We denote its restriction to $Y$ by $\overline{\mathbb V}$. Then $\mu^\ast \kappa_\ast{\mathbb V} \simeq \mu^\ast \kappa_\ast{\mathbb Q} \otimes \overline{\mathbb V}$ where ${\mathbb Q}$ denotes the constant variation of Hodge structure of weight zero with the fiber isomorphic to ${\mathbb Q}$. Denote by $D{\mathcal F}^\bullet$ the dual of ${\mathcal F}^\bullet$ in the derived category of MHMs. The spaces ${\mathcal M}_g[l] \cup Y$ and $Y$ are smooth, therefore $D{\mathbb Q} \simeq {\mathbb Q}[2n](n)$ and $D{\mathbb Q}_{Y_\beta} \simeq {\mathbb Q}_{Y_\beta}[2n-2](n-1)$. It follows that there is a string of isomorphisms in the derived category of MHMs: \begin{multline*} \mu^!{\mathbb Q} \simeq D_{Y_\beta}(\mu^\ast D{\mathbb Q})\simeq D_{Y_\beta}(\mu^\ast {\mathbb Q}[2n](n)) \simeq D_{Y_\beta}({\mathbb Q}_{Y_\beta}[2n](n)) \\ \simeq (D_{Y_\beta}{\mathbb Q}_{Y_\beta})[-2n](-n) \simeq {\mathbb Q}_{Y_\beta}[-2](-1). \end{multline*} Using this and the distinguished triangle \ecouple{\mu^!{\mathbb Q}} {} {{\mathbb Q}} {} {\mu^\ast \kappa_\ast{\mathbb Q}.} {[1]} one can deduce that $$ {\mathcal H}^0 \mu^\ast \kappa_\ast{\mathbb Q} \cong {\mathcal H}^0 {\mathbb Q}, \quad {\mathcal H}^1 \mu^\ast \kappa_\ast{\mathbb Q} \cong {\mathcal H}^2 \mu^!{\mathbb Q}, \quad \text{and} \quad {\mathcal H}^q \mu^\ast \kappa_\ast{\mathbb Q} = 0 $$ for $q\ge 2$. It follows that ${\mathcal H}^0 \mu^\ast \kappa_\ast{\mathbb V}$ is a pure Hodge module of weight $|{\lambda}|$, and ${\mathcal H}^1 \mu^\ast \kappa_\ast{\mathbb V}$ is a pure Hodge module of weight $2+|{\lambda}|$. According to \cite[1.20]{saito:intro}, there is a (perverse) spectral sequence in the category of MHMs: \begin{equation*} E_2^{p,q}= {\mathcal H}^p \bar{\pi}_\ast({\mathcal H}^q\mu^\ast \kappa_\ast{\mathbb V}) \Rightarrow {\mathcal H}^{p+q}\bar{\pi}_\ast \mu^\ast \kappa_\ast{\mathbb V}. \end{equation*} As all spaces involved are smooth the perverse spectral sequence coincides with the ordinary one. It has only two non-zero rows. Thus there is an exact sequence of MHMs $$ {\mathcal H}^2 \bar{\pi}_\ast ({\mathcal H}^0 \mu^\ast \kappa_\ast{\mathbb V}) \to {\mathcal H}^2 \bar{\pi}_\ast \mu^\ast \kappa_\ast{\mathbb V} \to {\mathcal H}^1 \bar{\pi}_\ast ({\mathcal H}^1\mu^\ast \kappa_\ast{\mathbb V}). $$ The map $\bar{\pi}$ is proper. Therefore ${\mathcal H}^2\bar{\pi}_\ast ({\mathcal H}^0 \mu^\ast \kappa_\ast{\mathbb V})$ is pure of weight $2+|{\lambda}|$, and ${\mathcal H}^1 \bar{\pi}_\ast ({\mathcal H}^1 \mu^\ast \kappa_\ast{\mathbb V})$ is pure of weight $3+|{\lambda}|$. Consequently, we have that $$ \Gr{W}{k} {\mathcal H}^2 \bar{\pi}_\ast \mu^\ast \kappa_\ast{\mathbb V}=0 $$ for $k> 3+|{\lambda}|$ and $k< 2+|{\lambda}|$. Since ${\mathbb V}$ is a variation of Hodge structure of geometric origin of weight $|{\lambda}|$, the intersection cohomology sheaf ${\mathcal S}^\bullet$ underlies a pure Hodge module of weight $|{\lambda}|$. Therefore $j^!{\mathcal S}^\bullet$ is a MHM of weight $\ge |{\lambda}|$ \cite[Prop.~1.7]{saito:intro}. It follows that $\Gr{W}{k} {\mathcal H}^3 j^!{\mathcal S}^\bullet=0$ for $k< 3+|{\lambda}|$. Combining the last two paragraphs, and isomorphisms \eqref{sheaves}, \eqref{links} one gets that ${\mathcal H}^3 j^! {\mathcal S}^\bullet$ is pure of weight $3+|{\lambda}|$. \end{proof} \begin{corollary} \label{cor:semipurity} Let ${\mathbb V}({\lambda})$ be a symplectic local system over ${\mathcal M}_g$ underlying a variation of Hodge structure of weight $|{\lambda}|$. If $g\ge 3$, then $\Gr{W}{k} H^2({\mathcal M}_g; {\mathbb V})=0$ for $k> 3+|{\lambda}|$ and $k<2+|{\lambda}|$. \end{corollary} \begin{proof} Choose $l\ge 3$. One has as isomorphism $$ H^2({\mathcal M}_g; {\mathbb V}) \cong H^2({\mathcal M}_g[l]; {\mathbb V})^{\operatorname{Sp}_{2g}({\mathbb Z}/l{\mathbb Z})} $$ in the category of MHSs. The weights of the right hand side are $2+|{\lambda}|$ and $3+|{\lambda}|$ according to the theorem above. Therefore the same is true for the left hand side. \end{proof}

9611

alg-geom/9611021

en

https://arxiv.org/abs/alg-geom/9611021

alg-geom/9611021

Yongbin Ruan

Yongbin Ruan

Virtual neighborhoods and pseudo-holomorphic curves

Latex, revised version, corrected some mistakes in section 2, 3, improve presentation

We use virtual neighborhood technique to establish GW-invariants, Quantum cohomology, equivariant GW-invariants, equivariant quantum cohomology and Floer cohomology for general symplectic manifold. We also establish GW-invariants for a family of symplectic manifolds. As a consequence, we prove Arnold conjecture for nondegenerate Hamiltonian symplectomorphisms.

alg-geom

\section{ Introduction} Since Gromov introduced his pseudo-holomorphic curve theory in the 80's, pseudo-holomorphic curve has soon become an eminent technique in symplectic topology. Many important theorems in this field have been proved by this technique, among them, the squeezing theorem \cite{Gr}, the rigidity theorem \cite{E}, the classification of rational and ruled symplectic 4-manifolds \cite{M2}, the proof of the existence of non-deformation equivalent symplectic structures \cite{R2}. The pseudo-holomorphic curve also plays a critical role in a number of new subjects such as Floer homology theory,etc. In the meantime of this development, a great deal of efforts has been made to solidify the foundation of pseudo-holomorphic curve theory, for examples, McDuff's transversality theorem for ``cusp curves'' \cite{M1} and the various proofs of Gromov compactness theorem. In the early day of Gromov theory, Gromov compactness theorem was enough for its applications to symplectic topology. However, it was insufficient for its potential applications in algebraic geometry, where a good compactification is often very important. For example, it is particularly desirable to tie Gromov-compactness theorem to the Deligne-Mumford compactification of the moduli space of stable curves. Gromov's original proof is geometric. Afterwards, many works were done to prove Gromov compactness theorem in the line of Uhlenbeck bubbling off. It was succeed by Parker-Wolfson \cite{PW} and Ye \cite{Ye}. One outcome of their work was a more delicate compactification of the moduli space of pseudo-holomorphic maps. But it didn't attract much attention until several years later when Kontsevich and Manin \cite{KM} rediscovered this new compactification in algebraic geometry and initiated an algebro-geometric approach to the same theory. Now this new compactification becomes known as the moduli space of stable maps. The moduli space of stable maps is one of the basic ingredients of this paper. During last several years, pseudo-holomorphic curve theory entered a period of rapid expansion. We has witnessed its intensive interactions with algebraic geometry, mathematical physics and recently with new Seiberg-Witten theory of 4-manifolds \cite{T2}. One should mention that those recent activities in pseudo-holomorphic curve theory did not come from the internal drive of symplectic topology. It was influenced mostly by mathematical physics, particularly, Witten's theory of topological sigma model. Around 1990, there were many activities in string theory about ``quantum cohomology'' and mirror symmetry. The core of quantum cohomology theory is so called ``counting the numbers of rational curves''. Many incredible predictions were made about those numbers in Calabi-Yau 3-folds, based on results from physics. But mathematicians were frustrated about the meaning of the so-called ``number of rational curves''. For example, the finiteness of such number is a well-known conjecture due to H. Clemens which concerns simplest Calabi-Yau 3-folds-Quintic hypersurface of ${\bf P}^4$. It was even worse that some Calabi-Yau 3-fold never has a finite number of rational curves. One of the basic difficulties at that time was that people usually restricted their attention to Kahler manifolds, where the complex structure is rigid. On the other hand, the advantage of pseudo-holomorphic curves is that we are allowed to choose almost complex structures, which are much more flexible. Unfortunately, the most of those exciting developments were little known to symplectic topologists. In \cite{R1}, the author brought the machinery of pseudo-holomorphic curves into quantum cohomology and mirror symmetry. Using ideas from Donaldson theory, the author provided a rigorous definition of the symplectic invariants corresponding the "numbers of rational curves" in string theory. Moreover, the author found many applications of new symplectic invariants in symplectic topology \cite{R1}, \cite{R2}, \cite{R3}. These new invariants are called ``Gromov-Witten invariants''. Gromov-Witten invariants are analogous of invariants in the enumerative geometry. However, the actual counting problem (like the numbers of higher degree rational curves in quintic three-fold) did not attract much of attention before the discovery of mirror symmetry. In general, these numbers are difficult to compute. Moreover, computing these number didn't seems to help our understanding of Calabi-Yau 3-folds themself. The introduction of quantum cohomology hence opened a new direction for enumerative geometry. According to quantum cohomology theory, these enumerative invariants are not isolated numbers; instead, they are encoded in a new cohomological structure of underline manifold. Note that the quantum cohomology structure is governed by the associativity law, which corresponds to the famous composition law of topological quantum field theory. Therefore, it would be very important to put quantum cohomology in a rigorous mathematical foundation. It was clear that the enumerative geometry is not a correct framework. (For example, the associativity or composition law of quantum cohomology computes certain higher genus invariants, which are always different from enumerative invariants). Based on \cite{R1}, a correct mathematical framework were layed down by the author and Tian \cite{RT1}, \cite{RT2} in terms of perturbed holomorphic maps. By proving the crucial associativity law, we put quantum cohomology in a solid mathematical ground. A corollary of the proof of associativity law is a computation of the number of rational curves in ${\bf P}^n$ and many Fano manifolds by recursion formulas. Such a formula for ${\bf P}^2$ was first derived by Kontsevich, based on associativity law predicted by physicists. It should be pointed out that the entire pseudo-holomorphic curve theory were only established for so-called semi-positive symplectic manifolds. They includes most of interesting examples like Fano and Calabi-Yau manifolds. But, semipositivity is a significant obstacle for some of its important applications like Arnold conjecture and birational geometry. Stimulated by the success of symplectic method, the progreses have been made on algebro-geometrical approach. An important step is Kontsevich-Manin's initiative of using stable (holomorphic) maps. The genus 0 stable map works nicely for homogeneous space. For example, the moduli spaces of genus $0$ stable maps always have expected dimension. Many of results in \cite{R1}, \cite{RT1} were redone in this category by \cite{KM}, \cite{LT1}. It was soon realized that moduli spaces of stable maps no longer have expected dimension for non-homogeneous spaces, for example, projective bundles \cite{QR}. To go beyond homogeneous spaces, one needs new ideas. A breakthrough came with the work of Li and Tian \cite{LT2}, where they employ a sophisticated excessive intersection theory (normal cone construction) (see another proof in \cite{B}). As a consequence, Li and Tian extended GW-invariant to arbitrary algebraic manifolds. In the light of these new developments, three obvious problems have emerged: (i) to remove semi-positivity condition in Gromov-Witten invariants; (ii) to remove semi-positive condition in Floer homology and solve Arnold conjecture. (iii) to prove that symplectic GW-invariants are the same as algebro-geometric GW-invariants for algebraic manifolds. We will deal with first two problems in this article and leave the last one to the future research. Recall that, the fundamental difficulty for pseudo-holomorphic curve theory on non-semi-positive symplectic manifolds is, that $\overline{{\cal M}}-{\cal M}$ may have arger dimension than that of ${\cal M}$, where ${\cal M}$ is the moduli space of pseudo-holomorphic maps and $\overline{{\cal M}}$ is a compactification. One view is that this is due to the reason that the almost complex structure is not generic at infinity. To deal with this non-generic situation, the author's idea \cite{R3} (Proposition 5.7) was to construct an open smooth manifold (virtual neighborhood ) to contain the moduli space. Then, we can work on virtual neighborhood, which is much easier to handle than the moduli space itself. In \cite{R4}, the author outlined a scheme to attack the non-generic problems in Donaldson-type theory using virtual neighborhood technique. Moreover, author applied virtual neighborhood technique to monopole equation under a group action. Further application can be found in \cite{RW}. But the case in \cite{R4} is too restricted for pseudo-holomorphic case. Recall that in \cite{R4}, we work with a compact-smooth triple $({\cal B}, {\cal F}, S)$ where ${\cal B}$ is a smooth Banach manifold (configuration space), ${\cal F}$ is a smooth Banach bundle and $S$ is a section of ${\cal F}$ such that the moduli space ${\cal M}=S^{-1}(0)$ is compact. Monopole equation can be interpreted as a smooth-compact triple. However, in the case of pseudo-holomorphic curve, $S^{-1}(0)$ is almost never compact in the configuration space. Furthermore, $({\cal B}, {\cal F})$ is often not smooth, but a pair of $V$-manifold and $V$-bundle. To overcome these difficulty, we need to generalize the virtual neighborhood technique to handle this situation. An outline of such a generalization were given in \cite{R4}. Another purpose of this paper is to construct an equivariant quantum cohomology theory. For this purpose, we need to study the GW-invariant for a family of symplectic manifolds. We shall work in this generality throughout the paper. Let's outline a definition of GW-invariant over a family of symplectic manifolds as follows. Let $P: Y\rightarrow X$ be a fiber bundle such that both the fiber $V$ and the base $X$ are smooth compact, oriented manifolds. Furthermore, we assume that $P: Y \rightarrow X$ is an oriented fibration. Then, $Y$ is also a smooth, compact, oriented manifold. Let $\omega$ be a closed 2-form on $Y$ such that $\omega$ restricts to a symplectic form over each fiber. A $\omega$-tamed almost complex structure $J$ is an automorphism of vertical tangent bundle such that $J^2=-Id$ and $\omega(X, JX)>0$ for vertical tangent vector $X\neq 0$. Let $A\in H_2(V, {\bf Z})\subset H_2(Y, {\bf Z})$. Let ${\cal M}_{g,k}$ be the moduli space of genus g Riemann surfaces with $k$-marked points such that $2g+k>2$ and $\overline{{\cal M}}_{g,k}$ be its Deligne-Mumford compactification. Suppose that $f: \Sigma\rightarrow Y$ ($\Sigma\in {\cal M}_{g,k}$) is a smooth map such that $im(f)$ is contained in a fiber and $f$ satisfies Cauchy Riemann equation $\partial_J f=0$ with $[f]=A$. Let ${\cal M}_A(Y, g,k,J)$ be the moduli space of such $f$. First we need a stable compactification of ${\cal M}_{A}(Y,g,k, J)$. Roughly speaking, {\em a compactification is stable if its local Kuranish model is the quotient of vector spaces by a finite group}. In our case, it is provided by the moduli space of stable holomorphic maps $\overline{{\cal M}}_{A}(Y,g,k,J)$. There are two technical difficulties to use virtual neighborhood technique to the case of pseudo-holomorphic curve. The first one is that there is a finite group action on its local Kuranish model. An indication is that we should work in the V-manifold and V-bundle category. As a matter of fact, it is easy to extend virtual neighborhood technique to this category. However, the finite dimensional virtual neighborhood constructed is a V-manifold in this case. It is well-known that the ordinary transversality theorem fails for V-manifolds. We will overcome this problem by using differential form and integration. We shall give a detail argument in section 2. The second problem is the failure of the compactness of ${\cal M}_A(Y,g,k)$. To include $\overline{{\cal M}}_A(Y,g,k)$, we have to enlarge our configuration space to $\overline{{\cal B}}_A(Y,g,k)$ of $C^{\infty}$-stable ( holomorphic or not) maps. Then, the obstruction bundle ${\cal F}_A(Y,g,k)$ extends to $\overline{{\cal F}}_A(Y,g,k)$ over $\overline{{\cal B}}_A(Y,g,k)$. Therefore, we obtained a compact triple $(\overline{{\cal B}}_A(Y,g,k), \overline{{\cal F}} _A(Y,g,k), {\cal S})$, where $S$ is Cauchy-Riemann equation. We want to generalize the virtual neighborhood technique to this enlarge space. Recall that for virtual neighborhood technique, we construct some stablizations of the equation ${\cal S}_e={\cal S}+s$, which must satisfy two crucial properties: (1) $\{x; Coker \delta_x({\cal S}+s)=0\}$ is open; (2)If ${\cal S}+s$ is a transverse section, $U=({\cal S}+s)^{-1}(0)$ is a finite dimensional smooth V-manifold. By using gluing argument, we can construct a local model of $U$ (local Kuranish model). (2) is equivalent to that the local Kuranish model is a quotient of vector spaces by a finite group. By definition, it means that our compactification has to be stable. Finally, we need an additional argument to prove that the local models patch together smoothly. We call a triple satisfying (1), (2) {\em virtual neighborhood technique admissible} or {\em VNA}. Suppose that ${\cal S}$ is already transverse. $\overline{{\cal M}}(Y,g,k)$ is naturally a stratified space whose stratification coincides with that of $\overline{{\cal B}}_A (Y,g,k)$. The attaching map of $\overline{{\cal B}}_A(Y,g,k)$ is defined by patching construction. The gluing theorem shows that if we restrict ourself to stable holomorphic maps one can deform this attaching map slightly such that the image of stable holomorphic maps is again holomorphic. The deformed attaching map gives a local smooth coordinate of $\overline{{\cal M}}_A(Y,g,k)$. Although it is not necessary in virtual neighborhood construction, one can also attempt to deform the whole attaching map by the same implicit funtion theorem argument. Then, it is attempting to think (as author did) that the deformed attaching map will give a smooth coordinate of $\overline{{\cal B}}_A(Y,g,k)$. It was Tian who pointed out the author that this is false. However, it is natural to ask if there is any general property for such an infinite dimensional object. Indeed, some elegant properties are formulated by Li and Tian \cite{LT3} and we refer reader to their paper for the detail. Applying virtual neighborhood technique, we construct a finite dimensional virtual neighborhood $(U,F, S)$. More precisely, $U$ is covered by finite many coordinate charts of the form $U_i/G_i$ ($i=1, \dots, m$) for $U_p \subset {\bf R}^{ind +m}$ and a finite group $G_p$. $F$ is a V-bundle over $U$ and $S: U\rightarrow F$ is a section. On the other hand, the evaluation maps over marked points define a map $$\Xi_{g,k}: \overline{{\cal B}}_A(Y,g,k) \rightarrow Y^k. \leqno(1.1)$$ We have another map $$\chi: \overline{{\cal B}}_A(Y,g,k) \rightarrow \overline{{\cal M}}_{g,k}.\leqno(1.2)$$ Recall that $\overline{{\cal M}}_{g,k}$ is a V-manifold. To define GW-invariant, choose a Thom form $\Theta$ supported in a neighborhood of zero section. The GW-invariant can be defined as $$\Psi^Y_{(A,g,k)}(K; \alpha_1, \cdots, \alpha_k)=\int_{U}\chi^*(K)\wedge \Xi^*_{g,k} \prod_i \alpha_i\wedge S^*\Theta. \leqno(1.4)$$ for $\alpha_i\in H^*(Y, {\bf R})$ and $K\in H^*(\overline{{\cal M}}_{g,k}, {\bf R})$ represented by differential form. Clearly, $\Psi^Y=0$ if $\sum deg(\alpha_i)+deg (K)\neq ind$. Recall that $H^*(Y, {\bf R})$ has a modular structure by $P^*\alpha$ for $\alpha\in H^*(X, {\bf R})$. In this paper, we prove the following, \vskip 0.1in \noindent {\bf Theorem A (Theorem 4.2): }{\it (i).$\Psi^Y_{(A,g,k)}(K; \alpha_1, \cdots, \alpha_k)$ is well-defined. \vskip 0.1in \noindent (ii). $\Psi^Y_{(A,g,k)}(K; \alpha_1, \cdots, \alpha_k)$ is independent of the choice of virtual neighborhoods. \vskip 0.1in \noindent (iii). $\Psi^Y_{(A,g,k)}(K; \alpha_1, \cdots, \alpha_k)$ is independent of $J$ and is a symplectic deformation invariant. \vskip 0.1in \noindent (iv). When $Y=V$ is semi-positive, $\Psi^Y_{(A,g,k)}(K; \alpha_1, \cdots, \alpha_k)$ agrees with the definition of \cite{RT2}. \vskip 0.1in \noindent (v). $\Psi^Y_{(A,g,k)}(K; \alpha_1, \cdots, \alpha_i\cup P^*\alpha,\cdots, \alpha_k)=\Psi^Y_{(A,g,k)}(K; \alpha_1, \cdots, \alpha_j \cup P^*\alpha, \cdots, \alpha_k)$} \vskip 0.1in Furthermore, we can show that $\Psi$ satisfies the composition law required by the theory of sigma model coupled with gravity. Assume $g=g_1+g_2$ and $k=k_1+k_2$ with $2g_i + k_i \ge 3$. Fix a decomposition $S=S_1\cup S_2$ of $\{1,\cdots , k\}$ with $|S_i|= k_i$. Then there is a canonical embedding $\theta _S: \overline {\cal M}_{g_1,k_1+1}\times \overline {\cal M}_{g_2,k_2+1} \mapsto \overline {\cal M}_{g,k}$, which assigns to marked curves $(\Sigma _i; x_1^i,\cdots ,x_{k_1+1}^i)$ ($i=1,2$), their union $\Sigma _1\cup \Sigma _2$ with $x^1_{k_1+1}$ identified to $x^2_{k_2+1}$ and remaining points renumbered by $\{1,\cdots,k\}$ according to $S$. There is another natural map $\mu : \overline {\cal M}_{g-1, k+2} \mapsto \overline {\cal M}_{g,k}$ by gluing together the last two marked points. Choose a homogeneous basis $\{\beta _b\}_{1\le b\le L}$ of $H_*(Y,{\bf Z})$ modulo torsion. Let $(\eta _{ab})$ be its intersection matrix. Note that $\eta _{ab} = \beta _a \cdot \beta _b =0$ if the dimensions of $\beta _a$ and $\beta _b$ are not complementary to each other. Put $(\eta ^{ab})$ to be the inverse of $(\eta _{ab})$. Now we can state the composition law, which consists of two formulas as follows. \vskip 0.1in \noindent {\bf Theorem B. (Theorem 4.7)} {\it Let $[K_i] \in H_*(\overline {\cal M}_{g_i, k_i+1}, {\bf Q})$ $(i=1,2)$ and $[K_0] \in H_*(\overline {\cal M}_{g-1, k +2}, {\bf Q})$. For any $\alpha _1,\cdots,\alpha _k$ in $H_*(V,{\bf Z})$. Then we have $$\begin{array}{rl} &\Psi ^Y_{(A,g,k)}(\theta _{S*}[K_1\times K_2];\{\alpha _i\})\\ =(-1)^{deg(K_2)\sum_{i=1}^{k_1}deg(\alpha_i)} ~& \sum \limits _{A=A_1+A_2} \sum \limits_{a,b} \Psi ^Y_{(A_1,g_1,k_1+1)}([K_1];\{\alpha _{i}\}_{i\le k_1}, \beta _a) \eta ^{ab} \Psi ^Y_{(A_2,g_2,k_2+1)}([K_2];\beta _b, \{\alpha _{j}\}_{j> k_1}) \\ \end{array} \leqno (1.5) $$ $$ \Psi ^Y_{(A,g,k)}(\mu_*[K_0];\alpha _1,\cdots, \alpha _k) =\sum _{a,b} \Psi ^Y_{(A,g-1,k+2)}([ K_0];\alpha _1,\cdots, \alpha _k, \beta _a,\beta _b) \eta ^{ab}\leqno (1.6) $$ } \vskip 0.1in There is a natural map $\pi: \overline{{\cal M}}_{g,k}\rightarrow \overline{{\cal M}}_{g, k-1}$ as follows: For $(\Sigma, x_1, \cdots, x_k)\in \overline{{\cal M}}_{g,k}$, if $x_k$ is not in any rational component of $\Sigma$ which contains only three special points, then we define $$\pi(\Sigma, x_1, \cdots, x_k)=(\Sigma, x_1, \cdots, x_{k-1}),$$ where a distinguished point of $\Sigma$ is either a singular point or a marked point. If $x_k$ is in one of such rational components, we contract this component and obtain a stable curve $(\Sigma', x_1, \cdots, x_{k-1})$ in $\overline{{\cal M}}_{g, k-1}$, and define $\pi(\Sigma, x_1, \cdots, x_k)=(\Sigma', x_1, \cdots, x_{k-1}).$ Clearly, $\pi$ is continuous. One should be aware that there are two exceptional cases $(g,k)=(0,3), (1,1)$ where $\pi$ is not well defined. Associated with $\pi$, we have two {\em k-reduction formula} for $\Psi^V_{(A, g, k)}$ as following: \vskip 0.1in \noindent {\bf Proposition C (Proposition 4.4). }{\it Suppose that $(g,k) \neq (0,3),(1,1)$. \vskip 0.1in \noindent (1) For any $\alpha _1, \cdots , \alpha _{k-1}$ in $H_*(Y, {\bf Z})$, we have} $$\Psi ^Y_{(A,g,k)}(K; \alpha _1, \cdots,\alpha _{k-1}, [V])~=~ \Psi ^Y_{(A,g,k-1)}([\pi_* (K)]; \alpha _1, \cdots,\alpha _{k-1}) \leqno (1.7)$$ \vskip 0.1in \noindent (2) Let $\alpha _k$ be in $H_{2n-2}(Y, {\bf Z})$, then $$\Psi ^Y_{(A,g,k)}(\pi^{*}(K); \alpha _1, \cdots,\alpha _{k-1}, \alpha _k)~=~\alpha^* _k (A) \Psi ^Y_{(A,g,k-1)}(K; \alpha _1, \cdots,\alpha _{k-1}) \leqno (1.8)$$ where $ \alpha^* _k$ is the Poincare dual of $\alpha _k$. \vskip 0.1in \noindent When $Y=V$, $\Psi^Y$ is the ordinary GW-invariants. Therefore, we establish a theory of topological sigma model couple with gravity over any symplectic manifolds. It is well-known that GW-invariant can be used to define a quantum multiplication. Let's briefly sketch it as follows. First we define a total 3-point function $$\Psi^V_{\omega}(\alpha_1, \alpha_2, \alpha_3)=\sum_A \Psi^V_{(A,0,3)}(pt; \alpha_1, \alpha_2, \alpha_3)q^A, \leqno(1.9)$$ where $q^A$ is an element of Novikov ring $\Lambda_{\omega}$ (see \cite{RT1}, \cite{MS}). Then, we define a quantum multiplication $\alpha\times_Q \beta$ over $H^*(V, \Lambda_{\omega})$ by the relation $$(\alpha\times_Q \beta)\cup \gamma[V]=\Psi^V_{\omega}(\alpha_1, \alpha_2, \alpha_3),\leqno(1.10)$$ where $\cup$ represents the ordinary cup product. As a consequence of Theorem B, we have \vskip 0.1in \noindent {\bf Proposition D: }{\it Quantum multiplication is associative over any symplectic manifolds. Hence, there is a quantum ring structure over any symplectic manifolds.} \vskip 0.1in Given a periodic Hamiltonian function $H: S^1\times V\rightarrow V$, we can define the Floer homology $HF(V, H)$, whose chain complex is generated by the periodic orbits of $H$ and the boundary maps are defined by the moduli spaces of flow lines. So far, Floer homology $HF(V, H)$ is only defined for semi-positive symplectic manifolds. Applying virtual neighborhood technique to Floer homology, we show \vskip 0.1in \noindent {\bf Theorem E: }{\it Floer homology $HF(V, H)$ is well-defined for any symplectic manifolds. Furthermore, $HF(V, H)$ is independent of $H$.} \vskip 0.1in Recall that Floer homology was invented to solve the \vskip 0.1in \noindent {\bf Arnold conjecture: }{Let $\phi$ be a non-degenerate Hamiltonian symplectomorphism. Then, the number of the fixed points of $\phi$ is greater than or equal to the sum of Betti number of $V$.} \vskip 0.1in As a corollary of Theorem E, we prove the Arnold conjecture \vskip 0.1in \noindent {\bf Theorem F: }{\it Arnold conjecture holds for any symplectic manifolds.} \vskip 0.1in In this paper, we give another application of our results in higher dimensional algebraic geometry. It was discovered in \cite{R3} that symplectic geometry has a strong connection with Mori's birational geometry. An important notion in birational geometry is uniruled variety, generalizing the notion of ruled surfaces in two dimension. An algebraic variety $V$ is uniruled iff $V$ is covered by rational curves. Kollar \cite{K1} proved that for 3-folds, uniruledness is a symplectic property. Namely, if a 3-fold $W$ is symplectic deformation equivalent to an uniruled variety $V$, $W$ is uniruled. To extend Kollar's result, we need a symplectic GW-invariants defined over any symplectic manifolds with certain property (Lemma 4.10). We will show that our invariant satisfies this properties. By combining with Kollar's result, we have \vskip 0.1in \noindent {\bf Proposition G: }{\it If a smooth Kahler manifold $W$ is symplectic deformation equivalent to a uniruled variety, $W$ is uniruled.} \vskip 0.1in An important topic in quantum cohomology theory is the equivariant quantum cohomology group $QH_G(V)$, which generalizes the notion of equivariant cohomology. Suppose that a compact Lie group $G$ acts on $V$ as symplectomorphisms. To define equivariant quantum cohomology, we first have to define equivariant GW-invariants. There are two approaches. The first approach is to choose a $G$-invariant tamed almost complex structure $J$ and construct an equivariant virtual neighborhood. Then, we can use finite dimensional equivariant technique to define equivariant GW-invariant. This approach indeed works. But a technically simpler approach is to consider equivariant GW-invariant as the limit of GW-invariant over the families of symplectic manifolds. This approach was advocated by Givental and Kim \cite{GK}. We shall use this approach here. Let $BG$ be the classifying space of $G$ and $EG\rightarrow BG$ be the universal $G$-bundle. Suppose that $$BG_1\subset BG_2\cdots\subset BG_m \subset BG \leqno(1.11)$$ such that $BG_i$ is a smooth oriented compact manifold and $BG=\cup_i BG_i$. Let $$EG_1\subset EG_2\cdots\subset EG_m \subset BG\leqno(1.12)$$ be the corresponding universal bundle. We can also form the approximation of homotopy quotient $V_G=V\times EG/G$ by $V^i_G=V\times EG_i/G$. Since $\omega$ is invariant under $G$, its pull-back on $V\times EG_i$ descends to $V^i_G$. So, we have a family of symplectic manifolds $P_i: V^i\rightarrow BG_i$. Applying our previous construction, we obtain GW-invariant $\Psi^{P_i}_{(A, g,k)}$. We define equivariant GW-invariant $$\Psi^G_{(A,g,k)}\in Hom((H^*(V_G, {\bf Z}))^{\otimes k}\otimes H^*(\overline{{\cal M}}_{g, k}, {\bf Z}), H^*(BG, {\bf Z})) \leqno(1.13)$$ as follow: For any $D\in H_*(BG, {\bf Z})$, $D\in H_*(BG_i, {\bf Z})$ for some $i$. Let $i_{V^i_G}: V^i_G\rightarrow V_G$. For $\alpha_i\in H^*_G(V)$, we define $$\Psi^G_{(A,g,k)}(K, \alpha_1, \cdots, \alpha_k)(D)=\Psi^{P_i}_{(A, g,k)}(K, i^*_{V^i_G}(\alpha_1), \cdots, i^*_{V^i_G}(\alpha_k); P^*_i(D^*_{BG_i})),\leqno(1.14)$$ where $D^*_{BG_i}$ is the Poincare dual of $D$ with respect to $BG_i$. \vskip 0.1in \noindent {\bf Theorem G: }{\it (i). $\Psi^G_{(A, g,k)}$ is independent of the choice of $BG_i$. \vskip 0.1in \noindent (ii). If $\omega_t$ is a family of $G$-invariant symplectic forms, $\Psi^G_{(A, g,k)}$ is independent of $\omega_t$.} \vskip 0.1in Recall that equivariant cohomology ring $H^*_G(X)$ is defined as $H^*(V_G)$. Notes that, for any equivariant cohomology class $\alpha\in H^*_G(V)$, $$\alpha [V]\in H^*(BG)\leqno(1.15)$$ instead of being a number in the case of the ordinary cohomology ring. Furthermore, there is a modulo structure by $H^*_G(pt)=H^*(BG)$, defined by using the projection map $$V_G\rightarrow BG.\leqno(1.16)$$ The equivariant quantum multiplication is a new multiplication structure over $H^*_G(V, \Lambda_{\omega})=H^*(V_G, \Lambda_{\omega})$ as follows. We first define a total 3-point function $$\Psi^G_{(V,\omega)}(\alpha_1, \alpha_2, \alpha_3)=\sum_A \Psi^G_{(A,0,3)}( pt; \alpha_1, \alpha_2, \alpha_3)q^A.\leqno(1.17)$$ Then, we define an equivariant quantum multiplication by $$(\alpha\times_{QG}\beta)\cup \gamma [V]=\Psi^G_{(V,\omega)}(\alpha_1, \alpha_2, \alpha_3).\leqno(1.18)$$ \vskip 0.1in \noindent {\bf Theorem I: }{\it (i) The equivariant quantum multiplication is commutative with the modulo structure of $H^*(BG)$. \vskip 0.1in \noindent (ii) The equivariant quantum multiplication is skew-symmetry. \vskip 0.1in \noindent (iii) The equivariant quantum multiplication is associative. Hence, there is a equivariant quantum ring structure for any $G$ and symplectic manifold $V$} \vskip 0.1in Equivariant quantum cohomology has already been defined for monotonic symplectic manifold by Lu \cite{Lu}. The paper is organized as follows: In section 2, we work out the detail of the virtual neighborhood technique for Banach V-manifolds. In section 3, we prove that the virtual neighborhood technique can be applied to pseudo-holomorphic maps. In the section 4, we prove Theorem A, B, C, D, H and I. We prove Theorem E, Corollary F in section 5 and Theorem G in section 6. The results of this paper was announced in a lecture at the IP Irvine conference in the end of March, 96. An outline of this paper was given in \cite{R4}. During the preparation of this paper, we received papers by Fukaya and Ono \cite{FO}, B. Seibert \cite{S}, Li-Tian \cite{LT3}, Liu-Tian, were informed by Hofer/Salamon that they obtained some of the results of this paper independently using different methods. The author would like to thank G. Tian and B. Siebert for pointing out errors in the first draft and B. Siebert for suggesting a fix (Appendix) of an error in Lemma 2.5. The author would like to thank An-Min Li and Bohui Chen for the valuable discussions. \section{Virtual neighborhoods for V-manifolds} As we mentioned in the introduction, the configuration space $\overline{{\cal B}}_A(Y,g,k)$ is not a smooth Banach V-manifold in general. But for the purpose of virtual neighborhood construction, we can treat it as a smooth Banach V-manifold. To simplify the notation, we will work in the category of Banach V-manifold in this section and refer to the next section for the proof that the construction of this section applies to $\overline{{\cal B}}_A(Y,g,k,J)$. V-manifold is a classical subject dated back at least to \cite{Sa1}. Let's have a briefly review about the basics of V-manifolds. \vskip 0.1in \noindent {\bf Definition 2.1: }{\it (i).A Hausdorff topological space $M$ is a n-dimensional V-manifold if for every point $x\in M$, there is an open neighborhood of the form $U_x/G_x$ where $U_x$ is a connected open subset of ${\bf R}^n$ and $G_x$ is a finite group acting on $U_x$ diffeomorphic-ally. Let $p_x: U_x \rightarrow U_x/G_x$ be the projection. We call $(U_x, G_x, p_x)$ a coordinate chart of $x$. If $y\in U_x/G_x$ and $(U_y, G_y, p_y)$ is a coordinate chart of $y$ such that $U_y/G_y\subset U_x/G_x$, there is an injective smooth map $U_y\rightarrow U_x$ covering the inclusion $U_y/G_y\rightarrow U_x/G_x$. \vskip 0.1in \noindent (ii). A map between V-manifolds $h: M\rightarrow M'$ is smooth if for every point $x\in M$, there are local charts $(U_x, G_x,p_x), (U'_{h(x)}, G'_{h(x)}, p'_{ h(x)})$ of $x, h(x)$ such that locally $h$ can be lift to a smooth map $$h: U_x\rightarrow U'_{h(x)}.$$ \vskip 0.1in \noindent (iii).$P: E\rightarrow M$ is a V-bundle if locally $P^{-1}(U_{\alpha}/ G_{\alpha})$ can be lift to $U_{\alpha}\times {\bf R}^k$. Furthermore, the lifting of a transition map is linear on ${\bf R}^k$. \vskip 0.1in Furthermore, we can define Banach V-manifold, Banach V-bundle in the same way.} \vskip 0.1in An easy observation is that we can always choose a local chart $(U_x, G_x, p_x)$ of $x$ such that $G_x$ is the stabilizer of $x$ by shrinking the size of $U_x$. Furthermore, we can assume that $G_x$ acts effectively and $U_x$ is an open disk neighborhood of the origin $x$ in a linear representation $(G_x, {\bf R}^n)$. We call such a chart {\em a good chart} and $G_x$ a {\em local group}. Notes that if $S$ is a transverse section of a V-bundle, then $S^{-1}(0)$ is a smooth V-sub-manifold. But, it is well-known that the ordinary transversality theorems fail for V-manifolds. However, the differential calculus (differential form, orientability, integration, de Rham theory) extends over V-manifolds. Moreover, the theory of characteristic classes and the index theory also extend over $V$ manifolds. We won't give any detail here. Readers can find a detail expository in \cite{Sa1}, \cite{Sa2}. In summary, if we use differential analysis, we can treat a V-manifold as an ordinary smooth manifold. To simplify the notation, we will omit the word ``V-manifold'' without confusion when we work on the differential form and the integration. \vskip 0.1in \noindent {\bf Definition 2.2: }{\it We call that $M$ to be a fine $V$-manifold if any local $V$-bundle is dominated by a global oriented $V$-bundle. Namely, Let $U_{\alpha}\times_{\rho_{ \alpha}} E/G_{\alpha}$ be a local V-bundle, where $\rho_{\alpha}: G_{\alpha} \rightarrow GL(E)$ is a representation. There is a global oriented V-bundle $E\rightarrow M$ such that $U_{\alpha}\times_{\rho_{ \alpha}} E_{\alpha}/G_{\alpha}$ is a subbundle of $E_{U_{\alpha}/G_{\alpha}}$.} \vskip 0.1in By a lemma of Siebert (Appendix), $\overline{{\cal B}}_A(Y,g,k)$ is fine. {\em In the rest of the section, we will assume that all the Banach V-manifolds are fine} Let ${\cal B}$ be a fine Banach V-manifold defined by specifying Sobolov norm of some geometric object. Let ${\cal F}\rightarrow {\cal B}$ be a Banach V-bundle equipped with a metric and ${\cal S}: {\cal B}\rightarrow {\cal F}$ be a smooth section defined by a nonlinear elliptic operator. \vskip 0.1in \noindent {\bf Definition 2.3: }{\it ${\cal S}$ is a proper section if $\{x; ||{\cal S}(x)||\leq C\}$ is compact for any constant $C$. We call ${\cal M}_S={\cal S}^{-1}(0)$ the moduli space of $F$. We call $({\cal B}, {\cal F}, {\cal S})$ a compact- V triple if ${\cal B}, {\cal F}$ is a Banach V-pair and ${\cal S}$ is proper.} \vskip 0.1in When ${\cal S}$ is proper, it is clear that ${\cal M}_{{\cal S}}$ is compact. \vskip 0.1in \noindent {\bf Definition 2.4: }{\it Let $M$ be a compact topological space. We call $(U, E, S)$ a virtual neighborhood of $M$ if $U$ is a finite dimensional oriented V-manifold (not necessarily compact), $E$ is a finite dimensional V-bundle of $U$ and $S$ is a smooth section of $E$ such that $S^{-1}(0)=M$. Suppose that $M_{(t)}=\bigcup_t M_t\times \{t\}$ is compact. We call $(U_{(t)}, S_{(t)}, E_{(t)})$ a virtual neighborhood cobordism if $U_{(t)}$ is a finite dimensional oriented V-manifold with boundary and $E_{(t)}$ is a finite dimensional V-bundle and $S_{(t)}$ is a smooth section such that $S^{-1}_{(t)}(0)=M_{(t)}$. } \vskip 0.1in Let $L_x$ be the linearlization $$\delta {\cal S}_x: T_x{\cal B}\rightarrow {\cal F}_x, \leqno(2.12)$$ where the tangent space of a V-manifold at $x$ means the tangent space of $U_{\alpha}$ at $x$ where $U_{\alpha}/G_{\alpha}$ is a coordinate chart at $x$. Then, $L_x$ is an elliptic operator. When $Coker L_x=0$ for every $x\in {\cal M}$, ${\cal S}$ is transverse to the zero section and ${\cal M}_{{\cal S}}={\cal S}^{-1}(0)$ is a smooth V-manifold of dimension $ind(L_x)$. The case we are interested in is the case that $Coker L_x\neq 0$ and it may even jump the dimension. The original version of following Lemma is erroneous. The new version is corrected by B. Siebert (appendix). \vskip 0.1in \noindent {\bf Lemma 2.5: }{\it Suppose that $({\cal B},{\cal F},{\cal S})$ is a compact-V triple. There exists an open set ${\cal U}$ such that ${\cal M}_{{\cal S}}\subset {\cal U} \subset {\cal B}$ and a finite dimensional oriented V-bundle ${\cal E}$ over ${\cal U}$ with a V-bundle map $s: {\cal E}\rightarrow {\cal F}_{{\cal U}}$ such that $$L_x+s(x, v): T_x {\cal U}\oplus {\cal E}\rightarrow {\cal F}\leqno(2.13)$$ is surjective for any $x\in {\cal U}$. Furthermore, the linearlization of $s$ is a compact operator.} \vskip 0.1in {\bf Proof: } For each $x\in {\cal M}_S$, there is a good chart $(\tilde{U}_x, G_x, p_x)$. Suppose that $\tilde{U}_x$ is open disk of radius 1 in $H$ for some Banach space $H$. Let $({\cal F}_{\tilde{U}}, G_x, \pi_x)$ be the corresponding chart of ${\cal F}$. Let $H_x=Coker L_x$. Then, $G_x$ acts on $H_x$. Since ${\cal M}_{{\cal S}}$ is compact, there is a finite cover $\{(\frac{1}{2}\tilde{U}_{x_i}, G_{x_i}, p_{x_i})\}^m_1 $. Each $\frac{1}{2}\tilde{U}_{x_i}\times H_{x_i}/G_{x_i}$ is a local V-bundle. Since ${\cal B}$ is fine, there exists an oriented global finite dimensional V-bundle ${\cal E}_i$ over ${\cal U}=\bigcup_i \frac{1}{2} U_{x_i}$ such that $\frac{1}{2}\tilde{U}_{x_i}\times H_{x_i}/G_{x_i}$ is a subbundle of $({\cal E}_i)|_{\frac{1}{2} \tilde{U}_{x_i}/G_{x_i}}$. Let $${\cal E}=\oplus_i {\cal E}_i.\leqno(2.14)$$ Next, we define $s$. Each element $w$ of $H_{x_i}$ can be extended to a local section of ${\cal F}_{\tilde{U}_{x_i}}$. Then one can multiply it by a cut-off function $\phi$ such that $\phi=0$ outside of the disk of radius $\frac{3}{4}$ and $\phi=1$ on $\frac{1}{2}\tilde{U}_{x_i}$. Then, we obtain a section supported over $\tilde{U}_{x_i}$ (still denoted it by $s$). Define $$\bar{s}_i(x, w)=s(x).\leqno(2.15)$$ Then, $$s_i(x,w)=\frac{1}{|G_{x_i}|}\sum_{g_i\in G_{x_i}} (g_i)^{-1}\bar{s}(g_i(x), g_i(w)). \leqno(2.16)$$ By the construction, $s_i$ descends to a map $U_{x_i}\times H_{x_i}/G_{x_i}\rightarrow {\cal F}_{ U_{x_i}}$. Clearly, $s_i$ can be viewed as a bundle map from ${\cal E}_i$ to ${\cal F}$ since it is supported in $U_{x_i}$. Moreover, $$s(x_i, w): ({\cal E}_i)_{x_i}\rightarrow H_{x_i}\subset {\cal F}_{x_i} \leqno(2.17)$$ is projection. Then, we define $$s=\sum s_i.$$ By (2.17), $L_x+s_i$ is surjective at $x_i$ and hence it is surjective at a neighborhood of $x_i$. By shrinking $U_{x_i}$, we can assume that $L_x+s_i$ is surjective over $\frac{1}{2}U_{x_i}$. Hence, $L_x+s$ is surjective over ${\cal U}$. We have finished the proof. $\Box$ Next we define the extended equation $${\cal S}_e: {\cal E} \rightarrow {\cal F}\leqno(2.18)$$ by $${\cal S}_e(x, w)={\cal S}(x)+s(x, w) \leqno(2.19)$$ for $w\in E_x$. We call that $s$ {\em a stabilization term} and ${\cal S}_e$ {\em a stabilization of ${\cal S}$}. ${\cal S}_e$ can be identified with a section of $\pi^*{\cal F}$ where $\pi: {\cal E}\rightarrow {\cal U}$ is the projection. We shall use the same ${\cal S}_e$ to denote the corresponding section. Notes that ${\cal M}_{{\cal S}}\subset {\cal S}^{-1}_v(0)$, where we view ${\cal U}$ as the zero section of ${\cal E}$. Moreover, its linearlization $$(\delta {\cal S}_e)_{(x, 0)}(\alpha, u)=L_{x}(\alpha)+s(x,u). \leqno(2.20)$$ By lemma 2.5, it is surjective. Hence, ${\cal S}_e$ is a transverse section over a neighborhood of ${\cal M}_{{\cal S}}$. Since we only want to construct a neighborhood of ${\cal M}_{{\cal S}}$, without the loss of generality, we can assume that ${\cal S}_e$ is transverse to the zero section of $\pi^*{\cal F}$. Therefore, $$U=({\cal S}+ s)^{-1}(0)\subset {\cal E} \leqno(2.21)$$ is a smooth V-manifold of dimension $ind(L_x)+dim {\cal E}$. Clearly, $${\cal M}_{{\cal S}}\subset U. \leqno(2.22)$$ \vskip 0.1in \noindent {\bf Lemma 2.5: }{\it If $det(L_A)$ has a nowhere vanishing section, it defines an orientation of $U$.} \vskip 0.1in {\bf Proof: } $T_{(x, w)}U=Ker (\delta {\cal S}_v)$ and $Coker (\delta {\cal S}_v)=0$ by the construction. Hence, an orientation of $U$ is equivalent to a nowhere vanishing section of $det(ind (\delta {\cal S}_v))$. $$(\delta {\cal S}_v)_{(x, w)}(\alpha, u)=L_{x}(\alpha)+s(x,u)+\delta s_{(x,w)}( \alpha). \leqno(2.13)$$ Let $$(\delta^t {\cal S}_v)_{(x, w)}(\alpha, u)=L_{x}(\alpha)+ts(x,u)+t\delta s_{(x,w)} (\alpha).\leqno(2.14)$$ Then, $$det(ind( \delta {\cal S}_v))=det(ind (\delta^t {\cal S}_v))=det(ind (\delta^0 {\cal S}_v))=det( ind(L_x))\otimes det({\cal E}).$$ Therefore, a nowhere vanishing section of $det(ind(L_A))$ decides an orientation of $U$. $\Box$ Furthermore, we have the inclusion map $$S: U \rightarrow {\cal E}, \leqno(2.25)$$ which can be viewed as a section of $E=\pi^*{\cal E}$. $S$ is proper since ${\cal S}$ is proper. Moreover, $$S^{-1}(0)={\cal M}_{{\cal S}}. \leqno(2.26)$$ Here, we construct a virtual neighborhood $(U, E, S)$ of ${\cal M}_{{\cal S}}$. To simplify the notation, we will often use the same notation to denote the bundle (form) and its pull-back, Notes that for any cohomology class $\alpha\in H^*({\cal B}, {\bf Z})$, we can pull back $\alpha$ over $U$. Suppose that it is represented by a closed differential form on $U$ (still denoted it by $\alpha$) \vskip 0.1in \noindent {\bf Definition 2.8: }{\it Suppose that $det(ind (L_A))$ has a nowhere vanishing section so that $U$ is oriented. \vskip 0.1in \noindent (1). If $deg(\alpha)\neq ind(L_A)$, we define virtual neighborhood invariant $\mu_{{\cal S}}$ to be zero. \vskip 0.1in \noindent (2).When $deg(\alpha)=ind(L_A)$, choose a Thom form $\Theta$ supported in a neighborhood of zero section of $E$. We define $$\mu_{{\cal S}}(\alpha)=\int_U\alpha\wedge S^*\Theta.$$} \vskip 0.1in \noindent {\bf Remark: }{\it In priori, $\mu_S$ is a real number. However, it was pointed to the author by S. Cappell that when $\alpha$ is a rational cohomology class, $\mu_S(\alpha)$ is a rational number. This is because both $U, E$ have fundamental classes in compacted supported rational homology. Then, $\mu_S(\alpha)$ can be interpreted as paring with the fundamental class in rational cohomology.} \vskip 0.1in \noindent {\bf Proposition 2.9: }{\it \noindent (1). $\mu_{{\cal S}}$ is independent of $\Theta, \alpha$. \vskip 0.1in \noindent (2). $\mu_{{\cal S}}$ is independent of the choice of $s$ and ${\cal E}$.} \vskip 0.1in {\bf Proof: }(1). If $\Theta'$ is another Thom-form supported in a neighborhood of zero section, there is a $(k-1)$-form $\theta$ supported a neighborhood of zero section such that $$\Theta-\Theta'=d\theta.\leqno(2.28)$$ Then, $$\int_U \alpha\wedge S^*\Theta-\int_U \alpha\wedge S^*\Theta'=\int_U\alpha \wedge d(S^*\theta)=\int_U d(\alpha\wedge S^*\theta)=0. \leqno(2.28)$$ If $\alpha'$ is another closed form representing the same cohomology class, it is the same proof to show $$\int_U \alpha\wedge S^*\Theta=\int_U \alpha'\wedge S^*\Theta. \leqno(2.29)$$ To prove (2), suppose that $({\cal E}', s')$ is another choice and $(U',E', S')$ is the virtual neighborhood constructed by $({\cal E}', s')$. Let $\Theta'$ be the Thom form of $E'$ supported in a neighborhood of zero section. Consider $${\cal S}^{(t)}_e={\cal S}+(1-t)s+ts': {\cal E}\oplus {\cal E}'\times [0,1]\rightarrow {\cal F}.\leqno(2.30)$$ Let $(U_{(t)}, {\cal E}\oplus {\cal E}', S_{(t)})$ be the virtual neighborhood cobordism constructed by ${\cal S}^{(t)}_e$. By Stokes theorem, $$\int _{U_0}\alpha\wedge S^*_0(\Theta\wedge \Theta')-\int_{U_1}\alpha \wedge S^*_1(\Theta\wedge \Theta')=\int_{U_{(t)}}d (\alpha\wedge S^*_{(t)}(\Theta\wedge \Theta'))=0,\leqno(2.31)$$ since both $\alpha$ and $\Theta\wedge\Theta'$ are closed. It is easy to check that $U_0=\pi^* E'$ where $\pi: E\rightarrow U$ is the projection, $S_0=S\times Id$. Therefore, $$\int _{U_0}\pi^*\alpha\wedge S^*_0(\Theta\wedge \Theta')=\int_U\alpha\wedge S^*( \Theta)=\int_U\alpha\wedge S^*(\Theta).\leqno(2.32)$$ In the same way, one can show that $$\int_{U_1}\alpha\wedge S^*_1(\Theta\wedge \Theta')=\int_{U'}\alpha\wedge (S')^*(\Theta').$$ We have finished the proof. $\Box$ \vskip 0.1in \noindent {\bf Proposition 2.9: }{\it Suppose that ${\cal S}_t$ is a family of elliptic operators over ${\cal F}_t\rightarrow {\cal B}_t$ such that ${\cal B}_{(t)}=\bigcup_t{\cal B}_t\times \{t\}$ is a smooth Banach V-cobordism and ${\cal F}_{(t)}=\bigcup_t {\cal F}_t \times \{t\}$ is a smooth V-bundle over ${\cal B}_{(t)}$. Furthermore, we assume that ${\cal M}_{{\cal S}_{(t)}}=\bigcup_t{\cal M}_{{\cal S}_t}\times \{t\}$ is compact. We call $({\cal B}_{(t)}, {\cal F}_{(t)}, {\cal S}_{(t)})$ a compact-V cobordism triple. Then $\mu_{{\cal S}_0}=\mu_{{\cal S}_1}$. } \vskip 0.1in {\bf Proof: } Choose $({\cal E}_{(t)}, s)$ of ${\cal F}_{(t)}\rightarrow {\cal U}_{(t)}$ such that $$\delta ({\cal S}^t+s) \leqno(2.33)$$ is surjective to ${\cal F}_{{\cal U}_{(t)}}$ where ${\cal M}_{{\cal S}_{(t)}}\subset {\cal U}_{(t)} \subset {\cal B}_{(t)}$. Repeating previous argument, we construct a virtual neighborhood cobordism $(U_{(t)}, E_{(t)}, S_{(t)})$. Then, it is easy to check that $(U_0, E_0, S_0)$ is a virtual neighborhood of ${\cal S}_0$ defined by $({\cal E}_0, s(0))$ and $(U_1, E_1, S_1)$ is a virtual neighborhood of ${\cal S}_1$ defined by $({\cal E}_1, s(1))$. Applying the Stokes theorem as before, we have $\mu_{{\cal S}_0}=\mu_{{\cal S}_1}$. $\Box$ Recall that by \cite{Sa2} one can define connections and curvatures on a V-bundle. Then, characteristic classes can be defined by Chern-Weil formula in the category of V-bundle. Next, we prove a proposition which is very useful to calculate $\mu_{{\cal S}}$. \vskip 0.1in \noindent {\bf Proposition 2.10: }{\it (1) If $F$ is a transverse section, $\mu_{{\cal S}}(\alpha)=\int_{{\cal M}_{{\cal S}}}\alpha$. \vskip 0.1in \noindent (2) If $Coker L_A$ is constant and ${\cal M}_{{\cal S}}$ is a smooth V-manifold such that $dim ({\cal M}_{{\cal S}})=ind(L_A)+\dim Coker L_A$, $Coker L_A$ forms an obstruction V-bundle ${\cal H}$ over ${\cal M}_{{\cal S}}$. In this case, $$\mu_{{\cal S}}(\alpha)=\int_{{\cal M}_{{\cal S}}}e({\cal H})\wedge \alpha.\leqno(2.34)$$} \vskip 0.1in Before we prove the proposition, we need following lemma \vskip 0.1in \noindent {\bf Lemma 2.11: }{\it Let $E\rightarrow M$ be a V-bundle over a V-manifold. Suppose that $s$ is a transverse section of $E$. Then the Euler class $e(E)$ is dual to $s^{-1}(0)$ in the following sense: for any compact supported form $\alpha$ with $deg(\alpha)=\dim M-\dim E$, $$\int_M e(E)\wedge \alpha=\int_{s^{-1}(0)} \alpha.\leqno(2.35)$$} \vskip 0.1in {\bf Proof: } When $\dim {\cal H}=\dim {\cal M}_{{\cal S}}$, it is essentially Chern's proof of Gauss-Bonnett theorem. By \cite{Sa2}, Chern's proof in smooth case holds for V-bundle. For general case, it is an easy generalization of Chern's proof using normal bundle. We omit it. $\Box$ {\bf Proof of Proposition 2.10: } (1) follows from the definition where we take $k=0$. To prove (2), let $F_{b}$ be the eigenspace of Laplacian $L_AL^*_A$ of an eigenvalue $b$. Since $rank(Coker L_A)$ is constant, there is a $a\not\in Spec(L_A)$ for $A\in {\cal M}_{{\cal S}}$ such that the eigenspaces $$F_{\leq a}=\oplus_{b\leq a} F_b=Coker L_A \leqno(2.36)$$ has dimension $dim Coker(L_A)$ over ${\cal M}_{{\cal S}}$. Then, the same is true for an open neighborhood of ${\cal M}_{{\cal S}}$. Without the loss of generality, we can assume that the open neighborhood is ${\cal U}$. Therefore $F_{\leq a}$ form a V-bundle (still denoted by $F_{\leq a}$) over ${\cal U}$ whose restriction over ${\cal M}_{{\cal S}}$ is ${\cal H}$. In this case, we can choose $s$ such that $s\in F_{\leq a}$ and $s$ satisfy Lemma 2.4. Let $(U, E, S)$ be the virtual neighborhood constructed from $s$. Recall that $$U=({\cal S}_e)^{-1}(0), \leqno(2.37)$$ where $${\cal S}_e={\cal S}+s.\leqno(2.38)$$ Let $$p_{\leq a}: {\cal F}\rightarrow F_{\leq a}\leqno(2.39)$$ be the projection. Then, $${\cal S}_e=p_{\leq a}({\cal S}+s)+(1-p_{\leq a})({\cal S}+s)=p_{\leq a}({\cal S}+s)+(1-p_{\leq a}) ({\cal S}). \leqno(2.40)$$ The last equation follows from the fact that $s\in F_{\leq a}$. So, ${\cal S}_e=0$ iff $$p_{\leq a}({\cal S}+ s)=0 \mbox{ and } (1-p_{\leq a})({\cal S})=0.\leqno(2.41)$$ By our assumption, $(1-p_{\leq a})({\cal S})$ is transverse to the zero section over ${\cal M}_{{\cal S}}$ since $Coker(L^A)= F_{\leq a}$. Therefore, we can assume that $(1-p_{\leq a})({\cal S})$ is transverse to the zero section over ${\cal U}$. Hence, $((1-p_{\leq a})({\cal S}))^{-1}(0)$ is a smooth V-manifold of dimension $ind(L_A)+\dim F_{\leq a}=ind(L_A)+\dim Coker(L_A).$ But $${\cal M}_{{\cal S}}\subset ((1-p_{\leq a})({\cal S}))^{-1}(0)\leqno(2.42)$$ is a compact submanifold of the same dimension. Then, ${\cal M}_{{\cal S}}$ consists of the components of $((1-p_{\leq a})({\cal S}))^{-1}(0)$. In particular, other components are disjoint from ${\cal M}_{{\cal S}}$. Therefore, we can choose smaller ${\cal U}$ to exclude those components. Without the loss of generality, we can assume that $$((1-p_{\leq a})({\cal S}))^{-1}(0)={\cal M}_{{\cal S}}.\leqno(2.43)$$ Since ${\cal S}=0$ over ${\cal M}_{{\cal S}}$, the first equation of (2.31)becomes $$p_{\leq a}(F+ s)= s=0.\leqno(2.44)$$ Therefore, $U\subset E_{{\cal M}_s}$ and $$U=s^{-1}(0).\leqno(2.45)$$ However, $s$ is a transverse section by the construction. By Lemma 2.11, $$\int_U\alpha \wedge S^*(\Theta)=\int_{E_{{\cal M}_{{\cal S}}}}\pi^*(e({\cal H})\wedge \alpha) \wedge \Theta=\int_{{\cal M}_{{\cal S}}}e({\cal H})\wedge \alpha, \leqno(2.46)$$ since $S: E_{{\cal M}_{{\cal S}}}\rightarrow E_{{\cal M}_{{\cal S}}}$ is identity. Then, we proved (2). $\Box$ \section{Virtual neighborhoods of Cauchy-Riemann equation} This is a technical section about the local structure of $\overline{{\cal B}}_A( Y,g,k)$ and Cauchy-Riemann equation. Roughly speaking, we will show that for all the applications of this article $\overline{{\cal B}}_A(Y,g,k)$ behaves like a Banach V-manifold. Namely, $\overline{{\cal B}}_A(Y,g,k)$ is VNA. If readers only want to get a sense of big picture, one can skip over this section. There are roughly two steps in the virtual neighborhood construction. First step is to define an extended equation ${\cal S}_e$ by the stabilization. Then, we need to prove that (i) The set ${\cal U}_{{\cal S}_e}=\{x, Coker D_x {\cal S}_v=\emptyset\}$ is open; (ii) ${\cal U}_{{\cal S}_e}\cap {\cal S}^{-1}_e(0)$ is a smooth, oriented V-manifold. Ideally, we would like to set up some Banach manifold structure on our configuration space and treat ${\cal U}_{{\cal S}_e} \cap{\cal S}^{-1}_e(0)$ as a smooth submanifold. However, there are some basic analytic difficulty against such an approach, which we will explain now. For ${\cal B}_A(Y,g,k)$, we allow the domain of the map to vary to accomendate the variation of complex structures of Riemann surfaces. Let's look at a simpler model. Suppose that $\pi: M\rightarrow N$ be a fiber bundle with fiber $F$. We want to put a Banach manifold structure on $\bigcup_{x\in N}C^k(\pi^{-1}(x))$. A natural way is to choose a local trivialization $\pi^{-1}(U)\cong U\times F$. It induces a trivialization $\bigcup_{x\in U}C^k(\pi^{-1}(x))\rightarrow U\times C^k(F)$. Then, we can use the natural Banach manifold structure on $C^k(F)$ to induce a Banach manifold structure on $\bigcup_{x\in U}C^k(\pi^{-1}(x))$. However, if we have a different local trivialization, the transition function is a map $g:U\rightarrow Diff(F)$. The problem is that $Diff(F)$ only acts on $C^k(F)$ continuously. For example, suppose that $\phi_t$ is a one-parameter family of diffeomorphisms generated by a vector field $v$. Then, the derivative of the path $f\circ g_t$ is $v(f)$, which decreases the differentiability of $f$ by one. So we do not have a natural Banach manifold structure on $\bigcup_{x\in N}C^k(\pi^{-1}(x))$ in general. It is obvious that we have a natural Frechet manifold structure on $\bigcup_{x\in N}C^{\infty}( \pi^{-1}(x))$. However, we only care about the zero set ${\cal M}$ of some elliptic operator ${\cal S}_e$ defined over Frechet manifold $\bigcup_{x\in N}C^{\infty}(\pi^{-1}(x))$. The crucial observation is that locally we can choose any local trivialization and use Banach manifold structure induced from the local trivialization to show that ${\cal M}_U={\cal M}\cap U\times C^k(F)$ is smooth. The elliptic regularity implies that ${\cal M}_U \subset U\times C^{\infty}(F)$. Although the transition map is not smooth for $C^k(F)$, but it is smooth on ${\cal M}_U$. Therefore, ${\cal M}_U$ patches together to form a smooth manifold. Our strategy is to define the extended equation ${\cal S}_e$ over the space of $C^{\infty}$-stable map. In each coordinate chart, we enlarge our space with Sobolev maps. Then, we can use usual analysis to show that the moduli space can be given a local coordinate chart of a smooth manifold. Elliptic regularity guarantees that every element of the moduli space is indeed smooth. Then, we show that the moduli space in each coordinate chart patches up to form a $C^1$-V-manifold. Suppose that $(Y, \omega)$ is a family of symplectic manifold and $J$ is a tamed almost complex structure. Choose a metric tamed with $J$. \vskip 0.1in \noindent { \bf Definition 3.1 ([PW], [Ye], [KM]). }{\it Let $(\Sigma, \{x_i\})$ be a stable Riemann surface. A stable holomorphic map (associated with $(\Sigma, \{x_i\})$) is an equivalence class of continuous maps $f$ from $\Sigma'$ to $Y$ such that $f$ has the image in a fiber of $Y\rightarrow X$ and is smooth at smooth points of $\Sigma'$, where the domain $\Sigma'$ is obtained by joining chains of ${\bf P}^1$'s at some double points of $\Sigma$ to separate the two components, and then attaching some trees of ${\bf P}^1$'s. We call components of $\Sigma$ {\em principal components} and others {\em bubble components}. Furthermore, \begin{description} \item[(1)] If we attach a tree of ${\bf P}^1$ at a marked point $x_i$, then $x_i$ will be replaced by a point different from intersection points on some component of the tree. Otherwise, the marked points do not change. \item[(2)] The singularities of $\Sigma'$ are normal crossing and there are at most two components intersecting at one point. \item[(3)] If the restriction of $f$ on a bubble component is constant, then it has at least three special points (intersection points or marked points). We call this component {\em a ghost bubble} \cite{PW}. \item[(4)] The restriction of $f$ to each component is $J$-holomorphic. \end{description} Two such maps are equivalent if one is the composition of the other with an automorphism of the domain of $f$. If we drop the condition (4), we simply call $f$ a stable map. Let $\overline{{\cal M}}_{A}(Y,g,k,J)$ be the moduli space of stable holomorphic maps and $\overline{{\cal B}}_A(Y,g,k)$ be the space of stable maps.} \vskip 0.1in \noindent {\bf Remark 3.2: }{\it There are two types of automorphism here. Let $Aut_f$ be the group of automorphisms $\phi$ of the domain of $f$ such that $f\circ \phi$ is also holomorphic. This is the group we need to module out when we define $\overline{{\cal M}}_{A}(Y,g,k,J)$ and $\overline{{\cal B}}_A(Y,g,k)$. It consists two kinds of elements. (1) When some bubble component is not stable with only one or two marked points, there is a continuous subgroup of $PSL_2{\bf C}$ preserving the marked points. (2) Another type of element comes from the automorphisms of domain interchanging different components, which form a finite group. Let $stb_f$ be the subgroup of $Aut_f$ preserving $f$. It is easy to see that $stb_f$ is always a finite group. Type (1) elements of $stb_f$ appear with multiple covered maps.} \vskip 0.1in \noindent {\bf Proposition 3.3: }{\it $\overline{{\cal B}}_A(Y,g,k)$ (whose topology is defined later) is a stratified Hausdorff Frechet V-manifold of finite many strata.} \vskip 0.1in The proof consists of several lemmas. \vskip 0.1in \noindent {\bf Lemma 3.4: }{\it ${\cal B}_A(Y,g,k)$ is a Hausdorff Frechet V-manifold for any $2g+k\geq 3$ or $g=0, k\leq 2, A\neq 0$.} \vskip 0.1in {\bf Proof: } Recall $${\cal B}_A(Y,g,k)=\{(f, \Sigma); \Sigma\in {\cal M}_{g,k}, f: \Sigma \stackrel{F}{ \rightarrow} Y \},\leqno(3.1)$$ where $\stackrel{F}{\rightarrow}$ means that the image is in a fiber. When $2g+k\geq 3$, $\Sigma$ is stable and ${\cal M}_{g,k}$ is a V-manifold. Hence, the automorphism group $Aut_{\Sigma}$ is finite. Furthermore, there is a $Aut_{\Sigma}$-equivariant holomorphic fiber bundle $$\pi_{\Sigma}: U_{\Sigma}\rightarrow O_{\Sigma}$$ such that $O_{\Sigma}/Aut_{\Sigma}$ is a neighborhood of $\Sigma$ in ${\cal M}_{g,k}$ and fiber $\pi^{-1}_{\Sigma}(b)=b$. Consider $${\cal U}_{\Sigma, f}=\{(b, h); h: b\stackrel{F}{\rightarrow} Y, h\in C^{\infty}.\} \leqno(3.2)$$ As we discussed in the beginning of this section, ${\cal U}_{\Sigma, f}$ has a natural Frechet manifold structure. Let $stb_f\subset Aut_{\Sigma}$ be the subgroup preserving $f$. One can observe that ${\cal U}_{\Sigma, f}/stb_f$ is a neighborhood of $(\Sigma, f)$ in ${\cal B}_A(Y,g,k)$. Hence, ${\cal B}_A(Y,g,k)$ is a Frechet V-manifold. Since only a finite group is involved, ${\cal B}_A(Y,g,k)$ is obviously Hausdorff. For the case $g=0, k\leq 2, A\neq 0$, $\Sigma$ is no longer stable and the automorphism group $Aut_{\Sigma}$ is infinite. Here, we fix our marked points at $0$ or $0,1$. First of all, $stb_f$ is finite for any $f\in Map^F_A(Y,0,k)$ with $A\neq 0$. $${\cal B}_A(Y,g, k)=Map^F_A(Y,0,k)/Aut_{\Sigma}.$$ We first show that $B_A(Y, g,k)$ is Hausdorff. It requires showing that the graph $$\Delta=\{(f, f\tau); f\in Map^F_A(Y,0,k), \kappa\in Aut_{\Sigma}\}\leqno(3.3)$$ is closed. Suppose that $(f_n, f_n\tau_n)$ converges to $(f,h)$ uniformly for all its derivatives. We claim that $\{\tau_n\}$ has a convergent subsequence. Suppose that $\infty$ is one of marked point which $\tau_n$ fixes. They, $\tau_n$ can be written as $a_nz+b_n$ for $a_n\neq 0$. Suppose that $\tau_n$ is degenerated. Then, (i)$ b_n \rightarrow \infty$, (ii) $a_n\rightarrow 0$ or (iii) $a_n\rightarrow \infty$. In each case, we observe that $\tau_n$ converges pointwisely to $\tau$ which is either a constant map taking value $\infty$ or a map taking two different values. Since $f_n$ converges uniformly, $f_n\tau_n$ converges to $f\tau$ pointwisely. Hence, $h=f\tau$ which is either a constant map or discontinous. We obtain a contradiction. Suppose that $\tau_n$ converges to $\tau$. Then, $f_n\tau_n$ converges to $f\tau$. Therefore, $\Delta$ is closed. Notes that $$||df||_{L^2}\geq \omega(A).\leqno(3.4)$$ Choose the standard metric on ${\bf P}^1$ with volume 1. Then, for a holomorphic map, there are points $p$ (hence a open set of them) such that $df(p)$ is of maximal rank and $|df(p)|\geq \frac{1}{2}\omega(A)$. Since we only want to construct a neighborhood and the condition above is an open condition. Without the loss of generality, we assume that it is true for any $f$. We marked extra points $e_i$ such that $df(e_i)$ is of maximal rank, $|df(e_i)|\geq \frac{1}{2}\omega(A)$ and $(\Sigma, e_i)$ has three marked points. Next we construct a slice $W_f$ of the action $Aut_{\Sigma}$. Note that $Map^F_A(Y,0,k)$ is only a Frechet manifold. We can not use implicit function theorem. Since $stb_f$!is finite, we can construct a $stb_f$ invariant metric on $f^*TY$ by averaging the existing metric. Using $stb_f$ invariant metric, the set $$\{w\in \Omega^0(f^*T_FY); ||w||_{L^p_1}< \epsilon\}\leqno(3.5)$$ is $stb_f$-invariant and open in $C^{\infty}$-topology. Now, we fix the $stb_f$-invariant metric. For each extra marked point $e_i$ constructed in previous paragraph, $df(e_i)$ is a $2$-dimensional vector space. Clearly, $$f_{e_i}=\oplus_{\tau\in stb_f} df(\tau(e_i))\subset (T_{f(e_i)} Y)^{|stb_f|}$$ is $stb_f$-invariant. Now we want to construct a 2-dimensional subspace $E_{e_i}\subset f_{e_i}$ which is the orbit of action $Aut_{\Sigma}$. For simplicity, we assume that we only need one extra marked point $e_1$ to stabilize $\Sigma$. The proof of the case with two extra marked points is the same. In this case, a neighborhood of $id$ in $Aut_{\Sigma}$ can be identified with a neighborhood of $e_1$ by the relation $\tau_x(e_1)=x$ for $x\in D^2(e_1)$. $\frac{d}{dx}\tau_x(f)(y)|_{x=e_1} =df(y)(v(y))$, where $v=\frac{d}{dx}\tau_x|_{x=e_1}$ is a holomorphic vector field. By our identification, $v$ is decided by its value $v(e_1)\in T_{e_1}S^2$. Given any $v\in T_{e_1}S^2$, we use $v_{e_1}\in T_{id} Aut_{\Sigma}$ to denote its extension. Therefore, $v$ decides $v_{e_1}(\tau(e_1))$. To get a precise relation, we can differentiate $\tau_x(\tau(e_1))=\tau (\tau^{-1} \tau_x \tau)( e_1)$ to obtain $$v_{e_1}(\tau(e_1))=D\tau Ad_\tau(v),\leqno(3.6)$$ where $Ad_\tau$ is the adjoint action. $$E_{e_i}=\{ \oplus_{\tau\in stb_f} df(\tau(e_1))(v_{e_1}(\tau(e_1))); v\in T_{e_1}S^2\}\leqno(3.7)$$ It is easy to check that $E_{e_1}$ is indeed $stb_f$-invariant. We can identify $E_{e_1}$ with $T_{e_1}S^2$ by $$v\rightarrow \oplus_{\tau\in stb_f} df(\tau(e_1))(D\tau Ad_\tau(v)),\leqno(3.8)$$ Hence, $E_{e_i}$ is 2-dimensional. Given any $w\in \Omega^0(f^*T_FY)$, we say that {\em $w\perp E_{e_i}$ if $\oplus_{\tau\in stb_f} w(\tau(e_i))$ is orthogonal to $E_{e_i}$}. The slice $W_f$ can be constructed as $$W_f=exp_f \{ w\in \Omega(f^*T_FY); ||w||_{L^p_1}< \epsilon, ||w||_{C^1(D_{\delta_0}(g(e_i)))}< \epsilon \mbox{ for } g\in stb_f, w\perp E_{e_i}\},\leqno(3.9)$$ where $T_FY$ is the direct sum of vertical tangent bundle and $P^*TX$ and $\delta_0$ is a small fixed constant. We need to show that \begin{description} \item[(1)] $W_f$ is invariant under $stb_f$. \item[(2)] If $h\tau\in W_f$ for $h\in W_f$, then $\tau\in Stb_f$. \item[(3)] There is a neighborhood $U$ of $id\in Aut$ such that the multiplication $F: U\times W_f\rightarrow Map^F_A(Y,0,k)$ is a homeomorphism onto a neighborhood of $f$. \end{description} (1) follows from the definition. For (2), we claim that the set of $\tau$ satisfying $(2)$ is close to an element of $stb_f$ for small $\epsilon$. If not, there is a neighborhood $U_0$ of $stb_f$ and a sequence of $(h_n ,\tau_n)$ such that $\tau_n\not \in U_0$, $h_n$ converges to $f$ and $h_n\tau_n$ converges to $f$. By the previous argument, $\tau_n$ has a convergent subsequence. Without the loss of generality, we can assume that $\tau_n$ converges to $\tau\not\in U_0$. Then, $h_n\tau_n$ converges to $f\tau=f$. This is a contradiction. By (1), we can assume that $\tau$ is close to identity. Then, (2) follows from (3). Next we prove (3). Consider the local model around $f(\tau(e_1))$. Since $df(\tau(e_1))$ is injective, we can choose a local coordinate system of $V$ such that $Im(f)$ is a ball of ${\bf C}_{\tau}\subset {\bf C}_{\tau}\times {\bf C}^{n-1}_{\tau}$ in which the origin corresponds to $f(\tau(e_1))$.. Furthermore, we may assume that the metric is standard. For any $w$, let $$P(w): \Omega^0(f^*T_FY)\rightarrow E_{e_1}.$$ be the projection Then, $w\in W_f$ iff $P(w)=0$. Suppose that $w$ is bounded. $$\tau_x(w)(\tau(e_1))=w(\tau_x(\tau(e_1)))+f(\tau_x(\tau(e_1)))-f(\tau(e_1))+O(r^2)= w(\tau_x(\tau(e_1)))+f(\tau_x(\tau(e_1)))+O(r^2),\leqno(3.10)$$ where $r=|\tau_x(\tau(e_1))|$. Then, $$P(\tau_x(w))=P(w\circ \tau_x)+P(f\circ \tau_x)+O(r^2).\leqno(3.10.1)$$ Hence $P(\tau_x(w))=0$ iff $-P(w\circ \tau_x)=P(f\circ \tau_x)+O(r^2)$, where $$P(w\circ \tau_x), P(f\circ \tau_x): D^2\rightarrow E_{e_1}.\leqno(3.10.2)$$ Notes that $P(f\circ \tau_0)=0$. $$dP(f\circ \tau_x)(v)|_{x=0}=P(df(v_{e_1})).\leqno(3.10.3)$$ Under the identification (3.8), $dP(f\circ \tau_x)_0$ is the identity. Let $\bar{f}=P(f\circ \tau_x)$. Then, $\bar{f}^{-1}$ exists and $d\bar{f}^{-1}$ is bounded on a small disc. Consider $\bar{w}(x)=\bar{f}^{-1}P(w\circ \tau_x+O(r^2))$. Then, $P(\tau_x(w))=0$ iff $x$ is a fixed point of $\bar{w}$. Suppose that $\epsilon <<1$. Since $||w||_{C^1(D_{\delta_0}(g(e_i)))}<\epsilon$, $|\bar{w}(0)|<C\epsilon$. Furthermore, $|dw|<\epsilon$. $\bar{w}: D^2_{\delta_0} \rightarrow D^2_{\delta_0}$ for fixed $\delta_0$. The small bound on the derivative also implies that $\bar{w}$ is a contraction mapping. Therefore, there is a unique fixed point $x(w)$ in $D_{\delta_0}$ and hence $\tau_w=\tau_{x}$. Moreover, $x(w)$ depends smoothly on $w$. Therefore, $\tau_w$ depends smoothly on $w$. We define $H(w)=(\tau^{-1}_w, f_w\tau_w)$. By our construction, $H$ is continuous and an inverse of $F$. $\Box$ $\overline{{\cal M}}_A(Y,g,k,J)$ has an obvious stratification indexed by the combinatorial type of the domain. The later can be viewed as the topological type of the domain as an abstract 2-manifold with marked points such that each component is associated with a nonzero integral 2-dimensional class $A_i$ unless this component is genus zero with at least three marked points. Furthermore, each component is represented by a $J$-holomorphic map with fundamental class $A_i$ and total energy is equal to $\omega(A)$. Suppose that ${\cal D}^{J,A}_{g,k}$ is the set of indices. \vskip 0.1in \noindent {\bf Lemma 3.5: }{\it ${\cal D}^{J,A}_{g,k}$ is a finite set.} \vskip 0.1in {\bf Proof:} Let $(A_1, \cdots, A_k)$ be the integral 2-dimensional nonzero classes associated with the components. The last condition implies that $$\omega(A_i)>0, \sum A_i=A. \leqno(3.11)$$ \vskip 0.1in In \cite{RT1}(Lemma 4.5), it was shown that the set of tuple (3.11) is finite. Therefore, the number of non-ghost components is bounded. We claim that the number of ghost bubbles is bounded by the number of non-ghost bubbles. Then, the finiteness of ${\cal D}^{J,A}_{g,k}$ follows automatically. We prove our claim by the induction on the number of non-ghost bubbles. It is easy to observe that any ghost bubble must lie in some bubble tree $T$. By the construction, this ghost bubble can not lie on the tip of any branch. Otherwise, it has at most two marked points. Choose $B$ to be the ghost bubble closest to the tip. We remove the subtree $T_B$ with base $B$. Then, we obtain an abstract 2-manifold with marked points. If it is the domain of another stable map, we denote it by $T'$. If not, $B$ is based on another ghost bubble $B'$ with only three marked points. Then, we remove $T_B$ and contract $B'$ to obtain $T'$ the domain of another stable map. Let $gh(T')$ be the number of ghost bubbles and $ngh(T')$ be the number of non-ghost bubbles. By the induction, $$gh(T')\leq ngh(T').\leqno(3.12)$$ However, $$gh(T)\leq gh(T')+2, ngh(T_B)\geq 2.$$ Therefore, $$gh(T)\leq ngh(T')+2\leq ngh(T)+ngh(T_B)=ngh(T).\leqno(3.13)$$ We finish the proof. $\Box$ \vskip 0.1in For any $D\in {\cal D}^{J,A}_{g,k}$, let ${\cal B}_D(Y,g,k) \subset \overline{{\cal B}}_A(Y,g,k)$ be the set of stable maps whose domain and the corresponding fundamental class of each component have type $D$. Then, ${\cal B}_D(Y,g,k)$ is a strata of ${\cal B}_A(Y,g,k)$. \vskip 0.1in \noindent {\bf Lemma 3.6: }{\it ${\cal B}_D(Y,g,k)$ is a Hausdorff Frechet V-manifold.} \vskip 0.1in {\bf Proof: } ${\cal B}_D(Y,g,k)$ is a subset of $\prod_i {\cal B}_{A_i}( Y, \Sigma_i)$ such that the components intersect each other according to the intersection pattern specified by $D$. Therefore, it is Hausdorff. For the simplicity, let's consider the case that $D$ has only two components. The general case is the same. Let $D=\Sigma_1\wedge \Sigma_2$ joining at $p\in \Sigma_1, q\in \Sigma_2$. Assume that $A_i$ is associated with $\Sigma_i$. Then, $${\cal B}_D(Y,g,k)=\{(f_1, f_2)\in {\cal B}_{A_1}(Y, g_1, k_1+1)\times {\cal B}_{A_2}(Y, g_2, k_2+1); f_1(p)=f_2(p)\}.\leqno(3.14)$$ It is straightforward to show that ${\cal B}_D(Y,g,k)$ is Frechet V-manifold with the tangent space $$T_{(f_1, f_2)}{\cal B}_D(Y,g,k)=\{(w_1, w_2)\in \Omega^0(f^*_1T_FV)\times \Omega^0 (f^*_2T_FV); w_1(p)=w_2(q)\} \leqno(3.15)$$ We leave it to readers. $\Box$ Next, we discuss how different strata fit together. It amounts to show how a stable map deforms when it changes domain. A natural starting point is the deformation theory of the domain of stable maps as abstract nodal Riemann surfaces. However, it is well-known that unstable components cause a problem in the deformation theory. For example, the moduli space will not be Hausdorff. To have a good deformation theory, we have to consider a map with its domain together for unstable components. Let $\overline{{\cal M}}_{g,k}$ be the space of stable Riemann surfaces. The important properties of $\overline{{\cal M}}_{g,k}$ are that (i) $\overline{{\cal M}}_{g,k}$ is a V-manifold; (ii) there is a local universal V-family in following sense: for each $\Sigma\in \overline{{\cal M}}_{g,k}$, let $stb_{\Sigma}$ be its automorphism group. There is a $stb_{\Sigma}$-equivariant (holomorphic) fibration $$\pi_{\Sigma}: U_{\Sigma} \rightarrow O_{\Sigma}\leqno(3.16)$$ such that $O_{\Sigma}/Aut_{\Sigma}$ is a neighborhood of $\Sigma$ in $\overline{{\cal M}}_{g,k}$ and the fiber $\pi^{-1}_{\Sigma}(b)=b$. Suppose that the components of $f$ are $(\Sigma_1, f_1), \cdots, (\Sigma_m, f_m)$, where $\Sigma_i\in {\cal M}_{g_i, k_i}$ is a marked Riemann surface. If $\Sigma_i$ is stable, locally ${\cal M}_{g_i. k_i}$ is a V-manifold and have a local universal V-family. Suppose that they are $$\pi: U_i\rightarrow O_i\leqno(3.17)$$ divided by the automorphism group $Aut_i$ of $\Sigma_i$ preserving the marked points. Stability means that $Aut_{\Sigma_i}$ is finite. However, the relevant group for our purpose is $stb_i=stb_{f_i}\subset Aut_i$. Suppose that $x_{i1}, \cdots, x_{ik_i}$ are the marked points. We choose a disc $D_{ij}$ around each marked point $x_{ij}$ invariant under $stab_{\Sigma_i}$. For each $\tilde{\Sigma}_i\in O_i$, $x_{ij}$ may vary. We can find a diffeomorphism $\phi_{\Sigma}: \Sigma \rightarrow \tilde{\Sigma}_i$ to carry $x_{ij}$ together with $D_{ij}$ to the corresponding marked point and its neighborhood on $\tilde{\Sigma}_i$. Pulling back the complex structures by $\phi_{\tilde{\Sigma}_i}$, we can view $O_i$ as the set complex structure on $\Sigma_i$ which have the same marked points and moreover are the same on $D_{ij}$. $\phi_{\tilde{\Sigma}_i}$ gives a local smooth trivialization $$\phi_{\Sigma}: U_i\rightarrow O_i\times \Sigma.\leqno(3.18)$$ When $\Sigma_i$ is unstable, $\Sigma_i$ is a sphere with one or two marked points and we have to divide it by the subgroup $Aut_i$ of ${\bf P}^1$ preserving the marked points. But to glue the Riemann surfaces, we have to choose a parameterization. Recall that ${\cal B}_{A_i}(\Sigma_i)=Map^F_{A_i}(\Sigma_i, Y)/Aut_{i}$. For any $f_i\in Map^F_{A_i}(\Sigma_i, Y)$, one constructs a slice $W_{f_i}$ (Lemma 3.4) at $f_i$ such that $W_{f_i}/stb_{f_i}$ is diffeomorphic to a neighborhood of $[f_i]$ in the quotient. Moreover, we only want to construct a neighborhood of $f$. To abuse notation, we identify ${\cal B}_{A_i}(\Sigma_i)$ with the slice $W_{f_i}/stb_{f_i}$. Then, we can proceed as before. Fix a standard ${\bf P}^1$. We choose a disc $D_{ij}$ ($j\leq 2$) around each marked point invariant under $stb_{f_i}$. Then, $O_i=pt, U_i={\bf P}^1$. Let ${\cal N}$ be the set of the nodal points of $\Sigma$. For each $x\in {\cal N}$, we associate a copy of ${\bf C}$ (gluing parameter) and denote it by ${\bf C}_x$. Let ${\bf C}_{f}=\prod_{x\in {\cal N}} {\bf C}_x$, which is a finite dimensional space. For each $v\in C_{f}$ with $|v|$ small and $\tilde{\Sigma}_i\in O_i$ , we can construct a Riemann surface $\tilde{\Sigma}_v.$ Suppose that $x$ is the intersection point of $\Sigma_i, \Sigma_j$ and $\Sigma_i, \Sigma_j$ intersect at $p\in \Sigma_i, q\in \Sigma_j$. For any small complex number $v_x=re^{iu}$. We construct $\Sigma_i \#_{v_x} \Sigma_j$ by cutting discs with radius $\frac{2r^2}{\rho}$-$D_p(\frac{2r^2}{\rho}), D_q(\frac{2r^2}{\rho})$, where $\rho$ is a small constant to be fixed later. Then, we identify two annulus $N_p(\frac{\rho r^2}{2} ,\frac{2r^2}{\rho}), N_q(\frac{\rho r^2}{2},\frac{2r^2}{\rho})$ by holomorphic map $$(e^{i\theta}, t)\cong (e^{i\theta}e^{iu}, \frac{r^4}{t}).\leqno(3.19)$$ Notes that (3.19) sends inner circle to outer circle and vis versus. Moreover, we identify the circle of radius $r^2$. Roughly speaking, we cut off the discs of radius $r^2$ and glue them together by rotating $e^{i\theta}$. When $v_x=0$, we define $\Sigma_i\#_0 \Sigma_j=\Sigma_i\wedge \Sigma_j$-the one point union at $p=q$. Given any metric $\lambda=(\lambda_1, \lambda_2)$ on $\Sigma$, we can patch it up on the gluing region as follows. Choose coordinate system of $N_p(\frac{\rho r^2} {2}, \frac{2r^2}{\rho})$. The metric of $\Sigma_1$ is $t(ds^2+dt^2)$ and the metric from $\Sigma_2$ is $\frac{r^4}{t}(ds^2+dt^2)$. Suppose that $\beta$ is a cut off function vanishing for $t<\frac{\rho r^2}{2}$ and equal to one for $t>\frac{2r^2}{\rho}$. We define a metric $\lambda_v$ which is equal to $\lambda$ outside the gluing region and $$\lambda_v=(\beta t+(1-\beta)\frac{r^4}{t})(ds^2+dr^2)\leqno(3.20)$$ over the gluing region. We observe that on the annulus $N_p(\frac{\rho r^2}{2}, \frac{2r^2}{\rho})$ the metric $g_v$ has the same order as standard metric. For any complex structure on $\Sigma_i$ which is fixed on the gluing region, it induces a complex structure on $\Sigma_i\#_{v_x} \Sigma_j$. If we start from the complex structure of $\tilde{\Sigma}$, by repeating above process for each nodal point we construct a marked Riemann surface $\tilde{\Sigma}_v$. Clearly, $\tilde{\Sigma}_0=\tilde{\Sigma}$. \vskip 0.1in \noindent {\bf Remark 3.7: }{\it The reader may wonder why we glue in a disc of radius $r^2$ instead of $r$. The reason is a technical one. If we use $r$, the gluing map is only continuous at $r=0$. Using $r^2$, we can show that the gluing map is $C^1$ at $r=0$.} \vskip 0.1in Let $$\tilde{O}_{f}=\prod_i O_i\times {\bf C}_{f}.\leqno(3.21)$$ The previous construction yields a universal family $$\tilde{U}_{f}=\cup \{ \tilde{\Sigma}_v; \tilde{\Sigma}\in \prod_i O_{ f}, v\in {\bf C}_{f} \mbox{ small }\}.\leqno(3.22)$$ The projection $$\pi_{f}: \tilde{U}_{f}\rightarrow \tilde{O}_{f}\leqno(3.23)$$ maps $\tilde{\Sigma}_v$ to $(\tilde{\Sigma},v)$. We still need to show that (3.23) is $stb_{f}$-equivariant. $\prod_i stb_i$ induces an obvious action on (3.23). There are other types of automorphisms of $\Sigma$ by switching the different components and $stb_{f}$ is a finite extension of $\prod_i stb_i$ by such automorphisms. The gluing construction with perhaps different gluing parameter is clearly commutative with such automorphisms. Hence, $stb_{f}$ acts on (3.23). $(\tilde{U}_{f}, \tilde{O}_{f})/stb_{f}$ is the local deformation of domain we need. After we stabilize the unstable component, $\tilde{\Sigma}_v$ should be viewed as an element of $\overline{{\cal M}}_{g,k+l}$, where $l$ is the number of extra marked points. Hence, $\tilde{O}_{f}\subset \overline{{\cal M}}_{g,k+l}$ and $\tilde{U}_{f}$ is just the local universal family of $\overline{{\cal M}}_{g,k+l}$. Forgetting the extra marked points, we map $\tilde{O}_{f}$ to $\overline{{\cal M}}_{g,k}$ by the map $$\pi_{k+l}: \overline{{\cal M}}_{g,k+l}\rightarrow \overline{{\cal M}}_{g,k}\leqno(3.24)$$ Suppose that the extra marked points are $e^v_1,\cdots, e^v_l$. Sometimes, we also use notation $e^f_1, \cdots e^f_l$. To describe a neighborhood of $f$, without the loss of generality, we can assume that $dom(f)=\Sigma_1 \wedge \Sigma_2$ and $f=(f_1, f_2)$, where $\Sigma_1, \Sigma_2$ are marked Riemann surfaces of genus $g_i$ and $k_i+1$ many marked points such that $g=g_1+g_2, k=k_1+k_2$. Furthermore, suppose that $\Sigma_1, \Sigma_2$ intersects at the last marked points $p, q$ of $\Sigma_1, \Sigma_2$ respectively. The general case is identical and we just repeat our construction for each nodal point. In this case, the gluing parameter $v$ is a complex number. We choose $v$ small enough such that marked points other than $p,q$ are away from the gluing region described above. Let $f_1(p)=f_2(q)=y_0\in V \subset Y$. Let $U_{P(y_0)}$ be a small neighborhood of $P(y_0)\in X$. We can assume that $P^{-1}(U_{y_0})=V\times U_{P(y_0)}$ and $y_0=(x_0, x_1)$. Suppose that the fiber exponential map $exp: T_{x_0} V\rightarrow V \times \{x\}$ is a diffeomorphism from $B_{\epsilon}( x_0, T_{x_0}V)$ onto its image for any $x\in U_{P(y_0)}$, where $B_{\epsilon}$ is a ball of radius $\epsilon$. Furthermore, we define $$f^w=exp_f w.\leqno(3.25)$$ Next, we construct attaching maps which define the topology of $\overline{{\cal B}}_A (Y,g,k)$. First we construct a neighborhood ${\cal U}_{f,D}/stb_{f}$ of $f\in {\cal B}_D(Y,g,k)$. Recall that if $dom(f)=\Sigma$ is an irreducible stable marked Riemann surface, then a neighborhood of $f$ can be described as $$O_{f}\times \{f^w; w\in \Omega^0(f^*T_FY), ||w||_{L^p_1}< \epsilon\} \leqno(3.26)$$ divided by $stb_{f}$. If $\Sigma$ is unstable, we needs to find a slice $W_{f}$. By lemma 3.4, we mark additional points $e^f_i$ on $\Sigma$ such that $\Sigma$ has three marked points. We call the resulting Riemann surface $\bar{\Sigma}$. Furthermore, we choose $e^f_i$ such that $df_{e^f_i}$ is of maximal rank. Then, $$W_f=\{f^w; w\in \Omega^0(f^*T_FY); ||w||_{L^p_1}<\epsilon, ||w||_{C^{1}(D_{\delta_0}(g(e_i)))}< \epsilon, g\in sbt_f, w\perp E_{e^f_i}\}.\leqno(3.28)$$ If $dom(f)=\Sigma_1\wedge \Sigma_2$ joining at $p\in\Sigma_1, q\in \Sigma_2$ and $f=f_1\wedge f_2$, We define $$\Omega^0(f^*T_FY)=\{(w_1, w_2)\in \Omega^0(f^*_1T_FY)\times \Omega^0(f^*_2T_FY); w_1(p)=w_2(q), w\perp E_{e^f_i}\}.\leqno(3.29)$$ A neighborhood of $f$ in ${\cal B}_D(Y,g,k)$ is $$\prod_i O_i\times \{f^w; w\in \Omega^0(f^*T_FY), ||w||_{L^p_1}<\epsilon, ||w||_{C^{1}(D_{\delta_0}(g(e_i)))}< \epsilon, g\in sbt_f, w\perp E_{e^f_i}\}/stb_f. \leqno(3.30)$$ If $dom(f)$ is an arbitrary configuration, we repeat above construction over each nodal point to define $\Omega^0(f^*T_FY)$. A neighborhood of $f$ in ${\cal B}_D(Y, g,k)$ is $${\cal U}_{f,D}=\prod_i O_i\times \{f^w; w\in \Omega^0(f^*T_FV), ||w||_{L^p_1}<\epsilon, ||w||_{C^{1}(D_{\delta_0}(g(e_i)))}< \epsilon, g\in sbt_f, w\perp E_{e^f_i}\}/stb_f.\leqno(3.31)$$ We want to construct an attaching map $$\bar{f}^{w,v}: {\cal U}_{f,D}\times {\bf C}^{\epsilon}_f\rightarrow \overline{{\cal B}}_A(Y,g,k)$$ invariant under $stb_f$, where ${\bf C}^{\epsilon}_f$ is a small $\epsilon$-ball around the origin of ${\bf C}_f$. We simply denote $$\bar{f}^v=\bar{f}^{0,v}.\leqno(3.32)$$ Again, let's focus on the case that $D=\Sigma_1\wedge \Sigma_2$ and the general case is similar. Recall the previous set-up. $f_1(p)=f_2(q)=y_0=(x_0, x_1)\in V \subset Y$. Let $U_{P(y_0)}$ be a small neighborhood of $P(y_0)\in X$. We can assume that $P^{-1}(U_{y_0})=V\times U_{P(y_0)}$ and $y_0=(x_0, x_1)$. Suppose that the fiber exponential map $exp: T_{x_0,x} V \rightarrow V \times \{x\}$ is a diffeomorphism from $B_{\epsilon}(x_0, T_{x_0}V)$ to its image for any $x\in U_{P(y_0)}$. In the construction of $dom(f)_v$, we can choose $r$ small enough such that $$f^w_1(D_p(\frac{2r^2}{\rho})), f^w_2(D_q(\frac{2r^2}{\rho})) \subset B_{\epsilon}(x_0, T_{x_0} V)\times P(y'_1),$$ for any $w\in \Omega^0(f^*T_FY)$ and $||w||_{C^{1}}<\epsilon$. Following \cite{MS}, we choose a special cut-off function as follows. Define $\beta_{\rho}$ to be the involution of the function $$1-\frac{log(t)}{log \rho}.\leqno(3.33)$$ for $t\in [\rho, 1]$ and equal to $0, 1$ for $t<\rho, t>1$ respectively. This function has the property that $$\int |\bigtriangledown \beta|^2<\frac{C}{-log \rho}.$$ Such a cut-off function was first introduced by Donaldson and Kroheimer \cite{DK} in 4-dimension case. We refer to \cite{DK}, \cite{MS} for the discussion of the importance of such a cut-off function. Then, we define $$\bar{\beta}_r(t)=\beta(\frac{2t}{r^2}), \leqno(3.34)$$ which is a cut-off function for the annulus $N_p(\frac{\rho r^2}{2}, \frac{r^2}{2}).$ Clearly, $\bar{\beta}_r$ is the convolution of the function $$1-\frac{log(\frac{2t}{r^2})}{log \rho}.\leqno(3.35)$$ Let $\Sigma^w=dom(f^w)$, where we have already marked the extra marked $e^v_1, \cdots, e^v_l$ to stabilize the unstable components. Then, we define $$f^{v,w}: \Sigma^w_v\rightarrow Y$$ as $$f^{v,w}=\left\{\begin{array}{ll} f^w_1(x); x\in \Sigma_1-D_p(\frac{2r^2}{\rho})\\ \bar{\beta}_r(t)(f^w_1(s, t)-y_w)+f_2(\theta+s, \frac{r^4}{t}); x=re^{i\theta}\in N_p(\frac{\rho r^2}{2}, \frac{r^2}{2})\cong N_q(2r^2, \frac{2r^2}{\rho})\\ f^w_1(s,t)+f^w_2(\theta+s, \frac{r^4}{t})-y_w; x=re^{i\theta}\in N_p(\frac{r^2}{2}, 2r^2)\cong N_q(\frac{r^2}{2}, 2r^2)\\ \bar{\beta}_r(t)(f^w_2(s, t)-y_w)+f_1(\theta+s, \frac{r^4}{t}); x=re^{i\theta}\in N_q(\frac{\rho r^2}{2}, r^2)\cong N_p(r^2, \frac{2r^2}{\rho})\\ f^w_2(x); x\in \Sigma_2-D_q(\frac{2r^2}{\rho}) \end{array}\right. \leqno(3.36)$$ where $y_w=f^w_1(p)=f^w_2(q).$ To get an element of $\overline{{\cal B}}_A(Y,g,k)$, we have to view $f^{w,v}$ as a function $\pi_{k+l}(\tilde{\Sigma}_v)$ by forgetting the extra marked points. We denote it by $\bar{f}^{w,v}$. There is a right inverse of the map $f^{v,w}$ defined as follows. Suppose that $$f: \Sigma^w_v\rightarrow Y.\leqno(3.37)$$ Let $\tilde{beta}_r(t)$ be a cut-off function on the interval $(\frac{r^2}{2}, 2r^2)$, which is symmetry with respect to $t=r^2$. Namely, $$\tilde{\beta}_r(t)=1-\tilde{\beta}_r(-2t+3r^2), \mbox{ for $t<r^2$ }.$$ We define $$f_v=(f^1_v, f^2_v): \Sigma^w_1\wedge \Sigma^w_2\rightarrow Y.\leqno(3.38)$$ by $$f^1_v=\left\{\begin{array}{lll} f(x); x\in \Sigma_1-D_p(2r^2)\\ \tilde{\beta}_r(f(x)-\frac{1}{2\pi r^2}\int_{S^1} f(s, r^2))+ \frac{1}{2\pi r^2}\int_{S^1} f(s, r^2); x\in D_p(2r^2) \end{array} \right. \leqno(3.39)$$ $$f^2_v=\left\{\begin{array}{lll} f(x); x\in \Sigma_2-D_q(2r^2)\\ \tilde{\beta}_r(f(x)-\frac{1}{2\pi r^2}\int_{S^1} f(s, r^2))+ \frac{1}{2\pi r^2}\int_{S^1} f(s, r^2); x\in D_q(2r^2) \end{array} \right. \leqno(3.40)$$ Roughly speaking, we cut the $f$ over the annulus with $\frac{r^2}{2}<t<2r^2.$ By the construction, the attaching map is really the composition of two maps. The intermediate object is $${\cal U}_f=\bigcup_{\tilde{\Sigma}_v\in \tilde{O}_{f}}\{exp_{f^v}\{w\in \Omega^0((f^v)^*T^*_FY); w\perp E_{e^f_i}, ||w||_{L^p_1}<\epsilon, ||w||_{C^{1}(D_{\delta_0}(g(e_i)))}< \epsilon, g\in sbt_f\}\}.\leqno(3.41)$$ ${\cal U}_f$ is clearly a stratified Frechet V-manifold. Then, $$f^{.,.}: {\cal U}_{f,D}\times {\bf C}^{\epsilon}_f\rightarrow {\cal U}_f\leqno(3.42)$$ and $$\bar{\{.\}}: {\cal U}_f\rightarrow \overline{{\cal B}}_A(Y,g,k).\leqno(3.43)$$ Let $\tilde{{\cal U}}_f=Im({\cal U}_f)$ under $\bar{\{.\}}$. The different gluing parameters give rise to different $\tilde{\Sigma}_v\in \overline{{\cal M}}_{g, k+l}$. However, we want to study the injectivity of attaching map, where we have to consider $\bar{f}^{w,v}$. It would be more convenient to construct $\pi_{k+l}( \tilde{\Sigma}_v)$ directly. We shall give such an equivalent description of gluing process. Recall that the domain of a stable map can be constructed by first adding a chain of ${\bf P}^1$'s to separate double point and then add trees of ${\bf P}^1$'s. Now we distinguish principal components and bubble components in our construction. We first glue the principal components. In this case, the different gluing parameters give rise to the different marked Riemann surfaces. Then, we glue the maps according to formula (3.36). When we glue a bubble component, we gives an equivalent description. Suppose that $\Sigma_i$ is a stable Riemann surface and $\Sigma_j$ is a bubble component. Moreover, $\Sigma_i, \Sigma_j$ intersects at $p\in \Sigma_i, q\in \Sigma_j$. Suppose that the gluing parameter is $v=re^{i\theta}$. We can view the previous construction as follow. We cut off the balls $D^p_i\subset \Sigma_i, D^q_j\subset \Sigma_j$ of radius $\frac{2r^2}{\rho}$ centered at marked points we want to glue. The complement $\Sigma_j-D_j$ is conformal equivalent to a ball of radius $\frac{2r^2}{\rho}$. Then, we glue back the disc along the annulus by rotating angel $\theta$. Clearly, this is just a different parameterization of $\Sigma_i$. But we do obtain a holomorphic map from $\Sigma_i\#_v \Sigma_j$ to $\Sigma_i$. Furthermore, we obtain a local universal family $$\bar{\tilde{U}}_{f}\rightarrow \bar{\tilde{O}}_{f}\leqno(3.44)$$ of $\Sigma=dom(f)$ as an element of $\overline{{\cal M}}_{g,k}$. Although $\Sigma_i\#_v \Sigma_j$ is just $\Sigma_i$ in our alternative gluing construction, the different gluing parameters may give different maps. Let $\tau_v$ be the composition of rescaling and rotation conformal transformations described above. Let $e_{i}$ be the marked points of $\Sigma_j$ other than $q$. We observe that $\tau_v$ rescaled $|df(e_i)|$ at the order $\frac{1}{r^2}$. Then, we repeat above construction for each bubble component. \vskip 0.1in \noindent {\bf Lemma 3.8: }{\it Suppose that $\bar{f}^{v,w}=\bar{f}^{v',w'}$. Then, $$v=v'. \mbox{ mod }(stb_f) \leqno(3.45)$$} \vskip 0.1in As we mentioned above, $\Sigma^w_v\neq \Sigma^{w'}_{v'}$ if $v\neq v'$. If $\pi_{k+l}(\Sigma^w_v)\neq \pi_{k+l}(\Sigma^{w'}_{v'})$, $$\bar{f}^{v,w}\neq \bar{f}^{ v', w'}\leqno(3.46)$$ by the definition. If $\pi_{k+l}(\Sigma^w_v)=\pi_{k+l}(\Sigma^{w'}_{v'})$, there are two possibilities. Since $\pi_{k+1}(\Sigma^w_v)$ is the quotient of $\bar{\Sigma}^w_v$ by $stb_{\bar{\Sigma}^w_v}$, either $\bar{\Sigma}^w_v=\bar{\Sigma}^{w'}_{v'}$ or they are different by an element of $stb_{\bar{\Sigma}^w_v}\subset stb_f$. Since the attaching map is invariant under $stb_f$, we can apply this element to $(w', v')$. Therefore, we can just simply assume that $\bar{\Sigma}^w_v=\bar{\Sigma}^{w'}_{v'}$. On the other hand, $\Sigma^w_v$ is just $\bar{\Sigma}^w_v$ with additional marked points $e^v_1, \cdots, e^v_l$. Then, it is enough to show that $$e^v_i=e^{v'}_i. \mbox{ mod }(stb_f) \leqno(3.47)$$ Suppose that $\Sigma_j$ contains extra marked point $e_s$. We choose small $r$ such that $$\frac{1}{r^2}>>\frac{max \{|df^w_1| |df^{w'}_1|\}}{min \{|df^{w}_2(e_s)|, |df^{w'}_2(e_{s})|\}}.$$ When $\epsilon$ is small, $|df^{w}_2(e_s)|, |df^{w'}_2(e_s)|>0$. Therefore, we can assume that $$|d(\tau_vf)^{w}_2(e_s)|, |d(\tau_v)f^{w'}_2(e_s)|>max \{|df^w_1|, |df^{w'}_1|\}. \leqno(3.48)$$ Hence, $\tau(e^v_{s}), \tau(e^{v'}_{s})\in D^p_i\cap D^{p'}_j, \tau\in stb_{f_i}$. Furthermore, $$\tau_v f^w=\tau_{v'}f^{w'}.\leqno(3.49)$$ on a smaller open subset $D_0$ of $D^p_i\cap D^{p'}_j$ containing $\tau(e^v_s), \tau(e^{v'}_s)$. Hence, $$f^{w'}=\tau^{-1}_{v'}\tau_v f^w.\leqno(3.50)$$ on an open set containing $e_s$. However, both $f^w, f^{w'}$ are in the slice $W_f$. Hence, (3.50) is valid for $f^w, f^{w'}$ over a component of $\Sigma_f$ containing $e^v_s, e^{v'}_s$. Hence $$\tau^{-1}_{v'}\tau_v\in stb_f.\leqno(3.51)$$ Therefore, $$e^v_s=e^{v'}_s. \mbox{ mod }(stb_f) \leqno(3.52)$$ Furthermore, we also observe that $$f^{w}=f^{w'} \mbox{ on } \Sigma-\bigcup D_{ij}. \leqno(3.53)$$ $\Box$ $\bar{.}$ is obviously invariant under $stb_f$. Moreover, \vskip 0.1in \noindent {\bf Lemma 3.9: }{\it The induced map of $\bar{\{.\}}$ from $ {\cal U}_f/stb_f$ to $\tilde{{\cal U}}_f\subset \overline{{\cal B}}_A(Y,g,k)$ is one-to-one. Furthermore, the intersection of $\tilde{{\cal U}}_f$ with each strata is open and homeomorphic to the corresponding strata of ${\cal U}_f$.} \vskip 0.1in {\bf Proof:} Let $${\cal V}_f={\cal U}_{f,D}\times {\bf C}_f.$$ By (3.39), (3.40), $f^{v,w}$ is onto. Suppose that $\bar{f}^{v,w}=\bar{f}^{v',w'}$. By the Lemma 3.8, $v=v'$ mod($stb_f$). Therefore, we can assume that $v=v'$. Moreover, we can assume that $\Sigma^w_v=\Sigma^{w'}_{v'}.$ However, it is obvious that $$\bar{.}: Map^F_A(\Sigma^w_v)\rightarrow \overline{{\cal B}}_A(Y,g,k)$$ is injective. So we show that $$f^{w,v}=f^{w',v'}.\leqno(3.54)$$ To prove the second statement, let $w_0\in \Omega^0(f^*T_FY)$ with $w_0\perp E_{e^f_i}$. For any map close to $\bar{f}^{v, w_0}$, it is of the form $f^{v, w_0+w}$ with $||w||_{L^p_1}< \epsilon, ||w||_{L^p_1}<\epsilon, ||w||_{C^{1}(D_{\delta_0}(g(e_i)))}< \epsilon$. We want to show that we can perturb $e^f_i$ such that $$w_0+w\perp E_{e^f_i}.\leqno(3.55)$$ The argument of Lemma 3.4 applies. Now we define the topology of $\overline{{\cal B}}_A(Y,g,k)$ by specifying the converging sequence. \vskip 0.1in \noindent {\bf Definition 3.10: }{\it A sequence of stable maps $f_n$ converges to $f$ if for any $\tilde{{\cal U}}_f$, there is $N>0$ such that if $n>N$ $f_n\in \tilde{{\cal U}}_f$. Furthermore, $f_n$ converges to $f$ in $C^{\infty}$-topology in any compact domain away from the gluing region.} \vskip 0.1in \noindent {\bf Proposition 3.11: }{\it If a sequence of stable holomorphic maps weakly converge to $f$ in the sense of \cite{RT1}, they converge to $f$ in the topology defined in the Definition 3.8. } \vskip 0.1in The proof is delayed after Lemma 3.18. \vskip 0.1in Define $$\chi: \overline{{\cal B}}_A(Y,g,k)\rightarrow \overline{{\cal M}}_{g,k} \leqno(3.56)$$ by $\chi(f)=\pi_{k+l}(dom(f)).$ \vskip 0.1in \noindent {\bf Corollary 3.12: }{\it $\chi$ is continuous.} \vskip 0.1in The proof follows from the definition of the topology of $\overline{{\cal B}}_A(Y,g,k)$. \vskip 0.1in \noindent {\bf Theorem 3.13: }{\it $\overline{{\cal B}}_A(Y,g,k)$ is Hausdorff.} \vskip 0.1in {\bf Proof: } Suppose that $f\neq f'$. By the corollary 3.12, we can assume that $\pi_{k+l}(dom(f))=\pi_{k+l}(dom(f')$. We want to show that $\tilde{{\cal U}}_f\cap \tilde{{\cal U}}_{f'}=\emptyset$ for some $\epsilon$. Suppose that it is false. We claim that $dom(f), dom(f')$ have the same topological type. Namely, $f,f'$ are in the same strata. We start from the underline stable Riemann surfaces $\pi_{k+l}(dom(f))=\pi_{k+l}(dom(f'))$ which are the same by the assumption. We want to show that they always have the same way to attach bubbles to obtain $dom(f), dom(f')$. Suppose that we attach a bubble to $\pi_{k+l}(dom(f))$ at $p$. Recall that the energy concentrates at $D_p(\frac{2r^2}{\rho})$,i.e., $\int_{D_p(\frac{2r^2}{\rho})}|df|^2\geq \epsilon_0$. The same is true for $f^{w,v}$ when $||w||_{L^p_1}<\epsilon$. On the other hand, we have the same property for $(f')^{w',v'}$ for some $||w'||_{L^p_1}, |v'|<\epsilon$. If $\bar{f}^{w,v}=\bar{f'}^{w',v'}$, $f'$ must have a bubbling point in $D_p(\frac{2r^2}{\rho})$. In fact, the bubbling point must be $p$. Otherwise, we can construct a small ball $D_p(\frac{2r^2}{\rho})$ containing no bubbling points of $f'$. Then, we proceed inductively on the next bubble. Now the energy concentrates at a ball of radius $r^2r^2_1$, where $r_1=|v_1|$ is the next gluing parameter. By the induction, we can show that $dom(f), dom(f')$ have the same topological type. In fact, we proved that $dom(f), dom(f')$ have the same bubbling points and hence the same holomorphic type. Suppose that $f, f'\in {\cal B}_D(Y,g,k)$. Then, some component of $f,f'$ are different. Suppose that the component $f_i\neq f'_i$, where $f_i, f'_i\in {\cal B}_{A_i}(Y,g,k)$. Note that $f^v_i$ is equal to $f$ outside the gluing region. $f^v\neq (f')^v$ for small $v$. By Lemma 3.4, ${\cal B}_{A_i}(Y,g,k)$ is Hausdorff and the neighborhoods of $f_i, f'_i$ are described by slice $W_{f_i}, W_{f'_i}$ for a small constant $\epsilon$. Add extra marked points to stabilize unstable components. $||f_i-f'_i||_{L^p_1}\geq 2\epsilon$ for small $\epsilon$. Then, it is obvious that $$W_{f_i}\cap W_{f'_i}=\emptyset.\leqno(3.57)$$ Note that $f^{w,v}(e_0)=f^w(e_0), f^{w',v'}(e_0)=f^{w'}(e_0)$. It is straightforward to check that $$\tilde{{\cal U}}_f \cap \tilde{{\cal U}}_{f'}=\emptyset\leqno(3.58)$$ for the same $\epsilon$. This is a contradiction. $\Box$ \vskip 0.1in \noindent {\bf Corollary 3.14: }{\it $\overline{{\cal M}}_A(Y,g,k)$ is Hausdorff.} \vskip 0.1in To construct the obstruction bundle $\overline{{\cal F}}_A(Y,g,k)$, we start from the top strata ${\cal B}_A(Y,g,k)$. Let ${\cal V}(Y)$ be vertical tangent bundle. With an almost complex structure $J$, we can view ${\cal V}(Y)$ as a complex vector bundle. Therefore, for each $f\in {\cal B}_A(Y,g,k)$ we can decompose $$\Omega^1(f^*{\cal V}(Y))=\Omega^{1,0}(f^*{\cal V}(Y))\oplus \Omega^{0,1}(f^*{\cal V}(Y)). \leqno(3.59)$$ Both bundles patch together to form Frechet V-bundles over ${\cal B}_A(Y,g,k)$. We denote them by $\Omega^{1,0}({\cal V}(Y)), \Omega^{0,1}({\cal V}(Y))$. Then, $${\cal F}_A(Y,g,k)=\Omega^{0,1}({\cal V}(Y)).\leqno(3.60)$$ For lower strata ${\cal B}_D(Y,g,k)$, ${\cal B}_D(Y,g,k)\subset \prod_i {\cal B}_{A_i}(Y,g_i, k_i)$, where ${\cal B}_{A_i}(Y,g_i,k_i)$ are components. When a component is stable, we already have an obstruction bundle ${\cal F}_{A_i}(Y,g,k)$. When the i-th component is unstable, we first form the obstruction bundle over $Map^F_{A_i}( Y,,0,k_i)$ in the same way and divide it by $Aut_i$. In the quotient, we obtain a V-bundle denoted by $\Omega^{0,1}({\cal V}(Y))$. Let $$i: {\cal B}_D(Y,g,k)\rightarrow \prod_i {\cal B}_{A_i}(Y,g_i,k_i)\leqno(3.61)$$ be inclusion. We define $${\cal F}_D(Y,g,k)=i^*\prod_i {\cal F}_{A_i}(Y,g_i,k_i).\leqno(3.62)$$ Finally, we define $$\overline{{\cal F}}_A(Y,g,k)|_{{\cal B}_D(Y,g,k)}={\cal F}_D(Y,g,k). \leqno(3.63)$$ For any $f\in {\cal B}_D(Y,g,k)$, consider a chart $({\cal U}_f, V_f, stb_f)$. Suppose that $D=\Sigma_1 \wedge \Sigma_2$. For $\eta^w\in \Omega^{0,1}((f^w)^*{\cal V}(Y))$, define $$\eta^{w,v}\in \Omega^{0,1}((f^{w,v})^*{\cal V}(Y))$$ by $$\eta^{w, v}=\left\{\begin{array}{ll} \eta_1(x); x\in \Sigma_1-D_p(\frac{2r^2}{\rho})\\ \bar{\beta}_r(t)\eta_1(s, t)+\eta_2(\theta+s, \frac{r^4}{t} ); x=te^{is}\in N_p(\frac{\rho r^2}{2},\frac{r^2}{2})\cong N_q(2r^2, \frac{2r^2}{\rho})\\ \eta_1(s,t)+\eta_2(\theta+s, \frac{r^4}{t}); x=te^{is}\in N_p(\frac{r^2}{2},2r^2) \cong N_q(\frac{r^2}{2},2r^2)\\ \bar{\beta}_r(t)\eta_2(s, t)+\eta_1(\theta+s, \frac{r^4}{t} ); x=te^{is}\in N_q(\frac{\rho r^2}{2}, \frac{r^2}{2})\cong N_p(2r^2, \frac{2r^2}{\rho})\\ \eta_2; x\in \Sigma_2-D_q(\frac{2r^2}{\rho}) \end{array}\right. \leqno(3.64)$$ $\bar{\partial}_J$ is clearly a continuous section of $\bar{{\cal F}}_A(Y,g,k,J)$. Let $\bar{\partial}_{J, D}$ be the restriction of $\bar{\partial}_J$ over ${\cal B}_D$. Next, we define the local sections by repeating the constructions in section 2. Let $f\in {\cal B}_D(Y,g,k)$. $Coker D_f \bar{\partial}_{J,D}$ is a finite dimensional subspace of $\Omega^{0,1}(f^*{\cal V}(Y))$ invariant under $stb_f$. We first choose a $stb_f$-invariant cut-off function vanishing in a small neighborhood of the intersection points. Then we multiple it to the element of $Coker D_f \bar{ \partial}_{J,D}$ and denote the resulting finite dimensional space as $F_f$. By the construction, $F_f$ is $stb_f$-invariant. When the support of the cut-off function is small, $F_f$ will have the same dimension as $Coker D_f \bar{\partial}_J$ and $$D_f\bar{\partial}_{J,D}+Id: \Omega^0(f^*T_FY)\oplus F_f\rightarrow \Omega^{0,1}( f^*{\cal V}(Y))$$ is surjective. We first extend each element $s$ of $F_f$ to a smooth section $s^w\in \Omega^{0,1}((f^w)^*{\cal V}(Y))$ of ${\cal F}_D(Y,g,k,J)$ supported in $U_{f,D}$ such that it's value vanishes in a neighborhood of the intersection points. Hence, $s^w$ can be naturally viewed as an element of $\Omega^{0,1}((f^{w,v})^*{\cal V}(Y))$ supported away from the gluing region. Let $\beta_f$ be a smooth cut-off function on a polydisc ${\bf C}_f$ vanishing outside of a polydisc of radius $2\delta_1$ and equal to 1 in the polydisc of radius $\delta_1$. One can construct $\beta_f$ by first constructing such $\beta$ over each copy of gluing parameter ${\bf C}_{x}$ and then multiple them together. We now extend $s^w$ over ${\cal U}_f$ by the map $$s^{v}_c(f^{w,v})=\beta_f(v)s^{w}.\leqno(3.65)$$ Then, we use the method of the section 2 (2.5) to extend the identity map of $F_f$ to a map $$s_f: F_f \rightarrow \overline{{\cal F}}_A(Y,g,k)|_{{\cal U}_f}.\leqno(3.65.1)$$ invariant under $stb_f$ and supported in ${\cal U}_f$. Then, it descends to a map over $\overline{{\cal B}}_A(Y,g,k)$. We will use $s_f$ to denote the induced map on $\overline{{\cal B}}_A(Y,g,k)$ as well. We call such $s_f$ {\em admissible}. Our new equation will be of the form $${\cal S}_e=\bar{\partial}_f+\sum_i s_{f_i}: {\cal E} \rightarrow \overline{{\cal F}}_A(Y,g,k,J), \leqno(3.66)$$ where $s_{f_i}$ is admissible. We observe that the restriction ${\cal S}_D$ of ${\cal S}$ over each strata is smooth. Let $U_{{\cal S}_e}=({\cal S}_e)^{-1}(0)$ and $$S: U_{{\cal S}_e}\rightarrow E.$$ \vskip 0.1in {\bf Lemma 3.15: }{\it $S$ is a proper map.} \vskip 0.1in {\bf Proof: } Since the value of $s_{f_i}$ is supported away from the gluing region, the proof of lemma is completely same as the case to show that the moduli space of stable holomorphic maps is compact. We omit it. $\Box$ For $f\in {\cal B}_D(Y,g,k)$, we define the tangent space $$T_f\overline{{\cal B}}_A(Y,g,k)=T_f {\cal B}_D(Y,g,k)\times {\bf C}_f$$ and the derivative $$D_{f,t}{\cal S}_e=D_{f,t} {\cal S}_e|_{{\cal B}_D(Y,g,k)}: T_f\overline{{\cal B}}_A(Y,g,k)\rightarrow \Omega^{0,1}(f^* {\cal V}(Y)).\leqno(3.67)$$ \vskip 0.1in \noindent {\bf Lemma 3.16: }{\it $$Ind D_{f,t}{\cal S}=2C_1(V)(A)+2(3-n)(g-1)+2k+\dim X+dim E.\leqno(3.68)$$ } \vskip 0.1in {\bf Proof:} $$D_{f,t}{\cal S}_D(W,u)=D_f\bar{\partial}_J(W)+\sum_i D_{f,t}s_{f_i}(W,u). \leqno(3.69)$$ $$Ind D_{f,t}{\cal S}_D=Ind D_f\bar{\partial}_J + dim E.$$ If $\Sigma_f=dom(f)$ is irreducible, the lemma follows from Riemann-Roch theorem. Suppose that $\Sigma_f=\Sigma_1\wedge \Sigma_2$ and $f=(f_1, f_2)$ with $f_1(p)=f_2(q)$. $$\begin{array}{lll} Ind D_f\bar{\partial}_J&=&Ind D_{f_1}\bar{\partial}_J+Ind D_{f_2}\bar{ \partial}_J-dim Y\\ &=&2C_1(V)([f_1])+2(3-n)(g_1-1)+2(k_1+1)+\dim X+2C_1(V)([f_2])+2(3-n)(g_2-1)\\ &&+2(k_2+1)+\dim X-\dim Y\\ &=&2C_1(V)(A)+2(3-n)(g-1)+2k+\dim X-6+2n+2-2n\\ &=&2C_1(V)(A)+2(3-n)(g-1)+2k+\dim X-2 \end{array}$$ Adding the dimension of gluing parameter, we derive Lemma 3.16. The general case can be proved inductively on the number of the components of $\Sigma_f$. We omit it. This is the end of the construction of the extended equation. Next, we shall prove that $$(\overline{{\cal B}}_A(Y,g,k), \overline{{\cal F}}_A(Y,g,k), \bar{\partial}_J)\leqno(3.70)$$ is VNA. The openness of ${\cal U}_S=\{(x,t); Coker D_{f,t}{\cal S}_e=\emptyset\}$ is a local property. To prove the second property, we first construct a local coordinate chart for each point of virtual neighborhood. Then, we prove that the local chart patches together to form a $C^1$-V-manifold. The construction of a local coordinate chart is basically a gluing theorem. The first gluing theorem for pseudo-holomorphic curve was given by \cite{RT1}. There were two new proofs by \cite{Liu}, \cite{MS} which are more suitable to the set-up we have here. Here we follow that of \cite{MS}. For reader's convenience, we outline the proof here. We need to enlarge our space to include Sobolev maps. Suppose that $f\in {\cal M}_D(Y,g,k), t_0\in {\bf R}^m$ such that ${\cal S}_e(f,t_0)=0$ and $Coker {\cal D}_{f,t_0}{\cal S}_e=0$. Choose metric $\lambda$ on $\Sigma_1\wedge \Sigma_2$. Using the trivialization of (3.18), we can define Sobolev norm on ${\cal U}_{f,D}$. Let $$L^p_1({\cal U}_{f,D})=U_{\Sigma}\times\{f^{w}; w\in \Omega^0(f^*T_FY), ||w||_{L^p_1}< \epsilon, ||w||_{C^1(D_{\delta_0}(g(e_i)))}< \epsilon, w\perp E_{e^f_i}\}.\leqno(3.71)$$ By choosing small $\delta_0$, we can assume that $D_{\delta_0}(e_i)$ is away from gluing region. For the rest of this section, we assume that $2<p<4$. Then, $L^p_1({\cal U}_{f,D})$ is a Banach manifold. To simplify the notation, we shall assume that $dom(f)=\Sigma_1\wedge \Sigma_2$ for the argument below. However, it is obvious that the same argument works for the general case. Let $\lambda_v$ be the metric on $\Sigma_v$ defined in (3.20). We use $L^p_v, L^p_{1,v}$ to denote the Sobolev norms on $\Sigma_v$, where $v$ is used to indicate the dependence on $v$. By \cite{MS} (Lemma A.3.1), the Sobolev constants of the metric $\lambda_v$ are independent of $v$. Let $$L^p_1({\cal U}_f)=\bigcup_{\tilde{\Sigma}_v}\{f^{v,w}; w\in \Omega^0((f^v)^*T_FY), w\perp E_{e^f_i}, ||w||_{L^p_{1,v}}<\epsilon, ||w||_{C^1(D_{\delta_0}(g(e_i)))}< \epsilon\}.\leqno(3.72)$$ First of all, the map $$f^{w,v}: {\cal U}_{f,D}\times {\bf C}_f \rightarrow {\cal U}_f$$ induces a natural map $$\phi_f: \Omega^0((f^w)^*T_FY)\rightarrow \Omega^0((f^{w,v})^*T_FY)$$ by the formula $$u^{w,v}=\phi_f(u)=\left\{\begin{array}{ll} u_1(x); x\in \Sigma_1-D_p(\frac{2r^2}{\rho})\\ \bar{\beta}_r(t)(u_1(s, t))-u_1(0))+u_2(\theta+s, \frac{ r^4}{t}); x=re^{is}\in N_p(\frac{\rho r^2}{2}, \frac{r^2}{2})\cong N_q(2r^2, \frac{2r^2}{\rho})\\ u_1(s,t)+u_2(\theta+s, \frac{r^4}{t})-u(0); x=re^{is}\in N_p(\frac{r^2}{2}, 2r^2)\cong N_q(\frac{r^2}{2}, 2r^2)\\ \bar{\beta}_r(t)(u_2(s, t))-u_2(0))+u_1(\theta+s, \frac{ r^2}{t}); x=re^{is}\in N_q(\frac{r}{2}, r)\cong N_p(r, 2r)\\ u_2(x); x\in \Sigma_2-D_q(2r) \end{array}\right. \leqno(3.73)$$ where $u=(u_1, u_2)\in \Omega^0((f^w)^*T_FY)$. Notes that $u_1(0)=u_2(0)$. One can construct an inverse of $\psi_f$. For any $u\in \Omega^0((f^{w,v})^*T_FY)$, we define $$u_v=(u^1_v, u^2_v)$$ by $$u^1_v=\left\{\begin{array}{lll} u(x); x\in \Sigma_1-D_p(2r^2)\\ \tilde{\beta}_r(u(x)-\frac{1}{2\pi r^2}\int_{S^1} u(s, r^2))+ \frac{1}{2\pi r^2}\int_{S^1} u(s, r^2); x\in D_p(2r^2) \end{array} \right. \leqno(3.74)$$ $$u^2_v=\left\{\begin{array}{lll} u(x); x\in \Sigma_2-D_q(2r^2)\\ \tilde{\beta}_r(u(x)-\frac{1}{2\pi r^2}\int_{S^1} u(s, r^2))+ \frac{1}{2\pi r^2}\int_{S^1} u(s, r^2); x\in D_q(2r^2) \end{array} \right. \leqno(3.75)$$ For any $\eta\in \Omega^{0,1}((f^{w,v})^*{\cal V}(Y))$, we cut $\eta$ along the circle of radius $r^2$ and extend as zero inside the $D_p(r^2), D_q(r^2)$. We denote resulting 1-form as $\eta^f_1\in \Omega^{0,1}((f^w)^* {\cal V}(Y)), \eta^f_2\in \Omega^{0,1}(f^w)^*{\cal V}(Y))$. Clearly, $(\eta^f_1, \eta^f_2)$ is an right inverse of $\eta^{w,v}$. \vskip 0.1in \noindent {\bf Lemma 3.17: }{\it Let $u$ be a 1-form over a disc of radius $\frac{2r^2}{\rho}<1$. Then, $$||\bigtriangledown \bar{\beta}_r (u-u(0))||_{L^p}\leq c|log \rho|^{1-\frac{4}{p}}||u||_{L^p_1}. \leqno(3.76)$$ } \vskip 0.1in The inequality is just the lemma A.1.2 of \cite{MS}, where we use $r^2$ instead of $r$. \vskip 0.1in \noindent {\bf Lemma 3.18: }{\it $||\phi_f(u^w)||_{L^p_{1,v}}\leq C ||u^w||_{L^p_1}, ||u^i_v||_{L^p_1}\leq C||u||_{L^p_{1,v}}$.} \vskip 0.1in {\bf Proof: } We only have to consider $u^w$ over $N_p(\frac{\rho r^2}{ 2}, \frac{r^2}{2})$, where $$\phi_f(u_w)=\bar{\beta}_r(t)(u^w_1(s,t)-u^w_1(0))+u^w_2(s+\theta, \frac{r^4}{t}).\leqno(3.77)$$ $$\begin{array}{lll} ||\phi_f(u^w)||_{L^p(N_p(\frac{\rho r^2}{2}, \frac{r^2}{2}))}&\leq &C(||u^w_1||_{L^p_1( N_p(\frac{\rho r^2}{2}, \frac{r^2}{2}))}+||u^w_2||_{L^p_1(N_q(2r^2,\frac{2r^2}{\rho}))}+|u^w_1(0)|)\\ &\leq& C(||u^w_1||_{L^p_1(N_p(\frac{\rho r^2}{2},\frac{r^2}{2}))}+||u^w_2||_{L^p_1(N_q(2r^2,\frac{2r^2}{\rho}))}). \end{array}\leqno(3.78)$$ $$\begin{array}{lll} &&||\bigtriangledown \phi_f(u^w)||_{L^p(N_p(\frac{\rho r^2}{2},\frac{r^2}{2}))}\\ &\leq &C(||\bigtriangledown u^w_1||_{L^p_ 1(N_p(\frac{\rho r^2}{2}, \frac{r^2}{2}))}+||\bigtriangledown u^w_2||_{L^q_1(N_q(2r^2,\frac{2r^2}{\rho}))}+ ||\bigtriangledown\bar{ \beta}_r (u^w_1-u^w_1(0))||_{L^p(N_p(\frac{\rho r^2}{2}, \frac{r^2}{2}))})\\ &\leq& C||u^w||_{L^p_1(N_p(\frac{r^2} {4},\frac{r^2}{2}))}, \end{array}\leqno(3.79)$$ where the last inequality follows from Lemma 3.17. The proof of the second inequality is the same and we omit it. $\Box$ {\bf Proof of Proposition 3.11: } Suppose that $f_n\rightarrow f$ as a weakly convergent sequence of holomorphic stable maps in the sense of \cite{RT1}. Then, $f_n$ converges to $f$ in $C^{\infty}$-norm in any compact domain outside the gluing region, in particular on $D_{\delta_0}(g(e_i))$. Now, we want to show that $f_n$ is in the open set ${\cal U}_{f, D}$ for $n>N$. Note that formula (3.74,3.75) is a left inverse of formula (3.73). By Lemma 3.18, the formula (3.73) preserves $L^p_1$ norm. Hence, it is enough to show that $f_n$ is close to $f^v$ when $n$ is large. Namely, we want to estimate $||f_n-f^v||_{L^p_{1,v}}$. Outside of gluing region, $f_n$ converges to $f^v$ in the $C^{\infty}$ norm. So $||(1-\beta)(f_n-f^v)||_{L^p_{1,v}}$ converges to zero, where $\beta$ is a cut-off function vanishing outside gluing region. Over the gluing region, it is enough to show that $||\beta(f_n-pt)||_{L^p_1}$ is small where $pt$ is the intersection point of two components of $f$. Here we assume that $f$ has only two components to simplify the notation. The argument for general case is the same. By the decay estimate in \cite{RT1}(Lemma 6.10), $||f_n-pt||_{C^0}$ converges to zero over the gluing region with cylindric metric. However, $C^0$-norm is independent of the metric of domain. Hence, we have a $C^0$ estimate for the metric in this paper. Furthermore, $f_n$ is holomopophic. By elliptic estimate, $$||\beta(f_n-pt)||\leq c(||\bar{\partial}_J(\beta(f_n-pt)||_{L^p_v}+||\beta( f_n-pt)||_{C^0}\leq c(||\bigtriangledown \beta(f_n-pt)||_{L^p_v}+||f_n-pt||_{ C^0}\leq c||f_n-pt||_{C^0}.$$ We will finish the argument by showing that the constant in elliptic estimate is independent of the gluing parameter $v$. The later is easy since our metric is essentially equivalent to the metric on the anulus N(1,r) in $R^2$, where $r=|v|$ and $\beta(f_n-pt)$ is compact supported. $\Box$ Suppose that $D_{f,t_0}{\cal S}_e$ is surjective. Since ${\cal S}_e$ is smooth over ${\cal B}_D(Y,g,k)$, $D_{f^w,t}{\cal S}_e$ is surjective for $||w||_{L^p_1}<\delta, |t-t_0|<\delta$ with some small $\delta$. We choose a family of right inverse $Q_{f^w,t}$. Then, $$||Q_{f^w,t}||\leq C.\leqno(3.79.1)$$ We want to construct right inverse of $D_{f^{w,v},t}{\cal S}_e$. \vskip 0.1in \noindent {\bf Definition 3.19: }{\it Define $AQ_{f^{w,v},t}(\eta)=\phi_fQ_{f^{w},t}(\eta^f_1, \eta^f_2)$.} \vskip 0.1in Then, it was shown in \cite{MS} that \vskip 0.1in \noindent {\bf Lemma 3.20:}{\it $$||AQ_{f^{w,v},t}||\leq C, ||D_{f^{w,v},t}AQ_{f^{w,v},t}- Id||<\frac{1}{2} \mbox{ for small } r, \rho. \leqno(3.80)$$} \vskip 0.1in Now, we fix a $\rho$ such that Lemma 3.20 holds. The right inverse of $D_{f^{w,v},t}$ is given by $$Q_{f^{w,v},t}=AQ_{f^{w,v},t}(D_{f^{w,v},t}AQ_{f^{w,v},t})^{-1}. \leqno(3.81)$$ Furthermore, $$||Q_{f^{w,v},t}||\leq C.\leqno(3.82)$$ Therefore, we show that \vskip 0.1in \noindent {\bf Corollary 3.21:}{\it $${\cal U}_{{\cal S}_e}=\{(x,t); Coker D_{f,t}{\cal S}_e=\emptyset\}$$ is open.} \vskip 0.1in Next, we have an estimate of error term. \vskip 0.1in \noindent {\bf Lemma 3.23: }{\it Suppose that ${\cal S}_e(f^w)=0$. Then, $$||{\cal S}_e(f^{v,w})||_{L^p_v}\leq C r^{\frac{4}{p}}.\leqno(3.83)$$} \vskip 0.1in {\bf Proof: } It is clear that ${\cal S}_e(f^{v,w})=0$ away from the gluing region. Notes that the value of $s_{f_i}$ is supported away from the gluing region. Hence, ${\cal S}_e=\bar{\partial}_J$ over the gluing region. Then, the lemma follows from \cite{MS} (Lemma A.4.3). $\Box$ Next we construct the coordinate charts of ${\cal M}_{{\cal S}_e} \cap {\cal U}_{{\cal S}_e}$. Suppose that $(f,t_0)\in {\cal M}_{{\cal S}_e}\cap {\cal U}_{{\cal S}_e}$. By the previous argument, we can assume that some neighborhood ${\cal U}_f\times B_{\delta}(t_0)\subset {\cal U}_{{\cal S}_e}$. To simplify the notation, we drop $t$-component. It is understood that $s_{f_i}$ will not affect the argument since it's value is supported away from the gluing region. Since $L^p_1({\cal U}_{f,D})$ is a Banach manifold and the restriction to ${\cal S}_e$ is a Fredholm map, ${\cal M}_{{\cal S}_e}\cap {\cal B}_D(Y,g,k)$ is a smooth V-manifold by ordinary transversality theorem. Let $$f\in E^D_f\subset {\cal M}_{{\cal S}_e} \cap {\cal B}_D(Y,g,k)\leqno(3.84)$$ be a small $stb_f$-invariant neighborhood. \vskip 0.1in \noindent {\bf Theorem 3.24: }{\it There is a one-to-one continuous map $$\alpha_f: E^D_f \times B_{\delta_f}({\bf C}_f) \rightarrow {\cal U}_f \leqno(3.85)$$ such that $im(\alpha_f)$ is an open neighborhood of $f\in {\cal M}_{{\cal S}_e}$, where $\delta_f$ is a small constant.} \vskip 0.1in {\bf Proof:} For any $w\in E^D_f$ and small $v$, we would like to find an element $\xi(w,v)\in \Omega^0((f^v)^*T_FY)$ with $\xi\perp E_{e_i}$ and $\xi(w,v)\in Im Q_{f^{w,v}}$ such that $${\cal S}_e((f^{v,w})^{\xi(w,v)})=0.\leqno(3.86)$$ Consider the Taylor expansion $${\cal S}_e((f^{v,w})^{\xi})={\cal S}(f^{w,v})+D_{f^{w,v}}(\xi)+N_{f^{w,v}}(\xi),$$ for $w\in E^D_f, \xi\in \Omega^0((f^v)^*T_FY)$ with $\xi(e^v_i)\perp df(e^v_i), ||w||_{L^p_{1,v}}, ||\xi||_{L^p_{1,v}}< \epsilon$. Then, $$\xi(w,v)=-Q_{f^{w,v}}(S(f^{w,v})+N_{f^{w,v}}(\xi(w,v)).\leqno(3.87)$$ Hence, $\xi(w,v)$ is a fixed point of the map $$H(w,v; \xi)=-Q_{f^{w,v}}(S(f^{w,v})+N_{f^{w,v}}(\xi)).\leqno(3.88)$$ Conversely, if $\xi(w,v)$ is a fixed point, $${\cal S}_e((f^{v,w})^{\xi(w,v)})=0.\leqno(3.89)$$ $N_{f^{w,v}}$ satisfies the condition $$||N_{f^{w,v}}(\eta_1)-N_{f^{w,v}}(\eta_2)||_{L^p_v}\leq C(||\eta_1||_{L^p_{1,v}}+|| \eta_2||_{L^p_{1,v}})||\eta_1-\eta_2||_{L^p_{1,v}}.\leqno(3.90)$$ Next, we show that $H$ is a contraction map on a ball of radius $\delta/4$ for some $\delta$. $$||H(w,v; \xi)||_{L^p_{1,v}}\leq C(||{\cal S}_e(f^{w,v})||_{L^p_v}+ ||N_{f^{w,v}}(\xi)||_{L^p_v})$$ $$\leq C(r^{\frac{4}{p}}+||\xi||^2_{L^p_{1,v}})\leq \frac{\delta}{4},\leqno(3.91)$$ for $ ||\xi||_{L^p_{1,v}}\leq \frac{\delta}{4}$ and $2C\delta<1, r<(\frac{\delta^2}{4})^{ -\frac{4}{p}}$. $$||H(w,v; \xi)-H(w,v; \eta)||_{L^p_{1,v}}\leq C ||N_{f^{w,v}}(\xi)-N_{f^{w,v}}(\eta)||_{L^p_v}$$ $$\leq C (||\xi||_{L^p_{1,v}}+||\eta||_{L^p_{1,v}})||\xi-\eta||_{L^p_{1,v}}<2\delta C ||\xi-\eta||_{L^p_{1,v}}.\leqno(3.92)$$ Therefore, $H$ is a contraction map on the ball of radius $\frac{\delta}{4}$. Then, there is a unique fixed point $\xi(w,v)$. Furthermore, $\xi(w,v)$ depends smoothly on $w$. Recall that $\xi(w,v)$ is obtained by iterating $H$. One can check that $$||\xi(w,v)||_{L^p_{1,v}}\leq C r^{\frac{4}{p}}.\leqno(3.93)$$ Our coordinate chart at $f$ is $(E^D_f\times B_{\delta_f}({\bf C}_f), \alpha_f(v,w))$ where $\delta_f=(\frac{\delta^2}{4})^{\frac{4}{p}}$. and $$\alpha_f(v,w)=(f^{v,w})^{\xi(w,v)}.\leqno(3.94)$$ Notes that all the construction is $stb_f$-invariant. Hence $\alpha_f$ is $stb_f$-invariant. It is clear that $\alpha_f$ is one-to-one by contraction mapping principal. Notes that ${\cal S}_e=\bar{\partial}_J$ over the gluing region. It follows from Proposition 3.11 and uniqueness of contraction mapping principal that $\alpha_f$ is surjective onto a neighborhood of $f$ in ${\cal M}_{{\cal S}_e}$. $\Box$ Furthermore, $E^D_f\times {\bf C}_f$ has a natural orientation induced by the orientation of $J$, ${\bf R}^m$ and ${\bf C}_f$. Next, we show that the transition map is a $C^1$-orientation preserving map. In the previous argument, we expand ${\cal S}_e$ up to the second order, which is given in \cite{F}, \cite{MS}. To prove the transition map is $C^1$, we need to expand ${\cal S}_e$ up to third order. Let $z=s+it$ be the complex coordinate of $\Sigma_v$. Let $\bigtriangledown^v \xi=\bigtriangledown_t\xi+ \bigtriangledown_s\xi$ to indicate the dependence on $v$. Let $$f^{w,v}=exp_{f^v} w^v.\leqno(3.95)$$ Let $ \xi\in L^p_{1,v}(\Omega^0((f^{v})^*T_FY))$ with $||w^v||_{L^p_{1,v}}, ||\xi||_{L^p_{1,v}}\leq \delta$ for small $\delta$. A similar calculation of \cite{MS} (Theorem 3.3.4) implies $$\bar{\partial}_J(f^{v})^{w^v+\xi})=\bar{\partial}_J(f^{v,w})+D_{f^{v,w}}(\xi)+D^2_{f^{v,w}}(\xi^2) +\tilde{N}_{f^{v,w}}(\xi),\leqno(3.96)$$ where $$D_{f^{w,v}}(\xi)=\bigtriangledown^v_s \xi+J \bigtriangledown^v_t\xi+(C_1 \bigtriangledown w^v+ C_2\bigtriangledown^v f^v+C_2 \bigtriangledown^v w)\xi,\leqno(3.97)$$ $$D^2_{f^{w,v}}\xi=(C_1 \bigtriangledown^v f^v+C_2 \bigtriangledown^v w^v)\xi^2+C_3 \xi \bigtriangledown^v \xi,\leqno(3.98)$$ $$\tilde{N}_{f^{w,v}}(\xi)=(C_1\bigtriangledown^v f^v+C_2 \bigtriangledown^v w^v)\xi^3+ C_3 (\bigtriangledown^v \xi)\xi^2,\leqno(3.99)$$ where $C_1, C_2, C_3$ are smooth bounded functions for each of the identities. Furthermore, we have $$D_{(f^v)^{w^v+\tilde{w}}}(\xi)=D_{f^{w,v}}\xi+(2C_1\bigtriangledown^v f^v+2C_2 \bigtriangledown^v w^v) \tilde{w}\xi+C_3 \tilde{w}\bigtriangledown^v \xi+C_4 \tilde{w}\bigtriangledown \xi+ O(\tilde{w}^2), \leqno(3.100)$$ where the coefficients of higher order terms are independent from $\tilde{w}$ by (3.99). \vskip 0.1in \noindent {\bf Lemma 3.25: }{\it The derivative with respect to $w$ $$||\frac{\partial}{\partial w}D_{f^{v,w}}(\tilde{w})(\xi)||_{L^p_v}\leq C (||f^v||_{L^p_{1,v}}+||w^v||_{ L^p_{1,v}})||\tilde{w}||_{L^p_{1,v}} ||\xi||_{L^p_{1,v}}.\leqno(3.101)$$ $$||\frac{\partial}{\partial w}N_{f^{v,w}}(\tilde{w})(\xi)||_{L^p_v}\leq C(||f^v||_{L^p_{1,v}} +||w^v||_{L^p_{1,v}})||\tilde{w}||_{L^p_{1,v}} ||\xi||^2_{L^p_{1,v}}.\leqno(3.102)$$} \vskip 0.1in {\bf Proof: } The first inequality follows from 3.100. To prove the second inequality, recall that $$N_{f^{v,w}}(\xi)={\cal S}_e((f^{v})^{w^v+\xi})-{\cal S}_e(f^{v,w})-D_{f^{v,w}}(\xi).\leqno(3.103)$$ Hence $$\begin{array}{lll} &&N_{(f^{v})^{w^v+\tilde{w}}}(\xi)-N_{f^{v,w}}(\xi)\\ &=&{\cal S}_e(f^{v,w^v+\tilde{w}+\xi})-{\cal S}_e((f^{v})^{w^v+\xi})-({\cal S}_e((f^{v})^{w^v+\tilde{w}})- {\cal S}_e(f^{v,w}))-(D_{(f^{v})^{w^v+\tilde{w}}}(\xi)-D_{f^{v,w}}(\xi))\\ &=&D_{(f^{v})^{w^v+\xi}}(\tilde{w})-D_{f^{v,w}}(\tilde{w})-\frac{\partial}{\partial w}D_{f^{v,w}}(\tilde{w})(\xi)+O(\tilde{w}^2)\\ &=&\frac{\partial}{\partial w} D_{f^{v,w}}(\xi)(\tilde{w})-\frac{\partial}{\partial w}D_{f^{v,w}}(\tilde{w})(\xi)+O(\tilde{w}^2) \end{array}. \leqno(3.104)$$ Therefore, the second inequality follows from the first one. Next, we consider the derivative of $D, N$ with respect to the $v$. First of all, \vskip 0.1in \noindent {\bf Lemma 3.26: }{\it Let $|v-v_0|<\delta$ for small $\delta$ and $j_v$ be the complex structure on $\Sigma_v$, there is a smooth family of diffeomorphism $\Phi_v: \Sigma_{v_0}\rightarrow \Sigma_v$ such that $\Phi_v=id$ outside gluing region and $$|\frac{\partial}{\partial v}|_{v=v_0}(\Phi_v j_v (\frac{\partial}{\partial t}))|\leq \frac{C}{r_0}. \leqno(3.105)$$ $$|\frac{\partial}{\partial v}|_{v=v_0}(\Phi_v j_v (\frac{\partial}{\partial s}))|\leq \frac{C}{r_0}. \leqno(3.106)$$} \vskip 0.1in {\bf Proof: } The complex structure outside the gluing region does not change. Over the gluing region, it is conformal equivalent to a cylinder. Constructing $\Phi_v$ in the cylindric model, we will obtain the estimate of Lemma 3.26.$\Box$ Suppose that we want to estimate the derivative at $v_0$. We fix $u=f^{v_0}$ and the trivialization given by $\Phi_v$. To abuse the notation, let $f^{v,w}=exp_{f^{v_0}} w^v$. We still have the same Taylor expansion (3.96)-(3.100). Furthermore, we can estimate $\frac{\partial}{\partial v}|_{v=v_0}\bigtriangledown^v \xi$ by the norms of $\bigtriangledown^{v_0} \xi$ and the derivative of $\Phi_v$. Hence, \vskip 0.1in \noindent {\bf Corollary 3.27: }{\it Under the same condition of Lemma 3.26, $$||\frac{\partial}{\partial v}|_{v=v_0} D_{f^{v,w}}(\xi)||_{L^p_v}\leq \frac{C}{|v_0|} (||f^{v_0}||_{L^p_{1,v}}+||w^{v_0}||_{L^p_{1,v}})||\frac{\partial}{\partial v}|_{v=v_0} w^v||_{ L^p_{1,v}} ||\xi||_{L^p_{1,v}}.\leqno(3.107)$$ $$||\frac{\partial}{\partial v}|_{v=v_0}N_{f^{v,w}}(\xi)||_{L^p_v}\leq \frac{C}{|v_0|} (||f^{v_0}||_{L^p_{1,v}}+||w^{v_0}||_{L^p_{1,v}})||\frac{\partial}{\partial v}|_{v=v_0} w^v||_{ L^p_{1,v}} ||\xi||^2_{L^p_{1,v}}.\leqno(3.108)$$} \vskip 0.1in Next, we compute the derivative of $Q_{f^{v,w}}$. Recall that $$Q_{f^{v,w}}=AQ_{f^{v,w}}(D_{f^{v,w}} Q_{f^{v,w}})^{-1}.\leqno(3.109)$$ Therefore, it is enough to compute $AQ_{f^{v,w}}=\phi_f Q_{f^w}$ and $((D_{f^{v,w}} Q_{f^{v,w}})^{-1})'$. Clearly, $$\frac{\partial}{\partial w}AQ_{f^{w,v}}=\phi_f(\frac{\partial}{\partial w}Q_{f^w}).\leqno(3.110)$$ $$\frac{\partial}{\partial v}AQ_{f^{w,v}}=\frac{\partial }{\partial v}(\phi_f)Q_{f^w}.\leqno(3.111)$$ Recall that in the gluing construction, only the cut-off function has variable $v$. Hence, we need to compute the derivative of the cut-off function with respect to $v$. \vskip 0.1in \noindent {\bf Lemma 3.28:}{\it $$|\frac{\partial}{\partial r} \bar{\beta}_r|<\frac{C}{r}.$$} \vskip 0.1in {\bf Proof: } $$\bar{\beta}_r(t)=\int^{\frac{r^2}{2}}_{\frac{r^2\rho}{2}}(1-\frac{log(u)-log(r^2)} {-log \rho})T(t-u)du+\int^{\infty}_{\frac{r^2}{2}} T(t-u)du.$$ $$\frac{\partial}{\partial r}\bar{\beta}_r(t)=\int \frac{\partial}{\partial r} \frac{log(u)-log(r^2)}{-log \rho}T(t-u)du+(1-\frac{log(\frac{r^2}{2})-log(r^2)} {-log \rho})T(t-\frac{r^2}{2})r$$ $$+(1-\frac{log(\frac{r^2\rho}{2})-log(r^2)} {-log \rho})T(t-\frac{r^2\rho}{2})r\rho+ T(t-\frac{r^2}{2})r,\leqno(3.112)$$ where $T(t-u)$ is a positive smooth function with compact supported and integral 1. Then, $$|\frac{\partial}{\partial r} \bar{\beta}_r|<\frac{C}{r}.\leqno(3.113)$$ $\Box$ Furthermore, we can choose $Q_{f^w}$ such that $\frac{\partial}{\partial w} Q_{f^w}$ is bounded. Therefore, $$||\frac{\partial}{\partial w}AQ_{f^{w,v}}||< C.\leqno(3.114)$$ $$||\frac{\partial}{\partial v}|_{v=v_0}AQ_{f^{w,v}}||< \frac{C}{|v_0|}.\leqno(3.115)$$ Notes that $$D_{f^{w,v}}AQ_{f^{v,w}}(D_{f^{w,v}}AQ_{f^{v,w}})^{-1}=Id.\leqno(3.116)$$ Hence, $$((D_{f^{w,v}}AQ_{f^{v,w}})^{-1})'=-(D_{f^{w,v}}AQ_{f^{v,w}})^{-1}(D_{f^{w,v}} AQ_{f^{v,w}})'(D_{f^{w,v}}AQ_{f^{v,w}})^{-1}.\leqno(3.117)$$ Combined (3.114)-(3.117), we obtain \vskip 0.1in \noindent {\bf Lemma 3.29: }{\it $$||\frac{\partial}{\partial w} Q_{f^{v,w}}||\leq C.\leqno(3.118)$$ $$||\frac{\partial}{\partial v}|_{v=v_0} Q_{f^{v,w}}||\leq \frac{C}{|v_0|}.\leqno(3.119)$$} Next, let's compute the derivative of ${\cal S}_e(f^{v,w})$. Let $w_{\mu}\in E^D_f$ be a smooth path such that $w_0=w$ and $\frac{d}{d\mu}|_{\mu=0} w_{\mu}=\tilde{w}.$ \vskip 0.1in \noindent {\bf Lemma 3.30: }{\it For $w\in E^D_f$, we view ${\cal S}_e(f^{v,w})$ as a map from $E^D_f\times B_{\delta_f}({\bf C}_f)$ to ${\cal U}_f$ where we use local trivialization given by $\Phi_v$ in Lemma 3.26. Then, $$||\frac{d}{d\mu}|_{\mu=0} {\cal S}_e(f^{v,w_{\mu}})||_{L^p_v}\leq C r^{\frac{4}{p}},\leqno(3.120)$$ $$||\frac{\partial}{\partial v}|_{v=v_0} {\cal S}_e(f^{v,w})||_{L^p_{v_0}}\leq Cr_0^{\frac{4}{p}-1}. \leqno(3.121)$$} \vskip 0.1in {\bf Proof: } ${\cal S}_e(f^{v,w_{\mu}})=0$ outside the gluing region and over $N_p(\frac{r^2}{2}, 2r^2)$. Therefore, the derivative is zero outside the gluing region and over $N_p(\frac{r^2}{2}, 2r^2)$. Here, we work over a slightly larger domain $N_p(\frac{\rho r^2_0}{2.1}, \frac{(2.1)r^2_0}{\rho})$ so that we can vary $r$ in a fixed domain. It is enough to work over $N_p(\frac{\rho r^2_0}{2.1}, \frac{r^2_0}{2})$, where $$f^{v,w_{\mu}}=\bar{\beta}_r(t)(f^{w_{\mu}}_1(s, t))-f^{w_{\mu}}_1(s,0))+ f^{w_{\mu}}_2(s+\theta,\frac{r^4}{t}).\leqno(3.124)$$ $${\cal S}_e(f^{v,w_{\mu}})=\bigtriangledown \bar{\beta}_r(t)(f^{w_{\mu}}_1(s, t))- f^{w_{\mu}}_1(s,0)).\leqno(3.125)$$ Therefore, $$\frac{d}{d\mu}|_{\mu=0}{\cal S}_e(f^{v,w_{\mu}})=\bigtriangledown \bar{\beta}_r(t)( \tilde{w}_1(s,t)-\tilde{w}_1(s,0)).\leqno(3.126)$$ $$||\frac{d}{d\mu}|_{\mu=0}{\cal S}_e(f^{v,w_{\mu}})||_{L^p_v}\leq C r^{\frac{4}{p}} ||\tilde{w}||_{C^1}.\leqno(3.127)$$ Since $\tilde{w}$ varies in a finite dimensional space and $\tilde{w}$ is smooth, we can replace $C^1$ norm by $L^p_1$-norm. Hence, $$||\frac{d}{d\mu}|_{\mu=0}{\cal S}_e(f^{v,w_{\mu}})||_{L^p_v}\leq C r^{\frac{2}{p}} ||\tilde{w}||_{L^p_1}.\leqno(3.128)$$ When we pull it back to the $\Sigma_{v_0}$ by $\Phi_v=(\Phi^1_v, \Phi^2_v)$, $${\cal S}_e(f^{v,w})=\bigtriangledown \bar{\beta}_r(\Phi^2_v(s,t)) (f^{w}(\Phi_v(t,s))-f^{w}_1(\Phi^1_v(t,s),0)).\leqno(3.129)$$ Using Lemma 3.26 and Lemma 3.28, it is easy to estimate that $$|\frac{\partial}{\partial v}|_{v=v_0}{\cal S}_e(f^{v,w})|\leq C\frac{1}{r_0}||f^{w}||_{C^1}.\leqno(3.130)$$ Hence, $$||\frac{\partial}{\partial v}|_{v=v_0}{\cal S}_e(f^{v,w})||_{L^p_{v_0}}\leq C\frac{1}{r_0} vol(N_p(\frac{\rho r^2_0}{2}, \frac{r^2_0}{2}))^{\frac{1}{p}}||f^w||_{C^1}\leq C r_0^{\frac{4}{p}-1}. \leqno(3.131)$$ Here, we use the fact that $f^w$ is smooth and varies in a finite dimension set $E^D_f$ with bounded $L^p_1$ norm. $\Box$. The same analysis will also implies that \vskip 0.1in \noindent {\bf Lemma 3.31: }{\it $$||\frac{\partial}{\partial v}|_{v=v_0} f^{v,w}||_{ L^p_{1,v_0}}\leq C||w||_{L^p_1}\leqno(3.132)$$ for $f^w\in E^D_f$.} \vskip 0.1in We leave it to readers to fill out the detail. Let $F$ be the inverse of $exp_{f^{v_0}}$. Then, $$||\frac{\partial}{\partial v}|_{v=v_0}w^v||\leq C(F)||\frac{ \partial}{\partial v}|_{v=v_0} f^{v,w}||_{L^p_{1,v}} \leq C(F)||w||_{L^p_1}.\leqno(3.133)$$ Putting all the estimate together, we obtain \vskip 0.1in \noindent {\bf Proposition 3.32: }{\it $$||\frac{d}{d\mu}|_{\mu=0} \xi(v, w_{\mu})||_{L^p_{1,v}} \leq Cr^{\frac{4}{p}-1}.\leqno(3.134)$$ $$||\frac{\partial}{\partial v}|_{v=v_0} \xi(v, w)||_{L^p_{1,v}} \leq C r_0^{\frac{4}{p}-1}. \leqno(3.135)$$} \vskip 0.1in {\bf Proof: } Recall that $$\xi(v,w)=H(v,w,\xi(v,w))=-Q_{f^{v,w}}{\cal S}_e(f^{v,w})-Q_{f^{v,w}}N_{f^{v,w}}(\xi(v,w)).\leqno(3.136)$$ By Lemma 3.25-3.32, we have bound derivatives for all the term of $H$. Moreover, the derivative of error term ${\cal S}_e(f^{v,w})$ is of the order $r^{\frac{4}{p}}$. Recall $\xi(v,w)$ is obtained by iterating $H$. Hence, the derivative of $\xi(v, w)$ is bounded by $\delta$ in (3.91) when $r$ is small. $$ \xi'(v,w)=Q'_{f^{v,w}}{\cal S}_e(f^{v, w}) -Q_{f^{v, w}}{\cal S}'_e(f^{v, w})-(Q'_{f^{v, w}}N_{f^{v,w}}+Q_{f^{v,w}}N'_{f^{v,w}})(\xi(v, w)) -Q_{f^{v, w}}N_{f^{v,w}}(\xi'(v,w)).\leqno(3.137)$$ By Lemma 3.25-3.32 and formula 3.133, $$||\frac{d}{d\mu}|_{\mu=0} \xi(v,w_{\mu})||_{L^p_{1,v}}\leq C_1 r^{\frac{4}{p}-1} +C_2 ||\xi(v,w)||_{L^p_{1,v}}+ C_3||\frac{d}{d\mu}|_{\mu=0} \xi(v,w_{\mu})||^2_{L^p_{1,v}}.\leqno(3.138)$$ $$||\frac{d}{d\mu}|_{\mu=0} \xi(v,w_{\mu})||_{L^p_{1,v}}\leq\frac{1}{1-\delta C_3} (C_1 r^{\frac{4}{p}-1}+C_2 ||\xi(v,w)||_{L^p_{1,v}}).\leqno(3.139)$$ Using (3.93), we obtain the inequality (3.134). The proof of the second inequality (3.135) is completely same. Only difference is that the derivative of $Q_{f^{w,v}}, N_{f^{v,w}}$ has a order $\frac{1}{r_0}$. However, we have ${\cal S}_e(f^{v,w}), \xi(v, w)$ in the formula, where both have order $r_0^{\frac{4}{p}}$. Hence, we obtain the order $r_0^{\frac{4}{p}-1}$. $\Box$ Let $u$ be a map over $\Sigma_v$ and $\xi\in \Omega^0(f^*T_FV)$. We define $u_v=(u^1_v, u^2_v)$ and $\xi_v=(\xi^1_v, \xi^2_v)$ as in (3.36), (3.73). Now, we want to embed ${\cal M}_{{\cal S}_e}\cap {\cal U}_f$ into ${\cal U}_{f,D}\times {\bf C}_f$ by the map $$exp_u \xi\rightarrow (exp_{u_v} \xi_v,v)\leqno(3.140)$$ for $u$ over $\Sigma_v$. Consider the composition of (3.140) with $\alpha_{v,w}$. $$\alpha(v,w)_v: E^D_f\times B_{\delta_f}({\bf C}_f)\rightarrow L^p_1({\cal U}_{f,D})\times {\bf C}_f.\leqno(3.141)$$ \vskip 0.1in \noindent {\bf Proposition 3.33: }{\it $\alpha(v,w)_v$ is $C^1$-smooth.} \vskip 0.1in {\bf Proof: } Our proof is motivated by the following observation. Suppose that $f$ is a continuous function over ${\bf R}$ such that $f(0)=0$ and $f$ is $C^1$ for $x\neq 0$. If $|f'(x)|\leq Cx^{\alpha}$ for $\alpha>0$, by mean value theorem $f'(0)=0$ and $f'$ is continuous at $x=0$. We first prove $$(f^w,v)\rightarrow f^{v,w}_v \leqno(3.142)$$ is a $C^1$-map. $f^{v,w}_v=f^w$ outside the gluing region. By symmetry, it is enough to consider $D_p(\frac{2r^2}{\rho})$. Over $D_p(2r^2)$, $$\begin{array}{lll} f^{v,w}_v&=&\tilde{\beta}_r(f^w_1(s,t)+f^w_2(s,t)-\frac{1}{2\pi r^2} \int_{S^1}(f^{w}_1(s,r^2)+f^w_2(\theta+s, r^2)))\\ &&+\frac{1}{2\pi r^2}\int_{S^1}(f^{w}_1(s,r^2)+f^w_2(\theta+s, r^2)) \end{array}. \leqno(3.143)$$ $$\begin{array}{lll} \frac{d}{d\mu}|_{\mu=0} (f^{v, w_{\mu}}_v-f^{w_{\mu}})&=& \tilde{\beta}_r(\tilde{w}_1(s,t)+\tilde{w}_2(s,t)-\frac{1}{2\pi r^2} \int_{S^1}(\tilde{w}_1(s,r^2)+\tilde{w}_2(\theta+s, r^2)))\\ &&+\frac{1}{2\pi r^2}\int_{S^1}(\tilde{w}_1(s,r^2)+ \tilde{w}_2(\theta+s, r^2)) \end{array}.\leqno(3.144)$$ Note that $$|\frac{1}{2\pi r^2}\int_{S^1}\tilde{w}(s,r^2)- \tilde{w}(s,0)| \leq C r^2 ||\tilde{w}||_{C^1}. \leqno(3.145).$$ By inserting the term $\tilde{w}(s,0)$ in the formula (3.144), $$||\frac{d}{d\mu}|_{\mu=0}( f^{v, w_{\mu}}-f^{w_{\mu}})||_{L^p(D_p(2r^2)}\leq C vol(D_p(2r^2))^{\frac{1}{p}} ||\tilde{w}||_{C^1}\leq C r^{\frac{4}{p}} ||\tilde{w}||_{L^p_1}.\leqno(3.146)$$ Here, we use the fact that $\tilde{w}$ varies in a finite dimensional space. $$\begin{array}{lll} &&||\bigtriangledown \frac{d}{d\mu}|_{\mu=0}( f^{v, w_{\mu}}-f^{w_{\mu}})||_{L^p(D_p(2r^2)}\\ &\leq & ||\bigtriangledown \tilde{\beta}_r (\tilde{w}_1(s,t)+\tilde{w}_2(s,t)-\frac{1}{2\pi r^2} \int_{S^1}(\tilde{w}_1(s,r^2)+\tilde{w}_2(\theta+s, r^2))||_{L^p}\\ &&+ ||\tilde{\beta}_r\bigtriangledown (\tilde{w}_1+\tilde{w}_2)||_{L^p(D_p(2r^2)}\\ &\leq & C (vol(D_p(2r^2)))^{\frac{1}{p}} ||\tilde{w}||_{C^1}\\ &\leq &C r^{\frac{4}{p}}||\tilde{w}||_{L^p_1} \end{array}. \leqno(3.147)$$ Over $N_p(\frac{\rho r^2}{2}, \frac{r^2}{2})$, $$f^{v,w}_v=f^w_2(s,t)+\bar{\beta}_r(f^w_1(s+\theta, t)-f^w_1(0)).\leqno(3.147.1)$$ $$\frac{d}{d\mu}|_{\mu=0}(f^{v, w_{\mu}}_v-f^{w_{\mu}})=\bar{\beta}_r(\tilde{w}_2(s,t)- \tilde{w}_2(0)).\leqno(3.147.2)$$ The same argument shows that $$||\frac{d}{d\mu}|_{\mu=0}(f^{v, w_{\mu}}_v-f^{w_{\mu}})||_{L^p_1(N_p(\frac{\rho r^2}{2}, \frac{r^2}{2}))}\leq Cr^{\frac{4}{p}}||\tilde{w}||_{L^p_1}.\leqno(3.147.3)$$ Using previous argument and Lemma 3.26, we can also show that $$||\frac{\partial}{\partial v}|_{v=v_0} (f^{v,w}_v-f^w)||_{L^p_1}\leq Cr_0^{\frac{4}{p}-1}. \leqno(3.148)$$ Therefore, $$||(f^{v,w}_v)'-(f^w)'||\leq Cr^{\frac{4}{p}-1}.\leqno(3.149).$$ $f^{v,w}_v$ is $C^1$ for $v\neq 0$. At $v=0$, the estimate (3.149) implies $$(f^{v,w}_v)'=(f^w)' \mbox{ at } v=0.\leqno(3.150)$$ Moreover, $(f^{v,w}_v)'$ is continuous. The same argument together with Proposition 3.32 shows that $$||(\xi(v,w)_v)'||\leq C r^{\frac{4}{p}-1}.\leqno(3.151)$$ Hence, $\xi(v,w)_v$ is a $C^1$-map and has derivative zero at $v=0$. In general, $$(exp_{f^{v,w}_v} \xi(v,w)_v)'=D_1exp_{f^{v,w}_v}\xi(v,w)_v (f^{v,w}_v)' +D_2exp_{f^{v,w}_v}\xi(v,w)_v (\xi(v,w)_v)',\leqno(3.152)$$ where $D_1, D_2$ are the partial derivatives of $exp$-function. $$\begin{array}{lll} &&||(exp_{f^{v,w}_v} \xi(v,w)_v)'-(f^w)'||\\ &\leq &||D_1exp_{f^{v,w}_v}\xi(v,w)_v (f^{v,w}_v)'-(D_1exp_{f^{v,w}_v} 0)(f^{v,w}_v)'||\\ &&+||(f^{v,w}_v)'-(f^w)'|| +||D_2exp_{f^{v,w}_v}\xi(v,w)_v (\xi(v,w)_v)'||\\ &\leq&C||\xi(v,w)_v||_{L^{\infty}}||(f^{v,w}_v)'||+Cr^{\frac{4}{p}-1}\\ & \leq &C||\xi(v,w)_v||_{L^p_1}||(f^w)'||+Cr^{\frac{4}{p}-1}\\ &\leq &C r^{\frac{4}{p}-1} \end{array}. \leqno(3.153)$$ Notes that $\alpha(v,w)_v$ is identity on ${\bf C}_f$-factor. Hence, we prove the proposition. Moreover, the derivative of $\alpha(v,w)_v$ is identity at $v=0$, since $\frac{\partial}{\partial w} f^w=id$. $\Box$ \vskip 0.1in \noindent {\bf Theorem 3.34: }{\it With the coordinate system given by $(E^D_f\times B_{\delta_f}({\bf C}_f), \alpha_f(v,w))$, ${\cal M}_{{\cal S}_e}\cap {\cal U}_{{\cal S}_e}$ is a $C^1$-oriented V-manifold.} \vskip 0.1in {\bf Proof: } Recall the definition 2.1. Suppose that $\alpha_{\bar{f}}(E^{\bar{D}}_{\bar{f}} \times B_{\delta_{\bar{f}}}({\bf C}_{\bar{f}}))\subset \alpha_f(E^D_f\times B_{\delta_f}({\bf C}_f))$. Then, $stb_{\bar{f}}\subset stb_f$ and we can assume that $E^{\bar{D}}_{\bar{f}}\subset {\cal U}_{\bar{f}} \subset {\cal U}_f$. It is clear that $\bar{D}$ is either a higher strata than $D$ or $D$. Let's consider the case that $\bar{D}$ is a higher strata. The proof for the second case is the same. To be more precise, let's consider the case that $D$ has three components $\Sigma_1\wedge \Sigma_2\wedge \Sigma_3$ and $\bar{D}$ has two components $\Sigma_1\wedge \Sigma_2\#_{v_2} \Sigma_3$ for $v_2\neq 0$. The general case is similar and we leave it to readers. Suppose that the gluing parameters are $(v_1, v_2)\in {\bf C}_1\times {\bf C}_2$. To construct Banach manifold $L^p_1({\cal U}_{\bar{f}})$, we need a trivialization of $\bigcup_{v_2}\Sigma_2\#_{v_2} \Sigma_3$. As we discuss in the beginning of this section, we can choose any trivialization. Here, we choose the one given by $\Phi_{v_2}$ Lemma 3.26. Clearly, $\alpha_f(v,w)$ maps an open subset of $E^D_f\times B_{\delta_f}({\bf C}_2)$ onto $E^{\bar{D}}_{\bar{f}}$ as a diffeomorphism. Now, we embed ${\cal M}_{{\cal S}_e}\cap {\cal U}_{\bar{f}}$ into ${\cal B}_{\bar{D}}$ by (3.140). By Proposition 3.33, both $$\alpha_f(v,w)_{v_1}, \alpha_{\bar{f}}(v_1, w)_{v_1}\leqno(3.154)$$ are injective $C^1$-map. Hence, we can view the image of ${\cal M}_{{\cal S}_e}\cap {\cal U}_{\bar{f}}$ as a $C^1$-submanifold of ${\cal B}_{\bar{D}}\times {\bf C}_{\bar{f}}$ and both $\alpha_f(v,w)_{v_1}, \alpha_{\bar{f}}(v_1, w)_{v_1}$ as $C^1$-diffeomorphisms to this submanifold. Hence, $$(\alpha_f(v,w))^{-1}\alpha_{\bar{f}}(v_1,w)=(\alpha_f(v,w)_{v_1})^{-1} \alpha_{\bar{f}}(v_1, w)_{v_1}.\leqno(3.155)$$ is a $C^1$-diffeomorphism. Next, we consider the orientation. First of all, it was proved in \cite{RT1} (Theorem 6.1) that both $\alpha_f(v,w)$ and $\alpha_{\bar{f}}(v_1, w)$ are orientation preserving diffeomorphism when $v_1\neq 0, v_2\neq 0$. Therefore, it is enough to consider the case $v_1=0$. By our argument in Proposition 3.33 (3.150, 3.151), $$(\alpha_f(v,w)_{v_1})'|_{v_1=0}=(\alpha_f(v_2, w))'|_{v_1=0}\times id_{C_1}, \ (\alpha_{\bar{f}}(v_1,w))'|_{v_1=0}=id. \leqno(3.157)$$ Moreover, $\alpha_f(v_2,w)$ is an orientation preserving diffeomorphism. Hence, the transition map is an orientation preserving diffeomorphism. We finish the proof. $\Box$. \section{GW-invariants of a family of symplectic manifolds} In this section, we shall give a detail construction of GW-invariants for a family of symplectic manifolds. Furthermore, we will prove composition law and $k$-reduction formula. Let's recall the construction in the introduction. Let $$p: Y\rightarrow M\leqno(4.1)$$ be an oriented fiber bundle such that the fiber $X$ and the base $M$ are smooth, compact, oriented manifolds. Then, $Y$ is also a smooth, compact, oriented manifold. Let $\omega$ be a closed 2-form on $Y$ such that $\omega$ restricts to a symplectic form over each fiber. Hence, we can view $Y$ as a family of symplectic manifolds. A $\omega$-tamed almost complex structure $J$ is an automorphism of the vertical tangent bundle $V(Y)$ such that $J^2=-Id$ and $\omega(w, Jw)>0$ for any vertical tangent vector $w\neq 0$. Suppose $A\in H_2( V, {\bf Z})\subset H_2(Y, {\bf Z})$. Let ${\cal M}_{g,k}$ be the moduli space of genus g Riemann surfaces with $k$-marked points such that $2g+k>2$ and $\overline{{\cal M}}_{g,k}$ be its Deligne-Mumford compactification. We shall use $$f: \Sigma\stackrel{F}{\rightarrow} Y$$ to indicate that the $im(f)$ is contained in a fiber. Consider its compactification- the moduli space of stable holomorphic maps $\overline{{\cal M}}_A(Y,g,k,J)$. Using the machinery of section 2 and 3, we can define a virtual neighborhood invariant $\mu_{{\cal S}}$. Here, we have to specify the cohomology class $\alpha$ in the definition of virtual neighborhood invariant $\mu_{{\cal S}}$. Recall that we have two natural maps $$\Xi_{g,k}: \overline{{\cal B}}_A(Y,g,k)\rightarrow Y^k\leqno(4.2)$$ defined by evaluating $f$ at marked points and $$\chi_{g,k}: \overline{{\cal B}}_A(Y,g,k)\rightarrow \overline{{\cal M}}_{g,k}\leqno(4.3)$$ defined by forgetting the map and contracting the unstable components of the domain. Notes that $\overline{{\cal M}}_{g,k}$ is a V-manifold. Suppose $K\in H^*(\overline{ {\cal M}}_{g,k}, {\bf R})$ and $\alpha_i\in H^*(V, {\bf R})$ are represented by differential forms. \vskip 0.1in \noindent {\bf Definition 4.1: }{\it We define $$\Psi^Y_{(A,g,k)}(K; \alpha_1, \cdots, \alpha_k)=\mu_{{\cal S}}(\chi^*_{g,k}(K) \wedge\Xi_{g,k}^*(\prod_i \alpha_i)).\leqno(4.4)$$} \vskip 0.1in \noindent {\bf Theorem 4.2 }{\it (i).$\Psi^Y_{(A,g,k)}(K; \alpha_1, \cdots, \alpha_k)$ is well-defined, multi-linear and skew symmetry. \vskip 0.1in \noindent (ii). $\Psi^Y_{(A,g,k)}(K; \alpha_1, \cdots, \alpha_k)$ is independent of the choice of forms $K, \alpha_i$ representing the cohomology classes $[K], [ \alpha_i]$, and the choice of virtual neighborhoods. \vskip 0.1in \noindent (iii). $\Psi^Y_{(A,g,k)}(K; \alpha_1, \cdots, \alpha_k)$ is independent of $J$ and is a symplectic deformation invariant. \vskip 0.1in \noindent (iv). When $Y=V$ is semi-positive and some multiple of $[K]$ is represented by an immersed V-submanifold, $\Psi^Y_{(A,g,k)}(K; \alpha_1, \cdots, \alpha_k)$ agrees with the definition of \cite{RT2}.} \vskip 0.1in {\bf Proof: } (i) follows from the definition and we omit it. (ii) follows from Proposition 2.7. To prove (iii), suppose that $\omega_t$ is a family of symplectic structures and $J_t$ is a family of almost complex structures such that $J_t$ is tamed with $\omega_t$. Then, we can construct a weakly smooth Banach cobordism $({\cal B}_{(t)}, {\cal F}_{(t)}, {\cal S}_{(t)})$ of $$\overline{{\cal M}}_{A}(Y, g,k, J_{(t)})=\cup_{t\in [0,1]}\overline{{\cal M}}_A(Y,g,k, J_t)\times \{t\}. \leqno(4.5)$$ Then, (iii) follows from Proposition 2.8 and section 3. To prove (iv), recall the construction of \cite{RT2}. To avoid the confusion, we will use $\Phi$ to denote the invariant defined in \cite{RT2}. The construction of \cite{RT1} starts from an inhomogeneous Cauchy-Riemann equation. It was known that $\overline{{\cal M}}_{g,k}$ does not admit a universal family, which causes a problem to define inhomogeneous term. To overcome this difficulty, Tian and the author choose a finite cover $$p_{\mu}: \overline{{\cal M}}^{\mu}_{g,k}\rightarrow \overline{{\cal M}}_{g,k}.\leqno(4.6)$$ such that $\overline{{\cal M}}^{\mu}_{g,k}$ admits a universal family. One can use the universal family of $\overline{{\cal M}}^{\mu}_{g,k}$ to define an inhomogeneous term $\nu$ and inhomogeneous Cauchy-Riemann equation $\bar{\partial}_J f=\nu$. Any $f$ satisfying this equation is called {\em a $(J,\nu)$-map}. Choose a generic $(J, \nu)$ such that the moduli space ${\cal M}^{\mu}_A(\mu, g,k,J, \nu)$ of $(J, \nu)$-map is smooth and the certain contraction $\overline{{\cal M}}^r_A(\mu, g,k,J,\nu)$ of $\overline{{\cal M}}_A(\mu, g,k,J,\nu)$ is of codimension 2 boundary. Define $$\Xi^{\mu, \nu}_{g,k}: \overline{{\cal M}}_A(\mu, g,k,J,\nu)\rightarrow X^k$$ and $$\chi^{\mu, \nu}_{g,k}: \overline{{\cal M}}_A(\mu,g,k,J,\nu)\rightarrow \overline{{\cal M} }^{\mu}_{g,k}$$ similarly. Then, we can choose Poincare duals (as pseudo-submanifolds) $K^*, \alpha^*$ of $K, \alpha_i$ such that $K^*, \alpha^*$ did not meet the image of $\overline{{\cal M}}_A(\mu,g,k,J,\nu)-{\cal M}_A(\mu,g,k,J,\nu)$ under the map $\chi^{\mu,\nu}_{g,k}\times \Xi^{\mu,\nu}_{g,k}$ and intersects transversely to the restriction of $\chi^{\mu,\nu}_{g,k}\times \Xi^{\mu, \nu}_{g,k}$ to ${\cal M}_A(\mu, g,k,J,\nu)$. Once this is done, $\Phi^X_{(A,g,k, \mu)}$ is defined as the number of the points of $(\chi^{\mu,\nu}_{g,k}\times \Xi^{\mu,\nu}_{g,k})^{-1}(K^*\times \prod_i \alpha^*_i)$, counted by the orientation. Then, we define $$\Phi^V_{(A,g,k)}(K; \alpha_1, \cdots, \alpha_k)=\frac{1}{\lambda^{\mu}_{g,k}} \Phi^V_{(A,g,k, \mu)}(p_{\mu}^*(K); \alpha_1, \cdots, \alpha_k),$$ where $\lambda^{\mu}_{g,k}$ is the order of cover map $p_{\mu}$ (4.6). The proof of (iv) is divided into 3-steps. First we observe that we can replace $\overline{{\cal M}}_{g,k}$ by $\overline{{\cal M}}^{\mu}_{g,k}$ in our construction. Let $\pi_{\mu}: \overline{{\cal B}}^{\mu}_{g,k}$ be the projection and $({\cal E}_{g,k}, s_{g,k})$ be the stablization terms for $\overline{{\cal M}}_{g,k}$. Then, we can choose $(\pi^*_{\mu}{\cal E}_{g,k}, \pi^*_{\mu}s_{g,k})$ to be the stablization term of $\overline{{\cal M}}^{\mu}_{g,k}$. Suppose that the resulting finite dimensional virtual neighborhoods are $(U, E, S), (U^{\mu}, E^{\mu}, S^{\mu})$ and invariant are $\Psi^Y_{(A,g,k)}, \Psi^Y_{(A,g,k,\mu)}$, respectively. Then, we have a commutative diagram $$\begin{array}{ccc} U^{\mu}&\rightarrow &E^{\mu} \\ \downarrow& & \downarrow \\ U &\rightarrow & E \end{array}\leqno(4.7)$$ and $$\begin{array}{ccc} U^{\mu} & \rightarrow & V^k\times \overline{{\cal M}}^{\mu}_{g,k}\\ \downarrow & &\downarrow \\ U & \rightarrow & V^k\times \overline{{\cal M}}_{g,k} \end{array}.\leqno(4.8)$$ Let $\lambda$ be the order of the cover $p_U: U^{\mu}\rightarrow U$ and $\lambda'$ be the order of the cover $p_G: E^{\mu}\rightarrow E$. One can check that $$\lambda=\lambda' \lambda^{\mu}_{g,k}.\leqno(4.9)$$ Let $\Theta$ be a Thom-form supported in a neighborhood of zero section of $E$. Then, $$\begin{array}{lll} \Psi^V_{(A,g,k)}(K; \alpha_1, \cdots, \alpha_k)&=&\int_{U}\chi_{g,k}^*(K)\wedge\Xi_{g,k}^*(\prod_i \alpha_i)\wedge S^*(\Theta)\\ &=&\frac{1}{\lambda}\int_{U^{\mu}}(\chi^{\mu}_{g,k})^*(p_{\mu}^*(K))\wedge (\Xi^{\mu}_{g,k})^*(\prod_i \alpha_i)\wedge (p_U S)^*(\Theta)\\ &=&\frac{1}{\lambda^{\mu}_{g,k}} \int_{U^{\mu}}(\chi^{\mu}_{g, k})^*(p^*_{\mu}(K))\wedge (\Xi^{\mu}_{g,k})^*(\prod_i \alpha_i)\wedge (S^{\mu})^*(\frac{1}{\lambda'}p_G^*(\Theta))\\ &=&\frac{1}{\lambda^{\mu}_{g,k}}\Psi^V_{(A,g,k,\mu)}(p_{\mu}^*(K); \alpha_1, \cdots, \alpha_k) \end{array}\leqno(4.10)$$ where $\frac{1}{\lambda'} p_G^*(\Theta)$ is a Thom form of $E^{\mu}$. Therefore, it is enough to show that $$\Psi^V_{(A,g,k,\mu)}=\Phi^V_{(A,g,k,\mu)}. $$ The second step is to deform Cauchy-Riemann equation $\bar{\partial}_Jf=0$ to inhomogeneous equation $\bar{\partial}_Jf=\nu$. Consider a family of equations $\bar{\partial}_J f=t\nu$. We can repeat the argument of (ii) to show that $\Psi^Y_{(A,g,k,\mu)}$ is independent of $t$. Let $({\cal B}^{\mu,\nu}_{g,k}, {\cal F}^{\mu, \nu}_{g,k}, {\cal S}^{\mu, \nu}_{g,k})$ be VNA smooth compact V-triple of $\overline{{\cal M}}^{\mu}_A(g,k,J,\nu)$ and define $\Xi^{\mu, \nu}_{g,k}, \chi^{\mu, \nu}_{g,k}$ similarly. For the same reason, the virtual neighborhood construction applies. The third step is to construct a particular finite dimensional virtual neighborhood $(U^{\mu}_{\nu}, E^{\mu}_{\nu}, S^{\mu}_{\nu})$ such that the restriction $$\chi^{\mu,\nu}_{g,k}\times \Xi^{\mu,\nu}_{g,k}: U^{\mu}_{\nu}\rightarrow X^k\times \overline{{\cal M}}^{\mu}_{g,k}\leqno(4.11)$$ is transverse to $K^*\times \prod_i \alpha^*_i$. First of all, since we work over ${\bf R}$, we can assume that each $\alpha^*$ is represented by a bordism class, and hence an immersed submanifold by ordinary transversality. By the linearity (i), we can assume that $K^*$ is represented by an immersed V-submanifold. Hence, $K^*\times \prod_i \alpha^*$ is represented by an immersed -submanifold (still denoted by $K^*\times \prod_i \alpha^*$). We first assume that $K^*\times \prod_i \alpha^*$ is an embedded V-submanifold. Recall that $K^*\times \prod_i\alpha^*$ does not meet the image of $\overline{{\cal M}}_A(\mu,g,k,J,\nu)-{\cal M}_A(\mu,g,k,J,\nu)$ and intersects transversely to the image ${\cal M}_A(\mu, g,k,J,\nu)$. Therefore, $$(\chi^{\mu,\nu}_{g,k}\times \Xi^{\mu,\nu}_{g,k})^{-1}(K^*\times \prod_i\alpha^*)\cap \overline{{\cal M}}^{\mu}_A(g,k,J,\nu)\leqno(4.12)$$ is a collection of the smooth points of ${\cal M}_A(g,k,J,\nu)$. It implies that $L_x$ is surjective at $x\in (\chi^{\mu,\nu}_{g,k}\times \Xi^{\mu,\nu}_{g,k})^{-1}( K^*\times \prod_i\alpha^*)\cap \overline{{\cal M}}^{\mu}_A(g,k,J,\nu)$ and $$\delta(\chi^{\mu,\nu}_{g,k}\times \Xi^{\mu}_{g,k}): Ker L_A\rightarrow X^k\times \overline{{\cal M}}^{\mu}_{g,k}\leqno(4.13)$$ is surjective onto the normal bundle of $K^*\times \prod_i \alpha_i$. Hence, the same is true over an open neighborhood ${\cal U}'$ of $(\chi^{\mu,\nu}_{g,k}\times \Xi^{\mu,\nu}_{ g,k})^{-1}(K^*\times \prod_i\alpha^*)\cap \overline{{\cal M}}_A(\mu,g,k,J,\nu)$. We cover $\overline{{\cal M}}_A(\mu,g,k,J,\nu)$ by ${\cal U}'$ and ${\cal U}''$ such that $$\bar{{\cal U}}''\cap (\chi^{\mu,\nu}_{g,k}\times \Xi^{\mu}_{g,k})^{-1}(K^*\times \prod_i \alpha^*_i)=\emptyset. \leqno(4.14)$$ Then, we construct $({\cal E}^{\mu}_{\nu}, s^{\mu}_{\nu})$ such that $s^{\mu}_{ \nu}=0$ over ${\cal U}'-\bar{{\cal U}}''$. Suppose that $(U^{\mu}_{\nu}, E^{\mu}_{\nu}, S^{\mu}_{\nu})$ is the finite dimensional virtual neighborhood constructed by $({\cal E}^{\mu}_{\nu}, s^{\mu}_{\nu})$. It is easy to check that $$(\chi^{\mu,\nu}_{g,k}\times \Xi^{\mu,\nu}_{g,k})^{-1}(K^*\times \prod_i\alpha^*_i)\cap U^{\mu}_{\nu}\subset {\cal U}'-\bar{{\cal U}}''.\leqno(4.15)$$ On the other hand, $$s^{\mu}_{\nu}=0 \mbox{ over } {\cal U}'-\bar{{\cal U}}''.\leqno(4.16)$$ It implies that $$(\chi^{\mu,\nu}_{g,k}\times \Xi^{\mu,\nu}_{g,k})^{-1}(K^*\times \prod_i\alpha^*)\cap U^{\mu}_{\nu}=E^{\mu}_{\nu}|_{((\chi^{\mu,\nu}_{g,k}\times \Xi^{\mu,\nu}_{g,k})^{-1}(K^*\times \prod_i \alpha^*_i)\cap \overline{{\cal M}}_A(\mu,g,k,J,\nu))}.\leqno(4.17)$$ It is easy to observe that the restriction of $\chi^{\mu,\nu}_{g,k}\times \Xi^{\mu,\nu}_{g,k}$ to $U^{\mu}_{\nu}$ is transverse to $K^*\times \prod_i\alpha^*_i$. Since $K^*\times \prod_i\alpha^*_i$ is Poincare dual to $K\times \prod_i\alpha_i$, $(\chi^{\mu,\nu}_{g,k}\times \Xi^{\mu,\nu}_{g,k} )^{-1}(K^*\times \prod_i\alpha^*)$ is Poincare dual to $(\chi^{\mu,\nu}_{g,k} \times \Xi^{\mu,\nu}_{g,k})^*(K\times \prod_i \alpha_i)$. Therefore, $$\begin{array}{lll} \Psi^V_{(A,g,k,\mu)}(K; \alpha_1, \cdots, \alpha_k)&=&\int_{U^{\mu}_{\nu}}( \Xi^{\mu,\nu}_{g,k}\times \chi^{\mu,\nu}_{g,k})^*(K\times\prod_i \alpha_i)\wedge (S^{\mu}_{\nu})^*(\Theta)\\ &=&\int_{(\Xi^{\mu,\nu}_{g,k}\times \chi^{\mu,\nu}_{g,k})^{-1}(K^*\times \prod_i\alpha^* )\cap U^{\mu}_{\nu}}(S^{\mu}_{\nu})^*(\Theta)\\ &=&\Phi^V_{(A,g,k,\mu)}(K; \alpha_1, \cdots, \alpha_k) \end{array}.$$ When $K^*\times \prod_i \alpha^*_i$ is an immersed V-submanifold, there is a V-manifold $N$ and a smooth map $$H:N\rightarrow X^k\times \overline{{\cal M}}^{\mu}_{g,k}$$ whose image is $K^*\times \prod_i \alpha^*_i$. Then, we replace $\chi^{\mu, \nu}_{g,k}\times \Xi^{\mu,\nu}_{g,k}$ by $\chi^{\mu,\nu}_{g,k}\times \Xi^{\mu,\nu}_{g,k}\times N$ and $K^*\times\prod_i \alpha^*_i$ by the diagonal of $(X^k\times \overline{{\cal M}}^{\mu}_{g,k})^2$ in the previous argument. It will implies (iv). $\Box$ It is well-known that the projection map $p: Y\rightarrow X$ defines a modular structure on $H^*(Y, {\bf R})$ by $H^*(M, {\bf R})$, defined by $$\alpha \cdot \beta=p^*(\alpha)\wedge \beta\leqno(4.18)$$ where $\alpha\in H^*(M, {\bf R})$ and $\beta\in H^*(Y, {\bf R})$. GW-invariant we defined behave nicely over this modulo structure, which is the basis of the modulo structure of equivariant quantum cohomology (Theorem I). \vskip 0.1in \noindent {\bf Proposition 4.3: }{\it Suppose that $\alpha_i\in H^*(Y, {\bf R}), \alpha\in H^*(M, {\bf R})$. Then $$\Psi^Y_{(A,g,k)}(K; \alpha_1, \cdots, \alpha\cdot \alpha_i,\cdots, \alpha_j, \cdots, \alpha_k)$$ $$=\Psi^Y_{(A,g,k)}(K; \alpha_1, \cdots, \alpha_i,\cdots, \alpha\cdot\alpha_j, \cdots, \alpha_k). \leqno(4.19)$$} \vskip 0.1in {\bf Proof: } By the definition, $$\Psi^Y_{(A,g,k)}(K; \alpha_1, \cdots, \alpha_i,\cdots, \alpha_j, \cdots, \alpha_k)$$ $$=\int_{U}\chi^*_{g,k}(K)\wedge \Xi^*_{g,k}(\prod_i \alpha_i)\wedge S^*( \Theta).$$ Let $$p: Y^k\rightarrow V^k$$ and $\Delta$ be the diagonal of $V^k$. A crucial observation is that $$\Xi^*_{g,k}: \overline{{\cal B}}_A(Y,g,k)\rightarrow Y^k$$ is factored through $${\cal B}_{g,k}\stackrel{\Xi'_{g,k}}{\rightarrow} p^{-1}(\Delta) \stackrel{i_{p^{-1} (\Delta)}}{\rightarrow} Y^k.\leqno(4.20)$$ Furthermore, for any $i$ $$\begin{array}{lll} &&i^*_{p^{-1}(\Delta)}(\alpha_1\times \cdots \times\alpha\cdot\alpha_i\times \cdots \alpha_k)\\ &=&p^*(i^*_{\Delta}(1\times\cdots\times \alpha^{(i)}\times\cdots\times 1 )) \wedge i^*_{p^{-1}(\Delta)}(\alpha_1\times \cdots \times\alpha_i\times \cdots \alpha_k) \end{array}\leqno(4.21)$$ where we use $\alpha^{(i)}$ to indicate that $\alpha$ is at the $i$-th component. However, $$i^*_{\Delta}(1\times\cdots\times \alpha^{(i)}\times\cdots\times 1 )=\alpha= i^*_{\Delta}(1\times\cdots\times \alpha^{(j)}\times\cdots\times 1 ).\leqno(4.22)$$ Hence, $$\Xi^*_{g,k}(\alpha_1\times \cdots \times\alpha\cdot\alpha_i\times \cdots \alpha_k)=\Xi^*_{g,k}(\alpha_1\times \cdots \times\alpha\cdot\alpha_j\times \cdots \alpha_k).\leqno(4.23)$$ Then, $$\Psi^Y_{(A,g,k)}(K; \alpha_1, \cdots, \alpha\cdot \alpha_i,\cdots, \alpha_j, \cdots, \alpha_k)$$ $$=\Psi^Y_{(A,g,k)}(K; \alpha_1, \cdots, \alpha_i,\cdots, \alpha\cdot\alpha_j, \cdots, \alpha_k).$$ $\Box$ As we mentioned in the introduction, there is a natural map $$\pi: \overline{{\cal M}}_{g,k}\rightarrow \overline{{\cal M}}_{g, k-1} \leqno(4.24)$$ by forgetting the last marked point and contracting the unstable rational component. One should be aware that there are two exceptional cases $(g,k)=(0,3), (1,1)$ where $\pi$ is not well defined. $\pi$ is not a fiber bundle, but a Lefschetz fibration. However, the integration over the fiber still holds for $\pi$. In another words, we have a map $$\pi_*: H^*(\overline{{\cal M}}_{g,k}, {\bf R})\rightarrow H^{*-2}(\overline{{\cal M}}_{g,k-1}, {\bf R}).\leqno(4.25)$$ For a stable $J$-map $f\in \overline{{\cal M}}_A(Y,g,k,J)$, let's also forget the last marked point $x_k$. If the resulting map is unstable, the unstable component is either a constant or non-constant map. If it is a constant map, we simply contract this component. If it is non-constant map, we divided it by the larger automorphism group. Then, we obtain a stable $J$-map in $\overline{{\cal M}}_A(Y, g,k-1,J)$. Furthermore, we have a commutative diagram $$\begin{array}{ccc} \chi_{g,k}: \overline{{\cal M}}_A(Y,g,k,J)&\rightarrow &\overline{{\cal M}}_{g,k}\\ \downarrow \pi & &\downarrow \pi\\ \chi_{g,k-1}: \overline{{\cal M}}_A(Y,g,k-1, J)&\rightarrow &\overline{{\cal M}}_{g,k-1} \end{array}\leqno(4.26)$$ Associated with $\pi$, we have two {\em k-reduction formulas} for $\Psi^Y_{(A, g, k)}$. \vskip 0.1in \noindent {\bf Proposition 4.4. }{\it Suppose that $(g,k)\neq (0,3),(1,1)$. \vskip 0.1in \noindent (1) For any $\alpha _1, \cdots , \alpha _{k-1}$ in $H^*(Y, {\bf R})$, we have} $$\Psi ^Y_{(A,g,k)}(K; \alpha _1, \cdots,\alpha _{k-1}, 1)~=~ \Psi ^Y_{(A,g,k-1)}(\pi_*(K); \alpha _1, \cdots,\alpha _{k-1}) \leqno(4.27)$$ \vskip 0.1in \noindent (2) Let $\alpha _k$ be in $H^2(Y, {\bf R})$, then $$\Psi ^Y_{(A,g,k)}(\pi^*(K); \alpha _1, \cdots,\alpha _{k-1}, \alpha _k)~=~\alpha_k (A) \Psi ^Y_{(A,g,k-1)}(K; \alpha _1, \cdots,\alpha _{k-1}) \leqno (4.28)$$ where $ \alpha^* _k$ is the Poincare dual of $\alpha _k$. \vskip 0.1in \noindent {\bf Proof:} Let $(\overline{{\cal B}}_A(Y,g,k), \overline{{\cal F}}_A(Y,g,k), {\cal S}^A_{g,k})$ be the VNA smooth Banach compact V-triple of $\overline{{\cal M}}_A(Y,g,k,J)$. Following from our construction of last section, we have commutative diagram $$\begin{array}{ccc} \chi_{g,k}: \overline{{\cal B}}_A(Y,g,k)&\rightarrow &\overline{{\cal M}}_{g,k}\\ \downarrow \pi & &\downarrow \pi\\ \chi_{g,k-1}:\overline{{\cal B}}_A(Y,g,k)&\rightarrow &\overline{{\cal M}}_{g,k-1} \end{array}\leqno(4.29)$$ Furthermore, $\overline{{\cal F}}_A(Y,g,k)=\pi^* \overline{{\cal F}}_A(Y,g,k-1)$. Using the virtual neighborhood technique, we construct $({\cal E}, s)$ and a finite dimensional virtual neighborhood $(U_{g,k-1}, E_{g,k-1}, S_{g,k-1})$ of $\overline{{\cal M}}_A(Y,g,k-1,J)$. We observe that the same $({\cal E}, s)$ also works in the construction of finite dimensional virtual neighborhood of $\overline{{\cal M}}_A(Y,g,k,J)$. Let $(U_{g,k}, E_{g,k}, S_{g, k})$ be the virtual neighborhood. Then, $E_{g,k}$ is the pull back of $E_{g,k-1}$ by $\pi: \overline{{\cal B}}_{g,k} \rightarrow \overline{{\cal B}}_{g,k-1}$. There is a projection $$\pi: U_{g,k}\rightarrow U_{g,k-1}. \leqno(4.30)$$ Then, $$S_{g,k}=S_{g,k-1}\circ \pi.\leqno(4.31)$$ Hence, $$S^*_{g,k}(\Theta)=\pi^* S^*_{g,k-1}(\Theta).\leqno(4.32)$$ Moreover, $$\Xi^*_{g,k}(\prod^{k-1}_1 \alpha_i\wedge 1)=(\Xi_{g,k-1}\pi)^*(\prod^{k-1}_1 \alpha_i)=\pi^*\Xi^*_{g,k-1}(\prod^{k-1}_1\alpha_i).\leqno(4.33)$$ Furthermore, $$\pi_*\chi_{g,k}^*(K)=\chi^*_{g,k-1}(\pi_*(K)).\leqno(4.34)$$ So, $$\begin{array}{lll} \Psi^Y_{(A,g,k)}(K; \alpha_1, \cdots,\alpha_{k-1}, 1)&=&\int_{U_{g,k}} \chi_{g, k}^*(K)\wedge \Xi^*_{g,k}(\prod^{k-1}_1 \alpha_i \wedge 1)\wedge S^*_{g,k}( \Theta)\\ &=&\int_{U_{g,k-1}}\pi_*( \chi_{g, k}^*(K)\wedge \pi^*(\Xi^*_{g,k}(\prod^{k-1}_1\alpha_i )\wedge S^*_{g,k-1}( \Theta)))\\ &=&\int_{U_{g,k-1}}\chi^*_{g,k-1}(\pi_*K)\wedge \Xi^*_{g, k-1}(\prod^{k-1}_1\alpha_i) \wedge S_*(\Theta)\\ &=&\Psi^Y_{(A,g,k)}(\pi_*(K); \alpha_1, \cdots, \alpha_{k-1}) \end{array}.\leqno(4.35)$$ On the other hand, for $\alpha_k\in H^2(Y, {\bf R})$, $$\Xi^*_{g,k}(\prod^{k-1}_1\alpha_i\wedge \alpha_k)=\pi^*\Xi^*_{g,k-1}(\prod^{ k-1}_1\alpha_i)\wedge e^*_k (\alpha_k),\leqno(4.36)$$ where $$e_i: \overline{{\cal B}}_A(Y,g,k)\rightarrow Y \leqno(4.37)$$ is the evaluation map at the marked point $x_k$. One can check that $$\pi_*(e^*_k(\alpha_k))=\alpha_k(A).\leqno(4.38)$$ Therefore, $$\begin{array}{lll} \Psi^Y_{(A,g,k)}(\pi^*(K); \alpha_1, \cdots, \alpha_{k-1}, \alpha_k)&=& \int_{U_{g,k}} \chi_{g, k}^*(\pi^*(K))\wedge \Xi^*_{g,k}(\prod^{k-1}_1 \alpha_i \wedge \alpha_k)\wedge S^*_{g,k}( \Theta)\\ &=&\int_{U_{g,k-1}}\pi_*( \chi_{g, k}^*(\pi^*(K))\wedge \Xi^*_{g,k}(\prod^{k}_1\alpha_i\wedge \alpha_k)\wedge S^*_{g,k}( \Theta))\\ &=&\int_{U_{g,k-1}}\chi^*_{g,k-1}(K)\wedge \Xi^*_{g,k-1}(\prod^{k-1}_1\alpha_i) \wedge S^*_{g,k-1}(\Theta)\wedge \pi_*(e^*_k(\alpha_k))\\ &=&\alpha_k(A)\Psi^Y_{(A,g,k-1)}(K; \alpha_1, \cdots, \alpha_{k-1}) \end{array}.\leqno(4.39)$$ $\Box$ Let $\overline {\cal U}_{g,k}$ be the universal curve over $\overline {\cal M}_{g,k}$. Then each marked point $x_i$ gives rise to a section, still denoted by $x_i$, of the fibration $\overline {\cal U}_{g,k} \mapsto \overline {\cal M}_{g,k}$. If ${\cal K}_{{\cal U} |{\cal M}}$ denotes the cotangent bundle to fibers of this fibration, we put ${\cal L}_{(i)} = x_i^* ( {\cal K}_{{\cal U} |{\cal M}})$. Following Witten, we put $$\langle \tau _{d_1,\alpha _1}\tau_{d_2,\alpha _2}\cdots \tau _{d_k,\alpha _k} \rangle _g (q) ~=~\sum _{A \in H_2(X,{\bf Z})} \Psi^X_{(A,g,k)}(K_{d_1,\cdots,d_k}; \{\alpha _i\}) \, q^A \leqno (4.40)$$ where $\alpha _i \in H_*(V,{\bf Q})$ and $[K_{d_1,\cdots,d_k}]=c_1({\cal L}_{(1)})^{d_1} \cup \cdots \cup c_1({\cal L}_{(k)})^{d_k}$ and $q$ is an element of Novikov ring. Symbolically, $\tau _{d,\alpha}$'s denote ``quantum field theory operators''. For simplicity, we only consider the cohomology classes of even degree. Choose a basis $\{\beta_a\}_{1\le a\le N}$ of $H^{*,\rm even}(V, {\bf Z})$ modulo torsion. We introduce formal variables $t_r^a$, where $r= 0, 1, 2, \cdots$ and $1\le a \le N$. Witten's generating function (cf. [W2]) is now simply defined to be $$F^X(t^a_r ; q) = \langle e^{\sum _{r,a} t^a_r \tau _{r, \beta _a}} \rangle (q)\lambda^{2g-2} =\sum _{n_{r,a}} \prod _{r,a} {(t^a_r)^{n_{r,a}} \over {n_{r,a}}!} \left \langle \prod _{r,a} \tau _{r,\beta _a}^{n_{r,a}} \right \rangle (q)\lambda^{2g-2} \leqno (4.41)$$ where $n_{r,a}$ are arbitrary collections of nonnegative integers, almost all zero, labeled by $r, a$. The summation in (4.40) is over all values of the genus $g$ and all homotopy classes $A$ of $J$-maps. Sometimes, we write $F_g^X$ to be the part of $F^X$ involving only GW-invariants of genus $g$. Using the argument of Lemma 6.1 (\cite{RT2}), Proposition 4.4 implies that the generating function satisfies several equation. \vskip 0.1in \noindent {\bf Corollary 4.5. }{\it Let $X$ be a symplectic manifold. $F^X$ satisfies the generalized string equation $${\partial F^X\over \partial t^1_0} = {1\over 2} \eta _{ab} t^a_0 t^b_0 + \sum \limits_{i=0}^\infty \sum \limits _{a} t^a_{i+1} {\partial F^X\over \partial t^a_i}.\leqno(4.42)$$ $F^X_g$ satisfies the dilaton equation $$\frac{\partial F^X_g}{\partial t^1_1}=(2g-2+\sum_{i=1}^{\infty}\sum_{a}t^a_i \frac{\partial }{\partial t^a_i})F^X_g+\frac{\chi(X)}{24}\delta_{g,1}, \leqno(4.43) $$ where $\chi(X)$ is the Euler characteristic of $X$.} \vskip 0.1in Next, we prove the composition law. Recall the construction in the introduction. Assume $g=g_1+g_2$ and $k=k_1+k_2$ with $2g_i + k_i \ge 3$. Fix a decomposition $S=S_1\cup S_2$ of $\{1,\cdots , k\}$ with $|S_i|= k_i$. Recall that $\theta _S: \overline {\cal M}_{g_1,k_1+1}\times \overline {\cal M}_{g_2,k_2+1} \mapsto \overline {\cal M}_{g,k}$, which assigns to marked curves $(\Sigma _i; x_1^i,\cdots ,x_{k_1+1}^i)$ ($i=1,2$), their union $\Sigma _1\cup \Sigma _2$ with $x^1_{k_1+1}$ identified to $x^2_1$ and remaining points renumbered by $\{1,\cdots,k\}$ according to $S$. Clearly, $im(\theta_S)$ is a V-submanifold of $\overline{{\cal M}}_{g,k}$, where the Poincare duality holds. Recall the transfer map \vskip 0.1in \noindent {\bf Definition 4.6: }{\it Suppose that $X, Y$ are two topological space such that Poincare duality holds over ${\bf R}$. Let $f: X\rightarrow Y$. Then, the transfer map $$f_{!}: H^*(X, {\bf R})\rightarrow H^*(Y, {\bf R})\leqno(4.44)$$ is defined by $f_{!}(K)=PD(f_*(PD(K)))$.} \vskip 0.1in We have another natural map defined in the introduction $\mu : \overline {\cal M}_{g-1, k+2} \mapsto \overline {\cal M}_{g,k}$ by gluing together the last two marked points. Clearly, $im(\mu)$ is also a V-submanifold of $\overline{{\cal M}}_{g,k}$. Choose a homogeneous basis $\{\beta _b\}_{1\le b\le L}$ of $H^*(Y,{\bf R})$. Let $(\eta _{ab})$ be its intersection matrix. Note that $\eta _{ab} = \beta _a \cdot \beta _b =0$ if the dimensions of $\beta _a$ and $\beta _b$ are not complementary to each other. Put $(\eta ^{ab})$ to be the inverse of $(\eta _{ab})$. Let $\delta\subset Y \times Y$ be the diagonal. Then, its Poincare dual $$\delta^*=\sum_{a,b} \eta^{ab} \beta_a\otimes \beta_b.\leqno(4.45)$$ Now we can state the composition law, which consists of two formulas. \vskip 0.1in \noindent {\bf Theorem 4.7: }{\it Let $K_i \in H_*(\overline {\cal M}_{g_i,k_i+1}, {\bf R})$ $(i=1, 2)$ and $K_0 \in H_*(\overline {\cal M}_{g-1,k +2}, {\bf R})$. For any $\alpha _1, \cdots,\alpha _k$ in $H^*(Y,{\bf R})$. Then we have (1).$$\begin{array}{rl} &\Psi ^Y_{(A,g,k)}((\theta _{S})_{!}(K_1\times K_2])\{\alpha _i\})\\ =(-1)^{deg(K_2)\sum^{k_1}_{i=1} deg (\alpha_i)} ~& \sum \limits _{A=A_1+A_2} \sum \limits_{a,b} \Psi ^Y_{(A_1,g_1,k_1+1)}(K_1;\{\alpha _{i}\}_{i\le k}, \beta _a) \eta ^{ab} \Psi ^Y_{(A_2,g_2,k_2+1)}(K_2;\beta _b, \{\alpha _{j}\}_{j> k}) \\ \end{array} \leqno (4.46) $$ (2).$$ \Psi ^Y_{(A,g,k)}(\mu_{!}(K_0);\alpha _1,\cdots, \alpha _k) =\sum _{a,b} \Psi ^Y_{(A,g-1,k+2)}( K_0;\alpha _1,\cdots, \alpha _k, \beta _a,\beta _b) \eta ^{ab}\leqno (4.47) $$ } \vskip 0.1in {\bf Proof: } The proof of the theorem is divided into two steps. First of all, $$\chi_{g,k}: \overline{{\cal B}}_A(Y,g,k)\rightarrow \overline{{\cal M}}_{g,k} \leqno(4.48)$$ is a submersion. ${\cal B}_{im(\theta_S)}=\chi^{-1}_{g,k}(Im(\theta_S))$ is a union of some lower strata of $\overline{{\cal B}}_A(Y,g,k)$. Moreover, it is also weakly smooth. Consider weakly smooth Banach compact-V triple $({\cal B}_{im(\theta_S)},{\cal F}_{im(\theta_S)}, S_{im(\theta_S)})$. We can use it to define invariant $\Psi_{(A, \theta_S)}$. The first step is to show that $$\Psi^Y_{(A,g,k)}(i_{!}(K); \alpha_1, \cdots, \alpha_k)=\Psi_{(A, \theta_S)}( K; \alpha_1, \cdots, \alpha_k),\leqno(4.49)$$ Let $(im(\theta_S))^*$ be the Poincare dual of $im(\theta_S)$. $(im(\theta_S))^ *$ can be chosen to be supported in a tubular neighborhood of $im(\theta_S)$, which can be identified with a neighborhood of zero section of normal bundle. For any $K\in H^*(im(\theta_S), {\bf R})$, we can pull it back to the total space of normal bundle (denoted by $K_{\overline{{\cal M}}_{g,k}}$). Then, $K_{\overline{ {\cal M}}_{g,k}}$ is defined over a tubular neighborhood of $im(\theta_S)$. Since $(im(\theta_S))^*$ is supported in the tubular neighborhood, $$(im(\theta_S))^*\wedge K_{\overline{{\cal M}}_{g,k}} \leqno(4.50)$$ is a closed differential form defined over $\overline{{\cal M}}_{g,k}$. In fact, $$i_{!}(K)=(im(\theta_S))^*\wedge K_{\overline{{\cal M}}_{g,k}}.\leqno(4.51)$$ First we construct that $({\cal E}, s)$ for $({\cal B}_{im(\theta_S)}, {\cal F}_{im(\theta_S)}, S_{im(\theta_S)})$. Suppose that the virtual neighborhood is $(U_{im(\theta_S)}, E_{im{\theta_S}}, S_{im(\theta_S)})$. We first extend $s$ over a neighborhood in $\overline{{\cal B}}_A(Y,g,k)$. Then, we construct $s'$ supported away from $im(\theta_S)$. Suppose that the stabilization term is $({\cal E}\oplus {\cal E}', s+s')$ such that $$L_x+s+s'+\delta(\chi_{g,k}): T_x{\cal B}_{g,k}\oplus {\cal E}\oplus {\cal E}' \rightarrow {\cal F}_x\times T_{\chi_{g,k}(x)}\overline{{\cal M}}_{g,k}$$ is surjective over ${\cal U}$ in the construction of (4.14-4.16). Suppose that the resulting finite dimensional virtual neighborhood is $(U_{g,k}, E\oplus E', S_{g,k})$. Then, $$\chi_{g,k}: U_{g,k}\rightarrow \overline{{\cal M}}_{g,k}\leqno(4.52)$$ is a submersion and $$\chi^{-1}_{g,k}(im(\theta_S))=E'_{U_{im(\theta_S)}}\subset U_{g,k}\leqno(4.53)$$ is a V-submanifold. Then, $\chi^*_{g,k}((im(\theta_S))^*)$ is Poincare dual to $E'_{U_{im(\theta_S)}}$. Choose Thom forms $\Theta_1, \Theta_2$ of $E,E'$ Therefore, $$\begin{array}{lll} \Psi^Y_{(A,g,k)}(i_{!}(K); \alpha_1, \cdots, \alpha_k)&=&\int_{U_{g,k}}(im(\theta_S))^*) \wedge \chi^*_{g,k}(K_{\overline{{\cal M}}_{g,k}} \wedge \Xi^*_{g,k}(\prod_i \alpha_i)\wedge S^*_{g,k}(\Theta_1\wedge\Theta_2)\\ &=&\int_{U_{im(\theta_S)}\times {\bf R}^{m'}/G'}\chi^*_{ g,k}(K_{\overline{{\cal M}}_{g,k}})\wedge \Xi^*_{g,k}(\prod_i \alpha_i)\wedge S^*_{g,k}(\Theta_1\wedge\Theta_2)\\ &=&\int_{U_{im(\theta_S)}}\chi^*_{g,k}(K)\wedge\Xi^*_{g,k}(\prod_i \alpha_i)\wedge S^*_{g,k}(\Theta_1)\\ &=&\Psi^Y_{(A, \theta_S)}(K; \alpha_1, \cdots, \alpha_k) \end{array}.\leqno(4.55)$$ The second step is to show that $\Psi^Y_{(A, \theta_S)}$ can be expressed by the formula (1). By the construction in the last section, we have a submersion $$e^{A_1}_{k_1+1}\times e^{A_2}_{k_2+1}: \overline{{\cal B}}_A(Y, g_1, k_1+1)\times \overline{{\cal B}}_A(Y,g_2, k_2+1)\rightarrow Y\times Y\leqno(4.56)$$ such that $$\bigcup_{A_1+A_2=A}(e^{A_1}_{k_1+1}\times e^{A_2}_{k_2+1})^{-1}(\Delta)= {\cal B}_{Im(\theta_S)}\leqno(4.57)$$ where $\delta\subset Y\times Y$ is the diagonal. By Gromov-compactness theorem, there are only finite many such pairs $(A_1, A_2)$ we need. Notes that $$(e^{A_1}_{k_1+1}\times e^{A_2}_{k_2+1})^{-1}(\Delta)\cap (e^{A'_1}_{k_1+1} \times e^{A'_2}_{k_2+1})^{-1}(\Delta)\leqno(4.58)$$ may be nonempty for some $(A_1, A_2)\neq (A'_1, A'_2)$. But it is in lower strata of ${\cal B}_{im(\theta_S)}$ of codimension at least two. Furthermore, by the construction of the section 3 $$\overline{{\cal F}}_A(Y,g,k)|_{{\cal B}_{im(\theta_S)}}=\overline{{\cal F}}_A(Y,g_1, k_1+1) \times \overline{{\cal F}}_A(Y, g_2, k_2+1)|_{{\cal B}_{ im(\theta_S)}}.\leqno(4.59)$$ We want to construct {\em a system of stabilization terms compatible with the stratification}. The idea is to start from the bottom strata and construct inductively the stabilization term supported away from lower strata. The same construction is crucial in the construction of Floer homology. We choose to wait until the last section to give the detail (called a system of stablization terms compatible with the corner structure in the last section). Let $s_1, s_2$ be the stablization terms for $$(\overline{{\cal B}}_{A_1}(Y, g_1, k_1+1), \overline{{\cal F}}_{A_1}(g_1, K_1+1), {\cal S}^{A_1}_{g_1, k_1+1}), (\overline{{\cal B}}_{A_2}(Y,g_1, k_1+1), \overline{{\cal F}}_{A_2} (Y,g_1, K_1+1), {\cal S}^{A_2}_{g_1, k_1+1}).$$ Suppose that the resulting virtual neighborhoods are $$(U^{A_1}_{g_1,k_1+1}, E, S^{A_1}_{g_1, k_1+1}), (U^{A_2}_{g_1,k_1+1}, E', S^{A_2 }_{g_1, k_1+1}).$$ By (4.56) and adding sections if necessary, we can assume that $$e^{A_1}_{k_1+1}\times e^{A_2}_{k_2+1}: U^{A_1}_{g_1, k_1+1}\times U^{A_2}_{ g_2, k_2+1}\rightarrow Y\times Y\leqno(4.60)$$ is a submersion. Let $$U_{A_1, A_2}=(e^{A_1}_{k_1+1}\times e^{A_2}_{k_2+1})^{-1}(\Delta)\subset U^{A_1}_{g_1, k_1+1}\times U^{A_2}_{ g_2, k_2+1}.\leqno(4.61)$$ One consequence of our system of stabilization compatible with the stratification is $$U_{A_1, A_2}\cap U_{A'_1, A'_2}$$ is a V-submanifold of codimension at least two for both $U_{A_1, A_2}$ and $U_{A'_1, A'_2}$ if $(A_1, A_2)\neq (A'_1, A'_2)$. Then, $$(\bigcup_{A_1+A_2=A} U_{A_1,A_2}, E\oplus E', S_{A_1}\times S_{A_2})\leqno(4.62)$$ is a finite dimensional virtual neighborhood of $({\cal B}_{im(\theta_S)}, {\cal F}_{im( \theta_S)}, {\cal S}_{im(\theta_S)})$. Moreover, we can choose stabilization term such that both $E$ and $E'$ are of even rank. Let $\delta^*$ be the Poincare dual of $\delta$. Then, $(e^{A_1}_{g_1, k_1+1} \times e^{A_2}_{g_2, k_2+1})^*(\delta^*)$ is Poincare dual to $U_{A_1, A_2}$. Therefore, $$\begin{array}{lll} &&\Psi^Y_{(A, \theta_S)}(K_1\times K_2; \{\alpha_i\})\\ &=&\int_{\cup_{A_1+A_2=A} U_{A_1,A_2}} \chi^*_{g,k}(K_1\times K_2)\wedge \Xi^*_{g,k}(\prod_i \alpha_i) \wedge S^*_{A_1}(\Theta_1)\wedge S^*_{A_2}(\Theta_2)\\ &=&\sum_{A_1+A_2=A}\int_{U_{A_1, A_2}} \chi^*_{g,k}(K_1\times K_2)\wedge \Xi^*_{g,k}(\prod_i \alpha_i) \wedge S^*_{A_1}(\Theta_1)\wedge S^*_{A_2}(\Theta_2)\\ &=&\sum_{A_1+A_2=A} \int_{U^{A_1}_{g_1, k_1+1}\times U^{A_2}_{g_2, k_2+1}} (e^{A_1}_{g_1, k_1+1}\times e^{A_2}_{g_2, k_2+1})^*(\delta^*)\wedge\chi^*_{g,k}(K_1\times K_2) \\ &&\wedge \Xi^*_{g,k}(\prod_i \alpha_i) \wedge S^*_{A_1}(\Theta_1)\wedge S^*_{A_2}(\Theta_2)\\ &=&\sum_{A_1+A_2=A}\sum_{a,b}\eta^{ab}\int_{U^{A_1}_{g_1, k_1+1}\times U^{A_2}_{g_2, k_2+1}} (e^{A_1}_{g_1, k_1+1})^*{\beta_a}\wedge (e^{A_2}_{g_2, k_2+1})^*(\beta_b)\\ &&\wedge \chi^*_{g,k}(K_1\times K_2)\wedge \Xi^*_{g,k}(\prod_i \alpha_i) \wedge S^*_{A_1}(\Theta_1)\wedge S^*_{A_2}(\Theta_2)\\ &=&(-1)^{deg(K_2)\sum^{k_1}{i=1} deg(\alpha_i)}\sum_{A_1+A_2=A}\\ &&\sum_{a,b}\eta^{ab} \int_{U^{A_1}_{g_1, k_1+1}} \chi^*_{g_1, k_1+1}(K_1)\wedge \Xi^*_{g_1, k_1}( \prod^{k_1}_i \alpha_i)(e^{A_1}_{g_1, k_1+1})^*{\beta_a}\wedge S^*_{A_1}(\Theta_1)\\ &&\int_{U^{A_2}_{g_2, k_2+1}}( \chi^*_{g_2, k_2+1}(K_2)e^{A_2}_{g_2, k_2+1})^*{\beta_b}\wedge \Xi^*_{ g_2, k_2}(\prod_{j>k_1} \alpha_j)\wedge S^*_{A_2}(\Theta_2)\\ &=&(-1)^{deg(K_2)\sum^{k_1}{i=1} deg(\alpha_i)}\sum_{A_1+A_2=A}\\ &&\sum_{a,b}\eta^{ab}\Psi^Y_{(A,g_1,k_1+1)}(K_1; \{\alpha_i\}_ {i\leq k_1}, \beta_a)\Psi^Y_{(A_2, g_2, k_2+1)}(K_2; \{\alpha_j\}_{j>k_1}, \beta_b). \end{array}\leqno(4.63)$$ The Proof of (2) is similar. We leave it to readers. \vskip 0.1in \noindent {\bf Corollary 4.8: }{\it Quantum multiplication is associative and hence there is a quantum ring structure over any symplectic manifolds.} \vskip 0.1in {\bf Proof: } The proof is well-known (see \cite{RT1}). We omit it. $\Box$ Here, we give another application to higher dimensional algebraic geometry. Recall that a Kahler manifold $W$ is called uniruled if $W$ is covered by rational curves. As we mentioned in the beginning, Kollar showed that if $W$ is a 3-fold, the uniruledness is a symplectic property \cite{K1}. Combined Kollar's argument with our construction, we generalize this result to general symplectic manifolds. \vskip 0.1in \noindent {\bf Proposition 4.9: }{If a smooth Kahler manifold $W$ is symplectic deformation equivalent to a uniruled manifold, $W$ is uniruled.} \vskip 0.1in First we need following \vskip 0.1in {\bf Lemma 4.10: }{\it Suppose that $N\subset Y$ is a submanifold such that for any $x\in {\cal M}_N=(\overline{{\cal M}}_A(Y,g,k,J)\cap e^{-1}_1(N))$ $$Coker L_x=0 \mbox{ and } \delta(e_1): L_x\rightarrow T_{e_1(x)}Y\leqno(4.64)$$ is surjective onto the normal bundle of $N$. Then, ${\cal M}_N$ is a smooth V-manifold of dimensional $ind-Cod(N)$ and $$\Psi^Y_{(A,g,k+1)}(K; N^*, \alpha_1, \cdots, \alpha_k)=(-1)^{deg(K)deg(N^*)} \int_{{\cal M}_N}\chi^*_{g,k+1}(K)\wedge \Xi^*_{g,k}(\prod_i\alpha_i).\leqno(4.65)$$} \vskip 0.1in {\bf Proof: } Since $e_1: \overline{{\cal B}}_{g,k+1}\rightarrow Y$ is a submersion, we can construct $({\cal E}, s)$ such that $s=0$ over a neighborhood of ${\cal M}_N$ and $$e_1|_U: U\rightarrow Y$$ is transverse to $N$, where $(U, E, S)$ is the virtual neighborhood constructed by $({\cal E}, s)$. Therefore, $$(e_1|_U)^{-1}(N)=E_{{\cal M}_N}\leqno(4.66)$$ is a smooth V-submanifold of $U$. Thus, $e^*_1(N^*)$ is Poincare dual to $E_{{\cal M}_N}$. $$\begin{array}{lll} \Psi^Y_{(A,g,k+1)}(K; N^*, \alpha_1, \cdots, \alpha_k)&=&\int_U \chi^*_{g,k+1} (K)\wedge e^*_{1}(N^*)\wedge\Xi^*_{g,k}(\prod_i\alpha_i)\wedge S^*(\Theta)\\ &=&(-1)^{deg(K)deg(N^*)}\int_{E_{{\cal M}_N}}\chi^*_{g,k+1} (K)\wedge\Xi^*_{g,k}(\prod_i\alpha_i)\wedge S^*(\Theta)\\ &=&(-1)^{deg(K)deg(N^*)}\int_{{\cal M}_N}\chi^*_{g,k+1}(K)\wedge\Xi^*_{g,k}(\prod_i\alpha_i) \end{array}.\leqno(4.67)$$ $\Box$ \vskip 0.1in {\bf Proof of Proposition 4.9:} If $\Psi^Y_{(A,0,k+1)}(K; pt, \cdots)\neq 0$, then $W$ is covered by rational curves. Otherwise, there is a point $x_0$ where there is no rational curve passing through $x_0$. $${\cal M}_N=\overline{{\cal M}}_A(Y,0,k,J)\cap e^{-1}_1(N)=\emptyset\leqno(4.68)$$ for any $A, k$. The condition of Lemma 4.10 is obviously satisfied. By Lemma 4.10, $$\Psi^Y_{(A,0,k+1)}(K; pt, \cdots)=0$$ and this is a contradiction. Since GW-invariant $\Psi^Y_{(A,0,k+1)}(K; pt, \cdots)$ is a symplectic deformation invariant property, it is enough to show that if $W$ is uniruled, $\Psi^Y_{(A,0,k+1)}(K; pt, \alpha_1, \cdots, \alpha_k)\neq 0$ for some $K$, $\alpha_1, \cdots, \alpha_k$. Assuming Lemma 4.10, Kollar showed some $\Psi^W_{(A,0,3)}(pt; pt, \alpha, \beta)$ is not zero for some $A$ and $\alpha, \beta$. His argument uses Mori's machinery. Here we give a more elementary argument to show that $$\Psi^Y_{(A,0,k+1)}(pt; pt, \alpha_1, \cdots, \alpha_k)\neq 0\leqno(4.69)$$ for some $A$ and some $\alpha_i$ with $k>>0$. Then, using the composition law we proved, we can derive Kollar's calculation. First, we repeat some of Kollar's argument. By \cite{K}, for a generic point $x_0$, ${\cal M}_A(W, 0,k,J)\cap e^{-1}_1(x_0)$ satisfies the condition of Lemma 4.10 for any $A$. Next choose $A_0$ such that $$H(A_0)=min_A\{H(A); {\cal M}_A(W,0,k,J)\cap e^{-1}_1(x_0)\neq \emptyset\}. \leqno(4.70)$$ where $H$ is an ample line bundle. Then, one can check that $$(\overline{{\cal M}}_A(W,0,k,J)-{\cal M}_A(W,0,k,J))\cap e^{-1}_1(x_0)=\emptyset.$$ Furthermore, ${\cal M}_{x_0}={\cal M}_A(W, 0,k,J)\cap e^{-1}_1(x_0)$ is a compact, smooth, complex manifold. In particular, it carries a fundamental class. Next, we show that $$\Xi_{0,k}: {\cal M}_{x_0}\rightarrow W^k \leqno(4.71)$$ is an immersion for large $k>>0$. For any $f\in {\cal M}_{x_0}$, $$T_f{\cal M}_{x_0}=\{v\in H^0(f^*TV); v(x_0)=0\}.\leqno(4.72)$$ Since $v_f\in H^0(f^*TV)$ is holomorphic, there are finite many points $x_2, \cdots, x_{k+1}$ such that if for any $v_f$ with $v_f(x_i)=0$ for every $i$, $v_f=0$ . One can check that $$\delta (\Xi_{0,k})_f(v)=(v(x_2), \cdots, v(x_k)).\leqno(4.73)$$ Therefore, $\delta(\Xi_{0,k})$ is injective. Since $\Xi_{0,k}$ is an immersion, $\Xi_{0,k}({\cal M}_{x_0})\subset W^k$ is a compact complex subvariety of the same dimension. Hence, it carries a nonzero homology class $[\Xi_{0,k}({\cal M}_{x_0})]$. Furthermore, $(\Xi_{0,k})_*([{\cal M}_{x_0}])=\lambda [\Xi_{0,k}({\cal M}_{x_0})]$ for some $\lambda>0$. By Poincare duality, there are $\alpha_1, \cdots, \alpha_k$ such that $$\prod_i \alpha_i([\Xi_{0,k}({\cal M}_{x_0})])\neq 0. \leqno(4.74)$$ By Lemma 4.10, $$\begin{array}{lll} \Psi^W_{(A,g,k+1)}(pt; pt, \alpha_1, \cdots, \alpha_k)&=& \int_{{\cal M}_{x_0}}\Xi^*_{0,k}(\prod_i \alpha_i)\\ &=&(\prod_i \alpha_i)(\Xi_*([{\cal M}_{x_0}]))\neq 0 \end{array}\leqno(4.75)$$ $\Box$ \section{Equivariant GW-invariants and Equivariant quantum cohomology} We will study the equivariant GW-invariants and the equivariant quantum cohomology in detail in this section. The equivariant theory is an important topic. It has been studied by several authors \cite{AB}, \cite{GK}. As we mentioned in the \cite{R4}, equivariant theory is the one that usual Donaldson method has trouble to deal with, where there are topological obstructions to choose a ``generic'' parameter. But our virtual neighborhood method is particularly suitable to study equivariant theory. In our case, one can attempt to choose an equivariant almost complex structure and apply the equivariant virtual neighborhood technique. However, a technically simpler approach is to view the equivariant GW-invariants as a limit of GW-invariants for the families of symplectic manifolds. This approach was advocated by \cite{GK}, where they formulated some conjectural properties for the equivariant GW-invariants and the equivariant quantum cohomology. First work to give a rigorous foundation of the equivariant GW-invariants and the equivariant quantum cohomology was given by Lu \cite{Lu} for monotonic symplectic manifolds, where he used the method of \cite{RT1}, \cite{RT2}. Here, we use the invariants we established in last section to establish the equivariant GW-invariants and the equivariant quantum cohomology for general symplectic manifolds. Let's recall the construction of the introduction. Suppose that $G$ acts on $(X, \omega)$ as symplectomorphisms. Let $BG$ be the classifying space of $G$ and $EG\rightarrow BG$ be the universal $G$-bundle. Suppose that $$BG_1\subset BG_2\cdots\subset BG_m \subset BG \leqno(5.1)$$ such that $BG_i$ is a smooth oriented compact manifold and $BG=\cup_i BG_i$. Let $$EG_1\subset EG_2\cdots\subset EG_m \subset BG\leqno(5.2)$$ be the corresponding universal bundles. We can also form the approximation of homotopy quotient $X_G=X\times EG/G$ by $X^i_G=X\times EG_i/G$. Since $\omega$ is invariant under $G$, its pull-back on $V\times EG_i$ descends to $V^i_G$. So, we have a family of symplectic manifold $P_i: X^i\rightarrow BG_i$. Applying our previous construction, we obtain GW-invariant $\Psi^{X^i_G}_{(A, g,k)}$. We define equivariant GW-invariant $$\Psi^G_{(A,g,k)}\in Hom(H^*(\overline{{\cal M}}_{g,k}, {\bf R})\otimes(H^*(V_G, {\bf R}))^{ \otimes k}, H^*(BG, {\bf R})) \leqno(5.3)$$ as follow: For any $D\in H_*(BG, {\bf Z})$, $D\in H_*(BG_i, {\bf Z})$ for some $i$. For any $K\in H^*(\overline{{\cal M}}_{g,k}, {\bf R})$, $\pi^*(K)\in H^*(\overline{{\cal M}}_{g,k+1}, {\bf R})$. Let $i_{X^i_G}: X^i_G\rightarrow X_G$. \vskip 0.1in \noindent {\bf Definition 5.1: }{\it For $\alpha_i\in H^*_G(X,{\bf R})$, we define $$\Psi^G_{(A,g,k)}(K, \alpha_1, \cdots, \alpha_k)(D)=\Psi^{X^i_G}_{(A, g,k+1)} (\pi^*(K); i^*_{X^i_G}(\alpha_1), \cdots, i^*_{X^i_G}(\alpha_k), P^*_i(D^*_{BG_i})),\leqno(5.4)$$ where $D^*_{BG_i}$ is the Poincare dual of $D$ with respect to $BG_i$.} \vskip 0.1in \noindent {\bf Theorem 5.2: }{\it (i). $\Psi^G_{(A, g,k)}$ is independent of the choice of $BG_i$. \vskip 0.1in \noindent (ii). If $\omega_t$ is a family of $G$-invariant symplectic forms, $\Psi^G_{(A, g,k)}$ is independent of $\omega_t$.} \vskip 0.1in {\bf Proof: } The proof is similar to the third step of the proof of Proposition 4.2(iv). Choose a $G$-invariant tamed almost complex structure $J$ on $X$. It induces a tamed almost complex structure (still denoted by $J$) over every $X^i_G$. Clearly, there is a natural inclusion map $$\overline{{\cal M}}_A(X^i_G, g,k, J)\subset \overline{{\cal M}}_A(X^j_G, g, k,J) \mbox{ for } i\leq j. \leqno(5.5)$$ Suppose that $({\cal B}_i, {\cal F}_i, {\cal S}_i)$ is the configuration space of $\overline{{\cal M}}_A(X^i_G,g,k,J)$. Then, there is a natural inclusion. $$({\cal B}_i, {\cal F}_i, {\cal S}_i)\subset ({\cal B}_j, {\cal F}_j, {\cal S}_j) \mbox{ for } i\leq j.\leqno(5.6)$$ We first construct $({\cal E}_i, s_i)$ for $({\cal B}_i, {\cal F}_i, {\cal S}_i)$. Suppose that the resulting finite dimensional virtual neighborhood is $(U_i, E_i, S_i)$. Then, we extend $s_i$ over ${\cal B}_j$. Since $L_A+s_i$ is surjective over ${\cal U}_i\subset{\cal B}_i$. We can construct $({\cal E}_j, s_j)$ such that $s_j=0$ over ${\cal U}_i$ and $L_A+s_i+s_j$ is surjective over ${\cal U}_j$. Suppose that the resulting finite dimensional virtual neighborhood is $(U_j, E_i\oplus E_j, S_j)$. Then, $$U_j\cap ({\cal E}_j)_{{\cal B}_i}=(E_j)_{U_i}\subset U_j$$ is a V-submanifold. Let $$e^j_{k+1}: {\cal B}_j\rightarrow X^j_B\leqno(5.7)$$ be the evaluation map at $x_{k+1}$. Then, we can choose $s_i, s_j$ such that the restriction of $e^j_{k+1}$ to $U_j$ is a submersion. Furthermore, since $(e^j_{k+1})^{-1}( X^i_G)={\cal B}_i$, $$(e^j_{k+1})^{-1}(X^i_G)\cap U_j= (E_j)_{U_i}.\leqno(5.8)$$ Notes that $$S_j\circ i=S_i, \leqno(5.9)$$ where $i:(E_j)_{U_i} \rightarrow U_j$ is the inclusion. Choose Thom forms $\Theta_i, \Theta_j$ of $E_i,E_j$. Let's use $I_{ij}$ to denote the inclusion ${\cal B}_i\subset {\cal B}_j$, $BG_i\subset BG_j$ and $X^i_G\subset X^j_G$ and define $\Xi^i_{g,k}, \chi^i_{g,k}$ similarly. Then $$\Xi^j_{g,k}\circ I_{ij}=I_{ij}\Xi^i_{g,k}, \mbox{ and } \chi^j_{g,k}\circ I_{ij}=\chi^i_{g,k}.\leqno(5.11)$$ Furthermore, $$D^*_{BG_j}=(I_{ij})_{!} D^*_{BG_i}.\leqno(5.12)$$ Let $(BG_i)^*_j$ be the Poincare dual of $BG_i$ in $BG_j$. Choose $(BG_i)^*_j$ supported in a tubular neighborhood of $BG_i$. By Lemma 2.10, $$D^*_{BG_j}=(D^*_{BG_i})_{BG_j}\wedge (BG_i)^*_j.\leqno(5.13)$$ Furthermore, $P^*_j((BG_i)^*_j)$ is Poincare dual to $X^i_G$ in $X^j_G$. Hence, $(e^j_{k+1})^*P^*_j((BG_i)^*_j)$ is Poincare dual to $(E_j)_{U_i}$. $$\begin{array}{lll} &&\Psi^{X^j_G}_{(A,g,k+1)}(\pi^*(K), i^*_{X^j_G}(\alpha_1),\cdots, i^*_{X^j_G}(\alpha_{ k}), P^*_j(D^*_{BG_j}))\\ &=&\int_{U_j} \chi^j_{g,k+1}(\pi^*(K))\wedge \Xi^i_{g,k}(\prod_m i^*_{X^j_G}( \alpha_m))\wedge (e^j_{k+1})^*P^*_j(D^*_{BG_j})\wedge S_j^*(\Theta_i\times \Theta_j)\\ &=&\int_{U_j}\chi^j_{g,k+1}(\pi^*(K))\wedge \Xi^j_{g,k}(\prod_m i^*_{X^j_G}( \alpha_m))\wedge (e^j_{k+1})^*P^*_j((D^*_{BG_i})_{BG_j})\wedge (e^j_{k+1})^*P^*_j( (BG_i)^*_j)\wedge S_j^*(\Theta_i\times \Theta_j)\\ &=&\int_{(E_j)_{U_i}}\chi^i_{g,k+1}(\pi^*(K))\wedge \Xi^i_{g,k}(\prod_m i^*_{X^i_G}( \alpha_m))\wedge (e^i_{k+1})^*P^*_i(D^*_{BG_i})\wedge S_j^*(\Theta_i\times \Theta_j)\\ &=&\int_{U_i}\chi^i_{g,k+1}(\pi^*(K))\wedge \Xi^i_{g,k}(\prod_m i^*_{X^i_G}( \alpha_m))\wedge (e^i_{k+1})^*P^*_i(D^*_{BG_i})\wedge S_i^*(\Theta_i)\\ &=&\Psi^{X^i_G}_{(A,g,k+1)}(\pi^*(K); i^*_{X^i_G}(\alpha_1),\cdots, i^*_{X^i_G} (\alpha_{k}), P^*_i(D^*_{BG_i})) \end{array}.\leqno(5.14)$$ (ii) follows from the same property of $\Psi^{X^i_G}$. $\Box$ As we discussed in the introduction, for any equivariant cohomology class $\alpha\in H^*_G(X)$, we can evaluate over the fundamental class of $X$ $$\alpha [X]\in H^*(BG).\leqno(5.15)$$ Furthermore, there is a modulo structure by $H^*_G(pt)=H^*(BG)$, defined by using the projection map $$X_G\rightarrow BG.\leqno(5.16)$$ The equivariant quantum multiplication is a new multiplication structure over $H^*_G(X, \Lambda_{\omega})=H^*(X_G, \Lambda_{\omega})$ as follows. We first define a total 3-point function $$\Psi^G_{(X,\omega)}(\alpha_1, \alpha_2, \alpha_3)=\sum_A \Psi^G_{(A,0,3)}( pt; \alpha_1, \alpha_2, \alpha_3)q^A.\leqno(5.17)$$ Then, we define an equivariant quantum multiplication by $$(\alpha\times_{QG}\beta)\cup \gamma [X]=\Psi^G_{(X,\omega)}(\alpha_1, \alpha_2, \alpha_3).\leqno(5.18)$$ \vskip 0.1in \noindent {\bf Theorem I: }{\it (i) The equivariant quantum multiplication is skew-symmetry. \vskip 0.1in \noindent (ii) The equivariant quantum multiplication is commutative with the modulo structure of $H^*(BG)$. \vskip 0.1in \noindent (iii) The equivariant quantum multiplication is associative. Hence, there is a equivariant quantum ring structure for any $G$ and any symplectic manifold $V$} \vskip 0.1in {\bf Proof: } (i) follows from the definition. By the proposition 5.2, for any $\alpha\in H^*(BG, {\bf R})$, $$\begin{array}{lll} &&\Psi^{X^i_G}_{(A,g,k+1)}(\pi^*(K); i_{X^i_G}^*(\alpha_1 ),\cdots, P^*_i(i_{ BG_i})^*(\alpha)\wedge i_{X^i_G}^*(\alpha_j), \cdots,i_{X^i_G}(\alpha_k), P^*_i(D^*_{BG_i}))\\ &=&\Psi^{X^i_G}_{(A,g,k+1)}(\pi^*(K); i_{X^i_G}^*(\alpha ), i_{X^i_G}^*(\alpha_2), \cdots,i_{X^i_G}(\alpha_k), P^*_i(i_{ BG_i})^*(\alpha)\wedge P^*_i(D^*_{BG_i}))\\ &=&\Psi^{X^i_G}_{(A,g,k+1)}(\pi^*(K); i_{X^i_G}^*(\alpha ), i_{X^i_G}^*(\alpha_2), \cdots,i_{X^i_G}(\alpha_k), P^*_i(i_{ BG_i})^*(\alpha\wedge D^*_{BG_i}))\\ &=&\Psi^{X^i_G}_{(A,g,k+1)}(\pi^*(K); i_{X^i_G}^*(\alpha ), i_{X^i_G}^*(\alpha_2), \cdots,i_{X^i_G}(\alpha_k), P^*_i(i_{ BG_i})^*((\alpha(D)^*_{BG_i}))\\ &=&\Psi^G_{(A,g,k)}(K, \alpha_1, \alpha_2, \cdots, \alpha_k)(\alpha(D)) \end{array}. \leqno(5.19)$$ Then, (ii) follows from the definition. The proof of (iii) is the same as the case of the ordinary quantum cohomology. We omit it. $\Box$ \section{Floer homology and Arnold conjecture} In this section, we will extend our construction of previous sections to Floer homology to remove the semi-positive condition. Floer homology was first introduced by Floer in an attempt to solve Arnold conjecture \cite{F}. The original Floer homology was only defined for monotonic symplectic manifolds. Floer solved Arnold conjecture in the same category. The Floer homology for semi-positive symplectic manifolds was defined by Hofer and Salamon \cite{HS}. Arnold conjecture for semi-positive symplectic manifolds were solved by \cite{HS} and \cite{O}. Roughly speaking, there are two difficulties to solve Arnold conjecture for general symplectic manifolds,i.e., (i) to extend Floer homology to general symplectic manifolds and (ii) to show that Floer homology is the same as ordinary homology. For the second problem, the traditional method is to deform a Hamiltonian function to a small Morse function and calculate its Floer homology directly. This approach involved some delicate analysis about the contribution of trajectories which are not gradient flow lines of a Morse function. It has only been carried out for semi-positive symplectic manifolds \cite{O}. But the author and Tian showed \cite{RT3} that this part of difficulties can be avoided by introducing a Bott-type Floer homology, where we can deform a Hamiltonian function to zero. The difficulty to extend Floer homology for a general symplectic manifold is the same as the difficulty to extend GW-invariant to a general symplectic manifold. Once we establish the GW-invariant for general symplectic manifolds, it is probably not surprising to experts that the same technique can work for Floer homology. Since many of the construction here is similar to that of last several sections, we shall be sketch in this section. Let's recall the set-up of \cite{HS}. Let $(X,\omega)$ be a closed symplectic manifold. Given any function $H$ on $X\times S^1$, we can associate a vector field $X_H$ as follow: $$ \omega (X_H(z,t), v) = v(H)(z,t),~~~~~{\rm for~any}~v~\in ~T_zV \leqno (6.1)$$ We call $H$ a periodic Hamiltonian and $X_H$ a Hamiltonian vector field associated to $H$. Let $\phi_t(H)$ be the integral flow of the Hamiltonian vector field $X_H$. Then $\phi _1(H)$ is a Hamiltonian symplectomorphism. \vskip 0.1in \noindent {\bf Definition 6.1.} {\it We call a periodic Hamiltonian $H$ to be non-degenerate if and only if the fixed-point set $F(\phi_1(H))$ of $\phi_1(H)$ is non-degenerate. } \vskip 0.1in \noindent Let ${\cal L}(X)$ be the space of contractible maps (sometimes called contractible loops) from $S^1$ into $X$ and $\tilde{{\cal L}}(X)$ be the universal cover of ${\cal L}(X)$, namely, $\tilde{{\cal L}}(X)$ is as follows: $$\tilde{{\cal L}}(X)=\{(x,u)|x\in {\cal L}(X), u:D^2\rightarrow X \mbox{ such that } x=u|_{\partial D^2}\}/\sim, \leqno(6.2)$$ where the equivalence relation $\sim$ is the homotopic equivalence of $x$. The covering group of $\tilde {{\cal L}}$ over ${\cal L}$ is $\pi_2(V)$. We can define a symplectic action functional on $\tilde{{\cal L}}(X)$, $$a_H((x,u))=\int_{D^2 }u^*\omega+\int_{S^1} H(t, x(t)) dt \leqno(6.3)$$ It follows from the closeness of $\omega $ that $a_H$ descends to the quotient space by $\sim$. The Euler-Lagrange equation of $a_H$ is $$\dot{u}-X_H(t, u(t))=0 \leqno(6.4)$$ Let ${\bf R}(H)$ be the critical point set of $a_H$, i.e., the set of smooth contractible loops satisfying the Euler-Lagrange equation. The image $\bar{{\bf R}}(H)$ of ${\bf R}(H)$ in ${\cal L}(V)$ one-to-one corresponds to the fixed points of $\phi_1(H)$ and hence is a finite set. Since $\phi_1(H)$ is non-degenerate, it implies that ${\bf R}(H)$ is the set of non-degenerate critical points of $a(H)$. But ${\bf R}(H)$ may have infinitely many points, which are generated by the covering transformation group $\pi_2(V)$. Given $(x, u)\in {\bf R}(H)$, choose a symplectic trivialization $$\Phi(t): {\bf R}^{2n}\rightarrow T_{x(t)}V$$ of $u^*TV$ which extends over the disc $D$. Linearizing the Hamiltonian differential equation along $x(t)$, we obtain a path of symplectic matrices $$A(t)=\Phi(t)^{-1}d\phi_t(x(0))\Phi(0)\in Sp(2n, {\bf R}).$$ Here the symplectomorphism $\phi_t: X\rightarrow X$ denotes the time-$t$-map of the Hamiltonian flow $$\dot{\phi_t}=\nabla H_t(\phi_t).$$ Then, $A(0)=Id$ and $A(1)$ is conjugate to $d\phi_1(x(0))$. Non-degeneracy means that $1$ is not an eigenvalue of $A(1)$. Then, we can assign a Conley Zehnder index for $A(t)$. We can decomposed ${\bf R}(H)$ as $${\bf R}(H)=\cup_i {\bf R}_i(H),$$ where ${\bf R}_i(H)$ consists of critical points in ${\bf R}(H)$ with the Conley-Zehnder index $i$. To define Floer homology, we first construct a chain complex and a boundary map $(C_*(X, H), \delta)$. The chain complex $$C^*(X, H)=\otimes_i C_i(X,H).\leqno(6.5)$$ where $C_i(X, H)$ is a ${\bf R}$-vector space consisting of $\sum_{\mu(\tilde{x})=i} \xi(\tilde{x})\tilde{x}$ where the coefficients $\xi(\tilde{x})$ satisfy the finiteness condition that $$\{\tilde{x}; \xi(\tilde{x})\neq 0, a_H(\tilde{x})>c\}$$ is a finite set for any $c\in {\bf R}$. We recall that the Novikov ring $\Lambda _ \omega$ is defined as the set of formal sum $\sum_{A\in \pi_2(X)}\lambda_A e^A$ such that for each $c>0$, the number of nonzero $\lambda_A$ with $\omega(A) \leq c$ is finite. For each $(x, u_x)\in {\bf R}(H)$, we define $$e^A(x, u_x)=(x, u_x\#A),$$ where $\#$ is the connected sum operation in the interior of disc $u_x$. It is easy to check that $$\mu(e^A(x, u_x))=2C_1(A)+\mu(x, u_x).\leqno(6.6)$$ It induces an action of Novikov ring $\Lambda_{\omega}$ on $C_*(V,H)$. Next we consider the boundary map, where we have to study the moduli space of trajectories. Let $J(x)$ be a compatible almost complex structure of $\omega$. We can consider the perturbed gradient flow equation of $a_H$: $${\cal F}(u(s,t))=\frac{\partial u}{\partial s}+ J(u)\frac{\partial u}{\partial t}+ \bigtriangledown H(t, u)=0,$$ where we use $s$ to denote the time variable and $t$ to denote the circle variable. At this point, we ignore the homotopic class of disc, which we will discuss later. Let $$\tilde{{\cal M}}=\{ u: S^1\times {\bf R}\rightarrow {\bf R}\,|\, {\cal F}(u)=0, E(u)=\int_{S^1\times {\bf R}}( |\frac{\partial u}{\partial s}|^2+|J(u)\frac{\partial u}{\partial t}+ \bigtriangledown H(t, u)|^2)dsdt<\infty\}.$$ Because $a(H)$ has only non-degenerate critical points, the following lemma is well-known. \vskip 0.1in \noindent {\bf Lemma 6.3.} {\it For every $u\in \tilde{{\cal M}}$, $u_s(t)=u(s, t)$ converges to $x_{\pm} (t)\in \bar{{\bf R}}(H)$ when $s\rightarrow \pm \infty$. If $H$ is non-degenerate, $u_s$ converges exponentially to its limit, i.e., $|u_s-u_{\pm \infty}|<C e^{-\delta |s|}$ for $s\geq |T|.$} \vskip 0.1in By this lemma, we can divide $\tilde{{\cal M}}$ into $$\tilde{{\cal M}}=\bigcup_{x^-, x^+\in \bar{{\bf R}}} {\cal M}(x^-, x^+; H, J),$$ where $$\tilde{{\cal M}}(x^-,x^+; H, J)=\{u\in \tilde{{\cal M}}; \lim_{s\rightarrow -\infty}u_s=x^-, \lim_{s \rightarrow \infty}u_s=x^+\}.$$ Clearly, ${\bf R} ^1$ acts on $\tilde{{\cal M}}(x^-,x^+; H, J)$ as translations in time. Let $${\cal M}(x^-,x^+; H, J)=\tilde{{\cal M}}(x^-,x^+; H, J)/{\bf R} ^1.\leqno(6.7)$$ ${\cal M}(x^-, x^+; H, J)$ consists of the different components of different dimensions. For each $(x^-, u^-), (x^+, u^+)\in {\bf R}(H)$, let ${\cal M}((x^-, u^-),(x^+, u^+); H,J)$ be the components of ${\cal M}(x^-,x^+; H,J)$ satisfying that $$(x^+, u^-\#u)\cong (x^+, u^+)$$ for any $u\in{\cal M}((x^-, u^-),(x^+, u^+); H,J)$. Then, the virtual dimension of ${\cal M}((x^-, u^-),(x^+, u^+); H,J)$ is $\mu(x^+, u^+)-\mu(x^-, u^-)-1$. Next, we need a stable compactification of ${\cal M}(x^-, x^+; H, J)$. \vskip 0.1in \noindent {\bf Definition 6.4: }{\it A stable trajectory (or symplectic gradient flow line) $u$ between $x^-, x^+$ consists of trajectories $u_0\in {\cal M}(x^-, x_1;H,J), u_1\in {\cal M}(x_1, x_2; H, J) \cdots, u_k\in {\cal M}(x_k, x^+)$ and finite many genus zero stable $J$-maps $f_, \cdots, f_m$ with one marked point such that the marked point is attached to the interior of some $u_i$. Furthermore, if $u_i$ is a constant trajectory, there is at least one stable map attaching to it (compare with ghost bubble). We call two stable trajectories to be equivalent if they are different by an automorphism of the domain. For each stable map $f$, we define $E(f)=\omega( A)$ and denote the sum of the energy from each component by $E(u)$. If we drop the perturbed Cauchy Riemann equation from the definition of trajectory and Cauchy Riemann equation from the definition of genus zero stable maps, we simply call it a flow line. } \vskip 0.1in Suppose that $\overline{{\cal M}}((x^-,u^-), (x^+, u^+); H,J)$ is the set of the equivalence classes of stable trajectories $u$ between $x^-, x^+$ such that $E(u)=a(x^+)- a(x^-)$ and $(x^+, u^-\# u)\cong (x^+, u^+)$ . Let $\overline{{\cal B}}((x^-,u^-), (x^+, u^+))$ be the space of corresponding flow lines. A slight modification of \cite{PW} shows that \vskip 0.1in {\bf Theorem 6.5:(\cite{PW})}{\it $\overline{{\cal M}}((x^-,u^-), (x^+, u^+); H,J)$ is compact.} \vskip 0.1in We will leave the proof to readers. The configuration space is $\overline{{\cal B}}_{\delta}((x^-,u^-), (x^+, u^+))$-the space of flow lines converging exponentially to the periodic orbits $(x^-,u^-), (x^+, u^+)$. Next, we construct a virtual neighborhood using the construction of section 3. Since the construction is similar, we shall outline the difference and leave to readers to fill out the detail. The unstable component is either a unstable bubble or a unstable trajectory $u\in {\cal B}_{\delta}((x^-,u^-), (x^+,u^+))$ where ${\cal B}_{\delta}((x^-,u^-), (x^+,u^+))$ is the space of $C^{\infty}$-map from $S^1\times (-\infty, \infty)$ converging expentially to the periodic orbits. When $u$ is a unstable trajectory, $u$ is a non-constant trajectory and has no intersection point in the interior. Theirfore, ${\bf R}$ acts freely on $Map_{\delta}((x^-,u^-), (x^+,u^+))$ We want to show that $${\cal B}_{\delta}((x^-,u^-), (x^+,u^+))=Map_{\delta}((x^-,u^-), (x^+,u^+))/{\bf R} \leqno(6.8)$$ is a Hausdorff Frechet manifold. Using the same method of Lemma 3.4, we can show that $${\cal B}_{\delta}((x^-,u^-), (x^+,u^+))$$ is Hausdorff. For any $u\in {\cal B}_{\delta}((x^-,u^-), (x^+,u^+))$, we can construct a slice $$W_u=\{u^w; w\in \Omega^0(u^*TV), w_s \mbox{ converges expentially to zero and } ||w||_{L^p_1}<\epsilon, ||w||_{C^1(D_{\delta_0}(e))}<\epsilon, w\perp \frac{\partial u}{\partial s}(e)\},\leqno(6.9)$$ where $\frac{\partial u}{\partial s}$ is injective at $e$. Let $u\in \overline{{\cal B}}_{\delta}((x^-, u^-), (x^+, u^+))$ be a stable trajectory. Recall that for closed case, the gluing parameter for each nodal point is ${\bf C}$. For the trajectory, it satisfies the perturbed Cauchy Riemann equation. In particular, the Hamiltonian perturbation term depends on the circle parameter. Therefore, the rotation along circle is not a automorphism of the equation. The gluing parameter is only a real number in ${\bf R}^+$. If we have more than two components of broken trajectories. The gluing parameter is a small ball of $$I_k=\{(v_1, \cdots, v_k); v_i\in {\bf R} \& v_i\geq 0\},\leqno(6.10)$$ where $k+1$ is the number of broken trajectories of $u$. We call $u$ {\em a corner point}. \vskip 0.1in \noindent {\bf Remark: }{\it A minor midification of Siebert's construction (Appendix) is needed in this case. For the trajectory component, $H^0, H^1$ should be understood as the space of sections which are exponentially decay at infinity. Recall that the vanishing theorem of $H^1$ was proved by certain Weitzenbock formula, which still holds in this case.} \vskip 0.1in The obstruction bundle $\overline{{\cal F}}_{\delta}((x^-, u^-), (x^+, u^+))$ can be constructed similarly. Sometimes, we shall drop $u^-, u^+$ from the notation without any confusion. For the corner point, a special care is need to construct stabilizing term $s_{x^-, x^+}$. The idea is to construct a stabilized term first in a neighborhood of bottom strata. Then, we process to the next strata until we reach to the top. Furthermore, we need to construct stabilization terms for all the moduli spaces of stable trajectories at the same time. We can do it by the induction on the energy. Since there is a minimal energy for all the stable trajectories, the set of the possible values of the energy of stable trajectories are discrete. We can first construct a stabilization term for the stable trajectories of the smallest energy and then proceed to next energy level. By the compactness theorem, there are only finite many topological type of stable trajectories below any energy level. To simplify the notation, let's assume that the maximal number of broken trajectories for the element of $\overline{{\cal B}}_{\delta}(x^-, x^+)$ is $3$ and there are three energy levels. We leave to readers to fill out the detail for general case. Suppose that $u=(u_1, u_2, u_3)$, where $u_i$ is a trajectory connecting $x^{i-1}$ to $x^i$ attached by some genus zero stable maps. Moreover, $x^0=x^-, x^1,x^2, x^3=x^+$. Since $u_i$ is not a corner point, we can construct $s_{u_i}$ in the same way as section 3. Here, we require the value of $s_{u_i}$ to be compactly supported away form the gluing region. Note that over $$\overline{{\cal B}}_{\delta}(x^-, x^1)\times \overline{{\cal B}}_{\delta}(x^1, x^2)\times \overline{{\cal B}}_{\delta}(x^2, x^+),$$ the obstruction bundle $\overline{{\cal F}}_{\delta}(x^-, x^+)$ is naturally decomposed as $$\overline{{\cal F}}_{\delta}(x^-, x^1)\times \overline{{\cal F}}_{\delta}(x^1, x^2)\times \overline{{\cal F}}_{\delta}(x^2, x^+).\leqno(6.11)$$ Then, $s_{u_1} \times s_{u_2}\times s_{u_3}$ is a section on $$\overline{{\cal B}}_{\delta}(x^-, x^1)\times \overline{{\cal B}}_{\delta}(x^1, x^2)\times \overline{{\cal B}}_{\delta}(x^2, x^+)$$ supported in a neighborhood of $u$. Since its value is supported away from the gluing region, it extends naturally over a neighborhood of $u$ in $\overline{{\cal B}}_{\delta}( x^-, x^+)$. We multiple it by a cut-off function as we did in the section 3. Then, we can treat $s_{u_1}\times s_{u_2}\times s_{u_3}$ as a section supported in a neighborhood of $u$ in $\overline{{\cal B}}_{\delta}( x^-, x^+)$. By the assumption, $$\overline{{\cal M}}(x^-, x^1)\times \overline{{\cal M}}(x^1, x^2)\times \overline{{\cal M}}(x^2, x^+)$$ is compact. We construct finite many such sections such that the linearlization of the extend equation $${\cal S}_e=\bar{\partial}_J+\bigtriangledown H+\sum s_{u_i}$$ is surjective over the bottom strata. Let $$s_3=\sum_i s_{u_i}$$ to indicate that it is supported in neighborhood of third strata. Next, let's consider the next strata $$\overline{{\cal M}}(x^-, x^1)\times \overline{{\cal M}}(x^1, x^+) \cup \overline{{\cal M}}(x^-, x^2)\times \overline{{\cal M}}(x^2, x^+).$$ Two components are not disjoint from each other. Then have a common boundary in the bottom strata. By our construction, the restriction of $s_3$ over next strata is naturally decomposed as $$s^3_{(x^-, x_1)}\times s^3_{(x_1, x^+)}, s^3_{(x^-, x_2)}\times s^3_{(x_2, x^+)}.$$ Then, we construct a section of the form $$s^2_{(x^-, x_1)}\times s^2_{(x_1, x^+)}, s^2_{(x^-, x_2)}\times s^2_{(x_2, x^+)}$$ supported away from the bottom strata. Then, we extend it over a neighborhood of the second strata in $\overline{{\cal B}}_{\delta}(x^-, x^+)$. Over the top strata, we construct a section supported away from the lower strata. In general, the stabilization term $s_{x^-, x^+}$ is the summation of $s_i$, where $s_i$ is supported in a neighborhood of $i$-th strata and away from the lower strata. Suppose that the corresponding vector spaces are $${\cal E}^{m_{x^-, x^+}}=\prod_i {\cal E}_i.\leqno(6.12)$$ We shall choose $$\Theta_{x^-, x^+}=\prod_i\Theta_i, \leqno(6.13)$$ where $\Theta_i$ is a Thom form supported in a neighborhood of zero section of $E_i$. with integral $1$. We call such $(s_{x^-, x^+}, \Theta_{x^-, x^+})$ {\em compatible with the corner structure} and the set of $(s_{x^-, x^+}, \Theta_{x^-, x^+})$ for all $x^-, x^+$ {\em a system of stabilization terms compatible with the corner structure}. Suppose that $(s_{x^-, x^+}, \Theta_{x^-, x^+})$ is compatible with the corner structure. It has following nice property. (i) $s_{x^-, x^+}=s^t+s_l$, where $s^t$ is supported away from lower strata and $s_l$ is supported in a neighborhood of strata. (ii) the restriction of $s_l$ to any boundary component preserves the product structure. Namely, we view $$\partial \overline{{\cal B}}_{\delta}(x^-, x^+)=\bigcup_x \overline{{\cal B}}_{\delta}(x^-, x)\times \overline{{\cal B}}_{ \delta}(x, x^+).\leqno(6.14)$$ The restriction of $s_l$ is of the form $$\bigcup_x s_{x^-, x}\times s_{x, x^+}\times \{0\}.\leqno(6.15)$$ Let $(U_{x^-, x^+}, {\cal E}^{x^-, x^+}, S_{x^-, x^+})$ be the virtual neighborhood. Then, $U_{x^-, x^+}$ is a finite dimensional V-manifold with the corner. $$\partial U_{x^-,x^+}=\bigcup_{x}E^{ot}_{U_{x^-,x}\times U_{x,x^+}},$$ where $U_{x^-,x}, U_{x, x^+}$ are the virtual neighborhoods constructed by $s_{x^-, x}, s_{x, x^+}$ and $E^{ot}$ is the product of other $E_i$ factors. When $\mu(x^+)=\mu(x^-)+1$, $dim U_{x^-, x^+}=deg \Theta_{x^-, x^+}$. We define $$<(x^+,u^+), (x^-,u^-)>=\int_{U_{x^-,x^+}} S^*_{x^-,x^+}\Theta_{x^-, x^+},$$ where $(s_{x^-, x^+}, \Theta_{x^-, x^+})$ is compatible with the corner structure. When $\mu(x^+)<\mu(x^-)+1$, $dim U_{x^-, x^+}<deg \Theta_{x^-, x^+}$, we define $$<(x^+,u^+), (x^-,u^-)>=\int_{U_{x^-,x^+}} S^*_{x^-,x^+}\Theta_{x^-, x^+}=0,\leqno(6.17)$$ For any $x\in C_k(X,H)$, we define a boundary operator as $$\delta x=\sum_{y\in C_{k-1}}<x,y>y.\leqno(6.18)$$ Novikov ring naturally acts on $C_*(V, H)$ by $e^A(x,u)=(x, u\#A)$ for $A\in \pi_2(X)$. Furthermore, it is commutative with the boundary operator. Next, we show that \vskip 0.1in \noindent {\bf Proposition 6.6: }{\it $\delta^2=0$.} \vskip 0.1in {\bf Proof: } $$\delta^2 x=\sum_{z\in C_{k-2}}\sum_{y\in C_{k-1}}<x y><y,z>z.\leqno(6.19)$$ Let $<x,z>^2=\sum_{y\in C_{k-1}}<x y><y,z>$. It is enough to show that $$<x, z>^2=0. \leqno(6.20)$$ Consider ${\cal M}(x, z; H, J)$. Its stable compactification $\overline{{\cal M}}(x,z; H,J)$ consists of broken trajectories of the form $(u_0, u_1; f_1, \dots, f_m)$ for $u_0\in \overline{{\cal M}}(x, y; H, J), u_1\in \overline{{\cal M}}(y, z; H, J)$. Choose compatible $(s_{x,z}, \Theta_{x, z})$. The boundary components $$\partial \overline{{\cal B}}_{\delta}(x,z)=\bigcup_{y} \overline{{\cal B}}_{x,y}\times \overline{{\cal B}}_{y, z},\leqno(6.21)$$ where $\overline{{\cal B}}_{x, y}, \overline{{\cal B}}_{y,z}$ are the configuration spaces of $\overline{{\cal M}}( x,y, H, J)$, $\overline{{\cal M}}(y,z; H, J)$, respectively. Furthermore, $\overline{ {\cal F}}_{x,z}$ is naturally decomposed,i.e., $$\overline{{\cal F}}_{x,z}|_{\overline{{\cal B}}_{x,y}\times \overline{{\cal B}}_{y, z}}= \overline{{\cal F}}_{x,y}\times \overline{{\cal F}}_{y,z}.\leqno(6.22)$$ Suppose that the resulting virtual neighborhood by $s_{x,z}$ is $(U_{x,z}, E^{x,z}, S_{x,z})$. Then, $$\partial{U_{x,z}}=\bigcup_{y}E^{ot}_{ U_{x,y}\times U_{y,z}}.\leqno(6.23)$$ Note that $dim U_{x,z}=deg \Theta_{x,z}+1$. $$\begin{array}{ccl} 0&=&\int_{U_{x,z}}S^*_{x,z}d(\Theta_{x,z})\\ &=&\int_{\partial U_{x, z}}S^*_{x,z}(\Theta_{x,z})\\ &=&\sum_{y}\int_{U_{x,y}\times U_{y,z}}(S_{x,y}\times S_{y,z})^* (\Theta_{x,y}\times \Theta_{y,z})\\ &=&\sum_y <x,y><y,z>\\ &=&\sum_{y\in C_{k-1}} <x,y><y,z>, \end{array} \leqno(6.24)$$ where the last equality comes from (6.17). We finish the proof. $\Box$ \vskip 0.1in \noindent {\bf Definition 6.7: }{\it We define Floer homology $HF_*(X,H)$ as the homology of chain complex $(C_*(X, H), \delta)$} \vskip 0.1in Since the action of Novikov ring $\Lambda_{\omega}$ is commutative with the boundary operation $\delta$, Novikov ring acts on $HF_*(X, H)$ and we can view $HF_*(X,H)$ as a $\Lambda_{\omega}$-module. \vskip 0.1in \noindent {\bf Remark 6.8: }{\it The boundary operator $\delta$ may depend on the choice of compatible $\Theta_{x^-, x^+}$. However, Floer homology is independent of such a choice.} \vskip 0.1in \noindent {\bf Proposition 6.9: }{\it $HF_*(X, H)$ is independent of $(H,J)$ and the construction of the virtual neighborhood and the choice of compatible $\Theta_{x^-, x^+}$.} \vskip 0.1in The proof is routine. We leave it to the readers. \vskip 0.1in \noindent {\bf Theorem 6.10: }{\it $HF_*(X, H)=H_*(X, \Lambda_{\omega})$ as a $\Lambda_{ \omega}$-module.} \vskip 0.1in \noindent {\bf Corollary 6.11: }{\it Arnold conjecture holds for any symplectic manifold.} \vskip 0.1in The basic idea is to view $HF_*(X, H)$ and $H_*(X, \Lambda_{\omega})$ as the special cases of the Bott-type Floer homology \cite{RT3}, where $H_*(X, \Lambda_{\omega})$ is Floer homology of zero Hamiltonian function. The isomorphism between them is interpreted as the independence of Bott-type Floer homology from Hamiltonian functions. Instead of giving the general construction of Bott type Floer homology, we shall construct the isomorphism between $HF_*(X, H)$ and $H_*(X, \Lambda_{\omega})$ directly. It consists of several lemmas. Let $\Omega_i(X)$ be the space of the differential forms of degree $i$. Let $C_m(V, \Lambda_{\omega})=\oplus_{i+j=m}\Omega^{2n-i}( X)\otimes \Lambda^j_{\omega}$, where we define $deg(e^A)=2C_1(X)(A)$. For $\alpha\in \Omega^{2n-i}(X)$, define $\delta(\alpha)=d\alpha\in \Omega^{2n- (i-1)}$. The boundary operator is defined by $$\delta(\alpha\otimes \lambda)=\delta(\alpha)\otimes \lambda \in C_{m-1}(V, \Lambda_{ \omega}).\leqno(6.25)$$ Clearly, its homology $$H(C_*(V, \Lambda_{\omega}), \delta)=H_*(V, \Lambda_{\omega}).\leqno(6.26)$$ Consider a family of Hamiltonian function $H_s$ such that $H_s=0$ for $s<-1$ and $H_s=H$ for $s<1$. Furthermore, we choose a family of compatible almost complex structures $J(s,x)$ such that $J_s=J$ for $s<-1$ is $H$-admissible. Moreover, $J_s=J_0$ for $s>1$. Consider the moduli space of the solutions of equation $${\cal F}((J_s), (H_s))=\frac{\partial u}{\partial s}+J(t,s, u(t,s))\frac{ \partial u}{\partial t}-\bigtriangledown H$$ $S^1\times (-\infty, +\infty)$ is conformal equivalent to ${\bf C}-0$ by the map $$e^z: S^1\times (-\infty, +\infty)\rightarrow {\bf C}.\leqno(6.27)$$ Hence, we can view $u$ as map from ${\bf C}-\{0\}$ to $V$ which is holomorphic near zero. By removable singularity theorem, $u$ extends to a map over ${\bf C}$ with a marked point at zero. In another words, $\lim_{s\rightarrow -\infty}u_s=pt$. Furthermore, when the energy $E(u)<\infty$, $u(s)$ converges to a periodic orbit when $s\rightarrow \infty$ by Lemma 6.3. Let ${\cal M}(pt,x^+)$ be the space of $u$ such that $\lim_{s\rightarrow \infty} u_s=x^+$. ${\cal M}(pt, x^+)$ has many components of different dimensions. We use ${\cal M}(pt, A; x^+, u^+)$ to denote the components satisfying $u\#u^+=A$. Consider the stable compactification $\overline{{\cal M}}(pt, A; x^+, u^+)$ in the same fashion. The virtual dimension of ${\cal M}(pt, A; x^+, u^+)$ is $\mu(x^+, u^+)-2C_1(V)(A)$. Choose the stabilization terms $(s_{pt, A, x^+}, \Theta_{pt, A, x^+})$ compatible with the corner structure. Its virtual neighborhood $(U(A; x^+, u^+), E(A; x^+,u^+), S(A; x^+, u^+))$ is a smooth V-manifold with corner. Notice $$\partial(\overline{{\cal B}}(A; x^+, u^+))= \bigcup_{(x, u)}\overline{{\cal B}}(pt,A; x,u) \times \overline{{\cal B}}((x,u); (x^+,u^+)).\leqno(6.28)$$ By our construction, $$\partial(U(A; x^+, u^+))\cong \bigcup_{(x,u)}E^{ot}_{U(A; x, u)\times U((x,u); (x^+,u^+))} .\leqno(6.29)$$ Moreover, $$S(A, x^+, u^+)|_{\partial(U(A; x^+, u^+))}=\bigcup_{(x,u)}S(A; x, u)\times S((x,u); (x^+,u^+)),\leqno(6.30)$$ Let $e_{-\infty}$ be the evaluation map at $-\infty$. We define a map $$\psi: C_m(V, \Lambda_{\omega})\rightarrow C_m(V, H)$$ by $$\psi(\alpha, A;x^+, u^+)=\sum_{i=\mu(x^+, u^+)-2C_1(V)(A)}<\alpha,A; x^+, \mu^+> (x^+, u^+),\leqno(6.31)$$ where $$<\alpha, A; x^+, \mu^+>=\int_{U(A; x^+, u^+)}e^*_{-\infty}\alpha\wedge S(A; x^+, u^+)^*\Theta(A; x^+, u^+).\leqno(6.32)$$ \vskip 0.1in \noindent {\bf Lemma 6.12:}{\it (i) $\delta\psi=\psi\delta$. \vskip 0.1in \noindent (ii)$\psi$ is independent of the virtual neighborhood compatible with the corner structure.} \vskip 0.1in {\bf Proof of Lemma:} The proof of (ii) is routine. We omit it. To prove (i), for $\alpha\in \Omega^{2n-(i+1)}(X)$, $$<\delta \alpha, A; x^+, \mu^+>=\int_{\partial U(A; x^+, u^+)}e^*_{-\infty}\alpha\wedge S(A; x^+, u^+)^*\Theta(A; x^+, u^+)\leqno(6.33)$$ $$=\sum_{(x,u)}\int_{ U(A; x,u)}e^*_{-\infty}(\alpha)\wedge S(A; x, u)^*\Theta(A; x, u)\int_{U((x,u); (x^+, u^+))} S((x,u); (x^+,u^+))^*\Theta(x,u); (x^+,u^+)).$$ However, $$\dim (U(A; x, u))-deg(\Theta(A; x,u))=\mu(x,u)-2C_1(V)(A)<deg(\alpha)$$ unless $\mu(x,u)=\mu(x^+,u^+)+1$. Hence, $$\begin{array}{lll} &&\int_{\partial U(A; x^+, u^+)}\beta\wedge S(A; x^+, u^+)^*\Theta(A; x^+, u^+)\\ &=&\sum_{\mu(x,u)\mu( x^+, u^+)+1}\int_{ U(A; x,u)}\alpha\wedge S(A; x, u)^*\Theta(A; x, u)\\ &&\int_{U((x,u); (x^+, u^+))} S((x,u); (x^+,u^+))^*\Theta(x,u); (x^+,u^+))\\ &=&\psi\delta(x^+,u^+). \end{array}\leqno(6.34)$$ $\Box$ Therefore, $\psi$ induces a homomorphism on Floer homology. Consider a family of Hamiltonian function $H_s$ such that $H_s=0$ for $s>1$ and $H_s=H$ for $s<-1$. Furthermore, we choose a family of compatible almost complex structures $J(s,x)$ such that $J_s=J$ for $s<-1$. Moreover, $J_s=J_0$ for $s>1$. Consider the moduli space of the solutions of equation $${\cal F}((J_s), (H_s))=\frac{\partial u}{\partial s}+J(t,s, u(t,s))\frac{\partial u}{ \partial t}-\bigtriangledown H$$ $S^1\times (-\infty, +\infty)$ is conformal equivalent to ${\bf C}-0$ by the map $$e^{-z}: S^1\times (-\infty, +\infty)\rightarrow {\bf C}.\leqno(6.35)$$ Hence, we can view $u$ as map from ${\bf C}-\{0\}$ to $V$ which is holomorphic near zero. By removable singularity theorem, $u$ extends to a map over ${\bf C}$ with a marked point at zero. In another words, $\lim_{s\rightarrow \infty}u_s=pt$. Furthermore, when the energy $E(u)<\infty$, $u(s)$ converges to a periodic orbit when $s\rightarrow -\infty$ by Lemma 6.3. Let ${\cal M}(pt,x^-)$ be the space of $u$ such that $\lim_{s\rightarrow -\infty} u_s=x^-$. ${\cal M}(pt,x^-)$ has many components of different dimension. We use ${\cal M}(x^-, u^-; pt,A)$ to denote the components satisfying $u^-\#u=A$. The virtual dimension of ${\cal M}(x^-, u^-)$ is $\mu(x^-, u^-)-2C_1(V)(A)$. Consider the stable compactification $\overline{{\cal M}}( x^-, u^-; pt,A)$ and its configuration space $\overline{{\cal B}}_{\delta}(x^-, u^-; pt, A)$. Choose the stabilization terms $(s_{x^-; pt}, \Theta_{x^-, pt})$ compatible with the corner structure. Furthermore, by adding more sections, we can assume that the evaluation map $e_{\infty}$ is a submersion. Then, we define $$\phi: C_m(V,H)\rightarrow C_m(V, \Lambda_{\omega})$$ by $$\phi(x^-,u^-)=\sum_{A}<x^-, u^-; A>e^A.\leqno(6.36)$$ where $$<x^-, u^-; A>=(e_{\infty})_*S(x^-,u^-;A)^*\Theta(x^-,u^-;A)\in \Omega^{2n-i} (X)\leqno(6.37)$$ for $i=\mu(x^-, u^-)-2C_1(X)(A)$. \vskip 0.1in \noindent {\bf Lemma 6.13:}{\it (i)$\phi\delta=\delta \phi$. (ii)$\phi$ is independent of the choice of the virtual neighborhood compatible with the corner structure.} \vskip 0.1in {\bf Proof:} The proof of (i) is routine and we omit it. To prove (i), $$\begin{array}{lll} &&d<x^-, u^-; A>\\ &=&(e_{\infty})_*dS(x^-,u^-;A)^*\Theta(x^-,u^-;A)=(e_{\infty}|_ {\partial U(x^-, u^-; A)})_*S(x^-,u^-;A)^*\Theta(x^-,u^-;A)\\ &=&\sum_{\mu(x,u)=\mu(x^-,u^-)-1}(e_{\infty})_*S(x,u;A)^*\Theta( x,u;A)\int_{U((x^-, u^-); (x, u))}S((x^-, u^-); (x, u))^*\Theta((x^-, u^-); (x, u))\\ &=&\phi\delta(x^-, u^-). \end{array} \leqno(6.38)$$ $\Box$ \vskip 0.1in \noindent {\bf Lemma 6.14:}{\it $\phi\psi=Id$ and $\psi\phi=Id$ as the homomorphisms on Floer homology.} \vskip 0.1in {\bf Proof: } The proof is tedious and routine. We omit it. \section{Appendix} This appendix is due to B. Seibert \cite{S1}. We use the notation of the section 2. \vskip 0.1in \noindent {\bf Lemma A1: }{\it Any local V-bundle of $\overline{{\cal B}}_A(Y,g, k)$ is dominated by a global $V$-bundle.} \vskip 0.1in {\bf Proof: } The construction of global $V$-bundle imitates the similar construction in algebraic geometry. First of all, we can slightly deform $\omega$ such that $[\omega]$ is a rational class. By taking multiple, we can assume $[\omega]$ is an integral class. Therefore, it is Poincare dual to a complex line bundle $L$. We choose a unitary connection $\bigtriangledown$ on $L$. There is a line bundle associated with the domain of stable maps called dualized tangent sheaf $\lambda$. The restriction of $\lambda_C$ on $C$ is $\lambda_C(x_1, \dots, x_k)$-the sheaf of meromorphic 1-form with simple pole at the intersection points $x_1, \dots, x_k$. $\lambda_C$ varied continuously the domain of $f$. For any $f\in \overline{{\cal B}}_A(Y,g,k)$, $f^*L$ is a line bundle over $dom(f)$ with a unitary connection. It is well-known in differential geometry that $f^*L$ has a holomophic structure compatible with the unitary connection. Note that $L$ doesn't have holomoprhic structure in general. Therefore, $f^*L\otimes \lambda_C$ is a holomorphic line bundle. Moreover, if $D$ is not a ghost component, $\omega(D)>0$ since it is represented by a $J$-map. Therefore, $C_1(f^*L)(D)>0$. For ghost component, $\lambda_C$ is positive. By taking the higher power of $f^*L\otimes \lambda_C$, we can assume that $f^*L\otimes\lambda_C$ is very ample. Hence, $H^1(f^*L\otimes \lambda_C)=0$. Therefore, ${\cal E}_f=H^0(f^*L\otimes \lambda_C)$ is of constant rank. It is easy to prove that ${\cal E}=\cup_f {\cal E}_f$ is bundle in terms of topology defined in Definition 3.10. To show that ${\cal E}$ dominates any local $V$-bundle, we recall that the group ring of any finite group will dominate (or map surjectively to) any of its irreducible representation. So it is enough to construct a copy of group ring from ${\cal E}_f$. However, $stb_f$ acts effectively on $dom(f)$. We can pick up a point $x\in dom(f)$ in the smooth part of $dom(f)$ such that $stb_f$ acts on $x$ effectively. Then, $stb_f(x)$ is of cardinality $|stb_f|$. By choose higher power of $f^*L\otimes \lambda_C$, we can assume that there is a section $v\in {\cal E}_f$ such that $v(x)=1, v(g(v))=0$ for $g\in stb_f, g\neq id$. Then, $stb_f(v)$ generates a copy of the group ring of $stb_f$.

9611

alg-geom/9611030

en

https://arxiv.org/abs/alg-geom/9611030

alg-geom/9611030

Rita Pardini

Margarida Mendes Lopes, Rita Pardini

Irregular canonical double surfaces

LaTeX 2.09, 22 pages, the proof of the main result has been simplified

We study minimal surfaces X of general type with $K^2_X=6p_g-14$ and $q(X)>0$ such that $K_X$ is ample, the image of the canonical map is a canonically embedded surface of general type and the canonical map is not birational. The main result states that if X satisfies the above assumptions and $q(X)\ge 2$, then, apart from a finite number of exceptions, X belongs to an infinite series of examples due to Beauville. The exceptions are described in detail and some new examples are given.

alg-geom

\section{Introduction} \setcounter{defn}{0} \setcounter{equation}{0} Let $X$ be a minimal surface of general type of geometric genus $p_g$, let $\Sigma\subset{\bf P}^{p_g-1}$ be the canonical image of $X$ and let $\phi:X\to\Sigma$ be the canonical map. If $\Sigma$ is a surface but $\phi$ is not birational, then by theorem $3.1$ of \cite{bea} either i) $p_g(\Sigma)=0$ or ii) $\Sigma$ is the canonical image of a surface of general type $S$ whose canonical map is birational (and then, of course, $p_g(\Sigma)=p_g$). Recall (cf. \cite{bea}, thm. $5.5$ or \cite{ha}, page $44$) that the Castelnuovo inequality $K^2\ge 3p_g-7$ holds for surfaces of general type with birational canonical map; so in case ii) $S$ satisfies $K^2_S\ge 3p_g-7$, and $K^2_X\ge 6p_g-14$, with equality holding if and only if the canonical system of $X$ is base point free and the minimal resolution $S$ of $\Sigma$ is on the Castelnuovo line $K^2_S=3p_g-7$ (cf. prop \ref{harris} and proof). Case ii) of the theorem quoted above was thought to be impossible for a long time. In fact only very few examples are known, and all but one series due to Beauville (cf. section \ref{esempi}, example $4$) have bounded invariants. The examples in this infinite series satisfy: $K^2_X=6p_g-14$, $\deg \phi=2$ and $q(X):=h^0(X,\Omega^1_X)=2$. The main purpose of this paper is to show that these are almost the only examples satisfying $K^2_X=6p_g-14$ and $q(X)\geq 2$. Therefore we make the following: \begin{assu}\label{ipointro} Let $X$ be a minimal surface of general type of geometric genus $p_g$, with $K^2_X=6p_g-14$, $q(X)>0$ and $K_X$ ample, let $\Sigma\subset{\bf P}^{p_g-1}$ be the canonical image of $X$ and let $\phi:X\to\Sigma$ be the canonical map: assume that $\Sigma$ is a canonical surface and that $\phi$ is not birational. Moreover, if $p_g=4,5,7$ assume that $\Sigma$ is isomorphic to a divisor with at most rational double points in a ${\bf P}^2$-bundle over ${\bf P}^1$, such that the fibres $F$ of the projection $\Sigma\to{\bf P}^1$ are plane quartics. \end{assu} If the above assumption is satisfied, then the minimal desingularization $S$ of $\Sigma$ satisfies $K^2_S=3p_g-7$. Surfaces with these numerical invariants have been described by Ashikaga and Konno in \cite{ak}: for $p_g=6$ or $p_g\ge 8$, those with birational canonical map are isomorphic to a divisor with at most rational double points in a ${\bf P}^2$-bundle over ${\bf P}^1$, such that the fibres $F$ of the projection $\Sigma\to{\bf P}^1$ are plane quartics. (This accounts for the somewhat funny-looking final part of assumption \ref{ipointro}.) We divide the surfaces $X$ in types $I$ and $I\!I$, according to whether, for a generic fibre $F$, $\phi^*F$ is connected or not. Surfaces of type $I$ are the ``general case'', and, if $q(X)\ge 2$, they correspond to Beauville's examples: \begin{thm}\label{mainone} Assume that \ref{ipointro} holds, that $q(X)\ge 2$ and that $X$ is of type $I$: then $p_g(X)\equiv 1\,\, (\mbox{mod}\,\,\, 3)$, $q(X)=2$, the Albanese surface $A$ of $X$ has an irreducible principal polarization, and $X$ can be constructed as in example $4$ of section \ref{esempi}, with $n=(p_g(X)+3)/4$. \end{thm} Let us remark that we do not know any example of type $I$ surfaces with $q(X)=1$. To establish whether such surfaces exist, and in case of existence whether they have bounded invariants is an interesting problem. On the other hand, surfaces of type $I\!I$ should be regarded as exceptions, and can be described completely: \begin{thm}\label{maintwo} Assume that \ref{ipointro} holds and that $X$ is of type $I\!I$, let $X\to B\to{\bf P}^1$ be the Stein factorization of the pencil $\phi^*|F|$ and let $g$ be the genus of $B$: then there exist integers $0\le a\le b \le c$ with $c\le g$ and $a+b+c=p_g-3$ such that $\Sigma$ is isomorphic to a divisor in $\pp_{a,b,c}:={\rm Proj}({\cal O}_{\pp^1}(a)\oplus{\cal O}_{\pp^1}(b)\oplus{\cal O}_{\pp^1}(c))$ with the following properties: i) $\Sigma$ is linearly equivalent to $4T-(a+b+c-2)L$, where $T$ is the tautological hyperplane section and $L$ is the fibre of $\pp_{a,b,c}$ (and $F=L|_{\Sigma}$), ii) the pencil $|F|$ on $\Sigma$ has precisely $2g+2$ double fibres, iii) the only singularities of $\Sigma$ are nodes and $\Sigma$ is smooth outside the double fibres of $|F|$. The double fibres of $|F|$ occur at the branch points of $B$ and each contains $8$ nodes . Conversely, given integers $0\le a\le b\le c$ and $g$, with $c\le g$, if $\Sigma\subset \pp_{a,b,c}$ is a divisor satisfying conditions i),ii),iii) above, then $\Sigma$ has $16g+16$ nodes and there exists a double cover $\phi:X\to\Sigma$ branched over the nodes such that $X$ is a surface of type $I\!I$ and $\phi$ is the canonical map of $X$. The numerical possibilities for the invariants of $X$ are the following: \smallskip \noindent a) $p_g=3g+3$,\quad $q=g$,\quad\quad $a=b=c=g$,\quad $0<g\le 26$; \noindent b) $p_g=3g+2$,\quad $q=g+1$, \quad $a=b=g-1$, $c=g$, \quad $0<g\le 17$; \noindent c) $p_g=3g+1$,\quad $q=g+2$,\,\,\,\quad $a=b=g-1$, $c=g$ or $a=g-2$, $b=c=g$, \quad $0<g\le 8$. \end{thm} In \cite{bea} it is also proven that if the canonical map $\phi:X\to\Sigma$ is not birational and $\Sigma$ is a canonically embedded surface then $\deg\phi\le 3$ for $\chi({\cal O}_X)\ge 14$, and if $\deg\phi=3$ then $q(X)\le 3$. Thm.\ref{mainone} and \ref{maintwo} imply in particular that the irregularity $q(X)$ is also bounded under assumption \ref{ipointro}. It would be interesting to know whether $q(X)$ is bounded in general for $\deg\phi=2$. Another interesting problem is to study {\em regular} surfaces $X$ such that the canonical map is not birational and the canonical image is a canonically embedded surface: only very few examples are known (cf. section \ref{esempi}) and, lacking the information given by the Albanese map, their structure is quite mysterious even when the invariants satisfy the ``minimal'' relation $K^2_X=6p_g-14$. The paper is organized as follows: in section $1$ we set the notation and recall some facts on double cover that will be used later. In section $2$ we describe the general set-up and establish various facts about $X$, $S$ and $\Sigma$. In particular we study the structure of degenerate fibres of $\phi^*F$ both for type $I$ and type $I\!I$. In section $3$ we describe the construction of all the examples known to us of surfaces of general type with $2$-$1$ canonical map onto a canonical surface. In section $4$ we look at the surfaces of type $I$ with $q(X)\ge 2$ and we show, using a fine analysis involving the Albanese map and the Prym variety of $\phi^*F\to F$ for general $F$, that these are exactly Beauville's examples. In section $5$ we describe the surfaces of type $I\!I$ in detail and determine the possible ranges for their invariants. Section $6$ contains a computation with Macaulay that shows that example $3$ of section $3$ actually exists. {\em Acknowledgements:} We are indebted to several people for useful conversations; we would like to mention in particular C. Birkenhake, F. Catanese, M. Manetti, G. Ottaviani, N. Shepherd-Barron, A. Verra. \section {Notation and conventions}\label{notation} \setcounter{defn}{0} \setcounter{equation}{0} All varieties are normal projective varieties over the complex numbers. The $n$-dimensional projective space is denoted by ${\bf P}^n$, and its group of automorphisms by ${\bf PGL}(n)$. As usual, ${\cal O}_Y$ is the structure sheaf of the variety $Y$, $\HH{i}(Y,{\cal F})$ is the $i$-th cohomology group of a sheaf ${\cal F}$ on $Y$, and $h^i(Y,{\cal F})$ is the dimension of $\HH{i}(Y,{\cal F})$; for a line bundle $M$ on $Y$, we denote by $|M|$ the complete linear system ${\bf P}(\HH{0}(Y,M))$. When dealing with smooth varieties, we do not distinguish between line bundles and divisors. If $Y$ is smooth, then $K_Y$ denotes a canonical divisor and ${\rm Pic}(Y)$ the Picard group of $Y$. If $Y$ is a surface, then $p_g(Y)=h^0(Y,K_Y)$ is the {\em geometric genus} and $q(Y)=h^1(Y,{\cal O}_Y)$ is the {\em irregularity}, $K_Y^2$ is the self-intersection of the canonical divisor; we denote by $\chi(Y)=1-q(Y)+p_g(Y)$ the Euler characteristic of ${\cal O}_Y$ and by $c_2(Y)$ the second Chern class of the tangent bundle of $Y$, or, which is the same, the topological Euler characteristic of $Y$. A surface $Y$ is said to be {\em irregular} if $q(Y)\ne 0$. The intersection number of two divisors $C$, $D$ on a smooth surface is denoted simply by $CD$, linear equivalence is denoted by $\equiv$. A {\em node} of a surface is a double point of type $A_1$, namely a hypersurface singularity that in suitable local analytic coordinates is defined by the equation $x^2+y^2+z^2=0$. A {\em double cover} is a finite map $f:X\to Y$ of degree $2$ between normal projective varieties; we denote by $i:X\to X$ the involution that interchanges the two points of a generic fibre of $f$. In this paper we will need to consider only the following two cases: a) both $X$ and $Y$ are smooth, and b) $X$ is a smooth surface, $Y$ is normal and $f$ is unramified in codimension $1$. In case a), $f$ is a flat map and $f_*{\cal O}_X$ splits under the action of $i$ as $O_Y\oplus {\cal L}^{-1}$, where ${\cal L}$ is a line bundle and $i$ acts on ${\cal L}^{-1}$ as multiplication by $-1$. The branch locus of $f$ is a smooth divisor $B\equiv 2{\cal L}$, the ramification locus is a divisor $R\equiv f^*{\cal L}$ and one has: \begin{eqnarray}\label{formuledoppi} K_X=f^*(K_Y+{\cal L})\quad K_X^2=2(K_Y+{\cal L})^2\quad f_*K_X=K_Y\oplus K_Y+{\cal L}\\ h^i({\cal O}_X)=h^i({\cal O}_Y)+h^i({\cal L}),\quad i=1,\ldots \dim Y \end{eqnarray} (Actually, the above formulas also hold if $Y$ is a surface with rational dpoble points and $B$ is a smooth divisor containing no singulairities of $Y$). The cover $\phi:X\to Y$ can be reconstructed from $Y$, ${\cal L}$, $B$ as follows. Let $p:{\cal L}\to Y$ be the projection, let $w$ be the tautological section of $p^*{\cal L}$ and let $\sigma\in \HH{0}(Y,{\cal L}^2)$ be a section vanishing on $B$: the zero locus in ${\cal L}$ of the section $w^2-p^*\sigma$ of $p^*{\cal L}^2$, together with the restriction of the map $p$, is a double cover of $Y$ isomorphic to $\phi:X\to Y$. Moreover, it is clear that, given a line bundle ${\cal L}$ on $Y$ and a divisor $B$ in the linear system $|{\cal L}^2|$, the above construction yields a finite degree $2$ map $\phi:X\to Y$. A {\em linearization} of a line bundle $N$ on $X$ is an involution $i_N:N\to N$ that lifts the involution $i:X\to X$. If $N$ is a linearized line bundle, we say that $\sigma\in \HH{0}(X, N)$ is {\em even\,} if ${i_N}_*\sigma=\sigma$ and {\em odd\,} if ${i_N}_*\sigma=-\sigma$. A divisor defined by an even (odd) section is called {\em symmetric\,}({\em antisymmetric}). The canonical bundle $K_X$ and the pull-backs of line bundles from $Y$ have natural linearizations: in these cases, unless otherwise stated, we consider the natural linearizations. Consider now case b): the singularities of $Y$ are nodes, that are the images of the fixed points of $i$. If $\nu$ is the number of nodes of $Y$, then one has (see \cite{becky} (0.6)): \begin{equation}\label{nodi} \chi({\cal O}_X)=2\chi({\cal O}_Y)-\frac{1}{4}\nu \end{equation} A set $J$ of nodes on a normal surface $Y$ is said to be {\em even} if there exists a double cover $\phi:X\to Y$ branched precisely over $J$. \begin{prop}\label{criterio} Let $W$ be a smooth $3$-fold, let $Y\subset W$ be a divisor whose only singularities are nodes; if there exists a divisors $D$, $D'$ in $W$ such that $D\equiv 2D'$ and $D$ restricted to $Y$ is equal to $2C$, where $C$ is a curve passing though all the nodes of $Y$, then the nodes of $Y$ are an even set. \end{prop} {\bf Proof:}\,\, Denote by $\eta:\hat{W}\to W$ the blow-up at the nodes of $Y$, by $\epsilon:\hat{Y}\to Y$ its restriction to the strict transform $\hat{Y}$ of $Y$, by $\hat{E_i}$, $E_i$ the exceptional divisors of $\eta$ and $\epsilon$ respectively, and by $\hat{D}$, $\hat{C}$ the strict transforms of $D$ on $\hat{W}$ and of $C$ on $\hat{W}$. One has the following linear equivalence on $\hat{W}$: $2\eta^*D'\equiv\eta^*D=\hat{D}+\sum E_i$, which restricts to $2\epsilon^*D'\equiv\epsilon^*D=2\hat{C}+\sum E_i$. So $\sum E_i\equiv 2(\epsilon^*D'-\hat{C})$, and there exists a smooth double cover $g:\hat{X}\to \hat{Y}$ branched over $\sum E_i$; the ramification divisor of $g$ is a union of disjoint $-1$ curves that can be contracted to yield $f:X\to Y$ branched over the nodes of $Y$. $\quad \diamond$\par\smallskip \section{The set-up}\label{setup} \setcounter{defn}{0} \setcounter{equation}{0} The notations and the assumptions introduced in this section will be maintained throughout all the paper. We start by making the following: \begin{assu}\label{ipotesi} Let $X$ be a minimal surface of general type of geometric genus $p_g$, with $K^2_X=6p_g-14$, $q(X)>0$ and $K_X$ ample, let $\Sigma\subset{\bf P}^{p_g-1}$ be the canonical image of $X$ and let $\phi:X\to\Sigma$ be the canonical map: assume that $\Sigma$ is a canonical surface and that $\phi$ is not birational. Moreover, if $p_g=4,5,7$ assume that $\Sigma$ is isomorphic to a divisor with at most rational double points in a ${\bf P}^2$-bundle over ${\bf P}^1$, such that the fibres $F$ of the projection $\Sigma\to{\bf P}^1$ are plane quartics. \end{assu} Let now $A$ be the Albanese variety of $X$, let $x_0\in X$ be a fixed point of $i$, let $\alpha:X\to A$ be the Albanese map with base point $x_0$ and let $K=A/<-1>$ be the Kummer variety of $A$. Since $\Sigma$ is regular, the involution $i$ on $X$ induces on $A$ the multiplication by $-1$, and there is an induced map $f:\Sigma\to K$. Thus we have the following {\em basic commutative diagram}, where $q:A\to K$ is the natural projection: \begin{equation}\label{diagram} \begin{array}{rcccl} \phantom{1} & X &\stackrel{\alpha}{\rightarrow} & A & \phantom{1} \\ \scriptstyle{\phi}\!\!\!\!\!\! & \downarrow & \phantom{1} & \downarrow & \!\!\!\!\!\! \scriptstyle{q} \\ \phantom{1} & \Sigma & \stackrel{f}{\rightarrow} & K & \phantom{1} \end{array} \end{equation} Remark that, since $\phi$ is finite, $\phi:X\to\Sigma$ is obtained from $q$ by base change and normalization. Assuming $K^2_X=6p_g-14$ is equivalent to considering the lowest possible value of $K^2$ in the above situation, as it appears from the next proposition and its proof. In order to state it we introduce the following \begin{notdef}\label{castelnuovo} Let $0\le a\le b \le c$ be integers: we write $\pp_{a,b,c} = \mbox{Proj}({\cal O}_{\pp^1}(a)\oplus{\cal O}_{\pp^1}(b)\oplus{\cal O}_{\pp^1}(c))$, and denote by $T$ the tautological hyperplane section and by $L$ the fibre of $\pp_{a,b,c}$. We define a {\rm Castelnuovo surface of type $(a,b,c)$} to be a divisor $\Sigma$ in $\pp_{a,b,c}$ linearly equivalent to $4T-(a+b+c-2)L$ with at most rational double points as singularities. Notice that $T$ restricts to the canonical divisor of $\Sigma$ and that the minimal desingularization $S$ of $\Sigma$ satisfies: $K_S^2=3(a+b+c)+2=3p_g(S)-7$, $q(S)=0$. \end{notdef} \begin{prop}\label{harris} Assume that \ref{ipotesi} holds: then $K_X$ is base point free, the degree of $\phi$ is equal to $2$, $K^2_S=3p_g-7$, the only singularities of $\Sigma$ are nodes, and there exist integers $0\le a\le b \le c$ with $a+b+c=p_g-3$ such that $\Sigma$ is a Castelnuovo surface of type $(a,b,c)$. \end{prop} {\bf Proof:}\,\, Write $K_X=M+Z$, where $M$ is the moving part and $Z$ is the fixed part of $K_X$ and denote by $d$ the degree of $\Sigma\subset{\bf P}^{p_g-1}$. By \cite{ha} page $44$, one has $d\ge 3p_g-7$ and thus: \begin{eqnarray*}\label{ksquare} 6p_g-14=K_X^2=K_XM+K_XZ\ge K_XM= M^2+MZ\ge M^2\ge \\ (\deg\phi)d\ge (\deg\phi)(3p_g-7). \end{eqnarray*} The first and the second inequality are consequences of the fact that $K_X$ and $M$ are nef. It follows that $\deg\phi$ is equal to $2$ and all the above inequalities are equalities. In particular, one has $K_XZ=0$ and $M^2=2d$, and so $Z$ is empty (recall that $K_X$ is ample) and $K_X=M$ is base point free. The surface $\Sigma$ satisfies $d=3p_g-7$: so it is Castelnuovo variety in the sense of \cite{ha}, and in particular (cf. \cite{ha}, page $66$) it is projectively normal, and therefore normal. So $\phi:X\to \Sigma$ is a double cover as defined in section \ref{notation}, the only singularities of $\Sigma$ are nodes, and therefore $\Sigma$ is the canonical model of $S$. Castelnuovo varieties are classified in \cite{ha} (apart from a small mistake corrected in \cite{mi}), and the complete list of those that are canonical surfaces is given in thm. $1.5$ of \cite{ak}, where they are studied in detail. In particular, for $p_g=6$ or $p_g\ge 8$ $\Sigma$ is a Castelnuovo surface of type $(a,b,c)$ for some $0\le a\le b\le c$ with $a+b+c+3=p_g$. $\quad \diamond$\par\smallskip Denote by $|F|$ the pencil on $\Sigma$ induced by the projection $\pp_{a,b,c}\to{\bf P}^1$ and by $|\tilde{F}|$ the pull-back $\phi^*|F|$. Surfaces $X$ as in assumption \ref{ipotesi} fall into two types according to the nature of $|\tilde{F}|$: \begin{defn} We say that a surface $X$ as in assumption \ref{ipotesi} is of type $I$ if $|\tilde{F}|$ is irreducible, and of type $I\!I$ if $|\tilde{F}|$ is reducible. \end{defn} Remark that if $X$ is of type $I$ then $|\tilde{F}|$ is a linear pencil of genus $5$. We will show later (proposition \ref{invariantsII}) that, if $X$ is of type $I\!I$, then there are only a finite number of numerical possibilities for the invariants of $X$, so that one should think of surfaces of type $I$ as of the ``general case''. We close this section by stating some general facts about $X$ and $\Sigma$. \begin{prop}\label{basic} Assume that assumption \ref{ipotesi} holds. Then the map $\phi:X\to\Sigma$ is ramified precisely over the singular locus of $\Sigma$, which consists of $4(1+p_g(X)+q(X))$ nodes. \end{prop} {\bf Proof:}\,\, We have already remarked in prop. \ref{harris} and its proof that $\phi:X\to \Sigma$ is a double cover. Moreover, $\phi$ is necessarily unramified in codimension $1$, since otherwise $K_X$ would have a fixed part. So the number of nodes of $\Sigma$ can be computed by means of formula \ref{nodi}. $\quad \diamond$\par\smallskip \begin{prop}\label{nodes} Every irreducible component of a fibre of the pencil $|F|$ on $\Sigma$ has multiplicity $\le 2$, and every double fibre contains $8$ singular points of $\Sigma$. Moreover, if $X$ is of type $I$, then $|F|$ and $\phi^*|F|$ have no multiple fibres. \end{prop} {\bf Proof:}\,\, The fibres $F$ on $\Sigma$ are (possibly singular) plane quartics. Since the only singularities of $\Sigma$ are nodes and the fibres $F$ are the restriction to $\Sigma$ of smooth Cartier divisors on the smooth threefold $\pp_{a,b,c}$, the fibres of $|F|$ have double points at the singular points of $\Sigma$. So, if $C$ is a component of a fibre of multiplicity $m>2$, then $C$ does not contain any singular point of $\Sigma$. $C$ is necessarily a line and thus, if $C'$ denotes the strict transform of $C$ on $S$, one has: $K_S C'=1$, $C'^2=-3$, $C'(\epsilon^*F-mC')=4-m$. (The last equality is a consequence of the fact that $C'$ contains no singular point of $\Sigma$ and $F$ is a plane quartic.) So it follows: $0=C'\epsilon^*F=mC'^2+C'(\epsilon^*F-mC')=-3m+4-m=4(1-m)$, a contradiction since $m>1$. Let now $2C$ be a double fibre on $\Sigma$, with $C$ an irreducible plane conic. Let $P_1,\ldots P_k$ be the nodes of $\Sigma$ that lie on $C$, let $E_1,\ldots E_k$ be the corresponding $-2$-curves on $S$ and let $C'$ be the strict transform of $C$ on $S$. The pull-back of the fibre $2C$ to $S$ is $\epsilon^*(2C)=2C'+E_1+\ldots+E_k$, and $C'E_i=1$, for $i=1,\ldots k$. If $C$ is irreducible, then we have: $K_SC'=2$, $C'^2=-4$, $0=C'\epsilon^*F=C'(2C'+E_1+\ldots+E_k)=-8+k$, $k=8$. If $C$ consists of a pair of distinct lines, then a similar computation shows that each line contains $4$ nodes of $\Sigma$. Assume now that the surface $X$ is of type $I$, and suppose that $2C$ is a double fibre on $\Sigma$. The pull-back of $2C$ to $X$ is a double fibre $\tilde{F}=2D$, where $D$ is either a smooth hyperelliptic curve of genus $3$, or the sum of two smooth elliptic curves meeting transversely at $2$ points, according to whether $C$ is irreducible or not. In both cases, $D$ is a curve of arithmetic genus $3$. Tensoring with $K_X+\tilde{F}$ the decomposition sequence $0\to{\cal O}_D(-D)\to {\cal O}_{\tilde{F}}\to{\cal O}_D\to 0$ and taking global sections, one obtains the following exact sequence: $$0\to \HH{0}(D,K_D)\to \HH{0}(\tilde{F},K_{\tilde{F}})\to \HH{0}(D,{\cal O}_D(K_X+\tilde{F})).$$ By the Ramanujan vanishing theorem, one has $\HH{1}(X,-\tilde{F})=0$ and, as a consequence, $h^0(\tilde{F},{\cal O}_{\tilde{F}})=1$. As the arithmetic genus of $\tilde{F}$ is equal to $5$, this implies that $h^0(\tilde{F},K_{\tilde{F}})=5$. Since $h^0(D,K_D)=3$, the image $V$ of $\HH{0}(\tilde{F},K_{\tilde{F}})\to \HH{0}(D, {\cal O}_D(K_X+\tilde{F}))$ has dimension $2$. On the other hand, $V$ contains the restriction to $D$ of $\HH{0}(X,K_X)$, that has dimension $3$ since $\phi$ maps $D$ to a conic. So we have a contradiction, and we must conclude that if $X$ is of type $I$, then $|F|$ ( and thus also $\phi^*|F|$) does not contain multiple fibres. $\quad \diamond$\par\smallskip \section{The examples} \label{esempi} \setcounter{defn}{0} \setcounter{equation}{0} We describe here the known examples of surfaces $X$ such that the canonical map of $X$ is $2$-$1$ on a canonically embedded surface $\Sigma$, and we also present some new ones. We collect at the end of the section some lemmas that are needed in the description of the examples. \bigskip \noindent {\bf $1.$ Examples with $X$ regular.} The first example of a surface $X$ mapped non-birationally onto a canonical surface by the canonical system was found independently by several authors (\cite{bea}, \cite{babbage}, \cite{van}). One of the possible descriptions of the canonical image $\Sigma$ is the following: $\Sigma$ is a quintic surface in ${\bf P}^3$, defined by the vanishing of the determinant of a generic symmetric $5\times 5$ matrix $M$ of linear forms. The singularities of $\Sigma$ are $20$ nodes, occurring precisely where the rank of $M$ drops by $2$, and they form an even set. The double cover of $\Sigma$ branched over the nodes is a regular surface $X$ with $p_g=4$. In \cite{ciro} p. 126 Ciliberto has remarked that the same method can be used to produce similar examples, with $\Sigma$ a canonical complete intersection in a projective space. Notice that, if $\Sigma$ is of type $(3,3)$, $(2,2,3)$ and $(2,2,2,2)$, then the examples thus obtained are not on the Castelnuovo line. \bigskip \noindent {\bf $2.$ Examples with $p_g(X)=5$ and $q(X)=2$.} Surfaces $\Sigma$ with $p_g=5$ and $K^2=8$ are on the Castelnuovo line and have been described in detail in \cite{ho}. If the canonical map is birational, then the canonical image is isomorphic to the canonical model, and it is the intersection of a quadric and a quartic in ${\bf P}^4$. If the quadric is singular, then $\Sigma$ is a Castelnuovo surface of type $(0,1,1)$ or $(0,0,2)$, according to whether the quadric is singular at one point or along a line. In the former case $\Sigma$ carries two different pencils of genus $3$, while in the latter case $\Sigma$ is necessarily singular at the points of intersection with the singular line of the quadric. (For a generic choice of the quartic, these singularities will be $4$ nodes). Let now $A$ be a principally polarized abelian surface, let $K$ be the Kummer surface of $A$ and let $q:A\to K$ be the projection onto the quotient. If $D$ is a symmetric theta divisor, then the linear system $|2D|$ is the pull-back from $K$ of a linear system $|H|$. If $D$ is irreducible, then $|H|$ embeds $K$ as a quartic surface in ${\bf P}^3$; if $D$ is reducible, then $|H|$ maps $K$ $2$-$1$ onto the smooth quadric in ${\bf P}^3$. Let $f:\Sigma\to K$ be the double cover branched on a curve $B$ of $|2H|$ not meeting the singular set of $K$. If $B$ is smooth, then the singularities of $\Sigma$ are $32$ nodes, which are the inverse image of the $16$ singular points of $K$. If $B$ has simple double points, then $\Sigma$ has extra rational double points above the singularities of $B$. The sheaf $f_*{\cal O}_{\Sigma}$ splits as ${\cal O}_K\oplus{\cal O}_K(-H)$ and, using \ref{formuledoppi}, one computes: $K^2_{\Sigma}=2H^2=8$, $p_g(\Sigma)=p_g(K)+h^0({\cal O}_K(H))=5$, $q(\Sigma)=h^1({\cal O}_K)+h^1({\cal O}_K(-H))=0$. Denote by $\phi:X\to\Sigma$ the map obtained from $q:A\to K$ by base-changing with $f$, and by $\alpha:X\to A$ the map that completes the square as in diagram \ref{diagram}. The map $\phi$ is branched precisely over the inverse image of the singularities of $K$, while $\alpha$ is branched on $q^*B\in |4D|$, and $X$ is singular only above the singularities of $q^*B$. One has: $\alpha_*{\cal O}_X={\cal O}_A\oplus{\cal O}_A(-2D)$, and thus one may compute the invariants of $X$ as above, and obtain: $p_g(X)=5=p_g(\Sigma)$, $K^2_X=16$, $q(X)=2$. (In fact, $\alpha$ is the Albanese map of $X$). So the canonical map of $X$ is the composition of $\phi$ with the canonical map of $\Sigma$. If $D$ is irreducible, then by our construction $\Sigma$ is isomorphic to the intersection in ${\bf P}^4$ of the cone over $K$ with a quadric not passing through the vertex of the cone, and so it is a canonical surface. As we have already explained at the beginning, when the quadric is singular, namely when $B$ is cut out on $K$ by a singular quadric of ${\bf P}^3$, $\Sigma$ is a Castelnuovo surface. In this case, it is easy to check that the genus $3$ fibres are mapped to plane sections of $K$ by $\phi$, and that their inverse images in $X$ are connected genus $5$ curves; thus $X$ is a surface of type $I$. Assume now that $D$ is reducible: then $A$ is isomorphic to the product $E_1\times E_2$ of two elliptic curves, with origins $O_i$ and, if $\pi_i:A\to E_i$, $i=1,2$, are the projections, then $D=\pi_1^{-1}(O_1)+\pi_2^{-1}(O_2)$. The map $\pi_1\circ\alpha:X\to E_1$, is an elliptic pencil of genus $3$ curves. We wish to show that, for a generic choice of $B\in|2H|$, the generic fibres of this pencil are not hyperelliptic. The subspace $V=q^*\HH{0}(K,2H)\subset \HH{0}(A,4D)$ is the subspace of even sections. It is possible to find a basis $\sigma^i_1,\ldots \sigma^i_3, \tau^i$ of $\HH{0}(E_i,4O_i)$, $i=1, 2$, such that the $\sigma^i_j$'s are even and the $\tau^i$'s are odd. So $V$ is spanned by the products $\sigma^1_j\sigma^2_k$, $i,j=1,2,3$ and by $\tau^1\tau^2$. It follows that the restriction of $V$ to a generic fibre of $\pi_1$ contains sections that are not even. So, for a generic choice of $B\in|2H|$, the inverse image in $X$ of a generic fibre of $\pi_1$ is not hyperelliptic by lemma \ref{hyperelliptic}. The maps $\pi_1$ and $\pi_1\circ \alpha$ are compatible with the involutions on $A$, on $X$ and on $E_1$, and so they induce linear pencils $p_1:K\to {\bf P}^1$ and $p_1\circ f:\Sigma\to{\bf P}^1$. The generic fibre of $p_1\circ f$ is the same as the generic fibre of $q_1\circ\alpha$, and so it is a non-hyperelliptic curve of genus $3$. By lemma 1.1 of \cite{ak}, the canonical map of $\Sigma$ is not composed with a pencil and it has degree $\le 2$; on the other hand, the restriction of $|K_{\Sigma}|$ to a smooth fibre $F$ of $p_1\circ f$ is a subsystem of $|K_F|$. So we must conclude that the restriction of the canonical map of $\Sigma$ to $F$ is an embedding. Moreover, the system $|K_{\Sigma}|$ contains the $f^*|2H|$, and so it separates the fibres of $p_1\circ f$. So we conclude that the canonical map of $\Sigma$ is birational and $\Sigma$ is a Castelnuovo surface (of type $(0,1,1)$). The pull-back of the genus $3$ pencil $p_1\circ f$ factors through the elliptic pencil $\pi_1\circ\alpha$, thus it is not connected and $X$ is a surface of type $I\!I$. \bigskip \noindent {\bf $3.$ Surfaces of type $I\!I$ with $p_g=6$, $q=1$.} From propositions \ref{constrII} and \ref{invariantsII}, it follows that these examples arise from divisors $\Sigma$ of bidegree $(4,3)$ in ${\bf P}^2\times {\bf P}^1$ with only nodes as singularities and having the following properties: 1) the pencil on $\Sigma$ induced by the projection $p:{\bf P}^2\times {\bf P}^1\to{\bf P}^1$ has $4$ double fibres, 2) $\Sigma$ is smooth away from the double fibres. Such a surface $\Sigma$ has $32$ nodes, $8$ on each of the double fibres, and these form an even set. The double cover $\phi:X\to\Sigma$ branched over the nodes is a surface of type $I\!I$. In section \ref{conto} we produce explicitly such an example. This enables us to describe a $16$-dimensional family of non isomorphic surfaces $X$ of type $I\!I$ with the above invariants. Let $U_1$ be the open subset of the $4$-fold product of ${\bf P}^1$ with itself consisting of the $4$-tuples $(z_1,z_2,z_3,z_4)$ such that $z_i\ne z_j$ for $i\ne j$; let $U_2$ be the open subset of the $4$-fold product of the space ${\bf P}^5$ of conics with itself consisting of the $4$-tuples $(Q_1,Q_2,Q_3,Q_4)$ such that $Q_i$ is reduced, $i=1,\ldots 4$, and $Q_1^2$, $Q_2^2$, $Q_3^2$, $Q_4^2$ represent independent points in the space ${\bf P}^{14}$ of quartics; finally, denote by $U_3\subset{\bf P}^{55}$ the open subset of irreducible divisors of bidegree $(4,3)$ in ${\bf P}^2\times{\bf P}^1$, and let $Z\subset U_1\times U_2\times U_3$ be the closed subset consisting of the points $(z_1,z_2,z_3,z_4;Q_1,Q_2,Q_3,Q_4; \Sigma)$ such that the fibre of $\Sigma$ over $z_i$ is $Q_i^2$, for $i=1,\ldots 4$. It is easy to check that the projection of $Z$ onto $U_1\times U_2$ is surjective, and that the fibre of this projection over a point of $U_1\times U_2$ is naturally isomorphic to ${\bf P}^3$ minus the coordinate planes. So $Z$ is a smooth quasiprojective variety of dimension $27$. Let now $U_0$ be the open subset of $Z$ consisting of the points such that the singularities of $\Sigma$ are only nodes: the example of section \ref{conto} shows that $U_0$ is nonempty. Moreover, the number $\nu(\Sigma)$ of nodes is a lower-semicontinuous function of $\Sigma\in U_0$ and, by lemma \ref{nodes}, one always has $\nu(\Sigma)\ge 32$ . This minimum is attained in the example, and so there is a nonempty open subset $U\subset U_0$ such that $\Sigma$ has precisely $32$ nodes, occurring on the double fibres. Notice that the restriction to $U$ of the projection onto $U_3$ is a Galois cover of its image with Galois group $S_4$, the group action consisting simply in changing the ordering of the double fibres of $\Sigma$. We abuse notation and also denote by $U$ the image of $U$ in $U_3$. The double covers of surfaces $\Sigma\in U$, branched over the nodes, form a $27$-dimensional family $W$ of surfaces of type $I\!I$ with $q=1$ and $p_g=6$. The group ${\bf PGL}(2)\times {\bf PGL}(1)$ acts naturally on $U$, and thus on $W$. On the other hand, it is easy to show that two surfaces $X$, $X'\in W$ are isomorphic iff they belong to the same orbit of the action of ${\bf PGL}(2)\times {\bf PGL}(1)$. Thus the geometric invariant theory quotient of $W$ by ${\bf PGL}(2)\times {\bf PGL}(1)$ is an irreducible variety of dimension $16$, parametrizing non-isomorphic surfaces. \bigskip \noindent {\bf $4.$ An infinite family of surfaces of type $I$ with q=2.} These examples are due to Beauville (see \cite{special}, $2.9$). Let $A$ be an abelian surface with an irreducible principal polarization $D$, let $K$ be the Kummer surface of $A$, $q:A\to K$ the projection onto the quotient, and let $|H|$ be the linear system on $K$ induced by $|2D|$. For $n\ge 2$, consider a smooth hypersurface $G$ of bidegree $(1,n)$ in ${\bf P}^3\times {\bf P}^1$: the projection onto ${\bf P}^1$ exhibits $G$ as a ${\bf P}^2$-bundle and the hypersurfaces of bidegree $(1,k)$, $k\ge 0$ induce effective tautological hyperplane sections of $G$. Assume that $G$ intersects the singular locus of $Y=K\times{\bf P}^1$ transversely, and the intersection at smooth points of $Y$ is transversal. (This certainly happens for a generic choice of $G$). Then the surface $\Sigma=G\cap Y$ has $16n$ nodes at the intersections with the singular locus of $\Sigma$ and is smooth elsewhere. Using adjunction on ${\bf P}^3\times{\bf P}^1$, one sees that the hypersurfaces of bidegree $(1, n-2)$ induce canonical curves of $\Sigma$; so for $n\ge 3$ the canonical system $|K_{\Sigma}|$ is very ample. It is not difficult to check that the same is true for $n=2$. A straightforward computation yields: $p_g(\Sigma)=4n-3$, $K^2_{\Sigma}=12n-16$. So $\Sigma$ is a Castelnuovo surface, and $G$, with the natural projection onto ${\bf P}^1$, is isomorphic to the ${\bf P}^2$-bundle $\pp_{a,b,c}$ containing it. Since the canonical divisor of $\Sigma$ is induced by the hypersurfaces of bidegree $(1,n-2)$, one has $a+b+c=p_g(\Sigma)-3=4n-6$, and $a\ge n-2$. Now denote by $X$ the pull-back of $\Sigma$ to $A\times{\bf P}^1$ via the map $q\times 1$: the surface $X$ is smooth, the projection $X\to A$ is the Albanese map, and $q\times 1$ restricts to a double cover $\phi:X\to\Sigma$ branched over the nodes $\Sigma$. Using adjunction on $A\times{\bf P}^1$ and $K\times{\bf P}^1$, one checks immediately that $p_g(X)=p_g(\Sigma)$. So $\phi$ is the canonical map of $X$. Moreover, it is easy to see that the inverse image of a fibre $F$ of the projection $\Sigma\to{\bf P}^1$ is connected, and thus $X$ is of type $I$. \bigskip The results that follow show that the surfaces $\Sigma$ of example $4$ exist for all the admissible values of $a$, $b$ and $c$. \begin{lem} Let $n\ge 3$ and $n-2\le a \le b\le c$ be integers such that $a+b+c=4n-6$; then there exists a smooth divisor $G\in{\bf P}^3\times{\bf P}^1$ of bidegree $(1,n)$, and an isomorphism $\pp_{a,b,c}\to G$ such that hypersurfaces of bidegree $(1,n-2)$ pull back to tautological hyperplane sections of $\pp_{a,b,c}$. Write $4n-6=3\epsilon+\rho$, with $\rho$ and $\epsilon$ integers, $0\le\rho <3$; a generic hypersurface $G$ of bidegree $(1,n)$, with the polarization given by hypersurfaces of bidegree $(1,n-2)$, is isomorphic to $\pp_{a,b,c}$ with the tautological hyperplane section, where $a,b,c$ are as follows: $\rho =0$, $a=b=c=\epsilon$; $\rho =1$, $a=b=\epsilon$, $c=\epsilon+1$, $\rho =1$, $a=\epsilon$, $b=c=\epsilon +1$. \end{lem} {\bf Proof:}\,\, Let $T$ be the tautological hyperplane section and $L$ be the fibre of the projection $p:\pp_{a,b,c}\to{\bf P}^1$; the divisor $T'=T-(n-2)L$ is base point free, and the corresponding morphism $g:\pp_{a,b,c}\to{\bf P}^{n+2}$ is birational. More precisely, if $a>n-2$ then $g$ is an embedding and if $a=n-2$ then the image of $g$ is a cone over ${\bf P}_{b,c}$. Let $h:\pp_{a,b,c}\to{\bf P}^3$ be the morphism associated to a generic $3$-dimensional subsystem of $|T'|$: $h$ has degree $n$ and maps the fibres of $\pp_{a,b,c}$ linearly to planes in ${\bf P}^3$. The morphism $h\times p:\pp_{a,b,c}\to{\bf P}^3\times{\bf P}^1$ embeds $\pp_{a,b,c}$ as a divisor of type $(1,n)$, and hypersurfaces of bidegree $(1, n-2)$ pull-back to elements of $|T|$ via $h\times p$. To prove the second part of the statement, consider the space ${\bf P}^{4n+3}$ of divisors of bidegree $(1,n)$, and the dense open subset $U\subset {\bf P}^{4n+3}$ consisting of the smooth divisors. If $k$ is an integer, then $h^0(G,{\cal O}_G(1,k))$ is a lower semi-continuous function of $G\in U$. If, say, $\rho=0$, then we have shown that there exists $G_0\in U$ such that $G_0$ with the polarization given by hypersurfaces of bidegree $(1, n-2)$ is isomorphic to ${\bf P}_{\epsilon,\epsilon,\epsilon}$ with the tautological hyperplane sections. This is equivalent to the condition $h^0(G_0,{\cal O}_{G_0}(1,n-\epsilon-3)=0$. Then, by semi-continuity, one has $h^0(G,{\cal O}(1,n-\epsilon-3)=0$ on a dense open set $U_1\subset U$, and so $G$ is isomorphic to ${\bf P}_{\epsilon,\epsilon,\epsilon}$ for every $G\in U_1$. The same argument shows the statement for $\rho=1,2$. $\quad \diamond$\par\smallskip \begin{lem} Let $A$ be an abelian surface with an irreducible principal polarization, let $K\subset {\bf P}^3$ be the corresponding Kummer quartic, and let $G$ be a smooth hypersurface of bidegree $(1,n)$ in ${\bf P}^3\times{\bf P}^1$: there exists $\gamma\in {\bf PGL}(3)$ such that $\Sigma_{\gamma}= G\cap (\gamma K\times {\bf P}^1)$ has $16n$ nodes, occurring at the intersections of $G$ with the singular locus of $K\times{\bf P}^1$, and is smooth elsewhere. \end{lem} {\bf Proof:}\,\, The proof consists simply in counting dimensions. Let $({\bf P}^3)^*$ be the space of planes in ${\bf P}^3$, let $K^*\subset({\bf P}^3)^*$ be the dual surface of $K$, and let $\psi: {\bf P}^1\to ({\bf P}^3)^*$ be the map that associates to $z\in {\bf P}^1$ the plane $G\cap ({\bf P}^3\times\{z\})$. We say that $\gamma\in {\bf PGL}(3)$ is ``good'' if $\gamma K^*$ and $\psi({\bf P}^1)$ intersect transversely at smooth points, and moreover the intersection points are regular values of $\psi$ and do not lie on the exceptional curves corresponding to the nodes of $K$. We are going to show that if $\gamma$ is ``good'', then it satisfies the claim. Remark first of all that the points of the curve $\psi({\bf P}^1)$ correspond to planes that are tangent to $\gamma K$ at most at one smooth point. So the surface $\Sigma_{\gamma}$ has nodes at the points of intersections with the singular set of $\gamma K\times {\bf P}^1$. To show that $\Sigma$ is smooth elsewhere, notice that $\Sigma_{\gamma}\cap({\bf P}^3\times\{z\})$ is just the intersection of $\gamma K$ with the plane $\psi(z)$, and so $\Sigma_{\gamma}$ can be singular only at points $(x,z)\in G$ such that the plane $\psi(z)$ is tangent to $\gamma K$ at $x$. A computation in local coordinates shows that these points are also smooth if $\psi$ is regular at $z$ and the curve $\psi({\bf P}^1)$ meets $\gamma K^*$ transversely at $\psi(z)$. In order to conclude the proof it is enough to remark that the $\gamma$'s that are not ``good'' form a subset of dimension at most $14$. This is a consequence of the following facts: 1) the subset of ${\bf PGL}(3)$ consisting of the elements that map a point $x_1\in {\bf P}^3$ to a point $x_2$ has dimension $12$, 2) the subset of ${\bf PGL}(3)$ consisting of the elements that map a chosen point $x_1\in {\bf P}^3$ and a line $L_1$ through $x_1$ to a point $x_2$ and a line $L_2$ through $x_2$ has dimension $10$ . $\quad \diamond$\par\smallskip The two previous lemmas together yield the following: \begin{prop} Let $A$ be an abelian surface with an irreducible principal polarization, and let $a\le b\le c$ be integers such that $a+b+c\equiv 2 (\mbox{mod}\,\,\, 4)$ and $a\ge n:=(a+b+c+6)/4$; then there exist $X$ and $\Sigma$ as in example $4$ such that $\Sigma$ is a Castelnuovo surface of type $(a,b,c)$ and $A$ is the Albanese variety of $X$. \end{prop} We close the section by proving the lemma needed in example $2$. \begin{lem}\label{hyperelliptic} Let $E$ be an elliptic curve with origin $O$; let $B\in |4O|$ be a reduced divisor and let $f:C\to E$ be the double cover branched on $B$, with ${\cal L}=2O$. Then $C$ is hyperelliptic if and only if $B$ is symmetric with respect to the elliptic involution. \end{lem} {\bf Proof:}\,\, As it is explained in section \ref{notation}, $C$ is isomorphic to a divisor $D\subset {\cal L}$; the line bundle ${\cal L}$ has a natural linearization and, if $B$ is symmetric, then $D$ is easily seen to be also symmetric. Thus the involution on ${\cal L}$ induces an involution of $C$, whose fixed points are the inverse images of points of order $2$ of $E$. Since $B$ is symmetric and reduced, it does not contain any point of order $2$, and so the involution has $8$ fixed points on $C$. By the Hurwitz formula, the quotient of $C$ by the involution is rational, and thus $C$ is hyperelliptic. Conversely, assume that $C$ is hyperelliptic and denote by $\phi:C\to\phi(C)$ the canonical map, with $\phi(C)$ a plane conic. If $g:E\to{\bf P}^1$ is the quotient map of the elliptic involution, then by \ref{formuledoppi} the canonical system $\HH{0}(C,K_C)$ contains $f^*\HH{0}(E, 20)= f^*g^*\HH{0}({\bf P}^1,{\cal O}_{{\bf P}^1}(1))$ as a subsystem. So one has a map $\bar{f}:\phi(C)\to{\bf P}^1$ such that the following diagram commutes: \begin{equation} \begin{array}{rcccl} \phantom{1} & C &\stackrel{f}{\rightarrow} & E & \phantom{1} \\ \scriptstyle{\phi}\!\!\!\!\!\!\!\! & \downarrow & \phantom{1} & \downarrow & \!\!\!\!\!\! \scriptstyle{g} \\ \phantom{1} & \phi(C) & \stackrel{\bar{f}}{\rightarrow} & {\bf P}^1 & \phantom{1} \end{array} \end{equation} If we denote by $i_1$ the hyperelliptic involution on $C$ and by $i_2$ the elliptic involution on $E$, then it follows immediately: $f\circ i_1=i_2\circ f$. In particular, if $R$ is the ramification divisor of $f$ , then $i_1( R)=R$. Applying the Hurwitz formula to $f$, one sees that $R$ is a canonical divisor of $C$; since $R$ is reduced, it contains no Weierstrass point. Thus, $R$ may be written as $x+i_1(x)+y+i_1(y)$, for some $x$, $y\in C$ and $B=f(x)+i_2(f(x))+f(y)+i_2(f(y))$ is a symmetric divisor. $\quad \diamond$\par\smallskip \section{Surfaces of type $I$ with $q(X)\ge 2$}\label{typeI} \setcounter{defn}{0} \setcounter{equation}{0} In this section we study surfaces of type $I$ with $q\ge 2$ and we show that they are all obtained as in example $4$ of section \ref{esempi}. More precisely we prove the following: \begin{thm}\label{main} Let $\phi:X\to\Sigma$ be as in assumption \ref{ipotesi}; if $X$ is of type $I$ and $q\ge 2$, then $p_g(X)\equiv 1 (\mbox{mod}\,\,\, 3)$, $q(X)=2$, the Albanese surface $A$ of $X$ has an irreducible principal polarization, and $X$ can be constructed as in example $4$ of section \ref{esempi}, with $n=(p_g(X)+3)/4$. \end{thm} The proof of theorem \ref{main} requires some preliminary steps: first we show that the Albanese variety $A$ of $X$ is a surface, and then we prove that $A$ is isomorphic to the Prym variety of the unramified double cover $\tilde{F}\to F$, with $F$ a generic fibre. Thus $A$ is principally polarized and there is a map $f:\Sigma\to K$, where $K$ is the Kummer quartic $K$ of $A$. Finally we show that $f$ can be extended to a morphism $g:\pp_{a,b,c}\to{\bf P}^3$. For the rest of the section, we will assume that $\phi:X\to \Sigma$ is as in assumption \ref{ipotesi}, that $X$ is of type $I$ and that $q(X)\ge 2$. In particular, the pull-back to $X$ of the genus $3$ pencil $|F|$ is a linear pencil $|\tilde{F}|$ of genus $5$. \begin{lem} The irregularity $q(X)$ is equal to $2$. \end{lem} {\bf Proof:}\,\, It suffices to show that $q(X)\le 2$. Notice that for a generic fibre $F$, the restriction map $\HH{0}(\Sigma,K_{\Sigma}+F)\to \HH{0}(F,K_F)$ is surjective, by the regularity of $\Sigma$. So $\phi^*\HH{0}(\Sigma,K_{\Sigma}+F)\subset \HH{0}(X,K_X+\tilde{F})\to\HH{0}(\tilde{F},K_{\tilde{F}})$ is a subspace whose image via the restriction map $\HH{0}(X,K_X+\tilde{F})\to\HH{0}(\tilde{F},K_{\tilde{F}})$ has dimension $3$. Since the pencil $|\tilde{F}|$ is linear, by Ramanujan vanishing one has $\HH{1}(X, K_X+\tilde{F})=0$ and therefore the cokernel of the above restriction map is isomorphic to $\HH{1}(X,K_X)$. Since $\tilde{F}$ has genus $5$, it follows $q(X)\le 2$. $\quad \diamond$\par\smallskip \begin{prop}\label{ppav} The Albanese variety $A$ of $X$ is a principally polarized abelian surface, and the polarization $D$ of $A$ is irreducible. \end{prop} The above proposition is a consequence of the following lemmas, that describe the Prym variety $Z$ of the cover $\tilde{F}\to F$, for a generic $F$ and show that $Z$ is naturally isomorphic to $A$. We start by reviewing quickly the properties of Prym varieties that we need; for more details and proofs the reader may consult chapter $12$ of \cite{lb}. Let $J$ be the Jacobian of $\tilde{F}$, and let $\gamma:\tilde{F}\to J$ be the period map with base point $x_0\in \tilde{F}$. The Abel-Prym map with base point $x_0$, $\beta:\tilde{F}\to Z$, is defined as the composition $\hat{i}\circ \gamma$, where $\hat{i}:J\to Z$ is a surjective morphism of abelian varieties with connected kernel. $Z$ is an abelian surface having a natural principal polarization $D$, the restriction of $\beta$ to $\tilde{F}$ is an embedding and the image of $F$ is a divisor algebraically equivalent to $2D$, by Welters criterion (\cite{lb}, page $373$). \begin{lem} The polarization $D$ on $Z$ is irreducible. \end{lem} {\bf Proof:}\,\, Assume by contradiction that $D$ is reducible: then $Z$ is a product $E_1\times E_2$ of elliptic curves and $D$ is algebraically equivalent to $\pi_1^{-1}(O_1)+\pi_2^{-1}(O_2)$, where $\pi_i$ is the projection onto $E_i$, $i=1,2$, and $O_i\in E_i$. For a suitable choice of $O_1$ and $O_2$, the curve $\beta(\tilde{F})$ in $E_1\times E_2$ is linearly equivalent to $\pi_1^{-1}(2O_1)+\pi_2^{-1}(2O_2)$ Denote by $j_i$ the involution on $E_i$ that fixes $O_i$, $i=1,2$: then $\beta(\tilde{F})$is invariant under $j_1\times 1$ and $1\times j_2$. The quotient of $\beta(\tilde{F})$ by the diagonal automorphism $j_1\times j_2$ is isomorphic to $F$, and, via this isomorphism, $j_1\times 1$ induces an involution of $F$ whose quotient is a plane section of the smooth quadric in ${\bf P}^3$. So $F$ is hyperelliptic, but this contradicts the assumption that the canonical map of $\Sigma$ is birational. $\quad \diamond$\par\smallskip Given any map $h:\tilde{F}\to Y$, with $Y$ a complex torus, there exist a unique morphism of tori $\bar{\psi}:J\to Y$ and a unique translation $\tau:Y\to Y$ such that $\tau\circ h=\bar{\psi}\circ \gamma$. If, moreover, the map $h$ is equivariant with respect to the ${\bf Z}/2$-actions given by the involution on $\tilde{F}$ and by multiplication by $-1$ on $Y$, then the kernel of $\hat{i}$ is mapped to $0$ by $\bar{\psi}$; thus $\bar{\psi}$ induces a morphism $\psi:Z\to Y$ such that $\tau\circ h=\bar{\psi}\circ \beta$. We are interested in the case in which $h$ is the restriction to $\tilde{F}$ of the Albanese map $\alpha:X\to A$. Consistently with the above notation, we denote by $\bar{\psi}:J\to A$ and $\psi:Z\to A$ the morphisms induced by the restriction of $\alpha$. \bigskip \begin{lem}\label{iso} The morphism $\psi:Z\to A$ is an isomorphism. \end{lem} By the above discussion, $\psi$ is an isomorphism iff $\bar{\psi}$ is surjective and has connected kernel. In turn, if we consider the dual map of $\bar{\psi}$, $\bar{\psi}^*:{\rm Pic}^0(X)\to{\rm Pic}^0(\tilde{F})$, then the above conditions are equivalent to $\bar{\psi}^*$ being injective. So assume that $\bar{\psi}^*$ is not injective and consider a torsion element $\xi\in\ker\bar{\psi}^*$ of order $m>1$. Let $r:X'\to X$ be the unramified ${\bf Z}/m$-cover given by $\xi$: the restriction of $r$ to a generic fibre $\tilde{F}$ is a disjoint union of $m$ components isomorphic to $\tilde{F}$. Using the Stein factorization of the pull-back to $X'$ of the pencil $\tilde{F}$, one gets the following commutative diagram, where the vertical arrows are pencils with fibre $\tilde{F}$ and the horizontal arrows are connected ${\bf Z}/m$-covers: \begin{equation} \begin{array}{rcccl} \phantom{1} &X' &\stackrel{r}{\rightarrow} & X & \phantom{1} \\ \phantom{1} & \downarrow & \phantom{1} & \downarrow &\phantom{1} \\ \phantom{1} & B & \stackrel{\bar{r}}{\rightarrow} & {\bf P}^1 & \phantom{1} \end{array} \end{equation} The map $\bar{r}$ is ramified at at least $2$ points, while $r$, which is obtained from $\bar{r}$ by base change and normalization, is unramified. This implies that the fibres of the pencil $|\tilde{F}|$ over the branch points of $\bar{r}$ are $m$-tuple fibres. But this contradicts proposition \ref{nodes}. $\quad \diamond$\par\smallskip Finally, we put all the previous results together and get: \smallskip {\bf Proof of theorem \ref{main}:} Consider the basic diagram \ref{diagram}: we wish to show that the map $f:\Sigma\to K$ can be extended to a map $\bar{f}:\pp_{a,b,c}\to{\bf P}^3$ that maps the fibres of $\pp_{a,b,c}$ linearly to planes of ${\bf P}^3$. By lemma \ref{iso}, the Albanese variety $A$ can be identified with the Prym variety $Z$ of a generic $\tilde{F}$. So, the fibres $\tilde{F}$ are mapped to divisors of $|2D|$, where $D$ is the principal polarization on $A$ (see proposition \ref{ppav}); as a consequence, the fibres $F$ on $\Sigma$ are mapped isomorphically to plane sections of $K$ with respect to the embedding as a quartic in ${\bf P}^3$. If $F$ is a smooth fibre of $\Sigma$, then there is a natural linear isomorphism between the fibre of $\pp_{a,b,c}$ containing $F$ and the plane in ${\bf P}^3$ containing $f(F)$. So we can define a rational map $\bar{f}:\Sigma\to{\bf P}^3$, such that its restriction to $\Sigma$ extends to the morphism $f$. Let now $F_0$ be a fibre of $\pp_{a,b,c}$ containing an indeterminacy point of $\bar{f}$: the restriction of $\bar{f}$ to $F_0$ is a degenerate projectivity, whose singular locus can either be a point or a line. In the former case, the scheme theoretic image via $f$ of the curve $\Sigma\cap F_0$ would be a $4$--tuple line. This is impossible, because a Kummer surface has no such plane section. If the indeterminacy locus of $\bar{f}$ on $F_0$ were a line not contained in $\Sigma$, then the curve $\Sigma\cap F_0$ would be contracted to a point, but this is of course impossible. So the only possibility left is that the indeterminacy locus of $\bar{f}$ on $F_0$ is a line $R$ contained in $\Sigma$. Remark that every other component of $\Sigma\cap F_0$ is contracted by $f$, and so the pull-back to $X$ of every component of $\Sigma\cap F_0$ different from $R$ is contracted by $\alpha$. Arguing as in the proof of proposition \ref{nodes}, one shows that $R$ can contain at most $4$ nodes of $\Sigma$, so the pull-back $\tilde{R}$ of $R$ to $X$ is a curve of genus $0$ or $1$. On the other hand, the scheme-theoretic image $\Delta$ of the fibre of $|\tilde{F}|$ containing $\tilde{R}$ is supported on $\alpha(\tilde{R})$, but this is impossible because $\Delta$ is an ample divisor on an abelian surface. So we conclude that $\bar{f}$ is indeed a regular map. If we denote by $p:\pp_{a,b,c}\to{\bf P}^1$ the projection map, then the map $\bar{f}\times p:\pp_{a,b,c}\to {\bf P}^3\times {\bf P}^1$ embeds $\pp_{a,b,c}$ as a divisor $G$ of bidegree $(1,n)$, for a suitable value of $n$, and $\Sigma$ is mapped isomorphically to $G\cap(K\times{\bf P}^1)$. To determine $n$, we use adjunction on ${\bf P}^3\times{\bf P}^1$ and remark that divisors of bidegree $(1, n-2)$ cut out canonical curves on $\Sigma$, and so $p_g(X)=p_g(\Sigma)=a+b+c+3=4n-3\equiv 1 (\mbox{mod} 4)$, $n=(p_g(X)+3)/4$. By lemma \ref{basic}, $\Sigma$ has $16n$ nodes, occurring at the intersections of $G$ with the singular locus of $K\times{\bf P}^1$. So the intersection of $G$ with the singular locus of $K\times{\bf P}^1$ is transversal. $\quad \diamond$\par\smallskip \section{Surfaces of type $I\!I$}\label{typeII} \setcounter{defn}{0} \setcounter{equation}{0} In this section we describe surfaces of type $I\!I$ in detail and we show that the invariants of these surfaces are bounded. So here $X$ and $\Sigma$ are as in assumption \ref{ipotesi}, and moreover the pull-back $\tilde{F}$ of a generic $F$ is disconnected. The Stein factorization of the pencil $|\tilde{F}|$ gives rise to the following commutative diagram, where $p$ denotes the pencil $|F|$ and $\tilde{p}$ denotes the connected fibration on $X$ through which $|\tilde{F}|$ factors: \begin{equation}\label{diagramII} \begin{array}{rcccl} \phantom{1} & X & \stackrel{\phi}{\rightarrow} & \Sigma \\ \scriptstyle{\tilde{p}}\!\!\!\!&\downarrow &\phantom{1} & \downarrow & \!\!\!\!\scriptstyle{p} \\ \phantom{1} & B & \stackrel{\bar{\phi}}{\rightarrow} & {\bf P}^1 &\phantom{1} \end{array} \end{equation} The curve $B$ is smooth and the map $\bar{\phi}$ is a double cover. We introduce a new invariant of $X$, the genus $g$ of $B$. Notice that $g\le q(X)$ \begin{thm}\label{constrII} The surface $\Sigma$ has precisely $2g+2$ double fibres, occurring at the branch points of $\bar{\phi}$, and $c\le g$. Conversely, if $\Sigma$ a Castelnuovo surface of type $(a, b,c)$ with only nodes as singularities, with $c\le g$, having $2g+2$ double fibres, and smooth outside the double fibres, then $\Sigma$ has $16g+16$ nodes which form an even set, and the double cover $\phi:X\to\Sigma$ branched over the nodes is a surface of type $I\!I$. \end{thm} {\bf Proof:}\,\, For $i=1,\ldots 2g+2$, denote by $x_i$ the ramification points of $\bar{\phi}$, by $y_i\in {\bf P}^1$ the image of $x_i$, and by $F_i$ and $\tilde{F}_i$ the fibres of $p$ and $\tilde{p}$ over $y_i$ and $x_i$ respectively. In diagram \ref{diagramII}, the map $\phi$ is obtained from $\bar{\phi}$ by base and normalization: since $\phi$ is unramified in codimension $1$, the $F_i$'s are double fibres, $i=1,\ldots 2g+2$,and they contain all the nodes of $\Sigma$. So, by proposition \ref{nodes}, $\Sigma$ has precisely $16g+16$ nodes. In order to show that $c\ge g$, we construct explicitly $X$ as the normalization of a divisor in a ${\bf P}^2$-bundle. Denote by $\tilde{\pp}_{a,b,c}$ the pull-back of $\pp_{a,b,c}$ to $B$; so $\tilde{\pp}_{a,b,c}={\rm Proj}(\bar{\phi}^*({\cal O}_{\pp^1}(a)\oplus\bar{\phi}^*({\cal O}_{\pp^1}(b) \oplus\bar{\phi}^*({\cal O}_{\pp^1}(c)$. If $T$ and $L$ are the tautological hyperplane section and a fibre of $\pp_{a,b,c}$, and $\tilde{T}$, $\tilde{L}$, are the pull-backs of $T$ and $L$ to $\tilde{\pp}_{a,b,c}$, then the fibre product $W$ of $p$ and $\bar{\phi}$ is a divisor in $\tilde{\pp}_{a,b,c}$ linearly equivalent to $4\tilde{T}\!-\!(a+b+c-2)\tilde{L}$. The singular locus of $W$ consists of $2g+2$ double curves, that are the intersections of $W$ with the fibres of $\tilde{\pp}_{a,b,c}$ over $x_1,\ldots x_{2g+2}$. One has: $K_{\tilde{\pp}_{a,b,c}}=-3\tilde{T}+\tilde{p}^*(K_B)+(a+b+c)\tilde{L}$ and $K_{\tilde{\pp}_{a,b,c}}+W=\tilde{T}+\tilde{p}^*(K_B)$. So the canonical curves of $X$ correspond to sections of $\tilde{T}+\tilde{p}^*(K_B+\bar{\phi}^*({\cal O}_{\pp^1}(2))$ vanishing on the double curves of $W$, namely to sections of $\tilde{T}+tilde{p}^*(K_B-(x_1+\ldots +x_{2g+2}+\bar{\phi}^*({\cal O}_{\pp^1}(2))$. The Hurwitz formula shows that $K_B$ is linearly equivalent to $x_1+\ldots+ x_{2g+2}-\bar{\phi}^*({\cal O}_{\pp^1}(2)$, and so the canonical system of $X$ is the pull-back of $\HH{0}(\tilde{\pp}_{a,b,c},\tilde{T})$. By assumption, we have $p_g(X)=p_g(\Sigma)=a+b+c+3$; on the other hand, $h^0(\tilde{\pp}_{a,b,c},\tilde{T})=h^0(B,\bar{\phi}^*({\cal O}_{\pp^1}(a))+ h^0(B,\bar{\phi}^*({\cal O}_{\pp^1}(b)) +h^0(B,\bar{\phi}^*({\cal O}_{\pp^1}(c))$. Applying \ref{formuledoppi} to the double cover $\bar{\phi}$ (with ${\cal L}={\cal O}_{\pp^1}(g+1)$) yields for any integer $k\ge 0$: $h^0(B,\bar{\phi}^*{\cal O}_{\pp^1}(k))=h^0({\bf P}^1,{\cal O}_{\pp^1}(k))+h^0({\bf P}^1,{\cal O}_{\pp^1}(k-g-1))= k+1+h^0({\bf P}^1,{\cal O}_{\pp^1}(k-g-1))$. So it follows that $c\le g$. Conversely, assume that $\Sigma$ is a divisor in $\pp_{a,b,c}$ linearly equivalent to $4T-(a+b+c-2)L$, with only nodes as singularities, with exactly $2g+2$ double fibres, $c\le g$, and assume that $\Sigma$ is smooth away from the double fibres. Then by proposition \ref{nodes} $\Sigma$ has $16g+16$ nodes. Let $D\subset \pp_{a,b,c}$ be the sum of the fibres of $p:\pp_{a,b,c}\to{\bf P}^1$ that are double for $\Sigma\to{\bf P}^1$: $D$ is smooth, it is linearly equivalent to $2D'$, where $D'=(g+1)F$, it contains all the nodes of $\Sigma$ and, finally, the restriction of $D$ to $\Sigma$ is a union of double curves. Therefore, by proposition \ref{criterio}, the nodes of $\Sigma$ form an even set. If $\phi:X\to\Sigma$ is the double cover branched over the nodes, then the above computations show that $X$ is a smooth surface such that $p_g(X)=p_g(\Sigma)$. Let $\epsilon:S\to\Sigma$ be the minimal resolution of the singularities of $\Sigma$, let $E$ be the exceptional divisor of $\epsilon$, let $Z$ be the sum of the strict transforms of supports of the double fibres of $\Sigma$ and let $\tilde{\phi}:\tilde{X}\to S$ be obtained from $\phi$ by base change with $\epsilon$: $\tilde{\phi}$ is a smooth double cover branched on $E$, and the line bundle ${\cal L}$ associated with the cover is equal $(g+1)\epsilon^*F-Z$. The restriction of ${\cal L}$ to a generic fibre $F$ is trivial, so the inverse image in $\tilde{X}$ (and in $X$) of a generic $F$ is disconnected, and $X$ is of type $I\! I$. $\quad \diamond$\par\smallskip We will now give bounds for the invariants of $X$ and relations between them. \begin{lem}\label{q-g} The following relations between the invariants of $X$ hold: $$0\le q-g=3g+3-p_g.$$ \end{lem} {\bf Proof:}\,\, The inequality $q-g\ge 0$ is a consequence of the fact that there exists a dominant map $\tilde{p}:X\to B$, with $B$ a curve of genus $g$. The equality $q-g=3g+3-p_g$ is equivalent to the formula \ref{nodes}, since by propositions \ref{constrII} $\Sigma$ has $16g+16$ nodes. $\quad \diamond$\par\smallskip \begin{prop}\label{invariantsII} The numerical possibilities for the invariants of $X$ are the following: \smallskip \noindent a) $p_g=3g+3$,\quad $q=g$,\quad\quad $a=b=c=g$,\quad $0<g\le 26$; \noindent b) $p_g=3g+2$,\quad $q=g+1$, \quad $a=g-1$, $b=c=g$, \quad $0<g\le 17$; \noindent c) $p_g=3g+1$,\quad $q=g+2$,\,\,\,\quad $a=b=g-1$, $c=g$ or $a=g-2$, $b=c=g$, \quad $0<g\le 8$. \end{prop} {\bf Proof:}\,\, The topological Euler characteristic of the minimal desingularization $S$ off $\Sigma$ can be computed from Noether's formula as follows: $$c_2(S)=12\chi({\cal O}_S)-K^2_S=9p_g(S)+19.$$ On the other hand the following formula (see \cite{bpv}, page 97), in which $e(D)$ represents the topological characteristic of the support of a divisor $D$, expresses $c_2(S)$ in terms of the base and of the singular fibres of the pull-back to $S$ of the fibration $F$ on $\Sigma$, that we also denote by $F$: $$c_2(S)=e({\bf P}^1)e(F)+\sum_{F'\, singular}e(F')-e(F).$$ (The term $e(F')-e(F)$ is always non-negative, see \cite{bpv}, page 97). From proposition \ref{nodes}, it follows that if $F'$ is the pull-back of a double conic on $\Sigma$, then $e(F)=10$ or $11$, according to whether the conic is smooth or not. So, recalling that by proposition \ref{constrII} there are $2g+2$ double fibres on $\Sigma$, and comparing the two expressions for $c_2(S)$ one obtains: $c_2(S)\ge 2(-4) +(2g+2)14$, namely $9p_g\ge 28g+1$. The statement now follows in view of lemma \ref{q-g}. $\quad \diamond$\par\smallskip \begin{rem} Notice that at least for $g=1$ possibilities a) and b) actually occur, as it is shown by examples $3$ and $2$ of section \ref{esempi}. We do not know examples for possibility c). \end{rem} \section{Appendix: a computation with Macaulay}\label{conto} We describe here how we have used Macaulay (\cite{macaulay}) to show the existence of a divisor $\Sigma$ of bidegree $(4,3)$ in ${\bf P}^2\times{\bf P}^1$ with the properties required in example $3$ of section \ref{esempi}. We use the notation introduced there. We consider homogeneous coordinates $(s,t)$ in ${\bf P}^1$ and $(x_0,x_1,x_2)$ in ${\bf P}^2$ and set: $z_1=(1,0)$, $z_2=(0,1)$, $z_3=(1,-1)$, $z_4=(1,1)$;\quad $Q_1=x_0^2+x_1^2+x_2^2$, $Q_2=x_0^2+x_1^2-x_2^2$, $Q_3=x_0x_1+x_0x_2+x_1x_2$, $Q_4=x_0^2+5x_0x_1+7x_0x_2+2x_1^2+11x_1x_2+3x_2^2$. We start by writing down the equation $h$ of $\Sigma$: \begin{verbatim} Macaulay version 3.0, created 12 September 1994 ! number of variables ? 5 ! 5 variables, please ? stx[0]-x[2] ! variable weights (if not all 1) ? ! monomial order (if not rev. lex.) ? largest degree of a monomial : 217 ! number of generators ? 1 ! (1,1) ? x[0]2+x[1]2+x[2]2 ! number of generators ? 1 ! (1,1) ? x[0]2+x[1]2-x[2]2 ! (1,1) ? x[0]x[1]+x[0]x[2]+x[1]x[2] ! (1,1) ? x[0]2+5x[0]x[1]+2x[1]2+7x[0]x[2]+11x[1]x[2]+3x[2]2 \end{verbatim} Next we show that the singularities of $\Sigma$ are at most nodes. This is a local computation, that has to be repeated fo each of the $6$ standard open affine subsets. Consider for instance $U=\{sx_0\ne 0\}\subset{\bf P}^1\times{\bf P}^2$: we identify $U$ with ${\bf A}^3\subset{\bf P}^3$ and then consider the closure in ${\bf P}^3$ of $\Sigma\cap U$, defined by the equation $h_{s0}$. The plane at infinity is $w=0$. The ideal $I$ of the locus in ${\bf P}^3$ of the singular points of $h_{s0}=0$ that are not nodes is generated by the derivatives of $h_{s0}$ and by the $3\times 3$ minors of the Hessian matrix $hh_{s0}$ of $h_{s0}$. Computing the standard basis of $I$ and using it to reduce $w^{30}$ one gets $0$, namely $w^{30}\in I$ and therefore the singularities of $\Sigma\cap U$ are at most nodes. Here is the transcript of the Macaulay session (slightly edited): \begin{verbatim} ! characteristic (if not 31991) ? ! number of variables ? 4 ! 4 variables, please ? tx[1]x[2]w ! variable weights (if not all 1) ? ! monomial order (if not rev. lex.) ? largest degree of a monomial : 512 ! s ---> ? w ! t ---> ? t ! x[0] ---> ? w ! x[1] ---> ?x[1] ! x[2] ---> ? x[2] [189k][252k] 6.7.8.[378k]9.10.11.[441k]12.[504k][567k]13.14.15.[630k][692k]16. [755k][818k][881k]17.[944k][1007k]18.[1070k]19.20.[1133k]21. [1196k] computation complete after degree 21 ! number of generators ? 1 ! (1,1) ? w30 \end{verbatim} The computation in the other $5$ affine open sets goes exactly in the same way. Now, to finish the computation it is enough to show that the singular locus of $\Sigma$, that we already know to be reduced and of dimension $0$, has length $32$. In fact, by prop. \ref{nodes}, each of the $4$ double fibres contains $8$ nodes and therefore $\Sigma$ is smooth outside the double fibres. First one embeds $\Sigma$ in ${\bf P}^5$ via the Segre embedding: \begin{verbatim} ! number of variables ? 11 [126k] ! 11 variables, please ? stx[0]-x[2]y[0]-y[5] ! variable weights (if not all 1) ? 1:5 2:6 ! monomial order (if not rev. lex.) ? 5 1 1 1 1 1 1 largest degree of a monomial : 217 512 512 512 512 512 512 ! number of generators ? 6 ! (1,1) ? sx[0]-y[0] ! (1,2) ? sx[1]-y[1] ! (1,3) ? sx[2]-y[2] ! (1,4) ? tx[0]-y[3] ! (1,5) ? tx[1]-y[4] ! (1,6) ? tx[2]-y[5] 23.4.5.6.7.8.9.10.11.12.13.14.[189k]15.16.17. computation complete after degree 17 ! characteristic (if not 31991) ? ! number of variables ? 6 ! 6 variables, please ? y[0]-y[5] ! variable weights (if not all 1) ? ! monomial order (if not rev. lex.) ? largest degree of a monomial : 117 \end{verbatim} Now $h$ is the ideal of $\Sigma$ in ${\bf P}^5$; the singular locus of $\Sigma$ is defined by the equations of $\Sigma$ and by the $3\times 3$ minors of the Jacobian matrix of the equations of $\Sigma$: \begin{verbatim} [252k][315k][378k][441k][504k][567k][630k] [692k][755k][818k][881k][944k] degree 8 codimension : 5 degree : 32 \end{verbatim} The last line shows that the singularities of $\Sigma$ are $32$, as required.

9611

alg-geom/9611009

en

https://arxiv.org/abs/alg-geom/9611009

alg-geom/9611009

Aleksandr V. Pukhlikov

Birational automorphisms of algebraic varieties with a pencil of cubic surfaces

29 pages, latex, to appear in Izvestia

It is proved that on a smooth algebraic variety, fibered into cubic surfaces over the projective line and sufficiently ``twisted'' over the base, there is only one pencil of rational surfaces -- that is, this very pencil of cubics. In particular, this variety is non-rational; moreover, it can not be fibered into rational curves. The proof is obtained by means of the method of maximal singularities.

alg-geom

\section*{Introduction} The present paper deals with birational maps of smooth algebraic threefolds, fibered into cubic surfaces -- that is, Del Pezzo surfaces of the degree 3, -- over a rational curve. In [1] V.A.Iskovskikh formulated a conjecture of ``birational rigidity'' of the pencils of Del Pezzo surfaces of the degrees 1,2 and 3, which are ``sufficiently twisted'' over the base. For the pencils of surfaces of the degrees 1 and 2 this conjecture (to be exact, its slight modification) was proved by means of the method of maximal singularities in [6]. The present paper is a direct continuation of [6]. It completes the proof of V.A.Iskovskikh's conjecture in full. The structure of the paper is analogous to [6], whereas most of it (Sections 4-6) is devoted to the study of infinitely near maximal singularities of linear systems. We are mostly concerned with the situation where the techniques developed in [6] do not work: when the maximal singularity which is to be excluded is contracted to a point lying on a line in a fiber of the pencil of cubic surfaces. To cope with this situation, we need to develop a new technique, based upon detailed analysis of effective 1-cycles on the certain specially chosen blow ups of the original variety. In the first section of the paper we describe the class of varieties, the birational type of which is to be studied. In the second section we formulate the principal result -- that is, that there is exactly one pencil of rational surfaces on the varieties from this class. Here we also start to prove it. The third section deals with maximal curves. Here we follow Yu.I.Manin [4] and V.A.Iskovskikh [1]. In the rest of the paper, which is actually its principal part, we study infinitely near singularities. Note that we do not repeat those proofs which are completely analogous to the proofs of the corresponding statements in [6]. We just make reference to the corresponding arguments in [6]. All the notations in [6] and in the present paper are compatible. Thus the proof of V.A.Iskovskikh's conjecture is now complete. This makes it possible to ``sum up'' the results which were obtained by means of the method of maximal singularities during the twenty five years of its existence, starting from the pioneer paper of V.A.Iskovskikh and Yu.I.Manin [2], where G.Fano's ideas were for the first time realized on the modern rigorous level. So: in the ``stable'', that is, sufficiently twisted over the base, cases we have the complete description of the birational type of conic bundles [8,9] and Del Pezzo fibrations. For certain Fano varieties, including some singular ones, analogous results were also obtained, see the survey [3]. There is some hope that in the nearest time M.Reid, A.Corti and the author will complete the joint paper in which it is to be proved that the weighted Fano hypersurfaces of the index 1 (there are 95 families of them) are birationally rigid. All this makes it not completely unrealistic to hope that the method of maximal singularities can be developed to be powerful enough to ensure the complete birational classification and description of the groups of birational automorphisms of varieties with negative Kodaira dimension, at least in dimension three. The paper was completed at the beginning of my stay at Max-Planck-Institut f\"ur Mathematik. I would like to express my gratitude to the staff of the Institute for their hospitality. \section{Varieties with a pencil of cubic surfaces} We study smooth three-dimensional projective varieties $V$ over the field of complex numbers ${\bf C}$, permitting a morphism $\pi:V\to{\bf P}^1$, every fiber of which is an irreducible reduced Del Pezzo surface of the degree 3, whereas its generic fiber $F_{\eta}$ is a smooth Del Pezzo surface of the degree 3 over the non-closed field ${\bf C}({\bf P}^1)$ with the Picard group $\mathop{\rm Pic} F_{\eta}={\bf Z} K_{F_{\eta}}$. In particular, $\mathop{\rm Pic} V={\bf Z} K_V\oplus {\bf Z} F$, where $F$ is a fiber of the morphism $\pi$. Obviously, $$ (K^2_V\cdot F)=3. $$ The relatively ample sheaf ${\cal O}_V(-K_V)$ is generated by its sections on each fiber, and thus determines the inclusion $$ V\hookrightarrow X, $$ where $X={\bf P}({\cal E})$, ${\cal E}={\cal O}\oplus{\cal O}(a_1)\oplus{\cal O}(a_2)\oplus{\cal O}(a_3)$ is a locally free sheaf of rank 4 on ${\bf P}^1$, $0\leq a_1\leq a_2\leq a_3.$ This sheaf coincides, up to twisting by an invertible sheaf, with $\pi_*{\cal O}_V(-K_V)$, see [1]. We denote the projection of $X$ onto ${\bf P}^1$ by $\pi$, too. Let $G$ be a fiber of the morphism $\pi:X\to{\bf P}^1$. Then $\mathop{\rm Pic} X={\bf Z} L\oplus{\bf Z} G$, where $L$ is the class of the tautological sheaf ${\cal O}_{X/{\bf P}^1}(1)$, that is, $\pi_*{\cal O}_X(L)={\cal E}$. Note that the sheaf ${\cal O}_X(L)$ is generated by global sections. In other words, $|L|$ is a free linear system. The class of the variety $V$ as a divisor on $X$ is equal to $3L+mG$. The canonical class of the variety $V$ is equal to the restriction onto $V$ of the class $$ -L+(m+a_1+a_2+a_3-2)G. $$ The group of 1-cycles on $V$ can be written down as $$ A^2(V)={\bf Z} s\oplus{\bf Z} f, $$ where $s$ is the class of a section, $f$ is the class of a line in a fiber: $$ (-K_V\cdot f)=(F\cdot s)=1, (F\cdot f)=0. $$ In the present paper we study those varieties, which satisfy the following condition: \newline\newline the class of 1-cycles $MK^2_V-f$ is not effective for any $M\in{\bf Z}$. \newline\newline We shall refer to this condition as the ``$K^2$-condition''. This requirement is satisfied, if $V$ is ``sufficiently twisted'' over the base ${\bf P}^1$. It is satisfied, anyway, if the following inequality holds: $$ (K^2_V\cdot L)\leq 0, $$ because the system $|L|$ is free. Easy computations lead us to the inequality $$ 5m\geq 12-3(a_1+a_2+a_3). $$ In particular, if $X={\bf P}^1\times{\bf P}^2$, i.e. $a_1=a_2=a_3=0$, then we get the condition $m\geq 3$, that is, $V$ is given in $X$ by an equation of the form $$ p_{3,m}(x_0,x_1,x_2,x_3;u,v)=0 $$ of bidegree $(3,m), m\geq 3$. Our $K^2$-condition essentially coincides with the hypothesis of V.A.Is\-kovskikh's conjecture [1], whereas for the case $X={\bf P}^1\times {\bf P}^2$ the coincidence is exact. For this reason we identify our theorem, which is proved below, with the above-mentioned conjecture. Besides, for some purely technical reasons, we assume that the following condition of general position holds: \newline\newline if the fiber $\pi^{-1}(t)\subset V$ over a point $t\in{\bf P}^1$ is a singular cubic surface, then it has exactly one singular point, which is a non-degenerate quadratic singularity. Moreover, exactly six lines on the cubic $\pi^{-1}(t)=F_t$ pass through this point.\newline\newline There is no doubt that the theorem proved below is true whether this assumption is satisfied or not. However, in the general case one has to consider a lot of particular geometric configurations and to adjust the constructions of Sections 4-6 to them. The author has not completed this job, that is why we restrict ourselves by the situation of general position. As it was done in [6], we call an irreducible curve $C\subset V$ a {\it horizontal} one, if $\pi(C)={\bf P}^1$, and a {\it vertical} one, if $\pi(C)$ is a point. We define a vertical and a horizontal effective 1-cycles, respectively. The {\it degree} of a horizontal 1-cycle $Z$ is equal to $(F\cdot Z)$, of a vertical one -- to $(-K_V\cdot Z)$. In both cases it is denoted by $\mathop{\rm deg} Z$. The fiber of $V$ over a point $t\in{\bf P}^1$ -- a cubic surface -- is denoted by the symbol $F_t$ or $F_x$, if $x\in F_t$. \begin{ppp} Assume $C\subset V$ to be an irreducible curve, $x\in C$ to be a point on it. Then $$ \mathop{\rm mult}\nolimits_xC\leq\mathop{\rm deg} C. $$ \end{ppp} {\bf Proof:} this is obvious. {\bf Definition.} A line $L\subset V$ (that is, a vertical curve of the degree 1; in other words, a true line in ${\bf P}^3$) is said to be {\it non-special}, if $L\cap\mathop{\rm Sing} F=\emptyset$, where $F\supset L$ is the fiber of the morphism $\pi$, containing $L$. It is said to be {\it special}, otherwise. In other words, in the special case the line contains the double point of the fiber. \begin{ppp} Let $L\subset V$ be a special line, $F\supset L$ be the corresponding fiber -- a singular cubic surface. Let $\sigma:\widetilde V\to V$ be the blowing up of $L$, $\widetilde F$ be the proper inverse image of the surface $F$ on $\widetilde V$. Then $\widetilde F$ is a smooth surface, isomorphic to the blow up of $F$ at the singular point. \end{ppp} {\bf Proof:} straightforward local computations. Let us also define the ``intersection index'' of two curves on a variety of arbitrary dimension. Assume $R$ to be a nonsingular projective curve on an algebraic variety $Y$, $R\cap\mathop{\rm Sing} Y=\emptyset$, and $C\subset Y$ to be an arbitrary irreducible curve, $C\neq R$. {\bf Definition.} The {\it intersection index} $$ (C\cdot R) $$ of the curves $C$ and $R$ is the integer given by one of the two equivalent constructions: 1) $(C\cdot R)=\sum\limits_{x\in C\cap R}\mathop{\rm mult}\nolimits_xC$, where the sum is taken over all the points of intersection of $C$ and $R$, {\it including infinitely near ones}; 2) $(C\cdot R)=(\widetilde C\cdot E)$, where $\sigma:\widetilde Y\to Y$ is the blowing up of the curve $R$ with the exceptional divisor $E\subset\widetilde Y$, and $\widetilde C\subset\widetilde Y$ is the proper inverse image of the curve $C$. We define the intersection index $(Z\cdot R)$ for any 1-cycle $Z$ on $Y$, which does not contain $R$ as a component, by linearity. If $S\subset Y$ is a smooth projective surface, containing $R$ and the support of $Z$, then we get the intersection index $(Z\cdot R)$ on $S$ in the usual sense. Let us introduce one more notation: the multiplicity of the curve $C$ in a 1-cycle $Z$ is denoted simply by $$ \mathop{\rm mult}\nolimits_C Z. $$ \section{Formulation of the principal result.\protect\\ Start of the proof} Now fix a variety $V$, for which the $K^2$-condition is satisfied. Let $V'$ be a projective threefold, nonsingular in codimension 1 and fibered into rational surfaces $$ \pi':V'\to{\bf P}^1, $$ Let $F'$ be a fiber of this pencil. Our principal result is the following \newline\newline {\bf Theorem.} {\it Any birational map} $$ \chi:V-\,-\,\to V' $$ {\it (provided there are any) is fiber-wise with respect to the pencils} $\pi,\pi'$, {\it that is,} $$ \chi^{-1}(|F'|)=|F|. $$ {\it In other words, there is an isomorphism of the base} $$ \alpha:{\bf P}^1\to{\bf P}^1, $$ {\it such that} $\pi'\circ\chi=\alpha\circ\pi$. \newline\newline {\bf Remark.} In fact, we could assume $|F'|$ to be any pencil of surfaces of negative Kodaira dimension. It is not necessary to make any changes at all in the proof given below to make it work in this case. But this, formally more general, statement gives actually no additional information about the birational type of the variety $V$. So we formulate our theorem in the same manner as it was done in [1]. {\bf Corollary.} (i) $V$ {\it has only one pencil of rational surfaces. In particular, } $V$ {\it is not rational and has no structures of a conic bundle.} (ii) {\it The quotient group of the group} $\mathop{\rm Bir} V$ {\it of birational automorphisms of the variety} $V$ {\it by the normal subgroup of birational automorphisms, preserving the fibers of} $\pi$ {\it (which is isomorphic to the group} $\mathop{\rm Bir} F_{\eta}$ {\it of birational automorphisms of the generic fiber),} {\it is finite, generically trivial.} \newline\newline The theorem was formulated by V.A.Iskovskikh as a conjecture [1]. The corollaries (i) and (ii) follow from the theorem in an obvious way. They were also formulated in the above-mentioned paper of V.A.Iskovskikh. \subsection*{Start of the proof} Let $|\chi|$ be the proper inverse image of the pencil $|F'|$ on $V$. There exist integers $n=n(\chi)\in{\bf Z}_+$ and $l\in{\bf Z}_+$ such that $$ |\chi|\subset|-n(\chi)K_V+lF|. $$ If $n(\chi)=0$, then $l=1$ and the theorem is true. Starting from this moment, we assume that $n(\chi)\geq 1$. Obviously, this is equivalent to $\chi$ being not fiber-wise. We show below that this assumption leads to a contradiction. The fact that the pencil $|\chi|$ has no fixed components implies, in accordance with the $K^2$-condition, that $l\geq 0$ (see [6]). \subsection*{Adjunction break condition} We use the language of discrete valuations in the form of [5,7]. The centre of a valuation $\nu\in{\cal N}(V)$ is denoted by $Z(V,\nu)$. If $Z(V,\nu)$ is a point, then the fiber of the pencil $|F|$, containing this point, is denoted by $F_{\nu}$. Abusing the notations, we sometimes write $T$ instead of $\nu$, if $\nu=\nu_T$, where $T$ is a prime Weyl divisor on some model $\widetilde V$ of the field ${\bf C}(V)$, $T\not\subset\mathop{\rm Sing}\widetilde V$, realizing the discrete valuation $\nu$. For a valuation $\nu\in{\cal N}(V)$ set $$ e(\nu)=\nu(|\chi|)-n\delta(\nu), $$ where $\delta(\nu)=K(V,\nu)$ is the canonical valuation (discrepancy) of $\nu, n=n(\chi)$. The valuations for which $\nu(|\chi|)>0$ are said to be {\it singularities} of the system $|\chi|$. A discrete valuation for which $e(\nu)>0$ is said to be a {\it maximal singularity}. In the assumptions of the theorem we get \begin{ppp} A maximal singularity does exist. Moreover, one of the following two cases takes place: (i) there is a maximal singularity $\nu\in{\cal N}(V)$ such that its centre $Z(V,\nu)$ on $V$ is a curve; (ii) there is a finite set of maximal singularities ${\cal M}\subset{\cal N}(V)$, the centres $Z(V,\nu)=x(\nu)$ of which are points on $V$, and, moreover, the following inequality holds $$ \sum\limits_{t\in{\bf P}^1} \left( \max\limits_{\{\nu\in{\cal M}| x(\nu)\in F_t=F_{\nu}\}} \frac{e(\nu)}{\nu(F_t)} \right)>l. $$ \end{ppp} The {\bf proof} was given in [6]. The structure of the rest of the paper is similar to [6]: first (in the next section), we study the maximal curves (the case (i) of the Proposition). The principal part of our proof is concentrated in Sections 4-6, where we show that the case (ii) of the Proposition is impossible -- it leads to a contradiction. \section{Maximal curves} Here we follow Yu.I.Manin [4] and V.A.Iskovskikh [1]. If the centre of a maximal singularity $\nu$ is a curve $C\subset V$, then it is easy to see that the curve $C$ itself is already maximal: $$ \mathop{\rm mult}\nolimits_C|\chi|>n(\chi). $$ \subsection*{Case 1: $C$ is horizontal} First of all, we restrict the linear system $|\chi|$ onto the generic fiber $F$ and obtain the inequality $\mathop{\rm deg} C\leq 2$ (by the arguments, similar to the corresponding ones in [6], Section 4). Let us start with the case $\mathop{\rm deg} C=1$, that is, $C$ is a section of the morphism $\pi$. For a general point $t\in{\bf P}^1$ take a general line $L\subset{\bf P}^3=G_t$, passing through the point $C\cap F_t$. This line intersects the cubic surface $F_t$ at two more different points $x,y$. Set $$ \tau_C(x)=y. $$ Obviously, by means of this construction the birational involution $\tau_C\in\mathop{\rm Bir} F_{\eta}\subset\mathop{\rm Bir} V$ is defined. Let $\alpha:V^*\to V$ be the blowing up of the curve $C$, $E=\alpha^{-1}(C)$ be the exceptional divisor, $\mathop{\rm Pic} V^*={\bf Z} h\oplus{\bf Z} e\oplus{\bf Z} F, h=-K_V$. \begin{lll} The birational involution $\tau_C$ extends to a biregular involution of an invariant open set $V^*\backslash Y$, $\mathop{\rm codim} Y\geq 2$. Its action on $\mathop{\rm Pic} V^*/{\bf Z} F\cong{\bf Z}\bar h\oplus{\bf Z}\bar e$ is given by the relations $$ \tau^*_C\bar h= 3\bar h-4\bar e, $$ $$ \tau^*_C\bar e= 2\bar h-3\bar e. $$ \end{lll} {\bf Proof.} See [4]. Now let us ``untwist'' the curve $C$. Consider the composition $\chi\circ\tau_C:V-\,-\,\to V'$. \begin{lll} The following relation holds $$ n(\chi\circ\tau_C)= 3n(\chi)-2\nu_C(\chi)<n(\chi), $$ Besides, the curve $C$ is no more maximal for the composition $\chi\circ\tau_C$. \end{lll} {\bf Proof.} The linear system $|\chi\circ\tau_C|$ is the proper inverse image of $|\chi|$ with respect to $\tau_C$. Respectively, the proper inverse image of the linear system $|\chi\circ\tau_C|$ on $V^*$ can be obtained by applying $\tau_C$ to the proper inverse image of the linear system $|\chi|$ on $V^*$. Thus $$ n(\chi\circ\tau_C)h- \nu_C(\chi\circ\tau_C)e= $$ $$ =\tau^*_C\left( n(\chi)h-\nu_C(\chi)e \right)= \left(3n(\chi)-2\nu_C(\chi) \right)h+\dots, $$ and we are done. Q.E.D. \newline\newline Now assume that $\mathop{\rm deg} C=2$, i.e. $C$ is a bisection of the morphism $\pi$. We define the involution $\tau_C$ by its action on the generic fiber $F$ in the following way (see [4]). Set $\{a,b\}=C\cap F$. Let $q=L_{ab}\cap F$ be the third point of intersection of the line in ${\bf P}^3$, joining the points $a$ and $b$, with the cubic surface $F$. These points $q$ sweep out a curve $C^*\subset V$, which is a section of the morphism $\pi$, i.e. $q=C^*\cap F$. The pencil of planes $P$ in ${\bf P}^3$, containing the line $L_{ab}$, generates the pencil of elliptic curves $Q_P=P\cap F$ on the surface $F$. Set $$ \tau_C\left|_{Q_P}(x)\right.=y, $$ where $$ x+y\sim 2q $$ on $Q_P$, i.e. $\tau_C$ is the reflection from the point $x$ on the elliptic curve $Q_P$. Thus the involution $\tau_C\in\mathop{\rm Bir} F_{\eta}\subset\mathop{\rm Bir} V$ is defined. Let $\alpha:V^*\to V$ be the blowing up of the curve $C$, $E=\alpha^{-1}(C)$ be the exceptional divisor, $\mathop{\rm Pic} V^*={\bf Z} h\oplus{\bf Z} e\oplus{\bf Z} F, h=-K_V$. \begin{lll} The birational involution $\tau_C$ extends to a biregular involution of an invariant open set $V^*\backslash Y$, $\mathop{\rm codim} Y\geq 2$. Its action on $\mathop{\rm Pic} V^*/{\bf Z} F\cong{\bf Z}\bar h\oplus{\bf Z}\bar e$ is given by the relations $$ \tau^*_C\bar h= 5\bar h-6\bar e, $$ $$ \tau^*_C\bar e= 4\bar h-5\bar e. $$ \end{lll} {\bf Proof:} straightforward computations. Now consider the composition $\chi\circ\tau_C:V-\,-\,\to V'$. \begin{lll} The following relation holds $$ n(\chi\circ\tau_C)= 5n(\chi)-4\nu_C(\chi)<n(\chi). $$ Moreover, the curve $C$ is no more maximal for the composition $\chi\circ\tau_C$. \end{lll} {\bf Proof: } straightforward computations, similar to the above ones. The computations which were just performed are none else but the well known constructions of the two-dimensional birational geometry over non-closed fields [4], translated into the language of a threefold, fibered over ${\bf P}^1$. \subsection*{Case 2: $C$ is vertical} It was proved in [1], that this case does not realize. The most easy way to show it is by means of the techniques developed in [5]. First of all, it is easy to see that $C\subset F\subset {\bf P}^3$ is a line or a conic. Furthermore, let $x\in {\bf P}^3$ be a general point, $S(x)$ be the cone over $C$ with the vertex $x$. Then $$ C\cup R(x)=F\cap S(x), $$ where $R(x)$ is the residual curve. It was shown in [5] that $(C\cdot R(x))=\mathop{\rm deg} R(x)$ (in the sense of the definition of the ``intersection index'', given in Section 1). This fact together with the inequality $\mathop{\rm mult}\nolimits_C|\chi|>n$ implies that $R(x)$ is a basic curve of the linear system $|\chi|$. Consequently, $F$ is a fixed component of the pencil $|\chi|$. Contradiction. Q.E.D. Summing up all these results, we get \begin{ppp} There exists a fiber-wise birational automorphism $$ \chi^*=\tau_{C_1}\circ\dots\circ\tau_{C_k} $$ such that the linear system $|\chi\circ\chi^*|$ has no maximal curves on $V$. \end{ppp} All that we need to show now in order to prove our Theorem, is that the second case (case (ii)) of Proposition 2.1 does not realize. Starting from this moment, we assume thus that this very case takes place. Let us show that this assumption leads to a contradiction. \section{Infinitely near singularities I.\protect\\ Existence of a line } \subsection*{The supermaximal singularity} The symbols $D_1,D_2\in|\chi|$ stand for generic divisors of our linear system. Consider the effective 1-cycle $$ Z=(D_1\bullet D_2). $$ It can be decomposed into the vertical and horizontal components: $$ Z=Z^v+Z^h, $$ whereas for the vertical cycle $Z^v$ we get the decomposition $$ Z^v= \sum_{t\in{\bf P}^1}Z^v_t, $$ where the support of the cycle $Z^v_t$ lies in the fiber $F_t$. \begin{ppp} There is a maximal singularity $\nu=\nu_T\in {\cal M}$ such that $x=x(\nu_T)=Z(V,\nu)$ is a point in a fiber $F=F_t$, lying on a line $L\subset F$. Moreover, if $$ Z^v_t=C+kL, $$ where the effective 1-cycle $C\subset F$ does not contain the line $L$, then $$ (C\cdot L)< \frac{4ne}{\nu_T(F)}. $$ \end{ppp} {\bf Proof.} It was proved in [6] that the fact that the linear system $|\chi|$ has no maximal curves implies the existence of a {\it supermaximal singularity} $\nu=\nu_T\in{\cal M}$, which satisfies the inequality $$ \mathop{\rm deg} Z^v_t< \frac{2dne(T)}{\nu(F_t)}, $$ where $x(\nu)\in F_t$ and $d$ is the degree of the generic fiber, that is, the Del Pezzo surface $F_{\eta}$. In our case $d=3$. Let us show that the supermaximal singularity $\nu$ satisfies our proposition. As it was done in [6], we write $x$ instead of $x(\nu_T)$, $e$ instead of $e(T)$, $F$ instead of $F_t$, $Z^v$ instead of $Z^v_t$. \begin{lll} The following inequality is true: $$ \mathop{\rm mult}\nolimits_xZ^v\geq \frac{4ne}{\nu_T(F)}. $$ \end{lll} {\bf Proof.} Assume the converse. Now, repeating the arguments of Section 5 in [6] word for word, we come to a contradiction, for the only fact, upon which they were based, was exactly the inequality $$ \mathop{\rm mult}\nolimits_xZ^v< \frac{4ne}{\nu_T(F)}. $$ If it is true, then the supermaximal singularity just cannot exist. Q.E.D. for the Lemma. \subsection*{Existence of a line} \begin{lll} There is at least one line $L\subset F\subset {\bf P}^3$ passing through the point $x\in F$. \end{lll} {\bf Proof.} Assume the converse. Then the point $x$ is a smooth point of the cubic surface $F\subset{\bf P}^3$. Moreover, the curve $R=T_xF\cap F$ is irreducible, its degree is equal to 3 and its multiplicity at the point $x$ is equal to 2 exactly. If $C\subset F$ is any other curve, then $$ \mathop{\rm deg} C=(C\cdot R)\geq (C\cdot R)_x\geq 2\mathop{\rm mult}\nolimits_xC. $$ Thus for any curve $Q\subset F$ we get the inequality $$ \mathop{\rm mult}\nolimits_xQ\leq \frac23\mathop{\rm deg} Q. $$ Consequently, $$ \mathop{\rm mult}\nolimits_xZ^v\leq \frac23\mathop{\rm deg} Z^v< \frac{4ne}{\nu(F)}. $$ Contradiction. Q.E.D. Now let $x\in F$ be a smooth point. There exist $k$ lines, lying on $F$, $1\leq k\leq 3$ and passing through $x$. If $\mathop{\rm mult}\nolimits_x(F\cap T_xF)=2$, then $k\leq 2$ and for any curve $C\subset F$, which is different from these $k$ lines, we get the inequality $$ 2\mathop{\rm mult}\nolimits_x C\leq \mathop{\rm deg} C. $$ If $\mathop{\rm mult}\nolimits_x(F\cap T_xF)=3$, then this inequality can be strengthened: $$ 3\mathop{\rm mult}\nolimits_xC\leq \mathop{\rm deg} C. $$ \subsection*{The case of a single line} Assume that there is only one line on $F$, passing through $x$. In this case the point $x$ is smooth on $F$ and $T_xF\cap F=L+Q$, where $Q\subset F$ is a smooth conic. The arguments given above show that for any curve $C\subset F$, $C\neq L$, the following inequality takes place: $$ \mathop{\rm mult}\nolimits_xC\leq \frac12\mathop{\rm deg} C. $$ Write down $Z^v=C+kL$, where $C$ is an effective 1-cycle, not containing $L$. Now $$ k+\frac12\mathop{\rm deg} C\geq \frac{4ne}{\nu(F)}, $$ $$ k+\mathop{\rm deg} C< \frac{6ne}{\nu(F)}. $$ This implies that $$ \mathop{\rm deg} C< \frac{4ne}{\nu(F)}. $$ Since $(C\cdot L)\leq \mathop{\rm deg} C$, Proposition 4.1 is proved in this case. \subsection*{The case of two lines} Assume that there are exactly two lines, $L_1$ and $L_2$ on $F$, passing through $x$. In this case $x$ is a smooth point of $F$ and $T_xF\cap F=L_1+L_2+L_3$, where $L_3$ is a line, different from $L_1$, $L_2$, $x\not\in L_3$. Note that one of the points $L_i\cap L_3$, $i=1,2$ can be singular on $F$. Assume at first that this is not the case. Write down $Z^v=Q+k_1L_1+k_2L_2+k_3L_3$ and set $d=\mathop{\rm deg} Q$, $d_i=(Q\cdot L_i)$, $m=\mathop{\rm mult}\nolimits_xQ$. We get the following four inequalities $$ 2m\leq d, $$ $$ k_1+k_2+k_3+d< \frac{6ne}{\nu(F)}, $$ $$ d_1+d_2+d_3=d, $$ $$ k_1+k_2+m\geq \frac{4ne}{\nu(F)}. $$ It is easy to see that there is $i\in\{1,2\}$ such that for $\{j\}=\{1,2\}\backslash\{i\}$ $$ k_j+k_3+d_i< \frac{4ne}{\nu(F)}. $$ Indeed, otherwise for any $i\in\{1,2\}$ we have the opposite inequalities. Put them up together and add the fourth inequality. We get $$ 2(k_1+k_2+k_3)+m+d_1+d_2\geq \frac{12ne}{\nu(F)}. $$ This contradicts the second inequality. So we may assume that $\nu(F)(k_2+k_3+d_1)<4ne.$ Setting $C=Q+k_2L_2+k_3L_3$, we get exactly our proposition. Finally, if the point $L_1\cap L_3$ is singular on $F$, then, instead of the equality $d_1+d_2+d_3=d$ one should use the inequality $d_1+d_2\leq d$, where $d_1=(Q\cdot L_1)$ is taken in the sense of Section 1 of the present paper. It is this very inequality upon which our arguments are actually based. \subsection*{The case of three lines} Assume that there are exactly three lines $L_i$, $i=1,2,3$, on $F$, passing through $x$. Again $x$ is smooth on $F$ and $T_xF\cap F=L_1+L_2+L_3$. Write out the 1-cycle $Z^v=Q+k_1L_1+k_2L_2+k_3L_3$ and set $d=\mathop{\rm deg} Q$, $d_i=(Q\cdot L_i)$, $m=\mathop{\rm mult}\nolimits_xQ$. Again we get a set of inequalities, $$ 3m\leq d, $$ $$ k_1+k_2+k_3+d< \frac{6ne}{\nu(F)}, $$ $$ d_1+d_2+d_3=d, $$ $$ k_1+k_2+k_3+m\geq \frac{4ne}{\nu(F)}. $$ By means of the same elementary arithmetic as above we get that for some $i\in\{1,2,3\}$ and $\{j,l\}=\{1,2,3\}\backslash\{i\}$ the following inequality is true: $$ k_j+k_l+d_i< \frac{4ne}{\nu(F)}. $$ Setting $L=L_i$, $C=k_jL_j+k_lL_l+Q$, we get our proposition. \subsection*{The case of six lines} In this case $x\in F$ is a double point (a non-degenerate elementary singularity). Here it is not enough just to make reference to Lemma 4.1. It is necessary to retrace the arguments of [6] (Section 5) in details. Recall our principal notations: $$ \begin{array}{cccc} \displaystyle \varphi_{i,i-1}: & V_i & \to & V_{i-1} \\ \displaystyle & \bigcup & & \bigcup \\ \displaystyle & E_i & \to & B_{i-1} \end{array} $$ is the resolution of $\nu$ [5,7], that is, $\varphi_{i,i-1}$ blows up the irreducible cycle $B_{i-1}$ -- the centre $Z(V_{i-1},\nu)$ of the valuation $\nu$ on $V_{i-1}$, $E_i=\varphi^{-1}_{i,i-1}(B_{i-1})$ is the exceptional divisor, $1\leq i\leq K$, $\nu_{E_K}=\nu$. The first $L\leq K$ centres of the blowing ups are points, after that we blow up curves covering one another. We equip the set of indices $\{1,\dots,K\}$ with the natural oriented graph structure [2,3,5-7], $p(i.j)$ stands for the number of paths from $i$ to $j$, if $i\neq j$, and $p(i,i)=1$. Among the six lines $L_i\ni x$ we choose one (let it be $L_1$), such that $$ B_1\not\in L^1_i $$ for $i\neq 1$, if $B_1$ is a point. Such a line does exist. Now write down explicitly $$ Z^v=kL_1+R+Q, $$ where $R$ consists of the multiple lines $L_i$, $i\neq 1$, whereas the 1-cycle $Q$ does not contain any line, passing through $x$. We define the integers $M$ and $N$, by requiring that $$ B_{i-1}\in L^{i-1}_1, i=1,\dots,M, $$ $$ B_{i-1}\in F^{i-1}, i=1,\dots,N, $$ $M\leq N\leq L$, and set $$ q_i= \mathop{\rm mult}\nolimits_{B_{i-1}}Q^{i-1} $$ for $i=1,\dots,N$. Obviously, $q_1\geq q_2\geq\dots $ and $$ d=\mathop{\rm deg} Q\geq q_1+q_2+\dots+q_N $$ (since $Q$ does not contain the line $L_1$ as a component). Now, if the following inequality holds: $$ \left( \sum^M_{i=1}p_i \right) k+ p_1\mathop{\rm deg} R+ \sum^N_{i=1}p_iq_i< 4ne, $$ then we immediately get a contradiction by means of the Iskovskikh-Manin's techniques, repeating the arguments of [6], Section 5 word for word. Consequently, the opposite inequality holds. Since $2q_2\leq d$, then the following inequality is true: $$ \left( \sum^M_{i=1}p_i \right) k+ \frac12\left(2p_1+ \sum^N_{i=2}p_i\right) (\mathop{\rm deg} R+\mathop{\rm deg} Q)\geq 4ne. $$ Comparing it with the inequality $$ \left( 2p_1+ \sum^N_{i=2}p_i\right) (k+\mathop{\rm deg} R+\mathop{\rm deg} Q)< 6ne, $$ which is true by definition of a supermaximal singularity, we get the following estimate: $$ k> \frac{2ne}{\nu(F)}. $$ Setting $C=R+Q$, we complete the proof of the proposition. Note that some more delicate arguments (which are based upon the properties of the integers $p_j$ only) make it possible to obtain a stronger estimate of the integer $(C\cdot L)$ in the last case. However, to prove our main result, we need just the inequality of Proposition 4.1. \section{Infinitely near maximal singularities II. \protect\\ The basic construction} \subsection*{Description of the basic construction} An infinite series of blow ups $$ \begin{array}{cccc} \displaystyle \sigma_i: & V^{(i)} & \to & V^{(i-1)} \\ \displaystyle & \bigcup & & \bigcup \\ \displaystyle & E^{(i)} & \to & L_{i-1}, \end{array} $$ $i\geq 1$, starting from $V^{(0)}=V$, where $L_{i-1}$ is the centre of the $i$-th blow up, and $E^{(i)}=\sigma^{-1}_i(L_{i-1})$ is its exceptional divisor, $L_0=L$, is said to be a {\it staircase, associated with the line} $L$, or, simply, an $L$-{\it staircase}, if the following conditions are satisfied:\newline\newline $L_i$ is a curve for all $i\in{\bf Z}_+$, $E^{(i)}$ is a ruled surface of the type ${\bf F}_1$ over $L_{i-1}$ and $L_i\subset E^{(i)}$ is the exceptional section (i.e. the (-1)-curve).\newline\newline Obviously, by this definition the staircase is unique. Just below we show that it exists. Its segment, consisting of the blow ups $\sigma_i$ for $1\leq i\leq M$, is said to be a (finite) staircase of the length $M$. It is convenient to prove the existence of the staircase together with some of its properties. For conveniency of notations set $E^{(0)}$ to be the fiber $F$ of the morphism $\pi$, which contains $L$. The operation of taking the proper inverse image on the $i$-th step (i.e. on $V^{(i)}$) is denoted by adding the bracketed upper index $i$. For instance, the proper inverse image of the surface $E^{(i)}$ on $V^{(j)}$ for $j\geq i$ is written down as $E^{(i,j)}$. Set also: $s_i$ to be the class of $L_i$ in $A^2(V^{(i)})$, $s_0=f$; $f_i\in A^2(V^{(i)})$ to be the class of the fiber of the ruled surface $E^{(i)}$ over a point $\in L_{i-1}$. Abusing our notations, we sometimes treat $s_i$ and $f_i$ as numerical classes of curves on the ruled surface $E^{(i)}$: $$ A^1E^{(i)}= \mathop{\rm Pic} E^{(i)}= {\bf Z} s_i\oplus{\bf Z} f_i, $$ so that, in particular, the formulas like $$ (s_i\cdot s_i)=-1, $$ $$ (s_i\cdot f_i)=1 $$ make sense. In these notations we have the following \begin{ppp} (i) For $i\geq 2$ the effective 1-cycle $(E^{(i-1,i)}\bullet E^{(i)})$ is just the irreducible curve $E^{(i-1,i)}\cap E^{(i)}$. Its numerical class is equal to $(s_i+f_i)$. In particular, this curve does not intersect $L_i\sim s_i$. If the line $L$ is non-special, then this statement is true for $i=1$, too. If, on the contrary, $L$ is special, then the 1-cycle $(F^{(1)}\bullet E^{(1)})$ is a reducible curve. More exactly, it is the sum of the exceptional section $L_1$ and the fiber over the singular point of the surface $F$. (ii) The following equalities hold: $$ (E^{(i)})^3=1, $$ $$ (E^{(i)}\cdot L_i)=0. $$ Taking into account the isomorphism $L_i\cong{\bf P}^1$, we can write down $$ {\cal N}_{L_i/V^{(i)}}\cong {\cal O}_{L_i}\oplus{\cal O}_{L_i}(-1). $$ In this representation the first component is uniquely determined. It corresponds to the exceptional section $L_{i+1}\subset E^{(i+1)}= {\bf P}({\cal N}_{L_i/V^{(i)}})$. For the second component we can take the one-dimensional subbundle, corresponding exactly to the curve $E^{(i,i+1)}\cap E^{(i+1)}$. (iii) The classes $s_i$ and $f_i$ satisfy the relations $$ \sigma^*s_{i-1}=s_i, $$ $$ \sigma_* f_i=0 $$ for $i\geq 1$. \end{ppp} \subsection*{Proof in the non-special case} Assuming that $L\cap\mathop{\rm Sing} F=\emptyset$, we prove simultaneously the existence of the staircase and Proposition 5.1. Let us consider the first step of the staircase, that is, the morphism $$ \sigma_1: V^{(1)}\to V^{(0)}=V, $$ blowing up the line $L_0=L\subset F$. We get the exact sequence $$ 0\to {\cal N}_{L/F}\to {\cal N}_{L/V}\to {\cal O}_V(F)\left|_L\right.\to 0, $$ which can be rewritten down in the following way: $$ 0\to {\cal O}_L(-1)\to {\cal N}_{L/V}\to {\cal O}_L\to 0. $$ Consequently, $E^{(1)}$ is a ruled surface of the type ${\bf F}_1$, $(E^{(1)})^3=1$, whence $(E^{(1)}\cdot E^{(1)})\sim (-s_1-f_1)$ and $(E^{(0,1)}\cdot L_1)$ $=((F-E^{(1)})\cdot s_1)=0$. Thus all the requirements (i)-(iii) of the definition of the staircase are satisfied for the first blow up. We proceed by induction on $i\geq 1$. We get the exact sequence $$ 0\to {\cal N}_{L_i/E^{(i)}}\to {\cal N}_{L_i/V^{(i)}}\to \left.{\cal O}_{V^{(i)}}(E^{(i)})\right|_{L_i}\to 0. $$ Taking into account the facts which were already proved, this sequence can be rewritten down as follows: $$ 0\to {\cal O}_{L_i}(-1)\to {\cal N}_{L_i/V^{(i)}}\to {\cal O}_{L_i}\to 0. $$ Again this implies that $E^{(i+1)}={\bf P}({\cal N}_{L_i/V^{(i)}})$ is a ruled surface of the type ${\bf F}_1$ and $(E^{(i+1)})^3=1$, so that $$ E^{(i+1)}\left|_{E^{(i+1)}}\right.\sim (-s_{i+1}-f_{i+1}). $$ Thus $(E^{(i+1)}\cdot L_{i+1})=0$, (i) and (iii) are satisfied in an obvious way. The proof is complete. Q.E.D. \subsection*{Proof in the special case} Assume that the line $L$ contains the double point $p\in F$ of the fiber. Again consider the first blow up: $$ \sigma_1:V^{(1)}\to V^{(0)}=V. $$ As we have mentioned, the proper inverse image of the fiber $F^{(1)}=E^{(0,1)}$ is already a non-singular surface. It is easy to see that $(F^{(1)}\bullet E^{(1)})=L_1+R$, where $R=\sigma^{-1}_1(p)$ is the fiber of $\sigma_1$ over the singular point, whereas $L_1$ is a certain section of the ruled surface $E^{(1)}$. Since $K_{F^{(1)}}=\sigma^*_1 K_F$, we get $$ (L_1\cdot L_1)_{F^{(1)}}=-1, $$ so that $(L_1\cdot E^{(1)})=0$ and $(L_1\cdot F^{(1)})=0$. Since $E^{(1)}\left|_{E^{(1)}}= -F^{(1)}\right|_{E^{(1)}}= -(L_1+R)$, we get $$ (L_1\cdot L_1)_{E^{(1)}}=-1, $$ so that $E^{(1)}$ is a ruled surface of the type ${\bf F}_1$ and the conditions (i)-(iii) of Proposition 5.1 are satisfied. The rest of the arguments (for $i\geq 2$) just repeat word for word the non-special case. The proof of the existence of the staircase and of Proposition 5.1 is complete. Q.E.D. {\bf Remarks.} (i) Since $E^{(i-1,i)}$ does not intersect $L_i$ (for $i\geq 1$ in the non-special and for $i\geq 2$ in the special case), we get $$ E^{(i-1,i)}= E^{(i-1,i+1)}= \dots =E^{(i-1,j)}= \dots $$ for any $j\geq i$. In particular, if $C\subset E^{(i-1)}$ is a curve, which is not the exceptional section $L_{i-1}$, then its proper inverse images on all the varieties $V^{(j)}$, $j\geq i$, are the same: $$ C^{(i)}= C^{(i+1)}= \dots= C^{(j)}. $$ (ii) Abusing our notations, we call an irreducible curve $C\subset E^{(i)}$, $i\geq 1$, a {\it horizontal } one, if $\sigma_i(C)=L_{i-1}$, and a {\it vertical} one, if $\sigma_i(C)$ is a point on $L_{i-1}$. Respectively, we define horizontal and vertical 1-cycles with the support in $E^{(i)}$. The {\it degree} of a horizontal curve $C$ is equal to $\mathop{\rm deg} C=$ $\mathop{\rm deg}\sigma_i\left|_C\right.$ $=(C\cdot f_i)$, the {\it degree } of a vertical curve $C$ is equal to $\mathop{\rm deg} C=(C\cdot L_i)=1$. We define the degree of a horizontal and a vertical 1-cycle with the support in $E^{(i)}$ as its intersection with $f_i$ and $L_i$, respectively. In particular, the degree of a vertical 1-cycle is just the number of lines (fibers) in it. Note that if an effective horizontal 1-cycle $C$ does not contain the exceptional section $L_i$ as a component, then its class in $A^1(E^{(i)})$ or $A^2(V^{(i)})$ is equal to $\alpha s_i+\beta f_i$, where $\alpha \geq 1$ and $\beta\geq\alpha$. (iii) Obviously, the graph of the sequence of the blow ups $\sigma_i$ is a chain. In particular, $$ K_{V^{(M)}}= \sigma^*_{M,0}K_V+ \sum^M_{i=1} \sigma^*_{M,i}E^{(i)} $$ (where $\sigma_{i,j}$, as always, stands for the composition $\sigma_{j+1}\circ\dots\circ\sigma_i$) and the canonical multiplicity of the valuation $\nu_{E^{(i)}}$ is equal to $i$. In the non-special case $$ \sigma^*_{M,0}F= F^{(M)}+ \sum^M_{i=1}E^{(i,M)}, $$ whereas in the special case for $M\geq 2$ $$ \sigma^*_{M,0}F= F^{(M)}+ E^{(1,M)}+ 2\sum^M_{i=2}E^{(i,M)}. $$ In the special case $F^{(1)}\cap E^{(1)}$ is equal to the reducible curve $L_1+R$, $R=\sigma^{-1}_1(p)$ is the fiber over the double point of the surface $F$. There are five more lines on $F$ besides $L$ (in accordance with the condition of general position), passing through $p$. Let $Q$ be one of them, $Q^{(1)}\subset F^{(1)}$ be its proper inverse image on $V^{(1)}$. It is easy to see that the point $$ \left( Q^{(1)}\cap E^{(1)} \right) \in R $$ does not lie on the exceptional section $L_1$. \section{Infinitely near maximal singularities III.\protect\\ Completing the proof} The present section is the key part of the paper. Here we exclude the supermaximal singularity. In accordance with what was proved in Section 4, the centre $x=Z(V,\nu)$ lies on a line $L\subset F$ and $$ (C\cdot L)< \frac{4ne}{\nu(F)}, $$ $$ Z^v=C+kL. $$ \begin{ppp} There exists a finite $L$-staircase of the length $M\geq 1$ satisfying the following conditions: (i) for $i=0,\dots,M-1$ the centre $Z(V^{(i)},\nu)$ of the valuation $\nu$ on $V^{(i)}$ is a point $x_i\in L_i$, $x_0=x$, (ii) the centre $Z(V^{(M)},\nu)$ is either: A) a point $x_M\not\in L_M, x_M\not\in E^{(M-1,M)}$; B) the line $B=\sigma^{-1}_M(x_{M-1})$, that is, a fiber of the ruled surface $E^{(M)}$; C) the point $x_M= E^{(M-1,M)}\cap \sigma^{-1}_M(x_{M-1})$. \end{ppp} {\bf Proof.} If $Z(V^{(i)},\nu)\subset E^{(i)}$, then $i=K(V,E^{(i)})\leq K(V,\nu)$. Consequently, there exists an integer $M\geq 1$, such that for $i=0,\dots,(M-1)$ the condition (i) is satisfied, whereas for $i=M$ it is not satisfied. Now the centre $Z(V^{(M)},\nu)$ is either a curve (and in this situation we get the case B)), or a point $x_M$, not lying on $L_M$ (one of the cases A) and C)). Q.E.D. for the Proposition. Let us fix the just constructed staircase of the length $M$. Let us introduce a new parameter $d_F$, setting it to be equal to 1, if the case is non-special, and 2, otherwise. Denote $E^{(M)}\subset V^{(M)}$ by $E$ in the cases A) and B), $E^{(M)}$ by $E_+$ and $E^{(M-1,M)}$ by $E_-$ in the case C). \subsection*{Noether-Fano inequality in terms of the staircase} Denote by $|\chi|^{(i)}$ the proper inverse image of the system $|\chi|$ on $V^{(i)}$ and set $$ \lambda_i= \mathop{\rm mult}\nolimits_{L_{i-1}}|\chi|^{(i-1)}, $$ $n\geq\lambda_1\geq\dots$. Let $$ \begin{array}{cccc} \displaystyle \varphi_{i,i-1}: & V_i & \to & V_{i-1} \\ \displaystyle & \bigcup & & \bigcup \\ \displaystyle & E_i & \to & B_{i-1}, \end{array} $$ $i=1,\dots,K$, $V_0=V^{(M)}$, be the resolution of the valuation $\nu\in{\cal N}(V^{(M)})$. Let us introduce some new notations: $$ \nu_i=\mathop{\rm mult}\nolimits_{B_{i-1}}|\chi|^{i-1} $$ is the multiplicity of the proper inverse image of the system $|\chi|$ on $V_{i-1}$ along the cycle which is to be blown up; $p_i=p(E_K,E_i)$ is the number of paths in the oriented graph of the valuation $\nu=\nu_{E_K}$, leading from $E_K$ to $E_i$ (here $\nu$ is considered as a discrete valuation on the variety $V_0=V^{(M)}$!); $N^*=\max\{i|1\leq i\leq K, B_{i-1}\subset E^{i-1}\}$ in the cases A) and B); $L=\max \{i|1\leq i\leq K, B_{i-1} \mbox{ is a point } \}$ (so that for $j\leq L$ $B_{j-1}$ is a point, whereas for $j\geq L+1$ $B_{j-1}$ is a curve, see [6,7]) in the cases A) and C); $N=\min\{N^*,L\}$ in the case A); $N=N^*$ in the case B); $N^*_{\pm}=\max\{i|1\leq i\leq K, B_{i-1}\subset E^{i-1}_{\pm}\}$ in the case C), where the signs $+$ or $-$ are chosen to be the same in the right-hand and in the left-hand parts; $N_{\pm}=\min\{N^*_{\pm},L\}$ (in the right-hand part there is the minimum of two integers); $\Sigma_0=\sum\limits^L_{i=1}p_i$, $\Sigma_1=\sum\limits^K_{i=L+1}p_i$ in the cases A) and C); $\Sigma=\sum\limits^K_{i=1}p_i$ in the case B); $\Sigma^*=\sum\limits^{N^*}_{i=1}p_i$, $\Sigma_*=\sum\limits^N_{i=1}p_i$ in the cases A) and B); $\Sigma^*_{\pm}=\sum\limits^{N^*_{\pm}}_{i=1}p_i$, $\Sigma_{\pm}=\sum\limits^{N_{\pm}}_{i=1}p_i$ in the case C). Obviously, in these notations we get $$ \nu(E)=\varepsilon=\Sigma^*, $$ $$ \nu(E_{\pm})=\varepsilon_{\pm}=\Sigma^*_{\pm}. $$ In the non-special case A) or B) for $M\geq 2$ and in the special case A) or B) for $M\geq 3$ $$ \nu(F)=d_F\varepsilon, $$ whereas in the case C) (under the same restrictions on $M$) $$ \nu(F)=d_F(\varepsilon_++\varepsilon_-). $$ In either of the cases A) or B) we get $$ \nu(|\chi|)= \nu_E(|\chi|)\nu(E)+ \nu(|\chi|^{(M)}) $$ and $$ K(V,\nu)= K(V,\nu_E)\nu(E)+ K(V^{(M)},\nu), $$ so that the Noether-Fano inequality takes the form $$ \sum^K_{i=1}p_i\nu_i= \varepsilon\sum^M_{i=1}(n-\lambda_i)+ n\sum^K_{i=1}p_i\delta_i+e. $$ In a similar way, in the case C) the Noether-Fano inequality takes the form $$ \sum^K_{i=1}p_i\nu_i= \varepsilon_+\sum^M_{i=1}(n-\lambda_i)+ \varepsilon_-\sum^{M-1}_{i=1}(n-\lambda_i)+ n\sum^K_{i=1}p_i\delta_i+e. $$ \subsection*{Iskovskikh-Manin's techniques} As always, let $D^{(M)}_i$, $i=1,2$, be the proper inverse images of general divisors from the pencil $|\chi|$. Let $$ Z^{(M)}= (D^{(M)}_1\bullet D^{(M)}_2) $$ be the effective 1-cycle of their scheme-theoretic intersection. Set $$ m_i= \mathop{\rm mult}\nolimits_{B_{i-1}}(Z^{(M)})^{i-1} $$ for $i\leq L$ in the cases A) and C). In accordance with the Iskovskikh-Manin's techniques [6,7], we obtain the following estimate for the case A): $$ \sum^L_{i=1}p_im_i\geq \frac{ (2\Sigma_0n+\Sigma_1n+ \varepsilon\sum\limits^M_{i=1}(n-\lambda_i)+ e)^2}{ \Sigma_0+\Sigma_1}. $$ For the case C) we get the estimate $$ \sum^L_{i=1}p_im_i\geq \frac{1}{\Sigma_0+\Sigma_1}\times $$ $$ \times \left( 2\Sigma_0n+\Sigma_1n+ \varepsilon_+\sum\limits^M_{i=1}(n-\lambda_i)+ \varepsilon_-\sum\limits^{M-1}_{i=1}(n-\lambda_i)+ e\right)^2. $$ In the case B) we can obviously assert that the line $B$ comes into the 1-cycle $Z^{(M)}$ with the multiplicity at least $$ \sum^K_{i=1}\nu^2_i, $$ whereas the multiplicities $\nu_i$ satisfy the inequalities $$ \nu_i\geq \sum_{j\to i}\nu_j $$ (here the resolution of the maximal singularity $\nu$ is just a sequence of blowing ups of curves, covering each other). Computing the minimum of this quadratic form under the restrictions specified above and taking into account the Noether-Fano inequality, we get $$ \mathop{\rm mult}\nolimits_BZ^{(M)}\geq \frac{ \left( \Sigma n+\varepsilon\sum\limits^M_{i=1}(n-\lambda_i)+e\right)^2}{ \sum\limits^K_{i=1}p^2_i}. $$ \subsection*{The cycle $Z^{(M)}$ in terms of the staircase} Now to complete the proof of our theorem we must get some estimates of the upper bounds of the left-hand parts of the three principal inequalities, which were obtained above. The computations to be performed are rather tiresome. However, they are quite clear geometrically. Coming back to our basic construction -- that is, the staircase,-- let us introduce some new terminology and notations, connected with the linear system $|\chi|$. First of all, set $$ z_i=(D^{(i)}_1\cdot D^{(i)}_2)\in A^2V^{(i)} $$ to be the class of the effective 1-cycle $$ Z^{(i)}=(D^{(i)}_1\bullet D^{(i)}_2). $$ On the ``zeroth'' step of our staircase we have the decomposition $$ Z=Z^v+Z^h. $$ Let us trace down the changes which the 1-cycle $Z^{(k)}$ undergoes when $k$ comes from $i-1$ to $i$. Naturally, instead of the components of the cycle $Z^{(i-1)}$, which are different from $L_{i-1}$, their proper inverse images come into the cycle $Z^{(i)}$. Instead of the curve $L_{i-1}$, which is present in $Z^{(i-1)}$ with some multiplicity $k_{i-1}$, the cycle $Z^{(i)}$ contains an effective sub-cycle with the support in the exceptional divisor $E^{(i)}$. Let us break this sub-cycle into three parts: 1) $C^{(i)}_h$ includes all the curves, which are horizontal with respect to the morphism $\sigma_i:E^{(i)}\to L_{i-1}$, and different from the exceptional section $L_i$, 2) $C^{(i)}_v$ includes all the vertical curves, that is, the fibers of $\sigma_i$ over points of the curve $L_{i-1}$, 3) the exceptional section $L_i$ with a certain multiplicity $k_i\in{\bf Z}_+$. To make our notations look uniform set $C^{(0)}_h$ to be the part of the cycle $Z^v$, which includes all the curves different from $L$. Set also $$ d^{(i)}_{h,v}=\mathop{\rm deg} C^{(i)}_{h,v} $$ (see Remark (ii) in the previous section). Now we get the following representation of the cycles $Z^{(i)}$: $$ Z^{(0)}=Z^h+Z^v=Z^h+C^{(0)}_h+k_0L, $$ $$ Z^{(1)}= (Z^h)^{(1)}+ C^{(0,1)}_h+ C^{(1)}_h+ C^{(1)}_v+ k_1L_1, $$ $$ \dots, $$ $$ Z^{(i)}= (Z^h)^{(i)}+ C^{(0,a)}_h+ C^{(1,2)}_h+ C^{(1,2)}_v+ \dots + $$ $$ + C^{(i-1,i)}_h+ C^{(i-1,i)}_v+ C^{(i)}_h+ C^{(i)}_v+ k_iL_i. $$ Here $a=1$ in the non-special and $a=2$ in the special case. We write, for instance, $C^{(1,2)}_h$ instead of $C^{(1,i)}_h$, in accordance with Remark (i) of Section 5. \subsection*{Computation of the class $z_M$} Obviously, the class of the cycle $C^{(i)}_v$ in $A^1V^{(i)}$ is equal to $d^{(i)}_vf_i$, and the class of the cycle $C^{(i)}_h$ is equal to $d^{(i)}_hs_i+\beta_if_i$, where the coefficients satisfy the important inequality $$ \beta_i\geq d^{(i)}_h $$ (see Remark (ii) in Section 5). Furthermore, the class of the cycle $C^{(i,i+1)}_v$ is equal to $$ d^{(i)}_v(f_i-f_{i+1}) $$ and the class of the cycle $C^{(i,i+1)}_h$ is equal to $$ d^{(i)}_hs_i+ \beta_if_i- (\beta_i-d^{(i)}_h)f_{i+1}. $$ Setting $$ \alpha_i= \left( (Z^h)^{(i-1)}\cdot L_{i-1} \right) $$ in the sense of the definition of the ``intersection index'', which was given at the beginning of the paper, we can write down $$ z^h_i=z^h_{i-1}-\alpha_if_i, $$ where $z^h_i$ is the numerical class of the horizontal cycle $(Z^h)^{(i)}$. \begin{lll} The following inequality is true: $$ \alpha_i\leq\mathop{\rm deg} Z^h=3n^2. $$ \end{lll} {\bf Proof.} Since $L\subset F$, and $\mathop{\rm deg} Z^h$ is equal to $(Z^h\cdot F)$, this is obvious. Q.E.D. \begin{ppp} The classes $z_i$ satisfy the following chain of relations: $$ z_i=z_{i-1}-(2\lambda_in+\lambda^2_i)f_i-\lambda^2_is_i. $$ \end{ppp} {\bf Proof.} We just compute: $$ z_i= (D^{(i)})^2= (D^{(i-1)}-\lambda_iE^{(i)})^2= $$ $$ = z_{i-1}-2\lambda_i(D^{(i-1)}\cdot L_{i-1})f_i- \lambda^2_i(s_i+f_i). $$ It follows from what was proved in Section 5 that for any $j\in{\bf Z}_+$ $(D^{(j)}\cdot L_j)=(D\cdot L)=n$. Q.E.D. \begin{ppp} For $i\geq 2$ in the non-special and for $i\geq 3$ in the special case the integers $k_i$, $\alpha_i$, $\beta_i$ and $d^{(i)}_{h,v}$ satisfy the following system of relations: $$ d^{(i)}_v+\beta_i= \alpha_i+ d^{(i-1)}_v+ (\beta_{i-1}- d^{(i-1)}_h)- 2\lambda_in-\lambda^2_i. $$ For $i=1$ both in the non-special and special cases we get $$ d^{(1)}_v+\beta_1= \alpha_1+ (C^{(0)}_h\cdot L)- 2\lambda_1n-\lambda^2_1, $$ whereas for $i=2$ in the special case we get $$ d^{(2)}_v+\beta_2= $$ $$ =\alpha_2+d^{(1)}_v+ (\beta_1- d^{(1)}_h)+ (C^{(0,1)}_h\cdot L_1)- 2\lambda_2n-\lambda^2_2. $$ \end{ppp} {\bf Proof.} To obtain this proposition, it is necessary to write out explicitly the class of the cycle $Z^{(i)}$ in terms of the parameters introduced above, and to use the previous proposition. The corresponding computations are elementary. \begin{ppp} For any $i\geq 1$ in the non-special case we get the inequality $$ d^{(i)}_v+\beta_i\leq (C^{(0)}_h\cdot L)+ \sum^i_{j=1}(3n^2-2\lambda_jn-\lambda^2_j). $$ In the special case for $i\geq 2$ we get $$ d^{(i)}_v+\beta_i\leq (C^{(0)}_h\cdot L)+ (C^{(0,1)}_h\cdot L_1)+ \sum^i_{j=1}(3n^2-2\lambda_jn-\lambda^2_j), $$ and $$ d^{(1)}_v+\beta_1\leq (C^{(0)}_h\cdot L)+ 3n^2-2\lambda_1n-\lambda^2_1. $$ \end{ppp} {\bf Proof.} It is necessary to apply the corresponding inequality of the previous proposition $i$ times and to use the last lemma. Q.E.D. \subsection*{Completing the proof: case A)} In the case A) it is clear that among all the curves, lying on the divisor $$ \mathop{\bigcup}\limits^M_{i=0}E^{(i,M)}, $$ only those can possibly contain the point $x_M$, which lie entirely in $E^{(M)}$ and are different from the exceptional section $L_M$. Consequently, we are justified in writing down $$ Z^{(M)}= (Z^h)^{(M)}+ C^{(M)}_v+ C^{(M)}_h+\dots, $$ where the dots stand for the sum of all the curves, which do not contain the point $x_M$. Set $$ W=C^{(M)}_v+C^{(M)}_h, $$ $$ m^v_i=\mathop{\rm mult}\nolimits_{B_{i-1}}W^{i-1}, $$ $$ m^h_i=\mathop{\rm mult}\nolimits_{B_{i-1}}(Z^h)^{(M),i-1} $$ for $i\leq L$, so that $m_i=m^v_i+m^h_i$. Obviously, the multiplicities $m^v_i$ vanish for $N+1\leq i\leq L$. Furthermore, $m^{h,v}_i\leq m^{h,v}_1$, and similarly to Lemma 6.1 we get $m^h_1\leq 3n^2$. Finally, $m^v_1\leq d^{(M)}_v+d^{(M)}_h\leq d^{(M)}_v+\beta_M$, so that, summing up our information, we get $$ 3n^2\Sigma_0+ \Sigma_*\left( (C^{(0)}_h\cdot L)+ \sum^M_{i=1}(3n^2-2\lambda_in-\lambda^2_i) \right)\geq $$ $$ \geq \sum^L_{i=1}p_im^h_i+ \sum^N_{i=1}p_im^v_i\geq $$ $$ \geq\frac{ \left(2\Sigma_0n+\Sigma_1n+ \varepsilon\sum\limits^M_{i=1}(n-\lambda_i)+ e\right)^2}{ \Sigma_0+\Sigma_1}, $$ if the case is the non-special one. In the special case for $M\geq 2$ one should add $(C^{(0,1)}_h\cdot L_1)$ to $(C^{(0)}_h\cdot L)$. Now let us consider the non-special case. Replacing $\Sigma_*$ by $\varepsilon=\Sigma^*$, we make our inequality stronger, and replacing $\varepsilon(C^{(0)}_h\cdot L)$ by $4ne$, we get a strict inequality. Subtract the left-hand side from the right-hand one and look at the expression just obtained as a quadratic form in $\lambda_i$ on the domain $0\leq\lambda_i\leq n$. By symmetry, its minimum is attained somewhere on the diagonal line, that is, at $\lambda_i=\lambda,$ $0\leq\lambda\leq n$. Replace all the $\lambda_i$'s by this value $\lambda$. Thus we get the strict inequality $$ \Phi<0, $$ where the expression $\Phi$ by means of elementary arithmetic can be transformed as follows: $$ \Phi=( \Sigma^2_0+ \Sigma_0 \Sigma_1+ \Sigma^2_1)n^2+M\varepsilon \Sigma_0(n-\lambda)^2- $$ $$ -M\varepsilon \Sigma_1(n-\lambda)(n+\lambda)+ $$ $$ +M^2\varepsilon^2(n-\lambda)^2-2e \Sigma_1n+2M\varepsilon e(n-\lambda)+e^2. $$ Since $\lambda\leq n$, we can replace $(n+\lambda)$ by $2n$, preserving the strict inequality. However, it is easy to check that the last expression is the sum of the complete square $$ (\Sigma_1n-M\varepsilon(n-\lambda)-e)^2 $$ and a few non-negative components. Thus it can not be negative. Our proof is complete (in the case under consideration). In the special case for $M\geq 2$ the arguments are to be produced in accordance with the same scheme. Just take into account that here $\nu(F)=2\varepsilon$, so that now we may replace the expression $$ \varepsilon\left((C^{(0)}_h\cdot L)+(C^{(0,1)}_h\cdot L_1)\right) $$ in the left-hand side by $4ne$. The rest part of the computations is the same as in the previous case. For $M=1$ the computations are much easier. Q.E.D. for the case A). \subsection*{Completing the proof: case B)} As above, we shall trace all the details in the non-special case only. On one hand, we have the inequality $$ d^{(M)}_v\leq (C^{(0)}_h\cdot L)+ \sum^M_{i=1}(3n^2-2\lambda_in-\lambda^2_i). $$ On the other hand, the following estimate holds: $$ d^{(M)}_v\geq \mathop{\rm mult}\nolimits_BZ^{(M)}, $$ whereas for the last multiplicity, in its turn, a lower bound was obtained above by means of the Iskovskikh-Manin's techniques. It is easy to see that $$ \sum^K_{i=1}p^2_i\leq p_1\Sigma\leq\varepsilon\Sigma. $$ As it was done above, we may assume that all the $\lambda_i$'s are equal to $\lambda$, $0\leq\lambda\leq n$. Replacing $\varepsilon(C^{(0)}_h\cdot L)$ by $4ne$, we get the strict inequality $$ 0> \Sigma^2n^2-2\Sigma ne+M^2\varepsilon^2(n-\lambda)^2+e^2+2M\varepsilon(n-\lambda)e- $$ $$ -M\varepsilon(n-\lambda)\Sigma(n+\lambda), $$ which will be still true when we replace $(n+\lambda)$ by $2n$. But the final expression is a complete square: $$ (\Sigma n-M\varepsilon(n-\lambda)-e)^2. $$ This contradiction proves the theorem (in the case under consideration). In the special case we proceed in the same manner. Here we must add $(C^{(0,1)}_h\cdot L_1)$ to $(C^{(0)}_h\cdot L)$. However, the multiplicity $\nu(F)=2\varepsilon$ is twice bigger now, so that eventually we come to the same strict inequality. This contradiction completes the proof in the case B). \subsection*{Completing the proof: case C)} Let us assume at first that either the case is the non-special one, either it is special and $M\geq 3$, or, finally, that it is special, $M=2$, but the point $x=Z(V,\nu)$ is not the singular point of the fiber $F$. Here it is clear that among the curves lying on the divisor $$ \mathop{\bigcup}\limits^M_{i=0}E^{(i,M)}, $$ only those ones can pass through the point $x_M$, which either lie entirely in $E^{(M)}$ and are different from the exceptional section $L_M$ (exactly as it was in the case A)), or lie entirely in $E^{(M-1,M)}$ and are different from the exceptional section $E^{(M-1,M)}\cap E^{(M)}$ (which was already counted in the first group). Thus $$ Z^{(M)}= (Z^h)^{(M)}+W_-+W_++\dots, $$ where $W_+$ stands for the sum of all the curves in $E_+=E^{(M)}$, which are different from $L_M$, $W_-$ stands for the 1-cycle $C^{(M-1,M)}_v+C^{(M-1,M)}_h$, and the dots stand for the sum of all the rest curves, which do not pass through $x_M$. Let the symbol $m^h_i$ mean the same as in the case A), and set $$ m^{\pm}_i= \mathop{\rm mult}\nolimits_{B_{i-1}}W^{i-1}_{\pm}, $$ $i=1,\dots,L$, so that $m_i=m^+_i+m^-_i+m^h_i$. Obviously, the multiplicities $m^{\pm}_i$ vanish for $N_{\pm}+1\leq i\leq L$. Similarly to the case A), we get $m^{h,\pm}_i\leq m^{h,\pm}_1$, $m^h_1\leq 3n^2$, $m^+_1\leq d^{(M)}_v+\beta_M$, $m^-_1\leq d^{(M-1)}_v+\beta_{M-1}$, so that finally we come to the following inequality: $$ 3n^2\Sigma_0+ \Sigma_+\left( (C^{(0)}_h\cdot L)+ \sum^M_{i=1}(3n^2-2\lambda_in-\lambda^2_i) \right)+ $$ $$ +\Sigma_-\left( (C^{(0)}_h\cdot L)+ \sum^{M-1}_{i=1}(3n^2-2\lambda_in-\lambda^2_i) \right)\geq $$ $$ \geq \sum^L_{i=1}p_im^h_i+ \sum^{N_+}_{i=1}p_im^+_i+ \sum^{N_-}_{i=1}p_im^-_i\geq $$ $$ \geq\frac{1}{\Sigma_0+\Sigma_1} \left(2\Sigma_0n+\Sigma_1n+ \varepsilon_-\sum\limits^{M-1}_{i=1}(n-\lambda_i)+ \varepsilon_+\sum\limits^M_{i=1}(n-\lambda_i)+ e\right)^2, $$ provided that our case is the non-special one. In the special case for $M\geq 2$ one should add $(C^{(0,1)}_h\cdot L_1)$ to $(C^{(0)}_h\cdot L)$. Let us consider the non-special case. Replacing $\Sigma_{\pm}$ by $\varepsilon_{\pm}=\Sigma^*_{\pm}$, we preserve the inequality, and replacing $(\varepsilon_++\varepsilon_-)(C^{(0)}_h\cdot L)$ by $4ne$, we make it into a strict one. Subtract the left-hand side from the right-hand side and look at the resulting expression as a quadratic form in the two groups of variables, that is, $\lambda^+_i$ and $\lambda^-_i$, where we replace $\lambda_i$ by $\lambda^{\pm}_i$ in accordance with the following rule: if a variable comes into the sum $\sum\limits^{M-1}_{i=1}$, then we replace it by $\lambda^-_i$, and if it comes into $\sum\limits^{M}_{i=1}$, then we replace it by $\lambda^+_i$. The new variables take their values in the domain $0\leq \lambda^{\pm}_i\leq n$. By symmetry, the minimum of this quadratic form is attained at some point on the diagonal plane, that is, at $\lambda^{\pm}_i=\lambda_{\pm}$, $0\leq \lambda_{\pm}\leq n$. Now replace all the $\lambda^{\pm}_i$'s by $\lambda_{\pm}$. The inequality is still strict. Thus we get $$ \Phi<0, $$ where the expression $\Phi$ can be transformed by means of elementary arithmetic in the following way, where we set for conveniency $M-1=M_-$, $M=M_+$: $$ \Phi=( \Sigma^2_0+ \Sigma_0 \Sigma_1+ \Sigma^2_1)n^2+M_-\varepsilon_- \Sigma_0(n-\lambda_-)^2+ M_+\varepsilon_+ \Sigma_0(n-\lambda_+)^2- $$ $$ -M_-\varepsilon_- \Sigma_1(n-\lambda_-)(n+\lambda_-) -M_+\varepsilon_+ \Sigma_1(n-\lambda_+)(n+\lambda_+)+ $$ $$ +(M_-\varepsilon_-(n-\lambda_-)+ M_+\varepsilon_+(n-\lambda_+))^2- $$ $$ -2e\Sigma_1n+ $$ $$ +2M_-\varepsilon_-(n-\lambda_-)e +2M_+\varepsilon_+(n-\lambda_+)e +e^2. $$ Since $\lambda_{\pm}\leq n$, we can replace $(n+\lambda_{\pm})$ by $2n$, preserving the strict inequality. Now it is easy to check, that the last expression is the sum of the complete square $$ (\Sigma_1n- M_-\varepsilon_-(n-\lambda_-)- M_+\varepsilon_+(n-\lambda_+) -e)^2 $$ and a few non-negative components. Again we come to a contradiction, completing our proof in the case under consideration. In the special case we use the same arguments, taking into account the equality $\nu(F)=2(\varepsilon_-+\varepsilon_+)$. If, finally, our case is the special one with $M=1$ and $x=Z(V,\nu)$ is not the singular point of the fiber $F$, then the previous arguments work with simplifications. The only case, which is yet to be considered, is the special one with $M=1$ or 2, when the point $x$ is the singularity of the fiber. The case $M=1$ is more simple. If $M=2$, then the point $x_M$ is the only common point of the following three divisors: $$ x_2=F^{(2)}\cap E^{(1,2)}\cap E^{(2)} $$ (the intersection is transversal). Respectively, $$ Z^{(2)}= (Z^h)^{(2)}+ C^{(0,2)}_h+ C^{(1,2)}_h+ C^{(1,2)}_v+ C^{(2)}_h+ C^{(2)}_v+ k_2L_2, $$ where all the 1-cycles but the last one can contain the point $x_2$. This case is the only one, when our previous arguments formally do not work (because of the additional input, which is given by the 1-cycle $C^{(0,2)}_h\subset F^{(2)}$). Nevertheless, the general scheme of arguments, which was used in the cases A)-C) above, works here, too. We just outline the principal steps of the proof. Preserving the previous notations, set $$ N^*=\max\{i|1\leq i\leq K,B_{i-1}\subset E^{i-1}\}, $$ $$ N=\min\{N^*,L\}, $$ $$ \Sigma^*=\varepsilon=\sum\limits^{N^*}_{i=1}p_i, $$ $$ \Sigma_*=\sum\limits^N_{i=1}p_i\leq\Sigma^*. $$ Set also $$ m^0_i= \mathop{\rm mult}\nolimits_{B_{i-1}}(C^{(0,2)}_h)^{i-1} $$ for $1\leq i\leq L$. Obviously, $m^0_i=0$ for $i\geq N+1$. Now we get the following representation: $$ \sum^{L}_{i=1}p_im_i= \sum^{L}_{i=1}p_im^h_i+ \sum^{N_-}_{i=1}p_im^-_i+ \sum^{N_+}_{i=1}p_im^+_i+ \sum^{N}_{i=1}p_im^0_i. $$ For the four components in the right-hand side we get the following upper bounds: $$ \leq 3\Sigma_0n^2, $$ $$ \leq\varepsilon_-\left( (C^{(0)}_h\cdot L)+(3n^2-2n\lambda_1-\lambda^2_1)\right), $$ $$ \leq\varepsilon_+\left( (C^{(0)}_h\cdot L)+ (C^{(0,1)}_h\cdot L_1)+ \sum^2_{i=1}(3n^2-2n\lambda_i-\lambda^2_i)\right), $$ $$ \leq\varepsilon m^0_1\leq\varepsilon (C^{(0,1)}_h\cdot L_1). $$ It is because of the fourth component that this case is not embraced formally by the previous arguments. However, here the multiplicity $$ \nu(F)=2\varepsilon_++\varepsilon_-+\varepsilon $$ increases, too. Thus we are able again to replace all the components, into which $(C^{(0)}_h\cdot L)$ and $(C^{(0,1)}_h\cdot L_1)$ come, by $4ne$. {}From now on we can just repeat the arguments, which were used in the ``regular'' case C). For $M=1$ our computations work with considerable simplifications. The proof of our theorem is complete. \newpage \centerline{\large \bf References} 1. Iskovskikh V.A., On the rationality problem for three-dimensional algebraic varieties, fibered into Del Pezzo surfaces,-- Proc. of Steklov Math. Inst., V. 208, 1995, 113-122. 2. Iskovskikh V.A. and Manin Yu.I., Three-dimensional quartics and counterexamples to the L\"uroth problem,-- Math. USSR Sb., V. 86, 1971, 140-166. 3. Iskovskikh V.A. and Pukhlikov A.V., Birational automorphisms of multi-dimensional algebraic varieties,-- Cont. Math. and Its Appl., V. 19, 1995, 3-96. 4. Manin Yu.I., Cubic forms: Algebra, geometry, arithmetic.-- Amsterdam: North Holland, 1986. 5. Pukhlikov A.V., A remark on the theorem of V.A.Iskovskikh and Yu.I.Manin on the three-dimensional quartic,-- Proc. of Steklov Math. Inst., V. 208, 1995, 244-254. 6. Pukhlikov A.V., Birational automorphisms of three-dimensional Del Pezzo fibrations,-- Warwick Preprint 30/1996. 7. Pukhlikov A.V., Essentials of the method of maximal singulariti\-es,-- Warwick Preprint 31/1996. 8. Sarkisov V.G., Birational automorphisms of conic bundles,-- Math. USSR Izvestia, V. 17, 1981, No. 1, 177-202. 9. Sarkisov V.G., On conic bundles structures,-- Math. USSR Izvestia, V. 20, 1982, No. 2, 355-390. \end{document}

9611

alg-geom/9611022

fr

https://arxiv.org/abs/alg-geom/9611022

alg-geom/9611022

Bas Edixhoven

Pierre Parent

Bornes effectives pour la torsion des courbes elliptiques sur les corps de nombres

LaTeX, 33 pages with 2 figures, hard copy available in a few days; send email to [email protected]

In this paper we give a detailed proof of a result we announced a year ago. This result is an effective version of the theorem of Mazur-Kamienny-Merel concerning uniform bounds for rational torsion points on elliptic curves over number fields.

alg-geom

\section{Pr\'esentation des r\'esultats.} \subsection{Introduction.} {\it La ``conjecture de borne uniforme pour les courbes elliptiques'', affirmant qu'il existe pour tout entier $d$ un entier $B(d)$ tel que, pour tout corps de nombres $K$ de degr\'{e} $d$ sur ${\bf Q} $ et pour toute courbe elliptique $E$ sur $K$, la partie de torsion $E(K)_{\rm tors}$ du groupe de Mordell-Weil $E(K)$ est de cardinal major\'{e} par $B(d)$, a \'{e}t\'{e} d\'{e}montr\'{e}e dans le cas g\'{e}n\'{e}ral en f\'{e}vrier 1994 par Lo\"{\i}c Merel. En fait, Merel (et Oesterl\'{e}) montrent que, si $P$ est un point d'ordre $p$ premier de $E(K)$, on a $p \leq (1+{3^{d/2}})^2 $ . Des travaux de Faltings et Frey permettent alors de conclure \`{a} l'existence des bornes $B(d)$, mais pas de mani\`{e}re effective : en effet, si on a bien major\'{e} les nombres premiers pouvant diviser les groupes $E(K)_{\rm tors}$, on ne sait pas en pratique quelles puissances de ces nombres premiers peuvent intervenir dans ces groupes. Le but de cet article est de d\'emontrer une forme explicite de la forme forte de la conjecture de borne uniforme (le corollaire 1.8 ci-dessous), en donnant une borne pour ces puissances de premiers qui peuvent diviser la torsion.} \subsection{Sch\'{e}ma de la preuve.} Soit $E$ une courbe elliptique sur un corps de nombres $K$ de degr\'{e} $d$ sur ${\bf Q} $, poss\'{e}dant un point $K$-rationnel $P$, tel que le cardinal du sous-groupe cyclique de $E(K)$ engendr\'{e} par $P$ soit une puissance d'un nombre premier : $|\langle P\rangle |={p^n}$. On cherche \`{a} majorer $p^n $ en fonction de $d$. Soit $l$ un nombre premier diff\'{e}rent de $2$ et de $p$ (on prendra dans la suite le plus petit possible, $i.e.\ l=5$ si $p=3$, $l=3$ dans tous les autres cas). Soit $\cal L$ un id\'eal maximal de ${\cal O}_K$ (l'anneau des entiers du corps $K$), contenant $l$. Examinons la fibre en $\cal L$ du mod\`{e}le de N\'{e}ron (qu'on note ${\cal E}$) de la courbe elliptique sur ${\cal O}_K$. \begin{prop} Si en $\cal L$ au-dessus de $l$, on a l'un des trois cas : \begin{enumerate} \item ${\cal E}$ a bonne r\'eduction ; \item ${\cal E}$ a r\'eduction additive ; \item ${\cal E}$ a r\'eduction multiplicative tordue ; \item ${\cal E}$ a r\'eduction multiplicative d\'eploy\'ee et $\langle \widetilde{P} \rangle$ (o\`u $\widetilde{P}$ est la r\'{e}duction de $P \bmod \cal L$) appartient \`a la composante neutre, \end{enumerate} alors $|\langle P\rangle |\leq 2.(1+l^d )$. \end{prop} {\bf Remarque.} {\it Cette proposition classique est explicit\'ee dans \cite{Bas} par exemple, ou dans \cite{kamienny}. Dans le cas 2., on peut borner % par $4$ l'ordre du groupe des composantes de la r\'eduction additive, donc la borne pour l'ordre de $P$ est $1$ si $p\geq 5$, $3$ si $p=3$, et $4$ si $p=2$. Dans les cas 1. et 4., la borne est en fait $(l^{d/2} +1)^2$ (borne de Weil) et $(l^d -1)$ (cardinal de groupe multiplicatif) respectivement, donc en $l^d$. Le cas 3. la porte \`a $(1+l^d )$ ou \`a $2(1+3^d )$ : dans ce cas en effet, soit $\widetilde{P}$ appartient \`a la composante neutre, et son ordre est major\'e par $(1+l^d)$ ; soit il est dans une composante non triviale. Le groupe de Galois d'une extension quadratique du corps r\'esiduel $k({\cal L})$ agit alors par la multiplication par $(\pm 1)$ sur le groupe des composantes. Mais $\widetilde{P}$ est $k({\cal L})$-rationnel, et donc la composante \`a laquelle il appartient \'egale son oppos\'ee, ce qui veut dire que $2\widetilde{P}$ est dans la composante neutre (chose qui ne peut arriver que si $p$ est $2$).} Supposons donc qu'en tout $\cal L$ au-dessus de $l$, ${\cal E}$ ait r\'eduction multiplicative d\'eploy\'ee et que $\widetilde{P}$ ne soit pas trivial dans le groupe des composantes de cette r\'eduction ${\cal E}_{k({\cal L})}$ de ${\cal E}_{/{\cal O}_K}$. Pour avoir une bonne interpr\'etation modulaire dans la suite, comme indiqu\'e plus bas, on aimerait que $\widetilde{P}$ soit d'ordre $p^n$ dans le groupe des composantes de ${\cal E}_{k({\cal L})}$ : d'o\`u l'id\'ee d'examiner le quotient de ${\cal E}_{k({\cal L})}$ par le plus gros sous-groupe de $\langle \widetilde{P} \rangle$ inclus dans sa composante neutre ${\cal E}^0_{k({\cal L})}$. Soit donc $n_{\cal L}$ le plus petit entier tel que ${p^{n_{\cal L}} }. \widetilde{P}$ tombe dans ${\cal E}^0_{k({\cal L})}$. Soit aussi $n'$ le plus petit des $n_{\cal L}$, pour ${\cal L}$ parcourant l'ensemble des places de ${\cal O}_K$ au-dessus de $l$. Oesterl\'{e} \'enonce : \begin{lemm} Soit $k$ \'{e}l\'{e}ment de ${\bf N} $ ; si en la place $\cal L$, $p^{k-1} .P$ ne se r\'{e}duit pas dans la composante neutre de ${\cal E}_{k({\cal L})}$, alors la r\'{e}duction de $\widetilde{P}$ dans la fibre en ${\cal L}$ du mod\`ele de N\'eron ${\cal E}'$ de $E/\langle {p^k}.P\rangle$ est d'ordre exactement $p^k$ dans son groupe des composantes. \end{lemm} L'interpr\'{e}tation modulaire de ce lemme, qui sera d\'emontr\'e dans la section~2, est donc la suivante : le point $K-$rationnel $j$ de la courbe modulaire $X_0 (p^{n'})$ que d\'efinit le couple $(E/\langle {p^{n'}}.P\rangle ,\langle P\rangle )$ se r\'{e}duit en la pointe $0$ modulo toute place $\cal L$ ; et l'image $j'$ de ce point par l'involution d'Atkin-Lehner, en la pointe infinie (voir \cite{mazur} , page 159). Voyons comment on peut alors utiliser les arguments de Kamienny (\cite{kamienny}, \cite{Bas}) pour d\'emontrer son crit\`{e}re.\\ {\bf Remarque.} {\it Dans toute la suite, on notera $E$ ce qui sera en r\'ealit\'e $(E/\langle {p^{n'}}.P\rangle )$ ; on notera aussi $p^n$ pour $p^{n'}$. La borne qu'on obtiendra pour $|\langle P\rangle |$ devra donc \^etre multipli\'ee par un facteur $(l^d -1)$ pour avoir une borne \`a notre $p^n$ ``originel'' (en effet, $(p^n /p^{n'})$ est inf\'erieur au cardinal du groupe multiplicatif ${{\bf G}}_{m,{{\bf F}}_l}$)}. Si les ${\sigma }_i,\ 1\leq i\leq d$ sont les plongements de $K$ dans $\overline{{\bf Q} }$, ${j'}^{(d)} :=({\sigma}_1 (j'),{\sigma }_2 (j'),\\ ..., {\sigma}_d (j'))$ d\'{e}finit un point ${\bf Q} $-rationnel du produit sym\'{e}trique $d$-i\`{e}me : ${X_0 (p^n)}^{(d)}$, de $X_0 (p^n)$. D\'efinissons comme Merel le quotient d'enroulement (\cite{merel}). On consi\-d\`{e}re les premiers groupes d'homologie singuli\`{e}re absolue : $H_1 (X_0 (p^n )\, ;\, {\bf Z} )$ et relative aux pointes : $H_1 (X_0 (p^n ),\, {\rm pointes}\, ;\, {\bf Z} )$, de $X_0(p^n )$, le premier \'etant vu comme un sous-groupe du second. Si $a$ et $b$ sont deux \'el\'ements de ${\bf P}^1({\bf Q})$, le {\it symbole modulaire} $\{a,b\}$ est l'\'{e}l\'{e}ment de $H_1 (X_0 (p^n ),\, {\rm pointes}\, ;\, {\bf Z} )$ d\'efini par l'image de n'importe quel chemin continu reliant $a$ \`{a} $b$ sur le demi-plan de Poincar\'{e} auquel on a ajout\'{e} l'ensemble ${{\bf P} }^1 ({\bf Q} )$ de ses pointes. L'int\'egration d\'efinit un isomorphisme classique d'espaces vectoriels r\'eels : % $$\left\{ \begin{array}{c} H_1 (X_0 (p^n ) \, ;\, {\bf Z} )\otimes {\bf R} \to {\rm Hom}_{{\bf C} } \left( H^0 (X_0 (p^n )\, ;\, {\Omega }^1 ), \ {\bf C} \right) \\ \gamma \otimes 1 \mapsto \left( \omega \mapsto \int_{\gamma } \omega \right). \end{array} \right. $$ Selon un th\'eor\`eme de Manin et Drinfeld, % l'image % r\'eciproque de la forme lin\'{e}aire $\omega \mapsto \int_{\{ 0,\infty \} } \omega $ dans $H_1 (X_0 (p^n )\, ;\, {\bf R} )$ est en r\'{e}alit\'{e} dans $H_1 (X_0 (p^n )\, ;\, {\bf Q} )$. C'est {\it l'\'el\'ement d'enroulement}, qu'on note $e$ (comme d'habitude). (En fait, le r\'esultat de Manin assure plus g\'en\'eralement que l'image r\'eciproque de l'int\'egration sur {\em tout} symbole modulaire est \`a coefficients dans ${\bf Q}$.) Notons (toujours comme d'habitude) ${\bf T}$ l'alg\`ebre engendr\'ee sur ${\bf Z}$ par les op\'erateurs de Hecke $T_i$ ($i\geq 1$, entier), agissant fid\`element entre autres sur $H_1 (X_0 (p^n )\, ;\, {\bf Q} )$ et sur la jacobienne $J_0 (p^n )$ de la courbe modulaire. Soit ${\cal A}_e$ l'id\'eal annulateur dans ${\bf T}$ de $e$ ({\it id\'eal d'enroulement}) ; on d\'{e}finit alors le {\em quotient d'enroulement} $J_0^e$ comme la vari\'et\'e ab\'elienne quotient $J_0 (p^n )/ {{\cal A}_e} J_0 (p^n )$. De fa\c{c}on similaire \`a Merel dans \cite{merel}, un th\'eor\`eme de Kolyvagin-Logachev nous permet de montrer dans la section~3 le : % \begin{theo} $J_0^e ({\bf Q} )$ est fini. \end{theo} Soit maintenant l'application naturelle $f_d$ : ${X_0 (p^n )}^{(d)}_{\rm lisse} \rightarrow J_0^e $, qu'on a normalis\'ee par ${\infty }^{(d)} \mapsto 0$. On montre ``par les arguments standards'' que le fait que $f_d$ soit une immersion formelle en $\infty _{{{\bf F}}_l}^{(d)}$ contredit l'existence de notre point $j'^{(d)}$ (voir la sous-section 4.12). Or on a le ``crit\`ere de Kamienny'' : \begin{theo} On a \'{e}quivalence entre : \begin{enumerate} \item $f_d$ est une immersion formelle en ${\infty}_{{{\bf F}}_l}^{(d)}$, et \item $T_1 e,...,T_d e$ sont ${\bf F} _l$-lin\'{e}airement ind\'{e}pendants dans ${\bf T} e/l{\bf T} e$. \end{enumerate} De plus, ces deux conditions sont satisfaites si l'est : \begin{enumerate} \item[3.] $T_1 \{ 0,\infty \} ,...,T_{d.s} \{ 0,\infty \}$ sont ${\bf F}_l$-lin\'{e}airement ind\'{e}pendants dans l'espace vectoriel $H_1 (X_0 (p^n ),\, {\mathrm {pointes }}\, ;\, {\bf Z} )\otimes {\bf F}_l$ (ici et pour la suite, $s$ d\'{e}signe le plus petit nombre premier diff\'erent de $p$). \end{enumerate} \end{theo} (Le fait que la derni\`{e}re condition implique les pr\'{e}c\'{e}dentes est une remarque d'Oesterl\'{e} en niveau premier, utilis\'ee d\'ej\`a par Merel ; ce th\'eor\`eme sera d\'emontr\'e dans la section~4). Il suffit donc maintenant de prouver : \begin{prop} Soit $C:=\sqrt{65}$, si $p$ est diff\'erent de $2$, et $C:=\sqrt{129}$ si $p$ est $2$. Si $p^n\geq C^2 .(sd)^6$, alors les $T_i \{ 0,\infty \} ,\ 1\leq i \leq sd$ sont ${\bf F}$-lin\'{e}airement ind\'{e}pendants (dans le ${\bf F}$-espace vectoriel $H_1 (X_0 (p^n ),\, {\mathrm {pointes }}\, ;\, {\bf Z} )\otimes {\bf F}$) pour tout corps ${\bf F}$. \end{prop} De cette proposition, dont la d\'emonstration occupe la section~5, d\'ecoule donc le : \begin{cor} Soit $E$ une courbe elliptique sur un corps $K$ de degr\'{e} $d$ sur ${\bf Q}$. Si $E(K)$ poss\`{e}de un point $P$ d'ordre une puissance $p^n$ d'un nombre premier $p$, on a : \begin{enumerate} \item $p^n \leq 65. (3^d -1).(2d)^6 \ ,$ si $p$ est diff\'{e}rent de $2$ et $3$ ; \item Si $p=3$, $p^n \leq 65.(5^d -1).(2d)^6 $ ; \item et pour $p=2$, $2^n \leq 129.(3^d -1).(3d)^6 .$ \end{enumerate} \end{cor} \noindent{\it {\bf Remarques.} Le th\'eor\`eme de Mordell-Weil assure que pour tous $K$ et $E$ comme ci-dessus, il existe deux entiers $n_1$ et $n_2$ tel que $E(K)_{\rm tors} \cong {{\bf Z} }/{{n_1}{\bf Z} }\times {{\bf Z} }/{{n_2} {\bf Z} }$. On borne donc ainsi le cardinal de tous les $p$-groupes de $E(K)_{\rm tors}$, et avec la borne de Merel-Oesterl\'e pour les nombres premiers $p$ pouvant intervenir on obtient une borne effective ``globale'' pour l'ensemble $\{ card( E(K)_{\rm tors})\ |\ K$ est un corps de nombres de degr\'e $d$ et $E$ est une courbe elliptique sur $K\}$. Une telle borne globale semble de toute fa\c{c}on assez facile \`{a} am\'{e}liorer en reprenant les arguments de ce papier directement en niveau $N$ entier quelconque.} La preuve dans son ensemble ayant \'{e}t\'{e} esquiss\'{e}e, nous allons montrer dans la suite, dans l'ordre indiqu\'e, les diff\'erentes propositions utilis\'ees. \section{Quotients de courbes elliptiques.} On prouve le lemme 1.4 : \begin{prop} Soit $E$ une courbe elliptique sur un corps de nombres $K$, notons ${\cal E}$ son mod\`ele de N\'eron sur ${\cal O}_K$, et soit $P$ un point $K$-rationnel de $E$, d'ordre une puissance $p^n$ d'un nombre premier $p$. Soit encore $k$ un entier, et ${\cal L}$ une place de ${\cal O}_K$ qui ne soit pas au-dessus de $p$. Supposons qu'en ${\cal L}$, ${\cal E}$ ait r\'eduction multiplicative d\'eploy\'ee, et que $p^{k-1} .P$ ne se r\'eduise pas dans la composante neutre de ${\cal E}_{k({\cal L})}$. Alors la r\'eduction de $P$ dans la fibre en ${\cal L}$ du mod\`ele de N\'eron ${\cal E}'$ de $E/\langle p^k .P\rangle$ est d'ordre exactement $p^k$ dans son groupe de composantes. \end{prop} {\bf Preuve.} Pour simplifier les notations, \'ecrivons ${\cal O}_K$ pour ce qui sera son localis\'e en ${\cal L}$, et $F$ son corps r\'esiduel (isomorphe \`a $k({\cal L})$, donc dont la caract\'eristique est $l\neq p$). On a la suite exacte de sch\'emas en groupes sur $F$ (comme faisceaux f.p.p.f.) : $$0\to {{\bf G}}_{m,F} \to {\cal E}_F \to ({\bf Z} /N{\bf Z} )_F \to0.$$ Soit $a$ l'entier positif tel que ${\langle p^a .P \rangle}_F =\langle p^a . \widetilde{P} \rangle =\langle \widetilde{P} \rangle \cap {\cal E}^0_F \, (\subseteq \langle p^k .\widetilde{P} \rangle ),$ o\`u on a not\'e ${\cal E}^0_F$ la composante neutre de la fibre sp\'eciale du mod\`ele de N\'eron de $E$. On a $a\geq k$, et $p^a$ divise $N$. Il suffit de montrer que ${\cal E}_F /{\langle p^k .P\rangle}_F$ est un ouvert de ${\cal E}'_F$, car alors le lemme est \'evident : on peut travailler dans ${\cal E}_F /{\langle p^k .P\rangle}_F$, o\`u il est clair par le choix de $k$ que l'image de $P$ est d'ordre $p^k$ dans le groupe des composantes.\\ % {\bf Premi\`ere \'etape.} Montrons qu'on a ${\cal E}_F /{\langle p^k .P \rangle}_F ={({\cal E}/\langle p^k .P\rangle )}_F$. Le groupe fini $G:=({\bf Z} /p^{n-k} {\bf Z} )$ agit sur ${\cal E}_{/{\cal O}_K }$ par addition de $p^k .P$. Comme ${\cal E}_{/{\cal O}_K }$ est quasi-projectif, l'orbite sous $G$ de chaque point est contenue dans un affine. Donc on sait que tout ouvert affine Spec($A$), stable par $G$, de ${\cal E}_{/{\cal O}_K}$ donne les ouverts Spec($A^G {\otimes}_{{\cal O}_K} F$) et Spec($(A {\otimes}_{{\cal O}_K} F)^G $) de ${({\cal E}/\langle p^k .P\rangle )}_F$ et ${\cal E}_F /{\langle p^k .P \rangle}_F$ respectivement (voir \cite{serre}, III \S 12, \cite{mumford}, III \S 12). On a donc juste \`a v\'erifier que le morphisme canonique entre les deux anneaux ci-dessus est un isomorphisme. Il suffit pour cela de remarquer que, $|G|$ \'etant inversible dans $A$, le projecteur $A\to A^G$ qui envoie $x$ sur $\frac{1}{|G|} \sum_{G} g(x)$ commute au changement de base.\\ {\bf Deuxi\`eme \'etape.} Pour pouvoir d\'eduire de la propri\'et\'e universelle des mod\`eles de N\'eron un prolongement \`a tout Spec(${\cal O}_K$) du morphisme sur la fibre g\'en\'erique $({\cal E}/{\langle p^k .P \rangle})_K \to {{\cal E}'}_K$, on doit v\'erifier que le premier sch\'ema est lisse sur Spec(${\cal O}_K$). Consid\'erons la suite exacte de sch\'emas : % $$ 0 \to ({\bf Z} /p^{n-k}{\bf Z})_{/{\cal O}_K} \to {\cal E}_{/{\cal O}_K} \to ({\cal E}/{\langle p^k .P\rangle})_{/{\cal O}_K}\to 0 \ ;$$ ils sont tous ici localement de pr\'esentation finie, puisque localement de type fini sur un anneau noeth\'erien. De plus, le premier sch\'ema de la suite est \'etale sur ${\cal O}_K$ ; puisque $({\bf Z} /p^{n-k} {\bf Z} )$ agit librement sur ${\cal E}_{/{\cal O}_K}$, la seconde fl\`eche est finie \'etale (voir \cite{Katz-Mazur}, th\'eor\`eme A7.1.1). \'Etant donn\'ees nos hypoth\`eses, la lissit\'e sur ${\cal O}_K$ du dernier sch\'ema r\'esulte par exemple de la proposition 17.7.7 de \cite{egaiv}.\\ {\bf Troisi\`eme \'etape.} On consid\`ere donc le morphisme prolong\'e $({\cal E}/{\langle p^k .P\rangle})_{/{\cal O}_K} \to {{\cal E}'}_{/{\cal O}_K}$. Le premier sch\'ema est s\'epar\'e sur la base. On peut donc appliquer la proposition 3.2 de l'expos\'e IX de \cite{SGA7I} : et dire que ce morphisme est une immersion ouverte. En se restreignant \`a la fibre sp\'eciale, et en se servant de la premi\`ere \'etape, on peut bien voir comme on le voulait plus haut ${\cal E}_F /{\langle p^k .P\rangle}_F$ comme un ouvert de ${\cal E}'_F$. $\Box$ \section{L'alg\`ebre de Hecke pour ${\Gamma}_1$ en niveau et poids quelconques.} On a introduit dans le ``sch\'ema de la preuve'' l'alg\`ebre de Hecke sur ${{\bf Z}}$ pour ${\Gamma}_0$. On va maintenant se placer dans un cadre un peu plus g\'en\'eral : ce qu'on notera dans cette section ${{\bf T}}_{{\bf Z}}$ ou simplement ${{\bf T}}$ d\'esignera l'alg\`ebre de Hecke ``pour ${\Gamma}_1$, en niveau et poids quelconques'' comme intitul\'e (pour plus de d\'etails, voir la sous-section suivante). On % notera ${{\bf T}}_{{\bf Q}}$ et ${{\bf T}}_{{\bf C}}$ les tensorisations de ${{\bf T}}$ avec ${{\bf Q}}$ et ${{\bf C}}$ respectivement. Le but de ce qui suit est d'expliciter matriciellement l'action de ${{\bf T}}_{{\bf Q}}$ sur l'espace des formes paraboliques de poids $\lambda$ pour ${\Gamma }_1 (N)$, o\`u $N$ est un entier quelconque. Pour cela, on se sert des r\'esultats de \cite{Atkin-Lehner} g\'en\'eralis\'es (voir par exemple \cite{Lang}, ou \cite{Diamond} ; ces r\'ef\'erences seront constamment utilis\'ees). On en d\'eduira la forme de ${{\bf T}}_{{\bf Q}}$ comme ${\bf Q}$-alg\`ebre abstrai\-te, ce qui permettra ensuite, en se restreignant au cas de poids $2$ et de caract\`ere trivial ({\it i.e.}, formes modulaires sur ${\Gamma}_0 (N)$), de prouver la finitude du quotient d'enroulement sur ${\bf Q}$ en niveau quelconque (th\'eor\`eme 3.9) - m\^eme si on n'a besoin pour le th\'eor\`eme 1.5 que des niveaux puissance d'un nombre premier. \subsection{Rappels.} % On commence par fixer les notations : soit $f=\sum_{n\geq 1} a_n x^n$ une s\'erie formelle \`a coefficients dans un corps, en une variable ; soit $\lambda$ et $N$ deux entiers, et $\varepsilon$ l'extension \`a ${\bf Z}$ d'un caract\`ere de Dirichlet % sur $({\bf Z} /N {\bf Z} )^{\times}$ (on pose $\varepsilon (n)=0$ si $(n \wedge N)\neq 1)$. On note $t_p$, $U_q$, $B_d$ ($p$, $q$, premiers, $d$ entier quelconque), les op\'erateurs d\'efinis formellement (voir \cite{Atkin-Lehner}) par : $$\left\{ \begin{array}{l} t_p (f) = \sum_{n\geq 1} (a_{np} +\varepsilon (p)p^{\lambda -1}\, a_{n/p} ) x^n \ ;\\ U_q (f)= \sum_{n\geq 1} a_{np} \, x^n \ ;\\ B_d (f)= \sum_{n\geq 1} a_{n} x^{nd} = \sum_{n\geq 1} a_{n/d} x^n \ , \end{array} \right.$$ o\`u on pose $a_{n/m} =0$ si $m$ ne divise pas $n$. On a les relations de commutation suivantes : $$\left\{ \begin{array}{l} B_d \circ B_{d'} =B_{d'} \circ B_d ,{\mathrm{\ pour\ tous\ }}d,d'{\mathrm{\ dans\ }}{\bf N} \ ;\\ t_p \circ B_d =B_d \circ t_p ,{\mathrm{\ si\ }}p\ {\mathrm{et}}\ d\ {\mathrm{sont\ premiers\ entre\ eux}}\ ;\\ t_p \circ t_{p'}=t_{p'} \circ t_{p} {\mathrm{\ pour\ tous}}\ p\ et\ p'\ {\mathrm{premiers\ }};\\ t_p \circ U_q=U_q\circ t_p \ {\mathrm{si}}\ p\neq q\ ;\\ U_q\circ U_{q'}=U_{q'}\circ U_{q} \ ,\ {\mathrm{pour\ tous}}\ q\ {\mathrm{et}} \ q'\ {\mathrm{premiers}}\ ;\\ U_q\circ B_d=B_d\circ U_q,\ {\mathrm{si}}\ d\ {\mathrm{et}}\ q'\ {\mathrm{sont \ premiers\ entre\ eux.}} \end{array} \right. $$ De plus, $$U_q \circ B_{q^k} =B_{q^{k-1}} .$$ Consid\'erons maintenant l'espace des formes modulaires de poids $\lambda$ pour $\Gamma_1 (N)$ \`a coefficients dans ${\bf Z}$, $S_{\lambda} ({\Gamma}_1 (N))$, et sa tensorisation par ${\bf Q}$, $S_{\lambda} ({\Gamma}_1 (N)) {\otimes}_{{\bf Z}} {\bf Q}$, qu'on \'ecrira $S_{\lambda} ({\Gamma}_1 (N))_{\bf Q}$ (de m\^eme, $S_{\lambda} ({\Gamma}_1 (N))_{\bf C}$). On peut d\'efinir l'alg\`ebre de Hecke d'endomorphismes de ce ${\bf Q}$-espace, comme on le disait pr\'ec\'edemment, qui est engendr\'ee par les op\'erateurs $T_p$ ($p$ premier), et les op\'erateurs diamants $\langle n\rangle$ (pour les $n$ premiers au niveau $N$). Un \'el\'ement de $S_{\lambda} ({\Gamma}_1 (N))_{\bf Q} \subset S_{\lambda} ({\Gamma}_1 (N))_{\bf C}$ peut \^etre vu comme une fonction $f(z)$ holomorphe sur le demi-plan de Poincar\'e, et on peut \'ecrire son d\'eveloppement de Fourier en l'infini pour arriver \`a l'expres\-sion :% $$f(x)=\sum_{n\geq 1} a_{n} \, x^n \ ,$$ % o\`u $x=e^{2i\pi z}$, et les $a_n$ sont dans ${\bf Q}$. L'action des op\'erateurs de Hecke $T_p$ sur $S_{\lambda} ({\Gamma}_1 (N))_{\bf Q}$ est alors pr\'ecis\'ement celle d\'ecrite plus haut par op\'erateurs $t_p$, si $p$ ne divise pas $N$, et $U_q$ si $q$ divise $N$ ; on conserve d\'esormais cette notation ``mixte'' pour nous des $T_p \, ,U_q$, qui est celle d'Atkin-Lehner. On rappelle alors le th\'eor\`eme principal de leur th\'eorie (voir \cite{Diamond}, \cite{Lang}) : \begin{theo} % {\bf (Atkin-Lehner)} L'espace $S_{\lambda} ({\Gamma}_1 (N))_{\bf C}$ se d\'ecompose en une somme directe :% $$S_{\lambda} ({\Gamma}_1 (N))_{\bf C} = \bigoplus_{\varepsilon} S_{\lambda} (N, \varepsilon ),$$ % o\`u $S_{\lambda} (N, \varepsilon )$ d\'esigne l'espace propre correspondant au caract\`ere de Dirichlet $\varepsilon$, et la somme est prise sur tous les tels $\varepsilon$ v\'erifiant $\varepsilon (-1)=(-1)^{\lambda }$. Chaque espace $S_{\lambda} (N, \varepsilon )_{{\bf C}}$ poss\`ede \`a son tour une base ${\cal B}^{\varepsilon}$ compos\'ee de sous-bases ${\cal B}_f^{\varepsilon}$ du type suivant :% $${\cal B}_f^{\varepsilon} =\{ f(kz),\ k|(N/M)\} =\{ B_{k} (f),\ k|(N/M)\} \ ,$$ % o\`u $f$ est une newform en niveau $M$, de caract\`ere $\chi$ (sur $({\bf Z} /M{\bf Z} )^{\times}$) tel que l'extension de $\chi$ \`a $({\bf Z} /N{\bf Z} )^{\times}$ \'egale $\varepsilon$ ; si $C$ est le conducteur de $\varepsilon$ (ou de $\chi$), on a : $C|M|N$. Si $N=M$, alors ${\cal B}_f^{\varepsilon}$ ne comporte qu'un \'el\'ement, et on l'appelle une {\em newclass} ; sinon, ${\cal B}_f^{\varepsilon}$ est une {\em oldclass}. On note $E_f^{\varepsilon}$ le sous ${\bf C}$-espace de $S_{\lambda} (N, \varepsilon )_{{\bf C}} \subset S_{\lambda} ({\Gamma}_1 (N))_{\bf C}$ engendr\'e par ${\cal B}_f^{\varepsilon}$. Pour $p$ premier \`a $M$, (respectivement, $q$ premier divisant $M$), $f$ est un vecteur propre de $T_p$ (respectivement $U_q$), et la valeur propre associ\'ee est le $p$-i\`eme coefficient $a_p$ de $f$, (qui, puisque newform, est suppos\'ee normalis\'ee par $a_1 =1$). Dans le cas o\`u $\varepsilon$ est trivial (formes modulaires pour ${\Gamma}_0 (N)$), si $q^2$ divise $M$, on a $U_q (f)=0$, tandis que si $q$ - mais pas son carr\'e - divise $M$ (ce qu'on note $q||M$), $U_q (f)=\pm f$. \end{theo}% On va donc maintenant \'ecrire matriciellement l'action de l'alg\`ebre de Hecke ${{\bf T} }_{\bf C}$ sur $S_{\lambda} ({\Gamma}_1 (N))_{\bf C}$, en \'ecrivant cet espace comme somme de facteurs $S_{\lambda} (N, \varepsilon )_{{\bf C}}$, qu'on d\'ecompose \`a leur tour en somme des $E_f^{\varepsilon}$ qui correspondent aux classes du th\'eor\`eme. (On en d\'eduira \`a chaque fois l'action de ${{\bf T}}_{{\bf Q}}$ sur $S_{\lambda} ({\Gamma}_1 (N))_{\bf Q}$, en le d\'ecomposant en sous-${\bf Q}$-espaces ${\cal E}_f^{\varepsilon}$ dont la tensorisation avec ${\bf C}$ donne la somme des conjugu\'es sous Galois des $E_f^{\varepsilon}$.) On commencera d'abord (3.3) par \'etudier le cas de ``co-niveau'' puissance de premier, c'est-\`a-dire le cas o\`u, avec les notations du th\'eor\`eme, $(N/M)=p^k$ pour $p$ premier - (et c'est encore une fois le seul cas dont on ait besoin dans le reste du papier). Puis on traitera le cas g\'en\'eral en 3.4. \subsection{Cas de co-niveau qu'un seul premier $p$ divise.} Soit donc $f\in S_{\lambda} (N, \varepsilon )_{{\bf C}}$, qui est une newform en niveau $M$, avec $N/M=~p^k$. On a alors ${\cal B}_f =\{ (f(z),\, f(pz),\, f(p^2 z),\, \dots ,f(p^{k} z)\} = \{ f, B_p (f),\, \dots ,B_{p^k} (f)\}$. Examinons l'action des diff\'erents op\'erateurs de Hecke sur $E_f$ : d'apr\`es les r\'esul\-tats pr\'ec\'edents, les op\'erateurs $T_l$, $l$ ne divisant pas $N$, et $U_q$, $q$ divisant $M$ et diff\'erent de $p$, de m\^eme que les op\'erateurs diamants, agissent diagonalement sur $E_f$, puisqu'ils ont $f$ comme vecteur propre et qu'ils commutent avec les $B_{p^j} ,\ 0\leq j \leq k$. Puisque ${{\bf T}}_{{\bf Z}}$ op\`ere sur $S_{\lambda} ({\Gamma}_1 (N) )_{\bf Z} ,$ qui est un ${\bf Z}$-module libre de rang fini, les valeurs propres des $T_l$ sont des entiers alg\'ebriques. Et il existe un corps de nombre $K_f$ contenant tous les $a_p$ (car l'alg\`ebre de Hecke est un ${\bf Z}$-module de type fini).\\ {\bf Remarque.} {\it Le corps de nombres engendr\'e par les valeurs propres associ\'ees \`a $f$ de ${{\bf T}}_{{\bf Q}}$ n'est pas plus grand que $K_f$ : en effet, m\^eme s'il contient, en plus des $a_n$, les valeurs propres des op\'erateurs diamants, la relation $p^{\lambda -1}\langle p\rangle =T_p^2 - T_{p^2}$ (pour tout $p$ premier ne divisant pas le niveau) montre que ces valeurs propres appartiennent \`a $K_f$.} En caract\`ere trivial, les $U_q$ sont triviaux, puisque leurs valeurs propres en $f$ sont soit $0$, soit $1$, soit $-1$ ; dans ce cas encore, les coefficients $a_n$ de $f$ sont de plus totalement r\'eels : en effet, les op\'erateurs de Hecke sont auto-adjoints pour le produit scalaire de Petersson :% $$\langle f,g\rangle =\int \int_{z =x+iy\, \in D} f(z)\, \overline{g(z)} \, y^{\lambda -2} \, dx\, dy\ ,$$% qui fait de $S_2 (N)_{\bf C}$ un espace de Hilbert (on a d\'esign\'e par $D$ dans l'int\'egrale un domaine fondamental du demi-plan de Poincar\'e pour $\Gamma_0 (N)$). (En fait, on d\'efinit un produit scalaire de Petersson pour tout $S_{\lambda} ({\Gamma}_1 (N))_{\bf C}$, pour lequel les $T_l$ sont normaux (sinon auto-adjoints), et pour lequel la d\'ecomposition en $E_f^{\varepsilon}$ est orthogonale - voir la preuve de 3.7.) \setlength{\unitlength}{0.7cm} Revenant au cas g\'en\'eral, on consid\`ere maintenant l'action de $U_p$. Pour cela, on distingue deux cas : selon que $p$ divise ou non $M$. \subsubsection{Si $p$ divise $M$.} % L'op\'erateur $U_p$ de l'alg\`ebre de Hecke en niveau $N$ est le m\^eme que celui de niveau $M$, et donc avec le lemme plus haut,% $$\left\{ \begin{array}{l} U_p (f)=a_p \, f,\ {\mathrm et}\\ U_p (B_{p^j} (f))=B_{p^{j-1}} (f)\ {\mathrm si}\ j\geq 1\ . \end{array} \right.$$ % Dans la base ${\cal B}_f$, $U_p$ est donc sous la forme $(M_1 )$ : $$U_p = \left( \begin{picture}(7,4)(0.6,4) \put(1,7){$a_p$} \put(2,7){1} \put(1,6){0} \put(2,6){0} \put(3,6){1} \multiput(3,7)(1,0){5}{.} \put(7,7){0} \multiput(1,6)(0,-1){5}{.} \put(1,1){0} \multiput(2,1)(1,0){5}{.} \put(7,1){0} \multiput(7,6)(0,-1){4}{.} \put(7,2){1} \multiput(3,5)(1,-1){4}{.} \multiput(4,5)(1,-1){3}{.} \end{picture} \right) .$$% Si $a_p =0$, la matrice pr\'ec\'edente n'a qu'un bloc de Jordan, et la restriction $R_f$ de l'alg\`ebre de Hecke ${{\bf T}}_{{\bf Q}}$ au ${\bf Q}$-espace ${\cal E}_f^{\varepsilon}$ qu'on a d\'efini plus haut (correspondant sur ${\bf C}$ \`a la somme des conjugu\'es par Galois de $E_f^{\varepsilon}$), est de forme :% $$R_f =K_f [U_p ]\simeq K_f [X]/(X^{k+1} ).$$ % Sinon, on a pour $U_p$ deux blocs de Jordan, un petit correspondant \`a la valeur propre $a_p$, et un gros, correspondant \`a $0$ ; en tant que ${\bf Q}$-alg\`ebres, on a donc l'isomorphisme $$R_f \simeq K_f \times K_f [X]/(X^k ).$$ % {\bf Remarque.} {\it Le fait que le corps du deuxi\`eme facteur de $R_f$ soit bien tout $K_f$, malgr\'e l'absence de la valeur propre $a_p$, peut se voir par un argument de dimension des espaces cotangents ; ou bien avec les r\'esultats d'Atkin-Lehner, qui disent moralement qu'une newform est caract\'eris\'ee par ``tous ses coefficients de Fourier moins un nombre fini''.} % \paragraph{Application au cas de caract\`ere trivial.} % Dans ce cas, si $p^2$ divise $M$, $a_p =0$, et $R_f \simeq K_f [X]/(X^{k+1})$ ; la base ${\cal B}_f^{\varepsilon}$ diagonalise $U_p$. Si en revanche $p$ - mais pas son carr\'e - divise $M$, on a $a_p =\pm 1$, donc $R_f \simeq K_f \times K_f [X]/(X^k )$, et on voit facilement que la nouvelle base de $E_f^{\varepsilon}$ :% $${\cal B}'_f=\{ f,\, B_p (f)-a_p f,\, B_{p^2} (f)-f,\, \ldots ,\, B_{p^{2j}} (f)-f,\, B_{p^{2j+1}} (f)-a_p f,\ldots \}$$ % est de Jordan pour $U_p$ et donc triangulise toute la restriction de l'alg\`ebre de Hecke \`a $E_f^{\varepsilon}$. \subsubsection{Si $p$ ne divise pas $M$.} Le fait nouveau par rapport au cas pr\'ec\'edent est que l'alg\`ebre de Hecke en niveau $N$, ${{\bf T}}_{N,{\bf C}}$, qu'on fait agir sur $E_f^{\varepsilon}$, n'est plus la restriction \`a cet espace de l'alg\`ebre de Hecke en niveau $M$ : le ``bon'' $p$-i\`eme op\'erateur de Hecke, celui pour lequel $f$ est un vecteur propre, est $T_p$, mais puisqu'on s'int\'eresse \`a ${\bf T}$ en niveau $N$, on doit consid\'erer \`a la place l'op\'erateur $U_p$. Puisque $f$ est propre pour $T_p$, on a :% $$T_p (f)=\sum_{n\geq 1} (a_{np} +\varepsilon (p)\, p^{\lambda -1}\, a_{n/p} ) \, x^n \, =a_p \, f\ ,$$ % et les coefficients de $f$ v\'erifient donc pour tout $n$ les relations :% $$a_p \, a_n = a_{np} +\varepsilon (p)\, p^{\lambda -1}\, a_{n/p} .$$ % On calcule alors :% \begin{eqnarray*} U_p (f) & = & \sum_{n\geq 1} a_{np} \, x^n \\ & = & \sum_{n\geq 1} (a_p a_n \, -\varepsilon (p)\, p^{\lambda -1} \, a_{n/p} )\, x^n \\ & = & a_p f -\varepsilon (p)\, p^{\lambda -1} \, B_{p} (f). \end{eqnarray*} La matrice de $U_p$ dans ${\cal B}_f$ est donc $(M_2 )$ : $$U_p = \left( \begin{picture}(9,4)(0.6,4) \put(2,7){$a_p$} \put(.6,6){$(-\varepsilon (p)\, p^{\lambda -1} )$} \put(2,5){0} \put(4,7){1} \put(4,6){0} \put(5,6){1} \put(5,5){0} \put(6,5){1} \multiput(5,7)(1,0){5}{.} \put(9,7){0} \multiput(2,4)(0,-1){3}{.} \put(2,1){0} \multiput(4,1)(1,0){5}{.} \put(9,1){0} \multiput(9,6)(0,-1){4}{.} \put(9,2){1} \multiput(6,4)(0.5,-0.5){4}{.} \multiput(7,4)(0.5,-0.5){3}{.} \end{picture} \right) .$$ Le polyn\^ome caract\'eristique de la restriction de $U_p$ \`a $E_f^{\varepsilon}$ est % $${P}_{U_p} (X)=(X^2 -a_p X+\varepsilon (p)\, p^{\lambda -1}).X^{k-1} \ ,$$ et les racines $\alpha_p$ et $\overline{\alpha}_p$ du premier facteur sont simples si ${a_p }^2 \neq 4\varepsilon (p)\, p^{\lambda -1}$, doubles (valant $\pm \sqrt{\varepsilon (p)} p^{\frac{\lambda -1}{2}}$) sinon ; dans ce deuxi\`eme cas on voit sur le rang de $U_p$ qu'il a deux blocs de Jordan, un pour chaque valeur propre ($0$ et $\pm \sqrt{\varepsilon (p)} p^{\frac{\lambda -1}{2}}$).\\ % {\bf Remarque.} {\it En caract\`ere trivial et poids $2$, ce dernier cas, (qui correspond \`a une valeur maximale de $a_p$ selon la borne de Weil, qui dit que $|a_p |\leq 2\sqrt{p}$), est en fait exclu, par un th\'eor\`eme non encore publi\'e de Coleman et Edixhoven (\cite{Coleman-Edixhoven}) ; on l'explicite quand m\^eme - en attendant.} \\ % \underline{Si ${a_p}^2 \neq 4\varepsilon (p)\, p^{\lambda -1}$,} dans une base convenable la restriction de $U_p$ \`a $E_f^{\varepsilon}$ s'\'ecrit $(M_3 )$ : % $$U_p = \left( \begin{picture}(8,4.5)(-0.4,3.5) \put(-0.5,5.7){\dashbox{.3}(2,1.8){ }} \put(0.7,6){$\overline{\alpha}_p$} \put(1,7){0} \put(0,6){0} \put(0,7){$\alpha_p$} \put(0.5,3){0} \put(4.5,6.5){0} \put(1.6,-0.4){\dashbox{.4}(6,6){ }} \put(2,5){0} \put(3,5){1} \put(3,4){0} \put(4,4){1} \multiput(4,5)(1,0){3}{.} \put(7,5){0} \multiput(2,4)(0,-1){4}{.} \put(2,0){0} \multiput(3,0)(1,0){4}{.} \put(7,0){0} \multiput(7,4)(0,-1){3}{.} \put(7,1){1} \multiput(4,3)(1,-1){3}{.} \multiput(5,3)(1,-1){2}{.} \end{picture} \right) .$$ Dans ce cas-l\`a, si on pose $K'_f =K_f [X]/(X^2 -a_p X+\varepsilon (p)\, p^{\lambda -1} )$, (qui n'est pas n\'ec\'essairement un corps), la restriction de ${{\bf T}}_{{\bf Q}}$ \`a ${\cal E}_f^{\varepsilon}$ est % $$R_f \simeq K'_f \times K_f [X]/(X^{k-1} ).$$ % On note (cela servira dans la suite) qu'un vecteur propre associ\'e \`a la valeur propre $0$ est $(B_{p^2} (f)-\frac{a_p}{\varepsilon (p)\,p^{\lambda -1}} B_p (f)+\frac{1}{\varepsilon (p)\, p^{\lambda -1}} f)$. \\ \underline{Si ${a_p}^2 =4\varepsilon (p)\, p^{\lambda -1},$} on a de m\^eme dans une base convenable l'\'ecriture matricielle $(M_4 )$ : % $${U_p}_{|E_f^{\varepsilon}} = \left( \begin{picture}(8,4.5)(-0.4,3.5) \put(-0.5,5.7){\dashbox{.3}(2,1.8){ }} \put(0.7,6){$\frac{a_p}{2}$} \put(1,7){1} \put(0,6){0} \put(0,7){$\frac{a_p}{2}$} \put(0.5,3){0} \put(4.5,6.5){0} \put(1.6,-0.4){\dashbox{.4}(6,6){ }} \put(2,5){0} \put(3,5){1} \put(3,4){0} \put(4,4){1} \multiput(4,5)(1,0){3}{.} \put(7,5){0} \multiput(2,4)(0,-1){4}{.} \put(2,0){0} \multiput(3,0)(1,0){4}{.} \put(7,0){0} \multiput(7,4)(0,-1){3}{.} \put(7,1){1} \multiput(4,3)(1,-1){3}{.} \multiput(5,3)(1,-1){2}{.} \end{picture} \right) .$$ % Alors avec les notations pr\'ec\'edentes, $$R_f \simeq K_f [X]/(X^2 )\times K_f [Y]/(Y^{k-1} ).$$ Une base de Ker$(U_p -\alpha_p$ Id) est $\{ a_p f-2\varepsilon (p)p^{\lambda -1} B_p (f)\}$, et bien s\^ur le m\^eme vecteur $(B_{p^2} (f)- \frac{a_p}{\varepsilon (p)\, p^{\lambda -1}} B_p (f)+\frac{1}{\varepsilon (p) \, p^{\lambda -1}} f)$ que pr\'ec\'edemment engendre Ker($U_p$). \subsection{Cas de co-niveau quelconque.} % Soit maintenant $f\in S_{\lambda} (N, \varepsilon )_{{\bf C}}$, qui est une newform en niveau $M$, avec $N/M=n$ quelconque cette fois. On a alors ${\cal B}_f =\{ B_d (f), d|n\}$. On examine \`a nouveau l'action des diff\'erents op\'erateurs de Hecke : les op\'erateurs $T_l$, $l$ ne divisant pas $N$, et $U_q$, $q$ ne divisant pas $n$, agissent toujours diagonalement sur $E_f^{\varepsilon}$, comme en co-niveau premier. Soit $q$ un premier divisant $n$. Notons $m$ la valuation en $q$ de $n$ : $q^m ||n$. Alors, on construit pour $U_q$ une base qui lui est appropri\'ee, en ordonnant partiellement la base ${\cal B}_f$ ainsi : pour $d$ parcourant les diviseurs de $n/{q^m}$, on note ${\cal B}_f^{q,d} =\{ B_{d} (f),\, B_{dq} (f),\, B_{dq^2} (f),\, \ldots ,\, B_{dq^m} (f)\}$, et on \'ecrit % $${\cal B}_f^q =\{ f, B_q (f), B_{q^2} (f), \ldots ,B_{q^m} (f), B_{d} (f), B_{dq} (f), B_{dq^2} (f), \ldots , B_{dq^m} (f), \ldots \}$$ comme la r\'eunion sur $d$ de ces ${\cal B}_f^{q,d}$. (Le nombre des diff\'erentes bases ${\cal B}_f^{q,d}$ est ${\sigma}_0 (n/{q^m})$ (nombre de diviseurs de $n/{q^m}$).) On d\'eduit des relations de commutation entre les op\'erateurs consid\'er\'es et de ce qui pr\'ec\`ede que, dans la base ${\cal B}_f^{q}$, on a : $${U_q}|_{E_f} =\left( \begin{array}{cccc} B^q_1 & 0 & \cdots & 0 \\ 0 & B^q_{d_2} & \cdots & 0 \\ \vdots & & \ddots & \\ 0 & \ldots & \ldots & B^q_{d_{{\sigma}_0 (n/{q^m})}} \end{array} \right) \ ,$$ o\`u chaque $B^q_d,\ d|(n/{q^m})$ est une matrice repr\'esentant la restriction de $U_q$ \`a l'espace engendr\'e par ${\cal B}_f^{q,d}$ - que $U_q$ laisse stable. La forme de $B^q_d$ ne d\'epend en fait pas de $d$, mais uniquement de $q$. En effet :\\ \underline{Si $q$ divise $M$}, alors chaque $B^q_d$ est une bloc de forme $(M_1 )$, et on peut trouver une nouvelle base de l'espace engendr\'e par ${\cal B}_f^{q,d}$, dans laquelle on a $B^q_d$ \'equivalente si $a_p \neq 0$ \`a % $$\left( \begin{picture}(7,4)(0.6,4) \put(0.5,6.7){\dashbox{.3}(0.8,0.8){$a_q$}} \put(1,3.5){0} \put(4.5,7){0} \put(1.6,0.6){\dashbox{.4}(6,6){ }} \put(2,6){0} \put(3,6){1} \put(3,5){0} \put(4,5){1} \multiput(4,6)(1,0){3}{.} \put(7,6){0} \multiput(2,5)(0,-1){4}{.} \put(2,1){0} \multiput(3,1)(1,0){4}{.} \put(7,1){0} \multiput(7,5)(0,-1){3}{.} \put(7,2){1} \multiput(4,4)(1,-1){3}{.} \multiput(5,4)(1,-1){2}{.} \end{picture} \right) ,$$ et sinon \`a un bloc de Jordan nilpotent.\\ % \underline{Si $q$ ne divise pas $M$}, on conclut toujours comme dans la sous-section pr\'ec\`edente que $B^q_d =M_2 $, \'equivalente si ${a_q}^2 \neq 4\varepsilon (q) q^{\lambda -1}$ \`a $(M_3 )$, et si ${a_q}^2 =4 \varepsilon (q)q^{\lambda -1}$ \`a $(M_4 )$. Maintenant, si on note $E_f^q$ le sous-espace de $S_{\lambda} (N, \varepsilon )_{{\bf C}}$ engendr\'e par ${\cal B}_f^{q,1} =\{ f,B_{q} (f),\, B_{q^2} (f),\, \ldots ,\, B_{q^m} (f)\}$, alors $U_q$ agit comme pr\'ecis\'e ci-dessus sur $E_f^q$, et on peut consid\'erer $E_f^{\varepsilon}$ comme le produit tensoriel sur ${\bf C}$ : % $$E_f^{q_1} \otimes E_f^{q_2} \otimes \cdots \otimes E_f^{q_m} ,$$ % o\`u les $q_i$ sont les nombres premiers divisant $n$. Et $S_{\lambda} (N, \varepsilon )_{{\bf C}}$, \`a son tour, est la somme directe des tels $E_f^{\varepsilon}$. On r\'esume tout \c{c}a :% \begin{theo} L'entier naturel $N$ est quelconque. L'alg\`ebre de Hecke ${{\bf T}}_{{\bf Q}}$, vue comme sous-alg\`ebre des endomorphismes de ${\bf Q}$-espace vectoriel de $S_{\lambda} ({\Gamma}_1 (N))_{\bf Q}$, s'\'ecrit comme un produit d'anneaux :% $${{\bf T}}_{{\bf Q}} =R_{f_1} \times R_{f_2} \times \cdots \times R_{f_k} ,$$ % o\`u chacun de ces $R_{f_i}$ correspond \`a l'orbite sous Galois d'une newform $f_i$ en niveau $M_i$ divisant $N$, et caract\`ere $\varepsilon$. Plus pr\'ecis\'ement, le facteur $R_{f_i}$ est la restriction de ${{\bf T}}_{{\bf Q}}$ au sous-espace ${\cal E}_{f_i}$ de $S_{\lambda} ({\Gamma}_1 (N))_{{\bf Q}}$, qui est engendr\'e sur ${\bf Q}$ par les orbites sous Galois des $B_d (f_i )$, $d$ parcourant les diviseurs de $N/M_i$. Si $K_{f_i}$ d\'esigne le corps de nombres engendr\'e par les coefficients $a_n$ du d\'eveloppement de Fourier de $f_i$ \`a l'infini, l'id\'eal $R_{f_i}$ est de forme un produit tensoriel d'anneaux des quatre types suivants : $$\begin{array}{l} \bullet \ K_{f_i} [X]/(X^{n} )\ ;\\ \bullet \ K_{f_i} \times K_{f_i} [X]/(X^n )\ ;\\ \bullet \ K_{f_i} [X]/(X^2 -a_q X+\varepsilon (q)q^{\lambda -1} )\times K_{f_i} [Y]/(Y^n )\ ;\\ ou\ encore\ :\\ \bullet \ K_{f_i} [X]/(X^2 )\times K_{f_i} [Y]/(Y^n ). \end{array}$$ \end{theo} Notons un corollaire partiel de ce th\'eor\`eme qui nous servira dans la suite : \begin{cor} Soit $A$ une alg\`ebre de dimension finie sur un corps $K$, et qui a un unique id\'eal minimal non trivial ; alors $A$ est de Gorenstein, i.e. son dual en tant que $K$-espace vectoriel, $A^{\vee}$, est un $A$-module libre de rang 1. En particulier, l'alg\`ebre de Hecke sur ${\bf Q}$ (pour ${\Gamma}_1$, en niveau et poids quelconques) ${{\bf T}}_{{\bf Q}}$, est de Gorenstein. \end{cor} {\bf Preuve.}\ Soit $A$ une alg\`ebre comme dans l'\'enonc\'e. Son id\'eal minimal est par d\'efinition principal, engendr\'e par n'importe lequel de ses \'el\'ements non nuls. Soit $g$ l'un de ceux-l\`a. Il existe un \'el\'ement $L$ de $A^{\vee}$ qui ne s'annule pas en $g$. On en d\'eduit donc que le noyau de l'application lin\'eaire : $$\left\{ \begin{array}{ccl} A & \to & A^{\vee} \\ a & \mapsto & (x\mapsto L(a.x)) \end{array} \right.$$ est un id\'eal trivial de $A$, puisqu'il ne contient pas l'id\'eal minimal. Donc cette application est injective, et en fait bijective puisque les deux espaces $A$ et $A^{\vee}$ ont m\^eme dimension. Pour ce qui est de ${{\bf T}}_{{\bf Q}}$, le th\'eor\`eme pr\'ec\'edent dit qu'elle se d\'ecompose en produits d'anneaux de la forme $K[X_1 ,X_2 ,\dots , X_n ]/(X_1^{m_1} X_2^{m_2} \dots X_n^{m_n} )$, o\`u $K$ est un corps de nombres (en effet, chaque $A_{f_i}$ se d\'ecompose encore ainsi). Puisque $K[X_1 ,X_2 ,\dots ,X_n ]/(X_1^{m_1} X_2^{m_2} \dots X_n^{m_n} )$ poss\`ede un seul id\'eal minimal, % et puisque la propri\'et\'e d'\^etre de Gorenstein est clairement stable par somme directe finie, on peut appliquer la premi\`ere partie de la proposition. $\Box$ Montrons aussi un r\'eultat dont on se sert dans la sous-section suivante (et qui sera red\'emontr\'e en partie en 4.7 gr\^ace \`a 3.6) : \begin{prop} Pour tous niveau $N$ et poids $\lambda$, $S_{\lambda} ({\Gamma}_1 (N))_{{\bf Q}}$ est un ${{\bf T}}_{{\bf Q}}$-module libre de rang 1. \end{prop} {\bf Preuve}.\ On a un accouplement parfait $[\ ,\ ]$, ${\bf C}$-bilin\'eaire : $$\left\{ \begin{array}{rcl} S_{\lambda} ({\Gamma}_1 (N))_{{\bf C}} \times {{\bf T}}_{{\bf C}} & \to & {\bf C} \\ (f=\sum_{n\geq 1} a_n q^n ,T) & \mapsto & a_1 (Tf) \ , \end{array} \right.$$ qui est d\'efini sur ${{\bf Q}}$, pour lequel tout op\'erateur de l'alg\`ebre de Hecke est auto-adjoint. Maintenant, notons $\langle \ ,\, \rangle_{\mathrm P}$ le produit scalaire de Petersson. Appelons % $\sigma$ l'involution sur $S_{\lambda} ({\Gamma}_1 (N))_{{\bf C}}$ d\'efinie par l'op\'eration de conjugaison complexe des coefficients de Fourier en l'infini d'une forme parabolique. Rappelons aussi l'op\'erateur (classique) $W_N$, dont l'op\'eration sur $S_{\lambda} ({\Gamma}_1 (N))_{{\bf C}}$ est d\'efinie par $W_N (f(z))=N^{-\lambda /2} z^{-\lambda} f(-1/Nz)$ ; il v\'erifie $W_N^2 =(-1)^{\lambda}$, et l'adjoint pour le produit scalaire de Petersson de tout op\'erateur de Hecke $T_n$ est $W_N T_n W_N^{-1}$ (de m\^eme, l'adjoint de l'op\'erateur diamant $\langle n \rangle$ pour $n$ premier \`a $N$ est $W_N \langle n\rangle W_N^{-1}$) (voir par exemple \cite{Diamond}, I.4). Consid\'erons alors l'alt\'eration suivante : $(f,g)\to \{ f,g\} = \langle f,\sigma \circ W_N (g)\rangle_{\mathrm P}$ du produit scalaire de Petersson, c'est-\`a-dire : $$\left\{ \begin{array}{rcl} S_{\lambda} ({\Gamma}_1 (N))_{{\bf C}} \times S_{\lambda} ({\Gamma}_1 (N))_{{\bf C}} & \to & {\bf C} \\ (f,g) & \mapsto & \int_{\Delta} (\sum_{n\geq 1} a_n q^n )\, W_N (\sum_{n\geq 1} b_n {\overline q}^n )\, y^{\lambda} \frac{dx\, dy}{y^2} \ , \end{array} \right.$$ o\`u $(\sum_{n\geq 1} a_n q^n )$ et $(\sum_{n\geq 1} b_n q^n )$ sont les d\'eveloppements de Fourier de $f$ et $g$ respectivement en l'infini, et $\Delta$ est un domaine fondamental du quotient du demi-plan de Poincar\'e par ${\Gamma}_1 (N)$. Par construction, ce produit est ${\bf C}$-bilin\'eaire, et pour lui \'egalement l'alg\`ebre de Hecke est auto-adjointe. Les deux accouplements $[\ ,\ ]$ et $\{ \ ,\ \}$ pr\'esentent donc respectivement ${{\bf T}}_{{\bf C}}$ et $S_{\lambda} ({\Gamma}_1 (N))_{{\bf C}}$ comme les duaux comme ${\bf C}$-espaces de $S_{\lambda} ({\Gamma}_1 (N))_{{\bf C}}$, ce qui donne un isomorphis\-me de ${\bf C}$-espaces vectoriels entre eux. Mais l'auto-adjonction de ${{\bf T}}_{{\bf C}}$ pour les deux montre que cette application est aussi un morphisme de ${{\bf T}}_{{\bf C}}$-module, donc que $S_{\lambda} ({\Gamma}_1 (N))_{{\bf C}}$ est un ${{\bf T}}_{{\bf C}}$-module libre de rang 1. Enfin, que ceci soit vrai sur ${{\bf Q}}$ est une cons\'equence du r\'esultat de g\'eom\'etrie alg\'ebrique \'el\'ementaire suivant : si $k$ est un corps infini dont $K$ est une extension, si $R$ est une $k$-alg\`ebre et $M$ un $R$-module qui est un $k$-espace vectoriel de dimension finie, et tel que $M\otimes_{k} K$ soit un $R\otimes_{k} K$-module libre de rang $1$, alors $M$ est soi-m\^eme un $R$-module libre de rang $1$. Ce r\'esultat se montre comme suit. Choisisons des \'el\'ements $r_i$ de $R$, $1\leq i\leq {\mathrm {dim}}_k M$, tel que les $r_i .y$ forment une $K$-base de $M\otimes_{k} K$ pour un $y$ de $M\otimes_{k} K$. La fonction $f$ qui \`a tout \'el\'ement $x$ de $M$ associe le d\'eterminant de $(r_i .x)$ est polyn\^omiale en les coordonn\'ees de $x$ dans une $k$-base de $M$, et n'est pas nulle sur $M\otimes_{k} K$. Ce qui veut dire que le polyn\^ome correspondant est non nul, et comme $k$ est infini, on a bijection entre polyn\^omes et fonctions polyn\^omiales sur $M$ : donc $f$ ne peut \^etre uniform\'ement nulle sur $M$, et $M$ est un $R$-module de rang 1. Qu'enfin $M$ soit $R$-libre provient de sa $R\otimes_{k} K$-fid\'elit\'e apr\`es extension des scalaires \`a $K$. $\Box$ \subsection{Finitude du quotient d'enroulement.} On la d\'emontre en niveau quelconque, avec les r\'esultats qui pr\'ec\`edent appliqu\'es au cas de poids $\lambda$ \'egal \`a $2$, le caract\`ere $\varepsilon$ \'etant trivial. Soit $N$ un entier ; on consid\`ere la courbe modulaire $X_0 (N)_{{\bf Q}}$ et sa jacobienne $J_0 (N)_{{\bf Q}}$ sur ${\bf Q}$. On rappelle les notations de l'introduction, dans ce cadre plus g\'en\'eral : $e$ d\'esigne l'\'el\'ement d'enroulement, c'est-\`a-dire l'\'el\'ement de $H_1 (X_0 (N)\, ;\, {\bf Q} )$ d\'efini par l'int\'egration entre z\'ero et l'infini sur notre courbe modulaire ; ${\cal A}_e$ d\'esigne l'id\'eal d'enroulement, c'est-\`a-dire l'id\'{e}al annulateur dans ${{\bf T}}_{{\bf Z}}$ de~$e$. Notons aussi ${\cal A}_{e,{\bf Q}} ={\cal A}_e \otimes_{{\bf Z}} {\bf Q}$. Le quotient d'enroulement $J_0^e$ est la vari\'et\'e ab\'{e}lienne quotient $J_0 (N)/ {\cal A}_e J_0 (N)$. % \begin{theo} $J_0^e ({\bf Q} )$ est fini. \end{theo} {\bf Preuve.}\ \ Notons d'abord $S_2 (N)_{{\bf Q}}$ le ${\bf Q}$-espace des formes paraboliques de poids deux pour $\Gamma _0 (N)$ \`a coefficients dans ${\bf Q}$, et ${\mathrm {Cot}}_0 (J_0 (N)_{{\bf Q}} )$ l'espace cotangent \`a $J_0 (N )_{{\bf Q}}$ en z\'ero ; on a les identifications suivantes :% $$S_2 (N)_{{\bf Q}} \simeq H^0 (X_0 (N)_{{\bf Q}} , \Omega^1 )\simeq H^0 (J_0 (N)_{{\bf Q}} , \Omega^1 )\simeq {\mathrm {Cot}}_0 (J_0 (N)_{{\bf Q}} ),$$ et $S_2 (N)_{{\bf Q}}$ est un ${\bf T} {\otimes }_{{\bf Z}} {\bf Q}$-module libre de rang 1 (3.7, ou la proposition 4.7 de la section suivante). L'alg\`ebre de Hecke \`a coefficients dans ${\bf Q}$ s'\'ecrit donc comme on l'a explicit\'e plus haut, {\em i.e.} comme un produit d'anneaux (ou comme une somme directe d'id\'eaux) $R_f$, correspondants aux restrictions de ${{\bf T}}_{{\bf Q}}$ aux orbites sous l'action du groupe de Galois des newforms $f$ de niveau $M$ divisant $N$. D'apr\`es le th\'eor\`eme d'Atkin-Lehner (th\'eor\`eme 3.2), si on note $S_2 (M)^{\mathrm {new}} _{{\bf C}}$ le sous-${\bf C}$-espace de $S_2 (M)_{{\bf C}}$ engendr\'e par les {\em newforms} de niveau $M$, on a en faisant la somme directe des op\'erateurs $B_d$ convenables un isomorphisme de ${\bf C}$-espaces vectoriels : $${\oplus}_{M|N} {\oplus}_{d|(N/M)} S_2 (M)^{\mathrm {new}} _{{\bf C}} \stackrel{\sim}{\to} S_2 (N)_{{\bf C}} .$$ Gr\^ace \`a l'interpr\'etation en termes d'espaces cotangents des espaces de formes paraboliques, on d\'eduit des op\'erateurs $B_d$ des isog\'enies $J_0 (M)_{{\bf Q}} \stackrel{B^*_d}{\rightarrow} J_0 (N)_{{\bf Q}}$ (les op\'erateurs $B_d$ sont d\'efinis sur ${{\bf Q}}$), et pour tout $M$ on d\'efinit $J_0 (M)_{\mathrm {new}} := J_0 (M)/(\sum_{d|M} {\mathrm {Im}}({B^*_d}))$. On d\'eduit alors de l'isomorphisme pr\'ec\'edent (entre espaces de formes paraboliques) l'isog\'enie : $$J_0 (N)_{{\bf Q}} \to {\oplus}_{M|N} {\oplus}_{d|(N/M)} J_0 (M)_{{\bf Q} ,{\mathrm {new}}} .$$ On a, encore et toujours selon la d\'ecomposition d'Atkin-Lehner, des isog\'enies $ J_0 (M)_{{\bf Q} ,{\mathrm {new}}} \to {\oplus}_{G_{{\bf Q}} f} J_f$, o\`u la somme est prise sur les orbites sous Galois $G_{{\bf Q}} f$ des newforms $f$ en niveau $M$, et $J_f$ est la vari\'et\'e ab\'elienne ``d\'ecoup\'ee'' dans $J_0 (M)_{{\bf Q}}$ avec la forme $f$ : avec les notations de 3.5, l'annulateur ${\mathrm {Ann}}_{{{\bf T}}_{{\bf Q}}} f$ de $f$ dans ${\bf T}_{{\bf Q}}$ est $R^f :=\oplus_{g\neq f} R_g$, donc $J_f$ est isog\`ene \`a $J_0 (M)_{{\bf Q}} /R^f J_0 (M)_{{\bf Q}}$. On a donc pour finir l'isog\'enie :% $$J_0 (N)_{{\bf Q}} \to {\oplus}_{G_{{\bf Q}} .f} (J_f )^{\sigma_0 (N/M)} ,$$ o\`u la somme est prise sur les orbites sous l'action de Galois de toutes les newforms en niveaux $M$ divisant $N$. L'accouplement bilin\'eaire : $$\left\{ \begin{array}{rcl} H_1 (X_0 (N )\, ;\, {\bf Q} )\times S_2 (N)_{{\bf C}} & \to &{\bf C} \\ (c,f) &\mapsto &\langle c,f\rangle =\int_{c} f(z) dz \end{array} \right. $$ est non d\'eg\'en\'er\'e, et pour lui les op\'erateurs de Hecke sont auto-adjoints. On va d'abord prouver : \begin{lemm} Soit $f$ une newform en niveau $M$ divisant $N$ tel que $\langle e,f \rangle \neq 0$. Alors ${\cal A}_e \cap R_f = \{ 0\} .$ \end{lemm} {\bf Preuve du lemme.} Calculons l'effet de $B_D$ ($D$ entier quelconque) sur l'accouplement : $$ \langle e, B_D (f)\rangle = \int_{e} f (D.z)dz=i\int_{0}^{\infty} f (iDz)dz =\frac{i}{D} \int_{0}^{\infty} f (iz)dz=\frac{1}{D} \langle e,f\rangle .$$ Avant de faire des calculs explicites, expliquons l'id\'ee de la preuve. On a vu que $S_2 (N)_{{\bf Q}}$ \'etait un ${{\bf T}}_{{\bf Q}}$-module libre de rang 1 : comme tel, selon le th\'eor\`eme 3.5, il est isomorphe \`a un produit de ${\bf Q}$-espaces vectoriels de type $K[X_1 ,\dots ,X_n ]/(X_1^{m_1} \dots X_n^{m_n} )$, sur lesquels la restriction de ${{\bf T}}_{{\bf Q}}$ a la m\^eme forme, et agit par simple multiplication. Si donc on prend un \'el\'ement $t=\sum_{j} \lambda_j \, X_{1}^{m_1^j} X_{2}^{m_2^j} \dots X_{m}^{m_n^j}$ de la ``partie'' $K[X_1 ,\dots ,X_n ]/(X_1^{m_1} \dots X_n^{m_n} )$ de ${{\bf T}}_{{\bf Q}}~\cap~{\cal A}_e$, en le multipliant par des mon\^omes judicieux de $K[X_1 ,\dots ,X_n ]/(X_1^{m_1} \dots X_n^{m_n} )$ (vu cette fois comme le sous-espace des formes modulaires sur lequel $t$ agit), on aura que l'accouplement de $e$ avec des mon\^omes de type $\lambda_j X_1^{\alpha_1} \dots X_n^{\alpha_n}$ sera nul, pour tous les $\lambda_j$. Mais de tels mon\^omes correspondront \`a des ``d\'ecalages'' par des op\'erateurs $B_d$ de notre newform originelle $f$, d\'ecalages qui, comme le montre l'int\'egration par partie ci-dessus, ne font que multiplier le produit $\langle e,f\rangle$ par des constantes non nulles. On d\'eduira donc de la non nullit\'e de ce produit $\langle e,f\rangle$ que tous les coefficients $\lambda_j$ sont nuls, et donc $t$. \'Ecrivons-le maintenant pr\'ecis\'ement. Soit $t$ \'el\'ement de $R_f$. On a les \'equivalen\-ces suivantes :% $$(t.e=0)\Leftrightarrow (\forall g \in S_2 (N)_{{\bf C}} , \langle t.e, g \rangle =0)\Leftrightarrow (\forall g \in {\cal B}_f, \langle e, t.g\rangle =0).$$ % Supposons de plus $t$ dans ${\cal A}_e$. On d\'ecompose $E_f$ comme somme directe des intersections des sous-espaces caract\'eristiques des op\'erateurs $U_q$ pour $q$ divisant $N$. Consid\'erons la restriction $\tilde{t}$ de $t$ \`a l'un de ces sous-espaces, ${\tilde{E}}_f$. On \'ecrit $\tilde{t}$ comme un polyn\^ome en les $U_q$, pour $q$ divisant $N$, \`a coefficients dans ${\bf Q} [T_l ,...]$ ($l$ ne divisant pas $N$), c'est-\`a-dire \`a coefficients dans (un corps isomorphe \`a) $K_f$. On peut ne consid\'erer que les $U_q$ qui n'agissent pas diagonalement sur ${\tilde{E}}_f$. Si on note $\alpha_q$ la valeur propre de $U_q$ sur ${\tilde{E}}_f$, alors $u_q :=(U_q -{\alpha}_q {\mathrm {Id}}|_{{\tilde{E}}_f} )$ est nilpotent. On \'ecrit donc $\tilde{t}$ sous forme d'un polyn\^ome : $$\tilde{t} =\sum_{j\geq 0} \lambda_j \, u_{q_1}^{\alpha_1^j} u_{q_2}^{\alpha_2^j} \dots u_{q_m}^{\alpha_m^j} ,$$ % o\`u les multi-indices $\alpha_k^j$ sont ordonn\'es par exemple de fa\c{c}on lexicographique ($j<j'$ si $\alpha_s^j < \alpha_s^{j'} ,$ o\`u $s:={\mathrm {inf}} \{ r$ tel que $\alpha_r^j \neq \alpha_r^{j'} \}$). % D'apr\`es la r\'eduction de Jordan de la partie pr\'ec\'edente, il existe des polyn\^omes en $B_{q_j}$ : $P_j (B_{q_j} )$, et un \'el\'ement : $$g_1 := [ P_1 (B_{q_1} )\otimes P_2 (B_{q_2} )\otimes \cdots P_m (B_{q_m} )] (f)\, \in S_2 (N)_{{\bf C}} \ ,$$ % tel que $u_{q_j}^{\alpha_j^1} (P_j (B_{q_j} )(f))=: {\cal P}_j (B_{q_j} (f)) \neq 0$ et $u_{q_j}^{\alpha_j^1 +1} (P_j (B_{q_j} )(f))=0.$ Pour \^etre plus explicite, ${\cal P}_j (B_{q_j} (f))$ a l'une des quatre formes suivantes : % \begin{enumerate} \item $(f)$ si on est sur l'espace caract\'eristique associ\'e \`a $0$ et $q_j^2 |M$ ; \item $(B_{q_j} (f) -a_{q_j} f)$ si on est sur l'espace caract\'eristique associ\'e \`a $0$ et $q_j ||M$ ; \item $(B_{q_j}^2 (f)-(a_{q_j} /{q_j} )B_{q_j} (f) +(1/q_j ) f)$ si on est sur l'espace caract\'eristique associ\'e \`a $0$ et $q_j$ ne divise pas $M$ ; \item $(-q_j B_{q_j} (f)+\alpha_{q_j} f)$ si $q_j$ ne divise pas $M$ et on est sur le sous-espace Ker$(U^2_{q_j} -a_{q_j} U_{q_j} +q_j )=$Ker$(U_{q_j} -\alpha_{q_j} {\mathrm{Id}})^2$. \end{enumerate} Alors % $$0=\langle e, t.g_1 \rangle =\lambda_1 \, \int_{0}^{i\infty} \prod_{j=1}^{m} {\cal P}_j (B_{q_j} (f))\, dz\, =\lambda_1 \prod_{j=1}^{m} {\cal P}_j (\frac{1}{q_j} ) \int_{0}^{i\infty} f\, dz$$ % $$=\lambda_1 \prod_{j=1}^{m} {\cal P}_j (\frac{1}{q_j} )\langle e,f\rangle \ ;$$ % or aucun des facteurs ${\cal P}_j (\frac{1}{q_j} )$ n'est nul, puisque respectivement de forme : % \begin{enumerate} \item 1, \item $(1/q_j -a_{q_j} )=(1/q_j \ \pm 1)$, \item $(1/{q_j^2}-a_{q_j}/{q_j^2} \, +1/q_j )=(1/q_j^2 )(q_j \, +1\, -a_{q_j} )$, ou encore : \item $(\alpha_{q_j} -1)=(\pm \sqrt{q_j} -1).$ \end{enumerate} (La seule non-nullit\'e qui ne soit pas triviale est la troisi\`eme : c'est la borne de Weil ($|a_{q_j} |\leq 2\sqrt{q_j} )$ qui la montre.) Donc $\lambda_1$ est nul, et en recommen\c{c}ant la m\^eme op\'eration, par r\'ecurrence tous les $\lambda_j$ sont nuls - donc $\tilde{t}$ l'est aussi. Ceci \'etant vrai pour toutes les restrictions $\tilde{t}$ de $t$ aux intersections de sous-espaces caract\'eristiques, $t$ est lui-m\^eme nul. $\Box$ \\ {\bf Fin de la preuve du th\'eor\`eme.} On a montr\'e que si $\langle e,f \rangle \neq 0$, ${\cal A}_{e,{\bf Q}} \cap R_f =0$, et r\'eciproquement il est \'evident que si $\langle e,f \rangle = 0$ alors ${\cal A}_{e,{\bf Q}} \cap R_f =R_f$. Comme $L(f,1)= 2\pi \langle e,f\rangle$, on peut donc \'ecrire : % $${\cal A}_{e,{\bf Q}} = \bigoplus_{G_{{\bf Q}} f /L(f,1)= 0} R_f \ ,$$ % et on a des isog\'enies : % $$J^e_0 ({\bf Q} )=(J_0 (N)/{\cal A}_e J_0 (N))({\bf Q} ) \to \prod_{R_f \not\subset {\cal A}_{e,{\bf Q}}} (J_0 (N)/({{\bf T}}_{{\bf Q}} /R_f )J_0 (N)) ({\bf Q} ) \rightarrow$$ % $$\rightarrow \prod_{G_{{\bf Q}} f /L(f,1)\neq 0} J_f ({\bf Q} ).$$ % Selon le th\'eor\`eme de Kolyvagin-Logachev, les $J_f ({\bf Q} )$ ci-dessus sont finies (\`a cause justement de la non-nullit\'e en 1 de la fonction $L(f,s)$, voir le th\'eor\`eme 0.3 de \cite{Kolyvagin}, compl\'et\'e par des r\'esultats de \cite{Murty-Murty} ou \cite{Bump-Friedberg-Hoffstein}). Et donc le quotient d'enroulement est bien de rang nul. $\Box$ \section{Espaces tangents et cotangents.} \subsection{Espace tangent \`a $J_0 (N)_{/{\bf Z} [1/N]}$ en z\'ero.} Soit $N$ un entier, fix\'e pour toute cette partie. On montre dans cette sous-section : \begin{theo} L'espace tangent \`a $J_0 (N)_{/{\bf Z} [1/N]}$ en z\'ero, ${\mathrm {Tan}}_0 (J_0 (N)_{/{\bf Z} [1/N]})$, est un ${{\bf T}}_{{\bf Z}} \otimes {\bf Z} [1/N]$-module libre de rang $1$, de base ${\frac{d}{dq} |}_0$. \end{theo} On va donner quatre lemmes pr\'eliminaires, desquels le th\'eor\`eme d\'ecoulera naturellement. Dans ce qui suit, pour all\'eger les notations, on notera $M$ le module ${\mathrm {Tan}}_0 (J_0 (N)_{/{\bf Z} [1/N]} )$, et $R$ l'anneau ${{\bf T}}_{{\bf Z}} \otimes {\bf Z} [1/N]$, qui est un ${\bf Z} [1/N]$-module libre de type fini (${{\bf T}}_{{\bf Z}}$ l'est sur ${{\bf Z}}$, puisqu'on peut le voir comme un sous-module des endomorphismes de $J_0 (N)_{/{\bf Z}}$). % \begin{lemm} Le $R$-module $M$ est fid\`ele. \end{lemm} {\bf Preuve.} Comme ni $R$ ni $M$ n'ont de ${\bf Z} [1/N]$-torsion, on peut voir le lemme en \'etendant les scalaires \`a ${\bf Q}$, et en remarquant qu'alors, le dual de $M\otimes {\bf Q}$ (comme ${\bf Q}$-espace vectoriel), qui est ${\mathrm {Cot}}_0 (J_0(N)_{{\bf Q}} )$, est bien ${\bf T} \otimes {\bf Q}$-fid\`ele, comme on le voit dans la partie pr\'ec\'edente. $\Box$ \begin{lemm} Pour tout id\'eal maximal ${\cal M}$ de $R$, on a $M/{\cal M}M\neq 0$. \end{lemm} {\bf Preuve.} Supposons qu'il existe un id\'eal maximal ${\cal M}$ tel que $M/{\cal M}M=0$. Puisque $M$ est un $R$-module fini, le faisceau associ\'e $\tilde{M}$ sur le sch\'ema affine noeth\'erien $X:=$Spec($R$) est coh\'erent ; notons $x$ le point correspondant \`a ${\cal M}$. On a : $$M/{\cal M}M=M\otimes_R R/{\cal M}=M\otimes_R R_{\cal M} /{\cal M}R_{\cal M}= M_{\cal M} /{\cal M}M_{\cal M} ,$$ donc le lemme de Nakayama dit que $M_{\cal M} =0$. Si $g$ est un point g\'en\'erique de $X$ (correspondant \`a l'id\'eal ${\cal P}$ de $R$) qui se sp\'ecialise en $x$, on a $\tilde{M}_g =0$ (comme localisation de $\tilde{M}_x$). Puisque $R$ est sans ${\bf Z}$-torsion, $g$ est au-dessus du point g\'en\'erique de ${\bf Z} [1/N]$ ; et puisque c'est un ${\bf Z} [1/N]$-module de type fini, Spec($R\otimes_{{\bf Z} [1/2N]} {\bf Q}$) est de dimension nulle. On a donc $R\otimes {\bf Q} \simeq \prod_{\cal Q} R_{\cal Q}$, o\`u le produit (fini) est pris sur les id\'eaux minimaux ${\cal Q}$ de $R$ : ce qui veut dire que $R_{\cal P}$ est isomorphe \`a un facteur de $R\otimes_{{\bf Z} [1/N]} {\bf Q}$. Mais alors la fid\'elit\'e de $M$ comme $R$-module contredit le fait que $\tilde{M}_g =0$. $\Box$ \begin{lemm} Pour tout id\'eal maximal ${\cal M}$ de $R$, l'\'element ${\frac{d}{dq} |}_0$ de $M$ est d'image non nulle dans le quotient $M/{\cal M}M$. \end{lemm} {\bf Preuve.} Soit ${\cal M}$ un id\'eal maximal de $R$. De la finitude comme ${\bf Z}$-module de ${{\bf T}}_{{\bf Z}}$ on d\'eduit d'abord que le sous corps premier de $F:=R/{\cal M}$ est n\'ec\'essairement fini, ${{\bf F}}_l$, et que $F$ en est une extension finie. Posons $\overline{R} :=R\otimes_{{\bf Z}} {\overline{{\bf F}}}_l$, et $\overline{M} :=M\otimes_{{\bf Z}} {\overline{{\bf F}}}_l$. On a la suite exacte : ${\cal M} \otimes_{{\bf Z}} {\overline{{\bf F}}}_l \to \overline{R} \to F\otimes_{{\bf Z}} {\overline{{\bf F}}}_l \to 0.$ Choisissons un plongement $i$ : $F \hookrightarrow {\overline{{\bf F}}}_l$, et d\'eduisons-en un morphisme de ${\overline{{\bf F}}}_l$-alg\`ebre : % $$i\otimes 1\, :\, F\otimes_{{\bf F}_l} {\overline{{\bf F}}}_l \to {\overline{{\bf F}}}_l .$$ Notons $\overline{{\cal M}}$ son noyau : on a la suite exacte $0\to \overline{{\cal M}} \to \overline{R} \to {\overline{{\bf F}}}_l \to 0$. On v\'erifie alors qu'on a : $$\left.\frac{d}{dq}\right|_0 \mapsto (M/{\cal M}M)\hookrightarrow (M /{\cal M}M) \otimes_{F} {\overline{{\bf F}}}_l \simeq \overline{M} / \overline{{\cal M}} \, \overline{M} .$$ Si $E$ est un ${\overline{{\bf F}}}_l$-espace vectoriel, notons ${E}^{\vee}$ son dual, et d\'esignons par ${\overline{M}}^{\vee} [\overline{{\cal M}} ]$ le sous-$\overline{R}$-module de ${\overline{M}}^{\vee}$ tu\'e par $\overline{{\cal M}}$. Avec ces notations, $$(\overline{M} /\overline{{\cal M}} \, \overline{M} )^{\vee} \simeq \overline{M}^{\vee} [\overline{{\cal M}} ]$$ en tant que $\overline{R}$-modules. (En effet, les formes ${\overline{{\bf F}}}_l$-lin\'eaires sur l'espace vectoriel quotient $(\overline{M} /\overline{{\cal M}} \, \overline{M} )$ s'identifient avec celles sur $\overline{M}$ dont le noyau contient $\overline{{\cal M}} \, \overline{M}$, et l'action de $\overline{R}$-module que l'on consid\`ere est : $(t.f)(\ )=f(t.\ ),$ ($t\in \overline{R}$, $f\in {\overline{M}}^{\vee}$).) On a donc enfin : $$\overline{M}^{\vee} [\overline{{\cal M}} ]=H^0 (X_0 (N )_{{\overline{{\bf F}}}_l}, {\Omega}^1 )[\overline{{\cal M}} ]\simeq S_2 (N )_{\overline {{\bf F}}_l} [\overline{{\cal M}} ].$$ On a montr\'e dans le lemme pr\'ec\'edent que $M/{\cal M}M$ \'etait non nul : soit donc $f$ une forme parabolique non nulle appartenant \`a $S_2 (N)_{\overline {{\bf F}}_l} [\overline{{\cal M}} ]$. Le corollaire III, 12.9 de \cite{Hartshorne} assure que $H^0 (X_0 (N)_{{\overline{{\bf F}}}_l} ,{\Omega}^1 )=H^0 (X_0 (N)_{/{\bf Z} [1/N]} ,{\Omega}^1 )\otimes {\overline{{\bf F}}}_l$, ce qui implique que, quitte \`a prendre un rel\`evement de $f$, on peut supposer qu'on est en caract\'eristique nulle, et m\^eme dans $S_2 (N)_{{\bf C}}$ : pour voir que les coefficients de $f$ v\'erifient la relation : % $$a_1 (T_n .f)=a_n (f)$$ % m\^eme si $f$ n'est pas une newform (cela r\'esulte par exemple de la formule (3.5.12) de \cite{Shimura}). On en d\'eduit que pour tout entier $n$, $\frac{d}{dq} |_0 (T_n .f) =a_n (f)$. Mais puisque $X_0 (N)_{{\overline {\bf F}}_l}$ est int\`egre et lisse, ses formes diff\'erentielles sont d\'efinies par leur d\'eveloppement de Fourier en l'infini : donc $\frac{d}{dq} |_0$ ne peut \^etre d'image nulle. $\Box$ % \begin{lemm} Pour tout id\'eal maximal ${\cal M}$ de $R$, $M/{\cal M}M$ est un $R/{\cal M}$-module libre de rang $1$, de base l'image de $\frac{d}{dq} |_0$. \end{lemm} {\bf Preuve.} Les r\'esultats pr\'ec\'edents montrent qu'il suffit maintenant de prouver que ${\mathrm{dim}}_{R/{\cal M}} (M/{\cal M}M)\leq 1$. Si $(l)$ est l'id\'eal premier de ${\bf Z}$ au-dessus duquel ${\cal M}$ se trouve, choisissons comme dans la preuve du pr\'ec\'edent lemme un plongement de $F=R/{\cal M}$ dans ${\overline{{\bf F}}}_l$. Toujours selon la preuve pr\'ec\'edente et avec les m\^emes notations, on a les \'equivalences : % $$((M/{\cal M}M)\ {\mathrm {est\ un\ }}F{\mathrm {-espace\ vectoriel\ de\ dimension\ }}1)$$ % \centerline{$\Updownarrow$} % $$((M/{\cal M}M)\otimes_{F} {\overline {{\bf F}}}_l \ {\mathrm {est\ un\ }} {\overline {\bf F}}_l {\mathrm {-espace\ vectoriel\ de\ dimension\ }} 1)$$ % \centerline{$\Updownarrow$} % $$((\overline{M} /\overline{{\cal M}} \, \overline{M})\ {\mathrm {est\ un\ }} {\overline {\bf F}}_l {\mathrm {-espace\ vectoriel\ de\ dimension\ }} 1)$$ % \centerline{$\Updownarrow$} % $$({\overline{M}}^{\vee} [\overline{{\cal M}} ]\ {\mathrm {est\ un\ }} {\overline {\bf F}}_l {\mathrm {-espace\ vectoriel\ de\ dimension\ }}1).$$ % Pour tout entier $n$, soit $a_n$ l'image dans $\overline{R} /\overline{{\cal M}} \simeq {\overline {\bf F}}_l$ de l'op\'erateur de Hecke $T_n$. Soit $f$ un \'el\'ement non nul de $H^0 (X_0 (N)_{{\overline {\bf F}}_l} , {\Omega}^1 )[\overline{{\cal M}} ]$ ; $f$ est une forme propre pour les op\'erateurs de Hecke : pour tout $n$, $T_n (f)=a_n f$. \`A une constante multiplicative non nulle pr\`es, le d\'eveloppement de Fourier de $f$ en l'infini est donc : $q+a_2 q^2 +a_3 q^3 +\cdots$. De m\^eme que plus haut, on d\'eduit du fait que $X_0 (N)_{{\overline {\bf F}}_l}$ soit int\`egre et lisse que toute forme diff\'erentielle sur $X_0 (N)_{{\overline {\bf F}}_l}$ est d\'efinie par son $q$-d\'eveloppement en l'infini ; donc $f$ engendre l'espace vectoriel $S_2(N)_{\overline {{\bf F}}_l}.\ \Box$ \\ % {\bf Preuve du th\'eor\`eme.} Montrons que le module $M':=M/(R.(\frac{d}{dq} ) |_0 )$ est nul. D'apr\`es le lemme pr\'ec\'edent, pour tout id\'eal maximal ${\cal M}$ de $R$, % $$M'/{\cal M}M'=M'\otimes_R R_{\cal M} /{\cal M}R_{\cal M}=0\ ;$$ % donc par le lemme de Nakayama, $M'_{\cal M}$ est nul. On peut alors appliquer le raisonnement de la preuve du lemme 4.4, et en conclure que le faisceau $\tilde{M'}$ sur Spec($R$) est de fibre nulle en chaque point, donc nul. Enfin, $M$ est un $R$-module libre puisque fid\`ele. $\Box$ % Notons au passage un corollaire de ce dernier th\'eor\`eme (voir aussi 3.7) : \begin{prop} L'espace des formes paraboliques $S_2 (N)_{{\bf Q}}$ est un ${\bf T}_{{\bf Q}}$~-~module libre de rang 1. \end{prop} {\bf Preuve.} Une application du corollaire 3.6 du chapitre pr\'ec\'edent donne que la ${\bf Q}$-alg\`ebre de Hecke ${\bf T}_{{\bf Q}}$ (pour ${\Gamma}_0 (N)$, en poids 2) est de Gorenstein. De plus, on vient de montrer que ${\mathrm {Tan}}_0 (J_0 (N)_{{\bf Q}} )$ \'etait comme ${\bf T}_{{\bf Q}}$-module isomorphe \`a ${\bf T}_{{\bf Q}}$ lui-m\^eme ; et comme ${\mathrm {Cot}}_0 (J_0 (N)_{{\bf Q}} )={\mathrm {Tan}}_0 (J_0 (N)_{{\bf Q}} )^{\vee}$, ${\mathrm {Cot}}_0 (J_0 (N)_{{\bf Q}} )$ est isomorphe \`a ${\bf T}_{{\bf Q}}$ soi-m\^eme comme ${\bf T}_{{\bf Q}}$-module. $\Box$ % \subsection{Espace tangent au quotient de la jacobienne.} Soit $I$ un id\'eal de ${\bf T}_{{\bf Z}}$, satur\'e ({\em i.e.} tel que ${\bf T}/I$ soit sans ${\bf Z}$-torsion). On d\'eduit de la jacobienne $J_0 (N)$ sur ${{\bf Z} [1/2N]}$ les sch\'emas ab\'eliens $J^I :=I.J_0 (N)$ et $J_I :=J_0 (N) /I.J_0 (N).$ Le but de cette section est de d\'emontrer le th\'eor\`eme suivant, voisin et corollaire de la celui de la section pr\'ec\'edente :% \begin{theo} L'espace tangent \`a $J_{I,{\bf Z} [1/2N]}$ en z\'ero, ${\mathrm {Tan}}_0 (J_{I,{\bf Z} [1/2N]})$, est un ${\bf T} /I \otimes {\bf Z} [1/2N]$-module libre de rang 1, de base l'image de $\frac{d}{dq} |_0$. \end{theo} On commence par travailler sur ${\bf Q}$ : \begin{prop} On a ${\mathrm {Tan}}_0 (J^I_{{\bf Q}} )=I.{\mathrm {Tan}}_0 (J_0 (N)_{{\bf Q}} )$. \end{prop} {\bf Preuve de la proposition.} Soit $(i_1 ,i_2 ,\dots ,i_n )$ un syst\`eme de g\'en\'erateurs de $I$ (${\bf T}$ est noeth\'erien puisque ${\bf Z}$-module de type fini) ; $J^I_{{\bf Q}}$ est donc l'image du morphisme : $$\oplus_{1\leq j\leq n} J_0 (N)_{\bf Q} \stackrel{\phi}{\rightarrow} J_0 (N)_{\bf Q}$$ d\'efini par la multiplication par ce syst\`eme. On en d\'eduit sur les espaces tangents :% $$\oplus_{1\leq j\leq n} {\mathrm {Tan}}_0 (J_0 (N)_{{\bf Q}} ) \stackrel{ \tilde{\phi}}{\rightarrow} {\mathrm {Tan}}_0 (J_0 (N)_{{\bf Q}} ).$$ La proposition d\'ecoule alors du lemme :% \begin{lemm} Pour tout morphisme $A \stackrel{\psi}{\rightarrow} B$ de vari\'et\'es ab\'eliennes sur ${\bf Q}$, on a ${\mathrm {Tan}}_0 (\psi (A)) =\tilde{\psi} ({\mathrm {Tan}}_0 (A)).$ \end{lemm} {\bf Preuve du lemme.} Le foncteur $A \mapsto {\mathrm {Tan}}_0 (A)$ de la cat\'egorie des groupes alg\'ebriques commutatifs sur ${\bf Q}$ dans celle des ${\bf Q}$-espaces vectoriels, est exact \`a gauche. Mais il est aussi exact \`a droite : de la suite exacte $0\to A\to B\to C\to 0$ se d\'eduit celle sur les tangents : $0\to {\mathrm {Tan}}_0 (A)\to {\mathrm {Tan}}_0 (B)\to {\mathrm {Tan}}_0 (C)$ ; si les dimensions des vari\'et\'es sont $a,\ b$ et $c$ respectivement, on a $a+c=b$, et comme les dimensions des espaces tangents associ\'es sont les m\^emes (lissit\'e des groupes alg\'ebriques sur ${\bf Q}$), la derni\`ere fl\`eche de la derni\`ere suite est surjective. Si donc on a un morphisme $\psi$ :% $$0\to {\mathrm {ker}}(\psi )\to A\stackrel{\psi}{\rightarrow} \psi (A)\to 0,$$ % on a bien % $${\mathrm {Tan}}_0 (\psi (A))={\mathrm {Tan}}_0 (A)/{\mathrm {Tan}}_0 ({\mathrm {ker}}(\psi ))= \tilde{\psi} ({\mathrm {Tan}}_0 (A) ).\ \Box$$ % Consid\'erons maintenant le probl\`eme sur ${\bf Z} [1/2N]$.\\ % {\bf Preuve du th\'eor\`eme.} De la suite exacte de vari\'et\'es ab\'eliennes sur ${\bf Q}$ :% $$0\to J^I_{{\bf Q}} \to J_0 (N)_{{\bf Q}} \to J_{I,{\bf Q}} \to 0,$$ % on d\'eduit par propri\'et\'e universelle des mod\`eles de N\'eron un complexe de sch\'emas ab\'eliens : % $$0\to J^I_{{\bf Z} [1/2N] } \to J_0 (N)_{{\bf Z} [1/2N]} \to J_{I,{\bf Z} [1/2N]} \to 0,$$ % qui est en r\'ealit\'e une {\em suite exacte}, d'apr\`es un r\'esultat de Raynaud (voir section 1, proposition 1.2 de \cite{rational}). Ce r\'esultat assure encore que la suite :% $$0\to {\mathrm {Tan}}_0 (J^I_{{\bf Z} [1/2N] }) \to {\mathrm {Tan}}_0 (J_0 (N)_{{\bf Z} [1/2N]} )\to {\mathrm {Tan}}_0 (J_{I,{\bf Z} [1/2N]} )\to 0$$ % est exacte elle aussi (m\^eme r\'ef\'erence, corollaire 1.1). Or on voit par la section pr\'ec\'edente que ${\mathrm {Tan}}_0 (J_0 (N)_{{\bf Z} [1/2N]} )$ est un ${\bf T} \otimes {\bf Z} [1/2N]$-module libre de rang 1, de base $\frac{d}{dq} |_0$ ; il existe donc un id\'eal $I'$ de ${\bf T} [1/2N]$ tel que la suite pr\'ec\'edente de ${\bf T} [1/2N]$-modules soit isomorphe \`a :% $$0\to I'\to {\bf T} \otimes {\bf Z} [1/2N] \to {\bf T} \otimes {\bf Z} [1/2N] /I'\to 0.$$ % Mais puisque sur ${\bf Q}$ on a $I_{{\bf Q}} =I'_{{\bf Q}}$, que $I$ est par hypoth\`ese satur\'e, et que $I'$ l'est aussi (puisque ${\mathrm {Tan}}_0 (J_{I,{\bf Z} [1/2N]} )\simeq {\bf T} [1/2N]/I'$ est sans torsion), on a bien $I'=I_{{\bf Z} [1/2N]}$. Donc % $${\mathrm {Tan}}_0 (J_{I,{\bf Z} [1/2N]} )\simeq {({\bf T} /I)}_{{\bf Z} [1/2N]} .\ \Box $$ % \subsection{Preuve de la proposition ``crit\`ere de Kamienny''.} On rappelle la situation dans laquelle on s'est mis avec notre probl\`eme initial de borne pour la torsion de courbes elliptiques (voir la section 1, dont on reprend les hypoth\`eses et notations). Rappelons le morphisme naturel $f_d$ : ${X_0 (p^n )}^{(d)}_{\rm lisse} \rightarrow J_0^e $, normalis\'e par ${\infty }^{(d)} \mapsto 0$. On a d\'efini en (1.2) un point $j'^{(d)}$, \`a valeur dans ${\bf Z}$, de $X_ 0 (p^n )^{(d)}_{\rm lisse}$, qui croise $\infty^{(d)}$ sur la fibre en $l$. On va d'abord montrer que ceci implique que $f_d$ n'est pas une immersion formelle au point ${\infty}^{(d)}_{{\bf F}_l}$, en suivant l'expos\'e de \cite{Oest}. Soit $S$ un sch\'ema quelconque. Si $\phi$ : $X\to Y$ est un morphisme de $S$-sch\'emas noeth\'eriens, on dit que c'est une {\em immersion formelle} en un point $x$ de $X$ si l'application ${\hat{\cal O}}_{Y,\phi (x)} \to {\hat{\cal O}}_{X,x}$ d\'eduite de $\phi$ entre compl\'et\'es d'anneaux locaux est surjective. Si $s$ est l'image dans $S$ de $x$ (et $y$), on montre que ceci \'equivaut \`a ce que la restriction de $\phi$ aux fibres en $s$ soit une immersion formelle en $x$. On d\'emontre ensuite dans \cite{Oest} % (lemme 5.1) : % \begin{lemm} Supposons que $X$ soit s\'epar\'e, que $\phi$ : $X\to Y$ soit une immersion formelle en $x$. Supposons qu'il existe un sch\'ema int\`egre noeth\'erien $T$ et deux points $p_1$ et $p_2$ de $X$ \`a valeur dans $T$, tel qu'en un point $t$ de $T$ on ait $x=p_1 (t)=p_2 (t)$. Si de plus on a $\phi \circ p_1 =\phi \circ p_2$, alors $p_1 =p_2$. \end{lemm} Ce qui nous permet de montrer : \begin{theo} Le morphisme $f_d$ n'est pas une immersion formelle en ${\infty}^{(d)}_{ {\bf F}_l}$. \end{theo} {\bf Preuve.} Puisque $j'^{(d)}$ croise $\infty^{(d)}$ sur la fibre en $l$, le point $f_d (j'^{(d)} )$ de $J_0^e ({{\bf Z}}_l )$ se r\'eduit en $0$ au-dessus de $l$. Ce point est de torsion, car $J_0^e ({\bf Q} )$ l'est. Puisque $l>2$, le ``lemme de sp\'ecialisation'' (voir par exemple \cite{Bas}, lemme 3.3) assure que $f_d (j'^{(d)} )$ est nul dans tout $J_0^e ({{\bf Z}}_{(l)} )$. Mais alors, on peut appliquer le lemme pr\'ec\'edent et en conclure que si $f_d$ \'etait une immersion formelle en ${\infty}^{(d)}_{{\bf F}_l}$, on aurait $j'^{(d)} = {\infty}^{(d)}$ - ce qui contredit \'evidemment l'interpr\'etation modulaire de ces points. $\Box$ La courbe $X_0 (N)$ a en % l'infini la coordonn\'ee formelle $q$ ; donc $X_0 (N)^d$ a les coordon\-n\'ees formelles $q_1,\dots ,\, q_d$ au point $(\infty ,\dots ,\, \infty )$, et les fonctions sym\'etriques \'el\'ementaires $\sigma_1 =q_1 + \cdots +q_d ,\, \dots ,\ \sigma_d =q_1 \cdots q_d$ sont des coordon\-n\'ees formelles au point ${\infty}^{(d)}$ de $X_0 (N)^{(d)}$, donc $d\sigma_1 ,\dots ,\, d\sigma_d$ forment une base de ${\mathrm {Cot}}_{\infty^{(d)}} (X_0 (N)^{(d)} )$. On a alors le lemme : % \begin{lemm} Soit $\omega$ un \'el\'ement de ${\mathrm {Cot}}_0 (J_0^e )$. Alors $f_1^* (\omega )$ est une forme diff\'e\-ren\-tielle sur $X_0 (N)_{/{\bf Z} [1/N]}$. Notons $(\sum_{n\geq 1} a_n q^n )(dq/q)$ son d\'eveloppement de Fourier \`a l'infini. On a : $$f_d^* (\omega )=a_1 d\sigma_1 -a_2 d\sigma_2 + \cdots +(-1)^{d-1} d\sigma_d \ \in {\mathrm {Cot}}_{\infty^{(d)}} (X_0 (N)^{(d)}_{/{\bf Z} [1/N]} ).$$ \end{lemm} Pour la {\bf preuve}, voir \cite{Bas}, lemme 4.2. On en d\'eduit : \begin{cor} L'application tangente \`a $f_d$ en $\infty^{(d)}$ envoie $\frac{d}{d\sigma_1} ,\, \dots ,\, \frac{d}{d\sigma_d}$ sur $T_1 (\frac{d}{dq} |_0 ) , -T_2 (\frac{d}{dq} |_0 ),\, \dots ,\, (-1)^{d-1} T_d (\frac{d}{dq} |_0 )$ respectivement. \end{cor} {\bf Preuve.} La relation $a_1 (T_n f)=a_n (f)$ pour toute forme modulaire (voir la preuve du lemme 4.5) et le lemme pr\'ec\'edent suffisent \`a conclure. $\Box$ Et on arrive au th\'eor\`eme qu'on veut finalement montrer : \begin{theo} On a \'{e}quivalence entre : \begin{enumerate} \item $f_d$ est une immersion formelle en ${\infty}_{{{\bf F}}_l}^{(d)}$, et \item $T_1 e,...,T_d e$ sont ${\bf F} _l$-lin\'{e}airement ind\'{e}pendants dans ${\bf T} e/l{\bf T} e$. \end{enumerate} De plus, ces deux conditions sont satisfaites si l'est : \begin{enumerate} \item[3.] $T_1 \{ 0,\infty \} ,...,T_{d.s} \{ 0,\infty \}$ sont ${\bf F}_l$-lin\'{e}airement ind\'{e}pendants dans l'espace vectoriel $H_1 (X_0 (p^n ),\, {\mathrm {pointes }}\, ;\, {\bf Z} )\otimes {\bf F}_l$ (o\`u $s$ d\'{e}signe le plus petit nombre premier diff\'erent de $p$). \end{enumerate} \end{theo} Avant de passer \`a la d\'emonstration, un lemme pr\'eliminaire :% \begin{lemm} Soit $N$ un entier positif strict, et $r$ un second, premier au premier. Alors, sur la courbe modulaire $X_0 (N)_{{\bf C}}$, on a $T_r .0={\sigma}_1 (r). 0$ et $T_r .\infty ={\sigma}_1 (r).\infty ,$ o\`u comme d'habitude ${\sigma}_1 (r)$ est la somme des diviseurs de $r$. \end{lemm} {\bf Preuve du lemme.} On se place ``en projectif'' : on identifie $X_0 (N)_{{\bf C}}$ avec $({\bf H} \cup {{\bf P}}^1 ({\bf Q} ))/{\Gamma}_0 (N)$ et on d\'esigne par $({x\atop y})$ l'\'el\'ement $x/y$ de ${{\bf P}}^1 ({\bf Q} )$. En particulier, les points $0$ et $\infty$ seront \'ecrits $({0\atop 1})$ et $({1\atop 0})$ respectivement. On peut voir les op\'erateurs de Hecke comme des correspondances sur $X_0 (N)_{{\bf C}}$ (voir la section 5), qui v\'erifient : % $$T_r .0=\sum_{\begin{array}{c} 1\leq \delta |r\\ 0\leq \beta <\delta \end{array} } \left( \begin{array}{cc} \frac{r}{\delta} & -\beta \\ 0 & \delta \end{array} \right) .\left( \begin{array}{c} 0 \\ 1 \end{array} \right) =\sum_{\begin{array}{c} 1\leq \delta |r\\ 0\leq \beta <\delta \end{array} } \left( \begin{array}{r} -\beta \\ \delta \end{array} \right).$$ Or, chacun de ces $({-\beta \atop \delta})$ est conjugu\'e par un \'el\'ement de ${\Gamma}_0 (N)$ \`a $({0\atop 1})$ : en effet, quitte \`a remplacer $\beta$ et $\delta$ par leur quotient par leur p.g.c.d., on peut les supposer premiers entre eux, et choisir des entiers $a$ et $b$ tels que $a\delta -bN \beta =1$ ; alors on a :% $$\left( \begin{array}{cc} a & -\beta \\ bN & \delta \end{array} \right).\left( \begin{array}{c} 0 \\ 1 \end{array} \right) =\left( \begin{array}{c} -\beta \\ \delta \end{array} \right) ,$$ % et la matrice de gauche est bien de ${\Gamma}_0 (N)$. De la m\^eme fa\c{c}on, % $$T_r .\infty =\sum_{\begin{array}{c} 1\leq \delta |r\\ 0\leq \beta <\delta \end{array} } \left( \begin{array}{cc} \frac{r}{\delta} & -\beta \\ 0 & \delta \end{array} \right) .\left( \begin{array}{c} 1 \\ 0 \end{array} \right) =\sum_{\begin{array}{c} 1\leq \delta |r\\ 0\leq \beta <\delta \end{array} } \left( \begin{array}{c} \frac{r}{\delta} \\ 0 \end{array} \right) = \sigma_1 (r) .\infty \ \Box$$ % {\bf Preuve du th\'eor\`eme.}\ D'abord l'\'equivalence des deux premiers points. % La propri\'et\'e pour un morphisme d'\^etre une immersion formelle en un point est, formulation duale de celle sur les espaces cotangents, que l'application induite par ce morphisme sur les tangents correspondants est une injection. Le corollaire 4.16 dit donc que $f_d$ est une immersion formelle en ${\infty}^{(d)}_{{\bf F}_l}$ si et seulement si les vecteurs $T_i ({\frac{d}{dq}}|_0 )$, $1\leq i\leq d$ sont lin\'eairement ind\'ependants % dans ${\mathrm {Tan}}_0 (J_0^e % ({\overline{{\bf F}}}_l ))$. Mais le th\'eor\`eme 4.9 assure que cet espace tangent est un ${\bf T} /{\cal A}_e \otimes_{{\bf Z}} {\overline{{\bf F}}}_l$-module libre de rang 1 de base $\frac{d}{dq} |_0$, d'o\`u l'\'equivalence des propri\'et\'es {\em 1.} et {\em 2.} du th\'eor\`eme. Prouvons la derni\`ere implication. On rappelle qu'il y a un isomorphisme de ${\bf R}$-espaces vectoriels : % $$\left\{ \begin{array}{rcl} H_1 (X_0 (p^n ) \, ;\, {\bf Z} )\otimes {\bf R} & \to & {\rm Hom}_{{\bf C} } \left( H^0 (X_0 (p^n )\, ;\, {\Omega }^1 ),\ {\bf C} \right) \\ \gamma \otimes 1 & \mapsto & \left( \omega \mapsto \int_{\gamma } \omega \right) . \end{array} \right.$$ % L'int\'egration permet d'en d\'eduire un morphisme de $H_1 (X_0 (p^n ),\, {\rm pointes}\, ;\, {\bf Z} )$ dans $H_1 (X_0 (p^n ) \, ;\, {\bf Z} )\otimes {\bf R}$, et comme on l'a dit, l'\'el\'ement d'enroulement $e$, qui est l'image par ce morphisme du symbole modulaire $\{ 0,\infty \}$, est \'el\'ement de $H_1 (X_0 (p^n ) \, ;\, {\bf Q} )$. Maintenant, c'est avec les symboles modulaires, vivant dans l'homologie relative aux pointes, qu'on sait travailler ; mais c'est dans l'homologie absolue qu'il nous faut une ind\'ependance ${\bf F}_l$-lin\'eaire. Pour s'y ramener, on fait agir l'op\'erateur $I_s :=T_s -{\sigma}_1 (s)$ sur $\{ 0, \infty\}$. En effet, puisque $s$ est premier au niveau, le lemme pr\'ec\'edent dit que $T_s$ envoie les pointes $0$ et $\infty$ sur $(s+1)$-fois elles-m\^emes, donc $I_s$ pousse bien $\{ 0,\infty \}$ dans $H_1 (X_0 (p^n ) \, ;\, {\bf Z} )$ vu comme sous-module de $H_1 (X_0 (p^n ),\, {\rm pointes} \, ;\, {\bf Z} )$. Consid\'erons aussi $H_1 (X_0 (p^n )\, ;\, {\bf Z} )$ comme un sous-module de $H_1 (X_0 (p^n )\, ;\, {\bf Q} )$. % Supposons que la propri\'et\'e {\em 2.} du th\'eor\`eme ne soit pas v\'erifi\'ee. Il existe donc une relation de d\'ependance : % $$\overline{\lambda_1} T_1 e+\cdots +\overline{\lambda_k} T_k e =0 \in {\bf T} e/l{\bf T} e,$$ % pour un $k\leq d$ tel que $\overline{\lambda_k}$ soit non nul. On la rel\`eve en % $$\lambda_1 T_1 e+\cdots +\lambda_k T_k e = l\, x \in l{\bf T} e \subseteq H_1 (X_0 (p^n )\, ;\, {\bf Q} ),$$ % on la multiplie par $I_s$ pour obtenir % $$I_s (\sum_{i=1}^{k} {\lambda}_i T_i e)= \sum_{i=1}^{k} {\lambda}_i T_i (I_s e)={\lambda}_k T_{sk} \, e+\sum_{i=1}^{sk-1} {\mu}_i T_i e =l\, I_s x \in l {\bf T} e.$$ % (En effet, la d\'efinition des op\'erateurs de Hecke (par exemple avec la s\'erie formelle) donne que $T_s .T_n =T_{s.n} +$ (combinaison lin\'eaire de termes d'indice plus petit).) Les formes lin\'eaires que d\'efinissent $I_s \{ 0, \infty \}$ et $I_s e$ par l'int\'egration sont les m\^emes, et puisqu'on peut voir ces deux \'el\'ements comme appartenants \`a $H_1 (X_0 (p^n ) \, ;\, {\bf R} )$, ils sont en fait \'egaux. On en d\'eduit une relation de d\'ependance lin\'eaire : % $${\lambda}_k T_{sk} \, \{ 0,\infty \} +(\sum_{i=1}^{sk-1} {\mu}_i T_i \{ 0, \infty \} )=l\, I_s x \in lH_1 (X_0 (p^n ),\, {\rm pointes} \, ;\, {\bf Z} ),$$ % et en consid\'erant son image dans $H_1 (X_0 (p^n ),\, {\rm pointes} \, ; \, {\bf Z} )\otimes {{\bf F}}_l$, une contradiction avec la propri\'et\'e {\em 3.} du th\'eor\`eme. $\Box$ \section{Lemme combinatoire.} Le but de cette partie est de prouver la proposition-cl\'e de la d\'emonstration g\'en\'erale :\\ {\bf Proposition 1.7}\ {\it Soit $p$ un nombre premier. Posons $C=\sqrt{65}$ si $p$ est diff\'erent de $2$, et $C=\sqrt{129}$ si $p$ est $2$. Notons $s$ le plus petit nombre premier diff\'erent de $p$. Si $p^n>C^2.(sd)^6$, alors les $T_i \{ 0,\infty \} ,\ 1\leq i\leq sd$ sont ${\bf F}$-lin\'{e}airement ind\'{e}pendants dans le ${\bf F}$-espace vectoriel $H_1 (X_0 (p^n ),\, {\mathrm {pointes }}\, ;\, {\bf Z} )\otimes {\bf F}$ pour tout corps ${\bf F}$.} \subsection{Notations et rappels.} On identifie $\Gamma_0(p^n) \backslash SL_2 ({\bf Z})$ \`a ${{\bf P}}^1 \left( {{\bf Z}}/{p^n}{{\bf Z}} \right)$ avec : % $$ {\Gamma }_0 ({p^n})\left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \mapsto (\overline{c} ,\overline{d} ) = (c\pmod{p^n}, d\pmod{p^n}).$$ % L'application de ${\Gamma }_0 (p^n )\backslash SL_2({\bf Z})$ vers $H_1 ({X_0} ({p^n}),\ {\rm pointes} ; {\bf Z} ) $ : $ g \mapsto \{ g\cdot 0,g\cdot \infty \} $ s'identifie alors a une application de ${{\bf P}}^1 \left( {\bf Z} / {{p^n}{\bf Z}} \right)$ vers la m\^{e}me chose, qu'on note $\xi $ : $$\xi (\overline{w} ,\overline{t} ) = \left\{ \overline{ \left( \begin{array}{cc} a & b \\ w & t \end{array} \right) \cdot 0 } , \overline{\left( \begin{array}{cc} a & b \\ w & t \end{array} \right) \cdot \infty } \right\} = \left\{ \frac{b}{t} , \frac{a}{w} \right\} , $$ avec $w,t$, rel\`{e}vements dans ${\bf Z}$ de $\overline{w} $ et $\overline{t} \, \in {\bf Z} / {p^n {\bf Z}} $ et $a,b\, \in {\bf Z} $ tels que $\left( \begin{array}{cc} a & b \\ w & t \end{array} \right)$ soit dans $SL_2 ({\bf Z})$. On sait de plus qu'on a : $$T_r \{ 0,\infty \} = \sum_{\begin{array}{c} 0 \leq w < t \\ 0 \leq v < u \\ ut-vw =r \end{array} } \xi (\overline{w} ,\overline{t} )\ ,$$ % o\`{u} on pose $\xi (\overline{w} ,\overline{t} ) = 0$ si pgcd$(w,t,p) >1$ (voir \cite{Artin}, th\'eor\`eme 2 et proposition 20, ou la d\'emonstration du lemme 2 de \cite{merel}). Soit $\sigma = \overline{\left( \begin{array}{rc} 0 & 1 \\ -1 & 0 \end{array} \right) } $ et $\tau = \overline{\left( \begin{array}{cc} 0 & -1 \\ 1 & -1 \end{array} \right) }$. On choisit pour repr\'{e}sentants de ${{\bf P} }^1 ({\bf Z} / {p^n} {\bf Z} )$ : $\{ (R_1 ,1),\ R_1 $ un syst\`{e}me de repr\'{e}sentants de ${\bf Z} /{p^n}{\bf Z} \}\ \cup \{ (1,p.R_2 ),\ R_2 $ un syst\`{e}me de repr\'{e}sentants de ${\bf Z} /{p^{n-1}} {\bf Z} \}$. On note $w/t $ au lieu de $(\overline{w} ,\overline{t} )$, souvent. On fait agir $SL_2 ({\bf Z} )$ sur ${{\bf P} }^1 ({\bf Z} /{p^n} {\bf Z} )$ \`{a} droite, comme d'habitude. En particulier : $$(\overline{w} ,\overline{t} ) \cdot \sigma = (\overline{w} ,\overline{t} ) \overline{\left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right) } =(\overline{-t} ,\overline{ w} ),$$ % et de m\^{e}me $(w / t )\cdot \tau = -t / (w+t ) $. On note encore ${\Sigma }_r = \{ (\overline{w} ,\overline{t} ),\ 0\leq w<t \ /\ $il existe $(v,u),\ 0\leq v<u\ ,\ 0\leq (ut-vw)\leq r\} \backslash \{ (\overline{1} , \overline{r} )\} $, et ${\bf Z} [ {{\bf P} }^1 ({\bf Z} /{p^n} {\bf Z} ) { ] }^{\sigma }$ (respectivement, ${\bf Z} [ {{\bf P} }^1 ({\bf Z} /{p^n} {\bf Z} ) {] }^{\tau } $) d\'{e}signe l'ensemble des \'{e}l\'{e}ments de ${\bf Z} [ {{\bf P} }^1 ({\bf Z} /{p^n} {\bf Z} ) ]$ stables par l'action de $\sigma$ (respectivement, $\tau $). Le symbole $\sum_{\sigma }^{ } $ d\'{e}signera une somme \`{a} valeurs dans le ${\bf Z}$-module ${\bf Z} [ {{\bf P} }^1 ({\bf Z} /{p^n} {\bf Z} ) { ] }^{\sigma }$, et de m\^{e}me avec~$\tau $. On a la suite exacte (voir par exemple \cite{Mer}) : $${\bf Z} [ {{\bf P}}^1 \left( {\bf Z} / {p^n}{\bf Z} \right) { ] }^{\sigma } \times {\bf Z} [{{\bf P}}^1 \left( {\bf Z} / {p^n}{\bf Z} \right) { ] }^{\tau } \stackrel{{\phi}_1}{\rightarrow} {\bf Z} [ {{\bf P}}^1 ({\bf Z} / {p^n} {\bf Z} ) ] \stackrel{{\phi}_2}{\rightarrow} H_1 (X_0 ({p^n}),\, {\rm pointes}\, ;\, {\bf Z} ) \to 0,$$ avec ${\phi}_1$ : $(\Sigma {\alpha}_x x,\Sigma {\beta }_x x ) \mapsto (\Sigma {\alpha }_x x + \Sigma {\beta }_x x),$ et ${\phi}_2$ : $\Sigma {\lambda}_x x \mapsto \Sigma {\lambda}_x .\xi (x)$. \subsection{Preuve de la proposition.} Posons $D=sd$. Supposons $p^n >C^2 .D^6$, et que pour un $r\leq D$, on ait une relation de liaison : $\sum_{i=1}^{r} {\lambda }_i {T_i} \{ 0,\infty \} =0$ dans $H_1 (X_0 ({p^n} ),\, {\rm pointes}\, ;\, {{\bf F} }_l )$. On va montrer que ${\lambda }_r$ est nul : ce qui suffira \`a la preuve de la proposition. Ce qui pr\'{e}c\`{e}de permet d'\'{e}crire : $$\sum_{i=1}^{r} {\lambda }_i T_i \{ 0,\infty \} =0 \iff {\lambda }_r .\left( 1/r \right) + \sum_{{\Sigma }_r} {\mu }_{(\overline{w} ,\overline{t} )} \left( w/t \right) = \sum_{\sigma } {\alpha }_x (x) -\sum_{\tau } {\beta }_{x} (x).$$ De m\^{e}me que dans \cite{merel} on prouve avec l'aide de Fouvry le : \begin{lemm} Soit $A$ et $B$ deux intervalles de $\{ 1,2,...,{p^n}-1 \}$ tel que $$|A|.|B| \geq C'.{p^{3n/2}} ,$$ o\`u $C'=8$ si $p$ est impair, et $C'=8\sqrt{2}$ si $p=2$. Alors il existe $y\in A$ et $z\in B$ tel que $y.z=\ -1\pmod{p^n}$. \end{lemm} (On a prouv\'e une version moins bonne de ce lemme, utilisant une constante $C=(512{\pi}^2 )/(2\sqrt{2} -1)$ ; la forme ici utilis\'ee de ce lemme est obtenue en optimisant les calculs par Oesterl\'e dans \cite{Oesterle}.) On consid\`{e}re le graphe de ${{\bf P} }^1 ({\bf Z} /{p^n}{\bf Z} )$ dont les ar\^{e}tes sont l'action de $\sigma $ et $\tau $ (voir {\bf Figure 1}) : on a $(\tau \sigma )= \overline{\left( \begin{array}{cc} -1 & 0 \\ -1 & -1 \end{array} \right)} ,$ donc $(\overline{w} ,\overline{t} )\cdot \tau \sigma = ( \overline{w} ,\overline{t} ) +\overline{1}$ (et de m\^eme $(\overline{w} , \overline{t} )\cdot \sigma {\tau }^2 =(\overline{w} ,\overline{t} )- \overline{1} ).$ \setlength{\unitlength}{0.7cm} \begin{picture}(26,20)(3,0) \thicklines \put (10,16){\framebox (2,2){$(1,r)$}} \thinlines \put (9.7,17){\vector (-1,-1){1.8}} \put (8,16){${\tau }^2$} \put (5,14){\framebox (3,1){$(-r-1,1)$}} \put (7.3,13.7){\vector (1,-1){1}} \put (8,13.2){$\sigma$} \put (8,11){\framebox (2.5,1){$(1,r+1)$}} \put (7,10.5){${\tau}^2$} \put (7.7,10.7){\vector (-1,-1){1}} \put(6.5,9){.} \put(7,8.5){.} \put(7.5,8){.} \put(7.9,7.5){.} \put(8,6.5){\vector (-1,-1){1}} \put (6.8,6.2){${\tau}^2$} \put (5,4){\framebox (2,1){($y,1)$}} \put(10,5.5){$\sigma$} \thicklines \put (8,4.5){\vector (4,1){4}} \thinlines \put (13.2,16.8){$\sigma$} \put (15,15){\framebox (2,1){$(-r,1)$}} \put (12.6,17){\vector (2,-1){2}} \put (17,14.7){\vector (-1,-2){1}} \put (16,11){\framebox(2,1) {...}} \put (18.3,10.7){\vector (1,-2){0.9}} \put (19,8.2){.} \put(19,7.7){.} \put(19,7.2){.} \put(18,7){\vector(-3,-1){2.7}} \put(13,5){\framebox(2,1){$(z,1)$}} \put (4,15){\vector(0,-1) {10.5}} \put(2.5,9){${\cal A}$} \put(4,3.5){.} \put(4,3){.} \put(4,2.5){.} \put(20.5,16){\vector (0,-1){11}} \put(21,10){${\cal B}$} \put(20.5,4.5){.} \put(20.5,4){.} \put(20.5,3.5){.} \put(11.5,2){\bf Figure 1} \end{picture} On va montrer qu'on a sur ce graphe, ``de part et d'autre'' de $(\overline{1} ,\overline{r} )$ (c'est-\`{a}-dire, contenant $(\overline{1} ,\overline{r} ) \cdot {\tau }^2 = (\overline{-r-1} ,\overline{1} )$ et $(\overline{1} ,\overline{r} )\cdot \sigma =(\overline{-r} ,\overline{1} )$ respectivement), deux chemins ${\cal A}$ et ${\cal B}$ ne rencontrant pas d'\'{e}l\'{e}ments de ${\Sigma }_r$, (les autres \'{e}l\'{e}ments du graphe intervenants dans $\sum {\lambda }_i T_i \{ 0,\infty \} )$, et qui contiennent des intervalles de cardinal sup\'{e}rieur \`{a} $({{p^n}/D})-D-2$ et $({{p^n}/{D^2}})-2$ respectivement. Alors par le lemme de th\'eorie analytique des nombres, pour $$(({{p^n}/D})-D-2).(({{p^n}/{D^2}})-2)\geq C'.{p^{3n/2}} ,\ i.e.$$ $${p^n}\geq C^2 . {D^6} ,$$ on aura dans ${\cal A}$ et dans ${\cal B}$ respectivement des \'el\'ements $y$ et $z$ tel que $y\cdot \sigma =\frac{-1 }{y} =z$ (on explicite ces calculs, et notament le passage de $C'$ \`a $C$, \`a la fin de la section). De plus $y\cdot \sigma $ sera un \'{e}l\'{e}ment de ${\cal A}$, puisque ce chemin est de forme : $$\begin{array}{rcl} ...\to (\overline{a} ,\overline{1} ) \stackrel{\tau }{\rightarrow } & ( \overline{-1} ,\overline{a+1} ) & \stackrel{\sigma }{\rightarrow} (\overline{a+1} ,\overline{1} ) \to... \\ \longrightarrow & +\overline{1} =\tau \sigma & \longrightarrow \end{array}$$ Les deux chemins se ``rencontrent'', donc. Or on a, pour tout \'el\'ement $x$ du graphe, ${\mu }_x = {\alpha }_x - {\beta }_x$ par ce qui pr\'{e}c\`{e}de. Puisque les deux chemins ne rencontrent pas $\Sigma _r$, si $x$ en est, on a ${\mu }_x =0$ donc ${\alpha }_x ={\beta }_x$. De plus, on parcourt ces chemins en faisant agir $\sigma $ ou $\tau $ ; donc si $x'=x\cdot \sigma $, ${\alpha}_{x'}={\alpha }_{x\cdot \sigma }={\alpha }_x ={\beta }_x = {\beta }_{x'}$, de m\^{e}me avec $\tau $ - donc ${\alpha }_x ={\beta }_x \equiv {\alpha }_{-r} $ sur les deux chemins. Mais $${\lambda }_{\overline{r} }= {\mu }_{\frac{1}{r} } = {\alpha }_{\frac{1}{r} } -{\beta }_{\frac{1}{r} } = {\alpha }_{\frac{1}{r} \cdot \sigma } - {\beta }_{\frac{1}{r} \cdot {\tau }^2} =0.$$ Montrons donc l'existence de ces chemins. (Dans tous les calculs qui suivent, on confond l'\'ecriture d'un entier $w$ et de sa r\'eduction $\overline{w}$, pour all\'{e}ger les notations). $${ }$$ \underline{Premier chemin : ${\cal A}$} partant de $\frac{1}{r} \ {\tau }^2 =-r-1.$ \\ 1) Si $\frac{w}{t} -(-r-1)= \overline{a}$, avec $a$ choisi dans $\{ -{p^n}+1,...,-1,0 \}$, et $\overline{a} $ : classe de $a$ mod $\ {p^n}$ ($\iff w+t(r+1)=at+b.{p^n}$, $b\in {\bf Z} $).\\ \underline{Si $b=0$} : $\ t(r+1)-at=-w $ : incompatibilit\'{e} de signes. \\ \underline{Si $b \neq 0$} : $\ |a|=\frac{1}{t} |b.{p^n}-t(r+1)-w|\geq \frac{({p^n}-D(D+1)-D)}{D} \geq \frac{p^n}{D} -D-2 $ (on a en effet : det$\left( \begin{array}{cc} u & v \\ w & t \end{array} \right)=k\leq r,\ u>v\geq 0,\ t>w\geq 0$, donc : $k=ut-vw\geq ut-(u-1)(t-1)$ et $u+t-1\leq r,\ t\leq r-u+1 \leq r \leq D$).\\ 2) Si $(\frac{w}{t} )\sigma -(-r-1)= \frac{-t}{w} +r+1=\overline{a} ,\ a\in \{-{p^n}+1,...,-1,0\}\ $; $\ -t+w(r+1)=aw+b{p^n}\ $; \\ \underline{Si $b=0$} : $\ w(r+1-a)=t\ ;$ mais $(r+1-a)\geq r+1,$ et $0\leq w<t \leq r$ : contradiction.\\ \underline{ Si $b\neq 0$}, $\ |a|\geq ( \frac{{p^n}-D(D+1)+D}{D} )\geq \frac{p^n}{D} -D$.\\ On peut donc ``reculer'' (...$\ \alpha \stackrel{\sigma }{\to } . \stackrel{\tau ^2}{\to } \alpha -1 \stackrel{\sigma }{\to } . \stackrel{\tau ^2}{\to } \alpha -2\ ...$) \`{a} partir de $(-r-1)$, et d\'{e}crire ainsi un chemin contenant un intervalle de cardinal sup\'{e}rieur \`{a} ${p^n}/{D}\ -D-2$.\\ \underline{Second chemin.} On doit l\`{a} distinguer deux cas, selon que $p$ divise ou non $r$ (voir {\bf Figure 2}).\\ {\bf a}) Si $p$ ne divise pas $r$, chemin ${\cal B}$ : on part de $(\frac{1}{r} )$ lui m\^{e}me, on recule de m\^{e}me : \\ 1) $\frac{w}{t} -\frac{1}{r} =\overline{a} ,\ -{p^n} < a\leq 0 \iff wr-t=art+ b.{p^n}\ $;\\ \underline{$b=0$}\ : $t=r(w-at) \Rightarrow a=0,\ w=1,\ t=r$ : c'est $\frac{1}{r} $ lui-m\^{e}me.\\ \underline{$b\neq 0$}\ : $|a|\geq \frac{{p^n}-{D^2}}{D^2}$ de m\^{e}me que plus haut.\\ 2) $-\frac{t}{w} -\frac{1}{r} =\overline{a} $ : $-rt-w=awr+b.{p^n}$\ ; \\ \underline{$b=0$} $\Rightarrow r|w,$ impossible (car $w\leq r-1,$ et : $w=0 \Rightarrow t=0$).\\ \underline{$b\neq 0$} $\Rightarrow |a|\geq \frac{p^n}{D^2} -2$.\\ {\bf b}) Si $p$ divise $r$, on a alors que le chemin ${\cal B}$ pr\'{e}c\'{e}dent : $\frac{1}{r} \stackrel{\sigma {\tau }^2}{\longrightarrow} \frac{1-r}{r} \stackrel{\sigma \tau ^2}{\longrightarrow } ...$ est bien de longueur sup\'{e}rieure \`{a} $(\frac{p^n}{D^2} )$ ; mais il ne contient pas cette fois d'intervalle, puisque $r$ n'est pas inversible modulo ${p^n}$, donc l'action de $\sigma \tau ^2 $ ne correspond plus \`{a} l'addition de $(-1)$ (les $k/pl$ ne sont plus relevables en \'{e}l\'{e}ments de ${\bf Z} /{p^n} {\bf Z}$). Cependant, le calcul pr\'{e}c\'{e}dent montre que $\frac{r}{r-1} = \frac{1}{r} \cdot \sigma {\tau ^2}\sigma $\ ; et $(r-1)$ est, cette fois, inversible modulo $p^n$, donc en ``avan\c{c}ant'' \`{a} partir de cet \'{e}l\'{e}ment et \`{a} l'aide de $(\tau \sigma )$, on aura bien un intervalle ; on note ${\cal B}'$ ce chemin. On minore encore une fois sa longueur : \\ 1) $\frac{w}{t} -\frac{r}{r-1}= \overline{a} \iff w(r-1) -rt= at(r-1)+b.{p^n}$ (on \'{e}crit cette fois cela avec $0\leq a\leq {p^n}-1$).\\ \underline{Si $b=0$} :\ $\ t\left( r+a(r-1) \right) =w(r-1)$ ; mais $w<t,$ et $a(r-1)+r>r-1$, contradiction. \\ \underline{Si $b\neq 0$} : $\ |a|\geq \frac{p^n}{D^2} -1.$\\ 2) $\frac{w}{t} \cdot \sigma -\frac{r}{r-1} =-\frac{t}{w} -\frac{r}{r-1} = \overline{a} \iff -t(r-1)-rw=aw(r-1) + b.{p^n}$. \\ \underline{$b=0$} : $\ t(r-1)=-w \left( a(r-1)+r \right) $, contradiction de signes. \\ \underline{ $b\neq 0$} : $\ |a| \geq \frac{p^n}{D^2} -2.$ \\ Dans chaque cas, on obtient bien que ce second chemin contient un intervalle de cardinal sup\'{e}rieur ou \'{e}gal \`{a} $({p^n}/{D^2}) -2$.$\Box$ \begin{picture}(20,20.5)(1.5,-2.5) \thinlines \put (3,16.3){\vector (0,1){1}} \put (3.2,16.5){${\tau }^2$} \put(3,17.5){.} \put(3,17.7){.} \put(3,17.9){.} \thicklines \put (2,14){\framebox (2,2){$(1,r)$}} \thinlines \put (3.6,13.4){$\sigma$} \put (4,13.8){\vector (1,-1){0.7}} \put (5,12){\framebox (2,1){$(-r,1)$}} \put (5,11.8){\vector (-1,-1){0.7}} \put (3.8,11.5){${\tau}^2$} \thicklines \put (2,10){\framebox (2.5,1){$(1-r,r)$}} \thinlines \put (4,9.8){\vector (1,-1){0.7}} \put (4.5,9.5){$\sigma$} \put (5,8){\framebox (2.4,1){$(r,r-1)$}} \put (5,7.8){\vector (-1,-1){0.7}} \put (4.3,7.5){${\tau}^2$} \put (1.5,6){\framebox (2.7,1){$(1-2r,r)$}} \put (4,5.8){\vector (1,-1){0.7}} \put (4,5){$\sigma$} \put (5.1,4){\framebox (2.7,1){$(r,2r-1)$}} \put (5,3.8){\vector (-1,-1){0.7}} \put (5,3){${\tau}^2$} \put (1.6,2){\framebox (2.7,1){$(1-3r,r)$}} \put (4,1.8){\vector (1,-1){0.7}} \put(5,0.5){. . . } \put (7.5,8.4){\vector (1,0){1}} \put (7.8,8.5){$\tau$} \put (8.9,8){\framebox (3.6,1){$(1-r,2r-1)$}} \put (10.4,7.8){\vector (0,-1){0.7}} \put (10.6,7.5){$\sigma$} \put (8.9,6){\framebox (3.6,1){$(2r-1,r-1)$}} \put (12.7,6.4){\vector (1,0){1}} \put (13,6.7){$\tau$} \put (13.9,6){\framebox (3.6,1){$(1-r,3r-2)$}} \put (15.3,5.8){\vector (0,-1){0.7}} \put (15.5,5.3){$\sigma$} \put (13.9,4){\framebox (3.6,1){$(3r-2,r-1)$}} \put (17.7,4.5){\vector (1,0){1}} \put (18,4.7){$\tau$} \put (19,4.3){. . . .} \put(1,14){\vector (0,-1){12.5}} \put(0,8){${\cal B}$} \put(7,7){\vector (1,-1){6}} \put(9,4){${\cal B}'$} \put(8,-1){\bf Figure 2} \end{picture} On montre pour finir comment on passe du lemme 5.3 \`{a} la proposition 1.7 : on a vu qu'on pouvait prendre $|A|\geq ({p^n}/{D^2} )-2$ et $|B|\geq ({p^n}/D) -D-2 $. On a suppos\'e qu'\'etait satisfaite la condition de la proposition : $p^n \geq (C^2).D^6$. On a $C^2\geq 65$, $s\geq 2$, et les bornes du corollaire 1.8 sont sup\'erieures (!) \`a celles qu'on connaissait d\'ej\`a pour les degr\'es 1 et 2, donc on peut supposer $D\geq 6$. Minorons la taille de $A$ : on a $p^n /D^2 \geq 65. D^4 \geq 84240=42120.(2)$, donc $|A|\geq (42119/42120).(p^n /D^2 )$. Pour $B$, on a : $D+2\leq (4/3).D$, et $p^n /D\geq 65.D^5 .D\geq 379080.(D+2)$ ; donc $|B|\geq (379079/379080).(p^n /D)$. Si on pose $\lambda :=(42119/42120).(379079/379080)$, on a donc : $$|A|.|B|\geq \lambda .p^{2n} /D^3 .$$ Pour que les conditions du lemme soient v\'erifi\'ees, il suffit qu'on ait $\lambda .p^{2n} /D^3 \geq C'.p^{3n/2}$, {\em i.e.} $p^n \geq (C'^2 / {\lambda}^2 ).D^6$, et \'evidemment $C$ a \'et\'e choisie pour que \c{c}a marche. $\Box$ % $${ }$$ {\bf Remerciements.}\ Je remercie ici Lo\"{\i}c Merel qui m'a sugg\'{e}r\'{e} l'id\'{e}e essentielle de ce papier, et Bas{ }Edixhoven qui encadre tout ce travail. Merci aussi \`{a} Joseph Oesterl\'{e} pour l'id\'ee du lemme 1.4, et \`{a} \'Etienne Fouvry pour le lemme de th\'{e}orie analytique des nombres.\\

9611

alg-geom/9611012

en

https://arxiv.org/abs/alg-geom/9611012

alg-geom/9611012

Lothar Goettsche

Lothar G\"ottsche and Rahul Pandharipande

The quantum cohomology of blow-ups of P^2 and enumerative geometry

AMS-LaTeX, 26 pages

We compute the Gromov-Witten invariants of the projective plane blown up in r general points. These are determined by associativity from r+1 intial values. Applications are given to the enumeration of rational plane curves with prescribed multiplicities at fixed general points. We show that the numbers are enumerative if at least one of the prescribed multiplicities is 1 or 2. In particular, all the invariants for r<=8 (the Del Pezzo case) are enumerative.

alg-geom

P(n){P(n)} \def{\text{\rom{\bf b}}}{{\text{\rom{\bf b}}}} \defS^{(\bb)}{S^{({\text{\rom{\bf b}}})}} \def{\text{\rom{\bf E}}}{{\text{\rom{\bf E}}}} \def\Inc#1{Z_{#1}} \def\inc#1{\zeta_{#1}} \def\alpha{\alpha} \def\Gamma{\Gamma} \def\bar \Gamma{\bar \Gamma} \def\gamma{\gamma} \def\bar \eta{\bar \eta} \def{\cal Z}{{\cal Z}} \def{\Bbb H}{{\Bbb H}} \def{\Bbb S}{{\Bbb S}} \def{pt}{{pt}} \def\alpha{\alpha} \def{S^{(n)}}{{S^{(n)}}} \def{S^{(m)}}{{S^{(m)}}} \def\<{\langle} \def\>{\rangle} \def{\text{\rom{Hilb}}}{{\text{\rom{Hilb}}}} \def{\cal W}{{\cal W}} \def{\hbox{\rom{Tor}}}{{\hbox{\rom{Tor}}}} \def{\hbox{\rom{Ext}}}{{\hbox{\rom{Ext}}}} \def\stil#1{\tilde S^#1} \def\pi{\pi} \defg{g} \def\phi{\phi} \def\tilde{\tilde} \def{\bar\al}{{\bar\alpha}} \def\mathbin{\text{\scriptsize$\cup$}}{\mathbin{\text{\scriptsize$\cup$}}} \def\mathbin{\text{\scriptsize$\cap$}}{\mathbin{\text{\scriptsize$\cap$}}} \def{_{\scriptscriptstyle(}\!\le_{\scriptscriptstyle)}}{{_{\scriptscriptstyle(}\!\le_{\scriptscriptstyle)}}} \def{<\atop (=)}{{<\atop (=)}} \def{<\atop (=)}{{<\atop (=)}} \setcounter{secnumdepth}{2} \setcounter{tocdepth}{2} \begin{document} \title[The quantum cohomology of blow-ups of ${\Bbb P}^2$] {The quantum cohomology of blow-ups of ${\Bbb P}^2$ and enumerative geometry} \keywords{Quantum cohomology, Gromov-Witten invariants} \author{L. G\"ottsche } \author{R. Pandharipande} \date{11 November 1996} \thanks{The second author was partially supported by an NSF post-doctoral fellowship.} \address{Mittag-Leffler-Institute\\ Aurav\"agen 17\\ Djursholm, Sweden} \email{gottsche@@ml.kva.se, pandhari@@ml.kva.se} \maketitle\ \section{Introduction} \label{intro} The enumerative geometry of curves in algebraic varieties has taken a new direction with the appearance of Gromov-Witten invariants and quantum cohomology. Gromov-Witten invariants originate in symplectic geometry and were first defined in terms of pseudo-holomorphic curves. In algebraic geometry, these invariants are defined using moduli spaces of stable maps. Let $X$ be a nonsingular projective variety over ${\Bbb C}$. Let $\beta\in H_2(X,{\Bbb Z})$. In [K-M], the moduli space $\overline M_{0,n}(X,\beta)$ of stable $n$-pointed genus $0$ maps is defined. This moduli space parametrizes the data $[\mu:C\rightarrow X,p_1,\dots,p_n]$ where $C$ is a connected, reduced, (at worst) nodal curve of genus 0, $p_1,\ldots,p_n$ are nonsingular points of $C$, and $\mu$ is a morphism. $\overline M_{0,n}(X,\beta)$ is equipped with $n$ morphisms $\rho_1,\ldots,\rho_n$ to $X$ where $\rho_i ([\mu:C\rightarrow X,p_1,\dots,p_n]) =\mu(p_i)$. $X$ is a convex variety if $H^1({\Bbb P}^1,f^*(T_X))=0$ for all maps $f:{\Bbb P}^1 \rightarrow X$. In this case, $\overline M_{0,n}(X,\beta)$ is a projective scheme of pure expected dimension equal to $$dim(X)+n-3+\int_\beta c_1(T_X)$$ with only finite quotient singularities. Given classes $\gamma_1,\dots,\gamma_n$ in $H^*(X,{\Bbb Z})$, the Gromov-Witten invariants $I_\beta(\gamma_1\dots\gamma_n)$ are defined by: $$ I_\beta(\gamma_1\dots\gamma_n)= \int_{\overline M_{0,n}(X,\beta)} \rho_1^*(\gamma_1)\mathbin{\text{\scriptsize$\cup$}} \ldots \mathbin{\text{\scriptsize$\cup$}} \rho_n^*(\gamma_n). $$ The intuition behind these invariants is as follows. If the $\gamma_i$ are the cohomology classes of subvarieties $Y_i\subset X$ in general position, then $I_\beta(\gamma_1\dots\gamma_n)$ should count the (possibly virtual) number of irreducible rational curves $C$ in $X$ of homology class $\beta$ which intersect all the $Y_i$. In case $X$ is a homogeneous space, a correspondence between the Gromov-Witten invariants and the enumerative geometry of rational curves in $X$ can be proven by transversality arguments (see [F-P]). One can use the Gromov-Witten invariants to define the big quantum cohomology ring $QH^*(X)$ of $X$. The associativity of this ring yields relations among the invariants $I_\beta(\gamma_1\dots\gamma_n)$ which often are sufficient to determine them all recursively from a few basic ones. The model case for this approach is the recursive determination of the numbers $N_d$ of nodal rational curves of degree $d$ in the projective plane [K-M], [R-T]. If $X$ is not convex, the moduli space $\overline M_{0,n}(X,\beta)$ will not in general have the expected dimension. Recently, Gromov-Witten invariants have been defined and proven to satisfy basic geometric properties via the construction of virtual fundamental classes of the expected dimension [B-F], [B], [L-T 2] and, in the symplectic context, [L-T 3], [F-O], [S]. In particular, these Gromov-Witten invariants have been proven to satisfy the axioms of [K-M], [B-M]. Therefore, they again define an associative quantum cohomology ring $QH^*(X)$. The aim of this paper is to study the Gromov-Witten invariants of the blow-up $X_r$ of ${\Bbb P}^2$ in a finite set $x_1,\ldots x_r$ of points and to give enumerative applications. $X_r$ is a particularly simple example of a nonconvex variety, so this study (at least in the context of algebraic geometry) neccessitates the use of the above constructions. Let $S$ be a nonsingular, rational, projective surface. $S$ is either deformation equivalent to ${\Bbb P}^1 \times {\Bbb P}^1$ or to $X_{r(S)}$ where $r(S)+1= rank( A^1(S))$. Together with the invariants of ${\Bbb P}^1 \times {\Bbb P}^1$, the Gromov-Witten invariants of $X_r$ therefore determine the invariants of all these rational surfaces (the invariants are constant in flat families of nonsingular varieties). For enumerative applications, it is necessary to consider the blow-up $X_r$ of ${\Bbb P}^2$ in a finite set of {\em general} points. Let $H$ be the pull-back to $X_r$ of the hyperplane class in ${\Bbb P}^2$, and let $E_1,\ldots,E_r$ be the exceptional divisors. Our aim is to count the number of irreducible rational curves $C$ in $X_r$ of class $dH-\sum_{i=1}^r a_iE_i$ passing through $3d-\sum_{i=1}^r a_i -1$ general points. By associating to a curve in ${\Bbb P}^2$ its strict transform in $X_r$, this number can also be interpreted as the number of irreducible rational curves in ${\Bbb P}^2$ having singularities of order $a_i$ at the (fixed) general points $x_i$ and passing through $3d-\sum_{i=1}^r a_i -1$ other general points. The paper is naturally divided into two parts. First, we use the associativity of the quantum product to show that the Gromov-Witten invariants of $X_r$ can be computed from simple initial values by means of explicit recursion relations. There are $r+1$ initial values required for $X_r$: \begin{enumerate} \item[(i)] The number of lines in the plane passing through 2 points, $N_{1,(0, \ldots, 0)}=1$. \item[(ii)] The number of curves in the exceptional class $E_i$, $N_{0, -[i]}=1$. \end{enumerate} The relations are then used to prove properties of these invariants. In the second half of the paper, the enumerative significance of the invariants is investigated. Our main tool is a degeneration argument in which the points $x_i$ are specialized to lie on a nonsingular cubic in ${\Bbb P}^2$. The idea of using such degenerations is due independently to J. Koll\'ar and, in joint work, to L. Caporaso and J. Harris [C-H]. For a general blow-up $X_r$, the Gromov-Witten invariants are proven to be a count (with possible multiplicities) of the finite number of solutions to the corresponding enumerative problem on $X_r$. Let $\beta=dH-\sum_{i=1}^{r} a_iE_i$ be a class in $H_2(X_r, {\Bbb Z})$. If the expected dimension of the moduli space $\overline M_{0,0}(X_r,\beta)$ is strictly positive or if there exists a multiplicity $a_i \in \{1,2\}$, then the corresponding Gromov-Witten invariant is proven to be an actual count of the number of irreducible, degree $d$, rational plane curves of multiplicity $a_i$ at the (fixed) general points $x_i$ which pass through $3d-\sum_{i=1}^{r} a_i -1$ other general points. In the Del Pezzo case ($r\leq 8$), all invariants are shown to be enumerative (see also [R-T]). A basic symmetry of the Gromov-Witten invariants of the spaces $X_r$ obtained from the classical Cremona transformation is discussed in section \ref{cremmy}. These considerations show that for $d\leq 10$, the Gromov-Witten invariants always coincide with enumerative geometry. Tables of these invariants in low degrees are given in section \ref{tbls}. In [K-M], an associativity equation for Del Pezzo surfaces (corresponding to our relation $R(m)$) is derived. The small quantum cohomology ring of Del Pezzo surfaces is studied in [C-M]. In section 11 of [C-M], the associativity of the small quantum product on $X_r$ is used to derive some relations among the Gromov-Witten invariants of these surfaces. The invariants of ${\Bbb P}^2$ blown-up in a point are computed in [C-H2], [G], and [K-P]. In [G], A. Gathmann computes more generally the invariants of the blow-up of ${\Bbb P}^n$ in a point and studies their enumerative significance. In [D-I], the Gromov-Witten invariants of $X_6$ are computed via associativity. Our recursive strategy for $X_6$ differs. The first author would like to thank K. Hulek who suggested to him the possibility of studying the quantum cohomology of blow-ups. He also thanks the University of Chicago and in particular W. Fulton for an invitation which made this collaboration possible. The second author would like to thank J. Koll\'ar for many remarkable insights on these questions. In particular, the main immersion result (Lemma \ref{lemmy}) was first observed by him. Thanks are due to J. Harris for conversations in which the degeneration argument was explained. Both authors thank the Mittag-Leffler-Institut for support. \section{Notation and Background material} \label{notabackg} Let $X$ be a nonsingular projective variety. Assume for simplicity that the Chow and homology rings of $X$ coincide. Let $dim(X)$ be the complex dimension. Denote by $\alpha\mathbin{\text{\scriptsize$\cup$}} \beta$ the cup product of classes $\alpha,\beta\in H^*(X,{\Bbb Z})$ and let $(\alpha\cdot \beta)=\int_X\alpha\mathbin{\text{\scriptsize$\cup$}}\beta$. By definition, $(\alpha\cdot \beta)$ is zero if $\alpha\in H^{2i}(X,{\Bbb Z})$, $\beta\in H^{2j}(X,{\Bbb Z})$, and $i+j\ne dim(X)$. We recall the definition of quantum cohomology from [K-M] in a slightly modified form for nonconvex varieties. Let $B\subset H_2(X,{\Bbb Z})$ be the semigroup of non-negative linear combinations of classes of algebraic curves. Let $\beta\in H_2(X, {\Bbb Z})$. Let $n_\beta=dim(X)+\int_{\beta} c_1(T_X) -3$. Let $n\geq 0$. For classes $\gamma_i\in H^{2j_i}(X,{\Bbb Z})$ with $\sum_{i=1}^{n} j_i=n_{\beta}+n$, let $I_\beta(\gamma_1\dots \gamma_n)$ be the corresponding Gromov-Witten invariant: $$ I_\beta(\gamma_1\dots\gamma_n)= \int_{[\overline M_{0,n}(X,\beta)]} \rho_1^*(\gamma_1)\mathbin{\text{\scriptsize$\cup$}} \ldots \mathbin{\text{\scriptsize$\cup$}} \rho_n^*(\gamma_n) $$ where $[\overline M_{0,n}(X,\beta)]$ is the virtual fundamental class. Note if $n_\beta=0$ and $n=0$, then $I_\beta$ is just the degree of the fundamental class. Kontsevich and Manin introduced a set of axioms for the Gromov-Witten invariants which have now been established for nonsingular varieties (see section \ref{intro}). If $\overline M_{0,n}(X,\beta)$ is empty, then $I_\beta(\gamma_1\dots \gamma_n)=0$. In particular, all invariants vanish for $\beta\not\in B$. Let $T_0=1,T_1,\ldots,T_m$ be a homogeneous ${\Bbb Z}$-basis for $H^*(X,{\Bbb Z})$. We assume that $T_1,\ldots,T_p$ form a basis of $H^2(X,{\Bbb Z})=Pic(X)$. We denote by $T_i^\vee$ the corresponding elements of the dual basis: $(T_i\cdot T_j^\vee)=\delta_{ij}$. Denote by $(g_{ij})$ the matrix of intersection numbers $(T_i\cdot T_j)$ and by $(g^{ij})$ the inverse matrix. For variables $y_0,q_1,\ldots,q_p,y_{p+1},\ldots,y_{m}$ (which we also abbreviate as $q,y$), define the formal power series \begin{equation} \label{potfun} \Gamma(q,y)=\sum_{n_{p+1}+\ldots+n_{m}\ge 0}\sum_{\beta\in B \setminus \{0\}} I_\beta(T_{p+1}^{n_{p+1}}\cdots T_m^{n_m}) q_1^{\int_\beta T_1}\cdots q_p^{\int_\beta T_p} \frac{y_{p+1}^{n_{p+1}}\cdots y_{m}^{n_m}} {n_{p+1}!\cdots n_{m}!} \end{equation} in the ring $${\Bbb Q}[[q, q^{-1}, y]]={\Bbb Q}[[y_0, q_1, \ldots, q_p, q_1^{-1}, \ldots, q_p^{-1}, y_{p+1}, \ldots, y_m]].$$ In case $X$ is a homogeneous space, the substitution $q_i=e^{y_i}$ in (\ref{potfun}) yields a formal power series which equals the quantum part of the potential function of [K-M] modulo a quadratic polynomial in the variables $y_1, \ldots, y_m$. The form (\ref{potfun}) of the potential function is chosen to avoid convergence issues in the nonconvex case. Let $$ \partial_i= \begin{cases} q_i\frac{\partial}{\partial q_i}& i=1,\ldots,p\\ \frac{\partial}{\partial y_i} & i=0,p+1,\ldots,m \end{cases} $$ and denote $f_{ijk}=\partial_i\partial_j\partial_k f$ for $f\in {\Bbb Q}[[q,q^{-1},y]]$. Define a ${\Bbb Q}[[q,q^{-1},y]]$-algebra structure on the free ${\Bbb Q}[[q,q^{-1},y]]$-module generated by $T_0,\ldots,T_m$ by: $$T_i*T_j=T_i\mathbin{\text{\scriptsize$\cup$}} T_j+\sum_{e,f=0}^m \Gamma_{ije} g^{ef}T_f.$$ By definition, this is the quantum cohomology ring of $X$, $QH^*(X)$. We sketch the proof of the associativity of this quantum product following [K-M] and [F-P]. First, a formal calculation (using the axiom of divisor) yields: \begin{equation} \label{partpotfun} \Gamma_{ijk}=\sum_{n \ge 0}\sum_{\beta\in B \setminus \{0\}} \frac{1}{n!} I_\beta( \gamma^n \cdot T_i T_j T_k) q_1^{\int_\beta T_1}\cdots q_p^{\int_\beta T_p}, \end{equation} where $\gamma= y_{p+1} T_{p+1} +\ldots +y_m T_m$ and the ${\Bbb Q}[[y_0,y_{p+1},\ldots,y_{m}]]$-linear extension of $I_\beta$ is used. Define the symbol $\Phi_{ijk}$ by $\Phi_{ijk}= I_0(T_i T_j T_k)+ \Gamma_{ijk}$. In case $X$ is homogeneous, $\Phi_{ijk}$ is the partial derivative of the full potential function. The $*$-product can be expressed by: $T_i * T_j= \sum_{e,f=0}^m \Phi_{ije} g^{ef} T_f.$ Let $$F(i,j|k,l)=\sum_{e,f=0}^{m}\Phi_{ije}g^{ef}\Phi_{fkl}.$$ Associativity is now equivalent to $F(i,j|k,l)=F(j,k|i,l)$. Following [F-P], we let \begin{equation} \label{ggdef} G(i,j|k,l)_{\beta,n}=\sum {n\choose n_1}g^{ef} I_{\beta_1}(\gamma^{n_1}\cdot T_i T_j T_e) I_{\beta_2}(\gamma^{n_2}\cdot T_k T_l T_f) \end{equation} where the sum runs over all $n_1,n_2\geq 0$ with $n_1+n_2=n$ and all $\beta_1,\beta_2\in B$ with $\beta_1+\beta_2=\beta$. As before, $\gamma=y_{p+1} T_{p+1} +\ldots +y_m T_m$. A calculation using equations (\ref{partpotfun}) and (\ref{ggdef}) yields: $$ F(i,j|k,l)=\sum_{\beta\in B} q_1^{\int_\beta T_1}\dots q_{p}^{\int_\beta T_p} \sum_{n\ge 0}\frac{1}{n!} G(i,j|k,l)_{\beta,n}. $$ On the other hand, we can use the splitting axiom and linear equivalence on $\overline M_{0,4}={\Bbb P}^1$ to see that $G(i,j|k,l)_{\beta,n}=G(j,k|i,l)_{\beta,n}$, and thus the associativity follows. \section{Quantum cohomology of blow-ups of ${\Bbb P}^2$} \label{qcb} \begin{nota} Let $r\geq 0$. Let $X_r$ be the blowup of ${\Bbb P}^2$ in $r$ general points $x_1,\ldots,x_r$. Denote by $H\in H_2(X,{\Bbb Z})$ the hyperplane class and by $E_i$, for $i=1,\ldots ,r$, the exceptional divisors. Let $m=r+2$. Let $T_0=1$. Let $T_1$, $T_{i+1}$ ( for $i=1,\ldots,r$), and $T_m$ be the Poincar\'e dual cohomology classes of $H$, $E_i$ and the class of a point respectively. Let $\epsilon_1=1$ and $\epsilon_i=-1$ for $i=2,\ldots ,r+1$. Then, $T_0^\vee=T_m$ and $T_i^\vee=\epsilon_iT_i$ for $i=1,\ldots ,r+1$. For an $r$-tuple $\alpha=(a_1,\ldots,a_r)$ of integers, denote by $(d,\alpha)$ the class $dH-\sum_{i=1}^r a_i E_i$. Let $|\alpha|=\sum_i a_i$, and let $n_{d,\alpha}=3d-|\alpha|-1$ be the expected dimension of the moduli space $\overline M_{0,0}(X_r,(d,\alpha))$. If $n_{d,\alpha}\geq 0$, let $$ N_{d,\alpha}=I_{(d,\alpha)}(T_m^{n_{d,\alpha}}) $$ be the corresponding Gromov-Witten invariant. When writing $N_{d,\alpha}$ for $\alpha$ a sequence of length $r$, we will always mean the Gromov-Witten invariant on $X_r$. The components of the finite sequences $\alpha$, $\beta$, $\gamma$ are denoted by the corresponding roman letters $a_i$, $b_i$, $c_i$. For any $r$, we write $[i]_r$ for the sequence $(j_1,\ldots,j_r)$ with $j_k=\delta_{ik}$. We just write $[i]$ if $r$ is understood. For a sequence $\beta=(b_1,\ldots,b_{r-1})$, we denote by $(\beta,k)$ the sequence obtained by adding $b_r=k$. For a permutation $\sigma$ of $\{1,\ldots,r\}$, denote by $\alpha_\sigma$ the sequence $(a_{\sigma(1)},\ldots,a_{\sigma(r)})$. For an integer $k$, we write $\alpha\ge k$ to mean that $a_i\ge k$ for all $i$. \end{nota} The invariants $N_{1,(0, \ldots, 0)}$ and $N_{0,-[i]_r}$ are first determined. A result relating virtual and actual fundmental classes is needed. Let $\overline{M}_{0,0}^*(X,\beta)$ denote the open locus of automorphism-free maps ($\overline{M}_{0,0}^*(X,\beta)$ is a fine moduli space). \begin{prop} \label{fb} If $\overline{M}_{0,0}(X, \beta) = \overline{M}_{0,0}^*(X,\beta)$ and the moduli space is of pure expected dimension, then the virtual fundamental class is the ordinary scheme theoretic fundamental class $[\overline{M}_{0,0}(X, \beta)]$. \end{prop} \noindent If, in addition, the expected dimension is 0, then the Gromov-Witten invariant $N_{\beta}$ equals the (scheme-theoretic) length of $\overline{M}_{0,0}(X, \beta)$. This result is a direct consequence of the construction in [B-F]. \begin{lem} $N_{1,(0, \ldots, 0)}=1$ and $N_{0,-[i]_r}=1$. \end{lem} \begin{pf} A simple check shows that $\overline M_{0,2}(X_r,H)=\overline M_{0,2}^*(X_r,H)$. Also, the moduli space is irreducible of dimension 4 and (at least) generically nonsingular. For two general points $p_1, p_2 \in X_r$, $\rho_1^{-1}(p_1)\mathbin{\text{\scriptsize$\cap$}} \rho_2^{-2}(p_2)$ consists of one reduced point corresponding to preimage of the unique line connecting the images of $p_1$ and $p_2$ in ${\Bbb P}^2$. Hence, $N_{1, (0,\ldots,0)}= 1$ by Proposition \ref{fb}. The moduli space $\overline M_{0,0}(X_r,(0,-[i])$ consists of one automorphism-free map $\mu: {\Bbb P}^1 \stackrel {\sim} {\rightarrow} E_i \subset X_r$. The Zariski tangent space to $\overline M_{0,0}(X_r,(0,-[i])$ at $[\mu]$ is $H^0({\Bbb P}^1, N_{X_r})=0$ where $N_{X_r}\stackrel{\sim}{=} {\cal O}_{{\Bbb P}^1}(-1)$ is the normal bundle of the map $\mu$. Hence, $\overline M_{0,0}(X_r,(0,-[i])$ is nonsingular and $N_{0,-[i]}=1$ by Proposition \ref{fb}. \end{pf} The invariants $N_{d,\alpha}$ will be determined by explicit recursions. In addition, these Gromov-Witten invariants will be shown to satisfy the following geometric properties. \begin{enumerate} \item[(P1)] $N_{0,\alpha}=0$ unless $\alpha=-[i]$ for some $i$. \item[(P2)] $N_{d,\alpha}=0$ if $d>0$ and any of the $a_i$ is negative. \item[(P3)] $N_{d,\alpha}=N_{d,\alpha_{\sigma}}$ for any permutation $\sigma$ of $\{1,\ldots,r\}$. \item[(P4)] $N_{d,\alpha}=N_{d,(\alpha,0)}$. In particular $N_{d,(0,\ldots,0)}$ is the number of rational curves on ${\Bbb P}^2$ passing through $3d-1$ general points computed by recursion in [K-M]. \item[(P5)] If $n_{d,\alpha}>0$, then $N_{d,\alpha}=N_{d,(\alpha,1)}$. \end{enumerate} \begin{rem}\label{p11} Let $Y$ be the blow-up of ${\Bbb P}^1\times{\Bbb P}^1$ in a point with exceptional divisor $E$, and let $F,G$ be the pullbacks of the classes of the fibres of the two projections to ${\Bbb P}^1$. There is an isomorphism $\phi:X_2\to Y$ with $\phi_*(H)=F+G-E,$ $\phi_*(E_1)=F-E$, $\phi_*(E_2)=G-E$. Let $(d,\alpha)$ be given with $r\ge 2$. If $d-a_1-a_2\ge 0$, then pushing down first to $X_2$ and then further to ${\Bbb P}^1\times {\Bbb P}^1$ gives a bijection between the irreducible rational curves in $|(d,\alpha)|$ on $X_r$ passing through $n_{d,\alpha}$ general points and the irreducible rational curves of bidegree $(d-a_1,d-a_2)$ on ${\Bbb P}^1\times{\Bbb P}^1$, with points of multiplicities $d-a_1-a_2, a_3, \ldots ,a_r$ at $r-1$ general points and passing through $n_{d,\alpha}$ other general points. \end{rem} We obtain recursion formulas determining the $N_{d,\alpha}$ from the associativity of the quantum product. All effective classes $(d,\alpha)$ on $X_r$ satisfy $\alpha\leq d$. Therefore, we can write $$\Gamma(q,y)= \sum_{(d,\alpha)} N_{d,\alpha} q_1^{d}q_2^{a_1}\dots q_{r+1}^{a_r} \frac{y_m^{n_{d,\alpha}}}{n_{d,\alpha}!}, $$ where the sum runs over all $(d,\alpha)\ne 0$ satisfying $n_{d,\alpha}\geq0$, $d\ge 0$, and $\alpha\leq d$. Let $\Gamma_{ijk}=\partial_i\partial_j\partial_k\Gamma$ (following the notation of section \ref{notabackg}) . The quantum product of $T_i$ and $T_j$ is given by $$ T_i*T_j=(T_i\cdot T_j)T_m+\sum_{k=1}^{r+1}\epsilon_k \Gamma_{ijk}T_k+ \Gamma_{ijm} T_0. $$ \begin{lem} For $i,j,k,l\in \{1,\ldots,m\}$, there is a relation: \begin{align*}\tag{$R_{i,j,k,l}$} (T_i\cdot T_j)&\Gamma_{klm}-(T_k\cdot T_j)\Gamma_{ilm}+ (T_k\cdot T_l)\Gamma_{ijm}-(T_i\cdot T_l)\Gamma_{kjm}=\\ &=\sum_{s=1}^{m-1} \epsilon_s(\Gamma_{jks}\Gamma_{isl}-\Gamma_{ijs}\Gamma_{ksl}). \end{align*} \end{lem} \begin{pf} We write $$ (T_i*T_j)*T_k-(T_k*T_j)*T_i= \sum_{l=0}^m r_{i,j,k,l} T_l^\vee. $$ By associativity, we obtain the relation $r_{i,j,k,l}=0$. We show this relation is equivalent to $(R_{i,j,k,l})$. We compute directly \begin{align*} (T_i&*T_j)*T_k= (T_i\cdot T_j)T_m*T_k+\sum_{s=1}^{m-1}\epsilon_s\Gamma_{ijs}T_s*T_k +\Gamma_{ijm}T_k\\ &=\sum_{l=1}^m(T_i\cdot T_j)\Gamma_{klm}T_l^\vee+ \sum_{s=1}^{m-1}\Bigg(\epsilon_s\Gamma_{ijs} (T_s\cdot T_k)T_m+\sum_{l=1}^m \epsilon_s\Gamma_{ijs}\Gamma_{ksl}T_l^\vee\Bigg)+\Gamma_{ijm}T_k. \end{align*} It is easy to see that \begin{align*} \Gamma_{ijm}T_k &= \sum_{l=1}^{m}\Gamma_{ijm} (T_k\cdot T_l)T_l^\vee +\Gamma_{ijm}\delta_{km} T_0^\vee,\\ \sum_{s=1}^{m-1}\epsilon_s\Gamma_{ijs}(T_s\cdot T_k) T_m&= \Gamma_{ijk}(1-\delta_{km})T_0^\vee. \end{align*} Therefore, the sum of these two terms is just $ \sum_{l=1}^{m}\Gamma_{ijm} \big((T_k\cdot T_l)T_l^\vee+\Gamma_{ijk}T_0^\vee.$ Thus $$(T_i*T_j)*T_k=\sum_{l=1}^m \Bigg((T_i\cdot T_j)\Gamma_{klm}+(T_k\cdot T_l)\Gamma_{ijm} +\sum_{s=1}^{m-1}\epsilon_s\Gamma_{ijs}\Gamma_{ksl}\Bigg)T_l^\vee +\Gamma_{ijk}T_0^\vee,$$ and the result follows by exchanging the role of $i$ and $k$ and subtracting. \end{pf} For the recursive determination of the $N_{d,\alpha}$, only the following relations are needed: \begin{equation} \tag{$R_{1,1,m,m}$} \Gamma_{mmm}=\sum_{s=1}^{m-1} \epsilon_s(\Gamma_{1sm}^2-\Gamma_{11s} \Gamma_{smm}), \end{equation} and for all $i=2\ldots r+1$ \begin{equation}\tag{$R_{1,1,i,i}$} \Gamma_{iim}-\Gamma_{11m}= \sum_{s=1}^{m-1} \epsilon_s(\Gamma_{1is}^2-\Gamma_{11s} \Gamma_{iis}). \end{equation} Note that in case $r=0$, only the relation $(R_{1,1,m,m})$ occurs and coincides with that of [K-M]. In the summations below, the following notation is used. Let the symbol $\vdash\!\! (d,\alpha)$ denote the set of pairs $\left( (d_1, \beta), (d_2, \gamma) \right)$ satisfying: \begin{enumerate} \item[(i)] $(d_1, \beta), (d_2, \gamma)\neq 0$, \item[(ii)] $(d_1, \beta)+(d_2,\gamma)=(d, \alpha)$, \item[(iii)] $n_{d_1,\beta}, n_{d_2, \gamma}\geq 0$, $d_1, d_2 \geq 0,$ $\beta\leq d_1$, and $\gamma\leq d_2$. \end{enumerate} The notation $\vdash\!\!(d,\alpha), d_i>0$ will be used to denote the subset of $\vdash\!\!(d,\alpha)$ satisfying $d_1, d_2>0$. The binomial coefficient $\binom{p}{q}$ is defined to be zero if $q<0$ or $p<q$. \begin{thm} \label{recurr} The $N_{d,\alpha}$ are determined by the initial values: \begin{enumerate} \item[(i)] $N_{1,(\underbrace{0,\ldots,0}_r)}=1$, for all $r$, \item[(ii)] $N_{0,-[i]_r}=1$, for $i\in \{1,\ldots,r\}$, \end{enumerate} and the following recursion relations. \vspace{+15pt} \noindent If $n_{d,\alpha}\ge 3$, then relation $R(m)$ holds: \begin{align*} N_{d,\alpha}= \sum_{\vdash(d,\alpha), d_i>0}&N_{d_1,\beta}N_{d_2,\gamma} \Big(d_1d_2 -\sum_{k=1}^{r}b_kc_k\Big) \left( d_1d_2\binom{n_{d,\alpha}-3}{n_{d_1,\beta}-1} -d_1^2\binom{n_{d,\alpha}-3}{n_{d_1,\beta}} \right). \end{align*} \vspace{+5pt} \noindent If $n_{d,\alpha}\ge 0$, then for any $i\in \{1,\ldots,r\}$ relation $R(i)$ holds: \begin{align*} d^2a_iN_{d,\alpha}&= (d^2-(a_i-1)^2)N_{d,\alpha-[i]} \\ &+\sum_{\vdash(d,\alpha-[i]), d_i>0} N_{d_1,\beta}N_{d_2,\gamma} \Big(d_1d_2 -\sum_{k=1}^{r}b_kc_k\Big) \left(d_1d_2b_ic_i- d_1^2c_i^2\right) \binom{n_{d,\alpha}}{n_{d_1,\beta}}. \end{align*} \noindent Furthermore, the properties (P1)-(P5) hold. \end{thm} \begin{pf} From the relation $(R_{1,1,i+1,i+1})$ above, we get immediately (for $n_{d,\alpha}\ge 1$) the recursion formula $R(i)^*$: \begin{align*} (a_i^2-d^2)N_{d,\alpha}= \sum_{\vdash(d,\alpha)} N_{d_1,\beta}N_{d_2,\gamma} \Big(d_1d_2 -\sum_{k=1}^{r}b_kc_k\Big) \left(d_1d_2b_ic_i- d_1^2c_i^2\right) \binom{n_{d,\alpha}-1}{n_{d_1,\beta}}. \end{align*} We now show property (P1). If $N_{0,\alpha}\ne 0$, then $(0,\alpha)$ is effective and therefore $\alpha\leq 0$. If $n_{0,\alpha}=0$ we get $\alpha=-[i]$ for some $i\in\{1,\ldots,r\}$. If $n_{0,\alpha}>0$, we apply $R(i)^*$ for an $i$ with $a_i\ne 0$. We see that all summands on the right side are divisible by $d_1=0$, and thus (P1) follows. The relation $R(m)$ is obtained from $R_{1,1,m,m}$ in two steps. The relation $R_{1,1,m,m}$ immediately yields a recursion relation identical to $R(m)$ except for the fact that the sum is over $\vdash\!\!(d,\alpha)$ instead of $\vdash\!\!(d,\alpha), d_i>0$. It will be shown that the terms with $d_1=0$ or $d_2=0$ vanish. Since all summands are divisible by $d_1$, only the case $d_2=0$ need be considered. By (P1), either $N_{0,\gamma}=0$ or $\gamma=-[i]$. In the second case, both binomal coefficients vanish. Thus, relation $R(m)$ follows. Now we show relation $R(i)$ holds. We apply relation $R(i)^*$ to $N_{d,\alpha-[i]}$. All summands on the right side of $R(i)^*$ are divisible by $d_1$, thus all nonvanishing summands have $d_1>0$. By (P1), $N_{0,\gamma}$ can only be nonzero if $\gamma=-[j]$ for some $j\in\{1,\ldots,r\}$. Since the right side of $R(i)^*$ is divisible by $c_i$, the only nonzero summand on the right side with $d_2=0$ occurs for $(d_2,\gamma)=(0,-[i])$ and is $-d^2a_iN_{d,\alpha}$. Bringing this term on the left side and bringing $((a_i-1)^2-d^2)N_{d,\alpha-[i]}$ to the right side, we obtain the relation $R(i)$. Note that $n_{d,\alpha}\ge 0$ implies $n_{d,\alpha-[i]}\ge 1.$ We now show that the invariants $N_{d,\alpha}$ are determined recursively by the relations $R(1), \ldots, R(r)$, $R(m)$ and the intial values. By (P1), all $d=0$ invariants are determined. Let $d>0$. If $n_{d,\alpha}\ge 3$, then relation $R(m)$ determines $N_{d,\alpha}$ in terms of $N_{e,\lambda}$ with $e<d$. Assume now that $0 \leq n_{d,\alpha}<3$. Either $(d,\alpha)=(1,(0, \ldots, 0))$ (and $N_{d,\alpha}=1$) or there exists an $i_0$ with $a_{i_0}\ne 0$. By relation $R(i_0)$, we can determine $N_{d,\alpha}$ in terms of $N_{e,\lambda}$ satisfying either $e<d$ or $e=d$ and $n_{d,\lambda}>n_{d,\alpha}$. After at most 3 applications of a suitable $R(i)$, $R(m)$ may be applied. $N_{d,\alpha}$ is then expressed in terms of the intial values and $N_{e, \lambda}$ with $e<d$. This completes the recursion. Finally, we verify (P2)--(P5). First, (P2) is proven. For $d=0$, the statement of (P2) is void. Let $d>0$, and assume by induction that (P2) holds for all $d_0<d$. Let $(d,\alpha)$ be given with $d>0$, $a_j<0$. If $n_{d,\alpha}\ge 3$, we can apply $R(m)$ to express $N_{d,\alpha}$ as a linear combination of products $N_{d_1,\beta}N_{d-d_1,\alpha-\beta}$ with $d_1, d-d_1>0$. Furthermore $a_j<0$ implies $b_j<0$ or $a_j-b_j<0$. Therefore, $N_{d,\alpha}=0$ by induction. If $0\leq n_{d,\alpha}<3$, we apply $R(j)$ to express $N_{d,\alpha}$ as a linear combination of $N_{d,\alpha-[j]}$ and terms of the form $N_{d_1,\beta}N_{d-d_1,\alpha-[j]-\beta}$ with $d_1, d-d_1>0$. These last terms vanish by induction. Thus $N_{d,\alpha}$ is just a multiple of $N_{d,\alpha-[j]}$. As $n_{d,\alpha-[j]}=n_{d,\alpha}+1$, we can repeat this process to reduce to the case $n_{d,\alpha}\ge 3$. (P3) is obvious, as the initial values and the set $R(1), \ldots R(r), R(m)$ of relations are symmetric. (P4) Let $(d,\alpha)$ be given. We will show that $N_{d,\alpha}=N_{d,(\alpha,0)}$. By (P1) and the intial values, the result holds for $d=0$. Let $d>0$ and assume by induction that the result holds for all $d_1<d$. Case 1: $n_{d,\alpha}\ge 3$. Apply $R(m)$ to express $N_{d,\alpha}$ as a linear combination of terms $N_{d_1,\beta}N_{d-d_1,\alpha-\beta}$ and to express $N_{d,(\alpha,0)}$ as a linear combination of terms $N_{d_1,\beta_0}N_{d-d_1,(\alpha,0)-\beta_0}$ with $d_1, d-d_1>0$. (P2) implies, for nonzero terms, that $\beta_0$ must be of the form $(\beta,0)$. Furthermore the coefficient of $N_{d_1,(\beta,0)}N_{d_2,(\gamma,0)}$ in the expression for $N_{d,(\alpha,0)}$ is the same as that of $N_{d_1,\beta}N_{d_2,\gamma}$ in the expression for $N_{d,\alpha}$. Thus the result follows by induction on $d$. Case 2: $0\leq n_{d,\alpha}<3$. If $\alpha\le 0$, then $(d,\alpha)$ must be $(1,(0\ldots,0))$ and $N_{d,\alpha}=N_{d,(\alpha,0)}=1$. If there exists an $i$ with $a_i<0$, then $N_{d,\alpha}=N_{d,(\alpha,0)}=0$ by (P2). Assume there exists a $j$ with $a_j>0$. We apply $R(j)$ both to $N_{d,\alpha}$ and $N_{d,(\alpha,0)}$. $N_{d,\alpha}$ is expressed as a linear combination of $N_{d,\alpha-[j]}$ and the $N_{d_1,\beta}N_{d-d_1,\alpha-[j]-\beta}$ with $d_1, d-d_1>0$. Using (P2), the expression for $N_{d,(\alpha,0)}$ is obtained by replacing $N_{d_1,\beta}N_{d_2,\gamma}$ by $N_{d_1,(\beta,0)}N_{d_2,(\gamma,0)}$ and $N_{d,\alpha-[i]}$ by $N_{d,(\alpha,0)-[i]}$. By induction on $d$, it is enough to show the result for $N_{d,\alpha-[i]}$. Iterating the argument we reduce to $n_{d,\alpha}\ge 3$ or to $\alpha\le 0$, where we already showed the result. (P5) Let $(d,\alpha)$ be given with $n_{d,\alpha}\ge 0$ and $a_j=1$ for some $j$. We show that $N_{d, \alpha}=N_{d, \alpha -[j]}$. By (P1), we can assume $d>0$. We apply relation $R(j)$ to express $N_{d,\alpha}$ as a linear combination of $N_{d,\alpha-[j]}$ and terms $N_{d_1,\beta}N_{d-d_1,\alpha-[j]-\beta}$ with $d_1, d-d_1>0$. Furthermore, by (P2), all nonzero terms have $b_j=c_j=0$. The coefficient of these terms is divisible by $c_j$. Therefore, $R(j)$ just reads $d^2N_{d,\alpha}=d^2N_{d,\alpha-[j]}$. \end{pf} \section{Moduli Analysis} \subsection{Results} As before, let $X_r$ be the blow-up of ${\Bbb P}^2$ at $r$ general points $x_1, \ldots, x_r$. In this section, the connection between Gromov-Witten invariants and the enumerative geometry of curves in $X_r$ is examined. Let $\alpha=(a_1, \ldots, a_r)$. Let $(d,\alpha)$ denote the class $dH- \sum_{i=1}^{r} a_i E_i$ in $H_2(X_r, {\Bbb Z})$. Let $n_{d,\alpha}= 3d-|\alpha|-1$ be the expected dimension of the moduli space of maps $\overline{M}_{0,0}(X_r, (d, \alpha))$. If $n_{d,\alpha}\geq 0$, let $N_{d,\alpha}$ be the corresponding Gromov-Witten invariant. In this case, the number of genus 0 stable maps of class $(d, \alpha)$ passing through $n_{d,\alpha}$ general points of $X_r$ is proven to be {\em finite}. $N_{d,\alpha}$ is then shown to be a count with (possible) multiplicities of the finite solutions to this enumerative problem. Hence, the Gromov-Witten invariant $N_{d,\alpha}$ is always non-negative. An analysis of the moduli space of maps yields a more precise enumerative result. \begin{thm} \label{numtheorem} Let $n_{d,\alpha}\geq 0$, $d>0 $, and $\alpha\geq 0$. Let (at least) one of the following two conditions hold for the class $(d,\alpha)$: \begin{enumerate} \item[(i)] $n_{d,\alpha} > 0$. \item[(ii)] $a_i \in \{1,2\}$ for some $i$. \end{enumerate} Then, $N_{d,\alpha}$ equals the number of genus 0 stable maps of class $(d,\alpha)$ passing through $n_{d,\alpha}$ general points in $X_r$. Moreover, in this case, each solution map is an immersion of ${\Bbb P}^1$ in $X_r$. \end{thm} \subsection{Dimension 0 Moduli} Three coarse moduli spaces of will be considered: $$ M_{0,0}^\#(X_r, (d, \alpha)) \subset M_{0,0}(X_r, (d,\alpha)) \subset \overline{M}_{0,0}(X_r, (d,\alpha)).$$ $M_{0,0}(X_r, (d,\alpha))$ is the open set of maps with domain ${\Bbb P}^1$. $M_{0,0}^\#(X_r,(d,\alpha))$ is the open set of maps with domain ${\Bbb P}^1$ that are {\em birational} onto their image. As a first step, these unpointed moduli spaces are shown to be empty when their expected dimensions are negative. As always, $X_r$ is general. \begin{lem} \label{lemone} Let $(d,\alpha)\neq 0$ satisfy $n_{d,\alpha}<0$. Then, $\overline{M}_{0,0}(X_r, (d, \alpha))$ is empty. \end{lem} \begin{pf} If $d<0$, $\overline{M}_{0,0}(X_r, (d, \alpha))$ is clearly empty. Next, the case $d=0$ is considered. The only classes $(0, \alpha)\neq 0$ that can be represented by a connected curve are the classes $(0, -k[i])$ for $k\geq 1$. Since $3\cdot 0 + k -1 \geq 0$, these classes are ruled out by the assumption $n_{d,\alpha}<0$. It can now be assumed that $d>0$. Let ${\cal B}_r$ be the open configuration space of $r$ distinct ordered points on ${\Bbb P}^2$. ${\cal B}_r$ is an open set of ${\Bbb P}^2 \times \cdots \times {\Bbb P}^2$ (with $r$ factors). Let $\pi:{\cal X}_r \rightarrow {\cal B}_r$ be the universal family of blown-up ${\Bbb P}^2$ 's. The fiber of $\pi$ over the point $b=(b_1, \ldots, b_r)\in {\cal B}_r$ is simply ${\Bbb P}^2$ blown-up at $b_1, \ldots, b_r$. The morphism $\pi$ is projective. Let $\tau:\overline{{M}}_{0,0}(\pi, (d,\alpha)) \rightarrow {\cal B}_r$ be the relative coarse moduli space of stable maps associated to the family $\pi$. The morphism $\tau$ is projective. The fiber $\tau^{-1}(b)$ is the corresponding moduli space of maps $\overline{M}_{0,0}(\pi^{-1}(b), (d,\alpha))$ to the fiber $\pi^{-1}(b)$. Assume that $\overline{M}_{0,0}(X_r, (d,\alpha))$ is nonempty for general $X_r$. It follows that $\tau$ is a dominant projective morphism and thus surjective onto ${\cal B}_r$. Let $b=(b_1,\ldots, b_r)\in {\cal B}_r$ be $r$ general points on a nonsingular plane cubic $E \subset {\Bbb P}^2$. Let $X_b=\pi^{-1}(b)$. Since $\tau$ is surjective, there exists a stable map $\mu: C \rightarrow X_b$. By the numerical assumption, $$C \cdot \mu^*(c_1(T_{X_b}))= 3d-|\alpha| =n_{d,\alpha}+1\leq 0.$$ Since the points $b_1, \ldots, b_r$ lie on $E$, the strict transform of $E$ is a representative of the divisor class $c_1(T_{X_b})$ on $X_b$. Moreover, since $E$ is elliptic, no component of $C$ surjects upon $E$. Let $C=\bigcup C_j$ be the decompositon of $C$ into irreducible components. For each $C_j$, $\mu(C_j)$ is either a point or an irreducible curve in $X_b$ not equal to $E$. Hence, $C_j \cdot \mu^*(E) \geq 0$. Since $$\sum_j C_j \cdot \mu^*(E) = C \cdot \mu^*(c_1(T_{X_b}))\leq 0,$$ $C_j \cdot \mu^{*}(E) =0$ for all components $C_j$. Since $d>0$, there exists a component $C_{l}$ such that $\mu(C_{l})$ is of class $(d_{l}, \alpha_{l})$ with $d_{l}>0$. Then, $\mu(C_{l})$ is curve and $\mu(C_{l}) \cap E= \emptyset$. Now consider the image of $\mu(C_{l})$ in ${\Bbb P}^2$ (using the natural blow-down map $X_b \rightarrow {\Bbb P}^2$). The image of $\mu(C_{l})$ is a degree $d_{l}>0$ plane curve meeting $E$ only at the points $b_1, \ldots, b_r$. Hence, there is an equality in the Picard group of $E$: $${\cal O}_{{\Bbb P}^2}(d_l)|_E \stackrel{\sim}{=} {\cal O}_E(\sum_{i=1}^{r} m_i b_i)$$ for some non-negative integers $m_1, \ldots, m_r$. Since $b_1, \ldots, b_r$ were chosen to be general points on $E$, no such equality can hold. A contradiction is reached and the Lemma is proven \end{pf} A map $\mu: {\Bbb P}^1 \rightarrow X_r$ is simply incident to a point $y\in X_r$ if $\mu^{-1}(y)$ is scheme theoretically a single point in ${\Bbb P}^1$. \begin{lem} \label{lemtwo} Let $(d,\alpha)$ satisfy $n_{d,\alpha}\geq 0$. Every map $[\mu]\in \overline{M}_{0,0}(X_r, (d,\alpha))$ incident to $n_{d, \alpha}$ general points in $X_r$ is a birational map with domain ${\Bbb P}^1$. Morever, every such map is simply incident to the $n_{d,\alpha}$ points. \end{lem} \begin{pf} Let $C$ be a reducible curve. Assume there exists a genus 0 (unpointed) stable map $\mu:C \rightarrow X_r$ representing the class $(d, \alpha)$ incident to $n_{d, \alpha}$ general points. It is first claimed that at least two irreducible components are mapped nontrivially by $\mu$. If no component is mapped to a point, the claim is trivial. Otherwise, let $K$ be a maximal connected component of $C$ that is mapped to a point. $K$ must meet the union of the irreducible components mapped nontrivially in at least 3 points. Since $C$ is a tree, these 3 points lie on {\em distinct} components of $C$. Let $C_1, \ldots, C_{s}$ be the irreducible components mapped nontrivially by $\mu$. Let $(d_1,\alpha_1), \ldots, (d_{s}, \alpha_{s})$ be the classes represented by these components. Let $p_i$ be the number of the $n_{d,\alpha}$ general points contained in $\mu(C_i)$. Since $$n_{d,\alpha}=s-1+\sum_{i=1}^{s}n_{d_i,\alpha_i}> \sum_{i=1}^{s}n_{d_i,\alpha_i},$$ and $\sum_{i=1}^{s} p_i \geq n_{d,\alpha}$, it follows that for some $j$, $p_j>n_{d_j,\alpha_j}$. Let $y_1, \ldots ,y_{p_j}$ be the general points contained in $\mu(C_j)$. Let $X_{r+p_j}$ be the blow-up of $X_r$ at these points. Consider the strict transform of the map $\mu$ to the map $\mu': C_j \rightarrow X_{r+p_j}$. The class represented by $\mu'$ is $\beta= (d_j, (\alpha_j, m_1, \ldots, m_{p_j}))$ where $m_i\geq 1$ for all $1\leq i \leq p_j$. Therefore $n_\beta \leq n_{d_j, \alpha_j}-p_j <0$. By Lemma (\ref{lemone}), $\overline{M}_{0,0}(X_{r+p_j}, \beta)$ is empty. A contradiction is reached. Hence, no stable maps in $\overline{M}_{0,0}(X_r, (d,\alpha))$ with reducible domains pass through $n_{d,\alpha}$ general points of $X_r$. Next, assume there exists a stable map $\mu: {\Bbb P}^1 \rightarrow X_r$ passing through $n_{d,\alpha}$ general points which is not birational onto its image. Let $\mu: {\Bbb P}^1 \rightarrow Im(\mu)$ be a generically $k$-sheeted cover for $k\geq2$. Let $\gamma: {\Bbb P}^1 \rightarrow Im(\mu)$ be a desingularization of the image. The map $\gamma$ represents the class $(d/k, \alpha/k) \neq (0,0)$ and is incident to the $n_{d,\alpha}$ general points. Note that $$n_{d/k, \alpha/k}=3\cdot {\frac{d}{k}} - {\frac{1}{k}} |\alpha| -1 < n_{d, \alpha}.$$ As before, a contradiction is reached. Hence, the stable maps in $\overline{M}_{0,0}(X_r, (d,\alpha))$ passing through $n_{d,\alpha}$ general points of $X_r$ are birational. Finally, assume there exists a stable map $\mu: {\Bbb P}^1 \rightarrow X_r$ passing through $n_{d,\alpha}$ general points $y_1, \ldots, y_{n_{d,\alpha}}$ which is not simply incident to the point $y_1$. Let $X_{r+n_{d, \alpha}}$ be the blow-up of $X_r$ at the general points. Then, the strict transform of $\mu$ to $X_{r+n_{d,\alpha}}$ represents the class $\beta=(d, (\alpha, m_1, \ldots, m_{n_{d,\alpha}}))$ where $m_i\geq 1$ for all $1\leq i \leq n_{d,\alpha}$ and $m_1\geq 2$. Again, $n_\beta \leq n_{d,\alpha}- n_{d, \alpha}-1 <0$ and a contradiction is reached. \end{pf} \begin{cor} \label{cortwo} Let $(d,\alpha)$ satisfy $n_{d,\alpha}=0$. Then $\overline{M}_{0,0}(X_r, (d,\alpha)) = {M}^\#_{0,0}(X_r, (d,\alpha))$. \end{cor} A scheme $Z$ is of {\em pure dimension 0} if every irreducible component is a point. $Z$ may be empty. \begin{lem} \label{lemthree} Let $(d,\alpha)$ satisfy $n_{d,\alpha}=0$. Then, $\overline{M}_{0,0}(X_r,(d,\alpha))$ is of pure dimension 0. \end{lem} \begin{pf} By Corollary (\ref{cortwo}), $\overline{M}_{0,0}(X_r, (d,\alpha)) = M_{0,0}^\#(X_r, (d,\alpha))$. Let $\mu: {\Bbb P}^1 \rightarrow X_r$ correspond to a point $[\mu]\in M_{0,0}^\#(X_r, (d,\alpha))$. Consider the normal (sheaf) sequence on ${\Bbb P}^1$ determined by $\mu$: $$0 \rightarrow T_{{\Bbb P}^1} \rightarrow \mu^* T_{X_r} \rightarrow N_{X_r} \rightarrow 0.$$ The sheaf $N_{X_r}$ has generic rank 1 and degree equal to $3d-|\alpha|-2=n_{d,\alpha}-1=-1$. There is a canonical torsion sequence: $$0 \rightarrow \tau \rightarrow N_{X_r} \rightarrow F \rightarrow 0.$$ The torsion subsheaf, $\tau$, is supported on the locus where $\mu$ fails to be an immersion. $F$ is a line bundle of degree equal to $-1-dim(\tau)$. It follows that \begin{equation} \label{torr} H^0({\Bbb P}^1, N_{X_r})= H^0({\Bbb P}^1, \tau). \end{equation} Let $\lambda:{\cal C} \rightarrow M_{0,0}^\#(X_r, (d,\alpha))$ be any morphism of an irreducible curve to the moduli space. It will be shown that the image of $\lambda$ is a point. It can be assumed that ${\cal C}$ is nonsingular. Since $M_{0,0}^\#(X_r, (d,\alpha))$ is contained in the automorphism-free locus, there exists a universal curve $\pi: {\cal P} \rightarrow M_{0,0}^\#(X_r, (d,\alpha))$ and a universal morphism $\mu: {\cal P} \rightarrow X_r$ (see [F-P]). Moreover, $\pi$ is a ${\Bbb P}^1$-fibration. Let $\pi:S \rightarrow {\cal C}$ be the pull-back of ${\cal P}$ via $\lambda$ and let $\mu:S \rightarrow X_r$ be the induced map. $S$ is a nonsingular surface. Let $d\mu:T_S \rightarrow \mu^* T_{X_r}$ be the differential of $\mu$. Let $T_V \subset T_S$ be the line bundle of $\pi$-vertical tangent vectors, and let $U\subset S$ be the open set where $d\mu: T_V \rightarrow T_{X_r}$ is a bundle injection. The torsion result (\ref{torr}) directly implies that the bundle map $d\mu: T_S \rightarrow T_{X_r}$ is of constant rank 1 on $U$. Hence, by the complex algebraic version of Sard's theorem, $\mu(S)$ is irreducible of dimension $1$. The $\mu$-image of $S$ must equal the $\mu$-image of each fiber of $\pi$. It now follows easily that the image of $\lambda$ is a point. \end{pf} \subsection{The Map $\mu$ Over $E_i$} The results of the previous section do not show that $\overline{M}_{0,0}(X_r, (d,\alpha))$ is a nonsingular collection of points when $n_{d,\alpha}=0$. Conditions for nonsingularity will be established in section (\ref{nsns}). Preliminary results concerning the the map $\mu$ over the exceptional divisors are required. First, the injectivity of the differential over $E_i$ is established. \begin{lem} \label{lemfour} Let $(d,\alpha)$ satisfy $n_{d,\alpha}=0$. Let $\mu: {\Bbb P}^1 \rightarrow X_r$ correspond to a point $[\mu]\in \overline{M}_{0,0}(X_r, (d,\alpha))$. Then $d\mu$ is injective at all points in $\mu^{-1} (E_i)$ (for all i). \end{lem} \begin{pf} Consider again the relative coarse moduli space $\tau:\overline{{M}}_{0,0}(\pi, (d,\alpha)) \rightarrow {\cal B}_r$ and the universal family of blown-up ${\Bbb P}^2$'s, $\pi:{\cal X}_r \rightarrow {\cal B}_r$. Let ${\cal U}_r\subset {\cal B}_r$ denote the open subset to which the conclusions of Corollary \ref{cortwo} and Lemma \ref{lemthree} apply. For $b=(b_1, \ldots, b_r)\in {\cal B}_r$, let $E_i$ in $\pi^{-1}(b)$ denote the exceptional divisor corresponding to the point $b_i$. Assume, for a general point $b\in {\cal U}_r$, there exists a map $\mu: {\Bbb P}^1 \rightarrow \pi^{-1}(b)$ satisfying: \begin{enumerate} \item[(i)] $[\mu] \in \overline{M}_{0,0}(\pi^{-1}(b), (d,\alpha))$. \item[(ii)] There exists a point $p\in{\Bbb P}^1$ such that $d\mu(p)=0$ and $\mu(p)\in E_i$ for some $i$. \end{enumerate} In this case, there must exist a fixed index $j$ such that for general $b\in {\cal U}_r$ the moduli space $\overline{M}_{0,0}(\pi^{-1}(b), (d,\alpha))$ contains a map with vanishing differential at some point over $E_j$. Let $Y \subset \tau^{-1}({\cal U}_r)$ denote the locus of maps with vanishing differential at some point over $E_j$. $Y$ is closed in $\tau^{-1}({\cal U}_r)$. Let $\overline{Y}$ denote the closure of $Y$ in $\overline{M}_{0,0}(\pi, (d,\alpha))$. Let $[\mu] \in \overline{Y}$ where $\mu: C\rightarrow \pi^{-1}(\tau ([\mu]))$. It is easily seen that one of the following two cases hold: \begin{enumerate} \item[(i)] There exists a point $p\in C_{nonsing}$ satisfying $d\mu(p)=0$ and $\mu(p)\in E_j$. \item[(ii)] There is a node of $C$ mapped to $E_j$. \end{enumerate} \noindent These are the two possible degenerations of the singular point of the morphism $\mu$ over $E_j$. Since $Y$ dominates ${\cal B}_r$, the map $\overline{Y} \rightarrow {\cal B}_r$ is surjective. Define a complete curve ${\cal F}\subset {\cal B}_r$ as follows. Let the points $e_1, \ldots, e_r$ be distinct points on a nonsingular cubic plane curve $F \subset {\Bbb P}^2$. Choose a zero for the group law on $F$. Let the curve ${\cal F}\subset {\cal B}_r$ be determined by elliptic translates of the tuple $(e_1, \ldots, e_r)$. There is a natural map $\epsilon_j:{\cal F} \rightarrow F$ given by $\epsilon_j(f=(f_1,\ldots, f_r))=f_j$. Consider the fibration of blown-up ${\Bbb P}^2$'s over ${\cal F}$, $\pi^{-1}({\cal F})\rightarrow {\cal F}$. Let $S\subset \pi^{-1}({\cal F})$ be the subfibration of ${\Bbb P}^1$'s determined by the exceptional divisor $E_j$. $$S \subset \pi^{-1}({\cal F}) \rightarrow {\cal F}.$$ Via composition with $\epsilon_j$, there is a natural projection $S\rightarrow F$. There is a canonical isomorphism $S \stackrel{\sim}{=} {\Bbb P}(T_{{\Bbb P}^2}|_F) \rightarrow F$ of varieties over $F$. Let $\gamma:{\cal D} \rightarrow \overline{Y}$ be an irreducible curve that surjects onto ${\cal F}$ via $\tau$. After a possible base change, a flat family of stable maps which induces the morphism $\gamma$ exists over ${\cal D}$. (In [F-P], the moduli space of maps is constructed locally as finite quotient of a fine moduli space of rigidified maps, so a base change with a universal family exists on an open set of ${\cal D}$. The properness of the functor of stable maps implies, after further base changes, that this family can be completed over ${\cal D}$.) Denote this family of stable maps over ${\cal D}$ by $\eta: {\cal C} \rightarrow {\cal D}$ and $\mu: {\cal C} \rightarrow \pi^{-1}({\cal F})$. Let $Z \subset {\cal C}$ be the locus of nodes of the fibers of $\eta$ union the locus of nonsingular points of the fibers where $d\mu$ vanishes on the tangent space to the fiber. $Z$ is a closed subvariety. Let $Z'\subset {\cal C}$ denote the (closed) intersection $Z\cap \mu^{-1}(S)$. The subvariety $T=\mu(Z')\subset S= {\Bbb P}(T_{{\Bbb P}^2}|_F)$ dominates $F$ by the properties of $\overline{Y}$. There is a natural section $F \rightarrow {\Bbb P}(T_{{\Bbb P}^2}|_F)$ given by the differential of $F$. By Lemma \ref{lemfive} below, $F\cap T$ is nonempty. Let $\zeta \in F \cap T$. There are now two cases. First, let $d\in {\cal D}$ be such that there exists a nonsingular point $p\in{\cal C}_d$ at which the differential of $\mu_d$ vanishes satisfing $\zeta=\mu_d(p)$. Consider the map $\mu_d$ from ${\cal C}_d$ to ${\Bbb P}^2$ blown-up at the points $f=(f_1, \ldots, f_r)$. Since $\zeta \in F \subset {\Bbb P}(T_{{\Bbb P}^2}|_F)$, the strict transform of $F$ in this blow-up passes through $\zeta= \mu_d(p) \in E_j$. If $p$ lies on a component of ${\cal C}_d$ not mapped to a point, then ${\cal C}_d \cdot \mu^*(F) \geq 2$ because of the vanishing differential at $p$. However, since $n_{d,\alpha}=0$ and $F$ represents the first Chern class of the surface, ${\cal C}_d \cdot \mu^*(F) = 1$. A contradiction is reached. If $p$ lies on a component mapped to a point, let $K$ be the maximal connected subcurve of ${\cal C}_d$ which contains $p$ and is mapped to a point. By stability of the map, $K$ must intersect the other components of ${\cal C}_d$ in at least 3 points. By maximality, these intersection points lie on components not mapped to a point by $\mu_d$. Hence, in this case, ${\cal C}_d \cdot \mu^*(F) \geq 3$. Again a contradiction is reached. Second, let $d\in {\cal D}$ be such that a node $p\in{\cal C}_d$ maps to $\zeta$. Again consider the map $\mu_d$ from ${\cal C}_d$ to ${\Bbb P}^2$ blown up at the points $f=(f_1, \ldots, f_r)$. The strict transform of $F$ in this blow-up passes through $\zeta= \mu_d(p) \in E_j$. If the node $p$ is an intersection of $2$ components of ${\cal C}_d$ neither of which is mapped to a point by $\mu_d$, then ${\cal C}_d \cdot \mu^*(F) \geq 2$ and a contradiction is reached. If the node is on a component that is mapped to a point, then ${\cal C}_d \cdot \mu^*(F) \geq 3$ as before and a contradiction is again reached. \end{pf} \begin{lem}\label{lemfive} Let $\iota:F\hookrightarrow {\Bbb P}^2$ be a nonsingular plane cubic. Let $F\rightarrow {\Bbb P}(T_{{\Bbb P}^2}|_F)$ be the canonical section induced by the differential. Then $F \cap V$ is nonempty for any curve $V \subset {\Bbb P}(T_{{\Bbb P}^2}|_F)$. \end{lem} \begin{pf} First the divisor class of the section $F$ is calculated. Consider the tangent sequence on the plane cubic $F$: \begin{equation} \label{fruit} 0 \rightarrow {\cal O}_F=T_F \rightarrow T_{{\Bbb P}^2}|_F \rightarrow {\cal O}_{{\Bbb P}^2}(3)|_F={\cal O}_F(3) \rightarrow 0. \end{equation} Let $S={\Bbb P}(T_{{\Bbb P}^2}|_F)$ and let $\rho:S \rightarrow F$ denote the projection. Let $L$ denote the line bundle ${\cal O}_{{\Bbb P}}(1)$ on $S$. Via a degeneracy locus computation, sequence (\ref{fruit}) implies that the section $F$ is a divisor in the linear series of the line bundle $L\otimes \rho^*{\cal O}_F(3)$. Note that: $$H^0(S, L\otimes \rho^*{\cal O}_F(3))= H^0(F, T^*_{{\Bbb P}^2}|_F (3)).$$ The dual of the Euler sequence tensored with ${\cal O}_{{\Bbb P}^2}(3)$ restricted to $F$ yields: $$0 \rightarrow T^*_{{\Bbb P}^2}|_F (3) \rightarrow \oplus_{1}^{3} {\cal O}_F(2) \rightarrow {\cal O}_F(3) \rightarrow 0.$$ It is easy to see the corresponding sequence on global sections is exact. Hence $H^0(S, L\otimes \rho^*{\cal O}_F(3))=9$. Therefore, for any $s\in S$, there exists a divisor linearly equivalent to $F$ passing through $s$. Also, it is easy to calculate $F\cdot F =9$. Let $V$ be an irreducible curve in $S$ and assume $V\cap F$ is empty. Hence, $V\cdot F=0$ and $V$ is not a fiber of $\rho$. Let $G$ be a divisor equivalent to $F$ meeting $V$. By the equation $V\cdot G=0$, $V$ must be a component of $G$. Write $G=c_VV +\sum_i c_iW_i$. Let $f$ be a general fiber of $\rho$. $$c_v V\cdot f + \sum_i c_iW_i \cdot f =G \cdot f=1.$$ $V \cdot f\geq 1$ since $V$ is not a fiber. Therefore, $V\cdot f=1$, $c_V=1$, and $W_i\cdot f=0$. This implies each $W_i$ is a fiber. Then, $$9=F\cdot F=F\cdot G = \sum_i F\cdot c_i W_i=\sum_i c_i.$$ $V$ is therefore a section of ${\cal O}_S(F) \otimes \rho^*N$ where $N$ is degree $-9$ line bundle on $F$. Again $H^0(S, {\cal O}_S(F) \otimes \rho^*N)= H^0(F, T^*_{{\Bbb P}^2}|_F \otimes {\cal O}_F(3) \otimes N)$. The latter is seen to be zero by the dual Euler sequence argument. No such $V$ exists. \end{pf} The Lemma (\ref{lemfour}) showed the branches of the image curve $\mu({\Bbb P}^1)$ are nonsingular at their intersections with the $E_i$. Next, it is shown that distinct branches of the image curve do not intersect in the exceptional divisors. \begin{lem} \label{lemextra} Let $(d,\alpha)$ satisfy $n_{d,\alpha}=0$. Let $\mu: {\Bbb P}^1 \rightarrow X_r$ correspond to a point $[\mu]\in \overline{M}_{0,0}(X_r, (d,\alpha))$. Let $I$ be the image curve $\mu({\Bbb P}^1)$. Then the set $I\cap E_i$ is contained in the nonsingular locus of $I$ (for all i). \end{lem} \begin{pf} The proof of this Lemma exactly follows the proof of Lemma (\ref{lemfour}). If the assertion is false, a quasi-projective subvariety $W\subset \overline{M}_{0,0}(\pi, (d, \alpha))$ can be found where the image curve has distinct branches meeting in $E_j$ (for a fixed index $j$). The closure $\overline{W}$ of $W$ then surjects upon ${\cal B}_r$. Let $\mu:C \rightarrow X_b$ be a limit map $[\mu]\in \overline{W}$. At least one of the following properties must be satisfied: \begin{enumerate} \item[(i)] Distincts point of $C$ are mapped by $\mu$ to the same point of $E_j$. \item[(ii)] There exists a point $p\in C_{nonsing}$ satisfying $d\mu(p)=0$ and $\mu(p)\in E_j$. \item[(iii)] There is a node of $C$ mapped to $E_j$. \end{enumerate} The same curve ${\cal F}\subset {\cal B}_r$ is considered. Let $\gamma:{\cal D} \rightarrow \overline{W}$ be an irreducible curve that surjects onto ${\cal F}$ via $\tau$. As before, a curve in $T\subset S= {\Bbb P}(T_{{\Bbb P}^2}|_F)$ can be found representing the points on $E_j$ where the singularities occur. Using Lemma (\ref{lemfive}), $F\cap T$ is non-empty. It is then deduced that stable maps exist satisfying $\mu^* c_1(T_{X_b}) \geq 2$ as before. A contradiction is reached. \end{pf} \subsection{Nonsingularity Conditions} \label{nsns} The main nonsingularity result needed for the proof of Theorem (\ref{numtheorem}) can now be proven. \begin{lem} \label{lemmy} Let $(d,\alpha)$ satisfy $d>0$, $\alpha\geq 0$, and $n_{d,\alpha}=0$. If there exists an index $i$ for which $a_i \in \{1,2\}$, then $\overline{M}_{0,0}(X_r, (d,\alpha))$ is nonsingular of pure dimension 0. Moreover, the points of $\overline{M}_{0,0}(X_r, (d,\alpha))$ correspond to immersions of ${\Bbb P}^1$ in $X_r$. \end{lem} \begin{pf} If $\overline{M}_{0,0}(X_r, (d,\alpha))$ is empty for generic $X_r$, the Lemma is trivially true. Let $\mu: {\Bbb P}^1 \rightarrow X_r$ be a map in $\overline{M}_{0,0}(X_r, (d,\alpha))$. By the genericity assumption, the natural map: \begin{equation} \label{sirr} d\tau: T_{ \overline{M}_{0,0}(\pi, (d,\alpha)), [\mu]} \rightarrow \tau^* T_{{\cal B}_r, \tau([\mu])} \end{equation} must be surjective. The Lemma is proved in two steps. First, the surjectivity of (\ref{sirr}) is translated into a condition on the global sections map of a normal sheaf sequence associated to $\mu$. The map $\mu$ is then shown to be an {\em immersion}. $N_{X_r}$ is therefore locally free of rank $1$ and degree $3d-|\alpha|-2=n_{d,\alpha}-1 <0$. The Zariski tangent space to $\overline{M}_{0,0}(X_r, (d,\alpha))$ at $[\mu]$ is $H^0({\Bbb P}^1, N_{X_r})=0$. Hence, $[\mu]$ is a nonsingular point of $\overline{M}_{0,0}(X_r, (d,\alpha))$. Let $X_r$ be the blow-up of ${\Bbb P}^2$ at the points $x_1, \ldots, x_r$. The deformation problem as the blown-up points $x_1, \ldots, x_r$ vary is considered. There is a projection $X_r \rightarrow {\Bbb P}^2$ which yields a sequence on $X_r$: \begin{equation} \label{www} 0 \rightarrow T_{X_r} \rightarrow T_{{\Bbb P}^2} \rightarrow Q \rightarrow 0. \end{equation} $Q$ is a sheaf supported on the exceptional curves $E_i$. $Q|_{E_i}$ is a line bundle on $E_i$. More precisely, if the point $e\in E_i$ corresponds to the tangent direction $T_e \subset T_{{\Bbb P}^2,x_i}$, then the fiber of $Q$ at $e$ is $T_{{\Bbb P}^2,x_i}/ T_e$. The space of deformations of the points $x_1, \ldots, x_r$ is $\oplus _{i=1}^{r} T_{{\Bbb P}^2, x_i} = H^0(X_r, Q)$. $\oplus_{i=1}^r T_{{\Bbb P}^2, x_i}$ is also canonically the tangent space to ${\cal B}_r$ at the point $x=(x_1,\ldots, x_r)$. Therefore a vector $0 \neq v\in \oplus_{i=1}^r T_{{\Bbb P}^2, x_i}$ defines a first order deformation of $X_r$ in the family ${\cal X}_r$. Let $\lambda:\bigtriangleup\rightarrow {\cal B}_r$ be a nonsingular curve in ${\cal B}_r$ passing through $x$ with tangent direction ${\Bbb C} v$. Let ${\cal X}_{\bigtriangleup}= \lambda^{-1} {\cal X}_r \rightarrow \bigtriangleup$. This deformation naturally yields a differential sequence on $X_r$: \begin{equation} \label{rrr} 0 \rightarrow T_{X_r} \rightarrow T_{{\cal X}_{\bigtriangleup}} \rightarrow {\cal O}_{X_r} \rightarrow 0. \end{equation} Sequences (\ref{www}) and (\ref{rrr}) are related by a commutative diagram: \begin{equation} \label{sss} \begin{CD} 0 @>>> T_{X_r} @>>> T_{{\cal X}_\bigtriangleup} @>{a}>> {\cal O}_{X_r} @>>> 0 \\ @VVV @VV{=}V @VV{b}V @VV{c}V @VVV \\ 0 @>>> T_{X_r} @>>> T_{{\Bbb P}^2} @>{d}>> Q @>>> 0. \end{CD} \end{equation} Moreover, it is easy to check that the image of $c:H^0(X_r,{\cal O}_{X_r}) \rightarrow H^0(X_r,Q)$ is simply ${\Bbb C} v$. Since $d\geq 1$, $Im(\mu)$ is not contained in any $E_i$. Therefore the above commutatitive diagram {\em stays exact} when pulled back to ${\Bbb P}^1$. Let $N_{{\Bbb P}^2}$ and $N_{{\cal X}_{\bigtriangleup}}$ denote the normal sheaves on ${\Bbb P}^1$ of the maps to ${\Bbb P}^2$ and ${\cal X}_{\bigtriangleup}$ induced by $\mu$. Consider the commutative diagram of exact sequences obtained by pulling back (\ref{sss}) to ${\Bbb P}^1$ and quotienting by the inclusion of sheaves induced by the differential $d\mu:T_{{\Bbb P}^1}\rightarrow \mu^* T_{X_r}$. \begin{equation*} \begin{CD} 0 @>>> N_{X_r} @>>> N_{{\cal X}_\bigtriangleup} @>{a}>> {\cal O}_{{\Bbb P}^1} @>>> 0 \\ @VVV @VV{=}V @VV{b}V @VV{c}V @VVV \\ 0 @>>> N_{X_r} @>>> N_{{\Bbb P}^2} @>{d}>> \mu^*Q @>>> 0. \end{CD} \end{equation*} $H^0({\Bbb P}^1, N_{{\cal X}_\bigtriangleup})$ is the space of first order deformation of the map $\mu$ considered as a map to ${\cal X}_{\bigtriangleup}$. By the surjectivity of (\ref{sirr}), there must exist a first order deformation of $[\mu]$ not contained in $X_r$. Therefore, the image of $a: H^0({\Bbb P}^1, N_{{\cal X}_{\bigtriangleup}}) \rightarrow H^0({\Bbb P}^1, {\cal O}_{{\Bbb P}^1})$ must be non-zero. This condition is equivalent to the splitting of the top sequence. Using this splitting and the morphism $b$, it is seen that the section $v \in H^0({\Bbb P}^1, Q)$ must be in the image of $d: H^0({\Bbb P}^1, N_{{\Bbb P}^2}) \rightarrow H^0({\Bbb P}^1, \mu^*Q)$. The conclusion of the above considerations is the following. For every element $v\in \oplus _{i=1}^{r} T_{{\Bbb P}^2,x_i}$, there exists a section of $H^0({\Bbb P}^1, N_{{\Bbb P}^2})$ which has image $v\in H^0({\Bbb P}^1, \mu^*Q)$. The map $\mu$ will now be shown to be an immersion. Suppose $p\in {\Bbb P}^1$ satisfies $\mu(p)\in E_i$. By Lemma (\ref{lemfour}), $d\mu(p)$ is injective. Let $m$ be the multiplicity of $\mu^* E_i$ at $p$. Local calculations show that the following hold in a neighborhood $U\subset {\Bbb P}^1$ of $p$ with local parameter $t$: \begin{enumerate} \item[(i)] $N_{{\Bbb P}^2}$ has torsion part ${\Bbb C}[t]/(t^{m-1})$ (where $t$ is a local parameter at $p$). \item[(ii)] $\mu^*(Q)$ is the torsion sheaf ${\Bbb C}[t]/(t^m)$. \item[(iii)] The map on torsion parts from $N_{{\Bbb P}^2}$ to $\mu^*(Q)$ is multiplication by $t$. \end{enumerate} Let $\tau$ be the torsion part of $N_{{\Bbb P}^2}$. By (iii), the natural map of sheaves on $U$: $$N_{{\Bbb P}^2}/\tau \rightarrow \mu^*(Q) \otimes {\cal O}_p={\Bbb C}$$ is surjective. Therefore, a section $\overline{s}$ of the line bundle $N_{{\Bbb P}^2}/\tau$ is zero at $p$ if and only if the image of $\overline{s}$ in $\mu^*(Q) \otimes {\cal O}_p$ is zero. Decompose $\tau= A\oplus B$ where $A$ is the torsion part supported at the points $\bigcup_i \mu^{-1}(E_i)$ and $B$ is the torsion part supported elsewhere. Let $n$ equal the set theoretic cardinality $|\bigcup_i \mu^{-1}(E_i)|$. For each point $z\in {\Bbb P}^1$ lying over an exceptional divisor $E$, let $m_z$ be the multiplicity of $\mu^* E$ at $z$. The equations are obtained: $$\sum_{z\in \bigcup_i \mu^{-1}(E_i)} m_z = \sum_i a_i,$$ $$degree(A)= \sum_{z\in \bigcup_i \mu^{-1}(E_i)} (m_z-1) = -n+ \sum_i a_i.$$ The degree of $N_{{\Bbb P}^2}$ is $3d-2$. The degree of $N_{{\Bbb P}^2}/A= 3d-2 +n -\sum_i a_i = n-1$. Let $b=degree(B)$. Then, the degree of $N_{{\Bbb P}^2}/\tau$ is $n-1-b$. Note that $\mu$ is an immersion if and only if $b=0$. Without loss of generality, let $a_1\in\{1,2\}$. First consider the case $a_1=1$. There is a unique point $z_1$ in $\mu^{-1}(E_1)$. Let $v=\oplus_i v_i$ where $v_i\in T_{{\Bbb P}^2, x_i}$ satisfy: \begin{enumerate} \item[(i)] $v_1\neq 0$ in $\mu^* Q \otimes {\cal O}_{z_1}$. \item[(ii)] $v_i=0$ for $i\geq 2$. \end{enumerate} Since there exists a section $s$ of $H^0({\Bbb P}^1, N_{{\Bbb P}^2})$ with image $v\in H^0({\Bbb P}^1, \mu^*(Q))$, there must exist a nonzero section $\overline{s}$ of $H^0({\Bbb P}^1, N_{{\Bbb P}^2}/\tau)$ vanishing at (at least) $n-1$ points (all the $z$'s except $z_1$) by (iii). Therefore, $degree(N_{{\Bbb P}^2}/\tau) \geq n-1$. It follows that $b=0$. Next, consider the case $a_1=2$. There are two possiblities. Either $\mu^{-1}(E_1)$ consists of two points or one point. If there is a unique point in $\mu^{-1}(E_1)$, the argument proceeds exactly as in the $a_1=1$ case and $b=0$. Now suppose $\mu^{-1} (E_1)= \{z_1, z_2\}$. By Lemma (\ref{lemextra}), $\mu(z_1) \neq \mu(z_2)$. Let $v=\oplus_i v_i$ satisfy: \begin{enumerate} \item[(i)] $v_1\neq 0$ in $\mu^* Q \otimes {\cal O}_{z_1}$. \item[(ii)] $v_1=0$ in $\mu^* Q \otimes {\cal O}_{z_2}$. \item[(iii)] $v_i=0$ for $i\geq 2$. \end{enumerate} Such a selection of $v_1$ is possible since $T_{{\Bbb P}^2, x_1}$ surjects upon $\mu^* Q \otimes {\cal O}_{z_1} \oplus \mu^* Q \otimes {\cal O}_{z_2}$ for $\mu(z_1)\neq \mu(z_2)$. As before, there must exist a nonzero section $\overline{s}$ of $H^0({\Bbb P}^1, N_{{\Bbb P}^2}/\tau)$ vanishing at least $n-1$ points (all the $z$'s except $z_1$) by (iv). Therefore, $degree(N_{{\Bbb P}^2}/\tau) \geq n-1$. It follows that $b=0$. \end{pf} \begin{lem} \label{bee} Let $d>0$, $\alpha\geq 0$, $r\leq 8$, and $n_{d, \alpha}=0$. Then, $\overline{M}_{0,0}(X_r,(d, \alpha))$ is nonsingular of pure dimension 0. Moreover, the points of $\overline{M}_{0,0}(X_r, (d,\alpha))$ correspond to immersions of ${\Bbb P}^1$ in $X_r$. \end{lem} \begin{pf} Let $\mu: {\Bbb P}^1 \rightarrow X_r$ be a map in $\overline{M}_{0,0}(X_r,(d,\alpha))$. By Lemma \ref{lemfour}, $\mu$ is an immersion at the points of ${\Bbb P}^1$ mapping to the exceptional curves $E_i$. Suppose $p\in {\Bbb P}^1$ is a point where $\mu$ is not an immersion ($\mu(p) \notin E_i$). Since the number of blown-up points $x_1, \ldots , x_r$ is at most $8$, there is curve in the linear series $3H-\sum_{i=1}^{8} E_i$ passing through $\mu(p)$. Let $F$ denote this cubic (which may be reducible). There are now two cases. If $\mu({\Bbb P}^1)$ is not contained in any component of $F$, then ${\Bbb P}^1 \cdot \mu^*(F) \geq2$ because $\mu$ is not an immersion at $p$. This is a contradiction since the numerical assumption implies ${\Bbb P}^1 \cdot \mu^*(F) =1$. If $\mu({\Bbb P}^1)$ is contained in a component of $F$, then $d$ must equal 1,2, or 3 (since $\mu$ is birational). For these low degree cases, $\overline{M}_{0,0}(X_r,(d,\alpha))$ is empty unless $a_i=1$ for some $i$. Then, Lemma \ref{lemmy} yields a contradiction. We conclude $\mu$ is an immersion and $\overline{M}_{0,0}(X_r, (d, \alpha))$ is nonsingular. \end{pf} \subsection{Proof of Theorem (\ref{numtheorem})} First, the case $n_{d, \alpha}=0$ is considered. Since $d>0$, $\alpha \geq 0$, and $a_i \in \{1, 2\}$ (for some $i$), Lemma \ref{lemmy} shows that $\overline{M}_{0,0}(X_r, (d,\alpha))$ is a nonsingular set of points. By Proposition \ref{fb}, $N_{d, \alpha}$ equals the number of points in $\overline{M}_{0,0}(X_r, (d,\alpha))$. Moreover, by Lemma \ref{lemmy}, the points of $\overline{M}_{0,0}(X_r, (d,\alpha))$ represent immersions of ${\Bbb P}^1$. Theorem (\ref{numtheorem}) is established for classes $(d,\alpha)$ satisfying $n_{d,\alpha}=0$. Proceed now by induction on $n=n_{d, \alpha}$. If $n_{d,\alpha}>0$, consider the class $(d, (\alpha,1))$ on ${\Bbb P}^2$ blown-up at $r+1$ points $x_1, \ldots, x_{r+1}$. Certainly, $n_{d,(\alpha,1)}=n -1$. By property (P5) of section \ref{qcb}, $$N_{d, \alpha} = N_{d, (\alpha,1)}.$$ The class $(d,(\alpha,1))$ satisfies condition (ii) in the hypotheses of Theorem (\ref{numtheorem}). By induction, $N_{d, (\alpha,1)}$ equals the number of genus 0 stable maps of class $(d, (\alpha,1))$ passing through $n_{d,\alpha}-1$ points $p_1, \ldots, p_{n-1}$ in $X_{r+1}$. This is precisely equal to the number of stable maps of class $(d, \alpha)$ passing through the $n_{d, \alpha}$ points $p_1, \ldots, p_{n-1}, x_{r+1}$ in $X_r$ by Lemma \ref{lemtwo}. Since the solution curves are immersions in $X_{r+1}$, it follows easily that the corresponding curves in $X_r$ are also immersions. The proof of Theorem \ref{numtheorem} is complete. \section{Symmetries and Computations} \subsection{The Cremona transformation} \label{cremmy} Let $p_1, p_2, p_3$ be 3 non-collinear points in ${\Bbb P}^2$. Let $L_1, L_2, L_3$ be the 3 lines determined by pairs of points where $p_i, p_j \in L_k$ for distinct indices $i,j,k$. Let $S$ be the blow-up of ${\Bbb P}^2$ at the points $p_1, p_2,p_3$. Let $E_1,E_2,E_3$ be the exceptional divisors of this blow-up. Let $F_1, F_2, F_3$ be the strict transforms of the lines $L_1, L_2,L_3$. The $F_k$ are disjoint $(-1)$-curves on $S$ and can be blown-down. The resulting surface is another projective plane $\overline{{\Bbb P}}^2$. The blow-down maps are: \begin{equation} \label{crem} {\Bbb P}^2 \stackrel{e}{\leftarrow} S \stackrel{f}{\rightarrow} \overline{{\Bbb P}}^2. \end{equation} This is the classical Cremona transformation of the plane. Let $q_1,q_2,q_3\in \overline{{\Bbb P}}^2$ be the points $f(F_1),f(F_2), f(F_3)$. Let $H$ and $\overline{H}$ denote the hyperplane classes in $A_1({\Bbb P}^2)$ and $A_1(\overline{{\Bbb P}}^2)$ respectively. There are now 2 bases of $A_1(S)$ corresponding to the two blow-downs: $H, E_1, E_2, E_3$ and $\overline{H}, F_1, F_2, F_3$. The relationship between these bases is: \begin{align*}&dH-a_1E_1-a_2E_2-a_3E_3 =\\ &(2d-a_1-a_2-a_3)\overline{H} - (d-a_2-a_3)F_1 - (d-a_1-a_3)F_2 -(d-a_1-a_2)F_3. \end{align*} Let $x_4, \ldots, x_r \in {\Bbb P}^2$ be additional general points on ${\Bbb P}^2$ which correspond via the maps (\ref{crem}) to general points $s_4,\ldots, s_r\in S$ and $y_4,\ldots, y_r\in \overline{{\Bbb P}}^2$. The blow-up of $S$ at the points $s_4, \ldots, s_r$ may be viewed as a general blow-up of ${\Bbb P}^2$ at $p_1,p_2,p_3, x_4, \ldots, x_r$ or as a general blow-up of $\overline{{\Bbb P}}^2$ at $q_1, q_2, q_3, y_4, \ldots, y_r$. Let $G_4, \ldots, G_r$ denote the exceptional divisors of the blow-up of $S$. Since the class $dH- a_1 E_1 - a_2 E_2 -a_3 E_3 -\sum_{i=4}^r a_iG_i$ equals the class $$(2d-a_1-a_2-a_3)\overline{H} - (d-a_2-a_3)F_1 - (d-a_1-a_3)F_2 -(d-a_1-a_2)F_3-\sum_{i=4}^r a_iG_i,$$ the Gromov-Witten invariant $N_{d, \alpha}$ on the blow-up of ${\Bbb P}^2$ equals the invariant $N_{d', \alpha'}$ on the blow-up of $\overline{{\Bbb P}}^2$ where $$(d',\alpha')= (2d-a_1-a_2-a_3, (d-a_2-a_3,d-a_1-a_3,d-a_1-a_2,a_4,\ldots, a_r)).$$ It follows that $\overline{M}_{0,0}(X_r, (d, \alpha))$ is nonsingular if and only if $\overline{M}_{0,0}(X_r, (d', \alpha'))$ is nonsingular. Therefore, $N_{d, \alpha}$ is enumerative if and only if $N_{d', \alpha'}$ is enumerative. The Cremona symmetry of the Gromov-Witten invariants of $X_r$ is discussed in [C-M] from a slightly different perspective. For example, let $(d, \alpha)= (10, (4,4,3,3,3,3,3,3,3))=(10,(4^2,3^7))$ where the last equality is just notational convenience. Then, $n_{10,(4^2, 3^7)}= 30 -29-1=0$. The class $(10, (4^2,3^7)$ does not satisfying condition (i) or (ii) of Theorem (\ref{numtheorem}). Applying the Cremona transformation, $(d', \alpha')= (9, (3,3,2,3^6))$. Theorem (\ref{numtheorem}) applies to $(d', \alpha')$. Therefore, the moduli space $\overline{M}_{0,0}(X_r, (10, (4^2,3^7))$ is nonsingular (and all points correspond to immersions). $N_{10, (4^2,3^7)}=520$ is enumerative in this case. \subsection{Tables} \label{tbls} The arithmetic genus of the class $(d, \alpha)$ on $X_r$ is determined by: $$g_a(d, \alpha)= \frac{(d-1)(d-2)}{2} - \sum_{i=1}^{r} \frac {a_i(a_i-1)}{2}.$$ The arithmetic genus of a reduced, irreducible curve is non-negative. By Corollary \ref{cortwo}, $\overline{M}_{0,0}(X_r, (d, \alpha))$ is empty when $g_a(d,\alpha)<0$ and $n_{d,\alpha}=0$. A simple reduction to the case of expected dimension zero shows that $N_{d, \alpha}=0$ if $g_a(d, \alpha)<0$. If $a_i+a_j>d$ for indicies $i\neq j$, then $N_{d, \alpha}=0$ unless $(d, \alpha)=(1,(1,1))$. This follows again by a reduction to the expected dimension zero case. Then, Corollary \ref{cortwo} shows that $\overline{M}_{0,0}(X_r, (d, \alpha))$ is empty (unless $(d, \alpha)=(1,(1,1))$) by considering the intersection of a map with the line in ${\Bbb P}^2$ connecting the points $x_i$ and $x_j$. In the first table below, Gromov-Witten invariants $N_{d, \alpha}$ for $d\leq 5$ and $\alpha\geq 0$ are listed. By properties (P3), (P4), and (P5), it suffices to list the invariants for ordered sequences $\alpha$ satisfying $\alpha\geq 2$. Moreover, if $g_a(d,\alpha)<0$ or if $a_i+a_j>d$, the invariant vanishes and is omitted from the table. The invariants were computed by a Maple program via the recursive algorithm of the proof of Theorem \ref{recurr}. \begin{tabular}{|l|l|l|l|l|l|} \hline $d=1$&2 & 3 & 4 & 5 & 5 \\ \hline $N_1=1$ & $N_2=1$ & $N_3=12$ & $N_4=620$ &$N_5=87304$ &$N_{5, (2^6)}=1$ \\ & & $N_{3,(2)}=1$ & $N_{4, (2)}=96$ & $N_{5, (2)}= 18132$ &$N_{5, (3)}=640$ \\ & & & $N_{4, (2^2)}=12$ & $N_{5, (2^2)}=3510$ &$N_{5, (3,2)}=96$ \\ & & & $N_{4, (2^3)}=1$ & $N_{5, (2^3)}=620$ &$N_{5, (3,2^2)}=12$ \\ & & &$N_{4,(3)}=1$ & $N_{5, (2^4)}=96$ &$N_{5, (3,2^3)}=1$ \\ & & & & $N_{5, (2^5)}=12$ & $N_{5, (4)}=1$ \\ \hline \end{tabular} \vspace{+15pt} \noindent The Cremona transformation applied to the class $(5,(2,2,2))$ yields $N_{5,(2,2,2)}=N_{4,(1,1,1)}$. By Property (P5), $N_{4,(1,1,1)}=N_{4}=620$. The following table lists all the Gromov-Witten invariants for degrees $6$ and $7$ which are not obtained from lower degree numbers by the Cremona transformation. \vspace{+15pt} \begin{tabular}{|l|l|l|} \hline $d=6$ & 7 & 7\\ \hline $N_6=26312976$ & $N_7= 14616808192$ & $N_{7,(3,2)}=90777600$\\ $N_{6,(2)}=6506400$ & $N_{7,(2)}=4059366000$& $N_{7,(3,2^2)}=23133696$ \\ $N_{6,(2^2)}=1558272$ & $N_{7,(2^2)}=1108152240$& $N_{7,(3,2^3)}= 5739856$ \\ $N_{6,(2^3)}=359640$ & $N_{7,(2^3)}=296849546$ &$N_{7,(3,2^4)}=1380648$\\ $N_{6,(2^4)}=79416$ & $N_{7,(2^4)}=77866800 $ &$N_{7,(3,2^5)}= 320160$ \\ $N_{6,(2^5)}=16608$ & $N_{7,(2^5)}= 19948176$&$N_{7,(3,2^6)}=71040$\\ $N_{6,(2^6)}=3240$ & $N_{7,(2^6)}=4974460 $&$N_{7,(3,2^7)}=14928$\\ $N_{6,(2^7)}=576$ & $N_{7,(2^7)}= 1202355$&$N_{7,(3,2^8)}=2928$\\ $N_{6,(2^8)}=90$ & $N_{7,(2^8)}=280128$& $N_{7,(3^2)}=6508640$\\ $N_{6,(3)}=401172$ & $N_{7,(2^9)}=62450$& $N_{7,(4)}= 7492040$\\ $N_{6,(3,2)}=87544$ & $N_{7,(2^{10})}=13188$& $N_{7,(4,2)}= 1763415$\\ $N_{6,(4)}=3840$ & $N_{7,(3)}=347987200$& $N_{7,(5)}=21504$\\ \hline \end{tabular} \vspace{+15pt} In [D-I], the Gromov-Witten invariants of $X_6$ are computed. Our computation $N_{6,(2^6)}=3240$ disagrees with [D-I]. We have checked our number using different recursive strategies. Let $(d, \alpha)$ be a class for which all the hypotheses of Theorem \ref{numtheorem} and Lemma \ref{bee} fail. Then, $r\geq 9$, $3d=|\alpha|+1$, and $\alpha \geq 3$. Hence, $d\geq 10$. If $d=10$, then there are only two possible values (up to reordering) for $\alpha$: $(4^2, 3^7)$ or $(5, 3^8)$. The invariant $N_{10, (4^2, 3^7)}$ was show to be enumerative by the Cremona transformation in section \ref{cremmy}. Applying the transformation to $(10, (5,3^8))$ yields $(9,(4,2^2,3^6))$. Hence, $N_{10,(5,3^8)}= N_{9,(4,2^2,3^6)}=90$ is enumerative by Theorem \ref{numtheorem}. We have shown all invariants of degree $d\leq 10$ are enumerative. The only invariants of degree $11$ not proven to be enumerative by the methods of this paper correspond to the classes $(11,(5, 3^9))$ and $(11,(4^2,3^8))$. $N_{11,(5,3^9)}=707328$ and $N_{11,(4^2,3^8)}=2350228$. It is not known to the authors whether non-trivial multiplicities arise.

9611

alg-geom/9611013

en

https://arxiv.org/abs/alg-geom/9611013

alg-geom/9611013

Bas Edixhoven

Robert F. Coleman and Bas Edixhoven

On the semi-simplicity of the $U_p$-operator on modular forms

10 pages, hard copy available in a few days; send email to [email protected] LaTeX

Let $p$ be a prime number and $N$ an integer prime to $p$. We show that the operator $U_p$ on the space of cuspidal modular forms of level $pN$ and weight two is semi-simple. It follows from this that the Hecke algebra acting on the space of weight two forms of level $M$ is reduced if $M$ is cube free. Assuming Tate's conjecture for cycles on smooth projective varieties over finite fields, we generalize these results to higher weights. The main point in the proof is that the crystalline Frobenius of the reduction mod $p$ of the motive associated to a newform of level prime to $p$ and weight at least two cannot be a scalar. Assuming Tate's conjecture, it follows that Ramanujan's inequality is strict. For $N$ prime, we relate the discriminant of the weight two Hecke algebra to the height of the modular curve $X_0(N)$, for which we get an upper bound.

alg-geom

\section{\@startsection {section}{1}{\z@}{-3.5ex plus -1ex minus -.2ex}{2.3ex plus .2ex}{\large\bf}} \def\subsection{\@startsection{subsection}{2}{\z@}{-3.25ex plus -1ex minus -.2ex}{1.5ex plus .2ex}{\normalsize\bf}} \makeatother \makeatletter \newenvironment{eqn}{\refstepcounter{subsection} $$}{\leqno{\rm(\thesubsection)}$$\global\@ignoretrue} \makeatother \makeatletter \newenvironment{subeqn}{\refstepcounter{subsubsection} $$}{\leqno{\rm(\thesubsubsection)}$$\global\@ignoretrue} \makeatother \makeatletter \newenvironment{subeqarray}{\renewcommand{\theequation}{\thesubsubsection} \let\c@equation\c@subsubsection\let\cl@equation\cl@subsubsection \begin{eqnarray}}{\end{eqnarray}} \makeatother \makeatletter \def\@rmkcounter#1{\noexpand\arabic{#1}} \def\@rmkcountersep{.} \def\@beginremark#1#2{\trivlist \item[\hskip \labelsep{\bf #2\ #1.}]} \def\@opargbeginremark#1#2#3{\trivlist \item[\hskip \labelsep{\bf #2\ #1\ (#3).}]} \def\@endremarkwithsquare{~\hspace{\fill}~$\Box$\endtrivlist} \makeatother \newenvironment{prf}[1]{\trivlist \item[\hskip \labelsep{\it #1.\hspace*{.3em}}]}{~\hspace{\fill}~$\Box$\endtrivlist} \newenvironment{proof}{\begin{prf}{\bf Proof}}{\end{prf}} \let\tempcirc=\mathord{\raise0.25ex\hbox{$\scriptscriptstyle\tempcirc$}} \def\mathord{\raise0.25ex\hbox{$\scriptscriptstyle\tempcirc$}}{\mathord{\raise0.25ex\hbox{$\scriptscriptstyle\tempcirc$}}} \def\nobreak\hskip.1111em\mathpunct{}\nonscript\mkern-\thinmuskip{:{\nobreak\hskip.1111em\mathpunct{}\nonscript\mkern-\thinmuskip{:} \hskip .3333emplus.0555em\relax} \setlength{\textheight}{250mm} \setlength{\textwidth}{170mm} \setlength{\evensidemargin}{0in} \setlength{\oddsidemargin}{0in} \setlength{\topmargin}{-2cm} \newcommand{{\bf Z}}{{\bf Z}} \newcommand{{\bf Q}}{{\bf Q}} \newcommand{{\bf R}}{{\bf R}} \newcommand{{\bf C}}{{\bf C}} \newcommand{{\bf F}}{{\bf F}} \newcommand{{\bf H}}{{\bf H}} \newcommand{{\bf T}}{{\bf T}} \newcommand{{\rm SL}}{{\rm SL}} \newcommand{{\rm GL}}{{\rm GL}} \newcommand{\overline{\QQ}}{\overline{{\bf Q}}} \newcommand{{\rm End}}{{\rm End}} \newcommand{{\rm Gal}}{{\rm Gal}} \newcommand{{\rm H}}{{\rm H}} \newcommand{{\rm et}}{{\rm et}} \newcommand{{\rm crys}}{{\rm crys}} \newcommand{\varepsilon}{\varepsilon} \newcommand{{\rm Frob}}{{\rm Frob}} \newcommand{{\rm Fil}}{{\rm Fil}} \newcommand{{\rm DR}}{{\rm DR}} \newcommand{\langle}{\langle} \newcommand{\rangle}{\rangle} \newcommand{{{\rm D}_\crys}}{{{\rm D}_{\rm crys}}} \newcommand{{\rm new}}{{\rm new}} \newcommand{{\rm old}}{{\rm old}} \newcommand{{\rm discr}}{{\rm discr}} \newcommand{{\rm trace}}{{\rm trace}} \newcommand{{\rm Ar}}{{\rm Ar}} \newcommand{{\rm vol}}{{\rm vol}} \newcommand{{\rm Spec}}{{\rm Spec}} \newtheorem{theorem}[subsection]{Theorem.} \newtheorem{proposition}[subsection]{Proposition.} \newtheorem{lemma}[subsection]{Lemma.} \newtheorem{corollary}[subsection]{Corollary.} \newtheorem{tabel}[subsection]{Table.} \newtheorem{conjecture}[subsection]{Conjecture.} \newtheorem{sublemma}[subsubsection]{Lemma.} \newtheorem{subtheorem}[subsubsection]{Theorem.} \newremark{remark}[subsection]{Remark} \newremark{subremark}[subsubsection]{Remark} \newremark{notation}[subsection]{Notation} \newremark{example}[subsection]{Example} \newremark{definition}[subsection]{Definition} \renewcommand{\baselinestretch}{1.2} \begin{document} \title{On the semi-simplicity of the $U_p$-operator on modular forms.} \author{Robert F.\ Coleman \and Bas Edixhoven\thanks{partially supported by the Institut Universitaire de France}} \maketitle \section{Introduction.}\label{section1} For $N$ and $k$ positive integers, let $M^0(N,k)_{\bf C}$ denote the ${\bf C}$-vector space of cuspidal modular forms of level $N$ and weight~$k$. This vector space is equipped with the usual Hecke operators $T_n$, $n\geq1$. If we need to consider several levels or weights at the same time, we will denote this $T_n$ by $T_n^N$, or~$T_n^{N,k}$. If $p$ is a prime number dividing $N$, our $T_p$ is also known under the name~$U_p$. One of our main results can be stated very easily: if $k=2$ and $p^3$ does not divide $N$, then the operator $T_p$ is semi-simple. We can prove the same result for weight $k\geq3$, under the assumption that certain crystalline Frobenius elements are semi-simple. Milne has shown in \cite[\S2]{Milne1} that this semi-simplicity is implied by Tate's conjecture claiming that for $X$ projective and smooth over a finite field of characteristic $p$, and $r\geq0$, $\dim_{\bf Q}({\rm CH}^r(X)/{\equiv}_{\rm num})$ equals the order of $\zeta(X,s)$ at~$r$. Ulmer proved in \cite{Ulmer1} that $T_p$ is semi-simple, for $k=3$ and $p^2$ not dividing~$N$, under the assumption of the Birch-Swinnerton-Dyer conjecture for elliptic curves over function fields in characteristic~$p$. His method is quite different from ours: assuming that $T_p$ is not semi-simple, he really shows that the Birch-Swinnerton-Dyer conjecture does not hold for an explicitly given elliptic curve. The structure of our proof is as follows. Using the theory of newforms, the problem is shown to be equivalent to the problem of showing that, for a normalized newform $f$ of weight $k$, prime-to-$p$ level and character $\varepsilon$, the polynomial $x^2-a_px+\varepsilon(p)p^{k-1}$ has no double root. This polynomial happens to be the characteristic polynomial of the Frobenius element at $p$ in the two-dimensional Galois representations associated to~$f$; it is also the characteristic polynomial of the crystalline Frobenius asociated to~$f$. We show that this crystalline Frobenius cannot be a scalar. In Sections~\ref{section2} and \ref{section3} we prove the results concerning these Frobenius elements for $k=2$ and $k\geq2$, respectively. Section~\ref{section2} is quite elementary, whereas in Section~\ref{section3} we use a lot of the machinary for comparing $p$-adic etale and crystalline cohomology. In Section~\ref{section4} we give some applications: the Ramanujan inequality is a strict inequality in certain cases, certain Hecke algebras are reduced, hence have non-zero discriminant. Section~\ref{section5} gives some results, due to Abbes and Ullmo, concerning the discriminants of certain Hecke algebras. To end this introduction, let us explain why the case $k=1$ is completely different. Consider a normalized cuspidal eigenform $f=\sum a_nq^n$ of some level $N$, of weight one and with some character~$\varepsilon$. Deligne and Serre have shown (\cite[\S4]{DeligneSerre}) that there exists a continuous representation $\rho_f$ from ${\rm Gal}(\overline{\QQ}/{\bf Q})$ to ${\rm GL}_2({\bf C})$, unramified outside $N$, such that, for all primes $p$ not dividing $N$, the characteristic polynomial of a Frobenius element at $p$ is $x^2-a_px+\varepsilon(p)$. Since the image of $\rho_f$ is finite, Chebotarev's density theorem gives the existence of primes $p$ not dividing $N$ such that the Frobenius element at $p$ is the identity element, hence has characteristic polynomial $(x-1)^2$. \section{An elementary proof in the case of weight two.}\label{section2} \begin{theorem}\label{thm2.1} Let $f=\sum a_nq^n$ be a cuspidal normalized eigenform of weight two, some level $N$ and character $\varepsilon\colon({\bf Z}/N{\bf Z})^*\to{\bf C}^*$. Let $p$ be a prime number not dividing~$N$. Then the polynomial $x^2-a_px+\varepsilon(p)p$ has simple roots. \end{theorem} \begin{proof} The proof is by contradiction, so we suppose that the polynomial has a double root~$\lambda$. Then of course we have $\lambda^2=\varepsilon(p)p$ and $2\lambda=a_p$. Let $K$ be the finite extension of ${\bf Q}$ generated by the $a_n$ and the $\varepsilon(a)$, and let $O_K$ be its ring of integers. Let $J$ denote the jacobian variety of the modular curve $X_1(N)$ over~${\bf Q}$. We identify the space $M^0(N,2)_{\bf Q}$ of weight two cuspforms of level $N$ and with coefficients in ${\bf Q}$ with the cotangent space at the origin of~$J$; this is compatible with the action of the Hecke operators. Let ${\bf T}$ be the subring of ${\rm End}(J)$ that is generated by the $T_n$, $n\geq1$, and the diamond operators $\langle a\rangle$, $a\in({\bf Z}/N{\bf Z})^*$. Let $I$ be the annihilator of $f$ in ${\bf T}$, and let $A'_{\bf Q}:=J/IJ$ be the quotient of $J$ by its subvariety generated by the images of all elements of~$I$. It is well known that $A'_{\bf Q}$ has dimension $[K:{\bf Q}]$, and that for every prime number $l$, the free $K\otimes{\bf Q}_l$-module $V_l(A'_{\bf Q})$ of rank two gives the $l$-adic Galois representation $\rho_{f,l}$ associated to~$f$. We prefer to work with an abelian variety $A_{\bf Q}$ that is isogeneous to $A'_{\bf Q}$ and on which we have an action of all of~$O_K$. This is easily done: define $A_{\bf Q}:=O_K\otimes_{\bf T} A'_{\bf Q}$, where the tensor product should be calculated by taking a presentation of~$O_K$. The abelian variety $A_{\bf Q}$ has good reduction at $p$; let $A_{{\bf Z}_p}$ denote the corresponding abelian scheme over~${\bf Z}_p$. Consider the first algebraic de Rham cohomology group $M:={\rm H}^1_{\rm DR}(A_{{\bf Z}_p}/{\bf Z}_p)$. It is a free ${\bf Z}_p$-module of rank $2[K:{\bf Q}]$, equipped with its Hodge filtration: \begin{eqn}\label{eqn2.2} M = {\rm Fil}^0M \supset {\rm Fil}^1 M = {\rm H}^0(A_{{\bf Z}_p},\Omega^1). \end{eqn} The submodule ${\rm Fil}^1M$ is free of rank $[K:{\bf Q}]$ as ${\bf Z}_p$-module, and has the property that ${\rm Fil}^0M/{\rm Fil}^1M$ is torsion free. The double root $\lambda$ of $x^2-a_px+\varepsilon(p)p$ is in $O_K$, since it is integral and $2\lambda$ is in~$K$. In the endomorphism ring of $A_{{\bf F}_p}$ we have the Eichler-Shimura congruence relation: \begin{eqn}\label{eqn2.3} 0 = ({\rm Frob}_p-{\rm Frob}_p)({\rm Frob}_p-{\rm Frob}_p') = {\rm Frob}_p^2-a_p{\rm Frob}_p+\varepsilon(p)p = ({\rm Frob}_p-\lambda)^2, \end{eqn} where ${\rm Frob}_p$ is the Frobenius endomorphism and ${\rm Frob}_p'$ the Verschiebung, multiplied by~$\varepsilon(p)$. The fact that every abelian variety over ${\bf F}_p$ is isogeneous to a product of simple ones implies that ${\rm Frob}_p$ is semi-simple in the sense that it satisfies an identity of the form $P({\rm Frob}_p)=0$, with $P$ a polynomial with coefficients in ${\bf Q}$ that has simple roots. It follows that ${\rm Frob}_p=\lambda$ in ${\rm End}(A_{{\bf F}_p})$. Since $O_K\otimes{\bf Z}_p$ is a product of a finite number of discrete valuation rings, ${\rm Fil}^1M$ is a locally free module over it; it is in fact free of rank one (consider ${\bf Q}\otimes{\rm Fil}^1M$). It follows that $\lambda$ does not annihilate ${\bf F}_p\otimes{\rm Fil}^1M$, since we have $\lambda^2=\varepsilon(p)p$. But ${\bf F}_p\otimes{\rm Fil}^1M$ is the same as ${\rm H}^0(A_{{\bf F}_p},\Omega^1)$, and on this module $\lambda$ acts as ${\rm Frob}_p^*$, hence it does annihilate. This contradiction finishes the proof. \end{proof} \section{The general case.}\label{section3} In this section we try to generalize Theorem~\ref{thm2.1} as much as we can to higher weights. For doing that we replace the module $M$ of Section~\ref{section2} by the $p$-adic crystalline realization of the motive associated to~$f$; this gives us a filtered $\phi$-module $M$ of rank two. The comparison theorem for crystalline and $p$-adic etale cohomology implies that this filtered $\phi$-module is weakly admissible, from which it follows immediately that the crystalline Frobenius $\phi$ cannot be a scalar. Unfortunately, it is not known that $\phi$ is semi-simple, so all we show is that semi-simplicity of $\phi$ implies that the polynomial $x^2-a_px+\varepsilon(p)p^{k-1}$ has simple roots. Let $f=\sum a_nq^n$ be a normalized cuspidal newform of some level $N$, weight $k\geq2$ and character~$\varepsilon$. Let $K$ be the field generated by the $a_n$ and the~$\varepsilon(a)$. Let $p$ be a prime number not dividing~$N$. Our first objective is to construct the $p$-adic crystalline realization of the motive associated to~$f$. In \cite{Scholl1}, Scholl constructs a Grothendieck motive $M(f)$ over ${\bf Q}$, with coefficients in $K$, such that for every prime number $l$ the Galois representation $\rho_{f,l}\colon G_{\bf Q}\to{\rm GL}_2({\bf Q}_l\otimes K)$ is the dual of the $l$-adic realization~${\rm H}_l(M(f))$. Concretely, he constructs a projector in the group ring of a finite group of automorphisms of the smooth and projective model $X$ of the $k{-}2$-fold fibered product of the universal elliptic curve over $Y(N')$ (with $N'$ a suitable multiple of $N$) constructed by Deligne in \cite[\S5]{Deligne1}, such that the $l$-adic and Betti realizations of the corresponding Chow motive are, in a way that is compatible with Hecke operators, the parabolic cohomology groups used in~\cite{Deligne1}. The Grothendieck motive associated to this Chow motive (i.e., one replaces rational equivalence by homological equivalence) has an action by the Hecke algebra of the space of cuspidal modular forms of weight $k$ on the modular curve $X(N')_{\bf Q}$; $M(f)$ is a suitable factor. The variety $X$ over ${\bf Q}$ has a smooth projective model over ${\bf Z}[1/N']$ (see \cite[4.2.1]{Scholl1} and \cite{Deligne1}), hence $M(f)$ has a crystalline realization $M:={\rm H}_{\rm crys}(M(f))$ which is a free ${\bf Q}_p\otimes K$-module of rank two equipped with an endomorphism $\phi$, the crystalline Frobenius, that is induced by the Frobenius endomorphism of the reduction mod $p$ of~$X$. The characteristic polynomial of $\phi$ is $x^2-a_px+\varepsilon(p)p^{k-1}$; this can be shown in the same way as one can show it for the $l$-adic realizations, or one invokes a result of Katz and Messing (see \cite[4.2.3]{Scholl1}). \begin{theorem}\label{thm3.1} Let $f=\sum a_nq^n$ be a normalized cuspidal newform of some level $N$, weight $k\geq2$ and character~$\varepsilon$. Let $p$ be a prime number not dividing~$N$. Then the Frobenius $\phi$ of the crystalline realization $M$ of the motive $M(f)$ is not scalar, i.e., it is not in ${\bf Q}_p\otimes K$. \end{theorem} \begin{proof} The proof is by contradiction. We suppose that $\phi$ is an element, $\lambda$ say, of ${\bf Q}_p\otimes K$. The comparison theorem for crystalline and de Rham cohomology for smooth proper ${\bf Z}_p$-schemes (see \cite[\S1.3]{Illusie1}) gives us an isomorphism between $M$ and ${\bf Q}_p\otimes {\rm H}_{\rm DR}(M(f))$, and hence a decreasing filtration (the Hodge filtration) ${\rm Fil}$ on~$M$. So $(M,\phi,{\rm Fil})$ is an object of the category of filtered $\phi$-modules. (A filtered $\phi$-module is a finite dimensional ${\bf Q}_p$-vector space $M$ with a decreasing, exhaustive and separating filtration ${\rm Fil}^i$, $i\in{\bf Z}$, and an endomorphism $\phi$; morphisms are linear maps respecting ${\rm Fil}$ and~$\phi$; see~\cite[\S2.3]{Illusie1}.) The Hodge filtration on ${\rm H}_{\rm DR}(M(f))$ induces the Hodge decomposition of ${\bf C}\otimes{\rm H}_{\rm B}(M(f))$, which is of type $(k{-}1,0),(0,k{-}1)$, hence ${\rm Fil}^0(M)=M$, ${\rm Fil}^1(M)={\rm Fil}^{k-1}(M)$ is free of rank one, and ${\rm Fil}^k(M)=0$. Fontaine has constructed Grothendieck's ``mysterious functor'' ${{\rm D}_\crys}$ from the category of finite dimension ${\bf Q}_p$-vector spaces with continuous ${\rm Gal}(\overline{\QQ}_p/{\bf Q}_p)$-action to the category of filtered $\phi$-modules (see the introduction of~\cite{Illusie1}). It is a theorem of Faltings (see \cite[\S3.2]{Illusie1}), of which a special case was proved earlier by Fontaine and Messing, that there is an isomorphism of filtered $\phi$-modules between $M$ and ${{\rm D}_\crys}({\rm H}_p(M(f)))$. In fact, the theorem is stated for varieties, but since the isomorphism is compatible with the multiplicative structure and with cycle classes, it also works for Grothendieck motives. The most important consequence of this theorem for us is that the filtered $\phi$-module $M$ is admissible, hence weakly admissible, in the sense of Fontaine, see \cite[4.4.6]{Fontaine1}. Recall that to a filtered $\phi$-module $M$ one associates two polygons: the Hodge polygon, depending only on the filtration, and the Newton polygon, depending only on~$\phi$. Weakly admissible means that for every subobject $M'$ of $M$ the Newton polygon lies above or on the Hodge polygon, and that the two polygons for $M$ itself have the same end-point. An equivalent formulation is the following. For $M$ a filtered $\phi$-module let $t_N(M)$ be the $p$-adic valuation of the determinant of $\phi$, and let $t_H(M)$ be the maximal $i$ such that ${\rm Fil}^i(\det M)\neq0$. Then $M$ is weakly admissible if and only if firstly: $t_N(M)=t_H(M)$, and secondly: for all subobjects $M'$ of $M$ one has $t_H(M')\leq t_N(M')$. Consider now our weakly admissible filtered $\phi$-module~$M$. Since $\phi$ is the element $\lambda$ of ${\bf Q}_p\otimes K$, we have the subobject $M':={\rm Fil}^{k-1}(M)$ of $M$ (we give it the induced filtration). Then $t_H(M')=[K:{\bf Q}](k-1)$, whereas $t_N(M')=[K:{\bf Q}](k-1)/2$ (recall that $M'$ is free of rank one over ${\bf Q}_p\otimes K$ and that $\lambda^2=\varepsilon(p)p^{k-1}$). Since $k\geq2$, this contradicts the weak admissibility of~$M$. \end{proof} \begin{corollary}\label{cor3.2} Let $f=\sum a_nq^n$ be a normalized cuspidal eigenform of some level $N$, weight $k\geq2$ and character~$\varepsilon$. Let $p$ be a prime number not dividing $N$ and suppose that the crystalline Frobenius $\phi$ of the ${\bf Q}_p$-vector space $M(f)$ is semi-simple. Then the polynomial $x^2-a_px+\varepsilon(p)p^{k-1}$ has simple roots. \end{corollary} \begin{proof} The proof is by contradiction: we suppose that $\lambda\in K$ is a double root. As we have already said above, the polynomial in question is the characteristic polynomial of the endomorphism $\phi$ of the free rank two ${\bf Q}_p\otimes K$-module~$M$. Hence it satisfies the identity $(\phi-\lambda)^2=0$. Now ${\bf Q}_p\otimes K$ is a finite product of fields, hence the semi-simplicity of $\phi$ implies that $\phi$ is multiplication by~$\lambda$. But this contradicts Theorem~\ref{thm3.1}. \end{proof} \begin{remark}\label{remark3.3} The first three lines of \cite[\S2]{Milne1} show that Tate's conjecture mentioned in Section~\ref{section1} implies the semi-simplicity of $l$-adic and crystalline Frobenius elements of smooth projective varieties over finite fields. \end{remark} \begin{remark}\label{remark3.4} Scholl remarks that his explicit construction of the crystalline realization $M$ of $M(f)$ in \cite{Scholl2} should show directly that $M$ is weakly admissible. \end{remark} \section{Applications.}\label{section4} \begin{theorem}\label{thm4.0} Let $N\geq1$ and $k\geq2$ be integers. Let $f=\sum a_nq^n$ be a normalized cuspidal eigenform of level $N$ and weight~$k$. Let $p$ be a prime number not dividing~$N$. If $k>2$ assume Tate's conjecture mentioned in Section~\ref{section1}. Then we have $|a_p|<2p^{(k-1)/2}$. \end{theorem} \begin{proof} Let $\varepsilon$ be the character of~$f$. Theorems~\ref{thm2.1} and~\ref{thm3.1} show that $x^2-a_px+\varepsilon(p)p^{k-1}$ has no double root. Deligne has shown (\cite[Thm.~6.1]{DeligneSerre} and \cite[Thm.~1.6]{Deligne2}) that the roots have absolute value $p^{(k-1)/2}$. \end{proof} \begin{theorem}\label{thm4.1} Let $N\geq1$ and $k\geq2$ be integers. Let $p$ be a prime number such that $p^3$ does not divide~$N$. Assume Tate's conjecture mentioned in Section~\ref{section1} if $k\geq3$ and $p|N$. Then the endomorphism $T_p$ of $M^0(N,k)_{\bf C}$ is semi-simple. \end{theorem} \begin{proof} For the sake of notation, let $N\ge1$ be an integer and $p$ a prime number not dividing~$N$. Let $k\geq1$. In this case $T_p$ is normal with respect to the Petersson scalar product on $M^0(N,k)_{\bf C}$. Hence $T_p$ is diagonalizable. Let us now consider $M^0(pN,k)_{\bf C}$, with $k\geq2$. By the theory of newforms, $M^0(pN,k)_{\bf C}$ is the direct sum of its $p$-new part $M^0(pN,k)_{\bf C}^{p{\rm new}}$ and its $p$-old part $M^0(pN,k)_{\bf C}^{p{\rm old}}$, this decomposition being respected by all Hecke operators. The restriction of $T^{pN,k}_p$ to $M^0(pN,k)_{\bf C}^{p{\rm new}}$ is normal, hence diagonalizable. The $p$-old part is isomorphic to the direct sum of two copies of $M^0(N,k)_{\bf C}$, via the map $(f(q),g(q))\mapsto f(q)+g(q^p)$. The restriction of $T^{pN,k}_p$ to $M^0(pN,k)_{\bf C}^{p{\rm old}}$ is then given by the following two by two matrix: \begin{subeqn}\label{eqn4.1.1} T^{pN,k}_p|_{M^0(pN,k)_{\bf C}^{p{\rm old}}} = \left( \begin{array}{cc} T^{N,k}_p & 1\\ -p^{k-1}\langle p\rangle & 0 \end{array} \right). \end{subeqn} We have already seen that $M^0(N,k)_{\bf C}$ is the direct sum of its common eigenspaces $V_{a_p,\varepsilon}$ for $T_p$ and the diamond operators. It follows that $M^0(pN,k)_{\bf C}^{p{\rm old}}$ decomposes as a direct sum of terms $V_{a_p,\varepsilon}^2$, and that the restriction of $T^{pN,k}_p$ to each of the $V_{a_p,\varepsilon}^2$ is annihilated by $x^2-a_px+\varepsilon(p)p^{k-1}$. Under the hypotheses of the theorem we are proving, these polynomials have simple roots by Theorems~\ref{thm2.1} and~\ref{thm3.1}. Let us now consider $M^0(p^2N,k)_{\bf C}$, with $k\geq2$. Here too this space is the direct sum of its $p$-old and $p$-new parts. On the $p$-new part $T^{p^2N,k}_p$ is self-adjoint, hence diagonalizable. The $p$-old part is now isomorphic to the direct sum of three copies of $M^0(N,k)_{\bf C}$ and two copies of $M^0(pN,k)_{\bf C}^{p{\rm new}}$. The restrictions of $T^{p^2N,k}_p$ to $M^0(N,k)_{\bf C}^3$ and $(M^0(pN,k)_{\bf C}^{p{\rm new}})^2$ are given by the following matrices: \begin{subeqn}\label{eqn4.1.2} \left( \begin{array}{ccc} T^{N,k}_p & 1 & 0 \\ -p^{k-1}\langle p\rangle & 0 & 1\\ 0 & 0 & 0 \end{array} \right), \qquad \left( \begin{array}{cc} T^{pN,k}_p & 1\\ 0 & 0 \end{array} \right) \end{subeqn} One can now repeat the same type of argument as above, invoking Theorems~\ref{thm2.1} and \ref{thm3.1} to see that, under the hypotheses of the theorem we are proving, $x(x^2-a_px+\varepsilon(p)p^{k-1})$, with $(a_p,\varepsilon(p))$ as before, has simple roots. The space $M^0(pN,k)_{\bf C}^{p{\rm new}}$ is a direct sum of eigenspaces for $T^{Np,k}_p$, and one knows that the eigenvalues are non-zero (see~\cite[\S1.8]{DeligneSerre}). It follows that also the restriction of $T^{p^2N,k}_p$ to $(M^0(pN,k)_{\bf C}^{p{\rm new}})^2$ is diagonalizable. \end{proof} \begin{corollary}\label{cor4.2} Let $N\geq1$ be cube free, and let $k\geq2$. Let ${\bf T}$ be the ${\bf Z}$-algebra generated by the endomorphisms $T_n$, $n\geq1$, and $\langle a\rangle$, $a\in({\bf Z}/N{\bf Z})^*$, of $M^0(N,k)_{\bf C}$. Assume Tate's conjecture mentioned in Section~\ref{section1} if $k>2$. Then the ring ${\bf T}$ is reduced. \end{corollary} \begin{proof} This is so because ${\bf T}$ is a subring of the ${\bf C}$-algebra ${\bf T}_{\bf C}:={\bf C}\otimes{\bf T}$ generated by the $T_n$ and~$\langle a\rangle$. Theorem~\ref{thm4.1} tells us that the $T_p$ and $\langle a\rangle$ can be simultaneously diagonalized. They generate ${\bf T}_{\bf C}$, hence ${\bf T}_{\bf C}$ is a product of copies of ${\bf C}$, hence reduced. \end{proof} \begin{remark}\label{rmk4.3} For general $N$ and $k$, the Hecke algebra ${\bf T}$ is well known to be a free ${\bf Z}$-module; ${\bf T}_{\bf Q}:={\bf Q}\otimes{\bf T}$ is well known to be Gorenstein, i.e., its ${\bf Q}$-linear dual $({\bf Q}\otimes{\bf T})^\vee$ is free of rank one as ${\bf Q}\otimes{\bf T}$-module, see for example~\cite[p.~481]{Wiles1}. One way to prove this is as follows. By the $q$-expansion principle, $M^0(N,k)_{\bf C}^\vee$ is free of rank one as ${\bf T}_{\bf C}$-module. Then one constructs a ${\bf T}_{\bf C}$-bilinear ${\bf C}$-valued pairing on $M^0(N,k)_{\bf C}$ to get an isomorphism of ${\bf T}_{\bf C}$-modules between $M^0(N,k)_{\bf C}$ and its dual. Another way to prove it is to use the theory of new forms. This last proof gives more information on how exactly ${\bf T}_{\bf C}$ decomposes as a product of ${\bf C}$-algebras. Parent needed such information and so he worked out the details in~\cite{Parent1}. His work made see that Theorem~\ref{thm4.1} should be stated for $N$ cube free instead of square free. Of course, statements that completions of ${\bf T}$ at certain of its maximal ideals are Gorenstein are much more subtle and harder to prove (see for example~\cite[\S2.1]{Wiles1}). \end{remark} \section{Discriminants of Hecke algebras.}\label{section5} According to Corollary~\ref{cor4.2}, certain Hecke algebras ${\bf T}$ are reduced. This means that the discriminants ${\rm discr}({\bf T})$ of their trace forms $(x,y)\mapsto {\rm trace}(xy)$ are non-zero. These discriminants ``count'' all congruences between different eigenforms of fixed level and weight, hence are quite useless for dealing with congruences with a fixed form (in particular, nothing interesting can be said on the degrees of modular parametrizations of elliptic curves over ${\bf Q}$). The following result, relating such discriminants to heights of modular curves, is due to Abbes and Ullmo (unpublished). \begin{theorem}[Abbes, Ullmo]\label{thm5.1} Let $p$ be a prime number, and let ${\bf T}$ be the Hecke algebra associated to $M^0(p,2)_{\bf C}={\rm H}^0(X_0(p)_{\bf C},\Omega)$. Then one has: $$ h(X_0(p)_{\bf Q}) = \frac{1}{2}\log|{\rm discr}({\bf T})| - \sum_{i=1}^g \log\|\omega_i\|, $$ where $h$ is the modular height of curves over ${\bf Q}$ (see \cite[\S3.3]{Szpiro1}), where $\omega_1,\ldots,\omega_g$ are the normalized eigenforms and $\|\cdot\|$ the norm of the scalar product $\langle\omega|\eta\rangle= (i/2)\int_{X_0(p)({\bf C})}\omega\wedge\overline{\eta}$ on ${\rm H}^0(X_0(p)_{\bf C},\Omega)$. \end{theorem} \begin{proof} We start by recalling the definition of~$h$. So let $X_{\bf Q}$ be a smooth proper geometrically irreducible curve over~${\bf Q}$, of some genus~$g$. Let $J_{\bf Q}$ be its jacobian and $J$ the N\'eron model over~${\bf Z}$. Then we have the free ${\bf Z}$-module of rank one $\omega_J:=\Lambda^g0^*\Omega^1_{J/{\bf Z}}$, with the scalar product on ${\bf C}\otimes\omega_J$ given by $\langle\omega|\eta\rangle= (i/2)^g(-1)^{g(g-1)/2}\int_{J({\bf C})}\omega\wedge\overline{\eta}$. The height $h(X_{\bf Q})$ is then defined to be the Arakelov degree of this metrized line bundle: $h(X_{\bf Q})=\deg_{\rm Ar}(\omega_J)=-\log\|\omega\|$, with $\omega$ a generator of~$\omega_J$. The ${\bf Z}$-module $\omega_J$ is equipped with the scalar product on ${\bf C}\otimes\omega_J={\rm H}^0(X_0(p)_{\bf C},\Omega)$ already mentioned in the theorem above. This scalar product induces a real scalar product, and hence a volume form, on ${\bf R}\otimes\omega_J$ (the volume form being determined by the condition that a cube with edges of length one has volume one). A calculation (see \cite[lemme~3.2.1]{Szpiro1}) shows that one has: \begin{subeqn}\label{eqn5.1.1} h(X_{\bf Q}) = -\log{\rm vol}({\bf R}\otimes\omega_J/\omega_J). \end{subeqn} Let now $X$ be the curve~$X_0(p)_{\bf Q}$. In that case, $\omega_J$ is the same as ${\rm H}^0(X_{\bf Z},\Omega)$, with $X_{\bf Z}$ the usual model over ${\bf Z}$ (semi-stable, $X_{{\bf F}_p}$ consisting of two irreducible components) and $\Omega$ its dualizing sheaf (\cite[Ch.~II, \S3]{Mazur1}). The fact that the two irreducible components of $X_{{\bf F}_p}$ are of genus zero implies that the pairing \begin{subeqn}\label{eqn5.1.2} {\bf T} \times \omega_J \to {\bf Z},\quad (t,\omega)\mapsto a_1(t\omega), \end{subeqn} with $a_1$ denoting the linear form on $\omega_J$ that takes the coefficient of $q$ in the $q$-expansion, is perfect, i.e., it induces an isomorphism between ${\bf T}^\vee$ of ${\bf T}$ and $\omega_J$. Let $\omega_1,\ldots,\omega_g$ be as in the theorem. Sending an element of ${\bf T}_{\bf R}$ to the eigenvalues of the $\omega_i$ for it is an isomorphism of ${\bf R}$-algebras: \begin{subeqn}\label{eqn5.1.3} {\bf T}_{\bf R} \to {\bf R}^g, \quad t\mapsto (a_1(t\omega_1,\ldots,a_1(t\omega_g)). \end{subeqn} The trace form on ${\bf T}$ corresponds to the standard scalar product on~${\bf R}^g$. Composing the dual of the isomorphism (\ref{eqn5.1.3}) with the isomorphism ${\bf T}_{\bf R}^\vee\to{\bf R}\otimes\omega_J$ from (\ref{eqn5.1.2}) gives an isomorphism from ${\bf R}^g$ to ${\bf R}\otimes\omega_J$ mapping the $i$th standard basis vector $e_i$ to~$\omega_i$. It follows that the volume form on ${\bf R}\otimes\omega_J$ corresponds to $\prod_i\|\omega_i\|$ times the one on ${\bf T}_{\bf R}^\vee$ corresponding to the trace form. We find: \begin{subeqn}\label{eqn5.1.4} {\rm vol}({\bf R}\otimes\omega_J/\omega_J) = \left(\prod_{i=1}^g \|\omega_i\|\right){\rm vol}({\bf T}_{\bf R}^\vee/{\bf T}) = \left(\prod_{i=1}^g \|\omega_i\|\right){\rm vol}({\bf T}_{\bf R}/{\bf T})^{-1}. \end{subeqn} The proof is finished since ${\rm vol}({\bf T}_{\bf R}/{\bf T})=|{\rm discr}({\bf T})|^{1/2}$. \end{proof} \begin{theorem}[Abbes, Ullmo]\label{thm5.2} For every $\varepsilon>0$ there exists $c(\varepsilon)$ in ${\bf R}$ such that for all prime numbers $p$ one has $h(X_0(p)_{\bf Q})\leq c(\varepsilon)p^{1+\varepsilon}$. \end{theorem} \begin{proof} Let ${\bf T}$ and $g$ be as above, and let ${\bf T}':=\sum_{i=1}^g{\bf Z} T_i$. Then ${\bf T}'$ is of finite index in ${\bf T}$ because $\infty$ is not a Weierstrass point of $X_0(p)$ (see \cite[\S3]{LehnerNewman} or \cite[\S4]{Edixhoven1}). The image of $T_i$ under the isomorphism~(\ref{eqn5.1.3}) is $(a_i(\omega_1),\ldots,a_i(\omega_g))$. It follows that we have the equalities: \begin{subeqn}\label{eqn5.2.1} {\rm discr}({\bf T}) = {\rm discr}({\bf T}') |{\bf T}/{\bf T}'|^{-2}, \quad |{\rm discr}({\bf T}')| = |\det_{1\leq i,j\leq g} a_i(\omega_j)|^2. \end{subeqn} Weil's theorem on absolute values of eigenvalues of Frobenius endomorphisms of abelian varieties over finite fields implies that $|a_i(\omega_j)|\leq \sigma(i)i^{1/2}$, where $\sigma(i)$ is the number of positive integers dividing~$i$. It follows that $|{\rm discr}({\bf T}')|\leq g!\prod_{i=1}^g\sigma(i)i^{1/2}$. The rest of the proof consists of applying Theorem~\ref{thm5.1} and standard estimates (including an absolute lower bound for the~$\|\omega_i\|$). \end{proof} \begin{remark}\label{rmk5.3} One knows (see \cite[Ch.~II, Prop.~10.6]{Mazur1}) that ${\rm Spec}({\bf T})$ (for ${\bf T}$ as above) is connected. This implies a lower bound for ${\rm discr}({\bf T})$ (use \cite{Odlyzko1}). On the other hand, the $\|\omega_i\|$ are bounded above by a constant times~$p$. Unfortunately, the lower bound for $h(X_0(p)_{\bf Q})$ obtained like this seems too weak to be useful. Assuming ${\bf T}$ to be Gorenstein does not significantly improve this lower bound. \end{remark} \begin{remark}\label{rmk5.4} Several problems arise when one wants to generalize the above results for $X_0(p)_{\bf Q}$ to more general~$X_0(N)_{\bf Q}$. First of all, ${\bf T}^\vee$ will not be the same as $\omega_J$, but it should be possible to estimate $|{\bf T}^\vee/\omega_J|$. Secondly, the comparison of the trace form on ${\bf T}$ and the scalar product on ${\bf R}\otimes\omega_J$ is more complicated. Note that the trace form can even be degenerate, if $N$ is not cube free. Thirdly, it seems to be unknown if $\infty$ can be a Weierstrass point on $X_0(N)_{\bf Q}$ when $N$ is square free (see \cite[\S3]{LehnerNewman}; it is known that $\infty$ is a Weierstrass point when $N$ is divisible by 4 or 9, for example). Suppose now that $N$ is square free. Before knowing the result that ${\bf T}$ is reduced, Abbes and Ullmo have related $h(X_0(N))$ to the discriminants of the ``new parts'' of the Hecke algebras of level dividing~$N$ (unpublished). The techniques they use come from \cite{AbbesUllmo1} and~\cite{AbbesUllmo2}. They show that Theorem~\ref{thm5.2} holds for square free $N$ such that $\infty$ is not a Weierstrass point. With the same techniques it is certainly possible to solve the first two of the three problems mentioned above. \end{remark} \vspace{1.5cm}\noindent {\bf Acknowledgements.} We thank Rutger Noot for a discussion during which we were led to a proof of Theorem~\ref{thm2.1}. Bas Edixhoven is grateful to Richard Taylor for a discussion concerning the case of general weights. He also thanks Pierre Berthelot for explaining to him certain results on Newton and Hodge polygons. Thanks also go to Ahmed Abbes and Emmanuel Ullmo for communicating to us their results mentioned in Section~\ref{section5}, and permitting us to publish them. Bas Edixhoven wants to thank the Centre for Research in Mathematics at the Institut d'Estudis Catalans in Barcelona where he finally found time to write this text. Robert Coleman thanks the University of Rennes for inviting him for a month in~1995.

9611

alg-geom/9611008

en

https://arxiv.org/abs/alg-geom/9611008

alg-geom/9611008

Alexander Schmitt

Alexander Schmitt

Projective Moduli for Hitchin Pairs

AmS-LaTex, 11 pages

We give an algebraic geometric compactification of certain moduli spaces of semistable E-pairs in the sense of Yokogawa. In particular, we obtain a compactification of the moduli spaces of semistable Higgs pairs on a curve which were constructed by N. Hitchin.

alg-geom

\section*{Introduction} In the paper \cite{Hi}, Hitchin studied pairs $(E,\phi)$, where $E$ is a vector bundle of rank two with a fixed determinant on a curve $C$ and $\phi\colon E\longrightarrow E\otimes K_C$ is a trace free homomorphism, and constructed a moduli space for them. This moduli space carries the structure of a non-complete, quasi-projective algebraic variety. Later, Nitsure \cite{Ni} gave an algebraic construction of moduli spaces of pairs $(E,\phi)$ over a curve $C$ consisting of a vector bundle $E$ of fixed degree and rank and a homomorphism $\phi\colon E\longrightarrow E\otimes L$ where $L$ is some previously chosen line bundle. He also obtained non-complete moduli spaces. The most general results were obtained by Yokogawa \cite{Yo}. In his paper, $C$ is replaced by a relative scheme $f\colon X\longrightarrow S$ where $f$ is a smooth, projective, geometrically integral morphism and $S$ is a scheme of finite type over a universally Japanese ring, and $L$ by a locally free sheaf $F$ on $X$. \par It is the aim of our paper to compactify some of the spaces obtained by Yokogawa, namely those where $S=\mathop{\rm Spec}{\Bbb C}$ and $F$ is again a line bundle. In order to avoid confusion with the objects studied e.g.\ by Simpson, we will call our objects \it (oriented) Hitchin pairs\rm . \par We shall also mention that, only recently, T.\ Hausel compactified the space of oriented Hitchin pairs of rank two with fixed determinant over a curve $C$, using methods from symplectic geometry. This result and a detailed investigation of the resulting spaces will appear in a forthcoming preprint of his. \par The structure of this note is as follows: In the first section we treat the case where $X$ is a point. This case shows how to define Hitchin pairs correctly and suggests the definition of (semi)stability. Then we prove a boundedness result following \cite{Ni}, construct a projective parameter space for semistable Hitchin pairs and a universal family on this parameter space, and finally define a linearized $\mathop{\rm SL}(V)$-action on this parameter space such that the moduli space is given as $\text{parameter space}/\hskip-3pt/ \mathop{\rm SL}(V)$. After these constructions, we prove the (semi)stability criterion. \par At some places, the techniques of our notes are similar to those in \cite{OST}. Hence, we often omit or sketch only briefly arguments which were carried out in detail in \cite{OST} in an analogous situation. \section*{Acknowledgements} I want to thank Professor Okonek and Dr.\ A.\ Teleman for suggesting the problem and discussing various details of the proof with me. The author acknowledges support by AGE --- Algebraic Geometry in Europe, contract No.\ ER-BCHRXCT 940557 (BBW 93.0187), and by SNF, Nr.\ 2000 -- 045209.95/1. \section{Compactifying the categorical quotient of a vector space} \label{Simple} Let $G$ be a reductive algebraic group acting linearly on a vector space $V$. Consider the categorical quotient $W:=V/\hskip-3pt/ G=\mathop{\rm Spec}{\Bbb C}[V]^G$. The torus ${\Bbb C}^*$ acts canonically on $V$, and this action commutes with the given action of $G$. Now, let $G$ act trivially on ${\Bbb C}$ and let ${\Bbb C}^*$ act on ${\Bbb C}$ by multiplication. We obtain a ($G\times{\Bbb C}^*$)-action on $V\oplus {\Bbb C}$. Observe that the equivalence relation induced by the given action is the following: $$ (v_1,\varepsilon_1)\quad\sim\quad (v_2,\varepsilon_2) \qquad\Leftrightarrow\qquad \exists z\in{\Bbb C}^*,\ g\in G: v_2= z\cdot (g\cdot v_1);\ \varepsilon_2=z\cdot\varepsilon_1. $$ The point is $(v,\varepsilon)\in V\oplus {\Bbb C}$ is semistable if and only if $[v,\varepsilon]\in\P(V\oplus {\Bbb C})$ is $G$-semistable. By the Hilbert criterion, the latter happens if and only if either $\varepsilon\neq 0$ or $v\in V$ is $G$-semistable. The space $(V\oplus{\Bbb C})/\hskip-3pt/ (G\times{\Bbb C}^*)=\P(V\oplus {\Bbb C})/\hskip-3pt/ G$ obviously is a projective variety containing $W$ as an open affine subvariety. Let $W^{ss}$ be the image of the $G$-semistable points in $V$. The ${\Bbb C}^*$-action on $V$ induces a ${\Bbb C}^*$-action on $W^{ss}$. We observe that we have compactified $W$ with $W^{ss}/\hskip-3pt/{\Bbb C}^*$. \par Applying the above discussion to the case $G=\mathop{\rm SL}_n({\Bbb C})$ and $V=M_n({\Bbb C})$ (this is the case of Hitchin pairs over a point) shows that $(m,\varepsilon)\in M_n({\Bbb C})\oplus{\Bbb C}$ is semistable if and only if either $\varepsilon\neq 0$ or $m$ is not nilpotent. \begin{Rem} Comparing this with \cite{Ni}, Thm.2.8, for $r=p$ and $N=1$ shows that the semistability criterion stated there is false for points at infinity. \end{Rem} Our general construction is basically a relative version of the above over a projective scheme. \section{Hitchin pairs} Throughout this paper, we will work over the field of complex numbers. Let $X$ be a smooth projective variety of dimension $n$. If $n>1$, we fix an ample divisor $H$ on $X$ whose associated line bundle will be denoted by $\O_X(1)$. We will use $H$ to compute degrees and Hilbert polynomials. The Hilbert polynomial of a coherent sheaf ${\cal F}$ will be denoted by $P_{\cal F}$. We also fix a line bundle $L$ and a Hilbert polynomial $P$. The degree and the rank given by $P$ will be denoted by $d$ and $r$, respectively. Let $\mathop{\rm Pic}(X)$ be the Picard scheme of $X$. We fix a Poincar\'e sheaf $\L$ on $\mathop{\rm Pic}(X)\times X$. Furthermore, for a coherent sheaf ${\cal E}$, set $\L[{\cal E}]:=\L_{|\{[\det{\cal E}]\}\times X}$. This sheaf depends only on the isomorphy class of ${\cal E}$. Unlike the situation in \cite{OST}, the sheaf $\L$ will play no essential r\^ole in our considerations. \subsection{Oriented Hitchin Pairs} \label{or} An \it oriented Hitchin pair of type $(\L,P,L)$ \rm is a triple $({\cal E},\sigma,\phi)$ consisting of a torsion free coherent sheaf ${\cal E}$ with $P_{\cal E}=P$, a homomorphism $\sigma\colon \det({\cal E})\longrightarrow \L[{\cal E}]$, and a homomorphism $\phi\colon {\cal E}\longrightarrow{\cal E}\otimes L$. Two oriented Hitchin pairs $({\cal E}_1,\sigma_1,\phi_1)$ and $({\cal E}_2,\sigma_2,\phi_2)$ of type $(P,\L,L)$ are called \it equivalent\rm , if there is an isomorphism $\psi\colon {\cal E}_1\longrightarrow{\cal E}_2$ such that $\phi_2\circ\psi=(\psi\otimes \mathop{\rm id}_L)\circ \phi_1$ and $\sigma_1=\sigma_2\circ\det\psi$. \begin{Rem} Of course, we can fix a line bundle $\L_0$ on $X$ and consider oriented Hitchin pairs $({\cal E},\sigma,\phi)$ such that $\det{\cal E}\cong\L_0$. Our proofs carry over to this situation. \end{Rem} Now, consider pairs $({\cal E},\phi)$ consisting of a torsion free coherent sheaf ${\cal E}$ with $P_{\cal E}=P$ and a homomorphism $\phi\colon {\cal E}\longrightarrow{\cal E}\otimes L$. We say that $({\cal E}_1,\phi_1)$ is \it equivalent to \rm $({\cal E}_2,\phi_2)$ if and only if there is an isomorphism $\psi\colon {\cal E}_1\longrightarrow{\cal E}_2$ fulfilling $\phi_2\circ\psi= (\psi\otimes\mathop{\rm id}_L)\circ\phi_1$. Given a pair $({\cal E},\phi)$, we can choose a non-zero orientation $\sigma\colon\det{\cal E}\longrightarrow\L[{\cal E}]$ in order to obtain an oriented Hitchin pair. We observe that the equivalence class of $({\cal E},\sigma,\phi)$ does not depend on the choice of the orientation $\sigma$. Therefore, we call a pair $({\cal E},\phi)$ as above \it an oriented Hitchin pair of type $(L,P)$\rm . Let $S$ be a noetherian scheme. \it A family of oriented Hitchin pairs of type $(L,P)$ parametrized by $S$ \rm is a pair $({\frak E}_S,\phi_S)$ where ${\frak E}_S$ is a coherent sheaf on $S\times X$ and $\phi_S$ is an element of $H^0(S\times X, \underline{\mathop{\rm End}}({\frak E}_S)\otimes \pi_X^*L)$ such that $({\frak E}_{S|\{s\}\times X},\phi_{S|\{s\}\times X})$ is an oriented Hitchin pair of type $(L,P)$ for any closed point $s\in S$. Two families $({\frak E}^i_S,\phi^i_S)$, $i=1,2$, are said to be \it equivalent\rm , if there is an isomorphism $\psi_S\colon {\frak E}^1_S \longrightarrow {\frak E}_S^2\otimes\pi_X^*L$ with $\phi_S^2\circ\psi_S=\bigl((\psi_S\otimes\mathop{\rm id}_{\pi_X^*L})\circ \phi_S^1\bigr)$. \subsection{Hitchin Pairs} A \it Hitchin pair of type $(L,P)$ \rm is a triple $({\cal E},\varepsilon,\phi)$ consisting of a torsion free coherent sheaf ${\cal E}$ with $P_{\cal E}=P$, a complex number $\varepsilon\in{\Bbb C}$, and a homomorphism $\phi\colon {\cal E}\longrightarrow{\cal E}\otimes L$. Two Hitchin pairs $({\cal E}_1,\varepsilon_1,\phi_1)$ and $({\cal E}_2,\varepsilon_2,\phi_2)$ are called \it equivalent\rm , if there are an isomorphism $\psi\colon {\cal E}_1\longrightarrow{\cal E}_2$ and a complex number $z\in{\Bbb C}^*$ such that $\phi_2\circ\psi=\bigl(\psi\otimes (z\cdot \mathop{\rm id}_L)\bigr)\circ \phi_1$ and $\varepsilon_2=z\varepsilon_1$. Let $S$ be a noetherian scheme. \it A family of Hitchin pairs of type $(L,P)$ parametrized by $S$ \rm is a quadruple $({\frak E}_S,\varepsilon_S,\phi_S,{\frak M}_S)$ consisting of a coherent sheaf ${\frak E}_S$ over $S\times X$, an invertible sheaf ${\frak M}_S$ over $S$, a section $\varepsilon_S\in H^0(S,{\frak M}_S)$, and an element $\phi_S\in H^0(S\times X, \underline{\mathop{\rm End}}({\frak E}_S)\otimes\pi_S^*{\frak M}_S\otimes \pi_X^*L)$ such that its restriction to $\{s\}\times X$ is a Hitchin pair of type $(L,P)$ for any closed point $s\in S$. The family $({\frak E}^1_S,\varepsilon^1_S,\phi^1_S,{\frak M}^1_S)$ is said to be \it equivalent to \rm the family $({\frak E}^2_S,\varepsilon^2_S,\phi^2_S,{\frak M}^2_S)$ if there are isomorphisms $\psi_S\colon {\frak E}_S^1\longrightarrow {\frak E}_S^2$ and $z_S\colon {\frak M}_S^1\longrightarrow {\frak M}_S^2$ with $\phi_S^2\circ \psi_S=\bigl(\psi_S\otimes \pi_S^*z_S\otimes \mathop{\rm id}_{\pi_X^*L}\bigr)\circ\phi_S^1$ and $\varepsilon_S^2=\varepsilon_S^1\circ z_S$. \subsection{(Semi)Stability} We call a Hitchin pair $({\cal E},\varepsilon,\phi)$ of type $(L,P)$ \it (semi)stable\rm , if the following two conditions are satisfied: \begin{enumerate} \item For any $\phi$-invariant subsheaf $0\neq {\cal F}\subset{\cal E}$ we have: $(P_{\cal F}/\mathop{\rm rk}{\cal F})\quad (\le)\quad (P/r)$. \item Either $\varepsilon\neq 0$, or $(\phi\otimes\mathop{\rm id}_{L^{\otimes r-1}})\circ\cdots\circ \phi\neq 0$. \end{enumerate} \begin{Rem} As usual, there are the corresponding notions of \it slope-(semi)stability\rm . Slope-stability implies stability and semistability implies slope-semistability. \end{Rem} We are now able to define the functors $M^{(s)s}_{(L,P)}$ of equivalence classes of families of (semi)stable Hitchin pairs of type $(L,P)$. The functors of families of (semi)stable oriented Hitchin pairs of type $(L,P)$ are the open subfunctors $\varepsilon\neq 0$. \section{Boundedness} It is the aim of this section to show that the family of isomorphy classes of torsion free coherent sheaves occuring in slope-semistable Hitchin pairs of type $(L,P)$ is bounded. We recall that any torsion free coherent sheaf ${\cal E}$ possesses a Harder-Narasimhan filtration $$0={\cal E}_0\subset{\cal E}_1\subset\cdots\subset{\cal E}_l={\cal E}$$ where ${\cal E}_i/{\cal E}_{i-1}$ is the subsheaf of maximal rank of ${\cal E}/{\cal E}_{i-1}$ for which $P_{{\cal E}_i/{\cal E}_{i-1}}$ is maximal. We have \begin{equation} \label{G1} \mu({\cal E}_i/{\cal E}_{i-1})\qquad \ge \qquad \mu({\cal E}_{i+1}/{\cal E}_i),\qquad i=1,...,l-1. \end{equation} A simple inductive argument shows that $\mu({\cal E}_i)\ >\ \mu({\cal E})$, $i=1,...,l$, when $\mu({\cal E}_1)\ >\ \mu({\cal E})$. By a theorem of Maruyama \cite{Ma}, it is enough to bound $\mu({\cal E}_1)$ for torsion free coherent sheaves occuring in slope-semistable Hitchin pairs of type $(L,P)$: \begin{Thm} \label{B1} For any torsion free coherent sheaf which is part of a slope-semistable Hitchin pair of type $(L,P)$, we have $$\mu({\cal E}_1)\qquad \le\qquad \max\Bigl\{\, \mu({\cal E}),\ \mu({\cal E})+{(r-1)^2\over r}\deg L\,\Bigr\}.$$ \end{Thm} \begin{pf} We follow the proof of \cite{Ni}, Prop.3.2, in the case of curves. Let $({\cal E},\varepsilon,\phi)$ be a slope-semistable Hitchin pair of type $(L,P)$. If $\mu({\cal E}_1)\ \le\ \mu({\cal E})$, there is nothing to show. Otherwise, as we have seen above, $\mu({\cal E}_i)\ >\ \mu({\cal E})$, $i=1,...,l$. By definition of slope-semistability, this means that the ${\cal E}_i$ are not $\phi$-invariant. Hence, the homomorphism $\phi_i\colon {\cal E}_i\longrightarrow {\cal E}/{\cal E}_i\otimes L$ is not trivial for $i=1,...,l$. Let $\iota\in\{\, 0,...,i-1\,\}$ be maximal with $\phi_i({\cal E}_{\iota})=0$ and $\kappa\in \{\, i+1,...,l\,\}$ minimal with $\phi_i({\cal E}_i)\subset ({\cal E}_\kappa/{\cal E}_i)\otimes L$. With these choices, the induced homomorphism from ${\cal E}_{\iota+1}/{\cal E}_\iota$ to ${\cal E}_\kappa/{\cal E}_{\kappa-1}\otimes L$ is non-trivial. Both of these sheaves are slope-semistable, so that $ \mu({\cal E}_{\iota+1}/{\cal E}_\iota) \ \le\ \mu({\cal E}_\kappa/{\cal E}_{\kappa-1}) + \deg L$. By (\ref{G1}), we get \begin{equation} \label{G2} \mu({\cal E}_i/{\cal E}_{i-1})\qquad \le\qquad \mu({\cal E}_{i+1}/{\cal E}_i)+\deg L. \end{equation} Summing these inequalities from $i=1$ to $i=l-1$ yields $$\mu({\cal E}_1)\quad \le\quad \mu({\cal E}/{\cal E}_{l-1})+(l-1)\deg L\quad \le\quad \mu({\cal E}/{\cal E}_{l-1})+(r-1)\deg L.$$ Since $\mu({\cal E}_1)+(r-1)\mu({\cal E}/{\cal E}_{l-1})\ \le\ r\mu({\cal E})$, i.e., $$ \mu({\cal E}/{\cal E}_{l-1})\qquad \le \qquad {d-\mu({\cal E}_1)\over r-1}, $$ the assertion of the theorem follows. \end{pf} \begin{Rem} \label{B2} Fix a number $m$ such that $L\subset\O_X(m)$. For any coherent sheaf ${\cal F}$, we obviously have $P_{{\cal F}\otimes L}\ \le\ P_{{\cal F}(m)}$. It is easy to see that there is a constant $C^\prime$ \sl depending only on $H$ and $m$ \rm with $P_{{\cal F}(m)}\ \le\ P_{\cal F}+C x^{n-1}$. \end{Rem} We can now carry out the proof of \ref{B1} for semistable Hitchin pairs and Hilbert polynomials, where we replace (\ref{G2}) by $$ {P_{{\cal E}_i/{\cal E}_{i-1}}\over \mathop{\rm rk}{\cal E}_i-\mathop{\rm rk}{\cal E}_{i-1}}\qquad \le\qquad {P_{{\cal E}_{i+1}/{\cal E}_i}+ C x^{n-1}\over \mathop{\rm rk}{\cal E}_i-\mathop{\rm rk}{\cal E}_{i-1}} \qquad\le\quad {P_{{\cal E}_{i+1}/{\cal E}_i}\over \mathop{\rm rk}{\cal E}_i-\mathop{\rm rk}{\cal E}_{i-1}}+C x^{n-1}. $$ This gives $$ {P_{{\cal E}_1}\over \mathop{\rm rk}{\cal E}_1}\qquad\le\qquad {P +(r-1)^2 C x^{n-1}\over r}. $$ \section{A parameter space for semistable Hitchin pairs} For $\mu\in{\Bbb N}$, we define $P_\mu$ by $P_\mu(x):=P(x+\mu)$. Twisting by $\O_X(\mu)$ yields an isomorphism between the functors $M^{(s)s}_{(L,P)}$ and $M^{(s)s}_{(L,P_\mu)}$. By Theorem~\ref{B1}, we may assume that any torsion free coherent sheaf ${\cal E}$ appearing in a semistable Hitchin pair of type $(L,P)$ fulfills the following conditions: \begin{enumerate} \item ${\cal E}$ is globally generated. \item $H^i(X,{\cal E})=0$ for every $i>0$. \end{enumerate} Let $p:=P(0)$, $V$ be a complex vector space of dimension $p$, and ${\frak Q}$ the projective Quot scheme of (all) quotients of $V\otimes\O_X$ with Hilbert polynomial $P$. On the product ${\frak Q}\times X$, there is a universal quotient $$q_{\frak Q}\colon V\otimes \O_{{\frak Q}\times X}\longrightarrow {\frak E}_{\frak Q}.$$ We choose $m$ large enough, so that $L\subset \O_X(m)$ and so that $\O_X(m)$ is globally generated. Furthermore, we choose $\nu$ large enough, so that $q_{\frak Q}(\nu)$ induces a closed embedding ${\frak Q}\subset {\frak G}:= \mathop{\rm Gr}\bigl(V\otimes H^0(\O_X(\nu)), P(\nu)\bigr)$ and so that the multiplication map $H^0(\O_X(\nu))\otimes H^0(\O_X(m))\longrightarrow H^0(\O_X(\nu m))$ is surjective. We set $N:=H^0(\O_X(\nu))$, $M:=H^0(\O_X(m))$, and $W:=V\otimes N$. By our choice of $\nu$, for any Hitchin pair $({\cal E},\varepsilon,\phi)$, the map $V\otimes N\longrightarrow H^0({\cal E}(\nu))$ is surjective. It follows that $\phi\otimes \mathop{\rm id}_{\O_X(\nu)}$ is induced by an element $f\in W^\vee\otimes W\otimes M$. Set ${\bf P}:=\P({\Bbb C}\oplus W^\vee\otimes W\otimes M^\vee)$, and let $${\bf s}\colon \O_{\bf P}\longrightarrow [{\Bbb C}\oplus W^\vee\otimes W\otimes M]\otimes \O_{\bf P}(1)$$ be the tautological section. First, we can construct a subscheme $\tilde{\frak P}\subset {\frak Q}\times {\bf P}$ whose closed points are those $s=([q],\tilde{s})\in {\frak Q}\times {\bf P}$ for which the second component of $\pi_{\bf P}^*{\bf s}$ induces a homomorphism ${\frak E}_{{\frak Q}|\{[q]\}\times X}(\nu) \longrightarrow {\frak E}_{{\frak Q}|\{[q]\}\times X}(\nu)\otimes H^m$. Let ${\frak E}_{\tilde{\frak P}}$ be the restriction of $\pi_{\frak Q}^*{\frak E}_{\frak Q}$ to $\tilde{\frak P}\times X$ and $$ h_{\tilde{\frak P}}\colon {\frak E}_{\tilde{\frak P}}\longrightarrow {\frak E}_{\tilde{\frak P}}\otimes \pi_X^*(\O_X(m)/L) $$ be the induced homomorphism. We then define ${\frak P}$ as the closed subscheme of $\tilde{\frak P}$ whose closed points are those $s\in \tilde{\frak P}$ for which $h_{|\{s\}\times X}\equiv 0$. The scheme ${\frak P}$ is a parameter space for pairs $([q\colon V\otimes\O_X \longrightarrow{\cal E}],[\varepsilon,\phi])$ with $[q]\in {\frak Q}$, $[\varepsilon,\phi]\in \P({\Bbb C}\oplus H^0(\underline{\mathop{\rm End}}{\cal E}\otimes L)^\vee)$. On ${\frak P}\times X$, there exists a universal family $({\frak E}_{\frak P},\varepsilon_{\frak P}, \phi_{\frak P}, {\frak M}_{\frak P})$. \par Denote by ${\frak P}^{\mathop{\rm iso}}$ the open set of pairs $([q\colon V\otimes\O_X\longrightarrow {\cal E}],[\varepsilon,\phi])$ for which $H^0(q)$ is an isomorphism. It is not hard to see that any family of semistable Hitchin pairs of type $(L,P)$ is locally induced by morphisms to ${\frak P}^{\mathop{\rm iso}}$. \section{The $\mathop{\rm SL}(V)$-action on ${\frak P}$} On the Quot scheme ${\frak Q}$, there is a natural action $\rho\colon {\frak Q}\times \mathop{\rm SL}(V)\longrightarrow {\frak Q}$. Furthermore, there is a natural action of $\mathop{\rm SL}(V)$ from the right on the vector space $W^\vee\otimes W\otimes M$. If we let $\mathop{\rm SL}(V)$ act trivially on ${\Bbb C}$, we get an action of $\mathop{\rm SL}(V)$ from the right on the scheme ${\frak Q}\times {\bf P}$. Finally, we remark that the $\mathop{\rm SL}(V)$-action leaves the parameter space ${\frak P}$ invariant. Hence, there is an action from the right of $\mathop{\rm SL}(V)$ on ${\frak P}$. We deduce \begin{Prop} \label{Op1} Let $S$ be a noetherian scheme and $\beta_i\colon S\longrightarrow {\frak P}^{\mathop{\rm iso}}$ two morphisms. Suppose that the pullbacks via the maps $(\beta_i\times\mathop{\rm id}_X)$ of the universal family $({\frak E}_{\frak P},\varepsilon_{\frak P},\phi_{\frak P}, {\frak M}_{\frak P})$ are equivalent. Then there exist an \'etale covering $\tau\colon T\longrightarrow S$ and a morphism $g\colon T\longrightarrow\mathop{\rm SL}(V)$ such that $\beta_1\circ \tau=(\beta_2\circ\tau)\cdot g$. \end{Prop} \section{The (semi)stable points in ${\frak P}$} Suppose we are given a projective scheme $S$ and an action of an algebraic group $G$, linearized in an invertible sheaf ${\frak M}$. For a point $s\in S$ and a one parameter subgroup $\lambda\colon {\Bbb C}^*\longrightarrow G$, set $s_\infty:=\lim_{z\longrightarrow\infty} \lambda(z)\cdot s$. Then $s_\infty$ is a fixed point of the ${\Bbb C}^*$-action given by $\lambda$, and ${\Bbb C}^*$ acts on ${\frak M}\otimes{\Bbb C}(s_\infty)$ with weight, say, $\gamma$. We set $\mu(s,\lambda):=-\gamma$. If $G$ is reductive and ${\frak M}$ is ample, then the Hilbert-Mumford criterion says that $s$ is (semi)stable if and only if $\mu(s,\lambda)\ (\ge)\ 0$ for every one parameter subgroup $\lambda$ of $G$. We will apply this criterion in our situation. \par A one parameter subgroup of $\mathop{\rm SL}(V)$ is determined by the following data: \begin{enumerate} \item A basis $v_1,...,v_p$ of $V$. \item Weights $\gamma_1\le\cdots\le\gamma_p$ with $\sum_i\gamma_i=0$. \end{enumerate} We recall that a weight vector $(\gamma_1,...,\gamma_p)$, satisfying $\gamma_1\le\cdots\le\gamma_p$ and $\sum_i\gamma_i=0$ is a $\Bbb Q$-linear combination with non-negative coefficients of the weight vectors $$ \gamma^{(i)}:= (\quad\underbrace{i-p,...,i-p}_{\text{$i$ times}},\underbrace{i,...,i}_{\text{$(p-i)$ times}}\ ). $$ More precisely, \begin{equation} \label{LinKomb} (\gamma_1,...,\gamma_p)=\sum_{i=1}^{p-1}{\gamma_{i+1}-\gamma_i\over p}\gamma^{(i)}. \end{equation} Let's return to our construction. Let $\O_{\frak Q}(1)$ be the restriction of the very ample line bundle on ${\frak G}$ giving the Pluecker embedding. We denote by $\O(a_1,a_2)$ the restriction of the bundle $\pi_{\frak Q}^*\O_{\frak Q}(a_1)\otimes \pi_{\bf P}^*\O_{\bf P}(a_2)$ to the parameter space ${\frak P}$. The $\mathop{\rm SL}(V)$-action on ${\frak P}$ can be linearized in any of these sheaves. We will choose $a_1,a_2>0$ with $a_1<(p-1) a_2$. For $[q\colon V\otimes \O_X\longrightarrow {\cal E}]\in {\frak Q}$ and a subspace $U\subset V$, ${\cal E}_U$ is defined to be the subsheaf of ${\cal E}$ which is generically generated by $q(U\otimes\O_X)$ and for which ${\cal E}/{\cal E}_U$ is torsion free. Given a basis $v_1,...,v_p$ of $V$, we set ${\cal E}_i:={\cal E}_{\langle\, v_1,...,v_i\,\rangle}$, so that we obtain a filtration $$ \mathop{\rm Tors}{\cal E}={\cal E}_0\subset{\cal E}_1\subset\cdots\subset{\cal E}_{p-1}\subset{\cal E}_p={\cal E}. $$ Now, either ${\cal E}_i={\cal E}_{i+1}$ or $\mathop{\rm rk}{\cal E}_{i+1}=\mathop{\rm rk}{\cal E}_i+1$. For $\rho=1,...,p$, we set $k_\rho:=\min_{i=1,...,p}\{\,\mathop{\rm rk}{\cal E}_i=\rho\,\}$ and $\underline{k}:=(k_1,...,k_p)$. Suppose we are given a one parameter subgroup $\lambda$ of $\mathop{\rm SL}(V)$. For a point $[q\colon V\otimes\O_X\longrightarrow {\cal E}]\in {\frak Q}$, set $[q_\infty\colon V\otimes\O_X\longrightarrow{\cal E}]:=\lim_{z\longrightarrow\infty}\lambda(z)\cdot [q]$. We denote the fibre of $\O_{\frak Q}(a_1)$ over $[q_\infty]$ by $\Lambda$. Let $v_1,...,v_p$ be a basis of $V$. If $\lambda$ is the one parameter subgroup which is described by the weight vector $(\gamma_1,...,\gamma_p)$, then $\lambda$ acts on $\Lambda$ with weight $a_1\gamma_{\underline{k}}$ (\cite{HL2}, p.309) where we set $$ \gamma_{\underline{k}}:=\gamma_{k_1}+\cdots+\gamma_{k_p}. $$ In particular $$ \gamma_{\underline{k}}^{(i)}=(p-i)\mathop{\rm rk}{\cal E}_i-i(\mathop{\rm rk}{\cal E}-\mathop{\rm rk}{\cal E}_i)=p\mathop{\rm rk}{\cal E}_i-i\mathop{\rm rk}{\cal E}. $$ Now, consider the $\mathop{\rm SL}(V)$-action on ${\bf P}$. For a point $\tilde{s}\in {\bf P}$ and a one parameter subgroup $\lambda$ of $\mathop{\rm SL}(V)$, define $\tilde{s}_\infty$ as above and let $E$ be the fibre of $\O_{\bf P}(a_2)$ over $\tilde{s}_\infty$. For the statement of the next lemma, we need the notion of a superinvariant subspace which will. Suppose we are given a homomorphism $f\colon V\otimes N\longrightarrow V\otimes N\otimes M$. A subspace $U\subset V$ is called \it $f$-superinvariant\rm , if $U\otimes N\subset\ker f$ and if the induced homomorphism $\overline{f}\colon (V/U)\otimes N\longrightarrow (V/U)\otimes N\otimes M$ is identically zero. From now on, given an element $s:=([q\colon V\otimes\O_X\longrightarrow {\cal E}],[\varepsilon,\phi])$ in ${\frak P}$, the associated homomorphism in $W^\vee\otimes W\otimes M$ will be denoted by $f$. We have the following obvious \begin{Lem} \label{SemStab0} Set $s:=([q\colon V\otimes\O_X\longrightarrow {\cal E}],[\varepsilon,\phi])$. The one parameter subgroup $\lambda$ which is given w.r.t.\ to the basis $v_1,...,v_p$ by the weight vector $\gamma^{(i)}$ acts on $E$ with weight \begin{enumerate} \item $-a_2p$ if $\langle\, v_1,...,v_i\,\rangle$ is not $f$-invariant. \item $a_2p$ if $\langle\, v_1,...,v_i\,\rangle$ is $f$-superinvariant. \item $0$ in all the other cases. \end{enumerate} \end{Lem} An immediate consequence is: \begin{Cor} \label{SemStab1} A necessary condition for a point $s:=([q\colon V\otimes\O_X\longrightarrow {\cal E}],[\varepsilon,\phi])$ to be (semi)stable is that for any $f$-invariant subspace $U\subset V$ $$\dim U\mathop{\rm rk}{\cal E}\quad (\le)\quad p\mathop{\rm rk}{\cal E}_U.$$ \end{Cor} \begin{Cor} \label{SemStab2} Let $s:=([q\colon V\otimes\O_X\longrightarrow {\cal E}],[\varepsilon,\phi])$ be a point in ${\frak P}$ and suppose that either $H^0(q)$ is not an isomorphism or that ${\cal E}$ is not torsion free. Then $s$ is not semistable. \end{Cor} \begin{pf} Set $U:=\ker H^0(q)$ in the first case and $U:=H^0(\mathop{\rm Tors}{\cal E})$ in the second case. Then $U$ clearly violates the condition in Corollary~\ref{SemStab1}. \end{pf} We now state the main result of this section: \begin{Thm} \label{MainRes} For $d$ sufficiently large the following assertion holds true: A point $s:=([q\colon V\otimes\O_X\longrightarrow {\cal E}],[\varepsilon,\phi])$ is (semi)stable if and only if $H^0(q)$ is an isomorphism, ${\cal E}$ is torsion free, and $({\cal E},\varepsilon,\phi)$ is a (semi)stable Hitchin pair. \end{Thm} We will need \begin{Prop} \label{Aux1} There is an integer $k_0$ such that for any semistable Hitchin pair, any subsheaf ${\cal F}\subset{\cal E}$, and any $k\ge k_0$: $$ rh^0({\cal F}(k))\quad <\quad (\mathop{\rm rk}{\cal F}+1) P(k). $$ \end{Prop} \begin{pf} As in the proof on page 305 in \cite{HL2}, we conclude that for any sufficiently large constant $\kappa$ there is an integer $k_0$ such that for any Hitchin pair $({\cal E},\varepsilon,\phi)$ and any subsheaf ${\cal F}\subset{\cal E}$ $$ \text{either}\quad |\deg{\cal F}-\mathop{\rm rk}{\cal F}\mu({\cal E})|\ \le\ \kappa\quad\text{or}\quad h^0({\cal F}(k))/\mathop{\rm rk}{\cal F}\ <\ P(k)/r\quad \forall k\ge k_0. $$ Let ${\frak S}$ be the family of all saturated submodules of torsion free sheaves ${\cal E}$ occuring in the family ${\frak E}_{\frak Q}$ which satisfy $|\deg{\cal F}-\mathop{\rm rk}{\cal F}\mu({\cal E})|\ \le\ \kappa$. Then this family is bounded (\cite{HL2}, Lemma 2.7). Hence, we may assume that all ${\cal F}\in {\frak S}$ are globally generated and without higher cohomology. By the discussions following Remark~\ref{B2} \begin{eqnarray*} rh^0({\cal F}(k)) &\le& \mathop{\rm rk}{\cal F}(P(k)+(r-1)^2Ck^{n-1})\\ &=& (\mathop{\rm rk}{\cal F}+1)P(k)+[\mathop{\rm rk}{\cal F}(r-1)^2Ck^{n-1}-P(k)]\\ &\le& (\mathop{\rm rk}{\cal F}+1)P(k)+[r(r-1)^2Ck^{n-1}-P(k)]. \end{eqnarray*} Since $C$ does not depend on $d$, we can achieve $[r(r-1)^2Ck^{n-1}-P(k)]<0$ for all $k\ge k_0$. \end{pf} We choose $d$ large enough so that $k_0=0$, and so that all modules ${\cal F}$ in the family ${\frak S}$ are globally generated and without higher cohomology. Since there are only finitely many possible Hilbert polynomials for sheaves in ${\frak S}$, the proof of \ref{Aux1} shows that we can assume that for any ${\cal F}\subset {\cal E}$, ${\cal E}$ being a torsion free member of the family ${\frak E}_{\frak Q}$, the inequality $P_{\cal F}/\mathop{\rm rk}{\cal F}\ (\le)\ P/r$ is equivalent to the inequality $h^0({\cal F})/\mathop{\rm rk}{\cal F}\ (\le)\ p/r$. \begin{pf*}{Proof of Theorem~\ref{MainRes}} First, let $([q\colon V\otimes\O_X\longrightarrow {\cal E}],[\varepsilon,\phi])$ be a (semi)stable point. Then, by \ref{SemStab2}, $H^0(q)$ is an isomorphism and ${\cal E}$ is torsion free. Furthermore, \ref{SemStab1} shows that $({\cal E},\varepsilon,\phi)$ is a (semi)stable Hitchin pair, provided $\varepsilon\neq 0$. We still have to show that $(\phi\otimes\mathop{\rm id}_{L^{\otimes r-1}})\circ\cdots\circ\phi$ is not zero if $\varepsilon=0$. For this, set ${\cal F}_i:=\ker[(\phi\otimes\mathop{\rm id}_{L^{\otimes i-1}})\circ\cdots\circ\phi]$, $i=1,...,r$. We get a filtration $$ 0=:{\cal F}_0\subset{\cal F}_1\subset\cdots\subset{\cal F}_{r-1}\subset{\cal F}_r={\cal E} $$ of ${\cal E}$. Choose a basis $v_1,...,v_p$ of $V$ such that there are $\iota_i$ with $\langle\, v_1,...,v_{\iota_i}\,\rangle= H^0({\cal F}_i)$, $i=1,...,r$. Let $\lambda$ be the one parameter subgroup which is given by the weight vector $\sum\gamma^{(i)}$. The assumption $\mu(s,\lambda)\ (\ge)\ 0$ implies that there is an index $i$ with $a_1(rh^0({\cal F})-\mathop{\rm rk}{\cal F} p)+a_2 p\ \le\ 0$, in particular $$ rh^0({\cal F}_i)\qquad <\qquad (\mathop{\rm rk}{\cal F}_i-1)p. $$ This implies $$ rh^0({\cal E}/{\cal F}_i)\qquad \ge\qquad (r-\mathop{\rm rk}{\cal F}_i+1)p. $$ Now, $(\phi\otimes\mathop{\rm id}_{L^{\otimes i-1}}\circ\cdots\circ\phi)$ maps ${\cal E}/{\cal F}_i$ isomorphically onto a $(\phi\otimes\mathop{\rm id}_{L^{\otimes i}})$-invariant subsheaf of ${\cal E}\otimes L^{\otimes i}$. This sheaf can be identified with a $(\phi\otimes\mathop{\rm id}_{H^{\otimes im}})$-invariant subsheaf of ${\cal E}\otimes H^{\otimes im}$. But ${\cal E}\otimes H^{\otimes i m}$ is also semistable, and the assumptions made before the beginning of the proof hold for this sheaf as well, so that $$ rh^0({\cal E}/{\cal F}_i)\qquad \le\qquad (r-\mathop{\rm rk}{\cal F}_i)P_{{\cal E}\otimes H^{\otimes i m}} =(r-\mathop{\rm rk}{\cal F}_i)P(im), $$ and, consequently, $$ (r-\mathop{\rm rk}{\cal F}_i)(P(im)-p)\qquad \le\qquad p. $$ But when $d$ is large, this is not possible. \par Now, we prove the opposite direction: Let $s:=([q\colon V\otimes\O_X\longrightarrow {\cal E}], [\varepsilon,\phi])$ be a point such that $H^0(q)$ is an isomorphism and $({\cal E},\varepsilon,\phi)$ is a (semi)stable Hitchin pair. First, suppose $\varepsilon\neq 0$. Let $v_1,...,v_p$ a basis of $V$. Let $\lambda$ be given by the weight vector $\gamma=\sum \alpha_i \gamma^{(i)}$. If all the spaces $\langle\, v_1,...,v_i\,\rangle$ for which $\alpha_i\neq 0$ are $f$-invariant, then the (semi)stability condition implies $\gamma_{\underline{k}}\ (\le)\ 0$. Together with~\ref{SemStab0}, this implies $\mu(s,\lambda)\ (\ge)\ 0$. In the other case, let $\alpha$ be the largest coeffictient of a $\gamma^{(i)}$ for which $\langle\, v_1,...,v_i\,\rangle$ is not $f$-invariant. By~\ref{Aux1}, $\gamma_{\underline{k}}^{(i)}\ \le\ p$ for $i=1,...,p-1$ and, thus, $$ \mu(s,\lambda)\ \ge\ -a_1\alpha(p-1)p+a_2\alpha p. $$ Now, the right hand expression is $>0$, by our choice of $a_1$ and $a_2$. \par Next, let $\varepsilon=0$. Since the definition of (semi)stability implies in that case $(\phi_{\mathop{\rm id}_{L^{\otimes r-1}}})\circ\cdots\circ\phi\neq 0$, every one parameter subgroup acts with weight $\le 0$ on the ``${\bf P}$-component'' of $s$. This allows us to argue in the same way as before. \end{pf*} \section{The moduli space of semistable Hitchin pairs} \subsection{S-equivalence and the main result} We define ${\cal M}^{(s)s}_{(L,P)}:={\frak P}^{(s)s}/\hskip-3pt/\mathop{\rm SL}(V)$. Then ${\cal M}^{ss}_{(L,P)}$ is a projective scheme. In order to describe its closed points, we have to introduce the notion of S-equivalence: For any semistable Hitchin pair, we can construct a Jordan-H\"older filtration of ${\cal E}$ $$0={\cal E}_0\subset{\cal E}_1\subset\cdots\subset{\cal E}_l={\cal E}$$ by $\phi$-invariant subsheaves. We obtain stable Hitchin pairs $({\cal E}_i/{\cal E}_{i-1},\varepsilon,\phi_i)$, $i=1,...,l$. The associated graded object $$ \mathop{\rm gr}({\cal E},\varepsilon,\phi):=\bigoplus ({\cal E}_i/{\cal E}_{i-1},\varepsilon,\phi_i) $$ is well-defined up to isomorphism. We say that two semistable Hitchin pairs $({\cal E}_1,\varepsilon_1,\phi_1)$ and $({\cal E}_2,\varepsilon_2,\phi_2)$ of type $(L,P)$ are \it S-equivalent\rm , if the associated graded objects are equivalent Hitchin pairs. One can show that any semistable Hitchin pair degenerates into its associated graded object and that the associated graded object is polystable. We summarize the results of our discussions in: \begin{Thm} {\rm i)} There is a natural transformation of functors $$ \tau\colon M^{ss}_{(L,P)}\longrightarrow h_{{\cal M}_{(L,P)}^{ss}} $$ such that for any other scheme $\tilde{\cal M}$ and any natural transformation $\tau^\prime\colon {\cal M}^{ss}_{(L,P)}\longrightarrow h_{\tilde{\cal M}}$ there is a uniquely determined morphism $\vartheta\colon {\cal M}^{ss}_{(L,P)}\longrightarrow \tilde{\cal M}$ with $\tau^\prime=h(\vartheta)\circ \tau$. \par {\rm ii)} ${\cal M}^s_{(L,P)}$ is a coarse moduli space for the functor $M^s_{(L,P)}$. \par {\rm iii)} The closed points of ${\cal M}^{ss}_{(L,P)}$ naturally correspond to the S-equivalence classes of semistable Hitchin pairs of type $(L,P)$. \end{Thm} \subsection{The ${\Bbb C}^*$-action on ${\cal M}_{(L,P)}^{ss}$} On the space ${\cal M}:={\cal M}_{(L,P)}^{ss}$ there is a natural ${\Bbb C}^*$-action given by multiplication of $\phi$ by a constant. The fixed point set is the union of the part which corresponds to the Hitchin pairs $({\cal E},\varepsilon,0)$, i.e., the Gieseker moduli space, and the part ${\cal M}_\infty$ which corresponds to pairs $({\cal E},0,\phi)$. The closed subset ${\cal M}_\infty$ is the part which compactifies the moduli space of semistable oriented Hitchin pairs. Let ${\cal M}_{\neq 0}$ be the ${\Bbb C}^*$-invariant open subscheme of semistable oriented Hitchin pairs, i.e., the set described by $\varepsilon\neq 0$. We observe that ${\cal M}_\infty={\cal M}_{\neq 0}/\hskip-3pt/ {\Bbb C}^*$. Here, we use that the GIT-quotient comes with a natural ample line bundle and that the ${\Bbb C}^*$ action is canonically linearized in this line bundle. \subsection{The Hitchin map} Suppose that $X$ is a curve. Let ${\frak P}^*$ be the open subset of the parameter space ${\frak P}$ parametrizing elements $([q\colon V\otimes \O_X\longrightarrow {\cal E}],[\varepsilon,\phi])$ for which ${\cal E}$ is torsion free and $H^0(q)$ is an isomorphism, and ${\frak P}^*_{\neq 0}$ the part of ${\frak P}^*$ lieing in ${\frak Q}\times (V\otimes N)^\vee\otimes (V\otimes N\otimes M)$, i.e., the part parametrizing pairs with $\varepsilon\neq 0$. Since the Quot scheme is reduced in this case, the restriction of ${\frak E}_{\frak P}$ to ${\frak P}^*_{\neq 0}\times X$ is locally free. This allows us to define the characteristical polynomial map associated to $\phi_{{\frak P}|{{\frak P}^*_{\neq 0}\times X}}$: $$ \chi_{{\frak P}^*_{\neq 0}}\colon {{\frak P}^*_{\neq 0}}\longrightarrow H^0(X,L^{\otimes r}) \oplus\cdots\oplus H^0(X,L). $$ The ${\Bbb C}^*$-action on $(V\otimes N)^\vee\otimes (V\otimes N\otimes L)$ induces a ${\Bbb C}^*$-action on the right hand vector space which is given on $H^0(X, L^{\otimes i})$ by multiplication with $z^i$, $i=1,...,r$. Let ${\Bbb C}^*$ act on ${\Bbb C}$ by multiplication and form the weighted projective space $$ \widehat{\P}:=[H^0(X,L^{\otimes r})\oplus \cdots\oplus H^0(X,L)\oplus{\Bbb C}]/\hskip-3pt/{\Bbb C}^*. $$ Then the map $\chi_{{\frak P}^*_{\neq 0}}$ can be extended to a map $ \chi_{\frak P}\colon {\frak P}^*\longrightarrow \widehat{\P} $ which is invariant under the $\mathop{\rm SL}(V)$-action. Thus we get a map $$ \chi_{{\cal M}}\colon {\cal M}\longrightarrow \widehat{\P}, $$ which we call \it the Hitchin map\rm . We oberve that $\chi_{{\cal M}}$ is proper by \cite{Ha}, II.4.8.(c), applied to $f=\chi_{{\cal M}}$ and $g\colon \widehat{\P}\longrightarrow \{\text{pt}\}$.

9611

alg-geom/9611033

fr

https://arxiv.org/abs/alg-geom/9611033

alg-geom/9611033

Olivier Debarre

O. Debarre and L. Manivel

Sch\'emas de Fano

PlainTeX v 1.2, 22 pages, in French. Run it twice to get cross-references right

Let X be a subvariety of $P^n$ defined by equations of degrees $ d =(d_1,...,d_s)$, over an algebraically closed field k of any characteristic. We study properties of the Fano scheme $F_r(X)$ that parametrizes linear subspaces of dimension r contained in X. We prove that $F_r(X)$ is connected and smooth of the expected dimension for n big enough (this was previously known in characteristic 0 or for r=1). Using Bott's theorem, we prove a vanishing theorem for certain bundles on the Grassmannian and use it to calculate the cohomology groups of $F_r(X)$ in degree $\le \dim X-2r$, and to prove that $F_r(X)$ is projectively normal in the Grassmannian. Finally, we prove that for n big enough, the rational Chow group $A_1(F_r(X))$ is of rank 1, and $F_r(X)$ is unirational. All bounds on n are effective.

alg-geom

\global\def\currenvir{subsection{\global\defth{subsection} \global\advance\prmno by1 {\hskip-.5truemm(\number\secno.\number\prmno)}\hskip 5truemm} \def\global\def\currenvir{formule{\global\defth{formule} \global\advance\prmno by1 {\hbox{\rm (\number\secno.\number\prmno)}}} \def\proclaim#1{\global\advance\prmno by 1 {\bf #1 \the\secno.\the\prmno.-- }} \def\ex#1{\medbreak\global\advance\prmno by 1 {\bf #1 \the\secno.\the\prmno.}} \def\qq#1{\medbreak\global\advance\prmno by 1 { #1 \the\prmno)}} \mathop{\rm long}\nolimits\def\th#1 \enonce#2\endth{\global\defth{th} \medbreak\proclaim{#1}{\it #2}\medskip} \mathop{\rm long}\nolimits\def\comment#1\endcomment{} \def\isinlabellist#1\of#2{\notfoundtrue% {\def\given{#1}% \def\\##1{\def\next{##1}% \lop\next\to\za\lop\next\to\zb% \ifx\za\given{\zb\global\notfoundfalse}\fi}#2}% \ifnotfound{\immediate\write16% {Warning - [Page \the\pageno] {#1} No reference found}}% \fi}% \def\ref#1{\ifx\empty{\immediate\write16 {Warning - No references found at all.}} \else{\isinlabellist{#1}\of}\fi} \def\newlabel#1#2{\rightappenditem{\\{#1}\\{#2}}\to} \def{} \def\label#1{% \def\given{th}% \ifx\giventh% {\hwrite\lbl{\string\newlabel{#1}{\number\secno.\number\prmno}}}\fi% \def\given{section}% \ifx\giventh% {\hwrite\lbl{\string\newlabel{#1}{\number\secno}}}\fi% \def\given{subsection}% \ifx\given\currenvi {\hwrite\lbl{\string\newlabel{#1}{\number\secno.\number\prmno}}}\fi% \def\given{formule}% \ifx\given\currenvi {\hwrite\lbl{\string\newlabel{#1}{\number\secno.\number\prmno}}}\fi% \def\given{subsubsection}% \ifx\giventh% {\hwrite\lbl{\string% \newlabel{#1}{\number\secno.\number\subsecno.\number\subsubsecno}}}\fi \ignorespaces} \newwrite\lbl \def\openout\lbl=\jobname.lbl{\openout\lbl=\jobname.lbl} \def\closeout\lbl{\closeout\lbl} \newread\testfile \def\lookatfile#1{\openin\testfile=\jobname.#1 \ifeof\testfile{\immediate\openout\nameuse{#1}\jobname.#1 \write\nameuse{#1}{} \immediate\closeout\nameuse{#1}}\fi% \immediate\closein\testfile}% \def\lookatfile{lbl{\lookatfile{lbl} \input\jobname.lbl \openout\lbl=\jobname.lbl} \let\bye\closeall\bye \def\closeall\bye{\closeout\lbl\bye} \font\eightrm=cmr8 \font\eighti=cmmi8 \font\eightsy=cmsy8 \font\eightbf=cmbx8 \font\eighttt=cmtt8 \font\eightit=cmti8 \font\eightsl=cmsl8 \font\sixrm=cmr6 \font\sixi=cmmi6 \font\sixsy=cmsy6 \font\sixbf=cmbx6 \font\tengoth=eufm10 \font\tenbboard=msbm10 \font\eightgoth=eufm8 \font\eightbboard=msbm8 \font\sevengoth=eufm7 \font\sevenbboard=msbm7 \font\sixgoth=eufm6 \font\fivegoth=eufm5 \font\tencyr=wncyr10 \font\eightcyr=wncyr8 \font\sevencyr=wncyr7 \font\sixcyr=wncyr6 \skewchar\eighti='177 \skewchar\sixi='177 \skewchar\eightsy='60 \skewchar\sixsy='60 \newfam\gothfam \newfam\bboardfam \newfam\cyrfam \def\tenpoint{% \textfont0=\tenrm \scriptfont0=\sevenrm \scriptscriptfont0=\fiverm \def\rm{\fam\z@\tenrm}% \textfont1=\teni \scriptfont1=\seveni \scriptscriptfont1=\fivei \def\oldstyle{\fam\@ne\teni}\let\old=\oldstyle \textfont2=\tensy \scriptfont2=\sevensy \scriptscriptfont2=\fivesy \textfont\gothfam=\tengoth \scriptfont\gothfam=\sevengoth \scriptscriptfont\gothfam=\fivegoth \def\goth{\fam\gothfam\tengoth}% \textfont\bboardfam=\tenbboard \scriptfont\bboardfam=\sevenbboard \scriptscriptfont\bboardfam=\sevenbboard \def\bb{\fam\bboardfam\tenbboard}% \textfont\cyrfam=\tencyr \scriptfont\cyrfam=\sevencyr \scriptscriptfont\cyrfam=\sixcyr \def\cyr{\fam\cyrfam\tencyr}% \textfont\itfam=\tenit \def\it{\fam\itfam\tenit}% \textfont\slfam=\tensl \def\sl{\fam\slfam\tensl}% \textfont\bffam=\tenbf \scriptfont\bffam=\sevenbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\tenbf}% \textfont\ttfam=\tentt \def\tt{\fam\ttfam\tentt}% \abovedisplayskip=12pt plus 3pt minus 9pt \belowdisplayskip=\abovedisplayskip \abovedisplayshortskip=0pt plus 3pt \belowdisplayshortskip=4pt plus 3pt \smallskipamount=3pt plus 1pt minus 1pt \medskipamount=6pt plus 2pt minus 2pt \bigskipamount=12pt plus 4pt minus 4pt \normalbaselineskip=12pt \setbox\strutbox=\hbox{\vrule height8.5pt depth3.5pt width0pt}% \let\bigf@nt=\tenrm \let\smallf@nt=\sevenrm \normalbaselines\rm} \def\eightpoint{% \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm \def\rm{\fam\z@\eightrm}% \textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei \def\oldstyle{\fam\@ne\eighti}\let\old=\oldstyle \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy \textfont\gothfam=\eightgoth \scriptfont\gothfam=\sixgoth \scriptscriptfont\gothfam=\fivegoth \def\goth{\fam\gothfam\eightgoth}% \textfont\cyrfam=\eightcyr \scriptfont\cyrfam=\sixcyr \scriptscriptfont\cyrfam=\sixcyr \def\cyr{\fam\cyrfam\eightcyr}% \textfont\bboardfam=\eightbboard \scriptfont\bboardfam=\sevenbboard \scriptscriptfont\bboardfam=\sevenbboard \def\bb{\fam\bboardfam}% \textfont\itfam=\eightit \def\it{\fam\itfam\eightit}% \textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl}% \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}% \textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt}% \abovedisplayskip=9pt plus 3pt minus 9pt \belowdisplayskip=\abovedisplayskip \abovedisplayshortskip=0pt plus 3pt \belowdisplayshortskip=3pt plus 3pt \smallskipamount=2pt plus 1pt minus 1pt \medskipamount=4pt plus 2pt minus 1pt \bigskipamount=9pt plus 3pt minus 3pt \normalbaselineskip=9pt \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}% \let\bigf@nt=\eightrm \let\smallf@nt=\sixrm \normalbaselines\rm} \tenpoint \def\pc#1{\bigf@nt#1\smallf@nt} \def\pd#1 {{\pc#1} } \catcode`\;=\active \def;{\relax\ifhmode\ifdim\lastskip>\z@\unskip\fi \kern\fontdimen2 \font\kern -1.2 \fontdimen3 \font\fi\string;} \catcode`\:=\active \def:{\relax\ifhmode\ifdim\lastskip>\z@\unskip\fi\penalty\@M\ \fi\string:} \catcode`\!=\active \def!{\relax\ifhmode\ifdim\lastskip>\z@ \unskip\fi\kern\fontdimen2 \font \kern -1.1 \fontdimen3 \font\fi\string!} \catcode`\?=\active \def?{\relax\ifhmode\ifdim\lastskip>\z@ \unskip\fi\kern\fontdimen2 \font \kern -1.1 \fontdimen3 \font\fi\string?} \def\^#1{\if#1i{\accent"5E\i}\else{\accent"5E #1}\fi} \def\"#1{\if#1i{\accent"7F\i}\else{\accent"7F #1}\fi} \frenchspacing \newtoks\auteurcourant \auteurcourant={\hfil} \newtoks\titrecourant \titrecourant={\hfil} \newtoks\hautpagetitre \hautpagetitre={\hfil} \newtoks\baspagetitre \baspagetitre={\hfil} \newtoks\hautpagegauche \hautpagegauche={\eightpoint\rlap{\folio}\hfil\the\auteurcourant\hfil} \newtoks\hautpagedroite \hautpagedroite={\eightpoint\hfil\the\titrecourant\hfil\llap{\folio}} \newtoks\baspagegauche \baspagegauche={\hfil} \newtoks\baspagedroite \baspagedroite={\hfil} \newif\ifpagetitre \pagetitretrue \def\nopagenumbers{\def\folio{\hfil}} \def\topskip 10pt plus 36pt{\rho}@ggedbottomtrue{\topskip 10pt plus 36pt{\rho}@ggedbottomtrue} \def\unskip . --- \ignorespaces{\unskip . --- \ignorespaces} \def\vskip-\lastskip\bigbreak{\vskip-\lastskip\bigbreak} \def\vskip-\lastskip\medbreak{\vskip-\lastskip\medbreak} \def\ctexte#1\endctexte{% \hbox{$\vcenter{\halign{\hfill##\hfill\crcr#1\crcr}}$}} \mathop{\rm long}\nolimits\def\ctitre#1\endctitre{% \ifdim\lastskip<24pt\vskip-\lastskip\bigbreak\bigbreak\fi \vbox{\parindent=0pt\leftskip=0pt plus 1fill \rightskip=\leftskip \parfillskip=0pt\bf#1\par} \bigskip\nobreak} \let\+=\tabalign \def\signature#1\endsignature{\vskip 15mm minus 5mm\rightline{\vtop{#1}}} \mathcode`A="7041 \mathcode`B="7042 \mathcode`C="7043 \mathcode`D="7044 \mathcode`E="7045 \mathcode`F="7046 \mathcode`G="7047 \mathcode`H="7048 \mathcode`I="7049 \mathcode`J="704A \mathcode`K="704B \mathcode`L="704C \mathcode`M="704D \mathcode`N="704E \mathcode`O="704F \mathcode`P="7050 \mathcode`Q="7051 \mathcode`R="7052 \mathcode`S="7053 \mathcode`T="7054 \mathcode`U="7055 \mathcode`V="7056 \mathcode`W="7057 \mathcode`X="7058 \mathcode`Y="7059 \mathcode`Z="705A \def\spacedmath#1{\def\packedmath##1${\bgroup\mathsurround=0pt ##1\egroup$}% \mathsurround#1 \everymath={\packedmath}\everydisplay={\mathsurround=0pt }} \def\mathsurround=0pt \everymath={}\everydisplay={} {\mathsurround=0pt \everymath={}\everydisplay={} } \def\decale#1{\smallbreak\hskip 28pt\llap{#1}\kern 5pt} \def\decaledecale#1{\smallbreak\hskip 34pt\llap{#1}\kern 5pt} \def\smallbreak\hskip 6pt{$\scriptstyle\bullet$}\kern 5pt{\smallbreak\hskip 6pt{$\scriptstyle\bullet$}\kern 5pt} \def\displaylinesno#1{\displ@y\halign{ \hbox to\displaywidth{$\@lign\hfil\displaystyle##\hfil$}& \llap{$##$}\crcr#1\crcr}} \def\ldisplaylinesno#1{\displ@y\halign{ \hbox to\displaywidth{$\@lign\hfil\displaystyle##\hfil$}& \kern-\displaywidth\rlap{$##$}\tabskip\displaywidth\crcr#1\crcr}} \def\eqalign#1{\null\,\vcenter{\openup\jot{\mu}@th\ialign{ \strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil &&\quad\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil \crcr#1\crcr}}\,} \def\system#1{\left\{\null\,\vcenter{\openup1\jot{\mu}@th \ialign{\strut$##$&\hfil$##$&$##$\hfil&& \enskip$##$\enskip&\hfil$##$&$##$\hfil\crcr#1\crcr}}\right.} \let\@ldmessage=\message \def\message#1{{\def\pc{\string\pc\space}% \def\'{\string'}\def\`{\string`}% \def\^{\string^}\def\"{\string"}% \@ldmessage{#1}}} \def\diagram#1{\def\normalbaselines{\baselineskip=0pt \lineskip=10pt\lineskiplimit=1pt} \matrix{#1}} \def\up#1{\raise 1ex\hbox{\smallf@nt#1}} \def{\it cf.\/}{{\it cf}} \def\etc{{\it etc}} \def\qed{\raise -2pt\hbox{\vrule\vbox to 10pt{\hrule width 4pt \vfill\hrule}\vrule}} \def\unskip\penalty 500\quad\vrule height 4pt depth 0pt width 4pt\medbreak{\unskip\penalty 500\quad\vrule height 4pt depth 0pt width 4pt\medbreak} \def\virg{\raise .4ex\hbox{,}} \def\pv{\ ;\enspace} \def\build#1_#2^#3{\mathrel{ \mathop{\kern 0pt#1}\limits_{#2}^{#3}}} \def\boxit#1#2{% \setbox1=\hbox{\kern#1{#2}\kern#1}% \dimen1=\ht1 \advance\dimen1 by #1 \dimen2=\dp1 \advance\dimen2 by #1 \setbox1=\hbox{\vrule height\dimen1 depth\dimen2\box1\vrule \setbox1=\vbox{\hrule\box1\hrule}% \advance\dimen1 by .4pt \ht1=\dimen1 \advance\dimen2 by .4pt \dp1=\dimen2 \box1\relax} \def\date{\the\day\ \ifcase\month\or janvier\or f\'evrier\or mars\or avril\or mai\or juin\or juillet\or ao\^ut\or septembre\or octobre\or novembre\or d\'ecembre\fi \ {\old \the\year}} \def\dateam{\ifcase\month\or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \ \the\day ,\ \the\year} \def\moins{\mathop{\hbox{\vrule height 3pt depth -2pt width 5pt}}} \def\crog{{\vrule height 2.57mm depth 0.85mm width 0.3mm}\kern -0.36mm [} \def\crod{]\kern -0.4mm{\vrule height 2.57mm depth 0.85mm width 0.3 mm}} \def\moins{\mathop{\hbox{\vrule height 3pt depth -2pt width 5pt}}} \def\kern 1pt{\scriptstyle\circ}\kern 1pt{\kern 1pt{\scriptstyle\circ}\kern 1pt} \def\diagram#1{\def\normalbaselines{\baselineskip=0pt \lineskip=10pt\lineskiplimit=1pt} \matrix{#1}} \def\hfl#1#2{\mathsurround=0pt \everymath={}\everydisplay={} \smash{\mathop{\hbox to 12mm{\rightarrowfill}}\limits^{\scriptstyle#1}_{\scriptstyle#2}}} \def\ghfl#1#2{\mathsurround=0pt \everymath={}\everydisplay={} \smash{\mathop{\hbox to 25mm{\rightarrowfill}}\limits^{\scriptstyle#1}_{\scriptstyle#2}}} \def\phfl#1#2{\mathsurround=0pt \everymath={}\everydisplay={} \smash{\mathop{\hbox to 8mm{\rightarrowfill}}\limits^{\scriptstyle#1}_{\scriptstyle#2}}} \def\vfl#1#2{\llap{$\scriptstyle#1$}\left\downarrow\vbox to 6mm{}\right.\rlap{$\scriptstyle#2$}} \def\pvfl#1#2{\llap{$\scriptstyle#1$}\left\downarrow\vbox to 4mm{}\right.\rlap{$\scriptstyle#2$}} \defvari\'et\'e ab\'elienne{vari\'et\'e ab\'elienne} \defvari\'et\'es ab\'eliennes{vari\'et\'es ab\'eliennes} \defvari\'et\'e ab\'elienne principalement polaris\'ee{vari\'et\'e ab\'elienne principalement polaris\'ee} \defvari\'et\'es ab\'eliennes principalement polaris\'ees{vari\'et\'es ab\'eliennes principalement polaris\'ees} \defvari\'et\'e ab\'elienne polaris\'ee{vari\'et\'e ab\'elienne polaris\'ee} \defvari\'et\'es ab\'eliennes polaris\'ees{vari\'et\'es ab\'eliennes polaris\'ees} \def\S\kern.15em{\S\kern.15em} \def\vskip 5mm plus 1mm minus 2mm{\vskip 5mm plus 1mm minus 2mm} \def\hbox{\bf Z}{\hbox{\bf Z}} \def\hbox{\bf N}{\hbox{\bf N}} \def\hbox{\bf P}{\hbox{\bf P}} \def\hbox{\bf R}{\hbox{\bf R}} \def\hbox{\bf Q}{\hbox{\bf Q}} \def\hbox{\bf C}{\hbox{\bf C}} \def A_{\goth p}{ A_{\goth p}} \def B_{\goth p}{ B_{\goth p}} \def M_{\goth p}{ M_{\goth p}} \def A_{\goth q}{ A_{\goth q}} \def B_{\goth q}{ B_{\goth q}} \def M_{\goth q}{ M_{\goth q}} \def A_{\goth m}{ A_{\goth m}} \def B_{\goth m}{ B_{\goth m}} \def M_{\goth m}{ M_{\goth m}} \def\mathop{\rm Alb}\nolimits{\mathop{\rm Alb}\nolimits} \def\mathop{\rm Ann}\nolimits{\mathop{\rm Ann}\nolimits} \def\mathop{\rm Arctan}\nolimits{\mathop{\rm Arctan}\nolimits} \def\mathop{\rm Ass}\nolimits{\mathop{\rm Ass}\nolimits} \def\mathop{\rm Aut}\nolimits{\mathop{\rm Aut}\nolimits} \defc'est-\`a-dire{c'est-\`a-dire} \def\mathop{\rm car}\nolimits{\mathop{\rm car}\nolimits} \def\mathop{\rm Card}\nolimits{\mathop{\rm Card}\nolimits} \def$\clubsuit${$\clubsuit$} \def{\it cf.\/}{{\it cf.\/}} \def\mathop{\rm Cl}\nolimits{\mathop{\rm Cl}\nolimits} \def\mathop{\rm codim}\nolimits{\mathop{\rm codim}\nolimits} \def\mathop{\rm Coker}\nolimits{\mathop{\rm Coker}\nolimits} \def\mathop{\rm Der}\nolimits{\mathop{\rm Der}\nolimits} \def\mathop{\rm dh}\nolimits{\mathop{\rm dh}\nolimits} \def\mathop{\rm di}\nolimits{\mathop{\rm di}\nolimits} \def\mathop{\rm dp}\nolimits{\mathop{\rm dp}\nolimits} \def\mathop{\rm Div}\nolimits{\mathop{\rm Div}\nolimits} \def\rightarrow\kern -3mm\rightarrow{\rightarrow\kern -3mm\rightarrow} \def\mathop{\rm End}\nolimits{\mathop{\rm End}\nolimits} \def\hbox{\rm Exp.}{\hbox{\rm Exp.}} \def\mathop{\rm Ext}\nolimits{\mathop{\rm Ext}\nolimits} \def\mathop{\rm GL}\nolimits{\mathop{\rm GL}\nolimits} \def\mathop{\rm gr}\nolimits{\mathop{\rm gr}\nolimits} \def\mathop{\rm ht}\nolimits{\mathop{\rm ht}\nolimits} \def\mathop{\rm Hom}\nolimits{\mathop{\rm Hom}\nolimits} \def\mathop{\rm Homgr}\nolimits{\mathop{\rm Homgr}\nolimits} \def\hbox{\rm Id}{\hbox{\rm Id}} \def\hbox{i.e.}{\hbox{i.e.}} \defil faut et il suffit qu{il faut et il suffit qu} \def\mathop{\rm Im}\nolimits{\mathop{\rm Im}\nolimits} \def\mathop{\rm im}\nolimits{\mathop{\rm im}\nolimits} \def\par\hskip0.5cm{\par\hskip0.5cm} \def\simeq{\simeq} \def\mathop{\rm Jac}\nolimits{\mathop{\rm Jac}\nolimits} \def\mathop{\rm Ker}\nolimits{\mathop{\rm Ker}\nolimits} \def\longrightarrow\kern -3mm\rightarrow{\longrightarrow\kern -3mm\rightarrow} \def\mathop{\oalign{lim\cr\hidewidth$\longleftarrow$\hidewidth\cr}}{\mathop{\oalign{lim\cr\hidewidth$\longleftarrow$\hidewidth\cr}}} \def\mathop{\oalign{lim\cr\hidewidth$\longrightarrow$\hidewidth\cr}}{\mathop{\oalign{lim\cr\hidewidth$\longrightarrow$\hidewidth\cr}}} \def{\it loc.cit.\/}{{\it loc.cit.\/}} \def\mathop{\rm long}\nolimits{\mathop{\rm long}\nolimits} \def\longrightarrow{\longrightarrow} \def\nospacedmath\hbox to 10mm{\rightarrowfill}{\mathsurround=0pt \everymath={}\everydisplay={} \hbox to 10mm{\rightarrowfill}} \def\nospacedmath\hbox to 15mm{\rightarrowfill}{\mathsurround=0pt \everymath={}\everydisplay={} \hbox to 15mm{\rightarrowfill}} \def\Leftrightarrow{\Leftrightarrow} \def\mathop{\rm mult}\nolimits{\mathop{\rm mult}\nolimits} \def\mathop{\rm Nm}\nolimits{\mathop{\rm Nm}\nolimits} \defn\up{o}{n\up{o}} \def\mathop{\rm NS}\nolimits{\mathop{\rm NS}\nolimits} \def\cal O{\cal O} \def\hbox{p.}{\hbox{p.}} \def\mathop{\rm PGL}\nolimits{\mathop{\rm PGL}\nolimits} \def\mathop{\rm Pic}\nolimits{\mathop{\rm Pic}\nolimits} \def\mathop{\rm prof}\nolimits{\mathop{\rm prof}\nolimits} \def\rightarrow{\rightarrow} \def\Rightarrow{\Rightarrow} \def\mathop{\rm rang}\nolimits{\mathop{\rm rang}\nolimits} \def\mathop{\rm rank}\nolimits{\mathop{\rm rank}\nolimits} \def\mathop{\rm Re}\nolimits{\mathop{\rm Re}\nolimits} \def\hbox{resp.}{\hbox{resp.}} \def\mathop{\rm rg}\nolimits{\mathop{\rm rg}\nolimits} \def\mathop{\rm Supp}\nolimits{\mathop{\rm Supp}\nolimits} \def\mathop{\rm Sing}\nolimits{\mathop{\rm Sing}\nolimits} \def\mathop{\rm Spec}\nolimits{\mathop{\rm Spec}\nolimits} \def\mathop{\rm Spec}\nolimits{\mathop{\rm Spec}\nolimits} \def\mathop{\rm Sym}\nolimits{\mathop{\rm Sym}\nolimits} \defsi et seulement si{si et seulement si} \defth\'eor\`eme{th\'eor\`eme} \def\mathop{\rm Tor}\nolimits{\mathop{\rm Tor}\nolimits} \def\mathop{\rm Tr}\nolimits{\mathop{\rm Tr}\nolimits} \def\vrule height 12pt depth 5pt width 0pt\vrule{\vrule height 12pt depth 5pt width 0pt\vrule} \def\vrule height 12pt depth 5pt width 0pt{\vrule height 12pt depth 5pt width 0pt} \def\kern -1.5pt -{\kern -1.5pt -} \def\cc#1{\hfill\kern .7em#1\kern .7em\hfill} \def\note#1#2{\footnote{\parindent 0.4cm$^#1$}{\vtop{\eightpoint\baselineskip12pt\hsize15.5truecm \noindent #2}}\parindent 0cm} \def\leavevmode\raise.3ex\hbox{$\scriptscriptstyle\langle\!\langle$}{\leavevmode\raise.3ex\hbox{$\scriptscriptstyle\langle\!\langle$}} \def\leavevmode\raise.3ex\hbox{$\scriptscriptstyle\,\rangle\!\rangle$}{\leavevmode\raise.3ex\hbox{$\scriptscriptstyle\,\rangle\!\rangle$}} \catcode`\@=12 \showboxbreadth=-1 \showboxdepth=-1 \spacedmath{1.5pt}\parskip=1mm \baselineskip=14pt \font\eightrm=cmr10 at 8pt \overfullrule=0pt \input amssym.def \input amssym \def\rightarrow\kern -3mm\rightarrow{\dashrightarrow} \def\widetilde{\widetilde} \def\epsilon{\epsilon} \def\varphi{\varphi} \def\otimes{\otimes} \def{\mathord{\otimes\cdots\otimes }}{{\mathord{\otimes\cdots\otimes }}} \def{\mathord{\times\cdots\times }}{{\mathord{\times\cdots\times }}} \def\bigwedge{\bigwedge} \def\emptyset{\emptyset} \def{\alpha}{{\alpha}} \def{\beta}{{\beta}} \def{\delta}{{\delta}} \def{\varepsilon}{{\varepsilon}} \def{\kappa}{{\kappa}} \def{\lambda}{{\lambda}} \def{\mu}{{\mu}} \def{\nu}{{\nu}} \def{\omega}{{\omega}} \def{\rho}{{\rho}} \def{\sigma}{{\sigma}} \def{\tau}{{\tau}} \def\Lambda{\Lambda} \def{\cal S}{{\cal S}} \def{\cal A}{{\cal A}} \def{\cal C}{{\cal C}} \def{\cal E}{{\cal E}} \def{\cal F}{{\cal F}} \def{\cal G}{{\cal G}} \def{\cal H}{{\cal H}} \def{\cal I}{{\cal I}} \def{\cal J}{{\cal J}} \def{\cal K}{{\cal K}} \def{\cal L}{{\cal L}} \def{\cal M}{{\cal M}} \def{\cal R}{{\cal R}} \def{\cal P}{{\cal P}} \def{\cal Q}{{\cal Q}} \def{\cal O}{{\cal O}} \def{\cal W}{{\cal W}} \def{\cal S}_n{{\cal S}_n} \def{\cal S}_m{{\cal S}_m} \defalg\'ebriquement clos{alg\'ebriquement clos} \deflin\'eaire{lin\'eaire} \lookatfile{lbl \null\bigskip \centerline {\bf SCH\'EMAS DE FANO} \medskip \centerline{{\bf Olivier Debarre\note{1}{\baselineskip=3truemm\rm Financ\'e en partie par le Projet Europ\'een HCM \leavevmode\raise.3ex\hbox{$\scriptscriptstyle\langle\!\langle$} Al\-ge\-braic Geometry in Europe\leavevmode\raise.3ex\hbox{$\scriptscriptstyle\,\rangle\!\rangle$}\ (AGE), Contrat CHRXCT-940557.} et Laurent Manivel}} \vskip 1cm \hskip 1cm\relax Soient $k$ un corps alg\'ebriquement clos\ et $X$ un sous-sch\'ema d'un espace projectif $\hbox{\bf P}^n_k$; on appelle sch\'ema de Fano, et l'on note $F_r(X)$, le sous-sch\'ema de la grassmannienne $G(r,\hbox{\bf P}^n_k)$ qui param\`etre les espaces lin\'eaire s de dimension $r$ contenus dans $X$. Ces sch\'emas ont une longue histoire ([F], [vW], [AK], [BVV], [B1], [PS], [K], [ELV], [BV]) mais il ne semble pas exister dans la litt\'erature d'\'enonc\'e g\'en\'eral sur leurs propri\'et\'es, m\^eme les plus simples comme la connexit\'e, valable en toute caract\'eristique. Un des buts de cet article est de rassembler sous une r\'ef\'erence commune des faits g\'en\'eraux sur ces sch\'emas. \hskip 1cm\relax Apr\`es un paragraphe de notations, on obtient dans le \S 2, en se basant sur les id\'ees de [K], notre premier r\'esultat principal: pour une intersection compl\`ete g\'en\'erale $X$, le sch\'ema de Fano $F_r(X)$ {\it est non vide et lisse de la dimension attendue ${\delta}$ lorsque celle-ci est positive, et connexe lorsque} ${\delta}>0$. Dans le \S 3, on applique des r\'esultats de [D] et [S] pour calculer certains groupes d'homotopie de $F_r(X)$. Par ailleurs, le sch\'ema $F_r(X)$ est le lieu des z\'eros d'une section d'un fibr\'e vectoriel sur la grassmannienne $G(r,\hbox{\bf P}^n)$; lorsqu'il a la dimension attendue $\delta$, son id\'eal admet une r\'esolution par un complexe de Koszul. Un th\'eor\`eme\ d'annulation pour certains fibr\'es vectoriels sur $G(r,\hbox{\bf P}^n)$ (prop. \ref{annul}) nous permet de montrer notre second r\'esultat principal, \`a savoir {\it un th\'eor\`eme\ de type Lefschetz, qui permet d'obtenir, pour $k=\hbox{\bf C}$, les nombres de Hodge $h^{p,q}(F_r(X))$ pour $p+q$ assez petit} (inf\'erieur \`a $\dim X-2r$ pour $n$ grand). Apr\`es avoir r\'edig\'e cette partie, nous nous sommes rendus compte que Borcea avait d\'ej\`a utilis\'e le th\'eor\`eme\ d'annulation de Bott dans ce cadre (il obtient entre autres les r\'esultats du \S 2 en caract\'eristique nulle). \hskip 1cm\relax Les m\^emes m\'ethodes permettent d'\'etudier dans le \S 4 la restriction\break $H^0(G(r,\hbox{\bf P}^n),{\cal O}(l))\longrightarrow H^0(F_r(X),{\cal O}(l))$; on montre que pour $n$ assez grand, $F_r(X)$ est projectivement normal dans $G(r,\hbox{\bf P}^n)$, et que toute \'equation de $F_r(X)$ est de degr\'e au moins \'egal \`a une \'equation de $X$ dans $\hbox{\bf P}^n$. On donne aussi une formule explicite pour le calcul du degr\'e des sch\'emas $F_r(X)$: c'est le coefficient d'un mon\^ome particulier dans un polyn\^ome explicite en $r+1$ variables. On donne quelques exemples de ce calcul pour des hypersurfaces de bas degr\'e. \hskip 1cm\relax On s'int\'eresse ensuite aux sous-sch\'emas de $F_r(X)$ qui param\`etrent les $r$\kern -1.5pt - plans contenant un $r_0$\kern -1.5pt - plan fix\'e; le th\'eor\`eme\ principal du \S 5 g\'en\'eralise les r\'esultats analogues du \S 2 dans ce cadre. On en d\'eduit que $F_r(X)$ est s\'eparablement unir\'egl\'e en droites pour $n$ assez grand, ce qui nous permet dans le \S 6 d'adapter des id\'ees de [K] pour montrer, toujours pour $n$ assez grand, que le groupe de Chow rationnel des $1$\kern -1.5pt - cycles sur un sch\'ema $F_r(X)$ est de rang $1$. Il est tentant de g\'en\'eraliser une conjecture de Srinivas et Paranjape ([P]) de la fa{\gamma} con suivante: pour $n$ assez grand, les groupes de Chow rationnels de basse dimension de $F_r(X)$ devraient \^etre ceux de la grassmannienne ambiante $G(r,\hbox{\bf P}^n)$. \hskip 1cm\relax Dans le \S 7, on d\'emontre, comme conjectur\'e dans [BVV], que {\it les sch\'emas de Fano g\'en\'eriques sont unirationnels pour $n$ assez grand}. On se ram\`ene pour cela \`a un r\'esultat de Predonzan ([Pr]), pr\'ecis\'e dans l'article [PS], qui fournit un crit\`ere explicite pour l'unirationa\-li\-t\'e d'une intersection compl\`ete dans un espace projectif. Les bornes obtenues sont explicites, mais tr\`es grandes; par exemple, on montre que la vari\'et\'e des droites contenues dans une hypersurface cubique de $\hbox{\bf P}^n$ est unirationnelle pour $n\ge 433$ (alors que c'est d\'ej\`a une vari\'et\'e de Fano pour $n\ge 6$). \section{Notations} \hskip 1cm\relax Soient $k$ un corps alg\'ebriquement clos et $V$ un $k$\kern -1.5pt - espace vectoriel de dimension $n+1$. Pour toute suite finie ${\bf d}=(d_1,\ldots,d_s)$ d'entiers positifs, et tout entier positif $r$, on note $|{\bf d}|=\sum_{i=1}^sd_i$, puis ${\bf d}+r=(d_1+r,\ldots,d_s+r)$ et ${{\bf d}\choose r}={\sum_{i=1}^s}{d_i\choose r} $. On pose $\mathop{\rm Sym}\nolimits ^{\bf d}V^*=\bigoplus_{i=1}^s\mathop{\rm Sym}\nolimits ^{d_i}V^*$, espace vectoriel que l'on notera aussi $\Gamma_{{\bf P} V}({\bf d})$. Enfin, si ${\bf f}=(f_1\ldots,f_s)$ est un \'el\'ement non nul de $\mathop{\rm Sym}\nolimits ^{\bf d}V^*$, on note $X_{\bf f}$ le sous-sch\'ema de $\hbox{\bf P} V$ d'\'equations $f_1=\cdots=f_s=0$; on dira d'un tel sch\'ema qu'il est {\it d\'efini par des \'equations de degr\'e} ${\bf d}$. \hskip 1cm\relax On pose ensuite $$ \delta(n,{\bf d},r)=(r+1)(n-r)- {{\bf d}+r\choose r}$$ et $\delta_-(n,{\bf d},r)=\min\{ \delta (n,{\bf d},r), n-2r-s\}$, que l'on \'ecrira simplement $\delta$ et $\delta_-$ lorsque qu'aucune confusion ne sera \`a craindre. \section{Dimension, lissit\'e et connexit\'e} \hskip 1cm\relax On montre dans ce num\'ero que les sch\'emas de Fano d'un sous-sch\'ema $X$ de $\hbox{\bf P}^n_k$ d\'efini par des \'equations de degr\'es ${\bf d}=(d_1,\ldots,d_s)$ sont lisses de la dimension attendue pour $X$ g\'en\'erale, et connexes lorsque cette dimension est strictement positive. Divers cas particuliers du th\'eor\`eme\ suivant \'etaient d\'ej\`a connus: citons par exemple [BVV], qui traite le cas $k=\hbox{\bf C}$ et $r=s=1$; [P], [Mu] et [PS], qui d\'emontrent b); [B1], qui d\'emontre le th\'eor\`eme\ lorsque $k$ est de caract\'eristique nulle; et [K], qui traite le cas $r=s=1$ (th. 4.3, p. 266), et dont nous empruntons les id\'ees. Lorsque $k=\hbox{\bf C}$, une d\'emonstration compl\`etement diff\'erente d\'ecoule de celle du th\'eor\`eme \ref{leff} ({\it cf.\/}\ rem. \ref{rem}.1). \hskip 1cm\relax Pour appliquer le th\'eor\`eme , il est utile de noter que {\it lorsque ${\bf d}\ne (2)$, l'entier $\delta(n,{\bf d},r)$ est positif (resp. strictement positif) si et seulement si $\delta_-(n,{\bf d},r)$ l'est.} \th \label{fano} Th\'eor\`eme \enonce Soient $X$ un sous-sch\'ema de $\hbox{\bf P}^n_k$ d\'efini par des \'equations de degr\'e ${\bf d}$, et $F_r(X)$ le sch\'ema de Fano des $r$\kern -1.5pt - plans contenus dans $X$. \par\hskip0.5cm {\rm a)} Lorsque $\delta_-(n,{\bf d},r)< 0$, le sch\'ema $F_r(X)$ est vide pour $X$ g\'en\'erale. \par\hskip0.5cm {\rm b)} Lorsque $\delta_-(n,{\bf d},r)\ge 0$, le sch\'ema $F_r(X)$ est non vide; il est lisse de dimension $\delta(n,{\bf d},r)$ pour $X$ g\'en\'erale. \par\hskip0.5cm {\rm c)} Lorsque $\delta_-(n,{\bf d},r)>0$, le sch\'ema $F_r(X)$ est connexe. \endth \hskip 1cm\relax Consid\'erons la vari\'et\'e d'incidence $$I_r=\{ ([{\bf f}],\Lambda)\in \hbox{\bf P} \mathop{\rm Sym}\nolimits ^{\bf d}V^*\times G(r,\hbox{\bf P}^n) \mid \Lambda\i X_{\bf f}\}\ ,$$ et les projections $p_r:I_r\rightarrow \hbox{\bf P} \mathop{\rm Sym}\nolimits ^{\bf d}V^*$ (dont la fibre au-dessus de $[{\bf f}]$ s'identifie \`a $F_r(X_{\bf f})$) et $q:I_r\rightarrow G(r,\hbox{\bf P}^n)$. Etant donn\'e un $r$\kern -1.5pt - plan $\Lambda=\hbox{\bf P} W$, la fibre $q^{-1}([\Lambda])$ est l'espace projectif associ\'e au noyau du morphisme surjectif $\mathop{\rm Sym}\nolimits ^{\bf d}V^*\rightarrow \mathop{\rm Sym}\nolimits ^{\bf d}W^*$. Elle est donc de codimension ${{\bf d}+r\choose r}$ dans $\hbox{\bf P} \mathop{\rm Sym}\nolimits ^{\bf d}V^*$, de sorte que $I_r$ est irr\'eductible lisse de codimension ${{\bf d}+r\choose r}$ dans $ \hbox{\bf P} \mathop{\rm Sym}\nolimits ^{\bf d}V^*\times G(r,\hbox{\bf P}^n)$. \hskip 1cm\relax On note $Z_r$ le ferm\'e des points de $I_r$ o\`u $p_r$ n'est pas lisse, et $\Delta_r$ l'image de $Z_r$ par $p_r$ (avec la convention $\Delta_{-1}=\varnothing$). Soit $\Lambda$ un $r$\kern -1.5pt - plan, d'\'equations $x_{r+1}=\cdots=x_n=0$ dans $\hbox{\bf P} V$; pour tout entier $m\ge 0$, on note ${\cal B}_m$ la base $\{ {\bf x}^J\mid J\i \{0,\ldots,r\}\ ,\ \mathop{\rm Card}\nolimits(J)=m\}$ de l'espace vectoriel $\Gamma_\Lambda(m)$; on note aussi ${\cal B}_{\bf d}$ la base $\iota_1({\cal B}_{d_1})\cup\cdots\cup\iota_s({\cal B}_{d_s})$ de l'espace vectoriel $\Gamma_\Lambda({\bf d})$ (o\`u $\iota_j$ est l'injection canonique de $\Gamma_\Lambda(d_j)$ dans $\Gamma_\Lambda({\bf d})$). \th \label{lisse} Lemme \enonce Pour qu'un point $([{\bf f}],[\Lambda])$ de $I_r$ soit dans $Z_r$, il faut et il suffit que le morphisme $\alpha:\Gamma_\Lambda(1)^{n-r}\rightarrow \Gamma_\Lambda({\bf d})$ d\'efini par $$\alpha(h_{r+1},\ldots,h_n)=\Bigl( \sum_{i=r+1}^nh_i \Bigl({\partial f_1\over\partial x_i}\Bigr) _{\displaystyle\vert_{\scriptstyle\Lambda}},\ldots, \sum_{i=r+1}^nh_i \Bigl({\partial f_s\over\partial x_i}\Bigr) _{\displaystyle\vert_{\scriptstyle\Lambda}}\Bigr)$$ ne soit pas surjectif. \endth \hskip 1cm\relax Cela r\'esulte d'un calcul explicite fait dans [BVV] dans le cas $r=s=1$.\unskip\penalty 500\quad\vrule height 4pt depth 0pt width 4pt\medbreak \hskip 1cm\relax {\it Lorsque $X$ est lisse de codimension $r$ le long de} $\Lambda$, on a une suite exacte $$0\longrightarrow N_{\Lambda/X}\longrightarrow {\cal O}_\Lambda(1)^{n-r}\buildrel{u}\over{\longrightarrow} \bigoplus_{i=1}^s{\cal O}_\Lambda(d_i)\longrightarrow 0 \ ,$$ et le morphisme $\alpha$ n'est autre que $H^0(u)$ ({\it cf.\/}\ [BVV], prop. 3 et [K], p. 267 dans le cas $r=s=1$). La condition du lemme est donc \'equivalente dans ce cas \`a l'annulation de $H^1(\Lambda,N_{\Lambda/X})$. \hskip 1cm\relax Soit $\mu: \Gamma_\Lambda(1)\times \Gamma_\Lambda({\bf d}-1)\rightarrow \Gamma_\Lambda({\bf d})$ le morphisme de multiplication, d\'efini par\break $\mu(h,g_1,\ldots,g_s)=(hg_1,\ldots,hg_s)$. Si $H$ est un hyperplan de $\Gamma_\Lambda({\bf d})$, on note $\mu^{-1}(H)$ l'ensemble $\{\ g\in \Gamma_\Lambda({\bf d}-1)\mid \mu(\Gamma_\Lambda(1)\times\{ g\} )\i H\ \}$. \hskip 1cm\relax On peut r\'e\'enoncer le lemme \ref{lisse} de la fa{\gamma} con suivante: soit ${\cal Z}$ le sous-ensemble de $q^{-1}([\Lambda])\times\hbox{\bf P} \Gamma_\Lambda({\bf d})$ form\'e des couples $([{\bf f}],[\ell])$ tels que $$\Bigl( \Bigl({\partial f_1\over\partial x_i}\Bigr) _{\displaystyle\vert_{\scriptstyle\Lambda}},\ldots, \Bigl({\partial f_s\over\partial x_i}\Bigr) _{\displaystyle\vert_{\scriptstyle\Lambda}}\Bigr)$$ soit dans $\mu^{-1}(\mathop{\rm Ker}\nolimits(\ell))$ pour tout $i=r+1,\ldots,n$; alors $Z_r\cap q^{-1}([\Lambda])$ s'identifie \`a la premi\`ere projection de ${\cal Z}$. Pour tout entier $h$, notons ${\cal L}_h$ l'ensemble des formes lin\'eaires $\ell$ sur $\Gamma_\Lambda({\bf d})$ v\'erifiant $\mathop{\rm codim}\nolimits \mu^{-1}(\mathop{\rm Ker}\nolimits(\ell)) =h$, et ${\cal Z}_h$ l'ensemble des \'el\'ements $([{\bf f}],[\ell])$ de ${\cal Z}$ avec $\ell\in {\cal L}_h$. On peut \'ecrire $f_i=\sum_{j=r+1}^nx_jf_{ij}$, avec $f_{ij}\vert_\Lambda=\bigl({\partial f_i\over\partial x_j}\bigr) _{\displaystyle\vert_{\scriptstyle\Lambda}}$, de sorte que $$\mathop{\rm codim}\nolimits_{q^{-1}([\Lambda])}pr_1({\cal Z}_h)\ge h(n-r)-\dim\hbox{\bf P} {\cal L}_h\leqno{\global\def\currenvir{formule}\hbox{\label{minor1}}$$ et $$\mathop{\rm codim}\nolimits_{I_r}Z_r=\mathop{\rm codim}\nolimits_{q^{-1}([\Lambda])}pr_1({\cal Z})\ge\min_{1\le h\le r+1}[ h(n-r)-\dim\hbox{\bf P} {\cal L}_h]\ .\leqno{\global\def\currenvir{formule}\hbox{\label{minor2}}$$ \global\def\currenvir{subsection Soit $\ell$ une forme lin\'eaire sur $\Gamma_\Lambda({\bf d})$. Soit $M$ la matrice \`a coefficients dans $\Gamma_\Lambda({\bf d})$ de la forme bilin\'eaire $\mu$ dans les bases ${\cal B}_1$ et ${\cal B}={\cal B}_{{\bf d}-1}$. Pour qu'un \'el\'ement $g=\sum_{b\in{\cal B}} g_bb$ de $\Gamma_\Lambda({\bf d}-1)$ soit dans $\mu^{-1}(\mathop{\rm Ker}\nolimits(\ell))$, il faut et il suffit que $\sum_b g_b\ell(x_ib)$ soit nul pour tout $i=0,\ldots,r$, de sorte que {\it la codimension de $\mu^{-1}(\mathop{\rm Ker}\nolimits(\ell))$ dans $\Gamma_\Lambda({\bf d}-1)$ est le rang de la matrice} $\ell(M)$.\label{forme} \th \label{propre} Lemme \enonce Soient $([{\bf f}],[\Lambda])$ un \'el\'ement de $Z_r\moins p_r^{-1}(\Delta_{r-1})$ et $\ell$ une forme lin\'eaire non nulle sur $\Gamma_\Lambda({\bf d})$, qui s'annule sur l'image de $\alpha$. Alors $\mu^{-1}(\mathop{\rm Ker}\nolimits(\ell))$ est de codimension $r+1$ dans $\Gamma_\Lambda({\bf d}-1)$. \endth \hskip 1cm\relax Proc\'edons par l'absurde en supposant que la matrice $\ell(M)$ d\'efinie ci-dessus ne soit pas de rang maximal. Quitte \`a effectuer un changement lin\'eaire de coordonn\'ees, on peut supposer $\ell(x_rb)=0$ pour tout $b$ dans ${\cal B}$, de sorte que si $\Lambda'$ est l'hyperplan de $\Lambda$ d\'efini par $x_r=0$, la forme lin\'eaire $\ell$ provient d'une forme lin\'eaire $\ell'$ sur $ \Gamma_{\Lambda'}({\bf d})$. Si $\alpha':\Gamma_{\Lambda'}(1)^{n-r+1}\rightarrow \Gamma_{\Lambda'}({\bf d})$ est le morphisme associ\'e au point $([{\bf f}],[\Lambda'])$ de $I_{r-1}$ d\'efini dans le lemme \ref{lisse}, $\ell'$ s'annule sur $\alpha'(\{ 0\}\times \Gamma_{\Lambda'}(1)^{n-r})$. Comme la restriction de ${\partial f_i\over \partial x_r}$ \`a $\Lambda'$ est nulle pour tout $i$, la forme lin\'eaire $\ell'$ s'annule sur toute l'image de $\alpha'$, ce qui contredit l'hypoth\`ese $[{\bf f}]\notin \Delta_{r-1}$.\unskip\penalty 500\quad\vrule height 4pt depth 0pt width 4pt\medbreak \hskip 1cm\relax En d'autres termes, $q^{-1}([\Lambda])\cap\bigl( Z_r\moins p_r^{-1}(\Delta_{r-1})\bigr)$ est contenu dans $pr_1({\cal Z}_{r+1})$, et (\ref{minor1}) entra\^ine $$\mathsurround=0pt \everymath={}\everydisplay={} \displaylines{\dim (\overline{Z_r\moins p_r^{-1}(\Delta_{r-1})})=\dim I_r-\mathop{\rm codim}\nolimits_{I_r} (\overline{Z_r\moins p_r^{-1}(\Delta_{r-1})})\cr \le\dim \hbox{\bf P} \mathop{\rm Sym}\nolimits ^dV^*+\dim G(r,\hbox{\bf P}^n)-{{\bf d}+r\choose r}-(r+1)(n-r)+\dim\hbox{\bf P}\Gamma_\Lambda({\bf d}) <\dim \hbox{\bf P} \mathop{\rm Sym}\nolimits ^dV^*\ .\cr} $$ \global\def\currenvir{subsection Il en r\'esulte $\Delta_r\moins\Delta_{r-1}\ne \hbox{\bf P} \mathop{\rm Sym}\nolimits ^dV^*$, d'o\`u $\Delta_r\ne\hbox{\bf P} \mathop{\rm Sym}\nolimits ^dV^*$ par r\'ecurrence sur $r$.\label{ferme} \hskip 1cm\relax On remarquera que nous avons en fait d\'emontr\'e que $\Delta_r$ a au plus une composante irr\'eductible de plus que $\Delta_{r-1}$, c'est-\`a-dire\ au plus $r+1$ composantes irr\'eductibles. \th \label{major} Lemme \enonce Pour $1\le h\le r+1$, la dimension de ${\cal L}_h$ est au plus $h(r-h+1)+{{\bf d}+h-1\choose h-1}$. \endth \hskip 1cm\relax On garde les notations de (\ref{forme}). Supposons les $h$ premi\`eres lignes de la matrice $\ell(M)$ lin\'eairement ind\'ependantes; on peut \'ecrire $\ell(x_jb)=\sum_{i=0}^{h-1}a_{ij}\ell(x_ib)$, pour tous $j=h,\ldots,r$ et $b\in{\cal B}$, de sorte que les $\ell(b_i)$, pour $b_i={\bf x}^I$ dans ${\cal B}_{d_i}$, peuvent s'exprimer en fonction de ceux pour lesquels $I\i \{0,\ldots,h-1\}$, et des $h(r-h+1)$ coefficients $a_{ij}$.\unskip\penalty 500\quad\vrule height 4pt depth 0pt width 4pt\medbreak \hskip 1cm\relax L'in\'egalit\'e (\ref{minor2}) donne $$\mathop{\rm codim}\nolimits_{I_r}Z_r\ge \min_{1\le h\le r+1}\bigl[ h(n-2r+h-1)-{{\bf d}+h-1\choose h-1}\bigr]+1\ .$$ \global\def\currenvir{subsection Lorsque ${\bf d}\ne (2)$, on v\'erifie que l'expression entre crochets est une fonction {\it concave} $\varphi$ de $h$ sur $[1,+\infty [$; lorsque ${\bf d}=(2)$ et $\delta_-\ge 0$, c'est une fonction {\it croissante}. On a dans chacun de ces cas\label{concave}\vskip-5mm $$\mathop{\rm codim}\nolimits_{I_r}Z_r\ge \min \{ \varphi(1),\varphi(r+1)\}+1=\delta_-+1\ .$$ \hskip 1cm\relax Supposons $\delta_-<0$. Si ${\bf d}=(2)$, cela signifie $2r\ge n$; si une quadrique $X$ contient un $r$\kern -1.5pt - plan $\Lambda$, \'ecrivons en gardant les m\^emes notations $f=x_{r+1}\ell_{r+1}+\cdots+x_n\ell_n$, o\`u les $\ell_i$ sont des formes lin\'eaire s. Comme $n-r\le r$, celles-ci ont un z\'ero commun sur $\Lambda$, qui est un point singulier de $X$, ce qui ne peut se produire pour $X$ g\'en\'erale. Lorsque ${\bf d}\ne (2)$, on a $\delta<0$, d'o\`u $\dim I_r<\dim \hbox{\bf P} \mathop{\rm Sym}\nolimits ^dV^*$, et $p_r$ n'est pas surjective; ceci montre a) dans tous les cas. \hskip 1cm\relax Supposons $\delta_-\ge 0$; il existe d'apr\`es (\ref{concave}) un point de $I_r$ en lequel $p_r$ est lisse. Cela entra\^ine que $p_r$ est surjective, et que $F_r(X)$ est de dimension $\delta$ pour $X$ g\'en\'erale. Par (\ref{ferme}), $p_r$ est lisse au-dessus d'un ouvert dense de $\hbox{\bf P} \mathop{\rm Sym}\nolimits ^dV^*$, ce qui montre b). \hskip 1cm\relax Supposons maintenant $\delta_->0$, et consid\'erons comme dans [BVV] la factorisation de Stein $p_r:I_r\longrightarrow S\buildrel{\pi}\over{\longrightarrow}\hbox{\bf P} \mathop{\rm Sym}\nolimits ^dV^*$ du morphisme propre $p_r$. Si le morphisme $\pi$ est ramifi\'e, le th\'eor\`eme\ de puret\'e entra\^ine que $Z_r$ contient l'image inverse d'un diviseur de $S$, ce qui contredit l'estimation de (\ref{concave}). Il s'ensuit que $\pi$ est \'etale, donc que c'est un isomorphisme puisque $\hbox{\bf P} \mathop{\rm Sym}\nolimits ^dV^*$ est simplement connexe. La vari\'et\'e $F_r(X)$ est donc connexe pour toute hypersurface $X$, ce qui montre c).\unskip\penalty 500\quad\vrule height 4pt depth 0pt width 4pt\medbreak \ex{Remarques} 1) Soit $S$ le sous-fibr\'e tautologique sur $G(r,{\bf P}V)$. Tout \'el\'ement ${\bf f}$ de $\mathop{\rm Sym}\nolimits ^{\bf d}V^*$ induit une section du fibr\'e $\mathop{\rm Sym}\nolimits ^dS$, dont le lieu des z\'eros est le sch\'ema $F_r(X_{\bf f})$. La partie b) du th\'eor\`eme\ montre que lorsque $\delta_-(n,{\bf d},r)\ge 0$, la classe de Chern $c_{\max}(\mathop{\rm Sym}\nolimits ^{\bf d}S^*)$ est non nulle. On verra dans le \S 4 comment expliciter cette classe de Chern dans l'anneau de Chow de la grassmannienne. On remarque que lorsque ${\bf d}=(2)$ et que $\delta_-<0\le{\delta}$, le rang de $\mathop{\rm Sym}\nolimits ^2S$ est plus petit que la dimension de $G(r,{\bf P}^n)$, mais sa classe de Chern d'ordre maximal $2^{r+1}\sigma_{r+1,r,\ldots,1}$ est nulle ({\it cf.\/}\ [Fu], ex. 14.7.15).\label{chern} 2) Toute quadrique lisse $X$ dans $\hbox{\bf P}^n$ est projectivement \'equivalente \`a la quadrique d'\'equation $x_0x_1+x_2x_3+\cdots+x_{n-1}x_n=0$ si $n$ est impair, \`a la quadrique d'\'equation $x_0x_1+x_2x_3+\cdots+x_{n-2}x_{n-1}+x_n^2=0$ si $n$ est pair. Le sch\'ema $F_r(X)$ est donc lisse connexe d\`es que $\delta_->0$, c'est-\`a-dire\ $n>2r+1$; on sait qu'il a deux composantes connexes si $n=2r+1$. \section{Groupes d'homotopie, groupes de cohomologie et groupe de Picard} \hskip 1cm\relax Les r\'esultats de [D] et [S] permettent de calculer les groupes d'homotopie des sch\'emas de Fano pour $n$ assez grand. \th\label{pij} Proposition \enonce Soit $X$ un sous-sch\'ema de $\hbox{\bf P}^n_k$ d\'efini par des \'equations de degr\'e ${\bf d}$. On suppose $F_r(X)$ irr\'eductible de dimension $\delta$. \hskip 1cm\relax {\rm a)} Si $n\ge {2\over r+1}{{\bf d}+r\choose r}+r+1$, le sch\'ema $F_r(X)$ est alg\'ebriquement simplement connexe, topologiquement simplement connexe lorsque $k=\hbox{\bf C}$. \hskip 1cm\relax {\rm b)} Lorsque $k=\hbox{\bf C}$ et que $F_r(X)$ est lisse, on a $\pi_j\bigl( G(r,{\bf P}^n),F_r(X)\bigr)=0$ pour $n\ge 2{{\bf d}+r\choose r}+j-1$. En particulier, si $n\ge 2{{\bf d}+r\choose r}+2 $, le groupe de Picard de $F_r(X)$ est isomorphe \`a $\hbox{\bf Z}$, engendr\'e par la classe de ${\cal O}(1)$. \endth \hskip 1cm\relax Le point b) est cons\'equence directe de [S]. Pour a), il suffit par [D], cor. 7.4 de montrer que $[F_r(X)]\cdot[F_r(X)]\cdot[G(r,\hbox{\bf P}^{n-1})]$ est non nul dans $A(G(r,\hbox{\bf P}^n))$. Par la remarque \ref{chern}, cette intersection est la classe de Chern de degr\'e maximal de $\mathop{\rm Sym}\nolimits ^{\bf d}S^*\oplus \mathop{\rm Sym}\nolimits ^{\bf d}S^*\oplus S^*$, et celle-ci est non nulle d\`es que $\delta(n-1,({\bf d},{\bf d}),r)$ est positif, condition qui d\'ecoule de l'hypoth\`ese.\unskip\penalty 500\quad\vrule height 4pt depth 0pt width 4pt\medbreak \ex{Remarques} 1) On rappelle que $\pi_j(G(r,{\bf P}^n))\simeq \pi_{j-1}(U(r+1))$ pour $j\le 2(n-r)$ ([H], chap. 7); si l'on suppose aussi $j\le 2(r+1)$, le th\'eor\`eme de p\'eriodicit\'e de Bott implique donc $\pi_j(G(r,{\bf P}^n))={\bf Z}$ ou $0$ selon que $j$ est pair ou impair. En g\'en\'eral, il peut cependant appara\^{\i}tre de la torsion (par exemple, $\pi_{11}(G(3,\hbox{\bf P}^n))= {\bf Z}_2\oplus {\bf Z}_{120}$ si $n\ge 9$).\label{rempi} 2) La remarque \ref{chern} montre que lorsque $F_r(X)$ est de dimension $\delta$, on a $$\omega_{F_r(X)}\simeq \omega_{G(r,{\bf P}^n)}\otimes \bigwedge^{\max}\mathop{\rm Sym}\nolimits ^{\bf d}S^*\vert_{F_r(X)} \simeq {\cal O}_{F_r(X)}({{\bf d}+r\choose r+1}-n-1)\ .$$ \hskip 1cm\relax En particulier, $F_r(X)$ est une vari\'et\'e de Fano lorsque $n\ge {{\bf d}+r\choose r+1}$, donc simplement connexe lorsque $k=\hbox{\bf C}$ ([C1], [KMM1]). Cette borne est n\'eanmoins moins bonne que celle de la prop. \ref{pij}.a) d\`es que l'un des $d_i$ est $\ge 3$.. \medskip \ex{Exemple} Soit $X$ une hypersurface cubique lisse dans $\hbox{\bf P}^n_k$; par [BVV], prop. 5, $F_1(X)$ est une vari\'et\'e {\it lisse connexe} de dimension $2n-6$. La proposition entra\^ine que $F_1(X)$ est simplement connexe pour $n\ge 6$. Lorsque $k=\hbox{\bf C}$, cela reste vrai pour $n=5$ ([BD], prop. 3), mais pas pour $n=4$ puisque $h^1(F_1(X),{\cal O}_{F_1(X)})=5$ ([AK], prop. 1.15).\label{BD} \bigskip \hskip 1cm\relax Passons maintenant au r\'esultat principal de ce num\'ero. On a vu en \ref{chern} que $F_r(X)$ est le lieu des z\'eros d'une section d'un fibr\'e vectoriel sur la grassmannienne; lorsqu'il a la codimension attendue, son id\'eal admet une r\'esolution par un complexe de Koszul. Lorsque $k=\hbox{\bf C}$, on montre \`a l'aide du th\'eor\`eme de Borel-Weil-Bott ([Bo], [De]) et des r\'esultats de [Ma1] et [Ma2] un th\'eor\`eme\ d'annulation (prop. \ref{annul}) qui nous permettra de d\'eterminer certains groupes de cohomologie des sch\'emas de Fano. \th \label{leff} Th\'eor\`eme \enonce Soit $X$ un sous-sch\'ema de $\hbox{\bf P}^n_{\bf C}$ d\'efini par des \'equations de degr\'e ${\bf d}$, tel que $F_r(X)$ soit lisse de dimension $\delta(n,{\bf d},r)$. Le morphisme de restriction\break $H^i(G(r,\hbox{\bf P}^n),\hbox{\bf Q})\rightarrow H^i(F_r(X),\hbox{\bf Q})$ est bijectif pour $i<\delta_-(n,{\bf d},r)$, injectif pour $i=\delta_-(n,{\bf d},r)$. \endth \hskip 1cm\relax En particulier, les nombres de Hodge $h^{p,q}(F_r(X))$ et $h^{p,q}(G(r,\hbox{\bf P}^n))$ sont \'egaux si $p+q<\delta_-$. Rappelons que ces derniers sont nuls pour $p\neq q$, et qu'ils sont \'egaux si $p=q$ au nombre de partitions de $p$ inscrites dans un rectangle de c\^ot\'es $r+1$ et $n-r$. On retrouve aussi un r\'esultat de [BV]: \th \label{pic} Corollaire \enonce Si de plus $\delta_-\ge 3$, le groupe de Picard de $F_r(X)$ est de rang $1$. \endth \ex{Remarques} 1) La borne du th\'eor\`eme\ est souvent la meilleure possible: pour une hypersurface cubique lisse $X$ dans $\hbox{\bf P}^5$ et $r=1$, on a pour $\delta=4$ et $\delta_-=2$, et $h^2(F_1(X),\hbox{\bf Q} )=23$ ([BD], prop. 3); pour une intersection compl\`ete g\'en\'erique $X$ de deux quadriques dans $\hbox{\bf P}^6$, on a $h^2(F_1(X),\hbox{\bf Q} )=8$ ([B2], th. 2.1).\label{rem} 2) Le th\'eor\`eme\ permet de retrouver, lorsque $k=\hbox{\bf C}$, les points b) et c) du th\'eor\`eme\ \ref{fano}; c'est la m\'ethode suivie dans [B1]. 3) Il r\'esulte du corollaire \ref{sepunir} et de [K], cor. 1.11, p. 189 et cor. 3.8, p. 202, que pour $n\ge {{\bf d}+r\choose r+1}+r+1$, les groupes $H^0(F,\Omega^m_F)$ et $H^0(F,(\Omega^1_F)^{\otimes m})$ sont nuls pour tout $m>0$. Lorsque $k$ est de caract\'eristique nulle, l'annulation de ces groupes peut se d\'eduire de la remarque \ref{rempi}.2) et du th\'eor\`eme\ 2.13 de [K], p. 254 ({\it cf.\/}\ [C2] et [KMM2]), sous l'hypoth\`ese plus faible $n\ge {{\bf d}+r\choose r+1}$. 4) Lorsque $n\ge {{\bf d}+r\choose r+1}$, $F_r(X)$ est une vari\'et\'e de Fano (remarque \ref{rempi}.2). Lorsque $k$ est de caract\'eristique nulle et que $F_r(X)$ est lisse, le th\'eor\`eme\ d'annulation de Kodaira entra\^ine que son groupe de Picard est sans torsion ({\it cf.\/}\ [K], (1.4.13), p. 242). Vue l'hypoth\`ese sur $n$, et sauf dans le cas ${\bf d}=(2,2)$ et $n=2r+4$, on a $\delta_-\ge 3$, d'o\`u $\mathop{\rm Pic}\nolimits (F_r(X))\simeq\hbox{\bf Z}$ par le corollaire (comparer avec la prop. \ref{pij}.b). Lorsque ${\bf d}=(2,2)$ et $n=2g+1$, la vari\'et\'e $F_{g-2}(X)$ est isomorphe \`a l'espace de modules des fibr\'es stables de rang $2$ et de d\'eterminant fix\'e de degr\'e impair sur une courbe hyperelliptique $C$ de genre $g$ ([DR], th. 1). L'isomorphisme $\mathop{\rm Pic}\nolimits (F_{g-2}(X))\simeq\hbox{\bf Z}[{\cal O}(1)]$ (on a $\delta_-=3$) est d\'emontr\'e dans [DR], (5.10) (II), p. 177 ({\it cf.\/}\ aussi [R]). En revanche, $F_{g-1}(X)$ est isomorphe \`a la jacobienne de $C$ ([DR], th. 2). \medskip {\it D\'emonstration du th\'eor\`eme\ \ref{leff}}. Sous les hypoth\`eses du th\'eor\`eme , $F_r(X)$ est le lieu des z\'eros d'une section du fibr\'e $\mathop{\rm Sym}\nolimits ^{\bf d}S^*$, et sa codimension dans la grassmannienne est le rang de ce fibr\'e. Il existe donc une suite exacte (complexe de Koszul): $$0\rightarrow\bigwedge^{\max}(\mathop{\rm Sym}\nolimits^{\bf d}S)\rightarrow\cdots\rightarrow \bigwedge^2(\mathop{\rm Sym}\nolimits^{\bf d}S) \rightarrow \mathop{\rm Sym}\nolimits^{\bf d}S \rightarrow {\cal I}_{F_r(X)}\rightarrow 0\ . \leqno{\global\def\currenvir{formule}\hbox{\label{kos}}$$ \hskip 1cm\relax Notre outil essentiel sera le th\'eor\`eme\ d'annulation suivant: \th \label{annul} Proposition \enonce Soient $a,b,i,j_1,\ldots ,j_s$ des entiers tels que $b<a+d_1j_1+\cdots +d_sj_s$ et $b+i<\delta_-$. Alors $$H^{j_1+\cdots +j_s+i}(G(r,\hbox{\bf P}^n_{\bf C}),\bigwedge^{j_1}(\mathop{\rm Sym}\nolimits^{d_1}S){\mathord{\otimes\cdots\otimes }} \bigwedge^{j_s}(\mathop{\rm Sym}\nolimits^{d_s}S) \otimes S^{\otimes a}\otimes S^{*\otimes b})=0\ .$$ \endth \hskip 1cm\relax Soit $\mathop{\rm Sym}\nolimits_{{\lambda}}S$ une composante de $\bigwedge^{j_1}(\mathop{\rm Sym}\nolimits^{d_1}S){\mathord{\otimes\cdots\otimes }} \bigwedge^{j_s}(\mathop{\rm Sym}\nolimits^{d_s}S)\otimes S^{\otimes a}\otimes S^{*\otimes b}$, o\`u\break ${\lambda}=({\lambda}_0,\ldots,{\lambda}_r)$ est une suite d\'ecroissante d'entiers relatifs. D'apr\`es le th\'eor\`eme de Bott ([De], [Ma1]), $H^{j+i}(G,\mathop{\rm Sym}\nolimits_{{\lambda}}S)$ ne peut \^etre non nul que s'il existe un entier $h$, avec $0\leq h\leq r+1$, tel que $j+i=h(n-r)$ et ${\lambda}_h\leq h$, ce qui implique en particulier que la somme des composantes de ${\lambda}$ d'indice sup\'erieur ou \'egal \`a $h$ v\'erifie $|{\lambda}|_{\geq h}\leq h(r+1-h)$. \hskip 1cm\relax Comme $|{\lambda}|=|{\lambda}|_{\geq 0}=d_1j_1+\cdots +d_sj_s+a-b>0$, le cas $h=0$ est exclu. De plus, $$|{\lambda}|_{\geq h}\geq j_1+\cdots +j_s-{{\bf d}+h-1 \choose h-1 }-b\ .$$ \hskip 1cm\relax En effet, supposons tout d'abord $a=b=0$. Admettons provisoirement le cas $s=1$; le cas o\`u $s$ est quelconque s'ensuit, puisque si $\mathop{\rm Sym}\nolimits_{{\lambda}}S$ est un facteur direct de $\mathop{\rm Sym}\nolimits_{{\lambda}_1}S{\mathord{\otimes\cdots\otimes }} \mathop{\rm Sym}\nolimits_{{\lambda}_s}S$, la r\`egle de Littlewood et Richardson implique $|{\lambda}|_{\ge h}\ge$\break$ |{\lambda}_1|_{\ge h}+\cdots +|{\lambda}_s|_{\ge h}$. Enfin, tensoriser par $S^{ \otimes a}$ ne peut qu'augmenter $|{\lambda}|_{\ge h}$, tandis que tensoriser par $S^{*\otimes b}$ fait diminuer $|{\lambda}|_{\geq h}$ au plus de $b$. \hskip 1cm\relax Pour conclure \`a une contradiction, il suffit donc de v\'erifier que pour $1\leq h\leq r+1$, $$h(n-2r+h-1)-{{\bf d}+h-1 \choose h-1 }>b+i\ .$$ \hskip 1cm\relax On retrouve au membre de gauche la fonction $\varphi$ de (\ref{concave}); comme $\delta_-$ est positif, le lemme r\'esulte de l'hypoth\`ese $\delta_->b+i$ comme en (\ref{concave}). Il reste \`a traiter le cas $a=b=0$ et $s=1$, qui r\'esulte du lemme suivant. \unskip\penalty 500\quad\vrule height 4pt depth 0pt width 4pt\medbreak \th Lemme \enonce Soient $V$ un espace vectoriel complexe, $m$ et $d$ des entiers, et $e$ la dimension de $\mathop{\rm Sym}\nolimits^dV^m$. Pour toute composante irr\'eductible $\mathop{\rm Sym}\nolimits_{\lambda}V$ de $\bigwedge^j(\mathop{\rm Sym}\nolimits^dV)$, on a $|\lambda |_{>m}\ge j-e$. \endth \hskip 1cm\relax Soit $X$ la grassmannienne des sous-espaces de codimension $m$ de $V$, soit $Y$ celle des sous-espaces de codimension $e$ de $\mathop{\rm Sym}\nolimits^dV$. On notera $S_X$ et $Q_X$ les fibr\'es tautologique et quotient sur $X$, de m\^eme que $S_Y$ et $Q_Y$ sur $Y$. On plonge $X$ dans $Y$ en associant au noyau du quotient $V\rightarrow Q$ celui du quotient induit $\mathop{\rm Sym}\nolimits^dV\rightarrow \mathop{\rm Sym}\nolimits^dQ$. \hskip 1cm\relax D'apr\`es le th\'eor\`eme de Borel-Weil, $\bigwedge^j(\mathop{\rm Sym}\nolimits^dV)$ est l'espace des sections globales du fibr\'e $E=\det Q_Y\otimes\bigwedge^{j-e}S_Y$. Notons $(\Gamma_l)_{l\ge 0}$ la filtration de cet espace de sections selon leur ordre d'annulation $l$ sur $X$. On dispose d'applications injectives $$\Gamma_l/\Gamma_{l+1}\hookrightarrow H^0(X,E\vert_X\otimes \mathop{\rm Sym}\nolimits^lN^*)\ ,$$ o\`u $N$ est le fibr\'e normal de $X$ dans $Y$. \hskip 1cm\relax Le membre de droite ne se d\'eduit pas directement du th\'eor\`eme de Borel-Weil. Cependant, tout fibr\'e homog\`ene ${\cal F}$ sur $X$ admet une filtration homog\`ene dont les quotients successifs dont irr\'eductibles, c'est-\`a-dire produits de puissances de Schur de $Q_X$ et $S_X$. La somme $\mathop{\rm gr}\nolimits {\cal F}$ de ces quotients ne d\'epend pas de la filtration choisie, et le lemme de Schur implique l'existence d'une injection $H^0(X,{\cal F})\hookrightarrow H^0(X,\mathop{\rm gr}\nolimits {\cal F})$ : le th\'eor\`eme de Borel-Weil explicite ce dernier espace de sections. Par exemple, $Q_Y\vert_X=\mathop{\rm Sym}\nolimits^dQ_X$ est irr\'eductible, et $$\mathop{\rm gr}\nolimits S_Y\vert_X=\bigoplus_{i=1}^d\mathop{\rm Sym}\nolimits^{d-i}Q_X\otimes \mathop{\rm Sym}\nolimits^iS_X$$ a tous ses termes de degr\'e sup\'erieur ou \'egal \`a $1$ en $S_X$. Cela implique que $\rm E_{|X}$ est somme de fibr\'es de la forme $\mathop{\rm Sym}\nolimits_{\alpha}Q_X\otimes \mathop{\rm Sym}\nolimits_{\beta}S_X$, avec $|\beta|\ge j-e$. L'espace des sections globales d'un tel fibr\'e est une puissance de Schur $\mathop{\rm Sym}\nolimits_{\lambda}V$, o\`u $\lambda=(\alpha,\beta)$ est la partition (si c'en est une) obtenue en concat\'enant $\alpha$ et $\beta$. En particulier, $|\lambda|_{>m}=|\beta|\ge j-e$, ce qui d\'emontre le lemme pour les composantes de $\bigwedge^j(\mathop{\rm Sym}\nolimits^dV)$ qui proviennent de $\Gamma_0/\Gamma_1$. \hskip 1cm\relax Pour \'etendre ce r\'esultat \`a celles qui proviennent de $\Gamma_l/\Gamma_{l+1}$ pour tout $l>0$, il suffit de s'assurer que toute composante irr\'eductible de ${\rm gr} N^*$ est de degr\'e positif ou nul en $S_X$. Mais c'est une cons\'equence imm\'ediate du fait que $N^*$ est un sous-fibr\'e homog\`ene de $\Omega^1_Y\vert_X=Q^*_Y\vert_X\otimes S_Y\vert_X$, puisque $Q_Y\vert_X$ est de degr\'e z\'ero, et chaque composante de $S_Y\vert_X$ de degr\'e positif en $S_X$.\unskip\penalty 500\quad\vrule height 4pt depth 0pt width 4pt\medbreak \hskip 1cm\relax Revenons \`a la d\'emonstration du th\'eor\`eme\ \ref{leff}; posons $G=G(r,{\bf P}V)$ et $F=F_r(X)$. Il suffit de le v\'erifier pour la cohomologie complexe, donc, via la d\'ecomposition de Hodge, de d\'emontrer que les morphismes $H^q(G,\Omega_G^p)\rightarrow H^q(F,\Omega_F^p)$ sont bijectifs pour $p+q<\delta_-$, et injectifs pour $p+q=\delta_-$. On va montrer que les morphismes $H^q(G,\Omega_G^p)\rightarrow H^q(F,\Omega_G^p\vert_F)$ et $H^q(F,\Omega_G^p\vert_F)\rightarrow H^q(F,\Omega_F^p)$ ont les m\^emes propri\'et\'es. \hskip 1cm\relax Pour les premiers, il s'agit de v\'erifier que $H^q(G,{\cal I}_F\otimes\Omega^p_G)=0$ pour $p+q\leq \delta_-$, donc, via le complexe de Koszul, que $$H^{q+j-1}(G,\Omega_G^p\otimes\bigwedge^j(\mathop{\rm Sym}\nolimits^{\bf d}S))=0 \quad \hbox{pour tout}\ \ j>0\ .$$ \hskip 1cm\relax Rappelons que si $Q$ est le fibr\'e quotient sur $G$, on dispose d'un isomorphisme $\Omega^1_G\simeq Q^*\otimes S$, d'o\`u la suite exacte $0\rightarrow\Omega^1_G\rightarrow V^*\otimes S \rightarrow S^*\otimes S \rightarrow 0$. Sa puissance ext\'erieure $p$\kern -1.5pt - i\`eme montre que l'annulation pr\'ec\'edente est cons\'equence de $$H^{q+j-i-1}(G,\bigwedge^j(\mathop{\rm Sym}\nolimits^{\bf d}S)\otimes \bigwedge^{p-i}(V^*\otimes S)\otimes \mathop{\rm Sym}\nolimits^i(S^*\otimes S))=0 \qquad\hbox{pour tout}\ \ j>0\ ,\ i\geq 0\ ,$$ ce qu'assure la proposition \ref{annul} d\`es que $q\leq\delta_-$. \hskip 1cm\relax Pour les seconds, la suite exacte normale montre qu'il suffit de s'assurer que $$H^{q+i}(F,\Omega_G^{p-i-1}\vert_F\otimes \mathop{\rm Sym}\nolimits^i(\mathop{\rm Sym}\nolimits^{\bf d}S))=0 \qquad\hbox{pour tout}\ \ i>0\ ,$$ donc, \`a cause encore une fois du complexe de Koszul, que $$H^{q+i+j}(G,\Omega_{G}^{p-i-1}\otimes \mathop{\rm Sym}\nolimits^i(\mathop{\rm Sym}\nolimits^{\bf d}S)\otimes\bigwedge^j(\mathop{\rm Sym}\nolimits^dS))=0 \qquad\hbox{pour tout}\ \ i>0\ , j\geq 0\ .$$ \hskip 1cm\relax En raisonnant comme on vient de le faire, on constate que cette annulation a lieu d\`es que $i+q<\delta_-$, ce qui conclut cette d\'emonstration puisque $i<p$. \unskip\penalty 500\quad\vrule height 4pt depth 0pt width 4pt\medbreak \section{Normalit\'e projective, \'equations, degr\'e des sch\'emas de Fano} \th Th\'eor\`eme \enonce Soit $X$ un sous-sch\'ema de $\hbox{\bf P}^n_{\bf C}$ d\'efini par des \'equations de degr\'e ${\bf d}$, tel que $F_r(X)$ soit de dimension $\delta$. Supposons $n\ge r+{ {\bf d}+r\choose r}$. Alors $F_r(X)$ est projectivement normale, autrement dit les morphismes de restriction $$\rho_l:H^0(G(r,\hbox{\bf P}^n),{\cal O}(l))\longrightarrow H^0(F_r(X),{\cal O}(l))$$ sont surjectifs pour tout $l\geq 0$. Par ailleurs, $\rho_l$ est injectif pour $l< d_-=\min \{ d_1,\ldots ,d_s\}$. \endth \hskip 1cm\relax Posons $G=G(r,\hbox{\bf P}^n)$; d'apr\`es le th\'eor\`eme de Bott, $$H^j(G,\bigwedge^j(\mathop{\rm Sym}\nolimits^{{\bf d}}S)(l))=0\qquad\hbox{pour tout}\ \ j>0\ \ \hbox{et tout}\ \ l\geq 0\ .$$ \hskip 1cm\relax En effet, si l'on raisonne comme dans la d\'emonstration de la proposition \ref{annul}, cet espace ne peut \^etre non nul que si $j$ est multiple de $n-r$; vue l'hypoth\`ese $n-r\ge \mathop{\rm codim}\nolimits F_r(X)$, la seule possibilit\'e est $j=n-r=\mathop{\rm codim}\nolimits F_r(X)$, auquel cas $\bigwedge^j(\mathop{\rm Sym}\nolimits^{\bf d}S)(l)$ est une puissance de ${\cal O}(1)$, et n'a donc pas non plus de cohomologie en degr\'e $n-r$. La normalit\'e projective s'ensuit, via le complexe de Koszul (\ref{kos}) tordu par ${\cal O}(l)$. \hskip 1cm\relax En fait, les arguments pr\'ec\'edents impliquent plus pr\'ecis\'ement que la suite spectrale associ\'ee \`a ce complexe de Koszul tordu d\'eg\'en\`ere en $E_2$, ce dont on d\'eduit que le complexe des sections globales $$\cdots\longrightarrow H^0(G,\bigwedge^2(\mathop{\rm Sym}\nolimits^{\bf d}S)(l))\longrightarrow H^0(G,\mathop{\rm Sym}\nolimits^{\bf d}S(l))\longrightarrow H^0(G,{\cal I}_{F_r(X)}(l))\longrightarrow 0$$ est exact. Mais pour $l<d_-$, on a $H^0(G,\mathop{\rm Sym}\nolimits^{\bf d}S(l))=0$ d'apr\`es le th\'eor\`eme de Bott, d'o\`u l'inexistence d'\'equations de $F_r(X)$ de degr\'e $l$. \unskip\penalty 500\quad\vrule height 4pt depth 0pt width 4pt\medbreak \ex{Remarques} 1) Ce dernier complexe implique au passage que $H^0(G,{\cal I}_{F_r(X)}(d_-))$ n'est pas nul, et l'on peut calculer explicitement sa dimension. 2) Les sch\'emas de Fano ne sont en g\'en\'eral pas projectivement normaux; si l'on revient au cas ${\bf d}=(2,2)$ et $n=2g+1$ ({\it cf.\/}\ rem. \ref{rem}.4), Laszlo a montr\'e dans [L] (par des m\'ethodes similaires) que $H^0(F_{g-2},{\cal O}(1))$ est de dimension $2^{g-1}(2^g-1)$. En particulier, $\rho_1$ n'est pas surjectif. 3) Le th\'eor\`eme\ d'annulation de Kodaira entra\^ine que les groupes $H^i(F_r(X),{\cal O}(l))$ sont nuls pour $i> 0$ et $l\ge -n+{{\bf d}+r\choose r+1}$. Si l'on raisonne comme dans la preuve de la proposition \ref{annul}, on montre facilement la m\^eme annulation pour $i< \min (\delta ,n-(l+2)r-s)$. A l'ext\'erieur du domaine d\'efini par ces in\'egalit\'es, il peut ne pas y avoir annulation: pour une sextique $X$ dans $\hbox{\bf P}^6$, on peut montrer que $ H^2(F_1(X),{\cal O} (6))$ est de dimension $\ge 10024$ (alors que $F_1(X)$ est de dimension $3$). \bigskip \hskip 1cm\relax Introduisons des polyn\^omes de $r+1$ variables, $e(x)=x_0+\cdots +x_r$, et $$Q_{r,d}(x)=\prod_{a_0+\cdots +a_r=d}(a_0x_0+\cdots +a_rx_r)\ ,$$ puis $Q_{r,{\bf d}}(x)=Q_{r,d_1}(x)\cdots Q_{r,d_s}(x)$. \th\label{calcul} Th\'eor\`eme \enonce Soit $X$ un sous-sch\'ema de $\hbox{\bf P}^n_k$ d\'efini par des \'equations de degr\'e ${\bf d}$, tel que $F_r(X)$ soit de dimension $\delta$. Le degr\'e de $F_r(X)$ pour le plongement de Pl\"ucker de $G(r,\hbox{\bf P}^n)$ est \'egal au coefficient du mon\^ome $x_0^nx_1^{n-1}\cdots x_r^{n-r}$ dans le produit du polyn\^ome $Q_{r,{\bf d}}\times e^{\delta}$ et du Vandermonde. \endth \hskip 1cm\relax Ce degr\'e est $$\deg (F)=\int_{G(r,{\bf P}^n)} c_{\max}(\mathop{\rm Sym}\nolimits^{\bf d}S^*)c_1({\cal O}(1))^{\delta}\ .$$ \hskip 1cm\relax Rappelons que l'anneau de Chow de $G(r,\hbox{\bf P}^n)$ est un quotient d'un anneau des polyn\^omes sym\'etriques de $r+1$ variables $x_0,\ldots ,x_r$, $e(x)$ relevant $c_1({\cal O}(1))$, et $Q(x)$ relevant $c_{\max}(\mathop{\rm Sym}\nolimits^dS^*)$ ([Fu], 14.7). De plus, int\'egrer sur $G$ revient, au niveau des polyn\^omes, \`a d\'ecomposer sur les polyn\^omes de Schur ([M]), et ne retenir que le coefficient de celui qui est associ\'e \`a la partition rectangle ayant $r+1$ parts \'egales \`a $n-r$, \`a savoir $(x_0\ldots x_r)^{n-r}$. \hskip 1cm\relax Il suffit donc de montrer que si $P$ est un polyn\^ome sym\'etrique, que l'on d\'ecompose sur les polyn\^omes de Schur, le coefficient du pr\'ec\'edent est \'egal \`a celui du mon\^ome $x_0^nx_1^{n-1}\cdots x_r^{n-r}$ dans le produit de $P$ et du Vandermonde. Mais par lin\'earit\'e, il suffit de le v\'erifier lorsque $P$ est lui-m\^eme un polyn\^ome de Schur, auquel cas c'est une cons\'equence imm\'ediate de l'expression de Jacobi de ces polyn\^omes ([FH], (A.23), p. 459). \unskip\penalty 500\quad\vrule height 4pt depth 0pt width 4pt\medbreak \hskip 1cm\relax Donnons quelques exemples num\'eriques, d'abord pour le cas des droites d'une hypersurface, qui est d\^u \`a Van der Waerden ([vW]), puis pour $r\geq 2$, toujours dans le cas d'une hypersurface. {\eightpoint $$\vbox{\offinterlineskip \halign{\vrule height 12pt depth 5pt width 0pt\vrule\hskip.5mm \vrule height 12pt depth 5pt width 0pt\vrule#&\cc{$#$}& \vrule height 12pt depth 5pt width 0pt\vrule#& \cc{$#$}& \vrule height 12pt depth 5pt width 0pt\vrule#&\cc{$#$}& \vrule height 12pt depth 5pt width 0pt\vrule#&\cc{$#$}& \vrule height 12pt depth 5pt width 0pt\vrule\hskip.5mm \vrule height 12pt depth 5pt width 0pt\vrule#&\cc{$#$}& \vrule height 12pt depth 5pt width 0pt\vrule#&\cc{$#$}& \vrule height 12pt depth 5pt width 0pt\vrule#&\cc{$#$}& \vrule height 12pt depth 5pt width 0pt\vrule#&\cc{$#$}& \vrule height 12pt depth 5pt width 0pt\vrule\hskip.5mm \vrule height 12pt depth 5pt width 0pt\vrule#\cr \noalign{\hrule}\vrule height 12pt depth 5pt width 0pt &d&& n && \dim F && \deg F &&d&& n && \dim F && \deg F &\cr \noalign{\hrule}\vrule height 12pt depth 5pt width 0pt &3&& 3 && 0&& 27 && 5&& 5 && 2&& 6\;125&\cr \noalign{\hrule}\vrule height 12pt depth 5pt width 0pt &3&& 4 && 2&& 45 && 5&& 6 && 4&& 16\;100&\cr \noalign{\hrule}\vrule height 12pt depth 5pt width 0pt &3&& 5 && 4&& 108 && 5&& 7 && 6&& 46\;625&\cr \noalign{\hrule}\vrule height 12pt depth 5pt width 0pt &4&& 4 && 1&& 320 && 6&& 5 && 1&& 60\;480&\cr \noalign{\hrule}\vrule height 12pt depth 5pt width 0pt &4&& 5 && 3&& 736 && 6&& 6 && 3&& 154\;224&\cr \noalign{\hrule}\vrule height 12pt depth 5pt width 0pt &4&& 6 && 5&& 1\;984 && 7&& 5 && 0&& 698\;005&\cr \noalign{\hrule}\vrule height 12pt depth 5pt width 0pt &4&& 7 && 7&& 5\;824 && 7&& 6 && 2&& 1\;707\;797&\cr \noalign{\hrule}\vrule height 12pt depth 5pt width 0pt &5&& 4 && 0&& 2\;875 && 9&& 6 && 0&& 305\;093\;061&\cr \noalign{\hrule} }}$$} \centerline{{\eightpoint 1. Degr\'es de sch\'emas de Fano de droites ($r=1$).}} \bigskip {\eightpoint $$\vbox{\offinterlineskip \halign{\vrule height 12pt depth 5pt width 0pt\vrule\hskip.5mm \vrule height 12pt depth 5pt width 0pt\vrule#&\cc{$#$}& \vrule height 12pt depth 5pt width 0pt\vrule#& \cc{$#$}& \vrule height 12pt depth 5pt width 0pt\vrule#& \cc{$#$}& \vrule height 12pt depth 5pt width 0pt\vrule#&\cc{$#$}& \vrule height 12pt depth 5pt width 0pt\vrule#&\cc{$#$}& \vrule height 12pt depth 5pt width 0pt\vrule\hskip.5mm \vrule height 12pt depth 5pt width 0pt\vrule#&\cc{$#$}& \vrule height 12pt depth 5pt width 0pt\vrule#&\cc{$#$}& \vrule height 12pt depth 5pt width 0pt\vrule#& \cc{$#$}& \vrule height 12pt depth 5pt width 0pt\vrule#&\cc{$#$}& \vrule height 12pt depth 5pt width 0pt\vrule#&\cc{$#$}& \vrule height 12pt depth 5pt width 0pt\vrule\hskip.5mm \vrule height 12pt depth 5pt width 0pt\vrule#\cr \noalign{\hrule}\vrule height 12pt depth 5pt width 0pt &r &&d&& n && \dim F && \deg F &&r&&d&& n && \dim F && \deg F &\cr \noalign{\hrule}\vrule height 12pt depth 5pt width 0pt &2&& 3 && 6&& 2 && 2\;835&& 2 && 5 && 9&& 0 && 2\;103\;798\;896\;875&\cr \noalign{\hrule}\vrule height 12pt depth 5pt width 0pt &2&& 3 && 7&& 5 && 24\;219&& 3 && 3 && 8&& 0 && 321\;489&\cr \noalign{\hrule}\vrule height 12pt depth 5pt width 0pt &2&& 3 && 8&& 8 && 274\;590&& 3 && 3 && 9&& 4 && 10\;344\;510&\cr \noalign{\hrule}\vrule height 12pt depth 5pt width 0pt &2&& 4 && 7&& 0 && 3\;297\;280&& 4 && 3 && 11&& 0 && 1\;812\;768\;336&\cr \noalign{\hrule} }}$$} \centerline{{\eightpoint 2. Degr\'es de sch\'emas de Fano pour $r=2,3,4$.}} \medskip \hskip 1cm\relax La m\^eme m\'ethode permet en fait de d\'eterminer la d\'ecomposition $$[F_r(X)]=\sum_{|{\lambda}|=\mathop{\rm codim}\nolimits F_r(X)}f_{{\lambda}}{\sigma}_{{\lambda}}$$ de la classe fondamentale de $F_r(X)$ sur les classes des cycles de Schubert de la grassmannienne, o\`u l'on note ${\sigma}_{{\lambda}}$ la classe du cycle de codimension $|{\lambda}|$ associ\'e \`a la partition ${\lambda}=({\lambda}_0,\ldots ,{\lambda}_r)$. \th Proposition \enonce Si l'on \'ecrit $Q_{r,{\bf d}}(x)=\sum_{{\alpha}}q_{{\alpha}}x^{{\alpha}}$, et si ${\kappa}$ d\'esigne la suite $(r,\ldots ,1,0)$, alors $$f_{{\lambda}}=\sum_{{\sigma}\in{\cal S}_{r+1}}{\varepsilon}({\sigma})q_{{\sigma}({\lambda}+{\kappa})-{\kappa}}\ .$$ \endth \hskip 1cm\relax Notons que si l'on adopte pour les cycles de Schubert la m\^eme convention que pour les polyn\^omes de Schur, \`a savoir que pour chaque suite d'entiers ${\alpha}$, on pose ${\sigma}_{{\alpha}}={\varepsilon}({\tau}){\sigma}_{{\lambda}}$ s'il existe une partition ${\lambda}$ et une permutation ${\tau}\in{\cal S}_{r+1}$ telles que ${\alpha}+{\kappa}={\tau}({\lambda}+{\kappa})$, et ${\sigma}_{{\alpha}}=0$ sinon, la proposition pr\'ec\'edente se traduit par la simple \'egalit\'e $$[F_r(X)]=\sum_{{\alpha}}q_{{\alpha}}{\sigma}_{{\alpha}}.$$ \hskip 1cm\relax Donnons par exemple les classes de quelques vari\'et\'es de Fano en bas degr\'e. $$\eqalign{ {\rm Si}\ \ {\bf d}=(2) \ ,\qquad [F_r] &= 2^{r+1}\sigma_{r+1,r,\ldots,1}\ ,\cr {\rm Si}\ \ {\bf d}=(3) \ ,\qquad [F_1] &= 9(2{\sigma}_{3,1}+3{\sigma}_{2,2})\ ,\cr [F_2] &= 27(8{\sigma}_{6,3,1}+12{\sigma}_{6,2,2}+20{\sigma}_{5,4,1} +50{\sigma}_{5,3,2}+42{\sigma}_{4,4,2}+35{\sigma}_{4,3,3})\ .\cr {\rm Si}\ \ {\bf d}=(4)\ ,\qquad [F_1] &= 32(3{\sigma}_{4,1}+10{\sigma}_{3,2})\ ,\cr [F_2] &= 512(54{\sigma}_{10,4,1}+180{\sigma}_{10,3,2}+369{\sigma}_{9,5,1}+ 1599{\sigma}_{9,4,2}+1230{\sigma}_{9,3,3}\cr &\qquad +819{\sigma}_{8,6,1}+5022{\sigma}_{8,5,2}+8459{\sigma}_{8,4,3}+504{\sigma}_{7,7,1}+ 6039{\sigma}_{7,6,2}\cr &\ \ +18889{\sigma}_{7,5,3}+13354{\sigma}_{7,4,4}+ 11660{\sigma}_{6,6,3}+23560{\sigma}_{6,5,4}+6440{\sigma}_{5,5,5})\ .\cr {\rm Si}\ \ {\bf d}=(5)\ ,\qquad [F_1] &= 25(24{\sigma}_{5,1}+130{\sigma}_{4,2}+91{\sigma}_{3,3})\ .\cr {\rm Si}\ \ {\bf d}=(2,2)\ ,\quad [F_1] &= 16({\sigma}_{4,2}+{\sigma}_{3,3})\ ,\cr [F_2] &= 64({\sigma}_{6,4,2}+{\sigma}_{6,3,3} +{\sigma}_{5,5,2}+2{\sigma}_{5,4,3} +{\sigma}_{4,4,4})\ .\cr}$$ \section {Espaces lin\'eaire s sur les sch\'emas de Fano} \hskip 1cm\relax Le but de ce paragraphe est de montrer que les sch\'emas de Fano sont s\'eparablement unir\'egl\'es en droites pour $n$ assez grand (corollaire \ref{sepunir}). Pour cela, nous commen{\gamma} cons par g\'en\'eraliser les r\'esultats du \S 2 aux sous-sch\'emas de $F_r(X)$ form\'es des $r$\kern -1.5pt - plans contenant un sous-espace lin\'eaire\ fixe de dimension $r_0<r$. Pour de tels entiers, on pose $$ \delta(n,{\bf d},r,r_0)=(r-r_0)(n-r)+{{\bf d}+r_0\choose r_0}- {{\bf d}+r\choose r} $$ et $$\delta_-(n,{\bf d},r,r_0)=\min\{ \delta (n,{\bf d},r,r_0), n-2r+r_0+1-{{\bf d}+r_0\choose r_0+1}\}\ ,$$de sorte que $\delta(n,{\bf d},r)=\delta_-(n,{\bf d},r,-1)$ et $\delta_-(n,{\bf d},r)=\delta_-(n,{\bf d},r,-1)$. De nouveau, il est utile de noter que lorsque ${\bf d}\ne (2)$, l'entier $\delta(n,{\bf d},r,r_0)$ est positif (resp. strictement positif) si et seulement si $\delta_-(n,{\bf d},r,r_0)$ l'est; cela r\'esulte de la convexit\'e de la fonction $\psi:r\mapsto {{\bf d}+r\choose r}-r^2$, qui entra\^ine l'in\'egalit\'e $\psi(r)-\psi(r_0)\ge (r-r_0)(\psi(r_0+1)-\psi(r_0))$ (puisque $r>r_0$). Le th\'eor\`eme\ suivant g\'en\'eralise le th\'eor\`eme\ \ref{fano}. \th \label{fano2} Th\'eor\`eme \enonce Soit $X$ un sous-sch\'ema de $\hbox{\bf P}^n_k$ d\'efini par des \'equations de degr\'e ${\bf d}$, soit $\Lambda_0$ un $r_0$\kern -1.5pt - plan contenu dans $X$, et soit $F_r(X,\Lambda_0)$, avec $r> r_0$, le sch\'ema de Hilbert des $r$\kern -1.5pt - plans contenus dans $X$ et contenant $\Lambda_0$. \par\hskip0.5cm {\rm a)} Lorsque $\delta_-(n,{\bf d},r,r_0)< 0$, le sch\'ema $F_r(X,\Lambda_0)$ est vide pour $X$ g\'en\'erale et $\Lambda_0$ g\'en\'eral contenu dans $X$. \par\hskip0.5cm {\rm b)} Lorsque $\delta_-(n,{\bf d},r,r_0)\ge 0$, le sch\'ema $F_r(X,\Lambda_0)$ est non vide; il est lisse de dimension $\delta(n,{\bf d},r,r_0)$ pour $X$ g\'en\'erale et $\Lambda_0$ g\'en\'eral contenu dans $X$. \par\hskip0.5cm {\rm c)} Lorsque $\delta_-(n,{\bf d},r,r_0)>0$, le sch\'ema $F_r(X,\Lambda_0)$ est connexe. \endth \hskip 1cm\relax En gardant les notations de la d\'emonstration du th\'eor\`eme\ \ref{fano}, on consid\`ere\break $G_0=\{ [\Lambda]\in G(r,{\bf P}^n)\mid \Lambda\supset \Lambda_0\}$. La dimension de $I_0=q^{-1}(G_0)$ est \'egale \`a $$\dim \hbox{\bf P} \mathop{\rm Sym}\nolimits^{\bf d}V^*-{{\bf d}+r\choose r}+(r-r_0)(n-r)\ .$$\hskip 1cm\relax Le c\^one $S_0$ dans $\mathop{\rm Sym}\nolimits^{\bf d}V^*$ correspondant aux sous-sch\'emas contenant $\Lambda_0$ est de codimension ${{\bf d}+r_0\choose r_0}$, de sorte que $\dim I_0 =\dim \hbox{\bf P} S_0 +\delta$. Supposons ${\delta}_-<0$; si ${\bf d}=(2)$, cela signifie $2r\ge n$, et on a d\'ej\`a vu qu'une quadrique lisse dans $\hbox{\bf P}^n$ ne contenait pas de $r$\kern -1.5pt - plan; si ${\bf d}\ne (2)$, on a ${\delta}<0$, et le morphisme $p_0:I_0\rightarrow \hbox{\bf P} S_0$ induit par $p$ n'est pas surjectif. \hskip 1cm\relax Cela montre a); on suppose maintenant ${\delta}_-\ge 0$. Fixons un $r$\kern -1.5pt - plan $\Lambda$ contenant $\Lambda_0$, et choisissons des coordonn\'ees de fa{\gamma} con que $\Lambda_0$ soit d\'efini par les \'equations $x_{r_0+1}=\cdots=x_n=0$, et $\Lambda$ par $x_{r+1}=\cdots=x_n=0$; pour tout entier positif $m$, on note $\Gamma_0(m)$ le noyau du morphisme $\Gamma_\Lambda (m)\rightarrow \Gamma_{\Lambda_0} (m)$. \hskip 1cm\relax La d\'emarche est enti\`erement analogue \`a celle de la d\'emonstration de \ref{fano}. Soit ${\bf f}$ un \'el\'ement de $S_0$; pour que $p_0$ soit lisse en un point $(X_{\bf f},\Lambda)$ de $I_0$, il faut et il suffit que le morphisme $\alpha_0:\Gamma_0(1)^{n-r}\rightarrow \Gamma_0({\bf d})$ induit par le morphisme $\alpha$ du lemme \ref{lisse} soit surjectif. \hskip 1cm\relax Soit $Z_0$ le lieu des points de $I_0$ o\`u $p_0$ n'est pas lisse; on montre comme en \ref{propre}--\ref{ferme}, par r\'ecurrence sur $r-r_0$, que $p_0(Z_0)$ est distinct de $\hbox{\bf P} S_0$. Soit $\mu_0: \Gamma_0(1)\times \Gamma_\Lambda({\bf d}-1)\rightarrow \Gamma_0({\bf d})$ le morphisme induit par la multiplication $\mu$. On montre de la m\^eme fa{\gamma} con que si $h$ est un entier compris entre $1$ et $r-r_0$, l'ensemble des formes lin\'eaires $\ell_0$ sur $\Gamma_0({\bf d})$ telles que $\mathop{\rm codim}\nolimits \mu_0^{-1}(\ell_0) =h$ est de dimension $$ \le h(r-r_0-h)+{{\bf d}+r_0+h\choose r_0+h}-{{\bf d}+r_0\choose r_0}\ .$$ \hskip 1cm\relax On en d\'eduit que la codimension de $Z_0$ dans $I_0$ est $$\eqalign{&\ge \min_{1\le h\le r-r_0}[h(n-r)-h(r-r_0-h)-{{\bf d}+r_0+h\choose r_0+h}+{{\bf d}+r_0\choose r_0}]+1\cr &=\min\{ n-2r+r_0+1-{{\bf d}+r_0\choose r_0+1}, \delta\}+1={\delta}_-+1\ ,\cr}$$ puisque la fonction entre crochets est une fonction concave de $h$ lorsque ${\bf d}\ne (2)$, et croissante lorsque ${\bf d}=(2)$ puisque ${\delta}_-$ est positif ({\it cf.\/}\ (\ref{concave})). La fin de la d\'emonstration est la m\^eme que celle du th\'eor\`eme\ \ref{fano}. \unskip\penalty 500\quad\vrule height 4pt depth 0pt width 4pt\medbreak \hskip 1cm\relax Soient $X$ un sous-sch\'ema de $\hbox{\bf P}^n_k$ d\'efini par des \'equations de degr\'e ${\bf d}$, et $\Lambda$ un $(r+1)$\kern -1.5pt - plan contenu dans $X$. Les $r$-plans contenus dans $\Lambda$ d\'efinissent une inclusion de $\Lambda^*$ dans $F_r(X)$, dont l'image par le plongement de Pl\"ucker est un $(r+1)$\kern -1.5pt - plan. \th \label{uni} Corollaire \enonce Soit $X$ un sous-sch\'ema de $\hbox{\bf P}^n_k$ d\'efini par des \'equations de degr\'e ${\bf d}$. \par\hskip0.5cm {\rm a)} Si ${\bf d}\ne 2$ et $n\ge {1\over r}{{\bf d}+r\choose r}+r-{s\over r}$, ou si ${\bf d}= 2$ et $n\ge 2r+1$, la vari\'et\'e $X$ est recouverte par des $r$\kern -1.5pt - plans. \par\hskip0.5cm {\rm b)} Si $n\ge {{\bf d}+r\choose r+1}+r+1$, la sous-vari\'et\'e $F_r(X)$ de $G(r,\hbox{\bf P}^n)$ est unir\'egl\'ee en droites. \endth \hskip 1cm\relax Le point a) r\'esulte du th\'eor\`eme\ avec $r_0=0$. Soit $\Lambda_0$ un $r$\kern -1.5pt - plan contenu dans $X$; sous les hypoth\`eses de b), le th\'eor\`eme\ \ref{fano2}.b) entra\^ine qu'il existe un $(r+1)$\kern -1.5pt - plan $\Lambda_1$ contenu dans $X$ et contenant $\Lambda_0$. Le $(r+1)$\kern -1.5pt - plan $\Lambda_1^*$, contenu dans $F_r(X)$, passe par $[\Lambda_0]$. En particulier, il passe une droite par tout point de $F_r(X)$.\unskip\penalty 500\quad\vrule height 4pt depth 0pt width 4pt\medbreak \th Th\'eor\`eme \enonce Soit $X$ un sous-sch\'ema g\'en\'eral de $\hbox{\bf P}^n_k$ d\'efini par des \'equations de degr\'e ${\bf d}$; on suppose $n\ge {{\bf d}+r\choose r+1}+r+1$. Soit $\Lambda$ un $(r+1)$\kern -1.5pt - plan g\'en\'eral contenu dans $X$. La restriction \`a une droite g\'en\'erale de $\Lambda^*$ du fibr\'e normal \`a $\Lambda^*$ dans $F_r(X)$ est isomorphe \`a $$ {\cal O}^{r(n-r-1)+{{\bf d}+r\choose r+1}-{{\bf d}+r\choose r}}\oplus{\cal O}(1)^{n-r-1 -{{\bf d}+r\choose r+1}}\ .$$ \endth \hskip 1cm\relax Soit $N$ le fibr\'e normal \`a $\Lambda^*$ dans $F_r(X)$; on a la suite exacte \vskip -5mm $$\diagram{ 0&\longrightarrow &N &\longrightarrow &N_{\Lambda^*/G} &\longrightarrow &\bigl( N_{F_r(X)/G}\bigr) \vert_{\Lambda^*} &\longrightarrow &0\cr &&&&||&&||\cr &&&&(S^*\vert_{\Lambda^*})^{ n-r-1 }&&\mathop{\rm Sym}\nolimits^dS^*\vert_{\Lambda^*}\cr}$$ \vskip -5mm dont la restriction \`a une droite $\ell$ contenue dans $\Lambda^*$ est $$ 0 \longrightarrow N \vert_\ell \longrightarrow ( S^*\vert_\ell)^{ n-r-1 } \buildrel{u}\over{\longrightarrow} \mathop{\rm Sym}\nolimits^d S^*\vert_\ell\longrightarrow 0\ . \leqno{\global\def\currenvir{formule}\hbox{\label{suite}}$$ \hskip 1cm\relax Comme $S^*\vert_\ell$ est isomorphe \`a ${\cal O} ^r\oplus{\cal O} (1)$, cela entra\^ine que $N_\ell$ est isomorphe \`a une somme directe $\bigoplus_j{\cal O}(a_j)$ avec $a_j\le 1$ pour tout $j$. On v\'erifie que $H^0(\ell,S^*\vert_\ell)$ s'identifie \`a $H^0(\Lambda,{\cal O} (1))$, c'est-\`a-dire\ \`a l'espace vectoriel not\'e $\Gamma_{\Lambda}(1)$ dans la d\'emonstration du th\'eor\`eme\ \ref{fano} et $H^0(\ell,\mathop{\rm Sym}\nolimits^dS^*\vert_\ell)$ \`a $\Gamma_{\Lambda}(d)$. Soient $x_0$ un point de $\ell$, et $\Lambda_0$ l'hyperplan de $\Lambda$ associ\'e. On a un diagramme commutatif $$\diagram{\Gamma_0(1)^{n-r-1} &\phfl{}{}&\Gamma_\Lambda(1)^{n-r-1}&\phfl{}{}&\Gamma_{\Lambda_0}(1)^{n-r}\cr \pvfl{\displaystyle\alpha_0}{}&&\pvfl{\displaystyle\alpha}{}&&\pvfl{}{}\cr \Gamma_0(d) &\phfl{}{}&\Gamma_\Lambda(d)&\phfl{}{}&\Gamma_{\Lambda_0}(d)\cr }$$ o\`u les notations sont celles de la d\'emonstration du th\'eor\`eme\ \ref{fano2}. On v\'erifie que $\alpha$ s'identifie \`a $H^0(u)$, et $\alpha_0$ \`a $H^0(u(-x_0)):H^0(\ell, ( S^*\vert_\ell)(-x_0)^{ n-r-1 } )\rightarrow H^0(\ell, \mathop{\rm Sym}\nolimits^d S^*\vert_\ell(-x_0) )$. Comme $$\delta_-(n,{\bf d}, r+1,r)= n-r-1+{{\bf d}+r\choose r+1} $$ est positif par hypoth\`ese, la d\'emonstration du th\'eor\`eme\ \ref{fano2} entra\^ine que $H^0(u(-x_0))$ est surjectif; il en r\'esulte que $H^1(\ell,N\vert_\ell(-x_0))$ est nul. Cela entra\^ine que les $a_j$ sont tous positifs. Le rang et le degr\'e de $N_\ell$ \'etant donn\'es par (\ref{suite}), cela d\'emontre le th\'eor\`eme .\unskip\penalty 500\quad\vrule height 4pt depth 0pt width 4pt\medbreak \hskip 1cm\relax Il n'est pas vrai en g\'en\'eral que le fibr\'e normal \`a $\Lambda^*$ dans $F_r(X)$ soit somme de fibr\'es en droites; cependant, c'est le cas lorsque $\delta(n,{\bf d},r+1)$ est nul ([BV], prop. 3). \th \label{sepunir} Corollaire \enonce Soit $X$ un sous-sch\'ema g\'en\'eral de $\hbox{\bf P}^n_k$ d\'efini par des \'equations de degr\'e ${\bf d}$; on suppose $n\ge {{\bf d}+r\choose r+1}+r+1$. La vari\'et\'e $F_r(X)$ est s\'eparablement unir\'egl\'ee en droites. \endth \hskip 1cm\relax L'hypoth\`ese sur $n$ entra\^ine que $\delta_-(n,{\bf d},r+1)$ est positif; soient $\Lambda_1$ un $(r+1)$\kern -1.5pt - plan g\'en\'eral contenu dans $X$, et $\ell$ une droite g\'en\'erale contenue dans $\Lambda_1^*$. Le th\'eor\`eme\ pr\'ec\'edent entra\^ine que le fibr\'e normal \`a $\ell$ dans $F_r(X)$ est somme de copies de ${\cal O}_\ell$ et ${\cal O}_\ell(1)$, donc que $\ell$ est {\it libre} au sens de [K], p. 113 (et m\^eme {\it minimale} au sens de {\it loc.cit.\/} , p. 195). Le corollaire r\'esulte alors de {\it loc.cit.\/} , p. 188.\unskip\penalty 500\quad\vrule height 4pt depth 0pt width 4pt\medbreak \section{Cycles alg\'ebriques} \hskip 1cm\relax On voudrait montrer que pour $n$ assez grand, les groupes de Chow rationnels de $F_r(X)$ sont les m\^emes que ceux de la grassmannienne ambiante $G(r,\hbox{\bf P}^n)$, g\'en\'eralisant ainsi des r\'esultats de [P], [K] p. 266, et [ELV], qui traitent le cas $r=0$. On n'obtient malheureusement de r\'esultats nouveaux que pour les groupes $A_1(F_r(X))_{\bf Q}$, en caract\'eristique nulle. Les id\'ees sont celles de [K]. \th \label{rat} Proposition \enonce Soit $X$ un sous-sch\'ema de $\hbox{\bf P}^n_k$ d\'efini par des \'equations de degr\'e ${\bf d}$; on suppose $n\ge {{\bf d}+r\choose r+1}$. Le sch\'ema $F_r(X)$ est connexe par cha\^ines rationnelles; en particulier, $A_0(F_r(X))\simeq\hbox{\bf Z}$. \endth \hskip 1cm\relax Lorsque $X$ est g\'en\'erale, il r\'esulte du th\'eor\`eme\ \ref{fano} et de la remarque \ref{rem}.4) que $F_r(X)$ est une vari\'et\'e de Fano lisse connexe, donc est connexe par cha\^ines rationnelles ([K], 2.13, p. 254). Le cas g\'en\'eral s'en d\'eduit comme dans [K], 4.9, p. 271.\unskip\penalty 500\quad\vrule height 4pt depth 0pt width 4pt\medbreak \hskip 1cm\relax On suppose maintenant $k=\hbox{\bf C}$ (pour g\'en\'eraliser les r\'esultats qui suivent en toute caract\'eristique, il suffirait de montrer que le groupe de N\'eron-Severi d'un sch\'ema de Fano g\'en\'eral est de rang $1$). \th \label{chaine} Proposition \enonce Soit $X$ un sous-sch\'ema de $\hbox{\bf P}^n_{\bf C}$ d\'efini par des \'equations de degr\'e ${\bf d}$; on suppose $n\ge {{\bf d}+r\choose r+1}+r+1$. Deux points quelconques de $F_r(X)$ peuvent \^etre joints par une courbe connexe de degr\'e $\delta(n,{\bf d},r)$, dont toutes les composantes sont des droites. \endth \hskip 1cm\relax On peut supposer $X$ g\'en\'erale, de sorte que $F_r(X)$ est une vari\'et\'e de Fano lisse unir\'egl\'ee en droites (cor. \ref{uni}.b)), de groupe de N\'eron-Severi de rang $1$ (cor. \ref{pic}). Le corollaire r\'esulte de [K], p. 252.\unskip\penalty 500\quad\vrule height 4pt depth 0pt width 4pt\medbreak \hskip 1cm\relax Soient $X$ un $k$\kern -1.5pt - sch\'ema et $m$ un entier positif; on note $A_m(X)$ (resp. $B_m(X)$) le groupe des classes d'\'equivalence rationnelle (resp. alg\'ebrique) de $m$\kern -1.5pt - cycles sur $X$ ({\it cf.\/}\ [K], p. 122). \th Th\'eor\`eme \enonce Soit $X$ un sous-sch\'ema de $\hbox{\bf P}^n_{\bf C}$ d\'efini par des \'equations de degr\'e ${\bf d}$. \par\hskip0.5cm {\rm a)} Si $n\ge {{\bf d}+r\choose r+1}+r+1$, l'espace vectoriel $B_1(F_r(X))_{\bf Q}$ est de rang $1$. \par\hskip0.5cm {\rm b)} Si $n\ge {{\bf d}+r+1\choose r+2}$, l'espace vectoriel $A_1(F_r(X))_{\bf Q}$ est de rang $1$. \endth \hskip 1cm\relax En utilisant IV.3.13.3 de [K], p. 206, et en raisonnant comme dans {\it loc.cit.\/} , p. 271, on voit que le corollaire \ref{chaine} entra\^ine que $A_1(F_r(X))_{\bf Q}$ est engendr\'e par les classes des droites. Ces droites sont param\'etr\'ees par un fibr\'e en $G(r-1,\hbox{\bf P}^{r+1})$ au-dessus de $F_{r+1}(X)$, de sorte qu'il existe un morphisme surjectif $A_0(F_{r+1}(X))_{\bf Q}\rightarrow A_1(F_r(X))_{\bf Q}$. Sous l'hypoth\`ese de a), $F_{r+1}(X)$ est connexe. Sous l'hypoth\`ese de b), il r\'esulte du cor. \ref{rat} que $A_0(F_{r+1}(X))_{\bf Q}$ est de dimension $1$.\unskip\penalty 500\quad\vrule height 4pt depth 0pt width 4pt\medbreak \hskip 1cm\relax Lorsque $F_r(X)$ est lisse, la conclusion de la partie a) du th\'eor\`eme\ pr\'ec\'edent reste valable sous l'hypoth\`ese plus faible $n\ge {{\bf d}+r\choose r+1}$; cela r\'esulte du corollaire \ref{pic} et de [BS] ({\it cf.\/}\ aussi [K], th. 3.14, p. 207) . \hskip 1cm\relax Lorsque $X$ contient un $(r+l)$\kern -1.5pt - plan $\Lambda$, le plongement $G(r,\Lambda)\i F_r(X)\i G(r,\hbox{\bf P}^n)$ induit un isomorphisme $A_i(G(r,\Lambda))\simeq A_i(G(r,\hbox{\bf P}^n))$ pour $i\le l$ ([Fu], p. 271), de sorte qu'on a une surjection $A_i(F_r(X))\twoheadrightarrow A_i(G(r,\hbox{\bf P}^n))$. \th Conjecture \enonce Soit $X$ un sous-sch\'ema de $\hbox{\bf P}^n_k$ d\'efini par des \'equations de degr\'e ${\bf d}$. Si $n\ge {{\bf d}+r+l\choose r+l+1}$, le morphisme $A_l(F_r(X))_{\bf Q}\rightarrow A_l(G(r,\hbox{\bf P}^n))_{\bf Q}$ induit par l'inclusion est bijectif. \endth \hskip 1cm\relax Lorsque $l=1$ et $k=\hbox{\bf C}$, c'est le th\'eor\`eme\ pr\'ec\'edent; pour $r=0$ c'est le th\'eor\`eme\ principal de [ELV]. \section{Unirationalit\'e} \hskip 1cm\relax Nous allons maintenant d\'emontrer l'unirationalit\'e de certains sch\'emas de Fano en nous ramenant \`a un r\'esultat de [PS], qui fournit un crit\`ere explicite pour l'unirationa\-li\-t\'e d'une intersection compl\`ete dans un espace projectif. \hskip 1cm\relax Ce crit\`ere est le suivant. On d\'efinit tout d'abord, pour toute suite ${\bf d}=(d_1,\ldots,d_s)$ d'entiers strictement positifs, des entiers $n({\bf d})$ et $r({\bf d})$ de la fa{\gamma} con suivante: on pose $n(1)=r(1)=0$ (dans [PS], on trouve $n(1)=1$, mais $n(1)=0$ suffit); si l'un des $d_i$ vaut $1$, on note ${\bf d'} $ la suite ${\bf d}$ priv\'ee de $d_i$, et on pose $n({\bf d})=n({\bf d'})+1$ et $r({\bf d})=r({\bf d'})$; enfin, si tous les $d_i$ sont $>1$, on pose $r({\bf d})=n({\bf d }-1)$ et $n({\bf d})=r({\bf d})+{{\bf d}+r({\bf d})-1\choose r({\bf d})}$. On a par exemple $$\mathsurround=0pt \everymath={}\everydisplay={} \displaylines{r(2,\ldots,2)=s-1\qquad\qquad r(3,\ldots,3)=s^2+s-1\cr r(4,\ldots,4 )= s^2+s-1+{s^2(s+1)(s^2+s+1)\over 2}\ .\cr}$$ \hskip 1cm\relax Les bornes donn\'ees dans [R] sont un peu meilleures, mais je ne sais pas extraire de cet article un crit\`ere effectif. \th\label{PS} Th\'eor\`eme ([Pr], [PS]) \enonce Soit $F$ une intersection compl\`ete dans $\hbox{\bf P}^N_k$ d\'efinie par des \'equations ${\bf g}=(g_1,\ldots,g_S)$ de degr\'e ${\bf D}$ et contenant un $r({\bf D})$\kern -1.5pt - plan $\Lambda$. On suppose $N\ge n({\bf D})$, que $F$ est irr\'eductible de codimension $S$ et lisse le long de $\Lambda$, et que l'application $\beta:k^{N+1}\rightarrow \Gamma_{\Lambda}({\bf D}-1)$ d\'efinie par $$\beta(a_0,\ldots,a_N)=\Bigl( \sum_{i=0}^Na_i \Bigl({\partial g_1\over\partial x_i}\Bigr) _{\displaystyle{\vert_\Lambda}},\ldots, \sum_{i=0}^Na_i \Bigl({\partial g_S\over\partial x_i}\Bigr) _{\displaystyle{\vert_\Lambda}}\Bigr)$$ est surjective. Alors $F$ est unirationnelle. \endth \hskip 1cm\relax On remarquera que la surjectivit\'e de $\beta$ entra\^ine celle de l'application $\alpha$ d\'efinie en \ref{lisse}, donc la lissit\'e de $F_{r({\bf D})}(F)$ en $\Lambda$. \th \label{unirat} Th\'eor\`eme \enonce Il existe une constante explicite $n({\bf d},r)$ telle que, pour $n\ge n({\bf d},r)$, le sch\'ema de Fano des $r$\kern -1.5pt - plans contenus dans un sous-sch\'ema g\'en\'erique $X$ de $\hbox{\bf P}^n_k$ d\'efini par des \'equations de degr\'e ${\bf d}$, est unirationnel. \endth \ex{Remarques} 1) La borne $n({\bf d},r)$ que l'on obtient est tr\`es grande. Elle est d\'efinie de la fa{\gamma} con suivante: soit ${\bf D}$ la suite d'entiers o\`u chaque $d_i$ est r\'ep\'et\'e ${d_i +r\choose r}$ fois; on pose $r_1=(r({\bf D})+1)(r+1)-1$ et $$n({\bf d},r)= r_1+{{\bf d}+r_1-1\choose r_1}\ .$$ \hskip 1cm\relax Pour le cas le plus simple $r=1$ et ${\bf d}=(3)$, c'est-\`a-dire\ pour le sch\'ema des droites contenues dans une hypersurface cubique, on a ${\bf D}=(3,3,3,3)$, $r(3,3,3,3)=19$ et $n((3),1)=859$. Dans ce cas pr\'ecis, il est facile d'am\'eliorer la borne de [PS] en $r(3,3,3,3)=13$ (il suffit de remarquer qu'une intersection de $4$ quadriques est rationnelle d\`es qu'elle contient un $3$\kern -1.5pt - plan dans son lieu lisse, en proc\'edant par exemple comme dans [CTSSD]); on obtient alors $n((3),1)=433$. \hskip 1cm\relax On obtient aussi $n((2,\ldots,2),r)=s(s+1){r+2\choose 2}(r+1)-1$. Rappelons que pour ${\bf d}=(2,2)$ et $n=2g+1$, la vari\'et\'e $F_r(X)$ est une vari\'et\'e ab\'elienne\ pour $r=g-1$ ({\it cf.\/}\ rem. \ref{rem}.4), et qu'elle est {\it rationnelle} pour $r=g-2$ ([N]), donc unirationnelle pour $r\le g-2$. 2) L'adjectif \leavevmode\raise.3ex\hbox{$\scriptscriptstyle\langle\!\langle$} g\'en\'erique\leavevmode\raise.3ex\hbox{$\scriptscriptstyle\,\rangle\!\rangle$}\ de l'\'enonc\'e du th\'eor\`eme\ peut \^etre pr\'ecis\'e: si $n\ge n({\bf d},r)$, le sch\'ema $F_r(X)$ est unirationnel s'il est irr\'eductible de dimension ${\delta} (n,{\bf d},r)$, si $X$ contient un $r_1$\kern -1.5pt - plan $\Lambda_1$ pour lequel l'application $\beta$ du th\'eor\`eme\ \ref{PS} est surjective, et si $F_r(X)$ est lisse le long de $G(r,\Lambda_1)$. {\it D\'emonstration du th\'eor\`eme }. Soit $V$ l'espace vectoriel $k^{n+1}$. On note $(x^{(0)},\ldots,x^{(r)})$, avec $x^{(j)}=(x^{(j)}_0,\ldots,x^{(j)}_n)$, les coordonn\'ees homog\`enes d'un point de l'espace projectif $\hbox{\bf P}=\hbox{\bf P} (V^{r+1})=\hbox{\bf P}^{(r+1)(n+1)-1}$. Soit $\Sigma$ la sous-vari\'et\'e de $\hbox{\bf P}$ d\'efinie comme le lieu des points $(x^{(0)},\ldots,x^{(r)})$ tels que les points $[x^{(0)}],\ldots,[x^{(r)}]$ de $\hbox{\bf P} V$ ne soient pas lin\'eairement ind\'ependants. L'application $$\rho:\hbox{\bf P}\moins\Sigma\longrightarrow G(r,\hbox{\bf P} V)$$ qui \`a $(x^{(0)},\ldots,x^{(r)})$ associe le $r$\kern -1.5pt - plan engendr\'e par les points $[x^{(0)}],\ldots,[x^{(r)}]$ de $\hbox{\bf P} V$ est une fibration lisse connexe localement triviale. \hskip 1cm\relax Soient ${\bf f}=(f_1\ldots,f_s)$ les \'equations d\'efinissant $X$. On note $F$ l'adh\'erence dans $\hbox{\bf P}$ de $\rho^{-1}(F_r(X))$; lorsque $\delta(n,{\bf d},r)\ge 0$, il ressort du th\'eor\`eme\ \ref{fano} que la vari\'et\'e $F$ est irr\'eductible de codimension ${{\bf d}+r\choose r}$ dans $\hbox{\bf P}$, lisse en dehors de $\Sigma$. \hskip 1cm\relax Pour tout entier $d$, on note ${\cal I}_d$ le sous ensemble de $\hbox{\bf N}^{r+1}$ form\'e des $(i_0,\ldots,i_r)$ tels que $\sum i_{\alpha}=d$; il a ${d+r\choose r}$ \'el\'ements. Pour tout \'el\'ement $f$ de $\mathop{\rm Sym}\nolimits^dV^*$ et tout \'el\'ement $I=(i_0,\ldots,i_r)$ de ${\cal I}_d$, on d\'efinit un polyn\^ome $f_I$ multihomog\`ene de mutidegr\'e $(i_0,\ldots,i_r)$ sur $\hbox{\bf P}$ en posant $$f({\lambda}_0x^{(0)}+\cdots+{\lambda}_rx^{(r)})=\sum_{I\in {\cal I}_d}{\lambda}^If_I(x^{(0)},\ldots,x^{(r)})\ , \leqno{\global\def\currenvir{formule}\hbox{\label{deffI}}$$ o\`u ${\lambda}^I={\lambda}_0^{i_0}\cdots{\lambda}_r^{i_r}$; on convient aussi que $f_I$ est nul si l'un des $i_{\alpha}$ est strictement n\'egatif. En dehors de $\Sigma$, la vari\'et\'e $F$ est d\'efinie par les \'equations $$f_i({\lambda}_0x^{(0)}+\cdots+{\lambda}_rx^{(r)})=0\qquad {\rm pour}\quad i=1,\ldots,s\quad{\rm et\ pour\ tout}\quad ({\lambda}_0,\ldots,{\lambda}_r)\in\hbox{\bf P}^r\ ,$$ c'est-\`a-dire\ par les ${{\bf d}+r\choose r}$ \'equations $f_{i,I}$, pour $i=1,\ldots,s$ et $I\in {\cal I}_{d_i }$. En fait, comme $\Sigma$ est de codimension $n-r$ dans $\hbox{\bf P}$, si on suppose $n-r>{{\bf d}+r\choose r}$, ces \'equations d\'efinissent $F$ dans $\hbox{\bf P}$; la vari\'et\'e $F$ est alors une intersection compl\`ete irr\'eductible, lisse en dehors de $\Sigma$. Son degr\'e est la suite ${\bf D}$ o\`u chaque $d_i$ est r\'ep\'et\'e ${d_i +r\choose r}$ fois. Posons $r_1=(r({\bf D})+1)(r+1)-1$; on suppose $ {\delta} (n,{\bf d},r_1) \ge 0$, de sorte qu'il existe un $r_1$\kern -1.5pt - plan $\Lambda_1=\hbox{\bf P} W_1$ contenu dans $X$; on le suppose d\'efini par les \'equations $x_{r_1+1}=\cdots =x_n=0$. On note $\Lambda^{r+1}_1$ le $((r_1+1)(r+1)-1)$\kern -1.5pt - plan $\hbox{\bf P} (W_1^{r+1})$ dans $\hbox{\bf P}$. \hskip 1cm\relax Soit $\Lambda $ un $r({\bf D})$\kern -1.5pt - plan contenu dans $\Lambda^{r+1}_1$ et disjoint de $\Sigma$ (on pr\'ecisera plus bas notre choix de $\Lambda$). En vue d'appliquer le th\'eor\`eme\ \ref{PS}, on veut v\'erifier que l'application $\beta:k^{(r+1)(n+1)}\rightarrow \Gamma_\Lambda({\bf D}-1) $ d\'efinie par $$\beta(a^{(0)},\ldots,a^{(r)}) = \Bigl( \sum_{j,l} a_l^{(j)} \Bigl({\partial f_{i,I}\over\partial z_l^{(j)}}\Bigr) _{\displaystyle{\vert_\Lambda}}\Bigr)_{1\le i\le s,\ I\in{\cal I}_{d_i }} $$ est surjective. D\'erivons l'\'egalit\'e \ref{deffI} par rapport \`a $x^{(j)}_l$; on obtient $${\lambda}_j{\partial f\over\partial z_l}({\lambda}_0x^{(0)}+\cdots+{\lambda}_rx^{(r)})=\sum_{I\in {\cal I}_d}{\lambda}^I {\partial f_I\over\partial z^{(j)}_l}(x^{(0)},\ldots,x^{(r)})\ ,$$ de sorte que si $\epsilon_j$ est l'\'el\'ement de ${\cal I}_1$ dont toutes les composantes sauf la $j$i\`eme sont nulles, on a $$\Bigl({\partial f\over\partial z_l}\Bigr)_{I-\epsilon_j}={\partial f_I\over\partial z^{(j)}_l}\ ,$$ pour tout $I$ dans ${\cal I}_d$ et tout $j=0,\ldots,r$. On peut donc \'ecrire $$\beta(a^{(0)},\ldots,a^{(r)}) = \Bigl( \sum_{j,l} a_l^{(j)} {\Bigl({\partial f_i\over\partial z_l}\Bigr)_{I-\epsilon_j}}_{\displaystyle{\vert_\Lambda}}\Bigr)_{1\le i\le s,\ I\in{\cal I}_{d_i }}\ , $$ ou encore, en posant $\partial_a f= \sum_l a_l \Bigl(\displaystyle{\partial f\over\partial z_l}\Bigr) _{\displaystyle{\vert_{\Lambda_1}}}$ pour tout $f$ dans $\mathop{\rm Sym}\nolimits^dV^*$, $$\beta(a^{(0)},\ldots,a^{(r)}) = \Bigl( \sum_j {(\partial_{a^{(j)}}f_i)_{I-\epsilon_j}}_{\displaystyle{\vert_\Lambda}}\Bigr)_{1\le i\le s,\ I\in{\cal I}_{d_i }} \ .$$ \th Lemme \enonce Pour $n\ge r_1+{{\bf d}+r_1-1\choose r_1}$ et ${\bf f}$ g\'en\'erique dans $\mathop{\rm Sym}\nolimits^{\bf d}V^*$ nul sur $\Lambda_1$, l'application $$\eqalign{ \beta_1 :k^{ n+1 } &\ \longrightarrow& \Gamma_{\Lambda_1}({\bf d}-1)\ \ \cr a\ \ \ &\ \longmapsto& ( \partial_af_1 ,\ldots, \partial_af_s ) \cr}$$ est surjective. \endth \hskip 1cm\relax Il suffit de trouver un ${\bf f}$ pour lequel les $\Bigl( \displaystyle{\partial f_1\over\partial z_l},\ldots,\displaystyle{\partial f_s\over\partial z_l}\Bigr)_{0\le l\le n}$ engendrent $\Gamma_{\Lambda_1}({\bf d}-1 )$. Soient $J_1,\ldots, J_s$ des sous-ensembles disjoints de $\{ r_1+1,\ldots,n\}$ tels que $\mathop{\rm Card}\nolimits J_i= {d_i+r_1-1\choose r_1}$, et soit $\{ g_j\}_{j\in J_i}$ une base de $ \Gamma_{\Lambda_1}(d_i-1 )$. Il suffit de prendre $f_i=\sum_{j\in J_i} x_jg_j$.\unskip\penalty 500\quad\vrule height 4pt depth 0pt width 4pt\medbreak \hskip 1cm\relax Puisque $\Lambda$ est contenu dans $\Lambda_1^{r+1}$, l'application $\beta$ se factorise par $(\beta_1)^{r+1}$, et il suffit de d\'emontrer que les applications $$\eqalign{\gamma_d:\bigl( \Gamma_{\Lambda_1}(d-1)\bigr)^{r+1} &\ \longrightarrow & \Gamma_\Lambda(d-1)^{{\cal I}_d}\ \ \ \cr ( g^{(0)},\ldots,g^{(r)} )\ \ &\ \longmapsto&\Bigl( \sum_j {g^{(j)}_{I-\epsilon_j}}_{\displaystyle{\vert_\Lambda}}\Bigr)_{I\in{\cal I}_d}\cr}$$ sont surjectives pour $d=d_1,\ldots, d_s$. Nous allons montrer qu'elles sont surjectives pour tout $d$, pour un choix convenable de $\Lambda$. Posons $x_{{\alpha}{\beta}}=x_{{\alpha} (r({\bf D})+1)+{\beta}}$, de sorte que les $x_{{\alpha}{\beta}}$, pour $0\le {\alpha}\le r$ et $0\le {\beta}\le r({\bf D})$, forment des coordonn\'ees sur $\Lambda_1$. Prenons pour $\Lambda$ le $r({\bf D})$\kern -1.5pt - plan de $\Lambda_1^{r+1}$ d\'efini par les \'equations $$ x^{(j)}_{{\alpha}{\beta}} =x^{(0)}_{0{\beta}}\delta_{{\alpha},j}\ ;$$ il est bien disjoint de $\Sigma$, et param\'etr\'e par les $y_{\beta}=x^{(0)}_{0{\beta}}$, pour ${\beta}=0,\ldots,r({\bf D}) $. \th Lemme \enonce Pour tout entier $d$, l'application $$\eqalign{\gamma'_{d,q} : \Gamma_{ \Lambda_1} (d-1) &\ \longrightarrow & \Gamma_\Lambda (d-1)^{{\cal I}_{d-1}}\cr g &\ \longmapsto&\bigl( {g_I}_{\displaystyle{\vert_\Lambda}}\bigr)_{I\in{\cal I}_{d-1}}\cr}$$ est surjective. \endth \hskip 1cm\relax Soit $g=\displaystyle\prod_{{\alpha},{\beta}} x_{{\alpha}{\beta}}^{n_{{\alpha}{\beta}}}$; on a $$g({\lambda}_0x^{(0)}+\cdots+{\lambda}_rx^{(r)})_{\displaystyle{\vert_\Lambda}}= \prod_{{\alpha},{\beta}}({\lambda}_0x^{(0)}_{{\alpha}{\beta}}+\cdots+{\lambda}_rx^{(r)}_{{\alpha}{\beta}} )^{n_{{\alpha}{\beta}}}_{\displaystyle{\vert_\Lambda}}= \prod_{{\alpha},{\beta}}({\lambda}_{\alpha} y_{\beta} )^{n_{{\alpha}{\beta}}}\ ,$$ de sorte que ${g_I}_{\displaystyle{\vert_\Lambda}}$ est le mon\^ome $\prod_{\beta} y_{\beta} ^{\sum_{\alpha} n_{{\alpha}{\beta}}}$ si $\sum_{\beta} n_{{\alpha}{\beta}}=i_{\alpha}$ pour tout ${\alpha}$, et est nul sinon. Il reste \`a montrer que si $I=(i_0,\ldots,i_r)$ est fix\'e dans ${\cal I}_{d-1}$, et si $n_0,\ldots,n_{r({\bf D})}$ sont des entiers positifs de somme $d-1$, il existe des entiers positifs $n_{{\alpha}{\beta}}$ avec $\sum_{\alpha} n_{{\alpha}{\beta}}= n_{\beta}$ et $\sum_{\beta} n_{{\alpha}{\beta}}= i_{\alpha}$ pour tous ${\alpha}$ et ${\beta}$, ce pour quoi il suffit de se donner deux partitions d'un ensemble \`a $n$ \'el\'ements en parties $(A_{{\alpha}})_{0\le{\alpha}\le r}$ et $(B_{{\beta}})_{0\le{\beta}\le r({\bf D})} $ de cardinaux respectifs $i_{{\alpha}}$ et $n_{{\beta}}$, et de prendre pour $n_{{\alpha}{\beta}}$ le cardinal de $A_{{\alpha}}\cap B_{{\beta}}$.\unskip\penalty 500\quad\vrule height 4pt depth 0pt width 4pt\medbreak \hskip 1cm\relax Pour montrer la surjectivit\'e de $\gamma_d$, il suffit donc de montrer celle de l'application $$\eqalign{ \bigl( E^{{\cal I}_{d-1}}\bigr)^{r+1}\ &\ \longrightarrow & E^{{\cal I}_d}\hskip 2cm\cr ( {(g^{(0)}_I)}_I,\ldots,{(g^{(r)}_I)}_I ) &\ \longmapsto&\bigl( g^{(0)}_{J-\epsilon_0}+\cdots+ g^{(r)}_{J-\epsilon_r} \bigr)_{J\in{\cal I}_d}\cr}$$ o\`u $E$ est l'espace vectoriel $ \Gamma_{ \Lambda_1} (d-1)$; cela se fait sans difficult\'e pour n'importe quel espace vectoriel $E$ par r\'ecurrence sur $r$. \hskip 1cm\relax On a montr\'e que toutes les applications $\gamma_{d_i}$, donc aussi l'application $\beta$, sont surjectives. Si $(r+1)(n+1)-1\ge n({\bf D})$, on peut appliquer le th\'eor\`eme\ \ref{PS} pour conclure que $F$ est unirationnelle, donc aussi $F_r(X)$; ceci termine la d\'emonstration du th\'eor\`eme .\unskip\penalty 500\quad\vrule height 4pt depth 0pt width 4pt\medbreak \vskip 1cm \centerline{\bf R\'EF\'ERENCES} \bigskip \hangindent=1cm [AK] A. Altman, S. Kleiman: {\sl Foundations of the theory of Fano schemes}, Comp. Math. {\bf 34} (1977), 3--47. \hangindent=1cm [BVV] W. Barth, A. Van de Ven: {\sl Fano varieties of lines on hypersurfaces}, Arch. Math. {\bf 31} (1978), 96-104. \hangindent=1cm [BD] A. Beauville, R. Donagi: {\sl La vari\'et\'e des droites d'une hypersurface cubique de dimension $4$}, C.R.A.S., t. 301, S\'erie I (1985), 703--706. \hangindent=1cm [BS] S. Bloch, V. Srinivas: {\sl Remarks on correspondences and algebraic cycles}, Am. J. of Math. {\bf 105} (1983), 1235--1253. \hangindent=1cm [BV] L. Bonavero, C. Voisin: {\sl Sch\'emas de Fano et vari\'et\'es de Moishezon}, C.R.A.S., \`a para\^\i tre. \hangindent=1cm [B1] C. Borcea: {\sl Deforming varieties of $k$\kern -1.5pt - planes of projective complete intersections}, Pacific J. Math. {\bf 143} (1990), 25--36. \hangindent=1cm [B2] C. Borcea: {\sl Homogeneous Vector Bundles and Families of Calabi-Yau Threefolds. II}, in Several Complex Variables and Complex Geometry (Santa Cruz 1989), Part 2, Proc. Symp. Pure Math. {\bf 52} (1991), 83--91. \hangindent=1cm [Bo] R. Bott: {\sl Homogeneous vector bundles}, Ann. Math. {\bf 66} (1957), 203--248. \hangindent=1cm [C1] F. Campana: {\sl Remarques sur le rev\^etement universel des vari\'et\'es k\"ahleriennes compactes}, Bull. S.M.F. {\bf 122} (1994), 255--284. \hangindent=1cm [C2] F. Campana: {\sl Connexit\'e rationnelle des vari\'et\'es de Fano}, Ann. Sci. E.N.S. {\bf 25} (1992), 539--545. \hangindent=1cm [CTSSD] J.-L. Colliot-Th\'el\`ene, J.-J. Sansuc, P. Swinnerton-Dyer: {\sl Intersection of two quadrics and Ch\^atelet surfaces. I}, J. reine angew. Math. {\bf 373} (1987), 37--107. \hangindent=1cm [D] O. Debarre: {\sl Th\'eor\`emes de connexit\'e pour les produits d'espaces projectifs et les grassmanniennes}, Am. J. of Math. {\bf 118} (1996), \`a para\^\i tre. \hangindent=1cm [De] M. Demazure: {\sl A very simple proof of Bott's theorem}, Invent. Math. {\bf 33} (1976), 271--272. \hangindent=1cm [DR] U. Desale, S. Ramanan: {\sl Classification of Vector Bundles of Rank $2$ on Hyperelliptic Curves}, Invent. Math. {\bf 38} (1976), 161--185. \hangindent=1cm [ELV] H. Esnault, M. Levine, E. Viehweg: {\sl Chow groups of projective varieties of very small degrees}, Duke Math. J., \`a para\^itre. \hangindent=1cm [F] G. Fano: {\sl Sul sistema $\infty^2$ di rette contenuto in una variet\`a cubica generale dello spazio a quattro dimensioni}, Atti Reale Accad. Sci. Torino {\bf 39} (1904), 778--792. \hangindent=1cm [Fu] W. Fulton: Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete {\bf 2}, Springer Verlag, Berlin, 1984. \hangindent=1cm [FH] W. Fulton, J. Harris: Representation theory, Graduate Text in Mathematics {\bf 129}, Springer Verlag, Berlin, 1991. \hangindent=1cm [H] D. Husemoller: Fibre bundles, 2nd ed., Graduate Text in Mathematics {\bf 20}, Springer Verlag, Berlin, 1966. \hangindent=1cm [K] J. Koll\'ar: Rational Curves on Algebraic Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete {\bf 32}, Springer Verlag, Berlin, 1996. \hangindent=1cm [KMM1] J. Koll\'ar, Y. Miyaoka, S. Mori: {\sl Rationally Connected Varieties}, J. Alg. Geom. {\bf 1} (1992), 429--448. \hangindent=1cm [KMM2] J. Koll\'ar, Y. Miyaoka, S. Mori: {\sl Rational Connectedness and Boundedness of Fano Manifolds}, J. Diff. Geom. {\bf 36} (1992), 765--769. \hangindent=1cm [L] Y. Laszlo: {\sl La dimension de l'espace des sections du diviseur th\^eta g\'en\'eralis\'e}, Bull. Soc. math. France {\bf 119} (1991), 293--306. \hangindent=1cm [M] I.G. Macdonald: Symmetric functions and Hall polynomials, Clarendon Press, Oxford, 1979. \hangindent=1cm [Ma1] L. Manivel: {\sl Th\'eor\`emes d'annulation pour les fibr\'es associ\'es \`a un fibr\'e ample}, Ann. Scuola Norm. Sup. Pisa {\bf 19} (1992), 515--565. \hangindent=1cm [Ma2] L. Manivel: {\sl Applications de Gauss et pl\'ethysme 2}, pr\'epublication de l'Institut Fourier {\bf 352}, 1996. \hangindent=1cm [Mu] J. Murre: {\sl Discussion of a theorem of Morin}, notes de s\'eminaire \leavevmode\raise.3ex\hbox{$\scriptscriptstyle\langle\!\langle$} Argomenti di Geometrica Algebrica\leavevmode\raise.3ex\hbox{$\scriptscriptstyle\,\rangle\!\rangle$} , Povo, Trento, 1979. \hangindent=1cm [N] P.E. Newstead: {\sl Rationality of moduli spaces of stable bundles}, Math. Ann. {\bf 215} (1975), 251--268. \hangindent=1cm [P] K. Paranjape: {\sl Cohomological and cycle-theoretic connectivity}, Ann. Math. {\bf 139} (1994), 641--660. \hangindent=1cm [PS] K. Paranjape, V. Srinivas: {\sl Unirationality of the general Complete Intersection of small multidegree}, in Flips and Abundance for Algebraic Threefolds, ed. J. Koll\'ar, Ast\'erisque {\bf 211} (1992), 241--248. \hangindent=1cm [Pr] A. Predonzan: {\sl Intorno agli $S_k$ giacenti sulla variet\`a intersezione completa di pi\`u forme}, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. {\bf 5} (1948), 238--242. \hangindent=1cm [R] S. Ramanan: {\sl The moduli spaces of vector bundles over an algebraic curve}, Math. Ann. {\bf 200} (1973), 69--84. \hangindent=1cm [S] A. Sommese: {\sl Complex Subspaces of Homogeneous Complex Manifolds II---Homotopy Results}, Nagoya J. Math. {\bf 86} (1982), 101--129. \hangindent=1cm [vW] B.L. van der Waerden: {\sl Zur algebraischen Geometrie 2. Die geraden Linien auf den Hyperfl\"achen des $\hbox{\bf P}_n$}, Math. Ann. {\bf 108} (1933), 253--259. \closeall\bye

9611

alg-geom/9611024

en

https://arxiv.org/abs/alg-geom/9611024

alg-geom/9611024

Mella Massimiliano

Massimiliano Mella

Existence of good divisors on Mukai manifolds

LaTex, 12 pages, nofig

A normal projective variety X is called Fano if a multiple of the anticanonical Weil divisor, -K_X, is an ample Cartier divisor, the index of a Fano variety is the number i(X):=sup{t: -K_X= tH, for some ample Cartier divisor H}. Mukai announced, the classification of smooth Fano manifolds X of index i(X)=n-2, under the assumption that the linear system |H| contains a smooth divisor. In this paper we prove that this assumption is always satisfied. Therefore the result of Mukai provide a complete classification of smooth Fano n-folds of index $i(X)=n-2$, Mukai manifolds.

alg-geom

\section*{Introduction} A normal projective variety $X$ is called {\sf Fano} if a multiple of the \hbox{anticanonical} Weil divisor, $-K_X$, is an ample Cartier divisor. The importance of Fano \hbox{varieties} is twofold, from one side they give, has predicted by Fano \cite{Fa}, \hbox{examples} of non rational varieties having plurigenera and irregularity all zero (cfr \cite{Is}); on the other hand they should be the building block of uniruled variety, indeed recently, Minimal Model Theory predicted that every uniruled variety is birational to a fiber space whose general fiber is a Fano variety with terminal singularities. The index of a Fano variety $X$ is the number $$i(X):=sup\{t\in {\bf Q}: -K_X\equiv tH,\mbox{\rm for some ample Cartier divisor $H \}$}.$$ It is known that $0<i(X)\leq dimX+1$ and if $i(X)\geq dim X$ then $X$ is either an hyperquadric or a projective space by the Kobayashi--Ochiai criterion, smooth Fano n-folds of index $i(X)=n-1$, {\sf del Pezzo n-folds}, have been classified by Fujita \cite{Fu} and terminal Fano n-folds of index $i(X)>n-2$ have been independently classified by Campana--Flenner \cite{CF} and Sano \cite{Sa}. If $X$ has log terminal singularities, then $Pic(X)$ is torsion free and therefore, the $H$ satisfying $-K_X\equiv i(X)H$ is uniquely determined and is called the {\sf fundamental divisor} of $X$. Mukai announced, \cite{Mu}, the classification of smooth Fano n-folds $X$ of index $i(X)=n-2$, under the assumption that the linear system $|H|$ contains a smooth divisor. The main result of this paper is the following \vspace{.5cm}\noindent {\bf Theorem 1} {\it Let $X$ be a smooth Fano n-fold of index $i(X)=n-2$. Then the general element in the fundamental divisor is smooth.} \vspace{.5cm} \noindent Therefore the result of Mukai \cite{Mu} provide a complete classification of smooth Fano n-folds of index $i(X)=n-2$, {\sf Mukai manifolds}. The ancestors of the theorem, and indeed the lighthouses that directed its proof, are Shokurov's proof for smooth Fano 3-folds, \cite{Sh} and Reid's extension to canonical Gorenstein 3-folds using the Kawamata's base point free technique \cite{Re}. This technique was then applied by Wilson in the case of smooth Fano \hbox{4-folds} of index 2, \cite{Wi}, afterwards Alexeev, \cite{Al} did it for log terminal Fano \hbox{n-folds} of index $i(X)>n-2$ and recently Prokhorov used it to prove Theorem 1 in dimension 4 and 5, \cite{Pr1} \cite{Pr2} \cite{Pr3}. As in Reid's construction we will first prove the existence of a section with canonical singularities. To do this we will use Kawamata's base point free technique and Kawamata's notion of centers of log canonical singularities, \cite{Ka1} and his subadjunction formula for codimension 2 minimal centers \cite{Ka2}. These tools, together with Helmke's inductive procedure, \cite{He}, allows to replace difficult non vanishing arguments by a simple Riemann--Roch calculation. Finally the Theorem is proved by an inductive argument that lowers the dimension of $X$. A natural extension of this problem, motivated by the Minimal Model Program, should be to ask if for a terminal Fano $X$ of index $n-2$, with fundamental divisor $H$ it is true that the general element in $|H|$ has terminal singularities. A first, small, step in this direction is the following. \vspace{.5cm}\noindent {\bf Theorem 2} {\it Let $X$ be a terminal Gorenstein Fano n-fold of index n-2. Then the general element in the fundamental divisor $H$ has canonical singularities.} \vspace{.5cm} While working on this subject I had several discussions with M. Andreatta, who suggested me the direction in which this problem could be tackled, I would like to express him my deep gratitude, I would also like to thank A. Corti for valuable comments. \section{Preliminary results} We use the standard notation from algebraic geometry. In particular it is compatible with that of \cite{KMM} to which we refer constantly, everything is defined over {\bf C}. A {\bf Q}-divisor $D$ is an element in $Z_{n-1}(X)\times {\bf Q}$, that is a finite formal sum of prime divisors with rational coefficients; $D$ is called {\bf Q}-Cartier if there is an integer $m$ such that $mD\in Div(X)$, where $Div(X)$ is the group of Cartier divisors of $X$. In the following $\equiv$ (respectively $\sim$, $\sim_{\scriptscriptstyle Q}$) will indicate numerical (respectively linear, {\bf Q}-linear) equivalence of divisors. Let $\mu:Y\rightarrow X$ a birational morphism of normal varieties. If $D$ is a {\bf Q}-divisor ({\bf Q}-Cartier) then is well defined the strict transform $\mu_*^{-1}D$ (the pull back $\mu^*D$). For a pair $(X,D)$ of a variety $X$ and a {\bf Q}-divisor $D$, a log resolution is a proper birational morphism $\mu:Y\rightarrow X$ from a smooth $Y$ such that the union of the support of $\mu_*^{-1}D$ and of the exceptional locus is a normal crossing divisor. \begin{Definition} Let $X$ be a normal variety and $D=\sum_id_iD_i$ an effective {\bf Q}-divisor such that $K_X+D$ is {\bf Q}-Cartier. If $\mu:Y\rightarrow X$ is a log resolution of the pair $(X,D)$, then we can write $$K_Y+\mu_*^{-1}D=\mu^*(K_X+D)+F$$ with $F=\sum_je_jE_j$ for the exceptional divisors $E_j$. We call $e_j\in {\bf Q}$ the discrepancy coefficient for $E_j$, and regard $-d_i$ as the discrepancy coefficient for $D_i$. The pair $(X,D)$ is said to have {\sf log canonical} (LC) (respectively {\sf purely log terminal} (pLT), {\sf Kawamata log terminal} (KLT)) singularities if $d_i\leq 1$ (resp. $d_i\leq 1$, $d_i< 1$) and $e_j\geq -1$ (resp. $e_j>-1$, $e_j>-1$) for any $i,j$ of a log resolution $\mu:Y\rightarrow X$. In particular if $X$ is smooth at the generic point of $Z$, with $cod_XZ=a$ and $D$ is a Weil divisor with $mult_ZD=r$, then $(X,\gamma D)$ is LC for some $\gamma\leq a/r$. \label{lc} \end{Definition} \begin{Definition} A {\sf log-Fano variety} is a pair $(X,\Delta)$ with KLT singularities and such that for some positive integer $m$, $m(K_X+\Delta)$ is an ample Cartier divisor. The index of a log-Fano variety $i(X,\Delta):=sup \{t\in {\bf Q}: -(K_X+\Delta)\equiv tH$ for some ample Cartier divisor $H \}$ and the $H$ satisfying $-(K_X+\Delta)\equiv i(X,\Delta)H$ is called fundamental divisor. In case $\Delta=0$ we have log terminal Fano variety. \end{Definition} We will start recalling some results on log Fano varieties, essentially due to the Kawamata--Viehweg vanishing theorem. \begin{Lemma}[\cite{Al}] Let $(X,\Delta)$ be a log-Fano n-fold of index r, $H$ the fundamental divisor in $X$ and $H^n=d$. Then \begin{itemize} \item[-] If $r>n-2$ then $dim |H|=n-2+d(r-n+3)/2>0$ \item[-] If $r=n-2$ and $X$ has canonical Gorenstein singularities, then $dim |H|=g+n-2\geq n$, where $2g-2=d$, $g\in {\bf Z}$, $g\geq 2$. \end{itemize} \label{al} \end{Lemma} Let us recall the notion and properties of minimal center of log canonical singularities as introduced in \cite{Ka1} \begin{Definition}[\cite{Ka1}] Let $X$ be a normal variety and $D=\sum d_iD_i$ an effective {\bf Q}-divisor such that $K_X+D$ is {\bf Q}-Cartier. A subvariety $W$ of $X$ is said to be a {\sf center of log canonical singularities} for the pair $(X,D)$, if there is a birational morphism from a normal variety $\mu:Y\rightarrow X$ and a prime divisor $E$ on $Y$ with the discrepancy coefficient $e\leq -1$ such that $\mu(E)=W$ The set of all the centers of log canonical singularities is denoted by $CLC(X,D)$, for a point $x\in X$, define $CLC(X,x,D):=\{ W\in CLC(X,D): x\in W\}$. The union of all the subvarieties in $CLC(X,D)$ is denoted by $LLC(X,D)$. \end{Definition} \begin{Theorem}[\cite{Ka1}] Let $X$ be a normal variety and $D$ an effective {\bf Q}-Cartier divisor such that $K_X+D$ is {\bf Q}-Cartier. Assume that $X$ is KLT and $(X,D)$ is LC. \begin{itemize} \item[i)] If $W_1,W_2\in CLC(X,D)$ and $W$ is an irreducible component of $W_1\cap W_2$, then $W\in CLC(X,D)$. In particular, if $(X,D)$ is not KLT at a point $x\in X$ then there exists a unique minimal element of $CLC(X,x,D)$. \item[ii)] If $W\in CLC(X,D)$ is a minimal center then $W$ is normal \item[iii)] Assume that $D\equiv cL$, with $c<1$, for some ample Cartier divisor $L$. If $\{x\}\in CLC(X,D)$ is a minimal center then there is a section of $K_X+L$ not vanishing at $x$. \end{itemize} \label{clc} \end{Theorem} \begin{Theorem}[\cite{Ka2}] Let $X$ be a normal variety which has only KLT singularities, $D$ and effective {\bf Q}-Cartier divisor such that $(X,D)$ is LC, and $W$ a minimal element of $CLC(X,D)$. Assume that $cod W=2$. Then there exist canonically determined effective {\bf Q}-divisors $M_W$ and $D_W$ on $W$ such that $(K_X+D)_{|W}\sim_{\scriptscriptstyle Q} K_W+M_W+D_W$. If $X$ is affine then there exists an effective {\bf Q}-divisor $M_W^{\prime}$ such that $M_W^{\prime}\sim_{\scriptscriptstyle Q} M_W$ and the pair $(W,M_W^{\prime}+D_W)$ is KLT. \label{cod2} \end{Theorem} \vspace{.2cm}\par\noindent{\bf Remark } Note that in particular a $cod 2$ minimal center has rational singularities and if $M_W+D_W\equiv 0$, then $W$ is KLT. In fact It is enough to choose an open affine covering $\{U_i\}$ of $X$, then for $V_i=W\cap U_i$ we have $(V_i,M_{V_i}^{\prime}+D_{V_i})$ is KLT and therefore $V_i$ has rational singularities, \cite{KMM}. Furthermore if \hbox{$M_W+D_W\equiv 0$} then $M_W\sim D_W\sim {\cal O}_W$ and therefore $M_{V_i}^{\prime}\sim{\cal O}_{V_i}$ and these glue together to give that globally $W$ is KLT. \begin{Definition}[\cite{He}] Let $X$ be smooth at $x$ and $(X,D)$ be log canonical at $x$, let $\pi:\tilde{X}\rightarrow X$ the blow up of $x$. Following Helmke, the {\sf local discrepancy} of $(X,D)$ is the rational number $$ b_x(X,D)=inf\left\{t\left\vert \begin{array}{c} \mbox{\rm There is a center of log canonical singularity} \\ \mbox{of $(\tilde{X},\pi^*D-(n-1)E+tE)$ contained in $E$}\end{array} \right.\right\} $$ \end{Definition} \claim{ Let $(X,D)$ be LC and $Z\in CLC(X,D)$ with $x\in Z$ and $Z$, $X$ smooth at $x$, then $b=b_x(X,D)\leq dim Z$.} {\it proof of the claim} Let $\pi:Y\rightarrow X$ the blow up of $x$, with exceptional divisor $E$ and $Z^{\prime}=\pi^{-1}_*Z\cap E$ since $Z$ is a center of log canonical singularities for $(X,D)$ then $\pi^*D$ has multiplicity at least $2cod Z$ along $Z^{\prime}$. Therefore by definition $$b\leq -(2cod Z-n+1)+cod Z^{\prime}=dim Z$$. \nopagebreak\par\nopagebreak\hfill {\Large $\diamond$}\par The following inductive procedure due to Helmke (this is a particular case of his more general Theorem) allows us to decrease the dimension of a minimal center. \begin{Proposition}[\cite{He}] Let $L$ an ample divisor on $X$ and $D$ an effective {\bf Q}-divisor with $D\equiv \gamma L$ for some rational number $0\leq \gamma<1$. Assume that $X$ is smooth at $x$ and $(X,D)$ is log canonical with local discrepancy $b=b_x(X,D)$ at $x$. Let $Z$ be the minimal center of $CLC(X,x,D)$ assume that $d=dim Z>0$ and $Z$, $X$ smooth at $x$. If \begin{equation} L^d\cdot Z>p^d,\hspace{1cm}\mbox{\rm where $p=\frac{b}{1-\gamma}$,} \label{heq} \end{equation} then there is a {\bf Q}-divisor $D_1\equiv \gamma_1L$, with $\gamma<\gamma_1<1$ such that $(X,D_1)$ is log canonical at $x$ with minimal center $Z_1$ properly contained in $Z$ and $$p_1=\frac{b_1}{1-\gamma_1}<p\hspace{1cm}\mbox{\rm where $b_1=b_x(X,D_1)$.}$$ \label{He} \end{Proposition} We will sketch the proof for reader's convenience. Step 1. Produce a section $D_0\in |kL_{|Z}|$, for $k\gg0$, with $mult_x D_0>pk$. This is accomplished by R--R theorem using inequality (\ref{heq}). Step 2. Using Serre's vanishing and Bertini Theorem extend $D_0$ to a section $D^{\prime} \in |kL|$ which is smooth away from $Z$. Let $\gamma^{\prime}=sup\{t|(X,D+tD^{\prime})\mbox{\rm is LC at $x$}\}$, then $D_1=D+\gamma^{\prime}D^{\prime}\equiv \gamma_1 L$. Step 3. Use the definition of $p$ and the minimality of $Z$ to prove that $\gamma_1<1$, and then a straightforward computation gives the assert. \section{Existence of a canonical section} For this step we will use Kawamata's base point free technique, as explained in Reid \cite{Re}. Let us start with some lemmas. \begin{Lemma} Let $X$ be a log terminal Fano n-fold, with $n\geq 3$ and $H$ an ample Cartier divisor with $-K_X\equiv (n-2)H$ and $G$ a {\bf Q}-Cartier divisor with $(X,G)$ LC. Assume that $Z\in CLC(X,G)$ is a minimal center and $G\equiv \gamma H$, with $\gamma< cod Z-1$ then there is a section of $H$ not vanishing identically on $Z$. \label{KV} \end{Lemma} \par \noindent{\bf Proof. }\nopagebreak We proceed as in \cite[Prop 2.3]{Ka1}. Let $M\in |mH|$, for $m\gg 0$, be a general member among Cartier divisors containing $Z$, let \hbox{$G_1= (1-\epsilon_1) (G+\epsilon_2 M)$,} for $\epsilon_i\ll 1/m$, then $G_1\equiv \gamma_1 H$, with $\gamma_1<cod Z-1$. Furthermore we may assume that $(X,G_1)$ is LC and $Z$ is an isolated element of $LLC(X,G_1)$. Let $\mu:Y\rightarrow X$ a log resolution of $(X,G_1)$, then $$K_Y+E-A+F=\mu^*(K_X+G_1),$$ where $\mu(E)=Z$, $A$ is an integral \hbox{$\mu$-exceptional} divisor and $\lfloor F\rfloor=0$. Let $$ N(t):= -K_Y-E-F+A+\mu^*(tH),$$ then $N(t)\equiv \mu^*(t+(n-2)-\gamma_1)H$ and $N(t)$ is nef and big whenever $t+(n-2)-\gamma_1> 0$, hence by hypothesis this is true whenever $t\geq -n+1+cod Z$. Thus K--V vanishing yields \begin{equation} H^i(Y,\mu^*(tH)-E+A)=0\hspace{.7cm} H^i(E,(\mu^*(tH)+A)_{|E_0})=0 \label{van} \end{equation} for $i>0$ and $t\geq -n+1+cod Z$, and consequently $$ H^0(Y,\mu^*H+A)\rightarrow H^0(E,\mu^*H+A)\rightarrow 0; $$ since $A$ is effective and $\mu$-exceptional, then any section in $H^0(Y,\mu^*H+A)$, not vanishing on $E$, pushes forward to give a section of $H$ not vanishing on $Z$. To conclude the proof it is, therefore, enough to prove that $h^0(E,N(1))>0$. Let $p(t)=\chi(E,N(t))$, then by equation (\ref{van}), $p(0)\geq 0$ and $p(t)=0$ for $0>t\geq -n+1+cod Z=-dim Z+1$. Since $deg p(t)= dim Z$ and $p(t)>0$ for $t\gg 0$ then $h^0(E,N(1))=p(1)>0$. \nopagebreak\par\nopagebreak\hfill {\Large $\diamond$}\par The above lemma allows us, essentially, to treat minimal centers of codimension $\geq 3$. In the next couple of lemmas we will treat codimension 2 minimal centers. \begin{Lemma} Let $X$ be a log terminal Fano n-fold, with $n\geq 3$, and $H$ an ample Cartier divisor with $-K_X\equiv (n-2)H$, let $L\sim (n-1)H$ and $D$ a {\bf Q}-Cartier divisor with $(X,D)$ LC. Assume that $D\equiv H$ and $Z\in CLC(X,D)$ a cod 2 minimal center, then for $k\gg 0$ and $\delta\geq 0$ $$h^0(Z,kL_{|Z})\geq \frac{(n-1)^{n-2}}{(n-2)!} k^{n-2}+\frac{(n-3+\delta)(n-1)^{n-3}}{2(n-3)!}k^{n-3}+\mbox{\rm lower terms in k}.$$ Furthermore, keeping the notation of Theorem \ref{cod2}, if $M_Z+D_Z\not\equiv 0$ then $\delta>0$. \label{K2} \end{Lemma} \par \noindent{\bf Proof. }\nopagebreak By Theorem \ref{cod2} there are effective {\bf Q}-divisors $M_Z$ and $D_Z$ such that $$ -(n-3)H_{|Z}\equiv (K_X+D)_{|Z}\sim_{\scriptscriptstyle Q} K_Z+M_Z+D_Z.$$ Let $f:Y\rightarrow Z$ a log resolution of $(Z,M_Z+D_Z)$ then $$K_Y+\Delta=f^*(K_Z+M_Z+D_Z)+\sum e_iE_i,$$ where $\Delta=f^{-1}_*(M_Z+D_Z)$ is effective and the $E_i$ are $f$-exceptional. In particular $$-K_Y\cdot (f^*H_{|Z})^{n-3}=-(K_X+D)_{|Z}\cdot H^{n-3}_{|Z}+\Delta\cdot f^*H_{|Z}^{n-3}\geq n-3+\delta$$ Since $Z$ has rational singularities and $L$ is ample then, for $k\gg 0$, $$h^0(Z,kL_{|Z})=\chi(Z,kL_{|Z})=\chi(Y,kf^*L_{|Z}).$$ Hence by R--R formula $$h^0(Z,kL_{|Z})\geq \frac{(n-1)^{n-2}}{(n-2)!} k^{n-2}+\frac{(n-3+\delta)(n-1)^{n-3}}{2(n-3)!}k^{n-3}+\mbox{\rm lower terms in k}.$$ \nopagebreak\par\nopagebreak\hfill {\Large $\diamond$}\par \begin{Lemma} Let $X$ be a log terminal Gorenstein Fano n-fold, with $n\geq 3$, and $H$ an ample Cartier divisor with $-K_X\sim (n-2)H$, let $L\sim (n-1)H$ and $D$ a {\bf Q}-Cartier divisor with $(X,D)$ LC. Assume that $D\equiv H$ and $Z\in CLC(X,D)$ a cod 2 minimal center with $Z\not\subset Sing(X)$. Then there exists a section of $H\sim K+L$ not vanishing identically on $Z$. \label{KH} \end{Lemma} \par \noindent{\bf Proof. }\nopagebreak By Theorem \ref{cod2} there are effective {\bf Q}-divisors $M_Z$ and $D_Z$ such that $$ -(n-3)H_{|Z}\equiv (K_X+D)_{|Z}\sim_{\scriptscriptstyle Q} K_Z+M_Z+D_Z.$$ If $n=3$ then $Z$ is a smooth curve, by Theorem \ref{clc}, and $g(Z)\leq 0$, thus $h^0(Z,H)>0$; if $n>3$ and $(Z,M_Z+D_Z)$ is KLT then $(Z,M_Z+D_Z)$ is a log-Fano variety of index $i(Z,M_Z+D_Z)=dimZ-1$, therefore by Lemma \ref{al}, $h^0(Z,H)>0$. As in the proof of Lemma \ref{KV} let us replace $D$ with $D_1$ such that $(X,D_1)$ is LC, $Z$ is isolated in $LLC(X,D_1)$ and $D_1\equiv \gamma_1H$, for $\gamma_1<1+\epsilon$, with $\epsilon\ll 1$. Let $\mu:Y\rightarrow X$ a log resolution of $(X,D_1)$ with $K_Y+E-A+F=\mu^*(K_X+D_1)$, where $f(E)=Z$, $A$ is an integral \hbox{$\mu$-exceptional} divisor and $\lfloor F\rfloor=0$. Let $N(t):=-K_Y-E-F+A+\mu^*(tH)$, then $N(1)\equiv \mu^*(1+(n-2)-\gamma_1)H$ is nef and big and consequently $$ H^0(Y,\mu^*H+ A)\rightarrow H^0(E,\mu^*H+ A)\rightarrow 0. $$ Therefore the sections in $H^0(Z,H)$ extends to sections of $H^0(X,H)$ not vanishing identically on $Z$. \noindent By the remark after Theorem \ref{cod2} we can, therefore assume that $M_Z+D_Z\not\equiv 0$. Fix a smooth point $x\in Z$ outside of $Sing(X)$, such that $Z$ is the minimal element of $CLC(X,x,D)$. Let us mimic Helmke's arguments; in the notation of Proposition \ref{He}, $\gamma=1/(n-1)$ and $$p=\frac{b}{1-\gamma}\leq \frac{(n-1)(n-2)}{n-2}\leq n-1.$$ The first step is accomplished using Lemma \ref{K2}; in fact $$h^0(Z,{\cal O}_Z/{\cal I}^k_{Z,x})=\frac{1}{(n-2)!}k^{n-2}+ \frac{(n-3)}{2(n-3)!}k^{n-3}+\mbox{\rm lower terms in k},$$ therefore by Lemma \ref{K2} there exists a section $D^{\prime}\in|kL_{|Z}|$, for $k\gg 0$, such that \hbox{$mult_xD^{\prime}>pk$}. It is now enough to follow word by word Helmke's arguments to conclude that there is a {\bf Q}-divisor $D_1\equiv \gamma_1L$, with $\gamma<\gamma_1<1$ such that $(X,D_1)$ is log canonical at $x$, with minimal center $Z_1\ni x$ properly contained in $Z$. Since $x\in Z_1$ and $p_1<p\leq n-1$ then $Z_1\not\subset Sing(X)$ and we can choose a smooth point $x_1\in Z_1$ and apply directly Proposition \ref{He} to $(X,D_1)$ and $x_1$. Inductively the dimension of the minimal center is lowered and we find a divisor $D_l\equiv \gamma_lL$, with $c_l<1$, which has zero dimensional minimal center. Conclude by Theorem \ref{clc} iii) that there exists a section of $H\sim K_X+L$ not vanishing on $Z$. \nopagebreak\par\nopagebreak\hfill {\Large $\diamond$}\par We will need the forthcoming lemma only in the next section, to be able to apply an inductive procedure on the Fano variety, but we place it here since the flavor and the proof are close to the previous one. \begin{Lemma} Let $X$ be a log terminal Gorenstein Fano n-fold, with $n>3$, and $H$ an ample Cartier divisor with $-K_X\sim (n-2)H$, let $L\sim (n-1)H$ and $D$ a {\bf Q}-Cartier divisor with $(X,D)$ LC. Assume that $D\equiv 2H$, $Z\in CLC(X,D)$ is a codimension 3 minimal center not contained in $Sing(X)$. Furthermore assume that there exist $S\in|H|$ and an effective {\bf Q}-divisor $D_S$ such that $(S,D_S)$ satisfy the hypothesis of Theorem \ref{cod2} and Lemma \ref{K2}. Then there is a section of $H\sim K_X+L$ not vanishing identically on $Z$. \label{KH2} \end{Lemma} \par \noindent{\bf Proof. }\nopagebreak Let us simply sketch the proof since it is similar to that of Lemma \ref{KH}. By Theorem \ref{cod2} there exist effective {\bf Q}-divisors $M_Z$ and $D_Z$ such that $$-(n-4)H_{|Z}\equiv K_Z+M_Z+D_Z. $$ Let us, again, replace $D$ with $D_1$ such that $(X,D_1)$ is LC, $Z$ is isolated in $LLC(X,D_1)$ and $D_1\equiv \gamma_1H$, for $\gamma_1<2+\epsilon$, with $\epsilon\ll 1$. Let $\mu:Y\rightarrow X$ a log resolution of $(X,D_1)$ with $K_Y+E-A+F=\mu^*(K_X+D_1)$, where $f(E)=Z$, $A$ is an integral $\mu$-exceptional divisor and $\lfloor F\rfloor=0$. Let \hbox{$N(t):=-K_Y-E-F+A+\mu^*(tH)$,} then $N(1)\equiv \mu^*(1+(n-2)-\gamma_1)H$ is nef and big \noindent If $n=4$ then $Z$ is a smooth curve of non positive genus, therefore $h^0(Z,H)>0$; if $n\geq 5$ and $(Z,M_Z+D_Z)$ is KLT then it is a log-Fano variety of index $i(Z,M_Z+D_Z)= dim Z-1$, therefore as above the sections in $H^0(Z,H)$ extends to sections of $H^0(X,H)$ not vanishing identically on $Z$. Again we can assume that $M_Z+D_Z\not\equiv 0$, and choose a smooth point in $Z$ with $x\not\in Sing(X)$, in Helmke's notations, $\gamma=2/(n-1)<1$ and $$p\leq \frac{(n-1)(n-3)}{n-1-2}\leq n-1$$ and by Lemma \ref{K2} for $k\gg 0$ there is a section $D^{\prime}\in |kL_{|Z}|$ with $mult_x D^{\prime}>pk$; then conclude as in Lemma \ref{KH} \nopagebreak\par\nopagebreak\hfill {\Large $\diamond$}\par \begin{Proposition} Let $X$ be a log terminal Gorenstein Fano n-fold and $H$ an ample Cartier divisor with $-K_X\sim (n-2)H$. Assume that $codSing(X)>2$ and $n\geq 3$. Then the general element in $|H|$ has at worst canonical singularities. \label{alto} \end{Proposition} \par \noindent{\bf Proof. }\nopagebreak By Lemma \ref{al} we know that $dim |H|\geq 1$. Let $S\in |H|$ a generic element and assume that $S$ has worse than canonical singularities. Since both $H$ and $K_X$ are Cartier divisors then $(X,S)$ is not pLT, that is there exists $\gamma\leq 1$ such that $(X,\gamma S)$, is LC with $Z$ a minimal center in $CLC(X,\gamma S)$, and by Bertini theorem $Z\subset Bsl|H|$. We will derive a contradiction , producing a section of $|H|$ not vanishing identically on $Z$. \noindent If either $dim Z\leq n-3$ or $dim Z=n-2$ and $\gamma<1$ then apply Lemma \ref{KV}. \noindent If $dim Z=n-2$ and $\gamma=1$, by hypothesis $Z\not\subset Sing(X)$ hence apply Lemma \ref{KH}. To conclude we have to exclude the case $dim Z=n-1$. Assume that $|H|$ has a fixed component $F$, by \cite[Prop 3.2]{Al} $F$ must have multiplicity 1, that is $\gamma=1$. Since $H$ is connected and movable then $S$ must be singular along a codimension 2 set $Z\subset F$, therefore $F$ is not minimal in $CLC(X,S)$, see Definition \ref{lc}. \nopagebreak\par\nopagebreak\hfill {\Large $\diamond$}\par \Remark{ In particular the above argument shows that $H$ is smooth in codimension 1 and there are not fixed component.} \par \noindent{\bf Proof. }\nopagebreak (of Theorem 2) By Lemma \ref{al}, $h^0(X,H)\geq 2$; furthermore terminal singularities are smooth in codimension 2. It is, therefore enough to apply Proposition \ref{alto}. \nopagebreak\par\nopagebreak\hfill {\Large $\diamond$}\par \vspace{.2cm}\par\noindent{\bf Remark } It is not true, in general, that terminal Fano $X$ of index $i(X)=n-2$ are Gorenstein; consider a terminal Fano 3-fold with an Enriques surface as section of the fundamental divisor, this varieties are studied by Conte--Murre \cite{CM}. In this case $X$ has 8 singular points, which are cones over the Veronese surface, and $X$ is 2-Gorenstein; nevertheless $H$ has a smooth (terminal) section. \section{Proof of the main Theorem} By a direct calculation, for instance Lemma \ref{al}, $h^0(X,H)\geq n$ therefore by Proposition \ref{alto} there exists a section $S\in |H|$ with canonical singularities. Our aim is to apply inductively Proposition \ref{alto}, to do this we have to prove that $S$ is smooth in codimension 2. Assume the contrary, in particular $S$ is not terminal and there is a center $Z\subset Bsl|H|$ of canonical singularities in $S$ with $dim Z=n-3$. {\sc Case 1} Assume that all sections of $|H|$ are singular at $Z$, let $H_i\in |H|$ generic elements and \hbox{$D=1/2(H_1+H_2)$}. \claim{$(X,\gamma D)$ is log canonical for some $\gamma\leq 3/2$ with a minimal center $W\subseteq Z$ of codimension $\geq 3$.} Observe that by the claim we can apply Lemma \ref{KV}, to produce a section of $|H|$ not vanishing on $Z$ and derive in this way a contradiction. \par \noindent{\bf Proof. }\nopagebreak(of the claim) Let $f:Y\rightarrow X$ the blow up of $Z$ let $f^*S=S^{\prime}+ rE$, since $X$ is smooth at the generic point of $Z$ then $K_Y=f^*K_X+2E$. By adjunction formula $$K_{S^{\prime}}=(K_Y+S^{\prime})_{|S^{\prime}}=f^*K_{S}+(2-r) E_{|S^{\prime}},$$ since $S$ is canonical and is singular at $Z$ then $r=2$. $|H|$ has not fixed components and its general element is smooth in codimension 1 therefore for some $\gamma\leq 3/2$, $(X,\gamma D)$ is log canonical with a minimal center $W\subseteq Z$ of codimension $\geq 3$. \nopagebreak\par\nopagebreak\hfill {\Large $\diamond$}\par {\sc Case 2} Assume that there are infinitely many such codimension 3 components $Z_i\subset Bsl|H|$ centers of canonical singularities for $H_i\in |H|$. Let $H_1$ a generic element in $|H|$, we can assume that $H_1$ is singular along $Z_1$, with $Z_1\subset Bsl|H|$ and $Z_1\not\subset Sing(X)$. Let $D=1/2(H_1+H_2)$, with $H_2\in |H|$, a general element; by construction $(X,\gamma D)$ is log canonical for some $\gamma \leq 2$ with a minimal center $Z$ of codimension$\geq 3$. If either $\gamma <2$ or $codZ>3$ then conclude by Lemma \ref{KV}. Assume that $\gamma=2$ and $cod Z=3$, we can assume without loss of generality that $Z=Z_1$, let $S\in |H|$ a generic element smooth at the generic point of $Z$ and $D_S=H_{1|S}$, then $(S,D_S)$ and $Z$ satisfy the hypothesis of Theorem \ref{cod2} and Lemma \ref{K2}, thus, we derive a contradiction by Lemma \ref{KH2} if $n\geq 4$. At each inductive step we loose only one section of $|H|$, therefore $|H_{|S}|$ is always movable; furthermore by K--V vanishing theorem $$H^0(X,H)\rightarrow H^0(S,H_{|S})\rightarrow 0,$$ hence it is possible to study the singularities of $S$ trough the linear system $H_{|S}$. To carry on induction in Case 1 we need the following \claim{ Let $S_0=X$ and $S_j=X\cap H_1\cap\ldots\cap H_j$, for $H_i\in |H|$ general elements. Assume that $S_j$ has canonical singularities and is singular at $Z_j$, with $cod_XZ_j=j+2$. If $X$ is smooth at $Z_j$ then $S_{j-1}$ is smooth at $Z_j$.} \par \noindent{\bf Proof. }\nopagebreak(of the claim) We will prove it by induction on $j$. If $j=1$ then it follows by hypothesis. Let $f:Y\rightarrow X$ the blow up of $Z_j$, with $f^*H_i=H_i^{\prime}+ rE$, since $X$ is smooth at the generic point of $Z_j$ then $K_Y=f^*K_X+(j+1)E$. By adjunction formula $$K_{S_j^{\prime}}=(K_Y+\sum_i H_i^{\prime})_{|S_j^{\prime}}=f^*K_{S_j}+(j+1-jr) E_{|S_j^{\prime}},$$ where $S_j^{\prime}=Y\cap H_1^{\prime}\cap\ldots\cap H_j^{\prime}$. Since $S_j$ has canonical singularities then \hbox{$j+1-jr\geq 0$,} and consequently $r=1$, that is the generic element in $|H|$ is smooth at $Z_j$. On the other hand $Z_j=S_1\cap H_2\cap\ldots\cap H_j$, where $S_1\in |H|$ is a general element smooth at $Z_j$, and $cod_{S_1}Z_j=j-1$, therefore by induction hypothesis $S_{j-1}$ is smooth at $Z_j$. \nopagebreak\par\nopagebreak\hfill {\Large $\diamond$}\par By the inductive process we are reduced to a canonical Gorenstein 3-fold smooth in codimension 2, $S_3=X\cap(\bigcap_{i=1}^{n-3}H_i)$ with a line bundle $H_3=H_{|S_3}$ satisfying the following conditions: \begin{itemize} \item[-] $h^0(S_3,H_3)\geq 3$ \nopagebreak\item[-] $dim Bsl|H_3|= 1$ \end{itemize} Let $H_1\in |H_3|$ a general element and $B$ a curve contained in the base locus of $|H_3|$. Assume, without loss of generalities that $x_1\in B\cap H_1$ is such that $x_1\not\in Sing(S_3)$ and $H_1$ singular at $x_1$. Let $A=\{ M\in|H|$ $|$ $M$ is singular at $x_1 \}$ since $h^0(S_3,H_3)\geq 3$ and $dim Bsl|H_3|=1$ then $dim A\geq 1$. Let $H_i\in A$, for $i=1,2$ be general elements and $D=1/2(D_1+D_2)$. Then $(X,\gamma D)$ is log canonical for some $\gamma\leq 3/2$ with zero dimensional minimal center thus Lemma \ref{KV} apply to derive a contradiction and the theorem is proved. \nopagebreak\par\nopagebreak\hfill {\Large $\diamond$}\par \small

9611

alg-geom/9611015

en

https://arxiv.org/abs/alg-geom/9611015

alg-geom/9611015

Misha S. Verbitsky

Misha Verbitsky

Desingularization of singular hyperkaehler varieties I

13 pages, LaTeX 2e LaTeX 2e

Math. Res. Lett. 4 (1997), no. 2-3, 259--271.

Let $M$ be a singular hyperkaehler variety, obtained as a moduli space of stable holomorphic bundles on a compact hyperkaehler manifold (alg-geom/9307008). Consider $M$ as a complex variety in one of the complex structures induced by the hyperkaehler structure. We show that normalization of $M$ is smooth, hyperkaehler and does not depend on the choice of induced complex structure.

alg-geom

\section{Introduction} \label{_Intro_Section_} \setcounter{footnote}{1} The structure of this paper is following. \begin{itemize} \item In the first section, we give a compendium of definitions and results from hyperk\"ahler geometry, all known from literature. \item Section \ref{_real_ana_Section_} deals with the real analytic varieties underlying complex varieties. We define almost complex structures on a real analytic variety. This notion is used in order to define hypercomplex varieties. We show that a hyperk\"ahler manifold is always hypercomplex. \item In Section \ref{_singu_hype_Section_}, we give a definition of a singular hyperk\"ahler variety, following \cite{_Verbitsky:Hyperholo_bundles_} and \cite{_Verbitsky:Deforma_}. We cite basic properties and list the examples of such manifolds. \item In Section \ref{_SLHS_Section_}, we define locally homogeneous singularities. A space with locally homogeneous singularities (SLHS) is an analytic space $X$ such that for all $x\in X$, the $x$-completion of a local ring ${\cal O}_xX$ is isomorphic to an $x$-completion of associated graded ring $({\cal O}_xX)_{gr}$. We show that a complex variety is SLHS if and only if the underlying real analytic variety is SLHS. This allows us to define invariantly the notion of a hyperk\"ahler SLHS. The natural examples of hyperk\"ahler SLHS include the moduli spaces of stable holomorphic bundles, considered in \cite{_Verbitsky:Hyperholo_bundles_}. \footnote{In \cite{_Verbitsky:Hyperholo_bundles_}, we proved that the moduli of stable bundles over a compact hyperk\"ahler manifold is a hyperk\"ahler variety, if we assume certain numerical restrictions on the bundle's Chern classes. The stable bundles satisfying these restrictions are called {\bf hyperholomorphic}.} We conjecture that every hyperk\"ahler variety is a space with locally homogeneous singularities. \item In Section \ref{_tange_cone_Section_}, we study the tangent cone of a singular hyperk\"ahler manifold $M$ in the point $x\in M$. We show that its reduction, which is a closed subvariety of $T_x M$, is a union of linear subspaces $L_i\subset T_x M$. These subspaces are invariant under the natural quaternion action in $T_x M$. This implies that a normalization of $(M,I)$ is smooth. Here, as usually, $(M, I)$ denotes $M$ considered as a complex variety, with $I$ a complex structure induced by the singular hyperk\"ahler structure on $M$. \item In Section \ref{_desingu_Section_}, we formulate and prove the desingularization theorem for hyperk\"ahler varieties with locally homogeneous singularities. For each such variety $M$ we construct a finite surjective morphism $\tilde M \stackrel n {\:\longrightarrow\:} M$ of hyperk\"ahler varieties, such that $\tilde M$ is smooth and $n$ is an isomorphism outside of singularities of $M$. The $\tilde M$ is obtained as a normalization of $M$; thus, our construction is canonical and functorial. \end{itemize} \section{Hyperk\"ahler manifolds} \subsection{Definitions} This subsection contains a compression of the basic definitions from hyperk\"ahler geometry, found, for instance, in \cite{_Besse:Einst_Manifo_} or in \cite{_Beauville_}. \hfill \definition \label{_hyperkahler_manifold_Definition_} (\cite{_Besse:Einst_Manifo_}) A {\bf hyperk\"ahler manifold} is a Riemannian manifold $M$ endowed with three complex structures $I$, $J$ and $K$, such that the following holds. \begin{description} \item[(i)] the metric on $M$ is K\"ahler with respect to these complex structures and \item[(ii)] $I$, $J$ and $K$, considered as endomorphisms of a real tangent bundle, satisfy the relation $I\circ J=-J\circ I = K$. \end{description} \hfill The notion of a hyperk\"ahler manifold was introduced by E. Calabi (\cite{_Calabi_}). \hfill Clearly, hyperk\"ahler manifold has the natural action of quaternion algebra ${\Bbb H}$ in its real tangent bundle $TM$. Therefore its complex dimension is even. For each quaternion $L\in \Bbb H$, $L^2=-1$, the corresponding automorphism of $TM$ is an almost complex structure. It is easy to check that this almost complex structure is integrable (\cite{_Besse:Einst_Manifo_}). \hfill \definition \label{_indu_comple_str_Definition_} Let $M$ be a hyperk\"ahler manifold, $L$ a quaternion satisfying $L^2=-1$. The corresponding complex structure on $M$ is called {\bf an induced complex structure}. The $M$ considered as a complex manifold is denoted by $(M, L)$. \hfill Let $M$ be a hyperk\"ahler manifold. We identify the group $SU(2)$ with the group of unitary quaternions. This gives a canonical action of $SU(2)$ on the tangent bundle, and all its tensor powers. In particular, we obtain a natural action of $SU(2)$ on the bundle of differential forms. \hfill \lemma \label{_SU(2)_commu_Laplace_Lemma_} The action of $SU(2)$ on differential forms commutes with the Laplacian. {\bf Proof:} This is Proposition 1.1 of \cite{_Verbitsky:Hyperholo_bundles_}. \blacksquare Thus, for compact $M$, we may speak of the natural action of $SU(2)$ in cohomology. \subsection{Trianalytic subvarieties in compact hyperk\"ahler manifolds.} In this subsection, we give a definition and a few basic properties of trianalytic subvarieties of hyperk\"ahler manifolds. We follow \cite{_Verbitsky:Symplectic_II_}. \hfill Let $M$ be a compact hyperk\"ahler manifold, $\dim_{\Bbb R} M =2m$. \hfill \definition\label{_trianalytic_Definition_} Let $N\subset M$ be a closed subset of $M$. Then $N$ is called {\bf trianalytic} if $N$ is a complex analytic subset of $(M,L)$ for any induced complex structure $L$. \hfill Let $I$ be an induced complex structure on $M$, and $N\subset(M,I)$ be a closed analytic subvariety of $(M,I)$, $dim_{\Bbb C} N= n$. Denote by $[N]\in H_{2n}(M)$ the homology class represented by $N$. Let $\inangles N\in H^{2m-2n}(M)$ denote the Poincare dual cohomology class. Recall that the hyperk\"ahler structure induces the action of the group $SU(2)$ on the space $H^{2m-2n}(M)$. \hfill \theorem\label{_G_M_invariant_implies_trianalytic_Theorem_} Assume that $\inangles N\in H^{2m-2n}(M)$ is invariant with respect to the action of $SU(2)$ on $H^{2m-2n}(M)$. Then $N$ is trianalytic. {\bf Proof:} This is Theorem 4.1 of \cite{_Verbitsky:Symplectic_II_}. \blacksquare \remark \label{_triana_dim_div_4_Remark_} Trianalytic subvarieties have an action of quaternion algebra in the tangent bundle. In particular, the real dimension of such subvarieties is divisible by 4. \subsection{Totally geodesic submanifolds.} \nopagebreak \hspace{5mm} \proposition \label{_comple_geodesi_basi_Proposition_ Let $X \stackrel \phi\hookrightarrow M$ be an embedding of Riemannian manifolds (not necessarily compact) compatible with the Riemannian structure. Then the following conditions are equivalent. \begin{description} \item[(i)] Every geodesic line in $X$ is geodesic in $M$. \item[(ii)] Consider the Levi-Civita connection $\nabla$ on $TM$, and restriction of $\nabla$ to $TM \restrict{X}$. Consider the orthogonal decomposition \begin{equation} \label{TM_decompo_Equation_} TM\restrict{X} = TX \oplus TX^\bot. \end{equation} Then, this decomposition is preserved by the connection $\nabla$. \end{description} {\bf Proof:} Well known; see, for instance, \cite{_Besse:Einst_Manifo_}. \hbox{\vrule width 4pt height 4pt depth 0pt} \hfill \proposition \label{_triana_comple_geo_Proposition_} Let $X\subset M$ be a trianalytic submanifold of a hyperk\"ahler manifold $M$, where $M$ is not necessarily compact. Then $X$ is totally geodesic. {\bf Proof:} This is \cite{_Verbitsky:Deforma_}, Corollary 5.4. \blacksquare \section{Real analytic varieties} \label{_real_ana_Section_} Let $X$ be a complex analytic variety. The ``real analytic variety underlying $X$'' (denoted by $X_{\Bbb R}$) is the following object. By definition, $X_{\Bbb R}$ is a ringed space with the same topology as $X$, but with a different structure sheaf, denoted by ${\cal O}_{X_{\Bbb R}}$. Let $C(X, {\Bbb R})$ be a sheaf of continous ${\Bbb R}$-valued functions on $X$. Then ${\cal O}_{X_{\Bbb R}}$ is a subsheaf of $C(X, {\Bbb R})$, defined as follows. Let $A\subset C(X, {\Bbb R})$ be an arbitrary subsheaf of $C(X, {\Bbb R})$. By $Ser(A)\subset C(X, {\Bbb R})$, we denote the sheaf of all functions which can be locally represented by the absolutely convergent series $\sum P_i(a_1,..., a_n)$, where $a_1,..., a_n$ are sections of $A$ and $P_i$ are polynomials with coefficients in ${\Bbb R}$. By definition, ${\cal O}_{X_{\Bbb R}}= Ser(Re {\cal O}_X)$, where $Re {\cal O}_X$ is a sheaf of real parts of holomorphic functions. Another interesting sheaf associated with $X_{\Bbb R}$ is a sheaf ${\cal O}_{X_{\Bbb R}}\otimes{\Bbb C}\subset C(X,{\Bbb C})$ of complex-valued real analytic functions. Consider the sheaf ${\cal O}_X$ of holomorphic functions on $X$ as a subsheaf of the sheaf $C(X,{\Bbb C})$ of continous ${\Bbb C}$-valued functions on $X$. The sheaf $C(X,{\Bbb C})$ has a natural authomorphism $f{\:\longrightarrow\:} \bar f$, where $\bar f$ is complex conjugation. By definition, the section $f$ of $C(X,{\Bbb C})$ is called {\bf antiholomorphic} if $\bar f$ is holomorphic. Let ${\cal O}_X$ be the sheaf of holomorphic functions, and $\bar {\cal O}_X$ be the sheaf of antiholomorphic functions on $X$. Let ${\cal O}_X \otimes_{\Bbb C} \bar{\cal O}_X \stackrel i{\:\longrightarrow\:} C(X, {\Bbb C})$ be the natural multiplication map. Clearly, the image of $i$ belongs to the subsheaf ${\cal O}_{X_{\Bbb R}}\otimes {\Bbb C}\subset C(X, {\Bbb C})$. \hfill \claim \label{_comple_real_ana_produ_Claim_} The sheaf homomorphism $i:\; {\cal O}_X \otimes_{\Bbb C} \bar{\cal O}_X {\:\longrightarrow\:} {\cal O}_{X_{\Bbb R}}\otimes {\Bbb C}\subset C(X, {\Bbb C})$ is injective. For each point $x\in X$, $i$ induces an isomorphism on $x$-completions of ${\cal O}_X \otimes_{\Bbb C} \bar{\cal O}_X$ and ${\cal O}_{X_{\Bbb R}}\otimes {\Bbb C}$. {\bf Proof:} Well known (see, for instance, \cite{_real_anal_spa:GMT_}). $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$ \hfill Let $\Omega^1({\cal O}_{X_{\Bbb R}})$, $\Omega^1({\cal O}_X \otimes_{\Bbb C} \bar{\cal O}_X)$, $\Omega^1({\cal O}_{X_{\Bbb R}}\otimes {\Bbb C})$ be the sheaves of K\"ahler differentials associated with the corresponding ring sheaves. There are natural sheaf maps \begin{equation} \label{_complexifi_of_Omega_Equation_} \Omega^1({\cal O}_{X_{\Bbb R}})\otimes {\Bbb C} {\:\longrightarrow\:} \Omega^1({\cal O}_{X_{\Bbb R}}\otimes {\Bbb C}) \end{equation} and \begin{equation} \label{_Omega_X_R_and_Omega_X_Equation_} \Omega^1({\cal O}_{X_{\Bbb R}}\otimes {\Bbb C}){\:\longrightarrow\:} \Omega^1({\cal O}_X\otimes_{\Bbb C} \bar {\cal O}_X), \end{equation} correspoding to the monomorphisms \[ {\cal O}_{X_{\Bbb R}}\hookrightarrow {\cal O}_{X_{\Bbb R}}\otimes {\Bbb C}, \;\; {\cal O}_X\otimes_{\Bbb C} \bar {\cal O}_X\hookrightarrow{\cal O}_{X_{\Bbb R}}\otimes {\Bbb C} \] \hfill \claim \label{_differe_real_ana_and_co_ana_Claim_} The map \eqref{_complexifi_of_Omega_Equation_} is an isomorphism. Tensoring both sides of \eqref{_Omega_X_R_and_Omega_X_Equation_} by ${\cal O}_{X_{\Bbb R}}\otimes {\Bbb C}$ produces an isomorphism \[ \Omega^1({\cal O}_X\otimes_{\Bbb C} \bar {\cal O}_X) \bigotimes_{{\cal O}_X\otimes_{\Bbb C} \bar {\cal O}_X}\bigg({\cal O}_{X_{\Bbb R}}\otimes {\Bbb C}\bigg) =\Omega^1({\cal O}_{X_{\Bbb R}}\otimes {\Bbb C}). \] {\bf Proof:} Clear. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$ \hfill According to the general results about differentials (see, for example, \cite{_Hartshorne:Alg_Geom_}, Chapter II, Ex. 8.3), the sheaf $\Omega^1({\cal O}_X\otimes_{\Bbb C} \bar {\cal O}_X)$ admits a canonical decomposition: \[ \Omega^1({\cal O}_X\otimes_{\Bbb C} \bar {\cal O}_X) = \Omega^1({\cal O}_X)\otimes_{\Bbb C} \bar {\cal O}_X \oplus{\cal O}_X\otimes_{\Bbb C}\Omega^1(\bar {\cal O}_X). \] Let $\tilde I$ be an endomorphism of $\Omega^1({\cal O}_X\otimes_{\Bbb C} \bar {\cal O}_X)$ which acts as a multiplication by $\sqrt{-1}\:$ on \[ \Omega^1({\cal O}_X)\otimes_{\Bbb C} \bar {\cal O}_X \subset \Omega^1({\cal O}_X\otimes_{\Bbb C} \bar {\cal O}_X) \] and as a multiplication by $-\sqrt{-1}\:$ on \[ {\cal O}_X\otimes_{\Bbb C}\Omega^1(\bar {\cal O}_X) \subset \Omega^1({\cal O}_X\otimes_{\Bbb C} \bar {\cal O}_X). \] Let $\underline I$ be the corresponding ${\cal O}_{X_{\Bbb R}}\otimes {\Bbb C}$-linear endomorphism of \[ \Omega^1({\cal O}_{X_{\Bbb R}})\otimes {\Bbb C} = \Omega^1({\cal O}_X\otimes_{\Bbb C} \bar {\cal O}_X) \otimes_{{\cal O}_X\otimes_{\Bbb C} \bar {\cal O}_X} \bigg({\cal O}_{X_{\Bbb R}}\otimes {\Bbb C}\bigg). \] As easy check ensures that $\underline I$ is {\it real}, that is, comes from the ${\cal O}_{X_{\Bbb R}}$-linear endomorphism of $\Omega^1({\cal O}_{X_{\Bbb R}})$. Denote this ${\cal O}_{X_{\Bbb R}}$-linear endomorphism by \[ I:\; \Omega^1({\cal O}_{X_{\Bbb R}}){\:\longrightarrow\:} \Omega^1({\cal O}_{X_{\Bbb R}}), \] $I^2=-1$. The endomorphism $I$ is called {\bf a complex structure operator}. In the case when $X$ is smooth, $I$ coinsides with the usual complex structure operator on the cotangent space. \hfill \definition\label{_commu_w_comple_str_Definition_} Let $X$, $Y$ be complex analytic varieties, and \[ f:\; X_{\Bbb R}{\:\longrightarrow\:} Y_{\Bbb R}\] be a morphism of underlying real analytic varieties. Let $f^* \Omega^1_{Y_{\Bbb R}} \stackrel P{\:\longrightarrow\:} \Omega^1_{X_{\Bbb R}}$ be the natural map of sheaves of differentials associated with $f$. Let \[ I_X:\; \Omega^1_{X_{\Bbb R}}{\:\longrightarrow\:} \Omega^1_{X_{\Bbb R}}, \;\;\; I_Y:\; \Omega^1_{Y_{\Bbb R}}{\:\longrightarrow\:} \Omega^1_{Y_{\Bbb R}} \] be the complex structure operators, and \[ f^* I_Y:\; f^*\Omega^1_{Y_{\Bbb R}}{\:\longrightarrow\:} f^*\Omega^1_{Y_{\Bbb R}} \] be ${\cal O}_{X_{\Bbb R}}$-linear automorphism of $f^*\Omega^1_{Y_{\Bbb R}}$ defined as a pullback of $I_Y$. We say that $f$ {\bf commutes with the complex structure} if \begin{equation}\label{_commu_w_comle_Equation_} P\circ f^* I_Y = I_X \circ P. \end{equation} \hfill \theorem \label{_commu_w_comple_str_Theorem_} Let $X$, $Y$ be complex analytic varieties, and \[ f_{\Bbb R}:\; X_{\Bbb R}{\:\longrightarrow\:} Y_{\Bbb R}\] be a morphism of underlying real analytic varieties, which commutes with the complex structure. Then there exist a morphism $f:\; X{\:\longrightarrow\:} Y$ of complex analytic varieties, such that $f_{\Bbb R}$ is its underlying morphism. \hfill {\bf Proof:} By Corollary 9.4, \cite{_Verbitsky:Deforma_}, the map $f$, defined on the sets of points of $X$ and $Y$, is meromorphic; to prove \ref{_commu_w_comple_str_Theorem_}, we need to show it is holomorphic. Let $\Gamma \subset X \times Y$ be the graph of $f$. Since $f$ is meromorphic, $\Gamma$ is a complex subvariety of $X\times Y$. It will suffice to show that the natural projections $\pi_1:\; \Gamma {\:\longrightarrow\:} X$, $\pi_2:\; \Gamma {\:\longrightarrow\:} Y$ are isomorphisms. By \cite{_Verbitsky:Deforma_}, Lemma 9.12, the morphisms $\pi_i$ are flat. Since $f_{\Bbb R}$ induces isomorphism of Zariski tangent spaces, same is true of $\pi_i$. Thus, $\pi_i$ are unramified. Therefore, the maps $\pi_i$ are etale. Since they are one-to-one on points, $\pi_i$ etale implies $\pi_i$ is an isomorphism. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$ \hfill \definition Let $M$ be a real analytic variety, and \[ I:\; \Omega^1({\cal O}_M){\:\longrightarrow\:}\Omega^1({\cal O}_M) \] be an endomorphism satisfying $I^2=-1$. Then $I$ is called {\bf an almost complex structure on $M$}. If there exist a complex analytic structure $\mathfrak C$ on $M$ such that $I$ appears as the complex structure operator associated with $\mathfrak C$, we say that $I$ is {\bf integrable}. \ref{_commu_w_comple_str_Theorem_} implies that this complex structure is unique if it exists. \hfill \definition \label{_hypercomplex_Definition_} (Hypercomplex variety) Let $M$ be a real analytic variety equipped with almost complex structures $I$, $J$ and $K$, such that $I\circ J = -J \circ I = K$. Assume that for all $a, b, c\in {\Bbb R}$, such that $a^2 + b^2 + c^2=1$, the almost complex structure $a I + b J + c K$ is integrable. Then $M$ is called {\bf a hypercomplex variety}. \hfill \noindent \claim \label{_hyperka_hyperco_Claim_} Let $M$ be a hyperk\"ahler manifold. Then $M$ is hypercomplex. {\bf Proof:} Let $I$, $J$ be induced complex structures. We need to identify $(M, I)_{\Bbb R}$ and $(M,J)_{\Bbb R}$ in a natural way. These varieties are canonically identified as $C^\infty$-manifolds; we need only to show that this identification is real analytic. This is \cite{_Verbitsky:Deforma_}, Proposition 6.5. \blacksquare \hfill The following proposition will be used further on in this paper. \hfill \proposition\label{_tange_cone_underly_Proposition_} Let $M$ be a complex variety, $x\in X$ a point, and $Z_xM\subset T_xM$ be the reduction of the Zariski tangent cone to $M$ in $x$, considered as a closed subvariety of the Zariski tangent space $T_xM$. Let $Z_x M_{\Bbb R} \subset T_x M_{\Bbb R}$ be the Zariski tangent cone for the underlying real analytic variety $M_{\Bbb R}$. Then $(Z_x M)_{\Bbb R} \subset (T_x M)_{\Bbb R} = T_x M_{\Bbb R}$ coinsides with $Z_x M_{\Bbb R}$. {\bf Proof:} For each $v\in T_x M$, the point $v$ belongs to $Z_x M$ if and only if there exist a real analytic path $\gamma:\; [0, 1] {\:\longrightarrow\:} M$, $\gamma(0)=x$ satisfying $\frac{d\gamma}{dt}=v$. The same holds true for $Z_x M_{\Bbb R}$. Thus, $v\in Z_x M$ if and only if $v\in Z_x M_{\Bbb R}$. \blacksquare \section{Singular hyperk\"ahler varieties.} \label{_singu_hype_Section_} In this section, we follow \cite{_Verbitsky:Deforma_}, Section 10. For more examples, motivations and reference, the reader is advised to check \cite{_Verbitsky:Deforma_}. \hfill \definition\label{_singu_hype_Definition_} (\cite{_Verbitsky:Hyperholo_bundles_}, Definition 6.5) Let $M$ be a hypercomplex variety (\ref{_hypercomplex_Definition_}). The following data define a structure of {\bf hyperk\"ahler variety} on $M$. \begin{description} \item[(i)] For every $x\in M$, we have an ${\Bbb R}$-linear symmetric positively defined bilinear form $s_x:\; T_x M \times T_x M {\:\longrightarrow\:} {\Bbb R}$ on the corresponding real Zariski tangent space. \item[(ii)] For each triple of induced complex structures $I$, $J$, $K$, such that $I\circ J = K$, we have a holomorphic differential 2-form $\Omega\in \Omega^2(M, I)$. \item[(iii)] Fix a triple of induced complex structure $I$, $J$, $K$, such that $I\circ J = K$. Consider the corresponding differential 2-form $\Omega$ of (ii). Let $J:\; T_x M {\:\longrightarrow\:} T_x M$ be an endomorphism of the real Zariski tangent spaces defined by $J$, and $Re\Omega\restrict x$ the real part of $\Omega$, considered as a bilinear form on $T_x M$. Let $r_x$ be a bilinear form $r_x:\; T_x M \times T_x M {\:\longrightarrow\:} {\Bbb R}$ defined by $r_x(a,b) = - Re\Omega\restrict x (a, J(b))$. Then $r_x$ is equal to the form $s_x$ of (i). In particular, $r_x$ is independent from the choice of $I$, $J$, $K$. \end{description} \noindent \remark \label{_singu_hype_Remark_} \nopagebreak \begin{description} \item[(a)] It is clear how to define a morphism of hyperk\"ahler varieties. \item[(b)] For $M$ non-singular, \ref{_singu_hype_Definition_} is equivalent to the usual one (\ref{_hyperkahler_manifold_Definition_}). If $M$ is non-singular, the form $s_x$ becomes the usual Riemann form, and $\Omega$ becomes the standard holomorphically symplectic form. \item[(c)] It is easy to check the following. Let $X$ be a hypercomplex subvariety of a hyperk\"ahler variety $M$. Then, restricting the forms $s_x$ and $\Omega$ to $X$, we obtain a hyperk\"ahler structure on $X$. In particular, trianalytic subvarieties of hyperk\"ahler manifolds are always hyperk\"ahler, in the sense of \ref{_singu_hype_Definition_}. \end{description} \hfill {\bf Caution:} Not everything which is seemingly hyperk\"ahler satisfies the conditions of \ref{_singu_hype_Definition_}. Take a quotient $M/G$ os a hyperk\"ahler manifold by an action of finite group $G$, acting in accordance with hyperk\"ahler structure. Then $M/G$ is, generally speaking, {\it not} hyperk\"ahler (see \cite{_Verbitsky:Deforma_}, Section 10 for details). \hfill The following theorem, proven in \cite{_Verbitsky:Hyperholo_bundles_} (Theorem 6.3), gives a convenient way to construct examples of hyperk\"ahler varieties. \hfill \theorem \label{_hyperho_defo_hyperka_Theorem_} Let $M$ be a compact hyperk\"ahler manifold, $I$ an induced complex structure and $B$ a stable holomorphic bundle over $(M, I)$. Let $\operatorname{Def}(B)$ be a reduction\footnote{The deformation space might have nilpotents in the structure sheaf. We take its reduction to avoid this.} of the deformation space of stable holomorphic structures on $B$. Assume that $c_1(B)$, $c_2(B)$ are $SU(2)$-invariant, with respect to the standard action of $SU(2)$ on $H^*(M)$. Then $\operatorname{Def}(B)$ has a natural structure of a hyperk\"ahler variety. \nopagebreak \blacksquare \section{Spaces with locally homogeneous singularities} \label{_SLHS_Section_} \noindent \definition (local rings with LHS) Let $A$ be a local ring. Denote by $\mathfrak m$ its maximal ideal. Let $A_{gr}$ be the corresponding associated graded ring. Let $\hat A$, $\hat A_{gr}$ be the $\mathfrak m$-adic completion of $A$, $A_{gr}$. Let $(\hat A)_{gr}$, $(\hat A_{gr})_{gr}$ be the associated graded rings, which are naturally isomorphic to $A_{gr}$. We say that $A$ {\bf has locally homogeneous singularities} (LHS) if there exists an isomorphism $\rho:\; \hat A {\:\longrightarrow\:} \hat A_{gr}$ which induces the standard isomorphism $i:\; (\hat A)_{gr}{\:\longrightarrow\:} (\hat A_{gr})_{gr}$ on associated graded rings. \hfill \definition (SLHS) Let $X$ be a complex or real analytic space. Then $X$ is called {be a space with locally homogeneous singularities} (SLHS) if for each $x\in M$, the local ring ${\cal O}_x M$ has locally homogeneous singularities. \hfill By {\bf system of coordinates} on a complex space $X$, defined in a neighbourhood $U$ of $x\in X$, we understand a closed embedding $U\hookrightarrow B$ where $B$ is an open subset of ${\Bbb C}^n$. Clearly, a system of coordinates can be considered as a set of functions $f_1, ..., f_n$ on $U$. Then $U \subset B$ is defined by a system of equations on $f_1, ... f_n$. \hfill \remark \label{_SLHS_term_expla_Remark_} Let $X$ be a complex space. Assume that for each $x\in X$, there exist a system of coordinates $f_1, ... , f_n$ in a neighbourhood $U$ of $x$, such that $U\subset B$ is defined by a system of homogeneous polynomial equations. Then $X$ is a space with locally homogeneous singularities. This explains the term. \hfill \claim \label{_redu_SLHS_Claim_} Let $X$ be a complex analytic space with locally homogeneous singularities, and $X_r$ its reduction (same space, with structure sheaf factorized by nilradical). Then $X_r$ is also a space with locally homogeneous singularities. {\bf Proof:} Clear. \blacksquare \hfill \lemma \label{_produ_rings_LHS_Lemma_} Let $A_1$, $A_2$ be local rings over ${\Bbb C}$, with $A_i/ {\mathfrak m}_i = {\Bbb C}$, where ${\mathfrak m}_i$ is the maximal ideal of $A_i$. Then $A_1\otimes_{\Bbb C} A_2$ is LHS if and only if $A_1$ and $A_2$ are LHS. \hfill {\bf Proof (``if'' part):} Let $\rho_i:\; \hat A_i {\:\longrightarrow\:} \widehat {(A_i)_{gr}}$ be the maps given by LHS condition. Consider the map \begin{equation} \label{_produ_of_LHS_Equation_} \rho_1 \otimes \rho_2:\; \hat A_i \otimes_{\Bbb C} \hat A_2 {\:\longrightarrow\:} \widehat{(A_i)_{gr}} \otimes_{\Bbb C} \widehat{(A_2)_{gr}}. \end{equation} Denote the functor of adic completions of local rings by $B {\:\longrightarrow\:} \widehat B$. Clearly, $\widehat{\hat A_i \otimes_{\Bbb C} \hat A_2} = \widehat{A_1\otimes_{\Bbb C} A_2}$, and $\widehat{(\hat A_i)_{gr} \otimes_{\Bbb C} (\hat A_2)_{gr}} = \widehat{(A_1)_{gr}\otimes_{\Bbb C} (A_2)_{gr}}$. Plugging these isomorphisms into the completion of both sides of \eqref{_produ_of_LHS_Equation_}, we obtain that a completion of $\rho_1 \otimes \rho_2$ provides an LHS map for $A_1\otimes_{\Bbb C} A_2$. \hfill {\bf ``only if'' part:} Let \[ \rho:\; \widehat{A_1\otimes_{\Bbb C} A_2} {\:\longrightarrow\:} \widehat{((A_1)\otimes_{\Bbb C} (A_2))_{gr}}\] be the LHS map for $A_1\otimes_{\Bbb C} A_2$. There are natural maps \[u:\; \hat A_1 {\:\longrightarrow\:} \widehat{A_1\otimes_{\Bbb C} A_2} \] and \[ v:\; \widehat{((A_1)\otimes_{\Bbb C} (A_2))_{gr}} {\:\longrightarrow\:} (\hat A_1)_{gr}.\] The $u$ comes from the natural embedding $a {\:\longrightarrow\:} a\otimes 1\in A_1\otimes_{\Bbb C} A_2$ and $v$ from the natural surjection $a\otimes b {\:\longrightarrow\:} a \otimes \pi(b) \in A_1\otimes_{\Bbb C} {\Bbb C}$, where $\pi:\; A_2 {\:\longrightarrow\:} {\Bbb C}$ is the standard quotient map. It is clear that $u\circ v$ induces identity on the associated graded ring of $A_1$. \ref{_produ_rings_LHS_Lemma_} is proven. \blacksquare \hfill \proposition\label{_comple_LHS<=>real_LHS_Proposition_} Let $M$ be a complex variety, $M_{\Bbb R}$ the underlying real analytic variety. Then $M_{\Bbb R}$ is a space with locally homogeneous singularities (SLHS) if and only if $M$ is SLHS. {\bf Proof:} By \ref{_comple_real_ana_produ_Claim_}, $\widehat{({\cal O}_x M_{\Bbb R})\otimes {\Bbb C}} = \widehat{{\cal O}_x M \otimes \bar{\cal O}_x M}$. Thus, \ref{_comple_LHS<=>real_LHS_Proposition_} is implied immediately by \ref{_produ_rings_LHS_Lemma_}. \blacksquare \hfill \corollary \label{_hype_SLHS_for_diff_indu_c_str_Corollary_} Let $M$ be a hyperk\"ahler variety, $I_1$, $I_2$ induced complex structures. Then $(M, I_1)$ is a space with locally homogeneous singularities if and only is $(M, I_2)$ is SLHS. {\bf Proof:} The real analytic variety underlying $(M, I_1)$ coinsides with that underlying $(M, I_2)$. Applying \ref{_comple_LHS<=>real_LHS_Proposition_}, we immediately obtain \ref{_hype_SLHS_for_diff_indu_c_str_Corollary_}. \blacksquare \hfill \definition Let $M$ be a hyperk\"ahler variety. Then $M$ is called a space with locally homogeneous singularities (SLHS) if the underlying real analytic variety is SLHS or, equivalently, for some induced complex structure $I$ the $(M, I)$ is SLHS. \hfill \theorem Let $M$ be a compact hyperk\"ahler manifold, $I$ an induced complex structure and $B$ a stable holomorphic bundle over $(M, I)$. Assume that $c_1(B)$, $c_2(B)$ are $SU(2)$-invariant, with respect to the standard action of $SU(2)$ on $H^*(M)$. Let $\operatorname{Def}(B)$ be a reduction of a deformation space of stable holomorphic structures on $B$, which is a hyperk\"ahler variety by \ref{_hyperho_defo_hyperka_Theorem_}. Then $\operatorname{Def}(B)$ is a space with locally homogeneous singularities (SLHS). \hfill {\bf Proof:} Let $x$ be a point of $\operatorname{Def}(B)$, corresponding to a stable holomorphic bundle $B$. In \cite{_Verbitsky:Hyperholo_bundles_}, Section 7, the neighbourhood $U$ of $x$ in $\operatorname{Def}(B)$ was described explicitely as follows. We constructed a locally closed holomorphic embedding $U\stackrel \phi\hookrightarrow H^1(\operatorname{End}(B))$. We proved that $v\in H^1(\operatorname{End}(B))$ belongs to the image of $\phi$ if and only if $v^2 =0$. Here $v^2 \in H^2(\operatorname{End}(B))$ is the square of $v$, taken with respect to the product \[ H^1(\operatorname{End}(B))\times H^1(\operatorname{End}(B)) {\:\longrightarrow\:} H^2(\operatorname{End}(B)) \] associated with the algebraic structure on $\operatorname{End}(B)$. Clearly, the relation $v^2 =0$ is homogeneous. This relation defines a locally closed SLHS subspace $Y$ of $H^1(\operatorname{End}(B))$, such that $\phi(U)$ is its reduction. Applying \ref{_redu_SLHS_Claim_}, we obtain that $\phi(U)$ is also a space with locally homogeneous singularities. \blacksquare \hfill \conjecture \label{_hype_SLHS_Conjecture_} Let $M$ be a hyperk\"ahler variety. Then $M$ is a space with locally homogeneous singularities. \hfill There is a rather convoluted argument which might prove \ref{_hype_SLHS_Conjecture_}. This argument will be a subject of forthcoming paper \cite{_Verbitsky:singuII_}. \section{Tangent cone of a hyperk\"ahler variety} \label{_tange_cone_Section_} Let $M$ be a hyperk\"ahler variety, $I$ an induced complex structure and $Z_x(M,I)$ be a reduction of a Zariski tangent cone to $(M,I)$ in $x\in M$. Consider $Z_x(M,I)$ as a closed subvariety in the Zariski tangent space $T_xM$. The space $T_xM$ has a natural metric and quaternionic structure. This makes $T_xM$ into a hyperk\"ahler manifold, isomorphic to ${\Bbb H}^n$. \hfill \noindent\theorem \label{_cone_hype_Theorem_} Under these assumptions, the following assertions hold: \begin{description} \item[(i)] The subvariety $Z_x(M,I)\subset T_x M$ is independent from the choice of induced complex structure $I$. \item [(ii)] Moreover, $Z_x(M,I)$ is a trianalytic subvariety of $T_x M$. \end{description} {\bf Proof:} \ref{_cone_hype_Theorem_} (i) is implied by \ref{_tange_cone_underly_Proposition_}. By \ref{_cone_hype_Theorem_} (i), the Zariski tangent cone $Z_x(M,I)$ is a hypercomplex subvariety of $TM$. According to \ref{_singu_hype_Remark_} (c), this implies that $Z_x(M)$ is hyperk\"ahler. \blacksquare \hfill Further on, we denote the Zariski tangent cone to a hyperk\"ahler variety by $Z_xM$. The Zariski tangent cone is equipped with a natural hyperk\"ahler structure. \hfill The following theorem shows that the Zariski tangent cone $Z_xM\subset T_x M$ is a union of planes $L_i\subset T_x M$. \hfill \theorem \label{_cone_flat_Theorem_} Let $M$ be a hyperk\"ahler variety, $I$ an induced complex structure and $x\in M$ a point. Consider the reduction of the Zariski tangent cone (denoted by $Z_x M$) as a subvariety of the quaternionic space $T_x M$. Let $Z_x(M, I)= \cup L_i$ be the irreducible decomposition of the complex variety $Z_x(M,I)$. Then \begin{description} \item[(i)] The decomposition $Z_x(M, I)= \cup L_i$ is independent from the choice of induced complex structure $I$. \item[(ii)] For every $i$, the variety $L_i$ is a linear subspace of $T_x M$, invariant under quaternion action. \end{description} {\bf Proof:} Let $L_i$ be an irreducible component of $Z_x(M, I)$, $Z_x^{ns}(M,I)$ be the non-singular part of $Z_x(M,I)$, and $L_i^{ns}:=Z_x^{ns}(M,I) \cap L_i$. Then $L_i$ is a closure of $L_i^{ns}$ in $T_xM$. Clearly from \ref{_cone_hype_Theorem_}, $L_i^{ns}(M)$ is a hyperk\"ahler submanifold in $T_xM$. By \ref{_triana_comple_geo_Proposition_}, $L_i^{ns}$ is totally geodesic. A totally geodesic submanifold of a flat manifold is again flat. Therefore, $L_i^{ns}$ is an open subset of a linear subspace $\tilde L_i\subset T_xM$. Since $L_i^{ns}$ is a hyperk\"ahler submanifold, $\tilde L_i$ is invariant with respect to quaternions. The closure $L_i$ of $L_i^{ns}$ is a complex analytic subvariety of $T_x(M,I)$. Therefore, $\tilde L_i = L_i$. This proves \ref{_cone_flat_Theorem_} (ii). From the above argument, it is clear that $Z_x^{ns}(M,I)= \coprod L_i^{ns}$ (disconnected sum). Taking connected components of $Z_x^{ns}M$ for each induced complex structure, we obtain the same decomposition $Z_x(M, I)= \cup L_i$, with $L_i$ being closured of connected components. This proves \ref{_cone_flat_Theorem_} (ii). \blacksquare \hfill \corollary\label{_normali_smooth_Corollary_} Let $M$ be a hyperk\"ahler variety, and $I$ an induced complex structure. Assume that $M$ is a space with locally homogeneous singularities. Then the normalization of $(M,I)$ is smooth. {\bf Proof:} The normalization of $Z_xM$ is smooth by \ref{_cone_flat_Theorem_}. The normalization is compatible with the adic completions (\cite{_Matsumura:Commu_Alge_}, Chapter 9, Proposition 24.E). Therefore, the integral closure of the completion of ${\cal O}_{Z_xM}$ is a regular ring. Now, from the definition of locally homogeneous intersections, it follows that the integral closure of ${\cal O}_xM\check{\;}$ is also a regular ring, where ${\cal O}_xM\check{\;}$ is an adic completion of the local ring of holomorphic functions on $(M, I)$ in a neighbourhood of $x$. Applying \cite{_Matsumura:Commu_Alge_}, Chapter 9, Proposition 24.E again, we obtain that the integral closure of ${\cal O}_x M$ is regular. This proves \ref{_normali_smooth_Corollary_} \blacksquare \section{Desingularization of hyperk\"ahler varieties} \label{_desingu_Section_} \noindent\theorem \label{_desingu_Theorem_} Let $M$ be a hyperk\"ahrler or a hypercomplex variety. Assume that $M$ is a space with locally homogeneous singularities, and $I$ an induced complex structure. Let \[ \widetilde{(M, I)}\stackrel n{\:\longrightarrow\:} (M,I)\] be the normalization of $(M,I)$. Then $\widetilde{(M, I)}$ is smooth and has a natural hyperk\"ahler structure $\c H$, such that the associated map $n:\; \widetilde{(M, I)} {\:\longrightarrow\:} (M,I)$ agrees with $\c H$. Moreover, the hyperk\"ahler manifold $\tilde M:= \widetilde{(M, I)}$ is independent from the choice of induced complex structure $I$. \hfill {\bf Proof:} The variety $\widetilde{(M, I)}$ is smooth by \ref{_normali_smooth_Corollary_}. Let $x\in M$, and $U\subset M$ be a neighbourhood of $x$. Let ${\mathfrak R}_x(U)$ be the set of irreducible components of $U$ which contain $x$. There is a natural map $\tau: {\mathfrak R}_x(U) {\:\longrightarrow\:} Irr(Spec {\cal O}_xM\check{\;})$, where $Irr(Spec {\cal O}_xM\check{\;})$ is a set of irreducible components of $Spec {\cal O}_xM\check{\;}$, where ${\cal O}_xM\check{\;}$ is a completion of ${\cal O}_xM$ in $x$. Since ${\cal O}_x M$ is Henselian (\cite{_Raynaud_}, VII.4), there exist a neighbourhood $U$ of $x$ such that $\tau: {\mathfrak R}_x(U) {\:\longrightarrow\:} Irr(Spec {\cal O}_xM\check{\;})$ is a bijection. Fix such an $U$. Since $M$ is a space locally with locally homogeneous singularities, the irreducible decomposition of $U$ coinsides with the irreducible decomposition of the tangent cone $Z_x M$. Let $\coprod U_i \stackrel u {\:\longrightarrow\:} U$ be the morphism mapping a disjoint union of irreducible components of $U$ to $U$. By \ref{_cone_flat_Theorem_}, the $x$-completion of ${\cal O}_{U_i}$ is regular. Shrinking $U_i$ if necessary, we may assume that $U_i$ is smooth. Then, the morphism $u$ coinsides with the normalization of $U$. For each variety $X$, we denote by $X^{ns}\subset X$ the set of non-singular points of $X$. Clearly, $u(U_i) \cap U^{ns}$ is a connected component of $U^{ns}$. Therefore, $u(U_i)$ is trianalytic in $U$. By \ref{_singu_hype_Remark_} (c), $U_i$ has a natural hyperk\"ahler structure, which agrees with the map $u$. This gives a hyperk\"ahler structure on the normalization $\tilde U := \coprod U_i$. Gluing these hyperk\"ahler structures, we obtain a hyperk\"ahler structure $\c H$ on the smooth manifold$\widetilde{(M, I)}$. Consider the normalization map $n:\; \widetilde{(M, I)} {\:\longrightarrow\:} M$, and let $\tilde M^{n}:= n^{-1}(M^{ns})$. Then, $n\restrict{\tilde M^{n}} \tilde M^{n}{\:\longrightarrow\:} M^{ns}$ is a finite covering which is compatible with the hyperk\"ahler structure. Thus, $\c H\restrict{\tilde M^{n}}$ can be obtained as a pullback from $M$. Clearly, a hyperk\"ahler structure on a manifold is uniquely defined by its restriction to an open dense subset. We obtain that $\c H$ is independent from the choice of $I$. \blacksquare \hfill \remark The desingularization argument works well for hypercomplex varieties. The word ``hyperk\"ahler'' in this article can be in most cases replaced by ``hypercomplex'', because we never use the metric structure. \hfill {\bf Acknowledgements:} It is a pleasure to acknowledge the help of P. Deligne, who pointed out an error in the original argument. Deligne also suggested the term ``locally homogeneous singularities''. I am grateful to A. Beilinson, D. Kaledin, D. Kazhdan, T. Pantev and S.-T. Yau for enlightening discussions.

9611

alg-geom/9611014

en

https://arxiv.org/abs/alg-geom/9611014

alg-geom/9611014

Klaus Altmann

Klaus Altmann and Arne B. Sletsjoe

Andre-Quillen cohomology of monoid algebras

LaTeX 2.09, uses pb-diagram.sty, 8 pages

We compute the Andre-Quillen cohomology of an affine toric variety. The best results are obtained either in the general case for the first three cohomology groups, or in the case of isolated singularities for all cohomology groups, respectively.

alg-geom

\section{Introduction}\label{Int} \neu{Int-1} Let $k$ be a field. For any finitely generated $k$-algebra $A$ the so-called cotangent complex yielding the Andr\'{e}-Quillen cohomology $T^n_A=T^n(A,A;k)$ ($n\geq 0$) may be defined. The first three of these $A$-modules are important for the deformation theory of $A$ or its geometric equivalent $\mbox{\rm Spec} A$: $T^1_A$ equals the set of infinitesimal deformations, $T^0_A$ describes their automorphisms, and $T^2_A$ contains the obstructions for lifting infinitesimal deformations to larger base spaces. Apart from occurring in long exact sequences no meaning of the higher cohomology groups seems to be known when studying the deformation theory of closed subsets of $\mbox{\rm Spec} A$. A very readable reference for the definition of Andr\'{e}-Quillen cohomology and its relations to Hochschild and Harrison cohomology is Loday's book \cite{Loday}. For applications in deformation theory see for instance \cite{LaudalSLN}, \cite{Pa}, or the summary of the properties one has to know (without proofs) in the first section of \cite{BeCh}. \par \neu{Int-2} For smooth $k$-algebras $A$, the higher $T^n_A$ (i.e.\ $n\geq 1$) vanish. For complete intersections the situation is still easy; $T^0_A$ and $T^1_A$ are well understood, and the remaining cohomology groups vanish. As far as we know, only few further examples exist where the cotangent complex or at least the Andr\'{e}-Quillen cohomology groups are known. Palamodov has told us that he has computed (unpublished) the cotangent complex of an embedded point on a line; it turned out that the Poincar\'{e} series $T(s):=\sum_{n\geq 0}(\dim_k T^n)\cdot s^n$ of this singularity is a rational function. It would be interesting to know whether this is always the case for isolated singularities.\\ The result of the present paper is a spectral sequence converging to the Harrison (or Andr\'{e}-Quillen) cohomology for affine toric varieties. In the case of an isolated singularity, this spectral sequence degenerates; this leads to a down to earth description of the modules $T_A^n$. Moreover, in the general case, the information is still sufficient to determine $T_A^0$, $T_A^1$, and $T^2_A$. That is, we use the methods of \cite{Sl} to obtain straight formulas generalizing part of the results of \cite{T2}. \par \neu{Int-3} The paper is organized as follows: We begin in \S \ref{iHc} with fixing notation and recalling those facts of \cite{Sl} that will be used in the following. The sections \S \ref{sps} and \S \ref{E1l} contain the main theorems of the paper: First, we state our spectral sequence; then, in \S \ref{E1l}, we calculate some of the $E_1$-terms and show the vanishing of others. Finally, in \S \ref{app}, we present the resulting $T^n_A$-formulas promised before. \par \neu{Int-4} {\em Acknowledgements:} The first author became interested in describing the algebra cohomology of monoid algebras during his visit at the Senter for h\o{}yere studier (Academy of Sciences) in Oslo. He liked the warm and stimulating atmosphere very much, and would like to thank those who made it possible for him to stay there. Moreover, we are grateful to R.-O.~Buchweitz, J.A.~Christophersen, and A.~Laudal for many helpful discussions. \par \section{Inhomogeneous Harrison cohomology}\label{iHc} \neu{iHc-1} Let $M$, $N$ be mutually dual, finitely generated, free Abelian groups; we denote by $M_{{I\!\!R}}$, $N_{{I\!\!R}}$ the associated real vector spaces obtained via base change with ${I\!\!R}$. Assume we are given a rational, polyhedral cone $\sigma=\langle a^1,\dots,a^m\rangle \subseteq N_{{I\!\!R}}$ with apex in $0$ and with $a^1,\dots,a^m\in N$ denoting its {\em primitive} fundamental generators (i.e.\ none of the $a^i$ is a proper multiple of an element of $N$). We define the dual cone $\sigma^{{\scriptscriptstyle\vee}}:= \{ r\in M_{{I\!\!R}}\,|\; \langle \sigma,\,r\rangle \geq 0\} \subseteq M_{{I\!\!R}}$ and denote by $\Lambda:=\sigma^{{\scriptscriptstyle\vee}}\cap M$ the resulting monoid of lattice points.\\ The corresponding monoid algebra $A:= k[\Lambda]$ will be the object of the upcoming investigations. It is the ring of regular functions on the toric variety $Y_\sigma= \mbox{\rm Spec} A$ associated to $\sigma$. The ring $A$ itself as well as most of its important modules (such as $T^n_A$) admit an $M$-(multi)grading. It is this grading which will make computations possible. For general facts concerning toric varieties see for instance \cite{Oda}. \par \neu{iHc-2} The following definitions are taken from \cite{Sl}, \S 2. We note that the original requirement that the monoids involved have no non-trivial subgroups is unnecessary for our purposes. \par {\bf Definition:} {\em $L\subseteq \Lambda$ is said to be monoid-like if for all elements $\lambda_1,\lambda_2\in L$ the relation $\lambda_1-\lambda_2\in \Lambda_+:=\Lambda\setminus\{0\}$ implies $\lambda_1-\lambda_2\in L$.\\ Moreover, a subset $L_0\subseteq L$ of a monoid-like set is called full if $(L_0 + \Lambda)\cap L =L_0$. } \par For any subset $P\subseteq \Lambda$ and $n\geq 1$ we introduce $S_n(P):= \{(\lambda_1,\dots,\lambda_n)\in P^n\,|\; \sum_v \lambda_v \in P\}$. If $L_0\subset L$ are as in the previous definition, then this gives rise to the following set: \[ C^n(L\setminus L_0,L;k):= \{\varphi: S_n(L)\to k\,|\; \varphi \mbox{ is shuffle invariant and vanishes on } S_n(L\setminus L_0)\}\,. \] ($\varphi$ is said to be shuffle invariant if it is invariant under $\sum_\pi \mbox{sgn}(\pi)\cdot \pi$ where $\pi$ runs through all shuffles of the set $\{1,\dots,n\}$.) These $k$-vector spaces turn into a complex with the differential $\delta^n: C^{n-1}(L\setminus L_0,L;k)\to C^n(L\setminus L_0,L;k)$ defined via \[ (\delta^n\varphi)(\lambda_1,\dots,\lambda_n):= \varphi(\lambda_2,\dots,\lambda_n) + \sum_{v=1}^{n-1} (-1)^v \varphi(\lambda_1,\dots,\lambda_v+\lambda_{v+1},\dots, \lambda_n) + (-1)^n \varphi(\lambda_1,\dots,\lambda_{n-1})\,. \] {\bf Definition:} {\em The $k$-vector space $H\!A^n(L\setminus L_0,L;k):=H^n\big(C^{\scriptscriptstyle\bullet}(L\setminus L_0,L;k)\big)$ is called the $n$-th inhomogeneous Harrison cohomology of the pair $(L,L_0)$. } \par \neu{iHc-3} {\bf Theorem} (\cite{Sl}): {\em Let $R\in M$. Then, defining $\Lambda_+:=\Lambda\setminus\{0\}$, the homogeneous part of $T^n_A$ in degree $-R$ equals \[ T^n_A(-R)= H\!A^{n+1}\big(\Lambda_+\setminus (R+\Lambda),\,\Lambda_+;\,k\big) \quad\mbox{ for } n\geq 0. \vspace{-3ex} \] } \par The proof of the theorem is spread throughout the first two sections of \cite{Sl}: First, in (1.13), (1.14), the calculation of $T^n_A$ has been reduced down to the monoid level. Then, Proposition (2.9) shows that the homogeneous pieces $T^n_A(-R)$ equal the so-called graded Harrison cohomology groups $\mbox{Harr}^{n+1,-R}(\Lambda,k[\Lambda])$, and, finally, Theorem (2.10) states the above result. \par \section{The spectral sequence}\label{sps} \neu{sps-1} With the notation of \zitat{iHc}{1} we define for any face $\tau\leq\sigma$ and any degree $R\in M$ the monoid-like set \[ K_\tau^R:= \Lambda_+ \cap \big(R-\mbox{\rm int}\, \tau^{\scriptscriptstyle\vee}\big)\,. \] These sets admit the following elementary properties: \begin{itemize}\vspace{-2ex} \item[(i)] $K_0^R=\Lambda_+$, and $K_i^R:=K_{a^i}^R=\{r\in\Lambda_+\,|\; \langle a^i,r\rangle <\langle a^i,R\rangle\}$ with $i=1,\dots,m$. \item[(ii)] For $\tau\neq 0$ the equality $K_\tau^R=\bigcap_{a^i\in\tau}K_i^R$ holds. Moreover, if $\sigma$ is a top-dimensional cone, $K_\sigma^R=\Lambda_+ \cap \big(R-\mbox{\rm int}\, \sigma^{\scriptscriptstyle\vee}\big)$ is a (diamond shaped) finite set. \item[(iii)] $\Lambda_+\setminus (R+\Lambda)=\bigcup_{i=1}^m K_i^R$. \vspace{-1ex}\end{itemize} \neu{sps-2} Let us fix an element $R\in M$. The dependence of the sets $K^R_\tau$ on $\tau$ is a contravariant functor. This gives rise to the complexes $C^q(K^R_{\scriptscriptstyle \bullet};k)$ ($q\geq 1$) defined as \[ C^q(K^R_p;k):= \oplus_{\,\tau\leq\sigma,\, \mbox{\footnotesize dim}\hspace{0.1em} \tau=p\,} C^q(K^R_\tau;k)\qquad (0\leq p\leq \dim \sigma) \] with $C^q(K^R_\tau;k):= C^q(K^R_\tau,K^R_\tau;k)$ and the obvious differentials $d^p:C^q(K^R_{p-1};k)\to C^q(K^R_p;k)$. (One has to use the maps $C^q(K^R_\tau;k)\to C^q(K^R_{\tau^\prime};k)$ for any pair $\tau\leq\tau^\prime$ of $(p-1)$- and $p$-dimensional faces, respectively. The only problem might be the sign; it arises from comparison of the (pre-fixed) orientations of $\tau$ and $\tau^\prime$.) Our complex begins as \[ 0\to C^q(\Lambda_+;k)\to \oplus_{i=1}^m C^q(K_i^R;k)\to \oplus_{\langle a^i,a^j\rangle\leq\sigma} C^q(K^R_{ij};k)\to \oplus_{\,\mbox{\footnotesize dim}\hspace{0.1em} \tau=3\,} C^q(K^R_\tau;k)\to \cdots\,. \] \par {\bf Lemma:} {\em The canonical $k$-linear map $C^q(\Lambda_+\setminus (R+\Lambda),\,\Lambda_+;\,k)\to C^q(K^R_{\scriptscriptstyle \bullet};k)$ is a quasiisomorphism, i.e.\ a resolution of the first vector space. } \par {\bf Proof:} For an $r\in\Lambda_+\subseteq M$ we define the $k$-vector space \[ V^q(r):= \big\{\varphi:\{\underline{\lambda}\in\Lambda_+^q\,|\, \mbox{$\sum_v$}\lambda_v=r\} \to k\,\big|\; \varphi \mbox{ is shuffle invariant}\big\}\,. \] Then, our complex splits into a direct product over $r\in\Lambda_+$. Its homogeneous factors equal \[ 0\to V^q(r) \to V^q(r)^{\{i\,|\; r\in K^R_i\}}\to V^q(r)^{\{\tau\leq\sigma\,|\; \mbox{\footnotesize dim}\hspace{0.1em}\tau=2;\,r\in K^R_\tau\}}\to V^q(r)^{\{\tau\leq\sigma\,|\; \mbox{\footnotesize dim}\hspace{0.1em}\tau=3;\,r\in K^R_\tau\}}\to\cdots\,. \] On the other hand, denoting by $H^+_{r,R}$ the halfspace $H^+_{r,R} := \{ a\in N_{{I\!\!R}}\,|\; \langle a,r\rangle < \langle a, R\rangle\} \subseteq N_{{I\!\!R}}$, the relation $r\in K^R_\tau$ is equivalent to $\tau \setminus \{0\} \subseteq H^+_{r,R}$. Hence, the complex for computing the reduced cohomology of the topological space $\bigcup_{\tau\setminus\{0\}\subseteq H^+_{r,R}} \!\big(\tau\setminus \{0\}\big)\subseteq \sigma$ equals \[ 0\to k \to k^{\{i\,|\; r\in K^R_i\}}\to k^{\{\tau\leq\sigma\,|\; \mbox{\footnotesize dim}\hspace{0.1em}\tau=2;\,r\in K^R_\tau\}}\to k^{\{\tau\leq\sigma\,|\; \mbox{\footnotesize dim}\hspace{0.1em}\tau=3;\,r\in K^R_\tau\}}\to\cdots \] if $\sigma\cap H^+_{r,R}\neq\emptyset$, i.e.\ $r\in\bigcup_i K^R_i$; it is trivial otherwise. Since $\bigcup_{\tau\setminus\{0\}\subseteq H^+_{r,R}} \!\big(\tau\setminus \{0\}\big)$ is contractible, this complex is always exact. Thus, $C^q(K^R_{\scriptscriptstyle \bullet};k)= \prod_{r\in\Lambda_+} V^q(r)^{\{\tau\leq\sigma\,|\; \mbox{\footnotesize dim}\hspace{0.1em}\tau={\scriptscriptstyle \bullet};\,r\in K^R_\tau\}}$ has $\prod_{r\in\Lambda_+\setminus(\cup_i K^R_i)}V^q(r)= C^q(\Lambda_+\setminus (R+\Lambda),\,\Lambda_+;\,k)$ as cohomology in $0$, and it is exact elsewhere. \hfill$\Box$ \par \neu{sps-3} Combining the differentials $d^p$ from \zitat{sps}{2} and $\delta^q$ from \zitat{iHc}{2}, we obtain a double complex $C^{\scriptscriptstyle \bullet}(K^R_{\scriptscriptstyle \bullet};k)$ ($0\leq p\leq \dim\sigma;\;q\geq 1$). \par {\bf Theorem:} {\em The Andr\'{e}-Quillen cohomology of $A=k[\Lambda]$ equals the cohomology of the total complex, that is \[ T^n_A(-R)= H^{n+1}\big(\mbox{\rm tot}^{\scriptscriptstyle \bullet} \big[C^{\scriptscriptstyle \bullet}(K^R_{\scriptscriptstyle \bullet};k)\big]\big) \quad\mbox{for } n\geq 0. \] Moreover, given an element $s\in\Lambda$, the multiplication $[\cdot x^s]:T^n_A(-R)\to T^n_A(-R+s)$ is given by the homomorphism $C^{\scriptscriptstyle \bullet}(K^R_{\scriptscriptstyle \bullet};k)\to C^{\scriptscriptstyle \bullet}(K^{R-s}_{\scriptscriptstyle \bullet};k)$ assigned to the inclusions $K^{R-s}_\tau\subseteq K^R_\tau$. } \par {\bf Proof:} The first part is a straightforward consequence of Theorem \zitat{iHc}{3} and the previous lemma -- just use the corresponding spectral sequence of the double complex. For the $A$-module structure of $T^n_A$, one easily observes from the proofs in \cite{Sl} that the multiplication with $x^s$ arises from the complex homomorphism $C^{\scriptscriptstyle\bullet}(\Lambda_+\setminus (R+\Lambda),\,\Lambda_+;\,k) \to C^{\scriptscriptstyle\bullet}(\Lambda_+\setminus ([R-s]+\Lambda),\,\Lambda_+;\,k)$ provided by the inclusion $R+\Lambda\subseteq [R-s]+\Lambda$. \hfill$\Box$ \par \neu{sps-4} \hspace*{-0.5em}{\bf Corollary:} {\em There is a spectral sequence $E_1^{p,q}=\oplus_{\mbox{\footnotesize dim}\hspace{0.1em}\tau=p} H\!A^q(K^R_\tau;k) \Longrightarrow T_A^{p+q-1}(-R)$. } \par {\bf Proof:} This is the other spectral sequence associated to the double complex $C^{\scriptscriptstyle \bullet}(K^R_{\scriptscriptstyle \bullet};k)$. \hfill$\Box$ \par \section{The $E_1$-level}\label{E1l} \neu{E1l-1} {\bf Definition:} Let $K\subseteq M$ be an arbitrary subset of the lattice $M$. A function $f:K\to k$ is called {\em quasilinear} if $f(r)+f(s)=f(r+s)$ for any $r$ and $s$ with $r, s, r+s\in K$. The vector space of quasilinear functions is denoted by $\ko{\mbox{\rm Hom}}\,(K,k)$. \par Recalling the differential $\delta^2:C^1(K^R_\tau)\to C^2(K^R_\tau)$ from \zitat{iHc}{2} shows that the $E_1^{{\scriptscriptstyle \bullet},1}$-summands $H\!A^1(K^R_\tau;k)$ equal $\ko{\mbox{\rm Hom}}\,(K^R_\tau,k)$. \par \neu{E1l-2} The orbits of the torus acting on $Y_\sigma$ are parametrized by the faces of $\sigma$; the singular locus of $Y_\sigma$ is the disjoint union of some of these orbits. We call a face $\tau\leq\sigma$ smooth if our toric variety is smooth along $\mbox{\rm orb}(\tau)$. It is one of the basic facts that smooth faces are characterized by being generated from a part of a ${Z\!\!\!Z}$-basis of $N$. In particular, $0$ and the one-dimensional faces are always smooth. \par {\bf Proposition:} {\em If $\tau\leq\sigma$ is a smooth face, then the injections $\mbox{\rm Hom}_k(\mbox{\rm span}_k K^R_\tau,\,k)\hookrightarrow \ko{\mbox{\rm Hom}}\,(K^R_\tau,k)$ are even isomorphisms. Moreover, $\mbox{\rm span}_k K^R_\tau =\bigcap_{a^i\in\tau}\mbox{\rm span}_k K^R_i$, and the latter vector spaces equal $\mbox{\rm span}_k K^R_i= M_k$, $(a^i)^\bot$, or $0\,$ if $\langle a^i,R\rangle \geq 2$, $=1$, or $\leq 0$, respectively. } \par {\bf Proof:} Let $f:K_\tau^R\to k$ be quasilinear; it suffices to extend $f$ to a ${Z\!\!\!Z}$-linear map defined on $\mbox{\rm span}_{Z\!\!\!Z} K^R_\tau\subseteq M$. If $R$ was non-positive on any of the generators of $\tau$, then $K^R_\tau$ would be empty anyway. Hence, if (w.l.o.g.) $\tau=\langle a^1,\dots,a^k\rangle$, we may assume that $\langle a^i,R\rangle \geq 2$ for $i=1,\dots,l$ and $\langle a^j,R\rangle = 1$ for $j=l+1,\dots,k$.\\ $K^R_\tau$ contains the easy part $\tau^\bot\cap\Lambda_+$; it is no problem at all to extend $f_{|\tau^\bot\cap\Lambda_+}$ to a ${Z\!\!\!Z}$-linear function defined on $\tau^\bot\cap M$. In general, we have to show that for elements $s^v\in K^R_\tau$ the value $\sum_v f(s^v)$ only depends on $s:=\sum_v s^v$, not on the summands themselves. (Then, $f(s)$ may be defined as this value.)\\ By smoothness of $\tau$ there exist elements $r^1,\dots,r^l\in K^R_\tau$ such that $\langle a^i, r^j\rangle =\delta_{ij}$ for $1\leq i\leqk$ and $1\leqj\leql$. Hence, quasilinearity of $f$ implies \[ f(s^v)=\sum_{i=1}^l \langle a^i, s^v\rangle f(r^i) + f(p^v) \quad \mbox{ with }\; p^v:= s^v-\mbox{$\sum_i$}\langle a^i,s^v\rangle r^i \in \tau^\bot\cap M\,. \] Summing up yields \[ \sum_v f(s^v)=\sum_{i=1}^l \langle a^i, s\rangle f(r^i) + \sum_v f(p^v) = \sum_{i=1}^l \langle a^i, s\rangle f(r^i) + f\big(s-\mbox{$\sum_i$}\langle a^i,s\rangle r^i\big)\,. \] Finally, the second claim follows by $\bigcap_{a^i\in\tau}\mbox{\rm span}_k K^R_i = \bigcap_{j=l+1}^k (a^j)^\bot =\mbox{\rm span}_k\big(\tau^\bot; r^1,\dots,r^l\big)= \mbox{\rm span}_k K^R_\tau$. \hfill$\Box$ \par \neu{E1l-3} We turn to the remaining part of the first level and show the vanishing of $E_1^{p,\geq 2}$ if $Y_\sigma$ is smooth in codimension $p$: \par {\bf Theorem:} {\em If $\tau\leq\sigma$ is a smooth face, then $H\!A^q(K^R_\tau;k)=0$ for $q\geq 2$. } {\bf Proof:} We proceed by induction on $\dim \tau$, i.e.\ we may assume that the vanishing holds for all proper faces of $\tau$. Let $r(\tau)$ be an arbitrary element of $\mbox{\rm int}(\sigma^{\scriptscriptstyle\vee}\cap\tau^\bot)\cap M$, i.e.\ $\tau=\sigma\cap [r(\tau)]^\bot$. Then, via $R_g:=R-g\cdot r(\tau)$ with $g\in{Z\!\!\!Z}$, one obtains an infinite (if $\tau\neq\sigma$) series of degrees admitting the following two properties: \begin{itemize}\vspace{-2ex} \item[(i)] $K^{R_g}_\tau=K^R_\tau$ for any $g\in{Z\!\!\!Z}$ (since $R_g=R$ on $\tau$), and \item[(ii)] $K^{R_g}_{\tau^\prime}\neq\emptyset$ implies $\tau^\prime\leq\tau$ for any face $\tau^\prime\leq\sigma$ and $g\gg 0$ (since $\langle a^j,R_g\rangle\leq 0$ if $a^j\notin\tau$). \vspace{-1ex}\end{itemize} In particular, in degree $-R_g$ with $g\gg 0$, the first level of our spectral sequence is shaped as follows: \begin{itemize}\vspace{-2ex} \item For $p<\dim\tau$ only $H\!A^q(K^R_{\tau^\prime};k)$ with $\tau^\prime\leq\tau$ appear as summands of $E_1^{p,q}$. By the induction hypothesis they even vanish for $q\geq 2$, \item for $p=\dim\tau$ it follows that $E_1^{p,q}=H\!A^q(K^R_\tau;k)$, and \item all vector spaces $E_1^{p,q}$ vanish beyond the $[p=\dim\tau]$--line. \vspace{-1ex}\end{itemize} Hence, the differentials $d_r:E_r^{p,q}\to E_r^{p+r,q-r+1}$ are trivial for $r\geq 1$, $q\geq 2$, and we obtain \[ T_A^{q+\dim\tau-1}(-R_g)=H\!A^q(K^R_\tau;k) \quad\mbox{ for }\; g\gg 0\,. \] Moreover, under this identification, the multiplication \[ [\cdot x^{r(\tau)}]: T_A^{q+\dim\tau-1}(-R_g) \to T_A^{q+\dim\tau-1}(-R_{g+1}) \] is just the identity map. On the other hand, we may restrict $T^n_\sigma:=T^n_A$ onto the {\em smooth}, open subset $Y_\tau:=\mbox{\rm Spec}\, k[\tau^{\scriptscriptstyle\vee}\cap M]\subseteq Y_\sigma$. Since $k[\tau^{\scriptscriptstyle\vee}\cap M]$ equals the localization of $k[\sigma^{\scriptscriptstyle\vee}\cap M]$ by the element $x^{r(\tau)}$, we obtain \[ T^n_\sigma \otimes_{k[\sigma^{\scriptscriptstyle\vee}\cap M]} k[\sigma^{\scriptscriptstyle\vee}\cap M]_{x^{r(\tau)}} =T^n_\tau=0\quad \mbox{ for }\; n\geq 1. \] In particular, any element of $T_A^{q+\dim\tau-1}(-R_g) \subseteq T_A^{q+\dim\tau-1}$ will be killed by some power of $x^{r(\tau)}$; but this means $H\!A^q(K^R_\tau;k)=0$. \hfill$\Box$ \par \neu{E1l-4} {\bf Corollary:} {\em $E_1^{0,q}=E_1^{1,q}=0\,$ for $q\geq 2$. } \par \section{Applications}\label{app} \neu{app-1} The main ingredient for describing the Andr\'{e}-Quillen cohomology of $Y_\sigma$ will be the complex $\ko{\mbox{\rm Hom}}\,(K^R_{\scriptscriptstyle \bullet},k) = (E_1^{{\scriptscriptstyle \bullet},1},\,d_1)$ which is built from the vector spaces $\ko{\mbox{\rm Hom}}\,(K^R_\tau,k)$ as $C^q(K^R_{\scriptscriptstyle \bullet};k)$ was from $C^q(K^R_\tau;k)$ in \zitat{sps}{2}. It contains $(\mbox{\rm span}_k K^R_{\scriptscriptstyle \bullet})^\ast$ as a subcomplex. \par \neu{app-2} First, we discuss the case of an {\em isolated singularity} $Y_\sigma$. Here, our spectral sequence degenerates completely. \par \begin{center} \unitlength=1.0mm \linethickness{0.4pt} \begin{picture}(150.00,60.00) \put(20.00,10.00){\line(0,1){50.00}} \put(20.00,10.00){\line(1,0){130.00}} \put(120.00,20.00){\line(0,1){40.00}} \put(100.00,60.00){\line(0,-1){40.00}} \put(120.00,20.00){\line(0,-1){10.00}} \put(20.00,20.00){\line(1,0){80.00}} \put(10.00,15.00){\makebox(0,0)[cc]{$q=1$}} \put(10.00,25.00){\makebox(0,0)[cc]{$q=2$}} \put(10.00,35.00){\makebox(0,0)[cc]{$q=3$}} \put(10.00,55.00){\makebox(0,0)[cc]{$\vdots$}} \put(30.00,5.00){\makebox(0,0)[cc]{$p=0$}} \put(31.00,15.00){\makebox(0,0)[cc]{$\ko{\mbox{\rm Hom}}\,(K^R_0,k)$}} \put(55.00,5.00){\makebox(0,0)[cc]{$p=1$}} \put(55.00,15.00){\makebox(0,0)[cc]{$\ko{\mbox{\rm Hom}}\,(K^R_1,k)$}} \put(110.00,5.00){\makebox(0,0)[cc]{$p=\dim\tau$}} \put(110.00,35.00){\makebox(0,0)[cc]{$\vdots$}} \put(80.00,5.00){\makebox(0,0)[cc]{$\dots$}} \put(80.00,15.00){\makebox(0,0)[cc]{$\dots$}} \put(110.00,55.00){\makebox(0,0)[cc]{$H\!A^q(K^R_\sigma;k)$}} \put(60.00,30.00){\makebox(0,0)[cc]{$0$}} \put(135.00,30.00){\makebox(0,0)[cc]{$0$}} \put(50.00,55.00){\vector(3,-1){40.00}} \put(73.00,51.00){\makebox(0,0)[cc]{$d_r$}} \end{picture} \vspace{-5ex} \end{center} {\bf Proposition:} {\em Let $Y_\sigma$ be an isolated singularity. Then, the Andr\'{e}-Quillen cohomology in degree $-R$ equals \vspace{-2ex} \[ T^n_A(-R)= \renewcommand{\arraystretch}{1.3} \left\{ \begin{array}{ll} H^n\big(\ko{\mbox{\rm Hom}}\,(K^R_{\scriptscriptstyle \bullet},k)\big)= H^n\big((\mbox{\rm span}_k K^R_{\scriptscriptstyle \bullet})^\ast\big) & \mbox{for }\, 0\leq n\leq \dim\sigma-1\\ H^{\dim \sigma}\big(\ko{\mbox{\rm Hom}}\,(K^R_{\scriptscriptstyle \bullet},k)\big) & \mbox{for }\, n= \dim\sigma\\ H\!A^{n-\dim\sigma +1}(K^R_\sigma;k) & \mbox{for }\, n\geq \dim\sigma+1\,. \end{array}\right. \vspace{-2ex} \] } \par {\bf Proof:} The first level of the spectral sequence is non-trivial only in $E_1^{p,1}$ with $0\leq p\leq \dim\sigma$ and $E_1^{\dim\sigma,q}$ with $q\geq 1$, respectively. Moreover, the complexes $\ko{\mbox{\rm Hom}}\,(K^R_{\scriptscriptstyle \bullet},k)$ and $(\mbox{\rm span}_k K^R_{\scriptscriptstyle \bullet})^\ast$ are equal up to position $p=\dim\sigma-1$. \hfill$\Box$ \par \neu{app-3} In the {\em general case}, we still have enough information to determine the deformation relevant modules $T^0_A$, $T^1_A$, and $T^2_A$. Under the additional hypothesis of smoothness in codimension two, these results have already been obtained in \cite{T2} with a different proof. \par {\bf Proposition:} {\em Let $\sigma$ be an arbitrary rational, polyhedral cone with apex in $0$. Then, for every $R\in M$, \vspace{-0.5ex} \[ T^n_A(-R) = H^n\big(\ko{\mbox{\rm Hom}}\,(K^R_{\scriptscriptstyle \bullet},k)\big) \quad\mbox{ for } n=0,1,2. \] Moreover, for $n=0,1$, this vector space equals $H^n\big((\mbox{\rm span}_k K^R_{\scriptscriptstyle \bullet})^\ast\big)$, too. } \par {\bf Proof:} This is a direct consequence of Corollary \zitat{E1l}{4}. \hfill$\Box$ \par \neu{app-4} Since the complex $(\mbox{\rm span}_k K^R_{\scriptscriptstyle \bullet})^\ast$ is much easier to handle than $\ko{\mbox{\rm Hom}}\,(K^R_{\scriptscriptstyle \bullet},k)$, it pays to look for sufficient conditions for $T^2_A= H^2\big((\mbox{\rm span}_k K^R_{\scriptscriptstyle \bullet})^\ast\big)$ to hold. \par {\bf Proposition:} {\em If $Y_\sigma$ is Gorenstein in codimension two, i.e.\ for every two-dimensional face $\langle a^i,a^j\rangle\le\sigma$ there is an element $r(i,j)\in M$ such that $\langle a^i,r(i,j)\rangle = \langle a^j,r(i,j)\rangle = 1$, then \[ T^2_A= H^2\big((\mbox{\rm span}_k K^R_{\scriptscriptstyle \bullet})^\ast\big). \vspace{-3ex} \] } \par {\bf Proof:} For edges $\langle a^i,a^j\rangle\leq\sigma$ we have to show that $(\mbox{\rm span}_k K^R_{ij})^\ast\hookrightarrow \ko{\mbox{\rm Hom}}\,(K^R_{ij},k)$ is an isomorphism. We will adapt the proof of \zitat{E1l}{2}.\\ It may be assumed that $\langle a^i,R\rangle; \langle a^j,R\rangle \geq 1$ and $r(i,j)\in K^R_{ij}$. Let $d:=|\det (a^i,a^j)|\in{Z\!\!\!Z}$; it is the smallest positive value of $a^i$ possible on elements of $\Lambda\cap (a^j)^\bot$. We choose an $r^i\in\Lambda\cap (a^j)^\bot$ with $\langle a^i,r^i\rangle=d$; together with $r(i,j)$ it will play the same role as $r^1,\dots,r^l$ did in \zitat{E1l}{2}. \vspace{0.5ex}\\ {\em Case 1:}\quad $\langle a^i,R\rangle >d$ and $\langle a^i,R\rangle\geq\langle a^j,R\rangle$ (in particular, $r^i\in K^R_{ij}$): Then, elements $s^v$ or $s$ (cf.\ \zitat{E1l}{2}) may be represented as \[ s=\langle a^j,s\rangle\cdot r(i,j) + \frac{\displaystyle \langle a^i,R\rangle -\langle a^j,R\rangle}{\displaystyle d}\, r^i + \big[ (a^i,a^j)^\bot-\mbox{elements}\big]\,. \] The difference $\langle a^i,R\rangle -\langle a^j,R\rangle$ is always divisible by $d$, i.e.\ the coefficients are integers. \vspace{0.5ex}\\ {\em Case 2:}\quad $\langle a^i,R\rangle; \langle a^j,R\rangle\leq d$: This implies $K^R_{ij}\subseteq (a^i,a^j)^\bot +{Z\!\!\!Z}\cdot r(i,j) = (a^i-a^j)^\bot$. In particular, we may use the representation $s=\langle a^i,s\rangle\cdot r(i,j) + \big[ (a^i,a^j)^\bot-\mbox{elements}\big]\,.$ \vspace{1ex} \hfill$\Box$ \par {\bf Corollary:} {\em Let $Y_\sigma$ be a three-dimensional, toric Gorenstein singularity, i.e.\ $\sigma=\langle a^1,\dots,a^m\rangle$ with $a^{m+1}:=a^1$ being the cone over a lattice polygon embedded in height one. Then \[ T^2_A(-R)^\ast=\;^{\displaystyle \mbox{$\bigcap$}_{i=1}^m \big(\mbox{\rm span}_k K^R_{i,i+1}\big)} \!\Big/ _{\displaystyle \mbox{\rm span}_k \big(\mbox{$\bigcap$}_{i=1}^m K^R_{i,i+1}\big)}\,. \] } \par \neu{app-5} Finally, we would like to mention an alternative to the complexes $(\mbox{\rm span}_k K^R_{\scriptscriptstyle \bullet})^\ast$ and $\ko{\mbox{\rm Hom}}\,(K^R_{\scriptscriptstyle \bullet},k)$. Let $E\subseteq\Lambda_+$ be the (finite) set of non-splittable elements in $\Lambda_+$; it is the minimal generator set of the monoid $\Lambda=\sigma^{\scriptscriptstyle\vee}\cap M$ and gives rise to a canonical surjection $\pi:{Z\!\!\!Z}^E\to M$. The relations among $E$-elements are gathered in the ${Z\!\!\!Z}$-module $L(E):=\ker \pi$. Every $q\in L(E)$ splits into a difference $q=q^+ -q^-$ with $q^+,q^-\in {I\!\!N}^E$ and $\sum_v q^+_v\,q^-_v=0$. We denote by $\bar{q}\in\Lambda$ the image $\bar{q}:=\pi(q^+)=\pi(q^-)$. \par {\bf Definition:} With $E^R_\tau:=E\cap K^R_\tau$, we define $L(E^R_\tau):=L(E)\cap {Z\!\!\!Z}^{E^R_\tau}$ and $\ko{L}(E^R_\tau)\subseteq L(E^R_\tau)$ to be the submodule generated by the relations $q\in L(E)$ such that $\bar{q}\in K^R_\tau$. (Notice that with $E^R_0=E$ the notation differs slightly from that in \cite{T2}.) \par As usual (cf.\ \zitat{sps}{2}), we may construct complexes from these finitely generated, free abelian groups. They fit into the following commutative diagram with exact rows: \[ \dgARROWLENGTH=0.8em \begin{diagram} \node{0} \arrow{e} \node{(\mbox{\rm span}_k K^R_{\scriptscriptstyle \bullet})^\ast} \arrow{e} \arrow{s} \node{k^{E^R_{\scriptscriptstyle \bullet}}} \arrow{e} \arrow{s,r}{\sim} \node{\mbox{\rm Hom}_{{Z\!\!\!Z}}(L(E^R_{\scriptscriptstyle \bullet}),k)} \arrow{e} \arrow{s} \node{0}\\ \node{0} \arrow{e} \node{\ko{\mbox{\rm Hom}}\,( K^R_{\scriptscriptstyle \bullet},k)} \arrow{e} \node{k^{E^R_{\scriptscriptstyle \bullet}}} \arrow{e} \node{\mbox{\rm Hom}_{{Z\!\!\!Z}}(\ko{L}(E^R_{\scriptscriptstyle \bullet}),k)} \end{diagram} \] The cokernel of the second row is $\mbox{\rm Ext}^1_{Z\!\!\!Z}(L/\ko{L},k)$. It vanishes if $\mbox{\rm char}\, k=0$. \par {\bf Proposition:} {\em If $\mbox{\rm char}\, k=0$, then the $k$-duals of $T^n_A(-R)$ equal \[ T^1_A(-R)^\ast=L_k\big(\mbox{$\bigcup$}_i E_i^R\big) \big/\mbox{$\sum$}_i L_k(E^R_i) \vspace{-2ex} \] and \[ T^2_A(-R)^\ast=\,\ker\big(\oplus_i L_k(E_i^R) \to L(E)\big)\Big/\, \mbox{\rm im}\,\big(\oplus_{ij} \ko{L}_k(E_{ij}^R)\to \oplus_i L_k(E_i^R)\big)\,. \vspace{-2ex} \] } \par {\bf Proof:} As in the proof of Lemma \zitat{sps}{2}, one obtains that the complex $k^{E^R_{\scriptscriptstyle \bullet}}$ has no cohomology except $H^0=k^{E\setminus\cup_i E^R_i}$. Hence, the long exact sequence for the second row in the above diagram yields isomorphisms \[ H^{p-1}\big(\mbox{\rm Hom}_{Z\!\!\!Z}(\ko{L}(E^R_{\scriptscriptstyle\bullet}),k)\big) \stackrel{\sim}{\longrightarrow} H^{p}\big(\ko{\mbox{\rm Hom}}\,(K^R_{\scriptscriptstyle\bullet},k)\big)\quad \mbox{ for }\;p\geq 2. \] Taking a closer look at the first terms shows that the same result is true for $p=1$ if $E^R_0$ is replaced by $\bigcup_i E^R_i$. Finally, we know that $\ko{L}(E^R_\tau)=L(E^R_\tau)$ if $\dim\tau\leq 1$. \hfill$\Box$ \par \neu{app-6} {\bf Remark:} The vector space $T^1_A(-R)$ has also a convex-geometric interpretation; it is related to the set of Minkowski summands of the cross cut $\sigma\cap [R=1]$. For details, we refer to \cite{Flip}. \par

9611

alg-geom/9611032

en

https://arxiv.org/abs/alg-geom/9611032

alg-geom/9611032

Wolfgang Eholzer

Y. Choie, W. Eholzer

Rankin-Cohen Operators for Jacobi and Siegel Forms

15 pages LaTeX2e using amssym.def

DAMTP-96-106

For any non-negative integer v we construct explicitly [v/2]+1 independent covariant bilinear differential operators from J_{k,m} x J_{k',m'} to J_{k+k'+v,m+m'}. As an application we construct a covariant bilinear differential operator mapping S_k^{(2)} x S^{(2)}_{k'} to S^{(2)}_{k+k'+v}. Here J_{k,m} denotes the space of Jacobi forms of weight k and index m and S^{(2)}_k the space of Siegel modular forms of degree 2 and weight k. The covariant bilinear differential operators constructed are analogous to operators already studied in the elliptic case by R. Rankin and H. Cohen and we call them Rankin-Cohen operators.

alg-geom

\section{Introduction and results} \label{Introduction} It has been known for some time how to obtain an elliptic modular form from the derivatives of $N$ elliptic modular forms. The case $N=1$ has already been studied in detail by R.\ Rankin in 1956 \cite{R}. For $N=2$ H.\ Cohen has constructed certain covariant bilinear operators which he used to obtain modular forms with interesting Fourier coefficients \cite{C}. Later, these operators were called Rankin-Cohen operators by D.\ Zagier who studied their algebraic relations \cite{Z}. The main result of this paper is the explicit description of covariant bilinear operators for Jacobi forms and Siegel modular forms of degree $2$. Since they are generalisations of the Rankin-Cohen operators in the elliptic case we will also call them Rankin-Cohen operators. The main theorem reads \begin{thm} \label{Jacobithm} Let $f$ and $f'$ be Jacobi forms of weight and index $k,m$ and $k',m'$, respectively. For any $X\in{\Bbb C}} \def\D{{\Bbb D}$ and any non-negative integer $v$ define $$ [f,f']_{X,v} = \sum_{r+s+p= \lfloor v/2\rfloor \atop i+j = v-2\lfloor v/2\rfloor} C_{r,s,p}(k,k') \ D_{r,s,i,j}(m,m',X)\ L_{m+m'}^p\left( L_m^r(\partial_z^i f)\, L_{m'}^s(\partial_z^j f') \right) $$ where \begin{eqnarray} D_{r,s,i,j}(m,m',X) &=& m^j (-m')^i (1+mX)^s (1-m'X)^r\, , \nonumber\\ C_{r,s,p}(k,k') &=& \frac{(\alpha +r+s+p)_{s+p} }{ r! } \cdot \frac{(\beta +r+s+p)_{r+p} }{ s! } \cdot \frac{(-(\gamma+r+s+p))_{r+s} } {p!} \nonumber\\ &&(\alpha = k-3/2,\ \beta = k'-3/2,\ \gamma = k+k'-3/2 +(v-2\lfloor v/2\rfloor) )\, , \nonumber \end{eqnarray} where $(x)_m = \prod_{0\le i \le m-1} (x-i)$, $\lfloor x\rfloor$ denotes the largest integer $\le x $, and, where $L_m(f) = (8 \pi i m \frac{\partial}{\partial \tau} - \frac{\partial^2}{\partial z^2})f$ for $f$ a Jacobi form of index $m$. \\ Then $[f,f']_{X,v}$ is a Jacobi form of weight $k+k'+v$ and index $m+m'$ and, even more, a Jacobi cusp form for $v>1$. \end{thm} Let us remark that some special cases of Theorem \ref{Jacobithm} have already been considered in the literature: Firstly, the bilinear operator $[\cdot,\cdot]_{X,1}$, which actually does not depend on $X$, has already been shown to map two Jacobi forms to a Jacobi form \cite[Theorem 9.5]{EZ}. Secondly, one bilinear operator for each even $v$ has already been constructed in ref.\ \cite{Ch1}: up to a scalar multiple this operator is equal to $$ \left(\frac{d}{dX}\right)^{v/2} [f,f']_{X,v} \qquad (v\in 2{\Bbb N}} \def\Z{{\Bbb Z} =\{0,2\dots \}).$$ For fixed $v$ and $k,m$ and $k',m'$ large enough the operators $[\cdot,\cdot ]_{X,v}$ ($X\in{\Bbb C}} \def\D{{\Bbb D}$) span a vector space of dimension $\lfloor \frac{v}{2}\rfloor+1$ This shows that the space of such Rankin-Cohen operators is, in general, at least $\lfloor\frac{v}{2}\rfloor+1$ dimensional. A result of B\"ocherer \cite{Boe}, obtained by using Maa{\ss} operators, shows that this dimension actually equals $\lfloor\frac{v}{2}\rfloor+1$ in general (cf.\ Theorem \ref{Boecherer}). \medskip One of the applications of our result is to Siegel modular forms of degree $2$: the bilinear operators $[\cdot,\cdot ]_{0,v}$ with even $v$ can be lifted to bilinear covariant differential operators for Siegel modular forms of degree $2$. More precisely one has the following theorem. \begin{thm} \label{Siegelthm} Let $F$ and $F'$ be Siegel modular forms of degree $2$ and weight $k$ and $k'$, respectively. Define, for any non-negative integer $l$, $$ [F,F']_{l} = \sum_{r+s+p=l} C_{r,s,p}(k,k')\ \D^p( \D^r(F)\, \D^s(F') ) $$ with $C_{r,s,p}(k,k')$ as in Theorem \ref{Jacobithm} and $\alpha = k-3/2$, $\beta = k'-3/2$, $\gamma = k+k'-3/2$ and where $\D= \frac{4\partial^2}{\partial_{\tau_1}\partial_{\tau_2}} -\frac{\partial^2}{\partial_{z}^2} $ with $Z=\left(\begin{array}{cc} \tau_1 & z \\ z &\tau_2 \end{array}\right)$ the variable in ${\Bbb H}} \def\P{{\Bbb P}_2$.\\ Then $[F,F']_{l}$ is a Siegel modular form of degree $2$ and weight $k+k'+2l$ and, even more, a Siegel cusp form for $l>0$. \end{thm} This paper is organised as follows. In \S\ref{Jacobiforms} we recall some results on Jacobi forms which will be needed in the proof of Theorems \ref{Jacobithm}. Section \ref{Siegelforms} contains the definition of Siegel modular forms and their relation to Jacobi forms in the degree $2$ case. In \S\ref{Jacobiproofcom} we give a (combinatoric) proof of Theorem \ref{Jacobithm} and \S\ref{Jacobiproofgen} contains a proof using generating functions. In \S\ref{SiegelproofJacobi} we prove Theorem \ref{Siegelthm} using Theorem \ref{Jacobithm} and in \S\ref{SiegelproofTheta} we give a second, independent proof using theta series with spherical coefficients and a general result of Ibukiyama (cf. Theorem \ref{Ibuthm}). We conclude with several remarks and some open questions in section \S\ref{Conclusion}. In particular we discuss the uniqueness of the Rankin-Cohen operators for Siegel modular froms of degree $2$. \section{Jacobi forms} \label{Jacobiforms} In this section we recall a few general results about Jacobi forms. We first give the definition of Jacobi forms and the heat operator (as a general reference for Jacobi forms we refer to \cite{EZ}). Denote by ${\Bbb H}} \def\P{{\Bbb P}$ the complex upper half plane and define, for holomorphic functions $f:{\Bbb H}} \def\P{{\Bbb P} \times {\Bbb C}} \def\D{{\Bbb D} \to {\Bbb C}} \def\D{{\Bbb D}$ and integers $k$ and $m$, \begin{eqnarray} (f |_{k, m} M )(\tau,z) &=& (c \tau + d)^{-k} e^{2 \pi i m (\frac{-c z^2}{c \tau + d})} f(\frac{a \tau + b}{c \tau +d}, \frac{z}{c \tau +d}), \nonumber\\ (f|_{m} Y) (\tau,z) &=& e^{2 \pi i m (\lambda^{2} \tau + 2\lambda z)} f(\tau, z+\lambda \tau + \nu) \nonumber \end{eqnarray} where $\tau\in{\Bbb H}} \def\P{{\Bbb P}$, $z\in{\Bbb C}} \def\D{{\Bbb D}$, $M = \left(\begin{array}{cc} a & b\\ c & d \end{array} \right) \in \mbox{SL}(2,\Z)$ and $Y=(\lambda,\nu)\in\Z^2$. Using these slash actions the definition of Jacobi forms is as follows. \begin{df} A Jacobi form of weight $k$ and index $m$ ($k, m \in {\Bbb N}} \def\Z{{\Bbb Z}$) is a holomorphic function $f:{\Bbb H}} \def\P{{\Bbb P}\times{\Bbb C}} \def\D{{\Bbb D}\to{\Bbb C}} \def\D{{\Bbb D}$ satisfying $$ (f |_{k, m} M)(\tau, z) = f(\tau,z), \qquad (f|_m Y) (\tau,z)= f(\tau,z) $$ for all $M\in\mbox{SL}(2,\Z)$ and $Y\in\Z^2$ and such that it has a Fourier expansion of the form $$ f(\tau, z) = \sum_{ \begin{array}{cc} n = 0 \\ r \in \Z, r^{2} \leq 4nm \end{array} }^{\infty} c(n, r) q^n\zeta^r, $$ where $q = e^{2 \pi i \tau}$ and $\zeta = e^{2 \pi i z}$. If $f$ has a Fourier expansion of the same form but with $r^2 < 4nm$ then $f$ is called a Jacobi cusp form of weight $k$ and index $m$. \end{df} We denote by $J_{k,m}$ the (finite dimensional) vector space of all Jacobi forms of weight $k$ and index $m$ and by $J_{k,m}^{cusp}$ the vector space of all Jacobi cusp forms of weight $k$ and index $m$. Our main result (Theorem \ref{Jacobithm}) involves the heat operator which has already been studied in \cite{EZ} to connect Jacobi forms and elliptic modular forms and in ref.\ \cite{Ch1,Ch2} in the context of bilinear differential operators ({\it cf.} the remark after Theorem \ref{Jacobithm}). \begin{df} For any non-negative integer $m$ the heat operator $L_m$ is defined by $$ L_m(f) = \left(8 \pi i m \frac{\partial}{\partial \tau} - \frac{\partial^2}{\partial z^2} \right)(f) \qquad(f\in J_{k,m}). $$ \end{df} Finally, let us mention a result of B\"ocherer \cite{Boe}. \begin{thm} \label{Boecherer} For fixed $v$ and $k,m,k',m'$ large enough the vector space of all covariant bilinear differential operators mapping $J_{k,m}\times J_{k',m'}$ to $J_{k+k'+v,m+m'}$ has dimension $\lfloor v/2\rfloor+1$. \end{thm} Note that Theorem \ref{Jacobithm} describes a basis of this space explicitly. \section{Siegel modular forms} \label{Siegelforms} In this section we recall a few basic facts about Siegel modular forms and, in particular, the construction of Siegel modular forms using theta series with spherical coefficients (Theorem \ref{thetathm}). Furthermore, we describe the connection between Siegel modular forms of degree $2$ and Jacobi forms (Theorem \ref{Siegelexp}). Finally, we mention (a special case of) a result of Ibukiyama (Theorem \ref{Ibuthm}) which will be needed in the proof of Theorem \ref{Siegelthm} in section \S\ref{SiegelproofTheta}. The reader may take ref.\ \cite{Frei} as a general reference for Siegel modular forms. For any holomorphic function $f$ on the Siegel upper half plane ${\Bbb H}} \def\P{{\Bbb P}_n$, {\it i.e.}\ the space of complex symmetric $n\times n$ matrices with positive definite imaginary part, and $M\in \mbox{{\rm{Sp}}}(2n,\Z)$ define $$ (f|_M^k)(Z) = f(MZ) \det(CZ+D)^{-k} $$ where $Z\in{\Bbb H}} \def\P{{\Bbb P}_n$, $M=\left( \begin{array}{cc} A & B \\ C & D \end{array}\right)$ with $n\times n$ matrices $A,B,C,D$, and, where $MZ = (AZ+B)(CZ+D)^{-1}$. Then the definition of Siegel modular forms is given as follows. \begin{df} Let $n$ be a positive integer greater than $1$. Then a holomorphic function $f$ on ${\Bbb H}} \def\P{{\Bbb P}_n$ is called a Siegel modular form of degree $n$ and weight $k$ if $$ (f|_M^k)(Z) = f(Z)\qquad {\rm{for}}\ {\rm{all}}\ M\in\mbox{{\rm{Sp}}}(2n,\Z). $$ \end{df} We denote the space of all Siegel modular forms of degree $n$ and weight $k$ by $S^{(n)}_k$. Note that for $n=1$ one has to add a further condition on $f$ in order to obtain the usual definition of modular forms. The connection between Jacobi forms and Siegel modular forms of degree two becomes clear by the following theorem (see {\it e.g.} \cite[Theorem 6.1]{EZ}). \begin{thm} \label{Siegelexp} Let $F$ be a Siegel modular for degree $2$ and weight $k$ and write the Fourier development of $F$ with respect to $\tau_2$ in the form $$ F(Z) = F(\tau_1,z,\tau_2) = \sum_{m=0}^\infty f_m(\tau_1,z) e^{2\pi i m\tau_2} $$ with $Z =\left( \begin{array}{cc} \tau_1& z \\ z & \tau_2 \end{array}\right)$ the variable in ${\Bbb H}} \def\P{{\Bbb P}_2$.\\ Then, for each non-negative integer $m$, the function $f_m$ is a Jacobi form of weight $k$ and index $m$. \end{thm} \begin{rem} \label{Siegelcusp} Note that for general degree $n$ a Siegel cusp form is a Siegel modular form which is contained in the kernel of the Siegel operator $\Phi$ (for more details see {\it e.g.} \cite{Frei}). In the case of degree $2$ a Siegel cusp form $F$ is a Siegel modular form whose Jacobi-Fourier expansion is of the form $F= \sum_{m>0} f_m(\tau_1,z) e^{2\pi i m\tau_2}$, {\it i.e.}\ the first coefficient in the Jacobi-Fourier expansion is identically zero. \end{rem} To recall some facts about theta series with spherical coefficients we introduce the notion of spherical polynomials first. \begin{df} A spherical polynomial $P$ of weight $w$ in a matrix variable $X\in M_{m,n}$ is a polynomial (in the matrix elements of $X$, {\it i.e.}\ a polynomial in ${\Bbb C}} \def\D{{\Bbb D}[x_{ij}]_{1\le i \le m \atop 1\le j \le n}$) satisfying \\ 1) $P(X A) = \det(A)^w\, P(X)$ for all $A\in M_{n,n}$, \\ 2) $\Delta P = \sum_{i,j} \frac{\partial^2}{(\partial x_{ij})^2} P =0$. \end{df} Then one has (see {\it e.g.} \cite[p.\ 161]{Frei}) \begin{thm} \label{thetathm} Let $P$ be a spherical polynomial of weight $k$ in a matrix variable $X\in M_{m,n}$ and let $S\in M_{m,m}(\Z)$ be a symmetric, positive, even and unimodular matrix. Then the function $$ \theta_{S,P}(Z) = \sum_{G\in M_{m,n}(\Z)} P(S^{1/2}G)\, e^{\pi i \, \mbox{tr}(G^t S G Z)} $$ is a Siegel modular form of degree $n$ and weight $m/2+k$. \end{thm} Finally, let us mention (a special case of) a result of Ibukiyama (Corollary 2 (2) of ref.\ \cite{Ibu} with $r=2$) which we use to prove Theorem \ref{Siegelthm} in \S\ref{SiegelproofTheta}. \begin{thm} \label{Ibuthm} Let $P$ be a spherical polynomial of even weight $d$ in the matrix variable $(X,X')^t \in M_{m+m',n}$ which can be written as $P(X,X') = \tilde Q(X^t X, {X'}^t X')$ for some polynomial $\tilde Q$. Set ${\cal D} = \tilde Q(\partial_{\nu,\mu},\partial_{\nu',\mu'}')$ where $\partial_{\nu,\mu} = (1+\delta_{\nu,\mu}) \frac{\partial}{\partial z_{\nu,\mu}}$ and $\partial_{\nu',\mu'}' = (1+\delta_{\nu',\mu'}) \frac{\partial}{\partial z'_{\nu',\mu'}}$ with $Z = (z_{\nu,\mu})$ and $Z' = (z'_{\nu',\mu'})$ for $1\le \nu,\mu,\nu',\mu' \le n$.\\ Then, for any two Siegel modular forms $F(Z)$ and $F'(Z')$ of degree $n$ and weight $k$ and $k'$, respectively, the function ${\cal D}(F(Z) F'(Z'))|_{Z=Z'}$ is a Siegel modular form of degree $n$ and weight $k+k'+d$. \end{thm} Note that Theorem \ref{Ibuthm} essentially says that if a bilinear differential operator maps all pairs of theta series to theta series with spherical coefficients then it even maps all pairs of Siegel modular forms to Siegel modular forms ({\it cf.} the discussion at the end of \S6). \section{A combinatorial proof of Theorem \ref{Jacobithm}} \label{Jacobiproofcom} Let us now give the proof of our main theorem which will, in particular, imply Theorem \ref{Siegelthm} (see \S\ref{SiegelproofJacobi} for details). To prepare the proof of Theorem \ref{Jacobithm} we need the following three lemmata. \begin{lem} \label{Lmdis} Let $f$ be a holomorphic function on ${\Bbb H}} \def\P{{\Bbb P}$ and $g$ a holomorphic function on ${\Bbb H}} \def\P{{\Bbb P}\times{\Bbb C}} \def\D{{\Bbb D}$. Then, for each non-negative integer $r$, one has $$ L_m^r(fg) = \sum_{j=0}^r(8\pi i m)^{r-j} \bin{r}{j}\, (\partial_{\tau}^{r-j} f)\, (L_m^j g) $$ where $\tau$ is the variable in ${\Bbb H}} \def\P{{\Bbb P}$. \end{lem} {\it Proof.} We prove the formula by induction. Firstly, note that for $r=1$ one has $$ L_m(fg) = (8\pi i m \partial_{\tau}-\partial_{z}^2)(fg) = 8\pi i m (\partial_{\tau}f)g + 8\pi i m f (\partial_{\tau}g) - f\partial_{z}^2 g = 8\pi i m (\partial_{\tau}f) g + f L_m g. $$ Secondly, assume that the formula is valid for some $r$. Then we find \begin{eqnarray} L_m^{r+1}(fg) &=& L_m\left( \sum_{j=0}^r (8\pi i m)^{r-j} \bin{r}{j}\, (\partial_{\tau}^{r-j} f)\, (L_m^j g)\right) \nonumber\\ &=& \sum_{j=0}^r \Big( (8\pi i m)^{r-j+1}\bin{r}{j}\, (\partial_{\tau}^{r-j+1} f)\, (L_m^j g) \nonumber\\ && \qquad+ (8\pi i m)^{r-j}\bin{r}{j}\, (\partial_{\tau}^{r-j} f)\, (L_m^{j+1} g) \Big) \nonumber\\ &=& \sum_{j=0}^{r+1} (8\pi i m)^{r+1-j}\bin{r+1}{j}\, (\partial_{\tau}^{r+1-j} f)\, (L_m^j g) \nonumber \end{eqnarray} and the lemma becomes obvious. \qed The second lemma we will need is Lemma 3.1 of ref.\ \cite{Ch1} (note that the normalisation of $L_m$ in {\it loc. cit.} differs from ours by a factor of $(2\pi i )^2$). \begin{lem} \label{Lmcom} Let $f$ be a holomorphic function on ${\Bbb H}} \def\P{{\Bbb P}\times{\Bbb C}} \def\D{{\Bbb D}$. Then, for any non-negative integer $r$, one has $$ L_m^r(f)|_{k+2r,m}M = \sum_{j=0}^r (8\pi i m)^{r-j} \frac{r!}{j!} \bin{k-3/2+r}{r-j}\, \left(\frac{c}{c\tau+d}\right)^{r-j}\, L_m^j(f|_{k,m} M) $$ for all $M=\left(\begin{array}{cc} a&b\\ c&d \end{array}\right)\in{\mbox{SL}}(2,\Z)$.\\ Furthermore, one has $$ L_m(f)|_m Y = L_m(f|_m Y) $$ for all $Y\in \Z^2$. \end{lem} {\it Proof.} The proof of the first formula is a simple exercise and can be found in \cite{Ch1}. The second formula becomes obvious after a short calculation using only the definition of the $|_m Y$ action \qed The third lemma is essentially equivalent to Theorem 9.5 of ref.\ \cite{EZ}. \begin{lem} \label{oddlem} Let $f$ and $f'$ be holomorphic functions on ${\Bbb H}} \def\P{{\Bbb P}\times{\Bbb C}} \def\D{{\Bbb D}$. Then, for $z\in {\Bbb C}} \def\D{{\Bbb D}$, one has \begin{eqnarray} &&\left( m' (\partial_z f)\, f' - m f\, (\partial_z f') \right)|_{k+k'+1,m+m'} M = \nonumber\\ &&\qquad\qquad\qquad \qquad m' (\partial_z (f|_{k,m} M))\, (f'|_{k',m'} M) - m (f|_{k,m} M)\, (\partial_z (f'|_{k',m'} M))\, , \nonumber \\ &&\left( m' (\partial_z f)\, f' - m f\, (\partial_z f') \right)|_{m+m'} Y = \nonumber \\ &&\qquad\qquad\qquad\qquad m' (\partial_z (f|_{m} Y))\, (f'|_{m'} Y) - m (f|_m Y)\, (\partial_z (f'|_{m'} Y)) \nonumber \end{eqnarray} for all $M\in\mbox{SL}(2,\Z)$ and all $Y\in\Z^2$. \end{lem} {\it Proof.} The two formulas can be obtained by a straightforward calculation. \qed \bigskip We are now ready to prove our main theorem.\\ {\it Proof of Theorem \ref{Jacobithm}.} The definition of Jacobi forms contains essentially two parts: the invariance under the slash actions and an expansion condition. If $f$ and $f'$ are Jacobi forms then a simple computation shows that the latter condition is always satisfied for $[f,f']_{X,v}$ and, even more, that $[f,f']_{X,v}$ satisfies the expansion condition of a Jacobi cusp form for $v>1$. Therefore, it only remains to check the invariance under the slash actions. We consider first the case of even $v=2v'$. Since, by Lemma \ref{Lmcom}, the $|_m Y$ action commutes with $L_m$ we only have to show that $$ ([f,g]_{X,v})|_{k+k'+v,m+m'} M = [f|_{k,m} M,g|_{k',m'} M]_{X,v}$$ for all $M\in \mbox{SL}(2,\Z)$. Using Lemma \ref{Lmdis} and \ref{Lmcom} to calculate the left hand side we obtain that this equation is equivalent to \begin{eqnarray} &&\sum_{r+s+p=v'} C_{r,s,p} \frac{r!s!p!}{j!j'!j''!} \bin{\alpha+r}{r-j'} \bin{\beta +s}{s-j''} \bin{\gamma+v'+j'+j''}{p-j} \nonumber\\ && \qquad \times \quad (1+mX)^s (1-m'X)^r m^{r-j'}m'^{s+p-j-j''} (1+m/m')^{p-j} \nonumber\\ && = \delta_{v',j+j'+j''}\, C_{j',j'',j} (1+mX)^{j''} (1-m'X)^{j'}. \nonumber \end{eqnarray} Some simple manipulations show that this equation is equivalent to \begin{eqnarray} && \sum_{r+s+p=v'} \frac{(v'-j')!(v'-j'')!(v'-p)!}{(r-j')!(s-j'')!(v'-j)!} \bin{-(\gamma+v')}{v'-p} \bin{\gamma+v'+j'+j''}{p-j} \nonumber\\ &&\qquad \times \quad m^{r-j'} m'^{s+p-j-j''}(1+m/m')^{p-j} (1+mX)^{s-j'} (1-m'X)^{r-j''} \nonumber\\ && = \delta_{v',j+j'+j''}\, \bin{-(\gamma+v')}{v'-j}. \nonumber \end{eqnarray} To show this equality we view it as an equation between polynomials in $\gamma$ (of degree $v'-j$). It is easy to see that both sides coincide for $\gamma+v'+j'+j''=0$. Therefore, it is enough to show that both sides agree for all $\gamma = -v'-x$ $(0\le x <v'-j$). Note that for these values of $\gamma$ the right hand side obviously vanishes. With $A = \frac{m(1-m'X)}{m'(1+mX)}$ the left hand side becomes \begin{eqnarray} && c\ \sum_{r,p} \bin{v'-p-j'-j''}{r-j'} \bin{x-j'-j''}{v'-p-j'-j''} \bin{j'+j''-x}{p-j} A^r (A+1)^{p-j} \nonumber\\ && \qquad = c'\ \sum_{i'}\sum_{r,p} \bin{j+j'+j''-v'}{x+p-v'} \bin{v'-j-j''-i'}{v'-r-p-j''} \bin{i'-j'}{r-j'} A^{i'} \nonumber \end{eqnarray} for some (non-zero) factors $c$ and $c'$. The product of the three binomial coefficients is the factor in front of $$ Z^{x+p-v'}\cdot Z^{v'-r-p-j''} \cdot Z^{r-j'} = Z^{x-j'-j''}$$ in $$(1+Z)^{j+j'+j''-v'} \cdot (1+Z)^{v'-j-j''-i'} \cdot (1+Z)^{i'-j'} = 1.$$ Hence we find that our expression is zero unless $x=j'+j''$. However, in this case our expression can only be non-zero if $v'=j+j'+j''$ which is not allowed since $x=j'+j''$ has to be strictly less than $v'-j=j'+j'' = x$. Hence also the left hand side is equal to zero for $\gamma = -v'-x$ $(0\le x <v'-j$) so that we have proved the desired equality. This proves the theorem for even $v=2v'$. For odd $v=2v'+1$ note that the summand for fixed $r,s,p$ is equal to $$ C_{r,s,p} (1+mX)^s (1-m'X)^r L_{m+m'}^p \left( m' (\partial_z L_m^r(f))\, L_{m'}^s(f') - m L_m^r(f)\, (\partial_z L_{m'}^s(f')) \right). $$ The expression inside the $L_{m+m'}^p$ is exactly of the form considered in Lemma \ref{oddlem} so that we obtain $$ ([f,f']_{X,v})|_{m+m'} Y = [f|_{m} Y,f'|_{m'} Y]_{X,v}.$$ Therefore, it only remains to show the invariance with respect to the other slash action as in the even case. Note, however, that using Lemma \ref{oddlem} we have to check exactly the same combinatorial identity as in the case of even $v$ but with $\gamma = k+k'-1/2$ instead of $\gamma = k+k'-3/2$. This completes the proof of Theorem \ref{Jacobithm}. \qed \section{A proof of Theorem \ref{Jacobithm} using generating functions} \label{Jacobiproofgen} In this section we give a second, independent proof of Theorem \ref{Jacobithm}. Instead of proving the theorem directly we use some results of ref.\ \cite{Ch2} (Theorem 3.1 and Corollary 3.1) on generating functions. Let us first recall these results. \begin{thm} \label{Choiethm} Let $\tilde f(\tau,z;W)$ be a formal power series in $W$, {\it i.e.} $\tilde f$ can be written as $\tilde f(\tau,z;W) = \sum_{\nu=0}^\infty \chi_\nu(\tau,z)\, W^\nu$, satisfying the functional equation $$ \tilde f(\frac{a\tau+b}{c\tau+d},\frac{z}{c\tau+d};\frac{W}{(c\tau+d)^2}) = (c\tau+d)^K e^{2\pi i M \frac{cz^2}{c\tau+d}} e^{8\pi i M\frac{cW}{c\tau+d}}\ \tilde f(\tau,z;W) $$ for some integers $K$ and $M$ and all $\left(\begin{array}{cc} a &b\\ c& c\end{array}\right)\in \mbox{SL}(2,\Z)$. Furthermore, assume that the coefficients $\chi_\nu$ are holomorphic functions on ${\Bbb H}} \def\P{{\Bbb P}\times{\Bbb C}} \def\D{{\Bbb D}$ with a Fourier expansion of the form $$ \chi_\nu(\tau,z) = \sum_{r,n\in\Z\atop r^2\le 4mn} c(n,r) q^n \xi^r \qquad( q = e^{2\pi i \tau}, \xi = e^{2\pi i z}) $$ satisfying $ \chi_\nu|_m Y = \chi_\nu$ for all $Y\in \Z^2$.\\ Then, for each non-negative integer $\nu$, the function $\zeta_\nu$ defined by $$ \zeta_\nu(\tau,z) = \sum_{j=0}^\nu \frac{(-(K-3/2+\nu))_{\nu-j}}{j!}\ L^{j}_M(\chi_{\nu-j}) $$ is a Jacobi form of weight $K+2\nu$ and index $M$. \end{thm} An immediate consequence of this theorem is the following corollary \cite[Corollary 3.1]{Ch2}. \begin{cor} \label{Choiecor} Let $f(\tau,z)$ be a Jacobi form of weight $k$ and index $m$. Then $$ \tilde f(\tau,z;W) = \sum_{\nu=0}^\infty \frac{1}{\nu!(K-3/2+\nu)!}\ L^\nu_M(f)\ W^\nu $$ satisfies the functional equation $$ \tilde f(\frac{a\tau+b}{c\tau+d},\frac{z}{c\tau+d};\frac{W}{(c\tau+d)^2}) = (c\tau+d)^k e^{2\pi i m \frac{cz^2}{c\tau+d}} e^{8\pi i m\frac{cW}{c\tau+d}}\ \tilde f(\tau,z;W) $$ for all $\left(\begin{array}{cc} a &b\\ c& c\end{array}\right)\in\mbox{SL}(2,\Z)$. \end{cor} Using these two results we are now able to prove Theorem \ref{Jacobithm}.\\ {\it Proof of Theorem \ref{Jacobithm}}. Let $f$ and $f'$ be Jacobi forms of weight and index $k,m$ and $k',m'$, respectively. Denote by $\tilde f(\tau,z;W)$ and $\tilde f'(\tau,z;W)$ the formal power series associated to $f$ and $f'$ as in Corollary \ref{Choiecor}, respectively. Then, for any fixed complex number $X$, the formal power series $\tilde F_X(\tau,z;W)$ defined by $$\tilde F_X(\tau,z;W) = \tilde f (\tau,z;(1+m'X) W)\ \tilde f'(\tau,z;(1-m X) W) $$ satisfies, by Corollary \ref{Choiecor}, the functional equation stated in Theorem \ref{Choiethm} with $K=k+k'$ and $M=m+m'$. Furthermore, it is simple to check that its coefficients satisfy the expansion condition assumed in Theorem \ref{Choiethm} and, by Lemma \ref{Lmcom}, are invariant under the $|_m Y$ action. Hence the function $\tilde F_X(\tau,z;W)$ satisfies all assumptions of Theorem \ref{Choiethm} so that the corresponding functions $\zeta_\nu(\tau,z)$ are Jacobi forms of weight $k+k'+2\nu$ and index $m+m'$. It is now a simple exercise to see that these functions are just constant multiples of the Rankin-Cohen operators $[f,f']_{X,2\nu}$. This proves Theorem \ref{Jacobithm} for even $v=2\nu$. For the case of odd $v=2\nu+1$ consider the function $\tilde G_X(\tau,z;W)$ defined by \begin{eqnarray} \tilde G_X(\tau,z;W) &=& \ m'(\partial_z \tilde f (\tau,z;(1+m'X) W))\ (\tilde f'(\tau,z;(1-m X) W)) \nonumber\\ && -m (\tilde f (\tau,z;(1+m'X) W))\ (\partial_z \tilde f'(\tau,z;(1-m X) W)) \nonumber \end{eqnarray} where $X$ is a fixed complex number. Using again Lemma \ref{Lmcom} and Corollary \ref{Choiecor} we find that the function $\tilde G_X(\tau,z;W)$ satisfies the functional equation of Theorem \ref{Choiethm} with $K=k+k'+1$ and $M=m+m'$. By the same calculation as in the case of even $v$ the coefficients of $\tilde G_X(\tau,z;W)$ satisfy the expansion condition and are, by Lemma \ref{Lmcom}, invariant under the $|_{m+m'} Y$ action. Therefore, the corresponding functions $\zeta_\nu(\tau,z)$ are Jacobi forms of weight $k+k'+2\nu+1$ and index $m+m'$. These functions are just constant multiples of $[f,f']_{X,2\nu+1}$. This completes the proof of Theorem \ref{Jacobithm}. \qed \section{A proof of Theorem \ref{Siegelthm} using Theorem \ref{Jacobithm}} \label{SiegelproofJacobi} In this section we give a proof of Theorem \ref{Siegelthm} using Theorem \ref{Jacobithm}. \smallskip\noindent {\it Proof of Theorem \ref{Siegelthm}.} First we recall some well known facts about Siegel modular forms of degree $2$ which we need in the proof. Using $$Z = \left(\begin{array}{cc} \tau_1 & z\\ z & \tau_2 \end{array}\right)$$ for a variable in ${\Bbb H}} \def\P{{\Bbb P}_2$ the `$|^k_M$' action for the whole group ${{\rm{Sp}}}(4,\Z)$ (introduced in \S3) is generated by the following three slash actions (see {\it e.g.} page 73 of \cite{EZ}) \begin{eqnarray} f|^k_A(\tau_1,z,\tau_2) &=& (c\tau_1+d)^{-k} f(\frac{a\tau_1+b}{c\tau_1+d},\frac{z}{c\tau_1+d}, \tau_2-\frac{cz^2}{c\tau_1+d}) \nonumber \\ f|_{(\lambda,\nu)}(\tau_1,z,\tau_2) &=& f(\tau_1,z+\lambda \tau_1+\nu,\tau_2+2\lambda z +\lambda^2\tau_1) \nonumber \\ f|_{t}(\tau_1,z,\tau_2) &=& f(\tau_2,z,\tau_1) \nonumber \end{eqnarray} where $A = \left(\begin{array}{cc} a&b\\ c&d \end{array}\right)$ is in ${\mbox{SL}}(2,\Z)$ and $\lambda,\nu\in\Z$. Here we have identified, for any function $f$ on ${\Bbb H}} \def\P{{\Bbb P}_2$, $f(\tau_1,z,\tau_2)$ with $f(Z)$. For the proof of the theorem note that a simple calculation, using only the definition of the slash actions, shows that the operator $\D$ commutes with the $|_{(\lambda,\nu)}$ and $|_{t}$ action, respectively. Secondly, it is essential to realise that for a Jacobi form $f$ of index $m$ $$ \D( f \tilde q^m) = L_m(f) \tilde q^m $$ where $\tilde q =e^{2\pi i \tau_2}$ and $f$ is considered to be a function of $\tau_1$ and $z$. Consider now two Siegel modular forms $F$ and $F'$ of weight $k$ and $k'$, respectively. By Theorem \ref{Siegelexp} we can write $F$ and $F'$ as $$ F = \sum_{m \ge 0} f_m \tilde q^m,\qquad F' = \sum_{m'\ge 0} f'_{m'} \tilde q^{m'} $$ where $f_m$ and $f'_{m'}$ are Jacobi forms of weight $k$ and $k'$ and index $m$ and $m'$, respectively. Then $$[F,F']_l = \sum_{m,m'\ge 0} [f_m,f_{m'}]_{X=0,2l}\ \tilde q^{m+m'}.$$ By Theorem \ref{Jacobithm} the functions $[f_m,f_{m'}]_{X=0,v}$ are Jacobi forms of weight $k+k'+2l$ and index $m+m'$ so that the right hand side looks like the expansion of a Siegel form of degree $2$ and of weight $k+k'+2l$. In order to show that this is indeed the case the only additional property to be checked is that $[F,F']_l$ is symmetric in $\tau_1$ and $\tau_2$. This, however, is obvious from the definition of $[F,F']_l$ and the fact that $F$ and $F'$ are themselves Siegel modular forms. Hence, we have proved that $[\cdot,\cdot]_l$ maps $S^{(2)}_k\times S^{(2)}_{k'}$ to $S^{(2)}_{k+k'+2l}$. Finally, note that the first coefficient in the Jacobi-Fourier expansion of $[F,F']_l$ is $[f_0,f_{0'}]_{X=0,2l}$ which is identically zero for $l>0$. This implies that the image of $[\cdot,\cdot]_l$ is contained in the space of Siegel cusp forms for $l>0$ ({\it cf.} Remark \ref{Siegelcusp}). \qed \section{A proof of Theorem \ref{Siegelthm} using theta series} \label{SiegelproofTheta} In this section we give a proof of Theorem \ref{Siegelthm} using theta series with spherical coefficients and Theorem \ref{Ibuthm}. This proof also shows that the Rankin-Cohen operator $[\cdot,\cdot]_l$ is (up to multiplication by a constant) the only covariant bilinear differential operator for Siegel modular forms of degree $2$ which can be written in terms of the differential operator $\D$. The proof does not use Theorem \ref{Jacobithm} but instead Theorem \ref{Ibuthm} which essentially allows to assume that $F$ and $F'$ both can be written as theta series with harmonic coefficients. The proof is of independent interest since part of the calculations are valid for arbitrary degree $n$ of the Siegel modular forms involved. Firstly, we define the differential operator $\D$ for general degree $n$ $$ \D = \det(\partial_{\nu,\mu}) $$ where $\partial_{\nu,\mu} = (1+\delta_{\nu,\mu}) \frac{\partial}{\partial z_{\nu,\mu}} $ with $Z = (z_{\nu,\mu})$ the usual variable in the Siegel half plane ${\Bbb H}} \def\P{{\Bbb P}_n$. This operator has already been considered in the context of Siegel modular forms (see {\it e.g.} \cite{Frei}). Secondly, we study the action of $\D$ on theta series. \begin{lem} \label{trilem} Let $A$ be a symmetric matrix in $M_{n,n}$ and $Z$ a variable in ${\Bbb H}} \def\P{{\Bbb P}_n$. Then one has $$ \D\left( e^{\pi i\, \mbox{tr}(A Z)} \right) = (2\pi i)^n \, \det(A) \ e^{\pi i\, \mbox{tr}(A Z)} $$ \end{lem} {\it Proof.} The equality follows directly from $$ \partial_{\nu,\mu}\left( e^{\pi i\, \mbox{tr}(A Z)}\right) = (2\pi i) A_{\nu,\mu}\, e^{\pi i\, \mbox{tr}(A Z)} $$ with $A = (A_{\nu,\mu})$. \qed The proof of Theorem \ref{Siegelthm} will follow directly from the following Proposition and Theorem \ref{Ibuthm}. \begin{prop} \label{thprop} Let $F$ and $F'$ be Siegel modular forms of degree $2$ and weight $k$ and $k'$, respectively. Assume that $F$ and $F'$ can be written as theta series with harmonic coefficients. Then $[F,F']_l$ (defined in Theorem \ref{Siegelthm}) is a Siegel modular form of degree $2$ and weight $k+k'+2l$. \end{prop} {\it Proof.} Let $F$ and $F'$ be Siegel modular forms of degree $n=2$ and weight $k$ and $k'$, respectively which can be written as theta series with spherical coefficients. Then one has $$ F = \sum_{G\in M_{2m,n}(\Z)} Q (S^ {1/2}G )\, e^{\pi i \, \mbox{tr}(G^t S G Z)}, \qquad F' = \sum_{G'\in M_{2m',n}(\Z)} Q'(S'^{1/2}G')\, e^{\pi i \, \mbox{tr}(G'^tS' G' Z)} $$ for some symmetric, positive, even and unimodular matrices $S$ and $S'$, and spherical polynomials $Q$ and $Q'$ of weight $d$ and $d'$, respectively and $k=m+d$ and $k'=m'+d'$. Then, from the very definition of $[\cdot,\cdot]_l$ and Lemma \ref{trilem}, we have $$ [F,F']_l = (2\pi i)^{n l} \sum_{G,G'} e^{\pi i\, \mbox{tr}((G^t S G+G'^t S' G')Z)} \ \tilde P(S^{1/2}G,S'^{1/2}G') $$ with $\tilde P(X,X')= Q(X)\, Q'(X')\, P(X,X')$ where $$P(X,X') = \sum_{r+s+p=l} C_{r,s,v}(k,k') \det(X^t X)^r \det(X'^t X')^s \det(X^t X + X'^t X)^p. $$ We will now show that the polynomial $\tilde P$ is spherical of weight $w=d+d'+2l$ in the matrix variable $$ Y = \left( \begin{array}{cc} X\\ X' \end{array}\right).$$ If this is the case Theorem \ref{thetathm} implies that $[F,F']_l$ is a Siegel modular form of degree $2$ and weight $k+k'+2l$. Note that $\tilde P$ clearly satisfies the first property in the definition of a spherical polynomial with $w=d+d'+2l$. Therefore, we only have to show that $$ \Delta_Y \tilde P = (\Delta_{S^{1/2} G} + \Delta_{S'^{1/2}G'}) \tilde P = 0.$$ To calculate $\Delta_Y \tilde P$ we use a change of basis such that $S$ and $S'$ become equal to ${\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_{2m}$ and ${\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_{2m'}$, respectively. To show that $\tilde P$ is spherical we need to know the following expressions which can easily be obtained by a straightforward calculation (here we use that $n=2$ for the first time) \begin{eqnarray} \Delta_G( \det(G^t G) ) &=& 4(m-1/2)\, \mbox{tr}(G^t G), \nonumber \\ (\nabla_G( \det(G^t G) ))^2 &=& 4\,\mbox{tr}(G^t G)\, \det(G^t G), \nonumber \\ \Delta_G( \det(G^t G+G'^t G')) &=& 4(m-1/2)\, \mbox{tr}(G^t G)+4k\,\mbox{tr}(G'^tG'), \nonumber \\ (\nabla_G( \det(G^t G+G'^t G')))^2 &=& 4\, \mbox{tr}(G^t G+G'^t G')\, (\det(G^t G) - \det(G'^t G')) \nonumber \\ && + 4\, \mbox{tr}(G'^t G')\, \det(G^t G+G'^t G'), \nonumber \\ \nabla_G( \det(G^t G) ) \cdot \nabla_G( \det(G^t G+G'^t G') &=& 4\, \mbox{tr}(G^t G)\, \det(G^t G), \nonumber\\ \nabla_G(Q(G)) \cdot \nabla_G( \det(G^t G) ) &= & 2d\, Q(G)\, \mbox{tr}(G^t G), \nonumber\\ \nabla_G(Q(G)) \cdot \nabla_G( \det(G^t G+G'^t G'))&= & 2d\, Q(G)\, \mbox{tr}(G^t G+G'^t G'). \nonumber \end{eqnarray} Note the for deriving the last two expressions one has to use the second property in the definition of spherical polynomials with $$ A=\left( \begin{array}{cc} \lambda &0\\ 0&\lambda \end{array}\right),\quad A=\left( \begin{array}{cc} 1&0\\ \lambda & 1\end{array}\right) \quad {\rm{and}}\quad A=\left( \begin{array}{cc} 1&\lambda\\ 0 & 1\end{array}\right). $$ Let us, for the moment, use $D=\det(G^t G)$, $D' =\det(G'^t G')$ and $D'' = \det(G^t G+G'^t G')$. Then the last equalities imply that \begin{eqnarray} \frac{1}{4}(\Delta_G + \Delta_{G'})& \Big( Q(G) Q'(G') D^r D'^s D''^p \Big)\qquad \quad= & \nonumber\\ &r(\alpha +r)\, \mbox{tr}(G^t G) & Q(G) Q'(G') D^{r-1} D'^s D''^p \nonumber\\ + &s(\beta +s)\, \mbox{tr}(H^t H) & Q(G) Q'(G') D^r D'^{s-1} D''^p \nonumber\\ + &p(\gamma+l+p)\, \mbox{tr}(G^t G+H^t H) & Q(G) Q'(G') D^r D'^s D''^{p-1}. \nonumber \end{eqnarray} Here we have used $\alpha = k-3/2$, $\beta = k'-3/2$ and $\gamma = k+k'-3/2$. With these expressions it is clear that the equation $\Delta_X \tilde P=0$ is certainly satisfied if the coefficients $C_{r,s,p}$ obey \begin{eqnarray} 0 &=& (r+1)(\alpha+r+1)\, C_{r+1,s,p}(k,k') + (p+1)(\gamma+l+p+1)\, C_{r,s,p+1}(k,k') \nonumber\\ 0 &=& (s+1)(\beta +s+1)\, C_{r,s+1,p}(k,k') + (p+1)(\gamma+l+p+1)\, C_{r,s,p+1}(k,k'). \nonumber \end{eqnarray} It is simple to check that these conditions are indeed satisfied by the $C_{r,s,p}$ given in the theorem. \qed \begin{rem} Note that a covariant bilinear differential operator which can be written in terms of the operator $\D$ is, by the recursion relations obtained at the end of the proof of Proposition \ref{thprop}, equal to a multiple of $[\cdot,\cdot]_l$. Furthermore, the recursion relations imply the explicit form of the combinatorial factors $C_{r,s,p}$. \end{rem} \smallskip\noindent {\it Proof of Theorem \ref{Siegelthm}.} Firstly, note that the proof of Proposition \ref{thprop} implies for $Q=Q'=1$ that the polynomial $\tilde P = P$ is spherical of weight $2l$. Secondly, it can be written as $$P(X,X') = \tilde Q(X^t X, X'^t X')$$ with $$\tilde Q(a,b) = \sum_{r+s+p=l} C_{r,s,v}(k,k') \det(a+b)^p \det(a)^r \det(b)^s $$ so that it satisfies the assumptions of Theorem \ref{Ibuthm}. Finally, note that $ [F,F']_l$ can be written as $$ [F,F']_l = {\cal D}(F(Z) F(Z'))|_{Z=Z'} $$ where ${\cal D} = \tilde Q(\partial_{\nu,\mu},\partial_{\nu',\mu'}')$. This implies the desired result. \qed \newpage \section{Concluding remarks and open questions} \label{Conclusion} Let us end with some remarks and mention some open questions. \begin{enumerate} \item It is obvious from the proofs of the Theorems \ref{Jacobithm} and \ref{Siegelthm} that they also hold true for the case of Jacobi forms on $\Gamma{\Bbb n}\Z^2\subset \mbox{SL}(2,\Z){\Bbb n}\Z^2$ and Siegel forms on $\Gamma'\subset{{\rm{Sp}}}(4,\Z)$ if $\Gamma$ and $\Gamma'$ are finite index subgroups of $\mbox{SL}(2,\Z)$ and ${{\rm{Sp}}}(4,\Z)$, respectively. \item In the generic case, {\it i.e.}\ where $k$ and $k'$ are large enough, the dimension of the space of Rankin-Cohen operators for Jacobi forms is given by B\"ocherer's result so that Theorem \ref{Jacobithm} describes a basis of this space explicitly. This implies that, for $k$ and $k'$ large enough, the dimension of the space of Rankin-Cohen operators from $S^{(2)}_k\times S^{(2)}_{k'}$ to $S^{(2)}_{k+k'+v}$ is one dimensional for even $v$ and zero otherwise. (This can be verified by noting that any Rankin-Cohen operator for Siegel modular forms induces a Rankin-Cohen operator for Jacobi forms via the Jacobi-Fourier expansion (Theorem \ref{Siegelexp}). Hence it is enough to show that there is only one (up to multiplication by a constant) Rankin-Cohen operator for Jacobi forms for even $v$ and none for odd $v$ which can be `lifted' to a Rankin-Cohen operator for Siegel modular forms. In the general case, this can indeed be done using Theorem \ref{Jacobithm}.) \item Using the relation between Jacobi forms and modular forms of half-integral weight one can show that the operators $(\frac{d}{dX})^{v/2}[\cdot,\cdot]_{X,v}$ ($v\in 2{\Bbb N}} \def\Z{{\Bbb Z}$) can be obtained from the Rankin-Cohen operators for elliptic modular forms (for more details see \cite{Ch1}). It seems that this is the only Rankin-Cohen operator for Jacobi forms for which such a result holds true. \item The operators $(\frac{d}{dX})^{v/2}[\cdot,\cdot]_{X,v}$ ($v\in 2{\Bbb N}} \def\Z{{\Bbb Z}$) can be used to define generalised Rankin-Cohen algebras \cite{CE} which have very similar properties to the Rankin-Cohen algebras in the elliptic case considered in ref.\ \cite{Z}. \item It would be interesting to understand how our constructions (via generating functions or theta series) can be generalised to higher Jacobi and Siegel modular forms and multilinear differential operators. We hope to discuss this in a future publication. \item Is it possible to obtain the explicit formulae for the Rankin-Cohen operators in the case of Siegel modular froms from the representation theory of $\mbox{Sp}(4,\R)$? In this context the Rankin-Cohen operators can be viewed as certain projection operators. \item Is it possible to obtain the dimension of the space of covariant bilinear operators from $S^{(2)}_k\times S^{(2)}_{k'}$ to $S^{(2)}_{k+k'+v}$ not using Jacobi forms? Of course one possibility would be to use theta series and rephrase the question in terms of the theory of invariants. \item Is there any connection between covariant bilinear operators for Jacobi forms and/or Siegel modular forms and automorphic pseudodifferential operators like in the case of elliptic modular forms considered in ref.\ \cite{CMZ}? \end{enumerate} \newpage {\bf Acknowledgements} We would like to thank R.E.\ Borcherds and D.\ Zagier for valuable discussions and S.\ B\"ocherer and T.\ Ibukiyama for making their results available prior to publication. This work was done during the first author was visiting the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge. Y.C. would like to thank the Department and, in particular, J.\ Coates for support and warm hospitality during her stay in Cambridge.

9611

alg-geom/9611005

en

https://arxiv.org/abs/alg-geom/9611005

alg-geom/9611005

Harry Tamvakis

Harry Tamvakis

Bott-Chern Forms and Arithmetic Intersections

22 pages, LaTex, fonts corrected, to appear in L'Enseignement Mathematique

Let \E : 0 --> S --> E --> Q --> 0 be a short exact sequence of hermitian vector bundles with metrics on S and Q induced from that on E. We compute the Bott-Chern form of \E corresponding to any characteristic class, assuming E is projectively flat. The result is used to obtain a new presentation of the Arakelov Chow ring of the arithmetic grassmannian.

alg-geom

\section{Introduction} Arakelov theory is an intersection theory for varieties over rings ${\cal O}_F$ of algebraic integers, analogous to the usual one over fields. The fundamental idea is that in order to have a good theory of intersection numbers, one has to include information at the infinite primes. The work of Arakelov in dimension two has been generalized by Gillet and Soul\'{e} to higher dimensional {\em arithmetic varieties} $X$, by which we mean regular, projective and flat schemes over $\mbox{Spec}{\Bbb Z}$. They define an {\em arithmetic Chow ring} $\widehat{CH}(X)_{{\Bbb Q}}$ whose elements are represented by cycles on $X$ together with Green currents on $X({\Bbb C})$. The theory is a blend of arithmetic, algebraic geometry and complex hermitian geometry. For example, the Faltings height of an arithmetic variety $X$ is realized as an `arithmetic degree' with respect to a hermitian line bundle over $X$. A {\em hermitian vector bundle} $\overline{E}=(E,h)$ over $X$ is an algebraic vector bundle $E$ on $X$ together with a hermitian metric $h$ on the corresponding holomorphic vector bundle $E({\Bbb C})$ on the complex manifold $X({\Bbb C})$. To such an object one associates arithmetic Chern classes $\widehat{c}(\overline{E})$ with values in $\widehat{CH}(X)$. These satisfy most of the usual properties of Chern classes, with one exception: given a short exact sequence of hermitian vector bundles \begin{equation} \label{seq} \overline{{\cal E}}:\ 0\rightarrow \overline{S}\rightarrow\overline{E}\rightarrow\overline{Q}\rightarrow 0 \end{equation} the class $\widehat{c}(\overline{S})\widehat{c}(\overline{Q})-\widehat{c}(\overline{E})$ vanishes when $\overline{E}$ is the orthogonal direct sum of $\overline{S}$ and $\overline{Q}$. In general however this difference is non-zero and is the image in $\widehat{CH}(X)$ of a differential form on $X({\Bbb C})$, the {\em Bott-Chern form} associated to the exact sequence $\overline{{\cal E}}$. These secondary characteristic classes were originally defined by Bott and Chern [BC] with applications to value distribution theory. They later occured in the work of Donaldson [Do] on Hermitian-Einstein metrics. Bismut, Gillet and Soul\'{e} [BiGS] gave a new axiomatic definition for Bott-Chern forms, suitable for use in arithmetic intersection theory. Given an exact sequence of hermitian holomorphic vector bundles as in (\ref{seq}), we have $c(\overline{S})c(\overline{Q})-c(\overline{E})=dd^c\eta$ for some form $\eta$; the Bott Chern form of $\overline{{\cal E}}$ is a natural choice of such an $\eta$. Calculating these forms is important because they give relations in the arithmetic Chow ring of an arithmetic variety. No systematic work has appeared on this; rather one finds scattered calculations throughout the literature (see for example [BC], [C1], [D], [GS2], [GSZ], [Ma], [Mo]). We confine ourselves to the case where the metrics on $S$ and $Q$ are induced from the one on $E$. Our goal is to give explicit formulas for the Bott-Chern forms corresponding to {\em any} characteristic class, when they can be expressed in terms of the characteristic classes of the bundles involved. This is not always possible as these forms are not closed in general; however the situation is completely understood when $E$ is a projectively flat bundle. The results build on the work of Bott, Chern, Cowen, Deligne, Gillet, Soul\'{e} and Maillot. Some of our calculations overlap with previous work, but with simpler proofs. The main application we give to arithmetic intersection theory is a new presentation of the Arakelov Chow ring of the grassmannian over $\mbox{Spec} {\Bbb Z}$. Maillot [Ma] gave a presentation of this ring and formulated an `arithmetic Schubert calculus'. We hope our work contributes towards a better understanding of these intersections. This paper is organized as follows. Section \ref{isf} is a review of some basic material on invariant and symmetric functions. In \S \ref{bcfs} we recall the hermitian geometry we will need, including the definition of Bott-Chern forms. The basic tool for calculating these forms is reviewed in \S \ref{cbcf}, with some applications that have appeared before in the literature. \S \ref{ait} is mainly an exposition of the arithmetic intersection theory that we require. The rest of the paper is new. In sections \ref{flatses} and \ref{projflat} we derive formulas for computing Bott-Chern forms of short exact sequences (with the induced metrics) for any characteristic class when $\overline{E}$ is flat or more generally projectively flat. We emphasize the central role played by the {\em power sum forms} in the results; to our knowledge this phenomenon has not been observed before. The combinatorial identities involving harmonic numbers that we encounter are also interesting. Sections 2-6 contain results in hermitian complex geometry and may be read without prior knowledge of Arakelov theory. \S \ref{grass} applies our calculations to obtain a presentation of the Arakelov Chow ring of the arithmetic grassmannian. This should be regarded as a companion paper to [T]; both papers will be part of the author's 1997 University of Chicago thesis. I wish to thank my advisor William Fulton for many useful conversations and exchanges of ideas. \section{Invariant and symmetric functions} \label{isf} The symmetric group $S_n$ acts on the polynomial ring ${\Bbb Z}[x_1,x_2,\ldots,x_n]$ by permuting the variables, and the ring of invariants ${\Lambda}(n)={\Bbb Z}[x_1,x_2,\ldots,x_n]^{S_n}$ is the ring of symmetric polynomials. For $B={\Bbb Q}$ or ${\Bbb C}$, let ${\Lambda}(n,B)={\Lambda}(n)\otimes_{{\Bbb Z}} B$. Let $e_k(x_1,\ldots,x_n)$ be the $k$-th elementary symmetric polynomial in the variables $x_1,\ldots,x_n$ and $\displaystyle p_k(x_1,\ldots,x_n)=\sum_i x_i^k$ the $k$-th power sum. The fundamental theorem on symmetric functions states that ${\Lambda}(n)={\Bbb Z}[e_1,\ldots,e_n]$ and that $e_1,\ldots,e_n$ are algebraically independent. For $\lambda$ a partition, i.e. a decreasing sequence $\lambda_1\geqslant \lambda_2\geqslant\cdots\geqslant\lambda_m$ of nonnegative integers, define $\displaystyle p_{\lambda}:=\prod_{i=1}^mp_{\lambda_i}$. It is well known that the $p_{\lambda}$'s form an additive ${\Bbb Q}$-basis for the ring of symmetric polynomials (cf. [M], \S 2). The two bases are related by Newton's identity: \begin{equation} \label{ni} p_k-e_1p_{k-1}+e_2p_{k-2}-\cdots+(-1)^kke_k=0. \end{equation} Another important set of symmetric functions related to the cohomology ring of grassmannians are the Schur polynomials. For a partition $\lambda$ as above, the Schur polynomial $s_{\lambda}$ is defined by \[ \displaystyle s_{\lambda}(x_1,\ldots,x_n)=\frac{1}{\Delta}\cdot \det(x_i^{\lambda_j+n-j})_{1\leqslant i,j\leqslant n}, \] where $\displaystyle\Delta=\prod_{1\leqslant i<j\leqslant n}(x_i-x_j)$ is the Vandermonde determinant. The $s_{\lambda}$ for all $\lambda$ of length $m\leqslant n$ form a ${\Bbb Z}$-basis of ${\Lambda}(n)$ (cf. [M], \S I.3). Let ${\Bbb C}[T_{ij}]$ $(1\leqslant i,j\leqslant n$) be the coordinate ring of the space $M_n({\Bbb C})$ of $n\times n$ matrices. $GL_n({\Bbb C})$ acts on matrices by conjugation, and we let $I(n)={\Bbb C}[T_{ij}]^{GL_n({\Bbb C})}$ denote the corresponding graded ring of invariants. There is an isomorphism $\tau : I(n)\rightarrow {\Lambda}(n,{\Bbb C})$ given by evaluating an invariant polynomial $\phi$ on the diagonal matrix diag$(x_1,\ldots,x_n)$. We will often identify $\phi$ with the the symmetric polynomial $\tau(\phi)$. We will need to consider invariant polynomials with rational coefficients; let $I(n,{\Bbb Q})\simeq{\Bbb Q}[x_1,x_2,\ldots,x_n]^{S_n}$ be the corresponding ring. Given $\phi\hspace{-2pt}\in\hspace{-2pt} I(n)$, let $\phi^{\prime}$ be a $k$-multilinear form on $M_n({\Bbb C})$ such that \[ \displaystyle \phi^{\prime}(gA_1g^{-1},\ldots,gA_kg^{-1})=\phi^{\prime}(A_1,\ldots,A_k) \] for $g\hspace{-2pt}\in\hspace{-2pt} GL(n,{\Bbb C})$ and $\phi(A)=\phi^{\prime}(A,A,\ldots,A)$. Such forms are most easily constructed for the power sums $p_k$ by setting \[ p_k^{\prime}(A_1,A_2,\ldots,A_k)=\mbox{Tr} (A_1A_2\cdots A_k). \] For $p_{\lambda}$ we can take $p_{\lambda}^{\prime}=\prod p_{\lambda_i}^{\prime}$. Since the $p_{\lambda}$'s are a basis of ${\Lambda}(n,{\Bbb Q})$, it follows that one can use the above constructions to find multilinear forms $\phi^{\prime}$ for any $\phi\hspace{-2pt}\in\hspace{-2pt} I(n)$. An explicit formula for $\phi^{\prime}$ is given by polarizing $\phi$: \[ \displaystyle \phi^{\prime}(A_1\ldots,A_k)=\frac{(-1)^k}{k!}\sum_{j=1}^k \sum_{i_1<\ldots<i_j}(-1)^j\phi(A_{i_1}+\ldots+A_{i_j}). \] Although above formula for $\phi^{\prime}$ is symmetric in $A_1\ldots,A_k$, this property is not needed for the applications that follow. \section{Hermitian differential geometry} \label{bcfs} Let $X$ be a complex manifold, $E$ a rank $n$ holomorphic vector bundle over $X$. Denote by $A^k(X,E)$ the $C^{\infty}$ sections of $\Lambda^kT^*X\otimes E$, where $T^*X$ denotes the cotangent bundle of $X$. In particular $A^k(X)$ is the space of smooth complex $k$-forms on $X$. Let $A^{p,q}(X)$ the space of smooth complex forms of type $(p,q)$ on $X$ and $A(X):=\bigoplus_pA^{p,p}(X)$. The decomposition $A^1(X,E)=A^{1,0}(X,E)\bigoplus A^{0,1}(X,E)$ induces a decomposition $D=D^{1,0}+D^{0,1}$ of each connection $D$ on $E$. Let $d=\partial+\overline{\partial}$ and $d^c=(\partial-\overline{\partial})/(4\pi i)$. Assume now that $E$ is equipped with a hermitian metric $h$. The pair $(E,h)$ is called a {\em hermitian vector bundle}. The metric $h$ induces a canonical connection $D=D(h)$ such that $D^{0,1}=\overline{\partial}_E$ and $D$ is {\em unitary}, i.e. \[ \displaystyle d\,h(s,t)=h(Ds,t)+h(s,Dt), \mbox{ for all } s,t\hspace{-2pt}\in\hspace{-2pt} A^0(X,E). \] The connection $D$ is called the {\em hermitian holomorphic connection} of $(E,h)$. $D$ can be extended to $E$-valued forms by using the Leibnitz rule: \[ \displaystyle D({\omega}\otimes s)=d{\omega}\otimes s+ (-1)^{\deg{\omega}}{\omega}\otimes Ds. \] The composite \[ \displaystyle K=D^2:A^0(X,E)\rightarrow A^2(X,E) \] is $A^0(X)$-linear; hence $K\hspace{-2pt}\in\hspace{-2pt} A^2(X,\mbox{End}(E))$. In fact $K=D^{1,1}\hspace{-2pt}\in\hspace{-2pt} A^{1,1}(X,\mbox{End}(E))$, because $D^{0,2}=\overline{\partial}^2_E=0$, so $D^{2,0}$ also vanishes by unitarity. $K$ is called the {\em curvature} of $D$. Given a hermitian vector bundle $\overline{E}=(E,h)$ and an invariant polynomial $\phi\hspace{-2pt}\in\hspace{-2pt} I(n)$ there is an associated differential form $\phi(\overline{E}):=\phi(\frac{i}{2\pi}K)$, defined locally by identifying $\mbox{End}(E)$ with $M_n({\Bbb C})$; $\phi(\overline{E})$ makes sense globally on $X$ since $\phi$ is invariant by conjugation. These differential forms are $d$ and $d^c$ closed and have the following properties (cf. [BC]): \noindent (i) The de Rham cohomology class of $\phi(\overline{E})$ is independent of the metric $h$ and coincides with the usual characteristic class from topology. \noindent (ii) For every holomorphic map $f:X\rightarrow Y$ of complex manifolds, \[ f^*(\phi(E,h))=\phi(f^*E,f^*h). \] One thus obtains the {\em Chern forms} $c_k(\overline{E})$ with $c_k=e_k(x_1, \ldots,x_n)$, the {\em power sum forms} $p_k(\overline{E})$, the {\em Chern character form} $ch(\overline{E})$ with $\displaystyle ch(x_1,\ldots,x_n)=\sum_i \exp(x_i)=\sum_k \frac{1}{k!}p_k$, etc. We fix some more notation: A direct sum $\overline{E}_1\bigoplus\overline{E}_2$ of hermitian vector bundles will always mean the orthogonal direct sum $(E_1\bigoplus E_2,h_1\oplus h_2)$. Let $\widetilde{A}(X)$ be the quotient of $A(X)$ by $\mbox{Im} \partial + \mbox{Im} \overline{\partial}$. If $\omega$ is a closed form in $A(X)$ the cup product $\wedge\omega: \widetilde{A}(X)\rightarrow\widetilde{A}(X)$ and the operator $dd^c:\widetilde{A}(X)\rightarrow A(X)$ are well defined. Let ${\cal E}:\ 0\rightarrow S \rightarrow E\rightarrow Q \rightarrow 0$ be an exact sequence of holomorphic vector bundles on $X$. Choose arbitrary hermitian metrics $h_S,h_E,h_Q$ on $S,E,Q$ respectively. Let \[ \displaystyle \overline{{\cal E}}=({\cal E},h_S,h_E,h_Q):\ 0\rightarrow \overline{S} \rightarrow \overline{E}\rightarrow \overline{Q}\rightarrow 0. \] Note that we do not in general assume that the metrics $h_S$ or $h_Q$ are induced from $h_E$. We say that $\overline{{\cal E}}$ is {\em split} when $(E,h_E)=(S\bigoplus Q,h_S\oplus h_Q)$ and ${\cal E}$ is the obvious exact sequence. Following [GS2], we have the following \begin{thm} \label{bc} Let $\phi\hspace{-2pt}\in\hspace{-2pt} I(n)$ be any invariant polynomial. There is a unique way to attach to every exact sequence $\overline{{\cal E}}$ a form $\widetilde{\phi}(\overline{{\cal E}})$ in $\widetilde{A}(X)$ in such a way that: \noindent {\em (i)} $dd^c\widetilde{\phi}(\overline{{\cal E}})=\phi(\overline{S}\bigoplus \overline{Q}) -\phi(\overline{E})$, \noindent {\em (ii)} For every map $f:X\rightarrow Y$ of complex manifolds, $\widetilde{\phi}(f^*(\overline{{\cal E}}))=f^*\widetilde{\phi}(\overline{{\cal E}})$, \noindent {\em (iii)} If $\overline{{\cal E}}$ is split, then $\widetilde{\phi}(\overline{{\cal E}})=0$. \end{thm} In [BC], Bott and Chern solved the equation $dd^c\widetilde{\phi}(\overline{{\cal E}})=\phi(\overline{S}\bigoplus \overline{Q}) -\phi(\overline{E})$ when the metrics on $S$ and $Q$ are induced from the metric on $E$. In [BiGS] a new axiomatic definition of these forms was given, more generally for an acyclic complex of holomorphic vector bundles on $X$. The following useful calculation is an immediate consequence of the definition ([GS2], Prop. 1.3.1): \begin{prop} \label{bcprop} Let $\phi$ and $\psi$ be two invariant polynomials. Then \[ \displaystyle \widetilde{\phi + \psi}(\overline{{\cal E}})=\widetilde{\phi}(\overline{{\cal E}})+\widetilde{\psi}(\overline{{\cal E}}). \] \[ \widetilde{\phi\psi}(\overline{{\cal E}})=\widetilde{\phi}(\overline{{\cal E}})\psi(\overline{E})+ \phi(\overline{S}\oplus \overline{Q})\widetilde{\psi}(\overline{{\cal E}})= \widetilde{\phi}(\overline{{\cal E}})\psi(\overline{S}\oplus \overline{Q})+ \phi(\overline{E})\widetilde{\psi}(\overline{{\cal E}}). \] \end{prop} {\bf Proof.} One checks that right hand side of these identities satisfies the three properties of Theorem \ref{bc} that characterize the left hand side. \hfill $\Box$ \medskip We will also need to know the behaviour of $\widetilde{c}$ when $\overline{{\cal E}}$ is twisted by a line bundle. The following is a consequence of [GS2], Prop. 1.3.3: \begin{prop} \label{twist} For any hermitian line bundle $\overline{L}$, \[ \displaystyle \widetilde{c_k}(\overline{{\cal E}}\otimes\overline{L})= \sum_{i=1}^k{n-i \choose k-i}\widetilde{c_i}(\overline{{\cal E}})c_1(\overline{L})^{k-i}. \] \end{prop} \section{Calculating Bott-Chern Forms} \label{cbcf} In this section we will consider an exact sequence \[ \displaystyle \overline{{\cal E}}:\ 0\rightarrow \overline{S} \rightarrow \overline{E}\rightarrow \overline{Q}\rightarrow 0. \] where the metrics on $\overline{S}$ and $\overline{Q}$ are induced from the metric on $E$. Let $r$, $n$ be the ranks of the bundles $S$ and $E$. Let $\phi\hspace{-2pt}\in\hspace{-2pt} I(n)$ be homogeneous of degree $k$. We will formulate a theorem for calculating the Bott-Chern form $\widetilde{\phi}(\overline{{\cal E}})$. This result follows from the work of Bott-Chern, M. Cowen, J. Bismut and Gillet-Soul\'{e}. Let $\phi^{\prime}$ be defined as in \S \ref{isf}. For any two matrices $A,B \hspace{-2pt}\in\hspace{-2pt} M_n({\Bbb C})$ set \[ \displaystyle \phi^{\prime}(A;B):=\sum_{i=1}^k\phi^{\prime}(A,A,\ldots,A,B_{(i)},A,\ldots,A), \] where the index $i$ means that $B$ is in the $i$-th position. Choose a local orthonormal frame $s=(s_1,s_2,\ldots,s_n)$ of $E$ such that the first $r$ elements generate $S$, and let $K(\overline{S})$, $K(\overline{E})$ and $K(\overline{Q})$ be the curvature matrices of $\overline{S}$, $\overline{Q}$ and $\overline{E}$ with respect to $s$. Let $K_S=\frac{i}{2\pi}K(\overline{S})$, $K_E=\frac{i}{2\pi}K(\overline{E})$ and $K_Q=\frac{i}{2\pi}K(\overline{Q})$. The matrix $K_E$ has the form \[ \displaystyle K_E= \left( \begin{array}{c|c} K_{11} & K_{12} \\ \hline K_{21} & K_{22} \end{array} \right) \] where $K_{11}$ is an $r\times r$ submatrix. Also consider the matrices \[ \displaystyle K_0= \left( \begin{array}{c|c} K_S & 0 \\ \hline K_{21} & K_Q \end{array} \right) \ \mbox{ and } \ J_r= \left( \begin{array}{c|c} Id_r & 0 \\ \hline 0 & 0 \end{array} \right). \] Let $u$ be a formal variable and $K(u):=uK_E+(1-u)K_0$. Finally, let $\phi^!(u)=\phi^{\prime}(K(u); J_r)$. We then have the following \begin{thm} \label{calc} \begin{equation} \label{cowen} \widetilde{\phi}(\overline{{\cal E}})=\int_0^1\frac{\phi^!(u)-\phi^!(0)}{u}\,du. \end{equation} \end{thm} \medskip {\bf Proof.} We prove that $\widetilde{\phi}(\overline{{\cal E}})$ as defined above satisfies axioms (i)-(iii) of Theorem \ref{bc}. The main step is the first axiom; this was essentially done in [BC]\, \S 4, when $\phi=c$ is the total Chern class. In the form (\ref{cowen}) (again for the total Chern class), the equation was given by M. J. Cowen in [C1] and [C2], while simplifying Bott and Chern's proof. We follow both sources in sketching a proof of this more general result. Let $h$ and $h_Q$ denote the metrics on $E$ and $Q$ respectively. Define the orthogonal projections $P_1:\overline{E}\rightarrow \overline{S}$ and $P_2:\overline{E}\rightarrow \overline{Q}$ and put $h_u(v,v^{\prime})=uh(P_1v,P_1v^{\prime})+h(P_2v,P_2v^{\prime})$ for $v,v^{\prime}\hspace{-2pt}\in\hspace{-2pt} E_x$ and $0 < u\leqslant 1$. Then $h_u$ is a hermitian norm, $h_1=h$ and $h_u\rightarrow h_Q$ as $u\rightarrow 0$. Let $K(E,h_u)$ be the curvature matrix of $(E,h_u)$ relative to the holomorphic frame $s$ defined above. Proposition 3.1 of [C2] proves that $\frac{i}{2\pi}K(E,h_u)=K(u)$. It follows from Proposition 3.28 of [BC] that for $0 < t \leqslant 1$, \[ \displaystyle \phi(E,h_t)-\phi(E,h)= dd^c\int_t^1 \frac{\phi^{\prime}(K(u); J_r)}{u}\,du. \] If we could set $t=0$ we would be done; however, the integral will not be convergent in general. Note that $K(u)=K_0+uK_1$, where $K_1\hspace{-2pt}\in\hspace{-2pt} A^{1,1}(X,\mbox{End}(E))$ is independent of $u$. Therefore it will suffice to show that $\phi^{\prime}(K_0; J_r)$ is a closed form, so that it can be deleted from the integral. For this we may assume that $\phi=p_{\lambda}$ is a product of power sums, $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_m)$ a partition. Then \[ \displaystyle p_{\lambda}^{\prime}(K_0; J_r)= \sum_{i=1}^m \mbox{Tr}(K_S)^{\lambda_i-1} \prod_{j\neq i} (\mbox{Tr}(K_S)+\mbox{Tr}(K_Q))^{\lambda_j}= \sum_{i=1}^m p_{\lambda_i-1}(\overline{S})\prod_{j\neq i} p_{\lambda_j}(\overline{S}\oplus\overline{Q}) \] is certainly a closed form. This proves axioms (i) and (iii); axiom (ii) is easily checked as well. \hfill $\Box$ \medskip \noindent {\bf Remark.} A similar deformation to the one in [C2] was used by Deligne in [D], 5.11 for a calculation involving the Chern character form. Special cases of Theorem \ref{calc} have been used in the literature before, see for example [GS2] Prop. 5.3, [GSZ] 2.2.3 and [Ma] Theorem 3.3.1. \medskip We deduce some simple but useful calculations: \begin{cor} \label{c1} {\em (a)} $\widetilde{c_1^k}(\overline{{\cal E}})=0$ for all $k\geqslant 1$ and $\widetilde{c}_m(\overline{{\cal E}})=0$ for all $m > ${\em rk}$ E$. \noindent {\em (b)} $\widetilde{p_2}(\overline{{\cal E}})=2(${\em Tr}$K_{11}-c_1(\overline{S}))$ and $\widetilde{c_2}(\overline{{\cal E}})=c_1(\overline{S})-${\em Tr}$K_{11}$. \end{cor} {\bf Proof.} (a) $c^!_1(u)$ is independent of $u$; hence $\widetilde{c_1}(\overline{{\cal E}})=0$. The result for higher powers of $c_1$ follows from Proposition \ref{bcprop}. In addition, $\widetilde{c}_m(\overline{{\cal E}})=0$ for $m > \mbox{rk} E$ is an immediate consequence of the definition. \noindent (b) Using the bilinear form $p_2^{\prime}$ described previously, we find $p^!_2(u)=2(u\mbox{Tr} K_{11}+(1-u)c_1(\overline{S}))$, so \[ \displaystyle \widetilde{p_2}(\overline{{\cal E}})=2\int_0^1\frac{u\mbox{Tr} K_{11} +(1-u)c_1(\overline{S})-c_1(\overline{S})}{u}\,du=2(\mbox{Tr} K_{11}-c_1(\overline{S})). \] To calculate $\widetilde{c_2}(\overline{{\cal E}})$, use the identity $2c_2=c_1^2-p_2$.\ \hfill $\Box$ \medskip Corollary \ref{c1}(b) agrees with an important calculation of Deligne's in [D], 10.1, which we now describe: Using the $C^{\infty}$ splitting of ${\cal E}$, we can write the $\overline{\partial}$ operator for $E$ in matrix form: \[ \displaystyle \overline{\partial}_E=\left( \begin{array}{cc} \overline{\partial}_S & \alpha \\ 0 & \overline{\partial}_Q \end{array} \right), \ \ \ \mbox{for some } \alpha\hspace{-2pt}\in\hspace{-2pt} A^{0,1}(X,\mbox{Hom}(Q,S)). \] Let $\alpha^*\hspace{-2pt}\in\hspace{-2pt} A^{1,0}(X,\mbox{Hom}(S,Q))$ be the transpose of $\alpha$, defined using complex conjugation of forms and the metrics $h_S$ and $h_Q$. If $\nabla$ is the induced connection on $\mbox{Hom}(Q,S)$, we can write \[ \displaystyle K_E=\left( \begin{array}{c|c} K_S-\frac{i}{2\pi}\alpha\alpha^* & \nabla^{1,0}\alpha \\ \hline -\nabla^{0,1}\alpha^* & K_Q-\frac{i}{2\pi}\alpha^*\alpha \end{array} \right). \] Thus Corollary \ref{c1}(b) implies that \[ \widetilde{c_2}(\overline{{\cal E}})=-\frac{1}{2\pi i}\mbox{Tr} (\alpha\alpha^*)= \frac{1}{2\pi i}\mbox{Tr} (\alpha^*\alpha), \] and we have recovered Deligne's result. In this form the calculation was used by A. Moriwaki and C. Soul\'{e} to obtain a Bogomolov-Gieseker type inequality and a Kodaira vanishing theorem on arithmetic surfaces, respectively (see [Mo] and [S]). The calculation of $\widetilde{c_2}$ shows that in general Bott-Chern forms are not closed. In fact, calculating $\widetilde{c_k}$ for $k \geqslant 3$ leads to much more complicated formulas, involving traces of products of curvature matrices, for which a clear geometric interpretation is lacking (unlike the matrix $\alpha$ above, whose negative transpose $-\alpha^*$ is the second fundamental form of $\overline{{\cal E}}$). In the next two sections we shall see that when $\overline{E}$ is a projectively flat bundle, the Bott-Chern forms are closed and can be calculated explicitly for any $\phi\hspace{-2pt}\in\hspace{-2pt} I(n)$. \section{$0\rightarrow \overline{S} \rightarrow \overline{E}\rightarrow \overline{Q}\rightarrow 0$ with $\overline{E}$ flat} \label{flatses} Throughout this section we will assume that the hermitian vector bundle $\overline{E}$ is {\em flat}, i.e. that $K_E=0$. As before, the metrics $h_S$ and $h_Q$ will be induced from the metric on $E$. Define the {\em harmonic numbers} $\displaystyle {\cal H}_k=\sum_{i=1}^k\frac{1}{i}$, ${\cal H}_0=0$. Let $\lambda$ be a partition of $k$ (we denote this by $\lambda\vdash k$). Recall that the polynomials $\{p_{\lambda} : \lambda\vdash k\}$ form a ${\Bbb Q}$-basis for the vector space of symmetric homogeneous polynomials in $x_1,\ldots,x_n$ of degree $k$. The following result computes the Bott-Chern form corresponding to any such invariant polynomial: \begin{thm} \label{bcflat} The Bott-Chern class $\widetilde{p_{\lambda}}(\overline{{\cal E}})$ in $\widetilde{A}(X)$ is the class of \ \ \ \ \ \ \ \ \ \ {\em (i)} $k{\cal H}_{k-1}p_{k-1}(\overline{Q})$, if $\lambda=k=(k,0,0,\ldots,0)$ \ \ \ \ \ \ \ \ \ \ {\em (ii)} 0, otherwise. \end{thm} {\bf Proof.} Let us first compute $\widetilde{p_k}(\overline{{\cal E}})$ for $p_k(A)= \mbox{Tr} (A^k)$. Since $K_E=0$, the deformed matrix $K(u)=(1-u)K_{S\oplus Q}$, where $K_{S\oplus Q}= \left( \begin{array}{c|c} K_S & 0 \\ \hline 0 & K_Q \end{array} \right)$. Since \[ \displaystyle \int_0^1\frac{(1-u)^{k-1}-1}{u}\,du=-\int_0^1\frac{t^{k-1}-1}{t-1}\,dt= -{\cal H}_{k-1}, \] we obtain \[ \displaystyle \widetilde{p_k}(\overline{{\cal E}})=-{\cal H}_{k-1}p_k^{\prime}(K_{S\oplus Q}; J_r)= -k{\cal H}_{k-1}\mbox{Tr} (K_S^{k-1})=-k{\cal H}_{k-1}p_{k-1}(\overline{S}). \] Now since $p_k(\overline{S}\bigoplus\overline{Q})-p_k(\overline{E})$ is exact, $p_k(\overline{E})=0$ and $p_k(\overline{S}\bigoplus\overline{Q})= p_k(\overline{S})+p_k(\overline{Q})$, we conclude that $p_k(\overline{S})=-p_k(\overline{Q})$ in $\widetilde{A}(X)$, for each $k \geqslant 1$. This proves (i). Let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_m)$ be a partition ($m\geqslant 2$). Proposition \ref{bcprop} implies that \[ \displaystyle \widetilde{p_{\lambda}}(\overline{{\cal E}})=\widetilde{p_{\lambda_1}}(\overline{{\cal E}}) p_{\lambda_2}\cdots p_{\lambda_m}(\overline{S}\bigoplus\overline{Q}). \] But $\widetilde{p_{\lambda_1}}(\overline{{\cal E}})$ is a closed form (by (i)), and $p_{\lambda_2}\cdots p_{\lambda_m}(\overline{S}\bigoplus\overline{Q})$ is an exact form. Thus $\widetilde{p_{\lambda}}(\overline{{\cal E}})$ is exact, and so vanishes in $\widetilde{A}(X)$.\ \hfill $\Box$ \medskip It follows from Theorem \ref{bcflat} that for any $\phi \hspace{-2pt}\in\hspace{-2pt} I(n)$, the Bott-Chern form $\widetilde{\phi}(\overline{{\cal E}})$ is a linear combination of homogeneous components of the Chern character form $ch(\overline{Q})$. In [Ma], Theorem 3.4.1 we find the calculation \begin{equation} \label{maillotcalc} \widetilde{c_k}(\overline{{\cal E}})={\cal H}_{k-1}\sum_{i=0}^{k-1}ic_i(\overline{S})c_{k-1-i}(\overline{Q}) \end{equation} for the Chern forms $\widetilde{c_k}$. Our result gives the following \begin{prop} \label{maillotprop} $\widetilde{c_k}(\overline{{\cal E}})=(-1)^{k-1}{\cal H}_{k-1}p_{k-1}(\overline{Q})$. \end{prop} {\bf Proof.} By Newton's identity (\ref{ni}) we have \begin{equation} \label{newton} \widetilde{p_k}-\widetilde{c_1p_{k-1}}+\widetilde{c_2p_{k-2}}-\cdots+(-1)^kk\widetilde{c_k}=0. \end{equation} Reasoning as in Theorem \ref{bcflat}, we see that if $\phi$ and $\psi$ are two homogeneous invariant polynomials of positive degree, then $\widetilde{\phi\psi}(\overline{{\cal E}})=0$ in $\widetilde{A}(X)$. Thus (\ref{newton}) gives $\displaystyle \widetilde{c_k}(\overline{{\cal E}})=\frac{(-1)^{k-1}}{k}\widetilde{p_k}(\overline{{\cal E}})= (-1)^{k-1}{\cal H}_{k-1}p_{k-1}(\overline{Q})$. \hfill $\Box$ \medskip \noindent {\bf Remark.} The result of Proposition \ref{maillotprop} agrees with (\ref{maillotcalc}), i.e. $\displaystyle (-1)^kp_k(\overline{Q})=\sum_{i=0}^k ic_i(\overline{S})c_{k-i}(\overline{Q})$ in $\widetilde{A}(X)$. To see this, let $h(t)=\sum c_i(\overline{S})t^i$, $g(t)=\sum c_j(\overline{Q})t^j$, and $f(t)=\sum ic_i(\overline{S})t^i$. Then $h(t)g(t)=1$ in $\widetilde{A}(X)[t]$, and $f(t)=th^{\prime}(t)$. Choose formal variables $\{x_{\alpha}\}_{1\leqslant\alpha\leqslant r}$ and set $c_i(\overline{S})= e_i(x_1,\ldots,x_r)$, so that $\displaystyle h(t)=\prod_{\alpha}(1+x_{\alpha}t)$. Then $\displaystyle f(t)=\sum_{\alpha}tx_{\alpha}\prod_{\beta\neq\alpha} (1+x_{\beta}t)$. Thus \[ \displaystyle f(t)g(t)=\frac{f(t)}{h(t)}=\sum_{\alpha}\frac{x_{\alpha}t}{1+x_{\alpha}t}= r-\sum_{\alpha}\frac{1}{1+x_{\alpha}t}= \] \[ \displaystyle r-\sum_{\alpha,i}(-1)^ix_{\alpha}^it^i= r-\sum_i(-1)^ip_i(\overline{S})t^i=r+\sum_i(-1)^ip_i(\overline{Q})t^i. \] Comparing coefficients of $t^k$ on both sides gives the result. \medskip We can use Theorem \ref{bcflat} to calculate $\widetilde{\phi} (\overline{{\cal E}})$ for $\phi\hspace{-2pt}\in\hspace{-2pt} I(n)_k$: it is enough to find the coefficient of the power sum $p_k$ when $\phi$ is expressed as a linear combination of $\{p_{\lambda}\}_{\lambda\vdash k}$ in ${\Lambda}(n,{\Bbb Q})$. For example, we have \begin{cor} \label{appl0} $\displaystyle \widetilde{ch}(\overline{{\cal E}})=\sum_k{\cal H}_k ch_k(\overline{Q})$, where $ch_k$ denotes the $k$-th homogeneous component of the Chern character form. \end{cor} \begin{cor} \label{appl1} Let $\lambda$ be a partition of $k$ and $s_{\lambda}$ the corresponding Schur polynomial in ${\Lambda}(n,{\Bbb Q})$. Then $\widetilde{s_{\lambda}}(\overline{{\cal E}})=0$ unless $\lambda$ is a hook $\lambda_i=(i,1,1,\ldots,1)$, in which case $\widetilde{s_{\lambda_i}}(\overline{{\cal E}})=(-1)^{k-i}{\cal H}_{k-1}p_{k-1}(\overline{Q})$. \end{cor} {\bf Proof.} The proof is based on the Frobenius formula \[ \displaystyle s_{\lambda}=\frac{1}{k!}\sum_{\sigma\in S_k}\chi_{\lambda}(\sigma) p_{(\sigma)} \] where $(\sigma)$ denotes the partition of $k$ determined by the cycle structure of $\sigma$ (cf. [M], \S I.7). By the above remark, $\widetilde{s_{\lambda}}(\overline{{\cal E}})=\chi_{\lambda}((12\ldots k)){\cal H}_{k-1} p_{k-1}(\overline{Q})$. Using the combinatorial rule for computing $\chi_{\lambda}$ found in [M], p. 117, Example 5, we obtain \[ \displaystyle \chi_{\lambda}((12\ldots k))= \left\{ \begin{array}{cl} (-1)^{k-i}, & \mbox{if } \lambda=\lambda_i \mbox{ is a hook} \\ 0, & \mbox{otherwise}. \end{array} \right. \] \hfill $\Box$ The most natural instance of a sequence $\overline{{\cal E}}$ with $\overline{E}$ flat is the classifying sequence over the grassmannian $G(r,n)$. As we shall see in \S \ref{grass}, the calculation of Bott-Chern forms for this sequence leads to a presentation of arithmetic intersection ring of the arithmetic grassmannian over $\mbox{Spec}{\Bbb Z}$. \section{Calculations when $\overline{E}$ is projectively flat} \label{projflat} We will now generalize the results of the last section to the case where $E$ is {\em projectively flat}, i.e. the curvature matrix $K_E$ of $\overline{E}$ is a multiple of the identity matrix: $K_E=\omega Id_n$. This is true if $\displaystyle E=\overline{L}^{\oplus n}$ for some hermitian line bundle $\overline{L}$, with $\omega=c_1(\overline{L})$ the first Chern form of $\overline{L}$. The Bott-Chern forms (for the induced metrics) are always closed in this case as well, and will be expressed in terms of characteristic classes of the bundles involved. However this seems to be the most general case where this phenomenon occurs. The key observation is that for projectively flat bundles, the curvature matrix $K_E=\omega Id_n$ in {\em any} local trivialization. Thus we have \[ \displaystyle K(u)= \left( \begin{array}{c|c} (1-u)K_S+u\omega Id_r & 0 \\ \hline 0 & (1-u)K_Q+u\omega Id_s \end{array} \right) \] where $s=n-r$ denotes the rank of $Q$. Now Theorem \ref{calc} gives \[ \displaystyle \widetilde{p_k}(\overline{{\cal E}})= k\int_0^1\frac{1}{u}\mbox{Tr}[(u\omega Id_r+(1-u)K_S)^{k-1}-K_S^{k-1}]\,du= \] \[ -k{\cal H}_{k-1}p_{k-1}(\overline{S})+k\sum_{j=1}^{k-1}{k-1 \choose j} \mbox{Tr}(\omega^jK_S^{k-1-j}) \int_0^1u^{j-1}(1-u)^{k-j-1}\,du. \] Integrating by parts gives $\displaystyle \int_0^1u^m(1-u)^n\,du=\frac{1}{m+n+1}{m+n \choose n}^{-1}$, thus \begin{equation} \label{pkS} \frac{1}{k}\widetilde{p_k}(\overline{{\cal E}})= -{\cal H}_{k-1}p_{k-1}(\overline{S})+\sum_{j=1}^{k-1}\frac{\omega^j}{j} p_{k-1-j}(\overline{S}). \end{equation} We can rewrite this as an equation involving power sums of the quotient bundle: since $p_k(\overline{S})+p_k(\overline{Q})-p_k(\overline{L}^{\oplus n})=0$ in $\widetilde{A}(X)$, we have $p_k(\overline{S})=n\omega^k-p_k(\overline{Q})$. Thus (\ref{pkS}) becomes \begin{equation} \label{psum} \frac{1}{k}\widetilde{p_k}(\overline{{\cal E}})= {\cal H}_{k-1}p_{k-1}(\overline{Q})-\sum_{j=1}^{k-1}\frac{\omega^j}{j} p_{k-1-j}(\overline{Q}). \end{equation} \begin{thm} \label{appl2} Let $X$ be a complex manifold, $\overline{E}$ a projectively flat hermitian vector bundle over $X$. Let $0\rightarrow \overline{S}\rightarrow\overline{E}\rightarrow\overline{Q}\rightarrow 0$ a short exact sequence of vector bundles over $X$ with metrics on $S$, $Q$ induced from $\overline{E}$. Then for any invariant polynomial $\phi\hspace{-2pt}\in\hspace{-2pt} I(n)$, $\phi(\overline{S}\bigoplus\overline{Q})=\phi(\overline{E})$ as differential forms on $X$. \end{thm} {\bf Proof.} Since the $p_{\lambda}$ form an additive basis for $I(n)$, it suffices to prove the result when $\phi=p_{\lambda}$. The above calculation shows that $\widetilde{p_k}$ is a closed form. This combined with Proposition \ref{bcprop} shows that $\widetilde{p_{\lambda}}$ is closed for any partition $\lambda$. Thus \[ \displaystyle p_{\lambda}(\overline{S}\bigoplus\overline{Q})-p_{\lambda}(\overline{E})= dd^c\widetilde{p_{\lambda}}=0. \] \hfill $\Box$ \noindent {\bf Remark.} If $E$ is a trivial vector bundle, this result follows by pulling back the exact sequence $\overline{{\cal E}}$ from the classifying sequence on the Grassmannian. The forms are equal there because they are invariant with respect to the $U(n)$ action, so harmonic. \medskip The Bott-Chern forms $\widetilde{p_{\lambda}}$ for a general partition $\lambda=(\lambda_1,\ldots,\lambda_m)$ can be computed by using Proposition \ref{bcprop}. If $|\lambda|=\sum\lambda_i=k$ then we have \begin{equation} \label{ppsum} \displaystyle \widetilde{p_{\lambda}}(\overline{{\cal E}})=\sum_{i=1}^m \widetilde{p_{\lambda_i}}(\overline{{\cal E}}) \prod_{j\neq i} p_{\lambda_j}(\overline{E})= n^{m-1}\sum_{i=1}^m\omega^{k-\lambda_i}\widetilde{p_{\lambda_i}}(\overline{{\cal E}}). \end{equation} In principle equations (\ref{psum}) and (\ref{ppsum}) can be used to compute $\widetilde{\phi}(\overline{{\cal E}})$ for any characteristic class $\phi$. We now find a more explicit formula for the Bott-Chern forms of Chern classes. The computation is not as straightforward, as the argument of Proposition \ref{maillotprop} does not apply. Since by Theorem \ref{calc} the calculation depends only on the curvature matrices $K_E$, $K_S$ and $K_Q$, we may assume \[ \displaystyle \overline{{\cal E}}\ : \ \ 0\rightarrow \overline{S} \rightarrow \overline{L}\otimes{\Bbb C}^n\rightarrow \overline{Q}\rightarrow 0 \] is our chosen sequence, and define a new sequence \[ \displaystyle \overline{{\cal E}^{\prime}}=\overline{{\cal E}}\otimes\overline{L}^*\ : \ 0\rightarrow \overline{S}\otimes\overline{L}^* \rightarrow{\Bbb C}^n\rightarrow \overline{Q}\otimes\overline{L}^* \rightarrow 0. \] The metrics on the bundles in $\overline{{\cal E}^{\prime}}$ are induced from the trivial metric on ${\Bbb C}^n$. Using Propositions \ref{twist} and \ref{maillotprop} now gives \[ \displaystyle \widetilde{c_k}(\overline{{\cal E}})= \widetilde{c_k}(\overline{{\cal E}^{\prime}}\otimes\overline{L})= \sum_{i=1}^k{n-i \choose k-i}\widetilde{c_i}(\overline{{\cal E}^{\prime}})c_1(\overline{L})^{k-i}= \] \[ \displaystyle \sum_{i=1}^k{n-i \choose k-i}(-1)^{i-1}{\cal H}_{i-1} p_{i-1}(\overline{Q}\otimes\overline{L}^*)\omega^{k-i}= \] \[ \displaystyle \sum_{i=1}^k\sum_{j=0}^{i-1}(-1)^j{n-i \choose k-i}{i-1 \choose j}{\cal H}_{i-1} \omega^{k-1-j}p_j(\overline{Q})= \] \[ \displaystyle \sum_{j=0}^{k-1}(-1)^jd_j\omega^{k-1-j}p_j(\overline{Q}), \] where \[ \displaystyle d_j=\sum_{i=j+1}^k{n-i \choose k-i}{i-1 \choose j}{\cal H}_{i-1}. \] To find a closed form for the sum $d_j$, we can use the general identity \begin{equation} \label{harm} \sum_{i=q-s}^{n-p}{n-i \choose p}{s+i \choose q}{\cal H}_{s+i}= {n+s+1 \choose p+q+1}({\cal H}_{n+s+1}-{\cal H}_{p+q+1}+{\cal H}_p). \end{equation} This is identity (10) in [Sp]. In passing we note that writing equation (\ref{harm}) without the harmonic number terms: \[ \displaystyle \sum_{i=q-s}^{n-p}{n-i \choose p}{s+i \choose q}={n+s+1 \choose p+q+1} \] gives a well known identity among binomial coefficients. Applying (\ref{harm}) to $d_j$ and replacing $k$ by $k+1$ and $j$ by $k-i$ we arrive at the formula \[ \displaystyle \widetilde{c_{k+1}}(\overline{{\cal E}})= \sum_{i=0}^k (-1)^{k-i} {n \choose i} {\frak H}_i \omega^i p_{k-i}(\overline{Q}), \] where ${\frak H}_i={\cal H}_n-{\cal H}_{n-i}+{\cal H}_{k-i}$. As remarked previously, this calculation is valid for any projectively flat bundle $\overline{E}$ with $c_1(\overline{E})=n\omega$. Of course one can use the above method to compute the Bott-Chern form $\widetilde{p_k}(\overline{{\cal E}})$ as well; however this leads to a more complicated formula than (\ref{psum}). Equating the two proves(!) the following interesting combinatorial identity (compare [Sp], identity (30)): \begin{equation} \label{identity} \sum_{i=0}^s(-1)^{i+1} {n \choose i,s-i} {\cal H}_{n-s+i}=\frac{1}{s}\ \ \ \ \ \ \ \ (n\geqslant s). \end{equation} Here $\displaystyle {n \choose i,j}$ is a trinomial coefficient. The following summarizes the calculations of this section: \begin{thm} \label{bccalcthm} Let $X$ be a complex manifold, $\overline{E}$ a projectively flat hermitian vector bundle over $X$, with $c_1(\overline{E})=n\omega$. Let $0\rightarrow \overline{S}\rightarrow\overline{E}\rightarrow\overline{Q}\rightarrow 0$ be a short exact sequence of vector bundles over $X$ with metrics on $S$, $Q$ induced from $\overline{E}$. Then \[ \displaystyle \widetilde{p_{k+1}}(\overline{{\cal E}})= (k+1){\cal H}_kp_k(\overline{Q})-(k+1)\sum_{i=1}^k \frac{\omega^i}{i} p_{k-i}(\overline{Q}) \] \[ \displaystyle \widetilde{c_{k+1}}(\overline{{\cal E}})= \sum_{i=0}^k (-1)^{k-i} {n \choose i} {\frak H}_i \omega^i p_{k-i}(\overline{Q}) \] where ${\frak H}_i={\cal H}_n-{\cal H}_{n-i}+{\cal H}_{k-i}$. \end{thm} Note that the formulas in Theorem \ref{bccalcthm} reduce to the ones of the previous section when $\omega=0$! \section{Arithmetic intersection theory} \label{ait} We recall here the generalization of Arakelov theory to higher dimensions due to H. Gillet and C. Soul\'{e}. Our main references are [GS1], [GS2] and the exposition in [SABK]. For $A$ an abelian group, $A_{{\Bbb Q}}$ denotes $A\otimes_{{\Bbb Z}}{\Bbb Q}$. Let $X$ be an {\em arithmetic scheme over ${\Bbb Z}$}, by which we mean a regular scheme, projective and flat over $\mbox{Spec}{\Bbb Z}$. For $p\geqslant 0$, let $X^{(p)}$ be the set of integral subschemes of $X$ of codimension $p$ and $Z^p(X)$ be the group of codimension $p$ cycles on $X$. The $p$-th Chow group of $X$: $CH^p(X):=Z^p(X)/R^p(X)$, where $R^p(X)$ is the subgroup of $Z^p(X)$ generated by the cycles $\mbox{div}f$, $f\hspace{-2pt}\in\hspace{-2pt} k(x)^*$, $x\hspace{-2pt}\in\hspace{-2pt} X^{(p-1)}$. Let $CH(X)=\bigoplus_pCH^p(X)$. If $X$ is smooth over $\mbox{Spec}{\Bbb Z}$, then the methods of [F] can be used to give $CH(X)$ the structure of a commutative ring. In general one has a product structure on $CH(X)_{{\Bbb Q}}$ after tensoring with ${\Bbb Q}$. Let $D^{p,p}(X({\Bbb C}))$ denote the space of complex currents of type $(p,p)$ on $X({\Bbb C})$, and $F_{\infty}:X({\Bbb C})\rightarrow X({\Bbb C})$ the involution induced by complex conjugation. Let $D^{p,p}(X_{{\Bbb R}})$ (resp. $A^{p,p}(X_{{\Bbb R}})$) be the subspace of $D^{p,p}(X({\Bbb C}))$ (resp. $A^{p,p}(X({\Bbb C}))$) generated by real currents (resp. forms) $T$ such that $F^*_{\infty}T=(-1)^pT$; denote by $\widetilde{D}^{p,p}(X_{{\Bbb R}})$ and $\widetilde{A}^{p,p}(X_{{\Bbb R}})$ the respective images in $\widetilde{D}^{p,p}(X({\Bbb C}))$ and $\widetilde{A}^{p,p}(X({\Bbb C}))$. An {\em arithmetic cycle} on $X$ of codimension $p$ is a pair $(Z,g_Z)$ in the group $Z^p(X)\bigoplus\widetilde{D}^{p-1,p-1}(X_{{\Bbb R}})$, where $g_Z$ is a {\em Green current} for $Z({\Bbb C})$, i.e. a current such that $dd^cg_Z+\delta_{Z({\Bbb C})}$ is represented by a smooth form. The group of arithmetic cycles is denoted by $\widehat{Z}^p(X)$. If $x\hspace{-2pt}\in\hspace{-2pt} X^{(p-1)}$ and $f\hspace{-2pt}\in\hspace{-2pt} k(x)^*$, we let $\widehat{\mbox{div}}f$ denote the arithmetic cycle $(\mbox{div}f, [-\log|f_{{\Bbb C}}|^2\cdot \delta_{x({\Bbb C})}])$. The {\em $p$-th arithmetic Chow group of $X$}: $\widehat{CH}^p(X):= \widehat{Z}^p(X)/\widehat{R}^p(X)$, where $\widehat{R}^p(X)$ is the subgroup of $\widehat{Z}^p(X)$ generated by the cycles $\widehat{\mbox{div}}f$, $f\hspace{-2pt}\in\hspace{-2pt} k(x)^*$, $x\hspace{-2pt}\in\hspace{-2pt} X^{(p-1)}$. Let $\widehat{CH}(X)=\bigoplus_p\widehat{CH}^p(X)$. We have the following canonical morphisms of abelian groups: \[ \displaystyle \zeta :\widehat{CH}^p(X) \longrightarrow CH^p(X), \ \ \ {[(Z,g_Z)]} \longmapsto {[Z]}, \] \[ \displaystyle \omega : \widehat{CH}^p(X) \longrightarrow \mbox{Ker} d\cap \mbox{Ker} d^c\hspace{0.2cm} (\subset A^{p,p}(X_{{\Bbb R}})), \ \ \ {[(Z,g_Z)]} \longmapsto dd^cg_Z+\delta_{Z({\Bbb C})}, \] \[ \displaystyle a : \widetilde{A}^{p-1,p-1}(X_{{\Bbb R}}) \longrightarrow \widehat{CH}^p(X), \ \ \ \eta \longmapsto {[(0,\eta)]}. \] One can define a pairing $\widehat{CH}^p(X)\otimes\widehat{CH}^q(X) \rightarrow \widehat{CH}^{p+q}(X)_{{\Bbb Q}}$ which turns $\widehat{CH}(X)_{{\Bbb Q}}$ into a commutative graded unitary ${\Bbb Q}$-algebra. The maps $\zeta$, $\omega$ are ${\Bbb Q}$-algebra homomorphisms. If $X$ is smooth over ${\Bbb Z}$ one does not have to tensor with ${\Bbb Q}$. The definition of this pairing is difficult; the construction uses the {\em star product} of Green currents, which in turn relies upon Hironaka's resolution of singularities to get to the case of divisors. The functor $\widehat{CH}^p(X)$ is contravariant in $X$, and covariant for proper maps which are smooth on the generic fiber. Choose a K\"{a}hler form $\omega_0$ on $X({\Bbb C})$ such that $F^*_{\infty}\omega_0= -\omega_0$ (this is equivalent to requiring that the corresponding K\"ahler metric is invariant under $F_{\infty}$). It is natural to utilize the theory of harmonic forms on $X$ in the study of Green currents on $X({\Bbb C})$. Following [GS1], we call the pair $\overline{X}=(X,\omega_0)$ an {\em Arakelov variety}. By the Hodge decomposition theorem, we have $A^{p,p}(X_{{\Bbb R}})={\cal H}^{p,p}(X_{{\Bbb R}})\oplus \mbox{Im}d\oplus \mbox{Im}d^*$, where ${\cal H}^{p,p}(X_{{\Bbb R}})=\mbox{Ker}\Delta\subset A^{p,p}(X)$ denotes the space of harmonic (with respect to $\omega_0$) $(p,p)$ forms $\alpha$ on $X({\Bbb C})$ such that $F_{\infty}^*\alpha=(-1)^p\alpha$. The subgroup $CH^p(\overline{X}):=\omega^{-1}({\cal H}^{p,p}(X_{{\Bbb R}}))$ of $\widehat{CH}^p(X)$ is called the {\em $p$-th Arakelov Chow group of $X$}. Let $CH(\overline{X})=\bigoplus_{p\geqslant 0} CH^p(\overline{X})$. $CH^p(\overline{X})$ is a direct summand of $\widehat{CH}^p(X)$, and there is an exact sequence \begin{equation} \label{ex1} CH^{p,p-1}(X) \stackrel{\rho}\longrightarrow {\cal H}^{p-1,p-1}(X_{{\Bbb R}}) \stackrel{a}\longrightarrow CH^p(\overline{X}) \stackrel{\zeta}\longrightarrow CH^p(X)\longrightarrow 0. \end{equation} In the above sequence the group $CH^{p,p-1}(X)$ is defined as the $E_2^{p,1-p}$ term of a certain spectral sequence used by Quillen to calculate the higher algebraic $K$-theory of $X$, and the map $\rho$ coincides with the Beilinson regulator map (cf. [G] and [GS1], 3.5). If ${\cal H}(X_{{\Bbb R}})=\bigoplus_p {\cal H}^{p,p}(X_{{\Bbb R}})$ is a subring of $\bigoplus_p A^{p,p}(X_{{\Bbb R}})$, for example if $X({\Bbb C})$ is a curve, an abelian variety or a hermitian symmetric space (e.g. a grassmannian), then $CH(\overline{X})_{{\Bbb Q}}$ is a subring of $\widehat{CH}(X)_{{\Bbb Q}}$. This is not the case in general; for example it fails to be true for the complete flag varieties. Arakelov [A] introduced the group $CH^1(\overline{X})$, where $\overline{X}=(X,g_0)$ is an arithmetic surface with the metric $g_0$ on the Riemann surface $X({\Bbb C})$ given by $ \frac{i}{2g}\sum \omega_j\wedge \overline{\omega}_j$. Here $g$ is the genus of $X({\Bbb C})$ and $\{\omega_j\}$ for $1\leqslant j \leqslant g$ is an orthonormal basis of the space of holomorphic one forms on $X({\Bbb C})$. A {\em hermitian vector bundle} $\overline{E}=(E,h)$ on an arithmetic scheme $X$ is an algebraic vector bundle $E$ on $X$ such that the induced holomorphic vector bundle $E({\Bbb C})$ on $X({\Bbb C})$ has a hermitian metric $h$, which is invariant under complex conjugation, i.e. $F_{\infty}^*(h)=h$. To any hermitian vector bundle one can attach characteristic classes $\widehat{\phi}(\overline{E})\hspace{-2pt}\in\hspace{-2pt} \widehat{CH}(X)_{{\Bbb Q}}$, for any $\phi\hspace{-2pt}\in\hspace{-2pt} I(n,{\Bbb Q})$, where $n=\mbox{rk} E$. For example, we have {\em arithmetic Chern classes} $\widehat{c}_k(\overline{E}) \hspace{-2pt}\in\hspace{-2pt} \widehat{CH}^k(X)$. Some basic properties of these classes are: \noindent (1) $\widehat{c}_0(\overline{E})=1$ and $\widehat{c}_k(\overline{E})=0$ for $k>\mbox{rk}E$. \noindent (2) The form $\omega(\widehat{c}_k(\overline{E}))=c_k(\overline{E})\hspace{-2pt}\in\hspace{-2pt} A^{k,k}(X_{{\Bbb R}})$ is the $k$-th Chern form of the hermitian bundle $\overline{E({\Bbb C})}$. \noindent (3) $\zeta(\widehat{c}_k(\overline{E}))=c_k(E)\hspace{-2pt}\in\hspace{-2pt} CH^k(X)$. \noindent (4) $f^*\widehat{c}_k(\overline{E})=\widehat{c}_k(f^*\overline{E})$, for every morphism $f:X\rightarrow Y$ of regular schemes, projective and flat over ${\Bbb Z}$. \noindent (5) If $\overline{L}$ is a hermitian line bundle, $\widehat{c}_1(\overline{L})$ is the class of $(\mbox{div}(s),-\log \|s\|^2)$ for any rational section $s$ of $L$. Analogous properties are satisfied by $\widehat{\phi}$ for any $\phi\hspace{-2pt}\in\hspace{-2pt} I(n,{\Bbb Q})$ (see [GS2], Th. 4.1). The most relevant property of these characteristic classes is their behaviour in short exact sequences: if \[ \displaystyle \overline{{\cal E}} :\ 0 \rightarrow \overline{S} \rightarrow \overline{E} \rightarrow \overline{Q} \rightarrow 0 \] is such a sequence of hermitian vector bundles over $X$, then \begin{equation} \label{key} \widehat{\phi}(\overline{S}\oplus\overline{Q})- \widehat{\phi}(\overline{E})=a(\widetilde{\phi}(\overline{{\cal E}})). \end{equation} Relation (\ref{key}) is the main tool for calculating intersection products of classes in $\widehat{CH}(X)$ that come from characteristic classes of vector bundles. Combining it with the results of \S 4 and \S 5 gives immediate consequences for such intersections. For example, we have \begin{cor} \label{appl3} Let $\overline{{\cal E}}: 0\rightarrow \overline{S}\rightarrow\overline{E}\rightarrow\overline{Q}\rightarrow 0$ be a short exact sequence of hermitian vector bundles over an arithmetic scheme $X$. Assume that the metrics on $S({\Bbb C})$, $Q({\Bbb C})$ are induced from that on $E({\Bbb C})$. \noindent (a) If $\overline{E({\Bbb C})}$ is flat, then \[ \displaystyle (1)\ \ \ \widehat{p_{\lambda}}(\overline{S}\oplus\overline{Q})=\widehat{p_{\lambda}}(\overline{E}), \ \mbox{ if } \lambda \mbox{ has length } >1, \mbox{ and} \] \[ \displaystyle (2)\ \ \ \widehat{p_k}(\overline{S})+\widehat{p_k}(\overline{Q})-\widehat{p_k}(\overline{E})= k{\cal H}_{k-1}a(p_{k-1}(\overline{Q})), \ \forall k\geqslant 1, \] in the arithmetic Chow group $\widehat{CH}(X)_{{\Bbb Q}}$. \noindent (b) If $\overline{E}=\overline{L}^{\oplus n}$ for some hermitian line bundle $\overline{L}$ and $\omega=c_1(\overline{L({\Bbb C})})$, then \[ \displaystyle \widehat{c}(\overline{S})\widehat{c}(\overline{Q})-\widehat{c}(\overline{E})= \sum_{i,j} (-1)^j {n \choose i} ({\cal H}_n-{\cal H}_{n-i}+{\cal H}_j) a(\omega^i p_j(\overline{Q})), \] in the arithmetic Chow group $\widehat{CH}(X)$. \end{cor} \section{Arakelov Chow rings of grassmannians} \label{grass} In this section $G=G(r,n)$ will denote the grassmannian over $\mbox{Spec}{\Bbb Z}$. Over any field $k$, $G$ parametrizes the $r$-dimensional linear subspaces of a vector space over $k$. Let \begin{equation} \label{mex} \overline{{\cal E}}:\ 0\rightarrow \overline{S} \rightarrow \overline{E}\rightarrow \overline{Q} \rightarrow 0 \end{equation} denote the universal exact sequence of vector bundles over $G$. Here the trivial bundle $E({\Bbb C})$ is given the trivial metric and the tautological subbundle $S({\Bbb C})$ and quotient bundle $Q({\Bbb C})$ the induced metrics. The homogeneous space $G({\Bbb C})\simeq U(n)/(U(r)\times U(n-r))$ is a complex manifold. $G({\Bbb C})$ is endowed with a natural $U(n)$-invariant metric coming from the K\"{a}hler form $\eta_G=c_1(\overline{Q({\Bbb C})})$. $G$ is a smooth arithmetic scheme and $G({\Bbb C})$ with the metric coming from $\eta_G$ is a hermitian symmetric space, so we have an Arakelov Chow ring $CH(\overline{G})$. Note that since the hermitian vector bundles in (\ref{mex}) are invariant under the action of $U(n)$, their Chern forms are harmonic, and thus the arithmetic characteristic classes obtained are all elements of $CH(\overline{G})$. V. Maillot [Ma] found a presentation of $CH(\overline{G})$, using the above observation and the short exact sequence (\ref{ex1}). We wish to offer another description of this ring, based on the calculations in this paper. First recall the geometric picture: for the ordinary Chow ring we have \begin{equation} \label{chowg} CH(G)=\frac{{\Bbb Z}[c(S),c(Q)]}{\left<c(S)c(Q)=1\right>}. \end{equation} (see for instance [F], Example 14.6.6). If $x_1,\ldots,x_r$ are the Chern roots of $S$, $y_1,\ldots,y_s$ are the Chern roots of $Q$, $H=S_r\times S_{n-r}$ is the product of two symmetric groups, and $t$ is a formal variable, then (\ref{chowg}) can be rewritten \begin{equation} \label{chow} CH(G)=\frac{{\Bbb Z}[x_1,\ldots,x_r,y_1,\ldots,y_s]^H} {\left<\prod_i(1+x_it)\prod_j(1+y_jt)=1\right>}. \end{equation} Maillot's presentation of $CH(\overline{G})$ is an analogue of (\ref{chowg}); ours will be an analogue of (\ref{chow}). We introduce $2n$ variables \[ \widehat{x}_1,\ldots,\widehat{x}_r,\widehat{y}_1,\ldots,\widehat{y}_s, x_1,\ldots,x_r,y_1,\ldots,y_s \] and consider the rings \[ A={\Bbb Z}[\widehat{x}_1,\ldots,\widehat{x}_r,\widehat{y}_1,\ldots,\widehat{y}_s]^H\ \ \mbox{ and } \ \ B={\Bbb R}[x_1,\ldots,x_r,y_1,\ldots,y_s]^H \] and the ring homomorphism $\omega:A\rightarrow B$ defined by $\omega(\widehat{x}_i)=x_i$ and $\omega(\widehat{y}_j)=y_j$. A ring structure is defined on the abelian group $A\oplus B$ by setting \[ \displaystyle (\widehat{x},x^{\prime})*(\widehat{y},y^{\prime})= (\widehat{x}\widehat{y},\omega(\widehat{x})y^{\prime}+ x^{\prime}\omega(\widehat{y})). \] We will adopt the convention that $\widehat{\alpha}$ denotes $(\widehat{\alpha},0)$, $\beta$ denotes $(0,\beta)$, and any product $\prod x_iy_j$ denotes $(0,\prod x_iy_j)$; the multiplication $*$ is thus characterized by the properties $\widehat{\alpha} * \beta=\alpha\beta$ and $\beta_1 * \beta_2=0$. We now define two sets of relations in $(A\oplus B)[t]$: \[ \displaystyle {\cal R}_1:\ \prod_i(1+x_it)\prod_j(1+y_jt)=1, \] \[ \displaystyle {\cal R}_2:\ \prod_i(1+\widehat{x}_it) * \prod_j(1+\widehat{y}_jt) * \left(1+t\sum_j\frac{\log(1+y_jt)}{1+y_jt}\right)=1. \] and let ${\cal A}$ denote the quotient of the graded ring $A\oplus B$ by these relations. Using this notation we can state \begin{thm} \label{presentation} There is a unique ring isomorphism $\Phi: {\cal A}\rightarrow CH(\overline{G})$ such that \[ \displaystyle \Phi(\prod_i (1+\widehat{x}_it^i))= \sum_i \widehat{c_i}(\overline{S})t^i,\ \ \ \ \Phi(\prod_j (1+\widehat{y}_jt^j))= \sum_j \widehat{c_j}(\overline{Q})t^j, \] \[ \displaystyle \Phi(\prod_i (1+x_it^i))= \sum_i a(c_i(\overline{S}))t^i,\ \ \ \ \Phi(\prod_j (1+y_jt^j))= \sum_j a(c_j(\overline{Q}))t^j. \] \end{thm} \medskip {\bf Proof.} The isomorphism $\Phi$ of ${\cal A}$ with $CH(\overline{G})$ is obtained exactly as in [Ma], Theorem 4.0.5. The key fact is that since $G$ has a cellular decomposition (in the sense of [F], Ex. 1.9.1.), it follows that $CH^{p,p-1}(G)=0$ for all $p$ (using the excision exact sequence for the groups $CH^{*,*}(G)$; cf. [G], \S 8). Summing the sequence (\ref{ex1}) over all $p$ gives \begin{equation} \label{equ} 0 \longrightarrow {\cal H}(G_{{\Bbb R}}) \stackrel{a}\longrightarrow CH(\overline{G}) \stackrel{\zeta}\longrightarrow CH(G)\longrightarrow 0. \end{equation} We can now use our knowledge of the rings ${\cal H}(G_{{\Bbb R}})$ and $CH(G)$ together with the five lemma, as in loc. cit. The multiplication $*$ is a consequence of the general identity $a(x)y=a(x\omega(y))$ in $\widehat{CH}(G)$. To complete the argument we must show that the relation $\widehat{c}(\overline{S})\widehat{c}(\overline{Q})=1+ a(\widetilde{c}(\overline{{\cal E}}))$ translates to the relation ${\cal R}_2$ above. Let $p_i(y)$ be the $i$-th power sum in the variables $y_1\ldots,y_s$, identified under $\Phi$ with the class $a(p_i(\overline{Q}))$ in $CH(\overline{G})$ (we will use such identifications freely in the sequel). We also define $\displaystyle p_a(t)=\sum_{i=0}^{\infty} (-1)^{i+1}{\cal H}_ip_i(y)t^{i+1}$. Proposition \ref{maillotprop} implies that \begin{equation} \label{star} \widehat{c}_t(\overline{S})\widehat{c}_t(\overline{Q})=1+a(\widetilde{c}_t(\overline{{\cal E}}))= 1-p_a(t), \end{equation} where the subscript $t$ denotes the corresponding Chern polynomial. Multiplying both sides of (\ref{star}) by $1+p_a(t)$ and using the properties of multiplication in ${\cal A}$ gives the equivalent form \begin{equation} \label{star1} \widehat{c}_t(\overline{S})*\widehat{c}_t(\overline{Q})*(1+p_a(t))=1. \end{equation} We now note that the {\em harmonic number generating function} \[ \displaystyle \sum_{i=0}^{\infty}{\cal H}_it^i=\frac{t}{1-t}+\frac{1}{2}\frac{t^2}{1-t}+ \frac{1}{3}\frac{t^3}{1-t}+\cdots= \frac{\log(1-t)}{t-1}. \] It follows that \[ \displaystyle p_a(-t)=\sum_{i=0}^{\infty}{\cal H}_ip_i(y)t^{i+1}= t\sum_{j=1}^s\sum_{i=0}^{\infty}{\cal H}_i(y_jt)^i= -t\sum_{j=1}^s\frac{\log(1-y_jt)}{1-y_jt} \] and thus \[ \displaystyle p_a(t)=t\sum_{j=1}^s\frac{\log(1+y_jt)}{1+y_jt}. \] Substituting this in equation (\ref{star1}) gives relation ${\cal R}_2$. \hfill $\Box$ \medskip Theorem \ref{presentation} shows that the relations in the Arakelov Chow ring of $G$ are the classical geometric ones perturbed by a new `arithmetic factor' of $1+p_a(t)$. While this factor is closely related to the power sums $p_i(\overline{Q})$, the most natural basis of symmetric functions for doing calculations in $CH(G)$ is the basis of Schur polynomials (corresponding to the Schubert classes; see for example [F], \S 14.7). The arithmetic analogues of the special Schubert classes involve the power sum perturbation above; multiplication formulas are thus quite complicated (see [Ma]). In geometry the Chern roots $x_i$ and $y_j$ all `live' on the complete flag variety above $G$. There are certainly natural line bundles on the flag variety whose first Chern classes correspond to the roots in Theorem \ref{presentation}. However on flag varieties the situation is more complicated and our knowledge is not as complete. We refer the reader to [T] for more details.

9611

alg-geom/9611006

en

https://arxiv.org/abs/alg-geom/9611006

alg-geom/9611006

Harry Tamvakis

Harry Tamvakis

Arithmetic Intersection Theory on Flag Varieties

26 pages, LaTex

Let F be the complete flag variety over Spec(Z) with the tautological filtration 0 \subset E_1 \subset E_2 \subset ... \subset E_n=E of the trivial bundle E over F. The trivial hermitian metric on E(\C) induces metrics on the quotient line bundles L_i(\C). Let \hat{c}_1(L_i) be the first Chern class of L_i in the arithmetic Chow ring \hat{CH}(F) and x_i = -\hat{c}_1(L_i). Let h(X_1,...,X_n) be a polynomial with integral coefficients in the ideal <e_1,...,e_n> generated by the elementary symmetric polynomials e_i. We give an effective algorithm for computing the arithmetic intersection h(x_1,...,x_n) in \hat{CH}(F), as the class of a SU(n)-invariant differential form on F(\C). In particular we show that all the arithmetic Chern numbers one obtains are rational numbers. The results are true for partial flag varieties and generalize those of Maillot for grassmannians. An `arithmetic Schubert calculus' is established for an `invariant arithmetic Chow ring' which specializes to the Arakelov Chow ring in the grassmannian case.

alg-geom

\section{Introduction} \label{intro} Arakelov theory is a way of `completing' a variety defined over the ring of integers of a number field by adding fibers over the archimedian places. In this way one obtains a theory of intersection numbers using an arithmetic degree map; these numbers are generally real valued. The work of Arakelov on arithmetic surfaces has been generalized to higher dimensions by H. Gillet and C. Soul\'{e}. This provides a link between number theory and hermitian complex geometry; the road is via arithmetic intersection theory. One of the difficulties with the higher dimensional theory is a lack of examples where explicit computations are available. The arithmetic Chow ring of projective space was studied by Gillet and Soul\'{e} ([GS2], \sec 5) and arithmetic intersections on the grassmannian by Maillot [Ma]. In this article we study arithmetic intersection theory on general flag varieties and solve two problems: (i) finding a method to compute products in the arithmetic Chow ring, and (ii) formulating an `arithmetic Schubert calculus' analogous to the geometric case. The grassmannian case is easier to work with because the fiber at infinity is a hermitian symmetric space. To the author's knowledge this work is the first to provide explicit calculations when the harmonic forms are not a subalgebra of the space of smooth forms. The question of computing arithmetic intersection numbers on flag manifolds was raised by C. Soul\'{e} in his 1995 Santa Cruz lectures [S]. We now describe our results in greater detail. The crucial case is that of complete flags, so we discuss that for simplicity. Let $F$ denote the complete flag variety over $\mbox{Spec}{\Bbb Z}$, parametrizing over any field $k$ the complete flags in a $k$-vector space of dimension $n$. Let $\overline{E}$ be the trivial vector bundle over $F$ equipped with a trivial hermitian metric on $E({\Bbb C})$. There is a tautological filtration \[ \displaystyle {\cal E}:\ E_0=0 \subset E_1 \subset E_2 \subset\cdots \subset E_n=E \] and the metric on $E$ induces metrics on all the subbundles $E_i$. We thus obtain a {\em hermitian filtration} $\overline{{\cal E}}$ with quotient line bundles $L_i=E_i/E_{i-1}$, which are also given induced metrics. Let $\widehat{CH}(F)$ be the arithmetic Chow ring of $F$ (see \sec \ref{ait} and [GS1], 4.2.3) and $\widehat{x}_i:=-\widehat{c}_1(\overline{L}_i)$, where $\widehat{c}_1(\overline{L}_i)$ is the arithmetic first Chern class of $\overline{L}_i$ ([GS2], 2.5). Let $h\hspace{-2pt}\in\hspace{-2pt}{\Bbb Z}[X_1,\ldots,X_n]$ be a polynomial in the ideal $\left<e_1,\ldots,e_n\right>$ generated by the elementary symmetric polynomials $e_i(X_1,\ldots,X_n)$. Our main result is a computation of the arithmetic intersection $h(\widehat{x}_1,\ldots,\widehat{x}_n)$ in $\widehat{CH}(F)$, as a class corresponding to a $SU(n)$-invariant differential form on $F({\Bbb C})$. This enables one to reduce the computation of any intersection product in $\widehat{CH}(F)$ to the level of smooth differential forms; we show how to do this explicitly for products of classes $\widehat{c}_i(\overline{E_l/E_k})$. In particular, we obtain the following result: Let $k_i$, $1\leqslant i \leqslant n$ be nonnegative integers with $\sum k_i= \dim{F}={n \choose 2}+1$. Then the arithmetic Chern number $ \displaystyle \widehat{\deg}(\widehat{x}_1^{k_1}\widehat{x}_2^{k_2}\cdots\widehat{x}_n^{k_n}) $ is a rational number. Let $CH(\overline{G_d})$ be the Arakelov Chow ring ([GS1], 5.1) of the grassmannian $G_d$ over $\mbox{Spec}{\Bbb Z}$ parametrizing $d$-planes in $E$, with the natural invariant K\"{a}hler metric on $G_d({\Bbb C})$. Maillot [Ma] gave a presentation of $CH(\overline{G_d})$ and constructed an `arithmetic Schubert calculus' in $CH(\overline{G_d})$. There are difficulties in extending his results to flag varieties, mainly because the Arakelov Chow group $CH(\overline{F})$ is not a {\em subring} of $\widehat{CH}(F)$. To overcome this problem we define, for any partial flag variety $F$, an `invariant arithmetic Chow ring' $\widehat{CH}_{inv}(F)$. This subring of $\widehat{CH}(F)$ specializes to the Arakelov Chow ring if $F({\Bbb C})$ is a hermitian symmetric space. We extend the notion of Bott-Chern forms for short exact sequences to filtered bundles. These forms give relations in $\widehat{CH}(F)$ (Theorem \ref{abc}); however they are generally not closed forms, and thus do not represent cohomology classes. This forces us to work on the level of differential forms in order to calculate arithmetic intersections. We compute the Bott-Chern forms on flag varieties $F$ by using a calculation of the curvature matrices of homogeneous vector bundles on generalized flag manifolds due to Griffiths and Schmid [GrS]. One thus obtains expressions for the Bott-Chern forms in terms of invariant forms on $F({\Bbb C})$. The Schubert polynomials of Lascoux and Sch\"{u}tzenberger provide a convenient basis to describe the product structure of $\widehat{CH}_{inv}(F)$. Using them we formulate an `arithmetic Schubert calculus' for flag varieties which generalizes that of Maillot [Ma] for grassmannians. However explicit general formulas are lacking, as we cannot do these computations using purely cohomological methods. This paper is organized as follows. In \sec \ref{prelim} we review some preliminary material on Bott-Chern forms, arithmetic intersection theory, flag varieties and Schubert polynomials. In \sec \ref{cbcfs} we state the main tool for computing Bott-Chern forms (for any characteristic class) in the case of induced metrics. The definition and construction the Bott-Chern forms associated to a hermitian filtration is the content of \sec \ref{bcfffb}. \sec \ref{hvb} is concerned with the explicit computation of the curvature matrices of the tautological vector bundles over flag varieties $F$. In \sec \ref{afvs} we define the invariant arithmetic Chow ring $\widehat{CH}_{inv}(F)$. This subring of $\widehat{CH}(F)$ is where all the intersections of interest take place. In \sec \ref{ai} we give an algorithm for calculating arithmetic intersection numbers on the complete flag variety $F$, in particular proving that they are all rational. In \sec \ref{asc} we describe the product structure of $\widehat{CH}_{inv}(F)$ in more detail, formulating an arithmetic Schubert calculus. Some applications of our results are given in \sec \ref{ex}. One has the Faltings height of the image of $F$ under its pluri-Pl\"{u}cker embedding; this is always a rational number. We give a table of the arithmetic Chern numbers for $F_{1,2,3}$. Finally \sec \ref{pfvs} shows how to generalize the previous results to partial flag varieties. This paper will be part of the author's 1997 University of Chicago thesis. I wish to thank my advisor William Fulton for many useful conversations and exchanges of ideas. The geometric aspects of this work generalize readily to other semisimple groups. We plan a sequel discussing arithmetic intersection theory on symplectic and orthogonal flag varieties. \section{Preliminaries} \label{prelim} \subsection{Bott-Chern forms} \label{bcfs} The main references for this section are [BC] and [GS2]. Consider the coordinate ring ${\Bbb C}[T_{ij}]$ $(1\leqslant i,j\leqslant n$) of the space $M_n({\Bbb C})$ of $n\times n$ matrices. $GL_n({\Bbb C})$ acts on matrices by conjugation; let $I(n)={\Bbb C}[T_{ij}]^{GL_n({\Bbb C})}$ denote the corresponding graded ring of invariants. There is an isomorphism $\sigma : I(n)\rightarrow {\Bbb C}[X_1,X_2,\ldots,X_n]^{S_n}$ obtained by evaluating an invariant polynomial $\phi$ on the diagonal matrix diag$(X_1,\ldots,X_n)$. We will often identify $\phi$ with the the symmetric polynomial $\sigma(\phi)$. We let $I(n,{\Bbb Q})=\sigma^{-1}({\Bbb Q}[X_1,X_2,\ldots,X_n]^{S_n})$. For $A$ an abelian group, $A_{{\Bbb Q}}$ denotes $A\otimes_{{\Bbb Z}}{\Bbb Q}$. Let $X$ be a complex manifold, and denote by $A^{p,q}(X)$ the space of differential forms of type $(p,q)$ on $X$. Let $A(X)=\bigoplus_p A^{p,p}(X)$ and $\widetilde{A}(X)$ be the quotient of $A(X)$ by $\mbox{Im} \partial + \mbox{Im} \overline{\partial}$. If $\omega$ is a closed form in $A(X)$ the cup product $\wedge\omega: \widetilde{A}(X)\rightarrow\widetilde{A}(X)$ and the operator $dd^c:\widetilde{A}(X)\rightarrow A(X)$ are well defined. Let $E$ be a rank $n$ holomorphic vector bundle over $X$, equipped with a hermitian metric $h$. The pair $\overline{E}=(E,h)$ is called a {\em hermitian vector bundle}. A direct sum $\overline{E}_1\bigoplus\overline{E}_2$ of hermitian vector bundles will always mean the orthogonal direct sum $(E_1\bigoplus E_2,h_1\oplus h_2)$. Let $D$ be the hermitian holomorphic connection of $\overline{E}$, with curvature $K=D^2\hspace{-2pt}\in\hspace{-2pt} A^{1,1}(X,\mbox{End}(E))$. If $\phi\hspace{-2pt}\in\hspace{-2pt} I(n)$ is any invariant polynomial, there is an associated differential form $\phi(\overline{E}):=\phi(\frac{i}{2\pi}K)$, defined locally by identifying $\mbox{End}(E)$ with $M_n({\Bbb C})$. These differential forms are $d$ and $d^c$ closed, have de Rham cohomology class independent of the metric $h$, and are functorial under pull back by holomorphic maps (cf. [BC]). In particular one obtains the {\em power sum forms} $p_k(\overline{E})$ with $\displaystyle p_k=\sum_i X_i^k$ and the {\em Chern forms} $c_k(\overline{E})$ with $c_k=e_k$ the $k$-th elementary symmetric polynomial. Let ${\cal E}:\ 0\rightarrow S \rightarrow E\rightarrow Q \rightarrow 0$ be an exact sequence of holomorphic vector bundles on $X$. Choose arbitrary hermitian metrics $h_S,h_E,h_Q$ on $S,E,Q$ respectively. Let \begin{equation} \label{ses} \overline{{\cal E}}=({\cal E},h_S,h_E,h_Q):\ 0\rightarrow \overline{S} \rightarrow \overline{E}\rightarrow \overline{Q}\rightarrow 0. \end{equation} We say that $\overline{{\cal E}}$ is {\em split} when $(E,h_E)=(S\bigoplus Q,h_S\oplus h_Q)$ and ${\cal E}$ is the obvious exact sequence. Let $\phi\hspace{-2pt}\in\hspace{-2pt} I(n)$ be any invariant polynomial. Then there is a unique way to attach to every exact sequence $\overline{{\cal E}}$ a form $\widetilde{\phi}(\overline{{\cal E}})$ in $\widetilde{A}(X)$, called the Bott-Chern form of $\overline{{\cal E}}$, in such a way that: \noindent (i) $dd^c\widetilde{\phi}(\overline{{\cal E}})=\phi(\overline{S}\bigoplus \overline{Q}) -\phi(\overline{E})$, \noindent (ii) For every map $f:X\rightarrow Y$ of complex manifolds, $\widetilde{\phi}(f^*(\overline{{\cal E}}))=f^*\widetilde{\phi}(\overline{{\cal E}})$, \noindent (iii) If $\overline{{\cal E}}$ is split, then $\widetilde{\phi}(\overline{{\cal E}})=0$. For $\phi$, $\psi \hspace{-2pt}\in\hspace{-2pt} I(n)$ one has the following useful relations in $\widetilde{A}(X)$: \begin{equation} \label{define} \widetilde{\phi + \psi}=\widetilde{\phi}+\widetilde{\psi}, \ \ \ \ \ \ \widetilde{\phi\psi}=\widetilde{\phi}\cdot\psi(\overline{S}\oplus \overline{Q})+ \phi(\overline{E})\cdot\widetilde{\psi}. \end{equation} \subsection{Arithmetic intersection theory} \label{ait} We recall here the generalization of Arakelov theory to higher dimensions due to H. Gillet and C. Soul\'{e}. For more details see [GS1], [GS2], [SABK]. Let $X$ be an {\em arithmetic scheme}, by which we mean a regular scheme, projective and flat over $\mbox{Spec}{\Bbb Z}$. For $p\geqslant 0$, we denote the Chow group of codimension $p$ cycles on $X$ modulo rational equivalence by $CH^p(X)$ and let $CH(X)=\bigoplus_p CH^p(X)$. $\widehat{CH}^p(X)$ will denote the $p$-th arithmetic Chow group of $X$. Recall that an element of $\widehat{CH}^p(X)$ is represented by an arithmetic cycle $(Z,g_Z)$; here $g_Z$ is a Green current for the codimension $p$ cycle $Z({\Bbb C})$. Let $\widehat{CH}(X)=\bigoplus_p \widehat{CH}^p(X)$. The involution of $X({\Bbb C})$ induced by complex conjugation is denoted by $F_{\infty}$. Let $A^{p,p}(X_{{\Bbb R}})$ be the subspace of $A^{p,p}(X({\Bbb C}))$ generated by real forms $\eta$ such that $F^*_{\infty}\eta=(-1)^p\eta$; denote by $\widetilde{A}^{p,p}(X_{{\Bbb R}})$ the image of $A^{p,p}(X_{{\Bbb R}})$ in $\widetilde{A}^{p,p}(X({\Bbb C}))$. Let $A(X_{{\Bbb R}})=\bigoplus_p A^{p,p}(X_{{\Bbb R}})$ and $\widetilde{A}(X_{{\Bbb R}})=\bigoplus_p \widetilde{A}^{p,p}(X_{{\Bbb R}})$. We have the following canonical morphisms of abelian groups: \[ \displaystyle \zeta :\widehat{CH}^p(X) \longrightarrow CH^p(X), \ \ \ {[(Z,g_Z)]} \longmapsto {[Z]}, \] \[ \displaystyle \omega : \widehat{CH}^p(X) \longrightarrow A^{p,p}(X_{{\Bbb R}}), \ \ \ {[(Z,g_Z)]} \longmapsto dd^cg_Z+\delta_{Z({\Bbb C})}, \] \[ \displaystyle a : \widetilde{A}^{p-1,p-1}(X_{{\Bbb R}}) \longrightarrow \widehat{CH}^p(X), \ \ \ \eta \longmapsto {[(0,\eta)]}. \] For convenience of notation, when we refer to a real differential form $\eta\hspace{-2pt}\in\hspace{-2pt} A(X_{{\Bbb R}})$ as an element of $\widehat{CH}(X)$, we shall always mean $a([\eta])$, where $[\eta]$ is the class of $\eta$ in $\widetilde{A}(X_{{\Bbb R}})$. There is an exact sequence \begin{equation} \label{ex1} CH^{p,p-1}(X) \longrightarrow \widetilde{A}^{p-1,p-1}(X_{{\Bbb R}}) \stackrel{a}\longrightarrow \widehat{CH}^p(X) \stackrel{\zeta}\longrightarrow CH^p(X)\longrightarrow 0 \end{equation} Here the group $CH^{p,p-1}(X)$ is the $E_2^{p,1-p}$ term of a spectral sequence used by Quillen to calculate the higher algebraic $K$-theory of $X$ (cf. [G]). One can define a pairing $\widehat{CH}^p(X)\otimes\widehat{CH}^q(X) \rightarrow \widehat{CH}^{p+q}(X)_{{\Bbb Q}}$ which turns $\widehat{CH}(X)_{{\Bbb Q}}$ into a commutative graded unitary ${\Bbb Q}$-algebra. The maps $\zeta$, $\omega$ are ${\Bbb Q}$-algebra homomorphisms. If $X$ is smooth over ${\Bbb Z}$ one does not have to tensor with ${\Bbb Q}$. The functor $\widehat{CH}^p(X)$ is contravariant in $X$, and covariant for proper maps which are smooth on the generic fiber. We also note the useful identity $a(x)y=a(x\omega(y))$ for $x,y\hspace{-2pt}\in\hspace{-2pt}\widehat{CH}(X)$. Choose a K\"{a}hler form ${\omega}_0$ on $X({\Bbb C})$ such that $F^*_{\infty}\omega_0=-{\omega}_0$ and let $\cal{H}^{p,p}(X_{{\Bbb R}})$ be the space of harmonic (with respect to $\omega_0$) $(p,p)$ forms on $X({\Bbb C})$ invariant under $F_{\infty}$. The $p$-th {\em Arakelov Chow group} of $\overline{X}=(X,{\omega}_0)$ is defined by $CH^p(\overline{X}):= \omega^{-1}(\cal{H}^{p,p}(X_{{\Bbb R}}))$. The group $CH(\overline{X})_{{\Bbb Q}}=\bigoplus_p CH^p(\overline{X})_{{\Bbb Q}}$ is generally not a subring of $\widehat{CH}(X)_{{\Bbb Q}}$, unless the harmonic forms $\cal{H}^*(X_{{\Bbb R}})$ are a subring of $A(X_{{\Bbb R}})$. This is true if $(X({\Bbb C}),{\omega}_0)$ is a hermitian symmetric space, such as a complex grassmannian, but fails for more general flag varieties. Let $f:X\rightarrow\mbox{Spec}{\Bbb Z}$ be the projection. If $X$ has relative dimension $d$ over ${\Bbb Z}$, then we have an arithmetic degree map $\widehat{\deg}:\widehat{CH}^{d+1}(X)\rightarrow{\Bbb R}$, obtained by composing the push-forward $f_*:\widehat{CH}^{d+1}(X)\rightarrow\widehat{CH}^1(\mbox{Spec}{\Bbb Z})$ with the isomorphism $\widehat{CH}^1(\mbox{Spec}{\Bbb Z})\stackrel{\sim}\rightarrow{\Bbb R}$. The latter maps the class of $(0,2\lambda)$ to the real number $\lambda$. A {\em hermitian vector bundle} $\overline{E}=(E,h)$ on an arithmetic scheme $X$ is an algebraic vector bundle $E$ on $X$ such that the induced holomorphic vector bundle $E({\Bbb C})$ on $X({\Bbb C})$ has a hermitian metric $h$ with $F_{\infty}^*(h)=h$. There are characteristic classes $\widehat{\phi}(\overline{E})\hspace{-2pt}\in\hspace{-2pt} \widehat{CH}(X)_{{\Bbb Q}}$ for any $\phi\hspace{-2pt}\in\hspace{-2pt} I(n,{\Bbb Q})$, where $n=\mbox{rk} E$. For example, we have {\em arithmetic Chern classes} $\widehat{c}_k(\overline{E}) \hspace{-2pt}\in\hspace{-2pt} \widehat{CH}^k(X)$. For the basic properties of these classes, see [GS2], Theorem 4.1. \subsection{Flag varieties and Schubert polynomials} \label{classgps} Let $k$ be a field, $E$ an $n$-dimensional vector space over $k$. Let \[ \displaystyle {\frak r}=(0<r_1<r_2<\ldots<r_m=n) \] be an increasing $m$-tuple of natural numbers. A {\em flag of type $r$} is a flag \begin{equation} \label{fil} {\cal E}:\ E_0=0 \subset E_1 \subset E_2 \subset\cdots \subset E_m=E \end{equation} with $\mbox{rank}{E_i}=r_i$, $1\leqslant i\leqslant m$. Let $F({\frak r})$ denote the arithmetic scheme parametrizing flags ${\cal E}$ of type ${\frak r}$ over any field $k$. (\ref{fil}) will also denote the tautological flag of vector bundles over $F({\frak r})$, and we call the resulting filtration of the bundle $E$ a {\em filtration of type ${\frak r}$}. The above {\em arithmetic flag variety} is smooth over $\mbox{Spec}{\Bbb Z}$. There is an isomorphism $F({\frak r})({\Bbb C})\simeq SL(n,{\Bbb C})/P$, where $P$ is the parabolic subgroup of $SL(n,{\Bbb C})$ stabilizing a fixed flag. In the extreme case $m=2$ (resp. $m=n$) $F({\frak r})$ is the grassmannian $G_d$ parametrizing $d$-planes in $E$ (resp. the complete flag variety $F$). Although the results of this paper are true for any partial flag variety $F({\frak r})$, for simplicity we will work with the complete flag variety $F$, leaving the discussion of the general case to \sec \ref{pfvs}. The notation for these varieties and the dimension $n$ will be fixed throughout this paper. We now recall the standard presentation of the Chow group $CH(F)$. Define the quotient line bundles $L_i=E_i/E_{i-1}$. Consider the polynomial ring $P_n={\Bbb Z}[X_1,\ldots,X_n]$ and the ideal $I_n$ generated by the elementary symmetric functions $e_i(X_1,\ldots,X_n)$. Then $CH(F)\simeq P_n/I_n$, where the inverse of this isomorphism sends $[X_i]$ to $-c_1(L_i)$. The ring $H_n=P_n/I_n$ has a free ${\Bbb Z}$-basis consisting of classes of monomials $X_1^{k_1}X_2^{k_2}\cdots X_n^{k_n}$, where the exponents $k_i$ satisfy $k_i\leqslant n-i$. The Schubert polynomials of Lascoux and Sch\"{u}tzenberger [LS] are a natural ${\Bbb Z}$-basis of $H_n$, corresponding to the classes of Schubert varieties in $CH(F)$. Our main reference for Schubert polynomials will be Macdonald's notes [M]. Let $S_{\infty}=\cup_nS_n$ and $P_{\infty}={\Bbb Z}[X_1,X_2,\ldots]$. For each $w\hspace{-2pt}\in\hspace{-2pt} S_{\infty}$, $l(w)$ denotes the {\em length} of $w$ and $\partial_w:P_{\infty}\rightarrow P_{\infty}$ the corresponding {\em divided difference operator}\ ([M] Chp. 2). If $w_0$ is the longest element of $S_n$ and $w\hspace{-2pt}\in\hspace{-2pt} S_n$ is arbitrary, the {\em Schubert polynomial} ${\frak S}_w$ is given by \[ \displaystyle {\frak S}_w=\partial_{w^{-1}w_0}(X_1^{n-1}X_2^{n-2}\cdots X_{n-1}). \] This definition is compatible with the natural inclusion $S_n\subset S_{n+1}$ (with $w(n+1)=n+1$). It follows that ${\frak S}_w$ is well defined for any $w\hspace{-2pt}\in\hspace{-2pt} S_{\infty}$. We let $\Lambda_n=P_n^{S_n}$ be the ring of symmetric polynomials. The set $\{{\frak S}_w\ | \ w\hspace{-2pt}\in\hspace{-2pt} S_n \}$ is both a free $\Lambda_n$-basis of $P_n$ and a free ${\Bbb Z}$-basis of $H_n$. Let $S^{(n)}$ denote the set of permutations $w\hspace{-2pt}\in\hspace{-2pt} S_{\infty}$ such that $w(n+1)<w(n+2)<\cdots$ Then $\{{\frak S}_w\ | \ w\hspace{-2pt}\in\hspace{-2pt} S^{(n)} \}$ is a free ${\Bbb Z}$-basis of $P_n$\ ([M], (4.13)). Define a $\Lambda_n$-valued scalar product on $P_n$ by $\left< f ,g \right>=\partial_{w_0}(fg)$, for $f,g\hspace{-2pt}\in\hspace{-2pt} P_n$. If $\{{\frak S}^w\}_{w\in S_n}$ is the $\Lambda_n$-basis of $P_n$ dual to the basis $\{{\frak S}_w\}$ relative to this product, then ${\frak S}^w(X)=w_0{\frak S}_{ww_0}(-X)$. \ ([M], (5.12)). For any $h\hspace{-2pt}\in\hspace{-2pt} I_n$ we have a decomposition $\displaystyle h=\sum_{w\in S_n}\left<h,{\frak S}^w\right>{\frak S}_w$, where each $\left<h,{\frak S}^w\right>$ is in $\Lambda_n\cap I_n$. \section{Calculating Bott-Chern forms} \label{cbcfs} Consider the short exact sequence $\overline{{\cal E}}$ in (\ref{ses}) and assume that the metrics on $\overline{S}$ and $\overline{Q}$ are induced from the metric on $E$. Let $r$, $n$ be the ranks of the bundles $S$ and $E$. For $\phi\hspace{-2pt}\in\hspace{-2pt} I(n)$ homogeneous of degree $k$ we let $\phi^{\prime}$ be a $k$-multilinear invariant form on $M_n({\Bbb C})$ such that $\phi(A)=\phi^{\prime}(A,A,\ldots,A)$. Such forms are most easily constructed for the power sums $p_k$, by defining \[ p_k^{\prime}(A_1,A_2,\ldots,A_k)=\mbox{Tr} (A_1A_2\cdots A_k). \] If $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_m)$ is a partition of $k$, define $\displaystyle p_{\lambda}:=\prod_{i=1}^mp_{\lambda_i}$. For $p_{\lambda}$ we can take $p_{\lambda}^{\prime}=\prod p_{\lambda_i}^{\prime}$. Since the $p_{\lambda}$'s are an additive ${\Bbb Q}$-basis for the ring of symmetric polynomials, we can use the above constructions to find multilinear forms $\phi^{\prime}$ for any $\phi\hspace{-2pt}\in\hspace{-2pt} I(n)$. For any two matrices $A,B \hspace{-2pt}\in\hspace{-2pt} M_n({\Bbb C})$ let \[ \displaystyle \phi^{\prime}(A;B):=\sum_{i=1}^k\phi^{\prime}(A,A,\ldots,A,B_{(i)},A,\ldots,A), \] where the index $i$ means that $B$ is in the $i$-th position. Consider a local orthonormal frame $s$ for $E$ such that the first $r$ elements generate $S$, and let $K(\overline{S})$, $K(\overline{E})$ and $K(\overline{Q})$ be the curvature matrices of $\overline{S}$, $\overline{Q}$ and $\overline{E}$ with respect to $s$. Let $K_S=\frac{i}{2\pi}K(\overline{S})$, $K_E=\frac{i}{2\pi}K(\overline{E})$ and $K_Q=\frac{i}{2\pi}K(\overline{Q})$. Write \[ \displaystyle K_E= \left( \begin{array}{c|c} K_{11} & K_{12} \\ \hline K_{21} & K_{22} \end{array} \right) \] where $K_{11}$ is an $r\times r$ submatrix, and consider the matrices \[ \displaystyle K_0= \left( \begin{array}{c|c} K_S & 0 \\ \hline K_{21} & K_Q \end{array} \right) \ \mbox{ and } \ J_r= \left( \begin{array}{c|c} Id_r & 0 \\ \hline 0 & 0 \end{array} \right). \] Let $u$ be a variable and define $K(u)=uK_E+(1-u)K_0$. We can then state the main computational \begin{prop} \label{calc} For $\phi\hspace{-2pt}\in\hspace{-2pt} I(n)$, we have \[ \displaystyle \widetilde{\phi}(\overline{{\cal E}})=\int_0^1\frac{\phi^{\prime}(K(u); J_r)- \phi^{\prime}(K(0); J_r)}{u}\,du. \] \end{prop} Proposition \ref{calc} is essentially a consequence of the work of Bott and Chern [BC], although we have not been able to find this general statement in the literature. For history and a complete proof, see [T]. \medskip What will prove most useful to us in the sequel is \begin{cor} \label{ratcor} For any $\phi\hspace{-2pt}\in\hspace{-2pt} I(n,{\Bbb Q})$ the Bott-Chern form $\widetilde{\phi}(\overline{{\cal E}})$ is a polynomial in the entries of the matrices $K_E$, $K_S$ and $K_Q$ with {\em rational} coefficients. \end{cor} {\bf Proof.} By the equations (\ref{define}) it suffices to prove this for $\phi=p_k$ a power sum. Using the bilinear form $p_k^{\prime}$ described previously in Proposition \ref{calc} gives \[ \displaystyle \widetilde{p}_k(\overline{{\cal E}})=k\int_0^1\frac{1}{u}\mbox{Tr}((K(u)^{k-1}-K(0)^{k-1})J_r)\,du, \] so the result is clear. \hfill $\Box$ \medskip Define the {\em harmonic numbers} $\displaystyle \cal{H}_k=\sum_{i=1}^k\frac{1}{i}$, $\cal{H}_0=0$. We will need the following useful calculations, which one can deduce from the definition and from Proposition \ref{calc}: \noindent (a) $\widetilde{c_1^k}(\overline{{\cal E}})=0$ for all $k$ and $\widetilde{c}_p(\overline{{\cal E}})=0$ for all $p > \mbox{rk} E$. \noindent (b) $\widetilde{c}_2(\overline{{\cal E}})=c_1(\overline{S})-\mbox{Tr} K_{11}$ (see [D], 10.1 and [T]). \noindent (c) If $E$ is flat, then $\displaystyle \widetilde{c}_k(\overline{{\cal E}})= \cal{H}_{k-1}\sum_{i=0}^{k-1}ic_i(\overline{S})c_{k-1-i}(\overline{Q})$ ([Ma], Th. 3.4.1). \section{Bott-Chern forms for filtered bundles} \label{bcfffb} In this section we will extend the definition of Bott-Chern forms for an exact sequence of bundles to the case of a filtered bundle. Let $X$ and $E$ be as in \sec \ref{bcfs}, and assume that $E$ has a filtration of type ${\frak r}$ \begin{equation} \label{fil2} {\cal E}:\ E_0=0 \subset E_1 \subset E_2 \subset\cdots \subset E_m=E \end{equation} by complex subbundles $E_i$, with ${\frak r}$ as in \sec \ref{classgps}. Let $Q_i=E_i/E_{i-1}$, $1\leqslant i\leqslant m$ be the quotient bundles. A {\em hermitian filtration $\overline{{\cal E}}$ of type ${\frak r}$} is a filtration (\ref{fil2}) together with a choice of hermitian metrics on $E$ and on each quotient bundle $Q_i$. Note that we do not assume that metrics have been chosen on the subbundles $E_1,\ldots, E_m$ or that the metrics on the quotients are induced from $E$ in any way. We say that $\overline{{\cal E}}$ is {\em split} if, when $E_i$ is given the induced metric from $E$ for each $i$, the sequence $\overline{{\cal E}_i} : 0\rightarrow \overline{E}_{i-1} \rightarrow \overline{E}_i \rightarrow \overline{Q}_i \rightarrow 0$ is split. In this case of course $\displaystyle \overline{E}=\bigoplus_i\overline{Q}_i$. \begin{thm} \label{bcf} Let $\phi\hspace{-2pt}\in\hspace{-2pt} I(n)$ be an invariant polynomial. There is a unique way to attach to every hermitian filtration of type ${\frak r}$ a form $\widetilde{\phi}(\overline{{\cal E}})$ in $\widetilde{A}(X)$ in such a way that: \noindent {\em (i)} $\displaystyle dd^c\widetilde{\phi}(\overline{{\cal E}})= \phi(\bigoplus_{i=1}^m \overline{Q}_i)-\phi(\overline{E})$, \noindent {\em (ii)} For every map $f:X\rightarrow Y$ of complex manifolds, $\widetilde{\phi}(f^*(\overline{{\cal E}}))=f^*\widetilde{\phi}(\overline{{\cal E}})$, \noindent {\em (iii)} If $\overline{{\cal E}}$ is {\em split}, then $\widetilde{\phi}(\overline{{\cal E}})=0$. If $m=2$, i.e. the filtration ${\cal E}$ has length 2, then $\widetilde{\phi}(\overline{{\cal E}})$ coincides with the Bott-Chern class $\widetilde{\phi}(0 \rightarrow \overline{Q}_1 \rightarrow \overline{E} \rightarrow \overline{Q}_2 \rightarrow 0)$ defined in {\em \sec \ref{bcfs}}. \end{thm} {\bf Proof.} The essential ideas are contained in [GS2], Th. 1.2.2 and sections 7.1.1, 7.1.2., so we will only sketch the argument. We first show that such forms exist. Given any hermitian filtration ${\cal E}$, equip each subbundle $E_i$ with the induced metric from $\overline{E}$ and consider the exact sequence \[ \displaystyle \overline{{\cal E}_i} : 0\rightarrow \overline{E}_{i-1} \rightarrow \overline{E}_i \rightarrow \overline{Q}_i \rightarrow 0. \] If $\widetilde{\phi}(\overline{{\cal E}})$ and $\widetilde{\psi}(\overline{{\cal E}})$ have already been defined then the equations \[ \displaystyle \widetilde{\phi + \psi}(\overline{{\cal E}})=\widetilde{\phi}(\overline{{\cal E}})+\widetilde{\psi}(\overline{{\cal E}}) \] \[ \widetilde{\phi\psi}(\overline{{\cal E}})=\widetilde{\phi}(\overline{{\cal E}})\psi(\bigoplus_{i=1}^m \overline{Q}_i)+ \phi(\overline{E})\widetilde{\psi}(\overline{{\cal E}}) \] can be used to define $\widetilde{\phi + \psi}$ and $\widetilde{\phi\psi}$ (see [GS2], Prop. 1.3.1 for the case $m=2$). Therefore it suffices to construct the Bott-Chern classes $\widetilde{p_k}$ for the power sums $p_k$. For this we simply let \begin{equation} \label{*} \widetilde{p_k}(\overline{{\cal E}}):=\sum_{i=1}^m\widetilde{p_k}(\overline{{\cal E}_i}). \end{equation} Since the $\widetilde{p_k}(\overline{{\cal E}_i})$ are functorial and additive on orthogonal direct sums, it is clear that (\ref{*}) satisfies (i)-(iii). The construction for $m=2$ gives the classes of \sec \ref{bcfs}. We will use a separate construction of the total Chern forms $\widetilde{c}(\overline{{\cal E}})$: For each $i$ with $1\leqslant i\leqslant m-1$, let $\overline{\cal{Q}}_i$ be the sequence $\displaystyle 0 \rightarrow 0 \rightarrow \bigoplus_{j=i+1}^m \overline{Q}_j \rightarrow \bigoplus_{j=i+1}^m \overline{Q}_j \rightarrow 0$, and let $\overline{{\cal E}_i^+}=\overline{{\cal E}_i}\bigoplus \overline{\cal{Q}}_i$. Let $\overline{{\cal E}_m^+}=\overline{{\cal E}_m}$. To each exact sequence $\overline{{\cal E}_i^+}$ we associate a Bott-Chern form $\widetilde{c}(\overline{{\cal E}_i^+})$. It follows from [GS2], Prop. 1.3.2 that \[ \displaystyle \widetilde{c}(\overline{{\cal E}_i^+})=\widetilde{c}(\overline{{\cal E}_i}\bigoplus \overline{\cal{Q}}_i) =\widetilde{c}(\overline{{\cal E}_i})c(\bigoplus_{j=i+1}^m\overline{Q_j}) =\widetilde{c}(\overline{{\cal E}_i})\wedge\bigwedge_{j=i+1}^mc(\overline{Q_j}). \] It is easy to see that $\displaystyle \widetilde{c}(\overline{{\cal E}}):= \sum_{i=1}^m \widetilde{c}(\overline{{\cal E}_i^+})$ satisfies (i)-(iii). To prove that the form $\widetilde{\phi}(\overline{{\cal E}})$ is unique, one constructs a deformation of the filtration $\overline{{\cal E}}$ to the split filtration, as in [GS2], \sec 7.1.2. Let $\overline{{\cal O}(1)}$ be the canonical line bundle on ${\Bbb P}^1={\Bbb P}^1({\Bbb C})$ with its natural Fubini-Study metric and let $\sigma$ be a section of ${\cal O}(1)$ vanishing only at $\infty$. Let $p_1$, $p_2$ be the projections from $X\times{\Bbb P}^1$ to $X$, ${\Bbb P}^1$ respectively. We denote by $E$, $E_i$, $Q_i$ and ${\cal O}(1)$ the bundles $p_1^*E$, $p_1^*E_i$, $p_1^*Q_i$, and $p_2^*{\cal O}(1)$ on $X\times{\Bbb P}^1$. For a bundle $F$ on $X\times{\Bbb P}^1$ we let $F(k):=F\otimes{\cal O}(1)^k$. For each $i\leqslant m-1$, we map $E_i(m-1-i)$ to $E_{i+1}(m-1-i)$ by the inclusion of $E_{i}\hookrightarrow E_{i+1}$ and to $E_i(m-i)$ by $\mbox{id}_{E_i}\otimes\sigma$. For $1\leqslant j\leqslant m$ let \[ \displaystyle \widetilde{E}_k:=\left( \bigoplus_{i=1}^k E_i(m-i)\right)\Bigl/ \left( \bigoplus_{i=1}^{k-1} E_i(m-1-i)\right). \] Setting $\widetilde{E}:=\widetilde{E}_m$ we get a filtration of type ${\frak r}$ over $X\times {\Bbb P}^1$ : \[ \widetilde{{\cal E}} :\ 0 \subset \widetilde{E}_1 \subset \widetilde{E}_2 \subset \cdots \subset \widetilde{E}_m=\widetilde{E}. \] The quotients of this filtration are $\widetilde{Q}_i=\widetilde{E}_i/\widetilde{E}_{i-1}=Q_i(m-i)$, $1\leqslant i\leqslant m$. For $z\hspace{-2pt}\in\hspace{-2pt}{\Bbb P}^1$, denote by $i_z:X\rightarrow X\times {\Bbb P}^1$ the map given by $i_z(x)=(x,z)$. When $z\hspace{-2pt}\neq\hspace{-2pt}\infty$, $i_z^*\widetilde{E}\simeq E$, while $\displaystyle i_{\infty}^*\widetilde{E}\simeq \bigoplus_{i=1}^m Q_i$. Using a partition of unity we can choose hermitian metrics $\widetilde{h}_i$ on $\widetilde{Q}_i$ and $\widetilde{h}$ on $\widetilde{E}$ such that the isomorphisms $i_z^*\widetilde{Q}_i\simeq Q_i$, $i_0^*\widetilde{E}\simeq E$ and $\displaystyle i_{\infty}^*\widetilde{E}\simeq \bigoplus_{i=1}^m Q_i$ all become isometries. We also let $(\widetilde{{\cal E}},\widetilde{h})$ denote the hermitian filtration of type ${\frak r}$ defined by these data. Then one shows (as in loc. cit.) that $\widetilde{\phi}(\overline{{\cal E}})$ is uniquely determined in $\widetilde{A}(X)$ by the formula \[ \displaystyle \widetilde{\phi}(\overline{{\cal E}})=-\int_{{\Bbb P}^1}\phi(\widetilde{E},\widetilde{h})\log |z|^2. \] \hfill $\Box$ \medskip \noindent {\bf Remark.} Gillet and Soul\'{e} used $\widetilde{ch}(\overline{{\cal E}})$ to give an explicit description of the Beilinson regulator map on $K_1(X)$, where $X$ is an arithmetic scheme ([GS2], 7.1). \medskip It is easy to prove that analogues of the properties of Bott-Chern forms for short exact sequences ([GS2], \sec 1.3) are true for the above generalization to filtered bundles. In particular the formulas (\ref{define}) take the form: \begin{equation} \label{sumprod} \widetilde{\phi + \psi}=\widetilde{\phi}+\widetilde{\psi}, \ \ \ \ \ \ \widetilde{\phi\psi}=\widetilde{\phi}\cdot\psi(\bigoplus_{i=1}^m \overline{Q}_i)+ \phi(\overline{E})\cdot\widetilde{\psi}. \end{equation} for any $\phi$, $\psi \hspace{-2pt}\in\hspace{-2pt} I(n)$. Using Theorem \ref{bcf} and the same argument as in the proof of Theorem 4.8(ii) in [GS2], we obtain \begin{thm} \label{abc} Let \[ \displaystyle \overline{{\cal E}} :\ 0\subset \overline{E}_1\subset \overline{E}_2 \subset\cdots\subset \overline{E}_m=\overline{E} \] be a hermitian filtration on an arithmetic scheme $X$, with quotient bundles $Q_i$, and let $\phi\hspace{-2pt}\in\hspace{-2pt} I(n,{\Bbb Q})$. Then \[ \displaystyle \widehat{\phi}(\bigoplus_{i=1}^m\overline{Q}_i)- \widehat{\phi}(\overline{E})=a(\widetilde{\phi}(\overline{{\cal E}})). \] \end{thm} Assume that the subbundles $E_i$ are given metrics induced from $E$ and the quotient bundles $Q_i$ are given the metrics induced from the exact sequences $\overline{{\cal E}_i}$. Define matrices $K_{E_i}=\frac{i}{2\pi}K(\overline{E}_i)$ and $K_{Q_i}=\frac{i}{2\pi} K(\overline{Q}_i)$ as in \sec \ref{cbcfs}. Then the constructions in Theorem \ref{bcf} and Corollary \ref{ratcor} immediately imply \begin{cor} \label{ratbcf} For any $\phi\hspace{-2pt}\in\hspace{-2pt} I(n,{\Bbb Q})$ the Bott-Chern form $\widetilde{\phi}(\overline{{\cal E}})$ is a polynomial in the entries of the matrices $K_{E_i}$ and $K_{Q_i}$, $1\leqslant i \leqslant m$, with {\em rational} coefficients. \end{cor} \section{Curvature of homogeneous vector bundles} \label{hvb} Let $G=SL(n,{\Bbb C})$, $K=SU(n)$ and $P$ be a parabolic subgroup of $G$, with Lie algebras ${\frak g}$, ${\frak k}$ and ${\frak p}$ respectively. Complex conjugation of ${\frak g}$ with respect to ${\frak k}$ is given by the map $\tau$ with $\tau(A)=-\overline{A}^t$. We let ${\frak v}={\frak p}\cap\tau({\frak p})$ and ${\frak n}$ be the unique maximal nilpotent ideal of ${\frak p}$, so that ${\frak g}={\frak v}\oplus{\frak n}\oplus\tau({\frak n})$. Let ${\frak h}=\{\mbox{diag}(z_1,\ldots,z_n)\ |\ \sum z_i=0\}$ be the Cartan subalgebra of diagonal matrices in ${\frak g}$. The set of roots $\Delta=\{z_i-z_j\ |\ 1\leqslant i\neq j \leqslant n \}$ is a subset of ${\frak h}^*$. We denote the root $z_i-z_j$ by the pair $ij$, and fix a system of positive roots $\Delta_+=\{ij\ |\ i>j\}$. The adjoint representation of ${\frak h}$ on ${\frak g}$ determines a decomposition $\displaystyle {\frak g}={\frak h}\oplus\sum_{{\alpha}\in\Delta}{\frak g}^{{\alpha}}$. Here the root space ${\frak g}^{{\alpha}}={\Bbb C} e_{{\alpha}}$, where $e_{{\alpha}}=e_{ij}=E_{ij}$ is the matrix with 1 at the $ij$-th entry and zeroes elsewhere. Set $\overline{e}_{ij}=\tau(e_{ij})=-E_{ji}$. Let $V=K\cap P$ and consider the complex manifold $X=G/P=K/V$. Let $p:K\rightarrow X$ be the quotient map, and let $\Psi\subset\Delta_+$ be such that $\displaystyle {\frak n}=\sum_{{\alpha}\in\Psi}{\frak g}^{-{\alpha}}$. For ${\alpha},{\beta}\hspace{-2pt}\in\hspace{-2pt}\Psi$, the equations \begin{eqnarray*} \omega^{{\alpha}}(e_{{\beta}})=\delta_{{\alpha}{\beta}}, & \omega^{{\alpha}}(\overline{e}_{{\beta}})=0, & \omega^{{\alpha}}({\frak v})=0 \\ \overline{\omega}^{{\alpha}}(e_{{\beta}})=0, & \overline{\omega}^{{\alpha}}(\overline{e}_{{\beta}})=\delta_{{\alpha}{\beta}}, & \overline{\omega}^{{\alpha}}({\frak v})=0 \end{eqnarray*} define elements of the dual space ${\frak g}^*$, which we shall regard as left invariant complex one-forms on $K$. A given differential form $\eta$ on $X$ pulls back to \begin{equation} \label{forms} p^*\eta=\sum_{a,b} f_{ab}\omega^{{\alpha}_1}\wedge\ldots\wedge \omega^{{\alpha}_r}\wedge \overline{\omega}^{{\beta}_1}\wedge\ldots\wedge\overline{\omega}^{{\beta}_s} \end{equation} on $K$, with coefficients $f_{ab}\hspace{-2pt}\in\hspace{-2pt} C^{\infty}(K)$. Conversely, every $V$-invariant element of $C^{\infty}(K)\otimes \bigwedge\tau({\frak n})^*\otimes\bigwedge{\frak n}^*$ is the pullback to $K$ of a differential form on $X$. A form $\eta$ on $X$ is of $(p,q)$ type precisely when every summand on the right hand side of (\ref{forms}) involves $p$ unbarred and $q$ barred terms. \begin{defn} \label{invdefn} {\em Inv}${}_{{\Bbb R}}(X)$ (respectively {\em Inv}${}_{{\Bbb Q}}(X)$) denotes the ring of $K$-invariant forms in the ${\Bbb R}$-subalgebra (respectively ${\Bbb Q}$-subalgebra) of $A(X)$ generated by $ \{\frac{i}{2\pi}{\omega}^{{\alpha}}\wedge\overline{{\omega}}^{{\beta}} \ | \ {\alpha},{\beta} \hspace{-2pt}\in\hspace{-2pt} \Psi \}. $ \end{defn} Suppose now that $\pi:V\rightarrow GL(E_0)$ is an irreducible unitary representation of $V$ on a complex vector space $E_0$. $\pi$ defines a homogeneous vector bundle $\overline{E}=K\times_VE_0\rightarrow X$ which has a $K$-invariant hermitian metric. Extend $\pi$ to a unique holomorphic representation $\pi:P\rightarrow GL(E_0)$, and denote the induced representation of ${\frak p}$ by the same letter. Then $\overline{E}=G\times_PE_0$ is a holomorphic hermitian vector bundle over $X$ which gives a complex structure to $K\times_VE_0$. In [GrS], equation $(4.4)_X$, Griffiths and Schmid calculate the $K$-invariant curvature matrix $K(\overline{E})$ explicitly in terms of the above data. Their result is \begin{equation} \label{grs} K(\overline{E})=\sum_{{\alpha},{\beta}\in\Psi}\pi([e_{{\alpha}},e_{-{\beta}}]_{{\frak v}})\otimes \omega^{{\alpha}}\wedge\overline{\omega}^{{\beta}}. \end{equation} The invariant differential forms giving the Chern classes of homogeneous line bundles were given by Borel [B]; see the introduction to [GrS] for more references. Let $Y=F({\Bbb C})\simeq SL(n,{\Bbb C})/B=SU(n)/S(U(1)^n)$ be the complex flag variety and let $\overline{E}$ denote the trivial hermitian vector bundle over $Y$, with the tautological hermitian filtration \[ \displaystyle \overline{{\cal E}}:\ 0 \subset \overline{E}_1 \subset \overline{E}_2 \subset\cdots \subset \overline{E}_n=\overline{E} \] with quotient line bundles $\overline{L}_i$ and all metrics induced from the metric on $\overline{E}$. All of these bundles are homogeneous, and we want to use equation (\ref{grs}) to compute their curvature matrices. Note that (\ref{grs}) applies directly only to the line bundles $\overline{L}_i$, as the higher rank bundles are not given by irreducible representations of the torus $S(U(1)^n)$. We can avoid this problem by considering the grassmannian $Y_k=G_k({\Bbb C})=SU(n)/S(U(k)\times U(n-k))$ and the natural projection $\rho:Y\rightarrow Y_k$. Now (\ref{grs}) can be applied to the universal bundle $\overline{E}_k$ over $Y_k$ and the curvature matrix $K(\overline{E}_k)$ pulls back via $\rho$ to the required matrix over $Y$. In fact by projecting to a partial flag variety one can compute the curvature matrix of any quotient bundle $E_l/E_k$. The representations $\pi$ of $V$ inducing these bundles are the obvious ones in each case. What remains is a straightforward application of equation (\ref{grs}), so we will describe the answer without belaboring the details. We have defined differential forms ${\omega}^{ij}$, $\overline{{\omega}}^{ij}$ on $K=SU(n)$ which we identify with corresponding forms on $Y$. With this notation, we can state (compare [GrS], $(4.13)_X$) : \begin{prop} \label{curvmat} Let $k<l$ and consider the vector bundle $Q_{lk}=E_l/E_k$ over $F({\Bbb C})$. Let the curvature matrix of $\overline{Q}_{lk}$ with its induced metric be $\Theta=\{\Theta_{{\alpha}{\beta}}\}_{k+1\leqslant {\alpha},{\beta}\leqslant l}$. Then \[ \displaystyle \Theta_{{\alpha}{\beta}}=\sum_{i\leqslant k}\omega^{{\alpha} i}\wedge\overline{\omega}^{{\beta} i}- \sum_{j>l}\omega^{j{\alpha}}\wedge\overline{\omega}^{j{\beta}}. \] \end{prop} For notational convenience we let ${\omega}_{ij}:=\gamma{\omega}^{ji}$, $\overline{{\omega}}_{ij}:=\gamma\overline{{\omega}}^{ji}$ and ${\Omega}_{ij}:={\omega}_{ij}\wedge\overline{{\omega}}_{ij}$,\ $(i<j)$, where $\gamma$ is a constant such that $\gamma^2=\frac{i}{2\pi}$. Then we have \begin{cor} \label{grscor1} \[ \displaystyle c_1(\overline{L}_k)=\sum_{i<k}{\Omega}_{ik}-\sum_{j>k}{\Omega}_{kj} \] \[ \displaystyle K_{\overline{E}_k}=\frac{i}{2\pi}K(\overline{E}_k)=-\left\{\sum_{j>k}{\omega}_{{\alpha} j} \wedge\overline{{\omega}}_{{\beta} j}\right\}_{1\leqslant {\alpha},{\beta} \leqslant k} \] \end{cor} \medskip Let $\displaystyle \Omega:=\bigwedge_{i<j}{\Omega}_{ij}$. To compute classical intersection numbers on the flag variety using the differential forms in Corollary \ref{grscor1} it suffices to know $\displaystyle\int_Y\Omega$. If $x_i=-c_1(\overline{L}_i)$, it is well known that $\eta={\frak S}_{w_0}(x)=x_1^{n-1}x_2^{n-2}\cdots x_{n-1}$ is dual to the class of a point in $Y$; thus $\displaystyle\int_Y\eta=1$. An easy calculation shows that $\displaystyle\eta=\prod_{k=1}^{n-1}k!\cdot \Omega$, thus $\displaystyle \int_Y\Omega=\prod_{k=1}^{n-1}\frac{1}{k!}$. \section{Invariant arithmetic Chow rings} \label{afvs} It is well known that the arithmetic variety $F$ has a cellular decomposition in the sense of [Fu1], Ex. 1.9.1. It follows that one can use the excision exact sequence for the groups $CH^{*,*}(F)$ (cf. [G], \sec 8) to show that $CH^{p,p-1}(F)=0$ (compare [Ma], Lemma 4.0.6). Therefore the exact sequence (\ref{ex1}) summed over $p$ gives \begin{equation} \label{flagex} 0 \longrightarrow \widetilde{A}(F_{{\Bbb R}}) \stackrel{a}\longrightarrow \widehat{CH}(F) \stackrel{\zeta}\longrightarrow CH(F)\longrightarrow 0. \end{equation} Recall that $\widetilde{A}(F_{{\Bbb R}})=\mbox{Ker}\zeta$ is an ideal of $\widehat{CH}(F)$ whose $\widehat{CH}(F)$-module structure is given as follows: if ${\alpha}\hspace{-2pt}\in\hspace{-2pt}\widehat{CH}(F)$ and $\eta\hspace{-2pt}\in\hspace{-2pt}\widetilde{A}(F_{{\Bbb R}})$, then ${\alpha}\cdot\eta={\omega}(a)\wedge\eta$. $\widetilde{A}(F_{{\Bbb R}})$ is not a square zero ideal, but its product is induced by $\theta\cdot \eta=(dd^c\theta)\wedge\eta$. This product is well defined and commutative ([GS1], 4.2.11). We equip $E({\Bbb C})$ with a trivial hermitian metric. This metric induces metrics on all the $L_i$, which thus become hermitian line bundles $\overline{L}_i$. Recall from \sec \ref{classgps} that $CH(F)$ has a free ${\Bbb Z}$-basis of monomials in the Chern classes $c_1(L_i)$. The unique map of abelian groups $\epsilon:CH(F)\rightarrow \widehat{CH}(F)$ sending $\prod c_1(L_i)^{k_i}$ to $\prod \widehat{c}_1(\overline{L}_i)^{k_i}$ when $k_i\leqslant n-i$ for all $i$ is then a splitting of (\ref{flagex}). Thus we have an isomorphism of abelian groups \begin{equation} \label{bigiso} \widehat{CH}(F)\simeq CH(F)\oplus \widetilde{A}(F_{{\Bbb R}}). \end{equation} \medskip As an analogue of the Arakelov Chow ring we define an {\em invariant arithmetic Chow ring} $\widehat{CH}_{inv}(F)$ as follows. Let $\mbox{Inv}^{p,p}(F_{{\Bbb R}})$ be the group of $(p,p)$-forms $\eta$ in $\mbox{Inv}_{{\Bbb R}}(F({\Bbb C}))$ satisfying $F_{\infty}^*\eta=(-1)^p\eta$, and set $\mbox{Inv}(F_{{\Bbb R}})=\oplus_p \mbox{Inv}^{p,p}(F_{{\Bbb R}})$. Let $\widetilde{\mbox{Inv}}(F_{{\Bbb R}})\subset \widetilde{A}(F_{{\Bbb R}})$ be the image of $\mbox{Inv}(F_{{\Bbb R}})$ in $\widetilde{A}(F_{{\Bbb R}})$. Define the rings $\mbox{Inv}(F_{{\Bbb Q}})$ and $\widetilde{\mbox{Inv}}(F_{{\Bbb Q}})$ similarly, replacing ${\Bbb R}$ by ${\Bbb Q}$ in the above. \begin{defn} \label{invdef} The invariant arithmetic Chow ring $\widehat{CH}_{inv}(F)$ is the subring of $\widehat{CH}(F)$ generated by $\epsilon(CH(F))$ and $a(\widetilde{\mbox{\em Inv}}(F_{{\Bbb R}}))$. \end{defn} Suppose that $x,y\hspace{-2pt}\in\hspace{-2pt} CH(F)$ and view $x$ and $y$ as elements of $\widehat{CH}(F)$ using the inclusion $\epsilon$. In \sec \ref{ai} we will see that under the isomorphism (\ref{bigiso}), $xy\hspace{-2pt}\in\hspace{-2pt} CH(F)\oplus \widetilde{\mbox{Inv}}(F_{{\Bbb Q}})$. It follows that there is an exact sequence of abelian groups \begin{equation} \label{invex} 0 \longrightarrow \widetilde{\mbox{Inv}}(F_{{\Bbb R}}) \stackrel{a}\longrightarrow \widehat{CH}_{inv}(F) \stackrel{\zeta}\longrightarrow CH(F)\longrightarrow 0 \end{equation} which splits as before, giving \begin{thm} \label{chinv} There is an isomorphism of abelian groups \[ \widehat{CH}_{inv}(F)\simeq CH(F)\oplus\widetilde{\mbox{\em Inv}}(F_{{\Bbb R}}). \] \end{thm} \medskip \noindent {\bf Remark 1:} One can define another `invariant arithmetic Chow ring' \[ \widehat{CH}_{inv}^{\prime}(F):={\omega}^{-1}(\mbox{Inv}(F_{{\Bbb R}})), \] where ${\omega}$ is the ring homomorphism defined in \sec \ref{ait}. There is a natural inclusion $\widehat{CH}_{inv}(F)\hookrightarrow\widehat{CH}_{inv}^{\prime}(F)$; we do not know if these two rings coincide. \medskip \noindent {\bf Remark 2:} The arithmetic Chern classes of the natural homogeneous vector bundles over $F$ are all contained in the ring $\widehat{CH}_{inv}(F)$. In fact one need not use real coefficients for this; it suffices to take $CH(F)\oplus \widetilde{\mbox{Inv}}(F_{{\Bbb Q}})$ with the induced product from $\widehat{CH}(F)$. As there are bounds on the denominators that occur, it follows that {\em the subring of $\widehat{CH}(F)$ generated by $\epsilon(CH(F))$ is a finitely generated abelian group}. However it seems that this group is too small to contain the characteristic classes of all the vector bundles of interest. \section{Calculating arithmetic intersections} \label{ai} In this section we describe an effective procedure for computing arithmetic intersection numbers on the complete flag variety $F$. One has a tautological hermitian filtration of the trivial bundle $\overline{E}$ over $F$ \[ \displaystyle \overline{{\cal E}}:\ 0 \subset \overline{E}_1 \subset \overline{E}_2 \subset\cdots \subset \overline{E}_n=\overline{E} \] as in \sec \ref{hvb}. Recall that the inverse of the isomorphism $CH(F)\simeq P_n/I_n$ sends $[X_i]$ to $-c_1(L_i)$. Let $x_i=-c_1(\overline{L}_i)$ and $\widehat{x}_i=-\widehat{c}_1(\overline{L}_i)$ for $1\leqslant i \leqslant n$. If $\phi\hspace{-2pt}\in\hspace{-2pt} \Lambda_n\otimes_{{\Bbb Z}}{\Bbb Q}$ is a homogeneous symmetric polynomial of positive degree then $\phi$ defines a characteristic class. Theorem \ref{abc} applied to the hermitian filtration $\overline{{\cal E}}$ shows that \[ \displaystyle \phi(\widehat{x}_1,\widehat{x}_2,\ldots,\widehat{x}_n)=(-1)^{\deg \phi}\,\widetilde{\phi}(\overline{{\cal E}}) \] in the arithmetic Chow ring $\widehat{CH}(F)$. In particular for $\phi=e_i$ an elementary symmetric polynomial this gives \[ \displaystyle e_i(\widehat{x}_1,\widehat{x}_2,\ldots,\widehat{x}_n)=(-1)^i\,\widetilde{c}_i(\overline{{\cal E}}). \] Let $h$ be a homogeneous polynomial in the ideal $I_n$. We will give an algorithm for computing $h(\widehat{x}_1,\widehat{x}_2,\ldots\widehat{x}_n)$ as a class in $\widetilde{\mbox{Inv}}(F_{{\Bbb Q}})$: \medskip \noindent {\bf\underline{Step 1}:} Decompose $h$ as a sum $h=\sum e_if_i$ for some polynomials $f_i$. More canonically one may use the equality \[ \displaystyle h=\sum_{w\in S_n} \left< h , {\frak S}^w \right> {\frak S}_w \] from \sec \ref{classgps}. Since $a(x)y=a(x\omega(y))$ in $\widehat{CH}(F)$ and $\omega(f_i(\widehat{x}_1,\ldots,\widehat{x}_n))= f_i(x_1,\ldots,x_n)$, we have \[ \displaystyle h(\widehat{x}_1,\widehat{x}_2,\ldots\widehat{x}_n)=\sum_{i=1}^n (-1)^i\,\widetilde{c}_i(\overline{{\cal E}})f_i(x_1,x_2,\ldots,x_n)= \] \[ \displaystyle =\sum_{w\in S_n} (-1)^{\deg h +l(w)}\,\widetilde{\left< h , {\frak S}^w \right>}(\overline{{\cal E}}) {\frak S}_w(x_1,x_2,\ldots,x_n). \] \noindent {\bf\underline{Step 2}:} By Corollary \ref{ratbcf}, we may express the forms $\widetilde{c}_i(\overline{{\cal E}})$ and $\displaystyle \widetilde{\left< h , {\frak S}^w \right>}(\overline{{\cal E}})$ as polynomials in the entries of the matrices $K_{E_i}$ and $K_{L_i}=c_1(\overline{L_i})$ with rational coefficients. In practice this may be done recursively for the Chern forms $\widetilde{c}_i$ as follows: Use equation (\ref{*}) and the construction in Corollary \ref{ratcor} to obtain the power sum forms $\widetilde{p}_i(\overline{{\cal E}})$, then apply the formulas (\ref{sumprod}) to Newton's identity \[ \displaystyle p_i-c_1p_{i-1}+c_2p_{i-2}-\cdots+(-1)^iic_i=0. \] On the other hand Corollary \ref{grscor1} gives explicit expressions for all the above curvature matrices in terms of differential forms on $F({\Bbb C})$. Thus we obtain formulas for $\widetilde{c}_i(\overline{{\cal E}})$ and $\displaystyle \widetilde{\left< h , {\frak S}^w \right>}(\overline{{\cal E}})$ in terms of these forms. For example, using the notation of \sec \ref{hvb}, we have \begin{prop} $\widetilde{c}_1(\overline{{\cal E}})=0$ and $\displaystyle\widetilde{c}_2(\overline{{\cal E}})=-\sum_{i<j}{\Omega}_{ij}$. \end{prop} {\bf Proof.} Use (\ref{*}), properties (a) and (b) at the end of \sec \ref{cbcfs}, and the identity $2c_2=c_1^2-p_2$.\ \ \ \hfill $\Box$ \medskip \noindent {\bf\underline{Step 3}:} Substitute the forms obtained in Step 2 into the formulas given in Step 1. Note that the result is the class of a form in $\mbox{Inv}(F_{{\Bbb Q}})$ since all the ingredients are functorial for the natural $U(n)$ action on $F({\Bbb C})$. \medskip In particular, if $k_i$ are nonnegative integers with $\sum k_i= \dim{F}={n \choose 2}+1$, the monomial $X_1^{k_1}\cdots X_n^{k_n}$ is in the ideal $I_n$. If $X_1^{k_1}\cdots X_n^{k_n}=\sum e_if_i$, then we have \[ \displaystyle \widehat{x}_1^{k_1}\widehat{x}_2^{k_2}\cdots\widehat{x}_n^{k_n}= \sum_i(-1)^i\,\widetilde{c}_i(\overline{{\cal E}})f_i(x_1,\ldots,x_n). \] Now if ${\Omega}=\bigwedge {\Omega}_{ij}$ is defined as in \sec \ref{hvb}, we have shown that \[ \displaystyle \widetilde{c}_i(\overline{{\cal E}})f_i(x_1,\ldots,x_n)=r_i{\Omega} \] for some rational number $r_i$. Therefore \[ \displaystyle \widehat{\deg}(\widehat{x}_1^{k_1}\widehat{x}_2^{k_2}\cdots\widehat{x}_n^{k_n})= \frac{1}{2}\sum_i(-1)^i\,r_i\int_{F({\Bbb C})}{\Omega}= \frac{1}{2}\sum_i(-1)^i\,r_i\prod_{k=1}^{n-1}\frac{1}{k!}\, . \] Of course this equation implies \begin{thm} \label{mainthm} The arithmetic Chern number $ \displaystyle \widehat{\deg}(\widehat{x}_1^{k_1}\widehat{x}_2^{k_2}\cdots\widehat{x}_n^{k_n}) $ is a rational number. \end{thm} \noindent {\bf Remark.}\ For $a<b$ let $\overline{Q}_{b,a}=E_b/E_a$, equipped with the induced metric. Then one can show that any intersection number $ \displaystyle \widehat{\deg}(\prod_i \widehat{c}_{m_i}(\overline{Q}_{b_ia_i})^{k_i})$ for $ \sum k_im_i(b_i-a_i)=\dim F $ is rational. This is done by using the hermitian filtrations \[ \displaystyle 0\subset \overline{Q}_{a+1,a}\subset\overline{Q}_{a+2,a}\subset\cdots\subset\overline{Q}_{b,a} \] and Theorem \ref{abc} to reduce the problem to the intersections occuring in Theorem \ref{mainthm}. To compute arithmetic intersections of the form $(0,\eta)\cdot (0,\eta^{\prime})$ with $\eta,\eta^{\prime}\hspace{-2pt}\in\hspace{-2pt} \widetilde{\mbox{Inv}}(F_{{\Bbb Q}})$, we need to know the value of $dd^c\eta$. For this one may use the Maurer-Cartan structure equations on $SU(n)$ (cf. [GrS], Chp. 1); all such intersections lie in $\widetilde{\mbox{Inv}}(F_{{\Bbb Q}})$. \medskip Although there is an effective algorithm for computing arithmetic Chern numbers, explicit general formulas seem difficult to obtain. There are some general facts we can deduce for those intersections that pull back from grassmannians, for instance that $\widehat{x}_1^{n+1}=\widehat{x}_n^{n+1}=0$. There is also a useful symmetry in these intersections: \begin{prop} \label{symprop} $ \widehat{x}_1^{k_1}\widehat{x}_2^{k_2}\cdots\widehat{x}_n^{k_n}= \widehat{x}_n^{k_1}\widehat{x}_{n-1}^{k_2}\cdots\widehat{x}_1^{k_n}, $ \ for all integers $k_i\geqslant 0$. \end{prop} {\bf Proof.} This is a consequence of the involution $\nu : F({\Bbb C}) \rightarrow F({\Bbb C})$ sending \[ \displaystyle \overline{{\cal E}} : \ 0 \subset \overline{E}_1 \subset \overline{E}_2 \subset \cdots \subset \overline{E}_n=\overline{E} \] to \[ \displaystyle \overline{{\cal E}}^{\bot} : \ 0=\overline{E}^{\bot} \subset \overline{E}_{n-1}^{\bot} \subset \overline{E}_{n-2}^{\bot} \subset \cdots \subset 0^{\bot}=\overline{E}. \] Over $\mbox{Spec}{\Bbb Z}$, $\nu$ corresponds to the map of flag varieties sending $E_i$ to the quotient $E/E_i$. If $\widehat{x}_i^{\bot}$ are the arithmetic Chern classes obtained from $\overline{{\cal E}}^{\bot}$, then using the split exact sequences $0\rightarrow \overline{E}_i \rightarrow \overline{E} \rightarrow \overline{E}_i^{\bot} \rightarrow 0$ we obtain \[ \displaystyle \widehat{x}_i^{\bot}=-\widehat{c}_1(\overline{L}_i^{\bot})= -\widehat{c}_1(\overline{E}_{n-i}^{\bot})+\widehat{c}_1(\overline{E}_{n+1-i}^{\bot})= \widehat{c}_1(\overline{E}_{n-i})-\widehat{c}_1(\overline{E}_{n+1-i})=\widehat{x}_{n+1-i}. \] Since $\nu$ is an isomorphism, the result follows. \hfill $\Box$ \section{Arithmetic Schubert calculus} \label{asc} Let $P_n$, $I_n$, $\Lambda_n$ and $S^{(n)}$ be as in \sec \ref{classgps}. The Chow ring $CH(F)$ is isomorphic to the quotient $H_n=P_n/I_n$. Recall that $H_n$ has a natural basis of Schubert polynomials $\{{\frak S}_w\ | \ w\hspace{-2pt}\in\hspace{-2pt} S_n \}$, and that the ${\frak S}_w$ for $w\hspace{-2pt}\in\hspace{-2pt} S^{(n)}$ form a free ${\Bbb Z}$-basis of $P_n$. We let $T_n=S^{(n)}\smallsetminus S_n$. The key property of Schubert polynomials that we require for the `arithmetic Schubert calculus' is described in \begin{lemma} \label{schlemma} If $w\hspace{-2pt}\in\hspace{-2pt} T_n$, then ${\frak S}_w\hspace{-2pt}\in\hspace{-2pt} I_n$. In fact we have a decomposition \[ \displaystyle {\frak S}_w=\sum_{v\in S_n}\left<{\frak S}_w,{\frak S}^v\right>{\frak S}_v, \] where $ \left<{\frak S}_w,{\frak S}^v\right>\hspace{-2pt}\in\hspace{-2pt} \Lambda_n\cap I_n$. \end{lemma} {\bf Proof.} Assume first that $w(1)>w(2)>\cdots >w(n)$, so that $w$ is {\em dominant}. Then by [M], (4.7) we have \[ \displaystyle {\frak S}_w=X_1^{w(1)-1}X_2^{w(2)-1}\cdots X_n^{w(n)-1}. \] If $w\notin S_n$ then clearly $w(1)>n$, so $X_1^{w(1)-1}\hspace{-2pt}\in\hspace{-2pt} I_n$ and thus ${\frak S}_w\hspace{-2pt}\in\hspace{-2pt} I_n$. If $w\hspace{-2pt}\in\hspace{-2pt} T_n$ is arbitrary, form $w^{\prime}\hspace{-2pt}\in\hspace{-2pt} T_n$ by rearranging $(w(1),w(2),\ldots,w(n))$ in decreasing order and letting $w^{\prime}(i)= w(i)$ for $i>n$. We have shown that ${\frak S}_{w^{\prime}}\hspace{-2pt}\in\hspace{-2pt} I_n$. There is an element $v\hspace{-2pt}\in\hspace{-2pt} S_n$ such that $wv=w^{\prime}$ and $l(v)=l(w^{\prime})-l(w)$. Note that since $\partial_v$ is $\Lambda_n$-linear, $\partial_v I_n\subset I_n$. Therefore ([M], (4.2)): ${\frak S}_w=\partial_v{\frak S}_{wv}=\partial_v{\frak S}_{w^{\prime}}\hspace{-2pt}\in\hspace{-2pt} I_n.$ The decomposition claimed now follows, as in \sec \ref{classgps}. \hfill $\Box$ \medskip It is well known that there is an equality in $P_{\infty}$ \begin{equation} \label{cuvws} {\frak S}_u{\frak S}_v=\sum_{w\in S_{\infty}}c_{uv}^w{\frak S}_w, \end{equation} where the $c_{uv}^w$ are nonegative integers that vanish whenever $l(w)\neq l(u)+l(v)$\ ([M], (A.6)). A particular case of this is {\em Monk's formula:} if $s_k$ denotes the transposition $(k,k+1)$, then \[ \displaystyle {\frak S}_{s_k}{\frak S}_w=\sum_t {\frak S}_{wt} \] summed over all transpositions $t=(i,j)$ such that $i\leqslant k <j$ and $l(wt)=l(w)+1$\ ([M], ($4.15^{\prime\pr}$)). \medskip We now express arithmetic intersections in $\widehat{CH}(F)$ using the basis of Schubert polynomials. Lemma \ref{schlemma} is the main reason why this basis facilitates our task. This property (for Schur functions) also plays a crucial role in the arithmetic Schubert calculus for grassmannians (see \sec \ref{pfvs} and [Ma], Th. 5.2.1). For each $w\hspace{-2pt}\in\hspace{-2pt} S^{(n)}$, let $\widehat{{\frak S}}_w={\frak S}_w(\widehat{x}_1, \ldots,\widehat{x}_n)$. If $w\hspace{-2pt}\in\hspace{-2pt} T_n$ then Lemma \ref{schlemma} and the discussion in \sec \ref{ai} imply that $\widehat{{\frak S}}_w\hspace{-2pt}\in\hspace{-2pt} \widetilde{\mbox{Inv}}(F_{{\Bbb Q}})$; we denote these classes by $\widetilde{{\frak S}}_w$. We have \[ \displaystyle \widetilde{{\frak S}}_w=\sum_{v\in S_n}(-1)^{l(v)+l(w)} \widetilde{\left<{\frak S}_w,{\frak S}^v\right>}(\overline{{\cal E}}){\frak S}_v(x_1,\ldots,x_n). \] We can now describe the multiplication in $\widehat{CH}_{inv}(F)$: \begin{thm} \label{slring} Any element of $\widehat{CH}_{inv}(F)$ can be expressed uniquely in the form $\displaystyle \sum_{w\in S_n}a_w\widehat{{\frak S}}_w+\eta$, where $a_w\hspace{-2pt}\in\hspace{-2pt}{\Bbb Z}$ and $\eta\in\widetilde{\mbox{\em Inv}}(F_{{\Bbb R}})$. For $u,v\hspace{-2pt}\in\hspace{-2pt} S_n$ we have \begin{equation} \label{punchline} \widehat{{\frak S}}_u\cdot\widehat{{\frak S}}_v=\sum_{w\in S_n}c_{uv}^w\widehat{{\frak S}}_w+ \sum_{w\in T_n}c_{uv}^w\widetilde{{\frak S}}_w, \end{equation} \[ \displaystyle \widehat{{\frak S}}_u\cdot \eta={\frak S}_u(x_1,\ldots,x_n)\wedge\eta, \ \ \ \ \mbox{and} \ \ \ \ \eta\cdot \eta^{\prime}=(dd^c\eta)\wedge\eta^{\prime}, \] where $\widetilde{{\frak S}}_w\in\widetilde{\mbox{\em Inv}}(F_{{\Bbb Q}})$, $\eta$, $\eta^{\prime}\in\widetilde{\mbox{\em Inv}}(F_{{\Bbb R}})$ and the $c_{uv}^w$ are as in {\em (\ref{cuvws})}. \end{thm} {\bf Proof.} The first statement is a corollary of Theorem \ref{chinv}. Equation (\ref{punchline}) follows immediately from the formal identity (\ref{cuvws}) and our definition of $\widetilde{{\frak S}}_w$. The rest is a consequence of properties of the multiplication in $\widehat{CH}(F)$ discussed in \sec \ref{afvs} and \sec \ref{ai}. \hfill $\Box$ \medskip \noindent {\bf Remark.} It is interesting to note that we also have, for $u,v\hspace{-2pt}\in\hspace{-2pt} T_n$, \[ \displaystyle \widetilde{{\frak S}}_u\cdot\widetilde{{\frak S}}_v=(dd^c\widetilde{{\frak S}}_u)\wedge\widetilde{{\frak S}}_v= \sum_{w\in T_n}c_{uv}^w\widetilde{{\frak S}}_w \] in $\widetilde{\mbox{Inv}} (F_{{\Bbb Q}})$. \medskip Applying (\ref{punchline}) when ${\frak S}_u={\frak S}_{s_k}$ is a special Schubert class gives \begin{cor} (Arithmetic Monk Formula): \[ \displaystyle \widehat{{\frak S}}_{s_k}\cdot\widehat{{\frak S}}_w=\sum_s \widehat{{\frak S}}_{ws} + \sum_t \widetilde{{\frak S}}_{wt}, \] where the first sum is over all transpositions $s=(i, j)\hspace{-2pt}\in\hspace{-2pt} S_n$ such that $i\leqslant k <j$ and $l(ws)=l(w)+1$, and the second over all transpositions $t=(i,n+1)$ with $i\leqslant k$ and $w(i)>w(j)$ for all $j$ with $i<j \leqslant n$. \end{cor} \section{Examples} \label{ex} \subsection{Heights} The flag variety $F$ has a natural pluri-Pl\"{u}cker embedding $j:F\hookrightarrow {\Bbb P}^N_{{\Bbb Z}}$. $j$ is defined as the composition of a product of Pl\"{u}cker embeddings followed by a Segre embedding; if $Q_i=E/E_i$, then $j$ is associated to the line bundle $\displaystyle Q=\bigotimes_{i=1}^{n-1} \det (Q_i)$. Let $\overline{{\cal O}}(1)$ denote the canonical line bundle over projective space, equipped with its canonical metric (so that $c_1(\overline{{\cal O}}(1))$ is the Fubini-Study form). The {\em height} of $F$ relative to $\overline{{\cal O}}(1)$ (cf. [Fa], [BoGS], [S]) is defined by \[ \displaystyle h_{\overline{{\cal O}}(1)}(F) =\widehat{\deg}(\widehat{c}_1(\overline{{\cal O}}(1))^{{n \choose 2}+1}\vert \ F). \] Since \[ \displaystyle j^*(\widehat{c}_1(\overline{{\cal O}}(1)))=\widehat{c}_1(\overline{Q})= -\sum_{i=1}^{n-1}\widehat{c}_1(\overline{E}_i)= \sum_{i=1}^{n-1} (n-i)\widehat{x}_i= \sum_{i=1}^{n-1} \widehat{{\frak S}}_{s_i}, \] we have that \[ \displaystyle h_{\overline{{\cal O}}(1)}(F)= \widehat{\deg}(\widehat{c}_1(\overline{Q})^{{n \choose 2}+1}\vert \ F)= \widehat{\deg}((\sum_{i=1}^{n-1} \widehat{{\frak S}}_{s_i})^{{n \choose 2}+1}). \] Now Theorems \ref{mainthm} and \ref{slring} immediately imply \begin{thm} \label{slheight} The height $h_{\overline{{\cal O}}(1)}(F)$ is a rational number. \end{thm} \subsection{Intersections in $F_{1,2,3}$} In this section we calculate the arithmetic intersection numbers for the classes $\widehat{x}_i$ in $\widehat{CH}(F)$ when $n=3$, so $F=F_{1,2,3}$. Over $F$ we have 3 exact sequences \[ \displaystyle \overline{{\cal E}}_i : \ 0\rightarrow \overline{E}_{i-1} \rightarrow \overline{E}_i \rightarrow \overline{L}_i \rightarrow 0 \ \ \ \ \ \ 1\leqslant i \leqslant 3. \] We adopt the notation of \sec \ref{hvb} and define ${\Omega}_{ij}= {\omega}_{ij}\wedge\overline{{\omega}}_{ij}$. Then Corollary \ref{grscor1} gives \[ \displaystyle x_1={\Omega}_{12}+{\Omega}_{13}, \ \ \ \ x_2=-{\Omega}_{12}+{\Omega}_{23}, \ \ \ \ x_3=-{\Omega}_{13}-{\Omega}_{23}, \] \[ \displaystyle K_{E_2}=-\left( \begin{array}{cc} {\Omega}_{13} & {\omega}_{13}\wedge\overline{{\omega}}_{23} \\ {\omega}_{23}\wedge\overline{{\omega}}_{13} & {\Omega}_{23} \end{array}\right). \] We refer now to the properties of the forms $\widetilde{c_k}$ mentioned at the end of \sec \ref{cbcfs}. By property (a) $\widetilde{c}(\overline{{\cal E}}_1)=0$, while (b) gives $\widetilde{c}(\overline{{\cal E}}_2)=-{\Omega}_{12}$. Property (c) applied to $\overline{{\cal E}}_3$ gives $\widetilde{c}(\overline{{\cal E}}_3)=-{\Omega}_{13}-{\Omega}_{23}+ 3{\Omega}_{13}{\Omega}_{23}$. Using the construction of the Bott-Chern form for the total Chern class given in the proof of Theorem \ref{bcf}, we find that \begin{equation} \label{firstkey} \widetilde{c}(\overline{{\cal E}})= -{\Omega}_{12}-{\Omega}_{13}-{\Omega}_{23}-{\Omega}_{12}{\Omega}_{13}-{\Omega}_{12}{\Omega}_{23}+ 3{\Omega}_{13}{\Omega}_{23}. \end{equation} Notice that this expression for $\widetilde{c}(\overline{{\cal E}})$ is not unique as a class in $\widetilde{\mbox{Inv}}(F_{{\Bbb R}})$. For instance, we can add the exact form $c_1(\overline{L}_1)c_1(\overline{L}_2)-c_2(\overline{E}_2)={\Omega}_{12}{\Omega}_{23}- {\Omega}_{12}{\Omega}_{13}-{\Omega}_{13}{\Omega}_{23}$ to get \begin{equation} \label{key} \widetilde{c}(\overline{{\cal E}})= -{\Omega}_{12}-{\Omega}_{13}-{\Omega}_{23}-2{\Omega}_{12}{\Omega}_{13}+2{\Omega}_{13}{\Omega}_{23}. \end{equation} The Bott-Chern form (\ref{key}) is the key to computing any intersection number $\widehat{\deg}(\widehat{x}_1^{k_1} \widehat{x}_2^{k_2}\widehat{x}_3^{k_3})$, following the prescription of \sec \ref{ai}. (Of course we can just as well use (\ref{firstkey}), with the same results.) For example, since $x_1^4=x_1^3e_1-x_1^2e_2+x_1e_3$, we have \[ \displaystyle \widehat{x}_1^4=x_1^2({\Omega}_{12}+{\Omega}_{13}+{\Omega}_{23})+ x_1(2{\Omega}_{12}{\Omega}_{13}-2{\Omega}_{13}{\Omega}_{23})= 2{\Omega}-2{\Omega}=0. \] On the other hand, a similar calculation for $\widehat{x}_2^4$ gives \[ \displaystyle \widehat{x}_2^4=-x_2^2\widetilde{c}_2(\overline{{\cal E}})- x_2\widetilde{c}_3(\overline{{\cal E}})=-2{\Omega}+4{\Omega}=2{\Omega}. \] Thus $\displaystyle\widehat{\deg}(\widehat{x}_2^4)=\int_{F({\Bbb C})}{\Omega}=\frac{1}{2}$. The following is a table of all the intersection numbers $\widehat{\deg}(\widehat{x}_1^{k_1}\widehat{x}_2^{k_2}\widehat{x}_3^{k_3})$ (multiplied by 4): \begin{center} \begin{tabular}{|cc|cc|cc|} \hline &&&&& \\ $k_1k_2k_3$ & $4\,\widehat{\deg}$ & $k_1k_2k_3$ & $4\,\widehat{\deg}$ & $k_1k_2k_3$ & $4\,\widehat{\deg}$ \\ \hline 400 & 0 & 004 & 0 & 040 & 2 \\ 310 & 5 & 013 & 5 & 121 & 2 \\ 301 & -5 & 103 & -5 & 202 & 9 \\ 220 & -1 & 022 & -1 && \\ 211 & -4 & 112 & -4 && \\ 130 & -1 & 031 & -1 && \\ \hline \end{tabular} \end{center} Note that the numbers in the first two columns are equal, in agreement with Proposition \ref{symprop}. We can use the table to compute the height of $F$ in its pluri-Pl\"{u}cker embedding in ${\Bbb P}^8_{{\Bbb Z}}$: \[ \displaystyle h_{\overline{{\cal O}}(1)}(F_{1,2,3})=\widehat{\deg}((2\widehat{x}_1+\widehat{x}_2)^4)=\frac{65}{2}. \] \section{Partial flag varieties} \label{pfvs} In this final section we show how to generalize the previous work to partial flags $F({\frak r})$. Our results may thus be regarded as an extension of those of Maillot [Ma] in the grassmannian case. As usual we have a tautological filtration of type ${\frak r}$ \begin{equation} \label{fil3} {\cal E}:\ 0 \subset E_1 \subset E_2 \subset\cdots \subset E_m=E \end{equation} of the trivial bundle over $F({\frak r})$, with quotient bundles $Q_i$. Equip $E({\Bbb C})$ with the trivial hermitian metric, inducing metrics on all the above bundles. The calculations of \sec \ref{hvb} apply equally well to $X_{{\frak r}}=F({\frak r})({\Bbb C})$. Proposition \ref{curvmat} describes the curvature matrices of all the relevant homogeneous vector bundles, and one can compute classical intersection numbers on $X_{{\frak r}}$ in a similar fashion. Call a permutation $w\hspace{-2pt}\in\hspace{-2pt} S_{\infty}$ an {\em ${\frak r}$-permutation} if $w(i)<w(i+1)$ for all $i$ not contained in $\{r_1,\ldots,r_{m-1}\}$. Let $S_{n,{\frak r}}$ and $T_{n,{\frak r}}$ be the set of ${\frak r}$-permutations in $S_n$ and $T_n$, respectively. For such $w$ one knows (cf. [Fu2], \sec 8) that the Schubert polynomial ${\frak S}_w$ is symmetric in the variables in each of the groups \begin{equation} \label{vars} X_1,\ldots,X_{r_1};X_{r_1+1},\ldots,X_{r_2};\ldots ; X_{r_{m-2}+1},\ldots,X_{r_{m-1}}. \end{equation} The product group $\displaystyle H=\prod_{i=1}^m S_{r_i-r_{i-1}}$ acts on $P_n$, the factors for $i<m$ by permuting the variables in the corresponding group of (\ref{vars}), while $S_{n-r_{m-1}}$ permutes the remaining variables $X_{r_{m-1}+1},\ldots ,X_n$. If $P_{n,{\frak r}}=P_n^H$ is the ring of invariants and $I_{n,{\frak r}}=P_{n,{\frak r}} \cap I_n$, then $CH(F({\frak r}))\simeq P_{n,{\frak r}}/I_{n,{\frak r}}$. The set of Schubert polynomials ${\frak S}_w$ for all $w\hspace{-2pt}\in\hspace{-2pt} S_{n,{\frak r}}$ is a free ${\Bbb Z}$-basis for $P_{n,{\frak r}}/I_{n,{\frak r}}$. Let $w\hspace{-2pt}\in\hspace{-2pt} S_{n,{\frak r}}$. If we regard each of the groups of variables (\ref{vars}) as the Chern roots of the bundles $Q_1,Q_2,\ldots,Q_{m-1}$, it follows that we may write ${\frak S}_w$ as a polynomial ${\frak S}_{w,{\frak r}}$ in the Chern classes of the $Q_i$, $1\leqslant i \leqslant m-1$. The class of ${\frak S}_{w,{\frak r}}$ in $CH(F({\frak r}))$ is that of corresponding Schubert variety in $F({\frak r})$ (see loc. cit. for the relative case). By putting `hats' on all the quotient bundles involved (with their induced metrics as in \sec \ref{asc}) we obtain classes $\widehat{{\frak S}}_{w,{\frak r}}$ in $\widehat{CH}_{inv}(F({\frak r}))$. The analysis of \sec \ref{afvs} remains valid; the map $\epsilon$ can be defined by $\epsilon({\frak S}_{w,{\frak r}})=\widehat{{\frak S}}_{w,{\frak r}}$. In particular we have an invariant arithmetic Chow ring $\widehat{CH}_{inv}(F({\frak r}))$ for which Theorem \ref{chinv} holds. If $F({\frak r})=G_d$ is a Grassmannian over $\mbox{Spec}{\Bbb Z}$, then $\widehat{CH}_{inv}(G_d)$ coincides with the Arakelov Chow ring $CH(\overline{G_d})$, where $G_d({\Bbb C})$ is given its natural invariant K\"{a}hler metric, as in [Ma]. Suppose that ${\frak r}^{\prime}$ is a refinement of ${\frak r}$, so we have a projection $p:F({\frak r}^{\prime})\rightarrow F({\frak r})$. In this case there are natural inclusions $\widetilde{\mbox{Inv}}(F({\frak r})_{{\Bbb R}})\hookrightarrow \widetilde{\mbox{Inv}}(F({\frak r}^{\prime})_{{\Bbb R}})$ and $CH(F({\frak r}))\hookrightarrow CH(F({\frak r}^{\prime}))$. Applying the five lemma to the two exact sequences (\ref{invex}) shows that the pullback $p^*:\widehat{CH}_{inv}(F({\frak r}))\hookrightarrow \widehat{CH}_{inv}(F({\frak r}^{\prime}))$ is an injection. Note however that this is not compatible with the splitting of Theorem \ref{chinv}. One can compute arithmetic intersections in $\widehat{CH}_{inv}(F({\frak r}))$ as in \sec \ref{ai}. Applying Theorem \ref{abc} to the filtration (\ref{fil3}) (with induced metrics as above) gives the key relation required for the calculation. In particular we see that all the arithmetic Chern numbers are rational, as is the Faltings height of $F({\frak r})$ in its natural pluri-Pl\"{u}cker embedding. Theorem \ref{slheight} thus generalizes the corresponding result of Maillot mentioned in \sec \ref{intro}. There is an arithmetic Schubert calculus in $\widehat{CH}_{inv}(F({\frak r}))$ analogous to that for complete flags. The analogue of Lemma \ref{schlemma} is true, that is ${\frak S}_w\hspace{-2pt}\in\hspace{-2pt} I_{n,{\frak r}}$ if $w\hspace{-2pt}\in\hspace{-2pt} T_{n,{\frak r}}$ (this is an easy consequence of Lemma \ref{schlemma} itself). It follows that for $w\hspace{-2pt}\in\hspace{-2pt} T_{n,{\frak r}}$, $\widehat{{\frak S}}_{w,{\frak r}}$ is a class $\widetilde{{\frak S}}_{w,{\frak r}}\hspace{-2pt}\in\hspace{-2pt} \widetilde{\mbox{Inv}}(F({\frak r})_{{\Bbb Q}})$. The analogue of (\ref{punchline}) in this context is \begin{equation} \label{partialpunch} \widehat{{\frak S}}_{u,{\frak r}}\cdot\widehat{{\frak S}}_{v,{\frak r}}=\sum_{w\in S_{n,{\frak r}}}c_{uv}^w\widehat{{\frak S}}_{w,{\frak r}}+ \sum_{w\in T_{n,{\frak r}}}c_{uv}^w\widetilde{{\frak S}}_{w,{\frak r}} \end{equation} where $u,v\hspace{-2pt}\in\hspace{-2pt} S_{n,{\frak r}}$ and the numbers $c_{uv}^w$ are as in (\ref{cuvws}). The remaining statements of Theorem \ref{slring} require no further change. \medskip \noindent {\bf Remark.} Equation (\ref{partialpunch}) is not a direct generalization of the analogous statement in [Ma], Theorem 5.2.1. However one can reformulate Maillot's results using the classes $\widehat{c}_*(\overline{S})$ instead of $\widehat{c}_*(\overline{Q}-\overline{{\cal E}})$ (notation as in [Ma], \sec 5.2). With this modification, the arithmetic Schubert calculus described above (for $m=2$) and that in [Ma] coincide. In the grassmannian case ${\frak S}_{w,{\frak r}}$ is a Schur polynomial and there are explicit formulas for $\widetilde{{\frak S}}_{w,{\frak r}}$ in terms of harmonic forms on $G_d({\Bbb C})$ (as in loc. cit.).

9611

alg-geom/9611019

en

https://arxiv.org/abs/alg-geom/9611019

alg-geom/9611019

Peter Magyar

Peter M. Magyar

Bott-Samelson Varieties and Configuration Spaces

email address [email protected] LaTeX

We give a new construction of the Bott-Samelson variety $Z$ as the closure of a $B$-orbit in a product of flag varieties $(G/B)^l$. This also gives an embedding of the projective coordinate ring of the variety into the function ring of a Borel subgroup: $\CC[Z] \subset \CC[B]$. In the case of the general linear group $G = GL(n)$, this identifies $Z$ as a configuration variety of multiple flags subject to certain inclusion conditions, closely related to the the matrix factorizations of Berenstein, Fomin and Zelevinsky. As an application, we give a geometric proof of the theorem of Kraskiewicz and Pragacz that Schubert polynomials are characters of Schubert modules. Our work leads on the one hand to a Demazure character formula for Schubert polynomials and other generalized Schur functions, and on the other hand to a Standard Monomial Theory for Bott-Samelson varieties.

alg-geom

\section{Bott-Samelson varieties} \subsection{Three constructions} \label{Three constructions} In this section, $G$ is a reductive algebraic group. Our constructions are all valid over an arbitrary field, or over the integers, but we will use the complex numbers ${\bf C}$ for convenience. Let $W$ denote the Weyl group generated by simple reflections $s_1, \ldots , s_r$, where $r$ is the rank of $G$. For $w \in W$, \, $\ell(w)$ denotes the length of a reduced (i\.e\. minimal) decompostion $w = s_{i_1} \ldots s_{i_l}$, and $w_0$ is the element of maximal length. We let $B$ be a Borel subgroup, $T \subset B$ a maximal torus (Cartan subgroup), and $U_{{\alpha}} \subset B$ the one-dimensional unipotent subgroup associated to the root ${\alpha}$. Let $P_k \supset B$ be the {\em minimal} parabolic associated to the simple reflection $s_k$, so that $P_i/B \cong {\bf P}^1$, the projective line. Also, take $ \widehat{P} _k \supset B$ to be the {\em maximal} parabolic associated to the reflections $s_1,\ldots, \widehat{s_k},\ldots, s_r$. Finally, we have the Schubert variety as a $B$-orbit closure inside the flag variety: $$ X_w = \overline{BwB} \subset G/B $$ For what follows, we fix a reduced decompostion of some $w \in W$, $$ w = s_{i_1} \ldots s_{i_l}, $$ and we denote $ {\bf i} = (i_1,\ldots,i_l)$. Now let $P \supset B$ be any parabolic subgroup of $G$, and $X$ any space with $B$-action. Then the {\em induced $P$-space} is the quotient $$ P \mathop{\times} ^B X \stackrel{\rm def}{=} (P \times X)/B $$ where the quotient is by the free action of $B$ on $P \times X$ given by $(p,x) \cdot b = (pb, b^{-1}x)$. (Thus $(pb,x) = (p,bx)$ in the quotient.) The key property of this construction is that $$ \begin{array}{ccc} X & \rightarrow & P \mathop{\times} ^B X \\ & & \downarrow \\ & & P/B \end{array} $$ is a fiber bundle with fiber $X$ and base $P/B$. We can iterate this construction for a sequence of parabolics $P, P',\ldots$, $$ P \mathop{\times} ^B P' \mathop{\times} ^B \cdots \stackrel{\rm def}{=} P \mathop{\times} ^B (\, P' \mathop{\times} ^B (\cdots)\,). $$ Then the {\bf quotient Bott-Samelson variety} of the reduced word $ {\bf i} $ is $$ \Zii^{\tiny quo} \stackrel{\rm def}{=} P_{i_1} \mathop{\times} ^B \cdots \mathop{\times} ^B P_{i_l} /B. $$ Because of the fiber-bundle property of induction, $ \Zii^{\tiny quo} $ is clearly a smooth, irreducible variety of dimension $l$. It is a subvariety of $$ X_l \stackrel{\rm def}{=} \underbrace{G \mathop{\times} ^B \cdots \mathop{\times} ^B G}_{l\ \mbox{\footnotesize factors}} /B. $$ $B$ acts on these spaces by multiplying the first coordinate: $$ b \cdot (p_1,p_2,\ldots,p_l) \stackrel{\rm def}{=} (bp_1, p_2, \ldots, p_l). $$ The original purpose of the Bott-Samelson variety was to desingularize the Schubert variety $X_w$ via the multiplication map: $$ \begin{array}{ccc} \Zii^{\tiny quo} & \rightarrow & X_w \subset G/B \\ (p_1,\ldots,p_l) & \mapsto & p_1 p_2 \cdots p_l B, \end{array} $$ a birational morphism. Next, consider the {\em fiber product} $$ G/B \mathop{\times} _{G/P} G/B \stackrel{\rm def}{=} \{(g_1,g_2) \in (G/B)^2 \mid g_1 P = g_2 P \}. $$ We may define the {\bf fiber product Bott-Samelson variety} $$ \Zii^{\tiny fib} \stackrel{\rm def}{=} eB \mathop{\times} _{G/P_{i_1}} G/B \mathop{\times} _{G/P_{i_2}} \cdots \mathop{\times} _{G/P_{i_l}} G/B \subset (G/B)^{l+1}. $$ We let $B$ act diagonally on $(G/B)^{l+1}$; that is, simultaneously on each factor: $$ b \cdot (g_0 B,g_1 B,\ldots,g_l B) \stackrel{\rm def}{=} (b g_0 B, b g_1 B, \ldots, b g_l B). $$ This action restricts to $ \Zii^{\tiny fib} $. The natural map to the flag variety is the projection to the last coordinate: $$ \begin{array}{ccc} \Zii^{\tiny fib} & \rightarrow & G/B \\ (e B, g_1 B, \ldots, g_l B) & \mapsto & g_l B \end{array} $$ Finally, let us define the {\bf $B$-orbit Bott-Samelson variety} as the closure (in either the Zariski or analytic topologies) of the orbit of a point $ z_{\ii} $: $$ \Zii^{\tiny orb} \stackrel{\rm def}{=} \overline{B \cdot z_{\ii} } \subset G/ \widehat{P} _{i_1} \times \cdots \times G/ \widehat{P} _{i_l}, $$ where $$ z_{\ii} = (s_{i_1} \widehat{P} _{i_1}, \, s_{i_1}\! s_{i_2} \widehat{P} _{i_2} \, , \ldots , \, s_{i_1}\!\! \cdots\! s_{i_l} \widehat{P} _{i_l}) $$ Again, $B$ acts diagonally. In this case the map to $G/B$ is more difficult to describe, but see Sec. \ref{Varieties and defining equations}. \subsection{Isomorphism theorem} \label{Isomorphism theorem} The three types of Bott-Samelson variety are isomorphic. \begin{thm} \label{isomorphism theorem} (i) Let $$ \begin{array}{cccc} \phi : & X_l & \rightarrow & (G/B)^{l+1} \\ & (g_1, g_2, \ldots , g_l) & \mapsto & (\overline{e}, \overline{g_1}, \, \overline{g_1 g_2} \, , \ldots , \, \overline{g_1 g_2\! \cdots\! g_l}) , \end{array} $$ where $\overline{g}$ means the coset of $g$. Then $\phi$ restricts to an isomorphism of $B$-varieties $$ \phi : \Zii^{\tiny quo} \stackrel{\sim}{\rightarrow} \Zii^{\tiny fib} . $$ (ii) Let $$ \begin{array}{cccccccccc} \psi : & X_l & \rightarrow & G/ \widehat{P} _{i_1} & \times & G/ \widehat{P} _{i_2} & \times & \cdots & \times & G/ \widehat{P} _{i_l} \\ & (g_0,g_1, \ldots , g_l) & \mapsto & (\ \ \overline{g_1}&,& \overline{g_1 g_2} &,& \ldots &,& \overline{g_1 g_2\! \cdots\! g_l}) , \end{array} $$ where $\overline{g}$ means the coset of $g$. Then $\psi$ restricts to an isomorphism of $B$-varieties $$ \psi : \Zii^{\tiny quo} \stackrel{\sim}{\rightarrow} \Zii^{\tiny orb} . $$ \end{thm} {\em Proof.} (i) It is trivial to verify that $\phi$ is a $B$-equivariant isomorphism from $X_l$ to $eB \times (G/B)^l$ and that $\phi( \Zii^{\tiny quo} ) \subset \Zii^{\tiny fib} $, so it suffices to show the reverse inclusion. Suppose $$ z_f = (eB, g_1 B, \ldots, g_l B) \in \Zii^{\tiny fib} . $$ Then $$ z_q = \phi^{-1}(z_f) = (g_1, g_1^{-1} g_2, g_2^{-1} g_3, \ldots ) \in X_l. $$ By definition, $e P_{i_1} = g_1 P_{i_1}$, so $g_1 \in P_{i_1}$. Also $g_1 P_{i_2} = g_2 P_{i_2}$, so $g_1^{-1} g_2 \in P_{i_2}$, and similarly $g_{k-1}^{-1} g_k \in P_{i_k}$. Hence $z_q \in \Zii^{\tiny quo} $, and $\phi(z_q) = z_f$. \\ (ii) First let us show that $\psi$ is injective on $ \Zii^{\tiny quo} $. Suppose $\psi(p_1, \ldots, p_l) = \psi(q_1, \ldots, q_l)$ for $p_k, q_k \in P_{i_k}$. Then $p_1 \widehat{P} _{i_1} = q_1 \widehat{P} _{i_1}$, so that $p_1^{-1} q_1 \in \widehat{P} _{i_1} \cap P_{i_1} = B$. Thus $q_1 = p_1 b_1$ for $b_1 \in B$. Next, we have $$ p_1 p_2 \widehat{P} _{i_2} = q_1 q_2 \widehat{P} _{i_2} = p_1 b_1 q_2 \widehat{P} _{i_2}, $$ so that $p_2^{-1} b_1 q_2 \in \widehat{P} _{i_2} \cap P_{i_2} = B$, and $q_2 = b_1^{-1} p_2 b_2$ for $b_2 \in B$. Continuing in this way, we find that \begin{eqnarray*} (q_1, q_2, \ldots , q_l) & = & (p_1 b_1, b_1^{-1} p_2 b_2, \ldots, b_{l-1}^{-1} p_l b_l) \\ & = & (p_1, p_2, \ldots, p_l) \in X_l \end{eqnarray*} Thus $ \psi$ is injective on $ \Zii^{\tiny quo} $. Since we are working with algebraic morphisms, we must also check that $ \psi$ is injective on tangent vectors of $ \Zii^{\tiny quo} $. Now, the degeneracy locus $$ \{z \in \Zii^{\tiny quo} \mid \mbox{\rm Ker}\ d\psi_z \neq 0 \} $$ is a $B$-invariant, closed subvariety of $ \Zii^{\tiny quo} $, and by Borel's Fixed Point Theorem it must contain a $B$-fixed point. But it is easily seen that the degenerate point $$ z_0 = (e,\ldots,e) \in X_l $$ is the only fixed point of $ \Zii^{\tiny quo} $. Thus if $d \psi$ is injective at $z_0$, then the degeneracy locus is empty, and $d \psi$ is injective on each tangent space. The injectivity at $z_0$ is easily shown by an argument completely analogous to that for global injectivity given above, but written additively in terms of Lie algebras instead of multiplicatively with Lie groups. Thus it remains to show surjectivity: that $\psi$ takes $ \Zii^{\tiny quo} $ onto $ \Zii^{\tiny orb} $. Consider $$ z_{\ii}^{\tiny quo} = (s_{i_1},\ldots, s_{i_l}) \in X_l, $$ a well-defined point in $ \Zii^{\tiny quo} $. Then $$ \psi( z_{\ii}^{\tiny quo} ) = z_{\ii} = (s_{i_1} \widehat{P} _{i_1}, s_{i_1} s_{i_2} \widehat{P} _{i_2}, \ldots), $$ and $\psi$ is $B$-equivariant, so that $\psi( \Zii^{\tiny quo} ) \supset \ \psi (\overline{B \cdot z_{\ii}^{\tiny quo} }) = \overline{B \cdot z_{\ii} } = \Zii^{\tiny orb} $. Now we need only show that $\psi( \Zii^{\tiny quo} ) \subset \Zii^{\tiny orb} $, which results from the following: \begin{lem} $B \cdot z_{\ii}^{\tiny quo} $ is an open dense orbit in $ \Zii^{\tiny quo} $. \end{lem} {\bf Proof.} Since $ \Zii^{\tiny quo} $ is irreducible of dimension $l$, it suffices to show that the orbit has (at least) the same dimension. We may see this by determining $ {\mbox{\rm Stab}} _B( z_{\ii}^{\tiny quo} )$. Suppose $$ (b s_{i_1},\ldots,s_{i_l}) = (s_{i_1}b_1,\, b_1^{-1}s_{i_2} b_2, \ldots,b_{l-1}^{-1} s_{i_l} b_l) \in \Zii^{\tiny quo} . $$ Then $s_{i_l} = b_{l-1}^{-1} s_{i_l} b_{l}$, and $b_{l-1} \in B \cap s_{i_l} B s_{i_l}$. Repeating this calculation leftward, we find that $b \in B \cap w B w^{-1}$, so that $ {\mbox{\rm Stab}} _B( z_{\ii} ) \subset B \cap w B w^{-1}$. (Recall $w = s_{i_1} \dots s_{i_l}$.) Thus, using some well-known facts (see \cite{Spr}) we have: \begin{eqnarray*} \dim( B \cdot z_{\ii}^{\tiny quo} ) & = & \dim(B) - \dim( {\mbox{\rm Stab}} _B( z_{\ii} )) \\ & \geq & \dim(B) - \dim(B \cap wBw^{-1}) \\ & = & \dim(B)-(\dim(B) - \ell(w)\,) \\ & = & \ell(w) \, = \, l. \end{eqnarray*} Since the orbit can have dimension no bigger than $l$, we must have equality. Thus the Lemma and the Theorem both follow. $\bullet$ \begin{cor} For $w = s_{i_1} \cdots s_{i_l}$, we have $$ {\mbox{\rm Stab}} _B( z_{\ii} \in Z_{\ii} ) = {\mbox{\rm Stab}} _B(wB \in G/B) = B \cap w B w^{-1}. $$ \end{cor} \subsection{Open cells} \label{Open cells} In view of the Theorem, we will let $ Z_{\ii} $ denote the abstract Bott-Samelson variety defined by any of our three versions. It contains the degenerate $B$-fixed point $z_0$ defined by: \begin{eqnarray*} z_0 & = & (e,e,\ldots) \in \Zii^{\tiny quo} \\ &=& (eB,eB,\ldots) \in \Zii^{\tiny fib} \\ &=& (e \widehat{P} _{i_1},e \widehat{P} _{i_2},\ldots) \in \Zii^{\tiny orb} \end{eqnarray*} as well as the generating $T$-fixed point whose $B$-orbit is dense in $ Z_{\ii} $: \begin{eqnarray*} z_{\ii} &=& (s_{i_1}, s_{i_2}, s_{i_3}, \ldots) \in \Zii^{\tiny quo} \\ &=& (e B, s_{i_1} B, s_{i_1} s_{i_2} B, \ldots) \in \Zii^{\tiny fib} \\ &=& (s_{i_1} \widehat{P} _{i_1}, s_{i_1} s_{i_2} \widehat{P} _{i_2}, \ldots) \in \Zii^{\tiny orb} \end{eqnarray*} We may parametrize the dense orbit $B \cdot z_{\ii} \subset Z_{\ii} $ by an affine cell. Consider the normal ordering of the positive roots associated to the reduced word $ {\bf i} $. That is, let $$ {\beta}_1 = {\alpha}_{i_1},\ {\beta}_2 = s_{i_1}( {\alpha}_{i_2}),\ {\beta}_3 = s_{i_1} s_{i_2} ({\alpha}_{i_3}),\ \cdots $$ Recall that $U_{{\beta}_k}$ is the one-dimensional unipotent subgroup of B corresponding to the positive root ${\beta}_k$. Then we have a direct product: $$ B = U_{{\beta}_1}\! \cdots U_{{\beta}_l}\cdot (B\cap wBw^{-1}), $$ so that the multiplication map $$ \begin{array}{ccc} U_{{\beta}_1} \times \cdots \times U_{{\beta}_l} & \rightarrow & B\cdot z_{\ii} \\ (u_1,\ldots,u_l) & \mapsto & u_1\cdots u_l \cdot z_{\ii} \end{array} $$ is injective, and an isomorphism of varieties. The left-hand side is isomorphic to an affine space ${\bf C}^l$. \\[1em] $ Z_{\ii} $ also contains an opposite big cell centered at $z_0$ which is not the orbit of a group. Consider the one-dimensional unipotent subgroups $U_{-{\alpha}_{i}}$ corresponding to the negative simple roots $-{\alpha}_i$. The map $$ \begin{array}{cccc} {\bf C}^l\ \cong \!\! & U_{-{\alpha}_{i_1}}\! \times \cdots \times U_{-{\alpha}_{i_l}} & \rightarrow & \Zii^{\tiny quo} \\ & (u_1,\ldots,u_l) & \mapsto & (u_1 , \dots , u_l) \end{array} $$ is an open embedding. In the case of $G = GL(n)$, $B = $ upper triangular matrices, we may write an element of $U_{-{\alpha}_{i_k}}$ as $u_k = I+t_k e_k$, where $I$ is the identity matrix, $e_k$ is the sub-diagonal coordinate matrix $e_{(i_k+1,i_k)}$, and $t_k \in {\bf C}$. If we further map $ \Zii^{\tiny quo} $ to $G/B$ via the natural multiplication map, we get $$ \begin{array}{ccc} (t_1,\ldots,t_l) & \mapsto & (I+t_1 e_1) \cdots (I+t_l e_l) \\ {\bf C}^l & \rightarrow & N_- \\ \cap && \cap \\ \Zii^{\tiny quo} &\rightarrow & G/B \\ (p_1,\ldots,p_l) & \mapsto & p_1\cdots p_l B \end{array} $$ where $N_-$ denotes the unipotent lower triangular matrices (mod $B$). Thus the multiplication on the bottom is a compactification of the matrix factorizations studied by Berenstein, Fomin, and Zelevinsky \cite{BFZ}. \section{Configuration varieties} We define a class of varieties (more general than the Schubert varieties) which are desingularized by Bott-Samelson varieties. \subsection{Definitions} \label{Definitions} We continue with the case of a general reductive group $G$. Given a sequence of Weyl group elements $ {\bf w} = (w_1,\ldots,w_k)$ and a sequence of indices $ {\bf j} = ({j_1},\ldots,{j_k})$, we consider the $T$-fixed point $$ z_{ {\bf w} {\bf j} } = (w_1 \widehat{P} _{j_1},\ldots,w_k \widehat{P} _{j_k}) \in G/ \widehat{P} _{j_1} \times \cdots \times G/ \widehat{P} _{j_k}, $$ and we define the {\em configuration variety} as the $G$-orbit closure $$ { \cal F } _{ {\bf w} {\bf j} } \stackrel{\rm def}{=} \overline{ G \cdot z_{ {\bf w} {\bf j} }} \subset G/ \widehat{P} _{j_1} \times \cdots \times G/ \widehat{P} _{j_k}. $$ $G$ acts on this variety by multiplying each factor simultaneously (the diagonal action). We may define a ``flagged'' version of this construction by replacing $G$ with $B$. The {\em flagged configuration variety} is the $B$-orbit closure $$ \FF^B _{ {\bf w} {\bf j} } \stackrel{\rm def}{=} \overline{ B \cdot z_{ {\bf w} {\bf j} }} \subset G/ \widehat{P} _{j_1} \times \cdots \times G/ \widehat{P} _{j_k}. $$ Again, $B$ acts diagonally. \\[1em] {\bf Examples.} (a) Take $ {\bf w} = (w,w,\ldots,w)$ for any $w \in W$ and $ {\bf j} = (1,2,\ldots,r)$ \, (where $r = \mathop{\rm rank} G$). Then the configuration variety is isomorphic to the flag variety of $G$, and the flagged configuration variety is isomorphic to the Schubert variety of $w$: $$ { \cal F } _{ {\bf w} {\bf j} } \cong G/B \ \ \ \ \ \ \ \ \ \ \ \FF^B _{ {\bf w} {\bf j} } \cong X_w \ . $$ (b) For $ {\bf j} = {\bf i} = (i_1,i_2,\ldots)$, a reduced word, and $ {\bf w} = (s_{i_1}, s_{i_1} s_{i_2},\ldots)$, the flagged configuration variety is exactly our orbit version of the Bott-Samelson variety: $ \FF^B _{ {\bf w} {\bf j} } = \Zii^{\tiny orb} = Z_{\ii} .$ \ $\bullet$ \\[1em] {\bf Remark.} For a given $G$, there are only finitely many configuration varieties up to isomorphism. In fact, suppose a list ($ {\bf w} $, $ {\bf j} $) has repetitions of some element of $ {\bf w} $ with identical corresponding entries in $ {\bf j} $. Then we may remove the repetitions and the configuration variety will not change (up to $G$-equivariant isomorphism), only the embedding. Thus, all configuration varieties are projections of a maximal variety. This holds for the flagged and unflagged cases. \\[1em] {\bf Example.} The maximal configuration variety for $G = GL(3)$ is the {\em space of triangles} \cite{MTri}, and corresponds to $$ \begin{array}{ccccccccl} {\bf w} & = & (e,& e,& s_1,& s_2,& s_2 s_1,& s_1 s_2 &) \\ {\bf j} & = & (1,& 2,& 1, & 2, & 1, & 2 &). \end{array} $$ Further entries would be redundant: for example, $s_1 \widehat{P} _2 = e \widehat{P} _2$. All other configuration varieties are obtained by omitting some entries of $ {\bf w} $ and the corresponding entries of $ {\bf j} $. Hence there are at most $2^6$ configuration varieties for $G$. $\bullet$ \\[1em] One might attempt to broaden the definition of configuration varieties by replacing the minimal homogeneous spaces $G/ \widehat{P} _j$ by $G/P$ for arbitrary parabolics $P \supset B$. This gives the same class of varieties, however, since any $G/P$ can be embedded equivariantly inside a product of $G/ \widehat{P} _j$'s, resulting in isomorphic orbit closures. Once again, this changes only the embeddings, not the varieties. Varieties similar to our $ { \cal F } _{ {\bf w} {\bf j} }$ are defined and some small cases are analyzed in Langlands' paper \cite{Langlands}. \subsection{Desingularization} \label{Desingularization} Very little is known about general configuration varieties. However, certain of them are well understood because they can be desingularized by Bott-Samelson varieties. Recall that a sequence $ {\bf w} = (w_1,\ldots,w_K)$ of Weyl group elements is increasing in the weak order on $W$ if there exist $u_1,u_2,\ldots, u_K$ such that $w_k = u_1 u_2 \cdots u_k$ and $\ell(w_k) = \ell(w_{k-1}) + \ell(u_k)$ for all $k$. For $ {\bf w} = (w_1,\ldots,w_K)$ and $ {\bf j} = (j_1,\ldots,j_K)$, let $ {\bf w} ^+ = (e,\ldots,e,w_1,\ldots,w_k)$ with $r$ added entries of $e$, and $ {\bf j} ^+ = (1,2,\ldots,r,j_1,\ldots,j_K)$. Clearly $$ \FF^B _{ {\bf w} , {\bf j} } \cong \FF^B _{ {\bf w} ^+ {\bf j} ^+}. $$ \begin{prop} \label{desingularization} If $ {\bf w} $ is increasing in the weak order and $ {\bf j} $ is arbitrary, then the flagged configuration variety $ \FF^B _{ {\bf w} {\bf j} }$ can be desingularized by a Bott-Samelson variety. That is, there exists a reduced word $ {\bf i} $ and a regular birational morphism $$ \pi: Z_{\ii} \rightarrow \FF^B _{ {\bf w} {\bf j} }. $$ Furthermore, the unflagged configuration variety $ { \cal F } _{ {\bf w} ^+ {\bf j} ^+}$ is desingularized by the composite map $$ G \mathop{\times} ^B Z_{\ii} \stackrel{\mbox{id} \times \pi}{\rightarrow} G \mathop{\times} ^B \FF^B _{ {\bf w} {\bf j} } \cong G \mathop{\times} ^B \FF^B _{ {\bf w} ^+ {\bf j} ^+} \stackrel{\mu}{\rightarrow} { \cal F } _{ {\bf w} ^+ {\bf j} ^+}, $$ where $\mbox{id} \times \pi$ is the map induced from $\pi$, and $\mu$ is the multiplication map $(g, v) \mapsto g \cdot v$. \end{prop} {\bf Remark.} The map $$ G \mathop{\times} ^B Z_{\ii} \rightarrow G \mathop{\times} ^B \FF^B _{ {\bf w} {\bf j} } \rightarrow { \cal F } _{ {\bf w} {\bf j} } $$ is a surjection from a smooth space to $ { \cal F } _{ {\bf w} {\bf j} }$, but it is not birational in general. We will see in Sec 4 that for the purposes of Borel-Weil theory, this map can substitute for a desingularization of $ { \cal F } _{ {\bf w} {\bf j} }$. $\bullet$ \\[1em] To prove the Proposition, we will need the following \begin{lem} (a) For any $w \in W$ and parabolic $P$ with Weyl group $W(P)$, we have a unique factorization $w = \tilde{w} y$, where $y \in W(P)$, $ \tilde{w} $ has minimal length in $ \tilde{w} W(P)$, and $\ell(w) = \ell( \tilde{w} )+\ell(y)$. \\ (b) Suppose $w \in W$ has minimum length in the coset $w W(P)$, and consider the points $wP \in G/P$ and $wB \in G/B$. Then $ {\mbox{\rm Stab}} _B(wP) = {\mbox{\rm Stab}} _B(wB)$. \end{lem} {\bf Proof of Lemma.} (a) Well-known (see \cite{Humphreys}, \cite{Hiller}). \\ (b) The $\supset$ containment is clear, so we prove the other. Let ${\Delta}$ denote the set of roots of $G$, ${\Delta}_+$ the positive roots, ${\Delta}(P)$ the roots of $P$, etc. From considering the corresponding Lie algebras we obtain: $$ \begin{array}{rcl} \dim {\mbox{\rm Stab}} _{B}(wB) & = & | {\Delta}_+ \cap w({\Delta}_+) | \\ \dim {\mbox{\rm Stab}} _{B}(wP) & = & | {\Delta}_+ \cap w({\Delta}_+ \!\cup {\Delta}(P)) | . \end{array} $$ But the two sets on the right are identical. In fact, if $w$ is minimal in $wP$, then ${\Delta}_+ \cap w({\Delta}_-(P)) = \emptyset$. (See \cite{Humphreys}, 5.5, 5.7.) $\bullet$. \mbox{}\\[1em] {\bf Proof of Proposition.} Denote $W_k \stackrel{\rm def}{=} W( \widehat{P} _{j_k})$, a parabolic subgroup of the Weyl group. Given $ {\bf w} $ and $ {\bf j} $, we define a new sequence $ \widetilde{\ww} = ( \tilde{w} _1,\ldots, \tilde{w} _K)$. Take $ \tilde{w} _k$ to be the minimum-length coset representative in $w_k W_k$, so that $w_k = \tilde{w} _k y_k$ for some $y_k \in W_k$. I claim the new sequence $ \widetilde{\ww} $ is still increasing in the weak order. In fact, if $w_k = u_1\cdots u_k$ and $ \tilde{u} _k$ is minimal in $u_k W_k$, then $ \tilde{w} _k = \tilde{w} _{k-1} y_k \tilde{u} _k$ and $\ell( \tilde{w} _k) = \ell( \tilde{w} _{k-1}) + \ell(y_k) + \ell( \tilde{u} _k)$. Note that it is possible that $ \tilde{u} _k = e$, and $ \tilde{w} _k = \tilde{w} _{k+1}$. Now let $ {\bf i} $ be any reduced decomposition of the increasing sequence $ \widetilde{\ww} $: that is, for each $k$ we have a reduced decompostion $ \tilde{w} _k = s_{i_1} s_{i_2} \cdots s_{i_{l(k)}}$, where $l(k) = \ell( \tilde{w} _k)$, so that $0 \leq l(1) \leq l(2) \cdots \leq l(K)=l$. Also, $i_{l(k)} = j_k$ for all $k$. Define a projection map from the Bott-Samelson variety to the configuration variety: $$ \begin{array}{cccc} \phi: & Z_{\ii} = \Zii^{\tiny orb} & \rightarrow & \FF^B _{ {\bf w} {\bf j} } \\ & (g_1 \widehat{P} _{i_1},\ldots,g_l \widehat{P} _{i_l}) & \mapsto & (g_{l(1)} \widehat{P} _{j_1} \, , \ldots, \, g_{l(K)} \widehat{P} _{j_K}). \end{array} $$ I claim $\phi$ is well-defined, $B$-equivariant, onto, regular, and birational. Now, $ \tilde{w} _k$ and $w_k$ are equal modulo $W_k$, so $ \tilde{w} _k \widehat{P} _{j_k} = w_k \widehat{P} _{j_k}$, and thus $$ \phi(z_{ {\bf i} }) = z_{ \widetilde{\ww} {\bf j} } = z_{ {\bf w} {\bf j} } \in \FF^B _{ {\bf w} {\bf j} }. $$ Since $ Z_{\ii} = \overline{B \cdot z_{ {\bf i} }}$, this implies that the image of $\phi$ lies inside $ { \cal F } _{ {\bf w} {\bf j} }$, and $\phi$ is well-defined. It is clearly $B$-equivariant and therefore onto (since $ \FF^B _{ {\bf w} {\bf j} }$ is a $B$-orbit closure). The map is regular, and to show it is birational we need only check that it is a bijection between the big $B$-orbits in the domain and image. That is, we must show equality of the stabilizers $$ {\mbox{\rm Stab}} _B( z_{\ii} ) = {\mbox{\rm Stab}} _B(z_{ \widetilde{\ww} {\bf j} }). $$ By the corollary in Section \ref{Isomorphism theorem}, we have $ {\mbox{\rm Stab}} _B( z_{\ii} )= {\mbox{\rm Stab}} _B(wB \in G/B)$ for $w=s_{i_1}\cdots s_{i_l(K)}= \tilde{w} _K$. Now we use induction on the length of the sequence $ {\bf w} $. If the length $K=1$, we have immediately that $ {\mbox{\rm Stab}} _B(z_{ {\bf w} {\bf j} }) = {\mbox{\rm Stab}} _B( \tilde{w} _K \widehat{P} _{j_K}) = {\mbox{\rm Stab}} _B( \tilde{w} _K B)$ by the above Lemma. Assuming the assertion for $ {\bf w} '=(w_1,\ldots,w_{K-1})$ and using the Lemma, we have $$ \begin{array}{rcl} {\mbox{\rm Stab}} _B(z_{ \widetilde{\ww} {\bf j} }) & = & {\mbox{\rm Stab}} _B(z_{ \widetilde{\ww} ' {\bf j} }) \cap {\mbox{\rm Stab}} _B( \tilde{w} _K \widehat{P} _{j_K}) \\ &=& {\mbox{\rm Stab}} _B( \tilde{w} _{K-1}B) \cap {\mbox{\rm Stab}} _B( \tilde{w} _K B) \\ &=& {\mbox{\rm Stab}} _B( \tilde{w} _K B) . \end{array} $$ \mbox{} \\[1em] The remaining assertions about the unflagged $ { \cal F } _{ {\bf w} ^+ {\bf j} ^+}$ follow easily. That is, the map of fiber bundles $$ G \mathop{\times} ^B Z_{\ii} \rightarrow G \mathop{\times} ^B \FF^B _{ {\bf w} ^+ {\bf j} ^+} $$ is $G$-equivariant, onto, and regular and birational by our results above, and so is the multiplication map $$ G \mathop{\times} ^B \FF^B _{ {\bf w} ^+ {\bf j} ^+} \rightarrow { \cal F } _{ {\bf w} ^+ {\bf j} ^+} $$ since $ {\mbox{\rm Stab}} _G(z_{ {\bf w} ^+ {\bf j} ^+}) = {\mbox{\rm Stab}} _B(z_{ {\bf w} ^+ {\bf j} ^+})$. $\bullet$ \section{The Case of $GL(n)$} We begin again, restating many of our results more explicitly for the general linear group $G = GL(n,{\bf C})$. In this case $B$ = upper triangular matrices, $T =$ diagonal matrices, $r = n-1$, $$ P_k = \{ (x_{ij}) \in GL(n) \mid x_{ij} = 0 \mbox{ if } i>j \mbox{ and } (i,j) \neq (k+1,k) \}, $$ $$ \widehat{P} _k = \{ (x_{ij}) \in GL(n) \mid x_{ij} = 0 \mbox{ if } i>k\geq j \}, $$ and $G/ \widehat{P} _k \cong \mbox{\rm Gr}(k,{\bf C}^n)$, the Grassmannian of $k$-dimensional subspaces of complex $n$-space. Also $W$ = permutation matrices, $\ell(w) =$ the number of inversions of a permutation $w$, $s_i =$ the transposition $(i,i+1)$, and the longest permutation is $w_0 = n, n-1, \ldots, 2,1$. We will frequently use the notation $$ [k] = \{1,2,3,\ldots,k\}. $$ \subsection{Subset families} \label{Subset families} First, we introduce some combinatorics. Define a {\em subset family} to be a collection $D = \{C_1,C_2,\ldots\}$ of subsets $C_k \subset [n]$. The order of the subsets is irrelevant in the family, and we do not allow subsets to be repeated. This relates to the previous sections as follows. To a list of permutations $ {\bf w} = (w_1,\ldots,w_K)$, $w_k \in W$, and a list of indices $ {\bf j} = (j_1, \ldots, j_K)$, $1 \leq j_k \leq n$, we associate a subset family: $$ D = D_{ {\bf w} {\bf j} } \stackrel{\rm def}{=} \{ w_1 [j_1], \ldots, w_K[j_K] \} . $$ Here $w[j] = \{ w(1), w(2), \ldots, w(j)\}$. Now suppose the list of indices $ {\bf i} = (i_1, i_2, \ldots , i_l)$ encodes a reduced decomposition $w = s_{i_1} s_{i_2} \cdots s_{i_l}$ of a permutation into a minimal number of simple transpositions. We let $ {\bf w} = (s_{i_1}, s_{i_1} s_{i_2}, \ldots, w)$ and $ {\bf j} = {\bf i} $, and we define the {\em reduced chamber family} $D_{ {\bf i} } \stackrel{\rm def}{=} D_{ {\bf w} {\bf j} }$. Further, define the {\em full chamber family} $$ D^+_{\ii} \stackrel{\rm def}{=} \{ [1], [2], \ldots, [n] \} \cup D_{\ii} , $$ (which is $D_{ {\bf w} ^+ {\bf j} ^+}$ in our previous notation). We tentatively connect these structures with geometry. Let ${\bf C}^n$ have the standard basis $e_1,\ldots,e_n$. For any subset $C = \{j_1,\ldots, j_k\} \subset [n]$, the coordinate subspace $$ E_C = {\mbox{\rm Span}} _{{\bf C}}\{e_{j_1},\ldots,e_{j_k}\}\, \in\, \mbox{\rm Gr}(k) $$ is a $T$-fixed point in a Grassmannian. A subset family corresponds to a $T$-fixed point in a product of Grassmannians $$ z_D = (E_{C_1},E_{C_2},\ldots) \in \mbox{\rm Gr}(D) \stackrel{\rm def}{=} \mbox{\rm Gr}(\,|C_1|\,) \times \mbox{\rm Gr}(\,|C_2|\,) \times \ldots. $$ This is consistent with our previous notation for an arbitrary $G$: for $D = D_{ {\bf w} {\bf j} }$, we have $z_D = z_{ {\bf w} {\bf j} }$. We defined configuration varieties and Bott-Samelson varieties as orbit closures of such points (see also below, Sec \ref{Varieties and defining equations}). \\[1em] {\bf Examples.} For $n = 3$, $G = GL(3)$, $ {\bf i} = {\bf j} = 121$, we have $ {\bf w} = (s_1,s_1\! s_2,\, s_1\! s_2 s_1)$, and the reduced chamber family \begin{eqnarray*} D_{121} & = & \{\, s_1[1],\, s_1 s_2[2],\, s_1 s_2 s_1[1]\, \}\\ &=& \{\,\{2\},\{2,3\},\{3\}\,\} \\ &=& \{2,23,3\} \end{eqnarray*} The full chamber family is $D^+_{121} = \{1,12,123,2,23,3\}$. The chamber family of the other reduced word $ {\bf i} = 212$ is $D_{212} = \{13, 3, 23\}$. \\[.5em] For $n=4$, let $ {\bf w} = (e,s_1,s_1,s_3 s_2,s_1)$, $ {\bf j} = (2,1,3,1,1)$. Then we have the subset family $$ \begin{array}{ccl} D_{\ww\jj} &=& \{ \, e[2],\, s_1[1],\, s_1[3],\, s_3 s_2[1],\, s_1[1] \,\} \\ &=& \{12,2,123,3,2\} = \{12,123,2,3\} \end{array} $$ Note that we remove repetitions in $D$. The associated $T$-fixed configuration is $$ z_D = (E_{12},E_{123},E_{2},E_{3}) \in \mbox{\rm Gr}(D) = \mbox{\rm Gr}(2) \times \mbox{\rm Gr}(3) \times \mbox{\rm Gr}(1) \times \mbox{\rm Gr}(1). $$ $\bullet$ \subsection{Chamber families} \label{Chamber families} Chamber families have a rich structure. (See \cite{LZ}, \cite{MFour}, \cite{RSNew}.) Given a full chamber family $ D^+_{\ii} $, we may omit some of its elements to get a subfamily $D \subset D^+_{\ii} $. The resulting {\em chamber subfamilies} can be characterized as follows. For two sets $S, S' \subset [n]$, we say $S$ is {\em elementwise less than} $S'$, $S \stackrel{e\!l\!t}{<} S'$, if $s < s'$ for all $s \in S$, $s' \in S'$. Now, a pair of subsets $C, C' \subset [n]$ is {\em strongly separtated} if $$ (C \setminus C') \stackrel{e\!l\!t}{<} (C' \setminus C) \ \ \ \mbox{or} \ \ \ (C' \setminus C) \stackrel{e\!l\!t}{<} (C \setminus C') \ , $$ where $C\setminus C'$ denotes the complement of $C'$ in $C$. A family of subsets is called strongly separated if each pair of subsets in it is strongly separated. \begin{prop}{(LeClerc-Zelevinsky \cite{LZ})\ } \label{LeClerc-Zelevinsky} A family $D$ of subsets of $[n]$ is a chamber subfamily, $D \subset D^+_{\ii} $ for some $ {\bf i} $, if and only if $D$ is strongly separated. \end{prop} {\bf Remarks.} (a) Reiner and Shimozono \cite{RSNew} give an equivalent description of strongly separated families. Place the subsets of the family into lexicographic order. Then $D = (C_1 \stackrel{l\!e\!x}{\leq} C_2 \stackrel{l\!e\!x}{\leq} \cdots)$ is strongly separated if and only if it is ``\%-avoiding'': that is, if $i_1 \in C_{j_1}$, $i_2 \in C_{j_2}$ with $i_1 > i_2$, $j_1 < j_2$, then $i_1 \in C_{j_2}$ or $i_2 \in C_{j_1}$. \\ (b) If $ {\bf i} = (i_1,\ldots,i_l)$ is an initial subword of $ {\bf i} ' = (i_1, \ldots, i_l, \ldots, i_N)$, then $ D_{\ii} \subset D_{ {\bf i} '}$. Thus the chamber families associated to decompositions of the longest permutation $w_0$ are the maximal strongly separated families.\\ (c) In \cite{MFour}, we describe the ``orthodontia'' algorithm to determine a reduced decomposition $ {\bf i} $ associated to a given strongly separated family. See also \cite{RSNew}. \\[1em] {\bf Examples.} (a) For $n=3$, the chamber families $D^+_{121} = \{1,12,123,2,23,3\}$ and $D^+_{212} = \{1,12,123,13,3,23\}$ are the only maximal strongly separated families. The sets $13$ and $2$ are the only pair not strongly separated from each other. \\ (b) For $n=4$, the strongly separted family $D=\{24,34,4\}$ is contained in the chamber sets of the reduced words $ {\bf i} = 312132$ and $ {\bf i} =123212$. $\bullet$ \\[1em] Chamber families can be represented pictorially in several ways, one of the most natural being due to Berenstein, Fomin, and Zelevinsky \cite{BFZ}. The {\em wiring diagram} or {\em braid diagram} of the permutation $w$ with respect to the reduced word $ {\bf i} $ is best defined via an example. Let $G = GL(4)$, $w = w_0$ (the longest permutation), and $ {\bf i} = 312132$. On the left and right ends of the wiring diagram are the points 1,2,3,4 in two columns. Each point $i$ on the left is connected to the point $w(i)$ on the right by a curve which is horizontal and disjoint from the other curves except for certain crossings. The crossings, read left to right, correspond to the entries of $ {\bf i} $. The first entry $i_1 = 3$ corresponds to a crossing of the curve on level 3 with the one on level 4. (The other curves continue horizontally.) The second entry $i_2 = 1$ crosses the curves on level 1 and 2, and so on. \\[1em] \centerline{\Large \bf FIGURE 1 } \\[1em] If we add crossings only up to the $l^{\mbox{\tiny th}}$ step, we obtain the wiring diagram of the truncated word $s_{i_1} s_{i_2} \cdots s_{i_l}$. Now we may construct the chamber family $$ D^+_{\ii} = (1,12,123,1234,124,2,24,4,234,34) $$ as follows. Label each of the curves of the wiring diagram by its point of origin on the left. Into each of the connected regions between the curves, write the numbers of those curves which pass above the region. Then the sets of numbers inscribed in these chambers are the members of the family $ D^+_{\ii} $. If we list the chambers from left to right, we recover the natural order in which these subsets appear in $ D^+_{\ii} $. Another way to picture a chamber family, or any subset family, is as follows. We may consider a subset $C = \{j_1,j_2,\ldots\} \subset [n]$ as a column of $k$ squares in the rows $j_1, j_2,\ldots$. For each subset $C_k$ in the chamber family, form the column associated to it, and place these columns next to each other. The result is an array of squares in the plane called a {\em generalized Young diagram}. For our word $ {\bf i} = 312132$, we draw the (reduced) chamber family as: $$ D_{\ii} \ = \ \begin{array}{ccccccc} 1 & \Box & & & & & \\ 2 & \Box & \Box & \Box & & \Box & \\ 3 & & & & & \Box & \Box \\ 4 & \Box & & \Box & \Box & \Box & \Box \end{array} $$ where the numbers on the left of the diagram indicate the level. See \cite{RS1}, \cite{MNW}, \cite{MFour}. \subsection{Varieties and defining equations} \label{Varieties and defining equations} To any subset family $D$ we have associated a $T$-fixed point in a product of Grassmannians, $z_D \in \mbox{\rm Gr}(D)$, and we may define as before the {\em configuration variety} of $D$ to be the closure of the $G$-orbit of $z_D$: $$ \FF_D = \overline{G\cdot z_D} \subset \mbox{\rm Gr}(D); $$ and the {\em flagged configuration variety} to be the closure of its $B$-orbit: $$ \FF^B_D = \overline{B\cdot z_D} \subset \mbox{\rm Gr}(D). $$ Furthermore, if $D = D_{\ii} $, a chamber family, then the {\em Bott-Samelson variety} is the flagged configuration variety of $ D_{\ii} $: $$ Z_{\ii} = \Zii^{\tiny orb} = \FF^B _{ D_{\ii} }. $$ (We could also use the full chamber family $ D^+_{\ii} $, since the extra coordinates correspond to the standard flag fixed under the $B$-action.) Thus $ \FF_D $, $ \FF^B_D $, and $ Z_{\ii} $ can be considered as varieties of configurations of subspaces in ${\bf C}^n$, like the flag and Schubert varieties. We will give defining equations for the Bott-Samelson varieties analogous to those for Schubert varieties. For a family $D$, define the {\em flagged inclusion variety} $$ {\cal I} ^B_D = \left\{ \begin{array}{c|c} (V_C)_{C\in D}\in \mbox{\rm Gr}(D) & \begin{array}{c} \forall\, C, C' \in D,\ \, C \subset C' \Rightarrow V_C \subset V_{C'} \\ \mbox{and }\ \forall\, [i] \in D, \ \ V_{[i]} = {\bf C}^i \end{array} \end{array} \right\}. $$ $B$ acts diagonally on $ {\cal I} ^B_D$. \\[1em] {\bf Example.} For $n=4$, $ {\bf i} =312132$, we may use the picture in the above example to write the inclusion variety $ {\cal I} ^B_{ D_{\ii} ^+}$ as the set of all 10-tuples of subspaces of ${\bf C}^4$ $$ (V_1,V_{12},V_{123},V_{1234},V_{124},V_2,V_{24},V_4, V_{234},V_{34}) $$ with $\dim( V_C\!) = |C|$ and satisfying the following inclusions: $$ \begin{array}{ccccccc} &&0&&&&\\ &\!\!\!\!\!\!\swarrow\!\!\!\!\!\!&\!\downarrow\!&\!\!\!\!\!\!\searrow\!\!\!\!\!\!&&&\\ {\bf C}^1=V_1&&V_2&&V_4&&\\ \!\downarrow\!&\!\!\!\!\!\!\swarrow\!\!\!\!\!\!&&\!\!\!\!\!\!\searrow\!\!\!\!\!\!&\downarrow&\!\!\!\!\!\!\searrow\!\!\!\!\!\!&\\ {\bf C}^2=V_{12}&&&&V_{24}&&V_{34}\\ \!\downarrow\!&\!\!\!\!\!\!\searrow\!\!\!\!\!\!&&\!\!\!\!\!\!\swarrow\!\!\!\!\!\!&\downarrow&\!\!\!\!\!\!\swarrow\!\!\!\!\!\!\\ {\bf C}^3=V_{123}&&V_{124}&&V_{234}&&\\ &\!\!\!\!\!\!\searrow\!\!\!\!\!\!&&\!\!\!\!\!\!\swarrow\!\!\!\!\!\!&&&\\ &&V_{1234}&&&& \\ && = {\bf C}^4 &&&& \end{array} $$ where the arrows indicate inclusion of subspaces. \begin{thm} For every reduced word $ {\bf i} $, we have $ Z_{\ii} \cong {\cal I} ^B_{ D_{\ii} ^+}$. \end{thm} {\bf Proof.} Note that the generating point $z_{ D_{\ii} ^+}$ lies in $ {\cal I} ^B_{ D_{\ii} ^+}$, and $ {\cal I} ^B_{ D_{\ii} ^+}$ is B-equivariant, so $ Z_{\ii} \subset {\cal I} ^B_{ D_{\ii} ^+}$. To show the reverse inclusion, we use our previous characterization $$ Z_{\ii} \cong \Zii^{\tiny fib} = e \mathop{\times} _{G/P_{i_1}} G/B \mathop{\times} _{G/P_{i_2}} G/B \mathop{\times} _{G/P_{i_3}} \cdots \mathop{\times} _{G/P_{i_l}} G/B. $$ We may write this variety as the $(l+1)$-tuples of flags $ (V_1^{(k)} \subset V_2^{(k)} \subset \cdots \subset {\bf C}^n) $, \, $k = 0,1,\ldots,l$, such that: $V_i^{(k)} = V_i^{(k+1)}$ for all $k$ and all $i \neq i_k$; and $V_i^{(0)} = {\bf C}^i$ for all $i$. Consider the map $$ \begin{array}{cccc} \theta: & \Zii^{\tiny fib} & \rightarrow & \mbox{\rm Gr}(D) \\[.1em] & (V_1^{(k)} \subset V_2^{(k)} \subset \cdots)_{k=0}^l & \mapsto & (V_{i_1}^{(1)}, V_{i_2}^{(2)}, \ldots) \end{array} $$ We have seen in Theorem \ref{isomorphism theorem} that $ Z_{\ii} = \Zii^{\tiny orb} = \mathop{\rm Im}(\theta)$, since $\theta = \psi \circ \phi^{-1}$. It remains to show that $ {\cal I} ^B_{ D_{\ii} ^+} \subset \mathop{\rm Im}(\theta)$. For each $k$, define $k^- = \max\{ m \mid m < k,\, i_m = i_k +1 \}$ and $k^+ = \min\{ m \mid m > k,\, i_m = i_k +1 \}$. Then it is easily seen that a configuration $(V_1,V_2,\ldots) \in \mbox{\rm Gr}(D)$ lies in $\mathop{\rm Im}(\\theta)$ exactly when: \\ (i) for each $k$, we have $V_k \subset V_{k^-}$ and $V_k \subset V_{k^+}$ provided $k^-$ or $k^+$ is defined; \\ (ii) for each $k$, if $k^-$ is not defined, then $V_k \subset {\bf C}^{i_k+1}$; and \\ (iii) for each $i$, if $k = \min\{m \mid i_m = i+1\}$, then ${\bf C}^i \subset V_k$. Note that for any $k$, the $k^{\mbox{\tiny th}}$ subset of $ D_{\ii} $ is $$ \begin{array}{rcl} C_k & = & s_{i_1} \cdots s_{i_k} [i_k] \\ & = & s_{i_1} \cdots s_{i_k} \cdots s_{i_{k^+}} [i_k] \\ & \subset & s_{i_1} \cdots s_{i_k} \cdots s_{i_{k^+}} [i_k+1] \\ & = & s_{i_1} \cdots s_{i_k} \cdots s_{i_{k^+}} [i_{k^+}] \\ &=& C_{k^+} \end{array} $$ We can write similar inclusions of subsets for the other conditions (i)-(iii). This shows that the inclusions defining $ {\cal I} ^B_{ D_{\ii} ^+}$ do indeed imply those defining $\mathop{\rm Im}(\theta)$, Q.E.D. $\bullet$ \begin{conj} For any subset family $D$, a configuration $(V_C)_{C\in D} \in \mbox{\rm Gr}(D)$ lies in $ { \cal F } _D$ exactly if, for every subfamily $D' \subset D$, $$ \dim(\bigcap_{C \in D'} V_C) \geq \, |\! \cap_{C \in D'} C | $$ $$ \dim(\sum_{C \in D'} V_C) \leq \, |\! \cup_{C \in D'} C | $$ \end{conj} Note that a configuration $(V_1,\ldots,V_l) \in \mbox{\rm Gr}(D)$ lies in the flagged configuration variety $ \FF^B _D$ if and only if $({\bf C}^1,\ldots, {\bf C}^n, V_1,\ldots, V_l)$ lies in the unflagged variety $ { \cal F } _{D^+}$ of the augmented diagram $D^+ \stackrel{\rm def}{=} \{[1], [2], \ldots [n]\} \cup D$. Hence the above conjecture gives conditions defining flagged configuration varieties as well as unflagged. \\[1em] {\bf Examples.} (a) If $D= D_{\ii} $ is a chamber family, the conjecture reduces to the previous Theorem. \\ (b) The conjecture is known if $D$ satisfies the ``northwest condition'' (see \cite{MNW}): that is, the elements of $D$ can be arranged in an order $C_1, C_2, \ldots$ such that if $i_1 \in C_{j_1}$, $i_2 \in C_{j_2}$, then $\min(i_1,i_2) \in C_{\min(j_1,j_2)}$. In fact, it suffices in this case to consider only the intersection conditions of the conjecture. $\bullet$ \\[1em] It would be interesting to know whether the determinantal equations implied by the conditions of the above Theorem and Conjecture define $ \FF_D \subset \mbox{\rm Gr}(D)$ scheme-theoretically. Now, let $D$ be a strongly separated family. We know by Proposition \ref{LeClerc-Zelevinsky} that $D$ is part of some chamber family $ D_{\ii} $, and by Theorem \ref{desingularization} we may take $ {\bf i} $ so that the projection map $ Z_{\ii} = \FF^B _{ D_{\ii} } \rightarrow \FF^B _D$ is birational. \\[1em] {\bf Example.} Let $n=7$, and consider the family $D$ consisting of the single subset $C = 12457$. Its configuration variety is the Grassmannian $ \FF_D = \mbox{\rm Gr}(5, {\bf C}^7)$, and its flagged configuration variety is the Schubert variety $$ \FF^B _D = X_{211} = \{V \in \mbox{\rm Gr}(5) \mid {\bf C}^2 \subset V,\, \dim({\bf C}^5 \cap V) \geq 4 \}. $$ By the orthodontia algorithm \cite{MFour}, we find that this is desingularized by the reduced word $ {\bf i} = 3465$, for which $ D_{\ii} = \{124,1245,123457, 12457\}$ and $$ Z_{\ii} = \left\{ \begin{array}{c} (V_{124},V_{1245},V_{123457},V_{12457}) \in \mbox{\rm Gr}(3) \times \mbox{\rm Gr}(4) \times \mbox{\rm Gr}(6) \times \mbox{\rm Gr}(5) \\[.1cm] \mbox{ such that }\ \ \ {\bf C}^2 \subset V_{124} \subset {\bf C}^4 \subset V_{123457}\ \ , \ \ V_{1245} \subset {\bf C}^5\ , \\ \mbox{} \ \ \ \ \ \ \ V_{124} \subset V_{1245} \subset V_{12457} \subset V_{123457} \end{array} \right\}. $$ The desingularization map is the projection $$ \pi:(V_{124},V_{1245},V_{123457},V_{12457}) \mapsto V_{12457} . $$ In \cite{MNW} and Zelevinsky's work \cite{Zel}, there are given several other desingularizations of Schubert varieties, all of them expressible as configuration varieties. $\bullet$ \section{Schur and Weyl modules} We relate generalized Schur and Weyl modules for $GL(n)$, which are defined in completely elementary terms, to the sections of line bundles on configuration varieties, and hence to the coordinate rings of these varieties. One the one hand, this yields an unexpected Demazure character formula for the Schur modules, including the skew Schur functions and Schubert polynomials. On the other hand, it gives an elementary construction for line-bundle sections on Bott-Samelson varieties. \subsection{Definitions} \label{Schur and Weyl Modules: Definitions} We have associated to any subset family $D = \{C_1,\ldots,C_k\}$ a configuration variety $ { \cal F } _D$ with $G$-action, and a flagged configuration variety $ \FF^B _D$ with $B$-action. Now, assign an integer multiplicity $ {\bf m} (C) \geq 0$ to each subset $C \in D$. For each pair $(D, {\bf m} )$, we define a $G$-module and a $B$-module, which will turn out to sections of a line bundle on $ { \cal F } _D$ and $ \FF^B _D$. In the spirit of DeRuyts \cite{F} and Desarmenien-Kung-Rota \cite{DKR}, we construct these ``Weyl modules'' $M_{D, {\bf m} }$ inside the coordinate ring of $n\times n$ matrices, and their flagged versions $ M^B _{D, {\bf m} }$ inside the coordinate ring of upper-triangular matrices. (I am grateful to Mark Shimozono for pointing out this form of the definition.) Let ${\bf C}[x_{ij}]$ (resp. ${\bf C}[x_{ij}]_{i\leq j} $ ) denote the polynomial functions in the variables $x_{ij}$ with $i,j \in [n]$\ (resp. $x_{ij}$ with $1 \leq i\leq j\leq n$). For $R, C \subset [n]$ with $|R|=|C|$, let $$ {\Delta}_C^R = \det(x_{ij})_{(i\in R, j\in C)} \in {\bf C}[x_{ij}] $$ be the minor determinant of the matrix $x = (x_{ij})$ on the rows $R$ and the columns $C$. Further, let $$ \widetilde{\Del}_C^R = {\Delta}_C^R |_{x_{ij} = 0 , \ \forall\, i>j} \in {\bf C}[x_{ij}]_{i\leq j} $$ be the same minor evaluated on an upper triangular matrix of variables. Now, for a subset family $D=\{C_1,\dots,C_l\}$, $ {\bf m} =(m_1,\ldots,m_l)$, define the {\em Weyl module} $$ M_{D, {\bf m} } = {\mbox{\rm Span}} _{{\bf C}}\left\{\ {\Delta}_{C_1}^{R_{11}}\cdots {\Delta}_{C_1}^{R_{1m_1}} {\Delta}_{C_2}^{R_{21}} \ldots {\Delta}_{C_l}^{R_{lm_l}} \left| \begin{array}{c} \forall\, k,\! m \ \ R_{km} \subset [n] \\[.1cm] \mbox{ and }\ |R_{km}|=|C_k| \end{array} \right. \right\}. $$ That is, a spanning vector is a product of minors with column indices equal to the elements of $D$ and row indices taken arbitrarily. For two sets $R = \{i_1,\ldots,i_c\}$, $C = \{j_1,\ldots,j_c\}$ we say $R \stackrel{c\!o\!m\!p}{\leq} C$ (componentwise inequality) if $i_1 \leq j_1$, $i_2\leq j_2$, \ldots. Define the {\em flagged Weyl module} $$ M^B _{D, {\bf m} } = {\mbox{\rm Span}} _{{\bf C}}\left\{\ \widetilde{\Del}_{C_1}^{R_{11}}\cdots \widetilde{\Del}_{C_1}^{R_{1m_1}} \widetilde{\Del}_{C_2}^{R_{21}} \ldots \widetilde{\Del}_{C_l}^{R_{lm_l}} \left| \begin{array}{c} \forall\, k,m \ \ R_{km} \subset [n] \\[.1cm] |R_{km}|=|C_k| ,\, R_{km} \stackrel{c\!o\!m\!p}{\leq} C_k \end{array} \right. \right\}. $$ For $f(x) \in {\bf C}[x_{ij}]$, a matrix $g \in G$ acts by left translation, $(g \cdot f)(x) = f(g^{-1}x)$. It is easily seen that this restricts to a $G$-action on $M_{D, {\bf m} }$ and similarly we get a $B$-action on $ M^B _{D, {\bf m} }$. We clearly have the diagram of $B$-modules: $$ \begin{array}{ccc} M_{D, {\bf m} } & \subset & {\bf C}[x_{ij}] \\ \downarrow & & \downarrow \\ M^B _{D, {\bf m} } & \subset & {\bf C}[x_{ij}]_{i\leq j} \end{array} $$ where the vertical maps ($x_{ij} \mapsto 0$ for $i>j$) are surjective. That is, $ M^B _{D, {\bf m} }$ is a quotient of $M_{D, {\bf m} }$. The {\em Schur modules} are defined to be the duals $$ S_{D, {\bf m} } \stackrel{\rm def}{=} (M_{D, {\bf m} })^* \ \ \ \ \ \ S^B_{D, {\bf m} } \stackrel{\rm def}{=} ( M^B _{D, {\bf m} })^*. $$ We will deal mostly with the Weyl modules, but everything we say will of course also apply to their duals. \\[1em] {\bf Example.} We adopt the ``Young diagram'' method for picturing subset families. (See Sec \ref{Chamber families}.) Let $n=4$, $D = \{234,34,4\}$, $m = (2,0,3)$. (That is, $m(234)=2$, $m(34)=0$, $m(4)=3$.) We picture this by writing each column repeatedly, according to its multiplicity. Zero multiplicity means we omit the column. Thus $$ (D, {\bf m} ) = \begin{array}{cccccc} 1 & & & & & \\ 2 & \Box & \Box & & & \\ 3 & \Box & \Box & & & \\ 4 & \Box & \Box & \Box & \Box & \Box \end{array} \ \ \ \ \tau = \begin{array}{c|ccccc} 1 & & & & & \\ 2 & 1 & 1 & & & \\ 3 & 3 & 2 & & & \\ 4 & 4 & 3 & 2 & 4 & 3 \end{array} $$ The spanning vectors for $M_{D, {\bf m} }$ correspond to all column-strict fillings of this diagram by indices in $[n]$. For example, the filling $\tau$ above corresponds to $$ {\Delta}_{234}^{134}\ {\Delta}_{234}^{123}\ {\Delta}_{4}^{2}\ {\Delta}_{4}^{4}\ {\Delta}_{4}^{3}\ $$ $$ = \left| \begin{array}{ccc} x_{12} & x_{13} & x_{14} \\ x_{32} & x_{33} & x_{34} \\ x_{42} & x_{43} & x_{44} \end{array} \right| \cdot \left| \begin{array}{ccc} x_{12} & x_{13} & x_{14} \\ x_{22} & x_{23} & x_{24} \\ x_{32} & x_{33} & x_{34} \end{array} \right| \cdot x_{24}\cdot x_{44}\cdot x_{34} $$ $$ = \left( \begin{array}{ccccc|ccccc} 1 & 1 & & & & 2 & 2 & & & \\ 3 & 2 & & & & 3 & 3 & & & \\ 4 & 3 & 2 & 4 & 3 & 4 & 4 & 4 & 4 & 4 \end{array} \right) $$ The last expression is in the letter-place notation of Rota et al \cite{DKR}. A basis may be extracted from this spanning set by considering only the row-decreasing fillings (a normalization of the semi-standard tableaux), and in fact the Weyl module is the dual of the classical Schur module $S_{ \lambda }$ associated to the shape $D$ considered as the Young diagram $ \lambda = (5,2,2,0)$. The spanning elements of the flagged Weyl module $ M^B _{D, {\bf m} }$ correspond to the ``flagged'' fillings of the diagram: those for which the number $i$ does not appear above the $i^{\mbox{\tiny th}}$ level. For the diagram above, all the column-strict fillings are flagged, and $M_{D, {\bf m} } \cong M^B _{D, {\bf m} }$. However, for $$ (D', {\bf m} ) = \begin{array}{cccccc} 1 & & & & & \\ 2 & \Box & \Box & & & \\ 3 & \Box & \Box & \Box & \Box & \Box \\ 4 & \Box & \Box & & & \end{array} $$ $$ \tau_1 = \begin{array}{c|ccccc} 1 & & & & & \\ 2 & 2 & 1 & & & \\ 3 & 3 & 2 & 4 & 3 & 4 \\ 4 & 4 & 3 & & & \end{array} \ \ \ \ \tau_2 = \begin{array}{c|ccccc} 1 & & & & & \\ 2 & 2 & 1 & & & \\ 3 & 3 & 2 & 3 & 2 & 3 \\ 4 & 4 & 4 & & & \end{array} $$ the filling $\tau_1$ is {\em not} flagged, since 4 appears on the 3rd level, but $\tau_2$ {\em is} flagged, and corresponds to the spanning element $$ \widetilde{\Del}_{234}^{234}\ \widetilde{\Del}_{234}^{124}\ \widetilde{\Del}_{3}^{3}\ \widetilde{\Del}_{3}^{2}\ \widetilde{\Del}_{3}^{3} = \left| \begin{array}{ccc} x_{22} & x_{23} & x_{24} \\ 0 & x_{33} & x_{34} \\ 0 & 0 & x_{44} \end{array} \right| \cdot \left| \begin{array}{ccc} x_{12} & x_{13} & x_{14} \\ x_{22} & x_{23} & x_{24} \\ 0 & 0 & x_{44} \end{array} \right| \cdot x_{33}\cdot x_{23}\cdot x_{33}. $$ We have $M_{D, {\bf m} } \cong M_{D', {\bf m} } \cong M^B _{D, {\bf m} } \cong S^*_{(5,2,2,0)}$, the dual of a classical (irreducible) Schur module for $GL(4)$, and $ M^B _{D', {\bf m} } \cong S^*_{(2,5,2,0)}$, the dual of the Demazure module with lowest weight $(0,2,5,2)$ and highest weight $(5,2,2,0)$. Cf. \cite{RS1}, \cite{MFour}. $\bullet$ \\[1em] {\bf Remarks.} (a) In \cite{LM} we make a general definition of ``standard tableaux'' giving bases of the Weyl modules for strongly separated families. \\ (b) We briefly indicate the equivalence between our definition of the Weyl modules and the tensor product definition given in \cite{ABW}, \cite{RS1}, \cite{MNW}. Let $Y = Y_{D, {\bf m} } \subset {\bf N} \times {\bf N}$ be the generalized Young diagram of squares in the plane associated to $(D, {\bf m} )$ as in the above examples, and let $U = ({\bf C}^n)^*$. One defines $ M^{\mbox{\tiny tensor}} _Y = U^{\otimes Y} \gamma_Y$, where $\gamma_Y$ is a generalized Young symmetrizer. The spanning vectors ${\Delta}_{\tau}$ of $M_{D, {\bf m} }$ correspond to the fillings $\tau:Y \rightarrow [n]$. Then the map $$ \begin{array}{ccc} M_{D, {\bf m} } & \rightarrow & M^{\mbox{\tiny tensor}} _{D, {\bf m} } \\ {\Delta}_{\tau} & \mapsto & \left( \bigotimes_{(i,j)\in Y} e^*_{\tau(i,j)} \right) \gamma_Y \end{array} $$ is a well-defined isomorphism of $G$-modules, and similarly for the flagged versions. This is easily seen from the definitions, and also follows from the Borel-Weil theorems proved below and in \cite{MNW}. \subsection{Borel-Weil theory} \label{Borel-Weil theory} A configuration variety $ { \cal F } _D \subset \mbox{\rm Gr}(D)$ has a natural family of line bundles defined by restricting the determinant or Plucker bundles on the factors of $\mbox{\rm Gr}(D)$. For $D = (C_1,C_2,\ldots)$, and multiplicities $ {\bf m} = (m_1, m_2, \ldots)$, we define $$ \begin{array}{ccc} \LL_{\mm} & \subset & {\cal O} (m_1,m_2,\ldots) \\ \downarrow & & \downarrow \\ { \cal F } _D & \subset & \mbox{\rm Gr}(D) = \mbox{\rm Gr}(|C_1|) \times \mbox{\rm Gr}(|C_2|) \times \cdots \end{array} $$ We denote by the same symbol $ \LL_{\mm} $ this line bundle restricted to $ \FF^B _D$. Note that in the case of a Bott-Samelson variety $ { \cal F } _D = Z_{\ii} $, this is the well-known line bundle $$ \LL_{\mm} \cong {P_{i_1} \times \cdots \times P_{i_l} \times {\bf C} \over B^l} $$ $$ (p_1,\ldots,p_l,v)\cdot (b_1,\ldots,b_l) \stackrel{\rm def}{=} (p_1 b_1,\ldots,b_{l-1}^{-1}p_l b_l, \, \varpi _{i_1}(b_1^{-1})^{m_1} \cdots \varpi _{i_l}(b_l^{-1})^{m_l}\,v), $$ $ \varpi _i$ denoting the fundamental weight $ \varpi _i(\mathop{\rm diag}(x_1,\ldots,x_n)) = x_1 x_2 \cdots x_i$. Note that if $m_k \geq 0$ for all $k$ (resp. $m_k >0$ for all $k$) then $ \LL_{\mm} $ is effective (resp. very ample). However, $ \LL_{\mm} $ may be effective even if some $m_k < 0$. See \cite{LM}. \begin{prop} Let $(D, {\bf m} )$ be a strongly separated subset family with multiplicity. Then we have \\ (i) $M_{D, {\bf m} } \cong H^0( { \cal F } _D, \LL_{\mm} )$ \\[.1em] and $H^i( { \cal F } _D, \LL_{\mm} ) = 0$ for $i>0$. \\[.2em] (ii) $ M^B _{D, {\bf m} } \cong H^0( \FF^B _D, \LL_{\mm} )$ \\[.1em] and $H^i( \FF^B _D, \LL_{\mm} ) = 0$ for $i>0$. \\[.2em] (iii) $ { \cal F } _D$ and $ \FF^B _D$ are normal varieties, projectively normal with respect to $ \LL_{\mm} $, and have rational singularities. \end{prop} {\bf Proof.} First, recall that we can identify the sections of a bundle over a single Grassmannian, $ {\cal O} (1) \rightarrow \mbox{\rm Gr}(i)$, with linear combinations of minors in the homogeneous Stiefel coordinates $$ x = \left(\begin{array}{ccc} x_{11} & \cdots & x_{1i} \\ \vdots & \ddots & \vdots \\ x_{n1} & \cdots & x_{ni} \\ \end{array} \right) \in \mbox{\rm Gr}(i), $$ namely the $i \times i$ minors ${\Delta}^R(x)$ on the rows $R \subset [n]$, $|R| = i$. Thus, a typical spanning element of $H^0(\mbox{\rm Gr}(D), {\cal O} ( {\bf m} ))$ is the section $$ {\Delta}^{R_{11}}(x^{(1)}) \cdots {\Delta}^{R_{11}}(x^{(1)})\ {\Delta}^{R_{21}}(x^{(2)}) \cdots {\Delta}^{R_{lm_l}}(x^{(l)}), $$ where $x^{(k)}$ represents the homogeneous coordinates on each factor $\mbox{\rm Gr}(|C_k|)$ of $\mbox{\rm Gr}(D)$, and $R_{km}$ are arbitrary subsets with $|R_{km}|=i_k$. Now, restrict this section to $ { \cal F } _D \subset \mbox{\rm Gr}(D)$ and then further to the dense $G$-orbit $G\cdot z_D \subset { \cal F } _D$. Parametrizing the orbit by $g \rightarrow g\cdot z_D$, we pull back the resulting sections of $H^0( { \cal F } _D, \LL_{\mm} )$ to certain functions on $G \subset \mbox{Mat}_{n\times n}({\bf C})$, which are precisely the products of minors defining the spanning set of $M_{D, {\bf m} }$. This shows that $$ M_{D, {\bf m} } \cong \mathop{\rm Im}\left[ H^0(\mbox{\rm Gr}(D), {\cal O} ( {\bf m} )) \ \stackrel{\mbox{\footnotesize rest}}{\rightarrow} H^0( { \cal F } _D, \LL_{\mm} ) \right]. $$ Similarly for $B$-orbits, we have $$ M^B _{D, {\bf m} } \cong \mathop{\rm Im}\left[ H^0(\mbox{\rm Gr}(D), {\cal O} ( {\bf m} )) \ \stackrel{\mbox{\footnotesize rest}}{\rightarrow} H^0( \FF^B _D, \LL_{\mm} ) \right]. $$ Now we invoke the key vanishing result, \cite{MNW} Prop. 28 (due to W. van der Kallen and S.P. Inamdar, based on the work of O. Mathieu \cite{Mat}, P. Polo, et.al.) The conditions $(\alpha)$ and $(\beta)$ of that Proposition apply to $ { \cal F } _D$ because $D$ is contained in a chamber family $D^+_{ {\bf i} }$ (Prop. \ref{LeClerc-Zelevinsky} above). Furthermore, the proof of \cite{MNW}, Prop. 28 goes through identically with $ \FF^B _D$ in place of $ { \cal F } _D$, merely replacing $ { \cal F } _{w_0;u_1,\ldots,u_r}$ by $ { \cal F } _{e;u_1,\ldots,u_r}$. All of the assertions of our Proposition now follow immediately from the corresponding parts of \cite{MNW}, Prop. 28. $\bullet$. \begin{prop} \label{extend by zero} Suppose $(D, {\bf m} )$, $( \widetilde{D} , \widetilde{\mm} )$ are strongly separated subset families with $D \subset \widetilde{D} $, $ \widetilde{\mm} (C) = {\bf m} (C)$ for $C\in D$, $ {\bf m} (C) = 0$ otherwise. Then the natural projection $\pi: \mbox{\rm Gr}( \widetilde{D} ) \rightarrow \mbox{\rm Gr}(D)$ restricts to a surjection $\pi: { \cal F } _{ \widetilde{D} } \rightarrow { \cal F } _D$, and induces an isomorphism $$ \pi^*: H^0( { \cal F } _D, \LL_{\mm} ) \stackrel{\sim}{\rightarrow} H^0( { \cal F } _{ \widetilde{D} }, {\cal L} _{ \widetilde{\mm} }), $$ and similarly for the flagged case. \end{prop} {\bf Proof.} For the unflagged case, this follows immediately from \cite{MNW}, Prop. 28. Again, the argument given there goes through for the flagged case as well. $\bullet$ \\[1em] {\bf Remarks.} (a) Note that the proposition holds even if $\dim { \cal F } _{ \widetilde{D} } > \dim { \cal F } _D$. \\ (b) The Proposition allows us to reduce Weyl modules for strongly separated families to those for {\em maximal} strongly separated families, that is chamber families. $\bullet$ \\[1em] We may conjecture that the results of this section hold not only in the strongly separated case, but for all subset families and configuration varieties. \subsection{Demazure's character formula} \label{Demazure's character formula} We now examine how the iterative structure of Bott-Samelson varieties influences the associated Weyl modules. Define Demazure's isobaric divided difference operator $ \Lambda _i : {\bf C}[x_1,\ldots,x_n] \rightarrow {\bf C}[x_1,\ldots,x_n] $, $$ \Lambda _i f = {x_i f - x_{i+1} s_i f \over x_i - x_{i+1}}. $$ For example for $f(x_1,x_2,x_3) = x_1^2 x_2^2 x_3$, $$ \begin{array}{rcl} \Lambda _2 f(x_1,x_2,x_3) &=& { x_2(x_1^2 x_2^2 x_3) - x_3(x_1^2 x_3^2 x_2) \over x_2 - x_3 } \\ &=& x_1^2 x_2 x_3 (x_2+x_3). \end{array} $$ For any permutation with a reduced decompostion $w = s_{i_1}\ldots s_{i_l}$, define $$ \Lambda _{w} \stackrel{\rm def}{=} \Lambda _{i_1} \cdots \Lambda _{i_l}, $$ which is known to be independent of the reduced decomposition chosen. By the (dual) character of a $G$- or $B$-module $M$, we mean $$ {\mbox{\rm \, char}^*\, } M = \mathop{\rm tr}(\mathop{\rm diag}(x_1,\ldots,x_n)|M^*)\, \in\, {\bf C}[x_1^{\pm 1},\ldots,x_n^{\pm 1}]. $$ (We must take duals to get polynomial functions as characters.) Let $ \varpi _i$ denote the $i$th fundamental weight, the multiplicative character of $B$ defined by $ \varpi _i(\mathop{\rm diag}(x_1,\ldots,x_n)) = x_1 x_2 \cdots x_i$. \begin{prop} \label{character formula} Suppose $(D, {\bf m} )$ is strongly separated, and $$ D \subset D_{\ii} ^+ = \{ [1],\ldots,[n],C_1,\ldots,C_l \}, $$ for some reduced word $ {\bf i} = (i_1,\ldots,i_l)$. Define $ \widetilde{\mm} = (k_1,\ldots,k_n, m_1,\ldots, m_l)$ by $ \widetilde{\mm} (C) = {\bf m} (C)$ for $C \in D$, $ \widetilde{\mm} (C) = 0$ otherwise. Then $$ {\mbox{\rm \, char}^*\, } M^B _{D, {\bf m} } = \varpi _1^{k_1} \cdots \varpi _n^{k_n} \Lambda _{i_1} \varpi _{i_1}^{m_1} \cdots \Lambda _{i_l} \varpi _{i_l}^{m_l}. $$ Furthermore, $$ {\mbox{\rm \, char}^*\, } M_{D, {\bf m} } = \Lambda _{w_0} {\mbox{\rm \, char}^*\, } M^B _{D, {\bf m} } , $$ where $w_0$ denotes the longest permutation. \end{prop} {\bf Remark.} We explain in \cite{LS} how one can recursively generate the standard tableaux for $ M^B _D$ (in \cite{LM}) by ``quantizing'' this character formula. See also \cite{MFour}. \\[1em] We devote the rest of this section to proving the Proposition. For a subset $C = \{j_1,j_2,\ldots\} \subset [n]$, and a permutation $w$, let $wC = \{w(j_1), w(j_2),\ldots\}$, and for a subset family $D = \{C_1,C_2,\ldots\}$, let $wD = \{wC_1, wC_2,\ldots\}$. Now, for $i \in [n-1]$, let $$ \Lambda _i D \stackrel{\rm def}{=} \{s_i[i]\} \cup s_i D , $$ where $s_i [i] = \{1,2,\ldots,i-1,i+1\}$. We say that $D$ is {\em $i$-free} for $i\in [n]$ if for every $C\in D$, we have $C \cap \{i,i+1\} \neq \{i+1\}$. \begin{lem} \label{i-free} Suppose $(D, {\bf m} )$ is strongly separated and $i$-free.\\ (i) $ \FF^B _{ \Lambda _i D} \cong P_i \mathop{\times} ^B \FF^B _D$ .\\ (ii) $ \FF^B _{s_i D} \cong P_i \cdot \FF^B _D \subset \mbox{\rm Gr}(D)$ .\\ (iii) The projection $ \FF^B _{ \Lambda _i D} \rightarrow \FF^B _{s_i D}$ is regular, surjective, and birational. \\ (iv) Let $ \widetilde{\mm} $ be the multiplicity on $ \Lambda _i D$ defined by $ \widetilde{\mm} (s_i C) = {\bf m} (C)$ for $C\in D$, $ \widetilde{\mm} (s_i[i]) = m_0$. The bundle $ {\cal L} _{ \widetilde{\mm} } \rightarrow \FF^B _{ \Lambda _i D}$ is isomorphic to $$ {\cal L} _{ \widetilde{\mm} } \cong P_i \mathop{\times} ^B \left( ( \varpi _i^{m_0})^* \otimes \LL_{\mm} \right), $$ where $( \varpi _i^{m_0})^* \otimes \LL_{\mm} $ indicates the bundle $ \LL_{\mm} \rightarrow \FF^B _D$ with its $B$-action twisted by the multiplicative character $( \varpi _i^{m_0})^* = \varpi _i^{-{m_0}}$. \end{lem} {\bf Proof.} (i) Since $D$ is $i$-free, we have $U_i z_D = z_D$, where $U_i$ is the one-dimensional unipotent subgroup corresponding to the simple root ${\alpha}_i$. We may factor $B$ into a direct product of subgroups, $B = U_i B' = B' U_i$. Then $$ \FF^B _D = \overline{B \cdot z_D} = \overline{B'\cdot z_D}. $$ Hence the $T$-fixed point $(s_i,z_D) \in P_i \mathop{\times} ^B \FF^B _D$ has a dense $B$-orbit: $$ \begin{array}{rcl} \overline{B \cdot (s_i,z_D)} &=& \overline{(U_i B' s_i, z_D)} \\[.1em] &=& (\overline{U_i s_i}, \overline{B' \cdot z_D}) \\[.1em] &=& P_i \mathop{\times} ^B \FF^B _D. \end{array} $$ Clearly, the injective map $$ \begin{array}{cccc} \psi: & P_i \times^B \mbox{\rm Gr}(D) & \rightarrow & \mbox{\rm Gr}(i) \times \mbox{\rm Gr}(D) \\ & (p,V) & \mapsto & (p {\bf C}^i, pV) \end{array} $$ takes $\psi(s_i,z_D) = z_{ \Lambda _i D}$, the $B$-generating point of $ \FF^B _{ \Lambda _i D}$. Thus $\psi:P_i \times^B \FF^B _D \\ \rightarrow \FF^B _{ \Lambda _i D}$ is an isomorphism. \\ (ii+iii) By the above, the projection is a bijection on the open B-orbit, and hence is birational. The image of the projection is $P_i \cdot \FF^B _D$, which must be closed since $P_i \mathop{\times} ^B \FF^B _D$ is a proper (i.e. compact variety). \\ (iv) Clear from the definitions. $\bullet$ \begin{lem} Let $(D, {\bf m} )$ be a strongly separated family and $i \in [n-1]$. Let $$ \begin{array}{rcl} { \cal F } ' & = & P_i \mathop{\times} ^B \FF^B _D \\ {\cal L} ' & = & P_i \mathop{\times} ^B \LL_{\mm} . \end{array} $$ so that $ {\cal L} ' \rightarrow { \cal F } '$ is a line bundle. Then $$ {\mbox{\rm \, char}^*\, } H^0( { \cal F } ', {\cal L} ') = \Lambda _i {\mbox{\rm \, char}^*\, } H^0( \FF^B _D, \LL_{\mm} ). $$ \end{lem} {\bf Proof.} By Demazure's analysis of induction to $P_i$ (see \cite{Dem1}, ``construction \'{e}l\'{e}mentaire'') we have $$ \Lambda _i {\mbox{\rm \, char}^*\, } H^0( \FF^B _D, \LL_{\mm} ) = {\mbox{\rm \, char}^*\, } H^0( { \cal F } ', {\cal L} ') - {\mbox{\rm \, char}^*\, } H^1(\, P_i/B, H^1( \FF^B _D, \LL_{\mm} )\,). $$ However, we know by \cite{MNW}, Prop.28 that $H^0( \FF^B _D, \LL_{\mm} )$ has a good filtration, so that the $H^1$ term above is zero. $\bullet$ \begin{cor} If $(D, {\bf m} )$ is strongly separated and $i$-free, and $( \Lambda _i D, \widetilde{\mm} )$ is a diagram with multiplicities $ \widetilde{\mm} (s_i C) = m(C)$ for $C\in D$, $ \widetilde{\mm} (s_i[i]) = m_0$, then $$ {\mbox{\rm \, char}^*\, } M^B _{ \Lambda _i D, \widetilde{\mm} } = \Lambda _i \varpi _i^{m_0} {\mbox{\rm \, char}^*\, } M^B_{D, {\bf m} }. $$ If $m_0 = 0$, then $$ {\mbox{\rm \, char}^*\, } M^B _{s_i D, {\bf m} } = {\mbox{\rm \, char}^*\, } M^B _{ \Lambda _i D, \widetilde{\mm} } = \Lambda _i {\mbox{\rm \, char}^*\, } M^B_{D, {\bf m} } $$ \end{cor} This follows immediately from the above Lemmas and Proposition \ref{extend by zero}. \mbox{}\\[1em] {\bf Proof of Proposition.} The first formula of the Proposition now follows from the above Lemmas and Prop \ref{extend by zero}. The second statement follows from Demazure's character formula, combined with the vanishing result of \cite{MNW} Prop.28. $\bullet$. \section{Schubert polynomials} In this section, we again work with $G =GL(n)$. As a general reference, see Fulton \cite{F}. \\[.1em] There are two classical computations of the singular cohomology ring $H^.(G/B, {\bf C})$ of the flag variety. That of Borel identifies the cohomology with a coinvariant algebra $$ c: H^.(G/B, {\bf C}) \stackrel{\sim}{\rightarrow} {\bf C}[x_1,\ldots,x_n]/I_+, $$ where $I_+$ is the the ideal generated by the non-constant symmetric polynomials. The map $c$ is an isomorphism of graded ${\bf C}$-algebras, and the generator $x_i$ represents the Chern class of the $i^{\mbox{\tiny th}}$ quotient of the tautological flag bundle. (This is not the dual of an effective divisor.) The alternative picture of Schubert gives as a linear basis for $H^.(G/B, {\bf C})$ the Schubert classes $\sigma_w = [X_{w_0 w}]$, the Poincare duals of the Schubert varieties. The isomorphism between these pictures was defined by Bernstein-Gelfand-Gelfand \cite{BGG} and by Demazure \cite{Dem1}, and given a precise combinatorial form by Lascoux and Schutzenberger \cite{LS}. It identifies certain {\em Schubert polynomials} $ {\cal S} (w) \in {\bf C}[x_1,\ldots,x_n]$ with $c(\sigma_w) = {\cal S} (w)\, (\mbox{mod } I_+)$, and enjoying many remarkable properties. They can be defined combinatorially by a descending recurrence, starting with the representative of the fundamental class of $G/B$. For any permutation $w$ with $ws_i < w$ in the Bruhat order, and $w_0$ the longest permutation, we have $$ {\cal S} (w_0) = x_1^{n-1} x_2^{n-2} \cdots x_{n-2}^2 x_{n-1} $$ $$ {\cal S} (w s_i) = \partial_i {\cal S} (w), $$ where we use the divided difference operator $\partial_i: {\bf C}[x_1,\ldots,x_n] \rightarrow {\bf C}[x_1,\ldots,x_n]$, $$ \partial_i f = {f - s_i f \over x_i - x_{i+1}}. $$ (Note that $ \Lambda _i = \partial_i x_i$. This is special to the root system of type $A_{n-1}$.) \\[.5em] {\bf Example.} For $G=GL(3)$, we have $ {\cal S} (w_0) = x_1^2 x_2$, $ {\cal S} (s_1 s_2) = x_1 x_2$, $ {\cal S} (s_2 s_1) = x_1^2$, $ {\cal S} (s_2) = x_1 + x_2$, $ {\cal S} (s_1) = x_1$, $ {\cal S} (e) = 1$. $\bullet$ \\[1em] To compute any $ {\cal S} (w)$, we write $w_0 = w s_{i_1} \cdots s_{i_r}$ for some reduced word $s_{i_1} \cdots s_{i_r}$, and we have $$ {\cal S} (w) = \partial_{i_1} \cdots \partial_{i_r} (x_1^{n-1} x_2^{n-2} \cdots x_{n-1}). $$ In particular, we may take $i_k$ to be the {\em first ascent} of $w_k = w s_{i_1} \cdots s_{i_{k-1}}$; that is, $i_k =$ the smallest $i$ such that $w_k(i+1)>w_k(i)$. We now give a completely different geometric interpretation of the polynomials $ {\cal S} (w)$ in terms of configuration varieties and Weyl modules. For a permutation $w$ define the {\em inversion family} $I(w) = \{C_1(w),\ldots,C_{n-1}(w)\}$ with $$ C_j(w) = \{ i \in [n] \ \mid\, i<j, \ w(i)>w(j) \} $$ We may write this in our usual form $(D, {\bf m} )$ by dropping any of the $C_j(w)$ which are empty, and counting identical sets with multiplicity. We use the same symbol $I(w)$ to denote this multiset $(D, {\bf m} )$, so that $I(w) - {C}$ means we decrease by one the multiplicity of the element $C \in I(w)$. It is well-known that $I(w)$ is strongly separated. (In fact, it is northwest. See \cite{RS1}, \cite{RS3}, \cite{MNW}) \begin{thm}{(Kraskeiwicz-Pragacz \cite{KP})} $$ {\mbox{\rm \, char}^*\, } M^B _{I(w)} = {\cal S} (w) . $$ \end{thm} {\bf Proof.} (Magyar-Reiner-Shimozono)\, Let $\chi(w) = {\mbox{\rm \, char}^*\, } M^B _{I(w)}$. We must show that $\chi(w)$ satisfies the defining relations of $ {\cal S} (w)$. First, $I(w_0) = \{[1],\ldots,[n-1]\}$, $$ M^B _{I(w_0)} = {\bf C} \cdot \widetilde{\Del}_1^1 \widetilde{\Del}_{12}^{12} \ldots \widetilde{\Del}_{[n-1]}^{[n-1]}, $$ a one-dimensional $B$-module, and $ \chi(w_0) = x_1^{n-1} x_2^{n-2} \cdots x_{n-1} $. Now, suppose $ws_i < w$, and $i$ is the first ascent of $w s_i$. Then the $w(i)^{\mbox{th}}$ element of $I(w)$ is $C_{w(i)}(w) = [i]$. Letting $$ I'(w) \stackrel{\rm def}{=} I(w) - \{\, [i]\, \}, $$ it is easily seen that: \\ (i) $I'(w)$ is $i$-free, \\ (ii) $I(w) = I'(w)\, \cup \{\, [i]\, \}$, and \\ (iii) $I(w s_i) = s_i I'(w) \, \cup \, \{\,[i-1]\,\} $.\\ (Set $[0] = \emptyset$.) Hence we obtain trivially: $$ \begin{array}{rcl} \chi(w) &=& x_1\! \cdots x_i \, {\mbox{\rm \, char}^*\, } M^B _{I'(w)} \\[.2em] \chi(ws_i) &=& x_1\! \cdots x_{i-1}\, {\mbox{\rm \, char}^*\, } M^B _{s_i I'(w)}. \end{array} $$ Since $I'(w)$ is strongly separated and $i$-free, Cor 14 implies that $$ {\mbox{\rm \, char}^*\, } M^B _{s_i I'(w)} = \Lambda _i {\mbox{\rm \, char}^*\, } M^B _{I'(w)}. $$ This is the key step of the proof. Thus we have $$ \begin{array}{rcl} \chi(ws_i) & = & (x_1 \cdots x_{i-1})\ \Lambda _i {\mbox{\rm \, char}^*\, } M^B _{I'(w)} \\ &=& \Lambda _i x_i^{-1} (x_1 \cdots x_i)\, {\mbox{\rm \, char}^*\, } M^B _{I'(w)} \\ &=& \Lambda _i x_i^{-1}\, \chi(w) \\ &=& \partial_i\, \chi(w) . \end{array} $$ But now, using the the first-ascent sequence to write $w_0 = w s_{i_1} \cdots s_{i_r}$, we compute $$ \chi(w) = \partial_{i_1} \cdots \partial_{i_r} (x_1^{n-1} x_2^{n-2} \cdots x_{n-1}) = {\cal S} (w). $$ $\bullet$ Our Demazure character formula (Prop \ref{character formula}) now allows us to compute Schubert polynomials by a completely different recursion from the usual one. In particular, the defining recursion goes from higher to lower degree, whereas our Demazure formula goes from lower to higher. \\[1em] {\bf Example.} For the permutation $w = 24153$ in $GL(5)$, we have $I(w) = \{ 12, 24 \}$ (neglecting the empty set). Then the first-ascent sequence gives us: $$ {\cal S} (w) = \partial_1 \partial_3 \partial_2 \partial_1 \partial_4 \partial_3 (x_1^4 x_2^3 x_3^2 x_4). $$ However, it is easier to compute that $I(w) \subset D_{\ii} ^+$ for a chamber family with $ {\bf i} = 132$ (= reduced word $s_1 s_3 s_2$), so that $ D_{\ii} = \{2, 124, 24\}$ and $$ \begin{array}{rcccccccl} D_{\ii} ^+ = & \{1,& 12,& 123,& 1234,& 12345,& 2,& 124 & 24\} \\ {\bf m} = & ( 0, & 1,& 0,& 0, & 0, & 0,& 0,& 1) \end{array} $$ $$ \begin{array}{rcl} {\cal S} (w) & = & x_1 x_2\, \Lambda _1\, \Lambda _3\, \Lambda _2\, (x_1 x_2) \\ & = & x_1 x_2\, (x_1 x_2 + x_1 x_3 + x_1 x_4 + x_2 x_3 + x_2 x_4) \end{array} $$ See \cite{MFour} for more examples of such computations. $\bullet$ \section{Appendix: Non-reduced words} Let $G$ again be an arbitrary reductive group of rank $r$. For future reference, we note that many of our results hold when the decomposition $w=s_{i_1}\cdots s_{i_l}$ is not of minimal length (that is, $\ell(w) < l$). We call the resulting $ {\bf i} = (i_1,\ldots,i_l)$ (with $i_k \in \{1,\ldots,r\}$) a {\em non-reduced} word. In this case the quotient and fiber product definitions of the Bott-Samelson variety apply without change, and we still have $ Z_{\ii} \cong \Zii^{\tiny quo} \cong \Zii^{\tiny fib} $, as shown in Thm 1(i). However, $ Z_{\ii} $ is no longer the $B$-orbit closure of a $T$-fixed point, so we can no longer define $ \Zii^{\tiny orb} $. Nevertheless, the map $$ \psi:X_l \rightarrow \mbox{\rm Gr}_G( {\bf i} ) \stackrel{\rm def}{=} G/ \widehat{P} _{i_1}\times \cdots G/ \widehat{P} _{i_l} $$ of Thm 1(ii) is still injective on $ \Zii^{\tiny quo} \subset X_l$ (the first part of the proof of Thm 1(ii) is unchanged). Thus we may define an ``embedded'' version of $ Z_{\ii} $, $$ \Zii^{\tiny emb} \stackrel{\rm def}{=} \psi( \Zii^{\tiny quo} ) \subset \mbox{\rm Gr}_G( {\bf i} ) , $$ so that $ \Zii^{\tiny emb} = \Zii^{\tiny orb} $ if $ {\bf i} $ is reduced. We can also define analogues of Weyl modules for a general $G$ and $ {\bf i} $. We once again have the minimal-degree line bundles $ {\cal O} (1)$ over the $G/ \widehat{P} _i$, and hence $ {\cal O} ( {\bf m} ) = {\cal O} (m_1,\ldots,m_l)$ over $\mbox{\rm Gr}_G( {\bf i} )$. Let $ \LL_{\mm} $ be the restriction of $ {\cal O} ( {\bf m} )$ to $ \Zii^{\tiny emb} $. Then define $$ M^B _{ {\bf i} , {\bf m} } \stackrel{\rm def}{=} H^0( Z_{\ii} , \LL_{\mm} ). $$ These modules no longer embed in ${\bf C}[B]$, but they do have a spanning set of Pl\"ucker coordinates, the restrictions of sections from the ambient space $\mbox{\rm Gr}_G( {\bf i} )$: \begin{prop} Let $ {\bf i} = (i_1,\ldots,i_l)$ be an arbitrary word (not necessarily reduced), and $ {\bf m} = (m_1,\ldots,m_l)$ with $m_j \geq 0$ for all $j$. Then the restriction map $$ H^0(\mbox{\rm Gr}_G( {\bf i} ), {\cal O} ( {\bf m} )) \rightarrow H^0( Z_{\ii} , \LL_{\mm} ) $$ is surjective. Furthermore, $H^i( Z_{\ii} , \LL_{\mm} ) = 0$ for $i>0$, and the Demazure character formula also holds: $$ {\mbox{\rm \, char}^*\, } M^B _{ {\bf i} , {\bf m} } = \Lambda _{i_1} \varpi _{i_1}^{m_1} \cdots \Lambda _{i_l} \varpi _{i_l}^{m_l}, $$ $ \varpi _i$ being the (multplicative) fundamental weights and $ \Lambda _i$ the Demazure operators on the ring of characters of $T$. \end{prop} Once again the proof goes through as before, making appeal to the arguments of \cite{MNW}, Prop 28. In the case of $G = GL(n)$, the $ Z_{\ii} $ for non-reduced $ {\bf i} $ again have an explicit interpretation as configuration varieties. This is clear from the fiber-product realization $ Z_{\ii} \cong \Zii^{\tiny fib} $: each extra factor in the Bott-Samelson variety corresponds to one new space in the data of the configuration variety. For example, for $G=GL(3)$ and $ {\bf i} = 2112$, the Bott-Samelson variety is: $$ Z_{\ii} \cong \left\{ \begin{array}{c} (V_2,V_1,V_1',V_2') \in \mbox{\rm Gr}(2)\times \mbox{\rm Gr}(1)\times \mbox{\rm Gr}(1)\times \mbox{\rm Gr}(2) \\[.1em] \mbox{ with }\ {\bf C}^1\! \subset\! V_2\! \supset V_1 \ \mbox{ and }\ V_2\! \supset\! V_1'\! \subset V_2' \end{array} \right\}. $$

9611

alg-geom/9611003

en

https://arxiv.org/abs/alg-geom/9611003

alg-geom/9611003

Ezra Getzler

Ezra Getzler (Northwestern University)

Resolving mixed Hodge modules on configuration spaces

25 pages. amslatex-1.2, pb-diagram and lamsarrow. There are a number of corrections from the first version

MPI 96-161

Given a mixed Hodge module E on a scheme X over the complex numbers, and a quasi-projective morphism f:X->S, we construct in this paper a natural resolution of the nth exterior tensor power of E restricted to the nth configuration space of f. The construction is reminiscent of techniques from the theory of hyperplane arrangements, and relies on Arnold's calculation of the cohomology of the configuration space of the complex line. This resolution is S_n-equivariant. We apply it to the universal elliptic curve with complete level structure of level N>=3 over the modular curve Y(N), obtaining a formula for the S_n-equivariant Serre polynomial (Euler characteristic of H^*_c(V,Q) in the Grothendieck group of the category of mixed Hodge structures) of the moduli space M_{1,n}. In a sequel to this paper, this is applied in the calculation of the S_n-equivariant Hodge polynomial of the compactication \bar{M}_{1,n}.

alg-geom

\subsection*{Outline of the paper} In Section 1, we explain the relationship between Arnold's calculation of the cohomology of the configuration spaces $\mathsf{F}(\mathbb{C},n)$ and the theory of Stirling numbers of the first and second kinds. Section 2 is devoted to the construction of the resolution in the simpler case of sheaves of abelian groups. This is generalized in Section 3 to the cases of perverse sheaves and of mixed Hodge modules. In Section 4, we apply the associated spectral sequence to generalize the formula of \cite{I} for the $\SS_n$-equivariant Serre characteristic\xspace of $\mathsf{F}(X,n)$ to the relative case. In Section 5, we apply the formulas of Section 4 to calculate the $\SS_n$-equivariant Serre characteristic\xspace of the moduli space $\mathcal{M}_{1,n}$. \subsection*{Acknowledgments} I wish to thank the Department of Mathematics at the Universit\'e de Paris-VII, MIT and the Max-Planck-Institut f\"ur Mathematik in Bonn for their hospitality during the inception, elaboration and completion of this paper, respectively. I am grateful to E. Looijenga for introducing me to the Eichler-Shimura theory, to D. Zagier for his help with the proof of Proposition \ref{don}, and to the anonymous referee for a number of excellent suggestions and corrections. The author is partially supported by a research grant of the NSF and a fellowship of the A.P. Sloan Foundation. \section{The combinatorics of partitions and Stirling numbers} In this section, we recall the cohomology of the configuration space $\mathsf{F}(\mathbb{C},n)$ and its relationship with the Stirling numbers. We give more detail on the theory of Stirling numbers than is necessary, since it illuminates the combinatorics which we will apply to construct our resolution. \subsection{Partitions} A partition $J$ of $n$ is a decomposition of the set $\{1,\dots,n\}$ into disjoint non-empty subsets: for example, the partitions of $\{1,2,3,4\}$ are \begin{gather*} \{1,2,3,4\}\quad\{12,3,4\}\quad\{13,2,4\}\quad\{14,2,3\}\quad\{23,1,4\} \quad\{24,1,3\}\quad\{34,1,2\} \\ \{123,4\}\quad\{124,3\}\quad\{134,2\}\quad\{234,1\}\quad\{12,34\}\quad\{13,24\} \quad\{14,23\}\quad\{1234\} , \end{gather*} where we abbreviate the subset $\{i_1,\dots,i_\ell\}$ to $i_1\dots i_\ell$. We denote the subsets of $J$ by $\{J_1,\dots,J_k\}$, in no particular order. Denote by $\S(n,k)$ the set of partitions of $n$ into $k$ non-empty subsets. Associated to a partition $J$ of $n$ is an equivalence relation on $\{1,\dots,n\}$, such that $i\sim_Jj$ iff $i$ and $j$ lie in the same part of $J$. The set of all partitions of $n$ is a poset: if $J$ and $K$ are partitions, $J\prec K$ iff $i\sim_Jj$ implies that $i\sim_Kj$, that is, iff $K$ is coarser than $J$. If $\a=(a_n\mid n\ge1)$ is a sequence of natural numbers, let $|\a|=\sum_{n=1}^\infty na_n$. \begin{lemma} \label{partitions} The exponential generating function of the number $p(\a)$ of partitions of $|\a|$ into $a_j$ subsets of size $j$, $j\ge1$, is $$ B(\t) = \sum_\a p(\a) \frac{\t^\a}{|\a|!} = \exp \biggl( \sum_{j=1}^\infty \frac{t_j}{j!} \biggr) . $$ \end{lemma} \begin{proof} Indeed, $p(\a)$ is the number of automorphisms of the set with $|\a|$ elements divided by the number of automorphisms of such a partition, namely $$ p(\a) = |\a|! \Bigm/ \prod_{j=1}^\infty j!^{a_j}a_j! , $$ from which the lemma follows. \end{proof} Let $f$ be a power series $$ f(t) = \sum_{k=1}^\infty \frac{f_kt^k}{k!} . $$ Define the partial Bell polynomials $B_{n,k}$ by the generating function $$ \exp(xf(t)) = \sum_{n=0}^\infty \frac{t^n}{n!} \sum_{k=0}^n B_{n,k}(f_1,\dots,f_n) x^k . $$ Setting $t_j=xt^jf_j$ in the generating function $B(\t)$ of Lemma \ref{partitions}, we obtain the explicit formula \begin{equation} \label{Bell} B_{n,k}(f_1,\dots,f_n) = \sum_{J\in\S(n,k)} \prod_{i=1}^k f_{|J_i|} . \end{equation} (See Ex.\ 2.11 of Macdonald \cite{Macdonald}.) In particular, the partial Bell polynomials have positive integral coefficients. \begin{proposition} \label{inverse} If $g$ is the inverse power series to $f$ (that is, $g(f(t))=t$), then the matrices $F_{nk}=B_{n,k}(f_1,\dots,f_n)$ and $G_{nk}=B_{n,k}(g_1,\dots,g_n)$ are inverse to each other. \end{proposition} \begin{proof} The matrix $F$ is the transition matrix between the bases $(t^k/k!\mid k\ge0)$ and $(f(t)^k/k!\mid k\ge0)$ of $\mathbb{Q}[t]$. Its inverse $F^{-1}$ is thus the transition matrix between the bases $(f(t)^k/k!\mid k\ge0)$ and $(t^k/k!\mid k\ge0)$. Changing variables from $t$ to $g(t)$, the result follows. \end{proof} \subsection{Stirling numbers of the first kind} The Stirling number of the first kind $s(n,k)$ may be defined as $(-1)^{n-k}$ times the number of permutations on $n$ letters with $k$ cycles. A permutation of the set $\{1,\dots,n\}$ is the same thing as a partition of $n$, together with a cyclic order on each part of the partition. Since a set of cardinality $i$ has $(i-1)!$ cyclic orders, we see that \begin{equation} \label{link} s(n,k) = \sum_{J\in\S(n,k)} \prod_{i=1}^n (-1)^{|J_i|-1} (|J_i|-1)! . \end{equation} Applying \eqref{Bell} with $f_k=(-1)^{k-1}(k-1)!$ (or $f(t)=\log(1+t)$), we see that \begin{equation} \label{first} 1 + \sum_{n=1}^\infty \sum_{k=1}^n s(n,k) \frac{t^nx^k}{n!} = (1+t)^x . \end{equation} In particular, for $n\ge1$, \begin{equation} \label{descending} \sum_{k=1}^n s(n,k) x^k = x(x-1)\dots(x-n+1) . \end{equation} \subsection{Stirling numbers of the second kind} The number $S(n,k)$ of partitions of $n$ with $k$ parts (i.e.\ the cardinality of $\S(n,k)$) is called a Stirling number of the second kind. The special case of \eqref{Bell} with $f(t)=e^t-1$ (and hence $f_k=1$ for all $k$) shows that the Stirling numbers of the second kind have generating function \begin{equation} \label{second} \sum_{n=1}^\infty \sum_{k=1}^n S(n,k) \frac{t^nx^k}{n!} = e^{x(e^t-1)} - 1 . \end{equation} For the reader's edification, we display the first few rows of the matrices of first and second Stirling numbers: $$ s = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 & 0 & 0 \\ 2 & -3 & 1 & 0 & 0 & 0 \\ -6 & 11 & -6 & 1 & 0 & 0 \\ 24 & -50 & 35 & -10 & 1 & 0 \\ \hdotsfor{5} & \ddots \end{bmatrix} \quad\text{and}\quad S = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 3 & 1 & 0 & 0 & 0 \\ 1 & 7 & 6 & 1 & 0 & 0 \\ 1 & 15 & 25 & 10 & 1 & 0 \\ \hdotsfor{5} & \ddots \end{bmatrix} . $$ Applying Proposition \ref{inverse} to the functions $f(t)=e^t-1$ and $g(t)=\log(1+t)$, we see that the matrices $s$ and $S$ formed from the numbers $s(n,k)$ and $S(n,k)$ are inverse to each other: \begin{equation} \label{Stirling} \sum_{n=1}^\infty s(j,n) S(n,k) = \delta(j,k) \end{equation} We may rewrite \eqref{descending} in the form $$ \sum_{j=1}^n s(j,n) x^{-n} = x^{-j} (1-x)\dots(1-(j-1)x) . $$ {}From \eqref{Stirling}, it now follows easily that $$ \sum_{n=k}^\infty S(n,k) x^n = \frac{x^k}{(1-x)(1-2x)\dots(1-kx)} , $$ or equivalently, $$ S(n,k) = \sum_{1\le i_1\le\dots\le i_k\le n} i_1\dots i_k . $$ This is also not difficult to prove directly, by induction on $n$. \subsection{The cohomology of the configuration spaces $\mathsf{F}(\mathbb{C},n)$} Let $H^\bullet(\mathsf{F}(\mathbb{C},n),\mathbb{Z})$ be the cohomology of the configuration space of the complex line. Given distinct $j$ and $k$ in $\{1,\dots,n\}$, let $\omega_{jk}\in H^1(\mathsf{F}(\mathbb{C},n),\mathbb{Z})$ be the integral cohomology class represented by the closed differential form $$ \Omega_{jk} = \frac{1}{2\pi i} \frac{d(z_j-z_k)}{z_j-z_k} . $$ By induction on $n$, Arnold shows in \cite{Arnold} that the cohomology ring $H^\bullet(\mathsf{F}(\mathbb{C},n),\mathbb{Z})$ is generated by the classes $\omega_{jk}$, subject to the relations $\omega_{jk}=\omega_{kj}$ and \begin{equation} \label{Arnold} \omega_{ij}\omega_{jk} + \omega_{jk}\omega_{ki} + \omega_{ki}\omega_{ij} = 0 . \end{equation} The action of the group $\SS_n$ on the configuration space $\mathsf{F}(\mathbb{C},n)$ induces an action on the cohomology ring $H^\bullet(\mathsf{F}(\mathbb{C},n),\mathbb{Z})$, which permutes the generators, by the formula $$ \sigma\*\omega_{ij} = \omega_{\sigma(i)\sigma(j)} . $$ Using the above presentation of $H^\bullet(\mathsf{F}(\mathbb{C},n),\mathbb{Z})$, Arnold shows that $H^{n-k}(\mathsf{F}(\mathbb{C},n),\mathbb{Z})$ is a free abelian group of rank $(-1)^{n-k}s(n,k)$. This motivates the definition of a graded $\SS_n$-module $\mathsf{s}(n,k)$, with $$ \mathsf{s}(n,k)^i = \begin{cases} H^i(\mathsf{F}(\mathbb{C},n),\mathbb{Z}) , & i=n-k , \\ 0 . & \text{otherwise,} \end{cases} $$ We may think of $\mathsf{s}(n,k)$ as a lift of the Stirling number $s(n,k)$ to the category of graded $\SS_n$-modules. Denote by $\l(n)$ the graded $\SS_n$-module $\mathsf{s}(n,1)$. More generally, if $\mathbf{n}$ is a finite set of cardinality $n$, let $\l(\mathbf{n})$ be the graded $\Aut(\mathbf{n})$-module defined in the same way as $\l(n)$ but with the set $\{1,\dots,n\}$ replaced by $\mathbf{n}$. It is isomorphic to $\l(n)$, but to obtain an isomorphism, we must choose a total order on $\mathbf{n}$. The following theorem is proved by Orlik and Solomon for general hyperplane arrangements \cite{OS} (see Theorem 4.21 of Orlik \cite{Orlik}). See also Lehrer-Solomon \cite{LS} for the special case which we consider. \begin{theorem} \label{Lehrer-Solomon} There is a natural decomposition \begin{equation} \label{decompose} \mathsf{s}(n,k) = \bigoplus_{J\in\S(n,k)} \mathsf{s}(n,J) , \end{equation} together with natural isomorphisms $$ \mathsf{s}(n,J) \cong \bigotimes_{i=1}^k \l(J_i) . $$ \end{theorem} \begin{proof} The graded $\SS_n$-module $H^\bullet(\mathsf{F}(\mathbb{C},n),\mathbb{Z})$ is spanned by monomials in the generators $\omega_{ij}$. To such a monomial, we associate a forest (a graph each of whose components is a tree) with vertices the set $\{1,\dots,n\}$, and with an edge between vertices $i<j$ if and only if the generator $\omega_{ij}$ occurs in the monomial. Such a forest determines a partition of the set $\{1,\dots,n\}$. Let $\mathsf{s}(n,J)$ be the span of the monomials associated to the partition $J$. Since all of the forests related by application of one of Arnold's relations \eqref{Arnold} give rise to the same partition, we see that $\mathsf{s}(n,J)$ is well-defined. If $J$ has $k$ parts, its associated forest has $n-k$ edges, and hence $\mathsf{s}(n,J)$ is a subgroup of $H^{n-k}(\mathsf{F}(\mathbb{C},n),\mathbb{Z})$. In particular, if $J$ is the unique partition in $\S(n,1)$, we see that $\mathsf{s}(n,J)$ is generated by all trees with $n$ labelled vertices, modulo the Arnold relations. From this, it is easy to see that $$ \mathsf{s}(n,J) \cong \bigotimes_{i=1}^k \mathsf{s}(J_i,1) . $$ \def{} \end{proof} The characters of the $\SS_n$-modules $\l(n)$ have been calculated by Hanlon \cite{Hanlon1} and Stanley \cite{Stanley}. From their formula, one may calculate the characters of $\mathsf{s}(n,k)$ for all $k$. \begin{lemma} \label{Hanlon-Stanley} The equivariant Euler characteristic of the graded $\SS_n$-module $\l(n)$, evaluated at $\sigma\in\SS_n$, is given by the formula $$ \chi_\sigma(\l(n)) = \begin{cases} \displaystyle - \frac{\mu(d)}{n}(-d)^{n/d}(n/d)! , & \text{if $\sigma$ has $n/d$ cycles of length $d$,} \\ 0 , & \text{otherwise.} \end{cases}$$ \end{lemma} \subsection{A differential on $H^\bullet(\mathsf{F}(\mathbb{C},n),\mathbb{Z})$} We now study the differential $$ \partial : H^\bullet(\mathsf{F}(\mathbb{C},n),\mathbb{Z}) \to H^{\bullet-1}(\mathsf{F}(\mathbb{C},n),\mathbb{Z}) $$ of the algebra $H^\bullet(\mathsf{F}(\mathbb{C},n),\mathbb{Z})$ associated to the diagonal action of the multiplicative group $\mathbb{C}^\times$ on $\mathsf{F}(\mathbb{C},n)$; it is given by capping with the fundamental class of the circle $\U(1)\subset\mathbb{C}^\times$. It follows from the definition of $\omega_{ij}$ that $\partial\omega_{ij}=1$. One can easily check that $\partial$ is well-defined, by showing that the differential of the relation \eqref{Arnold} vanishes: $$ \partial \bigl( \omega_{ij}\omega_{jk} + \omega_{jk}\omega_{ki} + \omega_{ki}\omega_{ij} \bigr) = \bigl( \omega_{jk} - \omega_{ij} \bigr) + \bigl( \omega_{ki} - \omega_{jk} \bigr) + \bigl( \omega_{ij} - \omega_{ki} \bigr) = 0 . $$ The following lemma reflects the fact that the action of $\mathbb{C}^\times$ on $\mathsf{F}(\mathbb{C},n)$ is free if $n>1$, and that the resulting principal fibration is trivial. \begin{lemma} \label{acyclic} If $n>1$, the complex $(H^\bullet(\mathsf{F}(\mathbb{C},n),\mathbb{Z}),\partial)$ is acyclic. \end{lemma} \begin{proof} Let $H$ denote the operator of multiplication by $\omega_{12}$. Since $\partial\omega_{12}=1$, we see that $\partial\*H+H\*\partial$ equals the identity operator, proving acyclicity. \end{proof} If $J\in\S(n,j)$ and $K\in\S(n,k)$ are partitions of $\{1,\dots,n\}$, denote by $\partial_{JK}$ the component of $\partial$ mapping from $\mathsf{s}(n,J)$ to $\mathsf{s}(n,K)$; thus, $\partial_{JK}$ vanishes unless $k=j+1$. \begin{lemma} The differential $\partial_{JK}$ vanishes unless $K\prec J$. \end{lemma} \begin{proof} Let $\alpha$ be a monomial in the generators $\omega_{ij}$ of $H^\bullet(\mathsf{F}(\mathbb{C},n),\mathbb{Z})$. By the definition of $\partial$, $\partial\alpha$ is a sum of terms, in each of which one of the factors $\omega_{ij}$ occurring in $\alpha$ is omitted. Such a term corresponds to partition of $\{1,\dots,n\}$ in which $i$ and $j$ are no longer equivalent: in other words, the partition associated to the new monomial is a refinement of the partition associated to $\alpha$. \end{proof} \section{Resolving sheaves on configuration spaces} Before turning to the construction of resolutions of sheaves over configuration spaces, we explain by an informal argument why one expects Stirling numbers to arise in the construction. Let $\pi:X\to S$ be a continuous map of locally compact topological spaces, and let $X^n/S$ be the $n$th fibred power of $X$ with itself, defined inductively by $X^0/S=S$ and $$ X^{n+1}/S = (X^n/S) \times_S X . $$ Denote by $\pi(n):X^n/S\to S$ the projection to $S$. The space $X^n/S$ has a stratification, with strata indexed by the poset of partitions $J$ of $\{1,\dots,n\}$: the stratum associated to a partition $J$ is given by $$ \mathsf{F}(X/S,J) = \{ (x_1,\dots,x_n) \in X^n/S \mid \text{$x_i=x_j$ iff $i\sim_Jj$} \} . $$ A stratum $\mathsf{F}(X/S,K)$ lies in the closure of $\mathsf{F}(X/S,J)$ if and only if $J\prec K$; the closure of $\mathsf{F}(X/S,J)$ is the diagonal $$ X^J/S = \{ (x_1,\dots,x_n) \in X^n/S \mid \text{$x_i=x_j$ if $i\sim_Jj$} \} . $$ If $J\in\S(n,k)$, denote by $i(J):X^J/S\hookrightarrow X^n/S$ the diagonal embedding. If $\mathcal{F}$ is a sheaf on $X^n/S$, denote by $\mathcal{F}(J)$ the sheaf $i(J)_!i(J)^*\mathcal{F}$ on $X^n/S$. If $J\in\S(n,k)$, $\mathsf{F}(X/S,J)$ is isomorphic to $\mathsf{F}(X/S,k)$; thus, we may represent the above stratification of $X^n/S$ (in)formally as $$ X^n/S = \coprod_{k=1}^n S(n,k) \* \mathsf{F}(X/S,k) . $$ Equation \eqref{Stirling} leads us to expect that there is a ``virtual stratification'' of $\mathsf{F}(X/S,n)$, of the form \begin{equation} \label{haha} \mathsf{F}(X/S,n) = \coprod_{k=1}^n s(n,k) \* X^k/S . \end{equation} Rewritten in terms of generating functions, this becomes $$ \sum_{n=0}^\infty \frac{t^n[\mathsf{F}(X/S,n)]}{n!} = \sum_{k=0}^\infty \frac{\log(1+t)^n[X^n/S]}{n!} = (1+t)^{[X/S]} , $$ where we think of the symbol $[X^n/S]$ as the $n$th power of $[X/S]$, as indeed it is in the Grothendieck group of motivic sheaves on $S$. We may make sense of \eqref{haha} using complexes of sheaves on $X^n/S$. Let $j(n):\mathsf{F}(X/S,n)\hookrightarrow X^n/S$ be the open embedding of the configuration space in $X^n/S$. There is a natural resolution $\mathcal{L}^\bullet(X/S,n,\mathcal{F})$ of $j(n)_!j(n)^*\mathcal{F}$, whose underlying graded sheaf has the form \begin{equation} \label{resolve} \mathcal{L}^{n-k}(X/S,n,\mathcal{F}) = \bigoplus_{J\in\S(n,k)} \Hom(\mathsf{s}(n,J),\mathcal{F}(J)) . \end{equation} For example, if $n=2$, we recover \eqref{diagonal} while if $n=3$, we obtain the resolution $$ 0 \rightarrow j(3)_!j(3)^*\mathcal{F} \rightarrow \mathcal{F}(1,2,3) \rightarrow \mathcal{F}(12,3) \oplus \mathcal{F}(13,2) \oplus \mathcal{F}(23,1) \rightarrow \mathcal{F}(123)\oplus\mathcal{F}(123) \rightarrow 0 . $$ In the special case that $\mathcal{F}$ is a constant sheaf, we may interpret $\mathcal{F}(J)$ as a copy of the diagonal $X^J$, which is isomorphic to $X^k$ when $J\in\S(n,k)$; replacing the $\SS_n$-module $\mathsf{s}(n,J)$ by its Euler characteristic and bearing in mind \eqref{decompose}, or its numerical version \eqref{link}, we are led to \eqref{haha}. \subsection{Construction of the resolution} If $J\prec K$ are partitions of $\{1,\dots,n\}$, denote by $i(J,K)$ the inclusion $X^K/S\hookrightarrow X^J/S$, and by $i(J,K)^*:\mathcal{F}(J)\to\mathcal{F}(K)$ the induced map of sheaves. Let $\mathcal{L}^\bullet(X/S,n,\mathcal{F})$ be the complex of sheaves \eqref{resolve}, with differential \begin{equation} \label{differential} d = \sum_{J\prec K} \partial_{KJ}^* \o i(J,K)^* . \end{equation} For example, $\mathcal{L}^0(X/S,n,\mathcal{F})\cong\mathcal{F}$, while $\mathcal{L}^1(X/S,n,\mathcal{F})$ is the direct sum $$ \mathcal{L}^1(X/S,n,\mathcal{F}) = \bigoplus_{1\le k<l\le n} \mathcal{F}(kl,1,\dots,\widehat{k},\dots,\widehat{l},\dots,n) , $$ since $\dim\mathsf{s}(n,J)=1$ for all $J\in\S(n,n-1)$. In particular, $j^*\mathcal{L}^1(X/S,n,\mathcal{F})=0$. Denote by $\eta:j(n)_!j(n)^*\To\text{id}$ the unit of the adjunction between $j(n)_!$ and $j(n)^*$; it induces a map, also denoted by $\eta$, from $j(n)_!j(n)^*\mathcal{F}$ to $\mathcal{F}=\mathcal{L}^0(X/S,n,\mathcal{F})$. The composition of arrows $$ j(n)_!j(n)^*\mathcal{F} \xrightarrow{\eta} \mathcal{L}^0(X/S,n,\mathcal{F}) \xrightarrow{d} \mathcal{L}^1(X/S,n,\mathcal{F}) $$ is zero, showing that $\eta:j(n)_!j(n)^*\mathcal{F}\to\mathcal{L}^\bullet(X/S,n,\mathcal{F})$ is a morphism of complexes. \begin{theorem} \label{Resolve} The morphism $\eta:j(n)_!j(n)^*\mathcal{F}\to\mathcal{L}(X/S,n,\mathcal{F})$ is a quasi-isomorphism. \end{theorem} \begin{proof} We apply the following lemma. \begin{lemma} \label{strata} Let $X$ be a locally compact stratified space with strata $\{X_J\}$, and let $j(J)$ be the locally closed embedding of the stratum $X_J$ in $X$. Then a map of complexes of sheaves $\eta:\mathcal{F}_1\to\mathcal{F}_2$ on $X$ is a quasi-isomorphism if and only if $\eta : j(J)_!j(J)^*\mathcal{F}_1 \to j(J)_!j(J)^* \mathcal{F}_2$ is a quasi-isomorphism for all strata. \end{lemma} \begin{proof} If there is only one stratum, the lemma is a tautology. We now argue by induction on the number of strata. Let $X_J$ be an open stratum of $X$, and let $Z$ be its complement in $X$, with closed embedding $i:Z\hookrightarrow X$. Consider the diagram $$\begin{diagram} \node{0} \arrow{e} \node{j(J)_!j(J)^*\mathcal{F}_1} \arrow{e} \arrow{s,l}{\eta} \node{\mathcal{F}_1} \arrow{e} \arrow{s,l}{\eta} \node{i_!i^*\mathcal{F}_1} \arrow{e} \arrow{s,l}{\eta} \node{0} \\ \node{0} \arrow{e} \node{j(J)_!j(J)^*\mathcal{F}_2} \arrow{e} \node{\mathcal{F}_1} \arrow{e} \node{i_!i^*\mathcal{F}_1} \arrow{e} \node{0} \end{diagram}$$ Since the rows are exact, we conclude by the five-lemma that $\eta:\mathcal{F}_1\to\mathcal{F}_2$ is a quasi-isomorphism if and only if $\eta:j(J)_!j(J)^*\mathcal{F}_1\to j(J)_!j(J)^*\mathcal{F}_2$ and $\eta:i_!i^*\mathcal{F}_1\to i_!i^*\mathcal{F}_2$ are. By the induction hypothesis, $\eta:i_!i^*\mathcal{F}_1\to i_!i^*\mathcal{F}_2$ is a quasi-isomorphism if and only if $\eta:j(K)_!j(K)^*\mathcal{F}_1\to j(K)_!j(K)^*\mathcal{F}_2$ are for all $K\ne J$; this proves the induction step. \end{proof} If $J$ is a partition of $\{1,\dots,n\}$, let $j(J):\mathsf{F}(X/S,J)\hookrightarrow X^n/S$ be the inclusion of the locally closed subspace $\mathsf{F}(X/S,J)$. By the base change theorem, $$ j(J)_!j(J)^*j(n)_!j(n)^*\mathcal{F} \cong \begin{cases} j(n)_!j(n)^*\mathcal{F} , & \text{if $J$ is the unique partition in $\S(n,n)$,} \\ 0 , & \text{otherwise.} \end{cases}$$ The base change theorem we apply here is (1.2.1) of Verdier \cite{Verdier2}. (In the language of Section 3 of this paper, it says that the natural transformation $\phi:g^*t_!\To s_!f^*$ of Definition \ref{MACKEY} (ii) is an isomorphism.) Let the parts of $J$ be $\{J_1,\dots,J_k\}$, in no particular order, and let $n_i$ be the cardinality of $J_i$. Applying Theorem \ref{Lehrer-Solomon}, we see that \begin{align*} j(J)_!j(J)^*\mathcal{L}^\bullet(X/S,n,\mathcal{F}) & \cong \Hom \biggl( \bigoplus_{K\prec J} \mathsf{s}(n,K) , j(J)_!j(J)^*\mathcal{F} \biggr) \\ & \cong \Hom \biggl( \bigotimes_{i=1}^k H^{n_i-\bullet}(\mathsf{F}(\mathbb{C},n_i),\mathbb{Z}) , j(J)_!j(J)^*\mathcal{F} \biggr) . \end{align*} The differential on this complex of sheaves is induced by the differentials on the factors $H^{n_i-\bullet}(\mathsf{F}(\mathbb{C},n_i),\mathbb{Z})$, and hence by Lemma \ref{acyclic} is acyclic if $n_i>1$ for any $i$. We see that the hypotheses of Lemma \ref{strata} are fulfilled: if $J$ is a partition of $\{1,\dots,n\}$, $\eta:j(J)_!j(J)^*\mathcal{F}\to j(J)_!j(J)^*\mathcal{L}(X/S,n,\mathcal{F})$ is a quasi-isomorphism, since the two complexes are equal if $J\in\S(n,n)$, while they are both acyclic otherwise. \end{proof} \section{Mackey $2$-functors} In this section, we axiomatize those properties of the $2$-functor associating to a variety its derived category of mixed Hodge modules which will be used in constructing the analogue of the resolution $\mathcal{L}^\bullet(X/S,n,\mathcal{F})$ for mixed Hodge modules. It turns out that we need a natural analogue for $2$-functors of Dress's Mackey functors \cite{Dress}. Impatient readers may skip to Section \ref{exact}: all they need to know about the Mackey $2$-functor underlying the theory of mixed Hodge modules is that the usual properties of the functors $f_!$ and $f^*$ for locally closed immersions\xspace hold, such as the base change theorem (in particular, we make no use of Verdier duality). In Section \ref{exact}, we impose sufficient additional hypotheses on these functors to allow us to construct \v{C}ech\xspace-type resolutions of $j_!j^*\mathcal{F}$ when $j$ is an open immersion\xspace and of $i_!i^*\mathcal{F}$ when $i$ is a closed immersion\xspace. The analogue of $\mathcal{L}(X/S,n,\mathcal{F})$ for mixed Hodge modules is defined by replacing the sheaf $i(J)_!i(J)^*\mathcal{F}$ by this \v{C}ech\xspace complex. In all $2$-categories which occur in this paper, the $2$-morphisms are invertible. We often consider $2$-functors which have a category as their domain, which is thought of as a $2$-category all of whose $2$-morphisms are identities. \subsection{Mackey functors} Mackey functors were introduced by Dress \cite{Dress} as an axiomatization of induction in the theory of group representations. The motivating example is the functor $G\mapsto R(G)$ on the category of finite groups, which assigns to a group $G$ its virtual representation ring. Given a morphism $f:G\to H$ of finite groups, there is a contravariant map $f^\bullet:R(H)\to R(G)$, pull-back along $f$, and a covariant map $f_\bullet:R(G)\to R(H)$ generalizing induction: $$ f_\bullet V = ( \mathbb{C}[G] \o V )^H . $$ These functors satisfy the Mackey double coset formula, which says that given a Cartesian square of finite groups $$\begin{diagram} \node{G_1} \arrow{e,t}{f} \arrow{s,l}{s} \node{G_2} \arrow{s,r}{t} \\ \node{H_1} \arrow{e,t}{g} \node{H_2} \end{diagram}$$ we have the equality $g^\bullet t_\bullet=s_\bullet f^\bullet$. \subsection{Mackey $2$-functors} The group of virtual representations $R(G)$ is the Gro\-then\-dieck group of the category ${\mathsf{Proj}}(G)$ of finite-dimensional representations. The $2$-functor $G\mapsto{\mathsf{Proj}}(G)$ satisfies axioms which are the natural analogue for $2$-functors of the notion of a Mackey functor. Broadly speaking, a Mackey $2$-functor is a $2$-functor satisfying these axioms, along with an additivity axiom whose analogue for Mackey functors is only meaningful if we allow $G$ to be a groupoid. The concept of a Mackey $2$-functor is not new: it was introduced by Deligne in Expos\'e XVII of SGA~4 \cite{SGA4} (though not under this name). \begin{definition} \label{MACKEY} A Mackey $2$-functor from a category ${\mathsf{Cat}}$ to a $2$-category ${\mathbb{T}}$ consists of a pair of $2$-functors $\D^\bullet:{\mathsf{Cat}}^\circ\to{\mathbb{T}}$ (where ${\mathsf{Cat}}^\circ$ is the opposite of ${\mathsf{Cat}}$) and $\D_\bullet:{\mathsf{Cat}}\to{\mathbb{T}}$, such that \begin{enumerate} \item if $X$ is an object of ${\mathsf{Cat}}$, the objects $\D^\bullet(X)$ and $\D_\bullet(X)$ are identical --- we denote this object by $\D(X)$, and if $f:X\to Y$ is a morphism of ${\mathsf{Cat}}$, we denote the $1$-morphism $\D^\bullet(f):\D(Y)\to\D(X)$ by $f^\bullet$ and the $1$-morphism $\D_\bullet(f):\D(X)\to\D(Y)$ by $f_\bullet$; \item (base change) to each Cartesian square $$\begin{diagram} \node{X_1} \arrow{e,t}{f} \arrow{s,l}{s} \node{X_2} \arrow{s,r}{t} \\ \node{Y_1} \arrow{e,t}{g} \node{Y_2} \end{diagram}$$ in ${\mathsf{Cat}}$ is associated a natural $2$-morphism $\phi:g^\bullet t_\bullet\To s_\bullet f^\bullet$, such that given a diagram each square of which is Cartesian $$\begin{diagram} \node{X_1} \arrow{e,t}{f} \arrow{s,l}{s} \node{X_2} \arrow{e,t}{f'} \arrow{s,l}{t} \node{X_3} \arrow{s,l}{u} \\ \node{Y_1} \arrow{e,t}{g} \arrow{s,l}{s'} \node{Y_2} \arrow{e,t}{g'} \arrow{s,l}{t'} \node{Y_3} \\ \node{Z_1} \arrow{e,t}{h} \node{Z_2} \end{diagram}$$ the $2$-morphism $\phi$ associated to the top (resp.\ left) pair of squares is the $2$-com\-po\-si\-ti\-on of the $2$-morphisms associated to the squares from which it is formed; \item (additivity) there are $2$-morphisms of bifunctors $\alpha:\D^\bullet(X\coprod Y)\To \D^\bullet(X)\oplus \D^\bullet(Y)$ and $\beta:\D_\bullet(X\coprod Y)\To \D_\bullet(X)\oplus \D_\bullet(Y)$, such that $$ \alpha_{X,Y} = \beta_{X,Y} : {\textstyle\D(X\coprod Y)} \To \D(X)\oplus \D(Y) . $$ \end{enumerate} \end{definition} \subsection{Exact Mackey $2$-functors} \label{exact} Let ${\mathsf{Var}}$ be the category of quasi-projective complex varieties, with morphisms the locally closed immersions\xspace. Let ${\mathbb{T}}$ be the $2$-category whose objects are $t$-categories (Beilinson-Bernstein-Deligne \cite{BBD}), whose $1$-morphisms are right $t$-exact functors possessing a right adjoint, and whose $2$-morphisms are natural isomorphisms. \begin{definition} An exact Mackey $2$-functor on ${\mathsf{Var}}$ is a Mackey $2$-functor with values in ${\mathbb{T}}$ such that \begin{enumerate} \item for closed immersions\xspace $i:Z\hookrightarrow X$, $i^\bullet$ has right adjoint $i_\bullet$, and $i_\bullet$ is fully faithful; \item for open immersions\xspace $j:U\hookrightarrow X$, $j_\bullet$ has right adjoint $j^\bullet$ and is fully faithful; \item if $i:Z\hookrightarrow X$ is a closed immersion\xspace, and $j:U\hookrightarrow X$ is the open immersion\xspace of the complement $U=X\setminus Z$, there is an exact triangle $(j_\bullet j^\bullet V,V,i_\bullet i^\bullet V)$ for each object $V$ of $\D(X)$. \item if $j$ is an affine open immersion\xspace, the functor $j_\bullet$ is $t$-exact. \end{enumerate} \end{definition} We have in mind three examples of exact Mackey $2$-functors. \begin{example} The $2$-functors $X\longmapsto(D^b(X),f_\bullet=f_!,f^\bullet=f^*)$ assigning to $X$ the derived category of sheaves of abelian groups with the usual $t$-structure. \end{example} \begin{example} The $2$-functors $X\longmapsto({}^p\!D^b_c(X),f_\bullet=f_!,f^\bullet=f^*)$ assigning to $X$ the derived category of bounded complexes of sheaves with constructible cohomology, together with the $t$-structure associated to the \emph{middle} perversity $p$ (axiom (iv) is Corollaire 4.1.3 of \cite{BBD}). \end{example} \begin{example} The $2$-functors $X\longmapsto(D^b({\mathsf{MHM}}(X)),f_\bullet=f_!,f^\bullet=f^*)$ assigning to $X$ the derived category of mixed Hodge modules with the natural $t$-structure. \end{example} The Grothendieck group $K_0(\DD)$ of a triangulated category $\DD$ is the abelian group generated by the isomorphism classes of objects of $\DD$ (where we assume that these form a set); we impose the relation $[V]\sim[U]+[W]$ for all exact triangles $(U,V,W)$ in $\D$. For example, if $\DD$ is the derived category $D^b({\mathsf{Ab}})$ of bounded complexes in an abelian category ${\mathsf{Ab}}$, then $K_0(\DD)$ may be identified with the Grothendieck group $K_0({\mathsf{Ab}})$ of the abelian category ${\mathsf{Ab}}$, which is the abelian group generated by the isomorphism classes of objects of ${\mathsf{Ab}}$, with the relation $[V]\sim[U]+[W]$ whenever $V$ is an extension of $W$ by $U$. Although we will not need it, the following result goes some way towards justifying our introduction of Mackey $2$-functors. \begin{proposition} Let $\D$ be an exact Mackey $2$-functor, and let $\K$ be the composition of $\D$ with the functor $K_0$ from ${\mathbb{T}}$ to ${\mathsf{Ab}}$. Then $\K$ is a Mackey functor on ${\mathsf{Var}}$. \end{proposition} \subsection{\v{C}ech\xspace complexes} Let $j:U\hookrightarrow X$ be an open immersion\xspace and let $\mathcal{U}=\{U_i\}_{1\le i\le d}$ be a finite cover of $U$ by affine open immersions\xspace $j_i:U_i\hookrightarrow X$. For example, we might take the $U_i$ to be complements of Cartier divisors in $X$. Using these data, we now define a \v{C}ech\xspace-type resolution of $j_\bullet j^\bullet \mathcal{F}$, where $\D$ is an exact Mackey $2$-functor. (See also Proposition 2.19 of Saito \cite{Saito:mixed} and Section 3.4 of Beilinson \cite{Beilinson}.) \begin{definition} The \v{C}ech\xspace-complex $\mathcal{C}_\bullet(X,\mathcal{U},\mathcal{F})$ is the graded object of $\D(X)$ $$ \mathcal{C}_k(X,\mathcal{U},\mathcal{F}) = \bigoplus_{i_0<\dots<i_k} (j_{i_0\dots i_k})_\bullet (j_{i_0\dots i_k})^\bullet \mathcal{F} , $$ where $j_{i_0\dots i_k}$ is the open immersion\xspace of $U_{i_0\dots i_k} = \bigcap_{\ell=0}^k U_i$ in $X$. Its differential is the sum of maps $$ \partial = \sum_{\ell=0}^k (-1)^\ell \partial_\ell : \mathcal{C}_k(X,\mathcal{U},\mathcal{F}) \to \mathcal{C}_{k-1}(X,\mathcal{U},\mathcal{F}) . $$ Here, $\partial_\ell : (j_{i_0\dots i_k})_\bullet(j_{i_0\dots i_k})^\bullet\mathcal{F} \to (j_{i_0\dots\widehat{\imath_\ell}\dots i_k})_\bullet (j_{i_0\dots\widehat{\imath_\ell}\dots i_k})^\bullet\mathcal{F}$ is induced by the adjunction $q_\bullet q^\bullet\To\text{id}$ associated to the open immersion\xspace $q:U_{i_0\dots i_k}\hookrightarrow U_{i_0\dots\widehat{\imath_\ell}\dots i_k}$. \end{definition} If $\DD$ is a $t$-category, let $H^0(\DD)$ be its heart. Recall from Section 3.1.9 of \cite{BBD} the realization functor $$ \real : D^b(H^0(\DD)) \to \DD ; $$ this is an exact functor mapping bounded complexes $\mathcal{C}_\bullet(H^0(\DD))$ to objects $\real(\mathcal{C}_\bullet)$ in $\DD$. \begin{proposition} \label{Cech} Let $\D$ be an exact Mackey $2$-functor, let $j:U\hookrightarrow X$ be an open immersion\xspace, let $i:Z\hookrightarrow X$ be the closed immersion\xspace of the complement $Z=X\setminus U$, and let $\mathcal{U}=\{U_i\}_{0\le i\le d}$ be a cover of $U$ by affine open immersions\xspace $j_i:U_i\hookrightarrow X$. Then \begin{enumerate} \item $\real\bigl(\mathcal{C}_\bullet(X,\mathcal{U},\mathcal{F})\bigr)\in\Ob\D(X)^{[-d,0]}$ is isomorphic to $j_\bullet j^\bullet \mathcal{F}$; \item $\real\bigl(\Cone\bigl(\mathcal{C}_\bullet(X,\mathcal{U},\mathcal{F})\to\mathcal{F}\bigr)\bigr) \in \Ob\D(X)^{[-d-1,0]}$ is isomorphic to $i_\bullet i^\bullet \mathcal{F}$. \end{enumerate} \end{proposition} \begin{proof} Note that (ii) is implied by (i) and the five-lemma: in the diagram $$\begin{diagram} \divide\dgARROWLENGTH by 2 \node{0} \arrow[1]{e} \node[1]{\mathcal{F}} \arrow[2]{e} \arrow[2]{s,l}{\simeq} \node[2]{i_\bullet i^\bullet\mathcal{F}} \arrow[2]{e} \arrow[2]{s} \node[2]{j_\bullet j^\bullet\mathcal{F}[1]} \arrow[1]{e} \arrow[2]{s,l}{\simeq} \node[1]{0} \\[2] \node{0} \arrow[1]{e} \node[1]{\mathcal{F}} \arrow[2]{e} \node[2]{\Cone\bigl(\mathcal{C}_\bullet(X,\mathcal{U},\mathcal{F})\to\mathcal{F}\bigr)} \arrow[2]{e} \node[2]{\mathcal{C}_\bullet(X,\mathcal{U},\mathcal{F})[1]} \arrow[1]{e} \node[1]{0} \end{diagram}$$ the top row is exact by axiom (iii) for exact Mackey $2$-functors, and the bottom row is obviously exact. We now prove (i) by induction on $d$: for $d=0$, $\mathcal{C}_\bullet(X,\mathcal{U},\mathcal{F})\cong j_\bullet j^\bullet\mathcal{F}$, and the proposition is a tautology. The open subset $U^0=\bigcup_{i=1}^dU_i$ of $X$ has cover $\mathcal{U}^0=\{U_i\}_{1\le i\le d}$. Let $j^0$ and $j_0$ be the open immersions\xspace of $U^0$ and $U_0$ in $X$, and define $\mathcal{F}^0=(j^0)_\bullet(j^0)^\bullet\mathcal{F}$ and $\mathcal{F}_0=(j_0)_\bullet(j_0)^\bullet\mathcal{F}$. Let $p$ be the locally closed immersion\xspace of $U\setminus U^0=U_0\setminus U^0$ in $X$. We now form the diagram $$\begin{diagram} \divide\dgARROWLENGTH by 4 \node{0} \arrow[2]{e} \node[2]{\mathcal{F}^0} \arrow[2]{e} \arrow[3]{s,l}{\simeq} \node[2]{j_\bullet j^\bullet\mathcal{F}} \arrow[3]{e} \arrow[3]{s} \node[3]{p_\bullet p^\bullet\mathcal{F}} \arrow[3]{e} \arrow[3]{s,l}{\simeq} \node[3]{0} \\[3] \node{0} \arrow[2]{e} \node[2]{\mathcal{C}_\bullet(X,\mathcal{U}^0,\mathcal{F}^0)} \arrow[2]{e} \node[2]{\mathcal{C}_\bullet(X,\mathcal{U},\mathcal{F})} \arrow[3]{e} \node[3]{\textstyle\Cone(\mathcal{C}_\bullet(X,\mathcal{U}^0\cap U_0,\mathcal{F}_0)\to\mathcal{F}_0)} \arrow[3]{e} \node[3]{0} \end{diagram}$$ The lower row is defined by dividing the summands $(j_{i_0\dots i_k})_\bullet (j_{i_0\dots i_k})^\bullet\mathcal{F}$ of $\mathcal{C}(X,\mathcal{U},\mathcal{F})$ into two classes: \begin{enumerate} \item if $i_0>0$, the term $(j_{i_0\dots i_k})_\bullet (j_{i_0\dots i_k})^\bullet\mathcal{F}$ is a summand of $\mathcal{C}_\bullet(\mathcal{U}^0,(j^0)_\bullet(j^0)^\bullet\mathcal{F})$; \item if $i_0=0$, the term $(j_{0i_1\dots i_k})_\bullet (j_{0i_1\dots i_k})^\bullet\mathcal{F}$ is a summand of $\Cone_k(\mathcal{C}_\bullet(\mathcal{U}^0\cap U_0,\mathcal{F}_0)\to\mathcal{F}_0)$. \end{enumerate} In particular, the bottom row is exact. The top row is exact by axiom (iii) for exact Mackey $2$-functors, applied to the closed immersion\xspace of $U\setminus U^0$ in $U$, while the outer vertical arrows are quasi-isomorphisms by the induction hypothesis. The proposition now follows by the five-lemma. \end{proof} \subsection{The resolution $\mathcal{L}_\mathcal{U}^\bullet(X/S,n,\mathcal{F})$ for mixed Hodge modules} Using the \v{C}ech\xspace-com\-plex\-es $\mathcal{C}_\bullet(X,\mathcal{U},\mathcal{F})$, we now construct a resolution of $j_!j^*\mathcal{F}$, where $\mathcal{F}$ is a mixed Hodge module. The functor $i^*$ is not $t$-exact on mixed Hodge modules for general closed immersions\xspace $i$; for this reason, our construction depends on the choice of an auxiliary cover $\mathcal{U}$ of $\mathsf{F}(X/S,2)$ by affine open immersions\xspace $j_i:U_i\hookrightarrow X$. (If $X/S$ is a smooth family of curves, we may take the cover to have one element $\mathcal{U}=\{\mathsf{F}(X/S,2)\}$, since in that case, the diagonal in $X^2/S$ is a Cartier divisor.) We will actually construct the resolution in the more general setting of exact Mackey $2$-functors. For $k,l\in\{1,\dots,n\}$ with $k\ne l$, let $\pi_{kl}:X^n/S\to X^2/S$ be the morphism which projects onto the $k$th and $l$th factors. Define a cover $\mathcal{U}(J)$ of the complement of the diagonal $i(J):X^J/S\hookrightarrow X^n/S$ by $$ \mathcal{U}(J) = \bigl\{ \pi_{kl}^{-1}(U_i) \mid \text{$k\sim_Jl$ and $U_i\in\mathcal{U}$} \bigr\} . $$ The open immersions\xspace $\pi_{kl}^{-1}(U_i)\hookrightarrow X^n/S$ are affine, since affine open immersions\xspace are preserved under base change (EGA II, 1.6.2 \cite{EGA}). By Proposition \ref{Cech}, the realization of the complex $\Cone(\mathcal{C}_\bullet(X^n/S,\mathcal{U}(J),\mathcal{F})\to\mathcal{F})$ is quasi-isomorphic to $i(J)_\bullet i(J)^\bullet \mathcal{F}$. If $J\prec K$, the morphism $$ i(J,K)^* : i(J)_\bullet i(J)^\bullet\mathcal{F} \to i(K)_\bullet i(K)^\bullet\mathcal{F} $$ is induced by an inclusion of complexes $$ i(J,K)^* : \mathcal{C}_\bullet(X^n/S,\mathcal{U}(J),\mathcal{F}) \to \mathcal{C}_\bullet(X^n/S,\mathcal{U}(K),\mathcal{F}) , $$ which exists because the cover $\mathcal{U}(K)$ contains the open cover $\mathcal{U}(J)$. As in the case of sheaves, our resolution of $j_\bullet j^\bullet\mathcal{F}$ is a sum over partitions \eqref{resolve}; unlike that case, the result is a double complex, and we must take the realization of its total complex to obtain an object of $\D(X^n/S)$. Let $$ \mathcal{L}_\mathcal{U}^{n-k,-j}(X/S,n,\mathcal{F}) = \bigoplus_{J\in\S(n,k)} \Hom(\mathsf{s}(n,J),\Cone_j(\mathcal{C}_\bullet(X^n/S,\mathcal{U}(J),\mathcal{F})\to\mathcal{F})) . $$ There are two differentials: the analogue of differential \eqref{differential}, $$ d = \sum_{J\prec K} \partial_{KJ}^* \o i(J,K)^* , $$ and the \v{C}ech\xspace-differential $\mathcal{L}_\mathcal{U}^{n-k,-j}(X/S,n,\mathcal{F}) \to \mathcal{L}_\mathcal{U}^{n-k,1-j}(X/S,n,\mathcal{F})$. We may identify $\mathcal{L}_\mathcal{U}^{0,\bullet}(X/S,n,\mathcal{F})$ with $\mathcal{F}$; thus, there is a natural coaugmentation $$ \eta : j_\bullet j^\bullet\mathcal{F} \to \mathcal{L}_\mathcal{U}^{0,\bullet}(X/S,n,\mathcal{F}) . $$ If $\sigma$ is a permutation, the cover $\mathcal{U}(J)$ is carried into $\mathcal{U}(\sigma\*J)$ by the action of $\sigma$, so that $\sigma$ maps $\mathcal{C}_\bullet(X^n/S,\mathcal{U}(J),\mathcal{F})$ isomorphically to $\mathcal{C}_\bullet(X^n/S,\mathcal{U}(\sigma\*J),\mathcal{F})$. The differential of $\mathcal{L}_\mathcal{U}^{\bullet,\bullet}(X/S,n,\mathcal{F})$ is invariant under the action of $\sigma$, showing that $\mathcal{L}_\mathcal{U}^{\bullet,\bullet}(X/S,n,\mathcal{F})$ carries an action of $\SS_n$. It is clear that the coaugmentation $\eta$ is $\SS_n$-equivariant. We now come to the main result of this paper; we omit the proof, since it is essentially identical to that of Theorem \ref{Resolve}. \begin{theorem} \label{RESOLVE} Let $\D$ be an exact Mackey $2$-functor. If $\pi:X\to S$ is a quasi-projective morphism of comples varieties, $\mathcal{F}$ is an object of $\D(X)$ and $\mathcal{U}$ is a cover of $\mathsf{F}(X/S,2)$ by affine open immersions\xspace, the coaugmentation $$ \eta:j_\bullet j^\bullet\mathcal{F} \to \mathcal{L}_\mathcal{U}^{\bullet,\bullet}(X/S,n,\mathcal{F}) $$ induces an $\SS_n$-equivariant quasi-isomorphism between the objects $\real\bigl(\Tot\mathcal{L}_\mathcal{U}^{\bullet,\bullet}(X/S,n,\mathcal{F})\bigr)$ and $j_\bullet j^\bullet\mathcal{F}$ in $\D(X^n/S)$. \end{theorem} \section{The $\SS_n$-equivariant relative Serre characteristic\xspace of $j(n)_\bullet j(n)^\bullet\mathcal{E}^{\boxtimes n}$} \subsection{The relative Serre characteristic\xspace} Let $\D$ be an exact Mackey $2$-functor, and let $\pi:X\to S$ be a morphism of quasi-projective complex varieties. If $\mathcal{F}$ is an object of $\D(X)$, the \emph{relative Serre characteristic\xspace} $\Serre_S(X,\mathcal{F})$ of $\mathcal{F}$ is the class of $\pi_\bullet\mathcal{F}$ in $\K(S)$, where $\K(S)$ is the Grothendieck group $K_0(\D(S))$ of the triangulated category $\D(S)$. This terminology is motivated by the special case in which $\D=D^b({\mathsf{MHM}})$ and $S=\Spec(\mathbb{C})$. If we apply to $\Serre(X,\mathcal{F})$ the homomorphism $\varepsilon:K_0\bigl({\mathsf{MHM}}(\Spec(\mathbb{C}))\bigr)\to\mathbb{Z}[t]$ defined by $$ \varepsilon(V) = \sum_{i,k} (-1)^i t^k \dim \gr^W_kV^i , $$ we obtain the Serre characteristic\xspace of $\mathcal{F}$, $$ \varepsilon\bigl(\Serre(X,\mathcal{F})\bigr) = \sum_{i,k} (-1)^i t^k \dim \gr^W_k H^i_c(X,\mathcal{F}) . $$ Now suppose a finite group $\Gamma$ acts on $X$ and $Y$, and the morphism $\pi$ is $\Gamma$-equivariant. If $\mathcal{F}$ is an object of $\D^\Gamma(X)$, the equivariant relative Serre characteristic\xspace $\Serre_S^\Gamma(X,\mathcal{F})$ is the class of $\pi_\bullet\mathcal{F}$ in $\K^\Gamma(S)$, where $\K^\Gamma(X)$ is the Grothendieck group of $\Gamma$-equivariant objects in $\D(X)$. When $\D$ is defined over a field $\k$ of characteristic $0$ and the categories $\D(X)$ have tensor products, the associated Grothendieck groups $\K^\Gamma(X)$ are $\lambda$-rings, by the arguments of \cite{I}. If in addition the action of $\Gamma$ on $X$ is trivial, the Peter-Weyl theorem (see Theorem 3.2 of \cite{I}) implies the isomorphism of $\lambda$-rings $$ \K^\Gamma(X) \cong R(\Gamma) \o \K(X) , $$ where $R(\Gamma)$ is the virtual representation ring of $\Gamma$ (the Grothendieck group of fin\-ite-di\-men\-sion\-al $\k[\Gamma]$-modules). In this section, we calculate $\Serre_S^{\SS_n}(\mathsf{F}(X/S,n),j(n)^\bullet\mathcal{E}^{\boxtimes n})$. We obtain a generalization of Theorem 5.6 of \cite{I}, which is the special case of Theorem \ref{push} where $\D={\mathsf{MHM}}$, $S=\Spec(\C)$ and $\mathcal{E}={1\!\!1}$ is the unit of $\D(X)$. The reader interested only in the case of mixed Hodge modules may skip the next paragraph, whose r\^ole is to axiomatize the projection axiom. \subsection{Green $2$-functors} If $G$ is a finite group, $R(G)$ is not only an abelian group, but also a commutative ring: furthermore, the functor $f^\bullet$ preserves this product, and we have the projection axiom, which says that if $f:G\to H$ is a morphism of finite groups, there is a commutative diagram $$\begin{diagram} \node[3]{R(G)\o R(H)} \arrow{wsw,t}{1\o f^\bullet} \arrow{ese,t}{f_\bullet\o1} \\ \node{R(G)\o R(G)} \arrow{s} \node[4]{R(H)\o R(H)} \arrow{s} \\ \node{R(G)} \arrow[4]{e,b}{f_\bullet} \node[4]{R(H)} \end{diagram}$$ where the vertical arrows are multiplication in $R(G)$ and $R(H)$. Dress calls a Mackey functor with these additional structures a Green functor. The following definition gives the analogous condition for exact Mackey $2$-functors. \begin{definition} An exact Green $2$-functor is an exact Mackey $2$-functor $\D$ on ${\mathsf{Var}}$ such that given a morphism $f:X\to Y$ in ${\mathsf{Var}}$, there is a natural $2$-morphism $$\divide\dgARROWLENGTH by2 \begin{diagram} \node[3]{\D(X\times Y)} \arrow{wsw,t}{(1\times f)^\bullet} \arrow{ese,t}{(f\times1)_\bullet} \\ \node{\D(X\times X)} \arrow{s,l}{i_X^\bullet} \node[2]{\stackrel{\psi_f}\To} \node[2]{\D(Y\times Y)} \arrow{s,r}{i_Y^\bullet} \\ \node{\D(X)} \arrow[4]{e,b}{f_\bullet} \node[4]{\D(Y)} \end{diagram}$$ where $i_X:X\to X\times X$ and $i_Y:Y\to Y\times Y$ are the diagonal immersions\xspace. These $2$-morphisms must satisfy the condition that for any pair of composable arrows $X\xrightarrow{f}Y\xrightarrow{g}Z$, the $2$-morphism $\psi_{gf}$ equals the $2$-pasting of the following diagram: $$ \divide\dgARROWLENGTH by5 \multiply\dgARROWLENGTH by3 \begin{diagram} \node[5]{\D(X\times Z)} \arrow{wsw,t}{(1\times g)^\bullet} \arrow{ese,t}{(f\times1)_\bullet} \\ \node[3]{\D(X\times Y)} \arrow{wsw,t}{(1\times f)^\bullet} \arrow{ese,t}{(f\times1)_\bullet} \node[2]{\overset{\phi}{\To}} \node[2]{\D(Y\times Z)} \arrow{wsw,t}{(1\times g)^\bullet} \arrow{ese,t}{(g\times1)_\bullet} \\ \node{\D(X\times X)} \arrow{s,l}{i_X^\bullet} \node[2]{\stackrel{\psi_f}\To} \node[2]{\D(Y\times Y)} \arrow{s,l}{i_Y^\bullet} \node[2]{\stackrel{\psi_g}\To} \node[2]{\D(Z\times Z)} \arrow{s,r}{i_Z^\bullet} \\ \node{\D(X)} \arrow[4]{e,b}{f_\bullet} \node[4]{\D(Y)} \arrow[4]{e,b}{g_\bullet} \node[4]{\D(Z)} \end{diagram}$$ \end{definition} \bigskip All three examples of exact Mackey $2$-functors which we gave are exact Green $2$-functors. \subsection{A formula for $\Serre_S^{\SS_n}\bigl(\mathsf{F}(X/S,n),j(n)^\bullet\mathcal{E}^{\boxtimes n}\bigr)$} Let $\D$ be an exact Green $2$-functor defined over $\mathbb{Q}$ (that is, all of the $t$-categories $\D(X)$ are defined over $\mathbb{Q}$). If $\pi:X\to S$ is a quasi-projective morphism of complex varieties and $\mathcal{E}$ is an object of $\D(X)$, we define $$ \mathcal{E}^{\boxtimes n} = \pi_1^\bullet\mathcal{E} \o \dots \o \pi_n^\bullet\mathcal{E} \in \Ob\D^{\SS_n}(X^n/S) , $$ where $\pi_i:X^n/S\to X$ is the $i$th projection. In calculating $\Serre_S^{\SS_n}(\mathsf{F}(X/S,n),j(n)^\bullet\mathcal{E}^{\boxtimes n})$, we make free use of the results of \cite{I}. Recall from loc.\ cit.\ that if $R$ is a complete $\lambda$-ring, with decreasing filtration $F_iR$, $i\ge0$, there is an operation $$ \Exp(x) = \sum_{n=0}^\infty \sigma_n(x) = \exp\Bigl( \sum_{n=1}^\infty \frac{1}{n} \psi_n(x) \Bigr) : F_1R \to 1+F_1R $$ an analogue of the exponential, with inverse $$ \Log(x) = \sum_{n=0}^\infty \frac{\mu(n)}{n} \log(\psi_n(x)) : 1 + F_1R \to F_1R . $$ Here, $\sigma_n$ is the $n$th $\sigma$-operation on $R$, and $\psi_n$ is the $n$th Adams operation. For example, if $R=\mathbb{Z}\[q\]$ has its standard $\lambda$-ring structure, then $$ \Exp(a_1q+a_2q^2+a_3q^3+\dots) = (1-q)^{-a_1}(1-q^2)^{-a_2}(1-q^3)^{-a_3} \dots. $$ The ring of symmetric functions $\Lambda$ is the graded ring $$ \Lambda = \varprojlim \mathbb{Z}[x_1,\dots,x_k]^{\SS_k} , $$ where the variables $x_i$ are assigned degree $1$. It is a polynomial ring in the complete symmetric functions $$ h_n = \sum_{i_1\le\dots\le i_n} x_{i_1}\dots x_{i_n} , $$ and may thus be identified with the free $\lambda$-ring on one generator $h_1$, such that $\sigma_n(h_1)=h_n$ is the $n$th complete symmetric function, and $$ \psi_n(h_1) = p_n = \sum_i x_i^n $$ is the $n$th power sum. Denote the abelian group of symmetric polynomials of degree $n$ by $\Lambda_n$: it is free of rank $p(n)$, the number of partitions of $n$. If $R$ is a $\lambda$-ring, denote by $\Lambda\Hat{\otimes} R$ the complete tensor product of $\Lambda$ with $R$, with filtration $$ F_i(\Lambda\Hat{\otimes} R) = \Bigl\{ \sum_{n=i}^\infty a_i\o r_i \Big| a_n\in\Lambda_n , r_n\in R \Bigr\} , \quad i\ge0 . $$ (In particular, $\Lambda\Hat{\otimes} R=F_0(\Lambda\Hat{\otimes} R)$.) If $V$ is an $\SS_n$-module defined over $\mathbb{Q}$, denote by $\ch(V)\in\Lambda$ its Frobenius characteristic; this is the degree $n$ symmetric function given by the explicit expression $$ \ch(V) = \frac{1}{n!} \sum_{\sigma\in\SS_n} \Tr_V(\sigma) p_\sigma , $$ where $p_\sigma$ is the monomial in the power sums obtained by taking one factor $p_k$ for each cycle of $\sigma$ of length $k$. By the Peter-Weyl theorem, this definition may be extended to $\SS_n$-modules in a $1$-ring $\mathcal{R}$ defined over $\mathbb{Q}$: there is a naturally defined isomorphism $\ch_n$ between the Grothendieck group of $\SS_n$-modules in $\mathcal{R}$ and $\Lambda_n\o K_0(\mathcal{R})$ (see Theorem 4.8 of \cite{I}). \begin{theorem} \label{push} Let $\D$ be an exact Green $2$-functor, and let $\pi:X\to S$ be a morphism of quasi-projective complex varieties. If $\mathcal{E}$ is an object of $\D(X)$, the following equality holds in $\Lambda\Hat{\otimes}\K(S)$: $$ \sum_{n=0}^\infty \Serre_S^{\SS_n}(\mathsf{F}(X/S,n),j(n)^\bullet\mathcal{E}^{\boxtimes n}) = \Exp\biggl( \sum_{n=1}^\infty \frac{\mu(n)}{n} \Serre\bigl( X/S, \log(1+p_n\o\mathcal{E}^{\o n}) \bigr) \biggr) . $$ \end{theorem} \begin{proof} By Theorem \ref{RESOLVE}, we know that \begin{align*} \Serre_S^{\SS_n}(\mathsf{F}(X/S,n),j(n)^\bullet\mathcal{E}^{\boxtimes n}) &= \Serre_S^{\SS_n}(X^n/S,j(n)_\bullet j(n)^\bullet\mathcal{E}^{\boxtimes n}) \\ &= \sum_{k=1}^n \Serre_S^{\SS_n}\biggl(X^n/S, \bigoplus_{J\in\S(n,k)} \Hom(\mathsf{s}(n,J),\mathcal{E}^{\boxtimes n}(J)) \biggr) \\ &= \sum_{k=1}^n \bigoplus_{J\in\S(n,k)} \ch(\mathsf{s}(n,J)^\vee)\o\Serre_S(X^n/S,\mathcal{E}^{\boxtimes n}(J)) \in \Lambda_n\o\K(S) . \end{align*} We may replace $\mathsf{s}(n,k)^\vee$ by $\mathsf{s}(n,k)$, since any $\SS_n$-module is isomorphic to its dual. Applying Theorem \ref{Lehrer-Solomon}, we obtain $$ \sum_{k=1}^n \bigoplus_{J\in\S(n,k)} \prod_{i=1}^k \ch(\l(J_i)) \o \Serre_S\bigl( X,\mathcal{E}^{\o|J_i|} \bigr) , $$ where the product is taken in the ring $\Lambda\Hat{\otimes}\K(S)$. Summing over $n$ gives $$ \sum_{n=0}^\infty \Serre_S^{\SS_n}(X^n/S,j(n)_\bullet j(n)^\bullet\mathcal{E}^{\boxtimes n}) = \Exp \biggl( \sum_{n=1}^\infty \ch(\l(n))\o\Serre_S(X,\mathcal{E}^{\o n}) \biggr) . $$ The theorem now follows from the character formula of Lemma \ref{Hanlon-Stanley}, which may be rewritten as $$ \ch(\l(n)) = \frac{1}{n} \sum_{d|n} (-1)^{n/d-1} \mu(d) p_d^{n/d} . $$ We see that \begin{align*} \sum_{n=1}^\infty \l(n)\o\Serre_S(X,\mathcal{E}^{\o n})) &= \sum_{n=1}^\infty \frac{1}{n} \sum_{d|n} (-1)^{n/d-1} \mu(d) p_d^{n/d} \Serre_S(X,\mathcal{E}^{\o n}) \\ &= \sum_{d=1}^\infty \frac{\mu(d)}{d} \sum_{e=1}^\infty \frac{(-1)^{e-1}}{e} p_d^e \Serre_S(X,\mathcal{E}^{\o de}) \\ &= \sum_{d=1}^\infty \frac{\mu(d)}{d} \Serre_S\bigl(X,\log(1+p_d\o\mathcal{E}^{\o d}) \bigr) . \end{align*} \def{} \end{proof} \begin{remark} Rewriting $\Exp$ in terms of Adams operations, Theorem \ref{push} becomes $$ \sum_{n=0}^\infty \Serre_S^{\SS_n}(\mathsf{F}(X/S,n),j(n)^\bullet\mathcal{E}^{\boxtimes n}) = \exp \biggl( \sum_{n=1}^\infty \frac{1}{n} \sum_{\ell=1}^\infty \frac{(-1)^{\ell-1}}{\ell} p_n^\ell \sum_{d|n} \mu(n/d) \psi_d\bigl(\Serre_S\bigl( X, \mathcal{E}^{\o\ell n/d} \bigr) \bigr) \biggr) . $$ For example, with the notation $\Serre(n)=\Serre_S(X,\mathcal{E}^{\o n})$, we have $$ \Serre_S^{\SS_n}(\mathsf{F}(X/S,n),j(n)^\bullet\mathcal{E}^{\boxtimes n}) = \sum_{\lambda\vdash n} s_\lambda\o \Phi_\lambda , $$ where $\Phi_{1^n}=\sigma_{1^n}(\Serre(1))$, while the other $\Phi_\lambda$ are as follows for $|\lambda|\le4$: \begin{align*} \Phi_2 &= \sigma_2(\Serre(1)) - \Serre(2) , \\ \Phi_3 &= \sigma_3(\Serre(1)) - \Serre(1)\Serre(2) , \quad \Phi_{21} = \sigma_{21}(\Serre(1)) - \Serre(1)\Serre(2) + \Serre(3) , \\ \Phi_4 &= \sigma_4(\Serre(1)) - \sigma_2(\Serre(1))\Serre(2) + \sigma_{1^2}(\Serre(2)) , \\ \Phi_{31} &= \sigma_{31}(\Serre(1)) - \sigma_2(\Serre(1))\Serre(2) - \sigma_{11}(\Serre(1))\Serre(2) + \Serre(1)\Serre(3) + \sigma_2(\Serre(2)) -\Serre(4) , \\ \Phi_{2^2} &= \sigma_{2^2}(\Serre(1)) - \sigma_2(\Serre(1))\Serre(2) + \Serre(1)\Serre(3) + \sigma_{1^2}(\Serre(2)) , \\ \Phi_{21^2} &= \sigma_{21^2}(\Serre(1)) - \sigma_{1^2}(\Serre(1))\Serre(2) + \Serre(1)\Serre(3) - \Serre(4) . \end{align*} Note that the operations $\Phi_\lambda$ of \cite{I} are the specializations of these polynomials obtained on setting $\Serre(n)=\Serre(1)$ for all $n\ge1$. \end{remark} \medskip Applying Theorem \ref{push} with $\mathcal{E}$ equal to the unit object ${1\!\!1}$ of $\D(X)$, and using that ${1\!\!1}^{\o n}={1\!\!1}$ for all $n$, we obtain the following corollary. Here, we abbreviate $\Serre_S(X,{1\!\!1})$ to $\Serre_S(X)$. \begin{corollary} $\displaystyle \sum_{n=0}^\infty \Serre_S^{\SS_n}(\mathsf{F}(X/S,n)) = \Exp \bigl( \Log(1+p_1) \Serre_S(X) \bigr) $ \end{corollary} Theorem \ref{push}, and its corollary, generalize immediately to the equivariant situation, in which a finite group $\Gamma$ acts on $X$ and $S$, and the morphism $\pi:X\to S$ and $\mathcal{E}$ are $G$-equivariant. The calculations now take place in the complete $\lambda$-ring $\Lambda\Hat{\otimes}\K^\Gamma(S)$, and the formulas do not change. \subsection{The configuration spaces of group schemes} If ${\mathbb{G}}$ is a group scheme over $S$ and $n>0$, the scheme $\mathsf{F}({\mathbb{G}}/S,n)$ is an $\SS_n$-equivariant ${\mathbb{G}}$-torsor, and we may consider the quotient scheme ${\mathbb{G}}\backslash\mathsf{F}({\mathbb{G}}/S,n)$. Imitating the above proof, we now calculate its $\SS_n$-equivariant relative Euler characteristic. There is also a $\Gamma$-equivariant generalization, when a finite group $\Gamma$ acts on all of the data; however, it is formally identical, so we simplify notation by only treating the case $\Gamma=1$. \begin{theorem} \label{relative-torsor} If in the setting of Theorem \ref{push} $X={\mathbb{G}}$ is a group scheme, then $$ \sum_{n=1}^\infty \Serre_S^{\SS_n}({\mathbb{G}}\backslash\mathsf{F}({\mathbb{G}}/S,n)) = \frac{\Exp \bigl( \Log(1+p_1) \Serre_S({\mathbb{G}}) \bigr) - 1}{\Serre_S({\mathbb{G}})} . $$ \end{theorem} \begin{proof} We must first choose a ${\mathbb{G}}$-equivariant cover $\mathcal{U}$ of $\mathsf{F}({\mathbb{G}}/S,2)$ by affine open immersions\xspace. Observe that the automorphism $(g,h)\mapsto(g,g^{-1}h)$ of ${\mathbb{G}}^2/S$ identifies $\mathsf{F}({\mathbb{G}}/S,2)$ with ${\mathbb{G}}\times_S{\mathbb{G}}_0$, where ${\mathbb{G}}_0$ is the complement of the identity section of ${\mathbb{G}}$. Under this identification, the action of ${\mathbb{G}}$ on $\mathsf{F}({\mathbb{G}}/S,2)$ corresponds to left translation in the first factor of ${\mathbb{G}}\times_S{\mathbb{G}}_0$. We now choose a cover $\{U_i\}$ of ${\mathbb{G}}_0$ by affine open immersions\xspace $j_i:U_i\hookrightarrow{\mathbb{G}}_0$; the cover $\mathcal{U}$ of $\mathsf{F}({\mathbb{G}}/S,2)$ is the pullback of this cover by the projection from $\mathsf{F}({\mathbb{G}}/S,2)\cong{\mathbb{G}}\times_S{\mathbb{G}}_0$ to ${\mathbb{G}}_0$. (Here, we use that affine open immersions\xspace are preserved under base change.) If $n>0$, the morphism $j(n): \mathsf{F}({\mathbb{G}}/S,n) \hookrightarrow {\mathbb{G}}^n/S$ is an $\SS_n$-equivariant immersion\xspace of ${\mathbb{G}}$-torsors. On quotienting by the action of the group scheme ${\mathbb{G}}$, we obtain an $\SS_n$-equivariant immersion\xspace $$ \bar\jmath(n): {\mathbb{G}}\backslash\mathsf{F}({\mathbb{G}}/S,n) \hookrightarrow {\mathbb{G}}\backslash({\mathbb{G}}^n/S) . $$ (Note that ${\mathbb{G}}\backslash {\mathbb{G}}^n/S$ is isomorphic to ${\mathbb{G}}^{n-1}/S$; however, this isomorphism obscures the action of the symmetric group $\SS_n$.) Since the cover $\mathcal{U}$ of $\mathsf{F}({\mathbb{G}}/S,2)$ is ${\mathbb{G}}$-equivariant, the resolution $\mathcal{L}_\mathcal{U}^\bullet({\mathbb{G}}/S,{1\!\!1},n)$ is $\SS_n\times{\mathbb{G}}$-equivariant, so descends to an $\SS_n$-equivariant resolution ${\mathbb{G}}\backslash\mathcal{L}_\mathcal{U}^\bullet({\mathbb{G}}/S,{1\!\!1},n)$ of $\bar\jmath(n)_\bullet\bar\jmath(n)^\bullet{1\!\!1}$. The theorem now follows by a proof which is entirely parallel to that of Theorem \ref{push} (in the special case that $\mathcal{E}={1\!\!1}$), provided we observe that \begin{align*} \Serre_S^{\SS_n}({\mathbb{G}}\backslash({\mathbb{G}}^n/S),{\mathbb{G}}\backslash \mathcal{L}_\mathcal{U}^{n-k}({\mathbb{G}}/S,{1\!\!1},n)) &= \sum_{k=1}^n \ch(\mathsf{s}(n,k)) \Serre_S({\mathbb{G}})^{k-1} \\ &= \frac{1}{\Serre_S({\mathbb{G}})} \Serre_S^{\SS_n}({\mathbb{G}}^n/S,\mathcal{L}^{n-k}({\mathbb{G}}/S,{1\!\!1},n)) . \end{align*} \def{} \end{proof} Note that if ${\mathbb{G}}$ is a family of elliptic curves, the proof of Theorem \ref{relative-torsor} simplifies, since we may take for the cover $\mathcal{U}$ of $\mathsf{F}({\mathbb{G}}/S,2)$ the canonical choice $\{\mathsf{F}({\mathbb{G}}/S,2)\}$. In fact, this is the case of Theorem \ref{relative-torsor} which we apply in the next section, to the universal family $E(N)$ of elliptic curves over the modular curve $Y(N)$. \section{The $\SS_n$-equivariant Serre characteristic\xspace of the moduli space $\mathcal{M}_{1,n}$} Let $\mathcal{M}_{1,n}(N)$ be the fine moduli space of smooth elliptic curves of level $N\ge3$ with $n$ marked points; it is a smooth quasi-projective variety. The finite group $\SL(2,\mathbb{Z}/N)$ acts on $\mathcal{M}_{1,n}(N)$, with quotient $\mathcal{M}_{1,n}$ the coarse moduli space of smooth elliptic curves. Let $Y(N)$ be the modular curve $\mathcal{M}_{1,1}(N)$, and let $E(N)\to Y(N)$ be the universal elliptic curve of level $N$. The relative configuration space $\mathsf{F}(E(N)/Y(N),n)$ is an $\SL(2,\mathbb{Z}/N)$-equivariant $E(N)$-torsor with base $Y(N)$. Denote by $\mathsf{H}$ the mixed Hodge module ${\mathsf{R}}^1f_!\mathbb{Q}$ on $Y(N)$; it is of course an $\SL(2,\mathbb{Z}/N)$-equivariant local system of rank $2$, known as the Hodge local system. The sub-$\lambda$-ring which it generates in $\K^{\SL(2,\mathbb{Z}/N)}({\mathsf{MHM}}(Y(N)))$ is isomorphic to the Grothendieck group of polynomial representations of the algebraic group $\GL(2)$; this is the polynomial ring $\mathbb{Z}[\mathsf{H},\mathsf{L}]$, with $\sigma_t(\mathsf{H})=(1-t\mathsf{H}+t^2\mathsf{L})^{-1}$ and $\sigma_t(\mathsf{L})=(1-t\mathsf{L})^{-1}$. In this notation, we have $$ \Serre_{Y(N)}^{\SL(2,\mathbb{Z}/N)}(E(N)) = 1 - \mathsf{H} + \mathsf{L} . $$ We may now apply the $\SL(2,\mathbb{Z}/N)$-equivariant version of Theorem \ref{relative-torsor}, obtaining the following formula. \begin{proposition} $$ \sum_{n=1}^\infty \Serre_{Y(N)}^{\SL(2,\mathbb{Z}/N)\times\SS_n}(\mathcal{M}_{1,n}(N)) = \frac{\displaystyle \biggl\{ \prod_{k=1}^\infty (1+p_k)^{\frac{1}{k}\sum_{d|k}\mu(k/d) (1-\psi_d(\mathsf{H})+\mathsf{L}^d)} \biggr\} - 1 } {1-\mathsf{H}+\mathsf{L}} $$ \end{proposition} Denote the $n$th symmetric power of $\mathsf{H}$ by $\mathsf{H}_n$; it is a rank $(n+1)$ $\SL(2,\mathbb{Z}/N)$-equivariant local system on $Y(N)$, given by the Chebyshev polynomial of the second kind% \footnote{These polynomials have generating function $\sum_{n=0}^\infty t^n U_n(x) = (1-2xt+t^2)^{-1}$.} $$ \mathsf{H}_n = U_n(\mathsf{H}/2) . $$ The following table gives $\Serre_{Y(N)}^{\SL(2,\mathbb{Z}/N)\times\SS_n}(\mathcal{M}_{1,n}(N))$ for $n\le5$. This table was calculated using J. Stembridge's symmetric function package $\texttt{SF}$ \cite{SF} for $\mathtt{maple}$. $$\begin{tabular}{|C|L|} \hline n & \Serre_{Y(N)}^{\SL(2,\mathbb{Z}/N)\times\SS_n}(\mathcal{M}_{1,n}(N)) \\ \hline 1 & \mathsf{H}_0 \\[5pt] 2 & \mathsf{H}_0 \mathsf{L} s_2 - \mathsf{H}_1 s_{1^2} \\[5pt] 3 & \mathsf{H}_0 \mathsf{L}^2 s_3 - \mathsf{H}_1 (\mathsf{L} s_{21} - s_{3}) + \mathsf{H}_2 s_{1^3} \\[5pt] 4 & \mathsf{H}_0 (\mathsf{L}^3-\mathsf{L})s_4 - \mathsf{H}_1(\mathsf{L}^2s_{31}-\mathsf{L}(s_4+s_{31})+s_{2^2}) + \mathsf{H}_2 (\mathsf{L} s_{21^2} - s_{31}) - \mathsf{H}_3 s_{1^4} \\[5pt] 5 & \mathsf{H}_0 (\mathsf{L}^4 s_5-\mathsf{L}^2(s_5+s_{41})+\mathsf{L} s_{32}) \\ & {} - \mathsf{H}_1 (\mathsf{L}^3 s_{41}-\mathsf{L}^2(s_5+s_{41}+s_{32})-\mathsf{L}(s_{32}+s_{2^21})+s_{31^2}) \\ & {}+ \mathsf{H}_2 (\mathsf{L}^2 s_{31^2}-\mathsf{L}(s_{41}-s_{32}-s_{31^2})+(s_5+s_{32}+s_{2^21})) - \mathsf{H}_3 (\mathsf{L} s_{21^3} - s_{31^2}) + \mathsf{H}_4 s_{1^5} \\[3pt] \hline \end{tabular}$$ \subsection{The Eichler-Shimura isomorphism} Let $S_\ell(N)$ be the spaces of cusp forms of weight $\ell$ for the congruence group $\Gamma(N) = \ker\bigl( \SL(2,\mathbb{Z}) \to \SL(2,\mathbb{Z}/N) \bigr)$. It is an $\SL(2,\mathbb{Z}/N)$-module, and its invariant subspace $S_\ell=S_\ell(1)$ is the space of cusp forms of level $1$. Let $E_\ell(N)$ be the space of Eisenstein series of weight $\ell$. If $\ell>2$, this is isomorphic as a $\SL(2,\mathbb{Z}/N)$-module to the induced representation $$ \Sigma_\ell(N) = \Ind^{\SL(2,\mathbb{Z}/N)}_{P(N)} \chi_\ell , $$ where $P(N)\subset\SL(2,\mathbb{Z}/N)$ is the parabolic subgroup of upper triangular matrices, with generators $T=\bigl[\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\bigr]$ and $-I$, and $\chi_\ell$ is the character of $P(N)$ which equals $1$ on $T$ and $(-1)^\ell$ on $-I$. The space $E_2(N)$ is smaller than $\Sigma_2(N)$: it is isomorphic to $H_0(\SL(2,\mathbb{Z}/N),\Sigma_2(N)\bigr)$. If $\ell$ is even, $\Sigma_\ell(N)$ is the permutation representation of $\SL(2,\mathbb{Z}/N)$ on the set of cusps, and the $\SL(2,\mathbb{Z}/N)$-invariant subspace is one-dimensional; the corresponding subspace of $E_\ell(N)$ is spanned by the level $1$ Eisenstein series $E_\ell$. If $\ell$ is odd, there are no $\SL(2,\mathbb{Z}/N)$-invariant elements of $E_\ell(N)$, reflecting the fact that there are no level $1$ Eisenstein series of odd weight. In all cases, $\Sigma_\ell(N)$ has dimension $$ [\SL(2,\mathbb{Z}/N):P(N)] = \frac{k^2}{2} \prod_{p|k}(1-p^{-2}) $$ equal to the number of cusps of the congruence subgroup $\Gamma(N)$. Eichler and Shimura have calculated the cohomology of the sheaves $\mathsf{H}_n$. This calculation is explained in Verdier \cite{Verdier} and Shimura \cite{Shimura}. The mixed Hodge structure on this cohomology may be calculated by the same technique that Deligne uses in \cite{Deligne} to calculate the action of the Frobenius operator on the \'etale cohomology groups. Define the Hodge structure $\mathsf{S}_{n+2}(N)$ to be $\gr^W_{n+1}H^1_c(Y(N),\mathsf{H}_n)$. \begin{theorem} \label{Shimura} The vector spaces $\gr_k^WH^i_c(Y(N),\mathsf{H}_n)$ associated to the weight filtration on the cohomology groups $H^\bullet_c(Y(N),\mathsf{H}_n)$ vanish, with the exception of \begin{align*} & \gr^W_0H^1_c(Y(N),\mathsf{H}_n)\cong E_{n+2}(N) , \\ & \gr^W_{n+1}H^1_c(Y(N),\mathsf{H}_n)\cong \mathsf{S}_{n+2}(N) , \quad\text{and} \\ & \gr^W_2H^2_c(Y(N),\mathsf{H}_0)=\mathsf{L} . \end{align*} The Hodge filtration of $\mathsf{S}_{n+2}(N)$ has two steps: $0\subset F^0\mathsf{S}_{n+2}(N)\subset \mathsf{S}_{n+2}(N)$, and the vector space $F^0\mathsf{S}_{n+2}(N)$ is naturally isomorphic to $S_{n+2}(N)$. \end{theorem} \begin{corollary} \label{final} The equivariant Serre characteristic\xspace $$ \Serre^{\SL(2,\mathbb{Z}/N)\times\SS_n}(\mathcal{M}_{1,n}(N)) \in \K^{\SL(2,\mathbb{Z}/N)}({\mathsf{MHM}}(\Spec(\C))) \o \Lambda_n $$ is obtained from the equivariant Serre characteristic\xspace $$ \Serre_{Y(N)}^{\SL(2,\mathbb{Z}/N)\times\SS_n}(\mathcal{M}_{1,n}(N)) \in \K^{\SL(2,\mathbb{Z}/N)}({\mathsf{MHM}}(Y(N))) \o \Lambda_n $$ by the substitution $\mathsf{H}_n\mapsto\delta_{n,0}(\mathsf{L}+1)-\Sigma_{n+2}(N)-\mathsf{S}_{n+2}(N)$. \end{corollary} We may now descend to level $1$ by applying the augmentation $$ \varepsilon : \K^{\SL(2,\mathbb{Z}/N)}({\mathsf{MHM}}(\Spec(\C))) \to \K({\mathsf{MHM}}(\Spec(\C))) , $$ given explicitly by $\varepsilon(\mathsf{S}_\ell(N))=\mathsf{S}_\ell$ and $$ \varepsilon(\Sigma_\ell(N)) = \begin{cases} \mathbb{Q} , & \text{$\ell$ even,} \\ 0 , & \text{$\ell$ odd.} \end{cases} $$ The following table gives the $\SS_n$-equivariant Serre characteristic\xspace $\Serre^{\SS_n}(\mathcal{M}_{1,n})$ for $n\le5$, together with the underlying Serre characteristic\xspace $\Serre(\mathcal{M}_{1,n})$ and Euler characteristic. $$\begin{tabular}{|C|L|L|L|} \hline n & \Serre^{\SS_n}(\mathcal{M}_{1,n}) & \Serre(\mathcal{M}_{1,n}) & \chi(\mathcal{M}_{1,n}) \\ \hline 1 & \mathsf{L} s_1 & \mathsf{L} & 1 \\[5pt] 2 & s_2 \mathsf{L}^2 & \mathsf{L}^2 & 1 \\[5pt] 3 & s_3 \mathsf{L}^3 - s_{1^3} & \mathsf{L}^3-1 & 0 \\[5pt] 4 & s_4 \mathsf{L}^4 - s_4 \mathsf{L}^2 - s_{21^2} \mathsf{L} + s_{31} & \mathsf{L}^4 - \mathsf{L}^2 - 3\,\mathsf{L} + 3 & 0 \\[5pt] 5 & s_5 \mathsf{L}^5 - (s_5+s_{41})\mathsf{L}^3 + (s_{32}-s_{31^2})\mathsf{L}^2 & \mathsf{L}^5 - 5\,\mathsf{L}^3 - \mathsf{L}^2 + 15\,\mathsf{L} - 12 & -2\\ & {}+ (s_{41}+s_{32}+s_{31^2})\mathsf{L} & & \\ & {}- (s_5+s_{32}+s_{2^21}+s_{1^5}) & & \\[3pt] \hline \end{tabular}$$ Corollary \ref{final} may be expressed in closed form: \begin{multline} \label{closed} \sum_{n=1}^\infty \Serre^{\SS_n}(\mathcal{M}_{1,n}) = \Res_0 \Biggl[ \left( \frac{\prod_{n=1}^\infty (1+p_n)^{\frac{1}{n}\sum_{d|n}\mu(n/d)(1-\omega^d-\mathsf{L}^d/\omega^d+\mathsf{L}^d)} - 1} {1-\omega-\mathsf{L}/\omega+\mathsf{L}} \right) \\ \times \left( \sum_{k=1}^\infty \biggl( \frac{\mathsf{S}_{2k+2}+1}{\mathsf{L}^{2k+1}} \biggr) \omega^{2k} - 1 \right) \bigl( \omega-\mathsf{L}/\omega \bigr) d\omega \Biggr] , \end{multline} where $\Res_0$ is the residue of the differential form at the origin. This is an easy consequence of the Weyl integration formula for $\SU(2)$, in the form $$ - \frac{1}{2} \Res_0\biggl[ \biggl( \frac{\omega^{k+1}-(\mathsf{L}/\omega)^{k+1}}{\omega-\mathsf{L}/\omega} \biggr) \biggl( \frac{\omega^{\ell+1}-(\mathsf{L}/\omega)^{\ell+1}}{\omega-\mathsf{L}/\omega} \biggr) \bigl( \omega-\mathsf{L}/\omega \bigr)^2 \frac{d\omega}{\omega} \biggr] = \mathsf{L}^{k+1} \delta_{k\ell} . $$ To obtain a formula for the non-equivariant Serre characteristics\xspace, we replace $p_n$, $n>1$, by $0$, and expand in $p_1$, which gives \begin{equation} \label{non-equi} \frac{\Serre(\mathcal{M}_{1,n+1})}{n!} = \Res_0 \Biggl[ \binom{\mathsf{L}-\omega-\mathsf{L}/\omega}{n} \left( \sum_{k=1}^\infty \biggl( \frac{\mathsf{S}_{2k+2}+1}{\mathsf{L}^{2k+1}} \biggr) \omega^{2k} - 1 \right) \bigl( \omega-\mathsf{L}/\omega \bigr) d\omega \Biggr] . \end{equation} {}From \eqref{non-equi}, we can calculate the Euler characteristic $\chi(\mathcal{M}_{1,n})$ directly. The following proof was shown to the author by D. Zagier. \begin{proposition} \label{don} If $n\ge5$, $\chi(\mathcal{M}_{1,n})=(-1)^n(n-1)!/12$. \end{proposition} \begin{proof} If in \eqref{non-equi}, we replace $\mathsf{L}=1$ and $\mathsf{S}_{k+2}$ by $2\dim(S_{k+2})$, we see that $$ \frac{\chi(\mathcal{M}_{1,n+1})}{n!} = \Res_0 \biggl[ \binom{1-\omega-\omega^{-1}}{n} \frac{(1-\omega^2-2\omega^4-\omega^6+\omega^8)}{(1+\omega^2)(1-\omega^6)} \frac{d\omega}{\omega} \biggr] . $$ The poles of this differential form are all simple, and are located at $\omega=0$ and $\omega=\infty$, and at values of $\omega$ such that $\omega+\omega^{-1}$ is an integer in the interval $[-2,2]$ (the latter poles are on the unit circle). Since it is invariant under $\omega\mapsto\omega^{-1}$, its residues at $0$ and $\infty$ are equal. By the residue theorem, it follows that $$ \frac{\chi(\mathcal{M}_{1,n+1})}{n!} = - \frac12 \sum_{z\in\{\pm1,\pm i,\pm\rho,\pm\rho^2\}} \Res_z \biggl[ \binom{1-\omega-\omega^{-1}}{n} \frac{(1-\omega^2-2\omega^4-\omega^6+\omega^8)}{(1+\omega^2)(1-\omega^6)} \frac{d\omega}{\omega} \biggr] , $$ where $\rho$ is a primitive sixth root of unity. The residues of this differential form on the unit circle are as follows: \begin{equation}\label{residues} \Res_z\biggl[ \dfrac{(1-\omega^2-2\omega^4-\omega^6+\omega^8)}{(1+\omega^2)(1-\omega^6)} \biggr] = \begin{cases} 1/6 , & |z+z^{-1}| = 2 , \\ -1/3 , & |z+z^{-1}|=1 , \\ -1/2 , & |z+z^{-1}|=0 . \end{cases} \end{equation} At each of these poles except $\omega=1$, the binomial coefficient $\binom{1-\omega-\omega^{-1}}{n}$ vanishes for $n\ge4$. This leaves the residue at $1$, which equals $(-1)^{n+1}/12$. \end{proof} We close the paper with a calculation of the Serre characteristics\xspace of the spaces $\mathcal{M}_{1,n}/\SS_n$. If we substitute $x^n$ for $p_n$ in \eqref{closed}, we obtain the generating function for the $\SS_n$-invariant parts of the local systems $\Serre^{\SS_n}(\mathcal{M}_{1,n}/\mathcal{M}_{1,1})$. By Corollary (5.7) of \cite{I}, we have \begin{multline*} \sum_{n=1}^\infty H^0\bigl( \SS_n,\Serre^{\SS_n}(\mathcal{M}_{1,n}/\mathcal{M}_{1,1}) \bigr) x^n = \frac{\displaystyle\prod_{n=1}^\infty (1+x^n)^{\frac{1}{n}\sum_{d|n}\mu(n/d)(1-\omega^d-\mathsf{L}^d/\omega^d+\mathsf{L}^d)} - 1} {1-\omega-\mathsf{L}/\omega+\mathsf{L}} \\ \begin{aligned} {} &= \frac{1}{1-\omega-\mathsf{L}/\omega+\mathsf{L}} \left\{ \frac{(1-\omega x) (1-\mathsf{L} x/\omega) (1-x^2)(1-\mathsf{L} x^2)} {(1-x)(1-\mathsf{L} x)(1-\omega x^2)(1-\mathsf{L} x^2/\omega)} - 1 \right\} \\ {} & = x \left( \frac{1-\mathsf{L} x^3}{1-\mathsf{L} x} \right) \frac{1}{1-(\omega+\mathsf{L}/\omega)x^2+\mathsf{L} x^4} \\ {} & = x \left( \frac{1-\mathsf{L} x^3}{1-\mathsf{L} x} \right) \sum_{k=0}^\infty \mathsf{H}_k x^{2k} . \end{aligned} \end{multline*} Applying the functor $H^\bullet_c(\mathcal{M}_{1,1},-)$, we see that: $$ \sum_{n=1}^\infty \Serre(\mathcal{M}_{1,n}/\SS_n) x^n = x \left( \frac{1-\mathsf{L} x^3} {1-\mathsf{L} x} \right) \sum_{k=0}^\infty \Serre(\mathcal{M}_{1,1},\mathsf{H}_{2k}) x^{4k} $$ Taking Euler characteristics gives \begin{align*} \sum_{n=1}^\infty \chi(\mathcal{M}_{1,n}/\SS_n) x^n &=(x+x^2+x^3) \sum_{n=0}^\infty \chi(\mathcal{M}_{1,1},\mathsf{H}_n) x^{4n} \\ & = (x+x^2+x^3) \frac{(1-x^4-2x^8-x^{12}+x^{16})}{(1-x^8)(1-x^{12})} . \end{align*} The corresponding formulas in genus $0$ are $\Serre(\mathcal{M}_{0,n}/\SS_n)=\mathsf{L}^{n-3}$ and $\chi(\mathcal{M}_{0,n}/{\SS_n})=1$, for all $n\ge3$.

9611

alg-geom/9611001

en

https://arxiv.org/abs/alg-geom/9611001

alg-geom/9611001

A. Matuschke

Andreas Matuschke

On framed instanton bundles and their deformations

19 pages, LaTeX

We consider a compact twistor space P and assume that there is a surface S in P, which has degree one on twistor fibres and contains a twistor fibre F, e.g. P a LeBrun twistor space. Similar to Donaldson and Buchdahl we examine the restriction of an instanton bundle V equipped with a fixed trivialisation along F to a framed vector bundle over (S,F). First we develope inspired by Huybrechts and Lehn a suitable deformation theory for vector bundles over an analytic space framed by a vector bundle over a subspace of arbitrary codimension. In the second section we describe the restriction as a smooth natural transformation into a fine moduli space. By considering framed U(r)-instanton bundles as a real structure on framed instanton bundles over P, we show that the bijection between isomorphism classes of framed U(r)-instanton bundles and isomorphism classes of framed vector bundles over (S,F) due to Buchdahl is actually an isomorphism of moduli spaces.

alg-geom

\section{Introduction} We consider the twistor fibration $\rm \pi : P \rightarrow M$ over a real four-dimensional compact manifold M with self-dual Riemannian metric. P is a three-dimensional complex manifold with an induced antiholomorphic fixpoint free involution $\rm \tau$, an antipodal map on the twistor fibers (cf. \cite{AHS}, \cite{buchdahl1}, \cite{friedrich}). A line on P is a complex submanifold $\rm L\subset P$ with $\rm L\cong \rm I\! P_{\rm C\!\!\! I\,}^1$ and normal bundle $\rm {\cal N}_{L|P} \cong {\cal O}_{\rm I\! P^1}( 1 ) ^{\oplus 2}$. In particular, twistor fibres are lines. We denote with $\rm \mu : Z \rightarrow H$ the universal Douady-family of lines in P. The involution $\rm \tau$ maps lines to lines and consequently induces an antiholomorphic involution on H. Then M appears as a set of fixpoints of $\rm \tau$ and moreover as a real-analytic submanifold of H (cf. \cite{AHS}, \cite{kurke2}): \[ \begin{array}{ccccc} &&\rm H \times P&&\\ &&\cup&&\\ \rm P=Z {\times}_H M& \longrightarrow & \rm Z& \stackrel{\nu}{\longrightarrow}& \rm P\\ \rm\big\downarrow {}^\pi&&\big\downarrow {}^\mu&&\\ \rm M& \rm \subset & \rm H && \end{array} \] Let $\rm S \subset P$ be a surface intersecting twistor fibres with multiplicity 1 and containing a twistor fibre F. By \cite{kurke1}, Prop.2.1, S is a smooth algebraic surface and F the only twistor fibre in S. With $\rm \bar{S} = \tau \left( S \right)$ we have $\rm F=S \cdot \bar{S}$. The linear system $\rm |F|$ defines a birational morphism $\rm S\rightarrow \rm I\! P_{\rm C\!\!\! I\,}^2$ and we have $\rm M\approx \,\stackrel{n}{\#} \left( -\rm I\! P^2_{\rm C\!\!\! I\,} \right)${} or $\rm S^4$. A well examined class of examples are the LeBrun twistor spaces, which fulfill the additional property $\rm dim|S|\ge 1$, cf. \cite{lebrun}, \cite{kurke1}. These twistor spaces are classified as modifications of conic bundles and are algebraic in the sense of M. Artin \cite{knutson}. A (mathematical) instanton bundle is an holomorphic vector bundle on P trivial on all twistor fibres. The Penrose-Ward transformation gives us an analytic equivalence between the categories of instanton bundles and of pairs $\rm \left( E, \nabla \right)$ of complex vector bundles on M with self-dual connection. The pair $\rm \left( E, \nabla \right)$ is associated to the $\rm C^{\infty}$-bundle $\rm \pi^{\ast}E$ together with the holomorphic structure defined by $\rm \bar{\partial}=\left( \pi^{\ast}\nabla \right)^{01}$. Conversely, an instanton bundle V gives rise to a pair $\rm \left( E, \nabla \right)$ by taking $\rm E=\left( \mu_{\ast}\nu^{\ast}V \right) \big{|}_M$ and $\rm \nabla$ as restriction of \[ \begin{array}{ccc} \rm \mu_{\ast}\left( {\cal O}_Z \otimes_{\nu^{-1}{\cal O}_P} \nu^{-1}V \right)& \stackrel{\rm \mu_{\ast}( d_{Z|P}\otimes id_{\nu^{-1}V} )}{-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!- \!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!\rightarrow}& \rm \mu_{\ast}( {\Omega}_{Z|P}^1 \otimes_{{\cal O}_Z} \nu^{\ast}V )\\ \Big\downarrow \nabla& & \Big\downarrow \cong\\ \rm {\Omega}_H^1 \otimes_{{\cal O}_H} \mu_{\ast}\nu^{\ast}V & \stackrel{\cong}{-\!\!\!-\!\!\!- \!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\! \!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!\rightarrow}& \rm \mu_{\ast}{\Omega}_{Z|P}^1 \otimes_{{\cal O}_H} \mu_{\ast}\nu^{\ast}V \end{array} \] (cf. \cite{AHS}, \cite{kurke2}). For G a linear group as for example $\rm U(r)$, $\rm SU(r)$, $\rm Sp(r)$ or $\rm SO(r)$, a G-instanton on M is given by a complex G-vector bundle E with self-dual connection $\rm \nabla$ compatible with the G-structure on E. The Penrose-Ward transformation associates a G-instanton to an instanton bundle with additional properties. In particular, the category of $\rm U(r)$-instantons is analytically equivalent to the category of instanton bundles V on P, for which there is an isomorphism $\rm \varphi : V {\cong} \tau^\ast \bar{V}^\vee$ with $\rm \tau^\ast \bar{\varphi}^\vee = \varphi$, where $\rm \bar{V}^\vee$ denotes the bundle of antilinear forms. We denote these instanton bundles as $\rm U(r)$-instanton bundles or as physical instanton bundles (cf. \cite{AHS}, \cite{buchdahl1}, \cite{kurke2}). For an instanton bundle V on P we can fix a trivialisation $\rm \alpha : V|_F \cong {\cal O}_F^r$ along our twistor fibre $\rm F=S\cdot\bar{S}$. The resulting pair $\rm \left( V,\alpha \right)$ is called a framed instanton bundle. The frame lives in codimension 2. By restricting V to S we obtain a framed vector bundle on a smooth rational surface, framed along a divisor that is big and nef. For this case the moduli problem is well examined by Lehn and Huybrechts in the algebraic-projective case (\cite{lehn}, \cite{huy-lehn1}) and by L\"ubke \cite{lubke} from the analytic point of view. It was an idea of Donaldson \cite{donaldson} to use this restriction map to discuss moduli of framed instantons. By the results of Buchdahl \cite{buchdahl2} it follows, that there is a bijection between the isomorphism classes of framed $\rm U(r)$-instantons and framed vector bundles on S. In the first section this paper we develop the deformation theory for framed vector bundles on analytic spaces, where the frame has support of arbitrary codimension. I.e. we answer questions like: what is the tangent space and when is the deformation functor formally smooth. Everything in this section can be done in the same way and with the same results for seperated algebraic spaces of finite type over some field of characteristic zero. In the second section we apply these results to framed instanton bundles and describe the mentioned restriction map infinitisimally. By describing the functor of framed $\rm U(r)$-instanton bundles as a real structure on the functor of framed mathematical instanton bundles, we show a real-analytic isomorphism between the moduli of framed $\rm U(r)$-instanton bundles and the moduli of framed vector bundles on S. This paper is based on my Diplomarbeit, which was produced in March 1994 under the supervision of Herbert Kurke in Berlin. \section{Deformations of framed vector bundles} \subsection{Families and deformations of framed vector bundles} The following notation is fixed through the whole section. Let $\rm X_0$ be an analytic space, $\rm Y_0$ a closed subspace with associated ideal sheaf $\rm {\cal J}_{Y|X}$ and ${\rm W_0}$ a fixed locally free sheaf of mo- dules over $\rm Y_0$ as in \cite{grauert}. Then a {\it framed vector bundle} to the data $\rm (X_0,Y_0,W_0)$ consists of a pair $\rm ( E_0, \alpha_0)$ with ${\rm E_0}$ a locally free module sheaf over ${\rm X_0}$ and ${\rm \alpha_0: E_0|_{Y_0} \rightarrow W_0}$ a framing isomorphism. A {\it morphism} f between two framed vector bundles $\rm \left(E_0,\alpha_0\right)$ and $\rm \left(E'_0,\alpha'_0\right) $ is a sheaf morphism ${\rm f:E_0\rightarrow E'_0}$ with ${\rm \alpha'_0\circ\left(f|_{Y_0}\right)=\alpha_0}$. For an analytic space S a {\it family} of framed vector bundles for the given data ${\rm \left(X_0,Y_0,W_0\right) }$ parametrized by S is a framed vector bundle ${\rm \left( E,\alpha \right) }$ for the data ${\rm \left( S\times X_0, S\times Y_0, p^* W_0 \right) }$, where ${\rm p:S\times Y_0 \rightarrow Y_0 }$ is the projection. We fix a framed vector bundle $\rm ( E_0, \alpha_0 )$. A {\it deformation} of $\rm ( E_0, \alpha_0 ) $ over a germ of an analytic space $\rm (S,s_0)$ is represented by a triple ${\rm (E,\alpha,\psi ) }$, where ${\rm \left( E,\alpha \right) }$ is a family of framed vector bundles over S and ${\rm \psi }$ is an isomorphism from ${\rm E|_{\left\{ s_0 \right\} \times X_0} }$ to ${\rm E_0 }$, such that the diagram \[ \begin{array}{ccc} {\rm E|_{\left\{ s_0 \right\} \times Y_0 } }& \begin{array}{c} {\cong}\\[-3ex] -\!\!\!-\!\!\!-\!\!\!-\!\!\!\rightarrow\\[-2.5ex] \psi \end{array}& {\rm E_0|_{Y_0} }\\ {\rm {\cong}{\Big\downarrow}{\alpha} }& & {\rm {\cong}{\Big\downarrow }{\alpha_0} }\\ {\rm W_0 }& \begin{array}{c} {=}\\[-3ex] -\!\!\!-\!\!\!-\!\!\!-\!\!\!\rightarrow\\[-2.5ex] {} \end{array}& {\rm W_0 } \end{array} \] commutes. Here, the pointed space $\rm (S,s_0)$ is any representative of our germ. If $\rm \eta : (T,t_0)\rightarrow (S,s_0)$ is a local analytic isomorphism, then ${\rm ((\eta\times id_{X_0})^{\ast}E, (\eta\times id_{X_0})^{\ast}\alpha,(\eta\times id_{X_0})^{\ast}\psi ) }$ represents the same deformation. Two deformations $\rm (E,\alpha,\psi )$ and $\rm (E',\alpha',\psi') $ of $\rm ( E_0, \alpha_0 ) $ over the same germ can be realized as families over the same pointed analytic space $\rm (S,s_0)$. Then a morphism ${\rm f\!: (E,\alpha,\psi ) \rightarrow\! (E',\alpha',\psi') }$ is represented by a local analytic isomorphism $\rm \eta :(T,t_0)\rightarrow (S,s_0)$ and a morphism \[{\rm f: ((\eta\times id_{X_0})^{\ast}E,(\eta\times id_{X_0})^{\ast}\alpha ) \rightarrow ((\eta\times id_{X_0})^{\ast}E',(\eta\times id_{X_0})^{\ast}\alpha') }\] of framed vector bundles with ${\rm (\eta\times id_{X_0})^{\ast}\psi' \circ ( f|_{s_0 \times X_0} ) = (\eta\times id_{X_0})^{\ast}\psi }$. We denote with ${\rm {\cal M}\left( X_0,Y_0,W_0 \right) }$ the functor ${\rm (analytic\; spaces ) \longrightarrow ( sets ) }$ defined by \[ \begin{array}{ccc} {\rm S}& \longrightarrow& \left\{ \begin{array}{c} {\rm isomorphism\; classes\; of\; families\; of}\\[-1.5ex] {\rm framed\; vector\; bundles\; to\; the\; data}\\[-1.5ex] {\rm \left( X_0,Y_0,W_0 \right)\; parametrized\; by\; S} \end{array} \right\} \end{array} \] and with ${\rm Def \left( E_0,\alpha_0 \right) }$ the functor ${\rm \left(germs\; of\; analytic\;spaces\right)\longrightarrow\left( sets \right) }$\\ defined by \[ \begin{array}{cccc} {\rm \left( S,s_0 \right)}& \longrightarrow& \left\{ \begin{array}{c} {\rm isomorphism\;classes\;of\;deformations}\\[-1.5ex] {\rm of\; \left( E_0,\alpha_0 \right)\;over\; \left( S,s_0 \right) } \end{array} \right\}& {\rm .} \end{array} \] For a local ring (R,m) we write often ${\rm Def \left( E_0,\alpha_0 \right)(R) }$ instead of ${\rm Def \left( E_0,\alpha_0 \right)(SpecR,m) }$. We note that in the case $\rm dim(S,s_0)=0$ the germ represented by $\rm (S,s_0)$ coincides with the isomorphism class of this pointed space. \subsection{Deformation-theoretical tangent spaces} Let $\rm \rm C\!\!\! I\, [ \varepsilon ]$ be the $\rm \rm C\!\!\! I\,$-algebra $\rm \rm C\!\!\! I\, [X] / ( X^2)$ with $\rm \varepsilon = X \,\, mod\, X^2$. According to Schlessinger \cite{schlessinger} or M. Artin (\cite{artin1}, \cite{artin2}) we define $\rm Def( E_0, \alpha_0 )(\rm C\!\!\! I\,[ \varepsilon ])$ to be the deformation-theoretical tangent space of $\rm {\cal M}( X_0,Y_0,W_0)$ at the point $\rm ( E_0, \alpha_0 ) \in {\cal M}( X_0,Y_0,W_0 )( \rm C\!\!\! I\, )$. $\rm Aut(E_0,\alpha_0)$ acts on $\rm Def(E_0,\alpha_0)$ by $\rm g\cdot (E,\alpha,\psi)=(E,\alpha,g\circ \psi)$. If this action is trivial and if there is a local moduli $\cal M$ for $\rm ( E_0, \alpha_0 ) $, then we have canonically $\rm T_{ ( E_0, \alpha_0 )}{\cal M} = Def( E_0, \alpha_0 )( \rm C\!\!\! I\, [ \varepsilon ])$. Due to \cite{schlessinger}, $\rm Def ( E_0, \alpha_0 )(\rm C\!\!\! I\, \left[ \varepsilon \right])$ has a natural structure as complex vector space if the canonical mapping \[ \rm Def( E_0, \alpha_0 )(\rm C\!\!\! I\, [ \varepsilon ]\times_{\rm C\!\!\! I\,} \rm C\!\!\! I\, [ \varepsilon ]) \longrightarrow Def ( E_0, \alpha_0) ( \rm C\!\!\! I\, [ \varepsilon ])\times Def ( E_0, \alpha_0)(\rm C\!\!\! I\, [\varepsilon ])\] is bijective. For $\rm \lambda \in \rm C\!\!\! I\,$ and $\rm mult(\lambda ) : \rm C\!\!\! I\, [ \varepsilon]\rightarrow \rm C\!\!\! I\, [ \varepsilon]$ defined by $\rm x+y\varepsilon \rightarrow x+\lambda y\varepsilon$, we obtain \[ \rm Def( E_0, \alpha_0)(mult ( \lambda )): Def( E_0, \alpha_0)(\rm C\!\!\! I\, [ \varepsilon])\longrightarrow Def( E_0, \alpha_0 )( \rm C\!\!\! I\, [ \varepsilon ]),\] which explains the scalar multiplication. The mapping $\rm add:\, \rm C\!\!\! I\, [ \varepsilon ] \times_{\rm C\!\!\! I\,} \rm C\!\!\! I\, [ \varepsilon ]\rightarrow \rm C\!\!\! I\, [ \varepsilon] $ defined by $\rm add( x+y\varepsilon, x+z\varepsilon)= x+( y+z)\varepsilon$ induces \[ \rm Def( E_0, \alpha_0 )(add): Def( E_0, \alpha_0)(\rm C\!\!\! I\, [ \varepsilon] \times_{\rm C\!\!\! I\,} \rm C\!\!\! I\,[ \varepsilon]) \longrightarrow Def( E_0, \alpha_0)(\rm C\!\!\! I\,[ \varepsilon]).\] Under the bijection above we obtain the additive structure on $\rm Def( E_0, \alpha_0)(\rm C\!\!\! I\, [ \varepsilon])$. The assumed bijection is a consequence of the following lemma. \begin{lemma} Let $\rm A'\rightarrow A$ be a small extension of Artin rings. Then for any morphism $\rm B \rightarrow A$ of Artin rings and for $\rm B'=B\times_{A}A'$ the canonical mapping \[ \rm Def( E_0, \alpha_0)( B') \longrightarrow Def( E_0, \alpha_0)(A')\times_{ Def(E_0,\alpha_0)(A)} Def( E_0, \alpha_0)(B)\] is bijective. \end{lemma} {\it Proof:} We recall that an {\it Artin ring} is a local $\rm \rm C\!\!\! I\,$-algebra of finite dimension. A {\it small extension} $\rm A'\rightarrow A$ is given, if there is an element $\rm t\in A'$, such that $\rm A= A'/tA'$ and $\rm t\cdot m_{A'}=0$. Through the whole paper, $\rm m_R$ means the maximal ideal of the local ring R. An element in $\rm Def(E_0, \alpha_0)(A') \times_{ Def(E_0,\alpha_0)(A)} Def( E_0, \alpha_0 )(B)$ is given by a pair $\rm \left( \left( E',\alpha ' \right) , \left( F,\beta \right) \right)$ with $\rm \left( E',\alpha ' \right) \in Def( E_0, \alpha_0)(A')$ and $\rm \left( F,\beta \right) \in Def( E_0, \alpha_0)(B)$, such that we have $\rm E'\otimes_{A'}A=F\otimes_{B}=E$ and $\rm \alpha '\otimes 1_A=\beta \otimes 1_A=\alpha$, where $\rm \left( E,\alpha \right) \in Def( E_0, \alpha_0)(A)$. With $\rm A'\rightarrow A$ small also $\rm B'=B\times_{A}A'\rightarrow B$ is a small extension. Moreover there is an element $\rm t'\in B'$, such that $\rm B=B'/t'B'$, $\rm t'\cdot m_{B'}=0$ and $\rm B' \rightarrow A'$ induces $\rm t'B'\cong tA'$. By taking $\rm F'=E'\times_E F$ we obtain a commutative diagramm \[ \begin{array}{ccccccccc} \rm 0&\rightarrow&\rm E_0& \stackrel{\rm t}{\longrightarrow}& \rm E'& \stackrel{\rm p}{\longrightarrow}& \rm E & \rightarrow & 0 \\ &&\:\big\uparrow{\scriptstyle=}&&\:\big\uparrow{\scriptstyle\rm g}&& \:\big\uparrow{\scriptstyle\rm f}&&\\ \rm 0&\rightarrow&\rm E_0& \stackrel{\rm t'}{\longrightarrow}& \rm F'& \stackrel{\rm q}{\longrightarrow}& \rm F&\rightarrow&0 \end{array} \] with exact lines. Since the induced mappings $\rm \bar{f}:\,F/m_BF\rightarrow E/m_AE$ and $\rm \bar{f}:\,E'/m_{A'}E'\rightarrow E/m_AE$ are isomorphisms and since $\rm t:\,E'/m_{A'}E'\rightarrow tE'$ with $\rm \left( x\; mod\; m_{A'}E' \right) \rightarrow tx$ is surjective, we have $\rm t'F'\cong tE'\cong tA'\otimes_{A'}E'\cong E_0\cong F/m_BF\cong t'B'\otimes_{B'}F'$, where the second isomorphism is given by the $\rm A'$-flatness of $\rm E'$. Hence, with F is flat over B also $\rm F'$ is flat over $\rm A'$. By restriction we obtain \[ \begin{array}{ccccccccc} \rm 0&\rightarrow&\rm W_0& \stackrel{\rm t}{-\!\!\!-\!\!\!\longrightarrow}& \rm A'\otimes W_0& -\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!\longrightarrow& \rm A\otimes W_0 & \rightarrow & 0 \\ &&\:\Big\uparrow{\scriptstyle =}&& \hspace{5em}\Big\uparrow\rm\scriptstyle\alpha'\circ g|_{SpecB'\times Y_0}&& \Big\uparrow &&\\ 0& \rightarrow& \rm W_0& \stackrel{\rm t'\circ \alpha_0^{-1}}{-\!\!\!-\!\!\!\longrightarrow}& \rm F'|_{SpecB'\times Y_0}& \stackrel{\rm \beta\circ q |_{SpecB'\times Y_0}} {-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!\longrightarrow}& \rm B\otimes W_0& \rightarrow& 0 \end{array} \] For h the projection $\rm A'\otimes W_0 \rightarrow W_0$ the lower line splits by $\rm h\circ \alpha`\circ g|_{Y_0}$. We consider the composition $\rm \beta '=\left( h\circ \alpha ' \circ g|_{SpecB'\times Y_0} \right) \oplus\left( \beta \circ q|_{SpecB'\times Y_0} \right) : F'|_{SpecB'\times Y_0}\cong B'\otimes W_0$ and obtain $\rm ( F',\beta )$ as an element in $\rm Def( E_0,\alpha _0 )( B')$. The restriction of $\rm F'$ to $\rm SpecB\times X_0$ yields F, the pullback of $\rm F'$ to $\rm SpecA'\times X_0$ yields $\rm E'$ and both is compatible with the framings. Thus the mapping $\rm ( ( E',\alpha ' ) ,( F,\beta )) \rightarrow ( F',\beta ' )$ is the inverse to \[ \rm Def( E_0, \alpha_0 )(B') \longrightarrow Def( E_0, \alpha_0)( A')\times_{ Def( E_0, \alpha_0)(A )} Def( E_0, \alpha_0)(B).\;\Box \] \subsection{Deformations and extensions} \begin{theorem} \parbox[t]{30em}{ Let $\rm (E_0,\alpha_0)\in{\cal M}(X_0,Y_0,W_0)(\rm C\!\!\! I\,)$. There is a canonical isomorphism $\rm Ext^1_{X_0}\left( E_0,E_0\otimes_{{\cal O}_{X_0}}{\cal J}_{Y_0|X_0}\right) =Def \left( E_0,\alpha_0\right) \left( \rm C\!\!\! I\, [\varepsilon] \right)$.} \end{theorem} {\it Proof:} For $\rm X=Spec\rm C\!\!\! I\, \left[ \varepsilon \right] \times X_0$ we have $\rm {\cal O}_X= {\cal O}_{X_0}\oplus \varepsilon{\cal O}_{X_0}$. For $\rm \left( E,\alpha, \psi\right)\in Def\left( E_0, \alpha{}_0 \right) \left( \rm C\!\!\! I\, \left[ \varepsilon \right] \right)$ the exact sequence $\rm 0\rightarrow \varepsilon E \rightarrow E \rightarrow E/\varepsilon E \rightarrow 0$ yields therefore an element \[ \rm \left( e \right) = \left( 0\rightarrow E_0 \stackrel{i}{\longrightarrow} E \stackrel{p}{\longrightarrow} E_0 \rightarrow 0 \right)\] in $\rm Ext^1_{X_0}\left( E_0,E_0 \right)$, where we have used $\rm \psi : E/\varepsilon E \cong E_0$ and $\rm \rm C\!\!\! I\, \left[ \varepsilon \right] / \varepsilon \rm C\!\!\! I\, \left[ \varepsilon \right] \cong \varepsilon \rm C\!\!\! I\, \left[ \varepsilon \right] $. By restriction to $\rm Y_0$ we obtain a commutative diagram with exact lines: \[ \begin{array}{cccccccccc} & 0& \rightarrow & \rm E_0|_{Y_0} & \longrightarrow & \rm E|_{Y_0} & \longrightarrow & \rm E_0|_{Y_0} & \rightarrow & 0\\ \rm \left( d \right) \hspace{1cm}& & & \:{\big\downarrow} \scriptstyle\alpha _0 & & {\big\downarrow} \scriptstyle\alpha & & \:{\big\downarrow} \scriptstyle\alpha _0 & & \\ & 0& \rightarrow & \rm W_0 & \longrightarrow & \rm W_0 \oplus \varepsilon W_0 & \longrightarrow & \rm W_0 & \rightarrow & 0 \end{array} \] We note that $\rm \left( e \right) \in ker \left( Ext^1_{X_0}\left( E_0,E_0 \right) \rightarrow Ext^1_{Y_0}\left( E_0|_{Y_0},E_0|_{Y_0} \right) \right) $. Conversely, the pair ((e),(d)) determines the deformation $\rm \left( E,\alpha, \psi\right) $ by defining the multiplication of a local section $\rm x \in E$ with $\varepsilon $ by $\rm \varepsilon x = \left( i \circ p \right) \left( x \right) $. Now we desribe a canonical map $\nu$ from $\rm Ext^1_{X_0}\left( E_0,E_0\otimes_{{\cal O}_{X_0}}{\cal J}_{Y_0|X_0}\right)$ to $\rm Def \left( E_0,\alpha_0\right) \left( \rm C\!\!\! I\, [\varepsilon] \right)$. Let $\rm (f)=\left( 0\rightarrow E_0\otimes_{{\cal O}_{X_0}} {\cal J}_{Y_0|X_0} \stackrel{i}{\longrightarrow} F \longrightarrow E_0 \rightarrow 0 \right)$. For j the embedding $\rm E_0\otimes_{{\cal O}_{X_0}}{\cal J}_{Y_0|X_0} \stackrel{j}{\hookrightarrow} E_0$ the fibre sum of i and j \[ \begin{array}{ccccccccc} 0& \rightarrow& \rm E_0\otimes_{{\cal O}_{X_0}}{\cal J}_{Y_0|X_0}& \rm \stackrel{i}{\longrightarrow}& \rm F& \longrightarrow & \rm E_0 & \rightarrow& 0 \\ & & \rm {\big \downarrow} \scriptstyle j& & {\big \downarrow}& & {\big \downarrow}& & \\ 0& \rightarrow& \rm E_0& \rm {\longrightarrow}& \rm E=F\amalg_{E_0\otimes {\cal J}_{Y_0|X_0}}E_0& \longrightarrow & \rm E_0 & \rightarrow& 0 \end{array} \] defines an element $\rm (e)=(0\rightarrow E_0 \rightarrow E \rightarrow E_0 \rightarrow 0)$ in $\rm Ext^1_{X_0}\left( E_0,E_0 \right)$. In particular we have $\rm E=F\amalg_{E_0\otimes {\cal J}_{Y_0|X_0}}E_0=\left( F\oplus E_0 \right) / N$, where N is the subsheaf consisting of all pairs (x,y) such that there is a $\rm z\in E_0\otimes {\cal J}_{Y_0|X_0}$ with i(z)=y and j(z)=x. Thus we have \[ \rm E|_{Y_0}=E/{\cal J}_{Y_0|X_0}E= (F\oplus E_0)/ \left( N+( {\cal J}_{Y_0|X_0}F \oplus {\cal J}_{Y_0|X_0}E_0 ) \right)\] and together with $\rm N+{\cal J}_{Y_0|X_0}E_0 =i\left( {\cal J}_{Y_0|X_0}E_0 \right) \oplus {\cal J}_{Y_0|X_0}E_0$ we obtain the composition \[ \begin{array}{ll} \rm E|_{Y_0}& \rm =\left( F{\Big /}i\left( {\cal J}_{Y_0|X_0}E_0\right)+{\cal J}_{Y_0|X_0}F \right) \rm \oplus \left( E_0 {\Big /} {\cal J}_{Y_0|X_0}E_0 \right)\\ & \rm \cong \left( \left( F / i\left( {\cal J}_{Y_0|X_0}E_0\right) \right) {\Big /} \rm {\cal J}_{Y_0|X_0}\left( F / i\left( {\cal J}_{Y_0|X_0}E_0 \right) \right) \right) \rm \oplus \left( E_0 {\Big /} {\cal J}_{Y_0|X_0}E_0 \right)\\ & \rm \cong \left( E_0 {\Big /} {\cal J}_{Y_0|X_0}E_0 \right) \rm \oplus \left( E_0 {\Big /} {\cal J}_{Y_0|X_0}E_0 \right)\\ & \rm \cong E_0|_{Y_0}\oplus E_0|_{Y_0} \begin{array}{c} \cong \\[-1.4em] -\!\!\!-\!\!\!-\!\!\!\longrightarrow\\[-1em] \scriptstyle (\alpha_0, \varepsilon\alpha_0) \end{array} \rm W_0\oplus \varepsilon W_0, \end{array} \] which yields a framing $\rm \alpha : E|_{Y_0} \cong W_0 \oplus \varepsilon W_0$ in a canonical way, whence we have found our map by taking $\rm \nu\left( f \right)=\left( E,\alpha \right)$. To find the inverse map of $\nu $, we associate for a given $\rm \left( E,\alpha \right)\! \in \! Def\! \left( E_0,\alpha_0\right) \left( \rm C\!\!\! I\, [\varepsilon] \right)$ a pair ((e),(d)) with $\rm (e)\in Ext^1_{X_0}\left( E_0,\alpha_0\right)$ as described above and denote with q the composition \[\rm E \rightarrow E|_{Y_0} \stackrel{\alpha}{\longrightarrow} W_0 \oplus \varepsilon W_0 \stackrel{2nd\; projection} {-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!\longrightarrow} \varepsilon W_0 \cong W_0.\] By defining $\rm F= ker(q)$ we obtain $\rm {\cal J}_{Y_0|X_0}E_0 \cong E_0 \times_E F$ and moreover, the map $\rm {\cal J}_{Y_0|X_0}E_0 \rightarrow F$ given by the fibre product is injective and its cokernel is isomorphic to $\rm E_0$. Thus we have obtained an element $\rm (f)=\left( 0\rightarrow {\cal J}_{Y_0|X_0}E_0 \rightarrow F \rightarrow E_0 \rightarrow 0\right)$ in $\rm Ext^1_{X_0}\left( E_0,{\cal J}_{Y_0|X_0}E_0 \right)$ and a commutative diagram \[ \begin{array}{ccccccccc} &&0&&0&&&&\\ &&\downarrow&&\downarrow&&&&\\ \rm 0& \rightarrow &\rm {\cal J}_{Y_0|X_0}E_0 & \longrightarrow &\rm F& \rm \longrightarrow&\rm E_0& \rightarrow& 0\\ &&\big\downarrow&&\big\downarrow&&\hspace{1em}\big\downarrow \scriptstyle =&&\\ \rm 0& \rightarrow &\rm E_0 & \longrightarrow & \rm E& \rm \longrightarrow&\rm E_0& \rightarrow& 0\\ \rm &&\big\downarrow&&\hspace{1ex}\big\downarrow\rm q&&&&\\ \rm &&\rm E_0|_{Y_0} & \stackrel{\alpha_0}{\longrightarrow}&\rm W_0&&&&\\ &&\downarrow&&\downarrow&&&&\\ &&0&&0&&&& \end{array} \] with exact lines and columns, from where we see that $\rm \nu (f) = (E,\alpha)$ and hence $\nu $ is a canonical bijection. It is not hard but some writing to describe the linear structure of $\rm Def ( E_0,\alpha_0)( \rm C\!\!\! I\, [\varepsilon])$ in terms of $\rm ((e),(d))$. From there it is obvious that $\nu$ is linear, too. $\Box$ \subsection{Smoothness of the deformation functor} As in \cite{schlessinger} or \cite{artin2}, the functor $\rm Def(E_0,\alpha_0)$ is called {\it formally smooth}, if for every small extension $\rm R\rightarrow R/tR =\bar{R}$ of Artin rings (i.e. $\rm t\in R$ with $\rm t\cdot m_R=0$) the induced mapping from $\rm Def(E_0,\alpha_0)(R)$ to $\rm Def(E_0,\alpha_0)(\bar{R})$ is surjective. In the following we fix an arbitrary small extension $\rm R\rightarrow R/tR =\bar{R}$ of Artin rings and define spaces $\rm X=SpecR\times X_0$, $\rm Y=SpecR\times Y_0$, $\rm \bar{X}=Spec\bar{R}\times X_0$ and $\rm \bar{Y}=Spec\bar{R}\times Y_0$. With W and $\rm \bar{W}$ we denote the pullbacks of $\rm W_0$ to Y and $\rm \bar{Y}$, respectively. The assertion shown by the next Proposition is known as {\it $\rm T^1$-lifting property}. \begin{proposition} Let $\rm ({E},{\alpha})\in Def(E_0,\alpha_0)({R})$ and $\rm (\bar{E},\bar{\alpha})\in Def(E_0,\alpha_0)(\bar{R})$. We may think of both as framed vector bundles to the framing data $\rm (X,Y,W)$ and $\rm (\bar{X},\bar{Y},\bar{W})$, respectively. If $\rm Ext^2_{X_0}(E_0,{\cal J}_{Y_0|X_0}\otimes E_0)=0$, then the natural homomorphism between the deformation-theoretical tangent spaces $\rm Def(E,\alpha)(\rm C\!\!\! I\,[\varepsilon])\rightarrow Def(\bar{E},\bar{\alpha})(\rm C\!\!\! I\,[\varepsilon])$ is surjective. \end{proposition} {\it Proof:} With E flat over X and deformation of $\rm E_0$ we have $\rm tE = tR\otimes E \cong E/m_RE = E_0$ and therefore a short exact sequence \[ \rm 0 \rightarrow E_0 \stackrel{t}{\longrightarrow} E \longrightarrow \bar{E} \rightarrow 0. \] With $\rm \bar{E}$ flat over $\rm X_0$ and $\rm {\cal J}_{Y|X}={\cal J}_{Y_0|X_0}\otimes {\cal O}_X$ we also have \[ \rm 0 \rightarrow {\cal J}_{Y|X}\otimes E_0 \stackrel{t}{\longrightarrow} {\cal J}_{Y|X}\otimes E \longrightarrow {\cal J}_{Y|X}\otimes \bar{E} \rightarrow 0. \] As part of the long Ext-sequence we obtain \[ \rm Ext^1_X(E,{\cal J}_{Y|X}\otimes E) \longrightarrow Ext^1_{\bar{X}}(\bar{E},{\cal J}_{\bar{Y}|\bar{X}}\otimes \bar{E}) \longrightarrow Ext^2_{X_0}(E_0,{\cal J}_{Y_0|X_0}\otimes E_0), \] where $\rm Ext^1_X(E,{\cal J}_{Y|X}\otimes \bar{E})= Ext^1_{\bar{X}}(\bar{E},{\cal J}_{\bar{Y}|\bar{X}}\otimes \bar{E})$ and $\rm Ext^2_X(E,{\cal J}_{Y|X}\otimes E_0)= Ext^2_{X_0}(E_0,{\cal J}_{Y_0|X_0}\otimes E_0)$ were used. The first homomorphism of the last exact sequence coincides with $\rm Def(E,\alpha)(\rm C\!\!\! I\,[\varepsilon])\rightarrow Def(\bar{E},\bar{\alpha})(\rm C\!\!\! I\,[\varepsilon])$ by Theorem 1.2 and we are done. $\Box$ \begin{theorem} If $\rm dim_{\rm C\!\!\! I\,}Ext^1_{X_0} (E_0,{\cal J}_{Y_0|X_0}E_0)<\infty$ and $\rm Ext^2_{X_0} (E_0,{\cal J}_{Y_0|X_0}E_0)=0$, then $\rm Def(E_0,\alpha_0)$ is formally smooth. \end{theorem} {\it Proof:} Because of Lemma 1.1 and since the dimension of $\rm Ext^1_{X_0} (E_0,{\cal J}_{Y_0|X_0}E_0)$ is finite, the functor $\rm Def(E_0,\alpha_0)$ is prorepresentable by the criterion of Schlessinger \cite{schlessinger}. The $\rm T^1$-lifting property holds by Proposition 1.3 and the application of the Kawamata-Ran principle (\cite{kawamata}, Theorem 1) yields our assertion. $\Box$ {\it Remark.} D. Huybrechts and M. Lehn consider in \cite{huy-lehn1} a framed module over an algebraic nonsingular projective variety $\rm X_0$ given by a coherent $\rm {\cal O}_X$-module $\rm E_0$ and a nonzero morphism from $\rm E_0$ to a fixed coherent $\rm {\cal O}_X$-module $\rm D_0$. They introduce a notion of stability in this situation and describe the deformation-theoretical tangent space of a stable framed module $\rm (E_0, E_0 \rightarrow D_0)$ as $\rm \rm I\! E xt^1(E_0, E_0 \rightarrow D_0)$ and the obstructions for smoothness as living in $\rm \rm I\! E xt^2(E_0, E_0 \rightarrow D_0)$. Here we think of a sheaf as a complex of sheaves concentrated in zero and of a morphism of sheaves as a complex of sheaves concentrated in zero and one. From the short exact sequence of complexes \[ \rm 0 \rightarrow ({\cal J}_{Y_0|X_0}E_0 \rightarrow 0) \rightarrow (E_0 \rightarrow E_0|_{Y_0}) \rightarrow (E_0|_{Y_0} \rightarrow E_0|_{Y_0}) \rightarrow 0 \] we obtain the long hyperext-sequence \[ \begin{array}{lr} \rm \ldots \rightarrow \rm I\! E xt^{i-1}(E_0, E_0|_{Y_0} \rightarrow E_0|_{Y_0}) \rightarrow \rm I\! E xt^i(E_0, {\cal J}_{Y_0|X_0}E_0)) \rightarrow & \\ & \rm \hspace{-14em} \rm I\! E xt^i(E_0,E_0 \rightarrow E_0|_{Y_0}) \rightarrow \rm I\! E xt^i(E_0, E_0|_{Y_0} \rightarrow E_0|_{Y_0}) \rightarrow \ldots \end{array} \] Since $\rm Hom^{\bullet}(E_0,(E_0|_{Y_0} \rightarrow E_0|_{Y_0}))$ is the exact complex $\rm Hom(E_0,E_0|_{Y_0})\rightarrow Hom(E_0,E_0|_{Y_0})$ we have $\rm \rm I\! E xt^i(E_0, E_0|_{Y_0} \rightarrow E_0|_{Y_0}) =H^i(\rm I\! R^{+}Hom^{\bullet}(E_0,(E_0|_{Y_0} \rightarrow E_0|_{Y_0})))=0$ (cf. \cite{hartshorne2},\S 1.6). With $\rm \rm I\! E xt^i(E_0, {\cal J}_{Y_0|X_0}E_0))= Ext^i(E_0, {\cal J}_{Y_0|X_0}E_0))$ the results above correspond to the results in \cite{huy-lehn1} in the smooth projective and stable case. \section{Framed instanton bundles} \subsection{The restriction $\rm {\cal M}(P,F,{\cal O}_F^r)\rightarrow {\cal M}(S,F,{\cal O}_F^r)$} Let $\rm X_0$, $\rm Y_0$, $\rm W_0$ and $\rm {\cal M}(X_0,Y_0,W_0)$ be as in section 1.1. Then the framing data $\rm (X_0,Y_0,W_0)$ are called {\it simplifying}, if for any two representatives $\rm (E_1,\alpha_1)$, $\rm (E_2,\alpha_2)$ of elements in $\rm {\cal M}(X_0,Y_0,W_0)(\rm C\!\!\! I\,)$ we have $\rm H^0(X_0,{\cal H}\hspace{-0.1em}{\it o\hspace{-0.1em}m}(E_1,E_2)\otimes{\cal J}_{Y_0|X_0})=0$. This definition of simplifying is analogous to the definition in \cite{lehn} or \cite{lubke}. We consider now a twistor fibration $\rm \pi : P\rightarrow M$ with M compact and $\rm S\subset P$ a surface of degree 1 on the twistor fibres containing a unique twistor fibre $\rm F=S\cdot \bar{S}\subset S$ as in the introduction. Moreover we fix some integer $\rm r>0$. \begin{lemma} {The framing data $\rm (S,F,{\cal O}_F^r)$ are simplifying.} \end{lemma} {\it Proof:} Consider $\rm (E_1,\alpha_1),(E_2,\alpha_2)\in {\cal M}(S,F,{\cal O}_F^r)(\rm C\!\!\! I\,)$ and let s be in $\rm H^0(S,{\cal H}\hspace{-0.1em}{\it o\hspace{-0.1em}m}(E_1,E_2)(-F))$. If H denotes the Douady space of lines on S, then there is an open neighbourhood $\rm F\in U$ in H, such that $\rm E_1$ and $\rm E_2$ and hence $\rm {\cal H}\hspace{-0.1em}{\it o\hspace{-0.1em}m}(E_1,E_2)$ are trivial along all lines $\rm L\in U$ and moreover $\rm (F\cdot L)=1$. Since s is a global section of $\rm {\cal H}\hspace{-0.1em}{\it o\hspace{-0.1em}m}(E_1,E_2)$ vanishing along F, for all $\rm L\in U$ we have $\rm s|_L$ as global section of a trivial vector bundle over a line vanishing in one point and therefore $\rm s|_L=0$. Thus s vanishes on an open subset of S and with S irreducible we obtain, that s vanishes everywhere, which proves our claim. $\Box$ \begin{lemma} If $\rm W$ is a vector bundle on $\rm S$ with $\rm W|_F\cong {\cal O}_F^r$, then the framing data $\rm (P,S,W)$ are simplifying. \end{lemma} {\it Proof:} Consider $\rm (E_1,\alpha_1),(E_2,\alpha_2)\in {\cal M}(P,S,W)(\rm C\!\!\! I\,)$, $\rm s\in H^0(P,{\cal H}\hspace{-0.1em}{\it o\hspace{-0.1em}m}(E_1,E_2)(-S))$ and let now H denotes the Douady space of lines on P. We find an open neighbourhood $\rm F\in U$ in H, such that $\rm {\cal H}\hspace{-0.1em}{\it o\hspace{-0.1em}m}(E_1,E_2)$ is trivial along all lines $\rm L\in U$ and moreover $\rm (S\cdot L)=1$. Since s vanishes along S, it vanishes also along all L in U, therefore on an open subset of P and therefore everywhere. $\Box$ Finally we also state \begin{lemma} The framing data $\rm (P,F,{\cal O}_F^r)$ are simplifying. \end{lemma} {\it Proof: } Let $\rm (E_1,\alpha_1),(E_2,\alpha_2)\in {\cal M}(S,F,{\cal O}_F^r)(\rm C\!\!\! I\,)$ and $\rm s\in H^0(P,{\cal H}\hspace{-0.1em}{\it o\hspace{-0.1em}m}(E_1,E_2)\otimes{\cal J}_{F|P})$. Then by lemma 2.1 s vanishes along S and thus everywhere on P by lemma 2.2. $\Box$\vspace{1ex} \noindent {\it Remark:} If the framing data $\rm (X_0,Y_0,W_0)$ are simplifying, then we have for all elements $\rm (E,\alpha)$ in $\rm {\cal M}(X_0,Y_0,W_0)(\rm C\!\!\! I\,)$ in particular $\rm Aut((E,\alpha))=\{ id_E\}$. Because, for $\rm s\in Aut((E,\alpha))$ we have $\rm s|_{Y_0}=id_E|_{Y_0}$ and therefore $\rm s-id_E\in H^0(X_0,{\cal H}\hspace{-0.1em}{\it o\hspace{-0.1em}m}(E,E)\otimes{\cal J}_{Y_0|X_0})=0$. Since by \cite{poon} F intersects every effective divisor on P positively, every morphism of framed vector bundles to the framing data $\rm (P,F,{\cal O}_F^r)$ is an isomorphism and moreover there is at most one morphism between two such framed vector bundles. \indent \begin{proposition} For $\rm {\cal M}(P,F,{\cal O}_F^r)\rightarrow {\cal M}(S,F,{\cal O}_F^r)$ the natural transformation given by restriction, the fibre over a point $\rm (W,\sigma)\in {\cal M}(S,F,{\cal O}_F^r)(\rm C\!\!\! I\, )$ is naturally equivalent to $\rm {\cal M}(P,S,W)$. \end{proposition} {\it Proof:} Let X be some analytic space and $\rm V_1$ and $\rm V_2$ two vector bundles on the space $\rm X\times P$ with $\rm V_1|_{X\times F}\cong V_2|_{X\times F}\cong {\cal O}_{X\times F}^r$. The exactness of \[ \rm H^0(S,{\cal H}\hspace{-0.1em}{\it o\hspace{-0.1em}m}(V_1|_S,V_2|_S)(-X\times F)) \rightarrow Hom(V_1|_{X\times S},V_2|_{X\times S}) \rightarrow Hom(V_1|_{X\times F},V_2|_{X\times F}) \] and lemma 2.1 yield the injectivity of \[ \rm Hom(V_1|_{X\times S},V_2|_{X\times S}) \rightarrow Hom(V_1|_{X\times F},V_2|_{X\times F}), \] a fact which will be denoted with $\rm (\ast)$ in the following. For W a vector bundle on S and $\rm \sigma : W|_F \cong {\cal O}_F^r$ we consider the natural transformation $\rm {\cal M}(P,S,W)\rightarrow {\cal M}(P,F,{\cal O}_F^r)$ given by restriction of the framing map. I.e. if $\rm (V,\alpha)$ is an element in $\rm {\cal M}(P,S,W)(X)$, then $\rm (V,\alpha)$ is associated to the element $\rm (V,(p^{\ast}_F\sigma) \circ (\alpha |_{X\times F}))$, where the morphism $\rm p_F: X\times F \rightarrow F$ is the projection and $\rm p^{\ast}_F\sigma: p^{\ast}_FW\cong {\cal O}^r_{X\times F}$ is the pullback. If $\rm (V_1,\alpha_1)$ and $\rm (V_2,\alpha_2)$ are in $\rm {\cal M}(P,S,W)(X)$, such that there is an isomorphism of framed vector bundles \[ \rm \varphi : (V_1,(p^{\ast}_F\sigma) \circ (\alpha_1|_{X\times F})) \cong (V_2,(p^{\ast}_F\sigma) \circ (\alpha_2|_{X\times F})), \] i.e. with $\rm (p^{\ast}_F\sigma) \circ (\alpha_2|_{X\times F})\circ (\varphi |_{X\times F}) = (p^{\ast}_F\sigma) \circ (\alpha_1|_{X\times F})$, then $\rm (\ast)$ implies, that $\rm \varphi |_{X\times S}= \alpha_2^{-1}\circ \alpha_1$ and therefore $\rm (V_1,\alpha_1)\cong (V_2,\alpha_2)$. Thus $\rm {\cal M}(P,S,W)(X) \rightarrow {\cal M}(P,F,{\cal O}_F^r)(X)$ is injective. Now we consider an element $\rm (V,\alpha )\in {\cal M}(P,F,{\cal O}_F^r)(X)$, such that there is an isomorphism of framed vector bundles $\rm \varphi : (V|_{X\times S},\alpha )\cong (p^{\ast}_SW, p^{\ast}_F\sigma )$ with $\rm p_S: X\times S \rightarrow S$ the projection. With $\rm (\ast)$ and with $\rm (p^{\ast}_F\sigma)\circ (\varphi |_{X\times F})=\alpha$, $\rm \varphi$ is well defined by $\rm (p^{\ast}_F\sigma)^{-1}\circ \alpha \in Hom(V|_{X\times F},p^{\ast}_FW)$ and hence $\rm (V,\alpha )$ uniquely determines an element $\rm (V,\varphi ) \in {\cal M}(P,S,W)(X)$. Conversely, the image of $\rm (V,\varphi )$ in $\rm {\cal M}(P,F,{\cal O}_F^r)(X)$ is again $\rm (V,\alpha )$, hence the natural transformation $\rm {\cal M}(P,S,W)\rightarrow {\cal M}(P,F,{\cal O}_F^r)$ induces a bijection between $\rm {\cal M}(P,S,W)(X)$ and the fibre of $\rm {\cal M}(P,F,{\cal O}_F^r)(X)\rightarrow {\cal M}(S,F,{\cal O}_F^r)(X)$ over $\rm (p^{\ast}_SW,p^{\ast}_F\sigma)$. These bijections correspond actually to a natural equivalence between $\rm {\cal M}(P,S,W)$ and the fibre of $\rm {\cal M}(P,F,{\cal O}_F^r)\rightarrow {\cal M}(S,F,{\cal O}_F^r)$ over $\rm (W,\sigma)$. So far everything was natural except the choice of $\rm (W,\sigma)$ as a representative of an element in $\rm {\cal M}(S,F,{\cal O}_F^r)(\rm C\!\!\! I\, )$. If $\rm (W',\sigma')$ is another representative of the same isomorphism class, then there is an isomorphism $\rm \psi : W' \rightarrow W$ with $\rm \sigma \circ \psi |_F = \sigma$. Because of $\rm (\ast)$ this isomorphism is unique and therefore there is a unique natural equivalence from $\rm {\cal M}(P,S,W')$ to $\rm {\cal M}(P,S,W)$, which moreover maps the fibre of $\rm {\cal M}(P,F,{\cal O}_F^r)\rightarrow {\cal M}(S,F,{\cal O}_F^r)$ over $\rm (W,\sigma)$ identically to itself. $\Box$ Since Chern classes are locally constant, the functors of families of framed vector bundles $\rm {\cal M}(P,F,{\cal O}_F^r)$, $\rm \:{\cal M}(S,F,{\cal O}_F^r)$ and $\rm \,{\cal M}(P,S,W)$ as above split into open and closed subfunctors $\rm {\cal M}(P,F,{\cal O}_F^r,c_{\bullet})$, $\rm {\cal M}(S,F,{\cal O}_F^r,c_{\bullet})$ and $\rm {\cal M}(P,S,W,c_{\bullet})$, where $\rm c_{\bullet}$ are fixed Chern classes in $\rm H^{\bullet}(P,\rm Z\!\! Z)$ and $\rm H^{\bullet}(S,\rm Z\!\! Z)$, respectively. In particular, our restriction $\rm {\cal M}(P,F,{\cal O}_F^r)\rightarrow {\cal M}(S,F,{\cal O}_F^r)$ splits into open and closed parts \[ \rm {\cal M}(P,F,{\cal O}_F^r,c_{\bullet})\rightarrow {\cal M}(S,F,{\cal O}_F^r,i^{\ast}c_{\bullet})\] with $\rm c_{\bullet}\in H^{\bullet}(P,\rm Z\!\! Z)$ and $\rm i: S \hookrightarrow P$. It is easy to see, that the fibre of $\rm {\cal M}(P,F,{\cal O}_F^r)\rightarrow {\cal M}(S,F,{\cal O}_F^r)$ over an $\rm (W,\sigma)\in {\cal M}(S,F,{\cal O}_F^r,i^{\ast}c_{\bullet})(\rm C\!\!\! I\,)$ is just $\rm {\cal M}(P,S,W,c_{\bullet})$. Later on, when we restrict ourself to the case of framed instanton bundles, we only have to consider Cern classes coming from $\rm H^{\bullet}(M,\rm Z\!\! Z)$, i.e. we will consider \[ \rm {\cal M}(P,F,{\cal O}_F^r,\pi^{\ast}c_{\bullet})\rightarrow {\cal M}(S,F,{\cal O}_F^r,\pi^{\ast}_Sc_{\bullet}), \] where $\rm c_{\bullet}\in H^{\bullet}(M,\rm Z\!\! Z)$ and $\rm \pi_S$ is the restriction of the twistor fibration to S. The image of our natural transformation has a fine moduli space by the following proposition: \begin{proposition} The functor $\rm {\cal M}(S,F,{\cal O}_F^r,c_{\bullet})$ is represented by a separated algebraic space of finite type over $\rm C\!\!\! I\,$. \end{proposition} {\it Proof:} This is a corollary of Satz 3.4.1 in Lehn's \cite{lehn}. By \cite{kurke1}, Prop. 2.1, S is a smooth projective surface and by Lemma 2.1 the framing data $\rm (S,F,{\cal O}_F^r)$ are simplifying. Therefore it is sufficient to show, that F is the support of a divisor, which is big and nef. Since the linear system $\rm |F|$ defines by \cite{kurke1} a blowing up $\rm S\rightarrow \rm I\! P^2_{\rm C\!\!\! I\,}$, F intersects all lines on S coming from $\rm I\! P^2_{\rm C\!\!\! I\,}$ with multiplicity 1 and all exceptional lines with multiplicity 0, which implies that F is numerically effective. The big-condition follows by $\rm {\cal N}_{F|S}\cong {\cal O}_F(1)$. $\Box$ \subsection{Smoothness and dimensions} By H. Kurke's article \cite{kurke1}, Prop.2.1, the canonical bundle of P is $\rm K_P={\cal O}_P(-2S-2\bar{S})$. Moreover, $\rm H^{\bullet}(P,\rm Z\!\! Z)$ is generated by elements $\rm \{ \omega,\eta_1,\ldots ,\eta_n\}\in H^2(P,\rm Z\!\! Z)$, where $\rm \{ \eta_1,\ldots ,\eta_n\}$ are an orthogonal basis of $\rm H^2(M,\rm Z\!\! Z)$. We put $\rm \eta = \sum \eta_i$. There are the relations $\rm \eta_i^2=-[F]$, $\rm \eta_i^3=0$, $\rm \eta_i\eta_j=0$ for $\rm i\not= j$, $\rm \omega^2 + \omega\eta = [F]$, $\rm \omega^2 \eta_i =1$, $\rm \omega \eta_i^2 = -1$, $\rm c_1(P)=4\omega + 2\eta$ and $\rm c_2(P)=3(e(M)-sgn(M))[F]$. We have $\rm [S]=\omega + \sum a_i\eta_i$ with $\rm a_i=0\; or\; 1$ and from $\rm (S\cdot \bar{S})=F$ we easily infer $\rm [\bar{S}]=\omega + \sum (1-a_i)\eta_i$. We put $\rm \sigma = \sum a_i\eta_i$ and $\rm \bar{\sigma}= \sum (1-a_i)\eta_i $. We fix a Chern class $\rm c_{\bullet}\in H^{\bullet}(M,\rm Z\!\! Z)$ and consider a vector bundle V on P of rank r with $\rm c_{\bullet}(V)=\pi^{\ast}c_{\bullet}$. \begin{lemma} We have the equality of Euler characteristics: $\rm \chi({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S))= \chi ({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-\bar{S}))$. \end{lemma} {\it Proof:} By \cite{hirzebruch}, Theorem 4.4.3 we have \[ \rm c_1({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} V)=c_3({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} V)=0 \; and\; c_2({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} V)=2rc_2(V)+(1-r)c_1^2(V).\] Since the Chern classes of V are coming from $\rm H^{\bullet}(M,\rm Z\!\! Z)$, we have $\rm c_2({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} V)$ as a multiple of $\rm [F]$. From the same theorem we obtain by computation that \[ \begin{array}{l} \rm c_1({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S))=r(\omega + \sigma),\\ \rm c_2({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S))=c_2({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} V)+ \displaystyle{r \choose 2}(\omega^2 + 2\omega\sigma +\sigma^2),\\ \rm c_3({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S))=(r-2)c_2({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} V)(\omega + \sigma )+ \displaystyle{r \choose 3} (\omega^3 + 3\omega^2\sigma + 3\omega\sigma^2),\\ \rm c_1({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-\bar{S}))=r(\omega + \bar{\sigma}),\\ \rm c_2({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-\bar{S}))=c_2({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} V)+ \displaystyle{r \choose 2} (\omega^2 + 2\omega\bar{\sigma} +\bar{\sigma}^2),\\ \rm c_3({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-\bar{S}))=(r-2)c_2({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} V)(\omega + \bar{\sigma} )+ \displaystyle{r \choose 3} (\omega^3 + 3\omega^2\bar{\sigma} + 3\omega\bar{\sigma}^2),\\. \end{array} \] The Hirzebruch-Riemann-Roch theorem yields \[ \begin{array}{ll} \rm \chi({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S))= & \rm \displaystyle \frac{r}{24}c_1(P)c_2(P) +\frac{r\omega + r\sigma}{12}(16\omega^2+4\eta^2 +16 \omega\eta +3(e(M)-sgn(M))[F]) \\[2ex] & \rm \displaystyle +\frac{2\omega + \eta}{2}\left( r^2\omega^2 + r^2\sigma^2 +2r^2\omega\sigma -2c_2({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} V)- (r^2-r)(\omega^2 + \sigma^2 +2\omega\sigma)\right) \\[2ex] & \rm \displaystyle +\frac{r^3}{6}(\omega^3 + 3\omega^2\sigma + 3\omega\sigma^2) -\frac{r\omega+r\sigma}{2}(c_2({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} V) + \frac{r^2-r}{2}(\omega^2 + \sigma^2 + 2\omega \sigma)) \\[2ex] & \rm \displaystyle +\frac{r-2}{6}c_2({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} V)(\omega + \sigma) +\frac{r^3-3r^2+2r}{36}(\omega^3 + 3\omega^2 \sigma + 3\omega\sigma^2) \end{array} \] and the analogous formula for $\rm \chi({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-\bar{S}))$, where just all $\sigma$'s are replaced by $\bar{\sigma}$. Thus we obtain \[ \begin{array}{lr} \rm \chi({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S))- \chi ({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-\bar{S})) = &\\ &\rm\hspace{-12em} \omega(\sigma^2 - \bar{\sigma}^2) (\frac{7}{6}r+\frac{1}{2}r^2-\frac{1}{6}r^3) +\omega^2(\sigma - \bar{\sigma}) (\frac{13}{6}r+\frac{1}{2}r^2-\frac{1}{6}r^3) +\omega\eta(\sigma - \bar{\sigma})r , \end{array} \] where we have used \[ \rm (\sigma - \bar{\sigma})\eta^2=(\sigma^2 - \bar{\sigma}^2)\eta =(\sigma - \bar{\sigma})[F]=(\sigma - \bar{\sigma})c_2({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} V) =(\sigma - \bar{\sigma})(\omega^2+\omega\eta)=0 .\] We had $\rm \sigma = \sum a_i\eta_i$ and $\rm \bar{\sigma} = \sum (1-a_i)\eta_i$ with $\rm a_i=0,1$ and put $\rm A=\sum a_i$. Thus $\rm \omega(\sigma^2 - \bar{\sigma}^2)=n-2A$ and $\rm \omega^2(\sigma - \bar{\sigma})=2A-n$. With $\rm \eta\sigma=\sigma^2$ and $\rm \eta\bar{\sigma}=\bar{\sigma}^2$ we also have $\rm \omega\eta(\sigma - \bar{\sigma})=\omega(\sigma^2 - \bar{\sigma}^2) =n-2A$. Inserting all this in the equation above we obtain \[ \rm \chi({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S))- \chi ({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-\bar{S})) = 0,\; \Box\] Due to Hitchin's article \cite{hitchin} we have for all $\rm U(r)-$instanton bundles V on P the vanishing $\rm H^1(P,V(-S-\bar{S}))=0$. If V is an $\rm U(r)-$instanton bundle, then so does $\rm {\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} V$ and we have also $\rm H^1(P,{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S-\bar{S}))=0$. We denote with $\rm {\cal M}_0(P,F,{\cal O}_F^r)$ the open subfunctor of $\rm {\cal M}(P,F,{\cal O}_F^r)$ with \[ \rm {\cal M}_0(P,F,{\cal O}_F^r)(\rm C\!\!\! I\,)=\{ (V,\alpha)\in {\cal M}(P,F,{\cal O}_F^r)(\rm C\!\!\! I\,) | \, H^1(P,{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S-\bar{S}))=0 \}. \] Analogously to our notations in Section 2.1 we put \[ \rm {\cal M}_0(P,F,{\cal O}_F^r,c_{\bullet}) ={\cal M}_0(P,F,{\cal O}_F^r)\cap {\cal M}(P,F,{\cal O}_F^r,c_{\bullet}). \] \begin{lemma} \begin{tabular}[t]{l} Consider $\rm c_{\bullet}\in H^{\bullet}(M,\rm Z\!\! Z)$ and $\rm (V,\alpha)\in {\cal M}_0(P,F,{\cal O}_F^r,\pi^{\ast}c_{\bullet})(\rm C\!\!\! I\,)$.\\[1ex] Then we have $ \rm dim_{\rm C\!\!\! I\,}H^i(P,{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S))=\left\{ \begin{array}{ll} \rm -\chi({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d}(V)(-S))& \rm for\; i=1\\ \rm 0 & \rm else. \end{array} \right. $ \end{tabular} \end{lemma} {\it Proof:} By Lemma 2.3 we have \[ \rm H^0(P,{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S))=0.\] With Serre duality and with $\rm K_P={\cal O}_P(-2S-2\bar{S})$ we obtain \[ \rm H^3(P,{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S))^{\vee}= H^0(P,{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S-2\bar{S}))\subset H^0(P,{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S)) =0 . \] For $\rm \bar{E}$ be the exceptional divisor of the blowing up $\rm |F|: \bar{S}\rightarrow \rm I\! P^2$ we have $\rm K_{\bar{S}}={\cal O}_{\bar{S}}(-3F+\bar{E})$. With $\rm K_{\bar{S}}={\cal O}_{\bar{S}}\otimes {\cal O}_P(\bar{S})\otimes K_P$ we infer the short exact sequence \[ \rm 0 \rightarrow {\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S-2\bar{S}) \rightarrow {\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S-\bar{S}) \rightarrow {\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V|_{\bar{S}})(-2F+\bar{E}) \rightarrow 0. \] Since the bundle $\rm {\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V|_{\bar{S}})$ is trivial on general lines of $\rm \bar{S}$ and $\rm \bar{E}$ is exceptional, also $\rm {\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V|_{\bar{S}})(\bar{E})$ is trivial on general lines and analogously to the proof of Lemma 2.1 we obtain the vanishing of $\rm H^0(\bar{S},{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V|_{\bar{S}})(-2F+\bar{E}))$. Hence, the cohomology sequence of the last short exact sequence implies an inclusion $\rm H^1(P, {\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S-2\bar{S}))\hookrightarrow H^1(P,{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S-\bar{S}))$ and our presumption gives us the vanishing $\rm H^1(P, {\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S-2\bar{S}))=0 $. Then Serre duality yields \[ \rm H^2(P, {\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S))^{\vee}=H^1(P, {\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S-2\bar{S}))=0.\; \Box \] By Theorem 1.4 the last lemma gives us: \begin{corollary} Around every closed point $\rm (V,\alpha)\in {\cal M}_0(P,F,{\cal O}_F^r,\pi^{\ast}c_{\bullet})(\rm C\!\!\! I\,)$, the fibres of the natural transformation \[ \rm {\cal M}_0(P,F,{\cal O}_F^r,\pi^{\ast}c_{\bullet}) \rightarrow {\cal M}(S,F,{\cal O}_F^r,\pi_S^{\ast}c_{\bullet}) \] are smooth and of constant dimension $\rm -\chi({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d}(V)(-S))$. \end{corollary} \begin{theorem} The natural transformation \[ \rm {\cal M}_0(P,F,{\cal O}_F^r,\pi^{\ast}c_{\bullet}) \rightarrow {\cal M}(S,F,{\cal O}_F^r,\pi_S^{\ast}c_{\bullet}) \] is smooth. More detailed we have the tangent space of the functor $\rm {\cal M}_0(P,F,{\cal O}_F^r,\pi^{\ast}c_{\bullet})$ in a point $\rm (V,\alpha)\in {\cal M}_0(P,F,{\cal O}_F^r,\pi^{\ast}c_{\bullet})(\rm C\!\!\! I\,)$ as a natural direct sum of the tangent space in direction of the fibre and the tangent space in direction of the image, i.e. \[ \rm T_{(V,\alpha)}{\cal M}_0(P,F,{\cal O}_F^r,\pi^{\ast}c_{\bullet})= T_{(V,id_{V|_S})}{\cal M}(P,S,V|_S,\pi^{\ast}c_{\bullet}) \oplus T_{(V|_S,\alpha)}{\cal M}(S,F,{\cal O}_F^r,\pi_S^{\ast}c_{\bullet}) .\] Both summands are of dimension $\rm -\chi({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d}(V)(-S))$. $\rm {\cal M}_0(P,F,{\cal O}_F^r,\pi^{\ast}c_{\bullet})$ and the fibres over $\rm {\cal M}(S,F,{\cal O}_F^r,\pi_S^{\ast}c_{\bullet})$ are smooth and equidimensional. The image of the natural transformation in $\rm {\cal M}(S,F,{\cal O}_F^r,\pi_S^{\ast}c_{\bullet})$ is an open and smooth algebraic subspace. \end{theorem} {\it Proof:} The diagram $\rm (\ast)$ \[ \begin{array}{ccccccccc} &&&&0&&0&&\\ &&&&\big\downarrow&&\big\downarrow&&\\ &&&&\rm {\cal O}_P(-S)&\rm =&\rm {\cal O}_P(-S)&&\\ &&&&\big\downarrow&&\big\downarrow&&\\ 0&\longrightarrow&\rm {\cal O}_P(-S-\bar{S})&\longrightarrow& \rm {\cal O}_P(-S)\oplus{\cal O}_P(-\bar{S})&\longrightarrow& \rm {\cal J}_{F|P}&\longrightarrow&0\\ &&\big\downarrow =&&\big\downarrow&&\big\downarrow&&\\ 0&\longrightarrow&\rm {\cal O}_P(-S-\bar{S})&\longrightarrow& \rm {\cal O}_P(-\bar{S})&\longrightarrow& \rm {\cal O}_S(-F)&\longrightarrow&0\\ &&&&\big\downarrow&&\big\downarrow&&\\ &&&&0&&0&& \end{array} \] is commutative with exact lines and columns. For $\rm (V,\alpha)\in {\cal M}_0(P,F,{\cal O}_F^r,\pi^{\ast}c_{\bullet})(\rm C\!\!\! I\,)$ and because of the vanishing of the neighboured cohomology groups we obtain \[ \begin{array}{ccc} 0&&0\\ \downarrow & & \downarrow \\ \rm H^1(P,{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S))&\stackrel{=}{\longrightarrow}&\rm H^1(P, {\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S))\\ \downarrow & & \downarrow \\ \begin{array}{c} \rm H^1(P,{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S))\\[-1ex] \oplus\\[-1ex] \rm H^1(P,{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-\bar{S})) \end{array}& \stackrel{\cong}{\longrightarrow}&\rm H^1(P, {\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)\otimes {\cal J}_{F|P})\\ \downarrow & & \downarrow \\ \rm H^1(P,{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-\bar{S}))&\stackrel{\cong}{\longrightarrow}& \rm H^1(S,{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V|_S)(-F))\\ \downarrow & & \downarrow \\ 0&&0 \end{array} \] as a commutative diagram with exact columns. The second column contains the tangent maps at $\rm (V,\alpha)$ corresponding to the composition \[ \rm {\cal M}(P,S,V|_S,\pi^{\ast}c_{\bullet}) \hookrightarrow {\cal M}_0(P,F,{\cal O}_F^r,\pi^{\ast}c_{\bullet}) \rightarrow {\cal M}(S,F,{\cal O}_F^r,\pi_S^{\ast}c_{\bullet}) \] Therefore the natural transformation \[ \rm {\cal M}_0(P,F,{\cal O}_F^r,\pi^{\ast}c_{\bullet}) \rightarrow {\cal M}(S,F,{\cal O}_F^r,\pi_S^{\ast}c_{\bullet}) \] is submersive and the tangent space of $\rm {\cal M}_0(P,F,{\cal O}_F^r,\pi^{\ast}c_{\bullet})$ in $\rm (V,\alpha)$ is a natural direct sum of the tangent spaces in direction of the fibre and in direction of the image. With Lemma 2.6 we have \[ \rm h^1\!({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S))=-\chi ({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S)) =-\chi ({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-\bar{S})) =h^1\!({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-\bar{S}))=h^1({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V|_S)(-F)), \] whence both direct summands are of the same dimension $\rm -\chi ({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S))$. From $\rm (\ast)$ we also obtain \[ \rm H^2(P, {\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)\otimes {\cal J}_{F|P})\cong H^2(S,{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V|_S)(-F)) \] and \[ \rm H^2(S,{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V|_S)(-F))\cong H^3(P, {\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S-\bar{S})). \] We have \[ \rm H^3(P, {\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S-\bar{S}))^{\vee} = H^0(P, {\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S-\bar{S})) \subset H^0(P, {\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S-\bar{S}))=0 \] and hence $\rm H^2(S,{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V|_S)(-F))=0 $ and $\rm H^2(P, {\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)\otimes {\cal J}_{F|P})=0$, which shows the smoothness by Theorem 1.4. $\Box$ \subsection{ Framed U(r)-instanton bundles} \begin{lemma} The property of a vector bundle on $\rm P$ to be a mathematical instanton bundle is open. \end{lemma} {\it Proof:} Let T be an analytical space and V a vector bundle over $\rm T\times P$. For $\rm t\in T$ we denote with $\rm V_t$ the vector bundle on P induced by $\rm V|_{\{ t \} \times P}$. We assume that for a point $\rm 0\in T$ the vector bundle $\rm V_0$ is an instanton bundle, i.e. trivial along twistor fibres. We have to show that there is a neighbourhood $\rm 0\in T_1 \subset T$, such that for all $\rm t\in T_1$ the vector bundle $\rm V_t$ is an instanton bundle. We consider the universal family \[ \begin{array}{ccc} \rm Z & \stackrel{\nu}{\longrightarrow} & \rm P \\ \rm \big\downarrow \scriptstyle\mu &&\\ \rm H && \end{array} \] of lines on P as in the introduction. We may restrict H to the open neighbourhood of M consisting of points, where the corresponding lines have intersection product one with S. We put $\rm \bar{\nu}=id_T \times \nu$, \linebreak[4] $\rm \bar{\mu}=id_T \times\mu$ and $\rm W=\bar{\nu}^{\ast} (V\otimes {\cal O}_{T\times P}(-\, T\times S))$. Recall that $\mu$ is a proper and flat morphism with projective lines as fibres. There are an covering $\rm \{ U^{\lambda}_0|\, \lambda \in \; some\; index\; set \}$ of H of sufficiently small open neighbourhoods and effective divisors $\rm D^{\lambda}\subset\bar{\mu}^{-1} (T\times U^{\lambda}_0)=U^{\lambda}$, such that $\rm R^i(\bar{\mu}|_{U^{\lambda}})_{\ast}(W|_{U^{\lambda}}(D^{\lambda}))=0$ for all $\rm i>0$ and all $\rm \lambda$. We can choose $\rm D^{\lambda}$ of the form $\rm T\times D_0^{\lambda}$, where $\rm D_0^{\lambda}$ is a sufficiently high multiple of a section over $\rm U_0^{\lambda}$ meeting the fibres of $\rm \mu$ transversally. Since M is compact we may assume that $\rm \{ U^{\lambda}_0|\, \lambda =1,\ldots ,n\}$ already covers M. The short exact sequence \[ \rm 0\rightarrow W|_{U^{\lambda}} \rightarrow W|_{U^{\lambda}}(D^{\lambda}) \rightarrow W|_{U^{\lambda}}(D^{\lambda})\otimes {\cal O}_{D^{\lambda}} \rightarrow 0 \] yields the exact sequence \[ \rm 0\rightarrow (\bar{\mu}|_{U^{\lambda}})_{\ast}W|_{U^{\lambda}} \rightarrow (\bar{\mu}|_{U^{\lambda}})_{\ast}W|_{U^{\lambda}}(D^{\lambda}) \stackrel{u^{\lambda}}{\longrightarrow} (\bar{\mu}|_{U^{\lambda}})_{\ast}W|_{U^{\lambda}}(D^{\lambda}) \otimes {\cal O}_{D^{\lambda}} \rightarrow R^1(\bar{\mu}|_{U^{\lambda}})_{\ast}W|_{U^{\lambda}} \rightarrow 0 .\] In the following h denotes a point on H as well as the corresponding line on P. If $\rm V_t$ is trivial along h, then $\rm h\cong \rm I\! P^1_{\rm C\!\!\! I\,}$ gives $\rm W|_{\bar{\mu}^{-1}(t,h)}\cong {\cal O}_{\rm I\! P^1_{\rm C\!\!\! I\,}}(-1)$, since we had assumed $\rm (h\cdot S)=1$. Since $\rm V_0$ is trivial along twistor fibres and since the triviality of a vector bundle on $\rm \rm I\! P^1_{\rm C\!\!\! I\,}$ is an open property, there is an open neighbourhood $\rm \{ 0 \} \times M \subset B \subset T\times H$, such that for all $\rm (t,h)\in B$ the vector bundle $\rm V_t|_h$ is trivial. Inparticular we have for all $\rm (t,h)\in B$ \[ \rm H^0(\bar{\mu}^{-1}(t,h),W|_{\bar{\mu}^{-1}(t,h)})= H^1(\bar{\mu}^{-1}(t,h),W|_{\bar{\mu}^{-1}(t,h)})=0. \] Since $\rm \bar{\mu}$ is flat, W is flat over $\rm T\times H$ and base change implies the vanishing of $\rm (\bar{\mu}|_{U^{\lambda}})_{\ast}W|_{U^{\lambda}}$ and of $\rm R^1(\bar{\mu}|_{U^{\lambda}})_{\ast}W|_{U^{\lambda}}$ over $\rm B\cap (T\times U^{\lambda}_0)$. Therefore $\rm u^{\lambda}$ is an isomorphism over $\rm B\cap (T\times U^{\lambda}_0)$ and hence $\rm det(u^{\lambda})$ is a not constantly zero section of \[ \rm det\! \left( ((\bar{\mu}|_{U^{\lambda}})_{\ast}\!W|_{U^{\lambda}}(D^{\lambda}))^{\vee}\! \otimes ((\bar{\mu}|_{U^{\lambda}})_{\ast}\!W|_{U^{\lambda}}(D^{\lambda}) \otimes {\cal O}_{D^{\lambda}}) \right) \cong det\! \left( ((\bar{\mu}|_{U^{\lambda}})_{\ast}\!W|_{U^{\lambda}})^{\vee}\! \otimes R^1(\bar{\mu}|_{U^{\lambda}})_{\ast}\!W|_{U^{\lambda}} \right). \] Moreover, all points $\rm (t,h)$ with $\rm V_t|_h$ non-trivial belong to the support of $\rm \Delta^{\lambda}=div(det\, u^{\lambda})$. For $\rm \lambda = 1,\ldots ,n\,$ there are open neighbourhoods $\rm 0\in T_0^{\lambda}\subset T$, such that $\rm \Delta^{\lambda}_t = \Delta^{\lambda}|_{\{ t\} \times U_0^{\lambda}}$ are divisors on $\rm U_0^{\lambda}$ for all $\rm t\in T^{\lambda}_0$. Thus $\rm \Delta^{\lambda}\subset T^{\lambda}_0 \times U_0^{\lambda}$ are analytic families of divisors on $\rm U_0^{\lambda}$ for all $\rm \lambda = 1,\ldots ,n$. Since $\rm V_0$ is an instanton bundle we have $\rm \Delta^{\lambda}_0\subset U_0^{\lambda}-M$. M is a closed subset of H and thus there exists open neighbourhoods $\rm 0\in T_1^{\lambda}\subset T_0^{\lambda}$, such that for all $\rm t\in T^{\lambda}_1$ we have $\rm \Delta^{\lambda}_t\subset U_0^{\lambda}-M$. We put $\rm T_1 = \bigcap_{\lambda = 1}^{n}T_1^{\lambda}$ and obtain that for all $\rm t\in T_1$ and for all $\rm h\in M$ the vector bundle $\rm V_t|_h$ is trivial. $\Box$ Warning: Contrarily to the most other parts of this paper, the last lemma does not hold in the algebraic setup, since M has not to be Zariski-closed. \noindent {\it Remark.} Let $\rm {\cal M}_{\cal I}(P,F,{\cal O}_F^r)$ be the functor of families of framed mathematical instanton bundles. By the previous Lemma we have have $\rm {\cal M}_{\cal I}(P,F,{\cal O}_F^r) \subset {\cal M}(P,F,{\cal O}_F^r)$ as an analytically open subfunctor. In particular, \[ \rm {\cal M}_1(P,F,{\cal O}_F^r,\pi^{\ast}c_{\bullet}) ={\cal M}_{\cal I}(P,F,{\cal O}_F^r) \cap {\cal M}_0(P,F,{\cal O}_F^r,\pi^{\ast}c_{\bullet}) \] is an analytically open subfunctor and the natural transformation \[ \rm {\cal M}_1(P,F,{\cal O}_F^r,\pi^{\ast}c_{\bullet}) \rightarrow {\cal M} (S,F,{\cal O}_F^r,\pi_S^{\ast}c_{\bullet}) \] is smooth of relative dimension $\rm -\chi({\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S))$ with all the details as in Theorem 2.9. \indent For T an algebraic space and $\rm (V,\alpha)\in {\cal M}_{\cal I}(P,F,{\cal O}_F^r)(T)$ we define $\rm \sigma(V,\alpha)$ to be the family of framed instanton bundles $\rm ((id_T\times \tau)^{\ast}\bar{V}^{\vee}, (\tau^{\ast}\bar{\alpha}^{\vee})^{-1})$, where $\tau$ is the antiholomorphic involution on P as in the introduction and the "bar" means complex conjugation. Thus $\sigma$ defines a real structure on $\rm {\cal M}_{\cal I}(P,F,{\cal O}_F^r)$ and the closed fixpoints $\rm {\cal M}_{\cal I}(P,F,{\cal O}_F^r)(\rm I\! R)$ is just the set of isomorphism classes of framed $\rm U(r)-instanton$ bundles on P. To be more specific, we define $\rm {\cal M}_{\cal I}^{\rm I\! R}(P,F,{\cal O}_F^r)$ as the functor that associates to an algebraic space T the set of isomorphism classes of families of framed instanton bundles $\rm (V,\alpha)$ on P parametrized by T, which have the additional property that there exists an isomorphism of framed vector bundles \[ \rm \varphi : (V,\alpha) \rightarrow ((id_T\times \tau)^{\ast}\bar{V}^{\vee}, (\tau^{\ast}\bar{\alpha}^{\vee})^{-1}) .\] Then $\sigma$ defines $\rm {\cal M}_{\cal I}^{\rm I\! R}(P,F,{\cal O}_F^r)$ as closed real-analytic subfunctor of $\rm {\cal M}_{\cal I}(P,F,{\cal O}_F^r)$. We have $\rm {\cal M}_{\cal I}^{\rm I\! R}(P,F,{\cal O}_F^r)$ embedded in $\rm {\cal M}_1(P,F,{\cal O}_F^r)$ and we can examine the restriction of the natural transformation described in Section 2.2 to $\rm {\cal M}_{\cal I}^{\rm I\! R}(P,F,{\cal O}_F^r)$. \begin{theorem} The natural transformation \[ \rm {\cal M}_{\cal I}^{\rm I\! R}(P,F,{\cal O}_F^r) \rightarrow {\cal M}(S,F,{\cal O}_F^r) \] given by restriction is an open real-analytic embedding. \end{theorem} {\it Proof:} We have to show that the restriction is injective for closed points and the associated tangent maps are isomorphisms. We denote $\rm \tau |_{S}$ and $\rm \tau |_{\bar{S}}$ again with $\tau$ and obtain a real structure $\sigma'$ on $\rm {\cal M}(S,F,{\cal O}_F^r)\times {\cal M}(\bar{S},F,{\cal O}_F^r)$ by mapping \[ \rm \left( (V,\alpha),(W,\beta) \right) \in {\cal M}(S,F,{\cal O}_F^r)(T)\times {\cal M}(\bar{S},F,{\cal O}_F^r)(T)\] to \[ \rm \left( ((id_T\times \tau)^{\ast}\bar{W}^{\vee}, (\tau^{\ast}\bar{\beta}^{\vee})^{-1}), ((id_T\times \tau)^{\ast}\bar{V}^{\vee}, (\tau^{\ast}\bar{\alpha}^{\vee})^{-1}) \right).\] Hence the natural transformation \[ \rm \Psi : {\cal M}_{\cal I}(P,F,{\cal O}_F^r) \rightarrow {\cal M}(S,F,{\cal O}_F^r)\times {\cal M}(\bar{S},F,{\cal O}_F^r) \] given by restriction to both factors is real, i.e. compatible with both real structures $\sigma$ and $\sigma'$. From the diagram $\rm (\ast)$ in the proof of Theorem 2.9 we obtained for $\rm (V,\alpha) \in {\cal M}_0 (P,F,{\cal O}_F^r)(\rm C\!\!\! I\,)$ the commutative diagram \[ \begin{array}{ccc} 0&&0\\ \downarrow & & \downarrow \\ \rm H^1(P,{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S))&\stackrel{=}{\longrightarrow}&\rm H^1(P, {\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S))\\ \downarrow & & \downarrow \\ \begin{array}{c} \rm H^1(P,{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S))\\[-1ex] \oplus\\[-1ex] \rm H^1(P,{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-\bar{S})) \end{array}& \stackrel{\cong}{\longrightarrow}&\rm H^1(P, {\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)\otimes {\cal J}_{F|P})\\ \downarrow & & \downarrow \\ \rm H^1(P,{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-\bar{S}))&\stackrel{\cong}{\longrightarrow}& \rm H^1(S,{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V|_S)(-F))\\ \downarrow & & \downarrow \\ 0&&0 \end{array} \] with exact columns. We may interchange $\rm S$ and $\rm \bar{S}$ and make use of the isomorphism \[ \rm H^1(P,{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)(-S)) \stackrel{\cong}{\longrightarrow} H^1(\bar{S},{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V|_{\bar{S}})(-F)) \] to obtain a natural splitting \[ \rm H^1(P, {\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V)\otimes {\cal J}_{F|P}) = H^1(S,{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V|_S)(-F)) \oplus H^1(\bar{S},{\cal E}\hspace{-0.1em}{\it n\hspace{-0.1em}d} (V|_{\bar{S}})(-F)), \] which is just the tangent map of $\rm \Psi$ at $\rm (V,\alpha)$. Now we assume that $\rm (V,\alpha)$ and $\rm (W,\beta)$ are two elements in $\rm {\cal M}_{\cal I}(P,F,{\cal O}_F^r)(\rm I\! R)$, such that $\rm \Psi(V,\alpha)\cong \Psi(W,\beta)$. This means that there are two isomorphisms of framed vector bundles $\rm \varphi_S : (V|_S,\alpha)\rightarrow (W|_S,\beta)$ and $\rm \varphi_{\bar{S}} : (V|_{\bar{S}},\alpha)\rightarrow (W|_{\bar{S}},\beta)$, which implies that $\rm \varphi_S|_F = \beta^{-1}\circ \alpha = \varphi_{\bar{S}}|_F$. Thus we have an isomorphism of framed vector bundles $\rm \varphi_{S\cup \bar{S}} : (V|_{S\cup \bar{S}},\alpha)\rightarrow (W|_{S\cup \bar{S}},\beta)$. With V and W U(r)-instanton bundles also $\rm {\cal H}\hspace{-0.1em}{\it o\hspace{-0.1em}m} (V,W)$ is an U(r)-instanton bundle and in particular $\rm H^1(P,{\cal H}\hspace{-0.1em}{\it o\hspace{-0.1em}m} (V,W)(-S-\bar{S}))=0$. Therefore the restriction map $\rm Hom(V,W)\rightarrow Hom(V|_{S\cup \bar{S}},W|_{S\cup \bar{S}})$ is surjective and we can find a morphism of framed vector bundles $\rm \varphi : (V,\alpha) \rightarrow (W,\beta)$ as extension of $\rm \varphi_{S\cup \bar{S}}$. By Lemma 2.3 $\rm \varphi$ is an isomorphism and hence $\Psi$ injective along $\rm {\cal M}_{\cal I}(P,F,{\cal O}^r_F)(\rm I\! R)$. Therefore $\rm \Psi$ is an open embedding in an open neighbourhood of $\rm {\cal M}_{\cal I}^{\rm I\! R}(P,F,{\cal O}^r_F)$ and hence we have a real-analytic open embedding \[ \rm {\cal M}_{\cal I}^{\rm I\! R}(P,F,{\cal O}^r_F) \rightarrow {\cal N}^{\rm I\! R}, \] where $\rm {\cal N}^{\rm I\! R}$ denotes the closed real-analytic subfunctor of $\rm {\cal M}(S,F,{\cal O}_F^r)\times {\cal M}(\bar{S},F,{\cal O}_F^r)$ defined by the real structure $\rm \sigma'$. Now we consider the projection \[ \Phi :\rm {\cal M}(S,F,{\cal O}_F^r)\times {\cal M}(\bar{S},F,{\cal O}_F^r) \rightarrow {\cal M}(S,F,{\cal O}_F^r) \] and notice that $\rm \Phi $ induces an isomorphism $\rm {\cal N}^{\rm I\! R} \rightarrow {\cal M}(S,F,{\cal O}_F^r)$ due to the inverse transformation \[ \begin{array}{ccc} \rm {\cal M}(S,F,{\cal O}_F^r)(T) & \longrightarrow & \rm {\cal N}^{\rm I\! R}(T)\hookrightarrow \left( {\cal M}(S,F,{\cal O}_F^r)\times {\cal M}(\bar{S},F,{\cal O}_F^r) \right) (T)\\ \rm (V,\alpha)& \longrightarrow & \rm \left( (V,\alpha), ((id_T\times \tau)^{\ast}\bar{V}^{\vee}, (\tau^{\ast}\bar{\alpha}^{\vee})^{-1}) \right). \end{array} \] Thus $\rm \Phi \circ \Psi$ defines an open real-analytic embedding \[ \rm {\cal M}_{\cal I}^{\rm I\! R}(P,F,{\cal O}_F^r) \rightarrow {\cal M}(S,F,{\cal O}_F^r) .\;\;\Box \] The result of N.P. Buchdahl's article \cite{buchdahl2} implies that $\rm {\cal M}_{\cal I}(P,F,{\cal O}_F^r)(\rm I\! R)$ and $\rm {\cal M}(S,F,{\cal O}_F^r)(\rm C\!\!\! I\,)$ are naturally bijective and we obtain \begin{corollary} The natural transformation $ \rm {\cal M}_{\cal I}^{\rm I\! R}(P,F,{\cal O}_F^r) \rightarrow {\cal M}(S,F,{\cal O}_F^r) $ given by restriction is a real-analytic isomorphism. For fixed $\rm c_{\bullet}\in H^{\bullet}(M,\rm Z\!\! Z)$ the functor $\rm {\cal M}_{\cal I}^{\rm I\! R}(P,F,{\cal O}_F^r,\pi^{\ast}c_{\bullet})$ is represented by a smooth separated algebraic space of finite type over $\rm C\!\!\! I\,$. \end{corollary} \renewcommand{\baselinestretch}{1.0} \small

9706

alg-geom/9706003

en

https://arxiv.org/abs/alg-geom/9706003

alg-geom/9706003

Takashi Kimura

Alexandre Kabanov and Takashi Kimura

Intersection Numbers and Rank One Cohomological Field Theories in Genus One

LaTeX2e, 31 pages, 15 postscript figs; Minor changes in revised version

Commun.Math.Phys. 194 (1998) 651-674

10.1007/s002200050373

Max Planck Preprint 97-61

We obtain a simple, recursive presentation of the tautological (\kappa, \psi, and \lambda) classes on the moduli space of curves in genus zero and one in terms of boundary strata (graphs). We derive differential equations for the generating functions for their intersection numbers which allow us to prove a simple relationship between the genus zero and genus one potentials. As an application, we describe the moduli space of normalized, even, rank one cohomological field theories in genus one in coordinates which are additive under taking tensor products. Our results simplify and generalize those of Kaufmann, Manin, and Zagier.

alg-geom

\section{Moduli Space of Curves} \label{kmz} \begin{nota} In this paper we always consider cohomology with the rational coefficients: $H^\bullet(X)$ stands for $H^\bullet(X;\nq)$. We denote the set $\{ 1, \ldots, n \}$ by $[n]$. If $I$ is a finite set we denote its cardinality by $|I|$. \end{nota} \subsection{Basic Definitions} Let $\MM_{g,n}$ be the moduli space of smooth curves of genus $g$ with $n$ marked points, where $2g-2+n >0$, i.e.\ $\MM_{g,n} = \{\, [\Sigma ; x_1,x_2,\ldots,x_n ]\, \}$ where $\Sigma$ is a genus $g$ Riemann surface and $x_1,x_2,\ldots,x_n$ are distinct marked points on $\Sigma$. Two such configurations are equivalent if they are related by a biholomorphic map. The moduli space $\MM_{g,n}$ has a natural compactification due to Deligne, Knudsen, and Mumford denoted by $\M_{g,n} = \{\,[C; x_1,x_2,\ldots, x_n] \,\}$ which is the moduli space of stable curves of genus $g$ with $n$ punctures in which $\MM_{g,n}$ sits as a dense open subset. The spaces $\M_{g,n}$ are connected, compact, complex orbifolds (in fact, stacks) with complex dimensions $3g-3+n$. The complement of $\MM_{g,n}$ in $\M_{g,n}$ is a divisor with normal crossings and consists of those stable curves which have double points. The moduli space $\M_{g,n}$ forms the base of a universal family. Let $\pi\,:\,\C_{g,n}\,\to\, \M_{g,n}$ be the \emph{universal curve} which can be identified with $\C_{g,n} = \M_{g,n+1}$ where $\pi$ is the projection obtained by forgetting the $(n+1)^{\text{st}}$ puncture and followed by collapsing any resulting unstable irreducible components of the curve, if any, to a point. The universal curve $\C_{g,n}\,\to\,\M_{g,n}$ is furthermore endowed with canonical sections $\sigma_1,\sigma_2,\ldots,\sigma_n$ such that $\sigma_i$ maps $[C;x_1,x_2,\ldots,x_n]\,\mapsto\,[C';x_1',x_2',\ldots,x_{n+1}']$ where $C'$ is obtained from $C$ by attaching a three punctured sphere to $x_i$ at one of its punctures to create a double point, then labeling the remaining two punctures on the sphere $x_i'$ and $x_{n+1}'$, and finally setting all other $x_j' = x_j$. The sections $\sigma_i$ are well-defined since $\M_{0,3}$ is a point. The image of $\sigma_i:\M_{g,n}\,\to\,\C_{g,n}$ gives rise to a divisor $D_{i}$ in $\C_{g,n}$ for all $i=1,2,\ldots,n$. \subsection{Natural Stratification} In the sequel it will be convenient to consider markings by arbitrary finite sets rather than by just $[n]$. If $I$ is a finite set we denote by $\M_{g,I} \cong \M_{g,|I|}$ the corresponding moduli space. The natural stratification of $\M_{g,n}$ is best described in terms of graphs, and therefore we start with fixing the notation concerning graphs. We will consider only connected graphs. Each graph $\Gamma$ can be described in terms of its set of vertices $V(\Gamma)$, set of edges $E(\Gamma)$, and set of tails $S(\Gamma)$. Each edge has two endpoints belonging to $V(\Gamma)$ which are allowed to be the same. Each tail has only one endpoint. If $v\in V(\Gamma)$,we denote by $n(v)$ the number of half-edges emanating from $v$, where each edge gives rise to two half-edges, and each tail to one half-edge. The natural stratification of $\M_{g,n}$ is determined by the type of the degeneration of the curve representing a point in the moduli space, and its strata can be labeled by stable graphs. A \emph{stable graph} consists of a triple $(\Gamma,g,\mu)$, where $\Gamma$ is a connected graph as above, $g: V(\Gamma) \to \nz_{\ge 0}$, and $\mu$ is a bijection between $S(\Gamma)$ and a given set $I$. Moreover, one requires that for each vertex $v$, the stability condition $2g(v)-2+ n(v)>0$ is satisfied. If $[C; x_1,x_2,\ldots, x_n]$ is a stable, $n$-pointed curve one obtains the corresponding stable graph, called its \emph{dual graph}, by collapsing each irreducible component to a point (vertex), connecting any two vertices if their corresponding components share a double point and attaching a tail to a vertex for each marked point on that component. We define the \emph{genus $g(\Gamma)$ of $\Gamma$} to be $b_1(\Gamma)+ \sum_{v\in V(\Gamma)} g(v)$, where $b_1(\Gamma)$ is the first Betti number of $\Gamma$. We denote by $\G_{g,n}$ the set of the equivalence classes of stable graphs of genus $g$ with $n$ tails labeled by $[n]$. There is a natural action of the symmetric group $S_n$ on $\G_{g,n}$. Associating a stable curve to its dual graph provides an $S_n$-equivariant bijection between the strata of the natural stratification of $\M_{g,n}$ and the elements of $\G_{g,n}$. Let $\M_\Gamma$ be (the closure of) the moduli space of stable curves whose dual graph is $\Gamma$. It is a closed irreducible subvariety of codimension $|E(\Gamma)|$ of $\M_{g(\Gamma), S(\Gamma)}$. Moreover, it is isomorphic to a quotient of the cartesian product \[ \prod_{v\in V(\Gamma)} \M_{g(v),n(v)} \] by $\operatorname{Aut}(\Gamma)$, where the automorphisms of a stable graph $(\Gamma,g,\mu)$ are required to preserve $g$ and $\mu$. This quotient morphism can be made canonical if one creates a pair of labels for each edge of $\Gamma$ and labels the $n(v)$ half-edges emanating from $v$ by the corresponding elements of $S(\Gamma)$ with the labels corresponding to the edges. Each $\M_\Gamma$ determines the fundamental class, in the sense of orbifolds, lying in $H^\bullet(\M_{g(\Gamma), n(\Gamma)})$. The pull back of this class under the morphism $\pi: \M_{g,n+1} \to \M_{g,n}$ is represented by a subvariety of $\M_{g,n+1}$ corresponding to the $|V(\Gamma)|$ graphs each of which is obtained by attaching a tail numbered $n+1$ to a vertex of $\Gamma$. It is also easy to push down these fundamental classes. Let $\M_{\Gamma'}$ represents an element of $H^\bullet(\M_{g,n+1})$. The image of this element under $\pi_* : H^{\bullet +2}(\C_{g,n})\,\to\,H^\bullet (\M_{g,n})$, induced by the fiber integration, is zero if after removing the $(n+1)^{\text{st}}$ tail from $\Gamma'$ the graph remains stable. In the other case, when the removal of the $(n+1)^{\text{st}}$ tail destabilizes $\Gamma'$, the image is obtained by stabilization, i.e., contracting the edge connecting the unstable vertex with the rest of the graph. \subsection{Tautological Classes} \label{tautological} We will now describe three types of tautological cohomology classes ($\psi$, $\kappa$, and $\lambda$) associated to the universal curve. Consider the universal curve $\C_{g,n}\,\to\,\M_{g,n}$. The cotangent bundle to its fibers (in the orbifold sense) forms the holomorphic line bundle $\omega_{g,n}$. Let $\LL_{(g,n),i}\,\to\,\M_{g,n}$ be given by the pullback $\LL_{(g,n),i} = \sigma_i^* \omega_{(g,n)}$. The tautological classes $\psi_{(g,n),i}$ in $H^2(\M_{g,n})$ are defined by \[ \psi_{(g,n),i} := c_1(\LL_{(g,n),i}) \] where $c_1$ denotes the first Chern class.\footnote{The Chern classes are in the sense of orbifolds and are therefore rational.} The tautological classes $\kappa_{(g,n),i}$ in $H^{2i}(\M_{g,n})$ for $i=0,1,\ldots, (3 g - 3+n)$ are defined as follows. Consider the bundle $\omega_{g,n}(D) \, \to\, \C_{g,n}$ consisting of $\omega_{g,n}$ twisted by the divisor $D = \sum_{i=1}^n D_i$, then \[ \kappa_{(g,n),i} := \pi_*(c_1(\omega_{g,n}(D))^{i+1}). \] In particular, $\kappa_{(g,n),0} = 2g-2+n \in H^0(\M_{g,n})$ is the negative of the Euler characteristic of a smooth curve of genus $g$ with $n$ points removed. We also have the equality $\omega_{g,n}(D) = \LL_{(g,n+1),n+1}$ \cite{AC,HL}. Therefore \begin{equation} \label{psi:kappa} \kappa_{(g,n),i} = \pi_*(\psi_{(g,n+1),n+1}^{i+1}). \end{equation} The tautological $\lambda$ classes are defined to be \[ \lambda_{(g,n),i} := c_l (\pi_*\, \omega_{g,n}) \in H^{2i} (\M_{g,n}), \] where $l=1, \ldots, g$ because $\pi_*\, \omega_{g,n}$ is an orbifold bundle of rank $g$. (There are no $\lambda$ classes in genus $0$ and we define $\lambda_{(0,n),i} := 0$.) One can easily see that $\lambda_{(g,n+1),i} = \pi^* \lambda_{(g,n),i}$. Therefore, all of the $\lambda$ classes are pull backs of the $\lambda$ classes on $\M_{1,1}$ and $\M_{g,0}$, $g\ge 2$. They can be expressed in terms of the $\kappa$ classes, the $\psi$ classes, and the cohomology classes lying at the boundary \cite{Mu}. In particular, \begin{equation} \label{lambda:kappa} \kappa_{(g,n),1} = 12 \lambda_{(g,n),1} -\delta_{(g,n)} + \sum_{i=1}^n \psi_{(g,n),i}, \end{equation} where $\delta_{(g,n)}$ is the fundamental class of $\M_{g,n} - \MM_{g,n}$ \cite{Co,Fa2,Mu}. (This formula was brought to our attention by E.~Getzler.) We will drop subscripts associated to the genus and the number of punctures if there is no ambiguity. \begin{nota} Let $\mathcal{S}_k$ be the set of infinite sequences of non-negative integers $\mf = (m_k, m_{k+1}, m_{k+2}, \ldots)$ such that $m_i=0$ for all $i$ sufficiently large. We denote by $\boldsymbol{\delta}_a$ the infinite sequence which has only one non-zero entry 1 at the $a^{\text{th}}$ place. For $\mf = (m_0, m_1, m_2, \ldots) \in \mathcal{S}_0$ and $\mathbf{t} = (t_0, t_1, t_2, \ldots)$, a family of independent formal variables, we will use notation of the type \[ |\mf|:= \sum_{i\ge 0} i\, m_i, \quad |\!|\mf|\!|:= \sum_{i\ge 0} m_i, \quad \mf! := \prod_{i\ge 0} m_i!, \quad \mathbf{t}^\mf :=\prod_{i\ge 0} t_i^{m_i}. \] We say that $\ll \le \mf$ if $l_i \le m_i$ for all $i$. If $\ll \le \mf$ we let \[ \binom{\mf}{\ll} := \prod_{i\ge 0} \binom{m_i}{l_i} \] We will use the same notation when $\mf \in \mathcal{S}_1$. \end{nota} \subsection{Generating Functions} Witten \cite{W2} defined a generating function which incorporates all of the information about the integrals of products of the $\psi$ classes. In order to describe this function we need to introduce the following notation. Let \[ \la \tau_{d_1} \tau_{d_2} \ldots \tau_{d_n} \ra := \int_{\M_{g,n}} \psi_1^{d_1} \psi_2^{d_2} \ldots \psi_n^{d_n}, \] where $g$ is determined by the equation $3g-3+n=d_1+d_2+\dots+d_n$. If there exists no such $g$, then the left hand side is by definition zero. In case we want to mention the genus explicitly we will write $\la \tau_{d_1} \tau_{d_2} \ldots \tau_{d_n} \ra_g$. Note that this expression is symmetric with respect to $d_1, d_2, \ldots, d_n$ since $\psi_i$'s are interchanged under the action of the symmetric group $S_n$. Therefore one can write it as $\la \tau_0^{m_0} \tau_1^{m_1} \tau_2^{m_2} \ldots \ra$, where the set $\{ d_1,\ldots,d_n \}$ contains $m_0$ zeros, $m_1$ ones, etc. The generating function is defined by \[ F(t_0,t_1,t_2,\ldots) := \la \exp \sum_{j=0}^\infty t_j \tau_j \ra = \sum_{ \{ m_i \} \in \mathcal{S}_0} \prod_{i=0}^\infty \la \tau_0^{m_0} \tau_1^{m_1} \tau_2^{m_2} \ldots \ra\, \frac{t_i^{m_i}}{m_i !}. \] We will also use the notation \[ \sum_{\mf \in \mathcal{S}_0} \la \boldsymbol{\tau}^\mf \ra \, \frac{\mathbf{t}^\mf}{\mf!} \] for the last expression. Note that one can also write $F(\mathbf{t}) = \sum_{g=0}^\infty F_g(\mathbf{t})$, where $F_g(\mathbf{t}) := \sum_{\mf \in \mathcal{S}_0} \la \boldsymbol{\tau}^\mf \ra_g\, \frac{\mathbf{t}^\mf}{\mf!}$. Witten conjectured in \cite{W} and Kontsevich proved in \cite{Ko} that $F$ is the logarithm of a $\tau$-function in the KdV-hierarchy. In \cite{KMZ} Kaufmann, Manin, and Zagier considered a similar generating function for $\kappa$ classes. If one defines \[ \la \kaf^\mathbf{p} \ra_g = \la \kappa_1^{p_1} \kappa_2^{p_2} \ldots \ra_g := \int_{\M_{g,n}} \kappa_1^{p_1} \kappa_2^{p_2} \ldots, \] then their generating function is \[ K_g (x;\sf) = K_g(x;s_1, s_2, \ldots) := \sum_{\mathbf{p} \in \mathcal{S}_1} \la \kaf^\mathbf{p} \ra_g\, \frac{x^{|\mathbf{p}|}}{|\mathbf{p}|!}\, \frac{\sf^\mathbf{p}}{\mathbf{p}!}. \] Note that here it is important to indicate the genus. The number of punctures $n$ is then determined from $3g-3+n= |\mathbf{p}|$. We introduce the generating function $H$ which incorporates both of the $\psi$ and $\kappa$ classes. We shall see that $F$ and $K$ enjoy similar properties which arise because $H$ obeys those properties. First we introduce the following notation. Let $\mf \in \mathcal{S}_0$ and $\mathbf{p} \in \mathcal{S}_1$. Define \[ \la \boldsymbol{\tau}^\mf \kaf^\mathbf{p} \ra := \int_{\M_{g,n}} \psi_1^{d_1} \psi_2^{d_2} \ldots \psi_n^{d_n} \kappa_1^{p_1} \kappa_2^{p_2} \ldots, \] where the set $\{ d_1, \ldots, d_n \}$ contains $m_0$ zeros, $m_1$ ones, etc., and $(g,n)$ is determined by the equations $n=|\!| m |\!|$, $3g-3+n= |\mf|+|\mathbf{p}|$. If no such $g$ exists we set the expression above to zero. As before, we write $\la \boldsymbol{\tau}^\mf \kaf^\mathbf{p} \ra_g$ when we want to fix $g$. \begin{df} Let $\mathbf{t}= (t_0,t_1,\ldots)$ and $\sf=(s_1,s_2,\ldots)$ be independent families of independent formal variables. We define \[ H(\mathbf{t};\sf) := \sum_{\mf \in \mathcal{S}_0,\mathbf{p} \in \mathcal{S}_1} \la \boldsymbol{\tau}^\mf \kaf^\mathbf{p} \ra\, \frac{\mathbf{t}^\mf}{\mf!}\, \frac{\sf^\mathbf{p}}{\mathbf{p}!}. \] \end{df} One can split $H$ into the sum of $H_g$, $g=0,1,\ldots$. Each $H_g$ lies in a kernel of a certain scaling differential operator, i.e., it satisfies the \emph{charge conservation equation}. The multiplication of $H$ by $|\!| \mf |\!|$ is equivalent to applying the operator $\EE := \sum t_i \dd_i$, and by $|\mf|$ is equivalent to applying $\sum i\, t_i \dd_i$. Therefore one has \begin{equation} \label{charge} [ 3(1-g) + \sum_{i=0}^\infty (i-1)\, t_i \dd_i + \sum_{i=1}^\infty i\, s_i d_i]\, H_g = 0, \end{equation} where $\dd_i:= \dd/\dd t_i$ and $d_i:= \dd/\dd s_i$. Clearly $H(\mathbf{t};\zef) = F(\mathbf{t})$. In order to relate $H$ to $K$ one fixes a genus $g$, sets $t_1=t_2=\cdots=0$, and $t_0=x$. The infinite sequence $\mf$ reduces to $m_0=n=|\mathbf{p}|+3-3g$. It follows that \[ H_g(x,\zef;\sf)= \sum_{\mathbf{p}\in\mathcal{S}_1} \la \kaf^\mathbf{p} \ra_g\, \frac{x^{|\mathbf{p}|+3-3g}}{(|\mathbf{p}|+3-3g)!}\, \frac{\sf^\mathbf{p}}{\mathbf{p}!}. \] In this paper we are primarily interested in genus 0 and 1. Then $K_0(x;\sf) = H_0'''(x,\zef;\sf)$ and $K_1(x;\sf) = H_1(x,\zef;\sf)$, where the prime denotes the partial derivative with respect to $x$. \section{Presentation of the Tautological Classes via Graphs} \label{graphs} In this section we will mainly focus on $H_0$ and $H_1$, the generating functions for the intersection numbers of the $\kappa$ and $\psi$ classes in genus $0$ and genus $1$. First we show that $H_0$ satisfies a system of nonlinear differential equations. These equations when, restricted to the $\psi$ classes, were first obtained by Witten \cite{W2,W}. When restricted to the $\kappa$ classes in genus zero, equivalent but much more complicated equations were obtained by Kaufmann, Manin, and Zagier \cite{KMZ}, the difference being accounted for by our simple, recursive presentation of powers of $\kappa$ and $\psi$ classes in genus zero and one in terms of boundary divisors. We prove that $H_0$ satisfies differential equations of the same form as those obtained by Witten for just the $\psi$ classes. We then obtain a system of differential equations relating $H_0$ and $H_1$, and the explicit formula \eqref{our:rel}. Here we present a geometric approach using the explicit presentation of the $\psi$ and $\kappa$ classes in terms of the boundary strata. \subsection{Basic Relation} \label{basic} In the beginning we want to state some general facts which hold for all genera. Let $\pi: \C_{g,n} = \M_{g,n+1} \to \M_{g,n}$ be the universal curve. To simplify the notation in this subsection, we denote $\psi_{(g,n),i}$ by $\psi_i$, $\psi_{(g,n+1),i}$ by $\widehat{\psi}} % psi classes on M_{g,n+1_i$, and we use the same convention for the $\kappa$ classes. Recall from Sec.~\ref{kmz} that $D_i$ denotes the image of the $i^{\text{th}}$ canonical section $\sigma_i: \M_{g,n} \to \M_{g,n+1}$. We will denote by the same letter its dual cohomology class in $H^2(\M_{g,n+1})$. \begin{lm} For each $k\ge 1$ \begin{equation} \label{up} \widehat{\psi}} % psi classes on M_{g,n+1_i^a = \pi^* \psi_i^a + \sigma_{i*} \psi_i^{a-1}, \quad k\ge 1. \end{equation} \end{lm} \begin{proof} It is easy to show that the equation $\widehat{\psi}} % psi classes on M_{g,n+1_i^a = \pi^* \psi_i^a + \pi^* \psi_i^{a-1} D_i$ from \cite{W2} can be rewritten as \begin{equation*} \widehat{\psi}} % psi classes on M_{g,n+1_i^a = \pi^* \psi_i^a + (-1)^{a-1} D_i^a, \quad k\ge 1. \end{equation*} Applying the functor $\sigma_{i*} \sigma_i^*$, which is the multiplication by $D_i$, to the above equation, and using that $\sigma_i^* \widehat{\psi}} % psi classes on M_{g,n+1_i=0$ one gets \begin{equation*} (-1)^{a} D_i^{a+1} = \sigma_{i*} \psi_i^{a}, \quad k\ge 0. \end{equation*} These two equalities imply the lemma. \end{proof} \subsection{Explicit Presentation in Genus 0, 1} In genus $0$ and genus $1$ equation \eqref{up} allows us to express inductively all powers of $\psi$ classes, and therefore $\kappa$ classes, in terms of the boundary strata. Because the $\psi$ classes are interchanged under the action of the symmetric group this is enough to compute $\psi_{(g,n),1}$, $g=0,1$, and all its powers. In the calculations below we will use the properties stated in Sec.~\ref{kmz} regarding the pull backs and push forwards of the cohomology classes represented by graphs. Let us introduce the following notation for the rest of this section. On graphs we denote by $\bullet$ vertices of genus $0$, and by $\circ$ vertices of genus $1$. We always assume that the $\psi$ classes are associated to the marked point labeled $1$, and subsequently omit it from the notation. We denote $\psi_{(1,n),1}$ by $\psi_{(n)}$, and we denote $\psi_{(0,J),1}$ by $\phi_{(J)}$, where $J$ is a finite set, $1\in J$. Similarly, we denote $\kappa_{(1,n),a}$ by $\kappa_{(n),a}$, and $\kappa_{(0,J),a}$ by $\omega_{(J),a}$. We also adopt the following convention. Let $\Gamma$ be a stable graph. According to Sec.~\ref{kmz}, $\Gamma$ determines a canonical finite quotient map from a product of moduli spaces to $\M_\Gamma$ provided certain choices have been made. We denote by $\rho_\Gamma$ the composition \begin{equation} \label{rho:gamma} \rho_\Gamma: \prod_{v\in V(\Gamma)} \M_{g(v),n(v)} \longrightarrow \M_\Gamma \longrightarrow \M_{g(\Gamma), S(\Gamma)}, \end{equation} where the first arrow is the quotient morphism, and the second arrow is the inclusion. Let $\gamma_v \in H^\bullet(\M_{g(v),n(v)})$. We denote $\frac{1}{|\operatorname{Aut}(\Gamma)|} \rho_{\Gamma*} (\otimes_{v} \gamma_v)$ by the picture of $\Gamma$ where each vertex $v$ is in addition labeled by the cohomology class $\gamma_v$. We may omit the label of $v$ if $\gamma(v)$ is the fundamental class of $\M_{g(v),n(v)}$. In particular, the fundamental class of $\M_\Gamma$ (in the orbifold sense) is represented by $\Gamma$ with all additional labels omitted. The only possible ambiguity would arise if $\otimes_{v} \gamma_v$ were not invariant under $\operatorname{Aut} (\Gamma)$, but this situation will not arise in this paper. In the pictures the dashed line with two arrows indicates the (sub)set of the tails emanating from a particular vertex. If $\phi$ or $\omega$ labels a vertex of a graph we will omit the subscript from the notation because it is determined by the graph. (We also assume that $\phi$ is associated to the marked point labeled by $1$.) The power $\phi^0$ represents the fundamental class. \begin{prop} \label{prop:zero:psi} If $n\ge 4$, $a\ge 1$, then the following holds in $H^\bullet(\M_{0,n})$ \begin{equation} \label{zero:psi} \lift{1500}{$\phi_{(n)}^a = \displaystyle{\sum_{\substack{I\sqcup J = [n-2]\\ 1\in J}}}$} \ \input pic/phi.pstex_t \end{equation} \end{prop} \begin{rem} The class $\phi_{(n)}^a$ is invariant under the subgroup $S_{n-1} \subset S_n$ whose elements fix $1$. Therefore instead of $n-1,n$ we can choose any two labels $a,b$, $2\le a<b \le n$ to be distinguished. \end{rem} \begin{proof} The statement is true when $n=4$ and $a=1$ because $\int_{\M_{0,4}} \psi_1 = 1$. We shall first prove by induction that the statement is true for all $n\ge 4$ and $a=1$. Let us consider the projection $\pi: \M_{0,n+1} \to \M_{0,n}$ which ``forgets'' the $n+1^{\text{st}}$ marked point. By the induction hypothesis we assume that the statement is true for some $n$ and $a=1$. Applying $\pi^*$ one gets: \begin{align*} \lift{1500}{$\phi_{(n+1)}\ - $} \quad \ \input pic/d1.pstex_t &\\ \lift{1500} {$= \displaystyle{\sum_{\substack{I\sqcup J = [n-2]\\ 1\in J}}} \bigg( $} \ \input pic/proof1.pstex_t & \lift{1500}{$+$} \quad \input pic/proof2.pstex_t \lift{1500}{$\!\!\!\! \bigg) $} \end{align*} The left hand side is equal to $\phi_{(n+1)} - D_1 = \pi^* \phi_{(n)}$. Moving $D_1$ to the right hand side and relabeling the tails marked $n-1, n$ by $n,n+1$ respectively one gets the statement in case of $n+1$, $a=1$. Now assume that $a\ge 2$. If $n=4$, then the statement of the proposition is trivially satisfied because all terms vanish by dimensional considerations. We assume that the statement is true for a pair $(n,a)$ and all pairs $(n',a')$, where $a'<a$, and prove it for $(n+1,a)$. Applying $\pi^*$ one gets \begin{align*} \lift{1500}{$\phi_{(n+1)}^a - \sigma_{1*} \phi_{(n)}^{a-1} = \displaystyle{\sum_{\substack{I\sqcup J = [n-2]\\ 1\in J}}} \bigg( $} \ \input pic/proof3.pstex_t & \\ \lift{1500}{$+$} \quad \input pic/proof4.pstex_t \lift{1500}{$-\ \sigma_{1*} \big( \ $} \input pic/sigma.pstex_t & \lift{1500}{$\!\!\!\! \big) \bigg) $} \end{align*} According to the induction hypothesis the terms with $\sigma_{1*}$ on the left hand side and the right hand side cancel each other. Thus, we get the statement of the proposition for the pair $(n+1,a)$. \end{proof} Let $\pi_1$ be the morphism $\M_{0,n+1} \to \M_{0,n}$ forgetting the first marked point. Applying it to \eqref{zero:psi} with $a+1$, using \eqref{psi:kappa}, and renumbering the labels $\{ 2,\ldots,n+1 \}$ by the elements of $[n]$ we get the following \begin{crl} If $n\ge 4$, $a\ge1$, then the following holds in $H^\bullet(\M_{0,n})$ \begin{equation} \label{zero:kappa} \lift{1500}{$\omega_{(n),a} = \displaystyle{\sum_{I\sqcup J = [n-2]}}$} \ \input pic/omega.pstex_t \qed \end{equation} \end{crl} In \eqref{zero:kappa} if $a=1$ one should use that $\omega_{(J\sqcup *),0}$ associated to the right vertex is equal to $|J|-1$ times the fundamental class (cf.~Sec.~\ref{kmz}). This agrees with $\pi_{1*}$ of \eqref{zero:psi} when $a=1$. Now we establish a relation between the $\psi$ and $\kappa$ classes in genus $0$ and $1$. The proof is virtually identical to that of Prop.~\ref{prop:zero:psi}, and we will not reproduce it. It is shown in \cite[VI.4]{DR} that \begin{equation*} \lift{850}{$\psi_{(1)} = \displaystyle{\frac{1}{12}}$} \ \input pic/psi2.pstex_t \end{equation*} Note that we take the coefficient is $\frac{1}{12}$ rather than $\frac{1}{24}$ due to the non-trivial automorphism of the graph. Using this and \eqref{up} one obtains the following \begin{prop} \label{prop:one:psi} If $n\ge 1$, $a\ge 1$, then the following holds in $H^\bullet(\M_{1,n})$ \begin{equation} \label{one:psi} \lift{1500}{$\psi_{(n)}^a = \displaystyle{\frac{1}{12}}$} \ \input pic/loopsi.pstex_t \lift{1500}{$\displaystyle{+ \sum_{\substack{I\sqcup J=[n]\\ 1\in J} }}$} \ \input pic/psi.pstex_t \qed \end{equation} \end{prop} Pushing down the above along $\pi_1: \M_{1,n+1} \to \M_{1,n}$, and renumbering the labels one gets \begin{crl} If $n\ge 1$, $a\ge 1$, then the following holds in $H^\bullet(\M_{1,n})$ \begin{equation} \label{one:kappa} \lift{1500}{$\kappa_{(n),a}= \displaystyle{\frac{1}{12}}$} \ \input pic/loopka.pstex_t \lift{1500}{$\displaystyle{+ \sum_{I\sqcup J=[n]}}$} \ \input pic/kappa.pstex_t \qed \end{equation} \end{crl} \subsection{Recursion Relations and Differential Equations} Now we derive the corresponding recursion relations and differential equations for the intersection numbers of the products of the $\psi$ and $\kappa$ classes using the explicit graph presentations above. In order to obtain the recursion relations we use a method from \cite{KMZ} to integrate the product of the $\psi$ and $\kappa$ classes over the Poincar\'e dual of a chosen $\psi$ or $\kappa$ class. In order to do this we need to know how the tautological classes restrict to the strata of the natural stratification. The restriction of a $\psi$ class to a boundary stratum is obvious. In order to restrict products of the $\kappa$ classes we use Lemma 1.3 from \cite{KMZ} where the authors show the following restriction property for the $\kappa$ classes. (They show it in the case of genus $0$, but their proof is in fact valid for all genera.) Let $(\Gamma,g,\mu)$ be a stable graph, $\rho_\Gamma$ is the corresponding morphism defined by \eqref{rho:gamma}, and $\kaf^\mathbf{p}$ is a product of the $\kappa$ classes on $\M_{g(\Gamma),S(\Gamma)}$. Then \[ \int_{\prod_{v\in V(\Gamma)} \M_{g(v),n(v)}} \frac{\rho_\Gamma^* (\kaf^\mathbf{p})}{\mathbf{p} !} = \sum_{\substack{\mathbf{p}^v:\,v\in V(\Gamma)\\ \sum \mathbf{p}^v = \mathbf{p}}}\; \prod_{v\in V(\Gamma)} \frac{\la \kaf^{\mathbf{p}^v} \ra_{g(v)}} {\mathbf{p}^v !} \] The argument uses a fact proved in \cite{AC} that the collection $\kappa_{(g,n),a}$ for each fixed $a$ forms a \emph{logarithmic cohomological field theory} (cf. Sec.~\ref{cft}), i.e., the $\kappa$ classes satisfy the relation \begin{equation} \label{eq:lcft} \rho_\Gamma^* (\kappa_a) = \sum_{v\in V(\Gamma)} \kappa_{g(v),n(v)}. \end{equation} We start with genus $0$. Recall that $H(\mathbf{t};\sf)$ is the generating function incorporating the intersection numbers for the products of the $\psi$ and $\kappa$ classes defined in Sec.~\ref{kmz}, $\dd_a$, $d_a$ are partial derivative with respect to $t_a$, $a\ge 0$, $s_a$, $a\ge 1$, and $\EE = \sum_{i=0}^\infty t_i\dd_i$. \begin{thm} \label{H0} For each $\mf \in \mathcal{S}_1$, $\mathbf{p} \in \mathcal{S}$, $k,l \ge 0$, and $a\ge 1$ one has \begin{align*} \la \boldsymbol{\tau}^{\mf + \boldsymbol{\delta}_k + \boldsymbol{\delta}_l + \boldsymbol{\delta}_a} \kaf^\mathbf{p} \ra_0 & \\ = \sum_{\substack{\mf'+\mf''=\mf\\ \mathbf{p}' + \mathbf{p}''= \mathbf{p}} } & \binom{\mf}{\mf'} \binom{\mathbf{p}}{\mathbf{p}'} \la \boldsymbol{\tau}^{\mf' + \boldsymbol{\delta}_k + \boldsymbol{\delta}_l + \boldsymbol{\delta}_0} \kaf^{\mathbf{p}'} \ra_0 \la \boldsymbol{\tau}^{\mf'' + \boldsymbol{\delta}_{a-1} + \boldsymbol{\delta}_0} \kaf^{\mathbf{p}''} \ra_0, \\ \la \boldsymbol{\tau}^{\mf + \boldsymbol{\delta}_k + \boldsymbol{\delta}_l} \kaf^{\mathbf{p} + \boldsymbol{\delta}_a} \ra_0 & \\ = \sum_{\substack{\mf'+\mf''=\mf\\ \mathbf{p}' + \mathbf{p}''= \mathbf{p}} } & \binom{\mf}{\mf'} \binom{\mathbf{p}}{\mathbf{p}'} \la \boldsymbol{\tau}^{\mf' + \boldsymbol{\delta}_k + \boldsymbol{\delta}_l + \boldsymbol{\delta}_0} \kaf^{\mathbf{p}'} \ra_0 \, \la \boldsymbol{\tau}^{\mf'' + \boldsymbol{\delta}_0} \kaf^{\mathbf{p}'' + \boldsymbol{\delta}_{a-1}} \ra_0. \end{align*} Equivalently, for each $k,l\ge0$ the function $H_0(\mathbf{t};\sf)$ satisfies \begin{align*} &\dd_a \dd_k \dd_l H_0 = (\dd_k \dd_l \dd_0 H_0) (\dd_{a-1} \dd_0 H_0) \quad \text{when $a\ge 1$}, \\ & d_1 \dd_k \dd_l H_0 = (\dd_k \dd_l \dd_0 H_0) ((\EE-1) \dd_0 H_0), \\ & d_a \dd_k \dd_l H_0 = (\dd_k \dd_l \dd_0 H_0) (d_{a-1} \dd_0 H_0) \quad \text{when $a\ge 2$}. \end{align*} This system together with $H_0(t_0,\zef;\zef)= \frac{t_0^3}{6}$ uniquely determines $H_0$. \qed \end{thm} \begin{proof} The recursion relations are a direct consequence of \eqref{zero:psi}, \eqref{zero:kappa}, and the restriction properties of the $\psi$ and $\kappa$ classes described above. In order to derive the differential equations from the recursion relations one notices that the increment of $m_a$ or $p_a$ by one in a recursion relation corresponds to taking the partial derivative with respect to $t_a$ or $s_a$. The operator $\EE$ appears because the second recursion relation when $a=1$ produces $\omega_0$. The corresponding moduli space is $\M_{0,J\sqcup *}$. As $|J\sqcup *| = |\!|\mf''|\!|+1$, it follows that $\omega_0 = |J|-1= |\!|\mf''|\!|-1$, and we use that the multiplication by $m_i$ can be expressed by $t_i\dd_i$. \end{proof} \begin{rem} Setting $\sf=\zef$ in the first equation one gets differential equations satisfied by $F_0$ (cf. \cite{W}). \end{rem} \begin{rem} Setting $k=l=0$, $t_0=x$, and $t_1=t_2=\ldots=0$ in the second equation one gets differential equations satisfied by $H(x,\zef;\sf)$ whose third derivative with respect to $x$ is $K_0 (x;\sf)$. These equations are a simple consequence of the results in \cite[Sec.~1]{KMZ}. \end{rem} Now we turn to genus $1$. We use the explicit presentations \eqref{one:psi} and \eqref{one:kappa} and take into account the automorphism groups of the graphs to obtain the following \begin{thm} \label{H1} For each $\mf \in \mathcal{S}_0$, $\mathbf{p} \in \mathcal{S}_1$, and $a\ge 1$ one has \begin{align*} \la \boldsymbol{\tau}^{\mf + \boldsymbol{\delta}_a} \kaf^\mathbf{p} \ra_1 = \frac{1}{24} & \la \boldsymbol{\tau}^{\mf + 2\boldsymbol{\delta}_0 + \boldsymbol{\delta}_{a-1}} \kaf^\mathbf{p} \ra_0 \\ + \sum_{\substack{\mf'+\mf''=\mf\\ \mathbf{p}' + \mathbf{p}''= \mathbf{p}} } & \binom{\mf}{\mf'} \binom{\mathbf{p}}{\mathbf{p}'} \la \boldsymbol{\tau}^{\mf' + \boldsymbol{\delta}_0} \kaf^{\mathbf{p}'} \ra_1 \la \boldsymbol{\tau}^{\mf'' + \boldsymbol{\delta}_0 + \boldsymbol{\delta}_{a-1}} \kaf^{\mathbf{p}''} \ra_0, \\ \la \boldsymbol{\tau}^{\mf} \kaf^{\mathbf{p} + \boldsymbol{\delta}_a} \ra_1 = \frac{1}{24} & \la \boldsymbol{\tau}^{\mf + 2\boldsymbol{\delta}_0} \kaf^{\mathbf{p} + \boldsymbol{\delta}_{a-1}} \ra_0 \\ + \sum_{\substack{\mf'+\mf''=\mf\\ \mathbf{p}' + \mathbf{p}''= \mathbf{p}} } & \binom{\mf}{\mf'} \binom{\mathbf{p}}{\mathbf{p}'} \la \boldsymbol{\tau}^{\mf' + \boldsymbol{\delta}_0} \kaf^{\mathbf{p}'} \ra_1 \la \boldsymbol{\tau}^{\mf'' + \boldsymbol{\delta}_0} \kaf^{\mathbf{p}'' + \boldsymbol{\delta}_{a-1}} \ra_0. \end{align*} Equivalently, the functions $H_1(\mathbf{t};\sf)$ and $H_0(\mathbf{t};\sf)$ satisfy \begin{align*} & \dd_a H_1 = \frac{1}{24} \dd_{a-1} \dd_0 \dd_0 H_0 + (\dd_0 H_1) (\dd_{a-1} \dd_0 H_0) \quad \text{when $a\ge 1$}, \\ & d_1 H_1 = \frac{1}{24} \EE \dd_0 \dd_0 H_0 + (\dd_0 H_1) ( (\EE-1) \dd_0 H_0), \\ & d_a H_1 = \frac{1}{24} d_{a-1} \dd_0 \dd_0 H_0 + (\dd_0 H_1) (d_{a-1} \dd_0 H_0) \quad \text{when $a\ge 2$}. \end{align*} This system together with the system and the initial conditions from Thm.~\ref{H0} uniquely determines the pair $H_0, H_1$. \qed \end{thm} \begin{rem} The first recursion relation when $\mathbf{p}=\zef$ was obtained by Witten in \cite{W}. \end{rem} The system of differential equations above can be solved explicitly for $H_1$ in terms of $H_0$ to derive \eqref{our:rel}. \begin{crl} The functions $H_1$ and $H_0$ are related by \begin{equation*} H_1 = \frac{1}{24} \log \dd_0^3 H_0. \end{equation*} \end{crl} \begin{proof} Because of uniqueness it is enough to check that $\frac{1}{24} \log \dd_0^3 H_0$ satisfies the differential equation in Thm.~\ref{H1}. This is a straight forward calculation which makes use of the differential equations from Thm.~\ref{H0}. \end{proof} \begin{rem} Setting $\sf=\zef$ we recover a result from \cite[Sec.~2.2]{Di}. Setting $t_0=x, t_1=t_2=\ldots=0$ we get $K_1 = \frac{1}{24} \log K_0$. \end{rem} \section{Puncture and Dilaton Equations} \label{punc} In this section we introduce an approach which does not use explicit presentations of $\psi$ and $\kappa$ classes in terms of graphs. Instead we introduce the analogues of the puncture and dilaton equations. These equations generalize the classical puncture and dilaton equations obtained by Witten \cite{W,W2}. (We shall explain these equations below.) This will allow us to write differential equations for $H$, and then, using these differential equations, prove that $H_0$ and $H_1$ satisfy \eqref{our:rel}. \subsection{Recursion Relations} In Sec.~\ref{kmz} we introduce the notation incorporating the intersection numbers of both of the $\psi$ and $\kappa$ classes. Now we shall to prove certain recursion relations for these numbers. \begin{lm} \label{punc:dil} The following recursion relations are satisfied: \begin{equation} \label{puncture} \la \boldsymbol{\tau}^{\mf+\boldsymbol{\delta}_0} \kaf^\mathbf{p} \ra = \sum_{i=1}^\infty m_i \la \boldsymbol{\tau}^{\mf+\boldsymbol{\delta}_{i-1}-\boldsymbol{\delta}_i} \kaf^\mathbf{p} \ra + \sum_{\substack{\mathbf{j}=\zef\\ |\mathbf{j}|>0}}^\mathbf{p} \binom{\mathbf{p}}{\mathbf{j}} \la \boldsymbol{\tau}^\mf \kaf^{\mathbf{p}-\mathbf{j}+\boldsymbol{\delta}_{|\mathbf{j}|-1}} \ra, \end{equation} and for each $a\ge 1$ \begin{equation} \label{dilaton} \la \boldsymbol{\tau}^{\mf+\boldsymbol{\delta}_a} \kaf^\mathbf{p} \ra = \sum_{\mathbf{j}=\zef}^\mathbf{p} \binom{\mathbf{p}}{\mathbf{j}} \la \boldsymbol{\tau}^\mf \kaf^{\mathbf{p}-\mathbf{j}+\boldsymbol{\delta}_{|\mathbf{j}|+a-1}} \ra. \end{equation} \end{lm} \begin{proof} We continue to use the notation from \ref{basic}, i.e., $\pi$ is the universal curve over $\M_{g,n}$, $(\widehat{\psi}} % psi classes on M_{g,n+1_i,\ \widehat{\kappa}} % kappa classes on M_{g,n+1_i)$ and $(\psi_i,\ \kappa_i)$ are classes upstairs and downstairs respectively, $\sigma_i$ is the $i^{\text{th}}$ canonical section of $\pi$, and $D_i$ is its image. It was shown in \cite{W2} that $\widehat{\psi}} % psi classes on M_{g,n+1_i^a = \pi^* \psi_i^a + \pi^* \psi_i^{a-1} D_i$ and in \cite{AC} that $\widehat{\kappa}} % kappa classes on M_{g,n+1_i = \pi^* \kappa_i + \widehat{\psi}} % psi classes on M_{g,n+1_{n+1}^i$. Note also that $\widehat{\psi}} % psi classes on M_{g,n+1_i D_i = 0$, $\widehat{\psi}} % psi classes on M_{g,n+1_{n+1} D_i = 0$ for $i=1,\ldots,n$, and $D_i D_j = 0$ when $i \neq j$. Using this one derives that \begin{align*} \pi_* (\widehat{\psi}} % psi classes on M_{g,n+1_1^{d_1} \ldots & \widehat{\psi}} % psi classes on M_{g,n+1_n^{d_n} \widehat{\kappa}} % kappa classes on M_{g,n+1_1^{p_1} \widehat{\kappa}} % kappa classes on M_{g,n+1_2^{p_2} \ldots) \\ &= \pi_* \big(\ (\pi^* \psi_1^{d_1} + \pi^* \psi_1^{d_1-1} D_1) \ldots (\pi^* \psi_n^{d_n} + \pi^* \psi_n^{d_n-1} D_n) \\ & \qquad\qquad \times (\pi^* \kappa_1 + \widehat{\psi}} % psi classes on M_{g,n+1_{n+1})^{p_1} (\pi^* \kappa_2 + \widehat{\psi}} % psi classes on M_{g,n+1_{n+1}^2)^{p_2} \ldots \big) \\ &= \sum_{i:d_i \neq 0} \psi_1^{d_1} \ldots \psi_i^{d_i-1} \ldots \psi_n^{d_n} \kappa_1^{p_1} \kappa_2^{p_2} \ldots \\ & \qquad\qquad + \sum_{\substack{\mathbf{j}=\zef\\ |\mathbf{j}|>0}}^{\mathbf{p}} \binom{\mathbf{p}}{\mathbf{j}} \psi_1^{d_1} \ldots \psi_n^{d_n} \kappa_{|\mathbf{j}|-1} \kappa_1^{p_1-j_1} \kappa_2^{p_2-j_2} \ldots \\ \intertext{Similarly one can show that} \pi_* (\widehat{\psi}} % psi classes on M_{g,n+1_1^{d_1} \ldots & \widehat{\psi}} % psi classes on M_{g,n+1_n^{d_n} \widehat{\psi}} % psi classes on M_{g,n+1_{n+1}^a \widehat{\kappa}} % kappa classes on M_{g,n+1_1^{p_1} \widehat{\kappa}} % kappa classes on M_{g,n+1_2^{p_2} \ldots) \\ & = \sum_{\mathbf{j}=\zef}^{\mathbf{p}} \binom{\mathbf{p}}{\mathbf{j}} \psi_1^{d_1} \ldots \psi_n^{d_n} \kappa_{|\mathbf{j}|+a-1} \kappa_1^{p_1-j_1} \kappa_2^{p_2-j_2} \ldots \end{align*} Recall that $\kappa_0 = 2g-2+n$. One can further integrate the push forward formulas above to obtain the statement of the lemma. \end{proof} \begin{rem} Recursion relations \eqref{puncture} and \eqref{dilaton} do not mix intersection numbers in different genera. \end{rem} \begin{rem} If $\mathbf{p} = \zef$, then the second sum in the first relation vanishes, and we obtain the classical puncture equation. If $\mathbf{p}=\zef$ and $a=1$ in the second relation, then we get the classical dilaton equation. Note that both classical equations involve only $\psi$ classes. \end{rem} \begin{rem} This is clear recursion relations \eqref{puncture} and \eqref{dilaton} allow to eliminate $\boldsymbol{\tau}$ from the intersection number, i.e., to express all mixed intersection numbers through the intersection numbers on $\M_{0,3}$, $\M_{1,1}$, and the intersection numbers of the $\kappa$ classes on $\M_{g,0}$, $g\ge 2$. In \cite[Cor.~2.3]{KMZ} the authors obtained an explicit expression for the intersection numbers of the $\kappa$ classes through the intersection numbers of the $\psi$ classes. This should be related to \eqref{dilaton}, but we do not know how to derive their formula from it. \end{rem} \subsection{Differential Operators} Now we derive differential equations for $H$ using recursions \eqref{puncture} and \eqref{dilaton}. Recall that $\dd_i$, $d_i$ denote the partial derivatives with respect to $t_i$, $s_i$. \begin{thm} The function $\exp (H(\mathbf{t};\sf))$ is annihilated by the following differential operators: \begin{align*} -\dd_0 \ + & \ \sum_{\mathbf{j}:\, |\mathbf{j}|\ge 2} \frac{\sf^\mathbf{j}}{\mathbf{j}!} d_{|\mathbf{j}|-1} + \sum_{i=0}^\infty t_i \dd_{i-1} \\ + & \ s_1 ( \sum_{i=0}^\infty \frac{2i+1}{3} t_i \dd_i + \sum_{i=1}^\infty \frac{2}{3} i\, s_i d_i) + \frac{1}{2} t_0^2 \delta_{g,0} + \frac{1}{24} s_1 \delta_{g,1},\\ -\dd_1 \ + & \ \sum_{\mathbf{j}:\, |\mathbf{j}|\ge 1} \frac{\sf^\mathbf{j}}{\mathbf{j}!} d_{|\mathbf{j}|} + (\sum_{i=0}^\infty \frac{2i+1}{3} t_i \dd_i + \sum_{i=1}^\infty \frac{2}{3} i\, s_i d_i) + \frac{1}{24} \delta_{g,1},\\ -\dd_a \ + & \ \sum_{\mathbf{j}} \frac{\sf^\mathbf{j}}{\mathbf{j}!} d_{|\mathbf{j}|+a-1} \quad \text{when $a\ge 2$.} \end{align*} \end{thm} \begin{rem} The differential operators above do not mix genus, and therefore they annihilate each $\exp (H_g (\mathbf{t};\sf) )$ separately. \end{rem} \begin{rem} The function $\exp (F(\mathbf{t}))$ is annihilated by differential operators $L_i$, $i\ge -1$, which, after a rescaling of variables, satisfy the Virasoro relations \cite{Di}. The first two differential operators in the statement of the theorem are analogues of $L_{-1}$ and $L_0$ respectively, which encode the puncture and dilaton equations, respectively. \end{rem} \begin{proof} The differential operators above are the direct translation of the recursion relations \eqref{puncture} and \eqref{dilaton}. The addition of $\boldsymbol{\delta}_i$ to $\mf$ or $\mathbf{p}$ translates into taking the corresponding partial derivative. The subtraction of $\boldsymbol{\delta}_i$ from $\mf$, and multiplying the term by $m_i$ translates into the multiplication by $t_i$. One should also change the summation index to obtain the second summand of each differential operator. The terms in parentheses in the first two equations come from the value of $\kappa_0= 2g-2+n$. We use \eqref{charge} in order to express this number in terms of differential operators. Finally, the last terms in the first two equations correspond to the initial conditions $\la \tau_0^3 \ra_0=1$, $\la \kappa_1 \ra_1 = \frac{1}{24}$, and $\la \tau_1 \ra_1= \frac{1}{24}$. \end{proof} The theorem above leads to another proof of Cor.~\ref{our:rel}. First one notes that $H_0$ and $H_1$ are uniquely determined by the differential operators above. Therefore it suffices to check that $\frac{1}{24} \dd_0^3 H_0$ satisfies the genus $1$ equations. This is a direct calculation. \section{Explicit Expressions} \label{explicit} In this section we write the closed form expressions for the intersection numbers in genus $1$ as sums of the multinomial coefficients. \begin{nota} If $\mathbf{b}= (b_1, \ldots, b_k)$ is a vector with integer entries we denote by $\boldsymbol{[}\mathbf{b}\boldsymbol{]}$ the multinomial coefficient $\frac{(b_1+ \cdots +b_k)!} {b_1! \ldots b_k!}$, and we set it to zero if at least one entry is negative. Recall also that $|\!| \mathbf{b} |\!|$ denotes the sum $b_1+ \ldots +b_k$. \end{nota} In genus $0$ the intersection numbers of the $\psi$ classes are very simple: $\la \tau_{b_1} \ldots \tau_{b_k} \ra_0 = \boldsymbol{[} \mathbf{b} \boldsymbol{]}$. In order to state our result in genus $1$ we define for each $k\ge 1$ a function $f_k: \nz_{\ge 1}^k \to \nz_{\ge 1}$ by \[ f_k(\mathbf{b}) = f_k(b_1, \ldots, b_k) := \la \tau_0^{|\!| \mathbf{b} |\!| - k} \tau_{b_1} \ldots \tau_{b_k} \ra_1. \] Clearly each $f_k$ is invariant under the permutations of its arguments. \begin{prop} For each $k\ge1$ \begin{equation} \label{expl:psi} f_k (\mathbf{b}) = \frac{1}{24} \boldsymbol{[} \mathbf{b} \boldsymbol{]} - \frac{1}{24} \sum_{\substack{\boldsymbol{\varepsilon} \in \{ 0,1 \}^k\\ |\!| \boldsymbol{\varepsilon} |\!|\ge 2}} (|\!| \boldsymbol{\varepsilon} |\!| - 2)!\, \boldsymbol{[} \mathbf{b} - \boldsymbol{\varepsilon} \boldsymbol{]}. \end{equation} \end{prop} \begin{proof} The intersection numbers of the $\psi$ classes in genus $1$ are determined by the classical puncture equation and the classical dilaton equation. (See the second remark after Lemma \ref{punc:dil}.) Reformulated in terms of the collection $\{ f_k \}$ these equations say that this collection is uniquely determined by the following properties: \begin{itemize} \item $f_1 (b_1) \equiv \frac{1}{24}$, \item $f_k$ is invariant under the permutation of the arguments, \item $f_k (\mathbf{b}) = \sum_{i=1}^k f_k (\mathbf{b} - \boldsymbol{\delta}_i)$ when $b_i \ge 2$ for all $i$, \item $f_k (b_1, \ldots, b_{k-1}, 1) = (b_1+ \cdots +b_{k-1}) f_{k-1} (b_1, \ldots, b_{k-1})$. \end{itemize} The first three properties are obviously satisfied by the expression given in the statement of the proposition. A direct computation verifies the last property. \end{proof} An explicit expression for the intersection numbers of the $\kappa$ classes in genus $1$ can be obtained by substitution of \eqref{expl:psi} into Cor.~2.3 from \cite{KMZ}. This expression is quite complicated. We do not know how to simplify this expression, and therefore we do not present the resulting formula for the $\kappa$ classes here. \section{Asymptotic Formulas for Volumes of $\M_{1,n}$} \label{asymp} In this section, we derive an asymptotic formula for the Weil-Petersson volumes of $\M_{1,n}$ in the limit that $n$ becomes very large extending the proof of a similar result for genus zero in \cite{KMZ}. We do so by using analytic properties of the generating function $K(x;\sf)$ in the case where all $s_i=0$ for all $i\geq 2$. In some sense, these results are complementary to those of Penner \cite{Pe} who obtains similar formulas for the case where the genus becomes very large. The class of the Weil-Petersson symplectic form on $\M_{g,n}$ is precisely $\frac{1}{2\pi^2} \kappa_{(g,n),1}$. The symplectic volume of $\M_{g,n}$ is called the \emph{Weil-Petersson volume of} $\M_{g,n}$. For this reason, the intersection numbers associated to the $\kappa$ classes are sometimes called \emph{higher Weil-Petersson volumes}. To avoid unnecessary factors, we shall work instead with the quantity \begin{df} $w_{g,n} := \int_{\M_{g,n}}\kappa_1^{3g-3+n}$. \end{df} \begin{thm}[\cite{KMZ}] The genus zero Weil-Petersson volumes satisfy the asymptotic relation as $n\,\to\,\infty$ \[ w_{0,n+3}\,\thicksim\, \frac{\gamma_0\, 2^\frac{3}{2}}{C\sqrt{\pi}} \frac{2^{2n} n^{2n + \frac{1}{2}}}{C^n e^{2n}}, \] where $\gamma_0\,\approx\,2.40482555777\ldots $ is the smallest zero of the Bessel function $J_0$ and $C = - 2\gamma_0 J_0'(\gamma_0)\,\approx\, 2.496918339\ldots$. \end{thm} They proved this by noticing that the function $H_0''(x,\zef;\sf)$ is invertible, and when all $s_i=0$ except for $s_1$ the inverse function satisfies Bessel's equation, after a change of variables. Combining their results with ours for genus one, we obtain the following. \begin{thm} The genus one Weil-Petersson volumes satisfy the asymptotic relation as $n\,\to\,\infty$ \[ w_{1,n}\,\thicksim\,\frac{\pi}{24}\frac{(2n)^{2n}}{C^n e^{2n}}, \] where $C$ is same constant as in the above. \end{thm} \begin{proof} One uses the asymptotic formulas for the genus zero case and our result that the generating functions are related by \eqref{our:rel} . \end{proof} \begin{rem} The theorem above supports a conjecture of Itzykson regarding the existence of such an asymptotic formula for all genera with a constant $C$ independent of the genus (cf.~\cite[p.~765]{KMZ}). \end{rem} \section{The Moduli Space of Cohomological Field Theories} \label{cft} The moduli space of normalized, rank one cohomological field theories of genus zero was described in Kontsevich, Manin, and Zagier \cite{KMZ}. The generating function associated to the $\kappa$ classes endows this moduli space with coordinates which behave nicely with respect to taking tensor products of cohomological field theories, a notion introduced in \cite{KMK} (see also \cite{Ka}). In this section, we introduce the notion of a restricted, normalized, cohomological field theory in genus one and describe the moduli space of rank one theories of this kind. Such CohFTs\ turn out to be almost completely determined by their genus zero part using the relations between the boundary strata of $\M_{1,n}$ recently obtained by Getzler \cite{G}. The analogous set of coordinates are constructed for this moduli space but to do so, we must introduce the $\lambda$ classes, as well. \subsection{Cohomological Field Theories} Consider $\G_{g,n}$, the set of stable graphs of genus $g$ and $n$ tails labeled with the set $[n]$ . Each $\G_{g,n}$ is acted upon by the permutation group $S_n$ which permutes the labels on the tails. There are composition maps \[ \G_{g_1,n_1} \times \G_{g_2,n_2}\,\to\,\G_{g_1+g_2,n_1+n_2-2}\] taking $(\Gamma,\Gamma') \, \mapsto \, \Gamma \circ_{(i_1,i_2)} \Gamma'$ for all $i_1$ in $[n_1]$ and $i_2$ in $[n_2]$ given by grafting the tail $i_1$ of $\Gamma$ with tail $i_2$ of $\Gamma'$ and then relabeling the remaining tails with elements of the set $[n_1+n_2-2]$ by inserting orders. There are another set of composition maps $\G_{g,n}\,\to\,\G_{g+1,n-2}$ taking $\Gamma\,\mapsto\,\tr_{(i_1,i_2)}\Gamma$ for all distinct pairs $i_1$ and $i_2$ in $[n]$ in which the tails $i_1$ and $i_2$ of $\Gamma$ are grafted together. These composition maps are equivariant with respect to the action of the permutation groups. Let $\GG{g,n}$ be the vector space over $\nc$ with a basis $\G_{g,n}$ then the compositions and permutations group actions can be extended $\nc$-linearly. Let $\GG{}$ denote the direct sum of $\GG{g,n}$ for all stable pairs $g,n$. The collection $\{\,\GG{g,n}\,\}$ (or, for that matter, $\{\,\G_{g,n}\,\}$) together with the composition maps and actions of the permutation groups described above forms an example of a \emph{modular operad}, a notion due to by Getzler and Kapranov \cite{GK}. By restricting to just the genus zero subcollection $\{\,\GG{0,n}\,\}$ and forgetting about the composition maps $\tr$, we obtain an example of a \emph{cyclic operad} \cite{GK2}. Similarly, the homology groups $H_\bullet(\M_{g,n})$ are endowed with an action of $S_n$ which relabels the punctures on the stable curve and there are composition maps \[ \circ_{i_1,i_2}\,:\,H_{p_1}(\M_{g_1,n_1})\,\otimes\, H_{p_2}(\M_{g_2,n_2})\,\to\,H_{p_1+p_2}(\M_{g_1+g_2,n_1+n_2-2}) \] for all $i_1$ in $[n_1]$, $i_2$ in $[n_2]$ and $\tr_{(i_1,i_2)}\,:\, H_{p}(\M_{g,n})\,\to\,H_{p}(\M_{g+1,n-2})$ for all distinct $i_1$ and $i_2$ in $[n]$, both of which are induced from the inclusion of strata. These composition maps are equivariant under the action of the permutation groups. The natural maps $\alpha_{g,n}\,:\,\GG{g,n}\,\to\, H_\bullet(\M_{g,n})$ mapping $\Gamma\,\mapsto\,[\M(\Gamma)]$ where $\M(\Gamma) := \prod_{v\in V(\Gamma)}\,\M_{g(v),n(v)}$ preserves the above structures and gives rise to the sequence of morphisms \begin{equation} \label{eq:boundary} 0\,\longrightarrow\,\left< \R_{g,n}\right> \,\longrightarrow\,\GG{g,n}\, \overset{\alpha_{g,n}}{\longrightarrow} \, H_\bullet(\M_{g,n}) \end{equation} where the kernel of $\alpha_{g,n}$ is denoted by $\left< \R_{g,n}\right>$, the ideal in $\GG{}$ generated by some space of relations $\R_{g,n}$. \begin{df} The modular operad $\HH\, :=\,\{\,\HH_{g,n} \,\}$ is the collection of \[ \HH_{g,n}\,:= \, \frac{\GG{g,n}}{\left< \R_{g,n}\right>} \] \end{df} The canonical diagonal maps $\M_{g,n}\,\to\,\M_{g,n}\times \M_{g,n}$ induce maps $\,H_\bullet(\M_{g,n})\,\to\, H_\bullet(\M_{g,n})\otimes H_\bullet(\M_{g,n})$ making $H_\bullet(\M_{g,n})$ into a Hopf modular operad in the natural way \cite{GK}. This endows $\HH_{g,n}$ with the structure of a Hopf modular operad, as well. In the case of $g=0$, the results of \cite{KM} and \cite{Ke} implies that $\alpha_{0,n}$ is surjective and $\HH_{0,n}$ is isomorphic to $H_\bullet(\M_{0,n})$. Furthermore, the relations $\R_{0,n}$ are those due to Keel \cite{Ke} which come from a lift of the basic codimension one relations $\R_{0,4}$ on $\M_{0,4}$ via the canonical forgetful map $\M_{0,n}\,\to\,\M_{0,4}$. In the case of $g=1$, $\alpha_{1,n}$ is known not to be surjective since $\M_{1,n}$ has odd dimensional homology classes. However, Getzler \cite{G} has shown that the space of relations $\R_{1,n}$, in addition to those coming from Keel's relations, contains the lifts of two other relations. The first is the lift of the basic codimension one relation on $\M_{1,2}$ which contains no genus one vertices -- this may be regarded as the image of Keel's relations under the self-sewing morphism $\tr_{(3,4)}\,:\,\GG{0,4}\,\to\,\GG{1,2}$. The second relation, which contains genus one vertices, is between codimension two strata and is of the form \begin{equation} 12 \delta_{2,2} - 4\delta_{2,3} - 2 \delta_{2,4} + 6\delta_{3,4} + \delta_{0,3} + \delta_{0,4} - 2\delta_\beta = 0 \label{eq:getzler} \end{equation} where each term is an $S_4$-invariant combination of graphs of a given topological type and each graph $\Gamma$ represents the homology class $[\M_\Gamma]$. (See \cite{G} for details.) Getzler also states \cite{G} that he has shown \cite{G2} that $\alpha_{1,n}$ maps surjectively onto the even dimensional homology of $\M_{1,n}$ and that the relations mentioned above do in fact generate all of $\R_{1,n}$. A cohomological field theory is essentially a representation, in the sense of operads, of $H_\bullet(\M_{g,n})$. In order to define such an object, we need to define the appropriate notion of the endomorphisms of a vector space is in this context. Let $V$ be a vector space over $\nc$ with a symmetric, nondegenerate bilinear form $h$ of degree zero. Let $\End{V}_{g,n}:=T^n V$ be the $n^{\rm th}$ tensor power of $V$ for all nonnegative integers $g,n$ such that $2g-2+n>0$ where $T^0 V$ is understood to be $\nc$. $S_n$ acts upon $\End{V}_{g,n}$ by permuting the tensor factors and the composition maps $\End{V}_{g_1,n_1} \otimes \End{V}_{g_2,n_2}\,\to\,\End{V}_{g_1+g_2,n_1+n_2-2}$ taking $(\mu,\mu') \, \mapsto \, \mu \circ_{(i_1,i_2)} \mu'$ for all $i_1$ in $[n_1]$ and $i_2$ in $[n_2]$ given by applying the inverse of $h$ to the the corresponding tensor factors of $\mu$ and $\mu'$, and inserting the remaining factors in the usual way. Similarly, the composition $\End{V}_{g,n}\,\to\,\End{V}_{g+1,n-2}$ taking $\mu\,\mapsto\,\tr_{(i_1,i_2)}\mu$ for all distinct pairs $i_1$ and $i_2$ in $[n]$ corresponds to applying the inverse of $h$ to the appropriate pair of tensor factors of $\mu$. \begin{df}[Cohomological Field Theory] A \emph{(complete) cohomological field theory (CohFT) of rank} $r$, $(V,h)$, is a morphism of modular operads $\mu_{g,n}\,:\,H_\bullet(\M_{g,n})\,\to\,\End{V}_{g,n}$ where $(V,h)$ is an $r$-dimensional vector space with an invariant, symmetric bilinear form $h$. A \emph{CohFT\ of genus $g$ } are maps $\mu_{g',n}\,:\, H_\bullet(\M_{g',n})\,\to\,\End{V}_{g',n}$ which are defined only for $g'\leq g$ which satisfy all the axioms of a CohFT\ in which no higher genus maps appear. A \emph{restricted CohFT\ } is a morphism $\mu_{g,n}\,:\,\HH_{g,n}\,\to\,\End{V}_{g,n}$. \end{df} A CohFT\ can also be described described dually in terms of maps $\End{V}_{g,n}\, \to\, H^\bullet(\M_{g,n})$. Notice that a restricted CohFT\ of genus zero is the same as a CohFT\ of genus zero since $H_\bullet(\M_{0,n})\,=\, \HH_{0,n}$. \begin{rem} In the language of \cite{W2,W}, a \emph{topological gravity (coupled to topological matter)} is a CohFT\ and the morphisms $\mu_{g,n}$ are the \emph{correlation functions} of the theory. The genus zero CohFT\ is said to be \emph{tree level} while a genus one CohFT\ is said to be \emph{one loop}. \end{rem} \begin{rem} The natural Hopf structure on $H_\bullet(\M_{g,n})$ endows the category of CohFTs\ with a tensor product as is usual in representation theory. \end{rem} An restricted CohFT\ is completely determined by a generating function called its potential. If the CohFT\ is not restricted then one can still define the notion of a potential (essentially since the modular operad $H_\bullet(\M_{g,n})$ is the quotient of some free modular operad) but we will not need to work in such generality. \begin{df} The \emph{potential} $\Phi = \sum_{g=0}^\infty \Phi_g$ of a restricted CohFT\ $\mu\,:\,\HH\,\to\,\End{V}$ of rank $r$ is defined by choosing a basis $\{\,e_1,\ldots,e_r\,\}$ for $V$ where $I_{g,n}(e_{a_1},e_{a_2},\ldots,e_{a_n})$ is the number obtained by using $h$ to pair $\mu_{g,n}([\M_{g,n}])$ with $e_{a_1}\otimes e_{a_2}\otimes\ldots\otimes e_{a_n}$ and \[ \Phi_g(\mathbf{x}) := \sum_{n=0}^\infty \,I_{g,n}(e_{a_1},e_{a_2},\ldots,e_{a_n}) \,\frac{x^{a_1} x^{a_2}\ldots x^{a_n}}{n!}. \] (where the summation convention has been used) which is regarded as an element in $\nc[[x^1,\ldots,x^r]]$. \end{df} \begin{thm} \label{thm:wdvvg} A element $\Phi_0$ in $\nc[[x^1,\ldots,x^r]]$ is the potential of a rank $r$, genus zero CohFT\ $(V,h)$ if and only if \cite{KM,Ma} it satisfies the WDVV equation \[ (\partial_{a} \partial_{b} \partial_{e}\Phi_0)\, h^{ef}\, (\partial_{f} \partial_{c} \partial_{d}\Phi_0)\, = \,(-1)^{|x_a|(|x_b| + |x_c|)}\, (\partial_{b} \partial_{c} \partial_{e} \Phi_0)\, h^{ef} \,(\partial_{f} \partial_{a} \partial_{d}\Phi_0), \] where $h_{a,b} := h(e_a,e_b)$, $h^{ab}$ is in inverse matrix to $h_{ab}$, $\partial_a$ is derivative with respect to $x^a$, and the summation convention has been used. If $(\Phi_0,\Phi_1)$ is the potential associated to a restricted, rank $r$ CohFT\ of genus one then $\Phi_0$ must satisfy the WDVV equation and $(\Phi_0,\Phi_1)$ must satisfy Getzler's equation from proposition $(3.14)$ in \cite{G}. \qed \end{thm} The WDVV equation can be read off from the basic codimension one relation on $\M_{0,4}$. Similarly, Getzler's equation can be seen from his relation (equation \ref{eq:getzler}). The second statement will become an if and only if after the proof in \cite{G2} appears. \subsection{Rank One Cohomological Field Theories} Let $(V,h)$ be a rank one CohFT\ with a fixed unit vector $e$. The morphisms $\HH_{g,n}\,\to\,\End{V}_{g,n}$ are completely determined by the collection of numbers $\{\,I_{g,n}\,\}$ where $I_{g,n} := \mu_{g,n}([\M_{g,n}])(\underbrace{e,e,\ldots,e}_n)$ which must satisfy relations between themselves reflecting the way that the boundary strata in $\M_{g,n}$ fit together. The potential in this case is \[ \Phi_g = \sum_{n=0}^\infty I_{g,n} \frac{x^n}{n!}, \] where $I_{g,n}$ is defined to vanish for pairs $(g,n)$ which are not stable. We will see that tautological classes on the moduli space of curves give rise to complete rank one CohFTs. In order to describe the moduli space of restricted, rank one CohFTs\ of genus one, we need to introduce a combination of the $\lambda$ classes which behave nicely with respect to restriction. \begin{df} For all stable pairs, $(g,n)$, let $\Lambda_{g,n}$ be an element in $H^\bullet(\M_{g,n})[\mathbf{s},\mathbf{u}]$ (where $\mathbf{s} = (s_1,s_2,\ldots)$ and $\mathbf{u} = (u_1,u_2,u_3,\ldots\,)$) then let \[ \Lambda_{g,n} := \exp(\,\sum_{i=1}^\infty (\,s_i\,\kappa_{(g,n),i}\, +\, u_{i}\,\gamma_{(g,n),i}\,)\,) \] where $\gamma_{(g,n),i} := \mathrm{ch}_{2i-1}( \pi_*\,\omega_{g,n})$. Here $\mathrm{ch}_i$ is the $i^{\mathrm{th}}$ Chern character, and $\pi_*\,\omega_{g,n}$ is the pushforward of the relative dualizing sheaf. (Notice that $\mathrm{ch}_{2i}( \pi_*\,\omega_{g,n})$ vanishes for all $i$ \cite{Ma,Fa}.) \end{df} The classes $\gamma_{(g,n),i}$ are polynomials in the $\lambda$ classes. In particular, $\gamma_1 = \lambda_1$. \begin{thm} The collection $\Lambda := \{\,\Lambda_{g,n}\,\}$ gives rise to a complete, rank one CohFT\ for all values of $\textbf{u}$ and $\textbf{s}$ by integrating the cohomology classes $\Lambda_{g,n}$ over the homology classes on $\M_{g,n}$. Furthermore, the tensor product of the CohFT\ associated to parameter values $(\mathbf{s_1},\mathbf{u_1})$ and $(\mathbf{s_2},\mathbf{u_2})$ is the CohFT\ associated to $(\mathbf{s_1}+\mathbf{s_2}, \mathbf{u_1}+ \mathbf{u_2})$. Similarly, \end{thm} \begin{proof} In the case where $u_i$ vanishes for all $i$, this was proven in \cite{KMK} where it was realized that the $\kappa$ classes form a logarithmic CohFT\ following the work of Arbarello and Cornalba \cite{AC} (see equation \ref{eq:lcft}). The proof for the case where all the $s_i$ vanish is as follows. Consider the bundles $E_{g,n} := \pi_*\, \omega_{g,n}$ on $\M_{g,n}$ from section \ref{tautological}. If $\Gamma$ is a graph of genus $g$ with $n$ tails, then it determines the morphism $\rho_\Gamma: \prod_{v\in V(\Gamma)} \M_{g(v),n(v)} \to \M_{g(\Gamma), S(\Gamma)}$. The pull back of $E_{g,n}$ under $\rho_\Gamma$ differs from $\oplus_{v\in V(\Gamma)} E_{g(v),n(v)}$ by a trivial bundle. It follows that for each $k\ge 1$ the collection of the Chern characters $\gamma_k = ch_{2k-1} E_{g,n}$ forms a logarithmic CohFT\ (see \cite{Fa,Fa2}). The first part of the theorem follows by combining these two results. The proof that the coordinates $(\mathbf{s},\mathbf{u})$ are additive with respect to taking tensor products follows from the definition of coproduct which is induced from the diagonal map. \end{proof} \begin{crl} The potential of the rank one CohFT\ associated to $\Lambda$ (for given values of $\mathbf{s}$ and $\mathbf{u}$) is precisely the generating function $\chi_g$ for the intersection numbers of $\kappa_i$ and $\gamma_{i}$ classes \[ \chi_g(x;\mathbf{s},u) := \left< \exp(x\,\tau_0\,+\, \sum_{i=1}^\infty\,(\,s_i\,\kappa_i + u_{i}\,\gamma_{i}\,))\right>_g = \sum_{n=0}^\infty\,I_{g,n}\, \frac{x^n}{n!}\] where \[ I_{g,n} = \sum_{\mathbf{r},\mathbf{m}}\, \,\frac{\mathbf{s}^\mathbf{m}}{\mathbf{m}!}\,\frac{\mathbf{u}^\mathbf{r} }{\mathbf{r}!} \left<\,\kaf^\mathbf{m}\,\tau_0^n\,\boldsymbol{\gamma}^\mathbf{r}\,\right>_g. \] It is understood that $I_{g,n}\,:=\,0$ for unstable pairs $(g,n)$, \end{crl} Notice that the CohFT\ arising $\Lambda$ have the property that $I_{0,3} = 1$ for all values of $\mathbf{u}$ and $\mathbf{s}$. This motivates the following definition which will play an important role in what follows. \begin{df} A rank one, CohFT\ of genus $g$ is said to be \emph{invertible} if $I_{0,3}$ is nonzero and \emph{normalized} if $I_{0,3}=1$. \end{df} \subsection{Cohomological Field Theories in Genus Zero and One} Let us recall the results of Kaufmann, Manin, and Zagier for rank one CohFTs\ of genus zero \cite{KMZ}. A rank one CohFT\ in genus zero is uniquely determined by its potential $\Phi_0(x) = \sum_{n=3}^\infty I_{0,n}\frac{x^n}{n!}$. Furthermore, any function $\Phi_0(x)$ in $x^3\,\nc[[x]]$ arises from some rank one, CohFT\ of genus zero since the WDVV equation is trivially satisfied for rank one theories. Therefore, the moduli space of CohFTs\ of genus zero are parameterized by the independent variables $I_{0,n}$ for $n\geq 3$. What is nontrivial, however, is the behavior of these potentials under tensor product. In particular, the coordinates $I_{0,n}$ do not behave nicely under tensor product. However, the generating function associated to the genus zero $\kappa$ classes $H_0(x,\mathbf{0};\mathbf{s})$ (which is equal to $\chi_0$ with $u=0$) allows them to introduce coordinates on the space of normalized, rank one CohFTs\ which behave nicely under tensor products. \begin{thm}[\cite{KMZ}] \label{thm:kmz} The moduli space of normalized, rank one CohFTs\ in genus zero are parameterized by $\mathbf{s}$ with potential $\Phi_0(x;\mathbf{s}) = H_0(x,\mathbf{0};\mathbf{s})$ in $\nc[\mathbf{s}][[x]]$, our generating function for the intersection numbers of $\kappa$ classes in genus zero. Furthermore, taking tensor products is additive with respect to the coordinates $\mathbf{s}$. \end{thm} We now treat the case of genus one and discover that the $\kappa$ classes are not sufficient to describe the entire moduli space of normalized, rank one CohFTs. We will see that one needs to introduce the first $\lambda$ classes. \begin{thm} If the pair $(\Phi_0,\Phi_1)$ is a potential associated to a restricted, rank one CohFT\ of genus one then the following equation holds in $\nc[[x]]$ \[ \label{eq:rk1getzler} -(\Phi_0^{(3)})^2\Phi_1^{(2)} + \Phi_0^{(3)} \Phi_0^{(4)} \Phi_1^{(1)} - \frac{1}{12} (\Phi_0^{(4)})^2 + \frac{1}{24}\Phi_0^{(3)}\Phi_0^{(5)} = 0 \] where $\Phi_g^{(l)}$ is the $l^{\rm th}$ derivative of $\Phi_g$. \end{thm} \begin{proof} Equation \ref{eq:rk1getzler} above is nothing more than the equation due to Getzler in theorem \ref{thm:wdvvg} for the case of rank one theories. Our equation can be seen from equation \ref{eq:getzler} directly by associating to each graph \[ \Gamma\,\mapsto\, \frac{1}{|\textrm{Aut}(\Gamma)|} \prod_{v\in V(\Gamma)}\,\frac{\partial^{n(v)} \Phi_{g(v)}}{\partial x^{n(v)}} \] and then extending linearly to linear combinations of graphs. One will obtain $-36$ times the equation above. \end{proof} Unlike the case of genus zero where the WDVV equation is trivially satisfied, solutions to this equation fall into two classes depending upon whether $\Phi_0^{(3)}$ is invertible in the ring of formal power series $\nc[[x]]$. \begin{thm} \label{thm:rk1solu} The pair $(\Phi_0,\Phi_1)$ is a potential associated to an invertible, restricted, rank one CohFT\ of genus one if and only if $\Phi_0(x)$ is of the form $I_{0,3}\,\frac{x^3}{6} + x^4\,\nc[[x]]$ for $I_{0,3}$ nonzero and \[ \label{eq:rk1solu} \Phi_1 = \frac{1}{24}\,\log\Phi_0''' + B \Phi_0'' \] where $B$ is an arbitrary constant. Therefore, an invertible, restricted CohFT\ of genus one is uniquely determined by arbitrary values of $I_{0,n}$ for all $n\geq 4$, $I_{0,3}\,\not=\,0$, and $I_{1,1}$. If the restricted, rank one, CohFT\ is not invertible then $\Phi_0\,=\,0$ and $\Phi_1$ obeys no constraints. Therefore, the space of such theories is parameterized by all values of $I_{1,n}$ for all $n\geq 1$. \end{thm} \begin{proof} If the pair $(\Phi_0,\Phi_1)$ is a potential associated to an invertible restricted, rank one CohFT\ then since $\Phi_0'''$ has an inverse in $\nc[[x]]$, one can solve equation \ref{eq:rk1getzler} explicitly. The converse is more difficult in the absence of the proof that the lifts of the relations described above genus $1$ span the entire space of relations $\R_{1,n}$. However, we will not need this statement but will explicitly construct restricted, normalized, rank one CohFTs\ in genus one which realize all solutions to equation \ref{eq:rk1solu} above. This we do in the next subsection. Since $I_{1,1} = I_{0,4} + B I_{0,3}$, when $I_{0,3}$ nonzero varying $B$ is the same as varying $I_{1,1}$ and leaving all of the $I_{0,n}$ unchanged. In the case that the restricted, rank one CohFT\ is not invertible then our result follows from the equation. \end{proof} We shall not discuss noninvertible CohFTs\ any further in this paper. From now on, we shall restrict ourselves to normalized CohFTs. It is worth observing that by incorporating the $\psi$ classes, one can use the previous result to obtain yet another proof of the formula $H_1 = \frac{1}{24} \log H_0^{'''}$. It is not clear which of these approaches will prove most useful in higher genera. \subsection{Potentials in Genus Zero and One} In this subsection, we construct potentials for a class of normalized, restricted, rank one CohFTs\ in genus one explicitly and show that they span the entire space of solutions to equation \ref{eq:rk1solu} completing the proof of that theorem. These potentials are generating functions associated to the $\kappa$ classes and $\lambda_1$. This will give rise to coordinates which are additive under tensor product in analogy with the case of genus zero in \cite{KMZ}. We begin with a useful lemma. \begin{lm} The tautological class $\lambda_1$ on $\M_{1,n}$ can be written in terms of boundary classes as follows: \begin{equation*} \lift{1500}{$\lambda_1\,=\, \frac{1}{12}$} \ \input pic/lambda.pstex_t \end{equation*} \end{lm} \begin{proof} The proof follows from the fact that $\lambda_{(1,n),1} = \pi^*\lambda_{(1,1),1}$ via the forgetful map $\pi:\,\M_{1,n}\to\,\M_{1,1}$. One uses \eqref{lambda:kappa} to express $\lambda_{(1,1),1}$ in terms of boundary classes. \end{proof} In the sequel, let $\chit_g(\mathbf{s},u)$ be equal to the generating function $\chi_g(\mathbf{s},\mathbf{u})$ where all values of $u_i$ are set to zero except for $u\,:=\,u_1$. \begin{thm} \label{thm:lambda} The intersection numbers above satisfy the following: \[ \chit_0(x;\mathbf{s},u) = H_0(x,\mathbf{0};\mathbf{s}) \] and \[ \chit_1(x;\mathbf{s},u) = \frac{u}{24} H_0''(x,\mathbf{0};\mathbf{s}) + \frac{1}{24} \log H_0'''(x,\mathbf{0};\mathbf{s}) \] where $'$ denotes differentiation with respect to $x$. \end{thm} \begin{proof} Using that $\lambda_1$ vanishes on $\M_{0,n}$, the presentation of $\lambda_1$ on $\M_{1,n}$ in terms of boundary strata above, and the fact that the $\kappa$ classes and $\lambda_1$ forms a logarithmic CohFT, we obtain the equations \[ \left< \kaf^\mathbf{m}\lambda_1^r\tau_0^n\right>_0 = 0. \] and \[ \left< \kaf^\mathbf{m}\lambda_1^r\tau_0^n\right>_1 = \begin{cases} \frac{1}{24}\left<\kaf^\mathbf{m}\tau_0^{n+2}\right>_0 & \text{if $r=1$,} \\ 0 & \text{if $r\geq 2$.} \end{cases} \] Rewriting these identities in terms of $\chit_g$, using the fact that $\chit_g(\mathbf{s},x,0) = H_g(\mathbf{s},x)$ and theorem \ref{our:rel}, we obtain the desired result. \end{proof} \begin{proof}{(completion of theorem \ref{thm:rk1solu})} By setting $B := \frac{u}{24}$ in the previous theorem and $\Phi_g = \chit_g$ for $g=0,1$, we conclude the proof of theorem \ref{thm:rk1solu} since theorem \ref{thm:kmz} implies that by forgetting $\Phi_1$, we obtain all possible CohFTs\ in genus zero. Furthermore, by varying $u$, one obtains all possible values of $I_{1,1}$ without changing the values of $I_{0,n}$. \end{proof} \begin{rem} The relations between the intersection numbers obtained in the previous proof can be encoded in the differential equations \begin{equation} \frac{\partial}{\partial u} \chit_0 = 0\qquad\textrm{and}\qquad \frac{\partial}{\partial u} \chit_1 = \frac{1}{24} \frac{\partial^2}{\partial x^2} \chit_0 \end{equation} \end{rem} Putting everything together, we arrive at the following theorem. \begin{thm} The moduli space of normalized, restricted, rank one CohFTs\ of genus one is parameterized by coordinates $(\mathbf{s},u)$ via potentials $(\chit_0,\chit_1)$ where $\chit_0(x)$ belongs to $\frac{x^3}{6}+x^4\,\nc[\mathbf{s},u][[x]]$ and $\chit_1(x)$ belongs to $x\,\nc[\mathbf{s},u][[x]]$ satisfying theorem \ref{thm:lambda}. The tensor product is additive in the coordinates $(\mathbf{s},u)$. \end{thm} Given two rank one, normalized CohFTs\ in genus zero, it is not obvious how to write down the potential of the tensor product CohFT\ explicitly in terms of the potentials of the tensor factors. In \cite{KMZ}, the authors show that the operation of tensor product corresponds to multiplication of the formal Laplace transforms of the two potentials associated to the tensor factors. Because a rank one, normalized, restricted CohFTs\ in genus one is determined by its genus zero potential and the value of $u$, an explicit expression for the potential associated to the tensor product of two such theories follows from the genus zero result of \cite{KMZ} and theorem \ref{thm:lambda}. \bibliographystyle{amsplain} \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}

9706

alg-geom/9706004

en

https://arxiv.org/abs/alg-geom/9706004

alg-geom/9706004

Frank Sottile

Birkett Huber (Texas A&M University, College Station, TX) and Frank Sottile (University of Toronto, Ontario, Canada) and Bernd Sturmfels (University of California, Berkeley)

Numerical Schubert calculus

24 pages, LaTeX 2e with 2 figures, used epsf.sty

J. Symb. Comp., 26 (1998), 767-788.

MSRI 1997-063

We develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that generically no paths diverge. For problems defined by hypersurface Schubert conditions we give two algorithms based on extrinsic deformations of the Grassmannian: one is derived from a Gr\"obner basis for the Pl\"ucker ideal of the Grassmannian and the other from a SAGBI basis for its projective coordinate ring. The more general case of special Schubert conditions is solved by delicate intrinsic deformations, called Pieri homotopies, which first arose in the study of enumerative geometry over the real numbers. Computational results are presented and applications to control theory are discussed.

alg-geom

\section{Introduction} Suppose we are given linear subspaces $K_1,\ldots,K_n$ of ${\bf C}^{m+p}$ with $\dim K_i = m+1-k_i$ and $k_1+\cdots+k_n = mp $. Our problem is to find all $p$-dimensional linear subspaces of ${\bf C}^{m+p}$ which meet each $K_i$ nontrivially. When the given linear subspaces are in general position, the condition $k_1+\cdots+k_n=mp$ guarantees that there is a finite number $\,d \,=\, d(m,p,k_1,\ldots,k_n)\,$ of such $p$-planes. The classical Schubert calculus~\cite{Kleiman_Laksov} gives the following recipe for computing the number $d$. Let $h_1,\ldots,h_m$ be indeterminates with $degree(h_i) = i$. For each integer sequence $\lambda_1\geq \cdots\geq \lambda_{p+1} $ we define the following polynomial: \begin{equation}\label{schur} S_\lambda\ :=\ \det ( h_{\lambda_i+j-i})_{1\leq i,j\leq p+1}. \end{equation} Here $h_0 := 1$ and $h_i:=0$ if $i<0$ or $i>m$. Let $I$ be the ideal in ${\bf Q}[h_1,\ldots,h_m]$ generated by those $S_\lambda$ with $m \geq \lambda_1 $ and $\lambda_{p+1} \geq 1$. The quotient ring ${\mathcal A}_{m,p}:= {\bf Q}[h_1,\ldots,h_m]/I$ is the cohomology ring of the Grassmannian of $p$-planes in ${\bf C}^{m+p}$. It is Artinian with one-dimensional socle in degree $mp$. In the socle we have the relation \begin{equation} \label{soclerel} d \cdot (h_m)^p \, - \, h_{k_1} h_{k_2} \cdots h_{k_n} \quad \in \quad I. \end{equation} Thus we can compute the number $d$ by normal form reduction modulo any Gr\"obner basis for $I$. More efficient methods for computing in the ring ${\mathcal A}_{m,p}$ are implemented in the Maple package SF~\cite{Stembridge_SF}. In the important special case $k_1=\cdots=k_n=1$ there is an explicit formula for $d$: \begin{equation}\label{grassdeg} d \quad = \quad \frac{1! \, 2! \, 3! \cdots (p\!- \!2) ! \, (p \!-\!1)! \cdot (mp)!}{m!\, (m \! + \! 1)! \, (m \! + \, 2)! \cdots(m \! + \! p \! - \! 1)!} . \end{equation} The integer on the right hand side is the degree of the Grassmannian in its Pl\"ucker embedding. This formula is due to~\cite{Schubert_degree}; see also~\cite[XIV.7.8]{Hodge_Pedoe} and Section 2.3 below. The objective of this paper is to present semi-numerical algorithms for computing all $d$ solution planes from the input data $K_1,\ldots,K_n$. This amounts to solving certain systems of polynomial equations. Our algorithms are based on the paradigm of {\it numerical homotopy methods}~\cite{Morgan_book,Allgower_Georg,Allgower_Georg_handbook}. Homotopy methods have been developed for the following classes of polynomial systems: \begin{enumerate} \item complete intersections in affine or projective spaces~\cite{Drexler,Garcia_Zangwill}, \item complete intersections in products of projective spaces~\cite{MS_m-homogeneous}, \item complete intersections in toric varieties~\cite{CVVerschelde,Huber_Sturmfels}. \end{enumerate} In these cases the number of paths to be traced is optimal and equal to the standard combinatorial bounds: \begin{enumerate} \item the B\'ezout number (= the product of the degrees of the equations) \item the generalized B\'ezout number for multihomogeneous systems \item the BKK bound~\cite{Bernstein,Kouchnirenko,Khovanskii_newton} (= mixed volume of the Newton polytopes) \end{enumerate} None of these known homotopy methods is applicable to our problem, as the following simple example shows: Take $m=3$, $p=2$, and $k_1=\cdots=k_6=1$, that is, we seek the $2$-planes in ${\bf C}^5$ which meet six general $3$-planes nontrivially. By formula (1.3) there are $d=5$ solutions. Formulating this in Pl\"ucker coordinates gives $11$ homogeneous equations in ten variables, the five quadrics in display~(\ref{I23}) below and six linear equations~(\ref{sixlin}). A formulation in local coordinates~(\ref{sagbieqs}) has 6 quadratic equations in 6 unknowns, giving a B\'ezout bound of 64. These 6 equations all have the same Newton polytope, which has normalized volume 17, giving a BKK bound of 17. In Section 2 we give two homotopy algorithms which solve our problem in the special case $k_1 = \cdots =k_n = 1$, when the number of solutions equals (1.3). The first algorithm is derived from a Gr\"obner basis for the Pl\"ucker ideal of a Grassmannian and the second from a SAGBI basis for its projective coordinate ring. (See~\cite{CHV} or~\cite[Ch.~11]{Sturmfels_GPCP} for an introduction to SAGBI bases). Both the {\it Gr\"obner homotopy} and {\it SAGBI homotopy} are techniques for finding linear sections of Grassmannians in their Pl\"ucker embedding. In Section 3 we address the general case of our problem, that is, we describe a numerical method for solving the polynomial equations defined by {\it special Schubert conditions}. This is accomplished by applying a sequence of delicate intrinsic deformations, called {\it Pieri homotopies}, which were introduced in~\cite{Sottile_explicit_pieri}. Pieri homotopies first arose in the study of enumerative geometry over the real numbers~\cite{Sottile_real_lines,Sottile_santa_cruz}. For the experts we remark that it is an open problem to find {\it Littlewood-Richardson homotopies}, which would be relevant for solving polynomial equations defined by general Schubert conditions. A main challenge in designing homotopies for the Schubert calculus is that one is not dealing with complete intersections: there are generally more equations than variables. In Section 4 we discuss some of the numerical issues arising from this challenge, and how we propose to resolve them. In Section 5 we discuss applications of these algorithms to control theory and real enumerative geometry. Finally, in Section 6 we present computational results. In closing the introduction let us emphasize that all homotopies described in this paper are optimal in the sense the number of paths to be traced equals the number $d$. This means that for generic input data $K_1,\ldots,K_n$ no paths diverge. \section{Linear equations in Pl\"ucker coordinates} The set of $p$-planes in ${\bf C}^{m+p}$, $\mbox{\em Grass}(p,m+p)$, is called the {\em Grassmannian of $p$-planes in ${\bf C}^{m+p}$}. This complex manifold of dimension $mp$ is naturally a subvariety of the complex projective space ${\bf P}^{\binom{m+p}{p}-1} $. To see this, represent a $p$-plane in ${\bf C}^{m+p}$ as the column space of an $(m+p)\times p$-matrix $ X = (x_{ij})$. The {\it Pl\"ucker coordinates} of that $p$-plane are the maximal minors of $X$, indexed by the set $\binom{[m+p]}{p}$ of sequences $\alpha: 1\leq \alpha_1<\alpha_2<\cdots<\alpha_p\leq m+p$: \begin{equation}\label{minors} [\alpha]\ \longrightarrow\ \det \left[\begin{array}{ccc} x_{\alpha_1\,1}&\cdots& x_{\alpha_1\,p}\\ \vdots&\ddots&\vdots\\ x_{\alpha_p\,1}&\cdots& x_{\alpha_p\,p}\end{array}\right] . \end{equation} This section deals with the ``$k_i = 1$'' case of the problem stated in the Introduction. Given $mp$ general $m$-planes $K_1,\ldots,K_{mp}$, we wish to find all $p$-planes $X$ which meet $K_1,\ldots,K_{mp}$ nontrivially. This geometric condition translates into linear equations in the Pl\"ucker coordinates: Represent $X$ as an $(m+p) \times p$-matrix as above, represent $K_i$ as an $(m+p)\times m$-matrix, and form the $(m+p) \times (m+p)$-matrix $ [\, X \mid K_i \,] $. Then $$ \,X \,\cap \,K_i \,\not= \,\{ 0 \} \qquad \hbox{if and only if} \qquad \det \,[\,X \mid K_i \,] \,\,\,= \,\,\, 0 . $$ Laplace expansion with respect to the first $p$ columns gives \begin{equation}\label{lineqs} \det \, [\, X \mid K_i \,] \quad = \quad \sum_{\alpha\in\binom{[m+p]}{p}} C_{\alpha}^{i} \cdot [\alpha], \end{equation} where $C_{\alpha}^{i}$ is the correctly signed maximal minor of $K_i$ complementary to $\alpha$. Hence our problem is to solve $mp$ linear equations (\ref{lineqs}) on the Grassmannian. The number of solutions is the degree of the Grassmannian in its Pl\"ucker embedding, which is given in (1.3). The Grassmannian is represented either {\it implicitly}, as the zero set of polynomials in the Pl\"ucker coordinates, or {\it parametrically}, as the image of the polynomial map (\ref{minors}). These two representations lead to two different numerical homotopies. The implicit representation gives the Gr\"obner homotopy in Section~\ref{grobnerhomotopy} while the parametric representation gives the SAGBI homotopy in Section~\ref{SAGBIhomotopy}. The first is conceptually simpler but the second is more efficient. In both methods the number of paths to be traced equals the optimal number in (\ref{grassdeg}). \subsection{An example} We describe the two approaches for the case $(m,p)=(3,2)$. The Grassmannian of 2-planes in ${\bf C}^5$ has dimension 6 and is embedded into ${\bf P}^9$. Its degree (\ref{grassdeg}) is five. The Gr\"obner homotopy works directly in the ten Pl\"ucker coordinates: $$ [12],\ [13],\ [14],\ [15],\ [23],\ [24],\ [25],\ [34],\ [35],\ [45]. $$ The ideal $I_{3,2}$ of the Grassmannian in the Pl\"ucker embedding is generated by five quadrics: \begin{equation}\label{I23} \begin{array}{c} \underline{[14][23]} \ -\ [13][24]\ +\ [12][34], \\ \underline{[15][23]}\ -\ [13][25]\ +\ [12][35], \\ \underline{[15][24]}\ -\ [14][25]\ +\ [12][45] , \\ \underline{[15][34]}\ -\ [14][35]\ +\ [13][45] , \\ \underline{[25][34]}\ -\ [24][35]\ +\ [23][45]. \end{array} \end{equation} This set is the reduced Gr\"obner basis for $I_{3,2}$ with respect to any term order which selects the underlined terms as leading terms (see Proposition 2.1 below). Our problem is to compute all $2$-planes which meet six sufficiently general $3$-planes $K_1,\ldots, K_6$ nontrivially. This amounts to solving (\ref{I23}) together with six linear equations \begin{equation}\label{sixlin} \begin{array}{c} C_{12}^{i} \cdot [12] \, + \, C_{13}^{i} \cdot [13] \, + \, C_{14}^{i} \cdot [14] \, + \, C_{15}^{i} \cdot [15] \, + \, C_{23}^{i} \cdot [23] \phantom{\, = \, 0} \\ \, + \, C_{24}^{i} \cdot [24] \, + \, C_{25}^{i} \cdot [25] \, + \, C_{34}^{i} \cdot [34] \, + \, C_{35}^{i} \cdot [35] \, + \, C_{45}^{i} \cdot [45] \,\,\, = \,\,\, 0 , \end{array} \end{equation} for $i=1,\ldots,6 $. This is an overdetermined system of $11$ equations in $10$ homogeneous variables. To solve it we introduce a parameter $t$ into (\ref{I23}) as follows: $$ \begin{array}{l} [14][23]\ -\ t\;\,\cdot [13][24]\ +\ t^2 \cdot [12][34] \quad =\quad 0, \\ {[15][23]\ -\ t^2 \cdot [13][25]\ +\ t^4 \cdot [12][35]}\quad = \quad 0, \\ {[15][24]\ -\ t\;\,\cdot [14][25]\ +\ t^5 \cdot [12][45]}\quad =\quad 0, \\ {[15][34]\ -\ t^2 \cdot [14][35]\ +\ t^4 \cdot [13][45]}\quad = \quad 0, \\ {[25][34]\ -\ t\;\,\cdot [24][35]\ +\ t^2 \cdot [23][45] \quad =\quad 0.} \end{array} \eqno (\ref{I23}') $$ We call (\ref{I23}$'$) the {\it Gr\"obner homotopy} because this is an instance of the flat deformation which exists for any Gr\"obner basis; see~\cite[Theorem 15.17]{Eisenbud_geometry}. The flatness of this deformation ensures that, for almost every complex number $t$, the combined system $(\ref{I23}') \& (\ref{sixlin})$ has five roots. For $t\!=\!0$ the equations (\ref{I23}$'$) are square-free monomials. We decompose their ideal: \begin{equation}\label{primedec} \begin{array}{c} \langle \,\, [14][23],\, [15][23],\, [15][24],\, [15][34],\, [25][34] \,\, \rangle \hspace{2.2in} \\\hspace{1.3in} = \quad \, \langle \, [23], [24], [34] \,\rangle \,\, \cap \,\, \langle \, [15], [23], [34] \,\rangle \,\, \cap \,\, \langle \, [15], [23], [25] \,\rangle \\ \hspace{1.8in} \,\, \cap \,\, \,\, \langle \, [14], [15], [34] \,\rangle \,\, \, \cap \,\,\, \langle \, [14], [15], [25] \,\rangle . \end{array} \end{equation} The five distinct solutions for $t=0$ are computed by setting each listed triple of variables to zero and then solving the six linear equations (\ref{sixlin}) in the remaining seven variables. Thereafter we trace the five solutions from $t=0$ to $t=1$ by numerical path continuation. At $t=1$ we get the five solutions to our original problem. We next describe the SAGBI homotopy. For this we choose the local coordinates $$ X \quad = \quad \left[\begin{array}{ccc} 1 & 0 \\ x_{21} & x_{22} \\ x_{31} & x_{32} \\ x_{41} & x_{42} \\ 0 & 1 \end{array} \right] $$ on the Grassmannian. Substituting $X$ into (\ref{sixlin}) we get six polynomials in six unknowns: \begin{equation}\label{sagbieqs} \begin{array}{ccc} C_{23}^{i} \cdot (x_{21} x_{32} - x_{22} x_{31}) \,+\, C_{24}^{i} \cdot (x_{21} x_{42} - x_{22} x_{41}) \,+\, C_{34}^{i} \cdot (x_{31} x_{42} - x_{32} x_{41}) \\ \,+\, C_{25}^{i} \cdot x_{21} + C_{35}^{i} \cdot x_{31} + C_{45}^{i} \cdot x_{41} + C_{12}^{i} \cdot x_{22} + C_{13}^{i} \cdot x_{32} + C_{14}^{i} \cdot x_{42} + C_{15}^{i}. \\ \end{array} \end{equation} To solve these six equations we introduce a parameter $t$ as follows: $$ \begin{array}{ccc} C_{23}^{i} (x_{21} x_{32} - t \cdot x_{22} x_{31}) + C_{24}^{i} (x_{21} x_{42} - t^2 \cdot x_{22} x_{41}) + C_{34}^{i} (x_{31} x_{42} - t \cdot x_{32} x_{41}) \!\!\! \\ + C_{25}^{i} \cdot x_{21} + C_{35}^{i} \cdot x_{31} + C_{45}^{i} \cdot x_{41} + C_{12}^{i} \cdot x_{22} + C_{13}^{i} \cdot x_{32} + C_{14}^{i} \cdot x_{42} + C_{15}^{i}. \\ \end{array} \eqno (\ref{sagbieqs}') $$ The system (\ref{sagbieqs}$'$) has five complex roots for almost all $t \in {\bf C}$. For $t=0$ we get a {\it generic unmixed sparse system} (in the sense of~\cite{Huber_Sturmfels}) with support $$ {\mathcal A} \quad = \quad \{\, 1, \,x_{21} , \, x_{22} \,,\, x_{31} \,, \, x_{32} \,, x_{41}\,, x_{42}\,,\, x_{21} x_{32} \,, \, x_{21} x_{42}\,, \,x_{31} x_{42} \,\}.$$ We identify ${\mathcal A}$ with a set of ten points in ${\bf Z}^6$. Their convex hull {\it conv}$({\mathcal A})$ is a $6$-dimensional polytope with normalized volume five. We can therefore solve $(\ref{sagbieqs}')$ for $t=0$ using the homotopy method in~\cite{Huber_Sturmfels} or~\cite{CVVerschelde}, provided the input data $K_1,\ldots,K_6$ are sufficiently generic. Tracing the five roots from $t=0$ to $t=1$ by numerical path continuation, we obtain the five solutions to our original problem. \subsection{Gr\"obner homotopy}\label{grobnerhomotopy} We next describe a quadratic Gr\"obner basis for the defining ideal of $\mbox{\em Grass}(p,m+p)$. Let $S$ be the polynomial ring over ${\bf C}$ in the variables $\,[\alpha]\,$ where $\alpha \in \binom{ [m+p] }{p}$. We define a partial order on these variables as follows: $\,[\alpha] \leq [\beta]\,$ if and only if $\alpha_i \leq \beta_i$ for $i=1,\ldots,p$. This partially ordered set is called {\it Young's poset}. Figure~\ref{youngs_poset} shows Young's poset for $(m,p) = (3,2)$. \begin{figure}[htb] $$ \epsfxsize=1.05in\epsfbox{figure1.eps} $$ \caption{Young's poset for $(m,p) = (3,2)$.\label{youngs_poset}} \end{figure} Fix any linear ordering on the variables in $S$ which refines the ordering in Young's poset, and let $\prec$ denote the induced degree reverse lexicographic term order on $S$. Let $I_{m,p}$ be the ideal of polynomials in $S$ which vanish on the Grassmannian, that is, $I_{m,p}$ is the ideal of algebraic relations among the maximal minors of a generic $(m+p) \times p$-matrix $X$. The Gr\"obner homotopy is based on the following well-known result. \begin{prop}\label{prop:inImp} The initial ideal $\,in_\prec(I_{m,p}) \,$ is generated by all quadratic monomials $\,[\,\alpha \,][\,\beta\,] \,$ where $\alpha_i < \beta_i$ and $\alpha_j > \beta_j \,$ for some $i,j \in \{1,\ldots,p\}$. \end{prop} In other words, $\,in_\prec(I_{m,p}) \,$ is generated by products of incomparable pairs in Young's poset. Let ${\mathcal C}_{m,p}$ denote the set of all maximal chains in Young's poset. For example, \begin{eqnarray*} {\mathcal C}_{3,2} &=& \left\{ \, \{[12],[13],[14],[15],[25],[35],[45]\} \,, \,\, \{[12],[13],[14],[24],[25],[35],[45]\}\,,\right. \\ &&\hspace{6.6pt}\{[12],[13],[14],[24],[34],[35],[45]\}\,,\,\, \{[12],[13],[23],[24],[34],[35],[45]\}\,,\\ &&\hspace{5.2pt}\left. \{[12],[13],[23],[24],[25],[35],[45]\}\,\right\} \end{eqnarray*} A standard result in combinatorics~\cite{Stanley_schubert} states that the cardinality of ${\mathcal C}_{m,p}$ equals the number (\ref{grassdeg}). {}From Proposition~\ref{prop:inImp} we read off the following prime decomposition which generalizes~(\ref{primedec}): \begin{equation}\label{genprimedec} in_\prec(I_{m,p}) \quad = \quad \bigcap_{C \in {\mathcal C}_{m,p}} \langle \,\, [\alpha]\,\,: \,\, [\alpha] \not\in C \,\rangle . \end{equation} For a proof of Proposition~\ref{prop:inImp} see~\cite[\S XIV.9]{Hodge_Pedoe} or~\cite[Theorem (4.3)]{Bruns_Vetter} or~\cite[\S 3.1]{Sturmfels_invariant}. In these references one finds an explicit minimal Gr\"obner basis for $I_{m,p}$, which is classically called the set of {\it straightening syzygies}. In the special case $p=2$ the straightening syzygies coincide with the reduced Gr\"obner basis: \begin{prop} If $p=2$ then the reduced Gr\"obner basis of $I_{m,p}$ consists of the three-term Pl\"ucker relations $\,\underline{[i l][k j]} - [i k][j l] + [i j][k l] \,$ where $1 \leq i < j < k < l \leq m+p = m+2$. \end{prop} For $p \geq 3$ the straightening syzygies and the reduced Gr\"obner basis do not coincide, and they are complicated to describe. For our purposes the following coarse description suffices. Let $Std$ be the set of all quadratic monomials in $S$ which do not lie in $in_\prec(I_{m,p})$. The reduced Gr\"obner basis consists of elements of the form \begin{equation}\label{redGB} [\alpha][\beta] \quad - \quad \sum_{[\gamma][\delta] \in Std} E^{\alpha,\beta}_{\gamma,\delta} \cdot [\gamma][\delta] \end{equation} where $[\alpha][\beta]$ runs over all generators of $in_\prec(I_{m,p})$. The constants $\,E^{\alpha,\beta}_{\gamma,\delta} \,$ are integers which can be computed by substituting (\ref{minors}) into (\ref{redGB}) and solving linear equations. The term order $\prec$ can be realized for the ideal $I_{m,p}$ by the following choices of weights. We define the {\it weight} of the variable $[\alpha] = [\alpha_1 \alpha_2 \cdots \alpha_p]$ to be \begin{equation}\label{weights} v_\alpha \quad := \quad - \frac{1}{2} \sum_{1 \leq i < j \leq p} ( \alpha_j - \alpha_i - 1)^2 . \end{equation} If we replace each variable $\,[\alpha]\,$ in (\ref{redGB}) by $\,[\alpha] \cdot t^{v_\alpha}\,$ and clear $t$-denominators afterwards, then we get the {\it Gr\"obner homotopy}: $$ [\alpha][\beta] \quad - \quad \sum_{[\gamma][\delta] \in Std} E^{\alpha,\beta}_{\gamma,\delta} \cdot [\gamma][\delta] \cdot t^{v_\gamma + v_\delta - v_\alpha - v_\beta} \eqno (\ref{redGB}') $$ It can be checked that all exponents $v_\gamma + v_\delta - v_\alpha - v_\beta$ appearing here are positive integers. The special case $(m,p)=(3,2)$ is presented in (\ref{I23}$'$). In the Gr{\"o}bner homotopy algorithm, we first solve systems of $mp$ linear equations, one for each chain $C \in {\mathcal C}_{m,p}$. These systems consist of the $mp$ equations~(\ref{lineqs}), one for each $K_i$, and the $\binom{m+p}{p}-mp-1$ equations $$ [\alpha]\qquad\mbox{for}\qquad [\alpha]\not\in C, $$ suggested by the prime decomposition~(\ref{genprimedec}). Once this is accomplished, we trace each of these $d$ solutions from $t=0$ to $t=1$ in the Gr\"obner homotopy~(\ref{I23}$'$). Clearly, the weights $v_\alpha$ of~(\ref{weights}) are not best possible for any specific value of $m$ and $p$. Smaller weights can be found using Linear Programming, as explained e.g.~in the proof of~\cite[Proposition 1.11]{Sturmfels_GPCP}. Another method would be to adapt the ``dynamic'' approach in~\cite{VGC} to our situation. This is possible since the Gr\"obner basis in (\ref{redGB}) is reverse lexicographic: first deform the lowest variable to zero, then deform the second lowest variable to zero, then the third lowest variable, and so on. \subsection{SAGBI homotopy}\label{SAGBIhomotopy} Let $X = (x_{ij})$ be an $(m+p) \times p$-matrix of indeterminates. We identify the coordinate ring of the Grassmannian with the ${\bf C}$-algebra generated by the $p \times p$-minors of $X$. Call this algebra $R$ and write $[\alpha]( x_{ij} )$ for the minor indexed by $\alpha$. Reinterpreting classical results in~\cite[\S XIV.9]{Hodge_Pedoe}, it was shown in~\cite[Theorem 3.2.9]{Sturmfels_invariant} that these generators form a {\it SAGBI basis} with respect to the lexicographic term order induced from $\,x_{11} > x_{12} > \cdots > x_{1p} > x_{21} > \cdots > x_{m+p,p}$. This means that the {\it initial algebra} ${\bf C}[\,in_>(f)\,:\, f \in R \,]\,$ is generated by the main diagonal terms of the $p \times p$-minors, \begin{equation}\label{inR} {\it in}_> \bigl([\alpha](x_{ij}) \bigr) \quad = \quad x_{\alpha_1,1} \,x_{\alpha_2,2} \,x_{\alpha_3,3} \cdots \cdots x_{\alpha_m,m} . \end{equation} The resulting flat deformation can be realized by replacing $x_{ij}$ with $\,x_{ij} t^{(i-1)(p-j)} \,$ for $t \rightarrow 0$ in the matrix $X$. If we expand $\,[\alpha]( x_{ij} t^{(i-1)(p-j)}) \,$ as a polynomial in $t$, then the lowest term equals $t^{w_a}$ times the main diagonal monomial (\ref{inR}), where $$ w_\alpha \quad := \quad \sum_{j=1}^p \, (\alpha_j - 1)(p - j) . $$ In what follows we multiply that polynomial by $\,t^{-w_\alpha}$. For any $t \in {\bf C}$ consider the algebra $$ R_t \quad := \quad {\bf C} \left[ \,\, t^{-w_\alpha} \cdot [\alpha]( x_{ij} t^{(i-1)(p-j) } ) \,\,:\,\ \alpha \in {\textstyle \binom{[m+p]}{p}} \,\right]. $$ Then $R_1$ is the coordinate ring of the Grassmannian, and $\,R_0\,$ is the algebra generated by the monomials (\ref{inR}). This is a flat deformation of ${\bf C}$-algebras; see~\cite{CHV} and~\cite[\S 11]{Sturmfels_GPCP}. The {\it SAGBI homotopy} is the following system of $mp$ equations: \begin{equation}\label{SAGBIh} \sum_{\alpha \in \binom{[m+p]}{p}} C^{i}_\alpha \cdot t^{-w_\alpha} \cdot [\alpha]( x_{ij} t^{(i-1)(p-j) } ) \quad = \quad 0 \quad \qquad (i=1,\ldots,mp). \end{equation} We reduce the number of variables to $mp$ by introducing local coordinates as follows: $\,x_{ii} = 1 \,$ for $i=1,\ldots,p$ and $\,x_{ij} = 0 \,$ for $i<j$ or $i > m+j $. Our original problem is to solve the system (\ref{SAGBIh}) for $t=1$. The flatness of the family of algebras $R_t$ guarantees that the system (\ref{SAGBIh}) has the same finite number of complex solutions (counting multiplicities) for almost every $t \in {\bf C}$. For $t=0$ we get a system of linear equations in $R_0$: \begin{equation}\label{R0eqs} \sum_{\alpha \in \binom{[m+p]}{p}} C^{i}_\alpha \cdot x_{\alpha_1,1} x_{\alpha_2,2} x_{\alpha_3,3} \cdots x_{\alpha_p,p} \quad \qquad (i=1,\ldots,mp). \end{equation} In order to solve these equations we apply the symbolic-numeric algorithm in~\cite{Huber_Sturmfels}, while taking advantage of the following combinatorial structures described in~\cite[Remark 11.11]{Sturmfels_GPCP}. The common Newton polytope of the equations (\ref{R0eqs}) equals the {\it order polytope} of the product of an $m$-chain and a $p$-chain (Sturmfels, Remark 11.11). We have the following combinatorial result. \begin{prop} The following five numbers coincide: \begin{itemize} \item the right hand side of (\ref{grassdeg}), \item the number of linear extensions of the product of a $m$-chain and a $p$-chain, \item the number of maximal chains in Young's poset, \item the normalized volume of the order polytope, and \item the number of roots in $({\bf C}^*)^{mp}$ of a generic system (\ref{R0eqs}). \end{itemize} \end{prop} The equality of (4) and (5) is a special case of Kouchnirenko's Theorem~\cite{Kouchnirenko}, as all the equations have the same Newton polytope. The order polytope has a distinguished unimodular regular triangulation with simplices indexed by the chains in Young's poset. This regular triangulation is induced by the system of weights given in (\ref{weights}). We may use these weights to define a numerical homotopy for finding all isolated solutions of (\ref{R0eqs}). Once this is accomplished, we trace these roots from $t=0$ to $t=1$ in the homotopy~(\ref{SAGBIh}). \section{Special Schubert conditions}\label{pieri-deformations} We first describe a purely combinatorial method for computing the number $d$ of solution planes. Instead of the algebraic relation (1.2) we shall make use of Young's poset which was introduced in Section 2.2. A cover $[\alpha]\lessdot [\beta]$ in Young's poset determines a unique index $j=j(\alpha,\beta)$ for which $$ \alpha_j +1 \ =\ \beta_j \quad\mbox{and} \quad\alpha_i\ =\ \beta_i \ \ \ \mbox{for} \ \ \ i\neq j. $$ A chain $ \,[\alpha^0]\lessdot [\alpha^1]\lessdot \cdots\lessdot[\alpha^l]\,$ in Young's poset is {\em increasing at $i$} if either $i=1$, or else $i>1$ and $j(\alpha^{i-2},\alpha^{i-1})\leq j(\alpha^{i-1},\alpha^i)$. For instance, $[12]\lessdot [13]\lessdot [14]\lessdot [24]$ is increasing at 1 and 2, but decreasing at 3. Given positive integers $r_0,\ldots,r_a$, the {\em Pieri tree} ${\mathcal T}(r_0,\ldots,r_a)$ consists of all chains of length $r_0+\cdots+r_a$ in Young's poset which start at the bottom element $[1,2,\ldots,p]$ and which increase everywhere, except possibly at $r_0+1,r_0+r_1+1,\ldots,r_0+\cdots+r_{a-1}+1$. Here we include all initial segments of such chains and we order the chains by inclusion. Label a node in the Pieri tree by the endpoint of the chain which that node represents. Then the sequence of labels from the root to that node is the chain which that node represents. For example, here is ${\mathcal T}(2,2)$ when $m=5$, $p=2$: $$ \epsfxsize=1.05in\epsfbox{figure2.eps} $$ To compute the number $d$, partition the integer sequence $k_1,\ldots,k_n$ into three parts $r_0,\ldots,r_a$, $r'_0,\ldots,r'_{a'}$, and $q$. Then $d$ is the number of pairs $(R,S)$ where $R$ is a leaf of ${\mathcal T}(r_0,\ldots,r_a)$, $S$ is a leaf of ${\mathcal T}(r'_0,\ldots,r'_{a'})$, and the endpoints $[\alpha]$ of $R$ and $[\alpha']$ of $S$ satisfy {\it Pieri's condition}: \begin{equation}\label{eq:pieri} \alpha'_1\leq m+p+1-\alpha_p <\alpha'_2\leq \cdots<\alpha'_p\leq m+p+1-\alpha_1 \end{equation} Call this set of pairs $\mbox{\em Sols} = \mbox{\em Sols}(r_0,\ldots,r_a;r'_0,\ldots,r'_{a'})$. For instance, $d=6$ for the sequence $2,2,2,2,2$ with $(m,p)=(5,2)$: Of the 9 pairs $(R,S)$ of leaves of ${\mathcal T}(2,2)$, only the following 6 satisfy~(\ref{eq:pieri}): (Here we represent a leaf by its label.) \begin{equation}\label{Sols} \left\{ ([25],[34]),\ ([34],[25]),\ ([25],[25]),\ ([25],[16]),\ ([16],[25]), \ ([16],[16]) \right\}. \end{equation} This combinatorial rule for the number $d$ gives the same answer as the algebraic rule (1.2) because the Pieri tree and Pieri's condition~(\ref{eq:pieri}) encode the structure of the cohomology ring ${\mathcal A}_{m,p}$ with respect to its Schur basis $\{S_\lambda \mid p\geq \lambda_1\geq \cdots\geq \lambda_{p+1}=0\}$; see~\cite[\S I]{Macdonald_symmetric} or~\cite[\S 9.4]{Fulton_tableaux}. Specifically, $$ \prod_{i=0}^a h_{r_i} \ =\ \sum_{\lambda_1+\cdots +\lambda_p=r_0+\cdots+r_a} K_{\lambda,(r_0,\ldots,r_a)}\cdot S_\lambda, $$ where $K_{\lambda,(r_0,\ldots,r_a)}$ is the number of leaves in ${\mathcal T}(r_0,\ldots,r_a)$ with label $$ [\alpha(\lambda)]\quad :=\quad [\lambda_p+1,\lambda_{p-1}+2,\ldots,\lambda_1+p]. $$ The numbers $K_{\lambda,(r_0,\ldots,r_a)}$ are called {\it Kostka numbers}. In ${\mathcal A}_{m,p}$ we calculate \begin{eqnarray*} \prod_{i=1}^n h_{k_i} &=& \left(\prod_{i=0}^a h_{r_i}\right)\cdot \left(\prod_{j=0}^{a'} h_{r'_j}\right)\cdot h_q\\ &=& \sum_{\lambda,\mu}K_{\lambda,(r_0,\ldots,r_a)} K_{\mu,(r'_0,\ldots,r'_{a'})}\cdot S_\lambda\cdot S_\mu\cdot h_q. \end{eqnarray*} We evaluate this expression with Pieri's formula (Proposition~\ref{prop:triple} below): If $\lambda_1+\cdots+\lambda_p+\mu_1+\cdots+\mu_p+q=mp$, then $S_\lambda\cdot S_\mu\cdot h_q$ is either $(h_m)^p$ or 0 depending upon whether or not $$ \lambda_p\leq m-\mu_1 \leq \lambda_2\leq\cdots \leq \lambda_1 \leq m-\mu_p. \eqno (\ref{eq:pieri}') $$ The condition~(\ref{eq:pieri}$'$) is equivalent to~(\ref{eq:pieri}) under the transformation $\,\lambda\leftrightarrow[\alpha(\lambda)]$. These methods correctly enumerate the $p$-planes which meet each $K_1,\ldots,K_n$ nontrivially because, under the isomorphism between ${\mathcal A}_{m,p}$ and the cohomology ring of $\mbox{\em Grass}(p,m+p)$, the indeterminate $h_{k_i}$ corresponds to the cohomology class Poincar\'e dual to $\Omega_{K_i}$, the set of $p$-planes which meet $K_i$ nontrivially. Moreover, $(h_m)^p$ represents the class dual to a point. The Pieri tree models certain intrinsic deformations (described in \S 3.3 and \S \ref{explicit_pieri}) of the Grassmannian which establish this isomorphism, and which we shall use for computing the $p$-planes which meet each $K_1,\ldots,K_n$ nontrivially. \subsection{Basics on Schubert varieties} For vectors $f_1,\ldots,f_j$ in ${\bf C}^{m+p}$, let $\langle f_1,\ldots,f_j\rangle$ be their linear span. Fix the columns $e_1,\ldots,e_{m+p}$ of the identity matrix as a standard basis for ${\bf C}^{m+p}$. For $\alpha\in\binom{[m+p]}{p}$, define $\alpha^\vee\in\binom{[m+p]}{p}$ by $\alpha^\vee_j := m+p+1-\alpha_{p+1-j}$. A sequence $\alpha\in\binom{[m+p]}{p}$ determines a {\em Schubert variety} $$ \Omega_\alpha\quad:=\quad \{X\in\mbox{\em Grass}(p,m+p)\, \, \mid \,\, \dim X\cap\langle e_1,\ldots,e_{\alpha^\vee_j}\rangle \geq j \,\,\, \hbox{for} \,\, 1\leq j\leq p\}. $$ This variety has complex codimension $|\alpha|:=\alpha_1-1+\alpha_2-2+\cdots+\alpha_p-p$. Similarly define $$ \Omega'_\alpha\quad:=\quad \{X \in\mbox{\em Grass}(p,m+p)\, \, \mid \,\, \dim X \cap\langle e_{\alpha_j},\ldots,e_{m+p}\rangle \geq p+1-j \,\,\,\, \hbox{for} \,\, 1\leq j\leq p\}. $$ For a linear subspace $N$ of ${\bf C}^{m+p}$ of dimension $m+1-q$, define the {\em special Schubert variety} $$ \Omega_N \quad := \quad \{X\in\mbox{\em Grass}(p,m+p) \,\, \mid \,\, \dim X \cap N \geq 1\}. $$ This has codimension $q$. If $N=\langle e_1,\ldots,e_{m+1-q}\rangle$, then $\Omega_N = \Omega_{[1,\ldots,p-1,p+q]}$. The special Schubert variety $\Omega_N$ is cut out by the system of $\binom{m+p}{q-1}$ polynomial equations: \begin{equation}\label{maxminors} X\ \in \ \Omega_N \quad \Longleftrightarrow \quad \mbox{all maximal minors of }\left[ X\mid N\right] \, \mbox{are zero,} \end{equation} where $X\in \mbox{\em Grass}(p,m+p)$ is represented by a $(m+p)\times p$-matrix. The Laplace expansion of these equations in terms of the Pl\"ucker coordinates of $X$ define $\Omega_N$ as a subscheme of $\mbox{\em Grass}(p,m+p)$. These equations are redundant: select $ m\! + \! 1 \! - \! q $ rows of $\left[ X\mid N\right]$ such that the corresponding maximal minor of $N$ is invertible. Consider the set of maximal minors of $\left[ X\mid N\right]$ which cover all the rows selected. This gives $\binom{p}{q-1}$ polynomial equations which generate the same ideal as all $\binom{m+p}{q-1}$ minors of $\left[ X\mid N\right]$. For a purely set-theoretic (but not scheme-theoretic) representation of $\Omega_N$ a further substantial reduction in the number of equations is possible using the results of~\cite{Bruns_Schwanzl}. An intersection $Y\cap Z$ of subvarieties is {\em generically transverse} if every component of $Y\cap Z$ has an open subset along which $Y$ and $Z$ meet transversally. In this case the following identity in the cohomology ring holds: $$ [Y \cap Z]\quad = \quad [Y]\cdot[Z], $$ where $[W]$ denotes the cycle class of a subvariety $W$. By Kleiman's Transversality Theorem~\cite{Kleiman}, subvarieties of ${\it Grass}(p,m+p)$ in general position meet generically transversally. Transversality and generic transversality coincide when $Y\cap Z$ is finite. \begin{prop}[Hodge and Pedoe, 1952, Theorem III in \S XIV.4]\label{prop:triple} \hfill \break Let $\alpha,\,\alpha'\in\binom{[m+p]}{p}$ with $|\alpha|+|\alpha'|+q=mp$ and let $N$ be a linear subspace of ${\bf C}^{m+p}$ with dimension $m+1-q$ none of whose Pl\"ucker coordinates vanish. Then the intersection \begin{equation}\label{triple_int} \Omega_\alpha\,\cap\,\Omega'_{\alpha'} \, \cap \, \Omega_N \end{equation} either is transverse consisting of a single $p$-plane or is empty, depending upon whether or not~(\ref{eq:pieri}) holds. \end{prop} \noindent{\sc Proof and Algorithm: } The intersection $\Omega_\alpha \cap\Omega'_{\alpha'}$ is nonempty if and only if $\alpha'_j\leq \alpha^\vee_j$ for $j = 1,\ldots,p$. These are the weak inequalities in~(\ref{eq:pieri}). We shall assume that they hold in what follows. The $p$-planes in $\Omega_\alpha \cap\Omega'_{\alpha'}$ are represented by $(m+p)\times p$-matrices $X=(x_{ij})$ such that \begin{equation}\label{eq:alpha} x_{i,j}\ =\ 0\qquad\mbox{for}\qquad i<\alpha'_j \quad \mbox{or} \quad \alpha^\vee_j < i . \end{equation} Consider the nonzero coordinate subspaces $\,C_j :=\langle e_{\alpha'_j},\ldots,e_{\alpha^\vee_j}\rangle$, set $\,C := C_1 + \cdots + C_p $, and note that $\, p+q = \sum_j \dim (C_j) \geq dim(C)$. {}From~(\ref{eq:alpha}) we see that $\,X \in \Omega_\alpha\cap \Omega'_{\alpha'}\,$ implies $\, X\subseteq C \,$ and hence $\, N \cap X \subseteq N \cap C$. Therefore the triple intersection~(\ref{triple_int}) is nonempty only if the following equivalent conditions hold: $$ dim(C\cap N) \geq 1 \,\, \iff\,\, dim(C) = p+q \,\, \iff \, \mbox{the sum} \,\,\,C = C_1 + \cdots + C_p \,\,\, \mbox{is direct} $$ $$ \iff \, \alpha^\vee_j < \alpha'_{j+1} \,\,\mbox{for} \,\, j=1,\ldots,p-1 \, \iff \, \mbox{(3.1) holds.} $$ In this case we determine$\,C \cap N \,$ by computing vectors $\,g_j\in C_j \,$ such that $\,C \cap N = \langle g_1\oplus g_2\oplus\cdots\oplus g_p \rangle $. (This computation is the ``algorithm'' part in this proof.) The desired $p$-plane $X$ satisfies $\, C \cap N = X \cap N$, and, in view of~(\ref{eq:alpha}), this implies $X = \langle g_1,\ldots,g_p\rangle$. Transversality of~(\ref{triple_int}) is verified in local coordinates for $\Omega_\alpha\cap \Omega'_{\alpha'}$ by considering $p+q-1$ independent linear forms which vanish on $N$.\qed For $\alpha\in \binom{[m+p]}{m}$ define $\lambda(\alpha)$ by $\lambda(\alpha)_j:= \alpha_j-j$ for $1\leq j\leq p$. Then $S_{\lambda(\alpha)}$ represents the cycle class of $\Omega_\alpha$ (equivalently, of $\Omega'_\alpha$). If $\dim N=m+1-q$, then $h_q$ is the cycle class of $\Omega_N$. Suppose $|\alpha|+|\alpha'|+q=mp$. Then Proposition~\ref{prop:triple} implies the following identity in ${\mathcal A}_{m,p} $: $$ S_{\lambda(\alpha)}\cdot S_{\lambda(\alpha')}\cdot h_q\ =\ \left \{\begin{array}{cl}(h_m)^p&\quad\mbox{if } (\ref{eq:pieri}) \mbox{ holds} \\ 0&\quad\mbox{otherwise}\end{array}\right.. $$ This identity implies (via Poincar\'e duality) that $$ S_{\lambda(\alpha)}\cdot h_q \quad = \quad \sum S_{\lambda(\beta)}, \label{pieri_product} $$ the sum over all $\beta$ with $|\beta|=|\alpha|+q$ for which $\alpha,\beta^\vee$ satisfy Pieri's condition~(\ref{eq:pieri}). Call this set $\alpha*q$, which is also the set of endpoints of increasing chains of length $q$ in Young's poset that begin at $\alpha$. This last form has geometric content. In~\cite{Sottile_explicit_pieri}, explicit deformations were given that transform the irreducible intersection $\Omega_\alpha \cap \Omega_N$ into the cycle $\sum_{\beta\in\alpha*q} \Omega_\beta$. Moreover, the branching of the components of the cycles in these deformations reflects the branching among these increasing chains above $\alpha$. This process may be iterated to transform an intersection of several special Schubert varieties into a sum of triple intersections of the form~(\ref{triple_int}), indexed by pairs $(R,S)\in\mbox{\em Sols}$. {}From this sum, we obtain a set of start solutions indexed by {\em Sols}. Also, every intermediate cycle in these deformations consists of the same number (counting multiplicities) of $p$-planes. The Pieri homotopy begins with one of the start solutions and uses numerical path continuation to trace the sequence of curves defined by these deformations which connect that start solution to a solution of the original problem. \subsection{Pieri homotopy algorithm}\label{pieri_alg} Given linear subspaces $K_1,\ldots,K_n$ in general position with $\dim K_i=m+1-k_i$ and $k_1+\ldots+k_n=mp$, first partition $K_1,\ldots,K_n$ into three lists: $$ L_0,\ldots,L_a,\qquad L'_0,\ldots,L'_{a'},\qquad N $$ where $\dim L_i=m+1-r_i$, $\dim L'_i=m+1-r'_i$, and $\dim N=m+1-q$. Construct the Pieri trees ${\mathcal T}(r_0,\ldots,r_a)$ and ${\mathcal T}(r'_0,\ldots,r'_{a'})$, and form the set {\it Sols}. Change coordinates so that $L_0=\langle e_1,\ldots,e_{m+1-r_0}\rangle$ and $L'_0 =\langle e_{p+r'_0},\ldots,e_{m+p}\rangle$. Set $\tau := \max \{r_1+\cdots+r_a,\ r'_1+\cdots+r'_{a'}\}$. Given a chain $R$ in the Pieri tree and a positive integer $k$, let $R(k)$ be the $k$th element in that chain, or, if $k$ exceeds the length of $R$, then let $R(k)$ be the endpoint of $R$. For each $(R,S)\in{\it Sols}$ and $k$ from $\tau$ to $0$ we shall construct (in Definition~\ref{def:ZRkt} below) one-parameter families $Z_{R,k}(t)$ and $Z'_{S,k}(t)$ of pure-dimensional subvarieties of ${\it Grass}(p,m+p)$ with the following properties: \begin{enumerate} \item $Z_{R,k}(t)\subset \Omega_{R(r_0+k)}$ and $Z'_{S,k}(t)\subset \Omega'_{S(r'_0+k)}$. \item For $t=0$ or $1$ and each $k$, $Z_{R,k}(t)\cap Z'_{S,k}(t)\cap \Omega_N$ is transverse and 0-dimensional. \item $Z_{R,\tau}(t) = \Omega_{R(r_0+\tau)}$ and $Z'_{S,\tau}(t) = \Omega'_{S(r_0+\tau)}$ . \item $Z_{R,k+1}(1)$ is a component of $Z_{R,k}(0)$. Likewise, $Z'_{S,k+1}(1)$ is a component of $Z'_{S,k}(0)$. \item $Z_{R,0}(1) = \Omega_{L_0}\cap\cdots\cap\Omega_{L_a}$ and $Z'_{S,0}(1) = \Omega_{L'_0}\cap\cdots\cap\Omega_{L'_{a'}}$. \end{enumerate} Property 4 is a consequence of Proposition~\ref{prop:pieri}, the others follow from the assumption of genericity and the definition (Definition~\ref{def:ZRkt}) of the families $Z_{R,k}(t)$ and $Z'_{S,k}(t)$. By 2, the family $W_{(R,S),k}(t)$ over ${\bf C}$ whose fibre at general $t$ (including $t=0$ and $t=1$) is $$ W_{(R,S),k}(t) \quad:=\quad Z_{R,k}(t)\cap Z'_{S,k}(t)\cap\Omega_N $$ consists of a finite number of curves. In fact, for general $t$ (including $t=0$ and $t=1$), $W_{(R,S),k}(t)$ has the following general form (see Definition~\ref{def:ZRkt} for the precise form): $$ W_{(R,S),k}(t)\ =\ \Omega_\alpha\cap\Omega'_{\alpha'}\cap \Omega_{M_1}\cap\cdots\cap\Omega_{M_s}, $$ where $M_1,\ldots, M_s$ are linear subspaces with $M_s=N$, which depend upon $R,S,k$, the subspaces $L_1,\ldots,L_a$, $L'_1,\ldots,L'_{a'}$, and at most two of the $M_i$ depend upon $t$. Also, $\alpha$ and $\alpha'$ depend upon $R,S$, and $k$ with the typical case being $\alpha= R(r_0+k)$ and $\alpha'= S(r'_0+k)$. The numerical homotopy defined by the curves $W_{(R,S),k}(t)$ may be expressed in a parameterization $X=(x_{i,j})$ of an open subset of $\Omega_\alpha\cap\Omega'_{\alpha'}$: \begin{equation}\label{loccord} x_{i,j}\ =\ 0\qquad\mbox{if}\qquad i<\alpha'_j \quad\mbox{or}\quad \alpha^\vee_j<i\quad\qquad\mbox{and}\quad\qquad x_{\delta_j,\,j}\ =\ 1, \end{equation} where $\delta:=S(r'_0+\tau)$. The equations for $W_{(R,S),k}(t)$ are then $$ \mbox{maximal minors }\left[ X\mid M_i\right]\ =\ 0 \qquad i=1,\ldots,s. $$ The curves of $W_{(R,S),k}(t)$ define the sequences of homotopies in the Pieri homotopy algorithm as follows: For $(R,S)\in {\it Sols}$, let $X_{(R,S),\tau}$ be the (unique by Proposition~\ref{prop:triple}) $p$-plane in $\Omega_{R(r_0+\tau)}\cap\Omega'_{S(r'_0+\tau)}\cap\Omega_N = W_{(R,S),\tau}(1)$. By 3 and 4, $X_{(R,S),\tau}\in W_{(R,S),\tau-1}(0)$ and hence lies on a unique curve in $W_{(R,S),\tau-1}(t)$. Use numerical path continuation to trace this curve from $t=0$ to $t=1$ to obtain $X_{(R,S),\tau-1}$, which is a point of $W_{(R,S),\tau-2}(0)$, by 4. Then $X_{(R,S),\tau-1}$ lies on a unique curve in $W_{(R,S),\tau-2}(t)$, which we trace to find $X_{(R,S),\tau-2}\in W_{(R,S),\tau-2}(1)$. Continuing this process, after tracing $\tau$ curves, we obtain $X_{(R,S),0}\in W_{(R,S),0}(1)$, which is a solution to the original system, by 5. We show shall prove in Theorem~\ref{allsols} that $\{X_{(R,S),0}\mid (R,S)\in {\it Sols}\}$ consists of all the solutions to the original system. \subsection{Definition of the moving cycles $Z_{R,k}(t)$} The cycle $Z_{R,k}(t)$ will depend upon the choice of a general upper triangular $(m+p)\times (m+p)$-matrix $F$ with 1's on its anti-diagonal, $$ \left(\begin{array}{ccc}*&*&1\\ {}*&\hspace{1pt}\raisebox{0pt}{.}\raisebox{2.2pt}{.}% \raisebox{4.4pt}{.} \\1&&0\end{array}\right), $$ the $k$th link in the chain $R$, and the data $L_1,\ldots,L_a$. The key ingredient of this definition of $Z_{R,k}(t)$ is the construction of a one-parameter family of linear subspaces $\Lambda_i(t)$ in Definition~\ref{def:Lambda}, which depends upon $F$. The matrix $F$ is fixed throughout the algorithm, its purpose is that $\langle e_{m+p-j},\ldots,e_{m+p}\rangle$ equals the span of the first $j$ columns of $F$, and these columns are in general position with $e_1,\ldots,e_{m+p}$. The subtle linear degeneracies of $\Lambda_i(t)$ as $t\rightarrow 0$ are at the heart of this homotopy algorithm, as well as the explicit proof of Pieri's formula~\cite[Theorem~3.6]{Sottile_explicit_pieri}, which we state below (Proposition~\ref{prop:pieri}). \begin{defn}\label{def:ZRkt}\mbox{\ } \begin{enumerate} \item If $r_1+\cdots+r_a\leq k$, then set \ $Z_{R,k}(t)\ :=\ \Omega_{R(r_0+\tau)}$. \item Otherwise, define $c$ by $r_1+\cdots+r_{c-1}\leq k<r_1+\cdots+r_c$, and set $i:= k-r_1-\cdots-r_{c-1}$, $\alpha:= R(r_0+r_1+\cdots+r_{c-1})$, and $\beta:= R(r_0+k)$. \begin{enumerate} \item If $i>0$ and $\beta_p>\alpha_p$, then $\beta+(0,\ldots,0,r_c-i) = R(r_0+\cdots+r_c)$, and we set $$ Z_{R,k}(t)\ :=\ \Omega_{R(r_0+\cdots+r_c)} \cap \Omega_{L_{c+1}}\cap\cdots\cap\Omega_{L_a}. $$ \item Otherwise, let $\Lambda_i(t)$ be the 1-parameter family of linear subspaces given by Definition~\ref{def:Lambda}, where we let $L:= L_c$ and $r:= r_c$. If $i=0$, then $\Lambda_0(1)=L_c$, $\alpha=\beta=R(r_0+k)$, and we set $$ Z_{R,k}(t)\ :=\ \Omega_{R(r_0+k)}\cap \Omega_{\Lambda_0(t)} \cap \Omega_{L_{c+1}}\cap\cdots\cap\Omega_{L_a}. $$ If $i>0$, let $j$ be maximal such that $\beta_j>\alpha_j$. Then $j<p$ as $j=p$ is case 2(a). Set $$ Z_{R,k}(t)\ :=\ \Omega_{R(r_0+k)}\cap \Omega_{\Lambda_i(t)\cap\langle e_1,\ldots,e_{\beta^\vee_{p+1-j}}\rangle} \cap \Omega_{L_{c+1}}\cap\cdots\cap\Omega_{L_a}. $$ \end{enumerate} \end{enumerate} We define $Z'_{S,k}(t)$ similarly, but with the matrix $F$ replaced by a lower triangular matrix with 1's on its diagonal, and $\langle e_1,\ldots,e_{\beta^\vee_{p+1-j}}\rangle$ replaced by $\langle e_{\beta_j},\ldots,e_{m+p}\rangle$. \end{defn} \begin{defn}\label{def:Lambda} Let $F$ be an upper triangular matrix with 1's on the anti-diagonal and $L$ be a general $(m+1-r)$-plane, represented as a $(m+p)\times(m+1-r)$-matrix with columns $l_1,\ldots,l_{m+1-r}$. Construct a $(m+p)\times p$-matrix $U=(u_1,\ldots,u_p)$ as follows: Reverse the last $m+p-\alpha_p$ columns of $F$, then remove the columns indexed by $\alpha^\vee_1,\ldots,\alpha^\vee_p$. For each $0\leq i< r$, define a one-parameter family of $(m+p)\times(m+1-r)$-matrices $\Lambda_i(t)$ for $t\in{\bf C}$ as follows: \begin{enumerate} \item If $i=0$, then the $b$th column of $\Lambda_0(t)$ is $t\cdot l_b + (1-t)\cdot u_b$. \item For $0<i< r$, the $b$th column of $\Lambda_i(t)$ is $$ \begin{array}{lcl} t\cdot u_{b+i-1} + (1-t)\cdot u_{b+i} &\ & b+i-1<\alpha_p-p\\ t\cdot u_{b+i-1} + (1-t)\cdot u_{p+1+i-r} && b+i-1=\alpha_p-p\\ u_{b+i-1} && b+i-1>\alpha_p-p \end{array} $$ \end{enumerate} \end{defn} \subsection{An example} We give an example illustrating these definitions and the Pieri homotopy algorithm. Let $L_0,L_1,L'_0,L'_1$, and $N$ be general 4-planes in ${\bf C}^7$. We give a sequence of homotopies $W_{(R,S),k}(t)$ for $k=2,1,0$ for finding one of the six $2$-planes which meet each of the five given 4-planes nontrivially. Here, $(m,p)=(5,2)$ and $k_1=\cdots=k_5=2$ so that $\tau=2$. Construct the set {\em Sols} as in (\ref{Sols}). Let $(R,S)\in\mbox{\em Sols}$ be the following two sequences: $$ R\ :=\ [12]\lessdot[13]\lessdot[14]\lessdot[24]\lessdot[25],\qquad S\ :=\ [12]\lessdot[13]\lessdot[14]\lessdot[15]\lessdot[16]. $$ Let $e_1,\ldots,e_7$ be the columns of a $7\times 7$-identity matrix, a basis for ${\bf C}^7$. Suppose that $L_0=\langle e_4,e_5,e_6,e_7\rangle$ and $L'_0=\langle e_1,e_2,e_3,e_4\rangle$ and represent $L_1$ and $L'_1$ as $7\times 4$-matrices. Then $\Omega_{[14]}\cap\Omega_L\cap\Omega'_{[14]}\cap \Omega_{L'}\cap\Omega_N$ is the set of 2-planes which meet all five linear subspaces nontrivially. We first find the plane $X_{(R,S),2}\in \Omega_{[25]}\cap\Omega'_{[16]}\cap\Omega_N$, using the algorithm in the proof of Proposition~\ref{prop:triple}. Suppose that $N$ has the form $$ \left[ \begin{array}{c} n\\\hline I\end{array}\right], $$ where $I$ is the $4\times 4$ identity matrix, and $n$ is a $3\times 4$-matrix. In this case, $C_1 = \langle e_1,e_2,e_3\rangle$ and $C_2= \langle e_6\rangle$, hence $C = \langle e_1,e_2,e_3,e_6\rangle$. Thus the intersection $C\cap N$ is generated by the third column of $N$, and so $X_{(R,S),2}$ is represented by the matrix: $$ X_{(R,S),2}\quad=\quad \left[\begin{array}{cc} n_{13} & 0 \\ n_{23} & 0 \\ n_{33} & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 1 \\ 0 & 0 \\ \end{array}\right]. $$ Following Definition~\ref{def:Lambda}, we have: \begin{eqnarray*} \Lambda_0(t) &=&\langle tl_1 + (1-t)u_1, tl_2 + (1-t)u_2, tl_3 + (1-t)u_3, tl_4+(1-t)u_4\rangle,\\ \Lambda_1(t) &=&\langle tu_1+(1-t)u_2,tu_2+(1-t)u_5,u_3,u_4\rangle. \end{eqnarray*} $\Lambda'_i(t)$ is defined similarly. We describe the families $W_{(R,S),k}(t)$ for $k=2,1,0$ in local coordinates for $\Omega_{[14]}\cap \Omega'_{[14]}$ determined by the sequence $[16]$: $$ X \quad = \quad \left[\begin{array}{cc} 1 & 0 \\ x_{21} & 0 \\ x_{31} & 0 \\ x_{41} & x_{42} \\ 0 & x_{52} \\ 0 & 1 \\ 0 & x_{72} \\ \end{array} \right] \ . $$ The family $W_{(R,S),2}(t)$ is the constant family $\{X_{(R,S),2}\} = \Omega_{25}\cap\Omega'_{16}\cap \Omega_N$. Assuming that $n_{13}$, which is the $[1457]$th Pl\"ucker coordinate of $N$, is non-zero, then $X_{(R,S),2}$ may be expressed in these local coordinates: $$ X_{(R,S),2} \quad = \quad \left[\begin{array}{cc} 1 & 0 \\ n_{23}/n_{13} & 0 \\ n_{33}/n_{13} & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 1 \\ 0 & 0 \\ \end{array} \right] \ . $$ When $k=1$, first consider the definition of $Z_{R,1}(t)$. Here we are in case 2(b) with $\beta=[24]$ and $i>0$, so that $\beta^\vee=[46]$. Since $\Lambda_1(t)\subset \langle e_1,\ldots,e_6\rangle$, we have $Z_{R,1}(t)\ =\ \Omega_{[24]}\cap \Omega_{\Lambda_1(t)}$. For the definition of $Z'_{S,1}(t)$, we are in case 2(a), so that $Z'_{S,1}(t)\ =\ \Omega'_{[16]}$. Hence $$ W_{(R,S),1}(t)\ =\ \Omega_{[24]}\cap\Omega_{\Lambda_1(t)} \cap\Omega'_{[16]}\cap \Omega_N. $$ This has 3 linear equations $x_{42} = x_{52} = x_{72} = 0$, which describe $\Omega_{[24]}\cap\Omega'_{[16]}$, and 7 non-trivial equations, the vanishing of the maximal minors of $[ X\mid \Lambda_1(t)]$ and $[ X\mid N]$, which describe $\Omega_{\Lambda_1(t)}\cap \Omega_N$. For $k=0$, $i=0$ and we are in case 2(b) for both $Z_{R,0}(t)$ and $Z'_{S,0}(t)$ so that $$ W_{(R,S),0}(t)\ =\ \Omega_{[14]}\cap\Omega_{\Lambda_0(t)} \cap\Omega'_{[14]}\cap\Omega_{\Lambda'_0(t)}\cap \Omega_N. $$ This has 21 non-trivial equations, the vanishing of the maximal minors of $\left[ X\mid \Lambda_0(t)\right]$, $\left[ X\mid \Lambda'_0(t)\right]$, and $\left[ X\mid N\right]$. \subsection{Proof of correctness}\label{explicit_pieri} We describe the Pieri deformations linking the families $Z_{R,k}(t)$ for $k$ from $r_1\!+\!\cdots\!+\!r_{c-1}$ to $r_1+\cdots+r_c$, which establishes Property 4 of $Z_{R,k}(t)$ in \S\ref{pieri_alg}. We also show that the set $\{X_{(R,S),0}\mid (R,S)\in {\it Sols}\}$ consists of all the solutions to the original system. Consider the dynamic part of $Z_{R,k}(t)$, namely whichever of $$ \Omega_{R(r_0+\cdots+r_c)},\quad \Omega_{R(r_0+k)}\cap \Omega_{\Lambda_0(t)}, \quad\mbox{or}\quad \Omega_{R(r_0+k)}\cap \Omega_{\Lambda_i(t)\cap\langle e_1,\ldots,e_{\beta^\vee_{p+1-j}}\rangle}, $$ appeared in the definition of $Z_{R,k}(t)$. We call this cycle $Y_{\alpha,\beta,L}(t)$, where $L:= L_c$ and $\alpha=R(r_0+\cdots+r_{c-1})$, and $\beta=R(r_0+k)$. For $\beta\in \alpha*i$ and $\gamma \in \alpha*(i+1)$ write $\beta\prec_\alpha\gamma$ if $\gamma$ covers $\beta$ and $j(\beta,\gamma)\geq j(\alpha,\beta):= \max\{j\mid\beta_j>\alpha_j\}$. This partitions $\alpha*(i+1)$ into sets $\{\gamma\mid \beta\prec_\alpha\gamma\}$ for $\beta\in\alpha*i$. \begin{prop}\cite[Theorem~3.6]{Sottile_explicit_pieri}\label{prop:pieri} \mbox{ } Let $\alpha,\beta,i,r,L,\Lambda_i(t)$, and $Y_{\alpha,\beta,L}(t)$ be as above. Then \begin{enumerate} \item For all $t$, $\Omega_\alpha\cap \Omega_{\Lambda_0(t)}$ is generically transverse. \item $Y_{\alpha,\beta,L}(t)$ is free of multiplicities for all $t$ and irreducible for $t\neq 0$. \item If $i\neq r-1$, then $Y_{\alpha,\beta,L}(0)=\sum_{\beta\prec_\alpha\gamma} Y_{\alpha,\gamma,L}(1)$. \item If $\beta\in \alpha*(r-1)$, then $Y_{\alpha,\beta,L}(0)=\sum_{\beta\prec_\alpha\gamma} \Omega_\gamma$. \end{enumerate} \end{prop} By 3, the cycle class of $\sum_{\beta\in \alpha*i}Y_{\alpha,\beta,L}(t)$ is independent of $i$ and $t$, and it equals the cycle class of $\Omega_\alpha\cap \Omega_{\Lambda_0(1)}=\Omega_\alpha\cap \Omega_L$. By 4, we see that the cycle classes of $\Omega_\alpha\cap \Omega_L$ and $\sum_{\beta\in \alpha*r}\Omega_\beta$ coincide, furnishing another proof of Pieri's formula. Property 4 of $Z_{R,k}(t)$ follows from assertion 3. \begin{thm}\label{allsols} When $K_1,\ldots,K_n$ are generic, the Pieri homotopy algorithm finds all $p$-planes which meet each $K_1,\ldots,K_n$ nontrivially. That is, $$ \{X_{(R,S),0}\mid (R,S)\in {\it Sols}\}\quad=\quad \Omega_{K_1}\cap\cdots\cap\Omega_{K_n}. $$ \end{thm} \noindent{\bf Proof. } Note that for any $R,S\in{\it Sols}$, the families $Z_{R,k}(t), Z'_{S,k}(t)$, and $W_{(R,S),k}(t)$ depend only upon the initial segments $R(0),\ldots,R(r_0+k)$ and $S(0),\ldots,S(r'_0+k)$ of $R$ and $S$. By construction, the original system $\Omega_{K_1}\cap\cdots\cap\Omega_{K_l}$ coincides with $W_{(R,S),0}(1)$, for any $(R,S)\in\mbox{\em Sols}$. We inductively construct chains $R\in {\mathcal T}(r_0,\ldots,r_a)$ and $S\in {\mathcal T}(r'_0,\ldots,r'_{a'})$, and $p$-planes $X_k$ for $0\leq k\leq \tau$ such that $$ X_{k+1}\ \in \ W_{(R,S),k}(0)\cap W_{(R,S),k+1}(1) $$ and $X_k, X_{k+1}$ lie on the same curve of $W_{(R,S),k}(t)$. Then $X_\tau$ is the start solution $X_{(R,S),\tau}$, which shows that $X_0\in\{X_{(R,S),0}\mid (R,S)\in {\it Sols}\}$. First set $R(0),\ldots, R(r_0)$ to be the unique chain from $[1,\ldots,p]$ to $[1,\ldots,p-1,p+r_0]$, and similarly for $S(0),\ldots,S(r'_0)$. Then $X_0\in W_{(R,S),0}(1)$ and hence lies on a unique curve in $W_{(R,S),0}(t)$. Let $X_1$ be the point on that curve with $t=0$. By Proposition~\ref{prop:pieri} (4), $$ X_1\ \in\ Y_{(R(r_0),R(r_0),L_1)}(0)\ =\ \sum_{\beta\in R(r_0)*1} Y_{(R(r_0),\beta,L_1)}(1). $$ Let $R(r_0+1)$ be the index $\beta$ such that $X_1\in Y_{(R(r_0),\beta,L_1)}(1)$. Define $S(r'_0+1)$ similarly. Then $X_1\in W_{(R,S),1}(1)$. In general, suppose that we have constructed $R(0),\ldots,R(r_0+k)$, $S(0),\ldots,S(r'_0+k)$, and $X_k\in W_{(R,S),k}(1)$. Then $X_k$ lies on a unique curve in $W_{(R,S),k}(t)$. Let $X_{k+1}$ be the point on that curve at $t=0$. Let $c$ be minimal subject to $k<r_1+\cdots+r_c$ and set $\alpha=R(r_0+\cdots+r_{c-1})$ and $\beta=R(r_0+k)$. If $k+1<r_1+\cdots+r_c$, then by Proposition~\ref{prop:pieri} (3), there is a unique index $\gamma\in \beta*1$ such that $X_{k+1}\in Y_{\alpha,\gamma,L_c}(1)$. If $k+1=r_1+\cdots+r_c$ then by Proposition~\ref{prop:pieri} (4), there is a unique index $\gamma\in \beta*1$ such that $X_{k+1}\in\Omega_\gamma$. Set $R(r_0+k+1)=\gamma$ and likewise define $S(r'_0+k+1)$. Continuing in this fashion, we construct the chains $R$ and $S$, and $X_j$ for $0\leq j\leq \tau$. We show that $R$ increases everywhere, except possibly at $1,r_0+1,\ldots,r_0+\cdots+r_{a-1}+1$, and hence $R\in{\mathcal T}(r_0,\ldots,r_a)$. Similar arguments show that $S\in{\mathcal T}(r'_0,\ldots,r'_{a'})$, which will complete the proof. Suppose $k+1\not\in\{1,r_1+1,\ldots,r_1+\cdots+r_{a-1}+1\}$. Let $c$ be minimal subject to $k<r_1+\cdots+r_c$ and let $\alpha=R(r_0+\cdots+r_{c-1})$, $\beta=R(r_0+k)$, and $\gamma=R(r_0+k+1)$. Then by Proposition~\ref{prop:pieri} (3) and (4), $\beta\prec_\alpha\gamma$. The condition $j(\beta,\gamma)\geq j(\alpha,\beta)$ in the definition of $\beta\prec_\alpha\gamma$ ensures that $R$ increases at $k+1$. \qed \section{Homotopy continuation of overdetermined systems} Numerical homotopy continuation is a method for finding the isolated solutions of a system \begin{equation}\label{system} F(X)\ =\ 0 \end{equation} where $F=(f_1,\ldots,f_n)$ are polynomials in the variables $X=(x_1,\ldots,x_N)$. First, a {\em homotopy} $H(X,t)$ is found with the following properties: \begin{enumerate} \item $H(X,1)= F(X)$. \item The isolated solutions of $H(X,0)=0$ are known. \item The system $H(X,t)=0$ defines finitely many (rational) curves $\sigma_i(t)$, and each isolated solution of~(\ref{system}) is connected to an isolated solution $\sigma_i(0)$ of $H(X,0)=0$ by one of these curves. \end{enumerate} Given such a homotopy, numerical path continuation is used to trace these curves from solutions of $H(X,0)=0$ to solutions of the original system~(\ref{system}). When there are fewer solutions to $F(X)=0$ than to $H(X,0)=0$, some curves will diverge or become singular as $t\rightarrow 1$, and it is expensive to trace such a curve. When $N=n$, the system~(\ref{system}) is {\em square} and the homotopy \begin{equation}\label{convex} H(X,t)\quad :=\quad tF(X)\ +\ (1-t)G(X), \end{equation} where $G(X)=(x_1^{d_1}-a_1,\ldots,x_N^{d_N}-a_N)$ with $d_i:=\deg(f_i)$ and $a_i\neq 0$, gives $\prod d_i$ curves. This is the B\'ezout bound for a generic dense system $F$. In practice, $F(X)=0$ may have fewer than $\prod d_i$ solutions and we desire a homotopy with no divergent curves. Methods for such deficient systems which reduce the number of divergent curves are developed in~\cite{LSY_deficient,LS_deficient,LW_deficient}. When the polynomials $f_1,\ldots,f_n$ have special forms~\cite{MS_m-homogeneous,MSW_product}, then such homotopies~(\ref{convex}) are constructed where $G(X)$ shares this special form. When the polynomials $f_1,\ldots,f_n$ are sparse, polyhedral methods~\cite{CVVerschelde,Huber_Sturmfels} give a homotopy. The SAGBI homotopy algorithm (\S 2.3) is in the same spirit. We exploit a special feature of the coordinate ring of the Grassmannian to obtain a homotopy between the system (2.2) we wish to solve and one (2.12) whose solution may be obtained using polyhedral methods. Moreover, there are generically no divergent curves to be followed. The overdetermined situation of $n>N$ is more delicate. One difficulty is finding a homotopy $H(X,t)$ for an overdetermined system as generic perturbations of $F$ have no solutions. In~\cite[\S 2]{Sommese_Wampler_NAG}, this difficulty is avoided as follows: The system $F=(f_1,\ldots,f_n)$ is replaced by $N$ random linear combinations of the $f_1,\ldots,f_n$ yielding a square system whose isolated solutions include all isolated solutions of $F(X)=0$, but typically many more. They then find all isolated solutions of this random square subsystem. For the Gr\"obner and Pieri homotopy algorithms, we gave (in \S\S 2.2 and 3.2--3) homotopies $H(X,t)=(h_1(X,t),\ldots,h_n(X,t))$ and solutions $\sigma_i(0)$ at $t=0$ as above. For these, there are generically no divergent curves. To efficiently follow the curves $\sigma_i(t)$, we select a square subsystem $MH(X,t)$ of $H(X,t)$ ($M$ is an $(N\times n)$-matrix). If the Jacobian of $MH$ at each $\sigma_i(0)$ has the same rank ($N$) as does the Jacobian of $H(X,t)$, then the curves $\sigma_i(t)$ remain components of the algebraic set defined by the equations $$ MH(X,t)\ =\ 0. $$ Moreover, other components of this set meet the curves $\sigma_i(t)$ in at most finitely many points $t$ in ${\bf C}-\{0\}$. Thus, we may use the square subsystem $MH(X,t)$ to trace the curves $\sigma_i(t)$ along some path from 0 to 1 in the complex plane. We remark that in practice, $M$ may be chosen at random. \section{Applications} The algorithms of Sections 2 and 3 are useful for studying both the pole assignment problem in systems theory~\cite{Byrnes} and real enumerative geometry~\cite{Sottile_santa_cruz}. We describe the connection to the control of linear systems following~\cite{Byrnes}. Suppose we have a system (for example, a mechanical linkage) with inputs $u\in {\bf R}^m$ and outputs $y\in {\bf R}^p$ for which there are internal states $x\in {\bf R}^n$ such that the evolution of the system is governed by the first order linear differential equation \begin{equation}\label{linsystem} \begin{array}{rcl} \dot{x}&=&Ax + Bu,\\ y&=&Cx. \end{array} \end{equation} If the input is controlled by constant output feedback, $u=Fy$, then we obtain $$ \dot{x}\ =\ (A+BFC)x. $$ The natural frequencies of the controlled system are the roots $s_1,\ldots,s_n$ of \begin{equation}\label{charpoly} \varphi(s)\ :=\ \det(sI-A-BFC). \end{equation} The pole assignment problem asks, given a system~(\ref{linsystem}) and a polynomial $\varphi(s)$ of degree $n$, which feedback laws $F$ satisfy~(\ref{charpoly})? A standard transformation~({\em cf.}~\cite[\S 2]{Byrnes}) transforms the input data $A,B,C$ into matrices $N(s)$, $D(s)$ of polynomials with $\det(D(s))= \det(sI-A)$ and $N(s)D(s)^{-1}=C(sI-A)^{-1}B$ such that \begin{equation}\label{schubert_form} \varphi(s)\ =\ \det\left[ \begin{array}{cc} F & D(s)\\I& N(s)\end{array}\right]. \end{equation} Here $I$ is the $p\times p$-identity matrix and the feedback law $F$ is an $m\times p$-matrix. If we let $$ X\ :=\ \left[\begin{array}{c} F\\I\end{array}\right] \qquad {\rm and}\qquad K(s)\ :=\ \left[\begin{array}{c} D(s)\\N(s)\end{array}\right], $$ then $F$ gives local coordinates on $\mbox{\em Grass}(p,m+p)$ and~(\ref{schubert_form}) is equivalent to $$ X\cap K(s_i)\ \neq \ \{0\}\quad\mbox{for}\quad i\ =\ 1,\ldots,n. $$ These conditions are independent for generic $A,B,C$ and distinct $s_i$, hence $n\leq mp$ is necessary for there to be any feedback laws $F$. The critical case of $n=mp$ is an instance of the situation in \S 2. In~\cite{Byrnes_Stevens_homotopy} homotopy continuation was used to solve a specific feedback problem when $(m,p)=(3,2)$. From this result, they deduced that the pole assignment problem is not in general solvable by radicals. Despite this success, only few non-trivial examples have been computed in the control theory literature~\cite{Rosenthal_Sottile}. An important question is whether a given system may be controlled by {\em real} output feedback~\cite{Willems_Hesselink,Byrnes_real,Rosenthal_Schumacher_Willems}. That is, if all roots of $\varphi(s)$ are real, are there real feedback laws $F$ satisfying~(\ref{schubert_form})? Real enumerative geometry~\cite{Sottile_santa_cruz} asks a similar question: Are there real linear subspaces $K_1,\ldots,K_n$ in general position with $\dim K_i=m+1-k_i$ and $k_1+\cdots+k_n=mp$ such that {\em all} $p$-planes meeting each $K_i$ nontrivially are real? When either $m$ or $p$ is 2~\cite{Sottile_real_lines}, $n\leq 5$~\cite{Sottile_explicit_pieri}, or when $m=p=3$ and the $k_i=1$~\cite{Sottile_santa_cruz}, the answer is yes. In fact, the Pieri homotopies arose from these investigations. B.~Shapiro and M.~Shapiro give a precise conjecture relating both applications. Suppose $$ K_i(s)\ :=\ [\gamma(s),\gamma'(s),\gamma''(s),\ldots,\gamma^{(m+1-k_i)}(s)], $$ where $\gamma(s)$ is a parameterization of a rational normal (non-degenerate) curve in ${\bf R}^{m+p}$ of degree $m+p-1$. One such choice is \begin{equation}\label{ratnorm} \gamma(s)\ =\ {\it transpose}[1,s,s^2,\ldots,s^{m+p-1}]. \end{equation} Geometrically, $K_i(s)$ is the $(m+1-k_i)$-plane which osculates the curve $\gamma(s)$ at $s$. Such osculating $m$-planes have been used to prove non-degeneracy results in control theory. \begin{conj}[B.~Shapiro and M.~Shapiro]\label{conj:SS} Let $s_1,\ldots,s_n$ be distinct real numbers and suppose $K_i(s_i)$ osculates $\gamma$ at $s_i$ and $k_1+\cdots+k_n=mp$. Then each of the finitely many $p$-planes $X\subset {\bf C}^{m+p}$ satisfying $X\cap K_i(s_i)\neq \{0\}$ for $i=1,\ldots,n$ is defined over the reals. \end{conj} \section{Computational results} Our algorithms have been tested successfully in MATLAB, finding all 14 solutions in the case $(m,p)=(4,2)$ for both the SABGI and Gr\"obner homotopy algorithm, and all 15 solutions when $(m,p)=(6,2)$ and $k_1=\cdots=k_6=2$ for the Pieri homotopy algorithm. At present, the SAGBI and Gr\"obner homotopy algorithms have been fully implemented. Some timings from trial runs of these algorithms on a Sparc 20 are displayed in Table~\ref{Sagbi_times}. The input for these were $mp$ random complex $(m+p)\times m$-matrices. We provide a comparison to methods based upon Gr\"obner bases. Table~\ref{Sagbi_times} also gives the time on the Sparc 20 for the system Singular~\cite{SINGULAR} to compute a degree reverse lexicographic Gr\"obner basis for the polynomial systems: $$ \det \left[ X\mid K_i\right]\ =\ 0, \quad\mbox{for}\quad i=3,\ldots,K_{mp}. $$ Here $X$ is expressed in local coordinates for $\,\Omega_{[13]} \,\cap \,\Omega'_{[13]} \,$ and the $K_3,\ldots,K_{mp}$ are $(m+p)\times m$-matrices with random integral entries between $-4$ and $4$. A degree reverse lexicographic Gr\"obner basis is the input for some alternative numerical polynomial systems solvers (e.g.~eigenvalue methods~\cite{Auzinger_Stetter}). We note that the Gr\"obner basis calculation did not terminate within one week in the case $(m,p)=(6,2)$. \begin{table}[htb] \begin{tabular}{cccccc}\hline\hline $m$ & $p$&$d(m,p)$&SABGI homotopy&Gr\"obner homotopy&Gr\"obner basis\\ \hline 3 & 2 & 5 & $ {<}1$ & $ <0.5 $&$ <0.5$\\ 4 & 2 & 14 & $ 47$ & $ 6 $&$ 19 $\\ 5 & 2 & 42 & $ 373$ & $ 408 $&$ 149,897$\\ 6 & 2 & 132 & $3,364$ & $ 8,626 $&$\infty$\\ \hline\hline \end{tabular} \caption{Time (in seconds) \label{Sagbi_times}} \end{table} The final version of this paper will include data from implementations of the Pieri homotopy algorithm, as well.

9706

alg-geom/9706008

en

https://arxiv.org/abs/alg-geom/9706008

alg-geom/9706008

Klaus Altmann

Klaus Altmann and Lutz Hille

A Vanishing Result for the Universal Bundle on a Toric Quiver Variety

13 pages with a couple of small figures LaTeX 2.09

Let Q be a finite quiver without oriented cycles. Denote by U --> M the fine moduli space of stable thin sincere representations of Q with respect to the canonical stability notion. We prove Ext^i(U,U) = 0 for all i >0 and compute the endomorphism algebra of the universal bundle U. Moreover, we obtain a necessary and sufficient condition for when this algebra is isomorphic to the path algebra kQ of the quiver Q. If so, then the bounded derived category of finitely generated right kQ-modules is embedded into that of coherent sheaves on M.

alg-geom

\section{Introduction}\label{Int} \neu{Int-1} Let $Q$ be a quiver (i.e.\ an oriented graph) without oriented cycles; denote by $Q_0$ the vertices and by $Q_1$ the arrows of $Q$. For a fixed dimension vector $d$, that is a map $d: Q_0 \rightarrow \Z_{\geq 0}$, we denote by $I\!\!H(d) := \{\theta: Q_0 \rightarrow \R\mid \sum_{q \in Q_0} \theta_qd_q = 0 \}$ the vector space of the so-called weights with respect to $d$. We fix an algebraically closed field $k$. To each $\theta\inI\!\!H(d)$ there exists the moduli space ${\cal{M}}^{\theta}(Q,d)$ of $\theta$-semistable $k$-representations of $Q$ with dimension vector $d$ (cf.\ \cite{King}). This space is known to be projective and, in case $\theta$ is in general position and $d$ is indivisible, also smooth. Moreover, if we restrict ourselves to thin sincere representations, that is $d_q = 1$ for all $q \in Q_0$, then ${{\cal M}^{\wt}(\kQ)}$ is also toric (cf.\ \cite{HilleTor}). In any case, each integral weight $\theta$ induces an ample line bundle ${\cal{L}}(\theta)$ on ${\cal{M}}^{\theta}(Q,d)$.\\ If $\theta$ is in general position and $d$ indivisible, then ${\cal{M}}^{\theta}(Q,d)$ is, in addition, a fine moduli space admitting a universal bundle ${\cal U}$. The universal bundle splits into a direct sum of vector bundles ${\cal U} = \oplus_{q \in Q_0} {\cal U}_q$, and the summands ${\cal U}_q$ have rank $d_q$ (cf.\ \cite{King}). All known examples suggest that the universal bundles on those moduli spaces have no self-extensions, i.e.\ $\mbox{\rm Ext}\,^{\, l}_{{\cal{M}}^{\theta}(Q,d)}({\cal U},{\cal U}) = 0$ for all $l >0$. The issue of this paper is to prove this formula in special cases. The meaning of this property and its relation to tilting theory will be discussed in \zitat{Int}{4}. \\ In this paper we restrict ourselves to thin sincere representations; the corresponding moduli spaces are called {\em toric quiver varieties}. Because $d = (1,\ldots,1)$ is fixed, we will omit it in all notation introduced above. The direct summands of the universal bundle are line bundles, and they are characterized, up to a common twist, by the following property: For any arrow $\alpha\in Q_1$ pointing from $p$ to $q$ ($p,q\in Q_0$) the invertible sheaf $\,{\cal U}_p^{-1} \otimes {\cal U}_q$ corresponds to the divisor of all representations assigning the zero map to $\alpha$. Furthermore, there exists a distinguished weight $\theta^c$ (see \zitat{Int}{2} and \zitat{Mod}{7} for a definition and first properties). We denote the corresponding moduli space by ${{\cal M}(\kQ)}$. \par \neu{Int-2} Polarized projective toric varieties may be constructed from lattice polytopes. If one wants to forget about the polarization, simply consider the inner normal fan of the polytope. In \S \ref{Mod} we give a detailed description of the moduli space ${{\cal M}^{\wt}(\kQ)}$ of thin sincere representations via its ``defining polytope'' $\Delta(\theta)$. The easiest way to obtain $\Delta(\theta)$ from the quiver is to imagine $Q$ as a one-way pipe system carrying liquid; a weight $\theta\inI\!\!H$ describes the input (possibly negative) into the system at each knot. Using this language, $\Delta(\theta)$ is simply given as the set of all possible flows respecting both the direction of the pipes and the given input $\theta$ (see \zitat{Mod}{3}).\\ Considering the opposite viewpoint, {\em each} flow through our pipe system requires a certain input, i.e.\ a weight. In particular, from the special flow that is constant $1$ at each pipe we obtain a special, so-called canonical, weight $\theta^c$. The corresponding $\Delta(\theta^c)$ is a reflexive polytope (in the sense of Batyrev, \cite{Ba}), i.e.\ the moduli space ${{\cal M}(\kQ)}$ is Fano (Proposition \zitat{Mod}{7}). \\ Fixing a weight $\theta$ in general position, i.e.\ ${{\cal M}^{\wt}(\kQ)}$ is smooth, flows and weights have still another meaning. Each flow defines an equivariant, with respect to the defining torus, effective divisor, and each weight $\theta'$ defines an element ${\cal{L}}(\theta')$ in the Picard group of ${{\cal M}^{\wt}(\kQ)}$. Assigning a flow its input weight corresponds to assigning a divisor its class in the Picard group (see \zitat{Uni}{1}). \par {\bf Example:} 1) In the special case $\theta' := \theta$ this recovers our ample line bundle introduced before. \\ 2) The line bundle ${\cal U}_p^{-1} \otimes {\cal U}_q$ corresponds to the weight with values $1$ at $p$, $-1$ at $q$, and zero at all other points. \par \neu{Int-3} Our first main result is Theorem \zitat{Uni}{6} stating the lack of self-extensions of ${\cal U}$ on the moduli space ${{\cal M}(\kQ)}$ with respect to the canonical weight, i.e.\ $\mbox{\rm Ext}\,^{l}_{{{\cal M}(\kQ)}}({\cal U},{\cal U}) = 0$ for all $l >0$. This is proved by using a slightly generalized Kodaira vanishing argument which works for toric varieties, cf.\ Theorem \zitat{Uni}{5}. As a Corollary of Theorem \zitat{Uni}{6} we conclude that we obtain a full and faithful functor from the bounded derived category of finitely generated right modules over the endomorphism algebra ${\cal{A}}$ of ${\cal U}$ into the bounded derived category of coherent sheaves on the moduli space ${{\cal M}(\kQ)}$ (Theorem \zitat{End}{4}). Moreover, in Theorem \zitat{End}{3} we provide a criterion for ${\cal{A}} = \mbox{\rm End}\,_{{{\cal M}(\kQ)}}({\cal U},{\cal U})$ to be isomorphic to the path algebra $\ck Q$ of the quiver $Q$. \\ Combining both results we obtain the following relation between the derived categories of right $\ck Q$-modules and of coherent sheaves on ${{\cal M}(\kQ)}$, respectively{.} \par {\bf Theorem:} {\em Assume $Q$ is a quiver lacking $(1,0)$- and $(t,t)$-walls (see \zitat{Mod}{2} for an explanation). Then, \[ -\otimes_{\ck Q}^{I\!\!L}{\cal U}: {\cal{D}}^b\Big(\mbox{mod--}\ck Q \Big) \longrightarrow {\cal{D}}^b \Big( \mbox{Coh}\big( {{\cal M}(\kQ)} \big)\Big) \] is a full and faithful functor from the bounded derived category of finitely generated right $\ck Q$-modules into the bounded derived category of coherent sheaves on ${{\cal M}(\kQ)}$. } \par \neu{Int-4} The result above is closely related to tilting theory. Since the fundamental paper \cite{Beilinson}, tilting theory has become a major tool in classifying vector bundles; a tilting sheaf induces an equivalence of bounded derived categories, as in the previous Theorem. To be precise we recall the definition of a tilting sheaf (\cite{Baer}). A sheaf ${\cal{T}}$ on a smooth projective variety is called a {\sl tilting sheaf} if \begin{itemize}\vspace{-2ex} \vspace{-1ex} \item[1)] it has no higher self-extensions, that is $\mbox{Ext}^l({\cal{T}},{\cal{T}})=0$ for all $l >0$, \vspace{-1.5ex} \item[2)] the direct summands generate the bounded derived category, and \vspace{-1.5ex} \item[3)] the endomorphism algebra ${\cal{A}}$ of ${\cal{T}}$ has finite global (homological) dimension. \vspace{-1.5ex} \vspace{-1ex}\end{itemize} Then, the functors $\R \mbox{Hom}({\cal{T}},-)$ and $-\otimes^{I\!\!L}_{{\cal{A}}}{\cal{T}}$ define mutually inverse equivalences of the bounded derived categories of coherent sheaves on the underlying variety of ${\cal{T}}$ and of the finitely generated right ${\cal{A}}$-modules, respectively. For constructions of tilting bundles and their relations to derived categories we refer to the following papers \cite{Kapranov}, \cite{Beilinson}, \cite{Rudakov}, \cite{Bondal}, and \cite{Orlov}. For the similar notion of a tilting module we refer to \cite{HRTilt}.\\ For our purpose, the notion of an exceptional sequence is more useful. Let ${\cal{C}}$ be any of the categories introduced above: the category of finitely generated right modules over a finite dimensional algebra, the category of coherent sheaves on a smooth projective variety, or one of its derived categories. Thus, ${\cal{C}}$ is either an abelian or a triangulated $\ck$-category. Each object in ${\cal{C}}$ has a unique, up to isomorphism and reordering, decomposition into indecomposable direct summands, i.e.\ ${\cal{C}}$ is a Krull-Schmidt category. Moreover, the extension groups are defined and globally bounded; they are finite-dimensional $\ck$-vector spaces. An object in ${\cal{C}}$ is called {\sl exceptional} if it has no self-extensions and its endomorphism ring is $\ck$. A sequence $({\cal{E}}_0,\ldots,{\cal{E}}_n)$ of objects in ${\cal{C}}$ is called {\em exceptional} if \begin{itemize}\vspace{-2ex} \vspace{-1ex} \item[1)] each object ${\cal{E}}_i$ for $i=0,\ldots,n$ is exceptional and \vspace{-1.5ex} \item[2)] $\mbox{Ext}^l({\cal{E}}_j,{\cal{E}}_i) = 0$ for all $l\geq 0$, and $j>i$. \vspace{-1.5ex} \vspace{-1ex}\end{itemize} Such a sequence is called {\sl strong exceptional} if, additionally, \begin{itemize}\vspace{-2ex} \vspace{-1ex} \item[3)] $\mbox{Ext}^l({\cal{E}}_i,{\cal{E}}_j) = 0$ for all $l>0$ and all $i,j =0,\ldots,n$. \vspace{-1.5ex} \vspace{-1ex}\end{itemize} Finally, it is called {\sl full} if in addition to 1), 2) and 3) \begin{itemize}\vspace{-2ex} \vspace{-1ex} \item[4)] the objects ${\cal{E}}_i$ for $i=0,\ldots,n$ generate the bounded derived category. \vspace{-1.5ex} \vspace{-1ex}\end{itemize} Thus, each full strong exceptional sequence defines a tilting bundle $\oplus_{i=0}^n {\cal{E}}_i$, because the endomorphism algebra of $\oplus_{i=0}^n {\cal{E}}_i$ has global dimension at most $n$. Vice versa, each tilting bundle whose direct summands are line bundles gives rise to a strong exceptional sequence.\\ Using this language, our vanishing result Theorem \zitat{Uni}{6} means that the direct summands of ${\cal U}$ form a strong exceptional sequence. \par \neu{Int-5} In general, this sequence cannot be full. Assume the contrary; then the bounded derived categories in the previous theorem are equivalent. The first one is a derived category of a hereditary abelian category, whose structure is well-known (\cite{RinHer}). In particular, the Serre functor (see \cite{BK} for the definition) coincides with the Auslander-Reiten translation and fixes objects up to translation only in case the category is tame or just semi-simple (cf.\ \cite{Happel} \S 1.4/5). On the other hand, the Serre functor in the bounded derived category of coherent sheaves fixes all skyscraper sheaves up to a translation. Consequently, an equivalence implies that ${\cal{M}}$ is a point or a projective line in case the algebra $\ck Q$ is semi-simple or tame, respectively. It follows that $Q$ is a point or the Kronecker quiver; the remaining tame cases may not appear (see \cite{Ringel} Theorem p.\ 158). \\ Nevertheless, there is some hope that one may find a complement $\overline{{\cal U}}$ such that ${\cal U} \oplus \overline{{\cal U}}$ is a tilting bundle. At least a class of very particular examples of tilting bundles on toric quiver varieties is known (\cite{HilleTor}, Theorem 3.9). \par \neu{Int-6} For an introduction to quivers and path algebras we refer the reader to \cite{Ringel} and \cite{ARS}; the theory of localizations may be found in \cite{Schofield}. For an introduction to moduli spaces we mention \cite{Newstead} and for moduli of representations of quivers we refer to the work of King \cite{King}. For results on triangulated categories we refer to \cite{Happel} and \cite{HartshorneRes}. Our standard reference for toric geometry is \cite{Ke}; for a short introduction to this area we also mention \cite{Fulton}.\\ We would like to thank G.~Hein, A.~King, and A.~Schofield for helpful discussions. \par \section{Moduli spaces of thin sincere representations}\label{Mod} \neu{Mod-1} Let $Q$ be a connected quiver without oriented cycles; it consists of a set $Q_0$ of vertices, a set $Q_1$ of arrows, and two functions $s,t:Q_1 \rightarrow Q_0$ assigning to each arrow $\alpha \in Q_1$ its source $s(\alpha)$ and its target $t(\alpha)$. A representation of $Q$ is a collection of finite dimensional $\ck$-vector spaces $x(q)$ for each vertex $q$ together with a collection of linear maps $x(\alpha): x(s(\alpha)) \rightarrow x(t(\alpha))$ for each arrow $\alpha\in Q_1$. The dimension vector $d = (d_q \mid q \in Q_0)$ of a representation $x$ is defined by $d_q = \dim x(q)$. A representation is called {\sl thin} if $\dim x(q) \leq 1$ for all $q \in Q_0$ and {\sl sincere} if $\dim x(q) \geq 1$ for all $q \in Q_0$. In this paper we consider only thin sincere representations. \\ We denote by ${\cal{R}} = \oplus_{\alpha \in Q_1} \ck$ the space of all thin sincere representations, that is $x(q) = \ck$. By $G = \times_{q\in Q_0} \ck^{\ast}$ we denote the torus acting via conjugation on ${\cal{R}}$. The orbits of this action are exactly the isomorphism classes of thin sincere representations, i.e.\ their moduli space may be obtained via GIT. Doing so, we have to deal with the notion of stability with respect to a given weight (cf. \cite{King}){.} \par {\bf Definition:} The elements of the real vector space $I\!\!H := \{\theta: Q_0 \rightarrow \R\mid \sum_{q \in Q_0} \theta_q = 0 \}$ are called {\sl weights} of the quiver $Q$.\\ Let $\theta\inI\!\!H$. A thin sincere representation $x$ of $Q$ is {\sl $\theta$-stable} ({\sl -semistable}) if for each proper non-trivial subrepresentation $y \subset x$ we have $\sum_{q \in Q_0\mid y(q) \not= 0} \theta_q <0$ ($\leq 0$ respectively). Two semistable representations $x$ and $y$ are called {\sl $S$-equivalent} with respect to $\theta$ if the factors of the stable Jordan-H\"older filtration coincide.\\ A subquiver $Q' \subseteq Q$ with $Q'_0=Q_0$ is {\sl $\theta$-stable} ({\sl $\theta$-semistable}) if it has a $\theta$-stable ($\theta$-semistable) representation. Two quivers are {\sl $S$-equivalent} with respect to $\theta$ if they admit $\theta$-semistable representations of the same $S$-equivalence classes.\\ Finally, we denote by ${\cal{T}}(\theta)$ the set of all $\theta$-semistable subtrees $T \subseteq Q$ with $T_0 = Q_0$ and by $Q_1(\theta)$ the set of all arrows $\alpha$ such that $Q \setminus \{\alpha\}$ is a $\theta$-stable subquiver. \par In other words, a representation $x$ is $\theta$-stable precisely when the subquiver $Q_0 \cup \{ \alpha \in Q_1 \mid x(\alpha) \not= 0 \}$ is $\theta$-stable. Moreover, a subquiver $Q'$ is $\theta$-stable if and only if for all non-trivial proper subsets $S \subset Q_0$ which are closed under successors in $Q'$ we have $\sum_{q \in S} \theta_q < 0$ (see \cite{HilleThin} \S 2 and \cite{HilleTor} Lemma 1.4). We also note that each $S$-equivalence class contains a unique minimal $\theta$-semistable subquiver -- just take the disjoint union of the support of the Jordan-H\"older factors. \par \neu{Mod-2} As already mentioned in the beginning, for any given weight $\theta$ the moduli space ${{\cal M}^{\wt}(\kQ)}$ exists; however, different weights $\theta$ may cause different moduli spaces. According to \cite{HilleThin} there is a chamber system in $I\!\!H$, and the type of ${{\cal M}^{\wt}(\kQ)}$ can only flip if $\theta$ crosses walls of the following type: \par {\bf Definition:} $W\subseteqI\!\!H$ is called a $(t^+,t^-)${\sl -wall} if \[ W=\Big\{ \theta \in I\!\!H \mid \sum_{q \in Q_0^+} \theta_q= -\sum_{q \in Q_0^-} \theta_q=0 \Big\} \] for some decomposition $Q_0 = Q_0^+ \sqcup Q_0^-$ such that the full subquivers $Q^+$ and $Q^-$ are both connected and such there are exactly $t^+$ arrows pointing from $Q_0^+$ to $Q_0^-$ and $t^-$ arrows the other way around. We say that $\theta$ is in {\em general position} if $\theta$ does not lie on any wall and if the moduli space is not empty. Assume $\theta$ is in general position, then ${{\cal M}^{\wt}(\kQ)}$ is smooth and has the (maximal) dimension $d = \#Q_1\, -\, \#Q_0\, + \,1$, see \cite{HilleThin}. Moreover, for those weights, every semistable thin sincere representation is stable. \par \neu{Mod-3} To describe the toric structure of ${{\cal M}^{\wt}(\kQ)}$ we introduce the real vector space of flows defined as $I\!\!F:= \{r: Q_1\to {I\!\!R}\}={I\!\!R}^{Q_1}$. A flow is called {\sl regular} if it has only non-negative values, i.e.\ if it respects the direction of the pipes. For any $\alpha\in Q_1$ we denote by $f^\alpha\inI\!\!F$ the characteristic flow mapping $\alpha$ to $1$ and keeping the remaining pipes dry. More generally, for each walk $w$ without cycles in $Q$ we define the characteristic flow $f^w$ mapping an arrow $\alpha \in w$ to $1$, an arrow $\beta$ with $\beta^{-1} \in w$ to $-1$, and the remaining arrows to $0$. This flow is regular if and only if $w$ is a path, i.e.\ respects the orientation in $Q$. \\ There is a canonical linear map $\pi:I\!\!F\toI\!\!H$ describing the input of flows; if $\alpha\in Q_1$ points from $p$ to $q$, then $\pi(f^\alpha)$ sends $p$ and $q$ onto $1$ and $-1$, respectively. Thus $$ (\pi(r))_q = \sum_{s(\alpha) = q} r_{\alpha} - \sum_{t(\alpha) = q} r_{\alpha}. $$ This leads to the following definition: \par {\bf Definition:} The convex {\em polytope of flows} $\Delta(\theta)$ assigned to a weight $\theta$ is defined as the intersection \[ \Delta(\theta):=\pi^{-1}(\theta)\cap {I\!\!R}^{Q_1}_{\geq 0}\,. \] This means that $\Delta(\theta)$ consists of exactly those regular flows respecting the prescribed input $\theta$. Moreover, $\Delta(\theta)$ is compact since $Q$ has no oriented cycles.\\ The vector spaces $I\!\!F$ and $I\!\!H$ contain the lattices $I\!\!F_{Z\!\!\!Z}$ and $I\!\!H_{Z\!\!\!Z}$ of integral flows and weights, respectively. For any integral weight $\theta:Q_0\to{Z\!\!\!Z}$ we define the affine lattice $M^\wt\subseteqI\!\!F$ as the fiber $I\!\!F_{Z\!\!\!Z}\cap\pi^{-1}(\theta)$, i.e.\ \[ M^\wt = \Big\{r\in{Z\!\!\!Z}^{Q_1}\,|\; \sum\limits_{s(\alpha)=q}r_\alpha - \sum\limits_{t(\alpha)=q}r_\alpha =\theta_q\;\mbox{ for all }q\inQ_0\Big\}\,. \] Any element $r \in M^\wt$ provides obviously an isomorphism $(+r): M:=M^0 \stackrel{\sim}{\longrightarrow} M^\wt$. \par \neu{Mod-4} The following lemma will be crucial for the understanding of our flow polytope as well as for proving the upcoming vanishing theorem in \S \ref{Uni}; it explicitly provides points of the lattices $M^\wt${.} Let $T\subseteq Q$ be an arbitrary maximal tree. Each arrow $\alpha\in T_1$ divides $Q_0$ into two disjoint subsets, the source $S_T(\alpha)$ and the target $T_T(\alpha)$. \par \begin{center} \setlength{\unitlength}{0.00600in}% \begingroup\makeatletter \def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}% \expandafter\x\fmtname xxxxxx\relax \def\y{splain}% \ifx\x\y \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def\x{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter\x \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \endgroup \begin{picture}(395,80)(55,680) \thicklines \put(300,720){\circle*{10}} \put(335,720){\circle*{10}} \put(400,750){\circle*{10}} \put(360,695){\circle*{10}} \put(320,750){\circle*{10}} \put(425,700){\circle*{10}} \put(395,690){\circle*{10}} \put(315,690){\circle*{10}} \put(360,750){\circle*{10}} \put(380,720){\circle*{10}} \put(105,745){\circle*{10}} \put(170,750){\circle*{10}} \put(170,690){\circle*{10}} \put(120,695){\circle*{10}} \put(145,725){\circle*{10}} \put(215,695){\circle*{10}} \put(220,720){\circle*{10}} \thinlines \put( 80,760){\line( 0,-1){ 80}} \put( 80,680){\line( 1, 0){160}} \put(240,680){\line( 0, 1){ 80}} \put(240,760){\line(-1, 0){160}} \put(280,760){\line( 0,-1){ 80}} \put(280,680){\line( 1, 0){160}} \put(440,680){\line( 0, 1){ 80}} \put(440,760){\line(-1, 0){160}} \thicklines \put(450,715){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}\scriptsize $T_T(\alpha)$}}} \put(230,720){\vector( 1, 0){ 60}} \put(260,730){\makebox(0,0)[cc]{\smash{\SetFigFont{12}{14.4}{rm}\scriptsize $\alpha$}}} \put(105,745){\line( 2,-1){ 40}} \put(145,725){\line( 1, 1){ 25}} \put(145,725){\line( 3,-4){ 25.800}} \put(170,690){\line( 6, 1){ 44.595}} \put(215,695){\line( 1, 5){ 5}} \put(120,695){\line( 5, 6){ 25}} \put(300,720){\line( 1, 0){ 35}} \put(335,720){\line( 1, 0){ 45}} \put(300,720){\line( 2, 3){ 20}} \put(320,750){\line( 1, 0){ 40}} \put(360,750){\line( 1, 0){ 40}} \put(300,720){\line( 1,-2){ 15}} \put(315,690){\line( 6, 1){ 44.595}} \put(360,695){\line( 6,-1){ 34.865}} \put(395,690){\line( 3, 1){ 30}} \put( 25,715){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}\scriptsize $S_T(\alpha)$}}} \end{picture} \end{center} \par {\bf Lemma:} {\em Fix a tree $T$ and let $\theta\inI\!\!H$. For any flow $\varepsilon$ there is a unique element $r = r^T \in \pi^{-1}(\theta) \subseteq I\!\!F$ satisfying $r_\alpha := \varepsilon(\alpha)$ for $\alpha \notin T_1$. Its remaining coordinates (i.e.\ for $\alpha\in T_1$) are given by \[ r_\alpha\,-\,\varepsilon(\alpha)\;=\; \sum_{q\in S_T(\alpha)}\!\!\theta_q\; + \sum_{T_T(\alpha)\stackrel{\beta}{\rightarrow}S_T(\alpha)} \!\!\!\!\varepsilon(\beta)\; - \sum_{S_T(\alpha)\stackrel{\beta}{\rightarrow}T_T(\alpha)} \!\!\!\!\varepsilon(\beta)\,. \vspace{-2ex} \] } \par {\bf Proof:} First, we should note that both $\varepsilon$ and $r$ are flows -- the different notation for their coordinates ($\varepsilon(\alpha)$ and $r_\alpha$, respectively) was chosen for psychological reasons only. Now, let $r$ be {\em some} element of $I\!\!F$. If $\pi(r)=\theta$, then for any subdivision $Q_0 = Q_0^+ \sqcup Q_0^-$ we obtain \[ \sum_{s(\beta) \in Q_0^+, t(\beta) \in Q_0^-} r_\beta - \sum_{s(\beta) \in Q_0^-, t(\beta) \in Q_0^+} r_\beta = \sum_{q \in Q_0^+}\theta_q = -\sum_{q \in Q_0^-}\theta_q \] just by summing up the $M^\wt$-equations with $q\in Q_0^+$. The reverse implication is also true, even if we restrict ourselves to the special subdivisions provided by arrows $\alpha\in T_1$ via $Q_0^+:= S_T(\alpha)$ and $Q_1^+:= T_T(\alpha)$. \\ On the other hand, these subdivisions have the important property that $\alpha$ is the only arrow that belongs to both the tree $T$ and to one of the index sets $\{\beta\,|\; s(\beta) \in Q_0^+, t(\beta) \in Q_0^-\}$ or $\{\beta\,|\; s(\beta) \in Q_0^-, t(\beta) \in Q_0^+\}$. In particular, by just taking care of this single exception, in the above equations we may always replace $r_\beta$ by $\varepsilon(\beta)$. \hfill$\Box$ \par If the weight $\theta$ and the flow are integral, then so is $r$, i.e.\ $r\in M^\wt$. \par \neu{Mod-5} {\bf Proposition:} {\em Let $\theta\inI\!\!H$ be an integral weight, then the polytope of flows $\Delta(\theta)\subseteqM^\wt$ is always a lattice polytope. The associated projective toric variety equals ${{\cal M}^{\wt}(\kQ)}$. Moreover, $\Delta(\theta)$ provides an ample, equivariant line bundle ${\cal{L}}(\theta)$ on ${{\cal M}^{\wt}(\kQ)}$.} \par {\bf Proof:} First, we establish a one-to-one correspondence between vertices of $\Delta(\theta)$ and $S$-equivalence classes of $\theta$-semistable trees. Faces of $\Delta(\theta)$ in any dimension are obtained by forcing certain coordinates of $I\!\!F$ to be zero. In particular, vertices are points with a maximal set of vanishing coordinates. Let $r\in\Delta(\theta)$ be a vertex and denote its support by \[ \mbox{\rm supp}\,\, r:=\left\{\alpha\in Q_1\,| \; r_\alpha >0\right\}\,. \] If $\mbox{\rm supp}\,\, r$ contained any cycle of $Q$, then we could replace $r$ by a different regular flow with the same weight and a smaller support. Hence, $\mbox{\rm supp}\,\, r$ is contained in maximal trees of $Q$. Moreover, we obtain that \begin{itemize}\vspace{-2ex} \item every maximal tree $T$ containing $\mbox{\rm supp}\,\, r$ is $\theta$-semistable, and \item those trees are stable if and only if $\mbox{\rm supp}\,\, r=T_1$ (which determines the tree uniquely). \vspace{-1ex}\end{itemize} To prove these facts, take a proper subset $Q_0^+\subset Q_0$ that is closed under successors in $T$; denoting $Q_0^-:=Q_0\setminus Q_0^+$ this means that there are no arrows pointing from $Q_0^+$ to $Q_0^-$ in $T$. Hence, using the formula mentioned in the previous proof, $$ \sum_{q \in Q_0^+}\theta_q = - \sum_{[Q_0^-\stackrel{\beta}{\to} Q_0^+]} r_\beta \leq 0. $$ Conversely, let $T$ be any maximal tree. The previous lemma tells us that there is exactly one $r\in\pi^{-1}(\theta)\subseteq I\!\!F$ such that $\mbox{\rm supp}\,\, r\subseteq T_1$; it has integer coordinates $r_\alpha=\sum_{S_T(\alpha)}\theta_q$ for $\alpha\in T_1$. Moreover, if $T$ is $\theta$-semistable, then these numbers are non-negative, meaning that $r\in\Delta(\theta)$. Thus, it must be a vertex. Moreover, two different trees define the same vertex if and only if they are $S$-equivalent. \par What does the inner normal fan ${\Sigma(\wt)}$ look like? Denote by ${C}$ the matrix describing the incidences of our quiver; ${C}$ consists of $\# Q_0 $ rows and $\# Q_1 $ columns, and for $q\in Q_0$, $\alpha\inQ_1$ we have \[ {C}_{q\alpha}:=\left\{ \begin{array}{cl} +1&\mbox{if $q = s(\alpha)$ }\\ -1&\mbox{if $q = t(\alpha)$ }\\ 0&\mbox{otherwise}\,. \end{array} \right. \] The free abelian group $N:={Z\!\!\!Z}^{Q_1}/ ({C}\mbox{--rows})$ is dual to the lattice $M:=M^0=\big({C}\mbox{--rows}\big)^\bot\subseteqI\!\!F$. Hence, writing $a^\alpha\in N$ for the images of the canonical vectors $e^\alpha\in{Z\!\!\!Z}^{Q_1}$, we obtain $$ {\Sigma(\wt)}=\Big\{ \langle a^{\alpha_1},\dots,a^{\alpha_k}\rangle \, \Big| \begin{array}[t]{l} Q \setminus \{\alpha_1,\dots,\alpha_k\} \mbox{ is the minimal element in some $S$-equivalence class }\\ \mbox{ of $\theta$-semistable subquivers } \Big\}. \end{array} $$ According to \cite{HilleTor}, Theorem 1.7 and (3.3), this is exactly the fan defining the moduli space ${{\cal M}^{\wt}(\kQ)}$. \hspace*{\fill}$\Box$ \par {\bf Example:} In the quiver below the canonical weight is $\theta^c = (2,1,-1,-2)$. $$ \setlength{\unitlength}{0.00600in}% \begingroup\makeatletter \def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}% \expandafter\x\fmtname xxxxxx\relax \def\y{splain}% \ifx\x\y \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def\x{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter\x \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \endgroup \begin{picture}(100,175)(150,590) \thicklines \put(160,680){\circle*{10}} \put(240,680){\circle*{10}} \put(200,600){\circle*{10}} \put(205,750){\vector( 1,-2){ 30}} \put(195,750){\vector(-1,-2){ 30}} \put(200,760){\circle*{10}} \put(170,680){\vector( 1, 0){ 60}} \put(210,590){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}4}}} \put(165,670){\vector( 1,-2){ 30}} \put(235,670){\vector(-1,-2){ 30}} \put(210,755){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}1}}} \put(150,690){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}2}}} \put(250,675){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}3}}} \end{picture} \hspace{3cm} \begin{array}{c} C = \left( \begin{array}{rrrrr} 1 & 1 & 0 & 0 & 0 \\ -1& 0 & 1 & 1 & 0 \\ 0 & -1& -1& 0 & 1 \\ 0 & 0 & 0 & -1& -1 \end{array} \right) \\ ~ \\ ~ \\ ~ \\ ~ \\ ~ \\ ~ \end{array} $$ \vspace*{-2cm} The corresponding fan ${\Sigma(\wt)}$, and the polytope $\Delta(\theta^c)$ look like the following: \begin{center} \setlength{\unitlength}{0.00600in}% \begingroup\makeatletter \def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}% \expandafter\x\fmtname xxxxxx\relax \def\y{splain}% \ifx\x\y \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def\x{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter\x \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \endgroup \begin{picture}(170,170)(235,475) \thicklines \put(280,560){\circle*{10}} \put(240,560){\circle*{10}} \put(320,520){\circle*{10}} \put(280,520){\circle*{10}} \put(240,520){\circle*{10}} \put(240,480){\circle*{10}} \put(280,480){\circle*{10}} \put(320,480){\circle*{10}} \put(360,480){\circle*{10}} \put(360,520){\circle*{10}} \put(360,560){\circle*{10}} \put(320,560){\circle*{10}} \put(360,600){\circle*{10}} \put(280,600){\circle*{10}} \put(320,600){\circle*{10}} \put(240,600){\circle*{10}} \put(400,640){\line(-1,-1){ 80}} \put(240,640){\circle*{10}} \put(280,640){\circle*{10}} \put(320,640){\circle*{10}} \put(360,640){\circle*{10}} \put(400,640){\circle*{10}} \put(400,600){\circle*{10}} \put(400,560){\circle*{10}} \put(400,520){\circle*{10}} \put(400,480){\circle*{10}} \put(320,560){\line(-1, 0){ 80}} \put(320,560){\line( 0,-1){ 80}} \put(320,560){\line( 1, 0){ 80}} \put(320,560){\line( 0, 1){ 80}} \end{picture} \hspace{3cm} \setlength{\unitlength}{0.01200in}% \begingroup\makeatletter \def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}% \expandafter\x\fmtname xxxxxx\relax \def\y{splain}% \ifx\x\y \gdef\SetFigFont#1#2#3{% \ifnum #1<17\tiny\else \ifnum #1<20\small\else \ifnum #1<24\normalsize\else \ifnum #1<29\large\else \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else \huge\fi\fi\fi\fi\fi\fi \csname #3\endcsname}% \else \gdef\SetFigFont#1#2#3{\begingroup \count@#1\relax \ifnum 25<\count@\count@25\fi \def\x{\endgroup\@setsize\SetFigFont{#2pt}}% \expandafter\x \csname \romannumeral\the\count@ pt\expandafter\endcsname \csname @\romannumeral\the\count@ pt\endcsname \csname #3\endcsname}% \fi \endgroup \begin{picture}(90,90)(275,515) \thicklines \put(280,560){\circle*{7}} \put(320,520){\circle*{7}} \put(280,520){\circle*{7}} \put(360,520){\circle*{7}} \put(360,560){\circle*{7}} \put(320,600){\circle*{7}} \put(320,560){\circle*{7}} \put(280,520){\line( 0, 1){ 80}} \put(280,600){\circle*{7}} \put(280,600){\line( 1, 0){ 40}} \put(320,600){\line( 1,-1){ 40}} \put(360,560){\line( 0,-1){ 40}} \put(360,520){\line(-1, 0){ 80}} \end{picture} \end{center} It is known that the toric variety of this fan is the blow up of the projective plane in two points, which is isomorphic to the blow up of the two-dimensional smooth quadric in one point. \par \neu{Mod-6} Equivariant (with respect to the torus action), invertible sheaves ${{\cal{L}}}$ on a toric variety $X(\Sigma)$ are completely determined by their order function $\mbox{ord}\,{{\cal{L}}}:\Sigma^{(1)}\to {Z\!\!\!Z}$ or its piecewise linear continuation $\mbox{ord}\,{{\cal{L}}}:N_{{I\!\!R}}\to{I\!\!R}$. If $r(\sigma) \in M$ provides a local generator $x^{r(\sigma)}$ of ${{\cal{L}}}$ on $U(\sigma) \subseteq X(\Sigma)$, then $\mbox{ord}\,{\cal L}(a)$ is defined as $\langle a, r(\sigma) \rangle$ if $a\in\sigma$ (cf.\ \cite{Oda}). Moreover, if ${\cal L}$ is an ample (or at least globally generated) invertible sheaf given by a lattice polytope $\Delta\subseteq M_{{I\!\!R}}$, then the local generators of ${{\cal{L}}}$ correspond to the vertices of $\Delta$. In particular, $\mbox{ord}\,{{\cal{L}}}(a)=\mbox{min}\,\langle a, \Delta\rangle$. Shifting the polytope $\Delta$ by a vector $r\in M$ means to replace ${{\cal{L}}}$ by $x^{r}\cdot {\cal L}$. The corresponding order functions differ by the globally linear function $\langle \bullet ,r\rangle$. \par {\bf Lemma:} {\em Let $r\inM^\wt$ be an arbitrary element. Then, the mapping $a^\alpha\mapsto -r_\alpha$ gives the order function of ${\cal{L}}(\theta)$ on ${{\cal M}^{\wt}(\kQ)}$. A different choice $r^\prime\inM^\wt$ just changes the order function by the linear summand $\langle \bullet, r' - r \rangle$.} \par {\bf Proof:} We may use $r\inM^\wt$ to carry $\Delta(\theta)$ into the ``right'' lattice $M$ (see the end of \zitat{Mod}{3}). Then, the order function applied to $a^\alpha\in{\Sigma(\wt)}^{(1)}\subseteq N_{{I\!\!R}}$ is \[ \mbox{ord}\,{\cal{L}}(\theta)(a^\alpha)=\mbox{min}\,\langle a^\alpha,\Delta(\theta)-r\rangle = \mbox{min}\,\langle e^\alpha,\Delta(\theta)\rangle -r_\alpha =-r_\alpha\,. \vspace{-4ex} \] \hspace*{\fill}$\Box$ \par \neu{Mod-7} Given the quiver $Q$, the canonical weight $\theta^c$ announced in the introduction is defined as the weight of the flow $r^c$ that is constant $1$ on every arrow. Explicitly, this means \[ \theta^c(q):=\#\{\mbox{arrows with source } q\} - \#\{\mbox{arrows with target } q\}\,. \] The advantage of $\theta^c$ is the existence of a unique interior lattice point in the polytope $\Delta(\theta^c)$: it is again the flow $r^c=[1,\dots,1]$. \par {\bf Proposition:} {\em The polytope $\Delta(\theta^c)$ is reflexive (in the sense of \cite{Ba}), its order function is $-1$ on the generators $a^\alpha\in {\Sigma(\cwt)}^{(1)}$, and the ample divisor ${\cal{L}}(\theta^c)$ is anti-canonical.} \par {\bf Proof:} The three claims are synonymous, i.e.\ we just have to look at the order function of ${\cal{L}}(\theta^c)$. Applying Lemma \zitat{Mod}{6} on $r^c = \underline{1}$ yields the result. \hfill$\Box$ \par \section{The cohomology of the universal bundle}\label{Uni} \neu{Uni-1} From now on we assume that $\theta$ is an integral weight in general position, i.e.\ ${{\cal M}^{\wt}(\kQ)}$ is a smooth variety. To each integral flow we associate a divisor as follows: \[ f^\alpha\inI\!\!F_{Z\!\!\!Z} \mapsto D_\alpha:=\{ x \in {{\cal M}^{\wt}(\kQ)} \mid x_{\alpha} = 0\}= \left\{ \begin{array}{ll} \overline{\mbox{\rm orb}\, (\alpha)} & \mbox{if } a^\alpha\in\Sigma^{(1)}=Q_1(\theta)\\ \emptyset & \mbox{otherwise} \end{array}\right. \] with $\overline{\mbox{\rm orb}\, (\alpha)}$ denoting the closed orbit corresponding to the one-dimensional cone $\alpha$. One obtains surjective maps $I\!\!F_{\Z} \longrightarrow\hspace{-1.5em}\longrightarrow\mbox{\rm Div}\,{{\cal M}^{\wt}(\kQ)}$ from the space of integral flows onto the equivariant divisors and, as a consequence, $I\!\!H_{\Z} \longrightarrow\hspace{-1.5em}\longrightarrow\mbox{\rm Pic}\,{{\cal M}^{\wt}(\kQ)}$, $\theta' \mapsto {\cal{L}}(\theta')$ from the integral weights to the Picard group (see also \cite{HilleTor}, Theorem 2.3). Applying the map $\pi: I\!\!F_{\Z} \longrightarrow\hspace{-1.5em}\longrightarrow I\!\!H_{\Z}$ means to assign a divisor its class in the Picard group. \\ Copying the definition of \zitat{Mod}{3}, every weight $\theta'$ gives rise to \[ \overline{\Delta}(\theta'):= \{r\in \pi^{-1}(\theta')\,|\; r_\alpha\geq 0 \mbox{ for } \alpha\in Q_1(\theta)\}\,. \] Even if ${\cal{L}}(\theta')$ is not ample on ${{\cal M}^{\wt}(\kQ)}$, the polytope $\overline{\Delta}(\theta')$ may still be used to describe the global sections: \par {\bf Proposition:} {\em The lattice points of $\overline{\Delta}(\theta')$ provide a basis of the global sections of ${\cal{L}}(\theta')$. Moreover, if $Q_1(\theta')\subseteq Q_1(\theta)$, then both polytopes $\Delta(\theta')$ and $\overline{\Delta}(\theta')$ coincide. } \par {\bf Proof:} Given $\theta'$, we choose a flow $r\in M^{\theta'}$ providing the order function of a divisor in the class defined by $\theta'$. The corresponding polytope of global sections is contained in $M = M^0$; via the isomorphism $(+r): M^0 \rightarrow M^{\theta'}$ it is mapped onto $\overline{\Delta}(\theta')$.\\ If $Q_1(\theta')\subseteq Q_1(\theta)$, then $\overline{\Delta}(\theta')$ sits between $\Delta(\theta')$ and $\{r\in \pi^{-1}(\theta')\,|\; r_\alpha\geq 0 \mbox{ for } \alpha\in Q_1(\theta')\}$. On the other hand, Proposition \zitat{Mod}{5} implies that the latter two polytopes are equal; its proof shows quite directly that the inequalities parametrized by $Q_1\setminus Q_1(\theta')$ are redundant for the definition of $\Delta(\theta')$. \hspace*{\fill}$\Box$ \par \neu{Uni-2} Since $\theta$ is in general position, there is a universal bundle ${\cal U}$ on ${{\cal M}^{\wt}(\kQ)}$; it splits into a direct sum ${\cal U}= \oplus_{q \in Q_0}{\cal U}_q$ of line bundles. The direct summands ${\cal U}_{p,q}:={\cal U}_p^{-1}\otimes {\cal U}_q$ of $\underline{End\,}({\cal U})$ have the following shape: Choose a walk from $p$ to $q$ along (possibly reversed) arrows $\alpha_1^{\varepsilon(1)},\dots,\alpha_m^{\varepsilon(m)}$, i.e.\ $\alpha_1,\dots,\alpha_m\in Q_1$ and $\varepsilon(i)\in\{\pm 1\}$. Then, denoting by ${\cal O}(\alpha):={\cal O}(D_{\alpha})$ the sheaf corresponding to the prime divisor $D_{\alpha}$, \[ {\cal U}_{p,q}={\cal U}_p^{-1}\otimes {\cal U}_q= \bigotimes\limits_{i=1}^m\,{\cal O}(\alpha_i)^{\varepsilon(i)} \,. \] In the Picard group of ${{\cal M}^{\wt}(\kQ)}$ this sheaf does not depend on the particular choice of the walk from $p$ to $q$: Using the language of \zitat{Uni}{1}, the sheaves $\otimes_i\,{\cal O}(\alpha_i)^{\varepsilon(i)}$ are induced from the flows $\sum_i\varepsilon(i)\cdot f^{\alpha_i}$, which all have the same weight. \par {\em Notation:} Setting $\varepsilon(\alpha^i):=\varepsilon(i)$ and $\varepsilon(\alpha):=0$ for $\alpha\notin\{\alpha^1,\dots,\alpha^m\}$ provides a function $\varepsilon:Q_1\to\{1,-1,0\}$ for every walk. This is the characteristic flow introduced in \zitat{Mod}{3}. Then, the sheaf ${\cal U}_{p,q}$ may be written as ${\cal U}_{p,q}={\cal U}(\varepsilon)=\otimes_{\alpha\inQ_1}{\cal O}(\alpha)^{\varepsilon(\alpha)}$; the corresponding weight $\theta_{p,q}:=\pi(\varepsilon)$ has value $1$ in $p$, $-1$ in $q$, and $0$ in all other vertices. \par \neu{Uni-3} {\bf Proposition:} {\em \begin{itemize}\vspace{-2ex} \item[(1)] Let $\theta\inI\!\!H_{Z\!\!\!Z}$ be an integral weight in general position. Then, the sheaves ${\cal U}_{p,q}\otimes {\cal{L}}(\theta)$ and ${\cal U}_{p,q}^{-1}\otimes {\cal{L}}(\theta)$ are generated by their global sections. \item[(2)] If, additionally, $\theta=\theta^c$, then the polytopes $\overline{\Delta}(\theta^c\pm \theta_{p,q})$ (describing the global sections) have the same dimension as $\Delta(\theta^c)$. \vspace{-1ex}\end{itemize}} \par {\bf Proof:} Since $\,{\cal U}_{q,p}={\cal U}_{p,q}^{-1}$, it is sufficient to consider the latter sheaf. The corresponding polytope $\overline{\Delta}(\theta-\theta_{p,q})$ may be studied in different level sets: \[ \begin{array}{rcl} \overline{\Delta}(\theta-\theta_{p,q}) &=& \{r\in\pi^{-1}(\theta-\theta_{p,q})\,|\; r_\alpha\geq 0 \mbox{ for } \alpha\in Q_1(\theta)\}\\ &\cong& \{r\in\pi^{-1}(\theta)\,|\; r_\alpha\geq \varepsilon(\alpha)\mbox{ for } \alpha\in Q_1(\theta)\}\,. \end{array} \] We will use the second description. \vspace{1ex}\\ (1) The vertices of $\Delta(\theta)$ and thus also the top-dimensional cones of ${\Sigma(\wt)}$ are in a one-to-one correspondence with the $\theta$-stable trees in $Q$. Let $T\in{\cal T}(\wt)$; the corresponding vertex $\Delta^T$ of $\Delta(\theta)$ provides a local generator of ${\cal{L}}(\theta)$. Since the $\Delta^T$ are characterized by the property $\Delta^T_\alpha=0$ for $\alpha\notin T$, we obtain the local generators of ${\cal U}_{p,q}^{-1}\otimes {\cal{L}}(\theta)$ from the lattice points $r^T\inM^\wt$ assigned via Lemma \zitat{Mod}{4} to the map $\varepsilon:Q_1\to{Z\!\!\!Z}$ describing a walk from $p$ to $q$.\\ We have to show that these local generators $r^{T}$ are regular on any open, affine subset $U_{T^\prime}\subseteq{{\cal M}^{\wt}(\kQ)}$ corresponding to some possibly different tree $T^\prime\in{\cal T}(\wt)$. That means, it remains to check that $r^T\in r^{T^\prime}+(\sigma_{T'})^{\scriptscriptstyle\vee}$ where $\sigma_{T'}$ denotes the cone corresponding to $T'$, i.e.\ $\sigma_{T'}$ is spanned by those arrows {\em not} contained in $T^\prime$. \vspace{1ex}\\ {\em Claim: Let $T\in{\cal T}(\wt)$ be a $\theta$-stable tree, and let $\alpha$ be any arrow in $Q$. Then $r^T_\alpha\geq \varepsilon(\alpha)$ is true.} \vspace{0.5ex}\\ Before we prove that claim, we remark that it solves our problem, as, for any tree, we know for $T^\prime$ that $r^{T^\prime}_\alpha=\varepsilon(\alpha)$ for $\alpha\notin T^\prime$. Hence, the claim implies $r^T_\alpha\geq r^{T^\prime}_\alpha$ for $\alpha\notin T^\prime$. On the other hand, $(\sigma_{T'})^{\scriptscriptstyle\vee}$ is just given by the inequalities $r_\alpha\geq 0$ for those $\alpha$. \vspace{1ex}\\ The claim is trivial for $\alpha\notin T$; if $\alpha\in T$, we use the formula for $r_\alpha-\varepsilon(\alpha)$ presented in Lemma \zitat{Mod}{4}. First, as already used in \zitat{Mod}{5}, stability of $T$ implies $\sum_{q\in S_T(\alpha)}\theta_q\geq 1$. Now, the point is to interpret the two remaining sums well: together they just count the number of arrows $\alpha_i^{\varepsilon(i)}$ in the walk from $p$ to $q$ pointing from $T_T(\alpha)$ to $S_T(\alpha)$ minus those from $S_T(\alpha)$ to $T_T(\alpha)$. In particular, \[ \sum\limits_{\makebox[4em]{$\scriptstyle T_T(\alpha)\stackrel{\beta}{\rightarrow} S_T(\alpha)$}} \varepsilon(\beta)\; - \sum\limits_{\makebox[4em]{$\scriptstyle S_T(\alpha)\stackrel{\beta}{\rightarrow} T_T(\alpha)$}} \varepsilon(\beta) \;=\; \left\{\begin{array}{rl} -1 & \mbox{if } p\in S_T(\alpha) \mbox{ and } q\in T_T(\alpha)\\ 1 & \mbox{if } q\in S_T(\alpha) \mbox{ and } p\in T_T(\alpha)\\ 0 & \mbox{if } p,q\in S_T(\alpha) \mbox{ or } p,q\in T_T(\alpha)\,. \end{array}\right. \] In any case, $r_\alpha-\varepsilon(\alpha)$ remains non-negative. \vspace{1ex}\\ (2) For the second part, we consider $\theta=\theta^c$. Since $\varepsilon$ has only $-1$, $0$, or $1$ as values, the canonical flow $r^c=\underline{1}\in\Delta(\theta^c)$ is also contained in $\overline{\Delta}(\theta^c-\theta_{p,q})\subseteq\pi^{-1}(\theta^c)$. Assuming $\dim \overline{\Delta}(\theta^c-\theta_{p,q}) < \dim \Delta(\theta^c)$, this means that there exist arrows $\alpha^1,\dots,\alpha^k\in Q_1(\theta^c)$ having the following two properties: \begin{itemize}\vspace{-2ex} \item[(i)] The flow $r^c$ satisfies the $\alpha^v$-inequalities of $\overline{\Delta}(\theta^c-\theta_{p,q})$ sharp, i.e.\ $1=\varepsilon(\alpha^v)$ for $v=1,\dots,k$. \item[(ii)] It is possible to represent $0\in N_{I\!\!R}$ as a positive linear combination of the vectors $a^{\alpha^v}\in N$, $v=1,\dots,k$. (Recall that $a^\alpha$ is the normal vector of the supporting hyperplane corresponding to the inequality ``$r_\alpha\geq \varepsilon(\alpha)$''.) \vspace{-1ex}\end{itemize} The first property means that, along the chosen walk $\varepsilon$ from $p$ to $q$, the arrows $\alpha^1,\dots,\alpha^k$ have all the same direction. On the other hand, the second property implies that there is a decomposition $Q_0 = Q_0^+ \sqcup Q_0^-$ with $\alpha^1,\dots,\alpha^k$ pointing from $Q_0^+$ to $Q_0^-$ and being the only arrows connecting these two parts. This yields a contradiction. \hfill $\Box$ \par \neu{Uni-4} Let $\Sigma$ be a complete fan in some $d$-dimensional vector space $N_{I\!\!R}$ with lattice $N$. Denote by $M$ the dual lattice. By \cite{Ke}, I/\S 3 we know that the cohomology groups of equivariant, invertible sheaves ${\cal{L}}$ are $M$-graded and how to calculate their summands as reduced topological cohomology groups of certain subsets of $N_{I\!\!R}${.} With $r\in M$ and $A_r:=\{a\in N_{I\!\!R}\, |\; \langle a,r\rangle < \mbox{ord}\,{\cal L}(a)\}$ it follows that $H^l(X_\Sigma,{\cal{L}})_{r}= \widetilde{H}^{l-1}(A_r,k)$ for $l\geq 1$. We would like to use this method to prove a generalization of Kodaira-vanishing which holds for toric varieties. We restrict the subsets $A_r$ in question to the $(d-1)$-dimensional unit sphere $S^{d-1} \subset N_{\R}$: \par {\bf Lemma:} {\em Let $\phi:N_{I\!\!R}\to{I\!\!R}$ be a continuous function which is linear on the cones of $\Sigma$. For $B_{r} := \{a \in S^{d-1} \subset N_{{I\!\!R}} \mid \langle a,r \rangle < \phi(a) \}$ we denote by $\Sigma_r\subseteq\Sigma$ the subfan consisting of all cones $\sigma\in\Sigma$ such that $\sigma\cap S^{d-1}\subseteq B_r$. Then, the sets $B_{r}$ and $|\Sigma_r| \cap S^{d-1}$ are homotopy equivalent. Moreover, the assertion remains true if we replace ``$<$'' by ``$\leq$'' in the definition of $B_{r}$. }\\ ($|\Sigma_r|$ denotes the union of all cones contained in $\Sigma_r$.) \par {\bf Proof:} If the fan $\Sigma$ is simplicial (for instance for smooth varieties $X_\Sigma$), then it is possible to prove in one strike that $|\Sigma_r| \cap S^{d-1}$ is a deformation retract of $B_r$. In the general case, however, it seems to be necessary to project $B_r$ successively down dimension by dimension. Using the following fact, each step can be worked out in the cones of $\Sigma$ separately:\\ {\em Let $P$ be a compact convex polytope and $H^+$ an open or closed half space. Then $\partial P\cap H^+$ is a deformation retract of $P\cap H^+$. }\\ We leave the proof of this obvious fact to the reader. \hfill$\Box$ \par \neu{Uni-5} {\bf Proposition:} (Kodaira-vanishing) {\em Let $X_\Sigma$ be a complete toric variety with at most Gorenstein singularities. Assume that ${\cal{L}}$ is an equivariant line bundle which is generated by its global sections. Then, if the lattice polytope $\Delta\subseteq M_{I\!\!R}$ describing $H^0(X_\Sigma,\,{\cal{L}})$ (as explained in \zitat{Mod}{6}) is full-dimensional, we have $H^l(X_\Sigma,\,{\cal{L}}\otimes\omega_X)=0$ for $l \geq 1$. } \par {\bf Proof:} Denote by $\psi$ the order function of ${\cal{L}}\otimes\omega_X$. The order function $\varphi_K$ of the canonical divisor equals $1$ on the skeleton $\Sigma^{(1)}$; ``Gorenstein'' means that $\varphi_K$ is linear on the cones. Thus, we obtain by the previous Lemma that the sets $B_r:=\{a\in S^{d-1} \subset N_{{I\!\!R}}\,|\; \langle a,r\rangle < \psi(a)\}$ and $C_r:=\{a\in S^{d-1} \subset N_{{I\!\!R}}\,|\; \langle a,r\rangle \leq \psi(a)-\varphi_K(a)\}$ are homotopy equivalent. The first one computes the $r$-th graded piece of the desired cohomology, and the latter is contractible since $\psi-\varphi_K$ is the order function of ${\cal{L}}$, i.e.\ $(\psi-\varphi_K)(a)=\mbox{\rm min}\langle a, \Delta\rangle$. \hfill$\Box$ \par {\bf Remark:} The assumption about the dimension of $\Delta$ means that ${\cal{L}}$ may not be obtained via pull back from some lower-dimensional variety. \par \neu{Uni-6} {\bf Theorem:} {\em Assume $\theta^c$ is in general position. Then $\mbox{\rm Ext}\,^{l}_{{{\cal M}(\kQ)}}({\cal U},{\cal U}) = 0 \mbox{ for all } l >0$. } \par {\bf Proof:} Recall that $\mbox{\rm Ext}\,^l_{{{\cal M}(\kQ)}}({\cal U},{\cal U})=H^l({{\cal M}(\kQ)},\,\underline{End\,}\,{\cal U})= \bigoplus_{p,q\in Q_0} H^l({{\cal M}(\kQ)},\,{\cal U}_{p,q})$. Then, since ${\cal U}_{p,q}\otimes \omega_{{\cal M}}^{-1} \simeq {\cal U}_{p,q}\otimes {\cal{L}}(\theta^c)$ is globally generated with $\dim\overline{\Delta}(\theta_{p,q}+\theta^c)=\dim\Delta(\theta^c)$ (cf.\ Propositions \zitat{Mod}{7} and \zitat{Uni}{3}), the result follows from Kodaira vanishing. \hfill $\Box$ \par \neu{Uni-7} The previous theorem asks for the canonical weight to be in general position. We would like to close this section with a criterion for this fact to hold. Moreover, we present a criterion for $Q_1 = Q_1(\theta^c)$. \par {\bf Proposition:} {\em The canonical weight $\theta^c$ is in general position if and only if there does not exist any $(t,t)$-wall. Moreover, $Q_1(\theta^c) = Q_1$ if and only if there are no $(1,0)$- or $(1,1)$-walls.} \par {\bf Proof: } Assume a $(t^+,t^-)$-wall is given. Then, $\sum_{q \in Q_0^+}\theta^c_q = t^+ - t^-$ by adding the $M^\wt$-equations. Consequently, $\theta^c$ lies on this wall precisely when $t^+ - t^- = 0$. This proves the first claim.\\ For the second claim, we include the case of a $(1,1)$-wall in brackets. Let $Q_0 = Q_0^+ \sqcup Q_0^-$ be the subdivision defining the wall (see \zitat{Mod}{2}), and denote by $\alpha$ the unique arrow with $s(\alpha) \in Q_0^+$ and $t(\alpha) \in Q_0^-$ [by $\beta$ the unique arrow with $s(\beta) \in Q_0^-$ and $t(\beta) \in Q_0^+$]. We show that $\alpha$ is not in $Q_1(\theta^c)$ [$\alpha$ and $\beta$ are not both in $Q_1(\theta^c)$]. The set $Q_0^+$ is closed under successors in $Q \setminus \{ \alpha \}$, but $\sum_{q \in Q_0^+}\theta^c_q = 1$ [$ = 0$]. Thus, $Q \setminus \{ \alpha \}$ is not stable [both $Q \setminus \{\alpha \}$ and $Q \setminus \{ \beta \}$ are not stable]. \\ It remains to show the converse. Assume we have no $(1,0)$-wall and no $(1,1)$-wall. Thus, $t^+ \geq 2$ for each wall $W$. Assume further that $W$ is a $(t^+,t^-)$-wall with $t^+ > t^-$. We define the open halfspace $W^+ := \{ \theta \in I\!\!H \mid \sum_{q \in Q_0^+} \theta_q > 0 \}$, i.e.\ $\theta^c \in W^+$. Using the wall crossing formula from \cite{HilleTor}, Lemma 3.4, we obtain $Q_1(\theta^c) = \cup_{\theta \in I\!\!H} Q_1(\theta)$. It remains to show that $Q_1 = \cup_{\theta \in I\!\!H} Q_1(\theta)$. By assumption, for each $\alpha \in Q_1$ there exists a tree in $Q \setminus \{\alpha\}$. But for each tree $T$ there exists a weight $\theta$ such that $T$ is $\theta$-stable (\cite{HilleThin}, Proposition 2.5). This finishes the proof. \hfill $\Box$ \par \section{The endomorphism algebra of the universal bundle}\label{End} \neu{End-1} In this section we always assume that $\theta$ is in general position, i.e.\ the universal bundle on ${{\cal M}^{\wt}(\kQ)}$ exists. We start this section with a result about the endomorphism algebra ${\cal{A}}$ of the universal bundle. This algebra is non-commutative and finite-dimensional; in order to formulate the statements in this section, we need some basic results about those algebras. We denote by $\mbox{rad}({\cal{A}})$ the radical of ${\cal{A}}$. It consists of all strongly nilpotent elements $a$ of ${\cal{A}}$, that is $(a {\cal{A}})^n = 0$ for $n$ sufficiently large. Thus ${\cal{A}}$ is isomorphic to the quotient of the tensor algebra of the ${\cal{A}}/\mbox{rad}({\cal{A}})$-bimodule $\mbox{rad}({\cal{A}})/\mbox{rad}^2({\cal{A}})$ by some admissible ideal $I$ $$ {\cal{A}} \simeq T_{({\cal{A}}/\mbox{rad}({\cal{A}}))}\big(\mbox{rad}({\cal{A}})/\mbox{rad}^2({\cal{A}})\big)/I. $$ Recall that an ideal $I$ is called {\sl admissible} if $$ T^n_{({\cal{A}}/\mbox{rad}({\cal{A}}))}(\mbox{rad}({\cal{A}})/\mbox{rad}^2({\cal{A}})) \subset I \subset T^2_{({\cal{A}}/\mbox{rad}({\cal{A}}))}(\mbox{rad}({\cal{A}})/\mbox{rad}^2({\cal{A}})) $$ for some $n$. In case ${\cal{A}}$ is the path algebra of a quiver, the radical of ${\cal{A}}$ is the ideal generated by paths of length at least one. \par A finite-dimensional algebra ${\cal{A}}$ is called {\sl basic} if the semisimple quotient ${\cal{A}}/\mbox{rad}({\cal{A}})$ is a product of fields. Because we deal with basic algebras over an algebraically closed field, this semisimple quotient is a product of copies of the ground field. It turns out that each basic finite-dimensional algebra is isomorphic to the quotient of a path algebra of a finite quiver by some admissible ideal. Moreover, each finite-dimensional algebra is Morita equivalent to a basic finite-dimensional algebra, that is the module categories of both algebras are isomorphic. Thus, if we are interested in module categories, we may restrict ourselves to modules over basic algebras. \\ The endomorphism ring of ${\cal U}$ is basic precisely when ${\cal U}$ contains only pairwise non-isomorphic direct summands. Consequently, for $\mbox{End}({\cal U})$ to be isomorphic to the path algebra of $Q$, it is necessary that the direct summands ${\cal U}_q$ are pairwise non-isomorphic. In fact, in the theorem below we will see that the converse is also true. \par \neu{End-2} In any case it would be desirable to know $\mbox{End}({\cal U})$ and its Morita equivalent basic algebra. This leads to the following definitions (cf.\ \cite{Schofield}): Let $Q$ be a quiver without oriented cycles. Thus, the path algebra $\ck Q$ is finite-dimensional. For an arrow $\alpha\in Q_1$ we define the {\em localization} $\ck Q[\alpha^{-1}]$ by formally adjoining the inverse $\alpha^{-1}$ of the arrow $\alpha$, i.e.\ $s(\alpha^{-1}) = t(\alpha)$, $t(\alpha^{-1}) = s(\alpha)$, and $\alpha^{-1} \alpha = e_{t(\alpha)}$, $\alpha \alpha^{-1} = e_{s(\alpha)}$ where $e_q$ is the idempotent in $\ck Q$ corresponding to the vertex $q$. In particular, $\alpha$ and $\alpha^{-1} \in \ck Q[\alpha^{-1}]$ are {\sl not} in the radical. Consequently, $\ck Q[\alpha^{-1}]$ is not basic, because $\ck Q[\alpha^{-1}]/\mbox{rad}(\ck Q[\alpha^{-1}])$ contains the two-by-two full matrix ring with basis $e_{s(\alpha)},e_{t(\alpha)},\alpha, \alpha^{-1}$. \\ We consider the {\em quotient quiver} $\overline{Q} := Q/(Q_1 \setminus Q_1(\theta))$ defined by killing the arrows from $Q_1 \setminus Q_1(\theta)$ while identifying their sources and targets, respectively. Each weight of $Q$ provides in a canonical way a weight of $\overline{Q}$ -- just add the values of the identified vertices. The corresponding moduli spaces are isomorphic. The localization $Q[\alpha^{-1}\mid \alpha \notin Q_1(\theta)]$ is Morita equivalent to $\ck \overline{Q}$. Moreover, this localization is finite-dimensional if and only if the quotient quiver $\overline{Q}$ contains no oriented cycle. \par {\bf Lemma}: {\em The line bundle ${\cal U}_q$ is isomorphic to the line bundle ${\cal U}_p$ if and only if there exists a walk from $p$ to $q$ consisting of arrows {\rm not} in $Q_1(\theta)$.} \par {\bf Proof: } The class of the bundle ${\cal U}_p^{-1} \otimes {\cal U}_q$ is trivial if and only if the divisor $\sum_{\alpha \in w}D_\alpha - \sum_{\alpha^{-1} \in w}D_{\alpha} $ is linearly equivalent to zero for a walk $w$ from $p$ to $q$ in $Q$. Moreover, this divisor is also equivalent to $\sum_{\alpha \in w \cap Q_1(\theta)}D_\alpha - \sum_{\alpha^{-1} \in w \cap Q_1(\theta)}D_{\alpha}$, because $D_{\alpha} \sim 0$ for $\alpha \notin Q_1(\theta)$. This divisor is linearly equivalent to $0$ if and only if $w \cap Q_1(\theta)$ is a cycle in $Q/(Q_1 \setminus Q_1(\theta))$. This is true if and only if $w \cap Q_1(\theta) = \emptyset $. \hfill $\Box$ \nopagebreak \par \nopagebreak In particular, the indecomposable direct summands of the universal bundles on ${{\cal M}^{\wt}(\kQ)}$ and ${\cal{M}}^{\overline{\theta}}(\overline{Q})$ are isomorphic; in ${\cal U}(\overline{Q})$ we have just cancelled multiple summands. \par \neu{End-3} {\bf Theorem:} {\em The endomorphism algebra ${\cal{A}}$ of the universal bundle ${\cal U}$ on ${{\cal M}^{\wt}(\kQ)}$ is isomorphic to the localization of the path algebra of the quiver by all arrows {\rm not} in $Q_1(\theta)$. If $Q_1(\theta) = Q_1$, then ${\cal{A}}$ is isomorphic to the path algebra of the quiver $Q$.} \par {\bf Proof: } As explained in the previous remarks, we may assume that $Q_1(\theta) = Q_1$. By Proposition \zitat{Uni}{1} we know that \[ \mbox{Hom}_{{{\cal M}^{\wt}(\kQ)}}({\cal U}_p,{\cal U}_q)= H^0({{\cal M}^{\wt}(\kQ)},{\cal U}_p^{-1}\otimes {\cal U}_q)= k\cdot\Big\{\mbox{lattice points of } \Delta(\theta_{p,q})\Big\} \] where $\theta_{p,q}$ is the weight introduced in \zitat{Uni}{2}. Since an integral flow in $\Delta(\theta_{p,q})$ has values in $\{0,1\}$, we obtain a bijection between the lattice points of $\Delta(\theta_{p,q})$ and the paths from $p$ to $q$ in $Q$; hence $\mbox{End}_{{{\cal M}^{\wt}(\kQ)}}({\cal U}) \simeq kQ$. \hfill$\Box$ \par \neu{End-4} Let ${\cal U}$ be a vector bundle on a smooth projective algebraic variety ${\cal{M}}$. Let ${\cal{A}}$ be the endomorphism algebra of ${\cal U}$, which is finite-dimensional. Moreover, let ${\cal U} = \oplus_{q \in Q_0}{\cal U}_q$ be a decomposition into indecomposable direct summands. Then ${\cal{A}} = \oplus_{q \in Q_0}e_q{\cal{A}}$ is a decomposition of ${\cal{A}}$ into indecomposable projective right ${\cal{A}}$-modules. We denote by $K^b({\cal U}_q \mid q \in Q_0 )$ and $K^b(e_q{\cal{A}} \mid q \in Q_0 )$ the homotopy category of bounded complexes $\{ C^i \}$, where each $C^i$ is a direct sum of copies of ${\cal U}_q$ or $e_q{\cal{A}}$, respectively. The functor induced by the map ${\cal U}_q \mapsto e_q{\cal{A}}$ is an equivalence between $K^b({\cal U}_q \mid q \in Q_0 )$ and $K^b(e_q{\cal{A}} \mid q \in Q_0 )$ because the endomorphism algebra of ${\cal{A}}$ viewed as {\sl right} ${\cal{A}}$-module is ${\cal{A}}$. \par {\bf Theorem:} {\em Assume $Q$ is a quiver without any $(t,t)$-wall. Then the equivalence above induces a full and faithful functor $$ {\cal{D}}^b\Big(\mbox{mod--} {\cal{A}} \Big) \longrightarrow {\cal{D}}^b\Big(\mbox{Coh}\big({{\cal M}(\kQ)}\big)\Big). $$} \par {\bf Proof:} We define $p \leq q$ if $\mbox{Hom}({\cal U}_p,{\cal U}_q) \not= 0$. This is a partial order on $Q_0$ because ${\cal U}_q$ is a line bundle for all $q \in Q_0$. Consequently, $\mbox{End}({\cal U})$ is a directed algebra (there is an order on $Q_0$ such that $\mbox{Hom}_{{\cal{A}}}(e_p{\cal{A}},e_q{\cal{A}}) = 0$ for $p > q$). A directed algebra is of finite global dimension, thus the bounded derived category ${\cal{D}}^b(\mbox{mod--} {\cal{A}})$ of finitely generated right ${\cal{A}}$-modules is equivalent to the bounded homotopy category $K^b(e_q{\cal{A}} \mid q \in Q_0 )$ (cf.\ \cite{Happel} \S 1, 3.3). Since ${\cal U}$ has no self-extension (Theorem \zitat{Uni}{6}), the natural functor $K^b({\cal U}_q \mid q\in Q_0 ) \rightarrow {\cal{D}}^b(\mbox{Coh}({{\cal M}(\kQ)}))$ is full and faithful. \hfill $\Box$ \par {\bf Proof of Theorem \zitat{Int}{3}: } If there exists no $(1,0)$-wall and no $(1,1)$-wall we have ${\cal{A}} \simeq \ck Q$ by Proposition \zitat{Uni}{7} and Theorem \zitat{End}{3}. Thus, the result follows from Theorem \zitat{End}{4} \hfill $\Box$ \par

9706

alg-geom/9706011

en

https://arxiv.org/abs/alg-geom/9706011

alg-geom/9706011

Donu Arapura

Donu Arapura and Madhav Nori

Solvable fundamental groups of algebraic varieties and K\"ahler manifolds

16 pages, latex2e

This is a long overdue write up of the following: If the fundamental group of a normal complex algebraic variety (respectively Zariski open subset of a compact K\"ahler manifold) is a solvable group of matrices over Q (respectively polycyclic group), then it is virtually nilpotent. This should be interpreted as saying that a large class of solvable groups will not occur as fundamental groups of such spaces. The essential strategy is to first "complete" these groups to algebraic groups (generalizing constructions of Malcev and Mostow), and then check that the identity component is unipotent. This is done by Galois theoretic methods in the algebraic case, and is reduced to homological properties of one dimensional characters in the analytic case.

alg-geom

\section{Preliminaries on Algebraic Groups} A general reference for this section is \cite{Bo}. Let $DG = D^1G$ be the derived subgroup of a group $G$, and set $D^iG= DD^{i-1}G$. If $F \subset E$ is an extension of fields, and $G$ is an algebraic group defined over $F$, let $G_{E} = G\times_{spec\, F}spec\, E$. For the remainder of this section, $G$ will denote an algebraic group over a field $F$ of characteristic $0$, and $U(G)$ will be its unipotent radical. We will make use of the following result of Mostow \cite{M56}: \begin{thm}\label{thm:splitting} If $G$ is as above, then the exact sequence: $$1\to U(G)\to G\to G/U(G)\to 1$$ is split, and any two splittings are conjugate by an $F$ rational point of $U(G)$. \end{thm} Let $V$ be the centralizer of $U(G)$ in $G$, and let $W= V\cap U(G)$. Clearly $W$ is the unipotent radical of $V$. It follows that $$1\to W\to V\to V/W\to 1$$ has a unique splitting $s: V/W\to V$, because $W$ central in $V$. Put $N(G) = s(V/W)$. By construction $N(G)$ is reductive and invariant under all automorphisms of $G$; in particular, it is normal in $G$. \begin{lemma}\cite[lemma 4.6]{M} Every normal reductive algebraic subgroup of $G$ is contained in $N(G)$. \end{lemma} \begin{defn} An algebraic group $G$ is minimal if $N(G)$ is trivial. Let $G_{min} = G/N(G)$; then this is a minimal algebraic group. An algebraic group $G$ is minimal if and only if the centralizer of $U(G)$ is contained in $U(G)$, as we see from the definition of $N(G)$. \end{defn} \begin{lemma}\label{lemma:a3} Let $f:G\to G'$ be a homomorphism of algebraic groups for which $f(G)$ is normal in $G'$. Then $f(N(G)) \subseteq N(G')$, and consequently, $f$ induces a homomorphism $f_{min}:G_{min}\to G_{min}'$. If $f$ is a injection (respectively surjection), then $f_{min}$ is also a injection (respectively surjection). \end{lemma} \begin{proof} As $f(N(G))$ is normal and reductive in $f(G)$, we see that $f(N(G))\subseteq N(f(G))$. As $N(f(G))$ is invariant under all automorphisms of $f(G)$, and in particular the restriction of inner automorphisms of $G'$ to $f(G)$, it follows that $N(f(G))\subseteq N(G')$. This proves the first assertion. Because $H = f(G)\cap N(G')$ is normal in the reductive group $N(G')$, we see that $H$ is reductive. Also, $H$ is normal in $f(G)$, and this implies that $H=N(f(G))$, Thus $ker(f) = \{1\}$ implies $\ker(f_{min}) =\{1\}$. \end{proof} \begin{lemma}\label{lemma:a4} Let $G$ be a minimal algebraic group defined over a field $F$ of characteristic $0$. There is an affine algebraic group $A$, defined over $F$, that acts on $G$ so that for all fields $E\supseteq F$, $A(E)\to Aut(G_E)$ is an isomorphism. \end{lemma} \begin{proof} If $G = U(G)$, the automorphism of $G$ are just automorphisms of its Lie algebra, and these evidently form an affine algebraic group, denoted by $Aut(G)$. In the general case, by \ref{thm:splitting} we can express $G$ as a semidirect product of $M$ and $U(G)$, where $M$ is reductive. The homomorphism $\rho: M\to Aut(U(G))$ is faithful, because $G$ is minimal. Let $X$ be the normalizer of $\rho(M)$ in $Aut\, U(G)$. Thus $X$ is an algebraic group, and $X$ acts naturally on $U(G)$ and $M$, and hence on $G$, their semidirect product. Now $G$ acts on itself by conjugation. Thus $Y$, the semidirect product of $X$ and $G$, acts on $G$. For any field $E\supset F$, we see $X(E)\to Aut G_E$ is one to one and its image equals $\{\phi\in Aut G_E\, |\, \phi(M_E) = M_E\}$. But, if $\phi\in Aut(G_E)$, $\phi(M_E)$ is a conjugate of $M_E$, and we deduce that $Y(E)\to Aut G_E$ is surjective for all fields $E$ containing $F$. Finally, the coordinate ring $R$ of $G$ is generated as an $F$-algebra by a finite dimensional $Y$-stable subspace $V\subset R$. Put $K = ker(Y\to GL(V))$ and let $A = Y/K$. We see that $A$ is the desired algebraic group. \end{proof} The algebraic group $A$ in the previous lemma will be denoted by $Aut\, G$ henceforth. \begin{lemma}\label{lemma:a5} If $G$ is a solvable minimal algebraic group, then the action of $Aut(G)^o$ on $G^o/U(G)$ is trivial. \end{lemma} \begin{proof} This follows immediately from the fact that the action of a connected algebraic group on a torus is trivial. \end{proof} \begin{lemma}\label{lemma:a6} Let $\Lambda$ be a directed set, and $\{G_\lambda\}_{\lambda\in \Lambda}$ a directed system of minimal algebraic groups, such that for each $\lambda\le \mu$, we have $G_\lambda \subseteq G_\mu$ and $U(G_\lambda) = U(G_\mu)$. Then there is a minimal algebraic group $G$ and a monomorphism $f_\lambda: G_\lambda \to G$ for each $\lambda\in \Lambda$, so that $f_\lambda = f_\mu |_{G_\lambda}$ whenever $\lambda \le \mu$. \end{lemma} \begin{proof} Let $U= U(G_\lambda)$ for all $\lambda\in \Lambda$. Let $$S_\lambda = \{M\subseteq G_\lambda\, |\, M\> is\> a\> closed\> subgroup\> and\> M\to G_\lambda/U \> is \> an \> isomorphism\}$$ Then $U$ acts transitively on $S_\lambda$. Also $M\to M\cap G_\lambda$ gives a $U$-equivariant morphism from $S_\mu$ to $S_\lambda$ whenever $\lambda \le \mu$. Now choose $\lambda_0\in \Lambda$ so that $dim\, S_{\lambda_0} \ge dim\, S_\lambda$ for all $\lambda \in \Lambda$. From the above, we see that $S_\mu \to S_{\mu_0}$ is a bijection if $\lambda_0 \le \mu$. It follows that there is a collection $\{M_\lambda\, | \, \lambda \in \Lambda \}$ with $M_\lambda \in S_\lambda$, and $M_\mu \cap G_\lambda = M_\lambda$ whenever $\lambda \le \mu$. If $\rho_\lambda : M_\lambda \to Aut U$ denotes the conjugation action of $M_\lambda$ on $U$, we have seen that $\rho_\lambda$ is one to one because $G_\lambda$ is minimal. Also, the inequality $\lambda \le \mu$ implies that $\rho_\lambda (M_\lambda) \subseteq \rho_\mu (M_\mu)$. Let $M$ be the Zariski closure of $\cup \{\rho_\lambda(M_\lambda)\, |\, \lambda \in \Lambda \} $. This is reductive. Let $G$ be the semidirect product of $M$ and $U$, and define $f_\lambda: G_\lambda \to M$ by $f_\lambda(u) = u$ for all $u\in U$, and $f_\lambda = \rho_\lambda$ on $M_\lambda$. This completes the proof. \end{proof} Let $G$ be a connected solvable group defined over $F$. Let $U=U(G)$ and $T$ a maximal torus. As noted previously, $Aut\,U$ is an algebraic group which coincides with the group of automorphisms of the Lie algebra $N$ of $U$. There is a homomorphism of algebraic groups $T\to Aut\, U$ given by conjugation. $G$ is the semidirect product of $T$ with $U$. \begin{lemma} With the previous notation, suppose that $T$ acts trivially (by conjugation) on $U/DU$ then $G$ is isomorphic to the product of $U$ and $T$, and is therefore nilpotent. \end{lemma} \begin{proof} We have to show that $T$ acts trivially on $U$. Let $S$ be the subgroup of $Aut(N)$ of automorphisms $\sigma$ satisfying $(1-\sigma)(N) \subseteq [N,N]$. The elements of $S$ are unipotent. By assumption, the image of the homomorphism $T \to Aut(U)= Aut(N)$ lies in $S$ and so the map must be trivial. \end{proof} There is a homomorphism of algebraic groups $G/DG\to Aut\, DG/D^{2}G$ given by conjugation. \begin{lemma}\label{lemma:unip} With the previous notation, suppose that image of $G/DG$ in $Aut\linebreak[0] DG/D^2G$ is unipotent. Then $G$ is the product of $U$ with $T$. \end{lemma} \begin{proof} Note that $G/DG$ is a product of a reductive group, which is isomorphic to $T$, and a unipotent group $U'$, which is isomorphic to the image of $U$ under the projection $G\to G/DG$. The action of $T$ on $U'$ by conjugation is of course trivial. Consider the exact sequence $$ DG/D^2G \to U/DU \to U' \to 0.$$ By assumption, the image of $T\subset G/DG$ in $Aut\, DG/D^2G$ is unipotent, and therefore trivial. Thus $T$ acts trivially on the image of $DG/D^2G$ in $U/DU$ as well as on it the cokernel. Therefore $T$ acts trivially on $U/DU$, and the lemma follows from the previous one. \end{proof} \section{ Solvable groups of finite rank} In this section, we shall associate to a solvable group $\Gamma$ of finite rank, an algebraic group of $H(\Gamma)$ defined over $\Q$ and a homomorphism $i(\Gamma):\Gamma\to H(\Gamma)(\Q)$ with Zariski dense image and a torsion subgroup as kernel. While $\Gamma \mapsto H(\Gamma)$ is a not a functor, automorphisms of $\Gamma$ will extend to automorphisms of $H(\Gamma)$. Furthermore when $\Gamma$ is finitely generated, automorphisms of its profinite completion $\hat \Gamma$ extend to automorphisms of $H(\Gamma)_{\Q_l}$ for all but finitely many primes $l$. Our construction of $H(\Gamma)$ is identical to Mostow's \cite{M} in the case where $\Gamma$ is polycyclic, but our proof that this construction works for solvable groups of finite rank is necessarily a bit more complicated. Also the results on the profinite completion are used crucially when Galois theory is applied. For these reasons, we have chosen to give all the details of the proofs. Let us recall the standard construction of the proalgebraic hull of $\HH(\pi, F)$ associated to a topological group $\pi$ and a topological field $F$. One considers the category $C(\pi, F)$ where the objects are pairs $(G,f)$ with $G$ an affine algebraic group defined over $F$, and $f:\pi \to G(F)$ a continuous homomorphism with Zariski dense image. A morphism $(G,f)\to (G', f')$ in our category is simply a commutative diagram $$\begin{array}{ccccc} G& &\stackrel{\phi}{\longrightarrow}& & G'\\ &\nwarrow & & \nearrow& \\ & & \pi & & \\ \end{array} $$ such that $\phi$ is homomorphism of algebraic groups. Such a $\phi$, if it exists is unique and necessarily an epimorphism because $f$ and $f'$ have Zariski dense images. In particular, by lemma \ref{lemma:a3}, $\phi_{min}: G_{min} \to G_{min}'$ is defined. Set $$\HH(\pi, F) = \lim_{\stackrel{\longleftarrow}{(G,f)}}\, G$$ and $$H(\pi, F) = \lim_{\stackrel{\longleftarrow}{(G,f)}}\, G_{min}.$$ Some easy observations follow. \begin{remark}\label{remark:1} $\pi \mapsto \HH(\pi, F)$ is a functor, but $\pi \mapsto H(\pi, F)$ is not. However, if $a:\pi\to \pi'$ is a continuous homomorphism with the closure of $a(\pi)$ normal in $\pi'$, we deduce, from lemma \ref{lemma:a3} that $H(a,F):H(\pi, F)\to H(\pi',F)$ is defined. In particular, if $\Gamma$ is a discrete group and $t: \Gamma\to \hat\Gamma$ is the homomorphism to its profinite completion, we have a natural epimorphism $$H(t,F): H(\Gamma, F) \to H(\hat \Gamma, F)$$ \end{remark} \begin{remark} If $F\to E$ is a continuous homomorphism of fields, we have a functor $C(\pi, F) \to C(\pi, E)$. And this induces an epimorphism $H(\pi,E)\to H(\pi, F)_E$. \end{remark} \begin{remark}\label{remark:3} By construction, we have a continuous homomorphism $i(\pi, F)$ from $\pi$ to the group of $F$-rational points of $H(\pi,F)$. With $a:\pi\to\pi'$ as in remark \ref{remark:1}, we have: $$H(a,F)\circ i(\pi, F) = i(\pi', F)\circ a$$ \end{remark} \begin{defn} A solvable group $\Gamma$ has finite rank, if there is a decreasing sequence $$\Gamma = \Gamma_0\supset \Gamma_1 \supset \ldots \supset \Gamma_{m+1} = \{1\}$$ of subgroups, each normal in its predecessor, such that $\Gamma_i/\Gamma_{i+1}$ is abelian and $\Q \otimes (\Gamma_i/\Gamma_{i+1})$ is finite dimensional for all $i$. The rank $$rk(\Gamma) = \sum_{i=0}^m\, rk(\Q \otimes (\Gamma_i/\Gamma_{i+1}))$$ is clearly independent of the choice of the sequence $\{\Gamma_i\}$. \end{defn} This is a weakening of the notion of a polycyclic group which, in the above terms, amounts to requiring that each $\Gamma_i/\Gamma_{i+1}$ is finitely generated. For the remainder of this section, all groups considered are solvable of finite rank with discrete topology unless indicated otherwise. We endow $\Q$ with discrete topology, and abbreviate $H(\pi, \Q), i(\pi,\Q), H(a, \Q)$ by $H(\pi)$ etcetera. The only fields $F$ considered have characteristic zero. \begin{thm}\label{thm:solv} Let $\Gamma$ be a solvable group of finite rank with discrete topology. Then (A) $H(\Gamma)$ is an algebraic group (and not just a proalgebraic group). (B) $rk(\Gamma) = dim \, U(H(\Gamma))$. (C) The kernel of $i(\Gamma) : \Gamma \to H(\Gamma)(\Q)$ is a torsion group. (D) The image of $i(\Gamma)$ is Zariski dense. \end{thm} Part (D) of the theorem is immediate from the construction. When $\Gamma$ is polycyclic, the theorem is due to Mostow \cite[4.9]{M}. As a corollary we obtain a natural characterization of these groups. \begin{cor} A solvable group $\Gamma$ has finite rank if and only if there exists a torsion normal subgroup $N\subset\Gamma$ such that $\Gamma/N$ possesses an embedding into an affine algebraic group defined over $\Q$. \end{cor} \begin{proof} One direction follows from the theorem. For the converse, suppose that $N\subseteq \Gamma$ is torsion subgroup and $i:\Gamma\to G(\Q)$ a homomorphism into an algebraic group with kernel $N$. We can assume that $G$ is solvable, after replacing it by the Zariski closure of $\Gamma$. Then the sequence $\Gamma_i= i^{-1}(D^iG(\Q))$ has the required properties. \end{proof} \begin{defn} $n(\Gamma, F) = sup\,\{ dim\, U(G) \,|\, (G,f) \in Obj\, C(\Gamma, F) \}.$ Thus $n(\Gamma, F) \in \N \cup \{\infty\}$. \end{defn} \begin{lemma} \label{lemma:s6} If $1\to \Gamma' \to \Gamma \to \Gamma'' \to 1$ is exact, $n(\Gamma, F) \le n(\Gamma', F) + n(\Gamma'', F)$. \end{lemma} \begin{proof} Let $(G,f)$ be an object of $C(\Gamma, F)$. Now let $G' $ be the Zariski closure of $f(\Gamma')$ and let $G'' = G/G'$. Then $f$ induces $\bar f:\Gamma'' \to G''(F)$ and both $(G',f|_{\Gamma'})$ and $(G'', \bar f)$ are in $C(\Gamma',F)$ and $C(\Gamma'', F)$ respectively. A short exact sequence of algebraic groups induces a short exact sequence of unipotent radicals, so the lemma follows. \end{proof} \begin{lemma}\label{lemma:s7} $n(\Gamma, F) \le rk(\Gamma)$, where $F$ is a field of characteristic zero. \end{lemma} \begin{proof} From the previous lemma, by induction on the length of the derived series of $\Gamma$, we are reduced to the case where $\Gamma$ is abelian. If $(G,f) \in Obj\, C(\Gamma, F)$, then the inclusion $U(G) \hookrightarrow G$ is split by a homomorphism $p:G \to U(G)$. But in this case, $U(G)$ is a vector space, spanned by $p(f(\Gamma))$, and so evidently $rk(\Q\otimes \Gamma) \ge dim\, U(G)$. \end{proof} \begin{lemma}\label{lemma:s8} If $(G, f)\in Obj\, C(\Gamma, F)$ satisfies $n(\Gamma, F) = dim\, U(G)$, then $H(\Gamma, F) \to G_{min}$ is an isomorphism. In particular, $H(\Gamma, F) $ is an algebraic group and $dim\, U(H(\Gamma, F)) = n(\Gamma, F)$. \end{lemma} \begin{proof} For the first assertion, we need to check that any morphism $\phi: (G', f') \to (G, f)$ induces an isomorphism $\phi_{min}: G_{min}' \to G_{min}$. Now, $\phi$ being as epimorphism, restricts to an epimorphism of unipotent radicals. This gives $$n(\Gamma, F) \ge dim\, U(G') \ge dim \, U(G) = n(\Gamma, F).$$ Therefore, $U(G') \to U(G)$ gives an isomorphism of Lie algebras, and is itself an isomorphism. Consequently, $ker\phi$ is reductive, and $\phi_{min} $ is an isomorphism. For the second assertion, we need to know that such a $(G,f)$ exists, and this is assured by the previous lemma. \end{proof} \begin{lemma}\label{lemma:s9} Let $1\to \Gamma'\to \Gamma \to \Gamma'' \to 1$ be exact. Assume that $i(\Gamma', F)$ extends to a homomorphism $j:\Gamma\to G(F)$ where $G$ is an algebraic group that contains $H(\Gamma', F)$. Then (A) $H(\Gamma', F)\to H(\Gamma, F)$ is a monomorphism. (B) $n(\Gamma, F) = n(\Gamma', F) + n(\Gamma'', F)$ (C) $1\to ker\, i(\Gamma', F)\to ker\, i(\Gamma, F)\to ker\, i(\Gamma'', F)$ is exact. \end{lemma} \begin{proof} Let $q:\Gamma' \to \Gamma$ and $p:\Gamma' \to \Gamma''$ denote the given homomorphisms. By remark \ref{remark:1}, we have $H(q,F):H(\Gamma',F)\to H(\Gamma, F)$ and $H(p,F):H(\Gamma,F)\to H(\Gamma'', F)$. Now $H(p,F)$ is an epimorphism whose kernel contains the normal subgroup $image(H(q,F))$. Thus, if we assume (A), we obtain $$ n(\Gamma, F) = dim\, U(H(\Gamma, F)) $$ $$ \ge dim\, U(H(\Gamma', F)) + dim\, U(H(\Gamma'', F))$$ $$ = n(\Gamma', F) + n(\Gamma'', F)$$ from the previous lemma. But lemma \ref{lemma:s6} gives the reverse inequality, and this proves part (B). That (A) implies (C) is clear by remark \ref{remark:3}, for we have $$H(p,F)\circ i(\Gamma, F) = i(\Gamma'', F)$$ and $$H(q, F) \circ i(\Gamma', F) = i(\Gamma, F) \circ q.$$ To check (A), replace $G$ in the lemma by the Zariski closure of $f(\Gamma)$, This makes $(G,j)$ an object of $C(\Gamma, F)$; denote by $k: H(\Gamma, F) \to G_{min}$ the natural homomorphism. Then $k \circ H(q, F)$ is the composite: $H(\Gamma',F)\to G\to G_{min}$. By lemma \ref{lemma:a3}, this is an inclusion, This completes the proof of the lemma. \end{proof} \begin{lemma}\label{lemma:s10} The $(G,j)$ in the previous lemma exists if (A) $\Gamma'' \cong \Z$, or (B) $\Gamma''$ is a abelian torsion group. \end{lemma} \begin {proof} Case (A): Here $\Gamma$ is a semidirect product. Choose $\gamma\in \Gamma$ that maps to a generator of $\Gamma''$. Then $\sigma(\delta) = \gamma\delta \gamma^{-1}$, for $\delta\in \Gamma'$ gives an automorphism of $\Gamma'$, and induces therefore an automorphism $H(\sigma, F)$ of the algebraic group $H(\Gamma', F)$. By lemma \ref{lemma:a4}, $A=Aut\,H(\Gamma, F)$ is an algebraic group. Let $G$ be the semidirect product of $A$ and $H(\Gamma, F)$, and define $j:\Gamma \to G(F)$ by $j(\delta) = i(\Gamma')(\delta)$ for $\delta \in \Gamma'$ and $j(\gamma) = H(\sigma,F)$. Case (B): First assume that $\Gamma'' $ is finite. Let $\rho:H(\Gamma', F) \to GL(V)$ be a faithful representation where $V$ is finite dimensional vector space defined over $F$. Consider the induced representation $W = F[\Gamma]\otimes_{F[\Gamma']}V$. Let $G = GL(W)$, and $j:\Gamma\to G$ be the action of $\Gamma$ on $W$. Clearly, there is a monomorphism $k: H(\Gamma', F) \to GL(W)$ so that $k\circ i(\Gamma', F) = j|_{\Gamma'}$, so the result follows. In the general case, let $${\mathcal S} = \{\pi \subseteq \Gamma \, |\, \pi \supseteq \Gamma', \, \pi/\Gamma'\> is \> finite\}$$ If $\pi_{1},\pi_{2}\in {\mathcal S}$ and $\pi_1 \subseteq \pi_2$, then $\pi_1/\pi_2$ is finite. Therefore, from the first part of the previous lemma, $H(\pi_1, F) \to H(\pi_2, F)$ has no kernel. From lemma \ref{lemma:a6}, we get: $$\Gamma = \lim_{\stackrel{\longrightarrow}{\pi\in {\mathcal S}}}\, \pi \to \lim_{\stackrel{\longrightarrow}{\pi\in {\mathcal S}}}\, H(\pi,F)(F) \to G(F)$$ and this completes the proof. \end{proof} We can now prove theorem \ref{thm:solv}: \begin{proof} Part (A) has already proved in lemma \ref{lemma:s8}. The theorem is certainly true if $\Gamma \cong \Z$, it is also true if $\Gamma$ is a commutative torsion group by lemma \ref{lemma:s7}. For the general case, we note that $\Gamma$ has a filtration: $$\Gamma = \Gamma_0\supset \Gamma_1 \supset \ldots \supset \Gamma_{m+1} = \{1\}$$ such that the successive quotients are either abelian torsion groups or isomorphic to $\Z$. We proceed by induction on $m$, and so we can assume the theorem for $\Gamma_{1}$. Now lemmas \ref{lemma:s9}, \ref{lemma:s10} give the theorem for $\Gamma$. \end{proof} \begin{lemma} If $\Gamma$ is solvable of finite rank, then $H(\Gamma,F)\to H(\Gamma)_{F}$ an isomorphism. \end{lemma} \begin{proof} Put $G = H(\Gamma)_F$ and let $f :\Gamma \to H(\Gamma)_F(F)$ be given by $f(\gamma) = i(\Gamma)(\Gamma)$ for $\gamma\in \Gamma$. Then $(G,f)\in Obj\, C(\Gamma, F)$ and $dim\, U(G) = dim\, U(H(\Gamma)) = rk\Gamma$ by the theorem. From lemma \ref{lemma:s8}, $H(\Gamma, F)\to G_{min} = G$ is an isomorphism, and so the lemma is proved. \end{proof} \begin{thm}\label{thm:compltn} Let $\Gamma$ be a finitely generated solvable group of finite rank. There is a finite set of prime numbers $S$ so that $$H(\Gamma)_{\Q_l} \to H(\hat\Gamma, \Q_l)$$ is an isomorphism for all primes $l\notin S$. \end{thm} \begin{proof} We may regard $H(\Gamma)$ as an algebraic subgroup of $(GL_n)_\Q$. Because $\Gamma$ is finitely generated, $i(\Gamma)(\Gamma) \subset GL_n(S^{-1}\Z)$ for some finite set of primes $S$. If $l$ is a prime not in $S$, then $GL_n(S^{-1}\Z) \subset GL_n(\Z_l)$, and the latter is a profinite group. This gives a continuous homomorphism $f_l:\hat \Gamma \to GL_n(\Z_l)$. Because $image(i(\Gamma))\subset H(\Gamma)(\Q_l)$ and the second group is closed in $GL_n(\Q_l)$, we see that $f_l(\hat \Gamma) \subset H(\Gamma)(\Q_l)$. The object $(H(\Gamma)_{\Q_l}, f_l)$ of $C(\hat \Gamma, \Q_l)$ gives an epimorphism $H(\hat \Gamma, \Q_l) \to H(\Gamma)_{\Q_l}$. By the previous lemma and remark \ref{remark:1}, we have $H(\Gamma)_{\Q_l} \to H(\hat \Gamma, \Q_l)$, and these arrows are inverses of each other. \end{proof} \begin{lemma}\label{lemma:s12} If $\Gamma$ is solvable of finite rank and if $H(\Gamma)^o$ is unipotent, then there are normal subgroups $\Gamma_1\supseteq \Gamma_2$ of $\Gamma$ so that (a) $\Gamma/\Gamma_1$ is finite, (b) $\Gamma_1/\Gamma_2$ is nilpotent, and (c) $\Gamma_2$ is torsion. \end{lemma} \begin{proof} We take $\Gamma_1 = i(\Gamma)^{-1}H(\Gamma)^o(\Q)$ and $\Gamma_2 = ker(i(\Gamma))$. Then $\Gamma_1/\Gamma_2 \subset H(\Gamma)^o(\Q)$ and the latter is nilpotent. From theorem \ref{thm:solv} $\Gamma_2$ is torsion, and $\Gamma/\Gamma_1 \subset H(\Gamma)/H(\Gamma)^o(\Q)$ is finite. \end{proof} \section{Fundamental groups of varieties} Let $Y$ be a normal variety defined over a subfield $K\subset \C$ with a $K$-rational point $y_0$. For any field extension $K'\supseteq K$, set $Y_{K'} = Y\times_{spec\,K} spec\,K'$. Let $\pi_1^{alg}(X)$ denote the algebraic fundamental group of a connected scheme $X$ (with an unspecified base point), and $\hat \pi$ the profinite completion of a group $\pi$. Then we have a split exact sequence $$1\to \hat \pi_1(Y_\C^{an})\to \pi_1^{alg}(Y_{K})\to Gal(\bar K/K)\to 1$$ (where the splitting depends on $y_0$) \cite[IX 6.4, XII 5.2]{sga1}. Thus $Gal(\bar K/K)$ acts continuously on $\hat \pi_1(Y_\C)$, and therefore also on the abelianization $H_1(Y_{\C}^{an},\Z)\otimes \hat\Z$ and its pro-$l$ part $ H_1(Y_\C^{an}, \Z)\otimes \Z_l$. \begin{lemma}\label{lemma:finiteness} Let $Y$ be a normal variety defined over a finitely generated field $K\subset \C$. Then $H_0(Gal(\bar K/K), H_1(Y_\C^{an}\otimes \Z_l))$ is finite. \end{lemma} \begin{proof} $H_1(Y)/(torsion)\, \otimes \Z_l$ is dual to $H_{et}^1(Y_{\bar K},\Z_l)$ as a $Gal(\bar K/K)$-module, thus it suffices to prove that the second group has no invariants. Let $p:\tilde Y\to Y_{K'}$ be a desingularization defined over a finite extension $K'\supseteq K$. By Zariski's main theorem the geometric fibers of $p$ are connected, thus $H_{et}^1(Y_{\bar K},\Z_l)$ injects into $H_{et}^1(\tilde Y_{\bar K},\Z_l)$, and this is compatible with the $Gal(\bar K/K')$-action. $H^1(\tilde Y_{\bar K})$ has no $Gal(\bar K/K')$-invariants, because the eigenvalues of the Frobenius at any prime of good reduction have absolute value $q^{1/2}$ or $q$ by \cite[sect. 3.3]{Deligne}. \end{proof} \begin{remark} The argument can be simplified (and lengthened) in a couple of ways. An appropriate Lefschetz type argument allows one to reduce to the case where $Y$ is a curve where the relevant estimate on eigenvalues of the Frobenius goes back to Weil. Alternatively, when $Y$ is curve, one can deduce the finiteness of $H_0(Gal(\bar K/K), H_1(Y_\C^{an}\otimes \Z_l))$ directly from class field theory. \end{remark} \begin{thm} Let $X$ be a normal (not necessarily complete) algebraic variety defined over $\C$. Let $\pi = D^0\pi \supseteq D^1\pi \supseteq \ldots$ be the derived series of $\pi = \pi_1(X,x_0)$. If there is a natural number $n$ so that $\pi/D^n\pi$ is solvable of finite rank, then there are normal subgroups $P\supseteq Q \supseteq D^n\pi$ of $\pi$ so that (a) $\pi /P$ is finite, (b) $P/Q$ is nilpotent, and (c) $Q/D^n\pi$ is a torsion group. \end{thm} \begin{proof} Put $\Gamma = \pi/D^n\pi$, and $T = H(\Gamma)^0/U(H(\Gamma))$. By lemma \ref{lemma:s12}, the theorem follows once it has been proved that $T$ is trivial. We may assume that $X$ and $x_0$ are defined over a finitely generated field $K\subset \C$. There is an action of $Gal(\bar K/K) $ on $\hat \pi$, and also on $\hat \Gamma$, because this is a quotient of $\hat \pi$ by the closure of $D^n(\hat \pi)$. Choose a prime $l$, so that $\Gamma\to H(\Gamma)(\Q)$ extends to a continuous homomorphism $\hat \Gamma \to H(\Gamma)(\Q_l)$. For such a prime, $H(\hat \Gamma, \Q_l) = H(\Gamma)_{\Q_l}$ by theorem \ref{thm:compltn}. Thus the Galois action on $\hat \Gamma$ yields a homomorphism from $Gal(\bar K/K)$ to the group of $\Q_l$-rational points of $G = Aut(H(\Gamma))$. After replacing $K$ by a finite extension, if necessary, we can assume $image(\rho) \subset G^0(\Q_l)$. By lemma \ref{lemma:a5}, the action of $Gal(\bar K/K)$ on $T_{Q_l}$ is trivial. Let $\pi' = ker[\pi \to (H(\Gamma)/H(\Gamma)^0)(\Q)]$ Then $\pi' = \pi_1(Y,y_0)$ where $Y$ is an etale cover of $X$. The composite $$\hat \pi' \hookrightarrow \hat \pi \to (H(\Gamma)/U(H(\Gamma))(\Q_l)$$ factors through $$H_1(Y,\Z)\otimes \hat \Z = \hat \pi_{ab}' \to T(\Q_l),$$ $T(\Q_l)$ contains an open pro-$l$-group, thus $\hat \pi'$ further factors through: $$h:H_1(Y)\otimes\hat \Z \to H_1(Y)\otimes \Z_l \oplus A \to T(\Q_l)$$ where $A$ is a finite group. However $H_0(Gal(\bar K/K), H_1(Y)\otimes \Z_l)$ is finite by lemma \ref{lemma:finiteness}. So we deduce that the image of $H_1(Y)\otimes \Z_l$ in $T(\Q_l)$ is finite, because $Gal(\bar K/K)$ acts trivially on $T(\Q_l)$. However the image $h$ is Zariski dense. Thus $T$ is finite and connected, and therefore trivial. This proves the theorem. \end{proof} \begin{cor} If the fundamental group of a normal complex variety is solvable and possesses a faithful representation into $GL_n(\Q)$, then it is virtually nilpotent i.e. it must contain a nilpotent subgroup of finite index. \end{cor} \begin{proof} We can assume that the fundamental group is torsion free after passing to a subgroup of finite index \cite[lemma 8]{S}. The theorem implies that this must contain a nilpotent group of finite index. \end{proof} \section{Fundamental groups of K\"ahler manifolds} A group $\Gamma$ will be called quasi-K\"ahler if it there exists a connected compact K\"ahler manifold $X$, and a divisor with normal crossings $D\subset X$, such that $\Gamma\cong \pi_1(X-D)$. (Note that by resolution of singularities \cite{AHV, BM}, it is enough to assume that $D$ is an analytic subset.) The proof of the following lemma will be given in the appendix. \begin{lemma} A subgroup of a {\qk} group of finite index is {\qk}. \end{lemma} \begin{lemma} Let $A$ be a finitely generated abelian group, and $M$ a nontrivial one dimensional $\C[A]$-module. Then $H^i(A,M)=0$ for all $i$. \end{lemma} \begin{proof} This is clear for cyclic groups by direct computation. In general, express $A$ as a product of cyclic groups $\prod_{i}{C_i}$ and $M$ as a tensor product of $C_i$-modules, and apply the K\"unneth formula. \end{proof} Set $\Gamma^{ab} = \Gamma/D\Gamma$. \begin{lemma}\label{lemma:h1} Suppose that $\Gamma$ is a finitely generated group and $A=\Gamma/N$ an abelian quotient. Suppose that $M$ is a nontrivial one dimensional $A$-module then $$H^1(\Gamma,M) \cong Hom_{\Z[A]}(N^{ab}, M)\cong Hom_{\Q[A]}(N^{ab}\otimes\Q, M).$$ \end{lemma} \begin{proof} From the Hochschild-Serre spectral sequence associated to the extension $1\to N \to \Gamma \to A\to 0$, we obtain an exact sequence $$0\to H^1(A, M)\to H^1(\Gamma, M)\to H^0(A,H^1(N,M))\to H^2(A,M).$$ By the previous lemma, this gives an isomorphism $$H^1(\Gamma,M)\cong H^0(A,H^1(N,M))$$ Furthermore $H^1(N,M) \cong Hom(N,M) = Hom(N^{ab},M)$ as $A$-modules. Therefore $H^0(A,H^1(N,M)) \cong Hom_{\Z[A]}(N^{ab}, M)$. \end{proof} Given a character $\rho\in Hom(\Gamma, \C^*)$, let $\C_\rho$ denote the associated $\Gamma$-module. We define $\Sigma^1(\Gamma) $ to be the set of characters $\rho\in Hom(\Gamma,\C^*)$ such that $H^1(\Gamma,\C_\rho)$ is nonzero. Let us say that a $\Gamma$-module $V$ is quasiunipotent if there is a subgroup $\Gamma'\subseteq \Gamma$ of finite index whose elements act unipotently on $V$. \begin{lemma} Let $A$ be a finitely generated abelian group and $V$ a finite dimensional $\C[A]$-module. Then $A$ acts quasiunipotently on $V$ if and if the only characters $\rho\in Hom(A,\C^*)$ for which $Hom_{\C[A]}(V,\C_\rho)\not= 0$ are torsion characters (i.e. elements of $Hom(\Gamma, \C^*)$ of finite order). \end{lemma} \begin{proof} Define $V_\rho$ to be the generalized eigenspace associated to a character $\rho$. In other words, $V_\rho$ is the maximal subspace on which $a-\rho(a)$ is nilpotent for all $a\in A$. $V$ is a direct sum of these eigenspaces, thus we can assume that $V=V_\rho\not= 0$. To complete the proof, observe that $Hom(V_\rho,\C_{\rho'})\not= 0$ if and only if $\rho=\rho'$, and that $V_\rho$ is quasiunipotent if and if $\rho$ is torsion. \end{proof} \begin{lemma}\label{lemma:torsion} Let $\Gamma$ be a finitely generated group and $\Gamma'\subseteq \Gamma$ a subgroup of finite index such that $V= (\Gamma'\cap D\Gamma)^{ab}\otimes \Q$ is finite dimensional. Then $\Gamma'$ acts quasiunipotently on $V$ if and only if $$\Sigma^1(\Gamma')\cap image(Hom(\Gamma,\C^*)\to Hom(\Gamma',\C^*))$$ consists of torsion characters. \end{lemma} \begin{proof} Set $N = \Gamma'\cap D\Gamma$. Then $\Gamma'/N$ is isomorphic to the image of $\Gamma'$ in $\Gamma^{ab}$, and therefore $Hom(\Gamma'/N,\C^*)$ is coincides with $image(Hom(\Gamma,\C^*)\to Hom(\Gamma',\C^*)).$ Thus, lemma \ref{lemma:h1} implies that $$S=(\Sigma^1(\Gamma')-\{1\})\cap image(Hom(\Gamma,\C^*)\to Hom(\Gamma',\C^*))$$ is the set of nontrivial characters $\rho\in Hom(\Gamma'/N,\C^*)$ for which $Hom_{\C[A]}(V\otimes\C,\C_\rho)\not= 0$. Thus $\Gamma'/N$ acts quasiunipotently on $V$ if and only if $S$ consists of torsion characters by the previous lemma. The $\Gamma'$ action on $V$ factors through $\Gamma'/N$, thus the lemma is proved. \end{proof} \begin{lemma}\label{lemma:kron} Let $K\supseteq \Q$ be a finite extension, and $O_K$ the ring of integers of $K$. Let $\{\sigma_1,\ldots\sigma_n\}$ be the set of all embeddings $K$ into $\C$. Then for any finitely generated group $\Gamma$, $$Hom(\Gamma, U(1))\cap \bigcap_{i}\sigma_i\circ Hom(\Gamma, O_K^*) $$ consists of torsion characters. \end{lemma} \begin{proof} This follows from Kronecker's theorem that an algebraic integer is a root of unity if and only if all its Galois conjugates have absolute value one. \end{proof} \begin{thm}\label{thm:qunip} Let $\Gamma$ be a {\qk} group such that $D\Gamma$ is a finitely generated. Then for any subgroup $\Gamma'\subseteq \Gamma$ of finite index, $\Gamma'$ acts quasiunipotently on the finite dimensional vector space $(\Gamma'\cap D\Gamma)^{ab}\otimes \Q$. \end{thm} \begin{proof} By lemma \ref{lemma:torsion}, it is enough to show that the intersection $S$ of $\Sigma^1(\Gamma')$ and the image of $Hom(\Gamma,\C^*)$ consists of torsion characters. The subgroup $\Gamma'\cap D\Gamma\subseteq D\Gamma$ has finite index, and is therefore finitely generated. Thus the set of characters of $\Gamma'$ which correspond to one dimensional quotients of $(\Gamma'\cap D\Gamma)^{ab}\otimes \C$ are defined over the ring of integers of a finite extension $K$ of $\Q$. It then follows from lemma \ref{lemma:h1} that $S$ is a finite subset of $Hom(\Gamma, O_K^*)$. $S$ is evidently stable under $Aut(\C)$ and thus lies in $\cap_{\sigma:K\hookrightarrow\C}\, \sigma\circ Hom(\Gamma,O_K^*)$. Theorem V.1.6 of \cite{A2} implies that $\Sigma^1(\Gamma')$ is a finite union of translates of subtori of $Hom(\Gamma')$ by unitary characters. $S$, which is the intersection of this set with $image(Hom(\Gamma,\C^*))$, must clearly inherit a similar structure. In particular, being finite, it follows that $S$ consists of unitary characters. Therefore the theorem follows from lemma \ref{lemma:kron}. \end{proof} \begin{lemma}\label{lemma:key} Let $\Gamma$ be a solvable group of finite rank. Suppose that every subgroup $\Gamma'\subseteq \Gamma$ of finite index acts quasiunipotently on the finite dimensional vector space $(\Gamma'\cap D\Gamma)^{ab}\otimes \Q$. Then there exists normal subgroups $\Gamma_1\supset \Gamma_2$ of $\Gamma$ so that (a) $\Gamma_1$ has finite index, (b) $\Gamma_1/\Gamma_2$ is nilpotent, and (c) $\Gamma_2$ is torsion. \end{lemma} \begin{proof} It suffices to prove that $G= H(\Gamma)^o$ is unipotent by lemma \ref{lemma:s12} (then $\Gamma_1$ can be be taken to be $i(\Gamma)^{-1}(G)$ and $\Gamma_2= ker\, i(\Gamma)$). The unipotency of $G$ will follow from lemma \ref{lemma:unip}, once we show that the action of $G/DG$ on $DG/D^2G$, by conjugation, is unipotent. The map $$(\Gamma_1\cap D\Gamma)^{ab}\otimes\Q \to DG(\Q)/D^2G(\Q)$$ is compatible with the $\Gamma_1$-actions, and is surjective, because the image of $\Gamma_1\cap D\Gamma$ is Zariski dense in $DG(\Q)$. By hypothesis, $\Gamma_1$ contains a finite index subgroup $\Gamma''$ which acts unipotently on $(\Gamma_1\cap D\Gamma)^{ab}\otimes\Q$. As $\Gamma''$ is Zariski dense in $G$, the lemma follows. \end{proof} \begin{thm} Let $\pi $ be a {\qk} group. Suppose that $D\pi$ is finitely generated and $\pi/D^n\pi$ is solvable of finite rank for some natural number $n$. Then there are normal subgroups $P\supseteq Q \supseteq D^n\pi$ of $\pi$ so that (a) $\pi /P$ is finite, (b) $P/Q$ is nilpotent, and (c) $Q/D^n\pi$ is a torsion group. \end{thm} \begin{proof} The theorem follows from theorem \ref{thm:qunip} and lemma \ref{lemma:key}. \end{proof} \begin{cor} A polycyclic {\qk} group is virtually nilpotent. \end{cor} \begin{proof} By \cite[4.6]{R}, a polycyclic group contains a torsion free subgroup $\pi$ of finite index. The theorem implies that $\pi$ must contain a nilpotent subgroup of finite index. \end{proof}

9706

alg-geom/9706010

en

https://arxiv.org/abs/alg-geom/9706010

alg-geom/9706010

Mikhail Olshanetsky

A.Levin and M.Olshanetsky

Painlev\'{e} - Calogero correpondence

17 pages, Latex

ITEP-TH29/97

It is proved that the Painlev\'{e} VI equation $(PVI_{\al,\be,\ga,\de})$ for the special values of constants $(\al=\frac{\nu^2}{4},\be=-\frac{\nu^2}{4}, \ga=\frac{\nu^2}{4},\de=\f1{2}-\frac{\nu^2}{4})$ is a reduced hamiltonian system. Its phase space is the set of flat $SL(2,C)$ connections over elliptic curves with a marked point and time of the system is given by the elliptic module. This equation can be derived via reduction procedure from the free infinite hamiltonian system. The phase space of later is the affine space of smooth connections and the "times are the Beltrami differentials. This approach allows to define the associate linear problem, whose isomonodromic deformations is provided by the Painlev\'{e} equation and the Lax pair. In addition, it leads to description of solutions by a linear procedure. This scheme can be generalized to $G$ bundles over Riemann curves with marked points, where $G$ is a simple complex Lie group. In some special limit such hamiltonian systems convert into the Hitchin systems. In particular, for $\SL$ bundles over elliptic curves with a marked point we obtain in this limit the elliptic Calogero N-body system. Relations to the classical limit of the Knizhnik- Zamolodchikov-Bernard equations is discussed.

alg-geom

\section{} command!!! \newcommand{\sect}[1]{\setcounter{equation}{0}\section{#1}} \renewcommand{\theequation}{\thesection.\arabic{equation}} \newtheorem{predl}{Proposition}[section] \newtheorem{defi}{Definition}[section] \newtheorem{rem}{Remark}[section] \newtheorem{cor}{Corollary}[section] \newtheorem{lem}{Lemma}[section] \newtheorem{theor}{Theorem}[section] \vspace{0.3in} \begin{flushright} ITEP-TH29/97\\ \end{flushright} \vspace{10mm} \begin{center} {\Large\bf Painlev\'{e} - Calogero correpondence.} \footnote{A talk given by M.O. on a workshop {\em "Calogero-Moser-Sutherland Models"}, March 97, CRM, Montreal} \\ \vspace{5mm} A.M.Levin\\ {\sf Institut of Oceanology, Moscow, Russia,} \\ {\em e-mail [email protected]}\\ M.A.Olshanetsky \\ {\sf Institut of Theoretical and Experimental Physics, Moscow, Russia,} \\ {\em e-mail [email protected]}\\ \vspace{5mm} \end{center} \begin{abstract} It is proved that the Painlev\'{e} VI equation $(PVI_{\alpha,\beta,\gamma,\delta})$ for the special values of constants $(\alpha=\frac{\nu^2}{4},\beta=-\frac{\nu^2}{4}, \gamma=\frac{\nu^2}{4},\delta=\f1{2}-\frac{\nu^2}{4})$ is a reduced hamiltonian system. Its phase space is the set of flat $SL(2,{\bf C})$ connections over elliptic curves with a marked point and time of the system is given by the elliptic module. This equation can be derived via reduction procedure from the free infinite hamiltonian system. The phase space of later is the affine space of smooth connections and the "times are the Beltrami differentials. This approach allows to define the associate linear problem, whose isomonodromic deformations is provided by the Painlev\'{e} equation and the Lax pair. In addition, it leads to description of solutions by a linear procedure. This scheme can be generalized to $G$ bundles over Riemann curves with marked points, where $G$ is a simple complex Lie group. In some special limit such hamiltonian systems convert into the Hitchin systems. In particular, for ${\rm SL}(N,{\bf C})$ bundles over elliptic curves with a marked point we obtain in this limit the elliptic Calogero $N$-body system. Relations to the classical limit of the Knizhnik- Zamolodchikov-Bernard equations is discussed. \end{abstract} \section {Introduction} \setcounter{equation}{0} {\bf 1}.We learned from Yu. Manin's lectures in MPI (Bonn, 1996) about elliptic form of the famous Painlev\'{e} VI equation (PVI) \cite{Ma}. In this representation PVI looks very similar to the elliptic Calogero-Inozemtsev-Treibich-Verdier (CITV) rank one system \cite{Ca,In,TV}. Namely, the both equations are hamiltonian with the same symplectic structure for two dynamical variables and the same Hamiltonians. The only difference is that the time in the PVI system is nothing else as the elliptic module. Therefore, it is non autonomous hamiltonian system, while the CITV Hamiltonian is independent of time. This similarity is not accidental and based on very closed geometric origin of the both systems, which we will elucidate in this talk. It should be confessed from the very beginning that at the present time our approach is cover only the one parametric family of $PVI_{\alpha,\beta,\gamma,\delta}$. This family corresponds to the standard two-body elliptic Calogero equation. \bigskip {\bf 2. Painlev\'{e} VI and Calogero equations.} The Painlev\'{e} VI $PVI_{\alpha,\beta,\gamma,\delta}$ equation depends of four free parameters and has the form $$ \frac{d^2X}{dt^2}=\frac{1}{2}(\frac{1}{X}+\frac{1}{X-1}+\frac{1}{X-t}) (\frac{dX}{dt})^2-(\frac{1}{t}+\frac{1}{t-1}+ \frac{1}{X-t})\frac{dX}{dt}+ $$ \beq{I.1} +\frac{X(X-1(X-t)}{t^2(t-1)^2}(\alpha+\beta\frac{t}{X^2}+ \gamma\frac{t-1}{(X-1)^2} +\delta\frac{t(t-1)}{X-t)^2}) \end{equation} It is a hamiltonian systems \cite{O1}, but we will write the symplectic form and the Hamiltonian below in another variables. Among some distinguish features of this equation we are interesting in its relation to the isomonodromic deformations of linear differential equations. This approach was investigated by Fuchs \cite{F}, while first $PVI_{\alpha,\beta,\gamma,\delta}$ was written down by Gambier \cite{G}. The equation has a lot of different applications (see \cite{PT}). We shortly present $PVI_{\alpha,\beta,\gamma,\delta}$ in terms of elliptic functions \cite{Ma}. Let $\wp(u|\tau)$ be the Weiershtrass function on the elliptic curve $T^2_{\tau}={\bf C}/({\bf Z}+{\bf Z}\tau)$, and\\ $e_i=\wp(\frac{T_i}{2}|\tau),~(i=1,2,3)~~ (T_0,\ldots,T_3)=(0,1,\tau,1+\tau).$ Consider instead of $(t,X)$ in (\ref{I.1}) the new variables \beq{I.2} (\tau,u)\rightarrow (t=\frac{e_3-e_1}{e_2-e_1},X= \frac{\wp(u|\tau)-e_1}{e_2-e_1}). \end{equation} Then $PVI_{\alpha,\beta,\gamma,\delta}$ takes the form. \beq{I.3} \frac{d^2u}{d\tau^2}=\partial_uU(u|\tau),~~ U(u|\tau)=\frac{1}{(2\pi i)^2}\sum_{j=0}^3\alpha_j \wp(u+\frac{T_j}{2}|\tau), \end{equation} $(\alpha_0,\ldots,\alpha_3)=(\alpha,-\beta,\gamma,\f1{2}-\delta)$. As usual in non autonomous case, the equations of motion (\ref{I.3}) are derived from the variations of the degenerated symplectic form \beq{I.4} \omega=\delta v\delta u-\delta H\delta\tau,~~H=\frac{v^2}{2}+U(u|\tau), \end{equation} which is defined over the extended phase space ${\cal P}=\{v,u,\tau\}$. The semidirect product of ${\bf Z}+{\bf Z}\tau$ and the modular group act on the dynamical variables $(v,u,\tau)$ preserving (\ref{I.4}). Let us introduce the new parameter $\kappa$ (the level) and instead of (\ref{I.4}) consider \beq{I.6} \omega=\delta v\delta u-\f1{\kappa}\delta H\delta\tau. \end{equation} It corresponds to the overall rescaling of constants $\alpha_j\rightarrow\frac{\alpha_j}{\kappa^2}$. Put $\tau=\tau_0+\kappa t^H$ and consider the system in the limit $\kappa\rightarrow 0$, which is called the critical level. We come to the equation \beq{I.7} \frac{d^2u}{(dt^H)^2}=\partial_uU(u|\tau_0), \end{equation} It is just the rank one $CITV_{\alpha,\beta,\gamma,\delta}$ equation. Thus, we have in this limit\\ $PVI_{\alpha,\beta,\gamma,\delta}\stackrel{\kappa\rightarrow 0}\longrightarrow CI_{\alpha,\beta,\gamma,\delta}.$ Consider one-parametric family $PVI_{\frac{\nu^2}{4},-\frac{\nu^2}{4}, \frac{\nu^2}{4},\f1{2}-\frac{\nu^2}{4}}$ The potential (\ref{I.3}) takes the form \beq{I.8} U(u|\tau)=\frac{1}{(4\pi i)^2}\nu^2\wp(2u|\tau), \end{equation} We will prove that (\ref{I.6}) with the potential (\ref{I.8}) describe the dynamic of flat connections of ${\rm SL}(2,{\bf C})$ bundles over elliptic curves $T_{\tau}$ with one marked point $\Sigma_{1,1}$. In fact, $u$ lies on the Jacobian of $T_{\tau}$, $(v,u)$ defines a flat bundle, and $\tau$ defines a point in the moduli space ${\cal M}_{1,1}=\{\Sigma_{1,1}\}$. The choice of the polarization of connections, in other words $v$ and $u$, depends on complex structure of $\Sigma_{1,1}$. The extended phase space ${\cal P}$ includes beside the dynamical variables $v$ and $u$ the "time" $\tau$. It is the bundle over ${\cal M}_{1,1}$ with the fibers ${\cal R}=\{v,u\}$, which is endowed by the degenerated symplectic structure $\omega$ (\ref{I.6}). This system is derived by a reduction procedure from some free, but infinite hamiltonian system. In this way we obtain the Lax equations, the linear system which monodromies preserve by (\ref{I.7}) with $U(u|\tau)$ (\ref{I.8}) and the explicit solutions of the Cauchy problem via the so-called projection method. The discrete symmetries of (\ref{I.4}) are nothing else as the remnant gauge symmetries. On the critical level it is just two-body elliptic Calogero system. The corresponding quantum system is identified with the KZB equation \cite{KZ,B} for the one-vertex correlator on $T_\tau$. In the similar way $PVI_{\frac{\nu^2}{4},-\frac{\nu^2}{4}, \frac{\nu^2}{4},\f1{2}-\frac{\nu^2}{4}}$ is the classical limit of the KZB for $\kappa\neq 0$. \bigskip {\bf 3.}This particular example has far-reaching generalizations. Consider a phase space, which is the moduli space of flat connections ${\cal A}$ of $G$ bundle over Riemann curve $\Sigma_{g,n}$ of genus $g$ with n marked points, where $G$ is a complex simple Lie group. While the flatness is the topological property of bundles, the polarization of connections ${\cal A}=(A,\bar{A})$ depends on the choice of complex structure on $\Sigma_{g,n}$. Therefore, we consider a bundle ${\cal P}$ over the moduli space ${\cal M}_{g,n}$ of curves with flat connections ${\cal R}=(A,\bar{A})$ as fibers. The fibers are supplemented by elements of coadjoint orbits ${\cal O}_a$ in the marked points $x_a$. There exists a closed degenerate two-form $\omega$ on ${\cal P}$, which is non degenerate on the fibers. The equations of motions are defined as variations of the dynamical variables along the null-leaves of this symplectic form. We call them as the {\sl hierarchies of isomonodromic deformations} (HID). They are attended by the {\sl Whitham hierarchies}, which has occurred earlier in \cite{Kr1} as a result of the averaging procedure, and then in \cite{DVV} as the classical limit of "string equations". Our approach is closed to the Hitchin construction of integrable systems, living on the cotangent bundles to the moduli space of holomorphic $G$ bundles \cite{H1}, generalized for singular curves in \cite{Ne,ER}. Namely, the connection $\bar{A}$ plays the same role as in the Hitchin scheme, while $A$ replaces the Higgs field. Essentially, our construction is local - we work over a vicinity of some fixed curve $\Sigma_{g,n}$ in ${\cal M}_{g,n}$. The coordinates of tangent vector to ${\cal M}_{g,n}$ in this point play role of times, while the Hitchin times have nothing to do with the moduli space. The Hamiltonians are the same quadratic Hitchin Hamiltonians, but now they are time dependent. There is a free parameter $\kappa$ ({\sl the level}) in our construction. On the critical level $(\kappa=0)$, after rescaling the times, our systems convert into the Hitchin systems. In concrete examples our work is based essentially on \cite{Ne}, which deals with the same systems on the critical level. As the Hitchin systems, HID can be derived by the symplectic reduction from a free infinite hamiltonian system. In our case the upstairs extended phase space is the space of the affine connections and the Beltrami differentials. We consider its symplectic quotient with respect of gauge action on the connections. In addition, to come to the moduli space ${\cal M}_{g,n}$ we need the subsequent factorization under the action of the diffeomorphisms, which effectively acts on the Beltrami differentials only. Apart from the last step, this derivation resembles the construction of the KZB systems in \cite{ADPW}, where they are derived as a quantization of the very similar symplectic quotient. Our approach allows to write down the Lax pairs, prove that the HID are consistency conditions of the isomonodromic deformations of the linear Lax equations, and, therefore, justify the notion HID. Moreover, we describe solutions via linear procedures (the projection method). HID are the quasi classical limit of the KZB equations for $(\kappa\neq 0$, as the Hitchin systems are the quasi classical limit of the KZB equations on the critical level \cite{Ne,I}. The quantum counterpart of the Whitham hierarchy is the flatness condition, which discussed in \cite{H2} within derivation of KZB. The interrelations between quantizations of isomonodromic deformations and the KZB eqs were discussed in \cite{R,Ha,Ko}. For genus zero our procedure leads to Schlesinger equations. We restrict ourselves to simplest cases with only simple poles of connections. Therefore, we don't include in the phase space the Stokes parameters. This phenomen was investigated in the rational case in detail in \cite{JMU}. For genus one we obtain a particular case of the Painlev\'{e} VI equation (for $SL(2,{\bf C}$ bundles with one marked point), generalization of this case on arbitrary simple groups and arbitrary number of marked points. \bigskip {\bf Acknowledgments.}\\ {\sl We are thankful to Yu.Manin - his lectures and discussions with him concerning PVI, stimulated our interests to these problems. We are grateful to the Max-Planck-Institut f\"{u}r Mathemamatik in Bonn for the hospitality, where this work was started. We would like to thank our collegues V.Fock, A.Losev, A.Morozov, N.Nekrasov, and A.Rosly for fruitfull discussions duering working on this subject. The work is supported in part by by Award No. RM1-265 of the US Civilian Research \& Development Foundation (CRDF) for the Independent States of the Former Soviet Union, and grant 96-15-96455 for support of scientific schools (A.L); grants RFBR-96-02-18046, Award No. RM2-150 of the US Civilian Research \& Development Foundation (CRDF) for the Independent States of the Former Soviet Union, INTAS 930166 extension, and grant 96-15-96455 for support of scientific schools (M.O).} \section{Symplectic reduction} \setcounter{equation}{0} {\bf 1. Upstairs extended phase space.} Let $\Sigma_{g,n}$ be a Riemann curve of genus $g$ with $n$ marked points. Let us fix the complex structure of $\Sigma_{g,n}$ defining local coordinates $(z,\bar{z})$ in open maps covering $\Sigma_{g,n}$. Assume that the marked points $(x_1,\ldots,x_n)$ are in the generic positions. The deformations of the basic complex structure are determined by the Beltrami differentials $\mu$, which are smooth $(-1,1)$ differentials on $\Sigma_{g,n}$, $\mu\in{\cal A}^{(-1,1)}(\Sigma_{g,n})$ . We identify this set with the space of times ${\cal N}'$. The Beltrami differentials can be defined in the following way. Consider the chiral diffeomorphisms of $\Sigma_{g,n}$ \beq{2a.2} w=z-\epsilon(z,\bar{z}),~~\bar{w}=\bar{z} \end{equation} and the corresponding one-form $dw$. Up to the conformal factor $1-\partial \epsilon(z,\bar{z})$, it is equal \beq{1a.2} dw=dz-\mu d\bar{z},~~\mu=\frac{\bar{\partial} \epsilon(z,\bar{z})}{1-\partial \epsilon(z,\bar{z})}. \end{equation} The new holomorphic structure is defined by the deformed antiholomorphic operator annihilating $dw$, while the antiholomorphic structure is kept unchanged $$\partial_{\bar{w}}=\bar{\partial}+\mu\partial,~~\partial_w=\partial.$$ In addition, assume that $\mu$ vanishes in the marked points $\mu(z,\bar{z})|_{x_a}=0.$ We consider small deformations of the basic complex structure $(z,\bar{z})$. It allows to replace (\ref{1a.2}) by \beq{2b.2} \mu=\bar{\partial} \epsilon(z,\bar{z}). \end{equation} Let ${\cal E}$ be a principle stable $G$ bundle over a Riemann curve $\Sigma_{g,n}$. Assume that $G$ is a complex simple Lie group. The phase space ${\cal R}'$ is recruited from the following data:\\ i)the affine space $\{{\cal A}\}$ of Lie$(G)$-valued connection on ${\cal E}$.\\ It has the following component description:\\ a) $C^{\infty}$ connection $\{\bar{A}\}$ , corresponding to the $d\bar{w}=d\bar{z}$ component of ${\cal A}$;\\ b)The dual to the previous space the space $\{A\}$ of $dw$ components of ${\cal A}$. $A$ can have simple poles in the marked points. Moreover, assume that $\bar{A}\mu$ is a $C^{\infty}$ object ;\\ ii)cotangent bundles $T^*G_a=\{(p_a,g_a),~p_a\in {\rm Lie}^*(G_a),~g_a\in G_a\}, ~(a=1,\ldots,n)$ in the points $(x_1,\ldots,x_n)$.\\ There is the canonical symplectic form on ${\cal R}'$ \beq{1.2} \omega_0=\int_{\Sigma}<\delta A,\delta\bar{A}>+ 2\pi i\sum_{a=1}^n\delta<p_a,g_a^{-1}\delta g_a>, \end{equation} where $<~,~>$ denotes the Killing form on Lie$(G)$. \bigskip Consider the bundle ${\cal P}'$ over ${\cal N}'$ with ${\cal R}'$ as the fibers. It plays role of the extended phase space. There exists the degenerate form on ${\cal P}'$ \beq{2.2} \omega=\omega_0- \frac{1}{\kappa} \int_{\Sigma}<\delta A, A>\delta \mu. \end{equation} Thus, we deal with the infinite set of Hamiltonians $< A, A>(z,\bar{z})$, parametrized by points of $\Sigma_{g,n}$ and corresponding set of times $\mu(z,\bar{z})$. The equations of motion. take the form \beq{3.2} \frac{\partial A}{\partial\mu}(z,\bar{z}) =0,~~ \kappa \frac{\partial \bar{A}}{\partial\mu}(z,\bar{z})=A(z,\bar{z}),~~ \frac{\partial p_b}{\partial\mu}=0,~~ \frac{\partial g_b}{\partial \mu}=0. \end{equation} We will apply the formalism of hamiltonian reduction to these systems. \bigskip {\bf 2. Symmetries}. The form $\omega$ (\ref{2.2}) is invariant with respect to the action of the group ${\cal G}_0$ of diffeomorphisms of $\Sigma_{g,n}$, which are trivial in vicinities ${\cal U}_a$ of marked points: \beq{7.2} {\cal G}_0=\{z\rightarrow N(z,\bar{z}),\bar{z}\rightarrow \bar{N}(z,\bar{z}),),~ N(z,\bar{z})=z+o(|z-x_a|),~z\in {\cal U}_a\}. \end{equation} Another infinite gauge symmetry of the form (\ref{2.2}) is the group ${\cal G}_1=\{f(z,\bar{z})\in C^\infty (\Sigma_{g},G)\}$ that acts on the dynamical fields as $$ A\rightarrow f(A+\kappa\partial)f^{-1},~~~\bar{A}\rightarrow f(\bar{A}+\bar{\partial}+\mu\partial)f^{-1}, $$ \beq{8.2} (\bar{A}'\rightarrow f(\bar{A}'+\bar{\partial})f^{-1}), \end{equation} $$ p_a\rightarrow f_ap_af^{-1}_a, ~~g_a\rightarrow g_af^{-1}_a,~~(f_a=\lim_{z\rightarrow x_a}f(z,\bar{z})), ~~ \mu\rightarrow\mu. $$ In other words, the gauge action of ${\cal G}_1$ does not touch the base ${\cal N}'$ and transforms only the fibers ${\cal R}'$. The whole gauge group is the semidirect product ${\cal G}_1\oslash{\cal G}_0.$ \bigskip {\bf 3. Symplectic reduction with respect to ${\cal G}_1$.} Since the symplectic form (\ref{2.2}) is closed (though is degenerated) one can consider the symplectic quotient of the extended phase space ${\cal P}'$ under the action of the gauge transformations (\ref{8.2}). They are generated by the moment constraints \beq{10.2} F_{A,\bar{A}}(z,\bar{z})-2\pi i\sum_{a=1}^n\delta^2(x_a)p_a=0, \end{equation} where $F_{A,\bar{A}}=(\bar{\partial} +\partial\mu)A-\kappa\partial\bar{A}+[\bar{A}.A].$ It means that we deal with the flat connection everywhere on $\Sigma_{g,n}$ except the marked points. The holonomies of $(A,\bar{A})$ around the marked points are conjugate to $\exp 2\pi ip_a$. Let $(L,{\bar L})$ be the gauge transformed connections \beq{11.2} \bar{A}=f(\bar{L}+\bar{\partial}+\mu\partial)f^{-1},~~A=f(L+\kappa\partial)f^{-1}, \end{equation} Then (\ref{10.2}) takes the form \beq{12a.2} (\bar{\partial}+\partial\mu)L-\kappa\partial {\bar L}+[\bar{L},L]=2\pi i\sum_{a=1}^n\delta^2(x_a)p_a. \end{equation} \begin{rem} The gauge fixing allows to choose $\bar{A}$ in a such way that $\partial\bar L=0$. Then (\ref{12a.2}) takes the form \beq{14.2} (\bar{\partial}+\partial\mu)L+[\bar{L},L]=\sum_{a=1}^n\delta^2(x_a)p_a. \end{equation} It coincides with the moment equation for the Hitchin systems on singular curves \cite{Ne}. \end{rem} We can rewrite (\ref{14.2}) as \beq{13.2} \partial_{\bar w}L+[\bar{L},L]= 2\pi i\sum_{a=1}^n\delta^2(x_a)p_a. \end{equation} Anyway, by choosing $\bar{L}$ we fix somehow the gauge in generic case. There is additional gauge freedom $h_a$ in the points $x_a$, which acts on $T^*G_a$ as \mbox{$p_a\rightarrow p_a$, $g_a\rightarrow h_ag_a$}. It allows to fix $p_a$ on some coadjoint orbit $p_a=g_a^{-1}p_a^{(0)}g_a$ and obtain the symplectic quotient ${\cal O}_a=T^*G_a//G_a$. Thus in (\ref{12a.2}) or (\ref{14.2}) $p_a$ are elements of ${\cal O}_a$. Let ${\cal I}_{g,n}$ be the equivalence classes of the connections $(A,\bar{A})$ with respect to the gauge action (\ref{11.2}) - the moduli space of stable flat $G$ bundles over $\Sigma_{g,n}$ . It is a smooth finite dimensional space. Fixing the conjugacy classes of holonomies $(L,\bar L)$ around marked points (\ref{12a.2}) amounts to choose a symplectic leave ${\cal R}$ in ${\cal I}_{g,n}$. Thereby we come to the symplectic quotient $$ {\cal R}={\cal R}'//{\cal G}_1= {\cal J}^{-1}_1(0)/{\cal G}_1\subset{\cal I}_{g,n}. $$ The connections $(L,\bar{L})$ in addition to ${\bf p}=(p_1,\ldots,p_n)$ depend on a finite even number of free parameters $2r$ $ ({\bf v},{\bf u}),~{\bf v}=(v_1,\ldots,v_{r}),~{\bf u}=(u_1,\ldots,u_{r}).$ $$ r= \left\{ \begin{array}{ll} 0&~g=0,\\ {\rm rank} G,&~g=1\\ (g-1)\dim G,&~g\geq 2.\\ \end{array} \right. $$ The fibers ${\cal R}$ are symplectic manifolds with the nondegenerate symplectic form which is the reduction of (\ref{2.2}) \beq{15.2} \omega_0=\int_{\Sigma}<\delta L,\delta \bar{L}>+ 2\pi i\sum_{a=1}^n\delta<p_a,g_a\delta g_a^{-1}>. \end{equation} On this stage we come to the bundle ${\cal P}''$ with the finite-dimensional fibers\\ ${\cal R}$ over the infinite-dimensional base ${\cal N}'$ with the symplectic form \beq{16.2} \omega=\omega_0- \f1{\kappa}\int_{\Sigma}<L,\delta L>\delta\mu. \end{equation} \bigskip {\bf 4. Factorization with respect to the diffeomorphisms} ${\cal G}_0$. We can utilize invariance of $\omega$ with respect to ${\cal G}_0$ and reduce ${\cal N}'$ to the finite-dimensional space ${\cal N}$, which is isomorphic to the moduli space ${\cal M}_{g,n}$. The crucial point is that for the flat connections the action of diffeomorphisms ${\cal G}_0$ on the connection fields is generated by the gauge transforms ${\cal G}_1$. But we already have performed the symplectic reduction with respect to ${\cal G}_1$. Therefore, we can push $\omega$ (\ref{16.2}) down on the factor space ${\cal P}''/{\cal G}_0$. Since ${\cal G}_0$ acts on ${\cal N}'$ only, it can be done by fixing the dependence of $\mu$ on the coordinates in the Teichm\"{u}ller space ${\cal T}_{g,n}$. According to (\ref{2b.2}) represent $\mu$ as \beq{17a.2} \mu=\sum_{s=1}^{l}\mu_s. \end{equation} The Beltrami differential (\ref{17a.2}) defines the tangent vector $ {\bf t}=(t_1,\ldots,t_l), $ to the Teichm\"{u}ller space ${\cal T}_{g,n}$ at the fixed point of ${\cal T}_{g,n}$. We specify the dependence of $\mu$ on the positions of the marked points in the following way. Let ${\cal U}'_a\supset{\cal U}_a$ be two vicinities of the marked point $x_a$ such that ${\cal U}'_a\cap{\cal U}'_b=\emptyset$ for $a\neq b$, and $\chi_a(z,\bar{z})$ is a smooth function $$ \chi_a(z,\bar{z})=\left\{ \begin{array}{cl} 1,&\mbox{$z\in{\cal U}_a$ }\\ 0,&\mbox{$z\in\Sigma_{g,n}\setminus {\cal U}'_a.$} \end{array} \right. $$ Introduce times related to the positions of the marked points $t_a=x_a-x_a^0$. Then \beq{17.2} \mu_a=t_a\bar{\partial} n_a(z,\bar{z}),~~n_a(z,\bar{z})=(1+c_a(z-x_a^0))\chi_a(z,\bar{z}). \end{equation} The action of ${\cal G}_0$ on the phase space ${\cal P}''$ reduces the infinite-dimensional component ${\cal N}'$ to ${\cal T}_{g,n}$. After the reduction we come to the bundle with base ${\cal T}_{g,n}$. The symplectic form (\ref{16.2}) is transformed as follows \beq{19.2} \omega=\omega_0({\bf v},{\bf u},{\bf p},{\bf t})- \frac{1}{\kappa}\sum_{s=1}^{l}\delta H_s({\bf v}, {\bf u},{\bf p},{\bf t})\delta t_s,~ H_s=\int_\Sigma <L,L>\partial_s\mu \end{equation} where $\omega_0$ is defined by (\ref{15.2}). In fact, we still have a redundant discrete symmetry, since $\omega$ is invariant under the mapping class group $\pi_0({\cal G}_0)$. Eventually, we come to the moduli space ${\cal M}_{g,n}={\cal T}_{g,n}/\pi_0({\cal G}_0)$. The extended phase space ${\cal P}$ is the result of the symplectic reduction with respect to the ${\cal G}_1$ action and subsequent factorization under the ${\cal G}_0$ action. We can write symbolically ${\cal P}=({\cal P}''//{\cal G}_1)/{\cal G}_0.$ It is endowed with the symplectic form (\ref{19.2}). \bigskip {\bf 5. The hierarchies of the isomonodromic deformations (HID)}. The equations of motion (HID) can be extracted from the symplectic form (\ref{19.2}). In terms of the local coordinates they take the form \beq{21.2} \kappa\partial_s{\bf v}=\{H_s,{\bf v}\}_{\omega_0},~~ \kappa\partial_s{\bf u}=\{H_s,{\bf u}\}_{\omega_0},~~ \kappa\partial_s{\bf p}=\{H_s,{\bf p}\}_{\omega_0}~~ \end{equation} The Poisson bracket $\{\cdot,\cdot\}_{\omega_0}$ is the inverse tensor to $\omega_0$. We also has the Whitham hierarchy accompanying (\ref{21.2}) \beq{22.2} \partial_sH_r-\partial_rH_s+\{H_r,H_s\}_{\omega_0}=0. \end{equation} There exists the one form on ${\cal M}_{g,n}$ defining {\sl the tau function} of the hierarchy of isomonodromic deformations \beq{22a.2} \delta \log\tau=\delta^{-1}\omega_0-\f1{\kappa}\sum H_sdt_s. \end{equation} The following three statements are valid for the HID (\ref{21.2}): \begin{predl} There exists the consistent system of linear equations \beq{23.2} (\kappa\partial+L)\Psi=0, \end{equation} \beq{24.2} (\partial_s+M_s)\Psi=0,~~(s=1,\ldots,l=\dim{\cal M}_{g,n}) \end{equation} \beq{25a.2} (\bar{\partial}+\mu\partial+{\bar L})\Psi=0 \end{equation} where $M_s$ is a solution to the linear equation \beq{25.2} \partial_{\bar w}M_s-[M_s,{\bar L}]=\partial_s\bar{L}-\frac{1}{\kappa}L\partial_s\mu. \end{equation} \end{predl} \begin{predl} The linear conditions (\ref{24.2}) provide the isomonodromic deformations of the linear system (\ref{23.2}), (\ref{25a.2}) with respect to change the "times" on ${\cal M}_{g,n}$. \end{predl} Therefore, the HID (\ref{21.2}) are the monodromy preserving conditions for the linear system (\ref{23.2}),(\ref{25a.2}). The presence of derivative with respect to the spectral parameter $w\in\Sigma_{g,n}$ in the linear equation (\ref{23.2}) is a distinguish feature of the monodromy preserving equations. It plagues the application of the inverse scattering method to these types of systems. Nevertheless, in our case we have in some sense the explicit form of solutions: \begin{predl}[The projection method.] The solution of the Cauchy problem of (\ref{21.2}) for the initial data ${\bf v}^0,{\bf u}^0,{\bf p}^0$ at the time ${\bf t}={\bf t}^0$ is defined in terms of the elements $L^0,{\bar L}^0$ as the gauge transform \beq{26a.2} {\bar L}({\bf t})=f^{-1}( L^0(\mu({\bf t})-\mu({\bf t}^0))+ ({\bar L}^0))f+ f^{-1}(\bar{\partial}+\mu({\bf t})\partial)f, \end{equation} \beq{26.2} L({\bf t})=f^{-1}(\partial+L^0)f,~~{\bf p}({\bf t})=f^{-1}({\bf p}^0)f, \end{equation} where $f$ is a smooth $G$-valued functions on $\Sigma_{g,n}$ fixing the gauge. \end{predl} \section{Relations to the Hitchin systems and the KZB equations.} \setcounter{equation}{0} {\bf 1. Scaling limit.} Consider the HID in the limit $\kappa\rightarrow 0$. We will prove that in this limit we come to the Hitchin systems, which are living on the cotangent bundles to the moduli space of holomorphic $G$-bundles over $\Sigma_{g,n}$ \cite{H1}. The critical value $\kappa=0$ looks singular (see (\ref{2.2}), (\ref{19.2})). To get around we rescale the times ${\bf t}=\kappa{\bf t}^H$, where ${\bf t}^H$ are the "Hitchin times". Therefore, $ \delta\mu({\bf t})=\kappa\sum_s\partial_s\mu({\bf t}^0)\delta t^H. $ After this rescaling the forms (\ref{2.2}),(\ref{19.2}) become regular in the critical limit. The rescaling procedure means that we blow up a vicinity of the fixed point corresponding to $(\mu=0)$, and the whole dynamic of the Hitchin systems is developed in this vicinity. \footnote{We are grateful to A.Losev for elucidating this point.} Denote $\partial_s\mu_o=\partial_s\mu({\bf t})|_{{\bf t}={\bf t}^0}$ Then we have instead of (\ref{2.2}) \beq{2.3} \omega=\int_{\Sigma}<\delta A,\delta\bar{A}>+ 2\pi i\sum_{a=1}^n\delta<p_a,g_a^{-1}\delta g_a>- \sum_s\int_{\Sigma}<\delta A, A>\partial_s\mu({\bf t}^0)\delta t^H. \end{equation} If $\kappa=0$ the connection $A$ behaves as the one-form $A\in {\cal A}^{(1,0)}(\Sigma_{g,n},{\rm Lie}(G))$ (see (\ref{8.2})). It is so called the Higgs field . An important point is that the Hamiltonians now become the times independent. The form (\ref{2.3}) is the starting point in the derivation of the Hitchin systems via the symplectic reduction \cite {H1,Ne}. Essentially, it is the same procedure as described above. Namely, we obtain the same moment constraint (\ref{14.2}) and the same gauge fixing (\ref{11.2}). But now we are sitting in a fixed point $\mu({\bf t}^0)=0$ of the moduli space ${\cal M}_{g,n}$ and don't need the factorization under the action of the diffeomorphisms. This only difference between the solutions $L$ and $\bar{L}$ in the Hitchin systems and the hierarchies of isomonodromic deformations. Propositions 2.1, 2.3 are valid for the Hitchin systems in a slightly modified form. \begin{predl} There exists the consistent system of linear equations \beq{3.3} (\lambda+L)\Psi=0,~~\lambda\in\bf{C} \end{equation} \beq{4.3} (\partial_s+M_s)\Psi=0,~~\partial_s=\partial_{t^H_s},~(s=1,\ldots,l=\dim{\cal M}_{g,n}) \end{equation} \beq{5.3} (\bar{\partial}+{\bar L})\Psi=0, ~\bar{\partial}=\partial_{\bar{z}}, \end{equation} where $M_s$ is a solution to the linear equation \beq{6.3} \bar{\partial} M_s-[M_s,{\bar L}]=\partial_s\bar{L}-L\partial_s\mu({\bf t}^0). \end{equation} \end{predl} The parameter $\lambda$ in (\ref{3.3}) can be considered as the symbol of $\partial$ (compare with (\ref{23.2})). When $L$ and $M$ can be find explicitly the simplified form of (\ref{14.2}) allows to apply "the inverse scattering method" to find solutions of the Hitchin hierarchy as it was done for ${\rm SL}(N,{\bf C})$ holomorphic bundles over $\Sigma_{1,1}$ \cite{Kr2}, corresponding to the elliptic Calogero system with spins. We present the alternative way to describe the solutions: \begin{predl}[The projection method.] $$ \bar{L}(t_s)=f^{-1}(L^0\partial_s\mu_o(t_s-t_s^0)+\bar{L}^0)f+f^{-1}\bar{\partial} f, $$ $$ L(t_s)=f^{-1}L^0f,~~p_a(t_s)=f^{-1}(p_a^0)f $$ \end{predl} The degenerate version of these expressions was known for a long time \cite{OP}. \bigskip \noindent {\bf 2. About KZB.} The Hitchin systems are the classical limit of the KZB equations on the critical level \cite{Ne,I}. The later has the form of the Schr\"{o}dinger equations, which is the result of geometric quantization of the moduli of flat $G$ bundles \cite{ADPW,H2}. The conformal blocks of the WZW theory on $\Sigma_{g,n}$ with vertex operators in marked points are ground state wave functions $$\hat {H}_sF=0,~~(s=1,\ldots,l=\dim{\cal M}_{g,n}).$$ The classical limit means that one replaces operators on their symbols and finite-dimensional representations in the vertex operators by the corresponding coadjoint orbits. The level $\kappa$ plays the role of the Planck constant, but in contrast with the limit considerd before, we don't adjust the moduli of complex structures. Generically, for $\kappa\neq 0$ the KZB equations can be written in the form of the nonstationar Schr\"{o}dinger equations $$ (\kappa\partial_s+\hat {H}_s)F=0,~~(s=1,\ldots,l=\dim{\cal M}_{g,n}). $$ The flatness of this connection (see \cite{H2}) is the quantum counterpart of the Whitham equations (\ref{22.2}). The classical limit in the described above sense leads to the HID. Summarizing, we arrange these quantum and classical systems in the diagram. The vertical arrows denote to the classical limit, while the limit $\kappa\rightarrow 0$ on the horizontal arrows includes also the rescaling of the moduli of complex structures. The examples in the bottom of the diagram will be considered in next sections. $$ \def\normalbaselines{\baselineskip20pt \lineskip3pt \lineskiplimit3pt} \def\mapright#1{\smash{ \mathop{\longrightarrow}\limits^{#1}}} \def\mapdown#1{\Big\downarrow\rlap {$\vcenter{\hbox{$\scriptstyle#1$}}$}} \begin{array}{ccc} \left\{ \begin{array}{c} \mbox{KZB eqs.},~(\kappa,{\cal M}_{g,n},G)\\ (\kappa\partial_{t_a}+\hat{H}_a)F=0,\\ (a=1,\ldots,\dim{\cal M}_{g,n}) \end{array} \right \} &\mapright{\kappa\rar0,~{\bf t}=\kappa{\bf t}^H} & \left\{ \begin{array}{c} \mbox{KZB eqs. on the critical level},~\\ ({\cal M}_{g,n},G),~(\hat{H}_a)F=0,\\ (a=1,\ldots,\dim{\cal M}_{g,n}) \end{array} \right\} \\ \mapdown{\kappa\rightarrow 0}& &\mapdown{\kappa\rightarrow 0} \\ \left\{ \begin{array}{c} \mbox{Hierarchies of Isomonodromic} ~\\ \mbox{deformations on}~{\cal M}_{g,n} \end{array} \right \} &\mapright{\kappa=0,~{\bf t}=\kappa{\bf t}^H} & \left\{ \begin{array}{c} \mbox{Hitchin systems} ~\\ \\ \end{array} \right \} \\ & & \\ & \mbox{\sl EXAMPLES}& \\ \left\{ \begin{array}{c} \mbox{Schlesinger eqs.} \\ \mbox{Painlev\'{e} type eqs.} \\ \mbox{Elliptic Schlesinger eqs.} \end{array} \right\} & \mapright{\kappa\rightarrow 0,~{\bf t}=\kappa{\bf t}^H} & \left\{ \begin{array}{c} \mbox{Classical Gaudin eqs.}\\\ \mbox{Calogero eqs.}\\ \mbox{Elliptic Gaudin eqs.} \end{array} \right\} \end{array} $$ \section{Genus zero - Schlesinger's equation.} \setcounter{equation}{0} Consider ${\bf C}P^1$ with $n$ punctures $(x_1,\ldots,x_n|x_a\neq x_b)$. The Beltrami differential $\mu$ is related only to the positions of marked points (\ref{17.2}). On ${\bf C}P^1$ the gauge transform (\ref{11.2}) allows to choose $\bar L$ to be identically zero. Let $A=f(L+\kappa\partial_w)f^{-1}$. Then the moment equation takes the form \beq{2.1} (\bar{\partial} +\partial\mu)L=2\pi i\sum_{a=1}^n\delta^2(x_a)p_a. \end{equation} It allows to find $L$ \beq{2.5} L=\sum_{a=1}^n\frac{p_a}{w-x_a}. \end{equation} On the symplectic quotient $\omega$ (\ref{19.2}) takes the form $$\omega=\delta\sum_{a=1}^n<p_ag_a^{-1}\delta g_a>- \frac{1}{\kappa}\sum_{b=1}^n(\delta H_{b,1}+\delta H_{b,0})\delta x_b. $$ \beq{3.5} H_{a,1}=\sum_{b\neq a}\frac{<p_a,p_b>}{x_a-x_b},~~H_{2,a}=c_a<p_a,p_a>. \end{equation} $H_{1,a}$ are precisely the Schlesinger's Hamiltonians. Note, that we still have a gauge freedom with respect to the $G$ action. The corresponding moment constraint means that the sum of residues of $L$ vanishes: \beq{VP} \sum_{a=1}^np_a=0. \end{equation} While $H_{2,a}$ are Casimirs and lead to trivial equations, the equation of motion for $H_{1,a}$ are the Schlesinger equations $$ \kappa\partial_bp_a=\frac{[p_a,p_b]}{x_a-x_b},~(a\neq b),~~ \kappa\partial_ap_a=-\sum_{b\neq a}\frac{[p_a,p_b]}{x_a-x_b}. $$ As by product, we obtain by this procedure the corresponding linear problem (\ref{23.2}),(\ref{24.2}) with $L$ (\ref{2.5}) and $$M_{a,1}=-\frac{p_a}{w-x_a}$$ as a solution to (\ref{25.2}). The tau-function for the Schlesinger equations has the form \cite{JMU} $$\delta\log\tau=\sum_{c\neq b}<p_b,p_c>\delta\log (x_c-x_b).$$ \section{Genus one - elliptic Schlesinger, Painleve VI...} \setcounter{equation}{0} {\bf 1. Deformations of elliptic curves.} In addition to the moduli coming from the positions of the marked points there is an elliptic module $\tau,~Im\tau>0$ on $\Sigma_{1,n}$. As in (\ref{17a.2}),(\ref{17.2}) we take the Beltrami differential in the form \mbox{$ \mu=\sum_{a=1}^n\mu_a+\mu_\tau,~~(\mu_s=t_s\bar{\partial} n_s), $} where $n_a(z,\bar{z})$ is the same as in (\ref{17.2}) and \beq{1.6} n_\tau=(\bar{z}-z)(1-\sum_{a=1}^n\chi_a(z,\bar{z})). \end{equation} Then \beq{2b.6} \mu_\tau= \ti{\mu}_\tau(1-\sum_{a=1}^n\chi_a(z,\bar{z})),~~ (\ti{\mu}_\tau=\frac{t_\tau}{\tau-\tau_0},~t_\tau=\tau-\tau_0) \end{equation} Here $\tau_0$ defines the reference comlex structure on the curve $$ T^2_{0}=\{0<x\leq 1,~0<y\leq 1, ~z=x+\tau_0y,~\bar{z}=x+\bar{\tau}_0y\}. $$ \bigskip {\bf 2. Flat bundles on a family of elliptic curves.} Note first, that $\bar{A}$ can be considered as a connection of holomorphic $G$ bundle ${\cal E}$ over $T^2_{\tau}$. For stable bundles $\bar{A}$ can be gauge transformed by (\ref{11.2}) to the Cartan $z$-independent form\\ $\bar{A}=f(\bar{L}+\bar{\partial}+\mu\partial)f^{-1}$, $ \bar{L}\in{\cal H}$- Cartan subalgebra of Lie$(G)$. Therefore, stable bundle ${\cal E}$ is decoposed into the direct sum of line bundles ${\cal E}=\oplus_{k=1}^r {\cal L}_k, ~~r=\mbox{rank}G$. The set of gauge equivalent connections represented by $\{\bar{L}\}$ can be identified with the $r$ power of the Jacobian of $T^2_{\tau}$, factorized by the action of the Weyl group $W$ of $G$. Put \beq{4.6} \bar{L}=2\pi i \frac{1-\ti{\mu}_\tau}{\rho}{\bf u}, ~~{\bf u}\in{\cal H}, ~~(\rho=\tau_0-\bar{\tau}_0). \end{equation} The moment constraints (\ref{13.2}) leading to the flatness condition take the form \beq{3.6} \partial_{\bar{w}}L+[\bar{L},L]=2\pi i\sum_{a=1}^n\delta^2(x_a,)p_a. \end{equation} Let $R=\{\alpha\}$ be the root system of of Lie$(G)={\cal G}$ and ${\cal G}={\cal H}\oplus_{\alpha\in R}{\cal G}_\alpha$ be the root decomposition. Impose the vanishing of the residues in (\ref{3.6}) \beq{11.6} \sum_{a=1}^n(p_a)_{\cal H}=0, \end{equation} where $p_a|_{\cal H}$ is the Cartan component and we have identified ${\cal G}$ with its dual space ${\cal G}^*$. We will parametrized the set of its solutions by two elements ${\bf v},{\bf u}\in {\cal H}$. Define the solutions $L$ to the moment equation (\ref{3.6}), which is double periodic on the deformed curve $T^2_\tau$. Let $E_1(w)$ be the Eisenstein function $$ E_1(z|\tau)=\partial_z\log\theta(z|\tau), $$ where $$ \theta(z|\tau)=q^{\frac {1}{8}}\sum_{n\in {\bf Z}}(-1)^ne^{\pi i(n(n+1)\tau+2nz)}. $$ It is connected with the Weirstrass zeta-function as $$ \zeta(z|\tau)=E_1(z|\tau)+2\eta_1(\tau)z,~~(\eta_1(\tau)= \zeta(\frac{1}{2})). $$ Another function we need is $$ \phi(u,z)=\frac{\theta(u+z)\theta'(0)}{\theta(u)\theta(z)}= \exp(-2\eta_1uz) \frac {\sigma(u+z)}{\sigma(y)\sigma(z)}, $$ where $\sigma(z)$ is the Weierstrass sigma function. \begin{lem} The solutions of the moment constraint equation have the form \beq{5.6} L=P+X,~~P\in{\cal H},~~X=\sum_{\alpha\in R}X_{\alpha}.\end{equation} \beq{6.6} P=2\pi i(\frac{{\bf v}}{1-\ti{\mu}_\tau}-\kappa\frac{{\bf u}}{\rho}+ \sum_{a=1}^n(p_a)_{\cal H}E_1(w-x_a)), \end{equation} \beq{7.6} X_{\alpha}=frac{2\pi i}{1-\ti{\mu}_\tau} \sum_{a=1}^n(p_a)_{\alpha}\exp 2\pi i\{ \frac{(w-x_a)-(\bar{w}-\bar{x}_a)}{\tau-\bar{\tau}_0} \alpha(u)\} \phi(\alpha(u),w-x_a). \end{equation} \end{lem} \bigskip {\bf 2. Symmetries.} The remnant gauge transforms preserve the chosen Cartan subalebra ${\cal H}\subset G$. These transformations are generated by the Weyl subgroup $W$ of $G$ and elements \mbox{$f(w,\bar{w})\in {\rm Map}(T^2_{\tau},{\rm Cartan}(G))$}. Let $\Pi$ be the system of simple roots, $R^{\vee}=\{\alpha^{\vee}=\frac{2\alpha}{(\alpha|\alpha)}\},$ is the dual root system, and ${\bf m}=\sum_{\alpha\in \Pi} m_{\alpha}\alpha^{\vee}$ be the element from the dual root lattice ${\bf Z}R^{\vee}$. Then the Cartan valued harmonics \beq{9.6} f_{{\bf m},{\bf n}}=\exp 2\pi i( {\bf m}\frac{w-\bar{w}}{\tau-\tau_0}+ {\bf n}\frac{\tau\bar{w} -\bar{\tau}_0w}{\tau-\tau_0}),~~ ({\bf m} ,{\bf n}\in R^{\vee}) \end{equation} generate the basis in the space of Cartan gauge transformations. In terms of the variables ${\bf v}$ and ${\bf u}$ they act as \beq{10.6} {\bf u}\rightarrow {\bf u} +{\bf m}-{\bf n}\tau,~~ {\bf v}\rightarrow {\bf v}-\kappa{\bf n},~~ (p_a)_{\alpha}\rightarrow\varphi(m_\alpha,n_\alpha)(p_a)_{\alpha}. \end{equation} Here $ \varphi(m_\alpha,n_\alpha)= \exp \frac{4\pi i}{\rho}[(m_\alpha-n_\alpha\bar{\tau}_0x_a^0)- (m_\alpha-n_\alpha\tau_0\bar{x}_a^0)].$ The whole discrete gauge symmetry is the semidirect product $\hat{W}$ of the Weyl group $W$ and the lattice ${\bf Z}R^{\vee}\oplus\tau{\bf Z}R^{\vee}$. It is the Bernstein-Schvartsmann complex crystallographic group. The factor space ${\cal H}/\hat{W}$ is the genuin space for the "coordinates" ${\bf u}$. According with (\ref{8.2}) the transformations (\ref{9.6}) act also on $p_a\in{\cal O}_a$. This action leads to the symplectic quotient ${\cal O}_a//H$ and generates the moment equation (\ref{11.6}). The modular group ${\rm PSL}_2({\bf Z})$ is a subgroup of mapping class group for the Teichm\"{u}ller space ${\cal T}_{1,n}$. Its action on $\tau$ is the M\"{o}bius transform. We summarise the action of symmetries on the dynamical variables: \bigskip \begin{center} \begin{tabular}{|c|c|c|c|}\hline &W=\{s\}&${\bf Z}R^{\vee}\oplus\tau{\bf Z}R^{\vee}$ &${\rm PSL}_2({\bf Z})$ \\ \hline \hline ${\bf v}$ &$s{\bf v}$& ${\bf v}-\kappa{\bf n} $ & ${\bf v}(c\tau+d)-\kappa c{\bf u} $ \\ \hline ${\bf u}$ &$s{\bf u}$ & ${\bf u}+{\bf m}-{\bf n}\tau$ &${\bf u}(c\tau+d)^{-1}$ \\ \hline $(p_a)_{\cal H}$ & $s(p_a)_{\cal H}$ & $(p_a)_{\cal H}$ & $(p_a)_{\cal H}$ \\ \hline $(p_a)_{\alpha}$ & $(p_a)_{s\alpha}$ & $\varphi(m_\alpha,n_\alpha)(p_a)_{\alpha}$ & $(p_a)_\alpha$ \\ \hline $\tau$ &$\tau$ &$\tau$ &$\frac{a\tau+b}{c\tau+d}$ \\ \hline $x_a$ & $x_a$ &$x_a$ & $\frac {x_a}{c\tau+d}$ \\ \hline \end{tabular} \end{center} \bigskip {\bf 3. Symplectic form.} The set $({\bf v},{\bf u}\in {\cal H},{\bf p}=(p_1,\ldots,p_n))$ of dynamical variables along with the times ${\bf t}=(t_\tau,t_1,\ldots,t_n)$ describe the local coordinates in the bundle ${\cal P}$. According with the general prescription, we can define the hamiltonian system on this set. The main statement, formulated in terms of the theta-functions and the Eisenstein functions \beq{A.2} E_2(z|\tau)=-\partial_zE_1(z|\tau)= \partial_z^2\log\theta(z|\tau)=\wp(z|\tau)+2\eta_1(\tau). \end{equation} It takes the form \begin{predl} The symplectic form $\omega$ (\ref{19.2}) on ${\cal P}$ is \beq{16.6} \frac{1}{4\pi^2}\omega=(\delta{\bf v},\delta{\bf u})+ \sum_{a=1}^n\delta<p_a,g_a^{-1}\delta g_a> -\frac{1}{\kappa}(\sum_{a=1}^n\delta H_{2,a}+\delta H_{1,a})\delta t_a- \frac{1}{\kappa}\delta H_{\tau}\delta\tau, \end{equation} with the Hamiltonians $$H_{2,a}=c_a<p_a,p_a>;$$ $$ H_{1,a}= =2(\frac{{\bf v}}{1-\ti{\mu}_\tau}-\kappa\frac{{\bf u}}{\rho},p_a|_{\cal H})+ \sum_{b\neq a}(p_a|_{\cal H},p_b|_{\cal H})E_1(x_a-x_b)+ $$ $$ \sum_{b\neq a}\sum_\alpha(p_a|_{\alpha},p_b|_{-\alpha}) \frac {\theta(-\alpha({\bf u})+x_a-x_b)\theta'(0)} {\theta(\alpha({\bf u}))\theta(x_a-x_b)}; $$ $$ H_{\tau}= $$ $$\frac{({\bf v},{\bf v})}{2}+\{\sum_{a=1}^n \sum_{\alpha}(p_a|_{\alpha},p_a|_{-\alpha})E_2(\alpha({\bf u}))+ \sum_{a\neq b}^n(p_a|_{\cal H},p_b|_{\cal H})(E_2(x_a-x_b)- E_1^2(x_a-x_b))+ $$ $$ \sum_{a\neq b}\sum_\alpha(p_a|_{\alpha},p_b|_{-\alpha}) \frac {\theta(-\alpha({\bf u})+x_a-x_b)\theta'(0)} {\theta(\alpha({\bf u}))\theta(x_a-x_b)} (E_1(\alpha({\bf u})-E_1(x_b-x_a+\alpha({\bf u}))-E_1(x_b-x_a)). $$ \end{predl} \bigskip {\bf Example 1.} Consider ${\rm SL}(2,{\bf C})$ bundles over the family of $\Sigma_{1,1}$. Then (\ref{4.6}) takes the form \beq{BL} \bar{L}=2\pi i \frac{1-\ti{\mu}_\tau}{\rho}{\rm diag} (u,-u). \end{equation} In this case the position of the maked point is no long the module and we put $x_1=0$. Since $\dim{\cal O}=2$ the orbit degrees of freedom can be gauged away by the hamiltonian action of the diagonal group. We assume that $p=\nu[(1,1)^T\otimes(1,1)-Id]$. Then we have from(\ref{5.6}),(\ref{6.6}),(\ref{7.6}) \beq{17.6} L=2\pi i \mat{\frac{v}{1-\ti{\mu}_\tau}-\kappa\frac{u}{\rho}} {x(u,w,\bar{w})}{x(-u,w,\bar{w})} {-\frac{v}{1-\ti{\mu}_\tau}+\kappa\frac{u}{\rho}}. \end{equation} $$ x(u,w,\bar{w})=\frac{\nu}{1-\ti{\mu}_\tau}\exp 4\pi i\{(w-\bar{w})u \frac{1-\ti{\mu}_{\tau}}{\rho}\}\phi(2u,w). $$ The symplectic form (\ref{16.6}) $$ \frac{1}{4\pi^2}\omega=(\delta v,\delta u)-\frac{1}{\kappa}\delta H_{\tau}\delta\tau, $$ and $$ H_{\tau}=v^2+U(u|\tau),~U(u|\tau)=-\nu^2E_2(2u|\tau). $$ It leads to the equation of motion \beq{20.6} \frac{\partial^2 u}{\partial\tau^2}=\frac{2\nu^2}{\kappa^2}\frac{\partial}{\partial u}E_2(2u|\tau). \end{equation} In fact, due to (\ref{A.2}) we can use $\wp(2u|\tau)$ instead of $E_2(2u|\tau)$ and after rescaling the coupling constant come to (\ref{I.3}) for special values of constants as in (\ref{I.8}). The equation (\ref{20.6}) is the isomonodromic deformation conditions for the linear system (\ref{23.2}),(\ref{25a.2}) with $L$ (\ref{17.6}) and $\bar{L}$ (\ref{BL}). The lax pair is given by $L$ (\ref{17.6}) and $M_\tau$ $$ M_\tau=\mat{0}{y(u,w,\bar{w})}{y(-u,w,\bar{w})}{0}, $$ where $y(u,w,\bar{w})$ is defined as the convolution integral on $T^2_\tau$ $$y(u,w,\bar{w})=-\f1{\kappa}x\ast x(u,w,\bar{w}).$$ The projection method determines solutions of (\ref{20.6}) as a result of diagonalization of $L$ (\ref{17.6}) by the gauge transform on the deformed curve $T^2_\tau$. On the critical level $(\kappa=0)$ we come to the two-body elliptic Calogero system. \bigskip {\bf Example 2.} For flat $G$ bundles over $\Sigma_{1,1}$ we obtain Painlev\'{e} type equations, related to arbitrary root system. They are described by the system of differential equations for the ${\bf u}=(u_1,\ldots,u_r),~(r=$rank$G)$ variables. In addition there are the orbit variables $p\in{\cal O}(G)$ satisfying the Euler top equations. For ${\rm SL}(N,{\bf C})$ bundles the most degenerate orbits ${\cal O}=T^*{\bf C}P^{N-1}$ has dimension $2N-2$. These variables are gauge away by the diagonal gauge transforms as in the previous example. On the critical level this Painlev\'{e} type system degenerates into $N$-body elliptic Calogero system. For generic orbits we obtain the generalized Calogero-Euler systems. \small{

9706

alg-geom/9706002

en

https://arxiv.org/abs/alg-geom/9706002

alg-geom/9706002

Alice Silverberg

A. Silverberg and Yu. G. Zarhin

Subgroups of inertia groups arising from abelian varieties

LaTeX 2e, updated version

Given an abelian variety over a field with a discrete valuation, Grothendieck defined a certain open normal subgroup of the absolute inertia group. This subgroup encodes information on the extensions over which the abelian variety acquires semistable reduction. We study this subgroup, and use it to obtain information on the extensions over which the abelian variety acquires semistable reduction.

alg-geom

\section{Introduction} Suppose $X$ is an abelian variety over a field $F$, and $v$ is a discrete valuation on $F$. Fix an extension ${\bar v}$ of $v$ to a separable closure $F^{s}$ of $F$, and write ${\mathcal I}_{v}$ for the inertia subgroup in $\mathrm{Gal}(F^{s}/F)$ for ${\bar v}$. In \cite{SGA} (see pp.~354--355), Grothendieck defined a subgroup ${\mathcal I}'$ of ${\mathcal I}_{v}$ with the property that $X$ has semistable reduction at the restriction $w$ of ${\bar v}$ to a finite separable extension of $F$ if and only if ${\mathcal I}_{w} \subseteq {\mathcal I}'$. In particular, if $F_{v}^{nr}$ denotes the maximal unramified extension of the completion of $F$ at $v$, then ${\mathcal I}'$ cuts out the smallest Galois extension of $F_{v}^{nr}$ over which $X$ has semistable reduction. We denote the group ${\mathcal I}'$ by ${{\mathcal I}_{v,X}}$ because of its dependence on $X$ and $v$. In \S\ref{properties} we give some properties of the group ${{\mathcal I}_{v,X}}$. We show that the Zariski closure of its image under the $\ell$-adic representation (for $\ell$ different from the residue characteristic) coincides with the identity connected component of the Zariski closure of the image of ${\mathcal I}_{v}$. The proofs of the results in \S\ref{properties} are in the spirit of \cite{Compositio}, where we dealt with connectedness questions for Zariski closures of images of $\ell$-adic representations. In \S\ref{bounds} we show that the finite group $G_{v,X} = {\mathcal I}_{v}/{{\mathcal I}_{v,X}}$ injects into $\mathrm{GL}_{t-t_{v}}({\mathbf Z}) \times \mathrm{Sp}_{2(a-a_{v})}({\mathbf Z}_{\ell})$ for all but finitely many primes $\ell$, where $t_{v}$ and $a_{v}$ (respectively, $t$ and $a$) are the toric and abelian ranks of the special fiber of the N\'eron model of $X$ at $v$ (respectively, at an extension of $v$ over which $X$ has semistable reduction). Here, the projection onto the first factor is independent of $\ell$, and the characteristic polynomial of the projection onto the second factor has integer coefficients independent of $\ell$. The group $G_{v,X}$ was introduced by Serre in the case of elliptic curves in \S 5.6 of \cite{Serre72} (where it was called $\Phi_{p}$). In \S\ref{order} we obtain divisibility bounds on the order of $G_{v,X}$, and in \S\ref{applic} we deduce results on semistable reduction of abelian varieties. Bounds on the prime divisors and the exponent of $\#G_{v,X}$ were obtained by Lorenzini (see Proposition 3.1 of \cite{Lorenzini}). In particular, the bound on $Q_{v,X}$ in Corollary \ref{hdcor} was essentially obtained by Lorenzini. The paper continues our earlier work on semistable reduction of abelian varieties (see \cite{semistab} and \cite{smalldeg}). The proofs are heavily influenced by the fundamental results of Grothendieck and Serre. The authors would like to thank Karl Rubin for helpful conversations, and NSA and NSF for financial support. \section{Notation and preliminaries} \label{prelim} If $K$ is a field, write $K^s$ for a separable closure. If $G$ is an algebraic group, let $G^{0}$ denote its identity connected component. Let $\varphi$ denote the Euler $\varphi$-function, let $\zeta_{M}$ denote a primitive $M$-th root of unity, and let ${\mathbf F}_{\ell}$ denote the finite field with $\ell$ elements. If $Y$ is a commutative algebraic group over a field $K$ (e.g., an abelian variety or an algebraic torus), let $Y_{n}$ denote the kernel of multiplication by $n$ in $Y(K^{s})$, let $T_{\ell}(Y) = {\displaystyle{\lim_{\leftarrow}}} Y_{\ell^{n}}$, and let $V_{\ell}(Y) = T_{\ell}(Y) \otimes_{{\mathbf Z}_{\ell}} {\mathbf Q}_{\ell}$. For example, if ${\mathbf G}_{m}$ is the multiplicative group, then ${\mathbf Z}_\ell(1):=T_{\ell}({\mathbf G}_{m})$ is a free ${\mathbf Z}_{\ell}$-module of rank $1$. Throughout this paper, $X$ is a $d$-dimensional abelian variety over a field $F$, $v$ is a discrete valuation on $F$ of residue characteristic $p \ge 0$, and $\ell$ is a prime not equal to $p$. Fix an extension ${\bar v}$ of $v$ to $F^s$. If $w$ is the restriction of ${\bar v}$ to a finite separable extension $L$ of $F$, let ${\mathcal I}_{w}$ denote the inertia subgroup in $\mathrm{Gal}(F^{s}/L)$ for ${\bar v}$, let $X_{w}$ denote the special fiber of the N\'eron model of $X$ at $w$, and let ${\mathbf T}_{w}$ denote the maximal subtorus of $X_{w}$. Let $$\rho_{\ell}:\mathrm{Gal}(F^s/F)\to\mathrm{GL}(T_\ell(X))$$ denote the $\ell$-adic representation. Let ${\mathfrak G}$ denote the Zariski closure of $\rho_{\ell}({\mathcal I}_{v})$ in $\mathrm{GL}(V_{\ell}(X))$. We will make repeated use of the following result. \begin{thm}[Galois Criterion for Semistable Reduction] \label{galcrit} $X$ has semistable reduction at $v$ if and only if ${\mathcal I}_{v}$ acts unipotently on $V_\ell(X)$. \end{thm} \begin{proof} See Proposition 3.5 and Corollaire 3.8 of \cite{SGA} and Theorem 6 on p.~184 of \cite{BLR}. \end{proof} \begin{thm} \label{conn} The Zariski closure ${\mathfrak G}$ of $\rho_{\ell}({\mathcal I}_v)$ is connected if and only if $X$ has semistable reduction at $v$. \end{thm} \begin{proof} See Theorem 5.2 of \cite{Compositio}; see also Remarque~1 on p.~396 of \cite{Motives}. \end{proof} Suppose $\lambda$ is a polarization on $X$. Then $\lambda$ gives rise to a non-degenerate, alternating, $\mathrm{Gal}(F^{s}/F)$-equi\-var\-i\-ant, ${\mathbf Z}_\ell(1)$-valued pairing on $T_{\ell}(X)$ (see \S\S 1.0 and 2.5 of \cite{SGA}). Since ${\mathcal I}_v$ acts trivially on ${\mathbf Z}_\ell(1)$, we obtain a non-degenerate, alternating, ${\mathcal I}_v$-invariant pairing $$E_\lambda : T_\ell(X) \times T_\ell(X) \to {\mathbf Z}_\ell.$$ Then $E_\lambda$ is perfect if and only if $\deg(\lambda)$ is not divisible by $\ell$ (see (2.5.1) and \S 1.0 of \cite{SGA}). Let $\perp$ denote the orthogonal complement with respect to $E_\lambda$. \begin{prop} \label{serretateeqn} \begin{enumerate} \item[(i)] We can identify $$T_\ell(X_v)=T_\ell(X)^{{\mathcal I}_v} \quad \text{ and } \quad V_\ell(X_v)=V_\ell(X)^{{\mathcal I}_v}.$$ \item[(ii)] $T_\ell({\mathbf T}_v)=T_\ell(X)^{{\mathcal I}_v} \cap (T_\ell(X)^{{\mathcal I}_v})^\perp = T_\ell(X_v) \cap T_\ell(X_v)^{\perp}$. \item[(iii)] $T_\ell(X_v)/T_\ell({\mathbf T}_v)$ is a free ${\mathbf Z}_\ell$-module. \end{enumerate} \end{prop} \begin{proof} For (i), see Lemma 2 on p.~495 of \cite{SerreTate}. For (ii), use (i) and Grothendieck's Orthogonality Theorem (see (2.5.2) of \cite{SGA}). For (iii), see (2.1.6) of \cite{SGA}. \end{proof} \begin{prop} \label{orthogeqn} Suppose $L$ is a finite separable extension of $F$, $w$ is the restriction of ${\bar v}$ to $L$, and $X$ has semistable reduction at $w$. Then$$ T_\ell({\mathbf T}_w) = T_\ell(X_w)^\perp = (T_\ell(X)^{{\mathcal I}_w})^\perp \subseteq T_\ell(X)^{{\mathcal I}_w} = T_\ell(X_w).$$ \end{prop} \begin{proof} See Proposition 3.5 of \cite{SGA}. \end{proof} \section{Linear Algebra} We will use the following linear algebra facts in Theorem \ref{hd} below. \begin{defn} Suppose $R$ is a principal ideal domain, and $\mathcal{M}_1$ and $\mathcal{M}_2$ are free $R$-modules. A bilinear form $e : \mathcal{M}_1 \times \mathcal{M}_2 \to R$ is called a {\it perfect} pairing if the natural homomorphisms $\mathcal{M}_1 \to \mathrm{Hom}(\mathcal{M}_2,R)$ and $\mathcal{M}_2 \to \mathrm{Hom}(\mathcal{M}_1,R)$ are bijective. If $\mathcal{L}$ is a submodule of $\mathcal{M}_1$ (resp., $\mathcal{M}_2$), we write $\mathcal{L}^{\perp}$ for the orthogonal complement of $\mathcal{L}$ with respect to $e$ in $\mathcal{M}_2$ (resp., $\mathcal{M}_1$). \end{defn} \begin{rem} \label{perf} Suppose $R$ is a principal ideal domain, $\mathcal{M}$ is a free $R$-module of rank $2n$, and $e : \mathcal{M} \times \mathcal{M} \to R$ is an alternating bilinear form. If $e$ is perfect, then $\mathrm{Aut}(\mathcal{M},e) \cong \mathrm{Sp}_{2n}(R)$, where $\mathrm{Sp}_{2n}(R)$ denotes the group of $2n \times 2n$ symplectic matrices over $R$ (see \S{5} of \cite{Bourbaki}). \end{rem} \begin{prop} \label{linalg} Suppose $\ell$ is a prime number, $G$ is a finite group whose order is not divisible by $\ell$, $V$ is a finite-dimensional ${\mathbf Q}_\ell$-vector space with a linear $G$-action, and $e:V \times V \to {\mathbf Q}_\ell$ is a $G$-invariant non-degenerate alternating (resp., symmetric) bilinear form. Suppose $\mathcal{M}$ is a $G$-stable ${\mathbf Z}_\ell$-lattice in $V$ (there always exist such). Then there exists a perfect $G$-invariant alternating (resp., symmetric) bilinear form $e':\mathcal{M} \times \mathcal{M} \to {\mathbf Z}_\ell$. \end{prop} \begin{proof} First note that the result is true when $e$ is symmetric and $\mathrm{dim}(V)=1$ and when $e$ is alternating and $\mathrm{dim}(V)=2$, by scaling $e$. We may assume that $e(\mathcal{M},\mathcal{M})={\mathbf Z}_{\ell}$ (by replacing $e$ by $\ell^i e$ for a suitable $i$, if necessary). Then $e$ induces a non-zero alternating (resp., symmetric) bilinear form $\bar{e}$ on $\mathcal{M}/\ell \mathcal{M}$. By Nakayama's Lemma, if $\bar{e}$ is non-degenerate then $e$ is perfect and we are done. Assume $\bar{e}$ is degenerate, and let $\bar{\mathcal{M}_1}=\ker(\bar{e})$. Then $\bar{\mathcal{M}_1}$ is a proper subset of $\mathcal{M}/\ell \mathcal{M}$, since $e(\mathcal{M},\mathcal{M})={\mathbf Z}_{\ell}$. Since $\#G$ is not divisible by $\ell$, there is a $G$-invariant splitting $\mathcal{M}/\ell \mathcal{M}= \bar{\mathcal{M}_1} \oplus \bar{\mathcal{M}_2}$, where $\bar{\mathcal{M}_2}$ is a non-zero $G$-invariant subspace of $\mathcal{M}/\ell \mathcal{M}$, which can be lifted to a $G$-invariant splitting of ${\mathbf Z}_{\ell}$-lattices $\mathcal{M}=\mathcal{M}_1 \oplus \mathcal{M}_2$ with $\mathcal{M}_1/\ell \mathcal{M}_1=\bar{\mathcal{M}}_1$ and $\mathcal{M}_2/\ell \mathcal{M}_2=\bar{\mathcal{M}}_2$ (see \S 15.5 and Corollary 1 of \S 14.4 of \cite{serrereps}). Denote by $e_2$ the restriction of $e$ to $V_2:=\mathcal{M}_2\otimes_{{\mathbf Z}_\ell} {\mathbf Q}_{\ell}$, and let $V_1$ denote the orthogonal complement of $V_2$ in $V$ with respect to $e$. The restriction of $e$ to $V_1$ is non-degenerate, and the restriction of $e$ to $\mathcal{M}_2$ is perfect. We obtain a $G$-invariant orthogonal splitting $V= V_1 \oplus V_2$. Replace $\mathcal{M}_1$ by its (isomorphic) image in $V_1$ under the projection map from $V_1 \oplus V_2$ to $V_1$. Since $\mathrm{dim}(V_1) < \mathrm{dim}(V)$, we obtain inductively a perfect $G$-invariant alternating (resp., symmetric) form $e_1$ on $\mathcal{M}_1$. Let $\mathcal{M}=\mathcal{M}_1 \oplus \mathcal{M}_2$ and $e'=e_1 \oplus e_2$. \end{proof} \begin{lem} \label{perp} Suppose $R$ is a principal ideal domain, $\mathcal{M}_1$ and $\mathcal{M}_2$ are free $R$-modules, $\mathcal{L}$ is a submodule of $\mathcal{M}_1$, $\mathcal{M}_1/\mathcal{L}$ is torsion-free, and $e : \mathcal{M}_1 \times \mathcal{M}_2 \to R$ is a {\it perfect} pairing. Then: \begin{enumerate} \item[(i)] $(\mathcal{L}^{\perp})^{\perp} = \mathcal{L}$, \item[(ii)] the natural map $\mathrm{Hom}(\mathcal{M}_1,R) \to \mathrm{Hom}(\mathcal{L},R)$ is surjective, \item[(iii)] the induced form $\mathcal{L} \times \mathcal{M}_2/\mathcal{L}^{\perp} \to R$ is a perfect pairing. \end{enumerate} \end{lem} \begin{proof} Clearly, $\mathcal{L} \subseteq (\mathcal{L}^{\perp})^{\perp}$. Let $K$ denote the fraction field of $R$. By dimension arguments we have $\mathcal{L} \otimes K = ((\mathcal{L} \otimes K)^{\perp})^{\perp} = (\mathcal{L}^{\perp})^{\perp} \otimes K$. Therefore, $(\mathcal{L}^{\perp})^{\perp}/\mathcal{L}$ is torsion. Since $\mathcal{M}_1/\mathcal{L}$ is torsion-free, we obtain (i). Further, since $\mathcal{M}_1/\mathcal{L}$ is torsion-free, $\mathcal{L}$ is a direct summand of $\mathcal{M}_1$, and therefore we have (ii). Since $e$ is perfect, we can identify $\mathrm{Hom}(\mathcal{M}_2,R)$ with $\mathcal{M}_1$. Under this identification, $\mathrm{Hom}(\mathcal{M}_2/\mathcal{L}^{\perp},R) = (\mathcal{L}^{\perp})^{\perp} = \mathcal{L}$, by (i). Since $e$ is perfect we can identify $\mathcal{M}_2$ with $\mathrm{Hom}(\mathcal{M}_1,R)$. The natural injection $\mathcal{M}_2/\mathcal{L}^{\perp} \hookrightarrow \mathrm{Hom}(\mathcal{L},R)$ is surjective since $\mathcal{M}_2 = \mathrm{Hom}(\mathcal{M}_1,R) \to \mathrm{Hom}(\mathcal{L},R)$ is surjective by (ii). \end{proof} \begin{prop} \label{linalg1} Suppose $R$ is a principal ideal domain, $\mathcal{M}$ is a free $R$-module of finite rank, $e : \mathcal{M} \times \mathcal{M} \to R$ is an alternating (resp., symmetric) perfect pairing, $\mathcal{N}$ is a submodule of $\mathcal{M}$, and $\mathcal{M}/\mathcal{N}$ is torsion-free. Assume that $\mathcal{N}^{\perp} \subseteq \mathcal{N}$. Then $${\tilde e}(a+\mathcal{N}^{\perp},b+\mathcal{N}^{\perp})=e(a,b)$$ defines an alternating (resp., symmetric) perfect pairing ${\tilde e} : \mathcal{N}/\mathcal{N}^{\perp} \times \mathcal{N}/\mathcal{N}^{\perp} \to R$. \end{prop} \begin{proof} Applying Lemma \ref{perp}iii with $\mathcal{M}_1 = \mathcal{M}_2 = \mathcal{M}$ and $\mathcal{L} = \mathcal{N}$ gives an alternating (resp., symmetric) perfect pairing $e' : \mathcal{N} \times \mathcal{M}/\mathcal{N}^{\perp} \to R$. Now applying Lemma \ref{perp}iii to $e'$ with $\mathcal{M}_1 = \mathcal{M}/\mathcal{N}^{\perp}$, $\mathcal{M}_2 = \mathcal{N}$, and $\mathcal{L} = \mathcal{N}/\mathcal{N}^{\perp}$, we obtain the desired result. \end{proof} \section{Properties of ${{\mathcal I}_{v,X}}$} \label{properties} We will give a different definition for ${{\mathcal I}_{v,X}}$ than Grothendieck did, and will then show that the two definitions are equivalent. Grothendieck's definition coincides with (ii) of Theorem \ref{main} below. \begin{defn} Define ${{\mathcal I}_{v,X}}$ to be the kernel of the natural surjective homomorphism ${\mathcal I}_{v} \to {\mathfrak G}/{\mathfrak G}^{0}$. Define $G_{v,X} = {\mathcal I}_{v}/{{\mathcal I}_{v,X}}$. \end{defn} \begin{thm} \label{main} ${{\mathcal I}_{v,X}}$ is an open normal subgroup of ${\mathcal I}_{v}$ which enjoys the following properties: \begin{enumerate} \item[(i)] ${{\mathcal I}_{v,X}}$ is the largest open subgroup of ${\mathcal I}_{v}$ such that the Zariski closure of its image under $\rho_{\ell}$ is ${\mathfrak G}^{0}$. \item[(ii)] ${{\mathcal I}_{v,X}} = \{\sigma \in {\mathcal I}_{v} : \sigma \text{ acts unipotently on } V_{\ell}(X) \}$. \item[(iii)] If $L$ is a finite separable extension of $F$ and $w$ is the restriction of ${\bar v}$ to $L$, then the following are equivalent: \begin{enumerate} \item[(a)] $X$ has semistable reduction at $w$, \item[(b)] ${\mathcal I}_{w} \subseteq {{\mathcal I}_{v,X}}$, \item[(c)] the Zariski closure of $\rho_{\ell}({\mathcal I}_{w})$ is ${\mathfrak G}^{0}$. \end{enumerate} \item[(iv)] ${{\mathcal I}_{v,X}} = {\mathcal I}_{v}$ if and only if $X$ has semistable reduction at $v$. \item[(v)] ${{\mathcal I}_{v,X}}$ is independent of the choice of $\ell$. \end{enumerate} \end{thm} \begin{proof} By the definition of ${{\mathcal I}_{v,X}}$, it is an open normal subgroup of ${\mathcal I}_{v}$, and it is the largest open subgroup of ${\mathcal I}_{v}$ such that the Zariski closure of its image under $\rho_{\ell}$ is ${\mathfrak G}^{0}$. Suppose $L$ is a finite separable extension of $F$, and $w$ is the restriction of ${\bar v}$ to $L$. Let ${\mathfrak G}_{w}$ denote the Zariski closure of $\rho_{\ell}({\mathcal I}_{w})$, and let ${V} = V_{\ell}(X)^{{\mathcal I}_{w}}$. Suppose first that $X$ has semistable reduction at $w$. Then ${\mathfrak G}_{w}$ is connected, by Theorem \ref{conn}. Therefore, ${\mathcal I}_{w} \subseteq {{\mathcal I}_{v,X}}$ and ${\mathfrak G}_{w} = {\mathfrak G}^{0}$. Since ${\mathfrak G}^{0}$ ($ = {\mathfrak G}_{w}$) is a normal subgroup of ${\mathfrak G}$, it follows that ${V}$ is stable under ${\mathcal I}_{v}$. Next we will show that $${{\mathcal I}_{v,X}} = \{\sigma \in {\mathcal I}_{v} : \sigma \text{ acts unipotently on } V_{\ell}(X) \}$$ $$= \{\sigma \in {\mathcal I}_{v} : \sigma \text{ acts unipotently on } {V} \}.$$ Since ${\mathfrak G}^{0} = {\mathfrak G}_{w}$, ${\mathfrak G}^{0}$ acts as the identity on ${V}$ and on $V_{\ell}(X)/{V}$. Therefore, every element of ${{\mathcal I}_{v,X}}$ acts unipotently on $V_{\ell}(X)$, and therefore on ${V}$. To show the reverse inclusions, suppose $g \in {\mathcal I}_{v}$ and $g$ acts unipotently on ${V}$. By Proposition \ref{orthogeqn} (and after tensoring with ${\mathbf Q}_\ell$), ${V}^{\perp} \subseteq {V}$. Since ${V}^{\perp}$ is the dual of $V_{\ell}(X)/{V}$, it follows that $g$ acts unipotently on $V_{\ell}(X)/{V}$, and therefore acts unipotently on $V_{\ell}(X)$. By Proposition 2.5 of \cite{Compositio}, $g \in {{\mathcal I}_{v,X}}$. We therefore obtain the desired equalities. By (i), if ${\mathfrak G}^{0} = {\mathfrak G}_{w}$, then ${\mathcal I}_{w} \subseteq {{\mathcal I}_{v,X}}$. By (ii) and Theorem \ref{galcrit}, if ${\mathcal I}_{w} \subseteq {{\mathcal I}_{v,X}}$ then $X$ has semistable reduction at $w$. We therefore have (iii). We easily deduce (iv) from (iii). By Th\'eor\`eme 4.3 of \cite{SGA}, if $\sigma \in {\mathcal I}_{v}$ then the characteristic polynomial of $\rho_{\ell}(\sigma)$ is independent of $\ell$. By (ii), $${{\mathcal I}_{v,X}} = \{\sigma \in {\mathcal I}_{v} : \text{ the characteristic polynomial of } \rho_{\ell}(\sigma) \text{ is } (x-1)^{2d}\}.$$ Therefore, ${{\mathcal I}_{v,X}}$ is independent of $\ell$. \end{proof} \begin{prop} \label{Wunipot} If $L$ is a finite separable extension of $F$, and $X$ has semistable reduction at the restriction $w$ of ${\bar v}$ to $L$, then \begin{enumerate} \item[(i)] ${{\mathcal I}_{v,X}} = \{\sigma \in {\mathcal I}_{v} : \sigma \text{ acts unipotently on } V_{\ell}(X)^{{\mathcal I}_{w}} \}$, \item[(ii)] $G_{v,X}$ acts faithfully on $T_{\ell}(X)^{{\mathcal I}_{w}}/T_{\ell}(X)^{{\mathcal I}_{v}}$, \item[(iii)] $T_{\ell}(X)^{{\mathcal I}_{w}} = T_{\ell}(X)^{{{\mathcal I}_{v,X}}}$. \end{enumerate} \end{prop} \begin{proof} The proof of Theorem \ref{main} included a proof of (i), and easily implies (ii). For (iii), let ${\mathfrak G}_{w}$ denote the Zariski closure of $\rho_{\ell}({\mathcal I}_{w})$, and note that $T_{\ell}(X)^{{{\mathcal I}_{v,X}}} = T_{\ell}(X)^{{\mathfrak G}^{0}} = T_{\ell}(X)^{{\mathfrak G}_{w}} = T_{\ell}(X)^{{\mathcal I}_{w}}$, by Theorem \ref{main}. \end{proof} \section{Restrictions on $G_{v,X}$} \label{bounds} \begin{rem} \label{cyclicrem} Note that $G_{v,X}$ is isomorphic to the group of connected components of ${\mathfrak G}$. If $p=0$ then $G_{v,X}$ is cyclic, and if $p>0$ then $G_{v,X}$ is an extension of a cyclic group of order prime to $p$ by a $p$-group (as can be seen by replacing $F$ by the maximal unramified extension of the completion of $F$ at $v$, looking at the extension cut out by ${{\mathcal I}_{v,X}}$, taking its maximal tamely ramified subextension, and applying \S8 of \cite{Frohlich}). In particular, $G_{v,X}$ is solvable, and has a unique Sylow-$p$-subgroup if $p>0$. \end{rem} The group $G_{v,X}$ does not change if we replace $F$ by an extension unramified at $v$. Therefore, in this section we can and do replace $F$ by the maximal unramified extension of the completion of $F$ at $v$. Then ${{\mathcal I}_{v,X}}$ cuts out the smallest Galois extension $L$ of $F$ over which $X$ has semistable reduction. Let $w$ denote the restriction of ${\bar v}$ to $L$. Then ${{\mathcal I}_{v,X}} = {\mathcal I}_{w}$. Therefore, ${\mathcal I}_w$ is an open normal subgroup of ${\mathcal I}_v$ of finite index, and $T_\ell(X_w)=T_\ell(X)^{{\mathcal I}_w}$ is ${\mathcal I}_v$-stable. By Proposition \ref{serretateeqn}i we have $$T_{\ell}(X_{v}) = T_{\ell}(X)^{{\mathcal I}_{v}} \subseteq T_{\ell}(X)^{{\mathcal I}_{w}} = T_{\ell}(X_{w})$$ as ${\mathcal I}_v$-modules. Over an algebraic closure of the residue field, there are exact sequences $$0 \to {\mathbf T}_{w} \to X_{w}^{0} \to {\mathbf B}_{w} \to 0, \qquad 0 \to U_{v} \times {\mathbf T}_{v} \to X_{v}^{0} \to {\mathbf B}_{v} \to 0,$$ where ${\mathbf B}_{w}$ and ${\mathbf B}_{v}$ are abelian varieties and $U_{v}$ is a unipotent group (see \S 2.1 of \cite{SGA}). Base change for N\'eron models induces a homomorphism $\iota:X_{v} \to X_{w}$ such that if $n$ is a positive integer not divisible by $p$, then the restriction of $\iota$ to the $n$-torsion $(X_{v})_n$ is injective (see Lemma 2 of \cite{SerreTate}; see also (3.1.1) of \cite{SGA}). Here, $X_v$ and $X_w$ are viewed as commutative algebraic groups over an algebraic closure of the residue field at $w$. The map $\iota$ induces homomorphisms ${\mathbf T}_{v} \to{\mathbf T}_{w}$ and ${\mathbf B}_{v} \to {\mathbf B}_{w}$ whose kernels are finite group schemes killed by appropriate powers of $p$. The image of ${\mathbf T}_{v}$ (resp., ${\mathbf B}_v$) is a subtorus (resp., abelian subvariety) in ${\mathbf T}_{w}$ (resp., ${\mathbf B}_w$), and we let ${\mathbf T}$ (resp., ${\mathbf B}$) denote the corresponding quotient. Let $a$ and $t$ (respectively, $a_{v}$ and $t_{v}$) denote the abelian and toric ranks of $X_{w}$ (respectively, $X_{v}$). Note that $a$ and $t$ are independent of the valuation $w$ above $v$ at which $X$ has semistable reduction. We have $\mathrm{rk}(T_\ell(X_w))=2a+t$ and $\mathrm{rk}(T_\ell(X_v))=2a_v+t_v$. By the functoriality of N\'eron models, $G_{v,X}$ acts on $X_{w}$ (see \S 4.2 of \cite{SGA}), and therefore acts on ${\mathbf T}_{w}$ and on ${\mathbf B}_w$. One may easily check that this action is trivial on the image of $X_v \to X_w$. It follows that $G_{v,X}$ acts on ${\mathbf T}$ and on ${\mathbf B}$. Fix a polarization $\lambda$ on $X$. Let $W_{\ell}$ (respectively, $S_{\ell}$) denote the orthogonal complement of $V_{\ell}(X_{v})/V_{\ell}({\mathbf T}_{v})$ (respectively, $T_{\ell}(X_{v})/T_{\ell}({\mathbf T}_{v})$) with respect to the pairing $e_{\lambda}$ on $V_{\ell}(X_{w})/V_{\ell}({\mathbf T}_{w})$ (respectively, $T_{\ell}(X_{w})/T_{\ell}({\mathbf T}_{w})$) induced by $E_{\lambda}$. Then $W_{\ell}$ is a $G_{v,X}$-stable ${\mathbf Q}_\ell$-vector space of dimension $2a-2a_{v}$ and $S_{\ell}$ is a $G_{v,X}$-stable ${\mathbf Z}_\ell$-sublattice of rank $2a-2a_{v}$. Recall that $\ell$ is always a prime not equal to $p$. \begin{thm} \label{hd} \begin{enumerate} \item[(i)] The form $e_\lambda: W_{\ell} \times W_{\ell} \to {\mathbf Q}_\ell$ is non-degenerate, alternating, and $G_{v,X}$-invariant. The vector space $W_{\ell}$ and the lattice $S_{\ell}$ do not depend on the choice of polarization $\lambda$. The natural actions of $G_{v,X}$ on ${\mathbf T}$, $W_{\ell}$, and $e_\lambda$ induce an injection $$G_{v,X} \hookrightarrow \mathrm{Aut}({\mathbf T}) \times \mathrm{Aut}(W_{\ell},e_{\lambda}) \cong \mathrm{GL}_{t-t_{v}}({\mathbf Z}) \times \mathrm{Sp}_{2(a-a_{v})}({\mathbf Q}_{\ell})$$ such that the projection onto the first factor is independent of $\ell$, and the characteristic polynomial of the projection onto the second factor has integer coefficients independent of $\ell$. \item[(ii)] If $\ell$ does not divide $\deg(\lambda)\#G_{v,X}$, then $e_\lambda: S_{\ell} \times S_{\ell} \to {\mathbf Z}_\ell$ is perfect and the above injection takes values in $$\mathrm{Aut}({\mathbf T}) \times \mathrm{Aut}(S_{\ell},e_{\lambda}) \cong \mathrm{GL}_{t-t_{v}}({\mathbf Z}) \times \mathrm{Sp}_{2(a-a_{v})}({\mathbf Z}_{\ell}).$$ \item[(iii)] Suppose $\ell$ does not divide $\#G_{v,X}$. Then for every $G_{v,X}$-stable ${\mathbf Z}_{\ell}$-lattice $\mathcal{M}$ in $W_{\ell}$, there exists a perfect ${\mathbf Z}_{\ell}$-valued $G_{v,X}$-invariant alternating pairing $e'$ on $\mathcal{M}$. I.e., there is an injection $$G_{v,X} \hookrightarrow \mathrm{Aut}({\mathbf T}) \times \mathrm{Aut}(\mathcal{M},e') \cong \mathrm{GL}_{t-t_{v}}({\mathbf Z}) \times \mathrm{Sp}_{2(a-a_{v})}({\mathbf Z}_{\ell}).$$ \end{enumerate} \end{thm} \begin{proof} By Proposition \ref{Wunipot}ii, $G_{v,X}$ acts faithfully on $T_{\ell}(X_{w})$. The natural homomorphism $$\varphi_{\ell} : G_{v,X} \to \mathrm{Aut}(T_{\ell}({\mathbf T}_{w})) \times \mathrm{Aut}(T_{\ell}(X_{w})/T_{\ell}({\mathbf T}_{w}))$$ is injective, since its kernel consists of unipotent operators of finite order in characteristic zero. The map $\varphi_{\ell}$ factors through $\mathrm{Aut}({\mathbf T}) \times \mathrm{Aut}(T_{\ell}(X_{w})/T_{\ell}({\mathbf T}_{w}))$, since $G_{v,X}$ acts trivially on ${\mathbf T}_{v}$. By Theorem \ref{serretateeqn}ii and Proposition \ref{orthogeqn}, $$T_{\ell}({\mathbf T}_{v}) \subseteq T_{\ell}(X_{v}) \cap T_{\ell}({\mathbf T}_{w}) = T_{\ell}(X_{v}) \cap T_{\ell}(X_{w})^{\perp} \subseteq T_{\ell}(X_{v}) \cap T_{\ell}(X_{v})^{\perp} = T_{\ell}({\mathbf T}_{v}).$$ Therefore, $$T_{\ell}({\mathbf T}_{v}) = T_{\ell}(X_{v}) \cap T_{\ell}({\mathbf T}_{w}) = T_{\ell}(X_{v}) \cap T_{\ell}(X_{v})^{\perp}.$$ By Proposition \ref{orthogeqn}, $T_{\ell}({\mathbf T}_{w}) = T_{\ell}(X_{w})^{\perp}$. It follows that $e_\lambda$ is non-degenerate on $T_\ell(X_w)/T_\ell({\mathbf T}_w)$ and on $T_\ell(X_v)/T_\ell({\mathbf T}_v)$, and therefore on $V_\ell(X_w)/V_\ell({\mathbf T}_w)$ and on $V_\ell(X_v)/V_\ell({\mathbf T}_v)$. Further, $e_\lambda$ is $G_{v,X}$-invariant. Let $r = \#G_{v,X}$ and let $u = \frac{1}{r}\sum_{g\in G_{v,X}}g \in {\mathbf Q}_\ell[G_{v,X}]$. Then $$u(V_\ell(X_w)) = V_\ell(X_w)^{G_{v,X}} = V_\ell(X_v), \quad u(V_\ell({\mathbf T}_w)) = V_\ell({\mathbf T}_w)^{G_{v,X}} = V_\ell({\mathbf T}_v),$$ $$W_{\ell} = (1-u)(V_\ell(X_w)/V_\ell({\mathbf T}_w)), \quad \text{ and } S_{\ell}=W_{\ell} \cap (T_\ell(X_w)/T_\ell({\mathbf T}_w)).$$ Therefore $W_{\ell}$ and $S_{\ell}$ are independent of the choice of $\lambda$, and we have a $G_{v,X}$-invariant $e_\lambda$-orthogonal splitting $$V_{\ell}(X_{w})/V_{\ell}({\mathbf T}_{w}) \cong V_{\ell}(X_{v})/V_{\ell}({\mathbf T}_{v}) \oplus W_{\ell}.$$ Since $G_{v,X}$ acts trivially on $V_{\ell}(X_{v})/V_{\ell}({\mathbf T}_{v})$, the map $\varphi_{\ell}$ factors through $\mathrm{Aut}({\mathbf T}) \times \mathrm{Aut}(W_{\ell})$. Further, since $e_{\lambda}$ is non-degenerate, alternating, and $G_{v,X}$-invariant, $\varphi_{\ell}$ factors through $\mathrm{Aut}({\mathbf T}) \times \mathrm{Aut}(W_{\ell},e_{\lambda}) \cong \mathrm{GL}_{t-t_{v}}({\mathbf Z}) \times \mathrm{Sp}_{2(a-a_{v})}({\mathbf Q}_{\ell})$. See p.~360 of \cite{SGA} for the integrality and $\ell$-independence. If $r$ is not divisible by $\ell$, then $u \in {\mathbf Z}_\ell[G_{v,X}]$, $$u(T_\ell(X_w)) = T_\ell(X_w)^{G_{v,X}} = T_\ell(X_v), \quad u(T_\ell({\mathbf T}_w)) = T_\ell({\mathbf T}_w)^{G_{v,X}} = T_\ell({\mathbf T}_v),$$ $$S_{\ell} = (1-u)(T_\ell(X_w)/T_\ell({\mathbf T}_w)),$$ and we have a $G_{v,X}$-invariant $e_\lambda$-orthogonal splitting $$T_{\ell}(X_{w})/T_{\ell}({\mathbf T}_{w}) \cong T_{\ell}(X_{v})/T_{\ell}({\mathbf T}_{v}) \oplus S_{\ell}.$$ To derive (iii), apply Proposition \ref{linalg} and Remark \ref{perf}. To derive (ii), suppose $\deg(\lambda)$ is not divisible by $\ell$. Then $E_{\lambda}$ is perfect. Applying Proposition \ref{linalg1} with $\mathcal{M} = T_\ell(X)$, $\mathcal{N} = T_\ell(X_w)$, $R = {\mathbf Z}_\ell$, and $e = E_\lambda$, we deduce that $e_{\lambda}$ is perfect, giving (ii). \end{proof} Next we give a variation of Theorem \ref{hd}, whose proof and statement are of independent interest. Retain the notation from the proof of Theorem \ref{hd}. Since $G_{v,X}$ is finite, there exists a $G_{v,X}$-invariant polarization $\delta$ on the abelian variety ${\mathbf B}$. \begin{thm} \label{hd3} If $\ell$ does not divide $\deg(\delta)$, then the action of $G_{v,X}$ on ${\mathbf T}$ and on ${\mathbf B}$ induces injections $$G_{v,X} \hookrightarrow \mathrm{Aut}({\mathbf T}) \times \mathrm{Aut}({\mathbf B},\delta) \hookrightarrow \mathrm{GL}_{t-t_{v}}({\mathbf Z}) \times \mathrm{Sp}_{2(a-a_{v})}({\mathbf Z}_{\ell})$$ where the projections onto the first factors are independent of $\ell$, and the characteristic polynomials of the projections onto the second factors have integer coefficients independent of $\ell$. In particular, the conclusion holds for all $\ell \ne p$, whenever $a-a_v=0$ or $1$ (e.g., if $X$ or ${\mathbf B}$ is an elliptic curve). \end{thm} \begin{proof} As ${\mathcal I}_{v}$-modules, $$T_{\ell}({\mathbf B}_{w}) \cong T_{\ell}(X_{w})/T_{\ell}({\mathbf T}_{w}), \quad T_{\ell}({\mathbf B}_{v}) \cong T_{\ell}(X_{v})/T_{\ell}({\mathbf T}_{v}),$$ $$T_{\ell}({\mathbf B}) \cong T_{\ell}({\mathbf B}_{w})/T_{\ell}({\mathbf B}_{v}), \quad \text{ and } \quad T_{\ell}({\mathbf T}) \cong T_{\ell}({\mathbf T}_{w})/T_{\ell}({\mathbf T}_{v}),$$ and we have $\mathrm{dim}({\mathbf T}) = t - t_{v}$ and $\mathrm{dim}({\mathbf B}) = a - a_{v}$. By considering the pairing $E_\delta$ on $T_{\ell}({\mathbf B})$ induced by $\delta$, it follows that the image of $G_{v,X}$ in $\mathrm{Aut}(T_{\ell}({\mathbf B}))$ lies in $\mathrm{Aut}(T_{\ell}({\mathbf B}),E_\delta)$. Since $\ell$ does not divide $\deg(\delta)$, then $E_\delta$ is perfect. The actions of $G_{v,X}$ on ${\mathbf T}$ and ${\mathbf B}$ therefore induce a homomorphism $$\eta_{\ell} : G_{v,X} \to \mathrm{Aut}({\mathbf T}) \times \mathrm{Aut}(T_\ell({\mathbf B}),E_\delta) \cong \mathrm{GL}_{t-t_{v}}({\mathbf Z}) \times \mathrm{Sp}_{2(a-a_{v})}({\mathbf Z}_{\ell}).$$ By Proposition \ref{Wunipot}ii, $G_{v,X}$ acts faithfully on $T_{\ell}(X_{w})$. The kernel of $\eta_{\ell}$ consists of elements in $G_{v,X} \subset \mathrm{Aut}(T_{\ell}(X_w))$ which act as the identity on $T_{\ell}({{\mathbf T}}_w)$ and on $T_{\ell}(X_w)/T_{\ell}({{\mathbf T}}_w)$. Therefore, these elements act as unipotent operators on $T_{\ell}(X_w)$. Since they also have finite order, it follows that $\eta_{\ell}$ is injective. As before, see p.~360 of \cite{SGA} for integrality and $\ell$-independence. When $\mathrm{dim}({\mathbf B})=0$ or $1$, we may suppose that $\deg(\delta)=1$. \end{proof} \begin{rem} \label{finrem} If $\ell \ne 2$, then reducing the second factor modulo $\ell$ in Theorem \ref{hd}iii or \ref{hd3} gives $$G_{v,X} \hookrightarrow \mathrm{GL}_{t-t_{v}}({\mathbf Z}) \times \mathrm{Sp}_{2(a-a_{v})}({\mathbf F}_{\ell}).$$ If either $p\#G_{v,X}$ or $p \deg(\delta)$ is odd, then $$G_{v,X} \hookrightarrow \mathrm{GL}_{t-t_{v}}({\mathbf Z}) \times \mathrm{Sp}_{2(a-a_{v})}({\mathbf Z}/4{\mathbf Z}).$$ \end{rem} \begin{rem} It follows from Corollary \ref{hdcor} that the statement in Theorem \ref{hd}iii holds true whenever $p \ne \ell > 1+\max\{t-t_v,2(a-a_v)\}$ or $p \ne \ell > 2d+1$. \end{rem} \begin{rem} Theorems \ref{hd} and \ref{hd3} and Remark \ref{finrem} have led us to the question of when a finite interia group embedded in $\mathrm{Sp}_{2D}({\mathbf Q}_{\ell})$ can also be embedded in $\mathrm{Sp}_{2D}({\mathbf Z}_{\ell})$ or in $\mathrm{Sp}_{2D}({\mathbf F}_{\ell})$ in such a way that the characteristic polynomials are preserved. Suppose $p$ is a prime number, and $G$ is a finite group with a normal Sylow-$p$ subgroup $P$ such that $G/P$ is cyclic. Suppose $\ell$ is a prime number, $\ell \ne p$, $V$ is a $2D$-dimensional ${\mathbf Q}_{\ell}$-vector space, $e: V \times V \to {\mathbf Q}_{\ell}$ is a non-degenerate alternating bilinear form, and $\varphi: G \hookrightarrow \mathrm{Aut}(V,e)$ is an injection. Does there always exist a $G$-stable ${\mathbf Z}_{\ell}$-lattice $S$ in $V$ with a perfect $G$-invariant alternating ${\mathbf Z}_{\ell}$-valued pairing? The answer is no. However, the answer is yes if $\ell > D+1$, and this bound is sharp. Does there always exist an injection $\eta: G \hookrightarrow \mathrm{Sp}_{2D}({\mathbf F}_{\ell})$ such that for all $g \in G$, the characteristic polynomial of $\eta(g)$ is equal to the characteristic polynomial of $\varphi(g)$ mod $\ell$? The answer is no, but is yes if $\ell > 3$, and this bound is sharp. Proofs and counterexamples (including counterexamples with inertia groups of the form $G_{v,X}$ for abelian surfaces $X$) will appear in a later paper. \end{rem} \begin{rem} Note that if $\#G_{v,X}$ is not divisible by $p$, then $X$ acquires semistable reduction over a tamely ramified extension; see \cite{Edixhoven} for a study of N\'eron models in this important case. \end{rem} \section{Bounds on the order of $G_{v,X}$} \label{order} One can use Theorem \ref{hd}, Remark \ref{cyclicrem}, and group theory to obtain more precise information about the finite group $G_{v,X}$. In Corollary \ref{hdcor} below we give one such result. The next two results, along with Theorem \ref{hd}, will be used to prove Corollary \ref{hdcor}. \begin{lem} \label{2bd} If $\ell$ is a prime number, $\ell \equiv 5 \pmod{8}$, $r$ and $m$ are positive integers, and $\mathrm{Sp}_{2m}({\mathbf F}_{\ell})$ has an element of (exact) order $2^{r}$, then $2^{r-1} \le 2m$. \end{lem} \begin{proof} We may assume $r \ge 3$. Let $\zeta$ be a primitive $2^{r}$-th root of unity in ${\bar {\mathbf F}_{\ell}}$. It is easy to check that the condition $\ell \equiv 5 \pmod{8}$ implies that $[{\mathbf F}_{\ell}(\zeta):{\mathbf F}_{\ell}] = 2^{r-2}$ and that $\zeta^{-1}$ is not a conjugate of $\zeta$. It follows that in ${\mathbf F}_{\ell}[x]$ we have $\Phi_{2^{r}} = fg$, where $\Phi_{2^{r}}$ is the $2^{r}$-th cyclotomic polynomial, $f$ and $g$ are irreducible polynomials in ${\mathbf F}_{\ell}[x]$ of degree $2^{r-2}$, and the roots of $g$ are the inverses of the roots of $f$. Let $\gamma$ denote an element of order $2^{r}$ in $\mathrm{Sp}_{2m}({\mathbf F}_{\ell})$. If $\alpha$ is an eigenvalue of $\gamma$, then so is $\alpha^{-1}$. It follows that the characteristic polynomial of $\gamma$ is divisible by $\Phi_{2^{r}}$. Therefore, $2^{r-1} \le 2m$. \end{proof} \begin{prop} \label{qbd} If $q$ is a prime number, $r$ and $m$ are positive integers, and for all prime numbers $\ell$ in a set of density $1$ the group $\mathrm{Sp}_{2m}({\mathbf F}_{\ell})$ contains an element of (exact) order $q^{r}$, then $\varphi(q^{r}) \le 2m$. \end{prop} \begin{proof} If $q = 2$ the result follows from Lemma \ref{2bd}. Suppose $q$ is odd. Then $({\mathbf Z}/q^{r}{\mathbf Z})^{\times}$ is cyclic, and by the Chebotarev density theorem there is a set of primes $\ell$ of positive density such that $\mathrm{Gal}({\mathbf F}_{\ell}(\zeta_{q^{r}})/{\mathbf F}_{\ell}) \cong ({\mathbf Z}/q^{r}{\mathbf Z})^{\times} \cong \mathrm{Gal}({\mathbf Q}(\zeta_{q^{r}})/{\mathbf Q})$. If $q \ne \ell$, then $\mathrm{GL}_{n}({\mathbf F}_{\ell})$ contains an element of order $q^{r}$ if and only if $[{\mathbf F}_{\ell}(\zeta_{q^{r}}):{\mathbf F}_{\ell}] \le n$ (see \cite{Volvacev}). Therefore, $\varphi(q^{r}) \le 2m$. \end{proof} An explicit description of all possible orders of elements of general linear groups over arbitrary fields is given in \cite{Volvacev}. Let $[\,\,\,]$ denote the greatest integer function, let $s(n,q) = \sum_{j=0}^\infty \left[\frac{n}{q^j(q-1)}\right]$, and let $J(n) = \prod q^{s(n,q)}$, where $q$ runs over the prime numbers. Note that the prime divisors of $J(n)$ are the primes $q \le n+1$. For example, $J(0) = 1$, $J(1) = 2$, and $J(2) = 24$. For $n \ge 1$, Theorem 3.2 of \cite{JPAA} shows that $J(n) < (4.462n)^{n}$ if $n$ is even and $J(n) < \sqrt{2}(4.462n)^{n}$ if $n$ is odd. The method of Minkowski and Serre (\cite{Mink} and pp.~119--121 of \cite{Serrearith}; see also Formula 3.1 of \cite{JPAA}) shows that, for all $N \ge 3$, $J(2m)$ is equal to the greatest common divisor of the orders of the groups $\mathrm{Sp}_{2m}({\mathbf F}_{\ell})$, for primes $\ell \ge N$. Further (\cite{SerreHarvard}), $J(n)$ is the least common multiple of the orders of the finite subgroups of $\mathrm{GL}_{n}({\mathbf Q})$ (or equivalently, of $\mathrm{GL}_{n}({\mathbf Z})$). While $J(n)$ is optimal from the point of view of divisibility, there are sharper upper bounds on the orders of finite subgroups of $\mathrm{GL}_{n}({\mathbf Q})$. The determination of the finite subgroups of maximum order for general linear groups over ${\mathbf Q}$ and over cyclotomic fields is given in \cite{Feit}. Let $$r_{p} = s(t-t_{v},p) + s(2(a-a_{v}),p), \qquad M_{v,X} = \max\{t-t_{v},2(a-a_{v})\},$$ and for all primes $q$ such that $p \ne q \le M_{v,X}+1$ let $$r_{q} = 1 + \left[\log_{q}\left(\frac{M_{v,X}}{q-1}\right)\right].$$ If $X$ has semistable reduction at $v$ let $N_{v,X} = 1$, and otherwise let $N_{v,X} = \prod q^{r_{q}}$, where the product runs over all prime numbers $q \le M_{v,X}+1$ (this might include $q=p$). Let $Q_{v,X}$ denote the largest prime divisor of $\#G_{v,X}$ (let $Q_{v,X} = 1$ if ${\mathcal I}_{v} = {{\mathcal I}_{v,X}}$). \begin{cor} \label{hdcor} The order of $G_{v,X}$ divides $N_{v,X}$. In particular, $Q_{v,X} \le M_{v,X} + 1 \le 2d + 1$, and $\#G_{v,X}$ divides $J(t-t_{v})J(2(a-a_{v}))$ and divides $J(2d)$. \end{cor} \begin{proof} By Theorem \ref{hd}ii, $\#G_{v,X}$ divides $J(t-t_{v})J(2(a-a_{v}))$, and therefore $Q_{v,X} \le M_{v,X}+1$. Note that $J(t-t_{v})J(2(a-a_{v}))$ divides $J(2d)$, since $t - t_{v} + 2(a - a_{v}) \le t + 2a = 2d - t \le 2d$. As noted in Remark \ref{cyclicrem}, the prime-to-$p$ part of $G_{v,X}$ is cyclic. Suppose $q$ is a prime divisor of $\#G_{v,X}$, and $q \ne p$. Then $\mathrm{GL}_{n}({\mathbf Q})$ contains an element of order $q^{r}$ if and only if $\varphi(q^{r}) \le n$, i.e., if and only if $r \le 1+\left[\log_{q}\left(\frac{n}{q-1}\right)\right]$. The result now follows from Proposition \ref{qbd}. \end{proof} \section{Applications} \label{applic} Retain the notation of the previous sections. The next result follows immediately from Corollary \ref{hdcor} and Theorem \ref{main}iii,iv. \begin{cor} \label{boundcor} Suppose $L$ is a finite separable extension of $F$, and $w$ is the restriction of ${\bar v}$ to $L$. Suppose $X$ has semistable reduction at $w$ but not at $v$. Then $[{\mathcal I}_{v}:{\mathcal I}_{w}]$ has a prime divisor $q$ such that $q \le Q_{v,X} \le M_{v,X} + 1 \le 2d + 1$. \end{cor} \begin{rem} \label{divby} Suppose $L$ is a finite separable extension of $F$, and $w$ is the restriction of ${\bar v}$ to $L$. Let $k_{w}$ and $k_{v}$ denote the residue fields and let $e(w/v) = [w(L^\times):v(F^\times)]$ ($=$ the ramification degree). Then $[{\mathcal I}_{v}:{\mathcal I}_{w}] = e(w/v)[k_{w}:k_{v}]_{i}$, where the subscript $i$ denotes the inseparable degree (see Proposition 21 on p.~32 of \cite {Corps} for the case where $L/F$ is Galois. In the non-Galois case, take a Galois extension $L'$ of $F$ which contains $L$, and apply the result to $L'/L$ and $L'/F$, to obtain the result for $L/F$). Taking completions, then $[L_{w}:F_{v}] = e(w/v)[k_{w}:k_{v}] = [{\mathcal I}_{v}:{\mathcal I}_{w}][k_{w}:k_{v}]_{s}$, where the subscript $s$ denotes the separable degree. Therefore, $[{\mathcal I}_{v}:{\mathcal I}_{w}]$ divides $[L_{w}:F_{v}]$. \end{rem} \begin{cor} \label{prpow} Suppose $L$ is a finite separable extension of $F$. Suppose in addition that either $F$ is complete with respect to $v$, or $L/F$ is Galois. Suppose $X$ has semistable reduction at the restriction $w$ of ${\bar v}$ to $L$, but does not have semistable reduction at $v$. Then $[L:F]$ has a prime divisor $q$ such that $q \le Q_{v,X}$. In particular, if $[L:F]$ is a power of a prime $q$, then $q \le Q_{v,X}$. \end{cor} \begin{proof} Under our assumptions on $L/F$, its degree is divisible by $[I_{v}:{\mathcal I}_{w}]$. The result now follows from Corollary \ref{boundcor}. \end{proof} Recall that there exists a finite Galois extension $L$ of $F$ such that $X$ has semistable reduction at the extensions of ${v}$ to $L$ (see Proposition 3.6 of \cite{SGA}). \begin{cor} \label{Gal} Let $r = \#G_{v,X}$ and let $\zeta_{r}$ denote a primitive $r$-th root of unity. Suppose that $r$ is not divisible by $p$. Then there is a cyclic degree $r$ extension $L$ of $F(\zeta_{r})$ such that $X$ acquires semistable reduction over every extension of $v$ to $L$. If either $F = F(\zeta_r)$ or $p > Q_{v,X}$, then there exists a finite Galois extension $L$ of $F$ of degree prime to $p$ such that $X$ acquires semistable reduction over every extension of $v$ to $L$. \end{cor} \begin{proof} Let $L$ be the field obtained by adjoining an $r$-th root of a uniformizing parameter to $F(\zeta_{r})$. Then $L/F$ is Galois, $F(\zeta_{r})/F$ is unramified above $v$, $L/F(\zeta_{r})$ is totally ramified, and $\mathrm{Gal}(L/F(\zeta_{r})) \cong G_{v,X} \cong {\mathbf Z}/r{\mathbf Z}$. Let $w$ be the restriction of ${\bar v}$ to $L$. By the construction of $L$, we have ${\mathcal I}_{w} \cong {{\mathcal I}_{v,X}}$. By Theorem \ref{main}iii, $X$ has semistable reduction at $w$. Since $L/F$ is Galois, $X$ has semistable reduction at every extension of $v$ to $L$ by Theorem \ref{galcrit}. If $p > Q_{v,X}$, then $p$ does not divide $\varphi(r)$ by Corollary \ref{hdcor}, and therefore $p$ does not divide $[F(\zeta_r):F]$. \end{proof} By Corollary \ref{hdcor}, $Q_{v,X}$ can be replaced by $2d+1$ (or by $M_{v,X} + 1$) in Corollaries \ref{prpow} and \ref{Gal}.

9706

alg-geom/9706009

en

https://arxiv.org/abs/alg-geom/9706009

alg-geom/9706009

Francisco Jose Plaza Martin

Francisco J. Plaza Mart\'in

Prym varieties and the infinite Grassmannian

21 pages, Latex, To apper in Internation Journal of Mathematics

In this paper we study Prym varieties and their moduli space using the well known techniques of the infinite Grassmannian. There are three main results of this paper: a new definition of the BKP hierarchy over an arbitrary base field (that generalizes the classical one over the complex numbers; a characterization of Prym varieties in terms of dynamical systems, and explicit equations for the moduli space of (certain) Prym varieties. For all of these problems the language of the infinite Grassmannian, in its algebro-geometric version, allows us to deal with these problems from the same point of view.

alg-geom

\section{Introduction} The aim of this paper is to generalize some results concerning the BKP hierarchy and geometric characterizations of Jacobians and Pryms proved in \cite{LiMu,Mul1,Sh,Sh2} and to study the moduli space of (certain) Prym varieties following similar ideas to those of \cite{MP}. I should remark that the techniques employed here are those of algebraic geometry, and most statements are therefore valid over an arbitrary base field. The organization of the paper is as follows: In \S2 some basic definitions, results and tools needed for the next sections are introduced. Some of them are known (e.g. infinite Grassmannian, Krichever functor, $\Gamma$ group, etc) and have their origin in the study of the moduli space of Riemann surfaces, Jacobian varieties and conformal field theory (\cite{BS,DKJM,KSU,Mul1,N,PS,SW} . To develop analogues for the theory of Prym varieties, we will define certain subschemes of the infinite Grassmannian and will avoid the introduction of the formalism of the n-component KP hierarchy. However, the analogue of the determinant bundle is a non-trivial problem. An important result ({\ref{thm:pfaffian}}) of this section is the existence of a square root of the determinant bundle (over a certain subscheme), which will be called Pfaffian, as well as an explicit construction of global sections of this Pfaffian bundle, improving previous results of \cite{B,PS}. Section \S3 contains a new definition of the BKP hierarchy as the defining equations of a suitable subscheme of the infinite Grassmannian ({\ref{defn:BKP}}); namely, the space of maximal totally isotropic subspaces. This new definition is quite natural since we should recall that the KP hierarchy is in fact equivalent to the Pl\"ucker equations, which are the defining equations of the infinite Grassmannian (\cite{SS}). The definition is therefore valid over an arbitrary base field, but is shown to be equivalent to the classical one when the base field is $\mathbb C$. We wish to point out that this new definition of the BKP hierarchy, together with the previously defined Pfaffian line bundle, will give a justification for some results relating solutions of the KP and BKP hierarchies (\cite{DKJM,DKJM2}) as well as for techniques involving pfaffians when computing $\tau$-functions for the BKP hierarchy (\cite{H,O}). Note that this (algebraic) approach allows us to introduce the BKP hierarchy with no mention of pseduodifferential operators. The standard way to relate the study of Jacobians to that of infinite Grassmannian is the Krichever functor (\cite{Mul2}). Two remarkable papers dealing with the case of Pryms are \cite{LiMu,Sh2}. The characterization given in \cite{LiMu} for a point of (a certain quotient of) the infinite Grassmannian to be associated to a Prym (via the Krichever functor) is that its orbit (under a suitable group) is finite dimensional (see Theorem 5.14 of \cite{LiMu} for the precise statement). Moreover, that action is interpreted as a dynamical system on that space. The idea of Shiota (\cite{Sh,Sh2}), related to the above one, is to characterize Jacobian varieties as compact solutions of the KP hierarchy through the (analytic) study of infinitesimal deformations of sheaves. The characterizations of Jacobians ({\ref{thm:shiota}}) and Pryms ({\ref{thm:shiota-prym}}) given in \S4 profit from both approaches. Recalling that the action of the formal Jacobian (formal group $\Gamma$, \S{\ref{subsec:gamma}}) on the Grassmannian is algebraic and studying the structure of the orbits ({\ref{lem:orbit}}), one can give an algebraic statement generalizing Theorem 6 of \cite{Sh} and Theorem 5.14 of \cite{LiMu}. But in our statement no quotient space is needed. Bearing this in mind, there is no problem in extending this result to the case of Pryms ({\ref{thm:shiota-prym}}). Nevertheless, we shall use the language of dynamical systems to express both characterizations. In the last section (\S5), and following the spirit of \cite{MP}, explicit equations for the moduli space of Pryms are given; first as Bilinear Identities ({\ref{thm:bil-ident}}), then as partial differential equations ({\ref{thm:pde-p0}}) when $char(k)=0$. This set of equations should not be confused with the BKP hierarchy. Here we make two considerations; firstly, that a formal trivilization is attached to each datum (e.g. a curve, a bundle, \dots~) since we wish to work in a uniform frame (e.g. $\gr(k((z)))$, $\Gamma$, \dots~) and, secondly, that not all Pryms are considered; only those coming from an integral curve together with an involution with at least one fixed smooth point are taken into account since, for technical reasons, the involution should correspond to an automorphism of $k((z))$ preserving $k[[z]]$. We address the reader to \cite{AMP} for a detailed discussion of the infinite Grassmannian and to \cite{Mum} for the basic facts on Prym varieties needed. \section{Preliminaries on the Infinite Grassmannian} \subsection{Basic Facts} Recall (\cite{AMP, BS}) that given a pair $(V,V_+)$ consisting of a $k$-vector space\footnote{For simplicity's sake we will assume that $k$ is an algebraically closed field} and a subspace of it, there exists a scheme $\gr(V,V_+)$ over $\spk$ whose rational points are: $$\left\{\begin{gathered} \text{ subspaces $L\subseteq \hat V$, such that $L\cap\hat V_+$ }\\ \text{ and ${\hat V}/{L+\hat V_+}$ are of finite dimension } \end{gathered}\right\}$$ where $\,\hat{}\,$ denotes the completion with respect to the topology given by the subspaces that are commensurable with $V_+$. The points of $\gr(V,V_+)$ will be called discrete subspaces. The essential fact for its existence is that there is a covering by open subfunctors $F_A$ (where $A\sim V_+$ are commensurable) representing the functor $\fu\hom(L_A,\hat A)$, where $L_A$ is a rational point of $\gr(V,V_+)$ such that $L_A\oplus \hat A\simeq \hat V$ (\cite{BS}). (See \cite{AMP} for the definition of the functor of points of $\gr(V,V_+)$). Let us denote this infinite Grassmannian simply by $\grv$, and let $\L$ be the universal discrete submodule of $\hat V_{\grv}$. We will assume that $V$ is complete with respect to the $V_+$-topology. From this construction it is easily deduced that $\grv$ is locally integral and separated. The connected components of $\grv$ are given by the Euler characteristic (index) of the complex: \beq \L\to (\hat V/\hat V_+)_{\grv} \label{eq:complex}\end{equation} and that of index $n$ will be denoted by $\gr^n(V,V_+)$. It is also shown, that $\grv$ carries a line bundle, $\det_V$, given by the determinant of the complex {\ref{eq:complex}} (see \cite{KM} for a general theory of determinants) whose stalk at a rational point $L$ is: $$\overset{max}\wedge (L\cap\hat V_+)\otimes\overset{max}\wedge ( {\hat V}/{(L+\hat V_+)})^\ast$$ This line bundle has no global sections but its dual does. Moreover, for each $A\sim V_+$ one can define a global section, $\Omega_A$ of $\detd_V$ such that it vanishes outside $F_A$. For every subspace $\Omega$ of $H^0(\gr^0(V),\detd_V)$ one has a sheaf homomorphism: $$\Omega\otimes_k\o_{\grv}\to\detd_V$$ If it is surjective ($\Omega$ is ``big enough''), it induces a scheme homomorphism: $${\frak p}_V: \gr^0(V) \to \check{\mathbb P}\Omega^* \,\overset{\text{\tiny{def}}}{=}\,\operatorname{Proj}S^\bullet\Omega$$ which is known as the Pl\"ucker morphism. Although $k((z))$ is not complete with respect to the $k[[z]]$-topology, it is easy to see that the functor: \beq S\rightsquigarrow\{ L\in\gr(k((z)),k[[z]])(S)\;\vert\; L\subseteq \o_S((z))\} \label{eq:grkz}\end{equation} is locally closed, and it is therefore representable by a closed subscheme, which will be denoted by $\gr(k((z)))$ again, when no confussion arises (\cite{P}). It can now be shown that $H^0(\gr^0(k((z))),\detd_V)$ has a dense subspace, $\Omega$, consisting of sections of the kind $\Omega_A$ and labelled by Young diagrams (see \cite{P,SS}), and that the Pl\"ucker morphism is a closed immersion. The general results given along this section are valid for subscheme {\ref{eq:grkz}}. \subsection{Important subschemes of the Infinite Grassmannian} Let us now introduce two closed subschemes of $\grv$ that will be useful in the next sections. Assume now that an automorphism $\sigma$ of $V$ (as $k$-vector space) is given, and that $\sigma(\hat V_+)=\hat V_+$; it then induces an automorphism of the scheme $\grv$. Since $\grv$ is separated, one has that: $$\gr_\sigma(V)=\left\{ L\in\grv\,\vert\, \sigma(L)=L\right\}$$ is a closed subscheme of $\grv$. For the second, recall the isomorphism $\gr(V,V_+)\iso\gr(V^*,V_+^\circ)$ given by incidence (see \cite{P}); that is, it sends a discrete subspace $L$ to $L^\circ$, the space of continuous linear forms that vanish on $L$. Let $p: V\iso V^*$ be the isomorphism of $V$ with its dual vector space $V^*$ induced by an irreducible hemisymmetric metric on $V$. Assume further that $p(\hat V_+)=\hat V_+^\circ$. It then induces an isomorphism $\gr(V^*,V_+^\circ)\iso\gr(V,V_+)$, which composed with the one given by incidence, gives rise to the following automorphism of $\grv$: $$\aligned R:\grv &\longrightarrow \grv \\ L &\longmapsto L^{\perp} \endaligned$$ (where $\perp$ denotes the orthogonal with respect to the metric). Straightforward calculation shows that $R^*\det_V\simeq \det_V$, and that the index of a point $L\in\grv$ is exactly the opposite of the index of $R(L)=L^\perp$. Given $\sigma\in\aut_{k-alg}k((z))$ such that $\sigma^2=Id$, consider the following irreducible hemisymmetric metric: \beq \begin{aligned} V\times V&\to k \\ (f,g)&\mapsto\res_{z=0} f(z)\cdot (\sigma^*g(z)) dz\end{aligned} \label{eq:metric}\end{equation} It is now clear that there exists a closed subscheme $\gr^I_\sigma(V)$ of $\gr^0(V)$ such that: {\small \beq\gr^I_\sigma(V)^\bullet(k)=\left\{ L\in\grv\,\vert\, \begin{gathered}L\text{ is maximal totally isotropic }\\ \text{with respect to the metric {\ref{eq:metric}}}\end{gathered} \right\}\label{defn:gri}\end{equation}} From now on, a subspace of $V$ will be called m.t.i. when it is maximal totally isotropic (compare with \S2.2 of \cite{Sh2}). \begin{rem} Whenever we use a hemisymmetric metric or consider the automorphism of $k((z))$ induced by $z\mapsto -z$, is assumed $char(k) \neq 2$. \end{rem} \begin{exam}\label{exam:metric} This is a fundamental example because it will be the situation when studying Prym varieties in terms of $\gr^I_\sigma(V)$. Let $V=k((z))$, $V_+=k[[z]]$ and $\sigma_0$ that given by $z\mapsto -z$. The metric is now: $<f(z),g(z)>=\res_{z=0} f(z)g(-z)dz$. It is then easy to prove that: {\small $$\gr_0(k((z)),k[[z]])\simeq\gr(k((z^2)),k[[z^2]]) \underset\spk\times\gr(z\cdot k((z^2)),z \cdot k[[z^2]])$$} (we simply write $0$ instead of $\sigma_0$). It is worth comparing the 2 component BKP hierarchy (as given in \S3 of \cite{Sh2}) with $\gr_0(k((z)),k[[z]])$. \end{exam} \subsection{Pffafian Line Bundle} \begin{thm}\label{thm:pfaffian} There exists a line bundle, $\Pf$ (called Pfaffian), over $\gr^I_\sigma(V)$ such that: $$\Pf^{\otimes 2}\simeq \detd_V\vert_{\gr^I_\sigma(V)}$$ \end{thm} \begin{pf} Note that $\detd_V$ is isomorphic to the determinant of the dual of the complex over $\gr^I_\sigma(V)$: $\hat V_+\to \hat V/\L$ ($\L$ being the universal m.t.i. submodule). Let us compute its cohomology. Let $p:V\to V^*$ be the isomorphism induced by the metric {\ref{eq:metric}}. Then, in the commutative diagram: $$\CD @. \L @>>> \hat V/\hat V_+ @>>> \frac{\hat V}{\hat V_++\L} @>>> 0 \\ @. @V{p}VV @V{p}VV @VVV \\ 0 \to (\frac{\hat V}{\hat V_++\L})^* @>>> (\hat V/\L)^* @>>> \hat V_+^* @>>> \frac{\hat V_+^*}{(\hat V/\L)^*}@>>> 0 \endCD$$ the two middle vertical arrows are isomorphisms, since $\L$ and $\hat V_+$ are m.t.i.~; and thus the right one is an isomorphism. One now has: $$\detd_V\iso {\left(\bigwedge \hat V/(\hat V_++\L)\right)^*}^{\otimes 2}$$ By the local structure of $\gr^I_\sigma(V)$ it is not difficult to show that $\hat V/(\hat V_++\L$ is locally free of finite type, and by \cite{KM} it makes sense to define: $$\Pf\,\overset{\text{\tiny{def}}}{=}\,(\bigwedge \hat V/(\hat V_++\L))^*$$ \end{pf} An analytic construction of the Pfaffian line bundle can be found in \cite{B}, but nevertheless we prefer to continue with the algebraic machinery. Another construction is given in \cite{PS}. We are now interested in building sections of this line bundle. First, observe that the covering $\{F_A\}$ ($A\sim V_+$) of $\grv$ induces another one of $\gr^I_\sigma(V)$ by open subsets of the form $\{\bar F_A=F_A\cap\gr^I_\sigma(V)\}$ where $F_A\subset\grv$ ($A\sim V_+$) and $A$ is m.t.i.~. The second ingredient is the following: \begin{lem} Let $A$ and $B$ be two m.t.i. subspaces of $V$. One then has a canonical isomorphism: $$B/(B\cap A)\iso (A/(B\cap A))^*$$ induced by the metric. \end{lem} \begin{pf} Note that the morphism $p:V\to V^*$ gives an isomorphism from $A$ (resp. $B$) to $A^\circ$ (resp. $B^\circ$). Therefore, from the diagram: $$\CD 0 @>>> A\cap B @>>> A @>>> A/(A\cap B) @>>> 0 \\ @. @V{p}VV @V{p}VV @VVV \\ 0 @>>> B^\circ @>>> V^* @>>> B^* @>>> 0 \endCD$$ one has an injection $A/(A\cap B) \to B^*$. The linear forms belonging to the image vanish on $A\cap B$, and hence: $$A/(A\cap B) \to (A\cap B)^\circ\cap B^*\simeq \left(B/(A\cap B)\right)^*$$ Analogously, one obtains another injection $B/(A\cap B) \to \left(A/(A\cap B)\right)^*$, and they are transposed of each other, and are therefore both isomorphisms. \end{pf} \begin{thm} To each $A\sim V_+$ m.t.i. one associates a section $\bar\Omega_A$ of $\Pf$ that vanishes outside $\bar F_A$, and hence: $$\Omega_A\vert_{\gr^I_\sigma}=\lambda\cdot\bar\Omega_A^2\qquad \lambda\in k^*$$ \end{thm} \begin{pf} Observe now that by similar arguments to those of \cite{AMP} one can construct sections of $\Pf$ for each $A\sim V_+$ m.t.i.~. The only remarkable aspect is that this is possible because $B/(B\cap A)$ and $(A/(B\cap A))^*$ are canonically isomorphic (for $A,B$ m.t.i. and commensurable with $V_+$). From this last property, and from the fact $\Pf^{\otimes 2}\simeq \detd_V$, the claim follows. \end{pf} \subsection{Krichever Functor} \begin{defn} Define the moduli functor of pointed curves $\cur$ over the category of $k$-schemes by the sheafication of: $$S\rightsquigarrow\{\text{ families $(C,D,z)$ over $S$ }\}/\text{equivalence}$$ where these families satisfy: \begin{enumerate} \item $\pi:C\to S$ is a proper flat morphism, whose geometric fibres are integral curves, \item $s:S\to C$ is a section of $\pi$, such that when considered as a Cartier Divisor $D$ over $C$ it is smooth and of relative degree 1, and flat over $S$. (We understand that $D\subset C$ is smooth over $S$, iff for every closed point $x\in D$ there exists an open neighborhood $U$ of $x$ in $C$ such that the morphism $U\to S$ is smooth). \item $z$ is a formal trivialization of $C$ along $D$; that is, a family of epimorphisms of rings: $$\o_C\longrightarrow s_*\left({\o_S[t]}/{t^m\,\o_S[t]}\right)\qquad m\in{\mathbb N}$$ compatible with respect to the canonical projections: $${\o_S[t]}/{t^m\,\o_S[t]}\to {\o_S[t]}/{t^{m'}\,\o_S[t]}\qquad m\ge m'$$ and such that that corresponding to $m=1$ equals $s$. \end{enumerate} and the families $(C,D,z)$ and $(C',D',z')$ are said to be equivalent, if there exists an isomorphism $C\to C'$ (over $S$) such that the first family goes to the second under the naturally induced morphisms. \end{defn} By \cite{MP} it is known that the so called ``Krichever map'' is in fact the following morphism of functors: $$\aligned K:\cur &\longrightarrow \gr(k((z)),k[[z]])\\ (C,D,z)&\longmapsto \limil{n}\pi_*\o_C(n) \endaligned$$ It is also known that $K$ is an immersion and that there exists a locally closed subscheme of $\grv$ representing $\cur$ (which we will denote again by $\cur$). Let us recall another construction very similar to the above one. Set $m=(C,p,z)\in\cur(\spk)$, and consider the functor: $$S\rightsquigarrow \tilde\pic(C,p)= \left\{(L,\phi)\,\vert\,\gathered L\in\pic(C)^{\bullet}(S) \text{ and $\phi$ is a} \\ \text{formal trivialization of $L$ around $p$} \endgathered\right\}$$ Define the morphism: $$\aligned K_m:\tilde\pic(C,p) &\longrightarrow \gr(k((z)),k[[z]])\\ (L,\phi)&\longmapsto \limil{n}\pi_* L(n) \endaligned$$ which is also usually called ``Krichever functor''. (For a more detailed study of $\tilde\pic(C,p)$ and $K_m$ see \cite{Al}). Note that while $K$ is very well adapted to study of the moduli space of curves (\cite{MP,N}), the other one, $K_m$, is good for the study of Jacobian varieties and their subvarieties (\cite{Sh}). See also \cite{Mul2}. \subsection{The Formal Group $\Gamma$}\label{subsec:gamma} Let us now recall some basic facts about the formal group $\Gamma$ (for a complete study and definitions see \cite{AMP}). $\Gamma$ is defined as the formal group scheme $\Gamma_-\times{\Bbb G}_m\times \Gamma_+$ over $\spk$, where $\Gamma_-$ is the formal scheme representing the functor on groups: $$S\rightsquigarrow \Gamma_-(S)=\left\{\gathered \text{ series }\,a_n\,z^{-n}+\dots+a_1\,z^{-1}+1\\ \text{ where }a_i\in H^0(S,\o_S)\text{ are }\\ \text{ nilpotents and $n$ is arbitrary } \endgathered\right\}$$ ${\Bbb G}_m$ is the multiplicative group, and the scheme $\Gamma_+$ represents: $$S\rightsquigarrow \Gamma_+(S)=\left\{\gathered \text{ series }1+a_1\,z+a_2\,z^2+\dots\\ \text{ where }a_i\in H^0(S,\o_S) \endgathered\right\}$$ The group laws of $\Gamma_-$ and $\Gamma_+$ are those induced by the multiplication of series. Note also that there exists a natural inclusion of $\Gamma$ in the identity connected component of: $$S\rightsquigarrow H^0(S,\o_S)((z))^\ast\,\overset{\text{\tiny\rm def}}{=}\,H^0(S,\o_S)[[z]][z^{-1}]^*$$ which is an isomorphism when $char(k)=0$. Further, $\Gamma_-$ is the inductive limit of the schemes: $$S\rightsquigarrow \Gamma^n_-(S)=\left\{\gathered \text{ series }\,a_n\,z^{-n}+\dots+a_1\,z^{-1}+1\\ \text{ where }a_i\in H^0(S,\o_S)\text{ and the $n^{\text{th}}$}\\ \text{ power of the ideal $(a_1,\dots,a_n)$ is zero } \endgathered\right\}$$ in the category of formal schemes. Observe now that there exist two actions of $g(z)\in\Gamma$ in $V$; namely, the one given by homotheties: $$\aligned H_g:V& \to V\\h(z)&\mapsto g(z)\cdot h(z)\endaligned$$ and the one defined by the automorphism of $k$-algebras: $$\aligned U_g:V& \to V\\z&\mapsto z\cdot g(z)\endaligned$$ \begin{rem}\label{rem:sigma} It is known that $U:g\mapsto U_g$ establishes a bijection $k[[z]]^*=\Gamma_+(k)\iso\operatorname{Aut}_{\text{$k$-alg}}(k((z)))$. Recall from \cite{Bo} (chapter III, \S4.4) that given a $k$-algebra automorphism $\sigma$ of $k((z))$, there exists a unique $g(z)\in\Gamma_+(k)$ such that $U_g\circ \sigma=\sigma_0$ (where $\sigma_0$ is the $k$-algebra automorphism of $k((z))$ given by $z\mapsto -z$; that is, it is possible to ``normalize'' $\sigma$ such that $\sigma^*(g(z))=g(-z)$. \end{rem} \begin{rem} Now, $g\in\Gamma_+$ acts on $\tilde\pic(C,p)$ sending $(L,\phi)$ to $(L,H_g\circ \phi)$. And hence the projection morphism: $$\tilde\pic(C,p) @>p_1>> \pic(C)$$ may be interpreted as a principal bundle of group $\Gamma_+$. Now comparing the zero locus of sections of $\detd_V$ and $\o(\Theta)$, one deduces: $$\detd_V\vert_{\tilde\pic(C,p)}\,\iso\, p_1^*\o_{\pic(C)}(\Theta)$$ which allows one to write the $\tau$-function of the point $U=K(C,p,z)$ (restricted to $\tilde\pic(C,p)$ via $K$) in terms of the theta function of the Jacobian of $C$. For explicit formulas relating $\tau$-functions and theta functions of Riemann surfaces, see \cite{Kr,Sh} (see also \cite{Sh2} for the case of Pryms). \end{rem} \begin{rem} For other constructions and properties of the group $\Gamma$ see \cite{AMP,C,KSU,PS,SW}. \end{rem} \section{Formal Prym variety and BKP hierarchy} Observe now that $\Gamma$, and hence all the above-mentioned subgroups, acts on $\grv$ by homotheties and that on the set of rational points ${\Bbb G}_m$ acts trivially, and $\Gamma_+$ freely. Recall from \cite{AMP,KSU} that $\Gamma$ behaves like the Jacobian of the formal curve. Our goal is then to achieve an anologous result for the case of Pryms. This arises from the answer of the following question: which is the maximal subgroup of $\Gamma$ acting on $\gr^I_\sigma(V)$?. \begin{thm} The maximal subgroup of $\Gamma$ acting on $\gr^I_\sigma(V)$ is: $$\Pi_\sigma\,=\,\{g(z)\in\Gamma\,\vert\, g(z)\cdot\sigma^*g(z)=1\,\}$$ which is a subscheme of $\Gamma$. \end{thm} \begin{pf} Observe that the homothety by $g(z)\in\Gamma$ restricts to a automorphism of $\gr^I_\sigma(V)$ if and only if: $$g(z)\cdot U\in\gr^I_\sigma(V)\qquad\text{for all }U\in\gr^I_\sigma(V)$$ or, what amounts to the same: $$ g(z)\cdot U \,=\, \left(g(z)\cdot U\right)^{\perp}\qquad \text{for all } U\in\gr^I_\sigma(V)$$ Recalling the definition of the metric: $(f,g)\mapsto \res_{z=0} f(z)\cdot\sigma^*g(z)dz$, one has: $$\left(g(z)\cdot U\right)^{\perp}\,=\, (\sigma^*g(z))^{-1}\cdot U^{\perp}$$ Note that $U=U^{\perp}$, since $U\in\gr^I_\sigma(V)$. And one concludes that $g(z)\cdot\sigma^*g(z)\cdot U=U$ for all $U\in\gr^I_\sigma(V)$ and hence $g(z)\cdot\sigma^*g(z)$ must be equal to 1. \end{pf} \begin{defn} The formal Prym variety is the formal group scheme: $$\Pi^\sigma_-\,\overset{\text{\tiny{def}}}{=}\,\Pi_\sigma\cap\Gamma_-$$ \end{defn} It is therefore natural to consider $\Pi^\sigma_-$ instead of $\Gamma_-$ in the study of $\gr^I_\sigma(V)$, and hence in the study of Pryms. Recall that the action of $\Gamma_-$ on $\grv$ is essential in the definition of the $\tau$-function and the Baker-Akhiezer function of a point $U\in\grv$. But for $\gr^I_\sigma(V)$ we must restrict this action to $\Pi^\sigma_-$. Denote by $\mu^\sigma_U$ the restriction of $\mu_U$ to $\Pi^\sigma_-$; that is: $$\begin{array}{ccccc} \mu_U:\Gamma_-\times\{U\} & \hookrightarrow & \Gamma_-\times\grv & \to & \grv \\ \phantom{xxx}\cup & & \cup\; & & \cup \\ \mu^\sigma_U:\Pi^\sigma_-\times\{U\} & \hookrightarrow & \Pi^\sigma_-\times\gr^I_\sigma(V) & \to & \gr^I_\sigma(V) \end{array}$$ These actions are the cornerstone of \S4, where they will be studied at the tangent space level. \begin{defn} The $\bar\tau$-function of a point $U\in\gr^I_\sigma(V)$ is the section $({\mu^\sigma_U})^*\bar\Omega_+$ of $({\mu^\sigma_U})^*\Pf$. \end{defn} \begin{thm}\label{thm:taus} $$\tau_U\vert_{\Pi^\sigma_-}\,=\,\lambda\cdot \bar\tau_U^2\qquad \lambda\in k^*$$ \end{thm} It is known that the KP hierarchy is a system of partial differential equations for the $\tau$-function of a point $U\in\check{\mathbb P}\Omega$. These are in fact equivalent to the Pl\"ucker equations for the coordinates of $U$. It is thus quite natural to give the following: \begin{defn}\label{defn:BKP}\hfill \begin{itemize} \item {\rm $char(k)$ arbitrary:} The BKP hierarchy is the set of algebraic equations defining $\gr^I_\sigma(V)$ inside $\check{\mathbb P}\Omega$; in particular it gives, \item {\rm $char(k)=0$:} The BKP hierarchy is the system of partial differential equations that characterizes when a function is a $\bar\tau$-function of a point of $\gr^I_\sigma(V)$. \end{itemize} \end{defn} The relationship between the BKP hierarchy and Pryms will be clear in {\ref{thm:shiota-prym}}. \begin{rem} Let us relate all the above claims to the classical results when $char(k)=0$. Classically, the BKP hierarchy is introduced as the system of equations obtained from the KP system making $t_i=0$ for all even $i$. Take the formal trivilization around $p$ equal to $\sigma_0$; that is: $\sigma_0(g(z))=g(-z)$ for all $g(z)\in\Gamma$. Then, using the isomorphism of $\Gamma$ with an additive group given by the exponential map, one has that the set of $A$-valued points of $\Pi^0_-$ is (we write only $0$ instead of $\sigma_0$): $$\Pi^0_-(\sp(A))\,=\,\left\{\gathered \text{ series }\exp\big(\sum^{n} \Sb i=1 \\ \text{$i$ odd} \endSb a_iz^{-i}\big) \text{ where $a_i\in A$}\\ \text{ is nilpotent and $n>0$ arbitrary}\endgathered\right\}$$ Some well known results, such as formula 1.9.8 of \cite{DKJM}, are now a consequence of the relationship between the $\tau$-function of $U\in\gr^I_0(V)$ as a point of $\grv$ and the $\bar\tau$-function as a point of $\gr^I_0(V)$ given in Theorem {\ref{thm:taus}}. These connections of KP and BKP, of $\grv$ and $\gr^I_0(V)$, of $\detd_V$ and $\Pf$, and of $\tau$ and $\bar\tau$ (given above) justify the expression given in \cite{DKJM2} of a tau-function for the BKP in terms of theta functions of a Prym variety\footnote{It is known that the restriction of a theta function of a Jacobian variety is the square of a theta function of the Prym when the involution has two fixed points (\cite{Mum}).}, as well as the methods of \cite{H} and \cite{O} based on pfaffians of matrices in order to construct solutions for the BKP. \end{rem} \section{Geometric Characterizations} Geometric characterizations of Jacobians and Pryms offered in several papers (\cite{Mul1,Sh,LiMu,Sh2}) are based on the study of an action of a group on a space at the tangent space level and are therefore suitable for being expressed in terms of dynamical systems. Roughly, the group is a subgroup of the linear group and the space is the Grassmannian (or a quotient of it), and the way to relate Jacobians and Pryms with the Grassmannian is through the Krichever functor. Since we aim to give a scheme-theoretic generalization of Theorem 6 of \cite{Sh}, \S2.5 of \cite{Sh2} and Theorem 5.14 of \cite{LiMu}, we should use only algebraic methods. In Shiota's paper, the action of the group is given by analytic techniques, while the characterization of Pryms given by Li and Mulase involves quotients of the Grassmannian that do not need to be algebraic. Our approach therefore needs to use the notion that the action of $\Gamma$ in $\grv$ ($\Pi_-^\sigma$ on $\gr_\sigma^I(V)$) is algebraic and that there is no need to use quotient spaces because of Lemma {\ref{lem:orbit}}. Essentially, this Lemma implies that the orbit of a point of $\grv/\Gamma$ under $\Gamma$ coincides with that of a preimage in $\grv$ under $\Gamma_-$. Although the methods are algebraic, it seems quite natural to use the language of dynamical systems when working at the tangent space level. \subsection{Dynamical systems and the Grassmannian} Observe that the action: $$\aligned \Gamma\times\grv& @>\mu>> \grv \\ (g,U)&\mapsto g\cdot U\endaligned$$ canonically induces a system of partial differential equations (p.d.e.) on $\grv$. Taking $\sp(k[\epsilon]/(\epsilon^2))$-valued points, and using the canonical identification: $$T\grv\,\iso\,\hom(\L,\hat V/\L)$$ we obtain a morphism of functors on groups: $$\aligned T_{\{1\}}\Gamma & @>d\mu>> \hom(\L,\hat V/\L) \\ 1+\epsilon g &\longmapsto (\L\hookrightarrow\hat V @>\cdot g>> \hat V\to\hat V/\L) \endaligned$$ Denote by $\mu_-$ ($d\mu_-$) the restriction $\mu\vert_{\Gamma_-}$ ($d\mu\vert_{\Gamma_-}$ respectively). Moreover, note that the map $g$ to $1+\epsilon g$ gives an isomorphism of functors $\hat V\simeq T_{\{1\}}\Gamma$, where $\hat V(S)=\lim (V/A\underset{k}\otimes\o_S)$. Also, the kernel of $d\mu_-$ at a point $U$ is the maximal sub-$k$-algebra of $V$ acting (by homotheties) on $U$. \begin{defn} Given a subbundle $E\subseteq T\grv$, a finite dimensional solution of the p.d.e. associated with $E$ at a point $U$ will be a finite dimensional subscheme $X\subseteq\grv$ containing $U$ and such that $E_U\simeq T_{U}X$. \end{defn} \begin{rem} It is convenient to consider not only finite dimensional subschemes as solutions but also algebraizable formal schemes. \end{rem} \subsection{Characterization of Jacobian varieties} The goal of this subsection is to prove the following generalization of the Theorem 6 of \cite{Sh}: \begin{thm}\label{thm:shiota} A necessary and sufficient condition for a rational point $U\in\grv$ to lie in the image of the Krichever map $K_m$ (for a point $m\in\cur$) is that there exists a finite dimensional solution of the p.d.e. $\im{d\mu_-}$ at the point $U$. \end{thm} Proof of the theorem is a direct consequence of the following two lemmas, that are quite akin to Mulase's and Shiota's ideas (\cite{Mul1,Sh}). \begin{lem}\label{lem:orbit} Let $U$ be rational point of $\grv$, and let $G(U)$ denote the orbit of $U$ under the action of a group $G$. Then: $$\Gamma(U)\simeq \Gamma_-(U)\times\Gamma_+$$ \end{lem} \begin{pf} Since $\Gamma=\limi(\Gamma^n_-\times{\Bbb G}_m\times\Gamma_+)$ (as formal schemes), it is then enough to show that the natural map: $$\aligned \Gamma^n_-(U)\times\Gamma_+&\to (\Gamma^n_-\times{\Bbb G}_m\times\Gamma_+)(U)\\ \left(f_-U,f_+\right)&\mapsto (f_-\cdot f_+)U \endaligned$$ is an isomorphism; or, what amounts to the same: if $f_-U=f_+U$, then $f_-=f_+=1$. Since $U$ is a rational point, there exists an element $g=\sum_{i\ge m}c_i\,z^i\in U$ ($c_m\ne 0$) with $m$ maximal (this $m$ will be called order of $g$); note also that we can assume $c_m=1$. Note that homotheties map elements of maximal order into elements of maximal order, and therefore $f_-\cdot g=f_+\cdot g$, from which one deduces $f_-=f_+$, and hence $f_-,f_+\in\Gamma_-\cap\Gamma_+=\{1\}$. Finally, the natural structure of the formal scheme of $\Gamma_-(U)$ is equal to $\cup_{n>0}\Gamma^n_-(U)$ (where $\Gamma^n_-(U)$ denotes the schematic image of $\Gamma^n_-\times\{U\}\to\grv$). Let $\Gamma^n_-$ be $\sp\left(k[x_1,\dots,x_n]/(x_1,\dots,x_n)^n\right)$, $\o$ the structural sheaf $\o_{\grv}$, $I_U^n$ the kernel of $\o\to k[x_1,\dots,x_n]/(x_1,\dots,x_n)^n$, and $u^{n'}_n$ the morphism $\o/I_U^{n'}\to\o/I_U^n$ ($n'>n$). In order to show that $\Gamma_-(U)=\sf(A)$ ($A$ being $\limp\o/I_U^n$), it remains only to check that (\cite{EGA} I.10.6.3) $u^{n'}_n$ is surjective and $\ker(u^{n'}_n)$ is nilpotent. However, both are obvious. Note that as a by-product we have that the topology of $A$ is given by the ideals $J_n=\ker(A\to \o/{I_U^n})$ and that the definition ideal is $J=\limpl{n}(x_1,\dots,x_n)$. \end{pf} \begin{lem}\label{lem:five} Let $U$ be a rational point of $\grv$. Then the following conditions are equivalent: \begin{enumerate} \item $\Gamma_-(U)$ is algebraizable, \item $\dim_k(J/J^2)<\infty$, \item $\dim_k(T_U\Gamma_-(U))<\infty$, \item $\dim_k\im(d\mu_-)<\infty$, \item there exists a rational point $m=(C,p,z)$ of $\cur$ and a pair $(L,\phi)$ of $\tilde\pic(C,p)(\spk)$ such that $K_m(L,\phi)=U$. \end{enumerate} \end{lem} \begin{pf} $1\implies 2$: Recall that algebraizable (\cite{H} II.9.3.2) means that the formal scheme is isomorphic to the completion of a noetherian scheme along a closed subscheme, but the completion of a noetherian ring with respect to an ideal is noetherian (\cite{At} 10.26); hence $A$ is noetherian. Recall now from \cite{EGA} 0.7.2.6 that if $A$ is an admissible linearly topologized ring, and $J$ a definition ideal, $A$ is noetherian if and only if $A/J$ is so and $J/J^2$ is a finite type $A/J$-module. Since $A/J\simeq k$, we conclude. $2\implies 3$: Note that $$\aligned T_U\Gamma_-(U)&= \hom_{\text{for-esq}}(\sp(k[\epsilon]/(\epsilon^2),\sf(A))=\\ &=\hom \Sb \text{topological} \\ \text{$k$-algebras} \endSb (A,k[\epsilon]/(\epsilon^2)) \subset \hom_{\text{$k$-algebras}}(A,k[\epsilon]/(\epsilon^2))\endaligned$$ which is isomorphic to $\left( J/J^2\right)^*$. $3 \iff 4$ By the very definition, the morphism $\Gamma_-\to\grv$ factorizes through $\Gamma_-(U)$, and therefore: $$\dim_k\im(d\mu_-)\,\leq\,\dim_k T_U\Gamma_-(U)<\infty$$ $4\implies 5$: From 4 we have $\dim_k\ker(d\mu_-)=\infty$. Note, moreover, that $B=\ker(d\mu_-)\subseteq V$ is a integral $k$-algebra of transcendence degree 1, since $B_{(0)}= V$ such that $U$ is a free $B$-module of rank 1. Standard results (see \cite{SW}) show how from the pair $(B,U)$ one can construct the data $(C,p,z)\in\P(\spk)$ and $(L,\phi)\in\tilde\pic(C,p)(\spk)$ such that $K_m(L,\phi)=U$. $5\implies 4$: Easy. $3\implies 1$: Let $\sp(A_n)$ be the schematic image of $$\Gamma_-^n=\sp(k[x_1,\dots,x_n]/(x_1,\dots,x_n)^n)\to\grv$$ By the properties of formal schemes, one has: $$ T_U\Gamma_-(U)\quad=\quad \bigcup_{n>0} T_U \sp(A_n)$$ And therefore, if $\dim_k (T_U\Gamma_-(U))=d$ then $\dim_k (T_U \sp(A_n))=d$ for all $n>>0$. Denote with $J_{(n)}$ the maximal ideal of $A_n$, since $T_U \sp(A_n)\simeq (J_{(n)}/J_{(n)}^2)^*$ and $J=\limp J_{(n)}@>\pi_n>> J_{(n)}$ is surjective, there exist elements $\bar y_1,\dots,\bar y_d\in J$ such that $<\{\pi_n(\bar y_1),\dots, \pi_n(\bar y_d)\}>=J_{(n)}/J_{(n)}^2$. By Nakayama's lemma one has an epimorphism: $$\aligned p_n:k[y_1,\dots,y_d]&\to A_n\\ y_i\quad&\mapsto\pi_n(\bar y_i)\endaligned$$ It is now straightforward to see that $\{p_n\}$ is compatible with the natural morphism $A_m\to A_n$ for $m>n$. One concludes that $A_n\simeq k[y_1,\dots,y_d]/I_n$ ($I_n$ being $\ker p_n$) and that $I_m\subset I_n$ for $m> n$. Recall now that $A=\limp A_n$ and thus $A\simeq k[[y_1,\dots,y_d]]$; the topology on $A$ induced by $\ker(A\to A_n)$ coincides with the $(y_1,\dots,y_d)$-adic; that is, the formal scheme $\sf(A)$ is algebraizable. \end{pf} \begin{rem} Assume that the conditions of Lemma {\ref{lem:five}} are satisfied for a point $U$. Then the orbit $\Gamma_-(U)$, which is a formal scheme, is the desired finite dimensional solution. Let us give another interpretation. Let $m=(C,p,z)\in\cur$ and $(L,\phi)\in\tilde\pic(C,p)$ such that $K_m(L,\phi)=U$. It is straightforward to check that the p.d.e. defined by $\im(d\mu_-)$ is in fact the KP hierarchy and hence $\tilde\pic(C,p)$ is a finite dimensional solution of the KP hierarchy, modulo the action of $\Gamma_+$. \end{rem} \begin{rem} Let $char(k)=0$, $U\in\grv$ satisfying one condition of the previous lemma and $(C,p,z,L,\phi)$ that given by the fifth condition. Then, the morphism $T_{\{1\}}\Gamma^n_-\to T_U\grv$ is essentially: $$H^0(C,\o_C(np)/\o_C)\to H^1(C,\o_C)\simeq T_L \pic(C)$$ which is deduced from the cohomology sequence of: $$0\to\o_C\to\o_C(np)\to \o_C(np)/\o_C\to 0$$ and from the proof of the Lemma it is easy to conclude that $\Gamma_-(U)$ is canonically isomorphic to the formal completion of the Jacobian of $C$ along $L$. When $k=\Bbb C$ and $C$ is a smooth curve, we can interpret the algebraic variety $\pic(C)$ as a compact Lie group. Now using the exponential map, it is not hard to see that our condition of ``finite dimensional solution of the p.d.e.'' is equivalent to Shiota's ``compact solution for the KP hierarchy'' (see Theorem 6 of \cite{Sh}), and that the morphism $T_{\{1\}}\Gamma^n_-\to T_U\grv$ is the one studied in depth in its \S2, especially in Lemmas 2 and 4 of \cite{Sh}. \end{rem} \begin{rem} Note further that the point $m=(C,p,z)\in\cur$ obtained by Lemma {\ref{lem:five}} is not unique. However, it has a characterizing property. From the construction, it is easily seen that the ring $B$ is the maximal subring of $\hat V$ such that $B\cdot U= U$. This implies that there is no morphism $f:C'\to C$ and line bundle $L'$ such that $f_*L'=L$ (unless $f$ is an isomorphism). See \cite{Mul2} for more details. \end{rem} \subsection{Characterization of Prym varieties} We shall say that a point $(C,p,z)\in\cur$ admits the automorphism $\sigma$ if $\sigma:k((z))\iso k((z))$ (as a $k$-algebra) restricts to $K(C,p,z)\iso K(C,p,z)$. Note that in this case $\sigma$ induces an automophism of $C$ with $p$ as a fixed point (it will also be denoted by $\sigma$). \begin{defn}\label{defn:p0} Let $\P_\sigma$ denote the subfunctor of $\cur$ consisting of the data that admit the automorphism $\sigma$. (In the next section we shall see that it is in fact a subscheme). \end{defn} In this setting, the Prym variety associated with the data $m=(C,p,z)\in\cur$ that admits an involution $\sigma$ is a subscheme of $\tilde\pic(C,p)$ whose rational points are: $$\tilde{\operatorname{Prym}}(C,p,\sigma)=\left\{ (L,\phi)\in\tilde\pic(C,p)\,\vert\, \sigma^*(L)\iso \omega_C\otimes L^{-1}\right\}$$ In this subsection we shall restrict ourselves to the situation addressed in Example {\ref{exam:metric}}; that is: $\sigma=\sigma_0$. Note, however, that this can always be achieved (see Remark {\ref{rem:sigma}}). Thus, we shall assume here that $\sigma_0^*(g(z))=g(-z))$ for all $g(z)\in\Gamma$, and shall remove the super/sub-script $\sigma$ in the notations. From the above discussion and recalling {\ref{defn:gri}}, one has the following cartesian diagram: \begin{eqnarray*} \tilde\pic(C,p)\phantom{xx} & \overset{\text{\scriptsize $K_m$}}\hookrightarrow & \,\grv \\ \cup\phantom{xxxx} & & \phantom{xx} \cup \\ \tilde{\operatorname{Prym}}(C,p,\sigma_0) & \overset{\text{\scriptsize $K_m$}}\hookrightarrow & \gr^I_0(V) \end{eqnarray*} Let $\mu^0_-$ be the restriction of $\mu_-$ to $\Pi^0_-$, and let $d\mu^0_-$ be that induced in the tangent spaces. Our version of the Theorem 5.14 of \cite{LiMu} is the following: \begin{thm}\label{thm:shiota-prym} A necessary and sufficient condition for a rational point $U\in\gr^I_0(V)$ to lie in the image of the Krichever map $K_m$ (for a point of $m\in\P_0$) is that there exists a finite dimensional solution of the p.d.e. $\im{d\mu^0_-}$ at the point $U$. \end{thm} \begin{pf} Observe that $\dim_k\im{d\mu^0_-}<\infty$ if and only if $\dim_k\im{d\mu_-}<\infty$, and therefore that it is only necessary to show that in the last condition of Lemma {\ref{lem:five}}, the constructed data $(C,p,z)\in\cur$ admits the involution given by $z\mapsto -z$. First, note that since $U$ is m.t.i. we have: $<f,u>=0\quad \forall \,u\in U\implies f\in U$. Now for an element $f\in\ker(d\mu^0_-)$ we have $f\cdot U\subseteq U$ and therefore $<f\cdot u,v>=0$ for all $u,v\in U$. We want to see that $f(-z)\in\ker(d\mu^0_-)$; that is, $<f(-z)u(z),v(z)>=0$ for all $u,v\in U$. Note, however, that: $$<f(-z)u(z),v(z)>=<u(z),f(z)v(z)>=-<f(z)v(z),u(z)>$$ and hence $f(-z)\in\ker(d\mu^0_-)$, as desired. \end{pf} \begin{rem} Analogously to the case of Jacobian varieties, one has that $\im(d\mu^0_-)$ is equivalent to the BKP hierarchy and therefore $\Pi^0_-(U)$ is a finite dimensional solution for the BKP hierarchy. As before, one can say that $\tilde{\operatorname{Prym}}(C,p,z)$ (modulo the action of $\Gamma_+$) is a finite dimensional solution too. \end{rem} \begin{rem} Note that in the proof of Lemma {\ref{lem:five}} a $k$-algebra $B$ is constructed that turns out to be the ring $H^0(C-p,\o_C)$. If we now assume that $U\in\gr^I_0(V)$, $B$ has an involution. It is easy to check that in this situation $B\in\gr_0(k((z)),k[[z]])$. Since we know the structure of this Grassmannian (see Example {\ref{exam:metric}}), the projection on the first factor is precisely: $$B'\,=\,B\cap k((z^2))\,\in\,\gr_0(k((z^2)),k[[z^2]])$$ Now, the curve $C'$ constructed from $B'$ is the quotient of $C$ with respect to the involution $\sigma$. Note that $U$ is a rank 2 free $B'$-module, and that the sheaf induced by $U$ over $C'$ is the direct image of $L$ by $C\to C'$. \end{rem} \section{Equations for the Moduli Space of Prym varieties} Although the notion of Prym variety is more general, here we shall restrict our study to those coming from a curve and an involution with at least one fixed point. \begin{defn} The functor of Prym varieties, $\P$, is the sheafication of the following functor on the category of $k$-schemes: $$S\rightsquigarrow\left\{(C,D,z,\sigma)\,\vert\, \gathered (C,D,z)\in\cur(S),\text{ $\sigma$ is an involution} \\ \text{that induces an automorphism of} \\ \text{the formal completion of $C$ along $D$} \endgathered \right\}$$ (up to isomorphisms). \end{defn} Given $(C,D,z,\sigma)\in\P(\spk)$, note that $\sigma$ induces an automorphism of $k((z))$, and that the action of $\Gamma_+$ on the group $\operatorname{Aut}_{\text{$k$-alg}}(k((z)))$ (via $U$) is transitive and free. One therefore has a bijection (set-theoretic) $\P\simeq\P_0\times\Gamma_+$. From Example {\ref{exam:metric}} and Definition {\ref{defn:p0}} one now has the following easy but fundamental result: \begin{thm}\label{thm:char-p0} Via the Krichever map, one has: $$\P_0\simeq \cur \underset{\grv}\times\gr_0(V)$$ \end{thm} \begin{cor} The functor $\P_0$ is representable by a locally closed subscheme of the infinite Grassmannian $\grv$; namely, that whose ($S$-valued) points $U$ satisfy: \begin{enumerate} \item $\o_S\subset U$, \item $U\cdot U\subseteq U$ ($\cdot$ being the product of $\o_S((z))$), \item the map $\o_S((z))\to \o_S((z))$ defined by $z\mapsto -z$ restricts to an isomorphism $U\iso U$. \end{enumerate} \end{cor} \begin{pf} Recall from \cite{Al,MP} that the first two conditions are locally closed. The third condition is closed since it is where the identity and the involution of $\grv$ given by $z\mapsto -z$ coincide; recall also that $\grv$ is separated. \end{pf} We can now state a theorem characterising the points of $\gr_0(V)$ in terms of bilinear identities; this is an analogous result to the characterization of $\grv$ of \cite{DKJM,F,MP}. \begin{thm}[Bilinear Identities]\label{thm:bil-ident} $$U\in\gr_0(V)\quad\iff\quad\left\{ \gathered \res_{z=0}\psi_U(z,t)\psi^*_U(z,t')\frac{dz}{z^2}=0 \\ \res_{z=0}\psi_U(-z,t)\psi^*_U(z,t')\frac{dz}{z^2}=0 \endgathered\right. \,\text{ for all $t,t'$}$$ \end{thm} \begin{pf} Note that the third condition in the corollary is equivalent (for the rational points) to saying that: $\psi_U(z,t)=-\psi_U(-z,t)$ since for a point $U\in\grv$ one has that the Baker-Akhiezer function, $\psi_U(z,t)$, may be written as a series of the form $z\cdot\sum_{i>0}\psi_U^{(i)}(z)p_i(t)$ where $p_i(t)$ are universal polynomials (they do not depend on $U$) and $\{\psi_U^{(i)}(z)\}_{i>0}$ is a basis of $U$ as a $k$-vector space (see \cite{MP}). Recalling the property $\res_{z=0}\psi_U(z,t)\psi^*_{U'}(z,t')\frac{dz}{z^2}=0$ if and only if $U=U'$, one concludes. \end{pf} This result holds when $char(k)\neq 2$; however, it can be translated in the language of differential equations when $char(k)=0$. We arrive at two sets of differential equations in terms of $\tau$-functions defining the set of rational points of $\gr_0(V)$ and $\P_0$. For a Young diagram $\lambda$, denote by $\chi_\lambda$ its associated Schur polynomial. If the diagram has only one row of length $\beta$, then denote the corresponding Schur polynomial by $p_\beta$. Let $t$ be $(t_1,t_2,\dots)$ and $\tilde\partial_t$ be $(\partial_{t_1},\frac12\partial_{t_2},\frac13\partial_{t_3},\dots)$. Let $D_{\lambda,\alpha}$ be $\sum_\mu\chi(\tilde\partial_t)$ where the sum is taken over the set of Young diagrams $\mu$ such that $\lambda-\mu$ is a horizontal $\alpha$-strip. \begin{thm}[P.D.E. for $\gr_0(V)$]\label{thm:pde-gr0} A function $\tau(t)$ is the $\tau$-function of a rational point $U\in\gr_0(V)$ if and only if it satisfies the following infinite set of differential equations (indexed by a pair of Young diagrams $\lambda_1,\lambda_2$): $$\left(\sum p_{\beta_1}(-\tilde\partial_t)D_{\lambda_1,\alpha_1}(\tilde\partial_t)\cdot p_{\beta_2}(-\tilde\partial_{t'})D_{\lambda_2,\alpha_2}(\tilde\partial_{t'}) \right)\vert_{t=t'=0}\tau_U(t)\cdot \tau_U(t')\,=\,0$$ where the sum is taken over the 4-tuples $\{\alpha_1,\beta_1,\alpha_2,\beta_2\}$ of integers such that $-\alpha_1+\beta_1-\alpha_2+\beta_2=1$, $-\alpha_1+\beta_1$ is even. \end{thm} \begin{pf} Recall from \cite{MP} how the KP hierarchy is deduced from the Residue Bilinear Identity. Apply the same procedure to the identities in the preceding Theorem, and add and subtract both identities. \end{pf} We finish with the partial differential equations for $\tau$-functions that characterize $\P_0$ as a subscheme of $\check{\mathbb P}\Omega$. \begin{thm}[P.D.E. for $\P_0$]\label{thm:pde-p0} A function $\tau(t)$ is the $\tau$-function of a point $U\in\P_0$ if and only if it satisfies the following infinite set of differential equations: \begin{enumerate} \item the p.d.e. of Theorem {\ref{thm:pde-gr0}}, \item $$P(\lambda_1,\lambda_2,\lambda_3)\vert \Sb t=0 \\ t'=0 \\ t''=0 \endSb \left(\tau_U(t)\cdot \tau_U(t')\cdot\tau_U(t'')\right)\,=\,0$$ for every three Young diagrams $\lambda_1,\lambda_2,\lambda_3$, where $P(\lambda_1,\lambda_2,\lambda_3)$ is the differential operator defined by: $$\sum p_{\beta_1}(\tilde\partial_t)D_{\lambda_1,\alpha_1}(-\tilde\partial_t)\cdot p_{\beta_2}(\tilde\partial_{t'})D_{\lambda_2,\alpha_2}(-\tilde\partial_{t'}) \cdot p_{\beta_3}(-\tilde\partial_{t''})D_{\lambda_3,\alpha_3}(\tilde\partial_{t''}) $$ where the sum is taken over the 6-tuples $\{\alpha_1,\beta_1,\alpha_2,\beta_2,\alpha_3,\beta_3\}$ of integers such that $-\alpha_1+\beta_1-\alpha_2+\beta_2-\alpha_3+\beta_3=2$, $-\alpha_1+\beta_1$ is even, \item the p.d.e.'s: $$\left(\sum_{\alpha\geq 0} p_{\alpha}(-\tilde\partial_t)D_{\lambda,\alpha}(\tilde\partial_t) \right)\vert_{t=0}\tau_U(t)=0\quad \text{for all Young diagram }\lambda$$ \end{enumerate} \end{thm} \begin{pf} By Theorems {\ref{thm:char-p0}} and {\ref{thm:pde-gr0}}, it is enough to recall from \cite{MP} the p.d.e. defining $\cur$ in the infinite Grassmannian. \end{pf} Note that a theta function of the Prym variety associated to a 4-uple $(C,p,z,\sigma)$ (where $\sigma$ is the given by $z\mapsto -z$) satisfy these differential equations which are not a consequence of the BKP hierarchy.

9706

alg-geom/9706005

fr

https://arxiv.org/abs/alg-geom/9706005

alg-geom/9706005

Vincent Maillot

V. Maillot (CNRS, France)

G\'eom\'etrie d'Arakelov des vari\'et\'es toriques et fibr\'es en droites int\'egrables

LaTeX2e, 118 pages, 6 figures

En nous appuyant sur une construction due \`a Bedford et Taylor, et certains r\'esultats r\'ecents de Demailly, nous pr\'esentons une extension (partielle) de la g\'eom\'etrie d'Arakelov aux fibr\'es en droites int\'egrables. (Ces derniers sont les fibr\'es en droites hermitiens sur une vari\'et\'e arithm\'etique $X$ pouvant se d\'ecomposer sous la forme $\ov{E} = \ov{E}_{1}\otimes (\ov{E}_{2})^{-1}$, o\`u $\ov{E}_{1}$ et $\ov{E}_{2}$ sont des fibr\'es en droites munis \`a l'infini d'une m\'etrique continue approchable uniform\'ement sur $X(C)$ par des m\'etriques positives $C^{\infty}$). Nous appliquons notre th\'eorie aux fibr\'es en droites sur une vari\'et\'e torique munis \`a l'infini de leur m\'etrique canonique. Nous en d\'eduisons, entre autres choses, la d\'emonstration d'un analogue arithm\'etique du th\'eor\`eme de Bernstein-Koushnirenko.

alg-geom

\section{Vari\'et\'es toriques complexes}~ \subsection{Vari\'et\'e \`a coin associ\'ee \`a une vari\'et\'e torique}~ Dans ce paragraphe, on suit essentiellement (\cite{11}, \S 4). On peut \'egalement consulter (\cite{20}, prop. 1.8). Soit $\Delta$ un \'eventail de $N$. Pour tout c\^one\ $\sigma$ de $\Delta$ on note~: \[ (U_{\sigma})_{\geqslant} = \op{Hom}_{\op{sg}}(\sigma^{\ast}\cap M, \M{R}^{+}) \] l'ensemble des morphismes de semi-groupe avec \'el\'ement neutre de $\sigma^{\ast}\cap M$ vers $(\M{R}^{+},\times)$. Du fait de l'inclusion $\M{R}^{+} \subset \M{C}$, l'ensemble $(U_{\sigma})_{\geqslant}$ peut \^etre vu comme sous-ensemble ferm\'e de $U_{\sigma}(\M{C})$. En effet, on a~: \begin{multline*} (U_{\sigma})_{\geqslant} = \op{Hom}_{\op{sg}}(\sigma^{\ast}\cap M, \M{R}^{+})\\ \subset \op{Hom}_{\op{sg}}(\sigma^{\ast}\cap M, \M{C}) = \op{Hom}_{\text{$\M{C}$-alg\`ebre}} \left(\M{C}[\sigma^{\ast}\cap M],\M{C}\right) = U_{\sigma}(\M{C}). \end{multline*} L'application module $|~|:\M{C} \rightarrow \M{R}^{+}$ induit par composition une r\'etraction~: \[ U_{\sigma}(\M{C}) = \op{Hom}_{\op{sg}}(\sigma^{\ast}\cap M, \M{C}) \longrightarrow \op{Hom}_{\op{sg}}(\sigma^{\ast}\cap M, \M{R}^{+}) = (U_{\sigma})_{\geqslant} \] que l'on note encore $|~|$. Si $\sigma$ et $\sigma'$ sont deux \'el\'ements de $\Delta$, alors $(U_{\sigma})_{\geqslant} \cap (U_{\sigma'})_{\geqslant} = (U_{\sigma \cap \sigma'})_{\geqslant}$. Il existe donc une partie ferm\'ee $\P_{\geqslant}$ de $\M{P}(\Delta)(\M{C})$ telle que pour tout $\sigma \in \Delta$ on ait $\P_{\geqslant} \cap U_{\sigma}(\M{C}) = (U_{\sigma})_{\geqslant}$. L'application $|~|$ s'\'etend en une r\'etraction continue~: \[ |~|: \M{P}(\Delta)(\M{C}) \longrightarrow \P_{\geqslant} \] \begin{prop} Si $\Delta$ est r\'egulier, $\P_{\geqslant}$ est une sous-vari\'et\'e \`a coins analytique r\'eelle de $\M{P}(\Delta)(\M{C})$, de dimension r\'eelle $d$. \end{prop} \noindent {\bf D\'emonstration.}\ C'est une cons\'equence directe des d\'efinitions (voir par exemple \cite{20}, prop. 1.8). \medskip \begin{expl} On note $T_{\geqslant} = (U_{\{0\}})_{\geqslant} = \op{Hom}_{\op{sg}}(M,\M{R}^{+})$. On v\'erifie que $T_{\geqslant}$ est dense dans $\P_{\geqslant}$. \end{expl} Soient $T(\M{C})$ le tore analytique complexe associ\'e \`a $T$ et $\C{S}_{N}$ le {\it tore compact\/}~: \[ \C{S}_{N} = \op{Hom}(M,\C{S}_{1}) \subset \op{Hom}(M,\M{C}^{\ast}) = T(\M{C}), \] o\`u $\C{S}_{1} \subset \M{C}$ est le cercle unit\'e. C'est le sous-groupe compact maximal de $T(\M{C})$; en fait on a~: \[ \C{S}_{N} = \op{Hom}(M,\C{S}_{1}) = N\otimes_{\M{Z}}\C{S}_{1} \cong {\C{S}_{1}}^{d} \subset T(\M{C}) = \op{Hom}(M,\M{C}^{\ast}) = N\otimes_{\M{Z}}\M{C}^{\ast} \cong (\M{C}^{\ast})^{d}. \] L'action naturelle de $T(\M{C})$ sur $\M{P}(\Delta)(\M{C})$ induit une action de $\C{S}_{N}$ sur $\M{P}(\Delta)(\M{C})$. La d\'ecomposition polaire $\M{C}^{\ast} = \C{S}_{1} \times \M{R}^{+\ast}$ et l'isomorphisme $\log: \M{R}^{+} \rightarrow \M{R}$ d\'eterminent un isomorphisme $\C{S}_{N}$-\'equivariant~: \[ \log : T(\M{C}) = \C{S}_{N} \times \op{Hom}_{\op{sg}}(M,\M{R}^{+}) \longrightarrow \C{S}_{N} \times \op{Hom}(M,\M{R}) = \C{S}_{N} \times N_{\M{R}}. \] On note $\op{pr_{2}} : \C{S}_{N} \times N_{\M{R}} \rightarrow N_{\M{R}}$ la seconde projection. Le diagramme suivant commute~: \begin{center} \mbox{ \xymatrix{ T(\M{C}) \ar[rr]^(0.4){\log} \ar[d]_{|~|} & &\C{S}_{N} \times N_{\M{R}} \ar[d]^{\op{pr_{2}}} \\ T_{\geqslant} \ar[rr]^{\log} & &N_{\M{R}} }} \end{center} et les fl\`eches horizontales sont des isomorphismes de groupes de Lie r\'eels.\\ On note $\op{orb}$ la surjection canonique~: \[ \op{orb}: \M{P}(\Delta)(\M{C}) \longrightarrow \M{P}(\Delta)(\M{C})/\C{S}_{N}. \] On remarque qu'on peut identifier~: \[ T_{\geqslant} = \op{Hom}_{\op{sg}}(M,\M{R}^{+}) = T(\M{C})/\C{S}_{N}. \] Plus g\'en\'eralement, on a le r\'esultat suivant~: \begin{thm} Il existe un unique hom\'eomorphisme $\kappa : \P_{\geqslant} \rightarrow \M{P}(\Delta)(\M{C})/\C{S}_{N}$ tel que le diagramme suivant commute~: \begin{center} \mbox{ \xymatrix{ & \M{P}(\Delta)(\M{C}) \ar[dl]_{|~|} \ar[dr]^{\op{orb}} & \\ \P_{\geqslant} \ar[rr]^{\kappa}& & \M{P}(\Delta)(\M{C})/\C{S}_{N}} } \end{center} De plus, la restriction de $\kappa$ \`a $T_{\geqslant}$ co\"\i ncide avec l'identification ci-dessus. \end{thm} \noindent {\bf D\'emonstration.}\ Voir (\cite{11}, p. 79) et (\cite{20}, prop. 1.8). \bigskip \subsection{Un recouvrement canonique de $\M{P}(\Delta)(\M{C})$}~ \begin{defn}{\bf (Batyrev et Tschinkel).} Pour tout $\sigma \in \Delta$, on d\'efinit $C_{\sigma} \subset \M{P}(\Delta)(\M{C})$ de la fa\c con suivante~: \[ C_{\sigma} = \{x \in \M{P}(\Delta)(\M{C}): \forall m \in \E{S}_{\sigma}= \sigma^{\ast}\cap M, \quad \chi^{m} \text{ est r\'egulier en $x$ et }|\chi^{m}(x)| \leqslant 1\}. \] \end{defn} \begin{prop} Pour tout $\sigma \in \Delta$, on a $C_{\sigma} \subset U_{\sigma}(\M{C})$ et $C_{\sigma}$ est compact. De plus, si $\tau, \tau' \in \Delta$, alors~: \[ C_{\tau} \cap C_{\tau'} = C_{\tau \cap \tau'} \] \end{prop} \noindent {\bf D\'emonstration.}\ Soit $x \in C_{\sigma}$, la fonction $\chi^{m}$ est d\'efinie en $x$ pour tout $m \in \E{S}_{\sigma} = \sigma^{\ast}\cap M$, et donc $x \in U_{\sigma}(\M{C})$. Soit $\{m_{1},\dots,m_{q}\}$ une famille g\'en\'eratrice du semi-groupe $\E{S}_{\sigma}$. On dispose de l'immersion ferm\'ee~: \begin{alignat*}{3} \varphi : \; U_{\sigma}&(\M{C}) & &\longrightarrow & \;&\M{C}^{q} \\ &x & &\longmapsto & &(\chi^{m_{1}}(x),\dots,\chi^{m_{q}}(x)). \end{alignat*} On remarque que $\varphi (C_{\sigma}) = \varphi(U_{\sigma}(\M{C})) \cap B(0,1)^{q}$. Comme $\varphi (U_{\sigma}(\M{C}))$ est un ferm\'e analytique, on conclut que $\varphi(C_{\sigma})$, et donc $C_{\sigma}$, sont compacts. Enfin la derni\`ere assertion r\'esulte de l'identit\'e $\tau^{\ast} + {\tau'}^{\ast} = (\tau \cap \tau')^{\ast}$ (Voir \cite{20}, th. A.1). \medskip \begin{prop} \label{recouvrement} Si $\Delta$ est complet, alors les compacts $C_{\sigma}$ forment un recouvrement de $\M{P}(\Delta)(\M{C})$ lorsque $\sigma$ parcoure $\Delta_{\op{max}}$. \end{prop} \noindent {\bf D\'emonstration.}\ On suit ici \cite{2}. Puisque $T(\M{C})$ est dense dans $\M{P}(\Delta)(\M{C})$, il suffit de d\'emontrer que les $C_{\sigma}$, lorsque $\sigma$ parcourt $\Delta_{\text{max}}$, recouvrent $T(\M{C})$. Soit $x \in T(\M{C})$; puisque $\Delta$ est complet, il existe $\sigma \in \Delta_{\text{max}}$ tel que $-\log|x| \in \sigma$. Pour tout $m \in \E{S}_{\sigma}$, on a~: \[ <m,-\log|x|> = -\log|\chi^{m}(x)| \geqslant 0. \] On en conclut que $|\chi^{m}(x)| \leqslant 1$ pour tout $m \in \E{S}_{\sigma}$, c'est-\`a-dire que $x \in C_{\sigma}$. \medskip Les compacts de la forme $C_{\sigma}$ sont globalement invariants sous l'action de $\C{S}_{N}$. Pour mieux comprendre la structure des $C_{\sigma}$, on peut \'etudier l'image de $C_{\sigma} \cap T(\M{C})$ dans $N_{\M{R}}$ par l'application $\log |.|$. C'est l'objet de la proposition suivante~: \begin{prop} \label{application_log} Pour tout $\sigma \in \Delta$, on note $\stackrel{\circ}{C}_{\sigma} = C_{\sigma} \cap T(\M{C})$. On a alors~: \[ -\log |\stackrel{\circ}{C}_{\sigma}| = \sigma. \] \end{prop} \noindent {\bf D\'emonstration.}\ Soit $x \in T(\M{C})$. Pour tout $m \in M$, on a $<m,-\log|x|> = -\log|\chi^{m}(x)|$, et donc $x \in \stackrel{\circ}{C}_{\sigma}$ si et seulement si~: \[ <m,-\log|x|> \geqslant 0, \qquad \forall m \in \E{S}_{\sigma}, \] ce qui est \'equivalent \`a~: \[ - \log|x| \in (\sigma^{\ast})^{\ast} = \sigma . \] \medskip L'\'etude des $C_{\sigma}$ est donc ramen\'ee \`a l'\'etude de $\Delta$ ``plong\'e'' dans $\P_{\geqslant}$. \begin{expl} On a repr\'esent\'e ici $\M{P}^{2}(\M{C})/\C{S}_{N}$~: \bigskip \begin{center} \input{figure1.pstex_t} \end{center} \bigskip \end{expl} \begin{prop} Soit $p$ un entier strictement positif. Le morphisme $[p]: \M{P}(\Delta)(\M{C})\rightarrow \M{P}(\Delta)(\M{C})$ envoie $\P_{\geqslant}$ dans lui-m\^eme. Pour tout $\sigma \in D$, le compact $C_{\sigma}$ est laiss\'e stable par $[p]$. Enfin le diagramme suivant commute~: \begin{center} \mbox{ \xymatrix{ T(\M{C}) \ar[rr]^{[p]} \ar[dd]_{-\log|~|} \ar[dr]^{i} & & T(\M{C}) \ar'[d] [dd]^(0.4){-\log|~|} \ar[dr]^{i} \\ & \M{P}(\Delta)(\M{C}) \ar[dd]_(0.35){\op{orb}} \ar[rr]^(0.4){[p]} & & \M{P}(\Delta)(\M{C}) \ar[dd]_(0.35){\op{orb}} \\ N_{\M{R}} \ar'[r] [rr]^(0.4){[p]} \ar[rd]_(0.4){\exp} & & N_{\M{R}} \ar[dr]_(0.4){\exp} \\ & \P_{\geqslant} \ar[rr]^{[p]} & & \P_{\geqslant} } } \end{center} \medskip o\`u $\exp$ est l'inverse du morphisme $\log: T_{\geqslant} \rightarrow N_{\M{R}}$, et $[p]: N_{\M{R}} \rightarrow N_{\M{R}}$ d\'esigne la multiplication par $p$. \end{prop} \noindent {\bf D\'emonstration.}\ La premi\`ere partie de la proposition est une cons\'equence de l'\'egalit\'e $[p](x) = x^{p}$ pour tout $x \in T(\M{C})$ (on peut \'egalement consulter \cite{11}, p. 80). \\ Soient $\sigma \in \Delta$ et $x \in C_{\sigma}$, pour tout $m \in \E{S}_{\sigma}$, on a $|\chi^{m}([p](x))| = |\chi^{m}(x)|^{p} \leqslant 1$, c'est-\`a-dire $[p](x) \in C_{\sigma}$; et donc $C_{\sigma}$ est laiss\'e stable par $[p]$. \\ Enfin la commutativit\'e du diagramme r\'esulte des propositions \'enonc\'ees au paragraphe pr\'ec\'edent. \medskip \begin{defn} Pour tout $\sigma \in \Delta$, on pose~: \[ C_{\sigma}^{\op{int}} = \{ x \in C_{\sigma}\,: \quad |\chi^{m}(x)| < 1, \quad \forall m \in (\sigma^{\ast} - \sigma^{\perp}) \cap M \}. \] \end{defn} La proposition suivante rassemble diverses propri\'et\'es des ensembles $C_{\sigma}$ et $C_{\sigma}^{\op{int}}$ \begin{prop}~ \begin{enumerate} \item{Pour tout $\sigma \in \Delta$, on a $- \log | C_{\sigma}^{\op{int}} \cap T(\M{C}) | = \stackrel{\circ}{\sigma}$, o\`u $\stackrel{\circ}{\sigma}$ est l'int\'erieur relatif du c\^one $\sigma$.} \item{Les ensembles $C_{\tau}^{\op{int}}$ sont deux \`a deux disjoints. Pour tout $\sigma \in \Delta$, on a~: \begin{equation} \label{an_eq_1} C_{\sigma} = \bigcup_{\tau < \sigma}C_{\sigma}^{\op{int}}. \end{equation} } \item{Si l'\'eventail $\Delta$ est complet, alors les $C_{\tau}^{\op{int}}$ pour $\tau$ parcourant $\Delta$ forment une partition de $\M{P}(\Delta)(\M{C})$.} \end{enumerate} \end{prop} \noindent {\bf D\'emonstration.}\ D\'emontrons tout d'abord l'assertion (1). Soit $x \in T(\M{C}) \cap C_{\sigma}^{\op{int}}$. Pour tout $m \in (\sigma^{\ast} - \sigma^{\perp}) \cap M$, on a $<m,-\log|x|> \; > 0$, ce qui \'equivaut \`a \'ecrire que $- \log |x| \in \stackrel{\circ}{\sigma}$. On en d\'eduit que $- \log |C_{\sigma}^{\op{int}} \cap T(\M{C})| = \stackrel{\circ}{\sigma}$. On s'int\'eresse maintenant \`a l'assertion (2). On reprend ici les notations de la remarque (\ref{rem_decomposition1}). Soient $\tau$ et $\tau'$ deux c\^ones distincts de $\Delta$. D'apr\`es l'assertion (1), on a $- \log |C_{\tau}^{\op{int}}\cap C_{\tau'}^{\op{int}}\cap T(\M{C}) | = \stackrel{\circ}{\tau} \cap \stackrel{\circ}{\tau}^{\lower7pt\hbox{$\scriptstyle '$}} = \emptyset$, ce dont on d\'eduit que les ensembles $C_{\tau}^{\op{int}}$ et $C_{\tau'}^{\op{int}}$ sont disjoints sur $T(\M{C})$. En remarquant que pour tout $\sigma \in \Delta$, on a~: \[ C_{\tau}^{\op{int}} \cap V(\sigma)(\M{C}) = C_{\ov{\tau}}^{\op{int}}, \qquad (\text{resp.} \quad C_{\tau'}^{\op{int}} \cap V(\sigma)(\M{C}) = C_{\ov{\tau}'}^{\op{int}}) \] o\`u $\ov{\tau} \in \Delta(\sigma)$ (resp. $\ov{\tau}' \in \Delta(\sigma)$) et o\`u $C_{\ov{\tau}}^{\op{int}} \subset \M{P}(\Delta)(\Delta(\sigma))(\M{C}) = V(\sigma)(\M{C})$ (resp. $C_{\ov{\tau}'}^{\op{int}} \subset V(\sigma)(\M{C})$), on obtient la relation~: \[ C_{\tau}^{\op{int}} \cap C_{\tau'}^{\op{int}} \cap V(\sigma)(\M{C}) = C_{\ov{\tau}}^{\op{int}} \cap C_{\ov{\tau}'}^{\op{int}}. \] Pa r\'ecurrence, on d\'eduit de cela, de la d\'ecomposition en tores disjoints donn\'ee \`a la proposition (\ref{decomposition1}) et de la remarque (\ref{rem_decomposition1}) que $C_{\tau}^{\op{int}}$ et $C_{\tau'}^{\op{int}}$ sont disjoints sur $\M{P}(\Delta)(\M{C})$. La d\'ecomposition (\ref{an_eq_1}) se montre par r\'ecurrence de fa\c con similaire en utilisant la relation $\sigma = \bigcup_{\tau < \sigma}\stackrel{\circ}{\tau}$ et la proposition (\ref{application_log}). Il reste \`a prouver l'assertion (3). D'apr\`es ce qui pr\'ec\`ede, il suffit de montrer que les $C_{\tau}^{\op{int}}$ pour $\tau \in \Delta$ forment un recouvrement de $\M{P}(\Delta)(\M{C})$; or ceci se d\'eduit directement de la d\'ecomposition (\ref{an_eq_1}) et de la proposition (\ref{recouvrement}). \medskip Dans le cas o\`u $\Delta$ est complet et r\'egulier, on peut pr\'eciser la structure des ensembles $C_{\sigma}$ et $C_{\sigma}^{\op{int}}$~: \begin{prop} \label{dissection} Soient $\Delta$ un \'eventail complet et r\'egulier et $\sigma$ un \'el\'ement de $\Delta$. \begin{enumerate} \item{ L'ensemble $C_{\sigma}^{\op{int}}$ est une sous-vari\'et\'e analytique r\'eelle lisse de dimension r\'eelle $d + \op{dim} \sigma$ de $\M{P}(\Delta)(\M{C})$.} \item{le compact $C_{\sigma}$ est une sous-vari\'et\'e r\'eelle \`a coins de dimension r\'eelle $d + \op{dim} \sigma$ de $\M{P}(\Delta)(\M{C})$, et son bord $\partial C_{\sigma}$ est une vari\'et\'e \`a coins v\'erifiant la relation~: \[ \partial C_{\sigma} = C_{\sigma} - C_{\sigma}^{\op{int}} = \bigcup_{\substack{ \tau < \sigma \\ \tau \not= \sigma}} C_{\sigma}^{\op{int}}\; . \] } \end{enumerate} \end{prop} \noindent {\bf D\'emonstration.}\ On pose $q = \dim \sigma$. Soit $\tau \in \Delta_{\op{max}}$ tel que $\sigma < \tau$. D'apr\`es la proposition (\ref{lissite}), on peut trouver $\{m_{1}, \dots, m_{d}\}$ une base de $M$ telle que $\{m_{1},\dots,m_{q}\}$ (resp. $\{m_{1}, \dots, m_{d}\}$) soit une famille g\'en\'eratrice du semi-groupe $\E{S}_{\sigma}$ (resp. $\E{S}_{\tau}$). Dans la carte affine $\varphi: U_{\tau}(\M{C}) \rightarrow \M{C}^{d}$ donn\'ee par $\varphi(x) = (\chi^{m_{1}}(x),\dots,\chi^{m_{d}}(x))$, les ensembles $C_{\sigma}$ et $C_{\sigma}^{\op{int}}$ sont d\'efinis par les conditions~: \begin{align*} C_{\sigma} &= \{x \in \M{C}^{d}: \quad |x_{1}| \leqslant 1, \dots, |x_{d-q}| \leqslant 1, |x_{d-q+1}| = 1, \dots, |x_{d}| = 1\} \\ \intertext{et} C_{\sigma}^{\op{int}} &= \{x \in \M{C}^{d}: \quad |x_{1}| < 1, \dots, |x_{d-q}| < 1, |x_{d-q+1}| = 1, \dots, |x_{d}| = 1\}. \end{align*} On en d\'eduit directement les assertions \'enonc\'ees. \bigskip \subsection{M\'etriques canoniques sur les faisceaux inversibles \'equivariants au-dessus de $\M{P}(\Delta)(\M{C})$}~ Dans toute cette section, $\Delta$ d\'esigne un \'eventail complet de $N$. Pour tout diviseur de Cartier $D$ horizontal et $T$-invariant sur $\M{P}(\Delta)$, on construit de mani\`ere canonique une m\'etrique sur le faisceau inversible $\C{O}(D)(\M{C})$. Plusieurs constructions \'equivalentes sont indiqu\'ees. \subsubsection{Construction de Batyrev et Tschinkel}~ On pr\'esente ici, sous une forme l\'eg\`erement diff\'erente de (\cite{2}, \S 2.1), une construction due \`a Batyrev et Tschinkel. \begin{prop_defn} \label{BT_construction} Soit $s$ une section holomorphe de $\C{O}(D)(\M{C})$ au-dessus d'un ouvert $\Omega \subset \M{P}(\Delta)(\M{C})$. Pour tout $x \in \Omega$, soit $\sigma \in \Delta$ tel que $x \in C_{\sigma} \subset U_{\sigma}(\M{C})$, et $m_{D,\sigma} \in M$ la restriction de $\psi_{D}$ \`a $\sigma$. Le quotient~: \begin{equation} \label{BT_definition} \| s(x) \|_{\op{BT}} = \left| \frac{s}{\chi^{m_{D,\sigma}}}(x) \right| \end{equation} est bien d\'efini et est appel\'e {\it norme de $s$ au point $x$ au sens de Batyrev-Tschinkel\/}. Cette norme d\'efinit une m\'etrique continue $\C{S}_{N}$-invariante sur $\C{O}(D)(\M{C})$, que l'on note $\| . \|_{\op{BT}}$. \end{prop_defn} \noindent {\bf D\'emonstration.}\ L'existence du quotient est une cons\'equence directe du lemme (\ref{locale}). Enfin, le second membre de (\ref{BT_definition}) est ind\'ependant du choix de $\sigma$ car~: \[ |\chi^{m_{D,\sigma}}(x)| = | \chi^{m_{D,\sigma'}}(x)| \] pour tout $x \in C_{\sigma} \cap C_{\sigma'}$ du fait de la continuit\'e de la fonction support $\psi_{D}$ sur $N_{\M{R}}$, et le m\^eme argument montre que $\| . \|_{\op{BT}}$ est bien continue sur $\M{P}(\Delta)(\M{C})$. \medskip On donne dans la proposition suivante quelques propri\'et\'es de la construction de Batyrev-Tschinkel~: \begin{prop}~ \label{BT_fonct} \begin{enumerate} \item{ {\rm (Multiplicativit\'e).} Soient $\Delta$ un \'eventail complet dans $N_{\M{R}}$ et $D_{1}$, $D_{2}$ deux diviseurs de Cartier horizontaux et $T$-invariants sur $\M{P}(\Delta)$. L'isomorphisme~: \[ \C{O}(D_{1}) \otimes \C{O}(D_{2}) \simeq \C{O}(D_{1} + D_{2}), \] est compatible aux m\'etriques de Batyrev-Tschinkel.} \item{ {\rm (Fonctorialit\'e).} Soient $\varphi: \Delta_{1} \rightarrow \Delta_{2}$ un morphisme d'\'eventails complets et $\varphi_{\ast}: \M{P}(\Delta_{1}) \rightarrow \M{P}(\Delta_{2})$ le morphisme de vari\'et\'es toriques associ\'e. Soient \'egalement $D_{2}$ un diviseur de Cartier horizontal $T$-invariant sur $\M{P}(\Delta_{2})$ et $\psi_{2}$ sa fonction support, et notons $D_{1} = (\varphi_{\ast})^{\ast}D_{2}$ le diviseur de Cartier $T$-invariant sur $\M{P}(\Delta_{1})$ dont la fonction support est donn\'ee par $\psi_{1} = \psi_{2}\circ \varphi$. L'isomorphisme~: \[ \C{O}(D_{1}) \simeq (\varphi_{\ast})^{\ast}\C{O}(D_{2}), \] est une isom\'etrie lorsque $\C{O}(D_{1})$ et $\C{O}(D_{2})$ sont munis de leur m\'etrique de Batyrev-Tschinkel.} \end{enumerate} \end{prop} \noindent {\bf D\'emonstration.}\ On d\'emontre tout d'abord l'assertion (1). Soient $s_{1}$ et $s_{2}$ des sections holomorphes, sur un ouvert $\Omega \subset \M{P}(\Delta)(\M{C})$, des faisceaux $\C{O}(D_{1})$ et $\C{O}(D_{2})$ respectivement. Pour tout $x \in \Omega$, soit $\sigma \in \Delta$ tel que $x \in C_{\sigma}$. On a~: \[ \|s_{1}(x)\|_{\op{BT}}\cdot \|s_{2}(x)\|_{\op{BT}} = \left| \frac{s_{1}\otimes s_{2}(x)}{\chi^{m_{D_{1},\sigma} + m_{D_{2},\sigma}}(x)}\right| = \left| \frac{s_{1}\otimes s_{2}(x)}{\chi^{m_{D_{1}+D_{2}},\sigma}(x)}\right| = \|s_{1} \otimes s_{2}(x)\|_{\op{BT}}, \] ce qui \'etablit l'\'enonc\'e recherch\'e. On s'int\'eresse maintenant \`a l'assertion (2). Soit $s_{2}$ une section holomorphe du faisceau $\C{O}(D_{2})$ sur un ouvert $\Omega \subset \M{P}(\Delta_{2})(\M{C})$ et notons $s_{1} = s_{2} \circ \varphi_{\ast}$ la section holomorphe de $\C{O}(D_{1})$ au-dessus de $(\varphi_{\ast})^{-1}(\Omega)$ obtenue par image r\'eciproque de $s_{2}$ par $(\varphi_{\ast})^{\ast}$. Pour tout $x \in (\varphi_{\ast})^{-1}(\Omega)$, soit $\sigma_{1} \in \Delta_{1}$ tel que $x \in C_{\sigma_{1}}$ et choisissons $\sigma_{2} \in \Delta_{2}$ tel que $\varphi(\sigma_{1}) \subset \sigma_{2}$. D'apr\`es la proposition (\ref{intro_inverse}), on sait que $\psi_{D_{1}} = \psi_{D_{2}}\circ \varphi$, et donc que $m_{D_{1},\sigma} = {}^{t}\varphi(m_{D_{2},\sigma})$. On d\'eduit de la proposition (\ref{application_log}) et du fait que $\varphi_{\ast}$ est $T$-\'equivariant, l'inclusion~: \[ \varphi_{\ast}(T_{1}(\M{C}) \cap C_{\sigma_{1}}) \subset T_{2}(\M{C}) \cap C_{\sigma_{2}}, \] ce qui, comme $\varphi_{\ast}$ est continue, montre que~: \[ \varphi_{\ast}(C_{\sigma_{1}}) \subset C_{\sigma_{2}}. \] On peut donc \'ecrire~: \[ \|s_{1}(x)\|_{\op{BT}} = \left|\frac{s_{1}(x)}{\chi^{m_{D_{1},\sigma}}(x)}\right| = \left| \frac{s_{2}\circ \varphi_{\ast}(x)}{\chi^{{}^t\varphi(m_{D_{2},\sigma})}(x)}\right| = \left| \frac{s_{2}\circ \varphi_{\ast}(x)}{\chi^{m_{D_{2},\sigma}}\circ \varphi_{\ast}(x)}\right| = \|s_{2}(\varphi_{\ast}(x))\|_{\op{BT}}, \] ce qui termine la d\'emonstration. \medskip \subsubsection{Construction d'apr\`es Zhang}~ On suit dans ce paragraphe (\cite{21}, th. 2.2). Soient $p$ un entier sup\'erieur ou \'egal \`a $2$ et $D$ un diviseur de Cartier horizontal $T$-invariant sur $\M{P}(\Delta)$. Comme $[p]$ est l'endomorphisme \'equivariant de $\M{P}(\Delta)$ associ\'e \`a l'endomorphisme de $\Delta$ d\'efini par la multiplication par $p$, on a d'apr\`es (\ref{intro_inverse}) l'\'egalit\'e des diviseurs de Cartier~: \[ [p]^{\ast}D = p\,D, \] et donc un isomorphisme de faisceaux sur $\M{P}(\Delta)$~: \[ \Phi_{D,p}: \C{O}(D)^{\otimes p} \simeq [p]^{\ast}\C{O}(D). \] Comme $\M{P}(\Delta)(\M{C})$ est compacte, on peut munir $\C{O}(D)(\M{C})$ d'une m\'etrique continue que l'on notera $\|.\|_{0}$. On d\'efinit alors par r\'ecurrence une suite de m\'etriques $(\|.\|_{n})_{n \in \M{N}}$ sur $\C{O}(D)(\M{C})$ de la fa\c con suivante $(n \geqslant 1)$~: \[ \| .\|_{n} = \left(\Phi^{\ast}_{D,p}\,[p]^{\ast}\,\|.\|_{n-1}\right)^{1/p} . \] On a alors le th\'eor\`eme suivant~: \begin{thm}~ \label{zhang} \begin{enumerate} \item{Les m\'etriques $\| .\|_{n}$ convergent uniform\'ement vers une m\'etrique $\|.\|_{\op{Zh},p}$ sur $\M{P}(\Delta)(\M{C})$ (i.e $\log \frac{\|.\|_{n}}{\|.\|_{0}}$ converge uniform\'ement sur $\M{P}(\Delta)(\M{C})$ vers $\log \frac{\|.\|_{\op{Zh},p}}{\|.\|_{0}}$).} \item{La m\'etrique $\|.\|_{\op{Zh},p}$ est l'unique m\'etrique continue sur $\C{O}(D)(\M{C})$ telle que~: \[ \|.\|_{\op{Zh},p} = \left(\Phi_{D,p}^{\ast}\,[p]^{\ast}\,\|.\|_{\op{Zh},p} \right)^{1/p}. \] } \end{enumerate} \end{thm} \noindent {\bf D\'emonstration.}\ C'est une cons\'equence directe de (\cite{21}, th. 2.2). \medskip \begin{rem} Zhang raisonne pour des vari\'et\'es projectives, mais son argument ne n\'ecessite en fait que la compacit\'e. \end{rem} Le th\'eor\`eme suivant nous permet d'identifier les deux m\'etriques introduites pr\'ec\'edemment~: \begin{thm} Soit $D$ un diviseur de Cartier horizontal et $T$-invariant sur $\M{P}(\Delta)$. Pour tout entier $p \geqslant 2$, on a l'\'egalit\'e des m\'etriques~: \[ \|.\|_{\op{BT}} = \|.\|_{\op{Zh},p} \] sur le faisceau inversible $\C{O}(D)(\M{C})$. \end{thm} \noindent {\bf D\'emonstration.}\ Il suffit de v\'erifier que $\| . \|_{\op{BT}}$ satisfait \`a la condition donn\'ee au (2) du th\'eor\`eme (\ref{zhang}). Cela d\'ecoule des propri\'et\'es de multiplicativit\'e et de fonctorialit\'e \'enonc\'ees \`a la proposition (\ref{BT_fonct}). \medskip \subsubsection{Construction par image inverse}~ \label{chap_image_inverse} Dans ce paragraphe, $\Delta$ d\'esigne un \'eventail poss\'edant une fonction support strictement concave relativement \`a $\Delta$. Par cons\'equent la vari\'et\'e torique $\M{P}(\Delta)$ est {\em projective}. Soit $D$ un diviseur de Cartier $T$-invariant sur $\M{P}(\Delta)$ et $\C{O}(D)$ le faisceau inversible associ\'e. On suppose dans un premier temps que $\C{O}(D)$ est engendr\'e par ses sections globales. Le choix d'un ordre sur les \'el\'ements de $K_{D} \cap M$ permet de d\'efinir un morphisme $T$-\'equivariant associ\'e \`a $D$, que l'on note $\phi_{D}$, de la fa\c con suivante~: \begin{alignat*}{3} \phi_{D}:\,\M{P} & (\Delta) & &\longrightarrow & & \;\M{P}_{\M{Z}}^{k_{D}} \\ &x & &\longrightarrow & &(\chi^{m}(x))_{m \in K_{D} \cap M} \end{alignat*} o\`u $k_{D}$ est un entier positif d\'efini par $k_{D} = \#(K_{D}\cap M) - 1$. On note $\C{O}(1)$ le fibr\'e de Serre sur $\M{P}_{\M{Z}}^{k_{D}}$ et on le munit de la m\'etrique d\'efinie pour toute section m\'eromorphe de $\C{O}(1)(\M{C})$ par~: \begin{equation} \label{image_inverse} \| s(x) \|_{\infty} = \frac{|s(x)|}{\sup_{1 \leqslant i \leqslant k_{D} + 1}|x_{i}|}\; . \end{equation} Cette m\'etrique est la m\'etrique de Batyrev-Tschinkel ou de Zhang pour le faisceau $\C{O}(1)$ sur $\M{P}_{\M{Z}}^{k_{D}}$ consid\'er\'ee comme vari\'et\'e torique comme dans l'exemple (\ref{exemple_intro_1}). On note $\overline{\C{O}(1)}_{\infty}$ le faisceau $\C{O}(1)$ muni de cette m\'etrique. Comme $\|.\|_{\infty}$ est invariante si l'on permute les \'el\'ements de $K_{D} \cap M$, la m\'etrique sur $\C{O}(D)$ d\'efinie par~: \[ \|.\|_{D,\infty} = \phi_{D}^{\ast}\,\|.\|_{\infty} \] est ind\'ependante du choix effectu\'e pour d\'efinir $\phi_{D}$. On note $\overline{\C{O}(D)}_{\infty}$ le faisceau $\C{O}(D)$ muni de la m\'etrique $\|.\|_{D,\infty}$. On a alors la proposition suivante~: \begin{prop} \label{decomposition_0} Soient $D$ et $E$ deux diviseurs $T$-invariants sur $\M{P}(\Delta)$ dont les faisceaux associ\'es sont engendr\'es par leurs sections globales, on a~: \[ \overline{\C{O}(D)}_{\infty} \otimes \ov{\C{O}(E)}_{\infty} = \ov{\C{O}(D + E)}_{\infty}. \] \end{prop} \noindent {\bf D\'emonstration.}\ On d\'emontre tout d'abord le lemme suivant~: \begin{lem} \label{metrique} Soit $D$ un diviseur $T$-invariant sur $\M{P}(\Delta)$. On notera $K_{D}(0)$ les points entiers extr\'emaux du polytope $K_{D}$. Pour tout $x \in \M{P}(\Delta)(\M{C})$, on a~: \[ \sup_{m \in K_{D} \cap M}|\chi^{m}(x)| = \sup_{m \in K_{D}(0)}|\chi^{m}(x)|. \] \end{lem} \noindent {\bf D\'emonstration.}\ Soient $m_{0}, \dots, m_{q}$ les \'el\'ements de $K_{D}(0)$ et $m$ un \'el\'ement quelconque de $K_{D} \cap M$. Comme $K_{D}$ est convexe, on peut trouver des r\'eels positifs $\alpha_{0}, \dots, \alpha_{q}$ tels que~: \[ \left\{ \begin{array}{l} m = \alpha_{0}m_{0} + \dots + \alpha_{q}m_{q} \\ 1 = \alpha_{0} + \dots + \alpha_{q}. \end{array} \right. \] On en d\'eduit l'in\'egalit\'e~: \[ |\chi^{m}(x)| \leqslant \sup_{0 \leqslant i \leqslant q}|\chi^{m_{i}}(x)|, \] ce qui joint \`a l'inclusion $K_{D}(0) \subset K_{D} \cap M$ donne le r\'esultat annonc\'e. \medskip On passe maintenant \`a la d\'emonstration de la proposition. Soit $s$ (resp. $r$) une section holomorphe de $\C{O}(D)(\M{C})$ (resp. de $\C{O}(E)(\M{C})$) au-dessus d'un ouvert $\Omega$ de $\M{P}(\Delta)(\M{C})$. Pour tout $x \in \Omega$, on a~: \[ \|s(x)\|_{D,\infty} = \frac{|s(x)|}{\sup_{m \in K_{D} \cap M}|\chi^{m}(x)|} \quad \left(\text{resp.} \quad \|r(x)\|_{E,\infty} = \frac{|r(x)|}{\sup_{m \in K_{E} \cap M}|\chi^{m}(x)|} \right). \] En utilisant le lemme (\ref{metrique}) et le fait que~: \[ K_{D + E}(0) = (K_{D} + K_{E})(0) \subseteq K_{D}(0) + K_{E}(0) \subseteq K_{D} + K_{E} = K_{D + E}, \] on obtient l'\'egalit\'e~: \[ \sup_{m \in K_{D + E}}|\chi^{m}(x)| = \left(\sup_{m \in K_{D}}|\chi^{m}(x)|\right) \left(\sup_{m \in K_{E}}|\chi^{m}(x)|\right) \] et la proposition est d\'emontr\'ee. \medskip \begin{prop_defn} \label{decomposition} Soit $D$ un diviseur de Cartier horizontal et $T$-invariant sur $\M{P}(\Delta)$. Il existe $E$ et $F$ des diviseurs de Cartier horizontaux et $T$-invariants sur $\M{P}(\Delta)$, dont le faisceau associ\'e est engendr\'e par ses sections globales, et tels que~: \[ D = E - F. \] La m\'etrique $\| .\|_{E,\infty}\, . \,\| .\|_{F,\infty}^{-1}$ induite sur $\C{O}(D) = \C{O}(E) \otimes \C{O}(F)^{-1}$ par les m\'etriques canoniques $\| .\|_{E,\infty}$ et $\| .\|_{F,\infty}$ est ind\'ependante de $E$ et de $F$. On note $\| .\|_{D,\infty}$ cette m\'etrique et on note $\ov{\C{O}(D)}_{\infty}$ le faisceau $\C{O}(D)$ muni de la m\'etrique $\| .\|_{D,\infty}$. \end{prop_defn} \noindent {\bf D\'emonstration.}\ La premi\`ere partie de l'\'enonc\'e est classique (Serre) et est tr\`es facile \`a g\'en\'eraliser dans le cadre \'equivariant (prendre $H$ diviseur de Cartier $T$-invariant tel que $\C{O}(H)$ est engendr\'e par ses sections globales et consid\'erer $E = nH$ et $F = - D + nH$ pour $n$ assez grand). Soient $E$, $F$, $E'$ et $F'$ des diviseurs de Cartier $T$-invariants dont les faisceaux associ\'es sont engendr\'es par leurs sections globales et tels que~: \[ D = E - F = E' - F'. \] On tire $E + F' = E' + F$, et donc des isomorphismes isom\'etriques~: \[ \ov{\C{O}(E)}_{\infty}\otimes \ov{\C{O}(F')}_{\infty} = \ov{\C{O}(E) \otimes \C{O}(F')}_{\infty} = \ov{\C{O}(E') \otimes \C{O}(F)}_{\infty} = \ov{\C{O}(E')}_{\infty} \otimes \ov{\C{O}(F)}_{\infty}, \] on conclut que~: \[ \ov{\C{O}(E)}_{\infty} \otimes (\ov{\C{O}(F)}_{\infty})^{-1} \simeq \ov{\C{O}(E')}_{\infty} \otimes (\ov{\C{O}(F')}_{\infty})^{-1}, \] ce qui termine la d\'emonstration. \medskip \begin{prop} Soient $E$ et $F$ deux diviseurs de Cartier horizontaux et $T$-invariants sur $\M{P}(\Delta)$. On a un isomorphisme isom\'etrique~: \[ \ov{\C{O}(E)}_{\infty} \otimes \ov{\C{O}(F)}_{\infty} \simeq \ov{\C{O}(E + F)}_{\infty}. \] \end{prop} \noindent {\bf D\'emonstration.}\ La proposition est vraie dans le cas o\`u $\C{O}(E)$ et $\C{O}(F)$ sont engendr\'es par leurs sections globales. Le cas g\'en\'eral s'obtient par diff\'erence \`a partir des propositions (\ref{decomposition_0}) et (\ref{decomposition}). \medskip La construction qui vient d'\^etre donn\'ee co\"\i ncide avec les deux autres pr\'esent\'ees pr\'ec\'edemment~: \begin{thm} Pour tout diviseur de Cartier horizontal $T$-invariant $D$ sur $\M{P}(\Delta)$, les m\'etriques $\|.\|_{\op{BT}}$, $\|.\|_{\op{Zh}}$ et $\|.\|_{D,\infty}$ sur $\C{O}(D)$ co\"\i ncident. \end{thm} \noindent {\bf D\'emonstration.}\ Cela d\'ecoule directement des propri\'et\'es de multiplicativit\'e et de fonctorialit\'e de la m\'etrique $\|.\|_{\op{BT}}$ \'enonc\'ees \`a la proposition (\ref{BT_fonct}), ajout\'e au fait que $\|.\|_{\op{BT}}$ et $\|.\|_{D,\infty}$ co\"\i ncident par construction lorsque $\M{P}(\Delta)$ est l'espace projectif $\M{P}^{n}_{\M{Z}}$ et que $\C{O}(D) = \C{O}(1)$. \medskip Dans toute la suite de ce paragraphe, $\M{P}(\Delta)$ d\'esigne une vari\'et\'e torique {\em projective} et {\em lisse}. On constate que les m\'etriques canoniques d\'efinies pr\'ec\'edemment ne sont pas $C^{\infty}$ en g\'en\'eral (consid\'erer par exemple $\M{P}(\Delta) = \M{P}^{n}_{\M{Z}}$ et $\ov{\C{O}(D)}_{\infty} = \ov{\C{O}(1)}_{\infty}$). On a n\'eanmoins le r\'esultat suivant~: \begin{prop} \label{psh} Soient $\M{P}(\Delta)$ une vari\'et\'e torique projective et lisse, et $D$ un diviseur de Cartier horizontal $T$-invariant sur $\M{P}(\Delta)$ tel que $\C{O}(D)$ soit engendr\'e par ses sections globales. Pour tout ouvert $\Omega \subset \M{P}(\Delta)(\M{C})$ et pour toute section holomorphe $s$ de $\C{O}(D)(\M{C})$ sur $\Omega$ ne s'annulant pas, la fonction $x \mapsto - \log \|s(x)\|_{D,\infty}^{2}$ est continue et plu\-ri\-sous\-har\-mo\-ni\-que\ sur $\Omega$. \end{prop} \noindent {\bf D\'emonstration.}\ La continuit\'e d\'ecoule des d\'efinitions. Comme la condition de plurisousharmonicit\'e est pr\'eserv\'ee par changement de variable holomorphe (voir par exemple \cite{7}, th. 1.5.9), il suffit de d\'emontrer le r\'esultat pour le faisceau $\ov{\C{O}(1)}_{\infty}$ sur $\M{P}^{n}_{\M{Z}}$; et d'apr\`es la formule (\ref{image_inverse}) cela r\'esulte de (\cite{7}, th. 1.5.6 et exemple 1.5.10). On peut \'egalement consulter (\cite{5}, \S 3). \medskip On montre maintenant un th\'eor\`eme d'approximation des m\'etriques canoniques par des m\'etriques $C^{\infty}$ sur $\M{P}(\Delta)(\M{C})$~: \begin{prop} \label{approximation} Soient $\M{P}(\Delta)$ une vari\'et\'e torique projective et lisse, et $D$ un diviseur de Cartier horizontal $T$-invariant sur $\M{P}(\Delta)$ tel que $\C{O}(D)$ soit engendr\'e par ses sections globales. Il existe une suite de m\'etriques $\left( \|.\|_{n}\right)_{n \in \M{N}}$ sur $\C{O}(D)(\M{C})$ convergeant uniform\'ement vers $\|.\|_{D,\infty}$ sur $\M{P}(\Delta)(\M{C})$ et v\'erifiant les conditions suivantes~: \begin{itemize} \item{Les m\'etriques $\|.\|_{n}$ sont $C^{\infty}$.} \item{Si $\Omega \subset \M{P}(\Delta)(\M{C})$ est un ouvert et $s$ une section holomorphe de $\C{O}(D)(\M{C})$ sur $\Omega$ ne s'annulant pas, la fonction $x \mapsto \log \|s(x)\|_{n}^{2}$ est continue et plu\-ri\-sous\-har\-mo\-ni\-que\ sur $\Omega$; en particulier le courant~: \[ c_{1}(\C{O}(D)(\M{C}),\|.\|_{n}) = - d d^{c}\,\log \|s(x)\|^{2}_{n}, \] est positif.} \item{Pour tout $x \in \Omega$, la suite $( - \log \|s(x)\|^{2}_{n})_{n \in \M{N}}$ est d\'ecroissante et converge vers $- \log \|s(x)\|^{2}_{D,\infty}$.} \end{itemize} \end{prop} \noindent {\bf D\'emonstration.}\ La condition de plurisousharmonicit\'e \'etant pr\'eserv\'ee par changement de variable holomorphe, il suffit d'apr\`es la construction par image inverse de d\'emontrer la proposition pour $\ov{\C{O}(1)}_{\infty}$ sur $\M{P}^{m}_{\M{Z}}$. On consid\`ere alors la famille de m\'etriques $\|.\|_{n}$ d\'efinies par~: \[ \|s(x)\|_{n} = \frac{|s(x)|}{\left( \sum_{i = 0}^{m}|x_{i}|^{n}\right)^{1/n}}\; , \] pour toute section locale holomorphe $s$ de $\C{O}(1)(\M{C})$. Les m\'etriques $\|.\|_{n}$ sont $C^{\infty}$ sur $\M{P}(\Delta)(\M{C})$. De plus, la fonction~: \[ (x_{0}, \dots, x_{m}) \longmapsto \log (e^{x_{0}} + \dots + e^{x_{m}}) \] \'etant convexe et croissante en chacun des $x_{i}$ et la fonction $t \mapsto \log |t|$ \'etant sousharmonique, la fonction $\log \left( \sum_{i=0}^{m}|x_{i}|^{n}\right)^{1/n}$ est plu\-ri\-sous\-har\-mo\-ni\-que\ (voir par exemple \cite{7}, th. 1.5.6 et \cite{5}, \S 3). Enfin la suite $\left(\sum_{i =0}^{m}|x_{i}|^{n}\right)^{1/n}$, $(n \in \M{N})$, est d\'ecroissante et converge vers $\sup_{0 \leqslant i \leqslant m}|x_{i}|$, et la positivit\'e de $c_{1}(\C{O}(1)(\M{C}),\|.\|_{n})$ d\'ecoule directement de (\cite{7}, exemple 3.1.18). \bigskip \subsection{M\'etriques canoniques sur les fibr\'es en droites sur $\M{P}(\Delta)$.}~ Dans cette section, $\Delta$ d\'esigne un \'eventail complet et r\'egulier de $N$ poss\'edant une fonction support strictement concave. En d'autres termes, on suppose que $\M{P}(\Delta)$ est une vari\'et\'e torique projective lisse. Pour tout fibr\'e en droites $L$ sur $\M{P}(\Delta)$, on construit de mani\`ere canonique une m\'etrique sur $L(\M{C})$. \begin{prop_defn} \label{metrique_ind} Soit $L$ un fibr\'e en droites sur $\M{P}(\Delta)$. Il existe un diviseur horizontal $T$-invariant $D$ sur $\M{P}(\Delta)$ et un isomorphisme~: \[ \Phi: L \longrightarrow \C{O}(D). \] La m\'etrique $\Phi^{\ast}\|.\|_{D,\infty}$ sur $L$ est ind\'ependante des choix de $D$ et $\Phi$. On l'appelle {\it m\'etrique canonique\/} sur $L$ et on la note $\|.\|_{L,\infty}$, ou plus simplement $\|.\|_{\infty}$ lorsqu'aucune confusion n'est \`a craindre. On note $\ov{L}_{\infty} = (L,\|.\|_{L,\infty})$ le fibr\'e $L$ muni de sa m\'etrique canonique. \end{prop_defn} \noindent {\bf D\'emonstration.}\ L'existence de $D$ et $\Phi$ est une reformulation de la surjectivit\'e de $s$ dans la proposition (\ref{picard}). Soit maintenant $D'$ un second diviseur horizontal $T$-invariant de $\M{P}(\Delta)$ tel qu'il existe un isomorphisme $\Phi': L \simeq \C{O}(D')$. Le diviseur $D-D'$ est principal, et il existe un unique \'el\'ement $m \in M$ tel que $D-D' = \op{div}\chi^{m}$ d'apr\`es la proposition (\ref{picard}). Comme les unit\'es globales sur $\M{P}(\Delta)$ sont $\{1,-1\}$, les isomorphismes $\Phi$ et $\Phi'$ sont uniques au signe pr\`es. Pour montrer que~: \[ \Phi^{\ast}\|.\|_{D,\infty} = {\Phi'}^{\ast}\|.\|_{D',\infty} \] il suffit donc de v\'erifier que l'isomorphisme~: \[ \C{O}(D') \simeq \C{O}(D) \] d\'efini par la multiplication par $\chi^{m}$ transporte $\|.\|_{D',\infty}$ sur $\|.\|_{D,\infty}$. Cela d\'ecoule de l'expression (\ref{BT_definition}) de Batyrev et Tschinkel pour ces m\'etriques. \medskip La proposition suivante est une cons\'equence imm\'ediate du th\'eor\`eme (\ref{zhang}). \begin{prop} \label{relation_zhang} Soit $p$ un entier sup\'erieur ou \'egal \`a $2$. Pour tout fibr\'e en droites $L$ sur $\M{P}(\Delta)$, on a un isomorphisme isom\'etrique~: \[ [p]^{\ast}(\ov{L}_{\infty}) \simeq (\ov{L}_{\infty})^{\otimes p}, \] et $\|.\|_{L,\infty}$ est l'unique m\'etrique continue telle que $\ov{L}_{\infty} = (L,\|.\|_{L,\infty})$ v\'erifie cette propri\'et\'e. \end{prop} \bigskip \section{Un th\'eor\`eme de Bernstein-Koushnirenko arithm\'etique}~ Ce chapitre est consacr\'e \`a \'etablir, comme application des r\'esultats des pages qui pr\'ec\`edent, un analogue arithm\'etique du th\'eor\`eme de Bernstein-Koushnirenko. Rappelons que ce th\'eor\`eme fournit une borne du nombre de z\'eros communs dans $(\M{C}^{\ast})^{n}$ \`a $n$ polyn\^omes de Laurent $P_{1}, \dots, P_{n}$ dans $\M{C}[X_{1},X_{1}^{-1}, \dots, X_{n}, X_{n}^{-1}]$ en terme de volumes mixtes associ\'es aux polyh\`edres de Newton de $P_{1},\dots,P_{n}$. Lorsque $P_{1},\dots,P_{n}$ appartiennent \`a $\M{Z}[X_{1},X_{1}^{-1}, \dots, X_{n}, X_{n}^{-1}]$ nous d\'emontrons une majoration sur la hauteur de leurs z\'eros communs (cf. corollaire (\ref{coro_BK})). Cette majoration fait intervenir un certain invariant r\'eel $L(\nabla)$, associ\'e \`a un polytope convexe $\nabla$ dans $M$, que nous d\'efinissons et \'etudions dans la section (\ref{invariant_polytope}). \medskip \subsection{A propos d'une constante associ\'ee \`a un polytope convexe}~ \label{invariant_polytope} \subsubsection{D\'efinitions et propri\'et\'es} Soit $\nabla$ un polytope convexe dans $M_{\M{R}}$ \`a sommet dans $M$ que l'on suppose d'int\'erieur non vide (si tel n'est pas le cas, on s'y ram\`ene en se pla\c cant dans $M' = M \cap (\M{R}\nabla+ \M{R}(-\nabla))$. On note $\Delta$ l'\'eventail dans $N$ associ\'e \`a $\nabla$ par le th\'eor\`eme (\ref{construction_inverse}) et $\M{P}(\nabla)$ la vari\'et\'e torique $\M{P}(\Delta)$. On note \'egalement $E$ l'unique diviseur de Cartier horizontal $T$-invariant sur $\M{P}(\nabla)$ tel que $K_{E} = \nabla$. D'apr\`es (\ref{construction_inverse}), on sait que le faisceau $\C{O}(E)$ est ample. On associe alors \`a $\nabla$ la constante r\'eelle $L(\nabla)$ d\'efinie de la mani\`ere suivante~: \[ L(\nabla) = \Sup_{s \in \Gamma(\M{P}(\nabla)(\M{C}), \C{O}(E))}\left(\Sup_{x \in \M{P}(\nabla)(\M{C})}\log \|s(x)\|_{E,\infty} - M(s) \right) . \] \begin{prop} La constante $L(\nabla)$ est bien d\'efinie et est positive. \end{prop} \noindent {\bf D\'emonstration.}\ La diff\'erence~: \[ \Sup_{x \in \M{P}(\nabla)(\M{C})}\log \|s(x)\|_{E,\infty} - \int_{S_{N}^{+}}\log |s(x)|\, d\mu^{+} \] ne change pas lorsque $s$ est multipli\'ee par une constante $\lambda \in \M{C}^{\ast}$. On a donc~: \[ L(\nabla) = \Sup_{s \in \M{P}(\Gamma(\M{P}(\nabla)(\M{C}), \C{O}(E)))}\left(\Sup_{x \in \M{P}(\nabla)(\M{C})}\log \|s(x)\|_{E,\infty} - M(s) \right) . \] Comme $\dim_{\M{C}}\Gamma(\M{P}(\nabla)(\M{C}),\C{O}(E))$ est finie, $\M{P}(\Gamma(\M{P}(\nabla)(\M{C}),\C{O}(E)))$ est compact et l'on d\'eduit de la proposition (\ref{mahler_remarque}) que $|L(\nabla)| < + \infty$. Enfin pour toute section $s \in \Gamma(\M{P}(\nabla)(\M{C}),\C{O}(E))$, on a~: \[ \int_{S_{N}^{+}}\log |s(x)| \, d\mu^{+} \leqslant \Sup_{x \in \M{P}(\nabla)(\M{C})}\log \|s(x)\|_{E,\infty} \int_{S_{N}^{+}}d\mu^{+} = \Sup_{x \in \M{P}(\nabla)(\M{C})}\log \|s(x)\|_{E,\infty}, \] et donc $L(\nabla) \geqslant 0$. \medskip \begin{prop} \label{universalite} Soient $\Delta'$ un \'eventail complet dans $N_{\M{R}}$ et $E_{1}$ un diviseur de Cartier $T$-invariant sur $\M{P}(\Delta')$ tel que $\C{O}(E_{1})$ soit engendr\'e par ses sections globales. Si l'on note $\nabla_{1} = K_{E_{1}} \subset M_{\M{R}}$ le polytope convexe \`a sommets dans $M$ associ\'e \`a $E_{1}$, alors on a~: \[ L(\nabla_{1}) = \Sup_{s \in \Gamma(\M{P}(\Delta')(\M{C}),\C{O}(E_{1}))} \left(\Sup_{x \in \M{P}(\Delta')(\M{C})}\log \|s(x)\|_{E_{1},\infty} - \int_{S_{N}^{+}}\log |s(x)|\,d\mu^{+}\right) . \] \end{prop} \noindent {\bf D\'emonstration.}\ On pose $V' = \M{R}\nabla_{1} + \M{R}(-\nabla_{1})$ et on note $M' = M \cap V'$. Soit $M''$ un sous-groupe de $M$ tel que l'on ait~: $M = M'\oplus M''$; on tire~: \[ T(\M{C}) = \op{Spec}(\M{C}[M']) \times \op{Spec}(\M{C}[M'']) = \M{G}_{\op{m}}^{\op{rg}M'}(\M{C}) \times \M{G}_{\op{m}}^{\op{rg}M''}(\M{C}). \] Notons $pr_{1}$ la premi\`ere projection et soit $m \in M'$. Par construction, $m$ induit un caract\`ere $\chi_{\nabla_{1}}^{m}$ sur $\M{G}_{\op{m}}^{\op{rg}M'}(\M{C}) \subset \M{P}(\nabla_{1})(\M{C})$. Du fait de l'inclusion $M' \subset M$, $m$ induit \'egalement un caract\`ere $\chi_{\Delta'}^{m}$ sur $T(\M{C}) \subset \M{P}(\Delta')(\M{C})$ et les deux caract\`eres sont li\'es par la relation~: \[ \chi_{\Delta'}^{m} = \chi_{\nabla_{1}}^{m}\circ pr_{1} = pr_{1}^{\ast}(\chi_{\nabla_{1}}^{m}). \] On note $E_{1}'$ le diviseur de Cartier $T$-invariant sur $\M{P}(\nabla_{1})$, dont l'existence est assur\'ee par le th\'eor\`eme (\ref{construction_inverse}), tel que $K_{E_{1}'} = \nabla_{1}$ et $E_{1}'$ est ample. D'apr\`es la proposition (\ref{sections_globales}), on dispose d'un isomorphisme canonique~: \[ pr_{1}^{\ast}: \Gamma(\M{P}(\nabla_{1})(\M{C}),\C{O}(E_{1}')) \longrightarrow \Gamma(\M{P}(\Delta')(\M{C}),\C{O}(E_{1})). \] Pour tout $s \in \Gamma(\M{P}(\nabla_{1})(\M{C}),\C{O}(E_{1}'))$ et $x \in T(\M{C})$, on a~: \[ \|pr_{1}^{\ast}(s)(x)\|_{E_{1},\infty} = \|s(pr_{1}(x))\|_{E_{1}',\infty}, \] et comme $\M{G}_{\op{m}}^{\op{rg}M'}(\M{C})$ (resp. $T(\M{C})$) est dense dans $\M{P}(\nabla_{1})(\M{C})$ (resp. dans $\M{P}(\Delta')(\M{C})$), on a~: \[ \Sup_{x \in \M{P}(\Delta')(\M{C})}\|pr_{1}^{\ast}(s)(x)\|_{E_{1},\infty} = \Sup_{x' \in \M{P}(\nabla_{1})(\M{C})}\|s(x')\|_{E_{1}',\infty}. \] De plus, pour tout $s \in \Gamma(\M{P}(\nabla_{1})(\M{C}),\C{O}(E_{1}'))$, on a~: \[ \int_{\C{S}_{N}^{+}} \log |pr_{1}^{\ast}(s)(x)|\,d\mu^{+} = \int_{\C{S}_{N'}^{+}}\log |s(x')|\,d\mu^{+}. \] On en d\'eduit que~: \begin{align*} L(\nabla_{1}) &= \Sup_{s \in \Gamma(\M{P}(\nabla_{1})(\M{C}),\C{O}(E_{1}'))} \left(\Sup_{x' \in \M{P}(\nabla_{1})(\M{C})}\log \|s(x')\|_{E_{1}',\infty} - \int_{S_{N'}^{+}}\log |s(x')|\,d\mu^{+}\right) \\ &= \Sup_{s \in \Gamma(\M{P}(\Delta')(\M{C}),\C{O}(E_{1}))} \left(\Sup_{x \in \M{P}(\Delta')(\M{C})}\log \|s(x)\|_{E_{1},\infty} - \int_{S_{N}^{+}}\log |s(x)|\,d\mu^{+}\right). \end{align*} \medskip \begin{prop}{\rm (Fonctorialit\'e).} Soient $\nabla_{1}$ et $\nabla_{2}$ deux polytopes convexes dans $M$ et soit $\nabla = \nabla_{1} + \nabla_{2}$. Pour $i \in \{1,2\}$, on a~: \[ L(\nabla_{i}) \leqslant L(\nabla). \] \end{prop} \noindent {\bf D\'emonstration.}\ Il suffit de d\'emontrer le r\'esultat pour $i =1$. On suppose que $\nabla$ est d'int\'erieur non vide (si tel n'est pas le cas, on s'y ram\`ene en se pla\c cant dans $M' = M \cap (\M{R}\nabla + \M{R}(-\nabla))$. D'apr\`es le th\'eor\`eme (\ref{construction_inverse2}), il existe des diviseurs de Cartier $T$-invariants $E_{1}$, $E_{2}$ sur $\M{P}(\nabla)$ tels que l'on ait $K_{E_{i}} = \nabla_{i}$ pour $i \in \{1,2\}$. Les faisceaux inversibles $\C{O}(E_{1})$ et $\C{O}(E_{2})$ sont engendr\'es par leurs sections globales et l'on a $E = E_{1} + E_{2}$. Soit $s_{1} \in \Gamma(\M{P}(\nabla)(\M{C}),\C{O}(E_{1}))$ et soit $x_{0} \in \M{P}(\nabla)(\M{C})$ tel que~: \[ \|s_{1}(x_{0})\|_{E_{1},\infty} = \Sup_{x \in \M{P}(\nabla)(\M{C})}\|s_{1}(x)\|_{E_{1},\infty}. \] On peut trouver $\sigma \in \Delta_{\op{max}}$ tel que $x_{0} \in C_{\sigma}$. On d\'eduit de l'\'egalit\'e $E = E_{1} + E_{2}$ que $m_{\nabla,\sigma} = m_{\nabla_{1},\sigma} + m_{\nabla_{2},\sigma}$. D'apr\`es le th\'eor\`eme (\ref{polytope_et_vt}) on a $m_{\nabla_{2},\sigma} \in \nabla_{2}$, et donc $s_{1}\otimes \chi^{m_{\nabla_{2},\sigma}} \in \Gamma(\M{P}(\nabla)(\M{C}),\C{O}(E))$. De plus gr\^ace \`a la proposition (\ref{BT_construction}) on peut affirmer que~: \[ \|s_{1}\otimes \chi^{m_{\nabla_{2},\sigma}}(x_{0})\|_{E,\infty} = \|s_{1}(x_{0})\|_{E_{1},\infty} \] et donc que~: \[ \Sup_{x \in \M{P}(\nabla)(\M{C})}\|s_{1}\otimes \chi^{m_{\nabla_{2},\sigma}}(x)\|_{E,\infty} \geqslant \Sup_{x \in \M{P}(\nabla)(\M{C})}\|s_{1}(x)\|_{E_{1},\infty}. \] Comme $\int_{S_{N}^{+}}\log |\chi^{m_{\nabla_{2},\sigma}}|\,d\mu^{+} = 0$, on d\'eduit finalement de la proposition (\ref{universalite}) la majoration~: \begin{align*} L(\nabla_{1}) &= \Sup_{s_{1} \in \Gamma(\M{P}(\nabla)(\M{C}),\C{O}(E_{1}))}\left( \Sup_{x \in \M{P}(\nabla)(\M{C})}\log \|s_{1}(x)\|_{E_{1},\infty} - \int_{S_{N}^{+}}\log |s_{1}(x)|\,d\mu^{+}\right) \\ &\leqslant \Sup_{s \in \Gamma(\M{P}(\nabla)(\M{C}),\C{O}(E))} \left(\Sup_{x \in \M{P}(\nabla)(\M{C})}\log \|s(x)\|_{E,\infty} - \int_{S_{N}^{+}}\log |s(x)|\, d\mu^{+}\right) \\ &= L(\nabla). \end{align*} \medskip \subsubsection{Majoration de $L(\nabla)$.} Dans ce paragraphe, $\nabla$ d\'esigne un polytope convexe dans $M_{\M{R}}$ \`a sommets dans $M$ et suppos\'e {\em d'int\'erieur non vide} (si tel n'est pas le cas, on s'y ram\`ene en se pla\c cant dans $M' = M\cap (\M{R} \nabla + \M{R} (- \nabla))$). On note $\nabla_{0}$ l'ensemble des sommets de $\nabla$ et $\Delta$ l'\'eventail complet associ\'e \`a $\nabla$ par la construction du th\'eor\`eme (\ref{construction_inverse}). On note $E$ le diviseur ample sur $\M{P}(\Delta)$ associ\'e \`a $\nabla$ par cette construction; on sait que $K_{E} = \nabla$. D'apr\`es la remarque (\ref{construction_inverse3}), il existe un raffinement $\Delta'$ de $\Delta$ tel que $\M{P}(\Delta')$ est projective et lisse. On note $i_{\ast}: \M{P}(\Delta') \rightarrow \M{P}(\Delta)$ le morphisme \'equivariant induit par l'inclusion $i: \Delta' \hookrightarrow \Delta$ et on pose $E' = (i_{\ast})^{\ast}(E)$. On sait alors que $K_{E'} = \nabla$ et le faisceau inversible $\C{O}(E')$ est engendr\'e par ses sections globales. Suivant Lelong (cf. \cite{18}) on d\'efinit pour $n$ un entier strictement positif les constantes suivantes~: \begin{align*} C_{n} &= \frac{1}{2}\sum_{i=1}^{n-1}\frac{1}{i}\qquad \text{et} \qquad C_{n} = 0 \quad \text{pour} \quad n = 1, \\ C_{n}' &= \sum_{i =1}^{2n-2}\frac{1}{i}+ \sum_{i = 2n-1}^{+\infty} \frac{1}{i2^{i}}\; . \end{align*} La proposition suivante est une cons\'equence directe des \'enonc\'es donn\'es dans \cite{18}~: \begin{prop} \label{majoration_lelong} Soit $P \in \M{C}[X_{1},\dots,X_{n}]$ un polyn\^ome, on a~: \[ \Sup_{\substack{|z_{i}| \leqslant 1 \\ 1 \leqslant i \leqslant n}} \log |P(z_{1}, \dots, z_{n})| - M(P) \leqslant (C_{n} + C_{n}')\op{deg}P. \] \end{prop} \noindent {\bf D\'emonstration.}\ Voir \cite{18}, th. 2, prop. 4 et \'equation (14). Voir aussi \cite{19}. \medskip A tout $\sigma \in \Delta_{\op{max}}'$ on attache un nombre r\'eel $L(\sigma)$ que l'on d\'efinit de la fa\c con suivante~: \[ L(\sigma) = \Sup_{s \in \Gamma(\M{P}(\Delta')(\M{C}),\C{O}(E'))} \left( \Sup_{x \in C_{\sigma}} \log \|s(x)\|_{E',\infty} - \int_{S_{N}^{+}}\log |s(x)|\, d\mu^{+} \right). \] L'\'eventail $\Delta'$ \'etant complet, on d\'eduit des propositions (\ref{recouvrement}) et (\ref{universalite}) l'\'egalit\'e~: \begin{equation} \label{eq_bern_1} L(\nabla) = \Sup_{\sigma \in \Delta'_{\op{max}}}L(\sigma). \end{equation} Pour tout $\sigma \in \Delta_{\op{max}}'$, on choisit $f_{1},\dots,f_{d}$ une famille g\'en\'eratrice de $\sigma$ (qui est donc une $\M{Z}$-base de $N$) et on note $f_{1}^{\ast}, \dots, f_{d}^{\ast}$ la base duale de $M$. Dans la carte affine $\varphi: U_{\sigma}(\M{C}) \rightarrow \M{C}^{d}$ donn\'ee par $\varphi(x) = (\chi^{f_{1}^{\ast}}(x), \dots, \chi^{f_{d}^{\ast}}(x))$, l'ensemble $C_{\sigma}$ est d\'efini par les conditions~: \[ C_{\sigma} = \{z \in \M{C}^{d}: \quad |z_{1}| \leqslant 1, \dots, |z_{d}| \leqslant 1\}. \] Soit $s \in \Gamma(\M{P}(\Delta')(\M{C}), \C{O}(E'))$. Dans la carte affine $U_{\sigma}(\M{C})$, la fonction rationnelle $Q = s \cdot \chi^{- m_{E,\sigma}}$ est un polyn\^ome. Comme \[ \int_{S_{N}^{+}}\log |Q|\, d \mu^{+} = \int_{S_{N}^{+}}\log |s \cdot \chi^{- m_{E,\sigma}}|\, d\mu^{+} = \int_{S_{N}^{+}}\log |s|\, d \mu^{+}, \] et que pour tout $x \in C_{\sigma}$ on a~: \[ \log \|s(x)\|_{E,\infty} = \log|Q(x)|, \] on obtient d'apr\`es la proposition (\ref{majoration_lelong}) la majoration~: \begin{align} \label{eq_bern_2} \Sup_{x \in C_{\sigma}}\log \|s(x)\|_{E,\infty} - \int_{S_{N}^{+}}\log |s|\, d\mu^{+} &= \Sup_{\substack{|z_{i}| \leqslant 1 \\ 1 \leqslant i \leqslant d}} \log |Q(z)| - M(Q) \notag \\ &\leqslant (C_{d} + C_{d}')\op{deg}(Q). \end{align} Au vu de la description des sections globales de $\C{O}(E')$ que l'on d\'eduit de la proposition (\ref{sections_globales}), il vient~: \[ \op{deg}Q \leqslant \Sup_{m \in (\nabla - m_{E',\sigma})}\|m\|_{1} = \Sup_{m \in (\nabla_{0} - m_{E',\sigma})}\|m\|_{1}, \] o\`u l'on a pos\'e $\|m\|_{1} = \sum_{i = 1}^{d}|\hbox{$<f_{i},m>$}|$. On d\'eduit de cela et de (\ref{eq_bern_2}) la majoration~: \begin{equation} \label{eq_bern_3} L(\sigma) \leqslant (C_{d} + C_{d}')\Sup_{m \in (\nabla_{0} - m_{E',\sigma})}\|m\|_{1}. \end{equation} Les relations (\ref{eq_bern_1}) et (\ref{eq_bern_3}) fournissent un proc\'ed\'e th\'eorique pour l'obtention d'une majoration de l'invariant $L(\nabla)$. Malheureusement, la d\'etermination de l'\'eventail $\Delta'$, et {\it a fortiori\/} de la base $f_{1},\dots,f_{d}$, repose non seulement sur la connaissance de la g\'eom\'etrie du polytope $\nabla$ mais \'egalement sur les propri\'et\'es arithm\'etiques de l'\'eventail $\Delta$. Il n'est donc pas possible, en g\'en\'eral, de trouver une majoration de $\op{deg}Q$ en fonction de la g\'eom\'etrie du polytope $\nabla$ uniquement. \smallskip On suppose d\'esormais que $\nabla$ est un polytope absolument simple, ce qui nous autorise \`a poser $\Delta' = \Delta$. On donne, dans ce cas, une majoration totalement explicite de l'invariant $L(\nabla)$ en terme de la combinatoire du polytope $\nabla$. Pour tout $S \in \nabla_{0}$ on note $l_{1}(S), \dots , l_{d}(S)$ la base de $M$ associ\'ee au sommet $S$ comme \`a la proposition (\ref{simplicite}). Pour tout $S \in \nabla_{0}$ et tout $P \in (\nabla \cap M)$ on peut trouver $a_{1}^{(S)}(P), \dots, a_{d}^{(S)}(P)$ des entiers positifs tels que~: \[ P-S = \sum_{i =1}^{d}a_{i}^{(S)}(P)l_{i}(S). \] On pose alors~: \[ N_{S}(P) = \sum_{i = 1}^{d} a_{i}^{(S)}(P) \in \M{N}. \] \begin{defn} On appelle {\it norme\/} du polytope convexe absolument simple $\nabla$ et on note $N(\nabla)$ l'entier strictement positif d\'efini par~: \[ N(\nabla) = \Sup_{S \in \nabla_{0}} \Sup_{S' \in \nabla_{0}\backslash \{S\} } N_{S}(S'). \] \end{defn} \begin{prop} \label{majoration_cas_AS} Soit $\nabla$ un polytope convexe absolument simple dans $M$. On a~: \[ L(\nabla) \leqslant (C_{d} + C_{d}')N(\nabla). \] \end{prop} \noindent {\bf D\'emonstration.}\ On d\'eduit des d\'efinitions l'in\'egalit\'e~: \[ \Sup_{m \in (\nabla_{0} - m_{E,\sigma})}\|m\|_{1} = \Sup_{S \in \nabla_{0}\backslash \{m_{E,\sigma}\}}N_{m_{E,\sigma}}(S) \leqslant N(\nabla), \] valable pour tout $\sigma \in \Delta_{\op{max}}$, ce qui joint aux relations (\ref{eq_bern_1}) et (\ref{eq_bern_3}) donne le r\'esultat annonc\'e. \bigskip \subsection{Un th\'eor\`eme de Bernstein-Koushnirenko arithm\'etique.}~ \subsubsection{Rappels.} Soient $X \in Z_{p}(\M{P}(\nabla))$ et $Y \in Z_{q}(\M{P}(\nabla))$ deux cycles effectifs tels que $p + q \geqslant d +1$. On note $W_{1}, \dots, W_{r}$ les composantes irr\'eductibles de l'intersection (ensembliste) $|X| \cap |Y|$, ce qui nous permet d'\'ecrire~: \[ |X| \cap |Y| = \bigcup_{1 \leqslant i \leqslant r} W_{i}\; . \] Pour tout $1 \leqslant i \leqslant r$, on a~: \begin{equation} \label{int_propre} \op{dim}W_{i} \geqslant p + q - d - 1. \end{equation} On dit que $W_{i}$ est {\it propre\/} si (\ref{int_propre}) est une \'egalit\'e, et que $W_{i}$ est {\it impropre\/} sinon. On d\'efinit alors la {\it partie propre\/} $(X\cdot Y)_{\op{pr}}$ de l'intersection de $X$ avec $Y$ par la formule~: \[ (X\cdot Y)_{\op{pr}} = \sum_{\text{$W_{i}$ propre}}m_{i}W_{i} \in Z_{p+q -d -1}(\M{P}(\nabla)), \] o\`u $m_{i}$ d\'esigne la multiplicit\'e d'intersection de $X$ avec $Y$ le long de $W_{i}$ donn\'ee par la formule des ``Tor'' de Serre. En d'autres termes, $(X \cdot Y)_{\op{pr}}$ est la somme des composantes propres de l'intersection de $X$ avec $Y$ compt\'ees avec leur multiplicit\'e. Si $X$, $Y$ et $Z$ sont trois cycles effectifs de $\M{P}(\nabla)$, alors on a~: \[ ((X\cdot Y)_{\op{pr}}\cdot Z)_{\op{pr}} = (X \cdot (Y\cdot Z)_{\op{pr}})_{\op{pr}}, \] et dans toute la suite on \'ecrira $(X\cdot Y \cdot Z)_{\op{pr}}$ pour d\'esigner l'un ou l'autre de ces cycles. Pour plus de d\'etails sur ces questions, on peut consulter (\cite{3}, \S 5.5) d'o\`u est extraite notre pr\'esentation. Ceci \'etant rappel\'e, nous \'enon\c cons les r\'esultats que nous avons en vue~: \subsubsection{Pr\'esentation des r\'esultats.} On se place sur $\M{P}(\Delta')$ une vari\'et\'e torique projective lisse de dimension relative $d$, associ\'ee \`a un \'eventail $\Delta' \subset N_{\M{R}}$. On consid\`ere $E_{1}, \dots, E_{d}$ des diviseurs de Cartier horizontaux $T$-invariants sur $\M{P}(\Delta')$ tels que les faisceaux inversibles $\C{O}(E_{1}), \dots, \C{O}(E_{d})$ soient engendr\'es par leurs sections globales. On pose $E = E_{1} + \dots + E_{d}$ et on note $\nabla = K_{E}$, $\nabla_{1} = K_{E_{1}}, \dots, \nabla_{d} = K_{E_{d}}$ les polytopes convexes \`a sommets dans $M$ associ\'es \`a $E$, $E_{1}, \dots, E_{d}$ respectivement; on sait d'apr\`es (\ref{additivite_1}) que $\nabla = \nabla_{1} + \dots + \nabla_{d}$. On supposera ici que le polytope $\nabla$ est {\em d'int\'erieur non vide\/}. \begin{thm} \label{B_K} Soient $s_{1},\dots,s_{d}$ des sections r\'eguli\`eres non nulles sur $\M{P}(\Delta')$ des faisceaux $\C{O}(E_{1}), \dots, \C{O}(E_{d})$; et notons $\op{div}s_{1}, \dots, \op{div}s_{d}$ respectivement le lieu de leurs z\'eros. On a l'in\'egalit\'e~: \begin{multline*} h_{\ov{\C{O}(E)}_{\infty}}((\op{div}s_{1} \dotsm \op{div}s_{d})_{\op{pr}}) \\ \leqslant \sum_{i=1}^{d}\frac{\op{deg}(E\cdot E_{1} \dotsm \widehat{E_{i}}\dotsm E_{d})}{\op{deg}(E^{d})}\, h_{\ov{\C{O}(E)}_{\infty}} (\op{div}s_{i}) \\ + \sum_{i =1}^{d}L(\nabla_{i})\op{deg}(E\cdot E_{1} \dotsm \widehat{E_{i}} \dotsm E_{d}), \end{multline*} o\`u le symbole $\widehat{E_{i}}$ signifie que l'on omet ce terme dans le produit d'intersection consid\'er\'e. \end{thm} En utilisant le fait que~: \[ \op{deg}(E\cdot E_{1}\dotsm \widehat{E_{i}} \dotsm E_{d}) = d!\, V(\nabla,\nabla_{1}, \dots, \widehat{\nabla_{i}}, \dots, \nabla_{d}), \] o\`u $V(\nabla,\nabla_{1},\dots, \widehat{\nabla_{i}}, \dots, \nabla_{d})$ d\'esigne le volume mixte des polytopes $\nabla,\nabla_{1}, \dots, \widehat{\nabla_{i}}, \linebreak[4] \dots, \nabla_{d}$ (voir par exemple \cite{20}, \S A.4 et p. 78-79, et aussi \cite{11}, \S 5.4) et l'\'egalit\'e~: \[ h_{\ov{\C{O}(E)}_{\infty}}(\op{div}s_{i}) = \op{deg}(E^{d})\, M(s_{i}), \] valable pour tout $1 \leqslant i \leqslant d$ d'apr\`es la proposition (\ref{hauteur_hypersurfaces}), on peut r\'e\'ecrire la majoration du th\'eor\`eme (\ref{B_K}) sous la forme~: \begin{multline} \label{BK_intermediaire} h_{\ov{\C{O}(E)}_{\infty}}((\op{div}s_{1} \dotsm \op{div}s_{d})_{\op{pr}}) \\ \leqslant d!\, \sum_{i =1}^{d}V(\nabla,\nabla_{1},\dots,\widehat{\nabla_{i}},\dots,\nabla_{d})\, M(s_{i}) \\ + d!\, \sum_{i=1}^{d}L(\nabla_{i})V(\nabla,\nabla_{1},\dots,\widehat{\nabla_{i}},\dots, \nabla_{d}). \end{multline} \medskip \begin{defn} Soit $P \in \M{Z}[X_{1},1/X_{1}, \dots, X_{d},1/X_{d}]$ un polyn\^ome de Laurent, et notons $(a_{m})_{m \in \M{Z}^{d}}$ la famille presque nulle de ses coefficients (i.e. d\'efinie par l'\'egalit\'e $P(X) = \sum_{m \in \M{Z}^{d}}a_{m}X^{m}$). On appelle {\it support\/} du polyn\^ome $P$ et on note $\op{Supp}P$ le sous-ensemble fini de $\M{Z}^{d}$ d\'efini par~: \[ \op{Supp}P = \{ m \in \M{Z}^{d}: \;\; a_{m} \neq 0\}. \] Le {\it polyh\`edre de Newton\/} du polyn\^ome $P$ est l'enveloppe convexe de son support $\op{Supp}P$. C'est un polytope convexe \`a sommets dans $\M{Z}^{d}$. \end{defn} Pour tout corps de nombres $K \subset \ov{\M{Q}}$, notons $S_{K}$ l'ensemble canonique des places de $K$ (voir par exemple \cite{31}, II \S 1). Pour tout $\nu \in S_{K}$, on note $|\cdot|_{\nu}$ la valeur absolue {\em normalis\'ee} sur $K$ en la place $\nu$, celle-ci \'etant d\'efinie par~: $|\cdot|_{\nu} = |\sigma(\cdot)|$ si $\nu$ est associ\'ee \`a un plongement r\'eel $\sigma: K \hookrightarrow \M{R}$, $|\cdot|_{\nu} = |\sigma(\cdot)|^{2}$ si $\nu$ est associ\'ee \`a un plongement complexe $\sigma: K \hookrightarrow \M{C}$, et $|\cdot|_{\nu} = (N\mathfrak{p})^{- v_{\mathfrak{p}}(\cdot)}$ si $\nu$ est une place non-archim\'edienne associ\'ee \`a un id\'eal premier $\mathfrak{p}$ de $\C{O}_{K}$. On peut alors \'enoncer le corollaire suivant, de forme plus \'el\'ementaire~: \begin{cor} \label{coro_BK} Soient $P_{1},\dots,P_{d} \in \M{Z}[X_{1},1/X_{1}, \dots, X_{d},1/X_{d}]$ des polyn\^omes de Laurent \`a coefficients entiers, et notons $\nabla_{1}, \dots ,\nabla_{d}$ respectivement leur polyh\`edre de Newton. Notons $\nabla = \nabla_{1} + \dots + \nabla_{d}$ leur somme de Minkowski, que nous supposons d'int\'erieur non vide. Notons $Z_{1}, \dots, Z_{d}$ le lieu des z\'eros de $P_{1},\dots,P_{d}$ respectivement dans $(\ov{\M{Q}}^{\ast})^{d}$ et $(Z_{1} \cap \dots \cap Z_{d})_{\op{pr}}$ l'ensemble des points isol\'es du sch\'ema $Z_{1}\cap \dots \cap Z_{d}$, et pour chaque point $x$ de cet ensemble notons $l(x)$ sa multiplicit\'e et $h_{\nabla}(x)$ sa hauteur d\'efinie par l'\'egalit\'e~: \[ h_{\nabla}(x) = \sum_{\nu \in S_{K}}\log \left(\max_{m \in \nabla\cap M}|\chi^{m}(x)|_{\nu}\right), \] o\`u $K$ d\'esigne un corps de nombres contenant les coordonn\'ees de $x$. On a~: \[ \frac{1}{d!}\sum_{x \in (Z_{1}\cap \dots \cap Z_{d})_{\op{pr}}}l(x)h_{\nabla}(x) \leqslant \sum_{i =1}^{d}V(\nabla,\nabla_{1}, \dots, \widehat{\nabla_{i}}, \dots, \nabla_{d}) (M(P_{i}) + L(\nabla_{i})). \] \end{cor} \noindent {\bf D\'emonstration.}\ Soit $\Delta$ l'\'eventail dans $N$ que l'on associe \`a $\nabla$ gr\^ace au th\'eor\`eme (\ref{construction_inverse}) et soit $E$ l'unique diviseur de Cartier horizontal $T$-invariant sur $\M{P}(\Delta)$ tel que $K_{E} = \nabla$ et $E$ est ample. D'apr\`es le th\'eor\`eme (\ref{construction_inverse2}) il existe des diviseurs de Cartier horizontaux $T$-invariants $E_{1},\dots,E_{d}$ sur $\M{P}(\Delta)$ tels que pour tout $1 \leqslant i \leqslant d$, $K_{E_{i}} = \nabla_{i}$ et le faisceau inversible $\C{O}(E_{i})$ est engendr\'e par ses sections globales. De plus on a $E = E_{1} + \dots + E_{d}$. D'apr\`es la remarque (\ref{construction_inverse3}), il existe un raffinement $\Delta'$ de $\Delta$ tel que $\M{P}(\Delta')$ est {\em projective} et {\em lisse}. On note $i_{\ast}: \M{P}(\Delta') \rightarrow \M{P}(\Delta)$ le morphisme \'equivariant induit par l'inclusion $i: \Delta' \hookrightarrow \Delta$ et on pose $E' = (i_{\ast})^{\ast}(E)$, $E_{1}' = (i_{\ast})^{\ast}(E_{1}), \dots, E_{d}' = (i_{\ast})^{\ast}(E_{d})$. On sait alors que $K_{E'} = \nabla$, $K_{E_{1}'} = \nabla_{1}, \dots, K_{E_{d}'} = \nabla_{d}$ et que les faisceaux inversibles $\C{O}(E')$, $\C{O}(E_{1}'), \dots, \C{O}(E_{d}')$ sont engendr\'es par leurs sections globales. Pour tout $1 \leqslant i \leqslant d$, on d\'eduit de l'inclusion $\op{Supp}P_{i} \subset \nabla_{i}$ que $P_{i}$ s'\'etend en une section r\'eguli\`ere non nulle $s'_{i}$ de $E'_{i}$ sur $\M{P}(\Delta')$. On peut donc appliquer l'in\'egalit\'e (\ref{BK_intermediaire}) sur $\M{P}(\Delta')$ et on trouve~: \begin{multline*} h_{\ov{\C{O}(E')}_{\infty}}((\op{div}s'_{1} \dotsm \op{div}s'_{d})_{\op{pr}}) \\ \leqslant d!\, \sum_{i =1}^{d}V(\nabla,\nabla_{1},\dots,\widehat{\nabla_{i}},\dots,\nabla_{d})\, M(s'_{i}) \\ + d!\, \sum_{i=1}^{d}L(\nabla_{i})V(\nabla,\nabla_{1},\dots,\widehat{\nabla_{i}},\dots, \nabla_{d}). \end{multline*} On se ram\`ene de $\M{P}(\Delta')(\ov{\M{Q}})$ \`a $T(\ov{\M{Q}}) = (\ov{\M{Q}}^{\ast})^{d}$ en utilisant la positivit\'e de $h_{\ov{\C{O}(E')}_{\infty}}$ (cf. exemple \ref{positivite_torique}). Enfin l'\'egalit\'e $h_{\ov{\C{O}(E')}_{\infty}}(x) = h_{\nabla}(x)$ pour tout $x \in (\ov{\M{Q}}^{\ast})^{d}$ est une cons\'equence directe de la construction par image inverse de $\|.\|_{E',\infty}$ (cf. \S\ref{chap_image_inverse}). \medskip \subsubsection{D\'emonstration du th\'eor\`eme.}~ On d\'emontre tout d'abord le lemme suivant~: \begin{lem} \label{lemme_BK} Soit $k$ un entier compris entre $1$ et $d$, et soit $Z \in Z_{k}(\M{P}(\Delta'))$ un cycle effectif tel que la section $s_{k}$ de $\C{O}(E_{k})$ ne soit identiquement nulle sur aucune des composantes irr\'eductibles de $Z$. On a~: \begin{multline*} h_{\ov{\C{O}(E)}_{\infty}, \ov{\C{O}(E_{1})}_{\infty}, \dots, \ov{\C{O}(E_{k-1})}_{\infty}}(Z \cdot \op{div}s_{k}) \\ \leqslant h_{\ov{\C{O}(E)}_{\infty}, \ov{\C{O}(E_{1})}_{\infty}, \dots, \ov{\C{O}(E_{k})}_{\infty}}(Z) \qquad \qquad \qquad \qquad\\ + \op{deg}(E\cdot E_{1}\dotsm E_{k-1}\cdot Z)\left( L(\nabla_{k}) + \frac{h_{\ov{\C{O}(E)}_{\infty}}(\op{div}s_{k})}{\op{deg}(E^{d})} \right). \end{multline*} \end{lem} \noindent {\bf D\'emonstration.}\ D'apr\`es le th\'eor\`eme (\ref{gdthm}) alin\'ea (6), on a~: \begin{multline*} h_{\ov{\C{O}(E)}_{\infty}, \ov{\C{O}(E_{1})}_{\infty}, \dots, \ov{\C{O}(E_{k-1})}_{\infty}}(Z \cdot \op{div}s_{k}) = h_{\ov{\C{O}(E)}_{\infty}, \ov{\C{O}(E_{1})}_{\infty}, \dots, \ov{\C{O}(E_{k})}_{\infty}}(Z) \\ + \int_{Z(\M{C})}\log \|s_{k}\|_{E_{k},\infty} c_{1}(\ov{\C{O}(E)}_{\infty}) c_{1}(\ov{\C{O}(E_{1})}_{\infty}) \dotsm c_{1}(\ov{\C{O}(E_{k-1})}_{\infty}). \end{multline*} On d\'eduit de la proposition (\ref{universalite}) que~: \[ \Sup_{x \in \M{P}(\Delta')(\M{C})}\log \|s_{k}(x)\|_{E_{k},\infty} \leqslant M(s_{k}) + L(\nabla_{k}). \] De l'in\'egalit\'e pr\'ec\'edente et de la positivit\'e des courants $c_{1}(\ov{\C{O}(E)}_{\infty}),c_{1}(\ov{\C{O}(E_{1})}_{\infty}), \linebreak[4] \dots, c_{1}(\ov{\C{O}(E_{k-1})}_{\infty})$ (cf. exemple \ref{exemple_adm1}) on tire que~: \begin{multline*} \int_{Z(\M{C})}\log \|s_{k}\|_{E_{k},\infty}\, c_{1}(\ov{\C{O}(E)}_{\infty}) c_{1}(\ov{\C{O}(E_{1})}_{\infty}) \dotsm c_{1}(\ov{\C{O}(E_{k-1})}_{\infty}) \\ \leqslant \left( \int_{Z(\M{C})} c_{1}(\ov{\C{O}(E)}_{\infty}) c_{1}(\ov{\C{O}(E_{1})}_{\infty}) \dotsm c_{1}(\ov{\C{O}(E_{k-1})}_{\infty}) \right) (M(s_{k}) + L(\nabla_{k})) \\ = \op{deg}(E\cdot E_{1} \dotsm E_{k-1})(M(s_{k}) + L(\nabla_{k})). \end{multline*} Enfin, on a d'apr\`es la proposition (\ref{hauteur_hypersurfaces}) l'\'egalit\'e~: \[ M(s_{k}) = \frac{h_{\ov{\C{O}(E)}_{\infty}}(\op{div}s_{k})}{\op{deg}(E^{d})}, \] ce qui suffit \`a \'etablir le r\'esultat. \medskip On passe maintenant \`a la d\'emonstration du th\'eor\`eme~: On construit par r\'ecurrence une suite finie $Z_{0}, Z_{1}, \dots, Z_{d}$ de cycles effectifs dans $\M{P}(\Delta')$ tels que $Z_{i} \in Z^{i}(\M{P}(\Delta'))$ pour tout $0 \leqslant i \leqslant d$ de la fa\c con suivante~: \begin{itemize} \item{On pose $Z_{0} = \M{P}(\Delta')$.} \item{Pour $i \geqslant 1$, le cycle $Z_{i}$ est d\'efini comme la somme avec multiplicit\'es des composantes de $Z_{i-1}\cdot \op{div}s_{d+1-i}$ dont l'intersection avec $\op{div}s_{d-i}$ est propre.} \end{itemize} Comme dans $\M{P}(\Delta')$ l'intersection d'un cycle de codimension $k$ par une hypersurface est soit vide, soit un cycle de codimension $k+1$, toute composante non vide de~: \[ ((|Z_{i-1}|\cap |\op{div}s_{d+1-i}|) - |Z_{i}|)\cap |\op{div}s_{d-i}| \cap \dotsm \cap |\op{div}s_{1}| \] est au moins de dimension $1$. On est donc assur\'e de l'\'egalit\'e~: \[ Z_{d} = (\op{div}s_{1} \dotsm \op{div}s_{d})_{\op{pr}}. \] Pour tout $1 \leqslant k \leqslant d$, le cycle $(Z_{d-k}\cdot \op{div}s_{k}) - Z_{d -k +1}$ est effectif. On d\'eduit de cela et de l'exemple (\ref{positivite_torique}) que~: \[ h_{\ov{\C{O}(E)}_{\infty}, \ov{\C{O}(E_{1})}_{\infty}, \dots, \ov{\C{O}(E_{k-1})}_{\infty}}(Z_{d-k+1}) \leqslant h_{\ov{\C{O}(E)}_{\infty}, \ov{\C{O}(E_{1})}_{\infty}, \dots, \ov{\C{O}(E_{k-1})}_{\infty}}(Z_{d-k}\cdot \op{div}s_{k}). \] Du fait de la positivit\'e des fibr\'es $\C{O}(E), \C{O}(E_{1}), \dots, \C{O}(E_{k-1})$ on a de m\^eme~: \[ \op{deg}(E\cdot E_{1} \dotsm E_{k-1}\cdot Z_{d-k}) \leqslant \op{deg}(E\cdot E_{1} \dotsm \widehat{E_{k}} \dotsm E_{d}). \] En appliquant le lemme (\ref{lemme_BK}), on obtient pour tout $1 \leqslant k \leqslant d$ l'in\'egalit\'e~: \begin{multline*} h_{\ov{\C{O}(E)}_{\infty}, \ov{\C{O}(E_{1})}_{\infty}, \dots, \ov{\C{O}(E_{k-1})}_{\infty}}(Z_{d-k+1}) \\ \leqslant h_{\ov{\C{O}(E)}_{\infty}, \ov{\C{O}(E_{1})}_{\infty}, \dots, \ov{\C{O}(E_{k})}_{\infty}}(Z_{d-k}) \\ + \op{deg}(E\cdot E_{1}\dotsm \widehat{E_{k}} \dotsm E_{d}) \left( L(\nabla_{k}) + \frac{h_{\ov{\C{O}(E)}_{\infty}}(\op{div}s_{k})}{\op{deg}(E^{d})} \right). \end{multline*} Une r\'ecurrence finie permet alors de conclure. \medskip \section{Courants de Chern canoniques sur les vari\'et\'es toriques}~ Dans cette partie et les suivantes, nous revenons \`a l'\'etude des vari\'et\'es toriques projectives lisses sur $\op{Spec}\M{Z}$. Soit $\Delta$ un \'eventail r\'egulier complet admettant une fonction support strictement concave et soient $\ov{L}_{1},\dots,\ov{L}_{q}$ des fibr\'es en droites sur $\M{P}(\Delta)$ munis \`a l'infini de leur m\'etrique canonique. On donne dans cette partie une expression explicite du courant~: \[ c_{1}(\ov{L}_{1})\dotsm c_{1}(\ov{L}_{q}). \] On en d\'eduit un premier r\'esultat de trivialit\'e~: Si l'on note~: \[ \ov{A}_{\op{f}}^{\ast}(\M{P}(\Delta)_{\M{R}}) = \op{Ker}\{d: \ov{A}_{\op{g}}^{\ast}(\M{P}(\Delta)_{\M{R}}) \longrightarrow \ov{A}_{\op{g}}^{\ast}(\M{P}(\Delta)_{\M{R}})\} \] et \[ [\cdot]: \ov{A}_{\op{f}}^{\ast}(\M{P}(\Delta)_{\M{R}}) \longrightarrow H^{2\ast}(\M{P}(\Delta)(\M{C}),\M{R}) \] la surjection canonique induite par l'application classe en cohomologie des courants, alors on peut construire de mani\`ere canonique une section (d'anneaux) du morphisme d'anneaux $[\cdot]$. \bigskip \subsection{Pr\'eliminaires}~ Soit $\op{Arg}: \M{C}^{\ast} \rightarrow \M{R} / 2\pi\M{Z}$ la fonction argument d\'efinie pour tout $z = \rho e^{i\alpha} \in \M{C}^{\ast}$ avec $\rho \in \M{R}^{+\ast}$ et $\alpha \in \M{R}$ par~: \[ \op{Arg}(z) = [\alpha] \in \M{R} / 2\pi \M{Z}. \] Pour tout $m \in M$, on note $\op{Arg}(\chi^{m})$ la fonction multiforme sur $T(\M{C})$ d\'efinie comme l'argument du caract\`ere $\chi^{m}$. La diff\'erentielle $d \op{Arg}(\chi^{m})$ a bien un sens sur $T(\M{C})$ et d\'efinit un \'el\'ement de $A^{1,0}(T(\M{C}))\oplus A^{0,1}(T(\M{C}))$. On dispose donc d'un morphisme injectif de $\M{Z}$-module~: \begin{alignat*}{3} \Theta:\; &M & &\longrightarrow& &A^{1,0}(T(\M{C}))\oplus A^{0,1}(T(\M{C})) \\ &m & &\longmapsto & &\frac{d \op{Arg}(\chi^{m})}{2\pi}. \end{alignat*} Ce morphisme s'\'etend en un morphisme d'anneaux (encore not\'e $\Theta$)~: \[ \Theta: {\textstyle\bigwedge\nolimits^{\ast}_{\M{Z}}}M \longrightarrow \oplus_{p,q}A^{p,q}(T(\M{C})), \] o\`u ${\textstyle\bigwedge\nolimits^{\ast}_{\M{Z}}}M$ d\'esigne l'alg\`ebre ext\'erieure sur $\M{Z}$ de $M$ et o\`u la structure d'anneau sur $\oplus_{p,q}A^{p,q}(T(\M{C}))$ est celle induite par le produit ext\'erieur. Le morphisme d'anneaux $\Theta$ est injectif. \begin{rem} Soit $f_{1},\dots,f_{d}$ une base de $N$ et $f_{1}^{\ast},\dots,f_{d}^{\ast}$ la base duale. Pour tout $m \in M$, on a~: \[ \Theta(m) = \sum_{i =1}^{d}f_{i}(m) \frac{d \op{Arg}(\chi^{f_{i}^{\ast}})}{2\pi}. \] \end{rem} \begin{rem} \label{remarque_theta} Soit $m \in M$. On a~: \[ d^{c} \log |\chi^{m}|^{2} = \Theta(m) \qquad \text{sur $T(\M{C})$}. \] \end{rem} Pour tout $\tau$ \'el\'ement de $\Delta$, le $\M{Z}$-module $\bigwedge^{\op{max}}_{\M{Z}}(\tau^{\perp}\cap M)$ est un $\M{Z}$-module libre de rang $1$. On choisit un g\'en\'erateur de $\bigwedge^{\op{max}}_{\M{Z}}(\tau^{\perp}\cap M)$ que l'on notera dans toute la suite $\C{M}_{\tau}$. Le choix de $\C{M}_{\tau}$ d\'efinit une orientation sur la vari\'et\'e r\'eelle $C_{\tau}^{\op{int}}$ de la mani\`ere suivante~: On note $h = \op{dim}\tau$. Soient $u_{1},\dots,u_{h}$ un syst\`eme de g\'en\'erateurs du semi-groupe $(\tau \cap N)$ que l'on compl\`ete en une base $u_{1},\dots,u_{d}$ de $N$, et $v_{h+1},\dots,v_{d}$ une base du $\M{Z}$-module $(\tau^{\perp}\cap M)$ telle que $v_{h+1}\wedge \dots \wedge v_{d} = \C{M}_{\tau}$. On note $u_{1}^{\ast},\dots,u_{d}^{\ast}$ la base duale de $u_{1},\dots,u_{d}$. On note enfin $B(0,1) \subset \M{C}$ la boule ouverte de centre $0$ et de rayon $1$ et $S_{1} \subset \M{C}$ le cercle unit\'e. On dispose alors du diff\'eomorphisme~: \begin{alignat*}{3} \eta_{\tau}:\; &C_{\tau}^{\op{int}}& & \longrightarrow & &B(0,1)^{h}\times S_{1}^{d-h}\\ &x & &\longmapsto & &(\chi^{u_{1}^{\ast}}(x),\dots,\chi^{u_{h}^{\ast}}(x),\chi^{v_{h+1}}(x),\dots,\chi^{v_{d}}(x)) \end{alignat*} On notera $C_{\tau}^{\op{int}, +}$ la vari\'et\'e r\'eelle $C_{\tau}^{\op{int}}$ munie de l'orientation produit des orientations naturelles sur $B(0,1)^{h}$ et $S_{1}^{d-h}$. Cette orientation ne d\'epend que du choix de $\C{M}_{\tau}$. On peut remarquer que pour tout $\sigma \in \Delta_{\op{max}}$, l'orientation de $C_{\sigma}^{\op{int},+}$ co\"\i ncide avec celle induite par sa structure complexe. Par ailleurs, on a pour tout $\tau \in \Delta$ l'\'egalit\'e $|\chi^{v_{h+1}}| = \dots = |\chi^{v_{d}}| = 1$ sur $C_{\tau}^{\op{int}}$, ce qui entra\^\i ne que la forme $\Theta(\C{M}_{\tau})$ est de classe $C^{\infty}$ sur un voisinage de $C_{\tau}^{\op{int}}$; on dispose donc du courant r\'eel~: \[ \omega \longmapsto \int_{C_{\tau}^{\op{int},+}}\Theta(\C{M}_{\tau})\wedge \omega. \] \bigskip \subsection{Calcul de $c_{1}(\ov{L})$}~ Soit $\ov{L}$ un fibr\'e en droites sur $\M{P}(\Delta)$ muni de sa m\'etrique canonique. On donne, dans cette section, une expression du courant $c_{1}(\ov{L})$. On rappelle tout d'abord un r\'esultat bien connu (voir par exemple \cite{7}, \S 3.3). \begin{prop} \label{Green} {\rm (Formule de Green).} Soit $X$ une vari\'et\'e complexe de dimension $d$ et $\Omega \subset X$ un ouvert relativement compact tel que $\ov{\Omega}$ soit une sous-vari\'et\'e r\'eelle \`a coins de $X$. Soient $f$ et $g$ des formes diff\'erentielles de classe $C^{2}$ sur un voisinage de $\ov{\Omega}$ et de bidegr\'es $(p,p)$ et $(q,q)$ avec $p+q = d-1$. On a~: \[ \int_{\Omega}\left(f \wedge dd^{c} g - dd^{c} f \wedge g \right) = \int_{\partial\Omega}\left(f \wedge d^{c} g - d^{c}f \wedge g\right). \] \end{prop} \medskip Soit $D$ un diviseur (horizontal) $T$-invariant sur $\M{P}(\Delta)$ tel que $\ov{L} = \ov{\C{O}(D)}_{\infty}$. On rappelle que pour tout $\sigma \in \Delta$, on note $m_{D,\sigma}$ l'\'el\'ement de $M$ donnant la restriction \`a $\sigma$ de la fonction support $\Psi_{D}$ de $D$. On a alors le th\'eor\`eme suivant~: \begin{thm} \label{calcul_c1} Pour toute forme test $\omega \in A^{d-1,d-1}(\M{P}(\Delta)(\M{C}))$, on a~: \[ \int_{\M{P}(\Delta)(\M{C})}c_{1}(\ov{L}) \wedge \omega = - \sum_{\sigma \in \Delta_{\op{max}}}\int_{\partial C_{\sigma}^{\op{int},+}}\Theta(m_{D,\sigma}) \wedge \omega. \] \end{thm} \noindent {\bf D\'emonstration.}\ On montre tout d'abord que les termes intervenant dans le second membre sont bien d\'efinis. Pour tout $\sigma \in \Delta_{\op{max}}$, la forme $\Theta(m_{D,\sigma})\wedge \omega$ est $L^{1}$ sur $\partial C_{\sigma}^{\op{int}}$ (cela provient du fait que la forme $d \theta = d \op{Arg}(z)$ est localement $L^{1}$ sur $\M{C}$). La diff\'erence $\partial C_{\sigma}^{\op{int}} \backslash (\partial C_{\sigma}^{\op{int}} \cap T(\M{C}))$ \'etant de codimension r\'eelle sup\'erieure ou \'egale \`a $2$, elle est n\'egligeable au sens de la th\'eorie de la mesure, ce qui entra\^\i ne que l'int\'egrale $\int_{\partial C_{\sigma}^{\op{int},+}}\Theta(m_{D,\sigma})\wedge \omega$ est bien d\'efinie. Soit maintenant $\boldsymbol{1}$ la section (rationnelle) canonique de $\C{O}(D)$. On a, d'apr\`es la formule de Poincar\'e-Lelong g\'en\'eralis\'ee (\ref{PL_generalisee}) l'\'egalit\'e des courants~: \[ c_{1}(\ov{L}) = c_{1}(\ov{\C{O}(D)}_{\infty}) = \delta_{D} + (dd^{c} (-\log \|\boldsymbol{1}\|^{2}_{D,\infty})). \] On en d\'eduit que~: \begin{equation} \label{equa_1} \int_{\M{P}(\Delta)(\M{C})}c_{1}(\ov{L})\wedge \omega = \int_{\M{P}(\Delta)(\M{C})} \delta_{D} \wedge \omega + \int_{\M{P}(\Delta)(\M{C})}(- \log \|\boldsymbol{1}\|^{2}_{D,\infty}) \wedge dd^{c} \omega. \end{equation} Comme de plus, d'apr\`es (\ref{BT_construction}), on a pour tout $\sigma \in \Delta_{\op{max}}$ et $x \in C_{\sigma}$ l'\'egalit\'e~: \[ - \log \|\boldsymbol{1}(x)\|^{2}_{D,\infty} = \log | \chi^{m_{D,\sigma}}(x)|^{2}, \] on tire de (\ref{equa_1}) et de (\ref{dissection}) la relation~: \begin{equation} \label{equa_2} \int_{\M{P}(\Delta)(\M{C})}c_{1}(\ov{L}) \wedge \omega = \int_{\M{P}(\Delta)(\M{C})}\delta_{D}\wedge \omega + \sum_{\sigma \in \Delta_{\op{max}}}\int_{C_{\sigma}^{\op{int},+}}\log|\chi^{m_{D,\sigma}}(x)|^{2}\wedge dd^{c} \omega. \end{equation} On fixe provisoirement $\sigma \in \Delta_{\op{max}}$. Soit $f_{1},\dots,f_{d}$ une famille g\'en\'eratrice de $\sigma$ (et donc une $\M{Z}$-base de $N$ d'apr\`es (\ref{lissite})); on note $f_{1}^{\ast},\dots,f_{d}^{\ast}$ la base duale de $M$. On a $m_{D,\sigma} = \sum_{i =1}^{d}f_{i}(m_{D,\sigma})f_{i}^{\ast}$, ce dont on tire~: \[ \log | \chi^{m_{D,\sigma}}(x)|^{2} = \sum_{i =1}^{d}f_{i}(m_{D,\sigma})\log | \chi^{f_{i}^{\ast}}(x)|^{2}. \] Par lin\'earit\'e, on ram\`ene ainsi le calcul de l'int\'egrale $\int_{C_{\sigma}^{\op{int},+}}\log |\chi^{m_{D,\sigma}}(x)|^{2} \wedge dd^{c} \omega$ \`a celui des int\'egrales $\int_{C_{\sigma}^{\op{int},+}}\log | \chi^{f_{i}^{\ast}}(x)|^{2}\wedge dd^{c} \omega$. On ne perd rien en g\'en\'eralit\'e en posant $i = 1$. Dans la carte affine $\varphi: U_{\sigma}(\M{C}) \rightarrow \M{C}^{d}$ donn\'ee par $\varphi(x) = (\chi^{f_{1}^{\ast}}(x),\dots,\chi^{f_{d}^{\ast}}(x))$, les ensembles $C_{\sigma}$ et $C_{\sigma}^{\op{int}}$ sont d\'efinis par les conditions~: \[ C_{\sigma} = \{x \in \M{C}^{d}: \quad |x_{1}| \leqslant 1, \dots, |x_{d}| \leqslant 1\}, \] et \[ C_{\sigma}^{\op{int}} = \{x \in \M{C}^{d}: \quad |x_{1}| < 1, \dots, |x_{d}| < 1\}. \] On remarque que~: \[ \partial C_{\sigma} = \partial C_{\sigma}^{\op{int}} = \{x \in C_{\sigma}: \quad \exists i \in \{1,\dots,d\}: \quad |x_{i}| = 1\}. \] Pour tout $\varepsilon \in \M{R}^{+\ast}$, on pose~: \begin{align*} &C_{\sigma}^{\varepsilon} = \{ x \in C_{\sigma}^{\op{int}}: \quad \varepsilon < |x_{1}| < 1\}, \\ &D_{\sigma}^{\varepsilon} = \{x \in \partial C_{\sigma}: \quad \varepsilon < |x_{1}| \leqslant 1\} \\ \intertext{et} &E_{\sigma}^{\varepsilon} = \{x \in C_{\sigma}: \quad |x_{1}| = \varepsilon\}. \end{align*} \bigskip \begin{center} \input{figure5.pstex_t} \end{center} \bigskip On a~: \begin{multline} \label{equaa_1} \int_{C_{\sigma}^{\op{int},+}}\log | \chi^{f_{1}^{\ast}}(x)|^{2} \wedge dd^{c} \omega \\ = \int_{C_{\sigma}^{\varepsilon}}\log | \chi^{f_{1}^{\ast}}(x)|^{2} \wedge dd^{c} \omega + \int_{C_{\sigma}^{\op{int}}\backslash C_{\sigma}^{\varepsilon}} \log | \chi^{f_{1}^{\ast}}(x)|^{2} \wedge dd^{c} \omega. \end{multline} Comme $\log | \chi^{f_{1}^{\ast}}(x)|^{2} = \log |x_{1}|^{2}$ est localement $L^{1}$ sur $U_{\sigma}(\M{C})$, on a~: \begin{equation} \label{equaa_2} \lim_{\varepsilon \rightarrow 0}\int_{C_{\sigma}^{\op{int}}\backslash C_{\sigma}^{\varepsilon}} \log | \chi^{f_{1}^{\ast}}(x)|^{2} \wedge dd^{c} \omega = 0. \end{equation} L'application $x \mapsto \log | \chi^{f_{1}^{\ast}}(x)|^{2}$ est $C^{\infty}$ sur un voisinage de $C_{\sigma}^{\varepsilon}$; il vient donc, d'apr\`es la formule de Green (\ref{Green})~: \begin{multline} \label{equaa_3} \int_{C_{\sigma}^{\varepsilon}} \log |\chi^{f_{1}^{\ast}}(x)|^{2}\wedge dd^{c} \omega = \int_{C_{\sigma}^{\varepsilon}}dd^{c} \log |\chi^{f_{1}^{\ast}}(x)|^{2}\wedge \omega \\ + \int_{\partial C_{\sigma}^{\varepsilon}}\left( \log | \chi^{f_{1}^{\ast}}(x)|^{2}\wedge d^{c}\omega - d^{c}\log | \chi^{f_{1}^{\ast}}(x)|^{2}\wedge \omega\right). \end{multline} On a sur $C_{\sigma}^{\varepsilon}$ l'\'egalit\'e $dd^{c} \log |\chi^{f_{1}^{\ast}}(x)|^{2} = dd^{c} \log |x_{1}|^{2} = 0$. De plus, d'apr\`es la remarque (\ref{remarque_theta}), on a~: \[ d^{c}\log |\chi^{f_{1}^{\ast}}(x)|^{2} = d^{c} \log |x_{1}|^{2} = \Theta(f_{1}^{\ast}). \] On tire des d\'efinitions l'\'egalit\'e des courants~: \[ \int_{\partial C_{\sigma}^{\varepsilon}} = \int_{D_{\sigma}^{\varepsilon}}\; + \; \int_{E_{\sigma}^{\varepsilon}} \] Comme $\log |\chi^{f_{1}^{\ast}}(x)|^{2} = \log |x_{1}|^{2}$ est localement $L^{1}$ sur $\partial C_{\sigma}$, il vient~: \[ \lim_{\varepsilon \rightarrow 0}\int_{D_{\sigma}^{\varepsilon}}\log |\chi^{f_{1}^{\ast}}(x)|^{2} \wedge d^{c}\omega = \int_{\partial C_{\sigma}^{\op{int},+}}\log |\chi^{f_{1}^{\ast}}(x)|^{2} \wedge d^{c}\omega. \] De plus $\log |\chi^{f_{1}^{\ast}}(x)|^{2} = 2\log \varepsilon$ sur $E_{\sigma}^{\varepsilon}$, et pour toute forme $\eta \in A^{2d-1}(\M{P}(\Delta)(\M{C}))$ on a $\int_{E_{\sigma}^{\varepsilon}}\eta = {O}(\varepsilon)$ quand $\varepsilon$ tend vers $0$; on en d\'eduit que~: \[ \lim_{\varepsilon \rightarrow 0}\int_{E_{\sigma}^{\varepsilon}} \log |\chi^{f_{1}^{\ast}}(x)|^{2} \wedge d^{c}\omega = 0. \] Enfin, on a les limites~: \[ \lim_{\varepsilon \rightarrow 0}\int_{D_{\sigma}^{\varepsilon}}d^{c} \log |\chi^{f_{1}^{\ast}}(x)|^{2} \wedge \omega = \lim_{\varepsilon \rightarrow 0} \int_{D_{\sigma}^{\varepsilon}}\Theta(f_{1}^{\ast}) \wedge \omega = \int_{\partial C_{\sigma}^{\op{int},+}}\Theta(f_{1}^{\ast}) \wedge \omega, \] et \[ \lim_{\varepsilon \rightarrow 0} \int_{E_{\sigma}^{\varepsilon}} d^{c} \log |\chi^{f_{1}^{\ast}}(x)|^{2} \wedge \omega = \lim_{\varepsilon \rightarrow 0} \int_{E_{\sigma}^{\varepsilon}} \frac{d\op{Arg} (x_{1})}{2\pi}\wedge \omega = \int_{C_{\sigma}^{\op{int},+}}\delta_{H_{1}}\wedge \omega, \] o\`u $H_{1}$ d\'esigne le diviseur d'\'equation $\chi^{f_{1}^{\ast}}(x) = 0$. En revenant au cas g\'en\'eral ($m_{D,\sigma}$ quelconque), on d\'eduit de (\ref{equaa_1}), (\ref{equaa_2}), (\ref{equaa_3}) et des calculs de limites ci-dessus que~: \begin{multline*} \int_{C_{\sigma}^{\op{int},+}}\log | \chi^{m_{D,\sigma}}(x)|^{2} \wedge dd^{c} \omega \\ = \int_{\partial C_{\sigma}^{\op{int},+}} \log | \chi^{m_{D,\sigma}}(x)|^{2} \wedge d^{c}\omega - \int _{\partial C_{\sigma}^{\op{int},+}} \Theta(m_{D,\sigma}) \wedge \omega - \int_{C_{\sigma}^{\op{int},+}}\delta_{D} \wedge \omega. \end{multline*} D'apr\`es (\ref{equa_2}), il vient~: \begin{multline*} \int_{\M{P}(\Delta)(\M{C})} c_{1}(\ov{L}) \wedge \omega = \sum_{\sigma \in \Delta_{\op{max}}}\int_{\partial C_{\sigma}^{\op{int},+}} \log | \chi^{m_{D,\sigma}}(x)|^{2} \wedge d^{c}\omega \\ - \sum_{\sigma \in \Delta_{\op{max}}}\int_{\partial C_{\sigma}^{\op{int},+}} \Theta(m_{D,\sigma}) \wedge \omega - \sum_{\sigma \in \Delta_{\op{max}}}\int_{C_{\sigma}^{\op{int},+}} \delta_{D}\wedge \omega + \int_{\M{P}(\Delta)(\M{C})}\delta_{D} \wedge \omega. \end{multline*} Soient $\sigma_{1}$ et $\sigma_{2}$ deux \'el\'ements de $\Delta_{\op{max}}$ et $\tau \in \Delta (d-1)$ une face commune de $\sigma_{1}$ et $\sigma_{2}$ (i.e. telle que $\tau < \sigma_{1}$ et $\tau < \sigma_{2}$). Du fait de la continuit\'e de la fonction support $\Psi_{D}$, on a~: \begin{equation} \label{eq_chern_1} \log | \chi^{m_{D,\sigma_{1}}}(x)|^{2} = \log | \chi^{m_{D,\sigma_{2}}}(x)|^{2}, \qquad (\forall x \in C_{\tau}). \end{equation} Puisque $\Delta$ est complet, on a par ailleurs~: \begin{multline*} \sum_{\sigma \in \Delta_{\op{max}}} \int_{\partial C_{\sigma}^{\op{int},+}}\log |\chi^{m_{D,\sigma}}(x)|^{2}\wedge d^{c} \omega = \sum_{\substack{\sigma \in \Delta_{\op{max}} \\ \tau \in \Delta(d-1) \\ \tau < \sigma}} \int_{C_{\tau}^{\op{int},+}} \varepsilon_{\tau}(\sigma) \log |\chi^{m_{D,\sigma}}(x)|^{2}\wedge d^{c} \omega = \\ \sum_{\substack{\{\sigma_{1},\sigma_{2}\} \subset \Delta_{\op{max}} \\ \tau = \sigma_{1} \cap \sigma_{2} \in \Delta(d-1)}} \varepsilon_{\tau}(\sigma_{1}) \left( \int_{C_{\tau}^{\op{int},+}}\log|\chi^{m_{D,\sigma_{1}}}(x)|^{2} \wedge d^{c} \omega - \int_{C_{\tau}^{\op{int},+}}\log|\chi^{m_{D,\sigma_{2}}}(x)|^{2} \wedge d^{c} \omega \right) \end{multline*} o\`u pour tout $\sigma \in \Delta_{\op{max}}$ et tout $\tau \in \Delta(d-1)$ tels que $\tau < \sigma$ on a pos\'e $\varepsilon_{\tau}(\sigma) = 1$ si les orientations de $\partial C_{\sigma}^{\op{int},+}$ et de $C_{\tau}^{\op{int},+}$ sont compatibles et $\varepsilon_{\tau}(\sigma) = -1$ sinon. On d\'eduit de cela et de (\ref{eq_chern_1}) que~: \[ \sum_{\sigma \in \Delta_{\op{max}}} \int_{\partial C_{\sigma}^{\op{int},+}}\log |\chi^{m_{D,\sigma}}(x)|^{2}\wedge d^{c} \omega = 0. \] Enfin, comme $\op{codim}_{\M{R}}(D\cap \partial C_{\sigma}) \geqslant 3$, pour tout $\sigma \in \Delta_{\op{max}}$, on a~: \[ \sum_{\sigma \in \Delta_{\op{max}}}\int_{C_{\sigma}^{\op{int},+}} \delta_{D}\wedge \omega = \int_{\M{P}(\Delta)(\M{C})}\delta_{D}\wedge \omega, \] et le th\'eor\`eme est d\'emontr\'e. \medskip Soient $\sigma_{1},\dots,\sigma_{s}$ les \'el\'ements de $\Delta_{\op{max}}$. Comme $\Delta$ est complet, on peut associer \`a tout \'el\'ement $\tau$ de $\Delta(d-1)$ deux entiers $i_{\tau} < j_{\tau}$ dans $\{1,\dots,s\}$ tels que $\tau$ soit la face commune de $\sigma_{i_{\tau}}$ et $\sigma_{j_{\tau}}$. On munit $C_{\tau}^{\op{int}}$ de l'orientation compatible avec celle de $\partial C_{\sigma_{i_{\tau}}}^{\op{int}}$ et on note $C_{\tau}^{\op{int},++}$ la vari\'et\'e r\'eelle $C_{\tau}^{\op{int}}$ munie de cette orientation. On peut reformuler le th\'eor\`eme (\ref{calcul_c1}) de la fa\c con suivante~: \begin{thm} \label{calcul2_c1} Pour toute forme test $\omega \in A^{d-1,d-1}(\M{P}(\Delta)(\M{C}))$, on a~: \[ \int_{\M{P}(\Delta)(\M{C})} c_{1}(\ov{L}) \wedge \omega = \sum_{\tau \in \Delta(d-1)}\int_{C_{\tau}^{\op{int},++}}\Theta(m_{D,\sigma_{j_{\tau}}} - m_{D,\sigma_{i_{\tau}}}) \wedge \omega. \] \end{thm} \bigskip \subsection{Calcul de $c_{1}(\ov{L}_{1})\dotsm c_{1}(\ov{L}_{q})$}~ Soient $\ov{L}_{1},\dots,\ov{L}_{q}$ des fibr\'es en droites au-dessus de $\M{P}(\Delta)$ et munis sur $\M{P}(\Delta)(\M{C})$ de leur m\'etrique canonique. Le th\'eor\`eme suivant donne une expression du produit $c_{1}(\ov{L}_{1})\dotsm c_{1}(\ov{L}_{q})$. \begin{thm} \label{calcul_prod} Soient $\ov{L}_{1},\dots,\ov{L}_{q}$ des fibr\'es en droites au-dessus de $\M{P}(\Delta)$ munis de leur m\'etrique canonique. \begin{enumerate} \item{Il existe une famille d'entiers $(a_{\tau}(\ov{L}_{1},\dots,\ov{L}_{q}))_{\tau \in \Delta(d-q)}$ telle que, pour toute forme test $\omega \in A^{d-q,d-q}(\M{P}(\Delta)(\M{C}))$, on ait~: \[ \int_{\M{P}(\Delta)(\M{C})}c_{1}(\ov{L}_{1})\wedge \dots \wedge c_{1}(\ov{L}_{q}) \wedge \omega = \sum_{\tau \in \Delta(d-q)}a_{\tau}(\ov{L}_{1},\dots, \ov{L}_{q}) \int_{C_{\tau}^{\op{int},+}}\Theta(\C{M}_{\tau})\wedge \omega. \] Les entiers $a_{\tau}(\ov{L}_{1},\dots, \ov{L}_{q})$ sont d\'efinis de mani\`ere unique par cette \'egalit\'e. } \item{Les entiers $a_{\tau}(\ov{L}_{1},\dots, \ov{L}_{q})$ v\'erifient les relations suivantes~: Pour tout $\sigma \in \Delta(d-q-1)$, on a~: \[ \sum_{\substack{\tau > \sigma \\ \tau \in \Delta(d-q)}} \varepsilon_{\sigma}(\tau)a_{\tau}(\ov{L}_{1},\dots, \ov{L}_{q})\C{M}_{\tau} = 0; \] o\`u $\varepsilon_{\sigma}(\tau) = 1$ si les orientations de $\partial C_{\tau}^{\op{int},+}$ et de $C_{\sigma}^{\op{int},+}$ sont compatibles, et $\varepsilon_{\sigma}(\tau) = -1$ sinon. } \item{ On suppose que $d < q$ et que $\ov{L}$ est un fibr\'e en droites au-dessus de $\M{P}(\Delta)$ muni de sa m\'etrique canonique. On note $D$ un diviseur horizontal $T$-invariant sur $\M{P}(\Delta)$ tel que $\ov{L} = \ov{\C{O}(D)}_{\infty}$. On a~: \[ a_{\sigma}(\ov{L},\ov{L}_{1},\dots,L_{q})\C{M}_{\sigma} = - \sum_{\substack{\tau > \sigma \\ \tau \in \Delta(d-q)}} \varepsilon_{\sigma}(\tau) a_{\tau}(\ov{L}_{1},\dots,\ov{L}_{q}) \; m_{D,\tau} \wedge \C{M}_{\tau}. \] } \end{enumerate} \end{thm} \noindent {\bf D\'emonstration.}\ On remarque tout de suite que l'unicit\'e des entiers $a_{\tau}(\ov{L}_{1},\dots,\ov{L}_{q})$ est une cons\'equence directe de la proposition (\ref{dissection}) et du fait que le support du courant r\'eel $\omega \mapsto \int_{C_{\tau}^{\op{int},+}}\Theta(\C{M}_{\tau})\wedge \omega$ est $C_{\tau}$. D'apr\`es le th\'eor\`eme (\ref{calcul2_c1}), les assertions (1) et (2) sont vraies pour $q=1$. On va montrer que si (1) et (2) sont vraies au rang $q$, alors (1) est vraie au rang $q+1$, (3) est vraie au rang $q$, et enfin (2) est vraie au rang $q+1$. D'apr\`es le (1), on peut trouver une famille d'entiers $(a_{\tau}(\ov{L}_{1},\dots,\ov{L}_{q}))_{\tau \in \Delta(d-q)}$ telle que~: \[ \int_{\M{P}(\Delta)(\M{C})}c_{1}(\ov{L}_{1})\wedge \dots \wedge c_{1}(\ov{L}_{q})\wedge \omega = \sum_{\tau \in \Delta(d-q)}a_{\tau}(\ov{L}_{1},\dots,\ov{L}_{q})\int_{C_{\tau}^{\op{int},+}} \Theta(\C{M}_{\tau})\wedge \omega. \] Comme d'apr\`es (\ref{BT_construction}), on a pour tout $\sigma \in D$~: \[ - \log \|\boldsymbol{1}(x)\|_{D,\infty}^{2} = \log | \chi^{m_{D,\sigma}}(x)|^{2}, \qquad (\forall x \in C_{\sigma}). \] On d\'eduit de la formule de Poincar\'e-Lelong g\'en\'eralis\'ee (\ref{PL_generalisee}) et de la formule de Green l'\'egalit\'e~: \begin{multline} \label{eq_chern_2} \int_{\M{P}(\Delta)(\M{C})}c_{1}(\ov{L})\wedge c_{1}(\ov{L}_{1}) \wedge \dots \wedge c_{1}(\ov{L}_{q})\wedge \omega \\ \quad = \int_{\M{P}(\Delta)(\M{C})}(dd^{c} (- \log \|\boldsymbol{1}(x)\|_{D,\infty}^{2}) + \delta_{D}) \wedge c_{1}(\ov{L}_{1}) \wedge \dots \wedge c_{1}(\ov{L}_{q}) \wedge \omega \\ = \int_{\M{P}(\Delta)(\M{C})}(-\log \|\boldsymbol{1}(x)\|_{D,\infty}^{2})\, c_{1}(\ov{L}_{1}) \wedge \dots \wedge c_{1}(\ov{L}_{q}) \wedge dd^{c} \omega \qquad \quad \\ \qquad \qquad + \int_{\M{P}(\Delta)(\M{C})} \delta_{D}\wedge c_{1}(\ov{L}_{1}) \wedge \dots \wedge c_{1}(\ov{L}_{q})\wedge \omega \\ = \sum_{\tau \in \Delta(d-q)} a_{\tau}(\ov{L}_{1},\dots,\ov{L}_{q}) \int_{C_{\tau}^{\op{int},+}} \log | \chi^{m_{D,\tau}}(x)|^{2}\; \Theta(\C{M}_{\tau}) \wedge dd^{c} \omega \\ + \sum_{\tau \in \Delta(d-q)}a_{\tau}(\ov{L}_{1},\dots,\ov{L}_{q}) \int_{C_{\tau}^{\op{int},+}}\delta_{D\cap C_{\tau}^{\op{int}}}\wedge \Theta(\C{M}_{\tau}) \wedge \omega. \end{multline} (Dans le dernier terme, on a utilis\'e le fait que $D$ et $C_{\tau}^{\op{int}}$ s'intersectent de mani\`ere transverse). Nous allons calculer les int\'egrales du type~: \[ I_{\tau}(m_{D,\tau}) = \int_{C_{\tau}^{\op{int},+}} \log | \chi^{m_{D,\tau}}(x)|^{2}\; \Theta(\C{M}_{\tau}) \wedge dd^{c} \omega. \] Fixons momentan\'ement $\tau \in \Delta(d-q)$ et consid\'erons $\sigma \in \Delta_{\op{max}}$ tel que $\tau < \sigma$. Soient $f_{1},\dots,f_{d}$ un syst\`eme de g\'en\'erateurs du semi-groupe $(\sigma \cap N)$ tels que $f_{1},\dots,f_{d-q}$ engendrent le semi-groupe $(\tau \cap N)$ et tels que $f_{d-q+1}^{\ast}\wedge \dots \wedge f_{d}^{\ast} = \C{M}_{\tau}$. Comme $\Delta$ est r\'egulier, la famille $f_{1}^{\ast},\dots,f_{d}^{\ast}$ forme une $\M{Z}$-base de $M$. Par lin\'earit\'e, il suffit de calculer $I_{\tau}(m_{D,\tau})$ pour $m_{D,\tau} = f_{i}^{\ast}$ avec $i \in \{1,\dots,d\}$. Pour $i \in \{d-q+1,\dots,d\}$, on a $|\chi^{f_{i}^{\ast}}(x)| = 1$ pour tout $x\in C_{\tau}$, et donc $I_{\tau}(f_{i}^{\ast}) = 0$. On choisit maintenant $i \in \{1,\dots,d-q\}$. On ne perd rien en g\'en\'eralit\'e en supposant $i=1$. Dans la carte affine $\varphi : U_{\sigma}(\M{C}) \rightarrow \M{C}^{d}$ donn\'ee par $\varphi(x) = (\chi^{f_{1}^{\ast}}(x),\dots, \chi^{f_{d}^{\ast}}(x))$, les ensembles $C_{\tau}$ et $C_{\tau}^{\op{int}}$ sont d\'efinis par les conditions~: \begin{align*} C_{\tau} &= \{x \in \M{C}^{d}: \quad |x_{1}| \leqslant 1, \dots, |x_{d-q}| \leqslant 1, |x_{d-q+1}| = 1, \dots, |x_{d}| = 1\} \\ \intertext{et} C_{\tau}^{\op{int}} &= \{x \in \M{C}^{d}: \quad |x_{1}| < 1, \dots, |x_{d-q}| < 1, |x_{d-q+1}| = 1, \dots, |x_{d}| = 1\}. \end{align*} On remarque que~: \[ \partial C_{\tau} = \partial C_{\tau}^{\op{int}} = \{x \in C_{\tau}: \quad \exists i \in \{1,\dots,d-q\}: \quad |x_{i}| = 1\}. \] Pour tout $\varepsilon \in \M{R}^{+\ast}$, on pose~: \begin{align*} C_{\tau}^{\varepsilon} &= \{x \in C_{\tau}^{\op{int}}: \quad \varepsilon < |x_{1}| < 1\}, \\ D_{\tau}^{\varepsilon} &= \{x \in \partial C_{\tau}: \quad \varepsilon < |x_{1}| \leqslant 1\} \\ \intertext{et} E_{\tau}^{\varepsilon} &= \{x \in C_{\tau}: \quad |x_{1}| = \varepsilon\}. \end{align*} On a~: \begin{multline} \label{eqq_1} I_{\tau}(f_{1}^{\ast}) = \int_{C_{\tau}^{\varepsilon}}\log | \chi^{f_{1}^{\ast}}(x)|^{2}\; \Theta(\C{M}_{\tau}) \wedge dd^{c} \omega \\ + \int_{C_{\tau}^{\op{int},+}\backslash C_{\tau}^{\varepsilon}} \log | \chi^{f_{1}^{\ast}}(x)|^{2}\; \Theta(\C{M}_{\tau}) \wedge dd^{c} \omega. \end{multline} Comme $\log | \chi^{f_{1}^{\ast}}(x)|^{2}\,\Theta(\C{M}_{\tau}) = \log |x_{1}|^{2} \,\Theta(\C{M}_{\tau})$ est localement $L^{1}$ sur $C_{\tau}$, on a~: \begin{equation} \label{eqq_2} \lim_{\varepsilon \rightarrow 0} \int_{C_{\tau}^{\op{int},+}\backslash C_{\tau}^{\varepsilon}} \log | \chi^{f_{1}^{\ast}}(x)|^{2}\; \Theta(\C{M}_{\tau}) \wedge dd^{c} \omega = 0. \end{equation} Puisque $|\chi^{f_{d-q+1}^{\ast}}(x)| = \dots = |\chi^{f_{d}^{\ast}}(x)| = 1$ pour tout $x \in C_{\tau}$ et que $\C{M}_{\tau} \in \bigwedge^{\op{max}}_{\M{Z}}(\tau^{\perp}\cap M)$, la forme $\Theta(\C{M}_{\tau})$, et donc l'application $x \mapsto \log | \chi^{f_{1}^{\ast}}(x)|^{2}\,\Theta(\C{M}_{\tau})$, sont $C^{\infty}$ sur un voisinage de $C_{\tau}^{\varepsilon}$. On peut donc appliquer la formule de Green (\ref{Green}) et en d\'eduire l'\'egalit\'e~: \begin{multline} \int_{C_{\tau}^{\varepsilon}} \log | \chi^{f_{1}^{\ast}}(x)|^{2}\; \Theta(\C{M}_{\tau}) \wedge dd^{c} \omega = \int_{C_{\tau}^{\varepsilon}} dd^{c} \left( \log | \chi^{f_{1}^{\ast}}(x)|^{2}\; \Theta(\C{M}_{\tau}) \right) \wedge \omega \\ + \int_{\partial C_{\tau}^{\varepsilon}} \left( \log | \chi^{f_{1}^{\ast}}(x)|^{2}\; \Theta(\C{M}_{\tau}) \wedge d^{c}\omega - d^{c}( \log | \chi^{f_{1}^{\ast}}(x)|^{2}\; \Theta(\C{M}_{\tau}) ) \wedge \omega \right). \end{multline} En remarquant que $d \Theta(\C{M}_{\tau}) = d^{c}\Theta(\C{M}_{\tau}) = 0$, on a sur $C_{\tau}^{\varepsilon}$~: \begin{align*} dd^{c} ( \log |\chi^{f_{1}^{\ast}}(x)|^{2}\wedge\Theta(\C{M}_{\tau})) &= dd^{c} (\log|\chi^{f_{1}^{\ast}}(x)|^{2}) \Theta(\C{M}_{\tau}) = 0 \\ \intertext{et} d^{c}(\log |\chi^{f_{1}^{\ast}}(x)|^{2}\;\Theta(\C{M}_{\tau})) &= d^{c}(\log|\chi^{f_{1}^{\ast}}(x)|^{2})\wedge \Theta(\C{M}_{\tau}) \\ &= \Theta(f_{1}^{\ast}) \wedge \Theta(\C{M}_{\tau}) = \Theta(f_{1}^{\ast}\wedge \C{M}_{\tau}). \end{align*} On tire des d\'efinitions l'\'egalit\'e des courants~: \begin{equation} \label{eqq_3} \int_{\partial C_{\tau}^{\varepsilon}} = \int_{D_{\tau}^{\varepsilon}}\; + \; \int_{E_{\tau}^{\varepsilon}} \end{equation} Comme $\log | \chi^{f_{1}^{\ast}}(x)|^{2}\,\Theta(\C{M}_{\tau})$ est localement $L^{1}$ sur $\partial C_{\tau}$, il vient~: \[ \lim_{\varepsilon \rightarrow 0} \int_{D_{\tau}^{\varepsilon}} \log | \chi^{f_{1}^{\ast}}(x)|^{2}\;\Theta(\C{M}_{\tau})\wedge d^{c}\omega = \int_{\partial C_{\tau}^{\op{int},+}} \log | \chi^{f_{1}^{\ast}}(x)|^{2}\;\Theta(\C{M}_{\tau})\wedge d^{c}\omega. \] De plus, $\log | \chi^{f_{1}^{\ast}}(x)|^{2} = 2 \log \varepsilon$ sur $E_{\tau}^{\varepsilon}$, la forme $\Theta(\C{M}_{\tau})$ est $C^{\infty}$ sur un voisinage de $C_{\tau}$, et pour toute forme $\eta \in A^{2d - q -1}(\M{P}(\Delta)(\M{C}))$ on a $\int_{E_{\tau}^{\varepsilon}}\eta = O(\varepsilon)$ quand $\varepsilon$ tend vers $0$; on en d\'eduit que~: \[ \lim_{\varepsilon \rightarrow 0} \int_{E_{\tau}^{\varepsilon}} \log | \chi^{f_{1}^{\ast}}(x)|^{2}\;\Theta(\C{M}_{\tau})\wedge d^{c}\omega = 0. \] Enfin, puisque $\Theta(f_{1}^{\ast}) = d\op{Arg}(x_{1})/2\pi$ et la forme $\Theta(\C{M}_{\tau})$ est $C^{\infty}$ sur un voisinage de $C_{\tau}$, on a les limites~: \begin{align*} \lim_{\varepsilon \rightarrow 0} \int_{D_{\tau}^{\varepsilon}} \Theta(f_{1}^{\ast}\wedge \C{M}_{\tau}) \wedge \omega &= \int_{\partial C_{\tau}^{\op{int},+}} \Theta(f_{1}^{\ast}\wedge \C{M}_{\tau}) \wedge \omega \\ \intertext{et} \lim_{\varepsilon \rightarrow 0} \int_{E_{\tau}^{\varepsilon}} \Theta(f_{1}^{\ast}\wedge \C{M}_{\tau}) \wedge \omega &= \int_{C_{\tau}^{\op{int},+}}\delta_{H_{1}\cap C_{\tau}^{\op{int}}} \wedge \Theta(\C{M}_{\tau})\wedge \omega, \end{align*} o\`u $H_{1}$ d\'esigne le diviseur d'\'equation $\chi^{f_{1}^{\ast}}(x) = 0$. En revenant au cas g\'en\'eral (i.e. $m_{D,\tau}$ quelconque), on d\'eduit de (\ref{eqq_1}), (\ref{eqq_2}), (\ref{eqq_3}) et des calculs de limites ci-dessus que~: \begin{multline*} I_{\tau}(m_{D,\tau}) = \int_{\partial C_{\tau}^{\op{int},+}} \log | \chi^{m_{D,\tau}}(x)|^{2}\; \Theta(\C{M}_{\tau})\wedge d^{c}\omega \\ - \int_{\partial C_{\tau}^{\op{int},+}}\Theta(m_{D,\tau}\wedge\C{M}_{\tau})\wedge \omega - \int_{C_{\tau}^{\op{int},+}} \delta_{D\cap C_{\tau}^{\op{int}}}\wedge \Theta(\C{M}_{\tau})\wedge \omega. \end{multline*} (On a utilis\'e ici le fait que pour tout $i \in \{d-q+1,\dots,d\}$, on a $f_{i}^{\ast}\wedge \C{M}_{\tau} = 0$). En utilisant (\ref{eq_chern_2}) il vient~: \begin{multline*} \int_{\M{P}(\Delta)(\M{C})}c_{1}(\ov{L})\wedge c_{1}(\ov{L}_{1}) \wedge \dots \wedge c_{1}(\ov{L}_{q})\wedge \omega \\ = \sum_{\tau \in \Delta(d-q)}a_{\tau}(\ov{L}_{1},\dots,\ov{L}_{1})\int_{\partial C_{\tau}^{\op{int},+}} \log | \chi^{m_{D,\tau}}(x)|^{2}\;\Theta(\C{M}_{\tau})\wedge d^{c}\omega \\ - \sum_{\tau \in \Delta(d-q)}a_{\tau}(\ov{L}_{1},\dots,\ov{L}_{1})\int_{\partial C_{\tau}^{\op{int},+}} \Theta(m_{D,\tau}\wedge \C{M}_{\tau}) \wedge \omega. \end{multline*} Enfin, on a~: \begin{multline*} \sum_{\tau \in \Delta(d-q)}a_{\tau}(\ov{L}_{1},\dots,\ov{L}_{1})\int_{\partial C_{\tau}^{\op{int},+}} \log | \chi^{m_{D,\tau}}(x)|^{2}\;\Theta(\C{M}_{\tau})\wedge d^{c}\omega \\ = \sum_{\sigma \in \Delta(d-q-1)}\int_{C_{\sigma}^{\op{int},+}} \Theta\left( \sum_{\substack{\tau > \sigma \\ \tau \in \Delta(d-q)}}\varepsilon_{\sigma}(\tau)a_{\tau}(\ov{L}_{1}, \dots, \ov{L}_{q})\;\C{M}_{\tau}\right) \wedge (\log | \chi^{m_{D,\sigma}}(x)|^{2}d^{c}\omega) = 0 \end{multline*} d'apr\`es l'assertion (2). On a donc~: \begin{multline*} \int_{\M{P}(\Delta)(\M{C})}c_{1}(\ov{L})\wedge c_{1}(\ov{L}_{1}) \wedge \dots \wedge c_{1}(\ov{L}_{q})\wedge \omega \\ = \sum_{\sigma \in \Delta(d-q-1)}\int_{C_{\sigma}^{\op{int},+}} \Theta \left( - \sum_{\substack{\tau > \sigma \\ \tau \in \Delta(d-q)}} \varepsilon_{\sigma}(\tau) a_{\tau}(\ov{L}_{1},\dots,\ov{L}_{q}) \; m_{D,\tau} \wedge \C{M}_{\tau} \right) \wedge \omega. \end{multline*} Pour \'etablir $(1)$ au rang $q+1$ et $(3)$ au rang $q$, il suffit maintenant de prouver que pour tout $\sigma \in \Delta(d-q-1)$ il existe un entier $a_{\sigma}(\ov{L},\ov{L}_{1},\dots,\ov{L}_{q})$ tel que~: \[ a_{\sigma}(\ov{L},\ov{L}_{1},\dots,\ov{L}_{q})\; \C{M}_{\sigma} = - \sum_{\substack{\tau > \sigma \\ \tau \in \Delta(d-q)}} \varepsilon_{\sigma}(\tau) a_{\tau}(\ov{L}_{1},\dots,\ov{L}_{q})\; m_{D,\tau} \wedge \C{M}_{\tau}. \] Soit $\sigma \in \Delta(d-q-1)$ et $\tau_{0} \in \Delta(d-q)$ tel que $\sigma < \tau_{0}$. On d\'eduit de l'assertion (2) au rang $q$ que l'on a~: \[ \sum_{\substack{\tau > \sigma \\ \tau \in \Delta(d-q)}} \varepsilon_{\sigma}(\tau) a_{\tau}(\ov{L}_{1},\dots,\ov{L}_{q})\; m_{D,\tau_{0}} \wedge \C{M}_{\tau} = 0. \] On a donc~: \begin{multline*} \sum_{\substack{\tau > \sigma \\ \tau \in \Delta(d-q)}} \varepsilon_{\sigma}(\tau) a_{\tau}(\ov{L}_{1},\dots,\ov{L}_{q})\; m_{D,\tau} \wedge \C{M}_{\tau} \\ = \sum_{\substack{\tau > \sigma \\ \tau \in \Delta(d-q)}} \varepsilon_{\sigma}(\tau) a_{\tau}(\ov{L}_{1},\dots,\ov{L}_{q})\; (m_{D,\tau} - m_{D,\tau_{0}})\wedge \C{M}_{\tau}. \end{multline*} Or, comme la fonction support $\Psi_{D}$ est continue en $\sigma$, on a $(m_{D,\tau} - m_{D,\tau_{0}}) \in \sigma^{\perp}\cap M$ pour tout $\tau \in \Delta(d-q)$ tel que $\tau > \sigma$, et donc $(m_{D,\tau}- m_{D,\tau_{0}})\wedge \C{M}_{\tau} \in \M{Z}\C{M}_{\sigma}$. Montrons enfin que (2) est vraie au rang $q+1$. Soit $\sigma' \in \Delta(d-q-2)$. D'apr\`es le (3), on a~: \begin{multline*} \sum_{\substack{\sigma > \sigma' \\ \sigma \in \Delta(d-q-1)}} \varepsilon_{\sigma'}(\sigma) a_{\sigma}(\ov{L},\ov{L}_{1},\dots,\ov{L}_{q})\;\C{M}_{\sigma} \\ = - \sum_{\substack{\sigma > \sigma' \\ \sigma \in \Delta(d-q-1)}} \sum_{\substack{\tau > \sigma \\ \tau \in \Delta(d-q)}} \varepsilon_{\sigma'}(\sigma)\varepsilon_{\sigma}(\tau)a_{\tau}(\ov{L},\ov{L}_{1},\dots,\ov{L}_{q})\; m_{D,\tau}\wedge \C{M}_{\tau} \\ = - \sum_{\substack{\tau > \sigma' \\ \tau \in \Delta(d-q)}} a_{\tau}(\ov{L},\ov{L}_{1},\dots,\ov{L}_{q}) \left( \sum_{\substack{\sigma \in \Delta(d-q-1) \\ \tau > \sigma > \sigma'}} \varepsilon_{\sigma'}(\sigma)\varepsilon_{\sigma}(\tau)\right) \; m_{D,\tau}\wedge \C{M}_{\tau}. \end{multline*} Or pour tout $\tau \in \Delta(d-q)$ tel que $\tau > \sigma'$, on a~: \[ \sum_{\substack{\sigma \in \Delta(d-q-1) \\ \tau > \sigma > \sigma'}} \varepsilon_{\sigma'}(\sigma)\varepsilon_{\sigma}(\tau) = 0, \] du fait de la relation $\partial \circ \partial = 0$ dans le complexe simplicial associ\'e \`a la triangulation de $\C{S}^{d-1}$ induite par $\Delta\backslash\{0\}$. Le th\'eor\`eme est donc d\'emontr\'e. \medskip \begin{rem} \label{rem_dani} Pour tout couple d'entiers positifs $(p,q)$, on introduit le $\M{Z}$-module~: \[ C^{q}(\Delta,p) = \bigoplus_{\tau \in \Delta(d-q)}{\textstyle\bigwedge\nolimits^{p}}(\tau^{\perp}). \] On d\'efinit \'egalement un cobord $d: C^{q}(\Delta,p) \rightarrow C^{q+1}(\Delta,p)$ de la fa\c con suivante~: \begin{alignat*}{3} d:\; &C^{q}(\Delta,p) & &\longrightarrow & &\quad C^{q+1}(\Delta,p) \\ &\bigoplus_{\tau \in \Delta(d-q)}x_{\tau}& &\longmapsto & &\bigoplus_{\sigma \in \Delta(d-q-1)} \left( \sum_{\substack{\tau > \sigma \\ \tau \in \Delta(d-q)}}\varepsilon_{\sigma}(\tau)x_{\tau} \right). \end{alignat*} Danilov a d\'emontr\'e (cf. \cite{4}, 12.4.1) que le $q^{\text{\`eme}}$ groupe de cohomologie du complexe $(C^{\ast}(\Delta,p)\otimes \M{C},d)$ est isomorphe \`a $H^{p,q}(\M{P}(\Delta)(\M{C}))$. En reprenant les notations du th\'eor\`eme (\ref{calcul_prod}), on peut associer au courant $c_{1}(\ov{L}_{1})\dotsm c_{1}(\ov{L}_{q})$ un \'el\'ement $C(\ov{L}_{1},\dots,\ov{L}_{q})$ de $C^{q}(\Delta,q)$ d\'efini par~: \[ C(\ov{L}_{1},\dots,\ov{L}_{q}) = \bigoplus_{\tau \in \Delta(d-q)}a_{\tau}(\ov{L}_{1},\dots,\ov{L}_{q})\,\C{M}_{\tau}. \] D'apr\`es l'alin\'ea (2) de (\ref{calcul_prod}), on a $d C(\ov{L}_{1},\dots,\ov{L}_{q}) = 0$; et donc $C(\ov{L}_{1},\dots,\ov{L}_{q})$ d\'efinit un \'el\'ement de $H^{q,q}(\M{P}(\Delta)(\M{C}))$ dont on peut montrer qu'il co\"\i ncide avec la classe $c_{1}(L_{1})\dotsm c_{1}(L_{q})$ par l'isomorphisme \'evoqu\'e pr\'ec\'edemment. On retrouve ainsi gr\^ace au (3) du th\'eor\`eme (\ref{calcul_prod}) la structure multiplicative de $H^{2\ast}(\M{P}(\Delta)(\M{C}))$. \end{rem} \begin{rem} Comme l'\'eventail $\Delta$ est r\'egulier, la famille d'entiers \linebreak[4] $(a_{\tau}(\ov{L}_{1}, \dots, \ov{L}_{q}))_{\tau \in \Delta(d-q)}$ est un {\it poids de Minkowski\/} de codimension $q$ de $\Delta$ au sens de \cite{23}. \end{rem} On d\'eduit imm\'ediatement du th\'eor\`eme (\ref{calcul_prod}) le corollaire~: \begin{cor} \label{support_courant} Soient $\ov{L}_{1},\dots,\ov{L}_{q}$ des fibr\'es en droites au-dessus de $\M{P}(\Delta)$ munis \`a l'infini de leur m\'etrique canonique; on a~: \[ \op{Supp}(c_{1}(\ov{L}_{1})\dotsm c_{1}(\ov{L}_{q})) \subset \bigcup_{\tau \in \Delta(d-q)}C_{\tau}. \] \end{cor} On note $\C{S}_{N}^{+} = C_{\{0\}}^{\op{int},+}$ le tore compact $\C{S}_{N}$ muni de l'orientation canonique induite par le choix de $\C{M}_{\{0\}}$, et $d \mu^{+}$ la forme volume canonique $\Theta(\C{M}_{\{0\}})$. On peut alors \'enoncer un second corollaire~: \begin{cor} \label{produit_chern_max} Soient $\ov{L}_{1},\dots,\ov{L}_{d}$ des fibr\'es en droites au-dessus de $\M{P}(\Delta)$ munis \`a l'infini de leur m\'etrique canonique; pour toute fonction $f$ de classe $C^{\infty}$ sur $\M{P}(\Delta)(\M{C})$, on a~: \[ \int_{\M{P}(\Delta)(\M{C})}f\, c_{1}(\ov{L}_{1})\wedge \dots \wedge c_{1}(\ov{L}_{d}) = \op{deg}(c_{1}(L_{1}) \dotsm c_{1}(L_{d})) \int_{\C{S}_{N}^{+}}f \, d\mu^{+}. \] \end{cor} \noindent {\bf D\'emonstration.}\ D'apr\`es le th\'eor\`eme (\ref{calcul_prod}), il existe une constante \linebreak[4] $a_{\{0\}}(\ov{L}_{1},\dots,\ov{L}_{d}) \in \M{Z}$ telle que~: \[ \int_{\M{P}(\Delta)(\M{C})}f\, c_{1}(\ov{L}_{1}) \wedge \dots \wedge c_{1}(\ov{L}_{d}) = a_{\{0\}}(\ov{L}_{1},\dots,\ov{L}_{d}) \int_{\C{S}_{N}^{+}}f \, d\mu^{+}. \] On prend $f = 1$ et le r\'esultat d\'ecoule alors directement de (\ref{coho_courant}). \medskip \begin{rem} En prenant $\ov{L} = \ov{L}_{1} = \dots = \ov{L}_{d}$ dans le corollaire (\ref{produit_chern_max}), on constate que la m\'etrique canonique $\|.\|_{L,\infty}$ est une solution au premier probl\`eme de Calabi pour la forme volume singuli\`ere $\delta_{S_{N}^{+}}\wedge d\mu^{+}$ sur $\M{P}(\Delta)(\M{C})$. \end{rem} \begin{rem} \label{algo_efficace} Soient $K_{1},\dots,K_{d}$ des polytopes convexes de $M_{\M{R}}$ \`a sommets dans $M$ tels que $K = K_{1} + \dots + K_{d}$ soit d'int\'erieur non vide. Soient $\Delta'$ un raffinement r\'egulier de $\Delta$ l'\'eventail associ\'e \`a $K$ comme au th\'eor\`eme (\ref{construction_inverse}) et \`a la remarque (\ref{construction_inverse3}), et $E_{1}',\dots,E_{d}'$ les diviseurs horizontaux invariants sur $\M{P}(\Delta')$ associ\'es \`a $K_{1},\dots,K_{d}$ respectivement comme au (\ref{construction_inverse3}). En remarquant que le calcul de $c_{1}(\ov{E}_{1,\infty}')\dotsm c_{1}(\ov{E}_{d,\infty}')$ ne n\'ecessite pas de conna\^\i tre $\Delta'$ mais seulement $\Delta$ et les fonctions supports $\psi_{K_{1}}, \dots, \psi_{K_{d}}$, on d\'eduit du th\'eor\`eme (\ref{calcul_prod}) un algorithme efficace pour le calcul du volume mixte~: \[ V(K_{1},\dots,K_{d}) = \frac{1}{d!}\op{deg}( c_{1}(\ov{E}_{1,\infty}')\dotsm c_{1}(\ov{E}_{d,\infty}')). \] \end{rem} \bigskip \subsection{Diviseurs \'el\'ementaires}~ Dans ce paragraphe on \'etablit un raffinement de (\ref{support_courant}) dans le cas o\`u les fibr\'es en droites consid\'er\'es sont des faisceaux associ\'es \`a des diviseurs invariants \'el\'ementaires. Cela nous permet de construire de mani\`ere canonique une section du morphisme d'anneaux~: $[\cdot]: \ov{A}_{\op{f}}^{\ast}(\M{P}(\Delta)_{\M{R}}) \rightarrow H^{2\ast}(\M{P}(\Delta)(\M{C}),\M{R})$. \begin{thm} \label{trivialite1} Soient $D_{1} = V(\tau_{1}), \dots, D_{q} = V(\tau_{q})$ pour $q \leqslant d$ des diviseurs invariants \'el\'ementaires; on a~: \[ \op{Supp}(c_{1}(\ov{\C{O}(D_{1})}_{\infty}) \dotsm c_{1}(\ov{\C{O}(D_{q})}_{\infty})) \subset \bigcup_{\substack{\tau \in \Delta(d-q) \\ \tau < \sigma \in \Delta_{\op{max}} \\ \sigma > \tau_{1}, \dots,\tau_{q}}}C_{\tau}. \] \end{thm} \noindent {\bf D\'emonstration.}\ On suppose dans un premier temps que $q = 1$. Par d\'efinition, la fonction support $\Psi_{D_{1}}$ de $D_{1}$ est nulle sur tout c\^one maximal $\sigma \in \Delta_{\op{max}}$ ne contenant pas $\tau_{1}$. L'\'enonc\'e pour $q=1$ est alors une simple cons\'equence du th\'eor\`eme (\ref{calcul2_c1}). On suppose maintenant que $q > 1$. Le support du courant $c_{1}(\ov{\C{O}(D_{1})}_{\infty})\dotsm \linebreak[0] c_{1}(\ov{\C{O}(D_{q})}_{\infty})$ est inclus dans celui de $c_{1}(\ov{\C{O}(D_{1})}_{\infty})$ (on peut voir cela en approchant la m\'etrique canonique sur $\C{O}(D_{1})$ par une m\'etrique $C^{\infty}$). On d\'eduit de ce qui pr\'ec\`ede et du th\'eor\`eme (\ref{calcul_prod}) que~: \[ \op{Supp}(c_{1}(\ov{\C{O}(D_{1})}_{\infty}) \dotsm c_{1}(\ov{\C{O}(D_{q})}_{\infty})) \subset \bigcup_{\substack{\tau \in \Delta(d-q) \\ \tau < \sigma \in \Delta_{\op{max}} \\ \sigma > \tau_{1}}}C_{\tau}. \] Le courant $c_{1}(\ov{\C{O}(D_{1})}_{\infty})\dotsm c_{1}(\ov{\C{O}(D_{q})}_{\infty})$ \'etant ind\'ependant de l'ordre des diviseurs $D_{1},\dots,D_{q}$, il vient~: \[ \op{Supp}(c_{1}(\ov{\C{O}(D_{1})}_{\infty}) \dotsm c_{1}(\ov{\C{O}(D_{q})}_{\infty})) \subset \bigcup_{\substack{\tau \in \Delta(d-q) \\ \tau < \sigma \in \Delta_{\op{max}} \\ \sigma > \tau_{1}, \dots,\tau_{q}}}C_{\tau}. \] \medskip On a alors le corollaire suivant~: \begin{cor} \label{trivialite2} Soient $D_{1} = V(\tau_{1}), \dots, D_{q} = V(\tau_{q})$ pour $q \leqslant d$, des diviseurs invariants \'el\'ementaires tels que $D_{1}\dotsm D_{q} = 0$ dans $CH^{\ast}(\M{P}(\Delta))$ (i.e. tels que le c\^one $\tau = \M{R}^{+}\tau_{1} + \dots + \M{R}^{+}\tau_{q}$ ne soit pas un \'el\'ement de $\Delta$). On a~: \[ c_{1}(\ov{\C{O}(D_{1})}_{\infty})\dotsm c_{1}(\ov{\C{O}(D_{q})}_{\infty}) = 0. \] \end{cor} \noindent {\bf D\'emonstration.}\ Comme $\tau \notin \Delta$, on ne peut pas trouver $\sigma \in \Delta_{\op{max}}$ tel que $\tau_{1},\dots,\tau_{q}$ soient des faces de $\sigma$ (sinon $\tau$ serait une face de $\sigma$, et donc un \'el\'ement de $\Delta$). On d\'eduit de (\ref{trivialite1}) que $\op{Supp}(c_{1}(\ov{\C{O}(D_{1})}_{\infty}) \dotsm c_{1}(\ov{\C{O}(D_{q})}_{\infty})) = \emptyset$, et donc que $ c_{1}(\ov{\C{O}(D_{1})}_{\infty})\dotsm c_{1}(\ov{\C{O}(D_{q})}_{\infty}) = 0$. \medskip On d\'eduit de (\ref{trivialite2}) et des th\'eor\`emes (\ref{anneau_chow}) et (\ref{calcul2_c1}) qu'il existe un morphisme d'anneaux~: \[ \varsigma: H^{2\ast}(\M{P}(\Delta)(\M{C}),\M{R}) \longrightarrow \ov{A}_{\op{f}}^{\ast}(\M{P}(\Delta)_{\M{R}}) \] tel que pour tout $q$-uplet $(D_{1},\dots,D_{q})$ de diviseurs $T$-invariants sur $\M{P}(\Delta)$, on ait~: \[ \varsigma(c_{1}(\C{O}(D_{1})) \dotsm c_{1}(\C{O}(D_{q}))) = c_{1}(\ov{\C{O}(D_{1})}_{\infty})\dotsm c_{1}(\ov{\C{O}(D_{q})}_{\infty}). \] D'apr\`es (\ref{coho_courant}), $\varsigma$ est une section du morphisme classe $[\cdot]$. \begin{rem} Soit $\C{A}^{q}(\Delta)$ le groupe des poids de Minkowski de codimension $q$ pour $\Delta$ tel qu'il est d\'efini dans \cite{23}. On note $\xi: \C{A}^{q}(\Delta) \rightarrow \ov{A}_{\op{f}}^{q}(\M{P}(\Delta)_{\M{R}})$ le morphisme de groupes d\'efini par l'identit\'e~: \[ \xi(\oplus_{\tau \in \Delta(d-q)}a_{\tau}) = \sum_{\tau \in \Delta(d-q)}a_{\tau} \int_{C_{\tau}^{\op{int},+}}\Theta(\C{M}_{\tau}) \wedge \cdot \] On note \'egalement $\xi$ le morphisme $\C{A}^{q}(\Delta) \otimes \M{R} \rightarrow \ov{A}_{\op{f}}^{q}(\M{P}(\Delta)_{\M{R}})$ obtenu par extension des scalaires \`a partir de $\xi$. On peut alors factoriser l'application $\varsigma$ de la fa\c con suivante~: \begin{center} \mbox{ \xymatrix{ \C{A}^{q}(\Delta)\otimes \M{R} \ar[r]^{\xi} & \ov{A}_{\op{f}}^{q}(\M{P}(\Delta)_{\M{R}}) \\ H^{2\ast}(\M{P}(\Delta)(\M{C}),\M{R}) \ar[u]^{\tilde{\varsigma}} \ar[ru]_{\varsigma} & } } \end{center} \medskip \noindent D'apr\`es la remarque (\ref{rem_dani}), le morphisme $\tilde{\varsigma}$ est un isomorphisme; de plus il induit par restriction un isomorphisme d'anneaux gradu\'es~: \[ \tilde{\varsigma}: H^{2\ast}(\M{P}(\Delta)(\M{C}),\M{Z}) \longrightarrow \C{A}^{\ast}(\Delta). \] L'alin\'ea (3) du th\'eor\`eme (\ref{calcul_prod}) redonne la structure multiplicative de $\C{A}^{\ast}(\Delta)$ induite comme dans \cite{23} par la structure d'anneau naturelle de l'anneau de Chow op\'eratoriel de $\M{P}(\Delta)$. On retrouve ainsi , sous une forme diff\'erente, certains r\'esultats de \cite{23}. \end{rem} \bigskip \section{Produits de courants}~ \subsection{Motivation}~ Soient $\M{P}(\Delta)$ une vari\'et\'e torique projective lisse et $\ov{L}_{\infty}$ un fibr\'e en droites engendr\'e par ses sections globales sur $\M{P}(\Delta)$ et muni de sa m\'etrique canonique. Soit $s$ une section holomorphe de $L$ sur un ouvert $U \subset \M{P}(\Delta)(\M{C})$. En g\'en\'eral la fonction $x \mapsto -\log \|s(x)\|^{2}_{L,\infty}$ n'est pas de classe $C^{\infty}$ mais seulement plu\-ri\-sous\-har\-mo\-ni\-que\ sur $U$ (cf. prop. (\ref{psh})); la premi\`ere ``forme'' de Chern $c_{1}(\ov{L}_{\infty}) = -dd^{c} \log \|s\|^{2}_{L,\infty}$ n'est alors d\'efinie qu'au sens des distributions (c'est un courant de bidegr\'e (1,1)). On ne peut esp\'erer d\'efinir en toute g\'en\'eralit\'e le produit de deux courants; ici n\'eanmoins, $c_{1}(\ov{L}_{\infty})$ est un courant positif (voir \cite{7}, prop. 3.1.18 et 3.1.14). Une construction due \`a Bedford et Taylor (cf. \cite{1}) permet d'associer \`a des fibr\'es en droites $\ov{L}_{1,\infty}, \dots, \ov{L}_{p,\infty}$ engendr\'es par leurs sections globales sur $\M{P}(\Delta)$ et munis de leur m\'etrique canonique, un produit~: \[ c_{1}(\ov{L}_{1,\infty}) \dotsm c_{1}(\ov{L}_{p,\infty}) \] poss\'edant des propri\'et\'es satisfaisantes. La construction pr\'esent\'ee dans ce qui suit \'etant en fait tout \`a fait g\'en\'erale, on abandonne dans cette partie et la suivante le point de vue particulier des vari\'et\'es toriques. \bigskip \subsection{Th\'eorie de Bedford-Taylor-Demailly}~ On suit l'expos\'e donn\'e dans (\cite{5}, \S 3; \cite{6}, \S 1 et \S 2; et \cite{7}, \S 3.3). Tous les r\'esultats pr\'esent\'es ici sont dus \`a Bedford et Taylor (cf. \cite{1}) et Demailly (cf. \cite{5}, \cite{6} et \cite{7}). On rappelle tout d'abord quelques d\'efinitions et notations. \medskip Dans toute cette partie, $X$ d\'esigne une vari\'et\'e analytique complexe de dimension complexe $d$. On note $A^{p,q}(X)$ (resp. $D^{p,q}(X)$) l'espace vectoriel des formes diff\'erentiables $C^{\infty}$ complexes (resp. l'espace vectoriel des courants complexes) de type $(p,q)$ sur $X$. Soit $Y \subset X$ un cycle analytique irr\'eductible de codimension $p$, on note $\delta_{Y} \in D^{p,p}(X)$ le courant d'int\'egration sur $Y$. Pour tout \'el\'ement $T \in D^{p,p}(X)$, on note $\op{Supp} T \subset X$ le support de $T$. Enfin pour tout ouvert $U \subset X$, on notera $\op{Psh}(U)$ l'ensemble des fonctions plu\-ri\-sous\-har\-mo\-ni\-ques\ de $U$ vers $[-\infty,+\infty[$ semi-continues sup\'erieurement. \begin{defn} {\bf (Lelong).}~ Soit $p$ un entier positif et $U$ un ouvert de $X$. Un courant $T \in D^{p,p}(U)$ est dit {\it positif\/} (ou {\it faiblement positif\/}) et on note $T \geqslant 0$, si et seulement si\ pour tout choix de $(1,0)$ formes $\alpha_{1}, \dots, \alpha_{d-p}$ de classe $C^{\infty}$ \`a support compact sur $U$, la distribution~: \[ T \wedge (i\alpha_{1}\wedge\ov{\alpha}_{1}) \wedge \dots \wedge (i\alpha_{d-p}\wedge\ov{\alpha}_{d-p}) \] est une mesure positive sur $U$. On note $D_{+}^{p,p}(U) \subset D^{p,p}(U)$ l'ensemble des courants positifs de type $(p,p)$ sur $U$. \end{defn} \medskip On pose alors~: \begin{defn} {\bf (Bedford-Taylor).}~ Soit $T \in D_{+}^{p,p}(U)$ un courant positif {\em fer\-m\'e} de type $(p,p)$ et $u \in \op{Psh}(U)$ une fonction plu\-ri\-sous\-har\-mo\-ni\-que\ localement born\'ee sur $U$ un ouvert de $X$. Le produit $(dd^{c} u) \wedge T$ est d\'efini par la formule~: \[ (dd^{c} u) \wedge T = dd^{c} (uT) \] \end{defn} \begin{rem} Le produit $uT$ est bien d\'efini puisque $u$ est localement born\'ee et $T$ est d'ordre $0$, i.e. \`a coefficients mesures (pour une d\'emonstration de ce fait, voir par exemple \cite{7}, 3.1.14). \end{rem} \begin{prop} Le produit $(dd^{c} u) \wedge T$ d\'efini ci-dessus est un courant positif ferm\'e de bidegr\'e $(p+1, p+1)$. Il \'etend la d\'efinition usuelle du produit $(dd^{c} u) \wedge T$ dans le cas o\`u $u$ est une fonction $C^{\infty}$ sur $U$. \end{prop} \noindent {\bf D\'emonstration.}\ Voir par exemple (\cite{5}, prop. 5.1; \cite{6}, prop. 3.3.2 ou \cite{7}, prop. 1.2). \medskip De mani\`ere plus g\'en\'erale, \'etant donn\'ees $T \in D_{+}^{p,p}(U)$ et $u_{1}, \dots, u_{q}$ des fonctions plu\-ri\-sous\-har\-mo\-ni\-ques\ localement born\'ees sur $U$, on peut d\'efinir par r\'ecurrence sur $q$ le courant~: \[ (dd^{c} u_{1}) \wedge \dots \wedge (dd^{c} u_{q}) \wedge T = dd^{c} (u_{1}(dd^{c} u_{2}) \wedge \dots \wedge (dd^{c} u_{q}) \wedge T). \] C'est encore un courant positif ferm\'e, de bidegr\'e $(p+q,p+q)$. \medskip Le produit ainsi d\'efini a un comportement agr\'eable vis-\`a-vis de la limite uniforme~: \begin{thm} \label{lim_uniforme} Soient $u_{1}, \dots, u_{q}$ des fonctions plu\-ri\-sous\-har\-mo\-ni\-ques\ continues sur $U$. Soient $(u_{1}^{(k)})_{k \in \M{N}}, \dots, (u_{q}^{(k)})_{k \in \M{N}}$, $q$ suites de fonctions plu\-ri\-sous\-har\-mo\-ni\-ques\ localement born\'ees sur $U$ convergeant uniform\'ement sur tout compact de $U$ vers $u_{1}, \dots, u_{q}$ respectivement, et $(T_{k})_{k \in \M{N}}$ une suite de courants positifs ferm\'es convergeant faiblement vers $T$ sur $U$. Alors~: \begin{alignat*}{3} u_{1}^{(k)}(dd^{c} u_{2}^{(k)}) \wedge \dots \wedge (dd^{c} u_{q}^{(k)}) \wedge &T_{k} & &\text{\quad tend vers\quad}& &u_{1}(dd^{c} u_{2}) \wedge \dots \wedge (dd^{c} u_{q}) \wedge T \\ \text{et \quad} (dd^{c} u_{1}^{(k)}) \wedge \dots \wedge (dd^{c} u_{q}^{(k)})\wedge &T_{k}& &\text{\quad vers\quad} & &(dd^{c} u_{1}) \wedge \dots \wedge (dd^{c} u_{q}) \wedge T, \end{alignat*} au sens de la convergence faible des courants. \end{thm} \noindent {\bf D\'emonstration.}\ Voir (\cite{6}, cor. 1.6) ou (\cite{7}, cor. 3.3.6). \medskip Au vue de la prop. (\ref{approximation}), on souhaite affaiblir les hypoth\`eses du th\'eor\`eme pr\'ec\'edent et remplacer la convergence uniforme par la convergence simple d\'ecroissante; c'est l'objet du th\'eor\`eme suivant~: \begin{thm}{\rm \bf (Bedford-Taylor).}~ \label{BeT} Soient $u_{1}, \dots, u_{q}$ appartenant \`a $\op{Psh}(U)$ et localement born\'ees sur $U$, et $u_{1}^{(k)}, \dots, u_{q}^{(k)}$, $q$ suites d\'ecroissantes de fonctions dans $\op{Psh}(U)$ localement born\'ees sur $U$ et convergeant simplement sur $U$ vers $u_{1}, \dots, u_{q}$ respectivement. On a~: \begin{alignat*}{3} u_{1}^{(k)}(dd^{c} u_{2}^{(k)}) \wedge \dots \wedge (dd^{c} u_{q}^{(k)}) \wedge &T & &\text{\quad tend vers\quad}& &u_{1}(dd^{c} u_{2}) \wedge \dots \wedge (dd^{c} u_{q}) \wedge T \\ \text{et \quad} (dd^{c} u_{1}^{(k)}) \wedge \dots \wedge (dd^{c} u_{q}^{(k)})\wedge &T & &\text{\quad vers\quad} & &(dd^{c} u_{1}) \wedge \dots \wedge (dd^{c} u_{q}) \wedge T, \end{alignat*} au sens de la convergence faible des courants. \end{thm} \noindent {\bf D\'emonstration.}\ Voir par exemple (\cite{6}, \S 1.7 ou \cite{7}, th. 3.3.7). \medskip Par r\'egularisation (voir par exemple \cite{7}, th. 1.5.5), on d\'eduit imm\'ediatement de ce th\'eor\`eme le corollaire~: \begin{cor} \label{BeT_commutativite} Soient $u_{1}, \dots, u_{q}$ des fonctions plu\-ri\-sous\-har\-mo\-ni\-ques\ localement born\'ees sur $U$ et $T \in D_{+}^{p,p}(U)$ un courant positif ferm\'e; le produit $u_{1}(dd^{c} u_{2}) \wedge \dots \wedge (dd^{c} u_{q}) \wedge T$ (resp. le produit $(dd^{c} u_{1}) \wedge \dots \wedge (dd^{c} u_{q}) \wedge T$) est ind\'ependant de l'ordre des $u_{2}, \dots, u_{q}$ (resp. de l'ordre des $u_{1}, \dots, u_{q}$). \end{cor} \begin{defn} Soit $U$ un ouvert de $X$ et $u \in \op{Psh}(U)$. On appelle {\it lieu non born\'e\/} de $u$ dans $U$ et on note $L(u)$ l'ensemble des points $x \in U$ tels que $u$ n'est born\'ee sur aucun voisinage de $x$ dans $U$. \end{defn} \medskip Le r\'esultat suivant, d\^u \`a Demailly, montre que sous certaines conditions, on peut abandonner l'hypoth\`ese selon laquelle les fonctions plu\-ri\-sous\-har\-mo\-ni\-ques\ $u_{1}, \dots, u_{q}$ dans les \'enonc\'es (\ref{BeT}) et (\ref{BeT_commutativite}) doivent \^etre choisies {\it localement born\'ees\/}. \begin{thm} {\rm \bf (Demailly).}~ \label{demailly} Soit $U$ un ouvert de $X$ et soient $T \in D_{+}^{p,p}(U)$ ferm\'e et $u_{1}, \dots, u_{q}$ appartenant \`a $\op{Psh}(U)$. On suppose que $q \leqslant d - p$ et que pour tout choix d'indices $j_{1} < \dots < j_{m}$ dans $\{1,\dots,q\}$ l'intersection $L(u_{j_{1}}) \cap \dots \cap L(u_{j_{m}}) \cap \op{Supp}T$ est contenue dans un ensemble analytique de dimension complexe inf\'erieure ou \'egale \`a $(d - p - m)$. On peut construire des courants $u_{1}(dd^{c} u_{2}) \wedge \dots \wedge (dd^{c} u_{q}) \wedge T$ et $(dd^{c} u_{1}) \wedge \dots \wedge (dd^{c} u_{q}) \wedge T$ de masse localement finie sur $U$ et caract\'eris\'es de mani\`ere unique par la propri\'et\'e suivante~: Pour toutes suites d\'ecroissantes $(u_{1}^{(k)})_{k \in \M{N}}, \dots, (u_{q}^{(k)})_{k \in \M{N}}$ de fonctions plu\-ri\-sous\-har\-mo\-ni\-ques\ convergeant simplement vers $u_{1}, \dots, u_{q}$ respectivement, on a~: \begin{alignat*}{3} u_{1}^{(k)}(dd^{c} u_{2}^{(k)}) \wedge \dots \wedge (dd^{c} u_{q}^{(k)}) \wedge &T & &\text{\quad tend vers\quad}& &u_{1}(dd^{c} u_{2}) \wedge \dots \wedge (dd^{c} u_{q}) \wedge T \\ \text{et \quad} (dd^{c} u_{1}^{(k)}) \wedge \dots \wedge (dd^{c} u_{q}^{(k)})\wedge &T & &\text{\quad vers\quad} & &(dd^{c} u_{1}) \wedge \dots \wedge (dd^{c} u_{q}) \wedge T, \end{alignat*} au sens de la convergence faible des courants sur $U$. \end{thm} \noindent {\bf D\'emonstration.}\ Voir (\cite{6}, th. 2.5 et prop. 2.9 et \cite{7}, th. 3.4.5 et prop. 3.4.9). On peut aussi consulter (\cite{5}, th. 5.4). \medskip On d\'eduit imm\'ediatement du th\'eor\`eme pr\'ec\'edent les corollaires suivants~: \begin{cor} Soient $u_{1}, \dots, u_{q}$ et $T$ comme au th\'eor\`eme (\ref{demailly}); le produit $u_{1}(dd^{c} u_{2})\wedge \dots \wedge (dd^{c} u_{q}) \wedge T$ (resp. le produit $(dd^{c} u_{1}) \wedge \dots \wedge (dd^{c} u_{q}) \wedge T$) ne d\'epend pas de l'ordre de $u_{2}, \dots, u_{q}$ (resp. de l'ordre de $u_{1}, \dots u_{q}$). \end{cor} \begin{cor} \label{produit} Soient $u_{1}, \dots, u_{q}$ plu\-ri\-sous\-har\-mo\-ni\-ques\ sur $U$ et telles que pour tout $1\leqslant i \leqslant q$, $L(u_{i})$ est contenu dans un ensemble analytique $A_{i} \subset U$. Si pour tout choix d'indices $j_{1} < \dots j_{m}$ dans $\{1,\dots, q\}$, on a~: \[ \op{codim}A_{j_{1}} \cap \dots \cap A_{j_{m}} \geqslant m, \] alors les courants $u_{1}(dd^{c} u_{2}) \wedge \dots \wedge (dd^{c} u_{q})$ et $(dd^{c} u_{1}) \wedge \dots \wedge (dd^{c} u_{q})$ sont bien d\'efinis et v\'erifient les propri\'et\'es d'approximation \'enonc\'ees au th\'eor\`eme (\ref{demailly}). \end{cor} On expose enfin une construction due \`a Gillet-Soul\'e (cf. \cite{13}, \S 2.1.5)~: \\ Soient $X$ une vari\'et\'e projective complexe, $Z$ un cycle de codimension $q$ de $X$ et $g_{Z}$ un courant de Green localement $L^{1}$ pour $Z$ (i.e. un \'el\'ement de $D^{p-1,p-1}(X)$, localement $L^{1}$ sur $X$, $C^{\infty}$ sur $X - |Z|$ et tel que $dd^{c} g_{Z} + \delta_{Z} = \omega_{Z}$ est $C^{\infty}$ sur $X$). Choisissons une m\'etrique sur $X$ et notons pour tout $\varepsilon \in \M{R}^{+\ast}$~: \[ N_{\varepsilon}(Z) = \{x \in X: \quad d(x,Z) < \varepsilon\}. \] Pour tout $\varepsilon \in \M{R}^{+\ast}$, soit $\rho_{\varepsilon}: X \rightarrow \M{R}$ une fonction $C^{\infty}$ telle que~: \begin{itemize} \item{$0 \leqslant \rho_{\varepsilon} \leqslant 1$ sur $X$,} \item{$\rho_{\varepsilon} = 1$ en dehors de $N_{\varepsilon}(Z)$,} \item{$\rho_{\varepsilon} = 0$ sur $N_{\varepsilon/2}(Z)$,} \end{itemize} et posons $g_{Z}^{(\varepsilon)} = \rho_{\varepsilon}g_{Z}$. \begin{prop} \label{construction_GS_lissage} Pour tout $\varepsilon \in \M{R}^{+\ast}$, les assertions suivantes sont v\'erifi\'ees~: \begin{itemize} \item{$g_{Z}^{(\varepsilon)}$ est une forme $C^{\infty}$ sur $X$.} \item{On a~: $dd^{c} g_{Z}^{(\varepsilon)} + \delta_{Z}^{(\varepsilon)} = \omega_{Z}$, o\`u $\delta_{Z}^{(\varepsilon)}$ est une forme ferm\'ee $C^{\infty}$ dont le support est contenu dans $\ov{N_{\varepsilon}(Z)}$.} \end{itemize} De plus, on a les limites suivantes~: \begin{itemize} \item{$\lim_{\varepsilon \rightarrow 0}g_{Z}^{(\varepsilon)} = g_{Z}$,} \item{$\lim_{\varepsilon \rightarrow 0}\delta_{Z}^{(\varepsilon)} = \delta_{Z}$,} \end{itemize} au sens de la topologie faible des courants. \end{prop} \noindent {\bf D\'emonstration.}\ Voir \cite{13}, \S 2.1.5. \bigskip \subsection{Formes diff\'erentielles g\'en\'eralis\'ees}~ Les r\'esultats \'enonc\'es dans la section pr\'ec\'edente motivent la d\'efinition suivante~: \begin{defn} Soit $U$ un ouvert de $X$ et $u_{1}, \dots, u_{q}$ des fonctions de $U \rightarrow [-\infty, +\infty [$. Le $q$-uplet $(u_{1}, \dots, u_{q})$ est dit {\it admissible sur $U$\/} si et seulement si~: \begin{enumerate} \item{Les fonctions $u_{1}, \dots, u_{q}$ sont des \'el\'ements de $\op{Psh}(U)$.} \item{Pour tout $1 \leqslant i \leqslant q$, l'ensemble $L(u_{i})$ est contenu dans un ensemble analytique $A_{i} \subset U$.} \item{Pour tout choix d'indices $j_{1} < \dots < j_{m}$ dans $\{1, \dots, q\}$, on a~: \[ \op{codim}A_{j_{1}} \cap \dots \cap A_{j_{m}} \geqslant m. \] } \end{enumerate} \end{defn} \medskip Pour tout $q$-uplet $(u_{1}, \dots, u_{q})$ admissible sur un ouvert $U \subset X$, le corollaire (\ref{produit}) nous permet de d\'efinir sur $U$ les courants $u_{1}(dd^{c} u_{2}) \wedge \dots \wedge (dd^{c} u_{q})$ et $(dd^{c} u_{1}) \wedge \dots \wedge (dd^{c} u_{q})$. On pose~: \begin{defn} Soit $p$ un entier positif. On note $\ov{\ov{A}}^{p,p}(X) \subset D^{p,p}(X)$ (resp. $\ov{\ov{A}}^{p,p}_{\op{log}}(X) \subset D^{p,p}(X)$) l'espace vectoriel complexe form\'e des \'el\'ements de $D^{p,p}(X)$ qui, sur tout ouvert $U$ d'un recouvrement suffisamment fin de $X$, peuvent s'\'ecrire sous la forme~: \begin{align*} & \sum_{i = 1}^{n}\omega_{i}(dd^{c} u_{i,1})\wedge \dots \wedge (dd^{c} u_{i,q_{i}}) \\ \big( \text{resp. \quad} &\sum_{i =1}^{n}\omega_{i}u_{i,1}(dd^{c} u_{i,2}) \wedge \dots \wedge (dd^{c} u_{i, q_{i}}) \big) , \end{align*} o\`u pour tout $1 \leqslant i \leqslant n$, on a $\omega_{i} \in A^{p-q_{i}, p - q_{i}}(U)$ (resp. $\omega_{i} \in A^{p -q_{i}+1, p - q_{i} + 1}(U)$) et le $q_{i}$-uplet $(u_{i,1}, \dots, u_{i,q_{i}})$ est admissible sur l'ouvert $U$ consid\'er\'e. \end{defn} \medskip On pose aussi la d\'efinition suivante~: \begin{defn} \label{formes_generalisees} Soit $p$ un entier positif. On dit que $\alpha \in \ov{\ov{A}}^{p,p}(X)$ est une {\it forme diff\'erentielle g\'en\'eralis\'ee\/} de type $(p,p)$ si et seulement si\ sur tout ouvert $U$ d'un recouvrement suffisamment fin de $X$, on peut \'ecrire la restriction de $\alpha$ \`a $U$ sous la forme~: \[ \alpha_{/U} = \sum_{i=1}^{n}\omega_{i}(dd^{c} u_{i,1}) \wedge \dots \wedge (dd^{c} u_{i,q_{i}}), \] o\`u pour tout $1 \leqslant i \leqslant n$, on a $\omega_{i} \in A^{p-q_{i},p - q_{i}}(U)$ et $u_{i,1}, \dots, u_{i,q_{i}}$ sont plu\-ri\-sous\-har\-mo\-ni\-ques\ et {\it localement born\'ees\/} sur $U$. On note $\ov{A}^{p,p}(X)$ l'espace vectoriel complexe des formes diff\'erentielles g\'en\'eralis\'ees de type $(p,p)$ sur $X$. \end{defn} \medskip On pose \'egalement~: \[ \ov{A}^{\ast}(X) = \bigoplus_{p \geqslant 0}\ov{A}^{p,p}(X), \qquad \ov{\ov{A}}^{\ast}(X) = \bigoplus_{p \geqslant 0} \ov{\ov{A}}^{p,p}(X), \text{\quad et \quad} \ov{\ov{A}}_{\op{log}}^{\ast}(X) = \bigoplus_{p \geqslant 0} \ov{\ov{A}}_{\op{log}}^{p,p}(X). \] \begin{rem} On a imm\'ediatement les inclusions suivantes~: \[ \ov{A}^{\ast}(X) \subset \ov{\ov{A}}^{\ast}(X) \subset \ov{\ov{A}}_{\op{log}}^{\ast}(X) \subset D^{\ast}(X). \] \end{rem} \begin{rem} L'op\'erateur $dd^{c}: D^{p,p}(X) \rightarrow D^{p+1,p+1}(X)$ induit par restriction une application $dd^{c}: \ov{\ov{A}}^{p,p}_{\log}(X) \rightarrow \ov{\ov{A}}^{p+1,p+1}_{\log}(X)$. \end{rem} \medskip Soient $x \in \ov{A}^{p,p}(X)$ et $y \in \ov{\ov{A}}^{q,q}_{\op{log}}(X)$. Sur tout ouvert $U \subset X$ assez petit, on peut \'ecrire~: \begin{align*} x &= \sum_{i = 1}^{n}\omega_{i}(dd^{c} u_{i,1}) \wedge \dots \wedge (dd^{c} u_{i,r_{i}}) \\ \text{et \quad} y &= \sum_{j = 1}^{m}\eta_{j}v_{j,1}(dd^{c} v_{j,2}) \wedge \dots \wedge (dd^{c} v_{j, s_{j}}), \end{align*} o\`u pour tout $1 \leqslant i \leqslant n$, on a $\omega_{i} \in A^{p-r_{i}, p - r_{i}}(U)$ et $u_{i,1}, \dots, u_{i, r_{i}}$ sont des fonctions plu\-ri\-sous\-har\-mo\-ni\-ques\ localement born\'ees sur $U$; et o\`u pour tout $1 \leqslant j \leqslant m$, on a $\eta_{j} \in A^{q -s_{j}+1, q - s_{j}+1}(U)$ et le multiplet $(v_{j,1}, \dots, v_{j, s_{j}})$ est admissible sur $U$. D'apr\`es le th\'eor\`eme (\ref{demailly}), l'expression~: \[ \sum_{i=1}^{n}\sum_{j =1}^{m} \omega_{i}\eta_{j} v_{j,1}(dd^{c} u_{i,1}) \wedge \dots \wedge (dd^{c} u_{i,r_{i}}) \wedge (dd^{c} v_{j,2}) \wedge \dots \wedge (dd^{c} v_{j, s_{j}}), \] a bien un sens et d\'efinit sur $U$ un courant de type $(p+q,p+q)$ que l'on note provisoirement $[x\cdot y](U)$. \begin{prop} \label{produit_formes} Le courant $[x\cdot y](U)$ d\'efini ci-dessus ne d\'epend que de $x$ et de $y$. \end{prop} \noindent {\bf D\'emonstration.}\ Soient $x'$ et $y'$ deux courants sur $U$ tels que $x' = x_{/U}$ et $y' = y_{/U}$, et tels qu'on puisse \'ecrire~: \begin{align*} x' &= \sum_{i = 1}^{n'}\omega_{i}' (dd^{c} u'_{i,1}) \wedge \dots \wedge (dd^{c} u'_{i, r'_{i}}) \\ \text{et \quad} y' &= \sum_{j = 1}^{m'}\eta'_{j} v'_{j,1} (dd^{c} v'_{j,2}) \wedge \dots \wedge (dd^{c} v'_{j, s'_{j}}), \end{align*} o\`u pour tout $1 \leqslant i \leqslant n'$, on a $\omega'_{i} \in A^{p-r'_{i}, p - r'_{i}}(U)$ et $u'_{i,1}, \dots, u'_{i, r'_{i}}$ sont des fonctions plu\-ri\-sous\-har\-mo\-ni\-ques\ localement born\'ees sur $U$; et o\`u pour tout $1 \leqslant j \leqslant m'$, on a $\eta'_{j} \in A^{q -s'_{j}, q - s'_{j}}(U)$ et le multiplet $(v'_{j,1}, \dots, v'_{j, s'_{j}})$ est admissible sur $U$. On d\'efinit les courants~: \begin{align*} &[x'\cdot y](U) = \sum_{i=1}^{n'}\sum_{j =1}^{m} \omega'_{i}\eta_{j} v_{j,1}(dd^{c} u'_{i,1}) \wedge \dots \wedge (dd^{c} u'_{i,r'_{i}}) \wedge (dd^{c} v_{j,2}) \wedge \dots \wedge (dd^{c} v_{j, s_{j}}) \\ &\text{et~}\\ &[x'\cdot y'](U) = \sum_{i=1}^{n'}\sum_{j =1}^{m'} \omega'_{i}\eta'_{j} v'_{j,1}(dd^{c} u'_{i,1}) \wedge \dots \wedge (dd^{c} u'_{i,r'_{i}}) \wedge (dd^{c} v'_{j,2}) \wedge \dots \wedge (dd^{c} v'_{j, s'_{j}}). \end{align*} On veut d\'emontrer que le courant $\delta$ d\'efini sur $U$ par l'\'egalit\'e~: \[ \delta = [x'\cdot y'](U) - [x\cdot y](U) = ([x'\cdot y'](U) - [x'\cdot y](U)) + ([x'\cdot y](U) - [x\cdot y](U)), \] est nul. On va prouver pour cela que $[x'\cdot y](U) - [x\cdot y](U)= 0$. On d\'emontrerait de m\^eme que $[x'\cdot y'](U) - [x'\cdot y](U) = 0$. Par r\'egularisation, on peut trouver pour tout $1 \leqslant j \leqslant m$ et $1 \leqslant k \leqslant s_{j}$, une suite d\'ecroissante $(v_{j,k}^{(n)})_{n \in \M{N}}$ d'\'el\'ements de $\op{Psh}(U) \cap C^{\infty}(U)$ convergeant simplement sur $U$ vers $v_{j,k}$. Pour tout $n \in \M{N}$, on note $y^{(n)}$ la forme diff\'erentielle $C^{\infty}$ de type $(q,q)$ sur $U$ d\'efinie par~: \[ y^{(n)} = \sum_{j=1}^{m}\eta_{j}v_{j,1}^{(n)}(dd^{c} v_{j,2}^{(n)})\wedge \dots \wedge (dd^{c} v_{j,s_{j}}^{(n)}). \] De l'\'egalit\'e $x = x'$, on d\'eduit que $x\cdot y^{(n)} = x' \cdot y^{(n)}$ pour tout $n \in \M{N}$, le produit \'etant le produit habituel d'un courant par une forme diff\'erentielle. Or, d'apr\`es (\ref{demailly}), on a~: \begin{alignat*}{3} &x\cdot y^{(n)}& &\text{\quad tend vers \quad}& &[x\cdot y](U) \\ \text{et \quad} &x'\cdot y^{(n)}& & \text{\quad tend vers \quad}& &[x'\cdot y](U), \end{alignat*} au sens de la topologie faible. On en conclut que $[x'\cdot y](U) - [x\cdot y](U) = 0$ et la proposition est d\'emontr\'ee. \medskip L'assertion suivante est une cons\'equence directe de ce qui pr\'ec\`ede. \begin{prop} \label{produit_generalise} Soient $x \in \ov{A}^{p,p}(X)$ et $y \in \ov{\ov{A}}^{q,q}_{\log}(X)$. Il existe un (unique) \'el\'ement de $\ov{\ov{A}}_{\log}^{p+q,p+q}(X)$ que l'on notera $x\cdot y$ et dont la restriction \`a tout ouvert $U \subset X$ assez petit co\"\i ncide avec le courant $[x\cdot y](U)$ d\'efini \`a la proposition (\ref{produit_formes}). \end{prop} \begin{defn} Soient $x \in \ov{A}^{p,p}(X)$ et $y \in \ov{\ov{A}}_{\log}^{q,q}(X)$. On appelle {\it produit de $x$ et de $y$\/} le courant $x\cdot y \in \ov{\ov{A}}_{\log}^{p+q,p+q}(X)$ d\'efini \`a la proposition pr\'ec\'edente. \end{defn} \medskip Plus g\'en\'eralement, soient $x \in \ov{A}^{\ast}(X)$ et $y \in \ov{\ov{A}}^{\ast}_{\log}(X)$, on appelle {\it produit de $x$ et de $y$\/} et l'on note encore $x\cdot y$ le produit gradu\'e de $x$ et de $y$. Les propositions suivantes sont des cons\'equences imm\'ediates des d\'efinitions~: \begin{prop} \label{produit2} Soient $x_{1} \in \ov{A}^{p,p}(X)$ et $x_{2} \in \ov{A}^{q,q}(X) \subset \ov{\ov{A}}^{q,q}_{\log}(X)$; on a~: $x_{1}\cdot x_{2} \in \ov{A}^{p+q,p+q}(X)$. \end{prop} \begin{prop} Soient $x_{1}$ et $x_{2}$ des \'el\'ements de $\ov{A}^{\ast}(X)$ et $y \in \ov{\ov{A}}_{\log}^{\ast}(X)$; on a les relations suivantes~: \begin{enumerate} \item{$x_{1}\cdot (x_{2} \cdot y) = (x_{1}\cdot x_{2}) \cdot y \quad \text{\rm (associativit\'e)}.$} \item{$x_{1}\cdot x_{2} = x_{2} \cdot x_{1} \qquad \qquad \text{\rm (commutativit\'e)}.$} \end{enumerate} \end{prop} \begin{prop} Le produit d\'efini \`a la proposition $($\ref{produit2}$)$ munit $\ov{A}^{\ast}(X)$ d'u\-ne structure d'alg\`ebre associative commutative unif\`ere et gradu\'ee et $\ov{\ov{A}}_{\log}^{\ast}(X)$ d'une structure de $\ov{A}^{\ast}(X)$-module, qui \'etend sa structure usuelle de $A^{\ast}(X)$-module. La restriction \`a $A^{\ast}(X)$ de la structure d'alg\`ebre sur $\ov{A}^{\ast}(X)$ co\"\i ncide avec la structure d'alg\`ebre usuelle sur $A^{\ast}(X)$. \end{prop} Soient $X$ et $Y$ deux vari\'et\'es complexes et $f: Y \rightarrow X$ un morphisme lisse de vari\'et\'es analytiques (i.e. une submersion holomorphe). On consid\`ere $U \subset X$ et $V \subset Y$ deux ouverts tels que $f(V) \subset U$. \\ Un r\'esultat classique (voir par exemple \cite{7}, th. 1.5.9), affirme que si $ u \in \op{Psh}(U)$ alors $f^{\ast}(u) = u \circ f \in \op{Psh}(V)$. Plus g\'en\'eralement, on a le r\'esultat suivant~: \begin{prop} Soit $(u_{1}, \dots, u_{q})$ un $q$-uplet admissible de fonctions r\'eelles sur $U$; le $q$-uplet $(u_{1}\circ f, \dots, u_{q}\circ f)$ est admissible sur $V$. \end{prop} \noindent {\bf D\'emonstration.}\ D'apr\`es la remarque ci-dessus, on sait que les fonctions $u_{1}\circ f, \dots, u_{q} \circ f$ sont plu\-ri\-sous\-har\-mo\-ni\-ques\ sur $V$. Pour tout $1 \leqslant i \leqslant q$, on a~: $L(u_{i}\circ f) = f^{-1}(L(u_{i})) \subset f^{-1}(A_{i})$. Enfin, du fait de la lissit\'e de $f$, les ensembles $f^{-1}(A_{i})$ sont analytiques et on a pour tout choix d'indices $j_{1} < \dots < j_{m}$ dans $\{1, \dots,q\}$~: \[ \op{codim} \, f^{-1}(A_{j_{1}}) \cap \dots \cap f^{-1}(A_{j_{m}}) = \op{codim} A_{j_{1}} \cap \dots \cap A_{j_{m}} \geqslant m , \] d\`es que les intersections consid\'er\'ees sont non vides. \medskip La proposition suivante montre que l'image inverse du courant $u_{1}dd^{c} u_{2} \wedge \dots \wedge dd^{c} u_{q}$ s'exprime simplement~: \begin{prop} \label{pullback_formes} Soit $(u_{1},\dots,u_{q})$ un $q$-uplet admissible de fonctions r\'eelles sur $U$; on a sur $V$ l'\'egalit\'e~: \[ f^{\ast}(u_{1}dd^{c} u_{2} \wedge \dots \wedge dd^{c} u_{q}) = (u_{1}\circ f)(dd^{c} u_{2}\circ f) \wedge \dots \wedge (dd^{c} u_{q} \circ f). \] \end{prop} \noindent {\bf D\'emonstration.}\ Par r\'egularisation, on peut trouver pour tout $1 \leqslant i \leqslant q$ une suite d\'ecroissante $\left(u_{i}^{(n)}\right)_{n \in \M{N}}$ d'\'el\'ements de $\op{Psh}(U) \cap C^{\infty}(U)$ convergeant simplement sur $U$ vers $u_{i}$. Pour tout $n \in \M{N}$ on note $x^{(n)}$ la forme diff\'erentielle $C^{\infty}$ de type $(q-1,q-1)$ sur $U$ d\'efinie par~: \[ x^{(n)} = u_{1}^{(n)}dd^{c} u_{2}^{(n)} \wedge \dots \wedge dd^{c} u_{q}^{(n)}. \] D'apr\`es (\ref{demailly}) la suite $x^{(n)}$ (resp. la suite $f^{\ast}(x^{(n)})$) tend vers $u_{1}dd^{c} u_{2} \wedge \dots \wedge dd^{c} u_{q}$ (resp. vers $(u_{1}\circ f) (dd^{c} u_{2}\circ f) \wedge \dots \wedge (dd^{c} u_{q} \circ f)$) au sens de la convergence faible des courants quand $n$ tend vers $+\infty$. Le r\'esultat d\'ecoule alors de la continuit\'e faible de $f^{\ast}$. \medskip On d\'eduit de ce r\'esultat que toute application analytique lisse $f : Y \rightarrow X$ induit un morphisme de groupes $f^{\ast}: \ov{\ov{A}}^{p,p}_{\log}(X) \rightarrow \ov{\ov{A}}_{\log}^{p,p}(Y)$, qui induit un morphisme gradu\'e $f^{\ast}: \ov{\ov{A}}_{\log}^{\ast}(X) \rightarrow \ov{\ov{A}}_{\log}^{\ast}(Y)$.\\ Les deux propositions suivantes sont des cons\'equences directes de (\ref{pullback_formes})~: \begin{prop} On a~: $f^{\ast}\left(\ov{\ov{A}}^{\ast}(X)\right) \subset \ov{\ov{A}}^{\ast}(Y)$ et $f^{\ast}\left(\ov{A}^{\ast}(X)\right) \subset \ov{A}^{\ast}(Y)$. \end{prop} \begin{prop} \label{prop_formes1} Soient $x \in \ov{A}^{\ast}(X)$ et $y \in \ov{\ov{A}}^{\ast}_{\log}(X)$, on a~: \[ f^{\ast}(x\cdot y) = f^{\ast}(x)\cdot f^{\ast}(y). \] En particulier, le morphisme $f^{\ast}: \ov{A}^{\ast}(X) \rightarrow \ov{A}^{\ast}(Y)$ est un morphisme d'alg\`ebres. \end{prop} \medskip Enfin les deux propositions suivantes sont valables en toute g\'en\'eralit\'e~: \begin{prop} \label{prop_formes2} Soient $g: Z \rightarrow Y$ et $f: Y \rightarrow X$ deux morphismes lisses de vari\'et\'es complexes, on a l'identit\'e~: $(f \circ g)^{\ast} = g^{\ast} \circ f^{\ast}$. \end{prop} \begin{prop} \label{prop_formes3} Soient $\alpha \in A_{c}^{\ast}(Y)$ et $x \in \ov{\ov{A}}_{\log}^{\ast}(X)$, on a~: \[ f_{\ast}\left(\alpha \cdot f^{\ast}(x)\right) = f_{\ast}(\alpha)\cdot x, \] l'image directe $f_{\ast}\left(\alpha \cdot f^{\ast}(x)\right)$ \'etant prise au sens des courants. \end{prop} \bigskip \subsection{Convergence au sens de Bedford-Taylor et image inverse}~ On introduit dans ce paragraphe de nouvelles classes remarquables de courants. \begin{defn} \label{def_formeB_1} Soit $p$ un entier positif et $Z$ un cycle de codimension $q$ de $X$. On note $B^{p,p}(X)$ (resp. $B_{\log}^{p,p}(X)$, resp. $B_{Z}^{p,p}(X)$, resp. $B_{Z,\log}^{p,p}(X)$) l'espace vectoriel complexe form\'e des \'el\'ements de $D^{p,p}(X)$, qui, sur tout ouvert $U$ d'un recouvrement suffisamment fin de $X$, peuvent s'\'ecrire sous la forme~: \begin{align*} &\sum_{i=1}^{n}\omega_{i}(dd^{c} u_{i,1}) \wedge \dotsm \wedge (dd^{c} u_{i,q_{i}}), \\ \bigg(resp. \qquad &\sum_{i=1}^{n}\omega_{i}u_{i,1}(dd^{c} u_{i,2}) \wedge \dotsm \wedge (dd^{c} u_{i,q_{i}}) \bigg), \\ \bigg(resp. \qquad &\sum_{i=1}^{n}\omega_{i}(dd^{c} u_{i,1}) \wedge \dotsm \wedge (dd^{c} u_{i,q_{i}}) \wedge \delta_{Z}, \bigg), \\ \bigg(resp. \qquad &\sum_{i=1}^{n}\omega_{i}u_{i,1}(dd^{c} u_{i,2}) \wedge \dotsm \wedge (dd^{c} u_{i,q_{i}}) \wedge \delta_{Z}, \bigg), \end{align*} o\`u pour tout $1 \leqslant i \leqslant n$, on a $\omega_{i} \in A^{p-q_{i},p-q_{i}}(U)$ (resp. $\omega_{i} \in A^{p-q_{i}+1,p-q_{i}+1}(U)$, resp. $\omega_{i} \in A^{p-q-q_{i},p-q-q_{i}}(U)$, resp. $\omega_{i} \in A^{p-q-q_{i}+1,p-q-q_{i}+1}(U)$) et $u_{i,1},\dots, u_{i,q_{i}}$ sont des fonctions plu\-ri\-sous\-har\-mo\-ni\-ques\ {\em continues} sur $U$. \end{defn} \begin{rem} \label{rem_BT1} On a imm\'ediatement les inclusions $B^{p,p}(X) \subset B_{\log}^{p,p}(X)$,\\ $B_{Z}^{p,p}(X) \subset B_{Z,\log}^{p,p}(X)$ et l'\'egalit\'e $B_{\log}^{p,p}(X) = B_{X,\log}^{p,p}(X)$. \end{rem} En reprenant les hypoth\`eses et les notations de la d\'efinition pr\'ec\'edente, on pose \'egalement~: \begin{defn} On note $B_{0}^{p,p}(X)$ (resp. $B_{\log,0}^{p,p}(X)$, resp. $B_{Z,0}^{p,p}(X)$, \\ resp. $B_{Z,\log,0}^{p,p}(X)$) le sous-espace vectoriel complexe de $B^{p,p}(X)$ (resp. de $B_{\log}^{p,p}(X)$, resp. de $B_{Z}^{p,p}(X)$, resp. de $B_{Z,\log}^{p,p}(X)$) form\'e des \'el\'ements qui, sur tout ouvert $U$ d'un recouvrement suffisamment fin de $X$, peuvent s'\'ecrire sous la forme~: \begin{align*} &\sum_{i=1}^{n}\omega_{i}(dd^{c} u_{i,1}) \wedge \dotsm \wedge (dd^{c} u_{i,q_{i}}), \\ \bigg(resp. \qquad &\sum_{i=1}^{n}\omega_{i}u_{i,1}(dd^{c} u_{i,2}) \wedge \dotsm \wedge (dd^{c} u_{i,q_{i}}) \bigg), \\ \bigg(resp. \qquad &\sum_{i=1}^{n}\omega_{i}(dd^{c} u_{i,1}) \wedge \dotsm \wedge (dd^{c} u_{i,q_{i}}) \wedge \delta_{Z}, \bigg), \\ \bigg(resp. \qquad &\sum_{i=1}^{n}\omega_{i}u_{i,1}(dd^{c} u_{i,2}) \wedge \dotsm \wedge (dd^{c} u_{i,q_{i}}) \wedge \delta_{Z}, \bigg), \end{align*} o\`u les $u_{i,j}$ et les $\omega_{i}$ sont comme \`a la d\'efinition (\ref{def_formeB_1}) et o\`u de plus, pour tout $1 \leqslant i \leqslant n$, la forme $\omega_{i}$ est {\em ferm\'ee}. \end{defn} \begin{rem} On a les inclusions $B_{0}^{p,p}(X) \subset B_{\log,0}^{p,p}(X) = B_{X,\log,0}^{p,p}(X)$ et $B_{Z,0}^{p,p}(X) \subset B_{Z,\log,0}^{p,p}(X)$. \end{rem} \begin{defn} \label{conv_BT1} Soit $(x^{(k)})_{k \in \M{N}}$ une suite d'\'el\'ements de $B_{Z,\log}^{p,p}(X)$. On dit que $(x^{(k)})_{k \in \M{N}}$ converge vers $x \in B_{Z,\log}^{p,p}(X)$ {\it au sens de Bedford-Taylor\/} (en abr\'eg\'e {\it au sens BT\/}) si pour tout ouvert $U$ d'un recouvrement suffisamment fin de $X$, on peut \'ecrire~: \begin{align*} x^{(k)} & = \sum_{i=1}^{n}\omega^{(k)}u_{i,1}^{(k)}(dd^{c} u_{i,2}^{(k)}) \wedge \dotsm \wedge (dd^{c} u_{i,q_{i}}^{(k)})\wedge \delta_{Z} \\ \text{et} \qquad x &= \sum_{i=1}^{n}\omega u_{i,1}(dd^{c} u_{i,2}) \wedge \dotsm \wedge (dd^{c} u_{i,q_{i}})\wedge \delta_{Z}, \end{align*} o\`u $n$ est ind\'ependant de $k$ et o\`u, pour tout $1 \leqslant i \leqslant n$ et tout $k \in \M{N}$, on a $\omega_{i}^{(k)} \in A^{p-q-q_{i}+1,p-q-q_{i}+1}(U)$, $\omega_{i} \in A^{p-q-q_{i}+1,p-q-q_{i}+1}(U)$ et $u_{i,1}^{(k)}, \dots,u_{i,q_{i}}^{(k)}, u_{i,1}, \dots,u_{i,q_{i}}$ sont des fonctions plu\-ri\-sous\-har\-mo\-ni\-ques\ {\em continues} sur $U$; que $\omega_{i}^{(k)}$ tend vers $\omega_{i}$ au sens des formes $C^{\infty}$ quand $k$ tend vers $+\infty$ et que de plus $(u_{i,1}^{(k)})_{k \in \M{N}}, \dots,(u_{i,q_{i}}^{(k)})_{k\in \M{N}}$ convergent uniform\'ement sur $U$ vers $u_{i,1}, \dots, u_{i,q_{i}}$ respectivement quand $k$ tend vers $+\infty$. \end{defn} \begin{defn} \label{conv_BT2} Soit $(x^{(k)})_{k \in \M{N}}$ une suite d'\'el\'ements de $B_{Z}^{p,p}(X)$. On dit que $(x^{(k)})_{k \in \M{N}}$ converge vers $x \in B_{Z}^{p,p}(X)$ {\it au sens BTR\/}, si pour tout ouvert $U$ d'un recouvrement suffisamment fin de $X$, on peut \'ecrire~: \begin{align*} x^{(k)} & = \sum_{i=1}^{n}\omega^{(k)}(dd^{c} u_{i,1}^{(k)}) \wedge \dotsm \wedge (dd^{c} u_{i,q_{i}}^{(k)})\wedge \delta_{Z} \\ \text{et} \qquad x &= \sum_{i=1}^{n}\omega (dd^{c} u_{i,1}) \wedge \dotsm \wedge (dd^{c} u_{i,q_{i}})\wedge \delta_{Z}, \end{align*} o\`u $n$ est ind\'ependant de $k$ et o\`u, pour tout $1 \leqslant i \leqslant n$ et tout $k \in \M{N}$, on a $\omega_{i}^{(k)} \in A^{p-q-q_{i},p-q-q_{i}}(U)$, $\omega_{i} \in A^{p-q-q_{i},p-q-q_{i}}(U)$ et $u_{i,1}^{(k)}, \dots,u_{i,q_{i}}^{(k)}, u_{i,1}, \dots,u_{i,q_{i}}$ sont des fonctions plu\-ri\-sous\-har\-mo\-ni\-ques\ {\em continues} sur $U$; que $\omega_{i}^{(k)}$ tend vers $\omega_{i}$ au sens des formes $C^{\infty}$ quand $k$ tend vers $+\infty$ et que de plus $(u_{i,1}^{(k)})_{k \in \M{N}}, \dots,(u_{i,q_{i}}^{(k)})_{k\in \M{N}}$ convergent uniform\'ement sur $U$ vers $u_{i,1}, \dots, u_{i,q_{i}}$ respectivement quand $k$ tend vers $+\infty$. \end{defn} En reprenant les hypoth\`eses et les notations de la d\'efinition pr\'ec\'edente, on pose \'egalement~: \begin{defn} \label{conv_BT3} Soit $(x^{(k)})_{k \in \M{N}}$ une suite d'\'el\'ements de $B_{Z,\log,0}^{p,p}(X)$. On dit que $(x^{(k)})_{k \in \M{N}}$ converge vers $x \in B_{Z,\log,0}^{p,p}(X)$ {\it fortement au sens BT\/} si pour tout ouvert $U$ d'un recouvrement suffisamment fin de $X$, il existe $n \in \M{N}^{\ast}$ tel que pour tout $k \in \M{N}$ on puisse \'ecrire~: \begin{align*} x^{(k)} & = \sum_{i=1}^{n}\omega^{(k)}u_{i,1}^{(k)}(dd^{c} u_{i,2}^{(k)}) \wedge \dotsm \wedge (dd^{c} u_{i,q_{i}}^{(k)})\wedge \delta_{Z} \\ \text{et} \qquad x &= \sum_{i=1}^{n}\omega u_{i,1}(dd^{c} u_{i,2}) \wedge \dotsm \wedge (dd^{c} u_{i,q_{i}})\wedge \delta_{Z}, \end{align*} o\`u pour tout $1 \leqslant i \leqslant n$, $\omega_{i}^{(k)}$ tend vers $\omega_{i}$ dans $A^{\ast}(U)$ quand $k$ tend vers $+\infty$ et $u_{i,1}^{(k)},\dots,u_{i,q_{i}}^{(k)}$ converge uniform\'ement sur $U$ vers $u_{i,1}, \dots, u_{i,q_{i}}$ respectivement quand $k$ tend vers $+\infty$, et o\`u pour tout $1 \leqslant i \leqslant n$ et tout $k \in \M{N}$ les formes $\omega_{i}^{(k)}$ et $\omega_{i}$ sont {\em ferm\'ees}. \end{defn} \begin{defn} \label{conv_BT4} Soit $(x^{(k)})_{k \in \M{N}}$ une suite d'\'el\'ements de $B_{Z,0}^{p,p}(X)$. On dit que $(x^{(k)})_{k \in \M{N}}$ converge vers $x \in B_{Z,0}^{p,p}(X)$ {\it fortement au sens BTR\/} si pour tout ouvert $U$ d'un recouvrement suffisamment fin de $X$, il existe $n \in \M{N}^{\ast}$ tel que pour tout $k \in \M{N}$ on puisse \'ecrire~: \begin{align*} x^{(k)} & = \sum_{i=1}^{n}\omega^{(k)}(dd^{c} u_{i,1}^{(k)}) \wedge \dotsm \wedge (dd^{c} u_{i,q_{i}}^{(k)})\wedge \delta_{Z} \\ \text{et} \qquad x &= \sum_{i=1}^{n}\omega (dd^{c} u_{i,1}) \wedge \dotsm \wedge (dd^{c} u_{i,q_{i}})\wedge \delta_{Z}, \end{align*} o\`u pour tout $1 \leqslant i \leqslant n$, $\omega_{i}^{(k)}$ tend vers $\omega_{i}$ dans $A^{\ast}(U)$ quand $k$ tend vers $+\infty$ et $u_{i,1}^{(k)},\dots,u_{i,q_{i}}^{(k)}$ converge uniform\'ement sur $U$ vers $u_{i,1}, \dots, u_{i,q_{i}}$ respectivement quand $k$ tend vers $+\infty$, et o\`u pour tout $1 \leqslant i \leqslant n$ et tout $k \in \M{N}$ les formes $\omega_{i}^{(k)}$ et $\omega_{i}$ sont {\em ferm\'ees}. \end{defn} \begin{rem} D'apr\`es le th\'eor\`eme (\ref{lim_uniforme}) les quatre notions de convergence introduites aux d\'efinitions (\ref{conv_BT1}), (\ref{conv_BT2}), (\ref{conv_BT3}) et (\ref{conv_BT4}) sont plus fortes que la convergence faible au sens des courants. \end{rem} \begin{rem} L'application $dd^{c}: B_{\log,0}^{p,p}(X) \rightarrow B_{0}^{p+1,p+1}(X)$ envoie l'ensemble des suites convergeant fortement au sens BT dans l'ensemble des suites convergeant fortement au sens BTR. \end{rem} \begin{rem} L'application $dd^{c}: B^{p,p}(X) \rightarrow B_{0}^{p+1,p+1}(X)$ envoie l'ensemble des suites convergeant au sens BTR dans l'ensemble des suites convergeant fortement au sens BTR. \end{rem} \begin{defn} \label{formes_adherentes} On note $C^{p,p}(X)$ (resp. $C^{p,p}_{\log}(X)$) l'ensemble des limites des suites de $A^{p,p}(X)$ convergeant dans $B^{p,p}(X)$ (resp. dans $B_{\log}^{p,p}(X)$) au sens BTR (resp. au sens BT). On note $C^{p,p}_{0}(X)$ (resp. $C^{p,p}_{\log,0}(X)$)l'ensemble des limites des suites de $A^{p,p}(X)$ convergeant fortement dans $B_{0}^{p,p}(X)$ (resp. dans $B_{\log,0}^{p,p}(X)$) au sens BTR (resp. au sens BT). \end{defn} Soient $x \in B^{p,p}(X)$ et $y \in B^{r,r}_{Z,\log}(X)$. Sur un ouvert $U \subset X$ assez petit, on peut \'ecrire~: \begin{align*} x &= \sum_{i=1}^{n}\omega_{i}(dd^{c} u_{i,1}) \wedge \dotsm \wedge (dd^{c} u_{i,q_{i}}) \\ \text{et}\qquad y &= \sum_{j=1}^{m}\eta_{j}v_{j,1}(dd^{c} v_{j,2}) \wedge \dotsm \wedge (dd^{c} v_{j,r_{j}})\wedge \delta_{Z}, \end{align*} o\`u pour tout $1\leqslant i \leqslant n$, $\omega_{i} \in A^{p-q_{i},p-q_{i}}(U)$ et $u_{i,1},\dots,u_{i,q_{i}}$ sont des fonctions plu\-ri\-sous\-har\-mo\-ni\-ques\ continues sur $U$; et o\`u pour tout $1 \leqslant j \leqslant n$, on a $\eta_{j} \in A^{p-q-r_{j}+1,p-q-r_{j}+1}(U)$ et $v_{j,1}, \dots , v_{j,r_{j}}$ sont des fonctions plu\-ri\-sous\-har\-mo\-ni\-ques\ continues sur $U$. A partir de ces donn\'ees, on d\'efinit un courant de $B_{Z,\log}^{p+r,p+r}(U)$ que l'on note provisoirement $[x\cdot y](U)$, par la formule~: \begin{multline*} [x\cdot y](U) = \\ \sum_{i=1}^{n}\sum_{j=1}^{m} \omega_{i}\eta_{j} v_{j,1} (dd^{c} u_{i,1}) \wedge \dotsm \wedge (dd^{c} u_{i,q_{i}}) \wedge (dd^{c} v_{j,2}) \wedge \dotsm \wedge (dd^{c} v_{j,r_{j}})\wedge \delta_{Z}. \end{multline*} \begin{prop} \label{produit_uniforme} Le courant $[x\cdot y](U)$ ainsi d\'efini ne d\'epend que de $x$ et de $y$. \end{prop} \noindent {\bf D\'emonstration.}\ Par lin\'earit\'e, on se ram\`ene au cas o\`u $Z$ est effectif. Le probl\`eme \'etant local, on se restreint \`a un ouvert $U' \subset U$ tel qu'il existe une suite $(\delta_{Z}^{(n)})_{n \in \M{N}}$ de formes $C^{\infty}$ ferm\'ees et positives sur $U'$ convergeant faiblement au sens des courants vers $\delta_{Z}$. On suit alors {\it mutatis mutandis\/} la d\'emonstration de la proposition (\ref{produit_formes}) en utilisant le th\'eor\`eme (\ref{lim_uniforme}). \medskip On d\'eduit imm\'ediatement de ce qui pr\'ec\`ede~: \begin{prop_defn} \label{produit_uniforme2} Soient $x \in B^{p,p}(X)$ et $y \in B_{Z,\log}^{r,r}(X)$. Il existe un unique \'el\'ement de $B_{Z,\log}^{p+r,p+r}(X)$ que l'on note $x\cdot y$ et qu'on appelle produit de $x$ et de $y$, dont la restriction \`a tout ouvert $U \subset X$ assez petit co\"\i ncide avec le courant $[x\cdot y](U)$ d\'efini \`a la proposition (\ref{produit_uniforme}). \end{prop_defn} Plus g\'en\'eralement, soient $x \in B^{\ast}(X)$ et $y \in B_{Z,\log}^{\ast}(X)$, on appelle {\it produit de x et de y\/} et on note encore $x\cdot y$ le produit gradu\'e de $x$ et de $y$. \begin{rem} On d\'efinit de fa\c con similaire un produit~: \[ (~\cdot~): B_{\log}^{\ast}(X) \times B_{Z}^{\ast}(X) \longrightarrow B_{Z,\log}^{\ast}(X). \] \end{rem} \begin{rem} Du fait de l'inclusion $B^{\ast}(X) \subset B_{\log}^{\ast}(X) = B_{X,\log}^{\ast}(X)$, on dispose d'un produit $(~\cdot~): B^{\ast}(X) \times B^{\ast}_{\log}(X) \rightarrow B_{\log}^{\ast}(X)$. Ce produit co\"\i ncide avec le produit d\'efini \`a la proposition (\ref{produit_generalise}). \end{rem} La proposition suivante est une cons\'equence directe des d\'efinitions~: \begin{prop} Le produit d\'efini \`a la proposition (\ref{produit_uniforme2}) munit $B^{\ast}(X)$ d'une structure d'alg\`ebre associative commutative unif\`ere et gradu\'ee et $B_{Z,\log}^{\ast}(X)$ d'une structure de $B^{\ast}(X)$-module qui \'etend sa structure usuelle de $A^{\ast}(X)$-module. \end{prop} La proposition suivante montre que le produit d\'efini \`a la proposition (\ref{produit_uniforme2}) se comporte bien vis-\`a-vis des convergences ordinaires ou fortes au sens BT et BTR~: \begin{prop} Le produit~: \[ (~\cdot~): B^{\ast}(X) \times B_{Z,\log}^{\ast}(X) \longrightarrow B_{Z,\log}^{\ast}(X), \] est compatible avec les convergences au sens BTR et BT sur $B^{\ast}(X)$ et $B_{Z,\log}^{\ast}(X)$ respectivement. Sa restriction~: \[ (~\cdot~): B_{0}^{\ast}(X) \times B_{Z,\log,0}^{\ast}(X) \longrightarrow B_{Z,\log,0}^{\ast}(X), \] est compatible avec les convergences fortes au sens BTR et BT sur $B^{\ast}_{0}(X)$ et $B_{Z,\log,0}^{\ast}(X)$ respectivement. \end{prop} \noindent {\bf D\'emonstration.}\ C'est une cons\'equence imm\'ediate des d\'efinitions. \medskip \begin{rem} Soient $\varphi \in C_{\log,0}^{p,p}(X)$, $\varphi' \in C_{\log,0}^{q,q}(X)$ et $\varphi'' \in C_{0}^{r,r}(X)$, on a $\varphi dd^{c} \varphi' \in C_{\log,0}^{p+q+1,p+q+1}(X)$ et $\varphi \varphi'' \in C_{\log,0}^{p+r,p+r}(X)$. \end{rem} \begin{thm} \label{th_image_inverse} Soit $f: X \rightarrow Y$ un morphisme de vari\'et\'es projectives complexes. Il existe un unique morphisme de groupes gradu\'es~: \[ f^{\ast}: B^{\ast}_{\log}(Y) \longrightarrow B_{\log}^{\ast}(X), \] tel que les assertions suivantes soient v\'erifi\'ees~: \begin{enumerate} \item{$f^{\ast}$ restreint \`a $A^{\ast}(Y)$ co\"\i ncide avec l'application image inverse usuelle.} \item{$f^{\ast}$ respecte la convergence au sens BT.} \item{On a $f^{\ast}(B^{\ast}(Y)) \subset B^{\ast}(X)$, $f^{\ast}(B_{0}^{\ast}(Y)) \subset B_{0}^{\ast}(X)$ et $f^{\ast}(B_{\log,0}^{\ast}(Y)) \subset B_{\log,0}^{\ast}(Y)$.} \item{$f^{\ast}$ restreint \`a $B^{\ast}(Y)$ (resp. \`a $B_{0}^{\ast}(Y)$, resp. \`a $B_{\log,0}^{\ast}(Y)$) respecte la convergence au sens BTR (resp. la convergence forte au sens BTR, resp. la convergence forte au sens BT).} \item{Si $x \in B^{\ast}(Y)$ et $y \in B_{\log}^{\ast}(Y)$, alors on a~: \[ f^{\ast}(x\cdot y) = f^{\ast}(x)\cdot f^{\ast}(y). \] } \item{Soit $g : V \rightarrow X$ un autre morphisme de vari\'et\'es projectives complexes. On a~: $(f \circ g)^{\ast} = g^{\ast}\circ f^{\ast}$.} \end{enumerate} \end{thm} \noindent {\bf D\'emonstration.}\ Le morphisme $f: X \rightarrow Y$ peut se factoriser comme la composition d'une immersion ferm\'ee $i: X \hookrightarrow \M{P}^{n}_{Y}$ pour $n$ assez grand, et de la projection $\pi: \M{P}_{Y}^{n} \rightarrow Y$ qui est un morphisme {\em lisse}. On va d\'efinir $\pi^{\ast}$ et $i^{\ast}$, puis poser $f^{\ast} = i^{\ast} \circ \pi^{\ast}$. Comme $\pi$ est lisse, $\pi^{\ast}: D^{\ast}(Y) \rightarrow D^{\ast}(X)$ est d\'efini en toute g\'en\'eralit\'e et sa restriction \`a $B_{\log}^{\ast}(Y)$ v\'erifie les assertions $(1)$ \`a $(6)$ d'apr\`es les propositions (\ref{pullback_formes}), (\ref{prop_formes1}), (\ref{prop_formes2}) et (\ref{prop_formes3}). On donne \`a pr\'esent une construction de $i^{\ast}$. Soit $ x \in B_{\log}^{\ast}(Y)$; sur tout ouvert $U$ d'un recouvrement suffisamment fin de $Y$, on peut \'ecrire~: \[ x = \sum_{j = 1}^{n}\omega_{j}u_{j,1}(dd^{c} u_{j,2})\wedge \dotsm \wedge (dd^{c} u_{j,q_{j}}), \] o\`u pour tout $1 \leqslant j \leqslant n$, $\omega_{j} \in A^{\ast}(U)$ et $u_{j,1}, \dots , u_{j,q_{j}}$ sont des fonctions plu\-ri\-sous\-har\-mo\-ni\-ques\ {\em continues} sur $U$. On pose~: \[ i^{\ast}(x)(i^{-1}(U)) = \sum_{i=1}^{n} i^{\ast}(\omega_{j})(u_{j,1}\circ i)(dd^{c} u_{j,2}\circ i)\wedge \dotsm \wedge (dd^{c} u_{j,q_{j}}\circ i) \in B_{\log}^{\ast}(i^{-1}(U)). \] Le morphisme image directe $i_{\ast}: D^{\ast}(X) \rightarrow D^{\ast}(Y)$ \'etant injectif, le courant $i^{\ast}(x)$ est l'unique courant tel que~: \[ i_{\ast}(i^{\ast}(x)) = x\cdot \delta_{i(X)}, \] ce qui montre que le morphisme $i^{\ast}$ est bien d\'efini. On d\'eduit ais\'ement des d\'efinitions que $i^{\ast}$ v\'erifie les assertions $(1)$ \`a $(6)$. Les morphismes $\pi^{\ast}$ et $i^{\ast}$ \'etant d\'efinis, on pose $f^{\ast} = i^{\ast}\circ \pi^{\ast}$. Par composition, $f^{\ast}$ v\'erifie les assertions $(1)$ \`a $(5)$. Il faut montrer que $f^{\ast}$ ne d\'epend pas de la d\'ecomposition $f = \pi \circ i $ utilis\'ee pour le d\'efinir. Pour cela, on montre que les assertions $(1)$ et $(2)$ caract\'erisent $f^{\ast}$ de mani\`ere unique. Soit $x \in B_{\log}^{\ast}(X)$. En utilisant une partition de l'unit\'e associ\'ee \`a un recouvrement suffisamment fin de $Y$, on peut supposer que le support de $x$ est contenu dans un ouvert $U \subset X$ tel que l'on puisse \'ecrire~: \[ x = \sum_{j=1}^{n}\omega_{j}u_{j,1}(dd^{c} u_{j,2})\wedge \dotsm \wedge (dd^{c} u_{j,q_{j}}), \] o\`u pour tout $1 \leqslant j \leqslant n$, $\omega_{j}$ est une forme $C^{\infty}$ dont le support est contenu dans $U$ et $u_{j,1},\dots,u_{j,q_{j}}$ sont des fonctions plu\-ri\-sous\-har\-mo\-ni\-ques\ continues sur $U$. Par r\'egularisation, on peut trouver pour tout $1 \leqslant j \leqslant n$ des suites $(u_{j,1}^{(k)})_{k \in \M{N}}, \dots, $$(u_{j,q_{j}}^{(k)})_{k \in \M{N}}$ de fonctions plu\-ri\-sous\-har\-mo\-ni\-ques\ $C^{\infty}$ convergeant uniform\'ement vers $u_{j,1},\dots,$$u_{j,q_{j}}$ respectivement sur $U$. Posons, pour tout $k \in \M{N}$, \[ x^{(k)} = \sum_{j=1}^{n} \omega_{j}u_{j,1}^{(k)}(dd^{c} u_{j,2}^{(k)})\wedge \dotsm \wedge (dd^{c} u_{j,q_{j}}^{(k)}). \] On d\'eduit de l'assertion (2) que la suite $(f^{\ast}(x^{(k)}))_{k \in \M{N}}$ converge faiblement vers $f^{\ast}(x)$. Comme d'autre part les formes images inverses $f^{\ast}(x^{(k)})$ pour $k \in \M{N}$ ne d\'ependent pas, d'apr\`es l'assertion $(1)$, de la d\'ecomposition choisie, il en est de m\^eme pour $f^{\ast}(x)$. L'assertion $(6)$ se montre de fa\c con similaire. \medskip \begin{rem} Les op\'erateurs $f^{\ast}$ et $dd^{c}$ commutent. \end{rem} \begin{rem} Si $\varphi \in B_{\log,0}^{0}(X)$, alors $f^{\ast}(\varphi) = \varphi \circ f$. \end{rem} \begin{rem} Si $i: X \hookrightarrow Y$ est une immersion ferm\'ee et $\varphi \in B_{\log}^{\ast}(Y)$, on d\'eduit de la construction donn\'ee au th\'eor\`eme (\ref{th_image_inverse}) que~: \[ i_{\ast}(i^{\ast}(\varphi)) = \varphi \cdot \delta_{i(X)}. \] \end{rem} \bigskip \subsection{M\'etriques admissibles}~ Dans cet article, on appellera {\it fibr\'e en droites hermitien\/} sur $X$ tout couple $\ov{L} = (L,\|.\|)$ form\'e d'un fibr\'e en droites holomorphe $L$ sur $X$ et d'une m\'etrique hermitienne {\em continue} sur $L$. On consid\`ere $U \subset X$ un ouvert et $s_{1}$ et $s_{2}$ deux sections holomorphes de $L$ ne s'annulant pas sur $U$. Il existe alors une fonction $h$ holomorphe et ne s'annulant pas sur $U$ telle que $s_{2} = h\cdot s_{1}$. On en d\'eduit que~: \[ dd^{c} (- \log \|s_{2}\|^{2}) = - dd^{c} \log |h|^{2} + dd^{c} ( - \log \|s_{1}\|^{2}) = dd^{c} ( - \log \|s_{1}\|^{2}). \] Cette remarque justifie la d\'efinition suivante~: \begin{defn} \label{classe_chern} Soit $\ov{L} = (L,\|.\|)$ un fibr\'e en droites hermitien sur $X$. On appelle {\it premier courant de Chern\/} de $\ov{L}$ et on note $c_{1}(\ov{L}) \in D^{1,1}(X)$ le courant d\'efini localement par l'\'egalit\'e~: \[ c_{1}(\ov{L}) = dd^{c} ( - \log \|s\|^{2}), \] o\`u $s$ est une section locale holomorphe et ne s'annulant pas du fibr\'e $L$. \end{defn} \begin{prop} \label{additivite_chern} Soient $\ov{L}_{1}$ et $\ov{L}_{2}$ des fibr\'es en droites hermitiens sur $X$. On a la relation~: \[ c_{1}(\ov{L}_{1}\otimes \ov{L}_{2}) = c_{1}(\ov{L}_{1}) + c_{1}(\ov{L}_{2}). \] \end{prop} \noindent {\bf D\'emonstration.}\ C'est une cons\'equence imm\'ediate de la d\'efinition (\ref{classe_chern}). \medskip L'\'enonc\'e suivant est une extension imm\'ediate de la formule de Poincar\'e-Lelong classique au cas des fibr\'es hermitiens quelconques~: \begin{thm}{\bf \rm (formule de Poincar\'e-Lelong g\'en\'eralis\'ee).}~ \label{PL_generalisee} Soit $\ov{L} = (L,\|.\|)$ un fibr\'e en droites hermitien sur $X$ et $s$ une section m\'eromorphe de $L$ sur $X$ non identiquement nulle sur chaque composante connexe de $X$. On a l'\'egalit\'e entre courants~: \[ dd^{c} (- \log \|s\|^{2}) + \delta_{\op{div}s} = c_{1}(\ov{L}). \] \end{thm} \begin{defn} Soit $\ov{L} = (L,\|.\|)$ un fibr\'e en droites hermitien. La m\'etrique $\|.\|$ est dite {\it positive\/} (resp. {\it strictement positive\/}) si et seulement si\ $c_{1}(L,\|.\|) \geqslant 0$ sur $X$ (resp. pour toute forme de K\"ahler $\alpha$ sur $X$ et tout ouvert $U$ d'un recouvrement suffisamment fin de $X$, il existe $\varepsilon \in \M{R}^{+\ast}$ tel que $c_{1}(L,\|.\|) \geqslant \varepsilon\alpha$ sur $U$). \end{defn} \begin{defn} \label{def_admissibilite1} Soit $L$ un fibr\'e en droites holomorphe sur $X$. Une m\'etrique $\|.\|$ continue et positive sur $L$ est dite {\it admissible\/} s'il existe une suite $\left(\|.\|_{n}\right)_{n \in \M{N}}$ de m\'etriques positives de classe $C^{\infty}$ sur $L$, convergeant uniform\'ement vers $\|.\|$ sur $X$.\\ On appelle {\it fibr\'e admissible sur $X$\/} un fibr\'e en droites holomorphe muni d'une m\'etrique admissible sur $X$. \end{defn} \begin{prop} \label{exemple_produit_formes} Soient $\ov{L}_{1} = (L_{1},\|.\|_{1}), \dots, \ov{L}_{q} = (L_{q}, \|.\|_{q})$ des fibr\'es en droites admissibles sur $X$ et $s_{1},\dots,s_{q}$ des sections m\'eromorphes non identiquement nulles, sur chaque composante connexe de $X$, de $L_{1},\dots,L_{q}$ respectivement, et telles que les cycles $\op{div}s_{1}, \dots, \op{div}s_{q}$ soient d'intersection propre (i.e. tels que $\op{codim}(\op{div}s_{j_{1}} \cap \dotsm \cap \op{div}s_{j_{m}}) \geqslant m$ pour tout choix d'indices $j_{1} < \dots < j_{m}$ dans $\{1,\dots,q\}$). Pour tout $1 \leqslant i \leqslant q$, le courant~: \[ (- \log \|s_{i}\|_{i}^{2})c_{1}(\ov{L}_{1})\dotsm c_{1}(\ov{L}_{i-1})\cdot \delta_{\op{div}s_{i+1}}\dotsm \delta_{\op{div}s_{q}}, \] est bien d\'efini et est un \'el\'ement de $\ov{\ov{A}}_{\log}^{q-1,q-1}(X)$. \end{prop} \noindent {\bf D\'emonstration.}\ Le probl\`eme \'etant local, on se ram\`ene par lin\'earit\'e au cas o\`u $s_{1}, \dots,s_{q}$ sont holomorphes au-dessus d'un ouvert $U$. La proposition est alors une cons\'equence directe de la formule de Poincar\'e-Lelong g\'en\'eralis\'ee (\ref{PL_generalisee}), du fait que pour tout $1 \leqslant i \leqslant q$, $c_{1}(\ov{L}_{i}) \in \ov{A}^{1,1}(X)$, et des d\'efinitions. \medskip \begin{prop} \label{approx_uniforme1} Soit $\ov{L} = (L,\|.\|)$ un fibr\'e admissible sur $X$ compacte. Il existe une suite croissante de m\'etriques $C^{\infty}$ positives convergeant uniform\'ement vers $\|.\|$. \end{prop} \noindent {\bf D\'emonstration.}\ Soit $(\|.\|_{n})_{n \in \M{N}}$ une suite de m\'etriques $C^{\infty}$ positives sur $L$ convergeant uniform\'ement vers $\|.\|$. Pour tout $n \in \M{N}$, on note $\varphi(n)$ le plus petit entier positif tel que~: \[ \left| \frac{\;\;\;\|.\|_{\varphi(n)}}{\|.\|\;\;} - 1 \right| < \frac{1}{2^{n+2}}. \] On choisit alors un r\'eel $\lambda_{n}$ dans $]0,1[$ tel que~: \[ \left(1 - \frac{1}{2^{n}}\right) < \lambda_{n} \frac{\;\;\;\|.\|_{\varphi(n)}}{\|.\|\;\;} < \left(1 - \frac{1}{2^{n+1}}\right). \] On a donc construit une application croissante $\varphi: \M{N} \rightarrow \M{N}$ et une suite de r\'eels $(\lambda_{n})_{n \in \M{N}}$ dans $]0,1[$ tendant vers $1$ tels que la suite des m\'etriques $\|.\|_{n}' = \lambda_{n}\|.\|_{\varphi(n)}$ pour $n \in \M{N}$ soit croissante sur $X$. Comme $c_{1}(L,\|.\|'_{n}) = c_{1}(L,\|.\|_{\varphi(n)}) \geqslant 0$ pour tout $n \in \M{N}$ et que $\left(\|.\|'_{n}\right)_{n \in \M{N}}$ converge uniform\'ement vers $\|.\|$ sur $X$, la proposition est d\'emontr\'ee. \medskip \begin{expl}~ \label{exemple_adm1} \begin{itemize} \item{Toute m\'etrique positive $C^{\infty}$ est admissible (prendre $\|.\|_{n} = \|.\|$).} \item{ Soient $\M{P}(\Delta)$ une vari\'et\'e torique projective lisse et $L$ un fibr\'e en droites engendr\'e par ses sections globales au-dessus de $\M{P}(\Delta)$; la m\'etrique canonique $\|.\|_{L,\infty}$ introduite \`a la proposition (\ref{metrique_ind}) est admissible (voir prop. \ref{approximation}).} \item{Plus g\'en\'eralement, soit $L$ un fibr\'e en droites holomorphe engendr\'e par ses sections globales holomorphes au-dessus d'une vari\'et\'e $X$ que l'on suppose compacte. Soit $\varphi: X \rightarrow X$ un morphisme surjectif tel que $\varphi^{\ast}(L) \stackrel{\Phi_{L}}{\simeq} L^{k}$, avec $k$ un entier $> 1$; on munit $L$ d'une m\'etrique de Zhang $\|.\|_{\op{Zh}}$ pour $\varphi$ (i.e. une m\'etrique continue telle que $\varphi^{\ast}\left(L,\|.\|_{\op{Zh}}\right) \stackrel{\Phi_{L}}{\simeq} \left(L,\|.\|_{\op{Zh}}\right)^{k}$). Le fibr\'e hermitien $\left(L,\|.\|_{\op{Zh}}\right)$ est admissible. (Prendre sur $L$ une m\'etrique $\|.\|_{0}$ positive $C^{\infty}$, puis consid\'erer la suite de m\'etriques $(\|.\|_{n})_{n \in \M{N}}$ d\'efinie par la r\'ecurrence~: \[ \|.\|_{n} = \left(\Phi_{L}^{\ast}\,\varphi^{\ast}\, \|.\|_{n-1}\right)^{1/k}. \] Les m\'etriques $\|.\|_{n}$ sont positives et $C^{\infty}$, et la suite $(\|.\|_{n})_{n \in \M{N}}$ converge uniform\'ement vers $\|.\|_{\op{Zh}}$ sur $X$ d'apr\`es (\cite{21}, th. 2.2)).} \end{itemize} \end{expl} \bigskip \subsection{Un th\'eor\`eme d'approximation globale}~ Le th\'eor\`eme suivant montre que pour un fibr\'e ample, les notions de m\'etriques positives et de m\'etriques admissibles sont \'equivalentes~: \begin{thm} \label{admi_positivite} Soit $X$ une vari\'et\'e complexe projective et $L$ un fibr\'e en droites ample sur $X$. Toute m\'etrique positive sur $L$ est admissible. \end{thm} Pour d\'emontrer le th\'eor\`eme (\ref{admi_positivite}) on utilise une m\'ethode de r\'egularisation et de recollement essentiellement due \`a Richberg (cf. \cite{22}). On s'inspire ici de la pr\'esentation donn\'ee dans (\cite{7}, \S 1.5.C). Soit $\theta : \M{R} \rightarrow \M{R}^{+}$ une fonction $C^{\infty}$ dont le support est inclus dans $[-1,1]$ et telle que $\int_{\M{R}}\theta(t) \, dt = 1$ et $\int_{\M{R}}t\theta(t)\, dt = 0$. \begin{lem} \label{approx_richberg} Pour tout $\eta = (\eta_{1}, \dots, \eta_{p}) \in ]0, + \infty[^{p}$, la fonction~: \[ M_{\eta}(t_{1}, \dots, t_{p}) = \int_{\M{R}^{p}}\op{max}\{t_{1}+h_{1}, \dots, t_{p} + h_{p}\} \prod_{1 \leqslant j \leqslant p}\theta\left(h_{j}/\eta_{j}\right)\, dh_{1}\dots dh_{p}, \] v\'erifie les propri\'et\'es suivantes~: \begin{enumerate} \item{$M_{\eta}(t_{1}, \dots, t_{p})$ est croissante en chacune des variables et est convexe et $C^{\infty}$ sur $\M{R}^{p}$.} \item{On a~: $\op{max}\{t_{1}, \dots, t_{p}\} \leqslant M_{\eta}(t_{1}, \dots, t_{p}) \leqslant \op{max}\{t_{1} + \eta_{1}, \dots, t_{p} + \eta_{p}\}$.} \item{On a~: $M_{\eta}(t_{1}, \dots, t_{p}) = M_{(\eta_{1}, \dots, \widehat{\eta_{j}}, \dots, t_{p})}(t_{1}, \dots, \widehat{t_{j}}, \dots, t_{p})$ d\`es que $t_{j} + \eta_{j} \leqslant \max_{k \neq j}\{t_{k} - \eta_{k}\}$.} \item{$M_{\eta}(t_{1}+a, \dots, t_{p} +a) = M_{\eta}(t_{1}, \dots, t_{p}) + a$ pour tout $a \in \M{R}$.} \item{Si $u_{1}, \dots, u_{p}$ sont plu\-ri\-sous\-har\-mo\-ni\-ques\ sur $X$ et v\'erifient $dd^{c} u_{j} \geqslant \alpha$, o\`u $\alpha \in A^{1,1}(X)$, alors $u = M_{\eta}(u_{1}, \dots, u_{p})$ est plu\-ri\-sous\-har\-mo\-ni\-que\ et satisfait \`a l'in\'egalit\'e $dd^{c} u \geqslant \alpha$.} \end{enumerate} \end{lem} \noindent {\bf D\'emonstration.}\ Voir (\cite{7}, Lemme 1.5.16). \medskip On peut passer \`a la d\'emonstration du th\'eor\`eme~: \\ On note $\|.\|$ la m\'etrique positive consid\'er\'ee. Quitte \`a consid\'erer une puissance tensorielle assez grande de $L$, on peut supposer $L$ tr\`es ample.\\ On suppose dans un premier temps que la m\'etrique $\|.\|$ est strictement positive. On note $\alpha$ une forme de K\"ahler sur $X$ telle que $c_{1}(L,\|.\|) \geqslant \alpha$. On choisit $S = \{s_{1}, \dots, s_{N}\}$ une $\M{Z}$-base des sections holomorphes de $L$ au-dessus de $X$. Pour tout $i \in \{1,\dots, N\}$ on note $u_{i} = - \log \|s_{i}\|^{2} \in \op{Psh}(X)$. Pour $i$ et $j$ \'el\'ements de $\{1,\dots, N\}$, on note $f_{i,j}$ la fonction m\'eromorphe d\'efinie par $s_{i} = f_{i,j}s_{j}$. Pour tout couple $(\Omega, x)$ form\'e d'un point $x \in X$ et d'un ouvert $\Omega \subset X$ le contenant, on dit que $\Omega$ est une boule centr\'ee en $x$ de rayon $r$ s'il existe une carte $(V,\varphi)$ de $X$ contenant $\Omega$ telle que $\varphi(x) = 0$ et que $\varphi(\Omega)$ soit une boule centr\'ee en $0$ de rayon $r$ dans $\M{C}^{\dim X}$. On choisit $\left(\Omega_{\alpha}\right)_{\alpha \in A}$ un recouvrement ouvert fini de $X$ v\'erifiant les propri\'et\'es suivantes~: \begin{enumerate} \item{Pour tout $\alpha \in A$, l'ouvert $\Omega_{\alpha}$ est une boule centr\'ee en un point $0_{\alpha} \in \Omega_{\alpha}$ de rayon $r_{\alpha}$ dans une carte $(V_{\alpha}, \varphi_{\alpha})$ de $X$.} \item{Pour tout $\alpha \in A$, il existe $s_{i_{\alpha}} \in S$ qui ne s'annule pas sur un voisinage $U_{\alpha}$ de $\ov{\Omega_{\alpha}}$.} \end{enumerate} Comme $L$ est tr\`es ample et donc engendr\'e par ses sections globales, un tel recouvrement existe toujours. On choisit $\lambda$ un \'el\'ement de $]0,1[$. Pour tout $\alpha \in A$, on peut construire par r\'egularisation (voir par exemple \cite{7}, th. 1.5.5) une famille $\left(u_{\alpha}^{(\varepsilon)}\right)_{\varepsilon \in \M{R}^{+\ast}}$ d'\'el\'ements de $\op{Psh}(\Omega_{\alpha}) \cap C^{\infty}(\Omega_{\alpha})$ telle que $u_{\alpha}^{(\varepsilon)}$ soit une fonction croissante de $\varepsilon$ et que~: \[ \forall \varepsilon \in \M{R}^{+\ast}, \qquad \|u_{i_{\alpha}} - u_{\alpha}^{(\varepsilon)}\|_{\ov{\Omega}_{\alpha}, \infty} < \varepsilon. \] Pour tout $\alpha \in A$, on choisit $\Omega_{\alpha}'' \subset \Omega_{\alpha}' \subset \Omega_{\alpha}$ des boules concentriques de rayon respectif $r''_{\alpha} < r'_{\alpha} < r_{\alpha}$ dans la carte $V_{\alpha}$, et telles que la famille $\left(\Omega''_{\alpha}\right)_{\alpha \in A}$ forme encore un recouvrement de $X$. Soient $\varepsilon_{\alpha}$ et $\gamma_{\alpha}$ deux nombres r\'eels strictement positifs que l'on fixera par la suite. Pour tout $z \in \ov{\Omega}_{\alpha}$, on pose (dans le syst\`eme de coordonn\'ees relatif \`a la carte $(V_{\alpha},\varphi_{\alpha})$), \[ v_{\alpha}^{(\varepsilon_{\alpha},\gamma_{\alpha})}(z) = u_{\alpha}^{(\varepsilon_{\alpha})}(z) + \gamma_{\alpha}({r'}_{\alpha}^{2} - |z|^{2}). \] Pour $\varepsilon_{\alpha} < \varepsilon_{\alpha,0}$ et $\gamma_{\alpha} < \gamma_{\alpha,0}$ assez petits, on a $v_{\alpha}^{(\varepsilon_{\alpha},\gamma_{\alpha})} \leqslant u_{i_{\alpha}} + \lambda / 2$ et $dd^{c} v_{\alpha}^{(\varepsilon_{\alpha},\gamma_{\alpha})} \geqslant (1-\lambda)\alpha$ sur $\ov{\Omega}_{\alpha}$. On pose~: \[ \eta_{\alpha} = \gamma_{\alpha} \min \left\{{r'}_{\alpha}^{2} - {r''}_{\alpha}^{2}, (r_{\alpha}^{2} - \left. {r'}_{\alpha}^{2})\right/ 2\right\}. \] On choisit tout d'abord $\gamma_{\alpha} < \gamma_{\alpha,0}$ tel que $\eta_{\alpha} < \lambda / 2$, puis on choisit $\varepsilon_{\alpha} < \varepsilon_{\alpha,0}$ suffisamment petit pour que l'on ait~: \[ u_{i_{\alpha}} \leqslant u_{\alpha}^{(\varepsilon_{\alpha})} < u_{i_{\alpha}} + \eta_{\alpha} \] sur $\ov{\Omega}_{\alpha}$. Comme $\gamma_{\alpha}\left({r'}_{\alpha}^{2} - |z|^{2}\right)$ est inf\'erieur \`a $-2\eta_{\alpha}$ sur $\partial\Omega_{\alpha}$ et strictement sup\'erieur \`a $\eta_{\alpha}$ sur $\ov{\Omega}''_{\alpha}$, on a~: $v_{\alpha}^{(\varepsilon_{\alpha},\gamma_{\alpha})} < u_{i_{\alpha}} - \eta_{\alpha}$ sur $\partial \Omega_{\alpha}$ et $v_{\alpha}^{(\varepsilon_{\alpha},\gamma_{\alpha})} > u_{i_{\alpha}} + \eta_{\alpha}$ sur $\ov{\Omega}''_{\alpha}$. Pour tout $i \in \{1,\dots,N\}$, on d\'efinit maintenant sur $\ov{\Omega}_{\alpha}$ la fonction~: \[ \widetilde{u}_{i,\alpha} = v_{\alpha}^{(\varepsilon_{\alpha},\gamma_{\alpha})} + \left( - \log |f_{i,i_{\alpha}}|^{2} \right). \] D'apr\`es ce qui pr\'ec\`ede, les fonctions $\widetilde{u}_{i,\alpha}$ v\'erifient les assertions suivantes~: \begin{enumerate} \item{$\widetilde{u}_{i,\alpha} \leqslant u_{i} + \lambda / 2$ sur $\ov{\Omega}_{\alpha}$.} \item{$\widetilde{u}_{i,\alpha} < u_{i} - \eta_{\alpha}$ sur $\partial\Omega_{\alpha}$.} \item{$\widetilde{u}_{i,\alpha} > u_{i} + \eta_{\alpha}$ sur $\ov{\Omega}''_{\alpha}$.} \item{$\widetilde{u}_{i,\alpha}$ est $C^{\infty}$ sur $\ov{\Omega}_{\alpha} - \op{div}s_{i}$, et on a~: $dd^{c} \widetilde{u}_{i,\alpha} \geqslant (1-\lambda)\alpha$.} \end{enumerate} Gr\^ace aux $(2)$ et $(3)$ ci-dessus et au $(3)$ du lemme (\ref{approx_richberg}), on peut d\'efinir pour tout $i \in \{1,\dots,N\}$ la fonction~: \begin{alignat*}{3} \widetilde{u}_{i}: X - &\op{div} s_{i}& &\longrightarrow & &\M{R} \\ &z & &\longrightarrow & &M_{(\eta_{\alpha})}\left(\widetilde{u}_{i,\alpha}(z)\right). \end{alignat*} La fonction $\widetilde{u}_{i}$ est $C^{\infty}$ d'apr\`es le $(1)$ du lemme (\ref{approx_richberg}), et plu\-ri\-sous\-har\-mo\-ni\-que\ d'apr\`es le $(5)$ de (\ref{approx_richberg}). De plus, elle v\'erifie l'encadrement~: \begin{equation} \label{encadrement1} u_{i} < u_{i} + \min_{\alpha \in A}\eta_{\alpha} < \widetilde{u}_{i} \leqslant u_{i} + \lambda. \end{equation} Enfin, pour tout $i$ et $j$ \'el\'ements de $\{1,\dots,N\}$, on a~: \begin{equation} \label{egalite1} \widetilde{u}_{j} = \widetilde{u}_{i} + \left(- \log | f_{j,i}|^{2}\right), \end{equation} d'apr\`es le $(4)$ du lemme (\ref{approx_richberg}). Soit $U$ un ouvert de $X$ suffisamment petit pour qu'il existe $s_{i} \in S$ ne s'annulant pas sur $U$. Soit $s$ une section r\'eguli\`ere de $L$ au-dessus de $U$. On pose~: \[ \|s\|_{\lambda} = \left| \frac{s}{s_{i}}\right| e^{-\widetilde{u}_{i}/2}. \] Gr\^ace \`a l'\'egalit\'e (\ref{egalite1}), la d\'efinition ci-dessus ne d\'epend pas du choix de $s_{i}$ et d\'efinit par recollement une norme $C^{\infty}$ positive sur $L$. De plus, on tire des d\'efinitions l'\'egalit\'e~: \[ \frac{\;\,\|s\|_{\lambda}}{\|s\|} = e^{\frac{u_{i} - \widetilde{u}_{i}}{2}}, \] ce qui donne l'encadrement~: \[ e^{- \lambda /2} \leqslant \frac{\;\,\|s\|_{\lambda}}{\|s\|} \leqslant 1, \] d'apr\`es l'encadrement (\ref{encadrement1}). En faisant tendre $\lambda$ vers $0$, on extrait de la famille $\left(\|.\|_{\lambda}\right)_{\lambda \in ]0,1[}$ une suite croissante de m\'etriques $C^{\infty}$ positives convergeant uniform\'ement vers $\|.\|$ sur $X$. Le th\'eor\`eme est donc d\'emontr\'e dans le cas o\`u $\|.\|$ est strictement positive sur $X$. On s'int\'eresse d\'esormais au cas o\`u $\|.\|$ est suppos\'ee simplement positive. Comme $L$ est tr\`es ample, il existe sur $L$ une m\'etrique $\|.\|'$ qui est $C^{\infty}$ et strictement positive. Pour tout $n \in \M{N}$, la m\'etrique $\|.\|_{n} = (\|.\|)^{1-1/n}\cdot (\|.\|')^{1/n}$ est strictement positive. D'apr\`es le r\'esultat que l'on vient de d\'emontrer, elle est donc approchable uniform\'ement par des m\'etriques $C^{\infty}$ positives sur $X$. Comme $\|.\|_{n}$ tend uniform\'ement vers $\|.\|$ quand $n$ tend vers $+ \infty$, on en d\'eduit que $\|.\|$ est approchable uniform\'ement par des m\'etriques $C^{\infty}$ positives. \bigskip \subsection{Fibr\'es en droites int\'egrables}~ Suivant Zhang, on introduit \`a pr\'esent une nouvelle classe de fibr\'es en droites plus g\'en\'erale que celle des fibr\'es admissibles. On conserve ici la terminologie de Zhang (cf. \cite{21}, \S 1.5)~: \begin{defn} \label{decomposable} Soit $\ov{L} = (L,\|.\|)$ un fibr\'e en droites hermitien sur $X$. Le fibr\'e $\ov{L}$ est dit {\it int\'egrable sur $X$\/} ou plus simplement {\it int\'egrable\/} si et seulement s'il existe $\ov{E}_{1}$ et $\ov{E}_{2}$ deux fibr\'es admissibles sur $X$ tels que~: \[ \ov{L} = \ov{E}_{1} \otimes (\ov{E}_{2})^{-1}. \] \end{defn} \begin{expl}~ \label{exemple_decomp1} \begin{itemize} \item{Supposons $X$ projective; tout fibr\'e en droites $\ov{L} = (L,\|.\|)$ muni d'une m\'etrique $C^{\infty}$ est int\'egrable (consid\'erer $\ov{L}\otimes\ov{H}^{n}$, o\`u $\ov{H}$ est un fibr\'e ample muni d'une m\'etrique $C^{\infty}$ strictement positive sur $X$).} \item{Soient $\M{P}(\Delta)$ une vari\'et\'e torique projective lisse et $\ov{L}_{\infty} = (L,\|.\|_{L,\infty})$ un fibr\'e en droites sur $\M{P}(\Delta)$ muni de sa m\'etrique canonique, $(L(\M{C}),\|.\|_{L,\infty})$ est int\'egrable sur $\M{P}(\Delta)(\M{C})$ (cf. prop. (\ref{decomposition}) et (\ref{approximation})).} \end{itemize} \end{expl} \begin{prop} \label{produit_decomposables} Soient $\ov{E}$ et $\ov{F}$ deux fibr\'es admissibles (resp. int\'egrables) sur $X$; leur produit tensoriel $\ov{E} \otimes \ov{F}$ est admissible (resp. int\'egrable) sur $X$. \end{prop} \noindent {\bf D\'emonstration.}\ On suppose tout d'abord que $\ov{E} = (E,\|.\|_{E})$ et $\ov{F} = (F,\|.\|_{F})$ sont admissibles. On a $c_{1}(\ov{E}\otimes\ov{F}) = c_{1}(\ov{E}) + c_{1}(\ov{F}) \geqslant 0$ d'apr\`es (\ref{additivite_chern}). De plus, si $\left(\|.\|_{E}^{(n)}\right)_{n \in \M{N}}$ et $\left(\|.\|_{F}^{(n)}\right)_{n \in \M{N}}$ sont deux suites de m\'etriques $C^{\infty}$ positives convergeant uniform\'ement sur $X$ vers $\|.\|_{E}$ et $\|.\|_{F}$ respectivement, alors la suite donn\'ee par $\left(\|.\|_{E}^{(n)} \otimes \|.\|_{F}^{(n)}\right)_{n \in \M{N}}$ est une suite de m\'etriques positives $C^{\infty}$ convergeant uniform\'ement sur $X$ vers $\|.\|_{E} \otimes \|.\|_{F}$; on en d\'eduit que $\ov{E}\otimes\ov{F}$ est admissible sur $X$. On suppose maintenant que $\ov{E}$ et $\ov{F}$ sont int\'egrables sur $X$. On peut donc trouver $\ov{E}_{1}$, $\ov{E}_{2}$, $\ov{F}_{1}$ et $\ov{F}_{2}$ des fibr\'es admissibles sur $X$ tels que $\ov{E} = \ov{E}_{1} \otimes (\ov{E}_{2})^{-1}$ et $\ov{F} = \ov{F}_{1} \otimes (\ov{F}_{2})^{-1}$. On tire~: \[ \ov{E} \otimes \ov{F} = \left(\ov{E}_{1} \otimes \ov{F}_{1}\right) \otimes \left(\ov{E}_{2} \otimes \ov{F}_{2}\right)^{-1}, \] et comme $\ov{E}_{1} \otimes \ov{F}_{1}$ et $\ov{E}_{2} \otimes \ov{F}_{2}$ sont admissibles d'apr\`es ce qui pr\'ec\`ede, on en d\'eduit que $\ov{E} \otimes \ov{F}$ est int\'egrable. \medskip \begin{prop} \label{image_reciproque} Soient $X$ et $Y$ deux vari\'et\'es complexes et $f: Y \rightarrow X$ une application holomorphe. Si $\ov{L}$ est un fibr\'e en droites admissible (resp. int\'egrable) sur $X$, alors $f^{\ast}(\ov{L})$ est admissible (resp. int\'egrable) sur $Y$. \end{prop} \noindent {\bf D\'emonstration.}\ C'est une cons\'equence directe des d\'efinitions. \medskip \begin{prop} Soit $\ov{L}$ un fibr\'e en droites hermitien int\'egrable sur $X$; on a~: \[ c_{1}(\ov{L}) \in C_{0}^{1,1}(X) \subset \ov{A}^{1,1}(X). \] \end{prop} \noindent {\bf D\'emonstration.}\ C'est une cons\'equence directe des d\'efinitions (\ref{formes_generalisees}), (\ref{formes_adherentes}) et (\ref{decomposable}). \medskip \begin{thm} \label{coho_courant} Supposons $X$ projective. Soient $\ov{L}_{1}, \dots, \ov{L}_{p}$ des fibr\'es en droi\-tes hermitiens int\'egrables sur $X$ et $\alpha \in \ov{\ov{A}}^{\ast}_{\log}(X)$ un courant ferm\'e; on a l'\'egalit\'e des classes~: \[ [c_{1}(\ov{L}_{1}) \dotsm c_{1}(\ov{L}_{p})\cdot \alpha] = c_{1}(L_{1}) \dotsm c_{1}(L_{p})\cdot [\alpha], \] en cohomologie de de Rham des courants, o\`u l'on a not\'e $[c_{1}(\ov{L}_{1}) \dotsm c_{1}(\ov{L}_{p})\cdot \alpha]$ et $[\alpha ]$ les classes des courants $c_{1}(\ov{L}_{1})\dotsm c_{1}(\ov{L}_{p})\cdot \alpha$ et $\alpha$, et o\`u $c_{1}(L_{1}), \dots, c_{1}(L_{p})$ d\'esignent les premi\`eres classes de Chern des fibr\'es $L_{1}, \dots, L_{q}$ respectivement. \end{thm} \noindent {\bf D\'emonstration.}\ Par polarisation, il suffit de d\'emontrer le r\'esultat pour $(L_{1},\|.\|_{1}),$ $\dots, (L_{p},\|.\|_{p})$ des fibr\'es en droites admissibles. Soient $\left(\|.\|_{1}^{(n)}\right)_{n \in \M{N}}, \dots, \left(\|.\|_{p}^{(n)}\right)_{n \in \M{N}}$ des suites croissantes de m\'etriques $C^{\infty}$ positives sur $L_{1}, \dots, L_{p}$ convergeant vers les m\'etriques $\|.\|_{1}, \dots, \|.\|_{p}$ respectivement. On a~: \[ c_{1}\left(L_{1},\|.\|_{1}^{(n)}\right)\dotsm c_{1}\left(L_{p},\|.\|_{p}^{(n)}\right)\cdot \alpha \text{\quad tend vers \quad} c_{1}(\ov{L}_{1})\dotsm c_{1}(\ov{L}_{p})\cdot \alpha \] au sens de la topologie faible des courants, et donc~: \[ [c_{1}\left(L_{1},\|.\|_{1}^{(n)}\right)\dotsm c_{1}\left(L_{p},\|.\|_{p}^{(n)}\right)\cdot \alpha] \text{\quad tend vers \quad} [c_{1}(\ov{L}_{1})\dotsm c_{1}(\ov{L}_{p})\cdot \alpha] \] pour la topologie naturelle sur $H_{\op{DR}}^{\ast}(X)$. On d\'eduit le r\'esultat de l'\'egalit\'e~: \[ [c_{1}\left(L_{1},\|.\|_{1}^{(n)}\right)\dotsm c_{1}\left(L_{p},\|.\|_{p}^{(n)}\right)\cdot \alpha] = c_{1}(L_{1})\dotsm c_{1}(L_{p})\cdot [\alpha] \] valable pour tout $n \in \M{N}$. \medskip \section{G\'eom\'etrie d'Arakelov des vari\'et\'es toriques}~ Dans toute cette partie, $\M{P}(\Delta)$ d\'esigne une vari\'et\'e torique projective lisse de dimension absolue $d+1$. On montre tout d'abord que les multihauteurs ``canoniques'' de $\M{P}(\Delta)$ (c'est-\`a-dire celles relatives \`a des fibr\'es en droites munis de leur m\'etrique canonique) sont nulles. On en d\'eduit un r\'esultat remarquable~: La hauteur d'une hypersurface dans $\M{P}(\Delta)$ relativement \`a un fibr\'e en droites muni de sa m\'etrique canonique est essentiellement donn\'ee par la {\em mesure de Mahler} du polyn\^ome qui la d\'efinit. On construit enfin de mani\`ere canonique une section du morphisme d'anneaux~: \[ \zeta: \widehat{CH}_{\op{int}}^{\ast}(X) \longrightarrow CH^{\ast}(X). \] L'existence d'une section canonique pour $\zeta$ \'etend les \'enonc\'es d'annulations de nombres arithm\'etiques obtenus dans un premier temps et conduit \`a une description de la structure de l'anneau $\widehat{CH}_{\op{int}}^{\ast}(X)$. \bigskip \subsection{Annulation des multihauteurs}~ L'\'enonc\'e suivant est un cas particulier de (\cite{21}, th. 2.4). On peut \'egalement consulter (\cite{21}, conj. 2.5, 2.6 et th. 2.9) pour des \'enonc\'es proches de celui-ci. \begin{prop} \label{annulation_hauteur} Soient $\ov{L}_{1,\infty}, \dots, \ov{L}_{d+1,\infty}$ des fibr\'es en droites sur $\M{P}(\Delta)$ munis de leur m\'etrique canonique. On a~: \[ h_{\ov{L}_{1,\infty}, \dots, \ov{L}_{d+1,\infty}}(\M{P}(\Delta)) = 0. \] En particulier, pour tout fibr\'e en droites sur $\M{P}(\Delta)$ muni de sa m\'etrique canonique $\ov{L}_{\infty}$, on a~: \[ h_{\ov{L}_{\infty}}(\M{P}(\Delta)) = 0. \] \end{prop} \noindent {\bf D\'emonstration.}\ Soit $p$ un entier strictement sup\'erieur \`a $1$ et consid\'erons l'endomorphisme $[p]: \M{P}(\Delta) \rightarrow \M{P}(\Delta)$ d\'efini au (\ref{definition_endo}). D'apr\`es la proposition (\ref{relation_zhang}), on a pour tout $1 \leqslant i \leqslant d+1$~: \[ [p]^{\ast}(\ov{L}_{i,\infty}) \simeq (\ov{L}_{i,\infty})^{p}. \] On tire de cela que pour tout $1 \leqslant i \leqslant d+1$, \[ \hat{c}_{1}([p]^{\ast}(\ov{L}_{i,\infty})) = \hat{c}_{1}((\ov{L}_{i,\infty})^{p}) = p\,\hat{c}_{1}(\ov{L}_{i,\infty}). \] On obtient donc~: \begin{multline} \label{eq_canonique1} \widehat{\op{deg}}(\hat{c}_{1}([p]^{\ast}(\ov{L}_{1,\infty})) \dotsm \hat{c}_{1}([p]^{\ast}(\ov{L}_{d+1,\infty}))) \\ = p^{d+1}\,\widehat{\op{deg}}(\hat{c}_{1}(\ov{L}_{1,\infty}) \dotsm \hat{c}_{1}(\ov{L}_{d+1,\infty})). \end{multline} D'autre part, on a d'apr\`es les alin\'eas (5) et (7) du th\'eor\`eme (\ref{gdthm})~: \begin{equation} \begin{split} \label{eq_canonique2} \widehat{\op{deg}}(\hat{c}_{1}([p]^{\ast}(\ov{L}_{1,\infty})) \dotsm \hat{c}_{1}([p]&^{\ast}(\ov{L}_{d+1,\infty}))) \\ &= h_{[p]^{\ast}(\ov{L}_{1,\infty}),\dots,[p]^{\ast}(\ov{L}_{d+1,\infty})} (\M{P}(\Delta)) \\ &= h_{\ov{L}_{1,\infty},\dots,\ov{L}_{d+1,\infty}}([p]_{\ast}\M{P}(\Delta)) \\ &= p^{d} h_{\ov{L}_{1,\infty},\dots,\ov{L}_{d+1,\infty}}(\M{P}(\Delta))\\ &= p^{d}\,\widehat{\op{deg}}(\hat{c}_{1}(\ov{L}_{1,\infty}) \dotsm \hat{c}_{1}(\ov{L}_{d+1,\infty})). \end{split} \end{equation} Comme $p > 1$, la conjonction de (\ref{eq_canonique1}) et de (\ref{eq_canonique2}) implique que~: \[ h_{\ov{L}_{1,\infty},\dots,\ov{L}_{d+1,\infty}}(\M{P}(\Delta)) = \widehat{\op{deg}}(\hat{c}_{1}(\ov{L}_{1,\infty}) \dotsm \hat{c}_{1}(\ov{L}_{d+1,\infty})) =0. \] \bigskip \subsection{Hauteurs canoniques des hypersurfaces de $\M{P}(\Delta)$}~ \begin{prop} \label{hauteur_hypersurfaces} Soient $\ov{L}_{1,\infty},\dots, \ov{L}_{d,\infty}$ des fibr\'es en droites sur $\M{P}(\Delta)$ munis de leur m\'etrique canonique et soit $s$ une section rationnelle non nulle d'un fibr\'e en droites $L$ sur $\M{P}(\Delta)$. Soit alors $D$ un diviseur $T$-invariant sur $\M{P}(\Delta)$ tel que $L \simeq \C{O}(D)$ et notons $s_{D}$ la fonction rationnelle sur $\M{P}(\Delta)$ correspondant \`a $s$ par cet isomorphisme. On a~: \[ h_{\ov{L}_{1,\infty},\dots, \ov{L}_{d,\infty}}(\op{div}s) = \op{deg} (c_{1}(L_{1}) \dotsm c_{1}(L_{d})) \int_{\C{S}_{N}^{+}}\log |s_{D}| \, d\mu^{+}. \] \end{prop} \noindent {\bf D\'emonstration.}\ Soit $\|.\|_{\infty}$ la m\'etrique canonique de $L$ et notons $\ov{L}_{\infty} = (L,\|.\|_{\infty})$. Suivant l'alin\'ea (6) du th\'eor\`eme (\ref{gdthm}) il vient~: \begin{multline*} h_{\ov{L}_{1,\infty},\dots, \ov{L}_{d,\infty}}(\op{div}s) \\ = h_{\ov{L}_{1,\infty},\dots, \ov{L}_{d,\infty},\ov{L}_{\infty}}(\M{P}(\Delta)) + \int_{\M{P}(\Delta)(\M{C})}\log \|s\|_{\infty}\,c_{1}(\ov{L}_{1,\infty})\dotsm c_{1}(\ov{L}_{d,\infty}); \end{multline*} d'autre part on sait que $h_{\ov{L}_{1,\infty},\dots, \ov{L}_{d,\infty},\ov{L}_{\infty}}(\M{P}(\Delta)) = 0$ du fait de la proposition (\ref{annulation_hauteur}). On termine alors la d\'emonstration en utilisant le corollaire (\ref{produit_chern_max}) et en remarquant que $\|s\|_{\infty} = |s_{D}|$ sur $\C{S}_{N}$. \medskip \begin{defn} Soit $s$ une fonction rationnelle non nulle sur $\M{P}(\Delta)$. On appelle {\it mesure de Mahler\/} de $s$ et on note $M(s)$ le nombre r\'eel~: \[ M(s) = \int_{\C{S}_{N}^{+}}\log|s|\,d\mu^{+}. \] \end{defn} La proposition suivante est une cons\'equence imm\'ediate de la d\'emonstration de (\cite{3}, prop. 1.5.1). \begin{prop} \label{mahler_remarque} Soient $n$ un entier positif et $U$ un ouvert de $\M{C}^{n}$. Si $F$ est une fonction m\'eromorphe sur $U \times \M{P}(\Delta)(\M{C})$ dont le diviseur $\op{div}F$ est plat sur $U$ (relativement \`a la premi\`ere projection), alors la mesure de Mahler $M(F(u,\cdot))$ d\'epend contin\^ument de $u$. \end{prop} \bigskip \subsection{Un exemple.}~ Pla\c cons nous sur $\M{P}^{n}_{\M{Z}}$ vu comme vari\'et\'e torique de fa\c con standard, et soit $P \in \M{Z}[X_{0},\dots,X_{n}]$ un polyn\^ome homog\`ene de degr\'e $k \in \M{N}^{\ast}$. Le polyn\^ome $P$ d\'efinit une section globale (encore not\'ee $P$) de $\C{O}(k)$ et la hauteur de l'hypersurface $\op{div}P$ est donn\'ee d'apr\`es la proposition (\ref{hauteur_hypersurfaces}) par~: \[ h_{\ov{\C{O}(1)}_{\infty}}(\op{div}P) = M(P) = \frac{1}{(2\pi)^{n+1}} \int_{0}^{2\pi}\dotsi \int_{0}^{2\pi}\log | P(e^{i\theta_{0}},\dots,e^{i\theta_{n}})|\,d\theta_{0} \dotsm d\theta_{n}. \] On peut trouver dans certains cas une formule explicite pour $M(P)$. Soit par exemple la forme lin\'eaire $P(X_{0},\dots,X_{n}) = a_{0}X_{0} + \dots + a_{n}X_{n}$ avec $(a_{0},\dots,a_{n}) \in \M{C}^{n+1}-\{0\}$ et notons $I(a_{0},\dots,a_{n}) = M(a_{0}X_{0} + \dots + a_{n}X_{n})$ sa mesure de Mahler. On d\'eduit de la formule de Jensen l'\'egalit\'e~: \[ I(a_{0},a_{1}) = \log \Sup (|a_{0}|,|a_{1}|). \] Le calcul de $I(a_{0},a_{1},a_{2})$ est plus d\'elicat et fait l'objet de l'\'enonc\'e suivant, obtenu en collaboration avec J. Cassaigne~: \begin{prop}~ \label{polylog} \begin{enumerate} \item{ Si $|a_{0}|$, $|a_{1}|$ et $|a_{2}|$ sont les longueurs des c\^ot\'es d'un triangle du plan (i.e. v\'erifient les in\'egalit\'es $|a_{i}| \leqslant |a_{j}| + |a_{k}|$ avec $\{i,j,k\} = \{0,1,2\}$) alors~: \[ I(a_{0},a_{1},a_{2}) = \frac{\alpha_{0}}{\pi}\log |a_{0}| + \frac{\alpha_{1}}{\pi}\log |a_{1}| + \frac{\alpha_{2}}{\pi}\log |a_{2}| + \frac{1}{\pi}\C{D}\left(\left|\frac{a_{1}}{a_{0}}\right| e^{i\alpha_{2}}\right), \] o\`u $\C{D}(\cdot)$ d\'esigne le dilogarithme de Bloch-Wigner d\'efini par~: \[ \C{D}(z) = \Im (\op{li_{2}}(z) + \log|z|\log(1-z)) \qquad \text{pour}\quad z \in \M{C}\backslash \{0,1\}, \] et o\`u $\alpha_{0},\alpha_{1}$ et $\alpha_{2}$ sont les mesures principales non orient\'ees des angles aux sommets $A_{0}$, $A_{1}$ et $A_{2}$ d'un triangle du plan $\C{T} = (A_{0},A_{1},A_{2})$ tel que $|a_{0}| = A_{1}A_{2}$, $|a_{1}| = A_{0}A_{2}$ et $|a_{2}| = A_{0}A_{1}$.} \item{Sinon, \[ I(a_{0},a_{1},a_{2}) = \log \Sup (|a_{0}|,|a_{1}|,|a_{2}|). \]} \end{enumerate} \end{prop} \noindent {\bf D\'emonstration.}\ D\'emontrons tout d'abord l'assertion (1). Les expressions consid\'er\'ees \'etant sym\'etriques du fait de l'\'equation fonctionnelle v\'erifi\'ee par $\C{D}(\cdot)$, on peut supposer que $|a_{0}| \geqslant |a_{1}| \geqslant |a_{2}|$. On a, en toute g\'en\'eralit\'e, les relations~: \begin{align} I(a_{0},a_{1},a_{2}) &= \frac{1}{(2\pi)^{3}}\int_{0}^{2\pi} \int_{0}^{2\pi}\int_{0}^{2\pi}\log \left|a_{0}e^{i\theta_{0}} + a_{1}e^{i\theta_{1}} + a_{2}e^{i\theta_{2}}\right|\, d\theta_{0}d\theta_{1}d\theta_{2} \notag \\ &= \frac{1}{(2\pi)^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi} \log\left| |a_{0}| + |a_{1}|e^{i\theta_{1}} + |a_{2}|e^{i\theta_{2}}\right| \, d\theta_{1}d\theta_{2} \notag \\ &= \frac{1}{2\pi}\int_{0}^{2\pi}\log \Sup \left(\left| |a_{0}| + |a_{1}|e^{i\theta_{1}}\right|, |a_{2}|\right)\, d\theta_{1}, \label{geo_1} \end{align} la derni\`ere \'egalit\'e \'etant obtenue par application de la formule de Jensen. On pose alors $\alpha = \pi - \alpha_{2}$ et on v\'erifie que $|\,|a_{0}| + |a_{1}|e^{i\alpha}| = |a_{2}|$. On d\'eduit de cela et de (\ref{geo_1}) la relation~: \[ I(a_{0},a_{1},a_{2}) = \frac{\alpha_{2}}{\pi}\log |a_{2}| + \frac{1}{2\pi}\int_{-\alpha}^{\alpha}\log \left||a_{0}| + |a_{1}|e^{i\theta_{1}}\right|\,d\theta_{1}, \] ce dont on tire~: \begin{equation} \begin{split} \label{geo_2} I(a_{0},a_{1},a_{2}) = \frac{\alpha_{2}}{\pi}\log |a_{2}| + \frac{\alpha_{0}}{\pi}\log |a_{0}| + \frac{\alpha_{1}}{\pi}\log |a_{0}| \qquad \qquad \\ ~\qquad \qquad \qquad \qquad + \frac{1}{2\pi}\int_{-\alpha}^{\alpha}\log \left| 1 + \left| \frac{a_{1}}{a_{0}}\right|e^{i \theta_{1}}\right| \, d\theta_{1}, \end{split} \end{equation} puisque $\alpha = \alpha_{0} + \alpha_{1}$. \bigskip \begin{center} \input{figure6.pstex_t} \end{center} \bigskip Enfin, \begin{align*} \frac{1}{2\pi}\int_{-\alpha}^{\alpha}\log \left| 1 + \left| \frac{a_{1}}{a_{0}}\right|e^{i \theta_{1}}\right| \, d\theta_{1} &= \frac{1}{2\pi}\Re\left(\int_{-\alpha}^{\alpha}\log \left(1 + \left|\frac{a_{1}}{a_{0}}\right|e^{i\theta_{1}}\right)\,d\theta_{1}\right) \\ &= \frac{1}{2\pi}\Im\left(\int_{L}\frac{\log(1-z)}{z}\,dz\right), \end{align*} o\`u $L$ est un chemin allant de $(- |a_{1}/a_{0}|e^{-i\alpha})$ \`a $(- |a_{1}/a_{0}|e^{i\alpha})$ d'indice z\'ero par rapport \`a $0$ et $1$. On en d\'eduit que~: \begin{align*} \frac{1}{2\pi}\int_{-\alpha}^{\alpha}\log \left| 1 + \left| \frac{a_{1}}{a_{0}}\right|e^{i \theta_{1}}\right| &\, d\theta_{1} \\ &= \frac{1}{2\pi} \Im\left(\op{li_{2}}\left(- \left|\frac{a_{1}}{a_{0}}\right| e^{- i \alpha}\right) - \op{li_{2}}\left(- \left|\frac{a_{1}}{a_{0}}\right| e^{i \alpha}\right) \right) \\ &= \frac{1}{2\pi}\Im\left(\op{li_{2}}\left(\left|\frac{a_{1}}{a_{0}}\right|e^{i\alpha_{2}} \right)\right) - \frac{1}{2\pi}\Im\left(\op{li_{2}}\left(\left|\frac{a_{1}}{a_{0}}\right|e^{-i\alpha_{2}} \right)\right) \\ &= \frac{1}{\pi} \Im\left(\op{li_{2}}\left(\left|\frac{a_{1}}{a_{0}}\right|e^{i\alpha_{2}} \right)\right) \\ &= \frac{1}{\pi}\C{D}\left(\left|\frac{a_{1}}{a_{0}}\right|e^{i\alpha_{2}}\right) - \frac{1}{\pi}\log \left| \frac{a_{1}}{a_{0}}\right| \Im\left(\log \left(1 - \left|\frac{a_{1}}{a_{0}}\right|e^{i\alpha_{2}}\right)\right) \\ &= \frac{1}{\pi}\C{D}\left(\left|\frac{a_{1}}{a_{0}}\right|e^{i\alpha_{2}}\right) + \frac{\alpha_{1}}{\pi}\log \left|\frac{a_{1}}{a_{0}}\right|, \end{align*} ce qui ajout\'e \`a (\ref{geo_2}) donne le r\'esultat annonc\'e. Pla\c cons nous maintenant sous les hypoth\`eses de l'assertion (2). Les expressions consid\'er\'ees \'etant sym\'etriques, on peut supposer que $|a_{0}| + |a_{1}| < |a_{2}|$. Le r\'esultat se d\'eduit alors directement de la relation (\ref{geo_1}). \medskip \begin{rem} La proposition (\ref{polylog}) g\'en\'eralise certains r\'esultats partiels dus \`a Smyth (cf. \cite{26}, voir \'egalement \cite{25}). Pour un point de vue moderne sur la mesure de Mahler, on peut consulter \cite{24}. On peut s'inspirer de l'approche de (\cite{24}, \S 3) pour donner une interpr\'etation de la proposition (\ref{polylog}) dans le langage de la K-th\'eorie et des structures de Hodge-Tate mixtes. \end{rem} \begin{rem} Si l'on consid\`ere le plan euclidien comme le bord du demi-espace de Poincar\'e $\C{H}_{3}$ muni de la m\'etrique hyperbolique usuelle, le nombre $\C{D}\left(\left|\frac{a_{1}}{a_{0}}\right|e^{i\alpha_{2}}\right)$ intervenant dans la proposition (\ref{polylog}) est le volume du t\'etra\`edre hyperbolique id\'eal dans $\C{H}_{3}$ de sommets $(A_{0},A_{1},A_{2},\infty)$. \end{rem} \bigskip \subsection{L'anneau de Chow arithm\'etique g\'en\'eralis\'e d'une vari\'et\'e torique.}~ L'objet de cette section est de prouver le r\'esultat suivant~: \begin{thm} \label{section_chow} Soit $\M{P}(\Delta)$ une vari\'et\'e torique projective lisse de dimension relative $d$ sur $\op{Spec}\M{Z}$. Il existe une unique section $\hat{\varsigma}$ du morphisme $\zeta: \widehat{CH}_{\op{int}}^{\ast}(\M{P}(\Delta)) \rightarrow CH^{\ast}(\M{P}(\Delta))$ telle que~: \begin{itemize} \item{$\hat{\varsigma}$ soit un morphisme d'anneaux;} \item{ pour tout fibr\'e en droites $L$ sur $\M{P}(\Delta)$, on ait $\hat{\varsigma}(c_{1}(L)) = \hat{c}_{1}(\ov{L}_{\infty})$.} \end{itemize} \end{thm} On montre tout d'abord le lemme suivant~: \begin{lem} \label{lemme_section_chow} Si $D_{1},\dots,D_{q}$ $(q \leqslant d)$ sont des diviseurs $T$-invariants \'el\'ementaires tels que $D_{1}\dotsm D_{q} = 0$ dans $CH^{\ast}(\M{P}(\Delta))$, alors~: \[ \hat{c}_{1}(\ov{\C{O}(D_{1})}_{\infty}) \dotsm \hat{c}_{1}(\ov{\C{O}(D_{q})}_{\infty}) = 0 \qquad \text{dans} \quad \widehat{CH}_{\op{int}}^{q}(\M{P}(\Delta)). \] \end{lem} \noindent {\bf D\'emonstration.}\ On a~: \[ \zeta(\hat{c}_{1}(\ov{\C{O}(D_{1})}_{\infty}) \dotsm \hat{c}_{1}(\ov{\C{O}(D_{q})}_{\infty})) = c_{1}(\C{O}(D_{1})) \dots c_{1}(\C{O}(D_{q})) = 0. \] De plus, \[ \omega(\hat{c}_{1}(\ov{\C{O}(D_{1})}_{\infty}) \dotsm \hat{c}_{1}(\ov{\C{O}(D_{q})}_{\infty})) = c_{1}(\ov{\C{O}(D_{1})}_{\infty}) \dots c_{1}(\ov{\C{O}(D_{q})}_{\infty}) = 0 \] d'apr\`es le corollaire (\ref{trivialite2}). Il existe donc $\alpha \in D^{q-1,q-1}(\M{P}(\Delta)_{\M{R}})$ tel que~: \begin{equation} \label{equa1_sect6} \hat{c}_{1}(\ov{\C{O}(D_{1})}_{\infty}) \dotsm \hat{c}_{1}(\ov{\C{O}(D_{q})}_{\infty}) = [(0,\alpha)] \qquad \text{dans} \quad \widehat{CH}_{\op{int}}(\M{P}(\Delta)), \end{equation} et comme $dd^{c} \alpha = 0$, on peut choisir $\alpha$ dans $Z^{q-1,q-1}(\M{P}(\Delta)_{\M{R}})$. Soient maintenant $\ov{L}_{q+1,\infty},\dots, \ov{L}_{d+1,\infty}$ des fibr\'es en droites sur $\M{P}(\Delta)$ munis de leur m\'etrique canonique. On d\'eduit de (\ref{equa1_sect6}) la relation~: \begin{multline*} \widehat{\op{deg}}(\hat{c}_{1}(\ov{\C{O}(D_{1})}_{\infty}) \dotsm \hat{c}_{1}(\ov{\C{O}(D_{q})}_{\infty}) \cdot \hat{c}_{1}(\ov{L}_{q+1,\infty})\dots \hat{c}_{1}(\ov{L}_{d+1,\infty})) \\ = \frac{1}{2} \int_{\M{P}(\Delta)(\M{C})}\alpha\, c_{1}(\ov{L}_{q+1,\infty}) \dotsm c_{1}(\ov{L}_{d+1,\infty}), \end{multline*} ce qui entra\^\i ne, d'apr\`es la proposition (\ref{annulation_hauteur}) que~: \begin{equation} \label{equa2_sect6} \int_{\M{P}(\Delta)(\M{C})}\alpha\, c_{1}(\ov{L}_{q+1,\infty}) \dotsm c_{1}(\ov{L}_{d+1,\infty}) = 0, \end{equation} quels que soient les fibr\'es en droites $L_{q+1},\dots,L_{d+1}$ choisis. Comme l'anneau de cohomologie de De Rham de $\M{P}(\Delta)$ est engendr\'e par les classes des diviseurs dans $\M{P}(\Delta)$ (cf. th\'eor\`eme \ref{anneau_chow}), on d\'eduit de (\ref{equa2_sect6}) et de (\ref{coho_courant}) que~: \[ \int_{\M{P}(\Delta)(\M{C})}\alpha \wedge \beta = 0, \] quel que soit $\beta \in H^{d-q,d-q}(\M{P}(\Delta)(\M{C})_{\M{R}})$ et ceci implique par dualit\'e de Poincar\'e que $\alpha = 0$ dans $\widetilde{D}^{q-1,q-1}(\M{P}(\Delta)(\M{C})_{\M{R}})$. On passe maintenant \`a la d\'emonstration du th\'eor\`eme (\ref{section_chow}). En reprenant les notations du th\'eor\`eme (\ref{anneau_chow}), soit $\hat{\varsigma}: \C{S} \rightarrow \widehat{CH}_{\op{int}}^{\ast}(\M{P}(\Delta))$ le morphisme d'anneaux d\'efini par $\hat{\varsigma}(t_{\tau}) = \hat{c}_{1}(\ov{\C{O}(V(\tau))}_{\infty})$ pour tout $\tau \in \Delta(1)$. On d\'eduit de la proposition (\ref{metrique_ind}) et du lemme (\ref{lemme_section_chow}) respectivement les inclusions $\C{J} \subset \op{Ker}\hat{\varsigma}$ et $\C{I} \subset \op{Ker}\hat{\varsigma}$. On conclut la d\'emonstration en utilisant le th\'eor\`eme (\ref{anneau_chow}). \bigskip \section{Groupes de Chow arithm\'etiques g\'en\'eralis\'es}~ \subsection{Th\'eorie classique de Gillet-Soul\'e}~ \label{theorie_classique} On suit ici (\cite{13} et \cite{3}, \S 2.1). Soit $K$ un corps de nombre, $\C{O}_{K}$ son anneau d'entier et $S = \op{Spec}(\C{O}_{K})$ le sch\'ema associ\'e. Pour tout plongement $\sigma: K \rightarrow \M{C}$ et pour tout sch\'ema $X$ sur $\op{Spec}K$ ou sur $S$, on note $X_{\sigma}$ le sch\'ema sur $\M{C}$ d\'eduit de $X$ par le changements de base $\op{Spec}\M{C} \rightarrow \op{Spec}K$. De m\^eme, si $f: X \rightarrow Y$ est un morphisme de sch\'emas sur $K$, on note $f_{\sigma}: X_{\sigma} \rightarrow Y_{\sigma}$ le morphisme de sch\'emas sur $\M{C}$ induit par changement de base. Une {\it vari\'et\'e arithm\'etique\/} est par d\'efinition un sch\'ema $\pi: X \rightarrow S$, plat, projectif, int\`egre et {\em r\'egulier} sur $S$. En particulier la fibre g\'en\'erique $X_{K} = X \times_{S}\; \op{Spec} K$ est lisse. On note $d = \dim_{K}X_{K}$. Pour toute vari\'et\'e arithm\'etique $X$ et tout entier $p$ positif, on note $Z_{p}(X)$ (resp. $Z^{p}(X)$) le groupe des cycles sur $X$ de dimension $p$ (resp. codimension $p$). Pour un tel cycle $Z$, on note $|Z| \subset X$ le support de $Z$. On note $X(\M{C})$ les points complexes du sch\'ema $X$; c'est la r\'eunion disjointe $\coprod_{\sigma: K \hookrightarrow \M{C}}X_{\sigma}(\M{C})$. Soit $F_{\infty}: X(\M{C}) \rightarrow X(\M{C})$ l'involution antiholomorphe provenant de l'action de la conjugaison complexe sur les coordonn\'ees des points complexes de $X$. On note $A^{p,p}(X_{\M{R}})$ (resp. $D^{p,p}(X_{\M{R}})$) l'ensemble des formes r\'eelles $\alpha \in A^{p,p}(X(\M{C}))$ (resp. des courants r\'eels $\alpha \in D^{p,p}(X(\M{C}))$) tels que $F_{\infty}^{\ast}(\alpha) = (-1)^{p}\alpha$. On note $\widetilde{A}^{p,p}(X_{\M{R}})$ (resp. $\widetilde{D}^{p,p}(X_{\M{R}})$) le quotient $\left. A^{p,p}(X_{\M{R}})\right/ (\op{Im}\partial + \op{Im}\ov{\partial})$ (resp. le quotient $\left. D^{p,p}(X_{\M{R}}) \right/ (\op{Im}\partial + \op{Im}\ov{\partial})$). On note \'egalement $Z^{p,p}(X_{\M{R}}) = \op{Ker}\{d: A^{p,p}(X_{\M{R}}) \rightarrow A^{2p+1}(X(\M{C}))\} \subset A^{p,p}(X_{\M{R}})$. Tout cycle irr\'eductible $Z \in Z^{p}(X)$ d\'efinit un courant $\delta_{Z} \in D^{p,p}(X_{\M{R}})$ par int\'egration sur l'ensemble de ses points complexes. Si $Z = \sum_{i \in I}n_{i}Z_{i}$ o\`u $Z_{i}$ est irr\'eductible, on pose~: \[ \delta_{Z} = \sum_{i \in I}n_{i}\delta_{Z_{i}(\M{C})}. \] Un {\it courant de Green\/} pour $Z$ est un courant $g \in D^{p-1,p-1}(X_{\M{R}})$ tel que $dd^{c} g + \delta_{Z}$ est une forme $C^{\infty}$. Soit $X$ une vari\'et\'e arithm\'etique. On note $\widehat{Z}^{p}(X)$ le groupe form\'e des couples de la forme $(Z,g)$, o\`u $Z \in Z^{p}(X)$ et $g$ est un courant de Green pour $Z$, muni de la loi d'addition composante par composante. Soit $\widehat{R}^{p}(X) \subset \widehat{Z}^{p}(X)$ le sous-groupe engendr\'e par les paires de la forme $(0,\partial u + \ov{\partial} v)$ et $\left( \op{div} (f), - \log |f|^{2}\right)$, o\`u $f \in k(Y)^{\ast}$ est une fonction rationnelle non identiquement nulle sur un sous sch\'ema int\`egre $Y \subset X$ de codimension $p-1$, et o\`u pour simplifier les notations on a \'ecrit $- \log |f|^{2}$ pour d\'esigner $[-\log |f|^{2}]_{Y(\M{C})}$, le courant sur $X(\M{C})$ dont l'action sur les formes diff\'erentielles est obtenue par restriction \`a la partie lisse de $Y(\M{C})$ puis l'int\'egration contre la fonction $- \log |f|^{2}$. Le {\it groupe de Chow arithm\'etique de codimension $p$ de $X$\/} est d\'efini par~: \[ \widehat{CH}^{p}(X) = \left. \widehat{Z}^{p}(X) \right/ \widehat{R}^{p}(X). \] On note $R_{\op{fin}}^{p}(X)_{\M{Q}}$ l'espace des $\M{Q}$-cycles de la forme $\sum_{i}q_{i}\op{div}(f_{i})$, o\`u $q_{i} \in \M{Q}$ et $f_{i} \in k(Y_{i})^{\ast}$ est une fonction rationnelle non identiquement nulle sur $Y_{i}$ un sous-sch\'ema int\`egre de codimension $p-1$ contenu dans une fibre ferm\'ee du morphisme $\pi:X \rightarrow S$. On remarque que pour tout $R \in R_{\op{fin}}^{p}(X)_{\M{Q}}$, la classe de $(R,0)$ dans $\widehat{CH}^{p}(X)_{\M{Q}}$ est nulle. Une {\it $K_{1}$-chaine\/} de codimension $p$ dans $X$ est un \'el\'ement du groupe \\ $\bigoplus_{x \in X^{(p-1)}}k(x)^{\ast}$, o\`u $X^{(p-1)}$ d\'esigne l'ensemble des sous-sch\'emas ferm\'es int\`egres de codimension $p-1$ dans $X$. Si $f = \sum_{i \in I}[f_{W_{i}}]$, o\`u $W_{i} \in X^{(p-1)}$ et $f_{W_{i}} \in k(W_{i})^{\ast}$, est une $K_{1}$-chaine de codimension $p$, on pose~: \begin{align*} \op{div}f &= \sum_{i \in I}\op{div}(f_{W_{i}}) \in Z^{p}(X), \\ - \log |f|^{2} &= \sum_{i \in I}[-\log|f_{W_{i}}|^{2}]_{W_{i}(\M{C})} \in D^{p-1,p-1}(X_{\M{R}}), \\ \qquad \qquad \qquad \qquad \qquad \qquad \op{\widehat{\op{div}}}f &= (\op{div}f, - \log |f|^{2}) \in \widehat{Z}^{p}(X);\qquad \qquad \qquad \qquad \end{align*} et on appelle {\it support\/} de $f$ la fermeture de la r\'eunion des $W_{i}$. Si $\E{Z} = \{Z_{1},\dots,Z_{n}\}$ est une famille de sous-sch\'emas ferm\'es int\`egres de $X$, on dit que la $K_{1}$-chaine $f = \sum_{i \in I}[f_{W_{i}}]$ et la famille $\E{Z}$ s'intersectent {\it presque proprement\/} si pour tout $W_{i}$ et tout $Z_{j} \in \E{Z}$, les cycles $\op{div}(W_{i})$ et $Z_{j}$ s'intersectent proprement. \medskip On dispose de trois morphismes de groupes d\'efinis comme suit~: \begin{alignat*}{3} \zeta :\; &\widehat{CH}^{p}(X) & &\longrightarrow & &CH^{p}(X) \\ &[(Z,g)] & &\longmapsto & &[Z], \\ & & & & & \\ \omega :\; &\widehat{CH}^{p}(X) & &\longrightarrow & &A^{p,p}(X_{\M{R}})\\ &[(Z,g)] & & \longmapsto & &dd^{c} g + \delta_{Z}, \end{alignat*} et \begin{alignat*}{3} a :\; &\widetilde{A}^{p-1,p-1}(X_{\M{R}}) & &\longrightarrow & &\widehat{CH}^{p}(X) \\ &\eta & & \longmapsto & &[(0,\eta)]. \end{alignat*} Ces morphismes donnent lieu \`a deux suites exactes~: \begin{align} \label{suite_exacte1} &CH^{p,p-1}(X) \stackrel{\rho}{\longrightarrow} \widetilde{A}^{p-1,p-1}(X_{\M{R}}) \stackrel{a}{\longrightarrow} \widehat{CH}^{p}(X) \stackrel{\zeta}{\longrightarrow} CH^{p}(X) \longrightarrow 0, \\ \label{suite_exacte2} &CH^{p,p-1}(X) \stackrel{\rho}{\longrightarrow} H^{p-1,p-1}(X_{\M{R}}) \stackrel{a}{\longrightarrow} \widehat{CH}^{p}(X) \\ & \qquad \qquad \qquad \qquad \qquad \stackrel{(\zeta,-\omega)}{\longrightarrow} CH^{p}(X)\oplus Z^{p,p}(X_{\M{R}}) \stackrel{cl + [.]}{\longrightarrow} H^{p,p}(X_{\M{R}}) \longrightarrow 0, \notag \end{align} o\`u $CH^{p,p-1}(X)$ est un groupe d\'efini dans (\cite{12}, \S 8), o\`u $\rho$ est $-2$ fois le r\'egulateur de Beilinson (cf. \cite{13}, \S 3.3.5 et 3.5), o\`u $[.]$ est le morphisme ``classe en cohomologie de De Rham'' et o\`u $cl$ est l'application cycle. Tout morphisme de sch\'emas $f: X \rightarrow Y$ entre vari\'et\'es arithm\'etiques induit un morphisme de groupes~: \[ f^{\ast}: \widehat{CH}^{p}(Y) \longrightarrow \widehat{CH}^{p}(X). \] De plus, on dispose d'un produit~: \[ \widehat{CH}^{p}(X) \otimes \widehat{CH}^{q}(X) \longrightarrow \widehat{CH}^{p+q}(X)_{\M{Q}}, \] que l'on peut en fait d\'efinir \`a valeur dans $\widehat{CH}^{p+q}(X)$ quand $X$ est lisse sur $S$. Les morphismes $\zeta$ et $\omega$ d\'efinis ci-dessus sont des morphismes d'anneaux. On a \'egalement $f^{\ast}(x\cdot y) = f^{\ast}(x)\cdot f^{\ast}(y)$. Enfin la formule suivante est utile dans la pratique. Soient $\eta \in \widetilde{A}^{\ast}(X_{\M{R}})$ et $x \in \widehat{CH}^{\ast}(X)$, on a~: \[ a(\eta)\cdot x = a (\eta \, \omega(x)). \] Pour plus de d\'etails sur ces d\'efinitions et cette th\'eorie, voir \cite{13}. \medskip \subsection{Fibr\'es en droites int\'egrables sur une vari\'et\'e arithm\'etique}~ Soit $\pi: X \rightarrow \op{Spec}\C{O}_{K}$ une vari\'et\'e arithm\'etique de dimension relative $d$ et $p$ un entier positif. On note $\ov{A}^{p,p}(X_{\R})$ (resp. $\ov{\ov{A}}^{p,p}(X_{\R})$, resp. $\ov{\ov{A}}^{p,p}_{\log}(X_{\R})$) l'intersection $\ov{A}^{p,p}(X(\M{C})) \cap D^{p,p}(X_{\R})$ (resp. l'intersection $\ov{\ov{A}}^{p,p}(X(\M{C})) \cap D^{p,p}(X_{\R})$, resp. l'intersection $\ov{\ov{A}}^{p,p}_{\log} (X(\M{C})) \cap D^{p,p}(X_{\R})$). Un {\it fibr\'e en droites hermitiens sur $X$\/} est un couple $\ov{E} = (E,\|.\|)$ form\'e d'un fibr\'e en droites $E$ sur $X$ et d'une m\'etrique continue $\|.\|$ sur $E$ invariante sous l'action de $F_{\infty}$. Un tel fibr\'e est dit {\it admissible\/} lorsque le fibr\'e holomorphe hermitien $(E_{\M{C}},\|.\|)$ est admissible au sens de (\ref{def_admissibilite1}). Un fibr\'e en droites hermitien $\ov{L}$ est dit {\it int\'egrable\/} si et seulement si\ il existe $\ov{E}_{1}$ et $\ov{E}_{2}$ deux fibr\'es en droites hermitiens admissibles sur $X$ tels que~: \[ \ov{L} = \ov{E}_{1} \otimes \left(\ov{E}_{2}\right)^{-1}. \] \begin{rem} Si $\ov{L} = (L,\|.\|)$ est un fibr\'e en droites hermitiens int\'egrable sur $X$, alors le fibr\'e holomorphe hermitien $(L(\M{C}),\|.\|)$ est int\'egrable sur $X(\M{C})$ au sens de (\ref{decomposable}). \end{rem} \begin{expl}~ \begin{enumerate} \item{Soit $\ov{L} = (L,\|.\|)$ un fibr\'e en droites hermitien sur $X$; si $\|.\|$ est $C^{\infty}$ sur $X(\M{C})$, alors $\ov{L}$ est int\'egrable sur $X$. (\'Ecrire $\ov{L} \simeq (\ov{L}\otimes \ov{H}^{n})\otimes (\ov{H}^{n})^{\ast}$, avec $\ov{H}$ ample muni d'une m\'etrique $C^{\infty}$ strictement positive sur $X(\M{C})$ et $n \gg 0$).} \item{Soit $L$ un fibr\'e en droites ample muni d'une m\'etrique $\|.\|$ continue et positive; le fibr\'e hermitien $(L,\|.\|)$ est admissible sur $X$ d'apr\`es (\ref{admi_positivite}).} \item{ Soient $\M{P}(\Delta)$ une vari\'et\'e torique projective lisse et $\ov{L}_{\infty}$ un fibr\'e en droites sur $\M{P}(\Delta)$ muni de sa m\'etrique canonique; $\ov{L}_{\infty}$ est int\'egrable sur $\M{P}(\Delta)$ d'apr\`es (\ref{exemple_decomp1}).} \end{enumerate} \end{expl} On appelle {\it forme diff\'erentielle g\'en\'eralis\'ee globale de type $(p,p)$\/} tout \'el\'ement $\alpha \in \ov{A}^{p,p}(X_{\R})$ pouvant s'\'ecrire sur $X(\M{C})$ sous la forme~: \[ \alpha = \sum_{i = 1}^{n}\omega_{i}c_{1}(\ov{L}_{i,1})\dots c_{1}(\ov{L}_{i,q_{i}}) \] o\`u pour tout $1 \leqslant i \leqslant n$, $\omega_{i} \in A^{\ast}(X_{\M{R}})$ et $\ov{L}_{i,1}, \dots, \ov{L}_{i,q_{i}}$ sont des fibr\'es en droites int\'egrables sur $X$. On note $\ov{A}_{\op{g}}^{p,p}(X_{\M{R}}) \subset \ov{A}^{p,p}(X_{\M{R}})$ l'ensemble des formes diff\'erentielles g\'en\'eralis\'ees globales de type $(p,p)$ sur $X$. C'est une $\M{R}$-alg\`ebre. On note $\widetilde{\ov{A}}_{\op{g}}^{p,p}(X_{\M{R}})$ l'image de $\ov{A}^{p,p}_{\op{g}}(X_{\M{R}})$ dans $\widetilde{D}^{p,p}(X_{\M{R}})$. On note \'egalement $\ov{A}_{\op{g}}^{\ast}(X_{\M{R}}) = \oplus_{p \geqslant 0}\ov{A}_{\op{g}}^{p,p}(X_{\M{R}})$ et $\widetilde{\ov{A}}_{\op{g}}^{\ast}(X_{\M{R}}) = \oplus_{p \geqslant 0}\widetilde{\ov{A}}_{\op{g}}^{p,p}(X_{\M{R}})$. \begin{rem} L'espace $\ov{A}_{\op{g}}^{\ast}(X_{\M{R}})$ est stable pour le produit d\'efini au (\ref{produit_formes}). La structure d'alg\`ebre sur $\ov{A}^{\ast}(X(\M{C}))$ induit donc une structure d'alg\`ebre associative commutative unif\`ere et gradu\'ee sur $\ov{A}^{\ast}_{\op{g}}(X_{\M{R}})$. \end{rem} \begin{prop} Si $\ov{E}$ est un fibr\'e en droites int\'egrable, alors $(\ov{E})^{-1}$ est int\'egrable. \end{prop} \noindent {\bf D\'emonstration.}\ C'est une cons\'equence imm\'ediate des d\'efinitions. \medskip \begin{prop} Soient $\ov{E}$ et $\ov{F}$ deux fibr\'es en droites admissibles (resp. int\'egrables); leur produit tensoriel $\ov{E}\otimes\ov{F}$ est admissible (resp. int\'egrable). \end{prop} \noindent {\bf D\'emonstration.}\ On suppose tout d'abord que $\ov{E} = (E,\|.\|_{E})$ et $\ov{F} = (F,\|.\|_{F})$ sont admissibles. Le fibr\'e holomorphe hermitien $((E\otimes F)({\M{C}}), \|.\|_{E}\otimes\|.\|_{F})$ est admissible sur $X(\M{C})$ d'apr\`es (\ref{produit_decomposables}), et donc $\ov{E}\otimes\ov{F}$ est admissible. On suppose maintenant que $\ov{E}$ et $\ov{F}$ sont int\'egrables. On peut donc trouver $\ov{E}_{1}$, $\ov{E}_{2}$, $\ov{F}_{1}$ et $\ov{F}_{2}$ des fibr\'es admissibles sur $X$ tels que $\ov{E} \simeq \ov{E}_{1}\otimes(\ov{E}_{2})^{-1}$ et $\ov{F} \simeq \ov{F}_{1}\otimes (\ov{F}_{2})^{-1}$; et donc $\ov{E}\otimes\ov{F} \simeq (\ov{E}_{1}\otimes\ov{F}_{1})\otimes(\ov{E}_{2}\otimes\ov{F}_{2})^{-1}$, et comme $\ov{E}_{1}\otimes\ov{F}_{1}$ et $\ov{E}_{2}\otimes\ov{F}_{2}$ sont admissibles d'apr\`es ce qui pr\'ec\`ede, on en d\'eduit que $\ov{E}\otimes\ov{F}$ est int\'egrable. \medskip Les deux propositions pr\'ec\'edentes nous autorisent \`a poser la d\'efinition suivante~: \begin{defn} Soit $X$ une vari\'et\'e arithm\'etique. On note $\widehat{\op{Pic}}_{\,\op{int}}(X)$ le groupe form\'e des classes d'isomorphie isom\'etrique des fibr\'es hermitiens int\'egrables sur $X$, et dont la structure de groupe est donn\'ee par le produit tensoriel. \end{defn} \begin{prop} Soient $X$ et $Y$ deux vari\'et\'es arithm\'etiques et $f: Y \rightarrow X$ un morphisme. Si $\ov{L}$ est un fibr\'e en droites admissible (resp. int\'egrable) sur $X$, alors $f^{\ast}(\ov{L})$ est admissible (resp. int\'egrable) sur $Y$. L'application $f^{\ast}: \ov{L} \mapsto f^{\ast}(\ov{L})$ d\'efinit un morphisme de groupes~: \[ f^{\ast}: \widehat{\op{Pic}}_{\,\op{int}}(X) \longrightarrow \widehat{\op{Pic}}_{\,\op{int}}(Y). \] \end{prop} \noindent {\bf D\'emonstration.}\ C'est une cons\'equence directe de (\ref{image_reciproque}) et des d\'efinitions. \bigskip \subsection{Groupes de Chow arithm\'etiques g\'en\'eralis\'es}~ \label{sous_section_def} Pour tout entier $p \geqslant 0$, on pose $\widehat{Z}_{\op{gen}}^{p}(X) = Z^{p}(X) \oplus D^{p-1,p-1}(X_{\M{R}})$; du fait de l'inclusion $\widehat{R}^{p}(X) \subset \widehat{Z}_{\op{gen}}^{p}(X)$, on peut d\'efinir le groupe quotient~: \[ \widetilde{CH}^{p}(X) = \left. \widehat{Z}_{\op{gen}}^{p}(X)\right/ \widehat{R}^{p}(X). \] On a imm\'ediatement l'inclusion $\widehat{CH}^{p}(X) \subset \widetilde{CH}^{p}(X)$. On dispose des morphismes suivants~: \begin{alignat*}{3} \zeta :\; &\widetilde{CH}^{p}(X) & &\longrightarrow & &CH^{p}(X) \\ &[(Z,g)] & &\longmapsto & &[Z], \\ & & & & & \\ \omega :\; &\widetilde{CH}^{p}(X) & &\longrightarrow & &D^{p,p}(X_{\M{R}})\\ &[(Z,g)] & & \longmapsto & &dd^{c} g + \delta_{Z}, \end{alignat*} et \begin{alignat*}{3} a :\; &\widetilde{\ov{A}}^{p-1,p-1}(X_{\M{R}}) & &\longrightarrow & &\widetilde{CH}^{p}(X) \\ &\eta & & \longmapsto & &[(0,\eta)], \end{alignat*} dont les restrictions \`a $\widehat{CH}^{p}(X)$ et $\widetilde{A}^{p-1,p-1}(X_{\M{R}})$ respectivement co\"\i ncident avec ceux d\'efinis au (\ref{theorie_classique}). \begin{defn} \label{classe_de_chern} Soit $\ov{L} = (L,\|.\|)$ un fibr\'e en droites hermitien int\'egrable sur $X$. On appelle {\it premi\`ere classe de Chern arithm\'etique de $\ov{L}$\/} et on note $\hat{c}_{1}(\ov{L})$ la classe dans $\widetilde{CH}^{1}(X)$ de~: \[ \op{\widehat{div}}s := (\op{div} s, - \log \|s\|^{2}), \] o\`u $s$ est une section rationnelle non nulle de $L$ au-dessus de $X$. \end{defn} \begin{rem} La classe $\hat{c}_{1}(\ov{L})$ est bien d\'efinie et ne d\'epend pas du choix de $s$. En effet, si $s'$ est une section rationnelle de $L$ au-dessus de $X$, il existe $f$ une fonction rationnelle sur $X$ telle que $s' = f\cdot s$, et l'on a~: \[ \op{\widehat{div}}s' = (f,-\log|f|^{2}) + \op{\widehat{div}}s. \] \end{rem} \begin{prop} Soit $\ov{L} = (L,\|.\|)$ un fibr\'e en droites int\'egrable sur $X$, on a~: \begin{alignat*}{3} &\omega(\hat{c}_{1}(\ov{L}))& &= c_{1}(\ov{L})& &\in \ov{A}^{1,1}_{\op{g}}(X_{\M{R}}) \cap C_{0}^{1,1}(X_{\M{R}}) \\ \text{et} \quad &\zeta(\hat{c}_{1}(\ov{L}))& &= c_{1}(L) & &\in CH^{1}(X). \end{alignat*} \end{prop} \noindent {\bf D\'emonstration.}\ La premi\`ere \'egalit\'e est une cons\'equence directe de la formule de Poincar\'e-Lelong g\'en\'eralis\'ee (\ref{PL_generalisee}); la seconde \'egalit\'e r\'esulte imm\'ediatement des d\'efinitions. \medskip Soient $p$ et $q$ deux entiers positifs (\'eventuellement nuls), et soient $Z \in Z^{p}(X)$ un cycle de codimension $p$, $g \in D^{p-1,p-1}(X_{\M{R}})$ un courant de Green pour $Z$ et $\ov{L}_{1} = (L_{1},\|.\|_{1}), \dots, \ov{L}_{q} = (L_{q}, \|.\|_{q})$ des fibr\'es en droites admissibles sur $X$. On choisit $s_{1}, \dots, s_{q}$ des sections rationnelles non identiquement nulles sur $X$ de $L_{1}, \dots, L_{q}$ respectivement, telles que les cycles $Y_{0} = Z$, $Y_{1}=\op{div}s_{1}, \dots, Y_{q} = \op{div}s_{q}$ soient d'intersection propre (i.e. tels que $\op{codim} (Y_{j_{1}} \cap \dots \cap Y_{j_{m}}) \geqslant \sum_{i = 1}^{m}\op{codim}Y_{j_{i}}$ pour tout choix d'indices $j_{1} < \dots < j_{m}$ dans $\{0,\dots,q\}$). On pose $\tilde{\omega} = dd^{c} g + \delta_{Z}$. Pour tout $1 \leqslant i \leqslant q$, on note \'egalement $\delta_{i} = \delta_{\op{div}s_{i}} \in \ov{\ov{A}}^{1,1}(X_{\M{R}})$, $g_{i} = - \log \|s_{i}\|_{i}^{2}$ et $\omega_{i} = c_{1}(\ov{L}_{i}) \in \ov{A}^{1,1}(X_{\M{R}})$. On d\'efinit \`a partir de ces donn\'ees un \'el\'ement de $D^{p+q-1,p+q-1}(X_{\M{R}})$ que l'on notera $\{g\ast (g_{1},s_{1})\ast \dots \ast (g_{q},s_{q})\}$, par l'\'egalit\'e~: \begin{multline*} \{g\ast (g_{1},s_{1})\ast \dots \ast (g_{q},s_{q})\} = g\cdot\delta_{\op{div}s_{1}\cap\dots\cap\op{div}s_{q}} + g_{1}\cdot\tilde{\omega}\delta_{2}\dots\delta_{q} + \dots \\ + g_{i}\cdot\tilde{\omega}\omega_{1}\dots \omega_{i-1}\delta_{i+1}\dots\delta_{q} + \dots + g_{q-1}\cdot\tilde{\omega}\omega_{1}\dots\omega_{q-2}\delta_{q} + g_{q}\cdot\tilde{\omega}\omega_{1}\dots\omega_{q-1}. \end{multline*} On remarque que cette expression a bien un sens; en effet le premier terme $g \cdot\delta_{\op{div}s_{1}\cap\dots\cap\op{div}s_{q}}$ est bien d\'efini (cf. \cite{13}, 2.1.3.2), et les autres le sont gr\^ace \`a la proposition (\ref{exemple_produit_formes}). \begin{prop} \label{approximation2} Soient $\left(\|.\|_{1}^{(n)}\right)_{n \in \M{N}}, \dots , \left(\|.\|_{q}^{(n)}\right)_{n \in \M{N}}$ des suites croissantes de m\'etriques positives $C^{\infty}$ sur $L_{1}, \dots, L_{q}$ convergeant vers $\|.\|_{1}, \dots, \|.\|_{q}$ respectivement. Si l'on note $g_{1}^{(n)} = - \log \left(\|s_{1}\|_{1}^{(n)}\right)^{2}, \dots, g_{q}^{(n)} = - \log \left(\|s_{q}\|_{q}^{(n)}\right)^{2}$, alors $\{g \ast (g_{1}^{(n)},s_{1})\ast \dots \ast (g_{q}^{(n)},s_{q})\}$ tend vers $\{g\ast (g_{1},s_{1}) \ast \dots \ast (g_{q},s_{q})\}$ dans $D^{p+q-1,p+q-1}(X_{\M{R}})$, au sens de la topologie faible, quand $n$ tend vers $+\infty$. \end{prop} \noindent {\bf D\'emonstration.}\ C'est une cons\'equence directe de (\ref{demailly}). \medskip \begin{prop} \label{naturalite2} Supposons que, pour un certain $k \in \{1,\dots,q\}$, les m\'etriques $\|.\|_{1},\dots,\|.\|_{k}$ soient $C^{\infty}$ sur $X(\M{C})$. Les courants $g_{1},\dots,g_{k}$ sont alors des courants de Green pour les cycles $\op{div}s_{1}, \dots, \op{div}s_{k}$ et on peut former le produit ~: \[ (g\ast g_{1}\ast \dots \ast g_{k}) := g \ast (g_{1} \ast ( g_{2} \ast ( \dotsm \ast (g_{k}) \dotsm))), \] pris au sens de Gillet-Soul\'e (cf. \cite{13}, \S 2.1), qui est un courant de Green pour le cycle intersection $Y_{0}\cap Y_{1} \cap \dots \cap Y_{q}$. On a alors l'\'egalit\'e dans $\widetilde{D}^{p+q-1,p+q-1}(X_{\M{R}})$ \[ \{(g\ast g_{1} \ast \dots \ast g_{k})\ast (g_{k+1},s_{k+1}) \ast \dots \ast (g_{q},s_{q})\} = \{g\ast (g_{1},s_{1})\ast \dots \ast (g_{q},s_{q})\}. \] \end{prop} \noindent {\bf D\'emonstration.}\ On montre tout d'abord le r\'esultat quand $k=q$. Pour tout $1 \leqslant i \leqslant q$, le produit $g_{i}\ast\dots\ast g_{q}$ est un courant de Green pour $\op{div}s_{i} \cap \dots \cap \op{div}s_{q}$ et on a $\omega( \op{div}s_{i} \cap \dots \cap \op{div}s_{q}, g_{i}\ast\dots\ast g_{q}) = \omega_{i}\dots\omega_{q}$. On a dans $\widetilde{D}^{p+q-1,p+q-1}(X_{\M{R}})$~: \[ \begin{split} g\ast g_{1}\ast\dots\ast g_{q} & = g\ast(g_{1}\ast(\dots \ast(g_{q-1}\ast g_{q})\dots )) \\ &= g\cdot \delta_{\op{div}s_{1} \cap \dots \cap \op{div}s_{q}} + \tilde{\omega}(g_{1}\ast(\dots \ast(g_{q-1}\ast g_{q})\dots ))\\ = g\cdot \delta_{\op{div}s_{1} \cap \dots \cap \op{div}s_{q}} & \\ + \tilde{\omega}(g_{1}\cdot &\delta_{\op{div}s_{2}\cap \dots \cap \op{div}s_{q}} + \omega_{1}g_{2}\cdot \delta_{\op{div}s_{3}\cap \dots\cap \op{div}s_{q}} + \dots + \omega_{1}\dots\omega_{q-1}g_{q}). \end{split} \] Il suffit alors de prouver que pour tout $1 \leqslant i \leqslant (q-1)$, on a dans $D^{q-i,q-i}(X_{\M{R}})$ l'\'egalit\'e~: \begin{equation} \label{eq_inter_1} g_{i}\cdot \delta_{i+1}\dots \delta_{q} = g_{i}\cdot \delta_{\op{div}s_{i+1}\cap \dots \cap \op{div}s_{q}}, \end{equation} o\`u le premier produit est pris au sens de (\ref{produit_generalise}) et le second au sens de Gillet-Soul\'e (cf. \cite{13}, 2.1.3.2). Le probl\`eme \'etant local, on peut supposer que le diviseur $\op{div}s_{i}$ est effectif. D'apr\`es (\cite{7}, prop. 3.4.12), on a~: \[ \delta_{i+1}\dots \delta_{q} = \delta_{\op{div}s_{i+1}\cap \dots \cap \op{div}s_{q}}. \] Pour tout $t \in \M{R}$, on pose $g_{i}^{(t)} = \max (g_{i}, t)$. Pour tout $t \in \M{R}$ fix\'e, la fonction $x \mapsto g_{i}^{(t)}(x)$ est plu\-ri\-sous\-har\-mo\-ni\-que\ et born\'ee sur $X(\M{C})$; on a donc~: \[ g_{i}^{(t)}\cdot \delta_{i+1}\dots \delta_{q} = g_{i}^{(t)}\cdot \delta_{\op{div}s_{i+1} \cap \dots \cap \op{div}s_{q}}. \] D'apr\`es (\ref{demailly}), le courant $g_{i}^{(t)}\cdot \delta_{i+1}\dots \delta_{q}$ tend vers $g_{i}\cdot \delta_{i+1}\dots \delta_{q}$ au sens de la topologie faible quand $t$ tend vers $-\infty$. Pour finir de montrer (\ref{eq_inter_1}), il suffit de prouver que $g_{i}^{(t)}\cdot\delta_{\op{div}s_{i+1} \cap \dots \cap \op{div}s_{q}}$ tend faiblement vers $g_{i} \cdot\delta_{\op{div}s_{i+1} \cap \dots \cap \op{div}s_{q}}$ quand $t$ tend vers $-\infty$. Pour cela \'ecrivons~: \[ (\op{div}s_{i+1})_{\M{C}} \cap \dots \cap (\op{div}s_{q})_{\M{C}} = \sum_{k}n_{k}\, C_{k}, \] o\`u les $C_{k}$ sont les composantes irr\'eductibles de $(\op{div}s_{i+1})_{\M{C}} \cap \dots \cap (\op{div}s_{q})_{\M{C}}$. Nous devons v\'erifier que pour chaque $k$, le courant $g_{i}^{(t)}\cdot \delta_{C_{k}}$ tend faiblement vers $g_{i}\cdot \delta_{C_{k}}$, o\`u le produit $g_{i}\cdot\delta_{C_{k}}$ est d\'efini comme dans Gillet-Soul\'e (\cite{13}, \S 2.1.3.2). Soit $\pi : \widetilde{C}_{k} \rightarrow C_{k}$ une r\'esolution des singularit\'es de $C_{k}$. Par d\'efinition, on a~: \[ \delta_{C_{k}} = \op{\pi_{\ast}} 1, \] et donc ~: \[ g_{i}^{(t)}\cdot \delta_{C_{k}} = \op{\pi_{\ast}}(\op{\pi^{\ast}}g_{i}^{(t)}), \] car $\delta_{C_{k}}$ est un courant d'ordre $0$ et $g_{i}^{(t)}$ est born\'ee. On a par ailleurs (cf. \cite{13}, \S 2.1.3.2)~: \[ g_{i}\cdot \delta_{C_{k}} = \op{\pi_{\ast}}(\op{\pi^{\ast}}g_{i}). \] Enfin $\op{\pi^{\ast}}g_{i}$ est plu\-ri\-sous\-har\-mo\-ni\-que\ et $\op{\pi^{\ast}}g_{i}^{(t)} = \op{max}(\op{\pi^{\ast}}g_{i}, t)$ converge faiblement vers $\op{\pi^{\ast}}g_{i}$ quand $t$ tend vers $-\infty$, d'o\`u le r\'esultat par continuit\'e faible de $\pi_{\ast}$. On s'int\'eresse maintenant au cas g\'en\'eral~: Soient $\left(\|.\|^{(n)}_{k+1}\right)_{n \in \M{N}}, \dots, \left(\|.\|^{(n)}_{q}\right)_{n\in \M{N}}$ des suites croissantes de m\'etriques positives $C^{\infty}$ convergent uniform\'ement sur $X(\M{C})$ vers $\|.\|_{k+1}, \dots, \|.\|_{q}$ respectivement. On note, pour tout $n \in \M{N}$, $g_{k+1}^{(n)}, \dots, g_{q}^{(n)}$ les courants $- \log \left(\|s_{k+1}\|^{(n)}_{k+1}\right)^{2}, \dots, - \log \left(\|s_{q}\|_{q}^{(n)}\right)^{2}$. D'apr\`es ce qui pr\'ec\`ede, on a dans $\widetilde{D}^{p+q-1,p+q-1}(X_{\M{R}})$ les \'egalit\'es~: \[ \begin{split} \{(g\ast g_{1}\ast \dots \ast g_{k})\ast (&g_{k+1}^{(n)},s_{k})\ast \dots \ast (g_{q}^{(n)},s_{q})\} \\ &= (g\ast g_{1}\ast \dots g_{k}) \ast g_{k+1}^{(n)} \ast \dots \ast g_{q}^{(n)} \\ &= g \ast g_{1} \ast \dots g_{k} \ast g_{k+1}^{(n)}\ast \dots g_{q}^{(n)} \\ &= \{g\ast (g_{1},s_{1})\ast \dots \ast (g_{k},s_{k})\ast (g_{k+1}^{(n)},s_{k+1})\ast \dots \ast (g_{q}^{(n)},s_{q})\}. \end{split} \] Comme $X(\M{C})$ est projective, les sous-espaces $\op{Im}\partial$ et $\op{Im}\ov{\partial}$ sont ferm\'es (puisque les morphismes $\partial$ et $\ov{\partial}$ sont continues et que leurs groupes de cohomologie sont de dimension finie). On en d\'eduit que $\widetilde{D}^{p+q-1,p+q-1}(X_{\M{R}})$, muni de la topologie quotient de la topologie faible sur ${D}^{p+q-1,p+q-1}(X_{\M{R}})$, est s\'epar\'e. Pour \'etablir l'\'egalit\'e recherch\'ee~: \[ \{(g\ast g_{1}\ast \dots \ast g_{k})\ast (g_{k+1},s_{k+1})\ast \dots \ast (g_{q},s_{q})] = [g \ast (g_{1},s_{1}) \ast \dots \ast (g_{q},s_{q})\}, \] dans $\widetilde{D}^{p+q-1,p+q-1}(X_{\M{R}})$, il suffit d'observer que d'apr\`es la proposition (\ref{approximation2}) on a, pour cette topologie, les limites~: \begin{multline*} \lim_{n \rightarrow +\infty}\{(g\ast g_{1}\ast \dots \ast g_{k})\ast (g_{k+1}^{(n)},s_{k+1})\ast \dots \ast (g_{q}^{(n)},s_{q})\} \\ = \{(g \ast g_{1} \ast \dots \ast g_{k})\ast (g_{k+1},s_{k+1})\ast \dots \ast (g_{q},s_{q})\} \end{multline*} et \begin{multline*} \lim_{n \rightarrow +\infty}\{g \ast (g_{1},s_{1})\ast \dots \ast (g_{k},s_{k})\ast (g_{k+1}^{(n)},s_{k+1})\ast \dots \ast (g_{q}^{(n)},s_{q})\}\\ = \{g \ast (g_{1},s_{1})\ast \dots \ast (g_{q},s_{q})\}. \end{multline*} \medskip On en d\'eduit aussit\^ot la proposition suivante~: \begin{prop} \label{commutativite2} La classe du courant $\{g\ast (g_{1},s_{1})\ast \dots \ast (g_{q},s_{q})\}$ dans\\ $\widetilde{D}^{p+q-1,p+q-1}(X_{\M{R}})$ est ind\'ependante de l'ordre des $\ov{L}_{1},\dots,\ov{L}_{q}$. \end{prop} \noindent {\bf D\'emonstration.}\ Soit $\sigma$ une permutation de $\{1,\dots,q\}$. Reprenant les notations de la proposition (\ref{approximation2}), on sait que~: \begin{multline*} \delta_{n}:= \{g\ast (g^{(n)}_{1},s_{1})\ast \dots \ast (g^{(n)}_{q},s_{q})\} \\ - \{g\ast (g^{(n)}_{\sigma(1)},s_{\sigma(1)})\ast \dots \ast (g^{(n)}_{\sigma(q)},s_{\sigma(q)})\} \in \left(\op{Im}\partial + \op{Im}\ov{\partial}\right). \end{multline*} De plus, $\delta_{n}$ tend vers $\{g\ast (g_{1},s_{1})\ast \dots \ast (g_{q},s_{q})\} - \{g\ast (g_{\sigma(1)},s_{\sigma(1)})\ast \dots \ast (g_{\sigma(q)},s_{\sigma(q)})\} $ au sens de la topologie faible quand $n$ tend vers $+\infty$ d'apr\`es (\ref{approximation2}). La proposition d\'ecoule alors imm\'ediatement du fait que $\op{Im}\partial + \op{Im}\ov{\partial}$ est ferm\'e dans $D^{p+q-1,p+q-1}(X_{\M{R}})$ puisque $X(\M{C})$ est projective. \medskip On d\'efinit un \'el\'ement de $\widetilde{CH}^{p+q}(X)$ que l'on note provisoirement\\ $[(Z,g)\cdot \hat{c}_{1}(\ov{L}_{1},s_{1})\dotsm \hat{c}_{1}(\ov{L}_{q},s_{q})]$ par l'\'egalit\'e~: \begin{multline} \label{definition_prod2} [(Z,g)\cdot \hat{c}_{1}(\ov{L}_{1},s_{1})\dotsm \hat{c}_{1}(\ov{L}_{q},s_{q})] \\ = [(Z \cdot \op{div}s_{1} \dotsm \op{div}s_{q}, \{g\ast(g_{1},s_{1})\ast \dots \ast(g_{q},s_{q})\})]. \end{multline} \begin{prop} \label{independance2} La classe de $[(Z,g)\cdot \hat{c}_{1}(\ov{L}_{1},s_{1})\dotsm \hat{c}_{1}(\ov{L}_{q},s_{q})]$ dans $\widetilde{CH}^{p+q}(X)$ ne d\'epend que de la classe de $(Z,g)$ dans $\widehat{CH}^{p}(X)$ et des classes d'isomorphie isom\'etrique des fibr\'es admissibles $\ov{L}_{1},\dots,\ov{L}_{q}$; elle ne d\'epend pas de l'ordre des fibr\'es $\ov{L}_{1},\dots,\ov{L}_{q}$. \end{prop} \noindent {\bf D\'emonstration.}\ Le fait que la classe de $[(Z,g)\cdot \hat{c}_{1}(\ov{L}_{1},s_{1})\dotsm \hat{c}_{1}(\ov{L}_{q},s_{q})]$ ne d\'epende pas de l'ordre des fibr\'es $ \ov{L}_{1},\dots,\ov{L}_{q}$ est une simple cons\'equence de (\ref{commutativite2}). On s'int\'eresse maintenant \`a la premi\`ere partie de l'\'enonc\'e. Il suffit, pour prouver le r\'esultat recherch\'e, de montrer que pour toute section $s_{1}'$ de $L_{1}$ au-dessus de $X$ et tout repr\'esentant $(Z',g')$ de $[(Z,g)]$ dans $\widehat{CH}^{p}(X)$ tels que les cycles $Z',\op{div}s_{1}, \dots, \op{div}s_{q}$ et les cycles $Z',\op{div}s_{1}',\op{div}s_{2}, \dots,\op{div}s_{q}$ soient d'intersection propre, on a l'\'egalit\'e~: \[ [(Z',g')\cdot \hat{c}_{1}(\ov{L}_{1},s_{1}')\cdot \hat{c}_{1}(\ov{L}_{2},s_{2})\dotsm \hat{c}_{1}(\ov{L}_{q},s_{q})] = [(Z,g)\cdot \hat{c}_{1}(\ov{L}_{1},s_{1})\dotsm \hat{c}_{1}(\ov{L}_{q},s_{q})]. \] On choisit $\left(\|.\|_{1}^{(n)}\right)_{n \in \M{N}}, \dots, \left(\|.\|_{q}^{(n)}\right)_{n \in \M{N}}$, $q$-suites croissantes de m\'etriques positives $C^{\infty}$ sur $L_{1},\dots,L_{q}$ convergeant uniform\'ement sur $X(\M{C})$ vers $\|.\|_{1},\dots,\|.\|_{q}$ respectivement. On note $f_{1}$ la fonction rationnelle sur $X$ telle que $s_{1}' = f_{1}\cdot s_{1}$. On note \'egalement $g_{1}' = -\log \|s_{1}'\|_{1}^{2}$ et pour tout $1 \leqslant i \leqslant q$ et tout $n \in \M{N}$, $g_{i}^{(n)} = - \log \left(\|s_{i}\|_{i}^{(n)}\right)^{2}$ et ${g_{1}'}^{(n)} = - \log \left(\|s_{1}'\|_{1}^{(n)}\right)^{2}$. On d\'eduit de l'\'egalit\'e $[Z'] = \zeta(Z',g') = \zeta(Z,g) = [Z]$ qu'il existe une $K_{1}$-chaine $f$ telle que~: \begin{equation} \label{eq_fin1} Z' = Z + \op{div}f. \end{equation} D'apr\`es le lemme de d\'eplacement pour les $K_{1}$-chaines, (cf. \cite{13}, \S 4.2.6), on peut de plus supposer que $f$ rencontre $\op{div}s_{1}, \dots, \op{div}s_{q}$ presque proprement sur $X_{K}$. Notons~: \[ Y = Z'\cdot \op{div}s_{1}'\cdot \op{div}s_{2}\dotsm \op{div}s_{q} - Z \cdot \op{div}s_{1}\dotsm \op{div}s_{q}. \] Il vient~: \begin{align*} Y &= Z'\cdot (\op{div}s_{1}' - \op{div}s_{1})\cdot \op{div}s_{2}\dotsm \op{div}s_{q} + (Z' - Z)\cdot \op{div}s_{1}\dotsm \op{div}s_{q}\\ &= \op{div}f_{1}\cdot Z'\cdot \op{div}s_{2} \dotsm \op{div}s_{q} + \op{div}f\cdot \op{div}s_{1}\dotsm \op{div}s_{q}, \end{align*} ce que, d'apr\`es (\cite{13}, \S 4.2.5) on peut r\'ecrire~: \begin{equation} \label{eq_fin3} Y = \op{div}(f_{1}\cdot (Z'\cdot \op{div}s_{2}\dotsm \op{div}s_{q})) + \op{div}(f\cdot (\op{div}s_{1}\dotsm \op{div}s_{q})) + R, \end{equation} o\`u $f_{1}\cdot (Z'\cdot \op{div}s_{2}\dotsm \op{div}s_{q})$ (resp. $f\cdot (\op{div}s_{1}\dotsm \op{div}s_{q})$) d\'esigne une $K_{1}$-chaine repr\'esentant l'intersection de la $K_{1}$-chaine $f_{1}$ et du cycle $Z'\cdot\op{div}s_{2}\dotsm \op{div}s_{q}$ (resp. de la $K_{1}$-chaine $f$ et du cycle $\op{div}s_{1}\dotsm\op{div}s_{q}$) et o\`u $R \in R_{\op{fin}}^{p+q}(X)$. Posons~: \[ \delta = \{g'\ast(g'_{1},s_{1}')\ast (g_{2},s_{2})\ast \dotsm \ast(g_{q},s_{q})\} - \{g\ast(g_{1},s_{1})\ast\dotsm \ast(g_{q},s_{q})\}, \] et pour tout $n \in \M{N}$, \[ \delta_{n} = \{g'\ast({g_{1}'}^{(n)},s_{1}')\ast(g_{2}^{(n)},s_{2})\ast \dotsm \ast(g_{q}^{(n)},s_{q})\} - \{g\ast (g_{1}^{(n)},s_{1})\ast \dotsm \ast (g_{q}^{(n)},s_{q})\}. \] D'apr\`es la proposition (\ref{naturalite2}), il vient~: \begin{align*} \delta_{n} &= g'\ast {g_{1}'}^{(n)}\ast g_{2}^{(n)}\ast \dotsm \ast g_{q}^{(n)} - g \ast g_{1}^{(n)}\ast \dotsm \ast g_{q}^{(n)}\\ &= (g'\ast {g_{1}'}^{(n)} - g \ast g_{1}^{(n)})\ast g_{2}^{(n)}\ast\dotsm \ast g_{q}^{(n)}. \end{align*} Par ailleurs, on tire de l'\'egalit\'e $[(Z',g')] = [(Z,g)]$ et de la relation (\ref{eq_fin1}) qu'il existe des courants $u$ et $v$ tels que~: \[ (Z',g') = (Z,g) + \op{\widehat{\op{div}}}f + (0,\partial u + \ov{\partial}v), \] ce dont on d\'eduit que~: \[ g' = g + (-\log |f|^{2}), \] dans $\widetilde{D}^{p-1,p-1}(X_{\M{R}})$. Ceci combin\'e \`a la relation~: \[ {g_{1}'}^{(n)} = g_{1}^{(n)} + (- \log |f_{1}|^{2}), \] et \`a (\cite{13}, \S 4.2.5.1) montre que~: \begin{align*} g'\ast {g_{1}'}^{(n)} - g\ast g_{1}^{(n)} &= g'\ast ({g'_{1}}^{(n)} - g_{1}^{(n)}) + (g' - g)\ast g_{1}^{(n)} \\ &= (- \log |f_{1}|^{2})\ast g' + (-\log|f|^{2})\ast g_{1}^{(n)} \\ &= (- \log |f_{1}|^{2})\wedge \delta_{Z'} + (- \log|f|^{2})\wedge \op{div}s_{1} \\ &= (- \log|f_{1}\cdot Z'|^{2}) + (- \log |f\cdot \op{div}s_{1}|^{2}); \end{align*} et donc que~: \begin{align*} \delta_{n} &= ((-\log |f_{1}\cdot Z'|^{2}) + (-\log |f \cdot \op{div}s_{1}|^{2})\ast g_{2}^{(n)} \ast \dotsm \ast g_{q}^{(n)} \\ &= (- \log |f_{1}\cdot Z'\cdot \op{div}s_{2}\dotsm \op{div}s_{q}|^{2}) + (- \log |f\cdot \op{div}s_{1} \dotsm \op{div}s_{q}|^{2}), \end{align*} dans $\widetilde{D}^{p+q-1,p+q-1}(X_{\M{R}})$. Comme $\delta = \lim_{n \rightarrow +\infty}\delta_{n}$, on a de m\^eme~: \begin{equation} \label{eq_fin2} \delta = (-\log|f_{1}\cdot Z'\cdot \op{div}s_{2}\dotsm \op{div}s_{q}|^{2}) + (- \log|f\cdot \op{div}s_{1}\dotsm \op{div}s_{q}|^{2}). \end{equation} Finalement, on d\'eduit de la d\'efinition (\ref{definition_prod2}) et des relations (\ref{eq_fin3}) et (\ref{eq_fin2}) que~: \begin{multline*} [(Z',g')\cdot \hat{c}_{1}(\ov{L}_{1},s_{1}')\cdot \hat{c}_{1}(\ov{L}_{2},s_{2}) \dotsm \hat{c}_{1}(\ov{L}_{q},s_{q})] - [(Z,g)\cdot \hat{c}_{1}(\ov{L}_{1},s_{1})\dotsm \hat{c}_{1}(\ov{L}_{q},s_{q})]\\ = [(Y,\delta)] = [\op{\widehat{\op{div}}}(f_{1}\cdot Z'\cdot \op{div}s_{2}\dotsm \op{div}s_{q})] + [\op{\widehat{\op{div}}}(f\cdot \op{div}s_{1}\dotsm \op{div}s_{q})] + [(R,0)] = 0. \end{multline*} \medskip La proposition pr\'ec\'edente nous autorise \`a noter d\'esormais $[(Z,g)]\cdot \hat{c}_{1}(\ov{L}_{1})\dotsm \hat{c}_{1}(\ov{L}_{q})$ la classe $[(Z,g)\cdot \hat{c}_{1}(\ov{L}_{1},s_{1})\dotsm \hat{c}_{1}(\ov{L}_{q},s_{q})]$. \begin{prop} \label{linearite2} Soient $\alpha \in \widehat{CH}^{p}(X)$ et $\ov{L}_{0}$, $\ov{L}_{1},\dots, \ov{L}_{q}$ des fibr\'es en droites admissibles sur $X$. Pour tout $1 \leqslant i \leqslant q$, on a dans $\widetilde{CH}^{p+q}(X)$ la relation~: \begin{multline*} \alpha \cdot \hat{c}_{1}(\ov{L}_{1})\dotsm \hat{c}_{1}(\ov{L}_{i-1})\hat{c}_{1} (\ov{L}_{i}\otimes \ov{L}_{0})\hat{c}_{1}(\ov{L}_{i+1})\dotsm \hat{c}_{1}(\ov{L}_{q}) \\ = \alpha \cdot\hat{c}_{1}(\ov{L}_{1})\dotsm \hat{c}_{1}(\ov{L}_{q}) + \alpha \cdot \hat{c}_{1}(\ov{L}_{1})\dotsm \hat{c}_{1}(\ov{L}_{i-1}) \hat{c}_{1}(\ov{L}_{0})\hat{c}_{1}(\ov{L}_{i+1})\dotsm \hat{c}_{1}(\ov{L}_{q}). \end{multline*} \end{prop} \noindent {\bf D\'emonstration.}\ D'apr\`es (\ref{independance2}), il suffit de d\'emontrer le cas $i=1$. Soit $(Z,g)$ un repr\'esentant de la classe $\alpha \in \widehat{CH}^{p}(X)$. Soient \'egalement $s_{1},\dots,s_{q}$ des sections rationnelles au-dessus de $X$ de $L_{1}, \dots, L_{q}$ respectivement, telles que $Z$, $\op{div}s_{1}, \dots, \op{div}s_{q}$ s'intersectent proprement; et $s_{0}$ une section rationnelle de $L_{0}$ au-dessus de $X$ telle que $Z$, $\op{div}s_{0}$, $\op{div}s_{2}, \dots, \op{div}s_{q}$ s'intersectent proprement. Pour tout $0 \leqslant i \leqslant q$, on note $\delta_{i} = \delta_{\op{div}s_{i}}$, $g_{i} = - \log\|s_{i}\|_{i}^{2}$ et $\omega_{i} = c_{1}(\ov{L}_{i})$. On note \'egalement $\tilde{\omega} = \omega(Z,g)$. On a~: \begin{multline*} [g \ast (g_{0} + g_{1},s_{0} + s_{1})\ast (g_{2},s_{2}) \ast \dots \ast (g_{q},s_{q})] = g\cdot \delta_{(\op{div}s_{0}\cup\op{div}s_{1})\cap \op{div}s_{2}\cap \dots \cap \op{div}s_{q}}\\ + \tilde{\omega}(g_{0}+g_{1})\cdot \delta_{2}\dots\delta_{q} + \tilde{\omega}(\omega_{0}+ \omega_{1})g_{2}\cdot \delta_{3}\dots \delta_{q} + \dots + \tilde{\omega}(\omega_{0} + \omega_{1})\omega_{2}\dots\omega_{q-1}g_{q} \\ = [g \ast (g_{1},s_{1})\ast \dots \ast (g_{k},s_{k})] + [g \ast (g_{0},s_{0})\ast (g_{2},s_{2})\ast \dots \ast (g_{q},s_{q})], \end{multline*} et comme~: \begin{multline*} Z\cdot (\op{div}s_{0} + \op{div}s_{1})\cdot \op{div}s_{2}\dotsm \op{div}s_{q} = Z \cdot \op{div}s_{1}\cdot \op{div}s_{2} \dotsm \op{div}s_{q} \\ + Z \cdot \op{div}s_{0} \cdot \op{div}s_{2} \dotsm \op{div}s_{q} \end{multline*} dans $Z^{p+q}(X)$, on a bien le r\'esultat cherch\'e. \medskip Plus g\'en\'eralement, soient $\alpha \in \widehat{CH}^{p}(X)$ et $\ov{L}_{1}, \dots, \ov{L}_{q}$ des fibr\'es en droites int\'egrables sur $X$. On choisit $\ov{E}_{1},\dots,\ov{E}_{q}$ et $\ov{F}_{1},\dots,\ov{F}_{q}$ des fibr\'es en droites admissibles sur $X$ tels que pour tout $1 \leqslant i \leqslant q$, on ait~: $\ov{L}_{i} = \ov{E}_{i}\otimes \left(\ov{F}_{i}\right)^{-1}$. \begin{prop_defn} \label{produit_general2} La classe dans $\widetilde{CH}^{p+q}(X)$ donn\'ee par~: \[ \sum_{\substack{S_{1},S_{2} \\ S_{1}\cup S_{2} = \{1,\dots,q\}}} (-1)^{\#S_{2}}\; \alpha \cdot \prod_{i \in S_{1}}\hat{c}_{1}(\ov{E}_{i})\cdot \prod_{j \in S_{2}}\hat{c}_{1}(\ov{F}_{j}) \] ne d\'epend que de la classe $\alpha \in \widehat{CH}^{p}(X)$ et des classes d'isomorphie isom\'etrique des fibr\'es $\ov{L}_{1}, \dots,\ov{L}_{q}$. On note $\alpha\cdot \hat{c}_{1}(\ov{L}_{1})\dotsm \hat{c}_{1}(\ov{L}_{q})$ cette classe. Si les fibr\'es $\ov{L}_{1}, \dots, \ov{L}_{q}$ sont admissibles, cet \'el\'ement co\"\i ncide avec celui d\'efini en (\ref{definition_prod2}). \end{prop_defn} \noindent {\bf D\'emonstration.}\ C'est une cons\'equence directe de la proposition (\ref{linearite2}) et du fait que toute application $q$-lin\'eaire $\varphi: \underbrace{A\times\dots\times A}_{\text{$q$ fois}} \rightarrow B$, o\`u $A$ est un semi-groupe ab\'elien et $B$ un groupe ab\'elien, s'\'etend de mani\`ere unique en une application $q$-lin\'eaire $\varphi_{s}: \underbrace{A_{s}\times \dots\times A_{s}}_{\text{$q$ fois}} \rightarrow B$, o\`u $A_{s}$ est le groupe sym\'etris\'e de $A$. \medskip Diverses propri\'et\'es de cette construction sont rassembl\'ees dans le th\'eor\`eme suivant~: \begin{thm} \label{th_general} Soient $\alpha$ un \'el\'ement de $\widehat{CH}^{p}(X)$ et $\ov{L}_{1}, \dots, \ov{L}_{q}$ des fibr\'es en droites int\'egrables sur $X$. On a les propri\'et\'es suivantes~: \begin{enumerate} \item{La classe $\alpha\cdot \hat{c}_{1}(\ov{L}_{1})\dotsm \hat{c}_{1}(\ov{L}_{q})$ ne d\'epend pas de l'ordre de $\ov{L}_{1},\dots,\ov{L}_{q}$.} \item{ L'application qui \`a $\alpha \in \widehat{CH}^{p}(X)$ et $\ov{L}_{1},\dots,\ov{L}_{q}$ des fibr\'es en droites hermitiens int\'egrables sur $X$ associe $\alpha\cdot \hat{c}_{1}(\ov{L}_{1})\dotsm \hat{c}_{1}(\ov{L}_{q})$, d\'efinit une application multilin\'eaire~: \[ \widehat{CH}^{p}(X)\times \widehat{\op{Pic}}_{\,\op{int}}(X)\times \dots \times \widehat{\op{Pic}}_{\,\op{int}}(X) \longrightarrow \widetilde{CH}^{p+q}(X). \] } \item{Si les m\'etriques des fibr\'es $\ov{L}_{1},\dots,\ov{L}_{k}$ pour $1 \leqslant k \leqslant q$ fix\'e, sont $C^{\infty}$ sur $X(\M{C})$, alors on a~: \[ \alpha \cdot \hat{c}_{1}(\ov{L}_{1})\dotsm \hat{c}_{1}(\ov{L}_{q}) = (\alpha \cdot\hat{c}_{1}(\ov{L}_{1})\dotsm \hat{c}_{1}(\ov{L}_{k})) \cdot\hat{c}_{1}(\ov{L}_{k+1})\dotsm\hat{c}_{1}(\ov{L}_{q}), \] o\`u le produit $(\alpha\cdot\hat{c}_{1}(\ov{L}_{1})\dotsm \hat{c}_{1}(\ov{L}_{k})) \in \widehat{CH}^{p+k}(X)$ dans le second membre est pris au sens de Gillet-Soul\'e. (Voir \cite{13}, th. 4.2.3). En particulier si les m\'etriques sur $\ov{L}_{1},\dots,\ov{L}_{q}$ sont $C^{\infty}$, le produit $\alpha\cdot \hat{c}_{1}(\ov{L}_{1})\dotsm \hat{c}_{1}(\ov{L}_{q})$ d\'efini par la proposition (\ref{produit_general2}) co\"\i ncide avec celui d\'efini par Gillet-Soul\'e.} \item{Soit $\boldsymbol{1} = [(X,0)] \in \widehat{CH}^{0}(X)$. On a~: \[ \boldsymbol{1}\cdot \hat{c}_{1}(\ov{L}_{1}) = \hat{c}_{1}(\ov{L}_{1}). \] } \item{On a les relations~: \begin{multline*} \qquad \quad \; \omega (\alpha\cdot \hat{c}_{1}(\ov{L}_{1})\dotsm \hat{c}_{1}(\ov{L}_{q})) \\ = \omega(\alpha)\cdot c_{1}(\ov{L}_{1})\dotsm c_{1}(\ov{L}_{q}) \in \ov{A}_{\op{g}}^{p+q,p+q}(X_{\M{R}}) \cap C_{0}^{p+q,p+q}(X_{\M{R}}) \end{multline*} et \[ \zeta (\alpha\cdot \hat{c}_{1}(\ov{L}_{1})\dotsm \hat{c}_{1}(\ov{L}_{q})) = \zeta(\alpha) \cdot c_{1}(L_{1})\dotsm c_{1}(L_{q}) \in CH^{p+q}(X). \] } \item{Enfin, si $\alpha = (0,\varphi)$, avec $\varphi \in A^{p-1,p-1}(X_{\M{R}})$, alors~: \[ \alpha \cdot \hat{c}_{1}(\ov{L}_{1})\dotsm \hat{c}_{1}(\ov{L}_{q}) = [(0,\varphi\cdot c_{1}(\ov{L}_{1})\dotsm c_{1}(\ov{L}_{q}))]. \] } \end{enumerate} \end{thm} \noindent {\bf D\'emonstration.}\ \begin{enumerate} \item{Cela d\'ecoule de la proposition (\ref{independance2}) et de la d\'efinition donn\'ee \`a la proposition (\ref{produit_general2}).} \item{Cela d\'ecoule des propositions (\ref{linearite2}) et (\ref{produit_general2}).} \item{On se ram\`ene gr\^ace au (2) au cas o\`u $\ov{L}_{1},\dots,\ov{L}_{q}$ sont des fibr\'es en droites admissibles, on applique alors (\ref{naturalite2}).} \item{Il suffit de le faire pour $\ov{L}_{1}$ admissible et c'est alors \'evident.} \item{D'apr\`es le (2), il suffit de d\'emontrer les relations pour $\ov{L}_{1}, \dots, \ov{L}_{q}$ admissibles. L'\'egalit\'e $\zeta (\alpha\cdot \hat{c}_{1}(\ov{L}_{1})\dotsm \hat{c}_{1}(\ov{L}_{q})) = \zeta(\alpha)\cdot c_{1}(L_{1}) \cdots c_{1}(L_{q})$ est une cons\'equence imm\'ediate de la d\'efinition (\ref{definition_prod2}). Soient $\left(\|.\|_{1}^{(n)}\right)_{n \in \M{N}},\dots,$ $\left(\|.\|_{q}^{(n)}\right)_{n \in \M{N}}$ des suites croissantes de m\'etriques positives $C^{\infty}$ convergeant uniform\'ement sur $X(\M{C})$ vers $\|.\|_{1},\dots,\|.\|_{q}$ respectivement. D'apr\`es le (3) et (\cite{13}, th. 4.2.9), on a~: \begin{equation} \begin{split} \label{relation_intermediaire} \omega (\alpha\cdot\hat{c}_{1}(L_{1},\|.\|_{1}^{(n)})\dotsm \hat{c}_{1}(L_{q},\|.&\|_{q}^{(n)})) \\ &= \omega(\alpha)\cdot c_{1}(L_{1},\|.\|_{1}^{(n)})\dotsm c_{1}(L_{q},\|.\|_{q}^{(n)}). \end{split} \end{equation} Soient $(Z,g)$ un repr\'esentant de la classe $\alpha \in \widehat{CH}^{p}(X)$ et $s_{1},\dots,s_{q}$ des sections rationnelles non identiquement nulles sur $X$ de $L_{1},\dots, L_{q}$ respectivement, telles que les cycles $Z$, $\op{div}s_{1}, \dots, \op{div}s_{q}$ s'intersectent proprement. Pour tout $1 \leqslant i \leqslant q$, on note $g_{i} = - \log \|s_{i}\|_{i}^{2}$ et pour tout $1 \leqslant i \leqslant q$ et tout $n \in \M{N}$, on note $g_{i}^{(n)} = - \log \left(\|s_{i}\|_{i}^{(n)}\right)^{2}$. On a, pour tout $n \in \M{N}$, l'\'egalit\'e de courants~: \begin{multline*} \qquad \omega (\alpha\cdot\hat{c}_{1}(L_{1},\|.\|_{1}^{(n)})\dotsm \hat{c}_{1}(L_{q},\|.\|_{q}^{(n)})) = \\ dd^{c}(\{g\ast (g_{1}^{(n)},s_{1})\ast \dots \ast (g_{q}^{(n)},s_{q})\}) + \delta_{Z \cap \op{div}s_{1} \cap \dots \cap \op{div}s_{q}}. \end{multline*} Comme $\{g\ast (g_{1}^{(n)},s_{1})\ast \dots \ast (g_{q}^{(n)},s_{q})\}$ tend vers $\{g\ast (g_{1},s_{1})\ast \dots \ast (g_{q},s_{q})\}$ quand $n$ tend vers $+\infty$ d'apr\`es (\ref{approximation2}), on d\'eduit de la continuit\'e faible de l'op\'erateur $dd^{c}$ et de la relation~: \begin{multline*} \qquad \omega (\alpha\cdot\hat{c}_{1}(L_{1},\|.\|_{1})\dotsm \hat{c}_{1}(L_{q},\|.\|_{q})) = \\ dd^{c}(\{g\ast (g_{1},s_{1})\ast \dots \ast (g_{q},s_{q})\}) + \delta_{Z \cap \op{div}s_{1} \cap \dots \cap \op{div}s_{q}}, \end{multline*} que la forme diff\'erentielle $\omega (\alpha\cdot\hat{c}_{1}(L_{1},\|.\|_{1}^{(n)})\dotsm \hat{c}_{1}(L_{q},\|.\|_{q}^{(n)}))$ tend vers \\ $\omega (\alpha\cdot\hat{c}_{1}(L_{1},\|.\|_{1})\dotsm \hat{c}_{1}(L_{q},\|.\|_{q}))$ au sens de la convergence faible quand $n$ tend vers $+\infty$. Enfin, d'apr\`es (\ref{demailly}), on a~: \[ \lim_{n \rightarrow +\infty} \omega(\alpha)\cdot c_{1}(L_{1},\|.\|_{1}^{(n)})\dotsm c_{1}(L_{q},\|.\|_{q}^{(n)}) = \omega(\alpha)\cdot c_{1}(\ov{L}_{1})\dotsm c_{1}(\ov{L}_{q}), \] ce qui joint \`a la relation (\ref{relation_intermediaire}) permet de conclure.} \item{D'apr\`es le (2), il suffit de d\'emontrer l'\'egalit\'e recherch\'ee pour $\ov{L}_{1},\dots,\ov{L}_{q}$ des fibr\'es en droites admissibles sur $X$. On reprend ici les notations de la d\'emonstration du (5). En utilisant la d\'efinition (\ref{definition_prod2}), on obtient la relation~: \begin{equation} \label{int_eq_5} [(0,\varphi)]\cdot \hat{c}_{1}(\ov{L}_{1})\dotsm \hat{c}_{1}(\ov{L}_{q}) = [(0,\{\varphi\ast (g_{1},s_{1})\ast \dotsm \ast (g_{q},s_{q})\})]. \end{equation} D'apr\`es la proposition (\ref{naturalite2}) et la th\'eorie d\'evelopp\'ee dans \cite{13}, on a l'\'egalit\'e suivante, valable pour tout $n \in \M{N}$~: \begin{align*} \{\varphi\ast (g_{1}^{(n)},s_{1})\ast \dotsm \ast (g_{q}^{(n)},s_{q})\} &= \varphi\ast (g_{1}^{(n)}\ast ( \dotsm \ast (g_{q-1}^{(n)} \ast g_{q}^{(n)})\dotsm )) \\ &= (\dotsm ((\varphi \ast g_{1}^{(n)})\ast g_{2}^{(n)}) \ast \dotsm ) \ast g_{q}^{(n)}. \end{align*} Par ailleurs, on sait d'apr\`es (\cite{13}, \S 2.2.9) que pour toute forme diff\'erentielle $\varphi' \in A^{\ast}(X_{\M{R}})$ et tout $1 \leqslant i \leqslant q$~: \[ \varphi' \ast g_{i}^{(n)} = g_{i}^{(n)} \ast \varphi' = \varphi'\cdot c_{1}(L_{i},\|.\|_{i}^{(n)}). \] On d\'eduit alors par r\'ecurrence de ce qui pr\'ec\`ede la relation~: \[ \{\varphi \ast (g_{1}^{(n)},s_{1})\ast \dotsm \ast (g_{q}^{(n)},s_{q})\} = \varphi\cdot c_{1}(L_{1},\|.\|_{1}^{(n)}) \dotsm c_{1}(L_{q},\|.\|_{q}^{(n)}) \] dans $\widetilde{D}^{p+q-1,p+q-1}(X_{\M{R}})$. En remarquant que d'une part, $\{\varphi \ast (g_{1}^{(n)},s_{1})\ast \dotsm \ast (g_{q}^{(n)},s_{q})\}$ tend vers $ \{\varphi\ast (g_{1},s_{1})\ast \dotsm \ast (g_{q},s_{q})\}$ au sens de la convergence faible des courants quand $n$ tend vers $+\infty$ d'apr\`es (\ref{approximation2}), et que d'autre part~: \[ \lim_{n \rightarrow +\infty}\varphi\cdot c_{1}(L_{1},\|.\|_{1}^{(n)}) \dotsm c_{1}(L_{q},\|.\|_{q}^{(n)}) = \varphi\cdot c_{1}(\ov{L}_{1}) \dotsm c_{1}(\ov{L}_{q}), \] \'egalement au sens de la topologie faible des courants, on conclut de ce qui pr\'ec\`ede que~: \[ \{\varphi \ast (g_{1},s_{1}) \ast \dotsm \ast (g_{q},s_{q})\} = \varphi\cdot c_{1}(\ov{L}_{1})\dotsm c_{1}(\ov{L}_{q}) \] dans $\widetilde{D}^{p+q-1,p+q-1}(X_{\M{R}})$, ce qui joint \`a la relation (\ref{int_eq_5}) prouve le r\'esultat \'enonc\'e.} \end{enumerate} \medskip \begin{defn} \label{def_gch_gen} Soit $p$ un entier positif. On appelle {\it groupe de Chow arithm\'etique g\'en\'eralis\'e de codimension $p$\/} et on note $\widehat{CH}_{\op{int}}^{p}(X)$ le $\M{Z}$ sous-module de $\widetilde{CH}^{p}(X)$ engendr\'e par les \'el\'ements de la forme $\alpha\cdot \hat{c}_{1}(\ov{L}_{1})\dotsm \hat{c}_{1}(\ov{L}_{q})$, o\`u $\alpha \in \widehat{CH}^{p-q}(X)$ et o\`u $\ov{L}_{1},\dots,\ov{L}_{q}$ sont des fibr\'es en droites int\'egrables sur $X$. On convient de noter $\widehat{CH}_{\op{int}}^{\ast}(X) = \bigoplus_{p \geqslant 0}\widehat{CH}_{\op{int}}^{p}(X)$. \end{defn} \begin{rem} Le groupe $\widehat{CH}_{\op{int}}^{p}(X)$ contient $\widehat{CH}^{p}(X)$. \end{rem} \begin{rem} Soit $\ov{L}$ un fibr\'e int\'egrable sur $X$; la premi\`ere classe de Chern $\hat{c}_{1}(\ov{L})$ de $\ov{L}$ appartient \`a $\widehat{CH}_{\op{int}}^{1}(X)$. \end{rem} \begin{prop} L'application qui \`a tout fibr\'e en droites hermitien int\'egrable $\ov{L}$ associe sa classe de Chern $\hat{c}_{1}(\ov{L}) \in \widehat{CH}_{\op{int}}^{1}(X)$, d\'efinit un isomorphisme de groupes~: \[ \hat{c}_{1}: \widehat{\op{Pic}}_{\,\op{int}}(X) \longrightarrow \widehat{CH}_{\op{int}}^{1}(X). \] \end{prop} \noindent {\bf D\'emonstration.}\ Il d\'ecoule des d\'efinitions que $\hat{c}_{1}(\cdot)$ est bien d\'efinie, surjective, et pr\'eserve la structure de groupe. Il nous reste \`a montrer que $\hat{c}_{1}(\cdot)$ est injective. Soit $\ov{L} = (L,\|.\|) \in \widehat{\op{Pic}}_{\,\op{int}}(X)$ tel que $\hat{c}_{1}(\ov{L}) = 0$. Comme $c_{1}(L) = \zeta(\hat{c}_{1}(\ov{L})) = 0$, le fibr\'e $L$ est isomorphe au fibr\'e trivial; on note $s$ une section constante non nulle de $L$ au-dessus de $X$. La relation~: \[ \hat{c}_{1}(\ov{L}) = [\op{\widehat{\op{div}}}s] = [(0,-\log \|s\|^{2})] = 0, \] jointe au fait que le groupe $CH^{1,0}(X)$ est toujours trivial, entra\^\i nent que $\|s\| = 1$ identiquement sur $X(\M{C})$, ce qui nous permet de conclure. \medskip Soient $\alpha \in A^{r-1,r-1}(X_{\M{R}})$ et $\ov{L}_{1},\dots,\ov{L}_{q}$ des fibr\'es en droites int\'egrables sur $X$, avec $q$ et $r$ deux entiers positifs tels que $q + r = p$. D'apr\`es le th\'eor\`eme (\ref{th_general}), on a~: \[ [(0,\alpha c_{1}(\ov{L}_{1}) \dotsm c_{1}(\ov{L}_{q}))] = [(0,\alpha)]\cdot \hat{c}_{1}(\ov{L}_{1})\dotsm \hat{c}_{1}(\ov{L}_{q}) \in \widehat{CH}_{\op{int}}^{p}(X). \] On en d\'eduit que $a\left(\ov{A}_{\op{g}}^{p-1,p-1}(X_{\M{R}})\right) \subset \widehat{CH}_{\op{int}}^{p}(X)$. On dispose donc du morphisme de groupes~: \begin{alignat*}{3} a: \ov{A}_{\op{g}}^{p-1,p-1}&(X_{\M{R}})& &\longrightarrow & &\widehat{CH}_{\op{int}}^{p}(X) \\ &\beta& &\longmapsto & & [(0,\beta)]. \end{alignat*} D'apr\`es le th\'eor\`eme (\ref{th_general}), on obtient par restriction \`a $ \widehat{CH}_{\op{int}}^{p}(X)$ des morphismes $\zeta$ et $\omega$, les morphismes de groupes suivants~: \begin{alignat*}{3} \zeta :\; &\widehat{CH}_{\op{int}}^{p}(X) & &\longrightarrow & &CH^{p}(X) \\ &[(Z,g)] & &\longmapsto & &Z, \\ \end{alignat*} et \begin{alignat*}{3} \qquad \qquad \quad \omega :\; &\widehat{CH}_{\op{int}}^{p}(X) & &\longrightarrow & &\ov{A}_{\op{g}}^{p,p}(X_{\M{R}})\\ &[(Z,g)] & & \longmapsto & &\omega(Z,g) = dd^{c} g + \delta_{Z}. \end{alignat*} \bigskip \begin{prop} \label{decomposition_classe} Soit $\alpha \in \widehat{CH}_{\op{int}}^{p}(X)$ et choisissons $(Z,g)$ un repr\'esentant de $\alpha$ dans $\widehat{CH}_{\op{int}}^{p}(X)$. Il existe $g_{Z} \in D^{p-1,p-1}(X_{\M{R}})$ un courant de Green pour $Z$ et $\varphi \in C_{\log,0}^{p-1,p-1}(X_{\M{R}})$ tels que l'on ait~: \[ g = g_{Z} + \varphi \quad \text{dans} \quad \widetilde{D}^{p-1,p-1}(X_{\M{R}}). \] \end{prop} \noindent {\bf D\'emonstration.}\ Par lin\'earit\'e, il suffit de d\'emontrer le r\'esultat pour $\alpha = \beta\cdot \hat{c}_{1}(\ov{L}_{1})\dotsm \hat{c}_{1}(\ov{L}_{q})$, o\`u $\beta \in \widehat{CH}^{p-q}(X)$ et $\ov{L}_{1} = (L_{1},\|.\|_{1}), \dots, \ov{L}_{q} = (L_{q},\|.\|_{q})$ sont des fibr\'es admissibles sur $X$. Soient $(Z',g')$ un repr\'esentant de la classe $\beta$ et $s_{1},\dots,s_{q}$ des sections rationnelles non identiquement nulles sur $X$ de $L_{1},\dots,L_{q}$ respectivement telles que $Z',\op{div}s_{1},\dots,\op{div}s_{q}$ s'intersectent proprement. Soient \'egalement $\|.\|_{1}^{(0)}, \dots, \|.\|_{q}^{(0)}$ des m\'etriques positives $C^{\infty}$ sur $L_{1},\dots,L_{q}$ respectivement. On pose $\tilde{\omega}' = dd^{c} g' + \delta_{Z'}$. Pour tout $1 \leqslant i \leqslant q$, on note \'egalement $\delta_{i} = \delta_{\op{div}s_{i}}$, $g_{i} = - \log \|s_{i}\|_{i}^{2}$, $\omega_{i} = c_{1}(\ov{L}_{i})$, $g_{i}^{(0)} = - \log (\|s_{i}\|_{i}^{(0)})^{2}$ et $\omega_{i}^{(0)} = c_{1}(L_{i},\|.\|_{i}^{(0)})$. En appliquant les d\'efinitions, il vient~: \begin{align*} \{g'\ast (g_{1},s_{1})\ast \dotsm \ast (g_{q},&s_{q})\} - \{g'\ast (g_{1},s_{1})\ast \dotsm \ast (g_{q-1},s_{q-1})\ast (g_{q}^{(0)},s_{q})\}\\ &= (g_{q} - g_{q}^{(0)})\, \tilde{\omega}'\omega_{1}\dotsm \omega_{q-1} \\ &= - \log \left(\frac{\|.\|_{q}}{\;\, \|.\|_{q}^{(0)}}\right)^{2} \tilde{\omega}'\omega_{1}\dots \omega_{q-1} \in C_{\log,0}^{p-1,p-1}(X_{\M{R}}). \end{align*} Or, d'apr\`es la proposition (\ref{naturalite2}), on a~: \begin{multline*} \{g'\ast(g_{1},s_{1})\ast \dotsm \ast (g_{q-1},s_{q-1})\ast(g_{q}^{(0)},s_{q})\}\\ = \{(g'\ast g_{q}^{(0)})\ast(g_{1},s_{1})\ast \dotsm \ast (g_{q-1},s_{q-1})\}, \end{multline*} dans $\widetilde{D}^{p-1,p-1}(X_{\M{R}})$. En it\'erant $q$ fois ce proc\'ed\'e, on trouve $\varphi \in C_{\log,0}^{p-1,p-1}(X)$ tel que~: \[ \{g'\ast (g_{1},s_{1})\ast \dotsm \ast (g_{q},s_{q})\} = g'\ast g_{q}^{(0)}\ast \dotsm \ast g_{1}^{(0)} + \varphi, \] dans $\widetilde{D}^{p-1,p-1}(X_{\M{R}})$. Il suffit alors de remarquer que $g_{Z} = g'\ast g^{(0)}_{q} \ast \dotsm \ast g_{1}^{(0)}$ est un courant de Green pour $Z'\cdot \op{div}s_{1}\dotsm \op{div}s_{q}$ et la proposition est d\'emontr\'ee. \medskip la proposition (\ref{decomposition_classe}) justifie la d\'efinition suivante~: \begin{defn} Soit $p$ un entier positif. On note $\widehat{CH}_{\op{gen}}^{p}(X)$ le $\M{Z}$ sous-module de $\widetilde{CH}^{p}(X)$ engendr\'e par les \'el\'ements de $\widehat{CH}^{p}(X)$ et ceux de la forme $a(\varphi)$ avec $\varphi \in C_{\log,0}^{p-1,p-1}(X_{\M{R}})$. On convient de noter $\widehat{CH}_{\op{gen}}^{\ast}(X) = \bigoplus_{p \geqslant 0}\widehat{CH}_{\op{gen}}^{p}(X)$. \end{defn} \begin{rem} D'apr\`es la proposition (\ref{decomposition_classe}) on a $\widehat{CH}_{\op{int}}^{p}(X) \subset \widehat{CH}_{\op{gen}}^{p}(X)$. \end{rem} \begin{rem} On dispose du morphisme de groupes~: \begin{alignat*}{3} \omega~: \;&\widehat{CH}_{\op{gen}}^{p}(X) & & \longrightarrow & \;&C_{0}^{p,p}(X_{\M{R}}) \\ &[(Z,g)]& &\longmapsto & & dd^{c} g + \delta_{Z}. \end{alignat*} \end{rem} \bigskip \subsection{L'accouplement $\widehat{CH}_{\op{int}}^{p}(X) \otimes \widehat{CH}_{\op{int}}^{q}(X) \rightarrow \widehat{CH}_{\op{int}}^{p+q}(X)_{\M{Q}}$}~ Soit $\pi: X \rightarrow S = \op{Spec}\C{O}_{K}$ une vari\'et\'e arithm\'etique de dimension de Krull $d+1$. On d\'efinit dans cette section un accouplement $ \widehat{CH}_{\op{int}}^{p}(X) \otimes \widehat{CH}_{\op{int}}^{q}(X) \rightarrow \linebreak[4] \widehat{CH}_{\op{int}}^{p+q}(X)_{\M{Q}}$ qui \'etend l'accouplement $\widehat{CH}^{p}(X) \otimes \widehat{CH}^{q}(X) \rightarrow \widehat{CH}^{p+q}(X)_{\M{Q}}$ d\'efini par Gillet-Soul\'e (voir \cite{13}, \S 4.2; on peut \'egalement consulter \cite{3}, \S 2.2). Soient $x \in \widehat{CH}_{\op{int}}^{p}(X)$ et $y \in \widehat{CH}_{\op{int}}^{q}(X)$. On peut \'ecrire $x$ et $y$ sous la forme~: \begin{align*} x &= \sum_{i = 1}^{n} \alpha_{i}\cdot \hat{c}_{1}(\ov{E}_{i,1}) \dotsm \hat{c}_{1}(\ov{E}_{i,r_{i}}) \\ \intertext{et} y &= \sum_{j =1}^{m} \beta_{j}\cdot \hat{c}_{1}(\ov{F}_{j,1}) \dotsm \hat{c}_{1}(\ov{F}_{j,s_{j}}), \end{align*} o\`u pour tout $1 \leqslant i \leqslant n$ (resp. tout $1 \leqslant j \leqslant m$), $\alpha_{i}$ est un \'el\'ement de $\widehat{CH}^{p - r_{i}}(X)$ et $\ov{E}_{i,1}, \dots \ov{E}_{i,r_{i}}$ sont des fibr\'es en droites int\'egrables sur $X$ (resp. $\beta_{j}$ est un \'el\'ement de $\widehat{CH}^{q-s_{j}}(X)$ et $\ov{F}_{j,1}, \dots, \ov{F}_{j,s_{j}}$ sont des fibr\'es en droites int\'egrables sur $X$). On d\'efinit alors le produit $(x\cdot y) \in \widehat{CH}_{\op{int}}^{p+q}(X)_{\M{Q}}$ par la formule~: \[ (x\cdot y) = \sum_{i =1}^{n}\sum_{j=1}^{m} (\alpha_{i}\cdot \beta_{j})\cdot \hat{c}_{1}(\ov{E}_{i,1})\dotsm \hat{c}_{1}(\ov{E}_{i,r_{i}})\cdot \hat{c}_{1}(\ov{F}_{j,1})\dotsm \hat{c}_{1}(\ov{F}_{j,s_{j}}), \] o\`u pour tout $1 \leqslant i \leqslant n$ et $1 \leqslant j \leqslant m$, on a not\'e $(\alpha_{i}\cdot \beta_{j}) \in \widehat{CH}^{p+q - r_{i} - s_{j}}(X)_{\M{Q}}$ le produit de $\alpha_{i}$ et $\beta_{j}$ au sens de Gillet-Soul\'e. Bien entendu, il faut montrer que cette d\'efinition ne d\'epend pas des choix effectu\'es pour repr\'esenter $x$ et $y$. C'est l'objet du th\'eor\`eme suivant~: \begin{thm} \label{produit_bien_defini} La d\'efinition ci-dessus ne d\'epend pas des choix effectu\'es. Elle d\'efinit un accouplement~: \begin{equation} \label{accouplement_1} \widehat{CH}_{\op{int}}^{p}(X) \otimes \widehat{CH}_{\op{int}}^{q}(X) \longrightarrow \widehat{CH}_{\op{int}}^{p+q}(X)_{\M{Q}}, \end{equation} qui prolonge celui d\'efini par Gillet-Soul\'e. Cet accouplement munit $\widehat{CH}_{\op{int}}^{\ast}(X)_{\M{Q}}$ d'une structure d'anneau commutatif, associatif et unif\`ere. \end{thm} \begin{rem} \label{rem_acc_1} Si $p =1$ ou $q =1$, on dispose d'un accouplement~: \[ \widehat{CH}_{\op{int}}^{p}(X) \otimes \widehat{CH}_{\op{int}}^{q}(X) \longrightarrow \widehat{CH}_{\op{int}}^{p+q}(X) \] qui induit l'accouplement (\ref{accouplement_1}) \`a valeur dans $\widehat{CH}_{\op{int}}^{p+q}(X)_{\M{Q}}$. \end{rem} \begin{rem} \label{rem_acc_2} Si $X$ est lisse sur $\op{Spec}\C{O}_{K}$, on peut d\'efinir les produits $(\alpha_{i}\cdot \beta_{j})$ dans $\widehat{CH}^{p+q-r_{i}-s_{j}}(X)$. La construction pr\'ec\'edente donne donc un accouplement~: \[ \widehat{CH}_{\op{int}}^{p}(X) \otimes \widehat{CH}_{\op{int}}^{q}(X) \longrightarrow \widehat{CH}_{\op{int}}^{p+q}(X) \] qui induit par produit tensoriel avec $\M{Q}$ l'accouplement (\ref{accouplement_1}) \`a valeur dans \linebreak[4] $\widehat{CH}_{\op{int}}^{p+q}(X)_{\M{Q}}$. \end{rem} \noindent {\bf D\'emonstration.}\ On montre tout d'abord que le produit $(x\cdot y)$ introduit plus haut est bien d\'efini. Soit $x = \sum_{i=1}^{n} \alpha_{i}\cdot \hat{c}_{1}(\ov{E}_{i,1})\dotsm \hat{c}_{1}(\ov{E}_{i,r_{i}})$ un \'el\'ement de $\widehat{CH}_{\op{int}}^{p}(X)$. Il faut montrer que si $x =0$ dans $\widehat{CH}_{\op{int}}^{p} (X)$, alors pour tout $y \in \widehat{CH}_{\op{int}}^{q}(X)$, on a $(x\cdot y) = 0$. On montre dans ce qui suit un r\'esultat plus g\'en\'eral~: Si $x = a(\varphi)$ avec $\varphi \in C_{\log,0}^{p-1,p-1}(X_{\M{R}})$, alors $x\cdot y = a(\varphi\, \omega(y))$. Par lin\'earit\'e, il suffit de d\'emontrer cet \'enonc\'e pour $y = \beta\cdot \hat{c}_{1}(\ov{F}_{1})\dotsm \hat{c}_{1}(\ov{F}_{s})$, o\`u $\beta$ est un \'el\'ement de $\widehat{CH}^{q-s}(X)$ et $\ov{F}_{1} = (F_{1}, \|.\|_{1}), \dots, \ov{F}_{s} = (F_{s},\|.\|_{s})$ sont des fibr\'es en droites admissibles sur $X$. Remarquons \'egalement que la proposition (\ref{produit_general2}) et le th\'eor\`eme (\ref{th_general}) nous permettent de supposer que pour tout $1 \leqslant i \leqslant n$ les fibr\'es en droites $\ov{E}_{i,1} = (E_{i,1},\|.\|_{i,1}), \dots, \ov{E}_{i,r_{i}} = (E_{i,r_{i}}, \|.\|_{i,r_{i}})$ sont admissibles sur $X$. Soient $\left(\|.\|_{1}^{(k)}\right)_{k \in \M{N}}, \dots, \left(\|.\|_{s}^{(k)}\right)_{k \in \M{N}}$ des suites croissantes de m\'etriques positives $C^{\infty}$ sur $F_{1},\dots,F_{s}$ convergeant vers $\|.\|_{1},\dots,\|.\|_{s}$ respectivement, et pour tout $1 \leqslant i \leqslant n$, soient $\left(\|.\|^{(k)}_{i,1}\right)_{k \in \M{N}}, \dots, \left(\|.\|^{(k)}_{i,r_{i}}\right)_{k \in \M{N}}$ des suites croissantes de m\'etriques positives $C^{\infty}$ sur $E_{i,1},\dots,E_{i,r_{i}}$ convergeant vers $\|.\|_{i,1},\dots,\|.\|_{i,r_{i}}$ respectivement. On choisit $(Z',g')$, $(Z_{1},g_{1}), \dots, (Z_{n},g_{n})$ des repr\'esentants des classes $\beta, \alpha_{1}, \dots,$ $\alpha_{n}$ respectivement, tels que $Z_{K}'$ soit d'intersection propre avec chacun des $(Z_{1})_{K},$ $\dots, (Z_{n})_{K}$, et tels que $g',g_{1},\dots,g_{n}$ soient des courants de type logarithmique. Pour tout $1 \leqslant i \leqslant n$, on note $Z'\cdot Z_{i}$ un repr\'esentant dans $Z^{p+q-r_{i}}(|Z'|\cap |Z_{i}|)_{\M{Q}}$ du produit $[Z']\cdot [Z_{i}] \in CH_{|Z'|\cap|Z_{i}|}^{p+q-r_{i}}(X)_{\M{Q}}$; et on choisit $s_{i,1},\dots,s_{i,r_{i}}$ des sections rationnelles non identiquement nulles sur $X$ de $E_{i,1},\dots,E_{i,r_{i}}$ respectivement et $s_{1},\dots,s_{s}$ des sections rationnelles non identiquement nulles sur $X$ de $F_{1},\dots,F_{s}$ respectivement telles que les cycles $Z_{i},\op{div}s_{i,1},\dots,\op{div}s_{i,r_{i}}, \op{div}s_{1},\dots,\op{div}s_{s}$ ainsi que les cycles $(Z'\cdot Z_{i}), \op{div}s_{i,1},\dots,\op{div}s_{i,r_{i}}, \op{div}s_{1},\dots,\op{div}s_{s}$ et $Z_{K}, Z_{K}', (\op{div}s_{i,1})_{K}, \dots,\linebreak[4] (\op{div}s_{i,r_{i}})_{K}, (\op{div}s_{1})_{K},\dots,(\op{div}s_{s})_{K}$ s'intersectent proprement. \\ On convient de noter $g_{i,1} = -\log \|s_{i,1}\|_{i,1}^{2}, \dots, g_{i,r_{i}} = - \log\|s_{i,r_{i}}\|^{2}_{i,r_{i}}$ et $g_{1} = - \log\|s_{1}\|_{1}^{2},\dots,g_{s} = - \log \|s_{s}\|_{s}^{2}$, et pour tout $k \in \M{N}$, $g_{i,1}^{(k)} = - \log \left(\|s_{i,1}\|_{i,1}^{(k)}\right)^{2}, \dots\linebreak[4], g_{i,r_{i}}^{(k)} = - \log \left(\|s_{i,r_{i}}\|_{i,r_{i}}^{(k)}\right)^{2}$ et $g_{1}^{(k)} = - \log \left(\|s_{1}\|_{1}^{(k)}\right)^{2}, \dots, g_{s}^{(k)} = - \log \left( \|s_{s}\|_{s}^{(k)}\right)^{2}$. D'apr\`es la d\'efinition (\ref{definition_prod2}) et (\cite{13}, \S 4.2.1 et 4.2.2), on a~: \begin{equation} \label{eq_prod1} x = \sum_{i = 1}^{n} [(\, Z_{i}\cdot \op{div} s_{i,1} \dotsm \op{div} s_{i,r_{i}}, \{g_{i}\ast (g_{i,1},s_{i,1}) \ast \dotsm \ast (g_{i,r_{i}},s_{i,r_{i}})\} \,)] \end{equation} et \begin{multline} \label{eq_prod2} x\cdot y = {\sum_{i=1}^{n} [(\, (Z'\cdot Z_{i})\cdot \op{div} s_{i,1}\dotsm \op{div} s_{i,r_{i}} \cdot \op{div} s_{1} \dotsm \op{div} s_{s}, }\\ \{(g'\ast g_{i})\ast (g_{i,1},s_{i,1})\ast \dotsm \ast (g_{i,r_{i}},s_{i,r_{i}})\ast (g_{1},s_{1})\ast \dotsm \ast (g_{s},s_{s}) \}\, )]. \end{multline} Puisque $\zeta(x) = \sum_{i=1}^{n} [Z_{i}\cdot \op{div} s_{i,1} \dotsm \op{div} s_{i,r_{i}}] = 0$, on peut trouver une $K_{1}$-chaine $f$ telle que~: \begin{equation} \label{eq_prod3} \sum_{i=1}^{n}Z_{i}\cdot \op{div} s_{i,1} \dotsm \op{div} s_{i,r_{i}} = \op{div} f. \end{equation} D'apr\`es le lemme de d\'eplacement pour les $K_{1}$-chaines (cf. \cite{13}, \S 4.2.6), on peut de plus supposer que $f$ rencontre $Z' \cdot \op{div} s_{1}\dotsm \op{div} s_{s}$ presque proprement dans $X_{K}$. Cela entra\^\i ne que d'une part~: \begin{multline} \label{eq_prod4} \sum_{i=1}^{n} (Z_{i}\cdot \op{div} s_{i,1} \dotsm \op{div} s_{i,r_{i}}, \{g_{i}\ast (g_{i,1},s_{i,1})\ast \dotsm \ast(g_{i,r_{i}},s_{i,r_{i}})\} )\\ = \op{\widehat{\op{div}}}f + (0,\gamma_{1} + \gamma_{2}), \end{multline} o\`u l'on a not\'e~: \[ \gamma_{1} = \sum_{i=1}^{n} \{g_{i}\ast (g_{i,1},s_{i,1}) \ast \dotsm \ast(g_{i,r_{i}},s_{i,r_{i}})\}, \qquad \gamma_{2} = \log |f|^{2}; \] et que d'autre part, d'apr\`es (\cite{13},\S 4.2.1 et 4.2.5), \begin{multline} \label{eq_prod5} \sum_{i=1}^{n} (\,(Z'\cdot Z_{i})\cdot \op{div} s_{i,1} \dotsm \op{div} s_{i,r_{i}} \cdot \op{div} s_{1}\dotsm \op{div} s_{s}, \\ \shoveright{\{(g'\ast g_{i})\ast (g_{i,1},s_{i,1})\ast \dotsm \ast (g_{i,r_{i}},s_{i,r_{i}}) \ast (g_{1},s_{1})\ast \dotsm \ast (g_{s},s_{s})\})}\\ = \op{\widehat{\op{div}}}(f\cdot (Z' \cdot \op{div} s_{1}\dotsm \op{div} s_{s})) + (0,\gamma') + (R,0), \quad \end{multline} o\`u l'on a not\'e~: \begin{multline} \gamma' = \sum_{i=1}^{n}\{ (g'\ast g_{i})\ast (g_{i,1},s_{i,1}) \ast \dotsm \ast (g_{i,r_{i}},s_{i,r_{i}})\ast (g_{1},s_{1})\ast \dotsm \ast (g_{s},s_{s})\} \\ + \log |f\cdot (Z'\cdot \op{div} s_{1}\dotsm \op{div} s_{s})|^{2}, \end{multline} o\`u $R \in R_{\op{fin}}^{p+q}(X)$ et o\`u $f\cdot (Z'\cdot \op{div} s_{1}\dotsm \op{div} s_{s})$ d\'esigne une $K_{1}$-chaine repr\'esentant l'intersection de la $K_{1}$-chaine $f$ et du cycle $Z'\cdot \op{div} s_{1}\dotsm \op{div} s_{s}$ (bien que $f\cdot (Z'\cdot \op{div} s_{1}\dotsm \op{div} s_{s})$ ne soit pas d\'efinie de mani\`ere univoque, les cycles $\op{div} (f\cdot (Z'\cdot \op{div} s_{1}\dotsm \op{div} s_{s}))$ et $\op{\widehat{\op{div}}}(f\cdot (Z'\cdot \op{div} s_{1}\dotsm \op{div} s_{s}))$ le sont, voir \cite{13}, \S 4.2.5). D'apr\`es (\ref{eq_prod3}) et (\cite{13}, \S 2.2.9 et 4.2.7), on a pour tout $k \in \M{N}$~: \begin{multline*} \sum_{i=1}^{n}(g' \ast g_{1}^{(k)} \ast \dotsm \ast g_{s}^{(k)}) \ast (g_{i}\ast g_{i,1}^{(k)}\ast \dotsm \ast g_{i,r_{i}}^{(k)}) + \log |f\cdot (Z'\cdot \op{div} s_{1}\dotsm \op{div} s_{s})|^{2} \\ \shoveleft{= (g'\ast g_{1}^{(k)} \ast \dotsm \ast g_{s}^{(k)})\cdot \delta_{\op{div} f}} \\ + \omega(\beta) \, c_{1}\big(F_{1},\|.\|_{1}^{(k)}\big) \dotsm c_{1}\big(F_{s},\|.\|_{s}^{(k)}\big)\cdot \left( \sum_{i=1}^{n}g_{i}\ast g_{i,1}^{(k)}\ast \dotsm \ast g_{i,r_{i}}^{(k)}\right) \\ \shoveright{ + \log |f\cdot (Z'\cdot \op{div} s_{1}\dotsm \op{div} s_{s})|^{2}} \\ \shoveleft{= \Big((g'\ast g_{1}^{(k)} \ast \dotsm \ast g_{s}^{(k)}) \cdot \delta_{\op{div} f} } - \omega(\beta)\, c_{1}\big(F_{1},\|.\|_{1}^{(k)}\big) \dotsm c_{1}\big(F_{s},\|.\|_{s}^{(k)}\big)\log|f|^{2} \\ \shoveright{+ \log |f\cdot (Z'\cdot \op{div} s_{1}\dotsm \op{div} s_{s})|^{2}\Big) }\\ \shoveright{+ \omega(\beta)\, c_{1}\big(F_{1},\|.\|_{1}^{(k)}\big) \dotsm c_{1}\big(F_{s},\|.\|_{s}^{(k)}\big) \left( \sum_{i=1}^{n}g_{i}\ast g_{i,1}^{(k)}\ast \dotsm \ast g_{i,r_{i}}^{(k)} + \log|f|^{2}\right)\;\,} \\ \shoveleft{= \omega(\beta)\, c_{1}\big(F_{1},\|.\|_{1}^{(k)}\big) \dotsm c_{1}\big(F_{s},\|.\|_{s}^{(k)}\big) \left( \sum_{i=1}^{n}g_{i}\ast g_{i,1}^{(k)}\ast \dotsm \ast g_{i,r_{i}}^{(k)} + \log|f|^{2}\right),} \end{multline*} dans $\widetilde{D}^{p+q-1,p+q-1}(X_{\M{R}})$, ce qui, en faisant tendre $k$ vers $+\infty$, entra\^\i ne que~: \[ \gamma' = \gamma_{1} \,\omega(y) +\gamma_{2} \,\omega(y) , \] dans $\widetilde{D}^{p+q-1,p+q-1}(X_{\M{R}})$ (une telle expression a bien un sens car $\omega(y) \in C_{0}^{q-1,q-1}(X_{\M{R}})$). De l'\'egalit\'e $x = [(0,\gamma_{1} + \gamma_{2})] = [(0,\varphi)]$ obtenue en combinant (\ref{eq_prod1}) et (\ref{eq_prod4}), on d\'eduit qu'il existe une $K_{1}$-chaine $g$, que l'on choisit d'intersection presque propre avec $Z' \cdot \op{div} s_{1}\dotsm \op{div} s_{s}$ dans $X_{K}$, telle que $\op{div} g = 0$ et $\gamma_{1} + \gamma_{2} - \varphi = - \log |g|^{2}$ dans $\widetilde{D}^{p-1,p-1}(X_{\M{R}})$. On tire de ce qui pr\'ec\`ede et de (\cite{13}, \S 4.2.5 et 4.2.7) la relation~: \begin{multline*} - \log |g\cdot (Z'\cdot \op{div}s_{1}\dotsm \op{div}s_{s})|^{2}\\ = (\gamma_{1} + \gamma_{2} - \varphi)\,\omega(\beta) \, c_{1}\big(F_{1},\|.\|_{1}^{(k)}\big) \dotsm c_{1}\big(F_{s},\|.\|_{s}^{(k)}\big), \end{multline*} valable pour tout $k\in \M{N}$; ce qui montre, en faisant tendre $k$ vers $+\infty$, que~: \[ \gamma_{1}\,\omega(y) + \gamma_{2}\,\omega(y) - \varphi\, \omega(y) = - \log |g\cdot (Z'\cdot \op{div}s_{1}\dotsm \op{div}s_{s})|^{2}, \] dans $\widetilde{D}^{p+q-1,p+q-1}(X_{\M{R}})$. Comme de plus $\op{div}(g\cdot (Z'\cdot \op{div}s_{1}\dotsm \op{div}s_{s}))_{K} = (\op{div} g)_{K}\cdot (Z'\cdot \op{div}s_{1}\dotsm \op{div}s_{s})_{K} = 0$, on peut affirmer que $[(0,\gamma')] = [(0,\gamma_{1}\,\omega(y) + \gamma_{2}\,\omega(y))] = [(0,\varphi\,\omega(y))]$, ce qui combin\'e \`a (\ref{eq_prod2}) et (\ref{eq_prod5}) montre que $x\cdot y = a(\varphi\,\omega(y))$. On a donc d\'emontr\'e que l'accouplement (\ref{accouplement_1}) est bien d\'efini. Les autres propri\'et\'es \'enonc\'ees se d\'eduisent ais\'ement du th\'eor\`eme (\ref{th_general}) et des propri\'et\'es analogues pour $\widehat{CH}^{\ast}(X)$. \medskip Au cours de la d\'emonstration pr\'ec\'edente, on a de plus prouv\'e le r\'esultat suivant~: \begin{prop} \label{produit_generalise_fin} Soient $x \in \widehat{CH}_{\op{int}}^{\ast}(X)$ et $\varphi \in C_{\log,0}^{\ast}(X_{\M{R}})$ tel que $a(\varphi) \in \widehat{CH}_{\op{int}}^{\ast}(X)$, on a~: \[ a(\varphi)\cdot x = a(\varphi\,\omega(x)). \] \end{prop} \begin{rem} Soient $x$ et $y$ deux \'el\'ements de $\widehat{CH}_{\op{gen}}^{\ast}(X)$, et soient $\alpha$, $\alpha' \in \widehat{CH}^{\ast}(X)$ et $\varphi$, $\varphi' \in C_{\log,0}^{\ast}(X_{\M{R}})$ tels que $x = \alpha + a (\varphi)$ et $y = \alpha' + a(\varphi')$. On d\'efinit le produit de $x$ et de $y$ par la formule~: \[ x\cdot y = \alpha\cdot \alpha' + a(\varphi\,\omega(\alpha') + \varphi'\,\omega(\alpha) + \varphi\,dd^{c} \varphi') \in \widehat{CH}_{\op{gen}}^{\ast}(X)_{\M{Q}}. \] On peut montrer que ce produit est bien d\'efini et qu'il munit $\widehat{CH}_{\op{gen}}^{\ast}(X)_{\M{Q}}$ d'une structure d'anneau commutatif, associatif et unif\`ere. D'apr\`es la proposition (\ref{produit_generalise_fin}), il co\"\i ncide sur $\widehat{CH}_{\op{int}}^{\ast}(X)_{\M{Q}}$ avec le produit d\'efini au th\'eor\`eme (\ref{produit_bien_defini}). \end{rem} \begin{thm} Soit $\pi: X \rightarrow \op{Spec}\C{O}_{K}$ une vari\'et\'e arithm\'etique. Les morphismes~: \begin{align*} \zeta&: \widehat{CH}_{\op{int}}^{\ast}(X)_{\M{Q}} \longrightarrow CH^{\ast}_{\M{Q}}(X) \\ \intertext{et} \omega&: \widehat{CH}_{\op{int}}^{\ast}(X)_{\M{Q}} \longrightarrow \ov{A}_{\op{g}}^{\ast}(X_{\M{R}}), \end{align*} d\'efinis \`a la section (\ref{sous_section_def}) sont des morphismes d'anneaux. De plus, le produit d\'efini \`a la remarque (\ref{rem_acc_1}) et (dans le cas o\`u $\pi: X \rightarrow \op{Spec}\C{O}_{K}$ est lisse) celui d\'efini \`a la remarque (\ref{rem_acc_2}) sont compatibles avec les morphismes $\zeta$ et $\omega$. \end{thm} \noindent {\bf D\'emonstration.}\ C'est une simple cons\'equence du th\'eor\`eme (\ref{th_general}), alin\'ea (5) et des propri\'et\'es analogues pour $\widehat{CH}^{\ast}(X)$. \medskip \begin{rem} Soient $\varphi \in \ov{A}_{\op{g}}^{\ast}(X_{\M{R}})$ et $x \in \widehat{CH}_{\op{int}}^{\ast}(X)$. On d\'eduit ais\'ement de la d\'efinition de $\ov{A}_{\op{g}}^{\ast}(X_{\M{R}})$ et des alin\'eas (5) et (6) du th\'eor\`eme (\ref{th_general}) la formule utile suivante~: \[ a(\varphi)\cdot x = a(\varphi\, \omega(x)). \] \end{rem} \medskip \subsection{Degr\'e arithm\'etique et hauteurs}~ \subsubsection{Degr\'e arithm\'etique}~ \label{sous_section_degre} Soit $\pi: X \rightarrow \op{Spec}\C{O}_{K} = S$ une vari\'et\'e arithm\'etique (projective) de dimension relative $d$. On rappelle (voir par exemple \cite{3}, \S 2.1.3) que l'on dispose des deux morphismes~: \[ \op{deg}_{K} : \widehat{CH}^{0}(S) = CH^{0}(S) \longrightarrow \M{Z} \] et \[ \widehat{\op{deg}} : \widehat{CH}^{1}(S) \longrightarrow \M{R}, \] qui induisent par composition avec $\pi_{\ast}: \widehat{CH}^{\ast}(X) \rightarrow \widehat{CH}^{\ast -d}(S)$ les morphismes~: \[ \op{deg}_{K} : \widehat{CH}^{d}(X) \longrightarrow \M{Z} \qquad \text{(degr\'e g\'eom\'etrique)} \] et \[ \widehat{\op{deg}} : \widehat{CH}^{d+1}(X) \longrightarrow \M{R} \qquad \text{(degr\'e arithm\'etique)}. \] Soit $i \in \{ 0,1\}$. Pour toute classe $\alpha \in \widehat{CH}_{\op{int}}^{d+i}(X)$, choisissons $(Z,g)$ un repr\'esentant de $\alpha$; on d\'eduit de (\cite{13}, \S 3.6) que la classe $[(\pi_{\ast}(Z),\pi_{\ast}(g))]$ ne d\'epend que de $\alpha$. On dispose donc d'un morphisme, encore not\'e $\pi_{\ast}$, qui est d\'efini comme suit~: \begin{alignat*}{3} \pi_{\ast}: \;&\widehat{CH}_{\op{int}}^{d+i}(X)& &\longrightarrow &&\;\widehat{CH}^{i}(S) \\ &\;\;[(Z,g)]&&\longmapsto &&[(\pi_{\ast}(Z),\pi_{\ast}(g))], \end{alignat*} et qui prolonge \`a $\widehat{CH}_{\op{int}}^{d+i}(X)$ le morphisme image directe usuel $\pi_{\ast}: \widehat{CH}^{d+i}(X) \rightarrow \widehat{CH}^{i}(S)$. En composant $\op{deg}_{K}$ et $\widehat{\op{deg}}$ avec $\pi_{\ast}$, on obtient deux nouveaux morphismes~: \[ \op{deg}_{K} : \widehat{CH}_{\op{int}}^{d}(X) \longrightarrow \M{Z} \] et \[ \widehat{\op{deg}} : \widehat{CH}_{\op{int}}^{d+1}(X) \longrightarrow \M{R}, \] qui prolongent ceux d\'efinis pr\'ec\'edemment. \subsubsection{Hauteurs}~ Soient $Z \in Z_{q}(X)$ un cycle de dimension $q$ et $\ov{L}_{1} = (L_{1}, \|.\|_{1}), \dots, \ov{L}_{q} = (L_{q},\|.\|_{q})$ des fibr\'es en droites admissibles sur $X$. Choisissons $s_{1},\dots,s_{q}$ des sections rationnelles non identiquement nulles sur $X$ de $L_{1},\dots,L_{q}$ respectivement, telles que les cycles $Z,\op{div} s_{1}, \dots, \op{div} s_{q}$ soient d'intersection propre. Pour tout $1 \leqslant i \leqslant q$, on note $g_{i} = - \log \|s_{i}\|_{i}^{2}$ et $\omega_{i} = c_{1}(\ov{L}_{i})$. On d\'efinit \`a partir de ces donn\'ees un \'el\'ement de $D^{p,p}(X_{\M{R}})$ que l'on note $\{(g_{1},s_{1})\ast \dotsm \ast (g_{q},s_{q})|\delta_{Z}\}$ par la formule suivante~: \begin{multline} \label{int_fin_eq1} \{(g_{1},s_{1})\ast \dotsm \ast (g_{q},s_{q})|\delta_{Z}\} = g_{1}\cdot \delta_{\op{div} s_{2} \cap \dotsm \cap \op{div} s_{q} \cap Z} + \omega_{1}g_{2}\cdot \delta_{\op{div} s_{3}\cap \dotsm \cap \op{div} s_{q} \cap Z}\\ + \dots + \omega_{1}\dotsm \omega_{i-1}g_{i}\cdot \delta_{\op{div} s_{i+1}\cap \dotsm \cap \op{div} s_{q}\cap Z} + \dots + \omega_{1} \dotsm \omega_{q-1}g_{q}\cdot \delta_{Z}. \end{multline} On v\'erifie que chacun des termes de la relation (\ref{int_fin_eq1}) est bien d\'efini. En effet $\omega_{1}\dotsm \omega_{i-1}\cdot \delta_{\op{div} s_{i+1} \cap \dotsm \cap \op{div} s_{q} \cap Z}$ est bien d\'efini comme le produit de $\omega_{1}\dotsm \omega_{i-1} \in B^{i-1,i-1}(X_{\M{R}})$ et de $\delta_{\op{div} s_{i+1} \cap \dotsm \cap \op{div} s_{q} \cap Z} \in B^{d-i+1,d-i+1}_{\op{div} s_{i+1} \cap \dotsm \cap \op{div} s_{q} \cap Z}(X_{\M{R}})$ d'apr\`es les propositions (\ref{produit_uniforme}) et (\ref{produit_uniforme2}). Par ailleurs le produit~: \[ \gamma_{i} = g_{i}\,\omega_{1}\dotsm \omega_{i-1}\cdot \delta_{\op{div} s_{i+1}\cap \dotsm \cap \op{div} s_{q}\cap Z}, \] a bien un sens d'apr\`es le th\'eor\`eme (\ref{demailly}); et si l'on pose $g_{i}^{(t)} = \op{max} (g_{i}, t)$ pour tout $t \in \M{R}$, la limite~: \[ \lim_{t \rightarrow - \infty} g_{i}^{(t)}\omega_{1}\dotsm \omega_{i-1}\cdot \delta_{\op{div} s_{i+1} \cap \dotsm \cap \op{div} s_{q} \cap Z} = g_{i} \,\omega_{1}\dotsm \omega_{i-1}\cdot \delta_{\op{div} s_{i+1} \cap \dotsm \cap \op{div} s_{q} \cap Z}, \] montre que le courant $\gamma_{i}$ ne d\'epend pas des choix effectu\'es pour le d\'efinir. Ceci \'etant \'etabli, on pose~: \[ h_{\ov{L}_{1}, \dots,\ov{L}_{q}}(Z) := \widehat{\op{deg}} (\pi_{\ast}(Z\cdot \op{div} s_{1} \dotsm \op{div} s_{q}), \pi_{\ast}(\{(g_{1},s_{1})\ast \dotsm \ast (g_{q},s_{q})|\delta_{Z}\})) \in \M{R}. \] \begin{prop_defn} \label{int_fin_prop1} Le nombre r\'eel $h_{\ov{L}_{1},\dots,\ov{L}_{q}}(Z)$ d\'efini ci-dessus ne d\'epend pas du choix des sections $s_{1},\dots,s_{q}$; on l'appelle {\rm hauteur de $Z$ relativement \`a $\ov{L}_{1},\dots,\ov{L}_{q}$\/}. \end{prop_defn} \begin{prop} \label{int_fin_prop2} Soient $Z \in Z_{q}(X)$ et $\ov{L}_{0}, \ov{L}_{1} = (L_{1}, \|.\|_{1}), \dots, \ov{L}_{q} = (L_{q},\|.\|_{q})$ des fibr\'es admissibles sur $X$. Les assertions suivantes sont v\'erifi\'ees~: \begin{enumerate} \item{La hauteur $h_{\ov{L}_{1}, \dots, \ov{L}_{q}}(Z)$ ne d\'epend que du cycle $Z$ et des classes d'isomorphie isom\'etrique des fibr\'es admissibles $\ov{L}_{1},\dots,\ov{L}_{q}$; elle ne d\'epend pas de l'ordre des fibr\'es $\ov{L}_{1},\dots,\ov{L}_{q}$.} \item{Soient $\Big(\|.\|_{1}^{(k)}\Big)_{k \in \M{N}}, \dots, \Big(\|.\|_{q}^{(k)}\Big)_{k \in \M{N}}$ des suites croissantes de m\'etriques positives convergeant vers $\|.\|_{1}, \dots,\|.\|_{q}$ sur $L_{1},\dots,L_{q}$ respectivement, on a~: \[ \lim_{k \rightarrow + \infty} h_{\big(L_{1},\|.\|_{1}^{(k)}\big), \dots, \big(L_{q},\|.\|_{q}^{(k)}\big)} (Z) = h_{\ov{L}_{1},\dots,\ov{L}_{q}}(Z). \] } \item{Si les m\'etriques $\|.\|_{1}, \dots, \|.\|_{q}$ sont $C^{\infty}$, alors on a~: \[ h_{\ov{L}_{1}, \dots, \ov{L}_{q}}(Z) = \widehat{\op{deg}} (\hat{c}_{1}(\ov{L}_{1})\dotsm \hat{c}_{1}(\ov{L}_{q})|Z), \] o\`u $(\cdot |\cdot)$ d\'esigne l'accouplement d\'efini dans (\cite{3}, \S 2.3).} \item{Pour tout $1 \leqslant i \leqslant q$, on a~: \[ h_{\ov{L}_{1}, \dots,\ov{L}_{i-1},\ov{L}_{i}\otimes \ov{L}_{0}, \ov{L}_{i+1}, \dots,\ov{L}_{q}}(Z) = h_{\ov{L}_{1},\dots,\ov{L}_{q}}(Z) + h_{\ov{L}_{1}, \dots, \ov{L}_{i-1}, \ov{L}_{0}, \ov{L}_{i+1}, \dots, \ov{L}_{q}}(Z). \] } \end{enumerate} \end{prop} {\bf D\'emonstration des propositions (\ref{int_fin_prop1}) et (\ref{int_fin_prop2}).} La proposition (\ref{int_fin_prop1}) et les assertions (1) et (4) de la proposition (\ref{int_fin_prop2}) se d\'eduisent imm\'ediatement des assertions (2) et (3) de la proposition (\ref{int_fin_prop2}) et des propri\'et\'es analogues dans le cas classique (cf. \cite{3}, \S 2.3). L'assertion (2) \'etant une cons\'equence du th\'eor\`eme (\ref{demailly}), il suffit de prouver l'assertion (3). En remarquant que $g_{1} \ast (g_{2}\ast (\dotsm \ast(g_{q})))$ est un courant de Green pour le cycle $\op{div} s_{1} \dotsm \op{div} s_{q}$, on d\'eduit de (\cite{13}, \S 1.2.4 et 1.3.5) qu'il existe $g$ un courant de Green de type logarithmique pour $\op{div} s_{1} \dotsm \op{div} s_{q}$ tel que~: \begin{equation} \label{int_fin_eq2} g = g_{1} \ast (g_{2}\ast (\dotsm \ast(g_{q}))) + \partial u + \ov{\partial}v. \end{equation} Pour tout $\varepsilon \in \M{R}^{+\ast}$, soit $\delta_{Z}^{(\varepsilon)}$ une r\'egularisation de $\delta_{Z}$ comme \`a la proposition (\ref{construction_GS_lissage}). En multipliant les deux membres de l'\'egalit\'e (\ref{int_fin_eq2}) par $\delta_{Z}^{(\varepsilon)}$, il vient~: \begin{equation} \label{int_fin_eq3} g \wedge \delta_{Z}^{(\varepsilon)} = g_{1} \ast (g_{2}\ast (\dotsm \ast(g_{q}))) \wedge \delta_{Z}^{(\varepsilon)} + \partial (u \delta_{Z}^{(\varepsilon)}) + \ov{\partial}(v \delta_{Z}^{(\varepsilon)}). \end{equation} Comme d'une part~: \begin{multline*} g_{1} \ast (g_{2}\ast (\dotsm \ast(g_{q}))) \\ = g_{1} \cdot \delta_{\op{div} s_{2} \cap \dotsm \cap \op{div} s_{q}} + \omega_{1} g_{2} \cdot \delta_{\op{div} s_{3} \cap \dotsm \cap \op{div} s_{q}} + \dots + \omega_{1} \dotsm \omega_{q-1}g_{q}, \end{multline*} et que d'autre part, d'apr\`es (\cite{13}, \S 2.2.12), on a les limites~: \[ \lim_{\varepsilon \rightarrow 0} g \wedge \delta_{Z}^{(\varepsilon)} = g \wedge \delta_{Z}, \] et pour tout $1 \leqslant i \leqslant q$, \[ \lim_{\varepsilon \rightarrow 0} (\omega_{1}\dotsm \omega_{i-1})g_{i}\cdot \delta_{\op{div} s_{i+1} \cap \dotsm \cap \op{div} s_{q}}\wedge \delta_{Z}^{(\varepsilon)} = (\omega_{1}\dotsm \omega_{i-1})g_{i}\cdot \delta_{\op{div} s_{i+1} \cap \dotsm \cap \op{div} s_{q}}\wedge \delta_{Z}, \] au sens de la convergence faible des courants, on tire de (\ref{int_fin_eq3}) l'\'egalit\'e~: \begin{multline*} g \wedge \delta_{Z} = \\ g_{1} \cdot \delta_{\op{div} s_{2} \cap \dotsm \cap \op{div} s_{q} \cap Z} + \omega_{1} g_{2} \cdot \delta_{\op{div} s_{3}\cap \dotsm \cap \op{div} s_{q} \cap Z} + \dots + \omega_{1}\dotsm \omega_{q-1}g_{q}\delta_{Z} \\ = \{(g_{1},s_{1})\ast \dotsm \ast (g_{q},s_{q})|\delta_{Z}\}, \end{multline*} dans $\widetilde{D}^{d,d}(X_{\M{R}})$, ce qui termine la d\'emonstration. \medskip Plus g\'en\'eralement, soient $Z \in Z_{q}(X)$ et $\ov{L}_{1}, \dots,\ov{L}_{q}$ des fibr\'es int\'egrables sur $X$. Choisissons $\ov{E}_{1}, \dots,\ov{E}_{q}$ et $\ov{F}_{1}, \dots,\ov{F}_{q}$ des fibr\'es en droites admissibles sur $X$ tels que pour tout $ 1 \leqslant i \leqslant q$, on ait~: $\ov{L}_{i} = \ov{E}_{i}\otimes (\ov{F}_{i})^{-1}$. \begin{prop_defn} \label{int_fin_prop3} On appelle {\rm hauteur de $Z$ relativement \`a $\ov{L}_{1}, \dots, \ov{L}_{q}$} et on note $h_{\ov{L}_{1}, \dots,\ov{L}_{q}}(Z)$ le nombre r\'eel d\'efini par la formule~: \[ h_{\ov{L}_{1},\dots,\ov{L}_{q}}(Z) = \sum _{\substack{S_{1}, S_{2} \\ S_{1} \cup S_{2} = \{1,\dots,q\}}} (-1)^{\# S_{2}}h_{ \{\ov{E}_{i}\}_{i \in S_{1}}, \{\ov{F}_{j}\}_{j \in S_{2}}}(Z). \] La hauteur $h_{\ov{L}_{1},\dots,\ov{L}_{q}}(Z)$ ainsi d\'efinie ne d\'epend que de $Z$ et des classes d'isomorphie isom\'etrique des fibr\'es $\ov{L}_{1},\dots,\ov{L}_{q}$. Si les fibr\'es $\ov{L}_{1},\dots,\ov{L}_{q}$ sont admissibles, elle co\"\i ncide avec la hauteur d\'efinie \`a la proposition (\ref{int_fin_prop1}). \end{prop_defn} \noindent {\bf D\'emonstration.}\ On suit {\it mutatis mutandis\/} la d\'emonstration de la proposition (\ref{produit_general2}) en utilisant la proposition (\ref{int_fin_prop2}). \medskip Lorsque $\ov{L} := \ov{L}_{1} = \dots = \ov{L}_{q}$, on convient de noter $h_{\ov{L}}(Z)$ le nombre $h_{\ov{L}_{1},\dots,\ov{L}_{q}}(Z)$ que l'on appelle alors {\it hauteur de $Z$ relativement \`a $\ov{L}$}. \begin{rem} Si $q=0$, la hauteur $h_{\ov{L}}(Z)$ n'est autre que $h(Z) = \widehat{\op{deg}}[(Z,0)]$. Dans ce cas, la fonction $h: Z_{0}(X) \rightarrow \M{R}$ est d\'efinie par $h(P) = \log \# k(P)$. \end{rem} \begin{rem} Ces d\'efinitions ont \'et\'e introduites pour la premi\`ere fois dans cette g\'en\'eralit\'e par Zhang (cf. \cite{21}, th. 1.4) sous une forme diff\'erente. \end{rem} Le th\'eor\`eme suivant rassemble diverses propri\'et\'es de la hauteur introduite \`a la proposition (\ref{int_fin_prop3})~: \begin{thm} \label{gdthm} Soient $Z \in Z_{q}(X)$ et $\ov{L}_{1},\dots, \ov{L}_{q-1},\ov{L}_{q} = (L_{q},\|.\|_{q})$ des fibr\'es en droites int\'egrables sur $X$. On a les propri\'et\'es suivantes~: \begin{enumerate} \item{La hauteur $h_{\ov{L}_{1},\dots,\ov{L}_{q}}(Z)$ ne d\'epend pas de l'ordre de $\ov{L}_{1},\dots,\ov{L}_{q}$.} \item{L'application qui \`a $Z \in Z_{q}(X)$ et $\ov{L}_{1},\dots,\ov{L}_{q}$ des fibr\'es en droites int\'egrables sur $X$ associe $h_{\ov{L}_{1},\dots, \ov{L}_{q}}(Z) \in \M{R}$ d\'efinit une forme multilin\'eaire~: \[ Z_{q}(X) \times \widehat{\op{Pic}}_{\op{int}}(X) \times \dotsm \times \widehat{\op{Pic}}_{\op{int}}(X) \longrightarrow \M{R}. \] } \item{Si les m\'etriques des fibr\'es $\ov{L}_{1},\dots,\ov{L}_{q}$ sont $C^{\infty}$, alors on a~: \[ h_{\ov{L}_{1},\dots,\ov{L}_{q}}(Z) = \widehat{\op{deg}}(\hat{c}_{1}(\ov{L}_{1}) \dotsm \hat{c}_{1}(\ov{L}_{q})|Z), \] o\`u $(\cdot | \cdot )$ d\'esigne l'accouplement d\'efini dans (\cite{3}, \S 2.3).} \item{Si $Z$ est le diviseur d'une fonction rationnelle sur un sous-sch\'ema int\`egre contenu dans une fibre ferm\'ee de $X$, alors $h_{\ov{L}_{1},\dots,\ov{L}_{q}}(Z) = 0$.} \item{Pour tout morphisme $f: X \rightarrow X'$ de vari\'et\'es arithm\'etiques, on a~: \[ h_{f^{\ast}(\ov{L}_{1}), \dots , f^{\ast}(\ov{L}_{q})}(Z) = h_{\ov{L}_{1},\dots,\ov{L}_{q}}(f_{\ast}(Z)). \] } \item{Soit $s_{q}$ une section rationnelle de $L_{q}$ au-dessus de $Z$ qui n'est identiquement nulle sur aucune des composantes irr\'eductibles de $Z$. On a la relation~: \[ h_{\ov{L}_{1},\dots,\ov{L}_{q-1}}(Z\cdot \op{div} s_{q}) = h_{\ov{L}_{1},\dots,\ov{L}_{q}}(Z) + \int_{X(\M{C})}\log \|s_{q}\|_{q}\, c_{1}(\ov{L}_{1})\dotsm c_{1}(\ov{L}_{q-1})\cdot \delta_{Z}. \] } \item{On suppose que $Z = X$, et donc que $q = d+1$. On a~: \[ h_{\ov{L}_{1},\dots,\ov{L}_{d+1}}(X) = \widehat{\op{deg}}( \hat{c}_{1}(\ov{L}_{1})\dotsm \hat{c}_{1}(\ov{L}_{d+1})), \] o\`u $\widehat{\op{deg}}$ d\'esigne le degr\'e arithm\'etique sur $\widehat{CH}_{\op{int}}^{d+1}(X)$ introduit au \S \ref{sous_section_degre}.} \end{enumerate} \end{thm} \noindent {\bf D\'emonstration.}\ Par (multi)lin\'earit\'e on se ram\`ene au cas o\`u $\ov{L}_{1},\dots,\ov{L}_{q}$ sont des fibr\'es en droites admissibles. Les assertions (1) \`a (3) sont alors une cons\'equence imm\'ediate de la proposition (\ref{int_fin_prop2}), et les assertions (4) \`a (6) se d\'eduisent directement de (\ref{int_fin_prop2}) alin\'ea (2) et des assertions analogues dans le cas classique (cf. \cite{3}, prop. 2.3.1 et 3.2.1). Enfin l'assertion (7) est une cons\'equence des propositions (\ref{approximation2}) et (\ref{naturalite2}), et des d\'efinitions. \medskip La proposition suivante \'etend \`a notre cadre un \'enonc\'e de Faltings (cf. \cite{32}, prop. 2.6) g\'en\'eralis\'e dans (\cite{3}, \S 3.2.3)~: \begin{prop}{\rm \bf (Positivit\'e).} Soient $\ov{L}_{1},\dots,\ov{L}_{q}$ des fibr\'es en droites {\em admissibles} sur $X$ tels que pour tout $1 \leqslant i \leqslant q$, il existe une puissance tensorielle positive $\ov{L}_{i}^{n_{i}}$ de $\ov{L}_{i}$ engendr\'ee par ses sections globales de norme sup inf\'erieure ou \'egale \`a $1$. Pour tout cycle {\em effectif} $Z \in Z_{q}(X)$, on a~: \begin{equation} \label{int_eq_6} h_{\ov{L}_{1},\dots,\ov{L}_{q}}(Z) \geqslant 0. \end{equation} \label{positivite_arithmetique} \end{prop} \noindent {\bf D\'emonstration.}\ On raisonne par r\'ecurrence sur la dimension de $Z$. Si $\op{dim}Z = 0$, le calcul se fait aux places finies et il n'y a pas de changement avec la situation classique (voir par exemple \cite{3}, prop. 3.2.4). On se place d\'esormais dans le cas o\`u $\op{dim} Z > 0$. On peut supposer que $Z$ est irr\'eductible et choisir une section $s_{q}$ de $L_{q}^{n_{q}}$ de norme sup inf\'erieure ou \'egale \`a $1$ qui ne s'annule pas identiquement sur $Z$. On d\'eduit alors de l'alin\'ea (6) du th\'eor\`eme (\ref{gdthm}) appliqu\'e aux fibr\'es $\ov{L}_{1},\dots,\ov{L}_{q-1},\ov{L}_{q}^{n_{q}}$ que~: \begin{alignat*}{2} h_{\ov{L}_{1},\dots,\ov{L}_{q-1}}(Z\cdot \op{div}s_{q}) &\leqslant n_{q}\,h_{\ov{L}_{1},\dots,\ov{L}_{q}}(Z)& \qquad &\text{si} \quad q \geqslant 2 \\ \text{et} \qquad \qquad \qquad \qquad \qquad h(Z \cdot \op{div}s_{1}) &\leqslant n_{1} \, h_{\ov{L}_{1}}(Z)& \qquad &\text{si} \quad q = 1.\qquad \qquad \qquad \qquad ~ \end{alignat*} Comme $Z\cdot \op{div}s_{q}$ est effectif de dimension $\op{dim}Z - 1$ et que $h$ prend des valeurs positives ou nulles sur les cycles effectifs dans $Z_{0}(X)$, cela implique l'in\'egalit\'e (\ref{int_eq_6}) par r\'ecurrence sur $\op{dim}Z$. \medskip \begin{expl} \label{positivite_torique} Soient $\M{P}(\Delta)$ une vari\'et\'e torique projective lisse et $\ov{E}_{1},\dots,\ov{E}_{q}$ des fibr\'es en droites sur $\M{P}(\Delta)$ engendr\'es par leurs sections globales et munis de leur m\'etrique canonique. D'apr\`es (\ref{exemple_adm1}) les fibr\'es $\ov{E}_{1},\dots,\ov{E}_{q}$ sont admissibles et d'apr\`es la proposition (\ref{sections_globales}) et l'\'egalit\'e (\ref{image_inverse}) ils sont engendr\'es par leurs sections globales de norme sup inf\'erieure ou \'egale \`a $1$. On d\'eduit de la proposition (\ref{positivite_arithmetique}) que pour tout cycle effectif $Z \in Z_{q}(\M{P}(\Delta))$, on a~: \[ h_{\ov{E}_{1},\dots,\ov{E}_{q}}(Z) \geqslant 0. \] \end{expl} \bigskip \section{Vari\'et\'es toriques sur $\op{Spec}\M{Z}$}~ \subsection{C\^ones et \'eventails}~ Pour plus de d\'etails sur les d\'efinitions et les d\'emonstrations des propositions \'enonc\'ees ici, on peut consulter (\cite{20}, \S 1.1) et aussi (\cite{11}, \S 1.1 et 1.2). Soit $N \simeq \M{Z}^{d}\label{lab1}$ un $\M{Z}$-module libre de rang $d$ dans lequel on a choisi une base $e_{1}, \dots, e_{d}$. On note $M = \operatorname{Hom}_{\M{Z}}(N,\M{Z})\label{lab2}$ son $\M{Z}$-module dual. On obtient un accouplement non d\'eg\'en\'er\'e~: \[ \label{lab3} <~\, ,~>\;: M\times N \longrightarrow \M{Z}. \] On pose $N_{\M{R}} = N \otimes_{\M{Z}} \M{R}$ et $M_{\M{R}} = M\otimes_{\M{Z}}\M{R}$. Ce sont des $\M{R}$-espaces vectoriels de dimension $d$. L'accouplement ci-dessus s'\'etend en une forme bilin\'eaire non-d\'eg\'en\'er\'ee~: \[ <~\, ,~>\;: M_{\M{R}}\times N_{\M{R}} \longrightarrow \M{R}. \] \begin{defn} On appelle {\it c\^one polyh\'edral rationnel dans $N$} ou plus simplement {\it c\^one} tout ensemble $\sigma \subseteq N_{\M{R}}$ de la forme~: \[ \sigma = \sum_{i \in I}\M{R}^{+}n_{i}, \] o\`u $(n_{i})_{i \in I}$ est une famille finie d'\'el\'ements de $N$. La {\it dimension} de $\sigma$ est d\'efinie comme la dimension de l'espace vectoriel r\'eel engendr\'e par les points du c\^one $\sigma$~: \[ \dim \sigma = \dim_{\M{R}}(\operatorname{Vect}(\sigma)) = \dim_{\M{R}}(\sigma + (- \sigma)). \] \end{defn} \begin{rem} On d\'efinit de la m\^eme fa\c con les {\it c\^ones polyh\'edraux rationnels dans $M$}. \end{rem} On d\'efinit le {\it dual} $\sigma^{\ast}$ (resp. {\it l'orthogonal} $\sigma^{\perp}$) du c\^one $\sigma$ de la fa\c con suivante~: \begin{alignat*}{3} &\sigma^{\ast}& &=& &\{v \in M_{\M{R}}\, :\; <v,x> \;\geqslant 0, \quad \forall x \in \sigma\} \subseteq M_{\M{R}}, \\ &\sigma^{\perp}& &=& &\{v \in M_{\M{R}}\, :\; <v,x> \;= 0, \quad \forall x \in \sigma\} \subseteq M_{\M{R}}. \end{alignat*} \begin{defn} On dit que $\tau \subseteq \sigma$ est une {\it face} de $\sigma$, et on note $\tau < \sigma$, si l'on peut trouver $v \in \sigma^{\ast}$ tel que~: \[ \tau = \sigma \cap \{v\}^{\perp} \] \end{defn} \begin{rem} Dans un tel cas, on peut toujours choisir $v \in \sigma^{\ast}\cap M$ (voir par exemple \cite{20}, prop. 1.3 ou \cite{11}, p. 13). \end{rem} \begin{defn} Un c\^one $\sigma$ est dit {\it strict} s'il ne contient aucune droite r\'eelle. \end{defn} Les assertions suivantes d\'ecoulent ais\'ement des d\'efinitions~: \begin{prop}~ \begin{itemize} \item Toute face d'un c\^one est un c\^one. \item Le dual d'un c\^one $\sigma$ est un c\^one, et de plus ${(\sigma^{\ast})}^{\ast} = \sigma$. \item Pour tout c\^one strict $\sigma$, $\dim \sigma^{\ast} = n$. \item On a $\dim \sigma + \dim \sigma^{\perp} = n$ pour tout c\^one $\sigma$. \end{itemize} \end{prop} \begin{defn} Un {\it \'eventail} de $N_{\M{R}}$ est une famille finie $\Delta = \{\sigma\}$ de c\^ones stricts de $N_{\M{R}}$ tels que~: \begin{itemize} \item Si $\sigma \in \Delta$, alors toute face $\tau$ de $\sigma$ appartient \`a $\Delta$. \item Si $\sigma, \sigma' \in \Delta$, alors $\sigma \cap \sigma'$ est une face \`a la fois de $\sigma$ et de $\sigma'$. \end{itemize} \end{defn} La r\'eunion $|\Delta| = \bigcup_{\sigma \in \Delta}\sigma$ est appel\'ee {\it support} de $\Delta$. On note~: \[ \Delta(j) = \{\sigma\}_{ \substack{ \sigma \in \Delta \\ \dim \sigma = j}} \] le {\it $j$-squelette} de $\Delta$. On notera \'egalement $\Delta_{\rm{max}} = \Delta(d)$. \begin{defn} Soit $\sigma$ un c\^one strict de $N_{\M{R}}$, on note $\E{S}_{\sigma} = M \cap \sigma^{\ast}$. \end{defn} La proposition suivante donne une caract\'erisation alg\'ebrique des ensembles $\E{S}_{\sigma}$. \begin{prop} Soit $\sigma$ un c\^one strict de $N_{\M{R}}$. L'ensemble $\E{S}_{\sigma}$ est un semi-groupe (additif), satur\'e, de type fini. De plus, $\E{S}_{\sigma}$ engendre $M$ en tant que groupe. R\'eciproquement, si $\E{S} \subset M$ est un tel semi-groupe, alors il existe $\sigma$ un c\^one strict de $N_{\M{R}}$ tel que $\E{S} = \E{S}_{\sigma}$. \end{prop} \medskip \begin{center} \fbox{ \begin{minipage}{12cm} Dans toute la suite du texte, $N$ est un $\M{Z}$-module libre de rang $d$ fix\'e une fois pour toute. \end{minipage} } \end{center} \bigskip \medskip \subsection{Construction des vari\'et\'es toriques sur Spec $\M{Z}$}~ On suit ici essentiellement (\cite{8}, \S 4). Demazure s'int\'eresse aux vari\'et\'es toriques lisses, mais les d\'emonstrations des propositions donn\'ees ici s'\'etendent imm\'ediatement au cas g\'en\'eral. On peut \'egalement consulter (\cite{20}, \S 1) et (\cite{11}, \S 1 et \S 2) pour les d\'emonstrations de ces propositions dans le cas o\`u le corps de base est $\M{C}$. \begin{defn} (Vari\'et\'es toriques affines associ\'ee \`a un c\^one $\sigma$). Soit $\sigma$ un c\^one strict de $N_{\M{R}}$, on note $U_{\sigma}$ le $\M{Z}$-sch\'ema~: \[ U_{\sigma} = \operatorname{Spec}\left(\M{Z}\left[\E{S}_{\sigma}\right]\right) \] \end{defn} \begin{prop} Le $\M{Z}$-sch\'ema $U_{\sigma}$ est affine, normal, plat sur $\operatorname{Spec}\M{Z}$, \`a fibres g\'eom\'etriquement int\`egres. \end{prop} \begin{prop} Soit $\tau < \sigma$, l'inclusion $\E{S}_{\sigma} \subseteq \E{S}_{\tau}$ induit un morphisme canonique $U_{\tau} \rightarrow U_{\sigma}$; c'est une immersion ouverte de $\M{Z}$-sch\'emas. \end{prop} \noindent {\bf D\'emonstration.}\ Voir (\cite{8}, \S 4, lemme 1), et aussi (\cite{20}, th. 1.4). \begin{expl} Prenons $\sigma = \{0\}$, on obtient $U_{\{0\} } = T$, le $\M{Z}$-tore dual de $N$. \end{expl} Soient $\sigma$ et $\sigma'$ deux c\^ones\ d'un m\^eme \'eventail $\Delta$. On a le diagramme suivant~: \begin{center} \mbox{ \xymatrix{ & U_{\sigma} \\ U_{\sigma\cap\sigma'} \ar@{^{(}->}[ur] \ar@{^{(}->}[dr]& \\ & U_{\sigma'} }} \end{center} La proposition pr\'ec\'edente permet de recoller $U_{\sigma}$ et $U_{\sigma'}$ le long de $U_{\sigma \cap \sigma'}$. Plus g\'en\'eralement on pose la d\'efinition suivante~: \begin{defn} Soit $\Delta$ un \'eventail. On appelle {\it vari\'et\'e torique associ\'ee \`a l'\'e\-ventail $\Delta$\/} et on note $\M{P}(\Delta)$ le sch\'ema obtenu par recollement des $U_{\sigma}$, $\sigma$ parcourant $\Delta$, \`a l'aide des immersions ouvertes $U_{\sigma\cap\sigma'} \hookrightarrow U_{\sigma}$ et $U_{\sigma\cap\sigma'} \hookrightarrow U_{\sigma'}$, pour $\sigma$, $\sigma' \in \Delta$. \end{defn} \begin{prop} Le $\M{Z}$-sch\'ema $\M{P}(\Delta)$ est plat sur $\op{Spec}\M{Z}$, normal, s\'epar\'e, int\`egre, de dimension absolue $d+1$, \`a fibres g\'eom\'etriquement int\`egres. \end{prop} \noindent {\bf D\'emonstration.}\ Voir (\cite{8}, \S 4, prop. 1), et aussi (\cite{20}, th. 1.4); on peut \'egalement consulter (\cite{11}, \S 1.4) pour certains aspects. \medskip La proposition suivante justifie le nom de {\it vari\'et\'e torique} pour un $\M{Z}$-sch\'ema de la forme $\M{P}(\Delta)$~: \begin{prop} Notons $T$ le $\M{Z}$-tore dual de $N$; l'action~: \[ T \otimes_{\op{Spec}\M{Z}} U_{\{0\}} \simeq T \otimes_{\op{Spec}\M{Z}} T \longrightarrow T \] de $T$ sur $U_{\{0\}}$, d\'efinie par la structure de sch\'ema en groupe de $T$, se prolonge en une action~: \[ T \otimes_{\op{Spec}\M{Z}} \M{P}(\Delta) \longrightarrow \M{P}(\Delta) \] de $T$ sur $\M{P}(\Delta)$. \end{prop} \noindent {\bf D\'emonstration.}\ Voir (\cite{8}, p. 559). On peut \'egalement consulter (\cite{20}, p. 9). \medskip Comme $T = U_{\{0\}}$ est un ouvert dense de $\M{P}(\Delta)$, tout mon\^ome $x^{m}$ pour $m \in M$ s'\'etend en une fonction rationnelle sur $\M{P}(\Delta)$. On note $\chi^{m}$ cette fonction que l'on appelle {\it caract\`ere} associ\'e \`a $m$. La proposition suivante est une cons\'equence imm\'ediate de la d\'efinition de $\M{P}(\Delta)$~: \begin{prop} Soit $\sigma \in \Delta$ et $m \in M \cap \sigma^{\ast} = \E{S}_{\sigma}$, alors $\chi^{m}$ est r\'eguli\`ere sur $U_{\sigma}$. De plus si $m'$, $m'' \in \E{S}_{\sigma}$ alors $\chi^{m' + m''} = \chi^{m'}\chi^{m''}$ sur $U_{\sigma}$. \end{prop} \begin{defn} On notera $\epsilon : \op{Spec}\M{Z} \rightarrow \M{P}(\Delta)$ la section nulle de $T$ vue comme section de $\M{P}(\Delta)$~: \begin{center} \mbox{ \xymatrix{ \qquad \quad \; T \subseteq \M{P}(\Delta) \\ \op{Spec}\M{Z} \ar@/^/[u]_{\epsilon}} } \end{center} \end{defn} Les deux propositions suivantes donnent des crit\`eres simples sur $\Delta$ pour que $\M{P}(\Delta)$ soit propre (resp. lisse)~: \begin{prop} La vari\'et\'e torique $\M{P}(\Delta)$ est propre sur $\op{Spec}\M{Z}$ si et seulement si $\Delta$ est {\it complet} (i.e. $|\Delta| = N_{\M{R}}$). \end{prop} \begin{prop} \label{lissite} La vari\'et\'e torique $\M{P}(\Delta)$ est lisse sur $\op{Spec}\M{Z}$ si et seulement si tout c\^one\ $\sigma \in \Delta$ est engendr\'e par une partie d'une base de $N$. Dans ce cas, $\Delta$ est dit {\it r\'egulier}. \end{prop} \noindent {\bf D\'emonstration.}\ Voir (\cite{8}, \S 4, prop. 4) et (\cite{8}, \S 4, def. 1 et prop. 1). On peut \'egalement consulter (\cite{20}, th. 1.10) et (\cite{11}, \S 2.1 et 2.4). \begin{rem} Soit $k$ un corps quelconque et notons $\M{P}(\Delta)_{k} = \M{P}(\Delta) \otimes_{\op{Spec}\M{Z}}\op{Spec}k$ la vari\'et\'e torique sur $k$ associ\'ee \`a $\Delta$ comme dans \cite{4}. Les trois assertions suivantes sont \'equivalentes~: \begin{itemize} \item {Le sch\'ema $\M{P}(\Delta)$ est lisse sur $\op{Spec}\M{Z}$.} \item {Le sch\'ema $\M{P}(\Delta)$ est r\'egulier.} \item {Le sch\'ema $\M{P}(\Delta)_{k}$ est lisse sur $k$.} \end{itemize} \end{rem} \begin{expl} \label{projectif} On prend $N = \M{Z}^{2}$. On note $e_{0} = - e_{1} - e_{2}$ et on pose $\sigma_{1} = \M{R}^{+}e_{1} + \M{R}^{+}e_{2}$, $\sigma_{2} = \M{R}^{+}e_{0} + \M{R}^{+}e_{2}$ et $\sigma_{3} = \M{R}^{+}e_{0} + \M{R}^{+}e_{1}$. \bigskip \begin{center} \input{figure2.pstex_t} \end{center} \bigskip En prenant $\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$ ainsi que leurs faces $\M{R}^{+}e_{0}$, $\M{R}^{+}e_{1}$, $\M{R}^{+}e_{2}$ et $\{0\}$, on obtient un \'eventail complet et r\'egulier $\Delta_{2}$. Dans $M$, les c\^ones\ duaux sont donn\'es par~: \bigskip \begin{center} \input{figure3.pstex_t} \end{center} \bigskip Les ouverts $U_{\sigma_{1}}$, $U_{\sigma_{2}}$ et $U_{\sigma_{3}}$ sont des plans affines que l'on recolle par les applications~: \begin{alignat*}{9} \Theta_{1,2}:\; & U_{\sigma_{1}}& &\rightarrow & &U_{\sigma_{2}}\qquad & \Theta_{1,3}:\; & U_{\sigma_{1}}& &\rightarrow & &U_{\sigma_{3}}\qquad & \Theta_{2,3}:\; & U_{\sigma_{2}}& &\rightarrow & &U_{\sigma_{3}}\qquad \\ (&x,y) & &\mapsto & \bigg( & \frac{y}{x},\frac{1}{x}\bigg) & (&x,y) & &\mapsto & \bigg( & \frac{x}{y},\frac{1}{y}\bigg) & (&x,y) & &\mapsto & \bigg( & \frac{1}{x},\frac{y}{x}\bigg) \end{alignat*} On en d\'eduit que $\M{P}(\Delta_{2})$ s'identifie au plan projectif $\M{P}^{2}_{\M{Z}}$. \end{expl} La proposition suivante donne une d\'ecomposition de $X$ sous forme d'une r\'eu\-nion disjointe de tores~: \begin{prop} \label{decomposition1} Soit $\Delta$ un \'eventail de $N_{\M{R}}$ et $\M{P}(\Delta)$ la vari\'et\'e torique associ\'ee. Pour tout $\sigma \in \Delta$ on consid\`ere le tore~: \[ \E{O}(\sigma) = \operatorname{Spec}\left(\M{Z}\left[ M \cap \sigma^{\perp} \right]\right). \] Le tore $\E{O}(\sigma)$ se plonge de mani\`ere canonique dans l'ouvert $U_{\sigma}$ (et donc dans $\M{P}(\Delta)$) par le morphisme~: \[ i_{\sigma}: \E{O}(\sigma) = \operatorname{Spec}\left(\M{Z}\left[ M \cap \sigma^{\perp} \right]\right) \lhook\joinrel\longrightarrow \operatorname{Spec}\left(\M{Z}\left[ M \cap \sigma^{\ast} \right]\right) = U_{\sigma}, \] obtenu par prolongement par z\'ero (i.e. induit par le morphisme $i_{\sigma}: \M{Z}\left[M\cap \sigma^{\ast}\right] \rightarrow \M{Z} \left[ M\cap\sigma^{\perp} \right]$ d\'efini par $i_{\sigma}(\chi^{m}) = \chi^{m}$ si $m \in \sigma^{\perp}$ et $i_{\sigma}(\chi^{m}) = 0$ sinon). De plus~: \begin{enumerate} \item {Soit $k$ un corps alg\'ebriquement clos, toute $T(k)$-orbite de $\M{P}(\Delta)(k)$ est de la forme $\E{O}(\sigma)(k)$ avec $\sigma \in \M{P}(\Delta)$.} \item {Le sch\'ema $\M{P}(\Delta)$ est r\'eunion disjointe des sous-sch\'emas $\E{O}(\sigma)$ pour $\sigma$ parcourant $\Delta$ et cette d\'ecomposition est respect\'ee par l'action de $T$.} \item {On a~: \begin{itemize} \item{$\E{O}\left(\{O\}\right) = U_{\{O\}} = T.$} \item{$\tau < \sigma \Leftrightarrow \E{O}(\sigma) \subset \overline{\E{O}(\tau)}.$} \item{$U_{\sigma} = \bigcup_{\tau < \sigma}\E{O}(\tau)$.} \end{itemize} } \item {Pour tout $\sigma \in \Delta$, notons $V(\sigma) = \overline{\E{O}(\sigma)}$; c'est une vari\'et\'e torique et l'on a~: \[ V(\sigma) = \bigcup_{\sigma < \tau}\E{O}(\tau). \] } \end{enumerate} \end{prop} \noindent {\bf D\'emonstration.}\ Voir (\cite{8}, \S 4, prop. 2). On pourra \'egalement consulter (\cite{20}, prop. 1.6) et (\cite{11}, \S 3.1). \medskip \begin{rem} \label{rem_decomposition1} Soient $\sigma \in \Delta$ et $N_{\sigma} \subset N$ le $\M{Z}$-module engendr\'e par $\sigma \cap N$. On note $N(\sigma)$ le $\M{Z}$-module quotient $N / N_{\sigma}$ et $M(\sigma) = \sigma^{\perp} \cap M$ son dual. On appelle {\it \'etoile\/} du c\^one $\sigma$ l'ensemble des c\^ones $\tau \in \Delta$ contenant $\sigma$. Soit $\tau$ un tel c\^one; on note $\ov{\tau}$ son image dans $N(\sigma)_{\M{R}}$, c'est-\`a-dire~: \[ \ov{\tau} = (\tau + (N_{\sigma})_{\M{R}})\left/ (N_{\sigma})_{\M{R}} \subset N(\sigma)_{\M{R}}\right. . \] L'ensemble $\{\ov{\tau}: \tau \in \Delta, \; \sigma < \tau\}$ forme un \'eventail de $N(\sigma)$ que l'on note $\Delta(\sigma)$. On dispose de l'inclusion canonique $\E{O}(\sigma) \subset \M{P}(\Delta(\sigma))$ correspondant au c\^one $\{0\} \in \Delta(\sigma)$. On peut montrer (voir \cite{11}, \S 3.1; ou \cite{20}, cor. 1.7) que le morphisme $i_{\sigma}: \E{O}(\sigma) \hookrightarrow U_{\sigma}$ introduit \`a la proposition (\ref{decomposition1}) s'\'etend de mani\`ere canonique en une immersion ferm\'ee $i_{\sigma}: \M{P}(\Delta(\sigma)) \hookrightarrow \M{P}(\Delta)$ qui a pour image $V(\sigma)$. Dans toute la suite, on identifie $\M{P}(\Delta(\sigma))$ \`a $V(\sigma)$ par cet isomorphisme canonique. \end{rem} La d\'efinition et la proposition suivantes d\'ecrivent les morphismes naturels entre vari\'et\'es toriques~: \begin{defn} Soient $(N,\Delta)$ et $(N',\Delta')$ deux \'eventails, avec $N \simeq \M{Z}^{d}$ et $N' \simeq \M{Z}^{d'}$; un {\it morphisme d'\'eventails\/} $\varphi: (N',\Delta') \rightarrow (N,\Delta)$ est un morphisme de $\M{Z}$-module $\varphi: N'\rightarrow N$ telle que l'application induite~: $\varphi_{\M{R}}: N'_{\M{R}} \rightarrow N_{\M{R}}$, d\'efinie par extension des scalaires \`a partir de $\varphi$, v\'erifie~: pour tout $\sigma'\in\Delta'$, il existe $\sigma\in\Delta$ tel que $\varphi(\sigma') \subset \sigma$. \end{defn} Soit $\varphi: (N',\Delta') \rightarrow (N,\Delta)$ un tel morphisme d'\'eventails. On construit \`a partir de $\varphi$ un morphisme \'equivariant $\varphi_{\ast}: \M{P}(\Delta') \rightarrow \M{P}(\Delta)$ de la fa\c con suivante~: On note $^{t}\varphi: M \rightarrow M'$ la transpos\'ee de $\varphi$ et $^{t}\varphi_{\M{R}}: M_{\M{R}} \rightarrow M'_{\M{R}}$ l'application d\'efinie par extension des scalaires \`a partir de $^{t}\varphi$. Soient $\sigma' \in \Delta'$ et $\sigma \in \Delta$ tels que $\varphi_{\M{R}}(\sigma') \subset \sigma$. De l'inclusion $^{t}\varphi_{\M{R}}(\sigma^{\ast}) \subset (\sigma')^{\ast}$, on tire que $^{t}\varphi(\E{S}_{\sigma}) \subset \E{S}_{\sigma'}$. On dispose donc d'une application $^{t}\varphi: \E{S}_{\sigma} \rightarrow \E{S}_{\sigma'}$ qui induit un morphisme \'equivariant $\varphi_{\ast}: U_{\sigma'} \rightarrow U_{\sigma}$. En particulier, si l'on prend $\sigma' = \{0'\}$ et $\sigma = \{0\}$, on obtient le morphisme de tore~: \[ \varphi_{\ast}: T' = \op{Spec}(\M{Z}[M']) \longrightarrow \op{Spec}(\M{Z}[M]) = T \] induit par l'application $^{t}\varphi : M \rightarrow M'$. La proposition suivante affirme qu'on peut recoller ces constructions locales pour obtenir un morphisme global \'equivariant $\varphi_{\ast}$ et donne une condition n\'ecessaire et suffisante sur $\varphi$ pour que $\varphi_{\ast}$ soit propre~: \begin{prop} Soit un morphisme d'\'eventails $\varphi:(N',\Delta')\rightarrow (N,\Delta)$. Le morphisme de tore alg\'ebrique~: \[ \varphi_{\ast}:\,T' = \op{Spec}\left(\M{Z}[M']\right) \longrightarrow \op{Spec}\left(\M{Z} [M]\right) = T, \] induit par l'application duale $^{t}\varphi : M \rightarrow M'$, se prolonge en un morphisme~: \[ \varphi_{\ast}:\, \M{P}(\Delta') \longrightarrow \M{P}(\Delta). \] Le morphisme $\varphi_{\ast}$ est \'equivariant sous l'action de $T'$ et $T$. De plus, $\varphi_{\ast}$ est propre si et seulement si~: \[ \varphi_{\M{R}}^{-1}\left(|\Delta|\right) = |\Delta'|. \] \end{prop} \noindent {\bf D\'emonstration.}\ On peut consulter (\cite{20}, prop. 1.13 et 1.15) et aussi (\cite{11}, \S 1.4 et 2.4). \begin{expl} \label{definition_endo} Soit $p$ un entier sup\'erieur ou \'egal \`a un et prenons $N' = N$ et $\Delta' = \Delta$, et soit~: \begin{alignat*}{3} [p]:\,&N& &\longrightarrow& &N \\ &n& &\longmapsto& &p\,n. \end{alignat*} On note encore $[p]$ l'endomorphisme de $\M{P}(\Delta)$ induit par $[p]$. D'apr\`es la proposition pr\'ec\'edente, le morphisme $[p]$ est propre. Sa restriction \`a chacun des tores $\E{O}(\sigma)$ est le morphisme {\it puissance $p$-i\`eme\/}~: \begin{alignat*}{3} [p]:\, \E{O}&(\sigma) &&\longrightarrow &\E{O}&(\sigma) \\ &x &&\longmapsto &&x^{p}. \end{alignat*} On en d\'eduit que pour tout corps $k$, le morphisme $[p]_{k}: \M{P}(\Delta)_{k} \rightarrow \M{P}(\Delta)_{k}$ obtenu par extension des scalaires est fini de degr\'e $p^{d}$. \end{expl} \begin{defn} \label{raffinement_1} Soient $\Delta$ et $\Delta'$ deux \'eventails de $N$. On dit que $\Delta'$ est {\em plus fin} que $\Delta$ ou encore que $\Delta'$ est un {\em raffinement} de $\Delta$ si pour tout $\sigma'\in\Delta'$ il existe $\sigma \in \Delta$ tel que $\sigma' \subset \sigma$, et si de plus $|\Delta'| = |\Delta|$. L'inclusion induit un morphisme \'equivariant propre canonique $i_{\ast}: \M{P}(\Delta') \rightarrow \M{P}(\Delta)$. \end{defn} \bigskip \subsection{Diviseurs invariants sur $\M{P}(\Delta)$}~ On consid\`ere $\Delta$ un \'eventail complet, de sorte que la vari\'et\'e torique associ\'ee $\M{P}(\Delta)$ est propre. On note $\tau_{1}, \dots, \tau_{r}$ les \'el\'ements de $\Delta(1)$, c'est-\`a-dire les demi-droites de $\Delta$ et $u_{1}, \dots, u_{r}$ leur g\'en\'erateur dans $N$, c'est-\`a-dire les \'el\'ements de $N$ tels que $\tau_{i} \cap N = \M{N}u_{i}$. A tout $\tau_{i}$ on a associ\'e pr\'ec\'edemment $V(\tau_{i}) = \overline{\E{O}(\tau_{i})}$ un sch\'ema irr\'eductible de codimension $1$ invariant sous l'action de $T$. \begin{defn} On appelle {\it diviseur invariant \'el\'ementaire\/} ou plus simplement {\it diviseur \'el\'ementaire\/} et on note $D_{i}$ le cycle donn\'e par $V(\tau_{i})$. \end{defn} \begin{prop} \label{inv_intro} Tout diviseur de Weil $D$ sur $\M{P}(\Delta)$ horizontal invariant par $T$ (i.e. dont la restriction \`a la fibre g\'en\'erique est laiss\'ee invariante par $T_{\M{Q}}$) est de la forme~: \[ D = \sum_{i = 1}^{r}a_{i}D_{i} \qquad (a_{i} \in \M{Z}). \] \end{prop} \noindent {\bf D\'emonstration.}\ Cela d\'ecoule directement de la d\'ecomposition donn\'ee dans la proposition (\ref{decomposition1}). \medskip \begin{prop} \label{ordre} Soit $m \in M$ et $\chi^{m}$ le caract\`ere associ\'e; l'ordre de $\chi^{m}$ en $D_{i}$ est donn\'e par~: \[ \operatorname{ord}_{D_{i}}(\chi^{m}) = <m,u_{i}>. \] \end{prop} \noindent {\bf D\'emonstration.}\ Voir par exemple (\cite{11}, lemme p. 61). \medskip \begin{defn} On dit qu'une fonction $\psi : N_{\M{R}} \rightarrow \M{R}$ est lin\'eaire par morceaux sur $\Delta$ (ou plus simplement lin\'eaire par morceaux) si la restriction de $\psi$ \`a chacun des c\^ones\ de $\Delta$ est d\'efinie par une forme lin\'eaire $m_{\psi,\sigma} \in M$. \end{defn} \begin{prop} Un diviseur de Weil horizontal $T$-invariant $D = \sum_{i = 1}^{r}a_{i}D_{i}$ sur $\M{P}(\Delta)$ provient d'un diviseur de Cartier si et seulement si\ il existe une fonction $\psi : N_{\M{R}} \rightarrow \M{R}$ continue et lin\'eaire par morceaux sur $\Delta$ telle que $\psi(u_{i}) = -a_{i}$ pour $(1 \leqslant i \leqslant r)$. Si elle existe, une telle fonction $\psi$ est unique. \end{prop} \noindent {\bf D\'emonstration.}\ Voir (\cite{20}, prop. 2.1) et (\cite{11}, p. 66). \medskip \begin{defn} Soit $D$ un diviseur de Cartier sur $\M{P}(\Delta)$ horizontal et $T$-inva\-riant, on appelle {\it fonction support associ\'ee \`a $D$\/} et on notera $\psi_{D}$ la fonction d\'efinie par la proposition ci-dessus. On notera $m_{D,\sigma}$ la forme lin\'eaire d\'efinissant $\psi_{D}$ sur $\sigma \in \Delta$. \end{defn} \begin{rem} Lorsque $\M{P}(\Delta)$ est lisse, tout diviseur de Weil est un diviseur de Cartier. On remarquera que sous cette hypoth\`ese, l'existence d'une fonction support d\'ecoule du crit\`ere de lissit\'e (\ref{lissite}). \end{rem} \begin{prop} \label{intro_inverse} Si $\varphi: (N',\Delta') \rightarrow (N,\Delta)$ est un morphisme d'\'eventails et $D$ un diviseur de Cartier $T$-invariant sur $\M{P}(\Delta)$ de fonction support $\psi_{D}$, alors le diviseur de Cartier $T$-invariant $(\varphi_{\ast})^{\ast}(D)$ sur $\M{P}(\Delta')$ admet $\psi_{d}\circ \varphi$ comme fonction support. \end{prop} \noindent {\bf D\'emonstration.}\ Soient $\sigma' \in \Delta'$ et $\sigma \in \Delta$ tels que $\varphi(\sigma') \subset \sigma$. On pose $D' = (\varphi_{\ast})^{\ast}(D)$ et on note $m_{D,\sigma}$ (resp. $m_{D',\sigma'}$) la forme lin\'eaire d\'efinissant $\psi_{D}$ sur $\sigma$ (resp. $\psi_{D} \circ \varphi$ sur $\sigma'$). D'apr\`es la proposition (\ref{ordre}) le diviseur $D$ est de la forme $\op{div}(\chi^{m_{D,\sigma}})$ sur $U_{\sigma}$, et donc $D'$ est de la forme $\op{div}(\chi^{m_{D,\sigma}}\circ \varphi_{\ast})$ sur $U_{\sigma'}$. Il suffit alors de remarquer que~: \[ \chi^{m_{D,\sigma}}\circ \varphi_{\ast} = \chi^{^t\varphi(m_{D,\sigma})} = \chi^{m_{D',\sigma'}}, \] et d'appliquer une nouvelle fois la proposition (\ref{ordre}) pour conclure. \medskip Le lemme suivant est une simple cons\'equence de la proposition (\ref{ordre}). \begin{lem} \label{locale} Soit $D = \sum_{i=1}^{r}a_{i}D_{i}$ un diviseur de Cartier horizontal $T$-invariant et $\C{O}(D)$ le faisceau inversible associ\'e \`a $D$. Pour tout $\sigma \in \Delta$, on pose~: \[ P_{D}(\sigma) = \{ v\in M_{\M{R}}:\quad <v,u> \geqslant \psi_{D}(u), \quad \forall u \in \sigma \} = \sigma^{\ast} + m_{D, \sigma}. \] Le $\M{Z}$-module des sections de $\C{O}(D)$ sur $U_{\sigma}$ est donn\'e par~: \[ \Gamma(U_{\sigma},\C{O}(D)) = \bigoplus_{m \in P_{D}(\sigma) \cap M}\M{Z} \chi^{m}. \] \end{lem} \medskip La proposition suivante, qui d\'ecrit les sections globales d'un diviseur de Cartier $T$-invariant, d\'ecoule directement du lemme pr\'ec\'edent~: \begin{prop} \label{sections_globales} Soit $D = \sum_{i =1}^{r}a_{i}D_{i}$ un diviseur de Cartier horizontal $T$-invariant et $\C{O}(D)$ le faisceau inversible associ\'e \`a $D$. On note $K_{D}$ le polytope convexe de $M_{\M{R}}$ d\'efini par les in\'equations suivantes~: \begin{align*} K_{D} = &\{v \in M_{\M{R}}, \quad <v,u_{i}> \geqslant -a_{i}, \quad 0 \leqslant i \leqslant r\} \\ = &\{v \in M_{\M{R}}, \quad <v,u> \geqslant \psi_{D}(u),\quad \forall u \in N_{\M{R}}\}. \end{align*} Le $\M{Z}$-module des sections globales de $\C{O}(D)$ est donn\'e par~: \[ \Gamma \left(\M{P}(\Delta),\C{O}(D)\right) = \bigoplus_{m \in K_{D} \cap M}\M{Z} \chi^{m}. \] \end{prop} Pour plus de d\'etails, on peut consulter (\cite{20}, lemme 2.3 et \cite{11}, p. 66), les arguments donn\'es s'\'etendant imm\'ediatement \`a la situation sur $\op{Spec}\M{Z}$. \medskip \begin{defn} Une fonction $\psi: N_{\M{R}} \rightarrow \M{R}$ est dite {\it concave} si~: \[ \psi(tx + (1-t)y) \geqslant t\psi(x) + (1-t)\psi(y), \qquad \forall t \in [0,1] \] pour tout $x$, $y \in N_{\M{R}}$. \end{defn} \begin{defn} {\bf (Minkowski).} Soit $K$ un compact convexe non vide de $M_{\M{R}}$. On appelle {\it fonction d'appui associ\'ee \`a $K$\/} et on note $\psi_{K}$ la fonction $\psi_{K}: N_{\M{R}} \rightarrow \M{R}$ d\'efinie par~: \[ \psi_{K}(u) = \op{inf}\{<v,u>, \quad v \in K\} \] pour tout $u \in N_{\M{R}}$. \end{defn} On a alors le r\'esultat suivant (voir par exemple \cite{20}, th. A. 18)~: \begin{thm} \label{correspondance} Soit $\C{C}(M_{\M{R}})$ l'ensemble des compacts convexes non vides de $M_{|R}$ et notons $\C{S}(N_{\M{R}})$ l'ensemble des fonctions $\psi: N_{\M{R}} \rightarrow \M{R}$ positivement homog\`enes (i.e. telles que $\psi(cu) = c\psi(u)$ pour tout $c \in \M{R}^{+}$ et $u \in N_{\M{R}}$) et concaves. Pour tout $\psi \in \C{S}(N_{\M{R}})$, on d\'efinit $K_{\psi}$ le {\it compact connexe associ\'e \`a $\psi$\/} par les in\'egalit\'es suivantes~: \[ K_{\psi} = \{v \in M_{\M{R}}:\quad <v,u> \geqslant \psi(u), \quad \forall u \in N_{\M{R}}\}. \] On a alors~: \begin{itemize} \item {Les applications $\C{C}(M_{\M{R}}) \rightarrow \C{S}(N_{\M{R}})$ et $\C{S}(N_{\M{R}}) \rightarrow \C{C}(M_{\M{R}})$ qui envoient respectivement $K$ sur $\psi_{K}$ et $\psi$ sur $K_{\psi}$ sont r\'eciproques l'une de l'autre.} \item {Par la correspondance biunivoque d\'ecrite ci-dessus, la somme de Minkowski $K + K'$ de deux compacts convexes $K, K' \in \C{C}(M_{\M{R}})$ (resp. le dilat\'e $cK$, $c \in \M{R}^{+}$) est associ\'ee \`a la somme $\psi_{K} + \psi_{K'}$ (resp. \`a $c\psi_{K}$).} \end{itemize} \end{thm} On peut alors \'enoncer~: \begin{thm} \label{polytope_et_vt} Soit $D$ un diviseur de Cartier horizontal $T$-invariant sur $\M{P}(\Delta)$. Le faisceau inversible $\C{O}(D)$ est engendr\'e par ses sections globales si et seulement si\ $\psi_{D}$ la fonction support de $D$ est concave. De plus, si pour tout c\^one\ $\sigma \in \Delta$ on note $m_{D, \sigma}$ l'\'el\'ement de $M$ d\'efinissant la restriction de $\psi_{D}$ \`a $\sigma$, le polytope convexe $K_{D}$ associ\'e \`a $D$ est alors l'enveloppe convexe des $m_{D, \sigma}$ dans $M_{\M{R}}$ pour $\sigma$ parcourant $\Delta_{\op{max}}$. En particulier, $K_{D}$ est \`a sommets entiers. Enfin $K_{D}$ et $\psi_{D}$ sont images l'un de l'autre par la correspondance d\'efinie \`a la proposition $($\ref{correspondance}$)$. \end{thm} \noindent {\bf D\'emonstration.}\ Voir (\cite{20}, th 2.7) ou (\cite{11}, p. 68). \medskip \begin{prop} \label{additivite_1} Soient $D$ et $D'$ deux diviseurs horizontaux $T$-invariants et engendr\'es par leurs sections globales. Soit $D'' = D + D'$, on a~: \begin{align*} \psi_{D''} &= \psi_{D} + \psi_{D'} \\ K_{D''} &= K_{D} + K_{D'}. \end{align*} \end{prop} \noindent {\bf D\'emonstration.}\ C'est une cons\'equence des \'enonc\'es pr\'ec\'edents. On peut consulter (\cite{11}, p. 69) pour plus de d\'etails. \medskip \begin{rem} On remarquera que l'on a pas n\'ecessairement~: \[ K_{D''} \cap M = \left(K_{D} \cap M\right) + \left(K_{D'} \cap M\right). \] \end{rem} \begin{defn} Soit $\psi$ une fonction concave lin\'eaire par morceaux sur un \'eventail $\Delta$ complet. On dit que $\psi$ est {\it strictement concave relativement \`a $\Delta$\/} (ou plus simplement {\it strictement concave\/} lorsque aucune confusion n'est \`a craindre) si et seulement si\ $\Delta$ est l'\'eventail complet le plus grossier dans $N$ tel que $\psi_{|\sigma}$ soit lin\'eaire pour tout $\sigma \in \Delta$. \end{defn} \begin{thm} Soit $D$ un diviseur de Cartier horizontal $T$-invariant sur $\M{P}(\Delta)$ et $\psi_{D}$ la fonction support associ\'ee. Les trois assertions suivantes sont \'equivalentes~: \begin{enumerate} \item{Le diviseur $D$ est ample.} \item{La fonction support $\psi_{D}$ est strictement concave relativement \`a $\Delta$.} \item{Le polytope $K_{D}$ est de dimension $d$, et si pour tout $\sigma \in \Delta$, on note $m_{D, \sigma}$ l'\'el\'ement de $M$ donnant la restriction de $\psi_{D}$ \`a $\sigma$, alors les sommets de $K_{D}$ sont donn\'e par $\{m_{D,\sigma}, \sigma \in \Delta_{\op{max}}\}$. De plus $m_{D,\sigma} \not= m_{D,\tau}$ pour $\sigma$, $\tau \in \Delta_{\op{max}}$ d\`es que $\sigma \not= \tau $.} \end{enumerate} \end{thm} \noindent {\bf D\'emonstration.}\ Comme $\op{Spec}\M{Z}$ est un sch\'ema affine, le diviseur $D$ est ample sur $\M{P}(\Delta)$ si et seulement s'il est ample relativement \`a $\op{Spec}\M{Z}$ d'apr\`es (\cite{27}, cor. 4.6.6). Par ailleurs, d'apr\`es (\cite{27}, cor. 4.6.4) et le crit\`ere d'amplitude donn\'e dans (\cite{28}, th. 4.7.1), l'amplitude de $D$ sur $\M{P}(\Delta)$ relativement \`a $\op{Spec}\M{Z}$ est \'equivalente \`a l'amplitude de $D$ sur $\M{P}(\Delta)_{\M{F}_{p}}$ pour tout nombre premier $p$. Enfin pour tout corps $k$, (\cite{20}, cor. 2.14) ou (\cite{11}, p. 70) modifi\'es de fa\c con \'evidente montrent que les trois assertions du th\'eor\`eme sont \'equivalentes sur $\M{P}(\Delta)_{k}$. \medskip On remarque en particulier que sur une vari\'et\'e torique propre, tout diviseur de Cartier horizontal $T$-invariant ample est engendr\'e par ses sections globales. Concernant les fibr\'es tr\`es amples, on a~: \begin{thm} Soit $D$ un diviseur de Cartier horizontal $T$-invariant sur $\M{P}(\Delta)$ et $\psi_{D}$ sa fonction support associ\'ee. Les trois assertions suivantes sont \'equivalentes~: \begin{enumerate} \item{Le diviseur $D$ est tr\`es ample (relativement \`a $\op{Spec}\M{Z}$).} \item{La fonction $\psi_{D}$ est strictement concave relativement \`a $\Delta$. De plus, pour tout $\sigma \in \Delta_{\op{max}}$, l'ensemble $\left(M \cap K_{D}\right) - m_{D,\sigma}$ engendre le semi-groupe $M\cap \sigma^{\ast} = \E{S}_{\sigma}$.} \item{Le polytope $K_{D}$ est de dimension $d$, l'ensemble de ses sommets est donn\'e par $\{m_{D,\sigma}, \sigma \in \Delta_{\op{max}}\}$. De plus, $\left( M\cap K_{D}\right) - m_{D, \sigma}$ engendre le semi-groupe $M \cap \sigma^{\ast} = \E{S}_{\sigma}$ pour tout $\sigma \in \Delta_{\op{max}}$.} \end{enumerate} \end{thm} \noindent {\bf D\'emonstration.}\ Il suffit de modifier de fa\c con \'evidente la preuve de (\cite{20}, th 2.13). \medskip On a enfin le th\'eor\`eme suivant~: \begin{thm} \label{simplicite} {\rm \bf (Demazure).} Soit $\M{P}(\Delta)$ une vari\'et\'e torique propre et lisse de dimension relative $d$. Soit $D$ un diviseur horizontal $T$-invariant sur $\M{P}(\Delta)$ et $\psi_{D}$ sa fonction support associ\'ee. Les quatre assertions suivantes sont \'equivalentes~: \begin{enumerate} \item{Le diviseur $D$ est ample.} \item{Le diviseur $D$ est tr\`es ample.} \item{La fonction $\psi_{D}$ est strictement concave relativement \`a $\Delta$.} \item{Le polytope $K_{D}$ est de dimension $d$, et si pour tout $\sigma \in \Delta$, on note $m_{D,\sigma}$ l'\'el\'ement de $M$ donnant la restriction de $\psi_{D}$ \`a $\sigma$, alors les sommets de $K_{D}$ sont donn\'es par $\{m_{D,\sigma}, \sigma \in \Delta_{\op{max}}\}$. De plus $m_{D,\sigma} \not= m_{D,\tau}$ pour $\sigma$, $\tau \in \Delta_{\op{max}}$ d\`es que $\sigma \not= \tau$.} \end{enumerate} Lorsque ces conditions sont satisfaites, le polytope $K_{D}$ est {\rm absolument simple} dans le sens o\`u chaque sommet $m_{D,\sigma}$ rencontre exactement $d$ ar\^etes et o\`u, si $m_{1,\sigma}, \dots, m_{d,\sigma}$ sont les points de $M$ les plus proches de $m_{D,\sigma}$ sur chacune de ces diff\'erentes ar\^etes, alors $\{m_{1,\sigma}-m_{D,\sigma}, \dots, m_{d,\sigma}-m_{D,\sigma}\}$ est une base de $M$. \end{thm} \noindent {\bf D\'emonstration.}\ Voir (\cite{8}, \S 4, th. 2 et cor. 1) et aussi (\cite{20}, cor. 2.15). \bigskip \subsection{Vari\'et\'e torique et fibr\'e en droites associ\'e \`a un polytope}~ Dans ce paragraphe, on suit \cite{17}; on peut \'egalement consulter (\cite{20}, th. 2.22, et \cite{11}, \S 5.5). On a vu comment associer \`a tout diviseur de Cartier horizontal $T$-invariant $D$ sur une vari\'et\'e torique propre $\M{P}(\Delta)$ un polytope convexe $K_{D} \subset M_{\M{R}}$. Inversement le probl\`eme suivant se pose~: \'etant donn\'e un polytope convexe \`a sommets entiers $K \subset M_{\M{R}}$, peut-on construire une vari\'et\'e torique propre $\M{P}(\Delta_{K})$ et un diviseur horizontal $T$-invariant $E$ sur $\M{P}(\Delta_{K})$ tels que le polytope $K_{E}$ soit \'egal \`a $K$ ? La r\'eponse est donn\'ee par le th\'eor\`eme suivant~: \begin{thm} \label{construction_inverse} Soit $K \subset M_{\M{R}}$ un polytope convexe d'int\'erieur non vide dont les sommets sont dans $M$. Il existe un unique \'eventail complet $\Delta$ dans $N_{\M{R}}$ et un unique diviseur de Cartier $E$ horizontal $T$-invariant sur $\M{P}(\Delta)$ tels que~: \begin{enumerate} \item{$K_{E} = K$.} \item{Le diviseur $E$ est ample.} \end{enumerate} L'\'eventail $\Delta$ est le plus petit \'eventail complet tel que la fonction d'appui $\psi_{K}$ est lin\'eaire par morceau relativement \`a $\Delta$. De plus, $\M{P}(\Delta)$ est lisse si et seulement si\ le polytope $K$ est absolument simple; dans ce cas, le diviseur $E$ est tr\`es ample. \end{thm} \noindent {\bf D\'emonstration.}\ On suit ici (\cite{20}, th. 2.22) et (\cite{17}, \S 3, th. 1). On d\'efinit l'\'eventail $\Delta$ comme dans le dernier alin\'ea de l'\'enonc\'e. La fonction d'appui $\psi_{K}$ est continue et concave. Soient $u_{1}, \dots, u_{r}$ les g\'en\'erateurs dans $N$ des demi-droites $\tau_{1},\dots, \tau_{r}$ de $\Delta(1)$; le diviseur~: \[ E = -\sum_{i=1}^{r}\psi_{K}(u_{i})V(\tau_{i}) \] est un diviseur de Cartier horizontal $T$-invariant engendr\'e par ses sections globales d'apr\`es (\ref{polytope_et_vt}). De plus, par construction, $\psi_{K}$ est strictement concave relativement \`a $\Delta$, donc $E$ est ample et $\Delta$ est l'unique \'eventail \`a satisfaire cette condition. On a $K_{E} = K_{\psi_{K}} = K$. Enfin, $\M{P}(\Delta)$ est lisse si et seulement si\ $K$ est absolument simple d'apr\`es (\ref{lissite}) et (\ref{simplicite}). \medskip \begin{expl} \label{exemple_intro_1} Lorsque $d = 2$, consid\'erons $K_{2} \subset M_{\M{R}} = \M{R}^{2}$ le polytope convexe absolument simple d\'efini par les in\'equations~: \[ x_{1} \geqslant 0, \quad x_{2} \geqslant 0 \quad \text{et} \quad x_{1} + x_{2} \leqslant 1. \] \bigskip \begin{center} \input{figure4.pstex_t} \end{center} \bigskip L'\'eventail complet de $N_{\M{R}}$ associ\'e \`a $K_{2}$ est l'\'eventail $\Delta_{2}$ d\'ej\`a consid\'er\'e \`a l'exemple (\ref{projectif}) dont la vari\'et\'e torique associ\'ee est le plan projectif $\M{P}_{\M{Z}}^{2}$. La fonction $\psi_{K_{2}}$ est d\'efinie sur $\sigma_{1}$, $\sigma_{2}$ et $\sigma_{3}$ par respectivement $m_{1} = (0,0)$, $m_{2} = (1,0)$ et $m_{3} = (0,1)$. En particulier $\psi_{K_{2}}(e_{1}) = \psi_{K_{2}}(e_{2}) = 0$ et $\psi_{K_{2}}(e_{0}) = -1$. On en d\'eduit que le diviseur $E$ sur $\M{P}_{\M{Z}}^{2}$ associ\'e \`a $K_{2}$ par le th\'eor\`eme (\ref{construction_inverse}) est un hyperplan coordonn\'e. On a notamment $\C{O}(E) \simeq \C{O}(1)$. Plus g\'en\'eralement, consid\'erons $K_{d} \subset M_{\M{R}} = \M{R}^{d}$ le simplexe standard d\'efini par les in\'equations~: \[ x_{1} \geqslant 0, \dots, x_{d} \geqslant 0 \quad {\text{et}} \quad x_{1} + \dots + x_{d} \leqslant 1. \] C'est un polytope convexe absolument simple. La vari\'et\'e torique associ\'ee $\M{P}(K_{d})$ s'identifie avec l'espace projectif $\M{P}_{\M{Z}}^{d}$. Le diviseur $E$ sur $\M{P}_{\M{Z}}^{d}$ associ\'e \`a $K_{d}$ par le th\'eor\`eme (\ref{construction_inverse}) est un hyperplan coordonn\'e. On a donc $\C{O}(E) \simeq \C{O}(1)$ (cf. \cite{20}, \S 2.4 et aussi \cite{11}, \S 1.4 et 1.5). \end{expl} Si l'on consid\`ere plusieurs polytopes convexes, on a le r\'esultat suivant~: \begin{thm} \label{construction_inverse2} Soient $K_{1},\dots,K_{m}$ des polytopes convexes de $M_{\M{R}}$ \`a sommets dans $M$. Posons $K = K_{1} + \dots + K_{m}$. On suppose que l'int\'erieur de $K$ est non vide. Soient $\Delta$ l'\'eventail de $N_{\M{R}}$ et $E$ le diviseur de Cartier horizontal $T$-invariant sur $\M{P}(\Delta)$ associ\'es \`a $K$. Il existe des diviseurs de Cartier horizontaux $T$-invariants $E_{j}$ pour $(1 \leqslant j \leqslant m)$ tels que $K_{E_{j}} = K_{j}$ et les faisceaux inversibles $\C{O}(E_{j})$ soient engendr\'es par leurs sections globales. On a de plus $E = E_{1} + \dots + E_{m}$. \end{thm} \noindent {\bf D\'emonstration.}\ Il suffit de remarquer que les fonctions $\psi_{K_{j}}$ sont lin\'eaires par morceaux sur $\Delta$. On pose alors~: \[ E_{j} = - \sum_{i = 1}^{r}\psi_{K_{j}}(u_{i})V(\tau_{i}) \qquad (1 \leqslant j \leqslant m). \] Comme les fonctions $\psi_{K_{j}}$ sont concaves, $E_{j}$ est un diviseur de Cartier horizontal $T$-invariant et engendr\'e par ses sections globales; de plus, $K_{\psi_{K_{j}}} = K_{j}$. Enfin, on a $\psi_{K} = \psi_{K_{1}} + \dots + \psi_{K_{m}}$, et donc $E = E_{1} + \dots + E_{m}$ d'apr\`es (\ref{additivite_1}). \medskip \begin{rem} \label{construction_inverse3} Reprenons les hypoth\`eses et les notations du th\'eor\`eme (\ref{construction_inverse2}). D'apr\`es la r\'esolution torique des singularit\'es (cf. \cite{29}, th. 11, p. 94; voir aussi \cite{30}, th. 11, p. 273) il existe un raffinement $\Delta'$ de $\Delta$ tel que $\M{P}(\Delta')$ est projective et lisse. Si l'on note $i_{\ast}: \M{P}(\Delta') \rightarrow \M{P}(\Delta)$ le morphisme propre \'equivariant induit par l'inclusion $i: \Delta' \hookrightarrow \Delta$ comme \`a la d\'efinition (\ref{raffinement_1}) alors les diviseurs $E' = (i_{\ast})^{\ast}(E)$, $E_{1}' = (i_{\ast})^{\ast}(E_{1}), \dots, E_{m}' = (i_{\ast})^{\ast}(E_{m})$ sont tels que $K_{E'} = K$, $K_{E'_{1}} = K_{1}, \dots, K_{E_{m}'} = K_{m}$ d'apr\`es (\ref{intro_inverse}), et les faisceaux inversibles $\C{O}(E')$, $\C{O}(E_{1}'), \dots, \C{O}(E_{m}')$ sont engendr\'es par leurs sections globales (mais $E'$ n'est pas n\'ecessairement ample). De plus $E' = E_{1}' + \dots + E_{m}'$. \end{rem} \bigskip \subsection{Groupe de Picard et anneau de Chow d'une vari\'et\'e torique projective lisse}~ \subsubsection{Pr\'eliminaires}~ On d\'emontre une l\'eg\`ere g\'en\'eralisation d'un th\'eor\`eme d\^u \`a Gillet et Soul\'e (\cite{14}, prop. 3.1.4). \begin{thm} \label{iso_cellulaire} Soit $X$ un sch\'ema quasi-projectif sur $\op{Spec}\M{Z}$ admettant une d\'ecomposition cellulaire, c'est-\`a-dire tel qu'il existe une suite~: \[ X = X_{n} \supset X_{n-1} \supset \dots \supset X_{0} \supset X_{-1} = \emptyset \] de sous-sch\'ema ferm\'es tels que $(X_{i} - X_{i-1})$ soit r\'eunion finie disjointe d'ouverts affines $U_{i,j}$ isomorphes \`a $\M{A}^{i}_{\M{Z}}$. \begin{enumerate} \item{Pour tout $l$ entier positif, on a des isomorphismes de groupes~: \[ CH^{l}(X) \stackrel{b}{\simeq} CH^{l}(X_{\M{Q}}) \stackrel{b'}{\simeq} CH^{l}(X_{\M{C}}) \stackrel{cl}{\simeq} H_{2n - 2l}(X(\M{C}),\M{Z}), \] o\`u les isomorphismes $b$ et $b'$ sont d\'eduits des morphismes de changement de base $X_{\M{Q}} \rightarrow X$ et $X_{\M{C}} \rightarrow X_{\M{Q}}$ et o\`u $cl$ est l'application cycle. De plus, $CH^{l}(X)$ est un $\M{Z}$-module libre de type fini, et $H_{\op{impair}}(X(\M{C}),\M{Z}) = 0$.} \item{Pour tout couple $(r,s)$ d'entiers positifs tels que $r > s$, on a $CH^{r,s}(X) = 0$, les groupes $CH^{r,s}(X)$ \'etant ceux d\'efinis dans (\cite{12},\S 8).} \end{enumerate} \end{thm} \noindent {\bf D\'emonstration.}\ On commence par d\'emontrer l'assertion 2. Du fait de l'invariance des groupes $CH^{r,s}$ par homotopie (cf. \cite{12}, th. 8.3), on a pour tout $r > s \geqslant 0$~: \[ CH^{r,s}(\M{A}_{\M{Z}}^{i}) = CH^{r,s}(\M{A}_{\M{Z}}^{0}) = 0, \] et donc pour tout $i \in \{1,\dots,n\}$, \begin{equation} \label{eq_intro_1} CH^{r,s}(X_{i} - X_{i-1}) = \bigoplus_{j}CH^{r,s}(U_{i,j}) = 0. \end{equation} Par ailleurs, d'apr\`es la longue suite exacte d'excision (\cite{12}, th. 8.1), on a~: \begin{multline*} \dots \longrightarrow CH^{r+1,s}(X_{i+1} - X_{i}) \stackrel{\partial}{\longrightarrow} CH^{r,s}(X_{i}) \longrightarrow \\ CH^{r,s}(X_{i+1}) \longrightarrow CH^{r,s}(X_{i+1} - X_{i}) \stackrel{\partial}{\longrightarrow} CH^{r-1,s}(X_{i}) \longrightarrow \dotsm \end{multline*} On tire de (\ref{eq_intro_1}) que pour tout $r > s \geqslant 0$, \[ CH^{r,s}(X_{i+1}) \simeq CH^{r,s}(X_{i}). \] Par r\'ecurrence, on est donc ramen\'e \`a montrer le r\'esultat pour $X_{0}$, ce qui est imm\'ediat car $X_{0}$ est r\'eunion finie disjointe de $\M{A}_{\M{Z}}^{0} = \op{Spec}\M{Z}$. On montre maintenant que $b$ est un isomorphisme. Pour cela, on suit (\cite{14}, prop. 3.1.4) et on effectue une r\'ecurrence sur la dimension. On remarque que le m\^eme raisonnement que pr\'ec\'edemment appliqu\'e \`a $X_{\M{Q}}$ et \`a sa d\'ecomposition cellulaire montre que $CH^{l+1,l}((X_{n} - X_{n-1})_{\M{Q}}) = 0$ pour tout entier positif $l$. On a le diagramme commutatif suivant, form\'e de deux suites exactes~: \medskip \mbox{ \xymatrix{ & CH(X_{n-1}) \ar[d]\ar[r] & CH(X_{n}) \ar[d] \ar[r]& CH(X_{n} - X_{n-1}) \ar[r]\ar[d] & 0 \\ (\ast) \quad 0 \ar[r] & CH((X_{n-1})_{\M{Q}})\ar[r] & CH((X_{n})_{\M{Q}})\ar[r] & CH((X_{n} - X_{n-1})_{\M{Q}})\ar[r] & 0 } } \medskip \noindent o\`u les fl\`eches verticales sont induites par le morphisme $X_{\M{Q}} \rightarrow X$ et ses restrictions et o\`u la ligne $(\ast)$ est une suite exacte du fait de la longue suite exacte d'excision (\cite{12}, th. 8.1) et de la nullit\'e du groupe $CH^{l+1,l}((X_{n} - X_{n-1})_{\M{Q}})$. Puisque $X_{n-1}$ est encore un sch\'ema sur $\op{Spec}\M{Z}$ admettant une d\'ecomposition cellulaire, la fl\`eche $CH(X_{n-1}) \rightarrow CH((X_{n-1})_{\M{Q}})$ est un isomorphisme par hypoth\`ese de r\'ecurrence. Comme les groupes $CH$ sont invariants par homotopie (\cite{12}, th. 8.1), la fl\`eche $CH(X_{n} - X_{n-1}) \rightarrow CH((X_{n} - X_{n-1})_{\M{Q}})$ est \'egalement un isomorphisme, et donc $CH(X_{n}) \rightarrow CH((X_{n})_{\M{Q}})$ est un isomorphisme. De la suite exacte $(\ast)$, on tire imm\'ediatement que $CH(X_{\M{Q}})$, et donc $CH(X)$, est un $\M{Z}$-module libre de type fini. Enfin $b'$ et $cl$ sont des isomorphismes d'apr\`es (\cite{10}, \S 1.9.1 et \S 19.1.11), ce r\'esultat pouvant par ailleurs se montrer par le m\^eme argument que pr\'ec\'edemment. \medskip Le r\'esultat suivant (voir \cite{9}, p. 144, ou encore \cite{11}, \S 5.2, la d\'emonstration sur $\M{C}$ s'\'etendant imm\'ediatement \`a la situation sur $\op{Spec}\M{Z}$) nous permet d'appliquer ce qui pr\'ec\`ede aux vari\'et\'es toriques projectives lisses~: \begin{thm} Toute vari\'et\'e torique projective lisse $\M{P}(\Delta)$ sur $\op{Spec}\M{Z}$ admet une d\'ecomposition cellulaire. \end{thm} \medskip On d\'eduit imm\'ediatement des deux th\'eor\`emes pr\'ec\'edents le corollaire suivant~: \begin{cor} \label{absolu} Soit $\M{P}(\Delta)$ une vari\'et\'e torique projective lisse. \begin{enumerate} \item{Pour tout entier $l$ positif, $CH^{l}(\M{P}(\Delta))$ est un $\M{Z}$-module libre de type fini et on a les isomorphismes de groupes suivants~: \[ CH^{l}(\M{P}(\Delta)) \stackrel{b}{\simeq} CH^{l}(\M{P}(\Delta)_{\M{Q}}) \stackrel{b'}{\simeq} CH^{l}(\M{P}(\Delta)_{\M{C}}) \stackrel{cl}{\simeq} H^{2l}(\M{P}(\Delta)(\M{C}), \M{Z}), \] o\`u $b$ et $b'$ sont donn\'es par changement de base de $\M{Z}$ \`a $\M{Q}$ et de $\M{Q}$ \`a $\M{C}$, et o\`u $cl$ est l'application cycle.} \item{Pour tout $r$ entier strictement positif, on a~: \[ CH^{r,r-1}(\M{P}(\Delta)) = 0. \] } \end{enumerate} \end{cor} \begin{rem} Tous les isomorphismes de groupes introduits au corollaire (\ref{absolu}) sont compatibles avec l'intersection; ce sont donc des isomorphismes pour les anneaux gradu\'es associ\'es aux groupes consid\'er\'es. \end{rem} \subsubsection{Groupe de Picard}~ La proposition suivante caract\'erise les diviseurs de Cartier horizontaux $T$-invariants principaux sur une vari\'et\'e torique compl\`ete~: \begin{prop} \label{picard_0} Soit $\M{P}(\Delta)$ une vari\'et\'e torique compl\`ete et soit $D$ un diviseur de Cartier horizontal $T$-invariant sur $\M{P}(\Delta)$. Le diviseur $D$ est principal si et seulement si\ $\psi_{D}$ est lin\'eaire sur tout $N_{\M{R}}$. \end{prop} \noindent {\bf D\'emonstration.}\ En reprenant les notations de (\ref{inv_intro}), on \'ecrit $D$ sous la forme $D = \sum_{i=1}^{r}a_{i}D_{i}$. Si $D$ est principal sur $\M{P}(\Delta)$, alors $D_{\M{C}} = \sum_{i=1}^{r}a_{i}(D_{i})_{\M{C}}$ est principal sur $\M{P}(\D)_{\M{C}}$ et d'apr\`es (\cite{20}, prop. 2.4) il existe donc $m \in M$ tel que $m(u_{i}) = - a_{i}$ pour tout $1 \leqslant i \leqslant r$. On en d\'eduit que $D = \op{div}(\chi^{m})$. La r\'eciproque est imm\'ediate. \medskip \begin{prop} \label{picard} Soit $\M{P}(\Delta)$ une vari\'et\'e torique lisse projective. On note $\tau_{1},\dots,$ $\tau_{r}$ les \'el\'ements de $\Delta(1)$ et $u_{1},\dots,u_{r}$ leur g\'en\'erateur dans $N$. On a la suite exacte~: \[ 0 \longrightarrow M \stackrel{\iota}{\longrightarrow} \bigoplus_{i =1}^{r}\M{Z} V(\tau_{i}) \stackrel{s}{\longrightarrow} \op{Pic}(\M{P}(\Delta)) \longrightarrow 0, \] la fl\`eche $\iota$ \'etant donn\'ee par $\iota: m \rightarrow \sum_{i=1}^{r}<m,u_{i}> V(\tau_{i})$ et la fl\`eche $s$ d\'esignant la surjection canonique sur $\op{Pic}(\M{P}(\Delta))$ vu comme groupe quotient par l'\'equivalence lin\'eaire. Donc $\op{Pic}(\M{P}(\Delta))$ est un $\M{Z}$-module libre de rang $(\#\Delta(1) - d)$. \end{prop} \noindent {\bf D\'emonstration.}\ C'est une cons\'equence directe de (\cite{20}, cor. 2.5) et et de l'isomorphisme $b: CH^{1}(\M{P}(\Delta)) \simeq CH^{1}(\M{P}(\Delta)_{\M{Q}})$ donn\'e par le corollaire (\ref{absolu}). On peut \'egalement consulter (\cite{11}, \S 3.4). \medskip \subsubsection{Anneau de Chow}~ On d\'ecrit maintenant un th\'eor\`eme de Jurkiewicz et Danilov (\cite{4}, th. 10.8 et rem. 10.9) donnant la structure de l'anneau de Chow (et donc de l'anneau d'homologie) d'une vari\'et\'e torique projective lisse~: \begin{thm} \label{anneau_chow} Soient $\M{P}(\Delta)$ une vari\'et\'e torique projective lisse, $\tau_{1},\dots,$ $\tau_{r}$ les \'el\'ements de $\Delta(1)$ et $u_{1},\dots,u_{r}$ leurs g\'en\'erateurs respectifs dans $N$. Consid\'erons l'anneau de polyn\^omes en $r$ ind\'etermin\'ees~: \[ \C{S} = \M{Z}\left[t_{\tau_{i}}\right]_{1 \leqslant i \leqslant r}. \] Soient $\C{I}$ l'id\'eal de $\C{S}$ engendr\'e par l'ensemble~: \[ \{t_{\rho_{1}}t_{\rho_{2}}\dots t_{\rho_{s}}: \quad \rho_{1},\dots,\rho_{s} \in \Delta(1)\text{ deux \`a deux distincts et }\rho_{1} + \dots + \rho_{s} \notin \Delta \} \] et $\C{J}$ l'id\'eal de $\C{S}$ engendr\'e par~: \[ \left\{ \sum_{i=1}^{r}<m,u_{i}> t_{\tau_{i}}, \quad m \in M \right\}. \] Soit $[~]: \C{S} \rightarrow CH^{\ast}(\M{P}(\Delta))$ le morphisme d'anneau d\'efini par~: \[ [t_{\tau}] := [V(\tau)] \in CH^{\ast}(\M{P}(\Delta)), \] pour tout $\tau \in \Delta(1)$. On a $\op{Ker}[~] = (\C{I} + \C{J})$ et le morphisme~: \[ \C{S}/(\C{I} + \C{J}) \stackrel{[~]}{\longrightarrow} CH^{\ast}(\M{P}(\Delta)), \] est un isomorphisme d'anneaux gradu\'es. \end{thm} \noindent {\bf D\'emonstration.}\ C'est une cons\'equence de (\cite{4}, th. 10.8) ou encore (\cite{20}, p. 134). On peut \'egalement consulter (\cite{11}, p. 106). \medskip \section{Introduction}~ Le but de cet article est l'\'etude, dans le cadre de la g\'eom\'etrie d'Arakelov, des vari\'et\'es toriques projectives et lisses. Chemin faisant, nous \'etendons sur certains points la th\'eorie d\'evelopp\'ee dans \cite{13}. \medskip Gr\^ace \`a Demazure (cf. \cite{8}), on sait associer \`a tout \'eventail $\Delta$ de $\M{Z}^{d}$ un sch\'ema $\pi: \M{P}(\Delta) \rightarrow \op{Spec}\M{Z}$ que l'on appelle {\it vari\'et\'e torique\/} associ\'ee \`a $\Delta$. On trouve dans la litt\'erature (cf. \cite{4}, \cite{20} et \cite{11} pour des r\'ef\'erences pr\'ecises) une description explicite de la vari\'et\'e complexe $\M{P}(\Delta)_{\M{C}}$ en terme des propri\'et\'es combinatoires de $\Delta$. Revenant au point de vue originel de Demazure, nous montrons que cette description s'\'etend sans difficult\'e \`a la situation sur $\M{Z}$. En particulier, on dispose lorsque $\M{P}(\Delta)$ est projective et lisse, d'une description agr\'eable bas\'ee sur un th\'eor\`eme de Jurkiewicz et Danilov, de l'anneau de Chow $CH^{\ast}(\M{P}(\Delta))$ en terme de g\'en\'erateurs et relations (th\'eor\`eme \ref{anneau_chow}). Plus pr\'ecis\'ement, nous montrons que $CH^{\ast}(\M{P}(\Delta))$ est engendr\'e en tant qu'anneau par la premi\`ere classe de Chern $c_{1}(L) \in CH^{1}(\M{P}(\Delta))$ des fibr\'es en droites $L$ sur $\M{P}(\Delta)$. \medskip Afin de donner une description analogue de l'anneau de Chow arithm\'etique $\widehat{CH}^{\ast}(\M{P}(\Delta))$, il nous est n\'ecessaire de munir tout fibr\'e en droites $L$ sur $\M{P}(\Delta)$ d'une m\'etrique $\|.\|_{L,\infty}$ ``canonique'' permettant entre autre chose le calcul explicite du produit~: \[ \hat{c}_{1}(L,\|.\|_{L,\infty})^{p} \in \widehat{CH}^{p}(\M{P}(\Delta)), \] o\`u $\hat{c}_{1}(L,\|.\|_{L,\infty})$ d\'esigne la premi\`ere classe de Chern arithm\'etique de $(L,\|.\|_{L,\infty})$ (comparer avec \cite{33} o\`u un point de vue analogue est d\'evelopp\'e pour les grassmanniennes). Nous donnons plusieurs constructions d'une m\'etrique $\|.\|_{L,\infty}$ qui est canonique dans le sens o\`u l'application $L \mapsto \|.\|_{L,\infty}$ poss\`ede des propri\'et\'es fonctorielles (proposition \ref{BT_fonct}) qui permettent de la caract\'eriser enti\`erement. Malheureusement, la m\'etrique ainsi construite n'est pas $C^{\infty}$ en g\'en\'eral. La premi\`ere ``forme'' de Chern $c_{1}(L,\|.\|_{L,\infty})$ est un courant r\'eel de bidegr\'e (1,1), et donc le produit $c_{1}(L,\|.\|_{L,\infty})^{p}$, et {\it a fortiori\/} le produit $\hat{c}_{1}(L,\|.\|_{L,\infty})^{p}$, ne sont pas d\'efinis. Nous sommes donc amen\'es dans un premier temps \`a \'etendre sur certains points la th\'eorie d\'evelopp\'ee dans \cite{13}, afin de pouvoir consid\'erer en g\'eom\'etrie d'Arakelov des fibr\'es en droites munis de m\'etriques non-n\'ecessairement $C^{\infty}$. Comme, pour autant qu'il nous soit permis d'en juger, ces d\'eveloppements poss\`edent un int\'er\^et propre, nous avons choisi d'en donner une exposition valable en toute g\'en\'eralit\'e. \medskip Quittons donc pour quelques temps l'univers torique, et consid\'erons $X$ une vari\'et\'e arithm\'etique quelconque de dimension absolue $d+1$. Reprenant une terminologie introduite par Zhang \cite{21}, nous dirons d'un couple $(L,\|.\|)$ form\'e d'un fibr\'e en droites $L$ sur $X$ et d'une m\'etrique hermitienne continue $\|.\|$ sur $L(\M{C})$ qu'il est {\it admissible\/} si $L$ est engendr\'e par ses sections globales, $\|.\|$ est positive et peut \^etre approch\'ee uniform\'ement sur $X(\M{C})$ par des m\'etriques positives $C^{\infty}$. Plus g\'en\'eralement, un fibr\'e en droites sur $X$ sera dit {\it int\'egrable\/} s'il est diff\'erence de deux fibr\'es en droites admissibles. Tout fibr\'e en droites $L$ sur $\M{P}(\Delta)$ muni de sa m\'etrique canonique $\|.\|_{L,\infty}$ est int\'egrable (exemple \ref{exemple_decomp1}). \medskip Nous d\'eveloppons alors sur $X(\M{C})$ un formalisme de {\it formes diff\'erentielles g\'en\'eralis\'ees\/}; en particulier la premi\`ere ``forme'' de Chern $c_{1}(\ov{L})$ d'un fibr\'e en droites int\'egrable $\ov{L}$ est une forme diff\'erentielle g\'en\'eralis\'ee \`a notre sens. En nous appuyant sur une th\'eorie d\'evelopp\'ee par Bedford-Taylor \cite{1} puis Demailly \cite{6}, nous montrons comment l'on peut donner un sens au produit de deux telles formes (proposition \ref{produit_generalise}). Nous construisons ensuite un groupe gradu\'e $\widehat{CH}^{\ast}_{\op{int}}(X)$ contenant l'anneau de Chow arithm\'etique usuel $\widehat{CH}^{\ast}(X)$, de telle sorte que pour tout fibr\'e en droites int\'egrable $\ov{L}$ sur $X$, on ait $\hat{c}_{1}(\ov{L}) \in \widehat{CH}^{1}_{\op{int}}(X)$. Nous \'etendons alors partiellement le formalisme d\'evelopp\'e dans \cite{13} au groupe $\widehat{CH}^{\ast}_{\op{int}}(X)$, ce qui nous permet de retrouver certains r\'esultats de Zhang \cite{21} concernant les fibr\'es int\'egrables. Notre principal r\'esultat dans cette direction est le suivant (th\'eor\`eme \ref{produit_bien_defini}). Il existe un accouplement~: \[ \widehat{CH}^{p}_{\op{int}}(X)\otimes \widehat{CH}^{q}_{\op{int}}(X) \longrightarrow \widehat{CH}^{p+q}_{\op{int}}(X)_{\M{Q}}, \] qui prolonge celui d\'efini par Gillet-Soul\'e sur $\widehat{CH}^{\ast}(X)$ et qui munit $\widehat{CH}^{\ast}_{\op{int}}(X)$ d'une structure d'anneau commutatif, associatif et unif\`ere. De plus, on dispose d'un morphisme $\op{\widehat{\op{deg}}}: \widehat{CH}_{\op{int}}^{d+1}(X) \rightarrow \M{R}$ qui prolonge celui d\'efini dans \cite{13}, et tel que (th\'eor\`eme \ref{gdthm}) si $\ov{L}_{1},\dots,\ov{L}_{d+1}$ sont des fibr\'es int\'egrables sur $X$ et $h_{\ov{L}_{1}, \dots,\ov{L}_{d+1}}(X)$ est la hauteur de $X$ relativement \`a $\ov{L}_{1},\dots,\ov{L}_{d+1}$ telle qu'elle est d\'efinie dans \cite{21}, alors~: \[ h_{\ov{L}_{1},\dots,\ov{L}_{d+1}}(X) = \widehat{\op{deg}}(\hat{c}_{1}(\ov{L}_{1}) \dotsm \hat{c}_{1}(\ov{L}_{d+1})). \] Revenons au cas particulier des vari\'et\'es toriques~: Soient $\ov{L}_{1},\dots,\ov{L}_{q}$ des fibr\'es en droites sur $\M{P}(\Delta)$ munis de leur m\'etrique canonique; nous donnons une formule explicite pour le produit g\'en\'eralis\'e $c_{1}(\ov{L}_{1})\dotsm c_{1}(\ov{L}_{q})$ en terme de la combinatoire de $\Delta$ (th\'eor\`eme \ref{calcul_prod}). Nous montrons que cette formule conduit \`a un algorithme particuli\`erement simple pour le calcul du volume mixte de polytopes convexes (remarque \ref{algo_efficace}). Lorsque $q = d+1$, nous montrons que la hauteur canonique $h_{\ov{L}_{1}, \dots, \ov{L}_{d+1}}(\M{P}(\Delta))$ est nulle (proposition \ref{annulation_hauteur}), puis nous approfondissons ce r\'esultat en montrant (th\'eor\`eme \ref{section_chow}) que le sous-anneau de $\widehat{CH}^{\ast}_{\op{int}}(\M{P}(\Delta))$ engendr\'e par les premi\`eres classes de Chern arithm\'etiques $\hat{c}_{1}(L,\|.\|_{L,\infty})$ des fibr\'es en droites $L$ sur $\M{P}(\Delta)$ est isomorphe canoniquement \`a $CH^{\ast}(\M{P}(\Delta))$. Ce r\'esultat est l'analogue arithm\'etique du th\'eor\`eme de Jurkiewicz et Danilov. Nous en d\'eduisons que la hauteur canonique d'une hypersurface dans $\M{P}(\Delta)$ (i.e. sa hauteur relativement \`a des fibr\'es en droites sur $\M{P}(\Delta)$ munis de leur m\'etrique canonique) est donn\'ee essentiellement par la mesure de Mahler du polyn\^ome qui la d\'efinit (proposition \ref{hauteur_hypersurfaces}). En nous appuyant sur les r\'esultats pr\'ec\'edents, nous concluons cet article par la d\'emonstration d'un analogue arithm\'etique du th\'eor\`eme de Bernstein-Koushniren\-ko (th\'eor\`eme \ref{B_K} et corollaire \ref{coro_BK}) donnant une majoration de la hauteur des points d'intersections de $d$ hypersurfaces de $\M{P}(\Delta)$ en fonction de leur hauteur canonique et de leur polyh\`edre de Newton. \medskip Passons maintenant en revue l'organisation de cet article. \medskip Le chapitre 2 est consacr\'e aux vari\'et\'es toriques sur $\op{Spec}\M{Z}$. Nous rappelons au 2.1 quelques propri\'et\'es simples des c\^ones et des \'eventails, et au 2.2 la construction d'apr\`es Demazure des vari\'et\'es toriques et de leurs morphismes canoniques. Nous introduisons au 2.3 la notion de diviseur invariant et de fonction support associ\'ee. Nous rappelons en les adaptant \`a la situation sur $\M{Z}$ certains crit\`eres pour l'amplitude d'un diviseur invariant. Au 2.4 nous montrons comment l'on peut construire de mani\`ere canonique une vari\'et\'e torique projective \`a partir d'un polytope convexe entier. Enfin le 2.5 est consacr\'e \`a la structure de l'anneau de Chow d'une vari\'et\'e torique projective lisse. En utilisant une d\'ecomposition cellulaire sur $\M{Z}$ d'une telle vari\'et\'e, nous montrons comment la plupart des r\'esultats classiques (sur $\M{C}$) s'\'etendent \`a la situation sur $\M{Z}$; nous en profitons pour prouver un th\'eor\`eme d'annulation des groupes $CH^{p,p-1}(\M{P}(\Delta))$. C'est l\`a le seul point vraiment nouveau de ce chapitre. \medskip Au chapitre 3 nous nous int\'eressons plus particuli\`erement aux vari\'et\'es toriques complexes. La vari\'et\'e torique \`a coin $\M{P}(\Delta)_{\geqslant}$ est d\'efinie au 3.1. Nous pr\'esentons au 3.2 un recouvrement canonique de $\M{P}(\Delta)(\M{C})$ introduit par Batyrev et Tschinkel \cite{2} puis \'etudions certaines de ses propri\'et\'es. Nous utilisons ce recouvrement pour construire au 3.3 la m\'etrique canonique $\|.\|_{L,\infty}$. Nous montrons que cette m\'etrique v\'erifie certaines propri\'et\'es de multiplicativit\'e et de fonctorialit\'e (proposition \ref{BT_fonct}). Nous en donnons ensuite deux autres constructions, l'une bas\'ee sur un th\'eor\`eme de Zhang (th\'eor\`eme \ref{zhang}), l'autre par image inverse (proposition \ref{decomposition}). Nous d\'eduisons de cette derni\`ere construction un th\'eor\`eme d'approximation globale pour $\|.\|_{L,\infty}$ (proposition \ref{approximation}). \medskip Le chapitre 4 est centr\'e sur l'\'etude des formes diff\'erentielles g\'en\'eralis\'ees sur une vari\'et\'e complexe arbitraire. Apr\`es avoir rappel\'e au 4.2 la th\'eorie d\'evelopp\'ee par Bedford-Taylor \cite{1} puis Demailly \cite{6}, nous d\'efinissons au 4.3 et au 4.4 plusieurs types remarquables de courants qui, gr\^ace \`a cette th\'eorie, peuvent \^etre multipli\'es. Nous introduisons aux sections 4.5 et 4.7 les notions de fibr\'es en droites admissibles et de fibr\'es en droites int\'egrables, et montrons au 4.6 que tout fibr\'e en droites ample muni d'une m\'etrique positive sur une vari\'et\'e projective est admissible. \medskip Nous abordons au chapitre 5 le c\oe ur de notre sujet, \`a savoir la construction de l'anneau $\widehat{CH}_{\op{int}}^{\ast}(X)$ pour toute vari\'et\'e arithm\'etique $X$. Apr\`es avoir rappel\'e au 5.1 la th\'eorie classique de Gillet-Soul\'e et introduit au 5.2 le groupe de Picard arithm\'etique g\'en\'eralis\'e $\widehat{\op{Pic}}_{\op{int}}(X)$, nous d\'efinissons au 5.3 les groupes de Chow arithm\'etiques g\'en\'eralis\'es $\widehat{CH}^{p}_{\op{int}}(X)$ (d\'efinition \ref{def_gch_gen}) puis nous construisons au 5.4 l'accouplement $\widehat{CH}^{p}_{\op{int}}(X) \otimes \widehat{CH}^{q}_{\op{int}}(X) \rightarrow \widehat{CH}^{p+q}_{\op{int}}(X)_{\M{Q}}$. Finalement nous relions au 5.5 cette construction \`a celle de Zhang \cite{21} (th\'eor\`eme \ref{gdthm}), puis nous d\'emontrons un r\'esultat g\'en\'eral de positivit\'e (proposition \ref{positivite_arithmetique}) qui nous servira au chapitre 8 lors de la d\'emonstration d'un analogue arithm\'etique du th\'eor\`eme de Bernstein-Koushnirenko. \medskip Nous retournons \`a partir du chapitre 6 aux vari\'et\'es toriques. Apr\`es avoir donn\'e au 6.2 une expression explicite, pour tout fibr\'e en droites $\ov{L}$ sur $\M{P}(\Delta)$ muni de sa m\'etrique canonique, du courant $c_{1}(\ov{L})$, nous g\'en\'eralisons ce r\'esultat en donnant au 6.3 une formule explicite pour le produit g\'en\'eralis\'e de tels courants de Chern (th\'eor\`eme \ref{calcul_prod}). Nous d\'emontrons enfin au 6.4 un r\'esultat d'annulation pour un tel produit (corollaire \ref{trivialite2}). \medskip Le chapitre 7 est consacr\'e \`a l'\'etude de la g\'eom\'etrie d'Arakelov des vari\'et\'es toriques. Nous d\'emontrons au 7.1 un th\'eor\`eme d'annulation des multihauteurs canoniques de $\M{P}(\Delta)$ (proposition \ref{annulation_hauteur}) et \'etablissons au 7.2 une formule reliant la hauteur canonique d'une hypersurface dans $\M{P}(\Delta)$ \`a la hauteur de Mahler du polyn\^ome qui la d\'efinit (proposition \ref{hauteur_hypersurfaces}). Le 7.3 pr\'esente un exemple int\'eressant~: le calcul de la mesure de Mahler d'une droite dans $\M{P}^{2}$. Nous exhibons au 7.4 une section canonique du morphisme d'anneaux application cycle $\zeta: \widehat{CH}^{\ast}_{\op{int}}(\M{P}(\Delta)) \rightarrow CH^{\ast}(\M{P}(\Delta))$. \medskip Au 8.1, nous associons de mani\`ere fonctorielle une constante r\'eelle positive $L(\nabla)$ \`a tout polytope convexe entier $\nabla$ de $\M{R}^{d}$, puis nous en donnons une majoration explicite dans le cas o\`u $\nabla$ est absolument simple (proposition \ref{majoration_cas_AS}). Finalement nous d\'emontrons par r\'ecurrence au 8.2 un analogue arithm\'etique du th\'eor\`eme de Bernstein-Koushnirenko faisant intervenir dans son \'enonc\'e les constantes $L(\nabla)$ pr\'ec\'edemment introduites (th\'eor\`eme \ref{B_K} et corollaire \ref{coro_BK}). \medskip Une partie des r\'esultats pr\'esent\'es ici avaient \'et\'e annonc\'es dans \cite{34}. \medskip L'auteur tient \`a exprimer sa profonde gratitude \`a J.-B. Bost sans l'aide duquel ce travail n'aurait jamais vu le jour. Il lui est \'egalement agr\'eable de remercier J. Cassaigne, A. Chambert-Loir, D. Harari, M. Laurent, E. Leichtnam, J. Pfeiffer, C. Soul\'e et B. Teissier pour des discussions int\'eressantes. \bigskip

9304

alg-geom/9304004

en

https://arxiv.org/abs/alg-geom/9304004

alg-geom/9304004

Reyer Sjamaar

Holomorphic Slices, Symplectic Reduction and Multiplicities of Representations

34 pages, AmS-LaTeX version 1.1

I prove the existence of slices for an action of a reductive complex Lie group on a K\"ahler manifold at certain orbits, namely those orbits that intersect the zero level set of a momentum map for the action of a compact real form of the group. I give applications of this result to symplectic reduction and geometric quantization at singular levels of the momentum map. In particular, I obtain a formula for the multiplicities of the irreducible representations occurring in the quantization in terms of symplectic invariants of reduced spaces, generalizing a result of Guillemin and Sternberg.

alg-geom

\section*{Introduction} There has recently been much interest in formulas for multiplicities of Lie group representations arising in various different ways from group actions on manifolds. Typically, one can think of the manifold as being the phase space $M$ of a classical physical system acted upon by a group $G$ of symmetries, from which one obtains a unitary representation of $G$ through some kind of ``quantization''. A prototype of such formulas is the multiplicity formula of Guillemin and Sternberg \cite{gu:ge}. In their set-up the space $M$ is a compact K\"ahler manifold on which the compact group $G$ acts by holomorphic transformations, and the associated representation of $G$ is the space of holomorphic sections of a certain $G$-equivariant line bundle over $M$ (``geometric quantization''). The main result of \cite{gu:ge} expresses the multiplicities of the irreducible components of this representation in terms of the Riemann-Roch numbers of the symplectic (or Marsden-Weinstein-Meyer) quotients $M_\lambda$ of $M$. The most important auxiliary result is that the symplectic quotient at the zero level, $M_0$, can be identified with a geometric quotient of $M$ by the complexified group $G\co$ as defined by Mumford \cite{mu:ge}. The purpose of this paper is to generalize these results to the case where the symplectic quotient $M_\lambda$ is singular, a case which is of some interest in applications, but was excluded by Guillemin and Sternberg. This involves a closer study of the orbit structure of the action of the reductive group $G\co$ on $M$. The main technical result, expounded in Section \ref{section:slices}, is that one can construct slices for the $G\co$-action at points that are in the zero level set of a momentum map. The proof of this holomorphic slice theorem utilizes H\"ormander's $L_2$-estimates for the Cauchy-Riemann operator. For affine algebraic manifolds it is a special case of Luna's \'etale slice theorem \cite{lu:sl}. Kirwan \cite{ki:coh} has introduced the notion of the K\"ahler quotient of $M$ by $G\co$, which is the K\"ahler analogue of Mumford's categorical quotient. It is roughly speaking defined as the space of closed $G\co$-orbits in $M$. Kirwan showed it is homeomorphic to the symplectic quotient $M_0$, generalizing the result of Guillemin and Sternberg referred to above. In \cite{sj:st} it was shown that $M_0$ is a so-called symplectic stratified space. In Section \ref{section:quotient} I exploit the holomorphic slice theorem to study the analytic structure of Kirwan's quotient and to compare it to the stratified symplectic structure of the symplectic quotient. In the case of a projective manifold $M$ endowed with the Fubini-Study symplectic form, Kirwan and, independently, Ness \cite{ne:st} showed that the symplectic quotient $M_0$ coincides with Mumford's categorical quotient, the Proj of the invariant part of the homogeneous coordinate ring of $M$. I show that the same conclusion holds when $M$ has an arbitrary integral K\"ahler structure. (Under this hypothesis $M$ has a unique algebraic structure by Kodaira's Embedding Theorem, but the symplectic structure is not necessarily one coming from a Fubini-Study metric.) This result allows me to carry through the geometric quantization of the symplectic quotient. Now a theorem of Boutot \cite{bo:si} asserts that the singularities of a quotient such as $M_0$ are {\em rational}. This basically says that the Riemann-Roch numbers of $M_0$ are equal to those of any blowup, which, finally, leads to a generalization of Guillemin and Sternberg's multiplicity formula. \begin{acknowledgement} I am grateful to Hans Duistermaat for suggesting to me the problem discussed in this paper and for showing me an unpublished manuscript of his, a succinct version of which appeared in \cite{du:mu}. Eugene Lerman has been a great help in carrying out this work. Part of it appears as a joint announcement in \cite{le:re}. I would also like to thank Eugenio Calabi and Charlie Epstein for their generous help with the material in Section \ref{subsection:totallyreal}. I have furthermore benefited from helpful discussions with Victor Guillemin and Viktor Ginzburg. \end{acknowledgement} \section{Holomorphic Slices}\label{section:slices} Let $X$ be a complex space and let $G\co$ be a reductive complex Lie group acting holomorpically on $X$. I think of $G\co$ as being the complexification of a compact real Lie group $G$. \begin{definition} A {\em slice\/} at $x$ for the $G\co$-action is a locally closed analytic subspace $S$ of $X$ with the following properties: \begin{enumerate} \item $x\in S$; \item $G\co S$ of $S$ is open in $X$; \item $S$ is invariant under the action of the stabilizer $(G\co)_x$; \item\label{bundle} the natural $G\co$-equivariant map from the associated bundle $G\co\times_{(G\co)_x}S$ into $X$, which sends an equivalence class $[g,y]$ to the point $gy$, is an analytic isomorphism onto $G\co S$. \end{enumerate} \end{definition} It follows from (\ref{bundle}) that for all $y$ in $S$ the stabilizer $(G\co)_y$ is contained in $(G\co)_x$. Furthermore, if $X$ is nonsingular at $x$, a slice $S$, if it exists, is nonsingular at $x$ and transverse to the orbit $G\co x$. The problem of constructing slices has been solved by Luna \cite{lu:sl} for affine varieties and by Snow \cite{sn:re} for Stein spaces. One difficulty of the problem lies in the fact that an action of $G\co$ is typically not proper, unless it is locally free. One therefore faces the challenge of controlling the behaviour of the action ``at infinity in the group''. Another snag is that there may be cohomological obstructions to analytically embedding the ``normal bundle'' $G\co\times_{(G\co)_x}S$ of the orbit $G\co x$ into $X$. These obstructions vanish if the orbit is (analytically isomorphic to) an affine variety. A theorem of Matsushima's \cite{ma:es} says that a $G\co$-orbit is affine if and only if the isotropy subgroup $(G\co)_x$ is reductive. But even if the isotropy of $x$ is reductive one cannot always construct a slice at $x$. (Cf.\ Richardson's example \cite{le:re,lu:sl,ri:de} of the standard action of $\SL(2)$ on homogeneous cubic polynomials, and also Trautman \cite{tr:or}.) The additional condition that Luna, resp.\ Snow, impose in the context of an affine variety $X$, resp. a Stein space $X$, in order to deduce the existence of a slice is that the orbit should be {\em closed\/} in $X$. The above notion of a slice is slightly weaker than that of Luna and Snow, who require the set $G\co S$ to be saturated with respect to a quotient mapping. In our context the definition of a quotient depends upon the choice of a momentum map. In the next section we shall see that for any choice of a momentum map there always exists slices $S$ such that $G\co S$ is saturated with respect to the corresponding quotient map (Proposition \ref{proposition:saturated}). In this section I demonstrate the existence of slices at certain affine orbits of a $G\co$-action on a K\"ahler manifold. I was led to this result by the striking resemblance between Luna's and Snow's slice theorems and the normal forms in symplectic geometry due to Marle \cite{ma:mo} and Guillemin and Sternberg \cite{gu:no}. Before formulating the theorem I have to state a number of definitions and auxiliary results. In Section \ref{subsection:orbits} I discuss momentum maps on K\"ahler manifolds and the notion of orbital convexity. Section \ref{subsection:totallyreal} contains a result concerning interpolation between K\"ahler metrics in the neighbourhood of a totally real submanifold of a complex manifold, which relies on H\"ormander's $\bar\partial$-estimates, and which is the main ingredient in the proof of the slice theorem. In Section \ref{subsection:slices} I prove the slice theorem and discuss some of its immediate consequences. \subsection{Orbital convexity and isotropic orbits}\label{subsection:orbits} Recall that the decomposition of the complexified Lie algebra $\frak g\co=\frak g\otimes\bold C$ into a direct sum $\frak g\co=\frak g\oplus\sq\,\frak g$ gives rise to the polar (or Cartan) decomposition $G\co=G\cdot\exp\sq\,\frak g$. The map $G\times\sq\,\frak g\to G\co$ sending $(k,\sq\,\xi)$ to $k\exp\sq\,\xi$ is a diffeomorphism onto, and every element $g$ of $G\co$ can be uniquely decomposed into a product $g=k\exp\sq\,\xi$, with $k\in G$ and $\xi\in\frak g$. \begin{definition}[Heinzner {\cite{he:ge}}] A subset $A$ of a $G\co$-space $X$ is called {\em orbitally convex\/} with respect to the $G\co$-action if it is $G$-invariant and if for all $x$ in $U$ and all $\xi$ in $\frak g$ the intersection of the curve $\bigl\{\,\exp(\sq\,t\xi)x:t\in\bold R\,\bigr\}$ with $A$ is connected. Equivalently, $A$ is orbitally convex if and only if it is $G$-invariant and for all $x$ in $A$ and all $\xi$ in $\frak g$ the fact that both $x$ and $\exp(\sq\,\xi)x$ are in $A$ implies that $\exp(\sq\,t\xi)x\in A$ for all $t\in[0,1]$. \end{definition} \begin{remark}\label{remark:trivial} If $f\colon X\to Y$ is a $G\co$-equivariant map between $G\co$-spaces $X$ and $Y$, and $C$ is an orbitally convex subset of $Y$, then it follows immediately from the definition that $f^{-1}(C)$ is orbitally convex in $X$. \end{remark} A $G$-equivariant map defined on an orbitally convex open set can be analytically continued to a $G\co$-equivariant map. \begin{proposition}[Heinzner {\cite{he:ge}}, Koras \cite{ko:li}]\label{proposition:orbit} Let $X$ and $Y$ be complex manifolds acted upon by $G\co$. If $A$ is an orbitally convex open subset of $X$ and $f\colon A\to Y$ is a $G$-equivariant holomorphic map\rom, then $f$ can be uniquely extended to a $G\co$-equivariant holomorphic map $f\co\colon G\co A\to Y$. Consequently, if the image $f(A)$ is open and orbitally convex in $Y$ and $f\colon A\to f(A)$ is biholomorphic\rom, then the extension $f\co\colon G\co A\to Y$ is biholomorphic onto the open subset $G\co f(A)$. \end{proposition} \begin{pf} The only way to extend $f$ equivariantly is by putting $f\co\bigl(g\exp(\sq\,\xi)x\bigr)=g\exp(\sq\,\xi)f(x)$ for all $x$ in $A$, $g$ in $G$ and $\xi$ in $\frak g$. We have to check this is well-defined. Let $x\in A$ and $\xi\in\frak g$ be such that $\exp(\sq\,\xi)x\in A$. Then by assumption $\exp(\sq\,t\xi)x\in A$ for all $t$ between 0 and 1. So $f\bigl(\exp(\sq\,t\xi)x\bigr)$ is well-defined for $0\leq t\leq 1$. Define the curves $\alpha(t)$ and $\beta(t)$ in $Y$ by $\alpha(t)=f\bigl(\exp(\sq\,t\xi)x\bigr)$ and $\beta(t)= \exp(\sq\,t\xi)f(x)$ for $0\leq t\leq 1$. Then $\alpha(t)$ and $\beta(t)$ are integral curves of the vector fields $f_*(\sq\,\xi)_X$ and $(\sq\,\xi)_Y$ respectively, both with the same initial value $f(x)$. Now since $f$ is $G$-equivariant we have $f_*\xi_X=\xi_Y$, and, because $f$ is also holomorphic, $f_*(\sq\,\xi)_X= f_*(J\xi_X)= Jf_*\xi_X= J\xi_Y= (\sq\,\xi)_Y$. Hence $\alpha(t)=\beta(t)$, in other words $f\bigl(\exp(\sq\,t\xi)x\bigr)=\exp(\sq\,t\xi)f(x)$ for $0\leq t\leq 1$. It follows that for all $x$ in $A$ and all $\xi$ in $\frak g$ such that $\exp(\sq\,\xi)x$ is in $A$ we have $f\bigl(\exp(\sq\,\xi)x\bigr)=\exp(\sq\,\xi)f(x)$. It is easy to deduce from this that $f\co$ is well-defined. Finally observe that if the image $f(A)$ is open and orbitally convex in $Y$ and $f\colon A\to f(A)$ is biholomorphic, then the inverse $f^{-1}$ also has a holomorphic extension $(f^{-1})\co\colon G\co f(A)\to G\co A$, and by uniqueness this must be the inverse of $f\co$. \end{pf} \begin{remark}\label{remark:local} Suppose we drop the assumption that $A$ is orbitally convex from the statement of the proposition. Then it is not necessarily true that $f\bigl(\exp(\sq\,t\xi)x\bigr)$ is equal to $\exp(\sq\,t\xi)f(x)$ for all $t$ such that $\exp(\sq\,t\xi)x\in A$. But if we put $I=\{\,t\in\bold R:\exp(\sq\,t\xi)x\in A\,\}$ and let $I^0$ be the connected component of $I$ containing $0$, then the above proof shows that $f\bigl(\exp(\sq\,t\xi)x\bigr)=\exp(\sq\,t\xi)f(x)$ for all $t$ in $I^0$. \end{remark} In the remainder of this section $M$ shall denote a K\"ahler manifold, not necessarily compact, with infinitely differentiable K\"ahler metric $ds^2$, K\"ahler form $\omega=-\Im ds^2$, and complex structure $J$. Then $\Re ds^2=\omega(\cdot,J\cdot)$ is the corresponding Riemannian metric. We may assume without loss of generality that $ds^2$ is invariant under the compact group $G$. So the transformations on $M$ defined by $G$ are holomorphic and they are isometries with respect to the K\"ahler metric. The action of $G$ is called {\em Hamiltonian\/} if for all $\xi$ in the Lie algebra $\frak g$ of $G$ the vector field $\xi_M$ on $M$ induced by $\xi$ is Hamiltonian. In this case we have a {\em momentum map\/} $\Phi$ from $M$ to the dual $\frak g^*$ of the Lie algebra of $G$ with the property that $$d\Phi^\xi=\iota_{\xi_M}\omega$$ for all $\xi$. Here $\Phi^\xi$ is the $\xi$-th component of $\Phi$, defined by $\Phi^\xi(m)=\bigl(\Phi(m)\bigr)(\xi)$. After averaging with respect to the given action on $M$ and the coadjoint action on $\frak g^*$ we may assume that the map $\Phi$ is $G$-equivariant. An equivariant momentum map is uniquely determined up to additive constants ranging over the $\Ad^*$-fixed vectors in $\frak g^*$. (So if $G$ is connected, the number of degrees of freedom is equal to the dimension of the centre of $G$.) It is easy to give sufficient conditions for the existence of a momentum map, for instance, the first Betti number of $M$ is zero, or the K\"ahler form $\omega$ is exact. (See e.g.\ \cite{gu:sy,we:le}.) More surprisingly, by a theorem of Frankel \cite{fr:fi} a momentum map always exists if the action has at least one fixed point and $M$ is {\em compact}. A necessary and sufficient condition for a holomorphic $G$-action on a compact K\"ahler manifold to be Hamiltonian is that for every vector $\xi\in\frak g$ the holomorphic vector field $\xi_M$ should be killed by every global holomorphic one-form $\alpha$ on $M$, $\alpha(\xi_M)=0$. (This follows from Frankel's theorem and a fixed point theorem of Sommese \cite{so:ex}.) Note that this condition is independent of the K\"ahler structure. If $M$ is $\bold C^n$ with the standard Hermitian structure $dS^2=\sum_idz_i\otimes d\bar z_i$ and the standard symplectic form $\Omega=\sq\big/2\,\sum_idz_i\wedge d\bar z_i$, then a momentum map $\Phi_{\bold C^n}$ is given by the formula \begin{equation}\label{equation:quadratic} \Phi_{\bold C^n}^\xi(v)=1/2\,\Omega\bigl(\xi_{\bold C^n}(v),v\bigr), \end{equation} where $\xi _{\bold C^n}$ denotes the image of $\xi\in\frak g$ in the Lie algebra $\frak s\frak p({\bold C^n},\Omega)$, and $v\in\bold C^n$. Because $G$ acts holomorphically on $M$, there is a natural way to define an action of the complexified Lie algebra $\frak g^*$: For any $\xi$ in $\frak g$ the vector field $(\sq\,\xi)_M$ induced by $\sq\,\xi$ is equal to $J\xi_M$. It follows easily from the definition of a momentum map that $J\xi_M$ is equal to the gradient vector field (with respect to the Riemannian metric $\Re ds^2$) of the $\xi$-th component of the momentum map, \begin{equation}\label{equation:grad} (\sq\,\xi)_M=J\xi_M=\grad\Phi^\xi. \end{equation} We will assume that these vector fields are {\em complete\/} for all $\xi$ in $\frak g$. This assumption implies that the action of $G$ extends uniquely to a holomorphic action of $G\co$. The assumption holds for instance if $M$ is compact, or if $M$ is the total space of a vector bundle over a compact manifold on which $G$ acts by vector bundle transformations. The identity (\ref{equation:grad}) will enable us to gain control over the behaviour of the action ``at infinity in the group''. For one thing, it implies that the trajectory $\gamma(t)$ of $\grad\Phi^\xi$ through a point $x$ in $M$ is given by $\gamma(t)=\exp(\sq\,t\xi)x$, which does not depend on the choice of the K\"ahler metric or the momentum map. Here is another application of (\ref{equation:grad}). A submanifold $X$ of $M$ is called {\em totally real\/} if $T_xX\cap J\bigl(T_xX\bigr)=\{0\}$ for all $x\in X$. \begin{proposition}\label{proposition:totallyreal} Assume $G$ is connected\rom. Consider the following conditions on a point $m\in M$\rom: \begin{enumerate} \item\label{fixed} $\Phi(m)$ is fixed under the coadjoint action of $G$ on $\frak g^*$\rom; \item\label{isotropic} The orbit $Gm$ is isotropic with respect to the K\"ahler form\rom; \item\label{stabilizer} The complex stabilizer $(G\co)_m$ of $m$ is equal to the complexification $(G_m)\co$ of the compact stabilizer $G_m$\rom, $(G_m)\co=(G\co)_m$\rom; \item\label{realorbit} The $G$-orbit through $m$ is totally real\rom. \end{enumerate} Conditions \rom(\ref{fixed}\rom) and \rom(\ref{isotropic}\rom) are equivalent\rom. Any one of these conditions implies \rom(\ref{stabilizer}\rom)\rom; and \rom(\ref{stabilizer}\rom) implies \rom(\ref{realorbit}\rom)\rom. \end{proposition} \begin{pf} Put $\mu=\Phi(m)$. Let $G_\mu$ be the stabilizer of $\mu$ with respect to the coadjoint action; it is well-known that $G_\mu$ is a connected subgroup of $G$. Let $G\mu$ be the coadjoint orbit through $\mu$. We regard $G\mu$ as a symplectic manifold with the Kirillov-Kostant-Souriau symplectic form. Denote the tangent space $T_m(G m)$ to the compact orbit by $\frak m$; then $\frak m$ is isomorphic to $\frak g/\frak g_m$ as an $H$-module. Similarly, let $\frak n$ denote the tangent space $T_m(G_\mu m)$ to the orbit $G_\mu m$; then $\frak n\cong\frak g_\mu/\frak g_m$. We have a fibration $$ G_\mu m\hookrightarrow G m\overset{\Phi}{\to}G\mu, $$ which on the tangent level leads to an exact sequence of vector spaces $$ 0\to\frak n\to\frak m\overset{d\Phi}{\to}T_\mu(G\mu)\to 0. $$ The restriction of the symplectic form $\omega$ to $\frak m$ is an alternating bilinear (``presymplectic'') form. It follows from the fact that $\Phi$ is a Poisson map that $d\Phi$ preserves the presymplectic forms. Since $T_\mu(G\mu)$ is symplectic, $\frak n$ is exactly the nullspace of $\omega|_{\frak m}$. Therefore, $\frak m$ is isotropic if and only if $T_\mu(G\mu)=0$, that is, $G_\mu=G$, in other words, $\mu$ is $G$-fixed. This shows that (\ref{fixed}) is equivalent to (\ref{isotropic}). We now prove (\ref{fixed}) implies (\ref{stabilizer}). It is easy to see that for any point $m$ in $M$ the complex stabilizer $(G\co)_m$ contains the complexification $(G_m)\co$ of the compact stabilizer $G_m$. Now suppose $\mu$ is fixed under the coadjoint action. Let $g\exp\sq\,\xi$ be an arbitrary element of $\dim(G\co)_m$, where $g\in G$ and $\xi\in\frak g$. We want to show that $g\in G_m$ and $\xi\in\frak g_m$. (Cf.\ Kirwan \cite{ki:coh} for this part of the argument.) By $G$-equivariance of the momentum map we have $$ \Phi\bigl(\exp(\sq\,\xi)m\bigr)= g^{-1}\Phi\bigl(g\exp(\sq\,\xi)m\bigr)= g^{-1}\Phi(m)=g^{-1}\mu=\mu, $$ and therefore $\Phi^\xi\bigl(\exp(\sq\,\xi)m\bigr)=\Phi^\xi(m)$. By (\ref{equation:grad}) the curve $\exp(\sq\,t\xi)m$ is the gradient trajectory of the vector field $(\sq\,\xi)_M$ through $m$. So the function $\Phi^\xi$ is increasing along this curve, and it is strictly increasing if and only if $m$ is not a fixed point of $(\sq\,\xi)_M$. But it takes on the same values at $t=0$ and $t=1$, so $m$ must be a fixed point of $(\sq\,\xi)_M$, that is, $\xi\in\frak g_m$. Hence $gm=g\exp(\sq\,\xi)m=m$, so $g\in G_m$. Lastly we show (\ref{stabilizer}) implies (\ref{realorbit}). If $(G_m)\co=(G\co)_m$, the (real) dimension of the complex orbit $G\co m$ equals twice the dimension of the compact orbit $Gm$. Since the tangent space at $m$ to $G\co m$ is equal to $T_m(Gm)+J\bigl(T_m(Gm)\bigr)$, the intersection $T_m(Gm)\cap J\bigl(T_m(Gm)\bigr)$ has to be 0, that is, $Gm$ is totally real. \end{pf} \begin{remark}\label{remark:connected} The converse of the implications in the proposition are wrong. See \cite{le:re} for a simple counterexample. If $G$ is not connected, then the proof of the proposition shows the following implications hold: (\ref{fixed}) $\Rightarrow$ (\ref{isotropic}) $\Rightarrow$ (\ref{stabilizer}) $\Rightarrow$ (\ref{realorbit}). Moreover, if $Gm$ is isotropic, then $\Phi(m)$ is fixed under $G^0$, where $G^0$ denotes the component of the identity of $G$. \end{remark} Let $V$ be the orthogonal complement of $T_m(G\co m)$. Then using the notation of the proposition we have an $H$-invariant orthogonal direct sum decomposition of the tangent space: \begin{equation}\label{equation:sum} T_mM=\frak n\oplus J\frak n\oplus T_\mu(G\mu)\oplus V. \end{equation} This decomposition is symplectic in the sense that the summands $\frak n\oplus J\frak n$, $T_\mu(G\mu)$ and $V$ are symplectic subspaces, but it is not a complex-linear decomposition, since $T_\mu(G\mu)$ need not be $J$-invariant. \subsection{Interpolation of K\"ahler metrics near totally real submanifolds}\label{subsection:totallyreal} Let $M$ be a complex manifold and let $\sigma$ be a real-valued $C^\infty$ closed $(1,1)$-form on $M$. On any sufficiently small open subset $O$ of $M$ we can find a potential for $\sigma$, that is, a smooth real-valued function $u$ defined on $O$ such that $\sigma= \sq\,\partial\bar\partial u$. Now let $X$ be any real-analytic totally real submanifold of $M$. \begin{theorem}\label{theorem:potential} If the form $\sigma$ is exact in a neighbourhood of $X$\rom, then there exists a potential for $\sigma$ defined in a \rom(possibly smaller\rom) neighbourhood of $X$. Assume $\sigma$ vanishes to $m$-th order on $X$\rom, that is\rom, in local coordinates its coefficients vanish to $m$-th order on $X$\rom. Then $\sigma$ is exact near $X$\rom, and there exists a potential for $\sigma$ defined near $X$ which vanishes to $(m+2)$-nd order on $X$. \end{theorem} \begin{pf} Because $X$ is totally real, it has a basis of Stein tubular neighbourhoods in $M$. (See Grauert \cite{gr:on} and Reese and Wells \cite[Theorem 2.2]{ha:ho}.) Without loss of generality we may replace $M$ by one of these tubular neighbourhoods. Let $\alpha$ be a solution to the equation \begin{equation}\label{equation:poincare} d\alpha=\sigma, \end{equation} and let $\beta=\alpha^{01}$ be the $(0,1)$-part of $\alpha$. It is evident from the fact that $\sigma$ is of bidegree $(1,1)$ that $\bar\partial\beta=0$. So we can solve the equation \begin{equation}\label{equation:dolbeault} \sq\,\bar\partial f=\beta, \end{equation} since $M$ is a Stein manifold. It is easy to check that the function $u=f+\bar f=2\Re f$ satisfies $\sigma=\sq\,\partial\bar\partial u$. This proves the first statement. Now suppose $\sigma$ vanishes to $m$-th order on $X$. Then, evidently, the restriction of $\sigma$ to $X$ is zero. Since $\sigma$ is also closed, it follows easily from De Rham's Theorem that it is exact on the tubular neighbourhood $M$. Let $C\colon M\times[0,1]\to M$ be the homotopy defined by $C(p,t)=tp$, which contracts the bundle $M$ to the zero section along the fibres. There exists a very special solution to (\ref{equation:poincare}), namely the form $\alpha=\int_{[0,1]}C^*\sigma$. It is not hard to check that this form vanishes to $(m+1)$-st order on $X$. Therefore its $(0,1)$-part $\beta=\alpha^{01}$ also vanishes to $(m+1)$-st order on $X$. We now want to solve (\ref{equation:dolbeault}) augmenting the order of vanishing by one. We do this in three steps. First, we solve the problem locally and formally. That is, we assume $X$ is an open set in $\bold R^k$ and $M$ a strictly pseudoconvex open neighbourhood of $X$ in $\bold C^n$ (where $n\geq k$), and find a function $g$ vanishing to $(m+2)$-nd order on $X$ such that the $(0,1)$-form $\beta'=\beta-\sq\,\bar\partial g$ also vanishes to order $m+2$ on $X$. Secondly, and this is the crucial point, we use H\"ormander's $L^2$-estimates for the Cauchy-Riemann operator \cite{ho:in} to show that locally there exists a smooth solution $g'$ to the problem $\sq\,\bar\partial g'=\beta'$ which vanishes to $(m+2)$-nd order on $X$. Then the function $f=g+g'$ satisfies (\ref{equation:dolbeault}) and vanishes to $(m+2)$-nd order on $X$. Thirdly, we show that the local solutions to the problem can be glued together to obtain a global solution, which amounts to solving a Cousin type problem. {\it Step} 1. A {\em complexification\/} $(X\co,i)$ of the real-analytic manifold $X$ is a complex manifold $X\co$ together with a real-analytic map $i\colon X\to X\co$ such that for every complex manifold $V$ and every real-analytic map $j\colon X\to V$ there exists, for a sufficiently small open neighbourhood $O$ of $i(X)$, a unique complex-analytic map $j\co\colon O\to V$ with $j\co\circ i=j$. The uniqueness of the complexification (more precisely, the uniqueness of the germ of $X\co$ at $i(X)$) is immediate from the definition; the existence was proven by Bruhat and Whitney \cite{wh:qu}. The map $i$ is actually a closed embedding. If $j\colon X\to V$ is an embedding and the image $j(X)$ is totally real, the complexified map $j\co$ is an embedding (near $i(X)$). So after shrinking the tube $M$ if necessary, we may assume that we have inclusions $X\subset X\co\subset M$. About every point of $X$ we can find an open neighbourhood that can be biholomorphically identified with a strictly pseudoconvex bounded open subset $U$ of $\bold C^k\times\bold C^l$, in such a manner that $U\cap X$ is given by the equations $w=y=0$ and $U\cap X\co$ by $w=0$. Here we write a point in $\bold C^k\times\bold C^l$ as a pair $(z,w)$ with $z=x+\sq\,y\in\bold C^k$ and $w\in\bold C^l$. We shall call a neighbourhood with such a coordinate system a {\em distinguished\/} neighbourhood. Write $\beta=\sum_{\lambda=1}^{k}\zeta^\lambda\,d\bar z_\lambda + \sum_{\lambda=1}^{l}\eta^\lambda\,d\bar w_\lambda$ and consider the Taylor expansions at $X$ of the components $\zeta^\lambda$ and $\eta^\lambda$: \begin{align*} \zeta^\lambda(x,y,w,\bar w)&\sim \sum_{|I|+|J|+|K|\geq m+2} \zeta^\lambda_{I,J,K}(x)y^Iw^J\bar w^K,\\ \eta^\lambda(x,y,w,\bar w)&\sim \sum_{|I|+|J|+|K|\geq m+2} \eta^\lambda_{I,J,K}(x)y^Iw^J\bar w^K, \end{align*} with coefficients $\zeta^\lambda_{I,J,K}$ and $\eta^\lambda_{I,J,K}$ in $C^\infty(X,\bold C)$. Here $I$, $J$ and $K$ are multi-indices and $|I|$ denotes the norm $\sum_\lambda i_\lambda$ of $I=(i_1,\dots,i_k)$. The fact that $\bar\partial\beta=0$ amounts to: $$ \frac{\partial\zeta^\lambda}{\partial\bar z_\mu}= \frac{\partial\zeta^\mu}{\partial\bar z_\lambda},\qquad \frac{\partial\eta^\lambda}{\partial\bar z_\mu}= \frac{\partial\zeta^\mu}{\partial\bar w_\lambda},\qquad \frac{\partial\eta^\lambda}{\partial\bar w_\mu}= \frac{\partial\eta^\mu}{\partial\bar w_\lambda}. $$ Plugging the Taylor expansions of $\zeta^\lambda$ and $\eta^\lambda$ into this system of equations and inspecting the lowest-order terms in the resulting equalities yields the following identities: \begin{align}\label{equation:compatibility} (i_\mu+1)\zeta^\lambda_{I+e_\mu,J,K}&= (i_\lambda+1)\zeta^\mu_{I+e_\lambda,J,K},\notag\\ \sq\,(i_\mu+1)\eta^\lambda_{I+e_\mu,J,K}&= (k_\lambda+1)\zeta^\mu_{I,J,K+e_\lambda},\\ (k_\mu+1)\eta^\lambda_{I,J,K+e_\mu}&= (k_\lambda+1)\eta^\mu_{I,J,K+e_\lambda},\notag \end{align} for all $I$, $J$ and $K$ such that the total degree $|I|+|J|+|K|$ equals $m+1$. Here $e_\lambda$ denotes the multi-index whose entries are all 0, except the $\lambda$-th, which is 1. (For higher-order terms there are similar identities, but we will not need them.) Our object is to find a smooth function $g$ such that $\sq\,\bar\partial g=\beta$ up to terms of total degree $\geq m+3$ in $y$, $w$ and $\bar w$. Upon substitution of the Taylor expansion of $g$, $$ g(x,y,w,\bar w)\sim\sum_{|I|+|J|+|K|\geq m+3} g_{I,J,K}(x)y^Iw^J\bar w^K, $$ we see this amounts to solving the equations \begin{equation}\label{equation:relations} g_{I+e_\lambda,J,K}=-\frac{2\zeta^\lambda_{I,J,K}}{i_\lambda+1},\qquad g_{I,J,K+e_\lambda}=\frac{2\eta^\lambda_{I,J,K}}{\sq(k_\lambda+1)}, \end{equation} for all $I$, $J$ and $K$ such that $|I|+|J|+|K|=m+2$. There are no conditions on the terms of degree $>m+3$ in the expansion of $g$. It is a straightforward exercise to check that the equations (\ref{equation:relations}) with coefficients subject to the compatibility relations (\ref{equation:compatibility}) admit solutions $g_{I,J,K}$, where $|I|+|J|+|K|=m+3$. So if we put $g(x,y,w,\bar w)=\sum_{|I|+|J|+|K|=m+3} g_{I,J,K}(x)y^Iw^J\bar w^K$, then $g$ is a smooth function defined on $U$ and vanishing to $(m+2)$-nd order on $X$, and the $(0,1)$-form $\beta'=\beta-\sq\,\bar\partial g$ also vanishes to order $m+2$ on $X$. {\it Step} 2. Obviously the form $\beta'$ is $\bar\partial$-closed. We now want to find a smooth solution $g'$ defined on $U$ to the problem $\sq\,\bar\partial g'=\beta'$ together with an order estimate. But first note that {\em every\/} locally square integrable solution to this equation is actually smooth. This follows from the fact that there exists a smooth solution (\cite[Corollary 4.2.6]{ho:in}), and that the difference of any two solutions is a $\bar\partial$-closed function, therefore harmonic, and therefore smooth by the ellipticity of the Laplacian on $\bold C^n$. Let $\rho$ be the distance squared to the submanifold $X$, $\rho(z,w)=\|y\|^2+\|w\|^2$. Because $\beta'$ vanishes to order $m+2$ on $X$, the integral $\int_U|\beta'|^2\rho^{-r}\,dzd\bar zdwd\bar w$ is finite for all $r<k+2l+m+3$. In H\"ormander's parlance $\beta'$ is an element of the weighted $L^2$-space $L^2_{(0,1)}(U,\phi)$ with weight $\phi=r\log\rho$. It is easy to check that for every positive $r$ the weight function $\phi$ is plurisubharmonic. Let us take $r=k+2l+m+2$. By Theorem 4.4.2 of H\"ormander \cite{ho:in}, we can find a solution to the equation $\sq\,\bar\partial g'=\beta'$ such that $$ \int_U|g'|^2e^{-\phi}\bigl(1+\|z\|^2+\|w\|^2\bigr)^{-2}\,dzd\bar zdwd\bar w\leq\int_U|\beta'|^2e^{-\phi}\,dzd\bar zdwd\bar w<\infty. $$ But $g'$ is smooth as noted before, so this is only possible if $g'$ vanishes to order $\geq r-k-2l=m+2$ on $X$. The function $f=g+g'$ defined on $U$ satisfies (\ref{equation:dolbeault}) and vanishes to order $m+2$ on $X$. {\it Step} 3. Let $\{U_i\}$ be a Stein cover of $M$. By the previous result we can find smooth functions $f_i$ defined on $U_i$, which vanish to $(m+2)$-nd order on $U_i\cap X$ and satisfy $\sq\,\bar\partial f_i=\beta|_{U_i}$. Put $f_{ij}=f_j-f_i$; then $\bar\partial f_{ij}=0$, so $f_{ij}$ is a holomorphic function on $U_{ij}=U_i\cap U_j$. It also vanishes to $(m+2)$-nd order on $U_{ij}\cap X$, so, by Lemma \ref{lemma:vanish} below, it has to vanish to order $m+2$ on $U_{ij}\cap V$, where $V$ denotes the complexification $V=X\co$ of $X$. In other words, the collection of $f_{ij}$'s defines a \v Cech 1-cocycle with coefficients in the coherent sheaf $\cal I_V^{m+3}$, where $\cal I_V$ denotes the ideal sheaf of the complex submanifold $V$. Since $M$ is Stein, Cartan's Theorem B implies this cocycle is a coboundary (cf.\ \cite[Theorem 7.4.3]{ho:in}), so there exist holomorphic functions $g_i\in\Gamma(U_i,\cal I_V^{m+3})$ such that $f_{ij}=g_j-g_i$. Consider the smooth functions $f_i+g_i$ defined on $U_i$. Clearly $f_i+g_i=f_j+g_j$ on $U_{ij}$, so $f_i+g_i=f|_{U_i}$ for a global smooth function $f$. By construction $f$ vanishes to $(m+2)$-nd order on $X$, and because the $g_i$'s are holomorphic, we have $\sq\,\bar\partial f=\beta$. \end{pf} The proof of the theorem used the following little lemma. \begin{lemma}\label{lemma:vanish} Suppose $f$ is a holomorphic function on $M$ vanishing to $m$-th order on the totally real submanifold $X$. Then $f$ vanishes to $m$-th order on the complexification $X\co\subset M$ of $X$. \end{lemma} \begin{pf} We compute in a distinguished system of coordinates $(z,w)$, writing $z=x+\sq\,y$, as in the proof of Theorem \ref{theorem:potential}. First we prove the statement for $m=0$. So suppose $f$ vanishes on $X$; we have to show it vanishes on $X\co$. The assumption implies that the partial derivatives of $f$ along $X$, $\partial^{|I|}f/\partial x_I$, vanish identically on $X$ for all multi-indices $I$. Since $\partial^{|I|}/\partial z_I= \partial^{|I|}/\partial x_I$ on holomorphic functions, we conclude that the power series of the restriction $f|_{X\co}$ of $f$ to $X\co$ is trivial at any point of $X$. By the identity principle $f|_{X\co}=0$. Now suppose $f$ vanishes to order $m\geq0$ on $X$. This means the holomorphic functions $\partial^{|I|+|J|}f/\partial z_I\partial w_J$ vanish identically on $X$ for all $I$ and $J$ with $|I|+|J|\leq m$. Then by the previous result $\partial^{|I|+|J|}f/\partial z_I\partial w_J=0$ on $X\co$ if $|I|+|J|\leq m$, so $f$ vanishes to order $m$ on $X\co$. \end{pf} The next result says one can ``interpolate'' between two K\"ahler metrics that agree along $X$. \begin{theorem}\label{theorem:interpolate} Let $dS^2$ and $ds^2$ be two smooth K\"ahler metrics on $M$. Assume that the real-analytic totally real submanifold $X$ is compact and that $dS^2_x=ds^2_x$ for all $x$ in $X$. Then there is an open neighbourhood $U$ of $X$ in $M$ with the following property\rom: For all open $U_1$ with $X\subset U_1\subset U$ there exist an open subset $U_2$ with $X\subset U_2\subset U_1$ and a smooth K\"ahler metric $d\tilde s^2$ on $M$ such that $d\tilde s^2=dS^2$ on $U_2$ and $d\tilde s^2=ds^2$ on $M\backslash\bar U_1$. In the presence of a compact group $G$ of holomorphic transformations on $M$ leaving the submanifold $X$ and the metrics $dS^2$ and $ds^2$ invariant\rom, the metric $d\tilde s^2$ may be taken to be invariant. If the $G$-action is Hamiltonian with respect to the K\"ahler form $-\Im ds^2$\rom, it is Hamiltonian with respect to the K\"ahler form $-\Im d\tilde s^2$. \end{theorem} \begin{pf} For the first part of the theorem we may again assume that $M$ is a Stein tubular neighbourhood of $X$. Let $\Omega=-\Im dS^2$ and $\omega=-\Im ds^2$ be the K\"ahler forms corresponding to $dS^2$ and $ds^2$, and put $\sigma=\omega-\Omega$. Then $\sigma$ vanishes to order 0 on $X$, so by Theorem \ref{theorem:potential} there exists a smooth function $u$ vanishing to second order on $X$ such that $\sigma=\sq\,\partial\bar\partial u$. Let $\rho$ be the square of some distance function on the tube $M$. Then $\rho$ vanishes to first order on $X$, so $u$ is of order $O(\rho^{3/2})$ as $\rho$ tends to zero. Let $\chi\colon\bold R\to[0,1]$ be a smooth function with $\chi(t)=0$ for $t\leq1$ and $\chi(t)=1$ for $t\geq2$. For $\lambda>0$ define a smooth function $\eps$ on $M$ by $\eps(x)=\chi\bigl(\rho(x)/\lambda^2\bigr)$. Put $M_r=\{\,x\in M:\rho(x)<r\,\}$ and define a smooth two-form $\tilde\omega$ on $M$ by $$ \tilde\omega= \begin{cases} \Omega+\sq\,\partial\bar\partial\eps u & \text{on $M_{3\lambda^2}$},\\ \omega & \text{on $M-M_{2\lambda^2}$.} \end{cases} $$ Then on $M_{\lambda^2}$ the form $\tilde\omega$ is equal to $\Omega$. On $M_{3\lambda^2}$ we have $\tilde\omega-\omega=\sq\,\partial\bar\partial(\eps-1)u$. In a distinguished neighbourhood $U$ of a point of $X$ with coordinates $v=(z,w)$ we can write $\partial\bar\partial(\eps-1)u= \sum_{\alpha,\beta=1}^nf_{\alpha\beta}\,dv_\alpha\wedge d\bar v_\beta$ with $$ f_{\alpha\beta}(v) = \frac{\partial^2}{\partial v_\alpha\partial\bar v_\beta}\Bigl(\bigl(\eps(v)-1\bigr)u(v)\Bigr). $$ By carrying out the differentiation one can check in a straightforward manner that for every compact subset $K$ of $U$ the supremum of $|f_{\alpha\beta}(v)|$ over all $v\in K\cap M_{3\lambda^2}$ is of order $O(\lambda)$ as $\lambda$ tends to zero. For instance, one of the terms involved in $f_{\alpha\beta}$ is $$ \frac{\chi''(\rho/\lambda^2)}{\lambda^4}\frac{\partial\rho}{\partial v_\alpha}\frac{\partial\rho}{\partial\bar v_\beta}u= \frac{\chi''(\rho/\lambda^2)}{\lambda^4}O(\rho^{5/2}), $$ where we used $u=O(\rho^{3/2})$ and the fact that the first derivatives of $\rho$ are of order $O(\rho^{1/2})$ as $\rho$ tends to zero, since they vanish on $X$. Since $\chi''(\rho/\lambda^2)=0$ for $\rho\geq2\lambda^2$, we have $$ \sup_{\rho\leq3\lambda^2}\Bigl( \frac{\chi''(\rho/\lambda^2)}{\lambda^4}\frac{\partial\rho}{\partial v_\alpha}\frac{\partial\rho}{\partial\bar v_\beta}u\Bigr)= \frac{1}{\lambda^4}O(\lambda^5)= O(\lambda). $$ The other terms can be dealt with similarly. From the compactness of $X$ it now follows that $\tilde\omega$ becomes arbitrarily close to $\omega$ uniformly on $M$ as $\lambda$ tends to zero. Hence, for $\lambda$ small enough the symmetric bilinear form $\tilde\omega(\cdot,J\cdot)$ is positive-definite, and therefore $\tilde\omega$ is the imaginary part of a K\"ahler metric $d\tilde s^2$. By construction $d\tilde s^2$ is equal to $dS^2$ on $M_{\lambda^2}$ and equal to $ds^2$ on $M-M_{2\lambda^2}$. In the proof of the second part of the theorem we denote the Stein tube around $X$ by $N$ to distinguish it from the whole of $M$. Suppose the compact Lie group $G$ acts on $M$ by holomorphic transformations leaving $X$, $dS^2$ and $ds^2$ invariant. After averaging over $G$ we may assume the potential $u$ is invariant. It is not hard to verify by inspecting the proof in \cite{ha:ho} that the tube $N$ can be chosen to be invariant. If we choose an invariant distance function, the shrunken tubes $N_r$ are also invariant. It is clear from the definition that the form $\tilde\omega$ is then also $G$-invariant. Now assume that there exists a momentum map $\Phi$ for the action with respect to the symplectic form $\omega$. By construction we have $\tilde\omega_x=\omega_x$ for all $x$ in $X$, so by the equivariant Darboux-Weinstein Theorem (see e.g.\ \cite[\S 22]{gu:sy}) for sufficiently small $\lambda$ there exists a $G$-equivariant diffeomorphism $\Gamma\colon N_{3\lambda^2}\to N_{3\lambda^2}$ fixing the manifold $X$ such that $\Gamma^*\tilde\omega=\omega$. Then the map $\tilde\Phi\colon N_{3\lambda^2}\to\frak g^*$ defined by $\Gamma^*\tilde\Phi=\Phi$ is a momentum map with respect to the form $\tilde\omega$. On $N_{3\lambda^2}-N_{2\lambda^2}$ we have $\tilde\omega=\omega$, so there $\tilde\Phi$ differs by a locally constant function $c$ from the $\omega$-momentum map $\Phi$. Let us assume, as we may, that the manifolds $M$ and $X$ are connected. If $X$ is of codimension greater than one, the subset $N_{3\lambda^2}-N_{2\lambda^2}$, which is homeomorphic to $N-X$, is connected. In this case $c$ is a constant, so after shifting $\tilde\Phi$ by $c$ we can paste $\Phi$ and $\tilde\Phi$ together to obtain a global $\tilde\omega$-momentum map for the $G$-action. The totally real submanifold $X$ can only be of codimension one if $\dim X=1$ and $\dim M=2$. The only Riemann surfaces $M$ that admit a continuous group of automorphisms are $\bold P^1$, $\bold C$, $\bold C^\times$, elliptic curves $\bold C/\Lambda$, the unit disc $\Delta$ and annuli $\Delta_r= \{\,z\in\bold C:r<|z|<1\,\}$, for $0\leq r<1$. (Cf.\ Farkas and Kra \cite[Section V.4]{fa:ri}.) No subgroup of $\Aut(\bold C/\Lambda)=\bold C/\Lambda$ acts on $\bold C/\Lambda$ in a Hamiltonian fashion, so elliptic curves are out. In the other examples the only compact connected group of automorphisms is the circle acting in the standard way. In each of these cases $X$ has to be a circle, the complement of $X$ in the tube $N$ consists of two components, and $M-X$ also consists of two components. We can therefore glue together the two momentum maps $\Phi$ and $\tilde\Phi$ by adding appropriate constants to $\Phi|_{M-N_{2\lambda^2}}$ on each of the two components of $M-N_{2\lambda^2}$. \end{pf} \begin{remark} If $X$ connected, then the $\tilde\omega$-momentum map $\tilde\Phi$ is equal to $\Phi$ on $X$, and if $\Phi$ is equivariant, then so is $\tilde\Phi$. \end{remark} \subsection{Holomorphic slices}\label{subsection:slices} We now come to the main result of Section \ref{section:slices}. \begin{theorem}[Holomorphic Slice Theorem]\label{theorem:slice} Let $M$ be a K\"ahler manifold and let $G\co$ act holomorphically on $M$. Assume the action of the compact real form $G$ is Hamiltonian. Let $m$ be any point in $M$ such that the $G$-orbit through $m$ is isotropic. Then there exists a slice at $m$ for the $G\co$-action. \end{theorem} If $S$ is a slice at $m$, then $gS$ is a slice at $gm$. So the theorem implies the existence of a slice at any point $m$ such that the $G\co$-orbit through $m$ contains an isotropic $G$-orbit. Moreover, if $G'$ is another compact real form of $G\co$, then $G'$ is conjugate to $G$ by some element $g$ of $G\co$, $G'=gGg^{-1}$. Then $G'$ leaves invariant the symplectic form $g_*\omega$, and a $G'$-momentum map is given by $\Phi'=(\Ad^*g)\circ\Phi\circ g^{-1}$, where $\Phi\colon M\to\frak g^*$ is an $\Ad^*$-equivariant momentum map for the $G$-action. So the choice of the compact real form is irrelevant. \begin{trivlist}\item[\hskip\labelsep{\em Proof of Theorem \ref{theorem:slice}}.] We divide the proof into several steps. Using the analytic continuation argument of Proposition \ref{proposition:orbit}, we shall first reduce the question of the existence of a slice to the existence of orbitally convex open neighbourhoods of the compact orbit $Gm$. Next we consider the special case where the compact orbit $Gm$ has a $G$-invariant neighbourhood that can be embedded in a holomorphic, $G$-equivariant and isometric fashion into a unitary representation space of $G$. The last step of the proof consists in showing that an arbitrary metric $ds^2$ can always be deformed to a metric which close to $Gm$ is the pullback of a flat metric via some embedding into a Euclidean space, and which is still compatible with all the relevant data. The details are as follows. By Remark \ref{remark:connected} the vector $\Phi(m)$ is $\Ad^*G^0$-fixed. After shifting the momentum map we may assume that $\Phi(m)=0$. (If $\Phi(m)$ is not fixed under the whole of $G$, then the shifted momentum map $\Phi-\Phi(m)$ is merely $G^0$-equivariant. This will however be sufficient in what follows.) The tangent space $T_mM$ at $m$ is a Hermitian vector space, which we shall identify with standard $\bold C^n$. Then the value of the K\"ahler form $\omega$ at $M$ is the standard symplectic form $\Omega$ on $\bold C^n$. Let $H$ be the stabilizer of $m$ with respect to the $G$-action. Then by Remark \ref{remark:connected} the stabilizer with respect to the $G\co$-action is the complexification $H\co$ of $H$. The tangent action of $H\co$ defines a linear representation $H\co\to\GL(n,\bold C)$, the restriction of which to $H$ is a unitary representation $H\to\U(n)$. Let $\phi\colon O\to M$ be a local holomorphic coordinate system on $M$ with $\phi(0)=m$ and $d\phi_0=\id_{\bold C^n}$, where $O$ is a small $H$-invariant open ball about the origin in $\bold C^n$. Then the pullback of the form $\omega$ is equal to $\Omega$ at the origin. Let $O'=\phi(O)$ and let $\psi\colon O'\to O$ be the inverse of $\phi$. After averaging over $H$ and shrinking $O$ if necessary we may assume that $\psi$ and hence $\phi$ are $H$-equivariant. The tangent space to the complex orbit $G\co m$ is a Hermitian subspace of $T_mM\cong\bold C^n$. Denote its orthogonal complement by $V$; then $V$ is an $H\co$-invariant subspace, which can be identified with $\bold C^l$ for some $l\leq n$. Now let $B$ be the intersection of the ball $O$ with $V$, and let $B'$ be the image of $B$ under $\phi$, $B'=\phi(B)$. We claim that if $B'$ is sufficiently small the $H\co$-saturation $S'=H\co B'$ of $B'$ is a slice at $m$. (In Snow's terminology \cite{sn:re} $B'$ is a {\em local\/} slice.) To verify this claim we have to show that the natural map from the associated bundle $G\co\times_{H\co}S'$ into $M$ is biholomorphic onto an open subset of $M$. We shall show this indirectly by proving that the map $\phi\colon B\to B'$ can be uniquely extended to a $G\co$-equivariant map from $G\co\times_{H\co}S$ into $M$, which is biholomorphic onto an open subset. Here $S$ is defined to be the open subset $H\co B$ of $V$. Let us define $E$ to be the associated bundle $$ E=G\co\times_{H\co}V, $$ and let $e$ be the ``base point'' $[1,0]\in E$. Consider the $G$-equivariant map $G\times\frak m\to G\co/H\co$ sending a pair $(g,\sq\,\xi)$ to $g\exp(\sq\,\xi)H\co$. This map descends to a $G$-equivariant map \begin{equation}\label{equation:mostow} G\times_H\sq\,\frak m\to G\co/H\co, \end{equation} which by a refinement of the Cartan decomposition due to Mostow \cite{mo:on1,mo:so,mo:on2} is a {\em diffeomorphism}. In other words, every element of $G\co$ can be written as a product $g\exp(\sq\,\xi)h$ with $g\in G$, $\xi\in\frak m$ and $h\in H\co$; and if $g\exp(\sq\,\xi)h=g'\exp(\sq\,\xi')h'$, then $g'=gk^{-1}$, $\xi'=(\Ad k)\xi$ and $h'=kh$ for some $k\in H$. It follows that the map $$ G\times_H(\sq\,\frak m\times V)\to G\co\times_{H\co}V $$ sending the equivalence class $[g,\sq\,\xi,v]$ to the equivalence class $[g\exp(\sq\,\xi),v]$ is likewise a diffeomorphism. We conclude that the sets $U=G\exp(\sq\,D)B\simeq G\times_H(\sq\,D\times B)$, for $D$, resp.\ $B$, ranging over all balls about the origin in $\frak m$, resp.\ $V$, form a basis of neighbourhoods of the compact orbit $G e$ inside the space $E$. Furthermore, we can extend the $H$-equivariant holomorphic map $\phi\colon B\to M$ to a $G$-equivariant holomorphic map $U\to M$ by defining \begin{equation}\label{equation:openset} [g,\sq\,\xi,v]\mapsto g\exp(\sq\,\xi)\phi(v), \end{equation} for $g\in G$, $\xi\in D$ and $v\in B$. For simplicity we still call this map $\phi$. From the decomposition (\ref{equation:sum}), where now $\mu=0$ and $\frak n=\frak m$, it is clear that $G\times_H(\sq\,\frak m\times V)$ is nothing but the normal bundle to the compact orbit $G m\cong G/H$ in $M$. Consequently, for $D$ and $B$ small enough $\phi\colon U\to M$ is a biholomorphic map onto an open neighbourhood of $G m$ in $M$. Clearly $G\co U=G\co\times_{H\co}H\co B=G\co\times_{H\co}S$. We will prove: \begin{claim}\label{claim:convex} \begin{enumerate} \item\label{model} The compact orbit $G e\subset E$ possesses a basis of orbitally convex open neighbourhoods\rom; and \item\label{space} The compact orbit $G m\subset M$ possesses a basis of orbitally convex open neighbourhoods. \end{enumerate} \end{claim} In view of Proposition \ref{proposition:orbit} this will imply there is a $G\co$-equivariant biholomorphic map $\phi\co\colon G\co\times_{H\co}S\to G\co S$ extending the map $\phi$, which will conclude the proof of Theorem \ref{theorem:slice}. Heinzner \cite{he:ge} gives a proof of (\ref{model}). We shall present an adapted version of his argument, which can be utilized to give a proof of (\ref{space}). The argument bears a certain similarity to a convexity argument of Kempf and Ness \cite{ke:le}. Let us start with the simple case where $e$ is a fixed point. Then $G=H$, the space $E$ is just $\bold C^n$ and $e$ is the origin. On $\bold C^n$ we have the constant K\"ahler metric denoted by $dS^2$, the standard symplectic form $\Omega$ and the quadratic momentum map $\Phi_{\bold C^n}\colon\bold C^n\to\frak g^*$ given by (\ref{equation:quadratic}). As above, $B$ is an open ball about the origin in $\bold C^n$. By $r(v)$ we denote the Riemannian distance of $v\in B$ to the origin, and by $\langle\cdot,\cdot\rangle$ the positive-definite inner product $\Re dS^2$. \begin{lemma}\label{lemma:angle} For all $\xi\in\frak g$ and $v\in B$ the momentum function $\Phi_{\bold C^n}^\xi$ measures the inner product of the outward pointing normal $\grad r^2$ to the metric sphere of radius $r$ about the origin and the vector field $J\xi_{\bold C^n}=\grad\Phi_{\bold C^n}^\xi$\rom, as follows\rom: \begin{equation}\label{equation:angle} \bigl\langle\grad r^2,\grad\Phi_{\bold C^n}^\xi\bigr\rangle=4\Phi_{\bold C^n}^\xi. \end{equation} It follows that $B$ is orbitally convex with respect to the $G\co$-action. \end{lemma} \begin{pf} The path $\delta(t)=\exp(\sq\,t\xi)v$ is the trajectory of the gradient of $\Phi_{\bold C^n}^\xi$ through $v$. On one hand, $$ \frac{d}{dt}r^2\bigl(\delta(t)\bigr) = \bigl\langle\grad r^2\bigl(\delta(t)\bigr),\delta'(t)\bigr\rangle = \bigl\langle\grad r^2\bigl(\delta(t)\bigr),\grad\Phi_{\bold C^n}^\xi\bigl(\delta(t)\bigr)\bigr\rangle. $$ On the other hand, \begin{align*} \frac{d}{dt}r^2\bigl(\delta(t)\bigr)&= \frac{d}{dt}\bigl\|\delta(t)\bigr\|^2 = \frac{d}{dt}\bigl\langle\delta(t),\delta(t)\bigr\rangle = 2\bigl\langle\delta'(t),\delta(t)\bigr\rangle = \\ &= 2\bigl\langle(\sq\,\xi)_{\bold C^n}\bigl(\delta(t)\bigr),\delta(t)\bigr\rangle = 2\Omega\bigl(\xi_{\bold C^n}\bigl(\delta(t)\bigr),\delta(t)\bigr) = 4\Phi_{\bold C^n}^\xi\bigl(\delta(t)\bigr), \end{align*} where we have used (\ref{equation:quadratic}) and (\ref{equation:grad}). Taking $t=0$ yields (\ref{equation:angle}). Now (\ref{equation:angle}) implies that the curve $\delta(t)$ can only enter $B$ at a point $p$ in the boundary $\partial B$ for which $\Phi_{\bold C^n}^\xi(p)\leq0$ and leave it at a point $q\in\partial B$ where $\Phi_{\bold C^n}^\xi(q)\geq0$. But $\delta(t)$ is also a gradient curve of the function $\Phi_{\bold C^n}^\xi$ and so $\Phi_{\bold C^n}^\xi$ is increasing along $\delta(t)$. If $\delta(t)$ is not constant, $\Phi_{\bold C^n}^\xi\bigl(\delta(t)\bigr)$ is strictly increasing. Therefore, if $\delta(t)$ leaves the ball $B$ at some point, it can never sneak back in. Consequently $\{\,\delta(t):t\in\bold R\,\}\cap B$ is connected. If $\delta(t)$ is constant it is trivially true that $\{\,\delta(t):t\in\bold R\,\}\cap B$ is connected. \end{pf} Observe that the proof does not use that the metric is flat on all of $\bold C^n$; it works for any K\"ahler metric that is flat in a neighbourhood of the origin. We shall make repeated use of the following result of Kempf and Ness \cite{ke:le}. (Cf.\ also Procesi and Schwarz \cite{pr:in}.) \begin{proposition}\label{proposition:kempfness} Suppose $G$ acts unitarily on $\bold C^N$. Consider the complexified representation $G\co\to\GL(N,\bold C)$. An orbit $\cal O$ of $G\co$ in $\bold C^N$ is closed if and only if the restriction $r|_{\cal O}$ of the length function $r$ has a stationary point. If $v\in\cal O$ is a stationary point of $r|_{\cal O}$\rom, then \begin{enumerate} \item $r|_{\cal O}$ takes on its minimum at $v$\rom, and for all $w\in\cal O$\rom, $r(w)=r(v)$ implies $w\in Gv$\rom; \item $v$ is in the zero level set of the momentum map $\Phi_{\bold C^N}$\rom; \item $(G\co)_v=(G_v)\co$.\qed \end{enumerate} \end{proposition} To jack up Lemma \ref{lemma:angle} we embed the homogeneous bundle $E$ equivariantly into a representation space. \begin{lemma}\label{lemma:embedding} There exists a $G\co$-equivariant\rom, holomorphic and proper embedding $\iota$ of $E$ into a finite-dimensional representation space $\bold C^N$ of $G\co$. Choose any $G$-invariant Hermitian inner product on $\bold C^N$. Then the sets $\iota^{-1}(B)$\rom, where $B$ ranges over the collection of open balls about the origin in $\bold C^N$\rom, form a basis of orbitally convex open neighbourhoods of the orbit $G e$ in $E$. \end{lemma} \begin{pf} It is not hard to find an orthogonal representation of $G$ on $\bold R^{N_1}$ for some $N_1$ containing a vector $w$ whose stabilizer is exactly $G_w=H$. (See \cite{ja:di}.) Then the map assigning to $gH$ the vector $gw$ is a real-analytic $G$-equivariant embedding of the homogeneous space $G/H$ into $\bold R^{N_1}$. Complexifying the representation $G\to\operatorname{O}(N_1)$ yields a unitary representation $G\to\U(N_1)$, which extends to a complex-linear representation $G\co\to\GL(N_1,\bold C)$. Consider the inclusions $Gw\subset\bold R^{N_1}$ and $G\co w\subset\bold C^{N_1}$. Since the $G$-representation on $\bold R^{N_1}$ is orthogonal, the tangent space to the orbit $T_w(Gw)$ is a subspace of the tangent space to the $(N_1-1)$-dimensional sphere about the origin in $\bold R^{N_1}$ containing $w$. It follows that the tangent space to the complexified orbit $T_w(G\co w)=T_w(Gw)+JT_w(Gw)$ is a subspace of the tangent space to the $(2N_1-1)$-dimensional sphere about the origin in $\bold C^{N_1}$ containing $w$. In other words, $w$ is a critical point of the function $r^2|_{G\co w}$, where $r^2$ is the distance to the origin in $\bold C^{N_1}$. Proposition \ref{proposition:kempfness} now implies that $(G_w)\co=(G\co)_w$, and that the orbit $G\co w$ is closed in $\bold C^{N_1}$. We conclude that the map $\iota_1\colon G\co/H\co\to\bold C^{N_1}$ sending $gH\co$ to $gw$ is an equivariant, holomorphic and proper embedding. Next we show how to find an embedding of the $H\co$-module $V$ into a finite-dimensional $G\co$-module $\bold C^{N_2}$, that is, an $H\co$-equivariant injective complex-linear map $\iota_2\colon V\to\bold C^{N_2}$. Let $V_1$, $V_2,\dots$, $V_r$ be the irreducible components of $V$. It is an easy consequence of the Peter-Weyl Theorem that every irreducible $H$-module can be embedded $H$-equivariantly into an irreducible $G$-module. (Consider the decomposition of the left-regular representation $L^2(G)=\bigoplus_iW_i$ into $G$-irreducibles. Decompose each of the $W_i$ into $H$-irreducibles, $W_i=\bigoplus_{ij}Z_{ij}$. Let $\chi\colon H\to\bold C$ be the character of some irreducible $H$-representation; pushing $\chi$ forward as a measure to $G$ gives a measure on $G$, and the convolution product $f\to\chi*f$ defines a non-zero $H$-equivariant projection operator $\pi$ in $L^2(G)$. Now $\pi|_{Z_{ij}}=\id$ or $\pi|_{Z_{ij}}=0$ depending on whether or not $Z_{ij}$ has character $\chi$. Since $\pi\neq0$ at least one of the $W_i$ has to contain a $Z_{ij}$ with character $\chi$.) So we can find irreducible $G$-modules $\bold C^{n_k}$ with $H$-equivariant injective complex-linear maps $j_k\colon V_k\to\bold C^{n_k}$. Each of the $j_k$'s is necessarily $H\co$-equivariant. We can take $\iota_2$ to be the direct sum of the $j_k$'s. It is now easy to check that the map $\iota\colon E=G\co\times_{H\co}V\to\bold C^{N_1+N_2}$ mapping $[g,v]$ to $gw+g\iota_2(v)$ is a $G\co$-equivariant, holomorphic and proper embedding. By Proposition \ref{proposition:kempfness} $Gw$ is precisely the subset of vectors in $G\co w$ of minimal length. From the inequality $\bigl\|\iota[g,v]\bigr\|^2= \bigl\|gw+g\iota_2(v)\bigr\|^2= \|gw\|^2+\bigl\|g\iota_2(v)\bigr\|^2 \geq\|gw\|^2$ it is clear that $Gw$ is also equal to the subset of vectors of minimal length in the submanifold $\iota(E)$. Because of this and the $G$-invariance of the metric on $\bold C^{N_1+N_2}$, any open ball $B$ about $0$ such that $B\cap\iota(E)$ is nonempty contains the orbit $Gw$, and the sets $B\cap\iota(E)$ are a basis of open neighbourhoods of the orbit $Gw=G\cdot\iota(e)$. The second assertion of the lemma now follows from Lemma \ref{lemma:angle} and Remark \ref{remark:trivial}. \end{pf} This proves part (\ref{model}) of Claim \ref{claim:convex}. Now consider the $G$-equivariant holomorphic embeddings $$ \bold C^N\overset{\iota}{\hookleftarrow} U\overset{\phi}{\hookrightarrow}M, $$ where $\phi$ is the map defined in (\ref{equation:openset}). Pulling back the metric $ds^2$ on $M$ via $\phi$ we obtain a metric on $E$ defined on the neighbourhood $U$ of the compact orbit $Ge$. The proof of Lemma \ref{lemma:embedding} allows us to deduce the following stronger assertion. \begin{lemma}\label{lemma:convex} Suppose the linear embedding $\iota$ is isometric\rom, that is\rom, $\iota^*dS^2=\phi^*ds^2$\rom, where $dS^2$ is the flat metric on $\bold C^N$. Then for any orbitally convex open subset of the form $\iota^{-1}(B)$ contained in $U$ the image $\phi\bigl(\iota^{-1}(B)\bigr)$ is orbitally convex in $M$. \end{lemma} \begin{pf} Put $U'=\phi(U)$ and let $\psi\colon U'\to U$ be the inverse of $\phi$. We have two $G$-invariant K\"ahler metrics on $U'$, namely $ds^2$ and $\psi^*\iota^*dS^2$, with corresponding momentum maps $\Phi$ and $\Phi'=\psi^*\iota^*\Phi_{\bold C^N}$. By assumption $ds^2$ is equal to $\psi^*\iota^*dS^2$. Moreover, $\Phi(m)=0$ and, by Proposition \ref{proposition:kempfness}, $\Phi_{\bold C^N}\bigl(\iota(e)\bigr)=0$. This implies $\Phi(m)=\Phi'(m)$, and so $\Phi=\Phi'$. Put $O=\phi\bigl(\iota^{-1}(B)\bigr)$ and pick any point $x$ in $O$. Let $\gamma(t)\subset M$ be the curve $\exp(\sq\,t\xi)x$; then $\gamma(t)$ is contained in $U'$ for small $t$. Put $v=\iota\psi(x)$ and $\delta(v)=\exp(\sq\,t\xi)v$. Let $I=\{\,t\in\bold R:\gamma(t)\in O\,\}$ and let $I^0$ be the connected component of $I$ containing $0$. Because the map $\iota\psi$ is $G$-equivariant and holomorphic, we have $\delta(t)= \exp(\sq\,t\xi)\iota\psi(x)=\iota\psi\bigl(\exp(\sq\,t\xi)x\bigr)= \iota\psi\bigl(\gamma(t)\bigr)$ for all $t\in I^0$. (See Remark \ref{remark:local}.) It now follows from the proof of Lemma \ref{lemma:angle} that the curve $\gamma(t)$ can only enter the set $O$ at time $t_0$ if $\Phi\bigl(\gamma(t_0)\bigr)\leq0$ and leave it at time $t_1$ if $\Phi\bigl(\gamma(t_1)\bigr)\geq0$. Because the function $\Phi^\xi$ is increasing along $\gamma$, it follows that $I=I^0$, i.e.\ $\gamma$ intersects $O$ in a connected set. \end{pf} Of course, in general the map $\iota$ will not be an isometry for the given metric $ds^2$ on $M$. We claim, however, that we can arrange for $\iota$ to be an isometry along the compact orbit $G e$. \begin{lemma}\label{lemma:isometry} The representation $G\co\to\GL(N,\bold C)$ and the $G$-invariant Hermitian inner product on $\bold C^N$ in Lemma \ref{lemma:embedding} can be chosen in such a way that the embedding $\iota$ is a K\"ahler isometry at all points of the orbit $Ge$\rom, that is\rom, $\iota^*dS^2=\phi^*ds^2$ on $T_xE$ for all $x\in G e$. \end{lemma} \begin{pf} We use the notation of the proof of Lemma \ref{lemma:embedding}. Moore \cite{mo:eq} has shown that the representation $G\to\operatorname{O}(N_1)$ can be chosen in such a way as to make the embedding $G/H\to\bold R^{N_1}$ an isometry of Riemannian manifolds. (Cf.\ also \cite{mo:on}.) The associated embedding $\iota_1$ of the complexified homogeneous space $G\co/H\co$ into $\bold C^{N_1}$ is holomorphic and the complex structure $J$ is an orthogonal map (at each point of $G\co/H\co$ and $\bold C^{N_1}$). So the differential $d\iota_1$ is a unitary map $T_x(G\co/H\co)\to\bold C^{N_1}$ for all $x\in G/H$. We can also arrange for the embedding $\iota_2\colon V\to\bold C^{N_2}$ to be a unitary map. Indeed, we obtained $\iota_2$ by embedding each irreducible component $V_k$ of $V$ into an irreducible unitary representation $\bold C^{n_k}$ of $G$. By Schur's Lemma the invariant Hermitian inner products on $V_k$ and $\bold C^{n_k}$ are unique up to constant multiples. By suitably rescaling the metric on each $\bold C^{n_k}$ the embedding $\iota_2\colon\bigoplus_k V_k\to\bigoplus_k\bold C^{n_k}$ becomes unitary. The embedding $\iota\colon E\to\bold C^N$ is now a K\"ahler isometry along the orbit $Ge$. \end{pf} With a choice of embedding as in this lemma Theorem \ref{theorem:interpolate} tells us we can deform the metric $ds^2$ in such a manner that $\iota$ becomes an isometry. Theorem \ref{theorem:interpolate} plus Lemmas \ref{lemma:convex} and \ref{lemma:isometry} therefore imply part (\ref{space}) of Claim \ref{claim:convex}. This finishes the proof of Theorem \ref{theorem:slice}. \qed\end{trivlist} Along the lines of \cite{lu:sl} one can deduce from the slice theorem many results on the local structure of a $G\co$-action. Let us list a few for the record. \begin{theorem}\label{theorem:stein} Every point in $M$ the $G$-orbit through which is isotropic possesses a $G\co$-invariant Stein open neighbourhood. \end{theorem} \begin{pf} Let $m\in M$ and suppose $Gm$ is isotropic. Put $H=G_m$. Let $S$ be a slice at $m$ as constructed in the proof of Theorem \ref{theorem:slice}. Then $S$ is biholomorphically equivalent to the set $H\co B$ swept out by a ball $B$ in the tangent space $T_mS$. It is not hard to show that $H\co B$ is Stein. We conclude $m$ has an open $G\co$-invariant neighbourhood that is biholomorphically equivalent to a bundle with affine base $G\co m$, Stein fibre $S$ and reductive structure group $(G_m)\co$. By a theorem of Matsushima the total space of this bundle is Stein. \end{pf} \begin{theorem}\label{theorem:subconjugate} Let $m$ be any point in $M$ such that the $G$-orbit through $m$ is isotropic. Then for every point $x$ nearby $m$ the stabilizer subgroup $(G\co)_x$ is conjugate to a subgroup of $(G\co)_m$.\qed \end{theorem} \begin{theorem}\label{theorem:linear} Let $m\in M$ be any fixed point of the $G\co$-action. Then the action of $G\co$ can be linearized in a neighbourhood of $m$ in the sense that there exist a $G\co$-invariant open neighbourhood $U$ of $m$ in $M$\rom, a $G\co$-invariant open neighbourhood $U'$ of the origin $0$ in the tangent space $T_mM$ and a biholomorphic $G\co$-equivariant map $U\to U'$. \end{theorem} \begin{pf} A fixed point is obviously isotropic. The result now follows immediately from the Holomorphic Slice Theorem. \end{pf} \begin{remark} This theorem was also stated by Koras \cite{ko:li}, but my proof is different from Koras', which I have trouble understanding in places. In particular, I fail to see a justification for his application of the curve selection lemma. \end{remark} Recall that the $G\co$-action is called {\em proper\/} at the point $m$ if for all sequences $(m_i)\subset M$ and $(g_i)\subset G$ the following holds: If $(m_i)$ converges to $m$ and $(g_im_i)$ converges to some point in $M$, then $(g_i)$ converges to some element of $G$. If the action is proper at $m$, the stabilizer $(G\co)_m$ is compact. \begin{theorem}\label{theorem:proper} Suppose the $G$-orbit through a point $m\in M$ is isotropic. Then the following conditions are equivalent\rom: \begin{enumerate} \item\label{proper} The action of $G\co$ is proper at $m$\rom; \item\label{finite} The stabilizer $(G\co)_m$ is finite\rom; \item\label{regular} $m$ is a regular point of the momentum map $\Phi$. \end{enumerate} \end{theorem} \begin{pf} First we show (\ref{proper}) is equivalent to (\ref{finite}). If the $G\co$-action is proper at $m$, the stabilizer $(G\co)_m$ is a compact complex submanifold of $G\co$, which is a Stein manifold. Therefore $(G\co)_m$ is finite. Conversely, assume $(G\co)_m$ is finite. Then it is easy to see that the left action of $G\co$ on the homogeneous space $G\co/(G\co)_m$ is proper. It follows the left $G\co$-action on the homogeneous vector bundle $G\co\times_{(G\co)_m}V$ is also proper, $V$ being the tangent space at $m$ to a slice at $m$. By the Holomorphic Slice Theorem the point $m$ has an invariant neighbourhood which is equivariantly isomorphic to an invariant open subset of $G\co\times_{(G\co)_m}V$, so the $G\co$-action on $M$ is proper at $m$. Next we show (\ref{finite}) is equivalent to (\ref{regular}). If $(G\co)_m$ is finite, obviously the real stabilizer $G_m$ is also finite, so the stabilizer subalgebra $\frak g_m$ is trivial. Now the annihilator of $\frak g_m$ in $\frak g^*$ is equal to the range of $d\Phi_m$ (see \cite[\S 26]{gu:sy}), so $d\Phi_m$ is surjective. Conversely, if $d\Phi_m$ is surjective, $\frak g_m$ is trivial, so $G_m$ is finite, so by Proposition \ref{proposition:totallyreal} $(G_m)\co=(G\co)_m$ is finite. \end{pf} \begin{theorem}\label{theorem:torus} Suppose $G$ is a torus. Then slices for the $G\co$-action exist at all points of $M$. \end{theorem} \begin{pf} If $G$ is a torus, the coadjoint representation of $G$ is trivial, so by Proposition \ref{proposition:totallyreal} all $G$-orbits in $M$ are isotropic. Now apply the Holomorphic Slice Theorem. \end{pf} Our results can also be used to give a short proof of a theorem of Snow's \cite{sn:re}. \begin{theorem}[Snow]\label{theorem:snow} Let $X$ be a Stein space on which $G\co$ acts holomorphically. Let $x$ be any point in $X$ such that the orbit $G\co x$ is closed. Then there exists a slice at $x$ for the $G\co$-action. \end{theorem} \begin{pf} The first part of the proof is the same as in \cite{sn:re}. Snow proves there exists a $G\co$-equivariant holomorphic map $h$ of $X$ into a $G\co$-representation space $\bold C^n$ that is an immersion at $x$ (and hence at all points of the orbit $G\co x$) and whose restriction to $G\co x$ is a proper embedding (\cite[Proposition 2.5]{sn:re}). It follows the orbit $G\co\cdot h(x)$ is closed in $\bold C^n$, and therefore by Proposition \ref{proposition:kempfness} the compact orbit $G\cdot h(x)$ is contained in the zero level set of the quadratic momentum map $\Phi_{\bold C^n}$. So by Lemma \ref{lemma:embedding} the orbit $G\cdot h(x)$ possesses a basis $\cal U$ of orbitally convex neighbourhoods in $\bold C^n$. From the fact that $h|_{Gx}$ is injective, that $h$ is an immersion at all points of $Gx$ and that $Gx$ is compact, we conclude $h$ is a diffeomorphism from a neighbourhood of $Gx$ onto a neighbourhood of $G\cdot h(x)$. It follows that the sets $h^{-1}(U)$ for $U\in\cal U$ form a basis of neighbourhoods of $Gx$. By Remark \ref{remark:trivial} they are also orbitally convex. The theorem now follows from Proposition \ref{proposition:orbit} (or rather, the generalization of Proposition \ref{proposition:orbit} to arbitrary complex spaces, which is just as easy to prove; see \cite{he:ge}). \end{pf} \begin{remark} One can talk of holomorphic actions and momentum maps in the setting of K\"ahler spaces (``K\"ahler manifolds with singularities'') in the sense of Grauert \cite[\S 3]{gr:ub}. It seems reasonable to expect that the Holomorphic Slice Theorem can be extended to this more general situation. \end{remark} \section{K\"ahler Quotients and Geometric Quantization}\label{section:quotient} In this section I apply the Holomorphic Slice Theorem to the study of symplectic quotients of a K\"ahler manifold $M$. The upshot is that such a quotient has a natural structure of an analytic space, and that if $M$ is integral, the quotient is a complex-projective variety. Of course, if $M$ is integral, it is a complex-projective manifold by Kodaira's Embedding Theorem, but the embedding given by Kodaira's theorem is usually not a symplectic embedding into projective space. (For a simple example where it is not, consider any non-singular $X\subset\bold CP^n$. Let $\Omega$ be the restriction of the Fubini-Study form to $X$. For any smooth function $f$ on $X$, put $\Omega_f=\Omega+\sq\,\partial\bar\partial f$. If $f$ is $C^2$-small, $\Omega_f$ is a K\"ahler form. But for most $f$, for instance, those $f$ that are not real-analytic, no holomorphic embedding of $(X,\Omega_f)$ into any projective space $\bold CP^N$ is an isometry.) Under the assumption that Kodaira's embedding {\em does\/} preserve the symplectic form Kirwan \cite{ki:coh} and Ness \cite{ne:st} proved that the symplectic quotient of $M$ agrees with a categorical quotient of a semistable subset of $M$ in the sense of geometric invariant theory. I show that this conclusion still holds even if Kodaira's map is not a symplectic embedding. Thus the result says roughly that the class of symplectic quotients of an integral K\"ahler manifold is not bigger than the class of its algebraic quotients. Alternatively, it says that there are many non-equivalent symplectic structures on the algebraic quotients of $M$. The abovementioned result of Kirwan and Ness is a generalization of earlier work of Guillemin and Sternberg \cite{gu:ge}, and Kempf and Ness \cite{ke:le}. Guillemin and Sternberg dealt with the case where the quotient of $M$ is non-singular. This case is technically simpler mainly owing to the fact that here the action of $G\co$ is proper at all points of the zero level set of the momentum map. (See Theorem \ref{theorem:proper}.) Kempf and Ness handled the case of a linear action on a Hermitian vector space. In fact, I shall reduce the general case to that of a linear action by locally ``flattening out'' the K\"ahler metric. Section \ref{subsection:reduction} is a discussion of quotients of K\"ahler manifolds in the general setting of Section \ref{section:slices}. Section \ref{subsection:integral} focuses on the case of integral K\"ahler manifolds, placing the results of Section \ref{subsection:reduction} in the context of geometric invariant theory. In Section \ref{subsection:multiplicity} I rephrase some of the results in the language of geometric quantization and show how they lead to formul\ae\ for multiplicities of representations. \subsection{Reduction of K\"ahler manifolds}\label{subsection:reduction} As in the previous section let us fix a connected K\"ahler manifold $(M,ds^2)$ on which $G\co$ acts holomorphically and assume there exists an equivariant momentum map $\Phi$ for the action of $G$. Let $\lambda\in\frak g^*$. The {\em symplectic quotient\/} or {\em reduced \rom(phase\rom) space\/} of $M$ at the level $\lambda$ is by definition the topological space $M_\lambda=\Phi^{-1}(G\lambda)/G$, where $G\lambda$ is the coadjoint orbit through $\lambda$. By the results of \cite{sj:st} $M_\lambda$ has the structure of a symplectic stratified space. Roughly speaking, this means that $M_\lambda$ is a disjoint union of symplectic manifolds that fit together in a nice way, and that there is a unique open stratum, which is connected and dense in $M_\lambda$. We want to endow $M_\lambda$ with an analytic structure and show its stratification is analytic. We shall carry this out only for $\lambda=0$; the general case follows from this by dint of the ``shifting trick''. (See \cite{cu:on,sj:st}.) Define a point $m$ in $M$ to be {\em \rom(analytically\rom) semistable\/} if the closure of the $G\co$-orbit through $m$ intersects the zero level set $\Phi^{-1}(0)$, and denote the set of semistable points by $M\sst$. The point $m$ is called {\em \rom(analytically\rom) stable\/} if the closure of the $G\co$-orbit through $m$ intersects the zero level set $\Phi^{-1}(0)$ at a point where $d\Phi$ is surjective. The set of stable points is denoted by $M\st$. The notions of analytic semistability and stability depend on the K\"ahler metric and on the momentum map. If $M$ is integral, they will turn out to be equivalent to semistability, resp.\ stability in the sense of Mumford \cite{mu:ge} with respect to a suitable projective embedding (Theorem \ref{theorem:gaga}). Introduce a $G$-invariant inner product on the Lie algebra of $G$. Let $\mu$ be the ``Yang-Mills functional'' $\|\Phi\|^2$ and let $F_t$ be the gradient flow of the function $-\mu$. Since $\mu$ is $G$-invariant, $F_t$ is $G$-equivariant. By Lemma 6.6 of Kirwan \cite{ki:coh} the gradient of $\mu$ is given by \begin{equation}\label{equation:yangmills} \grad\mu(m)=2J\Phi(m)_{M,m}, \end{equation} where we have identified $\Phi(m)\in\frak g^*$ with a vector in $\frak g$ using the inner product, and where $\bigl(\Phi(m)\bigr)_{M,m}$ is the vector field on $M$ induced by $\Phi(m)$, evaluated at the point $m$. In particular, $\grad\mu$ is tangent to the $G\co$-orbits, so these are preserved by the flow $F_t$. Let us call the momentum map {\em admissible\/} if for every $m\in M$ the path of steepest descent $F_t(m)$ through $m$ is contained in a compact set, as in Kirwan \cite[\S 9]{ki:coh}. If $\Phi$ is admissible, the flow $F_t$ is defined for all $t\geq0$. Kirwan has proved $M\sst$ is the set of points $m\in M$ with the property that the path $F_t(m)$ has a limit point in $\Phi^{-1}(0)$. Using the ideas of Neeman \cite{ne:to} one can show that for all $m\in M$ the limit $F_\infty(m)= \lim_{t\to\infty}F_t(m)$ actually exists and, moreover, that the restriction of the map $F_\infty$ to $M\sst$ is a continuous retraction of $M\sst$ onto $\Phi^{-1}(0)$. (See also Schwarz \cite{sc:to}.) All proper momentum maps are admissible. Here is another simple example, which will be important in what follows. \begin{example}\label{example:kempfness} Consider a linear action of $G\co$ on $\bold C^N$ with the standard momentum map $\Phi_{\bold C^N}$ given by (\ref{equation:quadratic}). Let $\mu_{\bold C^N}=\|\Phi_{\bold C^N}\|^2$. An easy computation using (\ref{equation:angle}) shows that $\bigl\langle\grad r^2,\grad\mu_{\bold C^N}\bigr\rangle=8\mu_{\bold C^N}$, where $r$ denotes the distance to the origin. Consequently, at all points of the sphere bounding a ball $B$ about the origin the vector field $-\grad\mu_{\bold C^N}$ points into $B$. Therefore $\Phi_{\bold C^N}$ is admissible. Let $(F_{\bold C^N})_t$ be the gradient flow of $-\mu_{\bold C^N}$. Then it is clear that the limit map $(F_{\bold C^N})_\infty$ retracts the set $G\co B$ onto $\Phi^{-1}_{\bold C^N}(0)\cap B$. \end{example} Throughout this section we will assume $\Phi$ to be admissible. We now collect a number of basic results on the orbit structure of $M\sst$, which are either due to Kirwan \cite{ki:coh}, or which are refinements of her results. \begin{proposition}\label{proposition:semistable} In the following \romquote closed\romunquote\ means \romquote closed in the relative topology of $M\sst$\romunquote\ and \romquote closure\romunquote\ means \romquote closure in $M\sst$\romunquote. \begin{enumerate} \item\label{open} The semistable set $M\sst$ is the smallest $G\co$-invariant open subset of $M$ containing $\Phi^{-1}(0)$\rom, and its complement is a complex-analytic subset\rom; \item\label{closed} A $G\co$-orbit in $M\sst$ is closed if and only if it intersects $\Phi^{-1}(0)$\rom; \item\label{intersection} The intersection of a closed $G\co$-orbit with $\Phi^{-1}(0)$ consists of precisely one $G$-orbit\rom; \item\label{limit} For every semistable point $y$ the set $F_\infty(G\co y)\subset\Phi^{-1}(0)$ consists of precisely one $G$-orbit\rom; \item\label{separate} For any pair of points $x,y\in\Phi^{-1}(0)$ that do not lie on the same $G$-orbit there exist disjoint $G\co$-invariant open subsets $U$ and $V$ of $M$ with $x\in U$ and $y\in V$\rom; \item\label{closure} The closure of every $G\co$-orbit in $M\sst$ contains exactly one closed $G\co$-orbit. \end{enumerate} \end{proposition} \begin{pf} See \cite[\S 4]{ki:coh} for a proof of (\ref{open}). We now prove (\ref{closed}). If $x$ is semistable and $G\co x$ is a closed subset of $M\sst$, then $F_\infty(x)\in G\co x$ because the flow $F_t$ preserves the $G\co$-orbits, and also $F_\infty(x)\in\Phi^{-1}(0)$, so $G\co x\cap\Phi^{-1}(0)$ is non-empty. Conversely, suppose $G\co x\cap\Phi^{-1}(0)\neq\emptyset$. Let $(y_i)$ be a sequence in $G\co x$ converging to $y\in M\sst$. We have to show $y\in G\co x$. Clearly $F_\infty(y)\in\overline{G\co y}$ and by continuity $F_\infty(y)=\lim_{i\to\infty}F_\infty(y_i)\in G\co x$. Therefore $G\co y$ intersects every open neighbourhood of $G\co x$. In particular, $y$ is contained in every $G\co$-invariant open neighbourhood of $x$. Since $G\co x$ intersects $\Phi^{-1}(0)$, the Holomorphic Slice Theorem tells us $x$ has a $G\co$-invariant tubular neighbourhood $U$. Evidently, $G\co x$ is a closed subset of $U$. Since $y\in U$, this implies $y=\lim_{i\to\infty}y_i\in G\co x$. See \cite[\S 6]{ki:coh} for a proof of (\ref{intersection}). For the proof of (\ref{limit}), let $x=F_\infty(y)$. Pick an arbitrary point $z\in G\co y$. We need to show $F_\infty(z)\in Gx$. If $G$ is connected, we can find a continuous path $\gamma\colon[0,1]\to M\sst$ with $\gamma(0)=x$, $\gamma(1)=z$ and $\gamma(t)\in G\co y$ for $t>0$. (If $G$ is not connected, we can still do this, provided we replace $x$ by a suitable translate $gx$, where $g\in G$.) Consider the path $F_\infty\circ\gamma$ contained in $\Phi^{-1}(0)$, and let $I$ be the set of all $t$ in the unit interval such that the point $x(t)$ defined by $x(t)= F_\infty\bigl(\gamma(t)\bigr)$ is contained in $Gx$. We claim $I$ is open in $[0,1]$. Indeed, suppose $t\in I$. Recall that by Lemmas \ref{lemma:embedding}--\ref{lemma:isometry} we have a slice $S$ at $x$ with the following special properties: There exists a $G\co$-equivariant embedding $\iota$ of $U=G\co S$ into a $G\co$-representation space $\bold C^N$, and $U$ is equal to the set $G\co O$, where $O$ is the inverse image $\iota^{-1}(B)$ of a Euclidean ball about the origin in $\bold C^N$. In order not to overburden the notation we shall identify $U$ with its image in $\bold C^N$. Now let $(F_{\bold C^N})_t$ be the gradient flow on $\bold C^N$ associated to the function $-\mu_{\bold C^N}$ of Example \ref{example:kempfness}. Then $(F_{\bold C^N})_\infty=\lim_{t\to\infty}(F_{\bold C^N})_t$ retracts $U$ onto $\Phi^{-1}_{\bold C^N}(0)\cap O$. By choosing $B$ sufficiently small we can arrange that $O$, and hence $U$, are contained in $M\sst$. Also, $G\co y\subset U$, since $x$ is in the closure of $G\co y$. Since $\gamma(s)\in G\co y$ we also have $x(s)\in\overline{G\co y}$. By (\ref{closed}) the orbit $G\co x(s)$ is closed in $M\sst$, and hence in $U$. Furthermore, $x(s)\in O$ for $s$ sufficiently close to $t$. Therefore, $(F_{\bold C^N})_\infty\bigl(x(s)\bigr)\in\Phi^{-1}_{\bold C^N}(0)\cap G\co x(s)$. It now follows from (\ref{closed}) (applied to the momentum map $\Phi_{\bold C^N}$) that for $s$ sufficiently close to $t$ the orbit $G\co x(s)$ is closed in $\bold C^N$. By construction each $x(s)$ is contained in the closure of the orbit $G\co y\subset U$. But in a $G\co$-representation space each orbit contains a {\em unique\/} closed orbit in its closure. (See e.g.\ Luna \cite[\S 1]{lu:sl}.) We conclude that $x(s)\in G\co x$ for all $s$ close enough to $t$. Since $\Phi\bigl(x(s)\bigr)=0$, it follows from (\ref{intersection}) that $x(s)\in Gx$, in other words $s\in I$. Thus we have shown $I$ is open. Obviously, $I$ is also closed and $0\in I$. It follows $I=[0,1]$, and therefore $F_\infty(z)= F_\infty\bigl(\gamma(1)\bigr)\in Gx$. This finishes the proof of (\ref{limit}). To prove (\ref{separate}), observe that (\ref{limit}) implies that for any $G$-invariant subset $A$ of $\Phi^{-1}(0)$ the preimage $F_\infty^{-1}(A)\subset M\sst$ is $G\co$-invariant. Now suppose $x,y\in\Phi^{-1}(0)$ and $y\not\in Gx$. Because $G$ is compact, there exist disjoint $G$-invariant open subsets $A$ and $B$ of $\Phi^{-1}(0)$ with $x\in A$ and $y\in B$. Then $F_\infty^{-1}(A)$ and $F_\infty^{-1}(B)$ are disjoint $G\co$-invariant open sets containing $x$, resp.\ $y$. Finally, (\ref{closure}) follows immediately from (\ref{closed}) and (\ref{separate}). \end{pf} Call two semistable points $x$ and $y$ {\em related\/} if the closures of the orbits $G\co x$ and $G\co y$ intersect. (Again, ``closure'' means ``closure in $M\sst$''.) Assertion (\ref{closure}) of Proposition \ref{proposition:semistable} implies this relation is an equivalence relation. Write $M\sst\qu G\co$ for the quotient space and $\Pi$ for the quotient map $M\sst\to M\sst\qu G\co$. By (\ref{separate}) above, the space $M\sst\qu G\co$ is Hausdorff. \begin{theorem}\label{theorem:homeomorphism} The inclusion $\Phi^{-1}(0)\subset M\sst$ induces a homeomorphism $M_0\to M\sst\qu G\co$. \end{theorem} \begin{pf} By (\ref{closed}) and (\ref{intersection}) of Proposition \ref{proposition:semistable} the map $M_0\to M\sst\qu G\co$ sending a $G$-orbit $Gm\subset\Phi^{-1}(0)$ to the equivalence class $\Pi(m)$ is a continuous injection. By (\ref{closure}) it is a bijection. Moreover, the inverse is induced by the retraction $F_\infty\colon M\sst\to\Phi^{-1}(0)$ and is therefore continuous. \end{pf} \begin{remark} Proposition \ref{proposition:semistable} shows that $M_0$ can also be identified with the space of closed $G\co$-orbits in $M\sst$. \end{remark} Let us say that a subset $A$ of $M\sst$ is {\em saturated\/} with respect to $\Pi$ if $\Pi^{-1}\Pi(A)=A$. This means that for every $x$ in $A$ the closure of $G\co x$ is contained in $A$. \begin{proposition}\label{proposition:saturated} At every point of $\Phi^{-1}(0)$ there exists a slice $S$ such that the set $G\co S$ is saturated with respect to the quotient mapping $\Pi$. \end{proposition} \begin{pf} Let $x$ be any point in $\Phi^{-1}(0)$. We use the notation of the proof of part (\ref{limit}) of Proposition \ref{proposition:semistable}. We shall show that, after shrinking $O$ if necessary, the set $U$ becomes $\Pi$-saturated. Choose a ball $B'\subset B$ so small that the $G$-invariant neighbourhood $O'=\iota^{-1}(B')$ of $Gx$ has the property that $F_\infty(O')\subset O$. This is possible because the sets $\iota^{-1}(B)$ form a basis of neighbourhoods of $Gx$ by Lemma \ref{lemma:embedding} and because $F_\infty$ is the identity on $\Phi^{-1}(0)$. Take any $y\in U'$. We claim that $\overline{G\co y}$ is a subset of $U'$. Since $F_\infty(O')\subset O$, part (\ref{limit}) of Proposition \ref{proposition:semistable} implies that $F_\infty(U')\subset O$, where $U'=G\co O'$. In particular $F_\infty(y)\in O$, and so $G\co\cdot F_\infty(y)\subset U$. Assertion (\ref{closed}) of Proposition \ref{proposition:semistable} implies $G\co\cdot F_\infty(y)$ is closed in $M\sst$, and hence in $U$. Now $(F_{\bold C^N})_\infty$ maps $U$ into $O$, so $(F_{\bold C^N})_\infty\bigl(F_\infty(y)\bigr)\in O$. Moreover, since $G\co\cdot F_\infty(y)$ is closed in $M\sst$, $(F_{\bold C^N})_\infty\bigl(F_\infty(y)\bigr)$ sits in $G\co\cdot F_\infty(y)$. Therefore $G\co\cdot F_\infty(y)$ is closed in $\bold C^N$ (by part (\ref{closed}) of Proposition \ref{proposition:semistable} applied to the momentum map $\Phi_{\bold C^N}$). But $(F_{\bold C^N})_\infty(y)$ is contained in $O'$, and $G\co\cdot(F_{\bold C^N})_\infty(y)$ is closed in $\bold C^N$. Moreover, both orbits $G\co\cdot(F_{\bold C^N})_\infty(y)$ and $G\co\cdot F_\infty(y)$ are contained in the closure of $G\co y$. It follows that $G\co\cdot F_\infty(y)=G\co\cdot(F_{\bold C^N})_\infty(y)\subset U'$, and so $F_\infty(y)\in U'$. We now conclude from part (\ref{limit}) of Proposition \ref{proposition:semistable} and the continuity of $F_\infty$ that $\overline{G\co y}\subset U'$. \end{pf} {}From now on we'll identify the spaces $M\sst\qu G\co$ and $M_0$. We want to furnish $M_0$ with a complex-analytic structure in such a way that the quotient map $\Pi$ becomes holomorphic. The richest possible such structure is obtained by declaring a function $f$ defined on an open subset $A$ of $M_0$ to be holomorphic if the pullback of $f$ to $\Pi^{-1}(A)\subset M\sst$ is holomorphic. Let $\cal O_{M_0}$ be the sheaf of holomorphic functions on $M_0$. We claim this indeed defines an analytic structure. \begin{theorem}\label{theorem:analytic} The ringed space $(M_0,\cal O_{M_0})$ is an analytic space. \end{theorem} \begin{pf} Let $p\in M_0$ and let $m$ be a point in $\Phi^{-1}(0)$ sitting over $p$. By the definition of $\cal O_{M_0}$ a neighbourhood of $p$ is isomorphic as a ringed space to a quotient $U\qu G\co$, where $U$ is a $\Pi$-saturated open set containing $m$, equipped with the sheaf of $G\co$-invariant holomorphic functions. By Proposition \ref{proposition:saturated} we may take $U$ to be of the form $G\co S$, where $S$ is a slice at $m$. Then $U$ can be identified with an invariant open subset of the bundle $E=G\co\times_{(G\co)_m}V$, where $V$ is the tangent space to a slice $S$ at $m$, and the quotient $U\qu G\co$ can be identified with an open subset of $E\qu G\co= V\qu(G\co)_m$. Now by a theorem of Luna \cite{lu:fo} every $(G\co)_m$-invariant holomorphic function on the $(G\co)_m$-representation space $V$ is a holomorphic function of the invariant {\em polynomials\/} on $V$. Picking a finite number of generators $(\sigma_1,\dots,\sigma_l)$ of the ring of invariant polynomials we get a Hilbert map $\sigma\colon V\to\bold C^l$, sending $v$ to $\bigl(\sigma_1(v),\dots,\sigma_l(v)\bigr)$. The Hilbert map descends to a map $V\qu(G\co)_m\to\bold C^l$, which by Luna's theorem is a closed embedding of the ringed space $V\qu(G\co)_m$. It follows $U\qu G\co$ is isomorphic as a ringed space to an analytic subset of an open subset of $\bold C^l$. Therefore $(M_0,\cal O_{M_0})$ is an analytic space. \end{pf} The proof of this theorem shows that in a neighbourhood of the point $p$ the quotient map $\Pi$ is equivalent to the quotient map $E\to E\qu G\co$ of the non-singular affine $G\co$-variety $E$. \begin{corollary}\label{corollary:affine} The quotient map $\Pi$ is locally biholomorphically equivalent to an affine map. In particular\rom, the fibres of $\Pi$ are affine varieties.\qed \end{corollary} The Holomorphic Slice Theorem implies that if the stabilizer of a point $m\in\Phi^{-1}(0)$ is finite, all $G\co$-orbits in an invariant neighbourhood of $m$ must have the same dimension. From this observation plus Theorem \ref{theorem:proper} and Proposition \ref{proposition:semistable} one can easily deduce the following result. \begin{theorem}\label{theorem:stable} If $x\in M$ is stable\rom, the orbit $G\co x$ is closed in $M\sst$ and the stabilizer $(G_x)\co$ is finite. Let $Z$ be the set of $m\in\Phi^{-1}(0)$ with the property that $d\Phi_m$ is surjective\rom; then the stable set $M\st$ is equal to $F_\infty^{-1}(Z)$. Every fibre of $\Pi|_{M\st}$ consists of a single orbit. In particular\rom, if $0$ is a regular value of $\Phi$\rom, $M\sst$ coincides with $M\st$ and $M_0\cong M\st/G\co$\rom, the space of stable orbits in $M$\rom, is a K\"ahler orbifold.\qed \end{theorem} Let $p$ be in $M_0$, let $x$ be a point in $\Phi^{-1}(0)$ mapping to $p$ and let $(H)$ be a conjugacy class of closed subgroups of $G$. Then $p$ is said to be of {\em $G$-orbit type\/} $(H)$ if the stabilizer $G_x$ is conjugate to $H$ in $G$. In \cite{sj:st} we showed that the set of all points of orbit type $(H)$ is a manifold carrying a natural symplectic structure and that the decomposition of $M_0$ into orbit type manifolds is a stratification. Now let $(L)$ be a conjugacy class of reductive subgroups of $G\co$. We may assume $L=H\co$ for some closed subgroup $H$ of $G$. By Proposition \ref{proposition:semistable} the fibre $\Pi^{-1}(p)$ contains a unique closed $G\co$-orbit, namely $G\co x$. Let us say $p$ is of {\em $G\co$-orbit type\/} $(H\co)$ if the stabilizer $(G\co)_x$ is conjugate to $H\co$ in $G\co$. \begin{theorem}\label{theorem:stratification} The stratification of $M_0$ by $G$-orbit types is identical to the stratification by $G\co$-orbit types. Each stratum $\cal S$ is a complex manifold and its closure is a complex-analytic subvariety of $M_0$. The reduced symplectic form on $\cal S$ is a K\"ahler form. \end{theorem} \begin{pf} The first assertion boils down to showing that if $H$ and $K$ are two closed subgroups of $G$, and $H\co$ and $K\co$ are conjugate in $G\co$, then $H$ and $K$ are conjugate in $G$. To say that $H\co$ and $K\co$ are conjugate in $G\co$ amounts to saying that there is a $G\co$-equivariant diffeomorphism of homogeneous spaces $f\colon G\co/H\co\to G\co/K\co$. By Mostow's decomposition (\ref{equation:mostow}), for every closed subgroup $R$ of $G$ the complexified homogeneous space $G\co/R\co$ is a homogeneous vector bundle over $G/R$, so there exist a $G$-equivariant embedding $\iota_R\colon G/R\to G\co/R\co$ and a $G$-equivariant retraction $\rho_R\colon G\co/R\co\to G/R$. So the composite $\rho_K\circ f\circ\iota_H$ is a $G$-equivariant map $G/H\to G/K$. Therefore $H$ is conjugate (in $G$) to a subgroup of $K$. Switching the r\^oles of $H$ and $K$, we see that $K$ is conjugate to a subgroup of $H$. Therefore, since $H$ and $K$ have finitely many components, $H$ is conjugate to $K$ in $G$. For quotients of affine $G\co$-varieties Luna proved in \cite[\S III.2]{lu:sl} that each stratum is non-singular and that its closure is a variety. In view of the fact that $\Pi$ is locally equivalent to a quotient map of an affine variety this implies the second statement of the theorem. (To be precise, Luna's stratification is not the same as ours, but it is easy to see that they are the same up to connected components.) Let $\cal S$ be the stratum of orbit type $(H)$. It is well-known that if $G$ acts freely on the zero level set $\Phi^{-1}(0)$ (which implies $0$ is a regular value of $\Phi$) the reduced symplectic form is K\"ahler. (See e.g.\ \cite{gu:ge}.) Therefore, to prove that the reduced symplectic form on $\cal S$ is K\"ahler, it suffices to show that $\cal S$ can be obtained by carrying out reduction at a regular level on some K\"ahler manifold with respect to some group action. Let $N=N_G(H)$ be the normalizer of $H$ in $G$ and let $M_H$ be the set of ``symmetry type'' $H$, that is, the collection of all points whose stabilizer (with respect to the $G$-action) is exactly $H$. Then $M_H$ is a complex submanifold of $M$, so it is K\"ahler. Moreover, $M_H$ is $N$-invariant and $H$ acts trivially on it. Let $L=N/H$. By Theorem 3.5 of \cite{sj:st} the momentum map maps $M_H$ into $\frak l^*$ and the restriction $\Phi_H$ of $\Phi$ to $M_H$ is a momentum map for the $N$-action on $M_H$. Moreover, $0$ is a regular value of $\Phi_H$ and the reduced space $\Phi_H^{-1}(0)$ and the stratum $\cal S$ are symplectically diffeomorphic in a natural way. Since $M_H$ is complex, we have a well-defined action of $L\co=N\co/H\co$ on $M_H$. Let $(M_H)\sst=(M_H)\st$ denote the set of points in $M_H$ stable with respect to the momentum map $\Phi_H$. By Theorem \ref{theorem:homeomorphism} and Theorem \ref{theorem:stable} we have a map $(M_H)\st/L\co\cong\Phi_H^{-1}(0)/L\to M_0$, which is a homeomorphism onto the image $\cal S$. To finish the proof, it suffices to show that this map is biholomorphic onto $\cal S$. Since $\Phi(M_H)$ is a subset of $\frak l^*$, (\ref{equation:yangmills}) implies that the flow of $\|\Phi\|^2$ leaves $M_H$ invariant. Because also $\Phi_H^{-1}(0)= \Phi^{-1}(0)\cap M_H$, we see that $(M_H)\st=M\sst\cap M_H$. Moreover, we have a commutative diagram: \begin{equation}\label{equation:stratum} \begin{CD} (M_H)\st @>\Pi>>(M_H)\st/L\co \\ @ViVV @VV{\bar\imath}V\\ M\sst @>\Pi_H>> M\sst\qu G\co, \end{CD} \end{equation} where the inclusion $i$ is biholomorphic onto its image. From the definition of the complex structures on $M\sst\qu G\co$ and $(M_H)\st/L\co$ it now follows that $\bar\imath$ is biholomorphic onto its image. \end{pf} \begin{remark}\label{remark:strata} The orbit type stratification is the minimal real-analytic Whitney stratification of $M_0$. However, it is not the minimal complex-analytic stratification. This is obvious from the following simple example. The {\em $(1,-1)$-resonance\/} is the $S^1$-action on $\bold C^2$ defined by $e^{\sq\,\theta}\cdot(z_1,z_2)= (e^{\sq\,\theta}z_1,e^{-\sq\,\theta}z_2)$. As a real-analytic space the reduced space is isomorphic to the cone in $\bold R^3$ given by $x_1^2 = x_2^2 + x_3^2$ and $x_1\geq 0$. (See \cite{cu:on}.) There are two strata: the vertex and the complement of the vertex. But from the complex-analytic point of view the singularity at the vertex is spurious: The ring of $\bold C^\times$-invariant polynomials is just $\bold C[z_1z_2]$, so the quotient is simply $\bold C$. \end{remark} \subsection{The integral case}\label{subsection:integral} The most important special case of the situation of the previous section is that of a positive holomorphic line bundle over a complex manifold $M$, that is, a holomorphic line bundle $\rho\colon L\to M$ with Hermitian fibre metric $\langle\cdot,\cdot\rangle$ and curvature form $\Theta$ such that the real $(1,1)$-form $\omega=-(2\pi\sq)^{-1}\Theta$ is K\"ahler. Recall that $\Theta$ is the unique two-form on $M$ satisfying $\rho^*\Theta=\bar\partial\partial\log r^2$, where $r\colon L\to\bold R$ is the length function, $r(l)=\langle l,l\rangle^{1/2}$. The K\"ahler class $[\omega]$ is the image of the Chern class $c_1(L)$ of $L$ under the natural map $H^2(M,\bold Z)\to H^2(M,\bold R)$. Now suppose that the compact group $G$ acts on $L$ by linear bundle transformations that leave the Hermitian metric invariant. Then the connection on $L$ is invariant and $\omega$ is invariant under the induced action on the base $M$. This implies that for each $\xi$ in $\frak g$ there exists a unique real-valued function $\Phi^\xi$ on $M$ such that the vector field $\xi_L$ is given by the following formula: \begin{equation}\label{equation:lift} \xi_L=\xi_{M,\text{hor}}+2\pi\Phi^\xi\nu_L. \end{equation} Here $\xi_{M,\text{hor}}$ is the horizontal lift of $\xi_M$ to $TL$ with respect to the Hermitian connection, and $\nu_L$ denotes the vector field on $L$ generating the circle action defined by fibrewise multiplication by complex numbers of length one. It is not hard to check that $\Phi^\xi$ is a Hamiltonian for the vector field $\xi_M$, and therefore the action on $M$ is Hamiltonian. The momentum map $\Phi\colon M\to\frak g^*$ is automatically equivariant. The infinitesimal action of $G$ on smooth sections $s$ of $L$ is given by: $$ \xi\cdot s= -\nabla_{\xi_M}s+2\pi\sq\,\Phi^\xi\cdot s. $$ As before, let us assume that the $G$-action on $M$ extends to a $G\co$-action and that the momentum map $\Phi$ is admissible. Then the $G$-actions on $L$ and on its smooth sections can both be uniquely extended to actions of $G\co$, and the projection $\rho$ is $G\co$-equivariant. (See \cite{gu:ge}.) Note that a holomorphic section of $L$ defined over a $G\co$-invariant open set is $G$-invariant if and only if it is $G\co$-invariant. Let $\cal L$ be the sheaf of holomorphic sections of $L$ and define a sheaf $\cal L_0$ on $M_0$, the {\em sheaf of invariant sections}, by putting $\cal L_0(O)=\cal L(\Pi^{-1}(O))^G$. According to Roberts \cite{ro:no}, $\cal L_0$ is a coherent $\cal O_{M_0}$-module. The following result says that $\cal L_0$ is ``almost'' a holomorphic line bundle over $M_0$. \begin{proposition}\label{proposition:vbundle} The sheaf $\cal L_0$ is \rom(the sheaf of sections of\rom) a holomorphic $V$-line bundle over $M_0=M\sst\qu G\co$. \end{proposition} \begin{pf} Let $p\in M_0$. We have to show there exist a neighbourhood $O$ of $p$ that can be written as a quotient of an analytic space $\tilde O$ by the action of a finite group $\Gamma$ and a locally free sheaf $\tilde{\cal L}_0$ of rank one over $\tilde O$ such that $\cal L_0|_O$ is isomorphic to the sheaf of $\Gamma$-invariant sections of $\tilde{\cal L}_0$. Let $m$ be a point in $\Phi^{-1}(0)$ mapping to $p$ and let $S$ be a slice at $m$ such that $U=G\co S$ is $\Pi$-saturated. Put $O=\Pi(U)$; then $O\cong S\qu H\co$ as analytic spaces. Here $H$ denotes the stabilizer $G_m$ of $m$. The group $H$ acts linearly on the fibre $L_m$. If $\eta\in\frak h$ and $l\in L_m$, then (\ref{equation:lift}) implies $\eta_L\cdot l=0$, since $\Phi(m)=0$. In other words, the identity component $H^0$ acts trivially on $L_m$. Now let $\cal N$ be the restriction of $\cal L$ to $S$ and let $s$ be a holomorphic section of $\cal N$ that does not vanish at $m$. Then the section $\int_{H^0}h\cdot s\,dh$ is holomorphic, $H\co$-invariant and does not vanish at $m$, because $H^0$ acts trivially on $L_m$. Hence we may assume $s$ to be $H^0$-invariant. Define $\tilde O=S\qu(H^0)\co$ and let $\tilde{\cal L}_0$ be the sheaf of $(H^0)\co$-invariant sections of $\cal N$. After shrinking $\tilde O$ if necessary, we may assume $\tilde s$ vanishes nowhere on $\tilde O$, so $\tilde{\cal L}_0$ is a free sheaf of rank one on $\tilde O$. By construction $O$ is the quotient of $\tilde O$ by the finite group $H\co/(H^0)\co=H/H^0$, and $\cal L_0|_O$ is isomorphic to the sheaf of $H/H^0$-invariant sections of $\tilde{\cal L}_0$. It follows $\cal L_0$ is the sheaf of sections of a holomorphic $V$-line bundle over $M_0$. \end{pf} In fact, the total space $L_0$ of this $V$-line bundle is simply the quotient of $L|_{\Phi^{-1}(0)}$ by $G$. Furthermore, as an analytic space $L_0$ can be identified with a quotient $L\sst\qu G\co$, where $L\sst$ is by definition the restriction of $L$ to $M\sst$. The proofs of these facts are sufficiently similar to the proofs in Section \ref{subsection:reduction} that I can omit them. To get a genuine line bundle on $M_0$, we have to replace $L$ by a suitable power. The proof of Proposition \ref{proposition:vbundle} shows that for every $m\in\Phi^{-1}(0)$ the image of $G_m\to\Aut(L_m)$ is a finite cyclic group. Let $q(m)$ be the order of this group. \begin{proposition}\label{proposition:bundle} Suppose $\Phi$ is proper. Let $q$ be the least common multiple of the $q(m)$ for $m$ ranging over $\Phi^{-1}(0)$. Then $(L^q)_0=(L^q)\sst\qu G\co$ is a line bundle over $M_0$ satisfying $\Pi^*(L^q)_0=L^q|_{M\sst}$. \end{proposition} \begin{pf} First note that since $\Phi$ is proper, its zero level set is compact and so contains only finitely many orbit types. Furthermore, $q(m)$ is not greater than the order of the component group $G_m/(G_m)^0$. Therefore the integer $q$ is well-defined. It has the property that $G_m$ acts trivially on the fibre $L^q_m$ for all $m\in\Phi^{-1}(0)$. As in the proof of Proposition \ref{proposition:vbundle} we conclude that at every semistable point there exists a non-vanishing invariant local holomorphic section of $L^q$. By means of these sections we can define local trivializations of $(L^q)_0$, so $(L^q)_0$ is a holomorphic line bundle over $M_0$. Moreover, the existence of these sections implies $G_x$ acts trivially on $L^q_x$ for {\em all\/} semistable $x$. Using this one can easily show that the commutative diagram $$ \begin{CD} L^q|_{M\sst} @>>> (L^q)_0 \\ @V\rho VV @VVV\\ M\sst @>\Pi >> M_0 \end{CD} $$ is a pullback diagram. Therefore $\Pi^*(L^q)_0=L^q|_{M\sst}$. \end{pf} Grauert \cite{gr:ub} has defined a (holomorphic) line bundle $E$ over a complex space to be {\em negative\/} if the zero section in $E$ has a strictly pseudoconvex open neighbourhood. He called a bundle {\em positive\/} if its dual is negative. To show that $(L^q)_0$ is positive, we first need to discuss potentials for the reduced K\"ahler structure. If $m\in\Phi^{-1}(0)$, a potential for the K\"ahler form $\omega$ on an open neighbourhood $U$ of $m$ is given by $u=-(2\pi)^{-1}\log\langle s,s\rangle$, where $s$ is a local holomorphic section of $L$ that does not vanish at $m$. If $G$ acts freely on $\Phi^{-1}(0)$, the proof of Proposition \ref{proposition:vbundle} shows we can find an invariant such section. Then $u$ is a $G$-invariant smooth potential near $m$, and its restriction to $U\cap\Phi^{-1}(0)$ pushes down to a smooth function $u_0$ defined on $U_0=\bigl(U\cap\Phi^{-1}(0)\bigr)\big/G$. It is easy to see that $u_0$ is a potential for the reduced symplectic form $\omega_0$. If $G$ does not act freely on $\Phi^{-1}(0)$, it may not be possible to find such a section, but we can certainly find an invariant local holomorphic section $s$ of the $q$-th power of $L$ that does not vanish at $m$. Then $u=-(2\pi q)^{-1}\log\langle s,s\rangle$ is a $G$-invariant potential for $\omega$ near $m$, and as before its restriction to $U\cap\Phi^{-1}(0)$ pushes down to a function $u_0$ on the reduced space. Unfortunately, $u_0$ is not necessarily smooth or even $C^1$ on $M_0$. (By a smooth function on $M_0$ we mean a function that can be locally written as a differentiable function of the holomorphic functions on $M_0$. This notion of smooth functions differs from the one introduced in \cite{sj:st}.) This is clear from the example in Remark \ref{remark:strata}, where $u(z_1,z_2)= \bigl(|z_1|^2+|z_2|^2\bigr)\big/2$ and $u_0(w)=|w|/2$, with $w=z_1z_2$. Nonetheless, we claim $u_0$ is strictly plurisubharmonic in the sense of distributions. Recall that a continuous function $f$ on $M_0$ is {\em plurisubharmonic\/} if for all discs $D\subset\bold C$ and all analytic maps $c\colon D\to M_0$ the distribution $\Delta(c^*f)$ is non-negative, where $\Delta$ is the standard Laplacian on $\bold C$. It is {\em strictly plurisubharmonic\/} if for all smooth $g$ with compact support the function $f+\eps g$ is plurisubharmonic for small $\eps$. (See Grauert and Remmert \cite{gr:pl} and Lelong \cite[p. 46]{le:pl}.) \begin{lemma}\label{lemma:plurisubharmonic} The continuous function $u_0$ is strictly plurisubharmonic on the open subset $U_0$ of the analytic space $M_0$. It is a potential for the reduced K\"ahler structure in the following sense\rom: Let $\cal S$ be any orbit type stratum in $M_0$ and let $\omega_{\cal S}$ be the reduced symplectic form on $\cal S$. Then $u_0$ is smooth on $U_0\cap\cal S$ and there it satisfies $\omega_{\cal S}=\sq\,\partial\bar\partial u_0$. \end{lemma} \begin{pf} The second statement is an immediate consequence of the observation that the stratum $\cal S$ in $M_0$ of orbit type $(H)$ can be written as a quotient $\cal S=\Phi_H^{-1}(0)/L=(M_H)\st/L\co$, where $L=N_G(H)/H$ and $L\co$ acts properly and freely on $(M_H)\st$, as in diagram (\ref{equation:stratum}). The function $u_0|_{\cal S}$ is equal to the pushforward of $u|_{M_H}$ under the map $\Phi_H^{-1}(0)\to\cal S$. Therefore it is smooth and satisfies $\omega_{\cal S}=\sq\,\partial\bar\partial u_0|_{\cal S}$. In particular, $u_0$ is strictly plurisubharmonic on every stratum of $M_0$. To see it is strictly plurisubharmonic as a function on $M_0$, we first consider the special case where $M=\bold C^n$ is a $G\co$-representation with standard momentum map $\Phi=\Phi_{\bold C^n}$ and standard flat metric with potential $u=\|z\|^2\big/2$. We embed the quotient $M_0=\bold C^n\qu G\co$ into $\bold C^l$ using homogeneous invariant complex polynomials $\sigma_1,\dots$, $\sigma_l$ as in the proof of Theorem \ref{theorem:analytic}. We shall identify $M_0$ with its image $\sigma(M_0)\subset\bold C^l$. Let $w=(w_1,\dots,w_l)$ be coordinates on $\bold C^l$. We claim $u_0$ is strictly plurisubharmonic at the ``vertex'' $0\in M_0\subset\bold C^l$. It suffices to show that for sufficiently small $\eps$ the function $u_0-\eps\|w\|^2$ is plurisubharmonic close to the vertex. (For simplicity we have written $\|w\|^2$ for the restriction of $\|w\|^2$ to $M_0$.) Observe $u_0-\eps\|w\|^2$ is continuous on $M_0$, so by the extension theorem for plurisubharmonic functions of Grauert and Remmert \cite{gr:pl} it suffices to show the restriction of $u_0-\eps\|w\|^2$ to the complement of a thin subset is plurisubharmonic close to the vertex. By Theorem \ref{theorem:stratification} the complement of the topdimensional stratum $\cal T$ is a thin subset of $M_0$. We now exploit the fact that the cone $M_0$ is quasi-homogeneous in $\bold C^l$. Consider the action $A$ of the positive real numbers on $\bold C^n$ defined by scalar multiplication, $A_\lambda z=\lambda z$. Let $d_1,\dots$, $d_l$ be the degrees of the homogeneous polynomials $\sigma_1,\dots$, $\sigma_l$, and define an action $A$ of $\bold R_{>0}$ on $\bold C^l$ by putting $A_\lambda(w_1,\dots,w_l)= (\lambda^{d_1}w_1,\dots,\lambda^{d_l}w_l)$. Then the Hilbert map $\sigma\colon\bold C^n\to\bold C^l$ is equivariant, $A_\lambda\circ\sigma=\sigma\circ A_\lambda$, and the stratum $\cal T$ is $A$-invariant. The Hermitian bilinear forms on $\cal T$ corresponding to the real $(1,1)$-forms $\sq\,\partial\bar\partial u_0|_{\cal T}$ and $\sq\,\partial\bar\partial \|w\|^2\big|_{\cal T}$ are positive definite. Moreover, the flat metric on $\bold C^n$ is conical, that is, $A_\lambda^*(\partial\bar\partial u)=\lambda^2\partial\bar\partial u$, and therefore the induced metric on the quotient is conical, $A_\lambda^*(\partial\bar\partial u_0)=\lambda^2\partial\bar\partial u_0$. On the other hand, $$ A_\lambda^*\bigl(\partial\bar\partial\|w\|^2\bigr)= A_\lambda^*\Bigl(\sum_{ij}dw_i\wedge d\bar w_j\Bigr)= \sum_{ij}\lambda^{d_i+d_j}\,dw_i\wedge d\bar w_j. $$ Since $d_i\geq1$ for all $i$, we see that for sufficiently small $\eps$ the bilinear form corresponding to $\sq\,\partial\bar\partial\bigl(u_0-\eps\|w\|^2\bigr)\big|_{\cal T}$ is positive semidefinite on $\cal T\cap B$, where $B$ is a small ball about the origin in $\bold C^l$. Consequently $u_0-\eps\|w\|^2$ is plurisubharmonic on $M_0\cap B$ for small $\eps$, and so $u_0$ is strictly plurisubharmonic at the vertex $0\in M_0$. Now let $M$ be arbitrary and consider any point $m\in\Phi^{-1}(0)$. Let $H=G_m$ and let $V=\bigl(T_m(G\co m)\bigr)^\perp$ be the tangent space to the holomorphic slice at $m$. Then $V$ is a Hermitian vector space and by the Holomorphic Slice Theorem we have an analytic isomorphism $V\qu H\co\to M\sst\qu G\co$ defined near the vertex $0$ of $V\qu H\co$ and mapping $0$ to $\Pi(m)$. We now have two K\"ahler metrics on the top stratum defined near $\Pi(m)$, namely the metric $ds_0^2$ with potential $u_0$ induced by the metric on $M$, and the metric $d\tilde s_0^2$ with potential $\tilde u_0$ induced by the flat metric on $V$. These metrics are not the same, but they are {\em quasi-isometric\/} near $\Pi(m)$ in the sense that there is an estimate of the type $C\Re ds_0^2\leq\Re d\tilde s_0^2\leq C^{-1}\Re ds_0^2$ in $O\cap\cal T$, where $O$ is a neighbourhood of $\Pi(m)$ in $M_0$. From this and from the fact proved above that $\tilde u_0$ is strictly plurisubharmonic at $\Pi(m)$, it follows that $u_0$ is also strictly plurisubharmonic at $\Pi(m)$. \end{pf} We conclude the analytic space $M_0$ is a K\"ahler space as defined by Grauert \cite{gr:ub}, if we extend Grauert's definition to include local potentials that are not $C^2$. \begin{theorem}\label{theorem:positive} Assume $\Phi$ is proper. Let $\bold L$ be the line bundle $(L^q)_0$\rom, where $q$ is as in Proposition \ref{proposition:bundle}. Then $\bold L$ is positive in the sense of Grauert. The reduced space $M_0$ is a complex-projective variety\rom, a projective embedding being given by the Kodaira map $M_0\to\bold P\bigl(H^0(M_0,\bold L^k)\bigr)$ for all sufficiently large $k$. \end{theorem} \begin{pf} Let $\bold L^*$ be the dual of $\bold L$. We have to show the zero section of $\bold L^*$ possesses a strictly pseudoconvex open neighbourhood. The fibre metric $\langle\cdot,\cdot\rangle$ on $L^q$ pushes down to a fibre metric $\langle\cdot,\cdot\rangle_0$ on $\bold L$. Let $\langle\cdot,\cdot\rangle_0^*$ be the fibre metric on $\bold L^*$ obtained by duality. On $L^q$ we have the distance function $r(l)=\langle l,l\rangle^{1/2}$. Let $r_0$ and $r_0^*$ be the corresponding functions on $\bold L$, resp.\ $\bold L^*$. Let $\Delta\subset\bold L^*$ be the tubular domain $\{\,l:r_0^*(l)\leq1\,\}$. In a local trivialization $(z,\zeta)$ of $\bold L$ over an open subset $O$ of $M_0$ we can write $r_0(z,\zeta)=h_0(z)|\zeta|^2$ for a certain positive function $h_0$ on $O$. We can use the coordinates $\bigl(z,\bar\zeta\bigr)$ to trivialize $\bold L^*$ over $O$; then $r_0^*\bigl(z,\bar\zeta\bigr)=h_0(z)^{-1}|\zeta|^2$. Also, $\Delta\cap(\rho_0^*)^{-1}(O)$ is given by $|\zeta|^2\leq h_0(z)$, where $\rho_0^*\colon\bold L^*\to M_0$ is the bundle projection. Up to a positive constant factor the function $u_0=-\log h_0$ is a local potential for the reduced K\"ahler structure, so by Lemma \ref{lemma:plurisubharmonic} it is strictly plurisubharmonic. It follows immediately that $\Delta\cap(\rho_0^*)^{-1}(O)$ is strictly pseudoconvex in $(\rho_0^*)^{-1}(O)$. Thus we have shown $\Delta$ is a strictly pseudoconvex subset of $\bold L^*$. For the second part of the theorem, apply Grauert's generalization of Kodaira's Embedding Theorem, \cite[\S 3]{gr:ub}, Satz 2. \end{pf} Let us call a point $x\in M$ {\em algebraically semistable\/} if there exists an invariant global holomorphic section $s\in\Gamma(M,L^{l})^G$ of some power $L^{l}$ of $L$ such that $s(x)\neq0$. The point $x$ is called {\em algebraically stable\/} if in addition $G\co$ acts properly on the open set $\{\,x\in M:s(x)\neq0\,\}$. If $M$ is algebraic, for instance, if $M$ is compact, these notions coincide with the ones introduced by Mumford \cite{mu:ge} (except that Mumford uses the term ``properly stable'' where most authors nowadays use ``stable''). \begin{theorem}\label{theorem:gaga} If $\Phi$ is proper\rom, the quotient map $\Pi\colon M\sst\to M_0$ and the inclusion $M\sst\subset M$ induce isomorphisms $\Gamma(M_0,L_0)\cong\Gamma(M\sst,L)^G\cong\Gamma(M,L)^G$. It follows that a point in $M$ is analytically \rom(semi\rom)stable if and only if it is algebraically \rom(semi\rom)stable. \end{theorem} \begin{pf} The first isomorpism follows from Proposition \ref{proposition:vbundle}. The second isomorphism follows from the observation, essentially due to Guillemin and Sternberg \cite{gu:ge}, that the norm of an invariant holomorphic section $s$ of $L$ is increasing along the trajectories of $-\grad\mu$. Indeed, for any invariant holomorphic section $s$ defined on a $G\co$-invariant open subset and any $\xi\in\frak g$ we have $J\xi_M\langle s,s\rangle=-4\pi\Phi^\xi\langle s,s\rangle$ (see \cite{gu:ge}), so using (\ref{equation:yangmills}) we get for any $x\in M$: \begin{align}\label{equation:maximum} \begin{split} \frac{d}{dt}\bigl\langle s(F_tx),s(F_tx)\bigr\rangle&= -\grad\mu\bigl(\langle s,s\rangle\bigr)(F_tx)=\\ &=-2J\Phi(F_tx)_M\bigl(\langle s,s\rangle\bigr)(F_tx)=\\ &=8\pi\bigl\|\Phi(F_tx)\bigr\|^2\bigl\langle s(F_tx),s(F_tx)\bigr\rangle=\\ &=8\pi\mu(F_tx)\bigl\langle s(F_tx),s(F_tx)\bigr\rangle\geq\\ &\geq0. \end{split} \end{align} It follows that for all $x\in M\sst$ the restriction of the function $\langle s,s\rangle$ to $\overline{G\co x}$ takes on its maximum at $F_\infty x$. Therefore, if $s$ is defined on all of $M\sst$, $\langle s,s\rangle$ is bounded on $M$, since $\Phi$ is proper. An application of Riemann's Extension Theorem now gives $\Gamma(M\sst,L)^G\cong\Gamma(M,L)^G$. Now suppose $x\in M$ is algebraically semistable. Then there exists $s\in\Gamma(M,L^{l})^G$ such that $s(x)\neq0$. As $t$ tends to infinity, $F_tx$ approaches the critical point $F_\infty x$, so $\grad\mu(F_tx)\to0$. Letting $t\to\infty$ in (\ref{equation:maximum}) we obtain $$ \mu(F_\infty x)\bigl\langle s(F_\infty x),s(F_\infty x)\bigr\rangle=0. $$ But since $F_\infty x\in\overline{G\co x}$, we have $s(F_\infty x)=s(x)\neq0$. Therefore $\mu(F_\infty x)=0$, in other words $x$ is analytically semistable. Conversely, suppose $x\in M$ is analytically semistable, that is, $\Phi(m)=0$, where $m=F_\infty x$. By Proposition \ref{proposition:bundle}, $\bold L=(L^q)_0$ is an ample bundle on $M_0$, so $\bold L^r$ is generated by global sections for big $r$. Let $s_0$ be a global section of $\bold L^r$ with $s_0\bigl(\Pi(x)\bigr)= s_0\bigl(\Pi(m)\bigr)\neq0$. By the first part of the theorem (applied to the bundle $L^{rq}$) we can lift $s_0$ to a global invariant section $s\in\Gamma(M,L^{rq})^G$. Evidently, $s(x)\neq0$, so $x$ is algebraically semistable. One proves the equivalence of analytic and algebraic stability in a similar way, using Theorem \ref{theorem:stable}. \end{pf} To summarize, the set $M-M\sst$ can be characterized as the collection of points where all invariant global holomorphic sections of all powers of $L$ vanish. Also, the algebraic structure of $M_0$ depends only on the line bundle $L$ and the lift of the $G$-action to $L$, not on the symplectic form $\omega$ or the momentum map $\Phi$. (The symplectic structure of course does depend on $\omega$ and $\Phi$.) If $M$ is compact, we conclude that as a projective variety $M_0$ is nothing but the quotient defined by Mumford \cite{mu:ge}. \subsection{Multiplicity formul\ae}\label{subsection:multiplicity} In this section I have a stab at the ``geometric multiplicity theory'' of singular symplectic quotients. Let $M$ be a K\"ahler manifold furnished with $G$-equivariant ``prequantum data'' $\bigl(L,\langle\cdot,\cdot\rangle\bigr)$ and momentum map $\Phi$ as in the previous section. As before, let us assume that the $G$-action on $M$ extends to a $G\co$-action. Let us also suppose for simplicity that the map $\Phi$ is proper and that the group $G$ is connected. Ideally, one would like to show that as a $G$-representation the space of sections $\Gamma(L)$, sometimes called a ``quantization'' of $M$, is a symplectic invariant of $M$, in other words, that it is independent of the choice of the complex structure and the line bundle on $M$. One way of doing this would be to express the multiplicities of the unitary irreducible representations occurring in $\Gamma(L)$ in terms of symplectic data involving the reduced phase spaces $M_\lambda=\Phi^{-1}(G\lambda)/G$, where $\lambda$ ranges over the positive weights in the dual of a maximal torus of $\frak g$. Guillemin and Sternberg \cite{gu:ge} carried this out for those weights $\lambda$ for which $G$ acts freely on $\Phi^{-1}(G\lambda)$. (Then the reduced space $M_\lambda$ is non-singular.) Heckman \cite{he:pr} had earlier obtained related results in the important special case where $M$ is a coadjoint orbit of a big group containing $G$ as a subgroup. Using the results of the previous section, we can generalize their results. If we regard the space of sections $\Gamma(M_0,L_0)$ of the $V$-line bundle $L_0$ as the quantization of the reduced space $M_0$, then Theorem \ref{theorem:gaga} bears out the principle that quantization should commute with reduction. \begin{remark} The theory of geometric quantization is usually phrased in terms of polarizations, that is, involutive Lagrangian subbundles of the complexified tangent bundle of $M$. The quantization of $M$ with respect to a polarization $P$ is then the space of polarized sections of $L$, that is, sections $s$ such that $\nabla_Xs=0$ for all vectors $X$ tangent to the conjugate subbundle $\bar P$. It appears to be difficult to make sense of the notion of a polarization on a singular space, such as a symplectic quotient. In the case of a K\"ahler quotient $M_0$, however, the sheaf of holomorphic functions $\cal O_{M_0}$ seems to be a workable substitute for a polarization. \end{remark} \begin{remark} Properly speaking, the quantization of $M$ is not just the space of holomorphic sections of $L$, but the virtual representation $\bigoplus_i(-1)^iH^i(M,L)$, including all cohomology groups with coefficients in $L$. One might wonder whether it is true that reduction commutes with quantization in this broader sense, that is, $\bigoplus_i(-1)^iH^i(M,L)^G=\bigoplus_i(-1)^iH^i(M_0,L_0)$. It is not hard to prove that $H^i(M_0,L_0)$ is isomorphic to $H^i(M\sst,L)^G$ for all $i$, but I don't know if $H^i(M\sst,L)^G$ is isomorphic to $H^i(M,L)^G$ for $i>0$. \end{remark} Theorem \ref{theorem:gaga} obviously implies that the dimension of $\Gamma(M_0,L_0)$ is equal to the multiplicity of the one-dimensional trivial representation in $\Gamma(M,L)$. Let us briefly recall from \cite{gu:ge} how this statement generalizes to arbitrary multiplicities by dint of the shifting trick. Choose a maximal torus $\frak t$ in $\frak g$ and a positive Weyl chamber $\frak t^*_+$ in $\frak g^*$. For every positive weight $\lambda\in\frak t^*_+$ the coadjoint orbit $G\lambda$ is a K\"ahler manifold carrying a naturally defined Hermitian line bundle $V_\lambda$, and the Borel-Weil Theorem asserts that $\Gamma(G\lambda,V_\lambda)$ is the unitary irreducible representation with highest weight $\lambda$. Let $G\lambda^-$ be the orbit $G\lambda$ with the opposite symplectic and complex structures and consider the K\"ahler manifold $M\times G\lambda^-$. Let $\pi_M$ and $\pi_{G\lambda}$ denote the projections of $M\times G\lambda^-$ on the respective factors and let $V_\lambda^*$ be the dual of $V_\lambda$. Then the Hermitian line bundle $\pi_M^*L\otimes\pi_{G\lambda}^*V_\lambda^*$ prequantizes $M\times G\lambda^-$. The reduced space at 0 of $M\times G\lambda^-$ can be identified with $M_\lambda$, the reduced space of $M$ at the orbit $G\lambda$, and it comes equipped with a $V$-line bundle $L_\lambda=L|_{\Phi^{-1}(G\lambda)}\bigm/G$. By Theorem \ref{theorem:gaga}, $\Gamma(M_\lambda,L_\lambda)$ is isomorphic to the space of $G$-invariants in $\Gamma(M\times G\lambda^-,\pi_M^*L\otimes\pi_{G\lambda}^*V_\lambda^*)$. The K\"unneth theorem for coherent sheaves \cite{sa:ku} now implies the following assertion. \begin{theorem}\label{theorem:shift} For every positive weight $\lambda$ of $G$ the space of sections $\Gamma(M_\lambda,L_\lambda)$ is naturally isomorphic to the space of intertwining operators $$ \Hom\bigl(\Gamma(G\lambda,V_\lambda),\Gamma(M,L)\bigr)^G. \qed $$ \end{theorem} \begin{corollary} If the orbit $G\lambda$ does not lie in the image of the momentum map, the irreducible representation corresponding to $\lambda$ does not occur in $\Gamma(M,L)$.\qed \end{corollary} Let us write $\mu(\lambda,L)$ for the multiplicity of the representation with highest weight $\lambda$ occurring in $\Gamma(M,L)$. By Theorem \ref{theorem:shift} $\mu(\lambda,L)$ is equal to the dimension of $\Gamma(M_\lambda,L_\lambda)$. By Theorem \ref{theorem:positive}, for certain $q$ (possibly depending on $\lambda$) the sheaf $\bold L=(L^q)_\lambda$ is an ample line bundle on the projective variety $X=M_\lambda$, so for all sufficiently large $r$ we have $H^i(X,\bold L^r)=0$ for $i>0$. Then $\mu(rq\lambda,\bold L^r)$ is equal to the Euler characteristic $\chi(X,\bold L^r)$, so by the Hirzebruch-Riemann-Roch Theorem of Baum, Fulton and MacPherson \cite{ba:ri1,ba:ri2} \begin{equation}\label{equation:multiplicity} \mu(rq\lambda,\bold L^r)=\chi(X,\bold L^r)=\eps\bigl(\ch\bold L^r\cap\tau(X)\bigr). \end{equation} Here $\tau(X)$ denotes the homological Todd class of $X$, $\ch\bold L^r$ denotes the Chern character of $\bold L^r$ and $\eps$ is the augmentation (the map $H_\bu(X)\to\bold C$ induced by mapping $X$ to a point). If $X$ is non-singular, (\ref{equation:multiplicity}) comes down to the classical Hirzebruch-Riemann-Roch Theorem. As was pointed out by Guillemin and Sternberg, $\mu(rq\lambda,\bold L^r)$ is then a symplectic invariant, i.e., independent of the complex structure and the line bundle $L$ on $M$. It seems likely that this is also true if $X$ is singular. I cannot quite prove this, but here follows some evidence. For arbitrary singular spaces, the Todd class appears to be intractable, but for spaces with quotient singularities, such as $X$, the situation is simpler. Namely, a theorem of Boutot \cite{bo:si} asserts that quotient singularities are {\em rational}, i.e., \begin{equation}\label{equation:rational} f_*\cal O_Y=\cal O_X \qquad\text{and}\qquad R^if_*\cal O_Y=0\quad\text{for }i>0, \end{equation} where $f\colon Y\to X$ is a resolution of singularities of $X$. Therefore, by the functorial properties of the Todd class, $\tau(X)=f_*\bigl(\tau(Y)\bigr)$, so by (\ref{equation:multiplicity}) $\mu(rq\lambda,\bold L^r)=\chi(X,\bold L^r)$ is equal to $\chi(Y,f^*\bold L^r)$. In general, it is difficult to write down a desingularization of $X$, but Kirwan \cite{ki:pa} has explicitly constructed a ``partial'' resolution $p\colon\tilde X\to X$. It has all the properties of a desingularization, except that it is not a smooth projective variety, but a complex-projective $V$-manifold (or orbifold). It is easy to see that the vanishing property (\ref{equation:rational}) also holds for the partial resolution $p\colon\tilde X\to X$, and therefore $\mu(rq\lambda,\bold L^r)=\chi(\tilde X,p^*\bold L^r)$. To construct a partial resolution, one performs a certain sequence of blowups on $M\times G\lambda^-$ at $G\co$-invariant submanifolds, yielding a projective manifold $\tilde M$ with a $G\co$-action. The symplectic form on $M\times G\lambda^-$ pulls back to a degenerate $(1,1)$-form $\tilde\omega$ on $\tilde M$, which descends to a degenerate $(1,1)$-form $\tilde\omega_\lambda$ on $\tilde X$. The class of $q\tilde\omega_\lambda$ is the Chern class of the pull-back $p^*\bold L$ of $\bold L$. To get a K\"ahler form on $\tilde M$, one adds to the pullback of $\omega$ at each stage in the sequence of blowups a small $(1,1)$-form $\sigma_\eps$ supported on a neighbourhood of the exceptional divisor, such that the class of $\sigma_\eps$ is equal to $\eps$ times the dual class of the exceptional divisor. One then obtains $\tilde X$ by taking the quotient of $\tilde M$; $\tilde X=\tilde M\sst\qu G\co\simeq\tilde M_0$. See \cite{ki:pa} for the details. Using Kawasaki's formula \cite{ka:ri}, we can now write the multiplicity as a sum of integrals (still assuming that $r$ is sufficiently large): \begin{equation}\label{equation:integral} \mu(rq\lambda,\bold L^r)=\int_{\tilde X} e^{rq\tilde\omega_\lambda}\wedge \det\frac{\sq\,R/2\pi}{1-e^{-\sq\,R/2\pi}} + \Sigma, \end{equation} where $R$ denotes the curvature two-form of $\tilde X$ with respect to the K\"ahler metric, and where $\Sigma$ denotes a sum of contributions from the singular strata in $\tilde X$. In a local $V$-manifold chart, $\Sigma$ can be written as a sum of integrals over fixed-point manifolds, as in the holomorphic Lefschetz formula. Since the right-hand side is a function of the cohomology class of $\tilde\omega_\lambda$ only, we conclude that it is a polynomial function of $r$, and hence $\mu(rq\lambda,\bold L^r)$ is polynomial in $r$ for $r$ large. If the dimension of $X$ is $2n$, the form $\tilde\omega_\lambda$ enters in the term $\Sigma$ with exponents less than $n$. The highest-order term in the multiplicity is therefore the term in $r^n$, and the coefficient is $q^n\int_{\tilde X}\tilde\omega_\lambda^n/n!$, which is equal to $q^n\vol X$, where $\vol X$ is the symplectic volume of the top-dimensional stratum of $X=M_\lambda$. In particular, we see that the highest-order term is a symplectic invariant. The following points seem to call for further clarification: (i) Is it really necessary in (\ref{equation:multiplicity}) and (\ref{equation:integral}) to replace the $V$-bundle $L_\lambda$ by the line bundle $\bold L=(L^q)_\lambda$? In other words, does the Riemann-Roch formula of Baum, Fulton and MacPherson work for $V$-bundles on $M_\lambda$? It seems reasonable to guess that $\chi(M_\lambda,L_\lambda)$ equals $\chi(\tilde M_\lambda,f^*L_\lambda)$, which can then be computed by Kawasaki's recipe. (ii) Under what conditions does vanishing of the cohomology of $M$ with coefficients in $L$ imply vanishing of the cohomology of the quotients? (iii) More importantly, the right-hand side of (\ref{equation:integral}) makes sense even if the symplectic manifold $M$ does not carry a complex structure: Kirwan's partial resolution can be defined for any singular symplectic quotient, and all one needs to write down the form representing the Todd class is an {\em almost}-complex structure compatible with the symplectic form. It would be interesting to find out in how far (\ref{equation:integral}) can be generalized to this more general situation. \ifx\undefined\leavevmode\hbox to3em{\hrulefill}\, \newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,} \fi

9304

alg-geom/9304006

en

https://arxiv.org/abs/alg-geom/9304006

alg-geom/9304006

Vassil Kanev

Vassil Kanev

Recovering of curves with involution by extended Prym data

31 p., LATEX 2.09

With every smooth, projective algebraic curve $\tilde{C}$ with involution $\sigma :\tilde{C}\longrightarrow \tilde{C}$ without fixed points is associated the Prym data which consists of the Prym variety $P:=(1-\sigma )J(\tilde{C})$ with principal polarization $\Xi$ such that $2\Xi$ is algebraically equivalent to the restriction on $P$ of the canonical polarization $\Theta $ of the Jacobian $J(\tilde{C})$. In contrast to the classical Torelli theorem the Prym data does not always determine uniquely the pair $(\tilde{C},\sigma )$ up to isomorphism. In this paper we introduce an extension of the Prym data as follows. We consider all symmetric theta divisors $\Theta $ of $J(\tilde{C})$ which have even multiplicity at every point of order 2 of $P$. It turns out that they form three $P_2$ orbits. The restrictions on $P$ of the divisors of one of the orbits form the orbit $\{ 2\Xi \} $, where $\Xi $ are the symmetric theta divisors of $P$. The other restrictions form two $P_2$-orbits $O_1,O_2\subset \mid 2\Xi \mid $. The extended Prym data consists of $(P,\Xi )$ together with $O_1,O_2$. We prove that it determines uniquely the pair $(\tilde{C} ,\sigma )$ up to isomorphism provided $g(\tilde{C})\geq 3$. The proof is analogous to Andreotti's proof of Torelli's theorem and uses the Gauss map for the divisors of $O_1,O_2$. The result is an analog in genus $>1$ of a classical theorem for elliptic curves.

alg-geom

\section*{Introduction} \footnotetext{Supported in part by the Bulgarian foundation "Scientific research" and by NSF under the US-Bulgarian project "Algebra and algebraic geometry".} The classical Torelli theorem states that every smooth, projective algebraic curve $X$ is determined uniquely up to isomorphism by its principally polarized Jacobian $(J(X),\Theta )$. In this paper we consider curves $\tilde{C}$ with involution without fixed points $\sigma :\tilde{C}\longrightarrow \tilde{C}$. We let $C=\tilde{C}/\sigma $ and denote by $\pi :\tilde{C}\longrightarrow C$ the factor map. One associates the principally polarized Prym variety $(P,\Xi )$ where $P=P(\tilde{C},\sigma )=(1-\sigma )J(\tilde{C})$ and $\Theta \mid _P$ is algebraically equivalent to $2\Xi$. The natural question, whether the Prym variety determines uniquely the pair $(\tilde{C},\sigma )$ up to isomorphism, has negative answer in general if the genus $g$ of $C$ is $\leq 6$ as well as in every genus $\geq 7$ for some special loci of curves, e.g for hyperelliptic $C$. The problem of spotting the pairs $(\tilde{C},\sigma )$ which are not determined uniquely by the Prym variety is still open. Some partial results have been obtained in \cite{f-s},\cite{kan},\cite{don},\cite{don1},\cite{deb1},\cite{deb2}, \cite{nar},\cite{verra}. In this paper we propose an extension of the Prym data $(P,\Xi )$ and prove that it determines uniquely up to isomorphism any pair $(\tilde{C},\sigma )$ for $g\geq 2$. Our extension originates from the following observation. Consider the case of $\tilde{C}$ of genus 1. Here the involution is a translation $t_{\mu }$ by some point $\mu $ of order 2 in $J(\tilde{C})$. Notice that $\{ 0,\mu \} =Ker(Nm_{\pi }: J(\tilde{C})\longrightarrow J(C))$. The Prym variety is equal to 0. There is a classically known data which determines uniquely the pair $(\tilde{C},t_{\mu })$ up to isomorphism (see e.g. \cite{mum2}, \cite{clem}). Namely, one can always represent $J(\tilde{C})$ as ${\bf C}/{\bf Z}\tau +{\bf Z}$, where $\tau $ belongs to the upper-half plane ${\cal H}$, so that $$ \mu =\frac{1}{2}\tau +\frac{1}{2}(mod {\bf Z}\tau +{\bf Z}). $$ So, the moduli space of pairs $(\tilde{C},\sigma )$ is isomorphic to $\Gamma _{1,2}\backslash {\cal H}$ where $\Gamma _{1,2}\subset PGL(2,{\bf Z})$ is the subgroup \[ \Gamma _{1,2}=\{ \left( \begin{array}{cc}a&b\\c&d \end{array} \right) \mid ad-bc=1,ab\equiv 0(mod 2),cd\equiv 0(mod 2) \} \] One considers the three even theta functions with characteristics : $\theta _{00}(z,\tau ),\theta _{01}(z,\tau )$ and $\theta _{10}(z,\tau )$. Just one of them vanishes on $\mu $, namely $\theta _{00}(z,\tau )$. Now, by the transformation law for theta functions one checks that for \begin{equation}\label{ii.1} \lambda (\tau )=-\theta _{01}(0,\tau )^4/\theta _{10}(0,\tau )^4 \end{equation} the set $\{ \lambda (\tau ),1/\lambda (\tau )\} $, or equivalently the function $k(\tau )=\lambda (\tau )+1/\lambda (\tau )$, remain invariant with respect to the action of $\Gamma _{1,2}$. Moreover the map $$ k:\Gamma _{1,2}\backslash {\cal H}\longrightarrow {\bf C}^*-\{ 0,2\} $$ is an analytic isomorphism (see Section~(\ref{s2})). Generalizing to genus greater then 1, first we have that $Ker(Nm_{\pi }:J(\tilde{C})\longrightarrow J(C))=P\cup P_{\_ }$ where $P_{\_ }$ is a translation of $P$ by a point of order 2. One considers the symmetric theta divisors $\Theta $ of $J(\tilde{C})$ which have the property that for every $\rho \in P_2$ either $\rho \not \in \Theta $ or $\rho \in \Theta $ and $mult_{\rho }\Theta $ is even. It turns out that there are three $P_2$-orbits of symmetric theta divisors with this property. The divisors of one of the orbits contain $P_{\_ }$ ; these are exactly the theta divisors which appear in Wirtinger's theorem \cite{fay},\cite{mum} and satisfy $\Theta .P=2\Xi $ for symmetric theta divisors $\Xi $ of $P$. None of the divisors of the other two orbits contains $P_{\_ }$. Restricting the latter to $P$ we obtain two $P_2$-orbits of divisors in the linear system $\mid 2\Xi \mid $ which we denote by $O_1,O_2$. The extended Prym data consists of $(P,\Xi )$ together with the two $P_2$-orbits $O_1,O_2\subset \mid 2\Xi \mid $. Our result is that for $g\geq 2$ it determines uniquely up to isomorphism any pair $(\tilde{C},\sigma )$. The proof is analogous to Andreotti's proof of Torelli's theorem and uses the Gauss map for the divisors of $O_i$. Special treatment is required if $C$ is hyperelliptic, or bi-elliptic, or g=3. In fact the bi-elliptic case has been already considered earlier by Naranjo in \cite{nar} who proved that for $g\geq 10$ the pair $(\tilde{C},\sigma )$ can be recovered by the ordinary Prym data $(P,\Xi )$. His arguments however do not work for $g=4$ or $5$. If $\eta $ is the point of order 2 in $Pic^0(C)$ which determines the covering $\pi:\tilde{C}\longrightarrow C$ then the divisors of $O_i$ are equal to translations of connected components of the set $$ Z=\{ L\in Pic^{2g-2}(\tilde{C})\mid Nm_{\pi }(L)\simeq K_C\otimes \eta ,\; h^0(\tilde{C},L)\geq 1\} $$ It is interesting that line bundles $L$ of this type appear also in the study of rank 2 vector bundles on $C$ with canonical determinant \cite{be} as well as in representing $(\tilde{C},\sigma )$ as the spectral curves associated to $sp(2n)$ -matrices with parameter. {\bf Acknowledgement.} Part of this work was done while the author was a visitor in the University of Michigan and the University of Utah. The hospitality of these institutions is gratefully acknowledged. \begin{center}{\bf Contents}\end{center} 0.Notation and preliminaries. 1.Double unramified coverings of elliptic curves. 2.Extended Prym data. 3.The semicanonical map and the Gauss map. 4.The hyperelliptic case, $g\geq 2$. 5.The bi-elliptic case, $g\geq 4$. 6. The case $g=3$. \section{Notation and preliminaries}\label{s1} We denote by $\equiv$ the linear equivalence of divisors. Let $X$ be an algebraic, smooth, irreducible curve. We denote by $J_d(X)$ the divisor classes of degree $d$ modulo linear equivalence and by $Pic^d(X)$ the isomorphism classes of invertible sheafs of degree $d$ on $X$. Abusing the notation we shall write by the same letter an element in $J_d(X)$ and the corresponding element in $Pic^d(X)$. If $D$ is a divisor of degree $d$ and $\xi = cl(D)$ its class in $J_d(X)$ we write $$ h^0(C,D) = h^0(C,\xi ) = dim\mid \xi \mid + 1 $$ If $L$ is an invertible sheaf of $X$, then $\mid L \mid $ is the linear system of divisors of sections of $H^0(X,L)$. If $\mid L \mid $ is without base points we denote by $\varphi _L = \varphi _{\mid L \mid } $ the map $X \longrightarrow \mid L \mid ^*$ defined by $\varphi _L(x) = \mid L(-x) \mid +x$. Let $A$ be a principally polarized abelian variety of dimension $g$ isomorphic to ${\bf C}^g/\Lambda _{\tau }$, where $\Lambda _{\tau } = {\bf Z}^g\tau + {\bf Z}^g$ with $\tau \in {\cal H}_g $, where ${\cal H}_g$ is the Siegel upper-half space. Any point $e \in {\cal C}^g$ has two characteristics $\epsilon ,\delta \in {\bf R}^g$ such that $e =\epsilon \tau + \delta $. We shall write sometimes $e = \left[ \begin{array}{c} \epsilon \\ \delta \end{array} \right] $. Let $x \in A$ and let $x = (x'\tau +x'')(mod \Lambda _{\tau})$ with $x',x''\in {\bf R}^g$. If no confusion arises we shall refer to $x',x''$ as the characteristics of $x$, keeping in mind that $x',x''$ are determined modulo ${\bf Z}^g$. We shall denote by $A_2$ the points of order 2 in $A$. Let $\lambda =\lambda '\tau +\lambda '' , \mu = \mu '\tau + \mu '' \in A_2 $. The Weyl pairing $e_2 : A_2 \times A_2 \longrightarrow {\bf Z}$ is defined by $$ e_2(\lambda ,\mu) = 4(\lambda '^t\mu '' - \mu '^t\lambda ''(mod 2) $$ Let $\Theta $ be a symmetric theta divisor. One defines a quadratic form $q_{\Theta } : A_2 \longrightarrow {\bf Z}_2 $ associated with $\Theta $ by \begin{equation}\label{e5.1} q_{\Theta }(\lambda ) = mult_0(\Theta +\lambda )(mod 2) \end{equation} Consider the theta function with characteristics $\lambda ',\lambda ''\in \frac{1}{2} {\bf Z}^g$ $$ \theta \left[ \begin{array}{c} \lambda '\\ \lambda'' \end{array} \right] (z,\tau ) = \sum_{n\in {\bf Z}^g}\exp {(\pi i(n+\lambda ')\tau ^t(n+\lambda ') + 2\pi i(n+\lambda '')^t(z+\lambda ''))} $$ Then $$ \theta \left[ \begin{array}{c} \lambda '\\ \lambda'' \end{array} \right] (-z,\tau ) = \theta \left[ \begin{array}{c} -\lambda '\\ -\lambda'' \end{array} \right] (z,\tau ) = (-1)^{4\lambda '^t\lambda ''} \theta \left[ \begin{array}{c} \lambda '\\ \lambda'' \end{array} \right] (z,\tau ) $$ Thus if $\Theta $ is the divisor of the theta function $\theta \left[ \begin{array}{c} 0\\ 0 \end{array} \right] (z,\tau )$ then for any $\lambda =\lambda '\tau +\lambda '' \in A_2$ one has \begin{equation}\label{e6.2} q_{\Theta }(\lambda ) = 4\lambda '^t\lambda ''(mod 2) \end{equation} For any symmetric theta divisor $\Theta $ of $A$ the bilinear form associated with $q_{\Theta }$ is $e_2$, i.e. $$ \Delta _{\lambda }\Delta_{\mu}q_{\Theta }(\xi ) = q_{\Theta }(\xi +\lambda +\mu)-q_{\Theta }(\xi +\lambda ) -q_{\Theta }(\xi +\mu )+q_{\Theta }(\xi ) $$ is independent of $\xi $ and equals $e_2(\lambda ,\mu )$. In particular one has the following formula \begin{equation}\label{e6.1} q_{\Theta }(\lambda +\alpha ) = q_{\Theta }(\lambda ) + e_2(\lambda ,\alpha ) + q_{\Theta }(\alpha ) - q_{\Theta }(0) \end{equation} The map $\Theta \longmapsto q_{\Theta }$ gives a bijective correspondence between the set of symmetric theta divisors of $A$ and the set of quadratic forms on $A_2$ whose associated bilinear form equals $e_2$ and which vanish on $2^{g-1}(2^g+1)$ points of $A_2$. Let $\tilde{C}$ be an algebraic, projective, smooth, irreducible curve. Let $\sigma : \tilde{C} \longrightarrow \tilde{C}$ be an involution without fixed points, let $C= \tilde{C}/\sigma$ and let $\pi : \tilde{C} \longrightarrow C$ be the factor map. Let $g = g(C), \tilde{g} = g(\tilde{C}) = 2g-1 $. We suppose that $g\geq2$. Let $\tilde{J} = J(\tilde{C}) , J= J(C) $ be the Jacobians of $\tilde{C}, C$ respectively and let $P = P(\tilde{C},\sigma) = (1-\sigma )\tilde{J}$ be the Prym variety. One has the maps $\pi ^* : J\longrightarrow \tilde{J} , Nm : \tilde{J}\longrightarrow J$ such that $\pi^* \circ Nm = 1+\sigma $. We denote by $j : P \longrightarrow \tilde{J} $ the embedding. The kernel of $\pi ^* : J \longrightarrow \tilde{J} $ is $\{0,\eta \}$ where $\eta \in J_2$. Conversely, given $C$ and $\eta \in J(C)_2 , \eta \neq 0$ one constructs a double unramified covering $\pi : \tilde{C} \longrightarrow C$ such that $\pi _*{\cal O}_{\tilde{C}} \simeq {\cal O}_C \oplus{\cal O}_C(\eta)$ and gets the set-up above. Let $J \simeq {\bf C}^g/\Lambda , \tilde{J} \simeq {\bf C}^{2g-1}/\tilde{\Lambda} , P\simeq {\bf C}^{g-1}/\Lambda_{\_}$. Choosing as in \cite{fay}, \cite{clem} symplectic bases $\{a_0,...,a_{g-1},b_0 ,...,b_{g-1}\} , \{\tilde{a}_0,...,\tilde{a}_{2g-2},\tilde{b}_0,...,\tilde{b}_{2g-2}\} , \{ \tilde{a}_1-\tilde{a}_g,...,\tilde{a}_{g-1}-\tilde{a}_{2g-2},\tilde{b} _1-\tilde{b}_g,...,\tilde{b}_{g-1}-\tilde{b}_{2g-2}\} $ of $\Lambda , \tilde{\Lambda }$ and $\Lambda_{\_}$ respectively we can assume that $\eta = \frac{1}{2}a_0(mod \Lambda)$. Let us denote by $\tau \in {\cal H}_g , \tilde{\tau} \in {\cal H}_{2g-1}, \Pi \in {\cal H}_{g-1}$ the corresponding period matrices. One has the following formulas : \begin{equation}\label{e8.1} \begin{array}{ccc} (\pi ^*)_*\left[ \begin{array}{cc}\alpha _0&\alpha \\\beta _0&\beta \end{array}\right] & = & \left[ \begin{array}{rcc}\alpha _0&\alpha &\alpha \\2\beta _0&\beta &\beta \end{array}\right] \\\mbox{}\\ Nm_*\left[ \begin{array}{ccc}\alpha _0&\alpha &\alpha '\\\beta _0&\beta &\beta ' \end{array}\right] & = & \left[ \begin{array}{rc} 2\alpha _0&\alpha +\alpha '\\\beta _0&\beta +\beta ' \end{array}\right] \\ \mbox{}\\ j_*\left[ \begin{array}{c}\alpha \\\beta \end{array}\right] &=&\left[ \begin{array}{ccc}0&\alpha &-\alpha \\0&\beta &-\beta \end{array} \right] \end{array} \end{equation} Here $\alpha ,\alpha ',\beta ,\beta ' \in {\bf R}^{g-1} ,\alpha _0,\beta _0\in {\bf R}$ and $(\pi ^*)_*, Nm_* , j_*$ are the linear maps which induce the homomorphisms $\pi^* , Nm , j$. Wirtinger's theorem \cite{fay} states that there is a symmetric theta divisor $\Theta _0$ of $\tilde{J}$ which is equal to $W_{\tilde{g}-1}-\pi ^*\Delta $ for a certain theta characteristic $\Delta $ of $C$, such that $\Theta_0\mid _P = 2\Xi $ for some symmetric theta divisor $\Xi$ of $P$. Moreover any point $L-\pi ^*\Delta $ of $\Theta _0\cap P$ satisfies the properties $$ Nm(L) \equiv K_C , h^0(\tilde{C},L) \equiv 0(mod 2) , h^0(\tilde{C}, L) \geq 2 $$ $\Theta _0$ is the divisor of the theta function $\theta[\lambda ] (z,\tilde{\tau})$, where $$ \lambda = \frac{1}{2}\tilde{ a}_0 = \left[ \begin{array}{ccc}0&0&0 \\\frac{1}{2}&0&0 \end{array}\right] $$ Next we recall and state in more general form some results of Welters \cite{wel}. Let $f : X \longrightarrow Y$ be a double covering of smooth, projective curves. It might be ramified. Let $\Lambda = \mid D \mid$ be a complete linear system on $Y$ and let $deg(D) = d$. One denotes by $S$ the subscheme of $X^{(d)}$ which is the pull-back of $\Lambda $ under $f^{(d)}$ $$ \begin{array}{ccl} S&\longrightarrow & X^{(d)}\\ \downarrow & &\downarrow \\ \Lambda &\longrightarrow & Y^{(d)} \end{array} $$ The arguments on pp.103-107 of \cite{wel} combined with Riemann-Roch's theorem give the following proposition \begin{prop}\label{p91.1} Let $\hat{D}$ be a closed point of $S$ and let $A$ be the maximal effective divisor of $C$ such that $\hat{D}=\pi ^*A + E$ with $A\geq 0,E\geq 0$. Let $D=f^{(d)}(\hat{D})=2A+E_1$. Then (i) $S$ is nonsingular at $\hat{D}$ if and only if $$ h^0(D-A) = h^0(D) - deg(A) $$ (ii) Suppose $S$ is nonsingular at $\hat{D}$. Then $f^{(d)}\mid _{S} : S\longrightarrow \Lambda $ is nondegenerate at $\hat{D}$ if and only if $f^{(d)}:X^{(d)}\longrightarrow Y^{(d)}$ is nondegenerate at $\hat{D}$ and this is the case if and only if $D$ contains no branch points of $f$ and $A=0$. \end{prop} Let $B\subset Y$ be the branch locus of $f$ and let $\delta$ be the invertible sheaf with $\delta ^{\otimes 2}\simeq {\cal O}_Y(B)$ which determines the covering by $f_*{\cal O}_X\simeq {\cal O}_Y\oplus \delta $. An effective divisor $E$ of $X$ is called $f$-simple if $E\not \geq f^*(y)$ for any $y\in Y$. The following lemma is due to Mumford \cite{mum}. \begin{lem}\label{l92.1} Let $A$ be a divisor of $Y$ and let $E$ be an effective $f$-simple divisor of $X$. Then there is an exact sequence $$ 0\longrightarrow {\cal O}_Y(A)\longrightarrow f_*{\cal O}_X(\pi^*A+E) \longrightarrow {\cal O}_Y(A+Nm_f(E))\otimes \delta ^{-1}\longrightarrow 0 $$ \end{lem} \begin{cor}\label{c92.2} Under the assumptions of the preceding lemma suppose that \\ \noindent $deg(A)+deg(E)<deg(\delta )$. Then $$ h^0(X,\pi ^*A+E) = h^0(Y,A) $$ \end{cor} \section{Double unramified coverings of elliptic curves}\label{s2} Let $\tilde{E}$ be an elliptic curve. Choosing a point $o\in \tilde{E}$ we shall sometimes identify $\tilde{E}$ with $J(\tilde{E})$ by the map $x\mapsto cl(x-o)$. Let $\sigma :\tilde{E}\longrightarrow \tilde{E}$ be an involution without fixed points. \begin{lem}\label{l10.1} There exists $\mu \in J(\tilde{E})$ of order $2$ such that $\sigma (x) = x+\mu $. Furthermore $Ker(\pi_*:J(\tilde{E})\longrightarrow J(E))=\{ 0,\mu \} $. \end{lem} {\bf Proof.} Let $\mu =\sigma (0)-0.$ Since $P(\tilde{E},\sigma )=(\sigma -1)J(\tilde{E})=0$ we have $\sigma (x-o)\equiv x-o$, thus $\sigma x=x+\mu.$ Furthermore $2(\sigma (o)-o)=(1-\sigma )(\sigma (o)-0)\equiv 0$ and $\pi _*(\sigma (o)-o)\equiv 0$, thus $Ker\pi _*=\{ 0,\mu \} $ since $\# Ker\pi _*=2.$ q.e.d. Using the notation of the Introduction let $\tilde{E}\simeq {\bf C}/\Lambda _{\tau }$ where $\Lambda _{\tau }={\bf Z}\tau +{\bf Z}$ with $Im(\tau )>0$ and $\mu = \frac{1}{2}(\tau +1)(mod \Lambda _{\tau })$. Consider $\lambda (\tau )$ defined by Eq.~(\ref{ii.1}). Then $\{ \lambda (\tau ),1/\lambda (\tau )\} $ is the set of the roots of the equation $x^2-k(\tau )x+1=0$ where $$ k(\tau )=-(\theta _{10}(0,\tau )^8+\theta _{01}(0,\tau )^8)/ \theta _{10}(0,\tau )^4\theta _{01}(0,\tau )^4 $$ \begin{prop}\label{p12.1} The map \begin{equation}\label{e12.1} k:\Gamma _{1,2}\backslash {\cal H}\longrightarrow {\bf C}-\{0,2\} \end{equation} is an analytic isomorphism. \end{prop} {\bf Proof.} Let $\Gamma _2$ be the level $2$ subgroup of $PSL(2,{\bf Z})$ $$ \Gamma _{2}=\{ \left( \begin{array}{cc}a&b\\c&d \end{array} \right) \equiv \left( \begin{array}{cc}1&0\\0&1 \end{array} \right) (mod 2)\} $$ Then $\mid \Gamma _{1,2}:\Gamma _{2}\mid =2$ and the element $S=\left( \begin{array}{cc}0&1\\-1&0 \end{array}\right) $, belongs to $\Gamma _{1,2}\backslash \Gamma _{2}$. It is well-known (see e.g. \cite{clem}) that the map $$ \lambda :\Gamma _{2}\backslash {\cal H}\longrightarrow {\bf C}-\{ 0,1\} $$ given by Eq.~(\ref{ii.1}) is an isomorphism. We have $S(\tau )=-1/\tau $ and $\lambda (-1/\tau )=1/\lambda (\tau )$. The factor of $\Gamma _{2}\backslash {\cal H}$ by the action of $S$ is $\Gamma _{1,2}\backslash {\cal H}$, thus $k$ is an analytic isomorphism. q.e.d. Explicitly, given $k\neq 0,2$ we find $\lambda $ such that $\lambda +1/\lambda =k$ and the corresponding pair $(\tilde{E},\mu \in J(\tilde{E})_2)$ is given by the equation $y^2=x(x-1)(x-\lambda )$ and the point $\mu =cl(p_1-p_2)$ where $p_1=(0,0),p_2=(1,0)$. \section{Extended Prym data}\label{s3} Let $\tilde{C},C$ etc. be as in Section~(\ref{s1}). Let $\Theta _{0}$ be the divisor of the theta function \[ \theta \left[ \begin{array}{ccc}0&0&0\\\frac{1}{2}&0&0 \end{array}\right] (z,\tilde{\tau })\] Let us denote by $q_{0}$ the quadratic form $q_{\Theta _0} : \tilde{J}_2\longrightarrow {\bf Z}_2$ defined by Eq.~(\ref{e5.1}). By Eq.~(\ref{e6.1}) and (\ref{e6.2}) one has \begin{equation}\label{e14.1} q_{0}\left( \left[ \begin{array}{cc}\alpha \\\beta \end{array} \right] \right)=4\alpha ^t\beta +2\alpha _0(mod 2) \end{equation} Hence $q_{0}(\rho )=0$ for any $\rho \in P_2$. This follows also from Wirtinger's theorem. The same property holds for any symmetric theta divisor of the orbit $\{ \Theta _{0}+\rho \mid \rho \in P_2\} $. Let us denote $\pi ^*(J)$ by $B$. By Eq.~(\ref{e8.1}) one has $B_2\supset P_2$. \begin{lem}\label{l14.1} A symmetric theta divisor $\Theta \subset \tilde{J}$ has the property that $q_{\Theta }$ vanishes on $P_2$ if and only if $\Theta =\Theta_ {0}+\alpha $ where $\alpha \in B_2$ and $q_{0}(\alpha )=0$. \end{lem} {\bf Proof.} Suppose $q_{\Theta }(P_2)=0$. Let $\Theta =\Theta_ {0}+\alpha $ for some $\alpha \in \tilde{J}_2$.Then for $\rho \in P_2$ one has by Eq.~(\ref{e6.1}) $$ q_{\Theta }(\rho )=q_{0}(\rho +\alpha )=q_{0}(\rho )+e_2(\rho ,\alpha )+q_{0}(\alpha )-q_{0}(0) $$ Setting $\rho =0$ we conclude that $q_{0}(\alpha )=0$. Thus $q_{\Theta }(P_2)=0$ implies that $e_2(P_2,\alpha )=0$. Eq.~(\ref{e8.1}) show that the latter holds if and only if $\alpha \in B_2$. Conversely, if $\alpha \in B_2$ and $q_{0}(\alpha )=0$, then $q_{\Theta }(P_2)=0$ by the formula for $q_{\Theta }$ above. q.e.d. The zeros of $q_{0}$ which belong to $B_2$ are the following three cosets with respect to $P_2$~: $P_2,\lambda _1+P_2,\lambda _2+P_2$ where $\lambda _1=\frac{1}{2}\tilde{a}_0(mod \tilde{\Lambda }), \lambda _2=\frac{1}{2}\tilde{a}_0+\frac{1}{2}\tilde{b}_0(mod \tilde{\Lambda })$. Let $\Theta _{1}=\Theta _{0}+\lambda _1,\Theta _{2}=\Theta _{0}+\lambda _2$. We conclude that there are three $P_2$ - orbits of symmetric theta divisors $\Theta $ such that $q_{\Theta }$ vanishes on $P_2$ : \[ \begin{array}{ccc} \{ \Theta _{0}+\rho \}\; ,&\{ \Theta _{1}+\rho \}\; ,&\{ \Theta _{2}+\rho \} \end{array}\] These are respectively the divisors of the theta functions \[ \begin{array}{ccc} \theta \left[ \begin{array}{ccc}0&\alpha &\alpha \\\frac{1}{2}&\beta &\beta \end{array}\right](z,\tilde{\tau })\; ,& \theta \left[ \begin{array}{ccc}0&\alpha &\alpha \\0&\beta &\beta \end{array}\right](z,\tilde{\tau })\; ,& \theta \left[ \begin{array}{ccc}\frac{1}{2}&\alpha &\alpha \\0&\beta &\beta \end{array}\right](z,\tilde{\tau }) \end{array}\] where $\alpha ,\beta \in \frac{1}{2}{\bf Z}^{g-1}$. \begin{lem}\label{16.2} Let $\mu = \frac{1}{2}\tilde{b}_0(mod \tilde{\Lambda })$. Then $Ker(Nm)=P\cup P_{\mu } , Ker(Nm)\cap B_2=P_2\cup (\mu +P_2)$ and for any $x\in \tilde{C}$ there exists a unique $\xi \in P$ such that $\sigma x-x\equiv \mu +\xi $. \end{lem} {\bf Proof.} It is well-known \cite{fay},\cite{mum} that $Ker(Nm)$ has two connected components $P$ and $P_{\_ }$. Since $\mu \in B_2\backslash P_2$ and $Nm(\mu )=0$ we get that $P_{\_ }=P_{\mu }$. Hence $(P\cup P_{\_ })\cap B_2=P_2\cup (P_2+\mu )$. The last statement of the lemma follows from the equality $\sigma x-x+P= P_{\_ }$ \cite{mum}. q.e.d. Let $\tilde{J}_{2g-2},J_{2g-2}$ be the divisor classes of degree $2g-2$ on $\tilde{C},C$ respectively and let $Nm:\tilde{J}_{2g-2}\longrightarrow J_{2g-2}$ be the norm map. The subvariety $Nm^{-1}(K_C+\eta )$ is a principal homogeneous space for $Nm^{-1}(0)=P\cup P_{\_ }$, thus it has two connected components. Let \begin{equation}\label{e165.1} Z=\{ L\in \tilde{J}_{2g-2}\mid Nm(L)=K_C+\eta ,\; h^0(L)\geq 1\} \end{equation} and let $Z=Z_1\cup Z_2$ where $Z_{i}$ are the intersections of $Z$ with the connected components of $Nm^{-1}(K_C+\eta )$. \begin{lem}\label{l1634.1} Let $M$ be an effective divisor of $\tilde{C}$ such that $Nm(M) \in \mid K_C+\eta \mid$. Suppose $x\in \tilde{C}$ and $x\not \in Bs\mid M\mid $. Then $$ h^0(M+\sigma x-x)=h^0(M)-1 $$ \end{lem} {\bf Proof.} By Riemann-Roch's theorem $x$ is a base point of $\mid K_{\tilde{C}}-M+x\mid $. Now, $K_{\tilde{C}}-M\equiv\sigma (M)$ , thus $\sigma (x)$ is a base point of $\mid M+\sigma (x)\mid $ which proves the lemma since $x\not \in Bs\mid M\mid $. q.e.d. \begin{prop}\label{p17.1} The theta divisors $\Theta _{1}$ and $\Theta _{2}$ do not contain $P$ . The restrictions $\Theta _i.P$ are connected and reduced divisors of $P$ which belong to the linear system $\mid 2\Xi \mid $. Furthermore, up to a possible reordering of $Z_i$ one has $$ \Theta _i.P=Z_i-\pi ^*\Delta -\lambda _i $$ The point $L-(\pi ^*\Delta +\lambda _i)$ is nonsingular if and only if $h^0(\tilde{C},L)=1$. The corresponding tangent hyperplane is equal to $Nm(\mid L\mid )\in \mid K_C\otimes \eta \mid $ via the identification $$ T_0(P)^*\simeq H^0(\tilde{C},K_{\tilde{C}})^-\simeq H^0(C,K_C\otimes \eta ) $$ \end{prop} {\bf Proof.} Let $\kappa _i=\pi ^*\Delta +\lambda _i,i=1,2$.We have $Nm(\lambda _i)=\eta $, thus $Nm(\kappa _i)=K_C+\eta $ and for $L$ with $h^0(L)\geq 1$ one has $L-\kappa _i\in\Theta _i\cap Ker(Nm)$ if and only if $Nm(\mid L\mid )\in \mid K_C+\eta \mid $. Since $dim\mid K_C+\eta \mid =g-2$ we conclude that neither $P$ nor $P_{\_ }$ are contained in $\Theta _i$. Furthermore $\Theta _i$ are ample, so $\Theta _i\cap P$ are not empty and $\Theta _i.P$ are connected divisors of $P$. Upon a possible reordering of $Z_1$ and $Z_2$ we have $Z_1-\kappa _1=\Theta _1\cap P,Z_2-\kappa _1=\Theta _1\cap P_{\_ }$. Since $P_{\_ }=P+\mu $ and $\lambda _2=\lambda _1+\mu $ we obtain $Z_2-\kappa _2=\Theta _2\cap P$. {\bf Claim.} {\it For every irreducible component $T$ of any $\Theta _i.P$ the general element $L-\kappa _i\in T$ satisfies $h^0(\tilde{C},L)=1$ }.\\ {\bf Proof.} Suppose the contrary. Then for any $x\in \tilde{C}$ the image of the map \[ \begin {array}{cc} \psi :T\times \tilde{C}\longrightarrow Ker(Nm)\; ,& \psi (L,x)=L+\sigma x-x \end{array}\] is contained in $Z$. This image must be of dimension $dimT+1=g-1$. Indeed, if $M=L+\sigma x-x$, then for every sufficiently general $L\in T$ and $x\in \tilde{C}$ one has by Lemma~(\ref{l1634.1}) that $h^0(M)=h^0(L)-1$. If $dim\psi (T\times \tilde{C})\leq dimT$, then for every sufficiently general $M\in Im(\psi ),x\in \tilde{C}$ one has $h^0(M-\sigma x+x)=h^0(L)=h^0(M)+1$ which is an absurd by Lemma~(\ref{l1634.1}). Now, $dimZ=g-2$, thus it is impossible that $dim\psi (T\times \tilde{C})=g-1$. q.e.d. Now, suppose that $L-\kappa _i\in\Theta _i\cap P$ is an element with $h^0(L)=1$ and let $D=\mid L\mid $. Since $Nm(D)\in\mid K_C+\eta\mid $ there is an anti invariant holomorphic differential $\omega $ of $\tilde{C}$ whose divisor of zeros is $\pi ^*D$. Since $h^0(L)=1$ the point $L-\kappa _i$ is nonsingular on $\Theta _i$ and the tangent space in $T_0\tilde{J}$ is given by the equation $\omega =0$. We see that $\Theta _i$ and $P$ intersect transversely at $L-\kappa _i$ and the tangent hyperplane of $\Theta _i.P$ at $L-\kappa _i$ is given by the same equation $\omega =0$ in $T_0P$ since $\omega $ is anti invariant. What we have proved implies also that $Sing(\Theta _i.P)=P\cap Sing\Theta _i$. This concludes the proof of the proposition. q.e.d. \begin{cor}\label{c19.1} All irreducible components of $Z$ are of dimension $g-2$. \end{cor} We see that the $P_2$-orbit $\{ \Theta _0+\rho \} $ is distinguished among the three $P_2$-orbits of symmetric theta divisors $\Theta$ which satisfy $q_{\Theta}\mid _{P_2}=0$ by the property that the restriction of any $\Theta _0+\rho $ on $P$ is equal to twice a theta divisor of $P$. It is also distinguished by the property that every $\{ \Theta _0+\rho \} $ contains $P_{\_ }=P+\mu $ . Indeed, the fact that $\Theta _0\supset P+\mu $ follows from the parity lemma \cite{tju} and is well-known \cite{mum1},\cite{fay}. If $\Theta _1+\rho $ contained $P_{\_ }=P+\mu $, then $\Theta _2$ would contain $P$ since $\Theta _2=\Theta _1+\mu $ and $\rho \in P_2$ which contradicts Proposition~(\ref{p17.1}). Notice that the latter distinction of $\{ \Theta _1+\rho \} $ and $\{ \Theta _2+\rho \} $ parallels the distinction of $\theta _{10}(z,\tau )$ and $\theta _{01}(z,\tau )$ in the elliptic case. We can now state our extension of the Prym data. {\sc Extended Prym data}.{\it One associates to every algebraic, smooth, irreducible, projective curve $\tilde{C}$ of genus $\geq 3$ with an involution $\sigma :\tilde{C}\longrightarrow \tilde{C}$ without fixed points, the principally polarized Prym variety $(P,\Xi )$ and the two $P_2$-orbits $O_1,O_2\subset \mid 2\Xi \mid $ which consist of the restrictions $\Theta .P$ of the symmetric theta divisors $\Theta \subset J(\tilde{C})$ such that $q_{\Theta }\mid _{P_2}=0$ and $\Theta \not \supset P_{\_ }$ where $Ker(Nm_{\pi }:J(\tilde{C})\longrightarrow J(C))=P\cup P_{\_ }$ } We can now state our result which is a kind of generalization of Proposition~(\ref{p12.1}) to curves of genus $>1$. \begin{theo}\label{t20.1} The pair $(\tilde{C},\sigma )$ is uniquely determined up to isomorphism by the extended Prym data $(P(\tilde{C},\sigma ),\Xi ),O_1,O_2\subset \mid 2\Xi \mid $. \end{theo} \section{The semicanonical map and the Gauss map}\label{s4} In this section $K_C\in Pic^{2g-2}(C)$ is the canonical sheaf of $C$ and $\eta \in Pic^0(C)_2$ is the sheaf with $\eta ^{\otimes 2}\simeq {\cal O}_C$ such that $\pi _*{\cal O}_{\tilde{C}}\simeq {\cal O}_{C}\oplus \eta $. We shall denote by $\varphi _K,\varphi _{K\otimes \eta }$ the canonical, respectively semicanonical map of the curves under consideration. Let $L=K_C\otimes \eta $. The following lemma follows elementary from Riemann-Roch's theorem. \begin{lem}\label{l21.1} Suppose $g(C)\geq 2$. Then $\mid K_C\otimes \eta \mid $ has base points if and only if $C$ is hyperelliptic and $\eta \simeq {\cal O}_{C}(p_1-p_2)$ where $p_1,p_2$ are ramification points for the double covering $f:C\longrightarrow {\bf P}^1$. In this case $p_1+p_2=Bs\mid K_C\otimes \eta \mid $ and $K_C\otimes \eta \simeq (f^*{\cal O}_{{\bf P}^1}(g-2))(p_1+p_2)$. \end{lem} \begin{lem}\label{l21.2} Suppose $g(C)\geq 3 $ and $\mid K_C\otimes \eta \mid $ is without base points. Then \\ $\varphi _L:C\longrightarrow \mid K_C\otimes \eta \mid ^*={\bf P}^{g-2}$ is a birational embedding except in the following two cases : (i) $g(C)=3$. Then $\varphi _L:C\longrightarrow {\bf P}^1$ is of degree $4$. (ii) $g(C)\geq 4$, $C$ is bi-elliptic, i.e. it is a double covering $f:C\longrightarrow E$ of an elliptic curve, and $\eta \simeq f^*(\epsilon )$ where $\epsilon \in Pic^0(E)_2$. Here $\varphi _L=\varphi _{\delta \otimes \epsilon }\circ f$ where $\delta $ is the invertible sheaf of $E$ which determines the covering, i.e. $\delta ^{\otimes 2}\simeq {\cal O}_{E}(x_1+...+x_{2g-2})$ for the branch points $x_1,...,x_{2g-2}$ and $f_*{\cal O}_{C}\simeq {\cal O}_{E}\oplus \delta $. \end{lem} {\bf Proof.} The case $g(C) =3$ is clear, so let us suppose that $g\geq 4$. Let $X=\varphi _L(C)$ and let $d$ be the degree of the map $\varphi _L:C\longrightarrow X$. We have $d.deg(X)=2g-2$ and $deg(X)\geq g-2$. This implies that the case $d\geq 3$ may occur only if $g=4,deg(X)=2,d=3$. Otherwise either $\varphi _L$ is a birational embedding or $d=2,deg(X)=g-1 $. In the latter case $p_a(X)=1$. Suppose $d=2$ and $X$ is singular. Then the normalization of $X$ is $\hat{X}\simeq {\bf P}^1$ and we can decompose $\varphi _L$ as $$ \varphi _L=g\circ f:C\longrightarrow \hat{X}\longrightarrow {\bf P}^{g-2} $$ Since $\varphi _L$ is obtained from a complete linear system, $g$ must have the same property, thus $g^*{\cal O}_{{\bf P}^{g-2}}(1)\simeq {\cal O}_{{\bf P}^1}(g-2)$. This is impossible since $deg(X)=g-1$. Consequently if $d=2$ then $X\subset {\bf P}^{g-2}$ is an elliptic curve. Let $E=X,f=\varphi _L:C\longrightarrow E.$ Since $K_C\simeq f^*(\delta )$ and $K_C\otimes \eta \simeq f^*{\cal O}_{E}(1)$ we conclude that $\eta \simeq f^*(\epsilon )$ for some $\epsilon \in Pic^0(E).$ The covering $f$ is ramified, so $f^*:Pic^0(E)\longrightarrow Pic^0(C)$ is an injection, hence $\epsilon \in Pic^0(E)_2$ and $\epsilon \not \simeq {\cal O}_{E}$. Conversely, if $f:C\longrightarrow E$ is a double covering of an elliptic curve, and $\eta =f^*(\epsilon )$ with $\epsilon \in Pic^0(E)_2,\epsilon \not \simeq {\cal O}_{E}$ then $$ H^0(C,K_C\otimes \eta)\simeq H^0(C,\pi ^*(\delta \otimes \epsilon ))=\pi ^*H^0(E,\delta \otimes \epsilon ) $$ Hence $\varphi _L=\varphi _{\delta \otimes \epsilon }\circ f$ and $d=2$. It remains to rule out the possibility $g=4,d=3,deg(X)=2$. Here \\ $f=\varphi _L:C\longrightarrow X\simeq {\bf P}^1$ so $L\simeq M^{\otimes 2}$, where $deg(M)=3,h^0(C,M)=2$, and $\mid M\mid $ is without base points. Thus $C$ is not hyperelliptic and $$ \varphi _K(C)=Q\cap F\subset {\bf P}^3 $$ where $Q$ is a quadric and $F$ is a cubic surface. We have $\mid L\mid =\mid M\mid +\mid M\mid $ since $dim\mid L\mid =2$. Let $l_1,l_2$ be lines in $Q$ such that $l_1+l_2=Q.H$ for a plane $H$. Let $M_i={\cal O}_{C}(C.l_i).$ We can assume that $M=M_1$. Then $\eta \simeq M_1\otimes M_2^{-1}.$ If $Q$ were singular, then $M_1\simeq M_2,$ so $\eta \simeq {\cal O}_{C}$ which is absurd. Suppose $Q$ is nonsingular. Then $\eta ^{\otimes 2}\simeq {\cal O}_{C}$ implies $\mid M_2^{\otimes 2}\mid =\mid M_1^{\otimes 2}\mid =\mid M_1\mid +\mid M_1\mid $. This is again impossible since any reduced divisor $x_1+x_2+x_3\in \mid M_2\mid $ can have only two common points with any two divisors $D_1,D_2\in \mid M_1\mid $. q.e.d. Suppose $g\geq 3$. Following Welters \cite{wel} let $S$ be the subscheme of $\tilde{C}^{2g-2}$ which is the pull-back of $\mid K_C\otimes \eta \mid \subset C^{2g-2}$ $$ \begin{array}{ccl}S&\longrightarrow &\tilde{C}^{2g-2}\\ \downarrow &\mbox{}&\downarrow Nm\\\mid K_C\otimes \eta \mid &\longrightarrow &C^{2g-2} \end{array} $$ It breaks naturally into two disjoint subschemes $S=S_1\cup S_2.$ The singularities of $S$ can be calculated by Proposition~(\ref{p91.1}) with $X=\tilde{C},Y=C,f=\pi ,\pi ^{(2g-2)}=Nm$. Since $S$ is a locally complete intersection and $\pi ^{(2g-2)}$ is a finite map every irreducible component of $S$ has dimension $g-2$. A Zariski open, dense subset of $\mid K_C\otimes \eta \mid $ consists of reduced divisors by Lemmas~(\ref{l21.1}) and (\ref{l21.2}), so according to Proposition~(\ref{p91.1}) $S$ is reduced. The subvarieties $S_1,S_2$ are connected provided $g\geq 3$ \cite{wel}. Let $T_1,T_2$ be divisors from $\mid 2\Xi \mid $ which belong to the orbits $O_1,O_2$ respectively. Suppose $g\geq 3$.The Gauss maps $G_i:T_i^{ns}\longrightarrow {\bf P}(T_0P)^*$ are defined on the nonsingular loci of $T_i$ and send a point $x\in T_i^{ns}$ to the translation of the tangent hyperplane $T_x(T_i)$ to $0\in P$. Let $T=T_1\sqcup T_2$ and let $G:T^{ns}\longrightarrow {\bf P}(T_0P)^*$ be the map whose restriction on $T_i^{ns}$ equals $G_i$. Let $S^0$ be the Zariski open subset of $S$ which consists of those $\hat{D}$ with $h^0(\tilde{C},\hat{D})=1$. By Proposition~(\ref{p17.1}) the map $cl:S^0\longrightarrow Z^{ns}$ is an isomorphism and moreover one can identify $T_i^{ns}$ with $Z_i^{ns}$ by translation and ${\bf P}(T_0P)^*$ with $\mid K_C\otimes \eta \mid $. Since the Gauss map does not depend on the translation one has for the Gauss map $G:Z^{ns}\longrightarrow \mid K_C\otimes \eta \mid $ and every $\hat{D}\in S^0$ the formula \begin{equation}\label{e26.1} G(cl(\hat{D}))=Nm(\hat{D}) \end{equation} \begin{prop}\label{p26.1} Let $L=K_C\otimes \eta $. Suppose $g\geq 4$. Let $R\subset T_i$ be the ramification locus of $G$ and let $B$ be the algebraic closure of $G(R)$. (i) If $C$ is hyperelliptic and $\eta \simeq {\cal O}_C(p_1-p_2)$, where $p_1,p_2$ are Weierstrass points of $C$, then $P(\tilde{C},\sigma )\simeq J(C_2)$ for a certain hyperelliptic curve $C_2$ (see Section~(\ref{s5})) and $$ B=\varphi _K(C_2)^*\cup \bigcup _{i=1}^{2g}\varphi _K(q_i)^* $$ where $\varphi _K(C_2)^*$ is the dual hypersurface of $\varphi _K(C_2)$ and $\varphi _K(q_i)^*$ are the stars of hyperplanes which contain $\varphi _K(q_i)$, where $q_i$ are the Weierstrass points of $C_2$. (ii) If $\mid K_C\otimes \eta \mid $ is without base points and $\varphi _L:C\longrightarrow \mid K_C\otimes \eta \mid ^*$ is a birational embedding, then $B$ has a unique irreducible component of dimension $g-3$ and degree $>1$. This component is equal to $\varphi _L(C)^*$. (iii) If $\mid K_C\otimes \eta \mid $ is without base points and $$ f=\varphi _L:C\longrightarrow \varphi _L(C)=E $$ is a map of degree $2$ onto an elliptic curve, then $$ B=E^*\cup \bigcup_{i=1}^{2g-2}x_i^* $$ where $x_i$ are the branch points of $\varphi _L$. \end{prop} {\bf Proof.} (i) In this case any $T_i$ is the union of two translates of the theta divisor $\Xi \subset P$ as it will be shown in Section~(\ref{s5}). So, Part(i) follows from the description of the branch locus of the Gauss map of the theta divisor of a hyperelliptic Jacobian \cite{and}. Now, let us assume that $\mid K_C\otimes \eta \mid $ is without base points. If $Nm:S^0\longrightarrow \mid K_C\otimes \eta \mid $ is degenerate at $\hat{D}\in S^0$, then by Proposition~(\ref{p91.1}) $\hat{D}=\pi ^*A+E$ for some $A>0$, so $D=Nm(\hat{D})=2~A+Nm(E)$. Let $H\subset \mid K_C\otimes \eta \mid ^*$ be the hyperplane which corresponds to $D$. Either $H$ contains an image of a ramification point of the map $\varphi _L:C\longrightarrow \mid K_C\otimes \eta \mid ^*$, or $H$ is tangent to a branch $\varphi _L(U)$ at a point $\varphi _L(p)$, where $p\in U\subset C$ and $\varphi _L$ is nondegenerate at $p$. The former case can happen only for finitely many points. This shows that any component of $B$ of dimension $g-3$ must be either a star of hyperplanes which contain a branch point $\varphi _L(p)$, or it is contained in the dual variety $\varphi _L(C)^*$. {\bf Proof of (ii).} It remains to show that $\varphi _L(C)^*\subset B$. Let the hyperplane $H\subset \mid K_C\otimes \eta \mid ^*$ be a sufficiently general element of $\varphi _L(C)^*$ and let $$ D=\varphi _L^*(H)=2p+p_3+...+p_{2g-2} $$ be the corresponding divisor of $\mid K_C\otimes \eta \mid $. Here $p,p_3,...,p_{2g-2}$ are distinct points of $C$ and $\varphi _L$ is not degenerate at $p$. {\bf Claim 1.} {\it Let $\pi ^{-1}(p)=p'+p''$ One can choose $p'_i\in \tilde{C}$ with $\pi (p'_i)=p_i$ so that $$ \hat{D}=p'+p''+p'_3+...+p''_{2g-2} $$ has the property $h^0(\hat{D})=1$.}\\ {\bf Proof.} Let us choose arbitrary points $q_i\in \tilde{C}$ such that $\pi (q_i)=p_i$. Let $$ \hat{D}_0=p'+p''+q_3+...+q_{2g-2} $$ If $h^0(\hat{D})\geq 2$, then at least one of the points $q_i$ is not a base point of $\mid \hat{D}_0\mid $. Indeed, otherwise $$ h^0({\cal O}_{\tilde{C}}(\pi ^*p))=h^0({\cal O}_{C}(p))+h^0({\cal O}_{C}(p)\otimes \eta )\geq 2 $$ which is possible only in the Case (i). If $q_i\not \in Bs\mid \hat{D}_0\mid $, let $\hat{D}_1=\hat{D}_0+\sigma (q_i)-q_i$. Then by Lemma~(\ref{l1634.1}) $h^0(\hat{D}_1)=h^0(\hat{D}_0)-1$. Repeating the same argument with $\hat{D}_1$ etc. we obtain eventually the required divisor $\hat{D}$. q.e.d. Now, $\hat{D}\in S^0$, the Gauss map at the point $cl(\hat{D})\in Z^{ns}$ equals $H$ by Eq.~(\ref{e26.1}) and it is ramified at $\hat{D}$ according to Proposition~(\ref{p91.1}). So, $\varphi _L(C)^*\subset B$.\\ {\bf Proof of (iii).} We have proved above that $$ B\subset E^*\cup \bigcup_{i=1}^{2g-2}x_i^* $$ Let $H$ be a sufficiently general element of $E^*$ and let $$ D=\varphi _L^*(H)=2p+2q+p_5+...+p_{2g-2} $$ be the corresponding divisor of $\mid K_C\otimes \eta \mid $. Here $p,q,p_5,...p_{2g-2}$ are distinct points of $C,\varphi _L$ is nondegenerate at $p,q$ and $\{ p,q\} =f^*(x)$ for some $x\in E$. {\bf Claim 2.} {\it Let $\pi ^{-1}(p)=\{ p',p''\} ,\pi ^{-1}(q)=\{ q',q''\} $. One can choose $p'_i\in \tilde{C}$ with $\pi (p'_i)=p_i$ so that $$ \hat{D}=p'+p''+2q'+p'_5+...+p'_{2g-2} $$ has the property $h^0(\hat{D})=1$.}\\ {\bf Proof.} Let $\eta = f^*(\epsilon )$ and let $y\in E $ be the point such that $\epsilon \simeq {\cal O}_{E}(y-x)$. Then one has canonical isomorphisms $$ \begin{array}{l} H^0({\cal O}_{\tilde{C}}(\pi ^*(p+q))\simeq \pi ^*(H^0({\cal O}_{C}(p+q))\oplus H^0({\cal O}_{C}(p+q)\otimes \eta))\\ \simeq (\pi ^*\circ f^*)(H^0({\cal O}_{E}(x))\oplus H^0({\cal O}_{E}(y)))\simeq {\bf C}^2 \end{array} $$ Thus $\mid p'+p''+q'+q''\mid $ is a pencil without base points. Using the same argument as in Claim~1 we conclude that one can choose $p'_i$ so that $$ \hat{D}'=p'+p''+q'+q''+p'_5+...+p'_{2g-2} $$ has the property $h^0(\hat{D}')=2$. Applying once more Lemma~(\ref{l1634.1}) we conclude that $\hat{D}=\hat{D}'+ q' -q''$ satisfies $h^0(\hat{D})=1$. q.e.d. We conclude as in Part (ii) that $E^*\subset B$. If $x_i=\varphi _L(q_i)$ is a branch point of $f$ and $H$ is a sufficiently general hyperplane in $\mid K_C\otimes \eta \mid ^*$ which contains $x_i$, then the corresponding divisor of $\mid K_C\otimes \eta \mid $ has the form $$ D=\varphi _L^*(H)=2q_i+p_3+...+p_{2g-2} $$ where $q_i,p_3,...,p_{2g-2}$ are distinct. The same argument as in Part (ii) proves that $H\in B$, so $x_i^*\subset B$. Proposition~(\ref{p26.1}) is proved. q.e.d. We see that if $g\geq 4$ then the branch locus $B$ of the Gauss map $G:T^{ns}\longrightarrow {\bf P}(T_0P)^*$ has a unique irreducible component $B_0$ of dimension $g-3$ and degree $\geq 2$. We have shown above that $B=X^*$ for a certain irreducible curve $X\subset {\bf P}(T_0P)^*$. So, for $g\geq 4$, by the equality $(X^*)^*=X$ \cite{kleim} we obtain that $B_0^*$ is a curve $X$. The following alternative takes place. \begin{list}{(\roman{bean})}{\usecounter{bean}} \item deg(X)=g-2 \item deg(X)=2g-2 \item deg(X)=g-1 \end{list} The three cases correspond to those in Proposition~(\ref{p26.1}). If Case (ii) occurs we prove Theorem~(\ref{t20.1}) as follows: $C$ is isomorphic to the normalization of $X$. The normalization map $f:C\longrightarrow X\subset {\bf P}(T_0P)^*$ is associated to the complete linear system $\mid K_C\otimes \eta \mid $. Thus we obtain $\eta \in J(C)_2$. Cases (i) and (iii) are considered respectively in Sections~(\ref{s5}) and (\ref{s6}). The case $g=3$ is treated in Section~(\ref{s7}) and the case $g=2$ in Section~(\ref{s5}). \section{The hyperelliptic case, $g\geq 2$}\label{s5} Throughout this section we suppose that $C$ is a hyperelliptic curve of genus $g\geq 2$,\\ $f:C\longrightarrow {\bf P}^1$ is the double covering and $\eta \simeq {\cal O}_{C}(p_1-p_2)$ where $p_1,p_2$ are ramification points of $f$ and $p_1\neq p_2$. Let $R=\{ p_1,p_2,p_3,...,p_{2g+2}\} $ be the set of ramification points of $f,R_1=\{ p_1,p_2\}, R_2=R\backslash R_1$. Let $B_i=f(R_i),i=1,2$. According to \cite{mum},\cite{dal} the covering \\$f\circ \pi:\tilde{C}\longrightarrow C\longrightarrow {\bf P}^1$ has Galois group ${\bf Z}_2\times {\bf Z}_2$. In the corresponding diagram of Fig.1 \begin{figure}\label{f35.1} \begin{center} \begin{picture}(46,54)(0,0) \put (19,0){\makebox(8,8){${\bf P}^1$}} \put (0,23){\makebox(8,8){$C_1$}} \put (19,23){\makebox(8,8){$C$}} \put (38,23){\makebox(8,8){$C_2$}} \put (19,46){\makebox(8,8){$\tilde{C}$}} \put (19,46){\vector(-1,-1){15}} \put (27,46){\vector(1,-1){15}} \put (23,46){\vector(0,-1){15}} \put (23,23){\vector(0,-1){15}} \put (4,23){\vector(1,-1){15}} \put (42,23){\vector(-1,-1){15}} \put (24,15){$f$} \put (24,38){$\pi $} \put (6,38){$\pi _1$} \put (37,38){$\pi _2$} \put (6,15) {$f_1$} \put (37,15) {$f_2$} \end{picture} \end{center} \caption{} \end{figure} $f_i:C_i\longrightarrow {\bf P}^1$ is branched at $B_i,i=1,2$. Furthermore $\pi_2^*:J(C_2)\longrightarrow P(\tilde{C},\sigma )$ is an isomorphism. Let $\Theta _0=W_{\tilde{g}-1}(\tilde{C})-\pi ^*\Delta $ be as in Section~(\ref{s3}). \begin{lem}\label{l36.1} Let $\Xi \subset P(\tilde{C},\sigma )=P$ be a symmetric theta divisor. Then there exists a unique $\rho \in P_2$ such that \begin{equation}\label{e36.1} \Xi = \pi _1^*(\zeta _1)+\pi _2^*W_{g-2}(C_2)-\pi ^*\Delta-\rho \end{equation} where $\zeta _1$ is the rational equivalence class of the points of the rational curve $C_1$. \end{lem} {\bf Proof.} By Wirtinger's theorem there is a unique translation $\Theta =\Theta _0+\rho $ with $\rho \in P_2$ such that $\Theta .P=2\Xi $. The points of $\Theta \cap P$ have the form $L-\pi ^*\Delta-\rho $ where $Nm(L)=K_C\; ,\; h^0(\tilde{C},L)\equiv 0(mod 2)$ and $h^0(\tilde{C},L)\geq 2$. Now, $\mid K_C\mid \simeq f^*\mid {\cal O}_{{\bf P}^1}(g-1)\mid $. One easily checks that if $\hat{D}$ is effective divisor of $\tilde{C}$ and $Nm(\hat{D})\in f^*\mid {\cal O}_{{\bf P}^1}(g-1)\mid $ then $$ \hat{D}=\pi _1^*E+\pi _2^*F $$ where $E$ and $F$ are effective divisors of $C_1,C_2$ respectively. We have $deg(E)+deg(F)=g-1$ which gives only two irreducible components of dimension $\geq g-2$ of $$ Nm^{-1}(K_C)\cap W_{2g-2}(\tilde{C}) $$ namely $\pi _2^*W_{g-1}(C_2)$ of dimension $g-1$ and $\pi _1^*(\zeta _1)+\pi _2^*W_{g-2}(C_2)$ of dimension $g-2$. On the other hand the above intersection has, by the general theory, two irreducible components: a translation of $\Xi $ and a translation of $P_{\_ }$. This shows Eq.~(\ref{e36.1}). q.e.d. Now, let us calculate the orbits $O_1,O_2\subset \mid 2\Xi \mid $. Let $T_i\in O_i,i=1,2$. By Proposition~(\ref{p17.1}) one has \begin{equation}\label{e38.1} T_i=Z_i-\pi ^*\Delta -\nu _i \end{equation} for some $\nu _i\in \lambda _i+P_2 , i=1,2.$ \begin{lem}\label{l38.1} One can enumerate $\pi ^{-1}(p_1)$ as $\{ p'_1,p''_1\} $ and $\pi ^{-1} (p_2)$ as $\{ p'_2,p''_2\} $ so that \[ \begin{array}{ccc}Z_1&=&\pi _2^*W_{g-2}(C_2)+p'_1+p'_2\cup \pi _2^*W_{g-2}(C_2)+p''_1+p''_2\\Z_2&=&\pi _2^*W_{g-2}(C_2)+p'_1+p''_2\cup \pi _2^*W_{g-2}(C_2)+p''_1+p'_2 \end{array}\] \end{lem} {\bf Proof.} From Lemma~(\ref{l21.1}) we have $$ \mid K_C\otimes \eta \mid =f^*\mid {\cal O}_{{\bf P}^1}(g-2)\mid +p_1+p_2 $$ One easily checks that if $\hat{D}$ is effective divisor of $\tilde{C}$ and $Nm(\hat{D})\in f^*\mid {\cal O}_{{\bf P}^1}(g-2)\mid $ then $$ \hat{D}=\pi _1^*E+\pi _2^*F $$ where $E,F$ are effective divisors of $C_1,C_2$ respectively. We have $deg(E)+deg(F)=g-2.$ Thus the only irreducible component of dimension $\geq g-2$ of $$ Nm^{-1}(f^*{\cal O}_{{\bf P}^1}(g-2))\cap W_{2g-4}(\tilde{C}) $$ is $\pi _2^*W_{g-2}(C_2)$. The irreducible components of $Z=Z_1\cup Z_2$ are of dimension $g-2$ by Corollary~(\ref{c19.1}) and the transformation $L\mapsto L+\sigma (p)-p$ interchanges the two components of $Nm^{-1}(K_C\otimes \eta)$. This shows that $Z_1$ and $Z_2$ have the form given in the lemma. q.e.d. Lemmas~(\ref{l36.1}),(\ref{l38.1}) and Eq.~(\ref{e38.1}) give the following corollary \begin{cor}\label{c39.1} Let $\Xi $ be an arbitrary symmetric theta divisor of $P(\tilde{C},\sigma )$ and let $T_1,T_2$ be two divisors of the orbits $O_1,O_2\subset \mid 2\Xi \mid $ respectively. Then \[ \begin{array}{c}T_1=\Xi +p'_1+p'_2-\pi _1^*(\zeta _1)-\mu _1\cup \Xi +p''_1+p''_2-\pi _1^*(\zeta _1)-\mu _1\\T_2=\Xi +p'_1+p''_2-\pi _1^*(\zeta _1)-\mu _2\cup\Xi +p''_1+p'_2-\pi _1^*(\zeta _1)-\mu _2 \end{array}\] for some $\mu _i\in \lambda _i+P_2,i=1,2$. \end{cor} Let us choose in an arbitrary way $\Xi ,T_1,T_2$ as above and let us denote by $x_1,y_1,x_2,y_2$ the elements of $P(\tilde{C},\sigma )$ such that $T_1=\Xi +x_1\cup \Xi +y_1,T_2=\Xi +x_2\cup \Xi +y_2$. There are two possible ways of representing the set $\{ x_1,y_1,x_2,y_2\} $ as union $A\cup B$, where $\# A=\# B=2$, and $A,B$ have one point of $\{ x_1,y_1\} $ and one point of $\{ x_2,y_2\} $. Namely as : \begin{equation}\label{e40.1} \begin{array}{ccl}\{ x_1,x_2\} &\cup &\{ y_1,y_2\} ,\\ \{ x_1,y_2\} &\cup &\{ y_1,x_2\} \end{array} \end{equation} Taking the sums of the sets in (\ref{e40.1}), using Corollary~(\ref{c39.1}) and taking into account that $\lambda _1+\lambda _2=\mu $ (Section~(\ref{s3})) we see that the extended Prym data determines the following $P_2$-orbit of quadruples of points in $P(\tilde{C},\sigma )$, each quadruple being split into a union of two pairs: \begin{equation}\label{e405.1} \begin{array}{c}\{ \{ 2p'_1+\pi ^*p_2-\pi _1^*(2\zeta _1)-\mu +\rho ,2p''_1+\pi ^*p_2-\pi _1^*(2\zeta _1)-\mu +\rho \} \\ \cup \{ 2p'_2+\pi ^*p_1-\pi _1^*(2\zeta _1)-\mu +\rho ,2p''_2+\pi ^*p_1-\pi _1^*(2\zeta _1)-\mu +\rho \} \} \end{array} \end{equation} where $\rho \in P_2$. The splitting of the quadruples is consistent with the action of $P_2$ on $Q$. Let $f_2^{-1}(f(p_i))=\{ q'_i,q''_i\} $ where $\pi _2^{-1}(q'_i)=p'_i,\pi _2^{-1}(q''_i)=p''_i,i=1,2$. Let $f_2^{-1}(f(p_j))=q_j$ for $3\leq j\leq 2g+2$. Since $\pi _2:C\longrightarrow C_2$ is branched at $\{ q'_1,q''_1,q'_2,q''_2\} $ one has \begin{equation}\label{e41.2} 2p'_i=\pi _2^*(q'_i)\; ,\; 2p''_i=\pi _2^*(q''_i) \end{equation} One has also that \begin{equation}\label{e41.3} \pi _1^*(2\zeta _1)=\pi _1^*f^*_1(f(p_i))=2\pi ^*(p_i) \end{equation} {\bf Claim} {\it For any $i$ with $1\leq i\leq 2$ and any $j$ with $3\leq j\leq 2g+2$ there is a point $\rho _{ij}\in P_2$ such that $\mu =-\pi ^*(p_i-p_j)-\rho _{ij}$.}\\ {\bf Proof.} By Eq.~(\ref{e8.1}) one has $\pi ^*J(C)_2=P_2\cup (\mu +P_2)$. Since $P(\tilde{C},\sigma )=\pi ^*J(C_2)$ one concludes by the description of the points of order 2 of the hyperelliptic Jacobian $J(C_2)$ \cite{mum2} that $$ P_2=\{ \pi_2^*(S_1-S_2)\mid S_1\cup S_2\subset R_2,S_1\cap S_2=\emptyset ,\# S_1=\# S_2\} $$ Using this one easily shows that $\pi ^*(p_i-p_j)\not \in P_2$. q.e.d. Now, using Eq.~(\ref{e41.2}),(\ref{e41.3}) and the Claim we have for any $j$ with $3\leq j\leq 2g+2$ \[\begin{array}{cccc}\mbox{}&2p'_1+\pi ^*p_2-\pi _1^*(2\zeta _1)-\mu +\rho &=&\pi _2^*(q'_1)-\pi ^*(p_2)-\mu +\rho \\ =&\pi _2^*(q'_1)-\pi ^*(p_j)+\rho _{2j}+\rho &=&\pi _2^*(q'_1-q_j)+\rho _{2j}+\rho \end{array}\] We obtain that the $P_2$-orbit $Q$ is equal to \begin{equation}\label{e425.1} \begin{array}{c}\{ \{ \pi _2^*(q'_1-q_j)+\rho \; ,\; \pi _2^*(q''_1-q_j)+\rho\} \\ \cup \{ \pi _2^*(q'_2-q_j)+\rho \; ,\; \pi _2^*(q''_2-q_j)+\rho \} \} \end{array} \end{equation} {\bf Reconstruction of $(C,\eta )$ in the hyperelliptic case, $g\geq 3 .$}\\ We have a polarized isomorphism $P(\tilde{C},\sigma )\simeq J(C_2)$. So, by Torelli's theorem \cite{acgh} one reconstructs the smooth, hyperelliptic curve $C_2$ of genus $g_2=g-1\geq 2$. It has a unique complete linear system $g^1_2$. Take a point $q\in C_2$ such that $2q\in g^1_2$. Consider the Abel map $\alpha :C_2\longrightarrow J(C_2)$ given by $\alpha (x)=cl(x-q)$. \begin{lem}\label{l435.1} There is a unique quadruple of $Q$ whose points belong to $\alpha (C_2)$. Any other quadruple has no points in common with $\alpha (C_2)$. \end{lem} {\bf Proof.} If $q=q_j$ we set $\rho =0$ in (\ref{e425.1}) and see that the quadruple \begin{equation}\label{e435.1} \{ \{ \alpha (q'_1),\alpha (q''_1)\}\cup \{ \alpha (q'_2),\alpha (q''_2)\} \} \end{equation} is contained in $\alpha (C_2)$. Suppose that for some $\rho \in J(C_2)_2,\rho \neq0$ one has $$ q'_1-q_j+\rho \equiv x-q_j. $$ Multiplying by 2 both sides of this equality we obtain $2q'_1\equiv 2x$. Since $C_2$ has a unique $g^1_2$ and $q'_1\neq x$ for $\rho \neq 0$ we conclude that $2q'_1\in g^1_2$ which is an absurd. This argument shows that none of the quadruples of $Q$ different from (\ref{e435.1}) can have points in common with $\alpha (C_2)$. q.e.d. Now, we choose a map $f_2:C_2\longrightarrow {\bf P}^1$ of degree 2 and observe that the quadruple of points of $C_2$ defined in the lemma is transformed by $f_2$ into a set of two points. Furthermore this set does not depend on the choice of the ramification point $q$ of $f_2$. Let us denote it by $B_1$. Let $B_2$ be the branch locus of $f_2$. Then $C$ is isomorphic to the hyperelliptic curve branched at $B=B_1\cup B_2$ and $\eta \in J(C)_2$ corresponds to this partition of $B$ \cite{mum2}. {\bf Reconstruction of $(C,\eta )$ in the case $g=2$.}\\ Here $P(\tilde{C},\sigma )$ is an elliptic curve $E$. Let $o\in E$ be the zero, let $\varphi _1,\varphi _2$ be a basis of $H^0(E,{\cal O}_{E}(2o))$ and let $f_2=(\varphi _1:\varphi _2):E\longrightarrow {\bf P}^1$. For any $\rho \in E_2$, if $t_{\rho }:E\longrightarrow E$ is the translation by $\rho $, there exists $\psi \in PGL(2)$ such that the following diagram is commutative \begin{equation}\label{e45.1} \begin{array}{rlcl}\mbox{}&E&\stackrel{t_{\rho }}{\longrightarrow } &E\\ f_2&\downarrow &\mbox{}&\downarrow \hspace{.25cm}f_2\\ \mbox{}&{\bf P}^1&\stackrel{\psi }{\longrightarrow }&{\bf P}^1 \end{array} \end{equation} Moreover $\psi $ permutes the branch points of $f_2$. Let $B_2$ be the branch locus of $f_2$. Take any of the quadruples of $Q$. Each of its two pairs is invariant under the action of $-id_E$. Thus the image of the quadruple is a set of two points which we denote by $B_1$. If we choose another quadruple of $Q$ with image $B'_1$, then (\ref{e45.1}) shows that there is a $\psi \in PGL(2)$ such that $\psi (B_1)=B'_1,\psi (B_2)=B_2$. This gives the reconstruction of $(C,\eta )$, up to isomorphism, as the hyperelliptic curve branched at $B=B_1\cup B_2$ and $\eta $ as the point of $J(C)_2$ which corresponds to this partition of $B$. \section{The bi-elliptic case, $g\geq 4$}\label{s6} Let $f:C\longrightarrow E$ be a double covering of an elliptic curve $E$ ramified at $B=\{ x_1,...,x_{2g-2}\} $ and determined by $\delta \in Pic^{g-1}(E)$ with $\delta ^{\otimes 2}\simeq {\cal O}_{E}(B)$. Suppose $\eta =f^*(\epsilon )$ for some $\epsilon \in Pic^0(E)_2$. Then the unramified covering $\pi :\tilde{C}\longrightarrow C$ determined by $\eta $ fits into the commutative diagram of Fig.2 \begin{figure}\label{f47.1} \begin{center} \begin{picture}(46,54)(0,0) \put (19,0){\makebox(8,8){$E$}} \put (0,23){\makebox(8,8){$C_1$}} \put (19,23){\makebox(8,8){$C$}} \put (38,23){\makebox(8,8){$C_2$}} \put (19,46){\makebox(8,8){$\tilde{C}$}} \put (19,46){\vector(-1,-1){15}} \put (27,46){\vector(1,-1){15}} \put (23,46){\vector(0,-1){15}} \put (23,23){\vector(0,-1){15}} \put (4,23){\vector(1,-1){15}} \put (42,23){\vector(-1,-1){15}} \put (24,15){$f$} \put (24,38){$\pi $} \put (6,38){$\pi _1$} \put (37,38){$\pi _2$} \put (6,15) {$f_1$} \put (37,15) {$f_2$} \end{picture} \end{center} \caption{} \end{figure} where $deg(f_i)=deg(\pi _i)=2,f_1:C_1\longrightarrow E$ is unramified, determined by $\epsilon ,f_2:C_2\longrightarrow E$ is ramified at $B$ and is determined by $\delta _2=\delta \otimes \epsilon $. Here we have the assumptions of Part (iii) of Proposition~(\ref{p26.1}) so the extended Prym data determines : \begin{itemize} \item $E$ as the curve isomorphic to the dual $X\subset {\bf P}(T_0P)$ of the unique irreducible component of degree $\geq 2$ of the branch locus $G(R)$ of the Gauss map $G:T^{ns}\longrightarrow {\bf P}(T_0P)^*$. \item The points $\{ x_i \mid i=1,...,2g-2\} $ as the duals of the remaining irreducible components of $G(R)$. \item $\delta _2\simeq \delta \otimes \epsilon $ as isomorphic to ${\cal O}_{X}(1)$. \end{itemize} So, it remains to reconstruct $\epsilon $ which is the content of the rest of this section. \begin{lem}\label{l48.1} Let $T_1=Z_1-\pi ^*\Delta -\mu _1,T_2 = Z_2-\pi ^*\Delta -\mu _2$ be arbitrary divisors of the orbits $O_1,O_2\subset \mid 2\Xi \mid$, where $\mu _i\in \lambda _i+P_2$ (Section~(\ref{s3})). Then $T_1,T_2$ are irreducible. Reordering, if necessary, $\{ \lambda _1,\lambda _2\} $, respectively $\{ O_1,O_2\} ,\{ Z_1,Z_2\} ,\{ T_1,T_2\} $ one has that the elements $e_1\in T_1,e_2\in T_2$ have the form (i) $e_1=\pi _1^*(\xi _1)+\pi _2^*(\xi _2)-\pi ^*\Delta-\mu _1$\\ \noindent where $\xi _1\in C_1,\xi _2\in W_{g-2}(C_2)$ and $Nm_{f_1}(\xi _1)+Nm_{f_2}(\xi _2)=\delta _2$. (ii) $e_2=\pi _2^*(\xi _2)-\pi ^*\Delta-\mu _2$\\ where $\xi _2\in W_{g-1}(C_2)$ and $Nm_{f_2}(\xi _2)=\delta _2$. \end{lem} {\bf Proof.} One has to calculate the irreducible components of $Z$ defined in (\ref{e165.1}). One has $$ H^0(C,K_C\otimes \eta ) \simeq H^0(C,f^*\delta _2)\simeq H^0(E,\delta _2)\oplus H^0(E,\delta _2\otimes \delta ^{-1})\simeq H^0(E,\delta _2) $$ Thus $\mid K_C\otimes \eta \mid =f^*\mid \delta _2\mid $. If $\hat{D}$ is an effective divisor of $\tilde{C}$ such that $Nm_{\pi }(\hat{D})\in f^*\mid \delta _2\mid $, then $$ \hat{D}=\pi _1^*E+\pi _2^*F $$ where $E$ and $F$ are effective divisors of $C_1,C_2$ respectively. One has $Nm_{\pi }\circ \pi ^*_i=f^*\circ Nm_{f_i}$, so $$ Nm_{f_1}(E)+Nm_{f_2}(F)\equiv \delta _2 $$ Corollary~(\ref{c19.1}) and a dimension count show that $cl(\hat{D})$ might be a general element of $Z$ if either $deg(E)=1,deg(F)=g-2$ or $E=0,deg(F)=g-1$. So, $Z=Z'\cup Z''$ where $$ Z'=\{ \pi _1^*(\xi _1)+\pi _2^*(\xi _2)\mid \xi _1\in C_1,\xi _2\in W_{g-2}(C_2),Nm_{f_1}(\xi _1)+Nm_{f_2}(\xi _2)\equiv \delta _2\} $$ and $$ Z''=\{ \pi _2^*(\xi _2)\mid \xi _2\in W_{g-1}(C_2),Nm_{f_2}(\xi _2)\equiv \delta _2\} $$ {\bf Claim 1.} {\it $Z'$ is irreducible.}\\ {\bf Proof.} We consider the map $h:C_2^{(g-2)}\longrightarrow E$ defined by $h(D)=\mid \delta _2-Nm_{f_2}(D)\mid $ and the pull-back diagram $$ \begin{array}{ccl}X&\longrightarrow &C_1\\ \downarrow &\mbox{}&\downarrow f_1\\ C_2^{(g-2)}&\stackrel{h}{\longrightarrow }&E \end{array} $$ Then $Z'$ is the image of $X$ under the map $$ (D,x)\longmapsto cl(\pi _1^*(x)+\pi _2^*(D)) $$ Now, $X$ might be reducible if $h_*\pi _1(C_2^{(g-2)})$ is contained in $f_{1*}\pi _1(C_1)$. This is impossible. Indeed, $f_{2*}:H_1(C_2)\longrightarrow H_1(E)$ is epimorphic since $f_2$ is ramified. Therefore $$ (cl\circ f_2^{(g-2)})_*:H_1(C_2^{(g-2)})\longrightarrow H_1(J_{g-2}(E)) $$ is epimorphic. Composing it with the isomorphism $J_{g-2}(E)\longrightarrow E$ given by $\xi \mapsto \mid \delta _2-\xi \mid $ one obtains that $h_*:H_1(C_2^{(g-2)})\longrightarrow H_1(E)$ is epimorphic. This proves that $X$ and therefore $Z'$ are irreducible. q.e.d. {\bf Claim 2.} {\it $Z''$ is irreducible. }\\ {\bf Proof.} With the same notation as above one considers the pull-back diagram $$ \begin{array}{ccl}Y&\longrightarrow &C_2\\ \downarrow &\mbox{}&\downarrow f_2\\ C_2^{(g-2)}&\stackrel{h}{\longrightarrow }&E \end{array} $$ Then $Z''$ is the image of $Y$ under the map $$ (D,y) \longmapsto cl(\pi _2^*(D+y)) $$ In order to prove that $Y$ is irreducible it suffices to verify that not every component of the branch divisor $h^*(B)$ has even multiplicity. Now, $h$ can be decomposed as $$ h=p\circ f_2^{(g-2)}:C_2^{(g-2)}\longrightarrow E^{(g-2)} \longrightarrow E $$ where $p$ is the fiber bundle map defined by $p(A)=\mid \delta _2-A\mid $. Let $x\in B$. Then $\mid \delta _2-x\mid $ is a linear system of degree $g-2\geq 2$ without base points. Let $A=p_1+...+p_{g-2}$ be an element with no points in common with $B$. Let $D\in C_2^{(g-2)}$ with $f_2^{(g-2)}(D)=A$. Then $p^{-1}(x)$ is smooth at $A$ and $f_2^{(g-2)}$ is nondegenerate at $D$, thus $h^*(x)$ is a divisor with multiplicity 1. q.e.d. Now, $Z$ has two connected components $Z_1$ and $Z_2$, enumerated as in Proposition~(\ref{p17.1}). So, $Z'\neq Z'', Z_i$ are irreducible and either $Z_1=Z',Z_2=Z''$ or $Z_1=Z'',Z_2=Z'$. Reordering $\{ Z_1,Z_2\} $ if necessary we can assume that the former case takes place. q.e.d. \begin{lem}\label{l51.1} The singular locus of $Z_1$ has codimension $\geq 2$. \end{lem} {\bf Proof.} Consider a divisor of $\tilde{C}$ of the form $\pi _1^*A+H$ where $H$ is effective, $\pi _1$-simple and $deg(A)\geq 1$. Then by Corollary~(\ref{c92.2}) one concludes that \begin{equation}\label{e515.1} h^0(\tilde{C},\pi _1^*A+H)=h^0(C_1,A) \end{equation} Let $\hat{D}=\pi _1^*(x)+\pi _2^*(F)$ where $x\in C_1,F$ is effective divisor of $C_2$ and $f_1(x)+Nm_{f_2}(F)\equiv \delta _2$. Proposition~(\ref{p17.1}) and (\ref{e515.1}) show that ${\cal O}_{\tilde{C}}(\hat{D})$ is a singular point of $Z_1$ if and only if $F$ is not $\pi _1$-simple. Now, if $\pi _1^*(y)\leq \pi _2^*(F)$, then one easily checks that $f_2^*(f_1(y))\leq F$. Thus $Sing(Z_1)$ consists of \begin{equation}\label{e52.1} cl(\pi _1^*(x+f_1^*(t))+\pi _2^*(G)) \end{equation} where $x\in C_1,t\in E,G\in C_2^{(g-4)}$ and $Nm_{f_2}(G)\in \mid \delta _2-f_1(x)-2t\mid $. For any $G\in C_2^{(g-4)}$ there are two different $\zeta _1\in J_3(C_1)$ such that $Nm_{f_1}(\zeta _1)\equiv \delta _2-Nm_{f_2}(G)$. Thus the elements of the type (\ref{e52.1}) form a sublocus of $Z_1$ of dimension $\leq g-4$. q.e.d. \begin{lem}\label{l53.1} $Sing(Z_2)$ has a unique irreducible component $V$ of codimension 1 in $Z_2$. A Zariski open, dense subset of $V$ consists of the elements \begin{equation}\label{e53.3} cl(\pi _2^*(f^*_2(x)+G)) \end{equation} where $x\in E$, and $G$ is an effective, $f_2$-simple divisor of $C_2$ such that $Nm_{f_2}(G)\in \mid \delta _2-2x\mid $. \end{lem} {\bf Proof.} We claim that \begin{equation}\label{e53.1} dimW^1_{g-1}(C_2)\cap Nm_{f_2}^{-1}(\delta _2)\leq g-4 \end{equation} This is clear if $C_2$ were not hyperelliptic. If $C_2$ were hyperelliptic, then $W^1_{g-2}(C_2)=g^1_2+W_{g-3}(C_2)$. This irreducible variety can not be contained in $Nm_{f_2}^{-1}(\delta _2)$. Indeed, otherwise its translation would be contained in the abelian hypersurface $Nm_{f_2}^{-1}(0)$ of $J(C_2)$ which is absurd since this translation generates $J(C_2)$. By (\ref{e53.1}) we conclude that the sublocus of $Sing(Z_2)$ : \begin{equation}\label{e54.1} \{ \pi _2^*(\xi _2)\mid Nm_{f_2}(\xi _2)=\delta _2,h^0(C,\xi _2)\geq 2\} \end{equation} has codimension $\geq 2$ in $Z_2$. Suppose $F$ is an effective, $f_2$-simple divisor of $C_2$ such that $Nm_{f_2}(F)\in \mid \delta _2\mid $. Assume that $cl(\pi _2^*F)\in SingZ_2$. By Proposition~(\ref{p17.1}) this is equivalent to $h^0(\tilde{C},\pi _2^*F)\geq 2$. Then we claim that $h^0(C,F)\geq 2 $, so $\pi _2^*F$ belongs to the locus (\ref{e54.1}). Indeed, since $\pi _2:\tilde{C}\longrightarrow C$ is a double unramified covering corresponding to $f^*_2(\epsilon )\in Pic^0(C_2)_2$ we have \begin{equation}\label{e53.2} h^0(\tilde{C},\pi _2^*F)=h^0(C_2,F)+h^0(C_2,f^*_2(\epsilon )(F)) \end{equation} By Lemma~(\ref{l92.1}) we conclude that $h^0(C_2,f^*_2(\epsilon )(F))=0$. So, $cl(\pi _2^*(F))$ belongs to $Sing(Z_2)$ if and only if $cl(F)\in W^1_{g-1}(C_2)$. Now, suppose that $F=f_2^*(x)+G$ where $x\in E,G$ is effective and $Nm_{f_2}(G)\in \mid \delta _2-2x\mid $. Let $t_{\epsilon }(x)$ be the translation of x by $\epsilon $. Then by Eq.~(\ref{e53.2}) we have \begin{equation}\label{e54.3} h^0(\tilde{C},\pi _2^*F)=h^0(C_2,f_2^*(x)+G)+h^0(C_2,f_2^*(t_{\epsilon }(x))+G) \end{equation} Thus $h^0(\tilde{C},\pi ^*F)\geq 2$ and $cl(\pi ^*F)\in Sing Z_2$. The sublocus of $SingZ_2$ $$ V=\{ cl(\pi _2^*(f_2^*(x)+G))\mid x\in E,G\geq 0,Nm_{f_2}(G)\in \mid \delta _2-2x\mid \} $$ is the image of $X$ where $X$ is defined by the pull-back diagram \begin{equation}\label{e55.1} \begin{array}{ccl}X&\longrightarrow &E\\ \downarrow &\mbox{}&\downarrow \beta \\ C_2^{(g-3)}&\stackrel{\alpha }{\longrightarrow }&J_2(E) \end{array} \end{equation} and $\alpha (G)=cl(\delta _2-Nm_{f_2}(G)),\beta (x)=cl(2x)$. The same argument as in Claim 1 of Lemma~(\ref{l48.1}) shows that $X$ is irreducible. This implies that $V$ is irreducible as well. Corollary~(\ref{c92.2}) and Eq.~(\ref{e54.3}) show that for $F=f _2^*(x)+G$ one has $h^0(\pi _2^*F)=2$ if and only if $G$ is $f_2$-simple. Thus the points (\ref{e53.3}) form a Zariski open, dense subset of $V$. Finally, $dimX=g-3$ and we claim that the map $X\longrightarrow V$ given by $(G,x)\longmapsto cl(\pi _2^*(f_2^*(x)+G))$ is of degree 2, hence $dimV=g-3$. Indeed, let $\sigma _2:\tilde{C}\longrightarrow \tilde{C}$ be the involution which interchanges the sheets of $\pi _2$. Then for any $\pi _2^*(\xi _2)\in V$ with $h^0(\tilde{C},\pi _2^*(\xi _2))=2$ according to Eq.~(\ref{e54.3}) there are exactly two $\sigma _2$-invariant divisors in $\mid \pi _2^*(\xi _2)\mid $ namely \begin{equation}\label{e56.1} \begin{array}{cc} \pi _2^*(f_2^*(x)+G)\; ,&\pi _2^*(f_2^*(t_{\epsilon }(x))+G) \end{array} \end{equation} with $x,G$ as in the lemma. q.e.d. Let $$ S_2=\{ F\in C_2^{(g-2)}\mid Nm_{f_2}(F)\in \mid \delta _2\mid \} $$ One has a surjective map $$ \varphi =cl\circ \pi _2^*:S_2\longrightarrow Z_2 $$ {}From the proof of Claim 2 of Lemma~(\ref{l48.1}) we see that $S_2$ is irreducible. Moreover $deg\varphi =1$ since $h^0(\tilde{C},L)=1$ for any sufficiently general $L\in Z_2$. Let us consider the Stein factorization \cite{hart} $$ \varphi =\psi \circ \alpha :S_2\longrightarrow \Gamma \longrightarrow Z_2 $$ where $\psi $ is a finite map and $\alpha $ has connected fibers. Let $$ W_1=\{ F\in S_2\mid h^0(C_2,F)\geq 2\}\; ,\; W_2=\{ f_2^*A+E\in S_2\mid A\geq 0,E\geq 0,deg(A)\geq 2\} $$ One has $codim_{S_2}(W_1)\geq 1$ and $codim_{S_2}(W_2)\geq 2$. Let $S_2^0=S_2\backslash (W_1\cup W_2)$ and let $\Gamma ^0=\alpha(S^0_2)$. \begin{lem}\label{l58.1} The points of $S^0_2,\Gamma^0$ are nonsingular in $S_2,\Gamma $ respectively, $codim_{\Gamma }(\Gamma \backslash \Gamma ^0)\geq 2$ and the map $\alpha:S^0_2\longrightarrow \Gamma ^0$ is an isomorphism. Let $n:N\longrightarrow Z_2$ be the normalization of $Z_2$. Then there exists a finite map $\beta :N\longrightarrow \Gamma $ such that $n=\psi \circ \beta $. If $N^0=\beta ^{-1}(\Gamma ^0)$ then $codim_N(N\backslash N^0)\geq 2$ and $\beta :N^0\longrightarrow \Gamma ^0$ is an isomorphism \end{lem} {\bf Proof.} The points of $S^0_2$ are nonsingular by Proposition~(\ref{p91.1}). For any $x\in S^0_2$ one has $\# \varphi ^{-1}(\varphi (x))=2$ as we have shown in the proof of Lemma~(\ref{l53.1}). Thus the map $\alpha :S^0_2\longrightarrow \Gamma ^0$ is bijective. The map $\varphi $ is nondegenerate at the points of $S^0_2$. Indeed $\varphi = cl\circ \pi _2^*=\pi _2^*\circ cl$, the map $cl:S_2\longrightarrow J_{g-1}(C_2)$ is nondegenerate at any $F\in S^0_2$ since $h^0(F)=1$, and the map $\pi _2^*:J_{g-1}(C_2)\longrightarrow J_{2g-2}(\tilde{C})$ is obviously nondegenerate. We conclude that $\alpha :S^0_2\longrightarrow \Gamma ^0$ is an isomorphism. One has $codim_{\Gamma }(\Gamma \backslash \Gamma ^0)\geq 2$ since $codim_{Z_2}\varphi (W_1)\geq 2$ by (\ref{e53.1}). The rest of the lemma is clear by the universal property of the normalization. q.e.d. {\bf Reconstruction of $(C,\eta )$ in the bi-elliptic case, $g\geq 4$.} \noindent In the beginning of this section we have seen how to reconstruct up to isomorphism $E$ and the covering $f_2:C_2\longrightarrow E$. Let $T_i\in O_i\subset \mid 2\Xi \mid ,i=1,2$. We have proved above that $T_i$ are irreducible and just one of $T_i$ has a singular locus of codimension 1. Reordering, if necessary, as in Lemma~(\ref{l48.1}) we can assume that this divisor is $T_2$. We can identify $T_2$ and $Z_2$ by translation. Let $n:N\longrightarrow T_2$ be the normalization of $T_2$. Let $R=n^{-1}(V)$. The Zariski open subset $R^0=R\cap N^0$ is dense in $R$ since $codim_N(N\backslash N^0)\geq 2.$ By the irreducibility of $X$ in (\ref{e55.1}) one concludes that $\alpha^{-1}\circ \beta (R^0)$ and $R$ are irreducible as well. We have an isomorphism $f^*:\mid \delta _2\mid \longrightarrow \mid K_C\otimes \eta \mid $. Consider the Gauss map $G:Z_2^{ns}\longrightarrow \mid K_C\otimes \eta\mid $. Then for every $F\in S_2$ with $h^0(\tilde{C},\pi _2^*F)=1$ one has $$ G(\varphi (F))=f^*(Nm_{f_2}(F)) $$ Shrinking $N_0$ from Lemma~(\ref{l58.1}) we can assume that the following properties hold \begin{itemize} \item $codim_N(N\backslash N_0)\geq 2$. \item The composition $C\circ n$ can be extended to a regular map on $N_0$. \item Every point of $\alpha ^{-1}\circ \beta (R^0)$ has the form $f_2^*(x)+A$ where $x$ is not a branch point of $f_2$, $A$ is reduced and $f_2$-simple, and $x\not \in Supp(A).$ \end{itemize} Now, we define a rational map $$ \gamma : R\longrightarrow E $$ as follows. For every $L\in R_0$ the hyperplane $G\circ n(L)$ belongs to the unique irreducible component of degree $>1$ of the branch locus of the Gauss map $G:T\longrightarrow {\bf P}(T_0P)^*$, namely $E^*$. By the conditions above this hyperplane is tangent to a unique point of $E$. We denote this point by $\gamma (L)$. Now, let $L=\beta ^{-1}\circ \alpha (f_2^*(x)+A)\in R_0$. Then $n^{-1}(n(L))=\{ L,L'\} $ where \begin{equation}\label{e62.1} L'=\beta ^{-1}\circ \alpha (f_2^*(t_{\epsilon }(x))+A) \end{equation} according to (\ref{e56.1}). This shows that the map $n:R\longrightarrow V\subset Sing(T_2)$ is of degree 2. By (\ref{e62.1}) the corresponding involution $\tau ^*:{\bf C}(R)\longrightarrow {\bf C}(R)$ of the field of rational functions on $R$ transforms $\gamma ^*{\bf C}(E)$ into itself and $\tau ^*:\gamma ^*{\bf C}(E)\longrightarrow \gamma ^*{\bf C}(E)$ is induced by the translation map $t_{\epsilon }:E\longrightarrow E$. This gives the reconstruction of $\epsilon \in J(E)_2$ and completes the reconstruction of $(C,\eta )$ from the extended Prym data. \section{The case $g=3$}\label{s7} Let $a\in \tilde{C}$. We define the Abel-Prym map $\phi :\tilde{C}\longrightarrow P(\tilde{C},\sigma )$ by $$ \phi (x)=cl(x-a-\sigma (x-a)) $$ \begin{lem}\label{l635.1} Suppose $g\geq 2$. The following alternative takes place \begin{list}{(\roman{bean})}{\usecounter{bean}} \item $\phi $ maps $\tilde{C}$ isomorphically onto its image $\phi (\tilde{C})$. \item The map $\phi :\tilde{C}\longrightarrow \phi (\tilde{C})$ has degree 2. \end{list} The second case occurs if and only if {\rm (*)} $C$ is hyperelliptic and $\eta \simeq {\cal O}_{C}(p_1-p_2)$ for some $p_1,p_2\in C$.\\ \noindent Here $\phi (\tilde{C})\simeq C_2$ and $\phi =\pi _2$ (see Fig.1) via this isomorphism. \end{lem} {\bf Proof.} Suppose $\phi (x)=\phi (y)$ for some $x\neq y$. Then $x+\sigma y\equiv y+\sigma x$, thus $\tilde{C}$ is hyperelliptic. It has a unique $g^1_2$, so $\sigma (g^1_2)=g^1_2$. Let $\sigma _1$ be the hyperelliptic involution of $\tilde{C}$. Then $\sigma \neq \sigma _1$ and we claim that $\sigma $ and $\sigma _1$ commute. Indeed, for any $z\in \tilde{C}$ \[ \begin{array}{cc} \sigma z+\sigma _1(\sigma z)\in g^1_2\; ,&\sigma (z+\sigma _1z)\in g^1_2 \end{array} \] Thus $\sigma \sigma _1z=\sigma _1\sigma z$. Let $\sigma _2=\sigma \sigma _1$. Let $C_1=\tilde{C}/\sigma _1$ and let $\overline{\sigma }:C_1\longrightarrow C_1$ be the involution induced by $\sigma $. Then $\overline{\sigma }$ has two fixed points since $C_1\simeq {\bf P}^1$. Thus $\pi :\tilde{C}\longrightarrow C$ fits into the commutative diagram of Fig.1 and condition (*) holds. If $\phi $ were degenerate at some point $x\in \tilde{C}$, then $\pi (x)$ would be a base point of $\mid K_C\otimes \eta\mid $, thus condition (*) holds according to Lemma~(\ref{l21.1}). Conversely, suppose condition (*) holds. Then by the argument above $\phi (x)=\phi (y)$ and $x\neq y$ if and only if $y$ belongs to the divisor $\sigma (x+\sigma _1x)$. Thus $y=\sigma _2(x)$. q.e.d. Further we suppose that $g=3$. Let $T_i\in O_i\subset \mid 2\Xi \mid ,i=1,2.$ The divisors $T_i$ are reduced, connected curves according to Proposition~(\ref{p17.1}). Let $S=Nm^{-1}(\mid K_C\otimes \eta \mid )\subset \tilde{C}^{(4)} $ and let $Z=Nm^{-1}(K_C\otimes \eta )\cap W_4(\tilde{C})\subset J_4(\tilde{C})$. Both $S$ and $Z$ break into two disjoint, connected components $S=S_1\cup S_2,Z=Z_1\cup Z_2$. We enumerate so that $cl(S_i)=Z_i$ and $T_i$ is translation of $Z_i,i=1,2$. \begin{lem}\label{l66.1} The curves $T_1,T_2$ are both singular if and only if condition (*) of Lemma~(\ref{l635.1}) holds. If only one of $T_i$ is singular then the nonsingular one is a translation of $\phi (\tilde{C})$. \end{lem} {\bf Proof.} If condition (*) holds, then both $T_1$ and $T_2$ are reducible and hence singular by Corollary~(\ref{c39.1}). Since $T_i$ is a translation of $Z_i,i=1,2$ we can work with $Z_i$. Suppose that condition (*) does not hold and $Z'\in \{ Z_1,Z_2\} $ is a singular curve with a singular point $L$. Let $\{ Z',Z''\} =\{ Z_1,Z_2\} $. Then $$ X=\{ L+x-\sigma x\mid x\in \tilde{C}\} \subset Z'' $$ Clearly $X$ is a translation of $\phi (\tilde{C})$. According to Lemma~(\ref{l635.1}) $X$ is isomorphic to $\tilde{C}$ and $X$ is algebraically equivalent to $2\Xi $ \cite{mas}. Thus $X=Z''$. q.e.d. {\bf Reconstruction of $(C,\eta ) $ in the case $g=3$.} {\bf Case 1.} {\it Both $T_1,T_2$ are singular.}\\ According to Lemma~(\ref{l66.1}) we are in the situation of Section~(\ref{s5}) where a procedure for the reconstruction of $(C,\eta )$ was described. {\bf Case 2.} {\it Just one of the curves $T'\in \{ T_1,T_2\} $ is singular.}\\ We take the other curve $T''$. It is isomorphic to $\tilde{C}$ according to Lemmas~(\ref{l66.1}) and (\ref{l635.1}). The involution $-id_P:T''\longrightarrow T''$ coincides with $\sigma $ via this isomorphism. We thus reconstruct $(\tilde{C},\sigma )$. {\bf Case 3.} {\it $T_1$ and $T_2$ are nonsingular.}\\ Consider the involutions $\sigma _i:T_i\longrightarrow T_i$ induced by $-id_P$, let $C_i=T_i/\sigma _i$ and let $\pi _i:T_i\longrightarrow C_i$ be the factor maps, $i=1,2$. By Proposition~(\ref{p17.1}) the map $cl:S_i\longrightarrow Z_i$ is bijective. Thus $S_i$ are nonsingular since $Z_i$ are nonsingular. Using Proposition~(\ref{p91.1}) one checks that the nonsingularity of $S_i$ implies that $\sigma ^{(4)}:S_i\longrightarrow S_i$ is without fixed points, thus $\sigma _i:T_i\longrightarrow T_i$ is without fixed points as well, $i=1,2$. Consider the Gauss maps $G_i:T_i\longrightarrow {\bf P}(T_0P)^*={\bf P}^1$. By Eq.~(\ref{e26.1}) one shows that $G_i=f_i\circ \pi _i$ where $f_i:C_i\longrightarrow {\bf P}^1$ are maps of degree 4 and concludes that the maps $G_i:T_i\longrightarrow {\bf P}^1$ are obtained from $\varphi _{ K_C\otimes \eta }\circ \pi :\tilde{C}\longrightarrow {\bf P}^1$ by the tetragonal construction of Donagi \cite{don},\cite{don1}. Now, take the pair $(T_1,\sigma _1) $ and apply the tetragonal construction to $G_1:T_1\longrightarrow {\bf P}^1$ (ibid.). One obtains two 8-sheeted coverings $g_i:X_i\longrightarrow {\bf P}^1$ with involutions $\tau _i:X_i\longrightarrow X_i$ such that $g_i\circ \tau _i=g_i$. For one of them, e.g. $X_2$, there is an isomorphism of the coverings $$ \begin{array}{rlcl}\mbox{}&X_2&\stackrel{\psi _2}{\longrightarrow }&T_2\\ g_2&\downarrow &\mbox{}&\downarrow \hspace{.25cm}G_2\\ \mbox{}&{\bf P}^1&=&{\bf P}^1 \end{array} $$ such that $\psi _2\circ \tau _2=\sigma _2\circ \psi _2$. Then the remaining pair $(X_1,\tau _1)$ is isomorphic to $(\tilde{C},\sigma )$.

9304

alg-geom/9304003

en

https://arxiv.org/abs/alg-geom/9304003

alg-geom/9304003

Dave Bayer and David Mumford

What can be computed in algebraic geometry?

56 pages, Latex

This paper is a survey of computational issues in algebraic geometry, with particular attention to the theory of Grobner bases and the regularity of an algebraic variety. 1. A geometric introduction to Grobner bases. 2. An algebraic introduction to Grobner bases. 3. Bounds in algebraic geometry, and regularity and complexity questions. 4. Applications.

alg-geom

\section{A Geometric Introduction} \label{geometry} Let $X$ be a subvariety or a subscheme of projective $n$-space $\proj n$, over a field $k$. Let ${\cal F}$ be a vector bundle or a coherent sheaf supported on $X$. We would like to be able to manipulate such objects by computer. From algebra we get finite descriptions, amenable to such manipulations: Let $S = k[x_0,\ldots, x_n]$ be the homogeneous coordinate ring of $\proj n$. Then $X$ can be taken to be the subscheme defined by a homogenous ideal $I \subset S$, and ${\cal F}$ can be taken to be the sheaf associated to a finitely generated $S$-module $M$. We can represent $I$ by a list of generators $(f_1, \ldots, f_r)$, and $M$ by a presentation matrix $F$, where $$M_1 \stackrel{F}{\longrightarrow} M_0 \longrightarrow M \longrightarrow 0$$ presents $M$ as a quotient of finitely generated free $S$-modules $M_0$, $M_1$. We concentrate on the case of an ideal $I$; by working with the submodule $J = {\rm Im}(F) \subset M_0$, the module case follows similarly. The heart of most computations in this setting is a deformation of the input data to simpler data, combinatorial in nature: We want to move through a family of linear transformations of $\proj n$ so that in the limit our objects are described by monomials. Via this family, we hope to pull back as much information as possible to the original objects of study. Choose a one-parameter subgroup $\lambda(t) \subset GL(n+1)$ of the diagonal form $$\lambda(t) = \left[ \begin{array}{cccc} t^{w_0} & & & \\ & t^{w_1} & & \\ & & \ldots & \\ & & & t^{w_n} \end{array} \right], $$ where $W = (w_0, \ldots, w_n)$ is a vector of integer weights. For each $t \ne 0$, $\lambda(t)$ acts on $X$ via a linear change of coordinates of $\proj n$, to yield the subscheme $X_t = \lambda(t) X \cong X$. The limit $$X_0 = \lim_{t \rightarrow 0} X_t$$ is usually a simpler object, preferable to $X$ for many computational purposes. Even if we start out by restricting $X$ to be a subvariety rather than a subscheme of $\proj n$, it does not suffice to take the limit $X_0$ set-theoretically; often all we will get pointwise in the limit is a linear subspace $L \subset \proj n$, reflecting little besides the dimension of the original variety $X$. By instead allowing this limit to acquire embedded components and a nonreduced structure, we can obtain an $X_0$ which reflects much more closely the character of $X$ itself. We compute explicitly with the generators $f_1, \ldots, f_r$ of $I$: Let $\lambda$ act on $S$ by mapping each $x_i$ to $t^{w_i} x_i$; $\lambda$ maps each monomial $\sliver{\bf x}\sliver^A = x_0^{a_0} \cdots x_n^{a_n}$ to $t^{W \cdot A} \sliver{\bf x}\sliver^A = t^{w_0 a_0 + \ldots + w_n a_n} x_0^{a_0} \cdots x_n^{a_n}$. If $f = a \sliver{\bf x}\sliver^A + b \sliver{\bf x}\sliver^B + \ldots$, then $\lambda f = a \,t^{W \cdot A} \sliver{\bf x}\sliver^A + b \,t^{W \cdot B} \sliver{\bf x}\sliver^B + \ldots$. We take the projective limit ${\rm in}(f) = \lim_{t \rightarrow 0} \lambda f$ by collecting the terms of $\lambda f$ involving the least power of $t$; ${\rm in}(f)$ is then the sum of the terms $a \sliver{\bf x}\sliver^A$ of $f$ so $W \cdot A$ is minimal. For a given $f$ and most choices of $\lambda$, ${\rm in}(f)$ consists of a single term. The limit $X_0$ we want is defined with all its scheme structure by the ideal ${\rm in}(I) = \lim_{t \rightarrow 0} \lambda I$, generated by the set $\setdef{{\rm in}(f)}{f \in I}$. For a given $I$ and most choices of $\lambda$, ${\rm in}(I)$ is generated by monomials. Unfortunately, this definition is computationally unworkable because $I$ is an infinite set, and ${\rm in}(I)$ need not equal $({\rm in}(f_1), \ldots, {\rm in}(f_r))$ for a given set of generators $f_1, \ldots, f_r$ of $I$. To understand how to compute ${\rm in}(I)$, we need to look more closely at the family of schemes $X_t$ defined by $\lambda$. Let $S[t]$ be the polynomial ring $k[x_0, \ldots, x_n, t]$; we view $S[t]$ as the coordinate ring of a one-parameter family of projective spaces ${\bf P}^n_t$ over the affine line with parameter $t$. For each generator $f_j$ of $I$, rescale $\lambda f_j$ so the lowest power of $t$ has exponent zero: Let $g_j = t^{-\ell} \lambda f_j$, where $\ell = W \cdot A$ is the least exponent of $t$ in $\lambda f_j$. Then $f_j = g_j\!\!\mid_{t=1}$ and ${\rm in}(f_j) = g_j\!\!\mid_{t=0}$. Now, let $J \subset S[t]$ be the ideal generated by $(g_1, \ldots, g_r)$; $J$ defines a family $Y$ over $\affine 1$ whose central fiber is cut out by $({\rm in}(f_1), \ldots, {\rm in}(f_r))$. What is wrong with the family $Y$? $Y$ can have extra components over $t = 0$, which bear no relation to its limiting behavior as $t \rightarrow 0$. Just as the set-theoretic limit $\lim_{t \rightarrow 0} X_t$ can be too small (we need the nonreduced structure), this algebraically defined limit can be too big; the natural limit lies somewhere in between. The notion of a {\em flat\,} family captures exactly what we are looking for here. For example, if $Y$ is flat, then there are no extra components over $t = 0$. While the various technical definitions of flatness can look daunting to the newcomer, intuitively flatness captures exactly the idea that every fiber of a family is the natural scheme-theoretic continuation of its neighboring fibers. In our setting, all the $X_t$ are isomorphic for $t \ne 0$, so we only need to consider flatness in a neighborhood of $t = 0$. Artin \cite{art76} gives a criterion for flatness applicable here: The {\em syzygies\,} of $g_1, \ldots, g_r$ are the relations $h_1 g_1 + \ldots + h_r g_r = 0$ for $h_1, \ldots, h_r \in S[t]$. Syzygies correspond to elements $(h_1, \ldots, h_r)$ of the $S[t]$-module $S[t]^r$; the set of all syzygies is a submodule of $S[t]^r$. $Y$ is a flat family at $t = 0$ if and only if the restrictions $(h_1\!\!\mid_{t=0}, \ldots, h_r\!\!\mid_{t=0})$ of these syzygies to the central fiber generate the $S$-module of syzygies of $g_1\!\!\mid_{t=0}, \ldots, g_r\!\!\mid_{t=0}$. When $g_1\!\!\mid_{t=0}, \ldots, g_r\!\!\mid_{t=0}$ are single terms, their syzygies take on a very simple form: The module of syzygies of two terms $a\sliver{\bf x}\sliver^A$, $b\sliver{\bf x}\sliver^B$ is generated by the syzygy $b\sliver{\bf x}\sliver^C (a\sliver{\bf x}\sliver^A) - a\sliver{\bf x}\sliver^D (b\sliver{\bf x}\sliver^B) = 0$, where $\sliver{\bf x}\sliver^E = \sliver{\bf x}\sliver^C \sliver{\bf x}\sliver^A = \sliver{\bf x}\sliver^D \sliver{\bf x}\sliver^B$ is the least common multiple of $\sliver{\bf x}\sliver^A$ and $\sliver{\bf x}\sliver^B$. The module of syzygies of $r$ such terms is generated (usually not minimally) by the syzygies on all such pairs. We want to lift these syzygies to syzygies of $g_1, \ldots, g_r$, working modulo increasing powers of $t$ until each syzygy lifts completely. Whenever we get stuck, we will find ourselves staring at a new polynomial $g_{r+1}$ so $t^\ell g_{r+1} \in J$ for some $\ell > 0$. Including $g_{r+1}$ in the definition of a new $J^\prime \supset J$ has no effect on the family defined away from $t = 0$, but will cut away unwanted portions of the central fiber; what we are doing is removing $t$-torsion. By iterating this process until every syzygy lifts, we obtain explicit generators $g_1, \ldots, g_r, g_{r+1}, \ldots, g_s$ for a flat family describing the degeneration of $X = X_1$ to a good central fiber $X_0$. The corresponding generators $g_1\!\!\mid_{t=1}, \ldots, g_s\!\!\mid_{t=1}$ of $I$ are known as a {\em Gr\"{o}bner\ basis} for $I$. This process is best illustrated by an example. Let $S = k[w,x,y,z]$ be the coordinate ring of $\proj 3$, and let $I = (f_1, f_2, f_3) \subset S$ for $$f_1= w^2-xy,\; f_2 = wy-xz,\; f_3 = wz-y^2.$$ $I$ defines a twisted cubic curve $X \subset \proj 3$; $X$ is the image of the map $(r,s) \mapsto (r^2s,r^3,rs^2,s^3)$. Let $$\lambda(t) = \left[ \begin{array}{cccc} t^{-16} & & & \\ & t^{-4} & & \\ & & t^{-1} & \\ & & & t^0 \end{array} \right]. $$ If $w^a x^b y^c z^d$ is a monomial of degree $< 4$, then $\lambda \cdot w^a x^b y^c z^d = t^{-\ell} w^a x^b y^c z^d$ where $\ell = 16a + 4b + c$. Thus, sorting the monomials of $S$ of each degree $< 4$ by increasing powers of $t$ with respect to the action of $\lambda$ is equivalent to sorting the monomials of each degree in lexicographic order. We have \begin{eqnarray*} g_1 & =\; t^{32}\lambda f_1 & =\; w^2-t^{27}xy, \\ g_2 & =\; t^{17}\lambda f_2 & =\; wy-t^{13}xz, \\ g_3 & =\; t^{16}\lambda f_3 & =\; wz-t^{14}y^2. \end{eqnarray*} The module of syzygies on $w^2$, $wy$, $wz$ is generated by the three possible pairwise syzygies; we start with the syzygy $y(w^2) - w(wy) = 0$. Substituting $g_1$, $g_2$ for the lead terms $w^2$, $wy$ we get $$y(w^2-t^{27}xy) - w(wy-t^{13}xz) = t^{13}wxz - t^{27}xy^2$$ which is a multiple $t^{13}x$ of $g_3$. Thus, the syzygy $$y g_1 - w g_2 - t^{13} x g_3 = 0$$ of $g_1$, $g_2$, $g_3$ restricts to the monomial syzygy $y(w^2) -w(wy) = 0$ when we substitute $t = 0$, as desired. Similarly, the syzygy $$z g_1 - t^{14}y g_2 - w g_3 = 0$$ restricts to the monomial syzygy $z(w^2) -w(wz) = 0$. When we attempt to lift $z(wy) - y(wz) = 0$, however, we find that $$z(wy-t^{13}xz) - y(wz-t^{14}y^2) = - t^{13}xz^2 + t^{14}y^3.$$ $xz^2$ is not a multiple of $w^2$, $wy$, or $wz$, so we cannot continue; $J = (g_1, g_2, g_3)$ does not define a flat family. Setting $t = 1$, the troublesome remainder is $- xz^2 + y^3$. Making this monic, let $f_4 = xz^2 - y^3$; $f_4 \in I$ and $$g_4 =\; t^{4}\lambda f_4 =\; xz^2 - ty^3.$$ Adjoin $g_4$ to the ideal $J$, redefining the family $Y$. Now, $$z g_2 - y g_3 + t^{13} g_4 = 0$$ restricts to $z(wy) - y(wz) = 0$ as desired. The module of syzygies of $w^2$, $wy$, $wz$, and $xz^2$ is generated by the pairwise syzygies we have already considered, and by the syzygy $xz(wz) - w(xz^2) = 0$, which is the restriction of $$-ty^2 g_2 +xz g_3 - w g_4 = 0.$$ Thus, $J = (g_1, g_2, g_3, g_4)$ defines a flat family $Y$, and $$w^2-xy, \;wy-xz, \;wz-y^2, \;xz^2-y^3$$ is a Gr\"{o}bner\ basis for $I$. The limit $X_0$ is cut out by the monomial ideal ${\rm in}(I) = (w^2, wy, wz, xz^2)$, which we shall see shares many properties with the original ideal $I$. Note that $xz^2 - y^3 = 0$ defines the projection of $X$ to the plane $\proj 2$ in $x$, $y$, and $z$. The scheme structure of $X_0$ is closely related to the combinatorial structure of the monomial $k$-basis for $S/{\rm in}(I)$: For each degree $d$ in our example, the monomials not belonging to ${\rm in}(I)$ consist of three sets $\{x^d, x^{d-1}y,\ldots,y^d\}$, $\{x^{d-1}z, x^{d-2}yz,\ldots,y^{d-1}z\}$, $\{y^d, y^{d-1}z,\ldots,z^d\}$, and a lone extra monomial $x^{d-1}w$. The first two sets correspond to a double line supported on $w = z = 0$, the third set to the line $w = x = 0$, and the extra monomial to an embedded point supported at $w = y = z = 0$. Together, this describes the scheme structure of $X_0$. The first two sets consist of $d+1$ and $d$ monomials, respectively; the third set adds $d-1$ new monomials, and overlaps two monomials we have already seen. With the extra monomial, we count $3d+1$ monomials in each degree, which agrees with the dimensions of the graded pieces of $S/I$. The embedded point is crucial; it makes this count come out right, and it alone keeps $X_0$ nonplanar like $X$. The new monomial generator $xz^2$ of ${\rm in}(I)$ excludes the line $w = y = 0$ from $X_0$; combinatorially, it excludes all but three monomials of the set $\{x^d, x^{d-1}z,\ldots,z^d\}$ from the monomial $k$-basis for each degree of the quotient $S/{\rm in}(I)$. We can see that this line is unwanted as follows: Away from $t = 0$, $Y$ is parametrized by $(r,s,t) \mapsto (t^{16}r^2s, t^4r^3,trs^2,s^3,t)$. Thus, fixing $r$ and $s$, the curve $(r,ts,t) \mapsto (t^{17}r^2s, t^4r^3,t^3rs^2,t^3s^3,t)$, with projective limit $(0,0,r,s,0)$ as $t \rightarrow 0$. Similarly, the curve $(r,t^3s,t^2)$ has as its limit $(0,r^2,s^2,0,0)$. These calculations show that the lines $w = z = 0$ and $w = x = 0$ indeed belong set-theoretically to the limit $X_0$. We can find no such curve whose limit is a general point on the line $w = y = 0$, for $(r, t^4s, t^3)$ doesn't work. Thus, the line $w = y = 0$ sticks out of the good total space $Y$. One usually computes Gr\"{o}bner\ bases by working directly in the ring $S$, dispensing with the parameter $t$. The one-parameter subgroup $\lambda$ is replaced by a total order on the monomials of each degree, satisfying the {\em multiplicative\,} property $\sliver{\bf x}\sliver^A > \sliver{\bf x}\sliver^B \Rightarrow \sliver{\bf x}\sliver^C \sliver{\bf x}\sliver^A > \sliver{\bf x}\sliver^C \sliver{\bf x}\sliver^B$ for all $\sliver{\bf x}\sliver^C$. In fact, for our purposes these are equivalent concepts: The weight vector $W$ associated with $\lambda$ induces the order $\sliver{\bf x}\sliver^A > \sliver{\bf x}\sliver^B \Longleftrightarrow W\cdot A < W\cdot B$, which is a total multiplicative order in low degrees as long as no two monomials have the same weight. Conversely, given any multiplicative order and a degree bound $d$, one can find many $\lambda$ which induce this order on all monomials of degree $< d$. See \cite{bay82}, \cite{rob85} for characterizations of such orders. We shall be particularly interested in two multiplicative orders, the {\em lexicographic\,} order used in our example, and the {\em reverse lexicographic\,} order. The lexicographic order simply expands out the monomials of each degree into words, and sorts them alphabetically, i.e. $\sliver{\bf x}\sliver^A > \sliver{\bf x}\sliver^B$ iff the first nonzero entry in $A-B$ is positive. The reverse lexicographic order pushes highest powers of $x_n$ in any expression back to the end, then within these groups pushes highest powers of $x_{n-1}$ to the end, etc., i.e. $\sliver{\bf x}\sliver^A > \sliver{\bf x}\sliver^B$ iff the last nonzero entry of $A-B$ is negative. What do these orders mean geometrically? The dominant effect of the lexicographic order is a projection from $\proj n$ to $\proj{n-1}$, eliminating $x_0$. A second order effect is a projection to $\proj{n-2}$, and so forth. We could compute the deformation from $X$ to $X_0$ with respect to the lexicographic order in stages carrying out these projections, first applying a $\lambda$ with $W = (-1,0,\ldots,0)$, then with $W = (-1,-1,0,\ldots,0)$, etc. Alternatively, for monomials of each degree $< d$, we can apply the single $\lambda$ with $W = (-d^{n-1},\ldots,-d,-1,0)$, generalizing the $\lambda$ used in our example. Use of the lexicographic order tends to muck up the family $Y$ more than necessary in most applications, because projections tend to complicate varieties. For the reverse lexicographic order, the dominant effect is a projection of $\proj n$ down to the last coordinate point $(0,\ldots,0,1)$. As a secondary effect, this order projects down to the last coordinate line, and so forth. In other words, this order first tries to make $X$ into a cone over the last coordinate point, and only then tries to squash the result down to or cone it over the last coordinate line, etc. For monomials of each degree $< d$, this can be realized by applying $\lambda$ with $W = (0,1,d,\ldots,d^{n-1})$. Like such cones, the reverse lexicographic order enjoys special properties with respect to taking linear sections of $X$ or $X_0$ by intersection with the spaces defined by the last variable(s) (see \cite{bs87a}). The preferred status of the reverse lexicographic order can be attributed to this relationship, because generic linear sections do not complicate varieties. For example, if we take $X$ to be three general points in $\proj 2$, then using the lexicographic order $X_0$ becomes a triple point on a line, because the first order effect is the projection of the three points to a line, and the second order limiting process keeps the points within this line. By contrast, if we use the reverse lexicographic order then $X_0$ becomes the complete first order neighborhood of a point (a point doubled in all directions). This is because the first order limiting process brings the three points together from distinct directions, tracing out a cone over the three points. The first order neighborhood of the vertex in this cone has multiplicity 3, and is the same as the complete first order neighborhood in the plane of this vertex. For those familiar with the theory of valuations in birational geometry \cite[Vol. II, Ch. VI]{zs76}, the lexicographic and reverse lexicographic orders have simple interpretations. Recall that if $X$ is a variety of dimension $n$, and $$F: X = Z_0 \supset Z_1 \supset Z_2 \supset \ldots \supset Z_n$$ is a flag of subvarieties, ${\rm codim}_X(Z_i) = i$, with $Z_i$ smooth at the generic point of $Z_{i+1}$, then we can define a rank $n$ valuation $v_F$ on $X$ as follows: For each $i = 1,\ldots, n-1$, fix $f_i$ to be a function on $Z_{i-1}$ with a $1\thuh{st}$ order zero on $Z_i$. Then for any function $f$, we can define $e_1 = {\rm ord}_{Z_1}(f)$, $e_2 = {\rm ord}_{Z_2}((f/f_1^{e_1})\!\mid_{Z_1})$, etc., and $v_F(f) = (e_1, \ldots, e_n) \in \mbox{\bf Z}^n$, where the value group $\mbox{\bf Z}^n$ is ordered lexicographically. The arbitrarily chosen $f_i$ are not needed to compare two functions $f$, $g$: We have $v_F(f) \succ v_F(g)$ if and only if ${\rm ord}_{Z_1}(f/g) > 0$, or if this order is zero and ${\rm ord}_{Z_2}((f/g)\!\mid_{Z_1}) > 0$, and so forth. Such a valuation also defines an order on each graded piece $S_d$ of the homogeneous coordinate ring: take any $f_0 \in S_d$ and say $f > g$ if and only if $v_F(f/f_0) \succ v_F(g/f_0)$. More generally, one may take the $Z_i$ to be subvarieties of a variety $X^\prime$ dominating $X$ and pull back functions to $X^\prime$ before computing $v_F$. The lexicographic order on monomials of each degree of $\proj n$ is now induced by the flag $$\proj n \supset V(x_0) \supset V(x_0,x_1) \supset \ldots \supset V(x_0,\ldots,x_{n-1}).$$ For example, the first step in the comparison defining $v_F(\sliver{\bf x}\sliver^A/f_0) \succ v_F(\sliver{\bf x}\sliver^B/f_0)$ has the effect of asking if $a_0 - b_0 > 0$. The reverse lexicographic order is induced by a flag on a blowup $X$ of $\proj n$: First blow up $V(x_0,\ldots,x_{n-1})$ and let $E_1$ be the exceptional divisor. Next blow up the proper transform of $V(x_0,\ldots,x_{n-2})$, and let $E_2$ be this exceptional divisor. Iterating, we can define a flag $$X \supset E_1 \supset E_1 \cap E_2 \supset \ldots \supset E_1 \cap \ldots \cap E_n$$ which induces the reverse lexicographic order on monomials in each degree. For example, looking at the affine piece of the first blow up obtained by substituting $x_0 = x_0^\prime x_{n-1}, \;\ldots, \;x_{n-2} = x_{n-2}^\prime x_{n-1}$, the power of $x_{n-1}$ in the transform of $\sliver{\bf x}\sliver^A$ is $a_0 + \ldots + a_{n-1}$, which is the order of vanishing of this monomial on $E_1$. Thus, the first step in the comparison defining $v_F(\sliver{\bf x}\sliver^A/f_0) \succ v_F(\sliver{\bf x}\sliver^B/f_0)$ has the effect of asking if $a_0 + \ldots + a_{n-1} - b_0 - \ldots - b_{n-1} > 0$, which is what we want. Taking into account the equivalence between multiplicative orders and one-parameter subgroups, the process we have described in $S[t]$ is exactly the usual algorithm for computing Gr\"{o}bner\ bases. It is computationally advantageous to set $t = 1$ and dismiss our extra structure as unnecessary scaffolding, but it is conceptually advantageous to treat our viewpoint as what is ``really'' going on; many techniques of algebraic geometry become applicable to the family $Y$, and assist in analyzing the complexity of Gr\"{o}bner\ bases. Moreover, this picture may help guide improvements to the basic algorithm. For example, for very large problems, it could be computationally more efficient to degenerate to $X_0$ in several stages; this has not been tried in practice. The coarsest measure of the complexity of a Gr\"{o}bner\ basis is its maximum degree, which is the highest degree of a generator of the ideal ${\rm in}(I)$ defining $X_0$. This quantity is bounded by the better-behaved {\em regularity\,} of ${\rm in}(I)$: The regularity of an ideal $I$ is the maximum over all $i$ of the degree minus $i$ of any minimal $i$\thuh{th} syzygy of $I$, treating generators as $0$\thuh{th} syzygies. When $I$ is the largest (the {\em saturated}) ideal defining a scheme $X$, we call this the regularity of $X$. We take up regularity in detail in \secref{bounds}; here it suffices to know that regularity is {\em upper semi-continuous\,} on flat families, i.e. the regularity can only stay the same or go up at special fibers. Let ${\rm reg}(I)$ denote the regularity of $I$, and ${\rm reg}_0(I)$ denote the highest degree of a generator of $I$. In our case, $t = 0$ is the only special fiber, and the above says that $${\rm reg}_0(I) \;\le\; {\rm reg}(I) \;\le\; {\rm reg}({\rm in}(I)) \;\ge\; {\rm reg}_0({\rm in}(I)),$$ where ${\rm reg}_0(I)$ can be immediately determined from the input data, and ${\rm reg}_0({\rm in}(I))$ is the degree-complexity of the Gr\"{o}bner\ basis computation. In practice, each of these inequalities are often strict. However when $k$ is infinite, then for any set of coordinates for $\proj n$ chosen from a dense open set $U \subset GL(n+1)$ of possibilities, Galligo (\cite{gal74}; see also \cite{bs87b}) has shown that the limiting ideal ${\rm in}(I)$ takes on a very special form: ${\rm in}(I)$ is invariant under the action of the Borel subgroup of upper triangular matrices in $GL(n+1)$. This imposes strong geometric conditions on $X_0$. In particular, the associated primes of ${\rm in}(I)$ are also Borel-fixed, so they are all of the form $(x_0, \ldots, x_i)$ for various $i$. This means that the components of $X_0$ are supported on members of a flag. In characteristic zero, it is shown in \cite{bs87a} that the regularity of a Borel-fixed ideal is exactly the maximum of the degrees of its generators, or in our notation, that ${\rm reg}({\rm in}(I)) = {\rm reg}_0({\rm in}(I))$ when ${\rm in}(I)$ is Borel-fixed. Thus, for generic coordinates in characteristic zero, the degree-complexity of computing Gr\"{o}bner\ bases breaks down into two effects: the gap ${\rm reg}_0(I) \le {\rm reg}(I)$ between the input degrees and the regularity of $X$, and the gap ${\rm reg}(I) \le {\rm reg}({\rm in}(I))$ allowed by upper-semicontinuity. A combination of theoretical results, hunches and experience guides the practitioner in assessing the first gap; what about the second? Does the regularity have to jump at all? One can easily find examples of ideals and total orders exhibiting such a jump, but in \cite{bs87a}, it is shown that for the reverse lexicographic order, in generic coordinates and any characteristic, there is no jump: ${\rm reg}(I) = {\rm reg}({\rm in}(I))$, so in characteristic zero we have $${\rm reg}_0({\rm in}(I)) = {\rm reg}(I).$$ In this sense, this order is an optimal choice: {\em For the reverse lexicographic order, the degree-complexity of a Gr\"{o}bner\ basis computation is exactly the regularity of the input data.} This agrees with experience; computations made on the same inputs using the lexicographic order can climb to much higher degrees than the reverse lexicographic order, in practice. For many applications, one is free to choose any order, but some problems restrict us to using orders satisfying combinatorial properties which the reverse lexicographic order fails to satisfy. An example, developed further in \secref{division}, is that of eliminating variables, or equivalently, of computing projections. To compute the intersection of $I$ with a subring $R = k[x_i,\ldots,x_n]$, it is necessary to use an order which in each degree sorts all monomials not in $R$ ahead of any monomial in $R$. The lexicographic order is an example of such an order, for each $i$ simultaneously. This strength comes at a cost; we are paying in regularity gaps for properties we may not need in a particular problem. An optimal order if you need one specific projection (in the same sense as above) is constructed by sorting monomials by total degree in the variables to be eliminated, and then breaking ties using the reverse lexicographic order. See \cite{bs87b} for this result, and a generalization to the problem of optimally refining any nonstrict order. Using this elimination order, one finds that the inherent degree-complexity of a computation is given not by the regularity of $X$ itself, but rather by the regularity of the {\em flat projection\,} $X^\prime$ of $X$, which is the central fiber of a flat family which animates the desired projection of $X$ as $t \rightarrow 0$. The jump in regularity between $X$ and $X^\prime$ is unavoidable; by choosing an optimal order, we avoid the penalty of a further jump in regularity between $X^\prime$ and $X_0$. The regularity of algebraic varieties or schemes $X$ is far from being well understood, but there is considerable interest in its study; this computational interpretation of regularity as the inherent degree-complexity of an ideal is but one more log on the fire. {}From a theoretical computer science perspective, the full complexity of computing Gr\"{o}bner\ bases is determined not merely by the highest degree ${\rm reg}_0(I)$ in the basis, but by the total number of arithmetic operations in the field $k$ required to compute this basis. This has not been analyzed in general, but for $0$-dimensional ideals $I$, Lakshman and Lazard (\cite{lak91}, \cite{ll91}) have shown that the complexity of computing reduced Gr\"{o}bner\ bases is bounded by a polynomial in $d^n$, where $d$ is the maximum degree of the generators, and $n$ is the number of variables. \section{Gr\"{o}bner\ Bases} \label{division} Let $S = k[x_0, \ldots, x_n]$ be a graded polynomial ring over the field $k$, and let $I \subset S$ be a homogeneous ideal. Let $S_d$ denote the finite vector space of all homogeneous, degree d polynomials in $S$, so $S = S_0 \oplus S_1 \oplus \ldots \oplus S_d \oplus \ldots $. Writing $I$ in the same manner as $I = I_0 \oplus I_1 \oplus \ldots \oplus I_d \oplus \ldots $, we have $I_d \subset S_d$ for each $d$. Recall that the Hilbert function of $I$ is defined to be the function $p(d) = \dim(I_d)$, for $d \ge 0$. A total order $>$ on the monomials of $S$ is said to be {\it multiplicative\,} if whenever $\sliver{\bf x}\sliver^A > \sliver{\bf x}\sliver^B$ for two monomials $\sliver{\bf x}\sliver^A$, $\sliver{\bf x}\sliver^B$, then $\sliver{\bf x}\sliver^C \sliver{\bf x}\sliver^A > \sliver{\bf x}\sliver^C \sliver{\bf x}\sliver^B$ for all monomials $\sliver{\bf x}\sliver^C$. This condition insures that if the terms of a polynomial are in order with respect to $>$, then they remain in order after multiplication by a monomial. \begin{defn} \label{id1} Let $>$ be a multiplicative order. For a homogeneous polynomial $f = c_1 \sliver{\bf x}\sliver^{A_1} + \ldots + c_m \sliver{\bf x}\sliver^{A_m}$ with $\sliver{\bf x}\sliver^{A_1} > \ldots > \sliver{\bf x}\sliver^{A_m}$, define the initial term ${\rm in}(f)$ to be the lead (that is, the largest) term $c_1 \sliver{\bf x}\sliver^{A_1}$ of $f$. For a homogeneous ideal $I \subset S$, define the initial ideal ${\rm in}(I)$ to be the monomial ideal generated by the lead terms of all elements of $I$. \end{defn} Note that the definitions of ${\rm in}(f)$ and ${\rm in}(I)$ depend on the choice of multiplicative order $>$. See \cite{bm88} and \cite{mr88} for characterizations of the finite set of ${\rm in}(I)$ realized as the order $>$ varies. Fix a multiplicative order $>$ on $S$. \begin{prop}[Macaulay] \label{id2} I and {\rm in}(I) have the same Hilbert function. \end{prop} \begin{proof} (\cite{mac27}) The lead terms of $I_d$ span ${\rm in}(I)_d$, because every monomial $\sliver{\bf x}\sliver^A \in {\rm in}(I)$ is itself the lead term ${\rm in}(f)$ of some polynomial $f \in I$: Since $\sliver{\bf x}\sliver^A = \sliver{\bf x}\sliver^C \sliver{\bf x}\sliver^B$ for some $\sliver{\bf x}\sliver^B = {\rm in}(g)$ with $g \in I$, we have $\sliver{\bf x}\sliver^A = {\rm in}(f)$ for $f = \sliver{\bf x}\sliver^C g$. Choose a $k$-basis $B_d \subset I_d$ with distinct lead terms, and let ${\rm in}(B_d)$ be the set of lead terms of $B_d$; ${\rm in}(B_d)$ has cardinality $p(d) = \dim(I_d)$. Since any element of $I_d$ is a linear combination of elements of $B_d$, any lead term of $I_d$ is a scalar multiple of an element of ${\rm in}(B_d)$. Thus, ${\rm in}(B_d)$ is a basis for ${\rm in}(I)_d$, so $p(d) = \dim({\rm in}(I)_d).$ \end{proof} One can compute the Hilbert function of $I$ by finding ${\rm in}(I)$ and applying this result; see \cite{mm83}, \cite{bcr91}, and \cite{bs92}. \begin{corollary} \label{id3} The monomials of S which don't belong to ${\rm in}(I)$ form a $k$-basis for $S /I$. \end{corollary} \begin{proof} These monomials are linearly independent in $S / I$, because any linear relation among them is a polynomial belonging to $I$, and all such polynomials have lead terms belonging to ${\rm in}(I)$. These monomials can be seen to span $S / I$ by a dimension count, applying \propref{id2}. \end{proof} Two examples of multiplicative orders are the lexicographic order and the reverse lexicographic order. $\sliver{\bf x}\sliver^A > \sliver{\bf x}\sliver^B$ in the lexicographic order if the first nonzero coordinate of $A-B$ is positive. For example, if $S = k[w, x, y, z]$, then $w > x > y > z$ in $S_1$, and $$w^2 > wx > wy > wz > x^2 > xy > xz > y^2 > yz > z^2$$ in $S^2$. $\sliver{\bf x}\sliver^A > \sliver{\bf x}\sliver^B$ in the reverse lexicographic order if the last nonzero coordinate of $A-B$ is negative. For example, if $S = k[w, x, y, z]$, then $w > x > y > z$ in $S_1$, and $$w^2 > wx > x^2 > wy > xy > y^2 > wz > xz > yz > z^2$$ in $S^2$. These two orders agree on $S_1$, but differ on the monomials of $S$ of degree $> 1$ when $n \ge 2$. The lexicographic order has the property that for each subring $k[x_i, \ldots, x_n] \subset S$ and each polynomial $f \in S$, $f \in k[x_i, \ldots, x_n]$ if and only if ${\rm in}(f) \in k[x_i, \ldots, x_n]$. The reverse lexicographic order has the property that for each $f \in k[x_0, \ldots, x_i]$, $x_i$ divides $f$ if and only if $x_i$ divides ${\rm in}(f)$. One can anticipate the applications of these properties by considering a $k$-basis $B_d \subset I_d$ with distinct lead terms, as in the proof of \propref{id2}. With respect to the lexicographic order, $B_d \cap k[x_i, \ldots, x_n]$ is then a $k$-basis for $I_d \cap k[x_i, \ldots, x_n]$ for each $i$. With respect to the reverse lexicographic order, $B_d \cap (x_n)$ is then a $k$-basis for $I_d \cap (x_n)$. Thus, these orders enable us to find polynomials in an ideal which do not involve certain variables, or which are divisible by a certain variable. For a given degree $d$, one could construct such a basis $B_d$ by applying Gaussian elimination to an arbitrary $k$-basis for $I_d$. However, this cannot be done for all $d$ at once; such a computation would be infinite. We will finesse this difficulty by instead constructing a finite set of elements of I whose monomial multiples yield polynomials in $I$ with every possible lead term. Such sets can be described as follows: \begin{defn} \label{id4} A list $F = [f_1, \ldots, f_r] \subset I$ is a (minimal) Gr\"{o}bner\ basis for $I$ if ${\rm in}(f_1), \ldots, {\rm in}(f_r)$ (minimally) generate ${\rm in}(I)$. \end{defn} ${\rm in}(I)$ is finitely generated because $S$ is Noetherian, so Gr\"{o}bner\ bases exist for any ideal I. The order of the elements of $F$ is immaterial to this definition, so $F$ can be thought of as a set. We are using list notation for $F$ because we are going to consider algorithms for which the order of the elements is significant. For convenience, we shall extend the notation of set intersections and containments to the lists $F$. A minimal set of generators for an ideal $I$ need not form a Gr\"{o}bner\ basis for I. For example, if $S = k[x, y]$ and $I = (x^2 + y^2, xy)$, then with respect to the lexicographic order, ${\rm in}(x^2 + y^2) = x^2$ and ${\rm in}(xy) = xy$. Yet $y(x^2 + y^2) - x(xy) = y^3 \in I$, so $y^3 \in {\rm in}(I)$. Thus, any Gr\"{o}bner\ basis for $I$ must include $y^3$; it can be shown that ${\rm in}(I) = (x^2, xy, y^3)$ and $[x^2 + y^2, xy, y^3]$ is a Gr\"{o}bner\ basis for $I$. On the other hand, \begin{lemma} \label{id5} If $F = [f_1, \ldots, f_r]$ is a Gr\"{o}bner\ basis for $I$, then $f_1, \ldots, f_r$ generate $I$. \end{lemma} \begin{proof} For each degree $d$, we can construct a $k$-basis $B_d \subset I_d$ with distinct lead terms, whose elements are monomial multiples of $f_1, \ldots, f_r$: For each $\sliver{\bf x}\sliver^A \in {\rm in}(I)_d$, $\sliver{\bf x}\sliver^A$ is a scalar multiple of $\sliver{\bf x}\sliver^C {\rm in}(f_i)$ for some $\sliver{\bf x}\sliver^C$ and some $i$; include $\sliver{\bf x}\sliver^C f_i$ in the set $B_d$. Thus, the monomial multiples of $f_1, \ldots, f_r$ span $I$. \end{proof} \begin{prop}[Spear, Trinks] \label{id6} Let $R \subset S$ be the subring $R = k[x_i, \ldots, x_n]$. If $F = [f_1, \ldots, f_r ]$ is a Gr\"{o}bner\ basis for the ideal $I$ with respect to the lexicographic order, then $F \cap R$ is a Gr\"{o}bner\ basis for the ideal $I \cap R$. In particular, $F \cap R$ generates $I \cap R$. \end{prop} \begin{proof} (\cite{spe77}, \cite{zac78}, \cite{tri78}) Let $f \in I \cap R$; ${\rm in}(f)$ is a multiple of ${\rm in}(f_i)$ for some $i$. Since ${\rm in}(f) \in R$, ${\rm in}(f_i) \in R$, so $f_i \in R$. Thus, $F \cap R$ is a Gr\"{o}bner\ basis for $I \cap R$. By \lemref{id5}, $F \cap R$ generates $I \cap R$. \end{proof} \propref{id6} has the following geometric application: If $I$ defines the subscheme $X \subset \mbox{\bf P}^n$, then $I \cap k[x_i, \ldots, x_n]$ defines the projection of $X$ to $\mbox{\bf P}^{n-i} = {\rm Proj}(k[x_i, \ldots, x_n])$. Recall that the saturation $\sat I$ of $I$ is defined to be the largest ideal defining the same subscheme $X \subset \mbox{\bf P}^n$ as $I$. $\sat I$ can be obtained by taking an irredundant primary decomposition for $I$, and removing the primary ideal whose associated prime is the irrelevant ideal $(x_0, \ldots, x_n)$. $I$ is saturated if $I = \sat I$. If the ideal $I$ is saturated, and defines a finite set of points $X \subset \mbox{\bf P}^n$, then $I \cap k[x_{n-1}, x_n]$ is a principal ideal $(f)$, where $\{f=0\}$ is the image of the projection of $X$ to $\mbox{\bf P}^1 = {\rm Proj}(k[x_{n-1}, x_n])$. Given a linear factor of $f$ of the form $(b x_{n-1} - a x_n)$, we can make the substitution $x_{n-1} = a z$, $x_n = b z$ for a new variable $z$, to obtain from $I$ an ideal $J \subset k[x_0, \ldots, x_{n-2}, z]$ defining a finite set of points in $\mbox{\bf P}^{n-1}$. For each point $(c_0, \ldots, c_{n-2}, d)$ in the zero locus of $J$, $(c_0, \ldots, c_{n-2}, ad, bd)$ is a point in the zero locus of $I$. If $X \subset \mbox{\bf P}^{n-1}$ is of dimension $1$ or greater, then in general $I \cap k[x_{n-1}, x_n] = (0)$, because a generic projection of $X$ to $\mbox{\bf P}^1$ is surjective. In this case, an arbitrary substitution $x_{n-1} = a z$, $x_n = b z$ can be made, and the process of projecting to $\mbox{\bf P}^1$ iterated. Thus, the lexicographic order can be used to find solutions to systems of polynomial equations. Recall that the ideal quotient $(I : f)$ is defined to be the ideal $\setdef{g \in S}{f g \in I}$. Since $S$ is Noetherian, the ascending chain of ideals $(I : f) \subset (I : f^2) \subset (I : f^3) \subset \ldots $ is stationary; call this stationary limit $(I : f^\infty) = \setdef{ g \in S }{ f^m g \in I \mbox{ for some } m }$. \begin{prop} \label{id7} If $[x_n^{a_1} f_1, \ldots, x_n^{a_r} f_r]$ is a Gr\"{o}bner\ basis for the ideal $I$ with respect to the reverse lexicographic order, and if none of $f_1, \ldots, f_r$ are divisible by $x_n$, then $F = [f_1, \ldots, f_r]$ is a Gr\"{o}bner\ basis for the ideal $(I : x_n^\infty)$. In particular, $f_1, \ldots, f_r$ generate $(I : x_n^\infty)$. \end{prop} \begin{proof} (\cite{bay82}, \cite{bs87a}) We have $F \subset (I : x_n^\infty)$. Let $f \in (I : x_n^\infty)$; $x_n^m f \in I$ for some $m$, so ${\rm in}(x_n^m f)$ is a multiple of ${\rm in}(x_n^{a_i} f_i)$ for some $i$. Since $f_i$ is not divisible by $x_n$, ${\rm in}(f_i)$ is not divisible by $x_n$, so ${\rm in}(f)$ is a multiple of ${\rm in}(f_i)$. Thus, $F$ is a Gr\"{o}bner\ basis for $(I : x_n^\infty)$. By \lemref{id5}, $f_1, \ldots, f_r$ generate $(I : x_n^\infty)$. \end{proof} If $I = \sliver{\bf q}\sliver_0 \cap \sliver{\bf q}\sliver_1 \cap \ldots \cap \sliver{\bf q}\sliver_t$ is a primary decomposition of $I$, then $(I : x_n^\infty) = ( \cap \sliver{\bf q}\sliver_i : x_n^\infty) = \cap (\sliver{\bf q}\sliver_i : x_n^\infty)$. We have $(\sliver{\bf q}\sliver_i : x_n^\infty) = (1)$ if the associated prime $\sliver{\bf p}\sliver_i$ of $\sliver{\bf q}\sliver_i$ contains $x_n$, and $(\sliver{\bf q}\sliver_i : x_n^\infty) = \sliver{\bf q}\sliver_i$ otherwise. Thus, if $I$ defines the subscheme $X \subset \mbox{\bf P}^n$, then $(I : x_n^\infty)$ defines the subscheme consisting of those primary components of $X$ not supported on the hyperplane $\{x_n = 0\}$. $(I : x_n^\infty)$ is saturated, because it cannot have $(x_0, \ldots, x_n)$ as an associated prime. If $x_n$ belongs to none of the associated primes of $I$ except $(x_0, \ldots, x_n)$, or equivalently if $\{x_n = 0\}$ is a generic hyperplane section of $X \subset \mbox{\bf P}^n$, then $(I : x_n^\infty) = \sat I$. Thus, the reverse lexicographic order can be used to find the saturation of $I$. One of the most important uses of Gr\"{o}bner\ bases is that they lead to canonical representations of polynomials modulo an ideal $I$, i.e. a division algorithm in which every $f \in S$ is written canonically as $f = \sum g_i f_i + h$, where $[f_1, \ldots, f_r]$ is a Gr\"{o}bner\ basis for $I$, and $h$ is the remainder after division. Recall the division algorithm for inhomogeneous, univariate polynomials $f(x)$, $g(x) \in k[x]$: Let ${\rm in}(f)$ denote the highest degree term of $f$. The remainder of $g$ under division by $f$ can be recursively defined by $$R_f(g) = R_f(g -c x^a f)$$ if ${\rm in}(f)$ divides ${\rm in}(g)$, where $c x^a = {\rm in}(g) / {\rm in}(f)$, and by $$R_f(g) = g$$ otherwise. Division can be generalized to homogeneous polynomials $f_1, \ldots, f_r, g \in S$, given a multiplicative order on $S$ (\cite{hir64}, \cite{bri73}, \cite{gal74}, \cite{sch80}): The remainder $R_F (g)$ of $g$ under division by the list of polynomials $F = [f_1, \ldots, f_r]$ can be recursively defined by $$R_F(g) = R_F(g - c \sliver{\bf x}\sliver^A f_i)$$ for the least $i$ so ${\rm in}(g)$ is a multiple $c \sliver{\bf x}\sliver^A$ of ${\rm in}(f_i)$, and by $$R_F(g) = {\rm in}(g) + R_F(g - {\rm in}(g))$$ if ${\rm in}(g)$ is not a multiple of any ${\rm in}(f_i)$. $R_F(g)$ is an element of $S$. Thus, the fate of ${\rm in}(g)$ depends on whether or not ${\rm in}(g) \in ({\rm in}(f _1), \ldots, {\rm in}(f_r))$. Let $I$ be the ideal generated by $f_1, \ldots, f_r$. If $F = [f_1, \ldots, f_r]$ fails to be a Gr\"{o}bner\ basis for $I$, then the remainder is poorly behaved. For example, with respect to the lexicographic order on $k[x, y]$, $$R_{[xy, x^2 + y^2 ]}(x^2 y) = x^2 y - x(xy) = 0,$$ but $$R_{[x^2 + y^2, xy]}(x^2 y) = x^2 y - y(x^2 + y^2) = -y^3,$$ so the remainder $R_F(g)$ is dependent on the order of the list $F$. Note that $x^2 y \in (x^2 + y^2, xy)$. If on the other hand, $F$ is a Gr\"{o}bner\ basis for the ideal $I$, then $R_F(g)$ is a $k$-linear combination of monomials not belonging to ${\rm in}(I)$. By \corref{id3}, these monomials form a $k$-basis for $S / I$, so each polynomial in $S$ has a unique representation in terms of this $k$-basis, modulo the ideal $I$. The remainder gives this unique representation, and is independent of the order of $F$ (but dependent on the multiplicative order chosen for the monomials of $S$). In particular, $R_F(g) = 0$ if and only if $g \in I$. An algorithm for computing a Gr\"{o}bner\ basis for $I$ from a set of generators for $I$ was first given by Buchberger (\cite{buc65}, \cite{buc76}). This algorithm was discovered independently by Spear (\cite{spe77}, \cite{zac78}), Bergman \cite{ber78}, and Schreyer \cite{sch80}. It was termed the division algorithm by Schreyer, after the division theorem of Hironaka (\cite{hir64}, \cite{bri73}, \cite{gal74}). Define $S(f_i, f_j)$ for $i < j$ by $$S(f_i, f_j) = b \sliver{\bf x}\sliver^B f_i - c \sliver{\bf x}\sliver^C f_j,$$ where $\sliver{\bf x}\sliver^A = b \sliver{\bf x}\sliver^B {\rm in}(f_i) = c \sliver{\bf x}\sliver^C {\rm in}(f_j)$ is the least common multiple of ${\rm in}(f_i)$ and ${\rm in}(f_j)$. $b \sliver{\bf x}\sliver^B f_i$ and $c \sliver{\bf x}\sliver^C f_j$ each have $\sliver{\bf x}\sliver^A$ as lead term, so $\sliver{\bf x}\sliver^A$ cancels out in $S(f_i, f_j)$, and $\sliver{\bf x}\sliver^A > {\rm in}(S(f_i, f_j))$. If $F$ is a Gr\"{o}bner\ basis for the ideal $I$, then $R_F(S(f_i, f_j)) = 0$ for each $i < j$, since $S(f_i, f_j) \in I$. Conversely, \begin{prop}[Buchberger] \label{id8} If $R_F(S(f_i, f_j)) = 0$ for each $i < j$, then $F = [f_1, \ldots, f_r]$ is a Gr\"{o}bner\ basis for the ideal $I = (f_1, \ldots, f_r)$. \end{prop} See \cite{buc65}, \cite{buc76}. We postpone a proof until the theory has been extended to $S$-modules. This result can also be thought of as an explicit converse to the assertion that if $F$ is a Gr\"{o}bner\ basis, then division is independent of the order of $F$: Whenever we have a choice in division between subtracting off a multiple of $f_i$ and a multiple of $f_j$, the difference is a multiple of $S(f_i, f_j)$. If division is independent of the order of $F$, then these differences must have remainder zero, so by \propref{id8}, $F$ is a Gr\"{o}bner\ basis. As sketched in \secref{geometry}, \propref{id8} can be used to compute a Gr\"{o}bner\ basis from a set of generators $f_1, \ldots, f_r$ for the ideal $I$: For each $i < j$ so $f_{r+1} = R_F(S(f_i, f_j)) \ne 0$, adjoin $f_{r+1}$ to the list $F = [f_1, \ldots, f_r]$. Note that $f_{r+1} \in I$. By iterating until no new polynomials are found, a Gr\"{o}bner\ basis $F$ is obtained for $I$. This process terminates because $S$ is Noetherian, and each new basis element corresponds to a monomial not in the ideal generated by the preceding lead terms. We now extend this theory to $S$-modules. Let $M$ be a graded, finitely generated $S$-module, given by the exact sequence of graded $S$-modules $$M_1 \stackrel{F}{\longrightarrow} M_0 \longrightarrow M \longrightarrow {\bf 0},$$ where $M_0 = S e_{01} \oplus \ldots \oplus S e_{0q}$ and $M_1 = S e_{11} \oplus \ldots \oplus S e_{1r}$ are free $S$-modules with $\deg(e_{ij}) = d_{ij}$ for each $i$, $j$. We now think of $F$ both as a list $[f_1, \ldots, f_r]$ of module elements, and as a map between free modules: Let $f_i = F(e_{1i}) \ne 0$ for $i =1, \ldots, r$, and let $I \subset M_0$ be the homogeneous submodule generated by $f_1, \ldots, f_r$. Thus, $M = M_0 / I$. A monomial of $M_0$ is an element of the form $\sliver{\bf x}\sliver^A e_{0i}$; such an element has degree $\deg(\sliver{\bf x}\sliver^A) + d_{0i}$. An order on the monomials of $M_0$ is multiplicative if whenever $\sliver{\bf x}\sliver^A e_{0i} > \sliver{\bf x}\sliver^B e_{0j}$, then $\sliver{\bf x}\sliver^C \sliver{\bf x}\sliver^A e_{0i} > \sliver{\bf x}\sliver^C \sliver{\bf x}\sliver^B e_{0j}$ for all $\sliver{\bf x}\sliver^C \in S$. For some applications, such as developing a theory of Gr\"{o}bner\ bases over quotients of $S$, one wants this order to be compatible with an order on $S$: If $\sliver{\bf x}\sliver^A > \sliver{\bf x}\sliver^B$, then one wants $\sliver{\bf x}\sliver^A e_{0i} > \sliver{\bf x}\sliver^B e_{0i}$ for $i = 1, \ldots, r$. The orders encountered in practice invariably satisfy this second condition, but it does not follow from the first, and we do not require it here. One way to extend a multiplicative order on $S$ to a compatible multiplicative order on $M_0$ is to declare $\sliver{\bf x}\sliver^A e_{0i} > \sliver{\bf x}\sliver^B e_{0j}$ if $i < j$, or if $i = j$ and $\sliver{\bf x}\sliver^A > \sliver{\bf x}\sliver^B$. Another way is to assign monomials $\sliver{\bf x}\sliver^{C_1}, \ldots, \sliver{\bf x}\sliver^{C_q}$ in $S$ to the basis elements $e_{01}, \ldots, e_{0q}$ of $M_0$, and to declare $\sliver{\bf x}\sliver^A e_{0i} > \sliver{\bf x}\sliver^B e_{0j}$ if $\sliver{\bf x}\sliver^{A+C_i} > \sliver{\bf x}\sliver^{B+C_j}$, or if $A+C_i = B+C_j$ and $i < j$. Fix a choice of a multiplicative order $>$ on $M_0$. The constructions developed for $S$ carry over intact to $M_0$, with the same proofs (\cite{gal79}, \cite{sch80}, \cite{bay82}): Given an element $f \in M_0$, define ${\rm in}(f)$ to be the lead term of $f$. Define ${\rm in}(I)$ to be the submodule generated by the lead terms of all elements of $I \subset M_0$; ${\rm in}(I)$ is a monomial submodule of $M_0$ with the same Hilbert function as $I$. Define $F = [f_1, \ldots, f_r] \subset I$ to be a Gr\"{o}bner\ basis for $I$ if ${\rm in}(f_1), \ldots, {\rm in}(f_r)$ generate ${\rm in}(I)$; a set of generators for $I$ need not be a Gr\"{o}bner\ basis for $I$, but a Gr\"{o}bner\ basis for $I$ generates $I$. Given an element $g \in M_0$, define $R_F(g) \in M_0$ exactly as was done for the free module $S$. If $F$ is a Gr\"{o}bner\ basis for $I$, then $R_F(g) = 0$ if and only if $g \in I$. The quotient of $g$ under division by $f_1, \ldots, f_r$ can be recursively defined by $$Q_F(g) = c \sliver{\bf x}\sliver^A e_{1i} + Q_F(g - c \sliver{\bf x}\sliver^A f_i)$$ for the least $i$ so ${\rm in}(g)$ is a multiple $c \sliver{\bf x}\sliver^A$ of ${\rm in}(f_i)$, and by $$Q_F(g) = Q_F(g - {\rm in}(g))$$ if ${\rm in}(g)$ is not a multiple of any ${\rm in}(f_i)$. The quotient is an element of $M_1$. Following the recursive definitions of the remainder and quotient, it can be inductively verified that $$g = F(Q_F(g)) + R_F(g).$$ If $F$ is a Gr\"{o}bner\ basis for $I$, and $g \in I$, then $R_F(g) = 0$, so the quotient lifts $g$ to $M_1$. In this case, the quotient can be thought of as expressing $g$ in terms of $f_1, \ldots, f_r$. Define $S(f_i, f_j)$ for $i < j$ by $$S(f_i, f_j) = b \sliver{\bf x}\sliver^B f_i - c \sliver{\bf x}\sliver^C f_j,$$ if ${\rm in}(f_i)$ and ${\rm in}(f_j)$ have a least common multiple $\sliver{\bf x}\sliver^A e_{0k} = b \sliver{\bf x}\sliver^B {\rm in}(f_i) = c \sliver{\bf x}\sliver^C {\rm in}(f_j)$. Leave $S(f_i, f_j)$ undefined if ${\rm in}(f_i)$ and ${\rm in}(f_j)$ lie in different summands of $M_0$, and so don't have common multiples. Recall that the module of syzygies of $f_1, \ldots, f_r$ is defined to be the kernel of the map $F$, which is the submodule of $M_1$ consisting of all $h \in M_1$ so $F(h) = 0$. Thus, if $h = h_1 e_{11} + \ldots + h_r e_{1r}$ is a syzygy, then $h_1 f_1 + \ldots + h_r f_r = 0$. Let $J \subset M_1$ denote the module of syzygies of $f_1, \ldots, f_r$, and let $K \subset M_1$ denote the module of syzygies of ${\rm in}(f_1), \ldots, {\rm in}(f_r)$. Define the map ${\rm in}(F): M_1 \rightarrow M_0$ by ${\rm in}(F)(e_{1i}) = {\rm in}(f_i)$; $K$ is the kernel of ${\rm in}(F)$. For each $i < j$ so $S(f_i, f_j)$ is defined, define $t_{ij}$ to be the element $$t_{ij} = b \sliver{\bf x}\sliver^B e_{1i} - c \sliver{\bf x}\sliver^C e_{1j} \in M_1,$$ where $\sliver{\bf x}\sliver^A e_{0k} = b \sliver{\bf x}\sliver^B {\rm in}(f_i) = c \sliver{\bf x}\sliver^C {\rm in}(f_j)$ is the least common multiple of ${\rm in}(f_i)$ and ${\rm in}(f_j)$, as before. ${\rm in}(F)(t_{ij}) = 0$, so each $t_{ij}$ belongs to the syzygy module $K$. Observe that $F(t_{ij}) = S(f_i, f_j)$. Assign the following multiplicative order on $M_1$, starting from the order on $M_0$ (\cite{sch80}; see also \cite{mm86}): Let $\sliver{\bf x}\sliver^A e_{1i} > \sliver{\bf x}\sliver^B e_{1j}$ if $\sliver{\bf x}\sliver^A {\rm in}(f_i) > \sliver{\bf x}\sliver^B {\rm in}(f_j)$, or if these terms are $k$-multiples of each other and $i < j$. If the order on $M_0$ is compatible with an order on $S$, then this order on $M_1$ is compatible with the same order on $S$. With respect to this order on $M_1$, we have \begin{lemma} \label{tlem} The list $[\,t_{ij}\,]$ is a Gr\"{o}bner\ basis for the module $K$ of syzygies of ${\rm in}(f_1), \ldots, {\rm in}(f_r)$. \end{lemma} \begin{proof} Let $h \in M_1$, so ${\rm in}(F)(h) = 0$. Then ${\rm in}(F)({\rm in}(h))$ is canceled by ${\rm in}(F)(h - {\rm in}(h))$ in $M_0$. Therefore, if ${\rm in}(h) = \sliver{\bf x}\sliver^A e_{1i}$, then $h$ has another term $\sliver{\bf x}\sliver^B e_{1j}$ so $\sliver{\bf x}\sliver^A {\rm in}(f_i)$ and $\sliver{\bf x}\sliver^B {\rm in}(f_j)$ are $k$-multiples of each other and $i < j$. Thus, $t_{ij}$ is defined and ${\rm in}(t_{ij})$ divides ${\rm in}(h)$, so $[\,t_{ij}\,]$ is a Gr\"{o}bner\ basis for $K$. \end{proof} Thus, the set $\{t_{ij}\}$ generates $K$. In general, the $[\,t_{ij}\,]$ are far from being a minimal Gr\"{o}bner\ basis for $K$; we consider the effects of trimming this list in \propref{id9} below. Define $$s_{ij} = t_{ij} - Q_F(S(f_i, f_j))$$ whenever $R_F(S(f_i, f_j)) = 0$. Note that ${\rm in}(s_{ij}) = {\rm in}(t_{ij})$. Each $s_{ij}$ is the difference of two distinct elements of $M_1$, each of which is mapped by $F$ to $S(f_i, f_j)$, so $F(s_{ij}) = 0$. In other words, $s_{ij}$ belongs to the syzygy module $J$. Conversely, \begin{prop}[Richman, Spear, Schreyer] \label{id9} Choose a set of pairs $T = \{(i, j)\}$ such that the set $\{t_{ij}\}_{(i, j) \in T}$ generates the module $K$ of syzygies of ${\rm in}(f_1), \ldots, {\rm in}(f_r)$. If $R_F(S(f_i, f_j)) = 0$ for each $(i, j)\in T$, then (a) $F = [f_1, \ldots, f_r]$ is a Gr\"{o}bner\ basis for $I$; (b) the set $\{s_{ij}\}_{(i, j) \in T}$ generates the module $J$ of syzygies of $f_1, \ldots, f_r$. \noindent Moreover, (c) if $[\,t_{ij}\,]_{(i,j)\in T}$ is a Gr\"{o}bner\ basis for $K$, then $[\,s_{ij}\,]_{(i,j)\in T}$ is a Gr\"{o}bner\ basis for $J$. \end{prop} \begin{proof} (\cite{ric74}, \cite{spe77}, \cite{zac78}, \cite{sch80}) First, suppose that $[\,t_{ij}\,]_{(i,j) \in T}$ is a Gr\"{o}bner\ basis for $K$. Let $h \in J$, so $F(h) = 0$. By the same reasoning as in the proof of \lemref{tlem}, we can find $(i,j) \in T$ so ${\rm in}(t_{ij})$ divides ${\rm in}(h)$. Since ${\rm in}(s_{ij}) = {\rm in}(t_{ij})$, ${\rm in}(s_{ij})$ also divides ${\rm in}(h)$, so $[\,s_{ij}\,]_{(i,j) \in T}$ is a Gr\"{o}bner\ basis for $J$, proving (c). Now, suppose that $\{t_{ij}\}_{(i,j) \in T}$ merely generates $K$. Let $T^\prime$ be a set of pairs so $[\,t_{\ell m}\,]_{(\ell,m) \in T^\prime}$ is a Gr\"{o}bner\ basis for $K$. It is enough to construct a list $[\,u_{\ell m}\,]_{(\ell,m) \in T^\prime}$ of elements of $J$, generated by $\{s_{ij}\}_{(i,j) \in T}$, so ${\rm in}(u_{\ell m}) = {\rm in}(t_{\ell m})$ for all $(\ell,m) \in T^\prime$. Then by the preceding argument, $[\,u_{\ell m}\,]_{(\ell,m) \in T^\prime}$ is a Gr\"{o}bner\ basis for $J$, so $\{s_{ij}\}_{(i,j) \in T}$ generates $J$. Write each $t_{\ell m} = \sum g_{\ell m i j} t_{ij}$, for $(\ell,m) \in T^\prime$ and $(i,j) \in T$, in such a way that the terms of $t_{\ell m}$ and each term of each product $g_{\ell m i j} t_{ij}$ map via ${\rm in}(F)$ to multiples of the same monomial in $M_0$. In other words, find a minimal expression for each $t_{\ell m}$, which avoids unnecessary cancellation. Then define $$u_{\ell m} = \sum g_{\ell m i j} s_{ij}.$$ We have ${\rm in}(u_{\ell m}) = {\rm in}(t_{\ell m})$, proving (b). Let $f \in I$, and choose $g \in M_1$ so $f = F(g)$. Let $h \in M_1$ be the remainder of $g$ under division by $[\,u_{\ell m}\,]_{(\ell,m) \in T^\prime}$; $f = F(h)$. Since ${\rm in}(h)$ is not a multiple of any ${\rm in}(u_{\ell m}) = {\rm in}(t_{\ell m})$, the lead term of $F({\rm in}(h))$ is not canceled by any term of $F(h - {\rm in}(h))$. Therefore, if ${\rm in}(h) = a \sliver{\bf x}\sliver^A e_{1i}$, then ${\rm in}(f_i)$ divides ${\rm in}(F)$. Thus, $F = [f_1, \ldots, f_r]$ is a Gr\"{o}bner\ basis for $I$, proving (a). \end{proof} \propref{id8} follows as a special case of this result. The above proof can be understood in terms of an intermediate initial form ${\rm in}_0(h)$ for $h \in M_1$: Apply the map ${\rm in}(F)$ separately to each term of $h$, and let $\sliver{\bf x}\sliver^A \in M_0$ be the greatest monomial that occurs in the set of image terms. Define ${\rm in}_0(h)$ to be the sum of all terms of $h$ which map via ${\rm in}(F)$ to multiples of $\sliver{\bf x}\sliver^A$. Then ${\rm in}$ refines ${\rm in}_0$, for according to the order we have defined on $M_1$, ${\rm in}(h)$ is the term of ${\rm in}_0(h)$ lying in the summand of $M_1$ whose basis element $e_i$ has the smallest index $i$. In this language, $t_{ij} = {\rm in}_0(t_{ij}) = {\rm in}_0(s_{ij})$. Our expressions for the $t_{\ell m}$ have the property that each $g_{\ell m i j} t_{ij} = {\rm in}_0(g_{\ell m i j} t_{ij})$, with each term of each product for a given $t_{\ell m}$ mapping via ${\rm in}(F)$ to multiples of the same monomial $\sliver{\bf x}\sliver^A$. Thus, each ${\rm in}_0(g_{\ell m i j} s_{ij}) = g_{\ell m i j} t_{ij}$; the tails $g_{\ell m i j} (s_{ij} - {\rm in}_0(s_{ij}))$ stay out of our way, mapping termwise via ${\rm in}(F)$ to monomials which are less than $\sliver{\bf x}\sliver^A$ with respect to the order on $M_0$. Observe that $Q_F(g)$ is a linear combination of monomials not belonging to ${\rm in}(J)$, for any $g \in M_0$. In \cite{buc79}, Buchberger gives a criterion for selecting a set $T$ of pairs $(i, j)$ in the case where $I$ is an ideal: If $(i_0, i_1), (i_1, i_2), \ldots, (i_{s-1}, i_s) \in T$, and the least common multiple of ${\rm in}(f_{i_0}), {\rm in}(f_{i_1}), \ldots, {\rm in}(f_{i_s})$ is equal to the least common multiple of ${\rm in}(f_{i_0})$ and ${\rm in}(f_{i_s})$, then $(i_0, i_s)$ need not belong to $T$. In other words, if $t_{i_0 i_s} \in (t_{i_0 i_1}, \ldots, t_{i_{s-1} i_s})$, then the pair $(i_0, i_s)$ is unnecessary; this condition is equivalent to the condition of \propref{id9}, for the case of an ideal. Suppose that we wish to compute the syzygies of a given set of elements $g_1, \ldots, g_s$ of $M_0$. To do this, compute a Gr\"{o}bner\ basis $f_1, \ldots, f_r$ for the submodule $I \subset M_0$ generated by $g_1, \ldots, g_s$. Keep track of how to write each $f_i$ in terms of $g_1, \ldots, g_s$. Using these expressions, each syzygy of $f_1, \ldots, f_r$ can be mapped to a syzygy of $g_1, \ldots, g_s$. These images generate the module of syzygies of $g_1, \ldots, g_s$; the set of syzygies obtained in this way is not in general minimal. Syzygies can be used to find a minimal set of generators for a submodule $I \subset M_0$ from a given set of generators $g_1, \ldots, g_s$: If $h_1 g_1 + \ldots + h_r g_r = 0$ is a syzygy of $g_1, \ldots, g_s$ with $h_1 \in k$, then $g_1 = (h_2 g_2 + \ldots + h_r g_r) / h_1$, so $g_1$ is not needed to generate $I$. All unnecessary generators can be removed in this way. Alternatively, a careful implementation of Gr\"{o}bner\ bases can directly find minimal sets of generators for submodules: Starting from an arbitrary set of generators, we can eliminate unnecessary generators degree by degree, by removing those which reduce to zero under division by a Gr\"{o}bner\ basis for the ideal generated by the preceding generators. Either way, we can trim the set of syzygies computed via Gr\"{o}bner\ bases for a given set of generators $g_1, \ldots, g_s$ of $I$, to obtain a minimal set of generators for the syzygy module $J$. By starting with a minimal generating set for $I$, and iterating this method, a minimal free resolution can be found for $I$. A beautiful application of these ideas yields a proof of the Hilbert syzygy theorem, that minimal free resolutions terminate (Schreyer \cite{sch80}, \cite{sch91}, for an exposition see also Eisenbud \cite{eis92}). At each stage of a resolution, order the Gr\"{o}bner\ basis $F$ for $I$ in such a way that for each $i < j$, letting ${\rm in}(f_i) = a \sliver{\bf x}\sliver^A e_{0k}$ and ${\rm in}(f_j) = b \sliver{\bf x}\sliver^B e_{0\ell}$, we have $\sliver{\bf x}\sliver^A > \sliver{\bf x}\sliver^B$ in the lexicographic order. If the variables $x_1,\ldots, x_m$ are missing from the initial terms of the $f_i$, then the variables $x_1,\ldots, x_{m+1}$ will be missing from the initial terms of the syzygies $s_{ij}$. Iterating, we run out of variables, so the resolution terminates. \section{Bounds} \label{bounds} How hard are the algorithms in algebraic geometry? We describe some key bounds. The best known example is the bound established by G. Hermann \cite{her26} for ideal membership: \begin{thm}[G. Hermann]\label{thm1} Let $k$ be any field, let $(f_1,...,f_k) \subset k[x_1,...,x_n]$ and let $d = \max(\deg(f_i))$. If $g \in (f_1,...,f_k)$, then there is an expression $$ g = \sum_{i=1}^k a_i f_i $$ where $\deg(a_i) \le \deg(g) + 2(kd)^{2^{n-1}}$. \end{thm} This type of bound is called \quotes{doubly exponential}. However, with the advent of the concept of coherent sheaf cohomology \cite{ser55} and the systematic study of vanishing theorems, it has become apparent that the vanishing of these groups in high degrees is almost always the most fundamental bound. The concept of an ideal being \quotes{m-regular} or \quotes{regular in degrees $\ge m$} was introduced by one of us \cite{mum66} by generalizing ideas of Castelnuovo: \begin{defn}\label{def1} \footnote{ The definition has been slightly modified so as to apply to ideals $I$ instead of the corresponding sheaf of ideals ${\cal I}$. } Let $k$ be any field, let $I \subset k[x_0,\ldots,x_n]$ be an ideal generated by homogeneous polyomials, let $I_d$ be the homogeneous elements in $I$ of degree $d$, let $I$ be the corresponding sheaf of ideals in ${\cal O}_{\proj n}$, and let $I(d)$ be the $d^{\it th}$ twist of $I$. Then the following properties are equivalent and define the term \quotes{m-regular}: (a) the natural map $I_m \rightarrow H^0({\cal I}(m))$ is an isomorphism and \mbox{$H^i({\cal I}(m-i))$} $ = (0)$, $1 \le i \le n$ (b) the natural maps $I_d \rightarrow H^0({\cal I}(d))$ are isomorphisms for all $d \ge m$ and $H^i({\cal I}(d))$ = $(0)$ if $d+i \ge m$, $i \ge 1$. (c) Take a minimal resolution of $I$ by free graded $k[X]$-modules: $$ 0 \, \rightarrow \, \Limits{\oplus}{\alpha=1}{r_n} \, k[\sliver{\bf x}\sliver] \cdot e_{\alpha,n} \, \stackrel{\phi_n}{\rightarrow} \, \ldots \stackrel{\phi_1}{\rightarrow} \, \Limits{\oplus}{\alpha=1}{r_0} \, k[\sliver{\bf x}\sliver] \cdot e_{\alpha,0} \, \stackrel{\phi_0}{\rightarrow} \, k[\sliver{\bf x}\sliver] \longrightarrow k[\sliver{\bf x}\sliver]/I \rightarrow 0. $$ Then $\deg(e_{\alpha,i}) \le m+i$ for all $\alpha$, $i$. (In particular, if $f_\alpha = \phi_0(e_{\alpha,0})$, then $f_1,\ldots,f_{r_0}$ are minimal generators of $I$, and $\deg(e_{\alpha,0}) = \deg(f_\alpha) \le m$.) \end{defn} The intuitive idea is that past degree $m$, nothing tricky happens in the ideal $I$. Unfortunately, neither (a), (b) nor (c) can be verified by any obvious finite algorithm. This lack of a finitely verifiable criterion for $m$-regularity has been remedied by a joint result of the first author and M. Stillman \cite{bs87a}: \begin{thm}[Bayer-Stillman]\label{thm2} $I$ is $m$-regular if and only if the degrees of the minimal set of generators of $I$ are at most $m$, and there exists a set $y_0,\ldots,y_\ell$ of linear combinations of $x_0,...,x_n$ such that for all homogeneous $f$ of degree $m$, \begin{eqnarray*} y_0 f \in I &\Rightarrow& f \in I \\ y_1 f \in I &\Rightarrow& f \in I + k[\sliver{\bf x}\sliver] \cdot y_0 \\ &\cdots& \\ y_\ell f \in I &\Rightarrow& f \in I + \sum_{i=0}^{\ell-1} k[\sliver{\bf x}\sliver] \cdot y_i \end{eqnarray*} and $$f \in I + \sum_{i=0}^{\ell} k[\sliver{\bf x}\sliver] \cdot y_i.$$ Moreover, if this holds at all, it holds for $y_0,\ldots,y_\ell$ taken arbitrarily from a Zariski-open set in the space of $\ell+1$ linear forms. \end{thm} To see why $m$-regularity is a key bound, we want to show that it controls some of the geometric features of the ideal $I$. Let's introduce several refined notions of the \quotes{degree} of $I$: \begin{defn}\label{def2} If $I = \sliver{\bf q}\sliver_0 \,\cap\, \sliver{\bf q}\sliver_1 \,\cap\, \ldots \,\cap\, \sliver{\bf q}\sliver_t$ is a primary decomposition of $I$, $\sqrt{\sliver{\bf q}\sliver_i} = \sliver{\bf p}\sliver_i$ is prime and $V(\sliver{\bf p}\sliver_i)$ is the subvariety $Z_i$ of $\proj n$ for $i \ge 1$, while $\sliver{\bf p}\sliver_0 = (x_0,\ldots,x_n)$ (so that $V(\sliver{\bf p}\sliver_0)= \emptyset$), then first let $\sliver{\bf q}\sliver_1,\ldots,\sliver{\bf q}\sliver_s$ be the isolated components, (i.e., $Z_i \not\subset Z_j$ if $1\le i\le s$, $1\le j \le t$, $i \ne j$, or equivalently, $V(I)=Z_1 \,\cup\, \ldots \,\cup\, Z_s$ is set-theoretically the minimal decomposition of $V(I)$ into varieties). Then let \begin{eqnarray*} {\rm mult}(\sliver{\bf q}\sliver_i) &=& \mbox{length $\ell$ of a maximal chain of $\sliver{\bf p}\sliver_i$-primary ideals:} \\ & &\sliver{\bf q}\sliver_i = J_\ell \ssubset J_{\ell-1} \ssubset \ldots \ssubset J_1 = \sliver{\bf p}\sliver_i \end{eqnarray*} (Equivalently, this is the length of the local ring $k[\sliver{\bf x}\sliver]_{\sliver{\bf p}\sliver_i}/I k[\sliver{\bf x}\sliver]_{\sliver{\bf p}\sliver_i}$, or, in the language of schemes, if $\eta$ is the generic point of $Z_i$, then this is the length of ${\cal O}_{\eta,\proj n}.$) \begin{eqnarray*} \deg (Z_i) &=& \mbox{usual geometric degree of $Z_i$:} \\ & &\mbox{the cardinality of $Z_i \cap L$ for almost all} \\ & & \mbox{linear spaces $L$ of complementary dimension.} \\ \mbox{\rm geom-deg}_r(I) &=& \sum_{\stackrel{\mbox{\it \scriptsize $i$ such that $\dim Z_i = r$}} {\mbox{\scriptsize $1 \le i \le s$}}} {\rm mult}(\sliver{\bf q}\sliver_i) \deg(Z_i) \end{eqnarray*} \end{defn} If $\sliver{\bf q}\sliver_i$ is one of the non-isolated, or embedded components, then we extend the concept of multiplicity more carefully: Let $$ I_i = \left\{ \cap \sliver{\bf q}\sliver_j \mbox{\Large $\mid$} j \mbox{ such that } \sliver{\bf p}\sliver_j \ssubset \sliver{\bf p}\sliver_i \mbox{ or equivalently } Z_j \ssupset Z_i \right\} \cap \sliver{\bf p}\sliver_i $$ and \begin{eqnarray*} {\rm mult}_I(\sliver{\bf q}\sliver_i) &=& \mbox{length $\ell$ of a maximal chain of ideals:} \\ & & \sliver{\bf q}\sliver_i \cap I_i = J_\ell \ssubset J_{\ell-1} \ssubset \ldots \ssubset J_0 = I_i \\ & & \mbox{where each $J_k$ satisfies: } ab \in J_k, a \not\in \sliver{\bf p}\sliver_i \Rightarrow b \in J_k. \end{eqnarray*} (Equivalently, $J_k$ equals $\sliver{\bf q}\sliver_k \cap I_i$ for some $\sliver{\bf p}\sliver_i$-primary ideal $\sliver{\bf q}\sliver_k$.) In particular: \begin{eqnarray*} I_0 &=& \Bigcap{j=1}{t} \sliver{\bf q}\sliver_j \mbox{ is known as } \sat I \mbox{, and} \\ {\rm mult}_I(\sliver{\bf q}\sliver_0) &=& \mbox{length $\ell$ of a maximal chain of ideals} \\ & & I = J_\ell \ssubset J_{\ell-1} \ssubset \ldots \ssubset J_0 = \sat I \\ &=& \dim_k(\sat I/I). \end{eqnarray*} For $s+1 \le i \le t$, an equivalent way to define ${\rm mult}_I(\sliver{\bf q}\sliver_i)$ is as the length of the module $$I_i k[\sliver{\bf x}\sliver]_{\sliver{\bf p}\sliver_i}/I k[\sliver{\bf x}\sliver]_{\sliver{\bf p}\sliver_i}$$ or, in the language of schemes, the length of $$I_i{{\cal O}}_{\eta,\proj n}/I{{\cal O}}_{\eta,\proj n}$$ where $\eta$ is the generic point of $Z_i$. Then write $$\mbox{\rm arith-deg}_r(I) = \sum_{\stackrel{\mbox{\it \scriptsize $i$ such that $\dim Z_i = r$}} {\mbox{\scriptsize $1 \le i \le s$}}} {\rm mult}_I(\sliver{\bf q}\sliver_i) \deg(Z_i)$$ and $$\mbox{\rm arith-deg}_{-1}(I) = {\rm mult}_I(\sliver{\bf q}\sliver_0).$$ The idea here is best illustrated by an example: let $$I = (x_1^2,x_1 x_2) \subset k[x_0,x_1,x_2].$$ Then \begin{eqnarray*} I &=& \sliver{\bf q}\sliver_1 \cap \sliver{\bf q}\sliver_2 \\ q_1 &=& (x_1),\; \sliver{\bf p}\sliver_1 = (x_1),\; Z_1 = \{\mbox{line } x_1=0\} \\ q_2 &=& (x_1^2,x_1x_2,x_2^2),\; \sliver{\bf p}\sliver_2 = (x_1,x_2), \; Z_2 = \{\mbox{point } (1,0,0)\}. \end{eqnarray*} Then $$\deg(Z_1)=1,\; {\rm mult}(q_1) = 1$$ so $$\mbox{\rm geom-deg}_1(I) = \mbox{\rm arith-deg}_1(I) = 1.$$ One might be tempted to simply define $${\rm mult}_I(\sliver{\bf q}\sliver_2) = \mbox{ length of chain of $\sliver{\bf p}\sliver_2$-primary ideals between } \sliver{\bf q}\sliver_2, \sliver{\bf p}\sliver_2$$ and since $$k[\sliver{\bf x}\sliver]_{\sliver{\bf q}\sliver_2}/\sliver{\bf q}\sliver_2 k[\sliver{\bf x}\sliver]_{\sliver{\bf p}\sliver_2} \cong K \cdot 1+K \cdot x_1 + K \cdot x_2, K=k(x_0)$$ this is $3$. But embedded components are not unique! In fact, \begin{eqnarray*} I &=& \sliver{\bf q}\sliver_1 \cap \sliver{\bf q}\sliver_2^\prime \\ \sliver{\bf q}\sliver_2^\prime &=& (x_1^2 x_2) \mbox{ also}, \end{eqnarray*} which leads to $$k[\sliver{\bf x}\sliver]_{\sliver{\bf p}\sliver_2}/\sliver{\bf q}\sliver_2^\prime k[\sliver{\bf x}\sliver]_{\sliver{\bf p}\sliver_2} \cong K \cdot 1+ K \cdot x_2$$ which has length $2$. The canonical object is not the local ring $k[\sliver{\bf x}\sliver]_{\sliver{\bf p}\sliver_2}/\sliver{\bf q}\sliver_2 k[\sliver{\bf x}\sliver]{\sliver{\bf p}\sliver_2}$ but the ideal $$\mbox{\rm Ker} \left( k[\sliver{\bf x}\sliver]_{\sliver{\bf p}\sliver_2} / I k[\sliver{\bf x}\sliver]_{\sliver{\bf p}\sliver_2} \rightarrow k[\sliver{\bf x}\sliver]_{\sliver{\bf p}\sliver_2} / \sliver{\bf p}\sliver_2 k[\sliver{\bf x}\sliver]_{\sliver{\bf p}\sliver_2} \right) \cong k \cdot x_1$$ which has length $1$. Thus, the correct numbers are $${\rm mult}_I(\sliver{\bf q}\sliver_2) = 1$$ and \begin{eqnarray*} \mbox{\rm geom-deg}_0(I) &=& 0 \\ \mbox{\rm arith-deg}_0(I) &=&1. \end{eqnarray*} Now the question arises: find bounds on these degrees in terms of generators of $I$. For geometric degrees, a straightforward extension of Bezout's theorem gives: \begin{prop}\label{prop3} Let $d(I)$ be the maximum of the degrees of a minimal set of generators of $I$. Then $$\mbox{\rm geom-deg}_r(I) \le d(I)^{n-r}.$$ \end{prop} A proof can be found in \cite{mw83}. The idea is clear from a simple case: Suppose $f, g, h \in K[x,y,z]$ and $f=g=h=0$ consists of a curve $C$ and $\ell$ points $P_i$ off $C$. We can bound $\ell$ like this: Choose $2$ generic combinations $f^\prime$, $g^\prime$ of $f$, $g$, $h$ so that $f^\prime=g^\prime=0$ does not contain a surface. It must be of the form $C \cup C^\prime$, $C'$ one-dimensional, containing all the $P_i$ but not the generic point of $C$. Then by the usual Bezout theorem $$\deg C^\prime \le \deg f^\prime \;\deg g^\prime=d(I)^2.$$ Let $h^\prime$ be a $3$\thuh{rd} generic combination of $f,g,h$. Then $C^\prime \cap \{h^\prime = 0\}$ consists of a finite set of points including the $P_i$'s. Thus \begin{eqnarray*} \ell &=& \# P_i \le \#(C^\prime \cap \{h^\prime = 0\}) \\ & \le & \deg C^\prime \cdot d(I) \mbox{ by Bezout's theorem} \\ & \le & d(I)^3. \end{eqnarray*} Can $\mbox{\rm arith-deg}(I)$ be bounded in the same way? In fact, it cannot, as we will show below. Instead, we have \begin{prop}\label{prop4} If $m(I)$ is the regularity of $I$, then for $-1 \le r \le n$, $$\mbox{\rm arith-deg}_r(I) \le {m(I)+n-r-1 \choose n-r} \le m(I)^{n-r}$$ \end{prop} which replaces $d(I)$ by the regularity of $I$. A proof is given in the technical appendix. We have introduced two measures of the complexity of a homogeneous ideal $I$. The first is $d(I)$, the maximum degree of a polynomial in a minimum set of generators of $I$. The second is $m(I)$, which bounds the degrees of generators and of all higher order syzygies in the resolution of $I$ (\defref{def1} (c)). Obviously, $$d(I) \le m(I).$$ A very important question is how much bigger can $m(I)$ be than $d(I)$? The nature of the answer was conjectured by one of us in his thesis \cite{bay82} and this conjecture is being borne out by subsequent investigations. This conjecture is that in the worst case $m(I)$ is roughly the $(2^n)$\thuh{th} power of $d(I)$ -- a bound like G. Hermann's. But that if $I = I(Z)$ where $Z$ is geometrically nice, e.g. is a smooth irreducible variety, then $m(I)$ is much smaller, like the $n$\thuh{th} power of $d(I)$ or better. This conjecture then has three aspects: \begin{tabular}{@{\hspace{5pt}}ll} (1) & a doubly exponential bound for $m(I)$ in terms of $d(I)$, \\ & which is always valid,\\ (2) & examples of $I$ where the bound in (1) is best possible, or nearly so,\\ (3) &much better bounds for $m(I)$ \\ & valid if $V(I)$ satisfies various conditions.\\ \end{tabular} All three aspects are partially proven, but none are completely clarified yet. We will take them up one at a time. A doubly exponential bound for $m(I)$ in terms of $d(I)$ may be deduced easily {\em in characteristic zero} from the work of M. Giusti \cite{giu84} and A. Galligo \cite{gal79}: \begin{thm}\label{thm4} If $\mbox{\rm char}(k) = 0$ and $I \subset k[x_0, \ldots x_n]$ is any homogeneous ideal, then $$m(I) \le (2 d(I))^{2^{n-1}}.$$ \end{thm} It seems likely that \thmref{thm4} holds in characteristic $p$, too. A weaker result can be derived quickly in any characteristic by straightforward cohomological methods: \begin{prop}\label{prop5} If $I \subset k[x_0, \ldots x_n]$ is any homogeneous ideal, then $$m(I) \le (2 d(I))^{n!}.$$ \end{prop} The proof is given in the technical appendix. Next, we ask whether \thmref{thm4} is the best possible, or nearly so. The answer is yes, because of a very remarkable example due to E. Mayr and A. Meyer \cite{mm82}. \begin{example} Let $I_n^A$ be the ideal in $10 n$ variables $S^{(m)}, F^{(m)}, C_i^{(m)}, B_i^{(m)}, 1 \le i \le 4, 1 \le m \le n$ defined by the $10 n - 6$ generators $$\begin{array}{rl} 2 \le m \le n \;\; &\left\{\;\; \begin{array}{l} S^{(m)} - S^{(m-1)} C_1^{(m-1)} \\ F^{(m)} - S^{(m-1)} C_4^{(m-1)} \\ C_i^{(m)}F^{(m-1)}B_2^{(m-1)} - C_i^{(m)}B_i^{(m)}F^{(m-1)}B_3^{(m-1)}, \;\;1 \le i \le 4 \end{array} \right.\\ \\ 1 \le m \le n-1 \;\; &\left\{\;\; \begin{array}{l} F^{(m)}C_1^{(m)}B_1^{(m)} - S^{(m)} C_2^{(m)} \\ F^{(m)}C_2^{(m)} - F^{(m)} C_3^{(m)} \\ S^{(m)}C_3^{(m)}B_1^{(m)} - S^{(m)}C_2^{(m)}B_4^{(m)} \\ S^{(m)}C_3^{(m)} - F^{(m)}C_4^{(m)}B_4^{(m)} \end{array} \right.\\ \\ &\mbox{\hskip 20pt}C_i^{(1)}S^{(1)} - C_i^{(1)}F^{(1)}(B_i^{(1)})^2, \;\;1 \le i \le 4 \end{array}$$ Let $I_n^H$ be the ideal gotten from $I_n^A$ by homogenizing with an extra variable $u$. Then Mayr and Meyer \cite[lemma 8, p. 318]{mm82} prove: \end{example} \begin{lemma} Let $e_n=2^{2^n}$. If $M$ is any monomial in these variables, $S^{(n)}C_i^{(n)}-F^{(n)} M \in I_n^A$ if and only if $$M=C_i^{(n)}(B_i^{(n)})^{e_n},$$ and $S^{(n)}C_i^{(n)} - S^{(n)} M \in I_n^A$ if and only if $$M=C_i^{(n)}.$$ \end{lemma} Now note that the generators of $I_n^A$ and $I_n^H$ are all of the very simple type given by a difference of two monomials. Quite generally, if \begin{eqnarray*} J &\subset& k[x_1,\ldots,x_n] \\ J &=& (\ldots, \sliver{\bf x}\sliver^{\alpha_i} - \sliver{\bf x}\sliver^{\beta_i}, \ldots )_{1 \le i \le k} \end{eqnarray*} then the quotient ring $k[\sliver{\bf x}\sliver]/J$ has a very simple form. In fact, we get an equivalence relation between monomials generated by $$\sliver{\bf x}\sliver^{\alpha_i+\gamma} \sim \sliver{\bf x}\sliver^{\beta_i+\gamma}, \mbox{ any } i, \gamma$$ and $$k[\sliver{\bf x}\sliver]/J \cong \oplus_\delta \; k \cdot \sliver{\bf x}\sliver^\delta$$ where $\delta$ runs over a set of representatives of each equivalence class. Bearing this in mind, let's look at the $1$\thuh{st} order syzygies for the homogeneous ideal: $$J_n^H = (S^{(n)},F^{(n)},I_n^H).$$ $S^{(n)}$ and $F^{(n)}$ are part of a minimal set of generators, and let $f_\alpha \in I_n^H$ complete them. Then syzygies are equations: $$p \, S^{(n)} \, + \, q \, F^{(n)} \, + \, \sum \, r_\alpha \, f_\alpha \, = \, 0.$$ One such is given by: $$\left[ u^{e_n+e} \, C_i^{(n)} \right] \, S^{(n)} \, + \, \left[ - u^e \, (B_i^{(n)})^{e_n} \, C_i^{(n)} \right] \, F^{(n)} \, + \, \sum \, R_\alpha \, f_\alpha \, = \, 0$$ for some $R_\alpha$, and some $e \ge 0$ (the extra power $u^e$ is necessary because some terms $R_\alpha f_\alpha$ have degree greater than $e_n+2$) whose degree is $2+e_n+e$. Now express this syzygy as a combination of a minimal set of syzygies. This gives us in particular: \begin{eqnarray*} u^{e_n+e} \, C_i^{(n)} \, &=& \, \sum \, a_\lambda \, p_\lambda \\ - u^e \, (B_i^{(n)})^{e_n} \, C_i^{(n)} \, &=& \, \sum \, a_\lambda \, q_\lambda \\ p_\lambda \, S^{(n)} \, + \, q_\lambda \, F^{(n)} \, + \, \sum \, R_{\alpha\lambda} \, f_\alpha \, &=& \, 0 \, . \end{eqnarray*} Then for some $\lambda$, $p_\lambda$ must have a term of the form $u^\ell$ or $u^\ell \, C_i^{(n)}$, hence the monomial $u_\ell \, S^{(n)}$ or $u_\ell \, C_i^{(n)} \, S^{(n)}$ occurs in $p_\lambda \, S^{(n)}$. But by the general remark on quotient rings by such simple ideals, this means that this term must equal some second term $M \, S^{(n)}$ ($M$ a monomial in $p_\lambda$) or $M \, F^{(n)}$ ($M$ a monomial in $q_\lambda$) mod $I_n^H$. By the lemma, the first doesn't happen and the second only happens if the term $u^\ell \, C_i^{(n)} \, (B_i^{(n)})^{e_n}$ occurs in $q_\lambda$, in which case $e_n+1\le \deg \, q_\lambda \,= \, \deg(\mbox{syzygy}(p_\lambda,q_\lambda,R_{\alpha\lambda})) - 1$. This proves: \begin{prop}\label{prop7} $J_n^H$ has for its bounds: \begin{eqnarray*} d(J) &=& 4 \\ m(J) &\ge& 2^{2^n}+1. \end{eqnarray*} \end{prop} Going on to the $3$\thuh{rd} aspect of the conjecture, consider results giving better bounds for $m(I)$ under restrictive hypotheses on $V(I)$. \begin{thm}\label{thm8} If $Z \subset \proj n$ is a reduced subscheme purely of dimension $r$, and $I=I(Z)$ is the full ideal of functions vanishing on $Z$, then (a) if $r \leq 1$, or $Z$ is smooth, $\hbox{char}(k)=0$ and $r \leq 3$, then: $$m(I) \le \deg Z - n + r + 1$$ (b) if char$(k)=0$ and $Z$ is smooth, $$m(I) \le (r+1)(\deg(Z)-2)+2.$$ \end{thm} Since $\deg(Z) \le d(I)^{n-r}$ (\propref{prop3}), these bound $m(I)$ in terms of $d(I).$ Part (a) of this are due to Gruson-Lazarsfeld-Peskine \cite{glp83} for $r \leq 1$, and to Pinkham \cite{pin86}, Lazarsfeld \cite{laz87}, and Ran \cite{ran90} for $r \leq 3$. It is {\it conjectured} by Eisenbud and Goto \cite{eg84}, and others, that the bound in (a) holds for all reduced irreducible $Z$, and it might well hold even for reduced equidimensional $Z$ which are connected in codimension 1. As this problem is now understood, the needed cohomological arguments follow formally, once one can control the singularities of a projection of the variety. These singularities become progressively harder to subdue as the dimension of the variety increases, and are what impedes definitive progress beyond dimension 3. Part (b) is due to the second author and is proven in the technical appendix. It has been generalized by Bertram, Ein, and Lazarsfeld \cite{bel91} to show that any smooth characteristic $0$ variety of codimension $e$ defined as a subscheme of ${\bf P}^n$ by hypersurfaces of degrees $d_1 \ge \ldots \ge d_m$ is $(d_1 + \ldots d_e - e +1)$-regular. Since we cannot decide the previous conjecture, this is a result of considerable practical importance, for it strongly bounds the complexity of computing Gr\"{o}bner\ bases of smooth characterisitic $0$ varieties in terms of the degrees of the input equations. The biggest missing link in this story is a decent bound on $m(I)$ for any reduced equidimensional ideal $I$. We would conjecture that if a linear bound as in part (a) doesn't hold, at the least a so-called ``single exponential'' bound, i.e. $m(I) \leq d^{0(n)}$ ought to hold. This is an essential ingredient in analyzing the worst-case behavior of all algorithms based on Gr\"{o}bner\ bases, and would complete the story about what causes the bad examples discussed above. At least in some cases Ravi \cite{rav90} has proven that the regularity of the radical of a scheme is no greater than the regularity of the scheme itself. There is a direct link between the bounds that we have given so far and the G. Hermann bound with which we started the section. This results from the following: \begin{prop}\label{prop9} Let $I^A \subset k[x_1,\ldots,x_n]$ have generators $f_1,\ldots,f_k$ and let $I^H \subset k[x_0,x_1,\ldots,x_n]$ be the ideal generated by homogenizations $f_1^h,\ldots,f_k^h$ of the $f_i$. Let $I^H = \sliver{\bf q}\sliver_0 \cap \ldots \cap \sliver{\bf q}\sliver_t$ be the primary decomposition of $I^H$, let $Z_i=V(\sliver{\bf q}\sliver_i)$ and let $${\rm mult}_\infty(I^H) = \max \left[ {\rm mult}_I(\sliver{\bf q}\sliver_{i_1}) +\ldots+{\rm mult}_I(\sliver{\bf q}\sliver_{i_k})+{\rm mult}_I(\sliver{\bf q}\sliver_0) \right]$$ where the $\max$ is taken over chains $V((x_0)) \supset Z_{i_1} \ssupset \ldots \ssupset Z_{i_k}$. If $g \in I^A$, then we can write: $$g = \sum_{i=1}^k \, a_i \, f_i$$ where $$\deg a_i \le \deg g + {\rm mult}_\infty(I^H).$$ \end{prop} The proof goes like this: Let $g^h$ be the homogenization of $g$. Consider the least integer $m$ such that $x_0^m g^h \in I^H$. Since $g \in I$, this $m$ is finite. Moreover, if $$x_0^{m} g^h = \sum x_0^{m_i} a_i^h f_i^h$$ then $$g= \sum a_i f_i$$ and $$\deg a_i = \deg(a^h) \le \deg(x_0^m g^h) - \deg f_j \le m+\deg(g).$$ Now in the primary decomposition of $I^H$, suppose that for some $k$, $$x_0^{k} g^h \in \Bigcap{i\in S}{} q_i, \mbox{ and } x_0^{k} g^h \not\in \sliver{\bf q}\sliver_j \mbox{ if } j \not\in S.$$ Choose $\ell \not\in S$ such that $V(q_\ell)$ is maximal. Since $g \in I^A$, we know $V(\sliver{\bf q}\sliver_\ell) \subset V((x_0))$, hence $x_0 \in \sliver{\bf p}\sliver_\ell$. Let $$I_S= \Bigcap{i \in S}{} \sliver{\bf q}\sliver_i.$$ Then ${\rm mult}_I(\sliver{\bf q}\sliver_\ell)$ is easily seen to be the length of a maximal chain of ideals between: $$I \cdot k[\sliver{\bf x}\sliver]_{p_\ell} \mbox{ and } I_S \cdot k[\sliver{\bf x}\sliver]_{p_\ell}.$$ But look at the ideals $J_p$, for $p \ge 0$, defined by $$I \, k[\sliver{\bf x}\sliver]_{\sliver{\bf p}\sliver_\ell} \subset \underbrace{(I, x_0^{k+p} g^h) \, k[\sliver{\bf x}\sliver]_{\sliver{\bf p}\sliver_\ell}}_{\textstyle J_p} \subset I_S \, k[\sliver{\bf x}\sliver]_{\sliver{\bf p}\sliver_\ell}.$$ If $J_p=J_{p+1}$, then \begin{eqnarray*} x_0^{k+p} g^h &\in& (I, x_0^{k+p+1} g^h) \\ \mbox{i.e.,} \;\; x_0^{k+p} g^h &=& a \, x_0^{k+p+1} g^h + b, \;\; b \in I. \end{eqnarray*} But $1- a x_0$ is a unit in $k[\sliver{\bf x}\sliver]_{\sliver{\bf p}\sliver_\ell}$, so $$J_p= x_0^{k+p} g^h= (1- a x_0)^{-1} b \in I \, k[\sliver{\bf x}\sliver]_{\sliver{\bf p}\sliver_\ell}.$$ This means that in any case $$x_0^{k+{\rm mult}_I(\sliver{\bf q}\sliver_\ell)} g^h \in I \cdot k[\sliver{\bf x}\sliver]_{\sliver{\bf p}\sliver_\ell}$$ hence, because $q_\ell$ is $p_\ell$-primary: $$x_0^{k+{\rm mult}_I(\sliver{\bf q}\sliver_\ell)} g^h \in \sliver{\bf q}\sliver_\ell$$ Induction now shows that $$x_0^{{\rm mult}_\infty(I^H)} g^h \in I^H$$ \begin{corollary}\label{cor10} Let $I^A$, $I^H$ be as above. If $g \in I^A$, then $$g= \sum a_i f_i$$ where $\deg(a_i) \le \deg(g)+{m(I)+n+1 \choose n+1}$. \end{corollary} \begin{proofn} Combine Propositions \ref{prop4} and \ref{prop9}. \end{proofn} If we further estimate $m(I)$ by \thmref{thm4} in characteristic $0$ or by \propref{prop5}, we get somewhat weaker versions of Hermann's \thmref{thm1}. But if $I=V(Z)$, $Z$ a good variety, we may expect the Corollary to give much better bounds than \thmref{thm1}. \corref{cor10} shows that any example which demonstrates the necessity of double exponential growth in Hermann's ideal membership bound (\thmref{thm1}) also demonstrates the necessity of double exponential growth in the bounds on $m(I)$ given in \thmref{thm4} and \propref{prop5}. Thus we can make use of the general arguments for the existence of such examples given in \cite{mm82}, rather than depending on the single example of \propref{prop7}, to show that the bounds on $m(I)$ inevitably grow double exponentially: Since in \corref{cor10}, the degrees of the $a_i$ are bounded by a single exponential function of $m(I)$, in all examples where the degrees of the $a_i$ grow double exponentially, $m(I)$ also grows double exponentially. This line of argument gives a geometric link between the ideal membership problem and $m(I)$: In \corref{cor10}, if $I^A$ exhibits $a_i$ of high degree, then $I^H$ has primary components of high multiplicity. These components force $m(I)$ to be large, and distinguish $I^H$ from good ideals considered in \thmref{thm8} and related conjectures. A major step in understanding the gap between the double exponential examples and the strong linear bounds on the regularity of many smooth varieties was taken by Brownawell \cite{bro87} and Koll{\'a}r \cite{kol88}. They discovered the beautiful and satisfying fact that if we replace membership in $I$ by membership in $\sqrt{I}$, then there are single exponential bounds on the degrees of $a_i$: \begin{thm}[Brownawell, Koll{\'a}r] Let $k$ be any field, let $I = (f_1,...,f_k) \subset k[x_1,...,x_n]$ and let $d = \max(\deg(f_i),i=1,\cdots,k; 3)$. If $n=1$, replace $d$ by $2d-1$. If $g \in \sqrt{I}$, then there is an expression $$ g^s = \sum_{i=1}^k a_i f_i $$ where $s \leq d^n$ and $\deg(a_i) \le (1 + \deg(g))d^n$. In particular: $$ \left( \sqrt{I} \right) ^{d^n} \subset I. $$ \end{thm} What this shows is that although the bad examples have to have primary components at infinity of high degree, nonetheless these primary ideals contain relatively small powers of $\sqrt{I^H}$. The picture you should have is that these embedded components at infinity are like strands of ivy that creep a long way out from the hyperplane at infinity, but only by clinging rather closely to the affine components. \vskip .25in {\bf\noindent Technical Appendix to \secref{bounds}} \vskip .15in \noindent {\bf 1.} Proof of the equivalence of the conditions in \defref{def1}: In \cite[pp. 99-101]{mum66}, it is proven that for any coherent sheaf ${\cal F}$ on $\proj n$, $H^i({\cal F}(-i))=(0)$, $i \ge 1$ implies that the same holds for ${\cal F}(d)$, all $d \ge 0$, and that $H^0({\cal F}(d))$ is generated by $H^0({\cal F}) \otimes H^0({\cal O}(d))$. In particular, if you apply this to ${\cal F} = {\cal I}(m)$, the equivalence of (a) and (b) follows. (Note the diagram: $$ \begin{array}{ccc} I_d & \longrightarrow & H^0({\cal I}(d)) \\ \bigcap & & \bigcap \\ k[\sliver{\bf x}\sliver]_d & \longrightarrow & H^0({\cal O}_{\proj n}(d)) \end{array} $$ which shows that $I_m \rightarrow H^0({\cal I}(m))$ is injective for every $d$). To show that (b) $\Rightarrow$ (c), first note that we may rephrase the reults in \cite{mum66} to say that if $H^i({\cal F}(-i))=(0)$, $i \ge 1$, then the degrees of the minimal generators of the $k[\sliver{\bf x}\sliver]$-module $$\stackrel{\textstyle\oplus}{_{d \in \mbox{\bf Z}}} \;\; H^0({\cal F}(d))$$ are all zero or less. So we may construct the resolution in (c) inductively: at the $k$\thuh{th} stage, say $$ \Limits{\oplus}{\alpha = 1}{r_k + 1} \; k[\sliver{\bf x}\sliver] \cdot e_{\alpha,k-1} \stackrel{\phi_{k-1}}\longrightarrow \cdots \longrightarrow k[\sliver{\bf x}\sliver] \longrightarrow k[\sliver{\bf x}\sliver]/I \longrightarrow 0 $$ has been constructed, let $M_k= \mbox{ ker}(\phi_{k-\ell})$ and let ${\cal F}_k$ be the corresponding sheaf of ideals. The induction hypothesis will say that $H^i({\cal F}_k(m+k-1))=(0)$, $i \ge1$. Therefore $M_k$ is generated by elements of degree $ \le m+k$, i.e., $d_\alpha=\deg e_{\alpha,k} \le m+k$, all $\alpha$. We get an exact sequence $$0 \;\longrightarrow\; M_{k+1} \;\longrightarrow \Limits{\oplus}{\alpha=1}{r_k} \left( k[\sliver{\bf x}\sliver] \cdot e_{\alpha,k} \right) \;\longrightarrow\; M_k \;\longrightarrow\; 0 $$ hence \begin{eqnarray} 0 \;\longrightarrow\; {\cal F}_{k+1} \;\longrightarrow \Limits{\oplus}{\alpha=1}{r_k} {\cal O}_{\proj n}(-d_\alpha) \;\longrightarrow\; {\cal F}_k \;\longrightarrow\; 0 \end{eqnarray} Therefore \begin{eqnarray} \Limits{\oplus}{\alpha=1}{r_k} H^i({\cal O}_{\proj n}(m\!+\!k\!-\!i\!-\!d_\alpha)) \longrightarrow H^i({\cal F}_k(m\!+\!k\!-\!i)) \longrightarrow \hskip 0.3in \\ \;\;\;\;H^{i+1}({\cal F}_{k+1}(m\!+\!(k\!+\!1)\!-\!(i\!+\!1))) \longrightarrow \Limits{\oplus}{\alpha=1}{r_k} H^{i+1}({\cal O}_{\proj n}(m\!+\!k\!-\!i\!-\!d_\alpha)) \nonumber \end{eqnarray} is exact. But $m+k-i-d_\alpha \ge -i$ so $H^{i+1}({\cal O}_{\proj n}(m+k-i-d_\alpha))=(0)$. This shows that ${\cal F}_{k+1}$ satisfies the induction hypothesis and we can continue. Thus (c) holds. To see that (c) $\Rightarrow$ (a), we just use the same exact sequences (1) and prove now by descending induction on $k$ that $H^i({\cal F}_k(m+k-i))=(0)$, $i \ge1$. Since $I={\cal F}_0$, this does it. The inductive step again uses (2), since $H^i({\cal O}_{\proj n}(m+k-i-d_\alpha))=(0)$ too. \vskip .15in \noindent {\bf 2.} Proof of \propref{prop4}: Look first at the case $r=0$. Let ${\cal I}$ be the sheaf of ideals defined by $I$ and let ${\cal I}^* \supset {\cal I}$ be the sheaf defined by omitting all $0$-dimensional primary components of $I$. Consider the exact sequence: $$0 \;\longrightarrow\; {\cal I}(m-1) \;\longrightarrow {\cal I}^*(m-1) \;\longrightarrow\; ({\cal I}^*/{\cal I})(m-1) \;\longrightarrow\; 0 $$ This gives us: $$ H^0({\cal I}^*(m-1)) \;\longrightarrow\; H^0(({\cal I}^*/{\cal I})(m-1)) \;\longrightarrow\; H^1({\cal I}(m-1)) $$ Now $H^1({\cal I}(m-1))=(0)$ by $m$-regularity, and $h^0(({\cal I}^*/{\cal I})(m-1)) = h^0({\cal I}^*/{\cal I}) = \mbox{ length}({\cal I}^*/{\cal I}) = \mbox{\rm arith-deg}_0(I)$ since ${\cal I}^*/{\cal I}$ has $0$-dimensional support. But $H^0({\cal I}^*(m-1)) \subset H^0({\cal O}_{\proj n}(m-1))$, so \begin{eqnarray*} \mbox{\rm arith-deg}_0(I) & \le & h^0({\cal I}^*(m-1)) \\ & \le & h^0({\cal O}_{\proj n}(m-1)) \\ & = & { m+ n -1 \choose n } \end{eqnarray*} If $r>0$, we can prove the Proposition by induction on $r$. Let $H$ be a generic hyperplance in $\proj n$, given by $h=0$. Let $I_H= (I,h) / (h) \subset k[x_0,\ldots,x_n]/(h) \cong k[x_0^\prime,\ldots,x_{n-1}^\prime]$ for suitable linear combinations $x_i^\prime$ of $x_i$. Then it is easy to check that: $$\mbox{\rm arith-deg}_r(I) = \mbox{\rm arith-deg}_{r-1}(I_H)$$ and that $I_H$ is also $m$-regular, so by induction \begin{eqnarray*} \mbox{\rm arith-deg}_{r-1}(I_H) & \le & {m+(n-1)-(r-1)-1 \choose (n-1)-(r-1)} \\ & = & { m+n-r-1 \choose n-r } \end{eqnarray*} If $r = -1$, we use the fact that $$0 \longrightarrow I_d \longrightarrow H^0({\cal I}(d)) \stackrel{\approx}{\longleftarrow} (I^{\mbox{sat}})_d$$ if $d \ge m$, hence $$\dim(I^{\mbox{sat}}/I) \le \dim k[\sliver{\bf x}\sliver]/(x_0,\ldots,x_n)^m = {m+n \choose n+1}. $$ \vskip .15in \noindent {\bf 3.} Proof of \propref{prop5}: Let $I \subset k[x_0, \ldots x_n]$ and assume, after a linear change of coordinates, that $x_n$ is not contained in any associated prime ideals of $I$. Let $\overline I \subset k[x_0, \ldots x_{n-1}]$ be the image of $I$. Then $d(\overline I) = d(I)$ and by induction we may assume $$m(\overline I) \le (2d(I))^{(n-1)!}.$$ We will prove, in fact, that \begin{eqnarray} m(I) \le m(\overline I) + { m(\overline I) - 1 + n \choose n } \end{eqnarray} and then we will be done by virtue of the elementary estimate: $$\mbox{if } m^* = (2d(I))^{(n-1)!}, \mbox{ and } d \ge 2, \mbox{ then } m^* + { m^* - 1 + n \choose n } \le (2d(I))^{n!}$$ To prove (3), we use the long exact sequence $$ \begin{array}{ccccccccc} 0 & \longrightarrow & (I:(x_0))_{k-1} & \stackrel{x_0}\longrightarrow & I_k & \longrightarrow & \overline I_k & \longrightarrow & 0 \\ & & \downarrow & & \downarrow & & \downarrow \\ 0 & \longrightarrow & H^0({\cal I}(k-1)) & \longrightarrow & H^0({\cal I}(k-1)) & \longrightarrow & H^0(\overline{\cal I}(k-1)) & \stackrel{\delta}\longrightarrow \\ \\ & \stackrel{\delta}\longrightarrow & H^1({\cal I}(k-1)) & \longrightarrow & H^1({\cal I}(k)) & \longrightarrow & H^1(\overline{\cal I}(k)) & \\ \end{array} $$ where $(I:(x_0)) = \setdef{f}{x_0 f \in I}$. Let $\overline m = m(\overline I)$. Note that $H^i(\overline{\cal I}(k-1)) = (0)$, $i \ge 1$, $k \ge \overline m$, hence $$H^i({\cal I}(k-1)) \rightarrow H^i({\cal I}(k))$$ is an isomorphism if $k \ge \overline m-1+1$ and $i \ge 2$. Since $H^i({\cal I}(k)) = (0)$, $k \gg 0$, this shows that $H^i({\cal I}(k)) = (0)$, $i \ge 2$, $k \ge \overline m -i$. Moreover $\overline I_k \rightarrow H^0(\overline{\cal I}(k))$ is an isomorphism if $k \ge \overline m$, hence $\delta = 0$ if $k \ge \overline m$, hence $H^1({\cal I}(k)) = (0)$, $k \ge \overline m - 1$. But now look at the surjectivity of $I_k \rightarrow H^0({\cal I}(k))$. For all $k$, let $M_k$ be the cokernel. Then $\oplus M_k$ is a $k[\sliver{\bf x}\sliver]$-module of finite dimension. Multiplication by $x_0$ induces a sequence: $$0 \; \longrightarrow \; { (I:(x_0))_{k-1} \over I_{k-1} } \; \longrightarrow \; M_{k-1} \; \stackrel{x_0}\longrightarrow \; M_k \; \longrightarrow \; 0$$ which is exact if $k \ge \overline m$. But if, for one value of $k \ge \overline m$, \begin{eqnarray} (I:(x_0))_k = I_k \end{eqnarray} then by \thmref{thm2}, $I$ is $k$-regular and (4) continues to hold for larger $k$, and $M_k$ must be $(0)$. In other words, $$\dim M_k, \;\; k \;\ge\; \overline m-1$$ is non increasing and monotone decreasing to zero when $k \ge \overline m$. Therefore \begin{eqnarray*} m(I) & \le & \overline m + \dim M_{\overline m-1} \\ & \le & \overline m + \dim k[\sliver{\bf x}\sliver]_{\overline m-1} \\ & \le & \overline m + {\overline m-1+n \choose n} \\ \end{eqnarray*} which proves (3). \vskip .15in \noindent {\bf 4.} Proof of \thmref{thm8}(b): Let $Z$ be a smooth $r$-dimensional subvariety of $\proj n$ and $d =$ degree of $Z$. We first consider linear projections of $Z$ to $\proj r$ and to $\proj{r-1}$. To get there, let $L_1 \subset \proj n$ be a linear subspace of dimension $n-r-1$ disjoint from $Z$ and $L_2 \subset L_1$ a linear subspace of dimension $n-r-2$. Take these as centers of projection: $$ \begin{array}{ccc} \proj{n} - L_1 & \supset & \;\;\;\; \;Z \;\;\; \stackrel{p_2}\longrightarrow \\ \downarrow & & \downarrow p_1 \\ \proj{r+1} - \{ P \} & \supset & Z_1 \;\;\; \\ \downarrow \\ \stackrel{p_2}\longrightarrow \; \proj{r} \;\;\;\;\; \end{array} $$ Let $x_0, \ldots x_{r+1}$ be coordinates on $\proj{r+1}$ so that $p = (0, \ldots , 0, 1)$, hence $x_0, \ldots x_r$ are coordinates on $\proj r$. Let $f(x_0, \ldots x_{r+1}) = 0$ be the equation of the hypersurface $Z_1$. Now there are two ways of getting $r$-forms on $Z$: by pullback of $r$-forms on $\proj r$ and by residues of $(r+1)$-forms on $\proj{r-1}$ with simple poles along $Z_1$. The first gives us a sheaf map $$p_2^* \;\Omega_{\proj r}^r \hookrightarrow \Omega_Z^r$$ whose image is $\Omega_Z^r(-B_1)$, $B_1$ the branch locus of $p_2$. Corresponding to this on divisor classes: \begin{eqnarray} K_Z & \equiv & p_2^*(K_{\proj r}) + B_1 \\ & \equiv & - (r+1)H + B_1, \nonumber \end{eqnarray} where $H =$ hyperplane divisor class on $Z$. The second is defined by \begin{eqnarray} a(\sliver{\bf x}\sliver) \cdot { dx_1 \wedge \ldots \wedge dx_{r+1} \over f } \;\longmapsto\; p_1^* \left(a(\sliver{\bf x}\sliver) \cdot { dx_1 \wedge \ldots \wedge dx_r \over \partial f/\partial x_{r+1}} \right) \end{eqnarray} and it gives us an isomorphism $$p_1^*(\Omega_{\proj{r+1}}^{r+1}(Z_1) {\Large\mid}_{Z_1} ) \;\cong\; \Omega_Z^r(B_2)$$ $B_2$ is a divisor which can be interpreted as the {\em conductor} of the affine rings of $Z$ over those of $Z_1$: i.e., $$f \in {\cal O}_Z(-B_2) \;\Longleftrightarrow\; f \cdot (p_{1,*} {\cal O}_Z) \subset {\cal O}_{Z_1}.$$ In particular, \begin{eqnarray} p_{1,*}({\cal O}_Z(-B_2)) \cong \mbox{ sheaf of } {\cal O}_{Z_1} - \mbox{ ideals } C \mbox{ in } {\cal O}_{Z_1}. \end{eqnarray} A classical reference for these basic facts is Zariski \cite{zar69}, Prop. 12.13 and Theorem 15.3. A modern reference is Lipman \cite{lip84} (apply Def. (2.1)b to $p_1$ and apply Cor. (13.6) to $Z_1 \subset \proj{r+1}$). (4) gives us the divisor class identity: \begin{eqnarray} K_Z + B_2 & \equiv & p_1^*(K_{\proj{r+1}} + Z_1) \\ & \equiv & (d-r-2) H. \nonumber \end{eqnarray} (5) and (8) together tell us that $$B_1 + B_2 \equiv (d-1)H.$$ In fact, the explicit description (6) of the residue tells us more: namely that if $y_1,\ldots,y_r$ are local coordinates on $Z$, then $${\partial (x_1,\ldots,x_r) \over \partial (y_1,\ldots,y_r)} \cdot {1 \over \partial f / \partial x_{r+1}} \; dy_1 \wedge \ldots \wedge dy_r$$ generates $\Omega_Z^r(B_2)$ locally. But ${\partial (x_1,\ldots,x_r) \over \partial (y_1,\ldots,y_r)} = 0$ is a local equation for $B_1$, so this means that $\partial f / \partial x_{r+1} = 0$ is a local equation for $B_1+B_2$. But $\partial f / \partial x_{r+1} = 0$ is a global hypersurface of degree $d-1$ in $\proj{r+1}$, hence globally: $$B_1 + B_2 = p_1^*(V({\partial f \over \partial x_{r_1}}))$$ (equality of divisors, not merely divisor classes). All this is standard classical material. (7) has an important cohomological consequence: let $C^* \subset {\cal O}_{\proj{r+1}}$ be the sheaf of ideals consisting of functions whose restriction to $Z_1$ lies in C. Then we get an exact sequence: $$0 \,\rightarrow\, {\cal O}_{\proj{r+1}}(-Z_1) \,\rightarrow\, C^* \, {\cal O}_{\proj{r+1}} \,\rightarrow\, C \, {\cal O}_{Z_1} \,\rightarrow\, 0$$ hence an exact sequence $$0 \,\rightarrow\, {\cal O}_{\proj{r+1}}(\ell-d) \,\rightarrow\, C^* {\cal O}_{\proj{r+1}}(\ell) \,\rightarrow\, p_{1,*}({\cal O}_Z(\ell H - B_2)) \,\rightarrow\, 0$$ for all integers $\ell$. But $H^1({\cal O}_{\proj{r+1}}(\ell-d)) = (0)$, hence $$H^0(C^* {\cal O}_{\proj{r+1}}(\ell)) \,\rightarrow\, H^0({\cal O}_Z(\ell H - B_2))$$ is surjective, hence \begin{eqnarray} H^0({\cal O}_Z(\ell H - B_2)) \,\subset\, \mbox{ Im}\left[ H^0({\cal O}_{\proj{r+1}}(\ell)) \,\rightarrow\, H^0({\cal O}_Z(\ell H)) \right]. \end{eqnarray} Now let us vary the projections $p_1$ and $p_2$. For each choice of $L_1$, we get a different $B_1$: call it $B_1(L_1)$, and for each choice of $L_2$, as different $B_2$: call it $B_2(L_2)$. By (5) and (8), all divisors $B_1(L_1)$ are linearly equivalent as are all divisors $B_2(L_2)$. Moreover: \begin{eqnarray*} \Bigcap{L_1}{} B_1(L_1) &=& \emptyset \\ \Bigcap{L_2}{} B_2(L_2) &=& \emptyset \end{eqnarray*} This is because, if $x \in Z$, then there is a choice of $L_1$ such that $p_1: Z \rightarrow \proj{r}$ is unramified at $y$; and a choice of $L_2$ such that $p_2(x) \in Z_1$ is smooth, hence $p_2$ is an isomorphism near $x$. Thus $$\abs{B_1(L_1)} = \abs{K_Z + (r+1) H}$$ and $$\abs{B_2(L_2)} = \abs{K_Z + (d-r-2) H}$$ are base point free linear systems. Next choose $(r+1)$ $L_2$'s, called $L_2^\alpha$, $1 \le \alpha \le r+1$, so that if $B_2^{(\alpha)} = B_2(L_2^{(\alpha)})$, then $\Bigcap{\alpha}{} B_2^{(\alpha)} = \emptyset$. Look at the Koszul complex: \begin{eqnarray*} 0 \,\rightarrow\, {\cal O}_Z(\ell H - \sum B_2^{(\alpha)}) &\rightarrow& \cdots \\ \rightarrow\, \sum_{\alpha,\beta} {\cal O}_Z(\ell H - B_2^{(\alpha)}-B_2^{(\beta)}) &\rightarrow& \sum_{\alpha}{\cal O}_Z(\ell H - B_2^{(\alpha)}) \,\rightarrow\, {\cal O}_Z(\ell H) \,\rightarrow\, 0. \end{eqnarray*} This is exact and diagram chasing gives the conclusion: $$ H^i({\cal O}_Z(\ell H - (i+1) B_2)) = (0), \mbox{ all } i \ge 1 $$ $$\Rightarrow \sum_\alpha H^0({\cal O}_Z(\ell H - B_2^{(\alpha)})) \rightarrow H^0({\cal O}_Z(\ell H)) \mbox{ surjective} $$ hence by (9) $$H^0({\cal O}_{\proj{n}}(\ell)) \rightarrow H^0({\cal O}_Z(\ell H)) \mbox{ surjective}$$ and $$H^{i+j}({\cal O}_Z(\ell H - (i+1)B_2)) = (0), \mbox{ all } i \ge 0$$ $$\Rightarrow \;\; H^j({\cal O}_Z(\ell H) = (0).$$ Now $I(Z)$ is $m$-regular if and only if $H^i({\cal I}_Z(m-i)) = (0)$, $i \ge 1$, hence if and only if $$H^0({\cal O}_{\proj{n}}(m-1)) \rightarrow H^0({\cal O}_Z(m-1)) \mbox{ surjective}$$ $$H^i({\cal O}_Z(m-i-1)) = (0), \; i \ge 1. $$ By the previous remark, this follows provided that $$H^{i+j}({\cal O}_Z((m-i-1)H - (j+1)B_2)) = (0), \mbox{ if } i,j \ge 0,\; i+j \ge 1.$$ But let us rewrite: $$(m-i-1)H-(j+1)B_2 \;\equiv\; K_Z +jB_1+(m-i-(j+1)(d-1)+r)H$$ using (5) and (8). Note that $j B_1 + \ell H$ is an ample divisor if $\ell \ge 1, j \ge 0$, because $\abs{B_1}$ is base point free. Therefore by the Kodaira Vanishing Theorem, $$H^i({\cal O}_Z (K_Z+j B_1 + \ell H)) = (0), \; i,j \ge 1, \; j \ge 0$$ and provided $m=(r+1)(d-2)+2$, this gives the required vanishing. \section{Applications} \label{applications} {}From some points of view, the first main problem of algebraic geometry is to reduce the study of a general ideal $I$ to that of prime ideals, or the study of arbitrary schemes to that of varieties. One way of doing this is to find a decomposition of the ideal into primary ideals: i.e. write it as an intersection of primary ideals. But even when non-redundancy is added, this is not unique, and usually one actually wants something less: to find its radical and perhaps write the radical as an intersection of prime ideals, or to find its top dimensional part, or to find its associated prime ideals and their multiplicities. There are really four computational problems involved here which should be treated separately: (i) eliminating the multiplicities in the ideal $I$, (ii) separating the pieces of different dimension, (iii) ``factoring'' the pieces of each dimension into irreducible components, and finally (iv) describing the original multiplicities, either numerically or by a primary ideal. Three of these four problems are the direct generalizations of the basic problems for factoring a single polynomial: we can eliminate multiple factors, getting a square-free polynomial, we can factor this into irreducible pieces and we can ask for the multiplicities with which each factor appeared in the original polynomial. There is a fifth question which arises when we work, as we always must do on a computer, over a non-algebarically closed field $k$: we can ask (v) for an extension field $k'$ of $k$ over which the irreducible components break up into absolutely irreducible components. Classical algorithms for all of these of these rely heavily on making explicit projections of $V(I)$ to lower dimensional projective spaces. This can be done either by multi-variable resultants if you want only the set-theoretic projection, or by Gr\"{o}bner\ bases with respect to the lexicographic order or an elimination order, to get the full ideal $I \cap k[X_0, \cdots, X_m]$. Recent treatments of multi-variable resultants can be found in \cite{can89}, \cite{cha91}, and a recent treatment of the basis method can be found in \cite{gtz88}. There is no evidence that either of these is an efficient method, however, and taking Gr\"{o}bner\ bases in the lexicographical order or an elimination order is often quite slow, certainly slow in the worst case. The general experience is that taking projections can be very time consuming. One reason is that the degree of the generators may go up substantially and that sparse defining polynomials may be replaced by more or less generic polynomials. A specific example is given by principally polarized abelian varieties of dimension $r$: they are defined by quadratic polynomials in $(4^r - 1)$-space, but their degree here (hence the degree of their generic projection to $\proj{r+1}$) is $4^r r!$ \cite{mum70a}. In fact, any variety is defined purely by quadratic relations in a suitable embedding \cite{mum70b}. Instead of using real computational experience, the fundamental method in theoretical computer science for analyzing complexity of algorithms is to count operations. For algebraic algorithms, the natural measure of complexity is not the number of bit operations, but the number of field operations, addition, subtraction, multiplication and (possibly) division that are used. In this sense, any methods that involve taking Gr\"{o}bner\ bases for any order on monomials will have a worst-case behavior whose complexity goes up with the regularity of the ideal hence will take ``double exponential time''. However, it appears that this worst-case behavior may in fact only concern problem (iv) -- finding the primary ideals -- and that problems (i), (ii) and (iii) may be solvable in ``single exponential time''. The idea that such algorithms should exist for finding $V(I)$ set-theoretically was proposed in the 1984 lecture on which this article is based, but turned out, in fact, to have been already proven by Chistov and Grigoriev, cf. their unpublished 1983 note \cite{cg83}. Their line of research led, in some sense, to the work of Brownawell and Koll{\'a}r, showing the single exponential bound $\left(\sqrt{I}\right)^m \subset I$ for $m = d^n$, where $d = \max(\hbox{ degrees of generators of }I)$. Based on this work, Giusti and Heintz \cite{gh91} give a singly exponentially bounded algorithm for computing ideals ${\bf q}_i$ such that $V({\bf q}_i)$ are the irreducible components of $V(I)$ (over the ground field $k$). The method depends on computing what is essentially the Chow form of each component, and leads to an ideal defining this variety but not its full ideal. In fact, their ${\bf q}_i$ may be guaranteed to be prime except for possible embedded components. A direct approach to constructing both $\sqrt{I}$ and the intersection of the top-dimensional primary components of $I$, denoted $\hbox{Top}(I)$, is given in a recent paper by Eisenbud, Huneke and Vasconcelos \cite{ehv92}. Their construction of the radical uses the Jacobian ideals, i.e. the ideals of minors of various sizes of the Jacobian matrix of generators of $I$. This is certainly the most direct approach, but, again they have trouble with possible embedded components, and must resort to ideal quotients, hence they need a Gr\"{o}bner\ basis of $I$ in the reverse lexicographic order. They compute $\hbox{Top}(I)$ as the annihilator of $\hbox{Ext}^{{\rm codim}(I)} (k[X_0,\cdots,X_n]/I, k[X_0,\cdots,X_n])$, which is readily found from a full resolution using Gr\"{o}bner\ bases. Their algorithm appears to be practical in some cases of interest, but still has double exponential time worst-case behavior. It may turn out to be most effective in practice to combine these ideas. Often an ideal under study has regularity far smaller than the geometric degree of its top dimensional components; projecting these components to a hypersurface requires computing in degrees up to the geometric degree, which is wasteful. On the other hand, methods such as those in \cite{ehv92} work better in low codimensions, if only because there are fewer minors to consider in the Jacobian matrix. Thus, projecting an arbitary scheme down to low codimension and then switching to direct methods may work best of all. This still does not settle the issue of the complexity of calculating $\sqrt{I}$, or, for that matter, calculating the full prime ideal of any subvariety of codimension greater than one. Chow form type methods give you an effective method of defining the set $V(I)$ but only of generating $I$ up to possible embedded components. For this reason, the two schools of research, one based on the algebra of $I$, the other based on subsets of ${\bf P}^n$ have diverged. If we knew, as discussed in the previous section, that the regularity of a reduced ideal could be bounded singly exponentially, then we could bound the degrees of the generators of $\sqrt{I}$, and, using Brownawell-Koll{\'a}r, we could determine $\sqrt{I}$ up to these degrees and get the whole ideal. But without such a bound, it is still not clear whether only $V(I)$ and not $\sqrt{I}$ can be found in worst-case single exponential time. Let's look at problem (iii). Assume you have found a reduced equidimensional $I$. To study splitting it into irreducible or absolutely irreducible pieces, we shall assume initially it is a hypersurface, i.e. $I=(f)$. Computationally, there may often be advantages to not projecting a general $I$ to a hypersurface, and we will discuss one such approach below. Geometrically, there is nothing very natural about irreducible but not absolutely irreducible varieties: from the standpoint of their properties, they behave like reducible varieties, except that, being conjugate over $k$, their components have very similar properties. If the ground field $k$ gets bigger or smaller, the set of absolutely irreducible components gets partitioned in finer or coarser ways into the $k$-components. If one has never done any calculations, one would therefore be inclined to say -- let's extend $k$ as far as needed to split our algebraic set up into absolutely irreducible components. {\it This is a very bad idea!} Unless this extension $k'$ happens to be something simple like a quadratic or cyclotomic extension of $k$, the splitting field $k'$ is usually gigantic. This is what happens if one component of $V(I)$ is defined over an extension field $k_1$ of $k$ of degree $e$, and the Galois group of $k_1/k$ is the full symmetric group, a very common occurence. Then $V(I)$ only splits completely over the Galois closure of $k_1/k$ and this has degree $e!$. The moral is: never factor unless you have to. In fact, unless you need to deal simultaneously with more than one of its irreducible components, you can proceed as follows: the function field $K = k[X_0,\cdots,X_n]/(f)$ contains as a subfield an isomorphic copy of $k_1$: you find that field as an extension $k_1 = k[y]/(p(y))$ of $k$, and solve for the equation of one irreducible component $f_1 \in k_1[X_0,\cdots,X_n]$ by the formula $\hbox{Norm}_{k_1/k}(f_1) = f$. Pursuing this point, why should one even factor the defining equation $f$ over $k$? Factoring, although it takes polynomial time \cite{lll82}, is often very slow in real time, and, unless the geometry dictates that the components be treated separately, why not leave them alone. In some situations, for instance, \cite{dd84} one may have an ideal, module or other algebraic structure defined by polynomials or matrices of polynomials over a {\it ground ring} $D = k[y]/(p(y))$, where $p$ is a square-free polynomial. Thus $D$ is a direct sum of extension fields, but there is no need to factor $p$ or split up $D$ until the calculations take different turns with the structures over different pieces of $Spec(D)$. The standard methods of factoring in computer algebra all depend on (i) writing the polynomial over a ring, finitely generated over $\bf Z$, and reducing modulo a maximal ideal $\bf m$ in that ring, obtaining a polynomial over a finite field; and (ii) restricting to a line $L$, i.e. substituting $X_i = a_i X_0 + b_i, i \geq 1$ for all but one variable, obtaining a polynomial in one variable over a finite field. This is then factored and then, using Hensel's lemma, one lifts this factorization modulo higher powers of $\bf m$ and of the linear space $L$. One then checks whether a coarsened version of this factorization works for $f$. This is all really the arithmetic of various small fields. Geometrically, every polynomial in one variable factors over a suitable extension field and the question of counting the absolutely irreducible components of a variety is really more elementary: it is fundamentally topological and not arithmetic. One should, therefore, expect there to be direct geometric ways of counting these components and separating them. Assuming $I$ is a reduced, equi-r-dimensional ideal, the direct way should be to use Serre duality, computing the cohomology $H^r(\Omega^r_{V(I)})$, where $\Omega^r_{V(I)} \subset \omega_{V(I)}$ is the subsheaf of the top-dimensional dualizing sheaf of $V(I)$ of absolutely regular $r$-forms. Its dimension will be the number of absolutely irreducible components into which $V(I)$ splits. Calculating this cohomology involves two things: algebraically resolving the ideal $I$ and geometrically resolving the singularities of $V(I)$ far enough to work out $\Omega^r_{V(I)}$. Classically, when $I = (f)$ was principal, $\Omega^r_{V(I)}$ was called its ideal of ``subadjoint'' polynomials. There is one case where this is quite elementary and has been carried out: this is for plane curves. One can see immediately what is happening by remarking that a non-singular plane curve is automatically absolutely irreducible, hence one should expect that its singularities control its decomposition into absolutely irreducible pieces. Indeed, if ${\bf C} \subset k[X_0,X_1,X_2]/(f)$ is the conductor ideal, then $\Omega^1_{V(f)}$ is given by the homogeneous ideal ${\bf C}$, but with degree 0 being shifted to be polynomials of degree $d-3$, $d$ the degree of $f$. To calculate $H^1$, assume $X_0$ is not zero at any singularity of $V(f)$ and look at the finite-dimensional vector space of all functions $k[X_1/X_0,X_2/X_0]/({\bf C} + (f))$ modulo the restrictions $g/(X_0^{d-3})$ for all homogeneous polynomials $g$ of degree $d-3$. This will be canonically the space of functions on the set of components of $V(f)$ with sum $0$. In particular, it is $(0)$ if and only if $V(f)$ is absolutely irreducible. This follows from standard exact sequences and duality theory. It was known classically as the Cayley-Bacharach theorem, for the special case where $V(f)$ was smooth except for a finite number of ordinary double points. It states that $V(f)$ is absolutely irreducible if and only if for every double point $P$, there is a curve of degree $d-3$ passing through all the double points except $P$. This example gives one instance where a deeper computational analysis of varieties requires a computation of its resolution of singularities. We believe that there will be many instances where practical problems will require such an analysis. In many ways, resolution theorems look quite algorithmic, and, for instance, Abhyankar and his school have been approaching the problem in this way \cite{abh82}, as have Bierstone and Milman \cite{bm91}. However, the only case of resolution of singularities to be fully analyzed in the sense of computational complexity is that of plane curves. This has been done by Teitelbaum \cite{tei89}, \cite{tei90}. His analysis is notable in various ways: he is extremely careful about not making unnecessary factorizations, let alone taking unnecessary field extensions, and uses the ``$D$'' formalism discussed above. He describes his algorithm so precisely that it would be trivial to convert it to code and, as a result, he gives excellent bounds on its complexity. \newcommand{\noopsort}[1]{} \newcommand{\printfirst}[2]{#1} \newcommand{\singleletter}[1]{#1} \newcommand{\switchargs}[2]{#2#1} \ifx\undefined\leavevmode\hbox to3em{\hrulefill}\, \newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,} \fi

9304

alg-geom/9304007

en

https://arxiv.org/abs/alg-geom/9304007

alg-geom/9304007

David R. Morrison

David R. Morrison

Compactifications of moduli spaces inspired by mirror symmetry

25 pp., LaTeX 2.09 with AmS-Fonts

Journ\'ees de G\'eom\'etrie Alg\'ebrique d'Orsay (Juillet 1992), Ast\'erisque, vol. 218, 1993, pp. 243-271

DUK-M-93-06

We study moduli spaces of nonlinear sigma-models on Calabi-Yau manifolds, using the one-loop semiclassical approximation. The data being parameterized includes a choice of complex structure on the manifold, as well as some ``extra structure'' described by means of classes in H^2. The expectation that this moduli space is well-behaved in these ``extra structure'' directions leads us to formulate a simple and compelling conjecture about the action of the automorphism group on the K\"ahler cone. If true, it allows one to apply Looijenga's ``semi-toric'' technique to construct a partial compactification of the moduli space. We explore the implications which this construction has concerning the properties of the moduli space of complex structures on a ``mirror partner'' of the original Calabi-Yau manifold. We also discuss how a similarity which might have been noticed between certain work of Mumford and of Mori from the 1970's produces (with hindsight) evidence for mirror symmetry which was available in 1979. [The author is willing to mail hardcopy preprints upon request.]

alg-geom

\section*{References\@mkboth {REFERENCES}{REFERENCES}}\small\list {\arabic{enumi}.}{\settowidth\labelwidth{[#1]}\leftmargin\labelwidth \advance\leftmargin\labelsep \usecounter{enumi}} \def\hskip .11em plus .33em minus .07em{\hskip .11em plus .33em minus .07em} \sloppy\clubpenalty4000\widowpenalty4000 \sfcode`\.=1000\relax} \let\endthebibliography=\endlist \renewcommand{\Large}{\large} \renewcommand{\LARGE}{\large\bf} \newcommand{\text}[1]{\mathchoice{\mbox{\rm #1}}{\mbox{\rm #1}} {\mbox{\scriptsize\rm #1}}{\mbox{\tiny\rm #1}}} \newcommand{\operatorname}[1]{\mathop{\rm #1}\nolimits} \newcommand{\stackrel}{\stackrel} \newcommand{\mbox{\ \ mod\ }}{\mbox{\ \ mod\ }} \newcommand{\pfit}[1]{{\it #1:\/}} \newcommand{Q.E.D.}{Q.E.D.} \newcommand{\eqref}[1]{(\ref{#1})} \newcommand{\cdots}{\cdots} \newcommand{\binom}[2]{{{#1}\choose{#2}}} \newcommand{\fracwithdelims}[4]{\left#1\frac{#3}{#4}\right#2} \newenvironment{pf}{\noindent\pfit{Proof} }{Q.E.D.\par\par\medskip\par} \newenvironment{pf*}{\noindent\pfit }{Q.E.D.\par\par\medskip\par} \newenvironment{Sb}{_\bgroup\baselineskip10pt \def\ccr{\cr}\let\\\ccr\vbox\bgroup\ialign\bgroup\hfil $\scriptstyle{##}$\hfil\crcr}{\crcr\egroup\egroup\egroup } \newenvironment{bmatrix}{\left[\begin{array}{ccccc}}{\end{array}\right]} \let\pmatrix\relax \newenvironment{pmatrix}{\left(\begin{array}{ccccc}}{\end{array}\right)} \def\cases{\left\{\def\arraystretch{1.2}\hskip-\arraycolsep \array{l@{\quad}l}} \def\endarray\hskip-\arraycolsep\right.{\endarray\hskip-\arraycolsep\right.} \def\maketitle{\par \begingroup \def\fnsymbol{footnote}{\fnsymbol{footnote}} \def\@makefnmark{\hbox to 0pt{$^{\@thefnmark}$\hss}} \if@twocolumn \twocolumn[\@maketitle] \else \newpage \global\@topnum\z@ \@maketitle \fi\thispagestyle{empty}\@thanks \endgroup \setcounter{footnote}{0} \let\maketitle\relax \let\@maketitle\relax \gdef\@thanks{}\gdef\@author{}\gdef\@title{}\let\thanks\relax} \def\@address{} \def\address#1{\gdef\@address{#1}} \def\@email{} \def\email#1{\gdef\@email{#1}} \def@{@} \def\bigskip\par\noindent \footnotesize {\sc \@address{\bigskip\par\noindent \footnotesize {\sc \@address} \par\noindent {\it E-mail address:} \@email} \date{} \def\dedicatory#1{\def\@date{\normalsize\it#1}} \def\subjclass#1{\def\@thefnmark{}\@footnotetext{1991 {\it Mathematics Subject Classification.} #1}} \def\keywords#1{\def\@thefnmark{}\@footnotetext{ {\it Key words and phrases.} #1}} \def\thanks#1{\def\@thefnmark{}\@footnotetext{#1}} \def\@begintheorem#1#2{\trivlist \item[\hskip \labelsep{\sc #1\ #2}]\it} \def\@opargbegintheorem#1#2#3{\trivlist \item[\hskip \labelsep{\sc #1\ #2\ (#3)}]\it} \makeatother \newtheorem{definition}{Definition} \newtheorem{proposition}{Proposition} \newtheorem{ConeConjecture}{The Cone Conjecture} \renewcommand{\theConeConjecture}{} \newtheorem{ConvergenceConjecture}{The Convergence Conjecture} \renewcommand{\theConvergenceConjecture}{} \newtheorem{mirrorconjecture}{The Mathematical Mirror Symmetry Conjecture} \renewcommand{\themirrorconjecture}{} \renewcommand{\mirrorconjecture}{\trivlist \item[\hskip \labelsep{\sc The Mathematical Mirror Symmetry Conjecture}]\it} \newtheorem{minconjecture}{The Minimal Compactification Conjecture} \renewcommand{\theminconjecture}{} \newcommand{\frak}[1]{\mathchoice{\mbox{\frakturfont #1}}{\mbox{\frakturfont #1}} {\mbox{\smallfrakturfont #1}}{\mbox{\smallfrakturfont #1}}} \newcommand{\Bbb}[1]{\mathchoice{\mbox{\bbbfont #1}}{\mbox{\bbbfont #1}} {\mbox{\smallbbbfont #1}}{\mbox{\tinybbbfont #1}}} \newcommand{\Bbb C}{\Bbb C} \newcommand{\Bbb E}{\Bbb E} \renewcommand{\P}{\Bbb P} \newcommand{\Bbb Q}{\Bbb Q} \newcommand{\Bbb R}{\Bbb R} \newcommand{\Bbb V}{\Bbb V} \newcommand{\Bbb Z}{\Bbb Z} \newcommand{{\cal B}}{{\cal B}} \newcommand{{\cal C}}{{\cal C}} \newcommand{{\cal D}}{{\cal D}} \newcommand{{\cal F}}{{\cal F}} \newcommand{{\cal K}}{{\cal K}} \newcommand{{\cal M}}{{\cal M}} \newcommand{{\cal O}}{{\cal O}} \newcommand{{\cal P}}{{\cal P}} \newcommand{{\cal S}}{{\cal S}} \newcommand{{\cal X}}{{\cal X}} \newcommand{{\cal Y}}{{\cal Y}} \renewcommand{\aa}{\frak A} \newcommand{\D}{\Delta} \newcommand{\h}{\frak H} \renewcommand{\O}{\frak O} \newcommand{\mathbin{\mbox{\bbbfont\char"6E}}}{\mathbin{\mbox{\bbbfont\char"6E}}} \newcommand{\mathbin{\mbox{\bbbfont\char"6F}}}{\mathbin{\mbox{\bbbfont\char"6F}}} \newcommand{\rtimes}{\mathbin{\mbox{\bbbfont\char"6F}}} \newcommand{d\mskip0.5mu\log}{d\mskip0.5mu\log} \newcommand{\nabla}{\nabla} \newcommand{-}{-} \newcommand{ : }{ : } \newcommand{\operatorname{Aff}}{\operatorname{Aff}} \newcommand{\operatorname{Aut}}{\operatorname{Aut}} \newcommand{\operatorname{DLP}}{\operatorname{DLP}} \newcommand{\operatorname{GL}}{\operatorname{GL}} \newcommand{\operatorname{PGL\mbox{\mathsurround=0pt$^+$}}}{\operatorname{PGL\mbox{\mathsurround=0pt$^+$}}} \newcommand{\operatorname{PSL}}{\operatorname{PSL}} \newcommand{\operatorname{SL}}{\operatorname{SL}} \newcommand{\operatorname{SU}}{\operatorname{SU}} \renewcommand{\Im}{\operatorname{Im}} \newcommand{\operatorname{Ker}}{\operatorname{Ker}} \newcommand{\operatorname{Lie}}{\operatorname{Lie}} \newcommand{\operatorname{rk}}{\operatorname{rk}} \newcommand{\operatorname{diag}}{\operatorname{diag}} \newcommand{locally rational polyhedral}{locally rational polyhedral} \newcommand{rational polyhedral}{rational polyhedral} \newcommand{Satake-Baily-Borel}{Satake-Baily-Borel} \newcommand{nonlinear sigma-model}{nonlinear sigma-model} \begin{document} \title{Compactifications of moduli spaces \\inspired by mirror symmetry} \author{David R. Morrison} \date{} \maketitle \address{Department of Mathematics, Duke University, Durham, NC 27708-0320 USA, and \\School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540 USA.} \email{drm@ math.duke.edu} The study of moduli spaces by means of the period mapping has found its greatest success for moduli spaces of varieties with trivial canonical bundle, or more generally, varieties with Kodaira dimension zero. Now these moduli spaces play a pivotal r\^ole in the classification theory of algebraic varieties, since varieties with nonnegative Kodaira dimension which are not of general type admit birational fibrations by varieties of Kodaira dimension zero. Since such fibrations typically include singular fibers as well as smooth ones, it is important to understand how to compactify the corresponding moduli spaces (and if possible, to give geometric interpretations to the boundary of the compactification). Note that because of the possibility of blowing up along the boundary, abstract compactifications of moduli spaces are far from unique. The hope that the period mapping could be used to construct compactifications of moduli spaces was given concrete expression in some conjectures of Griffiths \cite[\S 9]{gr-bull} and others in the late 1960's. In particular, Griffiths conjectured that there would be an analogue of the Satake-Baily-Borel\ compactifications of arithmetic quotients of bounded symmetric domains, with some kind of ``minimality'' property among compactifications. Although there has been much progress since \cite{gr-bull} in understanding the behavior of period mappings near the boundary of moduli, compactifications of this type have not been constructed, other than in special cases. In the case of algebraic K3 surfaces, the moduli spaces themselves are arithmetic quotients of bounded symmetric domains, so each has a minimal (Satake-Baily-Borel) compactification. In studying the moduli spaces for K3 surfaces of low degree in the early 1980's, Looijenga \cite{Looijenga} found that the Satake-Baily-Borel\ compactification needed to be blown up slightly in order to give a good geometric interpretation to the boundary. He introduced a class of compactifications, the {\em semi-toric compactifications}, which includes the ones with a good geometric interpretation. In higher dimension, the moduli spaces are not expected to be arithmetic quotients of symmetric domains, so different techniques are needed. The study of these moduli spaces has received renewed attention recently, due to the discovery by theoretical physicists of a phenomenon called ``mirror symmetry''. One of the predictions of mirror symmetry is that the moduli space for a variety with trivial canonical bundle, which parameterizes the possible complex structures on the underlying differentiable manifold, should {\em also}\/ serve as the parameter space for a very different kind of structure on a ``mirror partner''---another variety with trivial canonical bundle. This alternate description of the moduli space turns out to be well-adapted to analysis by Looijenga's techniques; we carry out that analysis here. In the physicists' formulation, one fixes a differentiable manifold $X$ which admits complex structures with trivial canonical bundle (a ``Calabi-Yau manifold''), and studies something called nonlinear sigma-model s on $X$. Such an object can be determined by specifying both a complex structure on $X$, and some ``extra structure'' (cf.~\cite{guide}); the moduli space of interest to the physicists parameterizes the choice of both. The r\^oles of the ``complex structure'' and ``extra structure'' subspaces of this parameter space are reversed when $X$ is replaced by a mirror partner. Most aspects of mirror symmetry must be regarded as conjectural by mathematicians at the moment, and in this paper we conjecture much more than we prove. In a companion paper \cite{htams}, we consider formally degenerating variations of Hodge structure near normal crossing boundary points of the moduli space, and describe a conjectural link to the numbers of rational curves of various degrees on a mirror partner. In the present paper, we extend these considerations to boundary points which are {\em not}\/ of normal crossing type, and formulate a mathematical mirror symmetry conjecture in greater generality. In addition, we find that when studied from the mirror perspective, a ``minimal'' partial compactification of the moduli space---analogous to the Satake-Baily-Borel\ compactification---appears very natural, provided that several conjectures about the mirror partner hold. One of our conjectures is a simple and compelling statement about the K\"ahler cone of Calabi-Yau varieties. If true, it clarifies the r\^ole of some of the ``infinite discrete'' structures on such a variety, which nevertheless seem to be finite modulo automorphisms. We have verified this conjecture in a nontrivial case in joint work with A.~Grassi \cite{grassi-morrison}. The plan of the paper is as follows. In the first several sections, we review Looijenga's compactifications, describe a concrete example, and add a refinement to the theory in the form of a flat connection on the holomorphic cotangent bundle of the moduli space. We then turn to the description of the larger moduli spaces of interest to physicists, and analyze certain boundary points of those spaces. Towards the end of the paper, we explore the mathematical implications of mirror symmetry in constructing compactifications of moduli spaces. We close by discussing some evidence for mirror symmetry which (in hindsight) was available in 1979. \section{Semi-toric compactifications} \label{STC} The first methods for compactifying arithmetic quotients of bounded symmetric domains were found by Satake \cite{satake1} and Baily-Borel \cite{baily-borel}. The compactification produced by their methods, often called the {\em Satake-Baily-Borel\ compactification}, adds a ``minimal'' amount to the quotient space in completing it to a compact complex analytic space. This minimality can be made quite precise, thanks to the Borel extension theorem \cite{borel-extension} which guarantees that for a given quotient of a bounded symmetric domain by an arithmetic group, any compactification whose boundary is a divisor with normal crossings will map to the Satake-Baily-Borel\ compactification (provided that the arithmetic group is torsion-free). Satake-Baily-Borel\ compactifications have rather bad singularities on their boundaries, so they are difficult to study in detail. Explicit resolutions of singularities for these compactifications were constructed in special cases by Igusa \cite{igusa}, Hemperly \cite{hemperly}, and Hirzebruch \cite{hirzebruch}; the general case was subsequently treated by Satake \cite{satake2} and Ash et.~al \cite{AMRT}. The methods of \cite{AMRT} produce what are usually called {\em Mumford compactifications}---these are smooth, and have a divisor with normal crossings on the boundary, but unfortunately many choices must be made in their construction. The Satake-Baily-Borel\ compactification, on the other hand, is canonical. Some years later, Looijenga \cite{Looijenga} generalized both the Satake-Baily-Borel\ and the Mumford compactifications by means of a construction which can be applied widely, not just in the case of arithmetic quotients of bounded symmetric domains. Looijenga's construction gives partial compactifications of certain quotients of tube domains by discrete group actions. A {\em tube domain}\/ is the set of points in a complex vector space whose imaginary parts are constrained to lie in a specified cone. Whereas Ash et al.\ \cite{AMRT} had only considered homogeneous self-adjoint cones, Looijenga showed that analogous constructions could be made in a more general context. The starting point is a free $\Bbb Z$-module $L$ of finite rank, and the real vector space $L_{\Bbb R}:=L\otimes\Bbb R$ which it spans. A convex cone $\sigma$ in $L_{\Bbb R}$ is {\em strongly convex}\/ if $\sigma\cap(-\sigma)\subset\{0\}$. A convex cone is {\em generated}\/ by the set $S$ if every element in the cone can be written as a linear combination of the elements of $S$ with nonnegative coefficients. And a convex cone is {\em rational polyhedral}\/ if it is generated by a finite subset of the rational vector space $L_{\Bbb Q}:=L\otimes\Bbb Q$. Let ${\cal C}\subset L_{\Bbb R}$ be an open strongly convex cone, and let $\Gamma\subset\operatorname{Aff}(L)$ be a group of affine-linear transformations of $L$ which contains the translation subgroup $L$ of $\operatorname{Aff}(L)$. If the linear part $\Gamma_0:=\Gamma/L\subset\operatorname{GL}(L)$ of $\Gamma$ preserves the cone ${\cal C}$, then the group $\Gamma$ acts on the tube domain ${\cal D}:=L_{\Bbb R}+i\,{\cal C}$. We wish to partially compactify the quotient space ${\cal D}/\Gamma$, including limit points for all paths moving out towards infinity in the tube domain. Looijenga formulated a condition which guarantees the existence of partial compactifications of this kind. Let ${\cal C}_+$ be the convex hull of $\overline{{\cal C}}\cap L_{\Bbb Q}$. Following \cite{Looijenga}, we say that $(L_{\Bbb Q},{\cal C},\Gamma_0)$ is {\em admissible}\/ if there exists a rational polyhedral cone $\Pi\subset{\cal C}_+$ such that $\Gamma_0.\Pi={\cal C}_+$. Given an admissible triple $(L_{\Bbb Q},{\cal C},\Gamma_0)$, the (somewhat cumbersome) data needed to specify one of Looijenga's partial compactifications is as follows.\footnote{We have modified Looijenga's definition slightly, so that the use of the term ``face'' is the standard one (cf.~\cite{rockafellar}): a subset ${\cal F}$ of a convex set ${\cal S}$ is a {\em face}\/ of ${\cal S}$ if every closed line segment in ${\cal S}$ which has one of its relative interior points lying in ${\cal F}$ also has both endpoints lying in ${\cal F}$.} \begin{definition} {\rm \cite{Looijenga}} A\/ {\em locally rational polyhedral\ decomposition} of ${\cal C}_+$ is a collection ${\cal P}$ of strongly convex cones such that \begin{itemize} \item[(i)] ${\cal C}_+$ is the disjoint union of the cones belonging to ${\cal P}$, \item[(ii)] for every $\sigma\in{\cal P}$, the $\Bbb R$-span of $\sigma$ is defined over $\Bbb Q$, \item[(iii)] if $\sigma\in{\cal P}$, if $\tau$ is the relative interior of a nonempty face of the closure of $\sigma$, and if $\tau\subset{\cal C}_+$, then $\tau\in{\cal P}$, and \item[(iv)] if $\Pi$ is a rational polyhedral\ cone in ${\cal C}_+$, then $\Pi$ meets only finitely many members of ${\cal P}$. \end{itemize} \end{definition} (The decomposition ${\cal P}$ is called {\em rational polyhedral}\/ if all the cones in ${\cal P}$ are relative interiors of rational polyhedral\ cones. This is the same notion which appears in toric geometry \cite{Fulton,Oda}, except that the cones appearing in ${\cal P}$ as formulated here are the relative interiors of the cones appearing in that theory.) For each $\Gamma_0$-invariant locally rational polyhedral\ decomposition ${\cal P}$ of ${\cal C}_+$, there is a partial compactification of ${\cal D}/\Gamma$ called the {\em semi-toric (partial) compactification associated to ${\cal P}$}. This partial compactification has the form $\widehat{{\cal D}}({\cal P})/\Gamma$, where $\widehat{{\cal D}}({\cal P})$ is the disjoint union of certain strata ${\cal D}(\sigma)$ associated to the cones $\sigma$ in the decomposition. The complex dimension of the stratum ${\cal D}(\sigma)$ coincides with the real codimension of the cone $\sigma$ in $L_{\Bbb R}$; in particular, the open cones in ${\cal P}$ correspond to the $0$-dimensional strata in $\widehat{{\cal D}}({\cal P})$. The delicate points in the construction are the specification of a topology on $\widehat{{\cal D}}({\cal P})$, and the proof that the quotient space $\widehat{{\cal D}}({\cal P})/\Gamma$ has a natural structure of a normal complex analytic space. For more details, we refer the reader to \cite{Looijenga} or \cite{Sterk}. The construction has the property that if ${\cal P}'$ is a refinement of ${\cal P}$, then there is a dominant morphism $\widehat{{\cal D}}({\cal P}')/\Gamma\to\widehat{{\cal D}}({\cal P})/\Gamma$. Blowups of the boundary can be realized in this way. A bit more generally, we can partially compactify finite covers ${\cal D}/\Gamma'$ of ${\cal D}/\Gamma$, built from $L'\subset L$ of finite index, $\Gamma_0'\subset\operatorname{GL}(L')\cap\Gamma_0$ of finite index in $\Gamma_0$, and $\Gamma':=L'\rtimes\Gamma_0'$, by specifying a $\Gamma_0'$-invariant locally rational polyhedral\ decomposition ${\cal P}'$ of ${\cal C}_+$. There are two extreme cases of a semi-toric compactification. The {\em Satake-Baily-Borel\ decomposition}\/ ${\cal P}_{\text{SBB}}$ consists of all relative interiors of nonempty faces of ${\cal C}_+$. The resulting (partial) compactification $\widehat{{\cal D}}({\cal P}_{\text{SBB}})/\Gamma$ is the {\em Satake-Baily-Borel-type compactification of ${\cal D}/\Gamma$}. This is ``minimal'' among semi-toric compactifications in an obvious combinatorial sense; I do not know whether a more precise analogue of the Borel extension theorem holds in this context. The strata added to ${\cal D}/\Gamma$ include a unique $0$-dimensional stratum ${\cal D}({\cal C})$, which serves as a distinguished boundary point. At the other extreme, if every cone $\sigma\in{\cal P}$ is the relative interior of a rational polyhedral\ cone $\overline{\sigma}$ which is generated by a subset of a basis of $L$, then the associated partial compactification is smooth, and the compactifying set is a divisor with normal crossings. We call this a {\em Mumford-type}\/ semi-toric compactification. We will spell out the structure of the compactification more explicitly in this case, giving an alternative description of $\widehat{{\cal D}}({\cal P})/\Gamma$. We can think of producing a Mumford-type semi-toric compactification in two steps. In the first step, we construct a partial compactification $\widehat{{\cal D}}({\cal P})/L$ of ${\cal D}/L$ which is $\Gamma_0$-equivariant; in the second step we recover $\widehat{{\cal D}}({\cal P})/\Gamma$ as the quotient of $\widehat{{\cal D}}({\cal P})/L$ by $\Gamma_0$. The first step is done one cone at a time. Given $\sigma\in{\cal P}$, there is a basis $\ell^1,\dots,\ell^r$ of $L$ such that \[\sigma=\Bbb R_{>0}\ell^1+\cdots+\Bbb R_{>0}\ell^k\quad\text{for some $k\le r$}.\] Let $\{z_j\}$ be complex coordinates dual to $\{\ell^j\}$, so that $z=\sum z_j\ell^j$ represents a general element of $L_{\Bbb C}$. Consider the set ${\cal D}_{\sigma}:=L_{\Bbb R}+i\,\sigma$. Translations by the lattice $L$ preserve ${\cal D}_{\sigma}$, and coordinates on the quotient ${\cal D}_{\sigma}/L\subset L_{\Bbb C}/L$ can be given by $w_j=\exp(2\pi i\,z_j)$. In terms of those coordinates, ${\cal D}_{\sigma}/L$ can be described as \[{\cal D}_{\sigma}/L=\{w\in\Bbb C^r : 0<|w_j|<1 \text{ for $j\le k$}, |w_j|=1 \text{ for $j>k$}\}.\] We partially compactify this to \[({\cal D}_{\sigma}/L)^-:=\{w\in\Bbb C^r : 0\le|w_j|<1 \text{ for $j\le k$}, |w_j|=1 \text{ for $j>k$}\}.\] (We have suppressed the $\sigma$-dependence of $\ell^j$, $z_j$, $w_j$ to avoid cluttering up the notation.) We call any $w\in({\cal D}_{\sigma}/L)^-$ with $w_j=0$ for $j\le k$ a {\em distinguished limit point}\/ of ${\cal D}_{\sigma}/L$. Note that any path in ${\cal D}_{\sigma}$ along which $\Im(z_j)\to\infty$ for all $j\le k$, maps to a path in ${\cal D}_{\sigma}/L$ which approaches such a distinguished limit point. The set $\operatorname{DLP}(\sigma)$ of distinguished limit points is a subset of the stratum $\widehat{{\cal D}}(\sigma)$, and is a compact real torus of dimension $\dim_{\Bbb R}\operatorname{DLP}(\sigma)=r-k=\dim_{\Bbb C}\widehat{{\cal D}}(\sigma)$. When $k=r$, the distinguished limit point is unique, and it coincides with the $0$-dimensional stratum ${\cal D}(\sigma)$ of $\widehat{{\cal D}}({\cal P})$. The partial compactification $\widehat{{\cal D}}({\cal P})/L$ can now be described as a disjoint union of the $({\cal D}_{\sigma}/L)^-$'s, with $({\cal D}_{\tau}/L)^-$ lying in the closure of $({\cal D}_{\sigma}/L)^-$ whenever $\tau$ is the relative interior of a face of $\overline{\sigma}$. This space $\widehat{{\cal D}}({\cal P})/L$ is smooth and simply-connected, and the induced action of $\Gamma_0$ on it has no fixed points. The action of $\Gamma_0$ permutes the various $({\cal D}_{\sigma}/L)^-$'s, a finite number of which serve to cover $\widehat{{\cal D}}({\cal P})/\Gamma$ after we take the quotient by $\Gamma_0$. The structure of $({\cal D}_\sigma/L)^-$ near the distinguished limit point when $k=r$ can be formalized in the following way. For a complex manifold $T$, we say that $p$ is a {\em maximal-depth normal crossing point of $B\subset T$}\/ if there is an open neighborhood $U$ of $p$ in $T$ and an isomorphism $\varphi:U\to\D^r$ such that $\varphi(U\cap(T{-}B))=(\D^*)^r$ and $\varphi(p)=(0,\dots,0)$, where $\D$ is the unit disk, and $\D^*:=\D{-}\{0\}$. There are thus $r$ local components $B_j:=\varphi^{-1}(\{v_j=0\})$ of $B\cap U$, with $p=B_1\cap\cdots\cap B_r$, where $v_j$ is a coordinate on the $j^{\text{th}}$ disk. \section{Cusps of Hilbert modular surfaces}\label{HMS} We now give an example to illustrate the construction in the previous section: the cusps of Hilbert modular surfaces, as analyzed by Hirzebruch \cite{hirzebruch} and by Mumford in the first chapter of \cite{AMRT}. Let $\operatorname{PGL\mbox{\mathsurround=0pt$^+$}}(2,\Bbb R)=\operatorname{PSL}(2,\Bbb R)$ act by fractional linear transformations on the upper half plane $\h$. Let $K$ be a real quadratic field with ring of integers $\O_K$, and let $\operatorname{PGL\mbox{\mathsurround=0pt$^+$}}(2,K)$ be the group of invertible $2\times2$ matrices with entries in $K$ whose determinant is mapped to a positive number under both embeddings of $K$ into $\Bbb R$, modulo scalar multiples of the identity matrix. The map $\Phi:K\to\Bbb R^2$ given by the two embeddings of $K$ into $\Bbb R$ induces an action of $\operatorname{PGL\mbox{\mathsurround=0pt$^+$}}(2,K)$ on $\h\times\h$. A {\em Hilbert modular surface}\/ is an algebraic surface of the form $\h\times\h/\Gamma$ for some arithmetic group $\Gamma\subset\operatorname{PGL\mbox{\mathsurround=0pt$^+$}}(2,K)$ (that is, a group commensurable with $\operatorname{PGL\mbox{\mathsurround=0pt$^+$}}(2,\O_K)$), often assumed to be torsion-free. The Satake-Baily-Borel\ compactification of a Hilbert modular surface adds a finite number of compactification points, called {\em cusps}. Small deleted neighborhoods of such points have inverse images in $\h\times\h$ whose $\Gamma$-stabilizer is a parabolic subgroup $\Gamma_{\text{par}}$ of the form \[\Gamma_{\text{par}}=\{\begin{pmatrix}\varepsilon^k & a \\ 0 & 1 \end{pmatrix} : k\in\Bbb Z,a\in\aa \},\] where $\aa\subset\O_K$ is an ideal, and $\varepsilon\in\O_K^\times$ is a totally positive unit such that $\varepsilon\,\aa=\aa$. We can analyze a neighborhood of a cusp by studying appropriate partial compactifications of $\h\times\h/\Gamma_{\text{par}}$. \setlength{\unitlength}{1 true cm} \begin{figure}[t] \begin{center} \begin{picture}(10,5)(0,0) \put(2,3){\mbox{Insert here the figure from p.~52 of \cite{AMRT}}} \end{picture} \end{center} \caption{} \end{figure} The elements in $\Gamma_{\text{par}}$ with $k=0$ form the translation subgroup, which we identify with $\aa$. This is a free abelian group of rank 2. Let $(\alpha,\alpha')$, $(\beta,\beta')$ be a $\Bbb Z$-basis of $\Phi(\aa)$. Define a map $\h\times\h\to\Bbb C^2$ by \[(w_1,w_2)\mapsto\frac1{\alpha\beta'-\alpha'\beta} (\beta'w_1-\beta w_2,-\alpha'w_1+\alpha w_2),\] and let ${\cal D}$ denote the image of $\h\times\h$ in $\Bbb C^2$. Under this map, $\Phi(\aa)$ is sent to the standard lattice $L:=\Bbb Z^2$, and $\Phi(\Gamma_{\text{par}})$ is sent to a subgroup of $\operatorname{Aff}(L)$ with the translation subgroup $\aa$ of $\Gamma_{\text{par}}$ mapped to the translation subgroup $L$ of $\operatorname{Aff}(L)$. As in section \ref{STC}, we form the quotient in two steps: first take the quotient $\h\times\h/\aa={\cal D}/L$, and then take the quotient of the resulting space by the group $\Gamma_0:=\Gamma_{\text{par}}/\aa$. Mumford shows how to partially compactify the space ${\cal D}/L\subset L\otimes\Bbb C^*=(\Bbb C^*)^2$ in a $\Gamma_0$-equivariant way, so that the quotient by $\Gamma_0$ gives the desired partial compactification of $\h\times\h/\Gamma_{\text{par}}$. The map of $\h\times\h\to\Bbb C^2$ was designed so that the image would be a tube domain ${\cal D}:=\Bbb R^2+i\,{\cal C}$, where ${\cal C}$ is the cone \[{\cal C}=\{(y_1,y_2) : \alpha y_1 + \beta y_2>0, \alpha'y_1+\beta'y_2>0\}.\] The boundary lines of the closure $\overline{{\cal C}}$ have irrational slope, and in fact ${\cal C}_+={\cal C}$ is an open convex cone. To construct a $\Gamma_0$-invariant rational polyhedral\ decomposition ${\cal P}$, let $\Sigma$ be the convex hull of ${\cal C}\cap\Phi(\aa)$. The vertices of $\Sigma$ form a countable set $\{v_j\}_{j\in\Bbb Z}$ which can be numbered so that the edges of $\Sigma$ are exactly the line segments $\overline{v_jv_{j+1}}$. If we let $\sigma_j$ be the relative interior of the cone on $\overline{v_jv_{j+1}}$, and let $\tau_j$ be the relative interior of the cone on $v_j$, then ${\cal P}:=\{\sigma_j\}_{j\in\Bbb Z}\cup\{\tau_j\}_{j\in\Bbb Z}$ is a $\Gamma_0$-invariant rational polyhedral\ decomposition. An explicit example of this construction is illustrated on p.~52 of \cite{AMRT}, reproduced as figure 1 of this paper. The resulting partial compactification of ${\cal D}/L$ adds a point $p_j$ for each $\sigma_j$, and a curve $B_j\cong\P^1$ for each $\tau_j$, with $B_j\cap B_{j+1}=p_j$. This can be pictured as an ``infinite chain'' of $\P^1$'s, as in the top of figure 2 (which is also reproduced from \cite{AMRT}, p.~46). The generator $[\operatorname{diag}(\varepsilon,1)]$ of $\Gamma_0=\Gamma_{\text{par}}/\aa$ acts by sending $v_j$ to $v_{j+m}$ for some fixed $m$. Taking the quotient by $\Gamma_0$ leaves us with a ``cycle'' of rational curves, of length $m$ (as depicted in the bottom of figure 2). We arrive at Hirzebruch's description of the resolution of the cusps. \begin{figure}[t] \begin{center} \begin{picture}(10,5)(0,0) \put(2,3){\mbox{Insert here the figure from p.~46 of \cite{AMRT}}} \end{picture} \end{center} \caption{} \end{figure} \bigskip Conversely, suppose we are given a normal surface singularity $p\in \overline{S}$ (with $\overline{S}$ a small neighborhood of $p$) which has a resolution of singularities $f:T\to \overline{S}$ such that $B:=f^{-1}(p)$ is a cycle of rational curves, that is, $B=B_1+\cdots+B_m$ is a divisor with normal crossings such that $B_j$ only meets $B_{j\pm1}$, with subscripts calculated mod $m$. Much of the structure above can be recovered from this information alone. In fact, by a theorem of Laufer \cite{laufer} these singularities are {\em taut}, which means that the isomorphism type is determined by the resolution data. We will work out in detail some aspects of this tautness, in preparation for a general construction in the next section. The starting point is Wagreich's calculation \cite{wagreich} of the local fundamental group $\pi_1(\overline{S}-p)$ for such singularities, which goes as follows. Let $S:=\overline{S}-p=T-B$. The natural map $\iota:\pi_1(S)\to\pi_1(T)$ induced by the inclusion $S\subset T$ is surjective. Since $T$ retracts onto a cycle of $\P^1$'s, the group $\pi_1(T)\cong\pi_1(S^1)$ is infinite cyclic, and the universal cover $\widehat{T}$ of $T$ contains an infinite chain $\widehat{B}=\cdots+\widehat{B}_j+\widehat{B}_{j+1}+\cdots$ of $\P^1$'s lying over the cycle $B$. The kernel of $\iota$ is $\pi_1(\widehat{T}-\widehat{B})$, and by a result of Mumford \cite{mumford} this is a free abelian group generated by loops around any pair of adjacent components $\widehat{B}_j$, $\widehat{B}_{j+1}$ of $\widehat{B}$. In this way, we recover the two steps of the quotient construction, and the compactification $\widehat{T}$ of the intermediate quotient $\widehat{T}-\widehat{B}$. Let $\widehat{S}$ be the universal cover of $S$ (and of $\widehat{T}-\widehat{B}$). To complete the discussion of tautness, we should exhibit an isomorphism between $\widehat{S}$ and an open subset of $\h\times\h$, which descends to a $\pi_1(T)$-equivariant map $(\widehat{T}-\widehat{B})\to(\h\times\h)/\aa$. The easiest way to do this is to consider an extra piece of structure on $p\in \overline{S}$: a flat connection on the holomorphic cotangent bundle $\Omega^1_{S}$. We discuss this structure, and how to use it to determine the mapping from $\widehat{S}$ to $\h\times\h={\cal D}$, in the next section. (To give a complete proof of Laufer's tautness result along these lines, we would also need to show how the connection is to be constructed; we will not attempt to do that here.) \section{The flat connection} \label{flat} Let $(L_{\Bbb Q},{\cal C},\Gamma_0)$ be an admissible triple, with associated tube domain ${\cal D}=L_{\Bbb R}+i\,{\cal C}$ and discrete group $\Gamma=L\rtimes\Gamma_0\subset\operatorname{Aff}(L)$. We will define a flat connection on the holomorphic cotangent bundle of the quotient space ${\cal D}/\Gamma$. The intermediate quotient space ${\cal D}/L$ is an open subset of the algebraic torus $L_{\Bbb C}/L=L\otimes_{\Bbb Z}\Bbb C^*\cong(\Bbb C^*)^{\operatorname{rk}(L)}$. We identify the dual of the Lie algebra $\operatorname{Lie}(L_{\Bbb C}/L)^*$ of that torus with the space of right-invariant one-forms on the group $L_{\Bbb C}/L$. Any basis of $\operatorname{Lie}(L_{\Bbb C}/L)^*$, when regarded as a subset of the space of global sections of the sheaf $\Omega^1_{L_{\Bbb C}/L}$, freely generates that sheaf at any point. We can therefore define a connection $\nabla_{\text{toric}}$ on $\Omega^1_{L_{\Bbb C}/L}$, the {\em toric connection}, by the requirement that $\nabla_{\text{toric}}(\alpha)=0$ for every $\alpha\in\operatorname{Lie}(L_{\Bbb C}/L)^*$. Since the group $L_{\Bbb C}/L$ is abelian, the connection $\nabla_{\text{toric}}$ is flat. The action of $\operatorname{Aff}(L)$ on $L_{\Bbb C}$ descends to an action of $\operatorname{GL}(L)$ on $L_{\Bbb C}/L$ which preserves the space of right-invariant one-forms. In particular, the $\operatorname{GL}(L)$-action will be compatible with the toric connection. Thus, if we restrict $\nabla_{\text{toric}}$ to ${\cal D}/L$, it commutes with the action of $\Gamma_0$ and induces a connection on the holomorphic cotangent bundle of $({\cal D}/L)/\Gamma_0={\cal D}/\Gamma$, still denoted by $\nabla_{\text{toric}}$. Let $\sigma\subset L_{\Bbb R}$ be the relative interior of a rational polyhedral cone which is generated by a basis $\ell^1,\dots,\ell^r$ of $L$, and let $z_1,\dots,z_r$ be the coordinates on $L_{\Bbb C}$ dual to $\{\ell^j\}$. The one-forms $d\mskip0.5mu\log w_j:=2\pi i\,dz_j$ are right-invariant one-forms on $L_{\Bbb C}/L$ which serve as a basis of $\operatorname{Lie}(L_{\Bbb C}/L)^*$. If we compactify the open set ${\cal D}_\sigma/L\subset L_{\Bbb C}/L$ to $U_\sigma:=({\cal D}_\sigma/L)^-$, then the forms $d\mskip0.5mu\log w_j$ extend to meromorphic one-forms on $U_\sigma$ with poles along the boundary $B_\sigma:=({\cal D}_\sigma/L)^--({\cal D}_\sigma/L)$. In fact, the forms $d\mskip0.5mu\log w_1,\dots,d\mskip0.5mu\log w_r$ freely generate the sheaf $\Omega^1_{U_\sigma}(\log B_\sigma)$ as an ${\cal O}_{U_\sigma}$-module. The flat connection $\nabla_{\text{toric}}$ therefore extends to a flat connection on $\Omega^1_{U_\sigma}(\log B_\sigma)$ for which the $d\mskip0.5mu\log w_j$ are flat sections. Note that the connection does {\em not}\/ acquire singularities along the boundary, but extends as a regular connection to the sheaf of logarithmic differentials. If ${\cal P}$ is a rational polyhedral\ decomposition of ${\cal C}_+$, we get in this way an extension of the flat connection $\nabla_{\text{toric}}$ from $\Omega^1_{{\cal D}/L}$ to the sheaf of logarithmic differentials on $\widehat{{\cal D}}({\cal P})/L$ with poles on the boundary $(\widehat{{\cal D}}({\cal P})/L)-({\cal D}/L)$. As this extended connection still commutes with $\Gamma_0$, there is an induced extension of $\nabla_{\text{toric}}$ from $\Omega^1_{{\cal D}/\Gamma}$ to $\Omega^1_{\widehat{{\cal D}}({\cal P})/\Gamma}(\log{\cal B})$, where ${\cal B}:= (\widehat{{\cal D}}({\cal P})/\Gamma)-({\cal D}/\Gamma)$. This holds for any Mumford-type semi-toric compactification. The existence of this toric connection on ${\cal D}/\Gamma$ depends in an essential way on $\Gamma$ being a group of affine-linear transformations of $L$. If ${\cal D}$ admits an action by a larger group $\Gamma_{\text{big}}$ which includes discrete symmetries that do not lie in $\operatorname{Aff}(L)$, then $\nabla_{\text{toric}}$ may fail to descend to the quotient ${\cal D}/\Gamma_{\text{big}}$. For example, if $L=\Bbb Z$ acts on the upper half plane $\h$ by translations, then the associated flat connection $\nabla_{\text{toric}}$ has the property that $\nabla_{\text{toric}}(d\tau)=0$, where $\tau$ is the standard coordinate on $\h$. The flat section $d\tau$ is invariant under translations $\tau\mapsto\tau+n$, but if we apply $\nabla_{\text{toric}}$ to the pullback of the flat section $d\tau$ under the inversion $\tau\mapsto{-1}/\tau$ we get \[\nabla_{\text{toric}}(\tau^{-2}\,d\tau)=-2\tau^{-3}\,d\tau\otimes d\tau,\] which is not $0$. In particular, the connection $\nabla_{\text{toric}}$ does not descend to the $j$-line $\h/\operatorname{SL}(2,\Bbb Z)$. \bigskip We now want to explain how the abstract knowledge of the flat connection $\nabla_{\text{toric}}$ and of a Mumford-type semi-toric compactification of ${\cal D}/\Gamma$ can be used to recover the structure of ${\cal D}$ and of $\Gamma$. Suppose we are given a complex manifold $T$, a divisor with normal crossings $B$ on $T$, and a flat connection $\nabla$ on $\Omega^1_T(\log B)$. By the usual equivalence between flat connections and local systems \cite{RSP}, the flat sections of $\nabla$ determine a local system $\Bbb E$ on T. Such a local system is specified by giving its fiber $E$ at a fixed base point $\star$ (which we choose to lie in $T-B$), together with a representation of $\pi_1(T,\star)$ in $\operatorname{GL}(E)$. We first restrict the connection and the local system to $T-B$. If we pass to the universal cover $\widehat{S}$ of $T-B$, the flat sections give a global trivialization of the bundle $\Bbb E\otimes{\cal O}_{\widehat{S}}=\Omega^1_{\widehat{S}}$. There is a natural map $\operatorname{int}_\star:\widehat{S}\to E^*$ which sends $s\in\widehat{S}$ to the functional \[\alpha\mapsto\int_\star^s\widehat{\alpha},\] where $\widehat{\alpha}$ is the unique flat section of $\Bbb E$ (a holomorphic 1-form on $\widehat{S}$) such that $\widehat{\alpha}|_\star=\alpha\in E$. (Notice that if we vary the basepoint $\star$, we simply shift the image of the map by some constant vector in $E^*$.) On the other hand, if we consider $\nabla$ on $T$ and pass to the universal cover $\widehat{T}$ of $T$, the flat sections of $\Bbb E$ will trivialize the bundle $\Omega^1_{\widehat{T}}(\log\widehat{B})$, where $\widehat{B}$ is a divisor with normal crossings in $\widehat{T}$, the inverse image of $B\subset T$. We once again encounter the intermediate quotient space $\widehat{T}-\widehat{B}$, and its partial compactification $\widehat{T}$. At any maximal-depth normal crossing point $p$ of $\widehat{B}\subset\widehat{T}$, let $v_{j}=0$ define the $j^{\text{th}}$ local component $B_j$ of the boundary at $p$. There is a unique flat section $\widehat{\alpha}_j$ of $\Omega^1_{\widehat{T}}(\log\widehat{B})$, defined locally near $p$, such that $\widehat{\alpha}_j-d\mskip0.5mu\log v_j$ vanishes at $p$. It follows that $\widehat{\alpha}_j-d\mskip0.5mu\log v_j$ is a holomorphic one-form in a neighborhood of $p$, and so that $\widehat{\alpha}_1,\dots,\widehat{\alpha}_r$ is a basis for (flat) local sections of $\Omega^1_{\widehat{T}}(\log\widehat{B})$. Using the global trivialization, we may regard each $\alpha_j:=\widehat{\alpha}_j|_\star$ as an element of $E$. We let $L_p\subset E^*$ be the lattice spanned by the dual basis $\ell^1,\dots,\ell^r$ to $\alpha_1,\dots,\alpha_r$, and let $\sigma_p\subset L_p\otimes\Bbb R$ be the relative interior of the cone generated by $\ell^1,\dots,\ell^r$. If we are to recover the structure of the semi-toric compactification, we need a certain compatibility among the $L_p$'s and the $\sigma_p$'s: they should be related to a common lattice and a common cone, independent of $p$. We formalize this as follows. \begin{definition} \label{compat} We call $(T,B,\nabla)$\/ {\em compatible} provided that \begin{enumerate} \item each component of $B$ contains at least one maximal-depth normal crossing point, \item the lattices $L_p$ for maximal-depth normal crossing points $p$ all coincide with a common lattice $L\subset E^*$, \item the natural map $\operatorname{int}_\star:\widehat{S}\to E^*=L_{\Bbb C}$ descends to a map $(\widehat{T}-\widehat{B})\to(L_{\Bbb C}/L)$ which induces an isomorphism of fundamental groups, and \item the collection ${\cal P}$ of relative interiors of faces of the $\sigma_p$'s is a locally rational polyhedral\ decomposition of a strongly convex cone ${\cal C}_+$. \end{enumerate} \end{definition} Suppose that $(T,B,\nabla)$ is compatible, let ${\cal C}$ be the interior of ${\cal C}_+$, and let ${\cal D}=L_{\Bbb R}+i\,{\cal C}$. The action of $\pi_1(T)$ on $L_{\Bbb C}$ permutes the set of maximal-depth normal crossing points of $B\subset T$, and so preserves ${\cal P}$ and ${\cal C}$. Thus, $\Gamma:=\pi_1(T-B)$ acts on ${\cal D}$, and there is an induced map $(T-B)\to({\cal D}/\Gamma)$. We can now recover the compactification $T$ from this data (or at least its structure in codimension one). For any maximal-depth normal crossing boundary point $p$ of $\widehat{B}\subset\widehat{T}$, there is a neighborhood $U_p$ of $p$ in $\widehat{T}$ and a natural extension of the induced map $U_p\cap(\widehat{T}-\widehat{B})\to L_{\Bbb C}/L$ to a map $U_p\to\widehat{{\cal D}}({\cal P})/L$. We cannot tell from the behavior of these extensions what happens at ``interior'' points of boundary components (those which do not lie in any $U_p$), but we {\em can}\/ conclude that there is a meromorphic map $\widehat{T}\to\widehat{{\cal D}}({\cal P})/L$ which does not blow down any boundary components. This map is $\pi_1(T)$-equivariant, so it descends to a map $T\to\widehat{{\cal D}}({\cal P})/\Gamma$. \section{Moduli spaces of sigma-models} \label{sigma} A {\em Calabi-Yau manifold}\/ is a compact connected orientable manifold $X$ of dimension $2n$ which admits Riemannian metrics whose holonomy is contained in $\operatorname{SU}(n)$.\footnote{There is some confusion in the literature about whether ``Calabi-Yau'' should mean that the holonomy is precisely $\operatorname{SU}(n)$, or simply contained in $\operatorname{SU}(n)$. In this paper, we adopt the latter interpretation.} Given such a metric, there exist complex structures on $X$ for which the metric is K\"ahler. The holonomy condition is equivalent to requiring that this K\"ahler metric be Ricci-flat (cf.~\cite{beauville}). On the other hand, if we are given a complex structure on a Calabi-Yau manifold, then by the theorems of Calabi \cite{calabi} and Yau \cite{yau}, for each K\"ahler metric $\widetilde{g}$ there is a unique Ricci-flat K\"ahler metric $g$ whose K\"ahler form is in the same de Rham cohomology class as that of $\widetilde{g}$. (We have implicitly used the topological consequence of Ricci-flatness: Calabi-Yau manifolds have vanishing first Chern class.) Examples of Calabi-Yau manifolds are provided by the differentiable manifolds underlying smooth complex projective varieties with trivial canonical bundle. One can apply Yau's theorem to a K\"ahler metric coming from a projective embedding in order to produce a metric with holonomy contained in $\operatorname{SU}(n)$, where $n$ is the complex dimension of the variety. As explained in \cite{beauville}, if the Hodge numbers $h^{p,0}$ vanish for $0<p<n$, then the holonomy of this metric is precisely $\operatorname{SU}(n)$. Physicists have constructed a class of conformal field theories called nonlinear sigma-model s on Calabi-Yau manifolds $X$ (cf.~\cite{GSW,hubsch}). We consider here an approximation to those theories, which should be called ``one-loop semiclassical nonlinear sigma-model s''. Such an object is determined by the data of a Riemannian metric $g$ on $X$ whose holonomy is contained in $\operatorname{SU}(n)$ together with the de~Rham cohomology class $[b]\in H^2(X,\Bbb R)$ of a real closed $2$-form $b$ on $X$. Two such pairs $(g,b)$ and $(g',b')$ will determine isomorphic conformal field theories if there is a diffeomorphism $\varphi:X\to X$ such that $\varphi^*(g')=g$, and $\varphi^*([b'])-[b]\in H^2(X,\Bbb Z)$. It is therefore natural to regard the class of $[b]$ in $H^2(X,\Bbb R)/H^2(X,\Bbb Z)$ as the fundamental datum. We denote this class by $[b]\bmod\Bbb Z $. The set of all isomorphism classes of such pairs we call the {\em one-loop semiclassical nonlinear sigma-model\ moduli space}, or simply the {\em sigma-model moduli space}\/ (for short). This may differ from the actual {\em conformal field theory moduli space}, both because there may be additional isomorphisms of conformal field theories which are not visible in this geometric interpretation, and also because there may be deformations of the nonlinear sigma-model\ as a conformal field theory which do not have a sigma-model interpretation on $X$ (cf.~\cite{mmm,phases}). For our present purposes, we ignore these more delicate questions about the conformal field theory moduli space, and concentrate on the sigma-model moduli space we have defined above. We focus attention in this paper on the case in which the holonomy of the metric $g$ is precisely $SU(n)$, $n\ne2$. For each such metric, there are exactly two complex structures on $X$ for which the metric is K\"ahler (complex conjugates of each other).\footnote{More generally, as we will show elsewhere, if $h^{2,0}(X)=0$ there are only a finite number of complex structures for which $g$ is K\"ahler. The number depends on the decomposition of the holonomy representation into irreducible pieces.} Thus, there is a natural map from a double cover of the sigma-model moduli space to the usual ``complex structure moduli space'', given by assigning to $(g,b)$ one of the two complex structures for which $g$ is K\"ahler. The fibers of this map can be described as follows. If we fix a complex structure on $X$, then the corresponding fiber consists of all $B+i\,J\bmod\Bbb Z\in H^2(X,\Bbb C)/H^2(X,\Bbb Z)$ (modulo diffeomorphism) with $B$ denoting the class $[b]$, for which $J$ is the cohomology class of a K\"ahler form. (The metric $g$ is uniquely determined by $J$, by Calabi's theorem.) This quantity $B+i\,J\bmod\Bbb Z$ describes the ``extra structure'' $S$ which was alluded to in \cite{guide}. This is often called the {\em complexified K\"ahler structure}\/ on $X$ determined by $(g,b)$. The natural double cover of the sigma-model moduli space will be locally a product near $(g,b)$, with the variations of complex structure and of complexified K\"ahler structure describing the factors in the product, provided that neither the K\"ahler cone nor the group of holomorphic automorphisms ``jumps'' when the complex structure varies. (The non-jumping of the K\"ahler cone was shown to hold by Wilson \cite{wilson} in the case of holonomy $\operatorname{SU}(3)$, when the complex structure is generic.) We will tacitly assume this local product structure, and separately study the parameter spaces for the variations of complexified K\"ahler structure and of complex structure. With a fixed complex structure on $X$, the parameter space for complexified K\"ahler structures on $X$ can be described in terms of the K\"ahler cone ${\cal K}$ of $X$, and the lattice $L=H^2(X,\Bbb Z)/(\text{torsion})$. We must identify any pair of complexified K\"ahler structures which differ by a diffeomorphism that fixes the complex structure, that is, by an element of the group $\Gamma_0=\operatorname{Aut}(X)$ of holomorphic automorphisms. The natural parameter space for pairs $(g,b)$ such that $g$ is K\"ahler for the given complex structure thus has the form ${\cal D}/\Gamma$, where ${\cal D}=\{B+i\,J : J\in{\cal K}\}$ and $\Gamma=L\rtimes\Gamma_0$ is the extension of $\Gamma_0$ by the lattice translations. This is exactly the kind of space encountered in the first part of this paper: a tube domain modulo a discrete symmetry group of affine-linear transformations which includes a lattice acting by translations. A common technique in the physics literature is to consider what happens along paths $\{tz\bmod\Gamma\}_{t\to\infty}$, which go from $z\in{\cal D}$ out towards infinity in the tube domain. Many aspects of the conformal field theory can be analyzed perturbatively in $t$ along such paths. It seems reasonable to hope that such limits can be described in a common framework, based on a single partial compactification of ${\cal D}/\Gamma$. This hope (together with a bit of evidence, discussed below) leads us to conjecture that $(L_{\Bbb Q},{\cal K},\operatorname{Aut}(X))$ is an admissible triple, in order that Looijenga's methods could be applied to construct compactifications of ${\cal D}/\Gamma$. We formulate this conjecture more explicitly as follows. \begin{ConeConjecture} Let $X$ be a Calabi-Yau manifold on which a complex structure has been chosen, and suppose that $h^{2,0}(X)=0$. Let $L:=H^{2}(X,\Bbb Z)/\text{torsion}$, let ${\cal K}$ be the K\"ahler cone of $X$, let ${\cal K}_+$ be the convex hull of $\overline{{\cal K}}\cap L_{\Bbb Q}$, and let $\operatorname{Aut}(X)$ be the group of holomorphic automorphisms of $X$. Then there exists a rational polyhedral cone $\Pi\subset{\cal K}_+$ such that $\operatorname{Aut}(X).\Pi={\cal K}_+$. \end{ConeConjecture} The K\"ahler cone of $X$ can have a rather complicated structure, analyzed in the case $n=3$ by Kawamata \cite{kawamata} and Wilson \cite{wilson}. Away from classes of triple-self-intersection zero, the closed cone $\overline{{\cal K}}$ is locally rational polyhedral, but the rational faces may accumulate towards points with vanishing triple-self-intersection. The cone conjecture predicts that while the closed cone $\overline{{\cal K}}$ of $X$ may have infinitely many edges, there will only be finitely many $\operatorname{Aut}(X)$-orbits of edges. Other finiteness predictions which follow from the cone conjecture include finiteness of the set of fiber space structures on $X$, modulo automorphisms. Many of the large classes of examples, such as toric hypersurfaces, have K\"ahler cones ${\cal K}$ such that ${\cal K}_+=\overline{{\cal K}}$ is a rational polyhedral cone. For these, the cone conjecture automatically holds. A nontrivial case of the cone conjecture---Calabi-Yau threefolds which are fiber products of generic rational elliptic surfaces with section (as studied by Schoen \cite{schoen})---has been checked by Grassi and the author \cite{grassi-morrison}. In addition, Borcea \cite{borcea} has verified the finiteness of $\operatorname{Aut}(X)$-orbits of edges of $\overline{{\cal K}}$ in another nontrivial example, and Oguiso \cite{oguiso} has discussed finiteness of $\operatorname{Aut}(X)$-orbits of fiber space structures in yet another example.\footnote{Neither of these constitutes a complete verification of the cone conjecture for the threefold in question.} All three examples involve cones with an infinite number of edges. For any $X$ for which the cone conjecture holds, the K\"ahler parameter space ${\cal D}/\Gamma$ will admit both a Satake-Baily-Borel-type ``minimal'' compactification, and smooth compactifications of Mumford type built out of many cones $\sigma\subset{\cal K}$ as above. \section{Additional structures on the moduli spaces} \label{additional} Of particular interest to the physicists studying nonlinear sigma-model s has been the ``large radius limit'' in the K\"ahler parameter space. This is typically analyzed in the physics literature as follows (cf.~\cite{tsm,topgrav}). The quantities of physical interest will be invariant under translation by $L$. Many such quantities vary holomorphically with parameters, and their Fourier expansions take the form \renewcommand{\theequation}{*} \begin{equation} \label{eq} \sum_{\eta\in L^*}c_\eta \,e^{2\pi i\,z\cdot \eta}. \end{equation} The coefficients $c_\eta$ for $\eta\ne0$ are called {\em instanton contributions}\/ to the quantity \eqref{eq}, and in many cases they can be given a geometric interpretation which shows that they vanish unless $\eta$ is the class of an effective curve on $X$. A ``large radius limit'' should be a point at which instanton contributions to quantities like \eqref{eq} are suppressed \cite{greene-plesser,al2}. If we pick a basis $\ell^1,\dots,\ell^r$ of $L$ consisting of vectors which lie in the closure of the K\"ahler cone, write $\eta=\sum \eta^j\ell_j$ in terms of the basis $\{\ell_j\}$ of $L^*$ dual to $\{\ell^j\}$, and express \eqref{eq} as a power series in $w_j:=\exp(2\pi i\,z_j)$, where $\{z_j\}$ are coordinates dual to $\{\ell^j\}$, then the series expansion \renewcommand{\theequation}{**} \begin{equation} \label{eqq} \sum_{\eta\in L^*}c_\eta \,w_1^{\eta^1}\cdots w_r^{\eta^r} \end{equation} involves only terms with nonnegative exponents \cite{asp-mor}. If convergent,\footnote{From a rigorous mathematical point of view, the Fourier coefficients $c_\eta$ can often be defined and calculated, but no convergence properties of the series \eqref{eq} or \eqref{eqq} are known.} this will define a function on $({\cal D}_\sigma/L)^-$, where $\sigma$ is the relative interior of the cone generated by $\ell^1,\dots,\ell^r$. Thus, approaching the distinguished limit point of ${\cal D}_\sigma/L$ (where all $w_j$'s approach $0$) suppresses the instanton contributions, so the distinguished limit point is a good candidate for the large radius limit. We can repeat this construction for any cone $\sigma\subset{\cal K}$ which is the relative interior of a cone generated by a basis of $L$, obtaining partial compactifications which include large radius limit points for paths that lie in various cones $\sigma$. Among the ``quantities of physical interest'' to which this analysis is applied are a collection of multilinear maps of cohomology groups called {\em three-point functions}. These maps should depend on the data $(g,b)$, and should vary holomorphically with both complex structure and complexified K\"ahler structure parameters. Certain of these three-point functions (related to Witten's ``$A$-model'' \cite{witten}) would depend only on the complexified K\"ahler structure, while others (related to Witten's ``$B$-model'') would depend only on the complex structure. The $B$-model three-point functions can be mathematically interpreted in terms of the variation of Hodge structure, or period mapping, induced by varying the complex structure on the Calabi-Yau manifold \cite{cecotti2,guide,gmp}. In \cite{htams}, we discuss a mathematical version of the $A$-model three-point functions, expressed as formal power series near the distinguished limit point associated to the relative interior $\sigma$ of a rational polyhedral cone generated by a basis of $L$. (The coefficients $c_\eta$ of this power series are derived from the numbers of rational curves on $X$ of various degrees.) The choice of $\sigma$ is an additional piece of data in the construction which we call a {\em framing}. These formal power series representations of $A$-model three-point functions can be regarded as defining a formal degenerating variation of Hodge structure, which we call the {\em framed $A$-variation of Hodge structure}\/ with framing $\sigma$. Now there are manipulations of these formal series which suggest that the underlying convergent three-point functions (if they exist) will not depend on the choice of $\sigma$ and will be invariant under the action of $\operatorname{Aut}(X)$. We must refer the reader to \cite{htams} for the precise definition of framed $A$-variation of Hodge structure. But for reference, we would like to state here a conjecture which suggests how the various framed $A$-variations of Hodge structure will fit together, along the lines being discussed in this paper. \begin{ConvergenceConjecture} Suppose that $X$ is a Calabi-Yau manifold with $h^{2,0}(X)=0$, endowed with a complex structure, which satisfies the cone conjecture. Let $L:=H^2(X,\Bbb Z)/\text{torsion}$, let ${\cal K}$ be the K\"ahler cone of $X$, let ${\cal D}:=L_{\Bbb R}+i\,{\cal K}$ be the associated tube domain, and let $\Gamma:=L\rtimes\operatorname{Aut}(X)$. Then there is a neighborhood $U$ of the $0$-dimensional stratum $\widehat{{\cal D}}({\cal K})$ in the Satake-Baily-Borel-type compactification $\widehat{{\cal D}}({\cal P}_{\text{SBB}})/\Gamma$, and a variation of Hodge structure on $U\cap({\cal D}/\Gamma)$, such that for any $\sigma\subset{\cal K}$ which is the relative interior of a rational polyhedral\ cone $\overline{\sigma}\subset{\cal K}_+$ generated by a basis of $L$, the induced formal degenerating variation of Hodge structure at the distinguished limit point of ${\cal D}_{\sigma}/L$ agrees with the framed $A$-variation of Hodge structure with framing $\sigma$. \end{ConvergenceConjecture} If this variation of Hodge structure exists, we call it the {\em $A$-variation of Hodge structure}\/ associated to $X$. \section{Maximally unipotent boundary points} \label{MUBP} In the previous section, we discussed how to let the complexified K\"ahler parameter $B+i\,J$ approach infinity, analyzing certain partial compactifications and boundary points of the sigma-model moduli space in the $B+i\,J$ directions. We now turn to compactifications and boundary points in the transverse directions---the directions obtained by varying the complex structure on the Calabi-Yau manifold. We consider what happens when the complex structure degenerates. The local moduli spaces of complex structures on Calabi-Yau manifolds are particularly well-behaved, thanks to a theorem of Bogomolov \cite{bogomolov}, Tian \cite{tian}, and Todorov \cite{todorov}, which guarantees that all first-order deformations are unobstructed. In particular, there will be a local family of deformations of a given complex structure for which the Kodaira-Spencer map is an isomorphism. More generally, we consider arbitrary families $\pi:{\cal Y}\to S$ of complex structures on a fixed Calabi-Yau manifold $Y$, by which we mean: $\pi$ is a proper and smooth map between connected complex manifolds, and all fibers $Y_s:=\pi^{-1}(s)$ are diffeomorphic to $Y$. We will often assume that the Kodaira-Spencer map is an isomorphism at every point $s\in S$, so that $S$ provides good local moduli spaces for the fibers $Y_s$. To study the behavior when the complex structure degenerates, we partially compactify the parameter space $S$ to $\overline{S}$. There is a class of boundary points on $\overline{S}$ of particular interest from the perspective of conformal field theory. According to the interpretation of \cite{guide,htams}, these points can be identified by the monodromy properties of the associated variation of Hodge structure\footnote{The variation of Hodge structure in question is the usual geometric one (cf.~\cite{transcendental}) associated to a variation of complex structure. These might be called ``$B$-variations of Hodge structure'' by analogy with the previous section.} near $p\in\overline{S}$. We first review from \cite{htams} these monodromy properties for normal crossing boundary points, and then extend the definition to a wider class of compactifications and boundary points. Let $p$ be a maximal-depth normal crossing point of $B\subset\overline{S}$, where $B:=\overline{S}-S$ is the boundary, assumed for the moment to be a divisor with normal crossings. Let $U$ be a small neighborhood of $p$ in $\overline{S}$, and write $B\cap U$ in the form $B_1+\cdots+ B_r$. If we fix a point $s\in U{-}B$, then each local divisor $B_j$ gives rise to an monodromy transformation $T^{(j)}:H^n(Y_s,\Bbb Q)\to H^n(Y_s,\Bbb Q)$, which is guaranteed to be quasi-unipotent by the monodromy theorem \cite{monodromy}. \begin{definition} A maximal-depth normal crossing point $p$ of $B\in\overline{S}$ is called a\/ {\em maximally unipotent point}\footnote{When $\dim({\cal M})=1$, this definition is equivalent to the one given in \cite{guide}.} under the following conditions. \begin{enumerate} \item The monodromy transformations $T^{(j)}$ around local boundary components $B_j$ near $p$ are all unipotent. \item Let $N^{(j)}:=\log T^{(j)}$, let $N :=\sum a_jN^{(j)}$ for some $a_j>0$, and define \begin{eqnarray*} W_0&:=&\Im(N ^n)\\ W_1&:=&\Im(N ^{n-1})\cap\operatorname{Ker} N \\ W_2&:=&\Im(N ^{n-2})\cap\operatorname{Ker}(N ^2). \end{eqnarray*} Then $\dim W_0=\dim W_1=1$ and $\dim W_2=1+\dim({\cal M})$. \item Let $g^0,g^1,\dots,g^r$ be a basis of $W_2$ such that $g^0$ spans $W_0$, and define $m^{jk}$ by $N^{(j)}g^k=m^{jk}g^0$ for $1\le j,k\le r$. Then $m:=(m^{jk})$ is an invertible matrix. \end{enumerate} (The spaces $W_0$ and $W_2$ are independent of the choice of coefficients $\{a_j\}$ {\rm\cite{CK,deligne}}, and the invertibility of $m$ is independent of the choice of basis $\{g^k\}$.) \end{definition} Given a maximally unipotent point $p\in\overline{S}$, we define the {\em canonical logarithmic one-forms}\/ $d\mskip0.5mu\log q_j\in\Gamma(U, \Omega_{\overline{S}}^1(\log B))$ at $p$ by \[\frac1{2\pi i}d\mskip0.5mu\log q_j:= d\left(\frac{\sum_{k=1}^r\langle g^k|\omega\rangle m_{kj}} {\langle g^0|\omega\rangle}\right)\] where $(m_{kj})$ is the inverse matrix of $(m^{jk})$, and $\omega$ is a section of the sheaf $\Omega^n_{{\cal Y}/S}$ of relative holomorphic $n$-forms on the family of complex structures parameterized by $S$. The elements $g^k\in H^n(Y_s,\Bbb Q)$ have been implicitly extended to multi-valued sections of the local system $R^n\pi^*(\Bbb Q_{{\cal Y}})$ in order to evaluate $\langle g^k|\omega\rangle$; the monodromy measures the multi-valuedness of the resulting (locally defined) holomorphic functions $\langle g^k | \omega \rangle$. The fact that each $d\mskip0.5mu\log q_j$ as defined above has a single-valued meromorphic extension to $U$ follows from the nilpotent orbit theorem \cite{schmid}. In \cite{htams} we show that the canonical one-forms are independent of the choice of basis $\{g^k\}$, and also of the choice of relative $n$-form $\omega$; that for any local defining equation $v_j=0$ of $B_j$, the one-form $d\mskip0.5mu\log q_j-d\mskip0.5mu\log v_j$ extends to a regular one-form on $U$; and that $d\mskip0.5mu\log q_1$, \dots, $d\mskip0.5mu\log q_r$ freely generate the locally free sheaf $\Omega_{\overline{S}}^1(\log B)$ near $p$. The canonical logarithmic one-forms can be integrated to produce {\em quasi-canonical coordinates}\/ $q_1$, \dots, $q_r$ near $p$, but due to constants of integration, these coordinates are not unique. That is, if we attempt to define \[q_j=\exp\left(2\pi i\, \frac{\sum_{k=1}^r\langle g^k|\omega\rangle m_{kj}} {\langle g^0|\omega\rangle}\right)\] we find that changing the basis $\{g^k\}$ will alter the $q_j$'s by multiplicative constants (cf.~\cite{picard-fuchs}). To specify truly canonical coordinates, further conditions on the basis $\{g^k\}$ must be imposed, as discussed in \cite{guide,htams}. For example, by demanding that $g^0$ span $W_0\cap H^n(Y_s,\Bbb Z)/\text{torsion}$ and that $g^0,\dots,g^r$ span $W_2\cap H^n(Y_s,\Bbb Z)/\text{torsion}$ we can reduce the ambiguity in the $q_j$'s to a finite number of choices. With no ambiguity, we can use the canonical logarithmic one-forms to produce a (canonical) flat connection $\nabla$ on the holomorphic vector bundle $\Omega_{U}^1(\log B)$ by declaring $d\mskip0.5mu\log q_1$, \dots, $d\mskip0.5mu\log q_r$ to be a basis for the $\nabla$-flat sections, that is, $\nabla(d\mskip0.5mu\log q_j)=0$. Notice that the connection $\nabla$ is regular along the boundary divisor $B$. This connection is what we will use to extend the definition of maximally unipotent to a more general case. We now consider partial compactifications $\overline{S}$ of $S$ which are not necessarily smooth, and whose boundary is not necessarily a divisor with normal crossings. \begin{definition} \label{def3} Let $\Xi\subset\overline{S}{-}S$ be a connected subset of the boundary. We say that $\Xi$ is\/ {\em maximally unipotent} if there is a neighborhood $V$ of $\Xi$ in $\overline{S}$ and a flat connection $\nabla_{\text{unip}}$ on $\Omega^1_{V\cap S}$ such that for some resolution of singularities $f:U\to V$ which is an isomorphism over $V\cap S$, we have \begin{enumerate} \item the new boundary $B:=U{-}f^{-1}(V\cap S)$ on $U$ is a divisor with normal crossings, \item the flat connection $\nabla_{\text{unip}}$ extends to a connection on $\Omega^1_U(\log B)$ (also denoted by $\nabla_{\text{unip}}$), \item for every maximal-depth normal crossing point $p$ of $B\subset U$, we have $\nabla_{\text{unip}}(d\mskip0.5mu\log q_j)=0$ for each canonical logarithmic one-form $d\mskip0.5mu\log q_j$ at $p$, and \item $(U,B,\nabla_{\text{unip}})$ is compatible in the sense of definition \ref{compat}. \end{enumerate} We call $\nabla_{\text{unip}}$ the\/ {\em maximally unipotent connection determined by $\Xi$}. \end{definition} Note that $d\mskip0.5mu\log q_1,\dots,d\mskip0.5mu\log q_r$ is a basis for the vector space of local solutions of $\nabla_{\text{unip}}e=0$ near $p$. By analytic continuation of solutions, the connection $\nabla_{\text{unip}}$ is unique if it exists. The requirement of compatibility is quite strong, essentially guaranteeing that the structure of $\overline{S}$ near $\Xi$ resembles that of a semi-toric compactification. \section{Implications of mirror symmetry} \label{mirror} Mirror symmetry \cite{greene-plesser} predicts that Calabi-Yau manifolds should come in pairs,\footnote{The most recent results \cite{mmm,phases} suggest that it is {\em birational equivalence classes}\/ of Calabi-Yau manifolds which come in pairs.} with the r\^oles of variation of complex structure and of complexified K\"ahler structure being reversed between mirror partners. We make a precise mathematical conjecture about mirror symmetry in \cite{htams}, which can be stated as follows. \begin{mirrorconjecture} {\rm (Normal Crossings Case)} Let $Y$ be a Calabi-Yau manifold with $h^{2,0}(Y)=0$, and let $\pi:{\cal Y}\to S$ be a family of complex structures on $Y$ such that the Kodaira-Spencer map is an isomorphism at every point. Let $S\subset\overline{S}$ be a partial compactification whose boundary is a divisor with normal crossings. To each maximally unipotent normal crossing boundary point $p$ in $\overline{S}$ there is associated the following: \begin{enumerate} \item a Calabi-Yau manifold $X$ with $h^{2,0}(X)=0$, \item a lattice $L$ of finite index\footnote{The reason for allowing such an $L$ rather than insisting $H^2(X,\mbox{\footbbbfont Z})/\text{torsion}$ itself is that our basic defining condition on the family $S$---that the Kodaira-Spencer map be an isomorphism at every point---is invariant under finite unramified base change. So we must allow finite unramified covers of the parameter spaces.} in $H^2(X,\Bbb Z)/\text{torsion}$, \item the relative interior $\sigma\subset H^2(X,\Bbb R)$ of a rational polyhedral\ cone which is generated by a basis $\ell^1,\dots,\ell^r$ of $L$, and \item a map $\mu$ from a neighborhood of $p$ in $\overline{S}$ to $((H^2(X,\Bbb R)+i\,\sigma)/L)^-$, determined up to constants of integration by the requirement that $\mu^*(d\mskip0.5mu\log w_j)$ is the canonical logarithmic one-form $d\mskip0.5mu\log q_j$ on $\overline{S}$ at $p$ (as defined in section \ref{MUBP}), where $z_1,\dots,z_r$ are coordinates dual to $\ell^1,\dots,\ell^r$, and $w_j:=\exp(2\pi i\,z_j)$, \end{enumerate} such that \begin{itemize} \item[a.] $\sigma$ is contained in the K\"ahler cone for some complex structure on $X$, and \item[b.] $\mu$ induces an isomorphism between the formally degenerating geometric variation of Hodge structure at $p$ and the $A$-variation of Hodge structure with framing $\sigma$ associated to $X$. \end{itemize} \end{mirrorconjecture} Put more concretely, if we calculate the geometric variation of Hodge structure near $p\in\overline{S}$ using appropriate quasi-canonical coordinates $q_j$, we should produce power series expansions for $B$-model three-point functions (for $Y$) whose coefficients agree with the $c_\eta$ which are derived from the numbers of rational curves on $X$. This is precisely the type of calculation pioneered by Candelas, de la Ossa, Green, and Parkes \cite{pair} in the case of the quintic threefold. Given a family $\pi:{\cal Y}\to S$ of complex structures on $Y$, and a partial compactification $\overline{S}$ of $S$, if we move from point to point along the boundary of $\overline{S}$, or if we vary the compactification $\overline{S}$ by blowing up the boundary, we can produce many maximally unipotent normal crossing boundary points. On the other hand, if $X$ is a mirror partner of $Y$ for which the cone and convergence conjectures hold, there are many framed $A$-variations of Hodge structure (with different framings) associated to $X$. Given framings $\sigma$ and $\sigma'$ which belong to rational polyhedral\ decompositions ${\cal P}$ and ${\cal P}'$, respectively, there is always a common refinement ${\cal P}''$ of these decompositions. Geometrically, the corresponding compactification $\widehat{{\cal D}}({\cal P}'')/\Gamma$ is a blowup of both $\widehat{{\cal D}}({\cal P})/\Gamma$ and $\widehat{{\cal D}}({\cal P}')/\Gamma$. Analytic continuation on the common blowup $\widehat{{\cal D}}({\cal P}'')/\Gamma$ from a point in the inverse image of ${\cal D}(\sigma)$ to one in the inverse image of ${\cal D}(\sigma')$ will give an isomorphism of the $A$-variations of Hodge structure. The various maximally unipotent normal crossing boundary points will (conjecturally) lead to many mirror isomorphisms. We wish to fit these various mirror isomorphisms together. In fact, the mirror symmetry isomorphism is expected by the physicists to extend to an isomorphism between the full conformal field theory moduli spaces, and so, presumably, to compactifications as well. Thus, the structure of the semi-toric compactifications which are natural from the point of view of variation of complexified K\"ahler structure on $X$ should be reflected in the structure of compactifications of the complex structure moduli space ${\cal M}_Y$ of $Y$. This philosophy suggests two things about the compactified parameter spaces $\overline{S}$ of complex structures on $Y$. First, there should be a compatibility between compactification points whose mirror families are associated to the {\em same}\/ space $X$, and the {\em same}\/ K\"ahler cone ${\cal K}$. In fact, we should be able to extend our mathematical mirror symmetry conjecture to arbitrary maximally unipotent subsets of the boundary for {\em any}\/ compactification, not just ones whose boundary is a divisor with normal crossings. And second, there should be some kind of {\em minimal}\/ compactification of the coarse moduli space ${\cal M}_Y$ of complex structures on $Y$, whose mirror compactified family would be the Satake-Baily-Borel-type compactification of ${\cal D}/\Gamma$. The compatibility between compactifications can be recognized by means of the flat connection $\nabla_{\text{unip}}$ which we used to identify maximally unipotent subsets of the boundary. We extend our mirror symmetry conjecture to the general case as follows. \begin{mirrorconjecture} {\rm (General Case)} Let $Y$ be a Calabi-Yau manifold with $h^{2,0}(Y)=0$, let $\pi:{\cal Y}\to S$ be a family of complex structures on $Y$ such that the Kodaira-Spencer map is an isomorphism at every point, and let $S\subset\overline{S}$ be a partial compactification. To each maximally unipotent connected subset $\Xi$ of the boundary $\overline{S}{-}S$ there is associated the following: \begin{enumerate} \item a Calabi-Yau manifold $X$ satisfying the cone and convergence conjectures, \item a subgroup $\Gamma\subset\operatorname{Aff}(H^2(X,\Bbb R))$ whose translation subgroup $L$ is a lattice of finite index in $H^2(X,\Bbb Z)/\text{torsion}$, \item a locally rational polyhedral\ decomposition ${\cal P}$ of a cone ${\cal C}_+$ (which coincides with the convex hull of $\overline{{\cal C}_+}\cap L_{\Bbb Q}$) that is invariant under the group $\Gamma_0:=\Gamma/L$, and \item a map $\mu$ from a neighborhood $U$ of $\Xi$ in $\overline{S}$ to $\widehat{{\cal D}}({\cal P})/\Gamma$, determined up to constants of integration by the requirement that the flat connection $\nabla_{\text{toric}}$ on ${\cal D}/\Gamma$ pulls back to $\nabla_{\text{unip}}$ on $U\cap S$, where $\nabla_{\text{unip}}$ is the maximally unipotent connection determined by $\Xi$, \end{enumerate} such that \begin{itemize} \item[a.] for some complex structure on $X$, the interior ${\cal C}$ of ${\cal C}_+$ is contained in the K\"ahler cone and $\Gamma_0$ is contained in the group of holomorphic automorphisms, and \item[b.] $\mu$ induces an isomorphism between the geometric variation of Hodge structure over $U\cap S$ and the $A$-variation of Hodge structure associated to $X$. \end{itemize} \end{mirrorconjecture} {\em A priori}, the map $\mu$ determined by compatibility of the connections would only be a meromorphic map; we are asserting that it is in fact regular, and a local isomorphism. There is one further refinement of this conjecture which could be made: we could demand that the map $\mu$ also respect the quasi-canonical coordinates determined by choosing integral bases $g^0,\dots,g^r$. This would reduce the ambiguity in the choice of $\mu$ to a finite number of choices, but would require a compatibility among such integral quasi-canonical coordinates at various boundary points. \bigskip Finally, suppose that ${\cal M}_Y$ is the coarse moduli space for complex structures on a Calabi-Yau variety $Y$ such that $h^{2,0}(Y)$. (This coarse moduli space is known to exist as a quasi-projective variety, once we have specified a polarization, thanks to a theorem of Viehweg \cite{viehweg}.) In this case, we conjecture the existence of a Satake-Baily-Borel-style compactification, as follows. \begin{minconjecture} There is a partial compactification $(\overline{{\cal M}_Y})_{\text{SBB}}$ of the coarse moduli space ${\cal M}_Y$ with distinguished boundary points $p_1,\dots,p_k$ which are maximally unipotent, such that the data associated by the mathematical mirror symmetry conjecture to $p_j$ consists of: (1) a Calabi-Yau manifold $X_j$ (with a complex structure specified that determines the group $\operatorname{Aut}(X_j)$ of holomorphic automorphisms and the K\"ahler cone ${\cal K}_j$ of $X_j$), (2) the group \[\Gamma_j:=(H^2(X_j,\Bbb Z)/\text{torsion})\rtimes\operatorname{Aut}(X_j),\] and (3) the locally rational polyhedral\ decomposition ${\cal P}_j$ which is the Satake-Baily-Borel\ decomposition ${\cal P}_{\text{SBB}}$ of the cone $({\cal K}_j)_+$ (the convex hull of $\overline{{\cal K}}_j\cap H^2(X_j,\Bbb Q)$). \end{minconjecture} \section{Mumford cones and Mori cones} \label{mumford-mori} In the fall of 1979, Mori lectured at Harvard on his then-new results \cite{mori-cone} on the cone of effective curves. In order to show that his theorem about local finiteness of extremal rays fail when the canonical bundle is numerically effective, he gave an example. (A similar example appears in a Japanese expository paper he wrote a few years later, which has since been translated into English \cite{mori-sugaku}.) The example was of an abelian surface with real multiplication, that is, one whose endomorphism algebra contains the ring of integers $\O_K$ of a real quadratic field $K$. For such a surface $X$, the N\'eron-Severi group $L:=H^{1,1}(X)\cap H^2(X,\Bbb Z)$ is a lattice of rank $2$. The K\"ahler cone of $X$ lies naturally in $L_{\Bbb R}$, and is an open cone ${\cal K}$ bounded by two rays whose slopes are irrational numbers in the field $K$ (cf.~\cite{kuga-satake,vandergeer}). Rays through classes of ample divisors $[D]\in L\cap{\cal K}$ can be found which are arbitrarily close to the boundary, but the boundary is never reached. This phenomenon indicated that Mori's results on the structure of the dual cone ${\cal K}^\vee\subset H_2(X,\Bbb R)$ could not be extended to the case of abelian surfaces. The picture Mori drew for this example was remarkably similar to figure~1. The Hilbert modular surfaces in fact serve as moduli spaces for abelian surfaces with endomorphisms of this type (cf.~\cite[Chap.~IX]{vandergeer}), although a bit more data must be specified, which determines the group $\Gamma$. Now Mumford's figure~1 was drawn in some auxiliary space being used to describe this ``complex structure moduli space'', while Mori's version of figure~1 depicted the K\"ahler cone in $H^{1,1}$, and so is related to ``complexified K\"ahler moduli'' of the surfaces. The setting is not quite the same as the one in the present paper, since $h^{2,0}\ne0$. However, mirror symmetry for complex tori does predict that each cusp in the complex structure moduli space will be related to the K\"ahler moduli space for the abelian varieties parametrized by some $\h\times\h/\Gamma$, with the $\Gamma$ determined by the cusp. (This is not completely clear from the literature; I will return to this point in a subsequent paper.) In fact, under this association the Mumford cone from figure 1 corresponds precisely to the (dualized) Mori cone. Mirror symmetry might have been anticipated by mathematicians had anyone noticed the striking similarity between these two pictures back in 1979! \section*{Acknowledgments} I would like to thank P.~Aspinwall, R.~Bryant, P.~Deligne, G.~Faltings, A.\ Grassi, B.~Greene, R.~Hain, S.~Katz, R.~Plesser, L.~Saper and E.~Witten for useful discussions. This work was partially supported by NSF Grant DMS-9103827 and by an American Mathematical Society Centennial Fellowship. \makeatletter \renewcommand{\@biblabel}[1]{\hfill#1.}\makeatother \newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,}

9504

alg-geom/9504001

en

https://arxiv.org/abs/alg-geom/9504001

alg-geom/9504001

Bruce Hunt

Bruce Hunt

Hyperbolic Planes

29 pages (11 pt), dvi file available from the author by request to [email protected] , LaTeX v 2.09

In this paper we consider a special class of arithmetic quotients of bounded symmetric domains which can roughly be described as higher- dimensional analogues of the Hilbert modular varities. The algebraic groups are defined as the unitary groups over two-dimensional right vector spaces over a division algebra with involution. If $d$ denotes the degree of the division algebra, then $d=1$ is essentially just case giving rise to Hilbert modular varieties. We determine the class number (number of cusps) of the arithmetic quotients, and find inter- esting modular subvarities whos existence derives from the algebraic structure of the division algebras. Also the moduli interpretation, given by Shimuras theory, is described.

alg-geom

\section{Cyclic algebras with involution} We will be considering matrix algebras $M_n(D)$, where $D$ is a simple division algebra with an involution. There are two kinds of involution to be considered. Fix, for the rest of the paper, a totally real number field $k$ of finite degree $f$ over ${\Bbb Q}$, and assume $D$ is a division algebra over $k$ with involution. The two cases are: \begin{itemize}\item[(i)] $D$ is central simple over $k$, and $k$ is a maximal field which is symmetric with respect to the involution on $D$ (involution of the first kind). \item[(ii)] $D$ is central simple over an imaginary quadratic extension $K|k$, and $k$ is the maximal subfield of the center which is symmetric with respect to the involution on $D$; the involution restricts on $K$ to the $K|k$ involution (involution of the second kind). \end{itemize} If one sets $K:=k$ in case (i), then in both cases one speaks of a $K|k$-involution. With these notations, let $d$ be the degree of $D$ over $K$ (i.e., $\dim_KD=d^2$). Specifically, we will be considering the following cases: \begin{equation}\label{e1.0}\begin{minipage}{16cm} \begin{itemize}\item[1)] $d=1,\ D=K$ with an involution of the second kind, namely the $K|k$ involution. \item[2)] $d=2,\ D$ a totally indefinite quaternion division algebra over $k$ with the canonical involution (involution of the first kind). \item[3)] $d\geq3,\ D$ a central simple division algebra of degree $d$ over $K$ with a $K|k$-involution (involution of the second kind). \end{itemize}\end{minipage}\end{equation} Next recall that any division algebra over $K$ is a {\it cyclic algebra} (\cite{A}, Thm.~9.21, 9.22). These algebras are constructed as follows. Let $L$ be a cyclic extension of degree $d$ over $K$ and let $\sigma$ denote a generator of the Galois group $Gal_{L/K}$. For any $\gamma\in K^*$, one forms the algebra generated by $L$ and an element $e$, \begin{equation}\label{e1.1}\begin{minipage}{15cm} $$A=(L/K,\sigma,\gamma):= L\oplus e L\oplus \cdots \oplus e^{d-1}L,$$ $$e^d=\gamma,\hspace*{2cm} e\cdot z = z^{\sigma}\cdot e,\ \ \forall_{z\in L}.$$ \end{minipage}\end{equation} This algebra can be constructed as a subalgebra of $M_d(L)$, by setting: \begin{equation}\label{e1.2} e=\left(\begin{array}{cccc} 0 & 1 & \cdots & 0 \\ \vdots & & \ddots & \vdots \\ & & & 1 \\ \gamma & 0 & \cdots & 0 \end{array}\right), \hspace*{2cm} z=\left(\begin{array}{cccc} z &\cdots &\cdots & 0 \\ & z^{\sigma} & & \\ & & \ddots & \\ 0 & \cdots & \cdots & z^{\sigma^{d-1}}\end{array}\right). \end{equation} One verifies easily that these matrices fulfill the relations (\ref{e1.1}), and letting $B$ be the algebra in $M_d(L)$ generated by $e$ and $z\in L$, we have $(L/K,\sigma,\gamma)\cong B$, and clearly $B\otimes_KL\cong M_n(L)$, giving the explicit splitting. It is known that the cyclic algebra $A=(L/K,\sigma,\gamma)$ is split if and only if $\gamma \in N_{L|K}(L^*)$ (\cite{A}, Thm.~5.14). We will assume that $A$ is a division algebra; we will denote it in the sequel by $D$. Now assuming $d\geq2$, we consider the question of involutions on $D$. It is well-known that for $d=2$, an involution of the second kind is the base change of an involution of the first kind (\cite{A}, Thm.~10.21), so the assumption above that for $d=2$ the involution is of the first kind is no real restriction\footnote{perhaps somewhat more pragmatic is the remark that because of the exceptional isomorphism between domains of type ${\bf I_{2,2}}$ and ${\bf IV_4}$, the case $d=2$ and involution of the second kind can be translated into bilinear forms in eight variables, hence does not fit in well in the framework of hyperbolic planes.}. Furthermore, if $d\geq3$, there are no involutions of the first kind, since an involution of the first kind gives an isomorphism $A\cong A^{op}$ ($A^{op}$ the opposite algebra), implying that the class $[A]$ in the Brauer group $Br_K$ is of exponent one or two. Hence, for $d\geq3$ the restriction that the involution is of the second kind is no restriction at all. Finally in case $d=1$, an involution of the first kind on $D$ is trivial, so this case is not hermitian, and we will not consider it. Let us now consider the quaternion algebras $D$. Recall that $D$ is said to be {\it totally definite} (respectively {\it totally indefinite}), if at all real primes $\nu$, the local algebra $D_{\nu}$ is the skew field over ${\Bbb R}$ of the Hamiltonian quaternions ${\Bbb H}$ (respectively, if for all real primes $\nu$, the local algebra $D_{\nu}$ is split). Recall also that $D$ {\it ramifies} at a finite prime $\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}$ if $D_{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}$ is a division algebra; the isomorphism class of $D$ is determined by the (finite) set of primes $\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}$ at which it ramifies and its isomorphism class at those primes (\cite{A}, Thm.~9.34). As a special case of cyclic algebras, quaternion algebras can be displayed as algebras in $M_2(\ell)$, where $\ell/k$ is a real quadratic extension, $\ell=k(\sqrt{a})$. Then there is some $b\in k^*$ such that $$e=\left(\begin{array}{cc} 0 & 1 \\ b & 0 \end{array}\right), \hspace*{2cm} z=\left(\begin{array}{cc} z & 0 \\ 0 & z^{\sigma}\end{array}\right), \ \ z\in \ell,$$ where $z^{\sigma}=(z_1+\sqrt{a}z_2)^{\sigma}=z_1-\sqrt{a}z_2$. In other words, we can write for any $\alpha\in D$, \begin{equation}\label{e2.1} \alpha=\left(\begin{array}{cc} a_0+a_1\sqrt{a} & a_2+a_3\sqrt{a} \\ b(a_2-a_3\sqrt{a}) & a_0-a_1\sqrt{a}\end{array}\right). \end{equation} Then the canonical involution is given by the involution \begin{equation}\label{e2.2} \left(\begin{array}{cc} a & b \\ c & d \end{array}\right) \mapsto \left( \begin{array}{cc} d & -b \\ -c & a\end{array}\right) \end{equation} on $M_2(\ell)$, and the norm and trace are just the determinant and trace of the matrix (\ref{e2.1}). We will use the notation $(a,b)$ to denote this algebra $(\ell/k,\sigma,b)$, $\ell=k(\sqrt{a})$, if no confusion can arise from this. We remark also that for quaternion algebras we have $$Tr_{D|k}(x)=x+\overline{x},\quad N_{D|k}(x)=x\overline{x},$$ where the trace and norm are the reduced traces and norms. All traces and norms occuring in this paper are the reduced ones unless stated otherwise. In the case of involutions of the second kind, note first that the $K|k$-conjugation extends to the splitting field $L$; its invariant subfield $\ell$ is then a totally real extension of $k$, also cyclic with Galois group generated by $\sigma$. We have the following diagram: \begin{equation}\label{e3.1} \unitlength1cm \begin{picture}(2,2) \put(.9,-.35){$k$} \put(-.2,1){$\ell$} \put(2,.7){$K$} \put(.9,1.85){$L$} \put(.1,1.1){\line(6,5){.8}} \put(.1,.9){\line(5,-6){.8}} \put(1.9,.65){\line(-6,-5){.8}} \put(1.9,.85){\line(-5,6){.8}} \end{picture} \end{equation} and the conjugations on $L$ and $K$ give the action of the Galois group on the extensions $L/\ell$ and $K/k$; these are ordinary imaginary quadratic extensions. There are precise relations known under which $D$ admits a $K|k$-involution of the second kind. \begin{theorem}[\cite{A}, Thm.~10.18]\label{t4A.1} A cyclic algebra $D=(L/K,\sigma,\gamma)$ has an involution of the second kind $\iff$ there is an element $\omega\in \ell$ such that $$\gamma\overline{\gamma}=N_{K|k}(\gamma)=N_{\ell|k}(\omega)=\omega\cdot \omega^{\sigma} \cdots \omega^{\sigma^{d-1}}.$$ \end{theorem} If this condition holds, then an involution is given explicitly by setting: \begin{equation} \label{e4A.1} (e^k)^J=\omega\cdots \omega^{\sigma^{k-1}}(e^k)^{-1}, \left(\sum e^iz_i\right)^J = \sum \overline{z}_i(e^i)^J, \end{equation} where $x\mapsto \overline{x}$ denotes the $L/\ell$-involution. In particular for $x\in L$ we have $$x^J=\overline{x}, \hbox{ and } x=x^J \iff x\in \ell.$$ Later it will be convenient to have a description for when $x+x^J=0$. This results from the following. \begin{theorem}[\cite{A}, Thm.~10.10]\label{t4A.2} Given an involution $J$ of the second kind on an algebra $A$, central simple of degree $d$ over $K$, there are elements $u_1,\ldots, u_d$, with $u_i=u_i^J$, such that $A$ is generated over $K$ by $u_1,\ldots, u_d$. Furthermore, there is an element $q\in A,\ q^J=-q,\ q^2\in k$, such that, as a $k$-vector space, $$A=A^++qA^+,$$ where $A^+=\{x\in A | x=x^J\}$. \end{theorem} If $x\in A$ is arbitrary, then ${1\over 2}(x+x^J)\in A^+$, while ${1\over 2}(x-x^J)\in A^-$. For example, we have $e^i+(e^i)^J=:E^i\in A^+$, and then we have an isomorphism \begin{equation}\label{e4A.2} A^+\cong \ell\oplus E\ell \oplus \cdots \oplus E^{d-1}\ell. \end{equation} If, as above, $K=k(\sqrt{-\eta})$, $L=\ell(\sqrt{-\eta})$, then we may take $\sqrt{-\eta}=q$ in the theorem above, and for elements in \begin{equation}\label{e4A.3} qA^+\cong \sqrt{-\eta}\ell \oplus E\sqrt{-\eta}\ell\oplus \cdots \oplus E^{d-1}\sqrt{-\eta}\ell, \end{equation} we have $y=-y^J$. In particular, the dimension of $qA^+$ is $d^2$ as a $k$-vector space, and the dimension of $A$ is $2d^2$. \section{Algebraic groups} Let $D$ be a central simple $K$-division algebra with a $K|k$-involution $x\mapsto \overline{x}$ as discussed above (denoted $x\mapsto x^J$ above, but unless it is a cause of confusion we fix the notation $x\mapsto \overline{x}$ for the remainder of the paper), with $K|k$ imaginary quadratic for $d=1,\geq3$ and $K=k$ for $d=2$. Consider the simple algebra $M_2(D)$; it is endowed with an involution, $M\mapsto {^t\overline{M}}=:M^*$. The group of units of $M_2(D)$ is denoted $GL(2,D)$, its derived group is denoted by $SL(2,D)$. Consider a non-degenerate hermitian form on $D^2$ given by \begin{equation}\label{e4.1} h({\bf x},{\bf y})= x_1\overline{y}_2+x_2\overline{y}_1, \end{equation} where ${\bf x}=(x_1,x_2), {\bf y}=(y_1,y_2) \in D^2$. This means the hermitian form is given by the matrix $H:={0\ 1 \choose 1\ 0}$. The unitary and special unitary groups for $h$ are \begin{equation}\label{e4.2} U(D^2,h)=\left\{g\in GL(2,D) \Big| gHg^*=H\right\}, \hspace*{1cm} SU(D^2,h)=U(D^2,h)\cap SL(2,D). \end{equation} The equations defining $U(D^2,h)$ are then \begin{equation}\label{e4.3} U(D^2,h)=\left\{g=\left(\begin{array}{cc}a & b \\ c & d \end{array}\right)\Big| a\overline{d}+b\overline{c}=1, a\overline{b}+b\overline{a}=c\overline{d}+d\overline{c}=0\right\}. \end{equation} The additional equation defining $SU(D^2,h)$ can be written in terms of determinants, using Dieudonn\'e's theory of determinants over skew fields, (see \cite{Ar}, p.~157) \begin{equation}\label{e4.4} \det(g)=N_{D|k}(ad-aca^{-1}b)=1. \end{equation} The center of $U(D^2,h)$, which we will denote by ${\cal C}$, is given by $${\cal C}=\left\{\left(\begin{array}{cc} a & 0 \\ 0 & a\end{array}\right) \Big| a\in K, a\overline{a}=1\right\}\cong U(1)\cap K,$$ where we view $K\subset} \def\nni{\supset} \def\und{\underline {\Bbb C}$ as a subfield of the complex numbers. In particular: \begin{lemma}\label{l4.1} For $d=2$, $K=k$ a real field, we have ${\cal C}=\{\pm1\}$. \end{lemma} In other words, for the case $d=2$, there is no essential difference between the unitary and special unitary groups. Otherwise, $U(D^2,h)$ is a reductive $K$-group, and $SU(D^2,h)$ is the corresponding simple group. \begin{proposition}\label{p4.1} Let $SG_D$ denote $SU(D^2,h)$, the simple algebraic $K$-group defined by the relations (\ref{e4.3}) and (\ref{e4.4}). Then $SG_D$ is simple with the following index (cf. \cite{tits}) \begin{itemize}\item[1)] $d=1$, ${^2A_{1,1}^1}$. \item[2)] $d=2$, $C_{2,1}^2$. \item[3)] $d\geq 3$, ${^2A_{2d-1,1}^d}$. \end{itemize} \end{proposition} {\bf Proof:} This is immediate from the description in \cite{tits} of these indices. \hfill $\Box$ \vskip0.25cm } \def\cD{\ifmmode {\cal D The unitary group for $d=1$ is familiar in different terms. If we set $A=\left(\begin{array}{cc}-1/2 & -1 \\ -1/2 & 1\end{array}\right)$, then $$A\left(\begin{array}{cc}0 & 1 \\ 1 & 0\end{array}\right){^tA}=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right).$$ In particular, these two hermitian forms are equivalent. For the form $H_1:={1\ \ 0 \choose \ 0\ -1}$, the corresponding unitary group is customarily denoted by $U(1,1)$. Hence for $K\subset} \def\nni{\supset} \def\und{\underline {\Bbb C}$ imaginary quadratic over a real field, we have an isomorphism \begin{eqnarray}\label{e4.5} U(K^2,h) & \stackrel{\sim}{\longrightarrow} \def\sura{\twoheadrightarrow} & U(1,1;K) \\ g & \mapsto & A g A^{-1}; \nonumber \end{eqnarray} the group $U(1,1;K)$ is given by the familiar conditions $$U(1,1;K)=\left\{ \left(\begin{array}{cc} \alpha c & \beta c \\ \overline{\beta}\overline{c} & \overline{\alpha}\overline{c}\end{array}\right) \Big| \alpha,\beta\in K, \alpha\neq 0, c\overline{c}=1, \alpha\overline{\alpha}-\beta\overline{\beta}=1\right\},$$ and the special unitary group is given by $$SU(1,1;K)=\left\{\left(\begin{array}{cc} \alpha & \beta \\ \overline{\beta} & \overline{\alpha} \end{array}\right)\in GL(2,K) \Big| \alpha\overline{\alpha}-\beta\overline{\beta}=1\right\}.$$ Recall that this latter group is isomorphic to a subgroup of $GL(2,{\Bbb R})$, in fact we have the well-known isomorphism \begin{eqnarray}\label{e5.1} SU(1,1;{\Bbb C}) & \stackrel{\sim}{\longrightarrow} \def\sura{\twoheadrightarrow} & SL(2,{\Bbb R}) \\ g & \mapsto & B g B^{-1}, \nonumber \end{eqnarray} where $B={-i\ \ i \choose \ \! 1\ \ 1}$ is the matrix of a fractional linear transformation mapping the disk to the upper half plane. If, instead of $B$, we use \begin{equation}\label{e5.2} B_{\eta}=\left(\begin{array}{cc} -i & i \\ \sqrt{\eta} & \sqrt{\eta} \end{array}\right),\hspace*{.75cm} K=k(\sqrt{-\eta}), \end{equation} then we have in fact \begin{proposition}\label{p5.1} The matrix $B_{\eta}$ gives an isomorphism $$\begin{array}{rcl} SU(1,1;K) & \stackrel{\sim}{\longrightarrow} \def\sura{\twoheadrightarrow} & SL_2(k) \\ \mapsto & B_{\eta} g B_{\eta}^{-1}. \end{array}$$ In particular, we have an isomorphism $$\begin{array}{rcl} SU(K^2,h) & \stackrel{\sim}{\longrightarrow} \def\sura{\twoheadrightarrow} & SL_2(k) \\ g & \mapsto & B_{\eta}A g A^{-1} B_{\eta}^{-1}.\end{array}$$ \end{proposition} {\bf Proof:} We only have to check that $B_{\eta}gB_{\eta}^{-1}$ is real, as it is clearly a matrix in $SL(2,K)$, and $SL(2,K)\cap SL(2,{\Bbb R})=SL_2(k)$. If $g={\alpha\ \beta \choose \overline{\beta}\ \overline{\alpha}}$ then $$B_{\eta}gB_{\eta}^{-1} = {1 \over -2\sqrt{-\eta}}\left(\begin{array}{cc} \sqrt{-\eta}(-\alpha-\overline{\alpha}+\beta+\overline{\beta}) & -\alpha-\beta+\overline{\alpha}+\overline{\beta} \\ \eta(\alpha-\beta+\overline{\beta}-\overline{\alpha}) & -\sqrt{-\eta}(\alpha+\beta+\overline{\alpha}+\overline{\beta}) \end{array}\right) =$$ $$\left(\begin{array}{cc} Re(\alpha)-Re(\beta) & Im(\alpha)+Im(\beta) \\ \eta(-Im(\alpha)+Im(\beta)) & Re(\alpha)+Re(\beta) \end{array}\right),$$ which is clearly real. \hfill $\Box$ \vskip0.25cm } \def\cD{\ifmmode {\cal D Note that the inverse $SL_2(k)\longrightarrow} \def\sura{\twoheadrightarrow SU(K^2,h)$ is given by conjugating with $C=(B_{\eta}A)^{-1}=\left(\begin{array}{cc} 0 & -2i \\ \sqrt{\eta} & 0 \end{array}\right)^{-1} = \left(\begin{array}{cc}0 & {1\over \sqrt{\eta}} \\ {i\over 2} & 0 \end{array}\right)$, so we may describe the latter group as: \begin{equation}\label{e5.3} SU(K^2,h)=\left\{\left(\begin{array}{cc} \delta & 2\gamma/\sqrt{-\eta} \\ \beta\sqrt{-\eta}/2 & \alpha \end{array}\right) \Big| \alpha,\beta,\gamma,\delta\in k, \alpha\delta-\gamma\beta=1\right\}. \end{equation} We might remark at this point that it is a bit messy to describe the unitary groups in the same fashion, as they are {\it not} isomorphic to subgroups of $GL(2,{\Bbb R})$. We have the following extensions: \begin{equation}\label{e5.4}\begin{array}{ccccccccc} 1 & \longrightarrow} \def\sura{\twoheadrightarrow & SU(1,1;K) & \longrightarrow} \def\sura{\twoheadrightarrow & U(1,1;K) & \longrightarrow} \def\sura{\twoheadrightarrow & U(1)\cap K & \longrightarrow} \def\sura{\twoheadrightarrow 1 \\ & & |\wr \\ 1 & \longrightarrow} \def\sura{\twoheadrightarrow & SL_2(k) & \longrightarrow} \def\sura{\twoheadrightarrow & GL(2,k) & \longrightarrow} \def\sura{\twoheadrightarrow & k^* & \longrightarrow} \def\sura{\twoheadrightarrow 1, \end{array} \end{equation} the first being an extension by a compact torus, the second by a split torus. That is why the map of Proposition \ref{p5.1}, if extended to $U(1,1;K)$, does not land in $GL(2,{\Bbb R})$. It is also appropriate to remark here that, for $d=2$, the {\it reductive} group associated with the problem is the group of symplectic similtudes, i.e., $g\in GL(2,D)$ such that $gHg^*=H\lambda$ for some non-zero $\lambda\in D$. This gives an extension of $SU(D^2,h)$ similar to the second sequence above. Next we describe some subgroups of $G_D=U(D^2,h)$ and $SG_D$. For this, recall that we are dealing with right vector spaces, and matrix multiplication is done from the right, i.e., $$(x_1,x_2)\left(\begin{array}{cc}a & b \\ c & d \end{array}\right)=(x_1a+x_2c, x_1b+x_2d).$$ First, the vectors $(1,0)$ and $(0,1)$ are isotropic; the stabilisers of the lines they span in $D^2$ are maximal (and minimal) parabolics. For example, the stabiliser of $(0,1)D$ is \begin{equation}\label{e6.1} P=\left\{g \in G_D \Big| (0,1)g=(0,1)\lambda, \lambda\in D^*\right\} = \left\{g=\left(\begin{array}{cc}a & b\\ 0 & \overline{a}^{-1}\end{array}\right) \Big| a\in D^*,\ b+\overline{b}=0 \right\}. \end{equation} Indeed, it is immediate that for $g={a\ b\choose c\ d}\in P$, we must have $c=0$, while the first relation of (\ref{e4.3}) reduces to $a\overline{d}=1$, or $d=\overline{a}^{-1}$. Since a parabolic is a semidirect product of a Levi factor and the unipotent radical, to calculate the latter we may assume $a=1$. Then the second relation of (\ref{e4.3}) is $a\overline{b}+b\overline{a}=\overline{b}+b=0$. The corresponding parabolic in $SG_D$ takes the form \begin{equation}\label{ePar1} SP=\left\{g\in SG_D \Big| g=\left(\begin{array}{cc} a & b \\ 0 & \overline{a}^{-1} \end{array}\right), \ a\in D^*,\ b+\overline{b}=0 \right\}. \end{equation} Note that $g$ in (\ref{ePar1}) has $K$-determinant $$\det_K(g)=N_{D|K}(a \overline{a}^{-1})=N_{D|K}(a)N_{D|K}(\overline{a})^{-1},$$ so the requirement on $g$ is that $a$ must satisfy $N_{K|k}(N_{D|K}(a)/N_{D|K}(\overline{a}))=1$. This is satisfied for all $a\in K^*$, where $a$ is viewed as an element of $D^*$; if $a\not\in K$, then the condition becomes $N_{D|K}(a)=\overline{N_{D|K}(a)}$, i.e., $N_{D|K}(a)\in k$. It follows that the Levi component is a product $$L=\left\{g=\left(\begin{array}{cc}a & 0 \\ 0 & \overline{a}^{-1}\end{array}\right) \Big| a\in D^*\right\}\cong \left(U(1)\cap K\right)\cdot \left\{ g = \left(\begin{array}{cc}a & 0 \\ 0 & \overline{a}^{-1}\end{array}\right) \Big| a\in D^*,\ N_{D|K}(a)\in k\right\}.$$ Now recall from (\ref{e4A.3}) that for $d\geq3$, the set of elements fulfilling $b+\overline{b}=0$ in $D$ is $\sqrt{-\eta}D^+\cong \sqrt{-\eta}\ell \oplus E\sqrt{-\eta}\ell \oplus \cdots \oplus E^{d-1}\sqrt{-\eta}\ell$, where $E^i=e^i+\overline{e^i}$. In particular, the unipotent radical of $SP$, generated by ${1\ b\choose 0\ 1}$ with $b+\overline{b}=0$, has dimension $d^2$ over $k$. We also have non-isotropic vectors $(1,1)$ and $(1,-1)$. Any element $g\in G_D$ preserving one preserves also the other (as they are orthogonal). Hence the stabiliser of $(1,1)$ is \begin{equation}\label{e6.2}\label{eCom} {\cal K}=\left\{g=\left(\begin{array}{cc} a & c \\ c & a\end{array}\right) \Big| a\overline{a}+c\overline{c}=1,\ a\overline{c}+c\overline{a}=0\right\}. \end{equation} For $d=1$, this can be more precisely described as $${\cal K}=\left\{ g=\left(\begin{array}{cc} a & {c \over \sqrt{-\eta}} \\ -{c\sqrt{-\eta}\over \eta} & a \end{array}\right),\ a,c\in k, a^2-{c^2\over \eta}=1\right\}.$$ For $d=2$, the relations give, viewed as equations over $k$, four relations, so the group ${\cal K}$ is four-dimensional. Finally, for $d\geq3$, we get (over $k$) $d$ relations (as elements $a\overline{a}$ are in $\ell$, so diagonal) from the first condition and $d^2-d+1$ conditions from the second, leaving $d^2+d+2$ parameters for the group ${\cal K}$. If $D$ has an involution of the second kind, we have the splitting field $L$ ($=K$ for $d=1$), which is an imaginary quadratic extension of the totally real field $\ell$ ($=k$ for $d=1$), see the diagram (\ref{e3.1}). Suppose then $d=2$, $D$ quaternionic, say $D=(\ell/k,\sigma,b)$, where $\ell=k(\sqrt{a}),\ e^2=b\in k^*$; in this case $\ell$ is a splitting field. Recall the conjugation $\sigma$ is given by $$(z_1+\sqrt{a}z_2)^{\sigma}=z_1-\sqrt{a}z_2,$$ so the element $c=diag(\sqrt{a},-\sqrt{a})$ representing $\sqrt{a}\in \ell$ satisfies $c^{\sigma}=-c$, while $e^{\sigma}=e$. Consequently, the relation (\ref{e1.1}) for $c$ is $ec=c^{\sigma}e=-ce$, and \begin{equation}\label{e7.1} (ec)^2=(ec)(-ce)=-ec^2e=-ab, \end{equation} so $k(ec)\cong k(\sqrt{-ab})$. If we assume, as we may, that $a>0, b>0$ (otherwise replace $-ab$ in what follows by the negative one of $a, b$), then $L:=\sqrt{-ab}$ is an imaginary quadratic extension of $k$ which is a subfield of $D$. So in all cases ($d=1,2,\geq3$) we have an imaginary quadratic extension of $\ell$ ($d=1,\geq3$) or $k$ ($d=2$), $L$, which is a subfield of $D$. We now claim that $U(L^2,h)$ is a natural subgroup of $U(D^2,h)$. In all cases $U(L^2,h)$ and $U(D^2,h)$ are defined by the same relations (\ref{e4.3}), so we have \begin{equation}\label{e7.2} U(L^2,h)=U(D^2,h)\cap M_2(L), \end{equation} which verifies the claim. In the case $d=1$, where we took $L=K$, this is in fact the whole group. But even in this case we can get subgroups in this manner. Consider taking, instead of $L=K$, $L={\Bbb Q}(\sqrt{-\eta})$ for $K=k(\sqrt{-\eta})$. More generally, for any ${\Bbb Q}\subseteq k' \subseteq k$, we have the subfield $k'(\sqrt{-\eta})\subset} \def\nni{\supset} \def\und{\underline K$, and we may consider the corresponding subgroup: $$U(k'(\sqrt{-\eta})^2,h) = U(K^2,h)\cap M_2(k'(\sqrt{-\eta})).$$ So, setting $L=k'(\sqrt{-\eta})$ in (\ref{e7.2}), these subgroups are defined by the same relations (\ref{e4.3}). Now recall the description (\ref{e5.3}) for groups of the type (\ref{e7.2}). \begin{proposition}\label{p7.1} We have the following subgroups of $U(D^2,h)$. \begin{itemize}\item[1)] $d=1$, for any $ {\Bbb Q}\subseteq k' \subseteq k$ we have the subgroup $$SL_2(k')\cong \left\{ \left(\begin{array}{cc} \alpha & 2\beta/\sqrt{-\eta} \\ \gamma\sqrt{-\eta}/2 & \delta \end{array}\right) \Big| \alpha,\beta,\gamma,\delta \in k', \alpha\delta-\gamma\beta=1\right\}\subset} \def\nni{\supset} \def\und{\underline U(K^2,h).$$ \item[2)] $d=2$, for the field $L=k(\sqrt{-ab})$ above, we have the subgroup $$SL_2(k)\cong \left\{ \left(\begin{array}{cc} \alpha & 2\beta/\sqrt{-ab} \\ \gamma\sqrt{-ab}/2 & \delta \end{array}\right) \Big| \alpha,\beta,\gamma,\delta \in k, \alpha\delta-\gamma\beta=1\right\}\subset} \def\nni{\supset} \def\und{\underline U(D^2,h).$$ \item[3)] $d\geq3$, for the degree $d$ cyclic extension $L/K$ we have the following subgroup $$SL_2(\ell)\cong \left\{ \left(\begin{array}{cc} \alpha & 2\beta/\sqrt{-\eta} \\ \gamma\sqrt{-\eta}/2 & \delta \end{array}\right) \Big| \alpha,\beta,\gamma,\delta \in \ell, \alpha\delta-\gamma\beta=1\right\}\subset} \def\nni{\supset} \def\und{\underline U(D^2,h).$$ \end{itemize} \end{proposition} These are interesting subgroups, whose existence is a natural part of the description of $D$ as a cyclic algebra. In all cases we may view these subgroups as stabilisers. Let $L$ denote one of the fields $k'(\sqrt{-\eta}), k(\sqrt{-ab}), \ell(\sqrt{-\eta})$ as in Proposition \ref{p7.1}; let $L^2\subset} \def\nni{\supset} \def\und{\underline D^2$ denote the $k$-vector subspace of $D^2$, viewing $D$ as a $k$-vector space, and consider the stabiliser in $G_D$. Noting that $L^2\subset} \def\nni{\supset} \def\und{\underline D^2$ may be spanned over $L$ by two non-isotropic vectors $(1,1)$ and $(1,-1)$, it is clear that $g\in U(L^2,h)$ preserves the $L$ span of these two vectors, that is, $L^2$, hence $U(L^2,h)$ is contained in the stabiliser. The converse is also true: if an endomorphism of $D^2$ preserves the $L$ span, then its matrix representation is in $M_2(L)$, so the stabiliser is contained in the intersection $U(D^2,h)\cap M_2(L)=U(L^2,h)$. Let us record this as \begin{observation} The subgroups of Proposition \ref{p7.1} are stabilisers (with determinant 1) of $k$-vector subspaces of the $k$-vector space $D^2$. \end{observation} \section{Domains} Recall that given an almost simple algebraic group $G$ over $k$, by taking the ${\Bbb R}$ points one gets a semisimple real Lie group $G({\Bbb R})$. For the three kinds of groups above we determine these now. \begin{proposition}\label{p9.1} The real groups $G_D({\Bbb R})$ are the following. \begin{itemize}\item[1)] $d=1$, $G_D({\Bbb R})=(U(1,1))^f, SG_D({\Bbb R})\cong (SL(2,{\Bbb R}))^f$. \item[2)] $d=2$, $G_D({\Bbb R})=SG_D({\Bbb R})=(Sp(4,{\Bbb R}))^f$. \item[3)] $d\geq3$, $G_D({\Bbb R})=(U(d,d))^f,\ SG_D({\Bbb R})=(SU(d,d))^f$. \end{itemize}\end{proposition} {\bf Proof:} The $d=1$ case is obvious. For $d=2$, note that since $D$ is totally indefinite, $D_{\nu}\cong M_2({\Bbb R})$ for all real primes, and consequently $(G_D)_{\nu}$ is a simple group of type $Sp$ for each $\nu$, not compact as $G_D$ is isotropic, of rank 2, hence $Sp(4,{\Bbb R})$. Suppose now that $d\geq3$. By Proposition \ref{p4.1} we know the index of $G_D$ is ${^2A_{2d-1,1}^d}$, and the index, as displayed in \cite{tits}, is \begin{equation}\label{e9.1} \setlength{\unitlength}{0.006500in}% \begin{picture}(894,94)(53,693) \thicklines \put(140,780){\circle*{14}} \put( 60,700){\circle*{14}} \put(140,700){\circle*{14}} \put(400,700){\circle*{14}} \put(400,780){\circle*{14}} \put(460,740){\circle{14}} \put(600,700){\circle*{14}} \put(600,780){\circle*{14}} \put(680,700){\circle*{14}} \put(680,780){\circle*{14}} \put(860,780){\circle*{14}} \put(860,700){\circle*{14}} \put(940,780){\circle{14}} \put(940,700){\circle{14}} \put( 65,700){\line( 1, 0){130}} \put( 60,780){\circle*{14}} \put( 65,780){\line( 1, 0){130}} \put(940,770){\line( 0,-1){ 65}} \put(330,780){\line( 1, 0){ 70}} \put(330,700){\line( 1, 0){ 70}} \put(405,780){\line( 3,-2){ 45}} \put(400,700){\line( 5, 3){ 50}} \put(605,700){\line( 1, 0){ 70}} \put(605,780){\line( 1, 0){ 75}} \put(680,780){\line( 1, 0){ 50}} \put(680,700){\line( 1, 0){ 45}} \put(800,780){\line( 1, 0){ 60}} \put(800,700){\line( 1, 0){ 60}} \put(865,780){\line( 1, 0){ 70}} \put(860,700){\line( 1, 0){ 75}} \put(230,695){$\cdots\cdots$} \put(750,695){$\cdots$} \put(230,775){$\cdots\cdots$} \put(750,775){$\cdots$} \put(230,680){$d$ even} \put(750,680){$d$ odd} \end{picture} \end{equation} \noindent where the number of black vertices is $2d-2$, $d-1$ on each branch. This shows in particular that the {\it isotropic} root is the one farthest to the right. If a root is isotropic over ${\Bbb Q}$, then all the more over ${\Bbb R}$. Hence, considering the Satake diagram of the real groups of type ${^2A}$, we see that the only possibility is $SU(d,d)$, as this is the only real group of this type for which the right most vertex is ${\Bbb R}$-isotropic. Actually, this only proves that {\it at least one} factor $(G_D)_{\nu}$ is $SU(d,d)$; but our hyperbolic form has signature $(d,d)$ at {\it any} real prime, so it holds for {\it all} (non-compact) factors. That there are no compact factors follows from the fact that $G_D$ is isotropic. In all cases there are $f$ factors, as this is the degree of $k|{\Bbb Q}$, so the ${\Bbb Q}$-group, $Res_{k|{\Bbb Q}}G_D$, has $f$ factors over ${\Bbb R}$. This verifies the proposition in all cases. \hfill $\Box$ \vskip0.25cm } \def\cD{\ifmmode {\cal D It is a consequence of this proposition that the symmetric spaces associated to the real groups $G_D({\Bbb R})$ are {\it hermitian symmetric}; indeed, this is why we have choosen those $D$ to deal with. More precisely, one has the following statement. \begin{theorem}\label{t9.1} The hermitian symmetric domains defined by the $G_D({\Bbb R})$ as in Proposition \ref{p9.1} \begin{itemize}\item are tube domains, products of irreducible components of the types $\bf I_{1,1}$, $\bf III_2$, $\bf I_{d,d}$ in the cases $d=1$, $d=2$ and $d\geq3$, respectively. \item For any rational parabolic $P\subset} \def\nni{\supset} \def\und{\underline G_D$, the corresponding boundary component $F$ such that $P({\Bbb R})=N(F)$ is a point. \end{itemize} \end{theorem} {\bf Proof:} The first statement follows immediately from Proposition \ref{p9.1}. The second is well known in the case $d=1$ from the study of Hilbert modular varieties. For $d=2$ the statement clearly holds for $k={\Bbb Q}$ (just look at the index, from which one sees the boundary components are points, not one-dimensional), and the general statement follows from this: all boundary components are of the type $(c_1,\ldots,c_f)\in \cD_1^*\times \cdots \times \cD_f^*,$ where $ c_i\in \cD_i^*$ is a point, which is a general fact about ${\Bbb Q}$-parabolics in a {\it simple} ${\Bbb Q}$-group (this applies directly to $Res_{k|{\Bbb Q}}SG_D$, but $G_D$ and $SG_D$ define the same domain and boundary components). Finally, for $d\geq3$, we point out that in the diagram (\ref{e9.1}), the isotropic vertex (farthest right) corresponds to a maximal parabolic which stabilises a point, establishing the statement for $k={\Bbb Q}$. Then the general statement follows as above from this. \hfill $\Box$ \vskip0.25cm } \def\cD{\ifmmode {\cal D We now consider the structure of the real parabolics for $d\geq2$ in more detail. For this we refer to Satake's book, \S III.4. \begin{itemize}\item $d=2$: Here we have the case denote $b=n$ in Satake's book, and the parabolic in $Sp(4,{\Bbb R})$ is $$P({\Bbb R})=(G^{(1)}\cdot G^{(2)})\rtimes U\cdot V = 1\cdot GL(2,{\Bbb R})\rtimes Sym_2({\Bbb R}),$$ where $Sym_2({\Bbb R})$ denotes the set of symmetric real $2\times 2$ matrices. Refering to (\ref{e6.1}), we may identify the Levi component of $P$ with $$L\cong \left\{\left(\begin{array}{cc}a & 0 \\ 0 & \overline{a}^{-1}\end{array}\right) \Big| a\in D^*\right\} \cong D^*,$$ and, as $D$ is indefinite, $D^*({\Bbb R})\cong GL(2,{\Bbb R})$; this is the factor $GL(2,{\Bbb R})$. The unipotent radical is $$U\cong \left\{\left(\begin{array}{cc} 1 & b \\ 0 & 1\end{array}\right) \Big| \ Tr(b)=0\right\} \cong D^0 \hbox{ (=totally imaginary elements)},$$ which is three-dimensional, and an isomorphism $U({\Bbb R})\cong Sym_2({\Bbb R})$ is given by $$\left(\begin{array}{cc}a_1\sqrt{a} & a_2+a_3\sqrt{a} \\ b(a_2-a_3\sqrt{a}) & -a_1\sqrt{a}\end{array}\right) \mapsto \left(\begin{array}{cc} a_1 & a_2 \\ a_2 & a_3\end{array}\right).$$ This describes the real parabolic completely in this case. \item $d\geq3$: Here we have $b=d$ in Satake's notation, $d=p=q$. The real parabolic is $$P({\Bbb R})=(G^{(1)}\cdot G^{(2)})\rtimes U\cdot V = (U(1)\cdot GL^0(d,{\Bbb C}))\rtimes {\cal H}_d({\Bbb C}),$$ where $GL^0(d,{\Bbb C})=\{g\in GL(d,{\Bbb C})|\det(g)\in {\Bbb R}^*\}$ and ${\cal H}_d({\Bbb C})$ denotes the space of hermitian $d\times d$ matrices. We must now consider the parabolic $SP$ as in (\ref{ePar1}). The Levi component is again $$L\cong \left\{\left(\begin{array}{cc} a & 0 \\ 0 & \overline{a}^{-1} \end{array}\right) \Big| a\in D^*,\ N_{D|k}(a\overline{a}^{-1})=1\right\},$$ the second condition because of determinant 1. From the multiplicativity of the norm, the second relation can be written $N_{D|K}(a)=N_{D|K}(\overline{a})$. Since the norm is given by the determinant, this means, in $D^*({\Bbb R})$, $$L({\Bbb R})\cong \left\{ a\in D^*({\Bbb R})\Big| a\not\in K\Ra \det(a)\in {\Bbb R}^*\right\} \cong U(1)\cdot GL^0(d,{\Bbb C}),$$ where the $U(1)$ factor arises from the elements in $U(1)\cap K$. Similarly, the unipotent radical takes the form $$U \cong \left\{\left(\begin{array}{cc} 1 & b \\ 0 & 1 \end{array}\right) \Big| b\in D,\ b+\overline{b}=0\right\} \cong \sqrt{-\eta}D^+,$$ which is $d^2$-dimensional. An explicit isomorphism $U({\Bbb R})\cong {\cal H}_d({\Bbb C})$ is given by the identity on $D^+({\Bbb R})$, since $x\in D^+\Ra x=\overline{x}$, hence as a real matrix, this element is hermitian. \end{itemize} The maximal compact subgroups of $G_D({\Bbb R})$ are the groups ${\cal K}({\Bbb R})$, where ${\cal K}$ is the group of (\ref{eCom}). Indeed, it is easy to see that the stabiliser of the non-isotropic vector $(1,1)$ is the stabiliser, in the domain $\cD$, of the base point, which is by definition the maximal compact subgroup. Now consider the subgroups $U(L^2,h)\subset} \def\nni{\supset} \def\und{\underline G_D$; let us introduce the notation $G_L$ for these subgroups. Then, since $G_L$ is of the type $d=1$ in Proposition \ref{p9.1}, it follows that Theorem \ref{t9.1} applies to $G_L({\Bbb R})$ as well as to $G_D({\Bbb R})$. \begin{proposition}\label{p10.1} We have a commutative diagram $$\begin{array}{ccc} G_L({\Bbb R}) & \hookrightarrow} \def\hla{\hookleftarrow & G_D({\Bbb R}) \\ \downarrow & & \downarrow \\ \cD_L & \hookrightarrow} \def\hla{\hookleftarrow & \cD_D, \end{array}$$ where $\cD_L$ and $\cD_D$ are hermitian symmetric domains, and the maps $G({\Bbb R})\longrightarrow} \def\sura{\twoheadrightarrow \cD$ are the natural projections. Moreover, the subdomains $\cD_L$ are as follows: \begin{itemize}\item[1)] $d=1$; We now assume that the extension $k/k'$ is {\it Galois}. Then, if $\hbox{deg}} \def\Pic{\hbox{Pic}} \def\Jac{\hbox{Jac}_{{\Bbb Q}}k'=f'$, $f/f'=m$, we have $\cD_L\cong (\hbox{\omegathic H}} \def\scI{\hbox{\script I})^{f'}$ and $\cD_D\cong ((\hbox{\omegathic H}} \def\scI{\hbox{\script I})^m)^{f'}$ and the embedding $\cD_L\subset} \def\nni{\supset} \def\und{\underline \cD_D$ is given by $\hbox{\omegathic H}} \def\scI{\hbox{\script I}\hookrightarrow} \def\hla{\hookleftarrow (\hbox{\omegathic H}} \def\scI{\hbox{\script I})^m$ diagonally, and the product of this $f'$ times. \item[2)] $d=2$; $\cD_L\cong \left(\begin{array}{cc}\tau_1 & 0 \\ 0 & b^{\zeta_1}\tau_1 \end{array}\right)\times \cdots \times \left(\begin{array}{cc}\tau_1 & 0 \\ 0 & b^{\zeta_f}\tau_1 \end{array}\right)$, where $\zeta_i:k\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb R}$ denote the distinct real embeddings of $k$. \item[3)] $d\geq 3$; $\cD_L\cong \left(\begin{array}{ccc}\tau_1 & & 0 \\ & \ddots & \\ 0 & & \tau_d\end{array}\right)^f$. \end{itemize} \end{proposition} {\bf Proof:} Considering the tube realisation of the domain $\cD_D$, we have the base point $$o=(\hbox{diag}} \def\det{\hbox{det}} \def\Ker{\hbox{Ker}(i,i,\ldots,i))^f,$$ and we just calculate the orbit of the ${\Bbb R}$-group $G_L({\Bbb R})$ of this point; this gives the corresponding subdomain. The commutativity of the diagram follows from this. \begin{itemize}\item[1)] $d=1$: Since $k/k'$ is Galois, we have that $G_L\subset} \def\nni{\supset} \def\und{\underline G_D$ is the subgroup fixed under the natural $Gal_{k|k'}$ action; hence lifting the ${\Bbb Q}$-groups to ${\Bbb R}$, we get the identification $$(Res_{k'|{\Bbb Q}}G_L)_{{\Bbb R}} \cong (Res_{k|{\Bbb Q}}G_D)_{{\Bbb R}}^{Gal_{k|k'}}.$$ In terms of the domains, if $\cD_D=(\hbox{\omegathic H}} \def\scI{\hbox{\script I})^f$, then $Gal_{k|k'}$ acts by permuting $m$ copies at a time, and the invariant part is the diagonally embedded $\hbox{\omegathic H}} \def\scI{\hbox{\script I}\hookrightarrow} \def\hla{\hookleftarrow (\hbox{\omegathic H}} \def\scI{\hbox{\script I})^m,\ z\mapsto (z,\ldots,z)$. Since $\cD_L\cong (\hbox{\omegathic H}} \def\scI{\hbox{\script I})^{f'}$, the result follows in this case. \item[2)] $d=2$: First we remark that $G_{{\Bbb R}}$ is conjugate to the {\it standard} $Sp(4,{\Bbb R})$ (by which we mean the symplectic group with respect to the symplectic form $J={0\ \ 1 \choose -1\ 0}$) under the element $$q=\left(\begin{array}{c|c}\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} & 0 \\ \hline 0 & \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\end{array}\right),$$ that is $qG_{{\Bbb R}}q^{-1}=Sp(4,{\Bbb R})$. To see this, we recall how the ($D$-valued) hyperbolic (hermitian) form is related to alternating forms. Since $D_{{\Bbb R}}\cong M_2({\Bbb R})$, our vector space $D^2$ over ${\Bbb R}$ is $V_{{\Bbb R}}=(M_2({\Bbb R}))^2$; the hyperbolic form gives a $M_2({\Bbb R})$-valued form on $V_{{\Bbb R}}$. We have matrix units $e_{ij},\ i,j=1,2$ in $M_2({\Bbb R})$, and setting $W_1=e_{11}V,\ W_2=e_{22}V$, the spaces $W_i$ are two-dimensional vector spaces over ${\Bbb R}$. The hermitian form $h({\bf x},{\bf y})$, restricted to $W_1$, turns out to have values in ${\Bbb R} e_{12}$: if $${\bf x}=\left(\left(\begin{array}{cc} x_{11} & x_{12} \\ x_{21} & x_{22}\end{array}\right),\left(\begin{array}{cc} x'_{11} & x'_{12} \\ x'_{21} & x'_{22}\end{array}\right)\right), \ \ \ {\bf y}=\left(\left(\begin{array}{cc} y_{11} & y_{12} \\ y_{21} & y_{22}\end{array}\right),\left(\begin{array}{cc} y'_{11} & y'_{12} \\ y'_{21} & y'_{22}\end{array}\right)\right),$$ then ${\bf x, y}\in W_1 \iff x_{2i}=x'_{2i}=y_{2i}=y'_{2i}=0$. Then we have\footnote{the involution on $M_2(k)$ is as in (\ref{e2.2}).} $$h({\bf x},{\bf y})=\left(\begin{array}{cc} x_{11} & x_{12} \\ 0 & 0 \end{array}\right)\left(\begin{array}{cc} 0 & -y'_{12} \\ 0 & y'_{11} \end{array} \right) + \left(\begin{array}{cc} x'_{11} & x'_{12} \\ 0 & 0 \end{array}\right) \left( \begin{array}{cc} 0 & -y_{12} \\ 0 & y_{11} \end{array} \right) $$ $$ = \left(\begin{array}{cc} 0 & -x_{11}y'_{12} + x_{12}y'_{11} -x'_{11}y_{12} + x'_{12}y_{11} \\ 0 & 0 \end{array}\right).$$ Then the alternating form $\langle {\bf x},{\bf y} \rangle$ is defined by $$h({\bf x},{\bf y})=\langle {\bf x}, {\bf y}\rangle\cdot e_{12},\ {\bf x,y}\in W_1.$$ In other words, viewing the symplectic group as a subgroup of $GL(2,M_2({\Bbb R}))$, the symplectic form will be given by $J_1={0\ \ J_2 \choose J_3\ 0}$, where $J_2$ and $J_3$ are symplectic forms on ${\Bbb R}^2$. It is clear that $q{0\ \ 1_2 \choose 1_2\ 0}{^tq}=J_1$ is of this form, and this shows $qG_{{\Bbb R}}q^{-1}=Sp(4,{\Bbb R})$. Next, recalling that we view $D$ as a cyclic algebra of $2\times 2$ matrices, an element $A={\alpha\ \beta \choose \gamma\ \delta}\in M_2(D)$ is actually a $4\times 4$ matrix; each of $\alpha,\beta,\gamma, \delta$ are of the form (\ref{e2.1}), and this $4\times 4$ matrix must be conjugated by $q$, then the element acts in the well known manner on the domain $\cD_D$: $$M=\left(\begin{array}{cc} A & B \\ C & D \end{array}\right) \in Sp(4,{\Bbb R}),\ \ \ \tau=\left(\begin{array}{cc} \tau_1 & \tau_{12} \\ \tau_{12} & \tau_2\end{array}\right)\in \cD_D,\ \ \ M\cdot \tau = (A\tau + B)(C\tau +D)^{-1}.$$ Finally we note that the subgroup $U(L^2,h)\subset} \def\nni{\supset} \def\und{\underline U(D^2,h)$ consists of the elements of the form as in Proposition \ref{p7.1} 2): $$x=\left(\begin{array}{cc} \alpha & 2\beta/\sqrt{-ab} \\ \gamma\sqrt{-ab}/2 & \delta \end{array}\right),\ \ \ \alpha,\beta,\gamma,\delta\in k,$$ and the element $\sqrt{-ab}=ec$ as in (\ref{e7.1}), so that $(\sqrt{-ab})^{-1}=(ec)^{-1}$. In other words, as a $4\times 4$ matrix, the element $x$ above is \begin{equation}\label{e10b.1} x=\left(\begin{array}{cc|cc} \alpha & 0 & 0 & {2\beta \over b\sqrt{a}} \\ 0 & \alpha & {2\beta \over -\sqrt{a}} & 0 \\ \hline 0 & {-\gamma\sqrt{a}\over 2} & \delta & 0 \\ {b\gamma\sqrt{a}\over 2} & 0 & 0 & \delta \\ \end{array}\right), \end{equation} and after conjugating with $q$ this becomes \begin{equation}\label{e10b.2} qxq^{-1}=\left(\begin{array}{cc|cc} \alpha & 0 & {2\beta \over b\sqrt{a}} & 0 \\ 0 & \alpha & 0 & {2\beta\over \sqrt{a}} \\ \hline {b\gamma\sqrt{a}\over 2} & 0 & \delta & 0 \\ 0 & {\gamma\sqrt{a}\over 2} & 0 & \delta\end{array}\right). \end{equation} {}From the form of the matrix (\ref{e10b.2}) we see that the corresponding subdomain is contained in the set of diagonal matrices ${\tau_1\ \ 0\choose 0\ \tau_2}^f$. Now calculating, setting $g=qxq^{-1}$, we have \begin{equation}\label{e10b.2a} g\left(\begin{array}{cc}\tau_1 & 0 \\ 0 & \tau_2\end{array}\right)^f=\left(\left(\begin{array}{cc} \left({\alpha b \tau_1 +{2\beta\sqrt{a}\over a} \over {\gamma\sqrt{a}\over 2}b\tau_1+\delta}\right){1\over b} & 0 \\ 0 & {\alpha \tau_2 +{2\beta\sqrt{a}\over a} \over {\gamma\sqrt{a}\over 2}\tau_2+\delta} \end{array}\right)^{\zeta_1},\ldots, \left(\begin{array}{cc} \left({\alpha b \tau_1 +{2\beta\sqrt{a}\over a} \over {\gamma\sqrt{a}\over 2}b\tau_1+\delta}\right){1\over b} & 0 \\ 0 & {\alpha \tau_2 +{2\beta\sqrt{a}\over a} \over {\gamma\sqrt{a}\over 2}\tau_2+\delta} \end{array}\right)^{\zeta_f} \right), \end{equation} So in particular, $G_L$ maps the set of matrices \begin{equation}\label{e10b.2b} \cD_L=\left(\begin{array}{cc}\tau_1 & 0 \\ 0 & b^{\zeta_1}\tau_1 \end{array}\right)\times \cdots \times \left(\begin{array}{cc}\tau_1 & 0 \\ 0 & b^{\zeta_f}\tau_1 \end{array}\right) \end{equation} into itself. (In the expression (\ref{e10b.2a}), the bracket $(...)^{\zeta_i}$ means $\zeta_i$ is applied to all matrix elements). This proves the theorem in the case $d=2$. \begin{remark} The description of the subdomain $\cD$ in terms of $b$, depends on the way we described $D$ as a cyclic algebra. We took $\ell=k(\sqrt{a}),\ D=(\ell/k,\sigma,b)$. But we could also consider $\ell'=k(\sqrt{b}),\ D'=(\ell'/k,\sigma',a)$; these two cyclic algebras are isomorphic, an isomorphism being given by extending the identity on the center $k$ by \begin{eqnarray*} D & \stackrel{\sim}{\longrightarrow} \def\sura{\twoheadrightarrow} & D' \\ e & \mapsto & c' \\ c & \mapsto & e',\end{eqnarray*} where $$e=\left(\begin{array}{cc} 0 & 1 \\ b & 0 \end{array}\right),\quad c=\left(\begin{array}{cc} \sqrt{a} & 0 \\ 0 & -\sqrt{a} \end{array}\right), \hspace*{2cm} e'=\left(\begin{array}{cc} 0 & 1 \\ a & 0\end{array}\right), \quad c'=\left( \begin{array}{cc} \sqrt{b} & 0 \\ 0 & -\sqrt{b}\end{array}\right).$$ Indeed, as one sees immediately, $(ec)^2=-ab=(e'c')^2$, so the map above is a morphism, which is clearly bijective. Consequently, it would be better to denote (\ref{e10b.2b}) $\cD_{L,b}$, since we {\it also} have a subdomain $$\cD_{L,a}=\left(\begin{array}{cc} \tau_1 & 0 \\ 0 & a^{\zeta_1}\tau_1\end{array}\right)\times \cdots \times \left(\begin{array}{cc} \tau_1 & 0 \\ 0 & a^{\zeta_f}\tau_1\end{array}\right),$$ neither of which is a priori privledged. \end{remark} \item[3)] $d\geq 3$: Here the situation is simpler; the matrix $H$ is the matrix of the hermitian form whose symmetry group acts on the usual unbounded realisation of the domain of type $\bf I_{d,d}$, hence no conjugation is necessary. Let $G_L$ be the subgroup we wish to consider, and recall from Proposition \ref{p7.1} that it consists of matrices $$x=\left(\begin{array}{cc} \alpha & 2\beta/\sqrt{-\eta} \\ \gamma\sqrt{-\eta}/2 & \delta \end{array}\right),\ \ \ \alpha,\beta,\gamma,\delta\in k,\ \alpha\delta-\gamma\beta=1.$$ Thinking of $D$ itself as $d\times d$ matrices, this element is \begin{equation}x=\left(\begin{array}{cccc|cccc} \alpha & & & 0 & {2\beta \over \sqrt{-\eta}} & & & 0\\ & \alpha^{\sigma} & & & & {2\beta^{\sigma} \over \sqrt{-\eta}} & & \\ & & \ddots & & & & \ddots & \\ 0 & & & \alpha^{\sigma^{d-1}} & 0 & & & {2\beta^{\sigma^{d-1}} \over \sqrt{-\eta}} \\ \hline {\gamma\sqrt{-\eta} \over 2} & & & 0 & \delta & & & 0 \\ & {\gamma^{\sigma}\sqrt{-\eta} \over 2} & & & & \delta^{\sigma} & & \\ & & \ddots & & & & \ddots & \\ 0 & & & {\gamma^{\sigma^{d-1}}\sqrt{-\eta} \over 2} & 0 & & & \delta^{\sigma^{d-1}} \end{array} \right), \end{equation} and the orbit of the base point $\hbox{diag}} \def\det{\hbox{det}} \def\Ker{\hbox{Ker}(i,\ldots, i)^f$ under elements as $x$ is clearly the diagonal subdomain $(\hbox{diag}} \def\det{\hbox{det}} \def\Ker{\hbox{Ker}(\tau_1,\ldots, \tau_d))^f$, and the action is, for $k={\Bbb Q}$, \begin{equation}\label{e10b.5} \hbox{diag}} \def\det{\hbox{det}} \def\Ker{\hbox{Ker}(\tau_1,\ldots,\tau_d) \mapsto \left( {\alpha\tau_1+{2\beta \over \sqrt{-\eta}} \over {\gamma\sqrt{-\eta}\over 2} \tau_1 + \delta},\ldots , {\alpha^{\sigma^{d-1}}\tau_{d}+{2\beta^{\sigma^{d-1}}\over \sqrt{-\eta}} \over {\gamma^{\sigma^{d-1}}\sqrt{-\eta}\over 2}\tau_d + \delta^{\sigma^{d-1}} }\right)=:\left(\begin{array}{cc} \alpha & {2\beta\over \sqrt{-\eta}} \\ {\gamma\sqrt{-\eta}\over 2} & \delta\end{array} \right)(\tau_1,\ldots,\tau_d), \end{equation} and for general $k$, letting as above $\zeta_1,\ldots,\zeta_f$ denote the $f$ embeddings of $k$, \begin{equation}\label{e10b.6} (\hbox{diag}} \def\det{\hbox{det}} \def\Ker{\hbox{Ker}(\tau_1,\ldots,\tau_d))^f \mapsto \left( \left(\begin{array}{cc} \alpha & {2\beta\over \sqrt{-\eta}} \\ {\gamma\sqrt{-\eta}\over 2} & \delta\end{array} \right)^{\zeta_1}(\tau_1,\ldots,\tau_d),\ldots, \left(\begin{array}{cc} \alpha & {2\beta\over \sqrt{-\eta}} \\ {\gamma\sqrt{-\eta}\over 2} & \delta\end{array} \right)^{\zeta_f}(\tau_1,\ldots,\tau_d) \right). \end{equation} \end{itemize} This establishes all statements of the theorem. \hfill $\Box$ \vskip0.25cm } \def\cD{\ifmmode {\cal D Finally, let $F$ be the rational boundary component $\in \cD_D^*$ corresponding to the isotropic vector $(0,1)$, of which the parabolic $P$ of (\ref{e6.1}) is the normaliser, $P({\Bbb R})=N(F)$. Note that by construction, $(0,1)$ is also isotropic for $G_L$; let $F_L$ be the rational boundary component $\in \cD_L^*$ corresponding to it, so the parabolic in $G_L$, $P_L$ of (\ref{e6.1}) is the normaliser, $P_L({\Bbb R})=N(F_L)$. Then clearly \begin{equation}\label{e10.2} P_L=G_L\cap P. \end{equation} Also we have \begin{proposition}\label{p10.2} Let $\cD_L\subset} \def\nni{\supset} \def\und{\underline \cD_D,\ F_L\in \cD_L^*,\ F\in \cD_D^*$ be as above, and let $i:\cD_L^*\hookrightarrow} \def\hla{\hookleftarrow \cD_D^*$ be the induced inclusion of Satake compactifications of the domains. Then $i(F_L)=F$. \end{proposition} We will refer to this as ``the subdomain $\cD_L$ contains $F$ as a rational boundary component.'' The proposition itself is, after unraveling the definitions, nothing by (\ref{e10.2}). \section{Arithmetic groups} It is well-known how to construct arithmetic groups $\Gamma\subset} \def\nni{\supset} \def\und{\underline G_D(K)$; we recall this briefly. View $D^2$ as a $D$-vector space, and let $\Delta\subset} \def\nni{\supset} \def\und{\underline D$ be a maximal order, that is an ${\cal O}_K$-module (where $D$ is central simple over $K$, $K=k$ if $d=2$), which spans $D$ as a $K$-vector space, is a sub{\it ring} of $D$, and is maximal with these properties. Then set \begin{equation} \label{e11.1}\Gamma_{\Delta}:=G_D(K)\cap M_2(\Delta). \end{equation} If we view $\Delta^2\subset} \def\nni{\supset} \def\und{\underline D^2$ as a lattice, then this is also described as $$\Gamma_{\Delta}=\left\{ g\in G_D(K) \Big| g(\Delta^2)\subset} \def\nni{\supset} \def\und{\underline \Delta^2 \right\}.$$ {}From the second description it is clear that $\Gamma_{\Delta}$ is an {\it arithmetic} subgroup. We will denote the quotient of the domain $\cD_D$ by this arithmetic subgroup by \begin{equation}\label{e11.2} X_{\Gamma_{\Delta}}=X_{\Delta}= \Gamma_{\Delta}\backslash \cD_D. \end{equation} If $\Gamma_{\Delta}$ were fixed-point free on $\cD_D$ this would be a non-compact complex manifold; if $\Gamma_{\Delta}$ has fixed points (as it usually does), these yield singularities of the space $X_{\Delta}$, which is still an analytic variety. It is a $V$-variety in the language of Satake. In fact it has a {\it global} finite smooth cover: let $\Gamma\subset} \def\nni{\supset} \def\und{\underline \Gamma_{\Delta}$ be a normal subgroup of finite index which does act freely (such exist by a theorem of Selberg). Then we have a Galois cover $$X_{\Gamma}\longrightarrow} \def\sura{\twoheadrightarrow X_{\Gamma_{\Delta}},$$ and $X_{\Gamma}$ is smooth. Hence the singularities of $X_{\Gamma_{\Delta}}$ are encoded in the action of the finite Galois group of the cover. The Baily-Borel compactification of $X_{\Gamma}$ ($\Gamma\subset} \def\nni{\supset} \def\und{\underline \Gamma_{\Delta}$) is an embedding $$X_{\Gamma}\subset} \def\nni{\supset} \def\und{\underline X_{\Gamma}^*\subset} \def\nni{\supset} \def\und{\underline {\Bbb P}^N,$$ where $ X_{\Gamma}^*$ is a normal algebraic variety. Since the boundary components of $\cD_D$ are points (Theorem \ref{t9.1}), it follows that the singularities of $ X_{\Gamma}^*$ contained in the boundary $ X_{\Gamma}^*-X_{\Gamma}$ are also {\it isolated points}. In particular, if $\Gamma\subset} \def\nni{\supset} \def\und{\underline \Gamma_{\Delta}$ has no elements of finite order, then $ X_{\Gamma}^*$ is smooth outside a finite set of isolated points, all contained in the compactification locus. The singularities of $ X_{\Gamma}^*$ may be resolved by means of toroidal compactifications; let us denote such a compactification by $\overline{X}_{\Gamma}$, for which we make the assumptions: \begin{itemize}\item[i)] $\overline{X}_{\Gamma}-X_{\Gamma}$ is a normal crossings divisor, \item[ii)] $\overline{X}_{\Gamma}\longrightarrow} \def\sura{\twoheadrightarrow X_{\Gamma}^*$ is a resolution of singularities and $\overline{X}_{\Gamma}$ is projective algebraic. \end{itemize} Such a compactification $\overline{X}_{\Gamma}$ depends on a set of polyhedral cones, and one can make choices such that i) and ii) are satisfied, by the results of \cite{SC}. We will not need these in detail; the mere existence will be sufficient. Recall that a cusp of an arithmetic group $\Gamma$ is a maximal flag of boundary components of $\Gamma$; in the case at hand this is just a point in $ X_{\Gamma}^*-X_{\Gamma}$. The {\it number of cusps} is the number of points in $ X_{\Gamma}^*- X_{\Gamma}$, and may be defined in the following ways: \begin{itemize}\item[a)] It is the number of $\Gamma$-equivalence classes of parabolic subgroups of $G_D$ or $SG_D$; here $\Gamma$ acts on the parabolic subgroups by conjugation. \item[b)] It is the number of $\Gamma$-equivalence classes of isotropic vectors of the hermitian plane $(D^2,h)$; here $\Gamma$ is acting as a subgroup of $G_D$ on the vector space. \end{itemize} In the $d=1$ case we are talking about the number of cusps of a Hilbert modular variety. Let us recall how these numbers are determined. For this we consider first the group $SL_2(k)$ (instead of $SU(K^2,h)$), which acts on a product of upper half planes, $\cD_D=\hbox{\omegathic H}} \def\scI{\hbox{\script I}^f$. The boundary is the product $({\Bbb P}^1({\Bbb R}))^f$, and the rational boundary components are the points of the image ${\Bbb P}^1(k)\hookrightarrow} \def\hla{\hookleftarrow ({\Bbb P}^1({\Bbb R}))^f$ given by $x\mapsto (x^{\zeta_1},\ldots,x^{\zeta_f})$. Hence we may denote the boundary components by $\xi=(\xi_1:\xi_2),\ \xi_i\in {\cal O}_k$. Let $\hbox{\omegathic a}_{\xi}$ denote the ideal generated by $\xi_1,\xi_2$. Then one has \begin{proposition}\label{p13.1} Two boundary components $\xi=(\xi_1:\xi_2)$ and $\eta=(\eta_1:\eta_2)$ are equivalent under $SL_2({\cal O}_k)$ $\iff$ the ideals $\hbox{\omegathic a}_{\xi}$ and $\hbox{\omegathic a}_{\eta}$ are in the same class. In particular, the number of cusps is the class number of $k$. \end{proposition} {\bf Proof:} One direction is trivial: if ${a\ b\choose c\ d}{\xi_1\choose \xi_2}={\eta_1\choose \eta_2},\ g={a\ b\choose c\ d}\in SL_2({\cal O}_k)$, then the ideals $\hbox{\omegathic a}_{\xi}$ and $\hbox{\omegathic a}_{\eta}$ are the {\it same}. This is because $g$ is unimodular, so affects just a change of base in the ideal $\hbox{\omegathic a}_{\xi}$. Conversely, suppose $\xi$ and $\eta$ are given, and suppose that $\hbox{\omegathic a}_{\xi}$ and $\hbox{\omegathic a}_{\eta}$ have the same class. After multiplication by an element $c\in k^*$, we may assume $\hbox{\omegathic a}_{\xi}=\hbox{\omegathic a}_{\eta}=\hbox{\omegathic a}$. Furthermore, writing ${\cal O}_k=\hbox{\omegathic a}\aa^{-1}=\xi_1\hbox{\omegathic a}^{-1}+\xi_2\hbox{\omegathic a}^{-1}$, we see that $1\in {\cal O}_k$ can be written \begin{equation}\label{e13.1} 1=\xi_1\xi_2'-\xi_2\xi_1'=\eta_1\eta_2'-\eta_2\eta_1',\ \ \xi_i', \eta_i'\in \hbox{\omegathic a}^{-1}. \end{equation} But this means the matrices (acting from the left) \begin{equation}\label{e13.2} M_{\xi}=\left(\begin{array}{cc} \xi_1 & \xi_1' \\ \xi_2 & \xi_2' \end{array}\right),\quad\quad M_{\eta}=\left(\begin{array}{cc} \eta_1 & \eta_1' \\ \eta_2 & \eta_2' \end{array}\right), \end{equation} are in $SL_2(k)$, and \begin{equation}\label{e13.3} M_{\xi}{ 1 \choose 0}={\xi_1\choose \xi_2}, \quad\quad M_{\eta}{1\choose 0}={\eta_1\choose \eta_2}. \end{equation} Of course $M_{\xi},\ M_{\eta}$ are not in $SL_2({\cal O}_k)$, but from the fact that $\xi_i',\eta_i'\in \hbox{\omegathic a}^{-1}$ it follows that $M_{\xi}M_{\eta}^{-1}\in SL_2({\cal O}_k)$. Hence $$(M_{\xi}M_{\eta}^{-1}){\eta_1\choose \eta_2} ={\xi_1\choose \xi_2},$$ and the cusps are conjugate under $SL_2({\cal O}_k)$. \hfill $\Box$ \vskip0.25cm } \def\cD{\ifmmode {\cal D We now translate this into the corresponding statement for the hyperbolic plane $(K^2,h)$, where $K|k$ is an imaginary quadratic extension. First note that if ${\xi_1\choose \xi_2}\in K^2$ is {\it isotropic}, then $$Tr_{K|k}(\xi_1\overline{\xi}_2)= \xi_1\overline{\xi}_2+\xi_2\overline{\xi}_1=0.$$ In fact, the converse also holds, \begin{lemma}\label{l14.1} A vector ${\xi_1\choose \xi_2}\in K^2$ is isotropic with respect to $h$ $\iff Tr(\xi_1\overline{\xi}_2)=Tr(\overline{\xi}_1\xi_2)=0$ $\iff$ $\xi_2=0$ or $Tr(\xi_1\xi_2^{-1})=0$. \end{lemma} {\bf Proof:} By definition, ${\xi_1\choose \xi_2}$ is isotropic $\iff$ $\xi_1\overline{\xi}_2+\xi_2\overline{\xi}_1=0$, but this is $Tr(\xi_1\overline{\xi}_2)=0$. Noting that $\xi_2^{-1}={1\over N(\xi_2)}\overline{\xi}_2$, if $\xi_2\neq0$, we get the second equivalence, from the linearity of the trace, $Tr(\xi_1\xi_2^{-1})=Tr(\xi_1{1\over N(\xi_2)}\overline{\xi}_2)= {1\over N(\xi_2)}Tr(\xi_1\overline{\xi}_2)$. If $\xi_2=0$, then ${\xi_1\choose\xi_2}={\xi_1\choose 0}$ is isotropic anyway, establishing both equivalences as stated. \hfill $\Box$ \vskip0.25cm } \def\cD{\ifmmode {\cal D Let $K^0$ denote the purely imaginary (traceless) elements of $K$, $K^0\cong \sqrt{-\eta}k$ for $K=k(\sqrt{-\eta})$. It follows from Lemma \ref{l14.1} that any $x\in K^0$ of the form $x=\xi_1\xi_2^{-1}$ or $x=\xi_1\overline{\xi}_2$ determines an isotropic vector ${\xi_1\choose \xi_2}$ of $K^2$. Consider in particular the case that ${\xi_1\choose \xi_2}\in {\cal O}_K^2$ is integral. Recall that (non-zero) $\xi_1$ and $\xi_2$ are relatively prime $\iff$ there are $x,y\in {\cal O}_K$ such that $x\xi_1+y\xi_2=1$. Define the set of relatively prime integral isotropic vectors, \begin{equation}\label{e14.2} \scI=\left\{ {\xi_1\choose \xi_2}\in {\cal O}_K^2-(0,0) \Big| Tr(\xi_1\overline{\xi}_2)=0; \hbox{ and if } \xi_i\neq 0, i=1,2,\ \hbox{ then } \exists_{x,y\in {\cal O}_K}\ x\xi_1+y\xi_2=1\right\}. \end{equation} Following the proof of Proposition \ref{p13.1}, we consider when two isotropic integral vectors are conjugate under $S\Gamma_{\Delta}=SU({\cal O}_K^2,h)$. Let now $\hbox{\omegathic a}_{\xi}$ be the ideal generated by $\xi_1,\xi_2$ in ${\cal O}_K$. \begin{proposition}\label{p14.1} Two isotropic vectors $\xi={\xi_1\choose \xi_2}$ and $\xi'={\xi'_1\choose \xi'_2}\in \scI$ are conjugate under $SU({\cal O}_K^2,h)$ $\iff$ the ideals $\hbox{\omegathic a}_{\xi}$ and $\hbox{\omegathic a}_{\xi'}$ are equivalent in $K$ $\iff$ the ideals $N(\hbox{\omegathic a}_{\xi})$ and $N(\hbox{\omegathic a}_{\xi'}) $ are equivalent in $k$. \end{proposition} {\bf Proof:} If $g{\xi_1\choose \xi_2}={\xi_1'\choose \xi_2'},\ g\in SU({\cal O}_K^2,h)$, then since $g$ is unimodular, the ideal classes coincide. Conversely, suppose ${\xi_1\choose \xi_2}$ and ${\xi_1'\choose{\xi'_2}}$ are equivalent; again we may assume $\hbox{\omegathic a}_{\xi}=\hbox{\omegathic a}_{\xi'}=\hbox{\omegathic a}$. Then, as above, there are $\rho,\ \rho'\in (\hbox{\omegathic a}^{-1})^2$ such that (\ref{e13.1}) holds. Consequently we have $M_{\xi}$, $M_{\xi'}$, but now in $SU(K^2,h)$, such that (\ref{e13.2}) and (\ref{e13.3}) hold. It follows that $M_{\xi}M_{\xi'}^{-1}\in SU({\cal O}_K^2,h)$ maps $\xi'$ to $\xi$. This completes verification of the first $\iff$. The second equivalence then follows from Proposition \ref{p5.1} and Proposition \ref{p13.1}. \hfill $\Box$ \vskip0.25cm } \def\cD{\ifmmode {\cal D We now derive an analogue of the above for general $D$ as in (\ref{e1.0}). We first note that Lemma \ref{l14.1} is true here also. \begin{lemma}\label{l15.1} Let $\xi=(\xi_1,\xi_2)\in D^2$ be given. Then $\xi$ is isotropic $\iff$ $\xi_1\overline{\xi}_2+\xi_2\overline{\xi}_1=0$. \end{lemma} {\bf Proof:} We have $h(\xi,\xi)=\xi_1\overline{\xi}_2+\xi_2\overline{\xi}_1=0$, so that $h(\xi,\xi)=0 $ if and only if $\xi_1\overline{\xi}_2+\xi_2\overline{\xi}_1=0$. \hfill $\Box$ \vskip0.25cm } \def\cD{\ifmmode {\cal D Next we note that any isotropic vector is conjugate in $G_D$ to the standard isotropic vector $(0,1)$. \begin{lemma}\label{l15.2} Let $\xi=(\xi_1,\xi_2)\in D^2$ be isotropic. Then there is a matrix $M_{\xi}\in G_D$ such that $(0,1)M_{\xi}=\xi$. \end{lemma} {\bf Proof:} If $M_{\xi}=\left(\begin{array}{cc} \xi_1' & \xi_2' \\ \xi_1 &\xi_2\end{array}\right)$, then $(0,1)M_{\xi}=\xi$. So we must show the existence of such an $M_{\xi}\in G_D$. The equations to be solved (for $\xi_i'$) are \begin{equation}\label{e15.1}\begin{minipage}{15cm}\begin{itemize}\item[1)] $\xi_1'\overline{\xi}_2+\xi_2'\overline{\xi}_1=1$. \item[2)] $\xi_1'\overline{\xi}'_2+\xi_2'\overline{\xi}'_1=0.$ \item[3)] $\xi_1\overline{\xi}_2+\xi_2\overline{\xi}_1=0.$ \end{itemize} \end{minipage}\end{equation} Since $\xi$ is isotropic, 3) is fulfilled by Lemma \ref{l15.1}. If $\xi_1\neq 0$, then setting $\xi_1'=0, \xi_2'=\overline{\xi}_1^{-1}$, $\xi'$ fulfills both 1) and 2). If $\xi_1=0$, then $\xi_2\neq 0$ and we set similarly $\xi_1'=\overline{\xi}_2^{-1}$ and $\xi_2'=0$. \hfill $\Box$ \vskip0.25cm } \def\cD{\ifmmode {\cal D Now we consider as above integral isotropic vectors. Let $\Delta\subset} \def\nni{\supset} \def\und{\underline D$ be a maximal order, and set $$\scI_{\Delta}=\left\{ (\xi_1,\xi_2)\in \Delta^2-\{(0,0)\} \Big| \xi_1\overline{\xi}_2+\xi_2\overline{\xi}_1=0, \xi_1, \xi_2\neq 0 \Ra \exists_{x,y\in \Delta}\ \xi_1x+\xi_2y=1\right\}.$$ \begin{lemma} \label{l15.3} Let $(\xi_1,\xi_2)\in \scI_{\Delta}$. Then there exists a matrix $$M_{\xi}=\left(\begin{array}{cc} \xi_1'' & \xi_2'' \\ \xi_1 &\xi_2\end{array}\right)\in \Gamma_{\Delta}.$$ \end{lemma} {\bf Proof:} If $\xi_2=0$, then $\xi''_2=1$, any $\xi''_1$ with $\xi''_1+\overline{\xi}''_1=0$ gives such a matrix, similarly, if $\xi_1=0$, then $\xi''_1=1$ gives a solution. So we assume $\xi_i\neq 0$; then by assumption we have $(x,y)\in \Delta^2$ such that $\xi_1x+\xi_2 y=1$, hence $\overline{x}\overline{\xi}_1+\overline{y}\overline{\xi}_2=1$. So setting $\xi_2'=\overline{x}, \xi_1'=\overline{y}$, the first relation of (\ref{e15.1}) is satisfied. Again 3) is satisfied by assumption, so we must verify 2). It turns out that we may have to alter $\xi_i'$ to achieve this. The relation 1) can be expressed as $h(\xi',\xi)=1$, where $\xi=(\xi_1,\xi_2),\ \xi'=(\xi_1',\xi_2')$. Since $\xi$ is isotropic by Lemma \ref{l15.1}, $h(\xi,\xi)=0$, and with respect to the base $<\xi,\xi'>$ of $D^2$, $h$ is given by a matrix $H_{\xi,\xi'}={0\ 1 \choose 1\ \epsilon}$, where $\epsilon=h(\xi',\xi')= \delta + \overline{\delta}$, $\delta=\xi_1'\overline{\xi}'_2$ (if $\epsilon\neq0$; otherwise we are done), in particular $\delta\in \Delta$. Now setting $$\xi''=(\xi_1'',\xi_2'')=(-\xi_1\overline{\delta}+\xi_1', -\xi_2\overline{\delta}+\xi_2')\in \Delta^2,$$ we can easily verify $h(\xi'',\xi'')=0,\ \ h(\xi,\xi'')=h(\xi'',\xi)=1$, so this vector $\xi''$ gives a matrix $M_{\xi} \in \Gamma_{\Delta}$, as was to be shown. \hfill $\Box$ \vskip0.25cm } \def\cD{\ifmmode {\cal D We require the following refinement of Lemma \ref{l15.2}. For this, given $(\xi_1,\xi_2)\in \Delta^2$, let $\hbox{\omegathic a}_{\xi}$ denote the left ideal in $\Delta$ generated by the elements $\xi_1, \xi_2$. \begin{lemma}\label{l16.1} Given $\xi=(\xi_1,\xi_2)\in \Delta^2$ isotropic, the entries $\xi_i'$ of the matrix $M_{\xi}$ of Lemma \ref{l15.2} are in $\hbox{\omegathic a}_{\xi}^{-1}$. \end{lemma} {\bf Proof:} This follows from the proof of \ref{l15.2}, as we took just inverses of elements $\xi_i$. \hfill $\Box$ \vskip0.25cm } \def\cD{\ifmmode {\cal D We can now consider when two integral isotropic vectors are equivalent under $\Gamma_{\Delta}$. \begin{proposition}\label{p16.1} Let $\xi=(\xi_1,\xi_2),\ \eta=(\eta_1,\eta_2)\in \Delta^2$ be two integral isotropic vectors, $\hbox{\omegathic a}_{\xi},\ \hbox{\omegathic a}_{\eta}$ the left ideals generated by the components of $\xi$ and $\eta$, respectively. Then $\xi$ and $\eta$ are equivalent under an element of $\Gamma_{\Delta}$ $\iff$ $\hbox{\omegathic a}_{\eta}\cong \hbox{\omegathic a}_{\xi}$ as $\Delta$-modules. \end{proposition} {\bf Proof:} If there is an element $g={a\ b\choose c\ d}\in \Gamma_{\Delta}$ with $(\xi_1,\xi_2)g=(\eta_1,\eta_2)$, then $\eta_1=\xi_1a+\xi_2c,\ \eta_2=\xi_1b+\xi_2d$, hence $\hbox{\omegathic a}_{\eta}\subset} \def\nni{\supset} \def\und{\underline \hbox{\omegathic a}_{\xi}$ and the map \begin{eqnarray*} \phi:\hbox{\omegathic a}_{\eta} & \longrightarrow} \def\sura{\twoheadrightarrow & \hbox{\omegathic a}_{\xi} \\ \eta_1\lambda_1+\eta_2\lambda_2 & \mapsto & \xi_1(a\lambda_1+b\lambda_2)+ \xi_2(c\lambda_1+d\lambda_2) \end{eqnarray*} is an isomorphism of $\Delta$-modules. The inverse is given is a similar manner by $g^{-1}$. Conversely, if $\hbox{\omegathic a}_{\eta}\cong \hbox{\omegathic a}_{\xi}$ we may assume $\hbox{\omegathic a}_{\xi}=\hbox{\omegathic a}_{\eta}$, hence also $\hbox{\omegathic a}_{\xi}^{-1}= \hbox{\omegathic a}_{\eta}^{-1}$. Applying Lemma \ref{l16.1} and making use of $\hbox{\omegathic a}_{\xi}\hbox{\omegathic a}_{\xi}^{-1}=\Delta$, which holds in central simple division algebras over number fields, we find for the matrices $M_{\xi},\ M_{\eta}$ associated by Lemma \ref{l15.2} to $\xi$ and $\eta$, respectively, that $M_{\xi}M_{\eta}^{-1}\in \Gamma_{\Delta}.$ For example, if $\xi_1\neq 0,\ \eta_1\neq 0$, then $$M_{\xi}M_{\eta}^{-1} = \left(\begin{array}{cc}0 & \overline{\xi}_1^{-1} \\ \xi_1 & \xi_2\end{array}\right)\left(\begin{array}{cc} \overline{\eta}_2 & \eta_1^{-1} \\ \overline{\eta}_1 & 0 \end{array}\right) = \left(\begin{array}{cc} \overline{\xi}_1^{-1}\overline{\eta}_1 & 0 \\ \xi_1\overline{\eta}_2+\xi_2\overline{\eta}_1 & \xi_1{\eta}_1^{-1}\end{array}\right).$$ This verifies ``$\Leftarrow} \def\bs{\ifmmode {\setminus} \else$\bs$\fi$'' and completes the proof. \hfill $\Box$ \vskip0.25cm } \def\cD{\ifmmode {\cal D Finally, as above we can now express this in terms of ideal classes. The ideals $\hbox{\omegathic a}_{\xi}$ and $\hbox{\omegathic a}_{\eta}$ of Proposition \ref{p16.1} are equivalent, according to the usual definition. For general division algebras, one requires a weaker notion of equivalence than isomorphism as $\Delta$-modules, the notion of {\it stable equivalence}. However it is standard that for central simple division algebras over number fields, excluding definite quaternion algebras, the stronger notion coincides with the weaker one (see \cite{R}, 35.13). This is: $\Delta$-modules $M$ and $N$ are stably equivalent if $$M+\Delta^r \cong N+\Delta^r,$$ for some $r\geq 0$, ``$\cong$'' being isomorphism of $\Delta$-modules. At any rate, one defines the {\it class ray group} (depending on $D$) as follows. Let $S\subset} \def\nni{\supset} \def\und{\underline \Sigma_{\infty}$ denote the set of infinite primes which ramify in $D$, and set $$\hbox{\script C}\ell_D({\cal O}_K):=\{ideals\}/\left\{\alpha{\cal O}_K,\ \alpha\in K^*, \alpha_{\nu}>0\hbox{ for all } \nu\in S\right\}.$$ Furthermore, let $\hbox{\script C}\ell(\Delta)$ denote the set of equivalence classes of left ideals in $\Delta$. The two are related by \begin{eqnarray*} {\hbox{\script C}}\ell(\Delta) & \stackrel{\sim}{\longrightarrow} \def\sura{\twoheadrightarrow} & {\hbox{\script C}}\ell_D({\cal O}_K) \\ {[M]} & \mapsto & {[N_{D|K}(M)]}. \end{eqnarray*} For details on these matters, see \cite{R}. Now notice that for the division algebras of (\ref{e1.0}), for $d\geq2$, we have $\hbox{\script C}\ell_D({\cal O}_K)=\hbox{\script C}\ell({\cal O}_K)$ (where again $K=k$ if $d=2$). Hence \begin{proposition}\label{p16a.1} For the division algebras $D$ we are considering, if $d\geq2$ and $\Delta\subset} \def\nni{\supset} \def\und{\underline D$ is a maximal order, then $$\hbox{\script C}\ell(\Delta)\stackrel{\sim}{\longrightarrow} \def\sura{\twoheadrightarrow}\hbox{\script C}\ell({\cal O}_K).$$ \end{proposition} We can now proceed to carry out the program sketched above for the division algebras $D$. \begin{theorem}\label{t16a.1} Let $D$ be a central simple division algebra over $K$ as in (\ref{e1.0}), $D^2$ the right vector space with the hyperbolic form (\ref{e4.1}). Let $\xi=(\xi_1,\xi_2)$ and $\epsilon=(\eta_1,\eta_2)$ be two isotropic vectors in $\scI_{\Delta}$. Then \begin{itemize}\item[(i)] If $d=2$, then $\xi$ and $\eta$ are equivalent under $\Gamma_{\Delta}$ $\iff$ the ideals $N_{D|k}(\hbox{\omegathic a}_{\xi})$ and $N_{D|k}(\hbox{\omegathic a}_{\eta})$ are equivalent in ${\cal O}_k$. \item[(ii)] If $d\geq 3$, then $\xi$ and $\eta$ are equivalent under $\Gamma_{\Delta}$ $\iff$ the ideals $N_{D|K}(\hbox{\omegathic a}_{\xi})$ and $N_{D|K}(\hbox{\omegathic a}_{\eta})$ are equivalent in ${\cal O}_K$. \end{itemize} \end{theorem} {\bf Proof:} By Proposition \ref{p16.1}, $\xi$ and $\eta$ are equivalent under $\Gamma_{\Delta}$ $\iff$ the ideals $\hbox{\omegathic a}_{\xi}$ and $\hbox{\omegathic a}_{\eta}$ are equivalent. By Proposition \ref{p16a.1} $\hbox{\omegathic a}_{\xi}\cong \hbox{\omegathic a}_{\eta}$ $\iff N_{D|K}(\hbox{\omegathic a}_{\xi})\cong N_{D|K}(\hbox{\omegathic a}_{\eta})$ (where $K=k$ for $d=2$). This is the statement of the theorem. \hfill $\Box$ \vskip0.25cm } \def\cD{\ifmmode {\cal D As a corollary of this \begin{corollary}\label{c16.1} The number of cusps of $\Gamma_{\Delta}$ is the class number of $K$ ($d\geq 3$) or the class number of $k$ ($d=1,2$). \end{corollary} {\bf Proof:} It follows from Theorem \ref{t16a.1} that the number of cusps is {\it at most} the class number in question; it remains to verify that it is at least the class number. If $\hbox{\omegathic a}_{\xi}$ denotes the ideal in $\Delta$, and $N_{D|K}(\hbox{\omegathic a}_{\xi})$ denotes the corresponding ideal in ${\cal O}_K$, then $\xi_1,\ \xi_2$ form a basis of $\hbox{\omegathic a}_{\xi}$, while $N_{D|K}(\xi_1),\ N_{D|K}(\xi_2)$ form an ${\cal O}_K$-basis of the norm ideal. So the question here is, given an ideal $(a_1,a_2)\subset} \def\nni{\supset} \def\und{\underline {\cal O}_K$ generated by two elements, is there an ideal $(b_1,b_2)$ in the same ideal class of $(a_1,a_2)$, such that $b_i=N_{D|K}(a_i')$ for some $a_i'\in \Delta$? But this is just the statement of Proposition \ref{p16a.1}: given any ideal $\hbox{\omegathic a}$, there is an ideal $\hbox{\omegathic b}} \def\cc{\hbox{\omegathic c}\subset} \def\nni{\supset} \def\und{\underline [\hbox{\omegathic a}]$, and an ideal $\hbox{\omegathic b}} \def\cc{\hbox{\omegathic c}'\subset} \def\nni{\supset} \def\und{\underline \Delta$ such that $\hbox{\omegathic b}} \def\cc{\hbox{\omegathic c}=N_{D|K}(\hbox{\omegathic b}} \def\cc{\hbox{\omegathic c}')$. We further require that $\hbox{\omegathic b}} \def\cc{\hbox{\omegathic c}'$ defines an isotropic vector in $\Delta^2$, in the above sense. There should be $b_i\in\hbox{\omegathic b}} \def\cc{\hbox{\omegathic c}'$ which generate $\hbox{\omegathic b}} \def\cc{\hbox{\omegathic c}'$, such that the vector $(b_1,b_2)\in \Delta^2$ is isotropic with respect to $h$. This in turn is a question about what happens to the relation $\xi_1\overline{\xi}_2+\xi_2\overline{\xi}_1=0$ under the norm map. This relation turns into $$N(\xi_1)\overline{N(\xi_2)}+N(\xi_2)\overline{N(\xi_1)}=0,$$ or setting $b_1=N(\xi_1),\ b_2=N(\xi_2),\ b_1\overline{b}_2+b_2\overline{b}_1=0$. This just says that $(b_1,b_2)$ is isotropic in the hyperbolic plane $K^2$, hence $(b_1,b_2)$ is isotropic in the hyperbolic plane $D^2$. In sum, for any isotropic vector $v\in K^2$, the vector is also an isotropic vector $v\in D^2$. Any isotropic vector of $D^2$ yields by the norm map an isotropic vector of $K^2$. It follows from Proposition \ref{p16a.1} that this gives an isomorphism on equivalence classes, or in other words, {\it the isomorphism of Proposition \ref{p16a.1} can be represented geometrically by isotropic vectors}. \hfill $\Box$ \vskip0.25cm } \def\cD{\ifmmode {\cal D \begin{remark} We have not been very precise about the groups, but the number of cusps of $\Gamma_{\Delta}$ and $S\Gamma_{\Delta}$ are the same, as is quite clear. \end{remark} Now assume $d\geq 2$ and recall the subfield $L\subset} \def\nni{\supset} \def\und{\underline D$ and subgroups $G_L\subset} \def\nni{\supset} \def\und{\underline G_D$ of Proposition \ref{p7.1}. By Proposition \ref{p10.1} these give rise to subdomains of the domain $\cD_D$. Consider, in $G_L$, the discrete group $\Gamma_{{\cal O}_L}\subset} \def\nni{\supset} \def\und{\underline G_L(K)$. \begin{lemma}\label{l17.1} We have for any maximal order $\Delta\subset} \def\nni{\supset} \def\und{\underline D$, $$\Gamma_{\Delta}\cap G_L=\Gamma_{{\cal O}_L}.$$ \end{lemma} {\bf Proof:} This follows from the definitions and the fact that $\Delta\cap L={\cal O}_L$. \hfill $\Box$ \vskip0.25cm } \def\cD{\ifmmode {\cal D Let $M_L$ denote the arithmetic quotient \begin{equation}\label{e17.1} M_L=\Gamma_{{\cal O}_L}\backslash \cD_L. \end{equation} It follows from Proposition \ref{p10.1} that we have a commutative diagram \begin{equation}\label{e17.2} \begin{array}{rcl} \cD_L & \hookrightarrow} \def\hla{\hookleftarrow & \cD_D \\ \downarrow & & \downarrow \\ M_L & \hookrightarrow} \def\hla{\hookleftarrow & X_{\Gamma_{\Delta}}. \end{array}\end{equation} Each $M_L$ has its own Baily-Borel compactification $M_L^*$, which is also affected by adding isolated points to $M_L$. Moreover, from Proposition \ref{p10.2}, we see that the embedding $M_L\hookrightarrow} \def\hla{\hookleftarrow X_{\Gamma_{\Delta}}$ can be extended to the cusp denoted $F_L\in \cD_L^*$ respectively $F\in \cD_D^*$ there. By Theorem 3 of \cite{S2}, we actually get embeddings of the Baily-Borel embeddings of $M_L$ and $X_{\Gamma_{\Delta}}$, respectively. \begin{theorem}\label{t17.1} Let $M_L^*\subset} \def\nni{\supset} \def\und{\underline {\Bbb P}^N,\ X_{\Gamma_{\Delta}}^*\subset} \def\nni{\supset} \def\und{\underline {\Bbb P}^{N'}$ be Baily-Borel embeddings. Then there is a linear injection ${\Bbb P}^N\hookrightarrow} \def\hla{\hookleftarrow {\Bbb P}^{N'}$ making the diagram $$\begin{array}{rcl} M_L^* & \hookrightarrow} \def\hla{\hookleftarrow & {\Bbb P}^N \\ \cap & & \cap \\ X_{\Gamma_{\Delta}}^* & \hookrightarrow} \def\hla{\hookleftarrow & {\Bbb P}^{N'}\end{array}$$ commute and making $M_L^*\in X_{\Gamma_{\Delta}}^*$ an algebraic subvariety. \end{theorem} {\bf Proof:} We have an injective holomorphic embedding $\cD_L\hookrightarrow} \def\hla{\hookleftarrow \cD_D$ which comes from a ${\Bbb Q}$-morphism $\rho:(G_L)_{{\Bbb C}}\hookrightarrow} \def\hla{\hookleftarrow (G_D)_{{\Bbb C}}$ (for this one takes the restriction of scalars from $k$ to ${\Bbb Q}$ yielding an injection $Res_{k|{\Bbb Q}}G_L\hookrightarrow} \def\hla{\hookleftarrow Res_{k|{\Bbb Q}}G_D$, then lifts this to ${\Bbb C}$) such that $\rho(\Gamma_{{\cal O}_L})\subset} \def\nni{\supset} \def\und{\underline \Gamma_{\Delta}$. Hence we map apply \cite{S2}, Thm.~3, and the theorem follows. \hfill $\Box$ \vskip0.25cm } \def\cD{\ifmmode {\cal D \begin{corollary}\label{c17.1} If $d=2$, then there are modular curves $M_L^*$ on the algebraic threefold $X_{\Gamma_{\Delta}}^*$ such that the cusps of $M_L^*$ are cusps of $X_{\Gamma_{\Delta}}^*$. If $d\geq 3$, then we have Hilbert modular varieties of dimension $d$, $M_L^*\subset} \def\nni{\supset} \def\und{\underline X_{\Gamma_{\Delta}}^*$ in the $d^2$-dimensional algebraic variety $X_{\Gamma_{\Delta}}^*$, such that the cusps of $M_L^*$ are cusps of $X_{\Gamma_{\Delta}}^*$. \end{corollary} The previous Theorem \ref{t17.1} applies to the cusp of $X_{\Gamma_{\Delta}}^*$ which represents the equivalence class of the isotropic vector $(0,1)$. We now consider the others. Given $(\xi_1,\xi_2)$, an isotropic vector representing a class of cusps, let $M_{\xi}$ be the matrix in $G_D$ of Lemma \ref{l15.2} which maps it to $(0,1)$. Then $$\Gamma_{\Delta,\xi}:=M_{\xi}\cdot \Gamma_{\Delta}\cdot M_{\xi}^{-1}\subset} \def\nni{\supset} \def\und{\underline U(D^2,h)$$ is a discrete subgroup of $G_D$, and the cusp $(\xi_1,\xi_2)$ of $\Gamma_{\Delta}$ is equivalent to the cusp $(0,1)$ of $\Gamma_{\Delta,\xi}$. Letting, as above, $\Gamma_{{\cal O}_L}\subset} \def\nni{\supset} \def\und{\underline \Gamma_{\Delta}$ denote the discrete subgroup defining the modular subvariety $M_L$ above, we have \begin{equation}\label{e17a.0} \Gamma_{{\cal O}_L,\xi}:=M_{\xi}\cdot \Gamma_{{\cal O}_L}\cdot M_{\xi}^{-1}\subset} \def\nni{\supset} \def\und{\underline \Gamma_{\Delta,\xi}, \end{equation} and without difficulty this gives a subdomain \begin{equation}\label{e17a.1} \cD_{L,\xi}\subset} \def\nni{\supset} \def\und{\underline \cD_D, \end{equation} such that, if $F_{\xi}$ denotes the boundary component corresponding to $\xi$, then $F_{\xi}\in \cD_{L,\xi}^*$. More precisely, $G_{L,\xi}:=M_{\xi}\cdot G_L\cdot M_{\xi}^{-1}$ is the $k$-group whose ${\Bbb R}$-points $G_{L,\xi}({\Bbb R})$ define $\cD_{L,\xi}$. Then the parabolic subgroup of $\Gamma_{\Delta,\xi}$ is $\Gamma_{\Delta,\xi}\cap P$ ($P$ as in (\ref{e6.1}) the normaliser of $(0,1)$), and similarly for $\Gamma_{{\cal O}_L,\xi}$. The corresponding modular subvariety \begin{equation}\label{e17a.2} M_{L,\xi}=\Gamma_{{\cal O}_L,\xi}\backslash \cD_{L,\xi}, \end{equation} is, as is easily checked, a Hilbert modular variety for the group $SL_2({\cal O}_{\ell},\hbox{\omegathic b}} \def\cc{\hbox{\omegathic c}^2)$, where for any ideal $\cc$ one defines \begin{equation}\label{e17a.3} SL_2({\cal O}_{\ell},\cc)=\left\{\left(\begin{array}{cc} a & b \\ c & d\end{array}\right) \Big| ad-cb=1,\ a,d\in {\cal O}_{\ell}, b\in \cc^{-1},\ c\in \cc\right\},\end{equation} and where $\hbox{\omegathic b}} \def\cc{\hbox{\omegathic c}$ is the intersection of the ideal $\hbox{\omegathic a}_{\xi}$ with ${\cal O}_{\ell}$. The same arguments as above then yield \begin{theorem}\label{t17a.1} Given any cusp $p\in X_{\Gamma_{\Delta}}^*\backslash X_{\Gamma_{\Delta}}$, there is a modular subvariety $M_{L,p}\subset} \def\nni{\supset} \def\und{\underline X_{\Gamma_{\Delta}}$ such that $p\in M_{L,p}^*$. \end{theorem} {\bf Proof:} If the cusp $p$ is represented by the isotropic vector $\xi=(\xi_1,\xi_2)$, then the modular subvariety is $M_{L,\xi}$ as in (\ref{e17a.2}). Just as in the proof of Theorem \ref{t17.1}, we get an embedding of the Baily-Borel embeddings, and as mentioned above, $F_{\xi}\in \cD_{L,\xi}^*$, and $p$ is the image of $F_{\xi}$ under the natural projection $\pi:\cD_{L,\xi}^*\longrightarrow} \def\sura{\twoheadrightarrow M_{L,\xi}^*=M_{L,p}^*$. \hfill $\Box$ \vskip0.25cm } \def\cD{\ifmmode {\cal D \begin{corollary}\label{c17a.1} If $d=2$, there are modular curves $M_{L,p}^*\subset} \def\nni{\supset} \def\und{\underline X_{\Gamma_{\Delta}}^*$ passing through each cusp of $X_{\Gamma_{\Delta}}^*$. If $d\geq3$, there are Hilbert modular varieties (to groups as in (\ref{e17a.3})) of dimension $d$ passing through any cusp of $X_{\Gamma_{\Delta}}^*$. \end{corollary} \section{An example} The theory of quaternion algebras is quite well established, and the corresponding arithmetic quotients have already been studied in \cite{Ara} and \cite{H}. However the $d\geq3$ case seems not to have drawn much attention as of yet, so we will give an example to illustrate the theory. Curiously enough, I ran across this example in the construction of Mumford's fake projective plane. Recall that this is an algebraic surface $S$ of general type with $c_1^2=3c_2,\ c_1^2=9, \ c_2=3$, just as for the projective plane. It then follows from Yau's theorem that $S$ is the quotient of the two-ball ${\Bbb B}_2$ by a discrete subgroup, \begin{equation}\label{e18.1} S=\Gamma\backslash {\Bbb B}_2, \end{equation} where $\Gamma$ is cocompact and fixed point free. Mumford's construction involves lifting a quotient surface from a $2$-adic field to the complex numbers, and while the $2$-adic group is clearly not arithmetic (as a subgroup of $SL(3,{\Bbb Q}_2)$, its elements are unbounded in the $2$-adic valuation), it is not clear from the construction whether $\Gamma$ is arithmetic. Now the {\it arithmetic} cocompact groups are known: these derive either from anisotropic groups (over ${\Bbb Q}$) which are of the form $U(1,D)$ or $SU(1,D)$, where $D$ is a central simple division algebra over an imaginary quadratic extension $K$ of a totally real field $k$, such that if the corresponding hermitian symmetric domain is irreducible, then $k={\Bbb Q}$, and such that $D$ has a $K|{\Bbb Q}$-involution, or from unitary groups over field extensions $k|{\Bbb Q}$ of degree $d\geq2$, such that, for all but one infinite prime, the real groups $G_{\nu}$ are compact. The strange thing is that in Mumford's construction such a $D$ comes up (implicitly) naturally, and it is this $D$ we will introduce. It is a fascinating question whether the $\Gamma$ of (\ref{e18.1}) occurs as a discrete subgroup of $U(1,D)$, a question which will be left unanswered here. This is a cyclic algebra, central simple over $K$, with splitting field $L$. The fields involved are \begin{equation}\label{e18.2} L={\Bbb Q}(\zeta),\ \ \zeta=\exp({2\pi i \over 7});\quad K={\Bbb Q}(\sqrt{-7}). \end{equation} \begin{lemma}\label{l18.1} $L$ is a cyclic extension of $K$, of degree three. The Galois group is generated by the transformation $$\sigma(\zeta)=\zeta^2,\ \sigma(\zeta^2)=\zeta^4,\ \sigma(\zeta^4)=\zeta,$$ where $1, \zeta, \zeta^2$ generate $L$ over $K$. \end{lemma} {\bf Proof:} Let $\gamma={-1+\sqrt{-7}\over 2}\in K$, so ${\cal O}_K={\Bbb Z}\oplus \gamma{\Bbb Z}$. The ring of integers of $L$ is generated by the powers of $\zeta$, $${\cal O}_L={\Bbb Z}\oplus\zeta{\Bbb Z}\oplus \cdots \oplus \zeta^5{\Bbb Z},$$ (remember that $\zeta^6=-1-\zeta-\cdots-\zeta^5$), with the inclusion ${\Bbb Z}\subset} \def\nni{\supset} \def\und{\underline {\cal O}_K, {\Bbb Z}\subset} \def\nni{\supset} \def\und{\underline {\cal O}_L$ on the first factors. The key identity is the following \begin{equation}\label{e18.3} \gamma=\zeta+\zeta^2+\zeta^4,\quad\quad \overline{\gamma}=\zeta^3+\zeta^5+\zeta^6. \end{equation} {}From this relation we see that ${\cal O}_L$ is generated over ${\cal O}_K$ by $\zeta$ and $\zeta^2$, $${\cal O}_L={\cal O}_K\oplus\zeta{\cal O}_K\oplus\zeta^2{\cal O}_K.$$ The conjugation on $K$ extends to (complex) conjugation on $L$, which we will denote by $x\mapsto \overline{x}$. Its action on ${\cal O}_K$ is clear, $\gamma\mapsto \overline{\gamma}$, and on ${\cal O}_L$ it affects $$\overline{\zeta}=\zeta^6,\quad \overline{\zeta}^2=\zeta^5,\quad \overline{\zeta}^3=\zeta^4.$$ It is clear that $L|K$ is cyclic, with a generator of the Galois group $$1\mapsto 1, \quad \zeta\mapsto \zeta^2,\quad \zeta^2\mapsto \gamma-\zeta-\zeta^2,$$ and by (\ref{e18.3}), this is the statement of the Lemma. \hfill $\Box$ \vskip0.25cm } \def\cD{\ifmmode {\cal D Now consider the fixed field in $L$ under conjugation, $\ell\subset} \def\nni{\supset} \def\und{\underline L$. \begin{lemma}\label{l19.1} $\ell$ is a degree three cyclic extension of ${\Bbb Q}$, with a generator of the Galois group being $$\sigma:\quad \zeta+\zeta^6\mapsto \zeta^2+\zeta^5 \mapsto \zeta^3+\zeta^4\mapsto \zeta+\zeta^6.$$ \end{lemma} {\bf Proof:} It is clear that $\eta_1=\zeta+\zeta^6, \eta_2=\zeta^2+\zeta^5,\ \eta_3=\zeta^3+\zeta^4$ are in $\ell$ (they are $\eta_1=Tr_{L|\ell}(\zeta),\ \eta_2=Tr_{L|\ell}(\zeta^2),\ \eta_3=Tr_{L|\ell}(\zeta^3)$). Moreover they generate the integral closure of ${\Bbb Z}$ in $\ell$, hence $${\cal O}_{\ell}={\Bbb Z}\oplus \eta_1{\Bbb Z}\oplus \eta_2{\Bbb Z},$$ and there is a relation $\eta_3=-\eta_1-\eta_2-1$. Now observe that $\sigma$ of Lemma \ref{l18.1} acts as follows $$\sigma(\eta_1)=\sigma(\zeta+\zeta^6)=\sigma(\zeta)+\sigma(\zeta^6)= \zeta^2+\zeta^5=\eta_2,\quad \sigma(\eta_2)=\eta_3, \quad \sigma(\eta_3)=\eta_1,$$ and this is the map specified in the lemma. \hfill $\Box$ \vskip0.25cm } \def\cD{\ifmmode {\cal D Next consider the cyclic algebra $D=(L/K,\sigma,\gamma)$ for the element $\gamma$ above. Then $D$ will be split $\iff$ $\gamma$ is the $L/K$ norm of some element. However, it seems difficult to verify this condition explicitly, so we show directly that $D$ is a division algebra. \begin{proposition}\label{p19.1} $D=(L/K,\sigma,\gamma)$, with $\gamma={-1+\sqrt{-7}\over 2}$, is a division algebra, central simple over $K$. \end{proposition} {\bf Proof:} Note first that since the degree of $D$ over $K$ is three, a prime, $D$ is a division algebra $\iff$ $D$ is not split. To show that $D$ is not split, it suffices to find a prime $\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}\in {\cal O}_K$ for which the local algebra $D_{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}$ is not split. This, it turns out, is easy to find. Quite generally we know that $D_{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}$ is split for almost all $\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}$, and non-split at the divisors of the discriminant. Note that $$N_{K|{\Bbb Q}}(\gamma)=\gamma\overline{\gamma}=2,$$ while $K$ ramifies over ${\Bbb Q}$ only at the prime $\sqrt{-7}$. These are the two primes where $D_{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}$ might ramify. To see that for $\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}=(\gamma)$ $D_{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}$ actually does ramify, we determine the Hasse invariant $inv_{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}(D)\in {\Bbb Q}/{\Bbb Z}$ of $D_{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}$. To do this we must first understand the action of the Frobenius acting as generator for the maximal unramified extension $L_{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}/K_{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}$. But Frobenius in characteristic two is just a squaring map, so is clearly the $\mod(\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q})$ reduction of the $\sigma$ of Lemma \ref{l18.1}. We denote this by $\sigma_{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}$. Finally, since we are at the prime $\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}=(\gamma)$, we may take the image of $\gamma$ as local uniformising element $\pi_{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}$. Then $$D_{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}=(L_{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}/K_{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}},\sigma_{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}},\pi_{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}),$$ which is the cyclic algebra with $inv_{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}(D)={1\over 3}$. From this it follows that $D_{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}$ is not split. That $D$ is central simple over $K$ is clear from construction. \hfill $\Box$ \vskip0.25cm } \def\cD{\ifmmode {\cal D Finally we require a $K|{\Bbb Q}$-involution on $D$. It is necessary and sufficient for the existence of such an involution $J$ that there exists an element $g\in \ell$, such that \begin{equation}\label{e20.1} N_{K|{\Bbb Q}}(\gamma)= \gamma\overline{\gamma}=N_{\ell|{\Bbb Q}}(g) =g g^{\sigma}g^{\sigma^2}. \end{equation} Given such an element $g$, the involution is given by (see (\ref{e4A.1})) $$e^J=ge^{-1},\ \ (e^2)^J=g g^{\sigma}(e^2)^{-1},\quad (\sum_0^2e^iz_i)^J:= \sum_0^2\overline{z}_i(e^i)^J.$$ An alternative to finding an explicit $g$ is to use a theorem of Landherr (\cite{Sch}, Thm.~10.2.4): $D$ admits a $K|{\Bbb Q}$-involution $\iff$ \begin{equation}\label{e20.2} \begin{minipage}{15cm} \begin{itemize}\item $inv_{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}(D)=0,\ \quad \forall_{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}=\overline{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}}$, and \item $inv_{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}(D)+inv_{\overline{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}}(D)=0, \quad \forall_{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}\neq \overline{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}}$. \end{itemize}\end{minipage}\end{equation} This condition is satisfied for all $\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}$ for which $D_{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}$ splits, so must be verified only for those primes which divide the discriminant. For us, this means at most $\pm\sqrt{-7}, \gamma,\ \overline{\gamma}$, which we will denote by $\pm \qq, \hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}, \overline{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}$, respectively. We showed above that $inv_{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}(D)={1\over 3}$. The same argument shows that $inv_{\overline{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}}(D)=-{1\over 3}$. The negative sign occurs because at the prime $\overline{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}$, $\overline{\gamma}$ is a local uniformising element $\pi_{\overline{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}}$, so localising $\gamma$ at $\overline{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}$ gives $\pi_{\overline{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}}^{-1}$. This verifies (\ref{e20.2}) for the primes $\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}\neq\overline{\hbox{\omegathic p}} \def\qq{\hbox{\omegathic q}}$ lying over 2. Consider the prime $\pm \qq$. Here it is easy to see that the image of $\gamma$ is actually {\it invertible}, and hence, that $D_{\qq}$ splits, so $inv_{\qq}(D)=0$, as was to be shown. \hfill $\Box$ \vskip0.25cm } \def\cD{\ifmmode {\cal D This completes the proof of the following. \begin{theorem}\label{t20.1} The cyclic algebra $D=(L/K,\sigma,\gamma)$ constructed above is a central simple division algebra over $K$ with a $K|{\Bbb Q}$-involution. It ramifies at the two primes $\gamma$ and $\overline{\gamma}$, and is split at all others. \end{theorem} This algebra gives rise to an anisotropic group $GL(1,D)\cong D^*$, and $D^*({\Bbb R})$ is a twisted real form of $GL_3$; since it cannot be compact, it must be $U(2,1)$. A maximal order $\Delta\subset} \def\nni{\supset} \def\und{\underline D$ gives rise to an arithmetic subgroup of $D^*({\Bbb R})$, and the quotient is then a compact ball quotient. As remarked above, this may be related with Mumford's fake projective plane. We can consider the hyperbolic space $(D^2,h)$, and the corresponding ${\Bbb Q}$-groups (here $k={\Bbb Q}$) $G_D$ and $SG_D$ as in (\ref{e4.2}) and (\ref{e4.4}), respectively, as well as the arithmetic subgroups $\Gamma_{\Delta}$ and $S\Gamma_{\Delta}$. By Corollary \ref{c16.1}, we see that $\Gamma_{\Delta}$ has $h(K)$ cusps, where $h(K)$ is the class number of $K$; this is known to be 1. So the arithmetic quotient has only 1 cusp. We also have the subgroups $\Gamma_{{\cal O}_L}$ as in Lemma \ref{l17.1}, as well as the modular subvarieties $M_L^*$ of $X_{\Gamma_{\Delta}}^*$. We note that these are Hilbert modular threefolds coming from the cyclic cubic extension $\ell/{\Bbb Q}$. Such threefolds have been considered in \cite{T}. \section{Moduli interpretation} The moduli interpretation of the arithmetic quotients $X_{\Gamma_{\Delta}}$, as well as of the modular subvarieties $M_L$, by which we mean the description of these spaces as moduli spaces, is a straightforward application of Shimura's theory. Fix a hyperbolic plane $(D^2,h)$, and consider the endomorphism algebra $M_2(D)$ of $D^2$, endowed with the involution $x\mapsto {^t\overline{x}}$, $x\in M_2(D)$, where the bar denotes the involution in $D$. Now view $D^2$ as a ${\Bbb Q}$-vector space, of dimension $4f$, $8f$ and $4d^2f$, in the cases $d=1$, $d=2$ and $d\geq3$, respectively. The data determining one of Shimura's moduli spaces (with no level structure) is $(D,\Phi,*),\ (T,{\cal M})$, where $D$ is a central simple division algebra over $K$, $\Phi$ is a representation of $D$ in $\Gg\hbox{\omegathic l}} \def\Gg{\hbox{\omegathic g}(n,{\Bbb C})$, $*$ is an involution on $D$, $T$ is a $*$-skew hermitian form (matrix) on a right $D$ vector space $V$ of dimension $m$, with $\Gg\hbox{\omegathic l}} \def\Gg{\hbox{\omegathic g}(n,{\Bbb C})\cong End(V,V)_{{\Bbb R}}$, and finally, ${\cal M}\subset} \def\nni{\supset} \def\und{\underline D^m$ is a ${\Bbb Z}$-lattice. Then for suitable $x_i\in V$, \begin{equation}\label{e21.1} \Lambda=\left\{\sum_{i=1}^m\Phi(a_i)x_i \Big| a_i\in {\cal M}\right\}\subset} \def\nni{\supset} \def\und{\underline {\Bbb C}^n \end{equation} is a lattice and ${\Bbb C}^n/\Lambda$ is abelian variety with multiplication by $D$. The data $(D,\Phi,*)$ will be given in our cases as follows. $D$ is our central simple division algebra over $K$ with a $K|k$-involution, and the representation $\Phi:D\longrightarrow} \def\sura{\twoheadrightarrow M_N({\Bbb C})$ is obtained by base change from the natural operation of $D$ on $D^2$ by right multiplication. Explicitly, \begin{equation}\label{e21.3} \Phi:D\longrightarrow} \def\sura{\twoheadrightarrow End_D(D^2,D^2) \otimes_{{\Bbb Q}}{\Bbb R} \cong M_2(D)\otimes_{{\Bbb Q}}{\Bbb R}\cong M_2(D\otimes_{{\Bbb Q}}{\Bbb R})\cong M_2({\Bbb R}^N)\cong M_N({\Bbb C}), \end{equation} where $N=\dim_{{\Bbb Q}}D=2f,\ 4f,\ 2d^2f$ in the cases $d=1,\ d=2$ and $d\geq3$, respectively. The involution $*$ on $D$ will be our involution, which we still denote by $x\mapsto \overline{x}$. Then a $^{-}$-skew hermitian matrix $T\in M_2(D)$ will be one such that $T=-T^*$, where $(t_{ij})^*=(\overline{t}_{ji})$, the canonical involution on $M_2(D)$ induced by the involution on $D$. Note that for any $c\in D^*$ such that $c=-\overline{c}$, the matrix $T=cH$ ($H$ our hyperbolic matrix ${0\ 1\choose 1\ 0}$) has this property. To be more specific, then, we set \begin{equation}\label{e21.4} \begin{minipage}{15cm}\begin{itemize}\item[1)] $d=1$: $T=\sqrt{-\eta}H=\left(\begin{array}{cc} 0 & \sqrt{-\eta} \\ \sqrt{-\eta} & 0 \end{array}\right)$. \item[2)] $d=2$: $T_a=\sqrt{a}H=\left(\begin{array}{cc} 0 & \sqrt{a} \\ \sqrt{a} & 0 \end{array}\right)$ or $T_b=eH={0\ e\choose e\ 0}$, where $e={0\ 1\choose b\ 0}$. \item[3)] $d\geq3$: $T=\sqrt{-\eta}H=\left(\begin{array}{cc} 0 & \sqrt{-\eta} \\ \sqrt{-\eta} & 0 \end{array}\right)$. \end{itemize}\end{minipage}\end{equation} Since two such forms $T$ are equivalent when they are scalar multiples of one another, assuming $T$ of the form in (\ref{e21.4}) is no real restriction. Finally the lattice ${\cal M}$ will be $\Delta^2\subset} \def\nni{\supset} \def\und{\underline D^2$. Then $D^2\otimes_{{\Bbb Q}}{\Bbb R}\cong {\Bbb C}^N$, $N$ as above, and for ``suitable'' vectors $x_1, x_2\in D^2$, the lattice \begin{equation}\label{e21.2} \Lambda_x=\{\Phi(a_1)x_1+\Phi(a_2)x_2 \big| (a_1,a_2)\in \Delta^2\} \end{equation} gives rise to an abelian variety $A_x={\Bbb C}^N/\Lambda_x$. Shimura has determined exactly what ``suitable'' means; the conditions determine certain unbounded realisations of hermitian symmetric spaces, in our cases just the domains $\cD_D$. The Riemann form on $A_x$ is given by the alternating form $E(x,y)$ on ${\Bbb C}^N$ defined by: \begin{equation}\label{ee21.6} E(\sum_1^2\Phi(\alpha_i)x_i, \sum_1^2\Phi(\beta_j)x_j)=Tr_{D|{\Bbb Q}}(\sum_{i,j=1}^2 \alpha_it_{ij}\overline{\beta}_j), \end{equation} for $\alpha_i,\beta_j\in D_{{\Bbb R}}$, and $(t_{ij})=T$ is the matrix above. In particular, in all cases dealt with here the abelian varieties are {\it principally polarised}. Shimura shows that for each $z\in \cD_D$, vectors $x_1,x_2\in D^2$ are uniquely determined, hence by (\ref{e21.2}) a lattice, denoted $\Lambda(z,T,{\cal M})$. The data determine an arithmetic group, which for our cases is just $\Gamma_{\Delta}$ defined above (not $S\Gamma_{\Delta}$), cf.~\cite{Sh2} (38). The basic result, applied to our concrete situation, is \begin{theorem}[\cite{Sh2}, Thm.~2]\label{t21.1} The arithmetic quotient $X_{\Gamma_{\Delta}}$ is the moduli space of isomorphism classes of abelian varieties determined by the data: $$(D,\Phi,^-),\ \ (T,\Delta^2),$$ where $\Phi$ is given in (\ref{e21.3}), $T$ in (\ref{e21.4}). \end{theorem} The corresponding classes of abelian varieties can be described as follows: \begin{itemize}\item[1)] $d=1$. Here we have {\it two} families, relating from the isomorphism of Proposition \ref{p5.1}. The first, for $D=k,\ D^2=k^2$ yields $D_{{\Bbb R}}^2\cong {\Bbb C}^f$, and we have abelian varieties of dimension $f$ with real multiplication by $k$. Secondly, for $D=K,\ D^2=K^2$, we have abelian varieties of dimension $2f$ with complex mulitplication by $K$, with signature $(1,1)$, that is, for each eigenvalue $\chi$ of the differential of the action, also $\overline{\chi}$ occurs. If $K=k(\sqrt{-\eta})$, then setting $K'={\Bbb Q}(\sqrt{-\eta})$ we have $k\otimes_{{\Bbb Q}}K'\cong K$, hence $k^2\otimes_{{\Bbb Q}}K'\cong K^2$ and $k^2_{{\Bbb R}}\otimes K'_{{\Bbb R}}\cong K_{{\Bbb R}}^2$, giving the relation between the ${\Bbb Q}$-vector spaces and their real points. Moreover, ${\cal O}_k\otimes_{{\Bbb Z}}{\cal O}_{K'}\cong {\cal O}_K$, and if \begin{equation}\label{e21.6} \Lambda_{x,k}=\{\sum_1^2\Phi(a_i)x_i \Big| (a_1,a_2)\in {\cal O}_k^2\} \end{equation} is a lattice giving an abelian variety with mulitiplication by $k$, $$A_x:={\Bbb C}^f/\Lambda_{x,k},$$ then $\Lambda_{x,k}\otimes_{{\Bbb Z}}{\cal O}_{K'}=\Lambda_{x,K}$ is a lattice in ${\Bbb C}^{2f}$, and determines an abelian variety \begin{equation}\label{e21.7} A_x':={\Bbb C}^{2f}/\Lambda_{x,K}. \end{equation} This abelian variety determines a point $x'$ in its moduli space, and the mapping $x\mapsto x'$ gives the isomorphism \begin{equation}\label{e21.5} \Gamma_{{\cal O}_k}\backslash \hbox{\omegathic H}} \def\scI{\hbox{\script I}^f \stackrel{\sim}{\longrightarrow} \def\sura{\twoheadrightarrow} \Gamma_{{\cal O}_K}\backslash \hbox{\omegathic H}} \def\scI{\hbox{\script I}^{f}. \end{equation} \begin{remark} It turns out that this case is one of the exceptions of Theorem 5 in \cite{Sh2}, denoted case d) there. The actual endomorphism ring of the generic member of the family is larger than $K$: \begin{theorem}[\cite{Sh2}, Prop.~18] The endomorphism ring $E$ of the generic element of the family (\ref{e21.5}) is a totally indefinite quaternion algebra over $k$, having $K$ as a quadratic subfield. \end{theorem} In our situation, the totally indefinite quaternion algebra $E$ over $k$ is constructed as the cyclic algebra $E=(K/k,\sigma,\lambda)$, where $\lambda=-u^{-1}v$, if the matrix $T$ of (\ref{e21.4}) is diagonalized $T={u\ 0\choose 0\ v}$. So in our case we have $\lambda=1$ and hence the algebra $E$ is split; the corresponding abelian variety is isogenous to a product of two copies of a simple abelian variety $B$ with real multiplication by $k$, as has been described already above. The conclusion follows from our choice of $T$, i.e., of hyperbolic form. It would seem one gets more interesting quaternion algebras by choosing different hermitian forms (which, by the way, will also lead to other polarisations). \end{remark} \item[2)] $d=2$. $D=(\ell/k,\sigma,b)=(a,b)$ is a totally indefinite quaternion algebra, cental simple over $k$, with canonical involution. We have $D_{\nu}\cong M_2({\Bbb R})$, and $D\otimes_{{\Bbb Q}}{\Bbb R}\cong {\Bbb R}^4$, while $M_2(D)\otimes_{{\Bbb Q}}{\Bbb R}\cong M_2({\Bbb R}^4)\cong M_4({\Bbb C})$. Let $\Delta\subset} \def\nni{\supset} \def\und{\underline D$ be a maximal order, $\Gamma_{\Delta} \subset} \def\nni{\supset} \def\und{\underline G_D$ the corresponding arithmetic group. Two vectors $x_1,x_2\in D^2$ arising from a point in the domain ${\Bbb S}_2$ (Siegel space of degree 2) determine a lattice $\Lambda_x$ as in (\ref{e21.6}), with $(a_1,a_2)\in \Delta^2$, and $A_x={\Bbb C}^{4f}/\Lambda_x$ is the corresponding abelian variety. \item[3)] $d\geq3$. In this case $D$ is the cyclic algebra of degree $d$ over $K$, and the abelian varieties are of dimension $2d^2f$. \end{itemize} Now we come to the most interesting point of the whole story -- the moduli interpretation of the modular subvarieties $M_L\subset} \def\nni{\supset} \def\und{\underline X_{\Gamma_{\Delta}}$ of (\ref{e17.2}). The moduli interpretation of the $M_L$ is as stated in Theorem \ref{t21.1}, $d=1$ case. Disregarding the case $d=1$, the subvarieties have the following interpretations. \begin{itemize}\item[2)] $d=2$: As $M_L$ arises from the group $U(L^2,h)$, where $L=k(\sqrt{-ab})$ in our notations above, this is the moduli space of abelian varieties of dimension $2f$ with complex multiplication by $L$. On the other hand, the space $X_{\Gamma_{\Delta}}$ parameterizes abelian varieties of dimension $4f$ with multiplication by $D$. The relation is given as follows. By definition we have $D=\ell\oplus e\ell$, which in terms of matrices, is $$D\cong \left\{\left(\begin{array}{cc} a_0+a_1\sqrt{a} & 0 \\ 0 & a_0-a_1\sqrt{a}\end{array}\right) \oplus\left(\begin{array}{cc} 0 & a_2+a_3\sqrt{a} \\ b(a_2-a_3\sqrt{a}) & 0 \end{array}\right)\right\}.$$ Now our subfield $L$ is the set of matrices of the form $$L\cong \left\{\left(\begin{array}{cc} a_0 & a_3\sqrt{a} \\ -ba_3\sqrt{a} & a_0\end{array}\right)\right\}.$$ If we let $c=\hbox{diag}} \def\det{\hbox{det}} \def\Ker{\hbox{Ker}(\sqrt{a},-\sqrt{a})$ be the element representing $\sqrt{a}$ and $e={0\ 1\choose b\ 0}$, then $c(ec)=c(-ce)=-c^2e=-ae$, so we can generate $D$ over ${\Bbb Q}$ by $c$ and $(ec)$. Recall that $L\cong k(ec)$, hence \begin{equation}\label{e22a.1} D\cong L\oplus cL.\end{equation} Now consider the representation $\Phi$; we have $\Phi(D)=\Phi(L\oplus cL) = \Phi_{|L}(L)\oplus \Phi_{|cL}(cL)$. The lattice $\Delta^2\subset} \def\nni{\supset} \def\und{\underline D^2$ gives rise to a lattice $\Lambda_x$ as in (\ref{e21.2}), and we would like to determine when the splitting (\ref{e22a.1}) gives rise to a splitting of the lattice $\Lambda_x$, hence of the abelian variety $A_x$. Consider the order \begin{equation}\label{e22a.2} \Delta':={\cal O}_L\oplus c{\cal O}_L\subset} \def\nni{\supset} \def\und{\underline \Delta; \end{equation} $\Delta'$ is in general not a maximal order, but it is of finite index in $\Delta$. Note that $\Delta'$ and a point $x$ (consisting of two vectors $x_1, x_2\in D^2$) determine a lattice $$\Lambda'_x=\{\sum \Phi(a_i)x_i \big| (a_1,a_2)\in \Delta'\},$$ which is also of finite index in $\Lambda_x$. Therefore $A'_x={\Bbb C}^4/\Lambda_x'$ and $A_x$ are isogenous. We now assume $x_i\in L^2$. From (\ref{e22a.2}) we can write $a_i=a_i^1+ca_i^2$ for $a_i\in \Delta'$, hence $\Phi(a_i)=\Phi(a_i^1+ca_i^2)= \Phi_{|{\cal O}_L}(a_i^1)+c\Phi_{|{\cal O}_L}(a_i^2). $ Then we have \begin{eqnarray}\label{e22.1} \Lambda_x'=\left\{\sum\Phi(a_i)x_i \Big| (a_1,a_2)\in \Delta'^2\right\} & = & \left\{\sum\left(\Phi_{|L}(a_i^1)+c\Phi_{|L}(a_i^2)\right) x_i \Big| (a_1,a_2)\in \Delta'^2 \right\} \\ \nonumber & = & \left\{ \Phi_{|L}(a_1^1)x_1+\Phi_{|L}(a_2^1)x_2+c\left(\Phi_{|L}(a_1^2)x_1 + \Phi_{|L}(a_2^2)x_2\right) \right\} \\ \nonumber & = & \Lambda^1\oplus c \Lambda^2,\nonumber \end{eqnarray} and each of $\Lambda^i$ has complex multiplication by $L$. It follows from this that $$A_x'\cong A_x'^1\times A_x'^2,\quad x=(x_1,x_2)\in L^2,$$ and each abelian variety $A_x'^i$, of dimension $2f$, has complex multiplication by $L$. Since $A_x'\longrightarrow} \def\sura{\twoheadrightarrow A_x$ is an isogeny, we have \begin{proposition}\label{p22.1} In case $d=2$, the abelian varieties parameterised by the modular subvariety $M_L$ are isogenous to products of two abelian varieties of dimensions $2f$ with complex multiplication by $L$. \end{proposition} \item[3)] $d\geq3$: Again the situation is somewhat simpler here. As above, the subvariety $M_L$ parameterises abelian varieties of dimension $2df$ with complex multiplication by $L$. Again we consider the order $$\Delta':={\cal O}_L\oplus e{\cal O}_L\oplus \cdots \oplus e^{d-1}{\cal O}_L\subset} \def\nni{\supset} \def\und{\underline \Delta,$$ where we assume $e$ is as in (\ref{e1.2}) and $\gamma\in {\cal O}_k$; then as above we can write the lattice $\Lambda'_x$ for $x_1, x_2\in L^2$, in terms of $\Phi_{|L}$. If we write $a_i=a_i^1+ea_i^2+\cdots +e^{d-1}a_i^d$ with $a_i^j\in {\cal O}_L$, then $\Phi(a_i)=\Phi_{|L}(a_i^1)+e\Phi_{|L}(a_i^2)+\cdots+e^{d-1} \Phi_{|L}(a_i^d)$, and consequently \begin{eqnarray*}\Phi(a_1)x_1+\Phi(a_2)x_2 & = & \left(\Phi_{|L}(a_1^1)x_1+\Phi_{|L}(a_2^1)x_2\right) + e\left(\Phi_{|L}(a_1^2)x_1+\Phi_{|L}(a_2^2)x_2\right) + \\ & & \ \ + \ldots + e^{d-1}\left(\Phi_{|L}(a_1^d)x_1+\Phi_{|L}(a_2^d)x_2\right), \\ \left\{\sum\Phi(a_i)x_i\Big| (a_1,a_2)\in {\Delta'}^2\right\} & = & \Lambda^1\oplus e\Lambda^2\oplus \cdots \oplus e^{d-1}\Lambda^d, \end{eqnarray*} and each sublattice $\Lambda^i$ has complex multiplication by $L$. Again, the abelian variety $A_x'$ so determined is isogenous to $A_x$, and hence \begin{proposition}\label{p22.2} In case $d\geq3$, the abelian varieties parameterised by the modular subvariety $M_L$ are isogenous to the product of $d$ abelian varieties of dimension $2df$ with complex multiplication by the field $L$. \end{proposition} \end{itemize}

9504

alg-geom/9504016

en

https://arxiv.org/abs/alg-geom/9504016

alg-geom/9504016

Christian Gantz

Christian Gantz and Brian Steer

Gauge fixing for logarithmic connections over curves and the Riemann-Hilbert-Problem

29 pages, Latex 2.09

We explain in detail the correspondence between algebraic connections over CP^{1}, logarithmic at X = { x_{1},...,x_{n} } \subset CP^{1}, and flat bundles over CP^{1}-X with integer weighted filtrations near each x_{j}. Included is a gauge fixing theorem for logarithmic connections. (Thus far, one could work over any Riemann surface.) We prove a bound on the splitting type of a semi-stable logarithmic connection over CP^{1}. Using this we extend and simplify some results on the Riemann-Hilbert-Problem, which asks for a logarithmic connection on a holomorphically trivial bundle over CP^{1}, extending a given flat bundle over CP^{1}-X. The work is self contained and elementary, using only basic knowledge of Gauge Theory and the Birkhoff-Grothendieck-Theorem.

alg-geom

\section{Introduction} {\sc Said briefly:} We explain in detail the correspondence $\mbox{$\cal F$}$ between algebraic connections over $\Proj^{1}$, logarithmic at $X =\{ x_{1},...,x_{n} \} \subseteq \Proj^{1}$, and flat bundles over $\Proj^{1} -X$ with integer weighted filtrations near each $x_{j}$. Included is a gauge fixing theorem for logarithmic connections. (Thus far, one could work over any Riemann surface.) We prove a bound on the splitting type of a semi-stable logarithmic connection over $\Proj^{1}$. Using this we extend some results on the Riemann-Hilbert-Problem and explain some others. The work is self contained and elementary, using only basic knowledge of gauge theory and the Birkhoff-Grothendieck-Theorem. \hfill {\sc The concepts:} A {\em logarithmic connection} over $(\Proj^{1},X)$ consists of a holomorphic vector bundle $E \rightarrow \Proj^{1}$ with an algebraic connection \[ \nabla: \Omega^{0}(E) \rightarrow \Omega^{0}(E) \otimes \Omega^{1}_{I\!\! P^{1}}(\log X), \] satisfying the Leibnitz rule, where $\Omega^{1}_{I\!\! P^{1}}(\log X)$ is the sheaf of holomorphic 1-forms generated near $x_{j}$ by $\mbox{d} z_{j}/z_{j}$ for a coordinate $z_{j}$ centred at $x_{j}$. $H:=(E,\nabla)|_{I\!\! P^{1}-X}$ is a flat bundle. Isomorphism classes of flat bundles of rank $r$ correspond to conjugacy classes of representations $\chi:\pi_{1}(\Proj^{1}-X) \rightarrow \mbox{Gl} \, (r,\C)$, \cite[p 200]{ati}, \cite[p 51-56]{ano}, \cite[p 4]{kob}. $\chi$ is called the monodromy (or holonomy) of $H$. If $E$ is trivial, one calls $(E,\nabla)$ a {\em Fuchsian system}. Choose a global coordinate $z$ on $\Proj^{1}$ such that $a_{j}:=z(x_{j})\neq \infty$. For any Fuchsian system $(\Proj^{1} \times \C^{r}, \nabla)$ there exist $B_{j} \in \mbox{ End }(\C^{r})$, \cite[p 4]{ano}, such that \begin{equation} \nabla=\mbox{d} + \sum_{j=1}^{n} \frac{B_{j}}{z-a_{j}}. \label{fz} \end{equation} \hfill {\sc The problem:} In 1900 Hilbert stated his twenty first problem: {\em Prove that for any given singularities $X$ and representation $\chi$ there exists a Fuchsian system realising $(X,\chi)$.} Literally, \cite{hil}, he said {\em Fuchsian equation}, i.e.\ higher order differential equations with prescribed singularities. But Anosov \& Bolibruch argue that he meant vector-valued linear equations, i.e.\ Fuchsian systems, because the alternative was already known to be wrong in 1900. Fuchsian equations induce Fuchsian systems, \cite[Ch. 7]{ano}. Since Riemann worked on the problem earlier, it is called the Riemann-Hilbert-Problem (RHP). For a comprehensive collection of known results and references to the RHP see \cite{ano}, also \cite{bea} and \cite{bol}. Much of the recent work is due to Bolibruch. (An approach different from most is Hain's, \cite{hai}.) Bolibruch discovered a pair $(X,\chi)$, of rank $r=3$ and with $n=4$, which cannot be realised by any Fuchsian system, \cite[p 74-76]{bol}, \cite[p 14]{ano}. Therefore, he modified the RHP to the question of which $(X,\chi)$ can occur on Fuchsian systems. Bolibruch shows that for fixed $\chi$ but varying $X$, the answer can be different. We do not consider the dependence on $X$ and concentrate on positive answers to the RHP. By the Birkhoff-Grothendieck-Theorem (BGT), \cite{oss}, any vector bundle $E \rightarrow \Proj^{1}$ is isomorphic to $\mbox{$\cal O$}(c_{1}) \oplus ... \oplus \mbox{$\cal O$}(c_{r})$ for unique integers $c_{1} \geq ... \geq c_{r}$, called the {\em splitting type} of $E$. So, considering the space of all logarithmic connections over $(\Proj^{1},X)$, the Fuchsian systems (\ref{fz}) constitute the connected component of the trivial connection. Fuchsian systems are clearly semi-stable. \hfill {\sc The approach:} We follow Deligne, \cite{del}. To each $x_{j}$, let $U_{j}$ be a small simply-connected neighbourhood and $U_{j}^{*}:= U_{j} - \{ x_{j} \}$. We show directly that a logarithmic connection $(E,\nabla)$ admits, over a small open neighbourhood of $x_{j}$, a {\em normal trivialisation} (Definition \ref{nt}). This is used to construct on $H=(E,\nabla)|_{I\!\! P^{1}-X}$ a filtration $0 \subset H_{j}^{1} \subset ... \subset H_{j}^{l_{j}}=H|_{U_{j}^{*}}$ by flat subbundles with integer weights $\Phi_{j}=\mbox{ diag }(\phi_{j}^{i})$, $(\phi_{j}^{1} \geq ... \geq \phi_{j}^{r}) \in \Z\, ^{r}$, for each $j=1,...,n$. Conversely, such data on a flat bundle $H \rightarrow (\Proj^{1}-X)$ induces a unique extension of $H$ to a logarithmic connection $(E,\nabla):= \mbox{$\cal F$}(H,H_{j}^{m},\Phi_{j})$ over $(\Proj^{1},X)$. Extending and restricting appropriate morphisms, $\mbox{$\cal F$}$ becomes an equivalence between the categories of weighted flat bundles $(H,H_{j}^{m},\Phi_{j})$ over $\Proj^{1}-X$ and the category of logarithmic connections $(E,\nabla) \rightarrow (\Proj^{1},X)$. The equivalence $\mbox{$\cal F$}$ has been constructed slightly differently by Manin, \cite{man}; Deligne, \cite{del}, and Simpson, \cite{sim}, and on objects partially by Anosov \& Bolibruch, see also \cite{for}. $\mbox{$\cal F$}$ prerves injections and surjections. The integer weights are used to define the degree of a weighted flat bundle. By Simpson, $\mbox{$\cal F$}$ preserves degrees and hence (semi-) stability (Definition \ref{fd}). If $\gamma_{j}$ is a loop in $U_{j}^{*}$ going once around $x_{j}$, the parallel transport in $H$ w.r.t. $\gamma_{j}$ is conjugation equivalent to an upper-triangular matrix. So, filtrations of $H|_{U_{j}^{*}}$ by flat subbundles exist. There is much freedom in choosing integer weights. Hence, any pair $(X,\chi)$ is realized by several logarithmic connections. (This even holds over Riemann surfaces, \cite{roe}.) If one is satisfied with any logarithmic connection realizing a given pair $(X,\chi)$, the problem is therefore solved; the difficulty is to decide when the underlying bundle is trivial. We seek, for given H, filtrations $H_{j}^{m}$ and integer weights $\Phi_{j}$ such that $\mbox{$\cal F$}(H,H_{j}^{m},\Phi_{j})$ is Fuchsian. To indicate the relation between our approach and previous ones, let $(X,\chi)$ be realised by $(E,\nabla)$. $E$ admits a system $W=(w_{1},...,w_{r})$ of global meromorphic section, holomorphic away from $x_{1}$, spanning $E$ off $x_{1}$. $W$ generates a flat bundle over $\Proj^{1}$. So, every pair $(X,\chi)$ is realized by a {\em regular system}, i.e.\ a singular algebraic connection on $\Proj^{1} \times \C^{r}$ such that the flat sections have at most polynomial growth. This has long been known, \cite{ple}, \cite{del}, and most attempts to find Fuchsian systems are by ``modifying'' (see \cite[p 77]{ano}) regular ones. Conversely, regular systems induce logarithmic connections. To see this, use the system of sections $V$ as in equation (2.2.21) of \cite{ano} to generate a free rank $r$ sheaf, i.e.\ vector bundle, and apply Levelt's result, \cite[p 28]{ano}, \cite[p 379]{lev}. The modification of regular systems does correspond to changing filtrations and integer weights on $H$. Bolibruch essentially introduced the approach, but worked himself mainly via regular systems. Bolibruch found that any irreducible representation is the monodromy of a Fuchsian system for any given singularities, \cite[p 83]{ano}. Having this, one attempts to apply induction on reducible ones. The difficulty is that the smaller subspaces in the local filtrations of $H$ have higher integer weights and tend to be contained in global subspaces. This restricts the choice of filtrations and weights which make $H$ into a semi-stable weighted flat bundle; which is neccessary should $H$ be a restriction of a Fuchsian system. This difficulty comes up in Theorems \ref{re} and \ref{p} and, in an extreme form in Proposition \ref{bt}. The results on reducible representations that we have, Lemma \ref{cc}, follow from the preservation of short exact sequences under $\mbox{$\cal F$}$ and the BGT. \hfill {\sc What is new:} We give a direct proof of a gauge fixing theorem for logarithmic connections over curves, Theorem \ref{nf}. The description of the inverse of $\mbox{$\cal F$}$ via this gauge fixing theorem seems new. We work on the RHP via logarithmic connections, avoiding regular systems. Instead of Bolibruch's {\em sum of exponents} of a regular system we use the degree of a bundle over $\Proj^{1}$ and the concept of semi-stability. In particular, the preservation of semi-stability under $\mbox{$\cal F$}$ is usefull because any Fuchsian system is semi-stable. Bolibruch does not mention the concept of semi-stability in relation to the RHP. Applying the properties of $\mbox{$\cal F$}$, explained in the first part of this article, several of Bolibruch's results on the RHP follow easily from the Birkhoff-Grothendieck-Theorem and the fact that $\mbox{$\Lambda$}^{0}(\mbox{$\cal O$}(c))$ equals $0$ if $c<0$ and $\C$ if $c=0$. We do not reprove this way as many results as possible, restricting to some signific ones, e.g. Theorem \ref{tw}, Proposition \ref{bt}, Lemma \ref{cc} here and \cite[Lem. 5.2.2]{ano} and \cite[Thm. 5.2.2]{ano}. Besides, perhaps, a more conceptual proof of known results, we have new ones. Bolibruch's first counter-example to the RHP implies that a semi-stable logarithmic connection is not neccessarily Fuchsian. However, we prove that any semi-stable logarithmic connection $(E,\nabla)$ has bounded splitting type, Theorem \ref{ne}. To be precise, $c_{i}-c_{i+1} \leq n-2$ for $i=1,...,r-1$ where $E=\mbox{$\cal O$}(c_{1}) \oplus ... \oplus \mbox{$\cal O$}(c_{r})$, $c_{1} \geq ... \geq c_{r}$, $n=\sharp X$. Bolibruch treats the special case of logarithmic connections with irreducible monodromy. His bound, Corollary \ref{bo} here, is weaker. Combined with a technical result of Bolibruch, namely Proposition \ref{bq} here, i.e.\ \cite[Lem. 4.1.3]{ano}, of which we provide a direct proof, our bound leads to the existence of a Fuchsian system with given monodromy $\chi:\pi_{1}(\Proj^{1}-X) \rightarrow \mbox{Aut}\, (\C^{r})$ if some $\chi(\gamma_{k})$ admits an eigenvector which is a cyclic vector of the $\pi_{1}(\Proj^{1}-X)$-module $\C^{r}$. Firstly, this implies Bolibruch's positive solution for irreducible representations. Secondly, it gives a shorter proof of his result that each $\chi$ is a subrepresentation of the monodromy of a Fuchsian system of double the rank. Thirdly, it leads to an alternative proof of his complete answer to the RHP in rank three. We have a new, sufficient condition for parabolic representations to come from Fuchsian systems, Theorem \ref{p}, and show that this is always satisfied in rank four. A new result for reducible representations is part (ii) of Theorem \ref{re}. \hfill {\sc Acknowledgements:} We thank Michael Thaddeus for bringing the RHP to our attention and showing us Bolibruch's work. \section{\sloppy Local logarithmic connections and weighted flat bundles} \subsection{Logarithmic connections} Let $U$ be a simply connected, open neighbourhood of $0 \in \C$, $z$ the natural coordinate. By $\Omega_{U}^{p}$ we denote the sheaves of holomorphic forms on $U$, Let \[ \Omega_{U}^{1}(\log 0):=\Omega_{U}^{0} \cdot (\frac{\dd z}{z}) \] be the free {\em sheaf of holomorphic 1-forms logarithmic at $0$}, \cite[p 449]{gah}. Naturally, this sheaf is isomorphic to $\Omega^{0}_{U}(\mbox{$\cal K$}_{U} \otimes [0])$, where $\mbox{$\cal K$}_{U}$ is the canonical bundle with section $\mbox{d} z$ and $[0]$ the line bundle on $U$ associated with the divisor $0$. The vector $(\mbox{d} z/z)(0) \in (\mbox{$\cal K$}_{U} \otimes [0])_{0}$ is independent of the choice of coordinate because, if $u:U \rightarrow \C$ is another coordinate with $u(0)=0$, \[ \frac{\mbox{d} u}{u}=\frac{z \, \mbox{d} u}{\mbox{d} z} \cdot \frac{\mbox{d} z}{z} =(1+o(z)) \frac{\mbox{d} z}{z}.\] \begin{defi} \showlabel{lc} A (local) connection logarithmic at 0 is a holomorphic vector bundle $E \rightarrow U$ and a $\C$-linear map \[ \nabla : \Omega^{0} (E) \rightarrow \Omega^{0} (E) \otimes \Omega_{U}^{1}(\log 0) \] satisfying the Leibnitz rule $\nabla (f v) =(\mbox{d} f ) v + f \nabla (v)$ for all $f \in \Omega^{0}$ and $v \in \Omega^{0} (E)$. A morphism $\tau: (E',\nabla') \rightarrow (E,\nabla)$ of logarithmic connections is a bundle map such that \[ \tau_{*} \circ \nabla' = \nabla \circ \tau_{*} \] for $\tau_{*}:\Omega^{0}(E') \rightarrow \Omega^{0}(E)$. \end{defi} A morphism is called {\em injective (surjective)} if it is as bundle map. A short sequence $ \threehorbb{(E',\nabla')}{(E,\nabla)}{(E'',\nabla'')} $ is called {\em exact} if it is as sequence of bundle maps. Simpson calls $(E,\nabla)$ a regular singular $D_{U}$-module, \cite{sim}. If $v \in \Omega^{0}(E)$ and $f \in \Omega^{0}$ then \[ \nabla(fv)=(\frac{\mbox{d} f}{\mbox{d} z})z v \frac{\dd z}{z} + f\nabla(v) \mbox{\ \ so \ \ } (\nabla(fv))(0) = f(0) (\nabla(v))(0) .\] Hence, $\nabla$ induces a canonical endomorphism \[ \rho':E_{0} \rightarrow E_{0} \otimes (\mbox{$\cal K$}_{U} \otimes [0])_{0}. \] \begin{defi} The canonical map $ \rho:E_{0} \rightarrow E_{0}$, determined by $ \rho'(w) = \rho(w) \cdot (\mbox{d} z/z)(0)$ for all $w \in E_{0}$, is called the residue of $\nabla$ at $0$. If $(\lambda^{1},...,\lambda^{r})$ denote the eigenvalues of $\rho$ and $ \phi^{i}:= [- \mbox{Re} (\lambda^{i})] \in \Z$ then $ \phi^{1} \geq ... \geq \phi^{r} $ are called the integer weights of $\nabla$ (at 0). \end{defi} We encode the integer weights as \[ \Phi:= \mbox{diag} (\phi^{i}) = \mbox{block-diag}(\psi^{m} I_{d^{m}}) \] with $\psi^{i} > \psi^{i+1}$ for all $i$. $\Phi$ induces a canonical block structure on all matrices which we will use a lot. If $\nabla_{0}$ is a second logarithmic connection on $E$, the Leibnitz rule implies that \[(\nabla_{0} - \nabla ) : E \rightarrow E \otimes \mbox{$\cal K$}_{U} \otimes [0] \] is a holomorphic bundle map. So, in a trivialisation $\theta:E \rightarrow U \times \C^{r}$, \[ \nabla_{\theta} := \theta \circ \nabla \circ \theta^{-1} = \mbox{d} + A(z) \frac{\mbox{d} z}{z} \] for holomorphic $A : U \rightarrow \mbox{End} (\C^{r})$. The converse clearly holds. Furthermore, $ \rho=\theta(0)^{-1} \circ A(0) \circ \theta(0)$. \subsection{Gauge fixing for logarithmic connections} If all eigenvalues $\mu$ of $K \in \mbox{End} (\C^{r})$ satisfy $\mbox{Re}\, (\mu) \in [0,1)$ then we say $K$ has {\em normalised eigenvalues}. If also $G=\exp(2 \pi i K)$ we call $K$ the {\em normalised logarithm} of $G$: $K = \mbox{norm log}\,\, G$. If $G$ is upper-triangular and has only one eigenvalue $\rho$ then, for $\mu= \mbox{norm log}\, \rho$, we have, \cite[p 376]{lev}, \[ \mbox{norm log }G=\mu I+\frac{1}{2 \pi i} \sum_{j=1}^{\infty} \frac{(-1)^{j}}{j} (\frac{1}{\rho} G -I)^{j}. \] \begin{defi} \showlabel{nt} A trivialisation $\theta:(E,\nabla) \rightarrow (U \times \C^{r},\nabla_{\theta} )$ is called normal (w.r.t. $z$) if \[ \nabla_{\theta} = \mbox{d} + z^{\Phi} (-K -\Phi )z^{- \Phi} \frac{\dd z}{z} \] for some constant, block-upper-triangular $K \in \mbox{End} (\C^{r} )$ with normalised eigenvalues, where $\Phi$ is the integer weights-matrix of $(E,\nabla)$. \end{defi} For a normal trivialisation $\theta$, integer weights $\Phi=\mbox{ block-diag }(\psi^{m} I_{d^{m}})$ and $(e_{1},...,e_{r})$ the standard frame of $U \times \C^{r}$, let \[ F^{m}:=\langle e_{d^{1}+...+d^{m-1}+1},...,e_{d^{1}+...+d^{m}} \rangle \subseteq U \times \C^{r} \] for $m=1,...,l$ and $F^{0}:= U \times \{ 0 \}$. Set \function{\phi}{U\times \C^{r}}{\Z \cup \{ + \infty \}} {v}% {\left\{ \begin{array}{lcl} \psi^{ m} & \mbox{if} & v\in (\oplus_{0}^{m} F^{k}) - (\oplus_{0}^{m-1} F^{k}) \\ + \infty & \mbox{if} & v=0. \end{array} \right.} Clearly, $\phi$ is invariant under parallel transport away from the singularity. \begin{rema} \em \showlabel{ag} For $v \in U^{*} \times \C^{r}$, $\phi(v)$ can be described as the integer part of the {\em asymptotic growth}, \cite[p 17]{ano}, \cite[p 374]{lev}, of the flat extension of $v$ over $U^{*}$. This follows from Lemma \ref{fs} and equations (2.2.8), (2.2.11) and (2.2.12) in \cite{ano}. Anosov \& Bolibruch use Levelt's work on asymptotic growth for {\em regular systems}, i.e.\ singular connections such that flat sections have at most polynomial growth. We avoid regular systems. \end{rema} Under the equivalence of asymptotic growth and $\phi$, (i) and the first part of (iii) of the following theorem correspond to results of Levelt, \cite[p 28]{ano}, \cite[p 60]{bol}, \cite[p 379]{lev}. Gantmacher has (i), \cite[p 185,191]{gat}. \begin{theo} \showlabel{nf} \begin{description} \item[(i)] For each logarithmic connection (and any coordinate), there exists a normal trivialisation in some small neighbourhood of the singularity. \item[(ii)] Let $\tau:(E',\nabla') \rightarrow (E,\nabla)$ and consider two normal trivialisations $\theta:(E,\nabla) \rightarrow (U \times \C^{r},\nabla_{\theta})$ w.r.t. $z$ and $\theta':(E',\nabla') \rightarrow (U \times \C^{r'},\nabla'_{\theta'})$ w.r.t. $u$. Then $M:=\theta \circ \tau \circ (\theta')^{-1}$ satisfies \[ \phi(M(v')) \geq \phi'(v') \] for all $v' \in U \times \C^{r'}$. \item[(iii)] If $\tau$ is injective then $\phi(M(v'))=\phi'(v')$ for all $v' \in U \times \C^{r'}$. If $\tau$ is surjective and $v \in U \times \C^{r}$ then there exists $v' \in \tau^{-1}(v)$ such that $\phi(v)=\phi'(v')$. \end{description} \end{theo} We give a direct proof. \noindent{\em Proof:} {\bf (i):} Start with any trivialisation and write the connection as $ \mbox{d} + A(z) dz/z$ for $A(z) = \sum_{0}^{\infty} A^{j} z^{j}$. Applying a constant gauge transformation, we can assume $ A^{0} = \mbox{ block-diag }(A_{m,m}^{0} )$ where each eigenvalue $\lambda$ of $A_{m,m}^{0}$ satisfies \begin{equation} [ - \mbox{Re}\, \, \lambda ] = \psi^{m}, \mbox{ \ \ i.e. \ \ } - \mbox{Re}\, \, \lambda - \psi^{m} \in [0,1). \label{three} \end{equation} Assume we could find $M(z)=\sum_{0}^{\infty} M^{j} z^{j} :U \rightarrow \mbox{Gl} (r,\C)$ and \newline $B(z)= \sum_{0}^{\infty} B^{j} z^{j} :U \rightarrow \mbox{End} (\C^{r})$ such that \begin{equation} \label{one} M^{-1} \circ (\mbox{d} + A(z) \frac{\dd z}{z} ) \circ M = \mbox{d} + B(z) \frac{\dd z}{z} \end{equation} with $M^{0} = I$, hence $B^{0} = A^{0}$, and $ B(z) = z^{\Phi} (-K -\Phi) z^{-\Phi}$ for $K$ constant, block-upper-triangular. Then the eigenvalues of $K$ would be those of $-B^{0} - \Phi = -A^{0} - \Phi $ and hence $K$ would have normalised eigenvalues by (\ref{three}). We would be done if the series of $M$ converges in a small neighbourhood of $0$. Eq. (\ref{one}) is equivalent to \[ \mbox{d} + M^{-1} (\mbox{d} M) + M^{-1} A M \frac{\dd z}{z} = \mbox{d} + B \frac{\dd z}{z} \] i.e.\ $ z \mbox{d} M = (M B - A M ) \mbox{d} z$. In the Taylor expansion we must have \[ j M^{j} = \sum_{k=0}^{j} \{ M^{k} B^{j-k} - A^{j-k} M^{k} \} \,\,\,\,\,\,\,\,\,\, \forall j\geq 0 .\] So (\ref{one}) is fulfilled if \begin{equation} \label{two} (j M^{j} + A^{0} M^{j} - M^{j} B^{0})-B^{j} = -A^{j} + \sum_{k=1}^{j-1} \{ M^{k} B^{j-k} - A^{j-k} M^{k} \} =:R^{j-1} \end{equation} for all $j \geq 1$. Work by induction on $j \geq 1$. For all $i,m=1, ..., l$, we need to satisfy, for $M^{0}=I $ ($A^{0}=B^{0}$), the equation on the block entries \[ (j+A^{0}_{i,i})M_{i,m}^{j} - M_{i,m}^{j} A_{m,m}^{0} - B_{i,m}^{j} = R^{j-1}_{i,m}. \] By (\ref{three}), any eigenvalue $\lambda''$ of $(j+A_{i,i}^{0})$ satisfies $[-\mbox{Re}\, \lambda'']=-j+\psi^{i}$. So, $\lambda''$ is not an eigenvalue of $A_{m,m}^{0}$ unless $\psi^{i}-j =\psi^{m}$. Hence there is a solution $(M_{i,m}^{j},B_{i,m}^{j})$ with $B_{i,m}^{j}=0$ if $\psi^{i} -j \neq \psi^{m}$. So we find $M$ and $B$ as required. \hfill To see that $\sum_{0}^{\infty} M^{j} z^{j}$ is absolutely convergent near $0$, set \[ c_{j}:= \| A^{j} \| + \| B^{j} \| \,\,\,\,\,\,\,\,\,\, \forall j \geq 1 \,\,\,\,\, \mbox{and} \,\,\,\,\, c_{0}:=2 [ \, \| A^{0} \| \, ] +2 \in \Z \] in operator norm. We can find $C > 1$ and $\varepsilon_{0} > 0$ such that \[ c_{j} \varepsilon^{j} < C \,\,\,\,\,\,\, \forall j\geq 0 \,\,\,\,\,\,\, \forall \,\, 0 \,\, \leq \varepsilon \leq \varepsilon_{0} \] since $A$ and $B$ are absolutely convergent. The equality (\ref{two}) implies \[ (j-c_{0}) \| M^{j} \| \leq \sum_{k=0}^{j-1} \| M^{k} \| \,\, c_{j-k}. \] Choose any $\delta \leq \varepsilon_{0}/2C$. Then $(2C\delta)^{j-k} c_{j-k} \leq C$ for all $j-k$ and so \[ (j - c_{0})\| M^{j} \| (2C \delta)^{j} \leq C \sum_{k=0}^{j-1} \| M^{k} \| (2C \delta) ^{k}. \] Hence \[ (j-c_{0}) \|M^{j} \| \delta^{j} \leq 2^{-j} \sum_{0}^{j-1} \| M^{k} \| (2\delta)^{k}. \] Let $D:= \sum_{k=0}^{c_{0}} \|M^{k} \| \delta^{k}$. Then we claim that \[ \| M^{j} \| \delta^{j} \leq D \,\, 2^{c_{0}-j} \,\,\,\,\,\,\,\,\,\, \forall j\geq c_{0} \] which would finish (i). The claim is clear for $j=c_{0}$ and for $j>c_{0}$ we use induction to find \begin{eqnarray*} \| M^{j} \| \delta^{j} & \leq & \frac{2^{-j}}{j-c_{0}} \left( \sum_{0}^{c_{0}} \|M^{k}\| \delta^{k} 2^{c_{0}} + \sum_{c_{0}+1}^{j-1} D 2^{c_{0}-k} 2^{k} \right) \\ & \leq & 2^{c_{0}-j} D \frac{1+(j-1) -c_{0}}{j-c_{0}} . \end{eqnarray*} \hfill {\bf (ii):} By hypothesis, \[ \nabla_{\theta'}=\mbox{d} +B(z) \frac{\mbox{d} z}{z}=\mbox{d} + u^{\Phi'}(-K'-\Phi') u^{-\Phi'} \frac{\mbox{d} u}{u}, \] \[ \nabla_{\theta}=\mbox{d} +A(z) \frac{\mbox{d} z}{z}=\mbox{d} + z^{\Phi}(-K-\Phi) z^{-\Phi} \frac{\mbox{d} z}{z} \] and $M=\sum_{0}^{\infty} M^{j}z^{j}:U \rightarrow \mbox{Hom}(\C^{r'},\C^{r})$ with $M \circ \nabla'_{\theta'}=\nabla_{\theta} \circ M$, i.e.\ (\ref{two}) above. Let $M=\oplus_{i,m} M_{i,m}$ for $M_{i,m}:F'^{m} \rightarrow F^{i}$ where $\C^{r'}=\oplus_{1}^{l'}F'^{m}$ and $\C^{r}=\oplus_{1}^{l} F^{i}$ according to $\Phi'$ and $\Phi$, respectively. If $M$ is block-upper-triangular, i.e.\ $M_{i,m}=0$ if $\psi'^{m} > \psi^{i}$, then (ii) follows. For $j=0$, (\ref{two}) gives $A^{0}M^{0}-M^{0}B^{0}=0$. Clearly, $B^{0}$ and $A^{0}$ are block-diagonal, any eigenvalue $\lambda'$ of $B_{m,m}^{0}$ satisfies $[-\mbox{Re}\, \lambda']=\psi'^{m}$ and any eigenvalue $\lambda$ of $A_{i,i}^{0}$ satisfies $[-\mbox{Re}\, \lambda ] = \psi^{i}$. This implies \begin{equation} \label{four} M_{i,m}^{0}=0 \mbox{\ \ if \ \ } \psi'^{m} \neq \psi^{i}. \end{equation} Since $A$ and $B$ are block-upper-triangular, (\ref{two}) implies by induction on $j \geq 0$ that $jM^{j} +A^{0}M^{j} -M^{j}B^{0}$ is block-upper-triangular, i.e. \[ (j+A_{i,i}^{0})M_{i,m}^{j} - M_{i,m}^{j}B^{0}_{m,m}=0 \] if $\psi'^{m} > \psi^{i} > \psi^{i}-j$. Hence, $M^{j}$ is block-upper-triangular. \hfill {\bf (iii):} Assume $\tau$ is injective and $v' \in U \times \C^{r'}$ has $\phi'(v')=\psi'^{m}$. By (\ref{four}), there exists $i$ such that $\psi^{i}=\psi'^{m}$ and $M^{0}_{i,m}$ has full rank. Since $\phi$ and $\phi'$ are invariant under parallel transport, $v'$ is, w.l.o.g., contained in the neighbourhood of $0$ where $M_{i,m}$ has full rank. Hence, $M(v')$ has a component in $F^{i}$ and so, $\phi(M(v')) \leq \psi^{i}$. Now assume $\tau$ is surjective and $v \in U \times \C^{r}$ with $\phi(v)=\psi^{i}$. Write $v=v^{1}+...+v^{i}$ according to the decomposition of $\C^{r}$. By the dual of the above argument, we find $v'=v'^{1}+...+v'^{i}$ with $v^{k}=M(v'^{k})$ and $\phi'(v'^{k}) =\psi^{k}$. Hence, $\phi'(v')=\psi^{i}$. \\ $\Box$ \begin{defi} Let $\theta$ be a normal trivialisation of $(E,\nabla)$. The integer weights filtration of $(E,\nabla)$ is \[ 0 \subset E^{1} \subset ... \subset E^{l} =E \mbox{ \ \ where \ \ } E^{m}:=\theta^{-1}(\oplus_{1}^{m} F^{k}). \] \end{defi} At first, the normal trivialisation $\theta$ and hence the filtration of $E$ exists only over a small neighbourhood of the singularity. But since each $E^{m}$ is invariant under $\nabla$, we can extend over all of $U$. This filtration, together with $\Phi$, is equivalent to $\phi \circ \theta:E \rightarrow \Z \cup \{ +\infty\}$. It is independent of the choice of $\theta$ by Theorem \ref{nf}. Let $\pi:\tilde{U}^{*} \rightarrow U^{*}$ be the universal covering and write $\tilde{z}$ for the coordinate over $z$. Let $\log \tilde{z}:\tilde{U}^{*} \rightarrow \C$ be a holomorphic function such that $\log \tilde{z} \equiv \log z \mbox{ \ mod \ } (2 \pi i)$. For $K \in \mbox{End}(\C^{r})$ let $\tilde{z}^{K}:= \exp (K \log \tilde{z})$; $\tilde{z}^{\Phi}=z^{\Phi}$. \begin{lemm} \showlabel{fs} For $\theta$ as in Definition \ref{nt}, $ \nabla_{\theta} (z^{\Phi} \tilde{z}^{K} ) =0$ on $\tilde{U}^{*}$, i.e.\ $z^{\Phi} \tilde{z}^{K}$ is a fundamental system of flat sections. Hence, $\exp (2 \pi i K)$ is the monodromy around $0$. \end{lemm} \noindent{\em Proof:} $ (\mbox{d} + z^{\Phi} (-K -\Phi) z^{-\Phi} \frac{dz}{z} )(z^{\Phi} \tilde{z}^{K}) = z^{\Phi} ( \Phi + K - K - \Phi ) \tilde{z}^{K} \frac{dz}{z}. \,\, \Box $ \subsection{Correspondence between local logarithmic connections and weighted flat bundles} \begin{defi}[Simpson, Deligne] A weighted flat bundle $(H,H^{m},\Phi)$ over $U^{*}$ consists of a holomorphic, rank $r$ vector bundle $H \rightarrow U^{*}$ with a holomorphic (i.e.\ flat and compatible) connection \[ \nabla:\Omega^{0}(H) \rightarrow \Omega^{0}(H) \otimes \Omega^{1}_{U^{*}},\] a filtration by proper subbundles $ 0 \subset H^{1} \subset ... \subset H^{l}=H$, invariant under $\nabla$, and an $r \times r$ matrix with integer entries $ \Phi= \mbox{diag }(\phi^{i})=\mbox{block-diag }(\psi^{m}I_{d^{m}}) $ where $\psi^{m} > \psi^{m+1}$ and $d^{m}=\mbox{rank}\, (H^{m}/H^{m-1})$. \end{defi} The integer $\psi^{m}$ is called the {\em integer weight} of $H^{m}$. The function \function{\phi}{H}{\Z \cup \{ + \infty \}}{v}{\left\{ \begin{array}{lcl} \psi^{m} & \mbox{if} & v \in H^{m} \setminus H^{m-1} \\ + \infty & \mbox{if} & v=0 \end{array} \right. } is equivalent to the filtration and weights; $(H,\phi):=(H,H^{m},\Phi)$. A {\em morphism} of weighted flat bundles $\eta :(H',\phi') \rightarrow (H,\phi)$ is a map of flat bundles such that \[ \phi(\eta(v')) \geq \phi'(v') \] for all $v' \in H'$. Equivalently, $\eta(H'^{k}) \subseteq H^{m-1}$ if $\psi'^{k} > \psi^{m}$. The morphism is called {\em injective} if it is as bundle map and satisfies $ \phi(\eta(v'))=\phi'(v') $ for all $v' \in H'$. It is called {\em surjective} if it is as bundle map and if for all $v \in H$ there exists $v' \in \eta^{-1}(v) $ such that $ \phi(v)=\phi'(v')$. A sequence \threehorss{(H',\phi')}{(H,\phi)}% {(H'',\phi)}{\eta}{\xi} is called {\em exact} if $\eta$ is injective, $\xi$ is surjective and Im$\, \eta = \,\,$Ker$\, \xi$. The {\em direct sum} $(H',\phi') \oplus (H'',\phi'')$ is defined by $(H' \oplus H'',\phi)$ where $\phi(h' \oplus h''):= \min (\phi'(h'),\phi''(h''))$. \begin{defi} Let $\mbox{$\cal F$}^{-1}$ be the functor from the category of logarithmic connections to the category of weighted flat bundles, given by restricting the integer weights filtration and the morphisms to $U^{*}$. \end{defi} By Theorem \ref{nf}, $\mbox{$\cal F$} ^{-1}$ sends injections, surjections and short exact sequences to such. To construct a functor $\mbox{$\cal F$}$, inverse to $\mbox{$\cal F$}^{-1}$, consider a weighted flat bundle $(H,\phi)$. Let $Y=(y_{1},...,y_{r})$ be a fundamental system of multivalued flat sections, such that $\langle y_{1},...,y_{d^{1}+...+d^{m}} \rangle = H^{m}$. Let $\mbox{$\gamma$}$ be a loop in $U^{*}$ going once around $0$ anticlockwise and write $\mbox{$\gamma$} ^{*}$ for the induced action on $\mbox{$\tilde{U}$}^{*}$; $\log (\tilde{z} \circ \gamma^{*})= (\log \tilde{z})+2 \pi i$. Then \[ Y \circ \mbox{$\gamma$}^{*} = Y G\] for constant block-upper-triangular $G \in \mbox{Gl} (r,\C)$. Put $K:= \mbox{norm log}\, G$ which is also block-upper-triangular. Since $\tilde{z}^{-K} z^{-\Phi}$ is invertible over $\tilde{U}^{*}$, $Y\tilde{z}^{-K} z^{-\Phi}$ is a trivialisation of $H$ over $\tilde{U}^{*}$. It is single valued, \cite[p 17]{ano}, since \[ (Y \tilde{z}^{-K} z^{-\Phi})\circ \mbox{$\gamma$}^{*} = Y G G^{-1} \tilde{z}^{-K} z^{-\Phi}.\] \begin{defi} \showlabel{f} Let $\mbox{$\cal F$}(H,\phi)$ be the extension of $H$ over $U$, whose stalk at $0$ is generated by the system of sections \[ V(z) := Y \tilde{z}^{-K} z^{-\Phi}:U^{*} \rightarrow H \times \cdots \times H .\] $\nabla$ becomes a singular connection on the extension of $H$. For a morphism $ \eta: (H',\phi') \rightarrow (H,\phi),$ let $\mbox{$\cal F$}(\eta): \mbox{$\cal F$}(H',\phi') \rightarrow \mbox{$\cal F$} (H,\phi)$ be the unique holomorphic extension of $\eta$. \end{defi} $\mbox{$\cal F$}(H,0)$ is called the {\em canonical extension}, \cite{nas}, of the flat bundle $H$. One checks that different choices of $Y$ and coordinate $z$ give extensions which are isomorphic via a map extending the identity of $H$. Anosov \& Bolibruch construct extensions of $H$ by choosing $Y$ such that $G$ is upper-triangular, but with $Y$ not requested to respect a fixed filtration. One can (in addition) choose $Y$ such that $G$ decomposes w.r.t.\ eigenvalues. Assuming then that $G$ has only one eigenvalue, it is easy to see that $\mbox{$\cal F$}$ equals the extension-functor of Manin, \cite[p 94]{del}, and is a special case of Simpson's extension functor, \cite[p 738]{sim}. They have the following lemma. \begin{lemm} $\mbox{$\cal F$} (H,\phi)$ is a logarithmic connection and $\mbox{$\cal F$}$ is inverse to $\mbox{$\cal F$}^{-1}$ on objects. \end{lemm} \noindent{\em Proof:} \hspace{0.5cm} $ \nabla(V(z)) =$ \[ Y \mbox{d} (\tilde{z}^{-K} z^{-\Phi} ) = Y \tilde{z}^{-K} (-K -\Phi) z^{-\Phi} \frac{\dd z}{z} = V(z) z^{\Phi} (-K -\Phi) z^{-\Phi} \frac{\dd z}{z}. \] So, $\nabla_{V} = \mbox{d} + z^{\Phi} (-K -\Phi ) z^{-\Phi} \frac{dz}{z}$ in the trivialisation given by the columns of $V$. Combine this with Lemma \ref{fs}. \\ $\Box$ \begin{lemm} \showlabel{gk} Let $G,G' \in \mbox{Gl} (r,\C)$ and put $K:=\mbox{norm log}\, G$, $K':=\mbox{norm log}\, G'$. If $C$ is an $r \times r'$-matrix such that $ GC=CG'$ then $KC=CK'$ and hence $ \tilde{z}^{K} C = C \tilde{z}^{K'}$. If $G G'=G' G$ then $KK'=K'K$. $\Box$ \end{lemm} \begin{lemm}[Simpson, Deligne] \showlabel{he} The holomorphic extension $\mbox{$\cal F$}(\eta)$ in Definition \ref{f} exists. It commutes with the logarithmic connections. If $\eta$ is injective (surjective) then so is $\mbox{$\cal F$}(\eta) $. $\mbox{$\cal F$}$ sends short exact sequences to such and is inverse to $\mbox{$\cal F$}^{-1}$ on morphisms. \end{lemm} \noindent{\em Proof:} Since $\eta$ maps flat sections to flat sections we can find a unique constant $r \times r'$-matrix $C$ such that $ \eta \circ Y'=Y C$. Because $\eta$ does not decrease weights, $z^{\Phi} C z^{-\Phi'}$ is holomorphic over $U$. Furthermore, \[(\eta \circ Y') \circ \gamma^{*} = (YC) \circ \gamma^{*} \mbox{\ \ so that \ \ } (\eta \circ Y')G'=YGC,\] i.e.\ $CG'=GC$. By Lemma \ref{gk} this implies $ \tilde{z}^{K} C=C\tilde{z}^{K'}$. We find \[ \mbox{$\cal F$}(\eta)(V') = (\eta \circ Y')\tilde{z}^{-K'} z^{-\Phi'} = V (z^{\Phi} \tilde{z}^{K} C \tilde{z}^{-K'} z^{-\Phi'}) = V(z^{\Phi} C z^{-\Phi'}) \] and $\mbox{$\cal F$}(\eta)$ is holomorphic. If $\eta$ is injective (surjective) then we can choose $Y$ ($Y'$) such that $C$ is a permutation matrix of full rank and $z^{\Phi} C z^{-\Phi'} = C$. The remainder of the statement follows from continuity. \\ $\Box$ \section{\sloppy Global logarithmic connections and weighted flat bundles} \subsection{Correspondence over $\Proj^{1}$} \showlabel{no} We extend the concepts to the Riemann sphere. Let $X = \{ x_{1},...,x_{n} \} \subseteq \Proj ^{1} $, put $S:= \Proj^{1} - X$ and fix a base point $s \in S$. For each $j=1,...,n$ choose a simply connected neighbourhood $U_{j} \subseteq \Proj^{1}$ of $x_{j}$ containing $s$ but no other $x_{k}$'s and a coordinate $z_{j}$ centered at $x_{j}$. ($U_{j}$ is easier to handle than a small neighbourhood around $x_{j}$ and a path from $x_{j}$ to $s$.) Let $\mbox{$\gamma$}_{j} \in \pi_{1}(U^{*}_{j},s)$ go once around $x_{j}$, anticlockwise, and $\mbox{$\tilde{U}$}^{*}_{j}$ be the universal covering of $U^{*}_{j}$. \hfill A {\em logarithmic connection} over $(\Proj^{1},X)$ consists of a holomorphic bundle $E \rightarrow \Proj^{1}$ and a $\C$-linear map $ \nabla:\Omega^{0}(E) \rightarrow \Omega^{0}(E) \otimes \Omega^{1}( \log X)$ satisfying the Leibnitz rule; where $\Omega^{1}( \log X)=\Omega^{0} (\mbox{$\cal K$}_{I\!\! P^{1}} \otimes [X])$. A weighted flat bundle over $S$ is a holomorphic flat bundle $H \rightarrow S$ together with filtrations by flat subbundles $ 0 \subset H_{j}^{1} \subset ... \subset H_{j}^{l_{j}} = H|_{U^{*}_{j}}$ and integer weights $ \Phi_{j}=\mbox{diag} (\phi^{i}_{j})= \mbox{block-diag} (\psi_{j}^{m} I_{d_{j}^{m}})$ for each $j=1,...,n$. Note that no compatibility is required for different $j$. Write $\phi=(\phi_{1},...,\phi_{n})$ for the weight functions $\phi_{j}:H|_{U_{j}^{*}} \rightarrow \Z \cup \{ +\infty \}$. We have $\pi_{1}(S,s)= \langle \gamma_{1}, ..., \gamma_{n} \,\, | \,\, \gamma_{1} \cdot ... \cdot \gamma_{n} = 1 \rangle$ where $\gamma_{1} \cdot \gamma_{2}$ means travelling along $\gamma_{1}$ first. A weighted flat bundle corresponds to a conjugacy class of representations $\chi:\pi_{1}(S,s) \rightarrow \mbox{Gl} (r,\C)$ with, for each $j=1,...,n$, a weighted filtration of $\C^{r}$ invariant under $\chi(\gamma_{j})$. \begin{coro}[\cite{sim}] \showlabel{fc} $\mbox{$\cal F$}$ induces an equivalence between the category of logarithmic connections over $\Proj^{1}$ and that of weighted flat bundles over $S$. It and its inverse preserve injections, surjections and short exact sequences. $\,\,\,\, \Box$ \end{coro} \begin{defi} \showlabel{fd} \begin{description} \item[(i)] $ \deg (H,\phi) := \sum_{j} \{ \mbox{Tr\,} \Phi_{j} + \mbox{Tr\,} (\mbox{norm log} \,\, \chi(\gamma_{j}))\} \in \Z $ \item[(ii)] A system (in one of the considered categories) of rank $r$ and degree $d$ is called stable if any proper subsystem of rank $r'$ and degree $d'$ satisfies $ d'/r' < d/r $. For semi-stability allow '$ \leq $'. The number $d/r$ is called the slope of the system. \end{description} \end{defi} Observe that $\deg (H,\phi) = \sum \{ \mbox{Tr\,} \Phi_{j} + \mbox{Re}\, \mbox{Tr\,} (\mbox{norm log} \,\, \chi(\gamma_{j})) \}$ and it is an integer because $\det \chi(\gamma_{1}) \cdot ... \cdot \det \chi(\gamma_{n})=1$. Consider a weighted flat bundle $(H,\phi)$. Choose a basis $Y_{s}$ of $H_{s}$ and denote its extensions by parallel transport over $\tilde{U}_{j}^{*}$ by $Y(\tilde{z}_{j})$. For each $j$, fix some $Z_{j} \in \mbox{Gl} (r,\C)$ such that $Y_{j}(\tilde{z}_{j}):=Y(\tilde{z}_{j})Z_{j}$ respects the filtration of $H|_{U^{*}_{j}}$. Let \[ Y_{j}(\tilde{z}_{j}) \circ \mbox{$\gamma$}_{j}^{*} = Y_{j}(\tilde{z}_{j}) G_{j}, \] $K_{j} := \mbox{norm log}\, G_{j}$. Set $(E,\nabla)=\mbox{$\cal F$} (H,\phi)$. By the Birkhoff-Grothendieck-Theorem (BGT), \cite{oss}, \cite{ano}, there is a system \[ W:\Proj^{1} \rightarrow E \times ... \times E \] of $r$ meromorphic sections such that $W|_{S}$ spans $H=E|_{S}$. We have \[ V_{j}(z_{j}):=Y_{j}(\tilde{z}_{j}) \tilde{z}_{j}^{-K_{j}} z_{j}^{-\Phi_{j}} = W(z_{j}) Q_{j}(z_{j}), \] for some meromorphic $Q_{j}:U_{j} \rightarrow \mbox{Gl} (r,\C)$ , holomorphic on $U_{j}^{*}$. Note that $W|_{U_{j}}$ spans $E|_{U_{j}}$ if and only if $Q_{j}$ is holomorphic at $x_{j}$. \begin{prop}[{\cite[p 32]{lev}}, {\cite[p 754]{sim}}] \showlabel{de} $\mbox{$\cal F$}$ preserves degrees and so \\ (semi-) stability. \end{prop} \noindent{\em Proof:} Let $\omega$ be the connection matrix of $\nabla|_{H}$ w.r.t.\ the trivialisation $W|_{H}$. $\mbox{Tr\,} \omega$ is a single valued holomorphic one-form on $S$. By the Residue-Theorem, $0=\sum_{1}^{n} \mbox{Res} _{x_{j}} (\mbox{Tr\,} \omega ).$ Since $ \nabla(W|_{U_{j}^{*}})= Y_{j}(\tilde{z}_{j}) \mbox{d} (\tilde{z}_{j}^{-K_{j}} z_{j}^{-\Phi_{j}} Q_{j}(z_{j})^{-1}),$ \begin{eqnarray*} \omega|_{U_{j}^{*}}& =& Q_{j} (z_{j}) z_{j}^{\Phi_{j}} \tilde{z}_{j}^{K_{j}} \mbox{d} ( \tilde{z}_{j}^{-K_{j}} z_{j}^{-\Phi_{j}} Q_{j}(z_{j})^{-1}) \\ & = & -Q_{j} [ z_{j}^{\Phi_{j}} K_{j} z_{j}^{-\Phi_{j}} + \Phi_{j} ] Q_{j}^{-1} \frac{ \mbox{d} z_{j}}{z_{j}} + Q_{j} \mbox{d} (Q_{j}^{-1}). \end{eqnarray*} Let $k_{j}$ be the order of vanishing of $\det Q_{j}$ at $x_{j}$. Then \[ \mbox{Tr\,} \omega|_{U_{j}^{*}} = - (\mbox{Tr\,} K_{j} + \mbox{Tr\,} \Phi_{j} + k_{j} ) \frac{\mbox{d} z_{j}}{z_{j}} + \alpha \] for a holomorphic 1-form $\alpha$. Hence $ -\sum_{j} k_{j} = \sum_{j} \mbox{Tr\,} K_{j} + \mbox{Tr\,} \Phi _{j} .$ The right-hand-side is the degree of the weighted flat bundle, while the left-hand-side is the sum of the orders of vanishing of $\det W$, the degree of $E$. \\ $\Box$ This result holds, in fact, over any Riemann surface. Note, that it implies Lemma 5.2.2 in \cite{ano}. Also, if $(E,\nabla) \rightarrow \Proj^{1}$ is logarithmic at $X$ and has residues $\rho_{j}:E_{x_{j}} \rightarrow E_{x_{j}}$ then $-\sum_{1}^{n} \mbox{Tr\,} \rho_{j}$ equals the degree of $E$. This is because in a normal trivialisation, using the notation of Definition \ref{nt}, $\rho_{j}=(z_{j}^{\Phi_{j}}(-K_{j}-\Phi_{j}) z_{j}^{-\Phi_{j}})(0)$. Hence, $-\sum \mbox{Tr\,} \rho_{j}=\sum (\mbox{Tr\,} K_{j}+\mbox{Tr\,} \Phi_{j})=\deg \mbox{$\cal F$}^{-1} (E,\nabla )$. \subsection{The splitting type of $E \rightarrow \Proj^{1}$} By the BGT, any holomorphic bundle $E \rightarrow \Proj^{1}$ has the form $ E \cong \mbox{$\cal O$} (c_{1}) \oplus ... \oplus \mbox{$\cal O$} (c_{r}) $ for unique integers $c_{1} \geq ... \geq c_{r}$. We call $C:= \mbox{diag}\, (c_{i})$ the {\em splitting type} of $E$ or $(E,\nabla)$. If $C=c_{1} I_{r}$ we say $E$ has {\em constant} splitting type. Recall that $\mbox{$\Lambda$}^{0}(\mbox{$\cal O$}(c))$ is zero if $c <0$ and equal to $\C$ if $c=0$. As $\mbox{$\cal F$}$ preserves subsystems and degrees, this implies Theorem 5.2.2 in \cite{ano}. \begin{theo} \showlabel{ne} If $(E,\nabla)$ is semi-stable, $n \geq 2$ and $C=\mbox{diag}\,(c_{i})$ the splitting type, then $ (0\leq ) \,\, c_{i} - c_{i+1} \leq n-2 $ for all $i=1,...,r-1.$ \end{theo} \noindent{\em Proof:} Fix a splitting $ E = \mbox{$\cal O$}(c_{1}) \oplus ... \oplus \mbox{$\cal O$}(c_{r}) . $ Suppose there exists an $i \in \{1,...,r-1 \}$ such that $c_{i} - c_{i+1} > n-2$. Let $z$ be a coordinate centred at $s \in S$. For all $k \in \{1,...,i \}$ we can find sections $v_{k}$ of $\mbox{$\cal O$} (c_{k})$ vanishing to order $c_{k}$ at $s$, i.e.\ $v_{k} z^{-c_{k}}$ spans $\mbox{$\cal O$} (c_{k})$ near $s$. For each $m \in \{ i+1,...,r\}$ we consider the natural projection \[ \pi_{m} : E \otimes \mbox{$\cal K$}_{I\!\! P^{1}} \otimes [X] \rightarrow \mbox{$\cal O$} (c_{m}) \otimes \mbox{$\cal K$}_{I\!\! P^{1}} \otimes [X] \] and obtain sections $\pi_{m} \circ \nabla (v_{k}) : \Proj^{1} \rightarrow \mbox{$\cal O$} (c_{m}) \otimes \mbox{$\cal K$}_{I\!\! P^{1}} \otimes [X].$ Near $s$ we have \begin{eqnarray*} \pi_{m} \circ \nabla (v_{k}) & = & \pi_{m} \circ \nabla (v_{k} z^{-c_{k}} z^{c_{k}} ) \\ & = & \pi_{m} ( \nabla ( v_{k} z^{-c_{k}}) z^{c_{k}} + (v_{k} z^{-c_{k}}) \mbox{d} (z^{c_{k}} )) \\ & = & \pi_{m} ( \nabla (v_{k} z^{-c_{k}} ) z^{c_{k}} ) \end{eqnarray*} since $k \neq m$. Either $\pi_{m} \circ \nabla (v_{k})$ is identically zero or of order at least $c_{k}$. The latter is equivalent to \[ c_{k} \leq \deg (\mbox{$\cal O$} (c_{m}) \otimes \mbox{$\cal K$}_{I\!\! P^{1}} \otimes [X]) = c_{m} +n - 2 . \] But from $k < i<m$ we see that $ c_{k} \geq c_{i} > c_{i+1} +n -2 \geq c_{m} +n -2 .$ Therefore, $\pi_{m} \circ \nabla (v_{k})$ is identically zero for all $k<i<m$. So, $\nabla$ preserves $\mbox{$\cal O$} (c_{1}) \oplus ... \oplus \mbox{$\cal O$}(c_{i})$. Semistability implies then that $c_{1} =c_{2}=...=c_{r}$. \\ $\Box$ This theorem easily extends to logarithmic connections with parabolic structure at the singularities, i.e.\ to filtered regular $D_{S}$-modules, c.f. \cite{sim}. Note, any logarithmic connection with irreducible monodromy is semi-stable, even stable. \begin{coro}[Bolibruch, {\cite[p 84]{ano}}] \showlabel{bo} If the monodromy of $(E,\nabla)$ is irreducible ($n \geq 2$) then $ \sum_{i=1}^{r} c_{1} - c_{i} \leq (n-2)r(r-1)/2$. $\Box$ \end{coro} \begin{lemm} \showlabel{ct} If $\mbox{$\cal F$} (H,H_{j}^{m},\Phi_{j})$ has splitting type $C$ and $\Phi'_{j}=\Phi_{j}+\lambda_{j}I_{r}$ then $\mbox{$\cal F$}(H,H_{j}^{m},\Phi'_{j}) $ has splitting type $C+(\sum_{1}^{n} \lambda_{j})I_{r}$. $\Box$ \end{lemm} \begin{lemm} \showlabel{cc} Let $ \threehorbb{(H',\phi')}{(H,\phi)}{(H'',\phi'')}$ be a short exact sequence and assume that two of them have the same slope. Then all three have the same slope and $\mbox{$\cal F$} (H,\phi)$ has constant splitting type $C=c I_{r}$ if and only if $\mbox{$\cal F$} (H',\phi')$ and $\mbox{$\cal F$} (H'',\phi'')$ have constant splitting types $C'=cI_{r'}$ and $C''=cI_{r''}$, respectively. $\Box$ \end{lemm} The proofs of these two lemmas are straightforward. \section{The Riemann-Hilbert-Problem} \subsection{Commutative and semi-simple representations} \begin{lemm}[\cite{bol}, {\cite[p 76]{ano}}] \showlabel{co} If $\chi:\pi_{1}(S) \rightarrow \mbox{Gl}\, (r,\C)$ factors through $\mbox{$\Lambda$}_{1}(S)$ then it is the monodromy of a Fuchsian system. \end{lemm} \noindent{\em Proof:} Since the $G_{j}=\chi(\gamma_{j})$ commute, each $G_{j}$ preserves $\ker (G_{k}-\mu I)^{t}$ of each $G_{k}$ and for each $t$. Assume then that each $G_{j}$ has only one eigenvalue $\rho_{j}$ and is upper-triangular. Let $\mu_{j}:=\mbox{norm log}\,\, \rho_{j}$ be the only eigenvalue of $K_{j}:=\mbox{norm log}\,\, G_{j}$ and $\xi:=\sum_{1}^{n} \mu_{j} \in \Z$. By Lemma \ref{gk} the $K_{j}$'s commute and $\exp(\xi \cdot I_{r} -\sum K_{j})=G_{1} \cdot ... \cdot G_{n} =I_{r}$. Since $\xi \cdot I_{r}-\sum K_{j}$ has only the eigenvalue $0$, it is the normalised logarithm of $I_{r}$, i.e. $0$. A short calculation then shows that \[ \nabla:= \mbox{d} + \left[ \frac{\xi}{z-x_{1}} - \sum_{j=1}^{n} \frac{K_{j}}{z-x_{j}} \right] \mbox{d} z \] is smooth at infinity. Over each $\tilde{U}_{k}^{*}$ we set $ Y:= (z-x_{1})^{-\xi} \prod_{1}^{n} \widetilde{(z-x_{j})}^{K_{j}}$ and find $\nabla (Y) = 0$ and $Y \circ \mbox{$\gamma$}_{j} = Y G_{j}$. \\ $\Box$ \begin{prop}[{\cite[p 80]{ano}}] \showlabel{bq} If $(E,\nabla)=\mbox{$\cal F$}(H,\phi)$ has splitting type $C$ and $k \in \{1,...,n \}$ is fixed then there exists a permutation $P$ and meromorphic $W:\Proj^{1} \rightarrow E \times ... \times E$ such that \begin{description} \item[(i)] $W$ is holomorphic except at $x_{k}$ and spans $E$ away from $x_{k}$, \item[(ii)] $Wz_{k}^{-C}$ spans $E$ near $x_{k}$ and \item[(iii)] $ Q_{k} (z_{k}) = \hat{Q}_{k}(z_{k}) z_{k}^{-C} P = \hat{Q}_{k}(z_{k}) P z_{k}^{-P^{-1}CP} $ for some invertible $\hat{Q}_{k}$. \end{description} \end{prop} We give a proof different from that in \cite{ano}. \noindent{\em Proof:} Let $W'=(w'_{1},...,w'_{r})$ where $w_{i}':\Proj^{1} \rightarrow \mbox{$\cal O$} (c_{i})$ vanishes to order $c_{i}$ at $s$ and the $\mbox{$\cal O$}(c_{i})$'s decompose $E$. Define $Q_{k}'$ by $V_{k}=W' Q'_{k}$ near $x_{k}$, $\det Q_{k} \neq 0$. Claim: For each permutation $P$ such that all bottom-right minors of $Q_{k}'(0) P^{-1}$ are non-singular, there exists a $W$ as in the proposition. (The existence of such $P$ follows by induction from the description of the determinant of a matrix in terms of co-rank one minors.) We may assume that $z_{k}(s)=\infty$. Suppose there exists $ b=((b_{i,j})):\Proj^{1} \rightarrow \mbox{Gl} (r,\C)$ such that \begin{equation} \label{i} b_{i,j} = \left\{ \begin{array}{cc} \sum_{0}^{c_{i}-c_{j}-1} b_{i,j}^{p} z_{k}^{p} & i<j \\ 1 & i=j \\ 0 & i>j, \end{array} \right. \end{equation} \begin{equation} \label{ii} z_{k}^{c_{i} -c_{m}} \,\, | \,\, (b \, Q'_{k} P^{-1})_{i,m} \,\, \ \ \ \forall \,\,\, i<m. \end{equation} Then $W=W' b z_{k}^{C}$ spans $E|_{I\!\! P^{1}-\{ x_{k} \}}$, $Wz_{k}^{-C}$ spans $E$ near $x_{k}$ and $W' Q_{k}'=V_{k}=W Q_{k}$ implies \[ Q_{k}=z_{k}^{-C} b Q'_{k} = z_{k}^{-C} (b Q'_{k} P^{-1})P =\hat{Q}_{k} z_{k}^{-C} P \] for some invertible $\hat{Q}_{k}$ and we would be done. To find $b$ as in (\ref{i}) satisfying (\ref{ii}) we need to solve a system of linear equations. With $Q'_{k}P^{-1}=((q_{j,m}))$, condition (\ref{ii}) is equivalent to \[ z_{k}^{c_{i}-c_{m}} \ | \ \sum_{j=i+1}^{r} b_{i,j} q_{j,m} + q_{i,m} \] for all $1 \leq i < m \leq r$. Writing $q_{j,m} = \sum_{0}^{\infty} q_{j,m}^{p} z_{k}^{p}$, (\ref{ii}) becomes \[ \sum_{j=i+1}^{r} \sum _{t=0}^{c_{i}-c_{j}-1} b_{i,j}^{t} q_{j,m}^{p-t} + q_{i,m}^{p} =0 \] for all $1\leq i < m \leq r$ and $0 \leq p <c_{i}-c_{m}$. We define \[\alpha(t):=\min \{ j \in \{i+1,...,r \} \ | \ t \leq c_{i} - c_{j} -1 \}. \] Then (\ref{ii}) is equal to \[ \sum_{t=0}^{p} \sum_{j=\alpha(t)}^{r} b_{i,j}^{t} q_{j,m}^{p-t} + q_{i,m}^{p}=0 \] for all $1 \leq i < m \leq r$ and $0 \leq p < c_{i}-c_{m}$. Note that $\alpha(p) \leq m $. To find the $b_{i,j}^{t}$'s we argue one row of $b$ at a time, i.e.\ fix $i$. Assume that $b_{i,j}^{t}$ is known by induction for all $t<p$. Then we have to fullfil \[ \sum_{j=\alpha(p)}^{r} b_{i,j}^{p} q_{j,m}^{0} = \mbox{known term} \] for $m=\alpha(p),...,r$. This system has a solution $(b_{i,\alpha(p)}^{p},...,b_{i,r}^{p})$ since the matrix $((q_{j,m}^{0}))_{\alpha(p) \leq j,m \leq r}$ is a right-bottom minor of $Q'_{k}(0)P^{-1}$. \\ $\Box$ \begin{coro}[Plemelj] \showlabel{ss} If there exists $k \in \{ 1,..., n\}$ such that $\chi (\mbox{$\gamma$}_{k})$ is semi-simple (i.e.\ diagonalizable) then $\chi$ is the monodromy of a Fuchsian system. \end{coro} \noindent{\em Proof:} If $\chi (\mbox{$\gamma$}_{k})$ is semi-simple we can split \[H|_{U_{k}^{*}} = H_{k,1} \oplus ... \oplus H_{k,r} \] into flat line bundles. Let $(E,\nabla)$ be the canonical extension of $H$ (i.e.\ $\Phi_{j}=0$ for all $j$) and choose $Z_{k}$ (see subsection \ref{no}) such that the i-th section in $Y_{k}(\tilde{z}_{k})= Y(\tilde{z}_{k}) Z_{k}$ spans $H_{k,i}$. Choose $W$ as in Proposition \ref{bq}. So $Q_{k}=\hat{Q}_{k}z_{k}^{-C} P$ for invertible $\hat{Q}_{k}$ and permutation $P$, where $C$ is the splitting type of $E$. Let $P'$ be the permutation with $P'_{i,j}=0$ if $i + j \neq n$ and $P'_{i,n-i}=1$ for $i=1,...,n$. Consider the filtration of $H|_{U_{k}^{*}}$, induced by the sections in $Y_{k}(\tilde{z}_{k})P^{-1} (P')^{-1}$. As $\chi(\gamma_{k})$ is semi-simple it will respect this filtration. Let $\Phi'_{k}:=-(P')^{-1} C P'$, which is diagonal with non-increasing entries, and $\Phi'_{j} =0$ for $j \neq k$. Put $(E',\nabla'):=\mbox{$\cal F$} (H,\Phi'_{j})$. Then $W$ trivializes $E'|_{I\!\! P^{1}-x_{k}}$. Furthermore, $Q_{k}'$ is invertible and hence, $W$ spans $E'$ globally, since \begin{eqnarray*} W Q'_{k} & = & Y_{k}(\tilde{z}_{k}) P^{-1} (P')^{-1} \tilde{z}_{k}^{-K_{k}'} z_{k}^{-\Phi_{k}'} = Y_{k}(\tilde{z}_{k}) \tilde{z}_{k}^{-K_{k}} P^{-1} (P')^{-1} z_{k}^{-\Phi'_{k}} \\ & = & W Q_{k} P^{-1} z_{k}^{C} (P')^{-1} = W \hat{Q}_{k} (P')^{-1}. \,\, \Box \end{eqnarray*} \subsection{The rank two case} For a representation $\chi:\pi_{1}(S,s) \rightarrow \mbox{Gl} \, (2,\C)$ with canonical extension $(E^{0},\nabla^{0})=\mbox{$\cal F$} (H,0)$ of splitting type $C^{0}=\mbox{ diag }(c_{1}^{0},c_{2}^{0})$, Bolibruch calls $c_{1}^{0}-c_{2}^{0}$ the {\em weight} of the canonical extension, \cite[p 102]{ano}. \begin{theo} \showlabel{tw} \begin{description} \item[(i) (Dekkers)] Any rank two representation $\chi$ is the holo\-nomy of a Fuchsian system. \item[(ii) (Bolibruch, {\cite[p 137]{ano}})] $ c_{1}^{0}-c_{2}^{0}= \min_{\phi} \sum_{1}^{n}(\phi_{j}^{1} - \phi_{j}^{2})$ where $\phi$ runs over all integer weight functions on $H$ such that $\mbox{$\cal F$} (H,\phi)$ is Fuchsian. \end{description} \end{theo} \noindent{\em Proof:} Assume $(E,\nabla)=\mbox{$\cal F$} (H,\phi)$ is Fuchsian. The identity of $H$ extends to a meromorphic map $\mbox{$\cal O$}(c_{1}^{0}) \oplus \mbox{$\cal O$}(c_{2}^{0})=E^{0} \rightarrow E$. By definition of $\mbox{$\cal F$}$ (or the proof of Lemma \ref{he}), the non-zero map $\mbox{$\cal O$}(c_{1}^{0}) \rightarrow E$ is of order greater or equal $\phi_{j}^{2}$ at $x_{j}$. Hence, $c_{1}^{0} \leq - \sum_{1}^{n} \phi_{j}^{2}$. By Proposition \ref{de}, $c_{1}^{0}+c_{2}^{0}+\sum (\phi_{j}^{1} + \phi_{j}^{2})=0$ and hence $c_{1}^{0}-c_{2}^{0} \leq \sum_{1}^{n} (\phi_{j}^{1} - \phi_{j}^{2})$. This proves (ii) in one direction. If there exists $k \in \{1,...,n\}$ such that $\chi(\gamma_{k})$ has two eigenvalues then we are done by the proof of Corollary \ref{ss}. Otherwise, $(E^{0},\nabla^{0})=\mbox{$\cal F$} (H,0)$ is semi-stable and we can argue much as in Theorem \ref{he}. If $v_{i}:\Proj^{1} \rightarrow \mbox{$\cal O$}(c_{i}^{0})$ vanishes to order $c_{i}^{0}$ at $s$ and $\pi_{2}:E^{0} \rightarrow \mbox{$\cal O$} (c_{2}^{0})$, then \[ \pi_{2} \circ \nabla^{0}(v_{1}):\Proj^{1} \rightarrow \mbox{$\cal O$} (c_{2}^{0} ) \otimes \mbox{$\cal K$}_{I\!\! P^{1}} \otimes [X] \] either vanishes identically, in which case $c_{1}^{0}=c_{2}^{0}$ and we are done, or it is of order at least $c_{1}^{0}$ at $s$. If it also vanishes at each $x_{j}$ then $n+c_{1}^{0} \leq c_{2}^{0}+n-2$. So, assume $(\pi_{2} \circ \nabla^{0} (v_{1})(x_{k}) \neq 0$. If $W=(v_{1},v_{2})$, the formula for the connection-matrix $\omega|_{U^{*}_{k}}$ in the proof of Proposition \ref{de} implies that $(Q_{k})_{1,2}(x_{k}) \neq 0$ ($\det Q_{k} (x_{k}) \neq 0$ by choice of $W$). Now apply the Claim at the beginning of the proof of Proposition \ref{bq} with $W'=(v_{1},v_{2})$ and $P$ the non-trivial rank two permutation. Then $\mbox{$\cal F$} (H,0,...,0,\Phi_{k},0,...,0)$ will be Fuchsian for $\Phi_{k}=-P^{-1} C^{0} P$. We have completed the proof of (ii) and also proved (i). \\ $\Box$ \subsection{The semi-stable case and implications} \begin{theo} \showlabel{ta} Let $H \rightarrow S$ be a flat bundle. If we can find filtrations $H_{j}^{m}$ and integer weights $\Phi_{j}$ such that \begin{description} \item[(a)] $(H,H_{j}^{m},\Phi_{j})$ is semi-stable and \item[(b)] there exists $k \in \{1,...,n\}$ with $\mbox{\rm rank} (H_{k}^{i+1} / H_{k}^{i})=1$ and \[ \phi_{k}^{i} - \phi_{k}^{i+1} \geq (r-1)(n-2) \,\,\, \forall \,\,\, i=1,...,r-1 \] \end{description} then we can find $\Phi_{k}'$ such that $ (E',\nabla'):=\mbox{$\cal F$} (H,H_{j}^{m},\Phi_{1},...,\Phi_{k}',...,\Phi_{n}) $ is Fuchsian. \end{theo} \noindent{\em Proof:} Let $C$ be the splitting type of $(E,\nabla) = \mbox{$\cal F$} (H,H_{j}^{m},\Phi_{j})$. Theorem \ref{ne} implies \[ c_{1} - c_{r} \leq (n-2)(r-1) \leq \phi_{k}^{i} - \phi_{k}^{i+1} \] for all $i=1,...,r-1$. Fix $W$ as in Proposition \ref{bq} and let, in that notation, $\Phi_{k}':= \Phi_{k}-P^{-1}CP$, which will have non-increasing entries. $W$ spans $E'$ off $x_{k}$ and \[WQ_{k}z_{k}^{\Phi_{k}} \tilde{z}^{K_{k}}_{k} = Y_{k}(\tilde{z}_{k}) = W Q'_{k} z_{k}^{\Phi_{k}'} \tilde{z}_{k}^{K_{k}} \] implies \[ Q_{k}'=Q_{k} z_{k}^{P^{-1}CP} = \hat{Q}_{k} z_{k}^{-C} P z_{k}^{P^{-1}CP} = \hat{Q}_{k}P. \] So, $Q_{k}'$ is invertible and $W$ a global trivialisation of $E'$. \\ $\Box$ \begin{prop} \showlabel{ca} Suppose there exists $k \in \{1,...,n \}$ and $h \in H_{s}$ such that $h$ is an eigenvector of $\chi(\mbox{$\gamma$}_{k})$ but a cyclic vector of the $\pi_{1}(S,s)$-module $H_{s}$ (i.e.\ $ \langle \,\, (\mbox{Im} \,\, \chi) (h) \,\, \rangle = H_{s}$). Let $N_{1},...,N_{n}$ be any integers. Then we can find filtrations $H_{j}^{m}$ and weights $\Phi_{j}$ such that $\mbox{$\cal F$} (H,H_{j}^{m},\Phi_{j})$ is Fuchsian. Moreover, we can arrange that \begin{description} \item[(1)] $\phi_{j}^{i} \geq N_{j}$ for all $j \neq k$, $i=1,...,r$ and \item[(2)] $\phi_{k}^{1}=\phi(h)\geq N_{k}$. \end{description} Hence, there are infinitely many Fuchsian systems with monodromy $\chi$. \end{prop} \noindent{\em Proof:} For each $j\neq k$ choose filtrations $H_{j}^{m}$ and weights $\Phi_{j}$ such that (1) is satisfied. Also choose a filtration $H_{k}^{m}$ and weights $\Phi_{k}$ such that $H_{k}^{1}=\langle h \rangle$, hypothesis (b) of Theorem \ref{ta} is satisfied and $\phi_{k}^{1} \geq N_{k}+(r-1)(n-2)$. These conditions remain satisfied if we increase $\phi_{k}^{1}$ or decrease $\phi_{k}^{r}$. Doing so we can assume that $ \deg (H,H_{j}^{m},\Phi_{j}) =0 $ and since no proper flat subbundle of $H$ contains $h$ we can also assume that $(H,H_{j}^{m},\Phi_{j})$ is semi-stable. Apply Theorem \ref{ta} to find $ \sum_{1}^{r} c_{i}=0 $ and $ c_{i}-c_{i+1} \leq (n-2), $ implying $|c_{i}| \leq (n-2)(r-1)$ for $i=1,...,r$. Hence, the first entry of $\Phi'_{k}=\Phi_{k}-P^{-1}CP$ is greater or equal to $N_{k}+(r-1)(n-2)-(r-1)(n-2)=N_{k}$ and we are done. \\ $\Box$ \begin{coro}[Bolibruch, {\cite[p 84]{bol}}, {\cite[p 83]{ano}}; Kostov, \cite{kos}] \showlabel{tb} Any \\ irreducible flat $H \rightarrow S$ is the restriction of a Fuchsian system. $\Box$ \end{coro} \begin{coro}[{\cite[p 114]{ano}}] Any $\chi: \pi_{1}(S,s) \rightarrow \mbox{Gl}(r,\C)$ is the subrepresentation of the monodromy of some Fuchsian system of double the rank. \end{coro} \noindent{\em Proof:} By Lemma \ref{co} we may assume that $n \geq 3$ and $r \geq 2$. Let $G_{j}:= \chi (\mbox{$\gamma$}_{j})$ for all $j=1,...,n$. By Corollary \ref{ss} we can assume that in canonical basis $e_{i}=(0,...,0,1,0,...,0)^{t}$ we have the equality of vector spaces $ \langle e_{r},G_{1}e_{r} \rangle = \langle e_{r-1},e_{r} \rangle$. We define $G'_{j}$ as follows \[ G_{1}':= \twomatrix{G_{1}}{M_{1}}{0}{I} \ \ \ \ M_{1}:= \left( \begin{array}{cccccc} 1 & & & & & \\ & \cdot & 0 & & & \\ & 0 & \cdot & & & \\ & & & 1 & & \\ & & & & 0 & 0 \\ & & & & 1 & 0 \end{array} \right) \] \[ G_{2}':= \twomatrix{G_{2}}{0}{0}{M_{2}} \ \ \ \ M_{2}:= \left( \begin{array}{ccccc} 1 & 1 & & & \\ & \cdot & \cdot & 0 & \\ & & \cdot & \cdot & \\ & 0 & & \cdot & 1 \\ & & & & 1 \end{array} \right) \] \[ G_{3}':= \twomatrix{G_{3}}{-G_{2}^{-1} G_{1}^{-1} M_{1}}{0}{M_{2}^{-1}} \mbox{\ \ and \ \ } G'_{j}:= \twomatrix{G_{j}}{0}{0}{I} \] for all $j \geq 4$. One checks that $G_{1}' \cdot ... \cdot G_{n}' = I_{2r}$ and so defines a representation $\chi':\pi_{1}(S,s) \rightarrow% \mbox{Gl} (2 r,\C)$. Furthermore, $e_{2r}$ is an eigenvector of $G_{1}'=\chi'(\gamma_{1})$ and \[ \langle e_{2r},G_{2}' e_{2r},...,(G_{2}')^{r-1} e_{2r} \rangle = \langle e_{r+1},...,e_{2r} \rangle ,\] \[ G_{1}' \langle e_{r+1},...,e_{2r} \rangle \oplus \langle e_{r+1},...,e_{2r} \rangle \supset \langle e_{1},...,e_{r-2},e_{r},...,e_{2r} \rangle \] and $ \langle e_{r},G_{1}' e_{r} \rangle = \langle e_{r-1},e_{r} \rangle .$ Apply Proposition \ref{ca} with $h:=e_{2r}$. \\ $\Box$ \subsection{Reducible representations} Let $H_{j}:=H|_{U_{j}^{*}}$. Part (i) of the following is due to Bolibruch, \cite[Cor. 5.4.1]{ano}, \cite[Thm. 3.8]{bol}, while (ii) is new and will be used to give an alternative proof of Bolibruch's answer to the RHP in rank three, Theorem \ref{dr} here. \begin{theo} \showlabel{re} Let \threehorbb{H'}{H}{H''} be a short exact sequence of flat bundles (without weights) and assume there exist filtrations and weights such that $(E'',\nabla'')=\mbox{$\cal F$} (H'',(H'')_{j}^{m},\Phi''_{j})$ is Fuchsian. Suppose at least one of the following conditions holds for some $k \in \{1,...,n\}$. \begin{description} \item[(i)] $\threehorbb{H'_{k}}{H_{k}}{H''_{k}}$ splits and there exist filtrations and weights such that $(E',\nabla')=\mbox{$\cal F$}(H',(H')_{j}^{m},\Phi'_{j})$ is Fuchsian. \item[(ii)] There exist splittings $ H'_{k}=H^{(3)} \oplus H^{(0)}$ and $H_{k}=H^{(3)} \oplus H^{(4)}$ where $\langle h \rangle =H^{(0)}=H'_{k} \cap H^{(4)}$ and $h$ is a cyclic vector of the $\pi_{1}(S,s)$ module $H'_{s}$. \end{description} Then $H$ is the restriction of a Fuchsian system. \end{theo} \noindent{\em Proof:} {\bf (i):} By Lemma \ref{ct} we can assume that for all $j \neq k$ the smallest diagonal entry in $\Phi'_{j}$ is greater then the largest one in $\Phi''_{j}$. Using \[ \threehorbb{H'_{j}}{H_{j}}{H''_{j}} \] for such $j$ we can therefore induce filtrations and weights on $H_{j}$ to make this local sequence a short exact one of weighted flat bundle. Let $\alpha:H''_{k} \rightarrow H_{k}$ be a splitting right inverse of $H_{k} \rightarrow H''_{k}$ and put the obvious weighted filtration on $\mbox{Im}\, \alpha$. Then use $H_{k}=H'_{k} \oplus \mbox{Im}\, \alpha$ to give $H_{k}$ the direct sum weighted filtration. $(H'_{k},\phi'_{k}) \rightarrow (H_{k},\phi_{k})$ becomes an injection. Since $(H_{k},\phi_{k}) \rightarrow (H''_{k},\phi''_{k})$ is the composition $ \threehorbs{H_{k}}{\mbox{Im}\, \alpha}{H''_{k}}{\alpha^{-1}} $, it is a surjection. Apply Lemma \ref{cc} to finish this case. \hfill {\bf (ii):} For $j=1,...,n$ let $N_{j}$ be the greatest diagonal entry in $\Phi''_{j}$. Then construct $(H',(H')_{j}^{m},\Phi'_{j})$ as in Proposition \ref{ca} so that $\mbox{$\cal F$}(H',(H')_{j}^{m},\Phi'_{j})$ is Fuchsian. If $j\neq k$ we induce weights on $H_{j}$ as in (i). Use the exact sequence of flat bundles $ \threehorbb{H^{(0)}}{H^{(4)}}{H''_{k}}$ to induce weights $\phi^{(4)}$ on $H^{(4)}$. Induce weights on $H_{k}$ using $ H_{k}=H^{(3)} \oplus H^{(4)}$. Then, $(H_{k},\phi_{k}) \rightarrow (H''_{k},\phi''_{k})$ is given by the composition \[ \threehorbb{(H_{k},\phi_{k})}{(H^{(4)},\phi^{(4)})}% {(H''_{k},\phi''_{k})} \] of two surjections and hence is one itself. For $h'=h^{(3)} + h^{(0)} \in H'_{k}$ we have \[ \phi'_{k}(h')=\min (\phi'_{k}(h^{(3)}),\phi'_{k}(h^{(0)})) = \phi(h') \] since $H^{(0)}$ is the highest weight subspace in the filtration of $H'_{k}$. Hence $(H'_{k},\phi'_{k}) \rightarrow (H_{k},\phi_{k})$ is an injection and we finish with Lemma \ref{cc}. \\ $\Box$ \begin{prop}[{\cite[p 83]{bol}}, {\cite[p 100]{ano}}] \showlabel{bt} If $H$ is reducible, the holonomy $\chi(\gamma_{j})$ has only one Jordan-block, for each $j=1,...,n$, and $\mbox{$\cal F$}(H,\phi)=(E,\nabla)$ has constant splitting type, then $\Phi_{j}=\phi_{j}^{1} \cdot I_{r}$ for all $j$. \end{prop} We give a more conceptual proof. \noindent{\em Proof:} If $G_{j}$ has only one Jordan block, there exists a canonical full flag of subsystems of $H_{j}$. If $H'$ is a proper subsystem of $H$ then it must contain the subbundles in the local filtrations of rank equal to rank $H'$. If some $\Phi_{k}$ has non equal diagonal entries then \[ \mbox{ slope}\,\, (H',\phi')>\, \mbox{slope}\,\, (H,\phi).\] But $(E,\nabla)$, and hence $(H,\phi)$, is semi-stable -- a contradiction. \\ $\Box$ Let $H$ be as in the previous proposition, $\rho_{j}$ the only eigenvalue of $\chi(\gamma_{j})$ and $\mu_{j}=\,\, \mbox{norm log}\,\, \rho_{j}$. If $(E,\nabla)=\mbox{$\cal F$}(H,\phi)$ is Fuchsian then $\deg (H,\phi)=0$. Since $r \, | \, \sum \mbox{Tr\,} \Phi_{j}$ we find $r \, | \, \sum \mbox{Tr\,} (\mbox{norm log } \chi (\gamma_{j}))$, i.e.\ $r \, | \, r \sum \mu_{j}$. So, $\sum \mu_{j}$ must be an integer. Bolibruch uses this to give an example of a representation with $r= 4$ and $n=3$ which can not be the monodromy of any Fuchsian system with three singularities, \cite[p 91]{bol}, \cite[p 105]{ano}. \subsection{Parabolic representations and the rank three case} Let $\mbox{B}\,(r,\C)$ be the group of invertible upper-triangular $r \times r$-matrices. \begin{theo} \showlabel{p} Let $\chi:\pi_{1}(S,s) \rightarrow \mbox{B}\,(r,\C)$ be a representation with \[ G_{j}=\chi(\gamma_{j})=\uppermatrix{\rho^{1}_{j}}{\rho_{j}^{r}} \] for $j=1,...,n$. Let $ \mu_{j}^{i}:=\mbox{\rm norm log} (\rho_{j}^{i}) $ and $\Lambda^{i}:=-\sum_{j=1}^{n} \mbox{Re}\, \mu_{j}^{i} \in \Z_{\leq 0}.$ Assume we can find $((\phi_{j}^{i}))$ such that \begin{description} \item[(a)] $\phi_{j}^{i} \geq \phi_{j}^{k}$ if ($i \leq k$ and $\rho_{j}^{i}=\rho_{j}^{k}$) and \item[(b)] $\Lambda^{i}=\sum_{j=1}^{n} \phi_{j}^{i}$ for all $i=1,...,r$. \end{description} Then $\chi$ is the monodromy of a Fuchsian system $(E,\nabla)$ and the integer weights of $(E,\nabla)$ equal $((\phi^{i}_{j}))$ as sets. \end{theo} \noindent{\em Proof:} The flat bundle $H$, associated to $\chi$, has a global natural filtration. We work by induction on the rank $r$ and extend the claim of the theorem by the fact that the integer weights function $\phi_{j}$ on $H_{j}$, which we construct, is the direct sum of its restrictions to the generalised eigenspaces of $G_{j}$, acting on $H_{j}$. For $r=1$ we let $\phi_{j}:H_{j}-\{ 0\} \rightarrow \Z \,\,\,$ have single value $\phi_{j}^{1}$. This implies $\deg (H,H_{j}^{m},\Phi_{j})=\Lambda^{1}-\Lambda^{1}=0$ and hence $\mbox{$\cal F$} (H,H_{j}^{m},\Phi_{j})$ is Fuchsian. For $r \geq 2$ write $H' \subseteq H$, $H'_{j} \subseteq H_{j}$ for the rank $(r-1)$ subbundles. We are given, by induction, \[ \phi_{j}':H_{j}' \rightarrow % \{ \phi_{j}^{1},...,\phi_{j}^{r-1} \} \cup \{ +\infty \} \] with the above described property. Consider one $j \in\{1,...,n\}$ at a time. Let $A \subseteq H_{j}$ ($A' \subseteq H_{j}'$) be the generalised eigenspace of $G_{j}$ ($G'_{j}$) of the eigenvalue $\rho_{j}^{r}$. Also let $B' \subseteq H_{j}'$ be the direct sum of the other generalised eigenspaces of $G'_{j}$. Then, the extended induction hypothesis implies $ \phi'_{j}(h_{A'}+h_{B'})= \min (\phi_{j}' (h_{A'}),\phi'_{j})(h_{B'})).$ Put \function{\phi_{A}}{A}{\Z \cup \{ + \infty\}}{h}{\left\{ \begin{array}{ccc} \phi'_{j}(h) & \mbox{if} & h \in A' \\ \phi_{j}^{r} & \mbox{if} & h \in A-A'. \end{array} \right.} We give $H_{j}=A \oplus B'$ the direct sum of the weighted filtrations. By construction, $(H'_{j},\phi_{j}') \rightarrow (H_{j},\phi_{j})$ is an injection. We give $H_{j}/H'_{j}$ the integer weight $\phi_{j}^{r}$. Then $h=h_{A}+h_{B'} \in H_{j}$ maps to zero under $\alpha:H_{j} \rightarrow H_{j}/H'_{j}$ unless $h_{A} \in A-A'$ in which case \[ \phi(h)=\min (\phi_{A}(h_{A}),\phi'_{j} (h_{B'})) \leq \phi_{A}(h_{A}) = \phi_{j}^{r}. \] $\alpha$ is surjective since $\phi(h_{A})=\phi_{j}^{r}$ for any $h_{A}\in A-A'$. We have constructed a short exact sequence \[ \threehorbb{(H',\phi')}{(H,\phi)}{(H/H',\phi^{H'/H})} \] where $H'$ is the restriction of a Fuchsian system by induction and $H/H'$ is so similar to the rank one case. Apply Lemma \ref{cc} to finish. \\ $\Box$ The following result is due to Bolibruch when $r=3$, \cite[p 133]{ano}. He has a counter example for $r=7$, $n=4$, \cite[p 106]{ano}. \begin{coro} \showlabel{th} For $\chi: \pi_{1}(S,s) \rightarrow \mbox{B}\,(r,\C)$ and $r\in \{1,2,3,4\}$ there exists a Fuchsian system with monodromy $\chi$. \end{coro} \noindent{\em Proof:} We want to find $((\phi_{j}^{i}))$ satisfying (b) of Theorem \ref{p} and \\ {\bf (a)':} $\phi_{j}^{i} =\phi_{j}^{k}$ if $\rho_{j}^{i}=\rho_{j}^{k}$. Claim : If there is $m \in \{1,...,n\}$ and $k \in \{1,...,r\}$ such that $\rho_{m}^{k} \neq \rho_{m}^{t}$ for all $t \neq k$ then the problem to find $((\phi_{j}^{i}))$, satisfying (b) and (a)', reduces to rank $(r-1)$. To see this just find $((\phi_{j}^{i}))_{i\neq k}$, choose $(\phi_{j}^{k})_{j \neq m}$ if they are not fixed by (a)' already and calculate $\phi_{m}^{k}$ using (b). The corollary is trivial for $r=1$. For $r=2,3$, we can either use the claim or have $\Lambda^{1}=...=\Lambda^{r}$ and finish easily. For $r=4$, if we can not use the claim there are two cases. Either, for each $j=1,...,n$, ($\rho_{j}^{1}=\rho_{j}^{3}$ and $\rho_{j}^{2}=\rho_{j}^{4}$) or ($\rho_{j}^{1}=\rho_{j}^{4}$ and $\rho_{j}^{2}=\rho_{j}^{3}$). Hence, $\Lambda^{1}+\Lambda^{2} -\Lambda^{3}=\Lambda^{4}$. Solve the system consisting of (a)' and (b) for $((\phi_{j}^{i}))_{j=1,..,n;i=1,2,3}$ and get a solution for the rank four problem by setting $\phi_{j}^{4}:= \phi_{j}^{1}+\phi_{j}^{2}-\phi_{j}^{3}$ for $j=1,...,n$. Or, $\rho_{m}^{1}=\rho_{m}^{2}\neq \rho_{m}^{3}=\rho_{m}^{4}$ for some $m \in \{1,...,n\}$. Solve two rank two problems, i.e.\ find $((\phi_{j}^{i}))_{i=1,2}$ satisfying (a)' and (b) and find $((\phi_{j}^{i}))_{i=3,4}$ satisfying (a)' and (b). Increasing all entries of $((\phi_{j}^{i}))_{i=1,2;j\neq m}$ by a sufficiently large integer $N$ and decreasing $\phi_{m}^{1}$ and $\phi_{m}^{2}$ by $(n-1)N$, we can satisfy (a) and (b) for the rank four problem. \\ $\Box$ \begin{theo}[{\cite[p 90]{bol}}, {\cite[p 133]{ano}}] \showlabel{dr} A rank three representation $\chi:\pi_{1}(S,s) \rightarrow \mbox{Gl} \, (3,\C)$ is the monodromy of a Fuchsian system if and only if one or more of the following holds. \begin{description} \item[(a)] $\chi$ is irreducible. \item[(b)] Some $\chi(\gamma_{k})=G_{k}$ has more than one Jordan block. \item[(c)] The canonical extension of $H=H(\chi)$ has constant splitting type. \end{description} \end{theo} The part of the proof which is left we do differently from Bolibruch. \noindent{\em Proof:} By Corollary \ref{tb}, Corollary \ref{ss}, Lemma \ref{ct} and Proposition \ref{bt} we are left to show that if $\chi(\gamma_{k})$ has two Jordan blocks for some $k$ then $\chi$ is the monodromy of a logarithmic connection on $\Proj^{1} \times \C^{3}$. Let $h_{1} \in \C^{3}$ ($h_{2} \in \C^{3}$) be the eigenvector corresponding to the rank one (rank two) Jordan block of $\chi(\gamma_{k})$. Consider the $\pi_{1}(S,s)$-submodules \[ F_{1}:=\langle (\mbox{Im}\, \chi)(h_{1}) \rangle \,\,\,\,\, \mbox{and} \,\,\,\,\, F_{2}:= \langle (\mbox{Im}\, \chi )(h_{2}) \rangle \] of $\C^{3}$. If $\mbox{rank}\, F_{1}=1$ we are in case (i) of Theorem \ref{re} because of the positive solvability of the RHP in rank two. If $\mbox{rank}\, F_{1}=3$ then we are in the case of Proposition \ref{ca}. So assume $\mbox{rank}\, F_{1}=2$ and hence $h_{2} \in F_{2} \subseteq F_{1}$ and $\mbox{rank}\, F_{2} <3$. If $\mbox{rank}\, F_{2}=1$ we are in the case of Corollay \ref{th} and if $\mbox{rank}\, F_{2}=2$ we are in case (ii) of Theorem \ref{re} with $H'=F_{1}$, $H^{(0)}=\langle h_{2} \rangle$ and $H^{(3)}=\langle h_{1} \rangle$. \\ $\Box$

9504

alg-geom/9504007

en

https://arxiv.org/abs/alg-geom/9504007

alg-geom/9504007

Stein A. Stromme

Geir Ellingsrud, Joseph Le Potier, and Stein A. Stromme

Some Donaldson invariants of CP^2

8 pages. Submitted to the proceedings of the Dec. 94 Taniguchi Symposium "Vector Bundles". AMSLaTeX v 1.2 (latex2e, amsart, amscd)

We compute the Donaldson numbers $q_{17}(CP^2)=2540$ and $q_{21}(CP^2)=233208$.

alg-geom

\section*{Introduction} For an integer $n\ge2$, let $q_{4n-3}$ be the coefficient of the Donaldson polynomial of degree $4n-3$ of $P={\mathbf C}{\mathbf P}^2$. An interpretation of $q_{4n-3}$ in an algebro-geometric context is the following. Let $M_n$ denote the Gieseker-Maruyama moduli space of semistable coherent sheaves on $P$ with rank 2 and Chern classes $c_1=0$ and $c_2=n$. For such a sheaf $F$, the Grauert-M\"ulich theorem implies that the restriction of $F$ to a general line $L\subseteq P$ splits as $F_L \simeq {\mathcal O}_L\oplus{\mathcal O}_L$, and that the exceptional lines form a curve $J(F)$ of degree $n$ in the dual projective plane $P^{\vee}$. The association $F\mapsto J(F)$ is induced from a morphism of algebraic varieties, called the Barth map, $f_n\: M_n \to P_n$. Here $P_n={\mathbf P}^{n(n+3)/2}$ is the linear system parameterizing all curves of degree $n$ in $P^{\vee}$. Let $H\in\operatorname{Pic}(P_n)$ be the hyperplane class and let $\alpha = f_n^*H$. The interpretation of the Donaldson invariant is: \[ q_{4n-3} = \int_{M_n} \alpha^{4n-3}. \] Thus $q_{4n-3}$ is the degree of $f_n$ times the degree of its image. From \cite{Bart-2} it follows that $f_n$ is generically finite for all $n\ge2$, that $f_2$ is an isomorphism and $q_5=1$, and that $f_3$ is of degree 3 and $q_9=3$. Le Potier \cite{LePo} proved that $f_4$ is birational onto its image and that $q_{13}=54$. The value of $q_{13}$ has also been computed independently by Tikhomirov and Tyurin \cite[prop.~4.1]{Tyur-1} and by Li and Qin \cite[thm.~6.29]{Li-Qin}. The main result in the present note is the following \begin{thm} \label{thm1} $q_{17}=2540$ and $q_{21}=233208$. \end{thm} The proof consists of two parts. The first part, treated in this note, is to express $q_{4n-3}$ in terms of certain classes on the Hilbert scheme of length-$(n+1)$ subschemes of $P$. This is theorems \ref{thm2} and \ref{thm3} below. The second part is to evaluate these classes numerically. This has been carried out in \cite[prop.~4.2]{Elli-Stro-5}. Let $H_{n+1}=\operatorname{Hilb}^{n+1}_P$ denote the Hilbert scheme parameterizing closed subschemes of $P$ of length $n+1$. There is a universal closed subscheme ${\mathcal Z}\subseteq H_{n+1}\times P$. Consider the vector bundles \[ {\mathcal E} = R^1{p_1}_* ({\mathcal I}_{{\mathcal Z}}\*{p_2}^* {\mathcal O}_{P}(-1))\text{ and } {\mathcal G} = R^1{p_1}_* {\mathcal I}_{{\mathcal Z}} \] on $H_{n+1}$ of ranks $n+1$ and $n$, respectively, and the linebundle \[{\mathcal L} = \det({\mathcal G}) \* \det({\mathcal E})^{-1}.\] \begin{thm} \label{thm2} Let the notation be as above. Then \[ q_{17} = \int_{H_6} s_{12}({\mathcal E}\*{\mathcal L}) \quad\text{and}\quad q_{21} = \dfrac25 \int_{H_7} s_{14}({\mathcal E}\*{\mathcal L}). \] \end{thm} This result was obtained both by Tikhomirov and Tyurin \cite{Tyur-Tikh}, using the method of ``geometric approximation procedure'' and by Le Potier \cite{LePo-3}, using ``coherent systems''. We present in this note what we believe is a considerably simplified proof, which is strongly hinted at on the last few pages of \cite{Tyur-Tikh}. The formula for $q_{17}$ is a special case of the following formula: \begin{thm}\label{thm3} For $2\le n\le 5$, we have \[q_{4n-3} = \dfrac1{2^{5-n}}\int_{H_{n+1}} c_1({\mathcal L})^{5-n} s_{3n-3}({\mathcal E}\*{\mathcal L}). \] \end{thm} With this it is also easy to recompute $q_5$, $q_9$, and $q_{13}$ using similar techniques as in \cite{Elli-Stro-5}. \begin{notation} We let $h$, $h^{\vee}$, and $H$ be the hyperplane classes in $P$, $P^{\vee}$, and $P_n$, respectively. In general, if $\omega$ is a divisor class, we denote by ${\mathcal O}(\omega)$ the corresponding linebundle and its natural pullbacks. \end{notation} \begin{ack} This work is heavily inspired by conversations with A.~Tyurin, and we thank him for generously sharing his ideas. We would also like to express our gratitude towards the Taniguchi Foundation. \end{ack} \section{Hulsbergen sheaves} Barth \cite{Bart-2} used the term Hulsbergen bundle to denote a stable rank-2 vector bundle $F$ on $P$ with $c_1(F)=0$ and $H^0(P,F(1))\ne0$. We modify this definition a little as follows: \begin{defn} A \emph{Hulsbergen sheaf} is a coherent sheaf $F$ on $P$ which admits a non-split short exact sequence (\emph{Hulsbergen sequence}) \begin{equation} \label{Hulsbergen} 0 \to {\mathcal O}_P \to F(1) \to {\mathcal I}_Z(2) \to 0, \end{equation} where $Z\subseteq P$ is a closed subscheme of finite length (equal to $c_2(F)+1$). \end{defn} Note that a Hulsbergen sheaf is not necessarily semistable or locally free. However: \begin{lem}\label{GM} Let $F$ be a Hulsbergen sheaf with $c_2(F)=n>0$. Then the set $J(F)\subseteq P^{\vee}$ of exceptional lines for $F$ is a curve of degree $n$, defined by the determinant of the bundle map \[ m\: H^1(P,F(-2))\*{\mathcal O}_{P^{\vee}}(-1) \to H^1(P,F(-1))\*{\mathcal O}_{P^{\vee}} \] induced by multiplication with a variable linear form. \end{lem} \begin{pf} First note from the Hulsbergen sequence that the two co\-ho\-mo\-logy groups have dimension $n$. It is easy to see that any Hulsbergen sheaf is slope semistable, in the sense that it does not contain any rank-1 subsheaf with positive first Chern class. Thus by \cite[thm.~1]{Bart-1}, $F_L \simeq {\mathcal O}_L \oplus {\mathcal O}_L$ for a general line $L$. On the other hand, it is clear that a line $L$ is exceptional if and only if $m$ is not an isomorphism at the point $[L]\in P^{\vee}$. \end{pf} It is straightforward to construct a moduli space for Hulsbergen sequences. For any length-$(n+1)$ subscheme $Z\subseteq P$, the isomorphism classes of extensions \eqref{Hulsbergen} are parameterized by ${\mathbf P}(\operatorname{Ext}^1_P({\mathcal I}_Z(2),{\mathcal O}_P)^{\vee})$. By Serre duality, \[ \operatorname{Ext}^1_P({\mathcal I}_Z(2),{\mathcal O}_P)^{\vee} \simeq H^1(P,{\mathcal I}_Z(-1)). \] For varying $Z$, these vector spaces glue together to form the vector bundle ${\mathcal E}$ over $H_{n+1}$, hence $D_n={\mathbf P}({\mathcal E})$ is the natural parameter space for Hulsbergen sequences. Let ${\mathcal O}(\tau)$ be the associated tautological quotient linebundle. For later use, note that for any divisor class $\omega$ on $H_{n+1}$, we have $\pi_*(\tau+\pi^*\omega)^{k+n} = s_k({\mathcal E}(\omega))$, where $\pi\: D_n \to H_{n+1}$ is the natural map \cite{IT}. The tautological quotient $\pi^*{\mathcal E} \to {\mathcal O}(\tau)$ gives rise to a short exact sequence on $D_n\times P$: \[ 0 \to {\mathcal O}(\tau) \to {\mathcal F}(h) \to (\pi\x1)^*{\mathcal I}_{{\mathcal Z}}(2h) \to 0 \] which defines a complete family ${\mathcal F}$ of Hulsbergen sheaves. As we noted earlier, a Hulsbergen sheaf is not necessarily semistable. On the other hand, the \emph{generic} Hulsbergen sheaf is stable if $n\ge 2$. It follows that the family ${\mathcal F}$ induces a \emph{rational} map $g_n\: D_n \to M_n$. By \lemref{GM} above, there is also a Barth map $b_n\: D_n \to P_n$, defined everywhere, and by construction, the following diagram commutes: \begin{equation} \begin{CD} D_n @>b_n>> P_n \\ @V{g_n}VV @VV{||}V \\ M_n @>>f_n> P_n \end{CD} \end{equation} \begin{prop} Put $\lambda=c_1(\pi^*{\mathcal L})$. Then $b_n^*H = \tau+\lambda$. \end{prop} \begin{pf} Let $L\subseteq P$ be a line. Twist the universal Hulsbergen sequence by $-2h$ and $-3h$ respectively. Multiplication by an equation for $L$ gives rise to the vertical arrows in a commutative diagram with exact rows on $D_n\times P$: \[ \begin{CD} 0 @>>> {\mathcal O}(\tau-3h) @>>> {\mathcal F}(-2h) @>>> (\pi\x1)^*{\mathcal I}_Z(-h) @>>>0 \\ @. @VVV @VVV @VVV @.\\ 0 @>>> {\mathcal O}(\tau-2h) @>>> {\mathcal F}(-h) @>>> (\pi\x1)^*{\mathcal I}_Z @>>>0 \end{CD} \] Pushing this down via the first projection, we get the following exact diagram on $D_n$: \[\begin{CD} 0@>>>R^1{p_1}_* {\mathcal F}(-2h) @>>> \pi^*{\mathcal E} @>>> {\mathcal O}(\tau)@>>> 0\\ @. @Vm_LVV @VVV @VVV\\ 0@>>> R^1{p_1}_* {\mathcal F}(-h) @>\simeq>> \pi^*{\mathcal G} @>>> 0 \end{CD} \] Here the last map of the top row is nothing but the tautological quotient map on ${\mathbf P}({\mathcal E})$. Let $A(L)\subseteq D_n$ be the set of Hulsbergen sequences where $L$ is an exceptional line for the middle term. Clearly, $A(L)$ is the degeneration locus of the left vertical map $m_L$ above. Hence the divisor class of $A(L)$ is \[ \begin{aligned} [A(L)]&= c_1(R^1{p_1}_* {\mathcal F}(-h)) - c_1(R^1{p_1}_* {\mathcal F}(-2h)) \\ &= \pi^*c_1({\mathcal G}) - \pi^*c_1({\mathcal E}) + \tau \\ &= \tau+\lambda. \end{aligned} \] On the other hand, $A(L)$ is the inverse image of a hyperplane in $P_n$ under $b_n$, so its divisor class is $b_n^*H$. \end{pf} \section{The case $n\le 5$} \begin{prop} For $2\le n\le 5$, the rational map $g_n$ is dominating, and the general fiber is isomorphic to ${\mathbf P}^{n-5}$. For $n\ge 5$, the map $g_n$ is generically injective with image of codimension $n-5$. In particular, $g_5$ is birational. \end{prop} \begin{pf} Everything follows from the observation that the fiber over a point $[F]\in M_n$ in the image of $g_n$ is the projectivization of $H^0(P,F(1))$, and that for general such $F$, this vector space has dimension $h^0(F(1))=\max(1,6-n)$, which is easily seen from \eqref{Hulsbergen}. The assertion about the codimension follows from a dimension count: $\dim(M_n)=4n-3$ and $\dim(D_n)=3n+2$. \end{pf} The first half of \thmref{thm2} now follows: First of all, since $g_5$ is birational, the two morphisms $f_5$ and $b_5$ have the same image and the same degree. Therefore $q_{17}$ can be computed as \[ q_{17} = \int_{D_5} H^{17} =\int_{D_5} (\tau+\lambda)^{17} = \int_{H_6} s_{12}({\mathcal E}\*{\mathcal L}). \] For \thmref{thm3}, let $L_1,\dots,L_{5-n}$ be general lines in $P$, and let $B_n\subseteq D_n$ be the locus of Hulsbergen sequences where the closed subscheme $Z$ meets all these $5-n$ lines. The cohomology class of $B_n$ in $H^*(D_n)$ is $\lambda^{5-n}$. \begin{lem} \label{cover} Let $2\le n\le5$. The general nonempty fiber of $g_n$ meets $B_n$ in $2^{5-n}$ points, hence the rational map $g_n|_{B_n}\: B_n \to M_n$ is dominating and generically finite, of degree $2^{5-n}$. \end{lem} \begin{pf} The general nonempty fiber is of the form ${\mathbf P}(H^0(P,F(1))^{\vee})$. It suffices to show that the restriction of ${\mathcal L}$ to this fiber has degree 2 (if $n<5$). For this, it suffices to consider a linear pencil in the fiber. So let $\sigma_0$ and $\sigma_1$ be two independent global sections of $F(1)$, and consider the pencil they span. Now $\sigma_0\wedge \sigma_1 \in H^0(P,\wedge^2F)=H^0(P,{\mathcal O}_P(2))$ is the equation of a conic $C\subseteq P$ which contains the zero scheme $V(t_0\sigma_0 + t_1\sigma_1)$ of each section in the pencil, $(t_0,t_1)\in{\mathbf P}^1$. Since $C$ meets a general line in two points, it follows that there are exactly two members of the pencil whose zero set meets a general line. \end{pf} To complete the proof of \thmref{thm3}, by \lemref{cover} we now have for $2\le n\le5$: \[\begin{aligned} 2^{5-n}\,q_{4n-3} &= 2^{5-n}\int_{M_n} H^{4n-3} \\ &=\int_{B_n} (\tau+\lambda)^{4n-3} \\ &=\int_{D_n} \lambda^{5-n}\,(\tau+\lambda)^{4n-3} \\ &=\int_{H_{n+1}}c_1({\mathcal L})^{5-n} \,s_{3n-3}({\mathcal E}\*{\mathcal L}). \end{aligned} \] This completes the proof of the theorems for $n\le 5$. \section{The case $n=6$} For $n\ge6$ the techniques above will say something about the restriction of the Barth map to the Brill-Noether locus $B\subseteq M_n$ of semistable sheaves whose first twist admit a global section. For general $n$ this locus is too small to carry enough information about $M_n$, but in the special case $n=6$, it is actually a divisor, whose divisor class $\beta=[B]$ we can determine. Now $\operatorname{Pic}(M_n)\*{\mathbf Q}$ has rank 2, generated by $\alpha$ and $\delta=[\Delta]$, the class of the locus $\Delta\subseteq M_n$ corresponding to non-locally free sheaves \cite{LePo-1}. \begin{prop} In $\operatorname{Pic}(M_6)\*{\mathbf Q}$, the following relation holds: \[ \beta = \frac52 \,\alpha - \frac12\,\delta. \] \end{prop} \begin{pf} Let $\xi\:X\to M_6$ be a morphism induced by a flat family ${\mathcal F}$ of semistable sheaves on $P$, parameterized by some variety $X$. For certain divisor classes $a$ and $d$ on $X$, the second and third Chern classes of ${\mathcal F}$ can be written in the form \[ c_2({\mathcal F}) = a\,h+6\,h^2, \quad c_3({\mathcal F}) = d\,h^2 \] modulo higher codimension classes on $X$. The Grothendieck Riemann-Roch theorem for the projection $p\: X\times P \to X$ easily gives (for example using \cite{schubert}) that \[ -c_1(p_!{\mathcal F}(h)) = \frac52\, a- \frac12\, d. \] The locus $\xi^{-1} B\subseteq X$ is set-theoretically the support of $R^1 p_*{\mathcal F}(h)$. It is not hard to see that one can take the family $X$ in such a way that the 0-th Fitting ideal of $R^1 p_*{\mathcal F}(h)$ is actually reduced. Therefore the left hand side of the equation above is $\xi^*\beta$. On the other hand, $a=\xi^*\alpha$ by the usual definition of the $\mu$ map of Donaldson \cite{Dona-1}, and $d=\xi^*\delta$. Since the family ${\mathcal F}/X$ was arbitrary, the required relation is actually universal, and so holds also in $\operatorname{Pic}(M_6)\*{\mathbf Q}$. (It suffices to take a family with the properties that (i) $\xi^*\:\operatorname{Pic}_{\mathbf Q}(M_6) \to \operatorname{Pic}_{\mathbf Q}(X)$ is injective, (ii) the Fitting ideal above is reduced, and (iii) the general non-locally free sheaf in the family has colength 1 in its double dual.) \end{pf} With this, we complete the proof of the second part of \thmref{thm2} in the following way. The general fiber of $f_6$ restricted to $\Delta$ has dimension 1, so $f_6(\Delta)$ has dimension 19, see e.g.~\cite{Stro-1}. Therefore we get \[\begin{aligned} \int_{H_7} s_{14}({\mathcal E}\*{\mathcal L}) &= \int_{D_6}(\lambda+\tau)^{20} \\ &= \int_{M_6} \beta\, \alpha^{20} \\ &= \int_{M_6} (\frac52\, \alpha - \frac12\,\delta)\,\alpha^{20} \\ &= \frac52\int_{M_6} \alpha^{21} -\frac12\int_{\Delta}\alpha^{20} = \frac52\, q_{21}. \end{aligned} \] \section{A geometric interpretation} \begin{defn} A \emph{Darboux configuration} in $P^{\vee}$ consists of a pair $(\Pi,C)$ where $\Pi\subseteq P^{\vee}$ is the union of $n+1$ distinct lines, no three concurrent, and $C\subseteq P^{\vee}$ is a curve of degree $n$ passing through all the nodes of $\Pi$. \end{defn} If we let $Z\subseteq P$ consist of the $n+1$ points dual to the components of $\Pi$, we have by Hulsbergen's theorem \cite[thm.~4]{Bart-2} a natural 1-1 correspondence between Hulsbergen sequences \eqref{Hulsbergen} and Darboux configurations $(\Pi,C)$, by letting $C=J(F)$. Therefore $D_n$ can be used as a compactification of the set of Darboux configurations, and the intersection number \[ \int_{D_n} \lambda^i (\tau+\lambda)^{3n+2-i} = \int_{H_{n+1}} c_1({\mathcal L})^i s_{2n+2-i}({\mathcal E}\*{\mathcal L}) \] can be interpreted as the number of Darboux configurations $(\Pi,C)$ where $\Pi$ passes through $i$ given points and $C$ passes through $3n+2-i$ given points. It is not known whether the Barth map has degree 1 for $n\ge5$. A related question is the following: Let $(\Pi,C)$ be a general Darboux configuration ($n\ge 5$). Is the inscribed polygon $\Pi$ uniquely determined by $C$?

9504

alg-geom/9504003

en

https://arxiv.org/abs/alg-geom/9504003

alg-geom/9504003

Rahul Pandharipande

J. Harris and R. Pandharipande

Severi Degrees in Cogenus 3

AMSLaTex 12 pages

In this short note, a new computation of the degree of the locus of 3-nodal plane curves in the linear system of degree d plane curves is given. The answer is expressed as a tautological class on a blow-up of the Hilbert scheme of 3 points in the plane. The class is evaluated by the Bott residue formula.

alg-geom

\section{{\bf Introduction}} \subsection{Summary} Denote by $\cal{P} (d)$ the linear system of degree $d$ curves in the complex projective plane $\Bbb P ^2$. $\cal{P} (d)$ is a projective space of dimension ${d+2 \choose 2}-1$. Let $\cal{N}(n,d) \subset \cal{P} (d)$ be the subset corresponding to reduced, nodal curves with exactly $n$ nodes. $\cal{N}(n,d)$ is empty unless $0\leq n \leq {d \choose 2}$. Points of $\cal{N}(n,d)$ may correspond to reducible curves. If nonempty, $\cal{N}(n,d)$ is a quasi-projective subvariety of pure codimension $n$ in $\cal{P} (d)$. Let $\overline{\cal{N}}(n,d)$ denote the closure of $\cal{N}(n,d)$ in $\cal{P} (d)$. In this paper, formulas for the the degree of $\overline{\cal{N}}(n,d)$ for $n=1,2,3$ are computed: \begin{equation} \label{one} f_1(d)= 3(d-1)^2, \end{equation} \begin{equation} \label{two} f_2(d)= {3 \over 2} (d-1)(d-2)(3d^2-3d-11), \end{equation} \begin{equation} \label{three} f_3(d)= {9 \over 2} d^6 - 27d^5+ {9 \over 2} d^4 + {423 \over 2 } d^3 - 229 d^2 - {829 \over 2 } d +525, \end{equation} $$\forall d \geq 1, \ \ degree(\overline{\cal{N}}(1,d))=f_1(d)$$ $$\forall d\geq 3, \ \ degree(\overline{\cal{N}}(2,d))= f_2(d)$$ $$\forall d\geq 3, \ \ degree(\overline{\cal{N}}(3,d))= f_3(d).$$ These formulas are classical. The computation presented here is new. The method involves the geometry of the Hilbert scheme of points in $\Bbb P^2$ and the Bott residue formula (following the technique developed in [E-S]). The most successful methods for obtaining cogenus formulas appear in [V]. Via a sophisticated singularity analysis, I. Vainsencher obtains the above results and the following further cogenus formulas: \begin{eqnarray*} f_4(d) & = & {27\over 8}d^8-27 d^7+{1809\over 4} d^5-642 d^4-2529 d^3 +{37881\over 8} d^2+{18057\over 4}d-8865, \\ f_5(d) & = & {81\over 40} d^{10}- {81\over 4} d^9 - {27\over 8}d^8+ {2349\over 4}d^7-1044 d^6-{127071\over 20} d^5 +{128859\over 8} d^4 \\ & & +{59097\over 2}d^3-{3528381\over 40}d^2 -{946929\over 20}d+153513, \\ f_6(d) & = & {81\over 80} d^{12}-{243\over 20}d^{11} -{81\over 20} d^{10} +{8667\over 16}d^9 -{9297\over 8} d^8 -{47727\over 5} d^7 \\ & & + {2458629\over 80} d^6 +{3243249\over 40}d^5 -{6577679\over 20}d^4-{25387481\over 80} d^3 +{6352577\over 4} d^2 \\ & & +{8290623\over 20}d -2699706. \\ \end{eqnarray*} The idea for the computation presented here originated in a conversation at the October 1994 Utah-UCLA-Chicago Algebraic Geometry Workshop. Discussions with W. Fulton were helpful. The second author benefitted from conversations with D. Edidin and W. Graham. Thanks are due to S. Stromme for pointing out I. Vainsencher's results. \subsection{The Method} \label{method} For each $n\geq 1$, let $H(n)$ be the Hilbert scheme of length $n$ subschemes of $\Bbb P ^2$. $H(n)$ is a nonsingular variety of dimension $2n$ with generic element corresponding to a subscheme of $n$ distinct points of $\Bbb P ^2$. There exists a rational map $$\psi_n: H(n) \ - \ - \rightarrow H(3n)$$ given by squaring the ideal sheaf. For $1\leq n \leq 3$, we will consider resolutions $$\overline{\psi}_n: X(n) \rightarrow H(3n)$$ of $\psi_n$. It is easily checked that $\psi_1$ and $\psi_2$ are everywhere defined. Let $X(1)=H(1)$, $\overline{\psi}_1=\psi_1$ and $X(2)=H(2)$, $\overline{\psi}_2=\psi_2$. Let $F \hookrightarrow H(3)$ be the locus of length $3$ subschemes isomorphic to $\Bbb C[x,y]/(x^2, xy, y^2)$. $F$ is a nonsingular subvariety abstractly isomorphic to $\Bbb P^2$. Since $(x^2,xy,y^2)^2$ is an ideal of length $10$, $\psi_3$ is not defined on $F$. Let $X(3)$ be the blow-up of $H(3)$ along $F$. In section (\ref{resmap}), it is shown that $\psi_3$ is defined on $H(3) \setminus F$ and extends to a morphism $$\overline{\psi}_3: X(3) \rightarrow H(9).$$ The degrees of $\overline{\cal{N}}(1,d)$, $\overline{\cal{N}}(2,d)$, and $\overline{\cal{N}}(3,d)$ will be expressed as Chern classes of certain tautological bundles over $X(1)$, $X(2)$, and $X(3)$ respectively. Let $U(n)\hookrightarrow H(n) \times \Bbb P^2 $ be the universal subscheme over $H(n)$. For $1\leq n \leq 3$, let $$Y(n)= X(n) \times_{H(3n)} U(3n) \hookrightarrow X(n) \times \Bbb P^2.$$ Let $\pi_n$, $\rho_n$ be the projections from $Y(n)$ to $X(n)$, $\Bbb P ^2$ respectively. For pairs $(n,d)$ where $1\leq n \leq 3$ and $\cal{N}(n,d)\neq \emptyset$, let $$E(n,d)= \pi_{n*} \rho_n^*({\cal{O}}_{\Bbb P^2}(d)).$$ $E(n,d)$ is easily seen to be a rank $3n$ vector bundle on $X(n)$. We claim $$c_{2n}(E(n,d))= degree(\overline{\cal{N}}(n,d)).$$ A sketch of the argument is as follows. Let $1\leq n \leq 3$ and let $d$ be such that $\cal{N}(n,d)\neq \emptyset$. Since $\overline{\cal{N}}(n,d)$ is of codimension $n$ in $\cal{P} (d)$, the degree is the cardinality of a generic $n$-plane slice. An $n$-plane, $\cal{L}\subset \cal{P} (d)$, is equivalent to an $n+1$-dimensional linear subspace of $L\subset H^0(\Bbb P^2,{\cal{O}}_{\Bbb P^2}(d))$. $L$ canonically yields an $n+1$ dimensional subspace $\overline{L}\subset H^0(X(n),E(n,d))$. A generic point $\xi \in X(n)$ corresponds to $n$ points of $\Bbb P^2$. It is checked that $\overline{L}$ drops rank at $\xi$ if and only if there exists an element of $\cal{L}$ corresponding to a plane curve singular at the $n$ points of $\xi$. It is further checked, for generic $\cal{L}$, the singular plane curve must be reduced, nodal with exactly $n$ nodes. The nongeneric points of $X(n)$ make no contribution to the degeneracy locus. The degree of $\overline{\cal{N}}(n,d)$ is thus equal to the cardinality of the degeneracy locus of $n+1$ sections of $E(n,d)$. The latter is the $c_{2n}(E(n,d))$. Section (\ref{loci}) contains the full argument. In principle, this approach may be attempted for $n\geq4$. Explicit resolutions of $\psi_n$ are needed. For $n\geq 4$, a correction term to $c_{2n}(E(n,d))$ for small $d$ is required to account for the contribution of nonreduced curves to the degeneracy locus. It remains to compute $c_{2n}(E(n,d))$. There is a diagonal torus action on $\Bbb P^2$ which can be lifted to $X(n)$ and $E(n,d)$. The Bott residue formula expresses the desired Chern class in terms of the differential data of the torus action at fixed points. This approach yields the degree formulas. It should be mentioned there are more direct ways of obtaining formulas (\ref{one}) and (\ref{two}). Since the residue calculations for formula (\ref{three}) contain those required for (\ref{one}) and (\ref{two}), we present a unified approach. The explicit residue computations are presented in section (\ref{bott}) \section{\bf $\psi_n$ And $\overline{\psi}_n$ For $n=1,2,3$} \label{resmap} Consider the universal subschemes $U(n) \hookrightarrow H(n) \times \Bbb P^2$. Let $U^2(n)$ be subscheme of $H(n)\times \Bbb P^2$ defined by the square of the ideal of $U(n)$. Let $G(n)\subset H(n)$ denote the generic locus corresponding to subschemes of $n$ distinct points of $\Bbb P^2$. $U^2(n)$ is flat over $G(n)$ of degree $3n$. Therefore there is map $$\psi_n: G(n) \rightarrow H(3n).$$ Certainly $G(1)=H(1)$. For $n=2$, the complement of $G(2)$ in $H(2)$ consists of linear double points (with ideals isomorphic to $(x,y^2)\subset \Bbb C[x,y]$). Since $(x,y^2)^2=(x^2, xy^2,y^4)$ has length $6$, $U^2(2)$ is flat over $H(2)$. The map $\psi_2$ extends to $H(2)$. For $n=3$, the situation is more complex. By the results for $n=2$, $\psi_3$ extends to all of $H(3)$ with the possible exception of the the triple points. The isomorphism classes of length $3$ subschemes of $\Bbb C^2$ supported at a point are given by the following ideals: \begin{enumerate} \item[(i.)] $(x,y^3)\subset \Bbb C[x,y]$. \item[(ii.)] $(x+y^2, x^2, xy) \subset \Bbb C[x,y]$. \item[(iii.)] $(x^2,xy,y^2) \subset \Bbb C[x,y]$. \end{enumerate} It is easy to check the squares of the ideas of type (i) and (ii) have length 9. $(x^2,xy,y^2)^2=(x^4,x^3y,x^2y^2,xy^3,y^4)$ has length 10. Let $F\hookrightarrow H(3)$ be the nonsingular subscheme corresponding to the points of type (iii). $U^2(3)$ is flat over $H(3) \setminus F$. Therefore $\psi_3$ extends to $\psi_3: H(3)\setminus F \rightarrow H(9).$ Let $V\subset \Bbb P^2$ be a coordinate affine chart. $V$ is a two dimensional complex vector space. Let $$A= \bigoplus_{k=0}^{\infty} Sym^k(V^*)$$ be the affine coordinate ring of $V$. Let $m\subset A$ be the maximal ideal corresponding to the point $0\in V$. The ideal $m^2$ is of type (iii). Let $[A/m^2]\in H(3)$ denote the Hilbert point corresponding to $A/m^2$. The tangent space to $H(3)$ at $[A/m^2]$ is canonically isomorphic the module of $A$-homomorphisms $Hom_A(m^2/m^4, A/m^2)$. It is easily seen there are canonical isomorphims $$Hom_A(m^2/m^4, A/m^2) \stackrel {\sim}{=} Hom_{\Bbb C}(m^2/m^3, m/m^2) \stackrel {\sim}{=} Hom_{\Bbb C}(Sym^2(V^*), V^*).$$ $V$ is canonically identified with the space of invariant vector fields on $V$. Therefore, there is a canonical map $$\mu: V \rightarrow Hom_{\Bbb C}(Sym^2(V^*), V^*)$$ given by differentiation of functions. Certainly, $[A/m^2]\in F$. The tangent space to $F$ at $[A/m^2]$ is canonically isomorphic to $V$. The map $\mu$ is the differential of the inclusion of $F$ in $H(3)$ at $[A/m^2]$. Consider the exact sequence \begin{equation*} 0\rightarrow K \rightarrow Sym^2(Sym^2(V^*)) \rightarrow Sym^4(V^*) \rightarrow 0 \end{equation*} given by multiplication. $K$ is a one dimension $\Bbb C$-vector space. There is a canonical map $\nu$ and an exact sequence: \begin{equation} \label{xact} 0 \rightarrow V \stackrel{\mu}{\rightarrow} Hom_{\Bbb C}(Sym^2(V^*), V^*) \stackrel{\nu}{\rightarrow} Hom_{\Bbb C}(K, Sym^3(V^*)) \rightarrow 0. \end{equation} Briefly, an element of $\gamma \in Hom_{\Bbb C}(Sym^2(V^*), V^*)$ yields a map $id+ \gamma: Sym^2(V^*) \rightarrow A$. Multiplication of $id+\gamma$ induces a map $Sym^2(Sym^2(V^*)) \rightarrow A$. The latter map takes $K$ to $Sym^3(V^*)$. The exactness of (\ref{xact}) is a simple exercise. Let $X(3)$ be the blow up of $H(3)$ along $F$. By sequence (\ref{xact}), there is a natural correspondence between the fiber of the projective normal bundle of $F$ in $H(3)$ at $[A/m^2]$ and the projective space $\Bbb P( Hom_{\Bbb C}(K, Sym^3(V^*)))$. The map $\psi_3$ can be extended to the projective normal bundle of $F$ by mapping an element $[\xi] \in \Bbb P(Hom_{\Bbb C}(K,Sym^3(V^*)))$ to the ideal of length $9$ given by $(m^4, image(\xi))\subset A$. We have defined a map $$\overline{\psi}_3: X(3) \rightarrow H(9).$$ It is not hard to check that $\overline{\psi}_3$ is an algebraic morphism. \section{\bf $E(n,d)$, Degeneracy Loci, and $degree(\overline{\cal{N}}(n,d))$.} \label{loci} Let $n=1,2,$ or $3$. Let $d$ be an integer such that $\cal{N}(n,d)$ is nonempty. Following the notation of section (\ref{method}), $$E(n,d)=\pi_{n*} \rho_n^*({\cal{O}}_{\Bbb P^2}(d)).$$ Let $\cal{L}\subset \cal{P} (d)$ be an $n$-plane corresponding to an $n+1$ dimensional subspace $L\subset H^0(\Bbb P^2,{\cal{O}}_{\Bbb P^2}(d))$. Let $\overline{L}$ denote the naturally induced $n+1$ dimensional subspace of $H^0(X(n), E(n,d))$. For any $\xi\in X(n)$, let $\overline{L}_{\xi} \subset E(n,d)_{\xi}$ be the subspace of the fiber generated by $L$. Let $I_{\xi}\subset {\cal{O}}_{\Bbb P^2}$ denote the ideal sheaf of subscheme corresponding to $\overline{\psi}_n(\xi)$. \begin{lm} \label{taut} Let $[\xi]\in X(n)$. Then, $dim(\overline{L}_{\xi}) < n+1$ if and only if there exists a nonzero $l \in L$ such that $l\in H^0(\Bbb P^2, I_{\xi}(d))$. \end{lm} \noindent Lemma (\ref{taut}) is a tautological statement. We abuse notation slightly to let $G(n)\subset X(n)$ denote the locus of points $\xi \in X(n)$ such that $\overline{\psi}_n(\xi)$ corresponds to a subscheme of $n$ distinct points of $\Bbb P^2$. \begin{lm} \label{nobound} For generic $\cal{L}$, the following holds: $$\forall \xi \in X(n)\setminus G(n), \ \ dim(\overline{L}_{\xi})=n+1.$$ \end{lm} \begin{pf} For $n=1$, the result is vacuous. For $n=2$, $d$ must be at least $3$ to ensure $\cal{N}(2,d) \neq \emptyset$. $X(2)\setminus G(2)$ consists of the $3$ dimensional locus of linear double points (with ideals isomorphic to $(x, y^2)\subset \Bbb C[x,y]$). The squared ideal $(x^2, xy^2, y^4)$ imposes 6 conditions on linear series of degree $d \geq 3$. Hence, the locus of elements $[l] \in \cal{P} (d)$ such that $l \in H^0(\Bbb P^2, I_{\xi}(d))$ for some $\xi \in X(2) \setminus G(2)$ is of codimension at least $6-3=3$. A generic $2$-plane has empty intersection with this locus. For $n=3$, $d$ must again be at least $3$ to ensure $\cal{N}(3,d)\neq \emptyset$. $X(3)\setminus G(3)$ consists of three loci: \begin{enumerate} \item [(a.)] The projective normal bundle of $F$, $\Bbb P(N_F)$. \item [(b.)] The quasi-projective locus, $B$, corresponding to ideals of type (i) and (ii) of section (\ref{resmap}). \item [(c.)] The quasi-projective locus, $C$, corresponding to a point $p$ union a linear double point supported at $q\neq p$. \end{enumerate} The ideal $I_{\xi}$ for any $\xi$ in $\Bbb P(N_F)$ certainly imposes $9$ conditions on linear series of degree $d\geq 3$. The dimensions of $\Bbb P(N_F)$ is $5$. The locus of elements $[l] \in \cal{P} (d)$ such that $l\in H^0(\Bbb P^2, I_{\xi}(d))$ for some $\xi \in \Bbb P(N_F)$ is codimension at least $9-5=4>3$. $B$ consists of a $3$ dimensional locus of ideals of type (i) and a $4$ dimensional locus of ideals of type (ii). Ideals of type (i) and (ii) are easily seen to impose $7$ and $9$ conditions respectively on linear series of degree $d\geq 3$. We see $7-3=4>3$ and $9-4=5>3$. If $J$ is the ideal of a linear double point supported at $q\in \Bbb P^2$, $J$ imposes $6$ conditions on linear series of degree $d\geq 3$. The generic element of $H^0(\Bbb P^2, J(d))$ is singular only at $q$. The condition that $l\in H^0(\Bbb P^2, J(d))$ be singular at some point in $\Bbb P^2 \setminus \{ q \}$ is therefore a codimension $1$ condition on $H^0(\Bbb P^2, J(d))$. The locus of elements $[l] \in \cal{P} (d)$ such that $l\in H^0(\Bbb P^2, I_{\xi}(d))$ for some $\xi \in T$ is therefore of codimension at least $7-3=4>3$. A generic $3$-plane in $\cal{P} (d)$ avoids the loci corresponding to subsets (a), (b), and (c) of $X(3) \setminus G(3)$. \end{pf} \begin{lm} \label{nsing} Let $Sing(n,d)\subset \cal{P}(d)$ be the quasi-projective locus of reduced curves with at least $n$ singular points. Then $Sing(n,d)\subset \overline{\cal{N}}(n,d)$. \noindent (This result holds for all $n$.) \end{lm} \begin{pf} Let $[C]\in Sing(n,d)$. We must show $[C]\in \overline{\cal{N}}(n,d)$. Since $\overline{\cal{N}}(m,d)\subset \overline{\cal{N}}(n,d)$ for $m>n$, we can reduce to the case were $C$ has exactly $n$ nodes. Let $\sum$ be the $n$ singular points of $C$. The projective tangent space to $Sing(n,d)$ at $[C]$ is given by the linear system of degree $d$ curves passing through $\sum$. By a study of the adjoint conditions ([ACGH], p.60), $\sum$ imposes $n$ independent conditions on degree $d$ curves. It also follows from the adjoint analysis that any subideal of $I(\sum)$ of index $1$ imposes independent conditions on degree $d$ curves. Since the condition of having exactly $n$ nodes is open in $Sing(n,d)$, $Sing(n,d)$ is nonsingular of codimension $n$ at $[C]$. Let $f(x,y)$ be the equation of a plane curve singularity at the origin $(0,0)$. Consider first order, {\em equisingular} deformations of the type $f(x,y)+\epsilon \cdot g(x,y)$. It is a fact ([fact]) that such $g(x,y)$ generate the the maximal ideal of $(0,0)$ if and only if the singularity of $f(x,y)$ at the origin is a node. Suppose $[C]$ is not nodal. The equisingular deformations of $[C]$ correspond at most to the linear system of degree $d$ curves passing through a subideal of $I(\sum)$ of index $1$. Hence the these equisingular deformations are of codimension at least $n+1$. Therefore, the generic member of each component must be nodal. \end{pf} \noindent Finally, a simple dimension analysis yields: \begin{lm} \label{reduced} Let $Nonred(d)\subset \cal{P} (d)$ be the closed locus of nonreduced curves. For all $d$, the codimension of $Nonred(d)$ is greater than $2$. For $d\geq 3$, the codimension of $Nonred(d)$ is greater than $3$. \end{lm} Let $\cal{L}\subset \cal{P}(d)$ be a generic $n$-plane. By Lemma (\ref{nsing}) the following sequence of inclusions hold: $$\cal{N}(n,d) \subset Sing(n,d) \subset \overline{\cal{N}}(n,d).$$ Therefore, we obtain \begin{equation} \label{nodes} \cal{L}\cap \cal{N}(n,d)= \cal{L}\cap Sing(n,d)= \cal{L}\cap \overline{\cal{N}}(n,d). \end{equation} The intersection is transverse and, \begin{equation*} |\cal{L}\cap {\cal{N}}(n,d) | = degree(\overline{\cal{N}}(n,d)). \end{equation*} The degeneracy locus $D$ of $\overline{L}$ is the subscheme where $\overline{L}$ drops rank. By Lemma (\ref{nobound}), $D$ is supported in $G(n)$. If $n=1,2,$ or $3$ and $d$ is such that $\cal{N}(n,d)\neq \emptyset$, $E(n,d)$ is generated on $G(n)$ by sections induced from $H^0(\Bbb P^2, {\cal{O}}_{\Bbb P^2}(d))$. (Note $E(n,d)$ need not be generated by these sections on $X(n)$.) Therefore, the canonical sections yield a map $$\lambda_n(d): G(n) \rightarrow Grassmanian(n,d).$$ The degeneracy locus $D$ on $G(n)$ is expressed as the $\lambda_n(d)$-intersection with a Schubert class determined by $\overline{L}$. By Kleiman's result on intersections in homogeneous spaces ([H], p.273), $D$ is a reduced dimension zero subscheme of $G(n)$ for generic $\cal{L}$. If $\xi \in D\subset G(n)$ is a point, by Lemma (\ref{taut}) there is a curve $[l]\in \cal{L}$ singular at the $n$ points of $\Bbb P^2$ corresponding to $\xi$. $[l]$ must be reduced by Lemma (\ref{reduced}). $[l]$ therefore corresponds to a point of the intersection (\ref{nodes}) and must be nodal with exactly $n$ nodes. $[l]$ must be unique or else there would be a pencil of curves of $\cal{L}$ singular at the $n$ points corresponding to $\xi$. We have defined a set map $\tau: D \rightarrow \cal{L}\cap \cal{N}(n,d)$. Since $D$ can be recovered from the nodes of $[l]$, $\tau$ is injective. By Lemma (\ref{taut}), $\tau$ is surjective. Hence $ |D| =| \cal{L} \cap \cal{N}(n,d)|$. By the Thom-Porteous formula, $|D|= c_{2n}(E(n,d))$. \begin{pr} \label{chern} For $n=1,2,3$, $degree(\overline{\cal{N}}(n,d))= c_{2n}(E(n,d))$. \end{pr} \section{\bf The Bott Residue Computation} \label{bott} \subsection{The Formula} We first state the form of the Bott Residue Formula ([B]) that will be used. Let $M$ be a nonsingular variety of dimension $m$ with an algebraic $\Bbb C^*$-action. Let $q\in M$ be a fixed point of the $\Bbb C^*$-action. The differential of the action naturally induces a $\Bbb C^*$-representation on the tangent space $T_q(M)$. Let $\alpha_1(q), \ldots, \alpha_m(q)$ be the $m$ weights of the $\Bbb C^*$-representation on $T_q(M)$. Let $\cal{Q}\subset M$ be the fixed point set. Assume \begin{enumerate} \item[(1.)] $\cal{Q}$ is discrete. \item[(2.)] $\forall q \in \cal{Q}$ and $\forall j$, $\alpha_j(q) \neq 0$. \end{enumerate} In fact, condition (2) is a consequence of condition (1). Suppose $E$ is an algebraic vector bundle of rank $r$ on $M$ with an equivarient $\Bbb C^*$-action. For each $q\in \cal {Q}$, there is a $\Bbb C^*$-representation on $E_q$. Let $\beta_1(q)$, $\ldots$, $\beta_r(q)$ be the weights of this $\Bbb C^*$-representation. Finally, let $\sigma_{i,j}(x_1, x_2, \ldots, x_j)$ be the $i^{th}$ elementary symmetric polynomial in the variables $x_1$, $x_2$, $\ldots$, $x_j$. If $i>j$, $\sigma_{i,j}=0$. The Bott Residue Formula expresses $c_m(E)$ in terms of the $\Bbb C^*$-weights at the fixed points: $$c_m(E)= \sum_{q\in \cal{Q}} {\sigma_{m,r}(\beta_1(q), \ldots, \beta_r(q)) \over \sigma_{m,m}(\alpha_1(q), \ldots, \alpha_m(q))}.$$ Since $\sigma_{m,m}$ is the product monomial and $\alpha_j(q)\neq 0$, the right hand side is well defined. \subsection{Torus Actions} Let $Z$ be a $3$ dimensional $\Bbb C$-vector with basis $\overline{z}=(z_0,z_1,z_2)$. Let $\overline{w}=(w_0,w_1,w_2)$ be a triple of integral weights. Let $\lambda(\overline{w})$ be the $\Bbb C^*$-representation with weights $(w_0,w_1,w_2)$ diagonal with respect to $\overline{z}$. Let $\Bbb P^2= \Bbb P(Z)$. The representation $\lambda(\overline{w})$ induces a $\Bbb C^*$-action on $\Bbb P(Z)$. There is an induced $\Bbb C^*$-action on $H(n)$, $X(n)$, and $Y(n)$. Recall the natural isomorphism $H^0(\Bbb P(Z), {\cal{O}}_{\Bbb P(Z)}(d)) \stackrel {\sim}{=} Sym^d(Z^*)$. There is a canonical equivariant lifting of $\lambda(\overline{w})$ to ${\cal{O}}_{\Bbb P(Z)}(d)$ such that the induced representation on global sections is $Sym^d(\lambda^*(\overline{w}))$. The canonical equivariant lifting of $\lambda(\overline{w})$ to ${\cal{O}}_{\Bbb P(Z)}(d)$ induces an equivariant $\Bbb C^*$-action on $E(n,d)= \pi_{n*}\rho^*({\cal{O}}_{\Bbb P(Z)}(d))$ over $X(n)$. \subsection{The Case $n=1$} $X(1)=H(1)=\Bbb P(Z)$. There are $3$ fixed points for distinct weights $\overline{w}=(w_0,w_1,w_2)$. Analysis of the fixed point $[z_0]$ yields \begin{equation*} \begin{array}{ll} \alpha_1([z_0])=& w_1-w_0 \\ \alpha_2([z_0])= &w_2-w_0 \end{array} \end{equation*} To simplify notation, let $Z_0, Z_1, Z_2$ be a basis of $Z^*$ dual to $\overline{z}$. To calculate the action on $E(1,d)$, observe that $\overline{\psi}_1([z_0])$ is the subscheme given by the ideal $(Z_1^2, Z_1Z_2, Z_2^2)$. Therefore the sections $Z_0^d$, $Z_0^{d-1}Z_1$, $Z_0^{d-1}Z_2$ generate the fibre of $E(1,d)$ at $[z_0]$. We obtain: \begin{equation*} \begin{array}{ll} \beta_1([z_0])= &-dw_0 \\ \beta_2([z_0])= &-(d-1)w_0-w_1 \\ \beta_3([z_0])= & -(d-1)w_0-w_2 \end{array} \end{equation*} The analysis for $[z_1]$ and $[z_2]$ is similar. The Bott Residue Formula now yields $c_2(E(1,d))= 3(d-1)^2$. Since the Chern class does not depend upon the weights, it is simplest to fix values $\overline{w}=(0,1,2)$ when using the residue formula. \subsection{The Case $n=2$} \label{case2} $X(2)=H(2)$. There are $9$ fixed points for distinct weights $\overline{w}=(w_0,w_1,w_2)$. Three fixed points correspond to the subschemes: $$[z_0]\cup [z_1], \ \ [z_0] \cup [z_2], \ \ [z_1]\cup [z_2].$$ For these points, the analysis of the $n=1$ suffices to yield the $\alpha$ and $\beta$-weights. There are $6$ fixed points given by the subschemes $D_{ij}= {\cal{O}}_{\Bbb P(Z)}/ (Z_i^2, Z_j)$ for ordered pairs $1 \leq i \neq j \leq 3$. We carry out the analysis at the fixed point $[D_{1,2}]$. In the affine open $Z_0\neq 0$, let $I=((Z_1/Z_0)^2, (Z_2/Z_0))$. Let $A=\Bbb C[(Z_1/Z_0), (Z_2/Z_0)]$. The tangent space to $H(2)$ at $[D_{1,2}]$ is canonically isomorphic to $Hom_A(I/I^2,A/I)$. $I/I^2$ is the free $A/I$ module with generator $(Z_1/Z_0)^2$ and $(Z_2/Z_0)$. $A/I$ is generated by $1$ and $(Z_1/Z_0)$. Since we know the $\Bbb C^*$-action on basis elements of $I/I^2$ and $A/I$, we obtain: \begin{equation*} \begin{array}{ll} \alpha_1([D_{1,2}])= & 2w_1-2w_0 \\ \alpha_2([D_{1,2}])=& w_1-w_0 \\ \alpha_3([D_{1,2}])=& w_2-w_0 \\ \alpha_4([D_{1,2}])=& w_2-w_1 \end{array} \end{equation*} Note $\overline{\psi}_2([D_{1,2}])$ is the subscheme defined by $(Z_1^4, Z_1^2Z_2,Z_2^2)$. Therefore the elements $Z_0^d$, $Z_0^{d-1}Z_1$, $Z_0^{d-1}Z_2$, $Z_0^{d-2}Z_1^2$, $Z_0^{d-2}Z_1Z_2$, and $Z_0^{d-3}Z_1^3$ yield a basis the fiber of $E(2,d)$ over $[D_{1,2}]$. The $\beta$-weights are therefore: \begin{equation*} \begin{array}{ll} \beta_1([D_{1,2}])= & -dw_0 \\ \beta_2([D_{1,2}])= & -(d-1)w_0-w_1 \\ \beta_3([D_{1,2}])= &-(d-1)w_0-w_2 \\ \beta_4([D_{1,2}])= &-(d-2)w_0-2w_1 \\ \beta_5([D_{1,2}])= &-(d-2)w_0-w_1-w_2 \\ \beta_6([D_{1,2}])= &-(d-3)-3w_1 \end{array} \end{equation*} The $\alpha$ and $\beta$-weights at the other points $[D_{i,j}]$ are obtained by appropriate permutations of $w_0$, $w_1$ and $w_2$ in the above formulas. After some algebra (MAPLE was used at this point), the Bott Residue formula yields $$c_4(E(2,d))={3\over 2} (d-1)(d-2)(3d^2-3d-11).$$ \subsection{The Case n=3} Consider the $\Bbb C^*$-action for weights $\overline{w}$ on $X(3)$. The analysis at nontriple points reduces to previous computations. There is one fixed point corresponding to the subscheme $[z_0]\cup[z_1] \cup [z_2]$. There are 12 fixed point of the type $$[z_i]\cup D_{j,k}$$ where $Supp(D_{j,k})\neq [z_i]$. The $\alpha$ and $\beta$-weights for these 1+12 points are easily obtained from the weights in the $n=1$ and $2$ cases. Next consider fixed points points $\xi \in X(3)\setminus \Bbb P_F(N)$ where $\overline{\psi}_3(\xi)$ is a triple point. If $(w_0,w_1,w_2)$ are distinct and no two weights sum to twice the third, then there are $6$ such $\xi$. They are given by the subschemes $T_{i,j}= {\cal{O}}_{\Bbb P(Z)}/ (Z_i^3,Z_j)$ for $1\leq i \neq j \leq 3$. We carry out the weight analysis at the point $[T_{1,2}]$. Following the notation of section (\ref{case2}), let $A=\Bbb C[(Z_1/Z_0), (Z_2/Z_0)]$ be the affine coordinate ring for $Z_0\neq 0$. Let $I=((Z_1/Z_0)^3, (Z_2/Z_0))$. Since $X(3)$ is isomorphic to $H(3)$ at $[T_{1,2}]$, the tangent space is given by $Hom_A(I/I^2, A/I)$. $I/I^2$ is seen to be a free $A/I$ module of rank $2$ with basis $(Z_1/Z_0)^3$ and $(Z_2/Z_0)$. We hence obtain the six $\alpha$-weights: $$3w_1-3w_0,\ \ \ 2w_1-2w_0,\ \ \ w_1-w_0,\ \ \ w_2-w_0,\ \ \ w_2-w_1,\ \ \ w_2+w_0-2w_1.$$ Since $(Z_1^3,Z_2)^2=(Z_1^6, Z_1^3Z_2, Z_2^2)$, the fiber of $E(3,d)$ at $[T_{1,2}]$ is spanned by the (possibly rational) sections $Z_0^d$, $Z_0^{d-1}Z_1$, $Z_0^{d-1}Z_2$, $Z_0^{d-2}Z_1^2$, $Z_0^{d-2}Z_1Z_2$, $Z_0^{d-3}Z_1^3$, $Z_0^{d-3}Z_1^2Z_2$, $Z_0^{d-4}Z_1^4$, $Z_0^{d-5}Z_1^5$. Therefore, the nine $\beta$-weights are: \begin{equation*} \begin{array}{lll} -dw_0 & -(d-1)w_0-w_1 & -(d-1)w_0-w_2 \\ -(d-2)w_0-2w_1 & -(d-2)w_0-w_1-w_2 & -(d-3)w_0-3w_1 \\ -(d-3)w_0-2w_1-w_2 & -(d-4)w_0-4w_1 & -(d-5)w_0-5w_1 \end{array} \end{equation*} Again, the weights at the other $[T_{i,j}]$ are obtained by appropriate permutations of $\overline{w}$. Finally, consider the fixed points $\xi \in \Bbb P_F(N)\subset X(3)$. Let $m_i$ be the ideal of the point $[z_i]$. Let $F_i$ for $1\leq i \leq 3$ be the subscheme ${\cal{O}}_{\Bbb P(Z)}/ m_i^2$. $[F_i]\in F$. The points $[F_i]$ are the 3 fixed points of the $\Bbb C^*$-action on $F$. The fixed points of $\Bbb P_F(N)$ must lie in the fibers of $\Bbb P_F(N)$ over the $[F_i]$. We analyze the case $i=1$. The fibered $\Bbb P^3$ of $\Bbb P_F(N)$ over $[F_1]$ was intrinsically described in section (\ref{resmap}). We see $$\Bbb P^3= \Bbb P(Hom_{\Bbb C}(K, Sym^3(V^*))) \stackrel {\sim}{=} \Bbb P(Sym^3(V^*))$$ where $V^*$ has a basis given by $(Z_1/Z_0)$ and $(Z_2/Z_0)$ with induced $\Bbb C^*$-weights $-w_1+w_0$ and $-w_2+w_0$ respectively. For weights $\overline{w}$ such that no two sum to twice the third, the $\Bbb C^*$ action on the fibered $\Bbb P^3$ has $4$ isolated fixed points. The total number of fixed points for the $\Bbb C^*$-action on $X(3)$ is $31=1+12+6+3\cdot 4$. The $\alpha$-weights at the $4$ fixed points in the fibered $\Bbb P^3$ are obtained in the following manner. First the differential of the $\Bbb C^*$-action on $H(3)$ at $F_1$ is determined. Then the blow-up along the locus $F$ is examined. The sequence (\ref{xact}) of section (\ref{resmap}) contains all the necessary data. The four fixed points in $\Bbb P(Sym^3(V^*))$ are $P_{r,s}=[(Z_1/Z_0)^r(Z_2/Z_0)^s]$ for non-negative integers $r$, $s$ with sum $r+s=3$. We tabulate the weight formulas for the four points $P_{r,s}$. First the six $\alpha$-weights: \begin{equation*} \begin{array}{lllllll} P_{3,0}: & w_1-w_0 & w_2-w_0 & 2w_2-w_1-w_0 & w_1-w_2 & 2w_1-2w_2 & 3w_1-3w_2 \\ P_{2,1}: & w_1-w_0 & w_2-w_0 & w_2-w_0 & w_2-w_1 & w_1-w_2 & 2w_1-2w_2 \\ P_{1,2}: & w_1-w_0 & w_2-w_0 & w_1-w_0 & 2w_2-2w_1 & w_2-w_1 & w_1-w_2 \\ P_{0,3}: & w_1-w_0 & w_2-w_0 & 2w_1-w_2-w_0 & 3w_2- 3w_1 & 2w_2-2w_1 & w_2-w_1 \end{array} \end{equation*} \noindent The $\beta$-weights for the four point $P_{r,s}$ all include the the six weights: \begin{equation*} \begin{array}{lll} -dw_0 & & \\ -(d-1)w_0-w_1 & -(d-1)w_0-w_2 & \\ -(d-2)w_0-2w_1 & -(d-2)w_0-w_1-w_2 & -(d-2)w_0-2w_2 \end{array} \end{equation*} The additional three $\beta$-weights at the points $P_{r,s}$ are: \begin{equation*} \begin{array}{llll} P_{3,0}: & -(d-3)w_0-2w_1-w_2 & -(d-3)w_0-w_1-2w_2 & -(d-3)w_0-3w_2 \\ P_{2,1}: & -(d-3)w_0-3w_1 & -(d-3)w_0-w_1-2w_2 & -(d-3)w_0-3w_2 \\ P_{1,2}: & -(d-3)w_0-3w_1 & -(d-3)w_0-2w_1-w_2 & -(d-3)w_0-3w_2 \\ P_{0,3}: & -(d-3)w_0-3w_1 & -(d-3)w_0-2w_1-w_2 & -(d-3)w_0-w_1-2w_2 \end{array} \end{equation*} As before, the $\alpha$ and $\beta$-weights at the fixed points on the other fibered $\Bbb P^3$'s can be obtained by permuting the weights $\overline{w}$ in the above formulas. The Bott Residue Formula now yields: $$c_6(E(3,d))={9\over2}d^6-27d^5+{9\over 2}d^4+{423\over 2}d^3-229d^2 -{829\over 2}d+525.$$ The algebraic computation was done on MAPLE with weights $\overline{w}= (0,1,3)$.

9504

alg-geom/9504012

en

https://arxiv.org/abs/alg-geom/9504012

alg-geom/9504012

Dieter Kotschick

D. Kotschick

On irreducible four--manifolds

AMSLaTeX

We show that minimal symplectic 4--manifolds with $b_2^+ >1$ and with residually finite fundamental groups are irreducible. We also give examples of irreducible orientable four--manifolds with indefinite intersection forms which are not almost complex with respect to either orientation.

alg-geom

\section{Introduction}\label{s:intro} For many years, four--manifold folklore suggested that all simply connected smooth four--manifolds should be connected sums of complex algebraic surfaces, with both their complex and non--complex orientations allowed\footnote{The $4$--sphere is the empty connected sum.}. The first counterexamples were constructed in 1990 by Gompf and Mrowka~\cite{GM}, and many others followed. Then, Gompf~\cite{gompf} showed that many, and possibly all, these counterexamples arise from symplectic four--manifolds. Having no indication to the contrary, many people have put forward the following: \begin{conj}\label{sc} Every smooth, closed, oriented and simply connected $4$--manifold is the connected sum of symplectic manifolds, with both the symplectic and the opposite orientations allowed. \end{conj} \noindent This conjecture is ambitious; it would imply the smooth Poincar\'e conjecture. Note that such a connected sum can have summands with definite intersection forms, for example copies of $\Bbb C P^2$. In fact, any definite summand has a diagonalizable intersection form by Donaldson's theorem~\cite{don}, and is therefore homeomorphic to $n\Bbb C P^2$ by Freedman's classification~\cite{freed}. When the manifolds under consideration are not simply connected, the situation is more complicated. Then there are obvious counterexamples to Conjecture~\ref{sc}, e.g. rational homology spheres which are not homotopy spheres. Thus, one has to allow definite summands which are more general than $n\Bbb C P^2$ or $n\overline{\bC P^2}$. The natural conjecture is: \begin{conj}\label{minimal} Every smooth, closed and oriented $4$--manifold is the connected sum of symplectic manifolds, with both the symplectic and the opposite orientations allowed, and of some manifolds with definite intersection forms. \end{conj} \noindent This has occurred to several people, especially in the light of the examples constructed in~\cite{KMT}, and has been dubbed the current ``minimal conjecture'' by Taubes. In this note we show that it is false\footnote{We do not have any proposal for a new ``minimal conjecture'' in the non--simply connected case.}. Conjecture~\ref{sc} remains open. Recall the following definition: \begin{defn} A smooth closed $4$--manifold $X$ is {\sl irreducible} if for every smooth connected sum decomposition $X\cong X_1\# X_2$ one of the summands $X_i$ must be a homotopy sphere. \end{defn} We will show: \begin{thm}\label{t:counter} There exist oriented irreducible $4$--manifolds $X$ with indefinite intersection forms and with $\pi_1 (X)=\Bbb Z_2$ and $b_2^+ (X)\equiv b_2^- (X)\equiv 0\pmod 2$. \end{thm} This follows from Proposition~\ref{p:irred1} in the next section. \begin{cor} There exist orientable irreducible $4$--manifolds $X$ with indefinite intersection forms, which are not almost complex (and therefore not complex and not symplectic) with respect to either orientation, and for which the Donaldson and Seiberg--Witten invariants are not defined (or must vanish by definition). \end{cor} \noindent If one drops the requirement that $X$ have indefinite intersection form, rational homology spheres give obvious examples. Recall that a symplectic $4$--manifold is called minimal if it contains no symplectically embedded $2$--sphere of selfintersection $-1$. Conjectures~\ref{sc} and~\ref{minimal} are complementary to Gompf's conjecture~\cite{gompf} that minimal symplectic $4$--manifolds are irreducible. In section~\ref{s:sympl} we deduce from the recent work of Taubes~\cite{swg} on the Gromov and Seiberg--Witten invariants that Gompf's conjecture is true in many cases, including all simply connected manifolds with $b_2^+ > 1$. We shall prove: \begin{thm}\label{t:sympl} Let $X$ be a minimal symplectic $4$--manifold with $b_2^+ (X)>1$. If $X\cong X_1\# X_2$ is a smooth connected sum decomposition of $X$, then one of the $X_i$ is an integral homology sphere whose fundamental group has no non--trivial finite quotient. \end{thm} \noindent This strengthening of Proposition 1 in~\cite{KMT} goes a long way towards confirming a conjecture made there. \begin{cor}\label{c:minirr} Minimal symplectic $4$--manifolds with $b_2^+ >1$ and with residually finite fundamental groups are irreducible. \end{cor} \noindent See section~\ref{s:sympl} for a result in and comments on the case $b_2^+=1$. For K\"ahler surfaces, Theorem~\ref{t:sympl} and Corollary~\ref{c:minirr} could be deduced easily from the reduction of the Seiberg--Witten equation to the K\"ahler vortex equation and the study of effective divisors on complex surfaces, due to Kronheimer--Mrowka and Witten~\cite{witten}. \section{Irreducibility of quotient manifolds}\label{s:irred} Theorem~\ref{t:counter} will follow from the following application of the covering trick introduced in~\cite{IMRN}: \begin{prop}\label{p:irred1} Let $X$ be a smooth, closed, simply connected and oriented spin $4$--manifold. If $b_2^+ (X)>1$, assume that $X$ has a non--trivial Donaldson or Seiberg--Witten invariant. Suppose a non--trivial finite group $G$ acts freely by orientation--preserving diffeomorphisms of $X$. Then the quotient $Y=X/G$ is an orientable irreducible $4$--manifold. \end{prop} \begin{pf} Let $Y\cong M\# N$ be a smooth connected sum decomposition. As $\pi_1 (Y)\cong G$ is finite, it cannot be a non--trivial free product and we may assume $\pi_1 (M)\cong G$ and $\pi_1 (N)\cong \{ 1\}$. Let $d$ be the order of $G$. The connected sum decomposition of $Y$ induces a connected sum decomposition $X\cong\overline{M} \# dN$, where $\overline{M}$ is the universal covering of $M$. As either $b_2^+ (X)\leq 1$ or $X$ is assumed to have a non--trivial Donaldson or Seiberg--Witten invariant, it follows that the intersection form of $N$ is negative definite. By Donaldson's theorem~\cite{don} it is diagonalizable over $\Bbb Z$, and therefore either trivial or odd. On the other hand, the intersection form of $N$ must be even, because it is a direct summand of the intersection form of $X$, which is even because $X$ is spin. We conclude $b_2 (Y)=0$. As $N$ is simply connected, it is a homotopy sphere and $Y$ is irreducible. \end{pf} To obtain examples as in the statement of Theorem~\ref{t:counter}, take for $X$ the Fermat surface of degree $d\equiv 2 \pmod 4$ in $\Bbb C P^3$ with $d\geq 6$. This is the surface defined in homogeneous coordinates by \begin{equation}\label{fermat} x^d+y^d+z^d+t^d=0\ . \end{equation} It is simply connected, and spin because its canonical class is the restriction of $(d-4)H$ to $X$, and therefore $2$--divisible. Being an algebraic surface with $b_2^+>1$, $X$ has non--trivial Donaldson and Seiberg--Witten invariants. Furthermore, the equation~\eqref{fermat} is invariant under complex conjugation on $\Bbb C P^3$, and has no non--trivial real solutions (because $d$ is even). Thus, complex conjugation acts freely on $X$ and the Proposition shows that $Y=X/\Bbb Z_2$ is irreducible. We have $\pi_1 (Y)=\Bbb Z_2$. Using the multiplicativity of the Euler characteristic and the signature under finite unramified coverings, one can calculate $b_2^{\pm}(Y)=\frac{1}{2}(b_2^{\pm}(X)-1)$ which are both positive (because $d\geq 6$) and even (because $d\equiv 2\bmod 4$). This completes the proof of Theorem~\ref{t:counter}. \begin{rem} Wang~\cite{wang} has shown that the quotients of simply connected minimal algebraic surfaces of general type by free anti--holomorphic involutions have trivial Seiberg--Witten invariants, even when the invariants do not vanish by definition as in the above examples. \end{rem} \medskip The assumptions in Proposition~\ref{p:irred1} are such that $X$ has to be irreducible, and then $Y$ turns out irreducible as well. Here is another result in the same spirit, but which does not require a spin condition and uses instead Corollary~\ref{c:minirr}. We could apply this to the examples discussed above, but Proposition~\ref{p:irred1} is much more elementary. \begin{prop}\label{p:irred2} Let $X$ be a closed, simply connected minimal symplectic $4$--manifold with $b_2^+(X)>1$. Suppose a non--trivial finite group $G$ acts freely by orientation--preserving diffeomorphisms of $X$. Then the quotient $Y=X/G$ is an orientable irreducible $4$--manifold. \end{prop} \begin{pf} If $Y\cong M\# N$ with $\pi_1 (M)\cong G$ and $\pi_1 (N) \cong \{ 1\}$, then $X\cong \overline{M}\# dN$. Thus Corollary~\ref{c:minirr} implies that $N$ is a homotopy sphere. Hence $Y$ is irreducible. \end{pf} \section{Irreducibility of symplectic manifolds}\label{s:sympl} In this section we give the proof of Theorem~\ref{t:sympl}. This requires some familiarity with Seiberg--Witten invariants, particularly the work of Taubes~\cite{taubes,taubes2,swg}. See also~\cite{KMT,KM,witten}. Let $X$ be a closed symplectic $4$--manifold with $b_2^+ (X)>1$. If $X$ splits as a connected sum $X\cong M\# N$, then by Proposition 1 of~\cite{KMT} we may assume that $N$ has a negative definite intersection form and that its fundamental group has no non--trivial finite quotient. In particular $H_1 (N,\Bbb Z ) =0$. This implies that the homology and cohomology of $N$ are torsion--free. Donaldson's theorem about (non--simply connected) definite manifolds~\cite{orient} implies that the intersection form of $N$ is diagonalizable over $\Bbb Z$. If $N$ is not an integral homology sphere, let $e_1,\ldots ,e_n \in H^2 (N,\Bbb Z )$ be a basis with respect to which the cup product form is the standard diagonal form. This basis is unique up to permutations and sign changes. It is a theorem of Taubes~\cite{taubes} that the Seiberg--Witten invariants of $X$ are non--trivial for the canonical $Spin^c$--structures with auxiliary line bundles $\pm K_X$. Note that we can write $$ K_X=K_M+\sum_{i=1}^n a_i e_i \ , $$ where $K_M\in H^2 (M,\Bbb Z )$ and the $a_i$ are odd integers because $a_i^2 = -1$ and $K_X$ is characteristic. Considering $-K_X$ and using a family of Riemannian metrics which stretches the neck connecting $M$ and $N$, we conclude that $M$ has a non--trivial Seiberg--Witten invariant for a $Spin^c$--structure with auxiliary line bundle $-K_M$. Now we can reverse the process and glue together solutions to the Seiberg--Witten equation for $-K_M$ on $M$ and reducible solutions on $N$ for the unique $Spin^c$--structure with auxiliary line bundle $e_1-\sum_{i\neq 1} e_i$, as in the proof of Proposition 2 in~\cite{KMT}. This gives a Seiberg--Witten invariant of $X$ which is equal (up to sign) to the Seiberg--Witten invariant of $M$ for $-K_M$, which is non--zero. This implies that $L=-K_M+e_1-\sum_{i\neq 1} e_i$ has selfintersection number $=K_X^2$ because for symplectic manifolds all the non--trivial Seiberg--Witten invariants come from zero--dimensional moduli spaces, see~\cite{swg}. Thus, $a_i = \pm 1$ for all $i\in\{ 1,\ldots ,n\}$. Without loss of generality we may assume $a_i =1$ for all $i$. The line bundle $L$ is obtained from $-K_X$ by twisting with $e_1$. Thus, by Taubes's main theorem in~\cite{swg}, the non--triviality of the Seiberg--Witten invariant of $X$ with respect to $L$ implies that $e_1$ can be represented by a symplectically embedded $2$--sphere in $X$. This contradicts the minimality of $X$. We conclude that $N$ must be an integral homology sphere. This completes the proof of Theorem~\ref{t:sympl}. \begin{rem} Gompf~\cite{gompf} has shown that all finitely presentable groups occur as fundamental groups of minimal symplectic $4$--manifolds, and conjecturally all these manifolds are irreducible. As was the case in~\cite{IMRN,KMT}, our arguments do not give an optimal result because we cannot deal with fundamental groups without non--trivial finite quotients. With regard to Theorem~\ref{t:sympl}, note that there are such groups which occur as fundamental groups of integral homology $4$--spheres. Let $G$ be the Higman $4$--group, an infinite group without non--trivial finite quotients, which has a presentation with $4$ generators and $4$ relations. Doing surgery on $4(S^1\times S^3)$ according to the relations produces an integral homology sphere with fundamental group $G$. \end{rem} \begin{rem} In another direction, the assumption $b_2^+ (X)>1$ can probably be removed from Theorem~\ref{t:sympl} and Corollary~\ref{c:minirr}. To do this one needs to understand how the neck--stretching in the proof of Theorem~\ref{t:sympl} and the perturbations in Taubes's arguments~\cite{taubes,swg} interact with the chamber structure of the Seiberg--Witten invariants for manifolds with $b_2^+ =1$. We will return to this question in the future. However, some results about the case when $b_2^+ =1$ can be deduced from Theorem~\ref{t:sympl}. For example, all manifolds with non--tivial finite fundamental groups are dealt with by the following: \end{rem} \begin{cor}\label{c:b=1} Let $X$ be a minimal symplectic $4$--manifold with $b_2^+ (X)=1$ and $b_1 (X)\leq 1$. If $\pi_1 (X)$ is a non--trivial residually finite group, then $X$ is irreducible. \end{cor} \begin{pf} Suppose $X\cong M\# N$. We may assume that $N$ has negative definite intersection form and its fundamental group has no non--trivial finite quotient. Residual finiteness then implies that $N$ is simply connected, and $\pi_1 (M)\cong\pi_1 (X)$. By assumption, $X$ has a finite cover $\overline{X}$ of degree $d>1$ which is diffeomorphic to $\overline{M}\# dN$, where $\overline{M}$ is a $d$--fold cover of $M$. But $\overline{X}$ is minimal symplectic because $X$ is, and so Corollary~\ref{c:minirr} implies that $N$ is a homotopy sphere whenever $b_2^+ (\overline{X})>1$. If $b_1 (X)=0$, the multiplicativity of the Euler characteristic and of the signature imply $b_2^+ (\overline{X})\geq 3$. If $b_1 (X)=1$, we can take $d\geq 3$ to obtain $b_2^+ (\overline{X})\geq 2$. \end{pf} \medskip\noindent {\sl Acknowledgements:} I would like to thank Shuguang Wang for correspondence and for his questions which this paper answers, and Cliff Taubes for conversations and correspondence. \bibliographystyle{amsplain}

9504

alg-geom/9504015

en

https://arxiv.org/abs/alg-geom/9504015

alg-geom/9504015

Ben Nasatyr and Brian Steer

Orbifold Riemann Surfaces and the Yang-Mills-Higgs Equations

Hard copies are available on request. This paper will appear in the "Annali della Scuola Normale Superiore di Pisa (classe di scienze)". LaTeX2.09 with AMS fonts

We extend Hitchin's results on "The self-duality equations on a Riemann surface" (Proc. LMS (3), vol. 55, 1987) to orbifold Riemann surfaces. We prove existence results for orbifold solutions of the Yang-Mills-Higgs equations and construct the moduli space of solutions. These moduli spaces provide interesting examples of non-compact hyper-Kahler manifolds in all dimensions divisible by 4 and of completely integrable Hamiltonian systems. We also reinterpret these moduli spaces as spaces of orbifold Higgs bundles and as representation varieties of Fuchsian groups.

alg-geom

\section{#1}\label{se:#2}\setcounter{equation}{0}\setcounter{paragraph}{0}\labels{ }{\mnote{#2}}} \newcommand{\bsu}[2]{\paragraph{#1.}\label{su:#2}\labels{}{\mnote{#2}}} \newcommand{\bss}[2]{\paragraph{#1.}\label{ss:#2}\labels{}{\mnote{#2}}} \newtheorem{proposition}{\sc Proposition}[section] \newtheorem{theorem}[proposition]{\sc Theorem} \newtheorem{lemma}[proposition]{\sc Lemma} \newtheorem{remark}[proposition]{\sc Remark} \newtheorem{definition}[proposition]{\sc Definition} \newtheorem{corollary}[proposition]{\sc Corollary} \newenvironment{proof}{\begin{list}{}{\leftmargin=0pt\rightmargin=0pt} \item {\em Proof.\ }}{\rule{6pt}{6pt}\end{list}\vspace{4pt plus 2pt minus 1pt}} \newenvironment{notation}{\begin{list}{}{\leftmargin=0pt\rightmargin=0pt} \item {\sc Notation\ }}{\end{list}} \newenvironment{convention}{\begin{list}{}{\leftmargin=0pt\rightmargin=0pt} \item {\sc Convention\ }}{\end{list}} \newcommand{\bth}[1]{\begin{theorem}\label{th:#1}\labels{}{\mnote{#1}}} \newcommand{\bthn}[2]{\begin{theorem}[#1]\label{th:#2}\labels{}{\mnote{#2}}} \renewcommand{\eth}{\end{theorem}} \newcommand{\bpr}[1]{\begin{proposition}\label{pr:#1}\labels{}{\mnote{#1}}} \newcommand{\end{proposition}}{\end{proposition}} \newcommand{\bco}[1]{\begin{corollary}\label{co:#1}\labels{}{\mnote{#1}}} \newcommand{\end{corollary}}{\end{corollary}} \newcommand{\ble}[1]{\begin{lemma}\label{le:#1}\labels{}{\mnote{#1}}} \newcommand{\end{lemma}}{\end{lemma}} \newcommand{\bre}[1]{\begin{remark}\label{re:#1}\labels{}{\mnote{#1}}\rm} \newcommand{\end{remark}}{\end{remark}} \newcommand{\bde}[1]{\begin{definition}\label{de:#1}\labels{}{\mnote{#1}}\rm} \newcommand{\end{definition}}{\end{definition}} \newcommand{\de}[1]{{\sc #1}} \newcommand{\begin{proof}}{\begin{proof}} \newcommand{\end{proof}}{\end{proof}} \newcommand{\refpr}[1]{Proposition \ref{pr:#1}} \newcommand{\refre}[1]{Remark \ref{re:#1}} \newcommand{\refth}[1]{Theorem \ref{th:#1}} \newcommand{\refco}[1]{Corollary \ref{co:#1}} \newcommand{\refle}[1]{Lemma \ref{le:#1}} \newcommand{\refde}[1]{Definition \ref{de:#1}} \newcommand{\refeq}[1]{(\ref{eq:#1})} \newcommand{\refse}[1]{\S\ref{se:#1}\,\,} \newcommand{\refsu}[1]{\S\ref{su:#1}\,\,} \newcommand{\refpa}[1]{part \ref{#1}} \newcommand{\beql}[1]{\labels{}{\mnote{#1}}\begin{eqnarray}\label{eq:#1}} \newcommand{\end{eqnarray}}{\end{eqnarray}} \newcommand{\begin{eqnarray*}}{\begin{eqnarray*}} \newcommand{\end{eqnarray*}}{\end{eqnarray*}} \newcommand{\alpha}{\alpha} \newcommand{\beta}{\beta} \newcommand{\epsilon}{\epsilon} \newcommand{\gamma}{\gamma} \newcommand{\lambda}{\lambda} \newcommand{\Gamma}{\Gamma} \renewcommand{\O}{\Omega} \renewcommand{\L}{\Lambda} \newcommand{\Lambda}{\Lambda} \newcommand{{\Bbb C}}{{\Bbb C}} \renewcommand{\H}{{\Bbb H}} \newcommand{{\Bbb N}}{{\Bbb N}} \renewcommand{\P}{{\Bbb P}} \newcommand{{\Bbb Q}}{{\Bbb Q}} \newcommand{{\Bbb R}}{{\Bbb R}} \newcommand{{\Bbb Z}}{{\Bbb Z}} \newcommand{{\cal D}}{{\cal D}} \newcommand{{\cal E}}{{\cal E}} \newcommand{{\cal F}}{{\cal F}} \newcommand{{\cal G}}{{\cal G}} \newcommand{{\cal H}}{{\cal H}} \newcommand{{\cal K}}{{\cal K}} \newcommand{{\cal L}}{{\cal L}} \newcommand{{\cal M}}{{\cal M}} \newcommand{{\cal N}}{{\cal N}} \newcommand{{\cal O}}{{\cal O}} \newcommand{{\cal P}}{{\cal P}} \newcommand{{\cal Q}}{{\cal Q}} \newcommand{{\cal S}}{{\cal S}} \newcommand{{\cal T}}{{\cal T}} \newcommand{Hermitian-Yang-Mills-Higgs}{Hermitian-Yang-Mills-Higgs} \newcommand{Hermitian-Yang-Mills}{Hermitian-Yang-Mills} \newcommand{Yang-Mills-Higgs}{Yang-Mills-Higgs} \newcommand{{\rm i.\,e.\ }}{{\rm i.\,e.\ }} \newcommand{{\rm e.\,g.\ }}{{\rm e.\,g.\ }} \newcommand{$V$-bundle}{$V$-bundle} \newcommand{\o\partial}{\o\partial} \renewcommand{\Im}{\mbox{Im\,}} \renewcommand{\deg}{\mbox{deg\,}} \newcommand{\mbox{trace\,}}{\mbox{trace\,}} \newcommand{\mbox{tr\,}}{\mbox{tr\,}} \newcommand{{\rm Herm\,}}{{\rm Herm\,}} \newcommand{{\rm End\,}}{{\rm End\,}} \newcommand{{\rm vol\,}}{{\rm vol\,}} \newcommand{{\rm Aut\,}}{{\rm Aut\,}} \newcommand{{\rm Hom\,}}{{\rm Hom\,}} \newcommand{\mbox{ad\,}}{\mbox{ad\,}} \renewcommand{\and}{\quad\mbox{ and }\quad} \renewcommand{\u}[1]{\underline{#1}} \renewcommand{\o}[1]{\overline{#1}} \newcommand{\oo}[1]{\overline{#1}'} \newcommand{\quad\mbox{ or }\quad}{\quad\mbox{ or }\quad} \renewcommand{\b}[1]{\breve{#1}} \renewcommand{\t}[1]{\tilde{#1}} \newcommand{{\rm i}}{{\rm i}} \newcommand{\sum_{i=1}^n}{\sum_{i=1}^n} \newcommand{\ilist}[1]{#1_1,\dots,#1_n} \newcommand{|}{|} \newcommand{\|}{\|} \newcommand{\supsetne}{\setbox0=\hbox{$\ne$}% \setbox1=\hbox{$ni$}% \supset\kern-\wd1 \lower1.2ex\box0} \newcommand{\surjarrow}{\setbox0=\hbox{$\rightarrow$}% \rightarrow\kern-\wd0 \longrightarrow} \newcommand{\bprn}[2]{\begin{proposition}[#1]\label{pr:#2}} \newcommand{\wo}[1]{\widetilde{#1}} \newcommand{\widehat}{\widehat} \begin{document} \title{{\bf Orbifold Riemann Surfaces and the Yang-Mills-Higgs\ Equations}} \author{{\bf Ben Nasatyr\thanks{Current address: Department of Mathematical Sciences, University of Aberdeen, Edward Wright Building, Dunbar Street, Old Aberdeen, AB9 2TY.}\, and Brian Steer,}\\ Peterhouse, Cambridge and Hertford College, Oxford} \date{Preprint, August 6, 1993,\\ revised, January 5 and April 26, 1995} \maketitle \pagenumbering{arabic} \setcounter{section}{-1} \bse{Introduction}{int} In this paper we study the $U(2)$ Yang-Mills-Higgs\ equations on orbifold Riemann surfaces. Among other aspects, we discuss existence theorems for solutions of the Yang-Mills-Higgs\ equations, the analytic construction of the moduli space of such solutions, the connectivity and topology of this space, its holomorphic symplectic structure and its reinterpretations as a space of orbifold Higgs bundles or $SL_2({\Bbb C})$-representations of (a central extension of) the orbifold fundamental group. We follow Hitchin's original paper for (ordinary) Riemann surfaces \cite{hi87} quite closely but there are many novelties in the orbifold situation. (There is some overlap with a recent paper of Boden and Yokogawa \cite{by}.) It may help to mention here a few of our motivations. \begin{enumerate} \item In studying the orbifold moduli space, we are also studying the parabolic moduli space (see \refsu{parhig}, also \cite{si90,by,ns93}). \item The moduli space provides interesting examples of non-compact hyper-K\"ahler manifolds in all dimensions divisible by 4. \item As a special case of the existence theorem for solutions of the Yang-Mills-Higgs\ equations we get the existence of metrics with conical singularities and constant sectional curvature on `marked' Riemann surfaces (see \refco{negative curvature}, \refth{conical} and compare \cite{ht92}). \item The orbifold fundamental groups we study are Fuchsian groups and their central extensions: these include the fundamental groups of elliptic surfaces and of Seifert manifolds. We obtain results on varieties of $SL_2({\Bbb C})$- and $SL_2({\Bbb R})$-representations of such groups (see \refse{rep} and compare {\rm e.\,g.\ } \cite{jn85}). In particular, we prove that Teichm\"uller space for a Fuchsian group or, equivalently, for a `marked' Riemann surface is homeomorphic to a ball (\refth{ball}). \item Moduli of parabolic Higgs bundles and of marked Riemann surfaces have potential applications in Witten's work on Chern-Simons gauge theory. \end{enumerate} Let $E$ be a Hermitian rank 2 $V$-bundle ({\rm i.\,e.\ } orbifold bundle) over an orbifold Riemann surface of negative Euler characteristic, equipped with a normalised volume form, $\Omega$. Let $A$ be a unitary connexion on $E$ and $\phi$ an ${\rm End\,}(E)$-valued $(1,0)$-form. Then the Yang-Mills-Higgs\ equations are \begin{eqnarray*} \begin{array}{rcl} F_A + [\phi,\phi^*] &=& - \pi {\rm i}\, c_1(E)\Omega I_E \quad{\rm and}\\ \o\partial_A\phi &=& 0. \end{array} \end{eqnarray*} See \refsu{ymhymh} for details. These equations arise by dimensional-reduction of the 4-dimensional Yang-Mills equations. Another interpretation is that they arise if we split projectively flat $SL_2({\Bbb C})$-connexions into compact and non-compact parts (see \refsu{repsta}). Just as for ordinary Riemann surfaces, the moduli space, $\cal M$, of solutions to the Yang-Mills-Higgs\ equations has an extremely rich geometric structure which we study in the later sections of this paper. Let us indicate the main results and outline the contents of each section. The first is devoted to preliminaries on orbifold Riemann surfaces and $V$-bundles ({\rm i.\,e.\ } orbifold bundles): \refsu{orbint} covers the very basics, for the sake of revision and in order to fix notation, while \refsu{orbdiv} deals with the correspondence between divisors and holomorphic line $V$-bundles on an orbifold Riemann surface (some of this may have been anticipated in unpublished work of B. Calpini). We particularly draw attention to the notational conventions concerning rank 2 $V$-bundles and their rank 1 sub-$V$-bundles established in \refsu{orbint} which are used throughout this paper. The second section introduces Higgs $V$-bundles and the appropriate stability condition (\refsu{highig}) and studies the basic algebraic-geometric properties of stable Higgs $V$-bundles (\refsu{higalg})---the principal result here is \refth{stable pairs}. This material roughly parallels \cite[\S 3]{hi87}, an important difference being that \cite[proposition 3.4]{hi87} does not generalise to the orbifold case. The third section introduces the Yang-Mills-Higgs\ equations (\refsu{ymhymh}), discusses the existence of solutions on stable Higgs $V$-bundles (\refsu{ymhexi}) and gives the analytic construction of $\cal M$ (\refsu{ymhmod}). These first three subsections parallel \cite[\S\S 4--5]{hi87} and only in \refsu{ymhexi} would any significant alteration to Hitchin's work necessary to allow for the orbifold structure. The main results are \refth{Narasimhan-Seshadri} and \refth{moduli}. The Riemannian structure of the moduli space (including the fact that the moduli space is hyper-K\"ahler) is also discussed briefly in \refsu{ymhmod}, following \cite[\S 6]{hi87}. There is one other subsection: \refsu{ymhequ} sketches alternative, equivariant, arguments that can be used for the existence theorem and the construction of $\cal M$. This last subsection also discusses the pull-back map between moduli spaces which arises when an orbifold Riemann surface is the base of a branched covering by a Riemann surface---see \refth{sub}. We stress that equivariant arguments {\em cannot} easily be applied throughout the paper---difficulties arise {\rm e.\,g.\ } in \refsu{higalg}, \refse{det} and \refse{rep}. The fourth section discusses the topology of $\cal M$, following \cite[\S 7]{hi87}. The results are \refth{Morse} and \refco{topology}. General formul\ae\ for the Betti numbers are not given but it is clear how to calculate the Poincar\'e polynomial in any given instance (however, see \cite{by}). The fifth section is devoted to the holomorphic symplectic structure on $\cal M$: following \cite[\S 8]{hi87}, $\cal M$ is described as a completely integrable Hamiltonian system via the determinant map $\det : {\cal M} \to H^0(K^2)$, defined by taking the determinant of the Higgs field. This result is given as \refth{determinant map} (we believe that a similar result was obtained by Peter Scheinost). There are a number of stages to the proof: first, it is simpler to use parabolic Higgs bundles and these are discussed in \refsu{parhig}; \refsu{gendet} contains the major part of the proof, with two special cases which arise in the orbifold case being dealt with separately in \refsu{detred} and \refsu{detspe}. Moreover, it is shown that with respect to the determinant map $\cal M$ is a fibrewise compactification of the cotangent bundle of the moduli space of stable $V$-bundles (\refsu{detnon}). The final section deals with the interpretation of the moduli space as a space of projectively flat connexions (\refsu{repsta}) or $SL_2({\Bbb C})$-representations of (a central extension of) the orbifold fundamental group (\refsu{reprep}), the identification of the submanifold of $SL_2({\Bbb R})$-representations (\refsu{reprea}) and the interpretation of one of the components as Teichm\"uller space (\refsu{reptei}), which leads to a proof that Teichm\"uller space is homeomorphic to ball. The proofs are much like those of \cite[\S\S 9--11]{hi87} and \cite{do87} and accordingly we concentrate on those aspects of the orbifold case which are less familiar. {\em Acknowledgements.} The great debt that the authors owe to the paper \cite{hi87} is obvious but they are also grateful to Nigel Hitchin for many useful conversations. Both authors would also like to thank Hans Boden, who pointed out an error in \refco{topology}, and Mikio Furuta. This work is an extension of part of Ben Nasatyr's doctoral thesis: he would like to thank Simon Donaldson for his patient supervision and Oscar Garc\'{\i}a-Prada, Peter Kronheimer and Michael Thaddeus for the contribution that their comments made to that thesis. At that time Ben Nasatyr was a College Lecturer at Lady Margaret Hall, Oxford, and he spent the following year as a Post-Doctoral Fellow at the University of British Columbia: he would like to thank LMH and NSERC of Canada for their generous financial support and Gabrielle Stoy at LMH and David Austin and Dale Rolfsen at UBC for their hospitality. He is currently the Sir Michael Sobell Research Fellow at Peterhouse, Cambridge. Part of Brian Steer's work on this paper took place during a sabbatical year that he spent in Bonn and Pisa: he would like to thank Friedrich Hirzebruch and the Max-Planck-Institut and Giuseppe Tomassini and the Scuola Normale Superiore di Pisa for their hospitality. \bse{Orbifold Riemann Surfaces}{orb} This section compiles some basic facts about orbifold Riemann surfaces and fixes some notations which we will need in the sequel. \bsu{Introduction to Orbifold Surfaces}{orbint} We start with the definition and basic properties of orbifold surfaces (or $V$-surfaces). The notion of a $V$-manifold was introduced by Satake \cite{sa56} and re-invented as `orbifold' by Thurston. By an \de{orbifold surface} (respectively \de{orbifold Riemann surface}) $M$ we mean a closed, connected, smooth, real 2-manifold (respectively complex 1-manifold) together with a finite number (assumed non-zero) of `marked' points with, at each marked point, an associated order of isotropy $\alpha$ (an integer greater than one). (See \cite{sa56} or \cite{sc'83} for full details of the definition.) Notice that $M$ has an `underlying' surface where we forget about the marked points and orders of isotropy. Although every point of a surface has a neighbourhood modeled on $D^2$ (the open unit disc), we think of a neighbourhood of a {\em marked} point as having the form $D^2/{\Bbb Z}_\alpha$, where ${\Bbb Z}_{\alpha}$ acts on ${\Bbb R}^2 \cong {\Bbb C}$ in the standard way as the $\alpha^{\rm th}$ roots of unity. We make this distinction because $M$ is to be thought of as an orbifold. Orbifold ideas do not seem to have been widely used in the study of `surfaces with marked points'. For instance the tangent $V$-bundle to $D^2/{\Bbb Z}_\alpha$ is $(D^2 \times {\Bbb R}^2)/{\Bbb Z}_\alpha$---this leads to an idea of an orbifold Riemannian metric on $M$ which corresponds to that of a metric on the underlying surface with conical singularities at the marked points (see \refsu{reptei}). We introduce the following notations, which will remain fixed throughout this paper. Let $M$ be an orbifold (Riemann) surface with topological genus $g$; denote by $\wo M$ the `underlying' (Riemann) surface obtained by forgetting the marked points and isotropy. Denote the number of marked points of $M$ by $n$, the points themselves by $p_1,\dots,p_n$ and the associated orders of isotropy by $\alpha_1,\dots,\alpha_n$. Let $\sigma_i$ denote the standard representation of ${\Bbb Z}_{\alpha_i}$, with generator $\zeta_i = e^{2\pi{\rm i}/\alpha_i}$. At a point where $M$ is locally $D^2$ or $D^2/\sigma_i$ use $z$ for the standard (holomorphic) coordinate on $D^2$; call this a local \de{uniformising} coordinate and at a marked point let $w=z^{\alpha_i}$ denote the associated local coordinate. When giving local arguments centred at a marked point, drop the subscript $i$'s; {\rm i.\,e.\ } use $p$ for $p_i$ and so on. Given a surface which is the base of a branched covering we naturally consider it to be an orbifold surface by marking a branch point with isotropy given by the ramification index. In this way we arrive at a definition of the \de{orbifold fundamental group} $\pi_1^V(M)$ (see \cite{sc'83}): it has the following presentation \beql{F-group} \begin{array}{rcl}\pi_1^V(M) &=& \langle a_1,b_1,\dots, a_g,b_g,q_1,\dots, q_n \quad |\\ && \quad q_i^{\alpha_i}=1,\ q_1\dots q_n[a_1,b_1]\dots[a_g,b_g]=1 \rangle. \end{array} \end{eqnarray} In this presentation $a_1,b_1,\dots,a_g,b_g$ generate the fundamental group of the underlying surface while $q_1,\dots,q_n$ are represented by small loops around the marked points. Similarly, in this situation, the Riemann-Hurwitz formula suggests the following definition of the \de{Euler characteristic} of an orbifold surface: \beql{Euler characteristic} \chi(M) = 2-2g-n+\sum_{i=1}^n \frac1{\alpha_i}.\end{eqnarray} We always work with orbifold surfaces with $\chi(M)<0$---note that this includes cases with $g=0$ or $g=1$ in contrast to the situation for ordinary surfaces. A \de{$V$-bundle}, $E$, with fibre ${\Bbb C}^r$, is as follows. We ask for a local trivialisation around each point of $M$ with smooth (or holomorphic) transition functions; at a marked point $p$ this should be of the form $E|_{D^2/\sigma} \stackrel{\simeq}{\to} (D^2 \times {\Bbb C}^r)/(\sigma \times \tau)$, where $\tau$ is an \de{isotropy representation} $\tau : {\Bbb Z}_\alpha \to GL_r({\Bbb C})$. We can always choose coordinates in a $V$-bundle which \de{respect the $V$-structure}: that is, if the isotropy representation is $\tau : {\Bbb Z}_\alpha \to GL_r({\Bbb C})$ then we can choose coordinates so that $\tau$ decomposes as $\tau = \sigma^{x_1} \oplus \sigma^{x_2} \oplus \cdots \oplus\sigma^{x_r}$, where, for $j=1,\dots,r$, $x_j$ is an integer with $0\le x_j < \alpha$ and the $x_j$ are increasing. We will mostly be interested in rank 2 and rank 1 $V$-bundles and for these we introduce particular notations for the isotropy, which will be fixed throughout: for a rank 2, respectively rank 1, $V$-bundle, denote the isotropy at a marked point by $x$ and $x'$, respectively by $y$, with $0 \le x,x',y <\alpha$. In the rank 2 case order $x$ and $x'$ so that $x\le x'$. If a rank 1 $V$-bundle is a sub-$V$-bundle of a rank 2 $V$-bundle then of course $y \in \{ x,x'\}$: in this case, let $\epsilon \in \{ -1,0,1 \}$ describe the isotropy of the sub-$V$-bundle, with $\epsilon = 0$ if $x=x'$, $\epsilon = -1$ if $y=x$ and $\epsilon = 1$ if $y=x'$. Add subscript $i$'s, when necessary, to indicate the marked point in question. Call a vector $(\epsilon_i)$ with $\epsilon_i = 0$ if $x_i = x'_i$ and $\epsilon_i\in\{\pm1\}$ if not an \de{isotropy vector}. For a rank 2 $V$-bundle let $n_0 = \# \{ i : x_i=x_i'\}$ and for a rank 1 sub-$V$-bundle let $n_\pm = \# \{ i : \epsilon_i = \pm 1\}$. If a $V$-bundle is, at a marked point, locally like $(D^2 \times {\Bbb C}^r)/(\sigma \times \tau)$ then by a Hermitian metric we mean, locally, a Hermitian metric on $D^2 \times {\Bbb C}^r$ which is equivariant with respect to the action of ${\Bbb Z}_\alpha$ via $\sigma\times\tau$. Considering the tangent $V$-bundle, we can also define the concepts of Riemannian metric and orientation for an orbifold surface (an orientation of an orbifold surface is just an orientation of the underlying surface). We introduce the notion of a connexion in a $V$-bundle in the obvious way. The first Chern class or degree of a $V$-bundle can be defined using Chern-Weil theory. Notice that the degree of a $V$-bundle is a {\em rational} number, congruent modulo the integers to the sum $\sum_{i=1}^n(y_i/\alpha_i)$, where $(y_i)$ is the isotropy of the determinant line $V$-bundle. When $E$ is a rank 2 $V$-bundle with isotropy $(x_i,x_i')$, as above then we write \begin{eqnarray*} c_1(\Lambda^2E) = l + \sum_{i=1}^n\frac{x'_i + x_i}{\alpha_i}, \end{eqnarray*} for $l \in {\Bbb Z}$. Similarly, if $L$ is a sub-$V$-bundle with isotropy given by an isotropy vector $(\epsilon_i)$ in the manner explained above then we write \begin{eqnarray*} c_1(L) = m + \sum_{i=1}^n \frac{\epsilon_i(x'_i - x_i) + (x'_i + x_i)}{2\alpha_i} \end{eqnarray*} for $m \in {\Bbb Z}$. These meanings of $l$ and $m$ will be fixed throughout. Topologically, $U(1)$ and $U(2)$ $V$-bundles are classified by their isotropy representations and first Chern class: we quote the following classification result from \cite{fs92}. \bprn{Furuta-Steer}{v-bundles} Let $M$ be an orbifold surface. Then, over $M$: \begin{enumerate} \item any complex line $V$-bundle is topologically determined by its isotropy representations and degree, \item any $SU(2)$ $V$-bundle is topologically determined by its isotropy representations (necessarily of the form $\sigma^{x}\oplus\sigma^{-x}$, where $0\le x\le [\alpha/2]$) and \item any $U(2)$ $V$-bundle is topologically determined by its isotropy representations and its determinant line $V$-bundle. \end{enumerate} \end{proposition} \bre{subbundles} Let $E$ be a $U(2)$ $V$-bundle with isotropy $(x_i,x_i')$ and let $(\epsilon_i)$ be any isotropy vector. Then there exists a $U(1)$ $V$-bundle $L$ with isotropy specified by $(\epsilon_i)$ (unique up to twisting by a $U(1)$-bundle {\rm i.\,e.\ } up to specifying the integer $m$, above) and, topologically, $E=L\oplus L^*\Lambda^2E$, by \refpr{v-bundles}. \end{remark} \bsu{Divisors and Line $V$-bundles}{orbdiv} The theory of divisors developed here has also been dealt with in the Geneva dissertation of B. Calpini written some time ago. Suppose $M$ is an orbifold Riemann surface. It is convenient to associate an order of isotropy $\alpha_p$ to every point $p$; it is 1 if the point is not one of the marked points (and $\alpha_i$ if $p=p_i$ for some $i$). A \de{divisor} is then a linear combination \begin{eqnarray*} D = \sum_{p\in M}\frac{n_p}{\alpha_p}.p \end{eqnarray*} with $n_p\in {\Bbb Z}$ and zero for all but a finite number of $p$. If $f$ is a non-zero meromorphic function on $M$ we define the \de{divisor of $f$} by $Df = \sum_p \nu_p(f).p$. Here $\nu_p(f)$ is defined in the usual way when $\alpha_p=1$. When $\alpha_p=\alpha > 1$ and $z$ is a local uniformising coordinate with $\rho : D^2 \surjarrow D^2/\sigma$ the associated projection, then on $D^2$ we find that $\rho^*f$ has a Laurent expansion of the form \begin{eqnarray*} \sum_{j\ge -N}a_j z^{\alpha j}\qquad\mbox{with $a_{-N}\ne 0$} \end{eqnarray*} and we set $\nu_p(f) = -N$. (The divisor of a meromorphic function is thus an {\em integral} divisor.) Two divisors $D$ and $D'$ are \de{linearly equivalent} if \begin{eqnarray*} D-D' = Df \end{eqnarray*} for some meromorphic f. The \de{degree} of a divisor $D=\sum_p(n_p/\alpha_p).p$ is defined to be $d(D)=\sum_p n_p/\alpha_p$. The correspondence between divisors and holomorphic line $V$-bundles goes through in exactly the same way as for Riemann surfaces without marked points. To a point $p$ with $\alpha_p=1$ we associate the point line bundle $L_p$ as in \cite{gu66}. If $\alpha_p=\alpha>1$ then to the divisor $p/\alpha$ we associate the following $V$-bundle. Let $z$ be a local uniformising coordinate; then, making the appropriate identification locally with $D^2/\sigma$, we define \begin{eqnarray*} L_{p/\alpha} = ((D^2\times {\Bbb C})/(\sigma\times\sigma)) \cup_\Phi ((M\setminus\{p\})\times{\Bbb C} ), \end{eqnarray*} where $\Phi: (D^2\setminus\{0\}\times {\Bbb C})/(\sigma\times\sigma) \to ((M\setminus\{p\})\times{\Bbb C} )$ is given by its ${\Bbb Z}_\alpha$-equivariant lifting \begin{eqnarray*} \widehat\Phi:(D^2\setminus\{0\})\times{\Bbb C} &\to& ((D^2/\sigma)\setminus\{0\})\times{\Bbb C}\\ (z,z')&\mapsto&(z^{\alpha},z^{-1}z'). \end{eqnarray*} This $V$-bundle has an obvious section `$z$'; this is given on $D^2\times {\Bbb C}$ by $z\mapsto(z,z)$ and extends by the constant map to the whole of $M$. So $L_{p/\alpha}$ is positive. We denote by $L_i$ the line $V$-bundle $L_{p_i/\alpha_i}$, associated to the divisor $p_i/\alpha_i$, and by $s_i$ the canonical section `$z$'. Finally for a general divisor \begin{eqnarray*} D = \sum_{p\in M}\frac{n_p}{\alpha_p}.p \end{eqnarray*} we set \begin{eqnarray*} L_D = \otimes_p(L_{p/\alpha_p})^{n_p}. \end{eqnarray*} As for a meromorphic function, we can define the divisor of a meromorphic section of a line $V$-bundle $L$. If $p$ has ramification index $\alpha_p=\alpha$ and we have a local uniformising coordinate $z$ and a corresponding local trivialisation $L|_{D^2/\sigma} \cong (D^2\times{\Bbb C}) /(\sigma\times\sigma^y) $, for some isotropy $y$ (with, by convention, $0\le y < \alpha$), then locally we have $ s(z) = \sum_{j\ge -N'}a'_j z^j$ with $a'_{-N'}\ne 0$. However, we have ${\Bbb Z}_\alpha$-equivariance which means that $s(\zeta.z)=\zeta^ys(z)$ (where $\zeta = e^{2\pi{\rm i}/\alpha}$ generates ${\Bbb Z}_\alpha$). It follows that $a'_j = 0$ unless $j\equiv y\pmod \alpha$ and hence \beql{Taylor} s(z) = z^{y}\sum_{j\ge -N}a_j z^{\alpha j}\qquad\mbox{with $a_{-N}\ne 0$,} \end{eqnarray} where $-N\alpha + y = -N'$. We define $\nu_p(s)=-N'/\alpha = -N + y/\alpha$: so for the canonical section $s_i$ of the line $V$-bundle $L_{i}$ we have $\nu_{p_i}(s_i) = 1/\alpha_i$. \bpr{divisors} The above describes a bijective correspondence between equivalence classes of divisors and of holomorphic line $V$-bundles. The degree $d(D)$ of a divisor $D$ is just $c_1(L_D)$, the first Chern class of the corresponding line $V$-bundle. \end{proposition} \begin{proof} Much of the proof is contained in \cite{fs92}. The correspondence has been defined above and it is clear that if we start from a divisor $D$ and pass to $L_D$ then taking the divisor associated to the tensor product of the canonical sections we get back $D$. We have to show that the correspondence behaves well with respect to equivalence classes. If $D_1\equiv D_2$, where $D_j = \sum (n^{(j)}_p/\alpha_p).p$ for $j=1,2$, then from what we know about divisors of meromorphic functions we see that $n^{(1)}_p \equiv n_p^{(2)} \pmod{\alpha_p}$. Now $L_{D_j}=\otimes_p(L_{p/\alpha_p})^{n^{(j)}_p}$. Since $n^{(1)}_p \equiv n_p^{(2)} \pmod{\alpha_p}$, we find that $L_{D_j}\otimes\bigotimes_{i=1}^n(L_{p_i/\alpha_i})^{-n^{(1)}_{p_i}}$ is a genuine line bundle for $j=1,2$. Moreover the two are equivalent because the corresponding divisors are. Hence $L_{D_1}\equiv L_{D_2}$. Similarly we show that two meromorphic sections of the same line $V$-bundle\ define equivalent divisors. \end{proof} \bco{divisors1} If $L$ is a holomorphic line $V$-bundle with $c_1(L)\le 0$ then $H^0(L) = 0$, unless $L$ is trivial. \end{corollary} Let $L$ be a holomorphic line $V$-bundle over $M$, with isotropy $y_i$ at $p_i$, and let ${\cal O}(L)$ be the associated sheaf of germs of holomorphic sections; we take the cohomology of $L$ over $M$ to be the sheaf cohomology of ${\cal O}(L)$ over $\wo M$. From \refeq{Taylor}, ${\cal O}(L)$ is locally free over ${\cal O}_M = {\cal O}_{\wo M}$ and hence there is a natural line bundle $\wo L$ over $\wo M$ with ${\cal O}(\wo L) \cong {\cal O}(L)$. If we define $\wo L = L\otimes L_1^{-y_1}\otimes \cdots\otimes L_n^{-y_n}$ then this gives the required isomorphism of sheaves. \bpr{parabolic} If $L$ is a holomorphic line $V$-bundle then, with $\wo L$ defined as above, there is a natural isomorphism of sheaves ${\cal O}(L)\cong {\cal O}(\wo L)$ given by tensoring with the canonical sections of the $L_i$. \end{proposition} \begin{proof} Recall that $\ilist{s}$ are the canonical sections of $\ilist{L}$. If $s$ is a holomorphic section of $L$ then $\wo s=s_1^{-y_1}\dots s_n^{-y_n}s$ will be a meromorphic section of $\wo L$, holomorphic save perhaps at $p_i$. In fact (by choosing a local coordinate) we see that $\wo s$ has removable singularities at $p_i$ and that $D(\wo s) = Ds - \sum_{i=1}^n ({y_i}/{\alpha_i})p_i$. Conversely, given a section $\wo s$ of $\wo L$, then $s_1^{y_1}\dots s_n^{y_n}\wo s$ is a section of $L$ and the correspondence is bijective. \end{proof} As corollaries we get the orbifold Riemann-Roch theorem, originally due to Kawasaki \cite{ka79} and an orbifold version of Serre duality. \bthn{Kawasaki-Riemann-Roch}{Riemann-Roch} Let $L$ be a holomorphic line $V$-bundle with the isotropy at $p_i$ given by ${y_i}$, with $0 \le y_i < \alpha_i$. Then $$ h^0(L) - h^1(L) = 1-g+c_1(L) - \sum_{i=1}^n \frac {y_i}{\alpha_i}, $$ where $h^i$ denotes the dimension of $H^i$. \eth \bth{Serre duality} If $L$ is a holomorphic line $V$-bundle\ and $K_M$ is the canonical $V$-bundle of the orbifold Riemann surface then \begin{eqnarray*} H^1(L) \cong H^{0}(L^* K_M)^*. \end{eqnarray*} \eth \begin{proof} By definition, $H^1(L)=H^1({\cal O}(L))=H^1(\wo L)$. So $H^1(L)\cong H^{0}((\wo L)^* K_{\wo M})^*$ by the standard duality. But $(\wo L)^* K_{\wo M}=\wo {L^* K_M}$ by a straightforward computation. \end{proof} \bse{Higgs $V$-Bundles}{hig} Throughout this section $E \to M$ is a holomorphic rank 2 $V$-bundle over an orbifold Riemann surface with $\chi(M)<0$ and we write $K=K_M$, the canonical $V$-bundle, and $\Lambda = \Lambda^2E$, the determinant line $V$-bundle. \bsu{Higgs $V$-Bundles}{highig} In this subsection we introduce Higgs $V$-bundles---this is a straightforward extension of the basic material in Hitchin's paper \cite{hi87} to orbifold Riemann surfaces. Define a \de{Higgs field}, $\phi$, to be a holomorphic section of ${\rm End}_0(E)\otimes K$ where ${\rm End}_0(E)$ denotes the trace-free endomorphisms of $E$. A \de{Higgs $V$-bundle} or \de{Higgs pair} is just a pair $({E},\phi)$. Let $({E}_1,\phi_1)$ and $({E}_2,\phi_2)$ be two Higgs $V$-bundles. A \de{homomorphism of Higgs $V$-bundles} is just a homomorphism of $V$-bundles $h : E_1 \to E_2$ such that $h$ is holomorphic and intertwines $\phi_1$ and $\phi_2$. The corresponding notion of an \de{isomorphism of Higgs $V$-bundles} is then clear. A holomorphic line sub-$V$-bundle $L$ of $E$ is called a \de{Higgs sub-$V$-bundle} (or `$\phi$-invariant sub-$V$-bundle') if $\phi(L) \subseteq KL$. A Higgs $V$-bundle $({E},\phi)$ is said to be \de{stable} if \beql{stable Higgs} c_1(L) < \frac12 c_1(E),\quad\mbox{for every Higgs sub-$V$-bundle, $L$.} \end{eqnarray} If we allow possible equality in \refeq{stable Higgs} then the Higgs $V$-bundle is called \de{semi-stable}. If a Higgs $V$-bundle is stable or a direct sum of two line $V$-bundles of equal degree with $\phi$ also decomposable then (it is certainly semi-stable and) it is called \de{polystable}. If $E$ is stable then certainly $(E,\phi)$ is stable for any Higgs field $\phi$. The following result, due to Hitchin in the smooth case \cite[proposition 3.15]{hi87}, goes over immediately. \bpr{stable regular} Let $({E}_1,\phi_1)$ and $({E}_2,\phi_2)$ be stable Higgs $V$-bundles with isomorphic holomorphic determinant line $V$-bundles, $\Lambda^2{E}_1 \cong \Lambda^2{E}_2$. Suppose that $\psi: E_1 \to E_2$ is a non-zero homomorphism of Higgs $V$-bundles. Then $\psi$ is an isomorphism of Higgs $V$-bundles. If $({E}_1,\phi_1)=({E}_2,\phi_2)$ then $\psi$ is scalar multiplication. \end{proposition} \bsu{Algebraic Geometry of Stable Higgs $V$-Bundles}{higalg} For applications in later sections we now develop some results on the possibilities for stable Higgs $V$-bundles. Higgs $V$-bundles are holomorphic $V$-bundles with an associated `Higgs field'; a holomorphic $(1,0)$-form-valued endomorphism of the $V$-bundle. We assume familiarity with \cite[\S 3]{hi87}. Given $E \to M$, we investigate whether there are any Higgs fields $\phi$ such that the Higgs pair $(E,\phi)$ is stable. Recall that the isotropy of $E$ at $p_i$ is denoted by $(x_i,x_i')$ and that $n_0=\#\{i\,:\,x_i = x_i'\}$. We will suppose throughout that $n_0 < n$---this is because the case $n=n_0$ is just that of a genuine bundle twisted by a line $V$-bundle and so essentially uninteresting (see also \refsu{detred}). The following lemma is a simple computation using the Kawasaki-Riemann-Roch theorem and Serre duality. \ble{} We have \[ h^0(K^2) = \chi(K^2) = 3g-3+n \and \chi({\rm End\,}_0(E)\otimes K)=3g-3+n-n_0. \] \end{lemma} If $E$ is stable we know that the only endomorphisms of $E$ are scalars and so $h^0({\rm End\,}_0(E))=0$; consequently if $3-3g-n+n_0>0$ (this only happens if $g=0$ and $n-n_0\le 2$) there are no stable $V$-bundles. Suppose that $L$ is a holomorphic sub-$V$-bundle of $E$. Then we have the short exact sequences \beql{b} &0\to L \stackrel{i}{\to} E \stackrel{j}{\to} L^* \Lambda\to 0&\and\nonumber\\ &0\to L \Lambda^* \stackrel{j^*}{\to} E^* \stackrel{i^*}{\to} L^*\to 0& \end{eqnarray} from which follows \beql{d} 0\to E^*\otimes KL \to {\rm End\,}_0(E) \otimes K\to KL^{-2} \Lambda\to 0. \end{eqnarray} Associated to \refeq{b} tensored by $KL$ is the long exact sequence in cohomology \beql{lesb} \begin{array}{c} 0 \to H^0(KL^2\Lambda^*) \to H^0(E^*\otimes KL) \to H^0(K)\stackrel{\delta}{\to}\qquad\\ \qquad\stackrel{\delta}{\to} H^1(KL^2\Lambda^*) \to H^1(E^*\otimes KL) \to H^1(K) \to 0 \end{array} \end{eqnarray} and associated to \refeq{d} we have \beql{les} \begin{array}{c} 0 \to H^0(E^*\otimes KL) \to H^0({\rm End\,}_0(E) \otimes K) \to H^0(KL^{-2} \Lambda)\stackrel{\delta}{\to}\qquad\\ \qquad\stackrel{\delta}{\to} H^1(E^*\otimes KL) \to H^1({\rm End\,}_0(E) \otimes K) \to H^1(KL^{-2} \Lambda) \to 0. \end{array} \end{eqnarray} Now let us review the strategy of the proof of \cite[proposition 3.3]{hi87}: if $E$ is stable then all pairs $(E,\phi)$ are certainly stable and we know something about stable $V$-bundles from \cite{fs92}. If $E$ is not stable then there is a destabilising sub-$V$-bundle $L_E$. Recall that $L_E$ is unique if $E$ is not semi-stable. Moreover, in the semi-stable case the assumption $n\ne n_0$ implies that $L_E \not\cong L_E^*\Lambda$ and so $L_E$ is unique if $E$ is not decomposable and if it is then $L_E$ and $L_E^*\Lambda$ are the only destabilising sub-$V$-bundles. Thus there will be some $\phi$ such that the pair $(E,\phi)$ is stable unless every Higgs field fixes $L_E$ (or $L_E^*\Lambda$, in the semi-stable, decomposable case). Moreover, the subspace of sections leaving $L$ invariant is $H^0(E^*\otimes KL) \subset H^0({\rm End\,}_0(E)\otimes K)$. It follows that a necessary and sufficient condition for $E$ to occur in a stable pair is $H^0(E^*\otimes KL_E)\ne H^0({\rm End\,}_0(E)\otimes K)$ (and similarly for $L_E^*\Lambda$, in the semi-stable, decomposable case). Considering \refeq{les} this amounts to non-injectivity of the Bockstein operator $\delta$, which we consider in the next lemma---proved as in the proof of \cite[proposition 3.3]{hi87}. From the lemma we obtain a version of \cite[proposition 3.3]{hi87}. \ble{extension'} If $L$ is a sub-$V$-bundle of $E$ with $\deg(L)\ge \deg(\Lambda)/2$ then \begin{enumerate} \item\label{ex1} $H^1(E^*\otimes KL)\cong{\Bbb C};$ \item\label{ex2} $H^0(KL^{-2} \Lambda)\stackrel{\delta}{\to}H^1(E^*\otimes KL)$ is surjective if and only if $e_E \ne 0$, where $e_E\in H^1(L^2 \Lambda^*)$ is the extension class. \end{enumerate} \end{lemma} \begin{proof} \begin{enumerate} \item Consider the long exact sequence in cohomology \refeq{lesb} for $L$, which includes the segment \beql{bit} \cdots \to H^1(KL^2 \Lambda^*) \stackrel{j^*}{\to} H^1(E^*\otimes KL) \stackrel{i^*}{\to} {\Bbb C} \to 0 . \end{eqnarray} Then the result follows from the fact that $h^1(KL^2 \Lambda^*)=0$, using Serre duality and the vanishing theorem. \item Consider \refeq{les} and let $i^*$ be the map on cohomology indicated in \refeq{bit}; then the result follows from the fact that $i^*.\delta$ is multiplication by the extension class $e_E$. \end{enumerate} \end{proof} \bpr{non-s} Let $E$ be a non-stable $V$-bundle. Then $E$ appears in a stable pair if and only if one of the following holds: \begin{enumerate} \item\label{ns1} $E$ is indecomposable with $h^0(KL_E^{-2}\Lambda)>1$; \item\label{ns2} $E$ is decomposable, not semi-stable with $h^0(KL_E^{-2}\Lambda)\ge 1$; \item\label{ns3} $E$ is decomposable, semi-stable with $h^0(KL_E^{-2}\Lambda)\ge 1$ and $h^0(KL_E^{2}\Lambda^*)\ge 1$. \end{enumerate} \end{proposition} To find more precise results in the case that $E$ is semi-stable we estimate $h^0(KL_E^{-2}\Lambda)$ and $h^0(KL_E^{2}\Lambda^*)$ using the following lemmas. For these recall the definitions of the integers $n_0$, $n_\pm$, $l$ and $m$ from \refsu{orbint}. \ble{chi} Suppose that $L$ is any sub-$V$-bundle of $E$. Then, with the notations established above, \begin{eqnarray*} \chi(KL^{-2} \Lambda) &=& l-2m +g - 1 + n_-\and\\ \chi(KL^{2} \Lambda^*) &=& 2m-l +g - 1 + n_+. \end{eqnarray*} Moreover: \begin{enumerate} \item\label{xpos} if $2c_1(L)-c_1(\Lambda) \ge 0$ then $h^0(KL^{2} \Lambda^*) = \chi(KL^{2} \Lambda^*) \ge g$ and $\chi(KL^{-2} \Lambda) \le g - 2 + n - n_0$; \item\label{xneg} if $2c_1(L)-c_1(\Lambda) \le 0$ then $h^0(KL^{-2} \Lambda) = \chi(KL^{-2} \Lambda) \ge g$ and $\chi(KL^{2} \Lambda^*) \le g - 2 + n - n_0$. \end{enumerate} \end{lemma} \begin{proof} The first part is just the Kawasaki-Riemann-Roch theorem. Now consider \refpa{xpos} (\refpa{xneg} is entirely similar): we have $H^1(KL^{2}\Lambda^*) \cong H^0(L^{-2}\Lambda)^*$ and this is zero (because the degree is non-positive and the isotropy is non-trivial as $n> n_0$). Let $\theta = \sum_{i=1}^{n}{\epsilon_i(x'_i-x_i)}/{\alpha_i}$ so that $2c_1(L)-c_1(\Lambda) \equiv \theta \pmod{{\Bbb Z}}$. Then $-n_- < \theta < n_+$ and the estimates on $\chi(KL^{2} \Lambda^*)$ and $\chi(KL^{-2} \Lambda)$ follow. \end{proof} \ble{bounds} For a given $M$ and $n-n_0$, an $E$ (with the given $n-n_0$) such that the bounds on $\chi(KL^{2} \Lambda^*)$ and $\chi(KL^{-2} \Lambda)$ in \refle{chi}, parts 1 and 2 are attained exists if and only if \begin{eqnarray*} \min_{ \{i_1,\dots,i_{n-n_0}\} \subseteq \{1,\dots,n \} }\left\{ \sum_{j=1}^{n-n_0} \frac{1}{\alpha_{i_j}} \right\} \le 1.\end{eqnarray*} For a given topological $E$ the bounds are attained for some holomorphic structure on $E$ if and only if \begin{eqnarray*} \min_{ \{(\epsilon_i)\ :\ n_+ + l\equiv 1 (2) \}} \left\{ n_+ - \sum_{i=1}^n\frac{\epsilon_i(x'_i-x_i)}{\alpha_i} \right\} \le 1,\end{eqnarray*} where $(\epsilon_i)$ varies over all isotropy vectors with $n_+ +l \equiv 1 (2)$. \end{lemma} \begin{proof} To see this we construct examples as follows. It is sufficient to consider only topological examples and therefore, given any $M$ and topological $E$, to choose $(\epsilon_i)$ and $m\in {\Bbb Z}$ to specify $L$ topologically. (Examples where $L$ is a topological sub-$V$-bundle of $E$ exist by \refre{subbundles}.) Now, given a choice of $(\epsilon_i)$ and $m$, we have $\chi(KL^{-2}\Lambda) = l - 2m + g - 1 + n_-$ and $\chi(KL^{2}\Lambda^*) = 2m - l + g - 1 + n_+$ from \refle{chi}. So, for $2c_1(L)-c_1(\Lambda) \ge 0$ (the case $2c_1(L)-c_1(\Lambda)\le 0$ is entirely similar), the bounds are attained provided $2m - l + n_+ = 1$ and $2m - l + \sum_{i=1}^{n}\epsilon_i(x'_i-x_i)/\alpha_i \ge 0$. Since we can vary $m$, the first equation just fixes the parity of $n_+$. Hence the problem reduces to finding $(\epsilon_i)$ such that \beql{succinct} \sum_{i=1}^{n}\frac{\epsilon_i(x'_i-x_i)}{\alpha_i} - n_+ &\ge& -1\and\\ n_+ + l &\equiv& 1 \pmod{2}.\label{eq:constraint} \end{eqnarray} This gives the desired result, for a given topological $E$. To see whether examples exist for a given $M$ and $n-n_0$ as we allow $E$ to vary over topological types with fixed $n-n_0$, we simply note that the maximum value of the left-hand side of \refeq{succinct} (subject to \refeq{constraint}) is \begin{eqnarray*} \max_{ \{i_1,\dots,i_{n-n_0}\} \subseteq \{1,\dots,n \} }\left\{ \sum_{j=1}^{n-n_0} \left( -\frac{1}{\alpha_{i_j}} \right) \right\}.\end{eqnarray*} Thus the bounds are certainly attained if the $\alpha_i$ are such that this is not less than $-1$.\end{proof} \bco{semi-stable'} If $L$ is a sub-$V$-bundle of $E$ with $c_1(L)=c_1(\Lambda)/2$ and $\epsilon_i$, $n_+$ and $n_-$ are defined by the isotropy of $L$, as before, then \beql{} h^0({\rm End\,}_0(E)\otimes K) &=& \left\{\begin{array}{ll} 3g-3+n-n_0 & \mbox{ if }0\to L \to E \to L^* \Lambda\to 0\mbox{ is non-trivial;}\\ 3g-2+n-n_0 & \mbox{ if it is trivial;} \end{array}\right.\label{h0end}\\ h^0(E^*\otimes KL) &=& 2g - 1 - \sum_{i=1}^{n}\frac{\epsilon_i(x'_i-x_i)}{\alpha_i} + n_+; \label{vkl}\\ h^0(KL^{-2} \Lambda) &=& g - 1 + \sum_{i=1}^{n} \frac{\epsilon_i(x'_i-x_i)}{\alpha_i} + n_-;\label{kl}\\ h^0(KL^{2} \Lambda^*) &=& g - 1 - \sum_{i=1}^{n} \frac{\epsilon_i(x'_i-x_i)}{\alpha_i} + n_+;\nonumber. \end{eqnarray} Moreover, \begin{eqnarray*} 2g \le &h^0(E^*\otimes KL)& \le n-n_0 + 2g -2,\\ g \le &h^0(KL^{-2} \Lambda) & \le n-n_0 + g -2\and\\ g \le &h^0(KL^{2} \Lambda^*) & \le n-n_0 + g -2. \end{eqnarray*} These estimates are attained for all values of $g$ and $n-n_0$ (but not necessarily for all $M$ or $E$). \end{corollary} \begin{proof} The results on $h^0(KL^{-2} \Lambda)$ and $h^0(KL^{2} \Lambda^*)$ follow from \refle{chi}. Moreover we know that $h^1(E^*\otimes KL)= 1$ from \refle{extension'}, \refpa{ex1} and so $h^0(E^*\otimes KL)$ follows from the Kawasaki-Riemann-Roch theorem. To calculate $h^0({\rm End\,}_0(E)\otimes K)$ we use \refeq{les} and \refle{extension'}, \refpa{ex2}. The estimates on $h^0(KL^{-2}\Lambda)$ and $h^0(KL^2\Lambda^*)$ are contained in \refle{chi} and the estimate on $h^0(E^*\otimes KL)$ follows (as $h^0(E^*\otimes KL) = - h^0(KL^{-2} \Lambda) + 3g -2 + n - n_0$). \end{proof} When $c_1(L)=c_1(\Lambda)/2$ it is not possible to have $n-n_0=1$ (because $c_1(L^2\Lambda^*)$ cannot be an integer if $n-n_0=1$ but, on the other hand, it is supposed zero). Applying these results to $L_E$ (and $L_E^*\Lambda$ in the semi-stable, decomposable case) we can strengthen \refpr{non-s} as far as it refers to semi-stable $V$-bundles. Adding in some necessary conditions on $g$ and $n-n_0$ derived from our estimates above we obtain the following theorem. \bth{stable pairs} A holomorphic rank 2 $V$-bundle $E$ occurs in a stable pair if and only if one of the following holds: \begin{enumerate} \item\label{s} $E$ is stable (if $g=0$ then necessarily $n-n_0\ge 3$); \item\label{ss} $E$ is semi-stable, not stable (necessarily $n-n_0 \ge 2$) with one of the following holding: \begin{enumerate} \item\label{ssin1} $E$ is indecomposable and $g>1$; \item\label{ssin2} $E$ is indecomposable, $g=0$ or 1 and $h^0(KL_E^{-2} \Lambda)>1$ (necessarily $g + n-n_0\ge 4$); \item\label{ssde1} $E$ is decomposable and $g>0$; \item\label{ssde2} $E$ is decomposable, $g=0$ and $1 \le h^0(KL_E^{-2} \Lambda) \le n -n_0 -3$ (necessarily $n-n_0\ge 4$); \end{enumerate} \item\label{not semi-stable} $E$ is not semi-stable with one of the following holding: \begin{enumerate} \item\label{nsin} $E$ is indecomposable and $h^0(KL_E^{-2} \Lambda)>1$ (necessarily $g\ge 2$ or $g + n-n_0 \ge 4$; if $g=2$ and $n-n_0 =1$ then $\wo{KL_E^{-2} \Lambda}$ is necessarily canonical); \item\label{nsde} $E$ is decomposable and $h^0(KL_E^{-2} \Lambda) \ge 1$ (necessarily $g\ge 1$ or $n-n_0\ge 3$; if $2g+n-n_0=3$ then $\wo{KL_E^{-2} \Lambda}$ is necessarily trivial). \end{enumerate} \end{enumerate} In all cases the necessary conditions are the best possible ones depending only on $g$ and $n-n_0$. \eth \begin{proof} In \refpa{ss} the first three items follow from \refco{semi-stable'} together with \refpr{non-s}, parts 1 and 3, while for the last item we note that when $g=0$, $h^0(KL_E^{2} \Lambda^*)\ge 1$ if and only if $h^0(KL_E^{-2} \Lambda)<n-n_0-2$ (from \refco{semi-stable'}) and apply \refpr{non-s}, \refpa{ns3}. Only the necessary conditions in \refpa{not semi-stable} need any additional comment. Using \refle{chi}, \refpa{xpos} we have that $\chi(KL_E^{-2} \Lambda) \le g - 2 + n - n_0$ and the bound is attained for some $M$ and $E$ by \refle{bounds}. Thus if $g>2$ there are cases with $\chi(KL_E^{-2} \Lambda)\ge 2$ and hence $h^0(KL_E^{-2} \Lambda)\ge 2$. If $g=2$ then there are cases with $\chi(KL_E^{-2} \Lambda)=n-n_0$, similarly. The only problem then occurs if $n-n_0 =1$ when $c_1(\wo{KL_E^{-2} \Lambda})=2$: in order to have $h^0(KL_E^{-2} \Lambda)>1$ we must have $\wo{KL_E^{-2} \Lambda} = K_{\wo M}$. Similarly, if $g=1$ we can suppose that $\chi(KL_E^{-2} \Lambda) = n - n_0 -1$. Then for $h^0(KL_E^{-2}\Lambda) >1$ we need $n-n_0 \ge 3$ and for $h^0(KL_E^{-2}\Lambda) \ge 1$ we need $n-n_0 \ge 1$ with $\wo{KL_E^{-2} \Lambda}$ trivial if $n-n_0 =1$. Finally, if $g=0$ we need $n-n_0 \ge 4$ for $h^0(KL_E^{-2}\Lambda) >1$ and $n-n_0 \ge 3$ (with $\wo{KL_E^{-2} \Lambda}$ trivial if $n-n_0 =3$) for $h^0(KL_E^{-2}\Lambda) \ge 1$. \end{proof} For each of the items of \refth{stable pairs} examples of such $V$-bundles do actually exist (see also \refse{top} and \refse{det}). Only items \ref{ssin2}, \ref{ssde2}, \ref{nsin} and \ref{nsde} pose any problem but it is fairly easy to construct the required examples using the ideas of \refsu{orbdiv} and \refle{bounds}. Of particular interest is \refpa{nsde} when $g=0$ and $n-n_0=3$: we have the following result (compare \refse{top}). \bpr{g0} There exist orbifold Riemann surfaces with $g=0$ with $V$-bundles with $n-n_0=3$ over them which are decomposable but not semi-stable and exist in stable pairs. Such a stable pair contributes an isolated point to the moduli space (which is nevertheless connected---see \refco{topology}). \end{proposition} \begin{proof} We set $E=L_E\oplus L_E^*\Lambda$ with $2c_1(L_E) > c_1(\Lambda)$. Now, according to \refth{stable pairs}, \refpa{nsde}, we get a stable pair if and only if $\wo{KL_E^{-2} \Lambda}$ is trivial. Moreover, applying \refsu{orbdiv} or \refle{bounds}, we see that examples certainly exist. We write the Higgs field according to the decomposition $\phi = \left( \begin{array}{cc}t & u\\ v & -t \end{array}\right)$. Now $h^0(KL_E^{-2} \Lambda)=1$ implies that $h^0(KL_E^2 \Lambda^*)=0$ and hence $u=0$. More simply, $g=0$ implies $t=0$ and so $\phi$ is given by $v$, with $v\in H^0(KL_E^{-2} \Lambda)\cong {\Bbb C}$ non-zero for a stable pair. Now we need to consider the action of $V$-bundle automorphisms: $\left( \begin{array}{cc}\lambda & 0\\ 0 & \lambda^{-1} \end{array}\right)$ acts on $ H^0(KL_E^{-2} \Lambda)\cong {\Bbb C}$ by $z \mapsto \lambda^2 z$ and hence there is a single orbit. \end{proof} Notice that \cite[proposition 3.4]{hi87} does {\em not} extend to orbifold Riemann surfaces with $\chi(M)<0$. To prove that result Hitchin uses Bertini's theorem to show that, for a given rank 2 holomorphic bundle over a Riemann surface with negative Euler characteristic, either the generic Higgs field leaves no subbundle invariant or there is a subbundle invariant under all Higgs fields; he then shows that the latter cannot happen when the bundle exists in a stable pair. Although we have not been able to enumerate all the cases in which this result is false in the orbifold case there are three things which can go wrong: \begin{enumerate} \item Bertini's theorem may not apply and the conclusion may be false: $E$ may be such that it exists in a stable pair, the generic Higgs field has an invariant sub-$V$-bundle and no sub-$V$-bundle is invariant by all Higgs fields; \item $E$ may be stable and have a sub-$V$-bundle invariant by all Higgs fields; \item $E$ may be non-stable, exist in a stable pair and have a sub-$V$-bundle invariant by all Higgs fields. \end{enumerate} We give counterexamples of the first and third types. Although we suspect that counterexamples of the second type also exist we have not been able to show this. For a counterexample where Bertini's theorem doesn't apply consider the following: if $g=1$ and $n-n_0=1$ then, anticipating \refle{invariants}, {\em every Higgs field has an invariant sub-$V$-bundle} and yet if $E$ is a non-stable $V$-bundle which exists in a stable pair (these exist by \refth{stable pairs}, \refpa{nsde}) then {\em no sub-$V$-bundle is invariant by all Higgs fields.} All counterexamples of the third type are given in the following proposition, which also has interesting applications in \refse{top}. \bpr{counter} A non-stable $V$-bundle $E$ exists in a stable pair and has a sub-$V$-bundle invariant by all Higgs fields if and only if $g=0$, $E=L_E \oplus L_E^*\Lambda$ with $2c_1(L_E) > c_1(\Lambda)$ and $L_E$ is such that the bounds in \refle{chi}, \refpa{xpos} are attained. Moreover, there exist orbifold Riemann surfaces with such $E$ over them, with $E$ having any given $n-n_0 \ge 3$. \end{proposition} \begin{proof} Suppose $E$ is non-stable, exists in a stable pair and has a sub-$V$-bundle invariant by all Higgs fields. Since $E$ is non-stable and exists in a stable pair the destabilising sub-$V$-bundle(s) cannot be invariant by all Higgs fields. Moreover, if $h^0(KL_E^2\Lambda^*)>0$ then, via the inclusions $H^0(KL_E^2\Lambda^*) \hookrightarrow H^0(E^*\otimes KL_E) \hookrightarrow H^0({\rm End\,}_0(E)\otimes K)$, we get a family of Higgs fields which leave no sub-$V$-bundle except $L_E$ invariant---hence we must have $h^0(KL_E^2\Lambda^*)=0$. By \refle{chi}, \refpa{xpos} this can only happen if the bounds there are attained and $g=0$. Now consideration of the long exact sequence \refeq{lesb} shows that $h^0(E^*\otimes KL_E)=g$ and hence \refle{extension'} and \refeq{les} together show that $E$ is decomposable. Considering the Higgs field according to the decomposition, in the manner of \refpr{g0}, we see that $L_E^*\Lambda$ is invariant under all Higgs fields: it follows that $2c_1(L_E)$ must be strictly greater than $c_1(\Lambda)$ for $E$ to form a stable pair. The converse is straightforward: we suppose that $g=0$, $2c_1(L_E) > c_1(\Lambda)$ and $L_E$ is such that the bounds in \refle{chi}, \refpa{xpos} are attained and, exactly as in \refpr{g0}, we set $E=L_E\oplus L_E^*\Lambda$. We write the Higgs field according to the decomposition as $\phi =\left( \begin{array}{cc}0 & u\\ v & -0\end{array}\right)$. Since $g=0$, the fact that $L_E$ is such that the bounds in \refle{chi}, \refpa{xpos} are attained means that $h^0(KL_E^{-2}\Lambda) = n - n_0 -2 \ge 1$ and $h^0(KL_E^{2} \Lambda^*) = 0$. Hence $v$ can be chosen non-zero so that $E$ exists in a stable pair and $u=0$ so that $L_E^*\Lambda$ is invariant by all $\phi$, as required. Finally, examples where the bounds in \refle{chi}, \refpa{xpos} are attained exist by \refle{bounds}. \end{proof} \bse{The Yang-Mills-Higgs Equations and Moduli}{ymh} We now prove an equivalence between stable Higgs $V$-bundles and the appropriate analytic objects---irreducible Yang-Mills-Higgs\ pairs---and use this to give an analytic construction of the moduli space. Throughout this section $M$ is an orbifold Riemann surface of negative Euler characteristic, equipped with a normalised volume form, $\Omega$, and $E$ is a smooth rank 2 $V$-bundle over $M$ with a fixed Hermitian metric. \bsu{The Yang-Mills-Higgs\ Equations}{ymhymh} Given the fixed Hermitian metric on $E$, holomorphic structures correspond to unitary connexions. Let $\phi$ be a Higgs field with respect to $A$, {\rm i.\,e.\ } a Higgs field on ${E}_A$ or satisfying $\o\partial_A\phi =0$. We call the pair $(A,\phi)$ a \de{Higgs pair}. (With the unitary structure understood Higgs pairs are entirely equivalent to the corresponding Higgs $V$-bundles and so we can talk about stable Higgs pairs, isomorphisms of Higgs pairs and so on.) (From some points of view it is more natural to consider the holomorphic structure as fixed and the unitary structure as varying. Of course the two approaches are equivalent.) We impose determinant-fixing conditions in what follows; they are not essential but they remove some redundancies associated with scalar automorphisms (see \refpr{stable regular}), tensoring by line $V$-bundles and so on. We have already made the assumption that the Higgs field $\phi$ fixes determinants in the sense that it is trace-free; the other determinant-fixing conditions are defined as follows. A unitary structure on $E$ induces one on the determinant line $V$-bundle $\Lambda$. With this fixed and a choice of isomorphism class of holomorphic structure on $\Lambda$, there is a unique (up to unitary gauge) unitary connexion on $\Lambda$ which is compatible with the class of holomorphic structure and is Yang-Mills, {\rm i.\,e.\ } has constant central curvature $-2\pi {\rm i}\,c_1(\Lambda)\Omega$. Fix one such connexion and denote it $A_\Lambda$. We say that a unitary connexion or holomorphic structure on $E$ has \de{fixed determinant} if it induces this fixed connexion or holomorphic structure in the determinant line $V$-bundle. (On the other hand if we fix the holomorphic structure then we can choose a Hermitian-Yang-Mills\ metric on the determinant line $V$-bundle and fix the determinant of our metrics by insisting that they induce this metric.) Given a unitary connexion $A$ the trace-free part of the curvature is $F_A^0 =_{\rm def} F_A + \pi {\rm i}\, c_1(\Lambda)\Omega I_E$, by the Chern-Weil theory. We say that a Higgs pair $(A,\phi)$ (with fixed determinants understood) is \de{Yang-Mills-Higgs} if \beql{hymh condition} \begin{array}{rcl} F_A^0 + [\phi,\phi^*] &=& 0 \quad{\rm and}\\ \o\partial_A\phi &=& 0. \end{array} \end{eqnarray} (For a Hermitian metric varying on a fixed Higgs $V$-bundle this is the condition for the metric to be Hermitian-Yang-Mills-Higgs .) The involution $\phi\mapsto \phi^*$ is a combination of the conjugation $dz\mapsto d\overline{z}$ and taking the adjoint of an endomorphism with respect to the metric. The second part of the condition merely reiterates the fact that $\phi$ is holomorphic with respect to the holomorphic structure induced by $A$. Of course if $\phi=0$ then \refeq{hymh condition} is just the Yang-Mills equation (see \cite{ab82,fs92}) and we say that $A$ is \de{Yang-Mills}. An existence theorem for Yang-Mills connexions in stable $V$-bundles, generalising the Narasimhan-Seshadri theorem from the smooth case \cite{do83}, is given in \cite{fs92}. The first half of our correspondence between stable Higgs $V$-bundles and Yang-Mills-Higgs\ pairs is not difficult; again a result of Hitchin \cite[theorem 2.1]{hi87} generalises easily. \bpr{stable} Let $M$ be an orbifold Riemann surface with negative Euler characteristic. If $(A,\phi)$ is a Yang-Mills-Higgs\ pair (with respect to the fixed unitary structure on $E$ and with fixed determinants) then the pair $(A,\phi)$ is stable unless it has a $U(1)$-reduction, in which case it is polystable. \end{proposition} We call a pair with a $U(1)$-reduction, {\em as a pair}, \de{reducible}; otherwise the pair is \de{irreducible}. Notice that a reducible pair is Yang-Mills-Higgs\ if and only if the connexions in the two line $V$-bundles are Yang-Mills. Define the \de{gauge group} ${\cal G}(E)$ to be the group of unitary automorphisms of $E$ (fixing the base). This acts on Higgs fields by conjugation and has a natural action on $\o\partial$-operators such that the corresponding Chern connexions transform in the standard way. Thus this action fixes the determinant line $V$-bundle, acts on the set of Higgs $V$-bundles by isomorphisms and takes one Yang-Mills-Higgs\ pair to another. We also consider the \de{complexified gauge group} ${\cal G}^c(E)$ of complex-linear automorphisms of $E$ (fixing the base). Again this acts on Higgs $V$-bundles by isomorphisms. Isomorphic Higgs $V$-bundle structures are precisely those that lie in the same ${\cal G}^c(E)$-orbit. Notice that \refpr{stable regular} implies that ${\cal G}^c(E)$ acts freely (modulo scalars) on the set of stable Higgs $V$-bundles. (If we think of the Higgs $V$-bundle $({E},\phi)$ as fixed and the Hermitian metric as variable then ${\cal G}^c(E)$ acts transitively on the space of Hermitian metrics.) Once again we easily obtain a uniqueness result due to Hitchin \cite[theorem 2.7]{hi87} in the smooth case. \bpr{regular pairs} Let $({E}_1,\phi_1)$ and $({E}_2,\phi_2)$ be isomorphic Higgs $V$-bundles with fixed determinants, with Chern connexions $A_1$ and $A_2$ and the same underlying rank 2 Hermitian $V$-bundle. Suppose that the Higgs pairs $(A_1,\phi_1)$ and $(A_2,\phi_2)$ are both Yang-Mills-Higgs. Then $({E}_1,\phi_1)$ and $({E}_2,\phi_2)$ are gauge-equivalent ({\rm i.\,e.\ } there is an element of ${\cal G}(E)$ taking one to the other). \end{proposition} \bsu{An Existence Theorem for Yang-Mills-Higgs\ Pairs}{ymhexi} A version of the Narasimhan-Seshadri theorem for stable Higgs $V$-bundles (essentially a converse to \refpr{stable}) can be proved directly for orbifolds, extending the arguments of \cite{do83,hi87}. \bth{Narasimhan-Seshadri} Let $E\to M$ be a fixed $U(2)$ $V$-bundle over an orbifold Riemann surface of negative Euler characteristic. If $(A,\phi)$ is a polystable Higgs pair with fixed determinant on $E$ then there exists an element $g \in {\cal G}^c$ of determinant 1, unique modulo elements of $\cal G$ of determinant 1, such that $g(A,\phi)$ is Yang-Mills-Higgs. \eth We shall deduce the theorem from the ordinary case by equivariant arguments in \refsu{ymhequ}, though there is some advantage to a direct proof, as an appeal to Fox's theorem is avoided and uniformisation results from the following corollary, proved as in \cite[corollary 4.23]{hi87}. \bco{negative curvature} If $M$ is an orbifold Riemann surface of negative Euler characteristic then $M$ admits a unique compatible metric of constant sectional curvature -4. \end{corollary} \begin{proof} We define a stable Higgs $V$-bundle by equipping $E=K\oplus 1$ with the Higgs field \begin{eqnarray*} \phi=\left(\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array}\right). \end{eqnarray*} We fix a Hermitian-Yang-Mills\ metric on $\Lambda^2E$. From \refth{Narasimhan-Seshadri} we have a Hermitian-Yang-Mills-Higgs\ metric $h$ on $E$. Exactly as in \cite[corollary 4.23]{hi87}, this must split and we obtain a metric on $K$ such that the dual metric in the tangent bundle has constant sectional curvature -4. \end{proof} \bsu{The Yang-Mills-Higgs\ Moduli Space}{ymhmod} We now construct the moduli space of irreducible Yang-Mills-Higgs\ pairs, beginning with a brief discussion of reducible Yang-Mills-Higgs\ pairs. Let $(A,\phi)$ be a reducible Yang-Mills-Higgs\ pair on $E$. The reduction means that there is a splitting of $E$ into a direct sum $E=L\oplus L^*\Lambda$, where $L$ and $L^*\Lambda$ have the same degree, with respect to which $A$ and $\phi$ are diagonal---the resulting Higgs $V$-bundle is polystable but not stable. The isotropy group of the pair $(A,\phi)$ is $S^1$ or $SU(2)$ according to whether the two summands are distinct or identical; since $\phi$ is trace-free the latter is only possible if $\phi=0$. Let us now consider the question of the existence of reductions. Obviously the essential prerequisite is that $L$ exists such that $L$ and $L^*\Lambda$ have the same degree. If $a$ denotes the least common multiple of the $\alpha_i$'s then the degrees of line $V$-bundles have the form $s/a$ for $s\in {\Bbb Z}$ and all $s$ occur. Thus a necessary condition for a reduction is that $c_1(\Lambda) = s/a$ with $s$ even. However, even when $s$ is even, there is a further constraint: the isotropy of $E$ is fixed and, as before, the isotropy of $L$ must be described by an isotropy vector $(\epsilon_i)$ with $c_1(L) \equiv \sum_{i=1}^n\{\epsilon_i(x'_i-x_i)+(x'_i+x_i)\}/2\alpha_i \pmod{{\Bbb Z}}$ and so the isotropy may imply a constraint to finding $L$ with appropriate $c_1(L)$. For general $M$ and $E$ it is impossible in \lq most' cases (see \cite{fs92} for details). {}From now on we make the assumption that the isotropy of $M$ and the degree and isotropy of $E$ are such that there are no reducible Yang-Mills-Higgs\ pairs on $E$. We outline the deformation theory to show that the moduli space is a finite-dimensional manifold. (For the purposes of this outline we suppress the use of Sobolev spaces---this is standard; see {\rm e.\,g.\ } \cite{pa'82}.) Fix an irreducible Yang-Mills-Higgs\ pair $(A,\phi)$. The `deformation complex' at $(A,\phi)$ is then the following elliptic complex: \beql{deformation} 0\to \Gamma(\frak{su}(E)) \stackrel{d_1}{\to} \Gamma(\frak{su}^1(E)) \oplus \Omega^{1,0}(\frak{sl}(E)) \stackrel{d_2}{\to} \Gamma(\frak{su}^2(E)) \oplus \Omega^{1,1}(\frak{sl}(E)) \to 0, \end{eqnarray} where $\frak{su}^k(E)$ denotes the bundle of skew-adjoint $k$-forms with values in the trace-free endomorphisms of $E$ and $\frak{sl}(E)$ denotes the bundle of trace-free endomorphisms of $E$. Here $d_1$, giving the linearisation of the action, is given by $$ d_1 : \psi \mapsto (d_A\psi,\ [\phi,\psi])) $$ and $d_2$, giving the linearisation of the Yang-Mills-Higgs equations, by $$ d_2 : (A',\phi') \mapsto (d_AA' + [\phi',\phi^*] +[\phi,\phi'^*],\ \o\partial_A\phi' + [(A')^{0,1},\phi]). $$ We use the orbifold Atiyah-Singer index theorem \cite{ka81} to calculate the index of \refeq{deformation} as $6(g-1) + 2(n-n_0)$. We note that the zeroeth and second cohomology groups, $H^{0}$ and $H^{2}$, of the complex vanish---for $H^{0}$ this follows from the irreducibility of $(A,\phi)$ and for $H^{2}$ the duality argument given by Hitchin will suffice. Hence the first cohomology group has dimension $ 6(g-1) + 2(n-n_0)$. Moreover the Kuranishi method shows that a neighbourhood of zero in $H^1$ is a local model for the moduli space and hence the moduli space is a smooth complex manifold of dimension $ 6(g-1) + 2(n-n_0)$. \bth{moduli} Let $M$ be an orbifold Riemann surface of negative Euler characteristic and $E\to M$ a fixed complex rank 2 $V$-bundle. \begin{enumerate} \item Suppose that $E$ is equipped with a Hermitian metric and admits no reducible Yang-Mills-Higgs\ pairs. Then the moduli space of Yang-Mills-Higgs\ pairs on $E$ with fixed determinants, ${\cal M}(E,A_\Lambda)$, is a complex manifold of dimension $ 6(g-1) + 2(n-n_0)$. \item Suppose that $E$ admits no Higgs $V$-bundle structures which are polystable but not stable. Then the moduli space of stable Higgs $V$-bundle structures on $E$ with fixed determinants is a complex manifold of dimension $6(g-1) + 2(n-n_0)$. \end{enumerate} \eth \bre{roots} In the smooth case there are essentially only two moduli spaces (of which only one is smooth), according to the parity of the degree. In the orbifold case, how many moduli spaces are there? Clearly it is sufficient to consider only one topological $\Lambda$ in each class under the equivalence $\Lambda \sim \Lambda L^2$, for any topological line $V$-bundle $L$---`square-free' representatives for each class will be discussed in \refsu{reprep}. A further subtlety in the orbifold case is the possibility of non-trivial topological square roots of the trivial line $V$-bundle, or simply \de{topological roots}: if $L$ is a topological root then there is a map on moduli ${\cal M}(E,A_\Lambda) \leftrightarrow {\cal M}(E\otimes L,A_\Lambda)$ by tensoring by $L$, which fixes $\Lambda$ but alters the topology of $E$. For $L$ to be a topological root necessarily $c_1(L)=0$ and $L$ has `half-trivial' isotropy, {\rm i.\,e.\ } the isotropy is 0 or $\alpha/2$ at each marked point. If we consider topological line $V$-bundles of the form $L= \otimes_{\alpha_i {\rm\ even}}L_i^{\delta_i\alpha_i/2}$, for $\delta_i \in {\Bbb Z}$ where the $L_i$ are the point $V$-bundles of \refsu{orbdiv}, then it is clear that $L$ is a topological root provided $c_1(L)=\sum\delta_i/2 =0$. If we let $n_2$ denote the number of marked points where the isotropy is even, then, provided $n_2 \ge 1$, there are $2^{n_2-1}$ topological roots. It follows that for each topological $\Lambda$, if $n_2 \ge 1$, there will be $2^{n_2-1}$ different topological $E$'s giving essentially the same moduli space. We will see another manifestation of this in \refsu{reprep}. \end{remark} Recall that the tangent space to the moduli space is given by the first cohomology of the deformation complex \refeq{deformation}, {\rm i.\,e.\ } by $\ker{(d_1^*)}\cap\ker{(d_2)}$. This space admits a natural $L^2$ metric and, just as in \cite[theorems 6.1 \& 6.7]{hi87}, we have the following result. \bpr{metric} Let $E$ be a fixed rank 2 Hermitian $V$-bundle over an orbifold Riemann surface of negative Euler characteristic and suppose that $E$ admits no reducible Yang-Mills-Higgs\ pairs. Then the natural $L^2$ metric on the moduli space ${\cal M}(E,A_\Lambda)$ is complete and hyper-K\"ahler. \end{proposition} \bsu{The Yang-Mills-Higgs\ equations and Equivariance}{ymhequ} Here we sketch how many {\em but not all} of the previous results of this section can be treated by equivariant arguments. Further details for this subsection can be found in \cite{na91}. An orbifold Riemann surface with negative Euler characteristic, $M$, has a topological orbifold covering by a surface \cite{sc'83} and so its universal covering is necessarily a surface with negative Euler characteristic. Pulling-back the complex structure we find that the universal covering is necessarily $D^2$, the unit disk, with $\pi_1^V(M)$ a group of automorphisms acting properly discontinuously. In other words $\pi_1^V(M)$ is a co-compact Fuchsian group or, in the terminology of \cite{fo52}, an \de{$F$-group}. Thinking of $D^2$ as the hyperbolic upper half-plane or Poincar\'e disk, the elements of $\pi_1^V(M)$ act by orientation-preserving isometries and so we get a compatible Riemannian metric of constant sectional curvature on $M$. This is just \refco{negative curvature}. In this context we need the following result of \cite{fo52}. \bprn{Fox}{Fox} If $\Gamma$ is an $F$-group then $\Gamma$ has a normal subgroup of finite index, containing no elements of finite order. \end{proposition} \bco{smooth covering} Let $M$ be an orbifold Riemann surface with negative Euler characteristic. Then there exists a smooth Riemann surface, $\widehat{M}$, with negative Euler characteristic, together with a finite group, $F$, of automorphisms of $\widehat M$, such that $M=F\backslash\widehat M$. \end{corollary} The important point here is that the covering is {\em finite} and hence $\widehat M$ is compact. The existence result of \refth{Narasimhan-Seshadri} follows from the corresponding result on $\widehat M$, \cite[theorem 4.3]{hi87}, using an averaging argument (compare \cite{gp91}). We will always use the notation that objects on $\widehat M$ pulled-back from $M$ under the covering map $\widehat M\to M$ will be denoted by a `hat'; $\widehat{\ \ }$. In this notation the pull-back of a $V$-bundle $E \to M$ becomes $\widehat E \to \widehat M$, and so on. For the equivariant argument it is easiest to fix the Higgs $V$-bundle structure on $E$ and vary the metric; therefore, rather than suppose that a {\em Hermitian} structure on $E$ is given, we temporarily suppose that a {\em holomorphic} structure on $E$ (and hence on $\widehat E$) is given. We will show that if $(E,\phi)$ is stable then $(\widehat E,\widehat\phi)$ is polystable and admits a Hermitian-Yang-Mills-Higgs\ metric which is $F$-invariant and so descends to the required metric on $E$. \bpr{polystable} Let $(E,\phi)$ be a stable Higgs $V$-bundle and let $(\widehat E,\widehat\phi)$ be the pull-back to $\widehat M$. Then $(\widehat E,\widehat\phi)$ is polystable. \end{proposition} \begin{proof} Suppose first that $(\widehat E,\widehat\phi)$ is {\em not semi-stable}. Then there is a unique destabilising Higgs sub-$V$-bundle $L=L_{\widehat E}$ and the action of $F$ cannot fix $L$. Therefore for some $f\in F$ we have that $f(L) \ne L$. However $f(L)$ is a Higgs sub-$V$-bundle of $(\widehat E,\widehat\phi)$ (because $\widehat\phi$ commutes with the action of $f\in F$) and has the same degree as $L$. This contradicts the uniqueness of $L$. So $(\widehat E,\widehat\phi)$ is semi-stable. Suppose it is {\em not stable}. Then again there is a destabilising Higgs sub-$V$-bundle $L= L_{\widehat E}$ (not necessarily unique). As before $L$ cannot be fixed by $F$ and so we obtain, for some $f\in F$, a Higgs sub-$V$-bundle $f(L) \ne L$ of the same degree as $L$. Let $g:f(L)\to \widehat E/L$ be the composition of the inclusion of $f(L)$ into $\widehat E$ with the projection onto $\widehat E/L$: $g$ is a homomorphism between two line bundles of the same degree and hence either zero or constant. Since $f(L) \ne L$ the map $g$ cannot be zero and hence $f(L)=\widehat E/L$. Since $f(L)$ is actually a Higgs sub-$V$-bundle, $(\widehat E,\widehat\phi)$ is a direct sum $(L \oplus f(L),\widehat\phi_{L}\oplus\widehat\phi_{f(L)})$ and so is polystable as claimed. \end{proof} \bpr{metrics exist} Let $(E,\phi)$ be a stable Higgs $V$-bundle and let $(\widehat E,\widehat\phi)$ be the pull-back to $\widehat M$. Then the polystable Higgs $V$-bundle $(\widehat E,\widehat\phi)$ admits a Hermitian-Yang-Mills-Higgs\ metric which is $F$-invariant (and unique up to scale). \end{proposition} \begin{proof} Certainly $(\widehat E,\widehat\phi)$ admits a Hermitian-Yang-Mills-Higgs\ metric (by \refpr{polystable} and \cite[theorem 4.3]{hi87}). By averaging, the Hermitian-Yang-Mills-Higgs\ metric can be supposed $F$-invariant. \end{proof} An $F$-invariant Hermitian-Yang-Mills-Higgs\ metric descends to $(E,\phi)$, where it trivially still satisfies the Hermitian-Yang-Mills-Higgs\ condition. We can satisfy the determinant-fixing condition by a choice of scalar multiple and so we obtain the desired existence result---\refth{Narasimhan-Seshadri}. Suppose again that a Hermitian, rather than holomorphic, structure on $E$ is given. We recall that Hitchin proves that if $\widehat E$ has odd degree then there is a smooth moduli space ${\cal M}(\widehat E,\widehat A_\Lambda)$ of complex dimension $6(\widehat g-1)$. The pull-back map $(A,\phi) \mapsto (\widehat A,\widehat\phi)$ defines a map from Higgs pairs on $E$ to $F$-invariant Higgs pairs on $\widehat E$---what can be said about the corresponding map on moduli? Suppose that $(A,\phi)$ is an irreducible Yang-Mills-Higgs\ pair on $E$. The first point to note is that $(\widehat A,\widehat\phi)$ may be reducible, by the analogue of \refpr{polystable} for pairs. For simplicity, we will ignore this possibility in our discussion---we suppose that there are topological obstructions to the existence of reducible Yang-Mills-Higgs\ pairs on $\widehat E$. \ble{regular lifts} Suppose that $(A,\phi)$ is an irreducible Yang-Mills-Higgs\ pair on $E$ with an irreducible lift. Suppose further that for some $g\in \widehat{\cal G}$, of determinant 1, $g(\widehat A,\widehat\phi)$ is $F$-invariant. Then $f^{-1}gf=\pm g$ for all $f\in F$. Conversely, given $g\in \widehat{\cal G}$ of determinant 1 such that $f^{-1}gf=\pm g$ for all $f\in F$, $g(\widehat A,\widehat\phi)$ is irreducible and $F$-invariant. \end{lemma} \begin{proof} Since $(\widehat A,\widehat\phi)$ is $F$-invariant we know that $fd_{\widehat A} = d_{\widehat A}f \and f\widehat\phi = \widehat\phi f $ for any $f\in F$. Since the same is also true of $g(\widehat A,\widehat\phi)$ It follows that $d_{\widehat A} = (g^{-1}f^{-1}gf)(d_{\widehat A})(f^{-1}g^{-1}fg)$ and similarly for the Higgs field. Since $(A,\phi)$ is a stable pair it follows (\refpr{stable regular}) that $\pm g = f^{-1}gf$. The converse is clear.\end{proof} Let $\widehat{\cal G}^F$ be the subgroup of $\widehat{\cal G}$ consisting of $F$-invariant elements of determinant 1 and let $\widehat{\cal G}^{\pm}$ denote that of elements $g\in \widehat{\cal G}$ of determinant 1 such that, for all $f\in F$, $f^{-1}gf=\pm g$. Clearly either $\widehat{\cal G}^{\pm}=\widehat{\cal G}^F$ or $\widehat{\cal G}^F < \widehat{\cal G}^{\pm}$ with even index. (In fact these groups will be equal under quite mild hypotheses, which amount to the vanishing of a certain equivariant ${\Bbb Z}_2$-characteristic class---see \cite{na91} and compare \cite[proposition 1.8, part iii)]{fs92}.) If these groups are unequal then $f^{-1}gf = -g$ for some $f\in F$ and $g\in \widehat{\cal G}$ of determinant 1---but such a $g$ cannot be close to $\pm 1$ and so does not enter the local description of the moduli space (compare \cite[theorem 4.1]{pa'82}). At an irreducible $F$-invariant pair $(\widehat A,\widehat\phi)$ the group $F$ acts on the deformation complex. The pull-back map induces a commutative diagram of deformation complexes and it follows immediately that ${\cal M}(E,A_\Lambda)$ covers a submanifold of ${\cal M}(\widehat E,\widehat A_\Lambda)$ with covering group $\widehat{\cal G}^{\pm}/\widehat{\cal G}^{F}$. \bth{sub} Let $M$ be an orbifold Riemann surface of negative Euler characteristic and $E\to M$ a fixed complex rank 2 $V$-bundle. Let $\widehat E$ be the pull-back of $E$ under the identification $M = F\backslash \widehat M$ of \refco{smooth covering}. \begin{enumerate} \item Suppose that $E$ is equipped with a Hermitian metric and $\widehat E$ with the pulled-back metric and that $E$ admits no reducible Yang-Mills-Higgs\ pairs. If $\widehat E$ has odd degree then, under pull-back, the moduli space of Yang-Mills-Higgs\ pairs with fixed determinants on $E$, ${\cal M}(E,A_\Lambda)$, covers a submanifold of the corresponding moduli space on $\widehat E$ with covering group $\widehat{\cal G}^{\pm}/\widehat{\cal G}^{F}$ (with $\widehat{\cal G}^\pm$ and $\widehat{\cal G}^F$ as above). If $\widehat E$ has even degree then this remains true for those classes of Higgs pairs which are irreducible on $\widehat E$. \item Suppose that $E$ admits no Higgs $V$-bundle structures which are polystable but not stable. If $\widehat E$ has odd degree then, under pull-back, the moduli space of stable Higgs $V$-bundle structures with fixed determinants on $E$ covers a submanifold of the corresponding moduli space on $\widehat E$ with covering group $\widehat{\cal G}^{\pm}/\widehat{\cal G}^{F}$ (with $\widehat{\cal G}^\pm$ and $\widehat{\cal G}^F$ as above). If $\widehat E$ has even degree then this remains true for those classes of Higgs $V$-bundle structure which are stable on $\widehat E$. \end{enumerate} \eth Notice that in the case when $\widehat M$ is a hyperelliptic surface of genus 2 branched over 6 points of the Riemann sphere then the dimensions of the two moduli spaces are equal (a simple arithmetic check shows that this is the only case where this happens). \bse{The topology of the moduli space}{top} We now give some results on the topology of the moduli space using the Morse function $(A,\phi)\stackrel{\mu}{\to}||\phi||_{L^2}^2$, following \cite[\S 7]{hi87}. Notation and assumptions remain as before; in particular, we suppose that $E$ admits no reducible Yang-Mills-Higgs\ pairs, so that the moduli space ${\cal M} = {\cal M} (E,A_\Lambda)$ is smooth and recall the definitions of the integers $n_\pm$ and $l$ from \refsu{orbint}. The function $(A,\phi)\stackrel{\mu}{\to}||\phi||_{L^2}^2=2{\rm i} \int \mbox{tr\,}(\phi \phi^*)$ is invariant with respect to the circle action $e^{{\rm i}\theta}(A,\phi)=(A,e^{{\rm i}\theta}\phi)$ and $d\mu(Y) = -2{\rm i} \omega_1(X,Y)$ where $X$ generates the $S^1$-action and $\omega_1$ is as in \cite[\S 6]{hi87}. The map $\mu$ is proper and there's an extension of \cite[proposition 7.1]{hi87}. To describe it we need to consider pairs $(m,(\epsilon_i))$ where $m$ is an integer and $(\epsilon_i)$ is an isotropy vector---such pairs describe topological sub-$V$-bundles of $E$, with isotropy described by $(\epsilon_i)$ and degree $m + \sum_{i=1}^n \{\epsilon_i(x'_i-x_i)+(x'_i+x_i)\}/(2\alpha_i)$ (see {\rm e.\,g.\ } \refre{subbundles}). \bth{Morse} Let $E$ be a fixed rank 2 Hermitian $V$-bundle over an orbifold Riemann surface of negative Euler characteristic and suppose that $E$ admits no reducible Yang-Mills-Higgs\ pairs. If $g=0$ then suppose that $n-n_0\ge 3$. Let $\mu$ be as above: then, with the notations established above, \begin{enumerate} \item\label{critical values} $\mu$ has critical values 0 and $2\pi\{ 2 m -l + \sum_{i=1}^n \{\epsilon_i(x'_i-x_i)/\alpha_i\} \}$ for an integer $m$ and isotropy vector $(\epsilon_i)$ with \begin{eqnarray*} l < 2m + \sum_{i=1}^n \frac{\epsilon_i(x'_i-x_i)}{\alpha_i} \le l + 2g - 2 + \sum_{i=1}^n\frac{\epsilon_i(x_i' - x_i)}{\alpha_i} + n_-; \end{eqnarray*} \item the minimum $\mu^{-1}(0)$ is a non-degenerate critical manifold of index 0 and is diffeomorphic to the space of stable $V$-bundles with fixed determinants and \item the other critical manifolds are also non-degenerate and are $2^{2g}$-fold coverings of $S^r\wo M$, where $r = l- 2m +2g -2 + n_-$. Moreover, they are of index $2\{2m -l + g - 1 + n_+ \}$. \end{enumerate} \eth \begin{proof} The critical points are the fixed points of the induced circle action on ${\cal M}$. Because we are taking quotients by the gauge group, these correspond to pairs $((A,\phi),\lambda)$ where $\lambda : S^1 \to {\cal G}$ such that, for all $\theta$, $\lambda(e^{{\rm i} \theta})d_A\lambda(e^{-{\rm i} \theta}) = d_A$ and $\lambda(e^{{\rm i} \theta})\phi\lambda(e^{-{\rm i} \theta}) = e^{{\rm i} \theta}\phi$. If $\phi=0$ then, holomorphically, we simply get stable $V$-bundles. If $\phi\ne 0$ then certainly $\lambda(e^{{\rm i} \theta})\ne 1$ for $\theta \not\equiv 0 \pmod{2\pi}$. The first equation now implies that the stabiliser ${\cal G}_A$ is non-trivial and $A$ is reducible to a $U(1)$-connexion. Consequently, as a holomorphic $V$-bundle, $E$ is decomposable (so, in particular, not stable) and can be written $L\oplus L^* \Lambda$. If we write $\phi = \left( \begin{array}{cc} t & u\\v & -t \end{array}\right)$ and $\lambda(e^{{\rm i} \theta})= \left( \begin{array}{cc} \mu_\theta & 0\\0 & \mu_\theta^{-1} \end{array}\right)$ with respect to this splitting then the second equation implies $t=0$ and either $u=0$ or $v=0$. Replacing $L$ by $L^* \Lambda$ if necessary, we can suppose that $u=0$ and that $v\in H^0(KL^{-2} \Lambda)$---$v$ is holomorphic from the self-duality equations. The remaining term of the Yang-Mills-Higgs\ equations is $*(F_A + [\phi,\phi^*] )= -\pi{\rm i} d I_E$. Writing $*F_{A_L} = *F-\pi{\rm i} d$, in terms of the above decomposition, so that $*F_{A_{L^*\Lambda}} = -*F-\pi{\rm i} d$, we find that $F = v \wedge \o v$ and \begin{eqnarray*} \deg L =\frac{{\rm i}}{2\pi}\int(F - *\pi{\rm i} c_1(\Lambda)) = \frac{{\rm i}}{2\pi}\int (v\wedge \o v) + \frac{c_1(\Lambda)}2 = \frac{\mu}{4\pi} + \frac{c_1(\Lambda)}{2}. \end{eqnarray*} Since $\mu> 0$ for $\phi\ne 0$, we have $2\deg L > c_1(\Lambda)$ and $L=L_E$, the destabilising sub-$V$-bundle of $E$. Moreover, because $v\ne 0$ we must have $h^0(KL^{-2}\Lambda)\ge 1$ (compare \refth{stable pairs}). Now, for any $(m,(\epsilon_i))$ let $L_{(m,(\epsilon_i))}$ be the corresponding topological sub-$V$-bundle of $E$. Consider pairs $(m,(\epsilon_i))$ with $2c_1(L_{(m,(\epsilon_i))})> c_1(\Lambda)$ and set $L=L_{(m,(\epsilon_i))}$ and $E=L\oplus L^*\Lambda$. This occurs as a stable pair $(E,\phi)$ provided $L$ admits a holomorphic structure with $h^0(KL^{-2}\Lambda) \ge 1$, and the Higgs field $\phi$ is then given by $v\in H^0(KL^{-2}\Lambda)\setminus\{0\}$ (compare \refth{stable pairs}, \refpa{nsde} and \refpr{counter}). To see whether a given topological $L=L_{(m,(\epsilon_i))}$ admits an appropriate holomorphic structure we use our results from \refsu{higalg}: by \refle{chi} we have $\chi(KL^{-2} \Lambda) = l - 2m + g - 1 + n_-$. It follows that $r= c_1 (\wo{ KL^{-2}\Lambda }) = l - 2m + 2g -2 + n_-$. Hence, supposing that $r \ge 0$, for each effective (integral) divisor of divisor order $r$ (if $r=0$ then for the empty divisor) we obtain a holomorphic structure on $\wo{K L^{-2}\Lambda}$ with a holomorphic section determining the divisor (determined up to multiplication by elements of ${\Bbb C}^*$). Hence we get a holomorphic structure on $K L^{-2}\Lambda$ with holomorphic section $v$ and all holomorphic sections arise in this way. Placing a corresponding holomorphic structure on $L$ requires a choice of holomorphic square root and there are $2^{2g}$ such choices. For each root $L$ the pair $(E,v) =(L\oplus L^*\Lambda,v)$ is clearly stable by construction. The section $v$ is determined by the divisor up to a multiplicative constant $\lambda\ne 0$ but $(L\oplus L^*\Lambda,v)$ and $(L\oplus L^*\Lambda,\lambda v)$ are in the same orbit under the action of the complexified gauge group and hence equivalent. Two distinct divisors determine distinct stable pairs so that we have the critical set is a $2^{2g}$-fold covering of the set of effective divisors of degree $r = l - 2m + 2g -2 + n_-$; that is, a $2^{2g}$-fold covering of $S^r\wo M$ (a point if $r=0$). Let $E=L\oplus L^*\Lambda$ for $L=L_{(m,(\epsilon_i))}$, as above. The subset $U=\left\{ \phi \in H^0({\rm End\,}_0(E)\otimes K)\ :\ (E,\phi)\mbox{ is stable } \right\}$ is acted upon freely by ${\rm Aut\,}_0(E)/\{\pm 1\}$, where ${\rm Aut\,}_0(E)$ are the holomorphic automorphisms of determinant 1 (see \refpr{stable regular}). The quotient $U/({\rm Aut\,}_0(E)/\{\pm 1\})$ is a complex manifold of dimension $3g-3+n-n_0$. So through each point $P\in {\cal M}$ there passes a $(3g-3+n-n_0)$-dimensional isotropic complex submanifold $U/({\rm Aut\,}_0(E)/\{\pm 1\})$, invariant under $S^1$: it is thus {\em Lagrangian}. Suppose $P\in {\cal M}$ is fixed under the $S^1$-action and $P=(E,\phi)$, where $E=L\oplus L^*\Lambda$, $\phi = \left( \begin{array}{cc}0 & 0 \\ v & 0 \end{array}\right)$, as above. The homomorphism $\lambda$ is given by $\lambda(\theta) = \left( \begin{array}{cc}e^{-{\rm i}\theta/2} & 0 \\ 0 & e^{{\rm i}\theta/2} \end{array}\right)$ with respect to this decomposition. Now ${\rm End\,}_0(E) = L^{-2}\Lambda\oplus L^2\Lambda^*\oplus {\Bbb C}$ and $\lambda(\theta)$ acts as $(e^{{\rm i}\theta},e^{-{\rm i}\theta},1)$. Hence $\lambda(\theta)$ acts with negative weight solely on $H^0(KL^2\Lambda^*) \subset H^0({\rm End\,}_0(E)\otimes K)$. As $\lambda(\theta)$ acts on $\phi$ by multiplication by $e^{{\rm i}\theta}$ there are no negative weights on $H^0({\rm End\,}_0(E)).\phi$ and hence we find, as in \cite{hi87}, that the index is $2 h^0(KL^2\Lambda^*) = 2\{2m -l + g - 1 + n_+ \}$, by \refle{chi}. \end{proof} {}From this, the work of \cite{fr59} and general Morse-Bott theory \cite{ab'99} we can, in principle, calculate the Betti numbers---see \cite{by}. We content ourselves with \refco{topology}, below, for which we need the following preliminary lemma. \ble{index0} There is exactly one critical manifold of index 0 and this is connected and simply-connected. \end{lemma} \begin{proof} \refth{Morse} shows that if $g>0$ then the space of stable $V$-bundles is the only index 0 critical manifold and this is connected and simply-connected (even when $g=0$) by \cite[theorem 7.11]{fs92}. When $g=0$, critical manifolds of index 0 other than the moduli of stable $V$-bundles may occur: these have the form $S^{r}\tilde M \cong {\Bbb C}\P^{r}$ and so are also connected and simply connected. It remains to show that exactly one of the possibilities is non-empty in each case. Making allowances for differences in notation, the following is implicit in \cite[theorem 4.7]{fs92}: the space of stable $V$-bundles is empty if and only if there exists a vector $(\epsilon_i)$ with $n_+ +l \equiv 1 (2)$ and \beql{empty} n_+ - \sum_{i=1}^n\frac{\epsilon_i(x'_i-x_i)}{\alpha_i} < 1-g. \end{eqnarray} Since the left-hand side of \refeq{empty} is clearly not less than zero we see that the space of stable $V$-bundles is non-empty whenever $g>0$. When $g=0$, \refth{Morse} shows that the critical manifolds of index 0 other than the moduli of stable $V$-bundles consist precisely of the $V$-bundles considered in \refpr{counter}. The number of such critical manifolds is the number of topological types $L_{(m,(\epsilon_i))}$ satisfying the criteria of \refpr{counter}, which, using the ideas of \refle{bounds}, is \beql{count} \# \left\{ (\epsilon_i)\ :\ n_+ + l\equiv 1 (2) \quad\mbox{and}\quad n_+ - \sum_{i=1}^n\frac{\epsilon_i(x'_i-x_i)}{\alpha_i} < 1 \right\}, \end{eqnarray} where $(\epsilon_i)$ varies over all isotropy vectors. Comparing \refeq{count} to \refeq{empty} we see that exactly one of the two types of critical manifold must occur. Moreover, we claim that the number in \refeq{count} is at most 1---this is sufficient to establish the lemma. To prove the claim suppose, without loss of generality, that $n_0=0$. Observe that it is an easy exercise to show that if $t_1, \dots t_{n}\in (0,1)$ are such that $\sum_{i=0}^{n} t_i <1$ then at most one $t_i$ can be replaced by $1-t_i$ with the sum remaining less than 1. Let \begin{eqnarray*} t_i = \frac{1+\epsilon_i}{2} - \frac{\epsilon_i(x'_i - x_i)}{\alpha_i} \end{eqnarray*} so that $\sum_{i=0}^n t_i = n_+ - \sum_{i=0}^n \epsilon_i(x'_i - x_i)/ \alpha_i$ and changing the sign of $\epsilon_i$ simply sends $t_i$ to $1-t_i$. The observation applies to show that this sum can be less than 1 for at most two vectors $(\epsilon_i)$ and these cannot have $n_+$ of the same parity. Hence the count in \refeq{count} is at most 1, as claimed. \end{proof} \bco{topology} The moduli space ${\cal M}$ is non-compact---except in the case $g=0$ and $n-n_0=3$ when it is a point---and connected and simply-connected. \end{corollary} \begin{proof} The non-compactness follows from the fact that the critical manifolds cannot be maxima except if $g=0$ and $n-n_0 = 3$. This is because the critical manifolds have index $ i = 2\left\{2m -l + g - 1 + n_+ \right\}$ and (real) dimension $2r = 2\left\{ l - 2m + 2g -2 + n_- \right\}$ and $2r+i = 6g - 6 +2(n-n_0)$, which is exactly half the (real) dimension of the moduli space. The connectedness and simple-connectedness follow from the analogous facts for the unique critical manifold of index 0 (\refle{index0}) and the fact that the other Morse indices are all even and strictly positive. \end{proof} \bse{The Determinant Map}{det} Recall that $M$ is an orbifold Riemann surface with negative Euler characteristic, with $E\to M$ a fixed $U(2)$-$V$-bundle. We assume that $E$ admits no reducible Yang-Mills-Higgs\ pairs so that the moduli space is smooth. Thinking of the moduli space as a space of stable Higgs $V$-bundles, there is a holomorphic gauge-invariant map $ (A,\phi) \mapsto \det(\phi) $ which descends to a holomorphic map $ \det : {\cal M}(E,A_\Lambda) \to H^0(K^2).$ Hitchin showed that in the smooth case this map is proper, surjective and makes ${\cal M}$ a completely integrable Hamiltonian system. Moreover he showed that when $q \in H^0(K^2)$ has simple zeros the fibre $\det^{-1}(q)$ is biholomorphic to the Prym variety of the double covering determined by $\sqrt{-q}$ \cite[theorem 8.1]{hi87}. We will see that things are similar but a little more involved in the orbifold case: the first significant observation is that $h^0(K^2) = 3g - 3 + n$---this is half the dimension of the moduli space exactly when $n_0 = 0$. For this reason it will be useful to suppose that $n_0 = 0$. (In \refsu{detred} we will show that the image of the determinant map is contained in a canonical $(3g - 3 +n-n_0)$-dimensional subspace of $H^0(K^2)$ and thus all cases can be reduced to the case $n_0 = 0$.) In addition, there are two special cases which we exclude: when $g=0$, $n=3$ the determinant map is identically zero, and when $g=1$, $n=1$ we have a special case which leads to a breakdown in our methods---this case is dealt with separately in \refsu{detspe}. We summarise our results in the following theorem (proofs are for the most part discussed in the remainder of this section; the details which have been omitted are exactly as in \cite[\S 8]{hi87}). We believe that a similar result was obtained by Peter Scheinost. \bth{determinant map} Let $E$ be a fixed rank 2 Hermitian $V$-bundle over an orbifold Riemann surface of negative Euler characteristic, with $n-n_0>3$ if $g=0$. Suppose further that $E$ admits no reducible Yang-Mills-Higgs\ pairs. Then the determinant map on the moduli space of Yang-Mills-Higgs\ pairs on $E$ with fixed determinants \begin{eqnarray*} \det : {\cal M}(E,A_\Lambda) \to H^0(K^2) \end{eqnarray*} has the following properties: \begin{enumerate} \item $\det$ is proper; \item the image of $\det$ lies in a canonical $(3g - 3 +n-n_0)$-dimensional subspace $H^0(\b M;K_{\b M}^2)\subseteq H^0(K^2)$ and $\det$ surjects onto $H^0(\b M;K_{\b M}^2)$; \item with respect to $\det : {\cal M}(E,A_\Lambda) \to H^0(\b M;K_{\b M}^2)$, ${\cal M}(E,A_\Lambda)$ is a completely integrable Hamiltonian system; \item for a generic $q$ in the image of $\det$, the fibre $\det^{-1}(q)$ is biholomorphic to a torus of dimension $3g-3+n-n_0$---this can be identified with the Prym variety of the covering determined by $q$ except when $g=n-n_0=1$, when it is identified with the Jacobian; \item ${\cal M}(E,A_\Lambda)$ is a fibrewise compactification of $T^*{\cal N}(E,A_\Lambda)$ with respect to the map $\det : T^*{\cal N}(E,A_\Lambda) \to H^0(\b M;K_{\b M}^2)$, where ${\cal N}(E,A_\Lambda)$ is the moduli space of Yang-Mills connexions on $E$ with fixed determinants. \end{enumerate} \eth It seems possible to obtain results arguing using orbifold methods but it is often simpler to translate this orbifold problem into one about parabolic bundles; we review the necessary results in the next subsection. \bsu{Parabolic Higgs bundles}{parhig} Recall the basic facts concerning the correspondence between $V$-bundles over $M$ and parabolic bundles over $\wo M$ \cite{fs92}. Let $\wo E$ be a rank $2$ holomorphic vector bundle over $\wo M$. A \de{quasi-parabolic structure} on $\wo E$ is, for each marked point $p \in \{ p_1,\dots,p_n\}$, a flag in $\wo E_p$ of the form \begin{eqnarray*} \wo E_{p} = {\Bbb C}^2 \supset {\Bbb C} \supset 0, &\mbox{ or }& \wo E_p = {\Bbb C}^2 \supset 0. \end{eqnarray*} A flag of the second form is said to be \de{degenerate}. A quasi-parabolic bundle $\wo E$ is a \de{parabolic bundle} if to each flag of the first form there is attached a pair of weights, $0\le \lambda < \lambda' < 1$ and to each of the second form there is a single (multiplicity 2) weight $0\le \lambda = \lambda' < 1$. There is a notion of parabolic degree involving the degree of $\wo E$ and the weights. A basis $\{ e,e' \}$ for the fibre at a parabolic point is said to \de{respect the quasi-parabolic structure} if either the flag is degenerate or $e'$ spans the intermediate subspace in the flag. An endomorphism of a parabolic bundle $\psi$ is a \de{parabolic endomorphism} if for each $p$, with respect to a basis which respects the quasi-parabolic structure, $\psi_p$ satisfies $(\psi_p)_{12} = 0$ whenever $\lambda < \lambda'$. Let $E$ be a rank $2$ holomorphic $V$-bundle over $M$. Recall that by convention $x \le x'$ (if we assume that $n_0 = 0 $ then there is strict inequality). For a line $V$-bundle $L$, we can consider the passage $L\mapsto \wo L$ (\refsu{orbdiv}) as a smoothing process and the construction of parabolic bundles follows similar lines: for a marked point $p$ we consider \begin{eqnarray*} (E|_{M \setminus\{ p \}}) \cup_\Psi D^2 \times {\Bbb C}^2, \end{eqnarray*} with clutching function $\Psi$ given, in local coordinates, by its ${\Bbb Z}_\alpha$-equivariant lifting \beql{patch} \begin{array}{rcl} \widehat\Psi : (D^2 \setminus\{0\}) \times {\Bbb C}^2 &\to& D^2 \times {\Bbb C}^2\\ (z,(z_1,z_2)) &\mapsto& (z^\alpha,(z^{-x}z_1,z^{-x'}z_2)). \end{array} \end{eqnarray} Now a holomorphic section of $(D^2 \times {\Bbb C}^2)/(\sigma \times \tau)$ is given by holomorphic maps $s_j : D^2\to {\Bbb C}$, for $j=1,2$, invariant under the action of ${\Bbb Z}_\alpha$. As with \refeq{Taylor}, Taylor's theorem implies that $s_j(z) = z^{x_j} \wo{s}_j(z^\alpha)$, where $\wo{s}_j$ is a holomorphic function $D^2 \to {\Bbb C}$ and we use the temporary notations $x_1=x$ and $x_2=x'$. Under the map $\Psi$ defined by \refeq{patch} we simply get a section of $(D^2 \setminus \{ 0 \})\times {\Bbb C}^2$ which is given by the functions $\wo{s}_j(w)$ and hence extends to a holomorphic section of $D^2\times {\Bbb C}^2$. In other words the map $\Psi$ is an isomorphism between the sheaves of germs of holomorphic sections. Repeating this construction about each marked point, we get a holomorphic bundle $\wo E\to \wo M$ corresponding to the holomorphic $V$-bundle $E\to M$. In fact $\wo E$ has a natural parabolic structure as follows: working in our local coordinates about a particular marked point (which respect the $V$-structure) we define weights $\lambda=x/\alpha$ and $\lambda'=x'/\alpha$. Define a flag in ${\Bbb C}^2$ so that the smallest proper flag space is the subspace of ${\Bbb C}^2$ on which $\tau$ acts like $\sigma^{x'}$. The corresponding quasi-parabolic structure on $\wo E_p$ is then given by the image of this flag---notice that this is degenerate if and only if $x = x'$. With the weights $\lambda,\lambda'$ it is clear that $\wo E$ is a parabolic bundle. (Whilst it is not true in general that $\Lambda^2\wo E = \wo{\Lambda}$, the bundle $\Lambda^2\wo E$ is determined by $\Lambda$ and the isotropy so that our determinant-fixing condition on $E$ translates to one on $\wo E$.) We quote the following result of \cite{fs92}. \bprn{Furuta-Steer}{V-parabolic} For a fixed orbifold Riemann surface $M$, the correspondence $E \mapsto \wo E$ gives a bijection between isomorphism classes of rank 2 holomorphic $V$-bundles and those of rank 2 parabolic bundles over $\wo M$ with rational weights of the form $x/\alpha$. Moreover, the induced map ${\cal O}(E) \mapsto {\cal O}(\wo E)$ is an isomorphism of analytic sheaves. \end{proposition} Now consider what happens to Higgs fields under the passage $E \mapsto \wo E$: we use a local uniformising coordinate $z$, centred on a given marked point, and let $w=z^\alpha$ be the local holomorphic coordinate on $\wo M$. There is a Taylor series expansion as before: if $\phi$ is a Higgs field on $E$ then in our local coordinates \beql{Taylor Higgs} \phi_{ij}dz &=&\left\{\begin{array}{ll} z^{x_i-x_j-1}\wo\phi_{ij}(z^\alpha)dz&\quad\mbox{ if }x_i>x_j\and\\ z^{\alpha + x_i-x_j-1}\wo\phi_{ij}(z^\alpha)dz&\quad\mbox{ if }x_i \le x_j,\\ \end{array}\right. \end{eqnarray} where $\wo\phi_{ij}$ are holomorphic functions and we again use the temporary notations $x_1=x$ and $x_2=x'$. To transfer this across to $\wo E$ simply notice that away from the marked point the clutching function $\Psi$ defined by \refeq{patch} is a bundle isomorphism and so acts on the Higgs field by conjugation. Conjugating by $\Psi$ we obtain \beql{Taylor Higgs 2} \phi_{ij}^\Psi dz &=& z^{x_j-x_i}\phi_{ij}dz\nonumber\\ &=& \left\{ \begin{array}{ll} \wo\phi_{ij}(w)\frac{dw}{\alpha w}&\quad\mbox{ if }x_i>x_j\and\\ \wo\phi_{ij}(w)\frac{dw}{\alpha}&\quad\mbox{ if }x_i\le x_j,\\ \end{array}\right. \end{eqnarray} with $x_1=x$ and $x_2=x'$. We take this to define a \de{parabolic Higgs field}. Denote the parabolic Higgs field constructed in this way by $\wo{\phi}$. In Simpson's language \cite{si90} is $\wo{\phi}$ just a filtered regular Higgs field. This defines a correspondence between Higgs $V$-bundles and parabolic Higgs bundles (with appropriate parabolic weights). In order to make this a correspondence between the stable objects we simply have to check that the invariant subbundles correspond---this is easy. Thus we can apply many of our preceding results to spaces of stable parabolic Higgs bundles. \bsu{Reduction to the case $n_0 = 0$}{detred} Suppose that at some marked points the $V$-bundle $E$ has $x = x'$ so that $n_0 > 0$. Number the marked points so that these are the last $n_0$. We can twist by a line $V$-bundle to make the isotropy zero at such points. Thus, as far as $E$ is concerned, the orbifold structure at these points is irrelevant and we suppose that $M$ only has $n-n_0$ marked points. More precisely, we can construct $\b M$ from $M$ using the smoothing process that gives $\wo M$ but only at the last $n_0$ marked points. We write $\b E$ for $E$ considered as a $V$-bundle over $\b M$. We also have to consider the canonical $V$-bundle $K$. Notice that $K = K_{\b M}\otimes_{i=n-n_0+1}^{n} L_{i}^{\alpha_i-1}$ so that there is a natural inclusion $H^0(K^2_{\b M}) \hookrightarrow H^0(K^2_M)$ given by $ s \mapsto s\otimes_{i=n-n_0+1}^{n} s_{i}^{2\alpha_i-2}$. (Here the $L_i$ are point $V$-bundles and $s_i$ are the canonical sections, as in \refsu{orbdiv}.) We identify $H^0(K^2_{\b M})$ with its image in $H^0(K^2_M)$. From \refeq{Taylor Higgs} it is clear that $\det(\phi)$ vanishes to order $2\alpha -2$ in $z$ at the last $n_0$ marked points (since $x = x'$ there). It follows that $\det(\phi)\in H^0(K^2_{\b M})$ for all Higgs fields $\phi$ on $E$. Moreover, if we pass from $\phi$ to $\b\phi$ by applying the smoothing process for Higgs fields at the last $n_0$ marked points, then it is clear that $(\b E,\b\phi)$ is a Higgs $V$-bundle over $\b M$. Notice that by \refeq{Taylor Higgs 2} $\b\phi$ is holomorphic at the last $n_0$ marked points because there we have $x = x'$. The process outlined above is invertible. For the proofs in the remainder of this section therefore, although we will be careful to state results for $q\in H^0(\b M,K_{\b M}^2)$ and $n_0\ge 0$, we can assume that $n_0 = 0$ without loss of generality. \bsu{Generic fibres of the determinant map}{gendet} We assume that $2g+n-n_0>3$. Let $q \in H^0(K_{\b M}^2)$ and consider the corresponding section $\wo q \in H^0(\wo{K_{\b M}^2})$. We want to suppose that $\wo q$ has simple zeros and that none of the zeros of $\wo q$ occurs at a marked point (of $\b M$) but first we would like to know that such behaviour is generic. \ble{generic} The generic section $\wo q \in H^0(\wo{K_{\b M}^2})$ has simple zeros, none of which is at a marked point of $\b M$, provided $2g+n-n_0>3$. \end{lemma} \begin{proof} We can assume that $n_0=0$. Notice that $\wo{K^2} = K_{\wo M}^2 \otimes_{i=1}^{n}L_{p_i}$, where $L_{p_i}=L_i^{\alpha_i}$ is the point bundle associated to a marked point $p_i$. We know that the $\wo q$ with simple zeros form a non-empty Zariski-open set in the complete linear system $|K_{\wo M}^2 \otimes_{i=1}^{n}L_{p_i}|$. The extra condition that none of the zeros is at a marked point is obviously also an open condition, so we only need to check that the resulting set is non-empty. If $n=1$ then we only need to show that the marked point is not a base-point of the linear system. Similarly, if there are several marked points then it suffices to show that none is a base point, because then the sections vanishing at a given marked point cut out a hyperplane in the projective space $|K_{\wo M}^2 \otimes_{i=1}^{n}L_{p_i}|$. Using \cite[IV, proposition 3.1]{ha77}, this is equivalent to showing that $h^0(K_{\wo M}^2 L_{p_j}^{*} \otimes_{i=1}^{n}L_{p_i}) = h^0(K_{\wo M}^2 \otimes_{i=1}^{n}L_{p_i}) - 1$ for each $j = 1,\dots,n$---this follows from an easy Riemann-Roch calculation, provided $2g+n>3$. \end{proof} \ble{generic2} Let $\phi$ be a Higgs field on $E$ with $\det(\phi)=q$ and $\wo q$ generic in the sense of \refle{generic}. Then $\wo q$ has simple zeros at each marked point where $x=x'$. Moreover, at every marked point of $M$ we have $\wo\phi_{21}\ne 0$ and $\wo\phi_{12}\ne 0$, where $\wo\phi_{21}$ and $\wo\phi_{12}$ are as in \refeq{Taylor Higgs}. \end{lemma} \begin{proof} Using \refeq{Taylor Higgs} we have that, in our local coordinates around a marked point, \beql{Taylor Higgs 3} \phi = \left( \begin{array}{rr} z^{\alpha-1}\wo\phi_{11}(z^\alpha) & z^{\alpha + x - x' -1}\wo\phi_{12}(z^\alpha) \\ z^{x' - x -1}\wo\phi_{21}(z^\alpha)& -z^{\alpha-1}\wo\phi_{11}(z^\alpha) \end{array}\right)dz, \end{eqnarray} assuming that $x \ne x'$. If $x' = x$ then the $(2,1)$-term is $z^{\alpha -1}\wo\phi_{21}(z^\alpha)dz$. Here the $\wo\phi_{ij}$ are holomorphic functions. If $\wo q$ is generic then it is non-zero at a marked point of $\b M$ and has at most a simple zero at a marked point where $x=x'$---in fact there will be a zero at such a point. It follows that we must have that $\det(\phi) = q$ vanishes exactly to order $\alpha -2$ in $z$ in the first case and order $2\alpha -2$ in the second. Hence $\wo\phi_{21}(0) \ne 0$ and $\wo\phi_{12}(0) \ne 0$ at each marked point of $M$. \end{proof} Henceforth we assume that $\wo q$ is a generic section, as in \refle{generic}, and construct $\det^{-1}(q)$. For the purposes of exposition we also assume that $n_0 = 0$. We face two problems in defining the spectral variety of $\phi$ or $\wo\phi$---the first is that $\wo\phi$ has simple poles at the marked points and the second is that $\wo q$ is not the determinant of $\wo\phi$. Let $s_{p_i}=s_i^{\alpha_i}$ be the canonical section of the point-bundle $L_{p_i}$ associated to a marked point $p_i$ and let $s_0 = \otimes_{i=1}^{n}s_{p_i}$ be the corresponding section of $\otimes_{i=1}^{n}L_{p_i}$. Define \begin{eqnarray*} \oo q = \wo q s_0 \in H^0(K_{\wo M}^2 \otimes_{i=1}^{n}L_{p_i}^2) &\mbox{and}& \oo\phi = \wo\phi s_0 \in \mbox{ParEnd}_0(\wo E)\otimes K_{\wo M} \otimes_{i=1}^{n}L_{p_i}.\label{eq:oophi} \end{eqnarray*} It is clear that $\det(\oo\phi)=\oo q$ and that $\oo q$ has simple zeros (including one at each marked point). Eventually we will need to reverse the construction of $\oo\phi$ from $\phi$; this can be done for a given $\oo\phi \in \mbox{ParEnd}_0(\wo E)\otimes K_{\wo M} \otimes_{i=1}^{n}L_{p_i}$ provided $\oo\phi$ obeys the obvious vanishing conditions at each marked point. The square root $\sqrt{-\oo q}$ defines a smooth Riemann surface $\widehat M$ with double-covering $\pi:\widehat M \to \wo M$ and branched at the zeros of $\oo q$. Therefore there are $4g-4 + 2n$ branch-points and the Riemann-Hurwitz formula gives the genus of $\widehat M$ as $\widehat g = 4g-3 + n$. We set $s=\sqrt{-\oo q}$---a section of $\pi^*(K_{\wo M}\otimes_{i=1}^{n}L_{p_i})$---and $\widehat\phi = \pi^*\oo\phi$. Moreover, if $\sigma$ is the involution interchanging the leaves of $\widehat M$ then $\sigma^* s = -s$ and $\widehat\phi$ is $\sigma$-invariant. In order to reverse the passage from $E$ to $\wo E$ we have to keep track of the quasi-parabolic data. The following lemma is useful here. (Applying the involution $\sigma$, the same result holds for $\sigma^*L =\ker(\widehat\phi - s)$.) \ble{quasi structure} If $\phi$ is a Higgs field on $E$ with $\det(\phi) = q$ and $\wo q$ generic in the sense of \refle{generic}, then the kernel of $\widehat\phi + s$ (with $s$, $\widehat\phi$ defined as above) is a line subbundle $L$ of $\pi^*\wo E$ and, at a marked point (of $\b M$) $p$, $0 \subsetneq L_{\pi^{-1}(p)} \subsetneq \pi^*\wo E_{\pi^{-1}(p)} = \wo E_p$ describes the quasi-parabolic structure. \end{lemma} \begin{proof} At a marked point, using \refeq{Taylor Higgs 2} and \refeq{oophi}, we write \beql{matrix oophi} \oo\phi = \left( \begin{array}{rr} w\wo\phi_{11}(w) & w\wo\phi_{12}(w) \\ \wo\phi_{21}(w)& -w\wo\phi_{11}(w) \end{array}\right)\frac{dw}\alpha, \end{eqnarray} with, from \refle{generic2}, $\wo\phi_{21}(0) \ne 0$ and $\wo\phi_{12}(0) \ne 0$. This means that $\oo\phi$ is not zero at a marked point. Similarly, using the fact that $\wo q$ has simple zeros, $\oo\phi$ is non-zero at {\em every} branch point. Now consider $\widehat\phi + s$: since $\det(\widehat\phi + s)\equiv 0$ this mapping has nullity 1 or 2 at every point. Because $\widehat\phi$ is trace-free and $s$ is scalar it follows that zeros of $\widehat\phi + s$ can only occur at zeros of $s$ {\rm i.\,e.\ } at the ramification points. However, since $\oo\phi$ is non-zero at a branch point $p$ it is impossible for $\widehat\phi + s$ to be zero at $\pi^{-1}(p)$. So $\widehat\phi + s$ is nowhere zero and the kernel is a line bundle. Finally, if $p$ is a marked point it is clear from \refeq{matrix oophi} that $\ker(\widehat\phi + s)_{\pi^{-1}(p)}$ is spanned by $\left( 0 , 1 \right)^T$ in our local coordinates. The result about the quasi-parabolic structure follows. \end{proof} \bth{fibres of det} Suppose that $2g+n-n_0>3$. Given $q \in H^0(\b M,K_{\b M}^2)$ such that $\wo q$ is generic in the sense of \refle{generic} the fibre of the determinant map $\det^{-1}(q)$ is biholomorphic to the Prym variety of the covering $\pi:\widehat M \to \wo M$, determined by $q$ (via $\wo q'$). \eth \begin{proof} Since the proof is familiar \cite[theorem 8.1]{hi87} we only sketch it. We assume $n_0=0$. Fix $q$ such that $\wo q$ is generic and $\widehat M$ as constructed above and also a line bundle $P$ over $\widehat M$ such that $P\sigma^* P = \pi^*(K_{\wo M}^*\Lambda^2\wo E\otimes_{i=1}^{n}L_{p_i}^* )$. Suppose that $(E,\phi)$ is a Higgs $V$-bundle over $M$ with $\det(\phi)=q$. Consider the parabolic bundle $\wo E$ and $\oo\phi\in \mbox{ParEnd}_0(\wo E)\otimes K_{\wo M} \otimes_{i=1}^{n}L_{p_i}$ with determinant $\oo q$ defined as above. Now set $L = \ker(\widehat\phi + s)$ and notice that $L\sigma^*L \cong \pi^*(K_{\wo M}^*\Lambda^2\wo E\otimes_{i=1}^{n}L_{p_i}^*)$. Since $P$ was chosen to have the same property $LP^*$ is an element of the Prym variety. Conversely, we consider $L$ such that $LP^*$ is a given point in the Prym variety. The push-forward sheaf $\pi_*{\cal O}(L)$ is locally free analytic of rank 2 and so defines a rank 2 holomorphic vector bundle $W$ over $\wo M$. There is a natural quasi-parabolic structure on $W^*$ at a branch point $p$ because $W_p = (J_1L)_{\pi^{-1}(p)}$ and there is a natural filtration of jets $ 0 \subset L^*_{\pi^{-1}(p)} \subset (J_1L)^*_{\pi^{-1}(p)}.$ The Hecke correspondence for quasi-parabolic bundles defines a rank 2 holomorphic bundle $W'^*$: that is, the quasi-parabolic structure on $W^*$ defines a natural surjective map ${\cal O}(W^*) \surjarrow {\cal S}$, where ${\cal S}$ is a sheaf supported at the branch points, and the kernel of this map is locally free analytic of rank 2 and so defines $W'^*$. This construction of $W'$ actually recovers $\wo E$: there is a natural map ${\cal O}(W) \to {\cal O}(W')$ which induces an inclusion $L \hookrightarrow \pi^*W'$. Similarly there is an inclusion $\sigma^*L \hookrightarrow \pi^*W'$. As subbundles of $\pi^*W'$, $L$ and $\sigma^*L$ coincide precisely on the ramification points so that there is a map $L\oplus \sigma^*L \to \pi^*W'$ which is an isomorphism away from the ramification points. It follows that $\Lambda^2W' = \Lambda^2\wo E$ and that $W' = \wo E$. Moreover, at a marked point $p$ the inclusion $L_{\pi^{-1}(p)} \hookrightarrow \pi^*\wo E_{\pi^{-1}(p)} = \wo E_p$ gives the quasi-parabolic structure and so we recover the original $V$-bundle $E$ (see \refpr{V-parabolic} and \refle{quasi structure}). We recover the Higgs field simply by defining $\widehat\phi : \pi^*\wo E \to \pi^*(\wo E\otimes K_{\wo M}\otimes_{i=1}^{n}L_{p_i})$ by $\widehat\phi(e) = \mp se$ according as $v \in L$ or $v \in \sigma^* L$. Since this is $\sigma$-invariant it descends to define $\oo\phi$ on $\wo M$---this is trace-free with determinant $\oo q$ and recovers the old $\oo\phi$. At a marked point $p$, we have $\ker(\widehat\phi_{\pi^{-1}(p)}) = L_{\pi^{-1}(p)}$ and hence, in coordinates which respect the quasi-parabolic structure, the $(1,2)$-, $(2,2)$- and $(1,1)$-components of $\oo\phi$ vanish at $p$ to first order in $w$. Of course this is exactly the condition for $\oo\phi$ to define $\wo\phi$ via \refeq{oophi} and to $\wo\phi$ there corresponds a Higgs field $\phi$ on the $V$-bundle $E$. Finally note that if there was an $\oo\phi$-invariant subbundle $L'$ then there would be a section $t \in H^0(K_{\wo M}\otimes_{i=1}^{n}L_{p_i})$ such that for any $l\in L'$, $\oo\phi(l) = tl$. Since $\oo\phi$ is trace-free it would follow that $\oo q = \det(\oo\phi) = -t^2$---contradicting the assumption that $\oo q$ has simple zeros. So $\oo\phi$ has no invariant subbundles and the same is therefore true of $\wo\phi$ and $\phi$. \end{proof} Notice that this shows that a Higgs field in the generic fibre of $\det$ leaves no sub-$V$-bundle invariant (compare \refsu{higalg}). \bsu{The case $g=n-n_0=1$}{detspe} We briefly indicate how the preceding arguments can be modified to identify the generic fibre of the determinant map when $g=n-n_0=1$. We outline the argument working with $V$-bundles although the proofs again require translation to the parabolic case. As before we simplify the exposition by supposing that $n_0=0$ so that there is a single marked point $p=p_1$. \ble{invariants} If $g=n-n_0=1$ then every Higgs field has an invariant sub-$V$-bundle. \end{lemma} \begin{proof} Since $h^0(K^2) = 1$ the natural squaring map $H^0(K) \to H^0(K^2)$ is surjective. Thus, given any Higgs field $\phi$, $\det(\phi) = -s^2$ for some $s \in H^0(K)$. Consider $\theta_\pm = \phi \pm s$: if $\phi\ne 0$ this is non-zero (if $\phi = 0$ then there is nothing to prove) but has determinant zero and so we have line $V$-bundles $L_\pm \hookrightarrow E$ with $L_\pm \subseteq \ker \theta_\pm$. Clearly $L_\pm$ are invariant, with $\phi$ acting on $L_\pm$ by multiplication by $\mp s$. \end{proof} Since the squaring map is surjective, \refle{generic} certainly can't hold in this case---we now consider any non-zero determinant to be `generic'. Using \refle{invariants} we see that any Higgs field with a generic ({\rm i.\,e.\ } non-zero) determinant has two invariant sub-$V$-bundles $L_+$ and $L_-$. Notice that $K = L_1^{\alpha_1 -1}$ and so sections of $K$ are multiples of the canonical section $s_1^{\alpha_1-1}$ and those of $K^2$ are multiples of $s_1^{2\alpha_1-2}$. Thus in \refeq{Taylor Higgs 3} $\wo\phi_{11}(z^{\alpha_1})$ and exactly one of $\wo\phi_{12}(z^{\alpha_1})$ and $\wo\phi_{21}(z^{\alpha_1})$ are non-zero at the marked point, while the other must vanish to first order in $w=z^{\alpha_1}$. A small local calculation using \refeq{Taylor Higgs 3} shows that $L_+$ and $L_-$ have the same isotropy; it is $x$ if $\wo\phi_{21}(0)=0$ and $x'$ if $\wo\phi_{12}(0) = 0$. Hence $L_+L_- \cong \Lambda L_{1}^{x - x'}$ or $\Lambda L_1^{x' - x - \alpha}$, where the isotropy of $L_\pm$ is $x$ in the first case and $x'$ in the second. Using these and stability, we calculate that $c_1(L_\pm) = r/2 + x/\alpha$ or $(r-1)/2 + x'/\alpha$, respectively, where $c_1(E) = r + (x + x')/\alpha$. Notice that the parity of $r$ determines the isotropy of $L_\pm$. Thus a point in the generic fibre gives a point not of a Prym variety but of the Jacobian $\mbox{Jac}_0{M}\cong T^2$ corresponding to $L_+$. Reversing the correspondence as in \refth{fibres of det} yields the following result. \bpr{special} If $g=n-n_0=1$ then for $q\in H^0(\b M,K^2_{\b M})\setminus \{ 0 \}$ the fibre $\det^{-1}(q)$ is biholomorphic to the Jacobian torus. \end{proposition} \bsu{Non-stable $V$-Bundles in Fibres of the Determinant Map}{detnon} We have a natural inclusion of the cotangent bundle to the moduli of stable $V$-bundles in to the moduli of stable Higgs $V$-bundles and we would like to show, following \cite[\S 8 ]{hi87}, that in fact we have a fibrewise compactification with respect to the determinant map. Thus we need to analyse the fibres of the determinant map and check that, generically, the non-stable $V$-bundles form subvarieties of codimension at least 1. We wish to adapt Hitchin's argument here but there are additional complications and a new variant of the argument is needed in the special case $g=n-n_0=1$. \bpr{} Suppose that $2g+n-n_0>3$. For fixed, generic, $q\in H^0(\b M,K_{\b M}^2)$ let ${\rm Prym}(\widehat M)$ be the Prym variety which is the fibre of the determinant map (\refth{fibres of det}). Then the points of ${\rm Prym}(\widehat M)$ corresponding to non-stable $V$-bundles form a finite union of subvarieties of codimension at least 1. \end{proposition} \begin{proof} Suppose $n_0=0$ and consider $L_E\hookrightarrow E$ a destabilising sub-$V$-bundle, with $\wo{L_E} \hookrightarrow \wo E$ {\em parabolically} destabilising (see \cite{fs92} and \refsu{parhig}) and $L' = \pi^*\wo{L_E}$. The outline of the argument is similar to that of \cite[\S 8]{hi87}---with which we assume familiarity---but there are two problems. Firstly, a sufficient condition for lifts from $H^0(L'^*L^*\pi^*\Lambda^2\wo E)$ to $H^0(L'^*\pi^*\wo E)$ to be unique is $H^0(L'^*L)=0$ but this is not always the case if $g=0$. However, {\em invariant} lifts will still be unique because $H^0(L'^*L)\hookrightarrow H^0(L'^*\pi^*\wo E)$ is moved by the involution $\sigma$. Secondly, because $\wo{L_E}$ is parabolic destabilising we can't fix the degree of $L'^*L^*\pi^*\Lambda^2\wo E$ in the same way that Hitchin does. Let the isotropy of $L_E$ be specified by an isotropy vector $(\epsilon_i)$. A small computation with the stability condition shows \begin{eqnarray*} c_1(L'^*L^*\pi^*\Lambda^2\wo E) \le \sum_{i=1}^n\frac{\epsilon_i(x_i'-x_i)}{\alpha_i} + 2g -2. \end{eqnarray*} Since $L_{\pi^{-1}(p)}$ gives the flag which describes the quasi-parabolic structure at a marked point $p$, by \refle{quasi structure}, the subset of $\pi^{-1}(\{p_1,\dots,p_n\})$ at which our section of $L'^*L^*\pi^*\Lambda^2\wo E$ vanishes is just $\pi^{-1}(\{p_i : \epsilon_i=1\})$. Hence, for given $(\epsilon_i)$, it is more natural to consider sections of $(\otimes_{\{i : \epsilon_i=1\}}L_i^*)L'^*L^*\pi^*\Lambda^2\wo E$ and these correspond to divisors of degree less than or equal to $\sum_{i=1}^n(\epsilon_i(x_i'-x_i)/\alpha_i) - n_+ +2g -2$. For each $(\epsilon_i)$ (a finite number) we obtain a subvariety of the variety of effective divisors and correspondingly a subvariety of the Prym variety of codimension at least 1. \end{proof} \bpr{} If $g=n-n_0=1$ then for $q\in H^0(K^2_{\b M})\setminus \{ 0 \}$ there are only a finite number of points in the fibre $\det^{-1}(q)$ corresponding to non-stable $V$-bundles. \end{proposition} \begin{proof} Again, we consider a destabilising sub-$V$-bundle $L_E \hookrightarrow E$ and the corresponding parabolic bundle $\wo{L_E}$. Since $\wo{L_E}$ is parabolic destabilising $2c_1(\wo{L_E}) \ge c_1(\wo E) + 1$ or $2c_1(\wo{L_E}) \ge c_1(\wo E)$, according to whether $L_E$ has isotropy $x$ or $x'$. Recall (from \refsu{detspe}) that $E$ has two $\phi$-invariant sub-$V$-bundles $L_\pm$ and so is an extension $0 \to L_\pm \to E \to L_\pm^*\Lambda \to 0$. Set $r=c_1(\wo E)$. The discussion in \refsu{detspe} also shows that if $r$ is even then $c_1(\wo{L_\pm})=r/2$, $L_\pm$ have isotropy $x$ and $\wo{L_+}\wo{L_-}\cong \Lambda^2\wo E$, while if $r$ is odd then $c_1(\wo{L_\pm}) = (r-1)/2$, $L_\pm$ have isotropy $x'$ and $\wo{L_+}\wo{L_-}\cong \Lambda^2\wo EL_{p_1}^*$. Consider the sequence of bundles \begin{eqnarray*} 0 \to \wo{L_E}^*\wo{L_\pm} \to \wo{L_E}^*\wo E \to \wo{L_E}^*\wo{L_\pm}^*\Lambda^2\wo E \to 0. \end{eqnarray*} and the first three terms of the associated cohomology long exact sequence. By assumption $H^0(\wo{L_E}^*\wo E)$ is non-zero so at least one of $\wo{L_E}^*\wo{L_+}$ and $\wo{L_E}^*\wo{L_+}^*\Lambda^2\wo E$ must have a non-zero section and the same is true with $L_-$ in place of $L_+$. If we had that $H^0(\wo{L_E}^*\wo{L_\pm})\ne 0$ and $H^0(\wo{L_E}^*\wo{L_\pm}^*\Lambda^2\wo E)=0$ then the inclusion of $\wo{L_E}$ in $\wo E$ would have to factor through that of $\wo{L_\pm}$, which is impossible as $\wo{L_\pm}$ does not destabilise. So $\wo{L_E}^*\wo{L_+}^*\Lambda^2\wo E$ and $\wo{L_E}^*\wo{L_-}^*\Lambda^2\wo E$ must have non-zero sections. However, considering cases according to the parity of $r$ and the isotropy of $L_E$, we see that $c_1(\wo{L_E}^*\wo{L_\pm}^*\Lambda^2\wo E)\le 0$. It follows that a non-stable $V$-bundle occurs only if $\wo{L_E} \cong \wo{L_\pm}^*\Lambda^2\wo E$. Since $\wo{L_+}\wo{L_-}\cong \Lambda^2\wo E$ or $\wo{L_+}\wo{L_-}\cong \Lambda^2\wo EL_{p_1}^*$, it follows that $\wo{L_+}^2\cong \Lambda^2\wo E$ or $\wo{L_+}^2\cong \Lambda^2\wo EL_{p_1}^*$. Hence, if a non-stable $V$-bundle occurs then $\wo{L_+}$ is one of the $2^{2g}=4$ possible square roots of a given line bundle. \end{proof} \bse{Representations and Higgs $V$-bundles}{rep} Throughout this section $E\to M$ is a complex rank 2 $V$-bundle over an orbifold Riemann surface of negative Euler characteristic. We also suppose that a fixed metric and Yang-Mills connexion, $A_\Lambda$, are given on $\Lambda$. \bsu{Stable Higgs $V$-bundles and Projectively Flat Connexions}{repsta} Suppose that $E$ is given a Higgs $V$-bundle structure with Higgs field $\phi$, compatible with $A_\Lambda$. Given a Hermitian metric on $E$ inducing the fixed metric on $\Lambda$, there is a unique Chern connexion $A$ compatible with the holomorphic and unitary structures and inducing $A_\Lambda$ on $\Lambda$. The metric also defines an adjoint of $\phi$, $\phi^*$. Set \begin{eqnarray*} D = \partial_A + \phi + \o\partial_A + \phi^*; \end{eqnarray*} this is a (non-unitary) connexion with curvature $F_D = F_A + [\phi,\phi^*]$ and $D$ is projectively flat if and only if the pair $(A,\phi)$ is Yang-Mills-Higgs. The determinant-fixing condition on $D$ is simply that it induces the fixed ({\em unitary}) Yang-Mills connexion $A_\Lambda$ in $\Lambda$. Conversely, given a connexion $D$ (with fixed determinant) and a Hermitian metric on $E$, inducing the fixed metric on $\Lambda$, we can decompose $D$ into its $(1,0)$- and $(0,1)$-parts; $D=\partial_1 + \o\partial_2$. There are then uniquely defined operators $\o\partial_1$ and $\partial_2$ (of types $(0,1)$ and $(1,0)$ respectively) such that $d_1=\partial_1 + \o\partial_1$ and $d_2=\partial_2 + \o\partial_2$ are unitary connexions. Define $\phi = (\partial_1-\partial_2)/2$ and $d_A=(d_1+d_2)/2$ so that $\o\partial_A = (\o\partial_1+ \o\partial_2)/2$. Clearly $(A,\phi)$ is a Higgs pair if and only if $\o\partial_A(\phi) = 0$, {\rm i.\,e.\ } $\phi$ is holomorphic; if we define $D''= \o\partial_A + \phi$ then this condition becomes $D''^2 = 0$. Here $D''$ is a first order operator which satisfies the appropriate $\o{\partial}$-Leibniz rule. Moreover, if $D''^2=0$ then $(A,\phi)$ is Yang-Mills-Higgs\ if and only if $D$ has curvature $- \pi {\rm i}\, c_1(\Lambda)\Omega I_E$. {}From now on suppose that $D$ has curvature $- \pi {\rm i}\,c_1(\Lambda)\Omega I_E$. We call a Hermitian metric (with fixed determinant) \de{twisted harmonic} with respect to $D$ if the resulting $D''$-operator satisfies $D''^2=0$. Using the fact that the curvature of $D$ is $- \pi {\rm i}\,c_1(\Lambda)\Omega I_E$, a small calculation shows that the condition for the metric to be twisted harmonic is $F_1 = F_2$, where $F_i$ is the curvature of $d_i$, for $i=1,2$. If the metric is twisted harmonic then $D''$ defines a Higgs $V$-bundle with respect to which the metric is Hermitian-Yang-Mills-Higgs. Clearly the processes of passing from a Higgs $V$-bundle to a projectively flat connexion and vice-versa are mutually inverse and respect the determinant-fixing conditions. We prove an existence result for twisted harmonic metrics, following \cite{do87}. The connexion $D$ on $E$ comes from a projectively flat connexion in the corresponding principal $GL_2({\Bbb C})$ $V$-bundle $P$ with $E=P\times_{GL_2({\Bbb C})}{\Bbb C}^2$. Hence $D$ determines a holonomy representation $\rho_D:\pi_1^V(M) \to PSL_2({\Bbb C})$. Let ${\rm Herm}^+_2$ denote the $2\times 2$ positive-definite Hermitian matrices (with the metric described in \cite[\S VI.1]{ko87}). The corresponding $V$-bundle of Hermitian metrics on $E$ is just $H'=P\times_{GL_2({\Bbb C})}{\rm Herm}_2^+$. Here $GL_2({\Bbb C})$ acts on ${\rm Herm}^+_2$ by $h \mapsto \o{g}^T h g$, for $h\in {\rm Herm}_2^+$ and $g\in GL_2({\Bbb C})$. This is an action of $PSL_2({\Bbb C})$ and so $H'$ is flat and can be written as $H'=H'_{\rho_D}={\cal H}^2 \times_{\rho_D} {\rm Herm}^+_2$ (where ${\cal H}^2$ is the universal cover of $M$). A choice of Hermitian metric on $E$ is a section of $H'_{\rho_D}$ or a $\pi_1^V(M)$-equivariant map ${\cal H}^2 \to {{\rm Herm\,}}_2^+$---is this map is harmonic in the sense that it minimises energy among such maps? Using the determinant-fixing condition, we suppose that the map to ${\rm Herm}^+_2$ has constant determinant 1. We identify the subspace of ${\rm Herm}_2^+ \cong GL_2({\Bbb C})/U(2)$ in which the image of the map lies with $SL_2({\Bbb C})/SU(2) \cong {\cal H}^3$. So we consider sections of the flat ${\cal H}^3$ $V$-bundle $H_{\rho_D}={\cal H}^2 \times_{\rho_D} {\cal H}^3$: the sections of $H_{\rho_D}$ are precisely the types of map considered by Donaldson in \cite{do87}. The condition that a metric $h$ be twisted harmonic will then be precisely that it is given by a harmonic $\pi_1^V(M)$-equivariant map $\widehat h: {\cal H}^2 \to {\cal H}^3$. Donaldson shows that the Euler-Lagrange condition for the map $\widehat h$ to be harmonic is just $d_A^*(\phi + \phi^*) = 0$ and moreover that, at least in the smooth case and when $\rho_D$ is irreducible, such a harmonic map always exists. This Euler-Lagrange condition agrees with our definition of a twisted harmonic metric. For the existence of such harmonic maps we either follow Donaldson's proof directly or argue equivariantly, as in \refsu{ymhequ}, obtaining the following results. \bpr{Donaldson} Let $\rho_D:\pi_1^V(M) \to PSL_2({\Bbb C})$ be an irreducible representation and $s_0$ a section of the flat ${\cal H}^3$ $V$-bundle $H_{\rho_D}={\cal H}^2 \times_{\rho_D} {\cal H}^3$. Then $H_{\rho_D}$ admits a twisted harmonic section homotopic to $s_0$. \end{proposition} \bco{twisted exists} Let $\Lambda$ have a fixed Hermitian metric and compatible Yang-Mills connexion. Given an irreducible $GL_2({\Bbb C})$-connexion $D$ on $E$ with curvature $- \pi {\rm i}\, c_1(\Lambda)\Omega I_E$ and fixed determinant, $E$ admits a Hermitian metric of fixed determinant which is twisted harmonic with respect to $D$. Hence $D$ determines a stable Higgs $V$-bundle structure on $E$ with fixed determinant, for which the metric is Hermitian-Yang-Mills-Higgs. \end{corollary} \bco{} Let $E$ have a fixed Hermitian metric and let $\Lambda$ have a compatible Yang-Mills connexion. Let $D$ be an irreducible $GL_2({\Bbb C})$-connexion on $E$ with curvature $- \pi {\rm i}\, c_1(\Lambda)\Omega I_E$ and fixed determinant. Then there is a complex gauge transformation $g \in {\cal G}^c$, of determinant 1, such that the {\em fixed} metric is twisted harmonic with respect to $g(D)$. Hence $g(D)$ determines a stable Higgs $V$-bundle structure on $E$ with fixed determinant. \end{corollary} To identify the space of such projectively flat connexions modulo gauge equivalence with our moduli space of Higgs $V$-bundles we have to consider the actions of the gauge groups and the question of irreducibility. We have the following result adapted from \cite[theorem 9.13 \& proposition 9.18]{hi87}. \bpr{irreducibles} Let $E\to M$ be a complex rank 2 $V$-bundle with a fixed Hermitian metric and compatible Yang-Mills connexion on the determinant line $V$-bundle $\Lambda$. Then the following hold. \begin{enumerate} \item\label{irreducible'} A Yang-Mills-Higgs\ pair $(A,\phi)$ (with fixed determinant) is irreducible if and only if the corresponding projectively flat $GL_2({\Bbb C})$-connexion $D=\partial_A + \o\partial_A + \phi + \phi^*$ is irreducible. \item\label{gauge} Two irreducible $GL_2({\Bbb C})$-connexions on $E$ with curvature $- \pi {\rm i}\, c_1(\Lambda)\Omega I_E$ (and fixed determinant), $D$ and $D'$, are equivalent under the action of ${\cal G}^c$ if and only if the corresponding Yang-Mills-Higgs\ pairs $(A,\phi)$ and $(A',\phi')$ are equivalent under the action of $\cal G$. \end{enumerate} \end{proposition} \bsu{Projectively Flat Connexions and Representations}{reprep} In the smooth case projectively flat connexions are described by representations of a universal central extension of the fundamental group (see \cite{hi87}, also \cite[\S 6]{ab82}). However over an orbifold Riemann surface there is in general no {\em one} central extension which will do \cite[\S 3]{fs92} but the determinant-fixing condition tells us that the appropriate central extension to use is the fundamental group of the circle $V$-bundle $S(\Lambda)$. Let $(y_i)$ ($0\le y_i \le \alpha_i-1$) denote the isotropy of a line $V$-bundle $L$ and let $b = c_1(L) - \sum_{i=1}^n (y_i/\alpha_i)$. The orbifold fundamental group of $S(L)$ is well-known (see, for instance, \cite[\S 2]{fs92}) and has presentation \beql{circle} \begin{array}{rcl} \pi_1^V(S(L)) &=& \langle a_1,b_1,\dots, a_g,b_g,q_1,\dots, q_n,h \quad | \\ & &\quad [a_j,h]=1, \ [b_j,h]=1,\ [q_i,h]=1,\ q_i^{\alpha_i} h^{y_i}=1,\ q_1\dots q_n[a_1,b_1]\dots[a_g,b_g]h^{-b}=1 \rangle. \end{array} \end{eqnarray} \bpr{representations} Let $\Lambda\to M$ be a line $V$-bundle with a fixed Hermitian metric and compatible Yang-Mills connexion. Let $S(\Lambda)$ be the corresponding circle $V$-bundle. Then there is a bijective correspondence between \begin{enumerate} \item conjugacy classes of irreducible representations $\pi_1^V(S(\Lambda)) \to SL_2({\Bbb C})$ such that the generator $h$ in \refeq{circle} is mapped to $-I_2\in SL_2({\Bbb C})$ and \item isomorphism classes of pairs $(E,D)$, where $E$ is a $GL_2({\Bbb C})$ $V$-bundle with $\Lambda^2E = \Lambda$ and $D$ is an irreducible $GL_2({\Bbb C})$ connexion on $E$ with curvature $- \pi {\rm i}\, c_1(\Lambda)\Omega I_E$ and inducing the fixed connexion on $\Lambda$. \end{enumerate} \end{proposition} \begin{proof} The proof can be carried over from \cite[theorem 4.1]{fs92} (compare also \cite[theorem 6.7]{ab82}) except that we need to replace $U(2)$ with $GL_2({\Bbb C})$ at each stage---only the unitary structure on the determinant line $V$-bundle is necessary for the proof. \end{proof} Since \refpr{representations} insists that $h$ maps to $-I_2$, it is sufficient to consider a central ${\Bbb Z}_2$-extension rather than the central ${\Bbb Z}$-extension of $\pi_1^V(M)$ given by the presentation \refeq{circle}---this is equivalent to adding the relation $h^2=1$ to that presentation. Then it is only the {\em parity} of the integers $y_i$ and $b$ that matters. Something a little subtler is true. Recall \refre{roots}: it is sufficient to consider topological $\Lambda$'s modulo the equivalence $\Lambda \sim \Lambda L^2$. Moreover, the topology of $\Lambda$ is specified by the $y_i$'s and $b$ (\refpr{v-bundles})---write $\Lambda=\Lambda_{(b,(y_i))}$ to emphasise this. Clearly if $(b,(y_i)) \equiv (b',(y'_i)) \pmod2$ (meaning that the congruence holds componentwise) then $\Lambda_{(b,(y_i))} \sim \Lambda_{(b',(y'_i))}$. However, if $\alpha_i$ is odd then $L$ can be chosen so that tensoring by $L^2$ brings about a change $y_i \mapsto y_i+1$; if any $\alpha_i$ is even then a change $b \mapsto b+1$ is possible similarly. These equivalences correspond to group isomorphisms between the corresponding presentations \refeq{circle}, with the added relation $h^2=1$. Thus we normalise the $y_i$'s and $b$ to find exactly one representative of each class, supposing that \beql{normal} y_i = \left\{\begin{array}{rl} 0 &\mbox{if $\alpha_i$ is odd;}\\ 0,1&\mbox{if $\alpha_i$ is even;} \end{array}\right. \quad b = \left\{\begin{array}{rl} 0 &\mbox{if at least one $\alpha_i$ is even;}\\ 0,1&\mbox{if no $\alpha_i$ is even.} \end{array}\right. \end{eqnarray} This is equivalent to considering only the following \de{square-free} topological $\Lambda$'s: \beql{normal2} \Lambda \in \left\{\begin{array}{rl} \{\otimes_{\alpha_i {\rm\ even}}L_i^{\delta_i} : \delta_i =0,1 \} &\mbox{if at least one $\alpha_i$ is even;}\\ \{L^{\delta_0} : \delta_0 = 0,1\} &\mbox{if no $\alpha_i$ is even,} \end{array}\right. \end{eqnarray} where $L$ has no isotropy with $c_1(L)=1$ and the $L_i$ are the point $V$-bundles of \refsu{orbdiv}. An alternative way to understand these ${\Bbb Z}_2$-extensions of the fundamental group is as follows. Since $SL_2({\Bbb C})$ double-covers $PSL_2({\Bbb C})$ any representation $\rho_D:\pi_1^V(M)\to PSL_2({\Bbb C})$ induces a central ${\Bbb Z}_2$-extension of $\pi_1^V(M)$: \beql{extensor} 0 \to {\Bbb Z}_2 \to \Gamma \to \pi_1^V(M) \to 0. \end{eqnarray} Since the group of central ${\Bbb Z}_2$-extensions of $\pi_1^V(M)$ is discrete, the $\Gamma$ thus induced is constant over connected components of the representation space. So, given any $\rho_D$, we obtain an extension $\Gamma$: what invariants $(b,(y_i))$ characterise these $\Gamma$'s and thus the central ${\Bbb Z}_2$-extensions of $\pi_1^V(M)$? The answer is that $(b,(y_i))$ can be supposed to have one of the normalised forms given by \refeq{normal} and so these parameterise the central ${\Bbb Z}_2$-extensions of $\pi_1^V(M)$. This is because the image of each generator of \refeq{circle} has exactly two possible lifts to $SL_2({\Bbb C})$ except that $h$ must map to $-I_2$: choosing lifts at random, the relations $q_i^{\alpha_i} h^{y_i}=1$ and $q_1\dots q_n[a_1,b_1]\dots[a_g,b_g]h^{-b}=1$ of \refeq{circle} will be satisfied for exactly one choice of normalised $(b,(y_i))$. By our previous discussion, this is exactly equivalent to considering only the square-free $\Lambda$'s of \refeq{normal2}. As well as topological types of determinant line $V$-bundles we need to consider topological types of rank 2 $V$-bundles with the {\em same} determinant line $V$-bundle---\refpr{representations} deals with all topological types of $V$-bundles with the same determinant line $V$-bundle simultaneously. These types can be determined following the ideas of \cite[\S 4]{fs92}, as follows. The various topological types are distinguished by the rotation numbers associated to the images of the elliptic generators $q_i$ of the presentation \refeq{circle}. By this we mean that the image of $q_i$ has conjugacy class described by the roots of its characteristic polynomial, necessarily of the form $e^{\pi{\rm i} r_i/\alpha_i}$, $e^{-\pi{\rm i} r_i/\alpha_i}$, for $0\le r_i \le \alpha_i$; these $r_i$ are the \de{rotation numbers}. Notice that the relation $q_i^{\alpha_i} h^{y_i}=1$ means that $r_i$ has the same parity as $y_i$ and this is the only {\em a priori} restriction on $r_i$. Call an abstract set of rotation numbers $(r_i)$ \de{compatible with $\Lambda$} if $r_i$ has the same parity as $y_i$. The result is the following and the proof, using \refpr{v-bundles}, is easy. \ble{rotation numbers} The topological types of $GL_2({\Bbb C})$ $V$-bundles $E$ with fixed determinant constructed in \refpr{representations} correspond to the rotation numbers $r_i$ associated to the images of the elliptic generators $q_i$ of the presentation \refeq{circle}. \end{lemma} Denote the space of representations of $\pi_1^V(S(\Lambda))$ into $SL_2({\Bbb C})$, sending the generator $h$ of \refeq{circle} to $-I_2$, by ${\rm Hom\,}^{-1}(\pi_1^V(S(\Lambda)),SL_2({\Bbb C}))$ and the {\em irreducible} representations by ${\rm Hom\,}^{*,-1}(\pi_1^V(S(\Lambda)),SL_2({\Bbb C}))$, for a fixed line $V$-bundle $\Lambda$. For any set of rotation numbers $(r_i)$ (with $0\le r_i \le \alpha_i$ and $r_i\equiv y_i \pmod2$) we have a corresponding subset ${\rm Hom\,}^{-1}_{(r_i)}(\pi_1^V(S(\Lambda)),SL_2({\Bbb C}))$ and, by \refpr{representations} and the results of \refsu{repsta}, a bijection between ${\rm Hom\,}^{*,-1}_{(r_i)}(\pi_1^V(S(\Lambda)),SL_2({\Bbb C}))/SL_2({\Bbb C})$ and the moduli space of stable Higgs $V$-bundles (with fixed determinants) on the topological $E$ corresponding to the rotation numbers (\refle{rotation numbers}). The representation space ${\rm Hom\,}^{*,-1}_{(r_i)}(\pi_1^V(S(\Lambda)),SL_2({\Bbb C}))/SL_2({\Bbb C})$ can be thought of as the quotient of a set of $2g+n$ matrices subject to conditions corresponding to the relations of \refeq{circle} and so has a natural topology; whether this description makes it into a smooth manifold is by no means immediate. Therefore we use the bijection with the moduli space of stable Higgs $V$-bundles, which is easily seen to be a homeomorphism, to define a manifold structure on this representation space. In summary we have the following theorem. \bth{} Let $M$ be an orbifold Riemann surface with negative Euler characteristic. Let $\Lambda$ be a fixed line $V$-bundle over $M$ and $(r_i)$ a set of rotation numbers compatible with $\Lambda$. Then the representation space ${\rm Hom\,}^{*,-1}_{(r_i)}(\pi_1^V(S(\Lambda)),SL_2({\Bbb C}))/SL_2({\Bbb C})$ is a complex manifold of dimension $6(g-1)+2(n-n_0)$, where $n_0$ is the number of rotation numbers congruent to 0 (mod $\alpha$). \eth \bre{twist again} In \refre{roots} we noted that twisting by a non-trivial topological root $L$ induces a map ${\cal M}(E,A_\Lambda) \leftrightarrow {\cal M}(E\otimes L,A_\Lambda)$, preserving the topology of $\Lambda$ but altering that of $E$. On the level of representations there is an equivalent map. Given any element $\widehat\rho_D \in {\rm Hom\,}^{-1}_{(r_i)}(\pi_1^V(S(\Lambda)),SL_2({\Bbb C}))$ we can obtain a representation with different rotation numbers and covering the same $PSL_2({\Bbb C})$-representation, by altering the signs of the images of certain of the generators of \refeq{circle}. We can change the sign of $\widehat\rho_D(q_i)$ (bringing about a change of rotation number $b_i \mapsto \alpha_i-b_i$) provided $\alpha_i$ is even and provided an even number of such changes is made---these conditions preserve the relations $q_i^{\alpha_i} h^{y_i}=1$ and $q_1\dots q_n[a_1,b_1]\dots[a_g,b_g]h^{-b}=1$. \end{remark} When there are no reducible points we can apply, among other results, \refpr{metric} and \refco{topology}. By \refle{rotation numbers} we can discuss the existence of reducible points in terms of the rotation numbers. (Either $\Lambda$ or a specific set of rotation numbers may provide an obstruction to the existence of reductions.) The discussion in \refsu{ymhmod} shows that reductions exist if and only if there exists an isotropy vector $(\epsilon_i)$ such that \begin{eqnarray*} \sum_{i=1}^n\frac{\epsilon_i(x'_i-x_i) + (x'_i + x_i)}{\alpha_i}\equiv c_1(\Lambda) \pmod 2. \end{eqnarray*} A small calculation expresses this in terms of the rotation numbers. Thus we obtain the following result. \bpr{} Let $M$ be an orbifold Riemann surface with negative Euler characteristic. Let $\Lambda$ be a fixed line $V$-bundle over $M$ with isotropy $(y_i)$ and $c_1(\Lambda) = b + \sum_{i=1}^n(y_i/\alpha_i)$. Let $(r_i)$ be a compatible set of rotation numbers. Then the representation space ${\rm Hom\,}^{-1}_{(r_i)}(\pi_1^V(S(\Lambda)),SL_2({\Bbb C}))/SL_2({\Bbb C})$ contains reducible points if and only if there exists an isotropy vector $(\epsilon_i)$ such that \begin{eqnarray*} \sum_{i=1}^n\frac{\epsilon_ir_i}{\alpha_i}\equiv b \pmod 2. \end{eqnarray*} When no reducible points exist the complex manifold ${\rm Hom\,}^{-1}_{(r_i)}(\pi_1^V(S(\Lambda)), SL_2({\Bbb C}))/SL_2({\Bbb C})$ \begin{enumerate} \item admits a complete hyper-K\"ahler metric and \item is connected and simply-connected. \end{enumerate} \end{proposition} \bsu{Real Representations}{reprea} In the previous subsection we discussed $SL_2({\Bbb C})$-representations of central extensions of the orbifold fundamental group. Here we study the submanifold of $SL_2({\Bbb R})$-representations. First notice that any irreducible representation into $SL_2({\Bbb C})$ can fix at most one disk ${\cal H}^2\subset {\cal H}^3$ because the intersection of two fixed disks would give a fixed line and hence define a reduction of the representation. Moreover, any representation which does fix a disk can be conjugated to a real representation and the conjugation action of $SL_2({\Bbb C})$ then reduces to that of $SL_2({\Bbb R})$. Now consider the action of complex conjugation on a representation. Recall that, via \refpr{representations} and \refco{twisted exists}, irreducible representations correspond to stable Higgs $V$-bundles. Note that $\pi_1^V(S(\Lambda))$ and $\pi_1^V(S(\o\Lambda))$ are isomorphic via the map $h \mapsto h^{-1}$: the following proposition follows, exactly as in \cite{si90}. \bpr{} Let $E$ be a complex rank 2 $V$-bundle such that $\Lambda$ has a fixed Hermitian metric and compatible Yang-Mills connexion. Let $\widehat\rho_D:\pi_1^V(S(\Lambda)) \to SL_2({\Bbb C})$ be an irreducible representation, sending $h$ to $-I_2$, with corresponding stable Higgs $V$-bundle structure on $E$, $(E_A,\phi)$. Then the complex conjugate representation (thought of as a representation of $\pi_1^V(S(\o\Lambda))$) determines a Higgs $V$-bundle structure on $\o E$, isomorphic to $(E_A,-\phi)^*$. \end{proposition} \bco{} Let $E$ be a complex rank 2 $V$-bundle such that $\Lambda$ has a fixed Hermitian metric and compatible Yang-Mills connexion. Let $\widehat\rho_D$ be an irreducible {\em real} representation $\widehat\rho_D:\pi_1^V(S(\Lambda)) \to SL_2({\Bbb R})$, sending $h$ to $-I_2$, with corresponding Higgs $V$-bundle structure $(E_A,\phi)$. Then there is an isomorphism of Higgs $V$-bundles $(E_A,\phi)\cong (E_A,-\phi)$. \end{corollary} Consider the involution on the moduli space of stable Higgs $V$-bundles (with fixed unitary structure and determinants) defined by $\sigma: (E,\phi) \mapsto (E,-\phi)$, where now $E$ denotes a holomorphic $V$-bundle and $(E,\phi)$ is a stable Higgs $V$-bundle. The fixed points of $\sigma$ can be determined much as the fixed points of the circle action were in the proof of \refth{Morse}. If $(E,\phi)$ is itself fixed then $\phi=0$ and $E$ is a stable $V$-bundle. Suppose now that $\phi\ne 0$. If $(E,\phi)$ is only fixed up to complex gauge-equivalence then we have an element $g \in {\cal G}^c$ such that $g(E,\phi) = (E,-\phi)$. Since $g$ fixes $E$ it must fix the Chern connexion $A$ and since $g$ cannot be a scalar it leads to a reduction of $A$ to a direct sum of $U(1)$-connexions. Hence we have a holomorphic decomposition $E = L \oplus L^*\Lambda$, where, without loss of generality, we may suppose that $2c_1(L) - c_1(\Lambda) \ge 0$. Since $(A,\phi)$ is an irreducible pair, $g$ must have order 2 in ${\cal G}^c$ and fix $A$. It follows that with respect to this decomposition (or, if $A$ has stabiliser $SU(2)$, {\em choosing} a decomposition) we can write \begin{eqnarray*} g =\pm\left( \begin{array}{ll} {\rm i} & 0\\ 0 & -{\rm i} \end{array} \right) &\mbox{and}& \phi = \phantom{\pm}\left( \begin{array}{rr} t & u\\ v & -t \end{array} \right). \end{eqnarray*} (Since our Higgs $V$-bundle is stable, we must have $v$ non-zero.) Calculating the conjugation-action of $g$ on $\phi$ we find that $t=0$. Recall that we chose $L$ with $2c_1(L) - c_1(\Lambda) \ge 0$ but to avoid semi-stable points (when $u=0$) we suppose that there is strict inequality. Exactly as in the proof of \refth{Morse} we consider the topological possibilities $L=L_{(m,(\epsilon_i))}$: we can have any $(m,(\epsilon_i))$ such that $2c_1(L) > c_1(\Lambda)$ and $c_1(\wo{KL^{-2}\Lambda}) = r \ge 0$. Then the possible holomorphic structures and the values of $v$, modulo the ${\Bbb C}^*$ automorphism group, are given by the effective (integral) divisors of order $r$ and taking square roots. A difference from \refth{Morse} is that $u$ needn't be zero; indeed, $u$ can take any value in $H^0(KL^2\Lambda^*)$. We obtain the following result, where $l$ is defined as in \refsu{orbint}. \bpr{real manifolds} Let $M$ be an orbifold Riemann surface of negative Euler characteristic and suppose that $E\to M$ admits no reducible Yang-Mills-Higgs\ pairs. Then the fixed points of the involution induced on ${\cal M}(E,A_\Lambda)$ by the mapping $(A,\phi) \mapsto (A,-\phi)$ consist of complex $(3g-3+n-n_0)$-dimensional submanifolds ${\cal M}_0$ and ${\cal M}_{(m,(\epsilon_i))}$, for every integer $m$ and isotropy vector $(\epsilon_i)$ such that \begin{eqnarray*} l < 2m + \sum_{i=1}^n \frac{\epsilon_i(x'_i-x_i)}{\alpha_i} \le l + 2g -2 + \sum_{i=1}^n\frac{\epsilon_i(x_i'-x_i)}{\alpha_i} + n_-. \end{eqnarray*} The manifold ${\cal M}_0$ is the moduli space of stable $V$-bundles with fixed determinants, while ${\cal M}_{(m,(\epsilon_i))}$ is a rank $(2m -l + g -1 +n_+)$ vector-bundle over a $2^{2g}$-fold covering of $S^r\wo M$, where $r = l -2m + 2g -2 + n_-$. \end{proposition} We interpret this as a result about $PSL_2({\Bbb R})$-representations of $\pi_1^V(M)$. Again, a representation $\rho_D$ of $\pi_1^V(M)$ into $PSL_2({\Bbb R})$ induces a central ${\Bbb Z}_2$-extension $\Gamma$ of $\pi_1^V(M)$, as in \refeq{extensor}, which is just $\pi_1^V(S(\Lambda))$ with the added relation $h^2=1$, for some square-free $\Lambda$. Consider the points of ${\rm Hom\,}^{-1}(\pi_1^V(S(\Lambda)),SL_2({\Bbb R}))$ covering $\rho_D$. On the level of representations there are $2^{2g+n_2-1}$ (or $2^{2g}$ if $n_2 = 0$) choices of sign for the images of certain generators and these correspond to twisting a stable Higgs $V$-bundle by any of the $2^{2g+n_2-1}$ (or $2^{2g}$) holomorphic roots of the trivial line $V$-bundle. In particular, if $n_2 \ge 1$ then the topology of the associated $E$ is only determined up to twisting by the $2^{n_2-1}$ non-trivial topological roots (see \refre{twist again}). Excluding the topologically non-trivial roots, we have an action of ${\Bbb Z}_2^{2g}$ on the fixed point submanifolds of \refpr{real manifolds} which is easily seen to be free if $E$ admits no reducible Yang-Mills-Higgs\ pairs. Moreover, even when $E$ admits reducibles there will be fixed submanifolds ${\cal M}_{(m,(\epsilon_i))}$ with \begin{eqnarray*} l \le 2m + \sum_{i=1}^n \frac{\epsilon_i(x'_i-x_i)}{\alpha_i} \le l + 2g -2 + \sum_{i=1}^n\frac{\epsilon_i(x_i'-x_i)}{\alpha_i} + n_-, \end{eqnarray*} exactly as in \refpr{real manifolds}, and the actions of ${\Bbb Z}_2^{2g}$ on these will be free provided the first inequality is strict. The quantity $2m -l + \sum_{i=1}^n \{\epsilon_i(x'_i-x_i)/\alpha_i\} = 2c_1(L_{(m,(\epsilon_i))}) - c_1(\Lambda)$ is just the Euler class of the flat ${\Bbb R}\P^1$ $V$-bundle $S(\rho_D) = S(L_{(m,(\epsilon_i))}^2 \Lambda^*)$ associated to the $PSL_2({\Bbb R})$-representation (this is well-defined as it is invariant under twisting $E$ by non-trivial topological roots). Note that, just as it is possible to have topologically distinct line $V$-bundles with the same Chern class, it is possible to have topologically distinct ${\Bbb R}\P^1$ $V$-bundles with the same Euler class---they are distinguished by their isotropy. The central ${\Bbb Z}$-extensions of $\pi_1^V(M)$ induced by the universal covering $\widetilde{PSL_2{\Bbb R}} \to PSL_2{\Bbb R}$ are just the (orbifold) fundamental groups of the flat ${\Bbb R}\P^1$ $V$-bundles $S(\rho_D)$ (see \cite{jn85}). Using the above discussion and the method of \refpr{real manifolds}, we obtain the following result (compare \cite{jn85}) and, as a corollary, a Milnor-Wood inequality. \bpr{psl2r reps} Let $M$ be an orbifold Riemann surface of negative Euler characteristic. For $\rho_D$ a $PSL_2({\Bbb R}))$-representation of $\pi_1^V(M)$ let ${\rm Hom\,}_{\rho_D}(\pi_1^V(M),PSL_2({\Bbb R}))$ denote the corresponding connected component. Let $(y_i)$ be the isotropy and $b + \sum_{i=1}^n (y_i/\alpha_i)$ the Euler class of the associated flat ${\Bbb R}\P^1$ $V$-bundle $S(\rho_D)$. Provided $b + \sum_{i=1}^n (y_i/\alpha_i) > 0$, ${\rm Hom\,}_{\rho_D}(\pi_1^V(M),PSL_2({\Bbb R}))/PSL_2({\Bbb R})$ is a smooth complex $(3g-3+n-n_0)$-dimensional manifold, diffeomorphic to a rank $(g - 1 + b + n -n_0)$ vector-bundle over $S^{2g-2-b}\wo M$. \end{proposition} \bco{milnor-wood} Let $M$ be an orbifold Riemann surface of negative Euler characteristic. Then the Euler class $b + \sum_{i=1}^n (y_i/\alpha_i)$ of any flat $PSL_2({\Bbb R})$ $V$-bundle satisfies \begin{eqnarray*} |b + \sum_{i=1}^n \frac{y_i}{\alpha_i}| \le 2g -2 + n -\sum_{i=1}^n\frac1{\alpha_i}. \end{eqnarray*} \end{corollary} \begin{proof} In \refpr{psl2r reps} we must have $b \le 2g-2$. The result follows since $y_i \le \alpha_i-1$. \end{proof} \bsu{Teichm\"uller Space for Orbifold Riemann Surfaces}{reptei} Assume, as usual, that $M$ is an orbifold Riemann surface of negative Euler characteristic. For a Fuchsian group such as $\pi_1^V(M)$, Teichm\"uller space, denoted ${\cal T}(M)$, is the space of faithful representations onto a discrete subgroup of $PSL_2{\Bbb R}$ modulo conjugation (see Bers's survey article \cite{be72}). Our previous results allow us to identify Teichm\"uller space with a submanifold of the moduli space. Let ${\cal T}_{-4}(M)$ denote the space of orbifold Riemannian metrics of constant sectional curvature -4, modulo the action of the group of diffeomorphisms homotopic to the identity, ${\cal D}_0(M)$. There is a bijection between ${\cal T}_{-4}$ and ${\cal T}$ as each metric of constant negative curvature determines an isometry between the universal cover of $M$ and ${\cal H}^2$ and hence a faithful representation of $\pi_1^V(M)$ onto a discrete subgroup of $PSL_2{\Bbb R}$ and conversely each such representation realises $M$ as a geometric quotient of ${\cal H}^2$. The results of \cite{jn85}, as well as those of \cite[\S 11]{hi87}, suggest that Teichm\"uller space is the component of the real representation space taking the extreme value in the Milnor-Wood inequality, \refco{milnor-wood}. Working with the holomorphic description, the results of the previous subsection show that the extreme is achieved when $E=L\oplus L^*\Lambda$ with $L^2\Lambda^*$ having the topology of $K$ and a holomorphic structure such that $\wo{KL^{-2}\Lambda}$ has sections: in other words we must have $L^2\Lambda^* = K$ (holomorphically). We suppose then that $E=K \oplus 1$ ($\Lambda^2E$ can be normalised to be square-free but this is not necessary). The corresponding Higgs field is just \begin{eqnarray*} \phi = \left( \begin{array}{cc} 0 & u\\ v & 0\\ \end{array}\right), \end{eqnarray*} where $u\in H^0(K^2)$ and $v\in {\Bbb C}\setminus \{0\}$. There is a ${\Bbb C}^*$-group of automorphisms so that we can normalise with $v=1$. Exactly as in \cite[theorem 11.2]{hi87}, we can identify Teichm\"uller space with the choices of $u$ {\rm i.\,e.\ } with $H^0(K^2)$. The two preliminaries which we need are the strong maximum principle for orbifolds (the proof is entirely local and generalises immediately; see \cite{jt80}) and the following orbifold version of a theorem of Sampson \cite{ee69}. \bpr{Sampson} Given two orbifold Riemannian metrics of constant sectional curvature -4 on $M$, $h$ and $h'$, there is a unique element of ${\cal D}_0$ which is a harmonic map between $(M,h)$ and $(M,h')$. \end{proposition} \begin{proof} This is a reformulation of \refpr{Donaldson}. The metrics $h$ and $h'$ give two discrete, faithful representations of $\pi_1^V(M)$ into $PSL_2{\Bbb R}$, one of which we consider fixed and the other we denote $\rho'$. The identity map on $M$ lifts to an orientation-preserving diffeomorphism $g$ of ${\cal H}^2$ which is equivariant with respect to the actions of the two representations. Taking this $g$ as an initial section of the $V$-bundle $H_{\rho_D}={\cal H}^2 \times_{\rho'} {\cal H}^3$ of \refpr{Donaldson} (via the inclusion ${\cal H}^2 \subset {\cal H}^3$) we obtain a harmonic section $g'$ homotopic to $g$. This is real and defines a harmonic diffeomorphism between $(M,h)$ and $(M,h')$. As $g'$ is homotopic to $g$ the resulting harmonic diffeomorphism is homotopic to the identity. Uniqueness follows either by a direct argument or from uniqueness over $\widehat M$, where $\widehat M$ is as in \refco{smooth covering}. \end{proof} We obtain the following theorem, which agrees with classical results due to Bers and others \cite{be72}. \bth{ball} Let $M$ be an orbifold Riemann surface of negative Euler characteristic. Let ${\cal T}(M)$ be the Teichm\"uller space of the Fuchsian group $\pi_1^V(M)$ and ${\cal T}_{-4}(M)$ the space of orbifold Riemannian metrics on $M$ of constant sectional curvature -4, modulo the action of the group of diffeomorphisms homotopic to the identity. Then ${\cal T}(M)$ and ${\cal T}_{-4}(M)$ are homeomorphic to $H^0(K^2)$, the space of holomorphic (orbifold) quadratic differentials on $M$. Hence Teichm\"uller space is homeomorphic to ${\Bbb C}^{3g-3+n}$. \eth We conclude by considering orbifold Riemannian metrics in greater detail. Considered as a metric on the underlying Riemann surface, $\wo M$, an orbifold Riemannian metric $h$ on $M$ has `conical singularities' at the marked points. To see this recall that locally $M$ is like $D^2/{\Bbb Z}_\alpha$ with $h$ a ${\Bbb Z}_\alpha$-equivariant metric on $D^2$. If $c_h(r)$ denotes the circumference of a geodesic circle of radius $r$ about the origin in $D^2$ (with respect to $h$), then $lim_{r\to 0}(c_h(r)/r)=2\pi$. Since this circle covers a circle in $D^2/{\Bbb Z}_\alpha$ exactly $\alpha$ times the metric on the quotient has a \de{conical singularity} at the origin, with \de{cone angle} $2\pi/\alpha$. Consider a Riemannian metric on $M$ which, near a marked point $D^2/{\Bbb Z}_\alpha$, is compatible with the complex structure and so has the form $h(z) dz \otimes d\o z$. If we set $w=z^\alpha$, then $w$ is a local holomorphic coordinate on $\wo M$. We find that the resulting `Riemannian metric' on $\wo M$ is given by \begin{eqnarray*} \frac{h(w^{1/\alpha})}{\alpha^2|w|^{2(1-1/\alpha)}} dw \otimes d\o w. \end{eqnarray*} Notice that $h(w^{1/\alpha})$ is well-defined by the ${\Bbb Z}_\alpha$-equivariance of $h$. This `Riemannian metric' has a singularity like $|w|^{-2(1-1/\alpha)}$ at the origin and is compatible with the complex structure away from there. Hence we obtain a compatible \lq singular Riemannian metric' on $\wo M$: the induced metric on $\wo M$ is continuous and induces the standard topology. How does such a singular Riemannian metric compares with a (smooth) Riemannian metric on $\wo M$? Suppose that $g$ is a fixed Riemannian metric on $\wo M$, compatible with the complex structure. Since $\wo M$ is compact any two Riemannian metrics give metrics on $\wo M$ which are mutually bounded and so will be equivalent for our purposes---we may as well use the Euclidean metric in any local chart. Now, $h$ and $g$ will give mutually bounded metrics on any compact subset of $M\setminus \{ p_1,\dots,p_n \}$. However, for small Euclidean distance $r$ from $p$, the singular metric has distance like $r^{1/\alpha}$. These are exactly the types of singularities of metrics considered by McOwen and Hulin-Troyanov in \cite{mo88,ht92}: they consider metrics which satisfy $h/g = O(r_g^{2k})$ as $r_g(z) = d_g(0,z) \to 0$, for some $k\in (-1,\infty)$. As McOwen points out, our \lq singular Riemannian metrics' have exactly this form with $k =-1 + 1/\alpha$. Interpreting \refco{negative curvature} in the light of this discussion we obtain the following result. (Our result is weaker than McOwen's since we consider only $k =-1 + 1/\alpha$ but the case of general $k\in (-1,\infty)$ can be obtained by a limiting argument as in \cite{ns93}.) \bthn{McOwen, Hulin-Troyanov}{conical} Let $\wo M$ be a Riemann surface with marked points $\{p_1,\dots,p_n\}$ with orders of isotropy $\{\alpha_1,\dots,\alpha_n\}$. If the genus $g$ and orders of isotropy satisfy \begin{eqnarray*} 2-2g-n+\sum_{i=1}^n 1/\alpha_i < 0 \end{eqnarray*} then $\wo M\setminus \{p_1,\dots,p_n\}$ admits a unique compatible Riemannian metric $h$ of constant sectional curvature -4 such that, for $i=1,\dots,n$, $h$ has a conical singularity at $p_i$ with cone angle $2\pi/\alpha_i$. \eth

9504

alg-geom/9504014

en

https://arxiv.org/abs/alg-geom/9504014

alg-geom/9504014

Yi Hu

Yi Hu

Relative Geometric Invariant Theory and Universal Moduli Spaces

31 pages, AMSLaTeX

We expose in detail the principle that the relative geometric invariant theory of equivariant morphisms is related to the GIT for linearizations near the boundary of the $G$-effective ample cone. We then apply this principle to construct and reconstruct various universal moduli spaces. In particular, we constructed the universal moduli space over $\overline{M_g}$ of Simpson's $p$-semistable coherent sheaves and a canonical dominating morphism from the universal Hilbert scheme over $\overline{M_g}$ to a compactified universal Picard.

alg-geom

\section{Introduction} \label{sec:introduction} \begin{say} {\sl Motivation: the universal moduli problems}. The motivation of this paper is to lay a GIT ground and to apply it to the so-called universal moduli problems such as the following ones. \begin{enumerate} \item The universal moduli space $\overline{FM_{g, n}} \rightarrow \overline{M_g}$ of Fulton-MacPherson configuration spaces of stable curves. That is, given a stable curve $[C]$ in $\overline{M_g}$, the fiber in the universal moduli space will be $C[n]/\hbox{Aut}(C)$, the Fulton-MacPherson configuration space for the curve $C$ modulo the automorphism group of $C$. (See also \cite{Pandharipande94b}) \item The compactified universal Picard $\overline{P^d_g} \rightarrow \overline{M_g}$ of degree $d$ line bundles (\cite{Caporaso94}). \item The universal moduli space $\overline{P_{g,m}(e, r, F, \alpha)} \rightarrow \overline{M_{g,n}}$ of $p$-semistable parabolic sheaves of degree $e$, rank $r$, type $F$, and weight $\alpha$ (\cite{Hu95}). \item The universal moduli space $M_{g}({\cal O}, P) \rightarrow \overline{M_g}$ of $p$-semistable coherent sheaves of pure dimension 1 with a fixed Hilbert polynomial $P$ such that the fiber over a stable curve $[C]$ is functorially identified with Simpson's moduli space $M_{C}({\cal O}_C, P)$ of $p$-semistable coherent sheaves over $C$ of the Hilbert polynomial $P$ modulo the automorphism group of $C$. The construction of the universal moduli space $M_{g}({\cal O}, P)$, as described above, bears a straightforward generalization to the universal moduli over the moduli spaces of higher dimensional varieties (e.g., surfaces of general type, Clabi-Yau 3-folds, etc, see \cite{Hu95}) because Simpson's construction \cite{Simpson94} works for any projective scheme. \item The universal Hilbert scheme $\hbox{Hilb}^n_g \rightarrow \overline{M_g}$ of 0-dimensional subschemes of length $n$ on the Mumford-Deligne stable curves such that the fiber over a stable curve $[C]$ is canonically the Hilbert scheme of 0-dimensional subschemes of length $n$ on $C$ modulo the automorphism group of $C$. Moreover, when $P(x) = x + n + 1 - g$ there exists a canonical dominating morphism $$\psi: \hbox{Hilb}^n_g \rightarrow M_{g}({\cal O}, P).$$ \end{enumerate} \end{say} \begin{say} {\sl Relative GIT}. All these universal moduli problems correspond to the following GIT setup which we shall call Relative GIT. We have a projective map $$\pi: Y \longrightarrow X$$ and an epimorphism of two reductive algebraic groups $$\rho: G' \longrightarrow G.$$ We assume that $\pi: Y \longrightarrow X$ is equivariant with respect to the homomorphism $\rho: G' \longrightarrow G$. \medskip\noindent {\bf Question RGIT}. Given a linearization $L$ on $X$ and the GIT quotient $X^{ss}(L)/\!/G$, find a linearization $M$ on $Y$ and the GIT quotient $Y^{ss}(M)/\!/G'$ so that $Y^{ss}(M)/\!/G'$ factors naturally to $X^{ss}(L)/\!/G$. \medskip Using the relation between GIT and moment maps, we solved in this paper the Question RGIT. In the case when $X^{ss} = X^s$ (which is the case for $\overline{M_g}$), the question has a particularly nice solution: the stable locus $X^{ss}=X^s$ downstairs essentially determines the stable locus $Y^{ss}(M) =Y^s(M)$ upstair. These results are contained in \S\S 3 and 4. The solution to Question RGIT leads to a unified and easy approach to all {\it those} universal moduli problems where the base moduli spaces have satisfactory GIT constructions (e.g., $\overline{M_g}$). In particular, with the aid of Simpson's approaches to the moduli of coherent sheaves over projective schemes (\cite{Simpson94}), our construction of the universal moduli space $$M_g({\cal O}, P) \longrightarrow \overline{M_g}$$ is much shorter than the approaches for the similar moduli problems in \cite{Caporaso94} and \cite{Pandharipande94a}. See \S\S 8 and 9. \end{say} \begin{say} {\sl Koll\'ar's Approaches (as opposed to the traditional GIT \cite{GIT})}. The solution of Question RGIT resembles Koll\'ar's approaches to algebraic quotient spaces (cf. Conjecture 1.1 of \cite{Kollar95a}). Now, one should ask how to do all of the above via Koll\'ar's approaches (\cite{Kollar95a} and \cite{Kollar90}). Theorem 2.14 of \cite{Kollar95a} sheds lights on this question. But the insistence on the \'etaleness of the the equivariant map is too restrictive for universal moduli. There are something more on the relative quotients that one can do with Koll\'ar's approaches and in many cases Koll\'ar's approaches are easier to apply than GIT. In light of the fact that many moduli spaces of higher dimensional varieties have no satisfactory GIT constructions, some alternatives are necessary in order to build universal moduli spaces effectively (for example, over the moduli space of surfaces of general type \cite{Kollar94}). This quest even includes the case of $\overline{M_{g,n}}$ which so far has no (satisfactory) GIT construction. We will return to these topics in an upcoming paper \cite{Hu95}. \end{say} \medskip\noindent {\bf Acknowledgements} The prototype of RGIT has appeared in \cite{GIT}, \cite{Simpson94}, \cite{Reichstein89}, and \cite{Pandharipande94a}. The author happily acknowledges some helpful conversations and/or correspondence with Aaron Bertram, Laurance Ein, David Gieseker, J\'anos Koll\'ar, Eyal Markman on quotients and/or moduli spaces, and useful comments from Lev Borisov. Thanks are due to Robert Lazarsfeld for informing him of the references on limit linear series. He also thanks John Millson for many interesting communications on moduli spaces of polygons. Finally, he appreciates the encouragement and support from Dan Burns, Ching-Li Chai, Igor Dolgachev, Victor Guillemin, Jun Li, Ralf Spatzier, Alex Uribe, and S.T. Yau during the course of this work. \bigskip \begin{center} {\bf I. Relative GIT} \end{center} \section{Equivariant theory of moment maps} \begin{say} A {\sl moment map} for a symplectic action of a compact Lie group $K$ on a symplectic manifold $(M, \omega)$ with a symplectic form $\omega$ is a smooth map $$\Phi:M \to {\frak k}^*$$ where ${\frak k}^*$ is the linear dual of the Lie algebra ${\frak k}$ of $K$, satisfying the following two properties: \begin{enumerate} \item $\Phi$ is equivariant with respect to the given action of $K$ on $M$ and the co-adjoint action of $K$ on ${\frak k}^*$; \item for any $a \in {\frak k}$, $d\Phi(\xi) \cdot a = \omega(\xi, \xi_a)$ for all vector fields $\xi$ on $M$, where $\xi_a$ is the vector field generated by $a$. \end{enumerate} The equation in (ii) determines the moment map by an additive constant in the set ${\frak z}^*$ of the {\sl central elements} of ${\frak k}^*$ (i.e., ${\frak z}^*$ = the set of invariants of the $K$-coadjoint action = the linear dual of the Lie algebra ${\frak z}$ of the center group of $K$). \end{say} When a moment map exists, we call the the symplectic action a Hamiltonian action and the symplectic form $\omega$ a ($K$-) Hamiltonian symplectic form. Fix a maximal torus $T_k$ of $K$. We shall use $T$ to denote the complexification of $T_k$. Let $\hbar^*_+$ is a positive Weyl chamber in the linear dual $\hbar^*$ of the Lie algebra $\hbar$ of the fixed maximal torus $T_k$ of $K$. Since $\hbar^*_+$ parametrizes the orbit space of the coadjoint action of $K$ on ${\frak k}^*$, we obtain, by quotienting the $K$-adjoint action, the so-called reduced moment map $$\Phi_{\text{red}}: M \to \hbar^*_+.$$ \begin{say} Although a Hamiltonian symplectic form $\omega$ only determines the moment map up to an additive central element in ${\frak k}^*$. However, as pointed out by Atiyah \cite{Atiyah82}, there is always a {\sl canonical choice}, $\Phi^\omega$, the one such that $$\int_M \Phi^\omega \omega^{\frac{1}{2}\dim M}= 0. $$ Such a moment map will be called the canonical moment map associated to $\omega$. All other moment maps defined by $\omega$ have the form $\Phi^{\widetilde{\omega}} = \Phi^\omega - \mu$ where $\widetilde{\omega} =(\omega, \mu)$ and $\mu \in {\frak z}$ (consult \ref{say:momentcone} in the sequel). \end{say} \begin{rem} \label{rem:momentmapforsingularspaces} For any invariant closed subset $Z$ in $M$ (possibly singular), one can simply define the moment map for the action of $K$ on $Z$ to be the restriction of the total moment map, for which we shall still use the same notation $\Phi$ if no confusion should emerge. \end{rem} \begin{say} Given any value $p$ of a moment map $\Phi$, the orbit space $\Phi^{-1}(\text{\sl O}_p)/K$, which is called the symplectic reduction at $p$, carries a natural induced symplectic structure away from singularities (\cite{MarsdenWeinstein}). Here $\text{\sl O}_p$ is the coadjoint orbit through $p$. \end{say} \begin{say} Now we shall consider equivariant maps and the relations between their induced moment maps. Let $Y$ and $X$ be smooth projective complex algebraic varieties acted on by complex reductive algbraic group $G'$ and $G$ respectively. Consider an algebraic surjection $$\pi: Y \longrightarrow X$$ that is equivariant with respect to a group epimorphism (surjective homomorphism) $$\rho: G' \longrightarrow G,$$ that is, $$\pi (g' \cdot y) = \rho (g') \cdot \pi (y) \; \text{for all} \; y \in Y \; \text{and}\; g' \in G'. $$ One can choose a maximal subgroup $K$ of $G$ and a maximal subgroup $K'$ of $G'$ so that $\rho$ restricts to a group homomorphism (still denoted by $\rho$) $\rho: K' \longrightarrow K$ and $\rho: G' \longrightarrow G$ is the complexification of $\rho: K' \longrightarrow K$. \end{say} \begin{say} Symplectic (K\"ahler) forms and line bundles on $Y$ will be denoted by letters $\eta$ and $M$, while symplectic (K\"ahler) forms and line bundles on $X$ will be denoted by $\omega$ and $L$. \end{say} \begin{say} \label{limitofsymplecticforms} Let $\omega$ be a symplectic (K\"ahler) form on $X$. Then $\eta_0 = \pi^* \omega$ is a closed two form on $Y$. We'd like to define a moment map for this degenerated two form $\eta_0$. The trick is to use nearby moment maps and then take the limit. Let $\eta (t), t \in (0, \epsilon]$ be a continuous path of symplectic (K\"ahler) forms on $Y$ such that $\eta_0 = \lim_{t \to 0} \eta (t)$. Here $\epsilon$ is a small positive number. (This can be done because $\eta_0 = \pi^* \omega$ lies on the boundary of the K\"ahler cone for $Y$.) Define $$ \Phi^{\eta_0} (y) = \lim_{t \to 0} \Phi^{\eta (t)} (y)$$ where $ \Phi^{\eta (t)}$ are the canonical moment maps for $\eta (t)$. \end{say} \begin{prop} \label{prop:onlimitforms} Keep the notations of \ref{limitofsymplecticforms}. Then we have \begin{enumerate} \item $\Phi^{\eta_0}$ is a $K'$-equivariant differentiable map; \item $\Phi^{\eta_0}$ satisfies the differential equation $$ d\Phi^{\eta_0} (\xi) \cdot a = \eta_0 (\xi, \xi^Y_a)$$ for every vector field $\xi \in TY$ and $a \in {\frak k}'$, where $\xi^Y_a$ is the vector field on $Y$ generated by the element $a$; \item $\Phi^{\eta_0}$ is constant on every fiber of $\pi$. \end{enumerate} \end{prop} \begin{pf} The equivariancy of the map $\Phi^{\eta_0}$ is obvious. Now since $X \times [0, \epsilon]$ is compact, we have that $\Phi^{\eta (t)}$ converges uniformly as $t \to 0$. Thus $ \Phi^{\eta_0} = \lim_{t \to 0} \Phi^{\eta (t)}$ is differentiable. This proves (i). Next, given any fixed vector field $\xi \in TY$ and fixed $a \in {\frak k}'$, we have $$ d\Phi^{\eta (t)} (\xi) \cdot a = \eta (t) (\xi, \xi_a)$$ for all $t \in (0, \epsilon]$. Passing to the limit as $t \to 0$, we obtain $$ d\Phi^{\eta_0} (\xi) \cdot a = \eta_0 (\xi, \xi_a).$$ This proves (ii). To show (iii), picking any $\xi \in T \pi^{-1}(x) \subset TY$ for any $x \in X$. Recall that $\eta_0 = \pi^* \omega$. Now we must have $$d \Phi^{\eta_0} (\xi) \cdot a = \eta_0 (\xi, \xi_a) = \pi^* \omega (\xi, \xi_a) =\omega (d\pi (\xi), d\pi (\xi_a)) = 0 $$ since $d\pi (\xi) = 0$. Because $a$ is an arbitrary element in ${\frak k}'$, we obtain $d \Phi^{\eta_0} (\xi) = 0$. This implies that $\Phi^{\eta_0}$ is constant along every fiber of $\pi$. The proposition is thus proved. \end{pf} \begin{rem} The map $\Phi^{\eta_0}$ may be considered as a moment map defined by the (pre-symplectic) form $\eta_0$. As the limit of some (true) moment maps, it enjoys a number of properties of the usual momentum mapping such as the convexity. In general, however, a moment map defined by a pre-symplectic form may have much more complicated image that needs not to be convex (see \cite{KarshonTollman93}). \end{rem} \begin{say} \label{say:killingmetrics} Choose a $K$-equivariant metric on ${\frak k}$ and a $K'$-equivariant metric on ${\frak k'}$ respectively so that $d\rho$ preserves the metrics. Then these two metrics lead two natural isomorphisms ${\frak k} \cong {\frak k}^*$ and ${\frak k'} \cong {\frak k'}^*$, making the following diagram commutes \newpage $${\frak k'}^*\;\;\;\;\;\;\;\;{\buildrel {\cong}\over\longrightarrow} \;\;\;{\frak k'} $$ $$\;\;\; \uparrow \! (d\rho)^* \;\;\;\;\; \;\;\;\downarrow \! d\rho$$ $${\frak k}^* \;\;\;\;\;\;\;\;{\buildrel {\cong}\over\longrightarrow} \;\;\;{\frak k}$$ where $d\rho$ is the differential of the map $\rho$ and $(d\rho)^*$ is the codifferential of the map $\rho$, i.e., the linear dual of the differential $d\rho$. \end{say} \begin{thm} \label{thm:descendingmomentmaps} The map $\Phi^{\eta_0}$ descends to the moment map $\Phi^{\omega}$ for the action of $K$ on $X$ with respect to the symplectic form $\omega$. In particular, we have the following commutative diagram $$Y \;\;\;{\buildrel {\Phi^{\eta_0}}\over\longrightarrow} \;\;\;{\frak k'}^* \;\;\;\;\;\;\;\; {\buildrel {\cong}\over\longrightarrow} \;\;\;{\frak k'} $$ $$\;\;\;\downarrow \!{\pi} \;\; {\buildrel {\Psi^{\eta_0}}\over\nearrow} \;\; \uparrow \! (d\rho)^* \;\;\;\;\; \;\;\;\downarrow \! d\rho$$ $$X\;\;\;{\buildrel{\Phi^{\omega}}\over\longrightarrow} \;\;\;{\frak k}^* \;\;\;\;\;\;\;\; {\buildrel {\cong}\over\longrightarrow} \;\;\;{\frak k}$$ \end{thm} \begin{pf} By Proposition \ref{prop:onlimitforms} (3), the map $\Phi^{\eta_0}$ desends to a well-defined differentiable map $\Psi^{\eta_0}$ from $X$ to ${\frak k'}^*$ such that $\Phi^{\eta_0} =\Psi^{\eta_0} \circ \pi$. Define $\Phi^{\omega}$ to be the composition: $$\Phi^{\omega}:X \;\;\stackrel{\Psi^{\eta_0}}{\longrightarrow}\;\; {\frak k'}^* \;\; \stackrel{\cong}{\longrightarrow}\;\; {\frak k'}\;\; \stackrel{d\rho}{\longrightarrow} \;\;{\frak k}\;\; \stackrel{\cong}{\longrightarrow}\;\; {\frak k}^*.$$ Clearly, this is a differentiable map, and we have that $\Psi^{\eta_0} = (d\rho)^* \Phi^\omega$. To show that $\Phi^{\omega}$ is equivariant with respect to the given $K$-action on $X$ and the coadjoint action on ${\frak k}^*$, take any element $k' \in K'$ and $y \in Y$, we then have $$\Phi^{\omega} (\rho(k') \cdot \pi(y)) = \Phi^{\omega} (\pi (k' \cdot y)) = d\rho \circ \Psi^{\eta_0} (\pi (k' \cdot y)) $$ $$= d\rho \Phi^{\eta_0}(k' \cdot y) = d\rho \text{Ad}(k') \cdot \Phi^{\eta_0}(y) = \text{Ad}(\rho(k')) d\rho \Phi^{\eta_0}(y) $$ $$=\text{Ad}(\rho(k')) d\rho \Psi^{\eta_0} (\pi (y)) = \text{Ad}(\rho(k')) \Phi^{\omega} (\pi (y)).$$ Here we have used the identity $d\rho \text{Ad}(k') = \text{Ad}(\rho(k')) d\rho$ coming from the equivariancy of the homomorphism $\rho$. Using the fact that both $\pi$ and $\rho$ are surjective, we obtain $$\Phi^{\omega} (k \cdot x) = \text{Ad}(k) \Phi^{\omega} (x), \; \text{for all} \; k \in K \; \text{and}\; x \in X.$$ To check that it satisfies the differential equation in the definition of a moment map (1.1 (2)), notice that $d\pi :TY \rightarrow TX$ is surjective everywhere but on a lower dimensional locus and $d\pi (\xi^Y_{a'}) = \xi^X_{d\rho (a')}$ for any $a' \in {\frak k'}$ because $\pi$ is $\rho$-equivariant, where again, $\xi^Y_{a'}$ is the vector field on $Y$ generated by $a' \in {\frak k'}$, while $\xi^X_{d\rho (a')}$ is the vector field on $X$ generated by $d\rho (a') \in {\frak k}$. Now for any $\xi \in TY$ and $a' \in {\frak k'}$, we have $$\omega (d\pi \xi, \xi^X_{d\rho (a')}) = \omega (d\pi \xi, d\pi (\xi^Y_{a'})) =\pi^* \omega (\xi, \xi^Y_{a'}) = \eta_0(\xi, \xi^Y_{a'}) $$ $$= d\Phi^{\eta_0} (\xi) \cdot a' \; \;\;(\text{because of \ref{prop:onlimitforms} (2)})$$ $$=d \Psi^{\eta_0} d \pi (\xi) \cdot a' \;\;\; (\text{because}\; \Phi^{\eta_0} = \Psi^{\eta_0} \circ \pi)$$ $$= (d \rho)^* d\Phi^{\omega} d \pi (\xi) \cdot a' \;\;\; (\text{because}\; \Psi^{\eta_0} = (d \rho)^* \Phi^{\omega})$$ $$=d\Phi^{\omega}(d\pi \xi) \cdot d \rho (a') \;\;\; (\text{because of \ref{say:killingmetrics}}).$$ By the surjectivity of $d\rho$, this means that the differential equation $$d\Phi^{\omega}(\xi^X) \cdot a = \omega (\xi^X, \xi^X_a)$$ holds for vector fields $\xi^X \in TX$ almost everywhere. Thus by continuity, it holds for all $\xi^X$ in $TX$ everywhere. That is, $\Phi^{\omega}$ is a moment map for the action of $K$ on $X$ with respect to the symplectic form $\omega$. Since $\Phi^{\eta (t)}$ are the canonical moment maps for $\eta (t)$, it is easy to check that $\Phi^{\omega}$, as the limit of $\Phi^{\eta (t)}$, is the canonical moment map for $\omega$. \end{pf} \begin{say} \label{deformationofmomentmaps} Finally, we remark that when $G' = G$, the above implies there is a deformation of moment maps $$\Phi^{\eta (t)}: Y \rightarrow {\frak k}^*$$ from $\Phi^{\eta (\epsilon)}: Y \rightarrow {\frak k}^*$ to $\Phi^{\omega}: X \rightarrow {\frak k}^*$, where $\Phi^{\omega} (X) = \Phi^{\eta_0} (Y)$. When $\dim G' > \dim G$, the dimension of $\Phi^{\omega} (X)$ is less than those of $\Phi^{\eta (t)}(Y) (t \in (0, \epsilon])$. In this case, we say that $\Phi^{\omega}$ is a degeneration of $\Phi^{\eta (t)} (t \in (0, \epsilon])$. \end{say} \section{$G$-Effective ample cone} Much of what follows is taken from \cite{DolgachevHu} and \cite{Hu94}. \begin{say} In this section we assume that $X$ is a smooth projective complex algebraic variety\footnote{We point out that much of results in this section can be extended to K\"ahler category (i.e., $X$ being K\"ahler manifolds only). Since our primary applications will be algebraic moduli spaces, we are content with working in the category of algebraic varieties.} over ${\Bbb C}$ acted on by a reductive group $G$ with a fixed maximal compact form $K$. \end{say} \begin{thm} \text{(cf. Theorem 2.3.6, \cite{DolgachevHu}.)} \label{thm:homologicalequivalence} Let $X$ be a smooth projective variety acted on by a reductive complex algebraic group $G$. Let $\omega$ and $\omega'$ be two $K$-equivariant K\"ahler forms. Suppose that $\omega$ is cohomological equivalent to $\omega'$. Then $\Phi^\omega = \Phi^{\omega'}$. \end{thm} \begin{pf} The proof is the same as that for Theorem 2.3.6, \cite{DolgachevHu}. \end{pf} \begin{say} Let $X$ is a smooth projective variety. We set ${\frak K}^G(X)$ to be the collection of all $K$-equivariant K\"ahler forms that are compatible with the algebraic $G$-action modulo cohomological equivalence. Theorem \ref{thm:homologicalequivalence} says that there is a well-defined canonical moment map $\Phi^{[\omega]}$ for each element $[\omega]$ of ${\frak K}^G(X)$. For notational simplicity, we shall omit the use the bracket ``[\;\;]'' when it is not likely to cause confusion. \end{say} \begin{say} \label{say:momentcone} When $X$ is smooth, set ${\frak M}^G(X)$ to be the collection of all pairs consisting of an element in ${\frak K}^G(X)$ and a moment map defined by it. Thus $${\frak M}^G(X) \cong {\frak K}^G(X) \times {\frak z}^*.$$ We shall adopt the following conventional scheme: {\sl $\widetilde{\omega}$ denotes an element in ${\frak M}^G(X)$ with its underlying (Hamiltonian) K\"ahler form symbolized by $\omega \in {\frak K}^G(X)$. Such a symbol $\widetilde{\omega}$ will be referred as an enriched symplectic K\"ahler form}. Thus an enriched symplectic (K\"ahler) form $\widetilde{\omega}$ in ${\frak M}^G(X)$ has the form $(\omega, \Phi^\omega - \mu)$ or simply $(\omega, \mu)$, where $\mu \in {\frak z}^*$. We frequently write $\Phi^\omega - \mu$ by $\Phi^{\widetilde{\omega}}$. \end{say} \begin{rem} All of the results in \S 1 are valid without modification if the symplectic (K\"ahler) forms ($\omega$ and $\eta (t)$, etc.) are replaced by the enriched symplectic (K\"ahler) forms ($\widetilde{\omega}$ and $\widetilde{\eta} (t)$, etc.) \end{rem} \begin{rem} If $X$ is (possibly) singular, we can set ${\frak M}^G(X)$ to be the cone spanned by the images of all linearized ample line bundles in $\hbox{NS}^G(X)\otimes_{\Bbb Z} {\Bbb R}$ (\cite{DolgachevHu}). \end{rem} \begin{defn} Assume that $X$ is smooth. The $G$-effective ample cone is a subcone of ${\frak M}^G(X)$ defined as follows: $${\frak E}^G(X) = \{(\omega, \mu) \in {\frak M}^G(X)\; |\; \mu \in \Phi^\omega(X) \};$$ \end{defn} \begin{rem} When $X$ is (possibly) singular, we define ${\frak E}^G(X)$ to be the subcone of ${\frak M}^G(X)$ that is spanned by the images of $G$-effective ample line bundles (that is, spanned by the ones such that $X^{ss}(L) \ne \emptyset$, see \cite{DolgachevHu}). When $X$ is actually smooth, the two definitions are equivalent (\cite{Hu94}). \end{rem} \begin{say} ${\frak E}^G(X)$ projects to ${\frak K}^G(X)$ whose fiber at $\omega \in {\frak K}^G(X)$ is the intersection of the moment map image $\Phi^\omega(X)$ with ${\frak z}^*$. This fiber $\Phi^\omega(X) \cap {\frak z}^* = \Phi_{red}^\omega(X) \cap {\frak z}^*$ is a convex compact polytope. In fact, it is not hard to see that $\Phi^\omega(X) \cap {\frak z}^*$ is the image of $X$ under the {\it canonical} moment map attached to the induced action of the center group of $K$. \end{say} \begin{rem} ${\frak E}^G(X)$ may be an empty subset of ${\frak M}^G(X)$ when $G$ is semisimple. For example, when $X=G/P$ is a generalized flag variety. However, ${\frak E}^G(X \times G/B)$ is never empty. In particular, ${\frak E}^G(X)$ is never empty when $G$ is a torus. \end{rem} \begin{say} Recall from \cite{Hu94} that for any point $x \in X$, we can define a (generalized Hilbert-Mumford) numerical function $$M^\bullet (x) : {\frak E}^G(X) \rightarrow {\Bbb R}$$ whose value $M^{\widetilde{\omega}}(x)$ at $ \widetilde{\omega} \in {\frak E}^G(X)$ is defined as the {\sl signed} distance from the origin to the boundary of $\Phi^{\widetilde{\omega}} (\overline{G \cdot x})$: it takes a positive value if $0$ is outside of $\Phi^{\widetilde{\omega}} (\overline{G \cdot x})$; it takes a nonpositive value otherwise. Using this numerical function we have the following criteria for (K\"ahler) semistabilities, \begin{enumerate} \item $X^{ss}(\widetilde{\omega})$ = the set of the semistable points with respect to $\widetilde{\omega}$ \\ = $\{x \in X| M^{\widetilde{\omega}}(x) \le 0\}$; \item $X^s(\widetilde{\omega})$ = the set of the stable points with respect to $\widetilde{\omega}$ \\ = $\{x \in X| M^{\widetilde{\omega}}(x) < 0\}$; \item $X^{us}(\widetilde{\omega})$ = the set of the unstable (or non-semistable) points with respect to $\widetilde{\omega}$ \\ = $\{x \in X| M^{\widetilde{\omega}}(x) > 0\}$; \item{(iv)} $X^{sss}(\widetilde{\omega})$ = the set of the strictly semistable points with respect to $\widetilde{\omega}$ \\ = $X^{ss}(\widetilde{\omega}) \setminus X^s(\widetilde{\omega})$. \end{enumerate} \end{say} \begin{rem} When a group is needed to be specified, we will write $X^{ss}_G(\widetilde{\omega})$, $X^{s}_G(\widetilde{\omega})$, etc. This applies especially when there is a redutive subgroup $H$ of $G$. In this case, the restriction map ${\frak M}^G(X) \to {\frak M}^H(X)$ (the $H$-moment map is obtained from the $G$-moment map by the orthogonal projection ${\frak k}^* \to {\frak h}^*$ where ${\frak h}$ is the Lie algebra of a suitable compact form of $H$) induces a linearization in ${\frak M}^H(X)$ for each linearization $\widetilde{\omega}$ in ${\frak M}^G(X)$. To specify this effect, we will write $X^{ss}_H(\widetilde{\omega})$ ($X^{s}_H(\widetilde{\omega})$, etc) for the set of semistable (stable, etc) points for the action of $H$. \end{rem} By the works of Kempf-Ness (for the algebro-geometric cases \cite{KempfNess78}) and Kirwan (for the K\"ahler generalizations \cite{Kirwan84}), there is a Hausdorff quotient topology on $X^{ss}(\widetilde{\omega})/\!/G$ such that it contains the orbit space $X^s(\widetilde{\omega})/G$ as a dense open subset. It is a K\"ahler space and has a K\"ahler form induced from $\omega$ away from the singularities. Moreover, it is homeomorphic to the symplectic reduction $(\Phi^{\widetilde{\omega}})^{-1}(0)/K$. \begin{thm} \text{(Kempf-Ness-Kirwan)} \label{thm:Kempf-Ness-Kirwan} Let $\widetilde{\omega}=(\omega, \mu)$ be an enriched K\"ahler form in ${\frak E}^G(X)$. Then $(\Phi^{\widetilde{\omega}})^{-1}(0) \subset X^{ss}(\widetilde{\omega})$ and the inclusion induces a homeomorphism $$(\Phi^{\widetilde{\omega}})^{-1}(0)/K {\buildrel {\cong}\over\longrightarrow} X^{ss}(\widetilde{\omega})/\!/G.$$ In case that $\widetilde{\omega}$ is integral (i.e., coming from a linearized ample line bundle), then $X^{ss}(\widetilde{\omega})/\!/G$ carries a projective structure. \end{thm} \begin{rem} The $G$-effective ample cone takes care of symplectic reductions at central values of ${\frak k}^*$ and identifies them with K\"ahler quotients of $X$ by the complex reductive group $G$. To include symplectic reductions at non-central values, one has to consider the so-called enlarged moment cone (see \cite{Hu94}) and use the so-called shifting trick to identify them with the K\"ahler quotients on $X \times G/P$ by the diagonal action of $G$. The relative GIT for the morphisms $X \times G/B \to X \times G/P$ has been studied and linked to degenerated quotients of $X \times G/B$ in \cite{Hu94}. So, in this paper, we stick with just the $G$-effective ample cone. \end{rem} \begin{say} The union of the zero sets of $M^\bullet(x): {\frak E}^G(X) \rightarrow {\Bbb R}$ for all $x$ with isotropy subgroups of positive dimensions equals the union ${\cal W}$ of all walls in ${\frak E}^G(X)$ (see \cite{DolgachevHu} and \cite{Hu94}). A connected component of ${\frak E}^G(X) \setminus {\cal W}$ is a chamber. Linearizations in the same chamber define the same notion of stabilities. \end{say} We shall need the following results in the sequel. \begin{say} Given any element $\widetilde{\omega} \in {\frak E}^G(X)$, hence a {\it unique} moment map $\Phi^{\widetilde{\omega}}= \Phi$. We have a stratification, the momentum Morse stratification with respect to $\Phi$, $X = \cup_{\beta \in {\bf B}} S_\beta$ induced by the norm square $|\!|\Phi|\!|^2$ of the moment map (\cite{Kirwan84}). The strata $S_\beta$ and their indexes $\beta$ can be described as follows: $$S_\beta = \{x\in X | \; \beta \ \hbox {is the unique closest point to 0 of }\ \Phi_{\operatorname{red}} (\overline {G\cdot x})\}.$$ \end{say} \begin{thm} \label{thm:finiteness} {\rm (\cite{DolgachevHu}, \cite{Hu94})} There are only finitely many momentum Morse stratifications. \end{thm} \begin{defn} {\rm (\cite{Hu94})} \label{defn:thinmorsestratification} For any momentum Morse stratification of $X$, we choose precisely one stratum from it. Then the intersection of all the chosen strata is called a {\it thin} momentum Morse stratum provided that it is not empty. There are only finitely many such strata. The {\it thin} momentum Morse strata form a stratification of $X$. \end{defn} \begin{prop} {\rm (\cite{Hu94})} Two points in the same {\it thin} momentum Morse stratum give rise to the same numerical function $M^\bullet (x): {\frak E}^G(X) \rightarrow {\Bbb R}$. \label{cor:tmm=snf} \end{prop} \begin{pf} The proof is short. So we repeat it here (see \cite{Hu94}). Let $x$ and $y$ be two points of a {\it thin} momentum Morse stratum. Then by definition, for any $\widetilde{\omega} \in {\frak E}^G(X)$, $x, y$ belong to the same momentum Morse stratum $S^{\widetilde{\omega}}_{\beta(\widetilde{\omega})}$ for some index $\beta(\widetilde{\omega})$. That is, $M^{\widetilde{\omega}}(x) = M^{\widetilde{\omega}}(y) = |\!|\beta(\widetilde{\omega})|\!|$ for all $\widetilde{\omega} \in {\frak E}^G(X)$. Hence $$M^\bullet(x) = M^\bullet(y) : {\frak E}^G(X) \rightarrow {\Bbb R}.$$ \end{pf} \begin{cor} \label{cor:finitenumericalfunctions} There are only finitely many numerical functions $$M^\bullet (x): {\frak E}^G(X) \rightarrow {\Bbb R}.$$ (And the points in the same {\it thin} momentum Morse stratum give rise to the same numerical function $M^\bullet (x)$.) \end{cor} \begin{rem} To close this section we make the following useful remark. We view the integral points in ${\frak K}^G(X)$ (${\frak E}^G(X)$) as being induced from the Hodge metrics of (linearized) ample line bundles. We shall think of {\it these integral points and (linearized) ample line bundles interchangably.} By default, {\sl the K\"ahler quotiens and maps associated to these integral points will be projective as in the traditional GIT cases.} \end{rem} \section{A relative GIT: first cases} \begin{say} In the rest of the paper, unless specified otherwise, we shall assume that $\pi:Y \rightarrow X$ is a projective morphism between two (possibly singular) quasi-projective algebaric varieties that is equivariant with respect to an epimorphism $\rho: G' \rightarrow G$ between two reductive complex algebraic groups having fixed maximal compact subgroups $K'$ and $K$, respectively. \end{say} \begin{say} \label{say:divideandconquer} Let $G_0$ be the kernel of $\rho$. Then we have $$\{1\} \rightarrow G_0 \rightarrow G' \rightarrow G \rightarrow \{1\} .$$ That is, up to a finite central extension, we may think of $G' = G_0 \times G$. Because GIT problems for a finite group is trivial (i.e., the orbit space is the natural solution to the quotient problems for a finite group action), to simplify the exposition, one may well assume that $G \cong G'/G_0$ is a subgroup of $G'$ and $G' = G_0 G$. Due to the above comments, the relative GIT problem for this case can be divided into two steps: \begin{enumerate} \item The fiberwise GIT problem: $G_0$ acts only on the fibers of $\pi: Y \to X$; (One may look at this from a slightly different point of view: $Y/X$ is projective over the base scheme $X$. $G_0/X$ is the trivial group scheme over $X$. $G_0/X$ acts on $Y/X$. This viewpoint is helpful for some universal moduli problems.) \item The $G$-equivariancy GIT problem: treat $\pi: Y \to X$ as a $G$-equivariant map alone. Here we identify the quotient group $G'/G_0$ with the group $G$ by the isomorphism induced by the epimorphism $\rho$ and $G$ acts on $Y$ via this identification. \end{enumerate} After those have been done, we can then put the two together and relate some properly chosen $Y$-quotient by $G'$ to a given $X$-quotients by $G$. \end{say} \begin{prop} \label{prop:fiberwisegit} Let $M$ be a $G_0$ linearized relatively ample line bundle with respect to the morphism $\pi: Y \to X$. Let $Y_x$ denote the fiber of $\pi$ over a point $x \in X$ and $M_x$ the restriction of $M$ to $Y_x$. Then we have $Y_x^{ss}(M_x) = Y_x \cap Y_{G_0}^{ss}(M)$. In particular $Y_{G_0}^{ss}(M)= \cup_{x \in X} Y_x^{ss}(M_x)$. \end{prop} \begin{pf} It follows from Proposition 1.19 of \cite{GIT}. See also \cite{Simpson94}. \end{pf} Consequently, since the categorical quotient is universal, one sees that the morphism $\pi: Y^{ss}_{G_0}(M) \to X$ descends to a map $$\widetilde{\pi}: Y_{G_0}^{ss}(M)/\!/G_0 \to X$$ with fibers $Y_x^{ss}(M_x)/\!/G_0$. \begin{prop} \label{G0quotientisGinvariant} If $M$ is a $G'$-linearized relatively ample line bundle with respect to the morphism $\pi: Y \to X$, then the $G'$-action on $Y$ descends to a $G'/G_0=G$-action on $Y_{G_0}^{ss}(M)/\!/G_0$ making the map $\widetilde{\pi}: Y_{G_0}^{ss}(M)/\!/G_0 \to X$ $G$-equivariant. \end{prop} \begin{pf} First we need to show that $Y_{G_0}^{ss}(M)$ is $G'/G_0=G$-invariant (recall that the quotient group $G'/G_0$ is identified with the group $G$ by the isomorphism induced by the epimorphism $\rho$). Take a point $y \in Y_{G_0}^{ss}(M)$ and an element $g \in G$. Let $s \in \Gamma(Y, M^{\otimes n})^{G_0}$ be an $G_0$-invariant section such that $Y_s:=\{y' \in Y | s(y') \ne 0 \}$ is affine and contains $y$ (see Defintion 1.7 in \S 4 of \cite{GIT}). Set $$^gs (y') = g s(g^{-1} y')$$ for all $y' \in Y$. We claim that $^gs \in \Gamma(Y, M^{\otimes n})^{G_0}$. To see this, pick any element $g_0 \in G_0$. We have $^{g_0} (^gs) (y') = g_0 (^gs) (g_0^{-1} y') = g_0 g s(g^{-1} g_0^{-1} y') = g \hat{g}_0 s (\hat{g}_0^{-1} g^{-1} y') = g (^{\hat{g}_0})s (g^{-1} y') = g s (g^{-1} y') = (^gs) (y')$ where $\hat{g}_0$ is an element in $G_0$ such that $g_0 g = g \hat{g}_0$ (note that $G_0$ is a normal subgroup). This implies that $^{g_0} (^gs) = ^gs$, i.e., $^gs \in \Gamma(Y, M^{\otimes n})^{G_0}$. Clearly $Y_{^gs} = g Y_s$ is affine and contains $g y$. That is, $g y \in Y_{G_0}^{ss}(M)$. Since $Y_{G_0}^{ss}(M) \subset Y$ factors to $X$ in a $G_0$-equivariant way ($G_0$ acts trivially on $X$), by the universality of categorical quotient, we see that there is a naturally induced morphism $\widetilde{\pi}: Y_{G_0}^{ss}(M)/\!/G_0 \to X$. Next it is straightforward to verify that the $G$-action on $Y_{G_0}^{ss}(M)$ passes naturally to the quotient and the $G$-equivariancy of $\pi: Y_{G_0}^{ss}(M) \to X$ implies the $G$-equivariancy of $\widetilde{\pi}: Y_{G_0}^{ss}(M)/\!/G_0 \to X$. \end{pf} \begin{say} Now let us consider the $G$-equivariancy GIT problems. That is, we are going to treat morphism $\pi: Y \to X$ as a $G$-equivariant map only. Our finest results in this section lie in the case when $X^{ss}(L) = X^s(L)$. This is what we shall sometimes refer as ``good cases'' . \end{say} \begin{say} \label{say:setupphieta} We begin with assuming that $\pi: Y \to X$ is a projective morphism between two smooth projective varieties. Let $\widetilde{\omega} = (\omega, \mu)$ be an enriched $K$-equivariant K\"ahler form on $X$ and $\widetilde{\eta}_0 = \pi^* \widetilde{\omega} = (\pi^* \omega, \mu)$ be an enriched closed $K$-equivariant two form on $Y$. Let $\widetilde{\eta}: (0, \epsilon] \rightarrow {\frak E}^G(Y)$ be a continuous path in ${\frak E}^G(Y)$ such that $ \widetilde{\eta}_0 = \lim_{t \to 0} \widetilde{\eta} (t)$. As in \ref{limitofsymplecticforms}, define $$ \Phi^{\widetilde{\eta}_0} (y) = \lim_{t \to 0} \Phi^{\widetilde{\eta} (t)} (y).$$ Then we have \end{say} \begin{thm} \label{thm:comparingstabilities} Let $Y \rightarrow X$ be a $G$-equivariant projective morphism between two smooth projective varieties. Then there exists $\delta > 0$ such that \begin{enumerate} \item If $x = \pi (y)$ is stable in $X$ w.r.t $\widetilde{\omega}$, then $y$ is stable in $Y$ w.r.t. $\widetilde{\eta} (t), t \in (0, \delta]$; \item If $x = \pi (y)$ is non-semistable in $X$ w.r.t $\widetilde{\omega}$ then $y$ is non-semistable in $Y$ w.r.t. $\widetilde{\eta} (t), t \in (0, \delta]$; \item $Y^{s}(\widetilde{\eta} (t)) \supset \pi^{-1}(X^{s}(\widetilde{\omega}))$ and $Y^{ss}(\widetilde{\eta} (t)) \subset \pi^{-1}(X^{ss}(\widetilde{\omega}))$ for $t \in (0, \delta]$. \end{enumerate} \end{thm} \begin{pf} Let $Y= \bigcup_{i=1, \cdots, d} S_i$ be the thin momentum Morse stratification of $Y$ (see Definition \ref{defn:thinmorsestratification}). By Corollary \ref{cor:finitenumericalfunctions}, it suffices to consider some fixed representing elements in $S_i, i=1, \cdots, d$. For any $1 \le i \le d$, if $\pi (S_i) \subset X^{sss}(\widetilde{\omega})$, set $\delta_i =1$. Otherwise, pick up some representing elements $y_i \in S_i, 1 \le i \le d$. If $x_i = \pi (y_i)$ is stable in $X$, then $0 \in \hbox{int}(\Phi^{\widetilde{\omega}}(\overline{G \cdot x_i}))$. By the remarks in \ref{deformationofmomentmaps}, a small deformation of $\Phi^{\widetilde{\omega}}(\overline{G \cdot x_i})$ should still contains the origin in its interior. That is, there exists $\delta_i > 0$ such that $0 \in \hbox{int}(\Phi^{\widetilde{\eta} (t)}(\overline{G \cdot y}))$ for $t \le \delta_i$. Thus $y_i$ is stable in $Y$ w.r.t. $\widetilde{\eta} (t)$ for $t \le \delta_i$. If $x_i = \pi (y_i)$ is non-semistable in $X$, then $0$ is outside of $\Phi^{\widetilde{\omega}}(\overline{G \cdot x_i})$. By the similar deformation argument as above, the same is true for its small deformations. This is,, there exists $\delta_i > 0$ such that $0 \notin \hbox{int}(\Phi^{\widetilde{\eta} (t)}(\overline{G \cdot y}))$ for $t \le \delta_i$. In other words, $y$ is non-semistable in $Y$ w.r.t $\widetilde{\eta} (t)$ for $t \le \delta_i$. Now choose $\delta= \hbox{min}\{\delta_1, \cdots, \delta_d, \epsilon \}$, the above implies both (1) and (2). (3) follows from (2). \end{pf} \begin{rem} When $x=\pi (y)$ is strictly semistable, i.e., the orgin is contained on the boundary of $\Phi^{\widetilde{\omega}}(\overline{G \cdot x})$, the stability of $y$ depends on the direction of deformation. So in general, all three possible situations (stable, strictly semistable, non-semistable) may happen. \end{rem} \begin{say} \label{say:traditional GIT} In keeping with the theme of the traditional GIT, consider the case that $\widetilde{\omega}$ is an integral form induced from an ample linearized line bundle $L$ on $X$. The pullback $\pi^* L$ is only a {\sl nef} linearized line bundle on $Y$ inducing the 2-form $\widetilde{\eta}_0 = \pi^* \widetilde{\omega}$. To get an ample linearized line bundle on $Y$, we need to choose an arbitrary {\it relatively} ample linearized line bundle $M$ on $Y$ and take a sufficiently large tensor power of $\pi^* L$. That is, $\pi^* L^n \otimes M$ is ample for $n \gg 0$ (\cite{EGA}). Set fractional linearizations $M_n = \frac{1}{n} (\pi^* L^n \otimes M)$. Clearly, $\lim_{n \to \infty} M_n = \pi^* L$. The purpose of this scheme is two fold: to get (fractional multiples of) {\it ample} linearized line bundles $M_n$ on $Y$; and to make $\pi^* L$ and $M_n = \frac{1}{n}(\pi^* L^n \otimes M)$ sufficiently close for sufficiently large $n$. \end{say} We need the following easy lemma \begin{lem} \label{lem:restriction} Let $G$ act on $X$ whose acton is linearized by $L$ and $i: Z \hookrightarrow X$ be a $G$-invariant closed embedding linearized by $i^* L$. Then $Z^{ss}(i^* L) = Z \cap X^{ss}(L)$ and $Z^{s}(i^* L) = Z \cap X^{s}(L)$. \end{lem} \begin{pf} This follows from Proposition 1.19 of \cite{GIT}. \end{pf} \begin{thm} \label{thm:generalizedReichstein} Let $\pi: Y \rightarrow X$ be a $G$-quivariant projective morphism between two (possibly singular) quasi-projective varieties. Given any linearized ample line bundle on $L$ on $X$, choose a {\it relatively} ample linearized line bundle $M$ on $Y$. Then there exists $n_0$ such that when $n \ge n_0$, we have \begin{enumerate} \item $Y^{ss}(\pi^*L^n \otimes M) \subset \pi^{-1}(X^{ss}(L))$; \item $Y^{s}(\pi^*L^n \otimes M) \supset \pi^{-1}(X^{s}(L))$ If in addition, $X^{ss}(L) = X^s(L)$, then \item $Y^{ss}(\pi^*L^n \otimes M)=Y^{s}(\pi^*L^n \otimes M)=\pi^{-1}(X^{s}(L))= \pi^{-1}(X^{ss}(L))$. In particular, $\pi^*L^n \otimes M$ lie in the same chamber of ${\frak E}^G(Y)$ for all $n \ge n_0$. \end{enumerate} \end{thm} \begin{pf} Given a linearized ample line bundle $L$ on $X$, and a relatively linearized ample line bundle $M$ for $\pi: Y \rightarrow X$, by \cite{EGA}, there exists $m_0$ such that when $m \ge m_0$, $L^m$ and $\pi^* L^{\otimes m} \otimes M$ are very ample. Consider some equivariant projective embeddings induced from $L^m$ and $\pi^* L^{\otimes m} \otimes M$ (for some $m \ge m_0$) $$Y \hookrightarrow X \times {\Bbb P}^{N'} \hookrightarrow {\Bbb P}^N \times {\Bbb P}^{N'}$$ $$\;\;\;\downarrow \!{\pi} \;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\; \;\;\;\;\;\;\; \downarrow \!\hbox{proj}$$ $$X \;\;\;\;\;\;\;\;\;\; \;\; \hookrightarrow \;\;\;\;\;\;\;\;\;\; \;\; {\Bbb P}^N $$ We may think $L^m$ as the pull back of a very ample line bundle ${\cal O}_{{\Bbb P}^N}(1)$. By \ref{say:traditional GIT}, there exists $d_0 > 0$ such that when $d \ge d_0$, Theorem \ref{thm:comparingstabilities} can be applied to the $G$-equivariant projection ${\Bbb P}^N \times {\Bbb P}^{N'} \rightarrow {\Bbb P}^N$ with respect to the linearizations ${\cal O}_{{\Bbb P}^N}(1)$ on ${\Bbb P}^N$ and $\frac{1}{d}({\cal O}_{{\Bbb P}^N}(d) \otimes {\cal O}_{{\Bbb P}^{N'}}(1))$ on $ {\Bbb P}^N \times {\Bbb P}^{N'}$, respectively . Now restricting everything to $\pi: Y \rightarrow X$ (see Lemma \ref{lem:restriction}), we see that $\pi: Y \rightarrow X$ shares all the properties as in Theorem \ref{thm:comparingstabilities}. This proves (1) and (2). All of (3) follows immediately from (1) and (2). \end{pf} \begin{rem} In the case that $M$ is ample and $X$ and $Y$ are projective, Theorem \ref{thm:generalizedReichstein} is largely due to Z. Reichstein (\cite{Reichstein89}) whose proof is completely different. But we only need to assume that $M$ is {\it relatively} ample and $X$ and $Y$ are quasi-projective. We emphasize the practical convenience that can result from the {\it relative} ampleness of the line bundle $M$, because in constructing moduli spaces, relative projective embeddings of relative Hilbert schemes have been constructed very explicitly by Grothendieck (hence are ready to use), whereas absolute projective embeddings will not be offhand. \end{rem} \begin{thm} \label{thm:inducedmapfromG-equivariancy} Keep the assumption as in Theorem \ref{thm:generalizedReichstein}. Then the inclusion $Y^{ss}(\pi^* L^n \otimes M) \subset \pi^{-1}(X^{ss}(L))$ induces a projective morphism $$\hat{\pi}: Y^{ss}(\pi^* L^n \otimes M)/\!/G \longrightarrow X^{ss}(L)/\!/G$$ such that \begin{enumerate} \item $\hat{\pi}^{-1}([G \cdot x]) \cong \pi^{-1}(x)/G_x$ for any $x \in X^s(L)$; \item if $\pi$ is a fibration, $X^{ss}(L) =X^s(L)$, and $G$ acts freely on $X^s(L)$, then $\hat{\pi}$ is also a fibration with the same fibers. \end{enumerate} \end{thm} \begin{pf} The morphism $\pi$ restricts to give a morphism $Y^{ss}(\pi^* L^n \otimes M) \rightarrow X^{ss}(L)$. This in turn induces an obvious morphism $$Y^{ss}(\pi^* L^n \otimes M)\longrightarrow X^{ss}(L) \longrightarrow X^{ss}(L)/\!/G.$$ Now by the universality of the categorical quotient, we get our desired induced morphism $$\hat{\pi}: Y^{ss}(\pi^* L^n \otimes M)/\!/G \longrightarrow X^{ss}(L)/\!/G.$$ To show (1), let $[G \cdot y_1]$ and $[G \cdot y_2]$ be arbitrary two points in $Y^{ss}(\pi^* L^n \otimes M)/\!/G$ that are mapped down to $[G \cdot x] \in X^s (L)/\!/G$. Then we must have that $\pi (y_1), \pi (y_2) \in X^s (L)$. By Theorem \ref{thm:generalizedReichstein} (2), $y_1, y_2 \in Y^{s}(\pi^* L^n \otimes M)$. Thus $\pi (G \cdot y_1) = \pi (G \cdot y_2) = G \cdot x$. Hence by applying some elements of $G$, we may assume that $y_1, y_2 \in \pi^{-1}(x)$. But, in this case, $[G \cdot y_1]=[G \cdot y_2]$ if and only if $y_2 = g\cdot y_1$ for some $g \in G$. This implies that $x =g \cdot x$. That is, $g \in G_x$. Hence we obtain that $\hat{\pi}^{-1}([G \cdot x]) \cong \pi^{-1}(x)/G_x$. (2) follows from (1) immediately. \end{pf} \section{A relative GIT: general cases} Now we begin to investigate the relation between $G'$-stability on $Y$ and the $G$-stability on $X$. \begin{defn} A linearized pair $(L, M)$ for the morphism $\pi: Y \to X$ \label{defn:linearizedpair} consists of a $G$-linearized ample line bundle $L$ over $X$ and a $G'$-linearized $\pi$-ample line bundle $M$ over $Y$. \end{defn} Out of a linearized pair $(L, M)$, we can have two Zariski open subsets $Y^{ss}_{G_0}(M)$ and $\pi^{-1}(X^{ss}(L))$. Proposition \ref{G0quotientisGinvariant} says that $Y^{ss}_{G_0}(M)$ is $G$-invariant and hence $G'$-invariant. Obviously $\pi^{-1}(X^{ss}(L))$ is $G_0$-invariant, hence also $G'$-invariant. \begin{lem} \label{triviallinearization} Let $(L, M)$ be a linearized pair for the morphism $\pi: Y \to X$. Then \begin{enumerate} \item $Y^{ss}_{G_0}(\pi^* L^n \otimes M) = Y^{ss}_{G_0}(M)$; \item $Y^{s}_{G_0}(\pi^* L^n \otimes M) = Y^{s}_{G_0}(M)$; \end{enumerate} \end{lem} \begin{pf} Notice that $G_0$ preserves the fibers of the morphism $\pi$. Thus we are in the position to apply Proposition \ref{prop:fiberwisegit}. Given any point $x \in X$. Since the restricted line bundle $\pi^* L^n|_{Y_x}$ is trivial, we obtain $(Y_x)_{G_0}^{ss}((\pi^* L^n \otimes M)|_{Y_x}) = (Y_x)^{ss}_{G_0}(M)$. Now Proposition \ref{prop:fiberwisegit} implies that $Y^{ss}_{G_0}(\pi^* L^n \otimes M) = Y^{ss}_{G_0}(M)$. (2) follows from (1) because of the fact: a point is stable if and only if its isotropy is finite and its orbit is closed in the semistable locus. \end{pf} Again by 4.6.13 (ii) of \cite{EGA}, there exists an positive integer $n_0$ such that $\pi^* L^n \otimes M$ is ample when $n \ge n_0$. \begin{lem} \label{G'=GG_0} Assume that $n \ge n_0$. Then \begin{enumerate} \item $Y^{ss}(\pi^* L^n \otimes M) = Y^{ss}_{G_0}(\pi^* L^n \otimes M) \cap Y^{ss}_{G}(\pi^* L^n \otimes M).$ \item $Y^{s}(\pi^* L^n \otimes M) = Y^{s}_{G_0}(\pi^* L^n \otimes M) \cap Y^{s}_{G}(\pi^* L^n \otimes M).$ \end{enumerate} \end{lem} \begin{pf} By general nonsense, we have $$Y^{ss}(\pi^* L^n \otimes M) \subset Y^{ss}_{G_0}(\pi^* L^n \otimes M) \cap Y^{ss}_{G}(\pi^* L^n \otimes M).$$ To show the other way inclusion, notice that $G'=G_0 G$ and any 1-PS of $G'$ can be written as $\lambda \lambda_0$ where $\lambda$ is a 1-PS of $G$ and $\lambda_0$ is is a 1-PS of $G_0$. Take a point $y \in Y^{ss}_{G_0}(\pi^* L^n \otimes M) \cap Y^{ss}_{G}(\pi^* L^n \otimes M)$. Now by Corollary 2.15 in \S 3 of \cite{GIT}, $$\mu^{\pi^* L^n \otimes M}(y, \lambda \lambda_0) \le \mu^{M \otimes \pi^* L^{n}}(y, \lambda) + \mu^{\pi^* L^n \otimes M}(y, \lambda_0) \le 0+0=0.$$ That is, $Y^{ss}(\pi^* L^n \otimes M) \supset Y^{ss}_{G_0}(\pi^* L^n \otimes M) \cap Y^{ss}_{G}(\pi^* L^n \otimes M)$. This shows (1). To show (2), again by general nonsense, $$Y^{s}(\pi^* L^n \otimes M) \subset Y^{s}_{G_0}(\pi^* L^n \otimes M) \cap Y^s_{G}(\pi^* L^n \otimes M).$$ Similar to the semistable case, the other way inclusion follows from the fact that $G'=GG_0$ and Corollary 2.15 in \S 3 of \cite{GIT}. \end{pf} Now we are in the position to state our main theorem on the comparisons of $G'$-semistabilities on $Y$ and $G$-semistabilities on $X$. \begin{thm} \label{thm:generalcomparisonofstabilities} Let $(L, M)$ be a linearized pair for the morphism $\pi: Y \to X$. Then for sufficiently large $n$ \begin{enumerate} \item $Y^{ss}(\pi^* L^n \otimes M) \subset Y^{ss}_{G_0}(M) \cap \pi^{-1}(X^{ss}(L))$. \item $Y^s(\pi^* L^n \otimes M) \supset Y^s_{G_0}(M) \cap \pi^{-1}(X^s(L))$ \\ Assume in addition that $X^{ss}(L)=X^s(L)$, then \item $Y^{ss}(\pi^* L^n \otimes M) = Y^{ss}_{G_0}(M) \cap \pi^{-1}(X^s(L))$. \item $Y^s(\pi^* L^n \otimes M) = Y^{s}_{G_0}(M) \cap \pi^{-1}(X^s(L))$. \end{enumerate} \end{thm} \begin{pf} {}From Lemma \ref{G'=GG_0} (1), $$Y^{ss}(\pi^* L^n \otimes M) = Y^{ss}_{G_0}(\pi^* L^n \otimes M) \cap Y^{ss}_{G}(\pi^* L^n \otimes M).$$ Then by Lemma \ref{triviallinearization} (1) and Theorem \ref{thm:generalizedReichstein} (1) (see also Theorem \ref{thm:comparingstabilities} (3)), we have $$Y^{ss}(\pi^* L^n \otimes M) \subset Y_{G_0}^{ss}(M) \cap \pi^{-1}(X^{ss}(L)).$$ This shows (1), as desired. (2) also follows in a similar way from the combination of Lemma \ref{G'=GG_0} (2), Lemma \ref{triviallinearization} (2), and Theorem \ref{thm:generalizedReichstein} (2). Now assume that $X^{ss}(L) =X^s (L)$. Then, similar to the above, (3) and (4) follow directly from the combination of Lemma \ref{triviallinearization}, Lemma \ref{G'=GG_0}, and Theorem \ref{thm:generalizedReichstein}. \end{pf} \begin{rem} When $\pi: Y \to \text{\{point\}}$ is the total contraction and $\rho$ is the homomorphism $G \to \{1\}$, we have $L = \text{\sl O}_{\text{\{point\}}}$. In this case, the relative $\pi$-ampleness is equivalent to the absolute ampleness and the effect of the a linearized pair $(L, M)$ is equivalent to that of the linearized ample line bundle $M$. Thus we recover the traditional (i.e., the absolute) GIT. \end{rem} \begin{say} The structure of the quotient of $Y^{ss}(\pi^* L^n \otimes M)$ by the large group $G'$ may be divided in two ways: mod out by the group $G_0$ first and then by the group $G$; or the other way around. In the end, we would have to prove that the quotient is independent of the order of the two procedures. But we shall take a unified approach as follows. \end{say} \begin{thm} \label{thm:generalrgit} Keep the assumption as in Theorem \ref{thm:generalcomparisonofstabilities}. Then the categorical quotient $Y^{ss}(\pi^* L^n \otimes M)/\!/G'$ exists and there is a naturally induced morphism $$\hat{\pi}: Y^{ss}(\pi^* L^n \otimes M)/\!/G' \rightarrow X^s(L)/\!/G.$$ Moreover, given an orbit $G \cdot x$ in $X^s(L)$, the fiber $\hat{\pi}^{-1}([ G \cdot x])$ can be identifibed with the quotient $((Y_x)^{ss}_{G_0}(M|_{Y_x})/\!/G_0)G_x = (Y_x)^{ss}_{G_0}(M|_{Y_x})/\!/G_0G_x$ where $G_x$ is the (finite) isotropy subgroup at the point $x$. \end{thm} \begin{pf} The idea of proof is similar to that of Theorem \ref{thm:inducedmapfromG-equivariancy}. We shall present the detail as follows. By \cite{GIT}, the categorical quotient $Y^{ss}(\pi^* L^n \otimes M)/\!/G'$ exists as a quasi-projective variety. By Theorem \ref{thm:generalcomparisonofstabilities} (1), the morphism $\pi$ restricts to a morphism (still denoted by $\pi$) $$\pi: Y^{ss}(\pi^* L^n \otimes M) \rightarrow X^{ss}(L).$$ Thus we get an obvious induced morphism $$\pi: Y^{ss}(\pi^* L^n \otimes M) \rightarrow X^{ss}(L) \rightarrow X^{ss}(L)/\!/G.$$ Since a categorical quotient is a universal quotient (\cite{GIT}), the above morphism passes to the quotient to give us the desired morphism $$\hat{\pi}: Y^{ss}(\pi^* L^n \otimes M)/\!/G' \rightarrow X^s(L)/\!/G.$$ To prove the rest of the statement, let $[G' \cdot y_1]$ and $[G' \cdot y_2]$ be arbitrary two points in $Y^{ss}(\pi^* L^n \otimes M)/\!/G'$ that are mapped down to $[G \cdot x] \in X^s (L)/\!/G$. Then we must have that $\pi (y_1), \pi (y_2) \in X^s (L)$. Hence $G \cdot \pi(y_1) = G \cdot \pi(y_2) = G \cdot x$. Thus we may well assume that $y_1, y_2 \in \pi^{-1}(x)$ by applying some elements of $G$ to $y_1$ and $y_2$ if necessary. But, in this case, $[G' \cdot y_1]=[G' \cdot y_2]$ if and only if $G_x [G_0 \cdot y_1]=G_x [G_0 \cdot y_2]$ where $G_x$ acts naturally on $(Y_x)^{ss}_{G_0}(M_x)/\!/G_0$ and $[G_0 \cdot y_1]$, $[G_0 \cdot y_2]$ are regarded as points in $(Y_x)^{ss}_{G_0}(M_x)/\!/G_0$. This implies that $\hat{\pi}^{-1}([G \cdot x]) \cong (Y_x)^{ss}_{G_0}(M_x)/\!/G_0G_x$. \end{pf} We shall need the following technical theorem in our later study of universal moduli problems. \begin{thm} \label{thm:Gmainthmforrelativemoduli} Let $\pi:Y \rightarrow X$ be a projective morphism between two algebraic varieties and equivariant with respect to $G$-actions on both $X$ and $Y$. Assume that with respect to some $G$-equivariant projective embedding, $X$ is \begin{enumerate} \item contained in the stable locus; and \item is closed in the semistable locus. \end{enumerate} Then there exists in addition a $G$-equivariant relative projective embedding for $\pi:Y \rightarrow X$ and an induced $G$-linearization such that $Y$ is \begin{enumerate} \item contained in the stable locus; and \item is closed in the semistable locus. Moreover \item $Y/G$ factors naturally to $X/G$ with the fiber over $[G \cdot x] \in X/G$ isomorphic to $\pi^{-1}(x)/G_x$ for $x \in X$. \end{enumerate} \end{thm} \begin{pf} It follows from Theorem \ref{thm:inducedmapfromG-equivariancy} by restricting everything to $\pi:Y \rightarrow X$. \end{pf} A more general technical theorem which we shall quote in studying moduli problems later is as follows: \begin{thm} \label{thm:G'toGmainthmforrelativemoduli} Let $\pi:Y \rightarrow X$ be a projective morphism between two algebraic varieties and equivariant with respect to a homomorphism $\rho:G' \rightarrow G$ where $G'$ acts $Y$ and $G$ acts on $X$. Assume that with respect to some $G$-equivariant projective embedding, $X$ is \begin{enumerate} \item contained in the stable locus; and \item is closed in the semistable locus. \end{enumerate} Then there exists in addition a $\rho$-equivariant relative projective embedding for $\pi:Y \rightarrow X$ and an induced $G'$-linearization such that $Y$ is \begin{enumerate} \item contained and closed in the $G_0$-semistable locus; \item contained and closed in the $G$-stable locus; \item contained and closed in the $G'$-semistable locus; \item $Y/\!/G'$ factors naturally to $X/G$ with the fiber over $[G \cdot x] \in X/G$ isomorphic to $\pi^{-1}(x)/\!/G_0G_x$ for $x \in X$. \end{enumerate} \end{thm} \begin{pf} It follows from Theorem \ref{thm:generalrgit} by restricting everything to $\pi:Y \rightarrow X$. \end{pf} \begin{rem} In studying moduli problems, the above two theorems will allow us to avoid actually writing down a relative projective embedding (for a universal curve or a relative Quot scheme) which is sometimes technical and time consuming. \end{rem} \begin{say} To close this section, let's look at an interesting special case. Assume that $G$ acts on two projective varieties $X$ and $X_0$ repectively and $G_0$ acts on $X_0$ only (acts on $X$ trivially). Set $Y=X_0 \times X$ acted on by $G'= G_0 \times G$ where $G_0$ operates on the first factor only, while $G$ acts diagonally. Thus the projection onto the first factor $\pi: Y \rightarrow X$ is equivariant with respect to the projection onto the second factor $G' \to G$. (This example has been worked out by R. Pandharipande in the case that $X_0 = {\bf P}(V)$ and $X= {\bf P}(W)$.) We begin with considering the diagonal action of $G$ on $Y=X_0 \times X$. The GIT problem for $G_0$ on Y is the same as the GIT problem for $G_0$ on $X_0$. We have the following inclusion $$\text{Pic}^G(X) \otimes \text{Pic}^G(X_0) \subset \text{Pic}^G(X_0 \times X).$$ We shall concentrate on the linearizations coming from $\text{Pic}^G(X) \otimes \text{Pic}^G(X_0)$. Take $L \in \text{Pic}^G(X)$ and $L_0 \in \text{Pic}^G(X_0)$. By Theorem \ref{thm:generalizedReichstein}, we have that there exists $m_0$ such that when $m \ge m_0$, \begin{enumerate} \item $Y^{ss}_G(L_0 \otimes L^{\otimes m}) \subset \pi^{-1}(X^{ss}(L))$. \item $Y^s_G(L_0 \otimes L^{\otimes m}) \supset \pi^{-1}(X^s(L))$. \end{enumerate} In case that $X^{ss}(L))=X^s(L)$, we have $$Y^{ss}_G(L_0 \otimes L^{\otimes m}) = Y^s_G(L_0 \otimes L^{\otimes m}) = \pi^{-1}(X^s(L)) = X_0 \times X^s(L).$$ Clearly, this implies that when $L_0 \in \text{Pic}^{G'}(X_0)$ $$Y^{ss}(L_0 \otimes L^{\otimes m}) = (X_0)_{G_0}^{ss}(L_0) \times X^s(L).$$ So we get the induced morphism $$\hat{\pi}: Y^{ss}(L_0 \otimes L^{\otimes m})/\!/G' \rightarrow X^s(L)/\!/G$$ with the fiber isomorphic to $(X_0)_{G_0}^{ss}(L_0)/\!/G_0G_x$ at the point $[G\cdot x] \in X^s(L)/\!/G$. This induced morphism $\hat{\pi}$ needs not to be a trivial fibration. \end{say} \section{Generalized Kempf-Ness's theorem} \begin{say} Again, we treat $\pi: Y \to X$ as a $G$-equivariant morphism alone. Place ourselves in the situation of \ref{say:setupphieta}. According to Theorem \ref{thm:comparingstabilities}, in the case that $X^{sss}(\widetilde{\omega}) = \emptyset$, $\widetilde{\omega}$ determines uniquely a semistable locus upstairs. This motivates: \end{say} \begin{defn} Let $\widetilde{\eta}_0$ be a polarization on the boundary of ${\frak E}^G(Y)$ which is the pullback of $\widetilde{\omega}$ by the morphism $\pi$. Assume that $X^{sss}(\widetilde{\omega}) = \emptyset$. Then we set $Y^{ss}(\widetilde{\eta}_0) = Y^s(\widetilde{\eta}_0)= \pi^{-1}(X^{ss}(\widetilde{\omega}))$. \end{defn} \begin{lem} \label{lem:forkempf-nessthm} If $y \in (\Phi^{\widetilde{\eta_0}})^{-1} (0)$, then $G \cdot y \cap (\Phi^{\widetilde{\eta_0}})^{-1} (0) = K \cdot y$. \end{lem} \begin{pf} First observe that $y \in (\Phi^{\widetilde{\eta_0}})^{-1} (0)$ implies $\Phi^{\widetilde{\omega}} (\pi (y)) = 0$. That is, $\pi (y) \in (\Phi^{\widetilde{\omega}})^{-1}(0)$. We shall adopt the proof of Lemma 7.2 of \cite{Kirwan84}. Suppose that $g \in G$ is such that $g \cdot y \in (\Phi^{\widetilde{\eta}_0})^{-1} (0)$. We want to show that there exists an element $k \in K$ such that $g \cdot y = k \cdot y$. Since $\Phi^{\eta_0}$ is $K$-equivariant and $G=K \text{exp}(i {\frak k})$, we may assume that $g = \text{exp}(i a), a \in {\frak k}$. or Lemma 7.2 of Set $$h(t) = \Phi^{\widetilde{\eta}_0} (\text{exp}(i at) \cdot y) \cdot a.$$ Then $$h(t) = \Phi^{\widetilde{\omega}} \circ \pi (\text{exp}(i at) \cdot y) \cdot a.$$ $h(t)$ vanishes at $t=0$ and $t=1$ because both $y$ and $\text{exp}(i a) \cdot y$ belong to $(\Phi^{\widetilde{\eta}_0})^{-1} (0)$. Thus there must be a point $t \in (0, 1)$ such that $$0 = h'(t) = d \Phi^{\widetilde{\omega}} \circ d \pi (i (\xi^Y_a)_z) \cdot a = \omega (d \pi (i (\xi^Y_a)_z), (\xi^X_a)_{\pi(z)})$$ $$= \omega (i (\xi^X_a)_{\pi(z)}, (\xi^X_a)_{\pi(z)}) =<(\xi^X_a)_{\pi(z)}, (\xi^X_a)_{\pi(z)}>$$ where $z = \text{exp}(i at) \cdot y$. Hence $(\xi^X_a)_{\pi(z)} = 0$. Or, $a \in \text{Lie}(G_{\pi(z)})$. But $$\pi(z) = \text{exp}(i at) \cdot \pi (y) \in G (\Phi^{\widetilde{\omega}})^{-1}(0).$$ So $a=0$ because $G_{\pi(z)}$ must be a finite group. This completes the proof. \end{pf} Now we have the following theorem which extends the Kempf-Ness theorem to the degenerated polarizations. \begin{thm} {\rm \text{(Generalized Kempf-Ness's Theorem)}} Assume that $X^{ss}(\widetilde{\omega})=X^s(\widetilde{\omega})$. Then \begin{enumerate} \item $Y^{ss}(\widetilde{\eta}_0) = G (\Phi^{\widetilde{\eta}_0})^{-1} (0)$; \item the topological quotient $Y^{ss}(\widetilde{\eta}_0)/G$ is Hausdorff; \item the inclusion $(\Phi^{\widetilde{\eta}_0})^{-1} (0) \hookrightarrow Y^{ss}(\widetilde{\eta}_0)$ induces a homeomorphism between \\ $(\Phi^{\widetilde{\eta}_0})^{-1} (0)/K$ and $Y^{ss}(\widetilde{\eta}_0)/G$. \item $Y^{ss}(\widetilde{\eta_0})/G = (\Phi^{\widetilde{\eta}_0})^{-1} (0)/K$ inherits from $\eta_0$ a closed 2-form (which may be degenerated somewhere) away from singularities. \end{enumerate} \end{thm} \begin{pf} (1). By definition, $$Y^{ss}(\widetilde{\eta}_0) = \pi^{-1}(X^s(\widetilde{\omega}) = \pi^{-1}(G (\Phi^{\widetilde{\omega}})^{-1}(0) = G \pi^{-1} (\Phi^{\widetilde{\omega}})^{-1}(0) $$ $$= G (\Phi^{\widetilde{\omega}} \circ \pi)^{-1}(0) = G (\Phi^{\widetilde{\eta_0}})^{-1} (0).$$ (2). This follows from the fact that $Y^{ss}(\widetilde{\eta}_0)= Y^{ss}(\widetilde{\eta}(t))$ for sufficiently small positive numbers $t$. (3). (1) implies that the induced map $(\Phi^{\eta_0})^{-1} (0)/K \to Y^{ss}(\widetilde{\eta}_0)/G$ is surjective. Lemma \ref{lem:forkempf-nessthm} implies that it is injective. Now as a continuous bijection between Hausdorff spaces, it must be a homeomorphism. (4) The proof is the same as the one for non-degenerated 2-forms (\cite{MarsdenWeinstein}). \end{pf} \begin{rem} In fact by Theorem \ref{thm:comparingstabilities}, the quotient $Y^{ss}(\widetilde{\eta}_0)/G$ admits a complex structure and many other non-degenerated 2-forms induced from $\widetilde{\eta}(t), t \in (0, \delta]$ such that the 2-form induced from $\widetilde{\eta}_0$ is the limit of the above. \end{rem} \begin{rem} When $X^{sss}(\widetilde{\omega}) \ne \emptyset$, there is a difficulty in defining that $$Y^{ss}(\widetilde{\eta}_0) := \{ y \in Y | 0 \in \Phi^{\widetilde{\eta}_0} (\overline{G \cdot y}) \} = \{ y \in Y | 0 \in \Phi^{\widetilde{\omega}} (\overline{G \cdot \pi (y)}) \} = \pi^{-1}(X^{ss}(\widetilde{\omega})).$$ The problem is as follows. Let $x \in X^{sss}(\widetilde{\omega})$ be a point such that $G \cdot x$ is closed in $X^{ss}(\widetilde{\omega})$. Then $G_x$ is reductive and acts on the fiber $\pi^{-1}(x) \subset Y^{ss}(\widetilde{\eta}_0)$. Since $\dim G_x > 0$, $\pi^{-1}(x)/ G_x$ is non-Hausdorff. Thus one would like to exclude some points in $\pi^{-1}(x)$ from $ Y^{ss}(\widetilde{\eta}_0)$, presumly by using a non-degenerate closed 2-form on $\pi^{-1}(x)$. That is, one would like to pick up some semistability on $\pi^{-1}(x)$ for the action of $G_x$. The form $\eta_0$ is helpless in this regard since the fiber $\pi^{-1}(x)$ is exactly where $\eta_0$ vanishes. Hence among many choices of the semistabilities on $\pi^{-1}(x)$ for the action of $G_x$, we do not know, a priori, which to choose. Notice that the same ambuguity does not happen when $X^{sss}(\widetilde{\omega}) = \emptyset$. In this case, the isotropy subgroup $G_x$ for every $x \in X^{ss}(\widetilde{\omega})$ is a finite group. $\pi^{-1}(x)/ G_x$ is always a good quotient. One of ways to solve (actually to pass) the problem of $X^{sss}(\widetilde{\omega}) \ne \emptyset$ is as follows. Assume that $\dim {\frak E}^G(X) > 1$ and it has at least one top chamber. Choose a linearization $\widetilde{\omega}'$ in a top chamber that contains $\widetilde{\omega}$ in its closure. We then have $X^{ss}(\widetilde{\omega}') = X^s(\widetilde{\omega}') \subset X^{ss}(\widetilde{\omega})$. Now $\pi^{-1}(X^{ss}(\widetilde{\omega}'))$ has a good complete quotient and one has natural maps $$\pi^{-1}(X^{ss}(\widetilde{\omega}'))/\!/G \rightarrow X^{ss}(\widetilde{\omega}')/\!/G \rightarrow X^{ss}(\widetilde{\omega})/\!/G .$$ This helps to reduce the case when $X^{sss}(\widetilde{\omega})\ne \emptyset$ to the nicer case when $X^{sss}(\widetilde{\omega}) = \emptyset$. \end{rem} \begin{rem} In general, the semistable set $Y^{ss}(\pi^*L^n \otimes M) (n \gg 0)$ may be recovered as follows: Let $X^{ss}_c (L)$ be the set of closed orbits in $X^{ss}(L)$. Then \begin{enumerate} \item $Y_c^{ss}(\pi^*L^n \otimes M) = \{ y \in Y | y \in \pi^{-1} (X_c^{ss}(L)) \cap (Y_{\pi (y)})_c^{ss} (M_{\pi (y)}) \}$. \item $Y^{ss}(\pi^*L^n \otimes M) = \{ y \in Y | \overline{G \cdot y} \cap Y_c^{ss}(\pi^*L^n \otimes M) \ne \emptyset \}$. \item $Y^s (\pi^*L^n \otimes M)= \{y \in Y | y \in Y_c^{ss}(\pi^*L^n \otimes M), G_y \; \text{ is finite} \}$. \end{enumerate} \end{rem} \begin{rem} There are nef linearized line bundles that are not the pull-backs for any algebraic contraction maps. In this case, we do not know yet how to make sense of RGIT in the algebraic category. One possible alternative is to allow the contractions to be just complex analytic and work in the complex analytic category. The price paid is then the loss of algebraicity and the possible validity over positive characteristics. \end{rem} \section{Configuration spaces and Grassmannians} In this section, we shall present an application of our various RGIT theorems to some elementary finitely dimensional settings (as opposed to the later applications to the moduli problems). \begin{say} Consider the Grassmannian of $n$-subspaces in ${\Bbb C}^m$, $\text{Gr}(n, {\Bbb C}^m)$, acted on by the maximal torus $T=({\Bbb C}^*)^{m-1}$. Since $\text{Pic Gr}(n, {\Bbb C}^m) \cong {\Bbb Z}$ and is generated by an ample line bundle, the stabilities of linearized actions are determined by the characters of $T=({\Bbb C}^*)^{m-1}$. In fact, let $\Phi: \text{Gr}(n, {\Bbb C}^m) \rightarrow {\Bbb R}^m$ be the standard moment map induced by the Pl\"uker embedding. Then the moment map image is the so called hypersimplex $\Delta^m_n$ $$\Delta^m_n = \{(\alpha_1, \cdots \alpha_m) | 0 \le \alpha_i \le 1, \sum_j \alpha_j = n\}.$$ Thus we have that the $T$-effective ample cone ${\frak E}^T(\text{Gr}(n, {\Bbb C}^m))$ can be identified with the cone over $\Phi(\text{Gr}(n, {\Bbb C}^m))=\Delta^m_n$. \end{say} \begin{say} On the other hand, consider the diagonal action of $G=\text{SL}(n)$ on $X=({\Bbb P}^{n-1})^m$ $$\text{SL}(n) \times ({\Bbb P}^{n-1})^m \rightarrow ({\Bbb P}^{n-1})^m.$$ We have $\text{Pic}^G(X) \cong \text{Pic}(X) \cong {\Bbb Z}^m$. A nef line bundle $$L \cong {\cal O}_{{\Bbb P}^{n-1}}(k_1) \otimes \cdots \otimes {\cal O}_{{\Bbb P}^{n-1}}(k_m), \;\; k_i \ge 0 $$ is $G$-effective (i.e., $X^{ss}(L) \ne \emptyset$) if and only if $n k_i \leq \sum_j k_j$ for any $i = 1, \cdots, m$. If we set $\alpha_i = n k_i/\sum_j k_j$, we can express this condition by the inequalities $ 0 \le \alpha_i \leq 1, \sum \alpha_i = n$. This shows that the $G$-effective ample cone ${\frak E}^G(X)$ is equivalent to the cone over the polytope $$\Delta^m_n = \{ (\alpha_1, \cdots, \alpha_m) \in {\Bbb R}^m: 0 \le \alpha_i \leq 1, \sum \alpha_i = n\}.$$ \end{say} \begin{say} \label{say:GMcorrespondence} Hence we obtain the identification ${\frak E}^T(\text{Gr}(n, {\Bbb C}^m)) = {\frak E}^G(({\Bbb P}^{n-1})^m)$. Applying the 1-1 correspondence between the $T$-orbits on $\text{Gr}(n, {\Bbb C}^m)$ and the $G$-orbits on $({\Bbb P}^{n-1})^m$ (\cite{GelfandMacPherson}), we can further obtain the identification bewteen the $T$-GIT quotients on the Grassmannian $\text{Gr}(n, {\Bbb C}^m)$ and the $G$-GIT quotients on $({\Bbb P}^{n-1})^m$. \end{say} \begin{say} Notice that the underlying line bundle of any element in ${\frak E}^T(\text{Gr}(n, {\Bbb C}^m))$ is ample. Thus the boundary of ${\frak E}^T(\text{Gr}(n, {\Bbb C}^m))$ consists of degenerating characters but with ample underlying line bundles. However, on the contrast, notice that the group $\text{SL}(n)$ has no characters and the boundary of ${\frak E}^G(X)$ consists of only {\sl nef} line bundles. This shows an interesting phenomenon that the two sorts of degenerations can sometimes be harmoniously linked. \end{say} \begin{say} In the following, we shall use the above identifications freely in the rest of paper. One will see that one point of view sometimes has advantage over the other (and vice versa). We will use the above to exhibit our theorems in \S\S 3 and 4. To simplify exhibition, we only consider the case when $n=2$, i.e., the action of $T=({\Bbb C}^*)^{m-1}$ on $\text{Gr}(2, {\Bbb C}^m)$ and the action of $G=\text{SL}(2)$ on $({\Bbb P}^1)^m$, leaving out some possible generalizations for $n > 2$ to a future paper. \end{say} \begin{say} {}From \ref{say:GMcorrespondence}, we have $${\frak E}^T(\text{Gr}(2, {\Bbb C}^m)) = {\frak E}^G(({\Bbb P}^1)^m) =\Delta^m_2.$$ The walls of $\Delta^m_2$ are of the form $$W_J = \{ (\alpha_1, \cdots, \alpha_m) \in \Delta^m_2 | \sum_{i \in J} \alpha_i = 1\}$$ where $J \subset \{1, \cdots, m\}$ is any proper subset. The faces of $\Delta^m_2$ but vertices can be obtained by setting some of the coordinates to be 0 or one coordinate to be 1. (If two coordinates are 1, we will get a vertex.) Thus they are divided into two types: the ones that are obtained by setting $k$ coordinates to be 0 are again hypersimplexes $\Delta^{m-k}_2$; the ones that are obtained by setting $k-1$ coordinates to be 0 but one coordinate to be 1 become simplexes $\Delta^{m-k}_1$. In particular, there are in general two different types of facets (faces of codimension 1): $\Delta^{m-1}_2$ and $\Delta^{m-1}_1$. Precisely, they are $$\Delta^{m-1}_2 [i] = \{ (\alpha_1, \cdots, \alpha_m) \in \Delta^m_2 | \alpha_i = 0, \sum_{j \ne i} \alpha_j = 2\}, \; \text{for all} \; 1 \le i \le m.$$ $$\Delta^{m-1}_1 [i] = \{ (\alpha_1, \cdots, \alpha_m) \in \Delta^m_2 | \alpha_i = 1, \sum_{j \ne i} \alpha_j = 1\}, \; \text{for all} \; 1 \le i \le m.$$ \end{say} \begin{say} Given an element $\alpha= (\alpha_1, \cdots, \alpha_m) \in \Delta^m_2$, we use ${\cal M}^m_\alpha$ to denote the corresponding (GIT or symplectic) quotient of either $({\Bbb P}^1)^m$ by $G=\text{SL}(2)$ or $\text{Gr}(2, {\Bbb C}^m)$ by $T=({\Bbb C}^*)^{m-1}$. \end{say} \begin{rem} \label{say:polygons} We point out that the (GIT or symplectic) quotients of these two actions admit some other interesting interpretations. First of all, they appear as the arithmetic quotients of complex balls (Deligne-Mostow). Secondly, they are also the moduli spaces of spatial polygons modulo orientation-preserving Euclidean motions (\cite{Klyachko92}, \cite{KapovichMillson94b}). Moreover, it is easy to see that the real points of these quotients are the trivial double covers (disjoint unions) of the moduli spaces of linkages in Euclidean palne modulo orientation-preserving Euclidean motions (\cite{KapovichMillson94a} and \cite{Hu92}). Thus, as the sets of the real points of the symplectic quotients ${\cal M}^m_\alpha$, the topological changes of the moduli spaces of linkages with prescribed side lengths in the Euclidean plane when crossing walls are governed by the changes of ${\cal M}^m_\alpha$ (cf. \cite{Hu92}). This is to say that two moduli spaces of linkages (without zero side lengths) in the Euclidean plane are related by a sequence of real blowup and blowdowns. \end{rem} \begin{say} There are $m$ many forgetful morphisms from $({\Bbb P}^1)^m$ to $({\Bbb P}^1)^{m-1}$. Use $f_i$ to denote the forgetful map obtained by forgetting the $i$th factor from $({\Bbb P}^1)^m$. These are $G$-equivariant {\sl trivial} fibrations with fiber ${\Bbb P}^1$. In terms of $G$-effective cones, $f_i$ corresponds to the inclusion of the facet $$\Delta^{m-1}_2 [i] \subset \Delta^m_2$$ where $\Delta^{m-1}_2 [i] \cong \Delta^{m-1}_2$ is obtained by setting the $i$th coordinate to be zero. Noting that ${\cal M}^m_{(\alpha_1, \cdots,\alpha_{i-1}, 0, \alpha_{i+1}, \cdots, \alpha_m)} = {\cal M}^{m-1}_{(\alpha_1, \cdots,\alpha_{i-1}, \alpha_{i+1}, \cdots, \alpha_m)}$, we have \end{say} \begin{thm} (cf. Theorem 2.1, \cite{Hu92}) \label{thm:Hu92thm2.1} Let $\alpha= (\alpha_1, \cdots,\alpha_{i-1}, \alpha_{i+1}, \cdots, \alpha_m) \in \Delta^{m-1}_2$ do not lie on any wall and set $$\widetilde{\alpha}_\epsilon = (\alpha_1 -{\epsilon \over {m-1}}, \cdots, \alpha_{i-1}-{\epsilon \over {m-1}}, \epsilon, \alpha_{i+1}-{\epsilon \over {m-1}}, \cdots, \alpha_m-{\epsilon \over {m-1}}) \in \Delta^m_2$$ where $\epsilon$ is a sufficiently small positive number. Then ${\cal M}^m_{\widetilde{\alpha}_\epsilon} = {\cal M}^{m-1}_\alpha \times {\Bbb P}^1$. \end{thm} \begin{pf} Applying Theorem \ref{thm:inducedmapfromG-equivariancy} (2) to the forgetful map $f_i: ({\Bbb P}^1)^m \rightarrow ({\Bbb P}^1)^{m-1}$, one sees immediately that ${\cal M}^m_{\widetilde{\alpha}_\epsilon}$ is a fibration over ${\cal M}^{m-1}_\alpha$ with fibers ${\Bbb P}^1$. The triviality follows from the fact that $f_i$ is equivariantly trivial and $\text{SL}(2)$ acts freely on any stable locus. \end{pf} \begin{rem} Using the remark in \ref{say:polygons} about the real loci of the quotients ${\cal M}^m_\alpha$, Theorem \ref{thm:Hu92thm2.1} gives an alternative justification for Corollary 15 of \cite{KapovichMillson94a}. \end{rem} \begin{rem} In terms of Grassmannians, the forgetting map $f_i$ corresponds to the (rational) projection from $\text{Gr}(2, {\Bbb C}^m)$ to $\text{Gr}(2, {\Bbb C}^{m-1})$ by projecting a 2-plane in ${\Bbb C}^m$ to the $i$th coordinate $(m-1)$-hyperplane (the hyperplane is obtained by setting the $i$th coordinate to be zero). \end{rem} \begin{say} To study the inclusion $$\Delta^{m-1}_1 [i] \subset \Delta^{m}_2$$ where $\Delta^{m-1}_1 [i]$ is obtained by setting the $i$th coordinate to be 1, we switch our point of view from configuation spaces of points on ${\Bbb P}^1$ to Grassmannians. There are $m$ many {\sl rational} facet maps ${\frak f}_i$ from $\text{Gr}(2, {\Bbb C}^m)$ to $\text{Gr}(1, {\Bbb C}^{m-1})={\Bbb P}^{m-2}$ by taking the intersection of a 2-plane in ${\Bbb C}^{m}$ with the $i$th coordinate $(m-1)$-hyperplane (the hyperplane is obtained by setting the $i$th coordinate to be zero). These are {\sl truly} rational maps but are equivariant with respect to the projection $\rho$ from the maximal torus $T= ({\Bbb C}^*)^{m-1} = ({\Bbb C}^*)^m/\hbox{(diagonal)}$ to the quotient group $T/T_1$ where the 1-PS $T_1 = \{(0, \cdots, \lambda,\cdots, 0) | \lambda \in {\Bbb C}^* \}$ (modulo the diagonal group) acts trivially on $\text{Gr}(1, {\Bbb C}^{m-1})={\Bbb P}^{m-2}$. Any element of $\Delta^{m-1}_1 [i]$ is of the form $\alpha = (\alpha_1, \cdots,\alpha_{i-1}, 1, \alpha_{i+1}, \cdots, \alpha_m)$. Since $T/T_1= ({\Bbb C}^*)^{m-2}$ acts on $\text{Gr}(1, {\Bbb C}^{m-1})={\Bbb P}^{m-2}$ with a dense open orbit, we see that ${\cal M}^m_{\alpha}$ is a point for such an element $\alpha$. \end{say} \begin{thm} (cf. Theorem 2.1, \cite{Hu92}) Let $\alpha = (\alpha_1, \cdots,\alpha_{i-1}, 1, \alpha_{i+1}, \cdots, \alpha_m)$ be any element in the interior of $\Delta^{m-1}_1 [i]$ and set $$\widetilde{\alpha}_{1-\epsilon} = (\alpha_1 +{\epsilon \over {m-1}}, \cdots,\alpha_{i-1}+{\epsilon \over {m-1}}, 1- \epsilon, \alpha_{i+1}+{\epsilon \over {m-1}}, \cdots, \alpha_{m+1}+{\epsilon \over {m-1}})$$ where $\epsilon$ is a sufficiently small positive number. Then ${\cal M}_{\widetilde{\alpha}_{1-\epsilon}}$ is isomorphic to ${\Bbb P}^{m-3}$. \end{thm} \begin{pf} By Bialynicki-Birula's decomposition theorem \cite{Bialynicki-Birula73}, the fibers of the map ${\frak f}_i$ have the structures of $T_1$-modules where $T_1$ acts with positive weights. Since $T$ acts quasi-freely (no finite isotropy subgroups except for the identity group), all of these weights are 1. The theorem now follows from Theorem \ref{thm:generalrgit} and the fact that ${\cal M}^m_{\alpha}$ is a point. \end{pf} \begin{rem} In terms of configurations of points on ${\Bbb P}^1$, the facet map ${\frak f}_i$ correspond to the projection onto the $i$th factor ($1 \le i \le m$). The reason that we hesitated working over $({\Bbb P}^1)^m$ is that the $i$th factor (${\Bbb P}^1$) has no stable points with respect to the action of $\hbox{SL}(2)$ although a quotient trivially exists. \end{rem} \vfill\eject \begin{center} {\bf II. Universal Moduli} \end{center} \section{Quest for universal moduli spaces} \begin{say} \label{say:moduli spaces for curves} Recall that one has the following coarse moduli spaces in the case of curves. \begin{enumerate} \item $M_{g}$, parameterizing nonsingular curves of genus $g\ge2$ and its compactification $\overline{M_{g}}$, parameterizing Mumford-Deligne stable curves. \item $M_{g,m}$, $2g-2+m>0$, parameterizing nonsingular $m$-pointed curves of genus $g\ge2$, and its compactification $\overline{M_{g,m}}$, parameterizing stable $m$-pointed curves. \end{enumerate} It is well known that $\overline{M_{g}}$ and $\overline{M_{g,m}}$ are projective. \end{say} Precisely, the Mumford-Deligne stable curves (resp. $m$-pointed curves) are defined as follows: \begin{defn} \label{defn:definitionofstablecurves} Let $S$ be a base scheme. A stable (resp. semistable) curve of genus $g \ge 2$ over $S$ is a proper flat morphism $f: C \to S$ such that for all $s \in S$ the geometric fiber $C_s$ of $f$ over $s$, satisfies: \begin{enumerate} \item $C_s$ is reduced, connected scheme of dimension 1 with $h^1(C_s, {\cal O}_{C_s}) =g$; \item every singular point of $C_s$ is an ordinary double point; \item if $C_1$ is an irreducible rational component of $C_s$ then $C_1$ meets the rest of $C_s$ in at least 3 points (resp. 2 points). \end{enumerate} \end{defn} \begin{defn} \label{defn:definitionofstablen-pointedcurves} A $m$-pointed curve stable (resp. semistable) curve of genus $g \ge 2$ over $S$ is a proper flat morphism $f: C \to S$ together with $m$-distinct sections $s_i:S \to C$ such that for all $s \in S$ the geometric fiber $C_s$ of $f$ over $s$, satisfies: \begin{enumerate} \item $C_s$ is reduced, connected scheme of dimension 1 with $h^1(C_s, {\cal O}_{C_s}) =g$; \item every singular point of $C_s$ is an ordinary double point; \item all $s_i(s)$ are smooth points of $C_s$ and $s_i(s)\ne s_j(s)$ for $i \ne j$ \item if $C_1$ is an irreducible rational component of $C$ then the number of points where $C_1$ meets the rest of $C_s$ plus the number of points $s_i(s)$ on $C_1$ is at least 3 (resp. 2). \end{enumerate} \end{defn} Rather than going through some abstract definitions of universal moduli spaces, let us go over (briefly) the concrete universal moduli problems listed in 0.1 of the introduction. \begin{say} \label{say:listofmoduli} \begin{enumerate} \item The universal moduli space $\overline{FM_{g, n}} \rightarrow \overline{M_g}$ of Fulton-MacPherson configuration spaces of stable curves. (see also \cite{Pandharipande94b}) \item The compactified universal Picard $\overline{P^d_g} \rightarrow \overline{M_g}$ of degree $d$ line bundles (\cite{Caporaso94}). \item The universal moduli space $\overline{P_{g,m}(e, r, F, \alpha)} \rightarrow \overline{M_{g,n}}$ of $p$-semistable parabolic sheaves of degree $e$, rank $r$, type $F$, and weight $\alpha$ (\cite{Hu95}). \item The universal moduli space $M_{g}({\cal O}, P) \rightarrow \overline{M_g}$ of $p$-semistable coherent sheaves with a fixed Hilbert polynomial $P$ To be more precise, for each $[C] \in \overline{M_{g}}$, one has a natural projective variety, $M_C({\cal O}_C, P)$, parameterizing (Simpson's) $p$-semistable coherent sheaves of a fixed Hilbert polynomial $P$. Our aim is to construct the universal moduli space $M_g({\cal O}, P) \rightarrow \overline{M_{g}}$ parametrizing the set of equivalence classes of pairs $(C, E)$ where $[C] \in \overline{M_{g}}$ and $[E] \in M_C({\cal O}_C, P)$ such that the fiber over the stable curve $C$ is $M_C({\cal O}_C, P)/\hbox{Aut}(C)$. \item The universal Hilbert scheme $\hbox{Hilb}^n_g \rightarrow \overline{M_g}$ of 0-dimensional schemes of length $n$ on the Mumford-Deligne stable curves. There exists a natural dominating morphism $$\psi: \hbox{Hilb}^n_g \longrightarrow M_g({\cal O}, P)$$ when $P(x) = x + n + 1 - g$. \end{enumerate} \end{say} \begin{rem} The last two cases in the above list are what we shall considered seriously in the sequel. The construction of the universal moduli space $M_{g}({\cal O}, P) \rightarrow \overline{M_g}$ will be done by using our RGIT. \end{rem} \begin{rem} The moduli problem in \ref{say:listofmoduli} (3) can be specified as follows. For each $[(C, p_1, \cdots, p_m)] \in M_{g,m}$, one has a natural projective variety, $P_C(e, r, F, \alpha)$, parameterizing $\alpha$-semistable vector bundles of degree $e$ and rank $r$ with quasi-parabolic structures of type $F$ at points $p_i$. Let $P_{g,m}(e, r, F, \alpha)$ be the set of equivalence classes of pairs $((C, p_1, \cdots, p_m), {\cal E})$ where $[(C, p_1, \cdots, p_m)] \in M_{g,m}$ and $[{\cal E}] \in P_C(e, r, F, \alpha)$. Now it is natural to ask for a compactification $\overline{P_{g,m}(e, r, F, \alpha)}$ of $P_{g,m}(e, r, F, \alpha)$ with the following desired properties (cf. \cite{Pandharipande94a}): \begin{enumerate} \item $\overline{P_{g,m}(e, r, F, \alpha)}$ is a projective variety parameterizing equivalence classes of algebro-geometric objects. \item $\overline{P_{g,m}(e, r, F, \alpha)}$ contains $P_{g,m}(e, r, F, \alpha)$ as an open dense subset. \item There is a natural morphism $\eta: \overline{P_{g,m}(e, r, F, \alpha)} \rightarrow \overline{M_{g,m}}$ such that the following natural diagram commutes: \begin{equation*} \begin{CD} P_{g,m}(e, r, F, \alpha) @>>> \overline{P_{g,m}(e, r, F, \alpha)} \\ @VVV @VV{\eta}V \\ M_{g, m} @>>> \overline{M_{g, m}} \end{CD} \end{equation*} \item For each $[(C, p_1, \cdots, p_m)] \in M_{g,m}$, there is an isomorphism $$\eta^{-1}([(C, p_1, \cdots, p_m)]) \cong P_C(e, r, F, \alpha)/\hbox{Aut}(C, p_1, \cdots, p_m).$$ \end{enumerate} Because of the lack of satisfactory GIT construction of $\overline{M_{g,m}}$, we postpone treating this problem in a later publication \cite{Hu95}. \end{rem} \section{The universal moduli space $M_g ({\cal O}, P) \rightarrow \overline{M_g}$} \label{section:simpson} \begin{say} {\sl Simpson's construction of the moduli space of $p$-semistable coherent sheaves.} Let $X$ be a projective scheme over ${\Bbb C}$ with a very ample invertible sheaf ${\cal O}_X(1)$. For any coherent sheaf ${\cal E}$ over $X$, let $p({\cal E}, n)$ be the Hilbert polynomial of ${\cal E}$ with $p({\cal E}, n)= \dim H^0(X, {\cal E}(n))$ for $n \gg 0$. Let $d=d({\cal E})$ be the dimension of the support of ${\cal E}$ which is also the degree of the Hilbert polynomial of ${\cal E}$. The leading coefficient is $r/d!$ where $r=r({\cal E})$ is an integer which is called the rank of ${\cal E}$. A coherent sheaf ${\cal E}$ is of pure dimension $d=d({\cal E})$ if for any non-zero subsheaf ${\cal F} \subset {\cal E}$, we have that $d({\cal E}) = d({\cal F})$. \end{say} \begin{defn} A coherent sheaf ${\cal E}$ is $p$-semistable (resp. $p$-stable) if it is of pure dimension, and if for any subsheaf ${\cal F} \subset {\cal E}$, there exists $N$ such that for $n \ge N$ $$\frac{p({\cal F}, n)}{r({\cal F})} \le \frac{p({\cal E}, n)}{r({\cal E})}.$$ \end{defn} As usual, a $p$-semistable sheaf ${\cal E}$ admits a filtration by subsheaves $$0 = {\cal E}_0, \subset {\cal E}_1 \subset \cdots \subset {\cal E}_k ={\cal E}$$ such that the quotient sheaves ${\cal E}_1/{\cal E}_{i-1}$ are $p$-stable. This filtration is not unique, but $gr({\cal E}) = \oplus {\cal E}_1/{\cal E}_{i-1}$ is. Two $p$-semistable sheaves ${\cal E}$ and ${\cal E}'$ are $s$-equivalent if $gr({\cal E}) = gr({\cal E}')$. Simpson also extends the above definition to the following relative version. \begin{say} Let $S$ be a base scheme of finite type over ${\Bbb C}$ and $X \rightarrow S$ a projective scheme over $S$. Fix a (Hilbert) polynomial $P$ of degree $d$. A $p$-semistable (resp. stable) sheaf ${\cal E}$ on $X/S$ with Hilbert polynomial $P$ is a coherent sheaf ${\cal E}$ on $X$, flat over $S$, such that for each closed point $s \rightarrow S$, ${\cal E}_s$ is a $p$-semistable (resp. stable) sheaf of pure dimension $d$ and Hilbert polynomial $P$ on the fiber $X_s$. \end{say} \begin{say} {\sl Hilbert schemes and Grassmannians}. \label{say:hilbertschemes} As above, let $X \rightarrow S$ be a projective scheme over $S$ with a relatively very ample invertible sheaf ${\cal O}_X(1)$. Fix a (Hilbert) polynomial $P(n)$. Suppose that ${\cal W}$ is a coherent sheaf on $X$ flat over $S$. The Hilbert scheme $\hbox{Hilb}({\cal W}, P)$ parametrizing quotients $${\cal W} \rightarrow {\cal F} \rightarrow 0$$ with Hilbert polynomial $P$. The fiber of $\hbox{Hilb}({\cal W}, P)$ over a closed point $s \in S$ is $\hbox{Hilb}({\cal W}_s, P)$. Grothendieck gives some very explicit relative projective embeddings of $\hbox{Hilb}({\cal W}, P)$ over $S$. There is an $M > 0$ such that for any $m \ge M$ we get a closed embedding $$\psi_m: \hbox{Hilb}({\cal W}, P) \rightarrow \hbox{Grass}(H^0(X/S, {\cal W}(m)), P(m)).$$ There is a canonical invertible sheaf ${\cal L}_m$ on the relative Grassmannian by the embedding $\psi_m$. Over any point in the Grassmannian represented by the quotient ${\cal W} \rightarrow {\cal F}$, the restriction of the invertible sheaf ${\cal L}_m$ is canonically identified with the $\wedge^{P(m)} H^0(X/S, {\cal F}(m))$. {\sl These invertible sheaves are the ones we shall use in place of the relatively ample line bundles $M$ in our various theorems in RGIT} (\S\S 3 and 4). \end{say} \begin{say} {\sl The construction of the moduli spaces of $p$-semistable coherent sheaves}. \label{say:Q2} Fix a large number $N$. Let ${\cal W}={\cal O}_X(-N)$ and $V={\Bbb C}^{P(N)}$. Let $Q_1 \subset \hbox{Hilb}(V \otimes {\cal W}, P)$ denote the open subset of $p$-semistable sheaves of pure dimension $d$. We can assume that $N$ is chosen large enough so that: every $p$-semistable coherent sheaf with Hilbert polynomial $P$ appears as a quotient corresponding to a point of $Q_1$. Now set $Q_2$ equal to the open subset in $Q_1$ such that $\alpha: V\otimes {\cal O}_S \rightarrow H^0(X/S, {\cal E}(N))$ is isomorphism where $\alpha$ is a morphism such that the sections in the image of $\alpha$ generate ${\cal E}(N)$. The group $SL(V)$ acts on $\hbox{Hilb}(V \otimes {\cal W}, P)$ and the line bundle ${\cal L}_m$. The open subset $Q_2$ is invariant under this action. \end{say} Let ${\bf M}_X^\sharp({\cal O}_X, P)$ be the functor for the moduli problem of $s$-equivalence classes of $p$-semistable coherent sheaves on $X$ of pure dimension $d$, Hilbert polynomial $P$, and flat over $S$. \begin{thm} \label{thm:simpsonmoduli} \text{{\rm (C. Simpson. \cite{Simpson94})}} $Q_2$ is contained in the semistable locus $\operatorname{Hilb} (V \otimes {\cal W}, P)^{ss}({\cal L}_m)$ with respect to the action of $SL(V)$ and the linearized line bundle ${\cal L}_m$ ($m \ge M$). And the categorical quotient ${\bf M}_X({\cal O}_X, P)= Q_2/SL(V)$ is a projective scheme over $S$ which coarsely represents the moduli functor ${\bf M}_X^\sharp({\cal O}_X, P)$. \end{thm} We also need to recall Gieseker's construction of $\overline{M_g}$. \begin{say} \label{say:Gieseker's construction} {\sl Gieseker's construction of $\overline{M_g}$}. Fix $g \ge 2$, $e= n(2g-2)$ $(n \ge 10)$, $I=e-g$, and a polynomial in $x$, $p(x) = ex -g +1$. Set the following Hilbert scheme of subschemes in ${\Bbb P}^I$ $$\hbox{Hilb}_I^{p(x)} :=\{\hbox{subschemes in} \; {\Bbb P}^I \; \hbox{with Hilbert polynomial} \; p(x) \}.$$ The group of projective linear transformations $PGL(I+1)$ acts on $\hbox{Hilb}_I^{p(x)}$ naturally. For the reason of lifting to a linear action, we take $G=SL(I+1)$. Now consider the locus $H_g$ of n-canonical stable curves in $\hbox{Hilb}_I^{p(x)}$, that is, $$H_g =\{[i_{\omega^n}(C)] \in \hbox{Hilb}_I^{p(x)} \}$$ where $C$ is a DM-stable curve, $i_{\omega^n}: C \rightarrow {\Bbb P}^I$ is the embedding induced by the nth power of the canonical line bundle $\omega$ over $C$, and $[i_{\omega^n}(C)]$ is the corresponding Hilbert point of the n-canonical curve $i_{\omega^n}(C)$. $H_g$ is a $G$-invariant, irreducible, nonsingular subscheme of $\hbox{Hilb}_I^{p(x)}$. By \cite{Gieseker82}, there can be chosen a $SL(I+1)$-linearization on the Hilbert scheme $\hbox{Hilb}_I^{p(x)}$ such that \begin{enumerate} \item $H_g$ is contained in the stable locus; \item $H_g$ is closed in the semistable locus; and \item the GIT quotient $H_g/SL(I+1)$ is the moduli space $\overline{M_g}$. \end{enumerate} For preciseness, we take $e=10(2g-2)$ and $I=10(2g-2) -g$, once and for all. \end{say} \begin{say} {\sl The construction of the universal moduli space ${\bf M}_g({\cal O}, P) \rightarrow \overline{M_g}$.} \label{say:construction} Let $\widehat{U_g}$ be the universal curve over $H_g$. Consider $X=\widehat{U_g} \rightarrow H_g=S$ as a projective scheme over the base scheme $S=H_g$. Fix a Hilbert polynomial $P (x)= r x + d + r(1-g)$. By Simpson, we get the coarse moduli space ${\bf M}_X({\cal O}_X, P)$ over the base scheme $S=H_g$ of $p$-semistable coherent sheaves (of pure dimension 1) with the Hilbert polynomial $P (x)= r x + d + r(1-g)$. The moduli space ${\bf M}({\cal O}_X, P)$ (as a projective scheme over $S=H_g$) is constructed as the GIT quotient of $Q_2/S$ by the group $SL(V) =SL(P(N))$ (see \ref{say:Q2} and Theorem \ref{thm:simpsonmoduli}). \end{say} \begin{thm} \label{thm:universalmoduliofsimpson'sconstructions} Fix the Hilbert polynomial $P (x)= r x + d + r(1-g)$. The projective categorical quotient ${\bf M}_g({\cal O}, P)=(Q_2/H_g)/(SL(V) \times SL(I+1)$ exists and factors naturally to $\overline{M_g}$ such that over each point $[C] \in \overline{M_g}$ the fiber is canonically identified with the moduli space ${\bf M}_C ({\cal O}_C, P)$ of $p$-semistable coherent sheaves (of pure dimension $1$) with the Hilbert polynomial $P$ modulo the automorphism group of $C$. \end{thm} \begin{pf} Consider the map $\pi: Q_2/H_g \rightarrow H_g$ equivariant with respect to the projection $\rho: SL(V) \times SL(I+1) \rightarrow SL(I+1)$. For $H_g$ we use a linearization $L$ as found by Gieseker (see \ref{say:Gieseker's construction}), while for $Q_2/H_g$ we use the linearization $\pi^* L^k \otimes {\cal L}_m$ (for $k \gg 0$ and some sufficiently large $m$, see \ref{say:hilbertschemes}). Now the theorem follows from Theorem \ref{thm:generalrgit} or Theorem \ref{thm:G'toGmainthmforrelativemoduli}. \end{pf} \begin{rem} \label{rem:thesameintheend} That is, every point of the moduli space ${\bf M}_g({\cal O}, P)$ represents an equivalence class of pairs $(C, {\cal E})$ up to automorphism group of $C$, where $C$ is a Mumford-Deligne stable curve and ${\cal E}$ is a $p$-semistable coherent sheaf of pure dimension $1$ over $C$ with the Hilbert polynomial $P(x) = rx + d+ r(1-g)$. When $P(x) = x + n + 1 - g$ the moduli space ${\bf M}_g({\cal O}, P)$ is a compactification of the universal Picard $P^n_g$. It would be interesting to compare our moduli spaces ${\bf M}_g({\cal O}, P)$ with those in \cite{Caporaso94} and \cite{Pandharipande94a}. \end{rem} \section{The universal Hilbert scheme $\hbox{Hilb}^n_g$} In this last section, we will give a GIT construction of the universal Hilbert scheme $\hbox{Hilb}^n_g$ over $\overline{M_g}$ of $0$-dimensional subschemes of length $n$ on Mumford-Deligne stable curves and a canonical morphism from $\hbox{Hilb}^n_g$ to the compactified universal Picard $M_g({\cal O}, P)$ where $P(x) = x + n + 1 - g$. \begin{say} Let $U_g \rightarrow \overline{M_g}$ be the (fake) universal curve of genus $g \ge 2$ over $\overline{M_g}$. $U_g$ has an obvious GIT construction as the quotient $\widehat{U_g}/SL(I+1)$ by our theorems for $G$-equivariancy RGIT (see \ref{say:construction} for the definition of $\widehat{U_g}$). Set $\hbox{Hilb}^n_g \rightarrow \overline{M_g}$ to be the relative Hilbert scheme over $\overline{M_g}$ of relatively 0-dimensional subschemes in $U_g$ of length $n$. Then the fiber of $\hbox{Hilb}^n_g$ over a point $[C] \in \overline{M_g}$ is the Hilbert scheme $\hbox{Hilb}^n_C$ of 0-dimensional subschemes in $C$ of length $n$ modulo the automorphism group $\hbox{Aut}(C)$ (we will give a GIT construction of $\hbox{Hilb}^n_g \rightarrow \overline{M_g}$ in the sequel). \end{say} \begin{thm} \label{thm:mapsfromHilbtopicard} Let $P(x) = x + n + 1 - g$. Then there exists a natural dominating morphism $\psi$ from $\operatorname{Hilb}^n_g$ to $M_g({\cal O}, P)$ such that the following diagram is commutative \begin{equation*} \begin{CD} \operatorname{Hilb}^n_g @>{\psi}>> M_g({\cal O}, P) \\ @VVV @VVV \\ \overline{M_g} @>{\operatorname{id}}>> \overline{M_g} \end{CD} \end{equation*} \end{thm} \begin{pf} To construct this morphism scheme-theoretically, we first need to give a GIT construction of $\hbox{Hilb}^n_g$ using our theory on $G$-equivariancy RGIT. Recall that $\overline{M_g}$ is constructed as a GIT quotient of a smooth irreducible scheme $H_g$ by the linear transformation $SL(I+1)$ (\ref{say:Gieseker's construction}). Let $\widehat{U_g}$ be the universal family over $H_g$. Set $\widehat{\hbox{Hilb}^n_g} \rightarrow H_g$ to be the relative Hilbert scheme over $H_g$ of relatively 0-dimensional subschemes in $\widehat{U_g}$ of length $n$. The group $SL(I+1)$ operates on $\widehat{\hbox{Hilb}^n_g}$ by moving the subschemes. Theorems \ref{thm:generalizedReichstein} and \ref{thm:inducedmapfromG-equivariancy} imply that the GIT quotient $\widehat{\hbox{Hilb}^n_g}/SL(I+1)$ exists and factors naturally to $H_g/SL(I+1) = \overline{M_g}$ with the fiber at a point $[C] \in \overline{M_g}$ isomorphic to $\operatorname{Hilb}^n_C/\hbox{Aut}(C)$. The quotient $\widehat{\hbox{Hilb}^n_g}/SL(I+1)$ is our universal Hilbert scheme $\hbox{Hilb}^n_g \rightarrow \overline{M_g}$. Now given any relatively 0-dimensional subschemes $Z$ in $\widehat{U_g}$ of length $n$. We can get a coherent sheave of rank 1 and pure dimension 1, ${\cal O}_{\widehat{U_g}/H_g}(Z) = {\cal O}_{\widehat{U_g}/H_g}(-Z)^* = (I_Z)^*$. This leads to a morphism $$\varphi: \widehat{\hbox{Hilb}^n_g} \longrightarrow Q_2/H_g$$ $$ \varphi: Z \longrightarrow {\cal O}_{\widehat{U_g}/H_g}(Z)$$ (see \ref{say:Q2} for the definition of $Q_2$). By passing the the quotient we get $$\varphi': \widehat{\hbox{Hilb}^n_g} \rightarrow Q_2/H_g \rightarrow (Q_2/H_g)/(SL(V) \times SL(I+1)) = M_g({\cal O}, P)$$ where $P=x + n + 1 -g$. One checks that $\varphi'$ is constant on the $SL(I+1)$-orbits. By the universality of categorical quotient, we obtain a canonical morphism $$\psi:\hbox{Hilb}^n_g =\widehat{\hbox{Hilb}^n_g}/SL(I+1) \longrightarrow M_g({\cal O}, P).$$ Both $\hbox{Hilb}^n_g$ and $M_g({\cal O}, P)$ are projective and $\psi$ maps surjectively to the universal Picard over nonsingular curves of genus $g$. Hnece $\psi$ is dominating. \end{pf} \begin{rem} We do not know if the morphism $\psi:\hbox{Hilb}^n_g \rightarrow M_g({\cal O}, P)$ (or its variants) can be useful in the study of limit linear series for stable curves (cf. \cite{EisenbudHarris86}). We do not know if there exists a canonical morphism from the universal FM configuration space over $\overline{M_g}$ to $ M_g({\cal O}, P)$ so that Theorem \ref{thm:mapsfromHilbtopicard} holds. The universal FM configuration space $\overline{FM_{g,n}} \rightarrow \overline{M_g}$ can be constructed by RGIT as follows. Again let $\widehat{U_g} \rightarrow H_g$ be the universal family of curves of genus $g$. Let $\widehat{U_g}[n] \rightarrow H_g $ be the relative FM configuration space over $H_g$ (cf. \cite{Pandharipande94b}). Then the action of $SL(I+1)$ lifts to a canonical action on $\widehat{U_g}[n]$. Now applying Theorem \ref{thm:Gmainthmforrelativemoduli} to $\widehat{U_g}[n] \rightarrow H_g$ and taking the quotients by the group $G=SL(I+1)$, we obtain the universal FM configuration space $$\overline{FM_{g,n}} = \widehat{U_g}[n] /SL(I+1) \rightarrow H_g/SL(I+1) = \overline{M_g}$$ whose fiber at a stable curve $[C]$ is isomorphic to $C[n]/\hbox{Aut}(C)$. \end{rem} \bibliographystyle{amsplain} \makeatletter \renewcommand{\@biblabel}[1]{\hfill#1.}\makeatother

9504

alg-geom/9504013

en

https://arxiv.org/abs/alg-geom/9504013

alg-geom/9504013

David R. Morrison

David R. Morrison

Making enumerative predictions by means of mirror symmetry

24 pages with 2 figures

Mirror Symmetry II (B. Greene and S.-T. Yau, eds.), International Press, Cambridge, 1997, pp. 457-482

Duke preprint DUK-M-94-05

Given two Calabi--Yau threefolds which are believed to constitute a mirror pair, there are very precise predictions about the enumerative geometry of rational curves on one of the manifolds which can be made by performing calculations on the other. We review the mechanics of making these predictions, including a discussion of two conjectures which specify how the elusive ``constants of integration'' in the mirror map should be fixed. Such predictions can be useful for checking whether or not various conjectural constructions of mirror manifolds are producing reasonable answers.

alg-geom

\section{Coordinates on the B-model moduli space} \label{s:bcoords} A {\em Calabi--Yau threefold}\/ is a compact oriented $6$-manifold $Y$ which admits Riemannian metrics whose (global) holonomy is contained in $\operatorname{SU}(3)$. For any such metric, there exists at least one complex structure with respect to which the metric is K\"ahler, and for each such complex structure ${\cal J}$ there is a nowhere-vanishing holomorphic $3$-form $\Omega$ on the complex manifold $Y_{\cal J}$. Given a Calabi--Yau threefold, the topological quantum field theory known as the {\em B-model of $Y$}\/ has as its essential parameters the choice of complex structure $\cal J$ on $Y$. In fact we should identify the {\em B-model moduli space}\/ with the usual moduli space of complex structures (with trivial canonical bundle) which is studied in algebraic geometry. There are some well-known technical difficulties in constructing such moduli spaces, but we are primarily concerned with two aspects of the moduli problem: we need to understand the moduli space {\em locally}, and we need to be sure that there are {\em good compactifications}\/ of the moduli space. For Calabi--Yau manifolds, the first aspect is covered by the theorem of Bogomolov, Tian and Todorov \cite{bogomolov,tian,todorov}, which says that the moduli space is smooth and that its tangent space at $\cal J$ can be naturally identified with $H^1(T^{(1,0)}_{Y_{\cal J}})$, the first cohomology group of the sheaf of holomorphic vector fields. The second aspect---the existence of a good compactification---follows from Viehweg's theorem \cite{viehweg} that the moduli space of {\em polarized}\/ Calabi--Yau manifolds is a quasi-projective variety. To compactify the moduli space, take a projective completion of one of Viehweg's spaces.\footnote{Because of the polarization condition, the resulting space is only a compactification of an {\em open subset}\/ of the original moduli space, but this is adequate for our purposes.} Other compactifications can then be found by blowing up the original one. The physically natural coordinates on this B-model moduli space are provided by ratios of periods of (any) holomorphic $3$-form $\Omega$. That is, if $\dim H^1(T^{(1,0)}_{Y_{\cal J}})=r$ then we choose $r{+}1$ elements $\gamma_0$, $\gamma_1$, \dots, $\gamma_r$ in $H_3(Y,{\bf Z})$, and use the ratios $\int_{\gamma_j} \Omega/\int_{\gamma_0} \Omega$ as local coordinates. (This form of the coordinates was arrived at empirically in \cite{CdGP} and explained in terms of conformal field theory in \cite{BCOV2}.) At each point in the moduli space, any generic choice of such ratios will provide good local coordinates, thanks to the local Torelli theorem, the Bogomolov--Tian--Todorov theorem cited above, and the analysis by Bryant and Griffiths \cite{bryant-griffiths} of the period map for such variations of Hodge structure. One important aspect, therefore, of the problem of making enumerative predictions will be to calculate such periods. This can sometimes be done directly, but a more common approach is an indirect one in which one first calculates the differential equations which the periods satisfy. Choose a family of holomorphic $3$-forms $\Omega(s)$ which depends on a parameter $s$ on the moduli space. (This can only be done locally on the moduli space.) The periods $\int_{\gamma} \Omega(s)$ can be differentiated with respect to parameters, and there must be differential operators ${\cal D}$ which annihilate the periods, that is \[ {\cal D}\left(\int_{\gamma} \Omega(s)\right)=0 \] for all $\gamma\in H_3(Y,{\bf C})$. (These form a differential ideal on the moduli space.) We call these differential operators the {\em Picard--Fuchs operators}, and call the resulting differential equations ${\cal D}\varphi=0$ the {\em Picard--Fuchs equations}\/ determined by $\Omega(s)$. In principle, the Picard--Fuchs equations are derived as follows. Let $\pi:{\cal Y}\to S$ be a proper holomorphic map such that each fiber $\pi^{-1}(s)$ is a complex manifold diffeomorphic to $Y$ which has the complex structure corresponding to $s\in S$. (Such ``universal families'' should at least exist locally over the moduli space.) Let ${\bf V}=R^3\pi_*{\bf C}_{\cal Y}$ be the local system of cohomology groups, and let ${\cal V}={\bf V}\otimes {\cal O}_S$ be the corresponding locally free sheaf, with flat connection \[\nabla:{\cal V}\to{\cal V}\otimes T^*_S\] which annihilates sections of ${\bf V}$. This {\em Gauss--Manin connection}\/ can actually be computed in purely algebraic terms \cite{Katz-Oda}. Doing so leads to the Picard--Fuchs equations indirectly: if we choose a basis $\gamma_0, \dots, \gamma_{2r+1}$ of $H_3(Y)$ then we can write \[\Omega(s)=\sum\left(\int_{\gamma_j}\Omega(s)\right)e^j,\] where $e^j$ is the dual basis of cohomology. Then \[\nabla \Omega(s)=\sum\left(d\int_{\gamma_j}\Omega(s)\right)e^j.\] Thus, we can calculate the effect of differential operators on the periods by calculating the effect of the Gauss--Manin connection on the cohomology itself, and thereby determine the Picard--Fuchs equations. In practice, calculating either the Gauss--Manin connection or the Picard--Fuchs equations is rather difficult. The cases in which these calculations have been carried out explicitly have involved one of two techniques: \begin{enumerate} \item In some cases it has been possible to explicitly evaluate some particular period integral, and expand its value in a power series. The Picard--Fuchs equations can then be found by finding which differential operators annihilate this known period. This method was pioneered in \cite{CdGP}, applied in \cite{CdGP,font,kt1,2param1,2param2,BK}, and reached its culmination in \cite{periods}. \item In somewhat greater generality, in many cases it has been possible to identify the periods with certain generalized hypergeometric functions; the Picard--Fuchs equations are then related to the differential equations of Gel'fand--Zelevinsky--Kapranov \cite{GZK:h}. This method was first suggested in \cite{Bat:vmhs}, developed in \cite{BvS,small}, and systematized in \cite{HKTY1,HKTY2}. \end{enumerate} We refer the interested reader to the cited papers for more details concerning this part of the calculation. In general, the Picard--Fuchs equations will have a $(2r{+}2)$-dimensional family of local solutions at any point of the moduli space, corresponding to the possible homology classes $\gamma$. The reduces the problem of identifying appropriate coordinates to the problem of selecting the ``correct'' homology classes $\gamma_0$ and other $\gamma_j$'s. We address this problem in the remainder of this paper. We will identify the ``correct'' homology classes by comparison with the behavior of the A-model, to which we now turn. \section{The large radius limit} \label{s:largeradius} The flat coordinates in the A-model---mirror to the ``ratio of periods'' coordinates discussed in the previous section---have an ambiguity in their definition which can be described in terms of an integral lattice. In order to explain this, we first review some of the mathematical aspects of the moduli spaces of nonlinear $\sigma$-models (cf.\ \cite{compact,icm}), which involve both A-model and B-model parameters. \subsection{The nonlinear $\sigma$-model} We briefly recall the Lagrangian formulation of nonlinear $\sigma$-models in dimension 2. The essential ingredients needed to describe a nonlinear $\sigma$-model consist of a compact manifold $X$, a Riemannian metric $g_{ij}$ on $X$, and a class $B\in H^2(X,{\bf R}/{\bf Z})$, all defined up to diffeomorphisms of $X$. (We represent $B$ as a closed, ${\bf R}/{\bf Z}$-valued $2$-form, that is, as a collection of locally defined closed real $2$-forms, the union of whose domains of definition is all of $X$, such that the difference between any two local representatives is ${\bf Z}$-valued wherever it is defined.) The nonlinear $\sigma$-model is then constructed from a ${\bf C}/{\bf Z}$-valued (Euclidean) action ${\cal S}$ whose bosonic part assigns to each sufficiently smooth map $\phi$ from an oriented Riemannian $2$-manifold $\Sigma$ to $X$ the quantity\footnote{We suppress the string coupling constant, and use a normalization in which the action appears as $\exp(2\pi i{\cal S})$ in the partition and correlation functions.} \[ {\cal S}_{\text{bosonic}}[\phi]:= i\int_\Sigma \|d\phi\|^2\,d\mu+\int_\Sigma \phi^*(B), \] where the norm $\|d\phi\|$ of $d\phi\in\operatorname{Hom}(T_\Sigma,\phi^*(T_X))$ is determined from the Riemannian metrics on $X$ and on $\Sigma$, and where $\int_\Sigma \phi^*(B)$ is a well-defined element of ${\bf R}/{\bf Z}$ by virtue of the canonical isomorphism $H^2(\Sigma,{\bf R}/{\bf Z})\cong{\bf R}/{\bf Z}$. (Additional fermionic terms must be added to ${\cal S}_{\text{bosonic}}$ in order to make the theory supersymmetric, but as they do not affect the essential parameters in the theory we suppress them here.) It is more customary to require $B$ to be a real $2$-form, in which case ${\cal S}_{\text{bosonic}}$ becomes ${\bf C}$-valued, and one observes that the physics is invariant under shifting $B$ by an integral cohomology class. (The possibility of a more general form of the action\footnote{The more general form of the action and its properties as described in this paragraph arose in discussion with Paul Aspinwall (cf.\ \cite{stable}).} ${\cal S}_{\text{bosonic}}$ which allows $B$ to be an ${\bf R}/{\bf Z}$-valued $2$-form is implicit in \cite{Vafa,DijkgraafWitten,chiral}.) To compare this more general form to the customary one, consider the exact sequence \[ 0\to H^2_{\text{DR}}(X,{\bf Z}) \to H^2(X,{\bf R}) \to H^2(X,{\bf R}/{\bf Z}) \to H^3(X,{\bf Z})_{\text{tors}} \to0, \] where $H^2_{\text{DR}}(X,{\bf Z})$ denotes the image of $H^2(X,{\bf Z})$ in de~Rham cohomology. The last term in this exact sequence is a finite group which labels the connected components of $H^2(X,{\bf R}/{\bf Z})$. If we only used real $2$-forms modulo integral $2$-forms to describe $B$, we would get only one connected component of that space. We are interested in a special case of this construction in which the theory has what is called $N{=}(2,2)$ supersymmetry and is in addition conformally invariant. To ensure the first property we assume that the Riemannian metric is K\"ahler with respect to some complex structure. The second property is somewhat problematic at present, but a necessary condition is that the K\"ahler form of the metric be in the same de~Rham cohomology class as the K\"ahler form of some Ricci-flat metric, and that the volume of the metric be sufficiently large. Let ${\cal J}$ be a complex structure on $X$ for which the metric $g_{ij}$ is K\"ahler. If we pick a complex structure on $\Sigma$ which makes its Riemannian metric K\"ahler, and which is compatible with its orientation, then the first term in the action can be rewritten using the formula: \[ \int_\Sigma \|d\phi\|^2\,d\mu=\int_\Sigma \|\bar\partial\phi\|^2\,d\mu +\int_\Sigma \phi^*(\omega), \] where $\bar\partial\phi\in\operatorname{Hom}(T_\Sigma^{(1,0)},\phi^*(T_{X_{\cal J}}^{(0,1)}))$ is determined by the complex structures, and where $\omega$ is the K\"ahler form of the metric $(g_{ij})$ on $X$. It follows that when the classical action is evaluated on a holomorphic map $\phi$ (i.e., one with $\bar\partial\phi\equiv0$), the result is simply $$\int_\Sigma \phi^*(B+i\omega)\in{\bf C}/{\bf Z}.$$ The ``topological'' correlation functions (of both A-model and B-model type)---when evaluated using $\sigma$-model perturbation theory---% depend only on these extrema of the classical action, and so ultimately will depend only on the choice of complex structure ${\cal J}$ and {\em complexified K\"ahler form}\/ $\beta:=B+i\omega\in H^2(X,{\bf C}/{\bf Z})$. \subsection{The A-model parameter space} Not every element of $H^2(X,{\bf C}/{\bf Z})$ corresponds to a complexified K\"ahler form; the ones which do, for a fixed complex structure ${\cal J}$ on $X$, constitute the {\em complexified K\"ahler cone} \[ {\cal K}_{{\bf C}}:= \{\beta\in H^2(X,{\bf C}/{\bf Z})\,|\, \Im(\beta) \text{ lies within the K\"ahler cone of }X_{\cal J}\}.\] The perturbative analysis of the $\sigma$-model is expected to be valid in some open subset of ${\cal K}_{{\bf C}}$ containing all metrics of sufficiently large volume, that is, in a set of the form \[ ({\cal K}_{{\bf C}})^\circ:= \{\beta\in H^2(X,{\bf C}/{\bf Z})\,|\, \Im(\beta) \text{ lies {\it deep}\/ within the K\"ahler cone of }X_{\cal J}\}.\] The actual parameter space for $\sigma$-models with complex structure ${\cal J}$ can then be represented as $({\cal K}_{{\bf C}})^\circ/\operatorname{Aut}(X_{\cal J})$, where $\operatorname{Aut}(X_{\cal J})$ is the group\footnote{Typically, the group $\operatorname{Aut}(X_{\cal J})$ acts discretely on $({\cal K}_{{\bf C}})^\circ$.} of diffeomorphisms of $X$ which preserve the complex structure ${\cal J}$. We refer to this as the {\it A-model parameter space}, since the correlation functions of the A-model are independent of the complex structure but do depend on the parameters being described here. A more global analysis \cite{catp,phases} reveals that the parameter space $({\cal K}_{{\bf C}})^\circ/\operatorname{Aut}(X_{\cal J})$ must often be enlarged if we wish to describe the full moduli space of $N{=}(2,2)$ conformal field theories. But for our present purposes, we are more concerned with the ``large radius limit'' which occurs at the boundary of $({\cal K}_{{\bf C}})^\circ/\operatorname{Aut}(X_{\cal J})$, and we need not worry about such enlargements. \subsection{Flat coordinates and the large radius limit} In order to put specific coordinates on the A-model parameter space $({\cal K}_{{\bf C}})^\circ/\operatorname{Aut}(X_{\cal J})$, we need to choose a presentation for $H_2(X,{\bf Z})$ with generators $e_1, \dots, e_r, f_1, \dots, f_s$ and relations $m_kf_k=0$, $k=1,\dots, s$, for some natural numbers $m_k>1$. Thus, $f_1, \dots, f_s$ generate the torsion subgroup, and $e_1, \dots, e_r$ form a basis for the free abelian group $H_2(X,{\bf Z})/(\text{torsion})$. We introduce the dual basis $e^1, \dots, e^r$ of $H^2_{\text{DR}}(X,{\bf Z})$, which will generate the integral lattice that provides the ambiguity in the flat coordinates. We make the crucial assumption that {\em each $e^j$ lies in the closure of the K\"ahler cone}. Since $H^2(X,{\bf C}/{\bf Z})$ is isomorphic to $\operatorname{Hom}(H_2(X,{\bf Z}),{\bf C}^*)$, each point $\beta\in({\cal K}_{{\bf C}})^\circ$ can be regarded as a homomorphism, and as such is determined by its values on a basis, i.e., by $q_j:=\beta(e_j)$ and $\tau_k:=\beta(f_k)$, which must be nonzero complex numbers. The latter are subject to the relations $\tau_k^{(m_k)}=1$; the choice of {\em which}\/ roots of unity to use for the $\tau_k$'s determines {\em which}\/ connected component of the parameter space we are working with. The $q_j$'s are exponentials of the components of the original $2$-form, that is, when $B+i\omega\in H^2(X,{\bf C})/H^2_{\text{DR}}(X,{\bf Z})$ we can write \[B+i\omega=\frac1{2\pi i}\sum_j(\log q_j)e^j.\] It is the logarithms $t_j:=\frac1{2\pi i}\log q_j$ which are the ``flat'' coordinates. These are multiple-valued, and can be shifted by independent integers. (This indicates how the lattice $H^2_{\text{DR}}(X,{\bf Z})$ specifies the ambiguity in the flat coordinates.) However, the corresponding vector fields and $1$-forms \[\frac{\partial}{\partial t_j}=2\pi i\, q_j\,\frac{\partial}{\partial q_j} \qquad \text{and} \qquad dt_j = \frac1{2\pi i}\,d\mskip0.5mu\log q_j \] are single-valued. In order to study the large radius limit, we restrict our attention to those K\"ahler classes which lie in the cone \[{\cal C}={\cal C}_{\vec{e}}:=\{\omega=\sum\omega_je^j\,|\, \omega_j>0\}\] spanned by the chosen basis vectors. If the action of $\operatorname{Aut}(X_{\cal J})$ on ${\cal K}_{{\bf C}}$ is discrete, then it will be possible to find such bases with the property that ${\cal C}_{\vec{e}}$ is disjoint from its translates under $\operatorname{Aut}(X_{\cal J})$. (In any case, we shall ignore the action of $\operatorname{Aut}(X_{\cal J})$ for the time being.) The corresponding complexified cone \[{\cal C}_{{\bf C}}:=\{\beta\in H^2(X,{\bf C}/{\bf Z})\,|\, \Im(\beta)\in{\cal C}\} \subset {\cal K}_{{\bf C}}\] is described in coordinates by the condition \[\Im(\frac1{2\pi i}\log q_j)=-\frac1{2\pi}\log|q_j|>0\text{ for all }j,\] or equivalently, \[0<|q_j|<1\text{ for all }j.\] To find the large radius limit, we should rescale $\omega\to \lambda\omega$, and let $\lambda$ grow to infinity. Under such a rescaling, we have \[q_j\mapsto |q_j|^{(\lambda-1)}\,q_j .\] Thus, all points in ${\cal C}_{{\bf C}}$ flow towards $q_j=0\ \forall j$ under this rescaling, and $q_j=0\ \forall j$ should be taken as the ``large radius limit.''\footnote{It is not yet precisely clear how one should interpret the torsion variables $\tau_k$ in the large radius limit, but see \cite{stable} for some steps in this direction. For the purposes of this paper, we work with the component in which $\tau_k=1$ for all $k$.} We form a partial compactification of our parameter space by enlarging it to include all $q$'s such that \[0\le |q_j|<1\text{ for all }j.\] On this enlarged space, the $q_j$'s occur as natural coordinates, and the ``boundary'' of the space is a divisor with normal crossings. There is a natural identification which can be made between the space of marginal operators for the A-model and the vector fields $\partial/\partial t_j$ corresponding to the flat coordinates. When we calculate three-point functions with respect to these coordinates, we find an expansion of the form \[\big\langle \frac{\partial}{\partial t_j} \frac{\partial}{\partial t_k} \frac{\partial}{\partial t_\ell} \big\rangle = e^j \cup e^k \cup e^\ell |_{[X]} + O(q), \] where $O(q)$ represents the instanton corrections to the classical value, which contain the data about the enumeration of rational curves. If we write this in terms of the (single-valued) coordinates $q_j$ at the large radius limit point, we find \[(2\pi i)^3\,q_jq_kq_\ell\, \big\langle \frac{\partial}{\partial q_j} \frac{\partial}{\partial q_k} \frac{\partial}{\partial q_\ell} \big\rangle = e^j \cup e^k \cup e^\ell |_{[X]} + O(q). \] In other words, the three-point function has poles along the boundary divisor in the $q$-coordinates. Moreover, the leading order term in a Laurent expansion of a three-point function picks out the corresponding cohomological quantity. The analysis we have given depends on a choice of basis; we defer to section \ref{s:ambiguity} a discussion of what happens when the basis is changed. \section{Maximally unipotent monodromy} \label{s:maxunip} The structure which we have found in the A-model---a partial compactification of the parameter space, with poles of the correlation functions along the compactification divisor---will now serve as a guide to making enumerative predictions by means of B-model calculations. In order to carry this out, we must the analyze compactifications of the B-model moduli space. Given an arbitrary compactification of the B-model moduli space, we are always free to blow up along the boundary until the boundary becomes a normal crossings divisor. The only remaining singularities of the space after such a blowup would lie in the interior of the moduli space (where there may well be quotient singularities associated with complex structures for which the automorphism group is larger than generic). Even those can be removed by passing to a finite cover. When the boundary is a normal crossings divisor, the monodromy theorem \cite{monodromy} guarantees that the monodromy of the periods around each component of the boundary is a quasi-unipotent transformation (unipotent after passing to a finite cover). Unipotent monodromy appears to be necessary in order to correctly reproduce the behavior of the A-model. We will therefore assume that the monodromy transformations near the points we seek are in fact unipotent. In order to analyze the boundary in detail and search for the mirrors of large radius limit points, we restrict our attention to a local situation in which a product of punctured disks (with coordinates $s_j$) has been embedded in the interior of our moduli space in such a way that the limit points $s_j\to0$ are mapped to the boundary. Let $T^{(j)}$ be the monodromy transformation about the $j^{\text{th}}$ coordinate (counterclockwise), with respect to some fixed basepoint $P$ near the origin. For discussions of monodromy, it is more convenient to represent each period $\int_{\gamma}\Omega(s)$ by means of cup product with a cohomology class $g\in H^3(Y_P,{\bf C})$, i.e., \[\int_{\gamma}\Omega(s)=\langle g\,|\,\Omega(s)\rangle:=\int_{Y_P}g\wedge\Omega(s).\] The cycle $g$ extends to a multi-valued section of the local system $R^3\pi_*{\bf C}_{\cal Y}$, and the corresponding period is also multi-valued. However, according to the nilpotent orbit theorem \cite{schmid}, the section \[\exp(-\frac1{2\pi i}\log s\cdot\log T)\,g\in\Gamma({\cal V})\] of the locally free sheaf ${\cal V}=R^3\pi_*{\bf C}_{\cal Y} \otimes{\cal O}_S$ is single-valued. We introduce\footnote{This sign convention differs from \cite{compact}, but agrees with \cite{Deligne}.} $N^{(j)}=-\log T^{(j)}$ so that the corresponding single-valued section can be written as \[\widetilde{g}:=\exp(\frac1{2\pi i}\sum(\log s_j)N^{(j)})\,g.\] We now consider the conditions on periods needed to match the behavior of the A-model. First, the period $\int_{\gamma_0}\Omega(s) =\langle g^0\,|\,\Omega(s)\rangle$ should be single-valued, so we need to find a cycle $g^0$ such that $N^{(j)} g^0=0$ for all $j$. Second, the monodromy on the period $\int_{\gamma_j}\Omega(s) =\langle g^j\,|\,\Omega(s)\rangle$ should only involve $\int_{\gamma_0}\Omega(s)$, in order that the multi-valuedness of the ratios shifts them by constants. Thus, we need for $N^{(j)} g^k$ to be a multiple of $g^0$ for every $j$ and $k$. In fact, if we write \[ N^{(j)}g^k=m^{jk}g^0,\] then the matrix $(m^{jk})$ must be invertible in order to solve for coordinates with the desired monodromy properties. Letting $(m_{k\ell})$ denote the inverse matrix of $(m^{jk})$, we have \[(T^{(j)}-I)(-\sum g^\ell m_{\ell k})=\sum N^{(j)}g^\ell m_{\ell k}=\delta^j_k\,g^0,\] so this cohomology class ``$-\sum g^\ell m_{\ell k}$'' determines the ratio of periods which has the correct monodromy properties. The corresponding (multi-valued) coordinates are then \[ \frac1{2\pi i}\,\log z_k=\frac{-1}{\langle g^0\,|\,\Omega\rangle} \sum_{\ell=1}^r\langle g^\ell\,|\,\Omega\rangle m_{\ell k}. \] As in the case of the A-model, by exponentiating these we obtain single-valued coordinates $z_k$ which extend across the boundary. The coordinates as written are not uniquely specified, and in fact we have not yet used all of the information available to us by comparison with the A-model. What we have not yet considered is the three-point functions of the B-model, which should be mirror to the three-point functions of the A-model. In order to describe these, we must fix a particular choice $\Omega(s)$ of holomorphic $3$-forms on the fibers, which can be thought of as fixing the gauge in the bundle $\pi_*\omega_{{\cal X}/S}$ whose fibers are the spaces of global holomorphic $3$-forms on the fibers of $\pi$. Moreover, as in the case of the A-model, the marginal operators whose correlation functions we wish to calculate can be identified with vector fields on the moduli space. Any system of coordinates $s_j$ has an associated collection of vector fields $\partial/\partial s_j$, and with respect to these, the three-point function can be written \[\big\langle \frac{\partial}{\partial s_j} \frac{\partial}{\partial s_k} \frac{\partial}{\partial s_\ell} \big\rangle := \int_Y\Omega(s)\wedge\nabla_{s_j}\nabla_{s_k}\nabla_{s_\ell}\Omega(s)\] where $\nabla_{s_j}\varphi$ represents the directional derivative $(\nabla\varphi)\lhk \frac{\partial}{\partial s_j}$. When we calculate these three-point functions near the boundary of the moduli space, we should expect to find poles (in order to replicate the behavior of the A-model moduli space), and indeed the presence of poles in the extension of $\nabla$ to the boundary is a well-known phenomenon in algebraic geometry (cf.\ \cite{regsings}). These poles arise from the behavior of the single-valued sections of ${\cal V}$ under differentiation: if we calculate using the single-valued section $\widetilde g$ introduced above, we find \[\nabla(\widetilde g)=\frac1{2\pi i}\sum(d\mskip0.5mu\log s_j)N^{(j)}\,\widetilde g.\] The coefficients $d\mskip0.5mu\log s_j$ in this expression are $1$-forms with poles along the boundary. Taking three directional derivatives $\nabla_{s_j}\nabla_{s_k}\nabla_{s_\ell}$, we see that the leading term in a Laurent expansion has coefficient proportional to something of the form \[\left(\frac1{2\pi i}\right)^3 N^{(j)}N^{(k)}N^{(\ell)}\,\widetilde g .\] Now the comparison with the A-model tells us that at least some of these coefficients must be nonzero, since they should reproduce the intersection numbers $e^j\cup e^k\cup e^\ell|_{[X]}$ of the mirror partner. (In fact, Poincar\'e duality on $X$ implies some rather strong conditions on these intersection numbers, which must be replicated by the coefficients we are calculating on the B-model side.) The simple fact that any of these numbers is nonzero, though, immediately implies that the order of unipotency of the monodromy transformations is in some sense ``maximal''. (Any quartic expressions in the $N$'s must vanish for dimension reasons, and so cubic expressions are the maximal possible degree for a non-vanishing expression.) Furthermore, the fact that the $N^{(j)}$'s define a limiting mixed Hodge structure in which $h^{(3,0)}=1$ implies that the images of all of the cubic expressions lies in a one-dimensional space $W_0$. Then, by including the relations deduced from Poincar\'e duality, we arrive at the following definition.\footnote{The original definition given in \cite{mirrorguide} only applied to the one-parameter case; this was extended to several parameters by Deligne \cite{Deligne} and the author \cite{compact}. (We follow the version given in \cite{compact}.)} \begin{definition} A normal crossings boundary point $P$ of $S$ is called a\/ {\em maximally unipotent point} under the following conditions. \begin{enumerate} \item $P$ lies at the intersection of $r=\dim S$ local boundary components $B_j$, and the monodromy transformations $T^{(j)}$ around these components are all unipotent. \item Let $N^{(j)}=-\log T^{(j)}$, let $N :=\sum a_jN^{(j)}$ for some $a_j>0$, and define \begin{eqnarray*} W_0&:=&\Im(N ^3)\\ W_1&:=&\Im(N ^{2})\cap\operatorname{Ker} N \\ W_2&:=&\left(\Im(N )\cap\operatorname{Ker}(N )\right)+\left( \Im(N ^{2})\cap\operatorname{Ker}(N ^2)\right). \end{eqnarray*} Then $\dim W_0=\dim W_1=1$ and $\dim W_2=1+\dim(S)$. \item Let $g^0,g^1,\dots,g^r$ be a basis of $W_2$ such that $g^0$ spans $W_0$, and define $m^{jk}$ by $N^{(j)}g^k=m^{jk}g^0$ for $1\le j,k\le r$. Then $m:=(m^{jk})$ is an invertible matrix. \end{enumerate} (The spaces $W_0$ and $W_2$ are independent of the choice of coefficients $\{a_j\}$ {\rm\cite{CK,deligne:weil2}}, and the invertibility of $m$ is independent of the choice of basis $\{g^k\}$.) \end{definition} When we restrict to maximally unipotent boundary points, the single-valued $1$-forms \[ \frac1{2\pi i}\,d\mskip0.5mu\log z_k=d\left(\frac{-1}{\langle g^0\,|\,\Omega\rangle} \sum_{\ell=1}^r\langle g^\ell\,|\,\Omega\rangle m_{\ell k}\right). \] are independent of the choice of basis $\{g^k\}$ and of $3$-form $\Omega$. The coordinates $z_k$ themselves {\em do}\/ depend on the choice of basis, but only through multiplicative constants: a change of basis replacing $g^k$ by $\sum_{\ell=0}^kc^k_\ell g^\ell$ will induce \[\frac1{2\pi i}\log z_k \mapsto \frac{c^k_0}{c^0_0} + \frac1{2\pi i}\log z_k\] and so \[z_k\mapsto e^{2\pi i(c^k_0/c^0_0)}\,z_k.\] Determining the ``constants of integration'' which specify $z_k$ once $d\mskip0.5mu\log z_k$ is known is the most delicate part of finding the mirror map. We will address this issue in section \ref{s:mirrormap}. In practice, computing the monodromies about all boundary components and locating which points on the boundary have maximally unipotent monodromy is a challenging task. In fact, this computation has only been carried out fully in a few examples \cite{CdGP,2param1,2param2}. In the special cases of complete intersections in toric varieties, there is another method which has been use to locate such points: one finds the natural ``toric'' limit points in the toric moduli space which correspond to K\"ahler cones of possible birational models of the mirror, and these turn out to have maximally unipotent monodromy (as follows from \cite{Bat:qcoho,BK}). This alternate method must be used with some caution, for {\em it is possible to have maximally unipotent boundary points which are not toric boundary points.} An explicit example of this phenomon was seen in \cite{2param2}, where there is a non-toric boundary point with maximally unipotent monodromy. Interestingly, in that example there is also an additional (non-toric) discrete symmetry of the toric moduli space by which one must quotient in order to obtain the true B-model moduli space. That additional discrete symmetry exchanges the toric and non-toric points with maximally unipotent monodromy. It would be interesting to know whether or not this is true in general: in the toric complete intersection case, given a boundary point with maximally unipotent monodromy, does there always exist a discrete symmetry of the moduli space which maps this point to a toric boundary point? \section{Equivalence among boundary points} \label{s:ambiguity} Our discussion in section \ref{s:largeradius} of coordinates near the large radius limit depended on the choice of basis for $H_2(X,{\bf Z})/(\text{torsion})$, or equivalently, on the choice of simplicial cone ${\cal C}\subset{\cal K}$. The fact that different choices of simplicial cone (always contained in the K\"ahler cone) lead to apparently different ``large radius limit'' points in the A-model parameter space should not be too surprising. The limit point we seek is actually a boundary point of our parameter space, and what we are finding is that there are different ways to compactify the space. Since we are using compactifications which (locally) have the structure of an algebraic variety, we should expect birational modifications along the boundary to provide a mechanism for passing between compactifications and indeed that is what happens with our choice of cones. Subdividing a given cone into smaller ones precisely corresponds to blowing up, as in toric geometry. For example, if we start from a basis $e^1, \dots, e^r$ and blowup the origin in the coordinate chart $(q^1,\dots,q^r)$, we find new coordinate charts after the blowup, with coordinates $(q_1,\frac{q_2}{q_1},\dots,\frac{q_r}{q_1})$, \dots, $(\frac{q_1}{q_r},\dots,\frac{q_{r-1}}{q_r},q_r)$, respectively. (The corresponding bases are $\{e^1+\cdots+e^r, e^2,\dots,e^r\}$, \dots, $\{e^1,\dots,e^{r-1}, e^1+\cdots+e^r\}$, respectively.) Rescaling the metric and taking $\lambda\to\infty$ sends $(q_1,\dots,q_r)$ to the origin in the first chart when $1>|q_1|>|q_j|$ ($\forall\ j\ne1$), sends it to the origin in the second chart when $1>|q_2|>|q_j|$ ($\forall\ j\ne2$), and so on. All of these ``origins'' can thus lay claim to being ``the large radius limit'' associated to at least {\it part}\/ of the A-model parameter space. Conversely, if we have a partial compactification of the A-model parameter space which includes more than one large radius limit point (each associated with a different basis $e^1,\dots,e^r$, and with a different domain inside the moduli space), we should attempt to blow down this space to produce a partial compactification with a single large radius limit point for the entire moduli space. These blowdowns are similar to those arising in toric geometry, and will often lead to singularities in the compactified space. The instanton contributions to correlation functions are still suppressed in such a limit, in spite of the singularities---we must accept the possibility that the ``true'' large radius limit point is not a smooth point. (Note that all of the large radius limit points under discussion are associated to a single K\"ahler cone. It is also possible to consider other large radius limit points associated to the K\"ahler cones of different birational models of $X$. This leads to topology-changing transitions \cite{catp}, and we would not expect to collapse those limit points to a single point by blowing down.) Comparison between different cones can be accomplished by considering the canonical $1$-forms $d\mskip0.5mu\log q_j$. These are intrinsically defined, and should only change by a constant change-of-basis matrix when moving from one large radius limit point to another (within the same K\"ahler cone). These $1$-forms will therefore define a local system $\cal L$ in a neighborhood of all of the exceptional divisors of a potential blowing-down map associated to the K\"ahler cone. We are thus led to introduce an equivalence relation among boundary points of the A-model parameter space: two boundary points $P$ and $Q$ are equivalent if there exists a connected subset $\Xi$ of the boundary containing both points and a local system ${\cal L}$ defined in a neighborhood of $\Xi$ which is spanned by the canonical $1$-forms $d\mskip0.5mu\log q_j$ at any maximally unipotent point within $\Xi$. (For further details about this construction, we refer the reader to \cite{compact}.) Even when we expect to be able to blow down and are willing to allow singularities, it may prove to be impossible to perform the desired blowing down, due to the presence of an infinite number of large radius limit points. We describe this phenomenon in an explicit example, following \cite{where}. Suppose that the K\"ahler cone is described as $\frac2{1-\sqrt5}\,y<x<\frac2{1+\sqrt5}\,y$. Then (as shown in figure 1) attempting to cover the cone using integral bases leads to a sequence of rays with slopes $\frac y x = \dots,-\frac58,-\frac23,-1,\frac10,2,\frac53,\frac{13}8,\dots$ which asymptotically approach the walls\footnote{The figure does not include these walls---the limiting rays with irrational slope $\frac{1\pm\sqrt5}2$ ---since they are less than a line-width's distance from the outer rays as shown (at the level of resolution of the figure).} of the cone. Each adjacent pair of rays in the sequence gives rise to a distinct large radius limit point. $$\vbox{\xpt \baselineskip=12pt \iffigs \centerline{\epsfxsize=8cm\epsfbox{f2.ps}} \else \vglue1cm \fi \centerline{Figure 1. Decomposing the cone $\frac2{1-\sqrt5}\,y<x<\frac2{1+\sqrt5}\,y$.}}$$ However, when we include the action of $\operatorname{Aut}(X_{\cal J})$ in our analysis, it may become possible to do the blowdowns---an infinite number of large radius limit points may turn into a finite number after these identifications \cite{compact,kcone}. In the example above, an automorphism acting on the cone as $(x,y)\mapsto(2x+3y,3x+5y)$ leads from an infinite number of large radius limit points on the original K\"ahler moduli space to two remaining points on the quotient space. There are two boundary divisors (after taking the quotient), and they meet in two large radius limit points. The quotient space can then be blown down explicitly using methods of Hirzebruch \cite{hirzebruch}, leading to a surface singularity with local equation $w^2=(u^3-v^2)(u^2-v^3)$. This is illustrated in figure 2. In general, $\Xi$ will be a subset of the compactified moduli space only after taking such a quotient, which is why we use a local system ${\cal L}$ rather than simply a trivial sheaf. $$\vbox{\xpt \baselineskip=12pt \iffigs \centerline{\epsfxsize=10cm\epsfbox{f1.ps}} \else\vglue1cm\fi \centerline{Figure 2. The blown down moduli space $w^2=(u^3-v^2)(u^2-v^3)$}} $$ Applying these ideas to the analysis of the B-model moduli space, we see that we should consider two large radius limit points to be equivalent when there exists a connected subset $\Xi$ of the boundary containing both points and a local system ${\cal L}$ defined in a neighborhood of $\Xi$ which is spanned by the canonical $1$-forms $d\mskip0.5mu\log z_k$ at any maximally unipotent point within $\Xi$. This can be effectively computed if we know the mirror map at each maximally unipotent point; in fact, it is enough to calculate the leading terms in Laurent expansions of $3$-point functions. Further details are in \cite{compact}. \section{Determining the mirror map (two conjectures)} \label{s:mirrormap} As pointed out in section \ref{s:maxunip}, the most delicate part of determining the mirror map is specifying the constants of integration, passing from the canonical $1$-forms $d\mskip0.5mu\log z_j$ to the actual multi-valued coordinates $z_j$ themselves. There are two conjectures which have been used to determine these constants. The first one (stated here in detail for the first time) is completely general, applying in principle to all Calabi--Yau threefolds, while the second one is special to the case of toric hypersurfaces. In addition to specifying the coordinates, one would like to specify a particular choice of holomorphic $3$-forms $\Omega(z)$ in order to determine the $3$-point functions precisely. This is something which the conjectures will also do. \subsection{A conjecture about integral cohomology} The first conjecture for determining the mirror map involves the integral cohomology groups of a Calabi--Yau threefold. The conjecture is quite natural, and there is a bit of evidence for it in a few specific examples. We will give additional evidence here. Simply put, we conjecture that the canonical coordinates and canonical gauge for $\Omega(z)$ should be given by periods over {\em integer-valued}\/ cohomology classes, in the following precise sense. The classes $g^0$, $g^1$, \dots, $g^r$ should be chosen from $H^3(Y,{\bf Z})$, in such a way that $g^0$ spans $W_0\cap H^3(Y,{\bf Z})$ and the entire set $g^0$, $g^1$, \dots, $g^r$ spans $W_2\cap H^3(Y,{\bf Z})$. We conjecture that if this is done, and if we write $N^{(j)}g^k=m^{jk}G^{(0)}$ as in the previous section, then the matrix $(m^{jk})$ {\em is invertible over the integers}. If this is true, then the mirror map will be uniquely specified by using the integral periods, and the gauge $\Omega(z)$ for which $\langle g^0\,|\,\Omega(z)\rangle=\pm1$ will be uniquely specified (up to sign) as well. (This is because in any change of basis $g^k\mapsto\sum_{\ell=0}^kc^k_\ell g^\ell$ preserving the integral structure, we will have $c^0_0=\pm1$ and $c^k_0\in{\bf Z}$ so that $\exp(2\pi i(c^k_0/c^0_0))=1$.) Our conjecture is motivated by the observation that the integral structure on $H^2(X)$ controls the choice of coordinates there. In fact, the action of $N^{(j)}$ on $H^3(Y)$ can be seen as mirroring the action ``cup product with $e^j$'' on $H^*(X)$, where $e^j\in H^2(X)$ is an integral class, part of the basis determining the coordinates. The idea that the integral structure on $H^3$ should mirror the integral structure on $H^0\oplus H^2\oplus H^4\oplus H^6$ is not a new one---it was explicitly mentioned in \cite{AL:qag}, for example, and it was used implicitly in the calculations of \cite{CdGP} (cf.\ also \cite{Cd2}). There is not a lot of evidence for this equivalence, however, other than the examples which we describe here. As a practical matter, our conjecture can be tested in the following way. Compute the monodromy matrices $T^{(j)}$ with respect to a basis of integral cohomology. There must then be a rank one matrix $M$ (with image $W_0$) such that $N^{(j)}N^{(k)}N^{(\ell)}=c^{jk\ell}M$ for all $j$, $k$, $\ell$, where $c^{jk\ell}:=e^j\cup e^k\cup e^\ell|_{[X]}$ are the intersection numbers on the mirror partner $Y$ of $X$. The conjecture states that $M$ should be a primitive integral matrix.\footnote{In this version of testing the conjecture, it is assumed that a mirror partner is known. One could test the conjecture without this assumption by finding the primitive integral matrix $M$ first, calculating the corresponding coefficients $c^{jk\ell}$, and checking to see if they have the numerical properties compatible with Poincar\'e duality over the integers.} The fact that the mirror map can perhaps be completely determined by looking at the integral structure was pointed out in \cite{mirrorguide}, where it was shown that this conjecture holds for the case of the quintic-mirror (using calculations from \cite{CdGP}), and that the integral basis leads to the correct mirror map. In the two-parameter examples of \cite{2param1} the same principle was used to determine the mirror map, with equal success.\footnote{The rank one matrices in those papers---denoted by $Y$ in eq.~(7.4) of \cite{2param1} and also by $Y$ preceding eq.~(6.4) in \cite{2param2}---have the property of being primitive integral matrices, although this was not pointed out in those papers.} We will give additional evidence for our conjecture by verifying it (and checking that it produces the ``correct'' mirror map) in the three further one-parameter examples studied in \cite{picard-fuchs,font,kt1}. We must repeat the verification made in Appendix C of \cite{mirrorguide} that the monodromy of the actual period functions has a certain form. In fact, we will find a somewhat better normalization of that form this time. We will use the explicit monodromy calculations from \cite{kt1}; a similar calculation could be done using \cite{font} (which uses a different normalization of the parameter, making it difficult to compare to the present approach). Our verification is displayed in table \ref{tab:mat}. We show in the second column of the table the monodromy matrix $A$ as calculated in \cite{kt1}. That matrix was not quite uniquely specified by the data with which those authors were working. In particular, there is freedom to replace $A$ by $A'=m'A(m')^{-1}$ for any matrix $m'\in\text{Sp}(4,\bf Z)$ whose second and fourth rows are the same as that of the identity matrix. We make a choice of $m'$, shown in the third column of the table, and calculate $A'$ in the fourth column. Notice that the result takes the form \[A'=\begin{bmatrix} \phantom{-}1 & -1 & \phantom{-}0 & \phantom{-}1 \\ \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & -1 \\ -\lambda & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 \\ -\lambda & \phantom{-}\mu & \phantom{-}1 & \phantom{-}{1-\mu} \end{bmatrix}\] where $(\lambda,\mu)$ are as given in the fifth column of the table. With $A'$ in the given form, we can calculate the monodromy around infinity as \[T_\infty:=T^{-1}\,{A'}^{-1}=\begin{bmatrix} \phantom{-}1 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 \\ \phantom{-}\lambda & \phantom{-}\lambda & \phantom{-}1 & \phantom{-}0 \\ \phantom{-}0 & -\mu & -1 & \phantom{-}1 \end{bmatrix},\] where \[T:=\begin{bmatrix}1&0&0&0\\0&1&0&1\\0&0&1&0\\0&0&0&1\end{bmatrix}.\] We can thus easily see that $T_\infty$ is unipotent, with $(T_\infty-I)^4=0$. It is then a straightforward computation to see that \[(-\log T_\infty)^3 =\begin{bmatrix} \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}\lambda & \phantom{-}0 & \phantom{-}0 \end{bmatrix}.\] Note that $\lambda$ is precisely the triple-self-intersection of an integral generator of $H^2$ of the mirror partner, verifying the conjecture in these cases. \begin{table} \begin{center} $\begin{array}{|c|c|c|c|c|} \hline \vphantom{\Big(} k & A & m' & A'=m'A(m')^{-1} & (\lambda,\mu) \\ \hline &&&&\\ 5 & \begin{bmatrix} -9 & -3 & \phantom{-}5 & \phantom{-}3 \\ \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & -1 \\ -20 & -5 & \phantom{-}11 & \phantom{-}5 \\ -15 & \phantom{-}5 & \phantom{-}8 & -4 \end{bmatrix} & \begin{bmatrix} \phantom{-}2 & \phantom{-}0 & -1 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 \\ -5 & \phantom{-}0 & \phantom{-}3 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 \end{bmatrix} & \begin{bmatrix} \phantom{-}1 & -1 & \phantom{-}0 & \phantom{-}1 \\ \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & -1 \\ -5 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 \\ -5 & \phantom{-}5 & \phantom{-}1 & -4 \end{bmatrix} & (5,5) \\ &&&&\\ \hline &&&&\\ 6 & \begin{bmatrix} \phantom{-}1 & -1 & \phantom{-}0 & \phantom{-}1 \\ \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & -1 \\ -3 & -3 & \phantom{-}1 & \phantom{-}3 \\ -6 & \phantom{-}4 & \phantom{-}1 & -3 \end{bmatrix} & \begin{bmatrix} \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 \\ -3 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 \end{bmatrix} & \begin{bmatrix} \phantom{-}1 & -1 & \phantom{-}0 & \phantom{-}1 \\ \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & -1 \\ -3 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 \\ -3 & \phantom{-}4 & \phantom{-}1 & -3 \end{bmatrix} & (3,4) \\ &&&&\\ \hline &&&&\\ 8 & \begin{bmatrix} \phantom{-}1 & -1 & \phantom{-}0 & \phantom{-}1 \\ \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & -1 \\ -2 & -2 & \phantom{-}1 & \phantom{-}2 \\ -4 & \phantom{-}4 & \phantom{-}1 & -3 \end{bmatrix} & \begin{bmatrix} \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 \\ -2 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 \end{bmatrix} & \begin{bmatrix} \phantom{-}1 & -1 & \phantom{-}0 & \phantom{-}1 \\ \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & -1 \\ -2 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 \\ -2 & \phantom{-}4 & \phantom{-}1 & -3 \end{bmatrix} & (2,4) \\ &&&&\\ \hline &&&&\\ 10 & \begin{bmatrix} \phantom{-}1 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & -1 \\ \phantom{-}0 & \phantom{-}1 & \phantom{-}1 & -1 \\ \phantom{-}1 & \phantom{-}3 & \phantom{-}1 & -2 \end{bmatrix} & \begin{bmatrix} \phantom{-}0 & \phantom{-}0 & -1 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 \\ \phantom{-}1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}0 \\ \phantom{-}0 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 \end{bmatrix} & \begin{bmatrix} \phantom{-}1 & -1 & \phantom{-}0 & \phantom{-}1 \\ \phantom{-}0 & \phantom{-}1 & \phantom{-}0 & -1 \\ -1 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 \\ -1 & \phantom{-}3 & \phantom{-}1 & -2 \end{bmatrix} & (1,3) \\ &&&&\\ \hline \end{array}$ \end{center} \medskip \caption{Monodromy calculations} \label{tab:mat} \end{table} \subsection{The monomial-divisor mirror map} In the case of toric hypersurfaces, there is an alternate conjectural method for specifying the mirror map, proposed in \cite{mondiv},\footnote{Some signs were left unspecified in \cite{mondiv}. The proposal for determining the signs given in \cite{small} is now in doubt; an alternate proposal \cite{summing} has much evidence in its favor.} and used with great success in \cite{catp,small,HKTY1} (see also \cite{2param1,2param2}). Briefly, the parameters on both the A-model and B-model sides can be described by remarkably similar combinatorics; this similarity is used to write a conjecture for the derivative of the mirror map, which specifies the constants of integration. The conjecture was extended in \cite{summing} to also specify the ``algebraic gauge'' which should be used as a starting point for determining the natural gauge $\Omega(z)$; in addition, much evidence was amassed in \cite{summing} in favor of this approach. We refer the reader to \cite{mondiv} and \cite{summing} for details. \section{Making enumerative predictions} \label{s:predictions} We are finally ready to put together all of the ingredients and describe the process of making enumerative predictions. The things which we are going to predict are the ``numbers of rational curves'' on a Calabi--Yau threefold, in the precise form of the ``Gromov--Witten invariants'' of the threefold. A mathematical version of these invariants has been extensively investigated \cite{MS,RT1} using Gromov's symplectic geometry techniques \cite{Gromov} which had inspired Witten's original work on the invariants \cite{tsm,W:aspects}. (An alternate proposed definition purely within algebraic geometry is currently under development by Kontsevich and Manin \cite{KM,Kontsevich}.) The steps in an enumerative prediction are these: given a proposed mirror pair $(X,Y)$, find the moduli space of complex structures on $Y$, blow up to obtain a model in which the boundary is a divisor with normal crossings, find the boundary points with maximally unipotent monodromy, and sort them into equivalence classes (as indicated in section \ref{s:ambiguity}). For one representative of each equivalence class, find the canonical coordinates $z_j$ and the canonical gauge $\Omega(z)$ (these are unique if the integral monodromy conjecture holds, otherwise make a choice) , calculate the three-point functions \[\big\langle \frac{\partial}{\partial z_j} \frac{\partial}{\partial z_k} \frac{\partial}{\partial z_\ell} \big\rangle := \int_Y\Omega(z)\wedge\nabla_{z_j}\nabla_{z_k}\nabla_{z_\ell}\Omega(z)\] in those coordinates and that gauge, and make a power series expansion \[(2\pi i)^3 z_j z_k z_\ell \int_Y\Omega(z)\wedge\nabla_{z_j} \nabla_{z_k}\nabla_{z_\ell}\Omega(z) = c^{jk\ell} + \sum_{\eta\in H_2} c_\eta^{jk\ell} z^\eta, \] where the leading term $c^{jk\ell}$ should coincide with the intersection numbers on the mirror partner, and where we use a kind of multi-index notation for monomials $z^\eta$. The coefficients $c_\eta^{jk\ell}$ themselves are not quite the predictions for ``numbers of rational curves.'' One must take into account the ``multiple cover formula'' for the A-model \cite{tftrc,Manin}\footnote{Although this multiple cover formula---first postulated in \cite{CdGP}---can be derived using path integral arguments in conformal field theory \cite{tftrc} and has been mathematically proved for the algebraic Gromov--Witten invariants \cite{Manin}, it is still not known for the symplectic Gromov--Witten invariants.} and write the three-point functions in the form \[(2\pi i)^3 z_j z_k z_\ell \int_Y\Omega(z)\wedge\nabla_{z_j} \nabla_{z_k}\nabla_{z_\ell}\Omega(z) = c^{jk\ell} + \sum_{\eta\in H_2} \eta_j\eta_k\eta_\ell\,\varphi_\eta \, \frac{z^\eta}{1-z^\eta}. \] The coefficients $\varphi_\eta$ are then the predicted number of rational curves in the homology class $\eta$. \ifx\undefined\leavevmode\hbox to3em{\hrulefill}\, \newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,} \fi

9504

alg-geom/9504004

en

https://arxiv.org/abs/alg-geom/9504004

alg-geom/9504004

Rahul Pandharipande

R. Pandharipande

Intersections of Q-Divisors on Kontsevich's Moduli Space $\bar{M}_{0,n}(P^r,d)$ and Enumerative Geometry

AMSLaTex 31 pages

The theory of Q-Cartier divisors on the space of n-pointed, genus 0, stable maps to projective space is considered. Generators and Picard numbers are computed. A recursive algorithm computing all top intersection products of Q-Divisors is established. As a corollary, an algorithm computing all characteristic numbers of rational curves in P^r is proven (including simple tangency conditions). Computations of these characteristic numbers are carried out in many examples. The degree of the 1-cuspidal rational locus in the linear system of degree d plane curves is explicitly evaluated.

alg-geom

\section{{\bf Introduction}} Let ${\Bbb C}$ be the field of complex numbers. Let $(C, p_1, \ldots, p_n)$ be a connected, reduced, projective, nodal curve over ${\Bbb C}$ with $n$ nonsingular marked points $(p_1, \ldots, p_n)$. Let $\omega_C$ be the dualizing sheaf of $C$. An algebraic map $\mu: (C, p_1,\ldots, p_n) \rightarrow {\Bbb P}^r$ is {\em Kontsevich stable} if $\omega_C(p_1+ \ldots +p_n) \otimes \mu^*({\cal{O}}_{{\Bbb P}^r}(3))$ is ample on $C$. Let $\barr{M}_{g,n}(r,d)$ be the coarse moduli space of degree $d$, Kontsevich stable maps from $n$-pointed, genus $g$ curves to ${\Bbb P}^r$. In the genus zero case, $\barr{M}_{0,n}(r,d)$ is an irreducible, projective variety with finite quotient singularities. Only the following cases will be considered here: $$d\geq 0,\ \ g=0,\ \ r\geq 2.$$ The stack of Kontsevich stable maps was first defined in [K-M] and [K]. A treatment of the corresponding coarse moduli spaces can also be found in [P] and [Al]. The dimension of $\barr{M}=\barr{M}_{0,n}(r,d)$ is $m=rd+d+r+n-3$. Let $Pic(\barr{M})$ be the Picard group of line bundles. Let $A_{m-1}(\barr{M})$ be the Chow group of Weil divisors modulo rational equivalence. Since $\barr{M}$ has finite quotient singularities, every Weil divisor is $\Bbb{Q}$-Cartier. Therefore, there is a canonical isomorphism: $$Pic(\barr{M})\otimes {\Bbb{Q}} \rightarrow A_{m-1}(\barr{M}) \otimes {\Bbb{Q}}\ .$$ $Pic(\barr{M})\otimes {\Bbb{Q}}$ is a finite dimensional vector space. An explicit set of generators is given below. Let $P=\{1,2,\ldots n\}$ be the set of markings ($P$ may be the empty set). The $n$ markings of the moduli problem yield $n$ canonical line bundles ${\cal{L}}_i= \nu_i^*({\cal{O}}_{{\Bbb P}^r}(1))$ on $\barr{M}$ via the $n$ evaluation maps $\forall i \in P, \ \ \nu_i: \barr{M} \rightarrow {\Bbb P}^r$. The {\em boundary} of $\barr{M}$ is the locus corresponding to maps with reducible domain curves. Since the boundary is of pure codimension $1$ in $\barr{M}$, each irreducible component is a Weil divisor. The irreducible components of the boundary are in bijective correspondence with data of weighted partitions $(A\cup B, d_A, d_B)$ where: \begin{enumerate} \item[(i.)] $A\cup B$ is a partition of $P$. \item[(ii.)] $d_A+d_B=d$, $d_A>0$, $d_B>0$. \item[(iii.)] If $d_A=0$ (resp. $d_B=0$), then $|A|\geq 2$ (resp. $|B| \geq 2$). \end{enumerate} For example, if $P=\emptyset$, then $A=B=\emptyset$ and the boundary components correspond to positive partitions $d_A+d_B=d$. Let $\bigtriangleup$ be the set of components of the boundary. In case $d\geq 1$, a Weil divisor is obtained on $\barr{M}$ by considering the locus of $\barr{M}$ corresponding to maps meeting a fixed $r-2$ dimensional linear subspace of ${\Bbb P}^r$ (note $r\geq 2$). It is shown in [P] that this incidence Weil divisor is actually Cartier. Denote the corresponding line bundle on $\barr{M}$ by ${\cal{H}}$. For convenience, let ${\cal{H}}=0$ in case $d=0$. \begin{pr} Results on generation: \begin{enumerate} \item[(i.)] If $d=0$, $g=0$, $r\geq 2$, $\{{\cal{L}}_i\}\cup \bigtriangleup$ generate $Pic(\barr{M})$. \item[(ii.)]If $d\geq 1$, $g=0$, $r\geq 2$, $\{{\cal{L}}_i\} \cup \bigtriangleup \cup \{{\cal{H}}\}$ generate $Pic(\barr{M})\otimes {\Bbb{Q}}$. \end{enumerate} \label{gen} \end{pr} \noindent If $d=0$, then (by stability) $n\geq 3$ and $\barr{M}_{0,n}(r,0) \stackrel {\sim}{=} \overline{M}_{0,n}\times {\Bbb P}^r$ where $\overline{M}_{0,n}$ is the Mumford-Knudsen space. In this case, ${\cal{L}}_i$ is the pull-back of ${\cal{O}}_{{\Bbb P}^r}(1)$ from the second factor. Therefore, part (i) is a consequence of the boundary generation of $Pic(\overline{M}_{0,n})$. There is an intersection pairing $A_1(\barr{M}) \otimes Pic(\barr{M}) \rightarrow \Bbb{Z}$. Let $Null \subset Pic(\barr{M})$ be the null space with respect to the intersection pairing. Define $$Num(\barr{M}) = Pic(\barr{M})/Null.$$ By Proposition (\ref{gen}), the classes $\{{\cal{L}}_i\} \cup \bigtriangleup \cup \{{\cal{H}}\}$ generate $Num(\barr{M})\otimes {\Bbb{Q}}$. The relations between these generators in $Num(\barr{M})\otimes {\Bbb{Q}}$ can be algorithmically determined by calculating intersections with curves. It will be shown that all the relations in $Num(\barr{M})\otimes {\Bbb{Q}}$ are obtained from linear equivalences in $Pic(\barr{M})\otimes {\Bbb{Q}}$. \begin{pr} \label{dimm} The canonical map $Pic(\barr{M})\otimes {\Bbb{Q}} \rightarrow Num(\barr{M})\otimes {\Bbb{Q}}$ is an isomorphism. The Picard numbers are: \begin{enumerate} \item[{}]$(n=0)$, $\ dim_{{\Bbb{Q}}}\ Pic(\barr{M})\otimes {\Bbb{Q}} = [{d\over 2}]+1.$ \item[{}]$(n\geq 1)$, $\ dim_{{\Bbb{Q}}} \ Pic(\barr{M}) \otimes {\Bbb{Q}}= (d+1)\cdot 2^{n-1}-{n\choose 2}.$ \end{enumerate} \end{pr} The main result of this paper concerns the computations of top intersection products in $Pic(\barr{M})\otimes {\Bbb{Q}}$. \begin{pr} \label{top} Let $d\geq 0$, $g=0$, $r\geq 2$. There exists an explicit algorithm for calculating the top dimensional intersection products of the ${\Bbb{Q}}$-Cartier divisors $\{{\cal{L}}_i\} \cup \bigtriangleup \cup \{{\cal{H}}\}$ on $\barr{M}$. \end{pr} Consider the space $R(d,r)$ of degree $d\geq 1$ rational curves in ${\Bbb P}^r$ ($r\geq 2$). The dimension of $R(d,r)$ is $rd+r+d-3$. Classically, the characteristic numbers of $R(d,r)$ are the numbers of degree $d$ rational curves in ${\Bbb P}^r$ passing through $\alpha_i$ general linear spaces of codimension $i$ (for $2\leq i \leq r$) and tangent to $\beta$ general hyperplanes where $$(i-1)\cdot \alpha_i + \beta = dim \ R(d,r).$$ The characteristic numbers of rational curves excluding tangencies ($\beta=0$) have been determined recursively by M. Kontsevich and Y. Manin in [K-M] (also by Y. Ruan and G. Tian in [R-T]). The divisor in $\barr{M}$ corresponding to the hyperplane tangency condition can be expressed as a linear combination of the classes $\{{\cal{L}}_i\} \cup \bigtriangleup \cup \{{\cal{H}}\}$. The characteristic numbers can then be expressed as top intersection products of $\{{\cal{L}}_i\} \cup \bigtriangleup \cup \{{\cal{H}}\}$ on suitably chosen Kontsevich spaces of maps $\barr{M}$. Therefore, all the characteristic numbers can be calculated by Proposition (\ref{top}). \begin{pr} \label{charny} There exists an explicit algorithm for calculating all the characteristic numbers of rational curves in projective space. \end{pr} P. Di Francesco and C. Itzykson have modified the methods of [K-M] to determine some ($\beta\neq 0$) characteristic numbers for rational plane curves ([D-I]). Unfortunately, the relations they obtain from the WDVV associativity equations do not suffice to recursively determine all the characteristic numbers for rational plane curves from a finite set of data. The structure of the paper is as follows. Propositions (\ref{gen}) and (\ref{dimm}) are proven in section (\ref{cone}). In section (\ref{calc}), several geometric classes are explicitly computed in $Pic(\barr{M})\otimes {\Bbb{Q}}$. These classes will be used in the algorithms of Propositions (\ref{top}) and (\ref{charny}). The algorithms are established in section (\ref{inter}). Section (\ref{exam}) is devoted to calculations of some characteristic numbers of rational curves for small values of $(d,r)$. As a final application, a new formula for cuspidal rational curves is derived in section (\ref{cusp}). The problem of calculating tangency characteristic numbers via Kontsevich's moduli space was suggested to the author by W. Fulton. Conversations with W. Fulton on related topics have been of significant aid. Thanks are due to S. Kleiman and P. Aluffi for mathematical and historical remarks. The remarkable ideas in [K-M] have been a source of inspiration. \section{$Pic(\barr{M})\otimes {\Bbb{Q}}$ And $Num(\barr{M})\otimes {\Bbb{Q}}$} \label{cone} \setcounter{subsection}{-1} \subsection{Summary} Propositions (\ref{gen}) and (\ref{dimm}) are established in sections (\ref{jenny}) and (\ref{realy}) respectively. Since these results are well known for $d=0$, $$\barr{M}_{0,n}(r,0) \stackrel {\sim}{=} \overline{M}_{0,n}\times {\Bbb P}^r,$$ the conditions $d\geq 1$, $g=0$, $r\geq2$ are assumed throughout sections (\ref{jenny}) and (\ref{realy}). \subsection{Generators} \label{jenny} The proof of Proposition (\ref{gen}) is divided into four cases depending upon the number $n$ of marked point. \begin{lm} If $n\geq 3$, then $Pic(\barr{M})\otimes {\Bbb{Q}}$ is generated by $\bigtriangleup \cup \{{\cal{H}}\}.$ \label{n3} \end{lm} \begin{pf} Let $V=\bigoplus_{0}^{r} H^0({\Bbb P}^1, {\cal{O}}_{{\Bbb P}^1}(d))$. Let $U\subset {\Bbb P}(V)$ be the Zariski open set corresponding to a well defined (basepoint free) degree $d$ map from ${\Bbb P}^1$ to ${\Bbb P}^r$. The complement of $U$ in ${\Bbb P}(V)$ is of codimension at least $r\geq 2$. There is a universal map $${\Bbb P}^1 \times U \rightarrow {\Bbb P}^r.$$ Fix the first three marked points to be $0$, $1$, $\infty\in {\Bbb P}^1$. Let $$W= {\Bbb P}^1 \times \ldots \times {\Bbb P}^1 \ \setminus \{D_{i,j}, \ S_{0,i}, \ S_{1,i}, \ S_{\infty, i} \}$$ where the product is taken over $n-3$ factors. $D_{i,j}$ is the large diagonal determined by factors $i$ and $j$. $S_{0,i}$ is the locus where the $i^{th}$ factor is $0\in {\Bbb P}^1$. $S_{1,i}$, $S_{\infty, i}$ are defined similarly. It follows there is a universal family of Kontsevich stable degree $d$ maps of $n$-pointed curves: $${\Bbb P}^1 \times W \times U \rightarrow {\Bbb P}^r.$$ The maps of the family are automorphism-free and distinct. By the universal property, there is an injection $W\times U \rightarrow \barr{M}$. A tangent space calculation shows $W\times U$ is an open set of $\barr{M}$. The complement of $W\times U$ is the boundary of $\barr{M}$. Hence $A_{m-1}(\barr{M})$ is generated by $\bigtriangleup$ and $A_{m-1}(W\times U)$. Information about $A_{m-1}(W\times U)$ is obtained from the open inclusion \begin{equation} \label{inclu} W\times U \subset {\Bbb P}^1 \times \ldots \times {\Bbb P}^1 \times {\Bbb P}(V). \end{equation} The Picard group of the right side of (\ref{inclu}) is generated by the pull-backs of ${\cal{O}}(1)$ from each factor. The pull-backs from the ${\Bbb P}^1$ factors are trivial on $W\times U$ because of the removal of the loci $S_{0,i}$. Hence, $A_{m-1}(W\times U)$ is generated by ${\cal{O}}_{{\Bbb P}(V)}(1)$. It is easily seen ${\cal{H}}$ restricted to $W\times U$ is the pull-back of a resultant hypersurface in ${\Bbb P}(V)$. Therefore, ${\cal{H}}$ restricted to $W\times U$ is linearly equivalent to a multiple of ${\cal{O}}_{{\Bbb P}(V)}(1)$. \end{pf} There are canonical morphisms $\barr{M}_{0,n}(r,d) \rightarrow \barr{M}_{0,n-1}(r,d)$ obtained by omitting the last marked point. Results for $0\leq n \leq 2$ are obtained via these morphisms. \begin{lm} If $n=2$, then $Pic(\barr{M})\otimes {\Bbb{Q}}$ is generated by $\bigtriangleup \cup \{{\cal{L}}_1, {\cal{L}}_2\}$. \end{lm} \begin{pf} Let $\barr{N}=\barr{M}_{0,3}(r,d)$ and $\barr{M}=\barr{M}_{0,2}(r,d)$. Fix a hyperplane $H_3\subset {\Bbb P}^r$. Let $X= \nu_3^{-1}(H_3)$ where $\nu_3$ is the third evaluation map, $\nu_3: \barr{N} \rightarrow {\Bbb P}^r.$ There is a map $\rho: X \rightarrow \barr{M}$ obtained by omitting the third point. The map $\rho$ is surjective and generically finite. Let $Z\subset \barr{M}$ be the open set corresponding to Kontsevich stable maps satisfying the following conditions: \begin{enumerate} \item[(i.)] The domain curve is ${\Bbb P}^1$. \item[(ii.)] The images of the marked points $\{1,2\}$ do not lie in $H_3$. \end{enumerate} It is clear the the complement of $Z$ is the boundary union $\nu_1^{-1}(H_3)$, $\nu_2^{-1}(H_3)$. By the definition of $Z$, $\rho^{-1}(Z) \rightarrow Z$ is a finite, projective morphism. If $A_{m-1}(\rho^{-1}(Z))=0$, then $A_{m-1}(Z)$ is torsion. To establish the Lemma, it therefore suffices to prove $A_{m-1}(\rho^{-1}(Z))=0$. In the notation of the proof of Lemma (\ref{n3}), $\rho^{-1}(Z) \subset U\subset \barr{M}_{0,3}(r,d)$. In fact, the following is easily seen: $$\rho^{-1}(Z) = \ U \cap L_{\infty}(H_3) \ \setminus \ \{L_0(H_3), L_1(H_3)\}.$$ $L_p(H_3)$ is the hyperplane in $U$ corresponding to maps sending the point $p\in {\Bbb P}^1$ to $H_3$. $U\cap L_{\infty}(H_3)$ is a an open set of $L_{\infty}(H_3)$ with complement of codimension at least $2$. Hence, $A_{m-1}(U\cap L_{\infty}(H_3))=\Bbb{Z}$ generated by the hyperplane class. Since $\rho^{-1}(Z)\subset U\cap L_{\infty}(H_3)$ is the complement of hyperplanes, the desired conclusion $A_{m-1}(\rho^{-1}(Z))=0$ is obtained. \end{pf} \begin{lm} If $n=1$, then $Pic(\barr{M})\otimes {\Bbb{Q}}$ is generated by $\bigtriangleup \cup \{{\cal{L}}_1, {\cal{H}}\}$. \end{lm} \begin{pf} Let $\barr{N}=\barr{M}_{0,3}(r,d)$ and $\barr{M}=\barr{M}_{0,1}(r,d)$. Fix two hyperplanes $H_2, H_3\subset {\Bbb P}^r$. Let $X= \nu_2^{-1}(H_2)\cap \nu_3^{-1}(H_3)$ where $\nu_2$, $\nu_3$ are the second and third evaluation maps on $\barr{N}$. There is a map $\rho: X \rightarrow \barr{M}$ obtained by omitting the second and third points. The map $\rho$ is surjective and generically finite. Let $Z\subset \barr{M}$ be the open set corresponding to Kontsevich stable maps satisfying the following conditions: \begin{enumerate} \item[(i.)] The domain curve is ${\Bbb P}^1$. \item[(ii.)] The image of the marked point $\{1\}$ does not lie in $H_2 \cup H_3$. \item[(iii.)] The map does not pass through the intersection $H_2 \cap H_3$. \end{enumerate} The complement of $Z$ is the boundary union $\nu_1^{-1}(H_2)$, $\nu_1^{-1}(H_3)$, and $D_{2,3}$. $D_{2,3}$ is the Cartier divisor of maps passing through $H_2 \cap H_3$. $D_{2,3}$ is a divisor in the linear series of ${\cal{H}}$. By the definition of $Z$, $\rho^{-1}(Z) \rightarrow Z$ is a finite, projective morphism. As before, it suffices to prove $A_{m-1}(\rho^{-1}(Z))=0$. Let $S\subset U$ be the union of the hyperplane sections $\{L_0(H_2), L_0(H_3)\}$ with the resultant hypersurface of maps meeting $H_2\cap H_3$. Conditions (i), (ii), and (iii) imply: $$\rho^{-1}(Z) = \ U \cap L_{1}(H_2) \cap L_{\infty}(H_3) \ \setminus \ S.$$ As before, $A_{m-1}(U\cap L_1(H_2) \cap L_{\infty}(H_3))=\Bbb{Z}$ generated by the hyperplane class. $S$ is a union of hyperplane classes and multiples of hyperplane classes. Hence, $A_{m-1}(\rho^{-1}(Z))=0$. \end{pf} \begin{lm} If $n=0$, then $Pic(\barr{M})\otimes {\Bbb{Q}}$ is generated by $\bigtriangleup \cup \{{\cal{H}}\}$. \label{n0} \end{lm} \begin{pf} Let $\barr{N}=\barr{M}_{0,3}(r,d)$ and $\barr{M}=\barr{M}_{0,0}(r,d)$. Fix three general hyperplanes $H_1, H_2, H_3\subset {\Bbb P}^r$. Let $X= \nu_1^{-1}(H_1) \cap \nu_2^{-1}(H_2)\cap \nu_3^{-1}(H_3)$ where the $\nu_i$ are evaluation maps on $\barr{N}$. There is a map $\rho: X \rightarrow \barr{M}$ obtained by omitting the marked points. The map $\rho$ is surjective and generically finite. Let $Z\subset \barr{M}$ be the open set corresponding to Kontsevich stable maps satisfying the following conditions: \begin{enumerate} \item[(i.)] The domain curve is ${\Bbb P}^1$. \item[(ii.)] The map does not pass through the intersections $H_1\cap H_2$, $H_1\cap H_3$, or $H_2 \cap H_3$. \end{enumerate} The complement of $Z$ is the boundary union $D_{1,2}$, $D_{1,3}$, and $D_{2,3}$. By the definition of $Z$, $\rho^{-1}(Z) \rightarrow Z$ is a finite, projective morphism. As before, it suffices to prove $A_{m-1}(\rho^{-1}(Z))=0$. Let $S\subset U$ be the union of the three resultant hypersurfaces of maps meeting $H_1\cap H_2$, $H_1\cap H_3$, and $H_2\cap H_3$. Let $I\subset U$ be the hyperplane intersection defined by $I=U\cap L_0(H_1)\cap L_1(H_2) \cap L_{\infty}(H_3)$. Conditions (i) and (ii) imply: $$\rho^{-1}(Z) = \ I \ \setminus \ S \cap I.$$ Note $S\cap I$ contains the intersections of the following hyperlanes with $I$: $$\{L_0(H_2), L_0(H_3), L_1(H_1), L_1(H_3), L_{\infty}(H_1) , L_{\infty}(H_2)\}.$$ As before, $A_{m-1}(U\cap I)=\Bbb{Z}$ generated by the hyperplane class. Since $S\cap I$ is a union of of hyperplane classes and multiples of hyperplane classes, $A_{m-1}(\rho^{-1}(Z))=0$. \end{pf} Lemmas (\ref{n3}) - (\ref{n0}) yield Proposition (\ref{gen}). \subsection{Relations} \label{realy} Curves in $\barr{M}=\barr{M}_{0,n}(r,d)$ are easily found. The following construction will be required for the calculations below. Let $C$ be a nonsingular, projective curve. Let $\pi: S={\Bbb P}^1 \times C \rightarrow C$. Select $n$ sections $s_1, \ldots, s_n$ of $\pi$. A point $x\in S$ is an {\em intersection point} if two or more sections contain $x$. Let ${\cal{N}}$ be a line bundle on $S$ of type $(d,k)$ where $k$ is very large. Let $z_l\in H^0(S, {\cal{N}})$ $(0\leq l \leq r)$ determine a rational map $\mu: S - \rightarrow {\Bbb P}^r$ with simple base points. A point $y\in S$ is a {\em simple base point} of degree $1\leq e\leq d$ if the blow-up of $S$ at $y$ resolves $\mu$ locally at $y$ and the resulting map is of degree $e$ on the exceptional divisor $E_y$. The set of {\em special points} of $S$ is the union of the intersection points and the simple base points. Three conditions are required: \begin{enumerate} \item [(1.)] There is at most one special point in each fiber of $\pi$. \item [(2.)] The sections through each intersection point $x$ have distinct tangent directions at $x$. \item [(3.)] If $n$ or $n-1$ sections pass through the point $x\in S$, then $x$ is not a simple base point of degree $d$. (If $n=0$ or $1$, there are no simple base points of degree $d$.) \end{enumerate} Let $\overline{S}$ be the blow-up of $S$ at the special points. It is easily seen $\overline{\mu} : \overline{S} \rightarrow {\Bbb P}^r$ is Kontsevich stable family of $n$-pointed, genus $0$ curves over $C$. Condition (2) ensures the strict transforms of the sections are disjoint. Condition (3) implies Kontsevich stability. There is a canonical morphism $C \rightarrow \barr{M}$. Condition (1) implies $C$ intersects the boundary components transversally. \begin{lm} \label{ihh} Results on the span of $\{{\cal{H}}, {\cal{L}}_1, {\cal{L}}_2\}$: \begin{enumerate} \item[(i.)] The element ${\cal{H}}$ is not contained in the linear span of $\bigtriangleup$ in $Pic(\barr{M})\otimes {\Bbb{Q}}$. \item[(ii.)] If $n=1$, $\{{\cal{H}}, {\cal{L}}_1\}$ are independent modulo $\bigtriangleup$. \item[(iii.)] If $n=2$, $\{{\cal{L}}_1, {\cal{L}}_2\}$ are independent modulo $\bigtriangleup$. \end{enumerate} \end{lm} \begin{pf} Consider $\pi: S={\Bbb P}^1 \times C \rightarrow C$ with $n$ trivial sections. There are no intersection points. Let ${\cal{N}}$, $z_l\in H^0(S, {\cal{N}})$ be such that $\mu$ has no base points (note: since $r\geq 2$, this is easily accomplished). ${\cal{N}}$ has degree type $(d,k)$. For each component $K\in \bigtriangleup$, $C\cdot K=0$. A simple calculation yields $C\cdot {\cal{H}}= {\cal{N}}\cdot {\cal{N}}=2dk$. Hence ${\cal{H}}$ is not contained in the span of $\bigtriangleup$. Consider $\pi: S={\Bbb P}^1 \times {\Bbb P}^1 \rightarrow {\Bbb P}^1$. Let $s$ be the trivial section; let $s'$ be the diagonal section. Let $\mu: S \rightarrow {\Bbb P}^r$ be a base point free map of type $(d,k)$. The two sections $s$, $s'$ determine two maps $\tau, \tau': {\Bbb P}^1 \rightarrow \barr{M}_{0,1}(r,d)$. Intersection via $\tau$ yields: $${\Bbb P}^1 \cdot {\cal{H}}= 2dk, \ \ {\Bbb P}^1 \cdot {\cal{L}}_1= k.$$ Intersection via $\tau'$ yields: $${\Bbb P}^1 \cdot {\cal{H}}=2dk, \ \ {\Bbb P}^1 \cdot {\cal{L}}_1= d+k.$$ In both cases ${\Bbb P}^1\cdot K=0$ for any $K\in \bigtriangleup$. Therefore $\{{\cal{H}}, {\cal{L}}_1\}$ are independent modulo $\bigtriangleup$ in $Pic(\barr{M}) \otimes {\Bbb{Q}}$ for $n=1$. In the $n=2$ case, twisted families must be considered. Let $E(a,b)$ be the rank two bundle ${\cal{O}}_{{\Bbb P}^1}(a)\oplus {\cal{O}}_{{\Bbb P}^1}(b)$ over ${\Bbb P}^1$. Let $S(a,b)= {\Bbb P}(E(a,b))$. Let $${\cal{N}}= {\cal{O}}_{{\Bbb P}(E)}(d)\otimes \pi^*({\cal{O}}_{{\Bbb P}^1}(k)).$$ For large $k$, let $\mu: S(a,b)\rightarrow {\Bbb P}^r$ be a base point free map. The sub-bundles ${\cal{O}}(a)$, ${\cal{O}}(b)$ define sections $s_1$ and $s_2$. There is an induced map ${\Bbb P}^1 \rightarrow \barr{M}_{0,2}(r,d)$. A calculation yields: $$ {\Bbb P}^1\cdot {\cal{L}}_1= -ad+k, \ \ {\Bbb P}^1 \cdot {\cal{L}}_2=-bd+k.$$ As before ${\Bbb P}^1\cdot K=0$ for any $K\in \bigtriangleup$. It follows $\{{\cal{L}}_1, {\cal{L}}_2\}$ are independent modulo $\bigtriangleup$ in $Pic(\barr{M}) \otimes {\Bbb{Q}}$ for $n=2$. \end{pf} If $n\geq 1$, let $\bigtriangleup_i\subset \bigtriangleup$ be the subset of boundary components $(A\cup B, d_A, d_B)$ with marking partition $|A|+|B|=n$ equal to the partition $i+(n-i)=n$. There is a disjoint union $$\bigtriangleup = \bigcup_{i=0}^{[{n\over 2}]} \bigtriangleup_i.$$ Let $\bigtriangleup'=\bigtriangleup \setminus( \bigtriangleup_0 \cup \bigtriangleup_1)$. \begin{lm} \label{i01} Results on the span of $\bigtriangleup_0$, $\bigtriangleup_1$: \begin{enumerate} \item[(i.)] If $n=0$, $\bigtriangleup_0=\bigtriangleup$ is a set of linearly independent elements of $Pic(\barr{M})\otimes {\Bbb{Q}}$. \item[(ii.)] If $n=1$, $\bigtriangleup_0=\bigtriangleup_1=\bigtriangleup$ is a set of linearly independent elements of $Pic(\barr{M})\otimes {\Bbb{Q}}$. \item[(iii.)] If $n\geq 2$, $\bigtriangleup_0 \cup \bigtriangleup_1$ is a set of linearly independent elements of $Pic(\barr{M})\otimes {\Bbb{Q}}$. Moreover, the span of $\bigtriangleup_0 \cup \bigtriangleup_1$ does not intersect the span of $\bigtriangleup'$ in $Pic(\barr{M})\otimes {\Bbb{Q}}$. \end{enumerate} \end{lm} \begin{pf} Let $\pi: S={\Bbb P}^1\times C \rightarrow C$ be as above with $n$ trivial sections. Let ${\cal{N}}$ be a line bundle on $S$ of degree type $(d,k)$. For large degrees $k$, the simple base points of $\mu$ of degree $1\leq e \leq d$ can be selected arbitrarily satisfying conditions (1) and (3). For suitable choices of simple base points and base point degrees on $S$, the classes in assertions (i-iii) can be seen to be independent in $Num(\barr{M}) \otimes {\Bbb{Q}}$. Therefore, the classes are independent in $Pic(\barr{M})\otimes {\Bbb{Q}}$. \end{pf} In case $n=0$, $\bigtriangleup \cup \{{\cal{H}}\}$ is a basis of $Pic(\barr{M})\otimes Q$ via Lemmas (\ref{n0}), (\ref{ihh}), and (\ref{i01}). For $1\leq n \leq 3$, $\bigtriangleup_0 \cup \bigtriangleup_1= \bigtriangleup$. Hence, the Lemmas show the generators of section (\ref{jenny}) are also bases for $1\leq n \leq 3$. The Picard numbers of Proposition (\ref{dimm}) can be verified for $0\leq n \leq 3$. For $n\geq 4$, let $\overline{M}_{0,n}$ be the Mumford-Knudsen moduli space of $n$-pointed, genus $0$ curves. The boundary components of $\overline{M}_{0,n}$ correspond bijectively to partitions $A\cup B$ of $P=\{1,2,\ldots, n\}$ such that $|A|, |B|\geq 2$. The boundary components generate $Pic(\overline{M}_{0,n})$. The three boundary components of $\overline{M}_{0,4}$ are linearly equivalent. A four element subset $Q\subset P$ induces a natural map $\overline{M}_{0,n} \rightarrow \overline{M}_{0,Q}$. The pull-backs of the basic boundary linear equivalences on $\overline{M}_{0,Q}$ induces boundary linear equivalences on $\overline{M}_{0,n}$. The relations among the boundary components of $\overline{M}_{0,n}$ are generated by these pull-back linear equivalences as $Q$ varies among all four element subsets of $P$. $Pic(\overline{M}_{0,n})$ is a free group of rank $$2^{n-1}-{n-1 \choose 2}-n.$$ Since there are $${2^{n}-2-2n \over 2}= 2^{n-1}-1-n$$ boundary components of $\overline{M}_{0,n}$, it follows there are ${n-1\choose 2}-1$ independent relations among the boundary components. Finally, $Pic(\overline{M}_{0,n}) \stackrel {\sim}{=} Num(\overline{M}_{0,n})$. See [Ke] for proofs of these results. Let $n\geq 4$. There is canonical morphism $\eta:\barr{M}=\barr{M}_{0,n}(r,d)\rightarrow \overline{M}_{0,n}$. The $\eta$ pull-back of a boundary component of $\overline{M}_{0,n}$ is a non-empty, multiplicity-free sum of boundary components $\bigtriangleup'$ of $M$: $$\eta^{-1}\big( (A\cup B) \big)= \sum_{d_A+d_B=d} (A\cup B, d_A, d_B).$$ \begin{lm} \label{pb} The relations among the boundary components $\bigtriangleup'$ in $Pic(\barr{M})\otimes {\Bbb{Q}}$ are the $\eta$ pull-backs of the relations among the boundary components of $\overline{M}_{0,n}$. \end{lm} \begin{pf} Let $\pi:S={\Bbb P}^1\times C \rightarrow C$ be a family with $n$ sections. Let $\mu: S \ - \rightarrow {\Bbb P}^r$ be a rational maps with simple base points obtained from a line bundle of degree type $(d,k)$. Suppose the special points satisfy (1), (2), and \begin{enumerate} \item[($3'$.)] An intersection point lies on at most $n-2$ sections. \item[(4.)] Every simple base point is an intersection point. \end{enumerate} Note condition ($3'$) implies condition (3). For large $k$, the simple base points may be selected arbitrarily (with arbitrary degree) among the intersection points. Let $\overline{S}$ be the blow-up of $S$ at the special points; let $\lambda: C\rightarrow \barr{M}$ be the induced curve. By condition ($3'$), the family $\overline{S} \rightarrow C$ with the strict transforms of the sections is flat family of stable, $n$-pointed, genus $0$ curves. The induced morphism $\gamma: C \rightarrow \overline{M}_{0,n}$ is simply $\gamma= \eta\circ \lambda$. Suppose $\sum_{K\in \bigtriangleup'} c_K K =0$ is a relation in $Pic(\barr{M})\otimes {\Bbb{Q}}$ ($c_B \in {\Bbb{Q}}$). Let $$K=(A\cup B, d_A,d_B)\in \bigtriangleup'.$$ Let $(A\cup B)$ be corresponding boundary component of $\overline{M}_{0,n}$. The set theoretic intersection $C\cdot K$ is the subset of $C \cdot (A\cup B)$ with simple base points of the correct degree. Since the simple base points can be assigned arbitrary degrees, the coefficient $c_K$ must depend only on the partition $(A\cup B)$ and not on the weights $d_A, d_B$. It now follows the relation $\sum_{K\in \bigtriangleup'} c_K \cdot K =0$ must be the $\eta$ pull-back of a boundary relation in $\overline{M}_{0,n}$. \end{pf} In particular, it follows there are ${n-1 \choose 2}-1$ independent relations among the boundary components $\bigtriangleup'$. For $n\geq 4$, $$|\bigtriangleup|= d+ dn+ |\bigtriangleup'|,$$ $$|\bigtriangleup'|= (d+1) \cdot (2^{n-1}-1-n).$$ By Lemmas (\ref{n3}), (\ref{ihh}), (\ref{i01}), (\ref{pb}), the Picard number of $\barr{M}$ ($n\geq4$) is: $$dim\ Pic(\barr{M})\otimes {\Bbb{Q}} = (d+1)\cdot 2^{n-1}-{n\choose 2}.$$ All the numerical relations are obtained from linear equivalences. The proof of Proposition (\ref{dimm}) is complete. \section{Computations in $Pic(\barr{M}) \otimes {\Bbb{Q}}$} \label{calc} \subsection{The Universal Curve and $\pi_*(c_1(\omega_{\pi})^2)$} \label{wclass} Classes of certain canonical elements in $Pic(\barr{M})\otimes {\Bbb{Q}}$ will be computed via intersections with curves. These computations will be used in the proof of Proposition (\ref{top}). In order to use the coarse moduli space throughout, an automorphism result is required. \begin{lm} \label{auto} Let $d\geq0$, $g=0$, $r\geq 2$. The locus of Kontsevich stable maps in $\barr{M}_{0,n}(r,d)$ with nontrivial automorphisms is of codimension at least $2$ except in one case: $\barr{M}_{0,0}(2,2)$. \end{lm} \begin{pf} The assertion follows from naive dimension estimates. If $d=0$ or $1$ , the are no stable maps with nontrivial automorphisms. Let $\barr{M}=\barr{M}_{0,n}(r,d)$, $(d\geq 2, r\geq 2)$. Recall $dim \barr{M} = rd+d+r+n-3$. Certainly the generic elements of the boundary components are automorphism-free. Let $A\subset \barr{M}$ be the locus of non-boundary, stable maps with nontrivial automorphisms. If a map $\mu: {\Bbb P}^1 \rightarrow {\Bbb P}^r$ with $n$ distinct marked points has an nontrivial automorphism, $\mu$ must be a $k\geq 2$ to $1$ map. For fixed $2\leq k\leq d$, the map $\mu$ moves in a family of dimension at most: $$(r+1)\cdot ({d\over k}+1)-1-3 + 2\cdot (k+1)-1-3= (rd+d)\cdot {1\over k}+r-3+2k-2.$$ The $n$ marked points must be fixed points of the nontrivial automorphism and hence move in a zero dimensional family for each $\mu$. A calculation yields: \begin{eqnarray*} dim \barr{M}- dim A & \geq & (rd+d)\cdot (1- {1\over k}) +n -2k+2 \\ & = & rd+d+n+2 - {rd+d+2k^2\over k}. \end{eqnarray*} A study of the function $(rd+d+2k^2)/k$ for $2\leq k \leq d$ shows the maximum value must be attained at the end points $k=2, d$. If $k=2$, $$ rd+d -{rd+d+8\over 2} = (r+1){d\over 2}-4 \geq 0$$ except when $r=2$, $d=2$. If $k=d$, $$ rd+d-{rd+d+2d^2\over d}= (r-1)(d-1)-2 \geq 0 $$ except when $r=2$, $d=2$. $A$ is of codimension at least 2 all cases except $\barr{M}_{0,0}(2,2)$. \end{pf} Since $\barr{M}_{0,0}(2,2)$ is isomorphic to the space of complete conics, its intersection theory is well known. In the sequel, it will be assumed $(g,n,r,d)\neq (0,0,2,2)$. Let $\barr{M}^*\subset \barr{M}$ denote the automorphism-free locus. There is a universal Kontsevich stable family of maps over $\barr{M}^*$: $$\pi: U^* \rightarrow \barr{M}^*$$ with sections $s_1, s_2 \ldots, s_n$ and a morphism $$\mu: U^* \rightarrow {\Bbb P}^r.$$ See [P] for details. Let $\omega_{\pi}$ be the relative dualizing sheaf of $\pi$. Since the complement of $\barr{M}^*$ is of codimension at least $2$ in $\barr{M}$, the following are well-defined elements of $Pic(\barr{M})\otimes {\Bbb{Q}}$ : \begin{equation} \label{fcla} \pi_*( c_1(\omega_{\pi})^2), \ \ \pi_*( s_i^2). \end{equation} Since $Pic(\barr{M}) \otimes {\Bbb{Q}} \stackrel {\sim}{=} Num(\barr{M}) \otimes {\Bbb{Q}}$, explicit expressions of the classes (\ref{fcla}) in terms of the generators $\{{\cal{L}}_i\} \cup \bigtriangleup \cup \{{\cal{H}}\}$ can be found by calculating intersection products with curves in $\barr{M}$. The methods of section (\ref{realy}) will be used to determine curves in $\barr{M}$. First consider $\pi_*(c_1(\omega_{\pi})^2)$: \begin{lm} \label{om} For $d\geq 0$, $g=0$, $r\geq 2$, $\ \pi_*( c_1(\omega_{\pi})^2) = - \sum_{K\in \bigtriangleup} K$ in $Pic(\barr{M})\otimes {\Bbb{Q}}$. \end{lm} \begin{pf} Let $\pi:S\rightarrow C$ be a projective bundle of rank $1$ over a nonsingular curve $C$. Let $\omega_{\pi}$ be the relative dualizing sheaf. A simple computation yields $$\pi_*(c_1(\omega_{\pi})^2)= 0$$ in $Num(C)$. Let $\rho: \overline{S}\rightarrow S$ be the blow-up at $k$ points in distinct fibers of $\pi$. Let $\overline{\pi}:\overline{S} \rightarrow C$ be the composition. $$\omega_{\overline{\pi}}= \rho^*(\omega_{\pi}) + \sum_{i=1}^{k} E_i$$ where the $E_i$ are the exceptional divisors of $\rho$. Hence, $$\pi_*(c_1(\omega_{\overline{\pi}})^2)= -k$$ in $Num(C)$. By considering curves $C\rightarrow \barr{M}^*$ and the pull-back of $U^*$, it follows $\pi_*( c_1(\omega_{\pi})^2) = -\sum_{K\in \bigtriangleup} K$ in $Num(\barr{M}) \otimes {\Bbb{Q}}$. By Proposition (\ref{dimm}), the Lemma is proven. \end{pf} \subsection{The Class $\pi_*(s_1^2)$} \label{sclass} The determination of the class $\pi_*(s_1^2)$ is surprisingly different in the cases $d=0$ and $d\geq 1$. If $d=0$, it suffices to determine $\pi_*(s_1^2)$ for the universal family over $\overline{M}_{0,n}$. Let $\bigtriangleup$ be the set of boundary components of $\overline{M}_{0,n}$. There is a partition of $\bigtriangleup$ with respect to the first marking. For $2\leq j \leq n-2$, let $\bigtriangleup^{1}_j \subset \bigtriangleup$ be defined by: $$(A\cup B) \in \bigtriangleup^{1}_j \ \ \ if \ and\ only \ if \ \ \ 1\in A, \ |A|=j.$$ There is a disjoint union $$\bigtriangleup= \bigcup_{j=2}^{n-2} \bigtriangleup^{1}_j.$$ Let $K^{1}_{j}=\sum_{K\in \bigtriangleup^{1}_{j}} K$. \begin{lm} \label{self0} The class $\pi_*(s_1^2)$ is expressed in $Pic(\overline{M}_{0,n})\otimes {\Bbb{Q}}$ by: \begin{equation} \label{exx0} \pi_*(s_1^2)= -{1\over {n-1\choose 2}} \cdot \sum_{j=2}^{n-2} {n-j\choose 2} K^{1}_{j}. \end{equation} \end{lm} \begin{pf} The proof is by intersections with curves in $\overline{M}_{0,n}$. Let $S={\Bbb P}^1\times C$ be a family with $n$ sections $s_1, \ldots, s_n$. Let $s_1$ be of degree type $(1,q)$. For $2\leq i \leq n$, let $s_i$ be of type $(1,p_i)$. As usual, assume the blow-up $\overline{S}$ up of $S$ at the intersection points yields a family of stable, $n$-pointed curves over $C$ with at most one exceptional divisor in each fiber. Let $\lambda: C \rightarrow \overline{M}_{0,n}$ be the induced map. It will be checked that the left and right sides of (\ref{exx0}) have the same intersection with $C$. A point of $C\cdot K^{1}_{j}$ can arise in exactly two cases. First an intersection point of $j$ sections including $s_1$ can be blown-up. Second, an intersection point of $n-j$ sections not including $s_1$ can be blown-up. Let $$C\cdot K^{1}_{j}= x_j + y_j$$ where $x_j$, $y_j$ are the number of instances of the first and second cases respectively. Let $\overline{s}_1$ be the strict transform of $s_1$ in $\overline{S}$. The intersection of $C$ with the left side of (\ref{exx0}) is : $$\overline{\pi}_*(\overline{s}_1^2)= 2q- \sum_{j=2}^{n-2} x_j.$$ For $2\leq i \leq n$, $s_i$ intersects $s_1$ in $q+p_i$ points. The following equation is easily obtained by analyzing intersection points contained in $s_1$: \begin{equation} \label{ff} (n-1) q + \sum_{i=2}^{n} p_i = \sum_{j=2}^{n-2} (j-1) x_j. \end{equation} Similarly, the number of intersections of the sections $2\leq i \leq n$ among themselves is $(n-2)\cdot \sum_{i=2}^{n} p_i$. Analysis of intersection points not contained in $s_1$ yields: \begin{equation} \label{ss} (n-2) \cdot \sum_{i=2}^{n} p_i = \sum_{j=2}^{n-2} {j-1 \choose 2} x_j + {n-j\choose 2} y_j. \end{equation} Via equations (\ref{ff}) and (\ref{ss}), \begin{eqnarray*} {n-1\choose 2} \cdot (2q-\sum_{j=2}^{n-2} x_j)& = & \sum_{j=2}^{n-2} \big((n-2)(j-1)-{j-1\choose 2}-{n-1\choose 2}\big) x_j - {n-j\choose 2} y_j \\ & = & - \sum_{j=2}^{n-2} {n-j\choose 2} (x_j+y_j). \end{eqnarray*} The Lemma is proved. \end{pf} Consider now the case $d\geq 1$. Let $\barr{M}=\barr{M}_{0,n}(r,d)$ where $d\geq1$, $n\geq 1$. Let $1$ be the first marking. There is another partition of $\bigtriangleup$ with respect to the first marking depending upon the degree. For $0\leq j \leq d$, let $\bigtriangleup^{1,j}\subset \bigtriangleup$ be defined by: $$(A\cup B, d_A, d_B) \in \bigtriangleup^{1,j} \ \ \ if\ and\ only\ if \ \ \ 1\in A,\ d_A=j.$$ Note if $n=1$, then $\bigtriangleup^{1,0}, \bigtriangleup^{1,d}=\emptyset$. If $n=2$, $\bigtriangleup^{1,d}=\emptyset$. In all other cases $\bigtriangleup^{1,j}\neq \emptyset$. There is a disjoint union $$\bigtriangleup= \bigcup_{j=0}^{d} \bigtriangleup^{1,j}.$$ Let $K^{1,j}= \sum_{K\in \bigtriangleup^{1,j}} K$. Let $K^{1,j}=0$ if $\bigtriangleup^{1,j}=\emptyset$. \begin{lm} \label{self} In case $d\geq 1$, The class $\pi_*(s_1^2)$ is expressed in $Pic(\barr{M})\otimes {\Bbb{Q}}$ by: \begin{equation} \label{exx} \pi_*(s_1^2)= -{1\over d^2}{\cal{H}} + {2\over d} {\cal{L}}_1 -\sum_{j=0}^{d} {(d-j)^2\over d^2} K^{1,j}. \end{equation} \end{lm} \begin{pf} The proof is by intersections with curves in $\barr{M}$. Let $\pi: S={\Bbb P}^1\times C \rightarrow {\Bbb P}^1$ be a family with $n$ sections $s_1, \ldots, s_n$. Let $s_1$ be of degree type $(1,q)$. Let $\mu:S - \ \rightarrow {\Bbb P}^r$ be a rational map with simple base points obtained from a line bundle of degree type $(d,k)$. Let conditions (1), (2), ($3'$), (4) of section (\ref{realy}) be satisfied. Let $\overline{S}\rightarrow S$ be the blow-up at the special points. Let $\lambda: C \rightarrow \barr{M}$ be the induced map. It will be checked that the left and right sides of (\ref{exx}) have the same intersection with $C$. A point of $C\cdot K^{1,j}$ can arise in exactly two cases. First, a simple base point of degree $j$ contained in $s_1$ can be blown-up. Second, a simple base point of degree $d-j$ not contained in $s_1$ can be blown-up. Let $$C \cdot K^{1,j}= x_j + y_j$$ where $x_j$, $y_j$ are the number of instances of the first and second cases respectively. Let $\overline{s}_1$ be the strict transform of the section $s_1$ to $\overline{S}$. The intersection of $C$ with the left side of (\ref{exx}) is given by: $$\overline{\pi}_*(\overline{s}_1^2) = 2q- \sum_{j=0}^{d} x_j.$$ A straightforward computation yields: $$C \cdot {\cal{H}}= 2dk - \sum_{j=0}^{d} j^2 x_j - \sum_{j=0}^{d} (d-j)^2 y_j,$$ $$C \cdot {\cal{L}}_1= dq+k - \sum_{j=0}^{d} j x_j.$$ The equality of the intersection of $C$ with the left and right sides of (\ref{exx}) is now a matter of simple algebra. \end{pf} \subsection{The Class ${\cal{T}}$} \label{tclass} Let $\barr{M}=\barr{M}_{0,n}(r,d)$, $d\geq 2$. Let $H\subset {\Bbb P}^r$ be a hyperplane. A tangency Weil divisor ${\cal{T}}_H \subset \barr{M}$ is defined as follows. Let $W_H\subset \barr{M}$ be the open locus of maps $\mu:C \rightarrow {\Bbb P}^r$ where $\mu^{-1}(H)$ is a subscheme of $d$ reduced points of $C_{nonsing}$. Let ${\cal{T}}_H$ be the complement of $W_H$. It must be shown that ${\cal{T}}_H$ is of pure codimension $1$ in $\barr{M}$. Let $M_H\subset \barr{M}$ be the open locus of maps $\mu: C \rightarrow {\Bbb P}^r$ satisfying : $$\forall x\in\mu^{-1}(H), \ \ \ x\in C_{nonsing} \ \ and \ \ d\mu_x\neq 0.$$ The intersection ${\cal{T}}_H\cap M_H$ corresponds to geometric tangencies and is certainly of pure codimension $1$ in $M_H$ ($d\geq 2$). The complement $\barr{M} \setminus M_H$ is of codimension $2$ in $\barr{M}$. It is not hard to see the closure of ${\cal{T}}_H\cap M_H$ in $\barr{M}$ contains the complement $\barr{M} \setminus M_H$. Therefore, ${\cal{T}}_H$ is a Weil divisor. Define for $0\leq j \leq [{d\over2}]$, $\bigtriangleup^j \subset \bigtriangleup$ as follows. A boundary component $(A\cup B, d_A, d_B)\in \bigtriangleup^j$ if and only if the degree partition $d_A+d_B=d$ equals the partition $j+(d-j)=d$. Let $K^j= \sum_{K\in \bigtriangleup^j} K$. \begin{lm} \label{tan} The class of ${\cal{T}}$ can be expressed in $Pic(\barr{M})\otimes {\Bbb{Q}}$ by: \begin{equation} \label{vexx} {\cal{T}} = {d-1\over d} {\cal{H}} + \sum_{j=0}^{[{d\over2}]} {j(d-j)\over d} K^j. \end{equation} \end{lm} \begin{pf} Let $S$, $\mu$, $\overline{S}$, $\lambda: C \rightarrow \barr{M}$ be exactly as in the proof of Lemma (\ref{self}). It will be checked that the left and right sides of (\ref{vexx}) have the same intersection with $C$. As before, a point of the intersection $C\cdot K^j$ can arise in two cases. A simple point of degree $j$ or $d-j$ can be blown-up. Let $$C\cdot K^j = x_j + y_j$$ where $x_j$ and $y_j$ are the number instances of the first and second case respectively. Let $E_{x_j}$ be the union of the $x_j$ exceptional divisors in $\overline{S}$ obtained from the $x_j$ points of $C\cdot K^j$. Let $E_{y_j}$ be defined similarly. First, the intersection $C\cdot {\cal{T}}$ is calculated. A general element of $\overline{\mu}^*({\cal{O}}_{{\Bbb P}}(1))$ is a nonsingular curve $D$ in the linear series $(d,k)- \sum_{j} j E_{x_j} - \sum_{j} (d-j) E_{y_j}$. Adjunction yields: $$2 g_D-2 = d(2g_C-2)+ 2dk-2k- \sum_{j=0}^{[{d\over2}]} j(j-1) x_j - \sum_{j=0}^{[{d\over2}]} (d-j)(d-j-1) y_j.$$ Since $D$ is a $d$ sheeted cover of $C$, the Riemann-Hurwitz formula determines the ramifications: $$C\cdot {\cal{T}} = 2dk-2k - \sum_{j=0}^{[{d\over 2}]} j(j-1)x_j+(d-j)(d-j-1)y_j.$$ $C\cdot {\cal{H}}$ is simply $D^2$. Hence $$C\cdot {\cal{H}}= 2dk - \sum_{j=0}^{[{d\over 2}]} j^2x_j+ (d-j)^2 y_j.$$ Again, an algebraic computation yields the equality of the intersections of the left and right sides with $C$. \end{pf} \section{Intersections of $\Bbb{Q}$-divisors} \label{inter} \subsection{Intersections of the classes $\{{\cal{L}}_i\}\cup \{{\cal{H}}\}$} The top dimensional intersection products on $\barr{M}=\barr{M}_{0,n}(r,d)$ of the classes $\{{\cal{L}}_i\}$ are algorithmically determined by the First Reconstruction Theorem [K-M]. These top classes are computed recursively in $d$ and $n$. The algorithm requires one initial value: the number of lines in ${\Bbb P}^r$ through two points. The top intersection products of $\{{\cal{L}}_i\}$ are exactly the characteristic numbers ($\beta=0$) of rational curves in ${\Bbb P}^r$. Top dimensional intersections of the classes $\{{\cal{L}}_i\} \cup \{{\cal{H}}\}$ are also characteristic numbers of rational curves in ${\Bbb P}^r$. Each factor of ${\cal{H}}$ is a codimension-2 characteristic condition. For example, if $\barr{M}=\barr{M}_{0,0}(2,3)$, then ${\cal{H}}^8$ equals the number of rational plane cubics through $8$ general points. If $\barr{M}=\barr{M}_{0,2}(3,4)$, then $c({\cal{L}}_1)^3\cdot c({\cal{L}}_2)^3 \cdot {\cal{H}}^{12}$ equals the number of rational space quartics passing through 2 general points and meeting 12 general lines. \subsection{Boundary Components} Let $K=(A\cup B, d_A, d_B)$ be a boundary component of $\barr{M}_{0,n}(r,d)$. Let $\barr{M}_A=\barr{M}_{0, |A|+1}(r,d_A)$ and $\barr{M}_B= \barr{M}_{0, |B|+1}(r, d_B)$. Let the additional markings be $p_A$ and $p_B$ respectively. Let $e_A: \barr{M}_A \rightarrow {\Bbb P}^r$ and $e_B: \barr{M}_B\rightarrow {\Bbb P}^r$ be the evaluation maps obtained from the markings $p_A$ and $p_B$. Let $\tau_A$, $\tau_B$ be the projections from $\barr{M}_A \times \barr{M}_B$ to the first and second factors respectively. Let $\tilde {K}= \barr{M}_A \times_{{\Bbb P}^r} \barr{M}_B$ be the fiber product with respect to the evaluation maps $e_A$, $e_B$. $\tilde{K}\subset \barr{M}_A \times_{{\Bbb C}} \barr{M}_B$ is the closed subvariety $(e_A\times_{{\Bbb C}} e_B)^{-1} (D)$ where $D\subset {\Bbb P}^r \times {\Bbb P}^r$ is the diagonal. $\tilde{K}$ is easily seen to be an irreducible, normal, projective variety with finite quotient singularities. These results follow, for example, from the local construction given in [P]. The class of $\tilde{K}$ in $\barr{M}_A \times \barr{M}_B$ can be computed by the pull-back of the K\"unneth decomposition of the diagonal in ${\Bbb P}^r \times {\Bbb P}^r$: \begin{equation} \label{f1} [\tilde{K}] = \sum_{i=0}^{r} \tau_A^*(c_1({\cal{L}}_A)^i)\cdot \tau_B^*(c_1({\cal{L}}_B)^{r-i}) \end{equation} where ${\cal{L}}_A$, ${\cal{L}}_B$ are the line bundles on $\barr{M}_A$, $\barr{M}_B$ induced by the marking $p_A$, $p_B$. There is a natural map $\psi: \tilde{K} \rightarrow K$. The set theoretic description of $\psi$ is clear: $\psi([\mu_A],[\mu_B])$ is the moduli point of the map obtained by gluing maps $\mu_A$, $\mu_B$ along the markings $p_A$, $p_B$. It is not hard to define $\psi$ algebraically. $\psi$ is a birational morphism except when $n=0$ and $d_A=d_B=d/2$. In the latter case, $\psi$ is generically 2-1. The pull-backs of the classes $\{{\cal{L}}_i\} \cup \bigtriangleup \cup \{{\cal{H}} \}$ on $\barr{M}$ to $\tilde{K}$ are determined in the following manner. Let ${\cal{H}}_A$, ${\cal{H}}_B$ be the codimension-$2$ plane incidence classes on $\barr{M}_A$, $\barr{M}_B$. Clearly, \begin{equation} \label{f2} \psi^*({\cal{H}})= (\tau_A^*({\cal{H}}_A) + \tau_B^*({\cal{H}}_B)) \ |_{\tilde{K}}. \end{equation} Let $P$ be the marking set of $\barr{M}$. For each $i\in P$, $i$ is either in $A$ or $B$. It follows \begin{equation} \label{f3} \psi^*({\cal{L}}_i)= \tau_A^*({\cal{L}}_i) \ |_{\tilde{K}}, \ \ \psi^*({\cal{L}}_i)= \tau_B^*({\cal{L}}_i)\ |_{\tilde{K}} \end{equation} in case $i\in A$, $i\in B$ respectively. Let $T=(A'\cup B', d_{A'}, d_{B'})$ be a boundary component of $\barr{M}$ {\em not} equal to $K$. $T$ intersects $K$ exactly when one of the following two conditions hold: \begin{enumerate} \item[(i.)] There exists a subset $C\subset A$ and an integer $d_C$ such that $$( (A\setminus C) \cup (B\cup C), d_A-d_C, d_B+d_C)= T.$$ \item[(ii.)] There exists a subset $C\subset B$ and an integer $d_C$ such that $$( (A\cup C) \cup (B\setminus C), d_A+d_C, d_B-d_C)=T.$$ \end{enumerate} \begin{equation} \label{f4} \psi^*(T) = \sum_{C, d_c} \tau_A^*(A\cup (C\cup \{p_A\}), d_A, d_C) \ |_{\tilde{K}} \end{equation} $$ + \ \ \sum_{C,d_c} \tau_B^* (B\cup (C\cup \{p_B\}), d_B, d_C) \ |_{\tilde{K}}.$$ The sums on the right are taken over subsets $C$ and degrees $d_c$ that satisfy (i) and (ii) above respectively. The main point is distinct boundary divisors have transverse (if nonempty) intersections in the stack $\overline{\cal{M}}_{0,n}(r,d)$. This can be seen an a property inherited from the Mumford-Knudsen space $\barr{M}_{0,m}$ by the local construction given in [P]. Since the automorphism loci of $\barr{M}_{0,n}(r,d)$ and the boundary component $(A\cup B, d_A, d_B)$ are of codimension at least two in $\barr{M}_{0,n}$, $(A\cup B, d_A, d_B)$ respectively, the transverse intersection property descends to the coarse moduli space. Let $\omega_{\pi A}$, $\omega_{\pi B}$ denote the relative dualizing sheaves of the the universal families over $\barr{M}^*_A$, $\barr{M}^*_B$ respectively. There are two universal curves over $\tilde{K}^*=\tilde{K}\cap (\barr{M}^*_A\times \barr{M}^*_B)$ obtained via pull-back of the universal families $U_A^*$ and $U_B^*$. These universal curves glue on the sections $s_{pA}$ and $s_{pB}$ to form a universal family $$\tilde{\pi}: U^*_{\tilde{K}^*} \rightarrow \tilde{K}^*$$ of maps for the moduli problem of $\barr{M}$. It follows, $$\omega_{U^*_{\tilde{K}^*}}\ |_ {\tau_A^*(U_A^*)}= \tau_A^*(\omega_{\pi A}) + s_{pA},$$ $$\omega_{U^*_{\tilde{K}^*}}\ |_ {\tau_B^*(U_B^*)}= \tau_B^*(\omega_{\pi B}) + s_{pB}.$$ Hence $$\psi^*(\pi_*(c_1(\omega_{\pi})^2))= \tau_A^*( \pi_{A*}((c_1(\omega_{\pi A})+s_{pA})^2)) + \tau_B^*(\pi_{B*}((c_1(\omega_{\pi B})+s_{pB})^2)).$$ A normal bundle calculation yields $c_1(\omega_{\pi A})\cdot s_{pA}= -s_{pA}^2$. Hence, $$(c_1(\omega_{\pi A})+s_{pA})^2 = c_1(\omega_{\pi A})^2 - s_{pA}^2$$ (similarly for $B$). Recall $\pi_*(c_1(\omega_{\pi})^2)= -\sum_{T\in \bigtriangleup} T$. Finally, \begin{equation} \label{f5} -\psi^*(K) = \sum_{T\in \bigtriangleup, T\neq K} \psi^*(T) \ \ +\ \ \tau_A^*(\pi_{A*}(c_1(\omega_{\pi A})^2 - s_{pA}^2)) \end{equation} $$+ \ \ \tau_B^*(\pi_{B*}(c_1(\omega_{\pi B})^2 - s_{pB}^2)).$$ Lemmas (\ref{om}), (\ref{self0}), and (\ref{self}) express $\pi_{A*}(c_1(\omega_{\pi A})^2)$, $\pi_{A*}(s_{pA}^2)$ explicitly in terms of the standard classes $\{{\cal{L}}_i\} \cup \bigtriangleup \cup \{{\cal{H}}\}$ on $\barr{M}_A$ (similarly for $\barr{M}_B$). Via equations (\ref{f2}) - (\ref{f5}), the $\psi$ pull-back of every standard class $\{{\cal{L}}_i\} \cup \bigtriangleup \cup \{{\cal{H}}\}$ on $\barr{M}$ has now been expressed as the restriction to $\tilde{K}$ of a linear combination of the $\tau_A$ and $\tau_B$ pull-backs of standard classes on $\barr{M}_A$ and $\barr{M}_B$. \subsection{The Algorithm} The inductive algorithm for computing top intersection products is now clear. All top monomials in the elements $\{{\cal{L}}_i\}\cup \{{\cal{H}}\}$ are known by the First Reconstruction theorem. If a monomial product on $\barr{M}$ includes a boundary class $K$, the intersection is carried out on $\tilde{K}$. By the above formulas (\ref{f1})-(\ref{f5}), the desired monomial can be expressed as a sum of top products of standard classes on $\barr{M}_A$ and $\barr{M}_B$. Since $\barr{M}_A$ is of lesser degree or of lesser marking number than $\barr{M}$ (similarly for $\barr{M}_B$), the inductive process terminates. \subsection{Characteristic Numbers} \label{cnum} Lemma (\ref{tan}) expresses the hyperplane tangency condition in terms of the standard classes. Hence all top products of the classes $\{{\cal{L}}_i\} \cup \{{\cal{H}}, {\cal{T}}\}$ can be effectively computed by the above algorithm. It remains to check the top intersections of $\{{\cal{L}}_i\} \cup \{{\cal{H}}, {\cal{T}}\}$ are the characteristic numbers of rational curves. Let \begin{equation} \label{cy} c_1({\cal{L}}_1)^{l_1} \cdots c_1({\cal{L}}_n)^{l_n} \cdot {\cal{H}}^{\alpha}\cdot {\cal{T}}^{\beta} \end{equation} be a top product on $\overline{M}=\overline{M}_{0,n}(r,d)$. Since the ${\cal{L}}_i$ are pull-backs of ${\cal{O}}_{{\Bbb P}^r}(1)$ via the evaluation maps, codimension $l_i$ linear spaces of ${\Bbb P}^r$ determine representatives of $c_1({\cal{L}}_i)^{l_i}$. The cycle ${\cal{H}}^{\alpha}$ is determined by $\alpha$ codimension $2$ linear spaces in ${\Bbb P}^r$. Finally, the cycle ${\cal{T}}^{\beta}$ is determined by $\beta$ hyperplanes in ${\Bbb P}^r$. When ($\beta\neq 0$), it is assumed $d\geq 2$. The first step is to show for general choices of all the linear spaces of ${\Bbb P}^r$ in question, the intersection cycle (\ref{cy}) in $\overline{M}$ is at most 0 dimensional and corresponds (set theoretically) to the correct geometric locus. The second step is to show the intersection cycle is multiplicity free. Let ${\Bbb P}^{r*}$ be the parameter space of hyperplanes in ${\Bbb P}^r$. Defined the universal tangency subvariety $${\cal{T}}_{univ} \subset \overline{M} \times {\Bbb P}^{r*}$$ as follows. Let $W_{univ}\subset \barr{M} \times {\Bbb P}^{r*}$ be the open locus of pairs $(\mu:C\rightarrow {\Bbb P}^r, H)$ where $\mu^{-1}(H)$ is a subscheme of $d$ reduced points of $C_{nonsing}$. Let ${\cal{T}}_{univ}$ be the complement of $W_{univ}$. Let ${\cal{T}}_H$ be the the fiber of ${\cal{T}}_{univ}$ over the parameter point of the hyperplane $H$. ${\cal{T}}_H$ is exactly the tangency Weil divisor defined in section (\ref{calc}). Similarly, let $${\cal{H}}_{univ} \subset \barr{M} \times {\Bbb G}({\Bbb P}^{r-2}, {\Bbb P}^r)$$ be the universal codimension 2 plane incidence subvariety. The fiber of ${\cal{H}}_{univ}$ over the parameter point of the codimension 2 plane $P$ is ${\cal{H}}_P$. Let $$I_{univ} \subset \barr{M} \ \times\ {\Bbb G}(r-l_1,r) \times \cdots \times {\Bbb G}(r-l_n,r)\ \times\ {\Bbb G}(r-2,r) \times \cdots \times {\Bbb G}(r-2,r)$$ $$\times \ {\Bbb P}^{r*}\times \cdots \times {\Bbb P}^{r*}$$ be the universal intersection cycle (\ref{cy}) defined by the universal divisors ${\cal{T}}_{univ}$, ${\cal{H}}_{univ}$ and the evaluation maps. $I_{univ}$ is closed subvariety. In the first step, slightly more than the dimensionality of the general intersection cycle will be established. A map $\mu: C \rightarrow {\Bbb P}^{r}$ is {\em simply tangent} to a hyperplane $H$ if \begin{enumerate} \item[(i.)] $\mu^{-1}(H)\subset C_{nonsing}$. \item[(ii.)] As a subscheme, $\mu^{-1}(H)$ consists of $1$ double and $d-2$ reduced points. \end{enumerate} A map $\mu: C \rightarrow {\Bbb P}^{r}$ has {\em simple intersection} with a codimension 2 plane $P$ if \begin{enumerate} \item[(i.)] $\mu^{-1}(P)$ consists of 1 point $x \in C_{nonsing}$. \item[(ii.)] $Im(d\mu(x))$ and the tangent space of $P$ span maximal rank. \end{enumerate} \begin{lm} For general choices of linear spaces \begin{equation} \label{lin} L_1, \ldots, L_n,\ P_1, \ldots, P_{\alpha},\ H_1, \ldots, H_{\beta} \end{equation} the intersection cycle (\ref{cy}) is at most 0 dimensional and set theoretically corresponds to maps $\mu: C \rightarrow {\Bbb P}^r$ satisfying: \begin{enumerate} \item[(1.)] $C\stackrel {\sim}{=} {\Bbb P}^1$, $\mu$ is an immersion/embedding ($r=2$ / $r\geq 3$). \item[(2.)] $\forall k$, $\mu$ is simply tangent to the hyperplanes $H_k$. \item[(3.)] $\forall j$, $\mu$ intersects the linear spaces $P_j$ simply. \item[(4.)] $\forall i$, the $\mu$-image of the $i^{th}$ marked point lies in $L_i$. \end{enumerate} \end{lm} \begin{pf} The intersection cycle $I$ determined by the linear spaces (\ref{lin}) is the fiber of $I_{univ}$ over the parameter points of the linear spaces. $dim(I)\leq 0$ is an open condition in the parameter space. It is first checked that general choice of the linear spaces (\ref{lin}) yields an intersection cycle of dimension at most $0$. Let $[\mu]\in \barr{M}$ be the moduli point of a map $\mu: C \rightarrow {\Bbb P}^r$. By Bertini's Theorem, the general hyperplane $H$ is transverse to $\mu$. Therefore, the general tangency divisor ${\cal{T}}_H$ satisfies $[\mu] \notin {\cal{T}}_H$. Similarly, the general incidence divisor ${\cal{H}}_P$ satisfies $[\mu]\notin {\cal{H}}_P$. By choosing at each stage tangency and incidence divisors that reduce the dimension of every component of the intersection, it follows $${\cal{H}}_{P_1} \cap \ldots \cap {\cal{H}}_{P_\alpha} \cap {\cal{T}}_{H_1} \cap \ldots \cap {\cal{T}}_{H_\beta}$$ has codimension at least $\alpha + \beta$. Since the remaining intersections are obtained from basepoint free linear series, the general intersection cycle has dimension at most $0$. If the general parameter point yields an empty cycle $I$, there is nothing more to prove. Let $W$ be the open set of the parameter space where $dim(I)=0$. The conditions (1-3) on $I$ determine open sets $W_1, W_2, W_3 \subset W$. Condition (iv) is automatic. It suffices to show $W_i$ is nonempty for $1\leq i \leq 3$. The subset $Y\subset \barr{M}$ of maps that are not immersion/embedding ($r=2$ / $r\geq 3$) is of codimension at least $1$. Hence, by the dimension reduction argument above, $Y\cap I=\emptyset$ for a general parameter point. Therefore, $W_1 \neq \emptyset$. Let $W_{2,k}, W_{3,j} \subset W$, be the set of parameter points that satisfy condition (2), (3) for the hyperplane $H_k$, linear space $P_j$ respectively. Since $W_2= \cap _{k=1}^{\beta} W_{2,k}$ and $W_3=\cap_{j=1}^{\alpha} W_{3,j}$, it suffices to show $W_{2,k}, W_{3,j} \neq \emptyset$. Let $H_k$ be any hyperplane. The locus of moduli points $[\mu]\in {\cal{T}}_{H_k}$ that are not simply tangent is of codimension at least 2 in $\barr{M}$. By the dimension reduction argument, $W_{2,k}\neq \emptyset$. Similarly, the locus of moduli points $[\mu]\in {\cal{H}}_{P_j}$ that do not intersect simply is of codimension at least 2 in $\barr{M}$. As before $W_{3,j} \neq \emptyset$. \end{pf} It must now be shown that the intersection cycle (\ref{cy}) is reduced for general linear spaces. This transversality is established by Kleiman's Bertini Theorem. Unfortunately, since the divisors ${\cal{T}}_H$, ${\cal{H}}_P$ need not move {\em linearly}, Bertini's Theorem can not be directly applied to $\overline{M}$. Instead, an auxiliary construction is undertaken. Kleiman's Bertini Theorem is applied to the universal curve over $\overline{M}$. It will be shown that suitable transversality on the universal curve implies transversality on $\overline{M}$. Let $\overline{M}^0\subset \overline{M}$ be the open set of immersed/embedded ($r=2$, $r\geq 3$) maps with irreducible domains. Since (for general linear spaces) the intersection cycle (\ref{cy}) lies in $\overline{M}^0$, transversality need only be established in $\overline{M}^0$. Note $\overline{M}^0$ is in the automorphism-free locus. Let $U \rightarrow \overline{M}^0$ be the universal curve. Let $\mu: U \rightarrow {\Bbb P}^r$ be the universal map. $U$, $\overline{M}^0$ are nonsingular. Let ${\Bbb P} T$ be the projective tangent bundle of ${\Bbb P}^r$. Since each point of $\overline{M}^0$ corresponds to an immersion/embedding, there is a natural algebraic map $\nu: U \rightarrow {\Bbb P} T$ given by the differential of $\mu$. The map $\nu$ is a lifting of $\mu$. By projectivizing tangent spaces, the hyperplanes $H_1, \ldots, H_{\beta}$ define nonsingular, codimension 2 subvarieties of ${\Bbb P} T$: $${\Bbb P} H_1, \ldots, {\Bbb P} H_{\beta}$$ Let $U_1, \ldots, U_{\beta}$ be $\beta$ copies of the universal curve $U$. Let $U'_1, \ldots, U'_{\alpha}$ be $\alpha$ more copies of $U$. Define the product: $$X \stackrel {\sim}{=} U'_1 \times_{\overline{M}^0} \ldots \times_{\overline{M}^0} U'_{\alpha} \times_{\overline{M}^0} U_1 \times_{\overline{M}^0} \ldots \times_{\overline{M}^0} U_{\beta}.$$ Let $\mu'_j: X \rightarrow {\Bbb P}^r$, $\nu_k:X \rightarrow {\Bbb P} T$ be the maps obtained by projection onto $U'_j$, $U_k$ and composition with $\mu$, $\nu$ respectively. Kleiman's Bertini Theorem may now be applied. The group $GL_{r+1}({\Bbb C})$ acts transitively on ${\Bbb P}^r$, ${\Bbb P} T$. Hence, the general intersection $$\mu'_1\ ^{-1}(P_1)\cap \ldots \cap \mu'_{\alpha}\ ^{-1}(P_{\alpha}) \cap \nu_1^{-1}({\Bbb P} H_1) \cap \ldots \cap \nu_{\beta}^{-1}({\Bbb P} H_{\beta}) \subset X$$ is nonsingular and of the correct codimension (if nonempty). It remains to obtain the corresponding result on $\overline{M}$. Consider the nonsingular, codimension 2 subvariety $\mu^{-1}(P_j) \subset U$. The projection $\mu^{-1}(P_j) \rightarrow {\cal{H}}_{P_j} \cap \overline{M}^0$ is \'etale and 1-1 over the locus of of maps meeting $P_j$ simply. Similarly, the projection $\nu^{-1}({\Bbb P} H_k) \rightarrow {\cal{T}}_{H_k}\cap \overline{M}^0$ is \'etale and 1-1 over the the locus of maps simply tangent to $H_k$. From Lemma (\ref{sim}) below, the projection $$\mu'_1\ ^{-1}(P_1)\cap \ldots \cap \mu'_{\alpha}\ ^{-1}(P_{\alpha}) \cap \nu_1^{-1}({\Bbb P} H_1) \cap \ldots \cap \nu_{\beta}^{-1}({\Bbb P} H_{\beta}) \longrightarrow\ \ \ \ $$ $$\ \ \ \ \ \ {\cal{H}}_{P_1}\cap \ldots {\cal{H}}_{P_\alpha} \cap {\cal{T}}_{H_1} \cap \ldots \cap {\cal{T}}_{H_\beta} \cap \overline{M}^0$$ is \'etale and 1-1 over the locus of points in ${\cal{H}}_{P_1}\cap \ldots {\cal{H}}_{P_\alpha} \cap {\cal{T}}_{H_1} \cap \ldots \cap {\cal{T}}_{H_\beta} \cap \overline{M}^0$ corresponding to simple intersection and tangency. It has therefore been proved, for general linear spaces, the locus of ${\cal{H}}_{P_1}\cap \ldots {\cal{H}}_{P_\alpha} \cap {\cal{T}}_{H_1} \cap \ldots \cap {\cal{T}}_{H_\beta} \cap \overline{M}^0$ corresponding to simple intersection and tangency is nonsingular and of the correct codimension (if nonempty). It was shown above, for general linear spaces, the intersection cycle (\ref{cy}) involves only maps that have simple intersection and tangency with the $P_j$, $H_k$. Since the intersections $c_1({\cal{L}}_i)^{l_i}$ are obtained from basepoint free linear series on $\overline{M}$, the further intersections yield a reduced intersection cycle (\ref{cy}) by Bertini's Theorem. \begin{lm} \label{sim} Let $M$ be a nonsingular base. Let $\pi:U\rightarrow M$ be smooth map of relative dimension 1. Let $D_1, D_2, \ldots, D_l\subset U$ be nonsingular, codimension $2$ subvarieties such that $D_i$ is \'etale and 1-1 over $\pi(D_i)$. Let $X\stackrel {\sim}{=} U_1\times_M \ldots \times_M U_l$ be the fiber product of copies of $U$. Let $\rho_i:X\rightarrow U_i$ be the projection. Then $$\rho_1^{-1}(D_1) \cap \ldots \cap \rho_l^{-1}(D_l)\subset X$$ is \'etale and 1-1 over the intersection $\pi(D_1) \cap \ldots \cap \pi(D_l) \subset M$. \end{lm} \begin{pf} The issue is local on $M$. Let $m\in M$ be in the intersection of the $\pi(D_i)$. Choose local defining equations $(f_i)$ of $\pi(D_i)$ near $m$. Let $u_i \in D_i$ be points over $m$. Locally (in the analytic topology) at $u_i$, $U_i$ is an open set of the trivial product ${\Bbb C}_i \times M$ and $D_i$ is the intersection of $(f_i)$ with a section $(z_i)$ of this product ($z_i$ is the coordinate on ${\Bbb C}_i$). It now follows local equations for for $\rho^{-1}_1 D_1\cap\ldots \cap \rho^{-1}_l D_l$ at $(u_1, \ldots, u_l)$ are $(z_1, \ldots, z_l, f_1, \ldots, f_l)$ in ${\Bbb C}_1 \times \ldots {\Bbb C}_l \times M$ which is certainly \'etale over $(f_1,\ldots, f_l) \subset M$. \end{pf} All the characteristic numbers of rational curves in projective space can be algorithmically computed. For example, the number of twisted cubics in ${\Bbb P}^3$ through $2$ points, $6$ lines, and tangent to $2$ planes can be expressed as $$c_1({\cal{L}}_1)^3 \cdot c_1({\cal{L}}_2)^3 \cdot c_1({\cal{L}}_3)^2 \cdots c_1({\cal{L}}_8)^2 \cdot {\cal{T}}^2$$ on $\barr{M}_{0,8}(3,3)$ or $$c_1({\cal{L}}_1)^3\cdot c_1({\cal{L}}_2)^3 \cdot {\cal{H}}^6\cdot {\cal{T}}^2$$ on $\barr{M}_{0,2}(3,3)$. \section{Examples} \label{exam} \subsection{Conics in ${\Bbb P}^2$ and ${\Bbb P}^3$} Since the Hilbert schemes of lines and conics are Grassmanians and projective bundles over Grassmanians, the $\beta=0$ characteristic numbers of rational curves in degrees $1$ and $2$ can be calculated directly via intersection theory on these Hilbert schemes. The tangency characteristic numbers for conics classically required the beautiful space of complete conics. $\overline{M}_{0,0}(2,2)$ is the space of complete conics. A new calculation of the characteristic numbers for plane conics is obtained by considering the pointed space $\overline{M}_{0,1}(2,2)$. Let $\barr{M}=\barr{M}_{0,1}(2,2)$. $Pic(\barr{M})\otimes {\Bbb{Q}}$ is freely generated by ${\cal{H}}$, ${\cal{L}}_1$, and the unique boundary component $K$ corresponding to the partition $(\{1\}\cup{\emptyset}, 1+1=2)$. The top intersection numbers are ($dim\barr{M}_{0,1}(2,2)=6$): $$\begin{array}{llrllllrllllr} {\cal{H}}^6 & & 0 & & & {\cal{H}}^5K & & 0 & & & {\cal{H}}^4K^2 & & 0 \\ {\cal{H}}^5 {\cal{L}}_1 & & +2 & & & {\cal{H}}^4K {\cal{L}}_1 & & +6 & & & {\cal{H}}^3 K^2 {\cal{L}}_1 & & +18\\ {\cal{H}}^4 {\cal{L}}_1^2 & & +1 & & & {\cal{H}}^3K {\cal{L}}_1^2 & & +3 & & & {\cal{H}}^2 K^2 {\cal{L}}_1^2 & & +9 \\ \end{array}$$ $$\begin{array}{llrllllrllllrllllr} {\cal{H}}^3K^3&&0&&&{\cal{H}}^2K^4&&0&&& {\cal{H}} K^5 &&0&&& K^6 &&0 \\ {\cal{H}}^2 K^3{\cal{L}}_1 &&-10 &&& {\cal{H}} K^4 {\cal{L}}_1 && -30 &&& K^5 {\cal{L}}_1 && +102 \\ {\cal{H}} K^3 {\cal{L}}_1^2 && -5 &&& K^4 {\cal{L}}_1^2 && -15 \end{array}$$ \noindent Note ${\cal{L}}_1^3=0$. The line tangency class ${\cal{T}}={1\over2}({\cal{H}}+K)$ is determined by Lemma (\ref{tan}). The characteristc number of plane conics through $\alpha$ points and tangent to $\beta$ lines is ${1 \over 2}{\cal{H}}^{\alpha}{\cal{T}}^{\beta}{\cal{L}}_1$: $$\begin{array}{llr} (1/ 2)\cdot{\cal{H}}^5 {\cal{L}}_1 & & 1 \\ (1/2)\cdot{\cal{H}}^4 {\cal{T}} {\cal{L}}_1 & & 2 \\ (1/2)\cdot{\cal{H}}^3 {\cal{T}}^2 {\cal{L}}_1 & & 4 \\ (1/2)\cdot{\cal{H}}^2 {\cal{T}}^3 {\cal{L}}_1 & & 4 \\ (1/2)\cdot{\cal{H}} {\cal{T}}^4 {\cal{L}}_1 & & 2 \\ (1/2)\cdot{\cal{T}}^5 {\cal{L}}_1 && 1 \end{array}$$ \noindent The class of maps tangent to a fixed conic can be easily calculated by the methods of Lemma (\ref{tan}). Let ${\cal{C}}\in Pic(\barr{M})\otimes {\Bbb{Q}}$ denote this conic tangency class. ${\cal{C}}=3{\cal{H}}+K$. The number of plane conics tangent to 5 fixed conics is therefore ${1\over 2}{\cal{C}}^5{\cal{L}}_1=3264$. For $r\geq 3$, $\overline{M}_{0,0}(r,2)$ differs from the space of complete conics and the algorithm described above yields a new computation of the characteristic numbers in these cases. Let $\barr{M}=\barr{M}_{0,0}(3,2)$. $Pic(\barr{M})\otimes {\Bbb{Q}}$ is freely generated by ${\cal{H}}$ and the unique boundary component $K$ corresponding to the degree partition $1+1=2$. $\tilde{K}\subset \barr{M}_{0,1}(3,1) \times \barr{M}_{0,1}(3,1)$. Since $\barr{M}_{0,1}(3,1)$ has no boundary, all top intersections are known. Using the formulas of section (\ref{inter}), the answers for the top intersections of ${\cal{H}}$ and $K$ on $\barr{M}_{0,0}(3,2)$ ($dim \barr{M}_{0,0}(3,2)=8$) are: $$\begin{array}{llr} {\cal{H}}^8 & & +92 \\ {\cal{H}}^7 K & & +140 \\ {\cal{H}}^6 K^2 & & +140 \\ {\cal{H}}^5 K^3 & & -100 \\ {\cal{H}}^4 K^4 & & -68 \\ {\cal{H}}^3 K^5 & & +172 \\ {\cal{H}}^2 K^6 & & -20 \\ {\cal{H}} K^7 & & -580 \\ K^8 & & +1820 \end{array}$$ By Lemma (\ref{tan}), ${\cal{T}}= {1\over 2} ({\cal{H}}+K)$. The characteristic number of space conics through $\alpha$ lines and tangent to $\beta$ planes is ${\cal{H}}^\alpha {\cal{T}}^\beta$: $$\begin{array}{llr} {\cal{H}}^8 & & 92 \\ {\cal{H}}^7 {\cal{T}} & & 116 \\ {\cal{H}}^6 {\cal{T}}^2 & & 128 \\ {\cal{H}}^5 {\cal{T}}^3 & & 104 \\ {\cal{H}}^4 {\cal{T}}^4 & & 64 \\ {\cal{H}}^3 {\cal{T}}^5 & & 32 \\ {\cal{H}}^2 {\cal{T}}^6 & & 16 \\ {\cal{H}} {\cal{T}}^7 & & 8 \\ {\cal{T}}^8 & & 4 \end{array}$$ These characteristic numbers (with complete proofs) were known classically. \subsection{ Rational Plane Cubics} Let $\barr{M}=\barr{M}_{0,0}(2,3)$. $Pic(\barr{M})\otimes {\Bbb{Q}}$ is freely generated by ${\cal{H}}$ and the unique boundary component $K$ corresponds to the degree partition $1+2=3$. The algorithm described above yields the top intersections of ${\cal{H}}$ and $K$ inductively. Since $\tilde{K}\subset \barr{M}_{0,1}(2,1)\times \barr{M}_{0,1}(2,2)$, first the top intersections on these Kontsevich spaces must be computed. $\barr{M}_{0,1}(2,1)$ has no boundary, hence all top products are known. There is a unique boundary component $B$ of $\barr{M}_{0,1}(2,2)$. $\overline{B}\subset \barr{M}_{0,2}(2,1) \times \barr{M}_{0,1}(2,1)$. Thus the top products on $\barr{M}_{0,2}(2,1)$ must be computed. Finally, the unique boundary component of $\barr{M}_{0,2}(2,1)$ requires knowledge of the top products on $\barr{M}_{0,3}(2,0)$ and $\barr{M}_{0,1}(2,1)$ which are known. The answers for the top intersections of ${\cal{H}}$ and $K$ on $\barr{M}_{0,0}(2,3)$ ($dim \barr{M}_{0,0}(2,3)=8$) are: $$\begin{array}{llr} {\cal{H}}^8 & & +12 \\ {\cal{H}}^7 K & & +42 \\ {\cal{H}}^6 K^2 & & +129 \\ {\cal{H}}^5 K^3 & & + 285 \\ {\cal{H}}^4 K^4 & & +336 \\ {\cal{H}}^3 K^5 & & -(2541/ 4) \\ {\cal{H}}^2 K^6 & & -(8259/ 16) \\ {\cal{H}} K^7 & &+ (19641/ 8) \\ K^8 & & - (44835/ 16) \end{array}$$ Note since $K$ is ${\Bbb{Q}}$-Cartier, the intersections ${\cal{H}}^i\cdot K^j$ need not be integers. By Lemma (\ref{tan}), ${\cal{T}}= {2\over 3} ({\cal{H}}+K)$. The characteristic number of plane cubics through $\alpha$ points and tangent to $\beta$ lines is ${\cal{H}}^\alpha {\cal{T}}^\beta$: $$\begin{array}{llr} {\cal{H}}^8 & & 12 \\ {\cal{H}}^7\cdot {\cal{T}} & & 36 \\ {\cal{H}}^6 \cdot {\cal{T}}^2 & & 100 \\ {\cal{H}}^5 \cdot {\cal{T}}^3 & & 240 \\ {\cal{H}}^4\cdot {\cal{T}}^4 & & 480 \\ {\cal{H}}^3\cdot {\cal{T}}^5 & & 712 \\ {\cal{H}}^2\cdot {\cal{T}}^6 & & 756 \\ {\cal{H}}\cdot {\cal{T}}^7 & & 600 \\ {\cal{T}}^8 & & 400 \end{array}$$ These characteristic numbers have been calculated by H. Zeuthen, S. Maillard, H. Schubert, G. Sacchiero, S. Kleiman, S. Speiser, and P. Aluffi. ([S], [Sa], [K-S], [A]). Complete proofs appear in [Sa], [K-S], and [A]. \subsection{ Twisted Cubics in ${\Bbb P}^3$} In case $\barr{M}=\barr{M}_{0,0}(3,3)$, $Pic(\barr{M})\otimes {\Bbb{Q}}$ is still generated freely by ${\cal{H}}$, $K$. A similar analysis yields the top intersections ($dim(\barr{M})=12$): $$\begin{array}{llr} {\cal{H}}^{12} & & +80160 \\ {\cal{H}}^{11} K & & +121440 \\ {\cal{H}}^{10} K^2 & & +148920 \\ {\cal{H}}^9 K^3 & & +112080 \\ {\cal{H}}^8 K^4 & & -7824 \\ {\cal{H}}^7 K^5 & & -104100 \\ {\cal{H}}^6 K^6 & & +35880 \\ {\cal{H}}^5 K^7 & & + (190095/2) \\ {\cal{H}}^4 K^8 & & - (222855/2) \\ {\cal{H}}^3 K^9 & & -(674007/ 16) \\ {\cal{H}}^2 K^{10} & & +(10112745/ 32) \\ {\cal{H}} K^{11} & & -(5995065/ 8) \\ K^{12} & & +(58086435/ 32) \end{array}$$ The hyperplane tangency class is again ${\cal{T}}={2\over 3}({\cal{H}}+K)$. The characteristic number of twisted cubics through $\alpha$ lines and tangent to $\beta$ planes is ${\cal{H}}^{\alpha} {\cal{T}}^{\beta}$: $$\begin{array}{llr} {\cal{H}}^{12} & & 80160 \\ {\cal{H}}^{11} {\cal{T}} & & 134400 \\ {\cal{H}}^{10} {\cal{T}}^2 & & 209760 \\ {\cal{H}}^9 {\cal{T}}^3 & & 297280 \\ {\cal{H}}^8 {\cal{T}}^4 & & 375296 \\ {\cal{H}}^7 {\cal{T}}^5 & & 415360 \\ {\cal{H}}^6 {\cal{T}}^6 & & 401920 \\ {\cal{H}}^5 {\cal{T}}^7 & & 343360 \\ {\cal{H}}^4 {\cal{T}}^8 & & 264320 \\ {\cal{H}}^3 {\cal{T}}^9 & & 188256 \\ {\cal{H}}^2 {\cal{T}}^{10} & &128160 \\ {\cal{H}} {\cal{T}}^{11} & & 85440 \\ {\cal{T}}^{12} & & 56960 \end{array}$$ These characteristic numbers have been calculated by H. Schubert and others ([S], [K-S-X]). Complete proofs appear in [K-S-X]. \subsection{Rational Plane Quartics} Let $\barr{M}=\barr{M}_{0,0}(2,4)$. $Pic(\barr{M})\otimes {\Bbb{Q}}$ is freely generated by ${\cal{H}}$ and the boundary components $J$, $K $ corresponding to the degree partitions $2+2=4$, $1+3=4$. The top intersection numbers are ($dim\barr{M}_{0,0}(2,4)=11$): $$\begin{array}{llrllllrllllr} {\cal{H}}^{11} && +620 \\ {\cal{H}}^{10} K &&+1620 &&& {\cal{H}}^{10}J &&+504 \\ {\cal{H}}^9 K^2 &&+3564 &&& {\cal{H}}^9JK && +1512 &&& {\cal{H}}^9J^2 && +0\\ {\cal{H}}^8 K^3 &&+4052 &&& {\cal{H}}^8JK^2&& +4536 &&& {\cal{H}}^8J^2K && +0 \\ {\cal{H}}^7 K^4 && -8340&&& {\cal{H}}^7JK^3&& +10920 &&& {\cal{H}}^7J^2K^2 && +672 \\ {\cal{H}}^6 K^5 && -48300 &&& {\cal{H}}^6JK^4 && +15480 &&& {\cal{H}}^6J^2K^3 && +4320 \\ {\cal{H}}^5 K^6 && +1260 &&& {\cal{H}}^5 J K^5 && -22296 &&& {\cal{H}}^5J^2K^4 && +17184 \\ {\cal{H}}^4K^7 && +153300 &&& {\cal{H}}^4JK^6 && -22728 &&& {\cal{H}}^4J^2K^5 && -11040 \\ {\cal{H}}^3K^8 && -(338620/3)&&& {\cal{H}}^3JK^7&& +70056 &&& {\cal{H}}^3J^2K^6 && -34560\\ {\cal{H}}^2K^9&& -(13690660/27)&&& {\cal{H}}^2JK^8&& +5880&&& {\cal{H}}^2J^2K^7&& +51072\\ {\cal{H}} K^{10}&& +(147582380/81)&&& {\cal{H}} JK^9&& -385560&&&{\cal{H}} J^2K^8 && +100800 \\ K^{11} && -(278947820/81)&&& JK^{10} && +1310904 &&& J^2K^9 && -616896 \\ &&&&&&&&&&&&\\ {\cal{H}}^8J^3&& -364\\ {\cal{H}}^7J^3K && -1260&&& {\cal{H}}^7J^4 && +630\\ {\cal{H}}^6J^3K^2&& -3852&&& {\cal{H}}^6J^4K && +1782&&& {\cal{H}}^6J^5 && -645\\ {\cal{H}}^5J^3K^3&& -8836&&& {\cal{H}}^5J^4K^2&& +3588&&& {\cal{H}}^5J^5K && -(2385/2)\\ {\cal{H}}^4J^3K^4&& +4980&&& {\cal{H}}^4J^4K^3&& -1788&&& {\cal{H}}^4J^5K^2&& +906\\ {\cal{H}}^3J^3K^5&& +16356&&& {\cal{H}}^3J^4K^4&& -7830&&& {\cal{H}}^3J^5K^3 && +(8241/2)\\ {\cal{H}}^2J^3K^6&& -22060&&& {\cal{H}}^2J^4K^5&& +7770&&& {\cal{H}}^2J^5K^4 && -1815\\ {\cal{H}} J^3K^7&& -46452&&& {\cal{H}} J^4K^6 && +22632&&& {\cal{H}} J^5 K^5 && -(22125/2)\\ J^3K^8 && +255444&&& J^4K^7 &&-92232 &&& J^5K^6&& +28920\\ &&&&&&&&&&&&\\ {\cal{H}}^5J^6 && +(2419/8)\\ {\cal{H}}^4J^6K && -(4743/8)&&& {\cal{H}}^4J^7 && +(765/2) \\ {\cal{H}}^3J^6K^2&&-(18549/8)&&& {\cal{H}}^3J^7K&& +1305&&& {\cal{H}}^3J^8&& -(5649/8)\\ {\cal{H}}^2J^6K^3&& -(3455/8)&&& {\cal{H}}^2J^7K^2&&+(1923/2)&&&{\cal{H}}^2J^8K&&-(6615/8)\\ {\cal{H}} J^6K^4&& +(39075/8)&&& {\cal{H}} J^7K^3&& -1680&&& {\cal{H}} J^8K^2&&+(2163/8)\\ J^6K^5&&-(56631/8)&&& J^7K^4&& +(1701/2)&&& J^8K^3&&+(2289/8)\\ &&&&&&&&&&&&\\ {\cal{H}}^2J^9&& +(4375/8)\\ {\cal{H}} J^9K&& +189&&& {\cal{H}} J^{10} && -(7875/32)\\ J^9K^2&& -189&&& J^{10}K && +0 &&& J^{11} && +(10143/128) \end{array}$$ The line tangency class is ${\cal{T}}={3\over 4}{\cal{H}}+J+{3\over 4}K$. The characteristic number of rational plane quartics through $\alpha$ points and tangent to $\beta$ lines is ${\cal{H}}^{\alpha} {\cal{T}}^{\beta}$: $$\begin{array}{llr} {\cal{H}}^{11} & & 620 \\ {\cal{H}}^{10} {\cal{T}} & & 2184 \\ {\cal{H}}^{9} {\cal{T}}^2 & & 7200 \\ {\cal{H}}^8 {\cal{T}}^3 & & 21776 \\ {\cal{H}}^7 {\cal{T}}^4 & & 59424 \\ {\cal{H}}^6 {\cal{T}}^5 & & 143040 \\ {\cal{H}}^5 {\cal{T}}^6 & & 295544 \\ {\cal{H}}^4 {\cal{T}}^7 & & 505320 \\ {\cal{H}}^3 {\cal{T}}^8 & & 699216 \\ {\cal{H}}^2 {\cal{T}}^9 & & 783584 \\ {\cal{H}}^1 {\cal{T}}^{10} & &728160 \\ {\cal{T}}^{11} & & 581904 \end{array}$$ \noindent The characteristic[ numbers of rational plane quartics have been calculated by H. Zeuthen in [Z]. \subsection{Cuspidal Rational Plane Curves} \label{cusp} For $d\geq 1$, let $N_d$ be the number of irreducible, nodal rational plane curves passing through $3d-1$ general points in ${\Bbb P}^2$. $N_d$ is a $\beta=0$ characteristic number. The numbers $N_d$ satisfy a beautiful recursion relation ([K-M]): $$N_1=1$$ $$\forall d\geq 2, \ \ N_d= \sum_{i+j=d,\ i,j>0} N_i N_j i^2j \Bigg( j{3d-4\choose 3i-2} - i {3d-4\choose 3i-1} \Bigg)\ \ \ .$$ The first few $N_d$'s are: $$N_1=1, \ N_2=1, \ N_3=12, \ N_4=620, \ N_5=87304, \ N_6=26312976, \ \ldots$$ As a final application, the enumerative geometry of cuspidal rational plane curves is considered. A rational plane curve, $C$, is {\em 1-cuspidal} if the singularities of $C$ consist of nodes and exactly 1 cusp. For $d\geq 3$, let $C_d$ be the number of irreducible, 1-cuspidal rational plane curves passing through $3d-2$ general points in ${\Bbb P}^2$. \begin{pr} \label{cuspr} The numbers $C_d$ can be expressed in terms of the $N_d$: $$\forall d\geq 3, \ \ C_d= {3d-3\over d} N_d \ + \ {1\over 2d}\cdot \sum_{i=1}^{d-1} {3d-2\choose 3i-1}N_iN_{d-i} \big( 3i^2(d-i)^2 - 2 di(d-i) \big) \ \ .$$ \end{pr} \noindent The first few $C_d$'s are: $$C_3=24, \ C_4=2304, \ C_5=435168, \ C_6=156153600, \ \ldots$$ $C_3$ is the degree of the locus of cuspidal cubics. $C_4$ has been computed by H. Zeuthen ([Z]). The 1-cuspidal numbers $C_d$ are evaluated by intersecting divisors on $\barr{M}_{0,0}(2,d)$. Let $d\geq 3$. Let $M_{0,0}(2,d)$ be $\barr{M}_{0,0}(2,d)$ minus the boundary. Let $Z \subset M_{0,0}(2,d)$ be the subvariety of maps that are not immersions. It is easily seen $Z$ is of pure codimension 1 and the generic element of every component corresponds to a 1-cuspidal rational plane curve. Let ${\cal{Z}}$ be the Weil divisor obtained by the closure of $Z$ in $\barr{M}_{0,0}(2,d)$. By the dimension reduction argument of section (\ref{cnum}), the intersection cycle on $\barr{M}_{0,0}(2,d)$ \begin{equation} \label{cy2} {\cal{Z}} \cap {\cal{H}}^{3d-2} \end{equation} determined by general points $P_1, \ldots, P_{3d-2}$ is of dimension (at most) 0 and lies in $Z$. A simple modification of the corresponding argument in section (\ref{cnum}) can be applied to show (\ref{cy2}) is reduced for general choices of $P_j$. Hence $C_d= {\cal{Z}}\cdot {\cal{H}}^{3d-2}$. The boundary of $\barr{M}_{0,0}(2,d)$ simply consists of the $[{d\over2}]$ Weil divisors $K^i$ ($1\leq i \leq [{d\over 2}]$). Recall $K^i$ is the boundary component corresponding to the degree partition $i+(d-i)=d$. By Lemmas (\ref{ihh}-\ref{i01}), the elements $\{{\cal{H}}\} \cup \{K^i\}$ span a basis of $Pic(\barr{M}_{0,0}(2,d)) \otimes {\Bbb{Q}}$. \begin{lm} \label{zz} The class of ${\cal{Z}}$ in $Pic(\barr{M}_{0,0}(2,d)) \otimes {\Bbb{Q}}$ is determined by ($d\geq 3$): \begin{equation} \label{laster} {\cal{Z}} = {3d-3\over d}{\cal{H}} \ + \ \sum_{i=1}^{[{d\over 2}]} {3i(d-i)-2d\over d} K^i \ . \end{equation} \end{lm} \begin{pf} Let $S$, $\mu$, $\overline{S}$, $\lambda: C \rightarrow \barr{M}_{0,0}(2,d)$ be exactly as in the proof of Lemma (\ref{tan}). It will be checked that the left and right sides of (\ref{laster}) have the same intersection with $C$. As before, a point of the intersection $C\cdot K^i$ can arise in two cases. A simple point of degree $i$ or $d-i$ can be blown-up. Let $$C\cdot K^i= x_i + y_i$$ where $x_i$ and $y_i$ are the number instances of the first and second case respectively. Let $E_{x_i}$ be the union of the $x_i$ exceptional divisors in $\overline{S}$ obtained from the $x_i$ points of $C\cdot K^i$. Let $E_{y_i}$ be defined similarly. First, the intersection $C\cdot {\cal{Z}}$ is calculated. Consider $\overline{\mu}:\overline{S} \rightarrow {\Bbb P}^2$. $\overline{\mu}^*({\cal{O}}_{{\Bbb P}}(1))$ is the element $(d,k) -\sum i E_{x_i} - \sum (d-i) E_{y_i}$. The differential map yields an injection of {\em sheaves}: \begin{equation} \label{seq} 0 \ \rightarrow\ T_{\overline{S}}\ \stackrel{d\overline{\mu}}{\rightarrow} \overline{\mu}^*(T_{{\Bbb P}^2}) \ \rightarrow\ Q\ \rightarrow \ 0. \end{equation} For general maps $\overline{\mu}$, $Q$ is line bundle supported on a nonsingular curve $D$. The restriction of the sequence (\ref{seq}) to $D$ yields an exact sequence of {\em bundles} on $D$: \begin{equation} \label{seq2} 0 \ \rightarrow \ L \ \rightarrow\ T_{\overline{S}}|_D\ \stackrel{d\overline{\mu}}{\rightarrow} \overline{\mu}^*(T_{{\Bbb P}^2})|_D \ \rightarrow\ Q|_D \ \rightarrow \ 0 \end{equation} where $L$ is a line bundle on $D$. Finally, there is exact sequence of bundles on $D$ obtained from the projection $\overline{\pi}: \overline{S} \rightarrow C$: \begin{equation} \label{seq3} 0 \ \rightarrow\ V \ \rightarrow\ T_{\overline{S}}|_D \ \stackrel{d\overline{\pi}}{\rightarrow} \ \overline{\pi}^*(T_C) \rightarrow 0 \end{equation} where $V$ is a line bundle on $D$. Maps in the family $\overline{\pi}$ have zero differential exactly at the points of intersection ${\Bbb P}(V) \cdot {\Bbb P}(L) \subset {\Bbb P}(T_{\overline{S}}|D)$. Hence $$C \cdot {\cal{Z}} = {\Bbb P}(V) \cdot {\Bbb P}(L)\ .$$ A lengthy, routine exercise in Chern classes and exact sequences now yields: $$C \cdot {\cal{Z}} = (6d-6)k + \sum_{i=1}^{[{d\over 2}]} (-3i^2+3i-2)x_i + \sum_{i=1}^{[{d\over 2}]} (-3(d-i)^2+3(d-i)-2)y_i.$$ Algebraic manipulation and the relation $$C\cdot {\cal{H}} = 2dk - \sum_{i=1}^{[{d\over 2}]} i^2 x_i - \sum_{i=1}^{[{d\over 2}]} (d-i)^2 y_i$$ yields the result. \end{pf} It remains to compute ${\cal{Z}}\cdot {\cal{H}}^{3d-2}$. By Lemma (\ref{zz}), it suffices to determine the products $K^i \cdot {\cal{H}}^{3d-2}$. If $i\neq d/2$, the result $$ K^i \cdot {\cal{H}}^{3d-2} = {3d-2 \choose 3i-1} i(d-i)N_i N_{d-i}$$ is obtained from a simple geometric argument. In case $i=d/2$, division by 2 is required to account for symmetry: $$ K^{d\over 2} \cdot {\cal{H}}^{3d-2}= {1\over 2} {3d-2\choose 3{d\over 2}-1} ({d\over 2})^2 N^2_{d\over 2}\ .$$ Evaluation of ${\cal{Z}} \cdot {\cal{H}}^{3d-2}$ yields the formula for $C_d$. The proof of Proposition (\ref{cuspr}) is complete.

9504

alg-geom/9504005

en

https://arxiv.org/abs/alg-geom/9504005

alg-geom/9504005

Carel Faber

Carel Faber

Intersection-theoretical computations on \Mgbar

13 pages, no figures. To appear in "Parameter Spaces", Banach Center Publications, volume in preparation. plain tex

We determine necessary conditions for ample divisors in arbitrary genus as well as for very ample divisors in genus 2 and 3. We also compute the intersection numbers $\lambda^9$ and $\lambda_{g-1}^3$ in genus 4. The latter number is relevant for counting curves of higher genus on manifolds, cf. the recent work of Bershadsky et al.

alg-geom

\section{Introduction.} In this paper we explore several concrete problems, all more or less related to the intersection theory of the moduli space of (stable) curves, introduced by Mumford [Mu 1]. \par In \S1 we only intersect divisors with curves. We find a collection of necessary conditions for ample divisors, but the question whether these conditions are also sufficient is very much open. \par The other sections are concerned with moduli spaces of curves of low genus, but we use the ring structure of the Chow ring. In \S\S2, 3 we find necessary conditions for very ample divisors on $\mbar2$ and $\mbar3$. \par The intersection numbers of the kappa-classes are the subject of the Witten conjecture, proven by Kontsevich. In \S4 we show how to compute these numbers for $g=3$ within the framework of algebraic geometry. \par Finally, in \S5 we compute $\lambda^9$ on $\mbar4$. This also gives the value of $\la{g-1}^3$ (for $g=4$), which is relevant for counting curves of higher genus on manifolds [BCOV]. Another corollary is a different computation of the class of the Jacobian locus in the moduli space of 4-dimensional principally polarized abelian varieties; in a sense this gives also a different proof that the Schottky locus is irreducible in dimension 4. \par {\sl Acknowledgement.\/} I would like to thank Gerard van der Geer for very useful discussions in connection with \S5. This research has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences. \section{1. Necessary conditions for ample divisors on ${\overline {\cal M}}_g$ .} Let $g\ge2$ be an integer and put $h=[g/2]$. Cornalba and Harris [C-H] determined which divisors on ${\overline {\cal M}}_g$ of the form $a\lambda-b\delta$ are ample: this is the case if and only if $a>11b>0$. Divisors of this form are numerically effective (nef) if and only if $a\ge11b\ge0$. (More generally, the ample cone is the interior of the nef cone and the nef cone is the closure of the ample cone ([Ha], p.~42)). Here $\delta=\sum_{i=0}^{h}\delta_i$ with $\delta_i=[\Delta_i]$ for $i\neq1$ and $\delta_1={1\over2}[\Delta_1]$. \par Arbarello and Cornalba [A-C] proved that the $h+2$ divisors $\lambda,\delta_0,\delta_1,\dots,\delta_h$ form for $g\ge3$ a ${\Bbb Z}$-basis of $Pic({\overline {\cal M}}_g)$ (the Picard group of the moduli functor), using the results of Harer and Mumford (we work over ${\Bbb C}$). As pointed out in [C-H] it would be interesting to determine the nef cone in $Pic({\overline {\cal M}}_g)$ for $g\ge3$. (For $g=2$ the answer is given by the result of [C-H], because of the relation $10\lambda-\delta_0-2\delta_1=0$.) \par In [Fa 1], Theorem 3.4, the author determined the nef cone for $g=3$. The answer is: $a\lambda-b_0\delta_0-b_1\delta_1$ is nef on $\mbar 3 $ if and only if $2b_0\ge b_1\ge0$ and $a-12b_0+b_1\ge0$. That a nef divisor necessarily satisfies these inequalities, follows from the existence of one-dimensional families of curves for which $({\rm deg\ }\lambda,{\rm deg\ }\delta_0,{\rm deg\ }\delta_1)$ equals $(1,12,-1)$ resp.~$(0,-2,1)$ resp.~$(0,0,-1)$. Such families are easily constructed: for the first family, take a simple elliptic pencil and attach it to a fixed one-pointed curve of genus $2$; for the second family, take a $4$-pointed rational curve with one point moving and attach a fixed two-pointed curve of genus $1$ to two of the points and identify the two other points; for the third family, take a $4$-pointed rational curve with one point moving and attach two fixed one-pointed curves of genus 1 to two of the points and identify the two other points. \par That a divisor on $\mbar 3$ satisfying the inequalities is nef, follows once we show that $\lambda$, $12\lambda-\delta_0$ and $10\lambda-\delta_0-2\delta_1$ are nef. It is well-known that $\lambda$ is nef. Using induction on the genus one shows that $12\lambda-\delta_0$ is nef: on $\mbar{1,1}$ it vanishes; for $g\ge2$, writing $12\lambda-\delta_0=\kappa_1+\sum_{i=1}^h\delta_i$ one sees that $12\lambda-\delta_0$ is positive on every one-dimensional family of curves where the generic fiber has at most nodes of type $\delta_0$; if on the other hand the generic fiber has a node of type $\delta_i$ for some $i>0$, one partially normalizes the family along a section of such nodes and uses the induction hypothesis (cf.~the proof of Proposition 3.3 in [Fa 1], which unfortunately proves the result only for $g=3$). Finally, the proof that $10\lambda-\delta_0-2\delta_1$ is nef on $\mbar 3$ is ad hoc (see the proof of Theorem 3.4 in [Fa 1]). \par All we do in this section is come up with a couple of one-dimensional families of stable curves for which we compute the degrees of the basic divisors. The naive hope is that at least some of these families are extremal (cf.~[C-H], p.~475), but the author hastens to add that there is at present very little evidence to support this. \par The method of producing families is a very simple one: we start out trying to write down all the families for which the generic fiber has $3g-4$ nodes. This turns out to be a bit complicated. However, the situation greatly simplifies as soon as one realizes that the only one-dimensional moduli spaces of stable pointed curves are $\mbar{0,4}$ and $\mbar{1,1}$: for the computation of the basic divisor classes on these families, one only needs to know the genera of the pointed curves attached to the moving $4$-pointed rational curve resp.~the moving one-pointed curve of genus $1$ as well as the types of the nodes one gets in this way. In other words, the fixed parts of the families can be taken to be general. \par We now consider the various types of families obtained in this way and compute on each family the degrees of the basic divisor classes. Each family gives a necessary condition for the divisor $a\lambda -\sum_{i=0}^hb_i\delta_i$ to be nef. In order to write this condition, it will be convenient to define $\delta_i=\delta_{g-i}$ and $b_i=b_{g-i}$ for $h<i<g$. \par A) In the case of $\mbar{1,1}$, there is very little choice: we can only attach a (general) one-pointed curve of genus $g-1$. Taking a simple elliptic pencil for the moving part, we get---as is well-known---the following degrees: ${\rm deg\ }\lambda=1$, ${\rm deg\ }\delta_0=12$, ${\rm deg\ }\delta_1=-1$ and ${\rm deg\ }\delta_i=0$ for $1<i\le h$. This gives the necessary condition $a-12b_0+b_1\ge0$. \par B) The other families are all constructed from a $4$-pointed smooth rational curve with one of the points moving and the other three fixed; when the moving point meets one of the fixed points, the curve breaks up into two 3-pointed smooth rational curves glued at one point. We have to examine the various ways of attaching general curves to this $4$-pointed rational curve. E.g., one can attach one curve, necessarily $4$-pointed and of genus $g-3$. All nodes are of type $\delta_0$ and the 3 degenerations have an extra such node. Therefore ${\rm deg\ }\delta_0=-4+3=-1$, while the other degrees are zero; one obtains the necessary condition $b_0\ge0$. \par C) Now attach a $3$-pointed curve of genus $i$ and a 1-pointed curve of genus $j\ge1$, with $i+j=g-2$. One checks ${\rm deg\ }\delta_0 =-3+3=0$ and ${\rm deg\ }\delta_j=-1$, the other degrees vanish. One obtains $b_j\ge0$ for $j\ge1$. Thus all $b_i$ are non-negative for a nef divisor. \remar{Remark.} If one uses the families above, one simplifies the proof of Theorem 1 in [A-C] a little bit. \vskip4pt plus2pt \par D) If we attach two-pointed curves of genus $i\ge1$ and $j\ge1$, with $i+j=g-2$, we find ${\rm deg\ }\delta_0=-4+2=-2$ and ${\rm deg\ }\delta_{i+1}=1$. So for $2\le k\le h$ we find the condition $2b_0-b_k\ge0$. \par E) Attaching a two-pointed curve of genus $i$ and two one-pointed curves of genus $j$ and $k$, with $i,j,k\ge1$ and $i+j+k=g-1$, we find that two of the degenerations have an extra node of type $\delta_0$ while the third has an extra node of type $\delta_{j+k}$. Therefore ${\rm deg\ }\delta_0=-2+2=0$. It is cumbersome to distinguish the various cases that occur for the other degrees, but is also unnecessary: one may simply write the resulting necessary condition in the form $b_j+b_k-b_{j+k}\ge0$, for $j,k$ with $1\le j\le k$ and $j+k\le g-2$. \par F) Attaching 4 one-pointed curves of genera $i,j,k,l\ge1$, with $i+j+k+l=g$, we get the necessary condition $b_i+b_j+b_k+b_l-b_{i+j}-b_{i+k}-b_{i+l}\ge0$. \par G) If we identify two of the 4 points to each other and attach a two-pointed curve of genus $g-2$ to the remaining two points, we obtain the necessary condition $2b_0-b_1\ge0$. \par H) As in G), but now we attach 1-pointed curves of genera $i,j\ge1$ to the remaining two points, with $i+j=g-1$. The resulting condition is $b_i+b_j-b_1\ge0$. \par The only other possibility is to identify the first with the second and the third with the fourth point. This gives a curve of genus 2, so this is irrelevant. We have proven the following theorem. \th{Theorem}{1.}{Assume $g\ge3$. A numerically effective divisor $a\lambda-\sum_{i=0}^hb_i\delta_i$ in $Pic(\mbar g)$ satisfies the following conditions: \item{a)} $a-12b_0+b_1\ge0\quad;$ \item{b)} for all $j\ge1$, $$2b_0\ge b_j\ge 0\quad;$$ \item{c)} for all $j,k$ with $1\le j\le k$ and $j+k\le g-1$, $$b_j+b_k\ge b_{j+k}\quad;$$ \item{d)} for all $i,j,k,l$ with $1\le i\le j\le k\le l$ and $i+j+k+l=g$, $$b_i+b_j+b_k+b_l\ge b_{i+j}+b_{i+k}+b_{i+l}\quad.$$} Here $b_i=b_{g-i}$ for $h<i<g$, as before. The conditions in the theorem are somewhat redundant. E.g., it is easy to see that condition (c) implies the non-negativity of the $b_i$ with $i\ge1$. \par As we have seen, the conditions in the theorem are sufficient for $g=3$. The proof proceeded by determining the extremal rays of the cone defined by the inequalities and analyzing the (three) extremal rays separately. It may therefore be of some interest to find (generators for) the extremal rays of the cone in the theorem. We have done this for low genus: $$ \eqalign{&g=4:\qquad \cases{\lambda\cr 12\lambda-\delta_0\cr 10\lambda-\delta_0-2\delta_1\cr 10\lambda-\delta_0-2\delta_1-2\delta_2\cr 21\lambda-2\delta_0-3\delta_1-4\delta_2\cr} \cr &g=5:\qquad\cases{\lambda\cr 12\lambda-\delta_0\cr 10\lambda-\delta_0-2\delta_1-\delta_2\cr 10\lambda-\delta_0-2\delta_1-2\delta_2\cr 32\lambda-3\delta_0-4\delta_1-6\delta_2\cr} \cr &g=6:\qquad\cases{\lambda\cr 12\lambda-\delta_0\cr 10\lambda-\delta_0-2\delta_1-2\delta_2\cr 10\lambda-\delta_0-2\delta_1-2\delta_3\cr 10\lambda-\delta_0-2\delta_1-2\delta_2-2\delta_3\cr 32\lambda-3\delta_0-4\delta_1-6\delta_2-6\delta_3\cr 98\lambda-9\delta_0-10\delta_1-16\delta_2-18\delta_3\cr} \cr}$$ \par Unfortunately, we have not been able to discover a general pattern. (There are 10 extremal divisors for $g=7$, 20 extremal divisors for $g=8$ and 21 extremal divisors for $g=9$.) It is easy to see that $\lambda$, $12\lambda-\delta_0$ and $10\lambda-2\delta+\delta_0$ are extremal in every genus. It should be interesting to know the answer to the following question. \th{Question.}{}{\item{a)} Is $10\lambda-2\delta+\delta_0$ nef for all $g\ge4$? \item{b)} Are the conditions in the theorem sufficient??} Note that an affirmative answer to the first question implies the result of [C-H] mentioned above, since $12\lambda-\delta_0$ is nef. Note also that a divisor satisfying the conditions in the theorem is non-negative on every one-dimensional family of curves whose general member is smooth. This follows easily from [C-H, (4.4) and Prop.~(4.7)]. (I would like to thank Maurizio Cornalba for reminding me of these results.) \section{2. Necessary conditions for very ample divisors on $\mbar2$ .} We know which divisors on $\mbar2$ are ample: it is easy to see that $\lambda$ and $\delta_1$ form a ${\Bbb Z}$-basis of the functorial Picard group $Pic(\mbar2)$; then $a\lambda+b\delta_1$ is ample if and only if $a>b>0$, as follows from the relation $10\lambda=\delta_0+2\delta_1$ and the fact that $\lambda$ and $12\lambda-\delta_0$ are nef. \par Therefore it might be worthwhile to study which divisors are very ample on the {\it space\/} $\mbar2$. Suppose that $D=a\lambda+b\delta_1$ is a very ample divisor. Then for every $k$-dimensional subvariety $V$ of $\mbar2$ the intersection product $D^k\cdot[V]$ is a positive integer, the degree of $[V]$ in the embedding of $\mbar2$ determined by $|D|$. We work this out for the subvarieties that we know; we use Mumford's computation [Mu~1] of the Chow ring (with ${\Bbb Q}$-coefficients) of $\mbar 2$. The result may be formulated as follows: $$ A^*(\mbar2)={\Bbb Q}[\lambda,\delta_1]/(\lambda(\lambda+\delta_1),\lambda^2 (5\lambda-\delta_1)). $$ The other piece of information we need is on p.~324 of [Mu 1]: $\lambda^3={1\over2880}p$. However, one should realize that the identity element in $A^*(\mbar2)$ is $[\mbar2]_Q={1\over2}[\mbar2]$, which means that $$\lambda^3\cdot[\mbar2]={1\over1440}\quad.$$ Therefore $$D^3\cdot[\mbar2]={{a^3+15a^2b-15ab^2+5b^3}\over1440}\quad.$$ One of the requirements is therefore that the integers $a$ and $b$ are such that the expression above is an integer. It is not hard to see that this is the case if and only if $$60|a\qquad\hbox{and}\qquad12|b\quad.$$ It turns out that these conditions imply that $D^2\cdot[\Delta_0]$ and $D^2\cdot[\Delta_1]$ are integers. Also $D^2\cdot4\lambda$ is an integer, but $D^2\cdot2\lambda$ is an integer if and only if $8|(a+b)$. Therefore, if for some integer $k$ the class $(4k+2)\lambda$ is the fundamental class of an effective $2$-cycle, then a very ample $D$ satisfies $8|(a+b)$. We don't know whether such a $k$ exists; clearly, $20\lambda= [\Delta_0]+[\Delta_1]$ is effective; the fundamental class of the bi-elliptic divisor turns out to be $60\lambda+3\Delta_1$. \par Turning next to one-dimensional subvarieties, the conditions $60|a$ and $12|b$ imply that $D\cdot[\Delta_{00}]$ and $D\cdot[\Delta_{01}]$ are integers as well. \th{Proposition}{2.}{A very ample divisor $a\lambda+b\delta_1$ on the moduli space $\mbar2$ satisfies the following conditions: \item{a)} $a,b\in{\Bbb Z}$ and $a>b>0\quad;$ \item{b)} $60|a$ and $12|b\quad.$} \th{Corollary}{3.}{The degree of a projective embedding of $\mbar2$ is at least $516$.} \Proof We need to determine for which $a$ and $b$ satisfying the conditions in the proposition the expression $5(b-a)^3+6a^3$ attains its minimum value. Clearly this happens exactly for $b=12$ and $a=60$. If $60\lambda+12\delta_1$ is very ample, the degree of $\mbar2$ in the corresponding embedding is $(5(b-a)^3+6a^3)/1440=516$. \vskip4pt plus2pt \remar{Remark.} It is interesting to compare the obtained necessary conditions with the explicit descriptions of $\mbar2$ given by Qing Liu ([Liu]). The computations we have done (in characteristic $0$) indicate that $60\lambda+60\delta_1$ maps $\mbar2$ to a copy of $\fam\frakfam\tenfrak X$ (loc.~cit., Th\'eor\`eme 2), that $60\lambda+36\delta_1$ maps $\mbar2$ to the blowing-up of $\fam\frakfam\tenfrak X$ with center ${\cal J}_{{\Bbb Q}}$ (loc.~cit., Corollaire 3.1) and that $60\lambda+48\delta_1$ is very ample, realizing $\mbar2$ as the blowing-up of $\fam\frakfam\tenfrak X$ with center the ideal generated by $I_4^3$, $J_{10}$, $H_6^2$ and $I_4^2H_6$ (loc.~cit., Corollaire 3.2). \vskip4pt plus2pt \section{3. Necessary conditions for very ample divisors on $\mbar3$ .} In this section we compute necessary conditions for very ample divisors on the moduli space $\mbar3$. As we mentioned in \S1, a divisor $D=a\lambda-b\d0-c\d1\in Pic(\mbar3)$ with $a,b,c\in{\Bbb Z}$ is ample if and only if $a-12b+c>0$ and $2b>c>0$. The necessary conditions for very ample $D$ are obtained as in \S2: for a $k$-dimensional subvariety $V$ of $\mbar3$, the intersection product $D^k\cdot[V]$ should be an integer. We use the computation of the Chow ring of $\mbar3$ in [Fa 1]. The computations are more involved than in the case of genus 2; also, we know the fundamental classes of more subvarieties. \par First we look at the degree of $\mbar3$: $$ \eqalign{ D^6=(a\lambda-b\d0-c\d1)^6 &=\textstyle{1\over90720}a^6-{1\over576}a^4c^2-{1\over18}a^3b^3 +{1\over48}a^3bc^2+{35\over3456}a^3c^3\cr &\qquad\textstyle +{5\over8}a^2b^2c^2-{43\over96}a^2bc^3+{13\over512}a^2c^4 +{203\over20}ab^5-{145\over12}ab^3c^2\cr &\qquad\textstyle +{25\over4}ab^2c^3-{31\over48}abc^4+{149\over7680}ac^5 -{4103\over72}b^6+{55}b^4c^2 \cr &\qquad\textstyle -{505\over18}b^3c^3+{65\over16}b^2c^4-{91\over384}bc^5+{5\over1024}c^6 \ , \cr} $$ as follows from [Fa 1], p.~418. The requirement that this is in ${\Bbb Z}_2$ implies, firstly, that $2|c$, secondly, that $2|a$ {\sl and\/} $4|c$, thirdly, that $2|b$. Looking in ${\Bbb Q}_3$ we get, firstly, that $3|a$, secondly, that $3|b$. Modulo 5 we get $5|a$ or $5|(a+3b+c)$. Finally, working modulo 7 we find that $7|a$ should hold. \par Writing $a=42a_1$, $b=6b_1$ and $c=4c_1$, with $a_1,b_1,c_1\in{\Bbb Z}$, the condition $D^6\cdot[\mbar3]\in{\Bbb Z}$ becomes $5|a_1$ or $5|(3a_1+2b_1+c_1)$. Interestingly, unlike the case of genus 2, these conditions are not the only necessary conditions we find. \par For instance, the condition $D^5\cdot\d0\in{\Bbb Z}$ translates in $3|c_1$; then $[\Delta_1]=2\d1$ gives no further conditions; but the hyperelliptic locus, with fundamental class $[{\cal H}_3]= 18\lambda-2\d0-6\d1$, improves the situation modulo 5: necessarily $5|(3a_1+2b_1+c_1)$. It follows that $D^5\cdot\lambda$ is an integer, so all divisors have integer-valued degrees. \par In codimension 2, writing $c_1=3c_2$ with $c_2\in{\Bbb Z}$, the condition $D^4\cdot[\Delta_{01a}]\in{\Bbb Z}$ translates in $$ 5|a_1\qquad\hbox{or}\qquad5|c_2\qquad\hbox{or}\qquad5|(a_1+c_2)\qquad\hbox{or} \qquad5|(a_1+3c_2)\ . $$ The (boundary) classes $[\D{00}]$, $[\D{01b}]$, $[\D{11}]$, $[\Xi_0]$, $[\Xi_1]$ and $[{\rm H}_1]$ ([Fa 1], pp.~340 sqq.) give no further conditions. \par In codimension 3, the class $[(i)]=8[(i)]_Q$ forces $2|a_1$. Write $a_1=2a_2$ with $a_2\in{\Bbb Z}$. Somewhat surprisingly, the class $[{\rm H}_{01a}]=4\eta_0$ (loc.~cit., pp.~386, 388) gives the condition $5|(a_2+2c_2)$. Consequently, combining the various conditions modulo 5, we obtain $$ 5|a_2\qquad\hbox{and}\qquad5|b_1\qquad\hbox{and}\qquad5|c_2\ . $$ Finally, we checked that the 12 cycles in codimension 4 and the 8 cycles in codimension 5 (loc.~cit., pp.~346 sq.) don't give extra conditions. \th{Proposition}{4.}{A very ample divisor $a\lambda-b\d0-c\d1$ on the moduli space $\mbar3$ satisfies the following conditions: \item{a)} $a,b,c\in{\Bbb Z}$ with $a-12b+c>0$ and $2b>c>0\quad;$ \item{b)} $420|a$ and $30|b$ and $60|c\quad.$} \th{Corollary}{5.} {The degree of a projective embedding of $\mbar3$ is at least $$ 650924662500=2^2\cdot3^2\cdot5^5\cdot7\cdot826571. $$} \Proof We need to minimize the expression given for the degree of $\mbar3$ while fulfilling the conditions in the proposition. Write $a=420A$, $b=30B$ and $c=60C$. One shows that in the cone given by $7A-6B+C\ge0$ and $B\ge C\ge0$ the degree is minimal along the (extremal) ray $(A,B,C)=(5x,7x,7x)$ (corresponding to $10\lambda-\d0-2\d1$). Comparing the value for $(A,B,C)=(5,7,7)$ with that for $(A,B,C)=(2,2,1)$, one concludes $A\le5$, $B\le7$ and $C\le7$. This leaves only a few triples in the interior of the cone; the minimum degree is obtained for $(A,B,C)=(2,2,1)$, corresponding to $840\lambda-60\d{}$. \vskip4pt plus2pt \remar{Remark.} In [Fa 1], Questions 5.3 and 5.4, we asked whether the classes $X$ (resp.~$Y$) are multiples of classes of complete subvarieties of $\mbar3$ of dimension 4 (resp.~3) having empty intersection with $\D1$ (resp.~$\D0$). We still don't know the answers, but we verified that $X$ and $-Y=504\la3$ are effective: $$ \eqalign{ X&=\textstyle{1\over15}\d{00}+{1\over6}\d{01a}+{11\over15}\d{01b} +8\d{11}+{3\over14}\xi_0+{48\over35}\xi_1+{40\over21}\eta_1\quad;\cr -Y&=\textstyle{1\over2}[(a)]_Q+[(b)]_Q+[(c)]_Q+{11\over30}[(d)]_Q +{2\over5}[(f)]_Q+2[(g)]_Q+{2\over3}\eta_0\quad.\cr } $$ (For the notations, see [Fa 1], pp.~343, 386, 388.) \vskip4pt plus2pt \section{4. Algebro-geometric calculation of the intersection numbers of the tautological classes on $\mbar3$ .} Here we show how to compute the intersection numbers of the classes $\ka{i}$ $(1\le i\le6)$ on $\mbar3$ in an algebro-geometric setting. These calculations were done originally in May 1990 to check the genus 3 case of Witten's conjecture [Wi], now proven by Kontsevich [Ko]. We believe that there is still interest, though, in finding methods within algebraic geometry that allow to compute the intersection numbers of the kappa- or tau-classes. For instance, the identity $$K^{3g-2}=\langle\tau_{3g-2}\rangle=\langle\ka{3g-3}\rangle ={1\over(24)^g\cdot g!} $$ (in cohomology) should be understood ([Wi], between (2.26) and (2.27)). \par In [Fa 1] the 4 intersection numbers of $\ka1$ and $\ka2$ were computed; using the identity $\ka1=12\lambda-\d0-\d1$, we can read these off from Table 10 on p.~418: $$ \textstyle \ka1^6={176557\over107520}\quad,\quad\ka1^4\ka2={75899\over322560}\quad, \quad\ka1^2\ka2^2={32941\over967680}\quad,\quad\ka2^3={14507\over2903040}\quad. $$ To compute the other intersection numbers, we need to express the other kappa-classes in terms of the bases introduced in [Fa 1]. The set-up is as in [Mu 1], \S8 (and \S6): if $C$ is a stable curve of genus 3, $\omega_C$ is generated by its global sections, unless \item{a)} $C$ has 1 or 2 nodes of type $\d1$, in which case the global sections generate the subsheaf of $\omega_C$ vanishing in these nodes; \item{b)} $C$ has 3 nodes of type $\d1$, i.e., $C$ is a ${\Bbb P}^1$ with 3 (possibly singular) elliptic tails, in which case $\Gamma(\omega_C)$ generates the subsheaf of $\omega_C$ of sections vanishing on the ${\Bbb P}^1$. \noindent (See [Mu 1], p.~308.) Let $Z\subset{\overline {\cal C}}_3$ be the closure of the locus of pointed curves with 3 nodes of type $\d1$ and with the point lying on the ${\Bbb P}^1$. Working over ${\overline {\cal C}}_3-Z$ we get $$ 0\to{\cal F}\to\pi^*\pi_* \omega_{{\overline {\cal C}}_3/\mbar3}\to I_{\D1^*}\cdot \omega_{{\overline {\cal C}}_3/\mbar3}\to0 $$ with ${\cal F}$ locally free of rank 2. Working this out as in [Fa 1], p.~367 we get $$ \eqalign{ 0=c_3({\cal F})&=\pi^*\la3-K\cdot\pi^*\la2+K^2\cdot\pi^*\la1-K^3\cr &\qquad-(\pi^*\la1-K)\cdot[\D1^*]_Q+i_{1,*}(K_1+K_2) \cr} $$ modulo $[Z]$. Multiplying this with $K$ and using that $\omega^2$ is trivial on $[\D1^*]$, we get $$0=K\cdot c_3({\cal F})=K\cdot\pi^*\la3-K^2\cdot\pi^*\la2 +K^3\cdot\pi^*\la1-K^4+*K\cdot[Z]\quad.\leqno(1) $$ It is easy to see that $K^2\cdot[Z]=0$, so we also get $$ \leqalignno{ 0&=K^2\cdot\pi^*\la3-K^3\cdot\pi^*\la2 +K^4\cdot\pi^*\la1-K^5\quad,&(2)\cr 0&=K^3\cdot\pi^*\la3-K^4\cdot\pi^*\la2 +K^5\cdot\pi^*\la1-K^6\quad,&(3)\cr 0&=K^4\cdot\pi^*\la3-K^5\cdot\pi^*\la2 +K^6\cdot\pi^*\la1-K^7\quad.&(4)\cr } $$ Pushing-down to $\mbar3$ we get $$ \leqalignno{ 0&=4\la3-\ka1\la2+\ka2\la1-\ka3+N\cdot[(i)]_Q\quad,&(1')\cr 0&=\ka1\la3-\ka2\la2+\ka3\la1-\ka4\quad,&(2')\cr 0&=\ka2\la3-\ka3\la2+\ka4\la1-\ka5\quad,&(3')\cr 0&=\ka3\la3-\ka4\la2+\ka5\la1-\ka6\quad.&(4')\cr } $$ To get $\ka3$ from $(1')$ we use two things. Firstly, one computes $Y=-504\la3$, as mentioned at the end of \S3. This follows since both $Y$ and $\la3$ are in the one-dimensional subspace of $A^3(\mbar3)$ of classes vanishing on all subvarieties of $\D0$. The factor $-504$ is computed using $\lambda^4=8\lambda\la3$ or $\la3\cdot[(i)]_Q={1\over6}\lambda^3 \cdot[(i)]_Q$. Secondly, to compute $N$, one uses that $\ka3$ vanishes on the classes $[(b)]_Q$, $[(c)]_Q$, $[(f)]_Q$, $[(g)]_Q$, $[(h)]_Q$ and $[(i)]_Q$. This gives 6 relations in $N$ of which 3 are identically zero; the other 3 all imply $N=1$. \par The formulas above allow one to express the kappa-classes in terms of the bases of the Chow groups given in [Fa 1]. We give the formula for $\ka3$ (from which the other formulas follow): $$\eqalign{\textstyle \ka3&\textstyle={1\over280}[(a)]_Q+{31\over840}[(b)]_Q+{19\over420}[(c)]_Q +{1\over1260}[(d)]_Q+{1\over35}[(e)]_Q\cr &\textstyle\qquad+{19\over840}[(f)]_Q+{29\over84}[(g)]_Q +{11\over35}[(h)]_Q+{93\over35}[(i)]_Q+{11\over252}\eta_0\quad. \cr } $$ This gives the following intersection numbers: $$ \displaylines{ \textstyle \ka1^3\ka3={4073\over161280}\quad,\quad\ka1\ka2\ka3={149\over40320}\quad,\quad \ka3^2={131\over322560}\quad,\quad\ka1^2\ka4={2173\over967680}\quad,\cr \textstyle \ka2\ka4={971\over2903040}\quad,\quad\ka1\ka5={1\over5760}\quad,\quad \ka6={1\over82944} \quad.\cr } $$ \section{5. A few intersection numbers in genus 4.} Kontsevich's proof of Witten's conjecture enables one to compute the intersection numbers of the kappa-classes on the moduli space of stable curves of arbitrary genus. There are many more intersection numbers that one would like to know, see e.g.~[BCOV], (5.54) and end of Appendix A. As a challenge, we pose the following problem: \th{Problem.}{}{Find an algorithm that computes the intersection numbers of the divisor classes $\lambda,\d0,\d1,\dots,\d{[g/2]}$ on $\mbar g$.} These numbers are known for $g=2$ [Mu 1] and $g=3$ [Fa 1]. Note that the problem includes the computation of $\ka1^{3g-3}$. \th{Proposition}{6.}{Denote by $h_g$ the intersection number $\lambda^{2g-1}\cdot[{\overline {\cal H}}_g]_Q$ , where ${\overline {\cal H}}_g$ is the closure in $\mbar g$ of the hyperelliptic locus. Then $$\eqalign{ h_1&= {1\over96}\quad;\cr h_g&= {2\over{2g+1}}\sum_{i=1}^{g-1}i(i+1)(g-i)(g-i+1){{2g-2}\choose{2i-1}} h_ih_{g-i}\qquad\hbox{for}\qquad g\ge2. \cr } $$} \Proof This follows from [C-H], Proposition 4.7, which expresses $\lambda$ on ${\overline {\cal H}}_g$ in terms of the classes of the components of the boundary ${\overline {\cal H}}_g-{\cal H}_g$. It is easy to see that $\lambda^{2g-2}\xi_i=0$ for $0\le i\le[(g-1)/2]$. Also, $$ \lambda^{2g-2}\d{j}[{\overline {\cal H}}_g]_Q =(2j+2)(2g-2j+2){{2g-2}\choose{2j-1}}h_jh_{g-j}\quad, $$ because $\lambda=\pi_j^*\lambda+\pi_{g-j}^*\lambda$ on $\D{j}\cap{\overline {\cal H}}_g$. Normalizing $h_1$ to ${1\over96}$, which reflects the identity $\lambda={1\over24}p$ on $\mbar{1,1}$ and the fact that an elliptic curve has four 2-torsion points, we get the formula. \vskip4pt plus2pt \par This gives for instance $h_2={1\over2880}$, $h_3={1\over10080}$ and $h_4={31\over362880}$. So this already gives the value of $\lambda^3$ on $\mbar2$, and the value of $\lambda^6$ on $\mbar3$ follows very easily: we only need that $[{\cal H}_3]_Q=9\lambda$ in $A^1({\cal M}_3)$, because clearly $\lambda^5\d0=\lambda^5\d1=0$. We get $\lambda^6={1\over90720}$. \th{Proposition}{7.}{$\lambda^9={1\over113400}$ on $\mbar4$ .} \Proof We need to know the class $[{\overline {\cal H}}_4]$ modulo the kernel in $A^2(\mbar4)$ of multiplication with $\lambda^7$. We computed this class using the test surfaces of [Fa 2]; of the 14 classes at the bottom of p.~432, only $\ka2$, $\lambda^2$ and $\d1^2$ are not in the kernel of $\cdot\lambda^7$, and the result is: $$ [{\overline {\cal H}}_4]\equiv\textstyle3\ka2-15\lambda^2+{27\over5}\d1^2 \pmod{\ker(\cdot\lambda^7)}. $$ We also have the relation ([Fa 2], p.~440) $$ 60\ka2-810\lambda^2+24\d1^2\equiv0\pmod{\ker(\cdot\lambda^7)}. $$ Thus $[{\overline {\cal H}}_4]\equiv{51\over2}\lambda^2+{21\over5}\d1^2$. We compute $$\eqalign{\textstyle \lambda^7\d1^2 &=\textstyle{7\choose1}(\lambda\cdot[\mbar{1,1}]_Q) (\lambda^6\cdot(-K_{\mbar{3,1}/\mbar3})\cdot[\mbar{3,1}])\cr &=\textstyle7\cdot{1\over24}\cdot{-4\over90720}\cr &=\textstyle{-1\over77760}\quad. } $$ Therefore $$ \lambda^9=\textstyle{2\over51}(2\cdot{31\over362880} +{21\over5}\cdot{1\over77760})={1\over113400}\quad. $$ Also $$ \lambda^7\ka2=\textstyle{169\over1360800}\quad. $$ The hardest part of this proof is the computation of (three of) the coefficients of the class $[{\overline {\cal H}}_4]$. We present the test surfaces we need to compute these coefficients. Write $$\eqalign{ [{\overline {\cal H}}_4]&=3\ka2-15\lambda^2+c\lambda\d0+d\lambda\d1+e\d0^2+f\d0\d1 +g\d0\d2\cr &\qquad+h\d1^2+i\d1\d2+j\d2^2+k\d{00}+l\d{01a}+m\gamma_1+n\d{11}\quad.\cr } $$ The class $[{\cal H}_4]\in A^2({\cal M}_4)$ was computed by Mumford ([Mu 1], p.~314). \item{a)} Take test surface $(\alpha)$ from [Fa 2], p.~433: two curves of genus 2 attached in one point; on both curves the point varies. We have $[{\overline {\cal H}}_4]_Q=6\cdot6=36$ and $\d2^2=8$. Thus $j=9$. \item{b)} Test surface $(\zeta)$: curves of type $\d{12}$, vary the elliptic tail and the point on the curve of genus 2. We have $[{\overline {\cal H}}_4]=0$, $\d0\d2=-24$ and $\d1\d2=2$. Thus $i=12g$. \item{c)} Test surface $(\mu)$: curves of type $\d{02}$, vary the elliptic curve in a simple pencil with 3 disjoint sections and vary the point on the curve of genus 2. Then $\d0\d2=-20$ and $\d2^2=4$. To compute $[{\overline {\cal H}}_4]$ we use a trick. Consider the pencil of curves of genus 3 which we get by replacing the one-pointed curve of genus 2 with a fixed one-pointed curve of genus 1. On that pencil $\lambda=1$, $\d0=12-1-1=10$, $\d1=-1$, thus $[{\overline {\cal H}}_3]_Q =9\lambda-\d0-3\d1=2$. So on the test surface we get $[{\overline {\cal H}}_4]_Q=2\cdot6=12$. Therefore $-20g+36=24$ so $g={3\over5}$ and $i={36\over5}$. \item{d)} This test surface is taken from [Fa 3], pp.~72 sq. We take the universal curve over a pencil of curves of genus 2 as in [A-C], p.~155, and we attach a fixed one-pointed curve of genus 2. As in [Fa 3] we have $\lambda=3(G-\Sigma)$, $\d0=30(G-\Sigma)$, $\d2=-2G+\Sigma$. Since $G^2=2$, $G\Sigma=0$ and $\Sigma^2=-2$ we have $\d0\d2=-60$ and $\d2^2=6$. To compute $\ka2$ we use the same trick as above: replacing the fixed one-pointed curve of genus 2 by one of genus 1, we get a test surface of curves of genus 3. This will not affect the computation of $\ka2$; using the formulas of [Fa 1] we find $\ka2=6$. Also $\d0\Sigma=2\gamma_1$ here, thus $\gamma_1=30$. Since $[{\overline {\cal H}}_4]=0$, we get $0=18-60g+6j+30m=30m+36$ so $m=-{6\over5}$. \item{e)} Test surface $(\lambda)$ from [Fa 2]: curves of type $\d{12}$, vary both the $j$-invariant of the middle elliptic curve and the (second) point on it. We have $\d0\d2=-12$, $\d1\d2=1$, $\d2^2=1$, $\ka2=1$, $\d{01a}=12$ and $\gamma_1=12$. Since $[{\overline {\cal H}}_4]=0$, we get $0=3-12g+i+j+12l+12m=12l-{12\over5}$ so $l={1\over5}$. \item{f)} Test surface $(\kappa)$: curves of type $\d{12}$, vary a point on the middle elliptic curve and vary the elliptic tail. Then $\d0\d2=-12$, $\d1\d2=1$, $\d{01a}=-12$, $\d{11}=-1$. Since $[{\overline {\cal H}}_4]=0$, we find $0=-12g+i-12l-n$ so $n=-{12\over5}$. \item{g)} The final test surface we need is $(\gamma)$ from [Fa 2]: we attach fixed elliptic tails to two varying points on a curve of genus 2. Then $\d1^2=16$, $\d2^2=-2$, $\ka2=2$, $\d{11}=6$. When the two varying points are distinct Weierstrass points, we get hyperelliptic curves. So $[{\overline {\cal H}}_4]_Q=6\cdot5=30$ and we get $60=6+16h-2j+6n=16h-{132\over5}$ so $h={27\over5}$, as claimed. \noindent This finishes the proof of Proposition 7. \vskip4pt plus2pt We can now evaluate the contribution from the constant maps for $g=4$ (cf.~[BCOV], \S5.13, (5.54)): \th{Corollary}{8.}{$\la3^3={1\over43545600}$ on $\mbar4$.} \Proof As explained in [Mu 1], \S5, we have on $\mbar4$ the identity $$ (1+\la1+\la2+\la3+\la4)(1-\la1+\la2-\la3+\la4)=1\quad.\leqno(*) $$ One checks that this implies $\la3^3={1\over384}\la1^9$, which finishes the proof. \vskip4pt plus2pt \th{Corollary}{9 (Schottky, Igusa).}{The class of ${\cal M}_4$ in ${\cal A}_4$ equals $8\lambda$.} \Proof Since $(*)$ holds also on the toroidal compactification $\widetilde{{\cal A}}_4$, we get $\la1^{10}=384\la1\la3^3=768\la1\la2\la3\la4$. But it follows from Hirzebruch's proportionality theorem [Hi 1, 2] that $$\la1\la2\la3\la4=\prod_{i=1}^4{{|B_{2i}|}\over{4i}}= {1\over1393459200}\quad, $$ hence $\lambda^{10}={1\over1814400}$ on $\widetilde{{\cal A}}_4$. Using Theorem 1.5 in [Mu 2] we see that the class of ${\cal M}_4$ in ${\cal A}_4$ is a multiple of $\lambda$. Denote by $t:{\cal M}_4\to{\cal A}_4$ the Torelli morphism and denote by ${\cal J}_4$ its image, the locus of Jacobians. Proposition 7 tells us that $t^*\lambda^9={1\over113400}$. Applying $t_*$ we get $[{\cal J}_4]\cdot\lambda^9={1\over113400}$, hence $[{\cal J}_4]=16\lambda$, hence $[{\cal J}_4]_Q=8\lambda$, as claimed. (The subtlety corresponding to the fact that a general curve of genus $g\ge3$ has only the trivial automorphism, while its Jacobian has two automorphisms, appears also in computing $\lambda^6$ on $\mbar3$ resp.~on $\widetilde{{\cal A}}_3$: we saw already that $t^*\lambda^6={1\over90720}$; applying $t_*$ we get $[{\cal J}_3]\cdot\lambda^6={1\over90720}$; since $[{\cal J}_3]=2[\widetilde{{\cal A}}_3]_Q$, we get $\lambda^6={1\over181440}$, which is also what one gets using the proportionality theorem.) \vskip4pt plus2pt \ref{BCOV}{ \item{[A-C]} E.~Arbarello and M.~Cornalba, {\it The Picard groups of the moduli spaces of curves}\/, Topology 26 (1987), 153--171. \item{[BCOV]} M.~Bershadsky, S.~Cecotti, H.~Ooguri and C.~Vafa, {\it Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes}\/, Commun.~Math.~Phys. 165 (1994), 311--428.\hfill \item{[C-H]} M.~Cornalba and J.~Harris, {\it Divisor classes associated to families of stable varieties, with applications to the moduli space of curves}\/, Ann.~scient.~\'Ec.~Norm.~Sup. 21 (1988), 455--475.\hfill \item{[Fa 1]} C.~Faber, {\it Chow rings of moduli spaces of curves I : The Chow ring of {${\overline {\cal M}}_3$\ }}\/, Ann.~of Math. 132 (1990), 331--419.\hfill \item{[Fa 2]} C.~Faber, {\it Chow rings of moduli spaces of curves II : Some results on the Chow ring of {${\overline {\cal M}}_4$\ }}\/, Ann.~of Math. 132 (1990), 421--449.\hfill \item{[Fa 3]} C.~Faber, {\it Some results on the codimension-two Chow group of the moduli space of curves}\/, in: Algebraic Curves and Projective Geometry (eds E.~Ballico and C.~Cili\-berto), Lecture Notes in Mathematics 1389, Springer, 66--75. \item{[Ha]} R.~Hartshorne, {\it Ample Subvarieties of Algebraic Varieties}\/, Lecture Notes in Mathematics 156, Springer.\hfill \item{[Hi 1]} F.~Hirzebruch, {\it Automorphe Formen und der Satz von Riemann-Roch}\/, Symposium Internacional de Topolog\'\i a Algebraica (M\'exico 1956), 129--144 = Ges.~Abh., Band I, 345--360.\hfill \item{[Hi 2]} F.~Hirzebruch, {\it Characteristic numbers of homogeneous domains}\/, Seminars on analytic functions, vol.~II, IAS, Princeton 1957, 92--104 = Ges.~Abh., Band I, 361--366.\hfill \item{[Ko]} M.~Kontsevich, {\it Intersection Theory on the Moduli Space of Curves and the Matrix Airy Function}\/, Commun.~Math.~Phys. 147 (1992), 1--23.\hfill \item{[Liu]} Qing Liu, {\it Courbes stables de genre 2 et leur sch\'ema de modules}\/, Math.~Ann. 295 (1993), 201--222.\hfill \item{[Mu 1]} D.~Mumford, {\it Towards an enumerative geometry of the moduli space of curves}\/, in: Arithmetic and Geometry II (eds M.~Artin and J.~Tate), Progress in Math. 36 (1983), Birkh\"auser, 271--328. \item{[Mu 2]} D.~Mumford, {\it On the Kodaira Dimension of the Siegel Modular Variety}\/, in: Algebraic Geometry---Open Problems (eds C.~Ciliberto, F.~Ghione and F.~Orecchia), Lecture Notes in Mathematics 997, Springer, 348--375. \item{[Wi]} E.~Witten, {\it Two dimensional gravity and intersection theory on moduli space}\/, Surveys in Diff.~Geom. 1 (1991), 243--310. } \end

9508

alg-geom/9508008

en

https://arxiv.org/abs/alg-geom/9508008

alg-geom/9508008

Alexander Goncharov

A.B. Goncharov, A.M. Levin

Zagier's conjecture on $L(E,2)$

this is the final version of our paper LaTeX

In this paper we introduce an elliptic analog of the Bloch-Suslin complex and prove that it (essentially) computes the weight two parts of the groups $K_2(E)$ and $K_1(E)$ for an elliptic curve $E$ over an arbitrary field $k$. Combining this with the results of Bloch and Beilinson we proved Zagier's conjecture on $L(E,2)$ for modular elliptic curves over $\Bbb Q$.

alg-geom

\section{Introduction} {\bf Summery}. In this paper we introduce an elliptic analog of the Bloch-Suslin complex and prove that it (essentially) computes the weight two parts of the groups $K_2(E)$ and $K_1(E)$ for an elliptic curve $E$ over an arbitrary field $k$. Combining this with the results of Bloch and Beilinson we proved Zagier's conjecture on $L(E,2)$ for modular elliptic curves over $\Bbb Q$. {\bf 1. The elliptic dilogarithm}. The dilogarithm is the following multivalued analytic function of on $\Bbb CP^1 \backslash \{0,1,\infty\})$: $$ Li_2(z) = -\int_0^z\log(1-t)\frac{dt}{t} $$ It has a single-valued version, the Bloch-Wigner function: $$ {\cal L}_2(z):= Im Li_2(z) + \arg(1-z)\cdot \log|z| $$ The elliptic analog of the dilogarithm was defined and studied by Spencer Bloch in his seminal paper [Bl1]. The story goes as follows. Let $E(\Bbb C) = \Bbb C^{\ast}/q^{\Bbb Z}$ be the complex points of an elliptic curve $E$. Here $q:= exp(2\pi i\tau ), Im \tau >0$. The function ${\cal L}_2(z)$ has a singularity of type $|z|\log|z|$ near $z=0$. It satisfies the relation ${\cal L}_2(z) = - {\cal L}_2(z^{-1})$. So averaging ${\cal L}_2(z)$ over the action of the group $\Bbb Z$ on $\Bbb C^{\ast}$ generated by $z \longmapsto qz$ we get the convergent series: $$ {\cal L}_{2,q}(z) := \sum_{n\in \Bbb Z} {\cal L}_2(q^nz),\qquad {\cal L}_{2,q}(z^{-1}) = -{\cal L}_{2,q}(z) $$ This function can be extended by linearity to the set of all divisors on $E(\Bbb C )$ setting ${\cal L}_{2,q}(P) := \sum_i n_i{\cal L}_{2,q}(P_i)$ for a divisor $P= \sum n_i (P_i)$. {\bf 2. The results on $L(E,2)$}. Let $L(E,s) = L(h^1(E),s)$ be the Hasse-Weil $L$-function of an elliptic curve $E$ over $\Bbb Q$. We will always suppose that an elliptic curve $E$ has at least one point over $\Bbb Q$: zero for the addition law. Let $v$ be a valuation of a number field $K$, and $h_v$ the corresponding canonical local height on $E(K)$. As usual $x \sim_{\Bbb Q^{\ast}} y$ means that $x = qy$ for a certain $q \in \Bbb Q^{\ast}$. Let $J = J(E)$ be the Jacobian of $E$. \begin{theorem} \label{zcc} Let $E$ be a modular elliptic curve over $\Bbb Q$. Then there exists a $\Bbb Q$-rational divisor $P = \sum n_j (P_j)$ over $\bar \Bbb Q$ which satisfy the conditions a)-c) listed below and such that \begin{equation} \label {resultaa} L(E,2) \sim_{\Bbb Q^{\ast}} \pi \cdot {\cal L}_{2,q}(P) \end{equation} The conditions on divisor $P$: \begin{equation} \label {condition1} a) \qquad\qquad \qquad \sum n_j P_j \otimes P_j \otimes P_j = 0 \quad \mbox{in} \quad S^3J({\bar \Bbb Q}) \qquad\qquad\qquad \end{equation} b) For any valuation $v$ of the field $\Bbb Q(P)$ generated by the coordinates of the points $P_j$ \begin{equation} \label {condition2} \sum n_j h_v(P_j)\cdot P_j =0 \quad \mbox{in} \quad J({\bar \Bbb Q})\otimes \Bbb R \end{equation} c) For every prime $p$ where $E$ has a split multiplicative reduction one has an integrality condition on $P$, see (\ref{icond121}) below. \end{theorem} {\it The integrality condition }. Suppose $E$ has a split multiplicative reduction at $p$ with $N$-gon as a special fibre. Let $L$ be a finite extention of $\Bbb Q_p$ of degree $n=ef$ and $ {\cal O}_L$ the ring of integers in $L$. Let $E^0$ be the connected component of the N\'eron model of $E$ over $ {\cal O}_L$. Let us fix an isomorphism $E^0_{F_{p^f}} = \Bbb G_m/{F_{p^f}}$. It provides a bijection between $\Bbb Z/{eN}\Bbb Z$ and the components of $E_{F_{p^f} }$. For a divisor $P$ such that all its points are defined over $L$ denote by $d(P;\nu)$ the degree of the restriction of the flat extension of a divisor $P$ to the $\nu$'th component of the $(eN)$-gon. Let $B_3(x):= x^3 - \frac{3}{2}x^2 + \frac{1}{2}x$ be the third Bernoulli polynomial. The integrality condition at $p$ is the following condition on a divisor $P$, provided by the work of Schappaher and Scholl ([SS]). For a certain (and hence for any, see s. 3.3) extention $L$ of $\Bbb Q_p$ such that all points of the divisor $P$ are defined over $L$ one has ($[L:\Bbb Q_p]= ef$): \begin{equation} \label {icond121} \sum_{\nu \in \Bbb Z/(eN) \Bbb Z}d(P;\nu)B_3(\frac{\nu}{eN}) =0 \end{equation} {\bf Remarks}. 1. For a $p$-adic valuation $v$ of the field $K(P)$ one has $(\log p)^{-1} h_v(P_j) \in \Bbb Q$. So the condition b) in this case looks as follows \begin{equation} \label {condition2nonar} (\log p)^{-1}\sum n_j h_v(P_j)\cdot P_j =0 \quad \mbox{in} \quad J(K(P))\otimes \Bbb Q \end{equation} In particular the right hand side is a finite dimensional $\Bbb Q$-vector space. 2. Lemma 1.5 below shows that, assuming (\ref{condition1}), if the condition (\ref{condition2}) is valid for all archimedean valuations but one then it is valid for all of them. In particular if $P \in \Bbb Z[E(\Bbb Q)]$ we can omit (\ref{condition2}) for the archimedean valuation. The proof of theorem (\ref{zcc}) is based on the results of S. Bloch [Bl1] on regulators on elliptic curves, a ``weak'' version of Beilinson's conjecture for modular curves proved by A.A. Beilinson in [B2] and the results presented in s.2-3 below. To prove the theorem we introduce for an elliptic curve $E$ over an {\it arbitrary} field $k$ a new complex (the elliptic motivic complex $B(E;3)$) and prove that its cohomology essentially computes the weight 2 parts of $K_2(E)$ and $K_1(E)$ (see theorems (\ref{mrezz}) and (\ref{zaza})). This complex mirrors the properties of the elliptic dilogarithm. It is an elliptic deformation of the famous Bloch-Suslin complex which computes $K_3^{ind}(F) \otimes \Bbb Q$ and $K_2(F)$ for an arbitrary field $F$ (see [DS], [S] and s.1.6). In particular we replace the ``arithmetical'' condition b) by its refined ``geometrical'' version (see s. 1.4), which is equivalent to the condition b) for curves over number fields. Our results imply \begin{theorem} \label {zccc} Let $E$ be an elliptic curve over $\Bbb Q$. Then i) For any element $\gamma \in K_2(E)$ there exists a $\Bbb Q$-rational divisor $P$ on $E$ satisfying the conditions a), b) from theorem (\ref{zcc}) such that the value of the Bloch-Beilinson regulator map $r_2: K_2(E) \longrightarrow \Bbb R$ on $\gamma$ is $ \sim_{\Bbb Q^{\ast}} {\cal L}_{2,q}(P)$ ii) For any $\Bbb Q$-rational divisor $P$ on $E$ satisfying the conditions a), b) there exists an element $\gamma \in K_2(E)\otimes \Bbb Q$ such that $r_2(\gamma) \sim_{\Bbb Q^{\ast}} {\cal L}_{2,q}(P)$. \end{theorem} Theorem (\ref{zccc}ii) implies immediately \begin{corollary} \label {zag11} Let $E$ be an elliptic curve over $\Bbb Q$. Let us assume that the image of $K_2(E)_{\Bbb Z} \otimes \Bbb Q$ under the regulator map is $L(E,2) \cdot \Bbb Q$. (This is a part of the Bloch-Beilinson conjecture). Then for any $\Bbb Q$-rational divisor $P $ on $E(\bar \Bbb Q)$ satisfying the conditions a) - c) of theorem (\ref{zcc}) one has $$ q\cdot L(E,2) = \pi \cdot {\cal L}_{2,q}(P) $$ where $q$ is a rational number, perhaps equal to $0$. \end{corollary} {\bf Remark}. Corollary (\ref{zag11}) has an analog for an elliptic curve over any number field. Its formulation is an easy exercise to the reader. Unlike in Zagier's conjecture on $\zeta$-functions of number fields one can not expect $P_i \in E(\Bbb Q)$: the Mordell-Weil group of an elliptic curve over $\Bbb Q$ could be trivial. The conditions a)-b) are obviously satisfied if $P$ is (a multiple of) a torsion divisor. Moreover, if $E$ is a curve with complex multiplication then $L(E,2)$ is the value of the elliptic dilogarithm on a torsion divisor ([Bl1]). However if $E$ is not a CM curve this should not be true in general. Thus one has to consider the non-torsion divisors, and so it is necessary to use the conditions a)-b) in full strength. The conditions a) and b) were guessed by D. Zagier several years ago after studying the results of the computer experiments with $\Bbb Q$-rational points on some elliptic curves, which he did with H. Cohen. {\bf 3. A numerical example}. $E$ is given by equation $y^2 -y = x^3 -x$. The discriminant $\Delta$ $=$ conductor $= 37$. So $E$ has split multiplicative reduction at $p=37$ with one irreducible component of the fiber of the N\'eron model. Therefore the integrality condition is empty. {\it Local nonarchimedean heights on $E$}. Let $P=[ a/p^{2\delta}, b/p^{3\delta}] \in E(\Bbb Q)$ where $a,b$ are prime to $p$. If $p$ is prime to $\Delta$ then $h_p(P) = 0$ if $\delta \leq 0$ and $h_p(P) = \delta \cdot \log p$ if $\delta > 0$. The local height at $p=37$ is given by $h_{37}(P) = -1/6 + 2 \delta$ (see the formula for the local height in s. 4.3 of or [Sil]). The Mordell-Weil group has rank one and is generated by the point $P =[0,0]$. Consider the following integral points on $E$: $$ P =[0,0], \quad 2P=[1,0], \quad 3P=[1,1],\quad 4P=[2,3],\quad 6P=[6,-14] $$ and also $$ 5P=[\frac{1}{4}, \frac{5}{8} ],\quad 10 P=[\frac{ 161}{16}, \frac{ 2065}{ 64}] $$ There are no height conditions at $p \not = 37$ for the integral points and there is just one at $p=2$ for the points $5P$ and $10P$. Consider the divisor $\sum n_k (kP)$. Notice that $S^3J(\Bbb Q) = \Bbb Z $ and the condition a) is $\sum n_k \cdot k^3 =0$. The height condition at $p=37$ gives $\sum n_k \cdot k =0$ provided that the coordinates of $(kP)$ are prime to $37$. The divisor $$ P_k = (kP) - k(P) - \frac{k^3 -k}{6}((2P) - 2(P)) $$ satisfies the conditions $\sum n_k \cdot k = \sum n_k \cdot k^3 =0$. Also $P_{10} - 4 \cdot P_5$ satisfies the height condition at $p=2$. The computer calculation (using PARI) shows $$ \frac{8 \pi \cdot{\cal L}_{2,q}( P_3)}{37 \cdot L(E,2) } = -8.0000..., \qquad \frac{8 \pi \cdot {\cal L}_{2,q}( P_4)}{37 \cdot L(E,2) } = -26.0000..., $$ $$ \frac{8 \pi \cdot {\cal L}_{2,q}( P_6)}{37 \cdot L(E,2) } = -90.0000..., \qquad \frac{8 \pi \cdot {\cal L}_{2,q}( P_{10} - 4 \cdot P_5)}{37 \cdot L(E,2) } = -248.0000... $$ {\bf 4. The group $B_2(E)$ and a refined version of conditions a) - b)}. Let $E$ be an elliptic curve over an arbitrary field $k$ and $J:= J(k)$ be the group of $k$-points of the Jacobian of $E$. Let $\Bbb Z[X]$ be the free abelian group generated by a set $X$. We will define in s. 2.1 a group $B_2(E/k) = B_2(E)$ such that a) one has an exact sequence \begin{equation} \label{exten} 0 \longrightarrow k^{\ast} \longrightarrow B_2(E/k) \stackrel{p}{\longrightarrow} S^2 J(k) \longrightarrow 0 \end{equation} b) one has a canonical (up to a choice of a sixth root of unity) surjective homomorphism \begin{equation} \label{homo} h: \Bbb Z[E(k) \backslash 0] \longrightarrow B_2(E/k) \end{equation} whose projection to $S^2J(k)$ is given by the formula $\{a\} \longmapsto a\cdot a$. c) if $K$ is a local field there is a canonical homomorphism $$ H: B_2(E/K) \longrightarrow \Bbb R $$ whose restriction to the subgroup $K^{\ast} \subset B_2(E/K))$ is given by $x \longmapsto \log |x|$, ( see s. 2.3). Moreover the canonical local height $h_K$ is given by the composition $$ \Bbb Z[E(K)\backslash 0] \stackrel{h}{\longrightarrow} B_2(E/K) \stackrel{H}{\longrightarrow} \Bbb R $$ The group $B_2(E)$ appears naturally as a version of the theory of biextensions. It is a ``motivic'' version of theta functions. Set $\{a\}_2:= h(\{a\}) \in B_2(E)$. The conditions a)-b) on a divisor $\sum_j n_j (P_j)$ are equivalent to the following single one: \begin{equation} \label {conditions1} \sum n_j \{P_j\}_2 \otimes P_j =0 \quad \mbox{in} \quad B_2(E( {\bar \Bbb Q}))\otimes J({\bar \Bbb Q}) \end{equation} More precisely, \begin{lemma} Let $K$ be a number field and $P_j \in E(K)$. Then $$ \sum n_j \{P_j\}_2 \otimes P_j =0 \quad \mbox{in} \quad B_2(E(K))\otimes J(K)\otimes \Bbb Q $$ if and only if the following two conditions hold: $$ \sum n_j P_j \otimes P_j \otimes P_j = 0 \quad \mbox{in} \quad S^3J(K) \otimes \Bbb Q $$ and for any valuation $v$ of the field $K$ $$ \sum n_j h_v(P_j)\cdot P_j =0 \quad \mbox{in} \quad J(K)\otimes_{\Bbb Q} \Bbb R $$ \end{lemma} {\bf Proof}. Multiplying the exact sequence (\ref{exten}) by $J(K)\otimes \Bbb Q$ we get $$ 0 \longrightarrow K^* \otimes J(K)\otimes \Bbb Q \longrightarrow B_2(E(K))\otimes J(K) \otimes \Bbb Q \stackrel{p \otimes id}{\longrightarrow} S^2J(K) \otimes J(K) \otimes \Bbb Q\longrightarrow 0 $$ and use the fact that the local norms $|\cdot |_v$ separate all the elements in $K^* \otimes \Bbb Q$. {\bf 5. The elliptic motivic complex}. Let us suppose first that $k$ is an algebraically closed field. In chapter 3 we define a subgroup $R_3(E) \subset \Bbb Z[E(k)]$. When $ k = \Bbb C$ it is a subgroup of all functional equations for the elliptic dilogarithm. In particular the homomorphism $$ {\cal L}_{2,q}: \Bbb Z[E(\Bbb C)] \longrightarrow \Bbb R, \quad \{a\} \longmapsto {\cal L}_{2,q}(a) $$ annihilates the subgroup $R_3(E/\Bbb C)$. Consider the homomorphism ($J:=J(k)$) $$ \delta_3: \Bbb Z[E(k)] \longrightarrow B_2(E) \otimes J, \quad \{a\} \longrightarrow -\frac{1}{2} \{a\}_2 \otimes a $$ An important result ( theorem (\ref{pro})) is that $\delta_3(R_3(E)) =0$ . Setting $$ B_3(E):= \frac{\Bbb Z[E(k)]}{R_3(E)} $$ we get a homomorphism $\delta_3: B_3(E) \longrightarrow B_2(E) \otimes J$. Let us consider the following complex \begin{equation} \label {comp221} B(E;3): \qquad B_3(E) \stackrel{\delta_3}{\longrightarrow} B_2(E) \otimes J \longrightarrow J \otimes \Lambda^2J \longrightarrow \Lambda^3J \end{equation} Here the middle arrow is $\{a\}_2 \otimes b \longmapsto a \otimes a\wedge b$ and the last one is the canonical projection. The complex is placed in degrees $[1,4]$. It is acyclic in the last two terms. This is our elliptic motivic complex. Let $I_E$ be the augmentaion ideal of the group algebra $\Bbb Z[E]$, and $I_E^4 $ its fourth pour. Let $B_3^{\ast}(E)$ be the quotient of $I_E^4$ by the subgroup generated by the elements $(f) \ast (1-f)^-$, where $\ast$ is the convolution in the group algebra $\Bbb Z[E]$, $f \in k(E)^*$, and $g^-(t):= g(-t)$. Then there is a homomorphism \begin{equation} \label{pp} \delta_3: B_3^*(E) \longrightarrow k^* \otimes J \end{equation} which fits the following commutative diagram $$ \begin{array}{ccccccc} 0&&0&&&&\\ \downarrow&&\downarrow&&&&\\ B_3^*(E)& \stackrel{ }{\longrightarrow} & k^* \otimes J &&&&\\ \downarrow &&\downarrow &&&&\\ B_3(E)&\stackrel{\delta_3}{\longrightarrow} &B_2(E) \otimes J &\longrightarrow &J \otimes \Lambda^2J&\longrightarrow& \Lambda^3J\\ \downarrow &&\downarrow &&\downarrow = &&\downarrow =\\ S^3J &\longrightarrow &S^2J \otimes J&\longrightarrow &J \otimes \Lambda^2J&\longrightarrow & \Lambda^3J\\ \downarrow&&\downarrow&&&&\\ 0&&0&&&& \end{array} $$ where the vertical sequences are exact, and the bottom one is the Koszul complex, and thus also exact. Let us denote by $B^*(E;3)$ the complex (\ref{pp}). It is canonically quasiisomorphic to the complex $B(E;3)$. The complex $B^*(E;3)$ looks simpler then $B(E;3)$. However a definition of the differential in $B^*(E;3)$ which does not use the embedding to $B(E;3)$ is rather awkward, see s. 4.6. If $k$ is not algebraically closed we postulate the Galois descent property: $$ B(E/k;3) := B(E/{\bar k};3) ^{Gal(\bar k/k)}; \qquad B^*(E/k,3) := B^*(E/{\bar k};3) ^{Gal(\bar k/k)} $$ {\bf 6. Relation with algebraic $K$-theory}. Let $k$ be an arbitrary field. Let ${\cal K}_2$ be the sheaf of $K_2$ groups in the Zariski topology on $E$. One has canonical inclusion $K_2(k) \hookrightarrow H^0(E,{\cal K}_2)$ and surjective projection \linebreak $H^1(E,{\cal K}_2) \to k^*$. \begin{theorem} \label{mrezz} Let $k = \bar k$. Then there is a sequence $$ Tor(k^*, J) \hookrightarrow \frac{H^0(E,{\cal K}_2)}{K_2(k)} \longrightarrow B_3^*(E) \longrightarrow k^* \otimes J \longrightarrow Ker (H^1(E,{\cal K}_2) \to k^*) \to 0 $$ It is exact in the term $k^* \otimes J$ and exact modulo $2$-torsion in the other terms. \end{theorem} For an abelian group $A(E)$ depending functorially on $E$ let $A(E)^-$ be the subgroup of skewinvariants under the involution $x \to -x$ of $E$. Recall that one has the $\gamma$-filtration on the Quillen $K$-groups. One can show that modulo $2$-torsion $$ \frac{H^0(E,{\cal K}_2)}{K_2(k)} = gr^{\gamma}_2K_2(E)^-, \qquad Ker (H^1(E,{\cal K}_2) \to k^*) = gr^{\gamma}_2K_1(E)^- $$ Recall that the Bloch-Suslin complex for an arbitrary field $k$ is defined as follows: $$ B_2(k) \stackrel{\delta}{\longrightarrow} \Lambda^2k^*; \qquad B_2(k) := \frac{\Bbb Z [k^*]}{R_2(F)} ; \quad \delta: \{x\} \longmapsto (1-x) \wedge x $$ Here $R_2(k)$ is the subgroup generated by the elements $\sum_i(-1)^i \{r(x_1,..., \hat x_i,... ,x_5)\}$, where $x_i$ runs through all $5$-tuples of distinct points over $k$ on the projective line and $r$ is the cross ratio. One should compare theorem (\ref{mrezz}) with the following exact sequence provided by Suslin's theorem on $K_3^{ind}(k)$ ([S]) and Matsumoto's theorem on $K_2(k)$ ([M]) (see also a closely related results by Dupont and Sah [DS]): $$ 0 \longrightarrow \tilde Tor(k^*,k^*) \longrightarrow K_3^{ind}(k) \longrightarrow B_2(k) \stackrel{\delta}{\longrightarrow} \Lambda^2k^* \longrightarrow K_2(k) \longrightarrow 0 $$ Here $ \tilde Tor(k^*,k^* ) $ is a nontrivial extension of $Tor(k^*,k^* )$ by $\Bbb Z/2\Bbb Z$. {\bf Remark}. To guess an elliptic analog of the Steinberg relation \linebreak $(1-x) \otimes x \in \Bbb C^* \otimes \Bbb C^*$ we might argue as follows. $E(\Bbb C) = \Bbb C^*/q^{\Bbb Z}$, so let us try to make sence out of $\sum_{n \in \Bbb Z}(1-q^nx) \otimes q^nx$. Let $p:\Bbb C^* \to E(\Bbb C)$. Projecting $\Bbb C^* \otimes \Bbb C^*$ to $\Bbb C^* \otimes J(\Bbb C) $ we get $\prod_{n \in \Bbb Z}(1-q^nx) \otimes x$, $x \in E(\Bbb C)$. Regularizing the infinite product we obtain $\theta(x) \otimes x$ where $$ \theta(x):= q^{1/12} z^{-\frac{1}{2}} \prod_{j \geq 0} (1-q^j z)\prod_{j > 0} (1-q^j z^{-1}) $$ Unfortunately this seems to make no sence: $\theta(x)$ is not a function on $E(\Bbb C)$ with values in $\Bbb C^*$, but a section of a line bundle. Only introducing the {\it group} $B_2(E/\Bbb C)$ and realizing that $\theta(x)$ {\it is a function on $E(\Bbb C)$, but with values in the group} $ B_2(E/\Bbb C)$, we find the elliptic analog of the Steinberg relation: $\theta(x)\otimes x \in B_2(E/\Bbb C) \otimes J(\Bbb C)$ (compare with s. 4.1, 4.2, 4.6). For an arbitrary base field $\{x\}_2 \in B_2(E)$ replaces $\theta(x)$. In [W] J.Wildeshaus, assuming standard conjectures about mixed motives, gave a conjectural inductive definition of groups similar to $B_{n}(E)$ and discuss an elliptic analog of weak version of Zagier's conjecture. {\bf Acknowledgement}. This paper was essentially written when the first author enjoyed the hospitality of MPI(Bonn) and IHES during the Summer of 1995, and completed when the authors met in 1996 at the MPI (Bonn). The generous support of both institutions gratefully acknowledged. A.G. was partially supported by the NSF Grant DMS-9500010 and A.L. by the Soros International Scientific Foundation. It is our pleasure to thank Don Zagier and Maxim Kontsevich for interesting discussions. We are very grateful to the referee who made a lot of useful remarks and pointed out some errors. \section {The group $B_2(E)$} {\bf 1. A construction of the group $B_2(X)$}. Let $k$ be an arbitrary field. For any two degree zero line bundles $L_1$ and $L_2$ on a regular curve $X$ over $k$ let us define, following Deligne ([De]), a $k^{\ast}$-torsor $[L_1,L_2]$. {\it Motivation}. Let $s_i$ be a section of a line bundle $L_i$. If $div (s_2)=\sum m_i P_i$ then $<s_1,s_2>$ is $s_1(div (s_2))\in {\otimes} L_1^{m_i}|_{p_i} = [L_1,L_2]$. Such a tensor product turns out to be symmetric and does not depend on the choice of $s_2$. If $L_1={\cal O}$ then $[L_1,L_2] =k^*$. For $f \in k(X)^*$ and a closed point $x \in X_1$ set $\bar f(x):= Nm_{k(x)/k}f(x)$. We extend $\bar f$ by linearity to the group of $k$-rational divisors on $E$. {\it Definition}. The elements of $[L_1,L_2]$ are pairs $<s_1,s_2>$, where $s_i$ is a section of the line bundle $L_i$ and the divisors $div(s_1)$ and $div(s_2)$ are disjoint. For a rational function $f,g$ such that $div(f)$ is disjoint from $div(s_2)$ and $div(g)$ is disjoint from $div(s_1)$, one has ($(s):= div (s)$): $$ <f\cdot s_1,s_2> = \bar f((s_2)) <s_1,s_2>, \quad < s_1,g\cdot s_2> = \bar g((s_1)) <s_1,s_2> $$ There are two {\it a priory} different expansions \begin{equation} \label{correct1} <f\cdot s_1,g\cdot s_2> = \bar f((g)) \bar f(s_2) < s_1,g\cdot s_2> = \bar f((g)) \bar f((s_2)) \bar g((s_1))<s_1,s_2> \end{equation} and \begin{equation} \label{correct2} <f\cdot s_1,g\cdot s_2> = \bar g((f)) \bar g((s_1)) < f\cdot s_1, s_2> = \bar g((f)) \bar g((s_1)) \bar f((s_2))<s_1,s_2> \end{equation} The right hand sides coincide thanks to the Weil reciprocity $ \bar f((g)) = \bar g((f))$. So the $k^{\ast}$-torsor $[L_1,L_2]$ is well defined. There is a canonical isomorphism of $k^{\ast}$-torsors $$ [L_1 \otimes L_2, M ] \longrightarrow [L_1, M] \otimes_{k^{\ast}} [L_2, M ] $$ and a similar additivity isomorphism for the second divisor. Further, the two possible natural isomorphisms $$ [L_1 \otimes L_2, M_1 \otimes M_2 ] \longrightarrow \otimes_{1 \leq i,j \leq 2 } [L_i , M_j] $$ coincide. Therefore for any element $s \in S^2J_X$ one gets a $k^{\ast}$-torsor $[s]$ defined up to an isomorphism. Moreover $[s_1+s_2] = [s_1] \otimes_{k^{\ast}} [s_2]$. The above facts just mean that the collection of $k^{\ast}$-torsors $[s]$ defines an extension of type (\ref{exten}). This is the definition of the group $B_2(X)$. To construct the homomorphism (\ref{homo}) we want to make sense of elements $<s_1,s_2>$ where the divisors of the sections $s_i$ {\it may not be disjoint}. The definition is suggested by the motivation given above. Namely \begin{equation} \label{disj} <s_1, s_2> \quad \in \quad [L_1,L_2] \otimes_{k^{\ast}} V(div(s_1),div(s_2)) \end{equation} where $V(L_1,L_2)$ is a $k^{\ast}$-torsor defined as follows. Recall that if $A \to B$ is a homomorphism of abelian groups and $T$ is an $A$-torsor, we can define a $B$-torsor $f_*T := T \otimes _A B$ where $A$ acts on $B$ via the homomorphism $f$. Thus for a $k(x)^*$-torsor $T^*_xX$ we define a $k^*$-torsor $N(T^*_xX)$ using the norm homomorphism $Nm: k(x)^* \to k^*$. For two arbitrary divisors $l_1$ and $l_2$ on $X$ consider the following $k^{\ast}$-torsor $$ V(l_1,l_2):= \otimes_{x \in X_1}N(T^*_xX)^{\otimes ord_x l_1 \cdot ord_x l_2} $$ Here $ord_x l$ is the multiplicity of the divisor $l$ at the point $x$. One has a canonical isomorphism $V(l_1 + l_2, m) \longrightarrow V(l_1,m) \otimes V(l_2,m)$. If the divisors $l_1$ and $l_2$ are disjoint then $V(l_1,l_2) = k^{\ast}$. Any rational function $f$ provides a canonical isomorphism $$ \varphi_f: V(l_1,l_2) \longrightarrow V((f) + l_1,l_2) \qquad v \longmapsto {\tilde f}(l_2) \cdot v $$ In this formula ${\tilde f}(x) \in N(T^*_xX)^{\otimes ord_x(f)}$ is the ``leading term'' of the function $f$ at $x$. It is defined as follows. Choose a local parameter $t$ at the point $x$. If $f(t) = at^k +$ higher order terms then ${\tilde f}(x) = a(dt)^k \otimes 1 \in N(T^*_xX)^k = (T^*_xX)^k \otimes k^*$. So ${\tilde f(x)}^{ord_x l_2} \in N(T^*_xX)^{\otimes ord_x(f) \cdot ord_x l_2}$ and the formula above makes sense. Let us recall the full version of the Weil reciprocity law. Let \begin{equation} \label{tame} \partial_x:\{f,g\} \longmapsto (-1)^{ord_x(f)\cdot ord_x(g) } \frac{ f(x)^{ord_x(g)}}{ g(x)^{ord_x(f)}} \end{equation} be the tame symbol. Then for {\it any} two rational functions $f,g$ on a curve over an algebraically closed field $k$ one has $\prod_{x \in X} \partial_x(f,g) =1$. There exists a canonical isomorphism of $k^*$-torsors $$ S: [L_1,L_2] \otimes V((s_1),(s_2)) \longrightarrow [L_2,L_1] \otimes V((s_2),(s_1)) $$ given on generators by the formula $$ S: \quad <s_1,s_2> \quad \longmapsto\quad (-1)^{deg L_1 \cdot deg L_2 + \sum_{x \in X_1} ord_xs_1 \cdot ord_xs_2}<s_2,s_1> $$ The defining properties of the torsor $[L_1,L_2]$ look as follows: $$ <f\cdot s_1,s_2> = \bar f((s_2)) <s_1,s_2>, \quad < s_1,g\cdot s_2> = S(<s_1,(g)>) <s_1,s_2> $$ The formula \begin{equation} \label{oioioi} S<(f),(g)> = <(f),(g)> \end{equation} is equivalent to the Weil reciprocity. Similar to (\ref{correct1}), (\ref{correct2}) and using the formula (\ref{oioioi}) we see that $S$ is well defined. {\bf 2. The homomorphism (\ref{homo})}. From now on $X=E$ is an elliptic curve over $k$, so there is canonical isomorphism $T_x^*E = T_0^*E$. For $c \in k^*$ one obviously has $<c\cdot s_1,s_2> = < s_1,s_2>$. Therefore we may consider $< s_1,s_2>$ when $s_1,s_2$ are divisors on $E$. Consider a map $$ \{a\} \longmapsto <(a) - (0), (a) - (0)> \in (T^{\ast}_0E)^{\otimes 2}\otimes_{k^{\ast}} [a\cdot a] $$ There is an almost canonical (up to a sixth root of unity) choice of an element in $(T^{\ast}_0E )^{\otimes 2}$. Namely, the quotient of $E$ by the involution $x \longmapsto -x$ is isomorphic to $P^1$. The image of $0$ on $E$ is the point $\infty$ on $P^1$. The images of the three nonzero 2-torsion points on $E$ gives 3 distinguished points on $P^1$. Their ordering = choice of a level 2 structure on $E $. A choice of ordering gives a {\it canonical} coordinate $t$ on $\Bbb A^1:= P^1 \backslash \infty$ for which $t=0$ is the first point and $t=1$ is the second. This coordinate provides a vector in $T^{\ast}_{\infty}P^1$ and so a vector in $(T^{\ast}_0E)^2$. Therefore we have six different trivializations of $(T^{\ast}_0E)^2$ and thus a canonical one of $(T^{\ast}_0E)^{12}$: their product. The sixth root of ($16 \times$ this trivialization) is the (almost) canonical element in $(T^{\ast}_0E)^2$ we need. Using this element in $(T^*_0E)^2$ we get a map (\ref{homo}). The composition $\Bbb Z[E(k) \backslash 0] \longrightarrow B_2(E) \longrightarrow S^2J$ is obviously given by $\{a\} \longmapsto a\cdot a$. If $E$ is written in the Weierstrass form $y^2 = (x-e_1) (x-e_2)(x-e_3)$ then $e_1, e_2, e_3$ are the coordinates of the distinguished points and $ \frac{x-e_i}{e_j -e_k}$ is the canonical coordinate corresponding to the ordering $e_i,e_j,e_k$. Let $\Delta$ be the discriminant of $E$. Then $\Delta = 16 \cdot \prod_{i<j}(e_i - e_j)^2$. The canonical trivialization is $\Delta^{1/6}(dx/2y)^2$. {\bf 3. The canonical height}. In this section we recall the construction of the canonical local heights via the biextension (compare with [Za] and [Bl2]). The canonical local height gives a homomorphism $B_2(K) \longrightarrow \Bbb Q$ (resp. to $\Bbb R$) when $K$ is a nonarchimedean (resp. archimedean) local field. The construction of the group $B_2(E)$ provides us with a collection of $k^{*}$-torsors $T_{(x,y)}$ where $(x,y)$ is a point of $J \times J$. The torsors $T_{(0,y)}$ and $T_{(x,0)}$ are trivialized: we have a distinguished element $<\emptyset,(y) - (0)> \in T_{(0,y)}$ and a similar one in $T_{(x,0)}$. The collection of torsors $T_{(x,y)}$ glue to a $k^{*}$-bundle on $J \times J$. It is isomorphic to the (rigidified) Poincar\'e line bundle minus zero section. Let $L$ be a degree zero line bundle on an elliptic curve over $k$. Let us denote by $L^{*}$ the complement of the zero section in $L$. It is a principal homogeneous space over a certain commutative algebraic group $A(L)$ over $k$ which is an extension $$ 0 \longrightarrow \Bbb G_m \longrightarrow A(L) \longrightarrow J \longrightarrow 0 $$ The group $A(L)$ is described as follows. For any $a \in E$ let $t_a: x \longmapsto a+x$ be the shift by $a$. Then $t_a^{*} L$ is isomorphic to $L$ (because $L$ is of degree zero). The set of isomorphisms from $L$ to $t_a^{*} L $ form a $k^{*}$-torsor. These torsors glue together to a $k^{*}$-torsor over $E$ which is isomorphic to $L^{*}$. On the other hand the collection of isomorphisms $ L \longrightarrow t_a^{*} L$ form a group. This is the group $A(L)$. It is commutative: the commutator provides a morphism from $E \times E$ to $\Bbb G_m$, which has to be a constant map. Now let $K$ be a local field and $E$ be an elliptic curve over $K$. We get a group extension $$ 0 \longrightarrow K^{\ast} \longrightarrow A(L)(K) \longrightarrow J(K)\longrightarrow 0 $$ Let $U(G)$ be the maximal compact subgroup of a locally compact commutative group $G$. One can show (use lemma 6.1, ch. 11 in [La]) that $U\Bigl(A(L)(K)\Bigr)$ projects surjectively onto $J(K)$ if $K$ is archimedean or $E$ has a good reduction over $K$. In the case of bad reduction the image is a subgroup of finite index. There is canonical homomorphism $$ A(L)(K) \longrightarrow A(L)(K)/U\Bigl(A(L)(K)\Bigr) =:H $$ The quotient $H$ is isomorphic to $\Bbb Z$ (resp. to a subgroup in $\Bbb Q$ which is an extension of $\Bbb Z$ by a finite group ) when $K$ is nonarchimedean and $E$ has good reduction (resp. bad reduction), and to $\Bbb R$ if $K= \Bbb C, \Bbb R$. Therefore we get a homomorphism $ A(L)(K) \longrightarrow \Bbb Z (\mbox{resp}\quad \Bbb Q ) $ for the nonarchimedean case and $ A(L)(K) \longrightarrow \Bbb R $ for the archimedean one. For a given $x$ the torsors $T_{(x,y)}$ form a group $T_{(x,\cdot)}$ which is isomorphic to the group $A(L|_{x \times J})$, and there is a similar statement for the torsors $T_{(x,y)}$ for a given $y$. Applying the homomorphism $ A(L|_{x \times J}) \longrightarrow H $ we will get a homomorphism of the group $T_{(x,\cdot)}$ to $H$. Similarly we have a homomorphism of the group $T_{(\cdot, y)}$ to $H$. Consider the map $$ U(T_{(x_1,\cdot)}) \times U(T_{(x_2,\cdot)}) \longrightarrow T_{(x_1+x_2,\cdot)} $$ induced by the multiplication $T_{(x_1,y)}\times T_{(x_2,y)} \longrightarrow T_{(x_1+x_2,y)}$. Its image is a subgroup. This follows from the commutativity of the diagram $$ \begin{array}{ccc} T_{(x_1,y_1)} \times T_{(x_2,y_1)}\times T_{(x_1,y_2)} \times T_{(x_2,y_2)}&\longrightarrow& T_{(x_1,y_1+y_2)} \times T_{(x_2,y_1+y_2)}\\ &&\\ \downarrow&&\downarrow\\ &&\\ T_{(x_1+x_2,y_1)} \times T_{(x_1+x_2,y_2)}&\longrightarrow&T_{(x_1+x_2,y_1+y_2)} \end{array} $$ It is therefore a compact subgroup, and so a maximal compact subgroup in $T_{(x_1+x_2,\cdot)}$. In particular the restrictions of the homomorphisms $T_{(x,\cdot)} \longrightarrow H$ and $T_{(\cdot,y)} \longrightarrow H$ to $T_{(x, y)}$ coincide. So we get a well defined homomorphism $B_2(E(K)) \longrightarrow H$. Now the composition \begin{equation} \label{heig} \Bbb Z[E(K)\backslash 0] \stackrel{h}{\longrightarrow} B_2(E(K)) \longrightarrow H \end{equation} is the canonical N\'eron height on $E(K)$. The restriction of the homomorphism (\ref{heig}) to the subgroup $K^{*} \subset B_2(E(K))$ coincides with the logarithm of the norm homomorphism. In the nonarchimedean case $H$ is a subgroup of $\log p \cdot \Bbb Q \subset \Bbb R$. {\bf Remark}. The homomorphism $h : \Bbb Z[E(K)\backslash 0] \longrightarrow B_2(E(K))$ was defined up to a sixth root of unity. This does not affect the definition of the height because the norm vanishes on roots of unity. \section{An elliptic analog of the Bloch-Suslin complex} {\bf 1. The group $B_3(E)$ and complex $B(E;3)$}. We will assume in s.3.1 - 3.3 that $k = \bar k$. We will always use notation $J := J(k)$. Set $g^-(t) := g(-t)$. Denote by $\ast$ the convolution in the group algebra $\Bbb Z[E(k)]$. \begin{definition} $R_3(E)$ is the subgroup of $\Bbb Z[E(k)]$ generated by the elements $ (f) \ast (1-f)^-, \quad f \in k(E)^{\ast} $, $\{0\}$, and the "distribution relations" $$ m\cdot(\{a\} - m\cdot \sum_{mb=a}\{b\}), \quad a \in E(k), \quad m = -1,2 $$ \end{definition} {\bf Remarks}. a) For $m=-1$ we get the elements $\{a\} + \{-a\} \in R_3(E)$. If we remove them from the definition of $R_3(E)$, we get the same group. b) It would be more natural to add to the subgroup $R_3(E)$ the distribution relations for all $m \in \Bbb Z \backslash 0$: we should get the same group (compare with lemma (\ref{destr})). But we will not need this. Consider the homomorphism $$ \beta: \otimes^2k(E)^{\ast} \longrightarrow \Bbb Z[E(k)] \qquad \beta: f\otimes g \longmapsto f*g^- := \sum n_im_j\{a_i - b_j\} $$ (the Bloch map), where $(f) = \sum n_i (a_i)$ and $(g) = \sum m_j(b_j)$. Recall that $$ \delta_3: \Bbb Z[E(k)] \longrightarrow B_2(E) \otimes J, \qquad \{a\} \longmapsto -\frac{1}{2} \{a\}_2 \otimes a $$ and $i: k^{\ast} \hookrightarrow B_2(E)$ is the canonical embedding (see (\ref{exten})). Let $I_E$ be the augmentation ideal of the group algebra $\Bbb Z[E(k)]$ and $p: I_E \to J$ the canonical projection. Recall that if $k = \bar k$ the tame symbol provides a homomorphism $$ \otimes^2 k(E)^* \stackrel{\partial}{\longrightarrow} k^{\ast} \otimes \Bbb Z[E] $$ The Weil reciprocity law shows that its image belong to $k^{\ast} \otimes I_E$. \begin{theorem} \label{commm} The following diagram is commutative $$ \begin{array}{ccc} \otimes^2 k(E)^* &\stackrel{\partial}{\longrightarrow}& k^{\ast} \otimes I_E \\ &&\\ \downarrow \beta&&\downarrow i \otimes p\\ &&\\ \Bbb Z[E(k)]&\stackrel{\delta_3}{\longrightarrow}& B_2(E) \otimes J \end{array} $$ \end{theorem} {\bf Proof}. Let $ (f) = \sum n_i(a_i), \quad (g) = \sum m_j (b_j)$. Then $$ \delta_3 \circ \beta (f\wedge g) = -\frac{1}{2} \cdot \sum_{i,j} m_in_j<a_i -b_j, a_i -b_j> \otimes (a_i -b_j) $$ The term $ \sum_{i,j} m_i n_j<(a_i -b_j), (a_i -b_j)> \otimes a_i $ equals to $$ \sum_{i,j} m_i n_j<(a_i) -(0), (a_i) -(0)> \otimes a_i + \sum_{i,j} m_i n_j<(b_j) -(0), (b_j) -(0)> \otimes a_i - $$ $$ 2\cdot \sum_{i,j} m_i n_j<(a_i) -(0), (b_j) -(0)> \otimes a_i $$ The first term here is zero because $\sum_{j} m_j =0$. The second is zero because $\sum_{i} n_ia_i =0$ in $J$. The last one can be written as $$ -2 \cdot \sum_{i} m_i <(a_i) -(0), (g)> \otimes a_i $$ So the theorem follows from the definition of the tame symbol. \begin{theorem} \label{pro} $\delta_3(R_3(E)) =0$. \end{theorem} {\bf Proof}. We will denote by $\{a\}_3$ projection of the generator $\{a\}$ onto the quotient $B_3(E)$. The map $\delta_3$ kills the distribution relations: $$ \delta_3 \Bigl(m(\{a\}_3 - m\sum_{mb=a}\{b\}_3))\Bigr) = m (\{a\}_2 \otimes a - m\sum_{mb=a}\{b\}_2 \otimes b) = $$ $$ m(\{a\}_2 - \sum_{mb=a}\{b\}_2)\otimes a =0 $$ The last equality is provided by corollary (\ref{divr}), which will be proved later, in s. 4.4. (The proof does not depend on any results or constructions in chapter 3). The fact that $\delta_3 (f \ast (1-f)) =0$ lies deeper and follows from the theorem (\ref{commm}). Theorem (\ref{pro}) is proved. Set $B_3(E):= \Bbb Z[E(k)]/R_3(E)$. We get a complex $$ B(E,3):= \qquad B_3(E) \longrightarrow B_2(E) \otimes J \longrightarrow J \otimes \Lambda^2J \longrightarrow \Lambda^3J $$ Let $I^k_E$ be $k$-th power of the augmentation ideal. \begin{lemma} \label{cor} $\beta( f\otimes g) \in I^4_E$. Moreover, $\beta$ is surjective onto $I^4_E$. \end{lemma} {\bf Proof}. A divisor $\quad \sum n_i (a_i) \quad$ is principal if and only if $\quad \sum n_i = 0 $ and $\sum n_i a_i =0$ in $J(E)$. So $I_E^2$ coincides with the subgroup of $\Bbb Z[E]$ given by the divisors of functions. So the convolution of two principal divisors belongs to $I^4_E$ and, moreover, generate it. Let $B_3^{\ast}(E)$ be the quotient of $I_E^4$ by the subgroup generated by the elements $(f) \ast (1-f)^-$. \begin{lemma} \label{tl} $\delta_3(I^4_E) \in k^{\ast}\otimes J \subset B_2(E)\otimes J$ \end{lemma} {\bf Proof}. It is easy to see that $\delta_3(I^4_E) \subset h(I_E^3) \otimes J$. Further, $h(I_E^3) \subset k^*$ because $p \circ h (I^3_E)=0$ (see the properties of the group $B_2(E)$ listed in s. 1.4). The lemma follows. So we get a complex \begin{equation} \label {comp2221} B^*(E;3): \quad B_3^{\ast}(E) \stackrel{\delta_3}{\longrightarrow} k^{\ast} \otimes J \end{equation} {\bf 2. Relation with algebraic $K$-theory}. Let us remind the long exact sequence of localization $$ K_3(k(E)) \stackrel{\partial_3}{\longrightarrow} \oplus_{x \in E_1} K_2(k(x)) \longrightarrow K_2(E) \longrightarrow $$ $$ K_2(k(E)) \stackrel{\partial_2}{\longrightarrow} \oplus_{x \in E_1} k(x)^* \longrightarrow K_1(E) \longrightarrow k(E)^* \stackrel{\partial_1}{\longrightarrow} \oplus_{x \in E_1} \Bbb Z $$ The group $K_1(E)$ has a subgroup $k^*$ which comes from the base. One can show that $$ H^0(E, {\cal K}_2) = gr^{\gamma}_2K_2(E), \qquad H^1(E, {\cal K}_2) = gr^{\gamma}_2K_1(E) = K_1(E)/k^* $$ \begin{lemma} \label{3.7} Modulo $2$-torsion one has $$ H^0(E, {\cal K}_2) = K_2(k) \oplus H^0(E, {\cal K}_2)^- $$ $$ Ker \Bigl( H^1(E,{\cal K}_2) \longrightarrow k^* \Bigr) = H^1(E,{\cal K}_2)^- = K_1(E)^- $$ \end{lemma} {\bf Proof}. It follows easily using the transfer related to the projection $E \to P^1$ given by factorization along the involution $x \to -x$. Recall that $H^3(B(E,3)) = H^4(B(E,3)) = 0$. \begin{theorem} \label{za} Let $k = \bar k$. Then the commutative diagram from theorem (\ref{commm}) provides a morphism of complexes $$ \begin{array}{ccc} \frac{K_2(k(E))}{k^{\ast} \cdot k(E)^{\ast}} &\stackrel{\partial}{\longrightarrow}& k^{\ast} \otimes J \\ &&\\ \tilde \beta \downarrow&&\downarrow id\\ &&\\ B^*_3(E)&\stackrel{\delta_3}{\longrightarrow}& k^* \otimes J \end{array} $$ where $ \tilde \beta$ is surjective and $Ker \tilde \beta = Tor(k^*,J)$ modulo $2$-torsion. \end{theorem} This theorem and lemma (\ref{3.7}) implies immediately \begin{theorem} \label{zaza} Let $k = \bar k$. Then there are an embedding $$ i: Tor(k^*, J(k)) \hookrightarrow \frac{H^0(E, {\cal K}_2)}{ K_2(k)} $$ and canonical isomorphisms \begin{equation} \label{098} \frac{H^0(E, {\cal K}_2)}{ Tor(k^*, J(k)) +K_2(k)} = H^1B^*(E;3) \quad \mbox{modulo $2$-torsion} \end{equation} \begin{equation} \label{099} Ker \Bigl( H^1(E,{\cal K}_2) \longrightarrow k^* \Bigr) = H^2B^*(E;3) \end{equation} For an arbitrary field $k$ (\ref{098}) and (\ref{099}) are isomorphisms modulo torsion. \end{theorem} {\bf Proof of theorem ({\ref{za})}}. We will first prove that we have a morphism of complexes and $Tor(k^*,J) \subset Ker \tilde \beta$, and then that $Tor(k^*,J) = Ker \tilde \beta$ modulo $2$-torsion. Recall that the tame symbol homomorphism $\partial$ maps $K_2(k(E))$ to $k^{\ast} \otimes I_E$. Therefore the complex computing the groups $H^0(E, {\cal K}_2)$ and $Ker \Bigl( H^1(E,{\cal K}_2) \longrightarrow k^* \Bigr)$ looks as follows $$ K_2(k(E)) \stackrel{\partial}{\longrightarrow} k^{\ast} \otimes I_E $$ Notice that $\partial(\{c,f\}) = c \otimes (f)$, so it has a subcomplex \begin{equation} \label{lllll} k^{\ast} \cdot k(E)^{\ast}\stackrel{\partial}{\longrightarrow} k^{\ast} \otimes I_E^2 \end{equation} where $\cdot$ is the product in $K$-theory. One obviously has $Ker \partial = K_2(k)$, and factorising by $K_2(k)$ we get the identity map $k^{\ast} \otimes I_E^2 \to k^{\ast} \otimes I_E^2$. The map $\beta: \otimes^2 k(E)^* \to I^4_E$ followed by the natural projection \linebreak $I^4_E \to B^*_3(E)$ leads to surjective map $$ \tilde \beta: \otimes^2 k(E)^* \longrightarrow B^*_3(E) $$ Notice that $\beta(k^{\ast} \otimes k(E)^{\ast}) =0$ and $\beta(f \otimes (1-f)) =0$ by the definition of the group $B^*_3(E)$. So we get the desired morphism of complexes. Tensoring the exact sequence $0 \to I^2_E \to I_E \to J \to 0$ by $k^*$ and using the fact that $I_E$ is a free abelian group, we get an exact sequence \begin{equation} \label{eexxs} 0 \longrightarrow Tor(k^*,J) \longrightarrow k^* \otimes I_E^2\longrightarrow k^* \otimes I_E \stackrel{id \otimes p}{\longrightarrow} k^* \otimes J\longrightarrow 0 \end{equation} So we get the following commutative diagram, where the vertical sequences are complexes, the complex on the right is the exact sequence (\ref{eexxs}), and $\alpha $ is injective: $$ \begin{array}{ccc} &&0\\ &&\downarrow \\ &&Tor(k^*,J)\\ &\swarrow&\downarrow \\ k^* \otimes I_E^2 &\stackrel{=}{\longrightarrow}&k^* \otimes I_E^2\\ \alpha \downarrow&&\downarrow\\ \frac{K_2(k(E))}{K_2(k)}&\stackrel{\partial }{\longrightarrow}&k^* \otimes I_E\\ &&\\ \tilde \beta \downarrow&&\downarrow \\ B_3^*(E)&\stackrel{\delta_3 }{\longrightarrow}&k^* \otimes J\\ &&\downarrow\\ &&0 \end{array} $$ From this diagram we see that $ Tor(k^*,J) \subset Ker \tilde \beta $. Notice that $$ Coker \Bigl(k^* \otimes I_E^2 \stackrel{\alpha}{\longrightarrow} \frac{K_2(k(E))}{K_2(k)}\Bigr) = \frac{\otimes^2I^2_E}{\{(1-f) \ast(f)^-\}} $$ \begin{theorem} \label{mmmm} The map $\bar \beta: \frac{\otimes^2I^2_E}{\{(1-f) \ast(f)^-\}}\longrightarrow B^{\ast}_3(E)$ is an isomorphism modulo $2$-torsion. \end{theorem} {\bf Proof of theorem (\ref{mmmm})}. It consists of three steps of quite different nature. {\it Step 1}. Set $t_a: x \to x+a$, $(t_af)(x):= f(x-a)$. \begin{proposition} \label{ref} Let $f$ and $g$ be rational functions on $E$. Then $$\{f,g\} - \{t_af,t_ag\} =0 \quad {\rm in } \quad \frac{ K_2(k(E))}{k^*\cdot (k(E))^*}$$ \end{proposition} Let $L/K$ be an extension of fields. Then one has a natural map $p^*: K_2(K) \to K_2(L)$ and the transfer map $p_* : K_2(L) \to K_2(K)$. We need the following result ([BT]). \begin{lemma} \label{BT} Let $L/K$ be a degree 2 extension of fields. Then $K_2(L)$ is generated by symbols $\{k,l\}$ with $k \in K, l \in L$ and $p_*(\{k,l\}) = \{k,Nm_{L/K}l\}$. \end{lemma} In particular $p^*p_* = Id + \sigma $ where $\sigma$ is a nontrivial element in $Gal(L/K)$, and thus modulo $2$-torsion any Galois invariant element in $K_2(L)$ belongs to $p^*K_2(K)$. \begin{lemma} \label {4.11} Assume $k = \bar k$. Then any rational function $f$ on an elliptic curve $E$ over $k$ can be decomposed into a product of functions with divisors of the following kind: $(a)-2(b)+(-a+2b)$. \end{lemma} {\bf Proof of the lemma \ref{4.11}}. Any function $f$ can be decomposed into a product of the functions with divisors of the following kind: $(a)-( b)-( c)+(-a+b+c)$. Indeed, let $(f)=\sum n_a (a)$. We will use induction on $\sum|n_a|$. Since $\sum n_a =0$, replacing if needed $f$ by $f^{-1}$ we can find points $a,b,c $ such that $n_a>0$, $n_b < 0$, $n_c < 0$ . Then $ (f)-[(a)-(b)-(c)+(-a+b+c)]$ is a principal divisor with smaller $\sum|n_a|$. Further $(a)-(b)-(c)+(-a+b+c)= [(a)-2(d)+(-a+b+c)]- [(b)-2(d)+(c)$ for $2d=b+c$. {\bf Proof of the proposition (\ref{ref})}. According to the lemma we can assume that $(f)=(b)-2(0)+(-b)$, $(g)=(c)-2(d)+(-c+2d)$. Therefore $(t_af)=(b+a)-2(a)+(-b+a)$, $(t_ag)=(c+a)-2(d+a)+(-c+2d+a)$. The quotient of $E$ under the involution $\sigma _a: x \to a-x$ is isomorphic to $\Bbb P^1$. The symbol $\{t_af,t_ag\}+\{\sigma _at_af,\sigma _at_ag\}$ is $\sigma _a$-invariant, so it comes from $K_2(k(\Bbb P^1) )$. It is known that $K_2 (k(\Bbb P^1) )$ is generated by $k(\Bbb P^1)^*\otimes k^*$. So $$\{f,g\}-\{t_af,t_ag\} \sim \{f,g\}+\{\sigma _at_af,\sigma _at_ag\}= \{f,g\cdot \sigma _0g\}\sim 0$$ where $x \sim y$ means $x-y =0$ in $\frac{ K_2(k(E))}{k^*\cdot (k(E))^*}$. Indeed, $\sigma _at_a =\sigma _0$, $\sigma _0f=f$, and the symbol $\{f,g\cdot \sigma _0g\}$ is $ \sigma_0 $-invariant. Therefore it is $\sim 0$ by lemma (\ref{BT}). {\it Step 2}. Let $A$ be an abelian group. Let $S_k(A) \subset S^kI_A $ be the subgroup generated by the elements $$ ((x_1)\ast X_1) \circ ... \circ ((x_k) \ast X_k) - X_1 \circ ... \circ X_k, \quad x_1 + ... + x_k =0, \quad x_i \in A, \quad X_i \in I_A $$ This subgroup clearly belongs to the kernel of the convolution map $$ S^kI_A \to I_A^k, \quad X_1 \circ ... \circ X_k \longmapsto X_1 \ast ... \ast X_k $$ So we get a homomorphism $ \alpha_k: S^kI_A/S_k(A) \longrightarrow I_A^k $. \begin{proposition} \label{schift} For any abelian group $A$ the homomorphism $\alpha_k$ is injective. \end{proposition} {\bf Proof}. We may assume that $A$ is finitely generated. Therefore the group ring $\Bbb Z[A]$ looks as follows: \begin{equation} \label{Lorant} \Bbb Z[A] = \Bbb Z[t_1,...,t_a, t_1^{-1},...,t_a^{-1}] \times \prod_{a+1 \leq j \leq m}\frac{\Bbb Z[t_j]}{(t_j^{N_j} -1)} \end{equation} Under this isomorphism the augmentation ideal $I_A$ goes to the maximal ideal $(t_1 - 1,...,t_{m}-1)$. The subgroup $S_k(A)$ is generated by the elements \begin{equation} \label{rant} (t_{i_1} - 1) f_1 \circ ... \circ (t_{i_{k-1}} - 1) f_{k-1} \circ (t_{i_k} - 1) f_k \quad - \end{equation} $$ (t_{i_1} - 1) \circ ... \circ (t_{i_{k-1}} - 1) \circ (t_{i_k} - 1) f_1 ... f_k $$ So any element of the quotient $S^kI_A / S_k(A)$ can be written as \begin{equation} \label{orant} (t_{i_1} - 1) \circ ... \circ (t_{i_{k-1}} - 1) \circ (t_{i_k} - 1) f \end{equation} The homomorphism $\alpha_k$ sends it to $(t_{i_1} - 1) ... (t_{i_{k-1}} - 1) (t_{i_k} - 1) f$. Let us suppose first that $a >0$, i.e. $A$ is an infinite group. We will use the induction on both $m$ and $k$. The case $m=1$ is trivial: if $(t-1)^k f =0$ then $(t-1) \circ ... \circ (t-1) f =0$, even in the case $f(t) \in \frac{\Bbb Z[t_j]}{(t_j^{N_j} -1)}$. Consider an element \begin{equation} \label{rantt} P= \sum_j(t_{i_1(j)} - 1) \circ ... \circ (t_{i_k(j)} - 1) f_j \in Ker \alpha_k \end{equation} Any element $f$ of the right hand side of (\ref{Lorant}) can be written as $ f'(t_2,...,t_m) + (t_1-1) f''(t_1,...,t_m)$, So writing $f_j = f'_j + (t_1-1)f''_j$ and setting $P': = \sum_j(t_{i_1(j)} - 1) \circ ... \circ (t_{i_k(j)} - 1) f_j'$ we get $P = P' + (t_1-1) \circ Q''$. Further, let $P'_{I}$ be the sum of all the terms of $P'$ where non of the indices $i_l(j)$ equal to $1$. Then $P' = P'_{I} + (t_1-1) \circ Q'_{II}$. The restriction of $P'_{I}$ to the divisor $t_1=1$ in $Spec$ $\Bbb Z[A]$ coincides with the restriction of $P$. Therefore it belongs to $Ker \alpha_k$ and thus is zero by the induction asumption for $(k,m-1)$. Since by the definition $P'_{I}$ does not depend on $t_1$, this implies $P'_{I}=0$. Thus $P = (t_1-1) \circ Q$ for some $Q \in S^{k-1}I_A / S_{k-1}(A)$. Therefore $\alpha_{k-1}(Q) =0$ since $t_1-1$ is not a divisor of zero (we have assumed $a>0$). Thus $Q=0$ by the induction assumption for $(k-1,m)$. Now let $A$ be a finite group. Decomposing $f(t)$ in (\ref{orant}) into a sum of monomials $(t_1-1)^{a_1} ... (t_q-1)^{a_q}$ we can write an element (\ref{orant}) as a sum \begin{equation} \label{iiii} P = \sum (t_1-1)^{b_1} \circ ... \circ (t_k-1)^{b_k} (t_{k+1}-1)^{b_{k+1}} ... (t_{k+l}-1)^{b_{k+l}} \end{equation} We will treat for a moment this sum as element of $S^k\Bbb Z[t_1,...,t_m]$, not $S^kI_A/S_k(A)$. Let $$ \tilde \alpha_k(P):= \sum (t_1-1)^{b_1} ... \cdot (t_{k+l}-1)^{b_{k+l}} \in \Bbb Z[t_1,...,t_m] $$ be its image in $\Bbb Z[t_1,...,t_m]$ under the product map. If $\tilde \alpha_k(P)=0$ then clearly $P=0$ modulo the relations (\ref{rant}) with $f_i \in \Bbb Z[t_1,...,t_m]$. Since $\alpha_k(P) =0$, $\tilde \alpha_k(P)$ is a linear combination of monomials of type $(t_i^{N_i}-1) \cdot Q(t)$. We are going to show that one can find another presentation $P'$ for the element in $S^kI_A/S_k(A)$ given by $P$ such that $\tilde \alpha_k(P') = \tilde \alpha_k(P) - (t_i^{N_i}-1) \cdot Q(t)$. Let us factorize $t_i^{N_i}-1 = (t_i -1)\cdot p_i(t_i-1)$, where $p_i(u) = \sum c_j u^j$ is a polynomial in one variable. Notice that $c_0 \not = 0$. We must have a monomial in the sum $P$ which goes under the map $\tilde \alpha_k$ to $c_0 \cdot (t_i -1) \cdot Q(t)$. It can be written (modulo $S_k(A)$) as $c_0 \cdot (t_i -1) \circ R$. Therefore the terms in (\ref{iiii}) which are maped by $\tilde \alpha_k$ to $ c_{j-1} (t_i -1)^j \cdot Q(t)$, $j>1$, can be written as $ c_{j-1}(t_i -1)^j \circ R$. Therefore the sum of these terms in (\ref{iiii}) can be written as $(t_i^{N_i}-1) \circ R$ and thus represent a zero element in $S^kI_A/S_k(A)$. The proposition is proved. {\it Step 3}. Let us write $\beta$ as a composition $\beta = i \circ \alpha$: $$ \otimes^2I^2_E \stackrel{i}{\longrightarrow} \otimes^2I^2_E \stackrel{\alpha}{\longrightarrow} I_E^4 $$ $$ i: A \otimes B \longmapsto A \otimes B^-; \quad \alpha: A \otimes B \longmapsto A \ast B $$ Let $S_E \subset \otimes^2I^2_E$ be the subgroup generated by the elements $(f) \otimes (1-f)^-$, $f \in k(E)^*$. One has $$ \otimes^2I^2_E = \frac{\otimes^2 k(E)^*}{k^*\otimes k(E)^* + k(E)^* \otimes k^*},\qquad \frac{\otimes^2I^2_E}{S_E} = \frac{ K_2(k(E))}{ k^* \cdot k(E)^*} $$ Let $A_+ $ (resp $A_- $) be the coinvariants of the involution $x \to -x$ on $E$ acting on a group $A$ functorially depending on $E$ (resp $A \otimes_{\Bbb Z}\lambda$, where $\lambda$ is the standard $\Bbb Z$-line where the involution acts by inverting the sign). \begin{lemma} \label{step13} $(Ker \alpha)_+ \subset S_E$. \end{lemma} {\bf Proof}. Let $x \in (Ker \alpha)_+$. Then $\partial (x) =0$, so $x$ defines an element of $ (H^0(E, {\cal K}_2)/K_2(k))_+$. But this group is zero modulo $2$-torsion by lemma (\ref{3.7}). \begin{lemma} \label{step14} $\Lambda^2I_E^2 \subset S_E$. \end{lemma} {\bf Proof}. $A \otimes B + B \otimes A$ belongs to the subgroup generated by the Steinberg relations $(f) \otimes (1-f)$. Thus $A \otimes B^- + B \otimes A^- \in S_E$. The lemma above states that $B \otimes A^- + B^- \otimes A \in S_E$. So $A \otimes B^- - B^- \otimes A \in S_E$. Let $\alpha': S^2I^2_E \to I_E^4, \quad A \circ B \longmapsto A \ast B$. To prove the theorem we need to show that $(Ker \alpha')_- \subset S_E$. Let $A_i \in I_E$. The element $$ < A_1,A_2,A_3,A_4>:= (A_1\ast A_2) \circ (A_3\ast A_4) - (A_1\ast A_3) \circ (A_2\ast A_4) $$ clearly belongs to $Ker \alpha $, and thus to $Ker \partial$. \begin{proposition} \label{step15} $<A_1,A_2,A_3,A_4 > \in S_E$. \end{proposition} To prove this we need the following lemma. \begin{lemma} \label{step16} If $A_i$ is a principal divisor for some $1 \leq i \leq 4$, then \linebreak $<A_1,A_2,A_3,A_4> \in S_E$. In particular we get a well defined homomorphism \begin{equation} \label{wdh} <\cdot,\cdot,\cdot,\cdot>: \otimes^4J \to S^2I^2_E/S_E \end{equation} \end{lemma} {\bf Proof}. Let us show that if $B_0,B_1 \in I_E$, then $<B_0 \ast B_1 ,A_2,A_3,A_4> \in S_E$ (the other cases are similar). We will write $A \stackrel{S_E}{=} B$ if $A -B \in S_E$. It follows from the proposition (\ref{ref}) that for $A,B \in I_E^2$ and $X \in \Bbb Z[E(k)]$ one has $(X\ast A) \circ B \stackrel{S_E}{=} A \circ (X\ast B)$. So $$ (B_0\ast B_1\ast A_2) \circ (A_3\ast A_4) \stackrel{S_E}{=} ( B_0\ast B_1) \circ ( A_2\ast A_3 \ast A_4) \stackrel{S_E}{=} (B_0\ast B_1\ast A_3) \circ (A_2\ast A_4) $$ Since $(a) -(0) + (b) -(0) - ((a+b) -(0)) \in I^2_E$, we get (\ref{wdh}). The lemma is proved. {\bf Proof of the proposition (\ref{step15})}. Let $(A)$ be the image of $A \in I_E$ in the Jacobian. Using $(A^-) = -(A)$ and the previous lemma we get $$ <A_1,A_2,A_3,A_4 > - <A_1^-,A_2^-,A_3^-,A_4^- >\in S_E $$ On the other hand according to the lemma (\ref{3.7}) $$ <A_1,A_2,A_3,A_4 > + <A_1^-,A_2^-,A_3^-,A_4^- > \in S_E $$ So $2<A_1,A_2,A_3,A_4 > \in S_E$. Using lemma ({\ref{step16}) and the divisibility (by $2$) of $J(k)$ we conclude that $<A_1,A_2,A_3,A_4 > \in S_E$. \begin{proposition} \label{11111} $Ker (S^2I^2_E \stackrel{\alpha'}{\longrightarrow} I^4_E)$ is generated by the elements $$ ((x)\ast A) \circ ((-x)\ast B) - A \circ B, \quad A,B \in I^2_E $$ and $ <A_1,A_2,A_3,A_4 >$, $A_i \in I_E$. \end{proposition} {\bf Proof}. Let $T \subset I^2_E$ be the subgroup generated by the elements \linebreak $<A_1,A_2,A_3,A_4 > $. There is a surjective homomorphism $$ S^4I_E \longrightarrow S^2(I^2_E) /T , \quad A_1 \otimes ... \otimes A_4 \longmapsto ( A_1\ast A_2) \circ (A_3 \ast A_4 ) $$ The proposition follows immediately from the proposition (\ref{schift}). Theorem (\ref{mmmm}) follows from proposition (\ref{11111}) (\ref{schift}) and (\ref{step16}). \begin{proposition} \label{qc} Assume that $k = \bar k$. Then a) There is an injective homomorphism of complexes $B^*(E;3) \to B(E;3)$. b)The quotient $B(E;3)/B^*(E;3)$ is isomorphic to the Koszul complex \begin{equation} \label{kc} S^3J \longrightarrow S^2J \otimes J \longrightarrow J \otimes \Lambda^2J \longrightarrow \Lambda^3J \end{equation} In particular the complexes $B(E;3)$ and $B^*(E;3)$ are quasiisomorphic. \end{proposition} We will need \begin{lemma} \label{destr} Suppose that $$ D:= m \sum_i(\{a_i\} - m \sum_{mb_i = a_i}\{b_i\}) \in I_E^4 $$ Then $D$ belongs to the subgroup generated by the elements $(f) \ast (1-f)^-$. \end{lemma} Let $[m]:E \to E$ be the isogeny of multiplication by $m$. \begin{lemma} \label{destr1} Let $f,g \in k(E)^*$, $k = \bar k$. Then $$ [m]^*\{f,g\} = m\{f,g\} \quad \mbox{in} \quad \frac{K_2(k(E)}{k^* \cdot k(E) } $$ \end{lemma} {\bf Proof of the lemma (\ref{destr1})}. One has an exact sequence $$ 0 \longrightarrow \frac{ H^0(E,{\cal K}_2)}{ K_2(k) + Tor(k^*,J)}\longrightarrow \frac{K_2(k(E))}{k^* \cdot k(E) } \longrightarrow k^* \otimes J $$ The operator $[m]_*$ acts on the left (different from $0$) and right groups by multiplication by $m$ (see [BL], ch. 5,6 where this was proved rationally; that proof works integrally). Further, $[m]_*$ has no Jordan blocks since $[m]_*[m]^* =m^2$. The lemma follows. {\bf Proof of the lemma (\ref{destr})}. There is an isomorphism $$ j: \Bbb Z[E(k)]/I^4_E = \Bbb Z \oplus J \oplus S^2J \oplus S^3J, \qquad \{a\} \longmapsto (1 , a , a\cdot a , a \cdot a \cdot a) $$ One has \begin{equation} \label{dizr} j(m(\{a\} - m\sum_{mb=a}\{b\})) = m((1-m^3), (1-m^2)a, (1-m)a \cdot a, 0) \end{equation} Using this we see that $\sum_i\{a_i\} \in I_E^4$. Thus $\sum_i\{a_i\} = \sum(f_j) \ast (g_j)^-$. It is easy to see that $\beta \Bigl([m]^*\sum_i\{f_i,g_i\} - m\{f_i,g_i\}\Bigr) = D$. The lemma is proved. {\bf Proof of the proposition (\ref{qc})}. a) We need only to show that $B^*_3(E)$ injects to $B_3(E)$. This boils down to the lemma (\ref{destr}) above, since in our definition of $R_3(E)$ we used only the distribution relations for $m=-1,2$. b) By definition $B_3(E)/B^*_3(E)$ is isomorphic to the quotient of $\Bbb Z[E(k)]/I^4_E$ modulo (the image of) the distribution relations. Using the computaion (\ref{dizr}) and divisibility of $J(k)$ we see that the map $j$ maps the subgroup generated by the distribution relations for any given $|m|>1$ surjectivly onto $2\Bbb Z \oplus J \oplus J^2$. Therefore $ B_3(E)/B^*_3(E) = S^3J $. Let us recall that $B_2(E)/k^* = S^2J$. So the terms of the quotient $B(E;3)/B^*(E;3)$ are the same as in (\ref{kc}). It is easy to see that the differentials coincide. The proposition is proved. {\bf 3. Zagier's conjecture on $L(E,2)$ for modular elliptic curves over $\Bbb Q$}. Let us recall that for a curve $X$ over $\Bbb R$ one has $H_{{\cal D}}^2(X/\Bbb R,\Bbb R(2)) = H^1(X/\Bbb R,\Bbb R(1))$. Let $\bar X := X \otimes \Bbb C$. The cup product with $\omega \in \Omega^1(\bar X)$ provides an isomorphism of vector spaces over $\Bbb R$: $$ H^1(X/\Bbb R,\Bbb R(1)) \longrightarrow H^0(\bar X, \Omega^1)^{\vee} $$ So we will present elements of $H_{{\cal D}}^2(X/\Bbb R,\Bbb R(2))$ as functionals on $H^0(\bar X, \Omega^1)$. Bloch constructed the regulator map $$ r_{{\cal D}}: K_2(E) \longrightarrow H^2_{{\cal D}}(E,\Bbb R(2)) $$ If we represent an element of $K_2(E)$ as $\sum_i\{f_i,g_i\}$ (with all the tame symbols vanish) then Beilinson's construction of the regulator looks as follows: $$ <r_{{\cal D}}\sum_i\{f_i,g_i\},\omega > = \frac{1}{2 \pi i}\sum_i \int_{E(\Bbb C)} \log|f_i|d \arg(g_i) \wedge \omega $$ Let $f$ and $g$ be rational functions on $E$ such that $$ (f)= \sum n_i(a_i), \quad (g)= \sum m_j(b_j) $$ Let $\Gamma = H_1(E(\Bbb C), \Bbb Z)$. We may assume that $\Gamma = \{\Bbb Z \oplus \Bbb Z\cdot \tau\} \subset \Bbb C$ and $z$ is the coordinate in $\Bbb C$. Let us briefly recall how the regulator integral $<r_{{\cal D}}\{f,g\},dz>$ is computed by means of the elliptic dilogarithm ([Bl1], see also [RSS]). The intersection form on $\Gamma$ provides a pairing \begin{equation} \label{myu} (\cdot,\cdot): E(\Bbb C) \times \Gamma \longrightarrow S^1; \qquad (z,\gamma):= exp(\frac{2\pi i (z{\bar \gamma} - {\bar z}\gamma )}{\tau - {\bar \tau}}) \end{equation} Let $$ K_{2,1}(z;\tau):= \frac{(Im \tau)^2}{ \pi }\sum_{\gamma \in \Gamma \backslash 0}\frac{(u,\gamma)}{\gamma^2{\bar \gamma}}, \qquad z = exp(2 \pi i u) $$ Then one has $$ \frac{1}{2 \pi i}\int_{E(\Bbb C)}\log|f| d \arg g \wedge dz = \frac{1}{i\pi }\sum_{a,b \in E(\Bbb C)}v_a(f)v_b(g)K_{2,1}(a - b;\tau) $$ To prove this one may use that $$ \int_{E(\Bbb C)}\log|f| d i \arg g \wedge dz = - \int_{E(\Bbb C)}\log|f| d \log|g| \wedge dz $$ together with the following lemma and the fact that the Fourier transform sends the convolution to the product. \begin{lemma} \label{weil} \begin {equation} \label{wei2} \log |f(z)| = -\frac{Im \tau}{2\pi}\sum_{\gamma \in \Gamma \backslash 0}v_a(f)\frac{(z-a,\gamma)}{|\gamma|^2} +C_f \end {equation} where $C_f$ is a certain constant. \end{lemma} {\bf Proof}. One can get a proof applying $\partial \bar \partial$ to the both parts of (\ref{wei2}). The constant $C_f$ can be computed from the decomposition of $f$ on the product of theta functions using the formula in s. 18 ch. VIII of [We]. It does not play any role in our considerations since $ \int_{E(\Bbb C)} C_f \cdot d\log|g| \wedge \omega =0 $ by the Stokes formula. The relation between the Eisenstein-Kronecker series $K_{2,1}(z)$ and the elliptic dilogarithm is the following ( [Bl1], [Z]): $ K_{2,1}(z;\tau)= {\cal L}_{2,q}(z) - iJ_q(z)$, where the function $J_q(z)$ is defined as follows. Let us average the function $J(z):= \log|z|\log|1-z|$ over the action of the group $\Bbb Z$ generated by the shift $z \longmapsto qz$ regularizing divergencies. We will get $$ J_q(z):= \sum_{n=0}^{\infty}J(q^nz) - \sum_{n=1}^{\infty}J(q^nz^{-1}) + \frac{1}{3}(\log |q|)^2 \cdot B_3(\frac{\log |z|}{\log |q|}) $$ Here $B_3(x)$ is the third Bernoulli polynomial. The function $J_q(z)$ is invariant under the shift $z \longmapsto qz$ and satisfies $J_q(z) = -J_q(z^{-1})$. It follows from the main result of Beilinson in [B2], see also [SS2], that for a modular elliptic curve $E$ over $\Bbb Q$ there always exists an element in $K_2(E)_{\Bbb Z}$ whose regulator gives (up to a standard nonzero factor) $L(E,2)$. So we get the formula $$ L(E,2) \sim_{\Bbb Q^*} \pi \cdot \sum_{i}\sum_{a,b \in E(\Bbb C)}v_a(f^{(i)})v_b(g^{(i)}){\cal L}_{2,q}(a -b) $$ Finally, the results of the present section implies that the element \linebreak $\sum_{i} \sum_{a,b \in E(\Bbb C)}v_a(f^{(i)})v_b(g^{(i)})\{a - b\}_3 \in I^4_{E(\bar \Bbb Q)}$ must satisfy all the conditions of the theorem(\ref{zcc}). Theorem (\ref{zccc}) follows from the surjectivity of the map $\beta$ and the arguments above. {\it The integrality condition} ([BG], [SS]). Let $E$ be an elliptic curve over $\Bbb Q$. Choose a minimal regular model $E_{\Bbb Z}$ of $E$ over $ \Bbb Z$. One has the exact sequence \begin{equation} \label {blgr} K_2(E_{ \Bbb Z}) \longrightarrow K_2(E_{\Bbb Q}) \stackrel{\partial}{\longrightarrow} \oplus_{p } K_1'(E_{p}) \end{equation} The group $K_1'(E_{p}) \otimes \Bbb Q$ is not zero if and only if $E_{p}$ has a split multiplicative reduction with special fibre a N\'eron $N$-gon. In this case $K_1'(E_{p}) \otimes \Bbb Q = \Bbb Q$. Consider an element $\sum_i \{f_i,g_i\} \in K_2(\Bbb Q(E))$ which has zero tame symbol at all points. It defines an element of $H^0(E,{\cal K}_2)$. Suppose first that the closure of the support of the divisors $f_i,g_i$ is contained on the smooth part of $E_{\Bbb Z}$. Then Schappaher and Scholl ([SS]) proved that the image of this element under the map $\partial$ is computed by the following formula: $$ \partial (\sum_i \{f_i,g_i \} = \pm \frac{1}{3N} \sum_{ \nu \in \Bbb Z/N\Bbb Z}d((f) \ast(g^-);\nu)B_3(\frac{\nu}{N})\cdot\Phi $$ here $\Phi$ is a generator in $K_1'(E_{ p}) \otimes \Bbb Q$. In general one should extend $\Bbb Q$ to $\Bbb Q((f_i),(g_i))$, which is the field of the definition of the divisors $(f_i), (g_i)$. After this we get precisely the condition (\ref{icond121}). Let us explane why the expression $\sum_{\nu \in \Bbb Z/(eN) \Bbb Z}d(P;\nu)B_3(\frac{\nu}{eN})$ does not depent on the field $L$. Let $L$ and $L'$ are two extentions of $\Bbb Q_p$ such that all the points are defined over them. We can assume that $L\subset L'$ (by taking the composit). Denote by $n_1=e_1 f_1$ degree of this extension. Looking at the Tate uniformization we see that points which intersect $\nu$-th component of special fiber corresponding to the field $L$ intersect $e_1\nu$-th component of special fiber corresponding to the field $L'$. {\bf 4. Main results from the motivic point of view}. Let ${\cal M}{\cal M}_X$ be the (hypothetical) abelian category of all mixed motivic sheaves over a regular scheme $X$ over a field $k$. Let $\Bbb Q(-1) := h^2(P^1)$, $\Bbb Q(n):= \Bbb Q(1)^{\otimes n}$ and ${\cal H}:= h^1(E)(1)$. The motivic refinement of our results is the following \begin{conjecture} \label{conjmainmot} There exists a canonical quasiisomorphism in the derived category $$ B(E,3) \otimes \Bbb Q= RHom_{{\cal M}{\cal M}_k}(\Bbb Q(0), {\cal H}(1)) $$ \end{conjecture} Let us explain how it fits with our results. Let $\pi: E \longrightarrow Spec (k)$ be the structure morphism. There are the Tate sheaves $\Bbb Q(n)_E := \pi^{\ast} \Bbb Q(n)$. Beilinson's description of Ext groups between the Tate sheaves over $E$ gives us \begin{conjecture} \label{ext1} $$ Ext^i_{{\cal M}{\cal M}(E)}(\Bbb Q(0)_E, \Bbb Q(2)_E) = gr^{\gamma}_2 K_{ 4-i}(E) \otimes \Bbb Q $$ \end{conjecture} \begin{lemma} \begin{equation} \label{mfor} RHom_{{\cal M}{\cal M}_k}(\Bbb Q(0), {\cal H}(1)) = RHom_{{\cal M}{\cal M}_{E}}(\Bbb Q(0),\Bbb Q(2))^{ -} \end{equation} \end{lemma} Indeed, let $p:E \longrightarrow Spec (k)$ be the canonical projection. Then we should have the motivic Leray spectral sequence $$ E_2^{p,q}= Ext^{p}_{{\cal M}{\cal M}_k}\Bigl(\Bbb Q(0), R^qp_{\ast}\Bbb Q( 2)\Bigr) $$ degenerating at $E_2$ and abutting to $Ext^{p+q}_{{\cal M}{\cal M}_{E }}\Bigl(\Bbb Q(0), \Bbb Q( 2)\Bigr)$. Noting that $$ h^{0}(E )^{ -} = h^{2}(E )^{ -} = 0; \qquad h^{ 1} (E )^{ -} = h^1(E) $$ we get (\ref{mfor}). Conjecture (\ref{ext1}) together with this lemma tell us that the cohomology of the elliptic motivic complexes are given by the formula \begin{equation} \label{gipo++} R^iHom_{{\cal M}{\cal M}_k}(\Bbb Q(0), {\cal H}( 1)) \otimes \Bbb Q = gr^{\gamma}_{ 2}K_{ 3-i}(E )^{ -}\otimes \Bbb Q \end{equation} Conjecture (\ref{conjmainmot}) follows from theorem (\ref{za}) and conjecture (\ref{ext1}), see [G2]. A very interesting and important is the following: {\bf Problem}. To construct explicitely the general elliptic motivic complexes $$ RHom_{{\cal M}{\cal M}_k}(\Bbb Q(0), Sym^n{\cal H}(m)) $$ For $m=1$ it is considered in [G2]. {\bf 5. Degeneration to the nodal curve (compare with [Bl1], [DS])}. Let $k$ be an algebraicly closed field. Denote by ${\cal I}$ the group of rational functions such that $f(0) = f(\infty) =0$. Let $I_{k^*}$ be the augmentation ideal of $\Bbb Z[k^*]$. For an element $f \otimes g \in (1+{\cal I} )\otimes k(t)^*$ consider the Bloch map $$ \beta(f\otimes g):= \sum_{x,y \in k^*} v_x(f) v_y(g) \{y/x\} + v_{\infty}(g)\cdot ((f) + (f^-)) \in I^2_{k^*} $$ Set \begin{equation} \Bbb Z[k^*] \stackrel{ \delta}{\longrightarrow} k^* \otimes k^*, \qquad \{x\} \longmapsto (1-x) \otimes x, \quad \{1\} \longmapsto 0 \end{equation} Let $p: I_{k^*} \to k^*$ be the natural projection $\{x\} \to x$. \begin{theorem} The following diagram is commutative: $$ \begin{array}{ccc} (1+{\cal I} )\otimes k(t)^*& \stackrel{ \partial}{\longrightarrow} & I_{k^*} \otimes k^*\\ &&\\ \beta \downarrow &&\downarrow p \otimes id\\ &&\\ I^2_{k^*}&\stackrel{\delta}{\longrightarrow}&k^* \otimes k^* \end{array} $$ \end{theorem} The proof is a direct calculation similar to (but simpler then) the proof of theorem (\ref{commm}). Let $S(k)$ be the subgroup generated by the elements $(1-f) \otimes f$ where $f \in {\cal I} $ and the subgroup $(1+ {\cal I})\otimes k^* $. Set $B^*_2(k) := I^2_{k^*} /\beta(S(k))$. \begin{lemma} $Ker \beta \subset S(k)$ \end{lemma} {\bf Proof}. An easy analog of theorem (\ref{ref}) for the nodal curve claims that the homomorphism $$ q: k(t)^* \longrightarrow (1+ {\cal I})\otimes k(t)^*, \quad f(t) \longmapsto f(x) \otimes (x-1) $$ is surjective. It is clear that $\beta \circ q$ is injective. The lemma is proved. This lemma implies that $\bar \beta: (1+{\cal I})\otimes k(t)^* \to B^*_2(k)$ is an isomorphism. So we get a morphism of complexes $$ \begin{array}{ccc} \frac{(1+{\cal I})\otimes k(t)^*}{ S(k)}& \stackrel{\partial}{\longrightarrow} & k^* \otimes k^*\\ &&\\ \bar \beta \downarrow &&\downarrow id\\ &&\\ B^*_2(k)&\stackrel{\delta}{\longrightarrow}&k^* \otimes k^* \end{array} $$ Such that $\bar \beta$ is surjective and $Ker \bar \beta = Tor(k^*,k^*)$, similar to the theorem (\ref{zaza}). So we see that when $E$ degenerates to a nodal curve the complex $B^*(E/k,3)$ degenerates to the complex $B_2^*( k ) \to k^* \otimes k^*$. Let $S_1(k)$ be the subgroup generated by the elements $(1-f) \otimes f$ where $f \in {\cal I} $. \begin{theorem} Let $k = \bar k$. The identity map on $\Bbb Z[k^*]$ provides a homomorphism of groups $B_2^*( k ) \to B_2(k)$. Its kernel is isomorphic to $S^2k^*$. This map provides a quasiisomorphism of complexes $$ \begin{array}{ccc} B_2^*( k ) & \longrightarrow & k^* \otimes k^*\\ \downarrow&&\downarrow\\ B_2( k ) & \longrightarrow & k^* \wedge k^* \end{array} $$ \end{theorem} One can show that the group $B_2^*( k )$ is isomorphic to a group defined by S. Lichtenbaum in [Li]. It is known ([Lev]) that $$ K_2(\Bbb P^1_{\{0,\infty\}}, \{0,\infty\}) = \frac{(1+{\cal I})\otimes k(t)^*}{ S_1(k)} $$ Using this and the results of this subsection we can get a proof of Suslin's theorem for $k = \bar k$. {\bf 5. On an elliptic analog of the $5$-term relation for the elliptic dilogarithm}. Notice that $$ B_2(F) = Coker (\Bbb Z[ M_{0,5}(k)] \stackrel{\partial}{ \longrightarrow} \Bbb Z[\Bbb G_m(k) ]) $$ Here $M_{0,5}$ is the configuration space of $5$ distict points on the projective line. So one may ask how to present the group $B_3(E)$ in a similar form: $$ B_3(E) \stackrel{?}{=} Coker (\Bbb Z[ X(k)] \stackrel{\partial}{ \longrightarrow} \Bbb Z[E(k) ]) $$ where $X$ is a {\it finite dimensional} variety. In the our definition we have an infinite dimensional $X$ (more precisely it is an inductive limit of finite dimensional varieties). Here is a guess. Let us realize $E$ as a cubic in $P^2$. Let $p$ be a point in $P^2$ and $l_1,l_2,l_3$ any three lines through this point. Set $A_i := l_i \cap E$. Let $\tilde R^*_3(E) \subset I^4_E$ be the subgroup generated by the elements \begin{equation} \label{lll} \{p;l_1,l_2,l_3\}:= A_1 \ast A_2^- + A_2 \ast A_3^- + A_3 \ast A_1^- \end{equation} and those linear combinations of the elements $\{a\}_3 + \{-a\}_3 $ which lie in $I^4_E$. ( Probably they belong to the subgroup generated (\ref{lll})). \begin{lemma} $\tilde R^*_3(E) \subset R^*_3(E)$. \end{lemma} {\bf Proof}. Let $f_i$ be a linear homogeneous equation of the line $l_i$. Since the lines $l_1,l_2,l_3$ intersect in a point, these equations are linearly dependent, so we may choose them in such a way that $f_1 + f_2 = f_3$. Thus $f_1/f_3 \wedge f_2/f_3 = f_1/f_3 \wedge (1- (f_1/f_3))$. Applying the map $\beta$ to this Steinberg relation and using the relations $\{a\}_3 + \{-a\}_3 =0$ we get the element $\{p;l_1,l_2,l_3\}$. One obviously has $\sum_{j=1}^4 (-1)^j \{p;l_1, ... \hat l_j, ..., l_4\} =0$, so we can assume that the line $l_3$ is, say, a vertical line. \begin{conjecture} $\tilde R^*_3(E) = R^*_3(E)$. \end{conjecture} \section{ $B_2(E)$, $\theta$-functions and action of isogenies } {\bf 1. Elliptic curves over $\Bbb C$}. Let us represent $E$ as a quotient $\Bbb C/{\Bbb Z}+{\Bbb Z}\tau$. Below $\xi$ denotes the coordinate on $\Bbb C$. The canonical trivialization of $(T^{*}_0E)^{12}$ is $-16\prod (e_j -e_i)(d\/\xi )^{12}$, which is equal to $ \Delta (\tau ) (d\/\xi)^{12}$, where $$ \Delta (\tau ) = (2\pi i)^{12}{\eta ^{24} (\tau )}, \quad \eta (\tau )= q^{\frac{1}{24}}\prod_{n=1}^{\infty}(1-q^n) \quad (q=\exp (2 \pi i \tau)) $$ So the trivialization of $(T^{*}_0E)^2$ is $ (2\pi i)^2{\eta ^4 (\tau )} (d\/\xi)^2$. Now we will give the analytic description of the Deligne pairing $[\ast ,\ast]$. Let $L_a$ be the line bundle corresponding to the divisor $(a)-(0)$. Choose a representative $\alpha \in \Bbb C$ of $a$ ($\alpha$ is defined up to ${\Bbb Z}+{\Bbb Z}\tau$). Let us define $L_{\alpha}$ on $E$ as the quotient of the trivial line bundle on $\Bbb C$ under the action of ${\Bbb Z}+{\Bbb Z}\tau$~: $1$ acts trivially and $\tau$ acts by multiplication by $\exp (2 \pi i \alpha )$. We identify $L_{\alpha +1 }$ with $L_{\alpha }$ trivially and $L_{\alpha +\tau }$ with $L_{\alpha}$ by multiplication by $\exp (2 \pi i \xi )$. The fiber of the Poincare line bundle $[L_{\alpha},L_{\beta}]$ on $J\times J$ over the point $(\alpha, \beta)$ is equal to $L_{\alpha}|_{\beta}\otimes L_{\alpha}^{-1}|_{0}$. This line bundle is described as a quotient $$ \frac{\Bbb C \times \Bbb C \times \Bbb C }{(\Bbb Z \oplus \Bbb Z \tau) \oplus (\Bbb Z \oplus \Bbb Z \tau) } $$ $$ (\alpha, \beta, \lambda) \longmapsto (\alpha + m + n \tau, \beta + m' + n' \tau, \lambda \cdot exp(2\pi i(n\beta + n'\alpha + nn' \tau))) $$ Let us show that $L_{\alpha}=L_{a}$. Consider a slight modification of the Jacobi $\theta$-function ($z = exp(2 \pi i \xi)$): $$ \theta (\xi) = \theta (\xi; \tau ) = q^{1/12} z^{-\frac{1}{2}} \prod_{j \geq 0} (1-q^j z)\prod_{j > 0} (1-q^j z^{-1}) $$ Then $\displaystyle \frac{\theta (\xi -\alpha )}{\theta (\xi )\theta (\alpha )}$ is a section of $L_{\alpha}$ with the required divisor $(a) - (0)$. Our recipe for the calculation of the element $$<(a) - (0), (a) - (0)> \in (T^{\ast}_0E)^{\otimes 2}\otimes_{k^{\ast}} [a\cdot a]$$ leads to the expression $$ \frac{d\/\xi \cdot \theta '(0)}{\theta ^2(\alpha )} ( \frac{\theta ( -\alpha )}{d\/\xi \cdot \theta '(0 )\theta (\alpha )})^{-1}= -(\frac{d\/\xi \cdot \theta '(0)}{\theta (\alpha )})^2.$$ Notice that $\theta '(0) =2 \pi i \eta ^2(\tau )$ and the chosen analytic trivialization of $(T^*_0E)^2$ is $ (2\pi i )^2{\eta (\tau )}^4 (d\/\xi)^2$. So the final answer is $-\theta(\alpha)^{-2}$. {\bf 2. The Tate curves}. Let $K$ be a field complete with respect to a discrete valuation $v$. Let ${\cal O}=\{a\in K|v(a)\leq 0\}$ be the ring of integers, $I=\{a\in K|v(a) < 0\}$ the maximal ideal, $k={\cal O}/I$ the residue field. Let $q\in I$. According to Tate, the group $K^{*}/ q^{\Bbb Z}$ is isomorphic to the group of points of the elliptic curve $E_{q}$ over $K$ given by equation $ y^2+xy = x^3 +a_4x +a_6$, where $$a_4 =-5\sum_{j\geq 1}\frac{j^3q^j}{1-q^j}; \quad a_6 =-\frac{1}{12}\sum_{j\geq 1} \frac{(7j^5+5j^3)q^j}{1-q^j}$$ The discriminant and $j$-invariant of this curve are: $\Delta =q\prod_{j\geq 1}(1-q^j)^{24}, \quad j=\frac{1}{q} + 744 +196884q+\cdots$. The map of $K^{*}/ q^{\Bbb Z}$ to the group of points of $E_q$ is defined by the following expressions: $$ x(u)=\sum_{j\in \Bbb Z }\frac{q^ju}{(1-q^j u)^2} -2\sum_{j\ge 1}\frac{jq^j}{1-q^j}; \quad y(u)=\sum_{j\in \Bbb Z}\frac{q^{2j}u^2}{(1-q^ju)^3} +\sum_{j\ge 1}\frac{jq^j}{1-q^j}. $$ The unity $1$ of $K^*$ maps to neutral element $0$ of the curve. Define a function $T(u)$ on $K^{*}$ by the formula: $$T(u)=\prod_{j\geq 0}(1-q^ju) \prod_{j> 0}(1-q^ju^{-1}) $$ This function vanishes on $\{q^{\Bbb Z}\}$. It is quasiperiodic: $T(uq) = -u^{-1}T(u)$. Let $a\in {\cal O}\setminus q{\cal O}$. Denote by the same symbol its image in $E_q$. Then a section $s$ of the bundle ${\cal O}_{E_q} ((a) -(0))$ can be represented by a function $f$ on $K^*$ such that $f(uq)= af(u)$. Indeed, the periodic function $f(u)T(u)(T(ua^{-1}))^{-1}$ has the required divisor. Like in the analytic case, the total space of the Poincar\'e line bundle $T_{(a,b)}$ is isomorphic to the quotient $K^*\times K^*\times K^*$ modulo the action of the group $\Bbb Z \oplus \Bbb Z$ generated by the following transformations: $$ (a, b, \lambda )\to (qa, b, b\lambda );\quad (a, b, \lambda )\to (a, qb, a\lambda ). $$ The corresponding group structure on the collection $T_{(a,\cdot)}$ is defined by the law: $(b_1,\lambda _1)\times (b_2,\lambda _2)=(b_1b_2, \lambda_1\lambda_2)$. Let us calculate the expression $<(a)-(0),(a)-(0)>$. The divisor of the section $T(u)^{-1}(T(ua^{-1}))$ equals $(a)-(0)$. The regularized value of this expression at the divisor $(a)-(0)$ is equal to: $$ \frac{\displaystyle -\frac{d\,u}{u} (\prod_{j> 0}(1-q^j ))^2} {\displaystyle\prod_{j\geq 0}(1-q^ja) \prod_{j> 0}(1-q^ja^{-1})}\times \left(\frac {\displaystyle\prod_{j\geq 0}(1-q^ja^{-1}) \prod_{j> 0}(1-q^ja)} {\displaystyle - \frac{d\,u}{u} (\prod_{j> 0}(1-q^j ))^2} \right)^{-1}= $$ $$ =-\left(\frac{\displaystyle \frac{d\,u}{u} (\prod_{j> 0}(1-q^j ))^2} {\displaystyle a^{-\frac{1}{2}}\prod_{j\geq 0}(1-q^ja) \prod_{j> 0}(1-q^ja^{-1})}\right)^2 . $$ In this calculation notice that $(1-u)\cdot u = -(u-1)$ modulo $(u-1)^2$, so the factor $1-u$ leads to $-\frac{du}{u}$. The trivialization of $(T^*_0E)^{\otimes 12}$ is defined by the section $$(\frac{d\,u}{u})^{12}\Delta= \left(\left( \frac{d\,u}{u}\right)^{2}q^{\frac{1}{6}} (\prod_{j> 0}(1-q^j ))^4\right)^6$$ Hence the needed expression is: $$-\left(\frac{\displaystyle\frac{d\,u}{u} (\prod_{j> 0}(1-q^j ))^2} {\displaystyle a^{-\frac{1}{2}}\prod_{j\geq 0}(1-q^ja) \prod_{j> 0}(1-q^ja^{-1})}\right)^2 \left(\left( \frac{d\,u}{u}\right)^{2}q^{\frac{1}{6}} (\prod_{j> 0}(1-q^j ))^4\right)^{-1}$$ $$= - \Bigl(q^{\frac{1}{12}} a^{-\frac{1}{2}}T(a) \Bigr)^{-2}. $$ {\bf 3. Calculation of the canonical height}. a) {\it Archimedean case}. Let us calculate the archimedean height. The torsor $T_{(a ,0)}$ is trivialized. So the group $T_{(a,\cdot)}$ is isomorphic to the quotient of $\Bbb C\times \Bbb C^*$ with coordinates $(\beta ,\lambda )$ by the action of the group $\Bbb Z +\Bbb Z \tau$: $$ 1\colon (\beta ,\lambda ) \to (\beta +1,\lambda ),\quad \tau\colon (\beta ,\lambda )\to (\beta +\tau,\lambda \times \exp ( 2\pi i \alpha )) $$ A homomorphism $|(\cdot,\cdot )|_{\alpha}: \Bbb C\times \Bbb C^* \to \Bbb R^+$ which is invariant under the action of $\Bbb Z +\Bbb Z \tau$ and coincides with the norm $|\cdot|$ on $\Bbb C^*$ if $\beta =0$ is given by $$ | (\beta, \lambda)|_{\alpha}= \exp(-\pi i \frac{({\alpha}-\bar{\alpha}) ({\beta}-\bar{\beta)}}{{\tau}-\bar{\tau}}) |\lambda | $$ It defines a homomorphism $T_{(\alpha ,\cdot)} \to \Bbb R^+$. Therefore the value of our height at $\alpha$ equals to $$ \log |(\alpha, -(\frac{ 1}{\theta (\alpha )})^2)|_{\alpha}= 2[-(\log|\theta (\alpha )| +\frac{\pi i}{2}\frac{(\alpha- \bar{\alpha})^2}{\tau -\bar{\tau}})] $$ It coincides with $2$ times the canonical N\'eron height from [Sil]. b) {\it The nonarchimedean case}. Let $q\in I^n\setminus I^{n+1}$. The Tate curve over $Spec({\cal O})$ has a singular fiber over $Spec(k)$. The singular fiber $C = \cup C_j$ of the minimal Neron model is an $n$-gon. The set $I^j \backslash I^{j+1}$ represents the points of the curve $E$ over $Spec (K)$ whose restrictions to the singular fiber belong to the $j$-th component $C_j$ of the $n$-gon. Consider the map $$\tilde{h}_v :K^*\times K^*\times K^*\to \Bbb Z; \quad \tilde{h}_v(a, b, \lambda )= v(q)v(\lambda )-v(a)v(b).$$ This map is invariant with respect to the action of $\Bbb Z\oplus \Bbb Z$; therefore it detemines a map $h_v$ from $T_{(\cdot ,\cdot) }$ to $\Bbb Z$. The map $h_v$ is a group homomorphism on $T_{(a,\cdot) }$, its image is discrete, hence its kernel is a maximal compact subgroup. Let $v(a)=j$. The value of this map on $<(a)-(0),(a)-(0)>$ equals to: $$v(q)v(\theta (a)^{-2})-v(a)v(a)= n\cdot (-2) \cdot(\frac{1}{12} n -\frac{1}{2}j+v(1-a))-j^2 $$ $$ =-\frac{1}{6}n^2+jn-j^2 -2nv(1-a) = -n^2B_2(\frac{j}{n}) - 2nv(1-a) $$ where $B_2(x) = x^2 - x + 1/6$ is the second Bernoulli polynomial. (Compare with the integrality condition given by the third Bernoulli polynomial). {\bf 4. Functoriality of the groups $B_2(E)_{\Bbb Q}$ under the isogenies}. Let $n$ be an integer prime to the characteristic of $ k$. Let $\lambda: E_1\to E_2$ be an isogeny of order $n$ between the elliptic curves $E_1$ and $E_2$. i) {\it Pull back } $\lambda^*_{2 }: B_2(E_2) \to B_2(E_1)[\frac{1}{n}] $. Let $\hat \lambda$ be the dual isogeny $J_{E_2} \to J_{E_1}$. The pull back of the basic extension $0 \to k^* \to B_2(E_1) \to S^2J_{E_1} \to 0$ (considered modulo $n$-torsion) under the homomorphism $\frac{1}{n}\hat \lambda \cdot \hat \lambda: S^2J_{E_2} \to S^2J_{E_1}$ provides the map $\lambda^*_{2 }$: $$ \begin{array}{ccccccccc} 0& \to &k^*&\to &B_2(E_2)&\to&S^2J_{E_2}&\to&0\\ &&&&&&&&\\ &&||&& \downarrow \lambda^*_{2 }&&\downarrow \frac{1}{n}\hat \lambda \cdot \hat \lambda&&\\ &&&&&&&&\\ 0&\to&k^*&\to&B_2(E_1)[\frac{1}{n}]&\to&S^2J_{E_1}[\frac{1}{n}]&\to&0 \end{array} $$ A more direct description of $\lambda^*_{2 }$ can be spelled as follows. We have a natural morphism of $k^*$- torsors: $$\lambda^{*}:[L,M]^{\otimes n}\to [\lambda^*L, \lambda^*M]; \quad <s_1,s_2>^{\otimes n}\quad \to \quad <\lambda^{-1}s_1,\lambda^{-1}s_2>$$ To check that it is a map of torsors notice that $$(<f\cdot s_1,s_2>)^{\otimes n} = f(s_2)^{ n} (<s_1,s_2>)^{\otimes n}$$ and $f(s_2)^{ n} = (\lambda^{-1}f)(\lambda^{-1}s_2)$. It is easy to see that these maps provide the pull-back map: $$ \lambda_2^*: B_2(E_2)\to B_2(E_1)[\frac{1}{n}];\quad <s_1,s_2>\to <\lambda^{-1}s_1,\lambda^{-1}s_2>^{\otimes \frac{1}{ n}}. $$ \begin{theorem} \label{eqtr} $n\cdot \Bigl(\lambda_2^*\{\lambda(a)\}_2 - \sum_{ \gamma \in Ker \lambda}\{a + \gamma\}_2\Bigr) =0$ \end{theorem} {\bf Proof}. Any isogeny can be presented as the composition of cyclic isogenies. So we can assume that $\lambda$ is cyclic. The expression $$ f_n(a):= n\cdot\Bigl(\lambda_2^*\{\lambda(a)\}_2 - \sum_{ \gamma \in Ker \lambda}\{a + \gamma \}_2\Bigr) $$ is a nonvanishing function on the noncompact curve $E_1\setminus Ker \lambda$. Let $p_n: {\cal E} \to X_0(n)$ be the universal family of elliptic curves over the modular curve $X_0(n)$, and $\Lambda : {\cal E}\to {\cal E}$ the universal cyclic $n$-isogeny. The curves $E_1$ and $E_2$ are the fibers of ${\cal E}$ over the points $(E_1, Ker \lambda)$ and $(E_2, E_2[n]/Ker \lambda)$ of $X_0(n)$, and $\lambda$ is the restriction of $\Lambda$. The construction above defines an algebraic function $F$ on the universal curve ${\cal E}_1$. Consider the punctured formal neighborhood of the cusp point in which the universal isogeny is totally ramified. The $j$-invariant of the restriction of the universal curves to this neighborhood has a pole at the cusp; hence, this restriction can be described by the {\it Tate} curves $E_{q}\to E_{q^n}$ ([Sil]). Let us prove that $F_n$ equals $1$ for the Tate curves. We are dealing with the isogeny $K^*/(q^n)^{\Bbb Z} \to K^*/q^{\Bbb Z}$. For the Tate curve we expressed in s. 4.2 the pairing $<(a) -(0),(a) -(0)>$ in terms of the $\theta$-function $$ \theta_q (a) := q^{\frac{1}{12}} a^{-\frac{1}{2}}T(a) = q^{1/12} a^{-\frac{1}{2}} \prod_{j \geq 0} (1-q^j a)\prod_{j > 0} (1-q^j a^{-1}). $$ {\bf Remark}. $\theta_q (a)$ is defined only up to a choice of sign, so only $(\theta_q (a))^2$ makes sense. So we need to prove the following proposition. \begin{proposition} \begin{equation} \label{TT} \left(\frac{\prod_{ 0 \leq k < n} \theta_q ( t \cdot q^k )}{\Bigl(\prod_{0 \leq k < n} \theta_{q^n} ( t \cdot q^{k })\Bigr)^{n} }\right)^2 =1 \end{equation} \end{proposition} It shows that the restriction of the function $F_n$ to the preimage of neighborhood of the cusp point equals $1$. Hence this function is equal to $1$ on all universal curve $X_0(N)$; the function $f_n$ is the restriction of $F_n$ to the fiber $E$; therefore $f_n=1$. {\bf Proof of the proposition}. The $\theta$-function has the following property: $$ \theta_q (t \cdot q^l) = (-1)^k a^{-k} q^{-k^2/2}\theta_q (t) $$ Therefore $ \prod_{0 \leq k < n } \theta_q ( t \cdot q^k )$ equals to \begin{equation} \label{raz} (-1)^{s_1(n)} t^{-s_1(n)} q^{- s_2(n)/2} q^{ n/12}t^{-n/2} \prod_{j \geq 0}( 1-q^{j} t )^{n}\prod_{j > 0}( 1-q^{j} t^{-1} )^{n} \end{equation} Using the definition and notations $$ s_1(n) := 1 + ... +(n-1) = \frac{n (n-1)}{2}; \quad s_2(n) := 1^2 + ... + (n-1)^2 = \frac{(n-1) n (2n-1)}{6} $$ we have $$ \theta_{q^n} (t \cdot q^{k}) = q^{ n/12} t^{-1/2} q^{ -k/2} \prod_{j \geq 0}\Bigl( 1-q^{nj}\cdot t q^{k}\Bigr)\prod_{j > 0}\Bigl( 1-q^{nj} \cdot t^{-1} q^{-k}\Bigr) $$ On the other hand $\prod_{0 \leq k < n }\theta_{q^n} ( t \cdot q^{k})$ is equal to $$ q^{\frac{n^2}{12} -\frac{s_1(n) }{2}} t^{\frac{-n}{2}} \cdot \prod_{0 \leq k <n}\prod_{j\geq 0}\Bigl( 1-q^{nj+k} t \Bigr)\prod_{0 \leq k <n}\prod_{j > 0}\Bigl( 1-q^{nj-k} t^{-1} \Bigr)= $$ \begin{equation} \label {dwa} q^{\frac{n^2}{12} - \frac{s_1(n) }{2} } t^{\frac{-n}{2}} \prod_{j' \geq 0} ( 1-q^{j'} t )\prod_{j' > 0} ( 1-q^{j'} t^{-1} ) \end{equation} Comparing (\ref{raz}) and (\ref{dwa}) we see that it remains to check that $$ \Bigl( q^{\frac{n^2}{12} -\frac{ s_1(n)}{2} } t^{-n/2}\Bigr)^{n} = q^{- s_2(n)/2 + \frac{n}{12}} \cdot t^{-\frac{n}{2} - s_1(n)} $$ The statement of the proposition follows. In particular when $\lambda [m]$ is the isogeny of multiplication by $m$ the theorem gives \begin{corollary} \label{divr} Suppose $\bar k = k$. Then for any $a \in E(k)$ one has the "distribution relation" \begin{equation} \label {dwadwa} m(\{a\}_2 - \sum_{mb=a}\{b\}_2) =0 \end{equation} \end{corollary} {\bf Remark}. In this case one can define $[m]^*: B_2(E_2) \to B_2(E_1)[\frac{1}{m}]$ using the map $\frac{1}{m}[m] \circ \frac{1}{m}[m]$ in the diagram defining $\lambda_2$. Thus we have the factor $m$ instead of $m^2$ in the formula (\ref{dwadwa}). ii) {\it The transfer map $\lambda_{2\ast}: B_2(E_1)[\frac{1}{n}] \to B_2(E_2)[\frac{1}{n}] $}. Notice that the group $B_2(E)_{\Bbb Q}$ does not satisfy the descent property. Namely, if $k \subset K$ is a finite Galois extension then $$ B_2(E/k)_{\Bbb Q} \hookrightarrow B_2(E/K)^{Gal(K/k)}_{\Bbb Q} $$ but this inclusion is not an isomorphism because the group $S^2J(k)_{\Bbb Q}$ does not have the descent property. Suppose $k = \bar k$. The transfer map $\lambda_{2\ast}$ should satisfy the projection formula \begin{equation} \label{fedka} \lambda^*_{2 } \circ \lambda_{2\ast } = n \cdot Id \end{equation} and should fit into the following diagram, considered modulo $n$-torsion: $$ \begin{array}{ccccccccc} \label{transfer} 0& \to &k^*&\to &B_2(E_1)&\to&S^2J_{E_1}&\to&0\\ &&&&&&&&\\ && \downarrow m_n&& \downarrow\lambda_{2\ast }&&\downarrow \lambda \cdot \lambda&&\\ &&&&&&&&\\ 0&\to&k^*&\to&B_2(E_2)&\to&S^2J_{E_2}&\to&0 \end{array} $$ Here $m_n: x \to x^n$. This is necessary in order to have (\ref{fedka}) on the subgroup $k^* \subset B_2(E)$. Let us define the transfer map as follows: \begin{equation} \label{tr2} \lambda_{2\ast }\{a\}_2:= \{\lambda(a)\}_2 - \Bigl(\lambda_2^*\{\lambda a\}_2 - n\{a\}_2\Bigr) \end{equation} {\bf Remark}. Projection of $\lambda_2^*\{\lambda a\}_2 - n\{a\}_2$ to $S^2J$ equals $\frac{1}{n}\hat \lambda \circ \lambda (a\cdot a) - n a\cdot a =0$, so $\lambda_2^*\{\lambda a\}_2 - n\{a\}_2 \in k^*$. Let us show that formula (\ref{tr2}) provides a transfer homomorphism. Suppose that $\sum\{a_i\}_2 +c=0$ in the group $B_2(E_1)$, where $c \in k^*$ (we write the group $B_2(E)$ additively). We have to prove that $$ \sum_i\{\lambda(a_i)\}_2 - \Bigl(\sum_i\lambda_2^*\{\lambda (a_i)\}_2 - \sum_i n \{a_i\}_2\Bigr) +nc =0 $$ By the assumption $\sum_i\{a_i\}_2 =c^{-1} \in k^*$. As we have shown before, the expression in brackets always belongs to the subgroup $k^* \subset B_2(E_2)$. Therefore $\sum_i\lambda_2^*\{\lambda (a_i)\}_2 \in k^*$. Notice that $\lambda^*_2$ is injective modulo $n$-torsion and is the identity on the subgroup $k^* \subset B_2(E)$. Therefore modulo $n$-torsion $\sum_i\{\lambda(a_i)\}_2 \in k^*$ and $ \sum_i\{\lambda(a_i)\}_2 = \sum_i\lambda_2^*\{\lambda (a_i)\}_2$. Using proposition (\ref{eqtr}) one can easyly see that \begin{equation} \label{tr1} \lambda_{2\ast }\{a\}_2:= \{\lambda(a)\}_2 - \frac{1}{n}\sum_{\gamma \in Ker \lambda}\Bigl( n\{a+ \gamma\}_2 - n\{a\}_2\Bigr) \end{equation} {\bf 5. A presentation of the group $B_2(E)$ by generators and relations}. Let ${\cal P}(a)$ (resp ${\cal P}^{'}(a)$) be the $x$-coordinate (resp. $y$-coordinate) of the point $a \in E$ in Tate's normal form of an elliptic curve over an arbitrary field $k$: $$ y^2 + a_1xy + a_3y = x^3 + a_2x^2 +a_4x + a_6 $$ \begin{proposition} \label{propo}. a) If $a \not = b$ then $$ <(a+b) - (0), (a+b) - (0)> \otimes <(a-b) - (0), (a-b) - (0)>$$ $$ \otimes <(a) - (0), (a) - (0)>^{-2} \otimes <(b)-(0),(b)-(0)>^{-2} = (\Delta^{-1/6}({\cal P}(a)-{\cal P}(b)))^{-2} $$ b). If $a=b$ but $2a \not = 0$ then the left hand side is equal to $(\Delta^{-1/4}{\cal P}'(a))^{-2}$. If $2a =0$ then we get $(\Delta^{-1/3}{\cal P}''(a))^{-2}$ \end{proposition} {\bf Proof}. We will prove part a). Part b) is similar. Let $L_a$ be the line bundle corresponding to the divisor $(a)-(0)$. Evidently $$ [L_{a +b}, L_{a +b}] \otimes [L_{a -b}, L_{a-b}] \otimes [L_a , L_a ]^{-2}\otimes [L_b, L_b]^{-2}=k^* $$ so $$ <(a+b) - (0), (a+b) - (0)> \otimes <(a-b) - (0), (a-b) - (0)> $$ $$ \otimes <(a) - (0), (a) - (0)>^{-2} \otimes <(b)-(0),(b)-(0)>^{-2} \in k^* \otimes (T^{\ast}_0E)^{\otimes -4} $$ We want to calculate this element. One has $$ <(a+b) - (0), (a+b) - (0)> \otimes <(a-b) - (0), (a-b) - (0)>$$ $$\otimes <(a) - (0), (a) - (0)>^{-2} \otimes <(b)-(0),(b)-(0)>^{-2} = $$ $$ <(a+b) - (a), (a+b) - (a)> \otimes <(a+b)-(a),(a)-(0)>^{ 2}$$ $$\otimes <(a-b) - (a), (a-b) - (a)>\otimes <(a-b)-(a),(a)-(0)>^{ 2}$$ $$\otimes <(b)-(0),(b)-(0)>^{-2}$$ We have \begin{equation} \label{lem} <(a+b)-(b),(a+b)-(b)>\otimes <(a)-(0),(a)-(0)>^{-1} = 1 \in k^* \end{equation} Indeed, the left hand side is a regular function in $b$ on the elliptic curve and so it is a constant; its value at $b=0$ is $1$. Therefore the first, third and last terms of the expression above that we need to compute cancel thanks to (\ref{lem}) and we get: $$<(a+b)+(a-b)-2(a),(a)-(0)>^{ 2}$$ Notice that $(a+b)+(a-b)-2(a)$ is the divisor of the function $\Delta^{-1/6}({\cal P}(\xi- a)-{\cal P}(b))$. Its value at the point $0$ is $\Delta^{-1/6}({\cal P}(a)-{\cal P}(b))$ and its generalized value at the point $a$ is the trivialization we have chosen. \begin{corollary} \label{surj} Assume $k = \bar k$. Then the homomorphism (\ref{homo}) is surjective. \end{corollary} Let us denote by $R_2(E)$ the kernel of the homomorphism (\ref{homo}). Then $$ B_2(E):= \frac{\Bbb Z[E(k) \backslash 0]}{R_2(E)} $$ Let ${\tilde R}$ be the subgroup of $\Bbb [E(k) \backslash 0]$ generated by the elements $$ \{a,b\}:= \{a+b\} + \{a-b\} - 2\{a\} -2 \{b\} $$ Notice that $\{a,a\} = \{2a\} - 4\{a\}$ and $\{a,a\} - \{a,-a\} = 2(\{a\} - \{-a\})$. \begin{lemma} \label{tors} For any abelian group $A$ the elements $\{a,b\}$ and $\{a\} - \{-a\}$ generate modulo $2$-torsion the kernel of the surjective homomorphism $\Bbb Z[A] \longrightarrow S^2A \quad \{a\} \longmapsto a\cdot a$. \end{lemma} We will not use this fact later, so a (simple) proof is omitted. Consider the homomorphism ${\tilde R} \longrightarrow k^{\ast}$ defined by the formulas $$ \{a,b\} \longmapsto \Delta^{-1/3}({\cal P}(a) - {\cal P}(b))^2, \quad a \not = b; \qquad \{a,a\} \longmapsto \Delta^{-1/2}({\cal P}^{'}(a))^2, \quad 2a \not = 0 $$ and $\{a,a\} \longmapsto (\Delta^{-1/3}{\cal P}^{''}(a))^2$ if $2a =0$. Thanks to corollary (\ref{surj}) this homomorphism is well defined. By definition the subgroup $R_2(E)$ is its kernel. {\bf Remark}. This is a homomorphism to the {\it multiplicative} group $k^*$ of the field $k$ defined via the additive structure of $k$. {\bf 6. A remark on the differential in the complex $B^*(E,3)$}. The restriction of $\delta_3$ to the subgroup $B^*_3(E)$ can be defined directly, without referring to the group $B_2(E)$ and the homomorphism $h$. A more complicated formula is the price we pay. Set for general $a_i \in E(k)$ \begin{equation} \label {pfan} \delta_3\Bigr( (\{a_1\} - \{0\}) \ast (\{a_2\} - \{0\}) \ast (\{a_3\} - \{0\}) \ast (\{a_4\} - \{0\})\Bigl) = \end{equation} $$ \frac{[{\cal P}(a_1+a_2 ) - {\cal P}(a_3-a_4)][{\cal P}(a_1+a_3) - {\cal P}(a_4)] [{\cal P}(a_1+a_4) - {\cal P}(a_3)]} {[{\cal P}(a_1 ) - {\cal P}(a_3-a_4)][{\cal P}(a_1+a_2+a_3) - {\cal P}(a_4)] [{\cal P}(a_1+a_2+a_4) - {\cal P}(a_3)]} \otimes -1/2 \cdot a_1 + ... $$ where ... means three other terms obtained by cyclic permutation of indices. Here ${\cal P}(a):= x(a)$ is the $x$-coordinate of a point $a$. The expression for (\ref{pfan}) is symmetric in $a_1,...,a_4$, which is not obvious from the formula. Over $\Bbb C$ one can rewrite the right hand side of (\ref{pfan}) in a more symmetric way using the $\theta$-function: $$ \frac{\theta(a_1+a_2 +a_3+a_4) \theta(a_1+a_2)\theta(a_1+a_3)\theta(a_1+a_4)}{\theta(a_1+a_2 +a_3)\theta(a_1+a_2 +a_4)\theta(a_1+a_3+a_4)\theta(a_1)} \otimes a_1 +... $$ Morally the differential $\delta_3$ is given by the ``formula'' $\{a\} \longmapsto - \frac{1}{2} \theta(a) \otimes a$ which, unfortunately, makes no sence if we don't use the group $B_2(E)$. The relation with (\ref{pfan}) is given by the classical formula $$ {\cal P}(a) - {\cal P}(b) = \frac{ \theta(a+b)\theta(a-b)}{\theta^2(a)\theta^2(b)}, \quad a \not = \pm b $$ \vskip 3mm \noindent {\bf REFERENCES} \begin{itemize} \item[{[BT]}] Bass H., Tate J., {\it The Milnor ring of a global field} Springer Lect. Notes in Math., vol. 342 p. 389-447. \item[{[B1]}] Beilinson A.A.: {\it Higher regulators and values of $L$-functions}, VINITI, 24 (1984), 181--238 (in Russian); English translation: J. Soviet Math. 30 (1985), 2036--2070. \item[{[B2]}] Beilinson A.A.: {\it Higher regulators of modular curves} Contemp. Math. 55, 1-34 (1986) \item[{[BD2]}] Beilinson A.A., Deligne P, {\it Interpr\'etation motivique de la conjecture de Zagier} in Symp. in Pure Math., v. 55, part 2, 1994, \item[{[BL]}] Beilinson A.A., Levin A.M.: {\it Elliptic polylogarithms}. Symposium in pure mathematics, 1994, vol 55, part 2, 101-156. \item[{[Bl1]}] Bloch S.: {\it Higher regulators, algebraic $K$- theory and zeta functions of elliptic curves}, Lect. Notes U.C. Irvine, 1977. \item[{[Bl2]}] Bloch S.: {\it A note on height pairings, Tamagawa numbers and the Birch and Swinnerton- Dyer conjecture} Invent. Math. 58 (1980) 65-76. \item[{[BG]}] Bloch S., Grayson D.: {\it $K_2$ and $L$-functions of elliptic curves} Cont. Math. 55, 79-88 (1986) \item[{[De]}] Deligne P. {\it La d\'eterminant de la cohomologie} Contemp. Math. 67 93-178, (1987). \item[{[DS]}] Dupont J., Sah S.H.: {\it Scissors congruences II}, J.\ Pure Appl.\ Algebra, v. 25, (1982), 159--195. \item[{[G1]}] Goncharov A.B.: {\it Polylogarithms and motivic Galois groups} Symposium in pure mathematics, 1994, vol 55, part 2, 43 - 96. \item[{[G2]}] Goncharov A.B.: {\it Mixed elliptic motives}. To appear. \item[{[La]}] Lang. S.: {\it Foundations of diophantian geometry} Springer Verlag \item[{[Lev]}] Levine M. {\it Relative Milnor K-theory} K-theory, 6 (1992). \item[{[Li]}] Lichtenbaum S.: {\it Groups related to scissor congruence groups} Contemporary math., 1989, vol. 83, 151-157. \item[{[M]}] Milnor J.: {\it Introduction to algebraic K-theory} Ann. Math. Studies, Princeton, (1971). \item[{[MT]}] Mazur B., Tate J.: {\it Biextensions and height pairings} Arithmetic and Geometry, Birkhauser 28, v.1, (1983) 195-237. \item[{[RSS]}] Rappoport M., Schappacher N., Schneider P.: {\it Beilinson's conjectures on special values of $L$-functions}, Perspectives in Mathematics, vol 4, New York: Academic Press, 1988. \item[{[SS]}] Schappacher N., Scholl, A.: {\it The boundary of the Eisenstein symbol} Math. Ann. 1991, 290, p. 303-321. \item[{[SS2]}] Schappacher N., Scholl, A.: {\it Beilinson's theorem on modular curves } in [RSS] \item[{[Sil]}] Silverman J.: {\it Advanced topics in the arithmetic of elliptic curves} Springer Verlag, 1994. \item[{[S]}] Suslin A.A.: {\it $K_{3}$ of a field and Bloch's group}, Proceedings of the Steklov Institute of Mathematics 1991, Issue 4. \item[{[We]}] Weil A. : {\it Elliptic functions according to Eisenstein and Kronecker} Ergebnisse der Mathematik, 88, Springer 77. \item[{[W]}] Wildeshaus J. : {\it On an elliptic analog of Zagier's conjecture} To appear in Duke Math. Journal. \item[{[Za]}] Zarkhin Yu.G. {\it N\'eron pairing and quasicharacters} Izvestia Acad. Nauk USSR 36, N2 (1972) 497-509. \item[{[Z]}] Zagier D.: {\it The Bloch-Wigner-Ramakrishnan polylogarithm function} Math. Ann. 286, 613-624 (1990) \item[{[Z2]}] Zagier D.: {\it Polylogarithms, Dedekind zeta functions and the algebraic $K$-theory of fields}, Arithmetic Algebraic Geometry (G.v.d.Geer, F.Oort, J.Steenbrink, eds.), Prog. Math., Vol 89, Birkhauser, Boston, 1991, pp. 391--430. \end{itemize} A.G.: Max-Planck-Institute fur mathematik, Bonn, 53115, Germany; [email protected] Dept. of Mathematics, Brown University, Providence, RI 02912, USA. A.L.: International Institute for Nonlinear Studies at Landau Institute Vorobjovskoe sh. 2 Moscow 117940, Russia; [email protected] \end{document}

9508

alg-geom/9508002

en

https://arxiv.org/abs/alg-geom/9508002

alg-geom/9508002

Ulrich P. Klein

Bruce Hunt

Modular subvarieties of arithmetic quotients of bounded symmetric domains

48 pages, also available at http://www.mathematik.uni-kl.de/~wwwagag/ LaTeX (e-mail: [email protected])

Arithmetic quotients are quotients of bounded symmetric domains by arithmetic groups, and modular subvarieties of arithmetic quotients are themselves arithmetic quotients of lower dimension which live on arithmetic quotients, by an embedding induced from an inclusion of groups of hermitian type. We show the existence of such modular subvarieties, drawing on earlier work of the author. If $\Gamma$ is a fixed arithmetic subgroup, maximal in some sense, then we introduce the notion of ``$\Gamma$-integral symmetric'' subgroups, which in turn defines a notion of ``integral modular subvarieties'', and we show that there are finitely many such on an (isotropic, i.e, non-compact) arithmetic variety.

alg-geom

\section{Rational groups of hermitian type} \subsection{Notations} We now fix some notations to be in effect for the rest of the paper. We will be dealing with algebraic groups defined over $\fQ$, which give rise to hermitian symmetric spaces, groups of {\it hermitian type}, as we will say. As we are interested in the automorphism groups of domains, we may, without restricting generality, assume the group is {\it centerless}, and {\it simple} over $\fQ$. We will also assume $G$ is Zariski connected. Henceforth, if not indicated otherwise (occasionally $G$ will denote a reductive group; in sections 2.1 and 2.2 $G$ will be a real Lie group) $G$ will denote such an algebraic group. To avoid complications, we exclude in this paper the following case: \vspace*{.2cm}{\bf Exclude:}\hspace*{2cm} All non-compact real factors of $G({\Bbb R}} \def\fH{{\Bbb H})$ are of type $SL_2({\Bbb R}} \def\fH{{\Bbb H})$. \vspace*{.2cm} \noindent Finally, we shall only consider {\it isotropic} groups. This implies the hermitian symmetric space ${\cal D} $ has no compact factors. By our assumptions, then, we have \begin{itemize}\item[(i)] $G=Res_{k|\fQ}G'$, $k$ a totally real number field, $G'$ absolutely simple over $k$. \item[(ii)] ${\cal D} ={\cal D} _1\times \cdots \times {\cal D} _f$, each ${\cal D} _i$ a non-compact irreducible hermitian symmetric space, $f=[k:\fQ]$. \end{itemize} \subsubsection{Real parabolics} This material is presented in detail in \cite{BB} and \cite{sym}, 1.2, so we just mention enough to fix notations. We work in this section in the category of real Lie groups. $G$ will denote a connected reductive real Lie group of hermitian type, such that the symmetric space ${\cal D} =G/K$ is irreducible. In a well-known manner one fixes a maximal set of strongly orthogonal (absolute) roots, defining a subalgebra $\aa\subset} \def\nni{\supset} \def\und{\underline \Gg$, such that $A=\exp(\aa)$ is a maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus which will be fixed throughout this discussion. The set of strongly orthogonal roots is ordered, defining an order on $A$, which determines a set of simple ${\Bbb R}} \def\fH{{\Bbb H}$-roots $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_{{\Bbb R}} \def\fH{{\Bbb H}}=\{\eta_1,\ldots, \eta_t\},\ t=\rank_{{\Bbb R}} \def\fH{{\Bbb H}}G=\dim(A)$, in the ${\Bbb R}} \def\fH{{\Bbb H}$-root system $\Phi_{{\Bbb R}} \def\fH{{\Bbb H}}:=\Phi(\aa,\Gg)$. For each $b\in \{1,\ldots,t\}$, the one-dimensional subtorus $A_b$ is defined: $\aa_b=\bigcap\limits_{i\neq b}\Ker(\eta_i),\ A_b=\exp(\aa_b)$. We also set $\frak n} \def\rr{\frak r=\sum\limits_{\eta\in \Phi_{{\Bbb R}} \def\fH{{\Bbb H}}^+} \Gg^{\eta},\ N=\exp(\frak n} \def\rr{\frak r)$. The {\it standard maximal ${\Bbb R}} \def\fH{{\Bbb H}$-parabolic}, $P_b,\ b=1,\ldots, t$, is the group generated by ${\cal Z} _G(A_b)$ and $N$; equivalently it is the semidirect product (Levi decomposition) \begin{equation}\label{e2.2} P_b={\cal Z} _G(A_b)\rtimes U_b, \end{equation} where $U_b$ denotes the unipotent radical. For real parabolics of hermitian type one has a very useful refinement of (\ref{e2.2}). This is explained in detail in \cite{SC} and especially in \cite{S}, \S III.3-4. First we have the decomposition of ${\cal Z} _G(A_b)$ \begin{equation}\label{e3.4} {\cal Z} _G(A_b)=M_b\cdot L_b \cdot {\cal R} _b, \end{equation} where $M_b$ is compact, $L_b$ is the {\it hermitian Levi factor}, ${\cal R} _b$ is reductive (of type $\bf A_{\hbox{\scriptsize \bf b-1}}$), and the product is almost direct (i.e., the factors have finite intersection). Secondly, the unipotent radical decomposes, \begin{equation}\label{e3.5} U_b={\cal Z} _b\cdot V_b, \end{equation} which is a direct product, ${\cal Z} _b$ being the center of $U_b$. For this decomposition the groups are both Zariski connected and connected in the real Lie groups. The action of ${\cal Z} _G(A_b)$ on $U_b$ can be explicitly described, and is the basis for the compactification theory of \cite{SC}. The main results can be found in \cite{S}, III \S3-4, and can be summed up as follows. \begin{theorem}\label{t4.1} In the decomposition of the standard parabolic $P_b$ (see (\ref{e3.4}) and (\ref{e3.5})) $$P_b=(M_b\cdot L_b \cdot {\cal R} _b)\rtimes {\cal Z} _b\cdot V_b,$$ the following statements hold. \begin{itemize}\item[(i)] The action of $M_b\cdot L_b$ is trivial on ${\cal Z} _b$, while on $V_b$ it is by means of a symplectic representation $\gr:M_b\cdot L_b \longrightarrow} \def\sura{\twoheadrightarrow Sp(V_b,J_b)$, for a symplectic form $J_b$ on $V_b$. \item[(ii)] ${\cal R} _b$ acts transitively on ${\cal Z} _b$ and defines a homogenous self-dual (with respect to a bilinear form) cone $C_b\subset} \def\nni{\supset} \def\und{\underline {\cal Z} _b$, while on $V_b$ it acts by means of a representation $\gs:{\cal R} _b\longrightarrow} \def\sura{\twoheadrightarrow GL(V_b,I_b)$ for some complex structure $I_b$ on $V_b$. \end{itemize} Furthermore the representations $\gr$ and $\gs$ are compatible in a natural sense. The decomposition and the representations in fact are valid for the corresponding real algebraic group $G$ and its algebraic subgroups. \end{theorem} Finally, there is a one to one correspondence between the maximal real parabolics $P$ (each of which is conjugate to a unique $P_b$) and the boundary components $F$ (each of which is the image of a unique standard boundary component $F_b$), given by $P\longleftrightarrow} \def\ra{\rightarrow} \def\Ra{\Rightarrow F$, where $P={\cal N} _G(F)$. In particular, $P_b={\cal N} _G(F_b)$. \subsubsection{Roots} We now return to the notation used above, $G=Res_{k|\fQ}G'$ the $\fQ$-simple group of hermitian type. and introduce a few notations concerning the root systems involved. Let $\gS_{\infty}$ denote the set of embeddings $\gs:k\longrightarrow} \def\sura{\twoheadrightarrow {\Bbb R}} \def\fH{{\Bbb H}$; this set is in bijective correspondence with the set of infinite places of $k$. We denote these places by $\nu$, and if necessary we denote the corresponding embedding by $\gs_{\nu}$. For each $\gs\in \gS_{\infty}$, the group $^{\gs}G'$ is the algebraic group defined over $\gs(k)$ by taking the set of elements $g^{\gs},\ g\in G'$. For each infinite prime $\nu$ we have $G_{k_{\nu}}\cong (^{\gs_{\nu}}G')_{{\Bbb R}} \def\fH{{\Bbb H}}$, and the decomposition of ${\cal D} $ above can be written $${\cal D} =\prod_{\gs\in \gS_{\infty}}{\cal D} _{\gs},\quad {\cal D} _{\gs}:=(^{\gs}G')_{{\Bbb R}} \def\fH{{\Bbb H}}/K_{(\gs)}=(^{\gs}G')_{{\Bbb R}} \def\fH{{\Bbb H}}^0/K_{(\gs)}^0.$$ We set $G_{\gs}=({^{\gs}G}')^0_{{\Bbb R}} \def\fH{{\Bbb H}}$ and note that the discussion of the last section applies to $G_{\gs}$ for each $\gs$. For convenience we now index the components ${\cal D} _{\gs}$ by $i\in \{1,\ldots, f\}$. For each ${\cal D} _i$ we have ${\Bbb R}} \def\fH{{\Bbb H}$-roots $\Phi_{i,{\Bbb R}} \def\fH{{\Bbb H}}$, of ${\Bbb R}} \def\fH{{\Bbb H}$-ranks $t_i$ and simple ${\Bbb R}} \def\fH{{\Bbb H}$-roots $\{\eta_{i,1},\ldots,\eta_{i,t_i}\},\ i=1,\ldots, d$. For each factor we have standard parabolics $P_{i,b_i}$ $(1\leq b_i\leq t_i$) and standard boundary components $F_{i,b_i}$. The standard parabolics of $G_{{\Bbb R}} \def\fH{{\Bbb H}}^0$ and boundary components of ${\cal D} $ are then products \begin{equation}\label{e3.2} P_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})^0=P_{1,b_1}\times \cdots \times P_{d,b_d},\quad F_{\hbox{\scsi \bf b}}=F_{1,b_1}\times \cdots \times F_{d,b_d},\quad ({\bf b}=(b_1,\ldots, b_d)), \end{equation} where $P_{i,b_i}\subset} \def\nni{\supset} \def\und{\underline G_{\gs_i}$, $P_{\hbox{\scsi \bf b}}\subset} \def\nni{\supset} \def\und{\underline G$ is a maximal $\fQ$-parabolic, and as above $P_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})^0={\cal N} _{G_{{\Bbb R}} \def\fH{{\Bbb H}}}(F_{\hbox{\scsi \bf b}})^0$. Furthermore, there is a $\fQ$-subgroup $L_{\hbox{\scsi \bf b}}\subset} \def\nni{\supset} \def\und{\underline G$ such that \begin{equation}\label{e3.3} \hbox{Aut}} \def\Im{\hbox{Im}} \def\mod{\hbox{mod}(F_{\hbox{\scsi \bf b}})^0=L_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})^0,\ L_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})^0=L_{1,b_1}\times \cdots\times L_{d,b_d}, \end{equation} where $L_{i,b_i}\subset} \def\nni{\supset} \def\und{\underline P_{i,b_i}$ is the hermitian Levi component as above. As far as the domains are concerned, any of the boundary components $F_{i,b_i}$ may be the {\it improper} boundary component ${\cal D} _i$, which is indicated by setting $b_i=0$. Consequently, $P_{i,0}=L_{i,0}=G_i$ and in (\ref{e3.2}) and (\ref{e3.3}) any ${\bf b}=(b_1,\ldots,b_d),\ 0\leq b_i\leq t_i$ is admissible. Since $G'$ is isotropic, there is a positive-dimensional $k$-split torus $S'\subset} \def\nni{\supset} \def\und{\underline G'$, which we fix. Then ${^{\gs}S}'$ is a maximal $\gs(k)$-split torus of $^{\gs}G'$ and there is a canonical isomorphism $S'\ra {^{\gs}S}'$ inducing an isomorphism $\Phi_k=\Phi(S',G')\longrightarrow} \def\sura{\twoheadrightarrow \Phi_{\gs(k)}({^{\gs}S}',{^{\gs}G}')=:\Phi_{k,\gs}$. The torus $Res_{k|\fQ}S'$ is defined over $\fQ$ and contains $S$ as maximal $\fQ$-split torus; in fact $S\cong S'$, diagonally embedded in $Res_{k|\fQ}S'$. This yields an isomorphism $\Phi(S,G)\cong \Phi_k$, and the root systems $\Phi_{\fQ}=\Phi(S,G)$, $\Phi_k$ and $\Phi_{k,\gs}$ (for all $\gs\in \gS_{\infty}$) are identified by means of the isomorphisms. In each group $^{\gs}G'$ one chooses a maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus $A_{\gs}\nni {^{\gs}S}'$, contained in a maximal torus defined over $\gs(k)$. Fixing an order on $X(S')$ induces one also on $X({^{\gs}S}')$ and $X(S)$. Then, for each $\gs$, one chooses an order on $X(A_{\gs})$ which is compatible with that on $X({^{\gs}S}')$, and $r:X(A_{\gs})\longrightarrow} \def\sura{\twoheadrightarrow X({^{\gs}S}')\cong X(S)$ denotes the restriction homomorphism. The canonical numbering on $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_{{\Bbb R}} \def\fH{{\Bbb H},\gs}$ of simple ${\Bbb R}} \def\fH{{\Bbb H}$-roots of $G$ with respect to $A_{\gs}$ is compatible by restriction with the canonical numbering of $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_{\fQ}$ (\cite{BB}, 2.8). Recall also that each $k$-root in $\Phi_k$ is the restriction of at most one simple ${\Bbb R}} \def\fH{{\Bbb H}$-root of $G'({\Bbb R}} \def\fH{{\Bbb H})$ (which is a simple Lie group). Let $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_k=\{\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda_1,\ldots,\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda_s\}$; for $1\leq i\leq s$ set $c(i,\gs)$:= index of the simple ${\Bbb R}} \def\fH{{\Bbb H}$-root of $^{\gs}G'$ restricting on $\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda_i$. Then $i<j$ implies $c(i,\gs)<c(j,\gs)$ for all $\gs\in \gS$. Each simple $k$-root defines a unique standard boundary component: for $b\in \{1,\ldots,s\}$, \begin{equation}\label{e9.1} P_{\hbox{\scsi \bf b}}:=\prod_{\gs\in \gS_{\infty}} P_{c(b,\gs)} \quad (\hbox{ resp. }F_{\hbox{\scsi \bf b}}:=\prod_{\gs\in \gS_{\infty}}F_{c(b,\gs)}), \end{equation} which is the product of standard (with respect to $A_{\gs}$ and $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_{{\Bbb R}} \def\fH{{\Bbb H},\gs}$) parabolics $P_{c(b,\gs)}\subset} \def\nni{\supset} \def\und{\underline G_{\gs}$ (resp. boundary components $F_{c(b,\gs)}$ of ${\cal D} _{\gs}$). It follows that $\overline{F}_{\hbox{\scriptsize\bf j}}\subset} \def\nni{\supset} \def\und{\underline \overline{F}_{\hbox{\scriptsize\bf i}}$ for $1\leq i\leq j\leq s$. Furthermore, setting $o_{\hbox{\scsi \bf b}}:=\prod o_{c(b,\gs)}$, then (\cite{BB}, p.~472) \begin{equation}\label{e9.2} F_{\hbox{\scsi \bf b}}=o_{\hbox{\scsi \bf b}}\cdot L_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})^0, \end{equation} where $L_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})^0$ denotes the hermitian Levi component (\ref{e3.3}). As these are the only boundary components of interest to us, we will henceforth refer to any conjugates of the $F_{\hbox{\scsi \bf b}}$ of (\ref{e9.1}) by elements of $G$ as {\it rational boundary components} (these should more precisely be called rational with respect to $G$), and to the conjugates of the parabolics $P_{\hbox{\scsi \bf b}}$ as the {\it rational parabolics}. \subsection{Classification}\label{classification} For the convenience of the reader we sketch the classification of rational groups of hermitian type. As $G=Res_{k|\fQ}G'$ for an absolutely simple $G'$ over $k$ we need only classify these. \subsubsection{Classical cases} By means of the correspondence given by Weil in \cite{W} if $G$ is of classical type, classifying the (semi)simple $k$-groups of interest to us is equivalent to classifying the central (semi)simple $k$-algebras with involution such that $\hbox{Aut}} \def\Im{\hbox{Im}} \def\mod{\hbox{mod}(A,*)$ is of hermitian type. We now just list the cases, the possible $k$-groups $G'$, the set of ${\Bbb R}} \def\fH{{\Bbb H}$-points of $G'$ as well as of the $\fQ$-group $G$, and the corresponding domains. Let $k$ be a totally real number field of degree $f$ over $\fQ$. For the bounded symmetric domains we shall use the notations $\bf I_{\hbf{p,q}},\ II_{\hbf{n}},\ III_{\hbf{n}},\ IV_{\hbf{n}},\ V,\ VI$. In what follows $G'$ will be simple but not necessarily centerless. \begin{itemize}\item[{\bf O}] Orthogonal type \begin{itemize}\item[{\bf O.1}] split case: $G'=SO(V,h)$, $V$ a $k$-vector space of dimension $n+2$, $h$ a symmetric bilinear form such that, at all real primes $\nu$, $h_{\nu}$ has signature $(n,2)$. $$G'({\Bbb R}} \def\fH{{\Bbb H})\cong SO(n,2),\quad G({\Bbb R}} \def\fH{{\Bbb H})\cong \prod^f_{i=1}SO(n,2)_i,\quad {\cal D} \cong {\bf IV_{\hbf{n}}}\times \cdots \times {\bf IV_{\hbf{n}}},\ f\hbox{ factors}.$$ \item[{\bf O.2}] non-split case: $G'=SU(V,h)$, $V$ a right $D$-vector space of dimension $n$, $h$ is a skew-hermitian form; here $D$ is a quaternion division algebra, central simple over $k$, and for all real primes, either \begin{itemize}\item $D_{\nu}\cong \fH,\ \ G_{\nu}'\cong SU(\fH^n,h)$, $h$ a skew-hermitian form on $\fH^n$, \item $D_{\nu}\cong M_2({\Bbb R}} \def\fH{{\Bbb H}),\ \ G_{\nu}'\cong SO(2n-2,2).$ \end{itemize} and in the first case $h$ has Witt index $[{n\over 2}]$. Number the primes such that for $\nu_1,\ldots, \nu_{f_1}$ the first case occurs and for $\nu_{f_1+1},\ldots, \nu_f$ the second occurs. Then $G'({\Bbb R}} \def\fH{{\Bbb H})\cong SU(\fH^n,h)$, and $$G({\Bbb R}} \def\fH{{\Bbb H})\cong \prod^{f_1}_{i=1}SU(\fH^n,h)_i\times \prod_{i=f_1+1}^f SO(2n-2,2)_i,\quad {\cal D} \cong \underbrace{\bf II_{\hbf{n}}\times \cdots \times {\bf II_{\hbf{n}}}}_{f_1\ \hbox{\small factors}}\times \underbrace{\bf IV_{\hbf{2n-2}}\times \cdots \times {\bf IV_{\hbf{2n-2}}}}_{f-f_1\ \hbox{\small factors}}.$$ \end{itemize} \item[{\bf S}] Symplectic type \begin{itemize}\item[{\bf S.1}] split case: $G'=Sp(2n,k),\ G'({\Bbb R}} \def\fH{{\Bbb H})\cong Sp(2n,{\Bbb R}} \def\fH{{\Bbb H}),\ G({\Bbb R}} \def\fH{{\Bbb H})\cong (Sp(2n,{\Bbb R}} \def\fH{{\Bbb H}))^f,\ {\cal D} \cong ({\bf III_{\hbf{n}}})^f.$ \item[{\bf S.2}] non-split case: $G'=SU(V,h)$, where $V$ is an $n$-dimensional right vector space over a quaternion division algebra $D$, central over $k$, which is however now required to be totally indefinite, and $h$ is a hermitian form on $V$. Then $G'({\Bbb R}} \def\fH{{\Bbb H})=Sp(2n,{\Bbb R}} \def\fH{{\Bbb H})$, and $$G({\Bbb R}} \def\fH{{\Bbb H})\cong (Sp(2n,{\Bbb R}} \def\fH{{\Bbb H}))^f,\ {\cal D} =({\bf III_{\hbf{n}}})^f.$$ \end{itemize} \item[{\bf U}] Unitary type \begin{itemize}\item[{\bf U.1}] split case: $G'=SU(V,h)$, where $V$ is an $n$-dimensional $K$-vector space, $K|k$ an imaginary quadratic extension, and $h$ is a hermitian form. Let for each real prime $\nu$ $(p_{\nu},q_{\nu})$ denote the signature of $h_{\nu}$. Then $$G({\Bbb R}} \def\fH{{\Bbb H})\cong \prod_{\nu}SU(p_{\nu},q_{\nu}),\ \ {\cal D} ={\bf I_{\hbf{p$_{\nu_1}$,q$_{\nu_1}$}}}\times \cdots \times {\bf I_{\hbf{p$_{\nu_f}$,q$_{\nu_f}$}}}.$$ \item[{\bf U.2}] non-split case: $G'=SU(V,h)$, where $D$ is a division algebra of degree $d$, central simple over $K$ ($K$ as in ${\bf U.1}$) with a $K|k$-involution and $V$ is an $n$-dimensional right $D$-vector space with hermitian form $h$. If $d=1$ this reduces to ${\bf U.1}$, so we may assume $d\geq 2$. Again letting $(p_{\nu},q_{\nu})$ denote the local signatures, we have $$G({\Bbb R}} \def\fH{{\Bbb H})\cong \prod_{\nu}SU(p_{\nu},q_{\nu}),\ \ {\cal D} \cong {\bf I_{\hbf{p$_{\nu_1}$,q$_{\nu_1}$}}}\times \cdots \times {\bf I_{\hbf{p$_{\nu_f}$,q$_{\nu_f}$}}}.$$ \end{itemize} \end{itemize} \subsubsection{Exceptional groups} The exceptional groups can be classified by results of Ferrar as we now describe. A general reference to non-associative algebra used here is \cite{shaffer}. See also \cite{faulk} for an excellent survey and further references. \begin{definition}\label{d46b.1} For an alternative algebra with involution $(\AA,-)$ let $\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta=(\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_1,\ldots,\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_n)$ be a diagonal matrix with coefficients in $k$, and set $ \LL(\AA^n,\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta):=\{g\in M_n(\AA) \Big| \gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta g^* \gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta^{-1}=g\}.$ One defines the Jordan algebra $J(\AA,\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta)$ by taking $n=3$, $$J(\AA,\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta):=\LL(\AA^3,\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta).$$ If $\AA$ is an octonion algebra we call $J(\AA,\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta)$ the {\em exceptional simple Jordan algebra} defined by $\AA$ and $\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta$. \end{definition} In particular, the following cases for exceptional simple Jordan algebras can occur over ${\Bbb R}} \def\fH{{\Bbb H}$: \begin{equation}\label{e46c.0}\begin{minipage}{12cm} \begin{itemize}\item[(i)] $J^c=J(\frak C,(1,1,1))$ (the compact form) \item[(ii)] $J^b=\eta J(\frak C,(1,-1,1))\eta^{-1},\ \eta=diag(1,i,1)$, \item[(iii)] $J^s=J({\Bbb O}} \def\fA{{\Bbb A},(1,1,1))$ (the split form). \end{itemize} \end{minipage} \end{equation} There is only one ${\Bbb R}} \def\fH{{\Bbb H}$-form for the split octonion algebra ${\Bbb O}} \def\fA{{\Bbb A}$. Furthermore, for an algebraic number field $k$, there are $3^t$ isomorphism classes of Jordan algebras, where $t$ denotes the number of real primes of $k$. The Jordan algebras $J^c, J^b,$ and $J^s$ have the following explicit matrix realisations (see \cite{druck}, p. 33) \begin{equation}\label{e46c.1} J^c \hbox{ (respectively $J^s$) } \ifmmode\ \cong\ \else$\isom$\fi} \def\storth{\underline{\perp} \left\{ g= \left(\begin{array}{ccc} \xi_1 & x_3 & \-x_2 \\ \-x_3 & \xi_2 & x_1 \\ x_2 & \-x_1 & \xi_3 \end{array}\right)\Big| \xi_i\in {\Bbb R}} \def\fH{{\Bbb H},\ x_i\in \frak C \hbox{ (respectively $x_i\in {\Bbb O}} \def\fA{{\Bbb A}$)} \right\} \end{equation} \begin{equation}\label{e46c.2} J^b\ifmmode\ \cong\ \else$\isom$\fi} \def\storth{\underline{\perp} \left\{g=\left(\begin{array}{ccc} \xi_1 & ix_3 & \-x_2 \\ i\-x_3 & \xi_2 & ix_1 \\ x_2 & i\-x_1 & \xi_3 \end{array}\right) \Big| \xi_i\in {\Bbb R}} \def\fH{{\Bbb H}, x_i \in \frak C \right\}, \end{equation} and the algebra of Definition \ref{d46b.1} is given explicitly as a matrix algebra as follows: \begin{equation}\label{e46c.3} J(\AA,(\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_1,\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_2,\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_3))=\left\{x=\left( \begin{array}{ccc} \xi_1 & \gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_2x_3 & \gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_3\-x_2 \\ \gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_1\-x_3 & \xi_2 & \gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_3x_1 \\ \gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_1x_2 & \gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_2\-x_1 & \xi_3 \end{array}\right) \Big| \xi_i\in k,\ x_i\in \AA\right\}. \end{equation} Utilizing composition algebras and Jordan algebras one can construct, with the following construction of {\it Tits algebras}, exceptional Lie algebras. \begin{definition}\label{d46f.1} Let $\AA$ be a composition algebra over $k$, and $\JJ=J(\BB,\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta)$ a Jordan algebra over another composition algebra $\BB$ as in Definition \ref{d46b.1}. Set: $$\hbox{{\script L}} (\AA,\JJ)=Der(\AA)\oplus (\AA_0\otimes\JJ_0)\oplus Der(\JJ).$$ One defines a multiplication $[\cdot,\cdot]$ on $\hbox{{\script L}} (\AA,\JJ)$, which extends the $[,]$ products on $Der(\AA)$ and $Der(\JJ)$, by the rules: \begin{itemize}\item[(a)] $[\cdot,\cdot]$ is bilinear and $[x,x]=0$ for all $x\in \hbox{{\script L}} (\AA,\JJ)$; \item[(b)] $[\cdot,\cdot]$ restricts to the usual commutator on $Der(\AA)$ and $Der(\JJ)$, and these are orthogonal with respect to $[\cdot,\cdot]$, i.e., $[D,E]=0$ for all $D\in Der(\AA),\ E\in Der(\JJ)$; \item[(c)] For $D\in Der(\AA), E\in Der(\JJ), D+E$ acts on $\AA_0\otimes \JJ_0$ by: $$[D+E,a\otimes x]=D(a)\otimes x + a\otimes E(x);$$ \item[(d)] $[\cdot,\cdot]$ is defined on $\AA_0\otimes \JJ_0$ by the formula: $$[a\otimes x,b\otimes y]={1\over 3}T(x\circ y)<a,b>+(a*b)\otimes (x*y) + {1\over 2} T(a\cdot b)<x,y>,$$ where the $*$ and $<,>$ products are defined as in \cite{druck} (in the cases which we require the definition simplifies somewhat and will be described below). \end{itemize} This makes $\hbox{{\script L}} (\AA,\JJ)$ a Lie algebra. \end{definition} For later use we mention that (\ref{e46c.3}) allows us to write elements in $J(\AA,\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta)$ in the following way: \begin{equation}\label{e46d.1} x=\sum_{i=1}^3 \xi_ie_{ii} + \sum_{i=1}^3x_i[j,k],\ x_i[j,k]:=\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_kxe_{jk} + \gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_j\-xe_{kj}, \end{equation} and the second sum is over cyclic permutations $(i,j,k)$ of $(1,2,3)$. In these terms the norm and trace forms are given by (see \cite{faulk}, 4.11) \begin{equation}\label{e46d.2} N(x)=\xi_1\xi_2\xi_3 + \xi_1\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_2\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_3n(x_1)+\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_1\xi_2\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_3n(x_2) + \gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_1\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_2\xi_3n(x_3), \end{equation} \begin{equation}\label{e46d.3} T(x)=\xi_1+\xi_2+\xi_3. \end{equation} In the first formula $n(a)=a\cdot\-a$ is the norm in $\AA$. The norm above is of course analogous to the determinant in a usual matrix algebra. In particular, $N(x)\neq 0$ is a neccessary and sufficient condition for $x$ to be {\em invertible} in $\JJ$, i.e., $N(x)\neq0 \iff \exists_{y\in \JJ}$ with $x\cdot y=1, x^2\cdot y=x$, and the inverse of $x$ is given by: \begin{equation}\label{e46d.4} x^{-1}={x^{\#}\over N(x)}, \end{equation} where $x^{\#}$ satisfies $x\cdot x^{\#}=N(x)\cdot 1$, or explicitly \begin{equation}\label{e46d.5} x^{\#}=\sum(\xi_j\xi_k-\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_j\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_kn(x_i))e_{ii} +\sum(\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_i(\-{x_jx_k})-\xi_ix_i)[j,k]. \end{equation} \paragraph{$\bf E_6$} There are two constructions leading to the real Lie algebra of hermitian type $\ee_{6(-14)}$. On the one hand there is the algebra $\hbox{{\script L}} (\fC,J^b)$ (see Definition \ref{d46f.1}), where $J^b$ is isomorphic to the Jordan algebra $\JJ(\frak C,(1,-1,1))$ of Definition \ref{d46b.1} and is given explicitly as a matrix algebra in (\ref{e46c.2}). Note that in this case the general definition of the algebra $\hbox{{\script L}} (\fC,J^b)$ simplifies to \begin{equation}\label{e54.1} \hbox{{\script L}} (\fC,J^b)\ifmmode\ \cong\ \else$\isom$\fi} \def\storth{\underline{\perp} i\cdot J^b_0\oplus Der(J^b), \end{equation} which, identifying $J^b_0$ with the right translations by traceless algebra elements ${\cal R} _{J^b_0}$, is nothing but Albert's twisted $\hbox{{\script L}} (\JJ)_{\gl}=\sqrt{\gl}{\cal R} _{\JJ_0}\oplus Der(\JJ)$, $\hbox{{\script L}} (\JJ) = {\cal R} _{\JJ_0}\oplus Der(\JJ)$, as mentioned in \cite{ferrar1}, p.~62. In our case $\gl=-1$ and $\JJ=J^b$, and this implies the $\bf \ee_6$-form is of {\em outer} type (see \cite{ferrar1}, \S4 and Theorem 5 b), p.~70). The Lie multiplication with respect to the decomposition in (\ref{e54.1}) is given as follows. Writing an element of $\hbox{{\script L}} (\fC,J^b)$ as $x=i\otimes A+D,\ A\in J^b_0$ and $D\in Der(J^b)$ and identifying $i\otimes J^b_0$ and $J^b_0$ so that $x=A+D$, the Lie multiplication is given by \begin{equation}\label{e54a.2} [A+D,A'+D']= \left(D(A')-D'(A)\right)+ \left([D,D']-[L(A),L(A')]\right), \end{equation} where $L(A)$ is left multiplication in $J^b$ by $A$ (cf.~\cite{druck}, 3.2, p.~46). It turns out that this construction is insufficient to describe all $k$-forms for number fields $k$. The other description of $\ee_{6(-14)}$ is as $\hbox{{\script L}} (\frak C, J^b_1)$, where $J^b_1$ is isomorphic to the Jordan algebra $J(\AA^c_1,(1,-1,1))=J(\fC,(1,-1,1))$ which can be explicitly described in matrix terms as \begin{eqnarray}\label{e54.3} J^b_1 & \cong & \hbox{{\script H}}_3(\fC,(1,-1,1)) \\ & = & \left\{ \left( \begin{array}{ccc} r_1 & \ga_3 & \-{\ga}_2 \\ -\-{\ga}_3 & r_2 & \ga_1 \\ \ga_2 & -\-{\ga}_1 & r_3 \end{array}\right) \Big| r_i\in {\Bbb R}} \def\fH{{\Bbb H}, \ga_i\in \fC \right\}. \nonumber \end{eqnarray} It is then clear that $J_1^b\ifmmode\ \cong\ \else$\isom$\fi} \def\storth{\underline{\perp} \frak D} \def\MM{\frak M^+$ for an associative algebra $\frak D} \def\MM{\frak M$ whose traceless elements form a Lie algebra of type $\frak s} \def\cc{\frak c\uu(2,1)$. With this information we can exhibit an explicit isomorphism: $$\hbox{{\script L}} (\fC,J^b) \stackrel{\sim}{\longrightarrow} \def\sura{\twoheadrightarrow}\hbox{{\script L}} (\frak C, J^b_1).$$ By means of the isomorphism (\ref{e54.1}) we may represent an element as a $k$-linear transformation of $J^b$, i.e., as an element of $\frak C\otimes M_3(k)$. Write an element in $\hbox{{\script L}} (\frak C,J^b_1)$ as follows: $D+c\otimes a + \ad y$, where $D\in Der(\frak C), c\in \frak C_0, a\in (J^b_1)_0, y\in M_3(k)$. The isomorphism is given by (\cite{ferrar2}, 2.1) \begin{eqnarray}\label{e54a.1} \psi:\hbox{{\script L}}(\frak C, J^b_1) & \longrightarrow} \def\sura{\twoheadrightarrow & \hbox{{\script L}}(\fC,J^b) \\ D+c\otimes a +\ad y & \mapsto & D\otimes 1 + (c\otimes a)_r + (\-c\otimes {^ta})_l + I \otimes (y_r+{^ty}_l). \nonumber \end{eqnarray} Now we have the following result of Ferrar concerning $k$-forms of $\ee_6$: \begin{theorem}[\cite{ferrar2}, p.~201]\label{t54.1} If $L$ is a Lie algebra of type $\bf \ee_{\hbf{6}}$ over an algebraic number field $k$, then $$L\ifmmode\ \cong\ \else$\isom$\fi} \def\storth{\underline{\perp} \hbox{{\script L}} (\AA_k,J(\BB,\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta))$$ as in Definition \ref{d46f.1} for some octonion algebra $\AA$ and Jordan algebra $J(\BB,\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta)$ as in Definition \ref{d46b.1}, with $\BB$ an alternative $k$-algebra of dimension two, and $\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta=diag(\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_1,\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_2,\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta_3)$ is a diagonal $k$-matrix. \end{theorem} For our situation of $k$-forms of the ${\Bbb R}} \def\fH{{\Bbb H}$-algebra $\ee_{6(-14)}$ this means: \begin{corollary}\label{c55.1} Any $k$-form of $\ee_{6(-14)}$ (with $k$ totally real) is of the form $$\hbox{{\script L}} (\frak C_k,(J^b_1)_k),$$ where $\frak C_k$ is an anisotropic octonion algebra over $k$ and $(J^b_1)_k$ is a $k$-form of the algebra (\ref{e54.3}). \end{corollary} As a corollary of this we get a classification of $k$-groups of hermitian $\bf E_6$ type: \begin{corollary}\label{c55.2} Let $G'$ be an absolutely almost simple $k$-group of hermitian type, in the class of structures of type $\bf E_6$. Then $(G')^0\sim \hbox{\em Aut}(\hbox{{\script L}} (\frak C_k,(J^b_1)_k))^0$ with the notations of the preceeding corollary, where ``$\sim$'' means isogenous. \end{corollary} Since an octonion algebra $\AA$ over $k$ is uniquely determined up to isomorphism by the set of real primes at which it ramifies, the totally definite (Cayley) algebra $\frak C_k$ is unique, and we need only apply the classification of $k$-forms of the Lie algebra $\frak s} \def\cc{\frak c\uu(2,1)$ to get a complete classification of $k$-forms of $J^b_1$, and hence a classification of the $k$-forms of $\ee_{6(-14)}$. There are essentially three cases which can occur (let $\frak D} \def\MM{\frak M$ denote the associative algebra with involution and $\frak D} \def\MM{\frak M^-$ the $k$-form of the Lie algebra $\uu(2,1)$): \begin{equation}\label{e55.1} \begin{minipage}{12cm} \begin{itemize}\item[(i)] $(V,h)$ is a $k$-vector space with hermitian form $h$ of Witt index 1, represented by a matrix $H$, and $\frak D} \def\MM{\frak M^-=\{g\in End(V)\big|gH-Hg=0\}$. \item[(ii)] $(V,h)$ is a $k$-vector space with {\em anisotropic} hermitian form $h$, represented by a matrix $H$, and $\frak D} \def\MM{\frak M^-=\{g\in End(V)\big|gH-Hg=0\}$. \item[(iii)] $D$ is a central simple division algebra of degree three over an imaginary quadratic extension $K$ of $k$ with a $K|k$-involution, and $\frak D} \def\MM{\frak M=D$. \end{itemize} \end{minipage} \end{equation} Considering the Tits index of these $\fQ$-groups, note that since a $\fQ$-split torus is all the more ${\Bbb R}} \def\fH{{\Bbb H}$-split, it follows that the set of split roots of the index of $G$ (usually drawn white in the Tits index) are a subset of the split roots of $G({\Bbb R}} \def\fH{{\Bbb H})$. This gives a simple criterion for deciding which indices may give rise to the given ${\Bbb R}} \def\fH{{\Bbb H}$-form. Looking now at the list of $\bf E_6$ indices (of outer type) in \cite{tits}, the following four possibilities arise for $\fQ$-forms of $E_{6(-14)}$: $^2E^{78}_{6,0},\ ^2E^{35}_{6,1},\ ^2E^{29}_{6,1},\ ^2E^{16'}_{6,2}$. However, as shown in \cite{kazdan}, the index $^2E^{29}_{6,1}$ does not give rise to a bounded symmetric domain, but rather has symmetric space $\bf E IV$ in the notation of \cite{Helg}. The argument is roughly as follows. If $H\subset} \def\nni{\supset} \def\und{\underline G$ is the anisotropic kernel, of type $\bf D_4$, then, since dim$[U,U]=8$ for a maximal unipotent subgroup (in the maximal $\fQ$-parabolic $P_{\ga_1}\cap P_{\ga_6}$), it follows that $H\subset} \def\nni{\supset} \def\und{\underline End_{\fQ}([U,U])$, a relation preserved upon tensoring with ${\Bbb R}} \def\fH{{\Bbb H}$, so that $P_{\ga_2}$ is still not defined over ${\Bbb R}} \def\fH{{\Bbb H}$; thus the index of $G({\Bbb R}} \def\fH{{\Bbb H})$ is $^1E^{28}_{6,2}$, giving rise to the symmetric space denoted $\bf E IV$ in \cite{Helg}. Hence there are only three possible Tits indices, namely $^2E^{16'}_{6,2}$, $^2E^{35}_{6,1}$ and $^2E^{78}_{6,0}$ for $k$-forms of $\ee_{6(-14)}$, and it may hold that the three possibilities in (\ref{e55.1}) coincide with the three possible indices. \paragraph{$\bf E_7$} There are two constructions utilizing the Tits algebra leading to the real form of type $\ee_{7(-25)}$. On the one hand there is the algebra $\hbox{{\script L}} (\AA,J^b)\ifmmode\ \cong\ \else$\isom$\fi} \def\storth{\underline{\perp} \hbox{{\script L}} (\AA,J^c)$ (see Definition \ref{d46f.1}), where $\AA\ifmmode\ \cong\ \else$\isom$\fi} \def\storth{\underline{\perp} M_2({\Bbb R}} \def\fH{{\Bbb H})$ and $J^b$ (respectively $J^c$) is isomorphic to the Jordan algebra $J(\frak C,(1,-1,1))$ (respectively is the Jordan algebra $J(\frak C,(1,1,1))$) in Definition \ref{d46b.1} and is given explicitly as a matrix algebra in (\ref{e46c.2}) (respectively in (\ref{e46c.1})). In this case the direct sum decomposition analogous to (\ref{e54.1}) is (\cite{druck}, 4.6, p. 50) \begin{equation}\label{e55a.1} \hbox{{\script L}} (\AA, J^b)\ifmmode\ \cong\ \else$\isom$\fi} \def\storth{\underline{\perp} (\AA_0\otimes J^b)\oplus Der(J^b).\end{equation} The multiplication is given by the rules \begin{equation} \begin{minipage}{14cm} \begin{itemize}\item[(i)] $[a\otimes A, b\otimes B]={1\over 2}[a,b]\otimes A\circ B + {1\over 2}Tr(ab)[L(A),L(B)],$ for $a,b\in \AA_0,\ A,B\in J^b$; \item[(ii)] $[D,b\otimes B]=b\otimes D(B),\ D\in Der(J^b), b\in \AA_0, B\in J^b$; \item[(iii)] $[D,D']=$ usual commutator of $D,D'\in Der(J^b)$. \end{itemize}\end{minipage}\end{equation} The other description of $\ee_{7(-25)}$ is as the algebra $\hbox{{\script L}} (\frak C,{\cal J} {\cal O} _6({\Bbb R}} \def\fH{{\Bbb H}))$, where ${\cal J} {\cal O} _6({\Bbb R}} \def\fH{{\Bbb H})$ is the Jordan algebra $\hbox{{\script H}} _3(M_2({\Bbb R}} \def\fH{{\Bbb H}),(1,1,1))$, and is given as a matrix algebra by (\ref{e46c.3}). Of course we could derive an explicit isomorphism as in (\ref{e54a.1}) between the two. But in this case it turns out that the first description is sufficient to get all $k$-forms. Namely, we have the following result of Ferrar: \begin{theorem}[\cite{ferrar3}, Theorem 4.3]\label{t55a.1} Let $k$ be an algebraic number field $k$ and let $L$ be a $k$-form of the Lie algebra $\ee_7$. Then $$L\ifmmode\ \cong\ \else$\isom$\fi} \def\storth{\underline{\perp} \hbox{{\script L}} (\AA,\JJ)$$ as in Definition \ref{d46f.1} for some quaternion algebra $\AA$ over $k$, and exceptional simple Jordan algebra $\JJ$ over $k$. \end{theorem} For the case at hand here, namely $k$-forms of the ${\Bbb R}} \def\fH{{\Bbb H}$-algebra $\ee_{7(-25)}$, this implies \begin{corollary}\label{t55a.2} Let $G'$ be an almost absolutely simple $k$-group of hermitian type, of type $\bf E_7$, and let $\Gg'$ be the Lie algebra. Then $$\Gg'\cong\hbox{{\script L}} (\AA_k,\JJ_k),$$ where $\JJ_k$ is exceptional simple such that for each real prime of $k$, $(\JJ_k)_{\nu}\ifmmode\ \cong\ \else$\isom$\fi} \def\storth{\underline{\perp} J^b$ or $J^c$, and $\AA_k$ is a quaternion algebra over $k$ which splits at all infinite primes $\nu$. \end{corollary} There are the following possibilities over $\fQ$: \begin{equation}\label{e55b.1} \begin{minipage}{14cm}\begin{itemize}\item[(i)] $\AA$ is split, $\JJ_{\fQ}$ is a $\fQ$-form of $J^b$; \item[(ii)] $\AA$ is split, $\JJ_{\fQ}$ is a $\fQ$-form of $J^c$; \item[(iii)] $\AA$ is division, $\JJ_{\fQ}$ is a $\fQ$-form of $J^b$; \item[(iv)] $\AA$ is division, $\JJ_{\fQ}$ is a $\fQ$-form of $J^c$. \end{itemize}\end{minipage}\end{equation} There are three possible Tits indices, namely $E^{28}_{7,3}, E^{31}_{7,2}$ and $E^{133}_{7,0}$. It is rather clear that the first (respectively the last) case above gives rise to $E^{28}_{7,3}$ (respectively to $E^{133}_{7,0}$), and it seems natural to expect the other two cases to give rise to $E^{31}_{7,2}$. \subsection{Boundary components} We briefly discuss the rational boundary components occuring in each of the cases. Again we tabulate this, giving the Tits index in each case and describing the boundary components. We also describe, in the classical cases, the corresponding isotropic subspaces of the vector space $V$. Throughout, ${\cal D} ^*$ denotes the union of ${\cal D} $ and the rational boundary components. \begin{itemize}\item[{\bf O.1}] The Tits index is $D_{n,s}$ (for $n\equiv2(4)$), ${^2D}_{n,s}$ (for $n\equiv0(4)$) or $B_{n,s}$ (for $n$ odd), where $s$ is the Witt index of $h$. The corresponding diagrams are (the top diagrams are for the case $s=2$, the lower ones giving the left ends for $s=1$): \setlength{\unitlength}{0.005500in}% \begin{picture}(1102,222)(64,609) \thicklines \put(160,760){\circle{22}} \put(315,760){\circle*{10}} \put(355,760){\circle*{10}} \put( 80,760){\circle{22}} \put( 90,760){\line( 1, 0){ 60}} \put(170,760){\line( 1, 0){ 60}} \put(250,760){\line( 1, 0){ 45}} \put(880,760){\circle*{10}} \put(910,760){\circle*{10}} \put(675,760){\circle{22}} \put(795,760){\circle{22}} \put(807,760){\line( 1, 0){ 53}} \put(687,760){\line( 1, 0){ 97}} \put(400,760){\circle*{10}} \put(580,820){\circle*{22}} \put(579,701){\circle*{20}} \put(235,620){\circle*{22}} \put(310,620){\circle*{10}} \put(350,620){\circle*{10}} \put( 75,620){\circle{22}} \put(240,760){\circle*{22}} \put(155,620){\circle*{22}} \put(687,620){\line( 1, 0){ 97}} \put(505,760){\circle*{22}} \put(935,760){\circle*{10}} \put(1015,760){\circle*{22}} \put(1155,760){\circle*{22}} \put(880,620){\circle*{10}} \put(910,620){\circle*{10}} \put(675,620){\circle{22}} \put(795,620){\circle*{22}} \put(425,760){\line( 1, 0){ 70}} \put(510,770){\line( 4, 3){ 60}} \put(511,752){\line( 4,-3){ 60}} \put( 85,620){\line( 1, 0){ 60}} \put(165,620){\line( 1, 0){ 60}} \put(245,620){\line( 1, 0){ 45}} \put(955,760){\line( 1, 0){ 45}} \put(1020,770){\line( 1, 0){110}} \put(1020,750){\line( 1, 0){110}} \put(1115,730){\line( 6, 5){ 28}} \put(1115,790){\line( 6,-5){ 28}} \put(807,620){\line( 1, 0){ 53}} \put(580,805){\vector( 0,-1){ 85}} \put(580,805){\vector( 0,1){ 0}} \end{picture} \noindent where the Galois action in the left-hand diagram is present only for $n\equiv 0(4)$. The boundary components of ${\cal D} '=G'({\Bbb R}} \def\fH{{\Bbb H})/K'$ are: \begin{itemize}\item $\{pt\}\subset} \def\nni{\supset} \def\und{\underline \{ \hbox{1-disc}\}^*$, ($s=2$) \item $\{pt\}$, ($s=1$). \end{itemize} \item[{\bf O.2}] The index in this case is $D_{{n\over 2},s}^{(2)}$ ($n$ even) or ${^2D}_{{n-1\over2},s}^{(2)}$ ($n$ odd), where $s$ is the Witt index of $h$. The corresponding diagrams are (with non-trivial Galois action identifying the two right most vertices for $n$ odd): \setlength{\unitlength}{0.005500in}% \begin{picture}(987,141)(14,670) \thicklines \put(435,740){\circle*{10}} \put(480,740){\circle*{10}} \put( 25,740){\circle*{22}} \put(185,740){\circle*{22}} \put(105,740){\circle{22}} \put(285,740){\circle{22}} \put(540,740){\circle{22}} \put(625,740){\circle*{22}} \put(740,740){\circle*{10}} \put(760,740){\circle*{10}} \put(720,740){\circle*{10}} \put(825,740){\circle*{22}} \put(395,740){\circle*{10}} \put(915,740){\circle*{22}} \put(921,732){\line( 4,-3){ 60}} \put(990,680){\circle*{20}} \put(990,800){\circle*{22}} \put( 35,740){\line( 1, 0){ 60}} \put(115,740){\line( 1, 0){ 60}} \put(195,740){\line( 1, 0){ 80}} \put(295,740){\line( 1, 0){ 75}} \put(500,740){\line( 1, 0){ 30}} \put(550,740){\line( 1, 0){ 65}} \put(635,740){\line( 1, 0){ 70}} \put(705,740){\line(-1, 0){ 5}} \put(785,740){\line( 1, 0){ 40}} \put(835,740){\line( 1, 0){ 70}} \put(920,750){\line( 4, 3){ 60}} \put(25,680){$\underbrace{\hspace*{7.2cm}}_{\hbox{$2s$}}$} \end{picture} \vspace*{.2cm} \noindent The corresponding boundary components are $\bf II_{\hbf{n-2}}^*\nni II_{\hbf{n-4}}^*\nni \cdots \nni II_{\hbf{n-2s}}$. \item[{\bf S.1}] The index is $C_{n,n}$, with the usual diagram and the following boundary components: $\{pt\}\subset} \def\nni{\supset} \def\und{\underline \bf III_{\hbf{1}}^*\subset} \def\nni{\supset} \def\und{\underline \cdots \subset} \def\nni{\supset} \def\und{\underline III_{\hbf{n-1}}^*.$ \item[{\bf S.2}] The index is $C_{n,s}^{(2)}$, with diagram \setlength{\unitlength}{0.005500in}% $$\begin{picture}(962,22)(14,729) \thicklines \put(435,740){\circle*{10}} \put(480,740){\circle*{10}} \put( 25,740){\circle*{22}} \put( 25,720){$\underbrace{\hspace*{7.5cm}}_{\hbox{$2s$}}$} \put(185,740){\circle*{22}} \put(105,740){\circle{22}} \put(285,740){\circle{22}} \put(565,740){\circle{22}} \put(655,740){\circle*{22}} \put(750,740){\circle*{10}} \put(780,740){\circle*{10}} \put(810,740){\circle*{10}} \put(395,740){\circle*{10}} \put(900,740){\circle*{22}} \put(901,750){\line( 1, 0){ 60}} \put(965,740){\circle*{22}} \put( 35,740){\line( 1, 0){ 60}} \put(115,740){\line( 1, 0){ 60}} \put(195,740){\line( 1, 0){ 80}} \put(295,740){\line( 1, 0){ 75}} \put(500,740){\line( 1, 0){ 55}} \put(575,740){\line( 1, 0){ 70}} \put(665,740){\line( 1, 0){ 70}} \put(735,740){\line(-1, 0){ 5}} \put(830,740){\line( 1, 0){ 60}} \put(905,730){\line( 1, 0){ 55}} \end{picture}$$ \vspace*{.2cm}\noindent The boundary components are then the following: $\bf III_{\hbf{n-2}}^*\nni \cdots \nni III_{\hbf{n-2s}}$. \item[{\bf U.1}] The index is ${^2A}_{n-1,s}$, with the diagram \setlength{\unitlength}{0.004500in}% \begin{picture}(734,190)(-30,610) \thicklines \put(260,785){\circle*{10}} \put(295,785){\circle*{10}} \put(220,625){\circle*{10}} \put(260,625){\circle*{10}} \put(295,625){\circle*{10}} \put(595,785){\circle*{10}} \put(635,785){\circle*{10}} \put(670,785){\circle*{10}} \put(595,625){\circle*{10}} \put(635,625){\circle*{10}} \put(670,625){\circle*{10}} \put(885,710){\circle*{28}} \put(785,785){\circle*{30}} \put(785,625){\circle*{30}} \put(785,785){\line( 4,-3){100}} \put(220,785){\circle*{10}} \put(785,625){\line( 6, 5){ 90}} \put( 95,625){\line( 1, 0){100}} \put(690,785){\line( 1, 0){100}} \put(690,625){\line( 1, 0){100}} \put(375,785){\circle{28}} \put(500,785){\circle*{30}} \put(310,785){\line( 1, 0){ 50}} \put(390,785){\line( 1, 0){ 95}} \put(515,785){\line( 1, 0){ 60}} \put(380,625){\circle{28}} \put(505,625){\circle*{30}} \put(310,625){\line( 1, 0){ 50}} \put(395,625){\line( 1, 0){ 95}} \put(520,625){\line( 1, 0){ 60}} \put( 80,785){\circle{30}} \put( 80,625){\circle{30}} \put( 95,785){\line( 1, 0){100}} \put(80,590){$\underbrace{\hspace*{3.5cm}}_{{\displaystyle s} \hbox{ vertices}}$} \end{picture} \vspace*{1cm}\noindent As above, let $(p_{\nu},q_{\nu})$ denote the signature of $h_{\nu}$, then in the factor ${\cal D} _{\nu}$ of ${\cal D} $ we have the boundary components of the type $\bf I_{\hbf{p$_{\nu}$-b,q$_{\nu}$-b}}$ for $1\leq b\leq s$. Hence a flag of boundary components will be \[ \prod \bf I_{\hbf{p$_{\nu}$-1,q$_{\nu}$-1}}^*\nni \prod I_{\hbf{p$_{\nu}$-2,q$_{\nu}$-2}}^* \nni \cdots \nni \prod I_{\hbf{p$_{\nu}$-s,q$_{\nu}$-s}}.\] \item[{\bf U.2}] The index is in this case ${^2A}_{nd-1,s}^{(d)}$, with diagram \vspace*{.5cm} $$\setlength{\unitlength}{0.004500in}% \begin{picture}(600,560)(400,235) \thicklines \put(280,780){\circle*{10}} \put(315,780){\circle*{10}} \put(105,780){\circle*{30}} \put(475,780){\circle*{30}} \put(120,780){\line( 1, 0){100}} \put(340,780){\line( 1, 0){120}} \put(845,780){\circle*{10}} \put(885,780){\circle*{10}} \put(920,780){\circle*{10}} \put(600,780){\circle{28}} \put(710,780){\circle*{28}} \put(1035,780){\circle*{30}} \put(490,780){\line( 1, 0){ 95}} \put(615,780){\line( 1, 0){ 80}} \put(720,780){\line( 1, 0){100}} \put(940,780){\line( 1, 0){100}} \put(240,620){\circle*{10}} \put(280,620){\circle*{10}} \put(315,620){\circle*{10}} \put(105,620){\circle*{30}} \put(475,620){\circle*{30}} \put(120,620){\line( 1, 0){100}} \put(340,620){\line( 1, 0){120}} \put(845,620){\circle*{10}} \put(885,620){\circle*{10}} \put(920,620){\circle*{10}} \put(600,620){\circle{28}} \put(710,620){\circle*{28}} \put(1035,620){\circle*{30}} \put(490,620){\line( 1, 0){ 95}} \put(615,620){\line( 1, 0){ 80}} \put(720,620){\line( 1, 0){100}} \put(940,620){\line( 1, 0){100}} \put(1135,705){\circle*{28}} \put(1035,780){\line( 4,-3){100}} \put(1035,620){\line( 6, 5){ 90}} \put(105,580){$\underbrace{\hspace*{4.4cm}}_{ \hbox{$d-1$ vertices}}$} \end{picture}$$ \vspace*{-3.5cm}\noindent where there are $2s$ white vertices altogether. Letting the notations be as for the case $\bf U.1$, we have the following boundary components: \[ \prod \bf I_{\hbf{p$_{\nu}$-d,q$_{\nu}$-d}}^*\nni \prod I_{\hbf{p$_{\nu}$-2d,q$_{\nu}$-2d}}^* \nni \cdots \nni \prod I_{\hbf{p$_{\nu}$-sd,q$_{\nu}$-sd}}.\] \end{itemize} We now describe briefly the parabolics in terms of the geometry of $(V,h)$ for all the cases above. Fixing a maximal $k$-split torus and an order on it amounts to fixing a maximal totally isotropic ($s$-dimensional) subspace $H_1\subset} \def\nni{\supset} \def\und{\underline V$ and a basis $v_1,\ldots, v_s$ of $H_1$. There are then $k$-vectors $v_1',\ldots, v_s'$ spanning a complementary totally isotropic subspace $H_2$ such that $h(v_i,v_j')=\gd_{ij}$. Then each pair $(v_i,v_i')$ spans a hyperbolic plane $V_i$ (over $D$), and $V$ decomposes: \begin{equation}\label{E8.a} V=V_1\oplus \cdots \oplus V_s\oplus V',\quad V'\hbox{ anisotropic for $h$}. \end{equation} Furthermore, $V_1\oplus \cdots \oplus V_s=H_1\oplus H_2$. With these notations, for $1\leq b\leq s$ the standard $k$-parabolic $P_b'\subset} \def\nni{\supset} \def\und{\underline G'$ is given as follows: \begin{equation}\label{E8.b} P_b'={\cal N} _{G'}(<v_1,\ldots, v_b>), \end{equation} where $<v_1,\ldots,v_b>$ denotes the span, a $b$-dimensional totally isotropic subspace. The hermitian Levi factor of $P_b'$ is \begin{equation}\label{E8.c} L_b'={\cal N} _{G'}(V_{b+1}\oplus \cdots \oplus V_s\oplus V')/{\cal Z} _{G'}(V_{b+1}\oplus \cdots \oplus V_s\oplus V'). \end{equation} It reduces to the $k$-anisotropic kernel for $b=s$. For the exceptional cases we have the following possibilities: \begin{itemize}\item {\bf $ E_{\hbf{6}}$:} Index: ${^2E}_{6,2}^{16'}$, boundary components: $\{pt\}\subset} \def\nni{\supset} \def\und{\underline \fB_5^*$. Index: ${^2E}_{6,1}^{35}$, boundary components: $\fB_5$. \item {\bf $E_{\hbf{7}}$:} Index: $E_{7,3}^{28}$, boundary components $\{pt\}\subset} \def\nni{\supset} \def\und{\underline \bf IV_{\hbf{1}}^*\subset} \def\nni{\supset} \def\und{\underline IV_{\hbf{10}}^*$. Index: $E_{7,2}^{31}$, boundary components $\bf IV_{\hbf{1}}\subset} \def\nni{\supset} \def\und{\underline IV_{\hbf{10}}^*$. \end{itemize} \section{Rational symmetric subgroups and incidence} \subsection{Holomorphic symmetric embeddings}\label{section2.1} Recall that an injection $i_{{\cal D} }:{\cal D} \hookrightarrow} \def\hla{\hookleftarrow {\cal D} '$ of symmetric spaces is said to be {\it strongly equivariant} if $i_{{\cal D} }$ is induced by an injection $i:\Gg\hookrightarrow} \def\hla{\hookleftarrow \Gg'$ of the Lie algebras $\Gg$ (resp. $\Gg'$) of the real Lie group $G=\hbox{Aut}} \def\Im{\hbox{Im}} \def\mod{\hbox{mod}({\cal D} )$ (resp. $G'=\hbox{Aut}} \def\Im{\hbox{Im}} \def\mod{\hbox{mod}({\cal D} ')$). This is equivalent to the condition that $i_{{\cal D} }({\cal D} )$ is totally geodesic in ${\cal D} '$ with respect to the $G'$-invariant metric on ${\cal D} '$. Assuming both ${\cal D} $ and ${\cal D} '$ are hermitian symmetric, there exist elements $\xi$ (resp. $\xi'$) in the center of the maximal compact subgroup $K$ (resp. $K'$) such that $J=\ad(\xi)$ (resp. $J'=\ad(\xi')$) gives the complex structure, and the condition that $i_{{\cal D} }$ be holomorphic is \vspace*{.2cm} $\hbox{(H$_1$)}\hspace*{5.8cm} i\circ \ad(\xi) = \ad(\xi')\circ i.$ \vspace*{.2cm} For any given hermitian symmetric space ${\cal D} '$, the possible hermitian symmetric subdomains $i_{{\cal D} }({\cal D} )$ have been classified by Satake and Ihara (see \cite{I} and \cite{S1}). Note in particular that the above applies to ${\cal D} _N$, where $N$ is a reductive subgroup of hermitian type and ${\cal D} _N$ is the associated hermitian symmetric space. We will refer to subgroups $N\subset} \def\nni{\supset} \def\und{\underline G$, where $G$ is the connected component of the automorphism group of ${\cal D} $, for which ${\cal D} _N\subset} \def\nni{\supset} \def\und{\underline {\cal D} $ is a hermitian symmetric subdomain, as {\it symmetric subgroups} $N\subset} \def\nni{\supset} \def\und{\underline G$\footnote{The term ``symmetric'' arises from the fact that in most cases, $N$ can be defined in terms of closed symmetric sets of roots.}. For this notion it is irrelevant whether $N$ is reductive, semisimple or even centerless. \subsection{Incidence over ${\Bbb R}} \def\fH{{\Bbb H}$} In this section let $G$ be a reductive Lie group of hermitian type such that the symmetric space ${\cal D} $ is irreducible, and let $A\subset} \def\nni{\supset} \def\und{\underline G$ be the maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus (with order) defined by the maximal set of strongly orthogonal roots of $G$ as in 1.1.1. Then we can speak of the standard parabolics $P_b,\ b=1,\ldots, t$, $t=\rank_{{\Bbb R}} \def\fH{{\Bbb H}}G$. We introduce the set of domains $({\cal E} {\cal D} )$ as follows. $({\cal E} {\cal D} )\hspace*{5cm} \bf I_{\hbox{\scriptsize\bf q,q}},\ II_{\hbox{\scriptsize\bf n}},\ n \hbox{ even},\ III_{\hbox{\scriptsize\bf n}}.$ With respect to a fixed $P_b$ we consider the following conditions on a symmetric subgroup $N\subset} \def\nni{\supset} \def\und{\underline G$ as in section \ref{section2.1}. \begin{itemize}\item[1)] $N$ has maximal ${\Bbb R}} \def\fH{{\Bbb H}$-rank, that is, $\rank_{{\Bbb R}} \def\fH{{\Bbb H}}N=\rank_{{\Bbb R}} \def\fH{{\Bbb H}}G$. \item[2)] $N$ is a maximal symmetric subgroup. \item[2')] $N$ is a maximal subgroup of tube type, i.e., such that ${\cal D} _N$ is a tube domain. \item[2'')] $N$ is {\it minimal}, subject to 1). \item[3)] $N=N_1\times N_2$, where $N_1\subset} \def\nni{\supset} \def\und{\underline P_b$ is a hermitian Levi factor of $P_b$ for some Levi decomposition. \item[3')] ${\cal D} _N^*$ contains $F$ as a boundary component. \end{itemize} \begin{definition}\label{d9.1} Let $G$ be a simple real Lie group of hermitian type, $A$ a fixed maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus defined as above by a maximal set of strongly orthogonal roots, $\eta_i,\ i=1,\ldots, t$ the simple ${\Bbb R}} \def\fH{{\Bbb H}$-roots, $F_{b}$ a standard boundary component and $P_{b}$ the corresponding standard maximal ${\Bbb R}} \def\fH{{\Bbb H}$-parabolic, $b\in \{1,\ldots, t\}$. A symmetric subgroup $N\subset} \def\nni{\supset} \def\und{\underline G$ (respectively the subdomain ${\cal D} _{N}\subset} \def\nni{\supset} \def\und{\underline {\cal D} $) will be called {\it incident} to $P_{b}$ (respectively to $F_{b}$), if $N$ fulfills: \begin{itemize}\item $b<t$, then $N$ satisfies 1), 2), 3). \item $b=t,\ {\cal D} \not\in ({\cal E} {\cal D} )$, then $N$ satisfies 1), 2) or 2'), 3'). \item $b=t,\ {\cal D} \in ({\cal E} {\cal D} )$, then $N$ satisfies 1), 2''), 3'). \end{itemize} For reducible ${\cal D} ={\cal D} _1\times \cdots \times {\cal D} _d$, we have the product subgroups $N_{b_1,1}\times \cdots \times N_{b_d,d}$, where ${\cal D} _{N_{b_i,i}}$ is incident to the standard boundary component $F_{{b_i}}$ of ${\cal D} _i$ (and $N_{0,i}=G_i$). \end{definition} This defines the notion of symmetric subgroups incident with a standard parabolic. Any maximal ${\Bbb R}} \def\fH{{\Bbb H}$-parabolic is conjugate to one and only one standard maximal parabolic, $P=gP_bg^{-1}$ for some $b$. Let $N_b$ be any symmetric subgroup incident with $P_b$. Then just as one has the pair $(P_b,N_b)$ one has the pair $(P,N)$, \begin{equation}\label{e10.3} P=gP_bg^{-1},\quad N=g N_b g^{-1}. \end{equation} \begin{definition} \label{d10.1} A pair $(P,N)$ consisting of a maximal ${\Bbb R}} \def\fH{{\Bbb H}$-parabolic $P$ and a symmetric subgroup $N$ is called {\it incident}, if the groups are conjugate by a common element $g$ as in (\ref{e10.3}) to a pair $(P_b,N_b)$ which is incident as in Definition \ref{d9.1}. \end{definition} The existence of the symmetric subgroups $N_b$ was proved in the above mentioned work of Ihara and Satake. Let $P_{b}$, $1\leq b< t$ (this means $\dim(F_b)>0$) be a standard parabolic and let $L_b$ be the ``standard'' hermitian Levi factor, i.e., such that $Lie(L_b)=\ll_b$; then \begin{equation}\label{e10.1} N_b:= L_b\times {\cal Z} _G(L_b) \end{equation} is a subgroup having the properties given in the definition, unique since $L_b$ is unique. We shall refer to this unique subgroup as the {\it standard} incident subgroup. As to uniqueness, the following was shown in \cite{sym}, Prop.~2.4. \begin{proposition} If $(N,P_b)$ are incident, there is $g\in V_b$ such that $N$ is conjugate by $g$ to the standard $N_b$ of (\ref{e10.1}), where $V_b$ is the factor of $P_b$ of Theorem \ref{t4.1}. \end{proposition} The situation for zero-dimensional boundary components was not considered in \cite{sym} in detail, so we take this up now. Consider first the case where ${\cal D} \not\in({\cal E} {\cal D} )$, so ${\cal D} $ is a product of factors of types: \[\bf I_{\hbf{p,q}}\ \hbox{ ($p>q$)}, II_{\hbf{n}}\ \hbox{ ($n$ even)},\ IV_{\hbf{n}},\ V\hbox{ or }\ \bf VI.\] The corresponding subgroups $N_t$ are: $\bf I_{\hbf{p-1,q}}\subset} \def\nni{\supset} \def\und{\underline I_{\hbf{p,q}},\ II_{\hbf{n-1}}\subset} \def\nni{\supset} \def\und{\underline II_{\hbf{n}},\ IV_{\hbf{n-1}}\subset} \def\nni{\supset} \def\und{\underline IV_{\hbf{n}},\ I_{\hbf{2,4}},\ II_{\hbf{5}}$ or $\bf IV_{\hbf{8}}\subset} \def\nni{\supset} \def\und{\underline V,\ I_{\hbf{3,3}}$ or $\bf II_{\hbf{6}}\subset} \def\nni{\supset} \def\und{\underline VI$. Next note that if $N$ is incident to $P_t$, so ${\cal D} _N$ is incident to $F_t$ (=pt.), then any other domain ${\cal D} _{N'}$ incident to $F_t$ (isomorphic to the given ${\cal D} _N$) will be the conjugate by some element of $G$ fixing $F_t$, that is by $g\in P_t$. If furthermore $g\in N$, then $g$ leaves ${\cal D} _N$ invariant. It follows that $N$ is unique (in its isomorphism class for type $\bf V$ and $\bf VI$) up to elements in $P_t$ modulo those in $N_t$. Hence we must find the intersection $N_t\cap P_t$. This can be done in the Lie algebras, i.e., we must find $\frak n} \def\rr{\frak r_t\cap \pp_t$. Ihara has shown that all the subalgebras $(\frak n} \def\rr{\frak r_t)_{\fC}$ (with the exception of $\bf IV_{\hbf{n-1}}\subset} \def\nni{\supset} \def\und{\underline IV_{\hbf{n}}$, $n$ even) are {\it regular} subalgebras, i.e., are generated by the Cartan subalgebra $\tt$ and the root spaces $\Gg^{\ga}$ for $\ga\in \Psi_{sym}$, where $\Psi_{sym}$ is a closed, {\it symmetric} set of roots. Similarly, $(\pp_t)_{\fC}$ is the subalgebra generated by $\tt$ and the root spaces $\Gg^{\ga}$ for $\ga\in \Psi_{par}$, where $\Psi_{par}$ is a closed, {\it parabolic} set of roots, $\Psi_{par}=\Phi^+\cup [\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi-\gt]$, where $\gt\subset} \def\nni{\supset} \def\und{\underline \Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi$ is some subset of the set of simple roots, and for any subset $\Xi\subset} \def\nni{\supset} \def\und{\underline \Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi$, $[\Xi]$ denotes the set of roots which are integral linear combinations of the elements of $\Xi$. Then the intersection of $(\pp_t)_{\fC}$ and $(\frak n} \def\rr{\frak r_t)_{\fC}$ is given by \[(\pp_t)_{\fC}\cap (\frak n} \def\rr{\frak r_t)_{\fC}=\tt+\left(\sum_{\ga\in \Psi_{sym}} \Gg^{\ga} \cap \sum_{\ga\in \Psi_{par}}\Gg^{\ga}\right)= \tt + \sum_{\ga\in \Psi_{sym}\cap \Psi_{par}}\Gg^{\ga}.\] {}From this it follows that the complement of $(\pp_t)_{\fC}\cap (\frak n} \def\rr{\frak r_t)_{\fC}$ in $(\pp_t)_{\fC}$ is given by \[\cc=\sum_{\ga\in \Psi_{par}-(\Psi_{sym}\cap \Psi_{par})} \Gg^{\ga}.\] This is of course not a subalgebra, but we can determine the dimension of the parameter space of non-trivial conjugates of $N_t$ incident with $P_t$. In other words, the homogenous space $P_t/(P_t\cap N_t)$ can be identified with the set of symmetric subgroups $N$ incident with $P_t$; its dimension is the cardinality of the set of roots $\Psi_{par}-\Psi_{sym}\cap \Psi_{par}$. To demonstrate this consider $SU(4,1)$. Let $\ga_1=\ge_1-\ge_2,\ldots, \ga_{4}=\ge_4-\ge_5$ denote the simple roots for $\Gg_{\fC}$, we have : \[\Psi_{sym}=\pm(\ge_2-\ge_3),\ \pm(\ge_2-\ge_4),\ \pm(\ge_2-\ge_5),\ \pm(\ge_3-\ge_4),\ \pm(\ge_3-\ge_5),\ \pm(\ge_4-\ge_5),\] \[\Psi_{par}=+(\ge_i-\ge_j),\ \hbox{ (10 of these) }, \pm(\ge_2-\ge_3),\ \pm(\ge_2-\ge_4),\ \pm(\ge_3-\ge_4),\] so that $\Psi_{par}-(\Psi_{sym}\cap \Psi_{par})=+(\ge_1-\ge_j),\ j=2,\ldots, 5$. Hence, taking the relation $\sum\ge_i=0$ into account, there are three effective parameters. Geometrically this can be seen as follows. The bounded symmetric domain is a four-dimensional ball, the boundary component is a point, and the symmetric subdomain ${\cal D} _N$ is an embedded three-ball passing through the point. Now think of the four-ball as embedded in ${\Bbb P}} \def\fB{{\Bbb B}^4$ via the Borel embedding; the three-ball is the intersection of $\fB_4\subset} \def\nni{\supset} \def\und{\underline {\Bbb P}} \def\fB{{\Bbb B}^4$ with a hyperplane ${\Bbb P}} \def\fB{{\Bbb B}^3$ passing through the given point. There is an infinitesimal ${\Bbb P}} \def\fB{{\Bbb B}^3$ of hyperplanes through the point, so we see three effective parameters. Now we turn to the embedding $\bf IV_{\hbf{n-1}}\subset} \def\nni{\supset} \def\und{\underline IV_{\hbf{n}}$, $n$ even. If $G=SO(V,h)$, $h$ symmetric of signature $(n,2)$, let $v\in V$ be an anisotropic vector. Then $v^{\perp}$ is of codimension one, $h_{|v^{\perp}}$ has signature $(n-1,2)$ and $N_t=N_G(v^{\perp})$. On the other hand the parabolic $P_t$ is the stabilizer of a (maximal) two-dimensional totally isotropic subspace $I\subset} \def\nni{\supset} \def\und{\underline V$. Then $V$ splits off two hyperbolic planes $H_1,\ H_2$, and $v$ is in the orthogonal complement of $H:=H_1\oplus H_2$. So the intersection $N_t\cap P_t$ is just the stabilizer of $v$ in $P_t$, i.e., \[ N_t\cap P_t =\{g\in G | g(I)\subseteq I,\ g(v)\in <v>\}.\] Finally we mention the case ${\cal D} \in ({\cal E} {\cal D} )$. Then ${\cal D} _N$ is a polydisc and it is easy to see that the intersection $N_t\cap P_t$ is just the parabolic in $N_t$ corresponding to the given point. Since $N_t\cong (SL_2)^t,\ t=\rank_{{\Bbb R}} \def\fH{{\Bbb H}}{\cal D} $, the parabolic is $(P_1)^t$, where $P_1\subset} \def\nni{\supset} \def\und{\underline SL_2$ is the standard one-dimensional parabolic. So the number of parameters in this case is the dimension of $P_t$ minus $t$. We now list the sets $\Psi_{sym}$, following Ihara, but we will use the notations of the root systems as in \cite{bour}. \begin{itemize}\item ${\bf I_{\hbf{p,q}}}$: $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi=\{\ga_1,\ldots, \ga_{p+q-1}\},\ \Psi_{sym}=[\ga_2,\ldots, \ga_{p+q-1}].$ \item ${\bf II_{\hbf{n}}}$: ($n$ even). $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi=\{\ga_1,\ldots, \ga_{[{n\over 2}]},\} \Psi_{sym}=[\ga_2,\ldots, \ga_{[{n\over2}]}]$. \item ${\bf IV_{\hbf{n}}}$: ($n=2\ell+1$), $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi=\{\ga_1,\ldots,\ga_{\ell}\},\ \beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda:=\ga_{\ell-1}+2\ga_{\ell}$. The following set of roots forms a diagram of type $D_{\ell}$ as indicated: $$ \setlength{\unitlength}{1cm}\begin{picture}(3,1.5) \put(0,1){$\ga_1$} \put(.5,1){$\ga_2$} \put(1,1){$\cdots$} \put(1.5,1){$\ga_{\ell-2}$} \put(2.5,1){$\ga_{\ell}$} \put(1.5,.5){$\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda$} \end{picture}$$ \vspace*{-1cm} \item ${\bf V}$: $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi=\{\ga_1,\ldots, \ga_6\},\ \beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda_1:=\ga_2+\ga_3+2\ga_4+\ga_5+\ga_6,\ \beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda_2:= \ga_2+\ga_4+\ga_5+\ga_6$. Then the subalgebras are determined by the following sets of roots: \[ \setlength{\unitlength}{1cm}\begin{picture}(10.5,1.5) \put(.75,1.4){$\bf I_{\hbf{2,4}}\times SU(2)$} \put(5.75,1.4){$\bf II_{\hbf{5}}$} \put(9.75,1.4){$\bf IV_{\hbf{8}}$} \put(0,1){$\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda_1$} \put(.5,1){$\ga_1$} \put(1,1){$\ga_3$} \put(1.5,1){$\ga_4$} \put(2,1){$\ga_2$} \put(2.5,1){$\cup$} \put(3,1){$\ga_6$} \put(5,1){$\ga_1$} \put(5.5,1){$\ga_3$} \put(6,1){$\ga_4$} \put(6.5,1){$\ga_5$} \put(5.5,.5){$\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda_2$} \put(9,1){$\ga_1$} \put(9.5,1){$\ga_3$} \put(10,1){$\ga_4$} \put(10.5,1){$\ga_5$} \put(10,.5){$\ga_2$} \end{picture}\] \vspace*{-1cm} \item $\bf VI$: $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi=\{\ga_1,\ldots, \ga_7\},\ \beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda_1:=\ga_6+2\ga_5+3\ga_4+2\ga_3+\ga_1+2\ga_2,\ \beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda_2:=\ga_5+2\ga_4+2\ga_3+\ga_1+\ga_2$. Then the subalgebras are determined by the following sets of roots: \[ \setlength{\unitlength}{1cm}\begin{picture}(8,1.5) \put(1,1.4){$\bf I_{\hbf{3,3}}$} \put(7,1.4){$\bf II_{\hbf{6}}$} \put(-.2,1){$(-\ga_2)$} \put(1,1){$\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda_1$} \put(1.5,1){$\ga_7$} \put(2,1){$\ga_6$} \put(2.5,1){$\ga_5$} \put(6,1){$\ga_7$} \put(6.5,1){$\ga_6$} \put(7,1){$\ga_5$} \put(7.5,1){$\ga_4$} \put(8,1){$\ga_2$} \put(6.5,.5){$\beta} \def\gk{\kappa} \def\gl{\lambda} \def\gL{\Lambda_2$} \end{picture} \] \end{itemize} \vspace*{-.75cm} We can also consider the converse question, i.e., given a symmetric subgroup, what is the set of parabolics to which it is indicent? The answer to this is easier: if $\dim(F)>0$, then for any other boundary component $F'$ of ${\cal D} _N$, conjugate to $F$, the parabolic $P_{F'}=N(F')$ is also incident to $N$. These boundary components are in 1-1 correspondence with the zero-dimensional boundary components of the second factors ${\cal D} _2$ of ${\cal D} _N={\cal D} _1\times {\cal D} _2$. If $\dim(F)=0$, then, assuming ${\cal D} _N$ is irreducible (i.e., ${\cal D} \not\in({\cal E} {\cal D} )$), then for any other zero-dimensional boundary component $F'$, the corresponding stabilizer $P_{F'}$ is incident with $N$. If ${\cal D} \in ({\cal E} {\cal D} )$, then we have the set of zero-dimensional boundary components of the polydisc. \subsection{Incidence over $\fQ$} We now return to the notations of section 1.1.2; $G$ is a simple $\fQ$-group of hermitian type. The following definition gives a $\fQ$-form of Definition \ref{d10.1}. \begin{definition}\label{d12.1} Let $P\subset} \def\nni{\supset} \def\und{\underline G$ be a maximal $\fQ$-parabolic, $N\subset} \def\nni{\supset} \def\und{\underline G$ a reductive $\fQ$-subgroup. Then we shall say that $(P,N)$ are {\it incident} (over $\fQ$), if $(P({\Bbb R}} \def\fH{{\Bbb H}),N({\Bbb R}} \def\fH{{\Bbb H}))$ are incident in the sense of Definition \ref{d10.1}. \end{definition} Note that in particular $N$ must itself be of hermitian type, and such that the Cartan involution of $G({\Bbb R}} \def\fH{{\Bbb H})$ restricts to the Cartan involution of $N({\Bbb R}} \def\fH{{\Bbb H})$. The main result of \cite{sym} is the following existence result. \begin{theorem}\label{t12.1} Let $G$ be $\fQ$-simple of hermitian type subject to the restrictions above ($G$ is isotropic and $G({\Bbb R}} \def\fH{{\Bbb H})$ is not a product of $SL_2({\Bbb R}} \def\fH{{\Bbb H})$'s), $P\subset} \def\nni{\supset} \def\und{\underline G$ a $\fQ$-parabolic. Then there exists a reductive $\fQ$-subgroup $N\subset} \def\nni{\supset} \def\und{\underline G$ such that $(P,N)$ are incident over $\fQ$, with the exception of the indices $C^{(2)}_{2n,n}$ for the zero-dimensional boundary components. \end{theorem} We now describe the standard symmetric subgroups $N_b'$ incident to $P_b'$ for the classical cases. For details, see \cite{sym}. We consider the vector space $V$ with the $\pm$symmetric/hermitian form $h$. In the notation of (\ref{E8.c}), if the standard hermitian Levi factor $L_b'$ is $L_b'={\cal N} _G(W)/{\cal Z} _G(W)$, $W=V_{b+1}\oplus \cdots \oplus V_s\oplus V'$ in the notations used there, then for $b<s$ or $c(s,\gs_i)<t_i$ for some $i=1,\ldots, f$, \begin{equation}\label{E12.1} N_b'={\cal N} _G(W). \end{equation} If $b=s$ and $c(s,\gs_i)=t_i$ for all $i=1,\ldots, f$, the boundary component is a point, and $L_s'$ is the anisotropic kernel, and $N_s'$ as in (\ref{E12.1}) is not the standard symmetric subgroup incident to $P_s'$ as we have defined it. Rather, these subgroups correspond to the following constructions. We consider first the case where ${\cal D} \not\in ({\cal E} {\cal D} )$. Pick an anisotropic vector $v\in V$ which is defined over $k$, and consider the subspace $W=v^{\perp}$, the space of vectors orthogonal to $v$. We describe the subgroup $N_s'={\cal N} _{G'}(W)$, which depends on the choice of $v$. \begin{itemize}\item[\ ] {\bf O.1:} $V$ is a $k$-vector space; the subgroup $N_s'$ gives rise to a subdomain ${\cal D} _{N_s'}$ of type $\bf IV_{\hbf{n-1}}\subset} \def\nni{\supset} \def\und{\underline IV_{\hbf{n}}$. \item[\ ] {\bf O.2:} Here $V$ is an $n$-dimensional $D$-vector space, and we have $n$ odd; $W\subset} \def\nni{\supset} \def\und{\underline V$ is of codimension one over $D$, giving rise to a subdomain of type $\bf II_{\hbf{n-1}}\subset} \def\nni{\supset} \def\und{\underline II_{\hbf{n}}$. \item[\ ] {\bf U.1:} In this case we get subdomains $\bf I_{\hbf{p-1,q}}\subset} \def\nni{\supset} \def\und{\underline I_{\hbf{p,q}}$. \item[\ ] {\bf U.2:} $V$ is $n$-dimensional over $D$, where $D$ has degree $d$ over $K$; the subspace $W$ gives rise to a subdomain of type $\bf I_{\hbf{p-d,q}}\subset} \def\nni{\supset} \def\und{\underline I_{\hbf{p,q}}$. Iteration of this gives subdomains of types $\bf I_{\hbf{p-jd,q}}\subset} \def\nni{\supset} \def\und{\underline I_{\hbf{p,q}}$, and for $j=s$ the boundary component will be a point $\iff$ $sd=q$, in which case $\bf I_{\hbf{q,q}}\subset} \def\nni{\supset} \def\und{\underline I_{\hbf{p,q}}$ is a maximal tube domain and fulfills 2'). \end{itemize} Finally we consider ${\cal D} \in ({\cal E} {\cal D} )$. In these cases, if the zero-dimensional boundary component is rational, then $V$ splits into a direct sum of hyperbolic planes (no anisotropic kernel). We can define a unique polydisc by the prescription: letting $V=\oplus_{i=1}^s V_i$ be the decomposition into hyperbolic planes as above, set: \[ N_s':=\{g\in G' | g(V_i)\subseteq V_i,\ i=1,\ldots,s\}. \] For the individual cases this gives rise to the following subdomains: \begin{itemize}\item[\ ] {$\bf I_{\hbf{q,q}}$:} ${\cal D} _{N_s'}\cong \bf I_{\hbf{d,d}} \times \cdots \times I_{\hbf{d,d}}$. In each of the factors $\bf I_{\hbf{d,d}}$ we can apply the results of \cite{hyp} to get a uniquely determined polydisc. \item[\ ] {$\bf II_{\hbf{n}}$, $n$ even:} In this case we get a subdomain $\bf II_{\hbf{2}}\times \cdots \times II_{\hbf{2}}$, which is a polydisc, as $\bf II_{\hbf{2}}$ is a disc. \item[\ ] {$\bf III_{\hbf{n}}$:} In case $\bf S.1$, the result is well known, giving just a polydisc. In case $\bf S.2$, we get as a subdomain $\bf III_{\hbf{2}}\times \cdots \times III_{\hbf{2}}$, and this case represents the exception in Theorem \ref{t12.1}; in general no polydisc (defined over $k$) can be found in each factor. \end{itemize} \section{Arithmetic groups}\label{s83.1} By definition, an arithmetic subgroup $\gG\subset} \def\nni{\supset} \def\und{\underline G(\fQ)$ is one which is commensurable with $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta^{-1}(GL(V_{\fZ}))\cap G(\fQ)$, for some ($\iff$ for any) faithful rational representation $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta:G\longrightarrow} \def\sura{\twoheadrightarrow GL(V)$, where $V$ is a finite-dimensional $\fQ$-vector space, and $V_{\fZ}$ is a $\fZ$-structure, i.e., a $\fZ$-lattice such that $V_{\fZ}\otimes_{\fZ}\fQ=V_{\fQ}$. In the classical cases, it is natural to take the fundamental representation as $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta$ (more precisely the fundamental representation $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta':G'\longrightarrow} \def\sura{\twoheadrightarrow GL_D(V)$ determines $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta:G\longrightarrow} \def\sura{\twoheadrightarrow Res_{k|\fQ}GL_D(V)$), and for the exceptional structures, one has either representations in exceptional Jordan algebras and related algebras, or simply the adjoint representation. Consider first the classical groups. For these, $D$ is a central simple division algebra over $K$, where $K$ is either the totally real number field $k$ or an imaginary quadratic extension of $k$, and $D$ has a $K|k$-involution. The rational vector space $V$ is an $n$-dimensional right $D$-vector space, $A=M_n(D)$ is a central simple algebra over $K$ with a $K|k$-involution extending the involution on $D$ by (\ref{e49.2}). We have a $\pm$symmetric/hermitian form $h:V\times V\longrightarrow} \def\sura{\twoheadrightarrow D$ such that \begin{equation}\label{E212} G'=\{g\in GL_D(V) \big| \forall_{x,y\in V},\ h(x,y)=h(gx,gy)\} \end{equation} is the unitary group of the situation. We take the natural inclusion given by (\ref{E212}), $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta':G'\longrightarrow} \def\sura{\twoheadrightarrow GL_D(V)$ and let the representation $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta:G\longrightarrow} \def\sura{\twoheadrightarrow Res_{k|\fQ}GL_D(V)$ determined by $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta'$ be our rational representation. We now consider $\fZ$-structures on $V$, for which we require an {\it order} $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi\subset} \def\nni{\supset} \def\und{\underline D$, i.e., a lattice that is a subring of $D$, and consider $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi$-lattices $\hbox{{\script L}} \subset} \def\nni{\supset} \def\und{\underline V$. The analog of (\ref{E212}), after fixing the $\fZ$-structure on $V$, is \begin{equation}\label{E212.1} \gG_{\hbox{\sscrpt L}}=\{g\in G \Big| g\hbox{{\script L}} \subseteq \hbox{{\script L}} \}. \end{equation} Then $\gG_{\hbox{\sscrpt L}}\subset} \def\nni{\supset} \def\und{\underline G(\fQ)$ is an arithmetic subgroup, as it is the set of elements which preserve the $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi$-structure on $V$ defined by $\hbox{{\script L}} $, which itself is a $\fZ$-lattice in the rational vector space $V$ (viewing $V$ as a $\fQ$-vector space). If, for example, $\gG_{\hbox{\sscrpt L}'} \subset} \def\nni{\supset} \def\und{\underline \gG_{\hbox{\sscrpt L}}$ is a normal subgroup of finite index, we get an induced representation of $\gG_{\hbox{\sscrpt L}}/\gG_{\hbox{\sscrpt L}'}$ in $\hbox{{\script L}} /\hbox{{\script L}} '$, where $\hbox{{\script L}} '$ is the sublattice of $\hbox{{\script L}} $ preserved by $\gG_{\hbox{\sscrpt L}'}$. This is the general formulation of an occurance which is well-known in specific cases. For example, if $\gG_{\hbox{\sscrpt L}'}=\gG(N)\subset} \def\nni{\supset} \def\und{\underline Sp(2n,\fZ)=\gG_{\hbox{\sscrpt L}}$ is the principal congruence subgroup of level $N$, there is a representation of $\gG/\gG(N)\cong Sp(2n,\fZ/N\fZ)$ in $(\fZ/N\fZ)^{2n}\ (=\hbox{{\script L}} /\hbox{{\script L}} ')$. Now consider the exceptional groups. In the case of $E_6$ we have the 27-dimensional representation in the exceptional Jordan algebra $\JJ$, while in the case of $E_7$ we have the 56-dimensional representation in the exceptional algebra of $2\times 2$ matrices over $\JJ$\footnote{This is what W.~Baily utilized in his beautiful paper \cite{Ba}.}. In both cases we can also use the adjoint representation, so we require a $\fZ$-structure on the Lie algebra itself. Such can be readily constructed, utilizing the Tits algebras, from lattices in the constituents, composition algebras and (exceptional) Jordan algebras. After these introductory remarks we proceed to give a few details, which in particular allow us to give some relevant references in each case. We start by discussing orders, then describe the arithmetic groups these give rise to. \subsection{Orders in associative algebras} A general reference for this section is \cite{reiner}. We first fix some notations. $k$ is a totally real Galois extension of degree $f$ over $\fQ$, and ${\cal O} _k$ will denote the ring of integers in $k$. $D$ will denote a division algebra (skew field), central simple of degree $d$ over $K$, with a $K|k$ involution ($K=k$ for involutions of the first kind, and $K$ is an imaginary quadratic extension of $k$ for involutions of the second kind). $V$ denotes an $n$-dimensional right $D$-vector space, so that $Hom_D(V,V)\ifmmode\ \cong\ \else$\isom$\fi} \def\storth{\underline{\perp} M_n(D)$. $A=M_n(D)$ is a central simple algebra over $K$ with involution extending the involution on $D$ by \begin{equation}\label{e49.2} M \mapsto M^*,\hbox{ where }(M^*)_{ij}=\overline{m}_{ji},\hbox{ for $M=(m_{ij})$}, \end{equation} where ``--'' denotes the involution in $D$. $(V,h)$ is a $\pm$-hermitian space with $\pm$-hermitian form $h$ (with respect to the involution on $D$). Hence $[D:K]=d^2,\ [A:K]=(nd)^2=t^2, t=nd$. Let $F$ be a number field, for example $F=K,k$ as above, and let $W$ be an $F$-vector space. A full ${\cal O} _F$-{\em lattice} ${\cal L} $ in $W$ is an ${\cal O} _F$-module, finitely generated, such that $F\cdot {\cal L} =W$. Usually we work with full lattices and delete the word full. If $W$ is an $F$-algebra, then an ${\cal O} _F$-lattice ${\cal L} $ is an ${\cal O} _F$-{\em order}, if ${\cal L} $ is a subring of $W$. In particular in $W=D$, an ${\cal O} _F$-order is a (full) lattice which is a subring. Let $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi\subset} \def\nni{\supset} \def\und{\underline D$ denote an order in $D$, and let $V$ be an $n$-dimensional vector space over $D$. Then a (full) $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi$-{\em lattice} in $V$ is a $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi$-module ${\cal M} $ with ${\cal M} \cdot D=V$; if again $A$ is the algebra $M_n(D)$, then a $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi$-lattice in $A$ is a $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi$-{\em order}, if it is a subring of $A$. Let an ${\cal O} _F$-lattice ${\cal L} \subset} \def\nni{\supset} \def\und{\underline A$ be given. ${\cal L} $ determines a right (respectively left) ${\cal O} _F$-order: \begin{equation}\label{e73.1}{\cal O} _r({\cal L} )=\{x\in A \Big| {\cal L} \cdot x\subset} \def\nni{\supset} \def\und{\underline {\cal L} \}, (\hbox{respectively } {\cal O} _l({\cal L} )=\{x\in A\Big| x\cdot {\cal L} \subset} \def\nni{\supset} \def\und{\underline {\cal L} \}). \end{equation} If ${\cal L} $ is a $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi$-lattice, then ${\cal O} _r({\cal L} )$ and ${\cal O} _l({\cal L} )$ are $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi$-orders. If an ${\cal O} _F$-order ${\cal O} \subset} \def\nni{\supset} \def\und{\underline A$ is given, and ${\cal L} \subset} \def\nni{\supset} \def\und{\underline A$ is a lattice with ${\cal O} ={\cal O} _r({\cal L} )$ (respectively ${\cal O} _l({\cal L} )$), then one also calls ${\cal L} $ an {\it ${\cal O} $-lattice}, and says that ${\cal L} $ and ${\cal O} $ are {\em associated}. An element $a\in A$ is called {\em integral}, if its characteristic polynomial has integer coefficients, $\chi_a\in{\cal O} _F[X]$. It is a basic result that every element $a\in {\cal O} $ is integral for any ${\cal O} _F$-order ${\cal O} $ in $A$. An order ${\cal O} $ is {\em maximal}, if it is not properly contained in any other order. It is a basic fact that maximal orders exist in $D$ and in $A$, and that any order is contained in a maximal one (\cite{reiner}, 10.4). One has the following description of maximal orders in $A$: \begin{theorem}[\cite{reiner}, 21.6]\label{t74.1} Notations as obove, let $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi\subset} \def\nni{\supset} \def\und{\underline D$ be a fixed maximal ${\cal O} _F$-order in $D$, and let ${\cal M} $ be any (full) right $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi$-lattice in $V$. Then $Hom_{\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi}({\cal M} ,{\cal M} )$ is a maximal ${\cal O} _F$-order in $A$, and for any maximal ${\cal O} _F$-order ${\cal O} $ in $A$, there exists a (full) right $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi$-lattice ${\cal N} \subset} \def\nni{\supset} \def\und{\underline V$ with ${\cal O} =Hom_{\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi}({\cal N} ,{\cal N} )$. \end{theorem} The following result of Chevally describes maximal orders in associative algebras. \begin{theorem}[\cite{reiner}, 27.6]\label{t78.1} Let $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi\subset} \def\nni{\supset} \def\und{\underline D$ be a maximal ${\cal O} _K$-order in $D$; for each right ideal $J\subset} \def\nni{\supset} \def\und{\underline \Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi$, set $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi'={\cal O} _l(J)$. Then every maximal order of $A=M_n(D)$ is of the form $${\cal O} _J=\left( \begin{array}{cccc} \Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi & \cdots & \Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi & J^{-1} \\ \vdots & \ddots & \vdots & \vdots \\ \Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi & \cdots & \Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi & J^{-1} \\ J & \cdots & J & \Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi' \end{array}\right),$$ for some right ideal $J$, and for each $J$, the lattice ${\cal O} _J$ above is a maximal order. \end{theorem} In other words, to give a maximal order in $A$ is the same as giving a maximal order $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi\subset} \def\nni{\supset} \def\und{\underline D$, together with a right ideal $J\subset} \def\nni{\supset} \def\und{\underline \Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi$, i.e., the same as giving a pair $(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi,J)$. In particular if the class number $h(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi)=1$ (note that $h(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi)$, which is defined as the number of left $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi$-ideal classes, is also equal to the number of right $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi$-ideal classes, see \cite{reiner}, Ex. 7 iii), p. 232), then up to $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi$-isomorphism there is a 1-1 correspondence between isomorphism classes of maximal orders in $D$ and $A$. \subsection{Orders in Jordan algebras} First recall the result on orders in the (definite) Cayley algebra from \cite{BS}. Let $e_0,\ldots,e_7$ be the base of $\frak C_{\fQ}$ given as follows \[\begin{minipage}{12cm} $\frak C_{\fQ}= e_0\fQ+ e_1\fQ+\ldots+ e_7\fQ,$ with center $e_0\fQ$ and relations:\hfill\break $e_i\cdot e_{i+1}=e_{i+3}, e_{i+1}\cdot e_{i+3}=e_i, e_{i+3}\cdot e_i=e_{i+1}, e_i^2=-e_0,\ i\in \fZ/7\fZ$. \end{minipage}\] Define \begin{equation}\label{e80a.1} \hbox{{\script M}} :=\{x=\sum\xi_ie_i\Big| 2\xi_i\in \fZ, \xi_i-\xi_j\in \fZ, \sum \xi_i\in 2\fZ\}. \end{equation} Then \begin{lemma}[\cite{BS}, 4.6]\label{l80a.1} $\hbox{{\script M}} $ is a maximal order in $\frak C_{\fQ}$, and any other maximal order is isomorphic to $\hbox{{\script M}} $. \end{lemma} (In \cite{BS} the authors call $\hbox{{\script M}} $ an octave-ring: a subring of $\frak C_{\fQ}$ containing 1, on which the norm form is integral, and maximal with these properties; we just call $\hbox{{\script M}} $ a maximal order.) A general reference for the remainder of this section is \cite{racine}. Let $R$ be a commutative ring. A {\em Jordan algebra} over $R$ is an $R$-module which is commutative and satisfies the relation \[(x^2\cdot y)\cdot x = x^2\cdot (y\cdot x),\ \forall_{x,y}.\] \vspace*{-.3cm} \begin{definition}\label{d74a.1} Let $\JJ$ be a Jordan algebra over a number field $K$, and let ${\cal O} _K$ denote the ring of integers in $K$. A full ${\cal O} _K$-lattice ${\cal L} \subset} \def\nni{\supset} \def\und{\underline \JJ$ is an {\em order}, if ${\cal L} $ is a Jordan algebra over ${\cal O} _K$. \end{definition} An element $x\in \JJ$ is {\em integral}, if the characteristic polynomial is integral, i.e., if $N(x), Q(x)$ and $T(x)$ are integral (see \cite{Jac}, pp.~91, also \cite{Jacob}, Chapter VI, for details). Let ${\cal L} $ be an order in $\JJ$, and $x\in {\cal L} $; then ${\cal O} _K[x]\subset} \def\nni{\supset} \def\und{\underline {\cal L} $ is an associative subalgebra, hence finitely generated, so $x$ is integral (\cite{racine} Prop.~1, p.~19). Conversely, any integral element of $\JJ$ is contained in an order ({\it loc.~cit.}~Prop.~2). Once again it is a basic fact that maximal orders exist ({\it loc.~cit.}~Thm.~2) and that an order is maximal if and only if it is maximal locally everywhere ({\it loc.~cit.}~Lemma 1). A maximal order ${\cal L} \subset} \def\nni{\supset} \def\und{\underline \JJ$ is said to be {\em distinguished}, if ${\cal L} $ is a maximal lattice of integral elements. For example, if ${\cal O} \subset} \def\nni{\supset} \def\und{\underline {\Bbb O}} \def\fA{{\Bbb A}$ is a maximal order in an octonion algebra, then $J({\cal O} ,\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta)\subset} \def\nni{\supset} \def\und{\underline J({\Bbb O}} \def\fA{{\Bbb A},\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta)$ (notations as in \ref{d46b.1}) is a distinguished maximal order. Conversely, for $\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta=1$, \begin{proposition}[\cite{racine}, Prop.~5, p.~115]\label{p74a.1} If $\JJ=J({\Bbb O}} \def\fA{{\Bbb A}_K,1)$ is the exceptional Jordan algebra over the totally indefinite octonion algebra ${\Bbb O}} \def\fA{{\Bbb A}_K$, then any distinguished order ${\cal P} \subset} \def\nni{\supset} \def\und{\underline \JJ$ is of the form $J({\cal O} ,1)\subset} \def\nni{\supset} \def\und{\underline \JJ$, with ${\cal O} $ a maximal order in ${\Bbb O}} \def\fA{{\Bbb A}_K$. \end{proposition} This may be considered in some sense as an analogue of Theorem \ref{t74.1} for orders in exceptional Jordan algebras. \subsection{Lattices in Tits algebras} Let $\AA$ be a composition algebra over $K$, and $\JJ=J(\AA',1)$ a Jordan algebra as in \ref{d46b.1} over a second composition algebra $\AA'$. For a totally indefinite octonion algebra over $K$, $\AA'$, and a maximal order $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi'\subset} \def\nni{\supset} \def\und{\underline \AA'$, then, as we have seen (Proposition \ref{p74a.1}), ${\cal L} =J(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi',1)$ is a distinguished order in $\JJ$ and conversely. More generally it is easy to see: \begin{lemma}\label{l74b.1} Let $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi\subset} \def\nni{\supset} \def\und{\underline \AA$ be a maximal order in the composition algebra $\AA$. Then $J(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi,\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta)$ is a maximal order in the Jordan algebra $J(\AA,\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta)$ of Definition \ref{d46b.1}. \end{lemma} {\bf Proof:} ${\cal L} :=J(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi,\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta)$ is clearly a Jordan algebra over ${\cal O} _K$, hence it is an order in $\JJ$. To see it is maximal, the method of \cite{racine} can be used. Let $L_1=\{a\in {\Bbb O}} \def\fA{{\Bbb A}_K \big| a[j,k]\in {\cal L} \}$ (notations as in (\ref{e46d.1})); this is a lattice in ${\Bbb O}} \def\fA{{\Bbb A}_K$, and, as can be checked, is the lattice $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi$ which we started with. If ${\cal L} $ is not maximal, then ${\cal L} \subsetneqq {\cal L} '$, and the corresponding $L_1'$ will be an ${\cal O} _K$-lattice in $\AA'$ with $L_1\subsetneqq L_1'$, contradicting the maximality of $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi$. \hfill $\Box$ \vskip0.25cm Let $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi\subset} \def\nni{\supset} \def\und{\underline \AA$ be a maximal order and ${\cal L} \subset} \def\nni{\supset} \def\und{\underline \JJ$ a maximal order. Consider the Tits algebra $\hbox{{\script L}} (\AA,\JJ)$ of Definition \ref{d46f.1}. Recall that the construction of Tits algebras requires, in addition to the algebras $\AA$ and $\JJ$, also the Lie algebras $Der(\AA)$ and $Der(\JJ)$. If we have maximal orders $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi\subset} \def\nni{\supset} \def\und{\underline \AA,\ {\cal L} \subset} \def\nni{\supset} \def\und{\underline \JJ$, then we {\it define}: \[Der(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi):= \{ D\in Der(\AA) \Big| D(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi)\subseteq \Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi\},\quad Der({\cal L} ):= \{ D\in Der(\JJ) \Big| D({\cal L} )\subseteq {\cal L} \}.\] Since we know that $Der(\AA)$ is a Lie algebra of type $G_2$ and $Der(\JJ)$ is a Lie algebra of type $F_4$, we are asking for $\fZ$-structures on these Lie algebras. Clearly $Der(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi)$ and $Der({\cal L} )$ are lattices in the corresponding Lie algebras, which are furthermore closed under the Lie bracket. It then is natural to consider the following lattice in the Tits algebra: \begin{equation}\label{E210A} \gL_{\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi,{\cal L} }:=Der(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi)\oplus \Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_0\otimes {\cal L} _0\oplus Der({\cal L} ), \end{equation} and the corresponding arithmetic group it defines (for $G=\hbox{Aut}} \def\Im{\hbox{Im}} \def\mod{\hbox{mod}(\hbox{{\script L}} (\AA,\JJ))^0$) \begin{equation}\label{E210B} \gG_{\gd,{\cal L} }:= \{ g\in G \Big| \ad(g)(\gL_{\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi,{\cal L} })\subseteq \gL_{\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi,{\cal L} }\}, \end{equation} where $G$ is acting by means of the adjoint representation on $\hbox{{\script L}} (\AA,\JJ)$. \subsection{Arithmetic groups -- classical cases}\label{sarithmetic} In this subsection $G'$ will denote an absolutely (almost) simple $k$-group ($k$ a totally real number field) which we assume is classical, $G=Res_{k|\fQ}G'$ the $\fQ$-simple group it defines, which we assume is of hermitian type. We let $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta':G'\longrightarrow} \def\sura{\twoheadrightarrow GL_D(V)$ be the natural inclusion and $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta:G\longrightarrow} \def\sura{\twoheadrightarrow Res_{k|\fQ}GL_D(V)$ be the natural representation of $G$ defined by $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta'$. Fix a maximal order $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi\subset} \def\nni{\supset} \def\und{\underline D$, and let ${\cal L} \subset} \def\nni{\supset} \def\und{\underline V$ be a $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi$-lattice (which is in particular a $\fZ$-lattice of the underlying $\fQ$-vector space). As above, ${\cal O} _r({\cal L} )$ (respectively ${\cal O} _l({\cal L} )$) will denote the right (respectively left) order of ${\cal L} $, given by the equation (\ref{e73.1}). First of all, we have the arithmetic subgroup $GL_{\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi}({\cal L} )\subset} \def\nni{\supset} \def\und{\underline GL_D(V)$, and we define the subgroup \[\gG_{{\cal L} }':= \{g\in G'(k) \Big| \varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta(g)({\cal L} )\subseteq {\cal L} \} = \varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta^{-1}(GL_{\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi}({\cal L} ))\subset} \def\nni{\supset} \def\und{\underline G'(k),\] and similarly $\gG_{{\cal L} }\subset} \def\nni{\supset} \def\und{\underline G(\fQ)$. By definition these are arithmetic subgroups of $G'(k)$ and $G(\fQ)$, respectively. Let us see how this is related to the orders ${\cal O} _r({\cal L} )$ and ${\cal O} _l({\cal L} )$. By Theorem \ref{t78.1} ${\cal O} _r({\cal L} )$ is of the form ${\cal O} _J$ for some right ideal $J\subset} \def\nni{\supset} \def\und{\underline {\cal O} $. Our central simple algebra is in this case $A=M_n(D)$, and ${\cal O} _r({\cal L} )$ is a maximal order in $A$. Recall how the group $G'$ and the algebra are related (\cite{W}, Thm.~2, p.~598). Let $U=\{z\in A | z z^*=1\}$, $U_0$ the connected component of $U$, ($(G')^0=$)$G_0:=(\hbox{Aut}} \def\Im{\hbox{Im}} \def\mod{\hbox{mod}(A))^0$, and let $C\subset} \def\nni{\supset} \def\und{\underline U_0$ be the center of $U_0$. Then we have an exact sequence \[1\longrightarrow} \def\sura{\twoheadrightarrow C\longrightarrow} \def\sura{\twoheadrightarrow U_0 \longrightarrow} \def\sura{\twoheadrightarrow G_0 \longrightarrow} \def\sura{\twoheadrightarrow 1.\] As a lattice in $A$ we consider ${\cal O} :={\cal O} _r({\cal L} )$ and its intersection with $U_0$, \[ {\cal O} _0={\cal O} \cap U_0.\] Similarly, ${\cal C} :=C\cap {\cal O} _0$ is the center of ${\cal O} _0$, and we have the sequence \[1\longrightarrow} \def\sura{\twoheadrightarrow {\cal C} \longrightarrow} \def\sura{\twoheadrightarrow {\cal O} _0 \longrightarrow} \def\sura{\twoheadrightarrow \gG'\longrightarrow} \def\sura{\twoheadrightarrow 1,\] where $\gG'\cong {\cal O} _0/{\cal C} $ is the arithmetic subgroup $\gG'\subset} \def\nni{\supset} \def\und{\underline G'(k)$, showing how the maximal orders are related to the arithmetic groups. In our situation here, $(G')^0$ plays the role of $G_0$, while $(U')^0=(\{z\in A \Big| z^*z=1\})^0$ plays the role of $U_0$. Let further $C'\subset} \def\nni{\supset} \def\und{\underline (U')^0$ be the center. We have ${\cal O} \cong {\cal O} _J$ for some right ideal $J$, and $({\cal O} _J)_0={\cal O} _J\cap U_0$ plays the role of ${\cal O} _0$. Then $C'\cap ({\cal O} _J)_0={\cal C} $ is the center of $({\cal O} _J)_0$, and we have sequences: \[\begin{array}{ccccccccc} 1 & \longrightarrow} \def\sura{\twoheadrightarrow & C' & \longrightarrow} \def\sura{\twoheadrightarrow & (U')^0 & \longrightarrow} \def\sura{\twoheadrightarrow & (G')^0 & \longrightarrow} \def\sura{\twoheadrightarrow & 1 \\ & & \cup & & \cup & & \cup \\ 1 & \longrightarrow} \def\sura{\twoheadrightarrow & {\cal C} & \longrightarrow} \def\sura{\twoheadrightarrow & ({\cal O} _J)_0 & \longrightarrow} \def\sura{\twoheadrightarrow & \gG_{{\cal L} }' & \longrightarrow} \def\sura{\twoheadrightarrow & 1. \end{array}\] In this sense, maximal orders give rise to arithmetic subgroups. Viewing the ${\cal O} _k$-lattice ${\cal L} $ as a $\fZ$-lattice gives the corresponding diagram for the $\fQ$-groups (with hopefully obvious notations) \[\begin{array}{ccccccccc} 1 & \longrightarrow} \def\sura{\twoheadrightarrow & C & \longrightarrow} \def\sura{\twoheadrightarrow & U^0 & \longrightarrow} \def\sura{\twoheadrightarrow & G^0 & \longrightarrow} \def\sura{\twoheadrightarrow & 1 \\ & & \cup & & \cup & & \cup \\ 1 & \longrightarrow} \def\sura{\twoheadrightarrow & {\cal Z} ({\cal O} _0) & \longrightarrow} \def\sura{\twoheadrightarrow & {\cal O} _0 & \longrightarrow} \def\sura{\twoheadrightarrow & \gG_{{\cal L} } & \longrightarrow} \def\sura{\twoheadrightarrow & 1. \end{array}\] We now describe this more precisely for the following special cases: \begin{itemize}\item[a)] Siegel modular groups. \item[b)] Picard modular groups. \item[c)] Hyperbolic plane modular groups. \end{itemize} These are examples of $\fQ$-groups which are of both inner type (for a)) and outer type (for b) and c)), of split over ${\Bbb R}} \def\fH{{\Bbb H}$-type, meaning the $\fQ$-rank is equal to the ${\Bbb R}} \def\fH{{\Bbb H}$-rank (for a) and b)) and more or less the {\it opposite} of split over ${\Bbb R}} \def\fH{{\Bbb H}$-type ($\fQ$-rank equal to one, ${\Bbb R}} \def\fH{{\Bbb H}$-rank unbounded) (for c)). Case a) is well-known, b) is also to a certain extent, while c) was introduced in \cite{hyp}. \begin{itemize}\item[a)] Siegel case: \begin{itemize}\item $A=M_{2n}(\fQ)$ with the involution $*:X\mapsto JX^tJ,\quad J=\left(\begin{array}{cc}0 & \hbox{\Large\boldmath $1_{\hbf{n}}$} \\ -\hbox{\Large\boldmath $1_{\hbf{n}}$} & 0 \end{array}\right)$. \item $\hbox{Aut}} \def\Im{\hbox{Im}} \def\mod{\hbox{mod}(A,*)\cong PSp(2n,\fQ),\quad V=\fQ^{2n}$. \item $D=\fQ$, a maximal order is $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi=\fZ,\ \ V_{\fZ}=\fZ^{2n}$. \item $\gG=PSp(2n,\fZ)$. \end{itemize} The sequence above becomes: \[ \begin{array}{ccccccccc} 1 & \longrightarrow} \def\sura{\twoheadrightarrow & \fZ/(2) & \longrightarrow} \def\sura{\twoheadrightarrow & Sp(2n,\fQ) & \longrightarrow} \def\sura{\twoheadrightarrow & PSp(2n,\fQ) & \longrightarrow} \def\sura{\twoheadrightarrow & 1 \\ & & || & & \cup & & \cup \\ 1 & \longrightarrow} \def\sura{\twoheadrightarrow & \fZ/(2) & \longrightarrow} \def\sura{\twoheadrightarrow & Sp(2n,\fZ) & \longrightarrow} \def\sura{\twoheadrightarrow & \gG & \longrightarrow} \def\sura{\twoheadrightarrow & 1. \end{array}\] \item[b)] Picard case: \begin{itemize}\item $A=M_n(K)$ with involution $*:X\mapsto HX^tH$, $H$ hermitian, where $K|\fQ$ is imaginary quadratic. \item $\hbox{Aut}} \def\Im{\hbox{Im}} \def\mod{\hbox{mod}(A,*)\cong PSU(K^n,h),\quad V=K^n$, $h$ is a hermitian form represented by $H$. \item $D=K$, a maximal order is $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi={\cal O} _K,\quad V_{\fZ}={\cal O} _K^n$. \item $\gG=PSU({\cal O} _K^n, h)$ (or $PU({\cal O} _K^n,h)$, which is not simple, but is often considered anyway). \end{itemize} The sequence above becomes: \[\begin{array}{ccccccccc} 1 & \longrightarrow} \def\sura{\twoheadrightarrow & {\cal C} & \longrightarrow} \def\sura{\twoheadrightarrow & SU(K^n,h) & \longrightarrow} \def\sura{\twoheadrightarrow & PSU(K^n,h) & \longrightarrow} \def\sura{\twoheadrightarrow & 1 \\ & & \cup & & \cup & & \cup \\ 1 & \longrightarrow} \def\sura{\twoheadrightarrow & \hbox{{\script C}} & \longrightarrow} \def\sura{\twoheadrightarrow & SU({\cal O} _K^n,h) & \longrightarrow} \def\sura{\twoheadrightarrow & PSU({\cal O} _K^n,h) & \longrightarrow} \def\sura{\twoheadrightarrow & 1. \end{array}\] Note that ${\cal C} $ is given essentially by ${\cal O} _K\cap U(1)$, which is $\pm1$ except for the two fields $K=\fQ(\sqrt{-1}),\ K=\fQ(\sqrt{-3})$ which contain fourth (respectively third) roots of unity. \item[c)] Hyperbolic plane case: \begin{itemize}\item $D$ is a division algebra, central simple of degree $d\geq2$ over $K$, with a $K|\fQ$-involution, $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi\subset} \def\nni{\supset} \def\und{\underline D$ is a maximal order. \item $A=M_2(D)$ with involution $*:X\mapsto {^t\overline{X}}$, where ${^t\overline{X}}=(\overline{x}_{ji})$, if $X=(x_{ij})$, and $\overline{x}$ denotes the involution in $D$. \item $\hbox{Aut}} \def\Im{\hbox{Im}} \def\mod{\hbox{mod}(A,*)$ is a $\fQ$-form of $PSU(d,d)$, and $V=D^2,$ with a hermitian form $h:V\times V\longrightarrow} \def\sura{\twoheadrightarrow D$ which is isotropic, $V_{\fZ}=\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi^2$. \item $\gG=PSU(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi^2,h)$. \end{itemize} The above sequence becomes in this case \[\begin{array}{ccccccccc} 1 & \longrightarrow} \def\sura{\twoheadrightarrow & {\cal C} & \longrightarrow} \def\sura{\twoheadrightarrow & SU(D^2,h) & \longrightarrow} \def\sura{\twoheadrightarrow & PSU(D^2,h) & \longrightarrow} \def\sura{\twoheadrightarrow & 1 \\ & & \cup & & \cup & & \cup \\ 1 & \longrightarrow} \def\sura{\twoheadrightarrow & {\cal C} \cap \Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi & \longrightarrow} \def\sura{\twoheadrightarrow & SU(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi^2,h) & \longrightarrow} \def\sura{\twoheadrightarrow & PSU(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi^2,h) & \longrightarrow} \def\sura{\twoheadrightarrow & 1. \end{array}\] As $D$ is central simple over $K$, the center is as in the last case, ${\cal C} \cong {\cal O} _K\cap U(1)$, hence it is $\pm 1$ except for the case $K=\fQ(\sqrt{-1})$ and $K=\fQ(\sqrt{-3})$ as above. \end{itemize} \subsection{Arithmetic groups -- exceptional cases} We mentioned above that for the exceptional cases, there are (at least) two natural types of representations we can consider: representations in algebras derived from exceptional Jordan algebras (Tits algebras), and the adjoint representation. These representations correspond to the following fundamental weights: \vspace*{.5cm} \[ \setlength{\unitlength}{0.006500in}% \begin{picture}(854,94)(33,693) \thicklines \put( 40,780){\circle{14}}\put( 30,800){$\go_1 (27)$} \put(120,780){\circle{14}}\put(110,800){$\go_3$} \put(200,780){\circle{14}}\put(190,800){$\go_4$} \put(280,780){\circle{14}}\put(270,800){$\go_5$} \put(360,780){\circle{14}}\put(350,800){$\go_6 (27)$} \put(200,700){\circle{14}}\put(190,670){$\go_2 (78)$} \put(480,780){\circle{14}}\put(470,800){$\go_7 (56)$} \put(560,780){\circle{14}}\put(550,800){$\go_6$} \put(640,780){\circle{14}}\put(630,800){$\go_5$} \put(720,780){\circle{14}}\put(710,800){$\go_4$} \put(800,780){\circle{14}}\put(790,800){$\go_3$} \put(880,780){\circle{14}}\put(870,800){$\go_1 (133)$} \put(720,700){\circle{14}}\put(710,670){$\go_2$} \put(200,773){\line( 0,-1){ 66}} \put(487,780){\line( 1, 0){ 66}} \put(567,780){\line( 1, 0){ 66}} \put(807,780){\line( 1, 0){ 66}} \put(720,773){\line( 0,-1){ 66}} \put(727,780){\line( 1, 0){ 66}} \put(647,780){\line( 1, 0){ 66}} \put( 47,780){\line( 1, 0){ 66}} \put(127,780){\line( 1, 0){ 66}} \put(207,780){\line( 1, 0){ 66}} \put(287,780){\line( 1, 0){ 66}} \end{picture}\] In the case of $E_6$, the 27-dimensional (respectively the adjoint, 78-dimensional) representation corresponds to the weights $\go_1$ and $\go_6$ (respectively to $\go_2$), while in the case of $E_7$, the 56-dimensional (respectively the adjoint, 133-dimensional) representation corresponds to the weight $\go_7$ (respectively to $\go_1$). We briefly discuss the arithmetic groups arising in this way. We first consider the 27-dimensional representation. For this we assume $G'$ has index ${^2E}_{6,2}^{16'}$ and we use the model \[\Gg'={\cal L} (\JJ)_{\gl}=\sqrt{\gl}R_{\JJ_0}\oplus Der(\JJ)\] (Albert's twisted ${\cal L} (\JJ)$), where $\gl<0, \gl\in k$. We then choose a maximal order ${\cal M} \subset} \def\nni{\supset} \def\und{\underline \JJ_k$ and set \[Der({\cal M} )=\{a\in Der(\JJ) \Big| a({\cal M} )\subseteq {\cal M} \}.\] Then we may consider the lattice \[{\cal L} ({\cal M} )_{\gl}:=\sqrt{\gl}R_{{\cal M} _0}\oplus Der({\cal M} ).\] This defines an arithmetic group: \[\gG_{{\cal M} }:=\{g\in G' \Big| \varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta(g)({\cal L} ({\cal M} )_{\gl})\subseteq {\cal L} ({\cal M} )_{\gl}\},\] where $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta$ is the 27-dimensional representation in ${\cal L} (\JJ)_{\gl}$. Next we consider the adjoint representation. For this we utilize the lattice in the Tits algebra constructed in (\ref{E210A}), and the corresponding arithmetic group (\ref{E210B}). That lattice depends on the choice of a maximal order $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi$ in the Cayley algebra, as well as on one in the algebra $\BB$. More explicitly, \begin{theorem}\label{t80b.1} Let $\Gg'$ be a $k$-form of $\ee_{6(-14)}$ as in Corollary \ref{c55.1}, $G'$ as in Corollary \ref{c55.2}, i.e., $$\Gg'\ifmmode\ \cong\ \else$\isom$\fi} \def\storth{\underline{\perp} \hbox{{\script L}} (\frak C_k, (J_1)_k^b),\quad (G')^0\ifmmode\ \cong\ \else$\isom$\fi} \def\storth{\underline{\perp} (\hbox{Aut}} \def\Im{\hbox{Im}} \def\mod{\hbox{mod}(\Gg'))^0.$$ Let $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi\subset} \def\nni{\supset} \def\und{\underline \frak C_k$ be a maximal order in the Cayley algebra $\frak C_k$ as above, let ${\cal L} \subset} \def\nni{\supset} \def\und{\underline (J_1)_k^b$ be a maximal order in the Jordan algebra (Definition \ref{d74a.1}), and set $$\Gg_{(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi,{\cal L} )}':=\hbox{{\script L}} (\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi,{\cal L} )=\left\{ X\in \hbox{{\script L}} (\frak C_k,(J_1)_k^b) \Big| \parbox{6cm}{$X=X_1+x\otimes y + Y_1: X_1\in Der(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi),$ $Y_1\in Der({\cal L} ), x\in \Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_0, y\in {\cal L} _0$ } \right\}.$$ Then $\Gg_{(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi,{\cal L} )}'$ is an ${\cal O} _k$-lattice in the $k$-vector space $\Gg'$. Set $$\gG_{(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi,{\cal L} )}':=\{g\in G'(k) \Big| \ad(g)(\Gg_{(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi,{\cal L} )}')\subset} \def\nni{\supset} \def\und{\underline \Gg_{(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi,{\cal L} )}'\}.$$ Then $\gG_{(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi,{\cal L} )}'\subset} \def\nni{\supset} \def\und{\underline G'(k)$ is an arithmetic subgroup. \end{theorem} Now consider type $E_7$. We first consider the 56-dimensional representaion. This is the situation considered by Baily in \cite{Ba}. In this example $k=\fQ$, and $\JJ$ is the exceptional Jordan algebra over $\fQ$, $\JJ_{\fQ}=J^b=J(\frak C_{\fQ},(1,-1,1))$ in the notation of Definition \ref{d46b.1}, and $\AA_{\fQ}=M_2(\fQ)$. Let $\hbox{{\script M}} \subset} \def\nni{\supset} \def\und{\underline \frak C_{\fQ}$ be the maximal order (\ref{e80a.1}). This determines, as in \ref{l74b.1}, a maximal order ${\cal L} $ in $\JJ_{\fQ}$. Also $\fZ\subset} \def\nni{\supset} \def\und{\underline \fQ$ defines the maximal order $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi=M_2(\fZ)\subset} \def\nni{\supset} \def\und{\underline M_2(\fQ)$. This then gives rise to an arithmetic group $G_{(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi,{\cal L} )}$, which Baily shows is maximal and has only one cusp. Again in this case we can also consider the adjoint representation. For this we again utilize the lattice (\ref{E210A}), and as above, this determines an arithmetic group as in (\ref{E210B}). This time, we need a lattice in the totally indefinite quaternion algebra $\AA$ as well as one in the Jordan algebra $(J_1)_k^b$. More explicitly, \begin{theorem}\label{t80b.2} Let $\Gg'$ be a $k$-form of $\ee_{7(-25)}$ as in Theorem \ref{t55a.2}, i.e., $$\Gg'\ifmmode\ \cong\ \else$\isom$\fi} \def\storth{\underline{\perp} \hbox{{\script L}} (\AA_k,\JJ_k),\quad (G')^0\ifmmode\ \cong\ \else$\isom$\fi} \def\storth{\underline{\perp} (\hbox{Aut}} \def\Im{\hbox{Im}} \def\mod{\hbox{mod}(\Gg'))^0.$$ Let $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi\subset} \def\nni{\supset} \def\und{\underline \AA_k$ be a maximal order in the indefinite quaternion algebra $\AA_k$ as in section 3.1, and let ${\cal L} \subset} \def\nni{\supset} \def\und{\underline \JJ_k$ be a maximal order in the Jordan algebra $\JJ_k$ as in \ref{d74a.1}, and set: $$\Gg_{(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi,{\cal L} )}':=\hbox{{\script L}} (\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi,{\cal L} ) \hbox{ as above }. $$ Then $\Gg_{(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi,{\cal L} )}'\subset} \def\nni{\supset} \def\und{\underline \Gg'$ is an ${\cal O} _k$-lattice, and $$G_{(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi,{\cal L} )}':=\{g\in \hbox{Aut}} \def\Im{\hbox{Im}} \def\mod{\hbox{mod}(\Gg')\Big| \ad(g)(\Gg_{(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi,{\cal L} )}')\subset} \def\nni{\supset} \def\und{\underline \Gg_{(\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi,{\cal L} )}'\}\cap (G')^0$$ is an arithmetic subgroup in $G'(k)$. \end{theorem} A more detailed discussion of these matters cn be found in \cite{new}. \section{Integral symmetric subgroups} Let $G$ be a $\fQ$-simple algebraic group of hermitian type, and let $A\subset} \def\nni{\supset} \def\und{\underline G$ be a maximal ${\Bbb R}} \def\fH{{\Bbb H}$-split torus defined by the set of strongly orthogonal roots as in section 1.1, given the canonical order. Let $S\subset} \def\nni{\supset} \def\und{\underline A$ be a maximal $\fQ$-split torus with the canonical order, compatible with the given order on $A$. Let further $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_{\fQ}=\{\eta_1,\ldots, \eta_s\}$ be the set of simple $\fQ$-roots, and let $F_{\hbox{\scsi \bf b}},\ P_{\hbox{\scsi \bf b}}$ be the standard boundary components and parabolics as explained above, ${\bf b}=(c(b,\gs_1),\ldots,c(b,\gs_f)), b=1,\ldots, s$. Finally, let $N_{\hbox{\scsi \bf b}}$ be the standard incident symmetric subgroup (i.e., given by (\ref{e10.1}) if $\dim(F_{\hbox{\scsi \bf b}})>0$, and in terms of root systems as explained in section 2.2 for $\dim(F_{\hbox{\scsi \bf b}})=0$). Since $N_{\hbox{\scsi \bf b}}$ is a reductive subgroup, it is not true that any $G$-conjugate $N'$ of $N_{\hbox{\scsi \bf b}}$ is already $G_{\fQ}$-conjugate. Therefore we make the following definition, yielding a proper subset of the set of $G$-conjugates of the given $N_{\hbox{\scsi \bf b}}$. \begin{definition}\label{D4.1} Let $G,\ S,\ P_{\hbox{\scsi \bf b}},\ N_{\hbox{\scsi \bf b}}$ be given as above. A symmetric subgroup $N'\subset} \def\nni{\supset} \def\und{\underline G$ which is conjugate to $N_{\hbox{\scsi \bf b}}$ by an element of $G(\fQ)$ is called a {\it rational symmetric} subgroup of $G$. \end{definition} The following well-known example illustrates the difference between rational and more general symmetric $\fQ$-subgroups. \begin{example}\label{example} Let $G'$ be the symplectic group $G'=Sp(V,h),\ G=Res_{k|\fQ}G'$, where $V$ is a $k$-vector space of dimension $2n$ and $h$ is skew-symmetric. If $n=2$, the corresponding domain is a product of copies of the Siegel space of degree 2 (type $\bf III_{\hbf{2}}$). The boundary components corresponding to $P_{\hbf{1}}$ (respectively $P_{\hbf{2}}$) are products of one-dimensional (respectively zero-dimensional) boundary components. Then $N_{\hbf{1}}$ is also a product of two factors, $N_{\hbf{1}}=N_{\hbf{1},1}\times N_{\hbf{1},2}$, and each $N_{\hbf{1},i}$ is a polydisc $({\bf H})^f$. If we consider the universal family of abelian varieties parameterized by the domain ${\cal D} $, say ${\cal A} \longrightarrow} \def\sura{\twoheadrightarrow {\cal D} $, we may consider the following conditions on the fibres $A_t\in {\cal A} $ ($t\in {\cal D} $): \begin{itemize}\item[1)] $A_t$ is isogenous to a product. \item[2)] $A_t$ is simple with real multiplication by some real quadratic extension $k'|k$. \end{itemize} We claim that the locus 1) is the locus of subdomains ${\cal D} _{N'}$, where $N'$ is rational symmetric, while the locus 2) is the union of ${\cal D} _{N'}$, where $N'$ is a $\fQ$-subgroup conjugate to $N_{\hbf{1}}$, but not in $G(\fQ)$. To see this, let us suppose $k=\fQ$; we have the familiar description for the domains ${\cal D} _{N'}$ of 2): in this case the standard symmetric subdomain is ${\Bbb S}} \def\fQ{{\Bbb Q}_1\times {\Bbb S}} \def\fQ{{\Bbb Q}_1 \subset} \def\nni{\supset} \def\und{\underline {\Bbb S}} \def\fQ{{\Bbb Q}_2$ (given by the diagonal $2\times 2$ matrices), and it is conjugated (in $G_{{\Bbb R}} \def\fH{{\Bbb H}}$) by the matrix $$\hbox{{\script S}} =\left(\begin{array}{cc} S^{-1} & 0 \\ 0 & {^tS} \end{array}\right),\quad S=\left(\begin{array}{cc} 1 & w \\ 1 & \overline{w} \end{array}\right),\ w\in {\cal O} _{k'}$$ for a real quadratic extension $k'|\fQ$. More precisely, the subdomains ${\cal D} _{N'}$ are given by the equations \begin{equation}\label{humbert} \HH_{(a,b,c,d,e)}:=\left\{ \tau=\left(\begin{array}{cc}\tau_{11} & \tau_{12} \\ \tau_{12} & \tau_{22} \end{array} \right) \Big| a\tau_{11}+b\tau_{12}+c\tau_{22}+ d(\tau_{12}^2-\tau_{11}\tau_{22})+e=0\right\}, \end{equation} for some integral tuple $(a,b,c,d,e)$ with $c,d\equiv0(p)$ for some prime $p$. Then the {\it discriminant} is $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi=b^2-4ac-4de$, and the field $k'=\fQ(\sqrt{\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi})$ is the field mentioned above; the element $w\in {\cal O} _{k'}$ can be taken here, for example, as $w={b+\sqrt{\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi} \over 2}$, yielding a Humbert surface with $a=1,\ d=e=0$. The standard symmetric subgroup $\ifmmode\ \cong\ \else$\isom$\fi} \def\storth{\underline{\perp} SL(2,\fQ) \times SL(2,\fQ)$ gets conjugated onto groups $\cong SL(2,k')$ by the elements $\hbox{{\script S}} \in Sp(4,k')\subset} \def\nni{\supset} \def\und{\underline Sp(4,{\Bbb R}} \def\fH{{\Bbb H})$. The rational boundary components of ${\cal D} _{N'}$ which are $SL_2(k')$-rational, are zero-dimensional, and are also rational boundary components of ${\cal D} $. Note that the domain ${\cal D} _{N'}$ defined by the subgroup $SL(2,k')$ also contains one-dimensional cusps of the domain ${\cal D} $, the normalizers of which are defined over $k'$, but not over $\fQ$ and these boundary components are consequently not rational (for either $G'$ or $N'$). It is clear that ${\cal D} _{\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi}$, the union of the subdomains of given discriminant $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi$, is the union of conjugates of the standard one by elements of $G(\fQ)$ if and only if $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi$ is a square, giving 1). If $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi$ is not a square, then $k'=\fQ(\sqrt{\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi})$, $N'$ is conjugate to $N_{\hbf{1}}$ by an element in $G(k')$, and these are the cases occuring in 2).\hfill $\Box$ \end{example} Next we note that the set of subgroups defined in Definition \ref{D4.1} is independent of the maximal $\fQ$-split torus used to define $N_{\hbf{1}}$; if $S'$ is another it is conjugate in $G(\fQ)$ to $S$, and $N'$ will be rational with respect to $S$ exactly when it is so with respect to $S'$. For a fixed $N$ the set of rational symmetric subgroups conjugate to $N$ is naturally identified with ${\cal H} =G(\fQ)/N(\fQ)$. Since $G(\fQ)$ acts on ${\cal H} $ so does any arithmetic subgroup $\gG\subset} \def\nni{\supset} \def\und{\underline G(\fQ)$ and one can consider the double coset space $\gG\backslash {\cal H} $. By definition, $N_{\hbox{\scsi \bf b}}^g$ and $N_{\hbox{\scsi \bf b}}^{g'}$ will be in the same $\gG$-orbit if $g(g')^{-1}\in \gG$, so the orbits are determined by the denominators occuring in $g$ and in $(g')^{-1}$, respectively. In the example above, for each prime $p$, the group $N_{\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi}$, $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi=p^2$ lies in a seperate $\gG$-orbit. In particular there are in general infinitely many orbits. It turns out that the following definition gives a convenient notion. For $b<t$ (by which we mean $\dim(F_{\hbox{\scsi \bf b}})>0$) let $N_{\hbox{\scsi \bf b}}=N_{\hbf{b,1}}\times N_{\hbf{b,2}}$ be the decomposition above, and for $N=N_{\hbox{\scsi \bf b}}^g$, let $N=N_1\times N_2$ denote the corresponding decomposition. If $b=t$ we set $N_{\hbox{\scsi \bf b}}=N_{\hbf{b,1}},\ N=N_1$. \begin{definition}\label{D5.1} Let $G,\ S,\ P_{\hbox{\scsi \bf b}},\ N_{\hbox{\scsi \bf b}}$ be fixed as above, $\gG\subset} \def\nni{\supset} \def\und{\underline G(\fQ)$ arithmetic. A rational symmetric subgroup $N\subset} \def\nni{\supset} \def\und{\underline G$, conjugate to $N_{\hbox{\scsi \bf b}}$ by $g\in G(\fQ)$, $N=N_{\hbox{\scsi \bf b}}^g:=gN_{\hbox{\scsi \bf b}}g^{-1}$ will be called $\gG$-{\it integral} (respectively {\it strongly $\gG$-integral}), if \[ N_1\cap \gG=g(N_{\hbf{b,1}}\cap \gG)g^{-1}\quad (\hbox{respectively } N \cap \gG = g(N_{\hbox{\scsi \bf b}}\cap \gG)g^{-1})\] for the element $g$ above. \end{definition} For $b=t$, both notions coincide, otherwise strongly $\gG$-integral implies $\gG$-integral. For our purposes, the weaker notion will be most important. Note that since $N=gN_{\hbox{\scsi \bf b}}g^{-1}$ the conditions are equivalent to \begin{equation}\label{E13.1} N_{\hbf{b,1}}\cap g^{-1}\gG g =N_{\hbf{b,1}}\cap \gG\quad \hbox{(respectively } N_{\hbox{\scsi \bf b}}\cap g^{-1}\gG g = N_{\hbox{\scsi \bf b}} \cap \gG). \end{equation} This in turn means that $g^{-1}\gG g$ is integral on $N_{\hbf{b,1}}$ (respectively integral on $N_{\hbox{\scsi \bf b}}$), in other words, that for some rational representation $\gr:G\longrightarrow} \def\sura{\twoheadrightarrow GL(V)$ we have $\gr_{|N_{\hbfs{b,1}}}(N_{\hbf{b,1}}\cap g^{-1}\gG g)\subset} \def\nni{\supset} \def\und{\underline GL(V_{\fZ})$ (respectively $\gr_{|N_{\hbfs{b}}}(N_{\hbox{\scsi \bf b}}\cap g^{-1}\gG g)\subset} \def\nni{\supset} \def\und{\underline GL(V_{\fZ})$). Note that this definition depends on the choosen maximal torus, as well as on $\gG$. If $S'=xSx^{-1}$ is another maximal $\fQ$-split torus, then $N_{\hbf{b}}'=xN_{\hbox{\scsi \bf b}}x^{-1}$ is the standard symmetric subgroup with respect to $S'$. If $N\subset} \def\nni{\supset} \def\und{\underline G$ is $\gG$-integral with respect to $N_{\hbox{\scsi \bf b}}$ (i.e., there is $g\in G(\fQ)$ such that $g N_{\hbox{\scsi \bf b}}g^{-1}=N$ and $N_{\hbf{b,1}}\cap g^{-1}\gG g = N_{\hbf{b,1}}\cap \gG$), then $N_{\hbf{b,1}}'\cap (gx^{-1})^{-1}\gG(gx^{-1})=N_{\hbf{b,1}}'\cap x\gG x^{-1}$; in other words when $N$ is $\gG$-integral with respect to $N_{\hbox{\scsi \bf b}}$, then $N$ is $x\gG x^{-1}$-integral with respect to $N_{\hbox{\scsi \bf b}}'$ (with similar statements for strongly $\gG$-integral). At least in the classical cases, when $G$ is a matrix group, there is a very canonical choice for $N_{\hbox{\scsi \bf b}}$, namely as a subgroup consisting of block matrices, so this dependence is not unreasonable. Let us now suppose $G$ is classical, $\gr':G'\longrightarrow} \def\sura{\twoheadrightarrow GL_D(V)$ the fundamental representation, $\gr:G\longrightarrow} \def\sura{\twoheadrightarrow Res_{k|\fQ}(GL_D(V))$ the corresponding representation of $G$. We have $P_{\hbox{\scsi \bf b}}=Res_{k|\fQ}P_b',\ N_{\hbox{\scsi \bf b}}=Res_{k|\fQ}N_b'$ and $P_b'$ (resp.~$N_b'$) are given in terms of $(V,h)$ by (\ref{E8.b}) (resp.~by (\ref{E12.1})). Clearly if $N=N_{\hbox{\scsi \bf b}}^g$ and $N_{\hbox{\scsi \bf b}}={\cal N} _G(W)$, then $N={\cal N} _G(g(W))$. \begin{lemma}\label{L13.1} $N={\cal N} _G(g(W))$ is $\gG$-integral $\iff$ $W_{\fZ}=W\cap V_{\fZ}=W\cap \gr(g^{-1})(V_{\fZ})$. \end{lemma} {\bf Proof:} By definition $N_1\cap \gG=gN_{\hbf{b,1}}g^{-1}\cap \gG= g(N_{\hbf{b,1}}\cap \gG)g^{-1}$, and this holds if and only if $N_{\hbf{b,1}}\cap g^{-1}\gG g = N_{\hbf{b,1}}\cap \gG$, i.e., $g^{-1}\gG g$ meets $N_{\hbf{b,1}}$ in the arithmetic group $\gG$. But this holds precisely when $g^{-1}\gG g$ maps $W_{\fZ}=W\cap V_{\fZ}$ into itself, and this is equivalent to $W_{\fZ}=W\cap \gr(g^{-1})(V_{\fZ})$, as $\gr(g^{-1}\gG g)$ maps $\gr(g^{-1})(V_{\fZ})$ into itself, and this is the statement of the lemma. \hfill $\Box$ \vskip0.25cm Recall also \begin{definition}\label{D13.2} A lattice $V_{\fZ}\subset} \def\nni{\supset} \def\und{\underline V_{\fQ}$ being given, a submodule $W_{\fZ}$ is {\it pure}, if $n\cdot x\in W_{\fZ},\ n\in \fZ\ \Ra x\in W_{\fZ}$. \end{definition} \begin{lemma}\label{L13.2} There is a 1-1 correspondence between rational subspaces $W_{\fQ}\subset} \def\nni{\supset} \def\und{\underline V_{\fQ}$ and pure $\fZ$-submodules $W_{\fZ}\subset} \def\nni{\supset} \def\und{\underline V_{\fZ}$, given by \[ W_{\fQ}\mapsto W_{\fQ}\cap V_{\fZ},\quad W_{\fZ}\mapsto W_{\fZ}\otimes_{\fZ}\fQ.\] \end{lemma} {\bf Proof:} Clear. \hfill $\Box$ \vskip0.25cm Note that the statement of Lemma \ref{L13.1} is also equivalent to $g(W_{\fZ})=g(W)\cap V_{\fZ}$, and the latter is by Lemma \ref{L13.2}, a pure submodule. This then yields: \begin{corollary}\label{C14.1} There is a 1-1 correspondence between the set of $\gG$-integral symmetric subgroups and pure submodules of the form $g(W_{\fZ})$, with $g\in G(\fQ)$ and $W_{\fZ}\subset} \def\nni{\supset} \def\und{\underline V_{\fZ}$ the submodule above (cf.~Lemma \ref{L13.1}). \end{corollary} {\bf Proof:} As we just remarked, $N$ is $\gG$-integral $\iff$ $g(W_{\fZ})$ is pure, and for any $g(W_{\fZ}),\ g\in G(\fQ)$, which is pure, $N_{\hbox{\scsi \bf b}}^g$ is clearly $\gG$-integral. \hfill $\Box$ \vskip0.25cm Next we consider, for $d=\dim(W)$, the representation \[ R=\bigwedge^d\gr:G\longrightarrow} \def\sura{\twoheadrightarrow GL({\cal V} ),\quad {\cal V} =\bigwedge^dV.\] Since $\gr(N_{\hbox{\scsi \bf b}})={\cal N} _G(W)$, it follows that $R(N_{\hbox{\scsi \bf b}})=\bigwedge^d\gr(N_{\hbox{\scsi \bf b}}) = {\cal N} _G(\bigwedge^dW)={\cal N} _G(W_{\hbox{\scsi \bf b}})$, where $W_{\hbox{\scsi \bf b}}$ is one-dimensional in ${\cal V} $, defined over $\fQ$, and we have slightly abused notation by denoting by ${\cal N} _G$ the inverse image under $R$ of the corresponding normalizer in $GL({\cal V} )$. Our lattice $V_{\fZ}$ produces a lattice in ${\cal V} ,\ {\cal V} _{\fZ}=\bigwedge^dV_{\fZ}$, and $\gG$ is commensurable with $R^{-1}(GL({\cal V} _{\fZ}))$. We now return to the general situation; $G$ is $\fQ$-simple of hermitian type, $P_{\hbox{\scsi \bf b}}$ is a standard parabolic and $N_{\hbox{\scsi \bf b}}$ is an incident symmetric subgroup, which we take to be the standard one. Since $N_{\hbox{\scsi \bf b}}$ is reductive, by \cite{BHC}, Theorem 3.8, there exists a rational representation $\pi:G\longrightarrow} \def\sura{\twoheadrightarrow GL({\cal V} )$, defined over $\fQ$, and an element $v\in {\cal V} _{\fQ}$, such that $v\cdot \pi(G)$ is a closed orbit and $N_{\hbox{\scsi \bf b}}=\pi^{-1}({\cal N} _{GL({\cal V} )}(v)$. For example, in the classical cases, the representation $R$ above is such a $\pi$. We now assume that ${\cal V} $ is given a $\fZ$-structure ${\cal V} _{\fZ}$ such that $\gG$ is given by \begin{equation} \label{E5.1} \gG=\pi^{-1}(GL({\cal V} _{\fZ})). \end{equation} Let $W_{\hbox{\scsi \bf b}}=\fQ\langle v\rangle$ be the one-dimensional vector subspace spanned by $v$; then we may choose a primitive integral vector $w\in W_{\hbox{\scsi \bf b}}$ such that $N_{\hbox{\scsi \bf b}}=\pi^{-1}({\cal N} _{GL({\cal V} )}(w))$. We consider the orbit $w\cdot \pi(G)$; as is well-known there is a natural isomorphism $w\cdot \pi(G)\cong G/N_{\hbox{\scsi \bf b}}$ given by $w\cdot \pi(g) \mapsto gN_{\hbox{\scsi \bf b}}$. We may consider the lattice ${\cal V} _{\fZ}$, defining {\it integral points} ${\cal V} _{\fZ}\cap w\cdot \pi(G) \stackrel{\sim}{\longrightarrow} \def\sura{\twoheadrightarrow} {\cal V} _{\fZ}\cap G/N_{\hbox{\scsi \bf b}}$. \begin{lemma}\label{L14.1} Assume $\gG$ fulfills (\ref{E5.1}). A subgroup $N$ given by $N=N_{\hbox{\scsi \bf b}}^g$ is $\gG$-integral $\iff$ under the isomorphism $w\cdot \pi(G) \stackrel{\sim}{\longrightarrow} \def\sura{\twoheadrightarrow} G/N_{\hbox{\scsi \bf b}}$, $N$ is given by an integral point $gN_{\hbox{\scsi \bf b}}$, i.e., $w\cdot \pi(g) \in {\cal V} _{\fZ}$. \end{lemma} {\bf Proof:} We have the following equivalences: \begin{eqnarray*} & & N_1\cap \gG = g(N_{\hbf{b,1}}\cap \gG)g^{-1} \\ & \stackrel{(\ref{E5.1})}{\iff} & N_1 \cap \pi^{-1}(GL({\cal V} _{\fZ})) = g(N_{\hbf{b,1}}\cap \pi^{-1}(GL({\cal V} _{\fZ})))g^{-1} \\ & \stackrel{\hbox{\footnotesize apply $\pi$}}{\iff} & \pi(N_1)\cap GL({\cal V} _{\fZ}) = \pi(g)(\pi(N_{\hbf{b,1}})\cap GL({\cal V} _{\fZ}))\pi(g^{-1}) \\ & \iff & \pi(N_{\hbf{b,1}})\cap \pi(g^{-1})GL({\cal V} _{\fZ})\pi(g) = \pi(N_{\hbf{b,1}})\cap GL({\cal V} _{\fZ}) \\ & \stackrel{\parbox{2cm}{\footnotesize$\pi(N_{\hbfs{b,1}})=$\\ ${\cal N} _{\pi(G)}(W_{\hbfs{b}})/ {\cal Z} _{\pi(G)}(W_{\hbfs{b}})$ }} {\iff} & \quad\quad\quad\quad{\cal N} _{\pi(G)}(W_{\hbox{\scsi \bf b}})/{\cal Z} _{\pi(G)}(W_{\hbox{\scsi \bf b}})\cap \pi(g^{-1})GL({\cal V} _{\fZ})\pi(g) \\ & & \quad \quad \quad \quad \quad \quad \quad = {\cal N} _{\pi(G)}(W_{\hbox{\scsi \bf b}})/{\cal Z} _{\pi(G)}(W_{\hbox{\scsi \bf b}}) \cap GL({\cal V} _{\fZ}) \\ & \iff & \pi(g)(W_{\hbox{\scsi \bf b}}\cap {\cal V} _{\fZ})\subset} \def\nni{\supset} \def\und{\underline {\cal V} _{\fZ} \\ & \iff & w\cdot \pi(g)\in {\cal V} _{\fZ} \end{eqnarray*} where the last equivalence follows from the fact that $w$ is primitive. The Lemma follows. \hfill $\Box$ \vskip0.25cm \begin{corollary}\label{C5.1} The set of $\gG$-integral symmetric subgroups is the set of subgroups corresponding to the integral points, \[ \left\{\parbox{5.3cm}{$\gG$-integral symmetric subgroups \\ conjugate to $N_{\hbox{\scsi \bf b}}$ }\right\} \cong G/N_{\hbox{\scsi \bf b}}\cap {\cal V} _{\fZ}.\] \end{corollary} {\bf Proof:} This follows immediately from the proceeding Lemma. \hfill $\Box$ \vskip0.25cm Utilizing Corollary \ref{C5.1}, we can prove finiteness of the set of $\gG'$-equivalence classes of $\gG$-integral symmetric subgroups, for any arithmetic subgroup $\gG'\subset} \def\nni{\supset} \def\und{\underline G(\fQ)$. Recall the basic finiteness result of \cite{BHC}. \begin{theorem}[\cite{BHC}, 6.9]\label{t15a.1} Let $G$ be a reductive algebraic group defined over $\fQ$, $\pi:G\longrightarrow} \def\sura{\twoheadrightarrow GL(V)$ a rational representation defined over $\fQ$, ${\cal L} \subset} \def\nni{\supset} \def\und{\underline V$ a lattice in $V_{\fQ}$ invariant under $G_{\fZ}$, and $X$ a closed orbit of $G$. Then $X\cap {\cal L} $ consists of a finite number of orbits of $G_{\fZ}$. \end{theorem} Here $G\subset} \def\nni{\supset} \def\und{\underline GL(n,\fC)$ and $G_{\fZ}=G\cap M_n(\fZ)$. \begin{corollary}\label{C6.1} Given $G,\ S,\ P_{\hbox{\scsi \bf b}},\ N_{\hbox{\scsi \bf b}}$ and $\gG$ as above ($\gG$ as in (\ref{E5.1})), there are finitely many $\gG'$-equivalence classes of $\gG$-integral symmetric subgroups, for any arithmetic subgroup $\gG'\subset} \def\nni{\supset} \def\und{\underline G(\fQ)$. \end{corollary} {\bf Proof:} Since $\gG$ satisfies (\ref{E5.1}), Corollary \ref{C5.1} holds and Theorem \ref{t15a.1} may be applied to $\gG$, hence finiteness holds for any $\gG'$. \hfill $\Box$ \vskip0.25cm Note that under the action of $\gG$ on $\gr(G)\cdot v\cap V_{\fZ}$, all orbits are bijective to $\gG/(\gG\cap N_{\hbox{\scsi \bf b}})$. Let the orbit decomposition with respect to $\gG'$ be \[ \gG'\backslash \gr(G)\cdot v \cap V_{\fZ}={\cal O} _1\cup \cdots \cup {\cal O} _q.\] Choose, in each orbit ${\cal O} _i$, a representative $x_i$, and let $N_{x_i}$ be the corresponding integral symmetric subgroup. The set $\{N_{x_i}\}$ serves as a finite set of $\gG$-integral symmetric subgroups representing all $\gG'$-equivalence classes of such. The following is then well defined. \begin{definition}\label{D6.1} Given $G,\ N_{\hbox{\scsi \bf b}},\ \gG$ as above, $\gG'\subset} \def\nni{\supset} \def\und{\underline G(\fQ)$ arithmetic, the {\it class number} of $\gG'$-equivalence classes of $\gG$-integral symmetric subgroups is the cardinality \[ \mu(G,N_{\hbox{\scsi \bf b}},\gG,\gG'):= |\gG'\backslash(G/N_{\hbox{\scsi \bf b}}\cap {\cal V} _{\fZ})|.\] If in a discussion $G$ and a maximal $\fQ$-split torus $S\subset} \def\nni{\supset} \def\und{\underline G$ are fixed, then $N_{\hbox{\scsi \bf b}}$ depends only on the integer $b\in \{1,\ldots, s\}$ ($s=\rank_{\fQ}G$)\footnote{here again with the two exceptions for $b=s=t$ and the two exceptional domains where there are three, resp.~two isomorphism classes of $N_{\hbox{\scsi \bf b}}$}, and we will denote this class number by $\mu_b(\gG,\gG')$. \end{definition} \section{Arithmetic quotients} In this section we keep the above notations. $G$ is $\fQ$-simple of hermitian type, ${\cal D} =G({\Bbb R}} \def\fH{{\Bbb H})/K=G({\Bbb R}} \def\fH{{\Bbb H})^0/K^0$ the hermitian symmetric space, $\gG\subset} \def\nni{\supset} \def\und{\underline G(\fQ)$ an arithmetic subgroup. The group $\gG$ acts on ${\cal D} $ by means of holomorphic isometries, preserving the natural Bergmann metric. \begin{definition}\label{d83.1} The quotient $X_{\gG}:=\gG\backslash {\cal D} $, where $\gG\subset} \def\nni{\supset} \def\und{\underline G({\Bbb R}} \def\fH{{\Bbb H})$ is arithmetic, is called an {\em arithmetic quotient}. \end{definition} If $\gG$ acts without fix points, then the quotient $X_{\gG}$ is a smooth complex manifold, not compact in general. If $\gG$ has fix points, then $X_{\gG}$ has certain quotient singularities, which can be described as follows. Let $\gG_1\subset} \def\nni{\supset} \def\und{\underline \gG$ be a normal subgroup of finite order without elements of finite order, so that $\gG_1$ acts freely and hence $X_{\gG_1}$ is smooth. We have a Galois cover, \begin{equation}\label{e83.1} X_{\gG_1}\longrightarrow} \def\sura{\twoheadrightarrow X_{\gG}, \end{equation} with $X_{\gG_1}$ smooth, and Galois group acting, yielding the singularities of $X_{\gG}$. It is clear that the Galois group, $\gG/\gG_1$, creates the singularities, so they are controlled by certain properties of $\gG/\gG_1$, such as the orders of the elements, etc. In particular, $X_{\gG}$ is {\em still} smooth if $\gG/\gG_1$ is generated by reflections, as the quotient is then smooth by Chevally's theorem. It is well-known that $X_{\gG}$ is compact $\iff$ $G$ is anisotropic. Suppose this is the case, and that in addition $\gG$ has no elements of finite order. Then, as Kodaira showed in 1954 as one application of his embedding theorem, $X_{\gG}$ is a smooth projective variety, the canonical bundle $K_{X_{\gG}}$ being ample. In this case one has Hirzebruch proportionality, which states that the ratios of the Chern numbers of $X_{\gG}$ are equal to the ratios of the corresponding Chern numbers of the compact hermitian symmetric spaces $\check{{\cal D} }$, and the overall factor of proportionality is just the volume of $X_{\gG}$, which is the same as the volume in ${\cal D} $ of a fundamental domain of $\gG$, where volume is taken with respect to the Bergmann metric. \subsection{Satake compactification and Baily-Borel embedding} In case $G$ is not anisotropic, $X_{\gG}$ is not compact. It has a topological compactification $X_{\gG}^*$, the so-called {\em Satake compactification}. This is constructed by putting an appropriate topology, the Satake topology, on ${\cal D} ^*:={\cal D} \cup \{\hbox{rational boundary components}\}$ (\cite{BB}, 4.8). With the Satake topology, the action of $\gG$ on ${\cal D} $ extends to one on ${\cal D} ^*$ (\cite{BB}, 4.9), and the quotient $\gG\backslash {\cal D} ^*=\ifmmode {X_{\gG}^*} \else$\xgs$\fi$ is the sought for compactification. It has the following property: \begin{proposition}[\cite{BB}, 4.11]\label{p84.1} $\ifmmode {X_{\gG}^*} \else$\xgs$\fi$ is a compact, Hausdorff space, and the complement $\ifmmode {X_{\gG}^*} \else$\xgs$\fi \backslash \xg$ is a finite disjoint union $\ifmmode {X_{\gG}^*} \else$\xgs$\fi\backslash\xg = V_1\cup\cdots \cup V_N$, with each $V_i$ an arithmetic quotient of dimension and $\fQ$-rank less than that of $\xg$. The length $k$ of a maximal chain $V_{i_1}\subsetneq V_{i_2}^*\subsetneq \cdots \subsetneq V_{i_k}^*$ is the $\fQ$-rank of $G$. \end{proposition} In our discussion of the $\fQ$-hermitian symmetric subgroups in section \ref{classification} we determined the rational boundary components for each $G$ giving rise to quotients $V_i$. One case of particular interest are the hyperbolic planes, discussed in detail in \cite{hyp}. We know that in this case all rational boundary components are zero-dimensional, i.e., points. Hence the finite union of \ref{p84.1} is a disjoint union of points; the number of such is just the number of cusps, defined as follows. Suppose again we have the fixed $\fQ$-split torus $S$ and the standard subgroups $P_{\hbox{\scsi \bf b}}$ and $N_{\hbox{\scsi \bf b}}$ with respect to $S$. \begin{definition}\label{D7.1} \begin{itemize}\item[(i)] For $b\in \{1,\ldots, s\}$, the number of boundary varieties, conjugate to the $b^{th}$ standard one, is the cardinality \[ \nu_b(\gG)=|\gG\backslash G(\fQ)/P_{\hbox{\scsi \bf b}}(\fQ)|.\] \item[(ii)] The {\it number of $\gG$-cusps} is the cardinality (where $B$ is a Borel subgroup) \[ h(\gG)=|\gG\backslash G(\fQ)/B(\fQ)|.\] \end{itemize} \end{definition} Note that $h(\gG)$ is also the number of {\it maximal flags} of boundary varieties, and it is often given by a class number, hence the notation. Since in the case of hyperbolic planes ($s=1$), $\nu_1(\gG)=h(\gG)$, both of these are given by the results of \cite{hyp} in terms of class numbers of certain fields. More generally, the number of components $N$ occurring in Proposition \ref{p84.1} is a sum $N=r_1+\ldots + r_s$, where $s=\fQ$-rank of $G$, $r_b$ = \# equivalence classes of boundary components conjugate to $F_{\hbf{b}}$. Then \begin{proposition}\label{p84.3} For any $\xg$, the number $N$ of Proposition \ref{p84.1} can be expressed: $N=r_1+\ldots + r_s,$ and $r_b=\nu_b(\gG)$ as in Definition \ref{D7.1}. \end{proposition} The term Baily-Borel embedding of $\ifmmode {X_{\gG}^*} \else$\xgs$\fi$ refers to the following result. \begin{theorem}[\cite{BB}, 10.11, 10.12]\label{l84.1} $\ifmmode {X_{\gG}^*} \else$\xgs$\fi$ can be embedded in projective space as a normal algebraic variety $V$. If $G$ has no normal $\fQ$-subgroups of dimension three, then the field of rational functions $K(V)$ is canonically isomorphic with the field of automorphic functions for $\gG$. \end{theorem} It follows in particular that $\xg$ is a normal, quasi-projective variety, which is even smooth if $\gG$ is torsion free. \subsection{Toroidal embeddings} Recall the decomposition (of algebraic groups over ${\Bbb R}} \def\fH{{\Bbb H}$) $P_{\hbox{\scsi \bf b}}=M_{\hbox{\scsi \bf b}}L_{\hbox{\scsi \bf b}}{\cal R} _{\hbox{\scsi \bf b}}\sdprod {\cal U} _{\hbox{\scsi \bf b}}$ for the real parabolic, with the exact sequence \[1\longrightarrow} \def\sura{\twoheadrightarrow M_{\hbox{\scsi \bf b}}L_{\hbox{\scsi \bf b}}{\cal R} _{\hbox{\scsi \bf b}} \longrightarrow} \def\sura{\twoheadrightarrow P_{\hbox{\scsi \bf b}} \longrightarrow} \def\sura{\twoheadrightarrow {\cal U} _{\hbox{\scsi \bf b}} \longrightarrow} \def\sura{\twoheadrightarrow 1;\] this gives rise to a similar sequence for $\gG_{\hbox{\scsi \bf b}}=\gG\cap P_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})$, \[ 1\longrightarrow} \def\sura{\twoheadrightarrow \gG_{\hbox{\scsi \bf b}}^{\ell} \longrightarrow} \def\sura{\twoheadrightarrow \gG_{\hbox{\scsi \bf b}} \longrightarrow} \def\sura{\twoheadrightarrow \gG_{\hbox{\scsi \bf b}}^r \longrightarrow} \def\sura{\twoheadrightarrow 1,\] with $\gG_{\hbox{\scsi \bf b}}^{\ell}$ being the intersection with the Levi factor and $\gG_{\hbox{\scsi \bf b}}^r$ the intersection with the radical of $P_{\hbox{\scsi \bf b}}$. Recall further that ${\cal U} _{\hbox{\scsi \bf b}}={\cal Z} _{\hbox{\scsi \bf b}}V_{\hbox{\scsi \bf b}}$, where ${\cal Z} _{\hbox{\scsi \bf b}}$ is the center, and $M_{\hbox{\scsi \bf b}}L_{\hbox{\scsi \bf b}}$ acts trivially on ${\cal Z} _{\hbox{\scsi \bf b}}$ and by means of a symplectic representation on $V_{\hbox{\scsi \bf b}}$, while ${\cal R} _{\hbox{\scsi \bf b}}$ acts transitively on ${\cal Z} _{\hbox{\scsi \bf b}}$ defining a homogenous self dual cone $C_{\hbox{\scsi \bf b}}\subset} \def\nni{\supset} \def\und{\underline {\cal Z} _{\hbox{\scsi \bf b}}$, and ${\cal R} _{\hbox{\scsi \bf b}}$ acts on $V_{\hbox{\scsi \bf b}}$ by means of complex linear transformations, see Theorem 1.1. This then gives us the following results about the factors of $\gG_{\hbox{\scsi \bf b}}$ (and similar results hold for any $\gG_F=N(F)\cap \gG$): \begin{itemize}\item[i)] $M_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})$ is compact, hence $\gG\cap M_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})$ is finite. In particular, if $\gG$ has no torsion, $\gG\cap M_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})=e$. \item[ii)] $\gG\cap L_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})$ is an arithmetic subgroup of $L_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})$, and $\gG\cap L_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})^0=:\gG_L^0$ acts on the boundary component $F_{\hbox{\scsi \bf b}}$, with the boundary variety $W_{\hbox{\scsi \bf b}}=\gG_L^0\backslash F_{\hbox{\scsi \bf b}}\subset} \def\nni{\supset} \def\und{\underline X_{\gG}^*$. \item[iii)] Let $V_{\fZ}=\gG\cap V_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})$. Then the group $\gG_L^0\sdprod V_{\fZ}$ acts on $F_{\hbox{\scsi \bf b}}\times V_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})$ (recall that $V_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})$ has the structure of complex vector space), and if $\gG$ is torsion free, the quotient is an analytic family of abelian varieties over the arithmetic quotient $W_{\hbox{\scsi \bf b}}=\gG_{L}^0 \backslash F_{\hbox{\scsi \bf b}}$. \item[iv)] {\bf ([SC], p.~248)} There is an exact sequence $$1 \longrightarrow} \def\sura{\twoheadrightarrow \gG^{\prime} \def\sdprod{\rtimes} \longrightarrow} \def\sura{\twoheadrightarrow \gG_{\hbox{\scsi \bf b}} \longrightarrow} \def\sura{\twoheadrightarrow \gG^{\prime} \def\sdprod{\rtimes\p} \longrightarrow} \def\sura{\twoheadrightarrow 1, $$ \begin{itemize}\item{} $\gG^{\prime} \def\sdprod{\rtimes}$= subgroup of elements in $\gG_{\hbox{\scsi \bf b}}$ acting trivially by conjugation on $Lie({\cal Z} _{\hbox{\scsi \bf b}})$, \item{} $\gG^{\prime} \def\sdprod{\rtimes\p}$= group of automorphisms of $Lie({\cal Z} _{\hbox{\scsi \bf b}})$ induced by $\gG_{\hbox{\scsi \bf b}}$; these map $C_{\hbox{\scsi \bf b}}$ into itself. \end{itemize} \end{itemize} The fourth point is important for the compactification theory, as one lets first $\gG''$ act, then $\gG'$. A sketch of the construction is as follows: fix a boundary component $F$, rational with respect to \gG\ (i.e., $\gG\cap N(F)$ is a lattice). Let ${\bf E}_F \subset} \def\nni{\supset} \def\und{\underline {\cal Z} (F)_{\fC}\times V(F) \times F$ be the realisation of ${\cal D} $ as a Siegel domain as in \cite{SC}, and let $1\longrightarrow} \def\sura{\twoheadrightarrow \gG'\longrightarrow} \def\sura{\twoheadrightarrow \gG_F\longrightarrow} \def\sura{\twoheadrightarrow \gG'' \longrightarrow} \def\sura{\twoheadrightarrow 1$ be the sequence above for $\gG_F=N(F)\cap \gG$. Furthermore, the objects denoted above by a subscript $?_{\hbox{\scsi \bf b}}$ will be denoted here by $?(F)$, for example ${\cal Z} (F)$ instead of ${\cal Z} _{\hbox{\scsi \bf b}}$, $C(F)$ instead of $C_{\hbox{\scsi \bf b}}$, etc. \begin{proposition}[\cite{SC}, p.249]\label{toroidal} A partial compactification along $F$ can be constructed as follows: \begin{itemize} \item[1)] Let ${\cal Z} (F)_{\fZ}$ act on ${\cal Z} (F)_{\fC}$ defining the algebraic torus $T_F$; do this in the fibration $${\bf E}_{\{1\}}={\bf E}_F/{\cal Z} (F)_{\fZ} \subset} \def\nni{\supset} \def\und{\underline {\cal Z} (F)_{\fC}/{\cal Z} (F)_{\fZ}\times V(F)\times F\longrightarrow} \def\sura{\twoheadrightarrow F.$$ More precisely, the map ${\cal Z} (F)_{\fC}\longrightarrow} \def\sura{\twoheadrightarrow {\cal Z} (F)_{\fC}/{\cal Z} (F)_{\fZ}$ is given by $\exp(2\pi i\gl_1),\ldots,\exp(2\pi i\gl_k)$, where one chooses a $\fZ$-base $\xi_1,\ldots,\xi_k$ of ${\cal Z} (F)_{\fZ}$, and $\gl_i:{\cal Z} (F)_{\fC}\longrightarrow} \def\sura{\twoheadrightarrow \fC$ is the dual base. \item[2)] Now compactify the algebraic torus $T_F$ by $T_F\subset} \def\nni{\supset} \def\und{\underline T_{F{\{\gs_{\ga}\}}},\ \{\gs_{\ga}\}$ a $\gG^{\prime} \def\sdprod{\rtimes\p}$-admissible polyhedral decomposition of $C(F)\subset} \def\nni{\supset} \def\und{\underline {\cal Z} (F)$ ($\gG''$ as in iv) above). $T_{F{\{\gs_{\ga}\}}}$ is locally of finite type, but will have infinitely many components corresponding to integral vectors $v\in {\cal Z} (F)_{\fZ}\cap C(F)$. The cones $\gs_{\ga}$ themselves correspond to orbits of highest codimension, i.e., to points. If $\gs_{\ga}\cap {\cal Z} (F)_{\fZ}$ is spanned by $v_1,\ldots, v_k$, then $\gs_{\ga}$ corresponds to $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_1\cap \cdots \cap \Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_k$, where $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_j$ is the divisor corresponding to $v_j$. \item[3)] Glue these into ${\bf E}_{\{1\}}$ by forming the fibre product $$({\bf E}_{\{1\}})\times^{T_F}(T_{F{\{\gs_{\ga}\}}})$$ and setting $({\bf E}_{\{1\}})_{\{\gs_{\ga}\}}$= interior of the closure of ${\bf E}_{\{1\}}$ in $({\bf E}_{\{1\}})\times^{T_F}(T_{F{\{\gs_{\ga}\}}})$. Hence $({\bf E}_{\{1\}})_{\{\gs_{\ga}\}}$ has a fibre structure over $F \times V(F)$ with fibres $T_{F{\{\gs_{\ga}\}}}$. If $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_i$ is the divisor corresponding to $\xi_i$ as in 1), then $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_i=\{z_i=0\},$ where $(z_1,\ldots,z_k)$ are local coordinates on $T_F$. \item[4)] $\gG_F$ still acts on $({\bf E}_{\{1\}})_{\{\gs_{\ga}\}}$, as follows. $\gG''$ now acts freely on $({\bf E}_{\{1\}})_{\{\gs_{\ga}\}}$, giving a fibre space over $F\times V(F)$ with fibre a finite $T_F$ compactification, i.e., modulo $\gG''$ there are only finitely many integral vectors, hence components, in the fibre. Now $\gG'$ acts on $F\times V(F)$; as ${\cal Z} _{\hbox{\scsi \bf b}}$ acts trivially this amounts to an action of $\gG_L^0\sdprod V_{\fZ}$ as in iii) above, and this action extends to $\gG''\backslash ({\bf E}_{\{1\}})_{\{\gs_{\ga}\}}$, hence an open neighborhood of $F$ will give an open neighborhood of the boundary variety $W(F)=\gG_L^0\backslash F$ in $\overline{\xg}$; this is the sought for partial compactification. \end{itemize} \end{proposition} Next one glues these partial compactifications together by means of $\{ \gs_{\ga,F}\}$, a \gG-admissible collection of polyhedral cones, one such collection for each boundary component. The main result is: \begin{theorem}[\cite{SC}, Main Theorem 1, p.~252] With \gG, {\cal D} \ as above, for every \gG-admissible collection of polyhedral cones $\{ \gs_{\ga,F}\}$, there is a unique compactification $\overline{\xg}\ (=(\overline{\xg})_{\{\gs_{\ga}\}})$ which is locally given at each $F$ (more precisely at $W(F)$) by the partial compactification above, corresponding to the given collection of cones. $\overline{\xg}$ is a compact Hausdorf, analytic variety, which is an algebraic space. Furthermore, for properly chosen \gG-admissible collections of polyhedral cones, the compactification is 1) a projective resolution of the Satake compactification: $\overline{\xg}\longrightarrow} \def\sura{\twoheadrightarrow \ifmmode {X_{\gG}^*} \else$\xgs$\fi$, hence a projective variety, and 2) smooth with $\Delta} \def\gd{\delta} \def\gG{\ifmmode {\Gamma} \else$\gG$\fi_{\gG}:=\ifmmode {\overline{X}_{\gG}} \else$\xgc$\fi-\xg$ a normal crossings divisor. \end{theorem} \section{Modular subvarieties} In this paragraph, the data $G,\ S,\ {\cal D} $ will be fixed as above, so that for each $b=1,\ldots, s=\rank_{\fQ}G$ we have the standard boundary components $F_{\hbox{\scsi \bf b}}$, the standard parabolic $P_{\hbox{\scsi \bf b}}$ and the standard incident symmetric subgroup $N_{\hbox{\scsi \bf b}}$. \subsection{Baily-Borel compactification} Let $N\subset} \def\nni{\supset} \def\und{\underline G$ be a reductive subgroup of hermitian type (this implies in particular that $N$ is defined over ${\Bbb R}} \def\fH{{\Bbb H}$, and we assume the inclusion $N\subset} \def\nni{\supset} \def\und{\underline G$ is also), ${\cal D} _N\subset} \def\nni{\supset} \def\und{\underline {\cal D} $ the subdomain (holomorphic symmetric embedding) determined by $N$. \begin{definition}\label{D9.1} The subdomain ${\cal D} _N\subset} \def\nni{\supset} \def\und{\underline {\cal D} $ will be said to be {\it defined over $\fQ$}, if $N$ is a $\fQ$-subgroup of $G$. \end{definition} Suppose a subdomain ${\cal D} _N\subset} \def\nni{\supset} \def\und{\underline {\cal D} $ is defined over $\fQ$, and consider an arithmetic subgroup $\gG\subset} \def\nni{\supset} \def\und{\underline G(\fQ)$ and the arithmetic quotient $X_{\gG}=\gG\backslash {\cal D} $. Note that for a reductive subgroup of hermitian type $N\subset} \def\nni{\supset} \def\und{\underline G$, the intersection $\gG_N:= \gG\cap N$ will be an arithmetic subgroup if and only if $N$ is defined over $\fQ$, and this is the case if and only if ${\cal D} _N$ is defined over $\fQ$. Hence the arithmetic quotient $X_{\gG_N}:=\gG_N\backslash {\cal D} _N$ is defined, and clearly fits into a commutative square \begin{equation}\label{E9.1} \begin{array}{ccc} {\cal D} _N & \hookrightarrow} \def\hla{\hookleftarrow & {\cal D} \\ \downarrow & & \downarrow \\ X_{\gG_N} & \hookrightarrow} \def\hla{\hookleftarrow & X_{\gG}. \end{array} \end{equation} \begin{definition}\label{D9.2a} A {\it modular subvariety} on $X_{\gG}$ is a sub-arithmetic quotient $X_{\gG_N}$ as in (\ref{E9.1}), where ${\cal D} _N$ is defined over $\fQ$. A modular subvariety $X_{\gG_N}\subset} \def\nni{\supset} \def\und{\underline X_{\gG}$ will be called {\it rational} (resp.~{\it $\gG$-integral}), if $N$ is a rational (resp.~$\gG$-integral) symmetric subgroup as in Definition \ref{D4.1} (resp.~\ref{D5.1}). \end{definition} The embedding of (\ref{E9.1}) turns out to extend to one of the Baily-Borel embeddings, legitimizing the terminology sub{\it variety}. This is given by the following result of Satake. \begin{theorem}\label{t17.1} Let $X_{\gG_N}^*\subset} \def\nni{\supset} \def\und{\underline {\Bbb P}^N,\ X_{\gG}^*\subset} \def\nni{\supset} \def\und{\underline {\Bbb P}^{N'}$ be Baily-Borel embeddings. Then there is a linear injection ${\Bbb P}^N\hookrightarrow} \def\hla{\hookleftarrow {\Bbb P}^{N'}$ making the diagram $$\begin{array}{ccl} X_{\gG_N}^* & \hookrightarrow} \def\hla{\hookleftarrow & {\Bbb P}^N \\ \cap & & \cap \\ X_{\gG}^* & \hookrightarrow} \def\hla{\hookleftarrow & {\Bbb P}^{N'}\end{array}$$ commute and making $X_{\gG_N}^*\subset} \def\nni{\supset} \def\und{\underline X_{\gG}^*$ an algebraic subvariety. \end{theorem} {\bf Proof:} We have an injective holomorphic embedding ${\cal D} _N\hookrightarrow} \def\hla{\hookleftarrow {\cal D} $ which comes from a ${\Bbb Q}$-morphism $\rho:(N)_{{\Bbb C}}\hookrightarrow} \def\hla{\hookleftarrow (G)_{{\Bbb C}}$ such that $\rho(\Gamma_{N}) \subset} \def\nni{\supset} \def\und{\underline \Gamma$. Hence we map apply \cite{S2}, Theorem 3, and the theorem follows from this. \hfill $\Box$ \vskip0.25cm \begin{definition}\label{D9.2} We say that $X_{\gG_N}$ and a boundary variety $W_i$ are {\it incident}, if in ${\cal D} ^*$ there is rational boundary component $F$ with parabolic $P=N(F)$ covering $W_i$, such that $N$ and the corresponding parabolic $P$ are incident. \end{definition} Note the following \begin{lemma}\label{L9.2} $X_{\gG_N}$ and $W_i$ are incident, if and only if $W_i^*\subset} \def\nni{\supset} \def\und{\underline X_{\gG_N}^*$ is a maximal-dimensional boundary component of $X_{\gG_N}^*$ (if $\dim(W_i)>0$), resp.~if and only if $W_i\subset} \def\nni{\supset} \def\und{\underline X_{\gG_N}^*$ (if $\dim(W_i)=0$). \end{lemma} {\bf Proof:} If $\dim(W_i)>0$, then the groups $P$ and $N$ are incident if $F\subset} \def\nni{\supset} \def\und{\underline {\cal D} _N^*$ and $F$ is maximal with this property, and if $F\in {\cal D} _N^*$ is rational and maximal with this property, then $P$ and $N$ are incident. If $\dim(W_i)=0$ and $N$ is an incident symmetric subgroup, then $W_i\subset} \def\nni{\supset} \def\und{\underline {\cal D} _N^*$ is a (point) rational boundary component, and conversely. \hfill $\Box$ \vskip0.25cm We now consider $\gG$-integral symmetric subgroups $N$ and arbitrary arithmetic subgroups $\gG'\subset} \def\nni{\supset} \def\und{\underline G(\fQ)$, let $\gG'_N=N\cap \gG'$ and consider the corresponding integral modular subvarieties they define, $X_{\gG'_N}\subset} \def\nni{\supset} \def\und{\underline X_{\gG'}$. As described above, the inclusion extends to the Baily-Borel embeddings $X_{\gG'_N}^*\subset} \def\nni{\supset} \def\und{\underline X_{\gG'}^*$. We now take $\gG$ to be $G_{\fZ}$ for some rational representation $\gr:G\longrightarrow} \def\sura{\twoheadrightarrow GL(V_{\fZ})$, that is $\gG=\gr^{-1}(GL(V_{\fZ}))$ for some $\fZ$-structure $V_{\fZ}$ on $V$. Recall the notations $\nu_b(\gG'),\ b=1,\ldots, s$ and $\mu_b(\gG,\gG'),\ b=1,\ldots, s$ of Definition \ref{D6.1} and \ref{D7.1}, respectively, for the number of $b^{th}$ boundary varieties and the number of $b^{th}$ integral modular subvarieties, respectively. We let $W_{b,i},\ b=1,\ldots, s,\ i=1,\ldots,\nu_b(\gG')$ be the corresponding boundary varieties on the Satake compactification, $Y_{b,j},\ b=1,\ldots, s,\ j=1,\ldots, \mu_b(\gG,\gG')$ the corresponding $\gG$-integral modular varieties, everything on the arithmetic quotient $X_{\gG'}$. Then the main result of the paper is the following. \begin{theorem} Let $\gG$ be as above, $\gG'\subset} \def\nni{\supset} \def\und{\underline G(\fQ)$ arithmetic, and $X_{\gG'}\subset} \def\nni{\supset} \def\und{\underline X_{\gG'}^*$ the Satake compactification, $X_{\gG'}^*-X_{\gG'}=\sum_{b,i}W_{b,i}$. Then $\Xi:=\sum_{b,j}Y_{b,j}$ is a complete (finite, non-empty) set of $\gG'$-equivalence classes of $\gG$-integral modular subvarieties, such that for each $W_{b,i}$, there is at least one $Y_{b,j}$ incident to $W_{b,i}$. \end{theorem} {\bf Proof:} There is for each $W_{b,i}$ an incident $\gG$-integral modular subvariety because for any representative parabolic there is a $\gG$-integral symmetric subgroup which is incident. The finiteness result Corollary \ref{C6.1} implies that for each $N_{\hbox{\scsi \bf b}}$ (of which there are finitely many) there are finitely many $\gG'$-equivalence classes of $\gG$-integral symmetric subgroups of $G$ conjugate to $N_{\hbox{\scsi \bf b}}$, so a complete set of $\gG'$-equivalence classes is finite. \hfill $\Box$ \vskip0.25cm This gives us a {\it well-defined, non-empty, finite} set of subvarieties of the Baily-Borel embedding $X_{\gG'}^*\subset} \def\nni{\supset} \def\und{\underline {\Bbb P}} \def\fB{{\Bbb B}^N$ for any subgroup $\gG'\subset} \def\nni{\supset} \def\und{\underline \gG$ of finite index. Furthermore these have a prescribed behavior near the cusps. For example, if $f:{\cal D} \longrightarrow} \def\sura{\twoheadrightarrow \fC$ is a modular form whose zero divisor $D_f$ on $X_{\gG'}^*$ contains the union of the integral modular subvarieties, then $f$ is a cusp form for $\gG'$. \subsection{Incidence} Consider a toroidal compactification $\ifmmode {\overline{X}_{\gG}} \else$\xgc$\fi$ which is smooth and projective; consider what incidence means here. Let $W_i$ be a rational boundary variety, and let $P$ be a parabolic $P={\cal N} _{G({\Bbb R}} \def\fH{{\Bbb H})}(F)$ for some rational boundary component $F$ which covers $W_i$. We have the decomposition $P=(ML{\cal R} )\sdprod {\cal Z} V$ of the parabolic. Recall from the construction \ref{toroidal} that the inverse image $\pi^{-1}(W_i)$ in $\ifmmode {\overline{X}_{\gG}} \else$\xgc$\fi$ of $W_i\subset} \def\nni{\supset} \def\und{\underline \ifmmode {X_{\gG}^*} \else$\xgs$\fi$ is a divisor which is a torus embedding bundle over the family of abelian varieties ``$V/V_{\fZ}$ over $W_i$''. On the other hand, if $X_{\gG_N}$ is an integral modular subvariety incident to $W_i$, and if $\dim(W_i)>0$, then the proper transform of $X_{\gG_N}^*$ on $\ifmmode {\overline{X}_{\gG}} \else$\xgc$\fi$ will meet $\pi^{-1}(W_i)$ in a {\it section} of the family of abelian varieties over $W_i$. In a sense, the standard one will meet in the zero-section, the others meet in certain sections associated with level structures (e.g. sections of torsion points). For $\dim(W_i)=0$ the situation is slightly different. We now discuss this in more detail. We consider first the case that $\dim(W_i)>0$. Then by (\ref{e10.1}), $N_{\hbox{\scsi \bf b}}$ (hence any $G(\fQ)$-conjugate) has the form $N_1\times N_2$, where $N_1\subset} \def\nni{\supset} \def\und{\underline P_{\hbox{\scsi \bf b}}$ is a hermitian Levi factor. Considering the arithmetic group $\gG_{N_{\hbox{\scsi \bf b}}}$ acting on ${\cal D} _{N_1}\times {\cal D} _{N_2}$, since the product is defined over $\fQ$, the quotient $\gG_{N_{\hbox{\scsi \bf b}}}\backslash {\cal D} _N$ is at most a finite quotient of a product itself. We assume that in fact $X_{\gG_{N}}$ is a product (we will show below in Lemma \ref{L6.4.2} that for $N$ $\gG$-integral this always holds); then $X_{\gG_N}=X_1\times X_2$, where $X_1$ is the arithmetic quotient $\gG_1\backslash {\cal D} _{N_1}$ and this is isomorphic to the boundary variety $W_i$. It follows that $X_2=\gG_1\backslash {\cal D} _{N_2}$ has rational boundary components which are zero-dimensional, say $w\in X_2^*$, such that with respect to the natural inclusion $i:X_{\gG_N}^*\subset} \def\nni{\supset} \def\und{\underline X_{\gG}^*$ we have \begin{equation}\label{E35} i(X_1\times \{w\})=W_i. \end{equation} Recall further that any two hermitian Levi factors are conjugate by an element $g\in V$, and that, modulo $\gG$, this means a point of the abelian variety ``$V/V_{\fZ}$''. This is of course true for any point $t\in W_i$, so we get \begin{lemma}\label{L11.1} Given $W_i$, $X_{\gG_N}^*$ any integral modular subvariety incident to it. Then the proper transform of $X_{\gG_N}^*$ in $\overline{X}_{\gG}$ determines a {\it section} of the family of abelian varieties of $\pi^{-1}(W_i)$ over $W_i$. \end{lemma} {\bf Proof:} Since $X_{\gG_N}^*$ is integral, by Lemma \ref{L6.4.2} below, $X_{\gG_N}^*$ is in fact a product $X_{\gG_N}^*=X_{\gG_1}^*\times X_{\gG_2}^*$. The boundary component $W_i$ is by (\ref{E35}) given by a zero-dimensional boundary component $w$ of $X_{\gG_2}^*$, which gets modified under $\pi$, $\pi^{-1}(X_{\gG_2}^*)=\overline{X}_{\gG_2}$. We have fibre spaces (at least locally over $W_i$) \[ \pi^{-1}(W_i)\stackrel{\eta}{\longrightarrow} \def\sura{\twoheadrightarrow} A_i \stackrel{\zeta}{\longrightarrow} \def\sura{\twoheadrightarrow} W_i,\] where $A_i=W_i\times V_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})/\gG_{\hbox{\scsi \bf b}}^0\sdprod V_{\hbox{\scsi \bf b}}(\fZ)$ is the natural family of abelian varieties parameterized by $W_i$. Note that the zero of $V_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})$ determines a zero section $\gs_0:W_i\longrightarrow} \def\sura{\twoheadrightarrow A_i,\ t\mapsto \hbox{ the image of } 0\in V_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})$ in $(A_i)_t=V_{\hbox{\scsi \bf b}}({\Bbb R}} \def\fH{{\Bbb H})/\gL_{\hbox{\scsi \bf b}}(t)$, where $\gL_{\hbox{\scsi \bf b}}(t)$ denotes the lattice at the point $t$, and any element $x\in V_{\hbox{\scsi \bf b}}(\fQ)$ determines locally a section $\gs_x=\gs_0+x$. Recall that $N$ (=$N_{\hbox{\scsi \bf b}}^g$) is determined by an element $g\in V_{\hbox{\scsi \bf b}}(\fQ)$, so the proper transform of $X_{\gG_N}^*$ in the abelian variety part of the exceptional locus is $\left[ X_{\gG_1}^*\right] = (\gs_0+g)(W_i)\subset} \def\nni{\supset} \def\und{\underline A_i$, which is a global section. \hfill $\Box$ \vskip0.25cm If $\dim(W_i)=0$, two different situations occur, depending on whether ${\cal D} $ is a tube domain or not. They are (assume for the moment that ${\cal D} $ is irreducible) \begin{itemize} \item ${\cal D} $ is a tube domain, then $V$ is trivial and there is no abelian variety; $\pi^{-1}(W_i)$ is a torus embedding. \item ${\cal D} $ is not a tube domain, $V$ is not trivial, and $\pi^{-1}(W_i)$ has an abelian variety factor and a torus embedding factor. \end{itemize} In the first case there is not much more to say than that each irreducible component $W_{ij}$ of $\pi^{-1}(W_i)$ meets the proper transform of $X_{\gG_N}^*$ in a divisor on $X_{\gG_N}^*$ (which gets itself blown up at the point). In the second case, the dimension of the abelian variety factors and of the corresponding integral symmetric subvarieties are given as follows: \begin{equation}\label{E11.1} \begin{array}{l|c|c|c|} & \bf I_{\hbf{p,q}} & \bf II_{\hbf{n}} & \bf V \\ \hline \dim(V) & q(p-q) & n-1 & 16 \\ \hline \dim({\cal D} _N) & q(p-1) & {n-2 \choose 2} & 8,\ 10,\ 8 \end{array} \end{equation} At any rate, we have the following result: \begin{theorem}\label{T11.1} The proper transform of $X_{\gG_N}^*$ on $\ifmmode {\overline{X}_{\gG}} \else$\xgc$\fi$ is $\overline{X}_{\gG_N}$, a partial compactification for some $\gG_N$-admissible collection of polyhedral cones. \end{theorem} {\bf Proof:} Let $P_N$ be the parabolic in $N$, $P$ the corresponding parabolic in $G$. Consider the decompositions (we omit the subscript ${\hbox{\scsi \bf b}}={\hbf{s}}$) \[ P_N=(M_NL_N{\cal R} _N)\sdprod {\cal Z} _NV_N,\quad P=(ML{\cal R} )\sdprod {\cal Z} V.\] Then $L_N=L$ is trivial (as the boundary component is a point), and there is a natural inclusion ${\cal Z} _N\subset} \def\nni{\supset} \def\und{\underline {\cal Z} $. Letting $C_N,\ C$ denote the corresponding homogenous self dual cones, we have $C_N\subset} \def\nni{\supset} \def\und{\underline C$, and both inclusions are defined over $\fQ$. Finally we have $\gG_N=\gG\cap N$ which implies $\gG_N\cap C_N=C_N\cap (C\cap \gG)$. We know by assumption that we have a $\gG$-admissible cone decomposition of $C$, and since $C_N\subset} \def\nni{\supset} \def\und{\underline C$ is defined over $\fQ$, this gives one also for $\gG_N$, as follows from \cite{oda}, Theorem 1.13. If $\{\gs\}$ is the cone decomposition of $C$, then $\{\gs_N\},\ \gs_N:=\gs\cap C_N$ gives a corresponding cone decomposition of $C_N$, and the theorem just mentioned applies. This argument applies to each boundary component of $X_{\gG_N}^*$, and it is clear that a $\gG$-admissible collection restricts to a $\gG_N$-admissible collection. \hfill $\Box$ \vskip0.25cm \subsection{Intersection} First note the following: \begin{lemma} Given $X_{\gG}$ and two modular subvarieties $X_1,X_2\subset} \def\nni{\supset} \def\und{\underline X_{\gG}$, the intersection, if of dimension $\geq1$, is again a modular subvariety. \end{lemma} {\bf Proof:} We are given two injections defined over $\fQ$, $i_1:N_1\hookrightarrow} \def\hla{\hookleftarrow G,\ i_2:N_2\hookrightarrow} \def\hla{\hookleftarrow G$, and commutative squares \[\begin{array}{ccccc}{\cal D} _{N_1} & \longrightarrow} \def\sura{\twoheadrightarrow & {\cal D} & \longleftarrow} \def\rar{\rightarrow & {\cal D} _{N_2} \\ \downarrow & & \downarrow & & \downarrow \\ X_1 & \longrightarrow} \def\sura{\twoheadrightarrow & X_{\gG} & \longleftarrow} \def\rar{\rightarrow & X_2; \end{array}\] it follows that $X_1\cap X_2$ is covered by ${\cal D} _{N_1}\cap {\cal D} _{N_2}$ with a corresponding injection $i_{12}:N_1\cap N_2 \hookrightarrow} \def\hla{\hookleftarrow G$, again defined over $\fQ$. Since $X_1$ and $X_2$ are modular subvarieties, ${\cal D} _{N_1}$ and ${\cal D} _{N_2}$ are by definition defined over $\fQ$, hence so is ${\cal D} _{N_1}\cap {\cal D} _{N_2}$. It is also a symmetric subspace since ${\cal D} _{N_1}\cap {\cal D} _{N_2}$ is totally geodesic in ${\cal D} $. Consequently $X_1\cap X_2$ is a modular subvariety. \hfill $\Box$ \vskip0.25cm This can be applied in particular to the integral modular subvarieties. Hence for any two integral modular subvarieties $X_i$, the intersection defines a (maybe empty) modular subvariety. As there are finitely many possible intersections, from the finite set of Corollary \ref{C6.1} we get a finite set of modular subvarieties. Note that if $X_1$ and $X_2$ are both integral, then also the intersection is, in the following sense: Let $N_1=N_{\hbox{\scsi \bf b}_{\hbfs{1}}}^{g_1},\ N_2=N_{\hbox{\scsi \bf b}_{\hbfs{2}}}^{g_2}$, then $\gG$-integral means: \[ N_{\hbf{b$_{\hbfs{1}}$,1}}\cap \gG =N_{\hbf{b$_{\hbfs{1}}$,1}}\cap g_1\gG g_1^{-1},\quad N_{\hbf{b$_{\hbfs{2}}$,2}}\cap \gG =N_{\hbf{b$_{\hbfs{2}}$,2}}\cap g_2\gG g_2^{-1},\] and $N_1\cap N_2=(g_1N_{\hbox{\scsi \bf b}_{\hbfs{1}}}g_1^{-1})\cap (g_2N_{\hbox{\scsi \bf b}_{\hbfs{2}}}g_2^{-1})$. Hence \begin{eqnarray*} (N_{\hbf{b$_{\hbfs{1}}$,1}}\cap N_{\hbf{b$_{\hbfs{2}}$,2}}) \cap \gG & = & (N_{\hbf{b$_{\hbfs{1}}$,1}}\cap \gG)\cap (N_{\hbf{b$_{\hbfs{2}}$,2}}\cap \gG) \\ & = & (N_{\hbf{b$_{\hbfs{1}}$,1}}\cap g_1\gG g_1^{-1})\cap (N_{\hbf{b$_{\hbfs{2}}$,2}}\cap g_2\gG g_2^{-1}) \\ & = & (N_{\hbf{b$_{\hbfs{1}}$,1}}\cap N_{\hbf{b$_{\hbfs{2}}$,2}})\cap (g_1\gG g_1^{-1}\cap g_2\gG g_2^{-1}). \end{eqnarray*} Note that adjoining these to the integral modular subvarieties implies that on an arithmetic quotient $X_{\gG'}$ for $\gG'\subset} \def\nni{\supset} \def\und{\underline \gG$ of finite index, there is a well-defined, finite, non-empty set of subvarieties, all of which are either integral modular subvarieties incident to rational cusps or intersections of such. Finally consider the boundary varieties $W_1$ and $W_2$ to which $X_1$ and $X_2$ are incident. Since $X_1\cap X_2$ is a modular subvariety, it is itself an arithmetic quotient (in general a product), and has a boundary variety $W_{12}^*=W_1^*\cap W_2^*$. In this sense, we make the \begin{definition} Let $X_1$ and $X_2$ be integral modular subvarieties, incident with $W_1$ and $W_2$, respectively. Then we say $X_{12}^*:=X_1^*\cap X_2^*$ is {\it incident to} $W_{12}^*:=W_1^*\cap W_2^*$. \end{definition} Next suppose that we are given the two parabolics, say $P_1$ and $P_2$, which are the stabilizers of the boundary components $F_1$ and $F_1$, of which $W_1$ and $W_2$ are the quotients, $W_1=\gG_1\backslash F_1,\ W_2=\gG_2\backslash F_2$. Assume that $F_1^*\cap F_2^*\subset} \def\nni{\supset} \def\und{\underline F_i^*,\ i=1,2$, is a maximal boundary component in $F_i^*$. Under this assumption, the intersection $P_1\cap P_2$ is a parabolic, associated with $F_1^*\cap F_2^*$. Either of the inclusions $F_1^*\cap F_2^* \subset} \def\nni{\supset} \def\und{\underline F_i^*$ determines the parabolic which is the (non-maximal) parabolic stabilizing a flag of two terms. Similarly, $X_1^*\cap X_2^*$ contains $F_1^*\cap F_2^*$ as a rational boundary component, and either of the inclusions $X_1^*\cap X_2^*\subset} \def\nni{\supset} \def\und{\underline X_i^*$ determines a symmetric subgroup, also the stabilizer of a flag with two terms. This is of course just $(N_1\cap N_2)\times {\cal Z} _G(N_1\cap N_2)$, where $N_i$ is the group giving rise to ${\cal D} _{N_i}$, covering $X_i$. So we have: $N_i$ incident with $P_i,\ i=1,2$, $P_{12}:=P_1\cap P_2$ a parabolic, then $(N_1\cap N_2)\times {\cal Z} _G(N_1\cap N_2)$ is incident to $P_{12}$. \subsection{Moduli interpretation} In this section we suppose the algebraic group $G$ comes from a moduli problem of {\sc Pel} structures, and will discuss the moduli-theoretic description of the modular subvarieties $X_{\gG_N}$ and the corresponding arithmetic quotients $\ifmmode {X_{\gG}^*} \else$\xgs$\fi$. We then also briefly describe the notion of incidence from this point of view. \subsubsection{{\sc Pel} structures} \paragraph{\ } Let $V$ be an abelian variety over $\fC$, $End(V)$ the endomorphism ring and $End_{\fQ}(V)=End(V)\otimes_{\fZ}\fQ$ the endomorphism algebra. A polarization, i.e., a linear equivalence class of ample divisors giving a projective embedding of $V$, gives rise to a positive involution on $End_{\fQ}(V)$, the so-called Rosatti involution: \begin{eqnarray}\label{e96.1} \varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta:End_{\fQ}(V) & \longrightarrow} \def\sura{\twoheadrightarrow & End_{\fQ}(V) \\ \phi & \mapsto & \phi^{\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta}. \nonumber \end{eqnarray} If $A$ is a central simple algebra over $\fQ$, an involution * on $A$ is called {\em positive}, if $tr_{A|\fQ}(x\cdot x^*)>0$ for all $x\in A,\ x\neq 0$, where $tr_{A|\fQ}$ denotes the reduced trace. Assuming $(A,*)$ to be simple with positive involution, the ${\Bbb R}} \def\fH{{\Bbb H}$-algebra $A({\Bbb R}} \def\fH{{\Bbb H})$ is isomorphic to one of the following (see \cite{shimura2}, Lemma 1) \begin{itemize}\item[(i)] $M_r({\Bbb R}} \def\fH{{\Bbb H})$ with involution $X^*={^tX}$; \item[(ii)] $M_r(\fC)$ with involution $X^*={^t\-X}$, where $^{-}$ is complex conjugation; \item[(iii)] $M_r(\fH)$ with involution $X^*={^t\-X}$, where $^{-}$ is quaternionic conjugation. \end{itemize} The algebras $A$ occuring in (i) and (iii) are central simple over ${\Bbb R}} \def\fH{{\Bbb H}$, while those of (ii) are central simple over $\fC$. The $\fQ$-algebra $A$ itself is a $\fQ$-form of one of these. The central simple algebras $A$ over $\fQ$ are known to be the $M_n(D)$, where $D$ is a division algebra over $\fQ$. If the algebra $A$ has a positive involution, the same holds for $D$. The division algebras $D$ which can occur are also known. \begin{proposition}\label{p96.1} Let $D$ be a division algebra over $\fQ$ with a positive involution. Then $D$ occurs in one of the following cases: \begin{itemize}\item[I.] A totally real algebraic number field $k$; \item[II.] $D$ a totally indefinite quaternion algebra over $k$; \item[III.] $D$ a totally definite quaternion algebra over $k$; \item[IV.] $D$ is central simple over $K$ with a $K|k$ involution of the second kind, where $K$ is an imaginary quadratic extension of $k$. \end{itemize} \end{proposition} In case III the canonical involution on $D$ is the unique positive involution, while in case II the positive involutions correspond to $x\in D$ such that $x^2$ is totally negative in $k$. If the algebra $D$ has an involution of the second kind it is easy to see that it admits a positive one. It follows from the fact that $End_{\fQ}(V)$ is a semisimple algebra over $\fQ$ with a positive involution that each simple factor is a total matrix algebra $M_n(D)$, with $D$ as in the proposition. \paragraph{ \ } Let $(A,*)$ be a semisimple algebra over $\fQ$ with positive involution, and let \begin{equation}\label{e96.2} \Phi:A\longrightarrow} \def\sura{\twoheadrightarrow GL(n,\fC) \end{equation} be a faithful representation. Shimura considers data ${\cal P} =(V,{\cal C} ,\gt)$ and $\{A,\Phi,*\}$ and defines the notion of {\em polarized abelian variety of type} $\{A,\Phi,*\}$ by the conditions: \begin{equation}\label{e96.3}\begin{minipage}{14cm} \begin{itemize}\item[(i)] $V$ is an abelian variety over $\fC$, ${\cal C} $ is a polarization; \item[(ii)] $\gt:A\stackrel{\sim}{\longrightarrow} \def\sura{\twoheadrightarrow} End_{\fQ}(V)$ is an algebra isomorphism, and for $\gt(x):\~V\longrightarrow} \def\sura{\twoheadrightarrow \~V$ (the $\~{ }$ denoting the universal cover, i.e., $\~V$ is a complex vector space) one has $\gt(x)=\Phi(x)$; \item[(iii)] the involution $\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta$ determined by ${\cal C} $ as in (\ref{e96.1}) coincides on $\gt(A)$ with the involution coming from $(A,*)$, i.e. $\gt(x)^{\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta}=\gt(x^*)$. \end{itemize} \end{minipage}\end{equation} The condition (ii) is to be understood as follows. Fixing an isomorphism \begin{equation}\label{e97.-1} \psi:V\ifmmode\ \cong\ \else$\isom$\fi} \def\storth{\underline{\perp} \fC^n/\gL, \end{equation} each $a\in End_{\fQ}(V)$ is represented by a linear transformation of $\fC^n$ preserving $\gL$; that is each $a$ can be represented by a matrix, and $\gt(x)=a$ is the matrix corresponding to $x\in A$ via $\gt$. Recall also that a complex torus $\fC^n/\gL$ is an abelian variety if and only if there exists a {\em Riemann form}: each positive (1,1) form $\go$ gives rise to a skew symmetric matrix $(q_{ij})$: $$\go=\sum q_{ij}dx_i\wedge dx_j,$$ where the $x_i$ are canonical coordinates on $\fC^n$. Hence if we fix a positive divisor $C\subset} \def\nni{\supset} \def\und{\underline V$, it determines an involution as in (\ref{e96.1}) {\em and} a Riemann form $E_C(x,y)$ on $\fC^n/\gL$, and these are related by \begin{equation}\label{e97.0} E_C(\psi(a)x,y)=E_C(x,\psi(a^{\varrho} \def\ge{\varepsilon} \def\gx{\xi} \def\gy{\eta})y), \end{equation} where for $a\in End_{\fQ}(V)$, $\psi(a)$ denotes the matrix representation for $a$ arising from the identification $\psi:V\ifmmode\ \cong\ \else$\isom$\fi} \def\storth{\underline{\perp} \fC^n/\gL$ in (\ref{e97.-1}). Let $(V,{\cal C} ,\gt)$ be an abelian variety of type $(D,\Phi,*)$ with $D$ a division algebra, so that $(D,*)$ is one of the algebras of Proposition \ref{p96.1}. In the notations used there, put \begin{equation}\label{e97.6} [k:\fQ]=f,\quad [D:K]=d^2,\hbox{ if $D$ is of type IV} \end{equation} defining the numbers $f$ and $d$. Let $n$=dim$(V)$; then, assuming $D$ to be a division algebra, $2n$ is a multiple of $[D:\fQ]$, i.e., $2n=[D:\fQ]m$. Note that $[D:\fQ]=f$ for type I, $[D:\fQ]=4f$ for types II and III, while $[D:\fQ]=2d^2f$ if $D$ is of type IV. For the existence of $(V,{\cal C} ,\gt)$ of type $(D,\Phi,*)$, certain restrictions are placed on $\Phi$; we assume these are fulfilled. So under the isomorphism $\gt$, each $x\in D$ is represented by the matrix $\Phi(x)$. This makes the lattice $\gL$ with $V\ifmmode\ \cong\ \else$\isom$\fi} \def\storth{\underline{\perp} \fC^n/\gL$, tensored with $\fQ$, a (left) $D$-module, i.e., \begin{equation}\label{e97.1} Q:=\fQ\cdot \gL = \sum_1^m\Phi(D)\cdot x_i \end{equation} for a suitable set of vectors $x_i$. But this is the same as saying there exists a $\fZ$-lattice ${\cal M} \subset} \def\nni{\supset} \def\und{\underline D$, such that \begin{equation}\label{e97.2} \gL=\{\sum_1^m\Phi(a_i)x_i \Big| (a_1,\ldots,a_m)\in {\cal M} \}. \end{equation} If $D$ is central over $K$, then ${\cal M} $ is clearly also an ${\cal O} _K$-lattice in $D$. The integrality of the Riemann form can be expressed in terms of $tr_{D|K}$: \begin{equation}\label{e97.3} E_C(\sum_1^m\Phi(a_i)x_i,\sum_1^m\Phi(b_j)x_j) = tr_{D|K}(\sum_{i,j}^ma_it_{ij}b_j^*), \end{equation} and $T=(t_{ij})\in M_m(D)$ is a skew-hermitian matrix: \begin{equation}\label{e97.4} T^*=-T, \end{equation} where $T^*$ denotes the matrix $(t_{ji}^*)$, where $*$ is the involution on $D$. For the lattice ${\cal M} $ one has \begin{equation}\label{e97.5} tr_{D|K}({\cal M} T {\cal M} ^*)\subset} \def\nni{\supset} \def\und{\underline \fZ. \end{equation} \paragraph{ \ } Hence to each $(V,{\cal C} ,\gt)$ of type $(D,\Phi,*)$ one gets a $*$-skew hermitian $T\in M_m(D)$ and a lattice ${\cal M} \subset} \def\nni{\supset} \def\und{\underline D$. To this situation there is a naturally associated $\fQ$-group. On the vector space $D^m$ we consider \begin{equation}\label{e98.1} G(D,T):=\{g\in GL(D^m) \Big| gTg^*=T\}, \end{equation} the symmetry group of the $*$-skew hermitian form determined by $T$. It is now easy to determine the ${\Bbb R}} \def\fH{{\Bbb H}$-group: \begin{equation}\label{e98.2} G(D,T)({\Bbb R}} \def\fH{{\Bbb H})=\left\{ \begin{minipage}{10cm} \begin{tabbing} Type I:\quad\quad \= $Sp(m,{\Bbb R}} \def\fH{{\Bbb H})\times \cdots \times Sp(m,{\Bbb R}} \def\fH{{\Bbb H})$ ($m$ is even) \\ Type II:\> $Sp(2m,{\Bbb R}} \def\fH{{\Bbb H})\times \cdots \times Sp(2m,{\Bbb R}} \def\fH{{\Bbb H})$ \\ Type III:\> $SO^*(2m)\times \cdots \times SO^*(2m)$ \\ Type IV: \> $U(p_1,q_1)\times \cdots \times U(p_g,q_g),$ \end{tabbing} \end{minipage} \right. \end{equation} where the number of factors is in each case $f$, and $p_{\nu}+q_{\nu}=md$, and $(p_{\nu},q_{\nu})$ is the signature corresponding to the $\nu^{th}$ real prime. For each $\nu$, there is a matrix $W_{\nu}$ which trasforms $T_{\nu}$ into the standard form, i.e., \begin{eqnarray}\label{e98.3} W_{\nu}T_{\nu}^{-1}{^tW_{\nu}} & = & \left(\begin{array}{cc} 0 & 1_l \\ -1_l & 0 \end{array} \right), l={m\over 2} \hbox{ for Type I, $l=m$ for Type II}; \\ \label{e98.4} W_{\nu}T_{\nu}^{-1}W^*_{\nu} & = & -i\left(\begin{array}{cc} -1_m & 0 \\ 0 & 1_m \end{array} \right), \hbox{ Type III}; \\ \label{e98.5} W_{\nu}(iT_{\nu}^{-1})W^*_{\nu} & = & \left(\begin{array}{cc} 1_{p_{\nu}} & 0 \\ 0 & -1_{q_{\nu}}\end{array}\right), \hbox{ Type IV}. \end{eqnarray} Let ${\cal D} ={\cal D} _{(D,T)}$ denote the domain determined by $G(D,T)({\Bbb R}} \def\fH{{\Bbb H})$ (actually a particular unbounded realisation of this domain, see \cite{shimura2}, 2.6). Then ${\cal D} =\prod{\cal D} _{\nu}$, and $z_{\nu}\in {\cal D} _{\nu}$ gives rise to a normalised period (i.e., one of the form $(1,\Omega} \def\go{\omega} \def\gm{\mu} \def\gn{\nu} \def\gr{\rho)$) for an abelian variety, by setting $X_{\nu}=Y_{\nu}\-{W}_{\nu}$, where \begin{eqnarray}\label{e98.6} Y_{\nu} & = & \left(\begin{array}{cc} z_{\nu} & 1_l \\ \-z_{\nu} & 1_l \end{array}\right), l={m \over 2}, \hbox{ Type I, $l=m$, Type II}; \\ \label{e98.7} Y_{\nu} & = & \left(\begin{array}{cc} -z_{\nu} & 1_m \\ 1_m & \-z_{\nu} \end{array}\right), \hbox{ Type III}; \\ \label{e98.8} Y_{\nu} & = & \left( \begin{array}{cc} 1_{p_{\nu}} & z_{\nu} \\ ^t\-z_{\nu} & 1_{q_{\nu}}\end{array}\right), \hbox{ Type IV.} \end{eqnarray} The matrix $X_{\nu}$ determine $m$ vectors $x_1,\ldots x_m$ of $\fC^n$ (in a rather complicated fashion, see formulas (17)-(20) in \cite{shimura2}), which determine a lattice $\gL=\gL(z,T,{\cal M} )$ by the formula in equations (\ref{e97.1})-(\ref{e97.2}) above. Note that the representation $\Phi$ contains the representations $\chi_{\nu}$= projection on the $\nu$th real factor with multiplicities. For Type IV, $p_{\nu}+q_{\nu}=md$, and $p_{\nu}$=multiplicity of $\chi_{\nu}$ while $q_{\nu}$=multiplicity of $\-{\chi}_{\nu}$. For things to work out one must therefore assume, in case of Type IV, that $iT^{-1}_{\nu}$ has the {\em same} signature $(p_{\nu},q_{\nu})$ as occurs in $\Phi$. With this restriction, the following holds: \begin{theorem}[\cite{shimura2}, Thm.~1]\label{t99.1} For every $z\in {\cal D} ={\cal D} _{(D,T)}$, and every lattice ${\cal M} \subset} \def\nni{\supset} \def\und{\underline D$, we get a polarized abelian variety $V_z=\fC^n/\gL(z,T,{\cal M} )$ of type $(D,\Phi,*)$, and conversely, every such $V$ is of the form $V=\fC^n/\gL(z,T,{\cal M} )$ for some $z\in {\cal D} _{(D,T)},\ {\cal M} \subset} \def\nni{\supset} \def\und{\underline D$ a lattice. \end{theorem} \paragraph{ \ } The lattice ${\cal M} \subset} \def\nni{\supset} \def\und{\underline D$ gives rise to an arithmetic subgroup \begin{equation}\label{e99.1} \gG=\gG_{(D,T,{\cal M} )}=\{g\in G(D,T) \Big| g{\cal M} \subset} \def\nni{\supset} \def\und{\underline {\cal M} \} \end{equation} as discussed in section \ref{s83.1}. If one defines an isomorphism $\phi:V_z\longrightarrow} \def\sura{\twoheadrightarrow V_{z'}$ of two abelian varieties of type $(D,\Phi,*)$ as an isomorphism of the underlying varieties, such that $\phi^{-1}({\cal C} ')={\cal C} $ and $\phi\gt(a)=\gt'(a)\phi,$ for all ${a\in D}$, then one has \begin{theorem}[\cite{shimura2}, Thm.~2]\label{t99.2} The arithmetic quotient $\xg=\gG\backslash{\cal D} _{(D,T)}$ is the moduli space of isomorphism classes of abelian varieties $V_z=\fC^n/\gL(z,T,{\cal M} )$ of type $\{(D,\Phi,*),(T,{\cal M} )\}$, where $\gG$ is the arithmetic group of (\ref{e99.1}). \end{theorem} Moreover, one calls two such pairs $(T_1,{\cal M} _1),\ (T_2,{\cal M} _2)$ {\em equivalent}, if $\exists_{U\in M_m(D)}$, such that $UT_2U^*=\gd T_1$ for some positive $\gd\in \fQ$ and ${\cal M} _1U={\cal M} _2$. Equivalent pairs give rise to isomorphic families of abelian varieties (\cite{shimura2}, Prop.~4). Summing up, $*$-skew hermitian matrices $T\in M_m(D)$ determine certain $\fQ$-groups, lattices ${\cal M} \subset} \def\nni{\supset} \def\und{\underline D$ determine certain arithmetic groups, and the corresponding arithmetic quotients are moduli spaces for certain families of abelian varieties. \begin{remark}\label{r99.1} The complex multiplication by ${\cal M} $ describes the endomorphism ring. The {\it automorphisms} determined by ${\cal M} $ are the invertible elements, i.e., $\hbox{Aut}} \def\Im{\hbox{Im}} \def\mod{\hbox{mod}(V)={\cal M} ^*$, the group of units. \end{remark} One can also accomodate level structures in this settup, introduced in \cite{shimura3}, cf.~also \cite{shimura4}. This is done by fixing $s$ points $y_1,\ldots,y_s$ in the $D$-module $Q$, as in (\ref{e97.1}), and $s$ points $t_1,\ldots,t_s$ of the abelian variety $V$. One requires that the map $\psi$ of (\ref{e97.-1}) maps the $y_i$ onto the $t_i$. More precisely, \begin{definition}\label{d99z.1} Let $Q$ be a $D$-vector space of dimension $m$, and $\hbox{{\script M}} \subset} \def\nni{\supset} \def\und{\underline Q$ a $\fZ$-lattice. Consider a conglomeration: $$\hbox{{\script T}} :=\{(D,\Phi,*),(Q,T,\hbox{{\script M}} );y_1,\ldots,y_s\},$$ where $(D,\Phi,*)$ is as above, $(Q,T,\hbox{{\script M}} )$ is a $D$-vector space with lattice $\hbox{{\script M}} $ and $*$-skew hermitian ($D$-valued) form $T$ on $Q$, and $y_i$ points in $Q$. This is called a {\sc Pel}{\em -type}. Consider a conglomeration: $$\hbox{{\script Q}} :=\{(V,{\cal C} ,\gt);t_1,\ldots,t_s\},$$ where $(V,{\cal C} ,\gt)$ is a polarized abelian variety with analytic coordinate $\gt$ as above and $t_i$ are points of {\em finite order} on $V$. This is called a {\sc Pel}-{\em structure}. Then $\hbox{{\script Q}} $ is {\em of type} $\hbox{{\script T}} $, if there exists a commutative diagram \begin{equation}\label{e99z.1} \begin{array}{ccccccccc} 0 & \longrightarrow} \def\sura{\twoheadrightarrow & \hbox{{\script M}} & \longrightarrow} \def\sura{\twoheadrightarrow & Q({\Bbb R}} \def\fH{{\Bbb H}) & \longrightarrow} \def\sura{\twoheadrightarrow & Q({\Bbb R}} \def\fH{{\Bbb H})/\hbox{{\script M}} & \longrightarrow} \def\sura{\twoheadrightarrow & 0 \\ & & \downarrow & & f\downarrow & & \downarrow & & \\ 0 & \longrightarrow} \def\sura{\twoheadrightarrow & \gL & \longrightarrow} \def\sura{\twoheadrightarrow & \fC^n & \stackrel{\psi}{\longrightarrow} \def\sura{\twoheadrightarrow} & V & \longrightarrow} \def\sura{\twoheadrightarrow & 0 \end{array}, \end{equation} satisfying the conditions\begin{itemize}\item[(i)] $\psi$ gives a holomorphic isomorphism (strictly speaking, this is the $\psi^{-1}$ of above);\item[(ii)] $f$ is an ${\Bbb R}} \def\fH{{\Bbb H}$-linear isomorphism, and $f(\hbox{{\script M}} )=\gL$; \item[(iii)] $f(ax)=\Phi(a)f(x)$, and $\Phi(a)$ defines $\gt(a)$ for every $a\in D$ as (\ref{e96.3}), (ii); \item[(iv)] $C\in{\cal C} $ determines a Riemann form $E_C$ as in (\ref{e97.0}). \end{itemize} \end{definition} Note that the finite set of points $y_i$ and $t_i$ come both equipped with a form; on the former the form $T$, and the Riemann form $E_C$ on the latter. These forms are preserved under the isomorphism. There is a natural notion of isomorphism of abelian varieties with {\sc Pel} structures. Given two {\sc Pel}-structures $\hbox{{\script Q}} $ and $\hbox{{\script Q}} '$, an isomorphism $\phi:V\longrightarrow} \def\sura{\twoheadrightarrow V'$ is an {\em isomorphism} from $\hbox{{\script Q}} $ to $\hbox{{\script Q}} '$, if $\phi\gt(a)=\gt'(a)\phi$ for all $a\in D$, and $\phi(t_i)=t_i'$ for all $i$. \begin{definition}\label{d99z.2} A {\sc Pel}-type $\hbox{{\script T}} $ is {\em equivalent} to a {\sc Pel}-type $\hbox{{\script T}} '$, if $D=D'$, $*=*'$, $s=s'$, $\Phi$ and $\Phi'$ are equivalent as representations of $D$, and there is a $D$-linear automorphism $\mu$ of $Q$ such that $T'(x\mu, y\mu)=T(x,y),\ \hbox{{\script M}} \mu=\hbox{{\script M}} ', y_i\mu\equiv y_i'$mod$\hbox{{\script M}} '$ for all $i$. If $\hbox{{\script Q}} $ is of type $\hbox{{\script T}} $, then $\hbox{{\script Q}} $ is also of type $\hbox{{\script T}} '$ if and only if $\hbox{{\script T}} $ and $\hbox{{\script T}} '$ are equivalent. A {\sc Pel}-type $\hbox{{\script T}} $ is called {\em admissible}, if there exists at least one {\sc Pel}-structure of that type. \end{definition} One has an anolgue of Theorems \ref{t99.1} and \ref{t99.2} in this situation also. \begin{theorem}[\cite{shimura4}, Thm.~3]\label{t99z.1} For every admissible {\sc Pel}-type $\hbox{{\script T}} $ there exists a bounded symmetric domain ${\cal D} $ (this is the same domain as in Theorem \ref{t99.1}) such that the statement of Theorem \ref{t99.1} holds in this situation, and every {\sc Pel}-structure $\hbox{{\script Q}} $ of type $\hbox{{\script T}} $ occurs in this family. \end{theorem} Now define a corresponding arithmetic group as follows: \begin{equation}\label{e99z.2} \gG=\{g\in G(D,T) \Big| \hbox{{\script M}} g=\hbox{{\script M}} ,\ y_i g\equiv y_i\hbox{mod}\hbox{{\script M}} ,\ i=1,\ldots,s\} \end{equation} Then the analogue of Theorem \ref{t99.2} is \begin{theorem}[\cite{shimura4}, Thm.~4]\label{t99z.2} Two members of the family of Theorem \ref{t99z.1} corresponding to points $z_1,z_2\in {\cal D} $ are isomorphic if and only if $z_1=\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta(z_2)$ for some $\gamma} \def\gs{\sigma} \def\gS{\Sigma} \def\gz{\zeta\in \gG$, $\gG$ as in (\ref{e99z.2}). \end{theorem} In Table \ref{table17} we list the data for each of the cases II, III and IV of (\ref{e98.2}). \begin{table}\caption{\label{table17} Rational groups for {\sc Pel}-structures.} \medskip\begin{center} \begin{tabular}{|c|c|c|c|}\hline & Type II & Type III & Type IV \\ \hline $D$ & \begin{minipage}{3.5cm} A totally \\ indefinite quaternion algebra over $\fQ$ \end{minipage} & \begin{minipage}{3.5cm} a totally \\ definite quaternion \\ algebra over $\fQ$ \end{minipage} & \parbox{4cm}{\footnotesize simple division algebra, central over $K$, an imaginary quadratic extension of $\fQ$, with an involution of the second kind. One may assume $D$ to be a cyclic algebra} \\ \hline $d$ & 2 & 2 & $d$ \\ \hline dim($V$) & $2m$ & $2m$ & $d^2m$ \\ \hline Tits index & $C_{m,s}^{(2)}$ & ${^iD}_{m,s}^{(2)}, i=1,2$ & $^2A^{(d)}_{dm-1,s}$ \\ \hline \end{tabular} \medskip The types listed are absolutely $\fQ$-simple, the $\fQ$-rank is $s$, and this is the Witt index of the $\pm$hermitian form. \end{center} \end{table} \subsubsection{Modular subvarieties} We continue with the notations above, $G,\ S,\ P_{\hbox{\scsi \bf b}}$ and $N_{\hbox{\scsi \bf b}}$ being fixed, $\gG$ an arithmetic group satisfying (\ref{E5.1}), and $\gG'\subset} \def\nni{\supset} \def\und{\underline G(\fQ)$ another arithmetic group. We consider the arithmetic quotient $X_{\gG'}$, its Baily-Borel embedding $X_{\gG'}^*$, and a smooth toroidal embedding $\overline{X}_{\gG'}$. Let $N\subset} \def\nni{\supset} \def\und{\underline G$ be a rational symmetric subgroup, $N=N_{\hbox{\scsi \bf b}}^g$ for some $b=1,\ldots, s$ and some $g\in G(\fQ)$. Let us first suppose for the boundary point in question $F_{\hbox{\scsi \bf b}}$ that $\dim(F_{\hbox{\scsi \bf b}})>0$. Under this assumption we know that $N_{\hbox{\scsi \bf b}}$ is a product \[ N_{\hbox{\scsi \bf b}}=L_{\hbox{\scsi \bf b}}\times {\cal Z} _G(L_{\hbox{\scsi \bf b}}),\] and $L_{\hbox{\scsi \bf b}} ({\Bbb R}} \def\fH{{\Bbb H})^0=(\hbox{Aut}} \def\Im{\hbox{Im}} \def\mod{\hbox{mod}(F_{\hbox{\scsi \bf b}}))^0$. This implies immediately that the domain ${\cal D} _{N_{\hbfs{b}}}$ is also a product, \[ {\cal D} _{N_{\hbfs{b}}}\cong {\cal D} _1\times {\cal D} _2.\] Let $\imath_i:{\cal D} _i\hookrightarrow} \def\hla{\hookleftarrow {\cal D} _1\times {\cal D} _2$ be the natural inclusion, and consider the inclusion $\eta:{\cal D} _{N_{\hbfs{b}}}\hookrightarrow} \def\hla{\hookleftarrow {\cal D} $. Then $\eta(\imath_i({\cal D} _i))$ is a symmetric subdomain, which itself has an interpretation in terms of {\sc Pel} structures, which is a sub-{\sc Pel} structure of that attached to ${\cal D} $. Let us now explain this for the individual cases. For {\sc Pel} structures, only the domains of type $\bf I_{\hbf{p,q}},\ II_{\hbf{n}},\ III_{\hbf{n}}$ occur, types {\bf U.1, U.2, O.2, S.1, S.2}. \begin{itemize}\item[{\bf U.1}]: If $b<t$, then $F_{\hbox{\scsi \bf b}}$ is of type $\bf I_{\hbf{p-b,q-b}}$, and ${\cal D} _{N_{\hbfs{b}}}$ is of the type $\bf I_{\hbf{p-b,q-b}}\times I_{\hbf{b,b}}$. The moduli interpretation is complex multiplication on abelian $(p+q)$-folds. In the locus $\bf I_{\hbf{p-b,q-b}}\times I_{\hbf{b,b}}$, the variety $A^{p+q}$ splits into $A^{p+q-2b}\times A^{2b}$, where the complex multiplication has signature $(p-b,q-b)$ and $(b,b)$, respectively. If $b=s=t$, $F_{\hbf{t}}$ is a point, ${\cal D} _{N_{\hbfs{b}}}$ is of type $\bf I_{\hbf{p-1,q}}$. Here the abelian variety $A^{p+q}$ splits off an elliptic curve, $A^{p+q}=A^{p+q-1}\times A^1$. Since $A^1$ has no moduli, only the moduli of $A^{p+q-1}$ contributes. \item[{\bf U.2}]: If $b<t$, then $F_{\hbox{\scsi \bf b}}$ is of type $\bf I_{\hbf{p-db,q-db}}$ and ${\cal D} _{N_{\hbfs{b}}}$ is of type $\bf I_{\hbf{p-db,q-db}}\times I_{\hbf{db,db}}$. The moduli involved here is a degree $d$ division algebra $D$, central simple over $K$ with $K|k$-involution, as endomorphism algebra. In the locus $\bf I_{\hbf{p-db,q-db}}\times I_{\hbf{db,db}}$ the abelian variety $A^{p+q}$ splits $A^{p+q}=A^{p+q-2db}\times A^{2db}$, and each factor retains the endomorphisms by $D$. If $b=s=t$, again $F_{\hbf{t}}$ is a point, ${\cal D} _{N_{\hbfs{b}}}$ is of type $\bf I_{\hbf{q,q}}\subset} \def\nni{\supset} \def\und{\underline I_{\hbf{p,q}}$. Here the abelian variety splits as $A^{2q}\times A^{p-q}$, where the endomorphism ring on $A^{p-q}$ is definite, and contributes no moduli. In this case the only moduli contributing is the modulus of $A^{2q}$. \item[{\bf O.2}]: If $b<s$ or $s<[{n\over 2}]$, then $F_{\hbox{\scsi \bf b}}={\bf II_{\hbf{n-2b}}},\ {\cal D} _{N_{\hbfs{b}}}=\bf II_{\hbf{n-2b}}\times II_{\hbf{2b}}$, and the splitting is evident. For $b=t$, $F_{\hbf{t}}=\bf II_{\hbf{0}}$ ($n$ even) or $\bf II_{\hbf{1}}$ ($n$ odd), both of which are points. Then for $n$ even, $N_{\hbf{t}}$ is a polydisc by definition, $\bf II_{\hbf{2}}\times \cdots \times II_{\hbf{2}}$, and again the splitting is evident, this time as a product of abelian surfaces with multiplication by the quaternion division algebra $D$. For $n$ odd, $N_{\hbf{t}}$ is of type $\bf II_{\hbf{n-1}}$, and the splitting of $A^{2n}$ is as $A^{2n}\cong A^{2n-s}\times A^2$. \item[{\bf S.1}]: If $b<t$, then $F_{\hbox{\scsi \bf b}}$ is of type $\bf III_{\hbf{n-b}}$, the subdomain ${\cal D} _{N_{\hbfs{b}}}$ is of type $\bf III_{\hbf{n-b}}\times III_{\hbf{b}}$, $A^n$ splits $A^n=A^{n-b}\times A^b$. If $b=s=t$, then $F_{\hbf{t}}$ is a point, ${\cal D} _{N_{\hbfs{t}}} \cong ({\bf III_{\hbf{1}}})^n$. Here the abelian variety splits into a product of elliptic curves. \item[{\bf S.2}]: If $b<t$, $F_{\hbox{\scsi \bf b}}={\bf III_{\hbf{n-2b}}}$,\ ${\cal D} _{N_{\hbfs{b}}}={\bf III_{\hbf{n-2b}}\times III_{\hbf{2b}}}$, where $b< [{n\over 2}]$. If $b=s=t$, ${\cal D} _{N_{\hbfs{t}}}=({\bf III_{\hbf{2}}})^{n\over 2}$, ($n$ even follows from $s=t$). Once again, in both cases the moduli-theoretic meaning is evident. \end{itemize} This explains the expression of sub-{\sc Pel} structures, and we have established \begin{proposition}\label{p6.4.1} For any subdomain ${\cal D} _{N_{\hbfs{b}}}\subset} \def\nni{\supset} \def\und{\underline {\cal D} $, the corresponding abelian varieties split in the manner described above. \end{proposition} {\bf Proof:} We prove a typical case and leave the verification of the other cases to the reader. We will do case {\bf U.2}. For this we consider the matrix $Y_{\nu}$ of (\ref{e98.8}). We may assume that the realisation of the domain ${\cal D} $ is such that for the subdomain ${\cal D} _N$, the corresponding $z_{\nu}$ splits, i.e., \begin{equation}\label{E161.1} z_{\nu}\in {\cal D} _N \Ra z_{\nu}=\left(\begin{array}{cc} z_{\nu,1} & 0 \\ 0 & z_{\nu,2} \end{array}\right), \end{equation} where $z_{\nu,1}\in {\cal D} _1$ and $z_{\nu,2}\in {\cal D} _2$ for the decomposition ${\cal D} _N={\cal D} _1\times {\cal D} _2$. In this case, ${\cal D} _1$ is of type $\bf I_{\hbf{p-jd,q-jd}}$, while ${\cal D} _2$ is of type $\bf I_{\hbf{jd,jd}}$. We need the vectors $x_1,\ldots, x_m$ determined by $X_{\nu}$, given in this case by the formula (20) in \cite{shimura2} \[ X_{\nu}=\left[\begin{array}{cccccccccc} u_{11}^{\nu} & \cdots & u_{m1}^{\nu} & u_{12}^{\nu} & \cdots & u_{m2}^{\nu} & \cdots & u_{1d}^{\nu} & \cdots & u_{md}^{\nu} \\ \overline{v}_{11}^{\nu} & \cdots & \overline{v}_{m1}^{\nu} & \overline{v}_{12}^{\nu} & \cdots & \overline{v}_{m2}^{\nu} & \cdots & \overline{v}_{1d}^{\nu} & \cdots & \overline{v}_{md}^{\nu} \end{array} \right]. \] Here the vectors $x_i$ are given by ${^tx}_i^{\nu}=({^tu}_{i1}^{\nu}\cdots {^tu}_{1d}^{\nu} {^tv}_{i1}^{\nu} \cdots {^tv}_{id}^{\nu})$ and $u_{ik}^{\nu}\in \fC^{p_{\nu}},\ v_{ik}^{\nu}\in \fC^{q_{\nu}}$. Now from the particular form of our $z_{\nu}$, we can conclude that also the vectors $x_i$ have a special form. Indeed, comparing the above with (\ref{E161.1}), we see that for $W_{\nu}=id$ we have \[ X_{\nu} = \left( \begin{array}{cc|cc}\hbox{\Large\bf 1}_{\hbf{p$_{\nu}$-jd}} & 0 & z_{\nu,1} & 0 \\ 0 & \hbox{\Large\bf 1}_{\hbf{jd}} & 0 & z_{\nu,2} \\ \hline {^t\overline{z}}_{\nu,1} & 0 & \hbox{\Large\bf 1}_{\hbf{q$_{\nu}$-jd}} & 0 \\ 0 & {^t\overline{z}}_{\nu,2} & 0 & \hbox{\Large\bf 1}_{\hbf{jd}} \end{array}\right), \] so that the vector ${^tx}_1$, for example, has the form\footnote{for convenience the transpostition $t$ is placed to the right of the vector in this expression} \[ {^tx}_1 = \left( \left( \begin{array}{c} 1 \\ 0 \\ \vdots \\ \vdots \\ \vdots \\ 0 \end{array}\right)^{t}\cdots \left(\begin{array}{c} 0 \\ \vdots \\ 1 \\ 0 \\ \vdots \\ 0 \end{array}\right)^{t} \left(\begin{array}{c} z_{\nu,1}^{(1)} \\ 0 \end{array} \right)^{t} \cdots \left(\begin{array}{c} z_{\nu,1}^{(p-(j-1)d+1)} \\ 0 \end{array} \right)^{t} \cdots \left( \begin{array}{c} 0 \\ z_{\nu,2}^{(1)} \end{array} \right)^{t} \cdots \left( \begin{array}{c} 0 \\ z_{\nu,2}^{((j-1)d+1)} \end{array} \right)^{t} \right),\] where $z_{\nu,j}^{(k)}$ denotes the $k^{th}$ column of $z_{\nu,j}$, and similarly for the other ${^tx}_i$. From this it follows that the lattice $\gL$ of (\ref{e97.2}) splits, $\gL=\gL_1\oplus \gL_2$, where $\gL_1$ and $\gL_2$ are orthogonal to each other, each being itself a normalized period matrix \[ \gL_1=\left(\begin{array}{c} 1 \\ 0 \\ \vdots \\ 0 \\ \vdots \\ 0 \end{array} \right)\fZ\oplus \cdots \oplus \left(\begin{array}{c} 0 \\ \vdots \\ 1 \\ 0 \\ \vdots \end{array} \right)\fZ \oplus z_{\nu,1}^{(1)}\fZ \oplus \cdots \oplus z_{\nu,1}^{(q-jd)}\fZ, \gL_2=\left(\begin{array}{c} 0 \\ \vdots \\ 1 \\ 0 \\ \vdots \\ 0 \end{array} \right) \fZ\oplus \cdots \oplus \left(\begin{array}{c} 0 \\ \vdots \\ 0 \\ 1 \end{array} \right) \fZ\oplus z_{\nu,2}^{(1)}\fZ\oplus \cdots \oplus z_{\nu,2}^{(jd)}\fZ. \] The proposition follows from this for the case that $W_{\nu}=id$. Finally we note that if $W_{\nu}\neq id$, this does not influence the reasoning above, and the splitting remains (only the polarization is no longer principal). In particular, the case $\dim(F_{\hbox{\scsi \bf b}})=0$, which occurs for $b=s,\ sd=q_{\nu}$, is covered by the above, \[ z_{\nu}=\left(\begin{array}{cc} \hbox{\Large\bf 1} & 0 \\ 0 & z_{\nu,2} \end{array} \right)\] with $z_{\nu,2}\in \bf I_{\hbf{sd,sd}}$. \hfill $\Box$ \vskip0.25cm We now consider conjugates $N=N_{\hbox{\scsi \bf b}}^g$. The following two lemmas apply to any $G$ as considered in this paper so we assume for the moment only that $G$ is $\fQ$-simple of hermitian type, $\gG$ fulfills (\ref{E5.1}) and $\gG'\subset} \def\nni{\supset} \def\und{\underline G(\fQ)$ is arithmetic. \begin{lemma}\label{L6.4.1} If $g\in G(\fQ)$, $N=N_{\hbox{\scsi \bf b}}^g$, then the modular subvariety $X_{\gG'_N}\subset} \def\nni{\supset} \def\und{\underline X_{\gG'}$ is a finite quotient of a product. Consequently, for all ${\cal D} _N$, $N$ rational symmetric, the arithmetic subvariety $X_{\gG'_N}$ is in the locus of isomorphism classes of abelian varieties which are isogenous to products, i.e., are not simple. \end{lemma} {\bf Proof:} Since $g\in G(\fQ)$, we see that ${\cal D} _N$ is $\fQ$-equivalent to ${\cal D} _{N_{\hbfs{b}}}$. For ${\cal D} _{N_{\hbfs{b}}}$ the statement follows from the fact that $N_{\hbox{\scsi \bf b}}=N_1\times N_2$ is a product over $\fQ$, so for $g\in G(\fQ)$ it is likewise true for $N=N_{\hbox{\scsi \bf b}}^g$. Consequently, the action of $\gG$ on ${\cal D} _N$ is up to a finite action a product action. The second statement follows from this, as on the finite cover which is a product, the splitting property follows as discussed above. \hfill $\Box$ \vskip0.25cm Now suppose $N$ is in fact $\gG$-integral, i.e., $N_1\cap \gG =g(N_{\hbf{b,1}}\cap \gG)g^{-1}$. \begin{lemma}\label{L6.4.2} If $N$ is $\gG$-integral, then the discrete subgroup $N\cap \gG$ is in fact a product, $\gG_N=N\cap \gG=\gG_1\times \gG_2$, $\gG_i\subset} \def\nni{\supset} \def\und{\underline \hbox{Aut}} \def\Im{\hbox{Im}} \def\mod{\hbox{mod}({\cal D} _i),\ i=1,2$. \end{lemma} {\bf Proof:} We note that there is a natural inclusion $N\cap \gG\subset} \def\nni{\supset} \def\und{\underline g\gG_{\hbf{b,1}}g^{-1}\times g\gG_{\hbf{b,2}}g^{-1}$, and since $N_1\cap \gG\subset} \def\nni{\supset} \def\und{\underline N\cap \gG$ is equal to $g\gG_{\hbf{b,1}}g^{-1}$ we get the exact diagram \[\begin{array}{ccccccccc} & & 1 & & 1 \\ & & \downarrow & & \downarrow \\ 1&\longrightarrow} \def\sura{\twoheadrightarrow & \gG_1 & \longrightarrow} \def\sura{\twoheadrightarrow & g\gG_{\hbf{b,1}}g^{-1} & \longrightarrow} \def\sura{\twoheadrightarrow & 1 \\ & & \downarrow & & \downarrow & & \downarrow \\ 1&\longrightarrow} \def\sura{\twoheadrightarrow & N\cap \gG & \longrightarrow} \def\sura{\twoheadrightarrow & g\gG_{\hbf{b,1}}g^{-1}\times g\gG_{\hbf{b,2}}g^{-1} & \longrightarrow} \def\sura{\twoheadrightarrow & K_1 & \longrightarrow} \def\sura{\twoheadrightarrow & 1 \\ & & \downarrow & & \downarrow & & \downarrow \\ 1&\longrightarrow} \def\sura{\twoheadrightarrow & Q & \longrightarrow} \def\sura{\twoheadrightarrow & g\gG_{\hbf{b,2}}g^{-1} & \longrightarrow} \def\sura{\twoheadrightarrow & K_2 & \longrightarrow} \def\sura{\twoheadrightarrow & 1 \\ & & \downarrow & & \downarrow & & \downarrow \\ & & 1& \longrightarrow} \def\sura{\twoheadrightarrow & 1 & \longrightarrow} \def\sura{\twoheadrightarrow & 1, \end{array} \] and the splitting $N\cap \gG\cong \gG_1\times Q$ follows from that of $g\gG_{\hbf{b,1}}g^{-1}\times g \gG_{\hbf{b,2}}g^{-1}$: $Q$ is a subgroup of finite index in $g\gG_{\hbf{b,2}}g^{-1}$, and giving the injection $\gG_1\times Q\hookrightarrow} \def\hla{\hookleftarrow g\gG_{\hbf{b,1}}g^{-1}\times g \gG_{\hbf{b,2}}g^{-1}$ is equivalent to giving the injection $Q\hookrightarrow} \def\hla{\hookleftarrow g\gG_{\hbf{b,2}}g^{-1}$. \hfill $\Box$ \vskip0.25cm It may well be that $N$ is in fact $\gG$-integral if and only if $X_{\gG_N}$ is a product, but I have no argument for this. At any rate, this can now be applied to derive the moduli interpretation of $X_{\gG'_N}$ for $N$ $\gG$-integral. Applying the two lemmas above again in the situation that $G$ corresponds to a {\sc Pel}-structure yields the following. \begin{theorem}\label{t6.4.1} Let $G,\ S,\ P_{\hbox{\scsi \bf b}},\ N_{\hbox{\scsi \bf b}}$ and $\gG$ be as above ($b<t$), $\gG'\in G(\fQ)$ arithmetic, and let $X_{\gG'_N}$ be a modular subvariety of $X_{\gG'}$ for $N$ rational symmetric, conjugate to $N_{\hbox{\scsi \bf b}}$. Then $X_{\gG'_N}$ is a finite quotient of a product, and the set of $\gG'$-equivalence classes of such modular subvarieties forms a locus in $X_{\gG'}$ where the corresponding abelian varieties are isogenous to products, i.e., are not simple. If $N$ is $\gG$-integral, then $X_{\gG'_N}$ is a product, and the set of $\gG'$-equivalence classes of such modular subvarieties forms a locus in $X_{\gG'}$ where the corresponding abelian varieties split while preserving the endomorphisms. \end{theorem} {\bf Proof:} The first statement follows immediately from Lemma \ref{L6.4.1}. By Lemma \ref{L6.4.2}, if $N$ is $\gG$-integral, the discrete subgroup $\gG'_N$ is a product, hence so is the quotient $X_{\gG'_N}$, giving the second assertion. We know by the discussion above the moduli interpretation upstairs in ${\cal D} $, given in Proposition \ref{p6.4.1}. Since $X_{\gG'_N}$ itself is a product, it follows that the abelian varieties $A^n$ also split as $A^n\cong A^q\times A^{n-q}$, where $\tau_q$, the modulus of $A^q$, defines a point in one of the factors of $X_{\gG'_N}$, while $\tau_{n-q}$, the modulus of $A^{n-q}$, defines a point in the second factor. That the endomorphisms are preserved was shown above in the proof of \ref{p6.4.1}. \hfill $\Box$ \vskip0.25cm We leave it to the reader to derive the correct result for $b=t$. Finally we briefly mention the moduli interpretation of incidence. For this, recall that one has the Satake compactification and the (smooth projective) toroidal compactifications. The former relate to degenerations of the abelian varieties as follows. A quasi-abelian variety $A'$ is an extension of an abelian variety by an algebraic torus \begin{equation}\label{E161.2} 1 \longrightarrow} \def\sura{\twoheadrightarrow (\fC^*)^h \longrightarrow} \def\sura{\twoheadrightarrow A' \longrightarrow} \def\sura{\twoheadrightarrow B \longrightarrow} \def\sura{\twoheadrightarrow 0. \end{equation} Thus $A'$ is still an abelian group. Let $c$ denote the dimension of the abelian variety $B$, $n=h+c$ the dimension of $A'$. We now suppose that $X_{\gG}$ is a moduli space of {\sc Pel} structures, and assume the notations used above in this case. Let $F_b$ be a standard boundary component of the domain ${\cal D} $, and $W_i$ a boundary variety which is covered by $F_b$. Let $n=\dim(A)$ for the abelian varieties parameterized by $X_{\gG}$, $m=\dim_D(V)$ so that $n=mg,\ g=f, 4f$ and $2d^2f$ in the respective cases. Since $F_b$ has rank $b$, it corresponds to a vector subspace $W\subset} \def\nni{\supset} \def\und{\underline V$ with $\dim_D(W)=\dim_D(V)-b=m-b\ (b=1,\ldots, s=\hbox{Witt index of the form},\ s\leq [{m\over 2}])$, and hence to abelian varieties $B$ with $\dim(B)=(m-b)g$ and $g$ as above. We observe that the sequence (\ref{E161.2}) is relevant here, with $h=bg,\ c=(m-b)g$. An extension as in (\ref{E161.2}) is far from being unique, and the precise degenerations have been constructed in many cases by utilizing methods from the theory of mixed variations of Hodge structures, and this can be brought into relation to the toroidal compactifications mentioned above, where the parameter spaces of the degenerations are divisors on $\xg$. For our purposes (\ref{E161.2}) is sufficient. We now consider an integral modular subvariety $X_{\gG_N}$ incident with a boundary variety $W_i$. As we have seen above, the abelian varieties parameterized by $X_{\gG_N}$ split, in this case as \begin{equation}\label{E161.3} 0\longrightarrow} \def\sura{\twoheadrightarrow A^h\longrightarrow} \def\sura{\twoheadrightarrow A' \longrightarrow} \def\sura{\twoheadrightarrow A^c\longrightarrow} \def\sura{\twoheadrightarrow 0, \end{equation} and the relation to (\ref{E161.2}) is obvious; the boundary varieties are the loci in $X_{\gG_N}$ where the $A^h$ of (\ref{E161.3}) totally degenerate.

9508

alg-geom/9508009

en

https://arxiv.org/abs/alg-geom/9508009

alg-geom/9508009

Niels Lauritzen

A. Buch, J. F. Thomsen, N. Lauritzen and V. B. Mehta

Frobenius morphisms over Z/p^2 and Bott vanishing

AMS-LaTeX, For a dvi-version of this preprint please check out http://www.mi.aau.dk/~niels/papers.html

Aarhus University Preprint Series no. 11, 1995

Let $X$ be a smooth projective algebraic variety over $Z/p$, which has a flat lift to a scheme $X'$ over $Z/p^2$. If the absolute Frobenius morphism $F$ on $X$ lifts to a morphism on $X'$, then an old trick by Mazur shows that push-down of the de Rham complex under $F$ decomposes. We show that the quasi-isomorphism in question is split. This is then applied to toric varieties (where a glueing argument gives lifting of Frobenius to $Z/p^2$) and we derive natural characteristic $p$ proofs of Bott vanishing and degeneration of the Danilov spectral sequence. For flag varieties we obtain generalizations of a result of Paranjape and Srinivas about non-lifting of Frobenius to the Witt vectors.

alg-geom

\section{Preliminaries} Let $k$ be a perfect field of characteristic $p>0$ and $X$ a smooth $k$-variety of dimension $n$. By $\Omega_X$ we denote the sheaf of $k$-differentials on $X$ and $\Omega^j_X=\wedge^j \Omega_X$. The (absolute) Frobenius morphism $F:X\rightarrow X$ is the morphism on $X$, which is the identity on the level of points and given by $F^\#(f)=f^p: \O_X(U)\rightarrow F_*\O_X(U)$ on the level of functions. If ${\E F}$ is an $\O_X$-module, then $F_* {\E F}={\E F}$ as sheaves of abelian groups, but the $\O_X$-module structure is changed according to the homomorphism $\O_X\rightarrow F_*\O_X$. \subsection{The Cartier operator} The universal derivation $d:\O_X \rightarrow \Omega_X$ gives rise to a family of $k$-homomorphisms $d^j: \Omega^j_X\rightarrow \Omega^{j+1}_X$ making $\Omega^\bullet_X$ into a complex of $k$-modules which is called the de Rham complex of $X$. By applying $F_*$ to the de Rham complex, we obtain a complex $F_*\Omega^\bullet_X$ of $\O_X$-modules. Let $B^i_X\subseteq Z^i_X\subseteq F_*\Omega^i_X$ denote the coboundaries and cocycles in degree $i$. There is the following very nice description of the cohomology of $F_*\Omega^\bullet_X$ due to Cartier. \begin{thm} There is a uniquely determined graded $\O_X$-algebra isomorphism $$ C^{-1}:\Omega_X^\bullet\rightarrow \cal H^\bullet(F_* \Omega^\bullet_X) $$ which in degree $1$ is given locally as $$ C^{-1}(da)= a^{p-1} da $$ \end{thm} \begin{pf} \cite{Katz}, Theorem 7.2. \end{pf} With some abuse of notation, we let $C$ denote the natural homomorphism $Z^i_X\rightarrow\Omega^i_X$, which after reduction modulo $B^i_X$ gives the inverse isomorphism to $C^{-1}$. The isomorphism $\bar{C}:Z^i_X/B^i_X\rightarrow \Omega^i_X$ is called the Cartier operator. \section{Liftings of Frobenius to $W_2(k)$} \label{flift} There is a very interesting connection between the Cartier operator and liftings of the Frobenius morphism to flat schemes of characteristic $p^2$. This beautiful observation was first made by Mazur in \cite{Maz}. We go on to explore this next. \subsection{Witt vectors of length two} The Witt vectors $W_2(k)$ (\cite{MumCu}, Lecture 26) of length $2$ over $k$ can be interpreted as the set $k\times k$, where multiplication and addition for $a=(a_0, a_1)$ and $b=(b_0, b_1)$ in $W_2(k)$ are defined by $$ a\, b=(a_0\, b_0, a_0^p b_1+ b_0^p a_1) $$ and $$ a+b=(a_0+b_0, a_1+b_1+\sum_{j=1}^{p-1} p^{-1}\binom{p}{j} a_0^j\, b_0^{p-j}) $$ In the case $k={\Bbb Z}/p$, one can prove that $W_2(k)\cong {\Bbb Z}/p^2$. The projection on the first coordinate $W_2(k)\rightarrow k$ corresponds to the reduction $W_2(k)\rightarrow W_2(k)/p\cong k$ modulo $p$. The ring homomorphism $F^{(2)}:W_2(k)\rightarrow W_2(k)$ given by $F^{(2)}(a_0, a_1)=(a_0^p, a_1^p)$ reduces to the Frobenius homomorphism $F$ on $k$ modulo $p$. \subsection{Splittings of the de Rham complex} The previous section shows that there is a canonical morphism $\operatorname{Spec} k\rightarrow \operatorname{Spec} W_2(k)$. Assume that there is a flat scheme $X^{(2)}$ over $\operatorname{Spec} W_2(k)$ such that \begin{equation} \label{modp} X\cong X^{(2)}\times_{\operatorname{Spec} W_2(k)}\operatorname{Spec} k \end{equation} We shall say that the Frobenius morphism $F$ lifts to $W_2(k)$ if there exists a morphism $F^{(2)}:X^{(2)}\rightarrow X^{(2)}$ covering the Frobenius homomorphism $F^{(2)}$ on $W_2(k)$, which reduces to $F$ via the isomorphism (\ref{modp}). When we use the statement that Frobenius lifts to $W_2(k)$ we will always implicitly assume the existence of the flat lift $X^{(2)}$. \begin{thm} \label{split} If the Frobenius morphism on $X$ lifts to $W_2(k)$ then there is a split quasi-isomorphism $$ 0 @>>> \bigoplus_{0\leq i}\Omega^i_X[-i] @>\sigma>> F_* \Omega^\bullet_X $$ \end{thm} \begin{pf} For an affine open subset $\operatorname{Spec} A^{(2)}\subseteq X^{(2)}$ there is a ring homomorphism $F^{(2)}: A^{(2)}\rightarrow A^{(2)}$ such that $$ F^{(2)}(b)=b^p + p\cdot \varphi(b) $$ where $\varphi: A^{(2)}\rightarrow A=A^{(2)}/ p A^{(2)}$ is some function and $p\, \cdot: A\rightarrow A^{(2)}$ is the $A^{(2)}$-homomorphism derived from tensoring the short exact sequence of $W_2(k)$-modules $$ \CD 0 @>>> p\,W_2(k)@>>> W_2(k) @>p\,\cdot>> p\,W_2(k) @>>> 0 \endCD $$ with the flat $W_2(k)$ module $A^{(2)}$ identifying $A\cong A^{(2)}/p A^{(2)}$ with $p\, A^{(2)}$. We get the following properties of $\varphi$: \begin{align*} \varphi(a+b)&=\varphi(a)+\varphi(b)-\sum_{j=1}^{p-1} p^{-1} \binom{p}{j} \bar{a}^j \bar{b}^{p-j} \\ \varphi(a\, b)&=\bar{a}^p \varphi(b)+ \bar{b}^p \varphi(a) \end{align*} where $\bar{\cdot}$ means reduction $\operatorname{mod} p$. Now it follows that $$ a\mapsto a^{p-1} da+ d \varphi(\tilde{a}) $$ where $\tilde{a}$ is any lift of $a$, is a well defined derivation $\delta:A\rightarrow Z^1_{\operatorname{Spec} A} \subset F_*\Omega^1_{\operatorname{Spec} A}$, which gives a homomorphism $\varphi:\Omega_{\operatorname{Spec} A}^1\rightarrow Z^1_{\operatorname{Spec} A} \subset F_* \Omega^1_{\operatorname{Spec} A}$. This homomorphism can be extended via the algebra structure to give an $A$-algebra homomorphism $\sigma: \oplus_i \Omega_{\operatorname{Spec} A}^i \rightarrow Z^\bullet_{\operatorname{Spec} A}\subseteq F_*\Omega_{\operatorname{Spec} A}^\bullet$, which composed with the canonical homomorphism $Z^\bullet_{\operatorname{Spec} A}\rightarrow \cal H^\bullet(F_* \Omega^\bullet_{\operatorname{Spec} A})$ gives the inverse Cartier operator. Since an affine open covering $\{\operatorname{Spec} A^{(2)}\}$ of $X^{(2)}$ gives rise to an affine open covering $\{\operatorname{Spec} A^{(2)}/p A^{(2)}\}$ of $X$, we have proved that $\sigma$ is a quasi-isomorphism of complexes inducing the inverse Cartier operator on cohomology. Now we give a splitting homomorphism of $\sigma_i:\Omega^i_X\rightarrow F_*\Omega^i_X$. Notice that $\sigma_0:\O_X\rightarrow F_*\O_X$ is the Frobenius homomorphism and that $\sigma_i$ ($i>0$) splits $C$ in the exact sequence $$ \CD 0 @>>> B^i_X @>>> Z^i_X @>C>> \Omega^i_X @>>> 0 \endCD $$ The natural perfect pairing $\Omega_X^i\otimes \Omega^{n-i}_X\rightarrow \Omega^n_X$ gives an isomorphism between ${\cal Hom}_X(\Omega^{n-i}_X, \Omega_X^n)$ and $\Omega^i_X$. It is easy to check that the homomorphism $$ F_*\Omega^i_X\rightarrow {\cal Hom}_X(\Omega^{n-i}_X, \Omega_X^n) \cong \Omega^i_X $$ given by $\omega\mapsto\varphi(\omega)$, where $\varphi(\omega)(z)= C(\sigma_{n-i}(z)\wedge \omega)$, splits $\sigma_i$. \end{pf} \subsection{Bott vanishing} Let $X$ be a normal variety and let $j$ denote the inclusion of the smooth locus $U\subseteq X$. If the Frobenius morphism lifts to $W_2(k)$ on $X$, then the Frobenius morphism on $U$ also lifts to $W_2(k)$. Define the Zariski sheaf $\tilde{\Omega}^i_X$ of $i$-forms on $X$ as $j_*\Omega^i_U$. Since $\operatorname{codim}(X-U)\geq 2$ it follows (\cite{Loc}, Proposition 5.10) that $\tilde{\Omega}^i_X$ is a coherent sheaf on $X$. \begin{thm} Let $X$ be a projective normal variety such that $F$ lifts to $W_2(k)$. Then $$ \H^s(X, \tilde{\Omega}^r_X\otimes L)=0 $$ for $s>0$ and $L$ an ample line bundle. \end{thm} \begin{pf} Let $U$ be the smooth locus of $X$ and let $j$ denote the inclusion of $U$ into $X$. On $U$ we have by Theorem \ref{split} a split sequence $$ 0\rightarrow \Omega^r_U\rightarrow F_*\Omega^r_U $$ which pushes down to the split sequence ($F$ commutes with $j$) $$ 0\rightarrow \tilde{\Omega}_X^r\rightarrow F_*\tilde{\Omega}_X^r $$ Now tensoring with $L$ and using the projection formula we get injections for $s>0$ $$ \H^s(X, \tilde{\Omega}^r_X\otimes L)\hookrightarrow \H^s(X, \tilde{\Omega}^r_X\otimes L^p) $$ Iterating these injections and using that the Zariski sheaves are coherent one gets the desired vanishing theorem by Serre's theorem. \end{pf} \subsection{Degeneration of the Hodge to de Rham spectral sequence} Let $X$ be a projective normal variety with smooth locus $U$. Associated with the complex $\tilde{\Omega}^\bullet_X$ there is a spectral sequence $$ E_1^{pq}=\H^q(X, \tilde{\Omega}^p_X)\implies \H^{p+q}(X, \tilde{\Omega}^\bullet_X) $$ where $\H^\bullet(X, \tilde{\Omega}^\bullet_X)$ denotes the hypercohomology of the complex $\tilde{\Omega}^\bullet_X$. This is the Hodge to de Rham spectral sequence for Zariski sheaves. \begin{thm} If the Frobenius morphism on $X$ lifts to $W_2(k)$, then the spectral sequence degenerates at the $E_1$-term. \end{thm} \begin{pf} As complexes of sheaves of abelian groups $\tilde{\Omega}^\bullet$ and $F_*\tilde{\Omega}^\bullet$ are the same so their hypercohomology agree. Applying hypercohomology to the split injection (Theorem \ref{split}) $$ \sigma:\bigoplus_{0\leq i}\tilde{\Omega}^i_{X/k}[-i]\rightarrow F_* \tilde{\Omega}^\bullet_X $$ we get \begin{eqnarray*} \sum_{p+q=n} \dim_k E_\infty^{pq}= \dim_k \H^n(X, \tilde{\Omega}^\bullet_X)&=& \dim_k \H^n(X, F_*\tilde{\Omega}^\bullet_X)\geq\\ \sum_{p+q=n} \dim_k \H^q(X, \tilde{\Omega}^p_X)&=& \sum_{p+q=n} \dim_k E_1^{pq} \end{eqnarray*} Since $E_\infty^{pq}$ is a subquotient of $E_1^{pq}$, it follows that $E_\infty^{pq}\cong E_1^{pq}$ so that the spectral sequence degenerates at $E_1$. \end{pf} \section{Toric varieties} In this section we briefly sketch the definition of toric varieties following Fulton \cite{Fulton} and demonstrate how the results of Section~\ref{flift} may be applied. \subsection{Convex geometry} Let $N$ be a lattice, $M = \operatorname{Hom}_{{\Bbb Z}}(N, {\Bbb Z})$ the dual lattice, and let $V$ be the real vector space $V = N \otimes_{{\Bbb Z}} {\Bbb R}$. It is natural to identify the dual space of $V$ with $M \otimes_{{\Bbb Z}} {\Bbb R}$, and we think of $N \subset V$ and $M \subset V^*$ as the subsets of integer points. By a cone in $N$ we will mean a subset $\sigma \subset V$ taking the form $\sigma = \{r_1 v_1 + \dots + r_s v_s ~ | ~ r_i \geq 0 \}$ for some $v_i \in N$. The vectors $v_1, \dots, v_s$ are called generators of $\sigma$. We define the dual cone to be $\sigma^{\vee} = \{ u \in V^* | \forall v \in \sigma: \left< u,v \right> \geq 0 \}$. One may show that $\sigma^{\vee}$ is a cone in $M$. A face of $\sigma$ is any set $\sigma \cap u^{\perp}$ for some $u \in \sigma^{\vee}$. Any face of $\sigma$ is clearly a cone in $N$, generated by the $v_i$ for which $\left< u, v_i \right> = 0$. Now let $\sigma$ be a strongly convex cone in $N$, this means that $\{0\}$ is a face of $\sigma$ or equivalently that no nontrivial subspace of $V$ is contained in $\sigma$. We define $S_{\sigma}$ to be the semi group $\sigma^{\vee} \cap M$. Since $\sigma^{\vee}$ is a cone in $M$, $S_{\sigma}$ is finitely generated. \subsection{Affine toric varieties} If $k$ is any commutative ring the semigroup ring $k[S_{\sigma}]$ is a finitely generated commutative $k$-algebra, and $U_{\sigma} = \operatorname{Spec} k[S_{\sigma}]$ is an affine scheme of finite type over $k$. Schemes of this form are called affine toric schemes. \subsection{Glueing affine toric varieties} Let $\tau = \sigma \cap u^{\perp}$ be a face of $\sigma$. One may assume that $u \in S_{\sigma}$. Then it follows that $S_{\tau} = S_{\sigma} + {\Bbb Z}_{\geq 0} \cdot (-u)$, so that $k[S_{\tau}] = k[S_{\sigma}]_{u}$. In this way $U_{\tau}$ becomes a principal open subscheme of $U_{\sigma}$. This may be used to glue affine toric schemes together. We define a fan in $N$ to be a nonempty set $\Delta$ of strongly convex cones in $N$ satisfying that the faces of any cone in $\Delta$ are also in $\Delta$ and the intersection of two cones in $\Delta$ is a face of each. The affine varieties arising from cones in $\Delta$ may be glued together to form a scheme $X_k(\Delta)$ as follows. If $\sigma, \tau \in \Delta$, then $\sigma \cap \tau \in \Delta$ is a face of both $\tau$ and $\sigma$, so $U_{\sigma \cap \tau}$ is isomorphic to open subsets $U_{\sigma\tau}$ in $U_{\sigma}$ and $U_{\tau\sigma}$ in $U_{\tau}$. Take the transition morphism $\phi_{\sigma\tau} : U_{\sigma\tau} \rightarrow U_{\tau\sigma}$ to be the one going through $U_{\sigma \cap \tau}$. A scheme $X_k(\Delta)$ arising from a fan $\Delta$ in some lattice is called a toric scheme. \subsection{Liftings of the Frobenius morphism on toric varieties} Let $X = X_k(\Delta)$ be a toric scheme over the commutative ring $k$ of characteristic $p > 0$. We are going to construct explicitly a lifting of the absolute Frobenius morphism on $X$ to $W = W_2(k)$. Define $X^{(2)}$ to be $X_W(\Delta)$. Since all the rings $W[S_{\sigma}]$ are free $W$-modules, this is clearly a flat scheme over $W_2(k)$. Moreover, the identities $W[S_{\sigma}] \otimes_W k \cong k[S_{\sigma}]$ immediately give an isomorphism $X^{(2)} \times_{\operatorname{Spec} W} \operatorname{Spec} k \cong X$. For $\sigma \in \Delta$, let $F_{\sigma}^{(2)} : W[S_{\sigma}] \rightarrow W[S_{\sigma}]$ be the ring homomorphism extending $F^{(2)} : W \rightarrow W$ and mapping $u \in S_{\sigma}$ to $u^p$. It is easy to see that these maps are compatible with the transition morphisms, so we may take $F^{(2)} : X^{(2)} \rightarrow X^{(2)}$ to be the morphism which is defined by $F_{\sigma}^{(2)}$ locally on $\operatorname{Spec} W[S_{\sigma}]$. This gives the lift of $F$ to $W_2(k)$ and completes the construction. \subsection{Bott vanishing and the Danilov spectral sequence} Since toric varieties are normal we get the following corollary of Section \ref{flift}: \begin{thm} Let $X$ be a projective toric variety over $k$. Then $$ \H^q(X, \tilde{\Omega}^p_X\otimes L)=0 $$ where $q>0$ and $L$ is an ample line bundle. Furthermore the Danilov spectral sequence $$ E_1^{pq}=\H^q(X, \tilde{\Omega}^p_X)\implies \H^{p+q}(X, \tilde{\Omega}^\bullet_X) $$ degenerates at the $E_1$-term. \end{thm} \begin{remark} One may use the above to prove similar results in characteristic zero. The key issue is that we have proved that Bott vanishing and degeneration of the Danilov spectral sequence holds in any prime characteristic. \end{remark} \section{Flag varieties} In this section we generalize Paranjape and Srinivas result on non-lifting of Frobenius on flag varieties not isomorphic to $\P^n$. The key issue is that one can reduce to flag varieties with rank $1$ Picard group. In many of these cases one can exhibit ample line bundles with Bott non-vanishing. We now set up notation. Let $G$ be a semisimple algebraic group over $k$ and fix a Borel subgroup $B$ in $G$. Recall that (reduced) parabolic subgroups $P\supseteq B$ are given by subsets of the simple root subgroups of $B$. These correspond bijectively to subsets of nodes in the Dynkin diagram associated with $G$. A parabolic subgroup $Q$ is contained in $P$ if and only if the simple root subgroups in $Q$ is a subset of the simple root subgroups in $P$. A maximal parabolic subgroup is the maximal parabolic subgroup not containing a specific simple root subgroup. We shall need the following result from the appendix to \cite{MeSri} \begin{prop} \label{splitimplieslift} If the sequence $$ 0 @>>> B^1_X @>>> Z^1_X @>C>> \Omega^1_X @>>> 0 $$ splits, then the Frobenius morphism on $X$ lifts to $W_2(k)$. \end{prop} We also need the following fact derived from (\cite{Hartshorne}, Proposition II.8.12 and Exercise II.5.16(d)) \begin{prop} \label{diffilt} Let $f:X\rightarrow Y$ be a smooth morphism between smooth varieties $X$ and $Y$. Then for every $n\in {\Bbb N}$ there is a filtration $F^0\supseteq F^1 \supseteq \dots$ of $\Omega^n_X$ such that $$ F^i/F^{i+1}\cong f^*\Omega_Y^i\otimes\Omega_{X/Y}^{n-i} $$ \end{prop} \begin{lemma} \label{fibrlemma} Let $f:X\rightarrow Y$ be a surjective, smooth and projective morphism between smooth varieties $X$ and $Y$ such that the fibers have no non-zero global $n$-forms, where $n>0$. Then there is a canonical isomorphism $$ \Omega_Y^\bullet\rightarrow f_*\Omega_X^\bullet $$ and a splitting $\sigma:\Omega^1_X\rightarrow Z^1_X$ of the Cartier operator $C: Z^1_X\rightarrow \Omega^1_X$ induces a splitting $f_*\sigma:\Omega^1_Y\rightarrow Z^1_Y$ of $C:Z^1_Y\rightarrow \Omega^1_Y$. \end{lemma} \begin{pf} Notice first that $\O_Y\rightarrow f_*\O_X$ is an isomorphism of rings as $f$ is projective and smooth. The assumption on the fibers translates into $f_*\Omega_{X/Y}^n\otimes k(y)\cong \H^0(X_y, \Omega^n_{X_y})=0$ for geometric points $y\in Y$, when $n>0$. So we get $f_*\Omega_{X/Y}^n=0$ for $n>0$. By Proposition \ref{diffilt} this means that all of the natural homomorphisms $\Omega_Y^n\rightarrow f_*\Omega^n_X$ induced by $\O_Y\rightarrow f_*\O_X\rightarrow f_*\Omega^1_X$ are isomorphisms giving an isomorphism of complexes $$ \CD 0 @>>> \O_Y @>>> \Omega^1_Y @>>> \Omega^2_Y @>>> \dots \\ @. @VVV @VVV @VVV \\ 0 @>>> f_*\O_X @>>> f_*\Omega^1_X @>>> f_*\Omega^2_X @>>> \dots \endCD $$ This means that the middle arrow in the commutative diagram $$ \CD 0 @>>> B^1_Y @>>> Z^1_Y @>C>> \Omega_Y @>>> 0\\ @. @VVV @VVV @VVV \\ 0 @>>> f_* B^1_X @>>> f_* Z^1_X @>f_* C>> f_*\Omega_X @>>> 0\\ \endCD $$ is an isomorphism and the result follows. \end{pf} \begin{cor} \label{maxparnonlift} Let $Q\subseteq P$ be two parabolic subgroups of $G$. If the Frobenius morphism on $G/Q$ lifts to $W_2(k)$, then the Frobenius morphism on $G/P$ lifts to $W_2(k)$. \end{cor} \begin{pf} It is well known that $G/Q\rightarrow G/P$ is a smooth projective fibration, where the fibers are isomorphic to $Z=P/Q$. Since $Z$ is a rational projective smooth variety it follows from (\cite{Hartshorne}, Exercise II.8.8) that $\H^0(Z, \Omega^n_Z)=0$ for $n>0$. Now the result follows from Lemma \ref{fibrlemma} and Proposition \ref{splitimplieslift}. \end{pf} In specific cases one can prove using the ``standard'' exact sequences that certain flag varieties do not have Bott vanishing. We go on to do this next. Let $Y$ be a smooth divisor in a smooth variety $X$. Suppose that $Y$ is defined by the sheaf of ideals $I\subseteq \O_X$. Then (\cite{Hartshorne}, Proposition II.8.17(2) and Exercise II.5.16(d)) gives for $n\in {\Bbb N}$ an exact sequence of $\O_Y$-modules $$ 0\rightarrow \Omega^{n-1}_Y\otimes I/I^2\rightarrow \Omega^n_X\otimes\O_Y\rightarrow \Omega^n_Y\rightarrow 0 $$ From this exact sequence and induction on $n$ it follows that $\H^0(\P^n, \Omega^j_{\P^n}\otimes\O(m))=0$, when $m\leq j$ and $j>0$. \subsection{Quadric hypersurfaces in $\P^n$} \label{quadric} Let $Y$ be a smooth quadric hypersurface in $\P^n$, where $n\geq 4$. There is an exact sequence $$ 0\rightarrow \O_Y(1-n)\rightarrow \Omega^1_{\P^n}\otimes \O(3-n)\otimes\O_Y \rightarrow \Omega^1_Y\otimes \O_Y(3-n)\rightarrow 0 $$ From this it is easy to deduce that $$ \H^{n-2}(Y, \Omega^1_Y\otimes \O_Y(3-n))\cong \H^1(Y, \Omega^{n-2}_Y\otimes \O_Y(n-3))\cong k $$ using that $\H^0(\P^n, \Omega^j_{\P^n}\otimes\O(m))=0$, when $m\leq j$ and $j>0$. \subsection{The incidence variety in $\P^n\times \P^n$} \label{inc} Let $X$ be the incidence variety of lines and hyperplanes in $\P^n\times \P^n$, where $n\geq 2$. Recall that $X$ is the zero set of $x_0 y_0+\dots+x_n y_n$, so that there is an exact sequence $$ 0\rightarrow \O(-1)\times \O(-1)\rightarrow \O_{\P^n}\times\O_{\P^n} \rightarrow \O_X\rightarrow 0 $$ Using K\"unneth it is easy to deduce that $$ \H^{2n-2}(X, \Omega^1_X\otimes \O(1-n)\times \O(1-n))\cong \H^1(X, \Omega^{2n-2}\otimes \O(n-1)\times \O(n-1))\cong k $$ \subsection{Bott non-vanishing for flag varieties} In this section we search for specific maximal parabolic subgroups $P$ and ample line bundles $L$ on $Y=G/P$, such that $$ \H^i(Y, \Omega_Y^j\otimes L)\neq 0 $$ where $i>0$. These are instances of Bott non-vanishing. This will be used in Section \ref{nonlift} to prove non-lifting of Frobenius for a large class of flag varieties. Let $\O(1)$ be the ample generator of $\operatorname{Pic} Y$. By flat base change one may produce examples of Bott non-vanishing for $Y$ for fields of arbitrary prime characteristic by restricting to the field of the complex numbers. This has been done in the setting of Hermitian symmetric spaces, where the cohomology groups $\H^p(Y, \Omega^q\otimes\O(n))$ have been thoroughly investigated by Sato \cite{Sato} and Snow \cite{Snow1}\cite{Snow2}. We now show that these examples exist. In each of the following subsections $Y$ will denote $G/P$, where $P$ is the maximal parabolic subgroup not containing the root subgroup corresponding to the marked simple root in the Dynkin diagram. These flag manifolds are the irreducible Hermitian symmetric spaces. \subsubsection{Type $A$} \label{A} \begin{picture}(30,30)(-20,30) \put( 0,0){\circle*{4}} \put(-4.5,-3){$\times$} \put( 1.5,0){\line(1,0){22}} \put( 25,0){\circle*{4}} \put( 30.6,-0.5){.\,.\,.} \put( 50,0){\circle*{4}} \put( 51.5,0){\line(1,0){22}} \put( 75,0){\circle*{4}} \put( 76.5,0){\line(1,0){22}} \put(100,0){\circle*{4}} \put(105.6,-0.5){.\,.\,.} \put(125,0){\circle*{4}} \put(126.5,0){\line(1,0){22}} \put(150,0){\circle*{4}} \put(145,-3){$\times$} \end{picture} \vskip 2.0truecm If $Y$ is a Grassmann variety not isomorphic to projective space ($Y=G/P$, where $P$ corresponds to leaving out a simple root which is not the left or right most one), one may prove (\cite{Snow1}, Theorem 3.3) that $$ \H^1(Y, \Omega_Y^3\otimes\O(2))\neq 0 $$ \subsubsection{Type $B$} \begin{picture}(30,30)(-20,30) \put( 0,0){\circle*{4}} \put( 1.5,0){\line(1,0){22}} \put( 25,0){\circle{3}} \put( 26.5,0){\line(1,0){22}} \put( 50,0){\circle{3}} \put( 55.6,-0.5){.\,.\,.} \put( 75,0){\circle{3}} \put( 76.5,0){\line(1,0){22}} \put(100,0){\circle{3}} \put(100, 1.5){\line(1,0){25}} \put(100,-1.5){\line(1,0){25}} \put(108.2,-3){$>$} \put(125,0){\circle{3}} \end{picture} \vskip 2.0truecm Here $Y$ is a smooth quadric hypersurface in $\P^n$, where $n\geq 4$ and Bott non-vanishing follows from Section \ref{quadric}. \subsubsection{Type $C$} \begin{picture}(30,30)(-20,30) \put( 0,0){\circle{3}} \put( 1.5,0){\line(1,0){22}} \put( 25,0){\circle{3}} \put( 26.5,0){\line(1,0){22}} \put( 50,0){\circle{3}} \put( 55.6,-0.5){.\,.\,.} \put( 75,0){\circle{3}} \put( 76.5,0){\line(1,0){22}} \put(100,0){\circle{3}} \put(100, 1.5){\line(1,0){25}} \put(100,-1.5){\line(1,0){25}} \put(108.2,-3){$<$} \put(125,0){\circle*{4}} \end{picture} \vskip 2.0truecm By (\cite{Snow2}, Theorem 2.2) it follows that $$ \H^1(Y, \Omega_Y^2\otimes\O(1))\neq 0 $$ \subsubsection{Type $D$} \begin{picture}(30,30)(-20,30) \put( 0,0){\circle*{4}} \put( 1.5,0){\line(1,0){22}} \put( 25,0){\circle{3}} \put( 26.5,0){\line(1,0){22}} \put( 50,0){\circle{3}} \put( 55.6,-0.5){.\,.\,.} \put( 75,0){\circle{3}} \put( 76.5,0){\line(1,0){22}} \put(100,0){\circle{3}} \put(101.5, 0.5){\line(2, 1){22}} \put(125,12){\circle*{4}} \put(101.5,-0.5){\line(2,-1){22}} \put(125,-12){\circle*{4}} \end{picture} \vskip 2.0truecm For the maximal parabolic $P$ corresponding to the leftmost marked simple root, Y=$G/P$ is a smooth quadric hypersurface in $\P^n$, where $n\geq 4$ and Bott non-vanishing follows from Section \ref{quadric}. For the maximal parabolic subgroup corresponding to one of the two rightmost marked simple roots we get by (\cite{Snow2}, Theorem 3.2) that $$ \H^2(Y, \Omega^4_Y\otimes\O(2))\neq 0 $$ \subsubsection{Type $E_6$} \begin{picture}(30,30)(-20,20) \put( 0,0){\circle*{4}} \put( 1.5,0){\line(1,0){22}} \put( 25,0){\circle{3}} \put( 26.5,0){\line(1,0){22}} \put( 50,0){\circle{3}} \put( 50,-1.5){\line(0,-1){22}} \put( 50,-25){\circle{3}} \put( 51.5,0){\line(1,0){22}} \put( 75,0){\circle{3}} \put( 76.5,0){\line(1,0){22}} \put(100,0){\circle*{4}} \end{picture} \vskip2.0truecm By (\cite{Snow2}, Table 4.4) it follows that $$ \H^3(Y, \Omega^5\otimes\O(2))\neq 0 $$ \subsubsection{Type $E_7$} \begin{picture}(30,30)(-20,20) \put( 0,0){\circle{3}} \put( 1.5,0){\line(1,0){22}} \put( 25,0){\circle{3}} \put( 26.5,0){\line(1,0){22}} \put( 50,0){\circle{3}} \put( 50,-1.5){\line(0,-1){22}} \put( 50,-25){\circle{3}} \put( 51.5,0){\line(1,0){22}} \put( 75,0){\circle{3}} \put( 76.5,0){\line(1,0){22}} \put(100,0){\circle{3}} \put(101.5,0){\line(1,0){22}} \put(125,0){\circle*{4}} \end{picture} \vskip 2.0truecm By (\cite{Snow2}, Table 4.5) it follows that $$ \H^4(Y, \Omega^6\otimes\O(2))\neq 0 $$ \subsubsection{Type $G_2$} \label{G} \begin{picture}(30,30)(-20,30) \put( 0,0){\circle*{4}} \put( 0, 1.5){\line(1,0){25}} \put( 1.5,0){\line(1,0){22}} \put( 0,-1.5){\line(1,0){25}} \put(8.2,-3){$<$} \put(25,0){\circle{3}} \end{picture} \vskip2.0truecm Here $Y$ is a smooth quadric hypersurface in $\P^6$ and Bott non-vanishing follows from Section \ref{quadric}. \subsection{Non-lifting of Frobenius for flag varieties} \label{nonlift} We now get the following \begin{thm} Let $Q$ be a parabolic subgroup contained in a maximal parabolic subgroup $P$ in the list \ref{A} - \ref{G}. Then the Frobenius morphism on $G/Q$ does not lift to $W_2(k)$. Furthermore if $G$ is of type $A$, then the Frobenius morphism on any flag variety $G/Q\not\cong \P^m$ does not lift to $W_2(k)$. \end{thm} \begin{pf} If $P$ is a maximal parabolic subgroup in the list \ref{A}-\ref{G}, then the Frobenius morphism on $G/P$ does not lift to $W_2(k)$. By Corollary \ref{maxparnonlift} we get that the Frobenius morphism on $G/Q$ does not lift to $W_2(k)$. In type $A$ the only flag variety not admitting a fibration to a Grassmann variety $\not\cong\P^m$ is the incidence variety. Non-lifting of Frobenius in this case follows from Section \ref{inc}. \end{pf} \begin{remark} The above case by case proof can be generalized to include projective homogeneous $G$-spaces with non-reduced stabilizers. It would be nice to prove in general that the only flag variety enjoying the Bott vanishing property is $\P^n$. We know of no other visible obstruction to lifting Frobenius to $W_2(k)$ for flag varieties than the non-vanishing Bott cohomology groups. \end{remark} \newpage \bibliographystyle{amsplain} \ifx\undefined\leavevmode\hbox to3em{\hrulefill}\, \newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,} \fi

9508

alg-geom/9508003

en

https://arxiv.org/abs/alg-geom/9508003

alg-geom/9508003

Joerg Jahnel

Joerg Jahnel

Heights for line bundles on arithmetic surfaces

Mathematica Gottingensis, Heft 16, 1995, revised version, LaTeX2.09

For line bundles on arithmetic varieties we construct height functions using arithmetic intersection theory. In the case of an arithmetic surface, generically of genus g, for line bundles of degree g equivalence is shown to the height on the Jacobian defined by the Theta divisor.

alg-geom

\section{Introduction} \footnotetext[1]{Mathematisches Institut, Universit\"at G\"ottingen, Bunsenstra{\ss}e 3-5, 37073 G\"ottingen, Germany \newline email: [email protected]} \thispagestyle{empty} In this paper we will suggest a construction for height functions for line bundles on arithmetic varieties. Following the philosophy of \cite{Bost/Gillet/Soule 93} heights should be objects in arithmetic geometry analogous to degrees in algebraic geometry. So let $K$ be a number field, ${\cal O}_{K}$ its ring of integers and ${\cal X} / {\cal O}_{K}$ an arithmetic variety, i.e. a regular scheme, projective and flat over ${\cal O}_{K}$, whose generic fiber $X / K$ we assume to be connected of dimension $d$. Then we have to fix a metrized line bundle $( {\cal T}, \| . \| )$ or, equivalently, its first Chern class $$\stackrel{\wedge}{{c}_{1}} ({\cal T} , \| . \| ) = (T,g_{T}) \in ~ \stackrel{\wedge}{{ \rm CH}^{1}} ({\cal X}) ~~.$$ The height of a line bundle ${\cal L}$ on ${\cal X}$ should be the arithmetic degree of the intersection of $\stackrel{\wedge}{{c}_{1}}({\cal L} )$ with $(T,g_{T})^{d}$. For this a natural hermitian metric has to be chosen on ${\cal L}$. We fix a K\"ahler metric $\omega_{0}$ on ${\cal X} ({\Bbb{C}})$, invariant under complex conjugation $F_{\infty}$, as in \cite{Arakelov 74}. Then it is well known that the condition on the Chern form to be harmonic defines $\| .\|$ up to a locally constant factor. In order to determine this factor we require $$\stackrel{\wedge}{\deg} \Big( \det R \pi_{*} {\cal L}, \| . \|_{Q} \Big) = 0 .$$ Here $\pi : {\cal X} \longrightarrow {\rm Spec}~ {\cal O}_{K}$ is the structural morphism and $\| . \|_{Q}$ is Quillen's metric (\cite{Quillen 85}, \cite{Bismut/Gillet/Soule 88}) at the infinite places of $K$. \begin{thm} {\bf Fact.} {\rm a)} If the Euler characteristic $\chi ({\cal L})$ does not vanish, such a metric exists. \newline {\rm b)} $\stackrel{\wedge}{{c}_{1}} ({\cal L}, \| .\|)$ is uniquely determined up to a summand $(0,C)$, where $C = (C_{\sigma})_{\sigma : K \hookrightarrow \Bbb{C}}$ is a system of constants on $X \times_{{\rm Spec K}, \sigma} {\rm Spec} ~ \Bbb{C}$ with $$\sum_{\sigma : K \hookrightarrow \Bbb{C}} C_{\sigma} = 0 ~~~~~(and {}~~C_{\sigma} = C_{\bar{\sigma}}).$$ \end{thm} \begin{thm} {\bf Fact.} Such $(0,C) \in ~ \stackrel{\wedge}{{\rm CH}^{1}} ({\cal X})$ are numerically trivial. \end{thm} \begin{thm} {\rm Now we can state our fundamental} \newline {\bf Definition.} The {\rm height} of the line bundle ${\cal L}$ is given by $$h_{{\cal T}, \omega_{0}} ({\cal L}) := ~ \stackrel{\wedge}{\deg} ~\pi_{*} \, \Big[ \stackrel{\wedge}{{c}_{1}} ({\cal L}, \| . \|) \cdot (T, g_{T})^{d} \Big] ,$$ where $\| . \|$ is one of the distinguished metrics specified above. \end{thm} \begin{thm}\label{main} {\rm In this paper we will analyze this definition in the case of arithmetic surfaces. Our main result is} \newline {\bf Theorem.} Let ${\cal C} / {\cal O}_{K}$ be a regular projective variety of dimension $2$, flat over ${\cal O}_{K}$ and generically connected of genus $g$, $x \in ({\cal C} \times_{{\rm Spec} {\cal O}_{K}} {\rm Spec} ~ K) (K)$ be a $K$-valued point and $\Theta$ be the Theta divisor on the Jacobian $J = {\rm Pic}^{g} (C)$ (defined using $x$). On $$\coprod_{\sigma: K \hookrightarrow \Bbb{C}} \Big( {\cal C} \times_{{\rm Spec} {\cal O}_{K}, \sigma} {\rm Spec} ~ \Bbb{C} \Big) (\Bbb{C})$$ let $\omega$ be a K\"ahler form invariant under $F_{\infty}$ and normalized by $$\int_{({\cal C} \times_{{\rm Spec} {\cal O}_{K}, \sigma} {\rm Spec} \Bbb{C})(\Bbb{C})} \omega = 1$$ for every $\sigma$. Then, for line bundles ${\cal L} / {\cal C}$, fiber-by-fiber of degree $g$ and of degree of absolute value less than $H$ on every irreducible component of the special fibers of ${\cal C}$ (with some constant $H \in \Bbb{N}$) $$h_{x,\omega} ({\cal L}) = h_{\Theta} ({\cal L}_{K}) + {\rm O}(1) ,$$ where $h_{\Theta}$ is the height on $J$ defined using the ample divisor $\Theta$. \end{thm} \begin{thm} {\bf Remark.} {\rm Another connection between heights on the Jacobian of a curve and arithmetic intersection theory was obtained by Faltings \cite{Faltings 84} and Hriljac \cite{Hriljac 85}. Recently it has been generalized to higher dimensions and higher codimension Chow groups by K\"unnemann \cite{Kunnemann 95}. They can write down an explicit formula for the N\'{e}ron-Tate height pairing on the Jacobian (higher Picard variety) in terms of arithmetic intersection theory. The main point is that they consider line bundles (cycles) algebraically equivalent to zero. So there is no need for them to scale a metric (to specify the infinite part of the arithmetic cycles occuring). Our approach, to the contrary, seems to work best for sufficiently ample algebraic equivalence classes of line bundles. A formal relationship between our approach and the other one is not known to the author.} \end{thm} \begin{thm} {\rm In order to prove the two facts above we will use the following simple} \newline {\bf Lemma.} Let $f: X \longrightarrow Y$ be a smooth proper map of complex manifolds, where $X$ has a K\"ahler structure $\omega$ and $Y$ is connected, and $E$ be a holomorphic vector bundle on $X$. For a hermitian metric $\| . \|$ on $E$ and a constant factor $D > 0$ we have $$ h_{Q,(E,D \cdot \| . \|)} = h_{Q,(E,\| . \|)} \cdot D^{\chi (E)} .$$ \end{thm} {\bf Proof.} The homomorphism \begin{eqnarray*} (E,\| . \|) & \longrightarrow & (E, D \cdot \| . \|) \\ s & \mapsto & \frac{1}{D} \cdot s \end{eqnarray*} is an isometry inducing the isometry \pagebreak \begin{eqnarray*} \Big( \det R\pi_{*} E, h_{Q,(E,\| . \|)} \Big) & \longrightarrow & \Big( \det R\pi_{*} E, h_{Q,(E,D \cdot \| . \|)} \Big) \\ x & \mapsto & D^{-\chi (E)} \cdot x ~~~~~~~~~~~~~~~~~. \end{eqnarray*} \begin{center} $\Box$ \end{center} \begin{thm} {\bf Proof of Fact 1. Existence:} {\rm Multiplication of $\| . \|$ by $D$ will change the Quillen metric by the factor $D^{\chi (E)}$ and therefore $\stackrel{\wedge}{\deg} \det R \pi_{*} E$ by the summand $[K:{\Bbb{Q}}] \chi(E) \log D$. \newline {\bf Uniqueness:} The harmonicity condition and invariance under $F_{\infty}$ determine $\| . \|$ up to constant factors $D_{\sigma} > 0$ for each $\sigma: K \hookrightarrow \Bbb{C}$ with $D_{\sigma} = D_{\bar{\sigma}}$. The scaling condition requires $$\prod_{\sigma: K \hookrightarrow \Bbb{C}} D_{\sigma}^{\chi(E)} = 1$$ or $\sum_{\sigma: K \hookrightarrow \Bbb{C}} \log D_{\sigma} = 0.$ } \begin{center} $\Box$ \end{center} \end{thm} \begin{thm} {\bf Proof of Fact 2.} {\rm Let $(Z,g_{Z}) \in ~ \stackrel{\wedge}{\rm CH}_{1}({\cal X})$. Then \begin{eqnarray*} (0,C) \cdot (Z,g_{Z}) & = & (0,g_{Z} \cdot \omega_{(0,C)} + C \cdot \delta_{Z}) \\ & = & (0,C \cdot \delta_{Z}) ~~~~~~~~. \end{eqnarray*} $Z$ is a zero-cycle on $X$, so $\delta_{Z}$ will have, independently on $\sigma$, always the integral $\deg Z$. Therefore \begin{eqnarray*} \stackrel{\wedge}{\deg} ~ \pi_{*} \Big[ (0,C) \cdot (Z,g_{Z}) \Big] & = & \frac{\scriptstyle 1}{\scriptstyle 2} ~ \sum_{\sigma} ~~ \left[ C_{\sigma} \int_{X \times_{{\rm Spec K}, \sigma} {\rm Spec} ~ {\Bbb{C}} (\Bbb{C})} \delta_{Z} \right] \\ & = & \frac{\scriptstyle 1}{\scriptstyle 2} ~ (\sum_{\sigma} C_{\sigma}) \cdot \deg Z \\ & = & 0 ~~. \end{eqnarray*} } \begin{center} $\Box$ \end{center} \end{thm} \section{Divisors versus points of the Jacobian} \begin{thm} {\rm The remainder of this paper is devoted to the proof of Theorem \ref{main}. So let $C/K$ be a regular proper algebraic curve of genus $g$ with $C(K) \neq \emptyset$. We consider a regular projective model ${\cal C} / {\cal O}_{K}$. Denote by $J = {\rm Pic}^{g}_{C/K}$ the Jacobian of $C$. When $x \in C(K)$ is chosen we have a canonical isomorphism ${\rm Pic}^{g-1}_{C/K} \longrightarrow {\rm Pic}^{g}_{C/K} = J$ and thus the divisor $\Theta$ on $J$. $\Theta$ induces a closed embedding $i^{'}: J \hookrightarrow {\bf P}^{N}_{K}$ and a "naive" height for $K$-valued points of $J$: $$h_{\Theta} (D) := \log ~ \left( \prod_{\nu \in M_{K}} \max \left\{ \| i(D)_{0} \|_{\nu}, ~ \ldots ~ , \| i(D)_{N} \|_{\nu} \right\} \right) ~~.$$ Accordingly $j^{*} (\Theta)$ induces a morphism $i: C^{g} \stackrel{j}{\longrightarrow} J \stackrel{i^{'}}{\longrightarrow} {\bf P}^{N}_{K}$ and a height function $h_{j^{*} (\Theta)}$ for $K$-valued points of $C^{g}$. Here $j$ denotes the natural map sending a divisor to its associated line bundle. A general construction for heights defined by a divisor, the "height machine", is given in [CS, Chapter VI, Theorem 3.3]. The underlying height $h$ for $K$-valued points of ${\bf P}^{N}_{K}$ is a height in the sense of Arakelov theory \cite{Bost/Gillet/Soule 93} as follows: We choose the regular projective model ${\bf P}^{N}_{{\cal O}_{K}} \supseteq {\bf P}^{N}_{K}$. Every \linebreak $K$-valued point $y$ of ${\bf P}^{N}_{K}$ can be extended uniquely to an ${\cal O}_{K}$-valued point $\underline{y}$ of ${\bf P}^{N}_{{\cal O}_{K}}$. Let $\overline{{\cal O} (1)}$ be the hermitian line bundle on ${\bf P}^{N}_{{\cal O}_{K}}$, where the hermitian metrics at the infinite places are given by $$\left\| x_{0} \right\| := \left( 1 + \left| \frac{x_{1}}{x_{0}} \right|^{2} + \ldots + \left| \frac{x_{N}}{x_{0}} \right|^{2} \right)^{- \frac{1}{2}} {}~~~~~~~~~ ({\rm i.e.~~} \left\| x_{i} \right\| := \left( \left| \frac{x_{0}}{x_{i}} \right|^{2} + \ldots + 1 + \ldots + \left| \frac{x_{N}}{x_{i}} \right|^{2} \right)^{- \frac{1}{2}} ) ~~.$$ Then $h = h_{\overline{{\cal O} (1)}}$ is the height defined by $\overline{{\cal O} (1)}$ in the sense of [BoGS, Definition 3.1.; \linebreak formula (3.1.6)]. } \end{thm} \begin{thm} {\bf Remark.} {\rm We need a better understanding of ${\cal O} (j^{*} (\Theta))$. By Riemann's Theorem [GH, Chapter 2, \S 7] one has $\Theta = \frac{1}{(g-1)!} j_{*} ((x) \times C^{g-1})$, where $j: C^{g} \stackrel{p}{\longrightarrow} C^{(g)} \stackrel{c}{\longrightarrow} J$ factors into a morphism finite flat of degree $g!$ and a birational morphism. So $j^{*} (\Theta)$ is an effective divisor containing the summands $\pi_{k}^{*} (x)$, where $\pi_{k}: C^{g} \longrightarrow C$ denotes $k$-th projection. $${\cal O} \Big( j^{*} \left( \Theta \right) \Big) = \bigotimes_{k=1}^{g} \pi_{k}^{*} \Big( {\cal O} (x) \Big) \otimes {\cal O} \Big( p^{*} (R) \Big)$$ Intuitively, the divisor $R$ on $C^{(g)}$ corresponds to the divisors on $C$ moving in a linear system. This can be made precise, but we will not need that here. } \end{thm} \begin{thm} {\bf Remark.} {\rm It is a difficulty that there are no regular projective models available for $J$ and $C^{g}$, such that arithmetic intersection theory does not work immediately. So we follow [BoGS, Remark after Proposition 3.2.1.] and consider a projective (not necessarily regular) model of $C^{g}$, namely ${\cal C}^{g} := {\cal C} \times_{{\rm Spec} {\cal O}_{K}} \ldots \times_{{\rm Spec} {\cal O}_{K}} {\cal C}$. Hereon let $\overline{\cal T}$ be a line bundle extending $\bigotimes_{k=1}^{g} \pi_{k}^{*} {\cal O} (x)$ equipped with a hermitian metric. One has to define a height $h_{\overline{\cal T}}$ induced by $\overline{\cal T}$. Consider more generally a projective (singular) arithmetic variety ${\cal X} / {\cal O}_{K}$ and a hermitian line bundle $\overline{\cal U}$ on $\cal X$. Then there is a morphism $\iota: {\cal X} \longrightarrow P$ into a projective variety $P$ smooth over ${\rm Spec} ~ {\cal O}_{K}$ and a line bundle ${\cal U_{P}}$ on $P$ such that $\iota^{*} (\cal U_{P}) = {\cal U}$ (see [Fu, Lemma 3.2.], cf. [BoGS, Remark 2.3.1.ii)]). We can even choose $\iota$ in such a way that the hermitian metric on $\overline{\cal{U}}$ is a pullback of one on ${\cal U}_{P}$ (e.g. as a closed embedding). $$\iota^{*} \Big( \overline{{\cal U}_{P}} \Big) = \overline{{\cal U}}$$ Then for an ${\cal O}_{K}$-valued point $\underline{y}$ of ${\cal X}$ one defines \begin{eqnarray*} h_{\overline{\cal U}} \Big( \underline{y} \Big) & := & h_{\overline{\cal U}_{P}} \Big( \iota_{*} (\underline{y}) \Big) \\ & = & \stackrel{\wedge}{\deg} \Big( \stackrel{\wedge}{c_{1}} (\overline{{\cal U}_{P}} ) \Big| \iota_{*} (y) \Big) ~~, \end{eqnarray*} where $( . | . )$ denotes the pairing $\stackrel{\wedge}{{\rm CH}^{1}} (P) \times {\rm Z}_{1} (P) \longrightarrow ~ \stackrel{\wedge}{{\rm CH}^{1}} ({\rm Spec} ~ {\cal O}_{K} )_{\Bbb{Q}}$ from [BoGS, 2.3.]. In [BoGS, Remark after Proposition 3.2.1.] independence of this definition of the $\iota$ chosen is shown. In particular it becomes clear at this point that the pairing $\Big( \stackrel{\wedge}{c_{1}} ( . ) \Big| . \Big)$ can be extended to arbitrary (singular) projective arithmetic varieties over ${\cal O}_{K}$ and satisfies the projection formula $$ \Big( \stackrel{\wedge}{c_{1}} (L) \Big| f_{*} (Z) \Big) = \Big( \stackrel{\wedge}{c_{1}} (f^{*} (L)) \Big| Z \Big) ~~ .$$ } \end{thm} \begin{thm} \label{sing} {\bf Remark.} {\rm If ${\cal X} / {\cal O}_{K}$ is a regular arithmetic variety, one has another pairing \begin{eqnarray*} [.,.]: \stackrel{\wedge}{{\rm CH}^{1}} ({\cal X}) ~ \times \stackrel{~^{\scriptstyle{\wedge}}}{{\rm CH}_{1}} ({\cal X}) & \longrightarrow & \stackrel{\wedge}{{\rm CH}^{1}} ({\rm Spec} ~ {\cal O}_{K})_{\Bbb Q} \\ (z, y) & \mapsto & \pi_{*} [z \cdot y] ~~. \end{eqnarray*} We note that also $\Big[ \stackrel{\wedge}{c_{1}} (.) , . \Big]$ can be extended to arbitrary (singular away from the generic fiber) projective arithmetic varieties. One has to represent $y$ by a cycle $(Y, g_{Y})$ and to put $$\Big[ \stackrel{\wedge}{c_{1}} (\overline{\cal U}) , Y \Big] := \Big( \stackrel{\wedge}{c_{1}} (\overline{\cal U}) \Big| Y \Big) + \Big(0, \Big( \int_{{\cal X} ({\Bbb C})} ~ g_{Y} \omega_{\stackrel{\wedge}{c_{1}} (\overline{\cal U})} \Big)_{\sigma: K \hookrightarrow {\Bbb C}} \Big)$$ obtaining a pairing satisfying the projection formula $$\Big[ \stackrel{\wedge}{c_{1}} (f^{*} (\overline{\cal U})) , y \Big] = \Big[ \stackrel{\wedge}{c_{1}} (\overline{\cal U}) , f_{*} (y) \Big]$$ for $f$ proper and smooth on the generic fiber. In particular, independence of the cycle chosen carries over from the regular case. Indeed, concerning a trivial arithmetic $1$-cycle one is automatically reduced to surfaces and resolution of singularities is known for two-dimensional schemes [CS, Chapter XI by M. Artin]. Let $f$ be one. Note that for cycles with $\omega_{y} = 0$ the push-forward $f_{*}$ makes sense for any proper $f$. If $f$ is a proper birational map inducing an isomorphism on the generic fibers one has $f_{*} f^{*} w = w$ for arithmetic one-cycles and therefore \begin{equation} \label{fun} \Big[ \stackrel{\wedge}{c_{1}} (f^{*} (\overline{\cal U})) , f^{*} (w) \Big] = \Big[ \stackrel{\wedge}{c_{1}} (\overline{\cal U}) , w \Big] ~~. \end{equation} This is useful for the special case of a (singular) projective arithmetic surface. There $[ . , . ]$ can be specialized to a pairing $\Big[ \stackrel{\wedge}{c_{1}} (.) , \stackrel{\wedge}{c_{1}} (.) \Big]$ between hermitian line bundles. This one is symmetric. Indeed, formula (\ref{fun}) tells us, that it is enough to show that after pullback by a birational morphism. But for regular arithmetic surfaces symmetry is clear. } \end{thm} \begin{thm} \label{class} {\bf Lemma.} Let ${\cal X} / {\cal O}_{K}$ be a (singular) projective arithmetic variety and $X/K$ its generic fiber which is assumed to be regular. Further, let $D$ be a divisor on $X$ and $\overline{\cal U}$ be a hermitian line bundle extending ${\cal O} (D)$. Then $h_{D} = h_{\overline{\cal U}} + {\rm O} (1)$ for $K$-valued points of $X.$ \newline {\bf Proof.} {\rm There is a very ample line bundle on $X$ that can be extended to ${\cal X}$. So we may assume $D$ to be basepoint-free (very ample). Then the two height functions arise from the situations $$\begin{array}{cccccccccccccc} & & & & \overline{{\cal O} (1)} & ~~ & & ~~ & & & {}~~~~\overline{\cal U}~~~~ & & \overline{{\cal U}_{P}}~~ & \\ & & & & | & ~~ & {\rm and} & ~~ & & & | & & | & \\ {\rm Spec} ~ {\cal O}_{K} & \stackrel{y}{\hookrightarrow} & {\cal X} & - \stackrel{i}{-} \rightarrow & {\bf P}^{N}_{{\cal O}_{K}} & ~~ & & ~~ & {\rm Spec} ~ {\cal O}_{K} & \stackrel{y}{\hookrightarrow} & {\cal X} & \stackrel{\iota}{\longrightarrow} & P & ~~. \end{array}$$ Here $i$ is the rational map defined by an extension ${\cal U}^{'}$ of ${\cal O} (D)$ over ${\cal X}$. In the generic fiber $i$ is defined everywhere. Note that $iy$ is a morphism by the valuative criterion. Of course, it comes from sections of the line bundle $y^{*} {\cal U}^{\prime}$. Note that ${\cal U}^{'}$ is equipped with a hermitian metric induced by that on ${\cal O} (1)$. $\iota: {\cal X} \longrightarrow P$ is a morphism into a smooth scheme as described above. Thus \begin{eqnarray*} h_{D} (y) & = & \stackrel{\wedge}{\deg} \left( \stackrel{\wedge}{c_{1}} \left( \overline{{\cal O} (1) } \right) \Big| (iy)_{*} ({\rm Spec} ~ {\cal O}_{K}) \right) \\ & = & \stackrel{\wedge}{\deg} \left( \stackrel{\wedge}{c_{1}} \left( (iy)^{*} \overline{{\cal O} (1) } \right) \Big| {\rm Spec} ~ {\cal O}_{K} \right) \\ & = & \stackrel{\wedge}{\deg} \left( \stackrel{\wedge}{c_{1}} \left( (iy)^{*} \overline{{\cal O} (1) } \right) \right) \\ & = & \stackrel{\wedge}{\deg} \left( \stackrel{\wedge}{c_{1}} \left( y^{*} \left( \overline{{\cal U}^{'}} \right) \right) \right) \end{eqnarray*} and, correspondingly, \begin{eqnarray*} h_{\overline{\cal T}} (y) & = & \stackrel{\wedge}{\deg} \left( \stackrel{\wedge}{c_{1}} \left( \overline{{\cal U}_{P}} \right) \Big| (\iota y)_{*} ({\rm Spec} ~ {\cal O}_{K}) \right) \\ & = & \stackrel{\wedge}{\deg} \left( \stackrel{\wedge}{c_{1}} \left( y^{*} \left( \overline{\cal U } \right) \right) \Big| {\rm Spec} ~ {\cal O}_{K} \right) \\ & = & \stackrel{\wedge}{\deg} \left( \stackrel{\wedge}{c_{1}} \left( y^{*} \left( \overline{\cal U } \right) \right) \right) ~~~ . \end{eqnarray*} But $\overline{{\cal U}^{'}}$ and $\overline{\cal U}$ coincide as line bundles on the generic fiber. As bundles their difference is some ${\cal O} (E)$ where $E$ is a divisor contained in the special fibers of ${\cal X}$, while the hermitian metrics differ by a continuous, hence bounded, factor. Therefore, the first arithmetic Chern classes of the pullbacks considered differ only at the infinite and a finite number of finite places by bounded summands. } \begin{center} $\Box$ \end{center} \end{thm} \begin{thm} {\bf Remark.} {\rm a) When one considers $L$-valued points instead of $K$-valued ones, where $L$ is a number field with $[L:K] = d$, the error term becomes ${\rm O}(d)$; i. e. there is a constant $C$ such that $$ \Big| h_{D} (x) - h_{\overline{\cal U}} (x) \Big| < C \cdot d$$ for $L$-valued points $x$ of $X$ and an arbitrary number field $L/K$. The reason for that is simply that the number of the critical places occuring grows as ${\rm O}(d)$. \newline b) The lemma can be applied to ${\cal X} = {\cal C}^{g}$ and $D = \sum_{k=1}^{g} \pi_{k}^{*} (x)$, since $\bigotimes_{k=1}^{g} \pi_{k}^{*} {\cal O} (\overline{x})$ extends ${\cal O} (D)$. } \end{thm} \begin{thm} \label{div} {\rm The height defined by an extension ${\cal U}$ of $\bigotimes_{k=1}^{g} \pi_{k}^{*} ({\cal O} (x))$ is understood by the following } \newline {\bf Proposition.} On ${\cal C} / {\cal O}_{K} $ let $\overline{\cal S}$ be the line bundle ${\cal O} (\overline{x})$, where $\overline{x}$ denotes the closure of $x$ in $\cal C$, equipped with a hermitian metric. For $L$-valued points $P = (P_{1}, \ldots ,P_{g})$ of $C^{g}$ we consider the divisor $\underline{P} := (P_{1}) + \ldots + (P_{g})$ on $C$. Then $$ h_{\overline{\cal S}} (\underline{P}) = h_{\overline{\cal U}} (P) + O(d) {}~~.$$ \end{thm} {\bf Proof.} By [BoGS, Proposition 3.2.2.ii)] we may assume ${\cal U} = {\cal O} \Big( \sum_{k=1}^{g} \pi_{k}^{*} (\overline{x}) \Big)$, where $\overline{x}$ is the closure of $x$ in ${\cal C}$. The extensions of $P$ and $\underline{P}$ over $\cal C$ and ${\cal C}^{g}$ will be denoted by $(\overline{P_{1}}) + \ldots + (\overline{P_{g}})$, respectively $(\overline{P_{1}}, \ldots , \overline{P_{g}})$. Then \begin{eqnarray*} h_{\overline{\cal S}} \Big( (\overline{P_{1}}) + \ldots + (\overline{P_{g}}) \Big) & = & \stackrel{\wedge}{\deg} \Big( \stackrel{\wedge}{c_{1}} (\overline{\cal S}) \Big| (\overline{P_{1}}) + \ldots + (\overline{P_{g}}) \Big) \\ & = & \sum_{k=1}^{g} \stackrel{\wedge}{\deg} \Big( \stackrel{\wedge}{c_{1}} (\overline{\cal S}) \Big| (\overline{P_{k}} ) \Big) \\ & = & \sum_{k=1}^{g} \stackrel{\wedge}{\deg} \Big( \stackrel{\wedge}{c_{1}} \Big( \pi_{k}^{*} (\overline{\cal S}) \Big) \Big| (\overline{P_{1}}, ~ \ldots ~ , \overline{P_{g}} ) \Big) ~~~~~~~~~~~{\rm "projection ~formula"}\\ & = & \stackrel{\wedge}{\deg} \left( \stackrel{\wedge}{c_{1}} \left( \bigotimes_{k=1}^{g} \pi_{k}^{*} \left( \overline{\cal S} \right) \right) \bigg| \left( \overline{P_{1}}, ~ \ldots ~ , \overline{P_{g}} \right) \right) {}~~. \end{eqnarray*} But by construction $\bigotimes_{k=1}^{g} \pi_{k}^{*} (\overline{\cal S})$ is the line bundle ${\cal U}$, equipped with a hermitian metric (and by definition the formula $\Big( \stackrel{\wedge}{c_{1}} (\bigotimes_{k} \overline{{\cal L}_{k}}) \Big| Z \Big) = {\displaystyle\sum}_{k} \Big( \stackrel{\wedge}{c_{1}} (\overline{{\cal L}_{k}}) \Big| Z \Big) $ holds in singular case, too). So we have $$h_{\overline{\cal S}} \Big( (\overline{P_{1}}) + \ldots + (\overline{P_{g}}) \Big) = h_{\overline{{\cal U}}^{'}} \Big( (\overline{P_{1}} , ~ \ldots ~ , \overline{P_{g}}) \Big) ~~~,$$ \newpage {}~ \newline where $\overline{{\cal U}}^{'}$ differs from $\overline{\cal U}$ only by the hermitian metric. The claim follows from [BoGS, Proposition 3.2.2.i)]. \begin{center} $\Box$ \end{center} \begin{thm} \label{two} {\bf Corollary.} Let $P \in C^{g}(L)$ and $\underline{P}$ be the associated divisor on $C$. Then $$h_{\Theta} ({\cal O} (\underline{P})) = h_{\overline{\cal S}} (\underline{P}) + h_{R} (p_{*} P) + {\rm O}(d) ~~,$$ where $h_{R}$ denotes the height for points of $C^{(g)}$ defined by $R$. \end{thm} \section{An observation concerning the tautological line \newline bundle} In this section we start analyzing the fundamental definition 1.3. First we will consider only varieties over number fields and forget about integral models. \begin{thm} {\bf Definition.} Let $\Delta$ be the diagonal in $C \times C$. Then $$\underline{\underline{\cal E}} := \bigotimes_{k=1}^{g} \pi_{k,g+1}^{*} \Big( {\cal O} (\Delta) \Big) $$ will be called the {\rm tautological line bundle} on $C^{g} \times C$. Note that the restriction of $\underline{\underline{\cal E}}$ to $\{ (P_{1}, \ldots , P_{g}) \} \times C$ equals ${\cal O} (P_{1} + \ldots + P_{g})$. By construction $\underline{\underline{\cal E}}$ is the pullback of some line bundle $\cal E$, said to be {\rm the tautological} one on $C^{(g)} \times C$. $$\underline{\underline{\cal E}} = (p \times id)^{*} ({\cal E})$$ \end{thm} \begin{thm} {\bf Proposition.} We have $\det R \pi_{*} {\cal E} = {\cal O}_{C^{(g)}} (-R)$. \end{thm} \begin{thm} {\rm This will be a direct consequence of the following} \newline {\bf Lemma.} Let $\underline{\cal E} := {\cal E} \otimes \pi_{C}^{*} ({\cal O} (-x))$ be a tautological line bundle fiber-by-fiber of degree $g-1$. Then $$\det R \pi_{*} \underline{\cal E} = {\cal O}_{C^{(g)}} \Big( -c^{*} (\Theta) \Big) ~~.$$ \end{thm} {\bf Proof.} The canonical map $c: C^{(g)} \longrightarrow J$ is given by ${\cal E}$ using Picard functoriality. So for a tautological line bundle ${\cal M}$, fiber-by-fiber of degree $g$ on $J \times C$, one has $${\cal E} = (c \times id)^{*} {\cal M} \otimes \pi^{*} {\cal H} ~~,$$ where $\cal H$ is a line bundle on $C^{(g)}$. Putting ${\cal M}_{0} := {\cal M} \otimes \pi_{C}^{*} {\cal O} (-x)$, where $\pi_{C}: J \times C \longrightarrow C$ denotes here the canonical projection from $J \times C$, we get $$\underline{\cal E} = (c \times id)^{*} {\cal M}_{0} \otimes \pi^{*} {\cal H} {}~~.$$ It follows \begin{eqnarray*} \det R \pi_{*} \underline{\cal E} & \cong & \det R \pi_{*} \Big[ (c \times id)^{*} {\cal M}_{0} \otimes \pi^{*} {\cal H} \Big] \\ & = & \det R \pi_{*} \Big[ (c \times id)^{*} {\cal M}_{0} \Big] \\ & = & c^{*} \det R \pi_{*} {\cal M}_{0} ~~, \end{eqnarray*} where we first used the projection formula, which is particularly simple here, since line bundles, fiber-by-fiber of degree $g-1$, have relative Euler characteristic $0$, and afterwords noted \pagebreak that the determinant of cohomology commutes with arbitrary base change \cite{Knudsen/Mumford 76}. But by [MB, Proposition 2.4.2] or [Fa, p. 396] we know $\det R \pi_{*} {\cal M}_{0} = {\cal O}_{J} (- \Theta)$. The assertion follows. \begin{center} $\Box$ \end{center} \begin{thm} {\bf Proof of the Proposition.} {\rm The short exact sequence $$0 \longrightarrow \underline{\cal E} \longrightarrow {\cal E} \longrightarrow {\cal E}|_{C^{(g)} \times \{ x \} } \longrightarrow 0 $$ gives \begin{eqnarray*} \det R \pi_{*} {\cal E} & = & \det R \pi_{*} \underline{\cal E} \otimes {\cal O} \left( {\frac{\scriptstyle 1}{\scriptstyle g!}} p_{*} \left( \sum_{k=1}^{g} \pi_{k}^{*} (x) \right) \right) \\ & = & {\cal O} \left( - c^{*} (\Theta) \right) \otimes {\cal O} \left( \frac{\scriptstyle 1}{\scriptstyle g!} p_{*} \left( \sum_{k=1}^{g} \pi_{k}^{*} (x) \right) \right) \\ & = & {\cal O} (-R) ~~. \end{eqnarray*} \begin{center} $\Box$ \end{center} } \end{thm} \begin{thm} {\bf Corollary.} \label{glatt} $\det R \pi_{*} ({\cal E} \otimes \pi^{*} {\cal O} (R)) = {\cal O}_{C^{(g)}}$. \newline {\bf Proof.} {\rm This is the projection formula for the determinant of cohomology.} \begin{center} $\Box$ \end{center} \end{thm} \begin{thm} {\rm Let $\cal J$ be the N\'eron model of the Jacobian $J$ of $C$. It is smooth over ${\cal O}_{K}$, consequently ${\cal J} \times_{{\rm Spec} {\cal O}_{K}} {\cal C}$ is smooth over ${\cal C}$ and therefore regular. We note that any $K$-valued point of $J$ can be extended uniquely to an ${\cal O}_{K}$-valued point of $\cal J$. On $J \times C$ we have a tautological line bundle ${\cal M}$, fiber-by-fiber of degree $g$. ${\cal M}$ can be extended over ${\cal J} \times_{{\rm Spec} {\cal O}_{K}} {\cal C}$. For this let ${\cal M} = {\cal O} (D)$ with some Weil divisor $D$ on $J \times C$. Its closure $\overline{D}$ in ${\cal J} \times_{{\rm Spec} {\cal O}_{K}} {\cal C}$ is obviously flat over ${\cal O}_{K}$ and therefore it has codimension $1$. We choose the extension ${\cal O} ({\overline{D}})$ and denote it by $\cal M$ again. $\cal M$ is a perfect complex of ${\cal O}_{{\cal J} \times_{{\rm Spec} {\cal O}_{K}} {\cal C}}$-modules. For the existence of the Knudsen-Mumford determinant we need that $$\pi: {\cal J} \times_{{\rm Spec} {\cal O}_{K}} {\cal C} \longrightarrow {\cal J}$$ has finite ${\rm Tor}$-dimension. For this there exists a closed embedding ${\cal C} \longrightarrow P$, where $P$ is smooth over ${\cal O}_{K}$. Thus $\pi$ factorizes as $${\cal J} \times_{{\rm Spec} {\cal O}_{K}} {\cal C} \stackrel{i}{\hookrightarrow} {\cal J} \times_{{\rm Spec} {\cal O}_{K}} P \stackrel{{\rm smooth}}{\twoheadrightarrow} {\cal J} ~~.$$ By [SGA 6, Expos\'e III, Proposition 3.6] it is enough to show that $i$ has finite ${\rm Tor}$-dimension. But $i_{*}$ is exact and ${\cal J} \times_{{\rm Spec} {\cal O}_{K}} P$ is regular implying quasi-coherent sheaves have locally finite free resolutions of bounded length. ${\cal M}$ has, relative to $\pi$, Euler characteristic $1$. Therefore $\cal M$ can be changed by an inverse image of a line bundle on ${\cal J}$, trivial on the generic fiber, in such a way, that we are allowed to assume $$\det R \pi_{*} {\cal M} \cong {\cal O}_{\cal J} ~~.$$ } \end{thm} \section{Choosing hermitian metrics continuously depending on moduli space} \setcounter{equation}{1} \begin{thm} {\bf Fact.} On ${\cal M}_{\Bbb{C}}$ there exists a hermitian metric $\underline{h}$ such that for every point $y \in J ({\Bbb{C}})$ the curvature form satisfies $$c_{1} ({\cal M}_{{\Bbb{C}},y}, \underline{h}_{y}) = g \omega$$ on $( \{ y \} \times C) ({\Bbb{C}}) \cong C (\Bbb{C})$. \newline {\bf Proof.} {\rm The statement is local in $C^{g} (\Bbb{C})$ by partition of unity. By the Theorem on cohomology and base change $R^{0} \pi_{*} {\cal M}_{\Bbb{C}} (g-1)$ is locally free and commutes with arbitrary base change. Hence there exists, locally on $J (\Bbb{C})$, a rational section $s$ of $\cal M$ that is neither undefined nor identically zero in any fiber. First we choose an arbitrary hermitian metric $\| . \|$ on ${\cal M}_{\Bbb{C}}$. Then \begin{equation} \omega^{'} := -d_{C} d_{C}^{c} \log \| s \|^{2} \end{equation} defines a smooth $(1,1)$-form on $(J \times C) ({\Bbb{C}}) \backslash {\rm div} (s)$, that is fiber-by-fiber the curvature form to be considered. Since construction (2) is independent of $s$ as soon as it makes sense at a point, we obtain $\omega^{'}$ as a smooth $(1,1)$-form on $(J \times C) (\Bbb{C})$ closed under $d_{C}$ and cohomologous to $g \omega$ on $\{ y \} \times C(\Bbb{C})$ for any $y \in C^{g} (\Bbb{C})$. The setup $\| . \|_{\underline{h}} = f \cdot \| . \|$ gives the equation \begin{equation} \omega^{'} - g \omega = d_{C} d_{C}^{c} \log | f |^{2} ~~. \end{equation} But $d d^{c}$ is an elliptic differential operator on the Riemann surface $C (\Bbb{C})$, so by Hodge theory it permits a Green`s operator $G$ compact with respect to every Sobolew norm $\| . \|_{\alpha}$. Consequently, there exists a solution $f$ of (3) being smooth on $(J \times C) (\Bbb{C})$. } \begin{center} $\Box$ \end{center} \end{thm} \begin{thm} {\rm We note, that $\det R \pi_{*} {\cal M} \cong {\cal O}_{\cal J}$ and the isomorphism is uniquely determined up to units of ${\cal O}_{K}$. Namely, one has ${\rm Aut}_{{\cal O}_{\cal J}} ({\cal O}_{\cal J}) = \Gamma ({\cal J}, {\cal O}_{\cal J}^{*})$ and already $\Gamma (J, {\cal O}_{J}^{*})$ consists of constants only. In particular, there is a unitary section, uniquely determined up to units of ${\cal O}_{K}$, $${\bf 1} \in \Gamma ({\cal J}, \det R \pi_{*} {\cal M}) ~~.$$ } \end{thm} \begin{thm} {\bf Corollary.} Let $R \in \Bbb{R}$. Then, on ${\cal M}_{\Bbb{C}}$ there exists exactly one hermitian metric $h$, such that for every point $y \in J(\Bbb{C})$ the curvature form $c_{1} ({\cal M}_{{\Bbb{C}}, y}, h_{y}) = \omega$ and for the Quillen metric one has $$h_{Q,h} ({\bf 1}) = R ~~,$$ where ${\bf 1} \in \Gamma (\{ y \} , \det R \pi_{*} {\cal M}_{{\Bbb{C}}, y} )$. \newline {\bf Proof.} {\rm Let $\underline{h}$ be the hermitian metric from the preceeding fact. We may replace $\underline{h}$ by $f \cdot \underline{h}$ with $f \in C^{\infty} (J(\Bbb{C}))$ without any effort on the curvature forms, since they are invariant under scalation. As $\cal M$ has relative Euler characteristic $1$, exactly $$h := \frac{R}{h_{Q, \underline{h}} ({\bf 1})} \cdot \underline{h} $$ satisfies the conditions required.} \begin{center} $\Box$ \end{center} \end{thm} \begin{thm} {\rm We have to consider ${\cal M}_{\Bbb{C}}$ on $J({\Bbb{C}}) \times (\coprod_{\sigma: K \hookrightarrow {\Bbb{C}}} C (\Bbb{C}))$. The metric $h$ on ${\cal M}_{\Bbb{C}}$ has to be invariant under $F_{\infty}$, its curvature form is required to be $g \omega$ and we want to realise \begin{equation} \prod_{\sigma: K \hookrightarrow \Bbb{C}} h_{Q,h} ({\bf 1}) = 1 \end{equation} simultaneously for all $y \in J(\Bbb{C})$. The first is possible since $\omega$ is invariant under $F_{\infty}$ and the corollary above already gives conditions uniquely determining $h$. (4) can be obtained by scalation with a constant factor over all $J({\Bbb{C}}) \times (\coprod_{\sigma: K \hookrightarrow {\Bbb{C}}} C({\Bbb{C}}))$. Altogether, for every line bundle of degree $g$ on $C$ we have found a distinguished hermitian metric and seen that it depends, in some sense, continuously on the moduli space $J$. One obtains } \end{thm} \begin{thm} {\bf Proposition.} \label{Ara} Let $K$ be a number field and $({\cal C} / {\cal O}_{K}, \omega)$ a regular connected Arakelov surface. Then, on the (non proper) Arakelov variety $({\cal J} \times_{{\rm Spec} {\cal O}_{K}} {\cal C}, \pi_{C}^{*} \omega)$ there is a hermitian line bundle $\overline{\cal M}$ with the following properties. \newline {\rm a)} $$(c \times id)^{*} ({\cal M} |_{J \times C}) = {\cal E} \otimes \pi^{*} {\cal O} (R)$$ is the modified tautological line bundle found in section 3. \newline {\rm b)} The hermitian metric $h$ on ${\cal M} |_{\coprod_{\sigma: K \hookrightarrow \Bbb{C}} (J \times C) ({\Bbb{C})}}$ is invariant under $F_{\infty}$ and has curvature form $g \omega$. \newline {\rm c)} For any $y \in J(K)$ one has $\overline{\{ y \}} \subseteq {\cal J}$ and $$\stackrel{\wedge}{\deg} \left( \det R \pi_{*} \left( {\cal M} |_{\overline{\{ y \} } \times_{{\rm Spec} {\cal O}_{K}} {\cal C}}, h_{{\cal M},y} \right) , \| . \|_{Q,h} \right) = 0 ~~.$$ {\bf Proof.} {\rm b) and c) are clear. For a) we know ${\cal E} = (c \times id)^{*} ({\cal M}|_{J \times C}) \otimes \pi^{*} {\cal H}$ from 4.3. But $\cal H$ is determined by $\det R \pi_{*} {\cal E} = {\cal O}_{C^{(g)}} (-R)$ and $\det R \pi_{*} ({\cal M}|_{J \times C}) = {\cal O}_{J}$. \begin{center} $\Box$ \end{center} } \end{thm} \section{An integral model of the symmetric power} \begin{thm} {\bf Remark.} {\rm It turns out here to be very inconvenient to work directly with the N\'eron model $\cal J$ of the Jacobian of $C$. When one considers the tautological line bundle ${\cal M} |_{J \times C}$, fiber-by-fiber of degree $g$ on $J \times C$ with $\det R \pi_{*} {\cal M}|_{J \times C} \cong {\cal O}_{J},$ then ${\cal M}|_{J \times C}$ will even have an (up to constant factor) canonical section. $$\pi_{*} {\cal M}|_{J \times C} \cong {\cal O}_{J}$$ But this section is zero over a codimension two subset of $J$ such that one is led to blow up this subset. } \end{thm} \begin{thm} {\bf Lemma.} {\rm a)} $C^{(g)}$ is a projective variety. \newline {\rm b)} The divisor $S := \frac{1}{g!} p_{*} \Big( \sum_{k=1}^{g} \pi_{k}^{*} (x) \Big) = \frac{1}{(g-1)!} p_{*} \Big( (x) \times C^{g-1} \Big)$ "one of the points is $x$" on $C^{(g)}$ is ample. \newline {\bf Proof.} {\rm a) There are at least two good reasons for that. First $C^{(g)}$ is proper as a quotient of the proper variety $C^{g}$ and b) gives an ample line bundle. On the other hand we can give a high-powered argument as follows. $C^{(g)}$ is the Hilbert scheme ${\rm Hilb}_{C/K}^{g}$ by [CS, Chapter VII by J. S. Milne, Theorem 3.13] and this is known to be projective for a long time [FGA, Expos\'e 221, Theorem 3.2]. \pagebreak \newline b) By [EGA III, Proposition 2.6.2] it is enough to show that $p^{*} {\cal O} (S) = \bigotimes_{k=1}^{g} \pi_{k}^{*} {\cal O} (x)$ is an ample line bundle on $C^{g}$, which is obvious. } \begin{center} $\Box$ \end{center} \end{thm} \begin{thm} {\bf Proposition.} The morphism $c: C^{(g)} \longrightarrow J$ is the blow-up of some ideal sheaf ${\cal I} \subseteq {\cal O}_{J}$ with $$c^{-1} {\cal I} = {\cal O} (-NR) ~~,$$ where $N$ is a positive integer. \newline {\bf Proof.} {\rm $c$ is birational and by the lemma it is a projective morphism. So it is a blow-up of some ideal sheaf ${\cal I} \subseteq {\cal O}_{J}$. Going through the lines of the proof of [Ha, Chapter II, Theorem 7.17] one sees that $c_{*} ({\cal O} (NS))$ for $N \gg 0$, up to tensor product with line bundles in order to make them ideal sheaves, can be used as $\cal I$. But $c^{*} {\cal O} (\Theta) = {\cal O} (S) \otimes {\cal O} (R)$ gives \begin{eqnarray*} c_{*} ({\cal O} (NS)) & = & c_{*} \Big( {\cal O} (-NR) \otimes c^{*} {\cal O} (N \Theta) \Big) \\ & = & c_{*} \Big( {\cal O} (-NR) \Big) \otimes {\cal O} (N \Theta) \end{eqnarray*} and therefore ${\cal I} = c_{*} {\cal O} (-NR)$ for some $N \gg 0$. We have a short exact sequence $$0 \longrightarrow {\cal O}_{C^{(g)}} (-NR) \longrightarrow {\cal O}_{C^{(g)}} \longrightarrow {\cal O}_{R_{N}} \longrightarrow 0 ~~,$$ where $R_{N}$ denotes the $N$-th infinitesimal neighbourhood of $R$. It follows exactness of $$0 \longrightarrow c_{*} {\cal O}_{C^{(g)}} (-NR) \longrightarrow {\cal O}_{J} \longrightarrow c_{*} {\cal O}_{R_{N}} ~~.$$ Now the image of ${\cal O}_{J} \longrightarrow c_{*} {\cal O}_{R_{N}}$ is the structure sheaf of the scheme-theoretic image $I_{N}$ of $R_{N}$ in $J$. So ${\cal I} = c_{*} {\cal O}_{C^{(g)}} (-NR) = {\cal I}_{I_{N}} \subseteq {\cal O}_{J}$. But ${\cal O}_{C^{(g)}} / c^{-1} {\cal I}_{I_{N}} = c^{*} ({\cal O}_{C^{g}} / {\cal I}_{I_{N}})$ and therefore $c^{-1} {\cal I} = c^{-1} {\cal I}_{I_{N}}$ is the ideal sheaf of $I_{N} \times_{J} C^{(g)}$ in ${\cal O}_{C^{(g)}}$. $c^{-1} {\cal I}$ is known to be invertible, so $I_{N} \times_{J} C^{(g)}$ is necessarily pure of codimension $1$ and by construction it contains the scheme $R_{N}$. But $R = c^{*} c_{*} S - S$ contains with one point its complete fiber in $c: C^{(g)} \longrightarrow J$. So $I_{N} \times_{J} C^{(g)}$ must be an infinitesimal thickening of $R_{N}$. On the other hand, when one replaces $R$ by $S$ and considers the scheme-theoretic image $\underline{I_{N}}$ of the $N$-th infinitesimal neighbourhood $S_{N}$, then $\underline{I_{N}} \times_{J} C^{(g)} \supseteq I_{N} \times_{J} C^{(g)}$ is a pure codimension $1$ subscheme not containing any thickening of $R_{N}$, but only other additional summands (it corresponds to the divisor $c^{*} (N \Theta)$). So, necessarily $I_{N} \times_{J} C^{(g)} = R_{N}$ and $$c^{-1} {\cal I} = {\cal O} (-NR) ~~.$$ } \begin{center} $\Box$ \end{center} \end{thm} \begin{thm} {\rm Denote by $\tilde{\cal J}$ the normalization of the blow-up of $\cal J$ with respect to some extension $\underline{\cal I}$ of the ideal sheaf $\cal I$ over $\cal J$. } \newline {\bf Facts.} {\rm a)} $\tilde{\cal J}$ is some (singular) arithmetic variety proper over $\cal J$. \newline {\rm b)} It is an integral model of $C^{(g)}$. \newline {\rm c)} On $\tilde{\cal J}$ one has the line bundle $${\cal R} := (c^{-1} \underline{\cal I})^{\vee}$$ extending ${\cal O} (NR)$ for some $N > 0$. \newline {\rm Note that we do not know whether we have an extension of ${\cal O} (R)$ over $\tilde{\cal J}$. } \end{thm} \begin{thm} {\rm On $\tilde{\cal J} \times_{{\rm Spec} {\cal O}_{K}} {\cal C}$ we will consider the hermitian line bundle $$\overline{\cal F} := (c \times id)^{*} \overline{\cal M} ~~,$$ where $c: \tilde{\cal J} \longrightarrow {\cal J}$ denotes here the extension of $C^{(g)} \longrightarrow J$ (the blow-down morphism). } \newline {\bf Facts.} {\rm a)} ${\cal F}|_{J \times C} = {\cal E} \otimes {\cal O} (R)$. \newline {\rm b)} One has $\det R \pi_{*} {\cal F} \cong {\cal O}_{\tilde{\cal J}}$. \newline {\rm Note here, $\tilde{\cal J}$ is not regular, so we do not know whether $\pi: \tilde{\cal J} \times_{{\rm Spec} {\cal O}_{K}} {\cal C} \longrightarrow \tilde{\cal J}$ has finite Tor-dimension. Thus $\det R \pi_{*}$ does may be not exist as a functor, but for line bundles, coming by base change from ${\cal J} \times_{{\rm Spec} {\cal O}_{K}} {\cal C}$, the definition makes sense. } \end{thm} \begin{thm} {\bf Remark.} {\rm All in all we obtain a decomposition $$\overline{\cal F}^{\otimes N} \cong \overline{\cal K} \otimes \pi^{*} \overline{\cal R}$$ of hermitian line bundles, where ${\cal K}$ extends ${\cal E}^{\otimes N}$, the $N$-th power of the tautological line bundle on $C^{(g)} \times C$. } \end{thm} \begin{thm} {\bf Remark.} {\rm Any line bundle of degree $g$ on $C$ gives an ${\cal O}_{K}$-valued point $y: {\rm Spec} ~ {\cal O}_{K} \longrightarrow {\cal J}$ and ${\rm Spec} ~ {\cal O}_{K} \times_{{\cal J}, y} \tilde{\cal J}$ will be proper over ${\rm Spec} ~ {\cal O}_{K}$. So at least for some finite field extension $L/K$ there will be an ${\cal O}_{L}$-valued point $\underline{y}: {\rm Spec} ~ {\cal O}_{L} \longrightarrow \tilde{\cal J}$ lifting $y$. Proposition \ref{Ara}.c) gives $$\stackrel{\wedge}{\deg} \left( \det R \pi_{*} \left( {\cal F} |_{\overline{\{ y \} } \times_{{\rm Spec} {\cal O}_{K}} {\cal C}}, h_{{\cal F},y} \right) , \| . \|_{Q,h} \right) = 0 ~~.$$ } \end{thm} \section{Decomposition into two summands} \begin{thm} {\rm In this section we will restrict to the case that ${\cal C}$ is {\it semistable}, i. e. $\pi: {\cal C} \longrightarrow {\rm Spec} ~ {\cal O}_{K}$ is smooth up to codimension } $2$. \newline {\bf Lemma.} {\rm a)} $\tilde{\cal J} \times_{{\rm Spec} {\cal O}_{K}} {\cal C}$ is a normal scheme. \newline {\rm b)} $\tilde{\cal J}$ is quasi-projective over ${\cal O}_{K}$. \newline {\bf Proof.} {\rm a) $\tilde{\cal J}$ is normal, so $\tilde{\cal J} \times_{{\rm Spec} {\cal O}_{K}} {\cal C}^{\rm smooth}$ is normal by [SGA1, Expos\'e I, Corollaire 9.10]. In particular $\tilde{\cal J} \times_{{\rm Spec} {\cal O}_{K}} {\cal C}$ is regular in codimension $1$. Further $\pi: \tilde{\cal J} \times_{{\rm Spec} {\cal O}_{K}} {\cal C} \longrightarrow \tilde{\cal J}$ is flat with one dimensional fibers. By [EGA IV, Corollaire 6.4.2] $\tilde{\cal J} \times_{{\rm Spec} {\cal O}_{K}} {\cal C}$ is Cohen-Macaulay in codimension $2$. \newline b) $\cal J$ is quasi-projective over ${\cal O}_{K}$ by [CS, Chapter VIII by M. Artin, \S 4] and blow-ups are projective morphisms. } \begin{center} $\Box$ \end{center} \end{thm} \begin{thm} {\bf Remark.} {\rm Let $\underline{y}: {\rm Spec} ~ {\cal O}_{L} \longrightarrow \tilde{\cal J}$ be an ${\cal O}_{L}$-valued point lifting an ${\cal O}_{K}$-valued point \linebreak $y: {\rm Spec} ~ {\cal O}_{K} \longrightarrow {\cal J}$. Then \begin{eqnarray*} h_{x, \omega} \left( {\cal M}|_{y \times_{{\rm Spec} {\cal O}_{K}} {\cal C}} \right) & = & \frac{\scriptstyle 1}{\scriptstyle [L:K] N} \stackrel{\wedge}{\deg} \pi_{*} \left[ \stackrel{\wedge}{c_{1}} \left( \overline{\cal F}^{\otimes N} |_{\underline{y} \times_{{\rm Spec} {\cal O}_{K}} {\cal C}} \right) \cdot (x,g_{x}) \right] \\ & = & \frac{\scriptstyle 1}{\scriptstyle [L:K] N} \stackrel{\wedge}{\deg} \pi_{*} \left[ \stackrel{\wedge}{c_{1}} \left( \overline{\cal K} |_{\underline{y} \times_{{\rm Spec} {\cal O}_{K}} {\cal C}} \right) \cdot (x,g_{x}) \right] \\ & + & \frac{\scriptstyle 1}{\scriptstyle [L:K] N} \stackrel{\wedge}{\deg} \pi_{*} \left[ \stackrel{\wedge}{c_{1}} \left( \pi^{*} \overline{\cal R} |_{\underline{y} \times_{{\rm Spec} {\cal O}_{K}} {\cal C}} \right) \cdot (x,g_{x}) \right] ~~. \end{eqnarray*} We note here, on ${\rm Spec} ~ {\cal O}_{L} \times_{{\rm Spec} ~ {\cal O}_{K}} {\cal C}$ being in general a singular scheme, $\pi_{*}$ of an intersection with an arithmetic Chern class is defined using an embedding into a regular scheme, where the line bundle comes from by base change (Remark \ref{sing}, \cite{Fulton 75}.) The first equality comes from projection formula. Note that $x$ means here an ${\cal O}_{L}$-valued point of ${\rm Spec} ~ {\cal O}_{L} \times_{{\rm Spec} ~ {\cal O}_{K}} {\cal C}$ whose push-forward to $\cal C$ is $[L : K] (x)$. } \end{thm} \begin{thm} {\rm Let us investigate the first summand. We have ${\cal K} |_{C^{(g)} \times C} = {\cal E}^{\otimes N}$ and this line bundle has a canonical section $s$, which can be extended over the finite places. Using this section we obtain the arithmetic cycle $({\rm div} ~ (s), -\log \| s \|^{2})$ representing $$\stackrel{\wedge}{c_{1}} (\overline{\cal K}) \in ~ \stackrel{\wedge}{CH^{1}} \left( \tilde{\cal J} \times_{{\rm Spec} {\cal O}_{K}} {\cal C} \right) ~~.$$ The scheme part ${\rm div} (s)$ of this cycle is an extension of the tautological divisor representing $c_{1} ({\cal E}^{\otimes N}) \in {\rm CH}^{1} (C^{(g)} \times C)$ (whose restriction to $\{ (x_{1}, \ldots, x_{g}) \} \times C$ is $N (x_{1}) + \ldots + N (x_{g})$). So ${\rm div} ~ (s)$ is the closure of that divisor, possibly plus a finite sum of divisors over the finite places. We note, that $\cal K$ is given by that divisor since $\tilde{\cal J} \times_{{\rm Spec} {\cal O}_{K}} {\cal C}$ is normal. Consequently, if $\underline{y}$ restricts to the $L$-valued point corresponding to the divisor $D$ on $C$, then $$c_{1} \left( {\cal K} |_{\underline{y} \times_{{\rm Spec} {\cal O}_{K}} {\cal C}} \right) = \left( \overline{D} \right) + ({\rm correction ~ terms}) ~~,$$ where $\overline{D}$ denotes the closure of $N D$ over $\cal C$ and the correction terms are vertical divisors which (over all the $y$) occur only over a finite amount of finite places. Their intersection numbers with $(x, g_{x})$, i. e. with the line bundle ${\cal O} (x)$, are bounded by ${\rm O} ([L:K])$. The infinite part $f$ of $\stackrel{\wedge}{c_{1}} (\overline{\cal K}) = ({\cal D}, f)$ is a function on $(C^{(g)} \times C) \backslash {\rm div} ~ (s)$ whose pullback to $C^{g} \times C$ satisfies all the assumptions of Lemma \ref{Int}. We obtain \begin{eqnarray*} \frac{\scriptstyle 1}{\scriptstyle [L:K] N} \stackrel{\wedge}{\deg} \pi_{*} \left[ \stackrel{\wedge}{c_{1}} \left( \overline{\cal K} |_{\underline{y} \times_{{\rm Spec} {\cal O}_{K}} {\cal C}} \right) \cdot (x,g_{x}) \right] & = & \frac{\scriptstyle 1}{\scriptstyle [L:K] N} \left[ \stackrel{\wedge}{\deg} \left( \stackrel{\wedge}{c_{1}} \left( \overline{{\cal O} (x)} \right) ~ \Big| {}~ \left( {\cal D} |_{\underline{y} \times_{{\rm Spec} ~ O_{K}} {\cal C}} \right) \right) ~~~ \ldots \right. \\ & ~ & ~~~\ldots ~~~ + \left. \frac{\scriptstyle 1}{\scriptstyle 2} \sum_{\sigma : L \hookrightarrow {\Bbb {C}}} \int_{C ({\Bbb{C}})} f_{D} \omega_{x} \right] \\ & = & \frac{\scriptstyle 1}{\scriptstyle N} h_{\overline{\cal S}} (N D) + {\rm O} (1) \\ & = & h_{\overline{\cal S}} (D) + {\rm O} (1) ~~. \end{eqnarray*} Note, for the first equation we used the symmetry of the intersection form for hermitian line bundles (Remark \ref{sing}). The denominator $[L:K]$ disappears by [BoGS, formula (3.1.8)]. } \end{thm} \begin{thm} {\rm The second summand is simpler. One has \begin{eqnarray*} \stackrel{\wedge}{c_{1}} \left( \pi^{*} \overline{\cal R} |_{\underline{y} \times_{{\rm Spec} {\cal O}_{K}} {\cal C}} \right) \cdot (x, g_{x}) & = & \pi^{*} \stackrel{\wedge}{c_{1}} \left( {\cal R} |_{\underline{y}} \right) \cdot (x, g_{x}) \\ & = & \stackrel{\wedge}{c_{1}} \left( {\cal R} |_{\underline{y}} \right) + \left( 0, g_{x} \omega_{R} (\underline{y}) \right) ~~, \end{eqnarray*} when one identifies $\tilde{\cal J} \times_{{\rm Spec} ~ {\cal O}_{K}} \overline{ \{ x \} }$ with $\tilde{\cal J}$. The integral $\int_{C ({\Bbb C})} ~ g_{x} \omega_{R} ( . )$ depends smoothly on the parameter, in particular it is bounded. So the push-forward of the right summand is bounded by ${\rm O} ([L:K])$. On the other hand $\pi_{*} \stackrel{\wedge}{c_{1}} \left( {\cal R} |_{\underline{y}} \right) = \left( \stackrel{\wedge}{c_{1}} (\overline{\cal R}) ~ \Big| ~ \underline{y} \right) $, where the last term is defined by embedding $\tilde{\cal J}$ into a scheme $P$ smooth and projective over ${\cal O}_{K}$ \cite{Fulton 75}. Note here we use $\tilde{\cal J}$ is quasi-projective. Thus Lemma \ref{class} gives $$\stackrel{\wedge}{\deg} \left( \stackrel{\wedge}{c_{1}} (\overline{\cal R}) ~ \Big| ~ \underline{y} \right) = \stackrel{\wedge}{\deg} \left( \stackrel{\wedge}{c_{1}} (\overline{{\cal R}_{P}}) ~ \Big| ~ \iota_{*} (\underline{y}) \right) = h_{\overline{{\cal R}_{P}}} \left( \iota_{*} (\underline{y}) \right) = h_{R_{P}} \left( \iota_{*} D \right) = h_{R} (D) {}~~,$$ where $D$ is the divisor corresponding to the restriction of $\underline{y}$ to $C^{(g)}$. } \end{thm} \begin{thm} \label{stab} {\rm We obtain} \newline {\bf Proposition.} Assume $\cal C$ is semistable and ${\cal L} = {\cal O} (\overline{D})$, where $\overline{D}$ is the closure of some divisor on $C$. Then Theorem \ref{main} is true. \newline {\bf Proof.} {\rm By Corollary \ref{two} this is now proven for line bundles coming by restriction from $\cal M$. This way one can realize the line bundles ${\cal O} (D)$ on the generic fiber $C$ for arbitrary divisors $D$ (defined over $K$) of degree $g$ over $C$. Consider the degrees $$\deg {\cal M} |_{y \times_{{\rm Spec} {\cal O}_{K}} {\cal C}_{{\goth p}, i}}$$ for ${\cal O}_{K}$-valued points $y$ of $\cal J$, where ${\cal C}_{{\goth p}, i}$ denote the irreducible components of the special fiber ${\cal C}_{\goth p}$. They are even defined for $\overline{{\cal O} / {\goth p}}$-valued points, where the bar denotes algebraic closure here, and are locally constant over the special fiber ${\cal J}_{\goth p}$. In particular they are bounded since the N\'eron model of an abelian variety is of finite type. The Proposition follows from Lemma \ref{degr}. } \begin{center} $\Box$ \end{center} \end{thm} \section{End of the proof} \begin{thm} \label{blow} {\bf Lemma.} {\rm Let ${\cal C} / {\cal O}_{K}$ be a regular projective arithmetic surface and $p: \tilde{\cal C} \longrightarrow {\cal C}$ be a blow-up of one point. Then $$h_{x, \omega} ({\cal L}) = h_{x, \omega} (p^{*} {\cal L}) ~~,$$ where $x \in {\cal C} ({\cal O}_{K}) = \tilde{\cal C} ({\cal O}_{K})$ and $\cal L$ is a line bundle with $\chi ({\cal L}) \neq 0$. \newline {\bf Proof.} Obviously $p_{*} p^{*} {\cal L} = {\cal L}$ and [SGA6, Expos\'e VII, Lemma 3.5] gives $R^{i} p_{*} p^{*} {\cal L} = 0$ for $i \geq 1.$ In particular $R p_{*} (p^{*} {\cal L}) = {\cal L}$, $R (\pi p)_{*} (p^{*} {\cal L}) = R \pi_{*} {\cal L}$ and $\det R (\pi p)_{*} (p^{*} {\cal L}) = \det R \pi_{*} {\cal L}$. \linebreak This means that $\cal L$ and $p^{*} {\cal L}$ get identical distinguished metrics and therefore \linebreak $\stackrel{\wedge}{c_{1}} \left( p^{*} {\cal L}, \| . \|_{p^{*} {\cal L}} \right) = p^{*} \stackrel{\wedge}{c_{1}} \left( {\cal L}, \| . \|_{{\cal L}} \right) $. On the other hand $p^{*} (x, g_{x}) = (x, g_{x}) + ({\rm exceptional ~ divisor})$, but an exceptional divisor intersects trivially with cycles coming from downstairs. Consequently, \begin{eqnarray*} h_{x, \omega} (p^{*} {\cal L}) & = & \stackrel{\wedge}{\deg} (\pi p)_{*} \left[ p^{*} \stackrel{\wedge}{c_{1}} ({\cal L}, \| . \|_{\cal L}) \cdot p^{*} (x, g_{x}) \right] \\ & = & \stackrel{\wedge}{\deg} (\pi p)_{*} p^{*} \left[ \stackrel{\wedge}{c_{1}} ({\cal L}, \| . \|_{\cal L}) \cdot (x, g_{x}) \right] \\ & = & \stackrel{\wedge}{\deg} \pi_{*} \left[ \stackrel{\wedge}{c_{1}} ({\cal L}, \| . \|_{\cal L}) \cdot (x, g_{x}) \right] ~~~~~~~~~~~~~~~~~~~~{\rm "projection ~ formula"}\\ & = & h_{x, \omega} ({\cal L}) ~~. \end{eqnarray*} } \begin{center} $\Box$ \end{center} \end{thm} \begin{thm} {\bf Corollary.} \label{model} {\rm (Change of model.)} \newline Let ${\cal C}_{1}, {\cal C}_{2} / {\cal O}_{K}$ be two regular projective models of the curve $C/K$ of genus $g$. Then, for divisors $D$ of degree $g$ on $C$ $$h_{x, \omega} \left( {\cal O}_{{\cal C}_{1}} (\overline{D}) \right) = h_{x, \omega} \left( {\cal O}_{{\cal C}_{2}} (\overline{D}) \right) + {\rm O} (1) {}~~,$$ where $\overline{D}$ denotes the closure of $D$ in ${\cal C}_{1}$, respectively ${\cal C}_{2}.$ \newline {\bf Proof.} {\rm By [Li, Theorem II.1.15] one is reduced to the case of the blow-up of one point $p: {\cal C}_{2} \longrightarrow {\cal C}_{1}$. By Lemma \ref{blow} we have to bound the difference $h_{x, \omega} \left( {\cal O}_{{\cal C}_{2}} (\overline{D}) \right) - h_{x, \omega} \left( p^{*} {\cal O}_{{\cal C}_{1}} (\overline{D}) \right)$. $\overline{D}$ will meet the point blown up $i$ times ($0 \leq i \leq g$). We get an exact sequence $$0 \longrightarrow {\cal O}_{{\cal C}_{2}} (\overline{D}) \longrightarrow p^{*} {\cal O}_{{\cal C}_{1}} (\overline{D}) \longrightarrow {\cal O}_{E^{i}} \longrightarrow 0 ~~,$$ where $E$ is the exceptional curve and $E^{i}$ denotes its $i$-th infinitesimal neighbourhood. But now the assertion is a direct consequence of Lemma \ref{red}. } \begin{center} $\Box$ \end{center} \end{thm} \begin{thm} {\bf Lemma.} {\rm (Change of base field.)} \newline Let ${\cal C} / {\cal O}_{K}$ be a regular arithmetic surface, generically of genus $g$, $L/K$ a finite field extension and $$p: {\cal C}^{'} = \overline{{\cal C} \times_{{\rm Spec} ~ {\cal O}_{K}} {\rm Spec} ~ {\cal O}_{L}} \longrightarrow {\cal C}$$ be some resolution of singularities of the base change to ${\cal O}_{L}$. Then, for divisors $D$ of degree $g$ on $C = {\cal C} \times_{{\rm Spec} ~ {\cal O}_{K}} {\rm Spec} ~ K$, $$h_{x, \omega} \left( {\cal O}_{{\cal C}^{'}} (\overline{p^{*} D}) \right) = [L:K] \cdot h_{x, \omega} \left( {\cal O}_{\cal C} (\overline{D}) \right) + {\rm O} (1) ~~.$$ \newline {\bf Proof.} {\rm $p$ is a composition of blow-ups and finite morphisms [CS, Chapter XI by M. Artin]. \linebreak Using the first formulas in the proof of Lemma \ref{blow} successively we obtain \linebreak $R p_{*} p^{*} {\cal O} (\overline{D}) = {\cal O} (\overline{D})$ and $\det R (\pi p)_{*} p^{*} \left( {\cal O} (\overline{D}) \right) = \det R \pi_{*} {\cal O} (\overline{D})$ such that ${\cal O} (\overline{D})$ and $p^{*} {\cal O} (\overline{D})$ get identical distinguished metrics. Here it follows $$h_{x, \omega} \left( p^{*} {\cal O} (\overline{D}) \right) = [L:K] \cdot h_{x, \omega} \left( {\cal O} (\overline{D}) \right) ~~,$$ since $p$ is a morphism of degree $[L:K]$ and the projection formula gives $p_{*} p^{*} Z = [L:K] Z$. $p^{*} {\cal O} (\overline{D})$ and ${\cal O} (\overline{p^{*} D})$ differ by a limited combination of the exceptional divisors such that the assertion follows from Lemma \ref{red}. } \begin{center} $\Box$ \end{center} \end{thm} \begin{thm} \label{hori} {\bf Proposition.} For line bundles ${\cal L} = {\cal O} (\overline{D})$, where $\overline{D}$ is the closure of some divisor of degree $g$ on $C$, Theorem \ref{main} is true. \newline {\bf Proof.} {\rm By [AW, Corollary 2.10] there is a stable model for $C \times_{{\rm Spec} ~ K} {\rm Spec} ~ L$ after some finite field extension $L/K$. The assertion follows from Proposition \ref{stab}. } \begin{center} $\Box$ \end{center} \end{thm} \begin{thm} {\bf Proposition.} Theorem \ref{main} is true. \newline {\bf Proof.} {\rm This is a direct consequence of Proposition \ref{hori} and Lemma \ref{degr}. } \begin{center} $\Box$ \end{center} \end{thm} \section{Some technical Lemmata} \begin{thm} \label{fibe} {\bf Lemma.} {\rm (Fibers do not change the height.)} \newline If $\cal L$ is a line bundle on ${\cal C} / {\cal O}_{K}$ with $\chi ({\cal L}) \neq 0$, then $$h_{x, \omega } \left( {\cal L} \otimes {\cal O} (\goth{p}) \right) = h_{x, \omega } ({\cal L})$$ for every prime ideal $\goth{p} \subseteq {\cal O}_{K}$. \newline {\bf Proof.} {\rm One has ${\cal O} (\goth{p}) = \pi^{*} (\goth{p}^{-1})$, hence by projection formula $$\det R \pi_{*} \left( {\cal L} \otimes {\cal O} (\goth{p}) \right) \cong \det R \pi_{*} {\cal L} \otimes {\cal O} ({\goth p})^{- \chi ({\cal L})} ~~.$$ Let $\| . \|$ be one of the distinguished metrics on the line bundle ${\cal L}_{\Bbb{C}}$ on $\coprod_{\sigma: K \hookrightarrow \Bbb{C}} C(\Bbb{C})$. \linebreak We put $\| . \|_\goth{p} = C \cdot \| . \|$ for a distinguished hermitian metric on $({\cal L} \otimes {\cal O} (\goth{p}))_{\Bbb{C}} = {\cal L}_{\Bbb{C}}$. It follows \linebreak $h_{Q, \det R \pi_{*} ({\cal L} \otimes {\cal O} (\goth{p}))} = C^{\chi ({\cal L})} \cdot h_{Q, \det R \pi_{*} {\cal L}}$ and \begin{eqnarray*} \stackrel{\wedge}{\deg} \left( \det R \pi_{*} ({\cal L} \otimes {\cal O} (\goth{p})), h_{Q, \det R \pi_{*} ({\cal L} \otimes {\cal O} (\goth{p}))} \right) & = & \stackrel{\wedge}{\deg} \Big( \det R \pi_{*} {\cal L}, h_{Q, \det R \pi_{*} {\cal L}} \Big) \\ & + & \chi ({\cal L}) \Big[ [K : {\Bbb{Q}}] \log C - \log (\sharp {\cal O} / \goth{p}) \Big] ~~. \end{eqnarray*} Thus a distinguished hermitian metric on $({\cal L} \otimes {\cal O} (\goth{p}))_{\Bbb{C}}$ can be given by $\| . \|_\goth{p} = ( \sharp {\cal O} / \goth{p})^{\frac{1}{[K : {\Bbb Q}]}} \cdot \| . \|$ and it follows $$\stackrel{\wedge}{c_{1}} ({\cal L} \otimes {\cal O} (\goth{p}), \| . \|_\goth{p}) = ~ \stackrel{\wedge}{c_{1}} ({\cal L}, \| . \|) + \pi^{*} \left( \goth{p} ; - \frac{\scriptstyle 2}{\scriptstyle [K : \Bbb{Q}]} \log (\sharp {\cal O} / \goth{p}) ,\ldots ,- \frac{\scriptstyle 2}{\scriptstyle [K : \Bbb{Q}]} \log (\sharp {\cal O} / \goth{p}) \right) ~~.$$ But the arithmetic cycle $\left( \goth{p} ; - \frac{2}{[K : \Bbb{Q}]} \log (\sharp {\cal O} / \goth{p}), \ldots, - \frac{2}{[K : \Bbb{Q}]} \log (\sharp {\cal O} / \goth{p}) \right) \in {}~ \stackrel{\wedge}{{\rm CH}^{1}}({\rm Spec} ~ {\cal O}_{K})$ vanishes after multiplication with the class number $\sharp {\rm Pic} ~ ({\rm Spec} ~ {\cal O}_{K})$, hence it is torsion and therefore numerically trivial. } \begin{center} $\Box$ \end{center} \end{thm} \begin{thm} \label{red} {\bf Lemma.} Let $F$ be some vertical divisor on ${\cal C} / {\cal O}_{K}$. Then, for line bundles ${\cal L} / {\cal C}$, fiber-by-fiber of degree $g$, $$h_{x, \omega} ({\cal L} (F)) = h_{x, \omega} ({\cal L}) + {\rm O} (1) ~~.$$ {\bf Proof.} {\rm By Lemma \ref{fibe} we may assume that $E := -F$ is effective. Using induction we are reduced to the case $E$ is an irreducible curve. We have a short exact sequence $$0 \longrightarrow {\cal L} (F) \longrightarrow {\cal L} \longrightarrow {\cal L}_{E} \longrightarrow 0$$ inducing the isomorphism $$\det R \pi_{*} {\cal L} (F) \cong \det R \pi_{*} {\cal L} \otimes (\det R \pi_{*} {\cal L}_{E})^{\vee} ~~.$$ But $\det R \pi_{*} {\cal L}_{E}$ depends only on the Euler characteristic of ${\cal L}_{E}$ and for the degree of that bundle there are only $g+1$ possibilities. So up to numerical equivalence there are only $g+1$ possibilities for $$\stackrel{\wedge}{c_{1}} \left( {\cal L} (F), \| . \|_{{\cal L} (F)} \right) - \stackrel{\wedge}{c_{1}} \Big( {\cal L}, \| . \|_{\cal L} \Big) ~~,$$ where $\| . \|_{\cal L}$ and $\| . \|_{{\cal L} (F)}$ denote distinguished hermitian metrics. } \begin{center} $\Box$ \end{center} \end{thm} \begin{thm} \label{degr} {\bf Lemma.} Consider line bundles ${\cal L}$, generically of degree $g$ on ${\cal C}$, equipped with a section $s \in \Gamma (C, {\cal L}_{C})$ over the generic fiber, and assume the degrees $\deg {\cal L} |_{{\cal C}_{{\goth p}, i}}$ of the restrictions of $\cal L$ to the irreducible components of the special fibers to be fixed. Then $$h_{x, \omega} ({\cal L}) = h_{x, \omega} \left( {\cal O} \left( \overline{{\rm div} (s)} \right) \right) + {\rm O} (1) ~~.$$ {\bf Proof.} {\rm We have ${\cal L} = {\cal L}^{'} (E)$, where ${\cal L}^{'} = {\cal O} \left( \overline{{\rm div} (s)} \right)$ is a line bundle induced by a horizontal divisor and $E$ is a vertical divisor. By Lemma \ref{fibe} we may assume $E$ to be concentrated in the reducible fibers of $\cal C$. So, using induction, let $E$ be in one such fiber ${\cal C}_{{\goth p}}$. Then for the degrees $\deg {\cal O} (E) |_{{\cal C}_{{\goth p}, i}}$ there are only finitely many possibilities. But by [Fa, Theorem 4.a)] the intersection form on ${\cal C}_{\goth p}$ \pagebreak is negative semi-definite where only multiples of the fiber have square $0$. \linebreak Hence, for $E$ there are only finitely many possibilities up to addition of the whole fiber, which does not change the height. Lemma \ref{red} gives the claim. } \begin{center} $\Box$ \end{center} \end{thm} \begin{thm} {\bf Lemma.} \label{Int} Let $X$ be a compact Riemann surface and $g \in \Bbb{N}$ be a natural number. Denote by $\Delta$ the diagonal in $X \times X$, by $\delta_{M}$ the $\delta$-distribution defined by $M$ and by $\pi_{i}: X^{g} \times X \longrightarrow X$ (resp. $\pi_{i,g+1}: X^{g} \times X \longrightarrow X \times X$) the canonical projection on the $i$-th component (resp. to the product of the $i$-th and $(g+1)$-th component.) Further let $$f: (X^{g} \times X) \backslash \bigcup_{i=1}^{g} \pi_{i,g+1}^{-1} (\Delta) \longrightarrow {\Bbb{C}}$$ be a smooth function such that the restriction of $$-d_{X} d_{X}^{c} f + \delta_{\Delta} \circ \pi_{1,g+1} + \ldots + \delta_{\Delta} \circ \pi_{g,g+1} = \rho ~~,$$ to $\{ (x_{1}, \ldots, x_{g}) \} \times X$ is a smooth $(1,1)$-form smoothly varying with $(x_{1}, \ldots, x_{g})$. Let $\omega$ be a smooth $(1,1)$-form on $X$. Then $$\int_{X} f(x_{1}, \ldots ,x_{g}, \cdot) \omega$$ depends smoothly on $(x_{1}, \ldots, x_{g}) \in X^{g}.$ \newline {\bf Proof.} {\rm Without restriction we may assume $\int_{X} \omega = 1$. Then, for any $x \in X$ there exists a function $h \in C^{\infty} (X \backslash \{ x \})$, having a logarithmic singularity in $x$, such that $\omega = -dd^{c} h + \delta_{x}$. It follows \begin{eqnarray*} \int_{X} f(x_{1}, \ldots, x_{g}, \cdot) \omega & = & -\int_{X} f(x_{1}, \ldots, x_{g}, \cdot) dd^{c} h + f(x_{1}, \ldots, x_{g}, x) \\ & = & -\int_{X} \Big( d_{X} d_{X}^{c} f(x_{1}, \ldots ,x_{g}, \cdot) \Big) h + f(x_{1}, \ldots, x_{g}, x) \\ & = & \int_{X} \rho (x_{1}, \ldots, x_{g}, \cdot) h - h(x_{1}) - \ldots - h(x_{g}) + f(x_{1}, \ldots, x_{g}, x) \\ & = & \int_{X} \rho (x_{1}, \ldots, x_{g}, \cdot) h - \Big[ h(x_{1}) - G(x, x_{1}) \Big] - \ldots - \Big[ h(x_{g}) - G(x, x_{g}) \Big] \\ & - & \Big[ G(x, x_{1}) + \ldots + G(x, x_{g}) - f(x_{1}, \ldots, x_{g}, x) \Big] ~~, \end{eqnarray*} where $G$ is the Green's function of $X$. Because $h$ has only a logarithmic singularity it is allowed to differentiate under the integral sign. So the integral is smooth. The other summands are solutions of equations of the form $dd^{c} F = \sigma$ with a smooth $(1,1)$-form $\sigma$ on $X$ satisfying $\int_{X} \sigma = 0$ (in $x_{1}, \ldots, x_{g}$, respectively $x$). Since $dd^{c}$ is elliptic, these solutions exist as smooth functions and are unique up to constants. In particular, also the last summand must depend smoothly on $(x_{1}, \ldots, x_{g})$, even when some of the $x_{i}$ equal $x$. Note that the symmetry of the Green's function is used here essentially. \begin{center} $\Box$ \end{center} } \end{thm} {}~ \newline {\footnotesize {\bf Acknowledgement.} When doing this work, the author had fruitful discussions with U. Bunke (Berlin), who explained him much of the analytic part of the theory. He thanks him warmly.} \newpage {}~ \vspace*{-1.20truecm} \thispagestyle{myheadings} \markright{\rm ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Jahnel} \small

9508

alg-geom/9508006

en

https://arxiv.org/abs/alg-geom/9508006

alg-geom/9508006

Martin Rainer

M. Rainer

Topological Classifying Spaces of Lie Algebras and the Natural Completion of Contractions

48 pages, latex, 2 figures

UniP-Math-95/03

The space K^n of all n-dimensional { Lie} algebras has a natural non-Hausdorff topology k^n, which has characteristic limits, called transitions, A -> B, between distinct Lie algebras A and B. The entity of these transitions are the natural transitive completion of the well known Inonu-Wigner contractions and their partial generalizations by Saletan. Algebras containing a common ideal of codimension 1 can be characterized by homothetically normalized Jordan normal forms of one generator of their adjoint representation. For such algebras, transitions A -> B can be described by limit transitions between corresponding normal forms. The topology k^n is presented in detail for n < 5. Regarding the orientation of the algebras as vector spaces has a non-trivial effect for the corresponding topological space K^n_or: There exist both, selfdual points and pairs of dual points w.r.t. orientation reflection.

alg-geom

\section{\bf Introduction} \setcounter{equation}{0} The main goal of this paper is to study the topological space of real {Lie} algebras of a given dimension $n\leq 4$. Extensive studies have been dedicated to generalizations of the classical {Lie} algebra structure. As an example think of the famous q-deformations or Santilli's Lie isotopic liftings \San. However, few work has been dedicated to pursue the theory of deformations and contractions of {Lie} algebras (or groups) within their category. {Smrz} \Smrz\ has considered the deformation of {Lie} algebras outside a fixed subgroup. This kind of deformation is in some sense complementary to a {In\"on\"u-Wigner} contraction \In, which consists in a parametric linear and isotropic contraction outside a given subgroup of a {Lie} algebra. A particularly interesting problem is to find all possible contractions and, more generally, all possible limit transitions between real or complex {Lie} algebras of fixed dimension $n$, and to uncover the natural topological structure of the space of all such {Lie} algebras. It is clear that this requires, as a precondition, to find all isomorphism classes of {Lie} algebras in the given dimension. Unfortunately, with increasing dimension $n$ the classification of real and complex {Lie} algebras becomes rapidly more complicated. For this goal, the {Levi} decomposition into a semidirect sum of a radical and a semisimple subalgebra proves to be useful. This way {Turkowski} has classified real {Lie} algebras which admit a nontrivial {Levi} decomposition, up to $n=8$ in \aTur\ and recently for $n=9$ in \cTur. In any dimension $n$, the classification of all nilpotent {Lie} algebras is an essential step required for a complete classification. For $n=7$, a complete list of all nilpotent, real and complex, {Lie} algebras has been given by {Romdhani} \Ro; the complex case has been considered first by {Ancochea-Bermudez} and {Goze} \An; complex decomposable algebras have been studied by {Charles} and {Diakite} \Ch. The variety of structure constants of complex {Lie} algebras has been examined for $n=4, 5, 6$ by {Kirillov} and {Neretin} \Ki. {Grunewald} and {O'Halloran} \Gru\ have investigated the complex, nilpotent {Lie} algebras for $n\leq 6$. For $n=6$, all real nilpotent {Lie} algebras are classified by {Morozov} \Mo; solvable, non nilpotent {Lie} algebras have been classified by {Mubarakzjanov} \cMu; and solvable real {Lie} algebras containing nilradicals are classified by {Turkovski} \bTur, thus completing the classification of the solvable ones. Both give reference to the early work of {Umlauf} \Um\ already classifying the nilpotent complex $6$-dimensional {Lie} algebras. {Mubarakzjanov} also classified real {Lie} algebras up to $n=5$ in \bMu. In \aMu\ he treats the case of real $n=4$, giving reference to the early works of {Lie} \bLie\ for complex algebras with $n\leq 4$ and, for the real $3$-dimensional case, to {Bianchi} \bBi\ and later equivalent classifications of {Lee} \Lee\ and {Vranceanu} \Vra. The $3$-dimensional real {Lie} algebras, are given by the so called {Bianchi} types, classified independently first by S. {Lie} \Lie\ and then by L. {Bianchi} \Bi. The original classification of {Bianchi} revealed 9 inequivalent types of $3$-parameter {Lie} groups $G_3$, numbered usually by the Roman numbers $\I,\ldots,\IX$. The types of number \VI\ and \VII\ are actually $1$-parameter sets of {Lie} algebras, \VIh\ resp. \mbox{${\rm VII}_h$} , with $h\geq 0$ all inequivalent. We will refer to the inequivalent $3$-dimensional real {Lie} algebras as the {Bianchi} types. Our choice of the parameter $h$ is according to {Landau-Lifschitz} \Lan, which agrees for \mbox{${\rm VII}_h$} with {Behr}'s choice in {\Est}. When the isomorphism classes of {Lie} algebras for a given real (or complex) dimension are known in a given dimension, one can start to compare their algebraic structure systematically and find their algebraic characteristics, i.e. the invariants. So {Paterea} and {Winternitz} \PaW\ determined subalgebra structures for real {Lie} algebras with $n\leq 4$. {Grigore} and {Popp} \Gri\ developed a general classification of subalgebras of {Lie} algebras with solvable ideal, and invariants of real {Lie} algebras have been calculated for $n\leq 5$ by {Patera, Sharp, Winternitz} and {Zassenhaus} {\PaSWZ}. But the algebraic properties of {Lie} algebras are also related to the topological structure of the space of all {Lie} algebras in a given dimension. On the space of all structure constants of real {Lie} algebras in $n$ dimensions {Segal} has introduced (see page 255 in \Se) the subspace topology induced from $\R^{n^3}$. The space $K^n$ of all isomorphism classes of real $n$-dimensional {Lie} algebras under general linear isomorphisms ${\GL}(n)$ of their generators has a natural weakly separating (i.e. $T_0$, not $T_1$) non-{Hausdorff} topology $\kappa^n$, induced as the quotient topology from the {Segal} topology by the equivalence relation given on the structure constants via the action of $\GL(n)$. This topology has been discovered and described explicitly by {Schmidt} for $n\leq 3$ in \aSch\ and more generally in \bSch. As a real vector space, a Lie algebra admits also a natural orientation. By the exponential map, for any {Lie} algebra there exists an associated {Lie} group which similarly admits the corresponding orientation as a differentiable manifold. Note that throughout the following any index or property concerning orientation is set in brackets $()$ iff the corresponding quantity can be considered optionally with or without reference to an orientation. The present paper is organized by the following sections. Sec. 2 resumes some well-known facts on {Lie} algebras and topology needed in the sequel. Sec. 3 describes the general construction of the topological spaces $(K^n,{\kappa}^n)$ and $(K^n_{or},{\kappa}^n_{or})$, respectively with and without orientation of the {Lie} algebras as vector spaces. Sec. 4 shows how those solvable elements of $K^n$ which contain all the same ideal $J_{n-1}$ can be characterized against each other by the normalized version (NJNF) of the {Jordan} normal form (JNF) of a single structure matrix. Correspondingly, an oriented normalized {Jordan} normal form (ONJNF) for the structure constants of oriented {Lie} algebras is defined. Hence transitions $A\to B$ between {Lie} algebras can be described by transitions between the corresponding normal forms. Sec. 5 resumes important general properties (see also {Schmidt} \bSch) of the topology ${\kappa}^n_{(or)}$ and shows up further features of orientation duality for arbitrary dimension $n$. The structure of $K^n_{or}$, the space of equivalence classes of oriented {Lie} algebras, as compared to its unoriented counterpart $K^n$, has also been described in {Rainer} \aRa. A generalization of {Schmidt}'s notion of atoms is made for arbitrary subsets of $K^n_{(or)}$. This is applied to the case of the non-selfdual subset $K^n_{or}\setminus K^n_{SD}$, decomposing it for $n=3$ and $n=4$ into its connected components $K^n_+$ and $K^n_-$. Sec. 6 is devoted to the topology of the non oriented $K^n$ for $n\leq 4$. The topological structure for $n\leq 4$ has also been described by {Rainer} \aRa. The $T_0$ topology $\kappa^n$ provides for $n\geq 3$ a rich local structure of $K^n$, which we describe for $n\leq 4$. In Sec. 6.1 the topological structure of $K^n$ for $n\leq 3$ is analysed by use of the NJNF. So, using a quite different method, we reproduce the results of {Schmidt} in \aSch\ and \bSch. Sec. 6.2 presents the detailed analysis of the components of $K^4$, their possible $\kappa^4$ limits, and transitions between them. Thereby the relation between the different classification schemes of {Mubarakzjanov} \aMu, {Patera, Winternitz} \PaW\ and {Petrov} \Pe\ is clarified. Sec. 6.3 determines the topological structure of $K^4$. Its parametrically connected components are related in a transitive network of $\kappa^4$ transitions. Sec. 7 is devoted to the topology of the oriented $K^n_{or}$ for $n\leq 4$, which is also described in {Rainer} \bRa. In Sec. 7.1 the topological structure of $K^n_{or}$ for $n\leq 3$ is analysed by use of the ONJNF, in correspondence with results listed by {Schmidt} \bSch. We give the connected components $K^3_\pm$ explicitly. Using the same method, Sec. 7.2 examines the orientation duality structure of $K^4_{or}$ in detail. In particular, we determine the connected components $K^4_\pm$. In Sec. 8 we discuss the present results. \section{\bf Preliminaries} \setcounter{equation}{0} In the following we remind shortly some of the notions needed throughout this paper. A (finite-dimensional) {Lie} algebra is a (finite-dimensional) vector space $V$, equipped with a skew symmetric bilinear product $[\cdot,\cdot]$ called {Lie} bracket, which maps $(X,Y)\in V\times V$ to $[X,Y]=-[Y,X]\in V$ and satisfies $\sum_{cycl.\atop X,Y,Z} [[X,Y],Z]=0, \forall X,Y,Z \in V$. The dimension of the {Lie} algebra is the dimension of the underlying vector space. Here and in the following all {Lie} algebras and vector spaces are assumed to be finite-dimensional. If the vector space is real resp. complex, we say that the {Lie} algebra is real resp. complex. If nothing else is specified in the following a {Lie} algebra or a vector space is assumed to be real. For a {Lie} algebra $A$ the descending central series of ideals is defined recursively by \begin{equation} C^0A:=A \quad{\rm and}\quad C^{i+1} A:=[A,C^{i}A]\subseteq C^{i}A. \end{equation} $A$ is called {\em nilpotent}, iff there exists a $p\in \N$, such that $C^pA=0$, i.e. the descending central series of ideals terminates at the zero ideal. Furthermore for a {Lie} algebra $A$ the derivative series of ideals is defined recursively by \begin{equation} A^{(0)}:=A \quad{\rm and}\quad A^{(i+1)}:=[A^{(i)},A^{(i)}]\subseteq A^{(i)}. \end{equation} $A$ is called {\em solvable}, iff there exists a finite $q\in \N$, such that $A^{(q)}=0$, i.e. the derivative series of ideals terminates at the zero ideal. In the following, we consider real {Lie} algebras of fixed finite dimension $n\geq 2$ (for $n=1$ there is only 1 type of {Lie} algebra, namely the {Abel}ian $A_1$), classified up to equivalence via real $\GL(n)$ transformations of their linear generators $\{e_i\}_{i=1,\ldots,n}$, which span an $n$-dimensional real vector space, which may in the following be identified with $\R^n$ or the tangent space $T_xM$ at any point $x$ of an $n$-dimensional smooth real manifold $M$. The {Lie} bracket $[\ ,\ ]$ is given by its action on the generators $e_i$, which is encoded in the structure constants $C^k_{ij}$, \begin{equation} [e_i,e_j]=C^k_{ij} e_k. \end{equation} (The sum convention is always understood implicitly, unless stated otherwise.) The bracket $[\ ,\ ]$ defines a {Lie} algebra, iff the structure constants satisfy the $n\{{n\choose 2}+{n\choose 1}\}$ antisymmetry conditions \begin{equation} C^k_{[ij]}=0, \end{equation} and the $n\cdot{n\choose 3}$ quadratic compatibility constraints \begin{equation} C^l_{[ij}C^m_{k]l}=0 \end{equation} with nondegenerate antisymmetric indices $i,j,k$. Here $_{[\quad ]}$ denotes antisymmetrization w.r.t. the indices included. Note that Eq. (2.5) is satisfied automatically by Eq. (2.4), if the bracket is derived via $[e_i,e_j]\equiv e_i\cdot e_j-e_j\cdot e_i$ from an associative multiplication $e_i\cdot e_j$. In this case Eq. (2.5) is an {\em identity}, called {Jacobi} identity. Otherwise Eq. (2.5) is an {\em axiom}, which might be called {Jacobi} axiom. If there is an (adjoint) matrix representation of the algebra, it is associative and hence satisfies the {Jacobi} axiom (2.5) trivially, i.e. as identity. We will not assume the existence of any matrix representation nor any associative algebra multiplication, because we want all the data for a {Lie} algebra to be encoded in the structure constants. Hence we take (2.5) as an axiom. The space of all sets $\{C^k_{ij}\}$ satisfying the {Lie} algebra conditions (2.4) and (2.5) can be viewed as a subvariety $W^n \subset \R^{n^3}$ of dimension \begin{equation} \dim W^n \leq n^3 - \frac{n^2(n+1)}{2} =\frac{n^2(n-1)}{2}. \end{equation} For $n=3$ the structure constants can be written as \begin{equation} C^k_{ij}= \mbox{$\varepsilon$} _{ijl}(n^{lk}+ \mbox{$\varepsilon$} ^{lkm}a_m), \end{equation} where $n^{ij}$ is symmetric and $ \mbox{$\varepsilon$} _{ijk}= \mbox{$\varepsilon$} ^{ijk}$ totally antisymmetric with $ \mbox{$\varepsilon$} _{123}=1$. With Eq. (2.7) the constraints Eq. (2.5) are equivalent to \begin{equation} n^{lm}a_m=0, \end{equation} which are $3$ independent relations. Actually, {Behr} has first classified the {Lie} algebras in $K^3$ according to their possible inequivalent eigenvalues of $n^{lm}$ and values of $a_m$ (see {Landau-Lifschitz} \Lan). With Eq. (2.8) also Eq. (2.5) is nontrivial for $n=3$. Therefore the inequality in Eq. (2.6) is strict for $n\geq 3$. Throughout the following, we will need the {\em separation axioms} from topology (for further reference see also {Rinow} \Ri). A given topology on a space $X$ is {\em separating} with increasing strength if it satisfies one or more of the following axioms. {\hfill \break} {\bf Axiom $T_0$}: For each pair of different points there is an open set containing only one of both. \hfill\mbox\break {\hfill \break} {\bf Axiom $T_1$}: Each pair of different points has a pair of open neighbourhoods with their intersection containing none of both points. \hfill\mbox\break {\hfill \break} {\bf Axiom $T_2$} ({Hausdorff}): Each pair of different points has a pair of disjoint neighbourhoods. \hfill$\Box$\break It holds: $T_2 {\Rightarrow } T_1 {\Rightarrow } T_0$. If a topology is only $T_0$, but not $T_1$, we say that it is only {\em weakly separating} and speak also shortly of the {\em weak} topology. (The present notion {\em weak} should not be confused with another one from functional analysis, which is not meant here. {\em Separability} of the topological space is defined here by the separation (german: Trennung) axioms $T_0, T_1$ or $T_2$. This should not be confused with a further notion related to the existence of a countable dense subset.) The separation axioms can equivalently be characterized in terms of sequences and their limits. {\bf Lemma.} For a topological space $X$ the following equivalences hold: {\hfill \break} a) $X$ is $T_0$ $ {\Leftrightarrow } $ For each pair of points there is a sequence converging only to one of them. {\hfill \break} b) $X$ is $T_1$ $ {\Leftrightarrow } $ Each constant sequence has at most one limit. {\hfill \break} c) $X$ is $T_2$ ({Hausdorff}) $ {\Leftrightarrow } $ Each {Moore-Smith}-sequence has at most one limit. {\hfill \break} \hfill\mbox\break (As a generalization of an ordinary sequence, a {Moore-Smith} sequence is a sequence indexed by a (directed) partially ordered set.) $T_1$ is equivalent to the requirement that each one-point set is closed. We define for the following the real {\em dimension} of a set as the largest number $k$ such that a subset homeomorphic to $\R^k$ exists. \section{\bf Spaces $K^n$ and $K^n_{or}$ of {Lie} algebras} \setcounter{equation}{0} The space of structure constants $W^n$ can also be considered as a subvariety of the fibrespace of the tensor bundle $\wedge^2T^*M\otimes TM$ over any point of some smooth $\GL(n)$-manifold $M$. If $M$ is oriented, the structure group of its tangent vector bundle $TM$ is reduced from $\GL(n)$ to its normal subgroup \begin{equation} \GL^+(n)=\{A\in \GL(n):\det A > 0\}. \end{equation} Then $W^n$ gets an additional structure induced from $\wedge^2T^*M\otimes TM$ by the orientation of $M$. {\hfill \break} $\GL(n)$ basis transformations induce $\GL(n)$ tensor transformations between equivalent structure constants. \begin{equation} C^k_{ij} \sim (A^{-1})^k_h\ C^h_{fg}\ A^f_i\ A^g_j \ \ \forall A \in \GL(n), \end{equation} where $\sim$ denotes the equivalence relation. This induces the space \begin{equation} K^n=W^n/\GL(n) \end{equation} of equivalence classes w.r.t. the nonlinear action of $\GL(n)$ on $W^n$. The analogous space for the oriented case is \begin{equation} K^n_{or}=W^n/\GL^+(n). \end{equation} The ${\GL}(n)$ action on $W^n$ is not free in general. It holds: \begin{equation} \dim W^n> \dim K^n_{(or)}\geq \dim W^n - n^2. \end{equation} The first inequality in Eq. (3.5) is a strict one, because the (positive) multiples of the unit matrix in $\GL^{(+)}(n)$ give rise to equivalent points of $K^n_{(or)}$. Eqs. (2.6) and (3.5) provide only insufficient information on $\dim K^n$. The latter is still unknown for general $n$. (For the analogous complex varieties {Neretin} \Ne\ has given an upper bound estimate.) Let $\phi_{(or)}: W^n\to K^n_{(or)}$ be the canonical map for the equivalence relation $\sim$ defined by the action of $\GL^{(+)}(n)$ in $W^n$. The natural topology $\kappa^n_{(or)}$ of $K^n_{(or)}$ is given as the quotient topology of the induced subspace topology of $W^n \subset \R^{n^3}$ w.r.t. the $\GL^{(+)}(n)$ equivalence relation. In the oriented case, orientation reversal of the basis yields a natural $Z_2$-action on $K^n_{or}$. This action is not free in general. Hence the fibres of the projection \begin{equation} \pi: K^n_{or}\to K^n = W^n/{\GL}(n)=K^n_{or}/Z_2 \end{equation} can be either $Z_2$ or $E$. In the first case there is a pair of dual points, i.e. points that transform into each other under the $Z_2$-action, in the latter case it is a selfdual point in $K^n_{or}$. The latter therefore decomposes into a selfdual part $K^n_{SD}$, on which $Z_2$ acts trivially, and 2 conjugate parts $K^n_{\pm}$. The latter are isomorphic to each other by that reflection in $GL(n)$ that is chosen to define $Z_2$ in Eq. (3.6). \begin{equation} K^n_{or}=K^n_{SD}\oplus K^n_{+} \oplus K^n_{-}, \end{equation} where $\oplus$ denotes the disjoint union of subvarieties. The projection $\pi$ has the property that its restriction to $K^n_{SD}$ is the identity. Therefore it is useful to make the following {\hfill \break} {\bf Definition 1.} A point $A\in K^n$ is called {\em selfdual} if $\pi^{-1}(A)\subset K^n_{or}$ consists of a single point, and {\em non-selfdual} if $\pi^{-1}(A)$ consists of a pair of dual points, denoted by $A^R$ and $A^L$ respectively. \hfill$\Box$\break In order to yield a more explicit notion of selfduality, we formulate {\hfill \break} {\bf Lemma.} {\em A {Lie} algebra $A$ is selfdual, $A\in K^n_{SD}$, if and only if there exist two different bases of $\R^n$ possessing different orientation such that all the structure constants $C^k_{ij}$ concerning both bases coincide. } \hfill$\Box$\break Obviously a direct sum of a selfdual algebra with any other algebra is selfdual. Let us mention already here that $K^n_{SD}$ is nonvoid for $n\geq 1$ while $K^n_\pm$ are nonvoid sets only for $n\geq 3$. We will see in Sec. 5 and 7 that the latter are actually nonvoid for $n=3,4,5$ and at least any further odd $n$. In any case $K^n_\pm$ are connected to $K^n_{SD}$. We will see in Sec. 7 that each of $K^n_\pm$ is connected for $n=3$ and $n=4$. Note that for each pair of conjugate {Lie} algebras $A^R$ and $A^L$ it is a priori completely arbitrary which one is assigned to $K^n_+$ and which one to $K^n_-$. In order to reduce this arbitrariness, in Sec. 4 we will minimize the number of connected components of $K^n_\pm$ to a single component each, thus making $K^n_+$ and $K^n_-$ disconnected to each other. However this requires first a better understanding of the topological structure $K^n_{or}$. When we do not want to care about effects of orientation, instead of {Schmidt}'s topological space $(G_n,\tau)\equiv (K^n_{or},\kappa^n_{or})$ from \bSch\ we will consider its projection to $(K^n,\kappa^n)$ by Eq. (3.6). Let us define now the notion of transitions $A\to B$ in $K^n_{(or)}$. {\hfill \break} {\bf Definition 2.} Consider $A, B \in K^n_{(or)}$ with $A\neq B$. If there is a sequence $\{A_i\}_{i\in \N}$ with $A_i=A$ for all $i\in \N$ which for $i\to \infty$ converges to $B$ in the topology $\kappa^n_{(or)}$, we say that there is a {\em transition} $A\to B$ in the topology $\kappa^n_{(or)}$. \hfill$\Box$ \break Note that this definition makes sense because $K^n$ is a $T_0$ but not a $T_1$ space. A transition is a special kind of limit characteristic for this topology. {\hfill \break} {\bf Convention.} We distinguish in notation between a concrete realization of a {Lie} algebra, $A$, and its equivalence class, $[A]$, where ever this is relevant. In the following, the former will an adjoint representation of the latter, sometimes also called abstract, {Lie} algebra. However for notational simplicity we prefer to denote a point in $K^n$ by $A$ rather than by $[A]$. If the context does not give the opportunity for confusion, $A$ is implicitly understood as a shorthand for the (abstract) {Lie} algebra $[A]$. \hfill$\Box$ \break In the topology $\kappa^n_{(or)}$, a transition $A\to B$ occurs if and only if $B\in \cl \{A\}$. For this transition the source $A$ is not closed, and the target $B$ is not open in any subset of $K^n_{(or)}$ containing both of them. In general, a point of $K^n_{(or)}$ will be neither open nor closed. Open points only appear as a source, and not as a target, of transitions. The structure of the rigid {Lie} algebras, which correspond just to these open points, is examined in {Charles} \Cha. Special kinds of transitions on a certain 2-point set $\{A,B\}$ of {Lie} algebra isomorphism classes are the contractions of {In\"on\"u-Wigner} \In\ and their generalization by {Saletan} \Sal. For convenience let us define these here. Consider a $1$-parameter set of matrices $A_t\in\GL(n)$ with $0<t\leq 1$, having a well defined matrix limit $A_0:=\lim_{t\to 0} A_t$ which is singular, i.e. $\det A_0=0$. For given structure constants $C^k_{ij}$ of a {Lie} algebra $A$ let us define for $0<t\leq 1$ further structure constants $C^k_{ij}(t):=(A^{-1}_t)^k_h\ C^h_{fg}\ (A_t)^f_i\ (A_t)^g_j$, which according to (3.2) all describe the same {Lie} algebra $A$. If there is a well defined limit $C^k_{ij}(0):=\lim_{t\to 0} C^k_{ij}(t)$ satisfying conditions (2.4) and (2.5) then this limit defines structure constants of a {Lie} algebra $B$, and the associated limit of {Lie} algebras $A\to B$ is called {\em contraction} according to {Saletan} \Sal\ or briefly {Saletan} {\em contraction}. Note that a {Saletan} contraction $A\to B$ might yield either $B=A$, then it is called {\em improper}, or $B\neq A$, then it is a transition of {Lie} algebras. A {Saletan} contraction is called {In\"on\"u-Wigner} {\em contraction} if there is a basis $\{e_i\}$ in which $$ A(t)= \left( \begin{array}{cc} E_m & 0 \\ 0 & t\cdot E_{n-m} \end{array} \right) \qquad \forall t\in [0,1], $$ where $E_k$ denotes the $k$-dimensional unit matrix. This definition closely follows {Conatser} \Co. Given the latter decomposition, {In\"on\"u} and {Wigner} \In\ have shown that the limit $C^k_{ij}(0)$ exists iff $e_i, i=1,\ldots,m$ span a subalgebra $W$ of $A$, which then characterizes the contraction. {Saletan} \Sal\ gives also a technical criterion for the existence of the limit $C^k_{ij}(0)$ defining his general contractions. We only remark here that, while a general {Saletan} contraction might be nontrivially iterated, the iteration of an {In\"on\"u-Wigner} contraction is always improper, i.e. no further contraction takes place. Not every transition $A\to B$ corresponds to an {In\"on\"u-Wigner} contraction. We will see some examples of transitions, which are given only by a more general {Saletan} contraction \Sal. However we will find also transitions $A\to B$, which are not even given by a {Saletan} contraction. Transitions $A\to B$ in the topology $\kappa^n$ reveal for $n\geq 3$ a more complicated structure of the underlying space $K^n$. In $K^n$ transitions $A\to B$ and $B\to C$ imply a transition $A\to C$; this means that transitions are transitive. There is a partial order, $A \geq B : {\Leftrightarrow } B\in \cl \{A\} {\Leftrightarrow } A\to B$ (which is also called the {\em specialization order}), which gives $K^n_{(or)}$ the structure of a transitive network of transitions. Since {Saletan} contractions \Sal\ are not transitive they do not exhaust all kinds of possible $\kappa^n$ transitions. Given the topology of $K^n$, on any 2-point subset $\{X,Y\}\subset K^n$ we can take the induced topology and consider the set $T^n:=\{\{X,Y\}\subset K^n\vert X\neq Y\}$ of all 2-point topological subspaces of $K^n$. Note that a $T_0$ topological space, like that of $K^n$ for $n\geq 3$, is in general not determined by the set $T^n$ of all its induced 2-point topological subspaces. However if the topological space under consideration is finite then $T^n$ determines already its topology, which is trivially true for $K^1$ and $K^2$. \section{\bf Normal forms of the structure constants} \setcounter{equation}{0} The structure constants of $A_n\in [A_n]\in K^n$ are given by the $n$ matrices $C_i:=(C^k_{ij})$, $i=1,\ldots,n$, with rows $k=1,\ldots,n$ and columns $j=1,\ldots,n$. $C_i$ is just the matrix of ad$e_i$ w.r.t. the basis $e_1,\ldots,e_n$. {\hfill \break} By Eq. (2.4), the column $j=i$ vanishes identically $\forall C_{i}$. Furthermore the diagonals $(C^j_{ij})$, $i=1,\ldots,n$ (no j-summation), determine the rows with $k=i$, since $(C^i_{ij})=(-C^i_{ji})$, $i=1,\ldots,n$ (no i-summation). Therefore $A_n$ is described completely by the $(n-1)\times (n-1)$-matrices $C_{<i>}:=(C^k_{ij})$, $i=1,\ldots,n$, with $k,j\neq i$ and $1\leq k,j\leq n$. {\hfill \break} In the special case where $A_n$ has an ideal $J_{n-1}\in [J_{n-1}]\in K^{n-1}$, we take without restriction $[A_n]/[J_{n-1}]=\mbox{span}(e_n)$. Then $A_n$ with a given $J_{n-1}$ is described completely by $C_{n}$ or $C_{<n>}$ only. {\hfill \break} {\bf Definition 3.} The {\em normalized} JNF (NJNF) of a matrix $C$ is given by the {Jordan} normal form, abbreviated JNF, of $C$ modulo $\R\setminus \{0\}$, i.e. given by the equivalence class of JNFs, which differ only by a common absolute scale and a common overall sign of their nonzero eigenvalues w.r.t. the eigenvalues of $C$. (The {Jordan} block structure and the multiplicities are the same for all of them.) \hfill $\Box$\break Thus a normalization convention for the JNF is the division of all nonzero eigenvalues by a fixed element of $\R {\setminus } \{0\}$. If not stated otherwise, we divide in the following just by the (absolutely) largest eigenvalue in order to represent the NJNF class of the JNF. {\hfill \break} Note that the $n^{th}$ row and column of $C_n$ add only an additional eigenvalue 0 (as {Jordan} block) to the JNF or NJNF of $C_{<n>}$. Since absolute scaling of all eigenvalues of a structure matrix $C_{<n>}$ by $\lambda \in \R {\setminus } \{0\}$ can be achieved by stretching the basis $\{e_i\}$ homogeneously by $\lambda^{-1}$, it is an equivalence transformation of the algebra. On the other hand it is evident that changing in $C_{<n>}$ the ratio $r$ of any 2 eigenvalues to $r'$, such that $r'$ is not a ratio of any original eigenvalues, changes the equivalence class. {\bf Theorem.} {\em Consider the set of algebras $A_n$ which have a common (abstract) ideal $J_{n-1}$. Then $A^{(1)}_n\sim A^{(2)}_n$, iff the matrices $C^{(1)}_{<n>}$ and $C^{(2)}_{<n>}$ have the same \NJNF. } {\hfill \break} {Proof:} $A^{(1)}_n\sim A^{(2)}_n$ iff $\exists M\in {\GL}(n): {C^{(1)}}^k_{ij} = (M^{-1})^k_h\ {C^{(2)}}^h_{fg}\ M^f_i\ M^g_j \sim M^f_i {C^{(2)}}^h_{fg}$. By linearity of $[\ ,\ ]$ in the second argument, the linearly independent recombinations ${\tilde{C}}^{(2)}_i:=M^f_i\ C^{(2)}_{f}$ describe still the same algebra as $C^{(2)}_{i}$. In particular, the (abstract) ideal $J_{n-1}$ is invariant under $M$. Since the algebras have the same ideal $J_{n-1}$, they are characterized by the matrices $C^{(1)}_{<n>}$ resp. $C^{(2)}_{<n>}$. They describe inequivalent algebras, iff $C^{(1)}_{<n>}$ is inequivalent (modulo overall scaling by $M=\lambda E_n,\ \lambda \in \R {\setminus } \{0\}$) to ${\tilde{C}}^{(2)}_{<n>}$ and therefore also to $C^{(2)}_{<n>}$. But the equivalence class of any structure matrix $C_{<n>}$ is described by its (real) JNF modulo homogeneous scaling of the eigenvalues with $\lambda \in \R {\setminus } \{0\}$. \hfill $\Box$ \break {\hfill \break} Already {Mubarakzyanov} \aMu\ had realized the advantage given by an ideal $J_{n-1}$ of codimension $1$. Since then also others, like {Magnin} \Mag\ within the nilpotent {Lie} algebras of dimension $\leq 7$, systematically cosidered subclasses of algebras which have a fixed {Lie} algebra of codimension $1$. In the following, we consider without restriction of generality the ideals $J_{(n-1)}$ in the normal form given by the NJNF of the structure constants. The equivalence class $[A_{n}]$ of any algebra $A_{n}$ with a fixed normal class ideal and additional structure constants from $C_{<n>}$ will be characterized in the following by the NJNF of $C_{<n>}$ and denoted by $$ \NJNF(A_n) := \NJNF(C_{<n>}). $$ Now we can define the ONJNF of structure matrices $C_{<n>}$ of oriented {Lie} algebras. {\hfill \break} {\bf Definition 4.} The ONJNF of the structure matrix $C_{<n>}$ of an oriented {Lie} algebra $A_n$ is set identical to its NJNF if $A_n$ is selfdual, and it is given as $ \mbox{${\rm ONJNF}$} (C_{<n>}):=\pm \NJNF(C_{<n>})$ for $A_n\in K^n_{\pm}$ respectively. \hfill $\Box$\break If $A_n$ is characterized by an ideal $J_{n-1}$ in normal form then we set $$ \mbox{${\rm ONJNF}$} (A_n) := \mbox{${\rm ONJNF}$} (C_{<n>}). $$ \section{\bf General properties of $\kappa^n$ and $\kappa^n_{or}$} \setcounter{equation}{0} In this section we describe the general topological properties of the topological space $(K^n_{(or)},\kappa^n_{(or)})$. Let us first remind some general properties from {Schmidt} \bSch\ (where also more details and proofs can be found). {\hfill \break} {\bf Proposition.} {\em $K^n_{(or)}$ has the following properties w.r.t. $\kappa^n_{(or)}$: } {\hfill \break} a) {\em The {Abel}ian algebra $\{nA_1\} \subset K^n_{(or)}$ is the only closed 1-point set and is contained in any nonempty closed subset of $K^n_{(or)}$. } {\hfill \break} b) {\em $K^n_{(or)}$ is connected and compact. } {\hfill \break} c) {\em $K^n_{*(,or)}:=K^n_{(or)} {\setminus } \{nA_1\}$ is a compact space, but $K^n_{*(,or)}$ is not a closed subset of $K^n_{(or)}$. } {\hfill \break} d) {\em $K^n_{*(,or)}$ is {Hausdorff} $(T_2)$ for $n=2$ only. } {\hfill \break} e) {\em For $n\geq 2$ (resp. $n\geq 3$) the separability of $K^n_{(or)}$ (resp. $K^n_{*(,or)}$) is only weak ($T_0$, i.e. for each pair of points there is a sequence converging to only one of them). } \hfill$\Box$\break d) and e) correspond to the fact that, though $K^n$ is still an algebraic variety (defined by purely algebraic relations (2.4), (2.5) and (3.2)), it can not be expected to be a (topological $T_1$) manifold. $K^n$ is the orbit space of $W^n$ w.r.t. the action of the noncompact group ${\GL}(n)$, which behaves algebraically badly on $W^n$ for $n\geq 2$. So some of the orbits (the elements of $K^n$) are closed in $K^n$, others are not. Strong separability ($T_1$, i.e. each constant sequence has at most one limit) would imply that there should not exist transitions $A \to B$ between inequivalent {Lie} algebra classes $A\not\sim B$, given by a sequence $\{A_i\}$ of {Lie} algebras of class $A$ converging to a {Lie} algebra of class $B$. But this is exactly what happens for dimension $n\geq 2$, as will be seen explicitly below. Obviously transitions $A\to B$ will be transitive, which decisively effects the topology of $K^n$. {\hfill \break} Transitions which are impossible in a given dimension $n$ can become possible after {Abel}ian embedding into dimension $n+1$. Therefore the following lemma holds. {\hfill \break} {\bf Lemma 1.} {\em The {Abel}ian embedding $\oplus \R$ of $K_n$ into $K_{n+1}$ is continuous, but for $n\geq 2$ not homeomorphic. } \hfill$\Box$ \break So we are led to the following {\hfill \break} {\bf Definition 5.} The {\em essential dimension} of an $n$-dimensional (oriented) {Lie} algebra $A_n$ is defined as the smallest possible number $n_e\leq n$, such that $A_n=A_{n_e}\oplus \R^{n-n_e}$. The essential-dimensional subset of $K^n$ is defined as $K^n_{de}=\{A\in K^n\vert n_e(A)=n\}$ \hfill$\Box$ \break {\bf Lemma 2.} {\em The subsets $\{A\in K^n\vert n_e(A)\leq m\}$ for any fixed $m\leq n$ need not to be closed. } \hfill$\Box$ \break This is due to the existence of transitions or limits of structure constants in NJNF such that one or more NJNF eigenvalues degenerate to another one (in Lemma 2 it is the eigenvalue $0$), initially distinct from them; in this case the algebraic multiplicity of this eigenvalue increases automatically, but its geometric multiplicity (expressed by its number of {Jordan} blocks) may remain constant, since an eigenvector of an eigenvalue different from the limit eigenvalue may converge to a principal (not necessarily eigen) vector of the limit eigenvalue ($0$ for Lemma 2). {\hfill \break} A {Lie} algebra characterized by structure constants $C^k_{ij}$ is called {\em unimodular} (on a corresponding {Lie} group) iff $ \mbox{${\rm tr }$} (C_i) = C^k_{ik} =0\ \forall i$, where the adjoint representation is generated by the matrices $C_i$. We denote the subset of all points in $K^n$ that correspond to unimodular {Lie} algebras by $U^n$, and set $U^n_*=U^n\cap K^n_*$. Since the zero set of a continuous function is always closed, we have {\hfill \break} {\bf Lemma 3.} {\em The unimodular subset $U^n\subset K^n$ is closed and compact. } \hfill$\Box$ \break For $n\geq 2$ the structure constants of any {Lie} algebra admit, as a tensor $C$, the irreducible decomposition \bSch \begin{equation} C^k_{ij}=D^k_{ij}+\delta^k_{[i}v_{j]} \end{equation} in a tracefree part $D$ with tensor components $D^k_{ij}$ (the trace free condition for $D$ can be written as $ \mbox{${\rm tr }$} (D_i)=D^k_{ik}=0\ \forall i$), and a vector part, constructed from a vector $v$ with components $v_i=C^j_{ij}/(1-n)$ and the Kronecker symbol of components $\delta^k_i$ (remind the sum convention over upper and lower indices and the convention to perform an antisymmetric sum over all permutations of the indices included in $_{[\quad ]}$). The {Lie} algebra is {\em unimodular} (like any associated connected {Lie} group), iff it is tracefree, $v\equiv 0$, and it is said to be of {\em pure vector type}, iff $D\equiv 0$. In this sense the unimodular and pure vector type are complementary. The class ${\V}^{(n)}$ of pure vector type is selfdual for all $n\geq 2$. It is the generalization of the unique non-{Abel}ian $2$-dimensional algebra $A_2$ (see Sec. 6) to arbitrary $n$. So for each $n$, there exists exactly one non-{Abel}ian pure vector type {Lie} algebra, denoted by $\V^{(n)}$ because for $n=3$ it is the Bianchi type \V. It has the {Abel}ian ideal $\I^{(n-1)}$ and $[\NJNF(\V^{(n)})]^k_j=\delta^k_j$. For convenience we mention explicitly the nonvanishing commutators of $\V^{(n)}$, for an adapted basis $\{e_1,\ldots,e_n\}$: \begin{equation} [e_n, e_i] =e_i,\ i=1,\ldots,{n-1}. \end{equation} The $3$-dimensional Heisenberg algebra (= Bianchi type \II) is defined in its NJNF by the nonvanishing commutators \begin{equation} [e_3, e_2] =e_1. \end{equation} By {Abel}ian embedding we define the class ${\II}^{(n)}:={\II}\oplus \R^{n-3}$ for $n\geq 3$. Like\II, it is unimodular and nilpotent of degree 2. ${\II}^{(n)}$ is non-selfdual for $n=3$ and selfdual for $n\geq 4$. Its nonvanishing structure constants for an adapted basis $\{e_1,\ldots,e_n\}$) are given by Eq. (5.3) with all indices increased by $n-3$. In {Schmidt} \bSch, an element $A_n\in K^n_*$ for which its closure in $K^n$ consists of 2 elements only, $\cl\{A_n\}=\{A_n,nA_1\}$, was called an {\em atom}. Here we will prefer to call equivalently $A_n$ an atom of $K^n_*$, iff its closure in $K^n_*$ is $\cl_{K^n_*}\{A_n\}=\{A_n\}$. Let us generalize this: {\hfill \break} {\bf Definition 6.} For any subset $S\subset K^n_{(or)}$, an element $A\in S$ is called an {\em $S$-atom}, iff it is closed w.r.t. $S$, i.e. $\cl_S\{A\}=\{A\}$. \hfill$\Box$ \break In the following, we call an $S$-atom also synonymously an {\em atom of $S$} and assume $S=K^n_*$ if not specified otherwise. Recall from {Schmidt} \bSch {\hfill \break} {\bf Theorem 1.} {\em For $n=2$ there is only 1 atom, $A_2\equiv {\V}^{(2)}$. {\hfill \break} For each $n\geq 3$ there exist exactly 2 atoms, the unimodular ${\II}^{(n)}$ and the pure vector type ${\V}^{(n)}$. } \hfill$\Box$ \break For $n\neq 3$ all atoms are selfdual. If we consider the corresponding atoms of $K^n_{*,or}$, then only for $n=3$ there is a difference to the nonoriented case. Instead of the unique non-selfdual atom $\II$ in $K^3$, there exist 2 non-selfdual atoms, $\II^R$ and $\II^L$, in $K^3_{or}$. For each $n\geq 3$ there is an algebra ${\IV}^{(n)}$, given by $[\NJNF(\IV^{(n)})]^k_j=\delta^k_j+\delta^k_{n-2}\delta^{n-1}_j$ w.r.t. to the {Abel}ian ideal $\I^{(n-1)}$. It is selfdual for $n\geq 4$ and non-selfdual for $n=3$. For convenience we mention explicitly the nonvanishing commutators of $\IV^{(n)}$, for an adapted basis $\{e_1,\ldots,e_n\}$ given by \begin{equation} [e_n, e_i] =e_i,\ i=1,\ldots,n-2, \quad [e_n, e_{n-1}] =e_{n-2}+e_{n-1}. \end{equation} $K^n_*$ is generated by infinitesimal deformations of the atoms; this means: $K^n_*$ itself is the only open subset of $K^n_*$ which contains all atoms. Since both, ${\IV}^{(n)} \to {\II}^{(n)}$ and ${\IV}^{(n)} \to {\V}^{(n)}$, it follows that $K^n_*$ is connected. {\hfill \break} Remark: Connectedness is trivial for $K^n$, but non-trivial for $K^n_*$. To understand better where the exceptionality of $n=3$ w.r.t. to duality comes from, realize that $n_e(\V^{(n)})=n$ but $n_e(\II^{(n)})=3$ for all $n\geq 3$. In particular, $\II^{(n)}$ has essential dimension $n_e=n$ only for $n=3$; for $n\geq 4$ it is decomposable and hence selfdual. More generally there holds {\hfill \break} {\bf Lemma 4.} {\em $K^n_{NSD}:=K^n {\setminus } K^n_{SD}$ is contained in the subset $K^n_{de}$ of $K^n$ for which $n_e=n$. } \hfill$\Box$ \break To overcome the difference in the essential dimension of the atoms for $n\geq 4$, let us search for atoms w.r.t. the subset $K^n_{de}$ of essential dimension $n_e=n$ in $K^n$. We find {\hfill \break} {\bf Theorem 3.} {\em The set $K^n_{de}$ has the following atoms: {\hfill \break} a) For $n\geq 2$ exactly $1$ pure vector type atom, called $\ve(n)$. } {\hfill \break} b) {\em For $n\geq 3$ a nilpotent unimodular atom, called $\ii(n)$, located in the subset of algebras with ideal $\I^{(n-1)}$. } {\hfill \break} c) {\em For $n\geq 5$ further $]\frac{2}{3}(n-4)[$ mixed type atoms, denoted $a_m(n)$, $m=2+[\frac{n-4}{3}],\ldots, n-3$, all located in the subset of algebras with ideal $\I^{(n-1)}$. {\hfill \break} (Here $[x]$ resp. $]x[$ denotes the largest/smallest integer less/greater or equal than x.) Within the subspace $K^n_{de\vert\I^{(n-1)}} \subset K^n_{de}$ given by $K^n_{de}$-algebras with ideal $\I^{(n-1)}$ there are no further $K^n_{de}$-atoms than that of {\rm a), b)} and {\rm c)}. $K^n_{de\vert\I^{(n-1)}}$ is connected. } {\hfill \break} Proof: a) By Theorem 1 the algebra $\V^{(n)}$ is an atom of $K^n_*$. Since $K^n_{de}\subset K^n_*$ and $n_e(\V^{(n)})=n$, it follows that $\V^{(n)}$ is an atom of $K^n_{de}$. Any algebra with only nonzero components $v_i$ in the vector $v$ of the decomposition (5.1) has a transition or limit to $\V^{(n)}$. Hence $\ve(n):=\V^{(n)}$ is the unique (pure) vector type $K^n_{de}$-atom. b) Some of the algebras with some vanishing component $v_i$ have transitions or limits to an algebra with $v\equiv 0$. Hence we have to search for unimodular $K^n_{de}$-atoms of essential dimension $n_e=n$. Such an atom is the nilpotent algebra $\ii(n)$ with $\NJNF(\ii(n))$ w.r.t. the ideal $I^{(n-1)}$ given for even $n$ as a direct sum of $1$ block of $\NJNF(\ii(4))$ and further blocks of $\NJNF(\ii(3))$, and for odd $n$ as a direct sum of $\NJNF(\ii(3))$ blocks only, where $$ \NJNF(\ii(3)):= \left( \begin{array}{cc} 0 & 1 \\ & 0 \end{array} \right) $$ and \begin{equation} \NJNF(\ii(4)):= \left( \begin{array}{ccc} 0 & 1 & \\ & 0 & 1 \\ & & 0 \end{array} \right). \end{equation} The algebra $\ii(n)$ is essential-dimensional, because any of its subalgebras $\ii(3)$ and $\ii(4)$ is so; it is an $K^n_{de}$-atom, because its only possible limits necessarily generate a $1\times 1$-block $(0)$ in its NJNF, thus decreasing $n_e$ at least by $1$. c) The mixed atoms can be characterized by their NJNF w.r.t. the ideal $\I^{(n-1)}$. Let us set $$ \NJNF(a_m(n)):=\NJNF(\ve(m+1))\oplus\NJNF(\ii(n-m)). $$ Since for $m=2+[\frac{n-4}{3}],\ldots, n-3$ the geometric multiplicity $m$ (= the number of {Jordan} blocks) of the eigenvalue $1$ is bigger than that of the eigenvalue $0$, any transition yields an additional {Jordan} block $0$ and hence leaves $K^n_{de}$. So, being essential-dimensional, $a_m(n)$ is an $K^n_{de}$-atom for $m=2+[\frac{n-4}{3}],\ldots, n-3$. Any algebra of $K^n_{de\vert\I^{(n-1)}}$ has a combination of transitions and parametric limits leading to at least one of the atoms from a), b) or c), depending on the degeneracy of its eigenvalues. The only nontrivial case, which remains to be checked, are the algebras with their NJNF w.r.t. an ideal $\I^{(n-1)}$ given as $\NJNF(\ve(m+1))\oplus\NJNF(\ii(n-m))$ where $m=1, \ldots, 1+[\frac{n-4}{3}]$ and $n\geq 4$. But any of these has a transition to $\ii(n)$. Let us now consider some algebra in $K^n_{de\vert\I^{(n-1)}}$ with only nondegenerate nonzero eigenvalues. By continuous deformation of its eigenvalues, such that every deformed algebra remains in $K^n_{de\vert\I^{(n-1)}}$, and transitions within $K^n_{de\vert\I^{(n-1)}}$ each of the atoms a), b) and c) can be reached. Since these have just been seen to be the only atoms of $K^n_{de\vert\I^{(n-1)}}$ it follows that $K^n_{de\vert\I^{(n-1)}}$ is connected. \hfill$\Box$ \break $K^n_{de}$ itself might have further atoms located in $K^n_{de}\setminus K^n_{de\vert\I^{(n-1)}}$. Since these are difficult to find, in general one cannot see whether $K^n_{de}$ is connected. The nonvanishing commutators of $\ii(n)$, $n\geq 3$, are given for an adapted basis $\{e_1,\ldots,e_n\}$ explicitly by $$ [e_n, e_2] =e_{1},\ [e_n, e_3] =e_{2},\ i=2,\ldots,n-1. $$ \begin{equation} [e_n, e_{2i+3}] =e_{2i+2},\quad i=1,\ldots,\frac{n-4}{2}. \end{equation} for $n$ even and by \begin{equation} [e_n, e_{2i}] =e_{2i-1},\quad i=1,\ldots,\frac{n-1}{2}. \end{equation} for $n$ odd. $\ii(3)\equiv \II$ is the {Heisenberg} algebra. The number of $\NJNF(\ii(3))$ blocks in its NJNF is even for $n\equiv 0 \,\mbox{mod}\, 4$ or $n\equiv 1 \,\mbox{mod}\, 4$, and it is odd for $n\equiv 2 \,\mbox{mod}\, 4$ or $n\equiv 3 \,\mbox{mod}\, 4$. The mixed types $a_m(n)$, $n\geq 5$, have respective algebraic and geometric multiplicities $m=2+[\frac{n-4}{3}],\ldots, n-3$ for the eigenvalue $1$. Their nonvanishing commutators are given w.r.t. an adapted basis {\hfill \break} $\{e_1,\ldots,e_n\}$ as $$ [e_n, e_1] =e_{1},\ldots, [e_n, e_m] =e_{m}, $$ $$ [e_n, e_{m+2}] =e_{m+1},\ [e_n, e_{m+3}] =e_{m+2},\quad i=2,\ldots,n-1, $$ \begin{equation} [e_n, e_{2i+m+3}] =e_{2i+m+2},\quad i=1,\ldots,\frac{n-m-4}{2}, \end{equation} for $n-m$ even, and by $$ [e_n, e_1] =e_{1},\ldots, [e_n, e_m] =e_{m}, $$ \begin{equation} [e_n, e_{2i+m}] =e_{2i+m-1},\quad i=1,\ldots,\frac{n-m-1}{2}, \end{equation} for $n-m$ odd. The reflection $e_1\to -e_1$ leaves $\ve(n)$ and any mixed type atom $a_m(n)$ invariant; hence all these atoms are selfdual. The nilpotent atom $\ii(n)$ remains as the only possibility for a non-selfdual $K^n_{de}$-atom within $K^n_{de\vert\I^{(n-1)}}$. Therefore, next we want to examine the orientation duality of $\ii(n)$. {\hfill \break} {\bf Theorem 4.} {\em For $n\geq 3$ the $K^n_{de}$-atom $\ii(n)$ is non-selfdual only if $n\equiv 3\, \mbox{\rm mod}\, 4$. {\hfill \break} $\ii(n)$ non-selfdual for $n\equiv 3\, \mbox{\rm mod}\, 4$ implies that $\ii(n)$ is a $K^n_{de}$-atom. } {\hfill \break} {Proof:} A combination of the reflections $e_n\to-e_n$ and $e_{2i}\to-e_{2i}$ for $i=1,\ldots,[\frac{n-1}{2}]$ leaves $\ii(n)$ invariant. The total number of these reflections is $[\frac{n+1}{2}]$, which is odd for $n\equiv 1 \,\mbox{mod}\, 4$ or $n\equiv 2 \,\mbox{mod}\, 4$. Furthermore for $n$ even, $e_i\to-e_i$, $i=1,\ldots,n-1$ yields a reflection keeping $\ii(n)$ invariant. So for all $n$ but $n\equiv 3 \,\mbox{mod}\, 4$ the algebra is selfdual. Any limit of $\ii(n)$ is selfdual, because it is a $K^n_{de}$-atom and any non-essential-dimensional algebra is decomposable and hence selfdual. Therefore non-selfduality for $n\equiv 3\, \mbox{\rm mod}\, 4$ implies that $\ii(n)$ is a $K^n_{de}$-atom. \hfill$\Box$ \break For $n\equiv 3 \,\mbox{mod}\, 4$ it was impossible to construct a reflection leaving $\ii(n)$ invariant. But when there is no such reflection the algebra is non-selfdual. Let us define for $n\geq 3$ an algebra $\iv(n)$ given for an adapted basis $\{e_1,\ldots,e_n\}$ by the nonvanishing commutators $$ [e_n, e_1] =e_{1},\ [e_n, e_2] =e_{1}+e_{2},\ [e_n, e_3] =e_{2}+e_{3},\ i=2,\ldots,n-1, $$ \begin{equation} [e_n, e_{2i+3}] =e_{2i+2}+e_{2i+3},\ i=1,\ldots,\frac{n-4}{2}, \end{equation} for $n$ even, and by \begin{equation} [e_n, e_{2i}] =e_{2i-1}+e_{2i},\ i=1,\ldots,\frac{n-1}{2}, \end{equation} for $n$ odd. By similar considerations as for $\ii(n)$ in Theorem $3$ one finds that $\iv(n)$ is non-selfdual for only for $n$ odd and selfdual for $n$ even. In any case it has an ideal $\I^{(n-1)}$ and for $n$ odd the geometric multiplicity of its eigenvalue $1$ of the NJNF w.r.t. $I^{(n-1)}$ can only be increased by yielding at least two $1\times 1$ blocks of that eigenvalue, hence the resulting algebra of such a transition is selfdual. Apart from limits which increase multiplicity, the only further limits of $\iv(n)$ are transitions with the eigenvalue becoming $0$, either to $\ii(n)$ or some limit thereof. But, according to Theorem 4, for $n\not\equiv\,\mbox{mod}\,4$, the algebra $\ii(n)$ is selfdual. Any limits of $\ii(n)$ are selfdual, because it is a $K^n_{de}$-atom and any non-essential-dimensional algebra is decomposable and hence selfdual. Hence non-selfduality of $\iv(n)$ for $n$ odd implies that $\iv(n)$ is a $K^n_{NSD}$-atom for $n\equiv 1 \,\mbox{mod}\, 4$. Non-selfduality of $\ii(n)$ for $n\equiv 3\, \mbox{\rm mod}\, 4$ implies further that $\iv(n)$ is no atom for $n\equiv 3\, \mbox{\rm mod}\, 4$. If for $n\equiv 3\, \mbox{\rm mod}\, 4$ resp. $n$ odd the algebras $\ii(n)$ resp. $\iv(n)$ are in fact non-selfdual, we get the {\hfill \break} {\bf Corollary.} {\em For odd $n\geq 3$ the set $K^n_{NSD}$ has an atom, located within the subspace $K^n_{NSD\vert\I^{(n-1)}}$ of non-selfdual algebras with ideal $I^{(n-1)}$. For $n\equiv 3\, \mbox{\rm mod}\, 4$ the atom is nilpotent unimodular, given by $\ii(n)$, and for $n\equiv 1\, \mbox{\rm mod}\, 4$ it is given by $\iv(n)$. } \hfill$\Box$ \break The selfduality of the $K^n_{de}$-atoms $a_m(n)$ and $\ve(n)$ excludes them as candidates for $K^n_{NSD}$-atoms. It remains an open problem to determine at least some $K^n_{NSD}$-atom for arbitrary even $n$, and all $K^n_{NSD}$-atoms for arbitrary $n$. For odd $n$, besides $\ii(n)$ or $\iv(n)$, there might be further $K^n_{NSD}$-atoms, even within $K^n_{NSD\vert\I^{(n-1)}}$. However, assume we succeed for some $n$ to determine all $K^n_{NSD}$-atoms and furthermore to show that $K^n_{NSD}$ is connected for that $n$. In Sec. 7 we will actually see that, for $n=3$ the {Heisenberg} algebra $\ii(3)$ is the only non-selfdual atom, hence $K^3_{NSD}$ is connected, and for $n=4$, with the topology of $K^4$ obtained in Sec. 6.3 and the non-selfdual algebras of Sec. 7.2.2, the resulting non-selfdual set $K^4_{NSD}$ will be connected, and its explicit structure will reveal the $K^4_{NSD}$-atoms. Let us assume in the following that for a given $n$ the space $K^n_{NSD}$ is connected. For $n\equiv 3 \,\mbox{mod}\, 4$ resp. $n\equiv 1 \,\mbox{mod}\, 4$ corresponding to the $K^n_{NSD}$-atom $\ii(n)$ resp. $\iv(n)$ there are in any case $2$ atoms of $K^n_{or,NSD}=K^n_+\oplus K^n_-$, either $\ii(n)^R$ and $\ii(n)^L$, or resp. $\iv(n)^R$ and $\iv(n)^L$. Similarly, we could pick for arbitrary $n$ any $K^n_{NSD}$-atom $a$ and will find a corresponding pair of $K^n_{or,NSD}$-atoms $a^R$ and $a^L$. At this place, let us make the convention to assign the right atom $a^R$ to $K^n_+$ and the left atom $a^L$ to $K^n_-$. Now consider all other pairs of dual points $A^R$ and $A^L$ in $K^n_{or,NSD}$, which constitute the preimage $\pi^{-1}(A)$ of a non-selfdual point $A\in K^n_{NSD}$. For any limit $A\to B$ or $C\to A$ in $K^n$ there exists a corresponding pair of limits $A^{R/L}\to B'$ or $C'\to A^{R/L}$ in $K^n_{or}$, with $B'\in \pi^{-1}(B)$ resp. $C'\in \pi^{-1}(C)$. Note however that there are no transitions or limits between conjugate points, neither $A^R\to A^L$ nor $A^L\to A^R$, because limits cannot reverse the orientation. If $B'$ or $C'$ is non-selfdual, we demand it, as the limit $B'=B^{R/L}$ resp. the prelimit $C'=C^{R/L}$ of $A^{R/L}$, to be contained in the same component of $K^n_{or,NSD}$ as $A^{R/L}$ itself. Under consideration of the transitivity of transitions in $K^n_{or}$ and use of the assumed connectedness of $K^n_{or,NSD}$, it follows from assignments for the non-selfdual atoms made above that, {\em all} right algebras have to be in $K^n_+$ and {\em all} left algebras have to be in $K^n_-$. If $K^n_{NSD}$ is connected, this choice is the only one which makes each of $K^n_+$ and $K^n_-$ connected and both disconnected to each other. Therefore it is the canonical assignment in the case of connected $K^n_{NSD}$. This will be the relevant situation in the following sections. {\hfill \break} For $n\geq 4$ let us define a selfdual algebra $A^a_{n,2}$ with {Abel}ian ideal $I^{(n-1)}$ by $\NJNF(A^a_{n,2}):=[a\cdot\NJNF(A_2)] \oplus \NJNF(\iv(n-1))$, where $\oplus$ denotes the direct sum of matrices. Now it is easy to prove {\hfill \break} {\bf Lemma 5.} {\em Within $K^n$ for $n\geq 3$, the subset $K^n_{SD}$ of selfdual elements in $K^n_{(or)}$ has the following properties: If there exists a non-selfdual algebra, which is the case at least for $n$ odd, then $K^n_{SD}$ is not open. For $n$ odd $K^n_{SD}$ is neither open nor closed. } {\hfill \break} {Proof:} Assume that there exists a non-selfdual algebra; such an algebra is given by $\iv(n)$ for $n$ odd. Then there is at least one $K^n_{NSD}$-atom. Any $K^n_{NSD}$-atom has a selfdual limit. Hence, there exists a selfdual limit from a non-selfdual sequence $ {\Rightarrow } K^n_{(or),NSD}$ not closed $ {\Rightarrow } K^n_{SD}$ not open. {\hfill \break} On the other hand, there are also non-selfdual limits from selfdual sequences, like $\VIo\to\II$ for $n=3$ and $A^a_{n,2} {\longrightarrow } \iv(n)$ with $a\to 1$ for odd $n>3$ $ {\Rightarrow } K^n_{SD}$ not closed for odd $n\geq 3$. \hfill$\Box$ \break Likewise, each of $K^n_\pm$ is neither open nor closed for $n$ odd. Note that $K^n_{SD}$ open would imply $K^n_{SD}=K^n$. In examination of duality of a given algebra, it is useful to remind the obvious {\hfill \break} {\bf Lemma 6.} {\em For an algebra $A\in K^n$, following assertions are equivalent: {\hfill \break} i) $A$ is selfdual. {\hfill \break} ii) The set $S(A)$ of all subalgebras of $A$ is selfdual. {\hfill \break} iii) The set $J(A)$ of all ideals of $A$ is selfdual. } \hfill$\Box$ \break Note that individual elements of $S(A)$ and $J(A)$ taken for themselves can be non-selfdual while $A$ is selfdual. Finally we deal with the case of simple {Lie} algebras. {\hfill \break} {\bf Lemma 7.} {\em Simple {Lie} algebras are not selfdual. } {\hfill \break} {Proof:} A simple $n$-dimensional {Lie} algebra $A_n$ can be characterized by a {Cartan-Weyl} basis. Such a basis consisting of generators $H_i$, $i=1,\ldots,l=\mbox{rank}A_n$, which span a maximal {Abel}ian subalgebra (usually called {Cartan} subalgebra) and $n-l$ generators $E_\alpha$, each satisfying, for any nonvanishing generator $H=\alpha^i H_i$ of the {Cartan} subalgebra, a root equation $[H, E_\alpha]=\alpha E_\alpha$ $\ (\ast)$ with root $\alpha=\alpha^i\alpha_i$. The commutators $[E_\alpha,E_\beta]=N_{\alpha\beta}E_{\alpha+\beta}$ $\ (\ast\ast)$ for $\alpha+\beta\neq 0$ and $[E_\alpha,E_{-\alpha}]=H$ $\ (\ast\ast\ast)$ are nonvanishing. {}From the root equations $(\ast)$ we see that for any nonvanishing {Cartan} subalgebra element $H$ (given by its coroots $\alpha^i$) the reflection $H\to-H$ changes the algebra. Furthermore by $(\ast\ast)$ and $(\ast\ast\ast)$ also none of the reflections $E_\alpha\to-E_\alpha$ keeps the algebra invariant. Since there is no reflection keeping the algebra invariant it can not be selfdual. \hfill$\Box$ \break For considerations of the topological structure of $K^3$ and $K^4$ in Sec. 6 and 7 respectively, we will define the notion of parametrical connectedness of points in $K^n$ like following: {\hfill \break} {\bf Definition 7.} $X, Y\in K^n$ are called {\em parametrically connected} iff there exists a continuous curve $c: [0,1] \to K^n$ with $c(0)=X$ and $c(1)=Y$ such that, for all $t_1\leq t_2\in [0,1]$ with $c(t_1)\neq c(t_2)$, there exist some $t_0\in [t_1,t_2]$ such that $c(t_1)\neq c(t_0)\neq c(t_2)$. Otherwise $X, Y\in K^n$ are said to be {\em parametrically disconnected}. \hfill $\Box$\break Note that, in the topology $\kappa^n$, arcwise connectedness does not imply parametrical connectedness as defined above. Furthermore, a set $S\subset K^n$ is called parametrically connected, iff any two points $X,Y\in S$ are parametrically connected in $S$. $S\subset K^n$ is a {\em parametrically connected component} iff $S$ is parametrically connected but not a proper subset of another parametrically connected set. {\hfill \break} \section{\bf Topology of $K^n$ for $n\leq 4$} \setcounter{equation}{0} Sec. 6.1 resumes already existing results for $n\leq 3$, Sec. 6.2 describes in detail the components and transitions of $K^4$, and Sec. 6.3 gives some overview over the topological structure of $K^4$. \subsection{\bf Structure of $K^n$ for $n\leq 3$} The {Lie} algebras with $n\leq 3$ are well known and listed, e.g. by {Patera} and {Winternitz} \PaW. $K^2$ contains only 2 elements, the {Abel}ian $2A_1$ and $A_2$ represented by the algebra with $[e_2,e_1]=e_1$ as only nonvanishing bracket. So $A_2$ has the ideal $J_1=A_1$ spanned by $\{e_1\}$, and is characterized by $C_{<2>}=(1)\neq 0$ in contrast to $2A_1$. Note that $A_2\equiv {\V}^{(2)}$. Obviously $\dim K^2_*=0$ and the unimodular subset $U^2_*\subset K^2_*$ is empty. The elements of $K^3$ correspond to the famous {Bianchi} (or {Bianchi-Behr}) types. They have been classified independently first by S. {Lie} \Lie\ and then by L. {Bianchi} \Bi. For their systematic derivation and explanation of their role for cosmological models see e.g. {Kramer, Stephani} et al. \Kr. For convenience of the reader we give for each of the Bianchi types I up to IX an explicit description by the commutators of its generators $e_1, e_2, e_3$ according to {Landau-Lifschitz} \Lan: Types I,II and VIII/IX are given by basic commutators $$ [e_1,e_2]=n_3 e_3, [e_2,e_3]=n_1 e_1, [e_3,e_1]=n_2 e_2, $$ with triplets $(n_1,n_2,n_3)$ respectively given by $(0,0,0), (1,0,0)$ and $(1,1,\mp 1)$. The 1-parameter families \VIh/ \mbox{${\rm VII}_h$} with $h\geq 0$ are given respectively by $$ [e_1,e_2]=e_3+he_2, [e_2,e_3]=0, [e_1,e_3]=\pm e_2+he_3, $$ and especially it is $\III={\rm VI}_1$. IV resp. V are given by $$ [e_1,e_2]=be_3+e_2, [e_2,e_3]=0, [e_1,e_3]=e_3 $$ with $b=1$ resp. $b=0$. The 3-dimensional real {Lie} algebras in the notation of {Patera} and {Winternitz} \PaW\ can be characterized by their NJNF, which is simultaneously the normal form (see Eqs. (6.1) up to (6.4) below) of the {Bianchi} types associated to them like in Table 1. \bigskip {\hfill \break} {\normalsize \begin{tabular}{ccccccccccc} $3A_1$&$A_1\oplus A_2$&$A_{3,1}$&$A_{3,2}$&$A_{3,3}$ &$A_{3,4}$&$A^a_{3,5}$&$A_{3,6}$&$A^a_{3,7}$&$A_{3,8}$&$A_{3,9}$\\ I&III&II&IV&V&\VIo&\VIh& \mbox{${\rm VII}_0$} & \mbox{${\rm VII}_h$} &VIII&IX \end{tabular} \begin{center} \begin{tabular}{ll} Table 1: &Inequivalent 3-dim. {Lie} algebras as denoted in \PaW\ (upper row)\\ &and corresponding {Bianchi} types (lower row). \end{tabular} \end{center} \smallskip } For convenience we explicitly give the nonvanishing commutators for the indecomposable algebras $A_{3,1}$ up to $A_{3,9}$ from \PaW: $$ A_{3,1}: [e_2,e_3]=e_1; $$ $$ A_{3,2}: [e_1,e_3]=e_1, [e_2,e_3]=e_1+e_2; $$ $$ A_{3,3}: [e_1,e_3]=e_1, [e_2,e_3]=e_2; $$ $$ A_{3,4}: [e_1,e_3]=e_1, [e_2,e_3]=-e_2; $$ $$ A^a_{3,5}: [e_1,e_3]=e_1, [e_2,e_3]=ae_2,\ 0<\vert a\vert<1; $$ $$ A_{3,6}: [e_1,e_3]=-e_2, [e_2,e_3]=e_1; $$ $$ A^a_{3,7}: [e_1,e_3]=ae_1-e_2, [e_2,e_3]=e_1+ae_2,\ 0<a; $$ $$ A_{3,8}: [e_1,e_2]=e_1, [e_2,e_3]=e_3, [e_3,e_1]=2e_2; $$ $$ A_{3,9}: [e_1,e_2]=e_3, [e_3,e_1]=e_2, [e_2,e_3]=e_1. $$ In $3$ dimensions, all solvable algebras contain the {Abel}ian ideal $J_2=2A_1$. Therefore they can be characterized by their NJNF. $$ \NJNF(\I) = \left( \begin{array}{cc} 0 & \\ & 0 \end{array} \right) , $$ $$ \NJNF(\III) = \left( \begin{array}{cc} 1 & \\ & 0 \end{array} \right) , $$ $$ \NJNF(\II) = \left( \begin{array}{cc} 0 & 1 \\ & 0 \end{array} \right) , \qquad \II={\II}^{(3)} , $$ $$ \NJNF(\IV) = \left( \begin{array}{cc} 1 & 1 \\ & 1 \end{array} \right) , \qquad \IV={\IV}^{(3)} , $$ \begin{equation} \NJNF(\V) = \left( \begin{array}{cc} 1 & \\ & 1 \end{array} \right) , \qquad \V={\V}^{(3)} . \end{equation} \begin{equation} \NJNF(\VIo) = \left( \begin{array}{cc} 1 & \\ & -1 \end{array} \right) , \ \NJNF(\VIh) = \left( \begin{array}{cc} 1 & \\ & a \end{array} \right) . \end{equation} In Eq. (6.2) the range $0< h< \infty, h\neq 1$ of the parameter according to {Landau-Lifschitz} \Lan, denoted here by $h$, is monotonously homeomorphic to the range $-1< a< 1, a\neq 0$. $h=1$ resp. $a=0$ yields a decomposable algebra, namely \III. \begin{equation} \NJNF( \mbox{${\rm VII}_0$} ) = \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right) , \ \NJNF( \mbox{${\rm VII}_h$} ) = \left( \begin{array}{cc} a & 1 \\ -1 & a \end{array} \right) . \end{equation} In Eq. (6.3) the range $0< h< \infty$ of the parameter according to {Landau-Lifschitz} \Lan, denoted here by $h$, is monotonously homeomorphic to the range $0< a< \infty$. {\hfill \break} Note that for a topological characterization of $K^3$ it is sufficient to know the relation of the parameters $a$ and $h$ in (6.2) and (6.3) at points of qualitative change in the NJNF and to ensure homeomorphisms of the ranges between these critical points. This is precisely the data we have given above. (Though the explicit relation of $a$ and $h$ follows from the equivalence transform to normal form, here we do not need to calculate it.) Both \VIh and \mbox{${\rm VII}_h$} are unimodular for $h=0$ and converge to \II for $0\leq h<\infty$ and to \IV for $h\to \infty$. {\hfill \break} The simple algebras $ \mbox{${\rm VIII}$} =su(1,1)$ and $\IX=su(2)$ are described respectively by the 3 matrices \begin{equation} C_{<3>} = \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right) , \qquad C_{<1>} = -C_{<2>} = \left( \begin{array}{cc} 0 & 1 \\ \pm 1 & 0 \end{array} \right) . \end{equation} $\NJNF(C_{<3>})=\NJNF( \mbox{${\rm VII}_0$} )$ for both $ \mbox{${\rm VIII}$} $ and $\IX$, {\hfill \break} but $\NJNF(C_{<1>})=\NJNF(C_{<2>})$ is equal to $\NJNF( \mbox{${\rm VII}_0$} )$ for $ \mbox{${\rm VIII}$} $ and to $\NJNF(\VIo)$ for $\IX$. Therefore $\IX\to \mbox{${\rm VII}_0$} $, but $\IX\not\to\VIo$, but both $ \mbox{${\rm VIII}$} \to\VIo$ and $ \mbox{${\rm VIII}$} \to \mbox{${\rm VII}_0$} $. {\hfill \break} Considering all components, their parametrical limits and transitions together, we get the full topological structure of $K^3$, which includes a transitive network of nearest neighbour transitions between different components. The network has been depicted already by {Mac Callum} \Mac\ and its transitivity was outlined by {Schmidt} \aSch. We have $\dim K^3=1$, since its largest parametrically connected components are 1-dimensional. {\hfill \break} For the unimodular subvariety $U^3_*\subset K^3_*$ it is $\dim U^3_* =0$, and $\{ \mbox{${\rm VIII}$} ,\IX\}\subset U^3_*$ is a minimal dense subset of isolated points. Fig. 1 shows the transitive network of transitions in $K^3_*$, with unimodular points encircled. \vspace*{12.7truecm} \begin{center} {\normalsize Fig. 1: Transitive network of transitions in $K^3_*$.} \end{center} {\newpage } \noindent \subsection {\bf Components of $K^4$, transitions and parametrical limits} The real 4-dimensional {Lie} algebras have been classified by {Mubarakzyanov} \bMu\ and listed by {Patera} and {Winternitz} \PaW. An early, somehow more coarse classification has been given by {Petrov} \Pe. For the convenience of the reader we explicitly give this classification in terms of nonvanishing basic commutators. In order to avoid confusion with the $3$-dimensional {Bianchi} types we alter the notation of {Petrov}'s classes \Pe\ from I,\ldots,VIII to $\wp_i,\ i=1,\ldots,8$. The subclasses ${\rm VI}_{1/4}$ are written as \PVIac respectively, and ${\rm VI}_2$ together with ${\rm VI}_3$ are resumed in a single class \PVIb in order to correspond to distinct classes of \PaW. With this notation {Petrov}'s classes are characterized like following: Solvable algebras, without {Abel}ian subgroup $3A_1$: $$ \PI: [e_2,e_3]=e_1, [e_1,e_4]=ce_1, [e_2,e_4]=e_2, [e_3,e_4]=(c-1)e_3,\ c\in\R; $$ $$ \PII: [e_2,e_3]=e_1, [e_1,e_4]=2e_1, [e_2,e_4]=e_2, [e_3,e_4]=e_2+e_3; $$ $$ \PIII: [e_2,e_3]=e_1, [e_1,e_4]=qe_1, [e_2,e_4]=e_3, [e_3,e_4]=-e_2+qe_3,\ q^2<4; $$ $$ \PIV: [e_2,e_3]=e_2, [e_1,e_4]=e_1; $$ $$ \PV: [e_2,e_3]=e_2, [e_3,e_1]=-e_1, [e_1,e_4]=e_2, [e_2,e_4]=-e_1; $$ Solvable algebras, with {Abel}ian subgroup $3A_1$: $$ \PVIa: [e_1,e_4]=ae_1+be_4, [e_2,e_4]=ce_2+de_4, [e_3,e_4]=ee_3+fe_4, $$ with real tuples $(a,b,c,d,e,f)$ of the form $(0,0,0,0,0,0)$, $(0,1,0,1,0,0)$, $(0,1,0,1,0,1)$, $(1,1,0,0,0,0)$ or $(1,0,c,0,e,0)$; $$ \PVIb: [e_1,e_4]=ke_1+e_2, [e_2,e_4]=ke_2+de_3, [e_3,e_4]=ee_3,\ k\in \R,\ d,e\in\{0,1\}; $$ $$ \PVIc: [e_1,e_4]=ke_1+e_2, [e_2,e_4]=-e_1+ke_2, [e_3,e_4]=le_3,\ k,l\in \R; $$ Non solvable algebras: $$ \PVII: [e_1,e_2]=e_1, [e_2,e_3]=e_3, [e_3,e_1]=-2e_2; $$ $$ \PVIII: [e_1,e_2]=e_3, [e_2,e_3]=e_1, [e_3,e_1]=e_2. $$ {\newpage } In the following we use the characterization of equivalence classes by their NJNF, according to Sec. 3, in order to find relative positions of the equivalence classes in $K^4$, possible transitions between them and parametrical limits of parametrically connected components of $K^4$. {\hfill \break} \nl {\bf 6.2.1 Decomposable {Lie} algebras} {\hfill \break} \nl A decomposable 4-dimensional {Lie} algebra can have the structures $4A_1$, $2A_1\oplus A_2$, $2A_2$ or $A_1\oplus A_3$. The first 3 possibilities are unique, since $A_1$ is the unique 1-dim. {Lie} algebra and $A_2$ is the unique non {Abel}ian 2-dim. {Lie} algebra. Note that $2A_2\equiv \PIV$ in {Petrov}'s classification \Pe. $A_1\oplus A_3$ consists of 9 classes, given by $\{A_{3,i}\}_{i=1,\ldots,9}$ listed in Table 1. It is $A_1\oplus \II\equiv {\II}^{(4)}$. $A_1\oplus \mbox{${\rm VIII}$} $ and $A_1\oplus\IX$ are the same as in \Pe\ the \PVII and \PVIII respectively. {\hfill \break} Transitions and limits: Besides the transitions and limits induced by {Abel}ian embedding $\oplus \R$ of transitions in $K^3$, there are further transitions, which prevent the embedding $\oplus \R$ to be a homeomorphism. So for example $\V\oplus \R\to \II\oplus \R$, but $\V\not\to \II$. This demonstrates that, while $\V$ is an atom, $\V\oplus \R$ is not. Furthermore $\VIo\oplus \R$ and $ \mbox{${\rm VII}_0$} \oplus \R$ both go first to $A_{4,1}$ and then to $\II\oplus \R$. $ \mbox{${\rm VIII}$} \oplus \R$ has a limit in the non decomposable $A_{4,8}$ and, like $\IX\oplus \R$, also in $A_{4,10}$, as described below. The algebra $2A_2$ in spite of being decomposable is not the limit of any other algebra in $K^4$. It has transitions to $\VIh\oplus \R$ with $h\geq 0$, to $A_{4,3}$ and to $A^0_{4,9}$. {\hfill \break} \nl \noindent {\bf 6.2.2 Indecomposable {Lie} algebras} {\hfill \break} \nl Coarsely these algebras have already been classified by {Petrov} \Pe. Table 2 relates his classification to that of {Patera} and {Winternitz} \PaW. \smallskip {\normalsize \begin{center} \begin{tabular}{ccccccc} $A_{4,1..4}$&$A_{4,5}$&$A_{4,6}$&$A_{4,7}$&$A_{4,8/9}$&$A_{4,10/11}$&$A_{4,12}$ \smallskip\\ \PVIb &\PVIa &\PVIc &\PII &\PI &\PIII &\PV \end{tabular}\smallskip\\ \begin{tabular}{ll} Table 2:&Classification of {Petrov} \Pe\ (lower row) and \PaW\ (upper row)\\ & of 4-dim. {Lie} algebras except decomposable ones. \end{tabular} \end{center} \smallskip } The algebra \PIII, $q=0$ is the same as $A_{4,10}$. It is the only indecomposable $4$-dimensional algebra that corresponds to a maximal isometry group of a $3$-dimensional homogeneous {Riemann}ian space (see Sec. 8, 9 below and {Bona} and {Coll} \Bo, Theorem 1). For convenience we explicitly give the nonvanishing commutators of the indecomposable algebras $A_{4,1}$ up to $A_{4,12}$ according to \PaW: $$ A_{4,1}: [e_2,e_4]=e_1, [e_3,e_4]=e_2; $$ $$ A^a_{4,2}: [e_1,e_4]=ae_1, [e_2,e_4]=e_2, [e_3,e_4]=e_2+e_3,\ a\neq 0; $$ $$ A_{4,3}: [e_1,e_4]=e_1, [e_3,e_4]=e_2; $$ $$ A_{4,4}: [e_1,e_4]=e_1, [e_2,e_4]=e_1+e_2, [e_3,e_4]=e_2+e_3; $$ $$ A^{a,b}_{4,5}: [e_1,e_4]=e_1, [e_2,e_4]=ae_2, [e_3,e_4]=be_3,\ -1\leq a\leq b\leq 1,\ ab\neq 0; $$ $$ A^{a,b}_{4,6}: [e_1,e_4]=ae_1, [e_2,e_4]=be_2-e_3, [e_3,e_4]=e_2+be_3,\ b\geq 0,\ a\neq 0; $$ $$ A_{4,7}: [e_1,e_4]=2e_1, [e_2,e_4]=e_2, [e_3,e_4]=e_2+e_3, [e_2,e_3]=e_1; $$ $$ A_{4,8}: [e_2,e_3]=e_1, [e_2,e_4]=e_2, [e_3,e_4]=-e_3; $$ $$ A^b_{4,9}: [e_2,e_3]=e_1, [e_1,e_4]=(1+b)e_1, [e_2,e_4]=e_2, [e_3,e_4]=be_3,\ -1<b\leq 1; $$ $A_{4,8}$ is the parametrical limit of $A^b_{4,9}$ for $b\to -1$; hence by {Mubarakzyanov} \aMu\ and {Petrov} \Pe\ $A_{4,8}$ and $A_{4,9}$ are subsumed in a single $1$-parameter set. $$ A_{4,10}: [e_2,e_3]=e_1, [e_2,e_4]=-e_3, [e_3,e_4]=e_2; $$ $$ A^a_{4,11}: [e_2,e_3]=e_1, [e_1,e_4]=2ae_1, [e_2,e_4]=ae_2-e_3, [e_3,e_4]=e_2+ae_3,\ 0<a; $$ $A_{4,10}$ is the parametrical limit of $A^a_{4,11}$ for $a\to 0$; hence by {Mubarakzyanov} \aMu\ and {Petrov} \Pe\ $A_{4,10}$ and $A_{4,11}$ are subsumed in a single $1$-parameter set. $$ A_{4,12}: [e_1,e_3]=e_1, [e_2,e_3]=e_2, [e_1,e_4]=-e_2, [e_2,e_4]=e_1. $$ The only difference of this classification to that of {Mubarakzyanov} \aMu\ is that, unlike there, here the endpoints $A_{4,8}$ and $A_{4,10}$ are distinguished against the rest of the $1$-parameter sets $A_{4,9}$ and $A_{4,11}$ respectively. In the following we reclassify these algebras by their NJNF. {\hfill \break} {\hfill \break} {\bf a) Algebras with an {Abel}ian ideal $J_3=3A_1\equiv \I$} {\hfill \break} {\hfill \break} These are the algebras of type \PVI. In the following the cases i) and ii) correspond to \PVIb, case iii) to \PVIa and case iv) to \PVIc. \medskip\hfill\break i) 1 eigenvalue with 1 Jordan block: {\hfill \break} Either the eigenvalue is $ {\lambda } = 0$ or otherwise it can be normalized to $ {\lambda } = 1$. {\newpage } \begin{equation} \NJNF(A_{4,1}) = \left( \begin{array}{ccc} 0 & 1 & \\ & 0 & 1 \\ & & 0 \end{array} \right) , \ \NJNF(A_{4,4}) = \left( \begin{array}{ccc} 1 & 1 & \\ & 1 & 1 \\ & & 1 \end{array} \right) . \end{equation} Transitions: Obviously $A_{4,4}\to A_{4,1}$ and, by increasing the geometric multiplicity, $A_{4,4}\to{\IV}^{(4)}\equiv A^1_{4,2}$ resp. $A_{4,1}\to{\II}^{(4)}\equiv \II\oplus \R$.\medskip\hfill\break ii) Maximally 2 eigenvalues with together 2 Jordan blocks: {\hfill \break} Here JNF$(A)$ consists of both a $1\times 1$ and a $2\times 2$ Jordan block, with eigenvalues $ {\lambda } _1$ and $ {\lambda } _2$ respectively. If $ {\lambda } _1 = 0$, the algebra would become decomposable ($\II \oplus \R$ if $ {\lambda } _2 = 0$, $\IV \oplus \R$ if $ {\lambda } _2 \neq 0$). Therefore assume $ {\lambda } _1 = a \neq 0$. Either $ {\lambda } _2 = 0$, then $ {\lambda } _1 = 1$ after normalization, or $ {\lambda } _2 \neq 0$, then it can be normalized to $ {\lambda } _2 = 1$. If $ {\lambda } _2 = {\lambda } _1 = a$, there is only 1 eigenvalue, which can be normalized to $a=1$. Note that $A^1_{4,2}\equiv {\IV}^{(4)}$, which is a case to be considered separately. \begin{equation} \NJNF(A_{4,3}) = \left( \begin{array}{ccc} 1 & & \\ & 0 & 1 \\ & & 0 \end{array} \right) , \ \NJNF(A^a_{4,2}) = \left( \begin{array}{ccc} a & & \\ & 1 & 1 \\ & & 1 \end{array} \right) . \end{equation} Transitions and limits: From $A^a_{4,2}$ with $a\neq 0,1$ to $\IV\oplus \R$ for $a\to 0$, to $A_{4,4}$ for $a\to 1$, and to $A_{4,3}$ for $\vert a\vert\to\infty$. By increasing the geometric multiplicity, to $A^{1,a}_{4,5}$ for $0<\vert a\vert <1$, to $A^{1,-1}_{4,5}=A^{-1,-1}_{4,5}$ for $a=-1$ and to $A^{\frac{1}{a},\frac{1}{a}}_{4,5}$ for $1<\vert a\vert <\infty$. Also generally $A^a_{4,2}\to A_{4,1}$. {\hfill \break} {}From ${\IV}^{(4)}\equiv A^{1}_{4,2}$ to ${\II}^{(4)}\equiv \II \oplus \R$ and ${\V}^{(4)}\equiv A^{1,1}_{4,5}$, according to the remark at the theorem in Sec. 4. {\hfill \break} {}From $A_{4,3}$ to $A_{4,1}$ and, by increasing of geometric multiplicity, to $\III\oplus \R$.\medskip\hfill\break \noindent iii) 3 real eigenvalues as Jordan blocks: {\hfill \break} Assuming the largest eigenvalue normalized to $ {\lambda } _1 = 1$, there remain $ {\lambda } _2=a$ and $ {\lambda } _3=b$ with $-1\leq b\leq a\leq 1$. If $a\cdot b = 0$, the algebra becomes decomposable ($a=b=0$ yields $\III \oplus \R$, for $a=1,b=0$ it is $\V \oplus \R$, for $a=0,b=-1$ it is $\VIo \oplus \R$ and otherwise $a=0$ or $b=0$ yields $\VIh \oplus \R$). Therefore assume $a\cdot b\neq 0$. The case $a=b=1$ (single 3-fold degenerate eigenvalue) corresponds to the pure vector type $A^{1,1}_{4,5} \equiv \Vv \neq \V \oplus \R$. In $A^{1,b}_{4,5}$ and $A^{a,a}_{4,5}$ there are 2 eigenvalues, one of them 2-fold degenerate. {\newpage } \noindent For the nondegenerate case, $-1\le b< a< 1$. Note that $A^{a,b}_{4,5}=A^{b,a}_{4,5}$, since permutations are in ${\GL}(4)$. \begin{equation} \NJNF(A^{a,b}_{4,5}) = \left( \begin{array}{ccc} 1 & & \\ & a & \\ & & b \end{array} \right) . \end{equation} Transitions and limits: From $A^{a,b}_{4,5}, a>b,$ to $A^{\frac{1}{a}}_{4,2}$ for ${b\to a}$, to $A^{b}_{4,2}$ for ${a\to 1}$. To $A_{4,4}$ for ${{a\to 1}\atop{b\to 1}}$, to $\IV\oplus \R$ for ${{a\to 1}\atop{b\to 0}}$, to $ \mbox{${\rm VII}_h$} \oplus \R, 1<h<\infty,$ for ${{a\not\to 0,1}\atop{b\to 0}}$, to $A_{4,3}$ for ${{a\to 0}\atop{b\to 0}}$, to $\VIh\oplus \R, 0\leq h<1,$ for ${{a\to 0}\atop{b\not\to 0}}$, and to $A^{-1}_{4,2}$ for ${b\to -1}$ and ${a\to \pm 1}$. Also generally $A^{a,b}_{4,5}\to A_{4,1}$. {\hfill \break} Note furthermore that $A^{a,-1}_{4,5}=A^{-a,-1}_{4,5}$. {\hfill \break} {}From $A^{1,b}_{4,5}$ to ${\IV}^{(4)}\equiv A^{1}_{4,2}$ for ${b\to 1}$, and to ${\V}\oplus \R$ for ${b\to 0}$. Generally $A^{1,b}_{4,5}\to {\II}^{(4)}$. {\hfill \break} {}From $A^{a,a}_{4,5}$ to ${\IV}^{(4)}\equiv A^{1}_{4,2}$ for ${a\to 1}$, and to ${\III}\oplus \R$ for ${a\to 0}$. Generally $A^{a,a}_{4,5}\to {\II}^{(4)}$. {\hfill \break} {}From ${\V}^{(4)}\equiv A^{1,1}_{4,5}$ only to $4A_1$, according to the theorem of Sec. 4.\medskip\hfill\break iv) 1 real eigenvalue and 2 complex conjugates: {\hfill \break} If $ {\lambda } _{2,3}=r(\cos\theta \pm i\sin\theta)$, by normalization $r\sin\theta = 1$ can be achieved, if $ {\lambda } _2 \neq {\lambda } _3$ is assured (otherwise the Jordan block becomes diagonal). Set then $r\cos\theta = b$ and $ {\lambda } _1=a$. Demand $a\neq 0$ to exclude decomposability ($a = 0$ yields $ \mbox{${\rm VII}_h$} \oplus \R$ and $b=0$ then corresponds to $h=0$) and without restriction $b\geq 0$. \begin{equation} \NJNF(A^{a,b}_{4,6}) = \left( \begin{array}{ccc} a & & \\ & b & 1 \\ &-1 & b \end{array} \right) . \end{equation} Transitions and limits: For $a\to 0$, $A^{a,b}_{4,6}\to \mbox{${\rm VII}_h$} \oplus \R$, with $0\leq h<\infty$ corresponding to $0\leq b<\infty$. For a fixed ratio $\frac{a}{b}$ and $b\to\infty$ there is a limit to $A^{\frac{a}{b}}_{4,2}$, if $a\neq b$, and to $A_{4,4}$, if $a=b$. $A^{a,b}_{4,6}\to A_{4,3}$ for $b$ finite (esp. $b=0$) and $\vert a\vert\to\infty$, and $A^{a,b}_{4,6}\to \IV\oplus \R$ for $a$ finite (esp. $a=0$) and $b\to\infty$. Also generally $A^{a,b}_{4,6}\to A_{4,1}$. {\hfill \break} Note furthermore that $A^{a,0}_{4,6}=A^{-a,0}_{4,6}$. {\hfill \break} {\hfill \break} {\newpage } \noindent {\bf b) Algebras with a nilpotent ideal $J_3=A_{3,1}\equiv{\rm II}$} {\hfill \break} {\hfill \break} In the following case i) corresponds to \PII, case ii) to \PI and case iii) to \PIII.\medskip\hfill\break i) 2 eigenvalues with together 2 Jordan blocks: {\hfill \break} \begin{equation} \NJNF(A_{4,7}) = \left( \begin{array}{ccc} 2 & & \\ & 1 & 1 \\ & & 1 \end{array} \right) . \end{equation} Transitions: $A_{4,7}\to A^{2}_{4,2}$ for $J_3\to \I$. Furthermore $A_{4,7}\to A^{1}_{4,9}$.\medskip\hfill\break ii) 3 real eigenvalues as Jordan blocks: {\hfill \break} $$ \NJNF(A^{b}_{4,9}) = \left( \begin{array}{ccc} 1+b & & \\ & 1 & \\ & & b \end{array} \right) , 0 < \vert b\vert < 1 , $$ $$ \NJNF(A^{0}_{4,9}) = \left( \begin{array}{ccc} 1 & & \\ & 1 & \\ & & 0 \end{array} \right) , \qquad \NJNF(A^{1}_{4,9}) = \left( \begin{array}{ccc} 2 & & \\ & 1 & \\ & & 1 \end{array} \right) , $$ \begin{equation} \NJNF(A_{4,8}) = \left( \begin{array}{ccc} 0 & & \\ & 1 & \\ & &-1 \end{array} \right) . \end{equation} Transitions: From $A^b_{4,9}$ to $A_{4,8}$ for $b\to -1$, to $A^0_{4,9}$ for $b\to 0$, and to $A_{4,7}$ for $b\to 1$. Furthermore, for $J_3\to \I$, to $A^{\frac{1}{1+b},\frac{b}{1+b}}_{4,5}$ if $0<b<1$, and to $A^{{1+b},{b}}_{4,5}$ if $-1<b<0$. {\hfill \break} For $J_3\to \I$, $A^1_{4,9}\to A^{\frac{1}{2},\frac{1}{2}}_{4,5}$ and $A_{4,8} \to \VIo\oplus \R$. $A^0_{4,9}$ goes to $\IV\oplus \R$ and further to $\V\oplus \R$. Since $ \mbox{${\rm VIII}$} \oplus \R\to A_{4,8}$, the latter is a limit from a decomposable algebra.\medskip\hfill\break iii) 1 real eigenvalue and 2 complex conjugates: {\hfill \break} $$ \NJNF(A^{a}_{4,11}) = \left( \begin{array}{ccc} 2a & & \\ & a & 1 \\ &-1 & a \end{array} \right) , a > 0 , $$ \begin{equation} \NJNF(A_{4,10}) = \left( \begin{array}{ccc} 0 & & \\ & 0 & 1 \\ &-1 & 0 \end{array} \right) . \end{equation} {\newpage } Transitions: From $A^a_{4,11}$ to $A_{4,10}$ for $a\to 0$, to $A^1_{4,9}$ for $a\to \infty$ and, for $J_3\to \I$, to $A^{{2a},{a}}_{4,6}$. {\hfill \break} For $J_3\to \I$, $A_{4,10}\to \mbox{${\rm VII}_0$} \oplus \R$. Furthermore both $ \mbox{${\rm VIII}$} \oplus \R\to A_{4,10}$ and $\IX\oplus \R\to A_{4,10}$. {\hfill \break} {\hfill \break} {\bf c) Algebras with a pure vector type ideal $J_3=A_{3,3}\equiv{\rm V}$} {\hfill \break} {\hfill \break} This case corresponds to type \PV. \begin{equation} \NJNF(A_{4,12}) = \left( \begin{array}{ccc} 0 & 1 & \\ -1& 0 & \\ & & 0 \end{array} \right) . \end{equation} Transitions: $A_{4,12}$ goes to $\V\oplus \R$, to $ \mbox{${\rm VII}_h$} \oplus \R$, especially for $J_3\to \I$ to $ \mbox{${\rm VII}_0$} \oplus \R$, and to $A_{4,9}$. {\hfill \break} {\hfill \break} \subsection{\bf The topological structure of $K^4$} Since we know all components, their parametrical limits and transitions in $K^4$, we can now put them together, in order to determine the full topological structure of $K^4$. Fig. 2 a), b) and c) show components of $K^4_*$, with $J_3$ equal to $\I$, $\II$ and $\V$ respectively, as parts of the transitive network of convergence. The dashed lines in Fig. 2 a) indicate the $\kappa^4$ limit lines from lines in Fig. 2 b). $\dim K^4=2$, since its largest (parametrically connected) components are 2-dimensional. {\hfill \break} For the unimodular subvariety $U^4_*\subset K^4_*$ it is $\dim U^4_* =1$. The union of $ \mbox{${\rm VIII}$} \oplus \R$, $\IX\oplus \R$, $\{A^{a,-a-1}_{4,5},-\frac{1}{2}<a<0\}$ and $\{A^{-2b,b}_{4,6},0<b<\infty\}$ is a dense subset of $ U^4_*$ and consists of a minimum number of parametrically connected components, namely 2 isolated points and 2 isolated line segments. In Fig. 2 the unimodular lines are dotted, and the unimodular points encircled. {\newpage } \vspace*{17.5truecm} \begin{center} {\normalsize Fig. 2 a: Transitions and limits at components of $K^4_*$ with ideal \I.} \end{center} \vspace*{8.5truecm} \begin{center} {\normalsize Fig. 2 b: Transitions at components of $K^4_*$ with ideal \II.} \end{center} \vspace{6.5truecm} \begin{center} {\normalsize Fig. 2 c: Transitions at components of $K^4_*$ with ideal \V.} \end{center} \section{\bf Orientation duality in $K^n_{(or)}$ for $n\leq 4$} \setcounter{equation}{0} In this section we examine in detail all points in $K^n_{or}$ for $n\leq 4$ under the aspect of orientation duality. In Sec. 7.1 the topological structure of $K^n_{or}$ for $n\leq 3$ is analysed by use of the (O)NJNF, thus reproducing the results listed by {Schmidt} \bSch. The connected components $K^3_\pm$ are determined explicitly. Using the same method, Sec. 7.2 analyses the orientation duality structure of $K^4_{or}$ in detail. Especially we determine the connected components $K^4_\pm$. \subsection{\bf Structure of $K^n_{or}$ for $n\leq 3$} The {Lie} algebras in $K^n$ for $n\leq 3$ have been classified in Sec. 6.1 using their $n-1$-dimensional ideals and the NJNF. Their orientation duality has already been listed by {Schmidt} \bSch. $K^2$ contains only 2 elements, the {Abel}ian $2A_1$ and $A_2$ represented by the algebra with $[e_2,e_1]=e_1$ as only nonvanishing bracket. Both are selfdual, because e.g. $e_1\to-e_1$ does not change the algebra. So $K^2_{or}=K^2_{SD}=K^2$ The elements of $K^3$ correspond to the familiar Bianchi types. In the following we analyse the orientation duality by looking at the NJNF in $K^3$ and for non-selfduality also considering the ONJNF, defining the elements of $K^3_\pm$. The solvable algebras in $K^3$ contain all the {Abel}ian ideal $J_2=2A_1$. In Sec. 6.1 they are classified according to their NJNF. Similarly the solvable algebras in $K^3_{or}$ can be classified according to their ONJNF, which agrees the NJNF in the case of selfduality. So the selfdual algebras in $K^3_{or}$ correspond to the following cases of NJNF w.r.t. the {Abel}ian ideal $J_2$: $$ \NJNF(\I) = \left( \begin{array}{cc} 0 & \\ & 0 \end{array} \right) , \quad \NJNF(\V) = \left( \begin{array}{cc} 1 & \\ & 1 \end{array} \right) , $$ $$ \NJNF(\III) = \left( \begin{array}{cc} 1 & \\ & 0 \end{array} \right) , $$ \begin{equation} \NJNF(\VIo) = \left( \begin{array}{cc} 1 & \\ & -1 \end{array} \right) , \ \NJNF(\VIh) = \left( \begin{array}{cc} 1 & \\ & a \end{array} \right) . \end{equation} The algebras $\I$ and $\III$ are selfdual, since they are decomposable. All algebras in Eq. (7.1) invariant under the reflection $e_1\to-e_1$, which guarantees their selfduality. The parameter range $0< h< \infty, h\neq 1$ ($h$ denoting the parameter of {Landau-Lifschitz} \Lan), corresponds monotonously to $-1< a< 1, a\neq 0$. $h=1$ resp. $a=0$ yields the decomposable \III. So $K^3_{SD}=\{\I,\V\}\cup\{\VIh,0\leq h<\infty\}$. The other solvable algebras which are not invariant under any reflection are non-selfdual. According to Sec. 3 and Sec. 4 we choose the reflection $e_3\to-e_3$ to characterize them as algebras in $K^3_\pm$, with their ONJNF respectively given like following: $$ \mbox{${\rm ONJNF}$} \{\IIRL\}=\pm\NJNF(\II) = \pm \left( \begin{array}{cc} 0 & 1 \\ & 0 \end{array} \right) , $$ $$ \mbox{${\rm ONJNF}$} \{\IVRL\}=\pm\NJNF(\IV) = \pm \left( \begin{array}{cc} 1 & 1 \\ & 1 \end{array} \right) , $$ $$ \mbox{${\rm ONJNF}$} \{ \mbox{${\rm VII}^{R/L}_0 $} \}=\pm\NJNF( \mbox{${\rm VII}_0$} ) = \pm\left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right) , $$ \begin{equation} \mbox{${\rm ONJNF}$} \{ \mbox{${\rm VII}^{R/L}_h $} \}=\pm\NJNF( \mbox{${\rm VII}_h$} ) = \pm\left( \begin{array}{cc} a & 1 \\ -1 & a \end{array} \right) . \end{equation} In Eq. (7.2) the parameter range $0< h< \infty$ ($h$ denoting the parameter of {Landau-Lifschitz} \Lan) corresponds monotonously to the range $0< a< \infty$. The simple algebras $ \mbox{${\rm VIII}$} =su(1,1)$ and $\IX=su(2)$ are described respectively by the 3 matrices $$ C_{<3>} = \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right) , \qquad C_{<1>} = -C_{<2>} = \left( \begin{array}{cc} 0 & 1 \\ \pm 1 & 0 \end{array} \right) . $$ $\NJNF(C_{<3>})=\NJNF( \mbox{${\rm VII}_0$} )$ for both $ \mbox{${\rm VIII}$} $ and $\IX$, {\hfill \break} but $\NJNF(C_{<1>})=\NJNF(C_{<2>})$ is equal to $\NJNF(\VIo)$ for $ \mbox{${\rm VIII}$} $ and to $\NJNF( \mbox{${\rm VII}_0$} )$ for $\IX$. In the {Cartan-Weyl} basis $H:=-ie_3$, $E_{\pm}:=e_1\pm i e_2$ the nonvanishing commutators are given as $[H,E_\pm]=\pm E_\pm$ and, for \mbox{${\rm VIII}$} or \IX respectively, $[E_+,E_-]=\pm 2H$. Note that the latter are different real sections in the same complex algebra. According to Lemma 4.5 neither $ \mbox{${\rm VIII}$} $ nor $\IX$ are selfdual. We discriminate the right and left algebra by the reflection $e_3\to -e_3$, defining both pairs $ \mbox{${\rm VIII}^{R/L} $} $ and $\IXRL$ of points in $K^3_\pm$. So it is \begin{equation} \mbox{${\rm ONJNF}$} \{C^{R/L}_{<3>}\}= \mbox{${\rm ONJNF}$} \{ \mbox{${\rm VII}^{R/L}_0 $} \} \end{equation} and $C_{<1>}$ and $C_{<2>}$ interchange under this reflection. {\hfill \break} The table below summarizes the duality properties of the Bianchi classes in $K^3$. Note that for any point $A \in K^3 {\setminus } K_{SD}$ there exists a pair $(A^R,A^L) \in K_+\oplus K_-$ of points in $K^3_{or} {\setminus } K_{SD}$ with right/left handed bases respectively. \medskip {\hfill \break} {\normalsize \begin{tabular}{ccccccccccc} $3A_1$&$A_1\oplus A_2$&$A_{3,1}$&$A_{3,2}$&$A_{3,3}$ &$A_{3,4}$&$A^a_{3,5}$&$A_{3,6}$&$A^a_{3,7}$&$A_{3,8}$&$A_{3,9}$\\ I&III&II&IV&V&\VIo&\VIh& \mbox{${\rm VII}_0$} & \mbox{${\rm VII}_h$} &VIII&IX\\ 1&1 &0 &0 &1&1 &1 &0 &0 &0 &0 \end{tabular} \begin{center} \begin{tabular}{ll} Table 1: &3-dimensional {Lie} algebra classes in $K^3$,\\ &corresponding Bianchi types and selfduality (yes=1/no=0) \end{tabular} \end{center} \medskip } The non-selfdual subset of $K^3_{or}$ has 2 connected $1$-dimensional components, $K^3_+$ and $K^3_-$, given respectively by \begin{equation} \mbox{${\rm VIII}^{R/L} $} /\IXRL\to \mbox{${\rm VII}^{R/L}_0 $} \gets \mbox{${\rm VII}^{R/L}_h $} \to\IVRL\to\IIRL , \end{equation} where $\IIRL$ is respectively the atom of $K^3_\pm$. \subsection{\bf Structure of $K^4_{or}$} In Sec. 6.2 we classified the real 4-dimensional {Lie} algebras. In this section they are reconsidered under the aspect of orientation duality. {\hfill \break} \nl {\bf 7.2.1 Selfdual {Lie} algebras} {\hfill \break} \nl There exist following types of selfdual algebras: a) all decomposable ones, b) indecomposable ones with ideal \I, and c) some indecomposable ones with ideal \II. {\hfill \break} {\hfill \break} {\bf a) Decomposable ones:} {\hfill \break} {\hfill \break} All decomposable {Lie} algebras are selfdual. A decomposable 4-dimensional {Lie} algebra can have the structures $4A_1$, $2A_1\oplus A_2$, $2A_2$ or $A_1\oplus A_3$. The first 3 possibilities are unique, since $A_1$ is the unique 1-dim. {Lie} algebra and $A_2$ is the unique non{Abel}ian 2-dim. {Lie} algebra. $A_1\oplus A_3$ consists of 9 classes, given by $\{A_{3,i}\}_{i=1,\ldots,9}$ listed in Table 1. Note that $A_1\oplus \II\equiv {\II}^{(4)}$. {\hfill \break} {\hfill \break} {\newpage } \noindent {\bf b) Indecomposable ones with ideal $J_3={\rm I}$:} {\hfill \break} {\hfill \break} Algebras with {Abel}ian ideal $J_3=3A_1\equiv \I$ are selfdual. They are given by the following cases: {\hfill \break} i) 1 Jordan block: {\hfill \break} These algebras are invariant under a combination of the $3$ reflections $e_i\to -e_i$, $i=1,\ldots,3$. Either the eigenvalue is $ {\lambda } = 0$ or otherwise it can be normalized to $ {\lambda } = 1$. $$ \NJNF(A_{4,1}) = \left( \begin{array}{ccc} 0 & 1 & \\ & 0 & 1 \\ & & 0 \end{array} \right) , $$ \begin{equation} \NJNF(A_{4,4}) = \left( \begin{array}{ccc} 1 & 1 & \\ & 1 & 1 \\ & & 1 \end{array} \right) . \end{equation} These algebras are a $4$-dimensional analogue to \II and \IV. While the latter are non-selfdual their even dimensional analogues are selfdual. These algebras are the essential dimensional ones, introduced in Sec. 5 and denoted by $\ii(4)$ and $\iv(4)$. $\ii(4)$ is an essential dimensional atom. {\hfill \break} {\hfill \break} ii) 2 Jordan blocks: {\hfill \break} All these algebras are all invariant under the reflection $e_1\to -e_1$. \begin{equation} \NJNF(A_{4,3}) = \left( \begin{array}{ccc} 1 & & \\ & 0 & 1 \\ & & 0 \end{array} \right) , \ \NJNF(A^a_{4,2}) = \left( \begin{array}{ccc} a & & \\ & 1 & 1 \\ & & 1 \end{array} \right) . \end{equation} In the latter case $a\neq 0$ and $A^1_{4,2}\equiv \IV^{(4)}$. {\hfill \break} {\hfill \break} iii) 3 real eigenvalues as Jordan blocks: {\hfill \break} All these algebras are all invariant under the reflection $e_1\to -e_1$. Assuming the largest eigenvalue normalized to $ {\lambda } _1 = 1$, there remain $ {\lambda } _2=a$ and $ {\lambda } _3=b$ with $-1\leq b\leq a\leq 1$. If $a\cdot b = 0$, the algebra becomes decomposable ($a=b=0$ yields $\III \oplus \R$, for $a=1,b=0$ it is $\V \oplus \R$, for $a=0,b=-1$ it is $\VIo \oplus \R$ and otherwise $a=0$ or $b=0$ yields $\VIh \oplus \R$). Therefore assume $a\cdot b\neq 0$. The case $a=b=1$ (single 3-fold degenerate eigenvalue) corresponds to the pure vector type $A^{1,1}_{4,5} \equiv \Vv \neq \V \oplus \R$. In $A^{1,b}_{4,5}$ and $A^{a,a}_{4,5}$ there are 2 eigenvalues, one of them 2-fold degenerate. For the nondegenerate case, $-1\le b< a< 1$. Note that $A^{a,b}_{4,5}=A^{b,a}_{4,5}$, since permutations are in ${\GL}(4)$. \begin{equation} \NJNF(A^{a,b}_{4,5}) = \left( \begin{array}{ccc} 1 & & \\ & a & \\ & & b \end{array} \right) . \end{equation} \noindent iv) 1 real eigenvalue and 2 complex conjugates: {\hfill \break} All these algebras are all invariant under the reflection $e_1\to -e_1$. If $ {\lambda } _{2,3}=r(\cos\theta \pm i\sin\theta)$, by normalization $r\sin\theta = 1$ can be achieved, if $ {\lambda } _2 \neq {\lambda } _3$ is assured (otherwise the Jordan block becomes diagonal). Set then $r\cos\theta = b$ and $ {\lambda } _1=a$. Demand $a\neq 0$ to exclude decomposability ($a = 0$ yields $ \mbox{${\rm VII}_h$} \oplus \R$ and $b=0$ then corresponds to $h=0$) and without restriction $b\geq 0$. \begin{equation} \NJNF(A^{a,b}_{4,6}) = \left( \begin{array}{ccc} a & & \\ & b & 1 \\ &-1 & b \end{array} \right) . \end{equation} {\hfill \break} {\hfill \break} {\bf c) Indecomposable ones with ideal $J_3={\rm II}$:} {\hfill \break} {\hfill \break} There exist algebras with non-selfdual ideal \II, which are selfdual. \begin{equation} \mbox{${\rm ONJNF}$} (A_{4,8})=\NJNF(A_{4,8}) = \left( \begin{array}{ccc} 0 & & \\ & 1 & \\ & &-1 \end{array} \right). \end{equation} This algebra is left invariant by a combination of reflections $e_4\to-e_4$, $e_1\to-e_1$ and $e_2\leftrightarrow e_3$. \begin{equation} \mbox{${\rm ONJNF}$} (A_{4,10})=\NJNF(A_{4,10}) = \pm\left( \begin{array}{ccc} 0 & & \\ & 0 & 1 \\ &-1 & 0 \end{array} \right) . \end{equation} This algebra is left invariant by a combination of reflections $e_4\to-e_4$, $e_1\to-e_1$ and $e_2\to -e_2$. {\newpage } \noindent {\bf 7.2.2 Non-selfdual {Lie} algebras} {\hfill \break} \nl This kind of algebras exists with a basic ideal $J_3$, given either by the non-selfdual \II or by the selfdual \V. For all of them we have dual pairs of right and left points in $K^n_{or}$, which transform to each other by $e_4\to -e_4$, constituting by Sec. 3 and Sec. 4 the connected components $K^4_\pm$ respectively. {\hfill \break} {\hfill \break} {\bf a) Indecomposable ones with ideal $J_3={\rm II}$:} {\hfill \break} {\hfill \break} The ideal \II is non-selfdual. For an algebra $A$ of the kinds listed below the there exists no reflection leaving the set $J(A)$ invariant. $$ \mbox{${\rm ONJNF}$} (A^{R/L}_{4,7})=\pm\NJNF(A_{4,7}) = \pm\left( \begin{array}{ccc} 2 & & \\ & 1 & 1 \\ & & 1 \end{array} \right) , $$ $$ \mbox{${\rm ONJNF}$} (A^{b,R/L}_{4,9})=\pm\NJNF(A^{b}_{4,9}) = \pm\left( \begin{array}{ccc} 1+b & & \\ & 1 & \\ & & b \end{array} \right) , 0 < \vert b\vert < 1 , $$ $$ \mbox{${\rm ONJNF}$} (A^{0,R/L}_{4,9})=\pm\NJNF(A^{0}_{4,9}) = \pm\left( \begin{array}{ccc} 1 & & \\ & 1 & \\ & & 0 \end{array} \right) , $$ $$ \mbox{${\rm ONJNF}$} (A^{1,R/L}_{4,9})=\pm\NJNF(A^{1}_{4,9}) = \pm\left( \begin{array}{ccc} 2 & & \\ & 1 & \\ & & 1 \end{array} \right) , $$ \begin{equation} \mbox{${\rm ONJNF}$} (A^{a,R/L}_{4,11})=\pm\NJNF(A^{a}_{4,11}) = \pm\left( \begin{array}{ccc} 2a & & \\ & a & 1 \\ &-1 & a \end{array} \right) , a > 0 . \end{equation} {\hfill \break} {\hfill \break} {\bf b) Indecomposable ones with ideal $J_3={\rm V}$:} {\hfill \break} {\hfill \break} The only case here is given by \begin{equation} \mbox{${\rm ONJNF}$} (A^{R/L}_{4,12})=\pm\NJNF(A_{4,12}) = \pm\left( \begin{array}{ccc} 0 & 1 & \\ -1& 0 & \\ & & 0 \end{array} \right) . \end{equation} Note that besides the selfdual ideal \V there is a second ideal \mbox{${\rm VII}_0$} which is not selfdual, causing here the subset of ideals $S(A_{4,12})$ to be non-selfdual. Hence $A_{4,12}$ itself is non-selfdual. {\hfill \break} {\hfill \break} {\bf c) The space $K^4_{NSD}$ and its components $K^4_\pm$:} {\hfill \break} {\hfill \break} Collecting the algebras of the previous subsections a) and b) and recalling transitions and parametrical limits of components in $K^4_{NSD}$ according to Sec. 6, we find that $K^4_{NSD}$ is connected, and so is each of $K^4_\pm$. There are $2$ pairs of $K^4_{or,NSD}$-atoms, given by $A^{0,R/L}_{4,9}$ and $A^{1,R/L}_{4,9}$. In $n=4$ all $K^4_{NSD}$-atoms have an ideal $\II$, and hence in the complement of the subspace $K^4_{NSD\vert\I}$ of $K^4_{NSD}$-algebras with ideal $\I$. Let us assign e.g. $A^{0,R/L}_{4,9}$ to $K^4_{\pm}$ respectively. Then the connectedness of $K^4_{\pm}$ and the orientation preservation of limits within $K^4_{or,NSD}$ imply the assignment $A^{R/L}$ to $K^4_{\pm}$ respectively. Note that with these assignments the component $K^4_+$ is given as \begin{equation} \begin{array}{lcrr} &A^{R}_{4,12}& \\ &\downarrow& \\ A^{-1<b<0,R}_{4,9}\to\!&A^{0,R}_{4,9}&\! \gets A^{0<b<1,R}_{4,9}\to A^{R}_{4,7}\to A^{1,R}_{4,9}\gets A^{a>0,R}_{4,11} \end{array} \end{equation} and the component $K^4_-$ as \begin{equation} \begin{array}{lcrr} &A^{L}_{4,12}& \\ &\downarrow& \\ A^{-1<b<0,L}_{4,9}\to\!&A^{0,L}_{4,9}&\! \gets A^{0<b<1,L}_{4,9}\to A^{L}_{4,7}\to A^{1,L}_{4,9} \gets A^{a>0,L}_{4,11} \end{array} \end{equation} So the non-selfdual components of $K^4_{(or)}$ are $1$-dimensional. Note that $A^{R/L}_{4,1}\equiv \ii(4)^{R/L}$ is the atom of $K^4_\pm$ respectively. \section{\bf Discussion and outlook} \setcounter{equation}{0} In Sec. 6 we determined {Lie} algebra transitions in $K^4$ as limits induced by the topology $\kappa^4$. Any {In\"on\"u-Wigner} contraction corresponds to a certain transition; explicitly any of the {In\"on\"u-Wigner} contractions listed in the tables of {Huddleston} \Hud\ for real $4$-dimensional {Lie} algebras corresponds to a transition in $K^4$. Since {In\"on\"u-Wigner} contractions are only a special case of the more general {Saletan} contractions, and since even the latter do not induce all possible transitions in $K^n$ with $n\geq 3$, it should not be surprising that we have obtained transitions, which do not correspond to any {In\"on\"u-Wigner} contraction, like e.g. transitions $\IX\oplus \R\to A_{4,10}$, $ \mbox{${\rm VIII}$} \oplus \R\to A_{4,10}$ and transitions from $A^{a,b}_{4,5}$, $A^{a,b}_{4,6}$, $\VIh\oplus \R$ and $ \mbox{${\rm VII}_h$} \oplus \R$ to $A_{4,1}$. The transition $\IX\oplus \R\to A_{4,10}$ corresponds to a {Lie} algebra contraction, which was given already in \Sal\ (see Eqs. (35') to (37)) as an example of a {Saletan} contraction, which can not be obtained as a {In\"on\"u-Wigner} contraction. It is also interesting to consider transitions in $K^3$ as obtained in Sec. 5. The {In\"on\"u-Wigner} contractions for real {Lie} algebras of dimension $d\leq 3$ are classified already by {Conatser} \Co. The sequence of transitions $ \mbox{${\rm VIII}$} \to\VIo\to\II\to\I$ is generated by an iterated {Saletan} contraction (see \Sal, Eqs. (30) and (31)), applied first to the {Lie} algebra $ \mbox{${\rm VIII}$} $, of the $3$-dimensional homogenous {Lorentz} group. On the 4-point subset $\{ \mbox{${\rm VIII}$} ,\VIo,\II,\I\}$ {Saletan} contractions are transitive. However this transitivity does not hold for {Saletan} transitions on general subsets of $K^3$. The sequence of transitions $\IX\to \mbox{${\rm VII}_0$} \to\II\to\I$, starting from the {Lie} algebra $\IX$ of the 3-dimensional {Euclid}ean group, can not be obtained by {Saletan} contractions. The only {Saletan} contractions starting from $\IX$ are in fact given by a {In\"on\"u-Wigner} contraction $\IX\to \mbox{${\rm VII}_0$} $ and the trivial contraction $\IX\to \I$. Though there exists a different {In\"on\"u-Wigner} contraction corresponding to the transitions $ \mbox{${\rm VII}_0$} \to\II$ there is no {Saletan} contraction corresponding to $\IX\to\II$ (for a proof see \Sal). This example shows that, on an arbitrary subset of $K^n$ with $n\geq 3$, in general not every transition can be obtained from a {Saletan} contraction. It implies that, even on a set of points connected by {In\"on\"u-Wigner} contractions, neither {Saletan} contractions nor {In\"on\"u-Wigner} contractions need to be transitive. Since we consider transitions between different points in $K^n$, improper contractions of an algebra to an equivalent one can not be seen by our method. For $n=4$ {Huddleston} \Hud\ identified two types of algebras which admit only trivial and improper contractions. These are precisely the two atoms of $K^4$, namely the unimodular ${\II}^{(4)}\equiv\II\oplus \R$ and the pure vector type ${\V}^{(4)}\equiv A^{1,1}_{4,5}$. For arbitrary dimension $n$, the atoms of $K^n$ have been introduced and described first by {Schmidt} {\bSch}. By now the topological properties of $K^n$ for $n\leq 4$ have been examined. It is natural to demand an investigation for arbitrary dimension $n$. Practically, this is obstructed by the rapidly increasing number of equivalence classes for increasing $n$. A classification for all nilpotent algebras has been done for $n=6$ by {Morozov} \Mo\ and for $n=7$ by {Ancochea-Bermudez} and {Goze} \An\ in the complex case and by {Romdhani} \Ro, who distinguishes $132$ components of real indecomposable nilpotent $7$-dimensional {Lie} algebras. For $K^5$ a full classification of all real {Lie} algebras still distinguishes $40$ components (compare {Mubarakzjanov} \bMu\ and {Patera} et al. \PaSWZ). A determination of all possible transitions would be a rather tidy work. However it is known by {\PaSWZ} that $\dim K^5=3$, because the maximal dimension of its components is 3. Unlike for the classification of subalgebra structures of each class in $K^n$ (see {Patera, Winternitz} \PaW, and {Grigore, Popp} \Gri), for the determination of all equivalence classes and transitions between them there exists no algorithm at present. However, a systematic exploitation of the NJNF, which has been defined for arbitrary $n$, may contribute some part to further progress. The NJNF has proven to be a useful tool in characterizing distinct $n$-dimensional {Lie} algebras with a common ideal $J_{n-1}$ as endomorphisms ad$e_n$ of a complementary generator $e_n$ on that ideal, with characteristic Jordan blocks of their eigenvalues normalized by an overall scale. In $4$ dimensions, besides decomposable algebras, only cases with ideal \I, \II, or \V appear. In $4$ dimensions, there are no simple algebras. In general for $n\geq 6$ further classes of simple {Lie} algebras arise, which lead to an additional further sophistication, as compared to $n=3$. A combination of the established knowledge on semisimple {Lie} algebras with the full classification of all {Lie} algebras would be desirable, but is practically far away, since the dimensionality of the simple {Lie} algebras increases rapidly with their rank. Note that all simple components belong to the unimodular subset $U^n_*$. In Sec. 7 we found that simple {Lie} algebras are non-selfdual w.r.t. orientation reflection. In general, we have neither a formula for $\dim K^n$ nor for $\dim U^n_*$. The $T_0$ topology allows components of different dimensions to converge pointwise to each other, i.e. such that any point of the first component converges to some point of the second component and any point of the second component is the limit of some point of the first. We have determined the topology of the space $K^n_{or}$ for $n\leq 4$. The essential difference to $K^n$ is that the single non-selfdual component of the latter is doubled to two components $K^n_+$ and $K^n_-$. For $n=3$ or $4$, the space $K^n_{NSD}$ is nonvoid and connected. For $n=3$ there is a unique $K^3_{NSD}$-atom $\ii(3)=\II$. $K^4_{NSD}$ has two atoms, $A^0_{4,9}$ and $A^1_{4,9}$, and $A^{0<b<1}_{4,9}$ has boundary limits to both of them. If $K^n_{NSD}$ is connected, the arbitrariness in assigning conjugate pairs of points to $K^n_\pm$ can be reduced to a single decision for one pair only, if we demand that both of $K^n_\pm$ are connected and to each other disconnected. At present, for general $n\geq 5$ it is not known whether $K_{NSD}$ is connected. {}From Eqs. (7.4) and (7.13-14), we see that the non-selfdual subset $K^n_{NSD(,or)}$ of $K^n_{(or)}$ is $1$-dimensional for both, $n=3$ and $n=4$. We have $\dim K^3_{(or)}=1$ and $\dim K^n_{(or)}\geq 2$ for dimension $n\geq 4$: In the latter case the contribution of the non-selfdual subset is of dimension less than that of the highest-dimensional component, while in the former case it is of highest dimension. Actually, the topology of the highest-dimensional component of $K^3_{or}$ differs from that of $K^3$ essentially. The question of dimensionality for general $n\geq 5$ remains open, for the non-selfdual subset as well as for $K^n_{(or)}$ itself. Partial progress has been made by determining a candidate of an atom of the non-selfdual subset in odd dimension $n$. We found an interesting periodicity in the structure of this $K^n_{NSD}$-atom: it is $\ii(n)$ for $n=3 \,\mbox{mod}\, 4$ and $\iv(n)$ for $n=1 \,\mbox{mod}\, 4$. However it remains an open problem to determine all atoms of $K^n_{NSD}$ for arbitrary $n\geq 5$. Presently we do not know how an atom for even $n\geq 6$ looks like in general. With Definition 6 the notion of an atom from {Schmidt} \bSch\ has been generalized to arbitrary subsets. Although the present work is on the case of real {Lie} algebras, we want to make some comments on the analogous complex cases to the pairs of algebras $ \mbox{${\rm VII}_h$} /\VIh$ ($h\geq 0$), $\IX/ \mbox{${\rm VIII}$} $ of $K^3$ and $A^{a,b}_{4,6}/A^{a,b}_{4,5}$, $A^{a}_{4,11}/A^{b}_{4,9}$ and $A_{4,10}/A_{4,8}$ of $K^4$. If one considers the analogous $3$- or $4$-dimensional {Lie} algebras over the complex basic field the group $\GL(n)$ is now correspondingly the group of nonsingular complex linear transformation. The pairs of complex conjugated eigenvalues associated to the $2\times 2$ Jordan block of each of the first algebras of the pairs above in the complex remain as $2$ Jordan blocks in a corresponding complex JNF. After introducing a similar normalization convention like for the real case, the complex analogues of the NJNF will be the same for members of any pair above. (For $n=3$ this had already been realized by {Bianchi} \Bi. The complex $4$-dimensional case was considered already by {Lie} \Lie). {\hfill \break} \nl {\Large {\bf Acknowledgments}} {\hfill \break} \nl I would like to express my gratitude to H.-J. { Schmidt} for valuable discussions on the present topic. {\newpage } \noindent {\Large {\bf References}} {\hfill \break} \nl \An\ { J. M. Ancochea-Bermudez} and { M. Goze}. Classification des algebres de { Lie} nilpotentes de dimension 7. Arch. Math. {\bf 52}, 175-185 (1989). \smallskip {\hfill \break} -, Sur la classification des algebres de { Lie} nilpotentes de dimension 7. Comptes Rendus Acad. Sci. Paris, Ser. I {\bf 302}, 611-613 (1986). \smallskip {\hfill \break} \Bi\ { L. Bianchi}, Sugli spazii a tre dimensioni che ammettono un gruppo continuo di movimenti. Soc. Ital. Sci. Mem. Mat., Ser. {IIIa}, {\bf 11}, 267 (1897). \smallskip {\hfill \break} \bBi\ -, Lezioni sulla teoria dei gruppi continui finiti di trasformazioni, Cap. XIII: Applicazioni alla teoria degli spazii pluridimensionali con un gruppo continuo di movimenti, \S\S 198, 199 (for a description of types \I-\VII and \mbox{${\rm VIII}$} /\IX) and \S\S 200-207 (for applications to homogeneous spaces), p. 550-578. Spoerri, Pisa 1918. \smallskip {\hfill \break} \Cha\ { R. Charles}, Sur la structure des algebres de { Lie} rigides. Ann. Inst. Fourier Grenoble {\bf 33}, 3, 65-82 (1984). \smallskip {\hfill \break} \Ch\ { R. Charles} and { Y. Diakite}, Sur les varietes d'algebres de { Lie} de dimension $\le 7$. J. Algebra {\bf 91}, 53-63 (1984). \smallskip {\hfill \break} \Co\ { C. W. Conatser}, Contractions of low-dimensional real { Lie} algebras. J. Math. Phys. {\bf 13}, 196-203 (1972). \smallskip {\hfill \break} \Est\ { F. B. Estabrook, H. D. Wahlquist} and { C. G. Behr}, Dyadic analysis of spatially homogeneous world models. J. Math. Phys. {\bf 9}, 497-504 (1968). \smallskip {\hfill \break} \Gri\ { D. R. Grigore} and { O. T. Popp}, On the classification of { Lie} subalgebras and { Lie} subgroups. J. Math. Phys. {\bf 32}, 33-39 (1991). \smallskip {\hfill \break} \Gru\ { F. Grunewald} and { J. O'Halloran}, Varieties of nilpotent lie algebras of dimension less than six. J. Algebra {\bf 112}, 315-325 (1988). \smallskip {\hfill \break} \Hud\ { P. L. Huddleston}, In\"on\"u-Wigner contractions of the real four-dimensional { Lie} algebras. J. Math. Phys. {\bf 19}, 1645-1649 (1978). \smallskip {\hfill \break} \In\ { E. In\"on\"u} and { E. P. Wigner}, On the contraction of groups and their representations. Proc. Nat. Acad. Sci. USA {\bf 39}, 510-524 (1953). \smallskip {\hfill \break} -, On a particular type of convergence to a singular matrix. Proc. Nat. Acad. Sci. USA {\bf 40}, 119-121 (1954). \smallskip {\hfill \break} \Ki\ { A. A. Kirillov} and { Yu. A. Neretin}, The variety $A_n$ of $n$-dimensional { Lie} algebra structures. Transl., Ser. (2), Amer. Math. Soc. {\bf 137}, 21-30 (1987); translation from: Some questions in modern analysis, Work. Collect., Moskva 1984, 42-56 (1984). \smallskip {\hfill \break} {\newpage } \noindent \Kr\ { D. Kramer, H. Stephani, M. Mac Callum} and { E. Herlt}, Exact Solutions of Einstein's Field Equations. Wissenschaften, Berlin 1980. \smallskip {\hfill \break} \Lan\ { L. D. Landau} and { E. M. Lifschitz}, Lehrb. d. theor. Phys., Bd. II, 12. Aufl., pp. 457 ff. Akademie-Verlag, Berlin 1992. \smallskip {\hfill \break} \Lee\ { H. C. Lee}, Sur les groupes de { Lie} r\'eels \`a trois param\`etres. J. Math. Pures et Appl. {\bf 26}, 251-267 (1947). \smallskip {\hfill \break} \Lie\ { S. Lie}, Differentialgleichungen, Kap. 21. Chelsea, Leipzig 1891. \smallskip {\hfill \break} \bLie\ -, Theorie der Transformationsgruppen, Bd. III, Abtheilung VI, Kap. 28: Allgemeines \"uber die Zusammensetzung $r$-gliedriger Gruppen, \S\S 136-137, p. 713-732. Teubner, Leipzig 1893. \smallskip {\hfill \break} \Mac\ { M. A. H. Mac Callum}, A class of homogeneous cosmological models III: Asymptotic behaviour. Commun. Math. Phys. {\bf 20}, 57-84 (1971). \smallskip {\hfill \break} \Mag\ { L. M. Magnin}, Sur les algebres de { Lie} nilpotent de dim $\leq 7$. J. G. P. {\bf 3}, 1, 119-134 (1986). \smallskip {\hfill \break} \Mo\ { V. V. Morozov}, Classification of nilpotent { Lie} algebras of 6$^{th}$ order. Izv. Vyssh. Uchebn. Zaved. Mat. {\bf 4}, 161-171 (1958). \smallskip {\hfill \break} \aMu\ { G. M. Mubarakzyanov}, On solvable { Lie} algebras. (Russian) Izv. Vys\v s. U\v cebn. Zavedeni\u\i\ Mat. 1963, no. 1 (32), 114-123. \smallskip {\hfill \break} \bMu\ -, Classification of real structures of { Lie} algebras of fifth order. (Russian) Izv. Vys\v s. U\v cebn. Zavedeni\u\i\ Mat. 1963, no. 3 (34), 99-106. \smallskip {\hfill \break} \cMu\ -, Classification of solvable { Lie} algebras of sixth order with a non-nilpotent basis element. (Russian) Izv. Vys\v s. U\v cebn. Zavedeni\u\i\ Mat. 1963, no. 4 (35), 104-116 (1963). \smallskip {\hfill \break} \Ne\ { Yu. A. Neretin}, Estimate of the number of parameters assigning an $n$-dimensional algebra. (Russian) Isv. Acad. Nauk SSSR, Ser. Mat. {\bf 51}, 306-318 (1987). \smallskip {\hfill \break} \PaSWZ\ { J. Patera, R. T. Sharp, P. Winternitz} and { H. Zassenhaus}, Invariants of real low dimension { Lie} algebras. J. Math. Phys. {\bf 17}, 986-994 (1976). \smallskip {\hfill \break} \PaW\ { J. Patera} and { P. Winternitz}, Subalgebras of real three- and four-dimensional { Lie} algebras. J. Math. Phys. {\bf 18}, 1449-1455 (1977). \smallskip {\hfill \break} \Pe\ { A. S. Petrov}, { Einstein}r\"aume. Akademie Verlag, Berlin 1964. \smallskip {\hfill \break} \aRa\ { M. Rainer}, Topology of the space of 4-dimensional real { Lie} Algebras. Preprint 93/2, Math. Inst. Univ. Potsdam (1993). \smallskip {\hfill \break} \bRa\ { M. Rainer}, Orientation Duality of 4-dimensional Real { Lie} Algebras. Preprint 93/8, Math. Inst. Univ. Potsdam (1993). \smallskip {\hfill \break} \Ri\ { W. Rinow}, Topologie. Wissenschaften, Berlin 1975. \smallskip {\hfill \break} \Ro\ { M. Romdhani}, Classification of real and complex nilpotent { Lie} algebras of dimension 7. Linear Multilin. Algebra {\bf 24}, 167-189 (1989). \smallskip {\hfill \break} \Sal\ { E. I. Saletan}, Contraction of { Lie} groups. J. Math. Phys. {\bf 2}, 1-21 (1961). \smallskip {\hfill \break} \San\ { R. M. Santilli}, Lie-isotopic lifting of Lie symmetries, I: General considerations. Hadronic J. {\bf 8} 25-35 (1985). \smallskip {\hfill \break} -, Lie-isotopic lifting of Lie symmetries, II: Lifting of Rotations. Hadronic J. {\bf 8} 36-51 (1985). \smallskip {\hfill \break} \aSch\ { H.-J. Schmidt}, Inhomogeneous cosmological models containing homogeneous inner hypersurface geometry. Changes of the { Bianchi} type. Astron. Nachr. {\bf 303}, 227-230 (1982). \smallskip {\hfill \break} \bSch\ -, On Geroch's limit of space-times and its relation to a new topology in the space of { Lie} groups. J. Math. Phys. {\bf 28}, 1928-1936 (1987); and Addendum in J. Math. Phys. {\bf 29}, 1264 (1988). \smallskip {\hfill \break} \Se\ { I. E. Segal}, A class of operator algebras which are determined by groups. Duke Math. J. {\bf 18}, 221-265 (1951). \smallskip {\hfill \break} \Smrz\ { P. K. Smrz}, Relativity and deformed { Lie} groups. J. Math. Phys. {\bf 19}, 2085-2088 (1978). \smallskip {\hfill \break} \aTur\ { P. Turkowski}, Low-dimensional { Lie} algebras. J. Math. Phys. {\bf 29}, 2139-2144 (1988). \smallskip {\hfill \break} \bTur\ -, Solvable { Lie} algebras of dimension six. J. Math. Phys. {\bf 31}, 1344-1350 (1990). \smallskip {\hfill \break} \cTur\ -, Structure of real { Lie} algebras. Linear Algebra Appl. {\bf 171}, 197-212 (1992). \smallskip {\hfill \break} \Um\ { K. A. Umlauf}, \"uber die Zusammensetzung der endlichen continuierlichen Transformationsgruppen, insbesondere der Gruppen vom Range Null. Breitkopf u. H\"artel, Leipzig 1891. \smallskip {\hfill \break} \Vra\ { G. Vranceanu}, Lecons de geom\'etrie diff\'erentielle, p. 105-111. Bucarest 1947. \smallskip {\hfill \break} \end{document}

9508

alg-geom/9508001

en

https://arxiv.org/abs/alg-geom/9508001

alg-geom/9508001

Dan Edidin

Dan Edidin and William Graham

Localization in equivariant intersection theory and the Bott residue formula

This paper is a substantially revised version of our preprint "Equivariant Chow groups and the Bott residue formula". Current email address for Dan Edidin is "[email protected]" and current email for William Graham is "[email protected]" Amslatex

The purpose of this paper is to prove the localization theorem for torus actions in equivariant intersection theory. Using the theorem we give another proof of the Bott residue formula for Chern numbers of bundles on smooth complete varieties. In addition, our techniques allow us to obtain residue formulas for bundles on a certain class of singular schemes which admit torus actions. This class is rather special, but it includes some interesting examples such as complete intersections and Schubert varieties.

alg-geom

\section{Introduction} The purpose of this paper is to prove the localization theorem for torus actions in equivariant intersection theory. Using the theorem we give another proof of the Bott residue formula for Chern numbers of bundles on smooth complete varieties. In addition, our techniques allow us to obtain residue formulas for bundles on a certain class of singular schemes which admit torus actions. This class is rather special, but it includes some interesting examples such as complete intersections (cf. \cite{BFQ}) and Schubert varieties. Let $T$ be a split torus acting on a scheme $X$. The $T$-equivariant Chow groups of $X$ are a module over $R_T = Sym(\hat{T})$, where $\hat{T}$ is the character group of $T$. The localization theorem states that up to $R_T$-torsion, the equivariant Chow groups of the fixed locus $X^T$ are isomorphic to those of $X$. Such a theorem is a hallmark of any equivariant theory. The earliest version (for equivariant cohomology) is due to Borel \cite{Borel}. Subsequently $K$-theory versions were proved by Segal \cite{Segal} (in topological $K$-theory), Quart \cite{Quart} (for actions of a cyclic group), and Thomason \cite{Thomason} (for algebraic $K$-theory \cite{Thomason}). For equivariant Chow groups, the localization isomorphism is given by the equivariant pushforward $i_*$ induced by the inclusion of $X^T$ to $X$. An interesting aspect of this theory is that the push-forward is naturally defined on the level of cycles, even in the singular case. The closest topological analogue of this is equivariant Borel-Moore homology (see \cite{E-G} for a definition), and a similar proof establishes localization in that theory. For smooth spaces, the inverse to the equivariant push-forward can be written explicitly. It was realized independently by several authors (\cite{I-N}, \cite{A-B}, \cite{B-V}) that for compact spaces, the formula for the inverse implies the Bott residue formula. In this paper, we prove the Bott residue formula for actions of split tori on smooth complete varieties defined over an arbitrary field, also by computing $(i_*)^{-1}$ explicitly. Bott's residue formula has been applied recently in enumerative geometry (cf. \cite{E-S}, \cite{K}) and there was interest in a purely intersection-theoretic proof. Another application of the explicit formula for $(i_*)^{-1}$ is given in \cite{E-G2}, where we prove (following Lerman \cite{L}) a residue formula due to Kalkman. An obvious problem, which should have applications to enumerative geometry (see e.g. \cite{K}), is to extend the Bott residue formula to complete singular schemes. Such a formula can be derived when we have an explicit description of $(i_*)^{-1}[X]_T$, where $[X]_T$ denotes the equivariant fundamental class of the whole scheme, as follows: Let $n = \mbox{dim }X$. If $[X]_T = i_*\alpha$ and $p(E)$ is a polynomial of weighted degree $n$ in Chern classes of equivariant vector bundles on $X$, then $\mbox{deg }(p(E) \cap [X])$ can be calculated as the residue of $\pi_*(i^*(p(E)) \cap \beta)$ where $\pi$ is the equviariant projection from the $X$ (or the fixed locus) to a point. This approach does not work for equivariant cohomology, because when $X$ is singular there is no pushforward from $H^*_G(X) \rightarrow H^*_G(M)$. However, in $K$-theory, where such pushforwards exist, similar ideas were used by \cite{BFQ} to obtain Lefschetz-Riemann-Roch formulas for the action of an automorphism of finite order. The problem of computing $(i_*)^{-1}$ is difficult, but we can do it in a certain class of singular examples, in particular, if there is an equivariant embedding $X \stackrel{f}\hookrightarrow M$ into a smooth variety, and every component of $X^T$ is a component of $M^T$. This condition is satisfied if $X \subset {\Bbb P}^r$ is an invariant subvariety where $T$ acts linearly with distinct weights (and thus isolated fixed points) or if $X$ is a Schubert variety in $G/B$. In this context we give a formula (Proposition \ref{xxxsing}) formula for $(i_*)^{-1}\alpha$ in terms of $f_*\alpha \in A_*^T(M)$. The case of Schubert of varieties is worked out in detail in Section \ref{schubs}. As a consequence it is possible to compute Chern numbers of bundles on $X$ provided we know $f_*[X]_T \in A_*^T(M)$. Thus for example, if $X$ is a $T$-invariant projective variety and $T$ acts linearly with distinct weights on ${\Bbb P}^n$, then we can calculate Chern numbers, provided we know the equivariant fundamental class of $X$. Rather than write down a general formula, we illustrate this with an example: in Section \ref{singex} we use a residue calculation to show that $$ \int_Q c_1(\pi^* T_{{\Bbb P}^2}) c_1(f^*T_{{\Bbb P}^3}) = 24 $$ where $Q \stackrel{f} \hookrightarrow {\Bbb P}^3$ is a (singular) quadric cone, and $\pi: Q \rightarrow {\Bbb P}^2$ is the projection from a point not on $Q$. The methods can be applied in other examples. {\bf Acknowledgements.} We thank Steven Kleiman for suggesting the problem of giving an algebraic proof of the Bott residue formula. We are also grateful to Robert Laterveer for discussions of Gillet's higher Chow groups. \section{Review of equivariant Chow groups} In this section we review some of the equivariant intersection theory developed in \cite{E-G}. The key to the theory is the definition of equivariant Chow groups for actions of linear algebraic groups. All schemes are assumed to be of finite type defined over a field of arbitrary characteristic. Let $G$ be a $g$-dimensional group, $X$ an $n$-dimensional scheme and $V$ a representation of $G$ of dimension $l$. Assume that there is an open set $U \subset V$ such that a principal bundle quotient $U \rightarrow U/G$ exist, and that $V-U$ has codimension more than $n-i$. Thus the group $G$ acts freely on the product $X \times U$. The group $G$ acts freely on $X \times U$, and if any one of a number of mild hypotheses is satisfied then a quotient scheme $X_G = (X \times U)/G$ exists (\cite{E-G}). In particular, if $G$ is special -- for example, if $G$ is a split torus, the case of interest in this paper -- a quotient scheme $X_G$ exists. \begin{defn} Set $A_i^G(X) = A_{i+l-g}(X_G)$, where $A_*$ is the usual Chow group. This definition is independent of the choice of $V$ and $U$ as long as $V-U$ has sufficiently high codimension. \end{defn} {\bf Remark:} Because $X \times U \rightarrow X \times^G U$ is a principal $G$-bundle, cycles on $X \times^G U$ exactly correspond to $G$-invariant cycles on $X \times U$. Since we only consider cycles of codimension smaller than the dimension of $X \times (V-U)$, we may in fact view these as $G$-invariant cycles on $X \times V$. Thus every class in $A_i^G(X)$ is represented by a cycle in $Z_{i+l}(X \times V)^G$, where $Z_*(X \times V)^G$ indicates the group of cycles generated by invariant subvarieties. Conversely, any cycle in $Z_{i+l}(X \times V)^G$ determines an equivariant class in $A_i^G(X)$. \medskip The properties of equivariant intersection Chow groups include the following. (1) Functoriality for equivariant maps: proper pushforward, flat pullback, l.c.i pullback, etc. (2) Chern classes of equivariant bundles operate on equivariant Chow groups. (3) If $X$ is smooth of dimension $n$, then we denote $A_{n-i}^G(X)$ as $A^i_G(X)$. In this case there is an intersection product $A^i_G(X) \times A^j_G(X) \rightarrow A_G^{i+j}(X)$, so the groups $\oplus_0^\infty A^i_G(X)$ form a graded ring which we call the equivariant Chow ring. Unlike, the ordinary case $A^i_G(X)$ can be non-zero for any $i \geq 0$. (The existence of an intersection product follows from (1), since the diagonal $X \hookrightarrow X \times X$ is an equivariant regular embedding when $X$ is smooth.) (4) Of particular use for this paper is the equivariant self-intersection formula. If $Y \stackrel{i} \hookrightarrow X$ is a regularly embedded invariant subvariety of codimension $d$, then $$i^*i_*(\alpha) = c_d^G(N_YX) \cap \alpha$$ for any $\alpha \in A_*^G(Y)$. \subsection{Equivariant higher Chow groups} Let $Y$ be a scheme. Denote by $A_i(Y,j)$ the higher Chow groups of Bloch \cite{Bl} (indexed by dimension) or the groups $CH_{i,i-j}(X)$ defined in \cite[Section 8]{Gillet}. Both theories agree with ordinary Chow groups when $j=0$, and both extend the localization short exact sequence for ordinary Chow groups. However, in the case of Bloch's Chow groups the localization exact sequence has only been proved for quasi-projective varieties. The advantage of his groups is that are naturally defined in terms of cycles on $X \times \Delta^j$ (where $\Delta^j$ is an algebraic $j$-simplex) and are rationally isomorphic to higher $K$-theory. Both these theories can be extended to the equivariant setting. We define the higher Chow groups $A_i^G(X,j)$ as $A_{i+l-g}(X_G,j)$ for an appropriate mixed space $X_G$. Because of the quasi-projective hypothesis in Bloch's work, Bloch's equivariant higher Chow groups are only defined for (quasi)-projective varieties with linearized actions. However, Gillet's are defined for arbitrary schemes with a $G$-action. We will use two properties of the higher equivariant theories.\\ (a) If $E \rightarrow X$ an equivariant vector bundle, then the equivariant Chern classes $c_i^G(E)$ operate on $A_*^G(X,j)$.\\ (b) If $U \subset X$ is an invariant open set, then there is a long exact sequence $$ \ldots \rightarrow A_i^G(U,1) \rightarrow A_i^G(X-U) \rightarrow A_i^G(X) \rightarrow A_i^G(U) \rightarrow 0.$$ \section{Localization} In this section we prove the main theorem of the paper, the localization theorem for equivariant Chow groups. For the remainder of the paper, all tori are assumed to be split, and the coefficients off all Chow groups are rational. Let $R_T$ denote the $T$-equivariant Chow ring of a point, and let $\hat{T}$ be the character group of $T$. \begin{prop} \cite[Lemma 4]{EGT} $R_T = Sym(\hat{T})\simeq {\Bbb Q}[t_1, \ldots , t_n]$. where $n$ is the rank of $T$. $\Box$ \medskip \end{prop} {\bf Remark.} The identification $R_T = Sym(\hat{T})$ is given explicitly as follows. If $\lambda \in \hat{T}$ is a character, let $k_\lambda$ be the corresponding 1-dimensional representation and let $L_\lambda$ denote the line bundle $U \times^T k_\lambda \rightarrow U/T$. The map $\hat{T} \rightarrow R^1_T$ given by $\lambda \mapsto c_1(L_{\lambda})$ extends to a ring isomorphism $Sym(\hat{T}) \rightarrow R_T$. \medskip \begin{prop} If $T$ acts trivially on $X$, then $A_*^T(X) = A_*(X) \otimes R_T$. \end{prop} Proof. If the action is trivial then $(U \times X)/T= U/T \times X$. The spaces $U/T$ can be taken to be products of projective spaces, so $A_*(U/T \times X) = A_*(X) \otimes A_*(U/T)$. $\Box$ \medskip \medskip If $T \stackrel{f} \rightarrow S$ is a homomorphism of tori, there is a pullback $\hat{S} \stackrel{f^*} \rightarrow \hat{T}$. This extends to a ring homomorphism $Sym(\hat{T}) \stackrel{f^*} \rightarrow Sym(\hat{S})$, or in other words, a map $f^*: R_S \rightarrow R_T$. \begin{lemma} \label{t-map}(cf. \cite{A-B}) Suppose there is a $T$-map $X \stackrel{\phi} \rightarrow S$. Then $t\cdot A_*^T(X,m)= 0$ for any $t=f^*s$ with $s \in R_S^+$. \end{lemma} Proof of Lemma \ref{t-map}. Since $A^*_S$ is generated in degree 1, we may assume that $s$ has degree 1. After clearing denominators we may assume that $s = c_1(L_s)$ for some line bundle on a space $U/S$. The action of $t=f^*s$ on $A_*(X_T)$ is just given by $c_1(\pi_T^*f^*L_s)$ where $\pi_T$ is the map $U \times^T X \rightarrow U/T$. To prove the lemma we will show that this bundle is trivial. First note that $L_s = U \times^S k$ for some action of $S$ on the one-dimensional vector space $k$. The pullback bundle on $X_T$ is the line bundle $$U \times^T(X \times k) \rightarrow X_T$$ where $T$ acts on $k$ by the composition of $f:T \rightarrow S$ with the original $S$-action. Now define a map $$\Phi: X_T \times k \rightarrow U \times^T(X \times k)$$ by the formula $$\Phi(e,x,v) = (e,x,\phi(x)\cdot v)$$ (where $\phi(x) \cdot v$ indicates the original $S$ action). This map is well defined since \begin{eqnarray*} \Phi(et,t^{-1}x,v) & = &(et, t^{-1}x, \phi(t^{-1}x) \cdot v)\\ & = & (et,t^{-1}x,t^{-1} \cdot(\phi(x) \cdot v)) \end{eqnarray*} as required. This map is easily seen to be an isomorphism with inverse $(e,x,v) \mapsto (e,x,\phi(x)^{-1} \cdot v)$. $\Box$ \medskip \begin{prop} \label{fix} If $T$ acts on $X$ without fixed points, then there exists $r \in R_T^+$ such that $r \cdot A_*^T(X,m)= 0$. (Recall that $A_*^T(X,m)$ refers to $T$-equivariant higher Chow groups.) \end{prop} \medskip Before we prove Proposition \ref{fix}, we state and prove a lemma. \begin{lemma} \label{porb} If $X$ is a variety with an action of a torus $T$, then there is an open $U \subset X$ so that the stabilizer is constant for all points of $U$. \end{lemma} Proof of Lemma \ref{porb}. Let $\tilde X \rightarrow X$ be the normalization map. This map is $T$-equivariant and is an isomorphism over an open set. Thus we may assume $X$ is normal. By Sumihiro's theorem, the $T$ action on $X$ is locally linearizable, so it suffices to prove the lemma when $X = V$ is a vector space and the action is diagonal. If $V = k^n$, then let $U = (k^*)^n$. The $n$-dimensional torus ${\bf G}_m^n$ acts transitively on $U$ in the obvious way. This action commutes with the given action of $T$. Thus the stabilizer at each closed point of $U$ is the same. $\Box$ \medskip Proof of Proposition \ref{fix}. Since $A_*^G(X) = A_*^G(X_{red})$ we may assume $X$ is reduced. Working with each component separately, we may assume $X$ is a variety. Let $X^0 \subset X$ be the ($G$-invariant) locus of smooth points. By Sumhiro's theorem \cite{Sumihiro}, the action of a torus on a normal variety is locally linearizable (i.e. every point has an affine invariant neighborhood). Using this theorem it is easy to see that the set $X(T_1) \subset X^0$ of points with stabilizer $T_1$ can be given the structure of a locally closed subscheme of $X$. By Lemma \ref{porb} there is some $T_1$ such that $U= X(T_1)$ is open in $X^0$, and thus in $X$. The torus $T'=T/T_1$ acts without stabilizers, but the action of $T'$ on $U$ is not a priori proper. However, by \cite[Proposition 4.10]{Th1}, we can replace $U$ by a sufficiently small open set so that $T'$ acts freely on $U$ and a principal bundle quotient $U \rightarrow U/T$ exists. Shrinking $U$ further, we can assume that this bundle is trivial, so there is a $T$ map $U \rightarrow T'$. Hence, by the lemma, $t \cdot A^T_*(U) = 0$ for any $t \in A_T^*$ which is pulled back from $A^*_{T'}$. Let $Z = X -U$. By induction on dimension, we may assume $p \cdot A^T_*Z = 0$ for some homogeneous polynomial $p \in R_T$. From the long exact sequence of higher Chow groups, $$\ldots A_*^T(Z,m) \rightarrow A_*^T(X,m) \rightarrow A_*^T(U,m) \rightarrow \ldots$$ it follows that $tp$ annihilates $A_*^T(X)$ where $t$ is the pullback of a homogeneous element of degree $1$ in $R_S$. $\Box$ \medskip If $X$ is a scheme with a $T$-action, we may put a closed subscheme structure on the locus $X^T$ of points fixed by $T$ (\cite{Iversen}). Now $R_T= Sym(\hat{T})$ is a polynomial ring. Set ${\cal Q}= (R_T^+)^{-1} \cdot R_T$, where $R_T^+$ is the multiplicative system of homogeneous elements of positive degree. \begin{thm} \label{lcztn}(localization) The map $i_*:A_*(X^T) \otimes {\cal Q} \rightarrow A_*^T(X) \otimes {\cal Q}$ is an isomorphism. \end{thm} \medskip Proof of Theorem \ref{lcztn}. By Proposition \ref{fix}, $A_*^T(X - X^T,m) \otimes {\cal Q} = 0$. Thus by the localization exact sequence $A_*^T(X^T) \otimes {\cal Q} = A_*^T(X) \otimes {\cal Q}$ as desired. $\Box$ \medskip {\bf Remark.} The strategy of the proof is similar to proofs in other theories, see for example for \cite[Chapter 3.2]{Hsiang}. \section{Explicit localization for smooth varieties} The localization theorem in equivariant cohomology has a more explicit version for smooth varieties because the fixed locus is regularly embedded. This yields an integration formula from which the Bott residue formula is easily deduced (\cite{A-B}, \cite{B-V}). In this section we prove the analogous results for equivariant Chow groups of smooth varieties. Because equivariant Chow theory has formal properties similar to equivariant cohomology, the arguments are almost the same as in \cite{A-B}. As before we assume that all tori are split. Let $F$ be a scheme with a trivial $T$-action. If $E \rightarrow F$ is a $T$-equivariant vector bundle on $F$, then $E$ splits canonically into a direct sum of vector subbundles $\oplus_{\lambda \in \hat{T}} E_{\lambda}$, where $E_{\lambda}$ consists of the subbundle of vectors in $E$ on which $T$ acts by the character $\lambda$. The equivariant Chern classes of an eigenbundle $E_{\lambda}$ are given by the following lemma. \begin{lemma} \label{l.trivchern} Let $F$ be a scheme with a trivial $T$-action, and let $E_{\lambda} \rightarrow F$ be a $T$-equivariant vector bundle of rank $r$ such that the action of $T$ on each vector in $E_{\lambda}$ is given by the character $\lambda$. Then for any $i$, $$ c^T_i(E_{\lambda}) = \sum_{j \leq i} \left( \begin{array}{c} r-j \\ i-j \end{array} \right) c_j(E_{\lambda}) \lambda^{i-j}. $$ In particular the component of $c_r^T(E_{\lambda})$ in $R^r_T$ is given by $\lambda^r$. $\Box$ \medskip \end{lemma} As noted above, $A^*_T(F) \supset A^*F \otimes R_T$. The lemma implies that $c^T_i(E)$ lies in the subring $A^*F \otimes R_T$. Because $A^N F = 0$ for $N > \mbox{dim }F$, elements of $A^i F$, for $i>0$, are nilpotent elements in the ring $A^*_T(F)$. Hence an element $\alpha \in A^d F \otimes R_T$ is invertible in $A^*_T(F)$ if its component in $A^0 F \otimes R^d_T \cong R^d_T$ is nonzero. For the remainder of this section $X$ will denote a smooth variety with a $T$ action. \begin{lemma} \cite{Iversen} If $X$ is smooth then the fixed locus $X^T$ is also smooth. $\Box$ \medskip \end{lemma} For each component $F$ of the fixed locus $X^T$ the normal bundle $N_FX$ is a $T$-equivariant vector bundle over $F$. Note that the action of $T$ on $N_FX$ is non-trivial. \begin{prop} If $F$ is a component of $X^T$ with codimension $d$ then $c_d^T(N_FX)$ is invertible in $A^*_T(F) \otimes {\cal Q}$. \end{prop} Proof: By (\cite[Proof of Proposition 1.3]{Iversen}), for each closed point $f \in F$, the tangent space $T_fF$ is equal to $(T_fX)^T$, so $T$ acts with non-zero weights on the normal space $N_f = T_fX/T_fF$. Hence the characters $\lambda_i$ occurring in the eigenbundle decomposition of $N_FX$ are all non-zero. By the preceding lemma, the component of $c_d^T(N_FX)$ in $R^d_T$ is nonzero. Hence $c_d^T(N_FX)$ is invertible in $A^*_T(F) \otimes {\cal Q}$, as desired. $\Box$ \medskip \medskip Using this result we can get, for $X$ smooth, the following more explicit version of the localization theorem. \begin{thm} \label{xxx}(Explicit localization) Let $X$ be a smooth variety with a torus action. Let $\alpha \in A_*^T(X) \otimes {\cal Q}$. Then $$\alpha = \sum_F i_{F*}\frac{i^*_F\alpha}{c_{d_F}^T(N_FX)},$$ where the sum is over the components $F$ of $X^T$ and $d_F$ is the codimension of $F$ in $X$. \end{thm} Proof: By the surjectivity part of the localization theorem, we can write $\alpha = \sum_F i_{F*}(\beta_F)$. Therefore, $i^*_F\alpha = i^*_Fi_{F*}(\beta_F)$ (the other components of $X^T$ do not contribute); by the self-intersection formula, this is equal to $ c_{d_F}^T(N_FX) \cdot \beta_F$. Hence $\beta_F = \frac{i^*_F\alpha}{c_{d_F}^T(N_FX)}$ as desired. $\Box$ \medskip {\bf Remark.} This formula is valid, using the virtual normal bundle, even if $X$ is singular, provided that the embedding of the fixed locus in $X$ is a local complete intersection morphism. Unfortunately, this condition is difficult to verify. However, if $X$ is cut by a regular sequence in a smooth variety, and the fixed points are isolated, then the methods of \cite[Section 3]{BFQ} can be used to give an explicit localization formula. A similar remark applies to the Bott residue formula below. \medskip If $X$ is complete, then the projection $\pi_X: X \rightarrow pt$ induces push-forward maps $\pi_{X*}: A^T_* X \rightarrow R_T$ and $\pi_{X*}: A^T_* X \otimes {\cal Q} \rightarrow {\cal Q}$. There are similar maps with $X$ replaced with any component $F$ of $X^T$. Applying $\pi_{X*}$ to both sides of the explicit localization theorem, and noting that $\pi_{X*} i_{F*} = \pi_{F*} $, we deduce the ``integration formula'' (cf. \cite[Equation (3.8)]{A-B}). \begin{cor} (Integration formula) Let $X$ be smooth and complete, and let $\alpha \in A_*^T(X) \otimes {\cal Q}$. Then $$\pi_{X*}(\alpha) = \sum_{F \subset X^T} \pi_{F*}\left( \frac{i^*_F\alpha}{c_{d_F}^T (N_FX)}\right)$$ as elements of ${\cal Q}$. $\Box$ \medskip \end{cor} \medskip {\bf Remark.} If $\alpha$ is in the image of the natural map $A_*^T(X) \rightarrow A_*^T(X) \otimes {\cal Q}$ (which need not be injective), then the equation above holds in the subring $R_T$ of ${\cal Q}$. The reason is that the left side actually lies in the subring $R_T$; hence so does the right side. In the results that follow, we will have expressions of the form $z = \sum z_j$, where the $z_j$ are degree zero elements of ${\cal Q}$ whose sum $z$ lies in the subring $R_T$. The pullback map from equivariant to ordinary Chow groups gives a map $i^*: R_T = A^T_* (pt) \rightarrow {\Bbb Q} = A_* (pt)$, which identifies the degree 0 part of $R_T$ with ${\Bbb Q}$. Since $\sum z_j$ is a degree 0 element of $R_T$, it is identified via $i^*$ with a rational number. Note that $i^*$ cannot be applied to each $z_j$ separately, but only to their sum. In the integration and residue formulas below we will identify the degree 0 part of $R_T$ with ${\Bbb Q}$ and suppress the map $i^*$. \medskip The preceding corollary yields an integration formula for an element $a$ of the ordinary Chow group $A_0 X$, provided that $a$ is the pullback of an element $\alpha \in A^T_0 X$. \begin{prop} Let $a \in A_0 X$, and suppose that $a = i^* \alpha$ for $\alpha \in A^T_0 X$. Then $$ \mbox{deg }(a) = \sum_F \pi_{F*}\{\frac{i^*_F\alpha}{c_{d_F}^T (N_FX)} \} $$ \end{prop} Proof: Consider the commutative diagram $$\begin{array}{ccc} X & \stackrel{i} \hookrightarrow & X_T\\ \downarrow\scriptsize{\pi_X} & & \downarrow\scriptsize{\pi^T_X}\\ \mbox{pt} & \stackrel{i} \rightarrow & U/T . \end{array}$$ We have $\pi_{X*}(a) = \pi_{X*} i^*(\alpha) = i^* \pi^T_{X*}(\alpha)$. Applying the integration formula gives the result. $\Box$ \medskip \subsection{The Bott residue formula} Let $E_1, \ldots , E_s$ be a $T$-equivariant vector bundles on a complete, smooth $n$-dimensional variety $X$. Let $p(x^1_1, \ldots x^1_s,\ldots , x^n_1, \ldots x^n_s)$ be a polynomial of weighted degree $n$, where $x^i_j$ has weighted degree $i$. Let $p(E_1, \ldots , E_s)$ denote the polynomial in the Chern classes of $E_1, \ldots , E_s$ obtained setting $x^i_j = c_i(E_g)$. The integration formula above will allow us to compute $\mbox{deg }(p(E_1, \ldots , E_s) \cap [X])$ in terms of the restriction of the $E_i$ to $X^T$. As a notational shorthand, write $p(E)$ for $p(E_1, \ldots, E_s)$ and $p^T(E)$ for the corresponding polynomial in the $T$-equivariant Chern classes of $E_1, \ldots , E_r$. Notice that $p(E) \cap [X] = i^* (p^T(E) \cap [X_T])$. We can therefore apply the preceding proposition to get the Bott residue formula. \begin{thm} \label{bott} (Bott residue formula) Let $E_1, \ldots , E_r$ be a $T$-equivariant vector bundles a complete, smooth $n$-dimensional variety. Then $$ \mbox{deg }(p(E) \cap [X]) = \sum_{F \subset X^T} \pi_{F*}\left(\frac{p^T(E|_{F}) \cap [F]_T}{c_{d_F}^T (N_FX)} \right). $$ $\Box$ \medskip \end{thm} {\bf Remark.} Using techniques of algebraic deRham homology, H\"ubl and Yekutieli \cite{H-Y} proved a version of the Bott residue formula, in characteristic 0, for the action of any algebraic vector field with isolated fixed points. \medskip By Lemma \ref{l.trivchern} the equivariant Chern classes $c^T_i(E_j|_{F})$ and $c_{d_F}^T (N_FX)$ can be computed in terms of the characters of the torus occurring in the eigenbundle decompositions of $E_j|_{F}$ and $N_FX$ and the Chern classes of the eigenbundles. The above formula can then be readily converted (cf. \cite{A-B}) to more familiar forms of the Bott residue formula not involving equivariant cohomology. We omit the details. If the torus $T$ is 1-dimensional, then degree zero elements of ${\cal Q}$ are rational numbers, and the right hand side of the formula is just a sum of rational numbers. This is the form of the Bott residue formula which is most familiar in practice. \section{Localization and residue formulas for singular varieties} \label{singex} In general, the problem of proving localization and residue formulas on singular varieties seems interesting and difficult. In this section we discuss what can be deduced from an equivariant embedding of a singular scheme $X$ into a smooth $M$. The results are not very general, but (as we show) they can be applied in some interesting examples, for example, if $X$ is a complete intersection in $M = {\Bbb P}^n$ and $T$ acts on $M$ with isolated fixed points, or if $X$ is a Schubert variety in $M = G/B$. The idea of using an embedding into a smooth variety to extract localization information is an old one. In the case of the action of an automorphism of finite order, the localization and Lefschetz Riemann-Roch formulas of \cite{Quart}, \cite{BFQ} on quasi-projective varieties are obtained by a calculation on ${\Bbb P}^n$. Moreover, as in our case, the best formulas on singular varieties are obtained when the embedding into a smooth variety is well understood. At least in principle, a localization theorem can be deduced if every component of $X^T$ is a component of $M^T$. This holds, for example, if the action of $T$ on $M$ has isolated fixed points; or if $X$ is a toric (resp. spherical) subvariety of a nonsingular toric (resp. spherical) variety $M$. In particular, the condition holds if $X$ is a Schubert variety and $M$ is the flag variety. We have the following proposition. \begin{prop} \label{xxxsing} Let $f: X \rightarrow M$ be an equivariant embedding of $X$ in a nonsingular variety $M$. Assume that every component of $M^T$ which intersects $X$ is contained in $X$. If $F$ is a component of $X^T$, write $i_F$ for the embedding of $F$ in $X$, and $j_F$ for the embedding of $F$ in $M$. Then: $(1)$ $f_*: A_*^T(X) \otimes {\cal Q} \rightarrow A_*^T(M) \otimes {\cal Q}$ is injective. $(2)$ Let $\alpha \in A_*^T(X) \otimes {\cal Q}$. Then $$\alpha = \sum_F i_{F*}\frac{j^*_F f_* \alpha}{c_{d_F}^T(N_FM)},$$ where the sum is over the components $F$ of $X^T$ and $d_F$ is the codimension of $F$ in $M$. \end{prop} Proof: (1) Since the components of $X^T$ are a subset of the components of $M^T$, $\oplus_{F \subset X^T} A_*^T(F)$ is an $R_T$-submodule of $\oplus _{F \subset M^T} A_*^T(F)$. By the localization theorem, $$\sum_{F \subset X^T} j_{F*}(A_*^T(F)) \otimes {\cal Q} \simeq A_*^T(X) \otimes {{\cal Q}}$$ and $$\sum_{F \subset M^T} i_{F*}(A_*^T(F)) \otimes {\cal Q} \simeq A_*^T(M) \otimes {{\cal Q}}.$$ Since $i_{F*} = f_* i_{F *}$, the result follows. (2) By (1) it suffices to prove that $$f_*(\alpha - \sum_F j_{F*}\frac{i^*_F f_* \alpha}{c_{d_F}^T(N_FM)}) = 0 \in A_*^T(X) \otimes {\cal Q}.$$ Since $i_{F*} = f_* j_{F*}$ the theorem follows from the explicit localization theorem applied to the class $f_* \alpha$ on the smooth variety $M$. $\Box$ \medskip To obtain a residue formula that computes Chern numbers of bundles on $X$, we only need to know an expansion $[X]_T =\sum_{F\subset X^T} i_{F*}(\beta_F)$, where $\beta_F \in A_*^T(F)$. In this case we obtain the formula $$ \mbox{deg }(p(E) \cap [X]) = \sum_{F \subset X^T} \pi_{F*}\left(\frac{p^T(i_F^*(E)) \cap \beta_F}{c_{d_F}^T(N_FM)}\right).$$ In the setting of Proposition \ref{xxxsing}, the classes $\beta_F$ are given by $i_F^* f_* [X]_T$. To obtain a useful residue formula, we need to make this expression more explicit. This is most easily done if we can express $f_* [X]_T$ in terms of Chern classes of naturally occuring equivariant bundles on $M$. The reason is that the pullback $i_F^*$ of such Chern classes is often easy to compute, particularly if $F$ is an isolated fixed point (cf. Lemma. \ref {l.trivchern}). Indeed, this is why the Bott residue formula is a good calculational tool in the non-singular case. Although the conditions to obtain localization and residue formulas are rather strong, they are satisfied in some interesting cases. We will consider in detail two examples: complete intersections in projective spaces, and Schubert varieties in $G/B$. For complete intersections some intrinsic formulas can be deduced using the virtual normal bundle (see the remark after Theorem \ref{xxx}). In this section our point of view for complete intersections is different. We do not use the virtual normal bundle, but instead use the fact that if $X \stackrel{f} \hookrightarrow M$ is a complete intersection, it is easy to calculate $f_*[X]_T \in A_*^T(M)$. As an example of our methods we do a localization and residue calculation on a singular quadric in ${\Bbb P}^3$. As a final remark, note that to compute Chern numbers of bundles on $X$ which are pulled back from $M$, it suffices to know $f_*[X]_T$, for then we can apply residue formulas on $M$. Information about the fixed locus in $X$ is irrelevant. The interesting case is when the bundles are not pulled back from $M$; see the example of the singular quadric below. \subsection{Complete intersections in projective space} For simplicity we consider the case where the dimension of $T$ is $1$. If $T$ acts on a vector space $V$ with weights $a_0, \ldots, a_n$ then $A^*_T({\Bbb P}(V)) = {\Bbb Z}[h,t] / \prod (h + a_i t)$. We are interested in complete intersections $X$ in ${\Bbb P}(V)$ where the functions $f_i$ defining $X$ are, up to scalars, preserved by the $T$-action, i.e., $t \cdot f_i = t^{a_i} f_i$. In this case we say $f_i$ has weight $a_i$. The following lemma is immediate. \begin{lemma} Suppose $X$ is a hypersurface in ${\Bbb P}(V)$ defined by a homogeneous polynomial $f$ of degree $d$ and weight $a$. Then $[X]_T = d h + a t \in A^*_T({\Bbb P}(V))$. Hence if $X$ is a complete intersection in ${\Bbb P}(V)$ defined by homogeneous polynomials $f_i$ of degree $d_i$ and weight $a_i$, then $[X]_T = \prod (d_i h + a_i t)$. $\Box$ \medskip \end{lemma} If $T$ acts on $V$ with distinct weights, then $T$ has isolated fixed points on $M = {\Bbb P}(V)$, and (trivially) every component of $X_T$ is a component of $M^T$; so by the preceding discussion there is a useful residue formula. In particular using a little linear algebra we can easily obtain a formula for $[X]_T$ in terms of the fixed points in $X$ and the weights of the action. We omit the details to avoid a notational quagmire, but the ideas are illustrated in the example of the singular quadric. \subsection{Schubert varieties in $G/B$} \label{schubs} In this section, we work over an algebraically closed field. For simplicity, we take Chow groups to have rational coefficients, and let $R = R_T \otimes {\Bbb Q}$ denote the rational equivariant Chow ring of a point. Let $G$ be a reductive group and $B$ a Borel subgroup, and ${\cal B} = G/B$ the flag variety. In the discussion below, the smooth variety ${\cal B}$ will play the role of $M$, and the Schubert variety $X_w$ the role of $X$. Let $T \subset B$ be a maximal torus. $T$ acts on ${\cal B}$ with finitely many fixed points, indexed by $w \in W$; denote the corresponding point by $p_w$. More precisely, if we let $w$ denote both an element of the Weyl group $W = N(T)/T$ and a representative in $N(T)$, then $p_w$ is the coset $wB$. The flag variety is a disjoint union of the $B$-orbits $X_w^0 = B \cdot p_w$. The $B$-orbit $X_w^0$ is called a Schubert cell and its closure $X_w$ a Schubert variety. If $e$ denotes the identity in $W$ and $w_0$ the longest element of $W$, then $X_e$ is a point and $X_{w_0} = {\cal B}$. We have $X_w = \cup_{u \leq w} X_u^0$. The $T$ equivariant Chow group of $X_w$ is a free $R_T$-module with basis $[X_u]_T$, for $u \leq w$. Let $j_u: p_u \hookrightarrow {\cal B}$. Fix $w \in W$ and let $f: X_w \hookrightarrow {\cal B}$. For $u \leq w$ let $i_u: p_u \hookrightarrow X_u$. If $v \leq w$ let $[X_v]_T$ denote the equivariant fundamental class of $X_v$ in $A_*^T(X_w)$. We want to make explicit the localization theorem for the variety $X_w$ (which is singular in general), i.e., to compute $[X_v]_T$ in terms of classes $i_{u*} \beta_u$. The (rational) equivariant Chow groups $A^*_T({\cal B})$ can be described as follows. We consider two maps $\rho_1, \rho_2: R \rightarrow A^*_T({\cal B})$. The map $\rho_1$ is the usual map $R \rightarrow A^*_T({\cal B})$ given by equivariant pullback from a point. The definition of $\rho_2$ is as follows. For each character $\lambda \in \hat{T}$ set $\rho_2(\lambda) = c_1^T(M_\lambda)$ where $M_{\lambda}$ is the line bundle $G \times^B k_{\lambda} \rightarrow {\cal B}$; extend $\rho_2$ to an algebra map $R \rightarrow A^*_T({\cal B})$. The map $R \otimes_{R^W} R \rightarrow A^*_T({\cal B})$ taking $r_1 \otimes r_2$ to $\rho_1(r_1) \rho_2(r_2)$ is an isomorphism (see e.g. \cite{Brion}). We adopt the convention that the Lie algebra of $B$ contains the positive root vectors. We can identify $T_{p_w}({\cal B})$ with ${\frak g} / (\mbox{Ad } w) {\frak b}$. This is a representation of $T$ corresponding to the $T$-equivariant normal bundle of the fixed point $p_w$. We identify $A^*_T(p_w) \cong R$. From our description of $T_{p_w}({\cal B})$, we see that (if $n$ denotes the dimension of ${\cal B}$) $c_n^T(N_{p_w}{\cal B})$ is the product of the roots in ${\frak g} /(\mbox{Ad } w) {\frak b}$, which is easily seen to give $$ c_n^T(N_{p_w}{\cal B}) = c_w := (-1)^n(-1)^w \prod_{\alpha > 0} \alpha. $$ where $n$ is the number of roots $\alpha >0$. To obtain a localization formula we also need to know the maps $j_u^* : A^*_T({\cal B}) \rightarrow A^*_T(p_u)$, where $j_u: p_u \hookrightarrow {\cal B}$ is the inclusion. We have identified $A^*_T({\cal B}) = R \otimes_S R$ and $A^*_T(p_w) = R$. Thus, we may view $j_u^*$ as a map $R \otimes_S R \rightarrow R$. There is a natural action of $W \times W$ on $R \otimes_S R$. Let $m: R \otimes_S R \rightarrow R$ denote the multiplication map. \begin{lemma} For $u \in W$, the map $j_u^*: R \otimes_S R \rightarrow R$ equals the composition $m \circ (1 \times u)$. \end{lemma} Proof: It suffices to show that $j_u^* \rho_1(\lambda) = \lambda$ and $j_u^* \rho_2(\lambda) = u \lambda$. Now, $j_u^* \rho_1$ is just the equivariant pullback by the map $p_u \rightarrow pt$. Since this equivariant pullback is how we identify $A^*_T(p_u) = A^*_T(pt) = R$, with these identifications, $j_u^*\rho_1$ is the identity map, $j_u^* \rho_1(\lambda) = \lambda$. Also, by definition $j_u^* \rho_2(\lambda) = c_1^T(M_{\lambda}|_{p_u})$. As a representation of $T$, $M_{\lambda}|_{p_u} \cong k_{u \lambda}$, so $c_1^T(M_{\lambda}|_{p_u}) = u \lambda$, as desired. $\Box$ \medskip If $F \in R \otimes_S R$ is a polynomial set $F(u) = j_u^*F \in R$. Recall that we have fixed $w$ and let $f : X_w \rightarrow {\cal B}$ denote the inclusion; for $v \leq w$, $[X_v]_T$ denotes a class in $A_*^T(X_w)$. By work of Fulton and Pragacz-Ratajski, for $G$ classical, it is known how to express $f_*[X_v]_T \in A^*_T({\cal B})$ in terms of the isomorphism $A^*_T({\cal B}) \cong R \otimes_{R^W} R$. More precisely, Fulton and Pragacz-Ratajski (\cite{Fu1}, \cite{P-R}) define elements in $R \otimes_{{\Bbb Q}} R$ which project to $f_*[X_v]_T$ in $R \otimes_{R^W} R$. Let $F_u$ denote either the polynomial defined by Fulton or that defined by Pragacz-Ratajski. Using these polynomials we can get an explicit localization formula for Schubert varieties. \begin{prop} With notation as above, the class $[X_v]_T$ in $A_*^T(X_w) \otimes {\cal Q}$ is given by $$ [X_v]_T = (-1)^n \frac{1}{\prod_{\alpha > 0} \alpha} \sum_{u \leq v} (-1)^u i_{u*} \left( F_v(u) \cap [p_u]_T \right). $$ \end{prop} Proof: This is an immediate consequence of the preceding discussion and Proposition \ref{xxxsing}. $\Box$ \medskip {\bf Remark.} Taking $w = w_0$, so $X_w = {\cal B}$, the above formula is an explicit inverse to the formula of \cite[Section 6.5, Proposition (ii)]{Brion}. This shows $\frac{f_w(u)}{\Pi_{\alpha > 0} \alpha}$ is Brion's equivariant multiplicity of $X_w$ at the fixed point $p_u$, and also links Brion's proposition to \cite[Theorem 1.1]{G}. \subsection{A singular quadric} In this section we consider the example of the singular quadric $Q \stackrel{f} \hookrightarrow {\Bbb P}^3$ defined by the equation $x_0x_1 + x_2^2 = 0$ (note that we allow the characteristic to be 2). Let ${\Bbb P}^2 \subset {\Bbb P}^3$ be the hyperplane defined the equation $x_2 = 0$ and let $\pi: Q \rightarrow {\Bbb P}^2$ be the projection from $(0,0,1,0)$. As a sample of the kinds of the residue calculations that are possible, we prove the following proposition. \begin{prop} $$\int_Q c_1(\pi^* T_{{\Bbb P}^2}) c_1(f^*T_{{\Bbb P}^3}) = 24$$. \end{prop} Proof. We will prove this by considering the following torus action. Let $T = {\Bbb G}_m$ act on ${\Bbb P}^3$ with weights $(1,-1,0,a)$, where $a \notin \{0,-1,1\}$. The quadric is invariant under this action, so $T$ acts on $Q$. Since $(0,0,1,0)$ is a fixed point, $\pi$ is an equivariant map where $T$ acts on ${\Bbb P}^2$ with weights $(1,-1,a)$. Thus $c_1^T(\pi^*T_{{\Bbb P}^2}) c_1^T(f^*T_{{\Bbb P}^3}) \cap [Q]_T$ defines an element of $A_*^T(Q) \otimes {\cal Q}$ which we will express as a residue in terms of the fixed points for the action of $T$ on $Q$. To do this we need to express $[Q]_T$ in terms of the fixed points. By Proposition \ref{xxxsing} this can be done if we know $f_*[Q]_T \in A_*^T({\Bbb P}^3)$. Since $Q$ is a quadric of weighted degree 0 with respect to the $T$-action, $f_*[Q]_T = 2h \in A_*^T({\Bbb P}^3)$. Since everything can be done explicitly, we will calculate more than we need and determine the entire $R_T$-module $A_*^T(Q)$ in terms of the fixed points. \medskip {\bf Explicit localization on the singular quadric.} The quadric has a decomposition into affine cells with one cell in dimensions 0,1 and 2. The open cell is $Q_0 = \{(1,-x^2,x,y) | (x,y) \in k^2\}$. In dimension 1 the cell is $l_0 = \{(0,1,0,x) | x \in k\}$, and in dimension 0, the cell is the singular point \footnote{In characteristic 2, this point is not an isolated singular point.} $p_s = \{(0,0,0,1)\}$. Thus, $A_i(Q) = {\Bbb Z}$ for $i = 0,1,2$ with generators $[Q]$, $[l]$ and $[p_s]$. Moreover, these cells are $T$-invariant, so their equivariant fundamental classes form a basis for $A_*^T(Q)$ as an $R_T = {\Bbb Z}[t]$ module. Let ${\Bbb I}$, $L$, and $P_s$ denote the corresponding equivariant fundamental classes $[Q]_T,[l]_T$, and $[p_s]_T$. There are three fixed points $p_s = (0,0,0,1)$, $p = (1,0,0,0)$ and $p' = (0,1,0,0)$. These points have equivariant fundamental classes in $A_*^T(Q^T)$ which we denote by $P_s$, $P$, and $P'$. By abuse of notation we will not distinguish between $P_s$ and $i_*(P_s)$. Both $A_*^T(Q^T)$ and $A_*^T(Q)$ are free $R_T$-modules of rank $3$, with respective ordered bases $\{P_s,P,P' \}$ and $\{ P_s,L,{\Bbb I} \}$. The map $i_*$ is a linear transformation of these $R_T$-modules, and we will compute its matrix with respect to these ordered bases. This matrix can be easily inverted, provided we invert $t$, and so we obtain $(i_*)^{-1}$. The equivariant Chow ring of ${\Bbb P}^3$ is given by $$ A^*_T({\Bbb P}^3) = {\Bbb Z}[t,h]/(h-t)(h+t)h(h+at) $$ so $A^*_T({\Bbb P}^3)$ is free of rank $4$ over $R_T$, with basis $\{1,h,h^2,h^3\}$. To compute $i_*P_s$, $i_*P$, and $i_*P'$, we take advantage of the fact that the pushforward $f_*: A_*^T(Q) \rightarrow A_*^T({\Bbb P}^3)$ is injective. Moreover it is straightforward to calculate the pushforward to ${\Bbb P}^3$ of all the classes in our story. To simplify the notation, we will use $f_*$ to denote either of the maps $A_*^T(Q) \rightarrow A_*^T({\Bbb P}^3)$ or $A_*^T(Q^T) \rightarrow A_*^T({\Bbb P}^3)$. We find: $$\begin{array}{l} f_*({\Bbb I}) = 2h\\ f_*(L)= (h-t)h\\ f_*(P_s) = h^3 - ht^2\\ f_*(P) = h^3 + (a-1) h^2 t - a h t^2\\ f_*(P') = h^3 + (a+1) h^2 t + a h t^2 \end{array} $$ This implies that $$\begin{array}{l} i_*(P_s) = P_s\\ i_*(P) = (a-1)t L + P_s\\ i_*(P') = (a+1)t^2 {\Bbb I} + (a+1)t L + P_s. \end{array} $$ So the matrix for $i_*^T$ is $$\left(\begin{array}{ccc} 1 & 1 & 1\\ 0 & (a-1)t & (a+1)t\\ 0 & 0 & (a + 1)t^2 \end{array} \right).$$ Inverting this matrix we obtain $$\left(\begin{array}{ccc} 1 & \frac{1}{t(- a)} & \frac{2}{t^2(a^2 -1)} \\ 0 & \frac{1}{t(a-1)} & \frac{1}{t^2(1-a)} \\ 0 & 0 & \frac{1}{t^2(a+1)} \end{array} \right) $$ Thus we can write (after supressing the $(i_*)^{-1}$ notation) $$\begin{array}{l} P_s = P_s\\ L = \frac{1}{t(a-1)}( -P_s + P)\\ {\Bbb I} =\frac{1}{t^2(a^2-1)}(2P_s - (a+1)P +(a -1)P') \end{array}. $$ \medskip {\bf Calculation of Chern numbers} We now return to the task of computing $c_1(\pi^*T_{{\Bbb P}^2}) c_1(f^*T_{{\Bbb P}^3}) \cap {\Bbb I}$. To simplify notation, set $\alpha_1 := c_1(\pi^*T_{{\Bbb P}^2})$ and $\alpha_2 := c_1(f^*T_{{\Bbb P}^3})$ and $\alpha := \alpha_1 \alpha_2$. By the calculations above $$ \alpha_1 \alpha_2 \cap {\Bbb I} = i_*(i^*\alpha_1 i^*\alpha_1 \cap \frac{t^{-2}}{a^2-1}(2P_s - (a+1)P +(a -1)P')).$$ To compute the class explicitly we must compute the restrictions of $\alpha_1$ and $\alpha_2$ to each of the fixed points $P_s$, $P$ and $P'$. The tangent space to $P_s$ in ${\Bbb P}^3$ has weights $(1-a, -1-a, -a)$. Thus $\alpha_{2}|_{P_s} = (1-a)t - (1+a)t - at = -3at$. To compute $\alpha_{1}|_{P_s}$ observe that $P_s$ is the inverse image of the fixed point $(0,0,1) \in {\Bbb P}^2$. Since $T_{{\Bbb P}^2}$ has weights $(1-a,-1-a)$ at this point, $c_1(\pi^*T_{{\Bbb P}_2})|_{P_s} = (1-a)t + (-1 -a)t = -2t$. The restrictions to the other two fixed points can be calculated similarly. In particular $$\begin{array}{ll} \alpha_{1}|_{P} = (a-3)t \mbox{ }& \alpha_{1}|_{P'} = (a+3)t\\ \alpha_{2}|_{P} = (a- 4)t & \alpha_{2}|_{P'} = (a+4)t \end{array} $$ Thus, $$\alpha \cap {\Bbb I} = \frac{12 a^2}{a^2-1}P_s - \frac{(a-3)(a-4)(a+1)}{a^2-1}P + \frac{(a+3)(a+4)(a-1)}{a^2-1}P' \in A_*^T(Q) \otimes {\Bbb Q}.$$ Thus, $$\begin{array}{ll} \mbox{deg }(c_1(\pi^*T_{{\Bbb P}^2}) c_1(f^{*}T_{{\Bbb P}^3}) \cap [Q]) & = \frac{12 a^2}{ a^2 -1 } \;- \;\frac{(a-3)(a-4)(a+1)}{a^2-1} \; + \; \frac{(a+3)(a+4)(a-1)}{a^2-1}\\ & = 24 \end{array}. $$ $\Box$ \medskip

9609

alg-geom/9609014

en

https://arxiv.org/abs/alg-geom/9609014

alg-geom/9609014

Carlos Simpson

Carlos Simpson

Algebraic (geometric) $n$-stacks

LaTeX

We propose a generalization of Artin's definition of algebraic stack, which we call {\em geometric $n$-stack}. The main observation is that there is an inductive structure to the definition whereby the ingredients for the definition of geometric $n$-stack involve only $n-1$-stacks and so are already previously defined. We use this inductive structure to obtain some basic properties. We look at maps from a projective variety into certain such $n$-stacks, and obtain an interpretation of the Brill-Noether locus as the set of points of a geometric $n$-stack. At the end we explain how this provides a context for looking at de Rham theory for higher nonabelian cohomology, how one can define the Hodge filtration and so on.

alg-geom

\section*{Algebraic (geometric) $n$-stacks} Carlos Simpson \bigskip In the introduction of Laumon-Moret-Bailly (\cite{LaumonMB} p. 2) they refer to a possible theory of algebraic $n$-stacks: \begin{inset} Signalons au passage que Grothendieck propose d'\'elargir \`a son tour le cadre pr\'ec\'edent en rempla\c{c}ant les $1$-champs par des $n$-champs (grosso modo, des faisceaux en $n$-cat\'egories sur $(Aff)$ ou sur un site arbitraire) et il ne fait gu\`ere de doute qu'il existe une notion utile de $n$-champs alg\'ebriques \ldots . \end{inset} The purpose of this paper is to propose such a theory. I guess that the main reason why Laumon and Moret-Bailly didn't want to get into this theory was for fear of getting caught up in a horribly technical discussion of $n$-stacks of groupoids over a general site. In this paper we simply {\em assume} that a theory of $n$-stacks of groupoids exists. This is not an unreasonable assumption, first of all because there is a relatively good substitute---the theory of simplicial presheaves or presheaves of spaces (\cite{Brown} \cite{BrownGersten} \cite{Joyal} \cite{Jardine1} \cite{kobe} \cite{flexible})---which should be equivalent, in an appropriate sense, to any eventual theory of $n$-stacks; and second of all because it seems likely that a real theory of $n$-stacks of $n$-groupoids could be developped in the near future (\cite{BreenAsterisque}, \cite{Tamsamani}). Once we decide to ignore the technical complications involved in theories of $n$-stacks, it is a relatively straightforward matter to generalize Artin's definition of algebraic $1$-stack. The main observation is that there is an inductive structure to the definition whereby the ingredients for the definition of algebraic $n$-stack involve only $n-1$-stacks and so are already previously defined. This definition came out of discussions with C. Walter in preparation for the Trento school on algebraic stacks (September 1996). He made the remark that the definition of algebraic stack made sense in any category where one has a reasonable notion of smooth morphism, and suggested a general terminology of ``geometric stack'' for this notion. One immediately realizes that the notion of smooth morphism makes sense---notably---in the ``category'' of algebraic stacks and therefore according to Walter's remark, one could define the notion of geometric stack in the category of algebraic stacks. This is the notion of algebraic $2$-stack. It is an easy step to go from there to the general inductive definition of algebraic $n$-stack. Walter informs me that he had also come upon the notion of algebraic $2$-stack at the same time (just before the Trento school). Now a note about terminology: I have chosen to write the paper using Walter's terminology ``geometric $n$-stack'' because this seems most closely to reflect what is going on: the definition is made so that we can ``do geometry'' on the $n$-stack, since in a rather strong sense it looks locally like a scheme. For the purposes of the introduction, the terminology ``algebraic $n$-stack'' would be better because this fits with Artin's terminology for $n=1$. There is another place where the terminology ``algebraic'' would seem to be useful, this is when we start to look at geometric stacks on the analytic site, which we call ``analytic $n$-stacks''. In fact one could interchange the terminologies and in case of confusion one could even say ``algebraic-geometric $n$-stack''. In \cite{RelativeLie} I proposed a notion of {\em presentable $n$-stack} stable under homotopy fiber products and truncation. One key part of the notion of algebraic stack is the smoothness of the morphism $X\rightarrow T$ from a scheme. This is lost under truncation (e.g. the sheaf $\pi _0$ of an algebraic stack may no longer be an algebraic stack); this indicates that the notion of ``geometric stack'' is something which combines together the various homotopy groups in a fairly intricate way. In particular, the notion of presentable $n$-stack is not the same as the notion of geometric $n$-stack (however a geometric $n$-stack will be presentable). This is a little bit analogous to the difference between constructible and locally closed subsets in the theory of schemes. We will work over the site ${\cal X}$ of schemes of finite type over $Spec (k)$ with the etale topology, and with the notion of smooth morphism. The definitions and basic properties should also work for any site in which fiber products exist, provided with a certain class of morphisms analogous to the smooth morphisms. Rather than carrying this generalization through in the discussion, we leave it to the reader. Note that there are several examples which readily come to mind: \newline ---the site of schemes of finite type with the etale topology and the class of etale morphisms: this gives a notion of what might be called a ``Deligne-Mumford $n$-stack''; \newline ---the site of schemes of finite type with the fppf topology and the class of flat morphisms: this gives a notion of what might be called a ``flat-geometric $n$-stack''; \newline ---the site of schemes of finite type with the qff topology and the class of quasi-finite flat morphisms: this gives a notion of what might be called a ``qff-geometric $n$-stack''. Whereas Artin proves \cite{ArtinInventiones} that flat-geometric $1$-stacks are also smooth-geometric stacks (i.e. those defined as we do here using smooth morphisms)---his proof is recounted in \cite{LaumonMB}---it seems unlikely that the same would be true for $n$-stacks. Artin's method also shows that qff-geometric $1$-stacks are Deligne-Mumford stacks. However it looks like Deligne-Mumford $n$-stacks are essentially just gerbs over Deligne-Mumford $1$-stacks, while on the other hand in characteristic $p$ one could apply Dold-Puppe (see below) to a complex of finite flat group schemes to get a fairly non-trivial qff-algebraic $n$-stack. This seems to show that the implication ``qff-geometric $\Rightarrow $ Deligne-Mumford'' no longer holds for $n$-stacks. This is why it seems unlikely that Artin's reasoning for the implication ``flat-geometric $\Rightarrow$ smooth-geometric'' will work for $n$-stacks. Here is the plan of the paper. In \S 1 we give the basic definitions of geometric $n$-stack and smooth morphism of geometric $n$-stacks. In \S 2 we give some basic properties which amount to having a good notion of geometric morphism between $n$-stacks (which are themselves not necessarily geometric). In \S 3 we briefly discuss some ways one could obtain geometric $n$-stacks by glueing. In \S 4 we show that geometric $n$-stacks are presentable in the sense of \cite{RelativeLie}. This is probably an important tool if one wants to do any sort of Postnikov induction, since presentable $n$-stacks are closed under the truncation processes which make up the Postnikov tower (whereas the notion of geometric $n$-stack is not closed under truncation). In \S 5 we do a preliminary version of what should be a more general Quillen theory. We treat only the $1$-connected case, and then go on in the subsection ``Dold-Puppe'' to treat the relative (i.e. over a base scheme or base $n$-stack) stable (in the sense of homotopy theory) case in a different way. It would be nice to have a unified version including a reasonable notion of differential graded Lie algebra over an $n$-stack $R$ giving an algebraic approach to relatively $1$-connected $n$-stacks over $R$, but this seems a bit far off in a technical sense. In \S 6 we look at maps from a projective variety (or a smooth formal category) into a geometric $n$-stack. Here again it would be nice to have a fairly general theory covering maps into any geometric $n$-stack but we can only say something interesting in the easiest case, that of maps into {\em connected very presentable $T$}, i.e. $n$-stacks with $\pi _0(T)=\ast$, $\pi _1(T)$ an affine algebraic group scheme and $\pi _i(T)$ a vector space for $i\geq 2$. (The terminology ``very presentable'' comes from \cite{RelativeLie}). At the end we speculate on how one might generalize to various other classes of $T$. In \S 7 we briefly present an approach to defining the tangent stack to a geometric $n$-stack. This is a generalization of certain results in the last chapter of \cite{LaumonMB} although we don't refer to the cotangent complex. In \S 8 we explain how to use geometric $n$-stacks as a framework for looking at de Rham theory for higher nonabelian cohomology. This is sort of a synthesis of things that are in \cite{SantaCruz} and \cite{kobe}. \bigskip We assume known an adequate theory of $n$-stacks of groupoids over a site ${\cal X}$. The main thing we will need is the notion of fiber product (which of course means---as it always shall below---what one would often call the ``homotopy fiber product''). We work over an algebraically closed field $k$ of characteristic zero, and sometimes directly over field $k={\bf C}$ of complex numbers. Note however that the definition makes sense over arbitrary base scheme and the ``Basic properties'' hold true there. The term ``connected'' when applied to an $n$-stack means that the sheaf $\pi _0$ is the final object $\ast$ (represented by $Spec (k)$). In the case of a $0$-stack represented by $Y$ this should not be confused with connectedness of the scheme $Y$ which is a different question. \numero{Definitions} Let ${\cal X}$ be the site of schemes of finite type over $Spec (k)$ with the etale topology. We will define the following notions: that an $n$-stack $T$ be {\em geometric}; and that a morphism $T\rightarrow Z$ from a geometric $n$-stack to a scheme be {\em smooth}. We define these notions together by induction on $n$. Start by saying that a $0$-stack (sheaf of sets) is {\em geometric} if it is represented by an algebraic space. Say that a morphism $T\rightarrow Z$ from a geometric $0$-stack to a scheme is {\em smooth} if it is smooth as a morphism of algebraic spaces. Now we give the inductive definitions: say that an $n$-stack $T$ is {\em geometric} if: \newline GS1---for any schemes $X$ and $Y$ and morphisms $X\rightarrow T$, $Y\rightarrow T$ the fiber product $X\times _TY$ (which is an $n-1$-stack) is geometric using the inductive definition; and \newline GS2---there is a scheme $X$ and a morphism of $n$-stacks $f:X\rightarrow T$ which is surjective on $\pi _0$ with the property that for any scheme $Y$ and morphism $Y\rightarrow T$, the morphism $$ X\times _TY\rightarrow Y $$ from a geometric $n-1$ stack to the scheme $Y$ is smooth (using the inductive definition). If $T$ is a geometric $n$-stack we say that a morphism $T\rightarrow Y$ to a scheme is {\em smooth} if for at least one morphism $X\rightarrow T$ as in GS2, the composed morphism $X\rightarrow Y$ is a smooth morphism of schemes. This completes our inductive pair of definitions. For $n=1$ we recover the notion of algebraic stack, and in fact our definition is a straightforward generalization to $n$-stacks of Artin's definition of algebraic stack. The following lemma shows that the phrase ``for at least one'' in the definition of smoothness can be replaced by the phrase ``for any''. \begin{lemma} \label{independence} Suppose $T\rightarrow Y$ is a morphism from an $n$-stack to a scheme which is smooth according to the previous definition, and suppose that $U\rightarrow T$ is a morphism from a scheme such that for any scheme $Z\rightarrow T$, $U\times _TZ\rightarrow Z$ is smooth (again according to the previous definition, as a morphism from an $n-1$-stack to a scheme). Then $U\rightarrow Y$ is a smooth morphism of schemes. \end{lemma} {\em Proof:} We prove this for $n$-stacks by induction on $n$. Let $X\rightarrow T$ be the morphism as in GS2 such that $X\rightarrow Y$ is smooth. Let $R= X\times _TU$. This is an $n-1$-stack and the morphisms $R\rightarrow X$ and $R\rightarrow U$ are both smooth as morphisms from $n-1$-stacks to schemes according to the above definition. Let $W\rightarrow R$ be a surjection from a scheme as in property GM2. By the present lemma applied inductively for $n-1$-stacks, the morphisms $W\rightarrow X$ and $W\rightarrow U$ are smooth morphisms of schemes. But the condition that $X\rightarrow Y$ is smooth implies that $W\rightarrow Y$ is smooth, and then since $W\rightarrow U$ is smooth and surjective we get that $U\rightarrow Y$ is smooth as desired. This argument doesn't work when $n=0$ but then $R$ is itself an algebraic space and the maps $R\rightarrow X$ (hence $R\rightarrow Y$) and $R\rightarrow U$ are smooth maps of algebraic spaces; this implies directly that $U\rightarrow Y$ is smooth. \hfill $\Box$\vspace{.1in} The following lemma shows that these definitions don't change if we think of an $n$-stack as an $n+1$-stack etc. \begin{lemma} \label{ntom} Suppose $T$ is an $n$-stack which, when considered as an $m$-stack for some $m\geq n$, is a geometric $m$-stack. Then $T$ is a geometric $n$-stack. Similarly smoothness of a morphism $T\rightarrow Y$ to a scheme when $T$ is considered as an $m$-stack implies smoothness when $T$ is considered as an $n$-stack. \end{lemma} {\em Proof:} We prove this by induction on $n$ and then $m$. The case $n=0$ and $m=0$ is clear. First treat the case $n=0$ and any $m$: suppose $T$ is a sheaf of sets which is a geometric $m$-stack. There is a morphism $X\rightarrow T$ with $X$ a scheme, such that if we set $R= X\times _TX$ then $R$ is an $m-1$-stack smooth over $X$. However $R$ is again a sheaf of sets so by the inductive statement for $n=0$ and $m-1$ we have that $R$ is an algebraic space. Furthermore the smoothness of the morphism $R\rightarrow X$ with $R$ considered as an $m-1$-stack implies smoothness with $R$ considered as a $0$-stack. In particular $R$ is an algebraic space with smooth maps to the projections. Since the quotient of an algebraic space by a smooth equivalence relation is again an algebraic space, we get that $T$ is an algebraic space i.e. a geometric $0$-stack (and note by the way that $X\rightarrow T$ is a smooth surjective map of algebraic spaces). This proves the first statement for $(0,m)$. For the second statement, suppose $T\rightarrow Y$ is a morphism to a scheme $Y$ which is smooth as a morphism from an $m$-stack. Then choose the smooth surjective morphism $X\rightarrow T$; as we have seen above this is a smooth morphism of algebraic spaces. The definition of smoothness now is that $X\rightarrow Y$ is smooth. But this implies that $T\rightarrow Y$ is smooth. This completes the inductive step for $(0,m)$. Now suppose we want to show the lemma for $(n,m)$ with $n\geq 1$ and suppose we know it for all $(n', m')$ with $n'<n$ or $n'=n$ and $m'<m$. Let $T$ be an $n$-stack which is geometric considered as an $m$-stack. If $X,Y\rightarrow T$ are maps from schemes then $X\times _TY$ is an $n-1$-stack which is geometric when considered as an $m-1$-stack; by the induction hypothesis it is geometric when considered as an $n-1$-stack, which verifies GS1. Choose a smooth surjection $X\rightarrow T$ from a scheme as in property GS2 for $m$-stacks. Suppose $Y\rightarrow T$ is any morphism from a scheme. Then $X\times _TY$ is an $n-1$-stack with a map to $Y$ which is smooth considered as a map from $m-1$-stacks. Again by the induction hypothesis it is smooth considered as a map from an $n-1$-stack to a scheme, so we get GS2 for $n$-stacks. This completes the proof that $T$ is geometric when considered as an $n$-stack. Finally suppose $T\rightarrow Y$ is a morphism from an $n$-stack to a scheme which is smooth considered as a morphism from an $m$-stack. Choose a surjection $X\rightarrow T$ as in property GS2 for $m$-stacks; we have seen above that it also satisfies the same property for $n$-stacks. By definition of smoothness of our original morphism from an $m$-stack, the morphism $X\rightarrow Y$ is smooth as a morphism of schemes; this gives smoothness of $T\rightarrow Y$ considered as a morphism from an $n$-stack to a scheme. This finishes the inductive proof of the lemma. \hfill $\Box$\vspace{.1in} {\em Remarks:} \newline (1)\, We can equally well make a definition of {\em Deligne-Mumford $n$-stack} by replacing ``smooth'' in the previous definition with ''etale''. This gives an $n$-stack whose homotopy group sheaves are finite... \newline (2)\, We could also make definitions of flat-geometric or qff-geometric $n$-stack, by replacing the smoothness conditoin by flatness or quasifinite flatness. If all of these notions are in question then we will denote the standard one by ``smooth-geometric $n$-stack''. Not to be confused with ``smooth geometric $n$-stack'' which means a smooth-geometric $n$-stack which is smooth! We now complete our collection of basic definitions in some obvious ways. We say that a morphism of $n$-stacks $R\rightarrow T$ is {\em geometric} if for any scheme $Y$ and map $Y\rightarrow T$ the fiber product $R\times _TY$ is a geometric $n$-stack. We say that a geometric morphism of $n$-stacks $R\rightarrow T$ is {\em smooth} if for any scheme $Y$ and map $Y\rightarrow T$ the morphism $R\times _TY\rightarrow Y$ is a smooth morphism in the sense of our inductive definition. \begin{lemma} If $T\rightarrow Z$ is a morphism from an $n$-stack to a scheme then it is smooth and geometric in the sense of the previous paragraph, if and only if $T$ is geometric and the morphism is smooth in the sense of our inductive definition. \end{lemma} {\em Proof:} Suppose that $T$ is geometric and the morphism is smooth in the sense of the previous paragraph. Then applying that to the scheme $Z$ itself we obtain that the morphism is smooth in the sense of the inductive definition. On the other hand, suppose the morphism is smooth in the sense of the inductive definition. Let $X\rightarrow T$ be a surjection as in GS2. Thus $X\rightarrow Z$ is smooth. For any scheme $Y\rightarrow Z$ we have that $X\times _ZY\rightarrow T\times _ZY$ is surjective and smooth in the sense of the previous paragraph; but in this case (and using the direction we have proved above) this is exactly the statement that it satisfies the conditions of GS2 for the stack $T\times _ZY$. On the other hand $X\times _ZY\rightarrow Y$ is smooth. This implies (via the independence of the choice in the original definition of smoothness which comes from \ref{independence}) that that $T\times _ZY\rightarrow Y$ is smooth in the original sense. As this works for all $Y$, we get that $T\rightarrow Z$ is smooth in the new sense. \hfill $\Box$\vspace{.1in} \numero{Basic properties} We assume that the propositions, lemmas and corollaries in this section are known for $n-1$-stacks and we are proving them all in a gigantic induction for $n$-stacks. On the other hand, in proving any statement we can use the {\em previous} statements for the same $n$, too. \begin{proposition} \label{fiberprod} If $R$, $S$ and $T$ are geometric $n$-stacks with morphisms $R,T\rightarrow S$ then the fiber product $R\times _ST$ is a geometric $n$-stack. \end{proposition} {\em Proof:} Suppose $R$, $S$ and $T$ are geometric $n$-stacks with morphisms $R\rightarrow S$ and $T\rightarrow S$. Let $X\rightarrow R$, $Y\rightarrow S$ and $Z\rightarrow T$ be smooth surjective morphisms from schemes. Choose a smooth surjective morphism $W\rightarrow X\times _SZ$ from a scheme (by axiom GS2 for $S$). By base change of the morphism $Z\rightarrow T$, the morphism $X\times _SZ\rightarrow X\times _ST$ is a geometric smooth surjective morphism. We first claim that the morphism $W\rightarrow X\times _ST$ is smooth. To prove this, suppose $A\rightarrow X\times _ST$ is a morphism from a scheme. Then $W\times _{X\times _ST}A\rightarrow A$ is the composition of $$ W\times _{X\times _ST}A \rightarrow (X\times _SZ)\times _{X\times _ST}A \rightarrow A. $$ Both morphisms are geometric and smooth, and all three terms are $n-1$-stacks (note that in the middle $(X\times _SZ)\times _{X\times _ST}A= A\times _TZ$). By the composition result for $n-1$-stacks (Corollary \ref{geocomposition} below with our global induction hypothesis) the composed morphism $W\times _{X\times _ST}A\rightarrow A$ is smooth, and this for any $A$. Thus $W\rightarrow X\times _ST$ is smooth. Next we claim that the morphism $W\rightarrow R\times _ST$ is smooth. Again suppose that $A\rightarrow R\times _ST$ is a morphism from a scheme. The two morphisms $$ W\times _{R\times _ST} A \rightarrow (X\times _ST)\times _{R\times _ST}A = X\times _RA \rightarrow A $$ are smooth and geometric by base change. Again this is a composition of morphisms of $n-1$-stacks so by Corollary \ref{smoothcomposition2} and our global induction hypothesis the composition is smooth and geometric. Finally the morphism $W\rightarrow R\times _ST$ is the composition of three surjective morphisms so it is surjective. We obtain a morphism as in GS2 for $R\times _ST$. We turn to GS1. Suppose $X\rightarrow R\times _ST$ and $Y\rightarrow R\times _ST$ are morphisms from schemes. We would like to check that $X\times _{R\times _ST}Y$ is a geometric $n-1$-stack. Note that calculating $X\times _{R\times _ST}Y$ is basically the same thing as calculating in usual homotopy theory the path space between two points $x$ and $y$ in a product of fibrations $r\times _st$. From this point of view we see that $$ X\times _{R\times _ST} Y = (X\times _RY)\times _{X\times _SY}(X\times _TY). $$ Note that the three components in the big fiber product on the right are geometric $n-1$-stacks, so by our inductive hypothesis (i.e. assuming the present proposition for $n-1$-stacks) we get that the right hand side is a geometric $n-1$-stack, this gives the desired statement for GS1. \hfill $\Box$\vspace{.1in} \begin{corollary} If $R$ and $T$ are geometric $n$-stacks then any morphism between them is geometric. In particular an $n$-stack $T$ is geometric if and only if the structural morphism $T\rightarrow \ast$ is geometric. \end{corollary} \hfill $\Box$\vspace{.1in} \begin{lemma} \label{smoothcomposition1} If $R\rightarrow S\rightarrow T$ are morphisms of geometric $n$-stacks and if each morphism is smooth then the composition is smooth. \end{lemma} {\em Proof:} We have already proved this for morphisms $X\rightarrow T \rightarrow Y$ where $X$ and $Y$ are schemes (see Lemma \ref{independence}). Suppose $U\rightarrow T\rightarrow Y$ are smooth morphisms of geometric $n$-stacks with $Y$ a scheme. We prove that the composition is smooth, by induction on $n$ (we already know it for $n=0$). If $Z\rightarrow T$ is a smooth surjective morphism from a scheme then the morphism $$ U\times _TZ \rightarrow Z $$ is smooth by the definition of smoothness of $U\rightarrow T$. Also the map $Z\rightarrow Y$ is smooth by definition of smoothness of $T\rightarrow Y$. Choose a smooth surjection $V\rightarrow U\times _TZ$ from a scheme $V$ and note that the map $V\rightarrow Z$ is smooth by definition, so (since these are morphisms of schemes) the composition $V\rightarrow Y$ is smooth. On the other hand $$ U\times _TZ \rightarrow U $$ is smooth and surjective, by base change from $Z\rightarrow T$. We claim that the morphism $V\rightarrow U$ is smooth and surjective---actually surjectivity is obvious. To prove that it is smooth, let $W\rightarrow U$ be a morphism from a scheme; then $$ W\times _UV \rightarrow W\times _U(U\times _TZ) = W\times _TZ \rightarrow W $$ is a composable pair of morphisms of $n-1$-stacks each of which is smooth by base change. By our induction hypothesis the composition is smooth. This shows by definition that $V\rightarrow U$ is smooth. In particular the map $V\rightarrow U$ is admissible as in GS2, and then we can conclude that the map $U\rightarrow Y$ is smooth by the original definition using $V$. This completes the proof in the current case. Suppose finally that $U\rightarrow T \rightarrow R$ are smooth morphisms of geometric $n$-stacks. Then for any scheme $Y$ the morphisms $U\times _RY\rightarrow T\times _RY \rightarrow Y$ are smooth by base change; thus from the case treated above their composition is smooth, and this is the definition of smoothness of $U\rightarrow R$. \hfill $\Box$\vspace{.1in} \begin{lemma} \label{descendgeometric} Suppose $S\rightarrow T$ is a geometric smooth surjective morphism of $n$-stacks, and suppose that $S$ is geometric. Then $T$ is geometric. \end{lemma} {\em Proof:} We first show GS2. Let $W\rightarrow S$ be a smooth geometric surjection from a scheme. We claim that the morphism $W\rightarrow T$ is surjective (easy), geometric and smooth. To show that it is geometric, suppose $Y\rightarrow T$ is a morphism from a scheme. Then since $S\rightarrow T$ is geometric we have that $Y\times _TS$ is a geometric $n$-stack. On the other hand, $$ Y\times _TW = (Y\times _TS)\times _SW, $$ so by Proposition \ref{fiberprod} $Y\times _TW$ is geometric. Finally to show that $W\rightarrow T$ is smooth, note that $$ Y\times _TW\rightarrow Y\times _TS \rightarrow Y $$ is a composable pair of smooth (by base change) morphisms of geometric $n$-stacks, so by the previous lemma the composition is smooth. The morphism $W\rightarrow T$ thus works for condition GS2. To show GS1, suppose $X,Y\rightarrow T$ are morphisms from schemes. Then $$ (X\times _TY)\times _TW = (X\times _TW)\times _W (Y\times _TW). $$ The geometricity of the morphism $W\rightarrow T$ implies that $X\times _TW$ and $Y\times _TW$ are geometric, whereas of course $W$ is geometric. Thus by Proposition \ref{fiberprod} we get that $(X\times _TY)\times _TW$ is geometric. Now note that the morphism $$ (X\times _TY)\times _TW \rightarrow X\times _TY $$ of $n-1$-stacks is geometric, smooth and surjective (by base change of the same properties for $W\rightarrow T$). By the inductive version of the present lemma for $n-1$ (noting that the lemma is automatically true for $n=0$) we obtain that $X\times _TY$ is geometric. This is GS1. \hfill $\Box$\vspace{.1in} \begin{corollary} \label{localization} Suppose $Y$ is a scheme and $T\rightarrow Y$ is a morphism from an $n$-stack. If there is a smooth surjection $Y' \rightarrow Y$ such that $T':=Y'\times _YT\rightarrow Y'$ is geometric then the original morphism is geometric. \end{corollary} {\em Proof:} The morphism $T'\rightarrow T$ is geometric, smooth and surjective (all by base-change from the morphism $Y'\rightarrow Y$). By \ref{descendgeometric}, the fact that $T'$ is geometric implies that $T$ is geometric. \hfill $\Box$\vspace{.1in} This corollary is particularly useful to do etale localization. It implies that the property of a morphism of $n$-stacks being geometric, is etale-local over the base. \begin{corollary} \label{fibration} Given a geometric morphism $R\rightarrow T$ of $n$-stacks such that $T$ is geometric, then $R$ is geometric. \end{corollary} {\em Proof:} Let $X\rightarrow T$ be the geometric smooth surjective morphism from a scheme given by GS2 for $T$. By base change, $X\times _TR \rightarrow R$ is a geometric smooth surjective morphism. However, by the geometricity of the morphism $R\rightarrow T$ the fiber product $X\times _TR$ is geometric; thus by the previous lemma, $R$ is geometric. \hfill $\Box$\vspace{.1in} \begin{corollary} \label{geocomposition} The composition of two geometric morphisms is again geometric. \end{corollary} {\em Proof:} Suppose $U\rightarrow T\rightarrow R$ are geometric morphisms, and suppose $Y\rightarrow R$ is a morphism from a scheme. Then $$ U\times _RY= U\times _T(T\times _RY). $$ By hypothesis $T\times _RY$ is geometric. On the other hand $U\times _RY\rightarrow T\times _RY$ is geometric (since the property of being geometric is obviously stable under base change). By the previous Proposition \ref{fiberprod} we get that $U\times _RY$ is geometric. Thus the morphism $U\rightarrow R$ is geometric. \hfill $\Box$\vspace{.1in} \begin{corollary} \label{smoothcomposition2} The composition of two geometric smooth morphisms is geometric and smooth. \end{corollary} {\em Proof:} Suppose $R\rightarrow S \rightarrow T$ is a pair of geometric smooth morphisms. Suppose $Y\rightarrow T$ is a morphism from a scheme. Then (noting by the previous corollary that $R\rightarrow T$ is geometric) $R\times _TY$ and $S\times _TY$ are geometric. The composable pair $$ R\times _TY \rightarrow S\times _TY \rightarrow Y $$ of smooth morphisms now falls into the hypotheses of Lemma \ref{smoothcomposition1} so the composition is smooth. This implies that our original composition was smooth. \hfill $\Box$\vspace{.1in} In a relative setting we get: \begin{corollary} Suppose $U\stackrel{a}{\rightarrow}T\stackrel{b}{\rightarrow}R$ is a composable pair of morphisms of $n$-stacks. If $a$ is geometric, smooth and surjective and $ba$ is geometric (resp. geometric and smooth) then $b$ is geometric (resp. geometric and smooth). \end{corollary} {\em Proof:} Suppose $Y\rightarrow R$ is a morphism from a scheme. Then $$ Y\times _RU = (Y\times _RT)\times _TU. $$ The map $Y\times _RU\rightarrow Y\times _RT$ is geometric, smooth and surjective (since those properties are obviously---from the form of their definitions---invariant under base change). The fact that $ba$ is geometric implies that $Y\times _RU$ is geometric, which by the previous lemma implies that $Y\times _RT$ is geometric. Suppose furthermore that $ba$ is smooth. Choose a smooth surjection $W\rightarrow Y\times _RT$ from a scheme. Then the morphism $$ W\times _{Y\times _RT} (Y\times _RU)\rightarrow Y\times _RU $$ is smooth by basechange and the morphism $Y\times _RU\rightarrow Y$ is smooth by hypothesis. Thus $W\times _{Y\times _RT} (Y\times _RU)\rightarrow Y$ is smooth. Choosing a smooth surjection from a scheme $$ V\rightarrow W\times _{Y\times _RT} (Y\times _RU) $$ we get that $V\rightarrow Y$ is a smooth morphism of schemes. On the other hand, the morphism $$ W\times _{Y\times _RT} (Y\times _RU)\rightarrow W $$ is smooth and surjective, so $V\rightarrow W$ is smooth and surjective. Therefore $W\rightarrow Y$ is smooth. This proves that if $ba$ is smooth then $b$ is smooth. \hfill $\Box$\vspace{.1in} {\em Examples:} Proposition \ref{fibration} allows us to construct many examples. The main examples we shall look at below are the {\em connected presentable $n$-stacks}. These are connected $n$-stacks $T$ with (choosing a basepoint $t\in T(Spec ({\bf C} ))$ $\pi _i (T, t)$ represented by group schemes of finite type. We apply \ref{fibration} inductively to show that such a $T$ is geometric. Let $T\rightarrow \tau _{\leq n-1}T$ be the truncation morphism. The fiber over a morphism $Y\rightarrow \tau _{\leq n-1}T$ is (locally in the etale topology of $Y$ where there exists a section---this is good enough by \ref{localization}) isomorphic to $K(G/Y, n)$ for a smooth group scheme of finite type $G$ over $Y$. Using the following lemma, by induction $T$ is geometric. \begin{lemma} \label{eilenbergExample} Fix $n$, suppose $Y$ is a scheme and suppose $G$ is a smooth group scheme over $Y$. If $n\geq 2$ require $G$ to be abelian. Then $K(G/Y, n)$ is a geometric $n$-stack and the morphism $K(G/Y,n)\rightarrow Y$ is smooth.. \end{lemma} {\em Proof:} We prove this by induction on $n$. For $n=0$ we simply have $K(G/Y,0)=G$ which is a scheme and hence geometric---also note that by hypothesis it is smooth over $Y$. Now for any $n$, consider the basepoint section $Y\rightarrow K(G/Y,n)$. We claim that this is a smooth geometric map. If $Z\rightarrow K(G/Y,n)$ is any morphism then it corresponds to a map $Z\rightarrow Y$ and a class in $H^n(Z,G|_Z)$. Since we are working with the etale topology, by definition this class vanishes on an etale surjection $Z'\rightarrow Z$ and for our claim it suffices to show that $Y\times _{K(G/Y,n)}Z'$ is smooth and geometric over $Z'$. Thus we may assume that our map $Z'\rightarrow K(G/Y,n)$ factors through the basepoint section $Y\rightarrow K(G/Y,n)$. In particular it suffices to prove that $Y\times _{K(G/Y,n)}Y\rightarrow Y$ is smooth and geometric. But $$ Y\times _{K(G/Y,n)}Y= K(G/Y,n-1) $$ so by our induction hypothesis this is geometric and smooth over $Y$. This shows that $K(G/Y,n)$ is geometric and furthermore the basepoint section is a choice of map as in GS2. Now the composed map $Y\rightarrow K(G/Y,n)\rightarrow Y$ is the identity, in particular smooth, so by definition $K(G/Y,n)\rightarrow Y$ is smooth. \hfill $\Box$\vspace{.1in} Note that stability under fiber products (Proposition \ref{fiberprod}) implies that if $T$ is a geometric $n$-stack then $Hom (K, T)$ is geometric for any finite CW complex $K$. See (\cite{kobe} Corollary 5.6) for the details of the argument---which was in the context of presentable $n$-stacks but the argument given there only depends on stability of our class of $n$-stacks under fiber product. We can apply this in particular to the geometric $n$-stacks constructed just above, to obtain some non-connected examples. If $T= BG$ for an algebraic group $G$ and $K$ is connected with basepoint $k$ then $Hom (K, T)$ is the moduli stack of representations $\pi _1(K,k)\rightarrow G$ up to conjugacy. \numero{Locally geometric $n$-stacks} The theory we have described up till now concerns objects {\em of finite type} since we have assumed that the scheme $X$ surjecting to our $n$-stack $T$ is of finite type. We can obtain a definition of ``locally geometric'' by relaxing this to the condition that $X$ be locally of finite type (or equivalently that $X$ be a disjoint union of schemes of finite type). To be precise we say that an $n$-stack $T$ is {\em locally geometric} if there exists a sheaf which is a disjoint union of schemes of finite type, with a morphism $$ \varphi : X=\coprod X_i \rightarrow T $$ such that $\varphi$ is smooth and geometric. Note that if $X$ and $Y$ are schemes of finite type mapping to $T$ we still have that $X\times _TY$ is geometric (GS1). All of the previous results about fibrations, fiber products, and so on still hold for locally geometric $n$-stacks. One might want also to relax the definition even further by only requiring that $X\times _TY$ be itself locally geometric (and so on) even for schemes of finite type. We can obtain a notion that we call {\em slightly geometric} by replacing ``scheme of finite type'' by ``scheme locally of finite type'' everywhere in the preceeding definitions. This notion may be useful in the sense that a lot more $n$-stacks will be ``slightly geometric''. However it seems to remove us somewhat from the realm where geometric reasoning will work very well. \numero{Glueing} We say that a morphism $U\rightarrow T$ of geometric stacks is a {\em Zariski open subset} (resp. {\em etale open subset}) if for every scheme $Z$ and $Z\rightarrow T$ the fiber product $Z\times _TU$ is a Zariski open subset of $Z$ (resp. an algebraic space with etale map to $Z$). If we have two geometric $n$-stacks $U$ and $V$ and a geometric $n$-stack $W$ with morphisms $W\rightarrow U$ and $W\rightarrow V$ each of which is a Zariski open subset, then we can glue $U$ and $V$ together along $W$ to get a geometric $n$-stack $T$ with Zariski open subsets $U\rightarrow T$ and $V\rightarrow T$ whose intersection is $W$. If one wants to glue several open sets it has to be done one at a time (this way we avoid having to talk about higher cocycles). As a more general result we have the following. Suppose $\Phi$ is a functor from the simplicial category $\Delta$ to the category of $n$-stacks (say a strict functor to the category of simplicial presheaves, for example). Suppose that each $\Phi _k$ is a geometric $n$-stack, and suppose that the two morphisms $\Phi _1 \rightarrow \Phi _0$ are smooth. Suppose furthermore that $\Phi $ satisfies the Segal condition that $$ \Phi _k \rightarrow \Phi _1\times _{\Phi _0} \ldots \times _{\Phi _0}\Phi _1 $$ is an equivalence (i.e. Illusie weak equivalence of simplicial presheaves). Finally suppose that for any element of $\Phi _1(X)$ there is, up to localization over $X$, an ``inverse'' (for the multiplication on $\Phi _1$ that comes from Segal's condition as in \cite{Segal}) up to homotopy. Let $T$ be the realization over the simplicial variable, into a presheaf of spaces (i.e. we obtain a bisimplicial presheaf, take the diagonal). \begin{proposition} In the above situation, $T$ is a geometric $n+1$-stack. \end{proposition} {\em Proof:} There is a surjective map $\Phi _0 \rightarrow T$ and we have by definition that $\Phi _0 \times _T\Phi _0 = \Phi _1$. From this one can see that $T$ is geometric. \hfill $\Box$\vspace{.1in} As an example of how to apply the above result, suppose $U$ is a geometric $n$-stack and suppose we have a geometric $n$-stack $R$ with $R\rightarrow U\times U$. Suppose furthermore that we have a multiplication $R\times _{p_2, U, p_1}R\rightarrow R$ which is associative and such that inverses exist up to homotopy. Then we can set $\Phi _k = R\times _U\ldots \times _UR$ with $\Phi _0 = U$. We are in the above situation, so we obtain the geometric $n$-stack $T$. We call this the {\em $n$-stack associated to the descent data $(U, R)$}. The original result about glueing over Zariski open subsets can be interpreted in this way. The simplicial version of this descent with any $\Phi$ satisfying Segal's condition is a way to avoid having to talk about strict associativity of the composition on $R$. \numero{Presentability} Recall first of all that the category of {\em vector sheaves} over a scheme $Y$ is the smallest abelian category of abelian sheaves on the big site of schemes over $Y$ containing the structure sheaf (these were called ``$U$-coherent sheaves'' by Hirschowitz in \cite{Hirschowitz}, who was the first to define them). A vector sheaf may be presented as the kernel of a sequence of $3$ coherent sheaves which is otherwise exact on the big site; or dually as the cokernel of an otherwise-exact sequence of $3$ {\em vector schemes} (i.e. duals of coherent sheaves). The nicest thing about the category of vector sheaves is that duality is involutive. Recall that we have defined in \cite{RelativeLie} a notion of {\em presentable group sheaf} over any base scheme $Y$. We will not repeat the definition here, but just remark (so as to give a rough idea of what is going on) that if $G$ is a presentable group sheaf over $Y$ then it admits a Lie algebra object $Lie (G)$ which is a vector sheaf with bilinear Lie bracket operation (satisfying Jacobi). In \cite{RelativeLie} a definition was then made of {\em presentable $n$-stack}; this involves a certain condition on $\pi _0$ (for which we refer to \cite{RelativeLie}) and the condition that the higher homotopy group sheaves (over any base scheme) be presentable group sheaves. For our purposes we shall often be interested in the slightly more restrictive notion of {\em very presentable $n$-stack}. An $n$-stack $T$ is defined (in \cite{RelativeLie}) to be very presentable if it is presentable, and if furthermore: \newline (1)\, for $i\geq 2$ and for any scheme $Y$ and $t\in T(Y)$ we have that $\pi _i (T|_{{\cal X} /Y}, t)$ is a vector sheaf over $Y$; and \newline (2)\, for any artinian scheme $Y$ and $t\in T(Y)$ the group of sections $\pi _1(T|_{{\cal X} /Y},t)(Y)$ (which is naturally an algebraic group scheme over $Spec (k)$) is affine. For our purposes here we will mostly stick to the case of connected $n$-stacks in the coefficients. Thus we review what the above definitions mean for $T$ connected (i.e. $\pi _0(T)=\ast $). Assume that $k$ is algebraically closed (otherwise one has to take what is said below possibly with some Galois twisting). In the connected case there is essentially a unique basepoint $t\in T(Spec (k))$. A group sheaf over $Spec (k)$ is presentable if and only if it is an algebraic group scheme (\cite{RelativeLie}), so $T$ is presentable if and only if $\pi _i(T,t)$ are represented by algebraic group schemes. Note that a vector sheaf over $Spec (k)$ is just a vector space, so $T$ is very presentable if and only if the $\pi _i (T,t)$ are vector spaces for $i\geq 2$ and $\pi _1(T,t)$ is an affine algebraic group scheme (which can of course act on the $\pi _i$ by a representation which---because we work over the big site---is automatically algebraic). \begin{proposition} \label{presentable} If $T$ is a geometric $n$-stack on ${\cal X}$ then $T$ is presentable in the sense of \cite{RelativeLie}. \end{proposition} {\em Proof:} Suppose $X\rightarrow R$ is a smooth morphism from a scheme $X$ to a geometric $n$-stack $R$. Note that the morphism $R\rightarrow \pi _0(R)$ satisfies the lifting properties $Lift _n(Y, Y_i)$, since by localizing in the etale topology we get rid of any cohomological obstructions to lifting coming from the higher homotopy groups. On the other hand the morphism $X\rightarrow R$ being smooth, it satisfies the lifting properties (for example one can say that the map $X\times _RY\rightarrow Y$ is smooth and admits a smooth surjection from a scheme smooth over $Y$; with this one gets the lifting properties, recalling of course that a smooth morphism between schemes is vertical. Thus we get that $X\rightarrow \pi _0(R)$ is vertical. Now suppose $T$ is geometric and choose a smooth surjection $u:X\rightarrow T$. We get from above that $X\rightarrow \pi _0(T)$ is vertical. Note that $$ X\times _{\pi _0(T)}X = im (X\times _TX \rightarrow X\times X). $$ Let $G$ denote the group sheaf $\pi _1(T|_{{\cal X} /X}, u)$ over $X$. We have that $G$ acts freely on $\pi _0(X\times _TX)$ (relative to the first projection $X\times _TX\rightarrow X$) and the quotient is the image $X\times _{\pi _0(T)}X$. Thus, locally over schemes mapping into the target, the morphism $$ \pi _0(X\times _TX) \rightarrow X\times _{\pi _0(T)}X $$ is the same as the morphism $$ G\times _X(X\times _{\pi _0(T)}X)\rightarrow X\times _{\pi _0(T)}X $$ obtained by base-change. Since $G\rightarrow X$ is a group sheaf it is an $X$-vertical morphism (\cite{RelativeLie} Theorem 2.2 (7)), therefore its base change is again an $X$-vertical morphism. Since verticality is local over schemes mapping into the target, we get that $$ \pi _0(X\times _TX) \rightarrow X\times _{\pi _0(T)}X $$ is an $X$-vertical morphism. On the other hand by the definition that $T$ is geometric we obtain a smooth surjection $R\rightarrow X\times _TX$ from a scheme $R$, and by the previous discussion this gives a $Spec({\bf C} )$-vertical surjection $$ R\rightarrow \pi _0(X\times _TX). $$ Composing we get the $X$-vertical surjection $R\rightarrow X\times _{\pi _0(T)}X$. We have now proven that $\pi _0(T)$ is $P3\frac{1}{2}$ in the terminology of \cite{RelativeLie}. Suppose now that $v:Y\rightarrow T$ is a point. Let $T':= Y\times _TY$. Then $\pi _0(T')= \pi _1(T|_{{\cal X} /Y}, v)$ is the group sheaf we are interested in looking at over $Y$. We will show that it is presentable. Note that $T'$ is geometric; we apply the same argument as above, choosing a smooth surjection $X\rightarrow T'$. Recall that this gives a $Spec ({\bf C} )$-vertical (and hence $Y$-vertical) surjection $X\rightarrow \pi _0(T')$. Choose a smooth surjection $R\rightarrow X\times _{T'}X$. In the previous proof the group sheaf denoted $G$ on $X$ is actually pulled back from a group sheaf $\pi _2(T|_{{\cal X} /Y}, v)$ on $Y$. Therefore the morphism $$ \pi _0(X\times _{T'}X)\rightarrow X\times _{\pi _0(T')}X $$ is a quotient by a group sheaf over $Y$, in particular it is $Y$-vertical. As usual the morphism $R\rightarrow \pi _0(X\times _{T'}X)$ is $Spec ({\bf C} )$-vertical so in particular $Y$-vertical. We obtain a $Y$-vertical surjection $$ R\rightarrow X\times _{\pi _0(T')}X. $$ This finishes the proof that $\pi _1(T|_{{\cal X} /Y}, v)$ satisfies property $P4$ (and since it is a group sheaf, $P5$ i.e. presentable) with respect to $Y$. Now note that $\pi _i(T|_{{\cal X} /Y}, v) = \pi _{i-1}(T'|_{{\cal X} /Y}, d)$ where $d: Y \rightarrow T' := Y\times _TY$ is the diagonal morphism. Hence (as $T'$ is itself geometric) we obtain by induction that all of the $\pi _i(T|_{{\cal X} /Y}, v)$ are presentable group sheaves over $Y$. This shows that $T$ is presentable in the terminology of \cite{RelativeLie}. \hfill $\Box$\vspace{.1in} Note that presentability in \cite{RelativeLie} is a slightly stronger condition than the condition of presentability as it is referred to in \cite{kobe} so all of the results stated in \cite{kobe} hold here; and of course all of the results of \cite{RelativeLie} concerning presentable $n$-stacks hold for geometric $n$-stacks. The example given below which shows that the class of geometric $n$-stacks is not closed under truncation, implies that the class of presentable $n$-stacks is strictly bigger than the class of geometric ones, since the class of presentable $n$-stacks is closed under truncation \cite{RelativeLie}. The results announced (some with sketches of proofs) in \cite{kobe} for presentable $n$-stacks hold for geometric $n$-stacks. Similarly the basic results of \cite{RelativeLie} hold for geometric $n$-stacks. For example, if $T$ is a geometric $n$-stack and $f:Y\rightarrow T$ is a morphism from a scheme then $\pi _i (T|_{{\cal X} /Y}, f)$ is a presentable group sheaf, so it has a Lie algebra object $Lie\, \pi _i (T|_{{\cal X} /Y}, f)$ which is a {\em vector sheaf} (or ``$U$-coherent sheaf'' in the terminology of \cite{Hirschowitz}) with Lie bracket operation. {\em Remark:} By Proposition \ref{presentable}, the condition of being geometric is stronger than the condition of being presentable given in \cite{RelativeLie}. Note from the example given below showing that geometricity is not compatible with truncation (whereas by definition presentability is compatible with truncation), the condition of being geometric is {\em strictly} stronger than the condition of being presentable. Of course in the connected case, presentability and geometricity are the same thing. \begin{corollary} A connected $n$-stack $T$ is geometric if and only if the $\pi _i(T,t)$ are group schemes of finite type for all $i$. \end{corollary} {\em Proof:} We show in \cite{RelativeLie} that presentable groups over $Spec (k)$ are just group schemes of finite type. Together with the previous result this shows that if $T$ is connected and geometric then the $\pi _i(T,t)$ are group schemes of finite type for all $i$. On the other hand, if $\pi _0(T)=\ast$ and the $\pi _i(T,t)$ are group schemes of finite type for all $i$ then by the Postnikov decomposition of $T$ and using \ref{fibration}, we conclude that $T$ is geometric (note that for a group scheme of finite type $G$, $K(G,n)$ is geometric). \hfill $\Box$\vspace{.1in} \numero{Quillen theory} Quillen in \cite{Quillen} associates to every $1$-connected rational space $U$ a {\em differential graded Lie algebra (DGL)} $L_{\cdot} = \lambda (U)$: a DGL is a graded Lie algebra (over ${\bf Q}$ for our purposes) $L_{\cdot} = \bigoplus _{p\geq 1}L_p$ (with all elements of strictly positive degree) with differential $\partial : L_p \rightarrow L_{p-1}$ compatible in the usual (graded) way with the Lie bracket. Note our conventions that the indexing is downstairs, by positive numbers and the differential has degree $-1$. The homology groups of $\lambda (U)$ are the homotopy groups of $U$ (shifted by one degree). This construction gives an equivalence between the homotopy theory of DGL's and that of rational spaces. Let $L_{\cdot} \mapsto |L_{\cdot}| $ denote the construction going in the other direction. We shall assume for our purposes that there exists such a realization functor from the category of DGL's to the category of $1$-connected spaces, compatible with finite direct products. Let $DGL_{{\bf C} , n}$ denote the category of $n$-truncated ${\bf C}$-DGL's of finite type (i.e. with homology groups which are finite dimensional vector spaces, vanishing in degree $\geq n$) and free as graded Lie algebras. We define a realization functor $\rho ^{\rm pre}$ from $DGL_{{\bf C} ,n}$ to the category of presheaves of spaces over ${\cal X}$. If $L_{\cdot}\in DGL_{{\bf C} ,n}$ then for any $Y\in {\cal X}$ let $$ \rho ^{\rm pre}(L_{\cdot})(Y):= | L_{\cdot} \otimes _{{\bf C}}{\cal O} (Y) |. $$ Then let $\rho (L_{\cdot})$ be the $n$-stack associated to the presheaf of spaces $\rho ^{\rm pre}(L_{\cdot})$. This construction is functorial and compatible with direct products (because we have assumed the same thing about the realization functor $|L_{\cdot}|$). Note that $\pi _0^{\rm pre}(\rho ^{\rm pre}(L_{\cdot}))=\ast$ and in fact we can choose a basepoint $x$ in $\rho ^{\rm pre}(L_{\cdot})(Spec ({\bf C} ))$. We have $$ \pi _i ^{\rm pre}(\rho ^{\rm pre}(L_{\cdot}), x)= H_{i-1}(L_{\cdot}) $$ (in other words the presheaf on the left is represented by the vector space on the right). This gives the same result on the level of associated stacks and sheaves: $$ \pi _i (\rho (L_{\cdot}), x)= H_{i-1}(L_{\cdot}). $$ In particular note that a morphism of DGL's induces an equivalence of $n$-stacks if and only if it is a quasiisomorphism. Note also that $\rho (L_{\cdot})$ is a $1$-connected $n$-stack whose higher homotopy groups are complex vector spaces, thus it is a very presentable geometric $n$-stack. \begin{theorem} The above construction gives an equivalence between the homotopy category $ho\, DGL_{{\bf C} , n}$ and the homotopy category of $1$-connected very presentable $n$-stacks. \end{theorem} {\em Proof:} Let $(L,M)$ denote the set of homotopy classes of maps from $L$ to $M$ (either in the world of DGL's or in the world of $n$-stacks on ${\cal X}$). Note that if $L$ and $M$ are DGL's then $L$ should be free as a Lie algebra (otherwise we have to replace it by a quasiisomorphic free one). We prove that the map $$ (L,M)\rightarrow (\rho (L), \rho (M)) $$ is an isomorphism. First we show this for the case where $L=V[n-1]$ and $M=U[n-1]$ are finite dimensional vector spaces in degrees $n-1$ and $m-1$. In this case (where unfortunately $L$ isn't free so has to be replaced by a free DGL) we have $$ (V[n-1], U[m-1])= Hom (Sym ^{m/n}(V), U) $$ where the symmetric product is in the graded sense (i.e. alternating or symmetric according to parity) and defined as zero when the exponent is not integral. Note that $\rho (V[n-1])= K(V, n)$ and $\rho (U[m-1])= K(U,m)$. The Breen calculations in characteristic zero (easier than the case treated in \cite{BreenIHES}) show that $$ (K(V,n), K(U,m))= Hom (Sym ^{m/n}(V), U) $$ so our claim holds in this case. Next we treat the case of arbitrary $L$ but $M= U[m-1]$ is again a vector space in degree $m-1$. In this case we are calculating the cohomology of $L$ or $\rho (L)$ in degree $m$ with coefficients in $U$. Using a Postnikov decomposition of $L$ and the appropriate analogues of the Leray spectral sequence on both sides we see that our functor induces an isomorphism on these cohomology groups. Finally we get to the case of arbitrary $L$ and arbitrary $M$. We proceed by induction on the truncation level $m$ of $M$. Let $M'=\tau _{\leq m-1}M$ be the truncation (coskeleton) with the natural morphism $M\rightarrow M'$. The fiber is of the form $U[m-1]$ (we index our truncation by the usual homotopy groups rather than the homology groups of the DGL's which are shifted by $1$). Note that $\rho (M')= \tau _{\leq m-1}\rho (M)$ (since the construction $\rho$ is compatible with homotopy groups so it is compatible with the truncation operations). The fibration $M\rightarrow M'$ is classified by a map $f:M'\rightarrow U[m]$ and the fibration $\rho (M)\rightarrow \rho (M')$ by the corresponding map $\rho (f): \rho (M')\rightarrow K(U, m+1)$. The image of $$ (L,M)\rightarrow (L, M') $$ consists of the morphisms whose composition into $U[m]$ is homotopic to the trivial morphism $L\rightarrow U[m]$. Similarly the image of $$ (\rho (L),\rho (M))\rightarrow (\rho (L),\rho (M')) $$ is the morphisms whose composition into $K(U,m+1)$ is homotopic to the trivial morphism. By our inductive hypothesis $$ \rho : (L,M')\rightarrow (\rho (L), \rho (M')) $$ is an isomorphism. The functor $\rho$ is an isomorphism on the images, because we know the statement for targets $U[m]$. Suppose we are given a map $a:L\rightarrow M'$ which is in the image. The inverse image of this homotopy class in $(L,M)$ is the quotient of the set of liftings of $a$ by the action of the group of self-homotopies of the map $a$. The set of liftings is a principal homogeneous space under $(L, U[m-1])$. Similarly the inverse image of the homotopy class of $\rho (a)$ in $(\rho (L), \rho (M))$ is the quotient of the set of liftings of $\rho (a)$ by the group of self-homotopies of $\rho (a)$. Again the set of liftings is a principal homogeneous space under $(\rho (L), K(U,m))$. The actions in the principal homogeneous spaces come from maps $$ U[m-1]\times M\rightarrow M $$ over $M'$ and $$ K(U, m)\times \rho (M)\rightarrow \rho (M) $$ over $\rho (M')$, the second of which is the image under $\rho$ of the first. Since $\rho : (L, U[m-1])\cong (\rho (L), K(U,m))$, we will get that $\rho$ gives an isomorphism of the fibers if we can show that the images of the actions of the groups of self-homotopy equivalences are the same. Notice that since these actions are on principal homogeneous spaces they factor through the abelianizations of the groups of self-homotopy equivalences. In general if $A$ and $B$ are spaces then $(A\times S^1, B)$ is the disjoint union over $(A,B)$ of the sets of conjugacy classes of the groups of self-homotopies of the maps from $A$ to $B$. On the other hand a map of groups $G\rightarrow G'$ which induces an isomorphism on sets of conjugacy classes is surjective on the level of abelianizations. Thus if we know that a certain functor gives an isomorphism on $(A,B)$ and on $(A\times S^1, B)$ then it is a surjection on the abelianizations of the groups of self-homotopies of the maps. Applying this principle in the above situation, and noting that we know by our induction hypothesis that $\rho$ induces isomorphisms on $(L, M')$ and $(L\times \lambda (S^1)\otimes _{{\bf Z}}k, M')$, we find that $\rho$ induces a surjection from the abelianization of the group of self-homotopies of the map $a:L\rightarrow M'$ to the abelianization of the group of self-homotopies of $\rho (a)$. This finally allows us to conclude that $\rho$ induces an isomorphism from the inverse image of the class of $a$ in $(L,M)$ to the inverse image of the class of $\rho (a)$ in $(\rho (L), \rho (M))$. We have completed our proof that $$ \rho : (L,M)\cong (\rho (L), \rho (M)). $$ In order to obtain that $\rho$ induces an isomorphism on homotopy categories we just have to see that any $1$-connected very presentable $n$-stack $T$ is of the form $\rho (L)$. We show this by induction on the truncation level. Put $T'=\tau _{\leq n-1}T$. By the induction hypothesis there is a DGL $L'$ with $\rho (L')\cong T'$ (and we may actually write $\rho (L')=T'$). Now the fibration $T\rightarrow T'$ is classified by a map $f:T'\rightarrow K(V,n+1)$. From the above proof this map comes from a map $b:L'\rightarrow V[n]$, that is $f=\rho (b)$. In turn this map classifies a new DGL $L$ over $L'$. The fibration $\rho (L)\rightarrow \rho (L')=T'$ is classified by the map $\rho (b)=f$ so $\rho (L)\cong T$. \hfill $\Box$\vspace{.1in} \subnumero{Dold-Puppe} Eventually it would be nice to have a relative version of the previous theory, over any $n$-stack $R$. The main problem in trying to do this is to have the right notion of complex of sheaves over an $n$-stack $R$. Instead of trying to do this, we will simply use the notion of {\em spectrum over $R$} (to be precise I will use the word ``spectrum'' for what is usually called an ``$\Omega$-spectrum''. For our purposes we are only interested in spectra with homotopy groups which are rational and vanish outside of a bounded interval. In absolute terms such a spectrum is equivalent to a complex of rational vector spaces, so in the relative case over a presheaf of spaces $R$ this gives a generalized notion of complex over $R$. For our spectra with homotopy groups nonvanishing only in a bounded region, we can replace the complicated general theory by the simple consideration of supposing that we are in the stable range. Thus we fix numbers $N, M$ with $M$ bigger than the length of any complex we want to consider and $N$ bigger than $2M+2$. For example if we are only interested in dealing with $n$-stacks then we cannot be interested in complexes of length bigger than $n$ so we could take $M>n$. An {\em spectrum} (in our setting) is then simply an $N$-truncated rational space with a basepoint, and which is $N-M-1$-connected. More generally if $R$ is an $n$-stack with $n \leq N$ then a {\em spectrum over $R$} is just an $N$-stack $S$ with morphism $p:S\rightarrow R$ and section denoted $\xi : R\rightarrow S$ such that $S$ is rational and $N-M-1$-connected relative to $R$. A morphism of spectra is a morphism of spaces (preserving the basepoint). Suppose $S$ is a spectrum; we define the {\em complex associated to $S$} by setting $\gamma (S)^i$ to be the singular $N-i$-chains on $S$. The differential $d: \gamma (S)^i\rightarrow \gamma (S)^{i+1}$ is the same as the boundary map on chains (which switches direction because of the change in indexing). Note that we have normalize things so that the complex starts in degree $0$. The homotopy theory of spectra is the same as that of complexes of rational vector spaces indexed in degrees $\geq 0$, with cohomology nonvanishing only in degrees $\leq M$. If $C^{\cdot}$ is a complex as above then let $\sigma (C^{\cdot})$ denote the corresponding spectrum. This can be generalized to the case where the base is a $0$-stack. If $Y$ is a $0$-stack (notably for example a scheme) and if $S$ is a spectrum over $Y$ then we obtain a complex of presheaves of rational vector spaces $\gamma (S/Y)$ over $Y$. Conversely if $C^{\cdot}$ is a complex of presheaves of rational vector spaces over $Y$ then we obtain a spectrum denoted $\sigma (C^{\cdot}/Y)$. These constructions are an equivalence in homotopy theories, where the weak equivalence between complexes means quasiisomorphism (i.e. morphisms inducing isomorphisms on associated cohomology sheaves). If $S$ is a spectrum and $n \leq N$ then we can define the {\em realization} $\kappa (S, n)$ to be the $N-n$-th loop space $\Omega ^{N-n}S$ (the loops are taken based at the given basepoint). Similarly if $S$ is a spectrum over an $n'$-stack $R$ then we obtain the {\em realization} $\kappa (S/R,n)\rightarrow R$ as the $N-n$-th relative loop space based at the given section $\xi$. Taken together we obtain the following construction: if $C^{\cdot}$ is a complex of vector spaces then $\kappa (\sigma (C^{\cdot}), n)$ is an $n$-stack. If $C^{\cdot}$ is a complex of presheaves of rational vector spaces over a $0$-stack (presheaf of sets) $Y$ then $\kappa (\sigma (C^{\cdot}/Y)/Y, n)$ is an $n$-stack over $Y$. These constructions are what is known as {\em Dold-Puppe}. They are compatible with the usual Eilenberg-MacLane constructions: if $V$ is a presheaf of rational vector spaces over $Y$ considered as a complex in degree $0$ then $$ \kappa (\sigma (V/Y)/Y, n)= K(V/Y, n). $$ The basic idea behind our notational system is that we think of spectra over $R$ as being complexes of rational presheaves over $R$ starting in degree $0$. The operation $\kappa (S/R, n)$ is the {\em Dold-Puppe} realization from a ``complex'' to a space relative to $R$. We can do higher direct images in this context. If $f:R\rightarrow T$ is a morphism of $n$-stacks and if $S$ is a spectrum over $R$ then define $f_{\ast}(S)$ to be the $N$-stack $\Gamma (R/T, S)$ of sections relative to $T$. This is compatible with realizations: we have $$ \Gamma (R/T, \kappa (S, n))= \kappa (f_{\ast}(S),n). $$ Suppose that $f: X\rightarrow Y$ is a morphism of $0$-stacks. Then for a complex of rational presheaves $C^{\cdot}$ on $X$ the direct image construction in terms of spectra is the same as the usual higher direct image of complexes of sheaves (applied to the sheafification of the complex): $$ f_{\ast}(\sigma (C^{\cdot}/X))= \sigma (({\bf R}f_{\ast}C^{\cdot})/Y). $$ We extend this just a little bit, in a special case in which it still makes sense to talk about complexes. Suppose $X$ is a $1$-stack and $Y$ is a $0$-stack, with $f: X\rightarrow Y$ a morphism. Suppose $V$ is a local system of presheaves on $X$ (i.e. for each $Z\in {\cal X}$, $V(Z)$ is a local system of rational vector spaces on $X(Z)$). Another way to put this is that $V$ is an abelian group object over $Z$. We can think of $V$ as being a complex of presheaves over $X$ (even though we have not defined this notion in general) and we obtain the spectrum which we denote by $\sigma (V/X)$ over $X$ (even though this doesn't quite fit in with the general definition of $\sigma$ above), and its realization $\kappa (\sigma (V/X)/X, n)\rightarrow X$ which is what we would otherwise denote as $K(V/X,n)$. The higher direct image ${\bf R}f_{\ast}(V)$ makes sense as a complex of presheaves on $Y$, and we have the compatibilities $$ f_{\ast} \sigma (V/X) = \sigma ({\bf R}f_{\ast}(V)) $$ and $$ \Gamma (X/Y, \kappa (\sigma (V/X)/X, n))= \kappa (\sigma ({\bf R}f_{\ast}(V)),n). $$ \begin{proposition} \label{ComplexOfVB} Suppose $R$ is an $n$-stack and $S$ is a spectrum over $R$ such that for every map $Y\rightarrow R$ from a scheme, there is (locally over $Y$ in the etale topology) a complex of vector bundles $E^{\cdot}_Y$ over $Y$ with $S\times _RY\cong \sigma (E^{\cdot}_Y/Y)$. Then the realization $\kappa (S/R,n)$ is geometric over $R$. In particular if $R$ is geoemtric then so is $\kappa (S/R,n)$. \end{proposition} {\em Proof:} In order to prove that the morphism $\kappa (S/R,n)\rightarrow R$ is geometric, it suffices to prove that for every base change to a scheme $Y\rightarrow R$, the fiber product $\kappa (S/R, n)\times _RY$ is geometric. But $$ \kappa (S/R,n)\times _RY= \kappa (\sigma (E^{\cdot}_Y/Y)/Y, n), $$ so it suffices to prove that for a scheme $Y$ and a complex of vector bundles $E^{\cdot}$ on $Y$, we have $\kappa (\sigma (E^{\cdot}/Y)/Y, n)$ geometric. Note that $\kappa (\sigma (E^{\cdot}), n)$ only depends on the part of the complex $$ E^0\rightarrow E^1 \rightarrow \ldots \rightarrow E^n\rightarrow E^{n+1} $$ so we assume that it stops there or earlier. Now we proceed by induction on the length of the complex. Define a complex $F^i= E^{i-1}$ for $i\geq 1$, which has length strictly smaller than that of $E^{\cdot}$. Let $E^0$ denote the first vector bundle of $E^{\cdot}$ considered as a complex in degree $0$ only. We have a morphism of complexes $E^0\rightarrow F^{\cdot}$ and $E^{\cdot}$ is the mapping cone. Thus $$ \sigma (E^{\cdot}/Y) = \sigma (E^0/Y)\times _{\sigma (F^{\cdot}/Y)}Y $$ with $Y\rightarrow \sigma (F^{\cdot}/Y)$ the basepoint section. We get $$ \kappa (\sigma (E^{\cdot}/Y)/Y,n) = K(E^0/Y,n)\times _{\kappa (\sigma (F^{\cdot}/Y)/Y, n)}Y. $$ By our induction hypothesis, $\kappa (\sigma (F^{\cdot}/Y)/Y, n)$ is geometric. Note that $E_0$ is a smooth group scheme over $Y$ so by Lemma \ref{eilenbergExample}, $K(E^0/Y,n)$ is geometric $Y$. By \ref{fiberprod}, \linebreak $\kappa (\sigma (E^{\cdot}/Y)/Y,n)$ is geometric. \hfill $\Box$\vspace{.1in} {\em Remark:} This proposition is a generalisation to $n$-stacks of (\cite{LaumonMB} Construction 9.19, Proposition 9.20). Note that if $E^{\cdot}$ is a complex where $E^i$ are vector bundles for $i<n$ and $E^n$ is a vector scheme (i.e. something of the form ${\bf V}({\cal M} )$ for a coherent sheaf ${\cal M}$ in the notation of \cite{LaumonMB}) then we can express $E^n$ as the kernel of a morphism $U^n\rightarrow U^{n+1}$ of vector bundles (this would be dual to the presentation of ${\cal M}$ if we write $E^n = {\bf V}({\cal M} )$). Setting $U^i= E^i$ for $i<n$ we get $\kappa (\sigma (E^{\cdot}), n)= \kappa (\sigma (U^{\cdot}),n)$. In this way we recover Laumon's and Moret-Bailly's construction in the case $n=1$. \begin{corollary} Suppose $f:X\rightarrow Y$ is a projective flat morphism of schemes, and suppose that $V$ is a vector bundle on $X$. Then $\Gamma (X/Y, K(V/X,n))$ is a geometric $n$-stack lying over $Y$. \end{corollary} {\em Proof:} By the discussion at the start of this subsection, $$ \Gamma (X/Y, K(V/X,n)) = \kappa (\sigma ({\bf R} f_{\ast}(V)/Y)/Y, n). $$ But by Mumford's method \cite{Mumford}, ${\bf R} f_{\ast}(V)$ is quasiisomorphic (locally over $Y$) to a complex of vector bundles. By Proposition \ref{ComplexOfVB} we get that $\Gamma (X/Y, K(V,n))$ is geometric over $Y$. \hfill $\Box$\vspace{.1in} Recall that a {\em formal groupoid} is a stack $X_{\Lambda}$ associated to a groupoid of formal schemes where the object object is a scheme $X$ and the morphism object is a formal scheme $\Lambda \rightarrow X\times X$ with support along the diagonal. We say it is {\em smooth} if the projections $\Lambda \rightarrow X$ are formally smooth. In this case the cohomology of the stack $X_{\Lambda}$ with coefficients in vector bundles over $X_{\Lambda}$ (i.e. vector bundles on $X$ with $\Lambda$-structure meaning isomorphisms between the two pullbacks to $\Lambda$ satisfying the cocycle condition on $\Lambda \times _X\Lambda$) is calculated by the {\em de Rham complex} $\Omega ^{\cdot}_{\Lambda}\otimes _{{\cal O}}V$ of locally free sheaves associated to the formal scheme \cite{Illusie} \cite{Berthelot}. We say that $X_{\Lambda}\rightarrow Y$ is a smooth formal groupoid over $Y$ if $X_{\Lambda}$ is a smooth fomal groupoid mapping to $Y$ and if $X$ is flat over $Y$. \begin{corollary} Suppose $f: X_{\Lambda} \rightarrow Y$ is a projective smooth formal groupoid over a scheme $Y$. Suppose that $V$ is a vector bundle on $X_{\Lambda}$ (i.e. a vector bundle on $X$ with $\Lambda$-structure). Then $\Gamma (X_{\Lambda}/Y, K(V/X_{\Lambda},n))$ is a geometric $n$-stack lying over $Y$. \end{corollary} {\em Proof:} By the ``slight extension'' in the discussion at the start of this subsection, $$ \Gamma (X_{\Lambda}/Y, K(V/X_{\Lambda},n)) = \kappa (\sigma ({\bf R} f_{\ast}(V)/Y)/Y, n). $$ But $$ {\bf R}f_{\ast}(V) = {\bf R} f'_{\ast}(\Omega ^{\cdot}_{\Lambda}\otimes _{{\cal O}}V) $$ where $f': X\rightarrow Y$ is the morphism on underlying schemes. Again by Mumford's method \cite{Mumford}, ${\bf R} f'_{\ast}(\Omega ^{\cdot}_{\Lambda}\otimes _{{\cal O}}V)$ is quasiisomorphic (locally over $Y$) to a complex of vector bundles. By the Proposition \ref{ComplexOfVB} we get that $\Gamma (X_{\Lambda}/Y, K(V/X_{\Lambda},n))$ is geometric over $Y$. \hfill $\Box$\vspace{.1in} \numero{Maps into geometric $n$-stacks} \begin{theorem} \label{maps} Suppose $X\rightarrow S$ is a projective flat morphism. Suppose $T$ is a connected $n$-stack which is very presentable (i.e. the fundamental group is represented by an affine group scheme of finite type denoted $G$ and the higher homotopy groups are represented by finite dimensional vector spaces). Then the morphism $Hom (X/S,T) \rightarrow Bun _G(X/S)= Hom (X/S, BG)$ is a geometric morphism. In particular $Hom (X/S, T)$ is a locally geometric $n$-stack. \end{theorem} {\em Proof:} Suppose $V$ is a finite dimensional vector space. Let $$ {\cal B} (V,n)= BAut (K(V,n)) $$ be the classifying $n+1$-stack for fibrations with fiber $K(V,n)$. It is connected with fundamental group $GL(V)$ and homotopy group $V$ in dimension $n+1$ and zero elsewhere. The truncation morphism $$ {\cal B} (V,n)\rightarrow B\, GL(V) $$ has fiber $K(V,n+1)$ and admits a canonical section $o: BGL(V)\rightarrow {\cal B} (V,n)$ (which corresponds to the trivial fibration with given action of $GL(V)$ on $V$---this fibration may itself be constructed as ${\cal B} (V, n-1)$ or in case $n= 2$ as $B(GL(V) \semidirect V)$). The fiber of the morphism $o$ is $K(V, n)$, and $BGL(V)$ is the universal object over ${\cal B} (V, n)$. Note that $BGL(V)$ is an geometric $1$-stack (i.e. algebraic stack) and by Proposition \ref{fibration} applied to the truncation fibration, ${\cal B} (V, n)$ is a geoemtric $n+1$-stack. If $X\rightarrow S$ is a projective flat morphism then $Hom (X/S, BGL(V))$ is a locally geometric $1$-stack (via the theory of Hilbert schemes). We show that $p:Hom (X/S, {\cal B} (V, n))\rightarrow Hom (X/S, BGL(V))$ is a geometric morphism. For this it suffices to consider a morphism $\zeta :Y\rightarrow Hom (X/S, BGL(V))$ from a scheme $Y/S$ which in turn corresponds to a vector bundle $V_{\zeta}$ on $X\times _SY$. The fiber of the map $p$ over $\eta$ is $\Gamma (X\times _SY/Y; K(V_{\zeta}, n+1))$ which as we have seen above is geometric over $Y$. This shows that $p$ is geometric. In particular $Hom (X/S, {\cal B} (V, n))$ is locally geometric. We now turn to the situation of a general connected geometric and very presentable $n$-stack $T$. Consider the truncation morphism $a:T\rightarrow T':=\tau _{\leq n-1}T$. We may assume that the theorem is known for the $n-1$-stack $T'$. The morphism $a$ is a fibration with fiber $K(V, n)$ so it comes from a map $b:T' \rightarrow {\cal B} (V, n)$ and more precisely we have $$ T = T' \times _{{\cal B} (V,n)} BGL(V). $$ Thus $$ Hom (X/S, T)= Hom (X/S, T') \times _{Hom (X/S,{\cal B} (V,n))} Hom (X/S,BGL(V)). $$ But we have just checked that $Hom (X/S,BGL(V))$ and $Hom (X/S,{\cal B} (V,n))$ are locally geometric, and by hypothesis $Hom (X/S, T')$ is locally geometric. Therefore by the version of \ref{fiberprod} for locally geometric $n$-stacks, the fiber product is locally geometric. This completes the proof. \hfill $\Box$\vspace{.1in} \begin{theorem} \label{smoothformal} Suppose $(X,\Lambda )\rightarrow S$ is a smooth projective morphism with smooth formal category structure relative to $S$. Let $X_{\Lambda}\rightarrow S$ be the resulting family of stacks. Suppose $T$ is a connected very presentable $n$-stack which is very presentable (with fundamental group scheme denoted $G$). Then the morphism $Hom (X_{\Lambda }/S,T) \rightarrow Hom (X_{\Lambda }/S, BG)$ is a geometric morphism. In particular $Hom (X_{\Lambda }/S, T)$ is a locally geometric $n$-stack. \end{theorem} {\em Proof:} The same as before. Note here also that $Hom (X_{\Lambda }/S, BG)$ is an algebraic stack locally of finite type. \hfill $\Box$\vspace{.1in} {\em Remark:} In the above theorems the base $S$ can be assumed to be any $n$-stack, one looks at morphisms with the required properties when base changed to any scheme $Y\rightarrow S$. \subnumero{Semistability} Suppose $X\rightarrow S$ is a projective flat morphism, with fixed ample class, and suppose $G$ is an affine algebraic group. We get a notion of semistability for $G$-bundles (for example, fix the convention that we speak of Gieseker semistability). Fix also a collection of Chern classes which we denote $c$. We get a Zariski open substack $$ Hom ^{\rm se}_c(X/S, BG)\subset Hom (X/S, BG) $$ (just the moduli $1$-stack of semistable $G$-bundles with Chern classes $c$). The boundedness property for semistable $G$-bundles with fixed Chern classes shows that $Hom ^{\rm se}_c(X/S, BG)$ is a geometric $1$-stack. Now if $T$ is a connected very presentable $n$-stack, let $G$ be the fundamental group scheme and let $c$ be a choice of Chern classes for $G$-bundles. Define $$ Hom ^{\rm se}_c(X/S, T):= Hom (X/S, T)\times _{Hom (X/S, BG)} Hom ^{\rm se}_c(X/S, BG). $$ Again it is a Zariski open substack of $Hom (X/S, T)$ and it is a geometric $n$-stack rather than just locally geometric. We can do the same in the case of a smooth formal category $X_{\Lambda} \rightarrow S$. Make the convention in this case that we ask the Chern classes to be zero (there is no mathematical need to do this, it is just to conserve indices, since practically speaking this is the only case we are interested in below). We obtain a Zariski open substack $$ Hom ^{\rm se}(X_{\Lambda}/S, BG)\subset Hom (X_{\Lambda}/S, BG), $$ the moduli stack for semistable $G$-bundles on $X_{\Lambda}$ with vanishing Chern classes. See \cite{Moduli} for the construction (again the methods given there suffice for the construction, although stacks are not explicitly mentionned). Again for any connected very presentable $T$ with fundamental group scheme $G$ we put $$ Hom ^{\rm se}(X_{\Lambda}/S, T):= Hom (X_{\Lambda}/S, T)\times _{Hom (X _{\Lambda}/S, BG)} Hom ^{\rm se}(X_{\Lambda}/S, BG). $$ It is a geometric $n$-stack. Finally we note that in the case of the relative de Rham formal category $X_{DR/S}$ semistability of principal $G$-bundles is automatic (as is the vanishing of the Chern classes). Thus $$ Hom ^{\rm se}(X_{DR/S}/S, T)= Hom (X_{DR/S}/S, T) $$ and $Hom (X_{DR/S}, T)$ is already a geometric $n$-stack. \subnumero{The Brill-Noether locus} Suppose $G$ is an algebraic group and $V$ is a representation. Define the $n$-stack $\kappa (G,V,n)$ as the fibration over $K(G,1)$ with fiber $K(V,n)$ where $G$ acts on $V$ by the given representation and such that there is a section. Let $X$ be a projective variety. We have a morphism $$ Hom (X, \kappa (G,V,n))\rightarrow Hom (X, K(G,1))= Bun _G(X). $$ The fiber over a point $S\rightarrow Bun _G(X)$ corresponding to a principal $G$-bundle $P$ on $X\times S$ is the relative section space $$ \Gamma (X\times S/S, K(P\times ^GV/X\times S, n)). $$ By the compatibilities given at the start of the section on Dold-Puppe, this relative section space is the $n$-stack corresponding to the direct image $Rp_{1,\ast}(P\times ^GV)$ which is a complex over $S$. Note that this complex is quasiisomorphic to a complex of vector bundles. Thus we have: \begin{corollary} \label{BN} The morphism $$ Hom (X, \kappa (G,V,n))\rightarrow Bun _G(X) $$ is a morphism of geometric $n$-stacks. \end{corollary} \hfill $\Box$\vspace{.1in} {\em Remark:} The $Spec ({\bf C} )$-valued points of $Hom (X, \kappa (G,V,n))$ are the pairs $(P, \eta )$ where $P$ is a principal $G$-bundle on $X$ and $\eta \in H^n(X, P\times ^GV)$. Thus $Hom (X, \kappa (G,V,n))$ is a geometric $n$-stack whose $Spec ({\bf C} )$-points are the Brill-Noether set of vector bundles with cohomology classes on $X$. \subnumero{Some conjectures} We give here some conjectures about the possible extension of the above results to any (not necessarily connected) geometric $n$-stacks $T$. \begin{conjecture} If $T$ is a geometric $n$-stack which is very presentable in the sense of \cite{RelativeLie} (i.e. the fundamental groups over artinian base are affine, and the higher homotopy groups are vector sheaves) then for any smooth (or just flat?) projective morphism $X\rightarrow S$ we have that $Hom (X/S, T)$ is locally geometric. \end{conjecture} \begin{conjecture} \label{KGm2} If $T= K({\bf G}_m , 2)$ then for a flat projective morphism $X\rightarrow S$, $Hom (X/S, T)$ is locally geometric. Similarly if $G$ is {\em any} group scheme of finite type (e.g. an abelian variety) then $Hom (X/S, BG)$ is locally geometric. \end{conjecture} Putting together with the previous conjecture we can make: \begin{conjecture} If $T$ is a geoemtric $n$-stack whose $\pi _i$ are vector sheaves for $i\geq 3$ then $Hom (X/S, T)$ is locally geometric. \end{conjecture} Note that Conjecture \ref{KGm2} cannot be true if $K({\bf G}_m, 2)$ is replaced by $K({\bf G}_m, i)$ for $i\geq 3$, for in that case the morphism stacks will themselves be only locally of finite type. Instead we will get a ``slightly geometric'' $n$-stack as discussed in \S 3. One could make the following conjecture: \begin{conjecture} If $T$ is any geometric (or even locally or slightly geometric) $n$-stack and $X\rightarrow S$ is a flat projective morphism then $Hom (X/S, T)$ is slightly geometric. \end{conjecture} After these somewhat improbable-sounding conjectures, let finish by making a more reasonable statement: \begin{conjecture} If $T$ is a very presentable geometric $n$-stack and $X$ is a smooth projective variety then $Hom (X_{DR}, T)$ is again geometric. \end{conjecture} Here, we have already announced the finite-type result in the statement that \linebreak $Hom (X_{DR}, T)$ is very presentable \cite{kobe} (I have not yet circulated the proof, still checking the details...). \subnumero{GAGA} Let ${\cal X} ^{\rm an}$ be the site of complex analytic spaces with the etale (or usual--its the same) topology. We can make similar definitions of geometric $n$-stack on ${\cal X} ^{\rm an}$ which we will now denote by {\em analytic $n$-stack} (in case of confusion...). There are similar definitions of smoothness and so on. There is a morphism of sites from the analytic to the algebraic sites. If $T$ is a geometric $n$-stack on ${\cal X}$ then its pullback by this morphism (cf \cite{realization}) is an analytic $n$-stack which we denote by $T^{\rm an}$. We have: \begin{theorem} \label{gaga} Suppose $T$ is a connected very presentable geometric $n$-stack. Suppose $X\rightarrow S$ is a flat projective morphism (resp. suppose $X_{\Lambda}\rightarrow S$ is the morphism associated to a smooth formal category over $S$). Then the natural morphism $$ Hom (X/S, T)^{\rm an} \rightarrow Hom (X^{\rm an}/S^{\rm an}, T^{\rm an}) $$ $$ \left( \mbox{resp.} Hom (X_{\Lambda}/S, T)^{\rm an} \rightarrow Hom (X_{\Lambda}^{\rm an}/S^{\rm an}, T^{\rm an}) \right) $$ is an isomorphism of analytic $n$-stacks. \end{theorem} {\em Proof:} Just following through the proof of the facts that $Hom (X/S, T)$ or $Hom (X_{\Lambda}/S, T)$ are geometric, we can keep track of the analytic case too and see that the morphisms are isomorphisms along with the main induction. \hfill $\Box$\vspace{.1in} {\em Remarks:} \newline (1)\, This GAGA theorem holds for $X_{DR}$ with coefficients in any very presentable $T$ (not necessarily connected) \cite{kobe}. \newline (2)\, In \cite{kobe} we also give a ``GFGA'' theorem for $X_{DR}$ with coefficients in a very presentable $n$-stack. \newline (3)\, The GAGA theorem does not hold with coefficients in $T= K({\bf G}_m , 2)$. Thus the condition that the higher homotopy group sheaves of $T$ be vector sheaves is essential. Maybe it could be weakened by requiring just that the fibers over artinian base schemes be unipotent (but this might also be equivalent to the vector sheaf condition). \newline (4)\, Similarly the GAGA theorem does not hold with coefficients in $T= BA$ for an abelian variety $A$; thus again the hypothesis that the fibers of the fundamental group sheaf over artinian base be affine group schemes, is essential. \numero{The tangent spectrum} We can treat a fairly simple case of the conjectures outlined above: maps from the spectrum of an Artin local algebra of finite type. \begin{theorem} \label{mapsFromArtinian} Let $X=Spec (A)$ where $A$ is artinian, local, and of finite type over $k$. Suppose $T$ is a geometric $n$-stack. Then $Hom (X,T)$ is a geometric $n$-stack. If $T\rightarrow T'$ is a geometric smooth morphism of $n$-stacks then $Hom (X, T)\rightarrow Hom (X, T')$ is a smooth geometric morphism of $n$-stacks. \end{theorem} {\em Proof:} We prove the following statement: if $Y$ is a scheme and $A$ as in the theorem, and if $T\rightarrow Y\times Spec (A)$ is a geometric (resp. smooth geometric) morphism of $n$-stacks then $\Gamma (Y\times Spec (A)/Y, T)$ is geometric (resp. smooth geometric) over $Y$. The proof is by induction on $n$; note that it works for $n=0$. Now in general choose a smooth surjection $X\rightarrow T$ from a scheme. Then $\Gamma (Y\times Spec (A)/Y, X)$ is a scheme over $Y$, and if $X$ is smooth over $Y$ then the section scheme is smooth over $Y$. We have a surjection $$ a:\Gamma (Y\times Spec (A)/Y, X)\rightarrow \Gamma (Y\times Spec (A)/Y, T), $$ and for $Z\rightarrow \Gamma (Y\times Spec (A)/Y, X)$ (which amounts to a section morphism $Z\times Spec (A)\rightarrow T$) the fiber product $$ \Gamma (Y\times Spec (A)/Y, X)\times _{ \Gamma (Y\times Spec (A)/Y, T)} Z $$ is equal to $$ \Gamma (Z\times Spec (A)/Z, X\times _T(Z\times Spec (A))). $$ But $X\times _T(Z\times Spec (A)$ is a smooth $n-1$-stack over $Z\times Spec (A)$ so by induction this section stack is geometric and smooth over $Z$. Thus our surjection $a$ is a smooth geometric morphism so $\Gamma (Y\times Spec (A)/Y, T)$ is geometric. The smoothness statement follows immediately. \hfill $\Box$\vspace{.1in} We apply this to define the {\em tangent spectrum} of a geometric $n$-stack. This is a generalization of the discussion at the end of (\cite{LaumonMB} \S 9), although we use a different approach because I don't have the courage to talk about cotangent complexes! Recall from \cite{Adams} Segal's infinite loop space machine: let $\Gamma$ be the category whose objects are finite sets and where the morphisms from $\sigma$ to $\tau$ are maps $P(\sigma )\rightarrow P(\tau )$ preserving disjoint unions (here $P(\sigma )$ is the set of subsets of $\sigma$). A morphism is determined, in fact, by the map $\sigma \rightarrow P(\tau )$ taking different elements of $\sigma$ to disjoint subsets of $\tau$ (note that the empty set must go to the empty set). Let $[n]$ denote the set with $n$ elements. There is a functor $s:\Delta \rightarrow \Gamma$ sending the the ordered set $\{ 0,\ldots , n\}$ to the finite set $\{ 1, \ldots , n\}$---see \cite{Adams} p. 64 for the formulas for the morphisms. Segal's version of an $E_{\infty}$-space (i.e. infinite loop space) is a contravariant functor $\Psi : \Gamma \rightarrow Top$ such that the associated simplicial space (the composition $\Psi \circ s$) satisfies Segal's condition \cite{Adams}. In order to really get an infinite loop space it is also required that $\Psi (\emptyset )$ be a point (although this condition seems to have been lost in Adams' very brief treatment). Segal's machine is then a classifying space functor $B$ from special $\Gamma$-spaces to special $\Gamma$-spaces. This actually works even without the condition that $\Phi (\emptyset )$ be a point, however the classifying space construction is the inverse to the {\em relative} loop space construction over $\Phi (\emptyset )$. Note that since $\emptyset$ is a final object in $\Gamma$ the components of a $\Gamma$-space are provided with a section from $\Phi (\emptyset )$. If $\Phi$ is a special $\Gamma$-space then $B^n\Phi$ is again a special $\Gamma$ space with $$ B^n\Phi (\emptyset )= \Phi (\emptyset ) $$ and $$ \Omega ^n(B^n\Phi ([1])/\Phi (\emptyset ))= \Phi ([1]). $$ The notion of $\Gamma$-space (say with $\Phi [1] $ rational over $\Phi (\emptyset )=R$) is another replacement for our notion of spectrum over $R$; we get to our notion as defined above by looking at $B^N\Phi ([1])$. The above discussion makes sense in the context of presheaves of spaces over ${\cal X}$ hence in the context of $n$-stacks. We now try to apply this in our situation to construct the tangent spectrum. For any object $\sigma \in \Gamma$ let ${\bf A}^\sigma$ be the affine space over $k$ with basis the set $\sigma$. An element of ${\bf A}^\sigma$ can be written as $\sum _{i\in \sigma} a_ie_i$ where $e_i$ are the basis elements and $a_i\in k$. Given a map $f:\sigma \rightarrow P(\tau )$ we define a map $$ {\bf A}^f : {\bf A}^{\sigma} \rightarrow {\bf A}^{\tau} $$ $$ \sum a_i e_i \mapsto \sum _{i\in \sigma} \sum _{j\in f(i)\subset \tau} a_i e_j. $$ For example there are four morphisms from $[1]$ to $[2]$, sending $1$ to $\emptyset$, $\{ 1\}$, $\{ 2\}$ and $\{ 1,2\}$ respectively. These correspond to the constant morphism, the two coordinate axes, and the diagonal from ${\bf A}^1$ to ${\bf A}^2$. We get a covariant functor from $\Gamma$ to the category of affine schemes. For a finite set $\sigma$ let $D^{\sigma}$ denote the subscheme of ${\bf A}^{\sigma}$ defined by the square of the maximal ideal defining the origin. These fit together into a covariant functor from $\Gamma$ to the category of artinian local schemes of finite type over $k$. If $T$ is a geometric $n$-stack thought of as a strict presheaf of spaces, then the functor $$ \Theta :\sigma \mapsto Hom (D^{\sigma}, T) $$ is a contravariant functor from $\Gamma$ to the category of geometric $n$-stacks, with $\Theta (\emptyset )=T$. To see that it satisfies Segal's condition we have to check that the map $$ Hom (D^n, T)\rightarrow Hom (D^1, R)\times _R \ldots \times _THom (D^1, T) $$ is an equivalence. Once this is checked we obtain a spectrum over $T$ whose interpretation in our terms is as the $N$-stack $B^N\Phi ([1])$. In the statement of the following theorem we will normalize our relationship between complexes and spectra in a different way from before---the most natural way for our present purposes. \begin{theorem} \label{tangent} Suppose $T$ is a geometric $n$-stack. The above construction gives a spectrum $\Theta (T)\rightarrow T$ which we call the {\em tangent spectrum of $T$}. If $Y\rightarrow T$ is a morphism from a scheme then $\Theta (T)\times _TY$ is equivalent to $\sigma (E^{\cdot}/Y)$ for a complex $$ E^{-n}\rightarrow \ldots \rightarrow E^0 $$ with $E^i$ vector bundles ($i<0$) and $E^0$ a vector scheme over $Y$. Furthermore if $T$ is smooth then $E^0$ can be assumed to be a vector bundle. In particular, $\kappa (\Theta (T)/T, n)$ is geometric, and if $T$ is smooth then $\Theta (T)$ is geometric. \end{theorem} {\em Proof of \ref{tangent}:} The first task is to check the above condition for $\Theta$ to be a special $\Gamma$-space. Suppose in general that $A,B\subset C$ are closed artinian subschemes of an artinian scheme with the extension property that for any scheme $Y$ the morphisms from $C$ to $Y$ are the same as the pairs of morphisms $A,B\rightarrow Y$ agreeing on $A\cap B$. We would like to show that for any geometric stack $T$, $$ Hom (C,T)\rightarrow Hom (A,T)\times _{Hom(A\cap B, T)}Hom (B,T) $$ is an equivalence. We have a similar relative statement for sections of a geometric morphism $T\rightarrow Y\times C$ for a scheme $Y$. We prove the relative statement by induction on the truncation level $n$, but for simplicity use the notation of the absolute statement. Let $X\rightarrow T$ be a smooth geometric morphism from a scheme. Then consider the diagram $$ \begin{array}{ccc} Hom (C,X) &\stackrel{\cong}{\rightarrow}& Hom (A,X)\times _{Hom(A\cap B, X)}Hom (B,X) \\ \downarrow && \downarrow \\ Hom (C,T)&\rightarrow &Hom (A,T)\times _{Hom(A\cap B, T)}Hom (B,T). \end{array} $$ It suffices to prove that for a map from a scheme $Y\rightarrow Hom (C,T)$ the morphism on fibers is an equivalence. The fiber on the left is $$ Hom (C,X)\times _{Hom (C,T)}Y= \Gamma (Y\times C, X\times _{T}(Y\times C)), $$ whereas the fiber on the right is $$ \Gamma (Y\times A, X\times _{T}(Y\times A)) \times _{\Gamma (Y\times (A\cap B), X\times _{T}(Y\times (A\cap B)))} \Gamma (Y\times B, X\times _{T}(Y\times B)). $$ By the relative version of the statement for the $n-1$-stack $X\times _{T}(Y\times C)$ over $Y\times C$, the map of fibers is an equivalence, so the map $$ Hom (C,T)\rightarrow Hom (A,T)\times _{Hom(A\cap B, T)}Hom (B,T) $$ is an equivalence. Apply this inductively with $C= D^n$, $A= D^1$ and $B= D^{n-1}$ (so $A\cap B=D^0$). We obtain the required statement, showing that $\Theta$ is a special $\Gamma$-space relative to $T$. It integrates to a spectrum which we denote $\Theta (T)\rightarrow T$. Note that if $T=X$ is a scheme considered as an $n$-stack then $\Theta (X)$ is just the spectrum associated to the complex consisting of the tangent vector scheme of $X$ in degree $0$. We obtain the desired statement in this case. If $R\rightarrow T$ is a morphism of geometric $n$-stacks then we obtain a morphism of spectra $$ \Theta (R) \rightarrow \Theta (T)\times _T R . $$ The cofiber (i.e. $B$ of the fiber) we denote by $\Theta (R/T)$. We prove more generally---by induction on $n$---that if $T\rightarrow Y$ is a geometric morphism from an $n$-stack to a scheme, and if $Y\rightarrow T$ is a section then $\Theta (T/Y)\times _TY$ is associated (locally on $Y$) to a complex of vector bundles and a vector scheme at the end; with the last vector scheme being a bundle if the morphism is smooth. Note that it is true for $n=0$. For any $n$ choose a smooth geometric morphism $X\rightarrow T$ and we may assume (by etale localization) that there is a lifting of the section to $Y\rightarrow X$. Now there is a triangle of spectra (i.e. associated to a triangle of complexes in the derived category) $$ \Theta (X)\times _XY \rightarrow \Theta (T)\times _TY \rightarrow B\Theta (X/T)\times _XY. $$ On the other hand, $$ B\Theta (X/T)\times _XY=B\Theta (X\times _TY/Y)\times _{X\times _TY}Y. $$ By induction this is associated to a complex as desired, and we know already that $\Theta (X)\times _XY$ is associated to a complex as desired. Therefore $\Theta(T)\times _TY$ is an extension of complexes of the desired form, so it has the desired form. Note that since $X\times _TY\rightarrow Y$ is smooth, by the induction hypothesis we get that $B\Theta (X/T)\times _XY$ is associated to a complex of bundles. If the morphism $T\rightarrow Y$ is smooth then the last term in the complex will be a bundle (again following through the same induction). \hfill $\Box$\vspace{.1in} If $T$ is a smooth geometric $n$-stack and $P: Spec (k)\rightarrow T$ is a point then we say that the {\em dimension of $T$ at $P$} is the alternating sum of the dimensions of the vector spaces in the complex making up the complex associated to $P^{\ast} (\Theta (T))$. This could, of course, be negative. For example if $G$ is an algebraic group then the dimension of $BG$ at any point is $-dim (G)$. More generally if $A$ is an abelian group scheme smooth over a base $Y$ then $$ dim (K(A/Y, n))= dim (Y) + (-1)^ndim (A). $$ \numero{De Rham theory} We will use certain geometric $n$-stacks as coefficients to look at the de Rham theory of a smooth projective variety. The answers come out to be geometric $n$-stacks. (One could also try to look at de Rham theory {\em for} geometric $n$-stacks, a very interesting problem but not what is meant by the title of the present section). If $X$ is a smooth projective variety let $X_{DR}$ be the stack (which is actually a sheaf of sets) associated to the formal category whose object object is $X$ and whose morphism object is the formal completion of the diagonal in $X\times X$. Another cheaper definition is just to say $$ X_{DR}(Y):= X(Y^{\rm red}). $$ If $f:X\rightarrow S$ is a smooth morphism, let $$ X_{DR/S}:= X_{DR}\times _{S_{DR}}S. $$ It is the stack associated to a smooth formal groupoid over $S$ (take the formal completion of the diagonal in $X\times _SX$). The cohomology of $X_{DR}$ with coefficients in an algebraic group scheme is the same as the de Rham cohomology of $X$ with those coefficients. We treat this in the case of coefficients in a vector space, or in case of $H^1$ with coefficients in an affine group scheme. Actually the statement is a more general one about formal categories. Suppose $(X,\Lambda )\rightarrow S$ is a smooth formal groupoid over $S$ which we can think of as a smooth scheme $X/S$ with a formal scheme $\Lambda$ mapping to $X\times _SX$ and provided with an associative product structure. There is an associated {\em de Rham complex} $\Omega ^{\cdot} _{\Lambda}$ on $X$ (cf \cite{Berthelot} \cite{Illusie})---whose components are locally free sheaves on $X$ and where the differentials are first order differential operators. Let $X_{\Lambda}$ denote the stack associated to the formal groupoid. It is the stack associated to the presheaf of groupoids which to $Y\in {\cal X}$ associates the groupoid whose objects are $X(Y)$ and whose morphisms are $\Lambda (Y)$. Suppose $V$ is a vector bundle over $X_{\Lambda}$, that is a vector bundle on $X$ together with isomorphisms $p_1^{\ast} V\cong p_2^{\ast} V$ on $\Lambda$ satisfying the cocycle condition on $\Lambda \times _X \Lambda$. We can define the cohomology sheaves on $S$, $H^i(X_{\Lambda}/S, V)$ which will be equal to $\pi _0(\Gamma (X_{\Lambda }/S; K(V, i))$ in our notations. These cohomology sheaves can be calculated using the de Rham complex: there is a twisted de Rham complex $\Omega ^{\cdot}_{\Lambda} \otimes _{{\cal O}}V$ whose hypercohomology is $H^i(X_{\Lambda}/S, V)$. When applied to the de Rham formal category (the trivial example introduced in \cite{Berthelot} in characteristic zero) whose associated stack is the sheaf of sets $X_{DR/S}$, we obtain the usual de Rham complex $\Omega ^{\cdot}_{X/S}$ relative to $S$. A vector bundle $V$ over $X_{DR/S}$ is the same thing as a vector bundle on $X$ with integrable connection, and the twisted de Rham complex is the usual one. Thus in this case we have $$ \pi _0(\Gamma (X_{DR/S}/S, K(V,i)))= {\bf H}^i(X/S, \Omega ^{\cdot}_{X/S}\otimes V). $$ We can describe more precisely $\Gamma (X_{DR/S}/S, K(V,i))$ as being the$i$-stack obtained by applying Dold-Puppe to the right derived direct image complex $Rf _{\ast} (\Omega ^{\cdot}_{X/S}\otimes V)[i]$ (appropriately shifted). For the first cohomology with coefficients in an affine algebraic group $G$, note that a principal $G$-bundle on $X_{DR}$ is the same thing as a principal $G$-bundle on $X$ with integrable connection. We have that the $1$-stack $\Gamma (X_{DR/S}/S, BG)$ on $S$ is the moduli stack of principal $G$-bundles with relative integrable connection on $X$ over $S$. For $X\rightarrow S$ projective this is constructed in \cite{Moduli} (in fact, there we construct the representation scheme of framed principal bundles; the moduli stack is immediately obtained as an algebraic stack, the quotient stack by the action of $G$ on the scheme of framed bundles). Of course we have seen in \ref{smoothformal} that for any smooth formal category $(X,\Lambda )$ over $S$ and any connected very presentable $n$-stack $T$, the morphism $n$-stack $Hom (X_{\Lambda} /S, T)$ is a locally geometric $n$-stack. Recall that we have defined the {\em semistable} morphism stack $Hom ^{\rm se}(X_{\Lambda} /S, T)$ which is geometric; but in our case all morphisms $X_{DR/S}\rightarrow BG$ (i.e. all principal $G$-bundles with integrable connection) are semistable, so in this case we find that $Hom (X_{DR/S}/S, T)$ is a geometric $n$-stack. In fact it is just a successive combination of the above discussions applied according to the Postnikov decomposition of $T$. \subnumero{De Rham theory on the analytic site} The same construction works for smooth objects in the analytic site. Suppose $f:X\rightarrow S$ is a smooth analytic morphism. Here we would like to consider any connected $n$-stack $R$ whose homotopy groups are represented by analytic Lie groups. Such an $R$ is automatically an analytic $n$-stack (by the analytic analogue of \ref{fibration}). We call these the ``good connected analytic $n$-stacks'' since we haven't yet proven that every connected analytic $n$-stack must be of this form (I suspect that to be true but don't have an argument). If $G$ is an analytic Lie group, a map $X_{DR/S}\rightarrow BG$ is a principal $G$-bundle $P$ on $X$ together with an integrable connection relative to $S$. Suppose $A$ is an analytic abelian Lie group with action of $G$. Then we can form the analytic $n$-stack $\kappa (G, A, n)$ with fundamental group $G$ and $\pi _n= A$. Given a map $X_{DR/S}\rightarrow BG$ corresponding to a principal bundle $P$, we would like to study the liftings into $\kappa (G,A,n)$. We obtain the twisted analytic Lie group $A_P:= P\times ^GA$ over $X$ with integrable connection relative to $S$. Let $V$ denote the universal covering group of $A$ (isomorphic to $Lie (A)$, thus $G$ acts here) and let $L\subset V$ denote the kernel of the map to $A$. Note that $V$ is a complex vector space and $L$ is a lattice isomorphic to $\pi _1(A)$. Again $G$ acts on $L$. We obtain after twisting $V_P$ and $L_P$. Note that $V_P$ is provided with an integrable connection relative to $S$. The following Deligne-type complex calculates the cohomology of $A_P$: $$ C^{\cdot}_{{\cal D}} (A_P):= \{ L_P \rightarrow V_P \rightarrow \Omega ^1_{X/S}\otimes _{{\cal O}} V_P \rightarrow \ldots \} . $$ The $n$-stack $\Gamma (X_{DR/S}/S, K(A_P,n))$ is again obtained by applying Dold-Puppe to the shifted right derived direct image complex $Rf_{\ast}(C^{\cdot}_{{\cal D}}(A_P))[n]$. We can write $ C^{\cdot}_{{\cal D}} (A_P)$ as the mapping cone of a map of complexes $L_P \rightarrow U^{\cdot}_P$. If $f$ is a projective morphism then applying GAGA and the argument of Mumford (actually I think there is an argument of Grauert which treats this for any proper map), we get that $$ Rf_{\ast}(C^{\cdot}_{{\cal D}}(U^{\cdot}_P)) $$ is quasiisomorphic to a complex of analytic Lie groups (vector bundles in this case). On the other hand, locally on the base the direct image $Rf_{\ast}(C^{\cdot}_{{\cal D}}(L_P)$ is a trivial complex so quasiisomorphic to a complex of (discrete) analytic Lie groups. The direct image $Rf_{\ast}(C^{\cdot}_{{\cal D}}(A_P))$ is the mapping cone of a map of these complexes, so the associated spectrum fits into a fibration sequence. The base and the fiber are analytic $N$-stack so the total space is also an analytic $N$-stack. Thus the spectrum associated to $Rf_{\ast}(C^{\cdot}_{{\cal D}}(A_P))$ is analytic over $S$. In particular its realization $\Gamma (X_{DR/S}/S, K(A_P,n))$ is a geometric $n$-stack over $S$. For $Hom (X_{DR/S}/S, BG)$ we can use the Riemann-Hilbert correspondence (see below) to see that it is an analytic $1$-stack. The same argument as in Theorem \ref{maps} now shows that for any good connected analytic $n$-stack $T$, the $n$-stack of morphisms $Hom (X_{DR/S}/S, T)$ is an analytic $n$-stack over $S$. If the base $S$ is a point we don't need to make use of Mumford's argument, so the same holds true for any proper smooth analytic space $X$. {\em Caution:} There is (at least) one gaping hole in the above argument, because we are applying Dold-Puppe for complexes of ${\bf Z}$-modules such as $L_P$ or its higher direct image, which are not complexes of rational vector spaces. Thus this doesn't fit into the previous discussion of Dold-Puppe, spectra etc. as we have set it up. In particular there may be problems with torsion, finite groups or subgroups of finite index in the above discussion. The reader is invited to try to figure out how to fill this in (and to let me know if he does). \subnumero{The Riemann-Hilbert correspondence} We can extend to our cohomology stacks the classical Riemann-Hilbert correspondence. We start with a statement purely in the analytic case. In order to avoid confusion between the analytic situation and the algebraic one, we will append the superscript {\em `an'} to objects in the analytic site, even if they don't come from objects in the algebraic site. We will make clear in the hypothesis whenever our analytic objects actually come from algebraic ones. \begin{theorem} \label{analyticRiemannHilbert} Suppose $T^{\rm an}$ is a good connected analytic $n$-stack, and suppose $X^{\rm an}$ is a smooth proper complex analytic space. Define $X^{\rm an}_{DR}$ as above. Let $X^{\rm an}_B$ denote the $n$-stack associated to the constant presheaf of spaces which to each $Y^{\rm an}$ associates the topological space $X^{\rm top}$. Then there is a natural equivalence of analytic $n$-stacks $Hom (X^{\rm an}_{DR}, T^{\rm an}) \cong Hom (X^{\rm an}_B, T^{\rm an})$. \end{theorem} {\em Proof:} By following the same outline as the argument given in \ref{maps}, it suffices to see this for the cases $T^{\rm an} = BG^{\rm an}$ for an analytic Lie group $G^{\rm an}$, and $T^{\rm an}= {\cal B} (A^{\rm an}, n)$ for an abelian analytic Lie group $A^{\rm an}$. In the second case we reduce to the case of cohomology with coefficients in a twisted version of $A^{\rm an}$. We now leave it to the reader to verify these cases (which are standard examples of using analytic de Rham cohomology to calculate singular cohomology). \hfill $\Box$\vspace{.1in} {\em Remark:} For convenience we have stated only the absolute version. We leave it to the reader to obtain a relative version for a smooth projective morphism $f: X\rightarrow S$. Now we turn to the algebraic situation. We can combine the above result with GAGA to obtain: \begin{theorem} \label{algebraicRiemannHilbert} Suppose $T$ is a connected very presentable algebraic $n$-stack, and suppose $X$ is a smooth projective variety. Define $X_{DR}$ as above. Let $X_B$ denote the $n$-stack associated to the constant presheaf of spaces which to each $Y$ associates the topological space $X^{\rm top}$. Then there is a natural equivalence of analytic $n$-stacks $Hom (X_{DR}, T)^{\rm an} \cong Hom (X_B, T)^{\rm an}$. \end{theorem} {\em Proof:} By GAGA we have $$ Hom (X_{DR}, T)^{\rm an} \cong Hom (X^{\rm an}_{DR}, T^{\rm an}). $$ Similarly the calculation of $Hom (X_B,T)$ using a cell decomposition of $X_B$ and fiber products yields the equivalence $$ Hom (X_B,T)^{\rm an}\cong Hom (X^{\rm an}_B, T^{\rm an}). $$ Putting these together we obtain the desired equivalence. \hfill $\Box$\vspace{.1in} \subnumero{The Hodge filtration} Let $H:= {\bf A}^1/{\bf G}_m$ be the quotient stack of the affine line modulo the action of the multiplicative group. This has a Zariski open substack which we denote $1\subset H$; note that $1\cong Spec ({\bf C} )$. There is a closed substack $0\subset H$ with $0\cong B{\bf G}_m$. As in \cite{SantaCruz} we can define a smooth formal category $X_{\rm Hod}\rightarrow H$ whose fiber over $1$ is $X_{DR}$ and whose fiber over $0$ is $X_{Dol}$. Suppose $T$ is a connected very presentable $n$-stack. Then we obtain the relative semistable morphism stack $$ Hom ^{\rm se}(X_{\rm Hod}/H, T) \rightarrow H. $$ In the case $T=BG$ this was interpreted as the {\em Hodge filtration on ${\cal M} _{DR}=Hom (X_{DR}, BG)$}. Following this interpretation, for any connected very presentable $T$ we call this relative morphism stack the {\em Hodge filtration on the higher nonabelian cohomology stack $Hom (X_{DR}, T)$}. Note that when $T= K({\cal O} , n)$ we recover the usual Hodge filtration on the algebraic de Rham cohomology, i.e. the cohomology of $X_{DR}$ with coefficients in ${\cal O}$. The above general definition is essentially just a mixture of the case $BG$ and the cases $K({\cal O} , n)$ but possibly with various twistings. {\em The analytic case:} The above discussion works equally well for a smooth proper analytic variety $X$. For any good connected analytic $n$-stack $T$ we obtain the relative morphism stack $$ Hom (X_{\rm Hod}/H^{\rm an}, T) \rightarrow H^{\rm an}. $$ Note that there is no question of semistability here. The moduli stack of flat principal $G$-bundles $Hom (X_{\rm Hod}/H^{\rm an}, BG)$ is still an analytic $n$-stack because in the analytic category there is no distinction between finite type and locally finite type. In case $X$ is projective and $G= \pi _1(T)$ affine algebraic we can put in the semistability condition and get $$ Hom ^{\rm se}(X_{\rm Hod}/H^{\rm an}, T) \rightarrow H^{\rm an}. $$ If $T$ is the analytic stack associated to an algebraic geometric $n$-stack then this analytic morphism stack is the analytic stack associated to the algebraic morphism stack. \subnumero{The Gauss-Manin connection} Suppose $X\rightarrow S$ is a smooth projective morphism and $T$ a connected very presentable $n$-stack. The formal category $X_{DR/S}\rightarrow S$ is pulled back from the morphism $X_{DR}\rightarrow S_{DR}$ via the map $S\rightarrow S_{DR}$. Thus $$ Hom (X_{DR/S}/S,T) = Hom (X_{DR}/S_{DR}, T)\times _{S_{DR}}S. $$ Thus the morphism stack $Hom (X_{DR/S}/S,T)$ descends down to an $n$-stack over $S_{DR}$. If $Y\rightarrow S_{DR}$ is a morphism from a scheme then locally in the etale topology it lifts to $Y\rightarrow S$. We have $$ Hom (X_{DR}/S_{DR},T)\times _{S_{DR}}Y= Hom (X_{DR}\times _{S_{DR}}Y/Y, T)= $$ $$ Hom (X_{DR/S}\times _SY/Y,T)= Hom (X_{DR/S}/S,T)\times _SY. $$ The right hand side is a geometric $n$-stack, so this shows that the morphism $$ Hom (X_{DR}/S_{DR},T)\rightarrow S_{DR} $$ is geometric. This descended structure is the {\em Gauss-Manin connection} on $Hom (X_{DR/S}/S,T)$. In the case $T=BG$ this gives the Gauss-Manin connection on the moduli stack of $G$-bundles with flat connection (cf \cite{Moduli}, \cite{SantaCruz}). In the case $T= K(V,n)$ this gives the Gauss-Manin connection on algebraic de Rham cohomology. In \cite{SantaCruz} we have indicated, for the case $T=BG$, how to obtain the analogues of {\em Griffiths transversality} and {\em regularity} for the Hodge filtration and Gauss-Manin connection. Exactly the same constructions work here. We briefly review how this works. Suppose $X\rightarrow S$ is a smooth projective family over a quasiprojective base (smooth, let's say) which extends to a family $\overline{X}\rightarrow \overline{S}$ over a normal crossings compactification of the base. Let $D= \overline{X}-X$ and $E= \overline{S}-S$. Recall that $\overline{X}_{\rm Hod}(\log D)\rightarrow H$ is the smooth formal category whose underlying space (stack, really, since we have replaced ${\bf A}^1$ by its quotient $H$) is $X\times H$ and whose associated de Rham complex is $(\Omega ^{\cdot}_{\overline{X}}(\log D), \lambda d)$ where $\lambda $ is the coordinate on $H$ (actually to be correct we have to twist everything by line bundles on $H$ to reflect the quotient by ${\bf G}_m$ but I won't put this into the notation). Similarly we obtain the formal category $\overline{S}_{\rm Hod}(\log E)\rightarrow H$, with a morphism $$ \overline{X}_{\rm Hod}(\log D) \rightarrow \overline{S}_{\rm Hod}(\log E). $$ If we pull back by $\overline{S}\rightarrow \overline{S}_{\rm Hod}(\log E)$ then we get a smooth formal category over $\overline{S}$. Thus by \ref{smoothformal} for any connected very presentable $n$-stack $T$ the morphism $$ Hom (\overline{X}_{\rm Hod}(\log D) / \overline{S}_{\rm Hod}(\log E), T)\rightarrow \overline{S}_{\rm Hod}(\log E), T) $$ is a geometric morphism. The existence of this extension (which over the open subset $S_{DR}\subset \overline{S}_{\rm Hod}(\log E), T)$ is just the Gauss-Manin family $Hom (X_{DR}/S_{DR}, T)$) combines the Griffiths transversality of the Hodge filtration and regularity of the Gauss-Manin connection. This is discussed in more detail in \cite{SantaCruz} in the case $T=BG$ or particularly $BGL(r)$---I just wanted to make the point here that the same thing goes through for any connected very presentable $n$-stack $T$. The same thing will work in an analytic setting, but in this case we can use any good connected analytic $n$-stack $T$ as coefficients.

9609

alg-geom/9609020

en

https://arxiv.org/abs/alg-geom/9609020

alg-geom/9609020

Teleman

Andrei Teleman

Non-abelian Seiberg-Witten theory and projectively stable pairs

TeX-Type: LaTeX, 31 pages, revised version

We introduce the concept of Spin^G-structure in a SO-bundle, where $G\subset U(V)$ is a compact Lie group containing $-id_V$. We study and classify $Spin^G(4)$-structures on 4-manifolds, we introduce the G-Monopole equations associated with a $Spin^G$-structure. On Kaehler surfaces a Kobayashi-Hitchin correspondence can be proved for the corresponding moduli spaces. Using this complex geometric interpretation, we determine explicitely a moduli space of "PU(2)-Monopoles" on $\P^2$, we describe its Uhlenbeck compactification, as well as the Donaldson- and the abelian locus.

alg-geom

\section{Introduction} The aim of this paper is to develop a systematic theory of non-abelian Seiberg-Witten equations. The equations we introduce and study are associated with a $Spin^G(4)$-structure on a 4-manifold, where $G$ is a closed subgroup of the unitary group $U(V)$ containing the central involution $-{\rm id}_V$. We call these equations the $G$-monopole equations. For $G=S^1$, one recovers the classical (abelian) Seiberg-Witten equations [W], and the case $G=Sp(1)$ corresponds to the "quaternionic monopole equations" introduced in [OT5]. Fixing the determinant of the connection component in the $U(2)$-monopole equations, one gets the so called $PU(2)$-monopole equations, which should be regarded as a twisted version of quaternionic monopole equations and will be extensively studied in the second part of this paper. It is known ([OT4], [OT5], [PT2]) that the most natural way to prove the equivalence between Donaldson theory and Seiberg-Witten theory is to consider a suitable moduli space of non-abelian monopoles. In [OT5] it was shown that an $S^1$-quotient of a moduli space of quaternionic monopoles should give an homological equivalence between a fibration over a union of Seiberg-Witten moduli spaces and a fibration over certain $Spin^c$-moduli spaces [PT1]. By the same method, but using moduli spaces of $PU(2)$-monopoles instead of quaternionic monopoles, one should be able to express any Donaldson polynomial invariant in terms of Seiberg-Witten invariants associated with the \underbar{twisted} abelian monopole equations of [OT6]. The idea can be extended to get information about the Donaldson theories associated with an arbitrary symmetry group $G$, by relating the corresponding polynomial invariants to Seiberg-Witten-type invariants associated with smaller symmetry groups. One has only to consider a suitable moduli space of $G$-monopoles and to notice that this moduli space contains distinguished closed subspaces of "reducible solutions". The reducible solutions with trivial spinor-component can be identified with $G$-instantons, and all the others reductions can be regarded as monopoles associated to a smaller group. It is important to point out that, if the base manifold is a K\"ahler surface one has Kobayashi-Hitchin-type correspondences (see [D], [DK], [K], [LT] for the instanton case) which give a complex geometric description of the moduli spaces of $SU(2)$, $U(2)$ or $PU(2)$-monopoles (see section 2). The first two cases were already studied in [OT5] and [OT1]. In the algebraic case one can explicitly compute such moduli spaces of non-abelian monopoles and prove the existence of a projective compactification. The points corresponding to instantons and abelian monopoles can be easily identified (see also [OST]).\\ The theory has interesting extensions to manifolds of other dimensions. On Riemann surfaces for instance, one can use moduli spaces of $PU(2)$-monopoles to reduce the computation of the volume or the Chern numbers of a moduli space of semistable rank 2- bundles to computations on the symmetric powers of the base, which occur in the moduli space of $PU(2)$-monopoles as subspaces of abelian reductions. \\ The present paper is divided into two parts: The first deals with the general theory of $Spin^G$-structures and $G$-monopole equations. We give classification theorems for $Spin^G$-structures in principal bundles, and an explicit description of the set of equivalence classes in the cases $G=SU(2)$, $U(2)$, $PU(2)$. Afterwards we introduce the $G$-monopole equations in a natural way by coupling the Dirac harmonicity condition for a pair formed by a connection and a spinor, with the vanishing condition for a generalized moment map. This first part ends with a section dedicated to the concept of reducible solutions of the $G$-monopole equations. Describing the moduli spaces of $G$-monopoles around the reducible loci is the first step in order to express the Donaldson invariants associated with the symmetry group $G$ in terms of Seiberg-Witten-type reductions.\\ In the second part of the paper, we give a complex geometric interpretation of the moduli spaces of $PU(2)$-monopoles in terms of stable oriented pairs, by proving a Kobayashi-Hitchin type correspondence. Using this result, we describe a simple example of moduli space of $PU(2)$-monopoles on ${\Bbb P}^2$, which illustrates in a concrete case how our moduli spaces can be used to relate Donaldson and Seiberg-Witten invariants. In order to be able to give general explicit formulas relating the Donaldson polynomial invariants to Seiberg-Witten invariants, it remains to construct $S^1$-equivariant smooth perturbations of the moduli spaces of $PU(2)$-monopoles, to construct an Uhlenbeck compactification of the perturbed moduli spaces, and finally to give explicit descriptions of the ends of the (perturbed) moduli spaces. The first two problems are treated in [T1], [T2]. Note that the proof of the corresponding transversality results for other moduli spaces of non-abelian connections coupled with harmonic spinors ([PT1], [PT2]) are not complete ([T1]). The third problem, as well as generalizations to larger symmetry groups will be treated in a future paper. I thank Prof. Christian Okonek for encouraging me to write this paper, as well as for the careful reading of the text and his valuable suggestions. \section{G-Monopoles on 4-manifolds} \subsection{The group $Spin^G$ and $Spin^G$-structures} \subsubsection{$Spin^G$-structures in principal bundles} Let $G\subset U(V)$ be a closed subgroup of the unitary group of a Hermitian vector space $V$, suppose that $G$ contains the central involution $-{\rm id}_V$, and denote by ${\germ g}\subset u(V)$ the Lie algebra of $G$. We put $$Spin^G :=Spin \times_{{\Bbb Z}_2} G \ . $$ By definition we get the following fundamental exact sequences: $$\begin{array}{c}1\longrightarrow Spin \longrightarrow Spin^G \stackrel{\delta}\longrightarrow \qmod{G}{{\Bbb Z}_2}\longrightarrow 1 \\ 1\longrightarrow G\longrightarrow Spin^G \stackrel{\pi}{\longrightarrow } SO \longrightarrow 1 \\ 1\longrightarrow {\Bbb Z}_2\longrightarrow Spin^G\textmap{(\pi,\delta)} SO\times \qmod{G}{{\Bbb Z}_2}\longrightarrow 1 \end{array}\eqno{(*)}$$ Note first that there are well defined morphisms $${\rm ad}_G:Spin^G\longrightarrow O({\germ g})\ ,\ {\rm Ad}_{G}:Spin^G \longrightarrow {\rm Aut}(G) $$ induced by the morphisms ${\rm ad}:G\longrightarrow O({\germ g})$ and ${\rm Ad}:G\longrightarrow {\rm Aut}( G)$. If $P^G$ is principal $Spin^G$-bundle, we denote by ${\Bbb G}(P^G)$, ${\scriptscriptstyle|}\hskip -4pt{{\germ g}}(P^G)$ the fibre bundles $P^G\times_{{\rm Ad}_G}G$, $P^G\times_{{\rm ad}_G}{\germ g}$. The group of sections $${\cal G}(P^G):=\Gamma({\Bbb G}(P^G)) $$ in ${\Bbb G}(P^G)$ can be identified with the group of bundle-automorphisms of $P^G$ over the $SO$-bundle $P^G\times_\pi SO$. After a suitable Sobolev completion ${\cal G}(P^G)$ becomes a Hilbert Lie group, whose Lie algebra is the corresponding Sobolev completion of $\Gamma({\scriptscriptstyle|}\hskip -4pt{\g}(P^G))$. We put also $$ \delta(P^G):=P^G\times_{\delta} \left(\qmod{G}{{\Bbb Z}_2}\right)\ .$$ Note that ${\Bbb G}(P^G)$ can be identified with the bundle $\delta(P^G)\times_{\bar{\rm Ad}} G$ associated with the $\qmod{G}{{\Bbb Z}_2}$-bundle $\delta(P^G)$. Let $P$ a principal $SO(n)$-bundle over a topological space $X$. A \underbar{$Spin^G(n)$}-\underbar{structure} in $P$ is a bundle morphisms $P^G\longrightarrow P$ of type $\pi$, where $P^G$ is a principal $Spin^G(n)$-bundle over $X$. Equivalently, a $Spin^G(n)$-structure in $P$ can be regarded as a pair consisting of a $Spin^G(n)$-bundle $P^G$ and an orientation preserving linear isometry $$\gamma:P\times_{SO(n)}{\Bbb R}^n\longrightarrow P^G\times_{\pi}{\Bbb R}^n $$ (called the \underbar{Clifford} \underbar{map} of the structure). Two $Spin^G$-structures $P^G_0\textmap{\sigma_0} P$, $P_1^G\textmap{\sigma_1} P$ in $P$ are called \underbar{equivalent}, if the $Spin^G$ bundles $P^G_0$, $P_1^G$ are isomorphic over $P$. If $(X,g)$ is an oriented Riemannian $n$-manifold, a $Spin^G(n)$-structure in $X$ is a $Spin^G(n)$-structure $P^G\longrightarrow P_g$ in the bundle $P_g$ of oriented $g$-orthonormal coframes of $ X $. This is equivalent with the data of a pair $(P^G,\gamma)$, where $P^G$ is a $Spin^G(n)$-bundle and $\gamma:\Lambda^1_X\stackrel{\simeq}{\longrightarrow} P^G\times_\pi{\Bbb R}^n$ is a linear orientation-preserving isometry. Here $\Lambda^1_X$ stands for the cotangent bundle of $X$, endowed with the dual $SO(n)$-structure. \vspace{5mm} Let $X$ be a fixed paracompact topological space. Note that there is a natural map $H^1(X,\underline {{G}/{{\Bbb Z}_2}}) {\longrightarrow} H^2(X,{\Bbb Z}_2)$, which we denote by $w$. If $G=Spin(k)$, $w$ coincides with the usual morphism $w_2$ defined on the set of $SO(k)$-bundles. By the third exact sequence in $(*)$ we get the following simple classification result \begin{pr} The map $P^G\longmapsto (P^G\times_\pi SO,\delta(P^G))$ defines a surjection of the set of isomorphism classes of $Spin^G$-bundles onto the set of isomorphism classes of pairs $(P,\Delta)$ consisting of an $SO$-bundle and a $\qmod{G}{{\Bbb Z}_2}$-bundle satisfying $w_2(P)+w(\Delta)=0$. Two $Spin^G$-bundles have the same image if and only if they are congruent modulo the natural action of $H^1(X,{\Bbb Z}_2)$ in $H^1(X,\underline{Spin^G})$. \end{pr} {\bf Proof: } Indeed, the natural morphism $H^1(X,\underline{SO_{}}\times\underline{G/{\Bbb Z}_2}) \longrightarrow H^2(X,{\Bbb Z}_2)$ is given by $(P,\Delta)\longmapsto (w_2(P)+w(\Delta))$. \hfill\vrule height6pt width6pt depth0pt \bigskip \\ For instance, we have the following result \begin{pr} Let $X$ be a 4-manifold. The group $H^1(X,{\Bbb Z}_2)$ acts trivially on the set of (equivalence classes of) $Spin^c(4)$-bundles over $X$. Equivalence classes of $Spin^c(4)$-bundles over $X$ are classified by pairs $(P,\Delta)$ consisting of an $SO(4)$-bundle $P$ and an $S^1$-bundle $\Delta$ with $w_2(P)+w_2(\Delta)=0$. \end{pr} {\bf Proof: } Using the identification (see [OT1], [OT3]) $$Spin^c(4)=\{(a,b)\in U(2)\times U(2)|\ \det a=\det b\}\ ,$$ we get an exact sequence $$1\longrightarrow Spin^c(4)\longrightarrow U(2)\times U(2)\longrightarrow S^1\longrightarrow 1\ . $$ Using this, one can prove that, on 4-manifolds, the data of an (equivalence class of) $Spin^c(4)$-bundles is equivalent to the data of a pair of $U(2)$-bundles having isomorphic determinant line bundles. The action of $H^1(X,{\Bbb Z}_2)$ is given by tensoring with flat line bundles with structure group ${\Bbb Z}_2$. The Chern class of such line bundles is 2-torsion, hence the assertion follows from the classification of unitary vector bundles on 4-manifolds in terms of Chern classes. \hfill\vrule height6pt width6pt depth0pt \bigskip The classification of the $Spin^G$-structures in a given $SO$-bundle $P$ is a more delicate problem. \begin{pr} Fix a $Spin^G$-structure $\sigma:P^G\longrightarrow P$ in $P$. Then the set of equivalence classes of $Spin^G$-structures in $P$ can be identified with the cohomology set $H^1(X,{\Bbb G}(P^G ))$ of the sheaf of sections in the bundle ${\Bbb G}(P^G )$. \end{pr} Recall that ${\Bbb G}(P^G)$ can be identified with the bundle $\delta(P^G) \times_{\bar{\rm Ad}} G$ associated with the $\qmod{G}{{\Bbb Z}_2}$-bundle $\delta(P^G)$. Therefore we get the exact sequence of bundles of groups $$1\longrightarrow {\Bbb Z}_2\longrightarrow{\Bbb G}(P^G)\longrightarrow \delta(P^G)\times_{{\rm Ad}} \left(\qmod{G}{{\Bbb Z}_2}\right)\longrightarrow 1\ . $$ The third term coincides with the gauge group of automorphisms of $\delta(P^G)$. The cohomology set $H^1\left(X,\delta(P^G)\times_{\bar{\rm Ad}} \left(\qmod{G}{{\Bbb Z}_2}\right)\right)$ of the associated sheaf coincides with the pointed set of (equivalence classes of) $\qmod{G}{{\Bbb Z}_2}$-bundles over $X$ with distinguished element $\delta(P^G)$. This shows that $\qmod{H^1(X,{\Bbb G}(P^G))}{H^1(X,{\Bbb Z}_2)}$ can be identified with the set of $\qmod{G}{{\Bbb Z}_2}$-bundles $\Delta$ with $w(\Delta)=w(\delta(P^G))$. Therefore \begin{pr} The map $$(\sigma:P^G\longrightarrow P) \longmapsto \delta(P^G)$$ is a surjection of the set of (equivalence classes of) $Spin^G$-structures in $P$ onto the set of $\qmod{G}{{\Bbb Z}_2}$-bundles $\Delta$ satisfying $w(\Delta)+w_2(P)=0$. Two $Spin^G$-structures have the same image if and only if they are congruent modulo the natural action of $H^1(X,{\Bbb Z}_2)$. \end{pr} Proposition 1.1.2 and the proposition below show that the classification of $Spin^G$-structures in the $SO$-bundle $P$ is in general different from the classification of $Spin^G$-bundles with associated $SO$-bundle isomorphic to $P$. \begin{pr}\hfill{\break} 1. If $G=S^1$ then $Spin^{S^1}=Spin^c$, ${\Bbb G}(P^G)=X\times S^1$, hence the set of $Spin^c$-structures in $P$ is a $H^1(X,\underline{S}^1)= H^2(X,{\Bbb Z})$-torsor if it is non-empty. The $H^1(X,{\Bbb Z}_2)$-action in the set of $Spin^c$-structures in $P$, factorizes through a ${\rm Tors}_2 H^2(X,{\Bbb Z})$-action, which is free and whose orbits coincide with the fibres of the determinant map $(\sigma:P^c\longrightarrow P)\longmapsto \delta(P^c)$.\\ 2. Suppose that $X$ is a 4-manifold, $P$ is an $SO$-bundle over $X$ and that $G$ is one of the following:\\ a) $SU(r)$, $r\geq 2$, b) $U(r)$, $r\geq 2$, $r$ even. c) $Sp(r)$, $r\geq 1$. Then $H^1(X,{\Bbb Z}_2)$ acts trivially in the set of $Spin^G$-structures in $P$, hence the classification of $Spin^G$-structures in $P$ reduces to the classification of $\qmod{G}{{\Bbb Z}_2}$-bundles over $X$. \end{pr} {\bf Proof: } \\ 1. The first assertion follows immediately from Propositions 1.1.3 and 1.1.4. \\ 2. Let $\sigma_i:P^G_i\longrightarrow P$, $i=0,\ 1$ be two $Spin^G$-structures in $P$. We consider the locally trivial bundle $Iso_P(P^G_1,P^G_0)$ whose fibre in $x\in X$ consists of isomorphism $\rho_x:(P^G_1)_x\longrightarrow (P^G_0)_x$ of right $Spin^G$-spaces which make the following diagram commutative. $$\begin{array}{rcl} (P^G_1)_x&\stackrel{\rho_x}{\longrightarrow }&{(P^G_0)_x}_{\phantom{X_{X_{X_X}}} }\\ % {\scriptstyle\sigma_{0x}}\searrow&&\swarrow{\scriptstyle \sigma_{\scriptscriptstyle 1x}}\\ & P_x& \end{array} $$ $Iso_P(P^G_1,P^G_0)$ is a principal bundle in the sense of Grothendieck with structure group bundle ${\Bbb G}(P^G_0)$. The $Spin^G$-structures $\sigma_i$ are equivalent if and only if $Iso_P(P^G_1,P^G_0)$ admits a section. Consider first the case $G=SU(r)$ ($r\geq 2$) or $Sp(r)$ ($r\geq 1$). Since $\pi_i\left([Iso_P(P^G_1,P^G_0)]_x\right)=0$ for $i\leq 2$ and $\pi_3\left([Iso_P(P^G_1,P^G_0)]_x\right)$ can be canonically identified with ${\Bbb Z}$, the obstruction $o(\sigma_1,\sigma_0)$ to the existence of such a section is an element in $H^4(X,{\Bbb Z})$. Assume now that $\sigma_1=\lambda\sigma_0$ for some $\lambda\in H^1(X,{\Bbb Z}_2)$ and let $p:\tilde X\longrightarrow X$ the cover associated to $\ker\lambda\subset\pi_1(X)$. It is easy to see that one has $o(p^*(\sigma_1),p^*(\sigma_0))=p^*(o(\sigma_1,\sigma_0))$. But, since $p^*(\lambda)=0$, we get $p^*(\sigma_1)=p^*(\sigma_0)$ hence $o(p^*(\sigma_1),p^*(\sigma_0))=0$. Since $p^*:H^4(X,{\Bbb Z})\longrightarrow H^4(\tilde X,{\Bbb Z})$ is injective for a 4-manifold $X$, the assertion follows immediately. Finally consider $G=U(r)$. When $r\geq 2$ is even, the determinant map $U(r)\longrightarrow S^1$ induces a morphism $Spin^{U(r)} \longrightarrow S^1$. If $\sigma_1=\lambda\sigma_0$, then there is a natural identification $P^G_1\times_{\det} S^1=P^G_0\times_{\det} S^1$, hence, denoting this line bundle by $L$, we get a subbundle $Iso_{P,L}(P^G_1,P^G_0)$ of $Iso_P(P^G_1,P^G_0)$ consisting fibrewise of isomorphisms $(P^G_1)_x\longrightarrow (P^G_0)_x$ over $P_x\times L_x$. Since the standard fibre of $Iso_{P,L}(P^G_1,P^G_0)$ is $SU(r)$, the same argument as above shows that this bundle admits a section, hence $\sigma_1$ and $\sigma_0$ are equivalent. \hfill\vrule height6pt width6pt depth0pt \bigskip \subsubsection{$Spin^G(4)$-structures on 4-manifolds and spinor bundles} Let ${\Bbb H}_{\pm}$ be two copies of the quaternionic skewfield, regarded as right quaternionic vector spaces. The canonical left actions of $Sp(1)$ in ${\Bbb H}_{\pm}$ define an orthogonal representation of the group $$Spin(4)=Sp(1)\times Sp(1)=SU(2)\times SU(2)$$ in ${\Bbb H}\simeq {\rm Hom}_{{\Bbb H}}({\Bbb H}_+,{\Bbb H}_-)$, which gives the standard identification $$\qmod{SU(2)\times SU(2)}{{\Bbb Z}_2}=SO({\Bbb H})=SO(4)$$ Therefore, the group $Spin^G(4)=\qmod{SU(2)\times SU(2)\times G}{{\Bbb Z}_2}$ comes with 2 unitary representations $$\lambda_\pm: Spin^G(4)\longrightarrow U({\Bbb H}_{\pm}\otimes_{\Bbb C} V)$$ obtained by coupling the natural representation of $G$ in $V$ with the spinorial representations $p_{\pm}:Spin(4)=SU(2)\times SU(2)\longrightarrow SU(2)$. There are well defined adjoint morphisms $$ {\rm ad}_{\pm}:Spin^G(4)\longrightarrow O(su(2))\ ,\ \ {\rm Ad}_{\pm}:Spin^G(4)\longrightarrow {\rm Aut}(SU(2))$$ induced by the projections $p_{\pm}$ and the corresponding adjoint representations associated with the Lie group $SU(2)$. If $P^G$ is a $Spin^G(4)$-bundle, we denote by ${\rm ad}_{\pm}(P^G)$, ${\rm Ad}_{\pm}(P^G)$ the corresponding bundles with fibres $su(2)$, $SU(2)$ associated with $P^G$. The \underbar{spinor} \underbar{vector} \underbar{bundles} associated with a $Spin^G(4)$-bundle $P^G$ are defined by $$\Sigma^{\pm}=\Sigma^{\pm}(P^G):=P^G\times_{\lambda_{\pm}} ({\Bbb H}_{\pm}\otimes_{\Bbb C} V)\ , $$ The bundles ${\rm ad}_{\pm}(P^G)$, ${\scriptscriptstyle|}\hskip -4pt{\g}(P^G)$ are real subbundles of the endomorphism bundle ${\rm End}_{\Bbb C}(\Sigma^{\pm})$. The bundle ${\Bbb G}(P^G)$ acts fibrewise unitarily in the bundles $\Sigma^{\pm}$. On the other hand, the identification ${\Bbb H}\simeq {\rm Hom}_{\Bbb H}({\Bbb H}_+,{\Bbb H}_-)$ defines a real embedding \begin{equation} P^G\times_\pi{\Bbb H}\stackrel{ }{\longrightarrow}{\rm Hom}_{{\Bbb G}(P^G)}(\Sigma^+, \Sigma^-)\subset{\rm Hom}_{\Bbb C}( \Sigma^+,\Sigma^-) \end{equation} of the $SO(4)$-vector bundle $P^G\times_\pi{\Bbb H}$ in the bundle ${\rm Hom}_{{\Bbb G}(P^G)}(\Sigma^+,\Sigma^-)$ of ${\Bbb C}$-linear morphisms $\Sigma^+\longrightarrow \Sigma^-$ which commute with the ${\Bbb G}(P^G)$-action. The data of a $Spin^G(4)$-structure with principal bundle $P^G$ on an oriented Riemannian 4-manifold $X$ is equivalent to the data of an orientation-preserving isomorphism $\Lambda^1_X\stackrel{\gamma}{\longrightarrow}P^G\times_\pi{\Bbb H}$, which defines (via the monomorphism in (1)) a \underbar{Clifford} \underbar{multiplication} $(\Lambda^1\otimes{\Bbb C})\otimes \Sigma^+ \longrightarrow \Sigma^-$ commuting with the ${{\Bbb G}(P^G)}$ actions in $\Sigma^{\pm}$. Moreover, as in the classical $Spin^c(4)$ case [OT1], [OT3], we also have induced identifications (which multiply the norms by 2) $$\Gamma:\Lambda^2_{\pm}\longrightarrow {\rm ad}_{\pm}(P^G)\ . $$ \subsubsection{ Examples} 1. $Spin^c(4)$-structures:\\ The group $Spin^c(4):=Spin^{U(1)}(4)$ can be identified with the subgroup $$G_2:=\{(a,b)\in U(2)\times U(2)|\ \det a=\det b\} $$ of $U(2)\times U(2)$. Via this identification, the map $\delta:Spin^c(4)\longrightarrow S^1\simeq\qmod{S^1}{{\Bbb Z}_2}$ in the exact sequence $$1\longrightarrow Spin(4)\longrightarrow Spin^c(4)\stackrel{\delta}{\longrightarrow} S^1\longrightarrow 1 $$ is given by the formula $\delta(a,b)=\det a=\det b$. The spinor bundles come with identifications $$\det\Sigma^+\stackrel{\simeq}{\rightarrow}\det\Sigma^- \stackrel{\simeq}{\rightarrow} P^c\times_\delta{\Bbb C}\ .$$ The $SO(4)$-vector bundle $P^c\times_\pi {\Bbb H}$ associated with a $Spin^c(4)$-bundle $P^c$ can be identified with the bundle ${\Bbb R} SU(\Sigma^+\Sigma^-)\subset{\rm Hom}(\Sigma^+,\Sigma^-)$ of real multiples of isometries of determinant 1. Using these facts, it easy to see that a $Spin^c(4)$-structure can be recovered from the data of the spinor bundles, the identification between the determinant line bundles and the Clifford map. More precisely \begin{pr} The data of a $Spin^c(4)$-structure in the $SO(4)$-bundle $P$ over $X$ is equivalent to the data of a triple consisting of:\\ i) A pair of $U(2)$-vector bundles $\Sigma^{\pm}$.\\ ii) A unitary isomorphism $\det\Sigma^+\stackrel{\iota}{\rightarrow}\det \Sigma^-$.\\ iii) An orientation-preserving linear isometry $$\gamma:P\times_{SO(4)}{\Bbb R}^4 \rightarrow{\Bbb R} SU(\Sigma^+,\Sigma^-)\ .$$ \end{pr} {\bf Proof: } Given a triple $(\Sigma^{\pm},\iota,\gamma)$, we define $P^c$ to be the manifold over $X$ $$ \begin{array}{cl} P^c:=\left\{ [x,(e_1^+,e_2^+),(e_1^-,e_2^-) ]|\right.& x\in X,\ (e_1^\pm,e_2^\pm)\ {\rm an\ orthonormal\ basis\ in\ }\Sigma^{\pm}_x,\\ &\left. \iota_*( e_1^+\wedge e_2^+)=e_1^-\wedge e_2^-\right\}\ . \end{array} $$ Every triple $[x,(e_1^+,e_2^+),(e_1^-,e_2^-)]\in P^c_x$ defines an orthonormal orientation-compatible basis in ${\Bbb R} SU(\Sigma^+_x,\Sigma^-_x)$ which is given with respect to the frames $(e_1^\pm,e_2^\pm)$ by the Pauli matrices. Using the isomorphism $\gamma$, we get a bundle morphism from $P^c$ onto the orthonormal oriented frame bundle of $P\times_{SO(4)}{\Bbb R}^4$, which can be canonically identified with $P$. \hfill\vrule height6pt width6pt depth0pt \bigskip Let $P$ be a principal $SO(4)$-bundle, $P^c\stackrel{{\germ c}_0} \longrightarrow P$ a fixed $Spin^c(4)$-structure in $P$, $\Sigma^{\pm}$ the associated spinor bundles, and $$\gamma_0:P\times_{SO(4)}{\Bbb R}^4\longrightarrow P^c\times_\pi{\Bbb H}= {\Bbb R} SU(\Sigma^+,\Sigma^-)$$ the corresponding Clifford map. For every $m\in H^2(X,{\Bbb Z})$ let $L_m$ be a Hermitian line bundle of Chern class $m$. The fixed identification $\det \Sigma^+\textmap{\simeq}\det \Sigma^-$ induces an identification $\det \Sigma^+\otimes L_m\textmap{\simeq}\det \Sigma^-\otimes L_m$, and the map $$\gamma_m: P\times_{SO(4)}{\Bbb R}^4\longrightarrow {\Bbb R} SU(\Sigma^+\otimes L_m,\Sigma^- \otimes L_m)\ ,\ \gamma_m(\eta):=\gamma_0(\eta)\otimes{\rm id}_{L_m}$$ is the Clifford map of a $Spin^c(4)$-structure ${\germ c}_m$ in $P$ whose spinor bundles are $\Sigma^{\pm} \otimes L_m$. Using the results in the previous section (see also [H], [OT1], [OT6]) we get \begin{pr} \hfill{\break} i) An $SO(4)$-bundle $P$ admits a $Spin^c(4)$-structure iff $w_2(P)$ admits integral lifts.\\ ii) The set of isomorphism classes of $Spin^c(4)$-structures in an $SO(4)$- bundle $P$ is either empty or is an $H^2(X,{\Bbb Z})$-torsor. If $\gamma_0$ is a fixed $Spin^c(4)$-structure in the $SO(4)$-bundle $P$, then the map $m\longmapsto {\germ c}_m$ defines a bijection between $H^2(X,{\Bbb Z})$ and the set of (equivalence classes of) $Spin^c(4)$-structures in $P$. \\ % iii) [HH] If $(X,g)$ is a compact oriented Riemannian 4-manifold, then $w_2(P_g)$ admits integral lifts. In particular any compact oriented Riemannian 4-manifold admits $Spin^c(4)$-structures\\ \end{pr} 2. $Spin^h(4)$-structures: \\ The quaternionic spin group is defined by $Spin^h:=Spin^{Sp(1)}$. By the classification results 1.1.4., 1.1.5 we get \begin{pr} Let $P$ be an $SO$-bundle over a compact oriented 4-manifold $X$. The map $$\left[\sigma:P^h\longrightarrow P\right]\longmapsto [\delta(P^h)]$$ defines a 1-1 correspondence between the set of isomorphism classes of $Spin^h$-structures in $P$ and the set of isomorphism classes of $PU(2)$-bundles $\bar P$ over $X$ with $w_2(\bar P)=w_2(P)$. The latter set can be identified ([DK], p.41) with $$\{p\in{\Bbb Z}|\ p\equiv w_2(P)^2\ {\rm mod}\ 4\}$$ via the Pontrjagin class-map. \end{pr} In dimension 4, the group $Spin^h(4)$ can be identified with the quotient $$\qmod{SU(2)\times SU(2) \times SU(2)}{\{\pm({\rm id},{\rm id},{\rm id})\}}\ ,$$ hence there is an exact sequence \begin{equation}1\longrightarrow {\Bbb Z}_2\longrightarrow SU(2)\times SU(2) \times SU(2)\longrightarrow Spin^h(4)\longrightarrow 1\ . \end{equation} Let $G_3$ be the group $$G_3:=\{(a,b,c)\in U(2)\times U(2)\times U(2)|\ \det a=\det b=\det c\} $$ We have an exact sequence $$1\longrightarrow S^1\longrightarrow G_3\longrightarrow Spin^h(4)\longrightarrow 1$$ extending the exact sequence (2). If $X$ is any manifold, the induced map $H^1(X,\underline{Spin^h(4)})\longrightarrow H^2(X,\underline{\phantom{(}S^1})= H^3(X,{\Bbb Z})$ factorizes as $$H^1(X,\underline{Spin^h(4)})\stackrel{\pi}{\longrightarrow} H^1(X,\underline{SO(4)})\stackrel{w_2}{\longrightarrow} H^2(X,{\Bbb Z}_2)\longrightarrow H^2(X,S^1)\ .$$ Therefore a $Spin^h(4)$-bundle $P^h$ admits an $G_3$-reduction iff the second Stiefel-Whitney class $w_2(P^h\times_\pi SO(4))$ of the associated $SO(4)$-bundle admits an integral lift. On the other hand, the data of a $G_3$-structure in a $SO(4)$-bundle $P$ is equivalent to the data of a triple consisting of a $Spin^c(4)$-structure $P^c\longrightarrow P$ in $P$, a $U(2)$-bundle $E$, and an isomorphism $$P^c\times_\delta{\Bbb C}\textmap{\simeq}\det E\ .$$ Therefore ( see [OT5]), \begin{pr} Let $P$ be a principal $SO(4)$-bundle whose second Stiefel-Whitney class $w_2(P)$ admits an integral lift. There is a 1-1 correspondence between isomorphism classes of $Spin^h(4)$-structures in $P$ and equivalence classes of triples consisting of a $Spin^c(4)$-structure $P^c\longrightarrow P$ in $P$, a $U(2)$-bundle $E$, and an isomorphism $P^c\times_\delta {\Bbb C}\textmap{\simeq}\det E$. Two triples are equivalent if, after tensoring the first with an $S^1$-bundle, they become isomorphic over $P$. \end{pr} Let us identify\ $\qmod{SU(2) \times SU(2)}{{\Bbb Z}_2}$ with $SO(4)=SO({\Bbb H})$ as explained above, and denote by $$\pi_{ij}:Spin^h\longrightarrow SO(4) \ \ \ 1\leq i<j\leq 3$$ the three epimorphisms associated with the three projections of the product $SU(2)\times SU(2)\times SU(2)$ onto $SU(2)\times SU(2)$. Note that $\pi_{12}=\pi$. The spinor bundles $\Sigma^{\pm}(P^h)$ associated with a principal $Spin^h(4)$-bundle $P^h$ are % $$\Sigma^+(P^h)=P^h\times_{\pi_{13}}{\Bbb C}^4\ ,\ \ \Sigma^-(P^h)= P^h\times_{\pi_{23}}{\Bbb C}^4 $$ This shows in particular that the Hermitian 4-bundles $\Sigma^{\pm}(P^h)$ come with \underbar{a} \underbar{real} \underbar{structure} and compatible trivializations of $\det(\Sigma^{\pm}(P^h))$. Suppose now that the \ $Spin^h(4)$-bundle $P^h$ \ admits a $G_3$-lifting\ , consider the associated triple $(P^c,E,P^c\times_\delta{\Bbb C}\textmap{\simeq}\det E)$, and let $\Sigma^{\pm}$ be the spinor bundles associated with $P^c$. The spinor bundles $\Sigma^{\pm}(P^h)$ of $P^h$ and the automorphism-bundle ${\Bbb G}(P^h)$ can be be expressed in terms of the $G_3$-reduction as follows $$\Sigma^{\pm}(P^h)=[\Sigma^{\pm}]^{\vee}\otimes E= \Sigma^{\pm}\otimes E^{\vee}\ ,\ \ {\Bbb G}(P^h)=SU(E) \ . $$ Moreover, the associated $PU(2)$-bundle $\delta(P^h)= P^h\times_\delta PU(2)$ is naturally isomorphic to the $S^1$-quotient of the unitary frame bundle $P_E$ of $E$. \vspace{0.5cm}\\ \\ 3. $Spin^{U(2)}$-structures: \\ Consider the $U(2)$ spin group $$Spin^{U(2)}:=Spin\times_{{\Bbb Z}_2} U(2)\ ,$$ and let $p:U(2)\longrightarrow PU(2)$ be the canonical projection. The map $$p\times\det: U(2)\longrightarrow PU(2)\times S^1$$ induces an isomorphism $\qmod{U(2)}{\{\pm{\rm id}\}}=PU(2)\times S^1$. Therefore the map $\delta:Spin^{U(2)}\longrightarrow \qmod{U(2)}{\{\pm{\rm id}\}}$ can be written as a pair $(\bar\delta,\det)$ consisting of a $PU(2)-$ and an $S^1$-valued morphism. We have exact sequences \begin{equation} \begin{array}{c}1\longrightarrow Spin \longrightarrow Spin^{U(2)}\textmap{(\bar\delta,\det)} PU(2)\times S^1\longrightarrow 1 \\ \\ 1\longrightarrow U(2)\longrightarrow Spin^{U(2)}\textmap {\pi} SO \longrightarrow 1 \\ \\ 1\longrightarrow {\Bbb Z}_2\longrightarrow Spin^{U(2)} \textmap{(\pi,\bar\delta,\det)} SO \times PU(2)\times S^1 \longrightarrow 1 \\ \\ 1 \longrightarrow SU(2)\longrightarrow Spin^{U(2)}\textmap{(\pi,\det)} SO \times S^1 \longrightarrow 1 \ . \end{array} \end{equation} Let $P^u\longrightarrow P $ be a $Spin^{U(2)}$-structure in a $SO$-bundle $P$ over $X$. An important role will be played by the subbundles $${\Bbb G}_0(P^u):=P^u\times_{{\rm Ad}_{U(2)}} SU(2)\ ,\ \ {\scriptscriptstyle|}\hskip -4pt{\g}_0(P^u):=P^u\times_{{\rm Ad}_{U(2)}}su(2)$$ of ${\Bbb G}(P^u)=P^u\times_{{\rm Ad}_{U(2)}}U(2)$, ${\scriptscriptstyle|}\hskip -4pt{\g}(P^u):=P^u\times_{{\rm Ad}_{U(2)}} u(2)$ respectively. The group of sections $${\cal G}_0(P^u):=\Gamma(X,{\Bbb G}_0(P^u))$$ in ${\Bbb G}_0(P^u)$ can be identified with the group of automorphisms of $P^u$ over the $SO\times S^1$-bundle $P\times_X (P^u\times_{\det} S^1)$. By Propositions 1.1.4, 1.1.5 we get \begin{pr} Let $P$ be a principal $SO$-bundle, $\bar P$ a $PU(2)$-bundle, and $L$ a Hermitian line bundle over $X$.\\ i) $P$ admits a $Spin^{U(2)}$-structure $P^u \rightarrow P$ with $$P^u\times_{\bar \delta}PU(2)\simeq\bar P\ ,\ \ P^u\times_{\det}{\Bbb C}\simeq L$$ iff $w_2(P)=w_2(\bar P)+\overline c_1(L)$, where $\overline c_1(L)$ is the mod 2 reduction of $c_1(L)$ .\\ ii) If the base $X$ is a compact oriented 4-manifold, then the map $$P^u\longmapsto \left([P^u\times_{\bar\delta} PU(2)] ,[P^u\times_{\det}{\Bbb C}]\right)$$ defines a 1-1 correspondence between the set of isomorphism classes of $Spin^{U(2)}$-struc\-tures in $P$ and the set of pairs of isomorphism classes $([\bar P],[L])$, where $\bar P$ is a $PU(2)$-bundle and $L$ an $S^1$-bundle with $w_2( P)=w_2(\bar P)+\overline c_1(L)$. The latter set can be identified with $$\{(p,c)\in H^4(X,{\Bbb Z})\times H^2(X,{\Bbb Z}) |\ p\equiv (w_2(P)+ \bar c)^2\ {\rm mod}\ 4\} $$ \end{pr} \hfill\vrule height6pt width6pt depth0pt \bigskip The group $Spin^{U(2)}(4)=\qmod{SU(2) \times SU(2) \times U(2)}{\{\pm({\rm id},{\rm id},{\rm id})\}}$ fits in the exact sequence $$1\longrightarrow S^1 \longrightarrow \tilde G_3\longrightarrow Spin^{U(2)}(4)\longrightarrow 1\ ,$$ where $$\tilde G_3:=\{(a,b,c)\in U(2)\times U(2)\times U(2)|\ \det a=\det b\}\ .$$ and a $Spin^{U(2)}(4)$-bundle $P^u$ admits a $\tilde G_3$-reduction iff $w_2(P^u\times_\pi SO(4))$ has integral lifts. Therefore, as in Proposition 1.1.9, we get \begin{pr} Let $P$ be an $SO(4)$-bundle whose second Stiefel-Whitney class admits integral lifts. There is a 1-1 correspondence between isomorphism classes of $Spin^{U(2)}$-structures in $P$ and equivalence classes of pairs consisting of a $Spin^c(4)$-structure $P^c\longrightarrow P$ in $P$ and a $U(2)$-bundle $E$. Two pairs are considered equivalent if, after tensoring the first one with a line bundle, they become isomorphic over $P$. \end{pr} Suppose that the $Spin^{U(2)}(4)$-bundle $P^u$ admits an $\tilde G_3$-lifting, let $(P^c,E)$ be the pair associated with this reduction, and let $\Sigma^{\pm}$ be the spinor bundles associated with $P^c$. Then the associated bundles $\Sigma^{\pm}(P^u)$, $\bar\delta(P^u)$, $\det(P^u)$, ${{\Bbb G}(P^u)}$, ${\Bbb G}_0(P^u)$ can be expressed in terms of the pair $(P^c,E)$ as follows: $$\Sigma^{\pm}(P^u)=[\Sigma^{\pm}]^{\vee}\otimes E=\Sigma^{\pm}\otimes (E^{\vee}\otimes[\det(P^u)]) \ ,\ \ \bar\delta(P^u)\simeq \qmod{P_E}{S^1}\ , $$ $$ \det(P^u)\simeq \det (P^c)^{-1}\otimes (\det E)\ , \ \ {\Bbb G}(P^u)=U(E),\ {\scriptscriptstyle|}\hskip -4pt{\g}(P^u)=u(E)\ ,$$ $$ \ {\Bbb G}_0(P^u)=SU(E),\ {\scriptscriptstyle|}\hskip -4pt{\g}_0(P^u)=su(E)\ . $$ \subsection{The G-monopole equations} \subsubsection{Moment maps for families of complex structures} Let $(M,g)$ be a Riemannian manifold, and ${\cal J}\subset A^0(so(T_M))$ a family of complex structures on $M$ with the property that $(M,g,J)$ is a K\"ahler manifold, for every $J\in {\cal J}$. We denote by $\omega_J$ the K\"ahler form of this K\"ahler manifold. Let $G$ be a compact Lie group acting on $M$ by isometries with are holomorphic with respect to any $J\in{\cal J}$. Let $U$ be a fixed subspace of $A^0(so(T_M))$ containing the family ${\cal J}$, and suppose for simplicity that $U$ is finite dimensional. We define the \underbar{total} \underbar{K\"ahler} \underbar{form} $\omega_{\cal J}\in A^2(U^{\vee})$ by the formula $$\langle\omega_{\cal J},u\rangle=g(u(\cdot),\cdot)\ .$$ \begin{dt} Suppose that the total K\"ahler form is closed and $G$-invariant. A map $\mu:M \longrightarrow {\rm Hom}({\germ g}, U^{\vee})$ for will be called a ${\cal J}$-moment map for the $G$-action in $X$ if the following two identities hold \\ 1. $\mu(ag)=({\rm ad}_g\otimes{\rm id}_{U^{\vee}})(\mu(a))$ $\forall\ a\in M,\ g\in G$.\\ 2. $d(\langle\mu,{\alpha}\rangle)=\iota_{\alpha^{\#}}\omega_{\cal J}$ in $A^1(U^{\vee})$ $\forall\ \alpha\in {\germ g}$, where $\alpha^{\#}$ denotes the vector field associated with $\alpha$. \end{dt} In many cases ${\germ g}$ comes with a natural ${\rm ad}$-invariant euclidean metric. A map $\mu:M \longrightarrow {\germ g}\otimes U$ will be called also a moment map if its composition with the morphism ${\germ g}\otimes U\longrightarrow {\germ g}^{\vee}\otimes U^{\vee}$ defined by the euclidean structures in ${\germ g}$ and $U$ is a moment map in the above sense. Similarly, the total K\"ahler form can be regarded (at least in the finite dimensional case) as an element in $\omega_{\cal J}\in A^2 (U)$. Note that if $\mu$ is a moment map with respect to ${\cal J}$, then for every $J\in{\cal J}$ the map $\mu_J:=\langle \mu,J\rangle:M\longrightarrow{\germ g}^{\vee}$ is a moment map for the $G$-action in $X$ with respect to the symplectic structure $\omega_J$. \ \begin{re} Suppose that the total K\"ahler form $\omega_{\cal J}$ is $G$-invariant and closed. Let $\mu:M\longrightarrow {\rm Hom}({\germ g}, U^{\vee})$ be a ${\cal J}$-moment map for a free $G$-action and suppose that $\mu$ is a submersion at every point in $\mu^{-1}(0)$. Then $\omega_{\cal J}$ descend to a closed $U^{\vee}$-valued 2-form on the quotient manifold $\qmod{\mu^{-1}(0)}{G}$. In particular, in this case, all the 2-forms $\omega_J$ descend to closed 2-formes on this quotient, but they may be degenerate. \end{re} \vspace{3mm} {\bf Examples:}\hfill{\break}\\ 1. Hyperk\"ahler manifolds: \\ Let $(M,g,(J_1,J_2,J_3))$ be a hyperk\"ahler manifold [HKLR]. The three complex structures $J_1,J_2,J_3$ span a sub-Lie algebra $U\subset A^0(so(T_M))$ naturally isomorphic to $su(2)$. Suppose for simplicity that $Vol(M)=1$. The sphere $S(U,\sqrt 2)\subset U$ of radius $\sqrt 2$ contains the three complex structures and for any $J\in S(U,\sqrt 2)$ we get a K\"ahler manifold $(M,g,J)$. Suppose that $G$ acts on $M$ preserving the hyperk\"ahler structure. A hyperk\"ahler moment map $\mu:M\longrightarrow {\germ g}\otimes su(2)$ in the sense of [HKLR] can be regarded as a moment map with respect to the family $S(U,\sqrt 2)$ in the sense above. If the assumptions in the Remark above are fulfilled, then the forms $(\omega_J)_{J\in S(U,\sqrt 2)}$ descend to \underbar{symplectic} forms on the quotient $\qmod{\mu^{-1}(0)}{G}$, which can be endowed with a natural hyperk\"ahler structure in this way [HKLR].\\ \\ 2. Linear hyperk\"ahler spaces:\\ Let $G$ be a compact Lie group and $G\subset U(W)$ a unitary representation of $G$. A moment map for the $G$-action on $W$ is given by $$\mu_G(w)=-{\rm Pr}_{\germ g}\left(\frac{i}{2}(w\otimes\bar w)\right) $$ where ${\rm Pr}_{\germ g}:u(W)\longrightarrow {\germ g}={\germ g}^{\vee}$ is the projection ${\germ g}\hookrightarrow u(V)$. Any other moment map can be obtained by adding a constant central element in ${\germ g}$. In the special case of the standard left action of $SU(2)$ in ${\Bbb C}^2$, we denote by $\mu_0$ the associated moment map. This is given by $$\mu_0(x)=-\frac{i}{2}(x\otimes\bar x)_0 \ , $$ where $(x\otimes\bar x)_0$ denotes the trace-free component of the Hermitian endomorphism $x\otimes\bar x$. Consider now the scalar extension $M:={\Bbb H}\otimes_{{\Bbb C}} W$. Left multiplications by quaternionic units define a $G$-invariant hyperk\"ahler structure in $M$. The corresponding family of complex structures is parametrized by the radius $\sqrt 2$-sphere $S$ in the space of imaginary quaternions identified with $su(2)$. Define the quadratic map $\mu_{0,G}:{\Bbb H}\otimes_{\Bbb C} W\longrightarrow su(2)\otimes {\germ g}$ by $$\mu_{0,G}(\Psi)={\rm Pr}_{[su(2)\otimes {\germ g}]}(\Psi\otimes\bar\Psi) \ . $$ It acts on tensor monomials by $$x\otimes w\stackrel{\mu_{0G}}{\longmapsto} -4\mu_0(x)\otimes\mu_G(w)\in su(2)\otimes {\germ g}\subset {\rm Herm}({\Bbb H}\otimes_{\Bbb C} W)\ . $$ It is easy to see that $-\frac{1}{2}\mu_{0,G}$ is a moment map for the $G$ action in $M$ with respect to the linear hyperk\"ahler structure in $M$ introduced above. \\ \\ 3. Spaces of spinors:\\ Let $P^G$ be $Spin^G(4)$-bundle over a compact Riemannian manifold $(X,g)$. The corresponding spinor bundles $\Sigma^{\pm}(P^G)$ have ${\Bbb H}_{\pm}\otimes_{\Bbb C} V$ as standard fibres. Any section $J\in \Gamma(X,S({\rm ad}_{\pm}(P^G),\sqrt 2))$ in the radius $\sqrt 2$-sphere bundle associated to $ad_{\pm}(P^G)$ gives a complex (and hence a K\"ahler) structure in $A^0(\Sigma^{\pm}(P^G))$. Therefore (after suitable Sobolev completions) the space of sections $$\Gamma(X,S({\rm ad}_{\pm}(P^G),\sqrt 2))$$ can be regarded as a family of K\"ahler structures in the space of sections $A^0(\Sigma^{\pm}(P^G))$ endowed with the standard $L^2$-Euclidean metric. Define a quadratic map $\mu_{0,{\cal G}}:A^0(\Sigma^{\pm}(P^G))\longrightarrow A^0(ad_{\pm}(P^G)\otimes {\scriptscriptstyle|}\hskip -4pt{\g})$ by sending an element $\Psi\in A^0(\Sigma^{\pm}(P^G))$ to the section in $ad_{\pm}(P^G)\otimes{\scriptscriptstyle|}\hskip -4pt{\g}$ given by the fibrewise projection of $\Psi\otimes\bar\Psi\in A^0({\rm Herm}(\Sigma^{\pm}(P^G)))$. Then $-\frac{1}{2}\mu_{0,{\cal G}}:A^0(\Sigma^{\pm}(P^G))\longrightarrow A^0({\rm ad}_{\pm}(P^G)\otimes {\scriptscriptstyle|}\hskip -4pt{\g})\subset{\rm Hom}(A^0({\scriptscriptstyle|}\hskip -4pt{\g}), A^0({\rm ad}_{\pm}(P^G)^{\vee})$ can be regarded as a $\Gamma(X,S({\rm ad}_{\pm}(P^G),\sqrt 2))$-moment map for the natural action of the gauge group ${\cal G}$. \\ \\ 4. Spaces of connections on a 4-manifold:\\ Let $(X,g)$ be a compact oriented Riemannian 4-manifold, $G\subset U(r)$ a compact Lie group, and $P$ a principal $G$-bundle over $X$. The space of connections ${\cal A}(P)$ is an euclidean affine space modelled on $A^1({\rm ad}(P))$, and the gauge group ${\cal G}:=\Gamma(X,P\times_{Ad}G)$ acts from the left by $L^2$-isometries. The space of almost complex structures in $X$ compatible with the metric and the orientation can be identified with space of sections in the sphere bundle $S(\Lambda^2_+,\sqrt 2)$ under the map which associates to an almost complex structure $J$ the K\"ahler form $\omega_J:=g(\cdot,J(\cdot))$ [AHS]. On the other hand any almost complex structure $J\in \Gamma(X,S(\Lambda^2_+,\sqrt 2))$ induces a gauge invariant \underbar{integrable} complex structure in the affine space ${\cal A}(P)$ by identifying $A^1({\rm ad}(P))$ with $A^{01}_J({\rm ad}(P)^{{\Bbb C}})$. The total K\"ahler form of this family is the element $\Omega\in A^2_{{\cal A}(P)}(A^2_{+,X})$ given by $$\Omega(\alpha,\beta) = {\rm Tr}(\alpha\wedge\beta)^+ \ , $$ where $\alpha$, $\beta\in A^1({\rm ad}(P))$. Consider the map $ F^+ :{\cal A}(P)\longrightarrow A^0({\rm ad}(P)\otimes \Lambda^2_{X,+})\subset {\rm Hom}(A^0({\rm ad}(P)),(A^2_+)^{\vee})$ given by $A\longmapsto F_A^+$. It satisfies the equivariance property 1. in Definition 1.2.1. Moreover, for every $A\in{\cal A}(P)$, $\alpha\in A^1({\rm ad}(P))=T_A({\cal A}(P))$, $\varphi\in A^0({\rm ad}(P))=Lie({\cal G})$ and $\omega\in A^2_+$ we have (denoting by $\delta$ the exterior derivative on ${\cal A}(P)$) $$\left\langle(\iota_{\varphi^{\#}}\Omega)(\alpha)- \langle\delta_A(F^+) (\alpha),\varphi\rangle,\omega\right\rangle =\langle d^+[{\rm Tr}(\varphi\wedge\alpha)],\omega\rangle= \int_X{\rm Tr}(\varphi\wedge\alpha)\wedge d\omega \ . $$ This formula means that the second condition in Definition 1.2.1. holds up to 1-forms on ${\cal A}(P)$ with values in the subspace ${\rm im}[d^+: A^1_X \rightarrow A^2_{X,+} ]$. Let $\bar\Omega$ be the image of $\Omega$ in $A^2_{{\cal A}(P)}\left[\qmod{A^2_{X,+}}{{\rm im}(d^+)}\right]$. Putting $${\cal A}^{ASD}_{reg}=\{A\in{\cal A}(P)|\ F_A^+=0,\ {\cal G}_A=Z(G),\ H^0_A=H^2_A=0\}$$ we see that $\bar\Omega$ descends to a closed $\left[\qmod{A^2_{X,+}}{{\rm im}(d^+)}\right]$-valued 2-form $[\bar\Omega]$ on the moduli space of regular anti-selfdual connections ${\cal M}^{ASD}_{reg}:=\qmod{{\cal A}^{ASD}_{reg}}{{\cal G}}$. Thus we may consider the map $F^+$ as a $\Gamma(X,S(\Lambda^2_+,\sqrt 2))$-moment map modulo $d^+$-exact forms for the action of the gauge group on ${\cal A}(P) $. Note that in the case $G=SU(2)$ taking $L^2$-scalar product of $\frac{1}{8\pi^2}[\bar\Omega]$ with a harmonic selfdual form $\omega\in {\Bbb H}^2_+$ defines a de Rham representant of Donaldson's $\mu$-class associated with the Poncar\'e dual of $[\omega]$. The following simple consequence of the above observations can be regarded as the starting point of Seiberg-Witten theory. \begin{re} The data of a $Spin^G(4)$-structure in the Riemannian manifold $(X,g)$ gives an isometric isomorphism $\frac{1}{2}\Gamma:\Lambda^2_{+}\longrightarrow {\rm ad}_{+}(P^G)$. In particular we get an identification between the two familes $\Gamma(X,S({\rm ad}_{+}(P^G),\sqrt 2))$ and $\Gamma(X,S(\Lambda^2_{+},\sqrt 2))$ of complex structures in $A^0(\Sigma^{+}(P^G))$ and ${\cal A}(\delta(P^G))$ studied before. Consider the action of the gauge group ${\cal G}:=\Gamma(X,{\Bbb G})$ on the product ${\cal A}(\delta(P^G))\times A^0(\Sigma^{+}(P^G))$ given by $$[(A,\Psi),f]\longmapsto (\delta(f)(A),f (\Psi))\ . $$ This action admits a (generalized) moment map modulo $d^+$-exact forms (with respect to the family $\Gamma(X,S({\rm ad}_{+}(P^G),\sqrt 2))$) which is given by the formula $$(A,\Psi)\longmapsto F_A^{+}-\Gamma^{-1}(\mu_{0,{\cal G}}(\Psi)) \ . $$ \end{re} \subsubsection{Dirac harmonicity and the $G$-monopole equations} Let $P^G$ be a $Spin^G$-bundle. Using the third exact sequence in $(*)$ sect. 1.1, we see that the data of a connection in $P^G$ is equivalent to the data of a pair consisting of a connection in $P^G\times_\pi SO$, and a connection in $\delta(P^G)$. In particular, if $P^G\longrightarrow P_g$ is a $Spin^G(n)$-structure in the frame bundle of an oriented Riemannian $n$-manifold $X$, then the data of a connection $A$ in $\delta(G)$ is equivalent to the data of a connection $B_A$ in $P^G$ lifting the Levi-Civita connection in $P_g$. Suppose now that $n=4$, and denote as usual by $\gamma: \Lambda^1\longrightarrow {\rm Hom}_{\Bbb G}(\Sigma^+(P^G)^+, \Sigma^-(P^G))$ the Clifford map of a fixed $Spin^G(4)$-structure $P^G\stackrel{\sigma}\longrightarrow P_g$, and by $\Gamma:\Lambda^2_{\pm}\longrightarrow {\rm ad}_{\pm}(P^G)$ the induced isomorphisms. We define the Dirac operators ${\raisebox{.17ex}{$\not$}}\hskip -0.4mm{D}_A^{\pm}$ associated with $A\in{\cal A}(\delta(P^G))$ as the composition $$A^0(\Sigma^{\pm}(P^G))\textmap{\nabla_{{B_A}}} A^1(\Sigma^{\pm}(P^G))\textmap{\cdot\gamma} A^0(\Sigma^{\mp}(P^G)) \ . $$ We put also $$\Sigma(P^G):=\Sigma^+(P^G)\oplus \Sigma^-(P^G)\ ,\ \ {\raisebox{.17ex}{$\not$}}\hskip -0.4mm{D}_A:={\raisebox{.17ex}{$\not$}}\hskip -0.4mm{D}_A^+\oplus{\raisebox{.17ex}{$\not$}}\hskip -0.4mm{D}_A^-:A^0(\Sigma(P^G))\longrightarrow A^0(\Sigma(P^G))\ .$$ Note that ${\raisebox{.17ex}{$\not$}}\hskip -0.4mm{D}_A$ is a self-adjoint first order elliptic operator. \begin{dt} A pair $(A,\Psi)\in{\cal A}(P^G)\times A^0(\Sigma(P^G))$ will be called (Dirac) harmonic if ${\raisebox{.17ex}{$\not$}}\hskip -0.4mm{D}_A\Psi=0$. \end{dt} The harmonicity condition is obviously invariant with respect to the gauge group ${\cal G}(P^G):=\Gamma(X,{\Bbb G}(P^G))$. The monopole equations associated to $\sigma$ couple the two gauge invariant equations we introduced above: the vanishing of the "moment map " (cf. 1.4.1) of the gauge action with respect to the family of complex structures $\Gamma(X,S({\rm ad}_+(P^G),\sqrt 2))$ in the affine space ${\cal A}(\delta(P^G))\times A^0(\Sigma^+(P^G))$ and the Dirac harmonicity. \begin{dt} Let $P^G\textmap{\sigma} P$ be a $Spin^G(4)$-structure on the compact oriented Riemannian 4-manifold $X$. The associated Seiberg-Witten equations for a pair $(A,\Psi)\in {\cal A}(\delta(P^G)) \times A^0(\Sigma^+(P^G))$ are $$\left\{\begin{array}{ccc} {\raisebox{.17ex}{$\not$}}\hskip -0.4mm{D}_A\Psi&=&0\\ \Gamma(F_A^+)&=&\mu_{0,{\cal G}}(\Psi) \end{array}\right. \eqno{(SW^\sigma)}$$ \end{dt} The solutions of these equations modulo the gauge group will be called $G$-monopoles. The case $G=S^1$ corresponds to the classical (abelian) Seiberg-Witten theory. The case $G=SU(2)$ was extensively studied in [OT5], and from a physical point view in [LM]. \begin{re} If the Lie algebra ${\germ g}$ of $G$ has non-trivial center $z({\germ g})$, then the moment map of the gauge action in $A^0(\Sigma^+(P^G))$ is not unique. In this case it is more natural to consider the family of equations $$\left\{\begin{array}{ccc} {\raisebox{.17ex}{$\not$}}\hskip -0.4mm{D}_A\Psi&=&0\\ \Gamma(F_A^+)&=&\mu_{0,{\cal G}}(\Psi)+\beta \ , \end{array}\right. \eqno{(SW^\sigma_\beta)}$$ obtained by adding in the second equation a section $$\beta\in A^0({\rm ad}_+(P^G)\otimes z({\germ g}))\simeq A^2_+(X,z({\germ g}))\ .$$ \end{re} In the case $G=S^1$ the equations of this form are called \underbar{twisted} \underbar{monopole} equations [OT6]. If $b_+(X)=1$, the invariants defined using moduli spaces of twisted monopoles depend in an essential way on the twisting term $\beta$ ([LL], [OT6]). The particular case $G=U(2)$ requires a separate discussion, since in this case $\delta(U(2))\simeq PU(2)\times S^1$ and, correspondingly, the bundle $\delta(P^u)$ associated with a $Spin^{U(2)}(4)$-structure $P^u\textmap{\sigma} P_g$ splits as the product $$\delta(P^u)=\bar\delta(P^u)\times_X\det(P^u)$$ of a $PU(2)$-bundle with a $U(1)$-bundle. The data of a connection in $P^u$ lifting the Levi-Civita connection in $P_g$ is equivalent to the data of a pair $A=(\bar A,a)$ formed by a connection $\bar A$ in $\bar\delta(P^u)$ and a connection $a$ in $\det(P^u)$. An alternative approach regards the connection $a\in {\cal A}(\det(P^u))$ as a parameter (not an unknown !) of the equations, and studies the corresponding monopole equations for a pair $(\bar A,\Psi)\in {\cal A}(\bar\delta(P^u))\times A^0(\Sigma^+)$. $$\left\{\begin{array}{ccc} {\raisebox{.17ex}{$\not$}}\hskip -0.4mm{D}_{\bar A,a}\Psi&=&0\\ \Gamma(F_{\bar{A}}^+)&=&\mu_{0,0}(\Psi) \end{array}\right. \eqno{(SW^\sigma_a)}$$ Here ${\raisebox{.17ex}{$\not$}}\hskip -0.4mm{D}_{\bar A,a}$ denotes the Dirac operator associated to the connection in $P^u$ which lifts the Levi-Civita connection in $P_g$, the connection $\bar A$ in the $PU(2)$-bundle $\bar\delta(P^u)$ and the connection $a$ in the $S^1$-bundle $\det P^u$; the quadratic map $\mu_{0,0}$ sends a spinor $\Psi\in A^0(\Sigma^+(P^u))$ to the projection of the endomorphism $(\Psi\otimes\bar\Psi)\in A^0({\rm Herm}(\Sigma^+(P^u)))$ on $A^0({\rm ad}_+(P^u)\otimes{\scriptscriptstyle|}\hskip -4pt{\g}_0(P^u))$. The natural gauge group which lets invariant the equations is the group ${\cal G}_0(P^u):= \Gamma(X,{\Bbb G}_0(P^u))$ of automorphisms of the bundle $P^u$ over the bundle-product $P_g\times_X \det(P^u)$,and $-\mu_{0,0}$ is the $\Gamma(X,S({\rm ad}_+,\sqrt 2))$-moment map for the ${\cal G}_0(P^u)$-action in the configuration space. There is no ambiguity in choosing the moment map of the ${\cal G}_0(P^u)$-action, so there is \underbar{no} natural way to perturb these equations besides varying the connection-parameter $a\in{\cal A}(\det(P^u))$. Since the connection-component of the unknown is a $PU(2)$-connection, these equations will be called the $PU(2)$-\underbar{monopole} \underbar{equations}, and its solutions modulo the gauge group ${\cal G}_0(P^u)$ will be called $PU(2)$-monopoles. Note that if the $Spin^{U(2)}(4)$-structure $P^u\longrightarrow P_g$ is associated with the pair $(P^c \longrightarrow P_g,E)$ (Proposition 1.1.11), the quadratic map $\mu_{0,0}$ sends a spinor $\Psi\in A^0\left(\Sigma^+(P^c)\otimes [E^{\vee}\otimes\det (P^u)]\right)$ to the projection of $$(\Psi\otimes\bar\Psi)\in A^0\left({\rm Herm}\left(\Sigma^+(P^c) \otimes[E^{\vee}\otimes\det (P^u)]\right)\right)$$ on $A^0\left(su(\Sigma^+)\otimes su([E^{\vee} \otimes\det (P^u)]\right)$. \begin{re} The data of a $Spin^h(4)$-structure in $X$ is equivalent to the data of $Spin^{U(2)}$-structure $P^u\textmap{\sigma} P$ together with a trivialization of the $S^1$-bundle $\det(P^u)$. The corresponding $SU(2)$-Seiberg-Witten equations coincide with the $PU(2)$-equations $SW^\sigma_\theta$ associated with the trivial connection $\theta$ in the trivial bundle $\det(P^u)$. \end{re} We shall always regard the $SU(2)$-monopole equations as special $PU(2)$-monopole equations. In particular we shall use the notation $\mu_{0,{\cal G}}=\mu_{0,0}$ if ${\cal G}$ is the gauge group associated with a $Spin^h(4)$-structure. \begin{re} The moduli space of $PU(2)$-monopoles of the form $(\bar A,0)$ can be identified with a moduli space of anti-selfdual $PU(2)$-connections, modulo the gauge group ${\cal G}_0$. The natural morphism of ${\cal G}_0$ into the usual $PU(2)$-gauge group of bundle automorphisms of $\bar\delta(P^u)$ is a local isomorphism but in general it is not surjective (see [LT] ). Therefore the space of $PU(2)$-monopoles of the form $(\bar A,0)$ is a finite cover of the corresponding Donaldson moduli space of $PU(2)$-instantons. \end{re} \begin{re} Let $G$ be a compact Lie group endowed with a central invlotion $\iota$ and an arbitrary unitary representation $\rho:G\longrightarrow U(V)$ with $\rho(\iota)=-{\rm id}_V$. One can associate to any $Spin^G(4)$-bundle the spinor bundles $\Sigma^{\pm}$ of standard fibre ${\Bbb H}_{\pm}\otimes V$. Endow the Lie algebra ${\germ g}$ with an ${\rm ad}$-invariant metric. Then one can define $\mu_{0,G}$ using the adjoint of the map ${\germ g}\longrightarrow u(V)$ instead of the orthogonal projection, and the $G$-monopole equations have sense in this more general framework. \end{re} \subsubsection{Reductions} Let $H\subset G\subset U(V)$ be a closed subgroup of $G$ with $-{\rm id}_V\in H$. Let $P^G\textmap{\sigma} P$ be a $Spin^G $-structure in the $SO$-bundle bundle $P$. \begin{dt} A $Spin^H$-reduction of $\sigma$ is a subbundle $P^H$ of $P^G$ with structure group $Spin^H\subset Spin^G$. \end{dt} Note that such a reduction $P^H\hookrightarrow P^G$ defines a reduction $\delta(P^H)\hookrightarrow\delta(P^G)$ of the structure group of the bundle $\delta(P^G)$ from $\qmod{G}{{\Bbb Z}_2}$ to $\qmod{H}{{\Bbb Z}_2}$, hence it defines in particular an injective linear morphism ${\cal A}(\delta(P^H))\hookrightarrow{\cal A}(\delta(P^G))$ between the associated affine spaces of connections. Let now $V_0$ be an $H$-invariant subspace of $V$. Consider a $Spin^G(4)$-structure $P^G\textmap{\sigma} P$ in the $SO(4)$-bundle $P$, and a $Spin^H(4)$-reduction $P^H\stackrel{\rho}{\hookrightarrow} P^G$ of $\sigma$. Let $\Sigma^{\pm}(P^H,V_0)$ be the spinor bundles associated with $P^H$ and the $Spin^H(4)$-representation in ${\Bbb H}^{\pm}\otimes_{\Bbb C} V_0$. The inclusion $V_0\subset V$ induces bundle inclusions of the associated spinor bundles $\Sigma^{\pm}(P^H,V_0)\hookrightarrow \Sigma^{\pm}(P^G)$. Suppose now that $P_g$ is the frame-bundle of a compact oriented Riemannian 4-manifold, choose $A\in{\cal A}(\delta(P^H))\subset {\cal A}(\delta(P^G))$, and let be $B_A\in{\cal A}(\delta(P^G))$ be the induced connection. Then the spinor bundles $\Sigma^{\pm}(P^H,V_0)$ become $B_A$-parallel subbundles of $\Sigma^{\pm}(P^G)$, and the Dirac operator $${\raisebox{.17ex}{$\not$}}\hskip -0.4mm{D}_A:\Sigma(P^G)\longrightarrow \Sigma(P^G)$$ maps $\Sigma(P^H,V_0)$ into itself. Therefore the set of Dirac-harmonic pairs associated with $(\sigma\circ\rho,V_0)$ can be identified with a subset of the set of Dirac-harmonic pairs associated with $(\sigma,V)$. The group $G$ acts on the set $$\{(H,V_0)|\ H\subset G\ {\rm closed\ subgroup},\ V_0\subset V\ {\rm is}\ H-{\rm invariant}\}\ . $$ of subpairs of $(G,V)$ by $[g,(H,V_0)]\longmapsto( Ad_g(H),g(V_0))$. Moreover, for any $Spin^H(4)$-reduction $P^H\hookrightarrow P^G$ of $\sigma$ and any element $g\in G$ we get a reduction $P^{{\rm Ad}_g(H)}\hookrightarrow P^G$ of $\sigma$ and subbundles $\Sigma^{\pm}(P^{{\rm Ad}_g(H)},g(V_0))$ of the spinor bundles $\Sigma^{\pm}(P^G)$. \begin{dt} A subpair $(H,V_0)$ of $(G,V)$ with $-{\rm id}_V\in H$ will be called admissible and $\mu_G|_{V_0}$ takes values in ${\germ h}$ or, equivalently, if $\langle ik(v), v\rangle=0$ for all $k\in{\germ h}^{\bot_{{\germ g}}}$ and $v\in V_0$. \end{dt} Therefore, if $(H,V_0)$ is admissible, then $\mu_G|_{V_0}$ can be identified with the moment map $\mu_H$ associated with the $H$-action in $V_0$ (with respect to the metric in {\germ h} induced from {\germ g} -- see Remark 1.2.9). If generally $E$ is a system of equations on a configuration space ${\cal A}$ we denote by ${\cal A}^E$ the space of solutions of this system, enowed with the induced topology. \begin{pr} Let $(H,V_0)$ be an \ admissible \ subpair of \ $(G,V)$. \ A \\ $Spin^H(4)$-reduction $P^H\textmap{\rho} P^G$ of the $Spin^G(4)$-structure $P^G\textmap{\sigma} P_g$ induces an inclusion $$\left[{\cal A}(\delta(P^H))\times A^0(\Sigma^+(P^H))\right]^{SW^{\sigma\circ\rho}}\subset \left[{\cal A}(\delta(P^G))\times A^0(\Sigma^+(P^G))\right]^{SW^{\sigma}}$$ which is equvariant with respect to the actions of the two gauge groups. \end{pr} \begin{dt} Let $(H,V_0)$ be an admissible subpair. A solution $(A,\Psi)\in \left[{\cal A}(\delta(P^G))\times A^0(\Sigma^+(P^G))\right]^{SW^\sigma}$ will be called \underbar{reducible} \underbar{of} \underbar{type} $(H,V_0)$, if it belongs to the image of such an inclusion, for a suitable reduction $P^H\textmap{\rho} P^G$. \end{dt} If $(H,V_0)$ is admissible, $H\subset H'$ and $V_0$ is $H'$-invariant, then $(H',V_0)$ is also admissible. An admissible pair $(H,V_0)$ will be called \underbar{minimal} if $H$ is minimal in the set of closed subgroups $H'\subset G$ such that $(H',V_0)$ is an admissible subpair of $(G,V)$. The sets of (minimal) admissible pairs is invariant under the natural $G$-action. We list the conjugacy classes of proper minimal admissible subpairs in the cases $(SU(2),{\Bbb C}^{\oplus 2})=(Sp(1),{\Bbb H})$, $(U(2),{\Bbb C}^{\oplus 2})$, $(Sp(2),{\Bbb H}^{\oplus 2})$. Fix first the maximal tori $$ T_{SU(2)}:=\left\{\left(\matrix{z&0\cr 0& z^{-1}}\right)|z\in S^1\right\}\ ,\ \ T_{U(2)}:=\left\{\left(\matrix{u&0\cr 0& v }\right)| u,v\in S^1\right\} $$ $$T_{Sp(2)}:=\left\{\left(\matrix{u&0\cr 0& v }\right)|u,v\in S^1\right\} $$ \\ On the right we list the minimal admissible subpairs of the pair on the left: $$\begin{array}{llcrl} (SU(2),{\Bbb C}^{\oplus 2}):\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ &(\{\pm1\}&, &\{0\}) &\\ \\ &(T_{SU(2)}&,&{\Bbb C}\oplus\{0\} )\\ \\ \end{array} $$ % $$\begin{array}{llcrl} (U(2),{\Bbb C}^{\oplus 2}):\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ &(\{\pm1\}&, &\{0\}) &\\ \\ &\left(\left\{\left(\matrix{\zeta&0\cr0&\pm1}\right)|\zeta\in S^1\right\}\right.&,&\left. \phantom{\matrix{1\cr1}}{\Bbb C}\times\{0\}\right) \\ \\ \end{array} $$ % $$\begin{array}{llcrl} (Sp(2),{\Bbb H}^{\oplus 2}):\ \ \ \ \ \ \ \ \ &(\{\pm 1\}&, &\{0\}) &\\ \\ &\left(\left\{\left(\matrix{\zeta&0\cr0&\pm1}\right)|\zeta\in T_{Sp(1)}\right\}\right.&,&\left. \phantom{\matrix{1\cr1}}{\Bbb C}\oplus\{0_{\Bbb H}\}\right)\\ &\left(\left\{\left(\matrix{\zeta&0\cr0&\pm1}\right)|\zeta\in\ \ Sp(1)\ \ \right\}\right.&,&\left. \phantom{\matrix{1\cr1}}{\Bbb H}\oplus\{0_{\Bbb H}\}\right)\\ \\ \end{array} $$ \begin{re} Fix a maximal torus $T$ of $G$ with Lie algebra ${\germ t}$, and let ${\fam\meuffam\tenmeuf W}\subset {\germ t}^{\vee}$ be the weights of the induced $T$-action in $V$. Let $V=\bigoplus\limits_{\alpha\in{\fam\meuffam\tenmeuf W}} V_\alpha$ be the corresponding decomposition of $V$ in weight spaces. If $(T,V')$ is a subpair of $(G,V)$, then $V'$ must be a sum of weight subspaces, i.e. there exist ${\fam\meuffam\tenmeuf W}'\subset {\fam\meuffam\tenmeuf W}$ such that $V'=\bigoplus\limits_{\alpha\in{\fam\meuffam\tenmeuf W}'} V'_\alpha$, with $0\ne V'_\alpha\subset V_\alpha$. When $G$ is one of the classical groups $SU(n)$, $U(n)$, $Sp(n)$ and $V$ the corresponding canonical $G$-module, it follows easily that $(T,V')$ is admissible iff $|{\fam\meuffam\tenmeuf W}'|=1$. Notice that there is a natural action of the Weil group $\qmod{N(T)}{T}$ in the set of abelian subpairs of the form $(T,V')$. \end{re} The case of the $PU(2)$-monopole equations needs a separate discussion: Fix a $Spin^{U(2)}(4)$-structure $\sigma:P^u\longrightarrow P_g$ in $P_g$ and a connection $a$ in the line bundle $\det (P^u)$. In this case the admissible pairs are by definition equivalent to one of $$(H,\{0\})\ ,\ \ H\subset U(2)\ {\rm with} -{\rm id}_V\in H\ ;\ \ (T_{U(2)},{\Bbb C}\oplus\{0\}) $$ An abelian reduction $P^{ T_{U(2)} }\stackrel{\rho}{\hookrightarrow} P^u$ of $\sigma$ gives rise to a pair of $Spin^c$-structures $({\germ c}_1:P^c_1\longrightarrow P_g, {\germ c}_2:P^c_2\longrightarrow P_g)$ whose determinant line bundles come with an isomorphism $\det(P^c_1)\otimes\det(P^c_2)=[\det (P^u)]^2$. Moreover, the $PU(2)$-bundle $\bar\delta(P^u)$ comes with an $S^1$-reduction $\bar\delta(P^u)= P^{S^1}\times_\alpha PU(2)$ where $[P^{S^1}]^2= \det(P^c_1)\otimes\det(P^c_2)^{-1}$ and $\alpha:S^1\longrightarrow PU(2)$ is the standard embedding $\zeta\longmapsto\left[\left(\matrix{\zeta&0\cr 0&1}\right)\right]$. Since we have fixed the connection $a$ in $\det (P^u)$, the data of a connection $\bar A\in{\cal A}(\bar\delta(P^u))$ which reduces to $P^{ T_{U(2)} }$ via $\rho$ is equivalent to the data of a connection $a_1\in{\cal A}(\det(P^c_1))$. Moreover, we have a natural parallel inclusion $\Sigma^{\pm}(P^c_1)\subset \Sigma^{\pm}(P^u)$. Consider the following twisted abelian monopole equations [OT6] for a pair $(A_1,\Psi_1)\in {\cal A}(\det (P^c_1))\times A^0(\Sigma^{\pm}(P^c_1))$ $$\left\{\begin{array}{ccc} {\raisebox{.17ex}{$\not$}}\hskip -0.4mm{D}_{A_1}\Psi_1&=&0\\ \Gamma(F_{A_1}^+)&=&(\Psi_1\bar\Psi_1)_0+\Gamma(F_a^+) \ . \end{array}\right. \eqno{(SW^{{\germ c}_1}_{\Gamma(F_a^+)})}$$ Taking in Remark 1.2.6 as twisting term the form $\beta=\Gamma(F_a^+)$, we get \begin{pr}A $Spin^{T_{U(2)}}$-reduction $P^{ T_{U(2)}}\stackrel{\rho}{\hookrightarrow} P^u$ of the $Spin^{U(2)}(4)$-structure $P^u\textmap{\sigma} P_g$ induces an inclusion $$\left[{\cal A}(\det(P^c_1))\times A^0(\Sigma^+(P^c_1))\right]^{SW^{{\germ c}_1}_{\Gamma(F_a^+)}}\subset \left[{\cal A}(\bar\delta(P^u))\times A^0(\Sigma^+(P^u))\right]^{SW^{\sigma}_a}$$ which is equivariant with respect to the actions of the two gauge groups. \end{pr} The fact that the Donaldson ($PU(2)$-) $SU(2)$-moduli space is contained in the space of ($PU(2)$-) $SU(2)$-monopoles, and that (twisted) abelian monopoles arise as abelian reductions in the space of ($PU(2)$) $SU(2)$-monopoles suggests that these moduli spaces can be used to prove the equivalence between the two theories [OT5]. This idea can be applied to get information about the Donaldson invariants associated with larger symmetry groups $G$ by relating these invariants to Seiberg-Witten type invariants associated with smaller symmetry groups. In order to do this, one has first to study invariants associated to the moduli spaces of reducible solutions of all possible types in a suitable moduli space of $G$-monopoles. \subsubsection{Moduli spaces of $G$-monopoles} Let ${\cal A}$ be the configuration space of one of the monopole equations $SW$ introduced in sect. 1.2.2.: For the equations $SW^\sigma_\beta$ associated with a $Spin^G(4)$-structure $\sigma:P^G \longrightarrow P_g$ in $(X,g)$ and a section $\beta\in A^0({\rm ad}_+(P^G)\otimes z({\germ g}))$, the space ${\cal A}$ coincides with ${\cal A}(\delta(P^G))\times A^0(\Sigma^+(P^G))$; in the case of $PU(2)$-monopole equations $SW^\sigma_a$ associated to a $Spin^{U(2)}(4)$-structure $\sigma:P^u\longrightarrow P_g$ and an abelian connection $a\in{\cal A}(\det(P^u))$ the configuration space is ${\cal A}(\bar\delta(P^u))\otimes A^0(\Sigma^+(P^u))$. In this section, we denote by ${\cal G}$ the gauge group corresponding to the monopole equation $SW$, i.e. ${\cal G}={\cal G}(P^G)$ if $SW=SW^\sigma_\beta$ and ${\cal G}={\cal G}_0(P^u)$ in the $PU(2)$-case $SW= SW^\sigma_a$. The Lie algebra $Lie({\cal G})$ of ${\cal G}$ is $\Gamma(X,{\scriptscriptstyle|}\hskip -4pt{\g}(P^G))$ in the first case and $\Gamma(X,{\scriptscriptstyle|}\hskip -4pt{\g}_0(P^u))$ in the second. The corresponding moduli space of $G$-monopoles is defined as a topological space by $${\cal M}:=\qmod{{\cal A}^{SW}}{{\cal G}}\ . $$ There is a standard way of describing the local structure of ${\cal M}$, which was extensively described in the cases $G=S^1$, $G=U(2)$ in [OT1] and in the case $G=SU(2)$ (which is similar to the $PU(2)$-case) in [OT5] (see [DK], [K], [LT], [M] for the instanton case and for the classical case of holomorphic bundles). We explain briefly the general strategy: Let $p=(A,\Psi)\in {\cal A}^{SW}$. The infinitesimal action of $Lie({\cal G})$ and the differential of $SW$ in $p$ define a "elliptic deformation complex" $$0\longrightarrow C^0_p \textmap{D^0_p} C^1_p\textmap{D^1_p}C^2_p \longrightarrow 0 \eqno{({\cal C}_p)} $$ where:\\ $C^0_p= Lie({\cal G})=\Gamma(X,{\germ g}(P^G))$ ( or $\Gamma(X,{\germ g}_0(P^u))$ in the $PU(2)$-case),\\ \\ ${\cal C}^1_p= T_p({\cal A})=A^1({\scriptscriptstyle|}\hskip -4pt{\g}(P^G))\oplus A^0(\Sigma^{+}(P^G))$ (or $A^1({\scriptscriptstyle|}\hskip -4pt{\g}_0(P^u))\oplus A^0(\Sigma^{+}(P^u))$ in the $PU(2)$-case),\\ \\ $C^2_p=A^0({\rm ad}_+(P^G)\otimes{\scriptscriptstyle|}\hskip -4pt{\g}(P^G))\oplus A^0(\Sigma^{-}(P^G))$ (or $A^0({\rm ad}_+(P^u)\otimes{\scriptscriptstyle|}\hskip -4pt{\g}_0(P^u))\oplus A^0(\Sigma^{-}(P^u))$ in the $PU(2)$-case),\\ \\ $D_p^0(f):=f^{\#}_p=(-d_A f, f\Psi)$, \\ \\ $D_p^1(\alpha,\psi):=d_pSW(\alpha,\psi)=\left(\Gamma(d_A^+\alpha)-m (\psi,\Psi)-m (\Psi,\psi),\gamma(\alpha)\Psi+{\raisebox{.17ex}{$\not$}}\hskip -0.4mm{D}_A(\psi)\right)\ .$ Here $m$ is the sesquilinear map associated with the quadratic map $\mu_{0,{\cal G}}$ (or $\mu_{0,0}$ in the $PU(2)$-case). The index $\chi$ of this elliptic complex is called the \underbar{expected} \underbar{dimension} of the moduli space and can be easily computed by Atiyah-Singer Index-Theorem [LMi] in terms of characteristic classes of $X$ and vector bundles associated with $P^G$. We give the result in the case of the $PU(2)$-monopole equations: $$\chi(SW^\sigma_a)=\frac{1}{2}\left(-3 p_1(\bar\delta(P^u))+ c_1(\det(P^u))^2\right)- \frac{1}{2}(3e(X)+4\sigma(X)) $$ The same methods as in [OT5] give: \begin{pr}\hfill{\break} 1. The stabilizer ${\cal G}_p$ of $p$ is a finite dimensional Lie group isomorphic to a subgroup of $G$ which acts in a natural way in the harmonic spaces ${\Bbb H}^i({\cal C}_p)$, $i=0,\ 1,\ 2$.\\ 2. There exists a neighbourhood $V_p$ of $P$ in ${\cal M}$, a ${\cal G}_p$-invariant neighbourhood $U_p$ of $0$ in ${\Bbb H}^1({\cal C}_p)$, a ${\cal G}_p$-equivariant real analytic map $K_p:U_p\longrightarrow {\Bbb H}^2({\cal C}_p)$ with $K_p(0)=0$, $dK_p(0)=0$ and a homeomorphism: $$V_p\simeq \qmod{Z(K_p)}{{\cal G}_p}\ . $$ \end{pr} The homeomorphisms in the proposition above define a structure of a smooth manifold of dimension $\chi$ in the open set $${\cal M}_{reg}=\{[p]\in{\cal M}|\ {\cal G}_p=\{1\},\ H^2({\cal C}_p)=0\}$$ of regular points, and a structure of a real analytic orbifold in the open set of points with finite stabilizers. Note that the stabilizer of a solution of the form $(A,0)$ contains always $\{\pm {\rm id}\}$, hence ${\cal M}$ has at least ${\Bbb Z}_2$-orbifold singularities in the Donaldson points (see Remark 1.2.8). As in the instanton case, the moduli space ${\cal M}$ is in general non-compact. The construction of an Uhlenbeck-type compactification is treated in [T1], [T2]. \section{$PU(2)$-Monopoles and stable oriented pairs} In this section we show that the moduli spaces of $PU(2)$-monopoles on a compact K\"ahler surface have a natural complex geometric description in terms of stable oriented pairs. We explain first briefly, following [OT5], the concept of oriented pair and we indicate how moduli space of simple oriented pairs are constructed. Next we restrict ourselves to the rank 2-case and we introduce the concept of stable oriented pair; the stability property we need [OT5] does \underbar{not} depend on a parameter and is an open property. An algebraic geometric approach can be found in [OST]. In section 2.2 we give a complex geometric description of the moduli spaces of irreducible $PU(2)$-monopoles on a K\"ahler surface in terms of moduli spaces of stable oriented pairs. This description is used to give an explicit description of a moduli space of $PU(2)$-monopoles on ${\Bbb P}^2$. \subsection{Simple, strongly simple and stable oriented pairs} Let $(X,g)$ be a compact K\"ahler manifold of dimension $n$, $E$ a differentiable vector bundle of rank $r$ on $X$, and ${\cal L}=(L,\bar\partial_{\cal L})$ a fixed holomorphic structure in the determinant line bundle $L:=\det E$. We recall (see [OT5]) the following fundamental definition: \begin{dt} An oriented pair of type $(E,{\cal L})$ is a pair $({\cal E},\varphi)$, where ${\cal E}$ is a holomorphic structure in $E$ such that $\det{\cal E}={\cal L}$, and $\varphi\in H^0({\cal E})$. Two oriented pairs $({\cal E}_1,\varphi_1)$, $({\cal E}_2,\varphi_2)$ of type $(E,{\cal L})$ are called isomorphic if they are congruent modulo the natural action of the group $\Gamma(X,SL(E))$ of differentiable automorphism of $E$ of determinant 1. \end{dt} Therefore we fix the underlying ${\cal C}^{\infty}$-bundle and the holomorphic determinant line bundle (not only its isomorphism type !) of the holomorphic bundles we consider. An oriented pair $p=({\cal E},\varphi)$ is called \underbar{simple} if its stabilizer $\Gamma(X,SL(E))_p$ is contained in the center ${\Bbb Z}_r{\rm id}_E$ of $\Gamma(X,SL(E))$, and is called \underbar{strongly} \underbar{simple} if its stabilizer is trivial. The first property has an equivalent infinitesimal formulation: the pair $({\cal E},\varphi)$ is simple if and only if any trace-free holomorphic endomorphism of ${\cal E}$ with $f(\varphi)=0$ vanishes. In [OT5] it was shown that \begin{pr} There exists a (possibly non-Hausdorff) complex analytic orbifold ${\cal M}^s(E,{\cal L} )$ parameterizing isomorphism classes of simple oriented pairs of type $(E,{\cal L})$. The open subset ${\cal M}^{ss}(E,{\cal L})\subset {\cal M}^{s}(E,{\cal L})$ consisting of strongly simple pairs is a complex analytic space, and the points in ${\cal M}^s(E,{\cal L})\setminus{\cal M}^{ss}(E,{\cal L})$ have neighbourhoods modeled on ${\Bbb Z}/r$-quotients. \end{pr} If ${\cal E}$ is holomorphic bundle we denote by ${\cal S}({\cal E})$ the set of reflexive subsheaves ${\cal F}\subset{\cal E}$ with $0<{\rm rk}({\cal F})<{\rm rk}({\cal E})$. Once we have fixed a section $\varphi\in H^0({\cal E})$, we put $${\cal S}_\varphi({\cal E}):=\{{\cal F}\in{\cal S}({\cal E})|\ \varphi\in H^0(X,{\cal F})\} \ .$$ We recall (see [B]) that ${\cal E}$ is called $\varphi$-\underbar{stable} if $$\max (\mu_g({\cal E}),\sup\limits_{{\cal F}'\in{\cal S} ({\cal E})} \mu_g({\cal F}'))< \inf\limits_{{\cal F}\in {\cal S}_\varphi({\cal E})} \mu_g(\qmod{{\cal E}}{{\cal F} })\ ,$$ where for a nontrivial torsion free coherent sheaf ${\cal F}$, $\mu_g({\cal F})$ denotes its slope with respect to the K\"ahler metric $g$. If the real number $\lambda$ belongs to the interval $\left(\max (\mu_g({\cal E}),\sup \limits_{{\cal F}'\in{\cal S} ({\cal E})} \mu_g({\cal F}')), \inf\limits_{{\cal F}\in {\cal S}_\varphi({\cal E})} \mu_g(\qmod{{\cal E}}{{\cal F} })\right)$, the pair $({\cal E},\varphi)$ is called $\lambda$-stable. If ${\cal M}$ is a holomorphic line bundle and $\varphi\in H^0({\cal M})\setminus\{0\}$, then $({\cal M},\varphi)$ is $\lambda$-stable iff $\mu_g({\cal M})<\lambda$. The correct definition of the stability property for oriented pairs of arbitrary rank is a delicate point [OST]. The definition must agree in the algebraic-projective case with the corresponding GIT-stability condition. On the other hand, in the case $r=2$ the definition simplifies considerably and this case is completely sufficient for our purposes. Therefore from now on we assume $r=2$, and we recall from [OT5] the following \begin{dt} \hfill{\break} An oriented pair $({\cal E},\varphi)$ of type $({ E},{\cal L})$ is called \underbar{stable} if one of the following conditions holds:\\ I. \ ${\cal E}$ is $\varphi$-stable, \\ II. $\varphi\ne 0$ and ${\cal E}$ splits in direct sum of line bundles ${\cal E}={\cal E}_1\oplus{\cal E}_2$, such that \hspace*{5mm} $\varphi\in H^0({\cal E}_1)$ and the pair $({\cal E}_1,\varphi)$ is $\mu_g({ E})$-stable.\\ A holomorphic pair $({\cal E},\varphi)$ of type $({ E},{\cal L})$ is called \underbar{polystable} if it is stable, or $\varphi=0$ and ${\cal E}$ is a polystable bundle. \end{dt} \begin{re} An oriented pair $({\cal E},\varphi)$ of type $(E,{\cal L})$ with $\varphi\ne 0$ is stable iff $\mu_g({\cal O}_X(D_\varphi))<\mu_g(E)$, where $D_\varphi$ is the divisorial component of the vanishing locus $Z(\varphi)$. An oriented pair of the form $({\cal E},0)$ is stable iff the holomorphic bundle ${\cal E}$ is stable. \end{re} \subsection{The projective vortex equation and stability of oriented pairs} The stability property for holomorphic bundles has a well known differential geometric characterization: an holomorphic bundle is stable if and only if it is simple and admits a Hermite-Einstein metric (see for instance [DK], [LT]). Similarly, an holomorphic pair $({\cal E},\varphi)$ is $\lambda$-stable if and only it is simple and ${\cal E}$ admits a Hermitian metric satisfying the vortex equation associated with the constant $t=\frac{4\pi\lambda}{Vol_g(X)}$ [B]. All these important results are infinite dimensional extensions of the {\it metric characterization of stability} (see [MFK], [DK]). The same approach gives in the case of oriented pairs the following differential geometric interpretation of stability [OT5]: Let $E$ be a differentiable rank 2 vector bundle over a compact K\"ahler manifold $(X,g)$, ${\cal L}$ a holomorphic structure in $L:=\det(E)$ and $l$ a fixed Hermitian metric in $L$. % \begin{thry} An holomorphic pair $({\cal E},\varphi)$ of type $(E,{\cal L})$ with ${\rm rk}({\cal E})=2$ is polystable iff ${\cal E}$ admits a Hermitian metric $h$ with $\det h=l$ which solves the following \underbar{projective} \underbar{vortex} \underbar{equation}: $$i\Lambda_g F_h^0 +\frac{1}{2}(\varphi\bar\varphi^h)_0=0\ .\eqno{(V)}$$ If $({\cal E},\varphi)$ is stable, then the metric $h$ is unique. \end{thry} \begin{re} With an appropriate definition of (poly)stability of oriented pairs [OST], the theorem holds for arbitrary rank $r$. \end{re} Denote by $\lambda\in{\cal A}(L)$ the the Chern connection of ${\cal L}$ associated with the metric $l$. Let $\bar{\cal A}_{\bar\partial_\lambda}$ be the space of semiconnections in $E$ which induce the fixed semiconnection ${\bar\partial_\lambda}$ in $L$. Fix a Hermitian metric $H$ in $E$ with $\det H=l$ and denote by ${\cal A}_\lambda$ the space of unitary connections in $E$ with induce the fixed connection $\lambda$ in $L$. There is an obvious identification ${\cal A}_\lambda\textmap{\simeq}\bar{\cal A}_{\bar\partial_\lambda}$, $C\longmapsto \bar\partial_C$ which endows the affine space ${\cal A}_\lambda$ with a complex structure compatible with the standard $L^2$ euclidean structure. Therefore, after suitable Sobolev completions, the product ${\cal A}_\lambda\times A^0(E)=\bar {\cal A}_{\bar\partial_\lambda}\times A^0(E)$ becomes a Hilbert K\"ahler manifold. Let ${\cal G}_0:=\Gamma(X,SU(E))$ be the gauge group of unitary automorphisms of determinant 1 in $(E,H)$ and let ${\cal G}_0^{\Bbb C}:=\Gamma(X,SL(E))$ be its complexification. \begin{re} The map $m:{\cal A}_\lambda\times A^0(E)\longrightarrow A^0(su(E))$ defined by $$m(C,\varphi)=\Lambda_g F_C^0 -\frac{i}{2}(\varphi\bar\varphi^H)_0 $$ is a moment map for the ${\cal G}_0$-action in the K\"ahler manifold ${\cal A}_\lambda\times A^0(E)$ \end{re} If ${\cal E}$ is a holomorphic structure in $E$ with $\det{\cal E}={\cal L}$ we denote by $C_{\cal E}\in {\cal A}_\lambda$ the Chern connection defined be ${\cal E}$ and the fixed metric $H$. The map $({\cal E},\varphi)\longmapsto (C_{\cal E},\varphi)$ identifies the set of oriented pairs of type $(E,{\cal L})$ with the subspace $Z(j)$ of the affine space ${\cal A}_\lambda\times A^0(E)$ which is cut-out by the integrability condition $$j(C,\varphi):=(F^{02}_C,\bar\partial_C\varphi)=0 $$ \begin{dt} A pair $(C,\varphi)\in {\cal A}_\lambda\times A^0(E)$ will be called \underbar{irreducible} if any $C$-parallel endomorphism $f\in A^0(su(E))$ with $f(\varphi)=0$ vanishes. \end{dt} This notion of (ir)reducibility must not be confused with that one introduced in section 1.2.3, which depends on the choice of an admissible pair. For instance, irreducible pairs can be abelian. The theorem above can now be reformulated as follows: \begin{pr} An oriented pair $({\cal E},\varphi)$ of type $(E,{\cal L})$ is polystable if and only if the complex orbit ${\cal G}_0^{\Bbb C}\cdot (C_{\cal E},\varphi)\subset Z(j)$ intersects the vanishing locus $Z(m)$ of the moment map $m$. $({\cal E},\varphi)$ is stable if and only if it is polystable and $(C_{\cal E},\varphi)$ is irreducible. \end{pr} It can be easily seen that the intersection $\left[{\cal G}_0^{\Bbb C}\cdot (C_{\cal E},\varphi)\right]\cap Z(m)$ of a complex orbit with the vanishing locus of the moment map is either empty or coincides with a \underbar{real} orbit. Moreover, using the proposition above one can prove that the set $Z(j)^{st}$ of stable oriented pairs is an \underbar{open} subset of the set $Z(j)^{s}$ of simple oriented pairs. The quotient $\qmod{Z(j)^s}{{\cal G}_0^{{\Bbb C}}}$ can be identified with the moduli space ${\cal M}^s(E,{\cal L})$ of simple oriented pairs of type $(E,{\cal L})$. The open subspace ${\cal M}^{st}(E,{\cal L}):=\qmod{Z(j)^{st}}{{\cal G}_0^{{\Bbb C}}}\subset {\cal M}^s(E,{\cal L})$ will be called the moduli space of stable oriented pairs, and comes with a natural structure of a \underbar{Hausdorff} complex space. The same methods as in [DK], [LT], [OT1] give finally the following \begin{thry} The identification map $(C,\varphi)\longmapsto (\bar\partial_C,\varphi)$ induces an isomorphism of real analytic spaces $\qmod{Z(j,m)^{ir}}{{\cal G}_0}\textmap{\simeq}\qmod{Z(j)^{st}}{{\cal G}_0^{{\Bbb C}}}={\cal M}^{st}(E,{\cal L})$, where $Z(j,m)^{ir}$ denotes the locally closed subspace consisting of irreducible oriented pairs solving the equations $j(C,\varphi)=0$, $m(C,\varphi)=0$. \end{thry} % \subsection{Decoupling the $PU(2)$-monopole equations} Let $(X,g)$ be a K\"ahler surface and let $P^{\rm can}\longrightarrow P_g$ be the associated \underbar{canonical} $Spin^c(4)$-\underbar{structure} whose spinor bundles are $\Sigma^+=\Lambda^{00}\oplus\Lambda^{02}$, $\Sigma^-=\Lambda^{01}$. By Propositions 1.1.11, 1.1.7 it follows that the data of a $Spin^{U(2)}(4)$-structure in $(X,g)$ is equivalent to the data of a Hermitian 2-bundle $E$. The bundles associated with the $Spin^{U(2)}(4)$-structure $\sigma:P^u\longrightarrow P_g$ corresponding to $E$ are: $$\det(P^u)=\det E\otimes K_X ,\ \bar\delta(P^u)=\qmod{P_E}{S^1} , $$ \ $$\Sigma^{\pm}(P^u)=\Sigma^{\pm}\otimes E^{\vee}\otimes\det(P^u)=\Sigma^{\pm}\otimes E\otimes K_X\ ;\ \ \Sigma^{+}(P^u)=E \otimes K_X\oplus E\ . $$ Suppose that $\det(P^u)\in NS(X)$ and fix an \underbar{integrable} connection $a\in{\cal A}(\det(P^u))$. Denote by $c\in {\cal A}(K_X)$ the Chern connection in $K_X$, by $\lambda:=a\otimes \bar c$ the induced connection in $\det(E)=\det(P^u)\otimes \bar K_X$ and by ${\cal L}$ the corresponding holomorphic structure in this line bundle. Identify the affine space ${\cal A}(\bar\delta(P^u))$ with ${\cal A}_{\lambda\otimes c^{\otimes 2}}(E\otimes K_X)$ and the space of spinors $A^0(\Sigma^+(P^u))$ with the direct sum $A^0(E \otimes K_X)\oplus A^0(E)=A^0(E \otimes K_X)\oplus A^{02}(E \otimes K_X)$. The same computations as in Proposition 4.1 [OT5] gives the following {\it decoupling theorem}: \begin{thry} A pair $$(C,\varphi+\alpha)\in {\cal A}_{\lambda\otimes c^{\otimes 2}}(E\otimes K_X)\times\left(A^0(E\otimes K_X)\oplus A^{02}(E \otimes K_X)\right)$$ solve the $PU(2)$-monopole equations $SW^\sigma_a$ if and only if the connection $C$ is integrable and one of the following conditions is fulfilled: $$1)\ \alpha=0,\ \bar\partial_C\varphi=0\ \ \ and\ \ \ i\Lambda_g F_C^0+\frac{1}{2}(\varphi\bar\varphi)_0=0\ , $$ $$\ 2)\ \varphi=0,\ \partial_C\alpha=0\ \ and\ \ i\Lambda_g F_C^0-\frac{1}{2}*(\alpha\wedge\bar\alpha)_0=0\ , $$ \end{thry} Using Theorem 2.2.5 we get \begin{re} The moduli space $({\cal M}^\sigma_a)_{\alpha=0}^{ir}$ of irreducible solutions of type 1) can be identified with the moduli space ${\cal M}^{st}(E\otimes K_X,{\cal L}\otimes{\cal K}_X^{\otimes 2})$. The moduli space $({\cal M}^\sigma_a)^{ir}_{\varphi=0}$ of irreducible solutions of type 2) can be identified with the moduli space ${\cal M}^{st}(E^{\vee},{\cal L}^{\vee})$ via the map $(C,\alpha)\longmapsto (\bar C\otimes c,\bar \alpha)$. \end{re} Concluding, we get the following simple description of the moduli space ${\cal M}^\sigma_a$ in terms of moduli spaces of stable oriented pairs. \begin{co} Suppose that the $Spin^{U(2)}(4)$-structure $\sigma:P^u\longrightarrow P_g$ is associated to the pair $(P^{\rm can}\longrightarrow P_g,E)$, where $P^{\rm can}\longrightarrow P_g$ is the canonical $Spin^c(4)$-structure of the K\"ahler surface $(X,g)$ and $E$ is a Hermitian rank 2 bundle. Let $a\in{\cal A}(\det(P^u))$ be an integrable connection and ${\cal L}$ the holomorphic structure in $\det E=\det(P^u)\otimes K_X^{\vee}$ defined by $a$ and the Chern connection in $K_X$. Then the moduli space ${\cal M}^\sigma_a$ decomposes as a union of two Zariski closed subspaces $${\cal M}^\sigma_a=({\cal M}^\sigma_a)_{\alpha=0}\mathop{\bigcup}({\cal M}^\sigma_a)_{\varphi=0} $$ which intersect along the Donaldson moduli space ${\cal D}(\delta(P^u)) \subset{\cal M}^\sigma_a$ (see Remark 1.2.8). There are canonical real analytic isomorphisms $$({\cal M}^\sigma_a)_{\alpha=0}^{ir}\simeq{\cal M}^{st} (E\otimes K_X,{\cal L}\otimes{\cal K}_X^{\otimes 2})\ ,\ \ ({\cal M}^\sigma_a)^{ir}_{\varphi=0}= {\cal M}^{st}(E^{\vee},{\cal L}^{\vee}) $$ \end{co} Using Remark 1.2.7, we recover the main result (Theorem 7.3) in [OT5] stated for quaternionic monopoles. \vspace{5mm}\\ {\bf Example:} (R. Plantiko) On ${\Bbb P}^2$ endowed with the standard Fubini-Study metric $g$ consider the $Spin^{U(2)}(4)$-structure $P^u\longrightarrow P_g$ with $c_1(\det(P^u))=4$, $p_1(\bar\delta(P^u))=-3$. It is easy to see that this $Spin^{U(2)}(4)$-structure is associated with the pair $(P^{\rm can}\longrightarrow P_g,E)$, where $E$ is a $U(2)$-bundle with $c_2(E)=13$, $c_1(E)=7$. Therefore $E\otimes K$ has $c_1(E\otimes K)=1$, $c_2( E\otimes K)=1$. Using Remark 2.1.4 it is easy to see that any stable oriented pair $({\cal F},\varphi)$ of type $(E\otimes K, {\cal O}(1))$ with $\varphi\ne 0$ fits in an exact sequence of the form $$ 0\longrightarrow {\cal O} \textmap{\varphi} {\cal F}\longrightarrow {\cal O}(1)\otimes J_{z_\varphi}\longrightarrow 0\ ,$$ where $z_\varphi\in{\Bbb P}^2$, $c\in{\Bbb C}$ and ${\cal F}={\cal T}_{{\Bbb P}^2}(-1)$ is the unique stable bundle with $c_1=c_2=1$. Moreover, two oriented pairs $({\cal F},\varphi)$, $({\cal F},\varphi')$ define the same point in the moduli space of stable oriented pairs of type $(E\otimes K,{\cal O}(1))$ if and only if $\varphi'=\pm \varphi$. Therefore $${\cal M}^{st}(E\otimes K,{\cal O}(1))=\qmod{H^0({\cal F})}{\pm {\rm id}}\simeq \qmod{{\Bbb C}^3}{\pm{\rm id}}$$ Studying the local models of the moduli space one can check that the above identification is a complex analytic isomorphism. On the other hand every polystable oriented pair of type $ (E\otimes K,{\cal O}(1))$ is stable and there is no polystable oriented pair of type $(E^{\vee},{\cal O}(-7))$. This shows that $${\cal M}^\sigma_a\simeq\qmod{{\Bbb C}^3}{\pm{\rm id}} \ $$ for every integrable connection $a\in{\cal A}(\det(P^u))$. The quotient $\qmod{{\Bbb C}^3}{\pm{\rm id}}$ has a natural compactification ${\cal C}:=\qmod{{\Bbb P}^3}{\langle\iota\rangle}$, where $\iota$ is the involution $$[x_0,x_1,x_2,x_3]\longmapsto [x_0,-x_1,-x_2,-x_3]\ .$$ ${\cal C}$ can be identified with cone over the image of ${\Bbb P}^2$ under the Veronese map $v_2:{\Bbb P}^2\longrightarrow {\Bbb P}^5$. This compactification coincides with the {\it Uhlenbeck compactification} of the moduli space [T1], [T2]. Let now $\sigma':P'^u\longrightarrow P_g$ be the $Spin^{U(2)}(4)$-structure in ${\Bbb P}^2$ with $\det(P'^u)=\det(P^u)$, $p_1(\delta(P'^u))=+1$. It is easy to see by the same method that ${\cal M}^{\sigma'}_a$ consists of only one point, which is the {\it abelian} solution associated with the {\it stable} oriented pair $({\cal O}\oplus{\cal O}(1),{\rm id}_{\cal O})$. Via the isomorphism explained in Proposition 1.2.15, ${\cal M}^{\sigma'}_a$ corresponds to the moduli space of solutions of the (abelian) twisted Seiberg-Witten equations associated with the canonical $Spin^c(4)$-structure and the positive chamber (see [OT6]). Therefore \begin{pr} The Uhlenbeck compactification of the moduli space ${\cal M}^\sigma_a$ can be identified with the cone ${\cal C}$ over the image of ${\Bbb P}^2$ under the Veronese map $v_2$. The vertex of the cone corresponds to the unique Donaldson point. The base of the cone corresponds to the space ${\cal M}^{\sigma'}_a\times{\Bbb P}^2$ of ideal solutions concentrated in one point. The moduli space ${\cal M}^{\sigma'}_a$ consists of only one abelian point. \end{pr} \newpage \centerline{\large{\bf References}} \vspace{6 mm} \parindent 0 cm [AHS] Atiyah M., Hitchin N. J., Singer I. M.: {\it Selfduality in four-dimensional Riemannian geometry}, Proc. R. Lond. A. 362, 425-461 (1978) [B] Bradlow, S. B.: {\it Special metrics and stability for holomorphic bundles with global sections}, J. Diff. Geom. 33, 169-214 (1991) [D] Donaldson, S.: {\it Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles}, Proc. London Math. Soc. 3, 1-26 (1985) [DK] Donaldson, S.; Kronheimer, P.B.: {\it The Geometry of four-manifolds}, Oxford Science Publications 1990 [FU] Freed D. S. ; Uhlenbeck, K.: {\it Instantons and Four-Manifolds.} Springer-Verlag 1984 [GS] Guillemin, V.; Sternberg, S.: {\it Birational equivalence in the symplectic category}, Inv. math. 97, 485-522 (1989) [HH] Hirzebruch, F.; Hopf, H.: {\it Felder von Fl\"achenelementen in 4-dimensiona\-len 4-Mannigfaltigkeiten}, Math. Ann. 136 (1958) [H] Hitchin, N.: {\it Harmonic spinors}, Adv. in Math. 14, 1-55 (1974) [HKLR] Hitchin, N.; Karlhede, A.; Lindstr\"om, U.; Ro\v cek, M.: {\it Hyperk\"ahler metrics and supersymmetry}, Commun.\ Math.\ Phys. (108), 535-589 (1987) [K] Kobayashi, S.: {\it Differential geometry of complex vector bundles}, Princeton University Press 1987 [KM] Kronheimer, P.; Mrowka, T.: {\it The genus of embedded surfaces in the projective plane}, Math. Res. Letters 1, 797-808 (1994) [LL] Li, T.; Liu, A.: {\it General wall crossing formula}, Math. Res. Lett. 2, 797-810 (1995). [LM] Labastida, J. M. F.; Marino, M.: {\it Non-abelian monopoles on four manifolds}, Preprint, Departamento de Fisica de Particulas, Santiago de Compostela, April (1995) [La] Larsen, R.: {\it Functional analysis, an introduction}, Marcel Dekker, Inc., New York, 1973 [LMi] Lawson, H. B. Jr.; Michelson, M. L.: {\it Spin Geometry}, Princeton University Press, New Jersey, 1989 [LT] L\"ubke, M.; Teleman, A.: {\it The Kobayashi-Hitchin correspondence}, World Scientific Publishing Co. 1995 [M] Miyajima, K.: {\it Kuranishi families of vector bundles and algebraic description of the moduli space of Einstein-Hermitian connections}, Publ. R.I.M.S. Kyoto Univ. 25, 301-320 (1989) [MFK] Mumford, D,; Fogarty, J.; Kirwan, F.: {\it Geometric invariant theory}, Springer Verlag, 1994 [OST] Okonek, Ch.; Schmitt, A.; Teleman, A.: {\it Master spaces for stable pairs}, Preprint, alg-geom/9607015 [OT1] Okonek, Ch.; Teleman, A.: {\it The Coupled Seiberg-Witten Equations, Vortices, and Moduli Spaces of Stable Pairs}, Int. J. Math. Vol. 6, No. 6, 893-910 (1995) [OT2] Okonek, Ch.; Teleman, A.: {\it Les invariants de Seiberg-Witten et la conjecture de Van De Ven}, Comptes Rendus Acad. Sci. Paris, t. 321, S\'erie I, 457-461 (1995) [OT3] Okonek, Ch.; Teleman, A.: {\it Seiberg-Witten invariants and rationality of complex surfaces}, Math. Z., to appear [OT4] Okonek, Ch.; Teleman, A.: {\it Quaternionic monopoles}, Comptes Rendus Acad. Sci. Paris, t. 321, S\'erie I, 601-606 (1995) [OT5] Ch, Okonek.; Teleman, A.: {\it Quaternionic monopoles}, Commun.\ Math.\ Phys., Vol. 180, Nr. 2, 363-388, (1996) [OT6] Ch, Okonek.; Teleman, A.: {\it Seiberg-Witten invariants for manifolds with $b_+=1$, and the universal wall crossing formula}, Int. J. Math., to appear [PT1] Pidstrigach, V.; Tyurin, A.: {\it Invariants of the smooth structure of an algebraic surface arising from the Dirac operator}, Russian Acad. Izv. Math., Vol. 40, No. 2, 267-351 (1993) [PT2] Pidstrigach, V.; Tyurin, A.: {\it Localisation of the Donaldson invariants along the Seiberg-Witten classes}, Russian Acad. Izv. , to appear [T1] Teleman, A. :{\it Non-abelian Seiberg-Witten theory}, Habilitationsschrift, Universit\"at Z\"urich, 1996 [T2] Teleman, A. :{\it Moduli spaces of $PU(2)$-monopoles}, Preprint, Universit\"at Z\"urich, 1996 [W] Witten, E.: {\it Monopoles and four-manifolds}, Math. Res. Letters 1, 769-796 (1994) \vspace{0.3cm}\\ Author's address : % Institut f\"ur Mathematik, Universit\"at Z\"urich, Winterthu\-rerstr. 190, CH-8057 Z\"urich, {\bf e-mail}: [email protected]\\ \hspace*{2.4cm} and Department of Mathematics, University of Bucharest.\\ \end{document}

9609

alg-geom/9609006

en

https://arxiv.org/abs/alg-geom/9609006

alg-geom/9609006

Klaus Altmann

Klaus Altmann

One-parameter families containing three-dimensional toric Gorenstein singularities

LaTeX 2.09; 16 pages; uses pb-diagram.sty

For affine toric varieties, the vector space T1 (containing the infinitesimal deformations) will be interpreted via Minkowski summands of cross cuts of the defining polyhedral cone. This result will be applied to study the deformation theory of (in particular non-isolated) three-dimensional Gorenstein singularities.

alg-geom

\section{Introduction}\label{Int} \neu{Int-1} Let $\sigma$ be a rational, polyhedral cone. It induces a (normal) affine toric variety $Y_\sigma$ which may have singularities. We would like to investigate its deformation theory. The vector space $T^1_Y$ of infinitesimal deformations is multigraded, and its homogeneous pieces can be determined by combinatorial formulas developed in \cite{T1}.\\ If $Y_\sigma$ only has an isolated Gorenstein singularity, then we can say even more (cf.\ \cite{Tohoku}, \cite{T2}): $T^1$ is concentrated in one single multidegree, the corresponding homogeneous piece allows an elementary geometric description in terms of Minkowski summands of a certain lattice polytope, and it is even possible (cf.\ \cite{versal}) to obtain the entire versal deformation of $Y_\sigma$.\\ \par \neu{Int-2} The first aim of the present paper is to provide a geometric interpretation of the $T^1$-formula for arbitrary toric singularities in every multidegree. This can be done again in terms of Minkowski summands of certain polyhedra. However, they neither need to be compact, nor do their vertices have to be contained in the lattice anymore (cf.\ \zitat{T1}{7}).\\ In \cite{Tohoku} we have studied so-called toric deformations only existing in negative (i.e.\ $\in -\sigma^{\scriptscriptstyle\vee}$) multidegrees. They are genuine deformations with smooth parameter space, and they are characterized by the fact that their total space is still toric. Now, having a new description of $T^1_Y$, we will describe in Theorem \zitat{Gd}{3} the Kodaira-Spencer map in these terms.\\ Moreover, using partial modifications of our singularity $Y_\sigma$, we extend in \zitat{Gd}{5} the construction of genuine deformations to non-negative degrees. Despite the fact that the total spaces are no longer toric, we can still describe them and their Kodaira-Spencer map combinatorially.\\ \par \neu{Int-3} Afterwards, we focus on three-dimensional, toric Gorenstein singularities. As already mentioned, everything is known in the isolated case. However, as soon as $Y_\sigma$ contains one-dimensional singularities (which then have to be of transversal type A$_k$), the situation changes dramatically. In general, $T^1_Y$ is spread into infinitely many multidegrees. Using our geometric description of the $T^1$-pieces, we detect in \zitat{3G}{3} all non-trivial ones and determine their dimension (which will be one in most cases). The easiest example of that kind is the cone over the weighted projective plane $I\!\!P(1,2,3)$ (cf.\ \zitat{3G}{4}).\\ At least at the moment, it seems to be impossible to describe the entire versal deformation; it is an infinite-dimensional space. However, the infinitesimal deformations corresponding to the one-dimensional homogeneous pieces of $T^1_Y$ are unobstructed, and we lift them in \zitat{3G}{5} to genuine one-parameter families. Since the corresponding multidegrees are in general non-negative, this can be done using the construction introduced in \zitat{Gd}{5}. See section \zitat{3G}{8} for a corresponding sequel of example $I\!\!P(1,2,3)$.\\ Those one-parameter families form a kind of skeleton of the entire versal deformation. The most important open questions are the following: Which of them belong to a common irreducible component of the base space? And, how could those families be combined to find a general fiber (a smoothing of $Y_\sigma$) of this component? The answers to these questions would provide important information about three-dimensional flips.\\ \par \section{Visualizing $T^1$}\label{T1} \neu{T1-1} {\em Notation:} As usual when dealing with toric varieties, denote by $N$, $M$ two mutually dual lattices (i.e.\ finitely generated, free abelian groups), by $\langle\,,\,\rangle:N\times M\toZ\!\!\!Z$ their perfect pairing, and by $N_{I\!\!R}$, $M_{I\!\!R}$ the corresponding $I\!\!R$-vector spaces obtained by extension of scalars.\\ Let $\sigma\subseteq N_{I\!\!R}$ be the polyhedral cone with apex in $0$ given by the fundamental generators $a^1,\dots,a^M\in N$. They are assumed to be primitive, i.e.\ they are not proper multiples of other elements from $N$. We will write $\sigma=\langle a^1,\dots,a^M\rangle$.\\ The dual cone $\sigma^{\scriptscriptstyle\vee}:=\{r\in M_{I\!\!R}\,|\; \langle\sigma,r\rangle\geq 0\}$ is given by the inequalities assigned to $a^1,\dots,a^M$. Intersecting $\sigma^{\scriptscriptstyle\vee}$ with the lattice $M$ yields a finitely generated semigroup. Denote by $E\subseteq\sigma^{\scriptscriptstyle\vee}\cap M$ its minimal generating set, the so-called Hilbert basis. Then, the affine toric variety $Y_\sigma:=\mbox{Spec}\,I\!\!\!\!C[\sigma^{\scriptscriptstyle\vee}\cap M]\subseteq\,I\!\!\!\!C^E$ is given by equations assigned to the linear dependencies among elements of $E$. See \cite{Oda} for a detailed introduction into the subject of toric varieties.\\ \par \neu{T1-2} Most of the relevant rings and modules for $Y_\sigma$ are $M$-(multi)graded. So are the modules $T^i_Y$, which are important for describing infinitesimal deformations and their obstructions. Let $R\in M$, then in \cite{T1} and \cite{T2} we have defined the sets \[ E_j^R:=\{ r\in E\,|\; \langle a^j,r\rangle<\langle a^j,R\rangle\} \quad (j=1,\dots,M)\,. \] They provide the main tool for building a complex $\mbox{span}(E^R)_{\bullet}$ of free Abelian groups with the usual differentials via \[ \mbox{span}(E^R)_{-k} := \!\!\bigoplus_{\begin{array}{c} \tau<\sigma \mbox{ face}\\ \mbox{dim}\, \tau=k \end{array}} \!\!\!\!\!\mbox{span}(E^R_{\tau})\quad \mbox{with} \quad \renewcommand{\arraystretch}{1.5} \begin{array}[t]{rcl} E_0^R &:=& \bigcup_{j=1}^N E_j^R\; ,\mbox{ and}\\ E^R_{\tau} &:=& \bigcap_{a^j \in \tau} E_j^R \; \mbox{ for faces } \tau < \sigma\,. \end{array} \] {\bf Theorem:} (cf.\ \cite{T1}, \cite{T2}) {\em For $i=1$ and, if $Y_\sigma$ is additionally smooth in codimension two, also for $i=2$, the homogeneous pieces of $T^i_Y$ in degree $-R$ are \[ T^i_Y(-R)=H^i\Big(\mbox{\em span}(E^R)_\bullet^\ast\otimes_{Z\!\!\!Z}\,I\!\!\!\!C\Big)\,. \vspace{-1ex} \] } \par In particular, to obtain $T^1_Y(-R)$, we need to determine the vector spaces $\mbox{span}_{\,I\!\!\!\!C}E_j^R$ and $\mbox{span}_{\,I\!\!\!\!C}E_{jk}^R$, where $a^j$, $a^k$ span a two-dimensional face of $\sigma$. The first one is easy to get: \[ \mbox{span}_{\,I\!\!\!\!C}E_j^R = \left\{ \begin{array}{ll} 0 & \mbox{if } \langle a^j ,R\rangle \leq 0\\ (a^j )^\bot & \mbox{if } \langle a^j ,R\rangle =1\\ M_{\,I\!\!\!\!C} & \mbox{if } \langle a^j ,R\rangle \geq 2\, . \end{array} \right. \] The latter is always contained in $(\mbox{span}_{\,I\!\!\!\!C}E_j^R)\cap(\mbox{span}_{\,I\!\!\!\!C}E_k^R)$ with codimension between $0$ and $2$. As we will see in the upcoming example, its actual size reflects the infinitesimal deformations of the two-dimensional cyclic quotient singularity assigned to the plane cone spanned by $a^j$, $a^k$. (These singularities are exactly the transversal types of the two-codimensional ones of $Y_\sigma$.)\\ \par \neu{T1-3} {\bf Example:} If $Y(n,q)$ denotes the two-dimensional quotient of $\,I\!\!\!\!C^2$ by the $^{\kdZ\!\!\!Z}\!\!/_{\!\!\displaystyle nZ\!\!\!Z}$-action via $\left(\!\begin{array}{cc}\xi& 0\\ 0& \xi^q\end{array}\!\right)$ ($\xi$ is a primitive $n$-th root of unity), then $Y(n,q)$ is a toric variety and may be given by the cone $\sigma=\langle(1,0); (-q,n)\rangle\subseteq I\!\!R^2$. The set $E\subseteq \sigma^{\scriptscriptstyle\vee}\cap Z\!\!\!Z^2$ consists of the lattice points $r^0,\ldots,r^w$ along the compact faces of the boundary of $\mbox{conv}\big((\sigma^{\scriptscriptstyle\vee}\setminus\{0\})\capZ\!\!\!Z^2\big)$. There are integers $a_v\geq 2$ such that $r^{v-1}+r^{v+1}=a_v\,r^v$ for $v=1,\dots,w-1$. They may be obtained by expanding $n/(n-q)$ into a negative continued fraction (cf.\ \cite{Oda}, \S (1.6)).\\ Assume $w\geq 2$, let $a^1=(1,0)$ and $a^2=(-q,n)$. Then, there are only two sets $E_1^R$ and $E_2^R$ involved, and the previous theorem states \[ T^1_Y(-R) = \left( \left. ^{\displaystyle (\mbox{span}_{\,I\!\!\!\!C}E^R_1)\cap (\mbox{span}_{\,I\!\!\!\!C}E^R_2)}\! \right/ \! {\displaystyle \mbox{span}_{\,I\!\!\!\!C}(E_1^R\cap E_2^R)}\right)^\ast\,. \] Only three different types of $R\inZ\!\!\!Z^2$ provide a non-trivial contribution to $T^1_Y$: \begin{itemize} \item[(i)] $R=r^1$ (or analogously $R=r^{w-1}):\;$ $\mbox{span}_{\,I\!\!\!\!C}E^R_1 =(a^1)^\bot$, $\mbox{span}_{\,I\!\!\!\!C}E^R_2 = \,I\!\!\!\!C^2 \;(\mbox{or }(a^2)^\bot,\,\mbox{if } w=2)$, and $\mbox{span}_{\,I\!\!\!\!C}E^R_{12}=0$. Hence, $\mbox{dim}\, T^1(-R)=1$ (or $=0$, if $w=2$). \item[(ii)] $R=r^v$ $(2\le v\le w-2)$:\quad $\mbox{span}_{\,I\!\!\!\!C}E^R_1 = \mbox{span}_{\,I\!\!\!\!C}E^R_2 = \,I\!\!\!\!C^2\,$, and $\mbox{span}_{\,I\!\!\!\!C}E^R_{12}=0$. Hence, we obtain $\mbox{dim}\, T^1(-R)=2$. \item[(iii)] $R=p\cdot r^v$ ($1\le v\le w-1$, $\;2\le p<a_v$ for $w\ge 3$; or $v=1=w-1$, $\;2\le p\le a_1$ for $w=2$):\quad $\mbox{span}_{\,I\!\!\!\!C}E^R_1 = \mbox{span}_{\,I\!\!\!\!C}E^R_2 = \,I\!\!\!\!C^2\,$, and $\mbox{span}_{\,I\!\!\!\!C}E^R_{12}=\,I\!\!\!\!C\cdot R\,$. In particular, $\mbox{dim}\, T^1(-R)=1$. \vspace{1ex} \end{itemize} \neu{T1-4} {\bf Definition:} {\em For two polyhedra $Q', Q''\subseteq I\!\!R^n$ we define their Minkowski sum as the polyhedron $Q'+Q'':= \{p'+p''\,|\; p'\in Q', p''\in Q''\}$. Obviously, this notion also makes sense for translation classes of polyhedra in arbitrary affine spaces.}\\ \par \begin{center} \unitlength=0.40mm \linethickness{0.4pt} \begin{picture}(330.00,40.00) \put(0.00,0.00){\line(1,0){20.00}} \put(20.00,0.00){\line(1,1){20.00}} \put(40.00,20.00){\line(0,1){20.00}} \put(40.00,40.00){\line(-1,0){20.00}} \put(20.00,40.00){\line(-1,-1){20.00}} \put(0.00,20.00){\line(0,-1){20.00}} \put(80.00,10.00){\line(1,0){20.00}} \put(100.00,10.00){\line(0,1){20.00}} \put(100.00,30.00){\line(-1,-1){20.00}} \put(140.00,30.00){\line(0,-1){20.00}} \put(140.00,10.00){\line(1,1){20.00}} \put(160.00,30.00){\line(-1,0){20.00}} \put(200.00,10.00){\line(1,0){20.00}} \put(260.00,10.00){\line(0,1){20.00}} \put(310.00,10.00){\line(1,1){20.00}} \put(60.00,20.00){\makebox(0,0)[cc]{$=$}} \put(120.00,20.00){\makebox(0,0)[cc]{$+$}} \put(240.00,20.00){\makebox(0,0)[cc]{$+$}} \put(180.00,20.00){\makebox(0,0)[cc]{$=$}} \put(290.00,20.00){\makebox(0,0)[cc]{$+$}} \end{picture} \end{center} Every polyhedron $Q$ is decomposable into the Minkowski sum $Q=Q^{\mbox{\footnotesize c}}+Q^{\infty}$ of a (compact) polytope $Q^{\mbox{\footnotesize c}}$ and the so-called cone of unbounded directions $Q^{\infty}$. The latter one is uniquely determined by $Q$, whereas the compact summand is not. However, we can take for $Q^{\mbox{\footnotesize c}}$ the minimal one - given as the convex hull of the vertices of $Q$ itself. If $Q$ was already compact, then $Q^{\mbox{\footnotesize c}}=Q$ and $Q^{\infty}=0$. \vspace{1ex}\\ {\em A polyhedron $Q'$ is called a Minkowski summand of $Q$ if there is a $Q''$ such that $Q=Q'+Q''$ and if, additionally, $(Q')^{\infty}= Q^{\infty}$.}\\ In particular, Minkowski summands always have the same cone of unbounded directions and, up to dilatation (the factor $0$ is allowed), the same compact edges as the original polyhedron.\\ \par \neu{T1-5} The {\em setup for the upcoming sections} is the following: Consider the cone $\sigma\subseteq N_{I\!\!R}$ and fix some element $R\in M$. Then $\A{R}{}:= [R=1]:= \{a\in N_{I\!\!R}\,|\; \langle a,R\rangle =1\}\subseteq N_{I\!\!R}$ is an affine space; if $R$ is primitive, then it comes with a lattice $\G{R}{}:= [R=1]\cap N$. The assigned vector space is $\A{R}{0}:=[R=0]$; it is always equipped with the lattice $\G{R}{0}:= [R=0]\cap N$. We define the cross cut of $\sigma$ in degree $R$ as the polyhedron \[ Q(R):= \sigma\cap [R=1]\subseteq \A{R}{}\,. \] It has the cone of unbounded directions $Q(R)^{\infty}=\sigma\cap \A{R}{0}\subseteq N_{I\!\!R}$. The compact part $Q(R)^{\mbox{\footnotesize c}}$ is given by its vertices $\bar{a}^j:=a^j/\langle a^j,R\rangle$, with $j$ meeting $\langle a^j,R\rangle\geq 1$. A trivial but nevertheless important observation is the following: The vertex $\bar{a}^j $ is a lattice point (i.e.\ $\bar{a}^j \in \G{R}{}$), if and only if $\langle a^j , R \rangle =1$.\\ Fundamental generators of $\sigma$ contained in $R^\bot$ can still be ``seen'' as edges in $Q(R)^{\infty}$, but those with $\langle \bullet, R\rangle <0$ are ``invisible'' in $Q(R)$. In particular, we can recover the cone $\sigma$ from $Q(R)$ if and only if $R\in \sigma^{\scriptscriptstyle\vee}$.\\ \par \neu{T1-6} Denote by $d^1,\dots,d^N\in R^\bot\subseteq N_{I\!\!R}$ the compact edges of $Q(R)$. Similar to \cite{versal}, \S 2, we assign to each compact 2-face $\varepsilon<Q(R)$ its sign vector $\underline{\varepsilon}\in \{0,\pm 1\}^N$ by \[ \varepsilon_i := \left\{ \begin{array}{cl} \pm 1 & \mbox{if $d^i$ is an edge of $\varepsilon$}\\ 0 & \mbox{otherwise} \end{array} \right. \] such that the oriented edges $\varepsilon_i\cdot d^i$ fit into a cycle along the boundary of $\varepsilon$. This determines $\underline{\varepsilon}$ up to sign, and any choice will do. In particular, $\sum_i \varepsilon_i d^i =0$.\\ \par {\bf Definition:} {\em For each $R\in M$ we define the vector spaces \vspace{-2ex} \[ \renewcommand{\arraystretch}{1.5} \begin{array}{rcl} V(R) &:=& \{ (t_1,\dots,t_N)\, |\; \sum_i t_i \,\varepsilon_i \,d^i =0\; \mbox{ for every compact 2-face } \varepsilon <Q(R)\}\\ W(R) &:=& I\!\!R^{\#\{\mbox{\footnotesize $Q(R)$-vertices not belonging to $N$}\}}\,. \end{array} \vspace{-2ex} \]} Measuring the dilatation of each compact edge, the cone $C(R):=V(R)\cap I\!\!R^N_{\geq 0}$ parametrizes exactly the Minkowski summands of positive multiples of $Q(R)$. Hence, we will call elements of $V(R)$ ``generalized Minkowski summands''; they may have edges of negative length. (See \cite{versal}, Lemma (2.2) for a discussion of the compact case.) The vector space $W(R)$ provides coordinates $s_j $ for each vertex $\bar{a}^j \in Q(R)\setminus N$, i.e.\ $\langle a^j ,R\rangle \geq 2$.\\ \par \neu{T1-7} To each compact edge $d^{jk}=\overline{\bar{a}^j \bar{a}^k }$ we assign a set of equations $G_{jk}$ which act on elements of $V(R)\oplus W(R)$. These sets are of one of the following three types: \begin{itemize} \item[(0)] $G_{jk}=\emptyset$, \item[(1)] $G_{jk} = \{ s_j -s_k=0\}$ provided both coordinates exist in $W(R)$, set $G_{jk}=\emptyset$ otherwise, or \item[(2)] $G_{jk} = \{t_{jk}-s_j=0 ,\; t_{jk}-s_k=0\}$, dropping equations that do not make sense. \end{itemize} Restricting $V(R)\oplus W(R)$ to the (at most) three coordinates $t_{jk}$, $s_j$, $s_k$, the actual choice of $G_{jk}$ is made such that these equations yield a subspace of dimension $1+\mbox{dim}\,T^1_{\langle a^j,a^k\rangle}(-R)$. Notice that the dimension of $T^1(-R)$ for the two-dimensional quotient singularity assigned to the plane cone $\langle a^j,a^k\rangle$ can be obtained from Example \zitat{T1}{3}.\\ \par {\bf Theorem:} {\em The infinitesimal deformations of $Y_\sigma$ in degree $-R$ equal \[ T^1_Y(-R)= \{ (\underline{t},\,\underline{s})\in V_{\,I\!\!\!\!C}(R)\oplus W_{\,I\!\!\!\!C}(R)\,|\; (\underline{t},\,\underline{s}) \mbox{ fulfills the equations } G_{jk}\} \;\big/\; \,I\!\!\!\!C\cdot (\underline{1},\, \underline{1})\,. \vspace{-1ex} \] } In some sense, the vector space $V(R)$ (encoding Minkowski summands) may be considered the main tool to describe infinitesimal deformations. The elements of $W(R)$ can (depending on the type of the $G_{jk}$'s) be either additional parameters, or they provide conditions excluding Minkowski summands not having some prescribed type.\\ If $Y$ is smooth in codimension two, then $G_{jk}$ is always of type (2). In particular, the variables $\underline{s}$ are completely determined by the $\underline{t}$'s, and we obtain the\\ \par {\bf Corollary:} {\em If $Y$ is smooth in codimension two, then $T^1_Y(-R)$ is contained in $V_{\,I\!\!\!\!C}(R) \big/ \,\,I\!\!\!\!C\cdot (\underline{1})$. It is built from those $\underline{t}$ such that $t_{jk}=t_{kl}$ whenever $d^{jk}$, $d^{kl}$ are compact edges with a common non-lattice vertex $\bar{a}^k$ of $Q(R)$.\\ Thus, $T^1_Y(-R)$ equals the set of equivalence classes of those Minkowski summands of $I\!\!R_{\geq 0}\cdot Q(R)$ that preserve up to homothety the stars of non-lattice vertices of $Q(R)$. }\\ \par \neu{T1-8} {\bf Proof:}\quad (of previous theorem)\\ {\em Step 1:}\quad From Theorem \zitat{T1}{2} we know that $T^1_Y(-R)$ equals the complexification of the cohomology of the complex \[ N_{I\!\!R}\rightarrow \oplus_j \left(\mbox{span}_{I\!\!R} E_j^R \right)^\ast \rightarrow \oplus_{\langle a^j ,a^k \rangle <\sigma} \left(\mbox{span}_{I\!\!R} E^R_{jk} \right)^\ast\,. \] According to \zitat{T1}{2}, elements of $\oplus_j \left(\mbox{span}_{I\!\!R} E_j^R \right)^\ast$ can be represented by a family of \[ b^j \in N_{I\!\!R}\; \mbox{ (if } \langle a^j ,R\rangle \geq 2)\quad \mbox{ and } \quad b^j \in N_{I\!\!R}\big/I\!\!R\cdot a^j\; \mbox{ (if } \langle a^j ,R\rangle =1). \] Dividing by the image of $N_{I\!\!R}$ means to shift this family by common vectors $b\in N_{I\!\!R}$. On the other hand, the family $\{b^j \}$ has to map onto $0$ in the complex, i.e.\ for each compact edge $\overline{\bar{a}^j ,\bar{a}^k }<Q$ the functions $b^j $ and $b^k $ are equal on $\mbox{span}_{I\!\!R}E_{jk}^R$. Since \[ (a^j ,a^k )^\bot \subseteq \mbox{span}_{I\!\!R}E_{jk}^R \subseteq (\mbox{span}_{I\!\!R}E_j^R) \cap (\mbox{span}_{I\!\!R}E_k^R)\,, \] we immediately obtain the necessary condition $b^j -b^k \in I\!\!R a^j + I\!\!R a^k $. However, the actual behavior of $\mbox{span}_{I\!\!R}E_{jk}^R$ will require a closer look (in the third step).\\ \par {\em Step 2:}\quad We introduce new ``coordinates'': \begin{itemize} \item $\bar{b}^j := b^j -\langle b^j , R \rangle \,\bar{a}^j \in R^\bot$, being well defined even in the case $\langle a^j , R \rangle =1$; \item $s_j :=-\langle b^j , R\rangle$ for $j$ meeting $\langle a^j , R \rangle \geq 2$ (inducing an element of $W(R)$). \end{itemize} The shift of the $b^j$ by an element $b\in N_{I\!\!R}$ (i.e.\ $(b^j )'=b^j +b$) appears in these new coordinates as \[ \renewcommand{\arraystretch}{1.5} \begin{array}{rcl} (\bar{b}^j )' &=& (b^j )' - \langle (b^j )', R \rangle \,\bar{a}^j \;=\; b^j +b - \langle b^j ,R \rangle \,\bar{a}^j - \langle b,R \rangle\,\bar{a}^j \vspace{-0.5ex}\\ &=& \bar{b}^j + b-\langle b,R \rangle \,\bar{a}^j \,,\\ s_j '&=& -\langle (b^j )',R \rangle \;=\; s_j -\langle b,R\rangle\,. \end{array} \] In particular, an element $b\in R^\bot$ does not change the $s_j$, but shifts the points $\bar{b}^j$ inside the hyperplane $R^\bot$. Hence, the set of the $\bar{b}^j $ should be considered modulo translation inside $R^\bot$ only.\\ On the other hand, the condition $b^j -b^k \in I\!\!R a^j + I\!\!R a^k $ changes into $\bar{b}^j -\bar{b}^k \in I\!\!R \bar{a}^j + I\!\!R \bar{a}^k $ or even $\bar{b}^j -\bar{b}^k \in I\!\!R (\bar{a}^j - \bar{a}^k )$ (consider the values of $R$). Hence, the $\bar{b}^j $'s form the vertices of an at least generalized Minkowski summand of $Q(R)$. Modulo translation, this summand is completely described by the dilatation factors $t_{jk}$ obtained from \[ \bar{b}^j -\bar{b}^k = t_{jk}\cdot (\bar{a}^j - \bar{a}^k )\,. \] Now, the remaining part of the action of $b\in N_{I\!\!R}$ comes down to an action of $\langle b,R\rangle\inI\!\!R$ only: \[ \begin{array}{rcl} t_{jk}' &=& t_{jk} - \langle b,R \rangle \quad \mbox{ and}\\ s_j ' &=& s_j - \langle b,R \rangle \,, \mbox{ as we already know}. \end{array} \] Up to now, we have found that $T^1_Y(-R)\subseteq V_{\,I\!\!\!\!C}(R)\oplus W_{\,I\!\!\!\!C}(R)/(\underline{1},\underline{1})$.\\ \par {\em Step 3:}\quad Actually, the elements $b^j $ and $b^k $ have to be equal on $\mbox{span}_{I\!\!R} E^R_{jk}$, which may be a larger space than just $(a^j ,a^k )^\bot$. To measure the difference we consider the factor $\mbox{span}_{I\!\!R} E^R_{jk}\big/ (a^j ,a^k )^\bot$ contained in the two-dimensional vector space $M_{I\!\!R}\big/ (a^j ,a^k )^\bot =\mbox{span}_{I\!\!R}(a^j ,a^k )^\ast$. Since this factor coincides with the set $\mbox{span}_{I\!\!R}E^{\bar{R}}_{jk}$ assigned to the two-dimensional cone $\langle a^j ,a^k \rangle \subseteq \mbox{span}_{I\!\!R}(a^j ,a^k )$, where $\bar{R}$ denotes the image of $R$ in $\mbox{span}_{I\!\!R}(a^j ,a^k )^\ast$, we may assume that $\sigma=\langle a^1, a^2\rangle$ (i.e.\ $j=1,\,k=2$) represents a two-dimensional cyclic quotient singularity. In particular, we only need to discuss the three cases (i)-(iii) from Example \zitat{T1}{3}:\\ In (i) and (ii) we have $\mbox{span}_{I\!\!R} E^R_{12}=0$, i.e.\ no additional equation is needed. This means $G_{12}=\emptyset$ is of type (0). On the other hand, if $T^1_Y=0$, then the vector space $I\!\!R^3_{(t_{12},s_1,s_2)}\big/I\!\!R\cdot (\underline{1})$ has to be killed by identifying the three variables $t_{12}$, $s_1$, and $s_2$; we obtain type (2).\\ Case (iii) provides $\mbox{span}_{I\!\!R} E^R_{12}=I\!\!R\cdot R$. Hence, as an additional condition we obtain that $b^1$ and $b^2$ have to be equal on $R$. By the definition of $s_j$ this means $s_1=s_2$, and $G_{12}$ has to be of type (1). \hfill$\Box$\\ \par \section{Genuine deformations}\label{Gd} \neu{Gd-1} In \cite{Tohoku} we have studied so-called toric deformations in a given multidegree $-R\in M$. They are genuine deformations in the sense that they are defined over smooth parameter spaces; they are characterized by the fact that the total spaces together with the embedding of the special fiber still belong to the toric category. Despite the fact they look so special, it seems that toric deformations cover a big part of the versal deformation of $Y_\sigma$. They do only exist in negative degrees (i.e.\ $R\in\sigma^{\scriptscriptstyle\vee}\cap M$), but here they form a kind of skeleton. If $Y_\sigma$ is an isolated toric Gorenstein singularity, then toric deformations even provide all irreducible components of the versal deformation (cf.\ \cite{versal}).\\ After a quick reminder of the idea of this construction, we describe the Kodaira-Spencer map of toric deformations in terms of the new $T^1_Y$-formula presented in \zitat{T1}{2}. It is followed by the investigation of non-negative degrees: If $R\notin\sigma^{\scriptscriptstyle\vee}\cap M$, then we are still able to construct genuine deformations of $Y_\sigma$; but they are no longer toric.\\ \par \neu{Gd-2} Let $R\in\sigma^{\scriptscriptstyle\vee}\cap M$. Then, following \cite{Tohoku} \S 3, toric $m$-parameter deformations of $Y_\sigma$ in degree $-R$ correspond to splittings of $Q(R)$ into a Minkowski sum \vspace{-0.5ex} \[ Q(R) \,=\, Q_0 + Q_1 + \dots +Q_m \vspace{-1ex} \] meeting the following conditions: \begin{itemize} \item[(i)] $Q_0\subseteq \A{R}{}$ and $Q_1,\dots,Q_m\in \A{R}{0}$ are polyhedra with $Q(R)^\infty$ as their common cone of unbounded directions. \item[(ii)] Each supporting hyperplane $t$ of $Q(R)$ defines faces $F(Q_0,t),\dots, F(Q_m,t)$ of the indicated polyhedra; their Minkowski sum equals $F\big(Q(R),t\big)$. With at most one exception (depending on $t$), these faces should contain lattice vertices, i.e.\ vertices belonging to $N$. \end{itemize} {\bf Remark:} In \cite{Tohoku} we have distinguished between the case of primitive and non-primitive elements $R\in M$: If $R$ is a multiple of some element of $M$, then $\A{R}{}$ does not contain lattice points at all. In particular, condition (ii) just means that $Q_1,\dots,Q_m$ have to be lattice polyhedra.\\ On the other hand, for primitive $R$, the $(m+1)$ summands $Q_i$ have equal rights and may be put into the same space $\A{R}{}$. Then, their Minkowski sum has to be interpreted inside this affine space.\\ \par If a Minkowski decomposition is given, {\em how do we obtain the assigned toric deformation?}\\ Defining $\tilde{N}:= N\oplus Z\!\!\!Z^m$ (and $\tilde{M}:=M\oplus Z\!\!\!Z^m$), we have to embed the summands as $(Q_0,\,0)$, $(Q_1,\,e^1),\dots, (Q_m,\,e^m)$ into the vector space $\tilde{N}_{I\!\!R}$; $\{e^1,\dots,e^m\}$ denotes the canonical basis of $Z\!\!\!Z^m$. Together with $(Q(R)^\infty,\,0)$, these polyhedra generate a cone ${\tilde{\sigma}}\subseteq \tilde{N}$ containing $\sigma$ via $N\hookrightarrow \tilde{N}$, $a\mapsto (a;\langle a,R\rangle,\dots,\langle a,R\rangle)$. Actually, $\sigma$ equals ${\tilde{\sigma}}\cap N_{I\!\!R}$, and we obtain an inclusion $Y_\sigma\hookrightarrow X_{{\tilde{\sigma}}}$ between the associated toric varieties.\\ On the other hand, $[R,0]:\tilde{N}\to Z\!\!\!Z$ and $\mbox{pr}_{Z\!\!\!Z^m}:\tilde{N}\toZ\!\!\!Z^m$ induce regular functions $f:X_{{\tilde{\sigma}}}\to \,I\!\!\!\!C$ and $(f^1,\dots,f^m):X_{{\tilde{\sigma}}}\to \,I\!\!\!\!C^m$, respectively. The resulting map $(f^1-f,\dots,f^m-f):X_{{\tilde{\sigma}}}\to \,I\!\!\!\!C^m$ is flat and has $Y_\sigma\hookrightarrow X_{{\tilde{\sigma}}}$ as special fiber.\\ \par \neu{Gd-3} Let $R\in\sigma^{\scriptscriptstyle\vee}\cap M$ and $Q(R) = Q_0 + \dots +Q_m$ be a decomposition satisfying (i) and (ii) mentioned above. Denote by $(\bar{a}^j)_i$ the vertex of $Q_i$ induced from $\bar{a}^j\in Q(R)$, i.e.\ $\bar{a}^j=(\bar{a}^j)_0 + \dots + (\bar{a}^j)_m$.\\ \par {\bf Theorem:} {\em The Kodaira-Spencer map of the corresponding toric deformation $X_{{\tilde{\sigma}}}\to \,I\!\!\!\!C^m$ is \[ \varrho: \,I\!\!\!\!C^m\,=\, T_{\,I\!\!\!\!C^m,0} \longrightarrow T^1_Y(-R) \subseteq V_{\,I\!\!\!\!C}(R)\oplus W_{\,I\!\!\!\!C}(R)\big/\,I\!\!\!\!C\cdot (\underline{1},\underline{1}) \] sending $e^i $ onto the pair $[Q_i ,\,\underline{s}^i ]\in V(R)\oplus W(R)$ ($i=1,\dots,m$) with \[ s^i_j :=\left\{ \begin{array}{ll} 0 & \mbox{ if the vertex } (\bar{a}^j)_i \mbox{ of } Q_i \mbox{ belongs to the lattice } N\\ 1 & \mbox{ if } (\bar{a}^j)_i \mbox{ is not a lattice point.} \end{array}\right. \] } \par {\bf Remark:} Setting $e^0:=-(e^1+\dots +e^m )$, we obtain $\varrho (e^0)= [Q_0,\, \underline{s}^0]$ with $\underline{s}^0$ defined similar to $\underline{s}^i $ in the previous theorem.\\ \par \neu{Gd-4} {\bf Proof} (of previous theorem): We would like to derive the above formula for the Kodaira-Spencer map from the more technical one presented in \cite{Tohoku}, Theorem (5.3). Under additional use of \cite{T2} (6.1), the latter one describes $\varrho(e^i)\in T^1_Y(-R)=H^1\big(\mbox{span}_{\,I\!\!\!\!C}(E^R)_\bullet^\ast\big)$ in the following way:\\ Let $E=\{r^1,\dots,r^w\}\subseteq \sigma^{\scriptscriptstyle\vee}\cap M$. Its elements may be lifted via $\tilde{M}\longrightarrow\hspace{-1.5em}\longrightarrow M$ to $\tilde{r}^v\in{\tilde{\sigma}}^{\scriptscriptstyle\vee}\cap\tilde{M}$ ($v=1,\dots,w$); denote their $i$-th entry of the $Z\!\!\!Z^m$-part by $\eta^v_i$, respectively. Then, given elements $v^j\in \mbox{span} E_j^R$, we may represent them as $v^j=\sum_v q^j_v\,r^v$ ($q^j\inZ\!\!\!Z^{E_j^R}$), and $\varrho(e^i)$ assigns to $v^j$ the integer $-\sum_v q^j_v\,\eta_i^v$. Using our notation from \zitat{T1}{8} for $\varrho(e^i)$, this means that $b^j$ sends elements $r^v\in E_j^R$ onto $-\eta_i^v\inZ\!\!\!Z$. \\ By construction of ${\tilde{\sigma}}$, we have inequalities \[ \Big\langle \big( (\bar{a}^j)_0,\,0\big),\, \tilde{r}^v \Big\rangle \geq 0 \quad\mbox{ and }\quad \Big\langle \big( (\bar{a}^j)_i,\,e^i\big),\, \tilde{r}^v \Big\rangle \geq 0 \quad (i=1,\dots,m) \] summing up to $\big\langle \bar{a}^j,\, r^v \big\rangle = \big\langle \big( \bar{a}^j,\,\underline{1}\big),\, \tilde{r}^v \big\rangle \geq 0$. On the other hand, the fact $r^v\in E_j^R$ is equivalent to $\big\langle \bar{a}^j,\, r^v \big\rangle <1$. Hence, whenever $(\bar{a}^j)_i\in Q_i$ belongs to the lattice, the corresponding inequality ($i=0,\dots,m$) becomes an equality. With at most one exception, this always has to be the case. Hence, \[ \Big\langle (\bar{a}^j)_i,\, r^v \Big\rangle + \eta_i^v = \; \left\{ \begin{array}{ll} 0& \mbox{ if } (\bar{a}^j)_i \in N\\ \langle \bar{a}^j,\, r^v \rangle & \mbox{ if } (\bar{a}^j)_i \notin N \end{array} \right. \quad(i=1,\dots,m) \] meaning that $b^j= (\bar{a}^j)_i\,$ or $\,b^j= (\bar{a}^j)_i - \bar{a}^j$, respectively. By the definitions of $\bar{b}^j$ and $s_j$ given in \zitat{T1}{8}, we are done. \hfill$\Box$\\ \par \neu{Gd-5} Now we treat the case of non-negative degrees; let $R\in M\setminus \sigma^{\scriptscriptstyle\vee}$. The easiest way to solve a problem is to change the question until there is no problem left. We can do so by changing our cone $\sigma$ into some $\tau^R$ such that the degree $-R$ becomes negative. We define \[ \tau:=\tau^R:= \sigma \cap \,[R\geq 0]\quad \mbox{ that is } \quad \tau^{\scriptscriptstyle\vee}=\sigma^{\scriptscriptstyle\vee}+I\!\!R_{\geq 0}\cdot R\,. \] The cone $\tau$ defines an affine toric variety $Y_\tau$. Since $\tau\subseteq\sigma$, it comes with a map $g:Y_\tau\to Y_\sigma$, i.e.\ $Y_\tau$ is an open part of a modification of $Y_\sigma$. The important observation is \[ \renewcommand{\arraystretch}{1.5} \begin{array}{rcccl} \tau \cap \,[R=0] &=& \sigma\cap\, [R=0] &=& Q(R)^\infty\quad \mbox{ and}\\ \tau \cap \,[R=1] &=& \sigma\cap\, [R=1] &=& Q(R)\;, \end{array} \] implying $T^1_{Y_\tau}(-R)=T^1_{Y_\sigma}(-R)$ by Theorem \zitat{T1}{7}. Moreover, even the genuine toric deformations $X_{\tilde{\tau}}\to\,I\!\!\!\!C^m$ of $Y_\tau$ carry over to $m$-parameter (non-toric) deformations $X\to\,I\!\!\!\!C^m$ of $Y_\sigma$:\\ \par {\bf Theorem:} {\em Each Minkowski decomposition $Q(R) = Q_0 + Q_1 + \dots +Q_m$ satisfying (i) and (ii) of \zitat{Gd}{2} provides an $m$-parameter deformation $X\to\,I\!\!\!\!C^m$ of $Y_\sigma$. Via some birational map $\tilde{g}:X_{\tilde{\tau}}\to X$ it is compatible with the toric deformation $X_{\tilde{\tau}}\to \,I\!\!\!\!C^m$ of $Y_\tau$ presented in \zitat{Gd}{2}. \[ \dgARROWLENGTH=0.4em \begin{diagram} \node[2]{\,I\!\!\!\!C^m} \arrow[3]{e,t}{\mbox{\footnotesize id}} \node[3]{\,I\!\!\!\!C^m}\\ \node{X_{\tilde{\tau}}} \arrow{ne} \arrow[3]{e,t}{\tilde{g}} \node[3]{X} \arrow[2]{e} \arrow{ne} \node[2]{Z_{{\tilde{\sigma}}}}\\[2] \node{Y_\tau} \arrow[2]{n} \arrow[3]{e,t}{g} \node[3]{Y_\sigma} \arrow[2]{n} \arrow[2]{ne} \end{diagram} \] The total space $X$ is not toric anymore, but it sits via birational maps between $X_{\tilde{\tau}}$ and some affine toric variety $Z_{{\tilde{\sigma}}}$ still containing $Y_\sigma$ as a closed subset. }\\ \par \neu{Gd-6} {\bf Proof:} First, we construct $\tilde{N}$, $\tilde{M}$, and ${\tilde{\tau}}\subseteq\tilde{N}_{I\!\!R}$ by the recipe of \zitat{Gd}{2}. In particular, $N$ is contained in $\tilde{N}$, and the projection $\pi:\tilde{M}\to M$ sends $[r;g_1,\dots,g_m]$ onto $r+(\sum_i g_i)\,R$. Defining ${\tilde{\sigma}}:= {\tilde{\tau}} + \sigma$ (hence ${\tilde{\sigma}}^{\scriptscriptstyle\vee}={\tilde{\tau}}^{\scriptscriptstyle\vee}\cap \pi^{-1}(\sigma^{\scriptscriptstyle\vee})$), we obtain the commutative diagram \[ \dgARROWLENGTH=0.5em \begin{diagram} \node{\,I\!\!\!\!C[{\tilde{\tau}}^{\scriptscriptstyle\vee}\cap \tilde{M}]} \arrow{s,r}{\pi} \node[3]{\,I\!\!\!\!C[{\tilde{\sigma}}^{\scriptscriptstyle\vee}\cap \tilde{M}]} \arrow{s,r}{\pi} \arrow[3]{w}\\ \node{\,I\!\!\!\!C[\tau^{\scriptscriptstyle\vee}\cap M]} \node[3]{\,I\!\!\!\!C[\sigma^{\scriptscriptstyle\vee}\cap M]} \arrow[3]{w} \end{diagram} \] with surjective vertical maps. The canonical elements $e_1,\dots,e_m\inZ\!\!\!Z^m\subseteq\tilde{M}$ together with $[R;0]\in\tilde{M}$ are preimages of $R\in M$. Hence, the corresponding monomials $x^{e_1},\dots,x^{e_m},x^{[R,0]}$ in the semigroup algebra $\,I\!\!\!\!C[{\tilde{\tau}}^{\scriptscriptstyle\vee}\cap \tilde{M}]$ (called $f^1,\dots,f^m,f$ in \zitat{Gd}{2}) map onto $x^R\in\,I\!\!\!\!C[\tau^{\scriptscriptstyle\vee}\cap M]$ which is not regular on $Y_\sigma$. We define $Z_{\tilde{\sigma}}$ as the affine toric variety assigned to ${\tilde{\sigma}}$ and $X$ as \[ X:=\mbox{Spec}\,B \quad \mbox{ with } \quad B:=\,I\!\!\!\!C[{\tilde{\sigma}}^{\scriptscriptstyle\vee}\cap\tilde{M}][f^1-f,\dots,f^m-f]\subseteq \,I\!\!\!\!C[{\tilde{\tau}}^{\scriptscriptstyle\vee}\cap\tilde{M}]\,. \] That means, $X$ arises from $X_{\tilde{\tau}}$ by eliminating all variables except those lifted from $Y_\sigma$ or the deformation parameters themselves. By construction of $B$, the vertical algebra homomorphisms $\pi$ induce a surjection $B\longrightarrow\hspace{-1.5em}\longrightarrow \,I\!\!\!\!C[\sigma^{\scriptscriptstyle\vee}\cap M]$.\\ \par {\em Lemma:} Elements of $\,I\!\!\!\!C[{\tilde{\tau}}^{\scriptscriptstyle\vee}\cap\tilde{M}]$ may uniquely be written as sums \vspace{-1ex} \[ \sum_{(v_1,\dots,v_m)\inI\!\!N^m} c_{v_1,\dots,v_m}\cdot (f^1-f)^{v_1}\cdot\dots\cdot (f^m-f)^{v_m} \vspace{-1.5ex} \] with $c_{v_1,\dots,v_m}\in\,I\!\!\!\!C[{\tilde{\tau}}^{\scriptscriptstyle\vee}\cap\tilde{M}]$ such that $s-e_i\notin {\tilde{\tau}}^{\scriptscriptstyle\vee}$ ($i=1,\dots,m$) for any of its monomial terms $x^s$. Moreover, those sums belong to the subalgebra $B$, if and only if their coefficients $c_{v_1,\dots,v_m}$ do. \vspace{1ex}\\ {\em Proof: (a) Existence.} Let $s-e_i\in{\tilde{\tau}}^{\scriptscriptstyle\vee}$ for some $s,i$. Then, with $s^\prime:=s-e_i+[R,0]$ we obtain \vspace{-1ex} \[ x^s = x^{s^\prime} + x^{s-e_i}\,(x^{e_i}-x^{[R,0]}) = x^{s^\prime} + x^{s-e_i}\,(f^i-f)\,. \vspace{-1ex} \] Since $e_i=1$ and $[R,0]=0$ if evaluated on $(Q_i,e^i)\subseteq{\tilde{\tau}}$, this process eventually stops. \vspace{1ex}\\ {\em (b) $B$-Membership.} For the previous reduction step we have to show that if $s\in\,I\!\!\!\!C[{\tilde{\sigma}}^{\scriptscriptstyle\vee}\cap\tilde{M}]$, then the same is true for $s^\prime$ and $s-e_i$. Since $\pi(s^\prime)=\pi(s)\in\sigma^{\scriptscriptstyle\vee}$, this is clear for $s^\prime$. It remains to check that $\pi(s-e_i)\in\sigma^{\scriptscriptstyle\vee}$. Let $a\in\sigma$ be an arbitrary test element; we distinguish two cases:\\ Case 1: $\langle a,R\rangle\geq 0$. Then $a$ belongs to the subcone $\tau$, and $\pi(s-e_i)\in \tau^{\scriptscriptstyle\vee}$ yields $\langle a, \pi(s-e_i)\rangle\geq 0$.\\ Case 2: $\langle a,R\rangle\leq 0$. This fact implies $\langle a, \pi(s-e_i)\rangle = \langle a, s\rangle - \langle a,R\rangle \geq \langle a, s\rangle \geq 0$. \vspace{1ex}\\ {\em (c) Uniqueness.} Let $p:=\sum c_{v_1,\dots,v_m}\cdot (f^1-f)^{v_1}\cdot\dots\cdot (f^m-f)^{v_m}$ (meeting the above conditions) be equal to $0$ in $\,I\!\!\!\!C[{\tilde{\tau}}^{\scriptscriptstyle\vee}\cap\tilde{M}]$. Using the projection $\pi:\tilde{M}\to M$, everything becomes $M$-graded. Since the factors $(f^i-f)$ are homogeneous (of degree $R$), we may assume this fact also for $p$, hence for its coefficients $c_{v_1,\dots,v_m}$. \vspace{0.5ex}\\ {\em Claim:} These coefficients are just monomials. Indeed, if $s,s^\prime\in{\tilde{\tau}}^{\scriptscriptstyle\vee}$ had the same image via $\pi$, then we could assume that some $e_i$-coordinate of $s^\prime$ would be smaller than that of $s$. Hence, $s-e_i$ would still be equal to $s$ on $(Q_0,0)$ and on any $(Q_j,e^j)$ ($j\neqi$), but even greater or equal than $s^\prime$ on $(Q_i,e^i)$. This would imply $s-e_i\in{\tilde{\tau}}^{\scriptscriptstyle\vee}$, contradicting our assumption for $p$. \vspace{0.5ex}\\ Say $c_{v_1,\dots,v_m}=\lambda_{v_1,\dots,v_m}\,x^\bullet$; we use the projection $\tilde{M}\toZ\!\!\!Z^m$ for carrying $p$ into the ring $\,I\!\!\!\!C[Z\!\!\!Z^m]=\,I\!\!\!\!C[y_1^{\pm 1},\dots,y_m^{\pm 1}]$. The elements $x^\bullet$, $f^i$, $f$ map onto $y^\bullet$, $y_i$, and $1$, respectively. Hence, $p$ turns into \vspace{-1ex} \[ \bar{p}=\sum_{(v_1,\dots,v_m)\inI\!\!N^m} \lambda_{v_1,\dots,v_m}\cdot y^\bullet\cdot(y_1-1)^{v_1}\cdot\dots\cdot (y_m-1)^{v_m}\,. \vspace{-1ex} \] By induction through $I\!\!N^m$, we obtain that vanishing of $\bar{p}$ implies the vanishing of its coefficients: Replace $y_i-1$ by $z_i$, and take partial derivatives. \hfill$(\Box)$\\ \par Now, we can easily see that $X\to\,I\!\!\!\!C^m$ is flat and has $Y_\sigma$ as special fiber: The previous lemma means that for $k=0,\dots,m$ we have inclusions \[ {\displaystyle B}\big/_{\displaystyle (f^1-f,\dots,f^k-f)} \quad\raisebox{-0.5ex}{$\hookrightarrow\;$}\quad {\displaystyle \,I\!\!\!\!C[{\tilde{\tau}}^{\scriptscriptstyle\vee}\cap\tilde{M}]\,}\big/_{\displaystyle (f^1-f,\dots,f^k-f)}\,. \] The values $k < m$ yield that $(f^1-f,\dots,f^m-f)$ forms a regular sequence even in the subring $B$, meaning that $X\to\,I\!\!\!\!C^m$ is flat. With $k=m$ we obtain that the surjective map $B/(f^1-f,\dots,f^m-f)\to\,I\!\!\!\!C[\sigma^{\scriptscriptstyle\vee}\cap M]$ is also injective. \hfill$\Box$\\ \par \section{Three-dimensional toric Gorenstein singularities}\label{3G} \neu{3G-1} By \cite{Ish}, Theorem (7.7), toric Gorenstein singularities always arise from the following construction: Assume we are given a {\em lattice polytope $P\subseteq I\!\!R^n$}. We embed the whole space (including $P$) into height one of $N_{I\!\!R}:=I\!\!R^n\oplusI\!\!R$ and take for $\sigma$ the cone generated by $P$; denote by $M_{I\!\!R}:=(I\!\!R^n)^\ast\oplusI\!\!R$ the dual space and by $N$, $M$ the natural lattices. Our polytope $P$ may be recovered from $\sigma$ as \[ P\, =\, Q(R^\ast)\subseteq\A{R^\ast}{}\quad \mbox{ with} \quad R^\ast:=[\underline{0},1]\in M\,. \hspace{-3em} \raisebox{-35mm}{ \unitlength=0.5mm \linethickness{0.6pt} \begin{picture}(130,80) \thinlines \put(0.00,30.00){\line(1,0){80.00}} \put(0.00,30.00){\line(5,3){42.00}} \put(42.00,55.33){\line(1,0){80.00}} \put(122.00,55.33){\line(-5,-3){42.00}} \put(10.00,5.00){\line(3,5){15.00}} \put(33.00,43.00){\line(3,5){19.67}} \put(10.00,5.00){\line(1,1){25.00}} \put(38.00,35.00){\line(1,1){27.00}} \put(10.00,5.00){\line(5,3){42.00}} \put(60.00,35.00){\line(5,3){35.00}} \put(10.00,5.00){\line(2,1){50.00}} \put(87.00,46.00){\line(2,1){35.00}} \put(10.00,5.00){\line(4,3){33.00}} \put(79.00,54.00){\line(4,3){30.00}} \put(57.00,42.00){\makebox(0,0)[cc]{$P$}} \put(91.00,16.00){\makebox(0,0)[cc]{$\mbox{cone}(P)$}} \put(33.00,43.00){\circle*{2.00}} \put(38.00,35.00){\circle*{2.00}} \put(60.00,35.00){\circle*{2.00}} \put(87.00,46.00){\circle*{2.00}} \put(79.00,54.00){\circle*{2.00}} \thicklines \put(32.67,43.00){\line(2,-3){5.33}} \put(38.00,35.00){\line(1,0){22.00}} \put(60.00,35.00){\line(5,2){27.00}} \put(87.00,46.00){\line(-1,1){8.00}} \put(79.00,54.00){\line(-4,-1){46.00}} \end{picture} } \] The fundamental generators $a^1,\dots,a^M\in\G{R^\ast}{}$ of $\sigma$ coincide with the vertices of $P$. (This involves a slight abuse of notation; we use the same symbol $a^j$ for both $a^j\inZ\!\!\!Z^n$ and $(a^j,1)\in M$.)\\ If $\overline{a^j a^k}$ forms an edge of the polytope, we denote by $\ell(j,k)\inZ\!\!\!Z$ its ``length'' induced from the lattice structure $Z\!\!\!Z^n\subseteqI\!\!R^n$. Every edge provides a two-codimensional singularity of $Y_\sigma$ with transversal type A$_{\ell(j,k)-1}$. In particular, $Y_\sigma$ is smooth in codimension two if and only if all edges of $P$ are primitive, i.e.\ have length $\ell=1$.\\ \par \neu{3G-2} As usual, we fix some element $R\in M$. From \zitat{T1}{6} we know what the vector spaces $V(R)$ and $W(R)$ are; we introduce the subspace \[ V^\prime(R):=\{\underline{t}\in V(R)\,|\; t_{jk}\neq 0 \mbox{ implies } 1\leq \langle a^j,R \rangle = \langle a^k,R \rangle \leq \ell(j,k)\} \] representing Minkowski summands of $Q(R)$ that have killed any compact edge {\em not} meeting the condition $\langle a^j,R \rangle = \langle a^k,R \rangle \leq \ell(j,k)$.\\ \par {\bf Theorem:} {\em For $T^1_Y(-R)$, there are two different types of $R\in M$ to distinguish: \begin{itemize} \item[(i)] If $R\leq 1$ on $P$ (or equivalently $\langle a^j,R\rangle\leq 1$ for $j=1,\dots,M$), then $T^1_Y(-R)=V_{\,I\!\!\!\!C}(R)\big/(\underline{1})$. Moreover, concerning Minkowski summands, we may replace the polyhedron $Q(R)$ by its compact part $P\cap [R=1]$ (being a face of $P$). \item[(ii)] If $R$ does not satisfy the previous condition, then $T^1_Y(-R)=V^\prime(R)$. \vspace{1ex} \end{itemize} } {\bf Proof:} The first case follows from Theorem \zitat{T1}{7} just because $W(R)=0$. For (ii), let us assume there are vertices $a^j$ contained in the affine half space $[R\geq 2]$. They are mutually connected inside this half space via paths along edges of $P$.\\ The two-dimensional cyclic quotient singularities corresponding to edges $\overline{a^j a^k}$ of $P$ are Gorenstein themselves. In the language of Example \zitat{T1}{3} this means $w=2$, and we obtain \[ \mbox{dim}\, T^1_{\langle a^j,a^k \rangle} (-R)\,=\, \left\{ \begin{array}{ll} 1 & \mbox{ if } \langle a^j,R \rangle = \langle a^k,R \rangle =2,\dots,\ell(j,k) \quad\mbox{(case (iii) in \zitat{T1}{3})}\\ 0 & \mbox{ otherwise.} \end{array}\right. \] In particular, $T^1_{\langle a^j,a^k \rangle} (-R)$ cannot be two-dimensional, and (using the notation of \zitat{T1}{7}) the equations $s_j -s_k=0$ belong to $G_{jk}$ whenever $\langle a^j,R \rangle ,\, \langle a^k,R \rangle\geq 2$. This means for elements of \[ T^1_Y\subseteq \Big(V_{\,I\!\!\!\!C}(R)\oplus W_{\,I\!\!\!\!C}(R)\Big)\Big/ \,I\!\!\!\!C\cdot (\underline{1},\underline{1}) \] that all entries of the $W_{\,I\!\!\!\!C}(R)$-part have to be mutually equal, or even zero after dividing by $\,I\!\!\!\!C\cdot (\underline{1},\underline{1})$. Moreover, if not both $\langle a^j,R \rangle$ and $\langle a^k,R \rangle$ equal one, vanishing of $T^1_{\langle a^j,a^k \rangle} (-R)$ implies that $G_{jk}$ also contains the equation $t_{jk} -s_\bullet=0$. \hfill$\Box$\\ \par {\bf Corollary:} {\em Condition \zitat{Gd}{2}(ii) to build genuine deformations becomes easier for toric Gorenstein singularities: $Q_1,\dots,Q_m$ just have to be lattice polyhedra. \vspace{-1ex}}\\ \par {\bf Proof:} If $R\leq 1$ on $P$, then $Q(R)$ itself is a lattice polyhedron. Hence, condition (ii) automatically comes down to this simpler form.\\ In the second case, there is some $W(R)$-part involved in $T^1_Y(-R)$. On the one hand, it indicates via the Kodaira-Spencer map which vertices of which polyhedron $Q_i$ belong to the lattice. On the other, we have observed in the previous proof that the entries of $W(R)$ are mutually equal. This implies exactly our claim. \hfill$\Box$\\ \par \neu{3G-3} In accordance with the title of the section, we focus now on {\em plane lattice polygons $P\subseteq I\!\!R^2$}. The vertices $a^1,\dots,a^M$ are arranged in a cycle. We denote by $d^j:=a^{j+1}-a^j\in \G{R^\ast}{0}$ the edge going from $a^j$ to $a^{j+1}$, and by $\ell(j):=\ell(j,j+1)$ its length ($j\inZ\!\!\!Z/\anZ\!\!\!Z$).\\ Let $s^1,\dots,s^M$ be the fundamental generators of the dual cone $\sigma^{\scriptscriptstyle\vee}$ such that $\sigma\cap(s^j)^\bot$ equals the face spanned by $a^j, a^{j+1}\in\sigma$. In particular, skipping the last coordinate of $s^j$ yields the (primitive) inner normal vector at the edge $d^j$ of $P$. \vspace{-1ex}\\ \par {\bf Remark:} Just for convenience of those who prefer living in $M$ instead of $N$, we show how to see the integers $\ell(j)$ in the dual world: Choose a fundamental generator $s^j$ and denote by $r, r^\prime\in M$ the closest (to $s^j$) elements from the Hilbert bases of the two adjacent faces of $\sigma^{\scriptscriptstyle\vee}$, respectively. Then, $\{R^\ast,s^j\}$ together with either $r$ or $r^\prime$ form a basis of the lattice $M$, and $(r+r^\prime)-\ell(j)\,R^\ast$ is a positive multiple of $s^j$. See the figure in \zitat{3G}{7}.\\ \par In the very special case of plane lattice polygons (or three-dimensional toric Gorenstein singularities), we can describe $T^1_Y$ and the genuine deformations (for fixed $R\in M$) explicitly. First, we can easily spot the degrees carrying infinitesimal deformations: \vspace{-1ex}\\ \par {\bf Theorem:} {\em In general (see the upcoming exceptions), $T^1_Y(-R)$ is non-trivial only for \begin{itemize} \item[(1)] $R=R^\ast\,$ with $\,\mbox{\em dim}\,T^1_Y(-R)=M-3$; \item[(2)] $R= qR^\ast$ ($q\geq 2)\,$ with $\,\mbox{\em dim}\,T^1_Y(-R)= \mbox{\em max}\,\{0\,;\; \#\{j\,|\; q\leq \ell(j)\}-2\,\}$, and \item[(3)] $R=qR^\ast - p\,s^j\,$ with $\,2\leq q\leq \ell(j)$ and $p\inZ\!\!\!Z$ sufficiently large such that $R\notin\mbox{\rm int}(\sigma^{\scriptscriptstyle\vee})$. In this case, $T^1_Y(-R)$ is one-dimensional. \end{itemize} Additional degrees \vspace{-1ex} exist only in the following two (overlapping) exceptional cases: \begin{itemize} \item[(4)] Assume $P$ contains a pair of parallel edges $d^j$, $d^k$, both longer than every other edge. Then $\mbox{\rm dim}\,T^1_Y(-q\,R^\ast)=1$ for $\mbox{\rm max}\{\ell(l)\,|\;l\neqj,k\}<q\leq \mbox{\rm min}\{\ell(j),\ell(k)\}$. \vspace{-1ex} \item[(5)] Assume $P$ contains a pair of parallel edges $d^j$, $d^k$ with distance $d$ ($d:=\langle a^j, s^k\rangle = \langle a^k,s^j\rangle$). If $\ell(k)>d \;(\geq \mbox{\rm max}\{\ell(l)\,|\;l\neqj,k\})$, then $\mbox{\rm dim}\,T^1_Y(-R)=1$ for $R=qR^\ast +p\,s^j$ with $1\leq q\leq\ell(j)$ and $1\leq p\leq \big(\ell(k)-q\big)/d$. \end{itemize} } \par The cases (1), (2), (4), and (5) yield at most finitely many (negative) $T^1_Y$-degrees. Type (3) consists of $\ell(j)\!-\!1$ infinite series to any vertex $a^j\in P$, respectively; up to maybe the leading elements ($R$ might sit on $\partial\sigma^{\scriptscriptstyle\vee}$), they contain only non-negative degrees.\\ \par {\bf Proof:} The previous claims are straight consequences of Theorem \zitat{3G}{2}. Hence, the following short remark should be sufficient: The condition $\langle a^j, R\rangle =\langle a^{j+1},R\rangle$ means $d^j\in R^\bot$. Moreover, if $R\notin Z\!\!\!Z\cdot R^\ast$, then there is at most one edge (or a pair of parallel ones) having this property. \hfill$\Box$\\ \par \neu{3G-4} {\bf Example:} A typical example of a non-isolated, three-dimensional toric Gorenstein singularity is the cone over the weighted projective space $I\!\!P(1,2,3)$. We will use it to demonstrate our calculations of $T^1$ as well as the upcoming construction of genuine one-parameter families. $P$ has the vertices $(-1,-1)$, $(2,-1)$, $(-1,1)$, i.e.\ $\sigma$ is generated from \[ a^1 =(-1,-1;1)\,,\quad a^2=(2,-1;1)\,,\quad a^3=(-1,1;1)\,. \] Since our singularity is a cone over a projective variety, $\sigma^{\scriptscriptstyle\vee}$ appears as a cone over some lattice polygon, too. Actually, in our example, $\sigma$ and $\sigma^{\scriptscriptstyle\vee}$ are even isomorphic. We obtain \[ \sigma^{\scriptscriptstyle\vee}=\langle s^1,s^2,s^3\rangle \quad \mbox{with}\quad s^1=[0,1;1]\,,\; s^2=[-2,-3;1]\,,\; s^3=[1,0;1]\,. \] The Hilbert basis $E\subseteq \sigma^{\scriptscriptstyle\vee}\capZ\!\!\!Z^3$ consists of these three fundamental generators together with \[ R^\ast=[0,0;1]\,, \quad v^1=[-1,-2;1]\,,\quad v^2=[0,-1;1]\,,\quad w=[-1,-1;1]\,. \vspace{1ex} \] \begin{center} \unitlength=0.7mm \linethickness{0.4pt} \begin{picture}(146.00,65.00) \put(10.00,20.00){\circle*{1.00}} \put(10.00,30.00){\circle*{1.00}} \put(10.00,40.00){\circle*{1.00}} \put(10.00,50.00){\circle*{1.00}} \put(10.00,60.00){\circle*{1.00}} \put(20.00,20.00){\circle*{1.00}} \put(20.00,30.00){\circle*{1.00}} \put(20.00,40.00){\circle*{1.00}} \put(20.00,50.00){\circle*{1.00}} \put(20.00,60.00){\circle*{1.00}} \put(30.00,20.00){\circle*{1.00}} \put(30.00,30.00){\circle*{1.00}} \put(30.00,40.00){\circle*{1.00}} \put(30.00,50.00){\circle*{1.00}} \put(30.00,60.00){\circle*{1.00}} \put(40.00,20.00){\circle*{1.00}} \put(40.00,30.00){\circle*{1.00}} \put(40.00,40.00){\circle*{1.00}} \put(40.00,50.00){\circle*{1.00}} \put(40.00,60.00){\circle*{1.00}} \put(50.00,20.00){\circle*{1.00}} \put(50.00,30.00){\circle*{1.00}} \put(50.00,40.00){\circle*{1.00}} \put(50.00,50.00){\circle*{1.00}} \put(50.00,60.00){\circle*{1.00}} \put(110.00,20.00){\circle*{1.00}} \put(110.00,30.00){\circle*{1.00}} \put(110.00,40.00){\circle*{1.00}} \put(110.00,50.00){\circle*{1.00}} \put(110.00,60.00){\circle*{1.00}} \put(120.00,20.00){\circle*{1.00}} \put(120.00,30.00){\circle*{1.00}} \put(120.00,40.00){\circle*{1.00}} \put(120.00,50.00){\circle*{1.00}} \put(120.00,60.00){\circle*{1.00}} \put(130.00,20.00){\circle*{1.00}} \put(130.00,30.00){\circle*{1.00}} \put(130.00,40.00){\circle*{1.00}} \put(130.00,50.00){\circle*{1.00}} \put(130.00,60.00){\circle*{1.00}} \put(140.00,20.00){\circle*{1.00}} \put(140.00,30.00){\circle*{1.00}} \put(140.00,40.00){\circle*{1.00}} \put(140.00,50.00){\circle*{1.00}} \put(140.00,60.00){\circle*{1.00}} \put(20.00,50.00){\line(0,-1){20.00}} \put(20.00,30.00){\line(1,0){30.00}} \put(50.00,30.00){\line(-3,2){30.00}} \put(110.00,20.00){\line(1,2){20.00}} \put(130.00,60.00){\line(1,-1){10.00}} \put(140.00,50.00){\line(-1,-1){30.00}} \put(15.00,25.00){\makebox(0,0)[cc]{$a^1$}} \put(55.00,25.00){\makebox(0,0)[cc]{$a^2$}} \put(25.00,55.00){\makebox(0,0)[cc]{$a^3$}} \put(104.00,15.00){\makebox(0,0)[cc]{$s^2$}} \put(146.00,50.00){\makebox(0,0)[cc]{$s^3$}} \put(130.00,65.00){\makebox(0,0)[cc]{$s^1$}} \put(116.00,42.00){\makebox(0,0)[cc]{$w$}} \put(124.00,27.00){\makebox(0,0)[cc]{$v^1$}} \put(134.00,37.00){\makebox(0,0)[cc]{$v^2$}} \put(128.00,46.00){\makebox(0,0)[cc]{$R^\ast$}} \put(30.00,8.00){\makebox(0,0)[cc]{$\sigma=\mbox{cone}\,(P)$}} \put(125.00,8.00){\makebox(0,0)[cc]{$\sigma^{\scriptscriptstyle\vee}$}} \end{picture} \vspace{-2ex} \end{center} In particular, $Y_\sigma$ has embedding dimension $7$. The edges of $P$ have length $\ell(1)=3$, $\ell(2)=1$, and $\ell(3)=2$. Hence, $Y_\sigma$ contains one-dimensional singularities of transversal type A$_2$ and A$_1$.\\ According to the previous theorem, $Y_\sigma$ admits only infinitesimal deformations of the third type. Their degrees come in three series: \begin{itemize} \item[($\alpha$)] $2R^\ast-p_\alpha\,s^3$ with $p_\alpha\geq 1$. Even the leading element $R^\alpha=[-1,0,1]$ is not contained in~$\sigma^{\scriptscriptstyle\vee}$. \item[($\beta$)] $2R^\ast-p_\beta\,s^1$ with $p_\beta\geq 1$. The leading element equals $R^\beta=v^2=[0,-1,1]$ and sits on the boundary of $\sigma^{\scriptscriptstyle\vee}$. \item[($\gamma$)] $3R^\ast-p_\gamma\,s^1$ with $p_\gamma\geq 2$. The leading element is $R^\gamma=[0,-2,1]\notin\sigma^{\scriptscriptstyle\vee}$. \end{itemize} \begin{center} \unitlength=0.70mm \linethickness{0.4pt} \begin{picture}(80.00,50.50) \put(15.00,10.00){\circle*{1.00}} \put(15.00,20.00){\circle*{1.00}} \put(15.00,30.00){\circle*{1.00}} \put(15.00,40.00){\circle*{1.00}} \put(15.00,50.00){\circle*{1.00}} \put(25.00,10.00){\circle*{1.00}} \put(25.00,20.00){\circle*{1.00}} \put(25.00,30.00){\circle*{1.00}} \put(25.00,40.00){\circle*{1.00}} \put(25.00,50.00){\circle*{1.00}} \put(35.00,10.00){\circle*{1.00}} \put(35.00,20.00){\circle*{1.00}} \put(35.00,30.00){\circle*{1.00}} \put(35.00,40.00){\circle*{1.00}} \put(35.00,50.00){\circle*{1.00}} \put(45.00,10.00){\circle*{1.00}} \put(45.00,20.00){\circle*{1.00}} \put(45.00,30.00){\circle*{1.00}} \put(45.00,40.00){\circle*{1.00}} \put(45.00,50.00){\circle*{1.00}} \put(15.00,10.00){\line(1,2){20.00}} \put(35.00,50.00){\line(1,-1){10.00}} \put(45.00,40.00){\line(-1,-1){30.00}} \put(90.00,30.00){\makebox(0,0)[cc]{$\sigma^{\scriptscriptstyle\vee}\subseteq M_{I\!\!R}$}} \put(25.00,40.00){\circle{2.00}} \put(35.00,30.00){\circle{2.00}} \put(35.00,20.00){\circle{2.00}} \put(21.00,43.00){\makebox(0,0)[cc]{$R^\alpha$}} \put(39.00,27.00){\makebox(0,0)[cc]{$R^\beta$}} \put(39.00,17.00){\makebox(0,0)[cc]{$R^\gamma$}} \end{picture} \vspace{-2ex} \end{center} \par \neu{3G-5} Each degree belonging to type (3) (i.e.\ $R=qR^\ast-p\,s^j$ with $2\leq q \leq \ell(j)$) provides an infinitesimal deformation. To show that they are unobstructed by describing how they lift to genuine one-parameter deformations should be no problem: Just split the polygon $Q(R)$ into a Minkowski sum meeting conditions (i) and (ii) of \zitat{Gd}{2}, then construct ${\tilde{\tau}}$, ${\tilde{\sigma}}$, and $(f^1-f)$ as in \zitat{Gd}{2} and \zitat{Gd}{5}. However, we prefer to present the result for our special case all at once by using new coordinates.\\ Let $P\subseteq \A{R^\ast}{}=I\!\!R^2\times\{1\}\subseteq I\!\!R^3=N_{I\!\!R}$ be a lattice polygon as in \zitat{3G}{3}, let $R=qR^\ast-p\,s^j$ be as just mentioned. Then $\sigma, \tau\subseteq N_{I\!\!R}$ are the cones over $P$ and $P\cap [R\geq 0]$, respectively, and the one-parameter family in degree $-R$ is obtained as follows:\\ \par {\bf Proposition:} {\em The cone ${\tilde{\tau}}\subseteq N_{I\!\!R}\oplusI\!\!R=I\!\!R^4$ is generated by the elements \begin{itemize} \item[(i)] $(a,0)-\langle a,R\rangle\, (\underline{0},1)$, if $a\in P\cap [R\geq 0]$ runs through the vertices from the $R^\bot$-line until $a^j$, \item[(ii)] $(a,0)-\langle a,R\rangle \,(d^j/\ell(j),1)$, if $a\in P\cap [R\geq 0]$ runs from $a^{j+1}$ until the $R^\bot$-line again, and \item[(iii)] $(\underline{0},1)$ and $(d^j/\ell(j),1)$. \end{itemize} The vector space $N_{I\!\!R}$ containing $\sigma$ sits in $N_{I\!\!R}\oplusI\!\!R$ as $N_{I\!\!R}\times\{0\}$. Via this embedding, one obtains ${\tilde{\sigma}}={\tilde{\tau}}+\sigma$ as usual. The monomials $f$ and $f^1$ are given by their exponents $[R,0], [R,1]\in M\oplusZ\!\!\!Z$, respectively. } \begin{center} \unitlength=0.7mm \linethickness{0.4pt} \begin{picture}(200.00,66.00) \thicklines \put(10.00,60.00){\line(1,-5){4.67}} \put(14.67,37.00){\line(6,-5){22.33}} \put(37.00,18.33){\line(3,-1){37.00}} \put(74.00,6.00){\line(6,1){40.00}} \put(114.00,12.67){\line(3,4){27.33}} \put(141.00,49.00){\line(0,1){13.00}} \put(141.00,62.00){\circle*{2.00}} \put(141.00,49.00){\circle*{2.00}} \put(114.00,13.00){\circle*{2.00}} \put(74.00,6.00){\circle*{2.00}} \put(37.00,18.00){\circle*{2.00}} \put(15.00,36.00){\circle*{2.00}} \put(96.00,4.00){\makebox(0,0)[cc]{$s^j$}} \put(70.00,2.00){\makebox(0,0)[cc]{$a^j$}} \put(119.00,8.00){\makebox(0,0)[cc]{$a^{j+1}$}} \put(158.00,66.00){\makebox(0,0)[cc]{$R^\bot$}} \put(36.00,63.00){\makebox(0,0)[cc]{$P\subseteq \A{R^\ast}{}$}} \put(46.00,33.00){\makebox(0,0)[cc]{$P\cap [R\geq 0]$}} \put(88.00,12.00){\makebox(0,0)[cc] {\raisebox{0.5ex}{$\frac{q}{\ell(j)}$}$\, \overline{a^j a^{j+1}}$}} \thinlines \put(5.00,40.00){\line(6,1){146.00}} \multiput(74.00,6.00)(2.04,12.24){4}{\line(1,6){1.53}} \multiput(81.67,53.00)(6.755,-11.258){4}{\line(3,-5){5.066}} \put(150.00,33.00){\makebox(0,0)[tl]{\parbox{11em}{ In $\A{R^\ast}{}$, the mutually parallel lines $R^\bot$ and $a^j a^{j+1}$ have distance $q/p$ from each other.}}} \end{picture} \end{center} Geometrically, one can think about ${\tilde{\tau}}$ as generated by the interval $I$ with vertices as in (iii) and by the polygon $P^\prime$ obtained as follows: ``Tighten'' $P\cap[R\geq 0]$ along $R^\bot$ by a cone with base $q/\ell(j)\cdot \overline{a^j a^{j+1}}$ and some top on the $R^\bot$-line; take $-\langle \bullet, R\rangle$ as an additional, fourth coordinate. Then, $[R^\ast,0]$ is still $1$ on $P^\prime$ and equals $0$ on $I$. Moreover, $[R,0]$ vanishes on $I$ and on the $R^\bot$-edge of $P^\prime$; $[R,1]$ vanishes on the whole $P^\prime$.\\ \par {\bf Proof:} We change coordinates. If $g:=\mbox{gcd}(p,q)$ denotes the ``length'' of $R$, then we can find an $s\in M$ such that $\{s,\,R/g\}$ forms a basis of $M\cap (d^j)^\bot$. Adding some $r\in M$ with $\langle d^j/\ell(j),r\rangle =1$ ($r$ from Remark \zitat{3G}{3} will do) yields a $Z\!\!\!Z$-basis for the whole lattice $M$. We consider the following commutative diagram: \[ \dgARROWLENGTH=0.3em \begin{diagram} \node{N} \arrow[4]{e,tb}{(s,r,R/g)}{\sim} \arrow{s,l}{(\mbox{\footnotesize id},\,0)} \node[4]{Z\!\!\!Z^3} \arrow{s,r}{(\mbox{\footnotesize id},\,g\cdot\mbox{\footnotesize pr}_3)}\\ \node{N\oplusZ\!\!\!Z} \arrow[4]{e,tb}{([s,0],\,[r,0],\,[R/g,0],\,[R,1])}{\sim} \node[4]{Z\!\!\!Z^3\oplusZ\!\!\!Z} \end{diagram} \] The left hand side contains the data being relevant for our proposition. Carrying them to the right yields: \begin{itemize} \item $[0,0,g]\in (Z\!\!\!Z^3)^\ast$ as the image of $R$; \item $[0,0,g,0], [0,0,0,1]\in (Z\!\!\!Z^4)^\ast$ as the images of $[R,0]$ and $[R,1]$, respectively; \item $\tau$ becomes a cone with affine cross cut \vspace{-1ex} \[ Q([0,0,g])=\mbox{conv}\Big(\big(\langle a,s\rangle/\langle a,R\rangle;\, \langle a,r\rangle/\langle a,R\rangle;\,1/g\big)\,\Big|\; a\in P\cap [R\geq 0]\Big)\,; \vspace{-1.2ex} \] \item $I$ changes into the unit interval $(Q_1,1)$ reaching from $(0,0,0,1)$ to $(0,1,0,1)$; \item finally, $\mbox{cone}(P^\prime)$ maps onto the cone spanned by the convex hull $(Q_0,0)$ of the points $\big(\langle a,s\rangle/\langle a,R\rangle;\, \langle a,r\rangle/\langle a,R\rangle;\,1/g;\,0\big)$ for $a\in P\cap [R\geq 0]$ on the $a^j$-side and\\ $\big(\langle a,s\rangle/\langle a,R\rangle;\, \langle a,r\rangle/\langle a,R\rangle-1;\,1/g;\,0\big)$ for $a$ on the $a^{j+1}$-side, respectively. \end{itemize} Since $Q([0,0,g])$ equals the Minkowski sum of the interval $Q_1\subseteq \A{[0,0,g]}{0}$ and the polygon $Q_0\subseteq\A{[0,0,g]}{}$, we are done by \zitat{Gd}{2}. \hfill$\Box$\\ \par \neu{3G-6} To see how the original equations of the singularity $Y_\sigma$ will be perturbed, it is useful to study first the dual cones ${\tilde{\tau}}^{\scriptscriptstyle\vee}$ or ${\tilde{\sigma}}^{\scriptscriptstyle\vee}={\tilde{\tau}}^{\scriptscriptstyle\vee}\cap\pi^{-1}(\sigma^{\scriptscriptstyle\vee})$: \vspace{-1ex}\\ \par {\bf Proposition:} {\em If $s\in\sigma^{\scriptscriptstyle\vee}\cap M$, then the $(M\oplusZ\!\!\!Z)$-element \[ S:= \left\{ \begin{array}{ll} [s,\,0] & \mbox{ if } \langle d^j,s\rangle\geq 0\\ {}[s,\, -\langle d^j/\ell(j),\,s\rangle] & \mbox{ if } \langle d^j,s\rangle\leq 0 \end{array} \right. \] is a lift of $s$ into ${\tilde{\sigma}}^{\scriptscriptstyle\vee}\cap (M\otimesZ\!\!\!Z)$. (Notice that it does not depend on $p,q$, but only on $j$.) Moreover, if $s^v$ runs through the edges of $P\cap[R\geq 0]$, the elements $S^v$ together with $[R,0]$ and $[R,1]$ form the fundamental generators of ${\tilde{\tau}}^{\scriptscriptstyle\vee}$. \vspace{-1ex} }\\ \par {\bf Proof:} Since we know ${\tilde{\tau}}$ from the previous proposition, the calculations are straightforward and will be omitted. \hfill$\Box$\\ \par \neu{3G-7} Recall from \zitat{T1}{1} that $E$ denotes the minimal set generating the semigroup $\sigma^{\scriptscriptstyle\vee}\cap M$. To any $s\in E$ there is a assigned variable $z_s$, and $Y_\sigma\subseteq \,I\!\!\!\!C^E$ is given by binomial equations arising from linear relations among elements of $E$. Everything will be clear by considering an example: A linear relation such as $s^1+2s^3=s^2+s^4$ transforms into $z_1\,z_3^2=z_2\,z_4$.\\ The fact that $\sigma$ defines a Gorenstein variety (i.e.\ $\sigma$ is a cone over a lattice polytope) implies that $E$ consists only of $R^\ast$ and elements of $\partial\sigma^{\scriptscriptstyle\vee}$ including the fundamental generators $s^v$. If $E\cap\partial\sigma^{\scriptscriptstyle\vee}$ is ordered clockwise, then any two adjacent elements form together with $R^\ast$ a $Z\!\!\!Z$-basis of the three-dimensional lattice $M$.\\ In particular, any three sequenced elements of $E\cap\partial\sigma^{\scriptscriptstyle\vee}$ provide a unique linear relation among them and $R^\ast$. (We met this fact already in Remark \zitat{3G}{3}; there $r$, $s^j$, and $r^\prime$ were those elements.) The resulting ``boundary'' equations do not generate the ideal of $Y_\sigma\subseteq\,I\!\!\!\!C^E$. Nevertheless, for describing a deformation of $Y_\sigma$, it is sufficient to know about perturbations of this subset only. Moreover, if one has to avoid boundary equations ``overlapping'' a certain spot on $\partial\sigma^{\scriptscriptstyle\vee}$, then it will even be possible to drop up to two of them from the list. \vspace{1ex} \begin{center} \unitlength=0.6mm \linethickness{0.4pt} \begin{picture}(122.00,68.00) \put(19.00,55.00){\line(-1,-3){11.67}} \put(7.33,20.00){\line(4,-1){42.67}} \put(50.00,9.33){\line(3,1){22.00}} \put(72.00,16.67){\line(1,1){15.00}} \put(19.00,55.00){\line(5,3){20.00}} \put(7.00,20.00){\circle*{2.00}} \put(13.00,37.00){\circle*{2.00}} \put(24.00,16.00){\circle*{2.00}} \put(46.00,68.00){\makebox(0,0)[cc]{$\dots$}} \put(88.00,39.00){\makebox(0,0)[cc]{$\vdots$}} \put(38.00,42.00){\makebox(0,0)[cc]{$R^\ast$}} \put(45.00,39.00){\circle*{2.00}} \multiput(7.00,20.00)(10.968,5.484){8}{\line(2,1){8.226}} \put(92.00,62.50){\circle*{2.00}} \put(94.00,68.00){\makebox(0,0)[cc]{$R$}} \put(7.00,40.00){\makebox(0,0)[cc]{$r$}} \put(20.00,8.00){\makebox(0,0)[cc]{$r^\prime$}} \put(3.00,13.00){\makebox(0,0)[cc]{$s^j$}} \put(13.00,58.00){\makebox(0,0)[cc]{$s^{j-1}$}} \put(50.00,3.00){\makebox(0,0)[cc]{$s^{j+1}$}} \put(122.00,21.00){\makebox(0,0)[cc]{$\sigma^{\scriptscriptstyle\vee}$}} \put(62.00,58.00){\makebox(0,0)[cc]{$[d^j\geq 0]$}} \put(78.00,46.00){\makebox(0,0)[cc]{$[d^j\leq 0]$}} \end{picture} \end{center} {\bf Theorem:} {\em The one-parameter deformation of $Y_\sigma$ in degree $-(q\,R^\ast-p\,s^j)$ is completely determined by the following perturbations: \begin{itemize} \item[(i)] (Boundary) equations involving only variables that are induced from $[d^j\geq 0]\subseteq\sigma^{\scriptscriptstyle\vee}$ remain unchanged. The same statement holds for $[d^j\leq 0]$. \item[(ii)] The boundary equation $z_r\,z_{r^\prime}-z_{R^\ast}^{\ell(j)}\,z_{s^j}^k=0$ assigned to the triple $\{r,s^j,r^\prime\}$ is perturbed into $\big(z_r\,z_{r^\prime}-z_{R^\ast}^{\ell(j)}\,z_{s^j}^k\big) - t\,z_{R^\ast}^{\ell(j)-q}\,z_{s^j}^{k+p}=0$. Divide everything by $z^k_{s^j}$ if $k<0$. \vspace{2ex} \end{itemize} } \par {\bf Proof:} Restricted to either $[d^j\geq 0]$ or $[d^j\leq 0]$, the map $s\mapsto S$ lifting $E$-elements into ${\tilde{\sigma}}\cap(M\oplusZ\!\!\!Z)$ is linear. Hence, any linear relation remains true, and part (i) is proven.\\ For the second part, we consider the boundary relation $r+r^\prime=\ell(j)\,R^\ast+k\,s^j$ with a suitable $k\inZ\!\!\!Z$. By Lemma \zitat{3G}{6}, the summands involved lift to the elements $[r,0]$, $[r^\prime,1]$, $[R^\ast,0]$, and $[s^j,0]$, respectively. In particular, the relation breaks down and has to be replaced by \[ \renewcommand{\arraystretch}{1.5} \begin{array}{rcl} [r,0]+[r^\prime,1]&=& [R,1] + \big(\ell(j)-q\big)\, [R^\ast,0] + (k+p)\, [s^j,0] \quad \mbox{ and}\\ \ell(j)\,[R^\ast,0]+k\,[s^j,0] &=& [R,0] + \big(\ell(j)-q\big)\, [R^\ast,0] + (k+p)\, [s^j,0]\,. \end{array} \] The monomials corresponding to $[R,1]$ and $[R,0]$ are $f^1$ and $f$, respectively. They are {\em not} regular on the total space $X$, but their difference $t:=f^1-f$ is. Hence, the difference of the monomial versions of both equations yields the result.\\ Finally, we should remark that (i) and (ii) cover all boundary equations except those overlapping the intersection of $\partial\sigma^{\scriptscriptstyle\vee}$ with $\overline{R^\ast R}$. \hfill$\Box$\\ \par \neu{3G-8} We return to Example \zitat{3G}{4} and discuss the one-parameter deformations occurring in degree $-R^\alpha$, $-R^\beta$, and $-R^\gamma$, respectively: \vspace{1ex}\\ {\em Case $\alpha$:}\quad $R^\alpha=[-1,0,1]=2R^\ast-s^3$ means $j=3$, $q=\ell(3)=2$, and $p=1$. Hence, the line $R^\bot$ has distance $q/p=2$ from its parallel through $a^3$ and $a^1$. In particular, $\tau=\langle a^1, c^1, c^3, a^3\rangle$ with $c^1=(1,-1,1)$ and $c^3=(3,-1,3)$. \begin{center} \unitlength=0.8mm \linethickness{0.4pt} \begin{picture}(100.00,47.00) \put(20.00,15.00){\circle*{1.00}} \put(20.00,25.00){\circle*{1.00}} \put(20.00,35.00){\circle*{1.00}} \put(30.00,15.00){\circle*{1.00}} \put(30.00,25.00){\circle*{1.00}} \put(40.00,15.00){\circle*{1.00}} \put(50.00,15.00){\circle*{1.00}} \put(20.00,35.00){\line(0,-1){20.00}} \put(20.00,15.00){\line(1,0){30.00}} \put(50.00,15.00){\line(-3,2){30.00}} \put(40.00,10.00){\line(0,1){30.00}} \put(44.00,41.00){\makebox(0,0)[cc]{$R^\bot$}} \put(44.00,24.00){\makebox(0,0)[cc]{$c^3$}} \put(15.00,10.00){\makebox(0,0)[cc]{$a^1$}} \put(37.00,12.00){\makebox(0,0)[cc]{$c^1$}} \put(55.00,10.00){\makebox(0,0)[cc]{$a^2$}} \put(15.00,40.00){\makebox(0,0)[cc]{$a^3$}} \put(90.00,25.00){\makebox(0,0)[cc]{$\tau\subseteq\sigma$}} \end{picture} \vspace{-2ex} \end{center} We construct the generators of ${\tilde{\tau}}$ by the recipe of Proposition \zitat{3G}{5}: $a^3$ treated via (i) and $a^1$ treated via (ii) yield the same element $A:=(-1,1,1,-2)$; from the $R^\bot$-line we obtain $C^1:=(1,-1,1,0)$ and $C^3:=(3,-1,3,0)$; finally (iii) provides $X:=(0,0,0,1)$ and $Y:=(0,-1,0,1)$. Hence, ${\tilde{\tau}}$ is the cone over the pyramid with plane base $X\,Y\,C^1\,C^3$ and $A$ as top. (The relation between the vertices of the quadrangle equals $3C^1+2X=C^3+2Y$.) Moreover, ${\tilde{\sigma}}$ equals ${\tilde{\sigma}}={\tilde{\tau}}+I\!\!R_{\geq 0}a^2$ with $a^2:=(a^2,0)$. Since $A+2X+2a^2=C^3$ and $A+2Y+2a^2=3C^1$, ${\tilde{\sigma}}$ is a simplex generated by $A$, $X$, $Y$, and $a^2$.\\ Denoting the variables assigned to $s^1, s^2, s^3, R^\ast, v^1, v^2, w \in E \subseteq \sigma^{\scriptscriptstyle\vee}\cap M$ by $Z_1$, $Z_2$, $Z_3$, $U$, $V_1$, $V_2$, and $W$, respectively, there are six boundary equations: \vspace{-1ex} \[ Z_3WZ_1-U^3\,=\, Z_1Z_2-W^2\,=\, WV_1-UZ_2\,=\, Z_2V_2-V_1^2\,=\, V_1Z_3-V_2^2\,=\, V_2Z_1-U^2\,=\,0\,. \vspace{-1ex} \] Only the four latter ones are covered by Theorem \zitat{3G}{7}. They will be perturbed into \vspace{-1ex} \[ WV_1-UZ_2 \,=\,Z_2V_2-V_1^2\,=\, V_1Z_3-V_2^2\,=\, V_2Z_1-U^2-t_\alpha Z_3\,=\,0\,. \vspace{1ex} \] \par {\em Case $\beta$:}\quad $R^\beta=[0,-1,1]=2R^\ast-s^1$ means $j=1$, $\ell(1)=3$, $q=2$, and $p=1$. Hence, $R^\bot$ still has distance $2$, but now from the line $a^1a^2$. \vspace{-2ex} \begin{center} \unitlength=0.8mm \linethickness{0.4pt} \begin{picture}(100.00,47.00) \put(20.00,15.00){\circle*{1.00}} \put(20.00,25.00){\circle*{1.00}} \put(20.00,35.00){\circle*{1.00}} \put(30.00,15.00){\circle*{1.00}} \put(30.00,25.00){\circle*{1.00}} \put(40.00,15.00){\circle*{1.00}} \put(50.00,15.00){\circle*{1.00}} \put(20.00,35.00){\line(0,-1){20.00}} \put(20.00,15.00){\line(1,0){30.00}} \put(50.00,15.00){\line(-3,2){30.00}} \put(10.00,35.00){\line(1,0){50.00}} \put(65.00,35.00){\makebox(0,0)[cc]{$R^\bot$}} \put(15.00,10.00){\makebox(0,0)[cc]{$a^1$}} \put(55.00,10.00){\makebox(0,0)[cc]{$a^2$}} \put(20.00,40.00){\makebox(0,0)[cc]{$a^3$}} \put(90.00,25.00){\makebox(0,0)[cc]{$\tau=\sigma$}} \end{picture} \vspace{-2ex} \end{center} We obtain ${\tilde{\tau}}=\langle (-1,-1,1,-2); (0,-1,1,-2); (-1,1,1,0); (0,0,0,1); (1,0,0,1) \rangle$.\\ The boundary equation corresponding to Theorem \zitat{3G}{7}(ii) is $Z_3WZ_1-U^3=0$; it perturbs into $Z_3WZ_1-U^3-t_\beta UZ_1=0$.\\ \par {\em Case $\gamma$:}\quad $R^\gamma=[0,-2,1]=3R^\ast-2s^1$ means $j=1$, $q=\ell(1)=3$, and $p=2$. \vspace{-2ex} \begin{center} \unitlength=0.8mm \linethickness{0.4pt} \begin{picture}(100.00,47.00) \put(20.00,15.00){\circle*{1.00}} \put(20.00,25.00){\circle*{1.00}} \put(20.00,35.00){\circle*{1.00}} \put(30.00,15.00){\circle*{1.00}} \put(30.00,25.00){\circle*{1.00}} \put(40.00,15.00){\circle*{1.00}} \put(50.00,15.00){\circle*{1.00}} \put(20.00,35.00){\line(0,-1){20.00}} \put(20.00,15.00){\line(1,0){30.00}} \put(50.00,15.00){\line(-3,2){30.00}} \put(10.00,30.00){\line(1,0){50.00}} \put(65.00,30.00){\makebox(0,0)[cc]{$R^\bot$}} \put(15.00,10.00){\makebox(0,0)[cc]{$a^1$}} \put(55.00,10.00){\makebox(0,0)[cc]{$a^2$}} \put(20.00,40.00){\makebox(0,0)[cc]{$a^3$}} \put(90.00,25.00){\makebox(0,0)[cc]{$\tau\subseteq\sigma$}} \end{picture} \vspace{-2ex} \end{center} Here, we have ${\tilde{\tau}}=\langle (-1,-1,1,-3); (-2,1,2,0); (-1,2,4,0); (0,0,0,1); (1,0,0,1) \rangle$, and the previous boundary equation provides $Z_3WZ_1-U^3-t_\gamma Z_1^2=0$.\\ \par

9609

alg-geom/9609003

en

https://arxiv.org/abs/alg-geom/9609003

alg-geom/9609003

Guillermo Matera

B. Bank, M. Giusti, J. Heintz, R. Mandel, G. M. Mbakop

Polar Varieties and Efficient Real Equation Solving: The Hypersurface Case

Latex

The objective of this paper is to show how the recently proposed method by Giusti, Heintz, Morais, Morgenstern, Pardo \cite{gihemorpar} can be applied to a case of real polynomial equation solving. Our main result concerns the problem of finding one representative point for each connected component of a real bounded smooth hypersurface. The algorithm in \cite{gihemorpar} yields a method for symbolically solving a zero-dimensional polynomial equation system in the affine (and toric) case. Its main feature is the use of adapted data structure: Arithmetical networks and straight-line programs. The algorithm solves any affine zero-dimensional equation system in non-uniform sequential time that is polynomial in the length of the input description and an adequately defined {\em affine degree} of the equation system. Replacing the affine degree of the equation system by a suitably defined {\em real degree} of certain polar varieties associated to the input equation, which describes the hypersurface under consideration, and using straight-line program codification of the input and intermediate results, we obtain a method for the problem introduced above that is polynomial in the input length and the real degree.

alg-geom

\section{Introduction} The present article is strongly related to the papers \cite{gihemorpar} and \cite{gihemopar}. Whereas the algorithms developed in these references are related to the algebraically closed case, here we are concerned with the real case. Finding a real solution of a polynomial equation $f(x)=0$ where $f$ is a polynomial of degree $d\ge 2$ with rational coefficients in $n$ variables is for practical applications more important than the algebraically closed case. Best known complexity bounds for the problem we deal with are of the form $d^{O(n)}$ due to \cite{hroy}, \cite{rene}, \cite{basu}, \cite{sole}. Related complexity results can be found in \cite{canny}, \cite{grigo1}. \par Solution methods for the algebraically closed case are not applicable to real equation solving normally. The aim of this paper is to show that certain {\em polar varieties\/} associated to an affine hypersurface possess a geometric invariant, {\em the real degree\/}, which permits an adaptation of the algorithms designed in the papers mentioned at the beginning. The algorithms there are of "intrinsic type", which means that they are able to distinguish between the semantical and the syntactical character of the input system in order to profit both for the improvement of the complexity estimates. Both papers \cite{gihemorpar} and \cite{gihemopar} show that the {\em affine degree\/} of an input system is associated with the complexity when measured in terms of the number of arithmetic operations. Whereas the algorithms in \cite{gihemorpar} still need algebraic parameters, those proposed in \cite{gihemopar} are completely rational. \par We will show that, under smoothness assumptions for the case of finding a real zero of a polynomial equation of degree $d$ with rational coefficients and $n$ variables, it is possible to design an algorithm of intrinsic type using the same data structure, namely straight-line programs without essential divisions and rational parameters for codifying the input system, intermediate results and the output, and replacing the affine degree by the real degree of the associated polar varieties to the input equation. \par The computation model we use will be an arithmetical network (compare to \cite{gihemorpar}). Our main result then consists in the following. {\em There is an arithmetical network of size $(nd\delta^*L)^{O(1)}$ with parameters in the field of rational numbers which finds a representative real point in every connected component of an affine variety given by a non-constant square-free $n$-variate polynomial $f$ with rational coefficients and degree $d\ge 2$ $($supposing that the affine variety is smooth in all real points that are contained in it.$)$. $L$ denotes the size of the straight-line program codifying the input and $\delta^*$ is the real degree associated to $f$.} \\ Close complexity results are the ones following the approach initiated in \cite{ShSm93a}, and further developed in \cite{ShSm93b}, \cite{ShSm93c}, \cite{ShSm93d}, \cite{ShSm1}, see also \cite{Dedieu1}, \cite{Dedieu2}. \par For more details we refer the reader to \cite{gihemorpar} and \cite{gihemopar} and the references cited there. \newpage \section{Polar Varieties and Algorithms } As usually, let $\mbox{}\; l\!\!\!Q, \; I\!\!R$ and $l\!\!\!C$ denote the field of rational, real and complex numbers, respectively. The affine n--spaces over these fields are denoted by $\mbox{}\; l\!\!\!Q^n, \; I\!\!R^n$ and $l\!\!\!C^n$, respectively. Further, let $l\!\!\!C^n$ be endowed with the Zariski--topology, where a closed set consists of all common zeros of a finite number of polynomials with coefficients in $\mbox{}\; l\!\!\!Q$. Let $W \subset {l\!\!\!C}^n$ be a closed subset with respect to this topology and let $W= C_1\cup\cdots \cup C_s$ be its decomposition into irreducible components with respect to the same topology. Thus $W, \; C_1,\ldots,C_s$ are algebraic subsets of ${l\!\!\!C}^n$. Let $1\le j \le s$, be arbitrarily fixed and consider the irreducible component $C_j$ of $W$. In the following we need the notion of degree of an affine algebraic variety. Let $W \subset l\!\!\!C^n$ be an algebraic subset given by a regular sequence $f_1, \cdots f_i \in \mbox{}\; l\!\!\!Q[x_1, \cdots, x_n]$ of degree at most $d$. If $W \subset l\!\!\!C^n$ ist zero--dimensional the {\it degree} of $W, \; deg W$, is defined to be the number of points in $W$ (neither multiplicities nor points at infinity are counted). If $W \subset l\!\!\!C^n$ is of dimension greater than zero (i.e. $dim W = n-i \ge 1$), then we consider the collection ${\cal M}$ of all affine varieties of dimension $i$ given as the solution set in $l\!\!\!C^n$ of a linear equation system $L_1 = 0, \; \cdots, L_{n-i} = 0$ with $L_{k} = \sum_{j=1}^{n} a_{kj} x_j + a_{k0}, \; a_{ki} \in \mbox{}\; l\!\!\!Q, 1 \le i \le n$. Let ${\cal M}_{W}$ be the subcollection of ${\cal M}$ formed by all varieties $H \in {\cal M}$ such that the affine variety $H \cap W$ satisfies $H \cap W \not= \emptyset$ and $dim(H \cap W) = 0$. Then the affine degree of $W$ is defined as $max\{ \delta | \delta = deg(H \cap W), \; H \in M_W \}$. \begin{definition}\label{def1} The component $C_j$ is called a {\rm real component} of $W$ if the real variety $C_j\cap I\!\!R^n$ contains a smooth point of $C_j$. \par \noindent If we denote \[ I = \{ j \in I\!\!N | 1 \le j \le s, \hskip 3pt C_j \hskip 3pt{\hbox {\rm is a real component of $W$}} \}. \] then the affine variety $W^\ast := \bigcup \limits_{j \in I} C_j \; \subset l\!\!\!C^n$ {\it is called the real part} of $W$. By $deg^{\ast} W := degW^{\ast} = \sum\limits_{j \in I}deg C_j$ we define the {\it real degree} of the set $W$. \end{definition} \begin{remark} {\rm Observe that $deg^{\ast} W= 0$ holds if and only if the real part $W^{\ast}$ of $W$ is empty.} \end{remark} \begin{proposition}\label{prop3} Let $f \in \mbox{}\; l\!\!\!Q[X_1,\cdots,X_n]$ be a non-constant and square-free polynomial and let $\widetilde{V}(f)$ be the set of real zeros of the equation $f(x) =0$. Assume $\widetilde{V}(f)$ to be bounded. Furthermore, let for every fixed $i, \; 0 \le i <n$, the real variety $$\widetilde{V}_i := \{ x \in I\!\!R^n | \; f(x) = {{\partial f(x)} \over {\partial X_1}} = \ldots,{{\partial f(x)} \over {\partial X_i}} = 0 \}$$ be non-empty (and $\widetilde{V}_0$ is understood to be $\widetilde{V}(f)$). Suppose the variables to be in generic position. Then any point of $\widetilde{V}_i$ that is a smooth point of $\widetilde{V}(f)$ is also a smooth point of $\widetilde{V}_i$. Moreover, for every such point the Jacobian of the equation system $f=\frac{\partial f}{\partial X_1} = \cdots = \frac{\partial f}{\partial X_i} =0$ has maximal rank. \end{proposition} \bigskip {\bf Proof} Consider the linear transformation $x \longleftarrow A^{(i)} y$, where the new variables are $y = (Y_1, \cdots, Y_n)$. Suppose that $A^{(i)}$ is given in the form \[ \left( \begin{array}{ll} I_{i,i} & 0_{i,n-i} \\ (a_{kl})_{n-i,i} & I_{n-i,n-i} \end{array} \right) , \] where $I$ and 0 define a unit and a zero matrix, respectively, and\\ $a_{kl} \in I\!\! R $ arbitrary if $k,l$ satisfy $i+1 \le k \le n, \;\; 1 \le l\le i$.\\ The transformation $x \longleftarrow A^{(i)} y$ defines a linear change of coordinates, since the square matrix $A^{(i)}$ has full rank. In the new coordinates, the variety $\widetilde{V}_i$ takes the form $$\widetilde{V}_i := \{ y \in I\!\!R^n | \; f(y) = {{\partial f(y)} \over {\partial Y_1}} + \sum_{j = i+1}^n a_{j1} {{\partial f(y)} \over {\partial Y_j}} = \ldots = {{\partial f(y)} \over {\partial Y_i}} +\sum_{j = i+1}^n a_{ji} {{\partial f(y)} \over {\partial Y_j}}= 0 \}$$ This transformation defines a map $\Phi_i \; : I\!\!R^n \times I\!\!R^{(n-i) i} \longrightarrow I\!\!R^{i+1}$ given by $$\Phi_i \left ( Y_1, \cdots, Y_i, \cdots, Y_n, a_{i+1, 1}, \cdots a_{n 1}, \cdots a_{i+1,i}, \cdots, a_{n, i} \right ) = $$ $$\left ( f,\; {{\partial f} \over {\partial Y_1}} + \sum_{j = i+1}^n a_{j 1} {{\partial f} \over {\partial Y_j}}, \ldots,\; {{\partial f} \over {\partial Y_i}} +\sum_{j = i+1}^n a_{j i} {{\partial f} \over {\partial Y_j}} \right )$$ For the moment let $$\alpha := (\alpha_1, \cdots, \alpha_{(n-i) i} ) := (Y_1, \cdots, Y_n, a_{i+1, \; 1}, \cdots a_{n,i}) \in I\!\!R^n \times I\!\!R^{(n-i) i}$$ Then the Jacobian matrix of $\Phi_i ( \alpha )$ is given by\newline $J \left (\Phi_i (\alpha ) \right ) = \left ( {{\partial \Phi_i ( \alpha )} \over {\partial \alpha_j}} \right )_{(i+1)\times (n + (n-i) i) } = $ $$\left ( \begin{array}{ccllclccl} {{\partial f}\over{\partial Y_1}}& \cdots & {{\partial f} \over {\partial Y_n}} & 0 & \cdots & 0 & \cdots & \cdots & 0 \\ \ast & \cdots& \ast & {{\partial f} \over {\partial Y_{i+1}}} & \cdots & {{\partial f} \over {\partial Y_n}} & 0 \cdots & \vdots & 0\\ \vdots & & \vdots & \ddots & \ddots & 0 & \cdots & \ddots & 0 \\ \ast & \cdots& \ast & 0 \cdots & 0 \cdots & \cdots &{{\partial f} \over {\partial Y_{i+1}}} & \cdots & {{\partial f} \over {\partial Y_n}} \end{array} \right ) $$ If $\alpha^0 = (Y_1^0, \cdots, Y_n^0, a_{i+1, \; 1}^0, \cdots a_{n, \; i}^0)$ belongs to the fibre $\Phi_i^{-1} (0)$, where $(Y_1^0, \cdots, Y_n^0)$ is a point of the hypersurface $\widetilde{V} (f)$ and if there is an index $j \in \{ i+1, \cdots, n \}$ such that ${{\partial f} \over {\partial Y_j}} \not= 0$ at this point, then the Jacobian matrix $J \left (\Phi_i (\alpha^0) \right )$ has the maximal rank $i + 1$.\\ Suppose now that for all points of $\widetilde{V} (f)$ $${{\partial f(y)} \over {\partial Y_{i+1}}} = \cdots = {{\partial f(y)} \over {\partial Y_n}} = 0$$ and let $C := I\!\!R^n \setminus \{ {{\partial f(y)} \over {\partial Y_1}} = \cdots = {{\partial f(y)} \over {\partial Y_n}} = 0 \}$, which is an open set. Then the restricted map \[ \Phi_i : C \times I\!\! R^{(n-i)i} \longrightarrow I\!\! R^{i+1} \] is transversal to the subvariety $\{ 0\}$ in $I\!\! R^{i+1}$.\\ By weak transversality due to {\sl Thom/Sard} (see e.g. \cite{golub}) applied to the diagram $$ \begin{array}{lcc} \Phi_i^{-1} (0) & \hookrightarrow & I\!\!R^n \times I\!\!R^{(n-i) i} \\ & \searrow & \downarrow \\ & & I\!\!R^{(n-i) i} \end{array} $$ \noindent one concludes that the set of all $A \in I\!\!R^{(n-i) i}$ for which transversality holds is dense in $I\!\!R^{(n-i) i}$. Since the hypersurface $\widetilde{V}(f)$ is bounded by assumption, there is an open and dense set of matrices $A$ such that the corresponding coordinate transformation leads to the desired smoothness. \hfill $\Box$\\ Let $f \in \mbox{}\; l\!\!\!Q[X_1,\cdots,X_n]$ be a non--constant squarefree polynomial and let $W := \{ x \in l\!\!\!C^n | \; f(x) = 0 \}$ be the hypersurface defined by $f$. Consider the real variety $V := W \cap I\!\!R^n$ and suppose: \begin{itemize} \item $V$ is non-empty and bounded, \item the gradient of $f$ is different from zero in all points of $V$\\ (i.e. $V$ is a compact smooth hypersurface in $I\!\!R^n$ and $f = 0$ is its regular equation) \item the variables are in generic position. \end{itemize} \begin{definition}[Polar variety corresponding to a linear space] Let $i, \; 0\leq i<n$, be arbitrarily fixed. Further, let $X^i := \{ x \in l\!\!\!C^n | \; X_{i+1} = \cdots = X_n = 0 \}$ be the corresponding linear subspace of $l\!\!\!C^n$. Then, $W_i$ defined to be the Zariski closure of $$ \{ x \in l\!\!\!C^n | \; f(x) = {{\partial f(x)} \over{\partial X_1}} = \cdots = {{\partial f(x)} \over {\partial X_i}} = 0, \; \Delta (x) := \sum_{j = 1}^{n} \left ( {{\partial f(x)} \over {\partial X_j}} \right )^2 \not= 0 \} $$ is called the {\it polar variety} of $W$ associated to the linear subspace $X^i$. The corresponding real variety of $W_i$ is denoted by $V_i := W_i \cap I\!\!R^n$. \end{definition} \bigskip \begin{remark} {\rm Because of the hypotheses that $V \not= \emptyset$ is a smooth hypersurface and that $W_i \not= \emptyset$ by the assumptions above, the real variety $V_i := W_i \cap I\!\!R^n, \; 0 \le i <n$, is not empty and by smoothness of $V$, it has the description $$V_i = \{ x \in I\!\!R^n | f(x) = {{\partial f(x)} \over {\partial X_1}} = \ldots = {{\partial f(x)} \over {\partial X_i}} = 0 \} .$$ ($V_0$ is understood to be $V$.)\\ According to Proposition 3, $V_i$ is smooth if the coordinates are chosen to be in generic position. Definition 4 of a polar variety is slightly different from the one introduced by L\^{e}/Teissier \cite{le}. } \end{remark} \bigskip \begin{theorem} Let $f \in \mbox{}\; l\!\!\!Q[X_1,\cdots,X_n]$ be a non--constant squarefree polynomial and let $W := \{ x \in l\!\!\!C^n | \; f(x) = 0 \}$ be the corresponding hypersurface. Further, let $V := W \cap I\!\!R^n$ be a non--empty, smooth, and bounded hypersurface in $I\!\!R^n$ whose regular equation is given by $f = 0$. Assume the variables $X_1, \cdots, X_n$ to be generic. Finally, for every $i, \; 0 \le i < n$, let the polar varieties $W_i$ of $W$ corresponding to the subspace $X^i$ be defined as above. Then it holds~: \begin{itemize} \item $V \subset W_0$, with $W_0 = W$ if and only if $f$ and $\Delta := \sum_{j=1}^n \left( \frac{\partial f}{\partial X_j} \right)^2$ are coprime, \item $W_i$ is a non--empty equidimensional affine variety of dimension $n-(i+1)$ that is smooth in all its points that are smooth points of $W$, \item the real part $W_i^\ast $ of the polar variety $W_i$ coincides with the Zariski closure in $l\!\!\!C^n$ of $$V_i = \left\{ x \in I\!\!R^n | f(x) = {{\partial f(x)} \over {\partial X_1}} = \ldots = {{\partial f(x)} \over {\partial X_i}} = 0 \right\} ,$$ \item for any $j$, $i<j \le n$ the ideal $$\left(f, {{\partial f} \over {\partial X_1}}, \ldots,{{\partial f} \over {\partial X_i}}\right)_{{{\partial f} \over {\partial X_j}}}$$ is radical. \end{itemize} \end{theorem} {\bf Proof:} Let $i, 0\le i < n$, be arbitrarily fixed. The first item is obvious since $W_0$ is the union of all irreducible components of $W$ on which $\Delta$ does not vanish identically. \\ Then $W_i$ is non-empty by the assumptions. The sequence $f, \frac{\partial f}{\partial X_1} , \ldots , \frac{\partial f}{\partial X_i}$ of polynomials of $\mbox{}\; l\!\!\!Q[X_1,\ldots , X_n]$ forms a local regular sequence with respect to the smooth points of $W_i$ since the affine varieties $\big\{ x \in l\!\!\!C^n | f(x) = \frac{\partial f(x)}{\partial X_1} = \cdots = \frac{\partial f(x)}{\partial X_k} = 0 \big\}$ and $\big\{ x \in l\!\!\!C^n | \frac{\partial f(x)}{\partial X_{k+1}} = 0\big\} $ are transversal for any $k, 0\le k\le i-1$, by the generic choice of the coordinates, and hence the sequence $f, \frac{\partial f}{\partial X_1} ,\cdots , \frac{\partial f}{\partial X_i}$ yields a local complete intersection with respect to the same points. This implies that $W_i$ are equidimensional and $dim_{l\!\!\!C} W_i = n-(i+1)$ holds. We observe that every smooth point of $W_i$ is a smooth point of $W$, which completes the proof of the second item.\\ The Zariski closure of $V_i$ is contained in $W_i^\ast$, which is a simple consequence of the smoothness of $V_i$. One obtains the reverse inclusion as follows. Let $x^\ast \in W_i^\ast$ be an arbitrary point, and let $C_{j\ast}$ be an irreducible component of $W_i^\ast$ containing this point, and $C_{j\ast} \cap V_i \not= \emptyset$. Then \[ \begin{array}{rl} n-i-1 &= dim_{I\!\! R}(C_{j\ast}\cap V_i)= dim_{I\!\! R} R(C_{j\ast}\cap V_i)=\\ &=dim_{l\!\!\!C} R((C_{j\ast}\cap V_i)')\le dim_{l\!\!\!C} C_{j\ast} = n-i-1, \end{array} \] where $R(\cdot)$ and $(\,\, )'$ denote the corresponding sets of smooth points contained in $( \cdot )$ and the associated complexification, respectively. Therefore, $dim_{l\!\!\!C} (C_{j\ast} \cap V_i)' = dim_{l\!\!\!C} C_{j\ast} = n-i-1$ and, hence, $C_{j\ast} = (C_{j\ast}\cap V_i)'$, and the latter set is contained in the Zariski closure of $V_i$.\\ We define the non-empty affine algebraic set \[ \widetilde{W}_i := \left\{ x\in l\!\!\!C^n | f(x) = \frac{\partial f(x)}{\partial X_1} = \cdots = \frac{\partial f(x)}{\partial X_i} = 0 \right\} . \] Let $j, i<j\le n,$ be arbitrarily fixed. Then one finds a smooth point $x^\ast$ in $\widetilde{W}_i$ such that $ \frac{\partial f(x^\ast)}{\partial X_j} \not= 0$; let $x^\ast$ be fixed in that way. The hypersurface $W = \{ x \in l\!\!\!C^n | f(x) =0 \}$ contains $x^\ast$ as a smooth point, too. Consider the local ring ${\cal O}_{W,x^\ast} $ of $x^\ast$ on the hypersurface $W$. (This is the ring of germs of functions on $W$ that are regular at $x^\ast$. The local ring ${\cal O}_{W,x^\ast}$ is obtained by dividing the ring $\mbox{} \; l\!\!\!C [X_1, \ldots , X_n ]$ of polynomials by the principal ideal $(f)$, which defines $W$ as an affine variety, and then by localizing at the maximal ideal $(X_1 - X_1^\ast,\ldots , X_n-X_n^\ast)$, of the point $x^\ast = (X_1^\ast, \ldots , X_n^\ast)$ considered as a single point affine variety.) Using now arguments from Commutative Algebra and Algebraic Geometry, see e.g. Brodmann \cite{brod}, one arrives at the fact that ${\cal O}_{W,x^\ast}$ is an integral regular local ring.\\ The integrality of ${\cal O}_{W,x^\ast}$ implies that there is a uniquely determined irreducible component $Y$ of $W$ containing the smooth point $x^\ast$ and locally this component corresponds to the zero ideal of ${\cal O}_{W,x^\ast}$, which is radical. Since the two varieties $\widetilde{W}_i \cap Y$ and $W\cap Y$ coincide locally, the variety $\widetilde{W}_i \cap Y$ corresponds locally to the same ideal. Thus, the desired radicality is shown. This completes the proof. \linebreak $\mbox{} \hfill \Box$\\ \begin{remark} {\rm If one localizes with respect to the function $ \Delta(x) = \sum\limits^n_{j=1} \big( \frac{\partial f(x)}{\partial X_j}\big)^2$, then one obtains, in the same way as shown in the proof above, that the ideal \[ \big( f, \frac{\partial f}{\partial X_1} , \ldots , \frac{\partial f}{\partial X_i} \big)_\Delta \] is also radical.} \end{remark} \begin{remark}\label{rem8} {\rm Under the assumptions of Theorem 6, for any $i,\;\; 0\le i<n$, we observe the following relations between the different non-empty varieties introduced up to now. \[ V_i \subset V,\quad V_i \subset W^\ast_i \subset W_i \subset \widetilde{W}_i , \] where $V$ is the considered real hypersurface, $V_i$ defined as in Remark 5, $W_i$ the polar variety due to Definition 4, $W^\ast_i$ its real part according to Definition 1, and $\widetilde{W}_i$ the affine variety introduced in the proof of Theorem 6. With respect to Theorem 6 our settings and assumptions imply that $n-i-1 = dim_{l\!\!\!C} \widetilde{W}_i = dim_{l\!\!\!C} W_i = dim_{l\!\!\!C} W^\ast_i = dim_{I\!\! R} V_i $ holds. By our smoothness assumption and the generic choice of the variables we have for the respective sets of smooth points (denoted as before by $R(\cdot ))$ \[ V_i = R(V_i)\subset R(W_i) \subset R(\widetilde{W}_i) \subset R(W) , \] where $W$ is the affine hypersurface.\\ For the following we use the notations as before, fix an $i$ arbitrarily,$ \;\; 0\le i<n$, denote by $\delta^\ast_i$ the real degree of the polar variety $W_i$ (compare with Definition 1, by smoothness one has that the real degree of the polar variety $W_i$ is equal to the real degree of the affine variety $\widetilde{W}_i$), put $\delta^\ast := \max \{ \delta^\ast_k | 0 \le k \le i \}$ and let $ d := \deg f$. Finally, we write for shortness $r := n-i-1$.\\ We say that the variables $X_1,\ldots , X_n$ are in Noether position with respect to a variety $\{ f_1= \cdots = f_s =0\}$ in $l\!\!\!C^n, \; f_1, \ldots , f_s \in \mbox{}\; l\!\!\!Q [ X_1, \ldots , X_n],$ if, for each $r<k \le n$, there exists a polynomial of $\mbox{}\; l\!\!\!Q [X_1, \ldots , X_r, X_k]$ that is monic in $X_k$ and vanishes on $\{ f_1=\cdots =f_s=0\}.$ \\ Then one can state the next, technical lemma according to \cite{gihemorpar}, \cite{gihemopar}, where the second reference is important in order to ensure that the occurring straight-line programs use parameters in $\mbox{}\; l\!\!\!Q$ only.} \end{remark} \begin{lemma}\label{lem9} Let the assumptions of Theorem 6 be satisfied. Further, suppose that the polynomials $f, \frac{\partial f}{\partial X_1}, \ldots , \frac{\partial f}{\partial X_i} \in \mbox{}\; l\!\!\!Q [ X_1,\ldots , X_n] $ are given by a straight-line program $\beta$ in $\mbox{}\; l\!\!\!Q [X_1, \ldots , X_n]$ without essential divisions, and let $ L$ be the size of $\beta$. Then there is an arithmetical network with parameters in $\mbox{}\; l\!\!\!Q$ that constructs the following items from the input $\beta$ \begin{itemize} \item a regular matrix of $\mbox{} \mbox{}\; l\!\!\!Q^{n\times n}$ given by its elements that transforms the variables $X_1, \ldots , X_n$ into new ones $Y_1, \ldots , Y_n$ \item a non-zero linear form $U \in \mbox{}\; l\!\!\!Q [Y_{r+1}, \ldots , Y_n]$ \item a division-free straight-line program $\gamma$ in $\mbox{}\; l\!\!\!Q[Y_1, \ldots ,Y_r, U]$ that represents non-zero polynomials $\varrho \in \mbox{}\; l\!\!\!Q[Y_1, \ldots , Y_r] $ and $q,p_1,\ldots , p_n \in \mbox{}\; l\!\!\!Q [Y_1,\ldots , Y_r, U]$. \end{itemize} These items have the following properties: \begin{itemize} \item[(i)] The variables $Y_1,\ldots ,Y_n$ are in Noether position with respect to the variety $W^\ast_{n-r}$, the variables $Y_1, \ldots , Y_r$ being free \item[(ii)] The non-empty open part $(W^\ast_{n-r})_\varrho$ is defined by the ideal $(q, \varrho X_1-p_1, \ldots ,$ $\varrho X_n-p_n)_\varrho$ in the localization $\mbox{}\; l\!\!\!Q[X_1,\ldots ,X_n]_\varrho$ . \item[(iii)] The polynomial $q$ is monic in $u$ and its degree is equal to\linebreak $\delta^\ast_{n-r}= \deg^\ast W_{n-r}= \deg W^\ast_{n-r} \le \delta^\ast$. \item[(iv)] $\max \{ \deg_up_k | 1\le k \le n \} < \delta^\ast_{n-r},\quad \max \{ \deg p_k | 1\le k \le n \} = (d \delta^\ast)^{0(1)},$\linebreak $\deg \varrho = (d \delta^\ast)^{0(1)}$. \item[(v)] The nonscalar size of the straight-line program $\gamma$ is given by $(sd\delta^\ast L)^{0(1)}$. \end{itemize} \end{lemma} The proof of Lemma 9 can be performed in a similar way as in \cite{gihemorpar}, \cite{gihemopar} for establishing the algorithm. For the case handled here, in the i-th step one has to apply the algorithm to the localized sequence $\left( f, \frac{\partial f}{\partial X_1}, \ldots , \frac{\partial f}{\partial X_i} \right)_{\Delta}$ as input. The only point we have to take care of is the process of cleaning extranious $\mbox{}\; l\!\!\!Q$-irreducible components. Whereas in the proofs of the algorithms we refer to it suffices to clean out components lying in a prefixed hypersurface (e.g. components at infinity), the cleaning process we need here is more subtile. We have to clean out all non-real $\mbox{}\; l\!\!\!Q$-irreducible components that appear during our algorithmic process. The idea of doing this is roughly as follows. Due to the generic position of the variables $X_1,\ldots,X_n$ all $\mbox{}\; l\!\!\!Q$-irreducible components of the variety $\widetilde{W}_{n-r}$ can be visualized as $\mbox{}\; l\!\!\!Q$-irreducible factors of the polynomial $q(X_1, \ldots,X_r,U)$. If we specialize {\em generically} the variables $X_1,\ldots,X_r$ to {\em rational} values $\eta_1,\ldots,\eta_r$, then by Hilbert's Irreducibility Theorem (in the version of \cite{lang}) the $\mbox{}\; l\!\!\!Q$-irreducible factors of the {\em multivariate} polynomial $q(\eta_1,\ldots,\eta_r,U)$ correspond to the $\mbox{}\; l\!\!\!Q$-irreducible factors of the {\em one--variate} polynomial $q(\eta_1,\ldots,\eta_r,U) \in \mbox{}\; l\!\!\!Q[U]$. In order to explain our idea simpler, we assume that we are able to choose our specialization of $X_1,\ldots,X_r$ into $\eta_1,\ldots,\eta_r$ in such a way that the hyperplanes $X_1 - \eta_1 = 0,\ldots,X_r - \eta_r = 0$ cut any {\em real} component of $\widetilde{W}_{n-r}$ (This condition is open in the strong topology and doesn't represent a fundamental restriction on the correctness of our algorithm. Moreover, our assumption doesn't affect the complexity). Under these assumptions the $\mbox{}\; l\!\!\!Q$-irreducible factors of $q(X_1,\ldots,X_r,U)$, which correspond to the real components of $\widetilde{W}_{n-r}$, reappear as $\mbox{}\; l\!\!\!Q$-irreducible factors of $q(\eta_1,\ldots,\eta_r,U)$ which contain a real zero. These $\mbox{}\; l\!\!\!Q$-irreducible factors of $q(\eta_1,\ldots,\eta_r,U)$ can be found by a factorization procedure and by a real zero test of standard features of polynomial complexity character. Multiplying these factors and applying to the result the lifting-fibre process of \cite{gihemorpar}, \cite{gihemopar} we find the product $q^*$ of the $\mbox{}\; l\!\!\!Q$-irreducible factors of $q( X_1,\ldots,X_r,U)$, which correspond to the union of the real components of the variety $\widetilde{W}_{n-r}$, i.e. to the real part of $\widetilde{W}_{n-r}$. The ideal $(q^*,\varrho X_1-p_1,\ldots,\varrho X_n-p_n)_\varrho$ describes the localization of the real part of $\widetilde{W}_{n-r}$ at $\varrho$. All we have pointed out is executible in polynomial time if a factorization of univariate polynomials over $\mbox{}\; l\!\!\!Q$ in polynomial time is available and if our geometric assumptions on the choice of the specialization is satisfied. \begin{theorem} Let the notations and assumptions be as in Theorem 6. Suppose that the polynomial $f$ is given by a straight-line program $\beta$ without essential divisions in $\mbox{}\; l\!\!\!Q [X_1,\ldots ,X_n]$, and let $L$ be the nonscalar size of $\beta$. Further, let $\delta^\ast_i := \deg^\ast W_i, \; \delta^\ast := \max \{ \delta^\ast_i | 0 \le i < n \}$ be the corresponding real degrees of the polar varieties in question, and let $d := \deg f$. Then there is an arithmetical network of size $(n d \delta^\ast L)^{0(1)}$ with parameters in $\mbox{}\; l\!\!\!Q$ which produces, from the input $\beta$, the coefficients of a non-zero linear form $u \in \mbox{}\; l\!\!\!Q [X_1, \ldots ,X_n]$ and non-zero polynomials $q,p_1, \ldots , p_n \in \mbox{}\; l\!\!\!Q[U]$ showing the following properties: \begin{enumerate} \item For any connected component $C$ of $V$ there is a point $\xi \in C$ and an element $ \tau \in I\!\! R$ such that $q (\tau)=0$ and $\xi = (p_1(\tau), \ldots, p_n(\tau))$ \item $\deg (q) = \delta^\ast_{n-1} \le \delta^\ast$ \item $\max \{ \deg (p_i) | 1 \le i \le n \} < \delta^\ast_{n-1}$. \end{enumerate} \end{theorem} \newpage

9609

alg-geom/9609021

en

https://arxiv.org/abs/alg-geom/9609021

alg-geom/9609021

David R. Morrison

David R. Morrison

Mathematical Aspects of Mirror Symmetry

74 pages, LaTeX2e with amsmath; minor changes, added table of contents

Complex Algebraic Geometry (J. Koll\'ar, ed.), IAS/Park City Math. Series, vol. 3, 1997, pp. 265-340

Lecture notes from 1993 Park City lectures and 1994 Trento lectures. The focus of these lectures is on giving a mathematical description of the A-model and B-model correlation functions on a Calabi--Yau manifold, and a precise mathematical conjecture relating these for a pair of mirror manifolds.

alg-geom

\chapter*{} \lecturename{Bibliography} \lecturestar{BIBLIOGRAPHY} \markboth{D. R. Morrison, Mathematical Aspects of Mirror Symmetry}{Bibliography} \bibliographystyle{amsplain} \ifx\undefined\leavevmode\hbox to3em{\hrulefill}\, \newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,} \fi \addvspace\linespacing \noindent{\large\bfseries Books}\par \addvspace{.5\linespacing} \chapter*{Mathematical Aspects of Mirror~Symmetry} \auth{David R. Morrison} \lecturename{Introduction} \markboth{D. R. Morrison, Mathematical Aspects of Mirror Symmetry}{Mathematical Aspects of Mirror Symmetry} \addcontentsline{toc}{chapter}{Introduction} \addvspace\linespacing \noindent{\large\bfseries Introduction}\par \addvspace{.5\linespacing} \noindent {\em Mirror symmetry}\/ is the remarkable discovery in string theory that certain ``mirror pairs'' of Calabi--Yau manifolds apparently produce isomorphic physical theories---related by an isomorphism which reverses the sign of a certain quantum number---when used as backgrounds for string propagation. The sign reversal in the isomorphism has profound effects on the geometric interpretation of the pair of physical theories. It leads to startling predictions that certain geometric invariants of one Calabi--Yau manifold (essentially the numbers of rational curves of various degrees) should be related to a completely different set of geometric invariants of the mirror partner (period integrals of holomorphic forms). The period integrals are much easier to calculate than the numbers of rational curves, so this idea has been used to make very specific predictions about numbers of curves on certain Calabi--Yau manifolds; hundreds of these predictions have now been explicitly verified. Why either the pair of manifolds, or these different invariants, should have anything to do with each other is a great mathematical mystery. The focus in these lectures will be on giving a precise mathematical description of two string-theoretic quantities which play a primary r\^ole in mirror symmetry: the so-called $A$-model and $B$-model correlation functions on a Calabi--Yau manifold. The first of these is related to the problem of counting rational curves while the second is related to period integrals and variations of Hodge structure. A natural mathematical consequence of mirror symmetry is the assertion that Calabi--Yau manifolds often come in pairs with the property that the $A$-model correlation function of the first manifold coincides with the $B$-model correlation function of the second, and {\em vice versa}. Our goal will be to formulate this statement as a precise mathematical conjecture. There are other recent mathematical expositions of mirror symmetry, by Voisin \cite{voisin} and by Cox and Katz \cite{coxkatz}, which concentrate on other aspects of the subject; the reader may wish to consult those as well in order to obtain a complete picture. I have only briefly touched on the physics which inspired mirror symmetry (in lectures one and eight), since there are a number of good places to read about some of the physics background: I recommend Witten's address at the International Congress in Berkeley \cite{witten:physgeom}, a book on ``Differential Topology and Quantum Field Theory'' by Nash \cite{nash}, and the first chapter of H\"ubsch's ``Calabi--Yau Manifolds: A Bestiary for Physicists'' \cite{hubsch}. There are, in addition, three collections of papers related to string theory and mirror symmetry which contain some very accessible expository material: ``Mathematical Aspects of String Theory'' (from a 1986 conference at U.C. San Diego) \cite{stringbook}, ``Essays on Mirror Manifolds'' (from a 1991 conference at MSRI) \cite{mirrorbook}, and its successor volume ``Mirror Symmetry II'' \cite{MSII}. I particularly recommend the paper by Greene and Plesser ``An introduction to mirror manifolds'' \cite{GP:intro} and the paper by Witten ``Mirror manifolds and topological field theory'' \cite{witten:mirror}, both in the MSRI volume. This is a revised version of the lecture notes which I prepared in conjunction with my July, 1993 Park City lectures, and which I supplemented when delivering a similar lecture series in Trento during June, 1994. The field of mirror symmetry is a rapidly developing one, and in finalizing these notes for publication I have elected to let them remain as a ``snapshot'' of the field as it was in 1993 or 1994, making only minor modifications to the main text to accommodate subsequent developments. I have, however, added a postscript that sketches the progress which has been made in a number of different directions since then. \addvspace\linespacing \noindent{\large\bfseries Acknowledgments}\par \addvspace{.5\linespacing} \noindent The ideas presented here concerning the mathematical aspects of mirror symmetry were largely shaped through conversations and collaborations I have had with Paul Aspinwall, Brian Greene, Sheldon Katz, Ronen Plesser, and Edward Witten. It is a pleasure to thank them all for their contributions. I am grateful to Antonella Grassi and Yiannis Vlassopoulos for providing me with copies of the notes they took during the lectures. I am also grateful to Grassi, Katz, Plesser, and Vlassopoulos, as well as to Michael Johnson, Lisa Traynor, and the referee of \cite{compact}, for pointing out errors in the first drafts of these notes. This research was supported in part by the National Science Foundation under grant number DMS-9103827. \chapter*{} \lecturename{Some Ideas From String Theory} \lecture \markboth{D. R. Morrison, Mathematical Aspects of Mirror Symmetry}{Lecture 1. Some Ideas From String Theory} \label{stringtheory} \section{String theory and quantum field theory} The origins of the startling calculations which have led to tremendous interest among mathematicians in the phenomenon of ``mirror symmetry'' lie in string theory. String theory is a proposed model of the physical world which idealizes its fundamental constituent particles as one-dimensional mathematical objects (``strings'') rather than zero-dimensional objects (``points''). In theories such as general relativity, one has traditionally imagined a point as tracing out what is known as a ``worldline'' in spacetime; the corresponding notion in string theory is of a ``worldsheet'' which will describe the trace of a {\em string}\/ in spacetime. We will consider here only ``closed string theory'' in which the string is a closed loop; the worldsheets are then (locally) closed surfaces: if we look at a portion of a worldsheet which represents the history of several interacting particles over a finite time interval, we will see a closed surface which has a boundary consisting of a finite number of closed loops. An early version of string theory was proposed as a model for nuclear processes in the 1960's. Those early investigations revealed a somewhat disturbing property: in order to get a sensible physical theory, the spacetime $M$ in which the string is propagating must have dimension twenty-six. Obviously, when we look around us, we do not see twenty-six dimensions. A later variant which incorporates supersymmetry\footnote{I shall not attempt to explain supersymmetry in these lectures.} is sensible exactly when the spacetime has dimension ten---again a bit larger than the four-dimensional spacetime which we observe. Partly for this reason, but primarily because a better model for nuclear processes was found, the original research activity in string theory largely died out in the early 1970's. String theory was subsequently revived in the 1980's when it was shown \cite{quantum:gravity} that if the ten-dimensional string theory were used to model things at much smaller distance scales, an apparently consistent quantum theory of gravity could be produced. (In fact, gravity is predicted as an essential ingredient of this theory.) This ``anomaly cancellation'' result explained how certain potential inconsistencies in the quantum theory are avoided through an interaction between gravity and the other forces present. Tremendous optimism and excitement pervaded this period, particularly since the new model contains a rich spectrum of elementary particles at low energies and exhibits many features one would expect of a ``grand unified field theory'' which could describe in a single theory all of the forces observed in nature. The ``problem'' of ten dimensions in this context can be resolved by assuming that the ten-dimensional spacetime is locally a product $M=M^{1,3}\times M^6$ of a macroscopic four-dimensional spacetime and a compact six-dimensional space whose size is on the order of the Planck length ($10^{-33}$ cm). Because this is so small compared to macroscopic lengths, one wouldn't expect to observe the compact space directly, but its effect on four-dimensional physics could be detectable in various indirect ways. The next step was even more remarkable for mathematicians---a group of string theorists \cite{CHSW} calculated that the compact six-dimensional space must have a Ricci-flat metric on it. (The physically relevant metric is actually a perturbation of this Ricci-flat one.) This is a very restrictive property---it implies, for example, that the six-manifold is a complex K\"ahler manifold of complex dimension three which has trivial canonical bundle; conversely, such K\"ahler manifolds always admit Ricci-flat metrics. (This had been conjectured by Calabi \cite{calabi} in the late 1950's and proved by Yau \cite{yau} in the mid 1970's.) These manifolds have since been named ``Calabi--Yau manifolds;'' finding and studying them become problems in algebraic geometry, thanks to Yau's theorem. The model being described here of a string propagating in a spacetime (with a specified metric) is generally regarded as a woefully inadequate description of the ``true'' string theory, a good formulation of which is as yet unknown. Indeed, if string theory is truly a theory of gravity as we observe it, then the theory should approximate general relativity when the distance scale approaches macroscopic levels. Since the metric on spacetime is part of general relativity, it should be a part of that ``approximation'' which is somehow to be deduced from a solution to the ultimate ``string equations,'' rather than being something which is put in by hand in advance. Even the {\em topology}\/ of spacetime should be dictated by the string theory. However, neither these ``string equations'' nor their exact solutions are known at present. The Calabi--Yau manifolds and their connections with string theory have been studied intensively for more than a decade. In the earliest period, these manifolds were analyzed using standard mathematical techniques, and the results were applied in a string-theoretic context. However, at the same time, other advances were being made in string theory which suggested other ways of looking at certain aspects of the theory of Calabi--Yau manifolds. This eventually led to the discovery of a surprising new phenomenon known as ``mirror symmetry,'' in which it was observed that different Calabi--Yau manifolds could lead to identical physical theories in a way that implied surprising connections between certain geometric features of the manifolds. To explain this mirror symmetry observation in more detail, we must first describe a few aspects of quantum field theory and its relationship to string theory. In classical mechanics, the worldline representing a particle is required to minimize the ``action'' (which is the energy integrated with respect to time), or more precisely, to be a stationary path for the action functional. Due to this ``stationary action principle,'' the location of the path in spacetime is completely determined by a knowledge of boundary conditions. Other physically measurable quantities associated to the particle (which are often represented as some kind of ``internal variables'') will also evolve from their boundary states in a completely predictable manner, again minimizing the action. In quantum field theory, however, this changes. Only the probability of various possible outcomes can be predicted with certainty, and {\em all}\/ trajectories---not just the action-minimizing ones---contribute to the measurement of this probability. The probability is calculated from an integral over the space of all possible paths with these initial and final states,\footnote{There are enormous mathematical difficulties in dealing with these ``path integrals'' or ``functional integrals,'' and they do not in general have a rigorous mathematical formulation. Nevertheless, in the hands of skilled practitioners they can be used to make predictions which agree with laboratory experiments to a remarkable degree of precision.} and the classical trajectory is recovered as the leading term in a stationary phase approximation to the path integral. Relativistic quantum field theories are frequently studied by treating the theory as a small perturbation of a simple type of theory---called a free field theory---whose functional integrals are well-understood. For example, the path integral describing the interaction of two charged particles can be expanded in a perturbative series whose terms are described by ``Feynman diagrams.'' The zeroth order term is the diagram \iffigs $$\vbox{\centerline{\epsfysize=2cm\epsfbox{feyn0.eps}} }$$ \else \vskip1in \noindent \fi which represents two particles which do not interact at all, the leading perturbative correction is described by the diagram \iffigs $$\vbox{\centerline{\epsfysize=2cm\epsfbox{feyn1.eps}} }$$ \else \input fakepic1.tex \noindent \fi which represents a transfer of momentum from one particle to the other via the emission and absorption of a third particle carrying the force, and higher order corrections involve diagrams with more complicated topologies---loops are allowed, for example. Such diagrams can be cut into simpler pieces, at the expense of performing an integral over all possible intermediate states. For example, the interacting Feynman diagram illustrated above can be decomposed into two more primitive pieces, \iffigs $$\vbox{\centerline{\epsfysize=2cm\epsfbox{feyn3.eps}} }$$ \else \input fakepic2.tex \noindent \fi each of which represents a fundamental ``interaction'' vertex. In string theory, the paths are replaced by surfaces: the interacting diagram might be represented as a sphere with four disks removed (or perhaps as something with more complicated topology), \iffigs $$\vbox{\centerline{\epsfysize=2cm\epsfbox{feyn4.eps}} }$$ \fi and this could also be decomposed into more primitive pieces \iffigs $$\vbox{\centerline{\epsfysize=2cm\epsfbox{feyn5.eps}} }$$ \fi (called ``pair of pants'' surfaces). One of the advantages of string theory is that this fundamental piece, the ``pair of pants'' surface, is a smooth surface, in contrast to the interaction vertex which introduced a singularity into the worldline. The methods of quantum field theory are applied to string theory in a rather interesting way. If we fix the spacetime and consider a string propagating through it, the location of the worldsheet can be viewed as a map from the worldsheet to the spacetime, and we can regard the coordinate functions on the spacetime as functions on the worldsheet. These spacetime coordinate functions are then treated as the ``internal variables'' of a two-dimensional quantum field theory---formulated on the worldsheet itself---which captures many of the important physical features of the string theory. (The functional integral in this theory involves an integration over all possible metrics on the worldsheet as well as all possible maps from the worldsheet to the spacetime.) The two-dimensional quantum field theories arising from string theory are of a particular type known as a {\em conformal field theory}; this means that a conformal change of metric on the worldsheet will act as an automorphism of the theory (typically acting linearly on various spaces of ``internal fields'' of the theory). The formulation in terms of conformal field theory has turned out to be a very fruitful viewpoint for the study of string theory. \section{Correlation functions and pseudo-holomorphic curves} The basic quantities which one needs to evaluate in any quantum field theory are the {\em correlation functions}\/ which determine the probabilities for a specified final state, given an initial state. Specifying the initial and final states means not only specifying positions, but also the values of any ``internal variables'' which form a part of the theory. The possible initial or final states in a conformal field theory can be represented as operators $\O_P$ on some fixed Hilbert space $\mathcal{H}$, often referred to as ``vertex operators.''\footnote{Generally, in quantum field theories states are represented as elements of a Hilbert space $\mathcal{H}$ but in conformal field theories there is also an operator interpretation.} (The label $P$ indicates the position; we should in principle be specifying initial conditions on an entire boundary circle, but in fact it suffices to consider a limit---within the conformal class of the given metric---in which the circles have been shrunk to zero size and the vertex operators are located at points.) The conjugate transpose of an initial state is a final state, so we often don't distinguish between those in our notation; with these conventions, the correlation function of a number of vertex operators is denoted by \[\langle\O_{P_1}\O_{P_2}\dots\O_{P_k}\rangle.\] (Note that the correlation functions are complex-valued and do not directly calculate probabilities, but also include the phase of the quantum-mechanical wavefunction.) If we fix the topology of the worldsheet we must in general integrate over the choice of metric on that worldsheet. A conformal change of metric leaves the correlation functions invariant, so we only need to integrate over the set of conformal classes of metrics, i.e., over the (finite-dimensional) moduli space ${\mathcal{M}}_{g,k}$ of $k$-punctured Riemann surfaces of genus $g$. The two-dimensional quantum field theories which are related to mirror symmetry have a subset of their correlation functions whose values do not depend on the position of the points $P_j$ on the worldsheet; these are called {\em topological}\/ correlation functions. (We don't need to consider an integral over ${\mathcal{M}}_{g,k}$ in this case.) They will be the primary objects of interest for us. In fact, due to the possibility of decomposing the worldsheet into more primitive pieces, the main case to consider is the case of three vertex operators on surfaces of genus zero, i.e., we take $\Sigma$ to be the ``pair of pants'' surface $\C\P^1-\{P_1,P_2,P_3\}$. To evaluate a correlation function \[\langle\O_{P_1}\,\O_{P_2}\,\O_{P_3}\rangle,\] however, we must still integrate over the infinite-dimensional space $\operatorname{Maps}(\Sigma,M)$ of maps from $\Sigma$ to the spacetime $M$. To proceed further, we need to introduce the ``action'' functional on the space of maps. We fix Riemannian metrics\footnote{This is a ``Euclidean'' version of the theory, whose correlation functions are related by analytic continuation to those of the ``Lorentzian'' version in which the worldsheet metric has signature $(1,1)$.} on both the worldsheet $\Sigma$ and the spacetime $M$ and define for any sufficiently smooth $\varphi\in\operatorname{Maps}(\Sigma,M)$ \[\mathcal{S}[\varphi]=\int_\Sigma\|d\varphi\|^2\,d\mu\] using the metrics to define the norm. (In practice, we take $M$ to be the compact six-dimensional manifold rather than the entire space.) The properties of the action functional are easier to analyze if we assume that $M$ has some additional structure---the minimal structure needed is a symplectic form $\omega$ and an almost-complex structure $J$ which is $\omega$-tamed. (We will review the definitions of these in lecture three.) When these have been chosen, there is a ``d-bar'' operator $\overline{\partial}_J$ on maps, and an alternate formula for the action \[\mathcal{S}[\varphi]=\int_\Sigma\|\overline{\partial}_J\varphi\|^2\,d\mu +\int_\Sigma\varphi^*(\omega).\] A lower bound for $\mathcal{S}[\varphi]$ in any homotopy class of maps is thus given by $\int_\Sigma\varphi^*(\omega)$; this bound will be achieved by the so-called pseudo-holomorphic maps---the ones for which $\overline{\partial}_J\varphi\equiv0$. These have been extensively studied by Gromov \cite{gromov} and others as a natural generalization of complex curves on K\"ahler manifolds. This action functional now appears in an integrand which is supposed to be integrated over the infinite-dimensional space of all maps. We will outline the standard manipulations which are made with these functional integrals in physics in order to express the correlation function as an infinite sum of finite-dimensional integrals. We will subsequently use the outcome of those manipulations to make mathematical definitions of the corresponding quantities in the form of a formal sum of these finite-dimensional integrals. The topological correlation functions we are studying are to be evaluated by a functional integral of the form \begin{equation}\label{eq:path} \begin{split} \langle\O_{P_1}\dots\O_{P_k}\rangle&= \int {\mathcal{D}}\varphi\,\O_{P_1}\dots\O_{P_k}\,e^{-2\pi \,{\mathcal{S}}[\varphi]}\\ &=e^{-2\pi\,\int_\Sigma\varphi^*(\omega)} \int {\mathcal{D}}\varphi\,\O_{P_1}\dots\O_{P_k}\,e^{-2\pi \,\int_\Sigma\|\overline{\partial}_J\varphi\|^2\,d\mu}. \end{split} \end{equation} (We are suppressing the ``fermionic'' part of this functional integral, which is actually very important, but explaining it would take us too far afield.) The ``topological'' property of these correlation functions turns out to imply \cite{tsm,witten:mirror} that if we introduce a parameter $t$ into the exponent of the last functional integral in eq.~\eqref{eq:path} to produce \[\int {\mathcal{D}}\varphi\,\O_{P_1}\dots\O_{P_k}\,e^{-2\pi t \,\int_\Sigma\|\overline{\partial}_J\varphi\|^2\,d\mu}, \] then the resulting expression is independent of $t$ and can be evaluated in a limit in which $t\to\infty$. In such a limit, the only contributions to the functional integral are the maps $\varphi$ for which $\overline{\partial}_J\varphi\equiv0$, i.e., the pseudo-holomorphic maps. (This trick for reducing to a finite-dimensional integral is known as the ``method of stationary phase.'') The space of pseudo-holomorphic maps in a given homotopy class is finite-dimensional, so we have reduced the evaluation of our correlation function to an infinite sum of finite-dimensional integrals, of the form \begin{equation}\label{eq:reduced} \langle\O_{P_1}\dots\O_{P_k}\rangle=\sum_{\text{homotopy classes}} e^{-2\pi\,\int_\Sigma\varphi^*(\omega)} \int_{\mathcal{M}}\mathcal{D}\varphi\,\O_{P_1}\dots\O_{P_k}, \end{equation} where $\mathcal{M}$ denotes the moduli space of pseudo-holomorphic maps in a fixed homotopy class. It is these finite-dimensional integrals on which we shall eventually base our definitions. The convergence of the infinite sum will remain an issue in our approach, and will lead us to (in some cases) assign a provisional interpretation to this formula as being a formal power series only. From the physics one expects convergence whenever the volume of the corresponding metric is sufficiently large. \section{A glimpse of mirror symmetry} If the target space $M$ for our maps is a Calabi--Yau manifold (equipped with a Ricci-flat metric), all of the vertex operators which participate in a given topological correlation functions must be of one of two distinct types. Correlation functions involving vertex operators of the first type are called {\em $A$-model correlation functions}\/ while those involving vertex operators of the second type are known as {\em $B$-model correlation functions}\/ \cite{witten:mirror}. (These are actually the correlation functions in two ``topological field theories'' \cite{tsm} which are closely related to the original quantum field theories.) For each type, the vertex operators $\O_P$ in the quantum field theory or topological field theory have a geometric interpretation; we will treat the correlation functions as functions of these geometric objects. The $A$-model correlation functions can be defined in a much broader context than Calabi--Yau manifolds: they can be defined for any semipositive symplectic manifold $M$ (where semipositive roughly means that $-c_1(M)$ is nonnegative---we will give the precise definition in lecture three). The vertex operators $\O_{P_j}$ in the topological field theory correspond to harmonic differential forms $\alpha_j$ on $M$, and the correlation functions $\langle\alpha_1\,\alpha_2\,\alpha_3\rangle$ take the form of an infinite series whose constant term---corresponding to homotopically trivial maps from $\Sigma$ to $M$---is the familiar trilinear function $\int_M\alpha_1\wedge\alpha_2\wedge\alpha_3$. To evaluate the non-constant terms we need an integral over the moduli space of pseudo-holomorphic two-spheres. In the Calabi--Yau case, those two-spheres are expected to be discrete (based on a formal dimension count), so there should be invariants which count the number of rational curves in a given homology class. (There are certain technical difficulties with this, as we shall see in lecture two.) More generally, the non-constant terms in the $A$-model correlation functions will be related to certain kinds of counting problems for pseudo-holomorphic curves on a semipositive symplectic manifold. The $B$-model correlation functions, on the other hand, require a choice of nonvanishing holomorphic $n$-form $\Omega$ on $M$ for their definition, so they are restricted to the Calabi--Yau case. The vertex operators in the topological correlation functions correspond to elements in the space $H^q(\Lambda^pT^{(1,0)}_M)$, where we use $T^{(1,0)}$ to denote the holomorphic tangent bundle of an almost-complex manifold. (More precisely, we should use Dolbeault cohomology to describe $H^q(\Lambda^pT^{(1,0)}_M)$, and take harmonic representatives to get the vertex operators in the topological field theory.) The ``first term'' in the correlation function is then defined as a composition of the standard map on cohomology groups \[H^{q_1}(\Lambda^{p_1}T^{(1,0)}_M)\times H^{q_2}(\Lambda^{p_2}T^{(1,0)}_M)\times H^{q_3}(\Lambda^{p_3}T^{(1,0)}_M) \to H^{n}(\Lambda^{n}T^{(1,0)}_M)\] (for $p_1+p_2+p_3=q_1+q_2+q_3=n$) with some isomorphisms depending on the choice of $\Omega^{\otimes2}$ \[H^n(\Lambda^n(T^{(1,0)}_M)) \overset{\lhk\,\Omega}{\longrightarrow} H^n(\O_M)\cong\left(H^0(K_M)\right)^* \overset{\otimes\Omega}{\longrightarrow} \mathbb{C},\] where the middle isomorphism is Serre duality. (This can be written as an integral over $M$, and so can be thought of as coming from integrating over the moduli space of homotopically trivial maps from $\Sigma$ to $M$---this is why we identify it with the first term in an expansion like eq.~\eqref{eq:reduced}.) Remarkably, all of the other terms in the expansion \eqref{eq:reduced} of a $B$-model correlation function are known to vanish on physical grounds \cite{DG:exact,witten:mirror}, so we can calculate these correlation functions exactly using geometry, and even use the geometric version of the correlation function as a mathematical definition. In brief, the idea of mirror symmetry is this. There could be pairs of complex manifolds $M$, $W$ (each with trivial canonical bundle) which produce identical physics when used for string compactification, except that the r\^oles of the $A$-model and $B$-model correlation functions are reversed. In particular, this would imply the existence of isomorphisms \[H^q(\Lambda^p(T_M^{(1,0)})^*)\cong H^q(\Lambda^p(T_W^{(1,0)}))\] (and vice versa), as well as formulas relating the $A$-model correlation functions on $M$ (which count the number of rational curves) to the $B$-model correlation functions on $W$ (which are related to period integrals of $\Omega$). \chapter*{} \lecturename{Counting Rational Curves} \lecture \markboth{D. R. Morrison, Mathematical Aspects of Mirror Symmetry}{Lecture 2. Counting Rational Curves} \noindent In this lecture we begin the discussion of the problem of counting rational curves on a complex threefold with trivial canonical bundle (a ``Calabi--Yau threefold''). These curve-counting invariants will eventually be used to formulate a mathematical version of the $A$-model correlation functions. In the present lecture, we focus on the problems one encounters in formulating these invariants purely algebraically; we give a number of examples. Consider the deformation theory of holomorphic maps from $\C\P^1\to M$, where $M$ is a complex projective variety. If we are given such a map $\varphi:\C\P^1\to M$, then a first order variation of that map can be described by specifying in which direction (and at what rate) each point of the image moves. That is, we need to specify a holomorphic tangent vector of $M$ for every point on $\C\P^1$, or in other words, a section of $H^0(\C\P^1,\varphi^*(T^{(1,0)}_M))$. As might be expected from other deformation problems, the obstruction group for these deformations is $H^1(\C\P^1,\varphi^*(T^{(1,0)}_M))$. The moduli problem for such maps will be best-behaved if the obstruction group vanishes, that is, if $h^1(\C\P^1,\varphi^*(T^{(1,0)}_M))=0$. When that is true, the moduli space will be a smooth complex manifold of complex dimension $h^0(\C\P^1,\varphi^*(T^{(1,0)}_M))$. More generally, the Euler--Poincar\'e characteristic \[\chi(\varphi^*(T^{(1,0)}_M))= h^0(\C\P^1,\varphi^*(T^{(1,0)}_M))-h^1(\C\P^1,\varphi^*(T^{(1,0)}_M))\] can be regarded as the ``expected complex dimension'' of the moduli space. Although the Euler--Poincar\'e characteristic can be easily computed from the Riemann--Roch theorem for vector bundles, we shall make a more elementary calculation, based on a structure theorem for bundles on $\C\P^1$. \begin{theorem}[Grothendieck] Every vector bundle ${\mathcal{E}}$ on $\C\P^1$ can be written as a direct sum of line bundles: \[{\mathcal{E}}\cong\O(a_1)\oplus\cdots\oplus\O(a_n).\] \end{theorem} Using a Grothendieck decomposition for $\varphi^*(T^{(1,0)}_M)$, we can calculate the cohomology directly. For if \[\varphi^*(T^{(1,0)}_M)\cong\O(a_1)\oplus\cdots\oplus\O(a_n)\] then using the fact the $h^0(\O(a))=1+a$ we find \[h^0(\varphi^*(T^{(1,0)}_M))=\sum_j \begin{cases} 1+a_j&\text{if } a_j\ge-1\\ 0&\text{if } a_j<-1 \end{cases} \] while since $H^1(\O(a_1)\oplus\cdots\oplus\O(a_m))\cong H^0(\O(-2-a_1)\oplus\cdots\oplus\O(-2-a_m))^*$ we have \[h^1(\varphi^*(T^{(1,0)}_M))=\sum_j \begin{cases} -(1+a_j)&\text{if } -2-a_j\ge0\\ 0&\text{if } -2-a_j<0 \end{cases} \] since $1+(-2-a_j)=-(1+a_j)$. Taking the difference, we find \[ \chi(\varphi^*(T^{(1,0)}_M)) =\sum_j(1+a_j)=n+\sum_ja_j=\dim_\mathbb{C} M+\deg\varphi^*(-K_M).\] Thus, the ``expected dimension'' is independent of the decomposition. The same result can be obtained from Riemann--Roch. But our calculation shows more---to ensure vanishing of the obstruction group, we must have $a_j\ge-1$ for all $j$. In addition to this condition on the $a_j$'s, they must also satisfy $\max\{a_j{-}2\}\ge0$, which is seen as follows. From the exact sequence \[0\toT^{(1,0)}_{\C\P^1}\to\varphi^*(T^{(1,0)}_M)\to N_\varphi\to0\] (where $N_\varphi$ denotes the normal bundle) and the fact that $T^{(1,0)}_{\C\P^1}\cong\O(2)$, we see that there must be a nontrivial homomorphism \[\O(2)\to\O(a_1)\oplus\cdots\oplus\O(a_m),\] which implies that $\max\{a_j{-}2\}\ge0$ as claimed. Without loss of generality, we may therefore assume that $a_1\ge2.$ In the case relevant to string theory ($K_M=0$, $\dim_\mathbb{C} M=3$) we then find that in order to have vanishing obstruction group we need \[0=a_1+a_2+a_3\ge2-1-1=0\] and so $a_1=2$, $a_2=a_3=-1$. In this case, the moduli space of holomorphic maps will be smooth of dimension three; if we mod out by the automorphism group $\operatorname{PGL}(2,\mathbb{C})$, the moduli space of unparameterized maps will be smooth of dimension $0$. The points in that space are what we would like to ``count.'' We discuss some examples, drawn largely from \cite{katz:mirror}, to which we refer the reader for more details. \begin{example} \label{exampleone} {\it Lines on the Fermat quintic threefold.} All of the lines on the Fermat quintic threefold \[\{x_0^5+x_1^5+x_2^5+x_3^5+x_4^5=0\}\subset\C\P^4\] can be described as follows.\footnote{We use the Fermat quintic because it is an easily-described nonsingular hypersurface, and because it will be related to a mirror symmetry construction later on, {\em not}\/ because Wiles announced a proof of Fermat's Last Theorem while the 1993 Park City Institute was underway!} \medskip \noindent {\it First type}\/ (375 lines): The line described by $x_0+x_1=x_2+x_3=x_4=0$, and others whose equations are obtained from these by permutations and multiplication by fifth roots of unity. \medskip \noindent {\it Second type}\/ (50 one-parameter families of lines): The lines described parametrically by \[(u,v)\mapsto(u,-u,av,bv,cv)\] for fixed constants $a$, $b$, $c$ satisfying $a^5+b^5+c^5=0$, and others whose parameterizations are obtained from these by permutations and multiplication by fifth roots of unity. \medskip \noindent So we see that the lines are not always finite in number, even for smooth hypersurfaces. (One might have suspected such a ``universal finiteness for smooth hypersurfaces'' based on experience with cubic surfaces---every smooth cubic surface in $\C\P^3$ has precisely twenty-seven lines.) \end{example} \begin{example} {\it Lines on the general quintic threefold.} However, if we deform from the Fermat quintic threefold to a general one, it is possible to show that the number of lines is finite. The generic number of lines can then be computed as follows. Start from the Grassmannian $\operatorname{Gr}(\C\P^1,\C\P^4)$ of lines in $\C\P^4$. Consider the universal bundle $U$ whose fiber at a line $L$ is the two-dimensional subspace $U_L\subset\mathbb{C}^5$ such that $\P(U_L)=L$. We define a bundle ${\mathcal{B}}=\operatorname{Sym}^5(U^*)$ whose fibers describe the quintic forms on the lines $L$. Then every quintic threefold $M$ determines a section $s_M\in\Gamma({\mathcal{B}})$: the equation of $M$ is restricted to $L$ to give a homogeneous quintic there. Clearly, the lines contained in $M$ are precisely those whose corresponding points in the Grassmannian are zeros of the section $s_M$. The Grassmannian $\operatorname{Gr}(\C\P^1,\C\P^4)$ has complex dimension six, and the bundle ${\mathcal{B}}$ has rank six; when things are generic, the section $s_M$ will have finitely many zeros, which can be counted by calculating \[\#\{L\ |\ s_M(L)=0\}=c_6({\mathcal{B}})=2875.\] \end{example} \begin{exampleonebis} Katz \cite{katz:mirror} has found a way to assign multiplicities to each of the isolated lines, and one-parameter families of lines, on the Fermat quintic threefold. His multiplicity assignment for each of the 375 isolated lines is ``5,'' and that for each of the 50 one-parameter families is ``20.'' Thus, the total count is \[5\cdot375+20\cdot50=2875.\] Katz's methods of assigning multiplicities are not yet completely general,\footnote{See the ``Postscript: Recent Developments'' section for the current status.} but they do hold out the hope that a ``count'' of rational curves might be made even in cases when the actual number of curves is not finite. \end{exampleonebis} \begin{example} {\it Conics on the general quintic threefold.} We can make a similar calculation for conics on the general quintic threefold. The key observation is that every conic spans a $\C\P^2$, so the starting point for describing them is the Grassmannian $\operatorname{Gr}(\C\P^2,\C\P^4)$. We need the bundle over the Grassmannian whose fiber is the set of conics in the $\C\P^2$ in question: this is described by $\P(\operatorname{Sym}^2(U^*))$, where $U$ is the universal subbundle as before. The space $\P(\operatorname{Sym}^2(U^*))$ contains degenerate conics (pairs of lines, and double lines) as well as smooth conics. However, if $M$ is sufficiently general, then the actual locus of conics which lie in $M$ will be finite in number, and contain only smooth conics. The vector bundle which will get a section $s_M$ for every quintic $M$ is the bundle ${\mathcal{B}}:=\operatorname{Sym}^5(U^*)/(\operatorname{Sym}^3(U^*)\oplus\O_\P(-1))$. This describes the effect of restricting the quintic equation to the conic: one gets a quintic equation on the $\C\P^2$, but must mod out by those quintics which can be written as the product of a cubic (the $\operatorname{Sym}^3(U^*)$ factor) and the given conic. We have $\dim_\mathbb{C}\P(\operatorname{Sym}^2(U^*))=\operatorname{rank}{\mathcal{B}}=11$, so the computation is made by calculating: \[\#\{C\ |\ s_M(C)=0\}=c_{11}({\mathcal{B}})=609250.\] \end{example} \begin{example} {\it Twisted cubics on the general quintic threefold.} The problem gets more difficult for twisted cubics. Again, we can look at the linear span (a $\C\P^3$) and begin by considering a Grassmannian $\operatorname{Gr}(\C\P^3,\C\P^4)$. But this time we must use a bundle ${\mathcal{H}}\to\operatorname{Gr}(\C\P^3,\C\P^4)$ whose fibers are isomorphic to the Hilbert scheme of twisted cubics in $\C\P^3$. That scheme contains limits which are quite complicated. (For example, there is a limit which is a nodal plane curve with an embedded point at the node which points out of the plane: \iffigs $$\vbox{\centerline{\epsfysize=2cm\epsfbox{cubics.eps}} }$$ \fi see Hartshorne \cite{Hartshorne}, pp.~259--260.) Although the bundle ${\mathcal{B}}$ and the section $s_M$ can be defined and understood at points representing smooth twisted cubics, their extension to the locus of degenerate cubics is by no means easy. Ellingsrud and Str\o mme \cite{ES} have, however, carried this out, and they find that the number of twisted cubics on the general quintic threefold is 317206375. \end{example} Clemens \cite{Clemens:AJ} has conjectured that the general quintic threefold will have only a finite number of rational curves of each degree, and that all of them will satisfy $\varphi^*(T^{(1,0)}_M)=\O(2)\oplus\O(-1)\oplus\O(-1)$. This has been verified up through degree nine by Katz \cite{katz:degree7}, Johnsen--Kleiman \cite{JK:nine} and Nijsse \cite{nijsse}, and the prospects are good for degrees as high as twenty-four \cite{JK:high}. However, as we have seen, making the calculation of the number becomes very difficult past degree two. In fact, for degree greater than three, effective techniques for calculating this number are not presently known. We now turn to another example which demonstrates that we cannot always expect finiteness, even for the generic deformation of a given threefold with trivial canonical bundle. \begin{example} {\it Rational curves on double solids.} We let $M$ be the double cover of $\C\P^3$, branched along a general surface $S$ of degree eight in $\C\P^3$; the double cover map is denoted by $\pi:M\to\C\P^3$. We let $\pi^*(H)$ be the pullback of a hyperplane $H$ from $\C\P^3$; the {\em degree}\/ of a rational curve $C$ will mean $\pi^*(H)\cdot C$. To find ``lines'' on $M$, that is, curves $L$ with $\pi^*(H)\cdot L=1$ we consider their images $\pi(L)$. Since $\pi^*(H)$ meets $L$ in a single point $P$, $H$ meets $\pi(L)$ in the single point $\pi(P)$. Thus, $\pi(L)$ must itself be a line. But its inverse image on $M$ will necessarily have two components: $\pi^{-1}(\pi(L))=L+L'$. In order to have this splitting into two components, the line $\pi(L)$ must be tangent to $S$ at every point of intersection with $S$, i.e., it must be four-times tangent to $S$. Now the Grassmannian $\operatorname{Gr}(\C\P^1,\C\P^3)$ has dimension four, and it is one condition to be tangent to a surface, so the dimension of the set of four-tangent lines is nonnegative, and can be expected to be equal to zero. (In fact, it turns out to equal zero as expected, when $S$ is general.) The number of such lines in the Grassmannian can be calculated with the Schubert calculus; it turns out to be 14752. The corresponding count of lines on $M$ is 29504. Finding ``conics'' on $M$ is a different story, as has been observed by Katz and by Koll\'ar. Given a curve $C$ with $\pi^*(H)\cdot L=2$, there are two possibilities for $\pi(C)$: it could be a line, or it could be a conic. In the latter case, the conic $\pi(C)$ must be eight-times tangent to $S$. But in the former case, in order to have an irreducible double cover with a rational normalization, the line $\pi(C)$ must be three-times tangent to $S$. By our previous dimension count, there is at least a one-parameter family of such lines for any choice of $S$. \end{example} So we won't always have a finite number of things to ``count,'' even if we perturb to a general member of a particular family. And there is an additional difficulty if we wish to count maps from $\C\P^1$ to $M$ when multiple covers are allowed, as the next example shows. \begin{example}\label{ex:multiple} {\it Multiple covers.} Suppose that $\varphi:\C\P^1\to M$ is generically one-to-one, but that we consider a map $\varphi':=u\circ\varphi$, where $u:\C\P^1\to\C\P^1$ is a covering of degree $\mu$. Even if $\varphi^*(T^{(1,0)}_M)=\O(2)\oplus\O(-1)\oplus\O(-1)$, we will get a bad splitting of the pullback via the new map: \[\varphi'{}^*(T^{(1,0)}_M)=\O(2\mu)\oplus\O(-\mu)\oplus\O(-\mu).\] Furthermore, the dimension of the moduli space can be calculated: the moduli space of maps $u:\C\P^1\to\C\P^1$ of degree $\mu$ has dimension $2\mu{+}1$. So we see that the dimension of the space of maps will go up and up. \end{example} To handle cases such as multiple covers, ``virtual'' numbers of curves must be introduced; Katz's approach to this is to use excess intersection theory \cite{fulton:intersection}. However, this introduction of ``virtual'' numbers leads to another complication, as our final example shows. \begin{example}\label{ex:negative}{\it Negative numbers of curves}\/ (see \cite{2param2}, section 8). There are cases in which the ``virtual'' number of curves is negative. In general, when the parameter space $B$ for a family of curves is smooth of dimension $b$, the virtual number of curves should be the top Chern class of the holomorphic cotangent bundle $c_b((T^{(1,0)}_B)^*)$. If $M$ is a complex threefold with $K_M=0$ which contains $\C\P^2$ as a submanifold (which can arise from resolving a $\mathbb{Z}/3\mathbb{Z}$-quotient singularity, for example), then the lines on $\C\P^2$ are parameterized by $\C\P^2$ and have virtual number $c_2((T^{(1,0)}_{\C\P^2})^*)=3$, but the conics on $\C\P^2$, being parameterized by $\C\P^5$, have virtual number $c_5((T^{(1,0)}_{\C\P^5})^*)=-6$. This negative value actually agrees with the predictions of mirror symmetry as shown in \cite{2param2}. \end{example} \chapter*{} \lecturename{Gromov--Witten Invariants} \lecture \markboth{D. R. Morrison, Mathematical Aspects of Mirror Symmetry}{Lecture 3. Gromov--Witten Invariants} \section{Counting curves via symplectic geometry} The difficulties we encountered in trying to count rational curves on a Calabi--Yau threefold can be avoided by enlarging the category we are considering, and using Gromov's theory of pseudo-holomorphic spheres in symplectic manifolds \cite{gromov}. This approach has the advantage that the almost-complex structure can be slightly perturbed to make the number of such spheres finite, and the finite number so obtained is independent of the choice of small perturbation. Let $(M,\omega)$ be a compact {\em symplectic manifold}\/ of dimension $2n$. This means that $M$ is a compact oriented differentiable manifold of (real) dimension $2n$ and $\omega$ is a closed real two-form on $M$ which is nondegenerate in the sense that its $n^{\text{th}}$ exterior power $\omega^{\wedge n}$ is nonzero at every point. An {\it almost complex structure}\/ on a manifold $M$ is a map $J:T_M\to T_M$ whose square is $-1$. If we complexify the tangent spaces, we get $T_{M,p}\otimes\mathbb{C}= T_{M,p}^{(1,0)}\oplus T_{M,p}^{(0,1)}$, the decomposition into $+i$ and $-i$ eigenspaces for $J$. If these subspaces are closed under Lie bracket, we say that the almost-complex structure is {\em integrable}; in this case, these subspaces give $M$ the structure of a complex manifold. If $(M,\omega)$ is a symplectic manifold, an almost-complex structure $J$ on $M$ is said to be {\em $\omega$-tamed}\/ if $\omega(\xi,J\xi)>0$ for all nonzero $\xi\in T_pM$. If we have fixed an ($\omega$-tamed) almost-complex structure $J$ on $M$, and $\varphi$ is a differentiable map from $S^2$ to $M$, we define \[\overline{\partial}_J\varphi=\frac12(d\varphi+J\,d\varphi\,J_0),\] where $J_0$ is the standard almost-complex structure on $S^2$. The main example we have in mind is this: $M$ is a compact K\"ahler manifold, $\omega$ is the K\"ahler form, and $J$ is an $\omega$-tamed perturbation of the original complex structure on $M$. \begin{definition}[McDuff \cite{McD:contact}] $(M,\omega)$ is {\em semipositive}\/ if there is no map $\varphi:S^2\to M$ satisfying \[\int_{S^2} \varphi^*(\omega)>0, \quad \text{and} \quad 3-n\le\int_{S^2}\varphi^*(-K_M)<0,\] where we are writing $-K_M$ as in algebraic geometry to indicate the first Chern class $c_1(M)$, which may be represented as a two-form. \end{definition} \begin{examples} Here are three ways of producing semipositive symplectic manifolds. \begin{enumerate} \item If $K_M=0$ (the Calabi--Yau case) then $(M,\omega)$ is semipositive for any $\omega$. \item If $M$ is a complex projective manifold with $|{-}K_M|$ ample (a ``Fano variety''), then we can take $\omega=-K_M$ to produce a semipositive $(M,\omega)$. \item If $n\le3$ then $M$ is automatically semipositive. \end{enumerate} \end{examples} Because it is sometimes difficult to check whether a homology class $\eta$ is represented as the image of a map $\varphi:S^2\to M$ we also introduce a variant of this property. \begin{definition} $(M,\omega)$ is {\em strongly semipositive}\/ if there is no class $\eta\in H_2(M,\mathbb{Z})$ satisfying \[\omega\cdot \eta>0, \quad \text{and} \quad 3-n\le(-K_M)\cdot \eta<0.\] All three of our examples satisfy this stronger property as well. \end{definition} Fix a homology class $\eta\in H_2(M,\mathbb{Z})$. As we saw in example \ref{ex:multiple}, there are technical problems caused by ``multiple-covered'' maps---maps whose degree onto the image is greater than one. Let us call a map {\em simple}\/ if its degree onto its image is one. We let $\operatorname{Maps}^*_\eta(S^2,M)\subset \operatorname{Maps}_\eta(S^2,M)$ denote the subset of simple maps with fundamental class $\eta$. We also let $\operatorname{Maps}^*_\eta(S^2,M)_{(p)}$ be the set of simple differentiable maps $S^2\to M$ with fundamental class $\eta$ whose derivative lies in $L_p$. Using an appropriate Sobolev norm, $\operatorname{Maps}^*_\eta(S^2,M)_{(p)}$ can be given the structure of a Banach manifold. We can then regard $\overline{\partial}_J$ as a section of the bundle $\mathcal{W}\to\operatorname{Maps}^*_\eta(S^2,M)_{(p)}$ whose fibers are \[\mathcal{W}_\varphi:=H^0_{(p)}(S^2,{\mathcal{A}}^{(0,1)}_{S^2} \otimes\varphi^*(T^{(1,0)}_M)),\] where the subscript $(p)$ denotes $L_p$-cohomology, and ${\mathcal{A}}^{(0,1)}_{S^2}$ denotes the sheaf of $(0,1)$-forms on $S^2$ (with respect to the complex structure $J_0$). The key technical properties we need are summarized in the following two theorems. \begin{theorem}[McDuff \cite{McD:examples}] If $J$ is generic, then \[\MMhol\eta:= \{\varphi\in\operatorname{Maps}^*_\eta(S^2,M)_{(p)}\ |\ \overline{\partial}_J\varphi=0\}\] is a smooth manifold of dimension \[\dim_\mathbb{R}\MMhol\eta=2\,\chi(\varphi^*(T^{(1,0)}_M)).\] (The dimension is calculated using the Atiyah--Singer index theorem, which yields the same result as the Riemann--Roch theorem did in algebraic geometry.) \end{theorem} \noindent (This theorem would have failed if we had allowed multiple-covered maps to be included.) The next theorem is due to Gromov \cite{gromov}, based on some techniques of Sacks--Uhlenbeck \cite{SacksUhlenbeck} and with further improvements by several authors \cite{PW,wolfson,rye}. (We refer the reader to those papers for a more precise statement.) \begin{theorem} $\MMhol\eta$ can be compactified by using limits of graphs of maps; this compactification has good properties. \end{theorem} In the case relevant to string theory in which $M$ is a projective manifold with $K_M=0$ of complex dimension three, we find that for generic $J$, the (real) dimension of $\MMhol\eta$ is six, and the dimension of \[\mathcal{M}^*_{(\eta,J)}:=\MMhol\eta/\operatorname{PGL}(2,\mathbb{C})\] is zero. The space $\mathcal{M}^*_{(\eta,J)}$ itself is already compact in this case; the number of points in that space is our desired invariant. (These points may need to be counted with multiplicity, or with signs.) This invariant counts the number of rational curves (of fixed topological type) on $M$ with respect to its original complex structure, if that number is finite; it can be used as a substitute for that count in the general case.\footnote{It has not yet been verified that Katz's method of assigning multiplicities to positive-dimensional components in the algebro-geometric context produces the same results as this method from symplectic geometry. Because of the need to include signs in certain circumstances, this invariant can even accommodate the ``negative virtual numbers'' which occurred in example \ref{ex:negative}.} To describe the invariants in situations more general than complex threefolds with trivial canonical bundle, we must introduce the oriented bordism group $\Omega_*(M)$. The elements of $\Omega_k(M)$ are equivalence classes of pairs $(B^k,F)$ consisting of a compact oriented differentiable manifold $B$ of dimension $k$ (but not necessarily connected), together with a differentiable map $F:B^k\to M$. We say that $(B^k,F)\sim0$ if there exists an {\em oriented bordism}\/ $(C^{k+1},H)$: i.e., a differentiable manifold $C$ of dimension $k{+}1$ and a differentiable map $H:C^{k+1}\to M$ with $\partial C^{k+1}=B^k$ and $H|_{B^k}=F$. We add elements of $\Omega_k(M)$ by means of disjoint union: $(B^k_1,F_1)+(B^k_2,F_2)=(B^k_1\cup B^k_2,F_1\cup F_2)$; the additive inverse is given by reversing orientation. The oriented bordism group $\Omega_*(M)$ is a module over the Thom bordism ring $\Omega$ (consisting of oriented manifolds modulo oriented bordisms, with no maps to target spaces) via \[N^j\cdot(B^k,F)=(N^j\times B^k,G)\] where $G(x,y)=F(y)$. \begin{theorem}[Thom \cite{Thom}, Conner--Floyd \cite{CF}]\quad \begin{enumerate} \item $\Omega_*(M)\otimes\mathbb{Q}\cong H_*(M,\mathbb{Q})\otimes\Omega$. \item If $H_*(M,\mathbb{Z})$ is torsion-free, then $\Omega_*(M)\cong H_*(M,\mathbb{Z})\otimes\Omega$. \end{enumerate} \end{theorem} To describe our basic invariants, we choose three classes $\alpha_1$, $\alpha_2$, $\alpha_3$ in $\Omega_*(M)$ represented by elements $(B^{k_1}_1,F_1)$, $(B^{k_2}_2,F_2)$, $(B^{k_3}_3,F_3)$, and let $Z_j=\operatorname{Image}(F_j)$. We call the invariants defined below the {\em Gromov--Witten invariants}, since it was Witten \cite{tsm} who pointed out how Gromov's study of $\MMhol\eta$ could be used in principle to describe invariants relevant in topological quantum field theory. The detailed construction of these invariants was recently carried out by Ruan \cite{ruan}. There are two cases to consider, with one being more technically challenging than the other.\footnote{To simplify the exposition, we have altered Ruan's description of the second case, ignored the necessity of passing to the inhomogeneous $\overline{\partial}$ equation (introduced already by Gromov \cite{gromov}), and built into our definition the so-called ``multiple cover formula'' expected from the physics \cite{CDGP,aspmor}. (This latter step is now justified thanks to a theorem of Voisin \cite{voisin:multiple}; there is also a related result of Manin \cite{Manin}.) We are also abusing notation somewhat by using $\Phi_\eta$ in both cases, since the second case is actually related to Ruan's $\widetilde\Phi_\eta$ invariant.} \begin{construction}[Ruan] Let $\eta\in H_2(M,\mathbb{Z})$, let $\alpha_j=(B^{k_j},F_j)$ be a bordism class, and let $Z_j=\operatorname{Image}(F_j)$, for $j=1,2,3$. Suppose that $\sum_{j=1}^3(2n-k_j)=2n-2K_M\cdot \eta$, where $\eta$ is the class of the image of $\varphi$, and suppose that the almost-complex structure $J$ is generic. \begin{itemize} \item[(a)] If $-K_M\cdot\eta>0$, then \[\{\varphi\in\MMhol\eta\ |\ \varphi(0)\in Z_1, \varphi(1)\in Z_2, \varphi(\infty)\in Z_3\}\] is a finite set. Let $\Phi_{\eta}(\alpha_1,\alpha_2,\alpha_3)$ denote the signed number of points in this set, with signs assigned according to orientations at the specified points of intersection. \item[(b)] If $-K_M\cdot\eta=0$, then there exists an integer $\Phi_{\eta}(\alpha_1,\alpha_2,\alpha_3)$ which agrees with \[\#\{\text{generically injective}\ \varphi\in\MMhol\eta\ |\ \varphi(0)\in Z_1, \varphi(1)\in Z_2, \varphi(\infty)\in Z_3\}\] (counted with signs) whenever the latter makes sense. (The signs are all positive if the almost-complex structure is integrable.) \end{itemize} These invariants $\Phi_{\eta}(\alpha_1,\alpha_2,\alpha_3)$ depend only on the bordism classes $\alpha_1$, $\alpha_2$, $\alpha_3$, and do not change under small variation of $J$. \end{construction} \section{Simple properties of Gromov--Witten invariants} In spite of the fact that we needed to pass to bordism to ensure that the Gromov--Witten invariants are well-defined, their dependence on bordism-related phenomena is minimal. In fact, Ruan checks that the invariants are trivial with respect to the $\Omega$-module structure on $\Omega_*(M)$, and so it follows from the theorem of Thom and Conner--Floyd that we get a well-defined $\mathbb{Q}$-valued invariant on rational homology $H_*(M,\mathbb{Q})$. If $M$ has no torsion in homology, we even get an integer-valued invariant on integral homology. The Gromov--Witten invariants $\Phi_\eta(\alpha_1,\alpha_2,\alpha_3)$ will vanish if $\alpha_1$ corresponds to a class of real codimension zero or one. This is easy to see---if there are any elements in the set \[\{\varphi\in\MMhol\eta\ |\ \varphi(0)\in Z_1, \varphi(1)\in Z_2, \varphi(\infty)\in Z_3\}\] then the intersection of the image of $\varphi$ with the image $Z_1$ of $F_1$ has real dimension two or one. By varying the location of $\varphi(0)$, we will produce a two-{} or one-parameter family of maps. This contradicts the set being finite; thus, the set must be empty and the invariant vanishes. Note what happens to the Gromov--Witten invariants in the case of interest to string theory ($\dim_\mathbb{C} M=3$, $K_M=0$): the only relevant invariants are those with $k_1=k_2=k_3=4$. (This is because $k_j\le 4$ to get a nonzero invariant, so that $6=\sum (6-k_j)\ge\sum_{j=1}^3 2=6$, which implies that each $k_j$ is $4$.) The possible location of $0$ under a generically injective map is easy to spot: the image curve $\varphi(S^2)$ is some rational curve on $M$, and meets the four-manifold $Z_1$ in precisely $\#(Z_1\cap \eta)=\alpha_1\cdot\eta$ points; we can choose any of these for the image of $0$. Similar remarks about the images of $1$ and $\infty$ lead to the calculation: \[\Phi_\eta(Z_1,Z_2,Z_3) =(\alpha_1\cdot\eta)(\alpha_1\cdot\eta)(\alpha_1\cdot\eta)\, \#(\mathcal{M}^*_{(\eta,J)}).\] \section{The $A$-model correlation functions} Although we have defined the Gromov--Witten invariants for oriented bordism classes, we will now use them in cohomology instead. As previously remarked, thanks to the triviality of the invariants under the $\Omega$-module structure, if we tensor with $\mathbb{Q}$ we can move the invariants to homology (and then by Poincar\'e duality, to cohomology). This is at the expense of possibly allowing them to become $\mathbb{Q}$-valued on integral classes. One hopes that they will remain integer valued on integer classes, but this has not yet been established. Therefore, we will give a presentation using $\mathbb{Q}$-coefficients, but the reader should bear in mind that most of the formulas are expected to be valid with integer coefficients if one uses integer cohomology classes. In brief, the bordism class of $\alpha=(B^k,F)$ gives rise to a homology class $[Z]\in H_k(M,\mathbb{Z})$ (using the image $Z$ of $F$ to represent the class), and by duality to a cohomology class $\zeta=[Z]^\vee\in H^{2n-k}(M,\mathbb{Q})$. (Our retreat to $\mathbb{Q}$-coefficients will be in part because we do not know that every integer cohomology class can be so represented.) We extend the definition of Gromov--Witten invariants to cohomology by defining \[\Phi_\eta(\zeta_1,\zeta_2,\zeta_3):=\Phi_\eta(\alpha_1,\alpha_2,\alpha_3)\] when $\alpha_j=(B^{k_j}_j,F_j)$ and $\zeta_j=[\operatorname{Image}(F_j)]^\vee$; then extend by linearity to all of $H^*(M,\mathbb{Q})$. Our ``$A$-model correlation functions'' are then built from the Gromov--Witten invariants, following a calculation from the physics literature \cite{strominger,DSWW,CDGP,aspmor}. There is a certain danger in using the {\em outcome}\/ of a physics calculation as a {\em definition}---later, the physicists may become interested in a slightly different problem, whose outcome is radically different from the original one, and we mathematicians will find that our definitions are inadequate. Nevertheless, we will go ahead and define the $A$-model correlation functions. These are trilinear functions on the cohomology $H^*(M,\mathbb{Q})$ defined by: \begin{equation}\label{A:correlation} \begin{split} \langle\zeta_1\,\zeta_2\,\zeta_3\rangle:= (\zeta_1\cup\zeta_2\cup\zeta_3)|_{[M]}\ \ &+\sum_{\substack{\eta\in H_2(M,\mathbb{Z}),\\{-}K_M\cdot\eta>0}} \Phi_\eta(\zeta_1,\zeta_2,\zeta_3)\,q^{\eta}\\ &+\sum_{\substack{0\ne\eta\in H_2(M,\mathbb{Z}),\\{-}K_M\cdot\eta=0}} \Phi_\eta(\zeta_1,\zeta_2,\zeta_3)\,\sum_{m=1}^\infty q^{m\eta} \end{split}\end{equation} It is sometimes convenient to formally sum the geometric series in the final term, and write $q^\eta/(1-q^\eta)$ in place of $\sum_{m=1}^\infty q^{m\eta}$, in which case eq.~\eqref{A:correlation} becomes \begin{equation}\label{A:correlationbis} \begin{split} \langle\zeta_1\,\zeta_2\,\zeta_3\rangle:= (\zeta_1\cup\zeta_2\cup\zeta_3)|_{[M]}\ \ &+\sum_{\substack{\eta\in H_2(M,\mathbb{Z}),\\{-}K_M\cdot\eta>0}} \Phi_\eta(\zeta_1,\zeta_2,\zeta_3)\,q^{\eta}\\ &+\sum_{\substack{0\ne\eta\in H_2(M,\mathbb{Z}),\\{-}K_M\cdot\eta=0}} \Phi_\eta(\zeta_1,\zeta_2,\zeta_3)\, \frac{q^{\eta}}{1-q^{\eta}} \end{split}\end{equation} The terms with $K_M\cdot\eta=0$ have been separated out because they are where the multiple-covered maps cause the greatest difficulty. Heuristically, the coefficients in these functions (as we have defined them) are expected to count the simple maps only. The symbol $q^\eta$ which appears in these formulas has not yet been defined. In fact, there are two natural interpretations of eq.~\eqref{A:correlation}, one algebraic and one geometric, and we consider them in turn in the next two lectures. \chapter*{} \lecturename{The Quantum Cohomology Ring} \lecture \markboth{D. R. Morrison, Mathematical Aspects of Mirror Symmetry}{Lecture 4. The Quantum Cohomology Ring} \section{Coefficient rings} There are several possible ways to interpret the ``$A$-model correlation functions'' defined by eq.~\eqref{A:correlation}. In this lecture, we will focus on the algebraic interpretation, in which the symbol $q^\eta$ can be regarded as an element of a group ring or semigroup ring.\footnote{I am grateful to A.~Givental for pointing out the relevance of group rings.} Recall that for any commutative semigroup ${\mathcal{S}}$ and any commutative ring $R$, the {\em semigroup ring of ${\mathcal{S}}$ with coefficients in $R$}\/ is the ring \[R[q;{\mathcal{S}}]:=\left\{\sum_{\eta\in{\mathcal{S}}}a_\eta q^\eta\ |\ a_\eta\in R\text{ and } \{\eta\ |\ a_\eta\ne0\}\text{ is finite}\right\}.\] The symbol $q$ serves as a placeholder, translating the semigroup operation (usually written additively) into a multiplicative structure on a set of monomials. If ${\mathcal{S}}$ is a group, this coincides with the usual ``group ring'' construction. In the case of a Fano variety, the sum in eq.~\eqref{A:correlation} defining the $A$-model correlation function is finite, and we can regard it as taking values in the rational group ring\footnote{If we knew that the Gromov--Witten invariants were integers, we could use the integral group ring $\mathbb{Z}[q;H_2(M,\mathbb{Z})]$. But when we passed from bordism to cohomology we lost control of the integer structure.}\ \ $\mathbb{Q}[q;H_2(M,\mathbb{Z})]$. To be more concrete, if we assume for simplicity that $H_2(M,\mathbb{Z})$ has no torsion, and choose a basis $e_1$,\dots,$e_r$ of $H_2(M,\mathbb{Z})$, then writing $\eta=\sum a^je_j$ we may associate to $\eta$ the rational monomial $q^{\eta}\in\mathbb{Q}(q_1,\dots,q_r)$ defined by \[\log q^{\eta}=\sum a^j\log q_j .\] (One can also write this multiplicatively: \[q^{\eta}=\prod (q_j)^{(a^j)}\] but then great care is required in distinguishing exponents from superscripts.) If we choose our basis so that the coefficients $a^j$ are nonnegative for all classes $\eta$ which have nonvanishing Gromov--Witten invariants $\Phi_\eta(\zeta_1,\zeta_2,\zeta_3)$ for some $\zeta_1$, $\zeta_2$, $\zeta_3$, then each $q^{\eta}$ occurring in eq.~\eqref{A:correlation} is a {\em regular}\/ monomial, i.e., $q^{\eta}$ belongs to the polynomial ring $\mathbb{Q}[q_1,\dots,q_r]$, and we can calculate eq.~\eqref{A:correlation} in that ring. In the Calabi--Yau case in which $K_M=0$, the sum in eq.~\eqref{A:correlation} is not finite and we must work harder. The simplest interpretation would be to simply allow infinite sums $\sum a_\eta q^\eta$ as formal expressions. However, in order to construct quantum cohomology (which we shall do in the next section) we need the values of the correlation function to lie in a {\em ring}. In the definition of semigroup rings one restricts to finite sums in order to ensure that the partial sums which occur in the expansion of a product will be finite. That finiteness can still be guaranteed for products of infinite sums if the semigroup satisfies a special property, given below. We say that a semigroup ${\mathcal{S}}$ has the {\em finite partition property}\/ if for every $\eta\in\mathcal{S}$ there are only finitely many pairs $(\eta_1,\eta_2)\in{\mathcal{S}}\times{\mathcal{S}}$ such that $\eta=\eta_1+\eta_2$. For such semigroups, any expression of the form \[\sum_{\substack{(\eta_1,\eta_2)\text{ s.t.}\\ \eta_1+\eta_2=\eta}} a_{\eta_1}a_{\eta_2}\] (for fixed $\eta$) will be finite. Thus, infinite sums can be multiplied. So if ${\mathcal{S}}$ is a semigroup with the finite partition property, we define the {\em formal semigroup ring of ${\mathcal{S}}$ with coefficients in $R$}\/ to be \[R[[q;{\mathcal{S}}]]:=\{\sum_{\eta\in{\mathcal{S}}}a_\eta q^\eta\},\] with the product defined by \[(\sum_{\eta_1\in{\mathcal{S}}}a_{\eta_1} q^{\eta_1})\cdot (\sum_{\eta_2\in{\mathcal{S}}}a_{\eta_2} q^{\eta_2})= \sum_{\eta\in{\mathcal{S}}} (\sum_{\substack{(\eta_1,\eta_2)\text{ s.t.}\\ \eta_1+\eta_2=\eta}} a_{\eta_1}a_{\eta_2}) q^\eta.\] The semigroup $H_2(M,\mathbb{Z})$ of interest to us is actually a {\em group}\/ with a nontrivial free abelian part, and so does not satisfy the finite partition property. However, in eq.~\eqref{A:correlation} we are only required to sum over classes which can be realized by pseudo-holomorphic curves---these generate a smaller semigroup. If we are using an integrable almost-complex structure $J$ on $M$ for which $M$ is a K\"ahler manifold, this smaller semigroup is the {\em integral Mori semigroup}\/ defined (in the case $h^{2,0}=0$, for simplicity) as \[\mathop{\overline{\text{NE}}}\nolimits(M,\mathbb{Z}):=\{\eta\in H_2(M,\mathbb{Z})\ |\ (\omega,\eta)\ge0\ \forall\ \omega\in\overline{\mathcal{K}}_J\},\] where $\mathcal{K}_J$ is the K\"ahler cone and $\overline{\mathcal{K}}_J$ is its closure. The Mori semigroup has the finite partition property (the free part lies in a strongly convex cone, and the torsion part is finite), so we can form the formal semigroup ring $R[[q;\mathop{\overline{\text{NE}}}\nolimits(M,\mathbb{Z})]]$. Presumably, by using the symplectic version of the K\"ahler cone, we would find a similar property for the analogous semigroup in the almost-complex case and could form a similar ring in that case. There is an important variant which we will have occasion to consider. Let $\operatorname{Aut}_J(M)$ be the image in $\operatorname{Aut} H_2(M,\mathbb{Z})$ of the group of diffeomorphisms of $M$ compatible with the almost-complex structure $J$. This group acts on the pseudo-holomorphic curves and so permutes the Gromov--Witten invariants. The values of the $A$-model correlation function are preserved by the group action, and can be regarded as lying in the ring of invariants \[R[[q;\mathop{\overline{\text{NE}}}\nolimits(M,\mathbb{Z})]]^{\operatorname{Aut}_J(M)}.\] As in the Fano variety case, if we choose an appropriate basis (and assume $H_2(M,\mathbb{Z})$ is torsion-free) then we can regard the correlation function defined in eq.~\eqref{A:correlation} as taking values in a formal power series ring $\mathbb{Q}[[q_1,\dots,q_r]]$. Note that if we set all $q_j$'s to $0$, we simply recover the topological trilinear function $(\zeta_1\cup\zeta_2\cup\zeta_3)|_{[M]}$. But although the formal series in eq.~\eqref{A:correlation} is expected by the physicists to converge near $q_j=0$, no convergence properties of the series (as we have defined it) are known at present. There is an alternative to using the semigroup rings: we could instead use the Novikov rings \cite{Novikov} which have played a r\^ole elsewhere in symplectic geometry \cite{HS}. For each K\"ahler class $\omega$, the {\em Novikov ring}\/ $\Lambda_\omega$ consists of all formal power series \[\sum_{\eta\in H_2(M,\mathbb{Z})} a_\eta q^\eta\] such that the set \[\{\eta\ |\ a_\eta\ne0 \ \text{and}\ (\omega,\eta) <c\}\] is finite for all $c\in\mathbb{R}$. (If it is necessary to specify the ring $R$ in which the coefficients $a_\eta$ take their values, the notation $\Lambda(\omega,R)$ is used.) The product of two elements of $\Lambda_\omega$ is well-defined, and also belongs to $\Lambda_\omega$. In the case $H_2(M,\mathbb{Z})=\mathbb{Z}^r$, $\Lambda_\omega$ is the ring of {\em generalized Laurent series}\/ \[\{\sum a_{\vec{k}} q^{\vec{k}}\ |\ \ \text{there are only finitely many terms with $\omega\cdot\vec{k}<c$ for any $c\in\mathbb{R}$}\}.\] \section{A new algebra structure} The correlation functions defined in the previous lecture can be used to describe a new algebra structure on the cohomology of $M$, in the following way. Let $R$ be an integral domain (usually we use $R=\mathbb{Z}$ or $R=\mathbb{Q}$), and choose a coefficient ring $\mathcal{R}$ from among \begin{enumerate} \item the group ring $R[q;H_2(M,\mathbb{Z})]$ (in the case of a Fano variety), \item the formal semigroup ring with coefficients in $R$ for the Mori semigroup $R[[q;\mathop{\overline{\text{NE}}}\nolimits(M,\mathbb{Z})]]$ (when this is well-defined, such as in the case of a K\"ahler manifold), \item the subring $R[[q;\mathop{\overline{\text{NE}}}\nolimits(M,\mathbb{Z})]]^{\operatorname{Aut}_J(M)}$ of $\operatorname{Aut}_J(M)$-invariants, or \item one of the Novikov rings $\Lambda(\omega,R)$. \end{enumerate} We introduce a binary operation $\zeta_1\star\zeta_2$ on $H^*(M,\mathcal{R})$ defined by the requirement \[((\zeta_1\star\zeta_2) \cup \zeta_3)|_{[M]} =\langle\zeta_1\,\zeta_2\,\zeta_3\rangle.\] (This is well-defined since the cup product is a perfect pairing.) The class $\mbox{\rm 1\kern-2.7pt l}:=[M]^\vee\in H^0(M)$ which is dual to the fundamental class $[M]\in H_{2n}(M)$ has the property that the Gromov--Witten invariants $\Phi_\eta(\mbox{\rm 1\kern-2.7pt l},\zeta_2,\zeta_3)$ all vanish, hence \[\langle\mbox{\rm 1\kern-2.7pt l}\,\zeta_1\,\zeta_2\rangle=(\zeta_2\cup\zeta_3)|_{[M]};\] it follows that $\mbox{\rm 1\kern-2.7pt l}$ serves as the identity element for the binary operation $\star$. This interpretation of the correlation function as a binary operation also comes from physics \cite{MooreSeiberg,topgrav}. Let us return to the picture we had of the ``pair of pants'' surface \iffigs $$\vbox{\centerline{\epsfysize=2cm\epsfbox{assoc1.eps}} }$$ \else \vglue2in\noindent \fi as describing a possible evolution between an initial state with two ``incoming'' vertex operators $\zeta_1$, $\zeta_2$ on the left and a final state with one ``outgoing'' vertex operator $\zeta_1\star\zeta_2$ on the right. This point of view leads to the remarkable expectation that the binary operation should be associative! A heuristic argument for this runs as follows: the product $(\zeta_1\star\zeta_2)\star\zeta_3$ is evaluated by means of the surface \iffigs $$\vbox{\centerline{\epsfysize=3.5cm\epsfbox{assoc2.eps}} }$$ \else \vglue2in\noindent \fi (with an outgoing vertex operator of one piece attached to an incoming vertex operator of the other) while the product $\zeta_1\star(\zeta_2\star\zeta_3)$ is evaluated by means of the surface \iffigs $$\vbox{\centerline{\epsfysize=3.5cm\epsfbox{assoc3.eps}} }$$ \else \vglue2in\noindent \fi which is a deformation of the first one. So long as the values of the resulting quadrilinear function do not depend on the location of the four points in $\C\P^1$ used in defining it, these two products will agree. In fact, as we pointed out in the introduction, the correlation functions we are studying are expected from the physics to be precisely of this ``topological'' nature which makes them independent of the location of the points \cite{tsm}. This associativity property of the binary operation $\star$ can be rewritten as a set of relations which must be satisfied among the Gromov--Witten invariants themselves. This turns out to be a very deep property, which had not been proved at the time these lectures were delivered (although proofs were given not too long thereafter \cite{RuanTian,Liu,MS}). We have formulated the Gromov--Witten invariants and the binary operation at this level of generality primarily because this associativity property is such an interesting one. However, as we will see in more detail below, for the case of primary interest in mirror symmetry---that of Calabi--Yau threefolds---the associativity is automatic, and there is nothing to prove. (Associativity {\it does}\/ say something interesting for Calabi--Yau manifolds of higher dimension.) The $\mathcal{R}$-module $H^*(M,\mathcal{R})$ equipped with the binary operation $\star$ is called the {\em quantum cohomology ring}\/ of $M$, or the {\em quantum cohomology algebra}\/ when we wish to emphasize the $\mathcal{R}$-module structure. We can give a more geometric description of the new binary operation, by turning each Gromov--Witten invariant itself into a kind of binary operation. Here is a heuristic description of what this construction should look like. We want a cohomology class $Q_\eta(\zeta_1,\zeta_2)$ with the property that \[(Q_\eta(\zeta_1,\zeta_2)\cup \zeta_3)|_{[M]}= \Phi_\eta(\zeta_1,\zeta_2,\zeta_3).\] Consider the set of pseudo-holomorphic curves which satisfy the conditions imposed by $\zeta_1$ and $\zeta_2$ only: \[\mathcal{M}_\eta(\zeta_1,\zeta_2):= \{\varphi\in\MMhol\eta\ |\ \varphi(0)\in Z_1,\varphi(1)\in Z_2\},\] where $\zeta_j=[Z_j]^\vee$. To count the maps contributing to $\Phi_\eta(\zeta_1,\zeta_2,\zeta_3)$, we must look for all maps in $\mathcal{M}_\eta(\zeta_1,\zeta_2)$ which also send $\infty$ into $Z_3$. What subset of $M$ has the property that its intersection with $Z_3$ is in one-to-one correspondence with such maps? It is the subset consisting of {\em all possible}\/ points $\varphi(\infty)$ which might be mapping to $Z_3$. In other words, we can write $Q_\eta(\zeta_1,\zeta_2)=[T_\eta(Z_1,Z_2)]^\vee$, where $T_\eta(Z_1,Z_2)$ is the cycle defined by \begin{align*} T_\eta(Z_1,Z_2)&:= \{P\in M\ |\ P=\varphi(\infty) \text{ for some }\varphi\in\mathcal{M}_\eta(\zeta_1,\zeta_2)\} \\&= \bigcup_{\varphi\in\mathcal{M}_\eta(\zeta_1,\zeta_2)}\operatorname{Image}(\varphi) .\end{align*} Then $T_\eta(Z_1,Z_2)\cap Z_3$ will correspond to the maps counted by $\Phi_\eta(\zeta_1,\zeta_2,\zeta_3)$, where $\zeta_3=[Z_3]^\vee$. Note that for this heuristic description to work, we need the set $T_\eta(Z_1,Z_2)$ to be of the expected dimension. A better formal definition of $Q_\eta(\zeta_1,\zeta_2)$ would be the pushforward under evaluation at $\infty$ of the pullback of $\mathcal{M}_\eta(\zeta_1,\zeta_2)$ to the universal family of maps. Expressed in these terms, then, the binary operation can be written: \begin{equation}\label{A:binary} \begin{split} \zeta_1\star\zeta_2:= \zeta_1\cup\zeta_2\ \ &+\sum_{\substack{\eta\in H_2(M,\mathbb{Z}),\\{-}K_M\cdot\eta>0}} q^{\eta}\,Q_\eta(\zeta_1,\zeta_2)\\ &+\sum_{\substack{0\ne\eta\in H_2(M,\mathbb{Z}),\\{-}K_M\cdot\eta=0}} \frac{q^{\eta}}{1-q^{\eta}}\,Q_\eta(\zeta_1,\zeta_2) \end{split}\end{equation} Recall that the Gromov--Witten invariant $\Phi_\eta(\zeta_1,\zeta_2,\zeta_3)$ with $\zeta_j\in H^{\ell_j}(M,\mathbb{Q})$ is zero unless \[\ell_1+\ell_2+\ell_3=2n+2({-}K_M\cdot\eta), \quad\text{and\ \ }\ell_j\ge2.\] It follows that if the cycle $Q_\eta(\zeta_1,\zeta_2)$ is nonzero, we have \[Q_\eta(\zeta_1,\zeta_2)\in H^{2n-\ell_3}(M,\mathbb{Q})= H^{\ell_1+\ell_2-2({-}K_M\cdot\eta)}(M,\mathbb{Q}).\] Thus, if $K_M=0$, then the binary operation $\star$ preserves the grading on cohomology, while if $-K_M\cdot\eta>0$ the grading is shifted down by $2({-}K_M\cdot\eta)$. But note that in any case, the $\mathbb{Z}/2\mathbb{Z}$-grading on cohomology is preserved. Note also that $\ell_j\le2n$ implies $\ell_1+\ell_2+\ell_3\le6n$ and hence $-K_M\cdot\eta\le2n$. \begin{exercise} Show that the semipositivity condition $3-n<-K_M\cdot \eta$ implies that the grading cannot shift up, it can only shift down. \end{exercise} \begin{example} (cf.\ \cite{example:pm,vafa}) We now compute an example of the quantum cohomology ring. Let $M=\C\P^n$ (with $\omega$ induced from the Fubini--Study metric, which will ensure semipositivity). The formal semigroup ring in this case can be written as $\mathcal{R}=\mathbb{Q}[[q]]$ (or we could use $\mathcal{R}=\mathbb{Q}[q]$ since we know the sums are finite, this being a Fano variety). If $C$ is any complex curve on $M$, then $-K_M\cdot C=d(n+1)$, where $d$ is the degree of the curve. Since $-K_M\cdot C\le2n$, we must have $d=1$. So only lines (and constant maps) will contribute to our correlation function. Now the predicted real dimension of the space of maps $\C\P^1\to M$ whose image $L$ has degree one is \[2n+2(-K_M\cdot L)=2n+2(n+1)=4n+2\] while the actual dimension is \[\dim_\mathbb{R}\operatorname{PGL}(2,\mathbb{C})+\dim_\mathbb{R}\operatorname{Gr}(\C\P^1,\C\P^n) =6+2\cdot2(n-1)=4n+2\] so we should be able to use the given complex structure to compute the invariants. The Gromov--Witten invariants are evaluated as follows. A basis for $H^*(M,\mathbb{Q})$ is given by the classes $\zeta^k\in H^{2k}(M,\mathbb{Q})$ where $\zeta$ is the class of a hyperplane. We choose $k_1$, $k_2$, $k_3$, satisfying \[2k_1+2k_2+2k_3=4n+2\] and find that there is a {\em unique}\/ line in $\C\P^m$ meeting three fixed linear spaces of codimensions $k_1$, $k_2$ and $k_3$. And there is a unique map sending $0$, $1$, $\infty$ to the intersection points with the three linear spaces. Thus, \[\Phi_L(\zeta^{k_1},\zeta^{k_2},\zeta^{k_3})=1.\] Expressed in terms of the binary operation, we find that \[\zeta^{k_1}\star\zeta^{k_2}= \begin{cases} \zeta^{k_1+k_2}&\text{if }k_1+k_2\le n\\ \zeta^{k_1+k_2-n-1}\,q&\text{if }k_1+k_2\ge n+1\\ \end{cases}.\] It follows that the quantum cohomology ring can be described as: \[\mathcal{R}[\zeta]/(\zeta^{\star (n+1)}-q).\] \end{example} \begin{example} If we consider the case relevant to string theory ($\dim_\mathbb{C}(M)=3$, $K_M=0$), we find that the only products which differ from the cup product are products $\zeta_1\star \zeta_2$, with $\zeta_1, \zeta_2\in H^2(M)$, and these are given by \begin{equation*} \zeta_1\star\zeta_2:= \zeta_1\cup\zeta_2\ \ + \sum_{0\ne\eta\in H_2(M,\mathbb{Z})} \left( \zeta_1(\eta)\cdot\zeta_2(\eta)\cdot \#(\mathcal{M}^*_{(\eta,J)})\right)\,\frac{q^{\eta}}{1-q^{\eta}}\,\eta \end{equation*} Here, $\#(\mathcal{M}^*_{(\eta,J)})$ denotes the number of curves in class $\eta$ (counted with appropriate multiplicity). \end{example} \begin{remark} Note that the associativity of the binary operation $\star$ is automatically satisfied by threefolds with trivial canonical bundle, since only one of the products being associated can be different from the cup product. \end{remark} \begin{example} \label{example43} Let $\lambda\in H^2(M)$ be represented by $L$, a submanifold of real codimension two. If we define \[\mathcal{M}_{\eta}(\zeta):= \{\varphi\in\MMhol\eta\ |\ \varphi(1)\in \zeta\},\] and \[\Gamma_\eta(\zeta):=[\{P\in M\ |\ P=\varphi(\infty)\text{ for some } \varphi\in\mathcal{M}_\eta(\zeta)\}]^\vee,\] then we can expect that \[Q_\eta(\lambda,\zeta)=\lambda(\eta)\cdot\Gamma_\eta(\zeta).\] This is because the image of each $\varphi$ should meet $L$ in precisely $\lambda(\eta)$ points, any of which may be chosen as $\varphi(0)$. The binary operation can then be written: \begin{equation*} \begin{split} \lambda \star\zeta := \lambda \cup\zeta \ \ &+\sum_{\substack{\eta\in H_2(M,\mathbb{Z}),\\{-}K_M\cdot\eta>0}} \lambda(\eta)\,q^{\eta}\,\Gamma_\eta(\zeta)\\ &+\sum_{\substack{0\ne\eta\in H_2(M,\mathbb{Z}),\\{-}K_M\cdot\eta=0}} \lambda(\eta)\, \frac{q^{\eta}}{1-q^{\eta}}\,\Gamma_\eta(\zeta) \end{split}\end{equation*} We regard $\Gamma_\eta$ as a map on cohomology, and call it the {\em Gromov--Witten map}. \end{example} \section{Algebraic properties of the correlation functions} Let $K$ be the field of fractions of our coefficient ring $\mathcal{R}$; tensoring the quantum cohomology ring with $K$ makes it into a $K$-algebra. This quantum cohomology algebra carries some additional structure which makes it into what is known as a {\em Frobenius algebra}.\footnote{We follow standard mathematical usage \cite{CR,Karp} and do not require a Frobenius algebra to be commutative; our definition therefore differs slightly from that in \cite{Dubrov}. However, we will primarily be interested in the even part $H^{ev}(M)$ of the cohomology of $M$, on which the quantum product will in fact be commutative.} By definition this is a $K$-algebra $A$ with a multiplicative identity element $\mbox{\rm 1\kern-2.7pt l}$, such that there exists a linear functional $\varepsilon:A\to K$ for which the induced bilinear pairing $(x,y)\mapsto\varepsilon(x\star y)$ is nondegenerate. There does not seem to be a standard name for such a functional; we call it an {\em expectation function}\/ (cf.~\cite{summing}). If an expectation function exists at all, then most linear functionals on $A$ can serve as expectation functions. If $A$ is $\mathbb{Z}$-graded, we call $\varepsilon$ a {\em graded expectation function}\/ when $\ker(\varepsilon)$ is a graded subalgebra of $A$ (and we call $A$ a {\em graded Frobenius algebra}\/ when such a function exists). There is much less freedom to choose graded expectation functions. The cohomology of a compact manifold $M$ has the structure of a graded Frobenius algebra, with multiplication given by cup product, $\mbox{\rm 1\kern-2.7pt l}$ given by the standard generator of $H^0(M)$, and a graded expectation function given by ``evaluation on the fundamental class.'' The quantum cohomology algebra is a deformation of this algebra, with the expectation function given by \[\varepsilon(\zeta)= \langle\zeta\,\mbox{\rm 1\kern-2.7pt l}\,\mbox{\rm 1\kern-2.7pt l}\rangle,\] which again can be interpreted as evaluation on the fundamental class. The induced bilinear pairing \[(\zeta_1,\zeta_1)\mapsto \varepsilon(\zeta_1\star\zeta_2) =\langle\zeta_1\,\zeta_2\,\mbox{\rm 1\kern-2.7pt l}\rangle\] coincides with the usual cup product pairing. Note that the correlation function is also determined by $\varepsilon$ and $\star$, via \[\langle\zeta_1\,\zeta_2\,\zeta_3\rangle= \varepsilon(\zeta_1\star\zeta_2\star\zeta_3).\] That is, rather than specifying the correlation function first and using it to determine the quantum product, we can simply work with the quantum product and the expectation function. For most symplectic manifolds, the Frobenius algebra structure on quantum cohomology is not graded; however, in the Calabi--Yau case we get the structure of a graded Frobenius algebra. Generally, given any associative $K$-algebra $A$ with multiplicative identity, and any linear functional $\varphi$ on $A$, the kernel of the bilinear form $(x,y)\mapsto\varphi(x*y)$ is an ideal ${\mathcal{J}_\varphi}$, and the quotient ring $A/{\mathcal{J}_\varphi}$ is a Frobenius algebra with expectation function induced by $\varphi$. If $A$ is itself a Frobenius algebra with an expectation function $\varepsilon$, then by a theorem of Nakayama \cite{Nakayama} (see \cite{Karp} for a modern discussion), $\varphi$ takes the form $\varphi(x)=\varepsilon(\alpha*x)$ for some fixed element $\alpha\in A$, and $\mathcal{J}_\varphi$ coincides with the annihilator of $\alpha$. Although the correlation functions determine the ring structure, the opposite does not hold in general---there can be many expectation functions on a given algebra. However, if $A$ is a graded Frobenius algebra of finite length as a $K$-module and all elements of $A$ have nonnegative degree, then the graded expectation functions on $A$ are in one-to-one correspondence with degree $0$ elements of $A$ which are not zero-divisors. (This is because they must all be of the form $\varphi(x)=\varepsilon(\alpha*x)$ for some $\alpha$ which is not a zero-divisor, but every element of degree ${}>0$ must be a zero-divisor.) In particular, in the case of the quantum cohomology algebra of a Calabi--Yau manifold $M$ (equipped with a symplectic structure), we have a graded Frobenius algebra of finite length in which the degree $0$ elements are just the one-dimensional vector space $H^0(M)$. This means that the graded expectation function is unique up to multiplication by an element of $K$, and that the ring structure determines the correlation functions up to this overall factor. (It is not hard to see in the Calabi--Yau case that the graded expectation function is nonzero precisely on the top degree piece $H^{2n}(M)$, where $n=\dim_{\mathbb{C}}M$, and that $H^{2n}(M)$ must also be one-dimensional.) We will see this structure again when we study the $B$-model correlation functions in lecture six. \chapter*{} \lecturename{Moduli Spaces of $\sigma$-Models} \lecture \markboth{D. R. Morrison, Mathematical Aspects of Mirror Symmetry}{Lecture 5. Moduli Spaces of $\sigma$-Models} \section{Calabi--Yau manifolds and nonlinear $\sigma$-models}\label{sec:51} In this lecture, we wish to give a more geometric interpretation to the $A$-model correlation functions as defined by eq.~\eqref{A:correlation}. This geometric interpretation is motivated in part by a study of the moduli spaces of the conformal field theories associated to Calabi--Yau manifolds, so we begin with a description of those moduli spaces. Let $M$ be a K\"ahler manifold with $K_M=0$. Underlying $M$ is a differentiable manifold $X$ of real dimension $2n$. We can regard $M$ as consisting of $X$ together with a chosen integrable almost-complex structure $J$ and a K\"ahler metric $g_{ij}$, such that $K_M=0$. (The complex manifold specified by $J$ will then be denoted $X_J$.) If $\omega$ denotes the K\"ahler form of the metric, then by a theorem of Calabi \cite{calabi} there is at most one Ricci-flat metric whose K\"ahler form is cohomologous to $\omega$; by a theorem of Yau \cite{yau} such a Ricci-flat metric always exists. The global holonomy of such a metric is necessarily contained in $\operatorname{SU}(n)$. (The metric being K\"ahler implies that its holonomy is contained in $\operatorname{U}(n)\subset SO(2n)$; the Ricci-flatness further restricts the holonomy to $\operatorname{SU}(n)$, and also implies that the canonical bundle is trivial. See \cite{beauville} for an account of these holonomy conditions.) We use the term {\em Calabi--Yau manifold}\/ to mean a compact connected orientable manifold $X$ of dimension $2n$ which admits Riemannian metrics whose (global) holonomy is contained in $\operatorname{SU}(n)$. You should be aware that there are some places in the literature (including papers of mine \cite{guide}) where ``Calabi--Yau manifold'' is used in the more restrictive sense of a Riemannian manifold with holonomy precisely $\operatorname{SU}(n)$. These alternate definitions will often also insist that a complex structure has been chosen on $X$. Given a Calabi--Yau manifold $X$ (in our sense) and a metric on it whose holonomy lies in $\operatorname{SU}(n)$, there always exist complex structures on $X$ for which the given metric is K\"ahler. If $h^{2,0}=0$, then there are only a finite number of such complex structures. (If the universal cover is a written as a product of indecomposable pieces, one may apply conjugation on the various factors to obtain other complex structures.) When $h^{2,0}>0$, however, the complex structures depend on parameters. There are some very interesting cases with $h^{2,0}>0$, including the famous K3 surfaces, but lack of time in these lectures forces us to assume---with regret---that $h^{2,0}=0$ henceforth. The physical model discussed in lecture one which considers maps from surfaces to a six-dimensional target space is a special case of a class of physical theories called ``nonlinear $\sigma$-models.'' One regards these as quantum field theories on the surfaces themselves, with various vertex operators and correlation functions derived from the space of maps from the surface to the target. The target should be a fixed Riemannian manifold, usually assumed to be compact. When the Riemannian metric on the target is (a particular perturbation of) one which has holonomy in $\operatorname{SU}(n)$, the resulting ``nonlinear $\sigma$-model'' is believed to be invariant under conformal transformations of the surface. It thus is a type of ``conformal field theory''---an even broader class of physical models which have a rich literature devoted to their study (see \cite{ginsparg} for an introduction and further references). Conformal field theories typically depend on finitely many parameters, and in the case of a nonlinear $\sigma$-model those parameters have a direct geometric interpretation. In the Lagrangian formulation of the theory, one must specify the metric $g_{ij}$ on the target $X$ together with an auxiliary harmonic two-form $B$ on $X$ called the ``$B$-field.'' (To simplify matters, we take our metrics to have holonomy in $\operatorname{SU}(n)$, even though the true metrics of interest in physics will be perturbations of those; we also assume that $H_2(X,\mathbb{Z})$ has no torsion.\footnote{The correct description of the moduli space will be slightly different if torsion is included---see section \ref{sec:53} below.}) The data consisting of the pair $(g_{ij},B)$ accounts for all local parameters in the conformal field theory moduli space, so we get at least a good local description of moduli if we specify such a pair. More details about these moduli spaces can be found in \cite{ICM}. Two pairs $(g_{ij},B)$ and $(g_{ij}',B')$ will determine isomorphic conformal field theories if there is a diffeomorphism $\varphi:X\to X$ such that $\varphi^*(g_{ij}')=g_{ij}$, and $\varphi^*(B')-B\in H^2_{\text{DR}}(X,\mathbb{Z})$. (We use the notation $H^k_{\text{DR}}(X,\mathbb{Z})$ to denote the image of integral cohomology in de Rham cohomology.) This second condition arises because the appearance of $B$ in the Lagrangian is always in the form $\int_\Sigma B$, and the Lagrangian is exponentiated (with an appropriate factor of $2\pi i$) in every physically observable quantity. We call the set of all isomorphism classes of such pairs the {\em semiclassical nonlinear $\sigma$-model moduli space}, or simply the {\em $\sigma$-model moduli space}\/ (for short). This may differ from the actual {\em conformal field theory moduli space}\/ for three reasons. \begin{enumerate} \item It may happen that the physical theory does not exist for all values of $g_{ij}$ and $B$. Most of the study of these theories uses perturbative methods, valid near a limit of ``large volume'' of the metric, but it may be that the theory breaks down when the volume (either of $X$, or of images of holomorphic maps into $X$) becomes too small. \item On the other hand, there may be a sort of analytic continuation of the theory beyond the region where the $\sigma$-model description is valid. (This was shown to occur in \cite{mmm,phases}.) It was only claimed above that the specification of $(g_{ij},B)$ gave good {\em local}\/ parameters for the moduli. \item Furthermore, there could be subtle isomorphisms between conformal field theories which do not show up in the $\sigma$-model interpretation. This is known to happen in the K3 surface case \cite{AM:K3}, for example (which we have no time to discuss here)---mirror symmetry provides a new identification of conformal field theories. \end{enumerate} We will ignore these phenomena for the present, and concentrate on the ``$\sigma$-model moduli space'' which parameterizes pairs $(g_{ij},B)$ modulo equivalence. To study this moduli space using the tools of algebraic geometry, we must choose a complex structure on $X$. In fact, if we consider the set of triples $(g_{ij},B,J)$ modulo equivalence, with $J$ being an integrable almost-complex structure for which the metric $g_{ij}$ is a Ricci-flat K\"ahler metric, then the map from the set of equivalence classes of triples to that of pairs is a finite map. (It is a map of degree two if the holonomy is precisely $SU(n)$.) On the other hand, we can map the set of triples $(g_{ij},B,J)$ to the moduli space $\MM_{\text{cx}}$ of complex structures on $X$. That moduli space is quite well-behaved, both locally and globally. The local structure is given by the theorem of Bogomolov--Tian--Todorov \cite{bogomolov,tian,todorov}, which says that all first-order deformations are unobstructed. (I recommend Bob Friedman's paper \cite{Friedman:threefolds} for a very readable account of this theorem.) Thus, the moduli space $\MM_{\text{cx}}$ will be smooth, and the tangent space at $[J]$ can be canonically identified with $H^1(T^{(1,0)}_{X_J})$. Globally, $\MM_{\text{cx}}$ is known to be a quasi-projective variety (if one specifies a ``polarization'') by a theorem of Viehweg \cite{viehweg}. We will study the moduli space $\MM_{\text{cx}}$ in more detail (using variations of Hodge structure) in the next section. The fibers of the map \begin{equation}\label{fibrebundle} \{(g_{ij},B,J)\}/{\sim}\ \to\ \MM_{\text{cx}}\end{equation} (from the set of equivalence classes of triples to the moduli space) are spaces of the form $\mathcal{D}/\Gamma$, with \begin{align*}\mathcal{D}&=H^2(X,\mathbb{R})+i\,\mathcal{K}_J\\ \Gamma&=H^2_{\text{DR}}(X,\mathbb{Z})\rtimes \operatorname{Aut}_J(X). \end{align*} One hopes that the map \eqref{fibrebundle} is some kind of fiber bundle (at least generically); this would require that both the family of K\"ahler cones and the family of automorphism groups are generically locally constant. This has been shown for the K\"ahler cones in the case of complex dimension three by Wilson \cite{wilson}. The tangent spaces to the fibers of the map \eqref{fibrebundle} can be canonically identified with $H^1((T^{(1,0)}_{X_J})^*)$. Mirror symmetry predicts that $X$ should have a mirror partner $Y$, such that the moduli spaces of conformal field theories on $X$ and $Y$ should be isomorphic, but with a reversal of r\^oles of $H^1(T^{(1,0)}_{X_J})$ and $H^1((T^{(1,0)}_{X_J})^*)$. That is, under the isomorphism between the conformal field theory moduli spaces, the part of the tangent space corresponding to $H^1((T^{(1,0)}_{X_J})^*)$ on $X$ should map to the part corresponding to $H^1(T^{(1,0)}_{Y_{J'}})$ on $Y$, and vice versa. In particular, the r\^oles of base and fiber in \eqref{fibrebundle} should be reversed. This is at first sight a rather peculiar statement, since the base and the fiber do not look much alike: the base $\MM_{\text{cx}}$ is a quasi-projective variety, whereas the fiber $\mathcal{D}/\Gamma$ looks much more like a Zariski open subset of a bounded domain---a typical model for the space is $(\Delta^*)^r$, where $\Delta^*$ is the punctured disk. This is in fact one of the indicators that the conformal field theory moduli space must be analytically continued beyond the realm of $\sigma$-models, as suggested in point 2 above. We will see further evidence of this at the end of lecture seven. \section{Geometric interpretation of the $A$-model correlation functions} We turn now to a geometric interpretation of the $A$-model correlation functions, which in the case of Calabi--Yau manifolds will turn out to be closely related to the spaces $\mathcal{D}/\Gamma$ described above. In the previous lecture, the symbols $q^\eta$ were treated purely formally, which allowed us to discuss some algebraic aspects of quantum cohomology. Now, however, we would like to make the new product more geometric by giving specific values to the $q^\eta$'s, thereby making the quantum cohomology ring into a deformation of the usual cohomology ring. Turning algebraic parameters into geometric data is a familiar task for algebraic geometers; however here, we only have formal parameters. We will describe a natural parameter space as a formal completion of a certain geometric space---if some day someone proves that the series \eqref{A:correlation} and \eqref{A:binary} are convergent power series, then the true parameter space will be a neighborhood (in the classical topology) of the completion point within the geometric space which we will construct. Let $\mathcal{R}=\mathbb{Q}[[q;\mathop{\overline{\text{NE}}}\nolimits(X_J,\mathbb{Z})]]$ be the formal semigroup ring of the integral Mori semigroup. If $\mathop{\overline{\text{NE}}}\nolimits(X_J,\mathbb{Z})$ is finitely generated, then we can take as the geometric space $\operatorname{Spec}\mathbb{C}[q;\mathop{\overline{\text{NE}}}\nolimits(X_J,\mathbb{Z})]$ (the spectrum of the semigroup ring), and as its completion the formal scheme $\operatorname{Spf} {\mathcal{R}}_{\mathbb{C}}$, where ${\mathcal{R}}_{\mathbb{C}}$ denotes ${\mathcal{R}}\otimes_{\mathbb{Q}}\mathbb{C}$ and $\operatorname{Spf}$ is the formal spectrum. More generally, if the ring of $\operatorname{Aut}_J(X)$-invariants ${\mathcal{R}}^{\operatorname{Aut}_J(X)}$ is the formal completion of a ring of finite type over $\mathbb{Q}$, we take our completed parameter space to be $\operatorname{Spf} ({\mathcal{R}}_{\mathbb{C}}^{\operatorname{Aut}_J(X)})$. In the finitely generated case, this geometric space $\operatorname{Spec}\mathbb{C}[q;\mathop{\overline{\text{NE}}}\nolimits(X_J,\mathbb{Z})]$ is in a natural way an affine toric variety, and as such admits a rather concrete description: the geometric points are in one-to-one correspondence with the set of semigroup homomorphisms $\operatorname{Hom}_{\text{sg}}(\mathop{\overline{\text{NE}}}\nolimits(X_J,\mathbb{Z}),\mathbb{C})$, where $\mathbb{C}$ is given the structure of a {\em multiplicative}\/ semigroup. Any geometric point $\xi$ in the parameter space---regarded as a semigroup homomorphism---specifies compatible values $\xi(q^\eta)$ for the symbols $q^\eta$. An important open problem is to decide for which $\xi$ the series expressions \eqref{A:correlation} for the correlation functions converge. If convergent, the correlation functions would become actual $\mathbb{C}$-valued functions on a parameter space (as expected by the physicists), which would be an open subset of $\operatorname{Spec}\mathbb{C}[q;\mathop{\overline{\text{NE}}}\nolimits(X_J,\mathbb{Z})]$ in the classical topology. To make this even more concrete, consider the case in which the Mori semigroup is freely generated by elements $e_1$, \dots, $e_r$ which also serve as a basis of the lattice $H_2(X,\mathbb{Z})$. In this case, we can define $q_j:=q^{e_j}$, and write the ring $\mathcal{R}$ as a formal power series ring ${\mathcal{R}}=\mathbb{Q}[[q_1,\dots,q_r]]$. The geometric space $\operatorname{Spec}\mathbb{C}[q_1,\dots,q_r]$ can then be identified as $\mathbb{C}^r$ with coordinates $q_1,\dots q_r$. One natural candidate for the open set on which the correlation functions might converge is \[\{(q_1,\dots,q_r)\in\mathbb{C}^r\ |\ 0\le|q_j|<1\}.\] More generally, still assuming that $H_2(X,\mathbb{Z})$ is torsion-free, suppose we choose a basis $e_1$, \dots, $e_r$ whose span as a semigroup {\em contains}\/ $\mathop{\overline{\text{NE}}}\nolimits(X_J,\mathbb{Z})$. Then the corresponding formal power series ring $\mathbb{Q}[[q_1,\dots,q_r]]$ contains our coefficient ring $\mathcal{R}$. If we let $\sigma$ denote the open real cone generated by the dual basis $e^1$, \dots, $e^r$, then that formal power series ring can be more canonically described as the formal semigroup ring $\mathcal{R}_\sigma:=\mathbb{Q}[[q;\check\sigma\cap H_2(X,\mathbb{Z})]]$. The same cone $\sigma$ can be used to give a canonical description of the open set specified by $0<|q_j|<1$ in the form \[(H^2(X,\mathbb{R})+i\sigma)/H^2(X,\mathbb{Z}).\] (To see this, write a general element of $H^2(X,\mathbb{C})$ modulo $H^2(X,\mathbb{Z})$ in the form \[\frac1{2\pi i}\sum(\log q_j)e^j,\] and note that the condition $0<|q_j|<1$ is equivalent to $\Im(\frac1{2\pi i}\log q_j)>0$.) The Mori semigroup $\mathop{\overline{\text{NE}}}\nolimits(X_J,\mathbb{Z})$ will be contained in the semigroup spanned by $\{e_j\}$ precisely when the cone $\sigma$ is contained in the K\"ahler cone of $X_J$. We will treat such a choice of cone $\sigma$ as specifying a coordinate chart on the geometric space we are trying to construct. For any such cone, we define \[\mathcal{D}_\sigma=H^2(X,\mathbb{R})+i\,\sigma\subset H^2(X,\mathbb{C})\] In terms of local coordinates, as pointed out above we have \[\mathcal{D}_\sigma/H^2(X,\mathbb{Z})=\{(q_1,\dots,q_r)\ |\ 0<|q_j|<1\}.\] The open subset of our desired geometric space will be a partial compactification of this, defined by \[(\mathcal{D}_\sigma/H^2(X,\mathbb{Z}))^-=\{(q_1,\dots,q_r)\ |\ 0\le|q_j|<1\}.\] We call the origin $0\in(\mathcal{D}_\sigma/H^2(X,\mathbb{Z}))^-$ the {\em distinguished limit point}\/ in this space. It is hoped that the expressions for the $A$-model correlation functions, or for the binary operation $\zeta_1\star\zeta_2$, will converge in a neighborhood of the distinguished limit point $0$ in $(\mathcal{D}_\sigma/H^2(X,\mathbb{Z}))^-$. The different possible choices of $\sigma$ will correspond to operations---such as blowing up the boundary---which change the compactification without changing the underlying space. Intrinsically, we can describe $\mathcal{R}_\sigma\otimes\mathbb{C}$ as the formal completion of the local ring of $(\mathcal{D}_\sigma/H^2(X,\mathbb{Z}))^-$ at its distinguished limit point $0$. The geometric space which is emerging from this discussion is very closely related to the space $\mathcal{D}/\Gamma$ which formed part of the nonlinear $\sigma$-model moduli space in the case of a Calabi--Yau manifold with $h^{2,0}=0$. In fact, if $\mathcal{K}_J$ is the K\"ahler cone of such a Calabi--Yau manifold which can be partitioned into cones $\sigma_\alpha$ which are spanned by various bases of $H^2(X,\mathbb{Z})$, then $\mathcal{D}/H^2(X,\mathbb{Z})$ is the interior of the closure of the union of the sets $\mathcal{D}_{\sigma_\alpha}/H^2(X,\mathbb{Z})$. Ideally, one could make such a partition in an $\operatorname{Aut}_J(X)$-equivariant way. This would be guaranteed by the following conjecture. \begin{ConeConjecture} Let $X$ be a Calabi--Yau manifold on which a complex structure $J$ has been chosen, and suppose that $h^{2,0}(X)=0$. Let $\mathcal{K}_J$ be the K\"ahler cone of $X$, let $(\mathcal{K}_J)_+$ be the convex hull of $\overline{\mathcal{K}}_J\cap H^2(X,\mathbb{Q})$, and let $\operatorname{Aut}_J(X)$ be the group of holomorphic automorphisms of $X$. Then there exists a rational polyhedral cone $\Pi\subset(\mathcal{K}_J)_+$ such that $\operatorname{Aut}_J(X).\Pi=(\mathcal{K}_J)_+$. \end{ConeConjecture} A nontrivial case of this conjecture---Calabi--Yau threefolds which are fiber products of generic rational elliptic surfaces with section (as studied by Schoen \cite{schoen})---has been checked by Grassi and the author \cite{GM}. There are some other pieces of supporting evidence in examples worked out by Borcea \cite{borcea} and Oguiso \cite{oguiso}. When this conjecture holds, there is a partial compactification of $\mathcal{D}/\Gamma$ constructed in \cite{compact} by gluing together the spaces $(\mathcal{D}_{\sigma_\alpha}/H^2(X,\mathbb{Z}))^-$ for an $\operatorname{Aut}_J(X)$-equivariant partitioning of $\mathcal{K}_J$, and modding out by $\operatorname{Aut}_J(X)$. This produces a ``semi-toric'' partial compactification of the type introduced by Looijenga \cite{Looijenga}. Because it is covered by explicit coordinate charts, this is a convenient type of compactification for making comparisons of correlation functions. There is also a ``minimal'' semi-tori compactification determined from the same data, which partially compactifies $\mathcal{D}/\Gamma$ more directly, adding several new strata but only a single stratum of maximal codimension (the analogue of the ``distinguished limit points''). When the cone conjecture holds, the ring of invariants $\mathcal{R}^{\operatorname{Aut}_J(X)}$ is the formal completion of a ring of finite type over $\mathbb{Q}$, and the completion of the local ring of the minimal semi-toric compactification at its distinguished point $P$ coincides with $\operatorname{Spf}(\mathcal{R}_{\mathbb{C}}^{\operatorname{Aut}_J(X)})$. On such a compactification, we will expect \begin{equation}\label{eq:limA} \lim_{Q\to P}\langle\zeta_1\,\zeta_2\,\zeta_3\rangle_Q =(\zeta_1\cup\zeta_2\cup\zeta_3)|_{[X]} \end{equation} (the ``$q_j=0$ values'' in coordinate charts). Such a point is called a ``semiclassical limit'' in the physics literature \cite{AL}. \section{The r\^ole of torsion in the moduli space}\label{sec:53} Up to this point, we have not considered the effects of possible torsion in $H_2(X,\mathbb{Z})$ and in fact we have explicitly assumed at several points that there was no torsion. If torsion is present, we can define the formal semigroup ring $\mathcal{R}=\mathbb{Q}[[q;\mathop{\overline{\text{NE}}}\nolimits(X_J,\mathbb{Z})]]$ as before, and it will have a torsion part ${\mathcal{R}}_{\text{torsion}}$ whose spectrum is a finite set of geometric points. This can be identified with the set of connected components of our parameter space. It can also be seen in the following description of the $\sigma$-model moduli space. The complete description of the $\sigma$-model moduli space (with the torsion included) considers the quantity $e^{2\pi i(B+i\omega)}$ to lie in $\operatorname{Hom}(H_2(X,\mathbb{Z}),\mathbb{C}^*)$. This can be thought of concretely as having a torsion part, together with a free part which lies in the space \[\operatorname{Hom}(H_2(X,\mathbb{Z})/\text{torsion},\mathbb{C}^*)\cong H^2(X,\mathbb{C}^*)\cong H^2(X,\mathbb{C})/H^2_{\text{DR}}(X,\mathbb{Z})\] where (as in section \ref{sec:51}) $H^2_{\text{DR}}(X,\mathbb{Z})$ is the image of $H^2(X,\mathbb{Z})$ in de Rham cohomology, isomorphic to $H^2(X,\mathbb{Z})/\text{torsion}$. A representative of the free part can be written as $B_{\text{free}}+i\omega\in H^2(X,\mathbb{C})$, where $\omega$ is the K\"ahler form and $B_{\text{free}}$ is the real two-form which appeared in section \ref{sec:51}. The torsion part of $e^{2\pi i(B+i\omega)}\in\operatorname{Hom}(H_2(X,\mathbb{Z}),\mathbb{C}^*)$ can be identified with the torsion part of our coefficient ring ${\mathcal{R}}_{\text{torsion}}$ from the algebraic interpretation. One way to interpret this ``$B$-field with torsion included'' is to regard it as an element of $H^2(X,\mathbb{R}/\mathbb{Z})$. \chapter*{} \lecturename{Variations of Hodge Structure} \lecture \markboth{D. R. Morrison, Mathematical Aspects of Mirror Symmetry}{Lecture 6. Variations of Hodge Structure} \section{The $B$-model correlation functions} Our goal in this lecture is to describe the $B$-model correlation functions and how they are related to variations of Hodge structure. We work with Calabi--Yau manifolds on which complex structures have been chosen. That is, we let $W$ be a complex manifold with $K_W=0$. The assumption of trivial canonical bundle is needed in order to define the $B$-model correlation functions. Let us define \[H^{-p,q}(W):=H^q(\Lambda^p(T^{(1,0)}_W)),\] and consider all of these groups together: \[H^{-*}(W):=\bigoplus_{p,q}H^{-p,q}(W).\] There is a natural ring structure on $H^{-*}(W)$ which can be thought of as a sheaf cohomology version of the cup product pairing: \[H^q(\Lambda^p(T^{(1,0)}_W))\otimes H^{q'}(\Lambda^{p'}(T^{(1,0)}_W)) \to H^{q+q'}(\Lambda^{p+p'}(T^{(1,0)}_W)).\] Note that since these are sheaf cohomology groups, this ring structure is not ``topological'' in nature; in fact, it depends heavily on the choice of complex structure on $W$. Recall that in the case of the $A$-model correlation functions on a symplectic manifold $M$, the expectation function which determined the Frobenius algebra structure was a very familiar object, given by evaluating a cohomology class on the fundamental class of $M$ (which determines a canonical map $H^{n,n}(M)\to\mathbb{C}$). By contrast, the ring structure on quantum cohomology was unusual. In this new ``$B$-model'' case, however, the ring structure is straightforward but the expectation function is more elusive. To define it, we must choose a nonvanishing global section $\Omega^{\otimes2}$ of $(K_W)^{\otimes2}$. This is then used in two steps to specify the expectation function: \[H^{-n,n}(W)=H^n(\Lambda^n(T^{(1,0)}_W)) \overset{\lhk\,\Omega}{\longrightarrow} H^n(\O_W)\cong\left(H^0(K_W)\right)^* \overset{\otimes\Omega}{\longrightarrow} \mathbb{C},\] where the middle isomorphism is Serre duality. Using this expectation function and the ``sheaf cup product'' binary operation, we define the $B$-model correlation functions (in the standard way from the Frobenius algebra structure): \[\langle\beta_1\,\beta_2\,\beta_3\rangle= ((\beta_1\cup\beta_2\cup\beta_3)\lhk\,\Omega)\otimes\Omega.\] (Once again we have a definition which is inspired by the outcome of a calculation in the physics literature \cite{strwit}.) This gives a map \[H^{-p,q}(W)\times H^{-p',q'}(W)\times H^{-(n-p-p'),n-q-q'}(W)\to\mathbb{C}.\] Note that as in the $A$-model case, we actually have a graded Frobenius algebra of finite length, so the expectation function is uniquely defined up to a scalar multiple (which can be absorbed in the choice of $\Omega^{\otimes 2}$.) In order to relate this correlation function to a more familiar mathematical object, we can proceed as follows: first use the two $\Omega$'s to transform two of the arguments, and then use the cup product: \[\langle\beta_1\,\beta_2\,\beta_3\rangle= ((\beta_1\lhk\,\Omega)\cup\beta_2\cup(\beta_3\lhk\,\Omega)).\] This variant of the correlation function can be regarded as a map \[H^{n-p,q}(W)\times H^{-p',q'}(W)\times H^{p+p',n-q-q'}(W)\to\mathbb{C},\] or, if we treat it as a modified ``binary operation,'' as a map \[H^{n-p,q}(W)\times H^{-p',q'}(W)\to H^{n-p-p',q+q'}(W).\] This version of the ``binary operation'' expresses the cohomology $H^*(W)$ as a module over the ring $H^{-*}(W)$. As we shall see, this variant has the pleasant property that it can be directly interpreted in terms of variations of Hodge structure and the differential of the period map. Of course, the original version of the correlation function can be recovered from this, once we have specified $\Omega^{\otimes 2}$. \section{Variations of Hodge structure} We now briefly review the theory of variations of Hodge structure, in order to explain the mathematical origin of the $B$-model correlation functions. Variations of Hodge structure were introduced as a tool for measuring how the complex structure on a differentiable manifold can vary. Good general references for this are Griffiths et al.~\cite{transcendental}, and Schmid \cite{schmid}. There are two primary ways one can view deformations of complex structure. In the first viewpoint, we fix a compact differentiable manifold $Y$, and consider various integrable almost-complex structures $J$ on $Y$. Then the set of such, modulo diffeomorphism, is known to be a finite-dimensional space. In the second viewpoint, we consider proper holomorphic maps $\pi:\mathcal{W}\to S$ with $W_s=\pi^{-1}(s)$ diffeomorphic to $Y$. Each fiber $W_s$ has an induced structure of a complex manifold. If $S$ is contractable, then $\pi$ can be trivialized in the $C^\infty$ category, and we can regard $\pi$ as specifying a family of complex structures. One wants to represent the functor \[S\mapsto\{\pi:\mathcal{W}\to S\}/(\text{isomorphism}),\] by maps to a moduli space which has a ``universal family.'' This is generally too much to hope for, but there are often ``coarse moduli spaces'' whose points are in one-to-one correspondence with the possible complex structures. (The appendices in \cite{MumfordFogarty} provide good background for moduli problems in general.) We will study complex structures on $Y$ by studying the Hodge decomposition induced on cohomology by each choice of complex structure. In general, if $W_s$ is a K\"ahler manifold there is a {\em Hodge decomposition}\/ of the cohomology: \begin{equation}\label{eq:hodge} H^k(W_s,\mathbb{C})\cong \bigoplus_{p+q=k}H^{p,q}(W_s). \end{equation} Now in a family over a contractable base, the bundle of $H^k(W_s,\mathbb{C})$'s may be canonically trivialized. Over more general bases $S$ (assumed to be connected), it is convenient to consider $R^k\pi_*\mathbb{C}_\mathcal{W}$, which is simply the sheaf whose local sections are topologically constant families of cohomology classes. This sheaf has the structure of a {\em local system}: it can be characterized by its fiber $H^k(W_s,\mathbb{C})$ at a particular point $s\in S$ together with a representation of the fundamental group \[\rho:\pi_1(S,s)\to\operatorname{Aut}(H^k(W_s,\mathbb{C}))\] which specifies what happens when the locally constant sections are followed around loops. There is useful dictionary \cite{rsp} between local systems and pairs $({\mathcal{H}},\nabla)$ consisting of a holomorphic vector bundle ${\mathcal{H}}$ on $S$ and a flat holomorphic connection \[\nabla:\mathcal{H}\to(T^{(1,0)}_S)^*\otimes\mathcal{H}.\] The way the dictionary works is this: given a local system $\mathbb{H}$, define $\mathcal{H}=\O_S\otimes\mathbb{H}$, and $\nabla(\sum \varphi_j h^j)=\sum d\varphi_j\otimes h^j$ for $\{h^j\}$ a local basis of sections of $\mathbb{H}$. Conversely, given $(\mathcal{H},\nabla)$, define $\Gamma(U,\mathbb{H})=\{h\in\Gamma(U,\mathcal{H})\ |\ \nabla(h)=0\}$ for every open set $U$. In the case of the cohomology local system $R^k\pi_*{\mathbb{C}}_{\mathcal{W}}$, the associated connection $\nabla$ on $\mathcal{H}^k$ is called the {\em Gauss--Manin connection}. An explicit version of this Gauss--Manin connection goes like this: if we choose a local basis $\alpha^1,\dots,\alpha^r$ for the space of sections $\Gamma(U,R^k\pi_*{\mathbb{C}}_{\mathcal{W}})$, then any $\beta(s)\in\Gamma(U,{\mathcal{H}}^k)$ can be written $\beta(s)=\sum f_j(s)\alpha^j$ for some coefficient functions $f_j\in\Gamma(U,{\O}_S)$. Then \[\nabla(\beta)=\sum df_j\otimes \alpha^j \in\Gamma(U,(T^{(1,0)}_S)^*\otimes{\mathcal{H}^k}).\] This can be given an interpretation in terms of classical ``period integrals'' as follows. The basis $\alpha^1,\dots,\alpha^r$ is dual to some basis $\gamma_1,\dots,\gamma_r\in H_k(W_{s_0},{\mathbb{C}})$. Then the coefficient functions are the period integrals $f_j(s)=\int_{\gamma_j}\beta(s)$. (We use integration to denote the pairing between homology and cohomology.) The great advantage of expressing everything in terms of the Gauss--Manin connection is that the Gauss--Manin connection can be computed algebraically, without knowing the topological cycles in advance. Although the sheaf $R^k\pi_*\mathbb{C}_{\mathcal{W}}$ of cohomology groups can be locally trivialized over the base $S$, the Hodge decomposition \eqref{eq:hodge} will vary as we vary the complex structure. The properties of this variation are more conveniently expressed using the {\em Hodge filtration}: \[F^p(W_s):=\bigoplus_{p'\ge p}H^{p',k-p'}(W_s)\subset H^k(W,\mathbb{C})\] rather than the Hodge groups $H^{p,q}(W_s)$ directly. The spaces $F^p(W_s)$ in the Hodge filtration vary holomorphically with parameters, fitting together to form a holomorphic subbundle $\mathcal{F}^p\subset\mathcal{H}^k$. One might also try to construct a bundle of $H^{p,q}$'s by the simple procedure \[{\mathcal{H}}^{p,q}_{C^\infty}:=\bigcup_{s\in S} H^{p,q}(W_s)\subset{\mathcal{H}^k}.\] As the notation indicates, this defines a $C^\infty$ bundle, but it is not in general holomorphic. There is a holomorphic bundle ${\mathcal{H}}^{p,k-p}$ defined by the exact sequence \begin{equation}\label{nonsplit} 0\to{\mathcal{F}}^{p+1}\to{\mathcal{F}}^p\to {\mathcal{H}}^{p,k-p}\to0, \end{equation} but this exact sequence {\em has no canonical splitting}, and $\mathcal{H}^k$ cannot in general be written as a direct sum of these holomorphic $\mathcal{H}^{p,k-p}$ bundles. The key property satisfied by the Hodge bundles is known as {\em Griffiths transversality}: when we differentiate with respect to parameters by using the Gauss--Manin connection, the Hodge filtration only shifts by one, i.e., \[\nabla(\mathcal{F}^p)\subset(T^{(1,0)}_W)^*\otimes \mathcal{F}^{p-1}.\] To study the totality of complex structures on $W$, we can map the moduli space, or any parameter space $S$ for a family, to the classifying space for Hodge structures. Each Hodge structure on a fixed vector space $H^k$ determines a point in a flag variety \[\operatorname{Flags}_{(f_j)}:=\{\{0\}\subset F^k\subset\cdots\subset F^0= H^k\ |\ \dim F^j=f_j\},\] with the $f_j$'s specifying the dimensions of the spaces making up the filtration. The group $\operatorname{GL}(f_0,\mathbb{C})$ acts transitively on such flags, and if we fix a reference flag $F_0^{\scriptscriptstyle\bullet}$, then the flag variety can be described as $\operatorname{GL}(f_0,\mathbb{C})/\operatorname{Stab}(F_0^{\scriptscriptstyle\bullet})$. (The stabilizer $\operatorname{Stab}(F_0^{\scriptscriptstyle\bullet})$ is the group of block lower triangular matrices.) There are some additional conditions which should be imposed to get a good Hodge structure (cf.~\cite{transcendental,schmid}); these restrict us to an open subset $\mathcal{U}$ of a subvariety\footnote{We must pass to a subvariety to restrict to the so-called {\it polarized}\/ Hodge structures---see \cite{transcendental} or \cite{deligne} for an explanation of this.} of the flag variety on which a discrete group $\Gamma$ acts, and the desired classifying space for Hodge structures is $\mathcal{U}/\Gamma$. The classifying map $S\to\mathcal{U}/\Gamma$ for a family is often referred to as the {\em period map}. The tangent space to the flag variety can be described as \[\bigoplus_j \operatorname{Hom}(F^j/F^{j+1},H^k/F^j).\] So another way of stating Griffiths transversality is to say that the differential of the period map $S\to\mathcal{U}/\Gamma$ sends $T^{(1,0)}_S$ to the subspace \[\bigoplus_j \operatorname{Hom}(F^j/F^{j+1},F^{j-1}/F^j) =\bigoplus_j \operatorname{Hom}(H^{j,k-j}(W_s),H^{j-1,k+1}(W_s))\] of the tangent space. The differential of the map $S\to\operatorname{Flags}_{(f_j)}$ factors through a map $T^{(1,0)}_S\to H^1(T^{(1,0)}_W)$ which describes the first-order deformations represented by $S$ at $[W]$. The map which then induces the differential is the map \begin{equation}\label{eq:differential} H^1(T^{(1,0)}_W)\to \bigoplus_j \operatorname{Hom}(H^{j,k-j}(W),H^{j-1,k+1}(W)) \end{equation} given by sheaf cup product. The success of this approach to studying the moduli of complex structures derives from the {\em local Torelli theorem}\/ for Calabi--Yau manifolds, which states that the map \eqref{eq:differential} is injective. This means that at least locally, the moduli space can be accurately described by using variations of Hodge structure. However, that same map can now be given a new interpretation, as a $B$-model correlation function. That is, {\em the $B$-model correlation function \[H^1(T^{(1,0)}_W)\times H^{j,k-j}(W)\to H^{j-1,k+1}(W)\] coincides with the differential of the period map!} We now restrict our attention to the middle-dimensional cohomology $H^n(W,\mathbb{C})$. Stated in terms of the Gauss--Manin connection, we find the following ``bundle version'' of our correlation function \cite{guide}: given a vector field $\theta$ on the moduli space and sections $\alpha\in\mathcal{F}^j$, $\beta\in\mathcal{F}^{j-1}$, the correlation function is \[\langle\theta\,\alpha\,\beta\rangle=\int_W\nabla_\theta(\alpha)\wedge\beta\] (where $\nabla_\theta=\theta\lhk\nabla$ denotes the directional derivative in direction $\theta$). However, as used in physics the correlation function is a specific function rather than a map between bundles. To find this interpretation, we will need to choose specific sections of these bundles on which to evaluate the map. It is this issue to which we now turn. \section{Splitting the Hodge filtration} Our method for specifying sections of the Hodge bundles will be given in terms of a choice of splitting for the Hodge filtration on the middle-dimensional cohomology $H^n(W,\mathbb{C})$, i.e., a set of splittings of the exact sequences (\ref{nonsplit}) (but defined only locally in the parameter space). We determine such a splitting by means of a filtration on {\em homology}, which we think of as specifying ``which periods to calculate.'' Let ${\mathbb{S}}_{\scriptscriptstyle\bullet}$ be a filtration of the homology local system $\operatorname{Hom}(R^n\pi_*{\mathbb{C}}_{\mathcal{W}},{\mathbb{C}}_S)$ by sub-local systems, and let \[{\mathbb{S}}^\ell:=\operatorname{Ann}({\mathbb{S}}_{\ell-1}):= \{\alpha\in \mathcal{H}^n\ |\ \int_\gamma\alpha=0\ \forall\ \gamma\in{\mathbb{S}}_{\ell-1}\}.\] be the associated filtration of annihilators of $\mathbb{S}_{\scriptscriptstyle\bullet}$ in cohomology. We say that $\mathbb{S}_{\scriptscriptstyle\bullet}$ is a {\em splitting filtration for $\mathcal{F}^{\scriptscriptstyle\bullet}$}\/ if $({\mathcal{H}}^n)_s \cong ({\mathcal{F}}^p)_s\oplus({\mathbb{S}}^{n-p+1})_s$ for every $s\in S$ and for every $0\le p\le n$. (In this case, $\mathbb{S}^{\scriptscriptstyle\bullet}$ and $\mathcal{F}^{\scriptscriptstyle\bullet}$ are called {\em opposite filtrations of weight $n$}\/ \cite{deligne}.) One way of producing examples of splitting filtrations is as follows: fix a point $s\in S$, and consider the conjugate of the Hodge filtration at $s_0$, namely, $\overline{F^q}_{s_0}$. The ``opposite'' property for these filtrations is easy to check: by definition \begin{align*}({\mathcal{F}}^p)_s&=H^{n,0}(W_s)\oplus\cdots\oplus H^{p,n-p}(W_s)\\ \intertext{and so} (\overline{{\mathcal{F}}^{n-p+1}})_s&= \overline{H^{n,0}(W_s)\oplus\cdots\oplus H^{n-p+1,p-1}(W_s)}\\ &=H^{0,n}(W_s)\oplus\cdots\oplus H^{p-1,n-p+1}(W_s), \end{align*} where we have used the fact that $\overline{H^{p,q}(W_s)}=H^{q,p}(W_s)$. The Gauss--Manin connection can be used to extend this from a filtration at one point to a filtration of the local system. Although this filtration only coincides with the conjugate of the Hodge filtration at one point in the parameter space, it remains opposite to the Hodge filtration at all points nearby. Given a splitting filtration ${\mathbb{S}}_{\scriptscriptstyle\bullet}$, we define \[{\mathcal{H}}^{p,q}_{\mathbb{S}}:={\mathcal{F}}^p\cap\operatorname{Ann}({\mathbb{S}}_{q-1}),\] on any open set on which $\mathbb{S}_{\scriptscriptstyle\bullet}$ is single-valued. Then \[{\mathcal{H}}=\bigoplus_{p=0}^n {\mathcal{H}}^{p,q}_{\mathbb{S}} \quad \text{and} \quad \mathcal{F}^p=\bigoplus_{p'\ge p}\mathcal{H}^{p',n-p'}_\mathbb{S}.\] (This is the promised splitting of the Hodge filtration.) More concretely, this space can be described in terms of conditions on the periods as follows. The sections of ${\mathcal{H}}^{p,q}_{\mathbb{S}}$ over $U$ are \[\Gamma(U,\mathcal{H}^{p,q}_{\mathbb{S}}):= \{\beta\in\Gamma(U,{\mathcal{F}}^p)\ |\ \int_\gamma\beta=0\ \forall\ \gamma\in{\mathbb{S}}_{q-1}\}.\] We also define a space of {\em distinguished sections}\/ of ${\mathcal{H}}^{p,q}_{\mathbb{S}}$ by \[\Gamma(U,{\mathcal{H}}^{p,q}_{\mathbb{S}})_{\text{dist}}:= \{\beta\in\Gamma(U,{\mathcal{H}}^{p,q}_{\mathbb{S}})\ |\ d\left(\int_\gamma\beta\right) =0\ \forall\ \gamma\in{\mathbb{S}}_{q}\}.\] (That is, the period integrals $\int_\gamma\beta$ are constant for all $\gamma\in{\mathbb{S}}_{q}$, and vanish for all $\gamma\in\mathbb{S}_{q-1}$.) For each ${\mathbb{S}}_{\scriptscriptstyle\bullet}$, then, we can define specific $B$-model correlation functions, using the $\Omega$ coming from the distinguished section of ${\mathcal{H}}^{n,0}_{\mathbb{S}}$ (which is well defined up to a complex scalar multiple). This has the advantage that the correlation functions have been turned into actual functions on a parameter space (in accord with the physicists' interpretation) rather than sections of a bundle. The disadvantage is that further parameters---in the form of a choice of splitting---have been introduced. However, the necessity of considering further parameters such as these, on which the correlation functions will depend anti-holomorphically rather than holomorphically, was recently realized in the physics literature \cite{t:tstar}. In addition to the distinguished $n$-form $\Omega$, our choice of splitting determines a family of distinguished vector fields which when contracted with $\Omega$ yield the distinguished sections of $\mathcal{H}^{n-1,1}_{\mathbb{S}}$. These vector fields can be integrated into {\em canonical coordinates}, well-defined up to a $\operatorname{GL}(r,{\mathbb{C}})$ transformation. (The flexibility of that final $\operatorname{GL}(r,\mathbb{C})$ choice comes from the constants of integration, which must also be specified in order to completely determine a set of canonical coordinates.) A bit more explicitly, if $\gamma_0$ spans ${\mathbb{S}}_0$ and $\gamma_0, \gamma_1,\dots,\gamma_r$ span ${\mathbb{S}}_1$, then the distinguished $\Omega$ satisfies $\int_{\gamma_0}\Omega=\text{constant}$, and the coordinates are given by \[\int_{\gamma_1}\Omega,\dots,\int_{\gamma_r}\Omega.\] If we start with an arbitrary $n$-form $\widetilde\Omega$, we can write the distinguished $n$-form as \[\Omega:=\frac{\widetilde\Omega}{\int_{\gamma_0}\widetilde\Omega}\] and the canonical coordinates as \[\frac{\int_{\gamma_1}\widetilde\Omega}{\int_{\gamma_0}\widetilde\Omega}, \dots, \frac{\int_{\gamma_r}\widetilde\Omega}{\int_{\gamma_0}\widetilde\Omega}.\] This is the most general possible form for canonical coordinates (and a distinguished $n$-form) needed for the physical theory, according to recent work in physics \cite{BCOV:KS}. Let us fix a splitting filtration ${\mathbb{S}_{\scriptscriptstyle\bullet}}$. Consider a basis $\{\beta^i\}$ of ${\mathcal{H}^n}$ consisting of distinguished sections of the bundles ${\mathcal{H}}_{\mathbb{S}}^{p,q}$ (ordered so that the basis is also adapted to the Hodge filtration ${\mathcal{F}}^{{\scriptscriptstyle\bullet}}$), and a multi-valued basis $\{\gamma_j\}$ of the homology local system $\operatorname{Hom}(R^n\pi_*{\mathbb{C}}_{\mathcal{W}},{\mathbb{C}}_S)$, adapted to the splitting filtration ${\mathbb{S}_{\scriptscriptstyle\bullet}}$. Then the period matrix $(\int_{\gamma_j}\beta^i)$ (which has multi-valued entries) will take a block upper triangular form with constant diagonal blocks. And if we calculate the connection matrix in the basis $\{\beta^i\}$, it takes the special form \[\begin{pmatrix} 0&A^1_0&0&&\cdots&0\\ &0&A^1_1&0&\cdots&0\\ &&\ddots&\ddots&&\vdots\\ \vdots&\vdots&&&0&A^1_{n-1}\\ 0&0&\cdots&&&0 \end{pmatrix}\] in which the only nonzero entries are in the first block superdiagonal of the matrix. The entries $A^1_j$ precisely contain the data for the $B$-model correlation functions, calculated in our distinguished basis. \chapter*{} \lecturename{The $A$-Variation of Hodge Structure} \lecture \markboth{D. R. Morrison, Mathematical Aspects of Mirror Symmetry}{Lecture 7. The $A$-Variation of Hodge Structure} \section{Variations of Hodge structure near the boundary of moduli} In this lecture, we begin by reviewing the asymptotic behavior of a variation of Hodge structure near the boundary of moduli space, and the behavior of the $B$-model correlation functions there. Comparing to the $A$-model correlation functions will reveal some similarities---this is one of the hints of mirror symmetry. We make the similarities even more apparent by using the $A$-model correlation functions to construct a new variation of Hodge structure, which we call the $A$-variation of Hodge structure. Let $S=(\Delta^*)^r\subset \overline{S}=\Delta^r$, and suppose we are given a family $\pi:\mathcal{W}\to S$ of complex manifolds. We will assume that there is a way to complete this to a family $\bar\pi:\overline{\mathcal{W}}\to\overline{S}$ in which $\bar\pi$ is still proper (but no longer smooth). Thus, $0\in\overline{S}$ is a boundary point in the parameter space. Pick a basepoint $s\in S$; then the fundamental group $\pi_1(S,s)$ is generated by loops $\gamma^{(1)}$,\dots,$\gamma^{(r)}$ with $\gamma^{(j)}$ homotopic to the standard generator of $\pi_1(\Delta^*_j)$, where $\Delta^*_j$ is the $j^{\text{th}}$ factor in $(\Delta^*)^r$. \begin{MonodromyTheorem}[Landman \cite{monodromy}] The action of each generator $\gamma^{(j)}$ gives a quasi-unipotent automorphism $T^{(j)}$ of $H^k(W_s,\mathbb{Q})$, i.e., $(((T^{(j)})^{b_j}-I)^{r_j}=0$. (This is called {\em unipotent}\/ if $b_j=1$.) \end{MonodromyTheorem} We will restrict attention to the unipotent case. This is partially for technical convenience, but in fact, in the examples which have been calculated for mirror symmetry purposes, only unipotent monodromy transformations have played a r\^ole. When $T^{(j)}$ is unipotent, its logarithm can be defined by the following sum (which is finite). \[N^{(j)}:=\log T^{(j)}:= (T^{(j)}-I) - \frac12\,(T^{(j)}-I)^2+\cdots.\] (Note that the $T^{(j)}$'s and $N^{(j)}$'s all commute.) Let $z_1,\dots,z_r$ be coordinates on $\overline{S}$, with $z_j$ a coordinate on the $j^{\text{th}}$ disk. Consider the operator \begin{multline*} \mathcal{N}:=\exp\left(-\frac1{2\pi i}\sum\,\log z_j\,N^{(j)}\right)=\\ I + \left(-\frac1{2\pi i}\sum\, \log z_j\,N^{(j)}\right) + \frac1{2!}\left(-\frac1{2\pi i}\sum\,\log z_j\,N^{(j)}\right)^2+\cdots \end{multline*} (also a finite sum). For any section $e$ of the local system $R^k\pi_*(\mathbb{C}_\mathcal{W})$, a simple calculation shows that \begin{equation}\label{eq:GMext} \nabla(\mathcal{N} (e)) = -\frac1{2\pi i}\sum\frac{dz_j}{z_j}\,N^{(j)}(e). \end{equation} The key facts about the asymptotic behavior are as follows. \begin{NilpotentOrbitTheorem}[Schmid \cite{schmid}] Assume that each monodromy transformation $T^{(j)}$ is unipotent. Let $e_1(s),\dots,e_r(s)$ be a multi-valued basis of $R^k\pi_*(\mathbb{C}_\mathcal{W})$, and let $\eta_\ell:=\mathcal{N}(e_\ell)$. Then each $\eta_\ell$ is a single-valued section of $\mathcal{H}^k$ on $S$, and together they can be used to generate an extension $\overline{\mathcal{H}}^k$ of $\mathcal{H}^k$ to $\overline{S}$. By eq.~\eqref{eq:GMext}, the Gauss--Manin connection extends to a connection on $\overline{\mathcal{H}}^k$ (again denoted by $\nabla$) with {\em regular singular points}, i.e., the extended connection is a map \[\nabla:\overline{\mathcal{H}}^k\to (T^{(1,0)}_{\overline{S}})^*(\log B)\otimes \overline{\mathcal{H}}^k\] where $(T^{(1,0)}_{\overline{S}})^*(\log B)$ is the free $\O_{\overline{S}}$-module generated by $\frac{dz_j}{z_j}$, $j=1,\dots,r$. Moreover, the Hodge bundles $\mathcal{F}^p$ have locally free extensions to subbundles $\overline{\mathcal{F}}^p\subset\overline{\mathcal{H}}^k$ such that \[\nabla(\overline{\mathcal{F}}^p)\subset (T^{(1,0)}_{\overline{S}})^*(\log B)\otimes \overline{\mathcal{F}}^{p-1}.\] \end{NilpotentOrbitTheorem} The asymptotic behavior as $z_j\to0$ of the $B$-model correlation functions \[\langle\theta\,\alpha\,\beta\rangle=\int_W\nabla_\theta(\alpha)\wedge\beta\] can be deduced from this theorem. If we let $\theta_j=2\pi i\,z_j\,\frac{d}{dz_j}$ (chosen to remove poles in the asymptotic expression for the correlation function) then the leading term in $\langle\theta_j\,\eta_\ell\,\beta\rangle$ is given by the monodromy: \begin{equation}\label{eq:limB} \lim_{z_j\to0}\langle\theta_j\,\eta_\ell\,\beta\rangle =-\int_W N^{(j)}(e_\ell)\wedge\beta. \end{equation} The essential properties of the monodromy are captured by the {\em monodromy weight filtration}\/ ${\mathbb{W}}_{{\scriptscriptstyle\bullet}}$ on the cohomology, which has the properties that $N^{(j)}{\mathbb{W}}_\ell\subset {\mathbb{W}}_{\ell-2}$, and that for any positive real numbers $a_1$, \dots, $a_r$, the operator $N:=\sum a_j N^{(j)}$ induces isomorphisms $N^\ell:\operatorname{Gr}^{\mathbb{W}}_{n+\ell}\to\operatorname{Gr}^{\mathbb{W}}_{n-\ell}$. Any splitting filtration which we use to make calculations of $B$-model correlation functions must be somehow compatible with this monodromy weight filtration, if those calculations are to make sense near the boundary. If mirror symmetry is going to hold, there must be a correspondence between the limiting behaviors described in eqs.~\eqref{eq:limA} and \eqref{eq:limB}. In fact, the first thing to notice is that the natural flat coordinates on the $A$-model moduli space are multiple-valued, with the ambiguity precisely specified by $H_{\text{DR}}^2(M,\mathbb{Z})$. So there must be some part of the monodromy weight filtration which matches that behavior. This motivated the following definition, first given in \cite{guide,compact} (cf.~also \cite{deligne}). We say that a boundary point is {\em maximally unipotent}\/ if \[{\mathcal{H}}_s=({\mathcal{F}}^n)_s\oplus({\mathbb{W}}_{2n-2})_s\] and \[{\mathcal{H}}_s=({\mathcal{F}}^{n-1})_s\oplus({\mathbb{W}}_{2n-4})_s\] for all $s$ near the point. With this definition, the distinguished holomorphic $n$-form and the canonical coordinates can be defined as in lecture six. There is an alternate version of this ``maximally unipotent monodromy'' condition, which agrees with the original one for Calabi--Yau threefolds, but is more restrictive in higher dimension. We say that a boundary point is {\em strongly maximally unipotent}\/ if the weight filtration ${\mathbb{W}}_{{\scriptscriptstyle\bullet}}$ has nontrivial graded pieces in even degree only, and if the induced filtration on homology defined by \[\mathbb{S}_\ell:=\operatorname{Ann}(\mathbb{W}_{2n-2\ell+2})\] is a splitting filtration. (Note that the corresponding filtration on cohomology is then \[\mathbb{S}^\ell:=\operatorname{Ann}(\mathbb{S}_{\ell-1})=\mathbb{W}_{2n-2\ell};\] this is the filtration which should be opposite to the Hodge filtration.) In this case, we will be able to use distinguished sections to calculate $B$-model correlation functions, as explained earlier. At the moment, only the original version of the definition has been justified to the satisfaction of physicists as an appropriate characterization of points which should be useful for mirror symmetry. To completely carry out a mirror symmetry type calculation, though, the second version would seem to be necessary. And as we shall see, that version has been extremely successful in examples. Actually, even just at the level of the monodromy action, the parallels between the structure of the Lefschetz operators on the cohomology and the action of monodromy are rather striking, as was first observed by Cattani, Kaplan and Schmid \cite{CKS}. The operators $\operatorname{ad}(e^j)$ describe Lefschetz decompositions of the cohomology of $M$, which have many structural parallels to the monodromy weight filtration at a maximally unipotent point. \section{Reinterpreting the $A$-model correlation functions} Let $M$ be a Calabi--Yau manifold on which a complex structure and K\"ahler metric have been fixed. Inspired by some of the similarities between the two different types of correlation functions, we wish to improve the analogy by translating the $A$-model correlation functions into data describing a variation of Hodge structure. Consider the moduli space $\mathcal{D}/\Gamma$ for $A$-model correlation functions, and a coordinate chart specified by a cone $\sigma$: \[\begin{array}{ccccc} \mathcal{D}/\Gamma&\leftarrow&\mathcal{D}_\sigma/H^2(M,\mathbb{Z})&\cong&(\Delta^*)^r\\ &&\cap\raise1pt\hbox{$\scriptstyle|$}&&\cap\raise1pt\hbox{$\scriptstyle|$}\\ &&(\mathcal{D}_\sigma/H^2(M,\mathbb{Z}))^-&\cong&\Delta^r \end{array}\] We assume that the cone $\sigma$ (which we call a {\em framing}\/) is generated by a basis $e^1$, \dots, $e^r$ of $H^2(M,\mathbb{Z})$. Let $t_1$, \dots, $t_r$ be coordinates on $H^2(M,\mathbb{C})$ dual to this basis (so that elements of $H^2(M,\mathbb{C})$ take the form $\sum t_j e^j$). The natural vector fields for making calculations of correlation functions which involve a term from the tangent space $H^2(M,\mathbb{C})$ are the vector fields $\partial/\partial t_j$. These are the analogues of the distinguished vector fields which we had on the $B$-model side. On the other hand, natural coordinates on $\mathcal{D}_\sigma/H^2(M,\mathbb{Z})$ are furnished by $q_j=\exp(2\pi i\,t_j)$. Then \[\frac{\partial}{\partial t_j}=2\pi i\,q_j\,\frac{\partial}{\partial q_j},\] from which we conclude that those correlation functions should naturally be evaluated on the basis $2\pi i\,q_j\,\partial/\partial q_j$ of the sheaf of logarithmic vector fields on the space $(\mathcal{D}_\sigma/H^2(M,\mathbb{Z}))^-$. We identify $\partial/\partial t_j$ with the operation of taking the quantum product with the basis element $e^j\in H^2(M,\mathbb{Q})$. The resulting map is determined by the correlation functions of the form $\langle e^j\, \alpha\,\beta\rangle$. We had a particularly simple form for these correlation functions, given in example \ref{example43}, in terms of the Gromov--Witten maps $\Gamma_\eta$. We now wish to reinterpret that formula in the following way. We will describe a holomorphic bundle\footnote{There are a few variants to this construction, in which one uses slightly different bundles. Essentially, one can restrict to any subbundle of $\bigoplus H^{\ell,\ell}(M)$ which is preserved by cup products with the part of $H^{1,1}(M)$ which it contains.} $\mathcal{E}:=\left(\bigoplus H^{\ell,\ell}(M)\right)\otimes \O_{(\mathcal{D}_\sigma/H^2(M,\mathbb{Z}))^-}$ with a connection\footnote{I am indebted to P. Deligne for advice which led to this form of the formula (cf.~\cite{deligne}).} (with regular singular points) \[\nabla:=\frac1{2\pi i}\,\left( \sum d\mskip0.5mu\log q_j\otimes\operatorname{ad}(e^j)+ \sum_{0\ne\eta\in H_2(M,\mathbb{Z})} d\mskip0.5mu\log\left(\frac1{1-q^{\eta}}\right)\otimes\Gamma_\eta \right)\] which was derived from the formulas for $e^j{\star}$, where $\operatorname{ad}(e^j):H^k(M)\to H^{k+2}(M)$ is defined by $\operatorname{ad}(e^j)(A)=e^j\cup A$. We also define a ``Hodge filtration'' \[\mathcal{E}^p:=\left(\bigoplus_{0\le\ell\le m-p} H^{\ell,\ell}(M)\right)\otimes\O_{(\mathcal{D}_\sigma/H^2(M,\mathbb{Z}))^-}.\] This describes a structure we call the {\em framed $A$-variation of Hodge structure with framing $\sigma$}. To be a bit more precise, we should study ``formally degenerating variations of Hodge structure,'' since the series used to define $\nabla$ is only formal. (We won't formulate that theory in detail here.) The connection $\nabla$ which we defined from the Gromov--Witten invariants is in fact a {\em flat}\/ holomorphic connection \cite{topgrav}. The flatness follows from the associativity (and commutativity) of the binary operation. In fact, since the directional derivatives with respect to $\nabla$ corresponded to binary products $e^j\star\zeta$ (where $e^j$ describes the direction of the derivative), iterated directional derivatives have the form $e^k\star(e^j\star\zeta)$. We would simply need to know that reversing the order of $j$ and $k$ produces the same result, and this is guaranteed by the commutativity and associativity. In particular, the flatness is automatic when $\dim_\mathbb{C} M=3$, a case in which there is no issue of associativity. The recent proofs of associativity of quantum cohomology \cite{RuanTian,Liu,MS} guarantee that this connection is flat in arbitrary dimension. As in the geometric case, there is an additional structure associated to this variation of Hodge structure: a local system. The local system on homology takes the simple form \[\mathbb{S}_\ell:=H_{0,0}\oplus H_{1,1}\oplus\dots\oplus H_{\ell,\ell},\] and the corresponding local system on cohomology then becomes \[\mathbb{S}^\ell=H^{\ell,\ell}\oplus H^{\ell+1,\ell+1}\oplus\dots\oplus H^{n,n}.\] The logarithms of the monodromy actions which define these local systems are specified by the topological pairings, and coincide with the cup-product maps \[H^2(M,\mathbb{Z})\otimes \mathbb{S}^\ell\to \mathbb{S}^{\ell+1}.\] In the next lecture, we will formulate a precise conjecture which equates this $A$-variation of Hodge structure with the geometric variation of Hodge structure on a mirror partner. \section{Beyond the K\"ahler cone}\label{sec73} We indicated in lecture five that the conformal field theory moduli space is actually {\em larger}\/ than the nonlinear $\sigma$-model moduli space. We can now explain how this comes about---it is due to an analysis of the effect of flops on the conformal field theory. Flops are birational transformations among Calabi--Yau threefolds which have been studied extensively as part of the minimal model program (see for example \cite{CKM}). The effect of flops on the K\"ahler cone of a Calabi--Yau threefold is as follows. Given a Calabi--Yau threefold $X$ with a complex structure $J$, and a linear system $|L|$ inducing a flopping contraction from $X_J$ to $\widehat X_{\widehat J}$, the K\"ahler cones $\mathcal{K}_J$ and $\widehat{\mathcal{K}}_{\widehat J}$ share a common wall, which contains the class of $|L|$, as depicted in figure \ref{fig0}. The K\"ahler cone has already occurred in our discussion of the moduli spaces of $\sigma$-models. The natural question arises: suppose we attempt to ``attach'' the moduli spaces $\mathcal{D}/\Gamma$ and $\widehat{\mathcal{D}}/\widehat{\Gamma}$ along (the images of) their common wall? In fact, it now appears likely that the conformal field theory moduli spaces of $X$ and $\widehat X$ are analytic continuations of each other, and that this ``attached space'' is a part of the full conformal field theory moduli space \cite{mmm,phases}. (This at least seems to happen in examples---the arguments for this rely on mirror symmetry, and involve finding regions in the mirror's moduli space which correspond to the $X_J$ and $\widehat X_{\widehat J}$ theories, respectively.) One of the consequences of this would be an analytic continuation of correlation functions from $\mathcal{D}/\Gamma$ to $\widehat{\mathcal{D}}/\widehat{\Gamma}$. \begin{figure} \iffigs $$\vbox{\centerline{\epsfysize=3cm\epsfbox{cones.eps}} }$$ \else \vglue2in\noindent \fi \caption{Adjacent K\"ahler cones} \label{fig0} \end{figure} Here is a formal calculation from \cite{phases,small} which supports this analytic continuation idea (see also \cite{beyond} for a more mathematical treatment). The union of all of the K\"ahler cones of birational models of $X_J$ is known as the {\em movable cone}\/ $\operatorname{Mov}{X_J}$ \cite{kawamata}. We compute in the formal semigroup ring $\mathbb{Q}[q;\operatorname{Mov}(X_J)^\vee]$ (which we identify canonically with the same ring for $\widehat X_{\widehat J}$), and so the computation is purely formal. Consider the simplest flop: the flop based on a collection of disjoint holomorphic rational curves $\Gamma_i\subset X_J$ (in a common homology class $[\Gamma]$) such that the normal bundle is $N_{\Gamma_i/X_J}=\O(-1)\oplus \O(-1)$. (These curves must be flopped simultaneously in order to ensure that the flopped variety is K\"ahler.) A reasonable genericity assumption about the {\em other}\/ rational curves on $X_J$ is this: all (pseudo-)holomorphic curves in classes $\eta\not\in\mathbb{R}_{>0}[\Gamma]$ are disjoint from the $\Gamma_i$'s. Since there is a proper transform map on divisors, the Gromov--Witten invariants (which in this case are determined entirely by intersection properties of $\eta$ and the number of elements in $\mathcal{M}^*_{(\eta,J)}$) do not change when passing from $X_J$ to $\widehat X_{\widehat J}$, except for the invariants $\Phi_{[\Gamma]}$ themselves. The cup product can also change. The $A$-model correlation functions on $X_J$ can be written in the form \begin{align*} \langle A\,B\,C\rangle=A\cdot B\cdot C &+\frac{q^{[\Gamma]}}{1-q^{[\Gamma]}}\,(A\cdot\Gamma)(B\cdot\Gamma) (C\cdot\Gamma)\,n_\Gamma \\ &+\sum_{\substack{\eta\in H_2(X,\mathbb{Z})\\ \eta\ne\lambda\Gamma}} \frac{q^\eta}{1-q^\eta}\,\Phi_\eta(A,B,C). \end{align*} Only the first terms change when passing to $\widehat X_{\widehat J}$ and in fact we claim that \begin{align*} A\cdot B\cdot C &+\frac{q^{[\Gamma]}}{1-q^{[\Gamma]}}\,(A\cdot\Gamma)(B\cdot\Gamma) (C\cdot\Gamma)\,n_\Gamma \\ = & {\widehat A}\cdot {\widehat B}\cdot {\widehat C} +\frac{q^{[\widehat \Gamma]}}{1-q^{[\widehat \Gamma]}}\, (\widehat A\cdot\widehat \Gamma)(\widehat B\cdot\widehat \Gamma) (\widehat C\cdot\widehat \Gamma)\,n_{\widehat \Gamma}, \end{align*} where $\widehat A$, $\widehat B$, and $\widehat C$ are the proper transforms of $A$, $B$, and $C$. (In other words, the change in the topological term is precisely compensated for by the change in the $q^{[\Gamma]}$ term.) We will check this formula in the case in which $A$ and $B$ meet one of the curves $\Gamma$ transversally at $a$ and $b$ points, respectively, and $\widehat C$ meets $\widehat\Gamma$ transversally at $c$ points. (The general case can be deduced from this one.) Then $C$ must contain $\Gamma$ with multiplicity $c$, and the configuration of divisors is as in figure \ref{fig1} (which illustrates the case $a=b=c=1$ for simplicity). $A$ and $B$ have no intersection points along $\Gamma$, but both $\widehat A$ and $\widehat B$ contain $\widehat \Gamma$, and they meet $\widehat C$. The total number of intersection points of $\widehat A$, $\widehat B$ and $\widehat C$ (counted with multiplicity) which lie in $\widehat \Gamma$ is thus $abc$. \refstepcounter{figure}\label{fig1} \begin{figure} \iffigs $$ \matrix\epsfxsize=2in\epsfbox{fig1a.eps} & \qquad & \epsfxsize=2in\epsfbox{fig1b.eps} \cr \quad & & \cr \hbox{{\footnotesize\bfseries Figure \ref{fig1}a}{.\footnotesize\mdseries\upshape\enspace Before the flop.}} & & \hbox{{\footnotesize\bfseries Figure \ref{fig1}b}{.\footnotesize\mdseries\upshape\enspace After the flop.}} \cr\endmatrix $$ \else \vglue3in\noindent \fi \end{figure} Since a similar thing happens for each curve $\Gamma_i$ in the numerical equivalence class, we see that \begin{equation} \widehat A\cdot\widehat B\cdot\widehat C -A\cdot B\cdot C = abc\,n_\Gamma =-(A\cdot\Gamma)(B\cdot\Gamma)(C\cdot\Gamma)\,n_\Gamma \label{eq:three} \end{equation} (using $A\cdot\Gamma=a$, $B\cdot\Gamma=b$, $C\cdot\Gamma=-c$). On the other hand, since $[\widehat\Gamma]=-[\Gamma]$ and $n_{\widehat\Gamma}=n_\Gamma$, we can compute: \begin{equation}\label{eq:four}\begin{split} \frac{q^{[\Gamma]}}{1-q^{[\Gamma]}}\,&(A\cdot\Gamma)(B\cdot\Gamma) (C\cdot\Gamma)\,n_\Gamma - \frac{q^{[\widehat \Gamma]}}{1-q^{[\widehat \Gamma]}}\, (\widehat A\cdot\widehat \Gamma)(\widehat B\cdot\widehat \Gamma) (\widehat C\cdot\widehat \Gamma)\,n_{\widehat \Gamma}\\ =&\frac{q^{[\Gamma]}}{1-q^{[\Gamma]}}\, (A\cdot\Gamma)(B\cdot\Gamma) (C\cdot\Gamma)\,n_\Gamma +\frac{q^{-[\Gamma]}}{1-q^{-[\Gamma]}} (A\cdot\Gamma)(B\cdot\Gamma) (C\cdot\Gamma)\,n_\Gamma \\ =&\left(\frac{q^{[\Gamma]}}{1-q^{[\Gamma]}} +\frac{1}{q^{[\Gamma]}-1}\right) (A\cdot\Gamma)(B\cdot\Gamma) (C\cdot\Gamma)\,n_\Gamma \\ =&-(A\cdot\Gamma)(B\cdot\Gamma) (C\cdot\Gamma)\,n_\Gamma. \end{split}\end{equation} Adding eqs.~\eqref{eq:three} and \eqref{eq:four} proves the desired formula. The conclusion from all of this should be that the mirror symmetry phenomenon is really about birational equivalence classes. For if there is any analytic continuation of the correlation function from the region associated to $\mathcal{K}_J$ out into the next cone $\widehat{\mathcal{K}}_{\widehat J}$, the calculation above shows that this analytic continuation must in fact reproduce the correlation function of the flopped model $\widehat X_{\widehat J}$. It is tempting to think that if we combined the $\sigma$-model moduli spaces for all birational models of $X$ we would fill out the entire conformal field theory moduli space. However, some examples that have been worked out by Witten \cite{phases} and by Aspinwall, Greene and the author \cite{catp} show that this is not the case. In those examples, there are other regions in the moduli space which correspond to rather different kinds of physical model, including some called {\em Landau--Ginzburg theories}\/ which will play a r\^ole again in the next lecture. \chapter*{} \lecturename{Mirror Symmetry} \lecture \markboth{D. R. Morrison, Mathematical Aspects of Mirror Symmetry}{Lecture 8. Mirror Symmetry} \section{Mirror manifold constructions} The original speculations about mirror symmetry were based on the appearance of arbitrariness of a choice that was made in identifying certain constituents of the conformal field theory associated to a Calabi--Yau manifold with geometric objects on the manifold. The distinction between vertex operators which appear in the $A$-model and $B$-model correlation functions is simply a difference in sign of a certain quantum number; if that sign is changed, the geometric interpretation is altered dramatically. This led Dixon \cite{Dixon} and Lerche--Vafa--Warner \cite{LVW} to propose that there might be a second Calabi--Yau manifold producing essentially the same physical theory as the first, but implementing this change of sign. Some time later,\footnote{At about the same time, another important piece of evidence for mirror symmetry was given by Candelas, Lynker, and Schimmrigk \cite{CLS}, who found an almost perfect symmetry under the exchange $h^{1,1}\leftrightarrow h^{2,1}$ on the set of Hodge numbers coming from Calabi--Yau threefolds which can be realized as weighted projective hypersurfaces.} an explicit construction was made by Greene and Plesser \cite{GreenePlesser} which showed that this phenomenon does indeed occur in physics. The construction rests on a chain of equivalences which are believed to hold among different physical models, as follows. \begin{enumerate} \item Certain $\sigma$-models on Calabi--Yau manifolds are believed to correspond to so-called Landau--Ginzburg theories \cite{GVW}. (It has recently been recognized \cite{phases} that this correspondence is not direct, but involves analytic continuation on the moduli space.) Roughly speaking, the class of Calabi--Yau manifolds for which this correspondence can be made is the class of ample anti-canonical hypersurfaces in toric varieties. Such a hypersurface will have an equation of the form $\Phi(x_1,\dots,x_{n+1})=0$ (in some appropriate coordinates on the torus), and this same polynomial is used as a ``superpotential'' in constructing the Landau--Ginzburg theory. \item Certain Landau--Ginzburg theories---quotients of the ones for which the superpotential is of ``Fermat type'' \[\Phi(x_1,\dots,x_{n+1})=x_1^{d_1}+\dots+x_{n+1}^{d_{n+1}}\] by certain finite groups $\Gamma$---are believed to correspond to yet another type of conformal field theory. This other theory is described in terms of discrete series representations $V^{(k)}$ of the ``$N{=}2$ superconformal algebra,'' and it takes the form \[\left(\bigotimes_j V^{(d_j+2)}\right)/G\] where $G$ is a slight enlargement of the group $\Gamma$. (Note that the case of $\Gamma$ being trivial is allowed, but then $G$ is not trivial.) The representation theory of the $N{=}2$ superconformal algebra is related to these things by analyzing the conformal field theory on an infinite cylinder. (The superconformal algebra can be described in terms of automorphisms of the cylinder.) \item By studying the representation theory, Greene and Plesser find a kind of duality among the finite groups $G$: there is a dual group $\widehat{G}$ and an isomorphism \[\left(\bigotimes_j V^{(d_j+2)}\right)/G\cong \left(\bigotimes_j V^{(d_j+2)}\right)/\widehat{G}\] which has the ``sign-reversing property'' of mirror symmetry. \item The duality can be extended to the groups $\Gamma$, and the mirror Landau--Ginzburg theory of $\Phi/\Gamma$ is $\Phi/\widehat{\Gamma}$. This looks a bit asymmetric, since for example the case $\Gamma$ trivial leads to a rather large group $\widehat{\Gamma}$. But the group $\Gamma$ continues to act as a group of ``quantum symmetries'' on the quotient theory, in a way that restores symmetry to this construction. \item Finally, the Calabi--Yau which is the quotient of the Fermat hypersurface by $\Gamma$ should have as its mirror the one which is the quotient by $\widehat{\Gamma}$. \end{enumerate} This is called the {\em Greene--Plesser orbifolding construction}. \medskip There is a conjectural generalization of this construction, which as of yet has no basis in conformal field theory---it is simply a mathematician's guess. This generalization would work for an arbitrary family of Calabi--Yau hypersurfaces in toric varieties. The construction is due to V.~Batyrev \cite{batyrev1}.\footnote{We restrict ourselves to the hypersurface case here; further generalizations---to complete intersections---were subsequently given by Borisov and Batyrev--Borisov \cite{borisov,BB:dual}.} Take an ample anticanonical hypersurface $M$ in a toric variety $V$, and let $\{M_t\}$ be the family of such. This family is determined by the Newton polygon of the corresponding equations---that is a polygon $P\subset L_\mathbb{R}:=L\otimes \mathbb{R}$, where $L$ is the {\em monomial lattice}\/ of the torus $T$ (of which $V$ is a compactification). Batyrev shows that the Calabi--Yau condition admits a particularly simple characterization in terms of $P$: the polyhedron $P$ is {\em reflexive}, which means that each hyperplane $H$ which supports a face of codimension one of $P$ can be written in the form \[H=\{y\in L_\mathbb{R}\ |\ (\ell,y)=-1\}\] for some appropriate vector $v\in \operatorname{Hom}(L,\mathbb{Z})$. (The key property here is the {\em integrality}\/ of the vector $v$---there would always be some $v\in \operatorname{Hom}(L,\mathbb{R})$ to define $H$.) \begin{lemma}[Batyrev] If $P$ is reflexive, then the {\em polar polyhedron}\/ \[P^o:=\{x\in \operatorname{Hom}(L,\mathbb{R})\ |\ (x,y)\ge-1\text{ for all }y\in P\}\] is also reflexive. \end{lemma} The conjectured generalization is that the mirror of the family $\{M_t\}$ of hypersurfaces determined by $P$ should be the family $\{W_s\}$ of hypersurfaces (in a compactification of the dual torus of $T$) determined by the polar polyhedron $P^o$. One of the pieces of evidence for this conjecture is \begin{theorem}[Batyrev]\label{batthm} \[\dim H^{\pm1,1}(\widehat{M})=\dim H^{\mp1,1}(\widehat{W}),\] where $\widehat{M}$ and $\widehat{W}$ are $\mathbb{Q}$-factorial terminalizations of $M$ and $W$ respectively. \end{theorem} \noindent Batyrev and collaborators have also explored the Hodge structures of these hypersurfaces in considerable detail \cite{Bat:vmhs,BvS,BatCox}. A refinement of Batyrev's theorem called the {\em monomial-divisor mirror map}\/ was introduced in \cite{mondiv}. This map gives an explicit combinatorial correspondence between (appropriate subspaces of) $H^{\pm1,1}(\widehat{M})$ and $H^{\mp1,1}(\widehat{W})$, and is expected to correctly determine the derivative of the mirror map near the large radius limit point. That derivative data is precisely what one needs in order to evaluate the ``constants of integration'' in finding the canonical coordinates $q_j$. \medskip There is another mirror manifold construction for a class of threefolds which has been proposed by Voisin \cite{Voisin:K3} and Borcea \cite{Borcea:K3}. Let $S$ be a K3 surface with an involution $\iota$ such that $\iota^*(\Omega)=-\Omega$ for any holomorphic two-form $\Omega$ on $S$, and let $E$ be an elliptic curve. The quotient $\overline{M}=(S\times E)/(\iota\times(-1))$ has singularities along the fixed curves of the involution $\iota\times(-1)$, but they can be resolved by a simple blowing up to produce a Calabi--Yau threefold $M$. Involutions of this type on K3 surfaces have been classified by Nikulin \cite{Nikulin:involutions}, who found that they fall into a pattern with a remarkable symmetry; when the Hodge numbers of the associated Calabi--Yau threefold are calculated, this symmetry becomes the expected mirror relation among Hodge numbers. The detailed knowledge which is available concerning the variations of Hodge structure on K3 surfaces can be used to study the correlation functions in detail for these models \cite{Voisin:K3}, which provides further evidence that mirror partners have been correctly identified. In fact, there is also a physics argument explaining why these pairs of conformal field theories are actually mirror to each other \cite{AM:K3}, based on the physics of mirror symmetry for K3 surfaces. \section{Hodge-theoretic mirror conjectures} We can now formulate the main conjecture in the mathematical study of mirror symmetry. \begin{HTmirrorconjecture} \quad Given a boundary point $P\in\overline{\mathcal{M}}_W$ with maximally unipotent monodromy (or perhaps with strongly maximally unipotent monodromy), there should exist a mirror partner $M$ of $W$, a framing $\sigma$ of $M$, a neighborhood $U$ of $P$ in $\mathcal{M}_W$, and a ``mirror map'' \[\mu:U \to (\mathcal{D}_\sigma/L)^-\] which is determined up to constants of integration by the property that \[\mu^*(d\mskip0.5mu\log q_j)= d\left(\frac{\int_{\gamma_j}\Omega}{\int_{\gamma_0}\Omega}\right),\] such that $\mu$ induces an isomorphism between appropriate sub-variations of Hodge structure of \begin{enumerate} \item the formal completion of the geometric variation of Hodge structure at $P$, and \item the framed $A$-variation of Hodge structure with framing $\sigma$. \end{enumerate} (The sub-variations of Hodge structure should contain the entire first two terms of the Hodge filtration on both sides.) \end{HTmirrorconjecture} There are additional conjectures one wants to make about the relationship between $M$ and $W$: there should also be isomorphisms \[H^{p,q}(W)\cong H^{-p,q}(M)\quad p\ge0,\] and these should preserve all correlation functions. (In particular, the ``reverse'' mirror isomorphism should hold, and there should also be isomorphisms between correlation functions which do not come from variations of Hodge structure.) Of course, such isomorphisms only make sense if we have specified the constants of integration. In fact, one wants to conjecture that the entire conformal field theory moduli spaces are isomorphic, but this is a difficult conjecture to make precisely at present since we do not have a complete mathematical understanding of conformal field theory moduli spaces. If we start with the $A$-variation of Hodge structure, there is another conjecture we can make. \begin{converse} Conversely, given $(M,\sigma)$, the corresponding $A$-variation of Hodge structure comes from geometry, in the sense that there is a family $\mathcal{Z}\to\overline{S}$ of varieties degenerating at $0\in\overline{S}$ such that the framed $A$-variation of Hodge structure is isomorphic to the formal completion at $0$ of a (Tate-twisted) sub-variation of Hodge structures of the variation of Hodge structures on some cohomology of $Z_s$. \end{converse} Due to the phenomenon of rigid Calabi--Yau manifolds, we can't assume any stronger properties about $Z_s$: Calabi--Yau threefolds with $h^{2,1}=0$ cannot have mirror partners in the usual sense, since such a mirror partner would satisfy $h^{1,1}=0$, which is absurd. However, there is an example in the physics literature of a rigid Calabi--Yau manifold, known as the ``$Z$-orbifold,'' which has a mirror physical theory that was worked out recently by Candelas, Derrick and Parkes \cite{CDP} (see also \cite{AspGr}). In this example, the variation of Hodge structure associated to the mirror theory can be described by the family of cubic sevenfolds in $\P^8$ (with a suitable Tate twist). \section{Some computations} We explain some of the evidence in favor of the mirror symmetry conjectures which has been accumulated through specific computations.\footnote{The computations presented here are taken from the original paper of Candelas, de la Ossa, Green and Parkes \cite{CDGP} on the quintic threefold, and a paper of Greene, Plesser and the present author \cite{GMP} on higher dimensional mirror manifolds. A survey of other calculations of this type (and the methods for making them) can be found in \cite{predictions}.} We will compute with Calabi--Yau hypersurfaces of dimension $n\ge3$ in ordinary projective space $\C\P^{n+1}$; the degree of the hypersurface must be $n+2$. The family of such hypersurface includes a Fermat hypersurface, which is part of the ``Dwork pencil'' with defining equation: \[x_0^{n+2}+\cdots+x_{n+1}^{n+2}-(n+2)\psi\,x_0{\cdots}x_{n+1}=0,\] where $\psi$ is a parameter. The group \[\Gamma:=\{(\alpha_0,\dots,\alpha_{n+1})\ |\ \alpha_j\in\mmu_{n+2}, \prod\alpha_j=1\}/\{(\alpha,\dots,\alpha)\}\] acts on the fibers of this family by componentwise multiplication on the coordinates. Using either the Greene--Plesser orbifolding construction, or Batyrev's polar polyhedron construction, one sees that the family $\{M\}$ of hypersurfaces of degree $n+2$ in $\C\P^{n+1}$ has as its predicted mirror the family $\{W\}$ described as the Dwork pencil modulo $\Gamma$ (living in the quotient space $\C\P^{n+1}/\Gamma$). In fact, we can describe the moduli space of this mirrored family in terms of the parameter $\psi^{n+2}$---the reason for passing to a power is the existence of an additional automorphism, acting on the family as a whole, generated by componentwise multiplication in the $x$'s by $(\alpha,1,\dots,1)$ while simultaneously multiplying $\psi$ by $\alpha^{-1}$. It is not difficult to compute where this family becomes singular. The partial derivatives of the defining equation are all of the form \[(n+2)\left(x_j^{n+1}-x_j^{-1}\psi\,x_0{\cdots}x_{n+1}\right)\] and for these to vanish simultaneously we must have $\psi^{n+2}=1$. Moreover, the additional automorphism of the family fixed the fiber $\psi=0$, and so causes additional singularities there. Thus, we can describe the moduli space as $\C\P^1-\{0,1,\infty\}$, with its natural compactification being $\C\P^1$. What is the monodromy behavior at the boundary points? (We label the monodromy transformations according to the point.) At $\psi^{n+2}=0$, we find that the monodromy has finite order, at $\psi^{n+2}=1$ it is unipotent but $(T_1-I)^2=0$ so the order is not maximal (since $n\ne1$), and at $\psi^{n+2}=\infty$ we find maximal order of unipotency $(T_\infty -I)^n\ne0$. In fact, this point is maximally unipotent, and even strongly maximally unipotent, in the terminology established earlier. To compute canonical coordinates and correlation functions near $\psi^{n+2}=\infty$ we need to know the period functions there. These can be found by studying the differential equations which they satisfy. In this case of toric hypersurfaces, we have a special method available---the representation of cohomology by means of residues of differential forms on the ambient space with poles along the hypersurface. A basis for the primitive cohomology can be written (in the affine chart $x_0=1$, say) as \[\beta_j:=\operatorname{Res}\left( \frac{\psi^{j+1}\,(x_1{\cdots}x_{n+1})^j\,dx_1\wedge\cdots\wedge dx_{n+1}} {\left(1+x_1^{n+2}+\cdots+x_{n+1}^{n+2} -(n+2)\psi\,x_1{\cdots}x_{n+1}\right)^{j+1}} \right)\] The connection matrix in this basis can then be found using Griffiths' ``reduction of pole order'' lemma \cite{Griffiths} to calculate coefficients $\theta_{ij}$ such that \[\nabla(\beta_i)=\sum\theta_{ij}\beta_j.\] To find the period matrix from the connection matrix, one must solve some differential equations. For if $\{e_k\}$ is a basis for the local system and we write $e_k=\sum\eta_{ki}\beta_i$ then \[0=\nabla(e_k)=\sum d\eta_{ki}\, \beta_i + \sum\eta_{ki}\theta_{ij}\beta_j\] gives differential equations for the unknown coefficient functions $\eta_{ki}$: \[d\eta_{ki}=-\sum\eta_{k\ell}\theta_{\ell i}.\] The flatness of $\nabla$ is equivalent to the integrability of these equations, which can therefore be solved. \begin{table} \begin{center} \begin{tabular}{|l|l|} \hline $n$&$n$-point function\\ \hline $3$& $5+2875\,q+4876875\,q^2+8564575000\,q^3+15517926796875\,q^4 $\\ &$\phantom{5} +28663236110956000\, q^5+53621944306062201000\,q^6 $\\ &$\phantom{5} +101216230345800061125625\,q^7+ 192323666400003538944396875\,q^8 $\\ &$\phantom{5} +367299732093982242625847031250\,q^9 $\\ &$\phantom{5} +704288164978454714776724365580000\,q^{10} $\\ &$\phantom{5} +1354842473951260627644461070753075500\,q^{11} $\\ &$\phantom{5} +2613295702542192770504516764304958585000\,q^{12} $\\ &$\phantom{5} +5051976384195377826370376750184667397150000\,q^{13} $\\ &$\phantom{5} +9784992122065556293839548184561593434114765625\,q^{14} $\\ &$\phantom{5} +18983216783256131050355758292004110332155634496875\,q^{15} $\\ &$\phantom{5} +36880398908911843175757970052077286676680907186572875\,q^{16} $\\ &$\phantom{5} +71739993072775923425756947313710004388338109828244718125\,q^{17} $\\ &$\phantom{5} +139702324572802672116486725324237666156179096139345867681250\,q^{18} $\\ &$\phantom{5} +\dots$\\[6pt] $4$& $6 + 120960 \,q \!+\! 4136832000 \,q^2 \!+\! 148146924602880 \,q^3 \!+ 5420219848911544320 \,q^4 $\\ &$\phantom{6} + 200623934537137119778560 \,q^5 + 7478994517395643259712737280 \,q^6 $\\ &$\phantom{6} + 280135301818357004749298146851840 \,q^7 $\\ &$\phantom{6} + 10528167289356385699173014219946393600 \,q^8 $\\ &$\phantom{6} + 396658819202496234945300681212382224722560 \,q^9 $\\ &$\phantom{6} + 14972930462574202465673643937107499992165427200 \,q^{10} $\\ &$\phantom{6} + 566037069767251121484562070892662863943365345190400 \,q^{11} $\\ &$\phantom{6} + 21424151141341932048068067497996096856724987411324108800 \,q^{12} \!+\! \dots$\\[6pt] $5$& $7 + 3727381 \,q + 2637885990187 \,q^2 + 1927092954108108787 \,q^3 $\\ &$\phantom{7} +1425153551321014327663291 \,q^4 + 1060347883438857662557634869906 \,q^5 $\\ &$\phantom{7} + 791661306374088776109692880989252173 \,q^6 $\\ &$\phantom{7} + 592348256908461616176898022359492565546566 \,q^7 $\\ &$\phantom{7} + 443865568545713063761643598030194801299861575595 \,q^8 $\\ &$\phantom{7} + 332947403131697202086626568381790256001850741509664373 \,q^9 +\dots$\\[6pt] $6$& $8 + 106975232 \,q + 1672023727001600 \,q^2 + 26611692333081695092736 \,q^3 $\\ &$\phantom{8} + 426129121674687823674948571136 \,q^4 $\\ &$\phantom{8} + 6842148599241293047857339542861643776 \,q^5 $\\ &$\phantom{8} + 110018992594692024449889564415904439556898816 \,q^6 $\\ &$\phantom{8} + 1770551943055574073245974844490813198478975912902656 \,q^7 $\\ &$\phantom{8} + 28508925683951911989843155602330000507452539542539447947264 \,q^8 $\\ &$\phantom{8} +\cdots$\\ \hline \end{tabular} \end{center} \medskip \caption{$n$-point functions in dimension $n$} \end{table} If we work in a local coordinate $z=\psi^{-n-2}$ near $\psi^{n+2}=\infty$, we find that a basis $e_0(z)$, \dots, $e_{n+1}(z)$ of local solutions can be found such that $e_0(z)$ is single-valued near $z=0$, and \[e_{j+1}(z) = (\log z)\, e_j(z) + \text{single-valued function}.\] (This is a consequence of the maximally unipotent monodromy.) The vectors $e_j(z)$ form the columns of the period matrix. One can then use row operations to put the period matrix in upper triangular form, with constant diagonal elements. (Let us choose the diagonal elements to all be $n+2$.) This implements the change of basis to a basis consisting of distinguished sections of $\mathcal{H}^{p,q}_\mathbb{S}$. The nonzero entries $A_j^1$ in the connection matrix are then calculated by differentiating rows of the period matrix, and writing the result as a multiple of a subsequent row. Each such entry takes the form \[A^1_j=Y^1_j\,\frac{dq}q,\] and the functions $Y^1_j$ represent correlation functions $\langle (\partial/\partial t)\,\beta_j\,\beta_{n-j-1}\rangle$. This can all be done very explicitly, using power series expansions of the unknown single-valued functions, in these examples. (I advise using {\sc maple} or {\sc mathematica} if you would like to try it for yourself.) We show two kinds of calculations in the tables. For the first, only the ``maximally unipotent'' assumption is required, since the calculation requires only the distinguished $n$-form and the canonical coordinates. What is computed in table 1 is the ``$n$-point function,'' which iterates the differential of the period map $n$ times. (This was introduced some years ago in the variation of Hodge structures context by Carlson, Green, Griffiths and Harris \cite{CGGH}.) \begin{table} \begin{center} \begin{tabular}{|l|} \hline $Y_1^1= 5+2875\,\cuone3+609250\,\cu23+317206375\,\cu33+242467530000\,\cu43 $\\$\phantom{Y_1^1=5} +229305888887625\,\cu53+ 248249742118022000\,\cu63 $\\$\phantom{Y_1^1=5} +295091050570845659250\,\cu73+375632160937476603550000\,\cu83 $\\$\phantom{Y_1^1=5} +503840510416985243645106250\,\cu93 $\\$\phantom{Y_1^1=5} +704288164978454686113488249750\,\cu{10}3 $\\$\phantom{Y_1^1=5} +1017913203569692432490203659468875\,\cu{11}3 $\\$\phantom{Y_1^1=5} +1512323901934139334751675234074638000 \,\cu{12}3 $\\$\phantom{Y_1^1=5} +2299488568136266648325160104772265542625\,\cu{13}3 $\\$\phantom{Y_1^1=5} +3565959228158001564810294084668822024070250\,\cu{14}3 $\\$\phantom{Y_1^1=5} +5624656824668483274179483938371579753751395250\,\cu{15}3 $\\$\phantom{Y_1^1=5} +9004003639871055462831535610291411200360685606000\,\cu{16}3+\dots $\\ \hline \end{tabular} \end{center} \medskip \caption{Three-point function in dimension three} \end{table} \begin{table} \begin{center} \begin{tabular}{|l|} \hline $Y_1^1=6+60480\,\cuone2+440884080\,\cu22+6255156277440\,\cu32$\\ $\phantom{Y_1^1=6}+117715791990353760\,\cu42 +2591176156368821985600\cdot5^2\,\cu52 +\dots$\\ \hline \end{tabular} \end{center} \medskip \caption{Three-point function in dimension four} \end{table} \begin{table} \begin{center} \begin{tabular}{|l|} \hline $Y_1^1=7+1009792\,\cuone2+122239786088\,\cu22 +30528671745480104\,\cu32$\\ $\phantom{Y_1^1=7}+10378199509395886153216\,\cu42 +\dots$\\[6pt] $Y_2^1=7+1707797\,\cuo1+510787745643\,\cuo2 +222548537108926490\,\cuo3$\\ $\phantom{Y_2^1=7}+113635631482486991647224\,\cuo4 +\dots$\\ \hline \end{tabular} \end{center} \medskip \caption{Three-point functions in dimension five} \end{table} \begin{table} \begin{center} \begin{tabular}{|l|} \hline $Y_1^1=8+15984640\,\cuone2+33397159706624\,\cu22 +154090254047541417984\,\cu32 $\\$\phantom{Y_1^1=8} +1000674891265872131899670528\,\cu42+\dots$\\[6pt] $Y_2^1=8+\!37502976\,\cuo1\!+\!224340704157696\,\cuo2 \!+\!2000750410187341381632\,\cuo3$\\ $\phantom{Y_2^1=8} +21122119007324663457380794368\,\cuo4+\dots$\\[6pt] $Y_2^2=8+\!59021312\,\cue{}\!+\!821654025830400\,\cue2 \!+\!\!12197109744970010814464\,\cue3$\\ $\phantom{Y_2^2=8} +186083410628492378226388631552\,\cue4+\dots$\\ \hline \end{tabular} \end{center} \medskip \caption{Three-point functions in dimension six} \end{table} The other computations, displayed in tables 2--5, are of three-point functions $Y^a_b$, read off of the connection matrix in a distinguished basis. (There is a symmetry $Y^a_b=Y^a_{n-a-b}=Y^b_{n-a-b}$ so we only show some of these.) The coefficients in the series expansions are the predicted values of the Gromov--Witten invariants. The three-point function $Y^1_0$ has the value $n+2$ (a constant, due to the definition of canonical coordinates) and is not shown in the tables. The other functions $Y^1_j$ come directly from the connection matrix. In dimension six, there is also a ``secondary'' function, which (by the $B$-model version of the associativity, which is simply the associativity of the ``sheaf cup product'' pairing) can be calculated as $Y^2_2 =(Y^1_2)^2/Y^1_1$. There is a relation between the computations in table 1, and those in tables 2--5, which can be explicitly verified from these tables: it is \[\text{$n$-point function } = \frac{Y^1_0\cdot Y^1_1\cdot {} \dotsm {} \cdot Y^1_{n-1}}{(n+2)^{n}}.\] The functions $Y^a_b$ are predicted to agree with quantum products on the mirror manifolds \[\zeta^a\star\zeta^b\star\zeta^{n-a-b},\] where $\zeta^j$ is the class of a linear space (in $\C\P^{n+1}$) of complex codimension $j$. In fact, we have displayed things in tables 2--5 with this in mind, writing series in terms of $q^k/(1-q^k)$. Also in tables 2--5, we have pulled out some factors of the degree of the rational curve. If there are $\ell$ occurrences of ``1'' among $\{a,b,n-a-b\}$, then there will be $\ell$ of the linear spaces of codimension one, and each meets a given rational curve $\Gamma$ in $\deg(\Gamma)$ points, giving rise to a factor of $(\deg(\Gamma))^{\ell}$ in the Gromov--Witten invariants. Pulling out those factors makes the comparison with ``counting'' problems more transparent. All of the predicted Gromov--Witten invariants in degrees one and two in these tables have been verified by Katz \cite{katz:verifying}; most of the invariants in degree three have been verified by Ellingsrud and Str{\o}mme \cite{ES,ESii}. \chapter*{} \lecturename{Postscript: Recent Developments} \lectureoptionstar{POSTSCRIPT:}{Recent Developments} \markboth{D. R. Morrison, Mathematical Aspects of Mirror Symmetry}{Postscript: Recent Developments} As mentioned in the introduction, the subject of mirror symmetry is a rapidly developing one, and much has happened since the lectures on which these notes are based were delivered. We will briefly sketch some of these developments in this postscript. The Gromov--Witten invariants and their generalizations have been studied particularly intensively. The definition of Ruan \cite{ruan} which we presented in the lectures has been supplanted by other definitions drawn from symplectic geometry (cf.~\cite{MS,RuanTian}) which work directly in cohomology (avoiding the bordism technicalities) and are also more general. In full generality these extended Gromov--Witten invariants are not only associated to curves of genus zero with three vertex operators, but also to curves of arbitrary genus $g$ with $k$ vertex operators (provided that $2g-2+k>0$) and even to some non-topological correlation functions.\footnote{There have also been investigations into the physical interpretation of these higher genus invariants, and how they should transform under mirror symmetry (in the case of Calabi--Yau threefolds) \cite{BCOV:anom,BCOV:KS}. At one time, it had been expected that for Calabi--Yau threefolds the genus zero topological correlation functions would completely determine the conformal field theory, but now it is known that higher genus invariants are needed as well \cite{chiral}.} There are at least three proofs of the associativity relations for these symplectic Gromov--Witten invariants \cite{RuanTian,Liu,MS}, including proofs of a stronger form of associativity known as the Witten--Dijkgraaf--Verlinde--Verlinde (WDVV) equations \cite{topgrav,DVV,Wit:twoDgrav,Dubrov} which are relevant in the case of higher genus. As in the genus zero case, these higher genus invariants can be used to encode a kind of quantum cohomology ring (somewhat larger than the one we studied here); it is also possible to interpret the WDVV associativity relation as the flatness of a certain connection \cite{Dubrov}. A very accessible exposition of this circle of ideas has been written by McDuff and Salamon \cite{MS}. Parallel to this development, Gromov--Witten invariants have also been defined purely within algebraic geometry. The methods of Katz described in the lectures were developed further (see \cite{katz:GW} and the appendix to \cite{BCOV:anom}), and similar methods based on the construction of a ``virtual moduli cycle'' were developed independently by Li and Tian \cite{LiTian}. The foundations for an algebraic theory of Gromov--Witten invariants were carefully laid by Kontsevich and Manin \cite{KM} (again, the higher genus invariants and the WDVV equations play an important r\^ole), and the program they initiated was ultimately carried out \cite{BehMan,BehFant,Beh}, producing a definition of Gromov--Witten invariants based on stable maps. (The work of Li--Tian mentioned above \cite{LiTian} is also closely related to this program.) Even before this program was complete, Kontsevich had applied it to obtain some spectacular results in enumerative geometry, including a verification of the predicted number $242467530000$ of rational quartics on the general quintic threefold \cite{Kontsevich}. The stable map theory is nicely explained, with further references, in \cite{FulPan}. Kontsevich has also formulated a ``homological'' version of the mirror conjecture \cite{kont:icm} involving what are known as $A^\infty$-categories (cf.~\cite{stasheff}), which is related to the ``extended moduli space'' introduced by Witten \cite{witten:mirror}. By a construction of Fukaya \cite{fukaya}, to every compact symplectic manifold $(Y,\omega)$ with vanishing first Chern class, one can associate an $A^\infty$-category whose objects are essentially the Lagrangian submanifolds of $Y$, and whose morphisms are determined by the intersections of pairs of submanifolds. Kontsevich's conjecture relates the bounded derived category of the Fukaya category of $Y$ (playing the r\^ole of the $B$-model) to the bounded derived category of the category of coherent sheaves on a mirror partner $X$ (playing the r\^ole of the $A$-model). I must refer the reader to \cite{kont:icm} for further details concerning this fascinating conjecture. The art of making predictions about enumerative geometry from calculations with the variation of Hodge structure on a candidate mirror partner has been considerably refined: see \cite{predictions} for a survey and references to the literature. The era of numerical experiments in mirror symmetry seems to be largely over, and has been supplanted by a more analytical period. Witten's analysis of the physics related to Calabi--Yau manifolds which are hypersurfaces in toric varieties \cite{phases} was further developed in \cite{summing}, where techniques were found---somewhat related to methods introduced by Batyrev \cite{Bat:qcoho} for the study of quantum cohomology of toric varieties---for precisely calculating a variant of the quantum cohomology ring of the Calabi--Yau manifold. (The variant is derived from enumerative problems on the ambient space rather than directly on the Calabi--Yau manifold.) There is a physics argument, but not a complete mathematics argument, which explains why this variant should coincide with the usual quantum cohomology ring after a change of coordinates in the coefficient ring. This variant {\em can}\/ be rigorously shown to agree with the correlation functions of the mirror Calabi--Yau manifold, again calculated in the ``wrong'' coordinates. In this way, the results of \cite{summing} provided the first analytical proof that some kind of enumerative problem on one side of the mirror could be related to a variation of Hodge structure calculation on the other side. Further development of these ideas in \cite{towards-duality} led to a preliminary argument to the effect that the physical theories associated to a Batyrev--Borisov pair should actually be mirror to each other. In a striking recent development, Givental has proved \cite{Givental:homological,Givental:ICM,Givental:equivariant} that for Calabi--Yau complete intersections in projective spaces, the ``predicted'' enumerative formulas which one calculates by using a Batyrev--Borisov candidate mirror partner are in fact correct evaluations of the Gromov--Witten invariants. This establishes, for example, the accuracy of {\em all}\/ of the predictions about the general quintic threefold made by Candelas et al. \cite{CDGP} (and which we listed in table 2). Givental's remarkable proof actually has very little to do with mirror symmetry {\em per se}: in studying an equivariant version of quantum cohomology, he finds enough structure to enable a calculation which is formally similar to (and certainly inspired by) the variation of Hodge structure calculations on the candidate mirror partner. The last several years have also been a period of dramatic developments in string theory. There are new techniques which go by the names of ``duality'' and ``nonperturbative methods,'' and a number of the recent results have been closely related to Calabi--Yau manifolds and mirror symmetry. One of the earliest nonperturbative results \cite{Str:,bhole} was the discovery\footnote{This had been anticipated some time earlier in the physics literature \cite{CDLS,GreenHubsch,CGH,Cd:con} based on the discovery of and speculations about conifold transitions in the mathematics literature \cite{Clemens:double,Friedman:simult,Hirzebruch:examples,tianyau,% reid,Friedman:threefolds}, but an understanding of the physical mechanism behind the attachment of the moduli spaces was lacking.} that the string theory moduli spaces associated to Calabi--Yau manifolds should be attached along loci corresponding to ``conifold transitions''---a process in which a collection of rational curves is contracted to ordinary double points and the resulting space is then smoothed to produce another Calabi--Yau manifold. This new attaching procedure supplements, but is rather different from, the gluing of K\"ahler cones which we discussed in section \ref{sec73}. In the new procedure, a moduli space of a different dimension (corresponding to a Calabi--Yau manifold with different Hodge numbers than the original) is cemented on at the same point where the two like-dimensional pieces (K\"ahler cones differing by a flop) have been glued together. The ``cement'' which holds these two spaces together (i.e., the physical process responsible) is a phase transition between charged black holes on one component of the moduli space and elementary particles on the other. The string theory moduli spaces mentioned above are actually somewhat larger than the conformal field theory moduli spaces which were one of the primary subjects of these lectures. There are two variants of string theory which are relevant, called type IIA and type IIB string theories, and the additional parameters which must be added to the conformal field theory moduli space differs between the two. In the case of type IIA, the extra parameters are a choice of holomorphic $3$-form and the choice of an element in the intermediate Jacobian of the Calabi--Yau threefold. (Some of the mathematical structure of these spaces related to the intermediate Jacobians was anticipated in work of Donagi and Markman \cite{DonMark}.) In the case of type IIB, the new parameters are similar, but related to the even cohomology of the manifold. These two types of parameters should be mapped to each other under mirror symmetry \cite{udual,mirrorII}. In fact, a large number of other related structures called ``D-brane moduli spaces'' should also correspond under mirror symmetry---the precise implications of this correspondence (which appears to be connected to Kontsevich's homological mirror symmetry conjecture) are still being worked out. Finally, in a very exciting recent development, a completely new geometric aspect of mirror symmetry has been discovered by Strominger, Yau and Zaslow \cite{SYZ}. A Calabi--Yau manifold $X$ of real dimension $2n$ on which a complex structure $J$ and K\"ahler form $\omega$ have been fixed has a natural class of $n$-dimensional submanifolds $M$ defined by the property that $\omega|_M\equiv0$ and $\Im(\Omega)|_M\equiv0$ for some choice of holomorphic $n$-form $\Omega$. These {\em special Lagrangian submanifolds}\/ were introduced by Harvey and Lawson \cite{HL} as a natural class of volume-minimizing submanifolds; they have many other interesting properties, including an exceptionally well-behaved deformation theory \cite{mclean}. Strominger, Yau and Zaslow argue on physical grounds (using the correspondence of D-brane moduli spaces mentioned above) that whenever $X$ has a mirror partner, then $X$ must admit a map $\rho:X^{2n}\to B^n$ whose generic fiber is a special Lagrangian $n$-torus, and which has a section $\sigma:B\to X$ whose image is itself a special Lagrangian submanifold. Given this structure, the mirror partner of $X$ is then predicted to be a compactification of the family of dual tori of the fibers of $\rho$. (The section specifies a point $p_b:=\sigma(b)$ on each torus $T_b:=\rho^{-1}(b)$; the dual torus is then $\operatorname{Hom}(\pi_1(T_b,p_b),\operatorname{U}(1))$.) There is also an argument---quite similar in nature to \cite{towards-duality}---that such a structure should suffice for producing a mirror isomorphism between the corresponding physical theories. A mathematical account of this construction can be found in \cite{underlying}, which attempts to make the mathematical implications of this story precise: given a ``special Lagrangian $m$-torus fibration,'' all of the structure we have seen relating the quantum cohomology and the variation of Hodge structure should (conjecturally) follow as a consequence. For the Voisin--Borcea threefolds, the structure of these special Lagrangian torus fibrations (using a mildly degenerate metric) has been worked out in complete detail by Gross and Wilson \cite{GrossWilson}, who find compatibility with the previously observed mirror phenomena in a beautiful geometric form.

9609

alg-geom/9609012

en

https://arxiv.org/abs/alg-geom/9609012

alg-geom/9609012

Dan Abramovich

Dan Abramovich

A linear lower bound on the gonality of modular curves

Latex2e in compatibility mode

The result in the title is proven, using the Selberg estimate on the leading eigenvalue of the non-Euclidean Laplacian, and the method of conformal volumes of Li and Yau.

alg-geom

\section{INTRODUCTION} \subsection{Statement of result} In this note we prove the following: \begin{th}\label{main} Let $\Gamma\subset PSL_2({\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}})$ be a congruence subgroup, and $X_\Gamma$ the corresponding modular curve. Let $D_\Gamma = [PSL_2({\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}):\Gamma]$ and let $d_\bfc(X_\Gamma)$ be the $\bfc$-gonality of $X_\Gamma$. Then $${7\over 800} D_\Gamma \leq d_\bfc(X_\Gamma).$$ For $\Gamma = \Gamma_0(N)$ we have that $d_\bfc(X_{\Gamma_0(N)})$ is bounded below by ${7\over {800}}\cdot N$. Similarly, we obtain a quadratic lower bound in $N$ for $d_\bfc(X_{\Gamma_1(N)})$. \end{th} \subsection{Remarks} The proof, which was included in the author's thesis \cite{thesis}, follows closely a suggestion of N. Elkies. In the exposition here many details were added to the argument in \cite{thesis}. We utilize the work \cite{liyau} of P. Li and S. T. Yau on conformal volumes, as well as the known bound on the leading nontrivial eigenvalue of the non-euclidean Laplacian $\lambda_1\geq {{21}\over {100}}$ \cite{lrs}. If Selberg's eigenvalue conjecture is true, the constant $7/800$ above may be replaced by $1/96$. Since, by the Gauss - Bonnet formula, the genus $g(X_\Gamma)$ is bounded by $D_\Gamma/12+1$ (indeed the difference is $o(D_\Gamma)$), we may rewrite the inequality above in the slightly weaker form $${{21}\over {200} } (g(X_\Gamma)-1) \leq d_\bfc(X_\Gamma).$$ For an analogous result about Shimura curves, see theorem \ref{shimura} below. It should be noted (as was pointed out by P. Sarnak) that the gonality has an {\em upper} bound of the same type. For the $\bfc$-gonality, by Brill-Noether theory \cite{kl} we have $d_\bfc(X_\Gamma) \leq 1+\left[{{g+1}\over 2}\right]$. If, instead, one is interested in the gonality over the field of definition of $X_\Ga$, one can use the canonical linear series to obtain the upper bound $2g-2$ if $g>1$, and in the few cases where $g=1$ one can use the morphism to $X(1)$ and get the upper bound $D_\Gamma$. \subsection{Acknowledgements} As mentioned above, I am indebted to Noam Elkies for the main idea. The question was first brought to my attention in a letter by S. Kamienny. The result first appeared in my thesis under the supervision of Prof. J. Harris. Thanks are due to David Rohrlich and Glenn Stevens who set me straight on some details, and to Peter Sarnak for helpful suggestions. \section{Setup and proof} \subsection{Gonality}\label{gonal} Let $C$ be a smooth, projective, absolutely irreducible algebraic curve over a field $K$. Define the $K$-{\bf gonality} $d_K(C)$ of $C$ to be the minimum degree of a finite $K$-morphism $f:C\rightarrow} \newcommand{\dar}{\downarrow \bfp^1_K$. Clearly if $K\subset L$ then $d_K(C)\geq d_L(C\times_KL)$, and equality must hold whenever $K$ is algebraically closed. \subsection{Congruence subgroups and modular curves} By a {\bf congruence subgroup} $\Gamma\subset PSL_2({\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}})$ we mean that for some $N$, $\Gamma$ contains the principal congruence subgroup $\Ga(N)$ of $2\times 2$ integer matrices congruent to the identity modulo $N$. Since $PSL_2({\Bbb{R}}} \newcommand{\bfh}{{\Bbb{H}})$ acts on $\bfh= \{z = x+iy|y>0\}$ via fractional linear transformations, we may let $Y_\Gamma = \Gamma\setminus \bfh$. It is well known that $Y_\Gamma$ may be compactified by adding finitely many points, called {\bf cusps}, to obtain a compact Riemann surface $X_\Ga$, which we call the {\bf modular curve} corresponding to $\Ga$. \subsection{The Poincar\'e metric} The upper half plane $\bfh$ carries the Poincar\'e metric $ds^2 = {{dx^2 + dy^2}\over {y^2}}$, which is $PSL_2({\Bbb{R}}} \newcommand{\bfh}{{\Bbb{H}})$ - invariant. The corresponding area form is given by ${{dx \, dy} \over {y^2}}$. Away from a finite set $T$ consisting the cusps and possibly some elliptic fixed points, the metric descends to a Riemannian metric on $X_\Ga {\,\,^{_\setminus}\,\,} T$, of finite area. We denote the area measure by $d\mu$. We will accordingly call a quadratic differential $ds^2$ a {\bf singular metric} if it is a Riemannian metric away from finitely many points, and has finite area. Thus the Poincar\'e metric gives rise to a singular metric on $X_\Ga$. \subsection{The Laplacian} It is natural to consider the Hilbert space $L_2(\Gamma\setminus \bfh)= L_2(X_\Gamma)$, where the $L_2$ pairing is taken with respect to the Poincar\'e metric. The Laplace-Beltrami operator associated with the metric $$ \Delta = -y^2({{\partial^2}\over {\partial x^2}} +{{\partial^2}\over {\partial y^2}})$$ gives rise to a self adjoint unbounded operator on $ L_2(X_\Gamma)$, which is in fact positive semidefinite. The kernel of $\Delta$ consists of the constant functions. In contrast with the case of a genuine Riemannian metric on a compact manifold, the spectrum of $\Delta$ is not discrete (see e.g. \cite{hejhal}, VI\S 9, VII\S 2, VIII\S 5). The continuous spectrum is $\{\lambda\geq 1/4\} \subset {\Bbb{R}}} \newcommand{\bfh}{{\Bbb{H}}$, and is fully accounted for by an integral formula involving Eisenstein series $E(z,s)$ for $Re(s) = 1/2$. The discrete part of the spectrum is given by $\lambda_0=0$ corresponding to the constants, and $0< \lambda_1 < \lambda_2...$ corresponding to the so called {\bf cuspidal} eigenvectors. \subsection{Selberg's conjecture} The question, what is $\lambda_1$ turns out to be a fundamental one. Selberg \cite{selberg} has shown that $\lambda_1\geq 3/16$ and conjectured that $\lambda_1\geq 1/4$. Recently, Luo, Rudnick and Sarnak \cite{lrs} showed that $\lambda_1\geq 0.21$ (note that $3/16 < 0.21 < 1/4$). Since the continuous spectrum is known to be $\lambda\geq 1/4$, denote by $\lambda_1' = \min(\lambda_1, 1/4)$. The value of $\lambda_1'$ has the following characterization: Let $g$ be a nonzero continuous, piecewise differentiable function on $X_\Ga$ such that $\nabla g$ is square integrable with respect to $\mu$, and $\int_{X_\Ga} g d\mu=0$. Then (identifying $X_\Ga$ with $\Gamma\setminus \bfh$) we have $$\int_{\Gamma\setminus \bfh} \left(\,\left({{\partial g}\over {\partial x}}\right)^2 +\left({{\partial g}\over {\partial y}}\right)^2 \,\right) dx \, dy \geq \lambda_1' \int_{\Gamma\setminus \bfh} g^2 {{dx \, dy} \over {y^2}}.$$ This is, in fact, the way Selberg originally stated his result. \subsection{Conformal area} Let $C$ be a compact Riemann surface. Following \cite{liyau}, we define the {\bf conformal area}, or the first conformal volume $A_c(C)$ to be the infimum of $\int_C f^* d\mu_0$, where $f:C\rightarrow} \newcommand{\dar}{\downarrow\bfp^1_\bfc$ runs over all nonconstant conformal mappings, and where $d\mu_0$ is the $SO_3$-invariant area element on the Riemann sphere. Using the conformal property of homotheties in $\bfp^1$, Li and Yau show easily that $$A_c(C) \leq 4\pi\cdot d_\bfc(C).$$ On the other hand, given a Riemannian metric on $C$, let $A(C)$ be the area of $C$. Using an elegant fixed point argument, Li and Yau obtain (\cite{liyau}, Theorem 1) $$\lambda_1 A(C) \leq 2A_c(C).$$ Their proof works word for word in the case of our singular metric on $X_\Ga$, once we replace $\lambda_1$ by $\lambda_1'$. All that is needed is, first, the characterization of $\lambda_1'$ discussed above, and second, the fact that differentiable functions on $X_\Gamma$ have a square-integrable gradient. The latter follows since $\int_{X_\Gamma}|\nabla g|^2d\mu$ is invariant under conformal change of the metric, therefore it may be calculated using a regular metric, and thus is finite. \subsection{Conclusion of the proof} Since the Poincar\'e metric on $X_\Ga$ is pulled back from $X_{PSL_2({\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}})}=X(1)$, we have $A(X_\Ga)=D_\Gamma \cdot A(X(1)) = D_\Gamma \cdot\pi/3$. Combine this with the inequalities of Li and Yau, and obtain the first part of the theorem. Now note that $[PSL_2({\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}):\Gamma_0(N)]$ is at least $N$, and similarly $[PSL_2({\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}}):\Gamma_1(N)]$ is quadratic in $N$ (between $6(N/\pi)^2$ and $N^2$), and obtain the second part. \qed \subsection{An analogous result for Shimura curves} As was pointed out by P. Sarnak, we have the following: \begin{th}\label{shimura} Let $D$ be an indefinite quaternion algebra over ${\Bbb{Q}}} \newcommand{\bff}{{\Bbb{F}}$, and let $G$ be the group of units of norm 1 in some order of $D$. Let $\Gamma\subset G$ be a subgroup of finite index, and let $X_\Gamma=\Gamma \setminus \bfh$ be the corresponding Shimura curve. Then $${{21}\over {200} } (g(X_\Gamma)-1) \leq d_\bfc(X_\Gamma).$$ \end{th} {\bf Proof.} Since $X_\Gamma$ is compact, every automorphic form $g$ appearing in $L^2(X_\Gamma)$ is cuspidal. It follows from the Jacquet - Langlands correspondence (see \cite{gelbart}, Theorem 10.1 and Remark 10.4) that unless $g$ is the constant function, there exists a cuspidal automorphic form for some congruence subgroup in $SL_2({\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}})$ which has the same eigenvalue with respect to the non-euclidean Laplacian. Therefore $\lambda_1 \geq 0.21$ holds for $X_\Gamma$. The results of Li and Yau give $\lambda_1 A(X_\Gamma) \leq 8\pi\cdot d_\bfc(X_\Gamma)$, and the Gauss - Bonnet formula gives $4\pi(g(X_\Gamma))-1) \leq A(X_\Gamma)$ (the difference coming from elliptic fixed points). Combining the three inequalities we obtain the result. \qed The author was informed that the results of \cite{lrs} were generalized by Rudnick and Sarnak to cuspidal automorphic forms on $GL_2$ over an arbitrary number field $F$. Therefore Theorem \ref{shimura} holds for $D$ a quaternion algebra over a totally real field, which is indefinite at exactly one infinite place. \section{Applications and remarks} \subsection{${\Bbb{Q}}} \newcommand{\bff}{{\Bbb{F}}$-gonality and rational torsion on elliptic curves} Let $C$ be a curve as in \ref{gonal}. Recall \cite{ah} that a point $P\in C$ is called {\bf a point of degree $d$} if $[K(P):K]=d$. Suppose $C$ has infinitely many points of degree $d$. By taking Galois orbits on the $d$-th symmetric power of $C$ we have that ${\operatorname{Sym}}^d(C)(K)$ is infinite. Let $W_d(C)\subset Pic^d(C)$ be the image of ${\operatorname{Sym}}^d(C)(K)$ by the Abel-Jacobi map. In \cite{ah} it was noted that in this situation either $d_K(C) \leq d$, or $W_d(C)(K)$ is infinite. Now assume $K$ is a number field. By a celebrated theorem of Faltings \cite{fal}, if $W_d(C)(K)$ is infinite then $W_d(C)\subset Pic^d(C)$ contains a positive dimensional translate of an abelian variety, and the simple lemma 1 of \cite{ah} implies that $d_K(C)\leq 2d$ (\cite{thesis}, theorem 9). The latter conclusion was also obtained by G. Frey in \cite{frey}. We now restrict attention to the case where $K={\Bbb{Q}}} \newcommand{\bff}{{\Bbb{F}}$ and $C = X_0(N)$. In \cite{thesis}, Theorem 12, as well as in \cite{frey}, it was noted that a lower bound on the ${\Bbb{Q}}} \newcommand{\bff}{{\Bbb{F}}$-gonality, such as given by theorem \ref{main}, implies that there exists a constant $m(d)$ (in fact, $m=230d$ will do), such that if $N> m(d)$ then $X_0(N)$ (and thus also $X_1(N)$) has finitely many points of degree $d$. In section 1 of \cite{km}, Kamienny and Mazur showd that this reduces the uniform boundedness conjecture on torsion points on elliptic curves to bounding rational torsion of prime degree. The conjecture was finally proved by L. Merel in \cite{merel}. It should be remarked that, since for this application one only needs a lower bound on the ${\Bbb{Q}}} \newcommand{\bff}{{\Bbb{F}}$-gonality of $X_0(N)$, one can use other methods, such as Ogg's method \cite{ogg}. This is indeed the method used by Frey in \cite{frey}, although the bound obtained is not linear. For points of low degree, one can use the main results of \cite{ah} with Ogg's method to slightly improve the bound on $N$ (see \cite{hs} and \cite{thesis}, 2.5). For another arithmetic application of the lower bound on teh $\bfc$-gonality, regarding pairs of elliptic curves with with isomorphic mod $N$ representations, see Frey \cite{frey2}. \subsection{Torsion points: the function field case} Recently, there has been renewed interest in the question of $\bfc$-gonality of modular curves. In their paper \cite{ns}, K. V. Nguyen and M.-H. Saito used algebraic techniques to give a lower bound on the gonality. Although their bound is a bit weaker than ours, their methods are of interest on their own right: they combine Ogg's method with a Castelnuovo type bound. They pointed out that given any such bound, one obtains a function field analogue of the strong uniform boundedness theorem about torsion on elliptic curves, namely: given a non-isotrivial elliptic curve over the function field of a complex curve $B$, the size of the torsion subgroup is bounded solely in terms of the gonality of $B$. This result is strikingly analogous to a recent result of P. Pacelli (\cite{p}, Theorem 1.3): assuming Lang's conjecture on rational curves on varieties of general type, the number of non-constant points on a curve $C$ of genus $>1$ over the function field of $B$ is bounded solely in terms of the genus of $C$ and the gonality of $B$.

9609

alg-geom/9609013

en

https://arxiv.org/abs/alg-geom/9609013

alg-geom/9609013

Dan Abramovich

Dan Abramovich and Jianhua Wang

Equivariant resolution of singularities in characteristic 0

Latex2e in compatibility mode

A new proof of equivariant resolution of singularities under a finite group action in characteristic 0 is provided. We assume we know how to resolve singularities without group action. We first prove equivariant resolution of toroidal singularities. Then we reduce the general case to the toroidal case.

alg-geom

\section{Introduction} We work over an algebraically closed field $k$ of characteristic 0. \subsection{Statement} In this paper, we use techniques of toric geometry to reprove the following theorem: \begin{th} Let $X$ be a projective variety of finite type over $k$, and let $Z\subset X$ be a proper closed subset. Let $G\subset {\operatorname{Aut}}_k(Z\subset X)$ be a finite group. Then there is a $G$-equivariant modification $r:X_1\to X$ such that $X_1$ is nonsingular projective variety, and $r^{-1}(Z_{{\mbox{\small red}}})$ is a $G$-strict divisor of normal crossings. \end{th} This theorem is a weak version of the equivariant case of Hironaka's well known theorem on resolution of singularities. It was announced by Hironaka, but a complete proof was not easily accessible for a long time. The situation was remedied by E. Bierstone and P. Milman \cite{bm1}, who gave a construction of completely canonical resolution of singularities. Their construction builds on a thorough understanding of the effect of blowing up. They carefully build up an invariant pointing to the next blowup. The proof we give in this paper takes a completely different approach. It uses two ingredients: first, we assume that we know the existence of resolution of singularities without group actions. The method of resolution is not important: any of \cite{hi}, \cite{bm}, \cite{aj} or \cite{bp} would do. Second, we use equivariant toroidal resolution of singularities. Unfortunately, in \cite{te} the authors do not treat the equivariang case. But proving this turns out to be straightforward given the methods of \cite{te}. To this end, section \ref{equitor} of this paper is devoted to proving the following: \begin{th}\label{equitorres} Let $U\subset X$ be a strict toroidal embedding, and let $G\subset {\operatorname{Aut}}(U\subset X)$ be a finite group acting toroidally. Then there is a $G$-equivariant toroidal ideal sheaf $\I$ such that the normalized blowup of $X$ along $\I$ is a nonsingular $G$-strict toroidal embedding. \end{th} \subsection{Acknowledgements} Thanks are due to A. J. de Jong, who was a source of inspiration for this paper. Thanks also to S. Katz and T. Pantev for helpful discussions relevant to this paper. Special thanks to S. Kleiman, who made this collaboraion possible. \section{Preliminaries} First recall some definitions. We restrict ourselves to the case of varieties over $k$. A large portion of the terminology is borrowed from \cite{aj}. A {\bf modification} is a proper birational morphism of irreducible varieties. Let a finite group $G$ act on a (possibly reducible) variety $Z$. Let $Z=\cup Z_i$ be the decomposition of $Z$ into irreducible components. We say that {\bf $Z$ is $G$-strict} if the union of translates $\cup_{g\in G} g(Z_i)$ of each component $Z_i$ is a normal variety. We simply say that $Z$ is {\bf strict} if it is $G$-strict for the trivial group, namely every $Z_i$ is normal. A divisor $D\subset X$ is called a {\bf divisor of normal crossings} if \'etale locally at every point it is the zero set of $u_1\cdots u_k$ where $u_1,\ldots,u_k$ is part of a regular system of parameters. Thus, in a strict divisor of normal crossings $D$, all components of $D$ are nonsigular. An open embedding $U\hookrightarrow X$ is called a {\bf toroidal embedding} if locally in the \'etale topology (or classical topology in case $k=\bfc$, or formally) it is isomorphic to a torus embedding $T \hookrightarrow V$, (see \cite{te}, II\S 1). Let $E_i, i\in I$ be the irreducible components of $X^{_\setminus} U$. A finite group action $G\subset {\operatorname{Aut}}(U\hookrightarrow X)$ is said to be {\bf toroidal} if the stabilizer of every point can be identified on the appropriate neighborhood with a subgroup of the torus $T$. We say that a toroidal action is {\bf $G$-strict} if $X^{_\setminus} U$ is $G$-strict. In particular the toroidal embedding itself is said to be strict if $X^{_\setminus} U$ is strict. This is the same as the notion of {\bf toroidal embedding without self-intersections} in \cite{te}. For any subset $J$ of $I$, the components of the sets ${\cap_{i\in J}}E_i - {\cup_{i\notin J}}E_i$ define a stratification of $X$. Each component is called a {\bf stratum}. Recall that in \cite{te}, p. 69-70 one defines the notion of a {\bf conical polyhedral complex} with {\bf integral structure}. As in \cite{te}, p. 71, to every strict toroidal embedding $U\subset X$ one canonically associates a conical polyhedral complex with integral structure. In the sequel, when we refer to a conical polhedral complex, it is understood that it is endowed with an integral structure. In \cite{te}, p. 86 (Definition 2) one defines {\bf a rational finite partial polyhedral decomposition} $\Delta'$ of a conical polyhedral complex $\Delta$. We will restrict attention to the case where $|\Delta'| = |\Delta|$, and we will call this simply a {\bf polyhedral decomposition} or {\bf subdivision}. The utility of polyhedral decompositions is given in Theorem 6* of \cite{te} (page 90), which establishes a correspondence between allowable modifications of a given strict toroidal embedding (which in our terminology are proper), and polyhedral decompositions of the conical polyhedral complex. In order to guarantee that a modification is projective, one needs a bit more. Following \cite{te}, p. 91, a function ${\operatorname{ord}}:\Delta\rightarrow} \newcommand{\dar}{\downarrow {\Bbb{R}}$ defined on a conical polydral complex with integral structure is called an {\bf order function} if: \parbox{4in}{ (1) ${\operatorname{ord}}(\lambda x)=\lambda\cdot{{\operatorname{ord}}(x)} , \lambda \in {\Bbb{R}}^+ $\\ (2) ${\operatorname{ord}}$ is continuous, piecewise-linear\\ (3) ${\operatorname{ord}} (N^Y\cap {\sigma^Y}) \subset {\Bbb{Z}}} \newcommand{\bfg}{{\Bbb{G}} $ for all strata $Y$.\\ (4) ${\operatorname{ord}}$ is convex on each cone $\sigma\subset \Delta$} $(*)$ For an order function on the conical polyhedral complex coresponding to $X$, we can define canonically a coherent sheaf of fractional ideals on $X$, and vice versa (see \cite{te}, I\S 2). The order function is positive if and only if the corresponding sheaf is a a genuine ideal sheaf. We have the following important theorem \cite{te}: \begin{th} Let $F$ be a coherent sheaf of ideals corresponding to a positive order function ${\operatorname{ord}}$, and let $B_{F}(X)$ be the normalized blowup of $X$ along $F$. Then $B_F(X)\rightarrow} \newcommand{\dar}{\downarrow X$ is an allowable modification of $X$, described by the decompostion of $|\Delta|$ obtained by subdividing the cones into the biggest subcones on which ${\operatorname{ord}}$ is linear. \end{th} A polyhedral decomposition is said to be {\bf projective} if it is obtained in such a way from an order function. Given a cone $\sigma$ and a rational ray $\tau\subset \sigma$, it is natural to defing the decomposition of $\sigma$ centered at $\tau$, whose cones are of the form $\sigma'+\tau$, where $\sigma'$ runs over faces of $\sigma$ disjoint from $\tau$. Given a polyhedral complex $\Delta$ and a rational ray $\tau$, we can take the subdivision of all cones containing $\tau$ centered at $\tau$, and again call the resulting decompositionion of $\Delta$, the subdivision centered at $\tau$. From \cite{te} I\S 2, lemmas 1-3, it follows that the subdivision centered at $\tau$ is projective. A very important decomposition is the {barycentric subdivision}. Let $\sigma$ be a cone with integral structure, $e_1,\ldots,e_k$ integral generators of its edges. The {\bf barycenter} of $\sigma$ is the ray $b(\sigma) = {\Bbb{R}}_{\geq 0}\sum e_i$. The {\bf barycentric subdivision} of a polyhedral complex $\Delta$ of dimension $m$ is the minimal subdivision $B(\Delta)$ in which the barycenters of all cones in $\Delta$ appear as cones in $B(\Delta)$. It may be obtained by first taking the subdivision centered at the barycenters of $m$ dimensional cones, then the decomposition of the resulting complex centered at the barycenters of the cones of dimension $m-1$ of the {\em original} complex $\Delta$, and so on. From the discussion above (or \cite{te} III \S 2 lemma 2.2), we have that the barycentric subdivision is projective. One can also obtain the barycentric subdivision inductively the other way: the barycentric subdivision of an $m$-dimensional cone $\delta$ is formed by first taking the barycentric subdivision of all its faces, and for each one of the resulting cones $\sigma$, including also the cone $\sigma + b(\delta)$. This way it is clear that $B(\Delta)$ is a simplicial subdivision. \section{Equivariant toroidal modifications}\label{equitor} \begin{lem}\label{equiact} Let $U\subset X$ be a strict toroidal embedding, $G\subset Aut(U\subset X)$ a finite group action. Then \begin{enumerate} \item The group $G$ acts linearly on $\Delta(X)$. \item Assume that the action of $G$ is strict toroidal. Let $g\in G$, and let $ \delta\subset \Delta(X)$ be a cone, such that $ g(\delta)= \delta$. Then $ g_{|\delta}=id $. \end{enumerate} \end{lem} {\bf Proof.} \begin{enumerate} \item Clearly, $G$ acts on the stratification of $U\subset X$. Note that, from Definition 3 of \cite{te}, page 59, $\Delta(X)$ is built up from the groups $M^Y$ of Cartier divisors on $Star(Y)$ supported on $Star(Y)^{_\setminus} U$, as $Y$ runs through the strata. As $g\in G$ canonically transforms $M^Y$ to $M^{g^{-1}Y}$ linearly, our claim follows. \item Assume $ g: \delta \rightarrow \delta$, and $g_{|\delta}\neq id $, then there exists an edge $e_1 \in \delta$ , s.t $g(e_1)\neq e_1$. Denote $g(e_1) =e_2$. Assume $e_1$ corresponds to a divisor $E_1$, and $e_2$ corresponds to a divisor $E_2$. Since $g(e_1) =e_2$ we have $g(E_1)=E_2$. As $e_1, e_2$ are both edges of $\delta$, $E_1\cap E_2 \neq \phi$. So $\cup g(E_1)$ can not be normal since it has two intersecting components. This is a contradiction to the fact that $G$ acts strictly on $X$.\qed \end{enumerate} \begin{lem} \label{equideco} Let $G\subset{\operatorname{Aut}}(U\subset X)$ act toroidally. Let $\Delta_1$ be a $G$-equivariant subdivision of $\Delta$, with corresponding modification $X_1\rightarrow} \newcommand{\dar}{\downarrow X$. Then $G$ acts toroidally on $X_1$. Moreover, if $G$ acts strictly on $X$, it also acts strictly on $X_1$. \end{lem} {\bf Proof.} The fact that $G$ acts on $X_1$ follows from the canonical manner in which $X_1$ is costructed from the decomposition $\Delta_1$, see Theorems 6* and 7* of \cite{te}, \S 2.2. Now for any point $a \in X_1$ and $g \in Stab_a$, we have $g\circ f(a)=f\circ g(a)=f(a)$ hence $g \in Stab_{f(a)}$, Thus $Stab_a$ is a subgroup of $Stab_{f(a)}$, which is identified with a subgroup of torus in a neigbourhood of $f(a)$. This proved that $Stab_a$ is identified with a subgroup of torus. We are left with showing that if $G$ acts strictly on $X$, then it acts strictly on $X_1$. Assume it is not the case. There exists two edges $\tau_1$,$\tau_2 $ in $\Delta_1$, which are both edges of a cone, $\delta'$, and $g(\tau_1)=\tau_2$. We choose the cone $\delta'$ of minimal dimension. Clearly, $\tau_1$ and $\tau_2 $ cannot be both edges in $\Delta $, since $G$ acts strictly on $X$. Let us assume $\tau_2$ is not an edge in $\Delta$. So $\tau_2$ must be in the interior of a cone $\delta$ in $\Delta$, which contains $\delta'$. Now since ${\delta'}\cap g({\delta'}) \supset \tau_2 \subset$ interior of $\delta$, we conclude: interior of $\delta \cap g(\delta) \neq \phi$, which means that $g(\delta)=\delta$. From the previous lemma, $g_{|\delta}=id$, so ${g_{|\delta'}}=id$ too, contradiction. \qed \begin{prp} \begin{enumerate} \item There is a 1 to 1 correspondence between edges $\tau_i$ in the barycentric subdivision $B(\Delta)$ and positive dimensional cones $\delta_i$ in $\Delta $. We denote this by $\tau\mapsto \delta_\tau$. \item Let $\tau_i\neq \tau_j$ be edges of a cone $\hat{\delta} \in B(\Delta)$. Then dim $ \delta_{\tau_i}\neq \delta_{\tau_j}. $ \item If $G$ is a finite group acting toroidally on a strict toroidal embedding $U\subset X$, then the action of $G$ on $ X_{B(\Delta)}$ is strict. \end{enumerate} \end{prp} {\bf Remark.} Using this proposition, the argument at the end of \cite{aj} can be significantly simplified: there is no need to show $G$-strictness of the toroidal embedding obtained there, since the barycentric subdivision automatically gives a $G$ strict modification. {\bf Proof.} 1. Define a map $b:$ positive dimension cones in $\Delta \rightarrow$ edges in $B(\Delta)$ by $$ b(\delta)=\mbox{ the barycenter of }(\delta) $$ and define $\delta:$ edges in $B(\Delta) \rightarrow $ cones in $\Delta$ by $$ \delta_\tau= \mbox{ the unique cone whose interior contains }\tau $$ then it is easy to see that $b$ and $\delta$ are invereses of each other. 2. We proceed by induction on $\dim \Delta$. The cone $\delta$ spanned by $\tau_i$ and $\tau_j$ must lie in some cone of $\Delta$, say $\delta^*$, which we may take of minimal dimension. We follow the second construction of the barycentric subdivision described in the preliminaries. Either $\dim \delta^* \leq m-1$, so $\delta$ is in the barycentric subdivision of the $m-1$-skeleton of $\Delta$, in which case the statement follows by the inductive assumption, or $\dim \delta^*=m$, in which case only one of $\tau_1$ and $\tau_2$ can be its barycenter, and the other is again a barycenter of a cone in th $m-1$ skeleton. 3. From lemma \ref{equideco}, since the decomposition $B(\Delta)$ of $\Delta$ is equivariant, $G$ acts toroidally on $X_B(\Delta)$. Let $E_1, E_2\subset X_B(\Delta)^{_\setminus} U$ be divisors corresponding to edges $ e_1, e_2 $ in $B(\Delta)$. Since $E_1 \cap E_2 \neq \phi$, there there is a cone in $B(\Delta)$ containing $e_1, e_2$ as edges. From part (2), $\dim \delta_{e_1} \neq \dim \delta_{e_2}$, so $ g(e_1)$ can not equal to $e_2$. This contradicts the fact that the morphism is equivariant and $g(E_1)=E_2$. \begin{prp} There is a positive $G$-equivariant order function on $B(\Delta)$ such that the associated ideal $\I$ induces a blowing up $B_{\I}X_{B(\Delta)}$, which is a nonsingular $G$-strict toroidal embedding, on which $G$ acts toroidally. \end{prp} {\bf Proof.} By the previous proposition, we know that $G$ acts toridally and strictly on $X_{B(\Delta)}$. It follows from Lemma \ref{equiact} that the quotient $B(\Delta)/{G}$ is a conical polyhedral complex, since no cone has two edges in $B(\Delta)$ which are identified in the quotient. We can use the argument of \cite{te}, I\S 2, lemmas 1-3, to get an order function ${\operatorname{ord}}:B(\Delta)/{G} \to {\Bbb{R}}$ which induces a simplicial subdivision with every cell of index 1. Denote by $\pi:B(\Delta) \to B(\Delta)/{G}$ the quotient map. Then $ord \circ \pi $ is an order function subdividing $B(\Delta)$ into simplicial cones of index 1. Let $\I$ be the corresponding ideal sheaf. The blow up $X_{B(\Delta)}$ along $\I$ is a nonsingular strict toroidal embedding $U\subset B_{\I}X_{B(\Delta)}$. By lemma \ref{equideco}, $G$ acts on $B_{\Gamma}X_{B(\Delta)}$ strictly and toroidally. \qed {\bf Proof of Theorem \ref{equitorres}.} Let $G\subset {\operatorname{Aut}}(U\subset X)$ be as in the theorem. The morphism $X_{B(\Delta)}\rightarrow} \newcommand{\dar}{\downarrow X$ is projective, and by the last two propositions there is a projective, totoidal $G$-equivariant morphism $Y \rightarrow} \newcommand{\dar}{\downarrow X$ where $Y$ is nonsingular and such that $G$ acts strictly and toroidally on $Y$. \qed {\bf Remark.} With a little more work we can obtain a {\bf canonical} choice of a toroidal equivariant resolution of singularities. One observes that the cones in the barycentric subdivision have canonically ordered coordinates, which agree on intersecting cones: for a cone $\delta$ choose the unit coordinate vectors $e_i$ to be primitive lattice vectors generating the edges $\tau$, where $i=\dim \delta_\tau$, the dimension of the cone of which $\tau$ is a barycenter. Recall that in order to resolve singularities, one successively takes the subdivisions centered at lattice points $w_j$ which are not integrally generated by the vectors $e_i$. These $w_j$ are partially ordered according to the lexicographic ordering of their canonical coordinates, in such a way that if $w_j\neq w_k$ have the same coordinates (e.g. if $g(w_1) = w_2$), they do not lie in a the same cone, and therefore we can take the centered subdivision simultaneousely. We conclude this section with a simple proposition which is implicitly used in \cite{aj}: \begin{prp} Let $U\subset X$ be a strict toroidal embedding, and let $G\subset {\operatorname{Aut}}(U\subset X)$ be a finite group acting strictly and toroidally. Then $ (X/{G},U/{G})$ is a strict toroidal embedding. \end{prp} {\bf Proof.} Since the quotient of a toric variety by a finite subgroup of the torus is toric, we conclude that $X/{G}$ is still a toroidal embedding, by the definition of toroidal embedding. We need to show that it is strict. Let $q:X\rightarrow} \newcommand{\dar}{\downarrow X/G$ be the quotient map. Let $Z \subset X^{_\setminus} U$ be a divisor. Then $q(Z) = q(\cup_g g(Z))$. Since the action is strict, we have $q(\cup_g g(Z)) \simeq Z/Stab(Z)$, which is normal. \section{Proof of the theorem} Given $Z,X$ with $G$ action , $G$ finite, let $Y=X/G$, $Z/G$ be the quotient, $B$ the branch locus. Define $ W = Z/G \cup B$. Let $(Y',W')\to (Y,W)$ be a resolution of singularities of $Y $ with $W'$ a strict divisor of normal crossings. Let $X'$ be the normalization of $Y'$ in $K(X)$, and $Z'$ the inverse image of $W'$. Let $U={X'}^{_\setminus} Z'$. Clearly $ U \subset X')$ is a strict toroidal embedding, on which $G$ acts toroidally (moreover, it is $G$-strict). Apply theorem \ref{equitorres} and obtain a nonsingular strict toroidal embedding $U\subset X_1 \rightarrow} \newcommand{\dar}{\downarrow X'$ as required. \qed

9609

alg-geom/9609002

en

https://arxiv.org/abs/alg-geom/9609002

alg-geom/9609002

Parusinski Adam

Adam Parusinski and Zbigniew Szafraniec

Algebraically Constructible Functions and Signs of Polynomials

AMS-LaTeX, 15 pages

Universite d'Angers prepublication no. 18

We show that on real algebraic sets algebraically constructible functions coincide with the finite sums of signs of polynomials. Then we give some applications.

alg-geom

\section{Introduction}\label{introduction} Let $f:X\to W$ be a regular morphism of real algebraic sets. Consider on $W$ an integer-valued function $\varphi(w) = \chi (X_w)$, which associates to $w\in W$ the Euler characateristic of the fibre $X_w=f^{-1}(w)$. The main purpose of this paper is to study the properties of such $\varphi$. Firstly, by stratification theory, $\varphi$ is (semialgebraically) constructible, that is there exists a semialgebraic stratification ${\mathcal S}$ of $W$ such that $\varphi$ is constant on strata of ${\mathcal S}$. Equivalently, we may express this property by saying that $\varphi$ is bounded and $\varphi^{-1}(n)$ is semialgebraic for every integer $n$. However it is well-known that not all semialgebraically constructible functions on $W$ are of the form $\chi (X_w)$ for a regular morphism $f:X\to W$. For instance, if $W$ is irreducible, then $\chi (X_w)$ has to be generically constant modulo 2, see for instance \cite[Proposition 2.3.2]{akbulutking}. Also in the case of $W$ irreducible, as shown in \cite{costekurdyka2}, there exists a real polynomial $g:W\to \mbox{$\mathbf R$}$ such that generically on $W$ $\varphi (w) \equiv \mbox{$\operatorname {sgn}$}\, g(w) \pmod 4$, where by $\mbox{$\operatorname {sgn}$}\, g$ we denote the sign of $g$. As we show in Theorem \ref{wtyczka} below, for any regular morphism $f:X\to W$ of real algebraic sets there exist real polynomials $g_1,\ldots, g_s$ on $W$ such that for every $w\in W$ \[ \chi (X_w)= \mbox{$\operatorname {sgn}$}\, g_1(w) + \cdots + \mbox{$\operatorname {sgn}$}\, g_s(w) .\] In particular, taking $g=g_1 \cdot sg_s$ we recover the result of \cite{costekurdyka2}. Constructible functions of the form $\varphi (w) = \chi (X_w)$, for proper regular morphisms $f:X\to W$, were studied in \cite{mccroryparusinski2} in a different context. Following \cite{mccroryparusinski2} we call them {\it algebraically constructible}. As shown in \cite{mccroryparusinski2} the family of algebraically constructible functions is preserved by various natural geometric operations such as, for instance, push-forward, duality, and specialization. In a way they behave similarly to constructible functions on complex algebraic varieties. However, unlike their complex counterparts, they cannot be defined neither in terms of stratifications nor as combinations of characteristic functions of real algebraic varieties, cf. \cite {mccroryparusinski2}. Algebraically constructible functions were used in \cite {mccroryparusinski2} to study local topological properties of real algebraic sets. In particular, Theorem \ref{wtyczka} below can be reformulated as follows. Algebraically constructible functions on a real algebraic set $W$ coincide with finite sums of signs of real polynomials on $W$, see Theorem \ref{key} below. Using this characterization, in section 6, we give new, alternative proofs of basic properties of algebraically constructible functions, without using the resolution of singularities as in \cite{mccroryparusinski2}. The main result of the paper, Theorem \ref{wtyczka}, is proven in section \ref{families}. In sections 2-4 we develop necessary techniques for the proof and recall basic results the proof is based on. In particular, in section \ref{preliminaries} we recall the Eisenbud-Levine Theorem \ref{el} and the Khimshiashvili formula \ref{deszczyk}. In section \ref{division} we review some basic facts on the Grauert-Hironaka formal division algorithm Theorem \ref{mrowka}, which we then use to obtain a parametrized version of the Eisenbud-Levine Theorem, Propositions \ref{trasa} and \ref{tara}, with parameter in a given algebraic set $w\in W$. In section \ref{vectorfields} we study polynomial families of polynomial vector fields $F_w:\Rn\mbox{$\longrightarrow$}\Rn$ parametrized by $w\in W$. The proof of Theorem \ref{wtyczka} can be sketched briefly as follows. First, by an argument similar to the Khimshiashvili Formula, we show that the Euler characteristic $\chi(X)$ of a real algebraic set $X$ can be calculated in terms of the local topological degree at the origin of a polynomial vector field, see Proposition \ref{guzik}. Then using the theory developed by the second named author, see e.g. \cite{szafraniec4,szafraniec17}, we generalize this observation in two directions. Firstly, we show that for a regular morphism $f:X\to W$, the Euler characteristic of the fibers $\chi(X_w)$ can be expressed in terms of the local topological degree $\deg_{0} G_w$ at the origin of a family $G_w:\Rn\mbox{$\longrightarrow$}\Rn$ of polynomial vector fields, which depends polynomially on $w$. Secondly, as shown in Lemma \ref{marazm}, we may choose all $G_w$ in such a way that they have algebraically isolated zero at the origin. Then, by the Eisenbud-Levine Theorem \ref{el}, each $\deg_{0} G_w$ can be calculated algebraically, that is $\deg_{0} G_w$ equals the signature of an associated symmetric bilinear form $\Psi_w$. By section 3, we may as well require that $\Psi_w$ depend "polynomially" on $w$. More precisely, there exists a symmetricmatrix $T(w)$ (representing $\Psi_w$) with entries polynomials in $w$, such that $\deg_{0} G_w$ equals the signature of $T(w)$, for all $w$ in a Zariski open subset of $W$. (See \ref{trasa} and \ref{tara} for the details.) Finally by Descartes' Lemma, we express the signature of $T(w)$ in terms of signs of polynomials in $w$, see Lemma \ref{rosa} and the proof of Lemma \ref{sloniczek},. For the definitions and properties of real algebraic sets and maps we refer the reader to \cite{benedettirisler}. By a real algebraic set we mean the locus of zeros of a finite set of polynomial functions on $\bf R^n$. \medskip \section{Preliminaries}\label{preliminaries} Let $f(x)=a_n x^n+a_{n-1}x^{n-1}\mbox{$\,+\cdots+\,$} a_0$ be a real polynomial. Let $\Lambda$ be the set of all pairs $(r,s)$ with $0\leq r<s\leq n$ such that $a_r\neq 0, a_s \neq 0$, and $a_i=0$ for $r<i<s$. Denote $\Lambda'=\{(r,s)\in\Lambda\mid r+s \mbox{ is odd }\}$. \begin{lemma}\label{rosa} Assume that all roots of $f(x)$ are real and $a_0\neq 0, a_n\neq 0$. Let $p_+$ (resp. $p_-$) denote the number of positive (resp. negative) roots counted with multiplicities. Then \[p_+-p_-=-\sum {\mbox{$\operatorname {sgn}$}\,}\, a_r a_s,\mbox{ where }(r,s)\in \Lambda',\] \[p_+-p_-\equiv n+1+(-1)^{n+1}{\mbox{$\operatorname {sgn}$}\,}\, a_0a_n\pmod{4}.\] \end{lemma} {\em Proof.\/} We say that the pair of real numbers $(a, b)$ changes sign if $ab<0$. If this is the case then $(1-\mbox{$\operatorname {sgn}$}\, ab)/2=1$, if $ab>0$ then $(1-\mbox{$\operatorname {sgn}$}\, ab)/2=0$. As a consequence of Descartes' lemma (see \cite [Theorem 6, p.232]{mostowskistark}, or \cite [Exercise 1.1.13 (4), p.16]{benedettirisler}), $p_+$ equals the number of sign changes in the sequence of non-zero coefficients of $f(x)$, that is \[ p_+=\sum(1-\mbox{$\operatorname {sgn}$}\, a_r a_s)/2,\mbox{ where } (r,s)\in\Lambda.\] According to the same fact, $p_-$ equals the number of sign changes in the sequence of non-zero coefficients of $f(-x)$, i.e. \[ p_-=\sum (1-(-1)^{r+s}\mbox{$\operatorname {sgn}$}\, a_r a_s)/2, \mbox{ where }(r,s)\in\Lambda.\] Hence \[p_+-p_-=-\sum \mbox{$\operatorname {sgn}$}\, a_r a_s, \mbox{ where } (r,s)\in\Lambda'.\] The sign of the product of all roots, that is $(-1)^{p_-}$, equals $(-1)^n\mbox{$\operatorname {sgn}$}\, a_0 a_n$. Thus $2p_-\equiv 3+(-1)^{p_-}=3+(-1)^n\mbox{$\operatorname {sgn}$}\, a_0 a_n\pmod{4}$. Finally, since $p_++p_-=n$, we conclude that \[p_+-p_-=n-2p_-\equiv n-3-(-1)^n\mbox{$\operatorname {sgn}$}\, a_0 a_n \equiv n+1+(-1)^{n+1}\mbox{$\operatorname {sgn}$}\, a_0 a_n\pmod{4}. \Box \] \smallskip Let $F:(\mbox{${\mathbf R}^{m}$},\mbox{$\mathbf 0$})\mbox{$\longrightarrow$}(\mbox{${\mathbf R}^{m}$},\mbox{$\mathbf 0$})$ be a germ of a continuous mapping with isolated zero at $\mbox{$\mathbf 0$}$. Then we denote by $\deg_0 F$ the local topological degree of $F$ at the origin. Suppose, in addition, that $F=(f_1\mbox{$\,,\ldots,\,$} f_m)$ is a real analytic germ. Let $\mbox{$\mathbf R$} [[x]]=\mbox{$\mathbf R$} [[x_1\mbox{$\,,\ldots,\,$} x_n]]$ denote the ring of formal power series and let $I$ denote the ideal in $\mbox{$\mathbf R$}[[x]]$ generated by $f_1\mbox{$\,,\ldots,\,$} f_m$. Then $Q=\mbox{$\mathbf R$}[[x]]/I$ is an $\mbox{$\mathbf R$}$-algebra. If $\dim_{\bf R} Q<\infty$, then $\mbox{$\mathbf 0$}$ is isolated in $F^{-1}(\mbox{$\mathbf 0$})$ and in this case we say that $F$ has {\em an algebraically isolated zero at\/} $\mbox{$\mathbf 0$}$. Let $J$ denote the residue class in $Q$ of the Jacobian determinant \[\frac{\partial(f_1\mbox{$\,,\ldots,\,$} f_m)}{\partial(x_1\mbox{$\,,\ldots,\,$} x_m)} . \] The next theorem is due to Eisenbud and Levine \cite{eisenbudlevine}, see also \cite{arnoldetal,beckeretal,khimshiashvili} for a proof. \smallskip \begin{theorem}[Eisenbud\&Levine Theorem] \label{el} Assume that $\dim_{ \bf R} Q<\infty$. Then \begin{enumerate} \item $J\neq 0$ in $Q$, \item for any $\mbox{$\mathbf R$}$-linear form $\varphi:Q\mbox{$\longrightarrow$}\mbox{$\mathbf R$}$ such that $\varphi(J)>0$, the corresponding symmetric bilinear form $\Phi: Q\times Q\mbox{$\longrightarrow$}\mbox{$\mathbf R$}$, $\Phi(f,g)=\varphi(fg)$, is non-degenerate and \[\mbox{$\operatorname {signature}$}\, \,\Phi=\deg_0 F.\quad \Box\] \end{enumerate}\end{theorem} The next formula was proved by Khimshiashvili \cite{khimshiashvili}, for other proofs see \cite{arnold,arnoldetal,wall}. \begin{theorem}[Khimshiashvili Formula] \label{deszczyk} Let $g:(\mbox{${\mathbf R}^{m}$},\mbox{${\mathbf 0}$})\mbox{$\longrightarrow$}(\mbox{$\mathbf R$},0)$ be a real analytic germ with isolated critical point at $\mbox{$\mathbf 0$}$. Let $S_\epsilon$ denote the sphere of a small radius $\epsilon$ centered at the origin and let $A_\epsilon=S_\epsilon \cap\{g\leq 0\}$. (Note that all $A_\epsilon$ are homeomorphic for $\epsilon>0$ small enough.) Then the gradient $\nabla g:\mbox{${\mathbf R}^{m}$}\mbox{$\longrightarrow$}\mbox{${\mathbf R}^{m}$}$ of $g$ has an isolated zero at $\mbox{$\mathbf 0$}$ and \[\chi(A_\epsilon)=1-\deg_0(\nabla g).\hspace{1.0cm}\Box\] \end{theorem} \medskip \begin{lemma} \label{biurko} Let $g:\Rn\times\mbox{$\mathbf R$}\mbox{$\longrightarrow$} \mbox{$\mathbf R$}$ be a polynomial vanishing at $\mbox{$\mathbf 0$}$ and such that if $g(x,t)\leq0$ then either $(x,t)=\mbox{${\mathbf 0}$}$ or $t>0$. Let $S_\epsilon\subset\Rn\times\mbox{$\mathbf R$}$ (resp. $B_\epsilon$) denote the sphere (resp. the open ball) of radius $\epsilon$ centered at the origin and let $A_\epsilon=S_\epsilon\cap\{g\leq 0\}$. Let $P_\eta=\Rn\times\{\eta\}$ and $M_{\epsilon,\eta}=P_\eta\cap\{g\leq 0\}\cap B_\epsilon$. Then, for $0<\eta \ll\epsilon\ll 1$, $A_\epsilon$ and $M_{\epsilon,\eta}$ have the same homotopy type. In particular, \[\chi(A_\epsilon)=\chi(M_{\epsilon,\eta}).\hspace{1.0cm} \] \end{lemma} {\em Proof.\/} Consider on $N=\{(x,t)\mid g(x,t)\leq0\}$, the functions $\omega_1(x,t)=\| x\| ^2+t^2$ and $\omega_2(x,t)=t$. Both $\omega_1$ and $\omega_2$ are non-negative on $N$ and $\omega_{1}^{-1}(0)\cap N=\omega_{2}^{-1}(0)\cap N=\{\mbox{$\mathbf 0$}\}$. Let $N_{i}^{y}=\{(x,t)\in N\mid 0<\omega_i(x,t)\leq y\}$. By the topological local triviality of semi-algebraic mappings, see for instance \cite[Theorem 9.3.1]{bochnaketal} or \cite{coste1,hardt}, there is $\delta>0$ such that $\omega_i:N_{i}^{\delta}\mbox{$\longrightarrow$} (0,\delta]$, $i=1,2$, are topologically trivial fibrations. For $0<y\leq \delta$ let $M_{2}^{y}$ denote the union of connected components of $N_{2}^{y}$ containing $\mbox{$\mathbf 0$}$ in their closures. Then $\omega_2:M_{2}^{\delta}\mbox{$\longrightarrow$} (0,\delta]$ is also topologically trivial. Hence there exist constants $0<\alpha<\beta<\gamma<\delta$ such that $M_{2}^{\alpha}\subset N_{1}^{\beta}\subset M_{2}^{\gamma} \subset N_{1}^{\delta}$. By the topological triviality, the inclusions $M_{2}^{\alpha}\subset M_{2}^{\gamma}$ and $N_{1}^{\beta}\subset N_{1}^{\delta}$ are homotopy equivalencies and hence so are $M_{2}^{\alpha}\subset N_{1}^{\beta}$ and $N_{1}^{\beta}\subset M_{2}^{\gamma}$. By the above, the total spaces of fibrations $\omega_1:N_{1}^{\delta}\mbox{$\longrightarrow$} (0,\delta]$, $\omega_2:M_{2}^{\delta}\mbox{$\longrightarrow$} (0,\delta]$ are homotopy equivalent to their fibers. Consequently the fibers of both fibrations are homotopy equivalent. Now, to complete the proof, it is enough to observe that these fibers are of the form $A_\epsilon$ and $M_{\epsilon,\eta}$, where $0<\eta\ll\epsilon\ll 1$. $\Box$ \begin{proposition} \label{guzik} Let $f:\Rn\times\mbox{$\mathbf R$}\mbox{$\longrightarrow$}\mbox{$\mathbf R$}$ be a non-negative homogeneous polynomial of degree $2d$ such that $f(x,0)=\| x\| ^{2d}$. Let $X=\{x\in\Rn\mid f(x,1)=0\}$ and define $g(x,t)=f(x,t)-t^{2d+1}$. Then $g$ has an isolated critical point at the origin and \[\chi(X)=1-\deg_0(\nabla g) \] \end{proposition} {\em Proof.\/} Let \[\Sigma=\left\{(x,t)\mid \frac{\partial f}{\partial x_1}= \cdots=\frac{\partial f}{\partial x_n}=0\right\},\] $P_\eta=\Rn\times\{\eta\}$, and $\Sigma_\eta=\Sigma\cap P_\eta$. Let $f_\eta$ (resp. $g_\eta$) denote the restriction of $f$ (resp. $g$) to $P_\eta$. Then $\Sigma_\eta$ is the set of critical points of both $f_\eta$ and $g_\eta$. We have $\Sigma_0=\{\mbox{${\mathbf 0}$}\}$. Since the set of critical values of any polynomial is finite, so is each $f_\eta(\Sigma_\eta)$. Moreover, since $f$ is non-negative homogeneous of degree $2d$ and $\Sigma$ is a homogeneous set, there is $D>0$ such that any $y\in f_\eta(\Sigma_\eta)$, if non-zero, satisfies $y >D\mid\eta\mid^{2d}$. If $\eta<0$, then $g_\eta>0$ and $0\in\mbox{$\mathbf R$}$ is a regular value of $g_\eta$. Clearly $g_0$ has a single critical point at the origin. Consider $0<\eta\ll 1$. Let $x\in\Sigma_\eta$. If $f_\eta(x)>0$ then \[g_\eta(x)=f_\eta(x)-\eta^{2d+1}>D\eta^{2d}-\eta^{2d+1}>0.\] If $f_\eta(x)= 0$ then $g_\eta(x)<0$. Thus $0\in\mbox{$\mathbf R$}$ is a regular value for $g_\eta$. Hence there is an open neighbourhood $U\subset\Rn\times\mbox{$\mathbf R$}$ of the origin such that $0\in\mbox{$\mathbf R$}$ is a regular value of $g$ on $U-\{\mbox{${\mathbf 0}$}\}$, i.e. $g$ has an isolated critical point at the origin. For $\eta$ fixed, $\lim f_\eta(x)=+\infty$ as $\| x\| \mbox{$\longrightarrow$} +\infty$. Denote $N_\eta=\{x\mid f_\eta(x)= 0\}$ and $M_\eta=\{x\mid g_\eta(x)\leq 0\}=\{x\mid f_\eta(x) \leq\eta^{2d+1}\}$. If $\eta<0$, then $M_\eta$ is empty and $M_0=\{\mbox{${\mathbf 0}$}\}$. If $\eta>0$, then $N_\eta\subset M_\eta$. As we have shown above, for $0<\eta\ll 1$ both $f_\eta$ and $g_\eta$ have no critical points in $M_\eta-N_\eta$. Hence $N_\eta$ is a deformation retract of $M_\eta$ and, in particular, $\chi(N_\eta)=\chi(M_\eta)$. Suppose $0<\eta\ll\epsilon$. Then, since $f_0=g_0= \|x\|^{2d}$, $M_\eta\subset B_\epsilon$, that is $M_\eta=M_{\epsilon,\eta}$ in the notation of Lemma \ref{biurko}. Moreover, let $A_\epsilon=S_\epsilon\cap\{g\leq 0\}$. By Lemma \ref {biurko}, $\chi(A_\epsilon)=\chi(M_\eta)$, and hence, by the above \[ \chi(A_\epsilon)=\chi(M_\eta)= \chi(N_\eta). \] Finally, by the Khimshiashvili formula \ref{deszczyk}, \[\chi(A_\epsilon)=1-\deg_0 (\nabla g),\] and the lemma follows since $\chi(X)=\chi(N_1)=\chi(N_\eta)$, for $\eta>0$. $\hspace{1.0cm}\Box$ \bigskip \section{The formal division algorithm}\label{division} \noindent In the first part of this section we review some basic facts on the Grauert-Hironaka formal division algorithm for formal power series with polynomial coefficients. In exposition and notation we follow closely \cite{bierstonemilman}. Then we apply the Grauert-Hironaka algorithm to derive a parametrized version of the Eisenbud-Levine Theorem \ref{el}, with parameter in a given algebraic set $W$. Let $A$ be an integral domain. Let $A[[y]]=A[[y_1,\ldots,y_n]]$ denote the ring of formal power series in $n$ variables with coefficients in $A$. If $\beta=(\beta^1,\ldots,\beta^n)\in\mbox{\( {\mathbf N}^{n} \)}$, put $\mid \beta\mid=\beta^1+\cdots+\beta^n$. We order the $(n+1)$-tuples $(\beta^1,\ldots,\beta^n,\mid\beta\mid)$ lexicographically from the right. This induces a total ordering of $\mbox{\( {\mathbf N}^{n} \)}$. Let $f\in A[[y]]$. Write $f=\sum_{\beta\in N^n}f_{\beta}y^{\beta}$, where $f_{\beta}\in A$ and $y^{\beta}$ denotes $y_{1}^{\beta^1}\cdots y_{n}^{\beta^n}$. Let $\mbox{$\operatorname {supp}$}\, (f)=\{\beta\in\mbox{\( {\mathbf N}^{n} \)}\mid f_\beta \neq 0\}$ and let $\nu (f)$ denote the smallest element of $\mbox{$\operatorname {supp}$}\, (f)$. Let in$(f)$ denote $f_{\nu(f)}y^{\nu(f)}$. Let $I$ be an ideal in $A[[y]].$ We define the diagram of initial exponents $\mbox{$\mathcal N$} (I)$ as $\{\nu(f)\mid f\in I\}$. Clearly, $\mbox{$\mathcal N$} (I)+\mbox{\( {\mathbf N}^{n} \)}=\mbox{$\mathcal N$} (I)$. There is a smallest finite subset $V(I)$ of $\mbox{$\mathcal N$} (I)$ such that $\mbox{$\mathcal N$} (I)=V(I)+\mbox{\( {\mathbf N}^{n} \)}$. We call the elements of $V(I)$ the vertices of $\mbox{$\mathcal N$} (I)$. Let $\beta_1,\ldots,\beta_t\in V(I)$ be the vertices of $\mbox{$\mathcal N$} (I)$ and choose $g^1\mbox{$\,,\ldots,\,$} g^t\in I$ so that $\beta_i=\nu(g^i)$, $i=1,\ldots,t$. The $\beta_1,\ldots,\beta_t$ induce the following decomposition of $\mbox{\( {\mathbf N}^{n} \)}$: Set $\Delta_0=\emptyset$ and then define $\Delta_i=(\beta_i+\mbox{\( {\mathbf N}^{n} \)})\setminus \Delta_0\cup\ldots\cup\Delta_{i-1},\ i=1,\ldots,t$. Put $\Delta=\mbox{\( {\mathbf N}^{n} \)}\setminus \Delta_1\cup\ldots\cup\Delta_t=\mbox{\( {\mathbf N}^{n} \)}\setminus \mbox{$\mathcal N$}(I)$. Let $\mbox{in}(g^i)=g_{\beta_i}^i y^{\beta_i}$. Then $g_{\beta_i}^{i}\neq 0$. Let $A_0$ denote the field of fractions of $A$. We denote by $S$ the multiplicative subset of $A$ generated by the $g_{\beta_i}^{i}$ and by $S^{-1}A$ the corresponding localization of $A$; i.e. the subring of $A_0$ comprising the quotients with denominators in $S$. Then $S^{-1}A[[y]]\subset A_0[[y]]$. \medskip \begin{theorem}[Grauert, Hironaka, \cite{arocaetal,bierstonemilman,briancon,grauert}]\label{mrowka} For every $f\in S^{-1}A[[y]]$ there exist unique $g_i\in S^{-1}A[[y]]$, $\ i=1,\ldots,t$ , and $r\in S^{-1}A[[y]]$ such that $\beta_i +\mbox{$\operatorname {supp}$}\, (g_i)\subset \Delta_i$, $\mbox{$\operatorname {supp}$}\, (r)\subset\Delta$, and \[f=\sum_{i=1}^{t}g_i g^i+r. \hspace{1.0cm}\Box\] \end{theorem} \begin{corollary}\label{rower} $\nu(f)\leq\nu(r)$. In particular, if $\Delta$ is finite and $\beta<\nu(f)$ for all $\beta\in\Delta$, then $r=0$ and $f$ belongs to the ideal in $S^{-1}A[[y]]$ generated by $g^1,\ldots,g^t.\Box$ \end{corollary} Let $S^{-1}I[[y]]$ denote the ideal in $S^{-1}A[[y]]$ generated by $I$. Then $S^{-1}A[[y]]/S^{-1}I[[y]]$ is finitely generated if and only if $\Delta$ is finite. If this is the case then $S^{-1}A[[y]]/S^{-1}I[[y]]$ is a free $S^{-1}A$ module and we take the monomials $y^\beta$, $\beta\in\Delta$, as a basis. Let $W\subset \Rn$ be an irreducible real algebraic set and let $\mbox{$\mathcal A$}$ denote the ring of real polynomial functions on $W$. Each $w\in W$ defines an evaluation homomorphism $h\mapsto h(w)$ of $\mbox{$\mathcal A$}$ onto $\mbox{$\mathbf R$}$. For $f=\sum_{\beta}f_{\beta}y^{\beta}\in\mbox{$\mathcal A$}[[y]]$ we write $f(x;y)=\sum_{\beta}f_{\beta}(x)y^{\beta}$, and $f(w;y)=\sum_{\beta}f_{\beta}(w)y^{\beta}$ when the coefficients are evaluated at $x=w$. Let $f^1\mbox{$\,,\ldots,\,$} f^s\in\mbox{$\mathcal A$}[[y]]$ and let $\mbox{$\mathcal I$}$ denote the ideal in $\mbox{$\mathcal A$}[[y]]$ generated by $f^1\mbox{$\,,\ldots,\,$} f^s$. Let $\mbox{$\mathcal N$} =\mbox{$\mathcal N$} ({\mathcal I})= \{\nu(g)\mid g\in\mbox{$\mathcal I$}\}$ denote the diagram of initial exponents (here $A=\mbox{$\mathcal A$}$). Given $w\in W$. We denote by$I_w$ the ideal in $\mbox{$\mathbf R$}[[y]]$ generated by $f^1(w;y)\mbox{$\,,\ldots,\,$} f^s(w;y)$ and by $\mbox{$\mathcal N$}_w=\mbox{$\mathcal N$} (I_w)$ the diagram of initial exponents of $I_w$ (so here $A=\mbox{$\mathbf R$}$). The next theorem was proved by Bierstone and Milman \cite{bierstonemilman}. \begin{theorem} \label{fotel} Assume that $W$ is irreducible (so that $\mbox{$\mathcal A$}$ is an integral domain). Let $\beta_1\mbox{$\,,\ldots,\,$}\beta_t$ denote the vertices of $\mbox{$\mathcal N$}$ and choose $g^i\in\mbox{$\mathcal I$}$ such that $\nu(g^i)=\beta_i$. Let \[\Sigma=\bigcup_{i=1}^{t}\{w\in W\mid g_{\beta_i}^{i}(w)=0\}.\] Then $\Sigma$ is a proper algebraic subset of $W$, $\mbox{$\mathcal N$}_w=\mbox{$\mathcal N$}$ for all $w\in W-\Sigma$, $\nu(g^i)=\beta_i=\nu(g^i(w;\, \cdot\, ))$ for every vertex $\beta_i\in \mbox{$\mathcal N$}$ and $w\in W \setminus \Sigma$. $\Box$ \end{theorem} \begin{corollary} Suppose that $\Delta_w=\mbox{\( {\mathbf N}^{n} \)}\setminus \mbox{$\mathcal N$}_w$ is finite for each $w\in W\setminus \Sigma$. Then $\Delta=\mbox{\( {\mathbf N}^{n} \)}\setminus \mbox{$\mathcal N$}$ is also finite and $\Delta=\Delta_w$ for all $w\in W\setminus \Sigma$. $\Box$ \end{corollary} Suppose that $\Delta$ is finite and let $\bar\beta$ denote the largest element in $\Delta$. Let $j=y^{\bar\beta}$. Then for $w\in W-\Sigma$, the residue class of $j$ in $Q_w=\mbox{$\mathbf R$}[[y]]/I_w$ is nonzero. \medskip {\em Definition.\/} Let $\varphi_w:Q_w\mbox{$\longrightarrow$}\mbox{$\mathbf R$}$ be the linear form given by $\varphi_w(j)=1$ and $\varphi_w(y^\beta)=0$ for $\beta\in\Delta-\{\bar\beta\}$. Let $\Phi_w:Q_w\times Q_w\mbox{$\longrightarrow$}\mbox{$\mathbf R$}$ be the corresponding symmetric bilinear form, $\Phi_w(f,g)=\varphi_w(fg)$. Let $M_w$ denote the matrix of $\Phi_w$ in the basis $y^\beta$, $\beta \in \Delta$. Let, as before, $S$ denote the multiplicative subset of $\mbox{$\mathcal A$}$ generated by $g_{\beta_i}^{i}$. \begin{lemma} \label{cukier} There is a symmetric matrix $M$ with entries in $S^{-1}\mbox{$\mathcal A$}$ such that $M_w=M(w)$ for $w\in W\setminus \Sigma$. $\Box$ \end{lemma} From now on we suppose that $F=(f_1\mbox{$\,,\ldots,\,$} f_n):W\times\Rn\mbox{$\longrightarrow$}\Rn$ is a polynomial mapping with $F(w;\mbox{${\mathbf 0}$})=\mbox{${\mathbf 0}$}$ for every $w\in W$. Denote \[J=\frac{\partial(f_1\mbox{$\,,\ldots,\,$} f_n)}{\partial(y_1\mbox{$\,,\ldots,\,$} y_n)} \mbox{ and }J_w=J(w;\, \cdot\, ).\] Let $\mbox{$\mathcal I$}$ be the ideal in $\mbox{$\mathcal A$}[[y]]$ generated by $f_1\mbox{$\,,\ldots,\,$} f_n$ and $I_w\subset\mbox{$\mathbf R$}[[y]]$ that generated by $f_1(w;\, \cdot\, )\mbox{$\,,\ldots,\,$} f_n(w;\, \cdot\, )$. We assume that $\dim_R Q_w<\infty$ for every $w\in W$. Hence, $\Delta$ and all $\Delta_w$ are finite. \begin{lemma} \label{kubek} If $w\in W\setminus \Sigma$ then there is $0\neq\lambda_w\in\mbox{$\mathbf R$}$ such that $J_w=\lambda_w j$ in $Q_w$. \end{lemma} {\em Proof.\/} By the Eisenbud-Levine Theorem \ref{el}, $J_w\neq 0$ in $Q_w$. By Theorem \ref{mrowka}, $J_w=\sum_{\beta\in\Delta}\lambda_\beta y^\beta$ in $Q_w$. Suppose, contrary to our claim, that $\lambda_{\beta'}\neq 0$ for a $\beta'<\bar\beta$. Then, define a linear form $\psi:Q_w\mbox{$\longrightarrow$}\mbox{$\mathbf R$}$ by the formula $\psi(f)=f_{\beta'}\lambda_{\beta'}$, where $f=\sum_{\beta\in\Delta}f_\beta y^\beta\in Q_w$. We show that the corresponding symmetric bilinear form $\Psi (f,g) = \psi (fg)$ is degenerate. For any $f\in Q_w$ we have $\nu(fj)=\nu(f)+\nu(j)\geq\nu(j)=\bar\beta$. Therefore, by Corollary \ref{rower}, $\psi(fj)=0$ for any $f\in Q_w$ and hence $\Psi(f,g)$ is degenerate. On the other hand $\psi(J_w)=\lambda_{\beta'}^{2}>0$, and hence the existence of $\psi$ contradicts Theorem \ref{el}. Thus $\lambda_{\beta'}=0$ for every $\beta' \in\Delta\setminus \{\bar\beta\}$ and we take $\lambda_w= \lambda_{\bar\beta}$. $\Box$ \medskip In particular, by Theorem \ref{mrowka}, there is $\lambda\in S^{-1}\mbox{$\mathcal A$}$ such that $\lambda_w=\lambda(w)$ for $w\in W\setminus \Sigma$.\\[0.7em] {\em Definition.\/} Let $\psi_w=\lambda_w\varphi_w:Q_w\mbox{$\longrightarrow$}\mbox{$\mathbf R$}$. Let $\Psi_w$ be the corresponding symmetric bilinear form. \begin{proposition} \label{trasa} The forms $\psi_x$ and $\Psi_w$ defined above satisfy \begin{enumerate} \item $\psi_w(J_w)>0$, \item $\Psi_w$ is non-degenerate, \item the entries and the determinant of the matrix of $\Psi_w$ in basis $y^\beta$, $\beta\in \Delta$, belong to $S^{-1}\mbox{$\mathcal A$}$. \end{enumerate} \end{proposition} {\em Proof.\/} $\psi_w(J_w)=\lambda_w\varphi_w(J_w)=\lambda_{w}^{2}\varphi_w(j)= \lambda_{w}^{2}>0$, so the statement follows from the Eisenbud-Levine Theorem \ref{el} and Lemma \ref{cukier}. $\Box$ \\ [0.4em] Clearly multiplication by a positive scalar does not change the signature of a symmetric matrix. So if we multiply the matrix of $\Psi_w$ by the product of squares of the denominators of its entries we get \begin{proposition} \label{tara} Assume that $W$ is irreducible. Then there are a symmetric matrix $T$ with entries polynomials in $w\in W$ and a proper algebraic subset $\Sigma\subset W$ such that for every $w\in W\setminus \Sigma$ \begin{enumerate} \item $T(w)$ is non-degenerate, \item $\mbox{$\operatorname {signature}$}\, \Psi_w=\mbox{$\operatorname {signature}$}\, \, T(w)$. $\Box$ \end{enumerate} \end{proposition} \bigskip \section{Families of vector fields}\label{vectorfields} \begin{lemma} \label{marazm} Let $F:W\times\Rn\mbox{$\longrightarrow$}\Rn$ be a polynomial mapping. For any $w\in W$ let $F_w=F(w;\,\cdot\, ):\Rn\mbox{$\longrightarrow$}\Rn$. Suppose that for all $w\in W$, $\mbox{$\mathbf 0$}\in\Rn$ is isolated in $F_{w}^{-1}(\mbox{$\mathbf 0$})$. (Hence $\deg_0 F_w$ is always well-defined.) Then there is a polynomial mapping $G:W\times\Rn\mbox{$\longrightarrow$}\Rn$ such that for every $w\in W$ \begin{enumerate} \item $G_w:(\Rn,\mbox{${\mathbf 0}$})\mbox{$\longrightarrow$}(\Rn,\mbox{${\mathbf 0}$})$ has an algebraically isolated zero at $\mbox{${\mathbf 0}$}$, \item $\deg_0F_w=\deg_0G_w$. \end{enumerate} \end{lemma} {\em Proof.\/} By the parametrized version of the {\L}ojasiewicz Inequality of \cite{fekak}, there is $\alpha>0$ such that \[\| F_w(y)\| \geq C\| y\| ^\alpha\] for every $w\in W$ and $\| y\| <\delta$, where $C=C(w)>0$ and $\delta=\delta(w)>0$ depend on $w$. Choose an integer $k\gg 0$. Define $G(w;y)=F(w;y)+(y_{1}^{k}\mbox{$\,,\ldots,\,$} y_{n}^{k})$. Let $G_{\bf C,w}:(\mbox{${\mathbf C}^{n}$},\mbox{${\mathbf 0}$})\mbox{$\longrightarrow$} (\mbox{${\mathbf C}^{n}$},\mbox{${\mathbf 0}$})$ denote the complexification of $G_w$. Then, for every $w\in W$, $G_{\bf C,w}^{-1}(\mbox{${\mathbf 0}$})$ is a bounded complex algebraic set and hence finite. So $\mbox{${\mathbf 0}$}$ is isolated in $G_{\bf C,w}^{-1}(\mbox{${\mathbf 0}$})$ and hence $G_w$ has an algebraically isolated zero at $\mbox{${\mathbf 0}$}$. We may assume that $k>\alpha$. So if $w\in W$ and $y$ is close enough to the origin then \[ \| tG_w(y)+(1-t)F_w(y)\| = \| F_w(y)+t(y_{1}^{k}\mbox{$\,,\ldots,\,$} y_{n}^{k})\| \geq \] \[ C\| y\|^{\alpha} - t \|(y_{1}^{k}\mbox{$\,,\ldots,\,$} y_{n}^{k})\| \geq \frac{C}{2} \|y\|^\alpha \] for $0\leq t\leq 1$. Hence $\deg_0 F_w=\deg_0 G_w$ as required. $\Box$\\[0.7em] \begin{lemma}\label{sloniczek} Under the assumptions of Lemma \ref{marazm}, if moreover $W$ is irreducible, then there exist a proper algebraic subset $\Sigma\subset W$, an integer $\mu$, and polynomials $q_1\mbox{$\,,\ldots,\,$} q_t,q$ nowhere vanishing in $W\setminus \Sigma$ such that for every $w\in W\setminus \Sigma$ \begin{enumerate} \item $\deg_0 F_w={\mbox{$\operatorname {sgn}$}\,}\, q_1(w)\mbox{$\,+\cdots+\,$} {\mbox{$\operatorname {sgn}$}\,}\, q_t(w)$, \item $\deg_0 F_w\equiv \mu+1\pmod{2}$, \item $\deg_0 F_w\equiv \mu+{\mbox{$\operatorname {sgn}$}\,}\,q(w)\pmod{4}$. \end{enumerate}\end{lemma} {\em Proof.\/} Let $F=(f_1\mbox{$\,,\ldots,\,$} f_n)$, let $I_w$ denote the ideal in $\mbox{$\mathbf R$}[[y]]$ generated by $f_1(w;\,\cdot\,),\ldots ,$ $f_n(w;\,\cdot\,)$ and let $Q_w=\mbox{$\mathbf R$}[[y]]/I_w$. By Lemma \ref{marazm} we may assume that each $F_w$ has an algebraically isolated zero at $\mbox{$\mathbf 0$}$. Let \[J=\frac{\partial(f_1\mbox{$\,,\ldots,\,$} f_n)}{\partial(y_1\mbox{$\,,\ldots,\,$} y_n)}\] and let $J_w$ denote the residue class of $J(w;\,\cdot\,)$ in $Q_w$. Let $\psi_w:Q_w \to \mbox{$\mathbf R$}$ be the linear form defined in section 3. By Proposition \ref{trasa}, $\psi_w$ satisfies the assumptions of the Eisenbud-Levine Theorem \ref{el}. Hence the corresponding symmetric bilinear form $\Psi_w$ is non-degenerate and $\deg_0 F_w=\mbox{signature}\,\Psi_w$. In particular, by Proposition \ref{tara}, there are a symmetric matrix $T$ with polynomial entries and a proper algebraic set $\Sigma'\subset W$ such that $T(w)$ is non-degenerate and $\deg_0 F_w=$ signature$\,T(w)$ for every $w\in W\setminus \Sigma'$. Let $P_w(\lambda)=a_N\lambda^N+ a_{N-1}(w)\lambda^{N-1}\mbox{$\,+\cdots+\,$} a_0(w)$, $a_N \equiv (-1)^N$, denote the characteristic polynomial of $T(w)$. Clearly its coefficients are polynomials in $w$ and $a_0(w)$ does not vanish in $W\setminus \Sigma'$. If $w\in W\setminus \Sigma'$ then all roots of $P_w$ are real and non-zero. Let$p_+(w)$ (resp. $p_-(w)$) denote the number of positive (resp. negative) roots. Then \[\mbox{signature}\,T(w)=p_+(w)-p_-(w), \] and, by Lemma \ref{rosa}, it is easy to see that there are a proper algebraic $\Sigma\subset W$, polynomials $q_1\mbox{$\,,\ldots,\,$} q_t,q$ nowhere vanishing on $W\setminus \Sigma$, and an integer $\mu$ such that \renewcommand{\labelenumi}{(\alph{enumi})}\begin{enumerate} \item $\deg_0 F_w=p_+(w)-p_-(w)=\mbox{$\operatorname {sgn}$}\, q_1(w)\mbox{$\,+\cdots+\,$} \mbox{$\operatorname {sgn}$}\, q_t(w)$, \item $\deg_0 F_w\equiv \mu+\mbox{$\operatorname {sgn}$}\, q(w)\pmod{4}$ \end{enumerate} for every $w\in W\setminus \Sigma$ which completes the proof. $\Box$\\[1em] \renewcommand{\labelenumi}{(\roman{enumi})} Let $P$ be any non-negative polynomial with $P^{-1}(0)\cap W=\Sigma$. Then \[\sum \mbox{$\operatorname {sgn}$}\, P(w)q_i(w)=\sum \mbox{$\operatorname {sgn}$}\, q_i(w) \] on $W\setminus \Sigma$ and \[\sum \mbox{$\operatorname {sgn}$}\, P(w)q_i(w)=0\] on $\Sigma$. Similarly, let $p_1\mbox{$\,,\ldots,\,$} p_r$ be another set of polynomials. Then \[\sum\mbox{$\operatorname {sgn}$}\, p_j(w)+\sum\mbox{$\operatorname {sgn}$}\, (-P(w)p_j(w))=0 \] on $W\setminus \Sigma$ and \[\sum\mbox{$\operatorname {sgn}$}\, p_j(w)+\sum\mbox{$\operatorname {sgn}$}\,(-P(w)p_j(w))=\sum\mbox{$\operatorname {sgn}$}\, p_j(w)\] on $\Sigma$. Hence, by induction on $\dim W$ we get \begin{theorem}\label{slonik} Let $W$ be a real algebraic set and let $F:W\times\Rn\mbox{$\longrightarrow$}\Rn$ be a polynomial mapping such that $\mbox{$\mathbf 0$}$ is isolated in $F_{w}^{-1}(\mbox{$\mathbf 0$})$ for all $w\in W$. Then there are polynomials $g_1\mbox{$\,,\ldots,\,$} g_s$ such that for every $w\in W$ \[\deg_0 F_w={\mbox{$\operatorname {sgn}$}\,}\, g_1(w)\mbox{$\,+\cdots+\,$} {\mbox{$\operatorname {sgn}$}\,}\, g_s(w). \hfil \Box\] \end{theorem} \bigskip \section{Families of algebraic sets}\label{families} Let $X\subset W\times\Rn$ be a real algebraic set such that $W\times\{\mbox{$\mathbf 0$}\}\subset X$. There is a non-negative polynomial $f:W\times\Rn\mbox{$\longrightarrow$}\mbox{$\mathbf R$}$ such that $X=f^{-1}(0)$. Denote $f_w(y)=f(w;y)$. Then $\mbox{$\mathbf 0$}$ is contained in the set of critical points of each $f_w.$ By the parametrized version of the {\L}ojasiewicz Inequality of \cite{fekak}, there is $\alpha>0$ such that for every $w\in W$ there are positive $C=C(w)$ and $\delta=\delta(w)$ such that \[ f_w(y)\geq C \|y\|^\alpha ,\] for all critical points $y$ of $f_w$ with $\|y\| <\delta$ and $f_w(y)\neq 0$. Let $k$ be an integer such that $2k>\alpha$. Define \[g(w;y)=f(w;y)-\|y\|^{2k}\] and let \[G=\left( \frac{\partial g}{\partial y_1}\mbox{$\,,\ldots,\,$} \frac{\partial g}{\partial y_n}\right):W\times\Rn\mbox{$\longrightarrow$}\Rn.\] Clearly, $G$ is a polynomial family of vector fields such that $G_w(\mbox{$\mathbf 0$})=\mbox{$\mathbf 0$}$. For every $w\in W$ let $L(w)=\{y\in S_{r}^{n-1}\mid (w;y)\in X\}$, where $r>0$ is small. It is well-known that $L(w)$ is well-defined up to a homeomorphism. Then $\chi(L(w))=1-\deg_0 G_w$. Indeed, this can be proven by an argument similar to that of proof of Lemma \ref{guzik}, if we replace $t^{2d+1}$ by $\|y\|^{2k}$, $P_\eta$ by the sphere $S_r$, and $\Sigma_\eta$ by the set of critical points of $f$ restricted to $S_r$, see \cite{szafraniec4} for the details. Therefore, Theorem \ref{slonik} implies \begin{theorem} \label{antracyt} For all $w\in W$, $\Rn\ni\mbox{$\mathbf 0$}$ is isolated in $G^{-1}_{w}(\mbox{$\mathbf 0$})$ and $\chi(L(w))=1-\deg_0 G_w$. In particular, there are polynomials $g_1\mbox{$\,,\ldots,\,$} g_s$ such that for every $w\in W$ \[ \chi(L(w)) ={\mbox{$\operatorname {sgn}$}\,}\, g_1(w)\mbox{$\,+\cdots+\,$} {\mbox{$\operatorname {sgn}$}\,}\, g_s(w). \quad \Box\] \end{theorem} Similarly, let $S(w)=\{y\in S_{R}^{n-1}\mid (w;y)\in X\}$, where $R>0$ is very large. $S(w)$ is well-defined up to a homeomorphism. \begin{corollary} \label{rubin} There is a polynomial family of vector fields $H_w:\Rn\mbox{$\longrightarrow$}\Rn$ such that $\Rn\ni\mbox{$\mathbf 0$}$ is isolated in $H_{w}^{-1}(\mbox{$\mathbf 0$})$ for all $w\in W$ and $\chi(S(w))=1-\deg_0 H_w$. \end{corollary} {\em Proof.\/} Let $d$ denote the degree of $f$, where as above, $f$ is a non-negative polynomial defining $X$. Then, there is a non-negative polynomial $h:W\times\Rn\mbox{$\longrightarrow$}\mbox{$\mathbf R$}$ such that $h(w;y)=\|y\|^{2d} f(w;y/\|y\|^2)$ for $y\neq \mbox{$\mathbf 0$}$. Clearly $h(w;\mbox{$\mathbf 0$})\equiv 0$ and $S(w)$ is homeomorphic to $L'(w)=\{y\in S_{r}^{n-1}\mid (w;y)\in h^{-1}(0)\},$ where $r>0$ is small. So the corollary follows from Theorem \ref{antracyt}. $\Box$ It is well-known (see, for instance, \cite{akbulutking4,benedettitognoli,bochnaketal}) that the single point Aleksandrov compactification of a real algebraic set is homeomorphic to a real algebraic set. We shall recall briefly the proof. Suppose $X=\{y\in \Rn\mid f_1(y)=\mbox{$\,\cdots\,$}=f_s(y)=0\}$, where $f_1\mbox{$\,,\ldots,\,$} f_s:\mbox{${\mathbf R}^{n}$$ are polynomials of degree $\leq p-1$. Set $h(y,y_{n+1})=y_{n+1}^{2}(f_{1}^{2}(y)\mbox{$\,+\cdots+\,$} f_{s}^{2}(y))+(y_{n+1}-1)^{2}$, so that $h^{-1}(0)$ is homeomorphic to $X$ and $h$ is a non-negative polynomial of degree $\leq 2p$. Put $y'=(y,y_{n+1})\in\Rn\times\mbox{$\mathbf R$}$ and $H(y')=\| y'\| ^{4p}h(y'/\| y'\| ^2)$. Then, it is easy to see that $H$ extends to a non-negative polynomial on $\Rn\times\mbox{$\mathbf R$}$ such that $H(\mbox{$\mathbf 0$},0)=0$ and $H(y')=\| y'\| ^{4p} +$ {\em monomials of lower degree\/}. Clearly $\tilde X=H^{-1}(0)$ is the single point compactification of $X$ (If $X$ is compact then $\tilde X=X\amalg\{${\em point\/}$\}$). Note that $t^{4p}H(y'/t)$ extends to a non-negative homogeneous polynomial $f(y',t)$ on $\Rn\times\mbox{$\mathbf R$}\times\mbox{$\mathbf R$}$ of degree $4p$ such that $f(y',0)=\|y'\|^{4p}$ and $\tilde X$ is homeomorphic to $\{y'\mid f(y',1)=0\}$. Proceeding exactly in the same way we may prove the following parametrized version of the above compactification method. \begin{lemma} \label{rusznica} Let $X\subset W\times\Rn$ be a real algebraic set. Then there is a non-negative polynomial $f:W\times\mbox{$\mathbf R$}^{n+1}\times\mbox{$\mathbf R$}\mbox{$\longrightarrow$}\mbox{$\mathbf R$}$ such that for every $w\in W$ \begin{enumerate} \item $f_w(y',t)=f(w;y',t):\mbox{$\mathbf R$}^{n+1}\times\mbox{$\mathbf R$}\mbox{$\longrightarrow$}\mbox{$\mathbf R$}$ is a non-negative homogeneous polynomial of degree $4p,$ \item $f_w(y',0)=\| y'\| ^{4p},$ \item $\tilde X_w=\{y'\in\mbox{$\mathbf R$}^{n+1}\mid f_w(y',1)=0\}$ is homeomorphic to the single point compactification of $X_w=\{y\in\Rn\mid (w;y)\in X\}$. $\Box$ \end{enumerate} \end{lemma} In particular, by Proposition \ref{guzik} we get \begin{proposition} \label{arogant} Let $X\subset W\times\Rn$ be a real algebraic set. Then there is a polynomial family of vector fields $F_w:\mbox{$ {\bf R}^{n} $$ such that for every $w\in W$ \begin{enumerate} \item $F_w(\mbox{$\mathbf 0$})=\mbox{$\mathbf 0$},$ \item $\mbox{$\mathbf 0$}$ is isolated in $F_{w}^{-1}(\mbox{$\mathbf 0$}),$ \item $\chi(\tilde X_w)=1-\deg_0 F_w .\ \Box$ \end{enumerate} \end{proposition} Let $S(w)=X_w\cap S_{R}^{n-1},$ where $R>0$ is sufficiently large. Then it is easy to check that \[\chi(X_w)=\chi(\tilde X_w)+\chi (S(w))-1.\] By \ref{arogant}, \ref{rubin}, \ref{slonik}, and since $\mbox{$\operatorname {sgn}$}\, a+\mbox{$\operatorname {sgn}$}\, b\equiv \mbox{$\operatorname {sgn}$}\, (ab)+1\pmod{4}$, provided $a\neq 0$ and $b\neq 0$, we get \begin{theorem} \label{wtyczka} Let $X\subset W\times \Rn$ be a real algebraic set. Then there are polynomials $g_1\mbox{$\,,\ldots,\,$} g_s$ on $W$ such that \[\chi(X_w)={\mbox{$\operatorname {sgn}$}\,}\,g_1(w)\mbox{$\,+\cdots+\,$} {\mbox{$\operatorname {sgn}$}\,}\,g_s(w).\] In particular, if $W$ is irreducible, then there are a proper algebraic subset $\Sigma\subset W$, an integer $\mu$, and a polynomial $g$ nowhere vanishing in $W-\Sigma$ such that for every $w\in W-\Sigma$ \[\chi(X_w)\equiv \mu+ {\mbox{$\operatorname {sgn}$}\,}\,g(w)\pmod{4},\] In particular $\chi(X_w)\equiv\mu+1\pmod{2}$. $\Box$ \end{theorem} \bigskip \section{Algebraically constructible functions}\label{constructible} Let $W$ be a real algebraic set. An integer-valued function $\varphi:W\to \mathbf Z$ is called ({\it semialgebraically}) {\it constructible} if it admits a presentation as a finite sum \begin{equation}\label{constr} \varphi = \sum m_i \mbox{$\mathbf 1$}_{W_i}, \end{equation} where for each $i$, $W_i$ is a semialgebraic subset of $W$, $\mbox{$\mathbf 1$}_{W_i}$ is the characteristic function of $W_i$, and $m_i$ is an integer. Constructible functions, well-known in complex domain, were studied in real algebraic set-up by Viro \cite{viro}, and in sub-analytic set-up by Kashiwara and Schapira \cite{kashiwaraschapira, schapira}. If the support of constructible function $\varphi$ is compact, then we may choose all $W_i$ in (\ref{constr}) compact. Then, cf. \cite{viro, schapira, mccroryparusinski2}, the {\it Euler integral} of $\varphi$ is defined as \[\int \varphi = \sum m_i \chi (W_i) . \] It follows from the additivity of Euler characteristic that the Euler integral is well-defined and does not depend on the presentation (\ref{constr}) of $\varphi$, provided all $W_i$ are compact. Let $f:W\to Y$ be a (continuous) semialgebraic map of real algebraic sets, $\varphi$ a constructible function on $W$ and suppose that $f:W\to Y$ restricted to the support of $\varphi$ is proper. Then the {\it direct image} $f_*\varphi$ is given by the formula \[f_*\varphi (y) = \int_{f^{-1}(y)} \varphi ,\] where by $\int_{f^{-1}(y)} \varphi$ we understand the Euler integral of $\varphi$ restricted to $f^{-1}(y)$. It follows from the existence of a stratification of $f$ that $f_*\varphi$ is a constructible function on $Y$. Another more restrictive class of constructible functions, was introduced in \cite{mccroryparusinski2} in order to study local topological properties of real algebraic sets. An integer-valued function $\varphi:W \to \mathbf Z$ is called {\it algebraically constructible} if there exists a finite collection of algebraic sets $Z_i$, regular proper morphisms ${f_i}:Z_i \to W$, and integers $m_i$, such that \[ \varphi = \sum m_i {f_i}_* \mbox{$\mathbf 1$}_{Z_i} \] It is obvious that every algebraically constructible function is semialgebraically constructible but the converse is false for $\dim W >0$. For instance, a constructible function on $\mathbf R$ is algebraically constructible if and only if it is is generically constant mod 2. The reader may consult \cite{mccroryparusinski2} for other examples. As a consequence of section \ref{families} we obtain the following simple decription of algebraically constructible functions. \begin{theorem}\label{key} Let $W$ be a real algebraic set. Then $\varphi:W\to \mathbf Z$ is algebraically constructible if and only if there exist polynomial functions $g_1\mbox{$\,,\ldots,\,$} g_s$ on $W$ such that \[\varphi(w)=\mbox{$\operatorname {sgn}$}\, g_1(w)\mbox{$\,+\cdots+\,$} \mbox{$\operatorname {sgn}$}\, g_s(w).\] \end{theorem} {\em Proof.\/} It is easy to see that the sign of a polynomial function $g$ on $W$ defines an algebraically constructible function. Indeed, let $\widetilde W = \{(w, t) \in W\times \mbox{$\mathbf R$} \ |\ g(w) = t^2\}$ and let $\pi : \widetilde W\to W$ denote the standard projection. Then $\mbox{$\operatorname {sgn}$}\, f = \pi_* \mbox{$\mathbf 1$}_{\widetilde W} - \mbox{$\mathbf 1$}_W$ is algebraically constructible. The opposite implication follows from Theorem \ref{wtyczka}. $\Box$ \par \begin{corollary} \label{gniazdko} \par \begin{enumerate} \item Let $F:W\times\Rn\mbox{$\longrightarrow$}\Rn$ be a polynomial mapping satisfying the assumptions of \ref{slonik}. Then $w\mbox{$\longrightarrow$} \deg_0 F_w$ is an algebraically constructible function on $W$. \item Let $X_w$ be an algebraic family of affine real algebraic sets parametrized by $w\in W$ as in \ref{wtyczka}. Then $w\mbox{$\longrightarrow$} \chi(X_w)$ is an algebraically constructible function on $W$. $\Box$ \end{enumerate} \end{corollary} The next corollary is virtually equivalent to the main result of \cite{costekurdyka2}. \begin{corollary}\label{discriminant} Let $\varphi$ be an algebraically constructible function on an irreducible real algebraic set $W$. Then there exist a proper real algebraic subset $\Sigma\subset W$, an integer $\mu$, and a polynomial $g$ on $W$, such that $g$ does not vanish on $W\setminus \Sigma$ and \[ \varphi(w)\equiv \mu+{\mbox{$\operatorname {sgn}$}\,}\,g(w) \pmod{4}\] for $w\in W-\Sigma$. In particular, for such $w$, $\varphi(w)\equiv\mu+1\pmod{2}$. \end{corollary} {\em Proof.\/} Let $g_1\mbox{$\,,\ldots,\,$} g_s$ be polynomials given by \ref{key}. We may suppose that all of them are not identically equal to zero. Since $\mbox{$\operatorname {sgn}$}\, a+\mbox{$\operatorname {sgn}$}\, b\equiv \mbox{$\operatorname {sgn}$}\, (ab)+1\pmod{4}$, for $a$ and $b$ non-zero, the polynomial $g= g_1\cdots g_s$ satisfies the statement. This ends the proof. $\Box$ \par Let $\varphi$ be a constructible function on $W$. Following \cite {mccroryparusinski2} we define the {\it link} of $\varphi$ as the constructible function on $W$ given by \[\Lambda \varphi (w) = \int_{S(w,\varepsilon)} \varphi , \] where $\varepsilon>0$ is sufficiently small, and $S(w, \varepsilon)$ denotes the $\varepsilon$-sphere centered at $w$. It is easy to see that $\Lambda\varphi$ is well defined and independent of the embedding of $W$ in $\bf R^n$. Then the duality operator $D$ on constructible functions, introduced by Kashiwara and Schapira in \cite {kashiwaraschapira, schapira}, satisfies \[ D \varphi = \varphi - \Lambda \varphi. \] As shown in \cite{mccroryparusinski2} the following general statement generalizes various previously known restrictions on local topological properties of real algebraic sets. In particular it implies Akbulut and King's numerical conditions of \cite{akbulutking} and the conditions modulo 4, 8, and 16 of Coste and Kurdyka \cite {coste2, costekurdyka1} generalized in \cite{mccroryparusinski1}. \begin{theorem}\label{link} Let $\varphi$ be an algebraically constructible function on a real algebraic set $W$. Then $\frac 1 2 \Lambda \varphi$ is integer-valued and algebraically constructible. \end{theorem} The above theorem was proven in \cite {mccroryparusinski2} using the resolution of singularites. As we show below it is a simple conseqence of Theorem \ref{key}. {\em Proof.\/} We begin the proof by some preparatory observations. \begin{lemma}\label{limits} $W$ be a real algebraic set and let $\gamma$ be an algebraically constructible function on $W\times \mathbf R$. Then \[ \psi_+ (w)= \lim_{t\to 0_+} \gamma(w,t), \, \, \psi_- (w)= \lim_{t\to 0_+} \gamma(w,-t), \, \, \psi (w)= \frac 1 2 (\psi_+ (w) - \psi_- (w)) \] are integer-valued and algebraically constructible on $W_0 = W\times \{0\}$. \end{lemma} {\em Proof.\/} We show the lemma for $\psi$. The proofs for $\psi_+$ and $\psi_-$ are similar. We proceed by induction on $\dim W$. Without loss of generality we may assume that that $W$ is affine and irreducible. We shall show that the statement of lemma holds generically on $W_0$, that is to say there exists a proper algebraic subset $W'_0$ of $W_0$ and an algebraically constructible function $\psi'$ on $W_0$ which equals $\psi$ in the complement of $W'_0$. Then the lemma follows from the inductive assumption since $\dim W'_0 < \dim W_0$. By Theorem \ref{key} we may assume that $\gamma = \mbox{$\operatorname {sgn}$}\, g$, where $g(w,t)$ is a polynomial function on $W\times \mathbf R$. We may also assume that $g$ does not vanish identically, and then there exists a nonnegative integer $k$ such that \[ g(w,t) = t^k h(w,t), \] where $h(w,t)$ is a polynomial function on $W\times \mathbf R$ not vanishing identically on $W\times \{0\}$. Then, in the complement of $W'_0 = \{w| h(w,0)=0\}$, either $\psi(w) = \mbox{$\operatorname {sgn}$}\, h(w,0)$ for $k$ odd or $\psi(w) =0$ for $k$ even satisfies the statement. This ends the proof of lemma. $\Box$ Let $\widetilde W = \{(w,y,t)\in W\times W\times\mathbf R|\, \|w-y\|^2=t \}$ and let $\pi :\widetilde W\to W\times \mathbf R$ be given by $\pi (w,y,t) = (w,t)$. Let $\tilde \varphi (w,y,t)= \varphi (y)$. Then $\tilde \varphi$ is algebraically constructible and hence $\gamma = \pi_* \tilde \varphi$ is an algebraically constructible function on $W\times \mathbf R$ and \[ \lim_{t\to 0_+} \gamma(w,t) = \Lambda \varphi (w) . \] Since $\gamma (w,t)=0$ for $t<0$ \[ \frac 1 2 \Lambda \varphi (w) = \frac 1 2 \lim_{t\to 0_+} (\gamma(w,t)-\gamma(w,-t)) \] is algebraically constructible by Lemma \ref{limits}. This ends the proof of Theorem \ref{link}. $\Box$ \medskip Suppose that $f:W\to \mbox{$\mathbf R$}$ is regular and let $w\in W_0 = f^{-1} (0)$. Then we define the {\it positive}, resp.~{\it negative}, {\it Milnor fibre} of $f$ at $w$ by \[ F_f^+(w) = B(w, \varepsilon )\cap f^{-1} (\delta) \] \[ F_f^-(w) = B(w, \varepsilon )\cap f^{-1} (-\delta) ,\] where $B(w, \varepsilon )$ is the ball of radius $\varepsilon$ centered at $w$ and $0<\delta\ll \varepsilon\ll 1$. Let $\varphi$ be an algebraically constructible function on $W$. Following \cite{mccroryparusinski2} we define the {\it positive} (resp.~{\it negative}) {\it specialization} of $\varphi$ with respect to $f$ by \[ (\Psi^+_f \varphi) (w) = \int_{F_f^+(w)} \varphi, \quad (\Psi^-_f \varphi) (w) = \int_{F_f^-(w)} \varphi . \] It is easy to see that both specializations are well-defined and that they are constructible functions supported in $W_0$. Moreover, as shown in \cite{mccroryparusinski2}, they are also algebraically costructible. We present below an alternative proof of this fact. \begin{theorem}\label{specialization} Let $f:W\to \mbox{$\mathbf R$}$ be a regular function on a real algebraic set $W$. Let $\varphi$ be an algebraically constructible function on $W$. Then $\Psi^+_f \varphi$, $\Psi^-_f \varphi$, and $\frac 1 2 (\Psi^+_f \varphi - \Psi^-_f \varphi)$ are integer valued and algebraically constructible. \end{theorem} {\em Proof.\/} The proof is similar to that of Theorem \ref{link}. Since the Milnor fibres are defined not only by equations but also by inequalities we use the following auxiliary construction. Let $\widetilde W = \{(w,y,t,r,s)\in W\times W\times\mathbf R^3|\, \|w-y\|^2 + t^2=r, f(y)=s \}$ and let $\pi :\widetilde W\to W\times \mbox{$\mathbf R$} ^2$ be given by $\pi (w,y,t,r,s) = (w,r,s)$. Note that for $w\in W_0$, $0<s\ll r\ll 1$, $\tilde F = \pi^{-1} (w,r,s)$ is a double cover of the Milnor fibre $F=F^+_f(w)$ branched along its boundary $\partial F = S(w,\sqrt r)\cap f^{-1}(s)$. Hence $\chi (\tilde F) = 2\chi (F) - \chi (\partial F)$. Let $\tilde \varphi (w,y,t,r,s)= \varphi (y)$. Then \[ \Psi_f^+ \varphi (w) = \frac 1 2 (\int_{\tilde F}\tilde \varphi + \int_{\partial F} \varphi) = \frac 1 2 \pi_* (\tilde \varphi + \tilde \varphi|_{t=0}) (w,r,s), \] for $0<s\ll r\ll 1$. Clearly an analogous formula holds for $\Psi_f^- \varphi (x)$. Let $\gamma = \pi_* (\tilde \varphi + \tilde \varphi|_{t=0})$. Then $\gamma (w,r,s)$ is algebraically constructible and $\gamma(w,r,s)=0$ for $r<0$. Hence, by Lemma \ref{limits}, the following functions are algebraically constructible \[ \Psi_f^{\pm} \varphi = \frac 1 2 \lim_{r\to 0_+} \lim_{s\to 0_+} \gamma(w,r,\pm s) , \] \[ \frac 1 2 (\Psi_f^{+} - \Psi_f^-) \varphi = \frac 1 4 \lim_{r\to 0_+} \lim_{s\to 0_+} (\gamma(w,r,s)-\gamma(w,r,-s)), \] as required. $\Box$ \medskip

9609

alg-geom/9609008

en

https://arxiv.org/abs/alg-geom/9609008

alg-geom/9609008

Elham Izadi

E. Izadi

Density and completeness of subvarieties of moduli spaces of curves or abelian varieties

AMS-LaTeX, 15 pages

Let $V$ be a subvariety of codimension $\leq g$ of the moduli space $\cA_g$ of principally polarized abelian varieties of dimension $g$ or of the moduli space $\tM_g$ of curves of compact type of genus $g$. We prove that the set $E_1(V)$ of elements of $V$ which map onto an elliptic curve is analytically dense in $V$. From this we deduce that if $V \subset \cA_g$ is complete, then $V$ has codimension equal to $g$ and the set of elements of $V$ isogenous to a product of $g$ elliptic curves is countable and analytically dense in $V$. We also prove a technical property of the conormal sheaf of $V$ if $V \subset \tM_g$ (or $\cA_g$) is complete of codimension $g$.

alg-geom

\section{The proofs} \label{sectpf} In this section we give the proof of Theorem 1 and its corollaries. We first consider the case where $V$ is contained in ${\cal M}_g$. We may and will replace $V$ with its inverse image in ${\cal M}_g'$. The relative jacobian of ${\cal C}_V$ gives us a family of ppav's on $V$. We can therefore apply Theorem (1) on page 162 of \cite{colpi}: to show that $E_1(V)$ is dense in $V$ it is enough to prove the following: There exists a Zariski dense (Zariski-)open subset $U$ of $V$, contained in the smooth locus $V_{sm}$ of $V$, such that, for all $t \in U$, there is a subvector space $W$ of $H^0(\omega_{C_t})$ which has dimension $1$ and is such that the composition \[W \otimes W^{\perp} \stackrel{\mu}{\hookrightarrow} H^0(\omega_{C_t}^{\otimes 2}) \stackrel{\pi}{\longrightarrow} T^*_tV \] is injective. Here $W^{\perp}$ is the orthogonal complement of $W$ with respect to the hermitian form on $H^0(\omega_{C_t})$ induced by the natural polarization of $JC_t$. We sketch briefly how this condition is obtained in the more general case where $W$ has dimension $q$ with $1 \leq q \leq g/2$ and $V \subset {\cal A}_g'$ has any dimension $\geq q(g-q)$. To prove the density of $E_q(V)$ in $V$, it is enough to show that there is a Zariski dense open subset $U$ of $V_{sm}$, such that, for all $t \in U$, there is an analytic neighborhood $U'$ of $t$, $U' \subset U$, such that $E_q(V) \cap U'$ is dense in $U'$. An abelian variety $A$ contains an abelian subvariety of dimension $q$ if and only if $H^0(\Omega^1_{A})$ contains a $q$-dimensional ${\Bbb C}$-subvector space which is the tensor product with ${\Bbb R}$ of a vector subspace of dimension $2q$ of $H^1(A, {\Bbb Q})$ (after identifying $H^0(\Omega^1_{A})$ with $H^1(A, {\Bbb R}) \cong H^1(A, {\Bbb Q}) \otimes {\Bbb R}$ as real vector spaces). Let $t$ be an element of $V_{sm}$. For a contractible analytically open set $U' \ni t$ contained in $V_{sm}$, let $F_{U'}$ be the Hodge bundle over $U'$. Then one can trivialize $F_{U'}$ as a real vector bundle. Therefore the grassmannian bundle of $2q$-dimensional real subvector spaces of $F_{U'}$ is isomorphic to $U' \times G_{\Bbb R}(2q,2g)$, where $G_{\Bbb R}(2q,2g)$ is the Grassmannian of $2q$-dimensional ${\Bbb R}$-subvector spaces of $H^1(A_t, {\Bbb R})$. Hence there is a well-defined map $\Phi : G(q,F_{U'}) \longrightarrow G_{\Bbb R}(2q,2g)$ where $G(q,F_{U'})$ is the Grassmannian of $q$-dimensional ${\Bbb C}$-subvector spaces of $F_{U'}$: The map $\Phi$ sends a $q$-dimensional complex subvector space of $H^0(\Omega^1_{A_s})$ (with $s \in U'$) to the image of its underlying real vector space under the isomorphism $H^1(A_s, {\Bbb R}) \stackrel{\cong}{\longrightarrow} H^1(A_t, {\Bbb R})$ obtained from the ${\Bbb R}$-trivialization of $F_{U'}$. Let $G_{{\Bbb Q}}(2q,2g) \subset G_{{\Bbb R}}(2q,2g)$ be the Grassmannian of $2q$-dimensional ${\Bbb Q}$-subvector spaces of $H^1(A_t, {\Bbb Q})$ and let $p : G(q,F_{U'}) \longrightarrow U'$ be the natural morphism. Then $E_q(V) \cap U' = p(\Phi^{-1}(G_{{\Bbb Q}}(2q,2g)))$. To prove the density of $E_q(V) \cap U'$ in $U'$, it is enough to prove that there is a subset ${\cal Y}$ of $G(q,F_{U'})$ such that $p({\cal Y}) = U'$ and $\Phi^{-1}(G_{{\Bbb Q}}(2q,2g)) \cap {\cal Y}$ is dense in ${\cal Y}$. Since $G_{{\Bbb Q}}(2q,2g)$ is dense in $G_{{\Bbb R}}(2q,2g)$, it is enough to find ${\cal Y}$ such that $p({\cal Y}) = U'$ and $\Phi |_{{\cal Y}}$ is an open map. If $\Phi$ has maximal rank (i.e., the differential $d \Phi$ of $\Phi$ is surjective) everywhere on ${\cal Y}$, then $\Phi |_{{\cal Y}}$ is an open map. Therefore $E_q(V) \cap U'$ is dense in $U'$ if for every $s \in U'$ there is a $q$-dimensional ${\Bbb C}$-subvector space $W$ of $H^0(\Omega^1_{A_s})$ such that $d \Phi$ is surjective at $(W,s) \in G(q,F_{U'})$ (then ${\cal Y}$ would be the set of such $(W,s)$). The tangent space $T_{(W,s)}G(q,F_{U'})$ is isomorphic to $W \otimes W^{\perp} \oplus T_sU'$, the tangent space to $G_{\Bbb R}(2q,2g)$ at $\Phi(W,s)$ is isomorphic to $W \otimes W^{\perp} \oplus \overline{W \otimes W^{\perp}} \cong W \otimes W^{\perp} \oplus (W \otimes W^{\perp})^*$ and the restriction of $d \Phi$ to the $W \otimes W^{\perp}$ summand of $T_{(W,s)}G(q,F_{U'})$ is an isomorphism onto the $W \otimes W^{\perp}$ summand of $T_{\Phi(W,s)}G_{\Bbb R}(2q,2g)$. Therefore $d \Phi$ is surjective if and only if the map it induces $T_sU' = \frac{T_{(W,s)}G(q,F_{U'})}{W \otimes W^{\perp}} \longrightarrow (W \otimes W^{\perp})^* = \frac{T_{\Phi(W,s)}G_{\Bbb R}(2q,2g)}{W \otimes W^{\perp}}$ is surjective, i.e., if and only if the dualized map $W \otimes W^{\perp} \longrightarrow T^*_sU'$ is injective. Let $F$ be the Hodge bundle over the Siegel upper half space ${\cal U}_g$. The inclusion $U' \hookrightarrow {\cal A}_g'$ lifts to an inclusion $U' \hookrightarrow {\cal U}_g$ because $U'$ is contractible and there is a family of ppav's on $U'$ (the restriction of ${\cal A}_V$). Factoring $\Phi$ through the Grassmannian of $q$-planes in $F$ over ${\cal U}_g$, the map $W \otimes W^{\perp} \longrightarrow T_s^*U'$ can be seen to be the composition \[ W \otimes W^{\perp} \stackrel{\rho}{\longrightarrow} S^2H^0(\Omega^1_{A_s}) \stackrel{\pi_a}{\longrightarrow} T_s^*U' = T_s^*V \: . \] For $V$ contained in ${\cal M}_g'$, we have $\pi_a = \pi m$. \vskip10pt Clearly, if $\pi \mu : W \otimes H^0(\omega_{C_t}) \longrightarrow T^*_tV$ is injective, then so is $\pi \mu : W \otimes W^{\perp} \longrightarrow T^*_tV$. In view of this (and also for use in the proof of Corollary 2) we show: \begin{proposition} Suppose that $g \geq 3$. Let $V$ be a subvariety of codimension at most $g$ of ${\cal M}_g'$. Let $t$ be a point of $V_{sm}$ and let $N$ be the kernel of $\pi : H^0(\omega_{C_t}^{\otimes 2}) \longrightarrow T^*_tV$. \begin{enumerate} \item Suppose that $C_t$ is non-hyperelliptic. Suppose that, for any one-dimensional subvector space $W$ of $H^0(\omega_{C_t})$, the map $\pi \mu : W \otimes H^0(\omega_{C_t}) \longrightarrow T^*_tV$ is {\em not} injective. Then $V$ has codimension exactly $g$ and there is a one-dimensional subvector space $W_N$ of $H^0(\omega_{C_t})$ such that $N = \mu (W_N \otimes H^0(\omega_{C_t}))$. \item Suppose that $C_t$ is hyperelliptic and that $V$ is {\em not} transverse to ${\cal H}_g' := s_c^{-1}({\cal H}_g)$ at $t$ (i.e., the sum $T_tV + T_t {\cal H}_g' \subset T_t {\cal M}_g'$ is {\em not} equal to $T_t {\cal M}_g'$). Then there exists a one-dimensional subvector space $W$ of $H^0(\omega_{C_t})$ such that the map $\pi \mu : W \otimes H^0(\omega_{C_t}) \longrightarrow T^*_tV$ is injective. \end{enumerate} \label{propNcurve} \end{proposition} {\em Proof :} Consider the composition \[ {\Bbb P}(H^0(\omega_{C_t})^{\otimes 2}) \stackrel{\overline{\rho}}{\longrightarrow} {\Bbb P}(S^2H^0(\omega_{C_t})) \stackrel{\overline{m}}{\longrightarrow} {\Bbb P}(H^0(\omega_{C_t}^{\otimes 2})) \: . \] The kernel of $m$ is the space $I_2(C_t)$ of quadratic forms vanishing on $\kappa C_t$. Hence the {\em rational} map $\overline{m}$ is the projection with center ${\Bbb P} (I_2(C_t))$. Let $\overline{\cal S}$ be the image by $\overline{\rho}$ of the Segre embedding ${\cal S}$ of ${\Bbb P}(H^0(\omega_{C_t})) \times {\Bbb P}(H^0(\omega_{C_t}))$ in ${\Bbb P}(H^0(\omega_{C_t})^{\otimes 2})$. Let $N'$ be the set of rank $2$ symmetric tensors in $S^2H^0(\omega_{C_t})$ which lie in $m^{-1}(N)$ (then ${\Bbb P}(N')$ is the {\em reduced} intersection of $\overline{\cal S}$ and $\overline{m}^{-1}({\Bbb P}(N))$). Suppose that for all $W \subset H^0(\omega_{C_t})$ of dimension $1$, the map $\mu : W \otimes H^0(\omega_{C_t}) \longrightarrow T_t^*V$ is not injective, i.e., for all $w \in H^0(\omega_{C_t})$, there is $w' \in H^0(\omega_{C_t})$ such that $\mu (w \otimes w') \in N$. This implies that the dimension of ${\Bbb P}(N')$ is at least $g-1$. We will show below that this does not happen if $C_t$ is hyperelliptic and $V$ is not transverse to ${\cal H}_g'$ at $t$. If $C_t$ is non-hyperelliptic, we will show that this implies that ${\Bbb P}(N')$ is a linear subspace of $\overline{\cal S}$ and that its inverse image in ${\cal S}$ is the union of two linear subspaces of ${\cal S}$ which are two fibers of the two projections of ${\cal S}$ onto ${\Bbb P}^{g-1}$ and are exchanged under the involution of ${\cal S}$ which interchanges the two ${\Bbb P}^{g-1}$-factors of ${\cal S}$. The proposition will then easily follow from this. Suppose first that $C_t$ is {\em non}-hyperelliptic. Then $m : S^2H^0(\omega_{C_t}) \longrightarrow H^0(\omega_{C_t}^{\otimes 2})$ is onto (see \cite{ACGH} page 117). We have \begin{lemma} Suppose $g= 2$ or $g \geq 3$ and $C_t$ is non-hyperelliptic. Suppose that for all $W \subset H^0(\omega_{C_t})$ of dimension $1$, the map $\mu : W \otimes H^0(\omega_{C_t}) \longrightarrow T_t^*V$ is not injective. Then the map ${\Bbb P}(N') \longrightarrow {\Bbb P}(N)$ is generically one-to-one. \label{N'isPN} \end{lemma} {\em Proof :} If not, then, for all $w \in H^0(\omega_{C_t})$, there exists $w', w_1, w_1'\in H^0(\omega_{C_t})$ such that $w w' := \rho(w \otimes w')$ and $w_1 w_1' := \rho(w_1 \otimes w_1')$ are not proportional but $m (w w')$ and $m (w_1 w_1')$ are proportional elements of $N$. Therefore, supposing $w$ general, there exits $\lambda \in {\Bbb C}, \lambda \neq 0$, such that the element $\lambda ww'-w_1w_1'$ of $S^2H^0(\omega_{C_t})$ lies in $I_2(C_t)$, i.e., defines a quadric $q(w)$ of rank $3$ or $4$ (in the canonical space $| \omega_{C_t} |^*$) which contains $\kappa C_t$ (the canonical curve $\kappa C_t$ is not contained in any quadric of rank $\leq 2$ since it is nondegenerate). If $g \leq 3$, this is impossible because in that case $I_2(C_t)=0$. If $g \geq 4$, the intersection $L$ of the two hyperplanes in $|\omega_{C_t}|^*$ with equations $w$ and $w_1$ is an element of a ruling of the quadric $q(w)$. Therefore $L$ cuts a divisor of a $g^1_d$ (a $g^1_d$ is a pencil of divisors of degree $d$) on $C_t$ with $d \leq g-1$ (see \cite{AM}, Lemmas 2 and 3 page 192). Therefore the divisor of zeros of $w$ on $C_t$ contains a divisor of a $g^1_d$. By the uniform position Theorem (see \cite{ACGH} Chapter $3, \S 1$) this does not happen for $w$ in some nonempty Zariski-open subset of $H^0(\omega_{C_t}) \setminus \{ 0 \}$. \hfill \qed \vskip20pt Therefore, since the dimension of ${\Bbb P}(N')$ is at least $g-1$ and the dimension of ${\Bbb P}(N)$ is at most $g-1$, the map ${\Bbb P}(N') \longrightarrow {\Bbb P} (N)$ is birational and ${\Bbb P}(N')$ and ${\Bbb P}(N)$ have both dimension $g-1$. {\em This proves, in particular, that $V$ has codimension {\em exactly} $g$.} Since no quadrics of rank $\leq 2$ contain $\kappa C_t$, the center ${\Bbb P} I_2(C_t)$ of the projection $\overline{m}$ does not intersect $\overline{\cal S}$. In particular, the space ${\Bbb P} I_2(C_t)$ does not intersect ${\Bbb P}(N')$. Therefore $\overline{m}$ restricts to a birational {\em morphism} ${\Bbb P}(N') \longrightarrow {\Bbb P}(N)$ and, since ${\Bbb P}(N)$ is a linear subspace of ${\Bbb P}(H^0(\omega_{C_t}^{\otimes 2}))$, the degree of ${\Bbb P}(N')$ (in the projective space ${\Bbb P} (S^2H^0(\omega_{C_t}))$) is equal to the (generic) degree of the map ${\Bbb P}(N') \longrightarrow {\Bbb P}(N)$. Hence ${\Bbb P}(N')$ is a linear subspace of ${\Bbb P}(S^2H^0(\omega_{C_t}))$ and $\overline{m}$ restricts to an {\em isomorphism} ${\Bbb P}(N') \stackrel{\cong}{\longrightarrow} {\Bbb P}(N)$. Let $N''$ be the cone of decomposable tensors in $H^0(\omega_{C_t})^{\otimes 2}$ which lie in $\mu^{-1}(N)$ (then ${\Bbb P}(N'')$ is the {\em reduced} inverse image of ${\Bbb P}(N')$ in ${\cal S} \subset {\Bbb P}(H^0(\omega_{C_t})^{\otimes 2})$). The map ${\cal S} \longrightarrow \overline{\cal S}$ is a finite {\em morphism} of degree $2$ ramified on the diagonal. Therefore the map ${\Bbb P}(N'') \longrightarrow {\Bbb P}(N')$ is a morphism of degree $\leq 2$. Since the diagonal of $\overline{\cal S} \cong S^2 {\Bbb P}(H^0(\omega_{C_t}))$ is irreducible of dimension $g-1$ and spans ${\Bbb P}(S^2H^0(\omega_{C_t}))$, the space ${\Bbb P}(N')$ intersects this diagonal in a subvariety of dimension at most $g-2$. Therefore the morphism ${\Bbb P}(N'') \longrightarrow {\Bbb P}(N')$ has degree $2$ and ${\Bbb P}(N'')$ has degree $2$ in ${\Bbb P}(H^0(\omega_{C_t})^{\otimes 2})$. If ${\Bbb P}(N'')$ is irreducible, it spans a linear subspace $\widetilde{\bP}$ of ${\Bbb P} (H^0(\omega_{C_t})^{\otimes 2})$ of dimension $g$. This implies that ${\Bbb P} (\Lambda^2H^0(\omega_{C_t}))$ intersects $\widetilde{\bP}$ in exactly one point. For $w_1, w_2 \in H^0(\omega_{C_t})$, let $w_1', w_2'$ be such that $\mu (w_1 \otimes w_1'), \mu (w_2 \otimes w_2') \in N$. For $w_i$ general, $w_i'$ is not proportional to $w_i$ since ${\Bbb P}(N'')$ intersects the diagonal of ${\cal S}$ in a subvariety of dimension at most $g-2$. Therefore the lines spanned by $w_1 \otimes w_1' - w_1' \otimes w_1$ and $w_2 \otimes w_2' - w_2' \otimes w_2$ give us elements of $\widetilde{\bP} \cap {\Bbb P}(\Lambda^2H^0(\omega_{C_t}))$ which is a point. Therefore, for all $w_1, w_2 \in H^0(\omega_{C_t})$ general there exists $\lambda \in {\Bbb C}, \lambda \neq 0$, such that \[ w_1 \otimes w_1' - w_1' \otimes w_1 = \lambda (w_2 \otimes w_2' -w_2' \otimes w_2) \] Complete $\{ w_1, w_2 \}$ to a general basis $\{ w_1, w_2, w_3, ..., w_g \}$ of $H^0(\omega_{C_t})$ and write $w_i' = \sum_{1 \leq j \leq g} a_{ij}w_j$ for $i = 1$ or $2$. Then from the equation above we deduce $a_{1j}=a_{2j}=0$ for $j > 2$. Therefore, $w_1'$ belongs to the span of $w_1$ and $w_2$. Repeating this argument with $w_1$ and $w_3$ instead of $w_1$ and $w_2$, we see that $w_1'$ also belongs to the span of $w_1$ and $w_3$. Hence $w_1'$ is proportional to $w_1$ (this is the only part in the proof of Proposition \ref{propNcurve} where we need $g \geq 3$). Contradiction. Therefore ${\Bbb P}(N'')$ is reducible, i.e., it is the union of two linear subspaces of dimension $g-1$. We have \begin{lemma} Suppose $g \geq 2$. All linear subspaces of dimension $g-1$ of ${\cal S} \subset {\Bbb P} ( H^0(\omega_{C_t})^{\otimes 2}) \cong {\Bbb P}^{g^2 -1}$ are elements of one of the two rulings of ${\cal S} \cong {\Bbb P} (H^0(\omega_{C_t})) \times {\Bbb P} (H^0(\omega_{C_t})) \cong {\Bbb P}^{g-1} \times {\Bbb P}^{g-1}$. \end{lemma} {\em Proof :} Let $T$ be a linear subspace of dimension $g-1$ of ${\cal S}$. Let $p_1$ and $p_2$ be the two projections of ${\cal S} \cong {\Bbb P}^{g-1} \times {\Bbb P}^{g-1}$ onto its two factors. Let $H_i$ be a general element of $p_i^* |{\cal O}_{{\Bbb P}^{g-1}}(1)|$ for $i= 1$ or $2$. Then $H_1 \cap T \neq H_2 \cap T$ and $H_i$ does not contain $T$. In particular, the intersection $H_i \cap T$ is either empty or of dimension $g-2$. The divisor $H_1 \cup H_2$ is the intersection of a hyperplane $H$ in ${\Bbb P}^{g^2 -1}$ with ${\cal S}$. Since $T$ is not contained in $H_1$ nor $H_2$, the hyperplane $H$ does not contain $T$ and hence $T \cap H$ is a linear space of dimension $g-2$. Since the two intersections $T \cap H_1 \neq T \cap H_2$ are both contained in the $(g-2)$-dimensional linear space $T \cap H$ and are either empty or have dimension $g-2$, we have either $H_1 \cap T = \emptyset$ or $H_2 \cap T = \emptyset$. Suppose, for instance, that $H_1 \cap T = \emptyset$. It is easily seen that $p_1^{-1}(p_1(H_1)) = H_1$ implies $p_1(H_1) \cap p_1(T) = p_1(H_1 \cap T)$. Therefore $p_1(T)$ does not intersect $p_1(H_1)$ which is a hyperplane in ${\Bbb P}^{g-1}$. Hence $p_1(T)$ is a point and $T$ is a fiber of $p_1$. \hfill \qed \bigskip We deduce from the above Lemma that ${\Bbb P}(N'') = {\Bbb P}(N_1) \cup {\Bbb P}(N_2)$ where ${\Bbb P}(N_1)$ and ${\Bbb P}(N_2)$ are elements of the two rulings of ${\cal S} \cong {\Bbb P}^{g-1} \times {\Bbb P}^{g-1}$. The spaces ${\Bbb P}(N_1)$ and ${\Bbb P}(N_2)$ are exchanged by the involution which exchanges the two factors of ${\cal S}$ because ${\Bbb P}(N'')$ is the inverse image of a linear subspace in $\overline{\cal S} \cong S^2 {\Bbb P}^{g-1}$. Therefore there exists a one-dimensional subvector space $W_N$ of $H^0(\omega_{C_t})$ such that, for instance, $N_1 = W_N \otimes H^0(\omega_{C_t})$ and $N_2 = H^0(\omega_{C_t}) \otimes W_N$. So $N = \mu(N_1) = \mu (W_N \otimes H^0(\omega_{C_t})$. This proves the Proposition in the non-hyperelliptic case.\\ Now suppose that $C_t$ is hyperelliptic and that $V$ is not transverse to ${\cal H}_g'$ at $t$, i.e., the subspaces $T_tV$ and $T_t {\cal H}_g'$ do {\em not} span $T_t {\cal M}_g'$. Let $\iota$ be the hyperelliptic involution of $C_t$. Let $H^0(\omega_{C_t}^{\otimes 2})^+$ and $H^0(\omega_{C_t}^{\otimes 2})^-$ be the subvector spaces of $H^0(\omega_{C_t}^{\otimes 2})$ of $\iota$-invariant and $\iota$-anti-invariant quadratic differentials respectively. Then $H^0(\omega_{C_t}^{\otimes 2})^+$ is the image of $S^2H^0(\omega_{C_t})$ by $m$ and the conormal space to ${\cal H}_g'$ at $t$ can be canonically identified with $H^0(\omega_{C_t}^{\otimes 2})^-$. The non-transversality of $V$ and ${\cal H}_g'$ means that $N \cap H^0(\omega_{C_t}^{\otimes 2})^- \neq \{ 0 \}$. This implies that $N$ is not contained in $H^0(\omega_{C_t}^{\otimes 2})^+$. Since $N$ has dimension at most $g$, the dimension of $N \cap H^0(\omega_{C_t}^{\otimes 2})^-$ is at most $g-1$. Hence the dimension of ${\Bbb P}(N \cap H^0(\omega_{C_t}^{\otimes 2})^+) = {\Bbb P}(N) \cap {\Bbb P}(H^0(\omega_{C_t}^{\otimes 2})^+) = {\Bbb P}(N) \cap \overline{m} ({\Bbb P}(S^2H^0(\omega_{C_t})))$ is at most $g-2$. We have \begin{lemma} Suppose $g \geq 2$ and $C_t$ hyperelliptic. The map $\overline{m} : \overline{\cal S} \longrightarrow \overline{\overline{\cal S}} := \overline{m} ( \overline{\cal S} )$ is a finite morphism of degree $ \frac{1}{2} \left( \begin{array}{c} 2g-2 \\ g-1 \end{array} \right) $. \label{oSooSfinite} \end{lemma} Note that the lemma finishes the proof of Proposition \ref{propNcurve}: we saw above that the dimension of ${\Bbb P}(N) \cap \overline{m} ({\Bbb P}(S^2H^0(\omega_{C_t})))$ is at most $g-2$. A fortiori, since $\overline{m} ({\Bbb P}(S^2H^0(\omega_{C_t}))) \supset \overline{\overline{\cal S}}$, the dimension of ${\Bbb P}(N) \cap \overline{\overline{\cal S}}$ is at most $g-2$ and the dimension of ${\Bbb P}(N')$ is at most $g-2$ which is what we needed to show (see the paragraphs preceding Lemma \ref{N'isPN}).\\ {\em Proof of lemma \ref{oSooSfinite}:} The map $\overline{m} : \overline{\cal S} \longrightarrow \overline{\overline{\cal S}}$ is a morphism if and only if the center ${\Bbb P}(I_2(C_t))$ of the projection $\overline{m}$ does not intersect $\overline{\cal S}$. This is the case because the canonical curve $\kappa C_t$ is nondegenerate and hence not contained in any quadrics of rank $\leq 2$. Fix a nonzero element $w w' = \rho (w \otimes w')$ of $S^2H^0(\omega_{C_t})$ and suppose that $w_1 w_1' \in S^2 H^0(\omega_{C_t})$ is not proportional to $w w'$ and $m (w_1 w_1') = \lambda. m (w w')$ for some $\lambda \in {\Bbb C}, \lambda \neq 0$. This is equivalent to $Z(w) + Z(w') = Z(w_1) + Z(w_1')$ where $Z(w)$, for instance, is the divisor of zeros of $w$ on the rational normal curve $\kappa C_t$. So there are only a finite number of possibilities for $Z(w_1)$ and $Z(w_1')$. This proves that $\overline{m} : \overline{\cal S} \longrightarrow \overline{\overline{\cal S}}$ is quasi-finite and hence finite since it is proper. Any divisor of degree $g-1$ on $\kappa C_t \cong {\Bbb P}^1$ is the divisor of zeros of some element of $H^0(\omega_{C_t}) = H^0({\cal O}_{{\Bbb P}^1}(g-1))$, hence, since there are $ \frac{1}{2} \left( \begin{array}{c} 2g-2 \\ g-1 \end{array} \right) $ ways to write a fixed reduced divisor of degree $2g-2$ as a sum of two divisors of degree $g-1$, the degree of $\overline{m} : \overline{\cal S} \longrightarrow \overline{\overline{\cal S}}$ is $ \frac{1}{2} \left( \begin{array}{c} 2g-2 \\ g-1 \end{array} \right) $. \hfill \qed \vskip15pt {\bf Proof of Theorem 1 in the case of curves:} As explained in the beginning of this section, we need to find a Zariski-dense open subset $U$ of $V_{sm}$, such that, for all $t \in U$, there exists $W \subset H^0(\omega_{C_t})$ ($W$ of dimension $1$) such that $\mu (W \otimes W^{\perp}) \cap N = \{ 0 \}$. First suppose $g \geq 3$. We may assume that $V$ is irreducible. If $V$ is contained in ${\cal H}_g'$, then $V$ is not transverse anywhere to ${\cal H}_g'$ and hence, by Proposition \ref{propNcurve}, we may take $U$ to be all of $V_{sm}$. If $V \not \subset {\cal H}_g'$, take $U = V_{sm} \setminus {\cal H}_g'$. Suppose that there exists $t \in U$ such that, for all $W \subset H^0(\omega_{C_t})$ of dimension $1$, we have $\mu (W \otimes W^{\perp}) \cap N \neq \{ 0 \}$. Then, a fortiori, the hypotheses of part 1 of Proposition \ref{propNcurve} are met and $N= \mu (W_N \otimes H^0(\omega_{C_t}))$. Then every element of $H^0(\omega_{C_t})$ is orthogonal to $W_N$. This is impossible given that the hermitian form on $H^0(\omega_{C_t})$ is positive definite. Now suppose $g=2$. Then $N$ has dimension $\leq 2$ and ${\Bbb P}(N)$ has dimension $\leq 1$. For each $W \subset H^0(\omega_{C_t})$ of dimension $1$, the space $W^{\perp}$ also has dimension $1$ and hence $W \otimes W^{\perp}$ has dimension $1$. The lines $W \otimes W^{\perp}$ form a real analytic subset of ${\Bbb P} (H^0(\omega_{C_t})^{\otimes 2})$ of real dimension $2$. Since $\overline{\rho} : {\cal S} \longrightarrow \overline{\cal S}$ is finite, we deduce that the lines $\rho(W \otimes W^{\perp}) = \mu (W \otimes W^{\perp})$ form a real analytic subset of ${\Bbb P} (S^2H^0(\omega_{C_t})) = {\Bbb P}(H^0(\omega_{C_t}^{\otimes 2})) \cong {\Bbb P}^2$ of real dimension $2$. An easy computation (with coordinates) will show that this subset is not contained in any projective line in ${\Bbb P}(H^0(\omega_{C_t}^{\otimes 2}))$ and hence is not contained in ${\Bbb P}(N)$. Hence there exists $W$ such that the line $\mu (W \otimes W^{\perp})$ is not contained in $N$, in other words $\mu(W \otimes W^{\perp}) \cap N = \{ 0 \}$. \hfill \qed \vskip15pt We now consider the case $V \subset {\cal A}_g'$. As before, we first prove \begin{proposition} Suppose that $g \geq 3$. Let $V$ be a subvariety of codimension at most $g$ of ${\cal A}_g'$. Let $t$ be a point of $V_{sm}$ and let $N$ be the kernel of $\pi_a : S^2H^0(\Omega_{A_t}^1) \longrightarrow T^*_tV$. Suppose that, for any one-dimensional subvector space $W$ of $H^0(\Omega_{A_t}^1)$, the map $\pi_a \rho : W \otimes H^0(\Omega_{A_t}^1) \longrightarrow T^*_tV$ is {\em not} injective. Then $V$ has codimension exactly $g$ and there is a one-dimensional subvector space $W_N$ of $H^0(\Omega_{A_t}^1)$ such that $N = \rho (W_N \otimes H^0(\Omega_{A_t}^1))$. \label{propNppav} \end{proposition} {\em Proof :} If the map $\pi_a \rho : W \otimes H^0(\Omega_{A_t}^1) \longrightarrow T^*_tV$ is {\em not} injective, then $\rho (W \otimes H^0(\Omega_{A_t}^1)) \cap N \neq \{ 0 \}$. If this holds for every $W \subset H^0(\Omega_{A_t}^1)$ of dimension $1$, then ${\Bbb P}(N)$ has dimension $g-1$ and is contained in $\overline{\cal S} \cong S^2{\Bbb P}(H^0(\Omega^1_A))$. It follows that $V$ has codimension $g$. The rest of the argument is now analogous to the proof of part $1$ of Proposition \ref{propNcurve} with $N' = N$. \hfill \qed \vskip15pt {\bf Proof of Theorem 1 in the case of abelian varieties:} This proof is now as in the case of curves. \hfill \qed \vskip15pt {\bf Proof of Corollary 1:} Let $V$ be a complete subvariety of codimension $g-d$ ($d \geq 0$) of ${\cal A}_g$. By Theorem 1, the set $E_1(V)$ is dense in $V$. In particular, it is nonempty. Let $Y$ be an irreducible component of $E_1(V)$. Let $r$ and $s$ be integers such that for every ppav $A$ with moduli point in $Y$ there is an elliptic curve $E$, a ppav $B$ and an isogeny $\nu : E \times B \longrightarrow A$ of degree at most $r$ such that the inverse image of the principal polarization of $A$ by $\nu$ is a polarization of degree at most $s$. Let $Y'$ be an irreducible component of the variety parametrizing such quadruples $(E,B,A, \nu)$. Then $Y'$ is a finite cover of $Y$. The morphism $Y' \longrightarrow {\cal A}_1$ which to $(E,B,A, \nu)$ associates the isomorphism class of $E$ is constant since $Y'$ is complete (and irreducible) and ${\cal A}_1$ is affine. For any irreducible component $Z$ of $E_1({\cal A}_g)$, there is a finite correspondance between $Z$ and ${\cal A}_{g-1} \times {\cal A}_1$. In particular, the codimension of $Z$ in ${\cal A}_g$ is $\frac{g(g+1)}{2} - (\frac{g(g-1)}{2} +1)= g-1$. The variety $Y$ is an irreducible component of the intersection of $V$ with such a $Z$, hence there is a nonnegative integer $e_0$ such that the codimension of $Y$ in $V$ is $g-1-e_0$. So the codimension of $Y$ in ${\cal A}_g$ is $g-d +g-1-e_0 = 2g-d-1-e_0$. Since $Y'$ maps to a point in ${\cal A}_1$, its image $V_1$ in ${\cal A}_{g-1}$ by the second projection has dimension equal to the dimension of $Y$. Therefore $V_1$ has dimension $g(g+1)/2-(2g-d-1-e_0)=(g-1)g/2-(g-1-d-e_0)$, i.e., codimension $g-1-d-e_0 \leq g-1$ in ${\cal A}_{g-1}$. By Theorem 1, the set $E_1(V_1)$ is dense in $V_1$. In particular, the set $E_1(V_1)$ is nonempty. Let $Y_1$ be an irreducible component of $E_1(V_1)$ and let $Y_1'$ be the analogue of $Y'$ for $Y_1$. Then, as before, the variety $Y_1$ has codimension $g-1-d-e_0+g-2-e_1$ in ${\cal A}_{g-1}$ (for some nonnegative integer $e_1$), the variety $Y_1'$ maps to a point in ${\cal A}_1$ and its image $V_2$ in ${\cal A}_{g-2}$ has codimension $g-2-d-e_0-e_1$. Repeating the argument, we obtain $V_i$ in ${\cal A}_{g-i}$ of codimension $g-i-d-e_0 - ... - e_{i-1}$ containing $Y_i$ of codimension $g-i-d-e_0 - ... -e_{i-1}+g-i-1- e_i$ in ${\cal A}_{g-i}$. For $i=g-2$, we can repeat the argument one last time for $V_{g-2} \subset {\cal A}_2$ to obtain $Y'_{g-2}$ with image $V_{g-1}$ in ${\cal A}_1$ with codimension $1-d-e_0-...-e_{g-2}$. Since ${\cal A}_1$ is affine, the variety $V_{g-1}$ is a point and $d=e_0=...=e_{g-2} = 0$. Therefore $Y$ has codimension $2g-1$ in ${\cal A}_g$, all the varieties $Y_i$ have codimension $g-i+g-i-1=2g-2i-1$ in ${\cal A}_{g-i}$, $V$ has codimension $g$ in ${\cal A}_g$ and $V_i$ has codimension $g-i$ in ${\cal A}_{g-i}$. In particular, the first part of Corollary $1$ is proved. For each $i$, there is an irreducible subvariety $Z_i$ of $V$ which parametrizes ppav's isogenous to the product of an element of $V_i$ and $i$ fixed elliptic curves ($Z_1 = Y$) because all the maps $Y_i' \longrightarrow {\cal A}_1$ (and also $Y' \longrightarrow {\cal A}_1$) are constant. It follows from the above that $Z_i$ has the expected dimension $\frac{(g-i)(g-i+1)}{2} +i -g$. Since our choices of the $Y_i$'s (and $Y$) and hence our choices of the $Z_i$'s were arbitrary, we have proved the second part of the Corollary as well. To prove the third part, first observe that a dimension count (similar to the case of $Y$) shows that the dimension of any irreducible component $X$ of $E_q(V)$ is at least $\frac{q(q+1)}{2} +\frac{(g-q)(g-q+1)}{2} -g$. Let $X'$ be the analogue of $Y'$ for $X$. Then the images $X_q$ and $X_{g-q}$ of $X'$ by the two projections to ${\cal A}_q$ and ${\cal A}_{g-q}$ are complete subvarieties of ${\cal A}_q$ and ${\cal A}_{g-q}$ whose codimensions are at least $q$ and $g-q$ respectively by part 1 of the Corollary. So we have \[ \begin{array}{c} \frac{q(q+1)}{2} + \frac{(g-q)(g-q+1)}{2} -g \leq \hbox{dim}(X) = \hbox{dim}(X') = \hbox{dim}(X_q) + \hbox{dim}(X_{g-q}) \leq \\ \leq \frac{q(q+1)}{2} -q + \frac{(g-q)(g-q+1)}{2} -(g-q) = \frac{q(q+1)}{2} + \frac{(g-q)(g-q+1)}{2} -g \: . \end{array} \] Therefore we have equality everywhere and part 3 is proved. Now let $V'$ be the analytic closure of $E_{1,g}(V)$ in $V$. Since, by Theorem $1$, the set $E_1(V_{g-2})$ is dense in $V_{g-2}$ (which is a curve), we see that $V'$ contains $Z_{g-2}$. Since all of our choices for the $Y_i$ and $Y$ (and hence for the $Z_i$) were arbitrary, we see that $V'$ contains $E_{1,g-2}(V)$. Repeating this reasoning, we see that $V'$ contains $E_{1,i}(V)$ for all $i$, hence $V'$ contains $E_1(V)$ and $V' = V$ by Theorem $1$. \hfill \qed \vskip15pt {\bf Proof of Corollary 2:} Let $V$ be a complete codimension $g$ subvariety of $\widetilde{\cal M}_g'$ or ${\cal A}_g'$. Again, by Theorem $1$, the set $E_1(V)$ is nonempty. Let $Y \subset V$ be an irreducible component of $E_1(V)$ and define $Y'$ as in the proof of Corollary $1$. As in loc. cit. the variety $Y$ is a complete subvariety of $V$, of codimension at most $g-1$ in $V$ (codimension exactly $g-1$ by Corollary $1$ if $V \subset {\cal A}_g'$). Suppose that $V \subset \widetilde{\cal M}_g'$. Again, since $Y'$ is irreducible and complete and ${\cal A}_1$ affine, the map $Y' \longrightarrow {\cal A}_1$ is constant, hence its differential has rank $0$ everywhere. It follows from \cite{colpi} pages 172-173 that, for all $t \in Y \cap V_0$ and every one-dimensional subvector space $W$ of $H^0(\omega_{C_t})$, the map $\mu : W \otimes H^0(\omega_{C_t}) \longrightarrow T^*_t V$ is {\em not} injective. Since this noninjectivity is a closed condition and $E_1(V_0)$ is dense in $V_0$, it follows that it holds for all $t \in V_0$. Therefore, by Proposition \ref{propNcurve} and with the notation there, for all $t \in V_0$, there is a one-dimensional subvector space $W_N$ of $H^0(\omega_{C_t})$ such that $N= \mu (W_N \otimes H^0(\omega_{C_t}))$. Let us globalize the constructions in the proof of Proposition \ref{propNcurve}. Let $F_0$ be the Hodge bundle on $V_0$ and let $S^2 {\Bbb P} (F_0)$ be the quotient of the fiber product ${\Bbb P} (F_0) \times_{V_0} {\Bbb P} (F_0)$ by the involution $\sigma$ exchanging the two factors of the fiber product. Let $T^* {\cal M}_g'$ be the cotangent bundle of ${\cal M}_g'$ and let ${\cal N}_0 \subset T^* {\cal M}_g' |_{V_0}$ be the conormal bundle to $V_0$. Denote by ${\cal N}''$ (resp. ${\cal N}'$) the subcone of decomposable tensors (resp. rank $2$ symmetric tensors) in $F_0 \otimes F_0$ (resp. $S^2 F_0$) lying in the inverse image of ${\cal N}_0$ by the multiplication map $S^2F_0 \longrightarrow T^* {\cal M}_g' |_{V_0}$. Then, by Proposition \ref{propNcurve} and with the notation there, the fibers of ${\cal N}''$, ${\cal N}'$, and ${\cal N}_0$ at $t$ are respectively $W_N \otimes H^0(\omega_{C_t}) \cup H^0(\omega_{C_t}) \otimes W_N$, $\rho(W_N \otimes H^0(\omega_{C_t}))$ and $\mu(W_N \otimes H^0(\omega_{C_t}))$. Hence the morphism $m : {\cal N}' \longrightarrow {\cal N}_0$ is an isomorphism because it is an isomorphism on each fiber and the map ${\Bbb P}({\cal N}'') \longrightarrow {\Bbb P}({\cal N}_0)$ is a double cover which splits on each fiber. Since the double cover of $V_0$ parametrizing the rulings of the fibers of ${\Bbb P} (F_0) \times_{V_0} {\Bbb P} (F_0)$ over $V_0$ is split, the double cover ${\Bbb P}({\cal N}'') \longrightarrow {\Bbb P}({\cal N}') \cong {\Bbb P}({\cal N}_0)$ is globally split and hence the variety ${\Bbb P}({\cal N}'')$ is the union of two subvarieties of ${\Bbb P} (F_0) \times_{V_0} {\Bbb P} (F_0)$ exchanged by $\sigma$ and both isomorphic to ${\Bbb P}({\cal N}')$ (by the quotient morphism ${\Bbb P}(F_0) \times_{V_0} {\Bbb P}(F_0) \longrightarrow S^2 {\Bbb P}(F_0)$ ) and to ${\Bbb P}(F_0)$ by either of the two projections ${\Bbb P}(F_0) \times_{V_0} {\Bbb P}(F_0) \longrightarrow {\Bbb P}(F_0)$. In particular, the two components of ${\Bbb P}({\cal N}'')$ are projective bundles on $V_0$ and ${\cal N}''$ is the union of two vector bundles ${\cal N}''_1$ and ${\cal N}''_2$ with respective fibers $W_N \otimes H^0(\omega_{C_t})$ and $H^0(\omega_{C_t}) \otimes W_N$ at $t$. Furthermore, we have ${\cal N}''_1 \stackrel{\cong}{\longrightarrow} {\cal N}_0 \stackrel{\cong}{\longleftarrow} {\cal N}''_2$ (checked on fibers again). Since ${\Bbb P}({\cal N}''_1)$ is isomorphic to ${\Bbb P} (F_0)$, there is a line bundle ${\cal W}$ such that ${\cal N}_1'' \cong {\cal W} \otimes F_0$. So ${\cal N}_0 \cong {\cal W} \otimes F_0$. From the injection ${\cal N}_1'' \hookrightarrow F_0 \otimes F_0$ we deduce the injection ${\cal W} \hookrightarrow F_0$ which is the composition of the morphism ${\cal W} \hookrightarrow F_0 \otimes F_0 \otimes F_0^*$ (obtained from ${\cal W} \otimes F_0 \cong {\cal N}''_1 \hookrightarrow F_0 \otimes F_0$) with the morphism $F_0 \otimes (F_0 \otimes F_0^*) \stackrel{id \otimes tr}{\longrightarrow} F_0$ which is the product of the identity $F_0 \stackrel{id}{\longrightarrow} F_0$ and the trace morphism $F_0 \otimes F_0^* \cong End(F_0) \stackrel{tr}{\longrightarrow} {\cal O}_{V_0}$. For $V \subset {\cal A}_g'$ the proof is similar to (and simpler than) the above and uses Proposition \ref{propNppav} instead of Proposition \ref{propNcurve}. \hfill \qed \section{Appendix: a remark on density in positive characteristic} \label{sectpb} In this section we use the notation of the introduction to denote moduli spaces of curves and abelian varieties over an algebraically closed field $k$ of characteristic $p > 0$. The subvariety $V_0$ of ${\cal A}_g$ parametrizing ppav's of $p$-rank $0$ is a complete (connected if $g > 1$ by \cite{oortnewton} (2.6)(c)) subvariety of codimension $g$ of ${\cal A}_g$ (see \cite{normoort}, (2) in the introduction and \cite{oortsubv}, the proof of Theorem 1.1a pages 98-99). We explain below how to deduce from the results of \cite{demazure}, \cite{katz}, \cite{manin} and \cite{oortnewton} that the moduli points of non-simple abelian varieties are contained in a proper closed subset of $V_0$ when $g \geq 3$. The formal group of an abelian variety $A$ of p-rank $0$ is isogenous to a sum \[ \sum_{1 \leq i \leq r} G_{m_i,n_i} \] where $m_i$ and $n_i$ are relatively prime positive integers for each $i$, the sum $m_i+n_i$ is less than or equal to $g$ for all $i$, the formal group $G_{m_i,n_i}$ has dimension $m_i$ and its dual is $G_{n_i,m_i}$ (see \cite{manin} chapter IV, $\S 2$). The decomposition is symmetric, i.e., the group $G_{m_i,n_i}$ appears as many times as $G_{n_i,m_i}$. We call the unordered $r$-tuple $\left( (m_i,n_i) \right)_{1 \leq i \leq r}$ the formal isogeny type of the abelian variety. As in \cite{oortnewton}, we define the Symmetric Newton Polygon of $A$ to be the lower convex polygon in the plane ${\Bbb R}^2$ which starts at $(0,0)$ and ends at $(2g,g)$, whose break-points have integer coordinates and whose slopes (arranged in increasing order because of lower convexity) are $\lambda_i = \frac{n_i}{m_i+n_i}$ with multiplicity $m_i+n_i$ (i.e., on the polygon, the $x$-coordinate grows by $m_i+n_i$ and the $y$-coordinate grows by $n_i$). The polygon is symmetric in the sense that if the slope $\lambda$ appears, then the slope $1- \lambda$ appears with the same multiplicity. Following \cite{oortnewton}, we shall say that the Newton Polygon $\beta$ is above the Newton Polygon $\alpha$ if for all real numbers $x \in [0,2g], y, z \in [0,g]$ such that $(x,z) \in \beta$, $(x,y) \in \alpha$, we have $z \geq y$. We shall say that $\beta$ is strictly above $\alpha$ if $\beta$ is above $\alpha$ and $\beta \neq \alpha$. Again as in \cite{oortnewton}, for a Symmetric Newton Polygon $\alpha$, we denote by $W_{\alpha}$ the set of points in ${\cal A}_g$ corresponding to abelian varieties whose Newton Polygon is above $\alpha$. By \cite{demazure} page 91, Newton polygons go up under specialization. By \cite{katz} page 143 Theorem 2.3.1 and Corollary 2.3.2 (see also \cite{oortnewton}, 2.4), for any Newton polygon $\alpha$, the set $W_{\alpha}$ is closed in $V_0$. By \cite{oortnewton} Theorem (2.6)(a) and Remark (3.3), the abelian variety $A_0$ with moduli point the generic point of $V_0$ has formal isogeny type $((1,g-1),(g-1,1))$. Therefore, since $g \geq 3$, the abelian variety $A_0$ is simple. Let $\alpha_0$ denote the Symmetric Newton Polygon of $A_0$. The moduli point of a non-simple ppav of $p$-rank $0$ is in $W_{\beta}$ for some Symmetric Newton Polygon $\beta$ strictly above $\alpha_0$. Therefore the set of non-simple ppav's in $V_0$ is contained in $\cup_{\beta \; strictly \; above \; \alpha_0} W_{\beta}$. Since there are only a finite number of Symmetric Newton Polygons (below the line $x=2y$ and) above $\alpha_0$, we deduce that all points of $V_0$ corresponding to nonsimple abelian varieties are in a proper closed subset of $V_0$ (which is $\cup_{\beta \; strictly \; above \; \alpha_0} W_{\beta}$). Therefore $V_0$ is an example of a subvariety $V$ of codimension $g$ of ${\cal A}_g$ (for all $g \geq 3$) or of $\widetilde{\cal M}_3$ such that $E_q(V)$ is not Zariski-dense in $V$ for any $q$. \bibliographystyle{amsplain} \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}

9609

alg-geom/9609018

en

https://arxiv.org/abs/alg-geom/9609018

alg-geom/9609018

Dan Edidin

Dan Edidin and William Graham

Equivariant intersection theory

Latex 40pp. In the current version, we have adapted the language of algebraic spaces. This avoids any hypothesis on the group action. The paper is otherwise (essentially) unchanged. This version will appear in Inventiones

This is a revised and shortened version of our paper "Equivariant intersection theory" (alg-geom/9603008). In particular, the sections on Riemann-Roch and localization are omitted. They will appear in separate papers, at which time alg-geom/9603008 will become obsolete. We have intsead added a section of examples, and have include a calculation of the integral Chow ring of the mdouli stack of elliptic curves.

alg-geom

\section{Introduction} The purpose of this paper is to develop an equivariant intersection theory for actions of linear algebraic groups on schemes and more generally algebraic spaces. The theory is based on our construction of equivariant Chow groups. These are algebraic analogues of equivariant cohomology groups which have all the functorial properties of ordinary Chow groups. In addition, they enjoy many of the properties of equivariant cohomology. Previous work (\cite{Br}, \cite{Gi}, \cite{Vi}) defined equivariant Chow groups using only invariant cycles on $X$. However, there are not enough invariant cycles on $X$ to define equivariant Chow groups with nice properties, such as being a homotopy invariant, or having an intersection product when $X$ is smooth (see Section \ref{noinprod}). The definition of this paper is modeled after Borel's definition of equivariant cohomology. It is made possible by Totaro's approximation of $EG$ by open subsets of representations (\cite{To}). Consequently, an equivariant class is represented by an invariant cycle on $X \times V$, where $V$ is a representation of $G$. By enlarging the definition of equivariant cycle, we obtain a rich theory, which is closely related to other aspects of group actions on schemes and algebraic spaces. After establishing the basic properties of equivariant Chow groups, this paper is mainly devoted to the relationship between equivariant Chow groups and Chow groups of quotient algebraic spaces and stacks. If $G$ is a linear algebraic group acting on a space $X$, denote by $A^G_{i}(X)$ the $i$-th equivariant Chow group of $X$. If $G$ acts properly on $X$ then a quotient $X/G$ exists in the category of algebraic spaces (\cite{Kollar}, \cite{KM}); under some additional hypotheses (see \cite{GIT}) if $X$ is a scheme then $X/G$ is a scheme. We prove that there is an isomorphism of $A^G_{i + \mbox{\small{dim}}G}(X) \otimes {\bf Q}$ and $A_i(X/G) \otimes {\bf Q}$. If $G$ acts with trivial stabilizers this holds without tensoring with ${\bf Q}$. For an action which is not proper, there need not be a quotient in the category of algebraic spaces. However, there is always an Artin quotient stack $[X/G]$. We prove that the equivariant groups $A^G_*(X)$ depend only on the stack $[X/G]$ and not on its presentation as a quotient. If $X$ is smooth, then $A^1_G(X)$ is isomorphic to Mumford's Picard group of the stack, and the ring $A^*_G(X)$ can naturally be identified as an integral Chow ring of $[X/G]$ (Section \ref{intstack}). These results imply that equivariant Chow groups are a useful tool for computing Chow groups of quotient spaces and stacks. For example, Pandharipande (\cite{Pa1}, \cite{Pa2}) has used equivariant methods to compute Chow rings of moduli spaces of maps of projective spaces as well as the Hilbert scheme of rational normal curves. In this paper, we compute the integral Chow rings of the stacks ${\cal M}_{1,1}$ and $\overline{{\cal M}}_{1,1}$ of elliptic curves, and obtain a simple proof of Mumford's result (\cite{Mu}) that $Pic_{fun}({\cal M}_{1,1}) = {\bf Z}/12{\bf Z}$. In an appendix to this paper, Angelo Vistoli computes the Chow ring of ${\cal M}_2$, the moduli stack of smooth curves of genus 2. Equivariant Chow groups are also useful in proving results about intersection theory on quotients. It is easy to show that if $X$ is smooth then there is an intersection product on $A^G_*(X)$. The theorem on quotients therefore implies that there exists an intersection product on the rational Chow groups of a quotient of a smooth algebraic space by a proper action. The existence of such an intersection product was shown by Gillet and Vistoli, but only under the assumption that the stabilizers are reduced. This is automatic in characteristic $0$, but typically fails in characteristic $p$. The equivariant approach does not require this assumption and therefore extends the work of Gillet and Vistoli to arbitrary characteristic. Furthermore, by avoiding the use of stacks, the proof becomes much simpler. Finally, equivariant Chow groups define invariants of quotient stacks which exist in arbitrary degree, and associate to a smooth quotient stack an integral intersection ring which when tensored with ${\bf Q}$ agrees with rings defined by Gillet and Vistoli. By analogy with quotient stacks, this suggests that there should be an integer intersection ring associated to an arbitrary smooth stack, which could be nonzero in degrees higher than the dimension of the stack. We remark that besides the properties mentioned above, the equivariant Chow groups we define are compatible with other equivariant theories such as cohomology and $K$-theory. For instance, if $X$ is smooth then there is a cycle map from equivariant Chow theory to equivariant cohomology (Section \ref{cycles}). In addition, there is a map from equivariant $K$-theory to equivariant Chow groups, which is an isomorphism after completion; and there is a localization theorem for torus actions, which can be used to give an intersection theoretic proof of residue formulas of Bott and Kalkman. These topics will be treated elsewhere. \tableofcontents \medskip {\bf Acknowledgments:} We thank William Fulton, Rahul Pandharipande and Angelo Vistoli for advice and encouragement. We also benefited from discussions with Burt Totaro, Amnon Yekutieli, Robert Laterveer and Ruth Edidin. Thanks to Holger Kley for suggesting the inclusion of the cycle map to equivariant cohomology, and to J\'anos Koll\'ar for emphasizing the algebraic space point of view. \section{Definitions and basic properties} \subsection{Conventions and Notation} This paper is written in the language of algebraic spaces. It is possible to work entirely in the category of schemes, provided the mild technical hypotheses of Proposition \ref{inap} are satisfied. These hypotheses insure that if $X$ is a scheme with a $G$-action, then the mixed spaces $X_G = (X \times U)/G$ are schemes. Here $G$ acts freely on $U$ (which is an open subset of a representation of $G$) and hence on $X \times U$. In the category of algebraic spaces, quotients of free actions always exist (\cite{D-M} or \cite{Artin}; see Proposition \ref{l.algspacequotient}). By working with algebraic spaces we can therefore avoid the hypotheses of Proposition \ref{inap}. Another reason to work with algebraic spaces comes from a theorem of Koll\'ar and Keel and Mori (\cite{Kollar}, \cite{KM}), generalizing the result about free actions mentioned above. They prove that if $G$ is a linear algebraic group acting properly on a separated algebraic space $X$, then a geometric quotient $X/G$ exists as a separated algebraic space. Such quotients arise frequently in moduli problems. However, even if $X$ is a scheme, the quotient $X/G$ need not be a scheme. By developing the theory for algebraic spaces, we can apply equivariant methods to study such quotients. For these reasons, the natural category for this theory is that of algebraic spaces. In Section \ref{algspace} we explain why the intersection theory of \cite{Fulton} remains unchanged in this category. Except in Section \ref{mixed}, all schemes and algebraic spaces are assumed to be quasi-separated and of finite type over a field $k$ which can have arbitrary characteristic. A {\em smooth} space is assumed to be separated, implying that the diagonal $X \rightarrow X \times X$ is a regular embedding. For brevity, we will use the term variety to mean integral algebraic space (rather than integral {\em scheme}, as is usual). An algebraic group is always assumed to be linear. For simplicity of exposition, we will usually assume that our spaces are equidimensional. \paragraph{Group actions} If an algebraic group $G$ acts on a scheme or algebraic space $X$ then the action is said to be {\it closed} if the orbits of geometric points are closed in $X$. It is {\it proper} if the action map $G \times X \rightarrow X \times X$ is proper. If every point has an invariant neighborhood such that the action is proper in the neighborhood then we say the action is {\em locally proper}. It is {\it free} if the action map is a closed embedding. If $G \times X \stackrel{j} \rightarrow X \times X$ is a group action, we will call $j^{-1}(\Delta_X) \rightarrow \Delta_X$ the stabilizer group scheme of $X$. Its fibers are the stabilizers of the points of $X$. A group action is said to have {\em finite stabilizer} if the map $j^{-1}(\Delta_X) \rightarrow \Delta_X$ is finite. Since a linear algebraic group is affine, the fibers of $j: G \times X \stackrel{j} \rightarrow X \times X$ are finite when $G$ acts properly. Hence, a proper action has finite stabilizer. The converse need not be true (\cite[Example 0.4]{GIT}). Note that a locally proper action has finite stabilizer as well. Finally we say the action is {\em set theoretically free} or {\em has trivial stabilizer} if the stabilizer of every point is trivial. In particular this means $j^{-1}(\Delta_X) \rightarrow \Delta_X$ is an isomorphism. If the action is proper and has trivial stabilizer then it is free (Lemma \ref{l.free}). A flat, surjective, equivariant map $X \stackrel{f}\rightarrow Y$ is called a principal bundle if $G$ acts trivially on $Y$, and the map $X \times G \rightarrow X \times_Y X$ is an isomorphism. This condition is equivalent to local triviality in the \'etale topology, i.e., there is an \'etale cover $U \rightarrow Y$ such that $U \times_Y X \simeq U \times G$. As noted above, if $G$ acts set-theoretically freely on $X$ then a geometric quotient $X/G$ always exists in the category of algebraic spaces, and moreover, $X$ is a principal $G$-bundle over $X/G$ (Proposition \ref{l.algspacequotient}). \subsection{Equivariant Chow groups} \label{basicdef} Let $X$ be an $n$-dimensional algebraic space. We will denote the $i$-th equivariant Chow group of $X$ by $A^G_i(X)$, and define it as follows. Let $G$ be a $g$-dimensional algebraic group. Choose an $l$-dimensional representation $V$ of $G$ such that $V$ has an open set $U$ on which $G$ acts freely and whose complement has codimension more than $n-i$. Let $U \rightarrow U/G$ be the principal bundle quotient. Such a quotient automatically exists as an algebraic space; moreover, for any algebraic group, representations exist so that $U/G$ is a scheme -- see Lemma \ref{q.exist} of Section \ref{appendix}. The principal bundle $U \rightarrow U/G$ is Totaro's finite dimensional approximation of the classifying bundle $EG \rightarrow BG$ (see \cite{To} and \cite{E-G}). The diagonal action on $X \times U$ is also free, so there is a quotient in the category of algebraic spaces $X \times U \rightarrow (X \times U)/G$ which is a principal $G$ bundle. We will usually denote this quotient by $X_G$. (See Proposition \ref{inap} for conditions that are sufficient to imply that $X_G$ is a scheme.) \begin{def-prop} \label{keydef} Set $A_i^G(X)$ (the $i$-th equivariant Chow group) to be $A_{i+l-g}(X_G)$, where $A_*$ is the usual Chow group. This group is independent of the representation as long as $V- U$ has sufficiently high codimension. \end{def-prop} \medskip Proof. As in \cite{To}, we will use Bogomolov's double fibration argument. Let $V_1$ be another representation of dimension $k$ such that there is an open subset $U_1$ with a principal bundle quotient $U_1 \rightarrow U_1/G$ and whose complement has codimension at least $n-i$. Let $G$ act diagonally on $V \oplus V_1$. Then $V \oplus V_1$ contains an open set $W$ which has a principal bundle quotient $W/G$ and contains both $U \oplus V_1$ and $V \oplus U_1$. Thus, $A_{i+k+l-g}(X \times^G W) = A_{i+k+l-g}(X \times^G (U \oplus V_1))$ since $(X \times^G W)-(X \times^G (U \oplus V_1)$ has dimension smaller than $i+k+l-g$. On the other hand, the projection $V \oplus V_1 \rightarrow V$ makes $X \times^G (U \oplus V_1)$ a vector bundle over $X \times^G U$ with fiber $V_1$ and structure group $G$. Thus, $A_{i+k+l-g}(X \times^G (U \oplus V_1)) = A_{i+l-g}(X \times^G U)$. Likewise, $A_{i+k+l-g}(X \times^G W) =A_{i+k-g}(X \times^G U_1)$, as desired. $\Box$ \medskip {\bf Remark.} In the sequel, the notation $U \subset V$ will refer to an open set in a representation on which the action is free, and $X_G$ will mean a mixed quotient $X \times^G U$ for any representation $V$ of $G$. If we write $A_{i+l-g}(X_G)$ then $V-U$ is assumed to have codimension more than $n-i$ in $V$. (As above $n=\mbox{dim }X$, $l=\mbox{dim }V$ and $g =\mbox{dim }G$.) \paragraph{Equivariant cycles} If $Y \subset X$ is an $m$-dimensional $G$-invariant subvariety (recall that variety means integral algebraic space), then it has a $G$-equivariant fundamental class $[Y]_G \in A_m^G(X)$. More generally, if $V$ is an $l$-dimensional representation and $S \subset X \times V$ is an $m+l$-dimensional subvariety, then $S$ has a $G$-equivariant fundamental class $[S]_G \in A_m^G(X)$. Thus, unlike ordinary Chow groups, $A_i^G(X)$ can be non-zero for any $i \leq n$, including negative $i$. \begin{prop} \label{eqcycle} If $\alpha \in A_m^G(X)$, then there exists a representation $V$ such that $\alpha = \sum a_i[S_i]_G$, where $S_i$ are $m +l$ invariant subvarieties of $X \times V$, where $l$ is the dimension of $V$. \end{prop} Proof. Cycles of dimension $m+l-g$ on $X_G$ correspond exactly to invariant cycles of dimension $m+l$ on $X \times U$. Since $V-U$ has high codimension, invariant $m+l$ cycles on $X \times U$ extend uniquely to invariant $m+l$ cycles on $X \times V$. $\Box$ \medskip The representation $V$ is not unique. For example, $[X]_G$ and $[X \times V]_G$ define the same equivariant class. The projection $X \times U \rightarrow U$ induces a flat map $X_G \rightarrow U$ with fiber $X$. Restriction to a fiber gives a map $i^*:A_*^G(X) \rightarrow A_*(X)$ from equivariant Chow groups to ordinary Chow groups. The map is independent of the choice of fiber because any two points of $U/G$ are rationally equivalent. For any $G$-invariant subvariety $Y \subset X$, $i^*([Y]_G) =[Y]$. Before reading further, the reader may want to skip to Section \ref{examples} for examples. \subsection{Functorial properties} In this section all maps $f: X \rightarrow Y$ are assumed to be $G$-equivariant. Let ${\bf P}$ be one of the following properties of morphisms of schemes: proper, flat, smooth, regular embedding or l.c.i. \begin{prop} If $f: X \rightarrow Y$ has property ${\bf P}$, then the induced map $f_G: X_G \rightarrow Y_G$ also has property ${\bf P}$. \end{prop} Proof. If $X \rightarrow Y$ has property ${\bf P}$, then, by base change, so does the map $X \times U \rightarrow Y \times U$. The morphism $Y \times U \rightarrow Y_G$ is flat and surjective (hence faithfully flat), and $X \times U \simeq X_G \times_{Y_G} Y \times U$. Thus by descent \cite[Section 8.4 - 5]{SGA1}, the morphism $X_G \rightarrow Y_G$ also has property ${\bf P}$. $\Box$ \begin{prop} Equivariant Chow groups have the same functoriality as ordinary Chow groups for equivariant morphisms with property ${\bf P}$. \end{prop} Proof. If $f:X \rightarrow Y$ has property ${\bf P}$, then so does $f_G:X_G \rightarrow Y_G$. Define pushforward $f_*$ or pullback $f^*$ on equivariant Chow groups as the pullback or pushforward on the ordinary Chow groups of $X_G$ and $Y_G$. The double fibration argument shows that this is independent of choice of representation. $\Box$ \subsection{Chern classes} Let $X$ be a scheme with a $G$-action, and let $E$ be an equivariant vector bundle (in the category of algebraic spaces) Consider the quotient $E \times U \rightarrow E_G$. \begin{lemma} $E_G \rightarrow X_G$ is a vector bundle. \end{lemma} Proof. The bundle $E_G \rightarrow X_G$ is an affine bundle which is locally trivial in the \'etale topology since it becomes locally trivial after the smooth base change $X \times U \rightarrow X_G$. Also, the transition functions are linear since they are linear when pulled back to $X \times U$. Hence, $E_G \rightarrow X_G$ is a vector bundle over $X_G$. $\Box$ \begin{defn} Define equivariant Chern classes $c_j^G(E):A_i^G(X) \rightarrow A_{i-j}^G(X)$ by $c_j^G(E)\cap \alpha= c_j(E_G) \cap \alpha \in A_{i-j+l-g}(X_G)$. \end{defn} By the double fibration argument, the definition does not depend on the choice of representation. Following \cite{GIT}, we denote by $Pic^G(X)$ the group of isomorphism classes of $G$-linearized locally free sheaves on $X$. \begin{thm} \label{piciscool} Let $X$ be a locally factorial variety of dimension $n$. Then the map $\mbox{Pic}^G(X) \rightarrow A_{n-1}^G(X)$ defined by $L \mapsto (c_1(L) \cap [X]_G)$ is an isomorphism. \end{thm} Proof. We know that the map $Pic(X_G) \stackrel{\cap c_1(L_G)} \rightarrow A_{n-g+l-g}(X_G)= A_{n-1}^G(X)$ is an isomorphism. Since $X \times U \rightarrow X_G$ is a principal bundle, $Pic(X_G) = Pic^G(X \times U)$. The theorem now follows from the following lemma. \begin{lemma} \label{units} Let $X$ be a locally factorial variety with a $G$-action. (a) Let $U \stackrel{j} \hookrightarrow X$ be an invariant subvariety such that $X-U$ has codimension more than 1. Then the restriction map $j^*:Pic^G(X) \rightarrow Pic^G(U)$ is an isomorphism. (b) Let $V$ be a representation and let $\pi: X \times V \rightarrow X$ be the projection. Then $\pi^*:Pic^G(X) \rightarrow Pic^G(X \times V)$ is an isomorphism. \end{lemma} Proof of Lemma \ref{units}. We first prove (a). Injectivity: Suppose $L \in \mbox{Pic}^G(X)$ and $j^*L$ is trivial. Since $\mbox{Pic}(X) \cong \mbox{Pic}(X-Y)$, this implies that as a bundle $L$ must be trivial. A linearization of the trivial bundle on $X$ is just a homomorphism $G \rightarrow \Gamma(X,{\cal O}^*_X)$. Since $X$ is a variety and $X-U$ has codimension more than one, $\Gamma(X,{\cal O}^*_X) = \Gamma(U,{\cal O}^*_U)$. Thus a linearization of the trivial bundle is trivial on $X$ if and only if it is trivial on $U$, proving injectivity. Surjectivity: A linearization of $L$ is a homomorphism of $G$ into the group of automorphisms of $L$ over $X$. To show that $j^*$ is surjective, we must show that if $L \in \mbox{Pic}(X)$ is linearizable on $U$ then it is linearizable on $X$. But any isomorphism $\alpha: L |_{U} \rightarrow g^*L|_{U}$ extends to an isomorphism over $X$. (To see this, pick an isomorphism $\beta: L \rightarrow g^*L$ ; we know one exists because $\mbox{Pic}(X) \cong \mbox{Pic}(U)$ and $L$ and $g^*L$ are isomorphic on $X-Y$. Then $\alpha = \beta \cdot f$, where $f \in \Gamma(U,{\cal O}^*(U))$, but $\Gamma(X,{\cal O}^*_X) = \Gamma(U, {\cal O}^*_U)$, so $\alpha$ extends to $X$.) Hence $L$ is linearizable on $X$. The proof of (b) is similar. The key point is that if $X$ is a variety and $V$ is a vector space, then $\Gamma(X \times V, {\cal O}^*_{X \times V}) = \Gamma(X, {\cal O}^*_X)$, because if $R$ is an integral domain, then the units in $R[t_1, \ldots , t_n]$ are the just the units of $R$. $\Box$ \subsection{Exterior Products} If $X$ and $Y$ have $G$-actions then there are exterior products $A_i^G(X) \otimes A_j^G(X) \rightarrow A^G_{i+j}(X \times Y)$. By Proposition \ref{eqcycle} any $\alpha \in A_*^G(X)$ can be written as $\alpha = \sum a_i [S_i]_G$ where the $S_i$'s are $G$-invariant subvarieties of $X \times V$ for some representation $V$. Let $V$, $W$ be representations of $G$ of dimensions $l$ and $k$ respectively. Let $S \subset X \times V$, $T \subset Y \times W$, be $G$-invariant subvarieties of dimensions $i+l$ and $j +k$ respectively. Let $s: X \times V \times Y \times W \rightarrow X \times Y \times (V \oplus W)$ be the isomorphism $(x,v,y,w) \mapsto (x,y,v \oplus w)$. \begin{def-prop} (Exterior products) The assignment $[S]_G \times [T]_G \mapsto [s(S \times T)]_G$ induces a well defined exterior product map of equivariant Chow groups $A_i^G(X) \otimes A_j^G(Y) \rightarrow A_{i+j}^G(X \times Y)$. \end{def-prop} Proof. The proof follows from \cite[Proposition 1.10]{Fulton} and the double fibration argument used above. $\Box$ \medskip Given the above propositions, equivariant Chow groups satisfy all the formal properties of ordinary Chow groups (\cite[Chapters 1-6]{Fulton}). In particular, if $X$ is smooth, there is an intersection product on the the equivariant Chow groups $A_*^G(X)$ which makes $\oplus A_*^G(X)$ into a graded ring. \subsection{Operational Chow groups} In this section we define equivariant operational Chow groups $A^i_G(X)$, and compare them with the operational Chow groups of $X_G$. Define equivariant operational Chow groups $A^i_G(X)$ as operations $c(Y \rightarrow X): A_*^G(Y) \rightarrow A_{*-i}^G(Y)$ for every $G$-map $Y \rightarrow X$. As for ordinary operational Chow groups (\cite[Chapter 17]{Fulton}), these operations should be compatible with the operations on equivariant Chow groups defined above (pullback for l.c.i. morphisms, proper pushforward, etc.). From this definition it is clear that for any $X$, $A^*_G(X)$ has a ring structure. The ring $A^*_G(X)$ is positively graded, and $A^i_G(X)$ can be non-zero for any $i \geq 0$. Note that by construction, the equivariant Chern classes defined above are elements of the equivariant operational Chow ring. \begin{prop} \label{opsmooth} If $X$ is smooth of dimension $n$, then $A^i_G(X) \simeq A_{n-i}^G(X)$. \end{prop} \begin{cor} (of Theorem \ref{piciscool}) If $X$ is a smooth variety with a $G$-action, then the map $Pic^G(X) \rightarrow A^1_G(X)$ defined by $L \mapsto c_1(L)$ is an isomorphism. $\Box$ \end{cor} Proof of Proposition \ref{opsmooth}. Define a map $A^i_G(X) \rightarrow A_{n-i}^G(X)$ by the formula $c \mapsto c \cap [X]_G$. Define a map $A_{n-i}^G(X)\rightarrow A^i_G(X)$, $\alpha \mapsto c_{\alpha}$ as follows. If $Y \stackrel{f} \rightarrow X$ is a $G$-map, then since $X$ is smooth, the graph $\gamma_f: Y \rightarrow Y \times X$ is a $G$-map which is a regular embedding. If $\beta \in A^G_*(Y)$ set $c_\alpha \cap \beta= \gamma_f^*(\beta \times \alpha)$. \medskip Claim (cf. \cite[Proposition 17.3.1]{Fulton}): $\beta \times (c \cap [X]_G) = c \cap (\beta \times [X]_G)$. \medskip Given the claim, the formal arguments of \cite[Proposition 17.4.2]{Fulton} show that the two maps are inverses. Proof of Claim: By Proposition \ref{eqcycle} and the linearity of of equivariant operations, we may assume there is a representation $V$ so that $\beta = [S]_G$ for a $G$-invariant subvariety $S \subset Y \times V$. Since $S$ is $G$-invariant, the projection $p: S \times X \rightarrow X$ is equivariant. Thus, $$[S]_G \times (c \cap [X]_G) = p^*(c \cap [X]_G)= c \cap p^*([X]_G) = c \cap ([S \times X]_G) = c \cap ([S]_G \times [X]_G).$$ $\Box$ \medskip Let $V$ be a representation such that $V- U$ has codimension more than $k$, and set $X_G = X \times^G U$. In the remainder of the subsection we will discuss the relation between $A^k_G(X)$ and $A^k(X_G)$ (ordinary operational Chow groups). Recall \cite[Definition 18.3]{Fulton} that an envelope $\pi:\tilde{X} \rightarrow X$ is a proper map such that for any subvariety $W \subset X$ there is a subvariety $\tilde{W}$ mapping birationally to $W$ via $\pi$. In the case of group actions, we will say that $\pi: \tilde{X} \rightarrow X$ is an {\it equivariant} envelope, if $\pi$ is $G$-equivariant, and if we can take $\tilde{W}$ to be $G$-invariant for $G$-invariant $W$. If there is an open set $X^0 \subset X$ over which $\pi$ is an isomorphism, then we say $\pi: \tilde{X} \rightarrow X$ is a {\it birational} envelope. \begin{lemma} If $\pi: \tilde{X} \rightarrow X$ is an equivariant (birational) envelope, then $p: \tilde{X}_G \rightarrow X_G$ is a (birational) envelope ($\tilde{X}_G$ and $X_G$ are constructed with respect to a fixed representation $V$). Furthermore, if $X^0$ is the open set over which $\pi$ is an isomorphism (necessarily $G$-invariant), then $p$ is an isomorphism over $X^0_G = X^0 \times^G U$. \end{lemma} Proof. Fulton \cite[Lemma 18.3]{Fulton} proves that the base extension of an envelope is an envelope. Thus $\tilde{X} \times U \stackrel{\pi \times id}\rightarrow X \times U$ is an envelope. Since the projection $X \times U \rightarrow X$ is equivariant, this envelope is also equivariant. If $W \subset X_G$ is a subvariety, let $W'$ be its inverse image (via the quotient map) in $X \times U$. Let $\tilde{W'}$ be an invariant subvariety of $\tilde{X} \times U$ mapping birationally to $W'$. Since $G$ acts freely on $\tilde{X} \times U$ it acts freely on $\tilde{W'}$, and $\tilde{W} = \tilde{W'}/G$ is a subvariety of $\tilde{X}_G$ mapping birationally to $W$. This shows that $\tilde{X}_G \rightarrow X_G$ is an envelope; it is clear that the induced map $\tilde{X}_G \rightarrow \tilde{X}$ is an isomorphism over $X_0^G$. $\Box$ \medskip Suppose $\tilde{X} \stackrel{\pi}\rightarrow X$ is an equivariant envelope which is an isomorphism over $U$. Let $\{S_i\}$ be the irreducible components of $S= X -X^0$, and let $E_i = \pi^{-1}(S_i)$. Then $\{S_{i G}\}$ are the irreducible components of $X_G - X^0_G$ and $E_{i G} = \pi^{-1}(S_{i G})$. \begin{thm} If $X$ has an equivariant smooth envelope $\pi: \tilde{X} \rightarrow X$ such that there is an open $X^0 \subset X$ over which $\pi$ is an isomorphism, and $V-U$ has codimension more than $k$, then $A^k_G(X) = A^k(X_G)$. \end{thm} Proof. If $\pi: \tilde{X} \rightarrow X$ is an equivariant non-singular envelope, then $p: \tilde{X}_G \rightarrow X_G$ is also an envelope and $\tilde{X}_G$ is non-singular. Thus, by \cite[Lemma 1.2]{Kimura} $p^*:A^*(X_G) \rightarrow A^*(\tilde{X}_G)$ is injective. The image of $p^*$ is described inductively in \cite[Theorem 3.1]{Kimura}. A class $\tilde{c} \in A^*(\tilde{X}_G)$ equals $p^*c$ if and only if for each $E_{i G}$ , $\tilde{c}_{| E_{i G}} = p^*c_i$ where $c_i \in A^*(E_i)$. This description follows from formal properties of operational Chow groups, and the exact sequence \cite[Theorem 2.3]{Kimura} $$A^*(X_G) \stackrel{p}\rightarrow A^*(\tilde{X}_G) \stackrel{p_1^* - p_2^*} \rightarrow A^*(\tilde{X}_G \times_{X_G} \tilde{X}_G)$$ where $p_1$ and $p_2$ are the two projections from $\tilde{X}_G \times_{X_G} \tilde{X}_G$. By Proposition \ref{opsmooth} above, we know that $A^k_G(\tilde{X}) = A^k(\tilde{X}_G)$. We will show that $A^k_G(X)$ and $A^k(X_G)$ have the same image in $A^k(\tilde{X}_G)$. By Noetherian induction we may assume that $A^k_G(S_i) = A^k((S_{i})_G)$. To prove the theorem, it suffices to show that there is also an exact sequence of equivariant operational Chow groups $$0 \rightarrow A^*_G(X) \stackrel{\pi^*}\rightarrow A^*_G(\tilde{X}) \stackrel{p_{1}^* -p_{2}^*}\rightarrow A^*_G(\tilde{X} \times_X \tilde{X})$$ This can be checked by working with the action of $A^*_G(X)$ on a fixed Chow group $A_{i}(X_G)$ and arguing as in Kimura's paper. $\Box$ \begin{cor} If $X$ is separated and has an equivariant resolution of singularities (in particular if the characteristic is 0), and $V-U$ has codimension more than $k$, then $A^k_G(X) = A^k(X_G).$ \end{cor} Proof (cf. \cite[Remark 3.2]{Kimura}). We must show the existence of an equivariant envelope $\pi:\tilde{X} \rightarrow X$. By equivariant resolution of singularities, there is a resolution $\pi_1:\tilde{X_1} \rightarrow X$ such that $\pi_1$ is an isomorphism outside some invariant subscheme $S \subset X$. By Noetherian induction, we may assume that we have constructed an equivariant envelope $\tilde{S} \rightarrow S$. Now set $\tilde{X} = \tilde{X_1} \cup \tilde{S}$. $\Box$ \medskip \subsection{Equivariant higher Chow groups} \label{higheq} In this section assume that $X$ is quasi-projective (a quasi-projective algebraic space is a scheme (\cite[p.140]{Knutson}). Bloch (\cite{Bl}) defined higher Chow groups $A^i(X,m)$ as $H_m(Z^i(X,\cdot))$ where $Z^i(X,\cdot)$ is a complex whose $k$-th term is the group of cycles of codimension $i$ in $X \times \Delta^k$ which intersect the faces properly. Since we prefer to think in terms of dimension rather than codimension we will define $A_p(X,m)$ as $H_m(Z_p(X,\cdot))$, where $Z_p(X,k)$ is the group of cycles of dimension $p+k$ in $X \times \Delta^k$ intersecting the faces properly. When $X$ is equidimensional of dimension $n$, then $A_p(X,m) = A^{n-p}(X,m)$. If $Y \subset X$ is closed, there is a localization long exact sequence. The advantage of indexing by dimension rather than codimension is that the sequence exists without assuming that $Y$ is equidimensional. \begin{lemma} Let $X$ be equidimensional, and let $Y \subset X$ be closed, then there is a long exact sequence of higher Chow groups $$\ldots \rightarrow A_p(Y,k) \rightarrow A_p(X,k) \rightarrow A_p(X-Y,k) \rightarrow \\ \ldots \rightarrow A_p(Y) \rightarrow A_p(X) \rightarrow A_p(X-Y) \rightarrow 0$$ (there is no requirement that $Y$ be equidimensional). \end{lemma} Proof. This is a simple consequence of the localization theorem of \cite{Bl}. We must show that the complex $Z_p(X - Y,\cdot)$ is quasi-isomorphic to the complex $\frac{Z_p(X,\cdot)}{Z_p(Y,\cdot)}$. By induction on the number of components, it suffices to verify the quasi-isomorphism when $Y$ is the union of two irreducible components $Y_1$ and $Y_2$. By the original localization theorem, $Z_p(X-(Y_1 \cup Y_2),\cdot) \simeq \frac{Z_p(X-Y_1,\cdot)}{Z_p(Y_2-(Y_1 \cap Y_2),\cdot)}$ and $Z_p(X-Y_1, \cdot) \simeq \frac{Z_p(X,\cdot)}{Z_p(Y_1)}$ (here $\simeq$ denotes quasi-isomorphism). By induction on dimension, we can assume that the lemma holds for schemes of smaller dimension, so $Z_p((Y_2 - (Y_1 \cap Y_2),\cdot) \simeq \frac{Z_p(Y_2,\cdot)}{Z_p(Y_1 \cap Y_2)}$. Finally note that $\frac{Z_p(Y_2,\cdot)}{Z_p(Y_1 \cap Y_2,\cdot)} \simeq \frac{Z_p(Y_1 \cup Y_2,\cdot)}{Z_p(Y_1,\cdot)}$. Combining all our quasi-isomorphisms we have $$Z_p(X-(Y_1 \cup Y_2), \cdot) \simeq \frac{\frac{Z_p(X,\cdot)} {Z_p(Y_1,\cdot)}}{\frac{Z_p(Y_1 \cup Y_2),\cdot}{Z_p(Y_1,\cdot)}} \simeq \frac{Z_p(X,\cdot)}{Z_p(Y_1 \cup Y_2,\cdot)}$$ as desired. $\Box$ \medskip If $X$ is quasi-projective with a linearized $G$-action, we can define equivariant higher Chow groups $A_{i}^G(X,m)$ as $A_{i+l-g}(X_G,m)$, where $X_G$ is formed from an $l$-dimensional representation $V$ such that $V-U$ has high codimension (note that $X_G$ is quasi-projective, by \cite[Prop. 7.1]{GIT}). The homotopy property of higher Chow groups shows that $A_{i}^G(X,m)$ is well defined. {\bf Warning.} Since the homotopy property of higher Chow groups has only been proved for quasi-projective varieties, our definition of higher equivariant Chow groups is only valid for quasi-projective varieties with a linearized action. However, if $G$ is connected and $X$ is quasi-projective and normal, then by Sumihiro's equivariant completion \cite{Sumihiro} and \cite[Corollary 1.6]{GIT}, any action is linearizable. \medskip One reason for constructing equivariant higher Chow groups is to obtain a localization exact sequence: \begin{prop} Let $X$ be equidimensional and quasi-projective with a linearized $G$-action, and let $Y \subset X$ be an invariant subscheme. There is a long exact sequence of higher equivariant Chow groups $$\ldots \rightarrow A_p^G(Y,k) \rightarrow A_p^G(X,k) \rightarrow A_p^G(X-Y,k) \rightarrow \\ \ldots \rightarrow A_p^G(Y) \rightarrow A_p^G(X) \rightarrow A_p^G(X-Y) \rightarrow 0. $$ $\Box$ \end{prop} \subsection{Cycle Maps} \label{cycles} If $X$ is a complex algebraic variety with the action of a complex algebraic group, then we can define equivariant Borel-Moore homology $H_{BM, i}^G(X)$ as $H_{BM,i+2l-2g}(X_G)$ for $X_G = X \times^G U$. As for Chow groups, the definition is independent of the representation, as long as $V -U$ has sufficiently high codimension, and we obtain a cycle map $$cl:A^G_i(X) \rightarrow H_{BM,2i}^G(X)$$ compatible with the usual operations on equivariant Chow groups (cf. \cite[Chapter 19]{Fulton}). Let $EG \rightarrow BG$ be the classifying bundle. The open subsets $U \subset V$ are topological approximations to $EG$. For, if $\phi$ is a map of the $j$-sphere $S^j$ to $U$, we may view $\phi$ as a map $S^j \rightarrow V$. Extend $\phi$ to a map $B^{j+1} \rightarrow V$. We may assume that the extended map is smooth and transversal to $V-U$. If $j+1 < 2i$, where $i$ is the complex codimension of $V - U$, then transversality implies that the extended map does not intersect $V-U$. Thus we have extended $\phi$ to a map $B^{j+1} \rightarrow U$. Hence $\pi_j(U) = 0$ for $j < 2i-1$. Note that $H_{BM,i}^G(X)$ is not the same as $H_i(X \times^G EG)$, However, if $X$ is smooth, then $X_G$ is also smooth, and $H_{BM,i}(X_G)$ is dual to $H^{2n-i}(X_G)=H^{2n-i}(X \times^G EG)=H^{2n-i}_G(X)$, where $n$ is the complex dimension of $X$. In this case we can interpret the cycle map as giving a map $$cl: A^i_G(X) \rightarrow H^{2i}_G(X).$$ If $X$ is compact, and the open sets $U \subset V$ can be chosen so that $U/G$ is projective, then Borel-Moore homology of $X_G$ coincides with ordinary homology, so $H^G_{BM*}(X)$ can be calculated with a compact model. In general, however, $U/G$ is only quasi-projective. If $G$ is finite, then $U/G$ is never projective. If $G$ is a torus, then $U/G$ can be taken to be a product of projective spaces. If $G = GL_n$, then $U/G$ can be taken to be a Grassmannian (see the example in Section \ref{s.subset}) If $G$ is semisimple, then $U/G$ cannot be chosen projective, for then the hyperplane class would be a nontorsion element in $A^1_G$, but by Proposition \ref{conred}, $A^*_G \otimes {\bf Q} \cong S(\hat{T})^W \otimes {\bf Q}$, which has no elements of degree 1. Nevertheless for semisimple (or reductive) groups we can obtain a cycle map $$cl: A_*^G(X)_{{\bf Q}} \rightarrow H_{BM*}^T(X;{\bf Q})^W$$ by identifying $A_*^G(X) \otimes {\bf Q}$ with $A_*^T(X)^W \otimes {\bf Q}$ and $H_{BM*}^G(X;{\bf Q})$ with $H_{BM*}^T(X;{\bf Q})^W$; if $X$ is compact then the last group can be calculated with a compact model. \section{Examples} \label{examples} In this section we calculate some examples of equivariant Chow groups, particularly for connected groups. The point of this is to show that computing {\it equivariant} Chow groups is no more difficult than computing ordinary Chow groups, and in the case of quotients, equivariant Chow theory gives a way of computing ordinary Chow groups. Moreover, since many of the varieties with computable Chow groups (such as $G/P$'s, Schubert varieties, spherical varieties, etc.) come with group actions, it is natural to study their equivariant Chow groups. \subsection{Representations and subsets} \label{s.subset} For some groups there is a convenient choice of representations and subsets. In the simplest case, if $G= {\bf G}_m$ then we can take $V$ to an $l$-dimensional representation with all weights $-1$, $U = V - \{0\}$, and $U/G = {\bf P}^{l-1}$. If $G=T$ is a (split) torus of rank $n$, then we can take $U = \oplus_1^n (V - \{0\})$ and $U/T = \Pi_1^n {\bf P}^{l-1}$. If $G = GL_n$, take $V$ to be the vector space of $n \times p$ matrices ($p>n$), with $GL_n$ acting by left multiplication, and let $U$ be the subset of matrices of maximal rank. Then $U/G$ is the Grassmannian $Gr(n,p)$. Likewise, if $G=SL(n)$, then $U/G$ fibers over $Gr(n,p)$ as the complement of the 0-section in the line bundle $\mbox{det}(S) \rightarrow Gr(n,p)$, where $S$ is the tautological rank $n$ subbundle on $Gr(n,p)$. \subsection{Equivariant Chow rings of points} The equivariant Chow ring of a point was introduced in \cite{To}. If $G$ is connected reductive, then $A^*_G \otimes {\bf Q}$ and (if $G$ is special) $A^*_G$ are computed in \cite{E-G}. The computation given there does not use a particular choice of representations and subsets. The result is that $A^*_G \otimes {\bf Q} \cong S(\hat{T})^W \otimes {\bf Q}$, where $T$ is a maximal torus of $G$, $S(\hat{T})$ the symmetric algebra on the group of characters $\hat{T}$, and $W$ the Weyl group. If $G$ is special this result holds without tensoring with ${\bf Q}$. \begin{prop} \label{conred} Let $G$ be a connected reductive group with split maximal torus $T$ and Weyl group $W$. Then $A_*^G(X) \otimes {\bf Q} = A_*^T(X)^W \otimes {\bf Q}$. If $G$ is special the isomorphism holds with integer coefficients. \end{prop} Proof. If $G$ acts freely on $U$, then so does $T$. Thus for a sufficiently large representation $V$, $A_{i}^T(X) = A_{i+l -t}((X \times U)/T)$ and $A_i^G(X) = A_{i+l-g}((X \times U)/G)$ (here $l$ is the dimension of $V$, $t$ the dimension of $T$ and $g$ the dimension of $G$). On the other hand, $(X \times U)/T$ is a $G/T$ bundle over $(X \times U)/G$. Thus $A_{k}((X \times U/T)) \otimes {\bf Q} =A_{k+g-t}((X \times U)/G)^W \otimes {\bf Q}$ and if $G$ is special, then the equality holds integrally (\cite{E-G}) and the proposition follows. $\Box$ \medskip For $G$ equal to ${\bf G}_m$ or $GL(n)$ the choice of representations in Section \ref{s.subset} makes it easy to compute $A^*_G$ directly, without appealing to the result of \cite{E-G}. If $l > i$, then $A^i_{{\bf G}_m} = A^i({\bf P}^{l-1}) = {\bf Z} \cdot t^i$, where $t= c_1({\cal O}(1))$. Thus, $A^*_{{\bf G}_m}(pt) = {\bf Z}[t]$. More generally, for a torus of rank $n$, $A^*_{T}(pt) = {\bf Z}[t_1, \ldots , t_n]$. Likewise, for $p$ sufficiently large, $A^*_{GL_{n}}(pt) = A^i(Gr(n,p))$ is the free abelian group of homogeneous symmetric polynomials of degree $i$ in $n$-variables (polynomials in the Chern classes of the rank $n$ tautological subbundle). Thus $A^*_{GL_{n}}(pt) = {\bf Z}[c_1, \ldots , c_n]$ where $c_i$ has degree $i$. Likewise $A^*_{SL_n}(pt) = {\bf Z}[c_2, \ldots c_n]$. There is a map $A^*_{GL_{n}} \rightarrow A^*_{T}$, where $T$ is a maximal torus. This is a special case of a general construction: if $G$ acts on $X$ and $H \subset G$ is a subgroup, then there is a pullback $A_*^G(X) \rightarrow A_*^H(X)$. This map is induced by pulling back along the flat map $X_H = X \times^H U \rightarrow X_G= X \times^G U$. We can identify the map $A^*_{GL_{n}} \rightarrow A^*_{T}$ concretely as the map ${\bf Z}[c_1, \ldots c_n] \rightarrow {\bf Z}[t_1, \ldots , t_n]$ given by $c_i \mapsto e_i(t_1, \ldots , t_n)$ (here $e_i$ denotes the $i$-th symmetric polynomial), so $A^*_{GL_n}(pt)$ can be identified with the subring of symmetric polynomials in ${\bf Z}[t_1, \ldots , t_n]$. This is a special case of the result of Proposition \ref{conred}. More elaborate computations are required to compute $A^*_G$ for other reductive groups (if one does not tensor with ${\bf Q}$). The cases $G = O(n)$ and $G = SO(2n+1)$ have been worked out by Pandharipande \cite{Pa2} and Totaro. There is a conjectural answer for $G = SO(2n)$, verified by Pandharipande for $n=2$. \paragraph{Equivariant Chern classes over a point} An equivariant vector bundle over a point is a representation of $G$. If $T = {\bf G}_m$, equivariant line bundles correspond to the 1-dimensional representation $L_a$ where $T$ acts by weight $a$. If (as above) we approximate $BT$ by $(V - \{0 \})/T = {\bf P}(V)$, where $T$ acts on $V$ with all weights $-1$, then the tautological subbundle corresponds to the representation $L_{-1}$. Hence $c_T(L_a) = at$. \subsection{Equivariant Chow rings of ${\bf P}^n$} We calculate $A^*_T({\bf P}^n)$, where $T = {\bf G}_m$ acts diagonally on ${\bf P}^n$ with weights $a_0, \ldots , a_n$ (i.e., $g \cdot (x_0:x_1 \ldots :x_n) = (g^{a_0}x_0: g^{a_1}x_1 : \ldots : g^{a_n}x_n)$). In this case, $X_T \rightarrow U/T$ is the ${\bf P}^n$ bundle $$ {\bf P}({\cal O}(a_0) \oplus \ldots \oplus {\cal O}(a_n)) \rightarrow {\bf P}^{l-1}. $$ Thus $A^*(X_T) = A^*({\bf P}^{l-1})[h]/(p(h,t))$ where $t$ is the generator for $A^1({\bf P}^{l-1})$ and $$ p(h,t) = \sum_{i = 0}^{n} h^i e_i(a_0t, \ldots , a_n t). $$ Letting the dimension of the representation go to infinity we see that $A^*_T({\bf P}^n) = {\bf Z}[t,h]/p(h,t)$. Note that $A^*_T({\bf P}^n)$ is a module of rank $n+1$ over the $T$-equivariant Chow ring of a point. Assume that the weights of the $T$-action are distinct. Then the fixed point set $({\bf P}^n)^T$ consists of the points $p_0, \ldots , p_n,$ where $p_r \in {\bf P}^n$ is the point which is non-zero only in the $r$-th coordinate. The inclusion $i_r: p_r \hookrightarrow {\bf P}^n$ is a regular embedding. The equivariant normal bundle is the equivariant vector bundle over the point $p_r$ corresponding to the representation $V_r = \oplus_{s \neq r} L_{a_s}$. The equivariant pushforward $i_{r*}$ is readily calculated. For example, if $n = 1$ then $i_{r*}$ takes $\alpha$ to $\alpha \cdot (h + a_s t)$ (where $s \neq r$). Hence the map $i_*: A_*^T(({\bf P}^1)^T) \rightarrow A^*_T({\bf P}^1)$ becomes an isomorphism after inverting $t$ (and tensoring with ${\bf Q}$ if $a_0$ and $a_1$ are not relatively prime). This is a special case of the localization theorem for torus actions (\cite{EG38}). We remark that the calculation of $A^*_T({\bf P}^n)$ can be viewed as a special case of the projective bundle theorem for equivariant Chow groups (which follows from the projective bundle theorem for ordinary Chow groups), since ${\bf P}^n$ is a projective bundle over a point, which is trivial but not equivariantly trivial. \subsection{Computing Chow rings of quotients} By Theorem \ref{quotient} the rational Chow groups of the quotient of a variety by a group acting with finite stabilizers can be identified with the equivariant Chow groups of the original variety. If the original variety is smooth then the rational Chow groups of the quotient inherit a canonical ring structure (Theorem \ref{quotient.cor}). For example, let $W$ be a representation of a split torus $T$ and let $X \subset W$ be the open set on which $T$ acts properly. Since representations of $T$ split into a direct sum of invariant lines, it easy to show that $W - X$ is a finite union of invariant linear subspaces $L_1, \ldots ,L_r$. When $T = {\bf G}_m$ and $X = W-\{0\}$ then the quotient is a twisted projective space. Let ${\cal R} \subset \hat{T}$ be the set of weights of $T$ on $V$. If $L \subset V$ is an invariant linear subspace, set $\chi_L = \Pi_{\chi \in {\cal R}} \chi^{d(L,{\chi})}$, where $d(L,{\chi})$ is the dimension of the $\chi$-weight space of $V/L$. \begin{prop} There is a ring isomorphism $A^*(X/T)_{{\bf Q}} \simeq S(\hat{T})/(\chi_{L_1}, \ldots \chi_{L_r})$. \end{prop} Proof. By Theorem \ref{quotient}, $A^*(X/T)_{{\bf Q}} = A^T(X)_{{\bf Q}}$. Since $W-X$ is a union of linear subspaces $L_1, \ldots L_r$, we have an exact sequence (ignoring the shifts in degrees) $$ \oplus A^*_T(L_i) \stackrel{i_*} \rightarrow A^*_T(V) \rightarrow A^*_T(U) \rightarrow 0 . $$ Identifying $A^*_T(W)$ with $A^*_T(pt) = S(\hat{T})$, we see that $A^T(U) = S(\hat{T})/im(i_*)$. Since each invariant linear subspace is the intersection of invariant hypersurfaces, the image of $A^*_T(L_i)$ in $A^*_T(W)=S(\hat{T})$ is the ideal $\chi^{c(L,\chi)}S(\hat{T})$. $\Box$ \medskip {\bf Remark}. The preceding proposition is a simpler presentation of a computation in \cite[Section 4]{Vi2}. In addition, we do not need to assume that the stabilizers are reduced, so there is no restriction on the characteristic. In \cite{E-S1}, Ellingsrud and Str{\o}mme considered representations $V$ of $G$ for which all $G$-semistable points are stable for a maximal torus of $G$, and $G$ acts freely on the set $V^s(G) $ of $G$-stable points. In this case they gave a presentation for $A^*(V^s/G)$. Using Theorem \ref{quotient}, it can be shown that their presentation is valid (with ${\bf Q}$ coefficients) even if $G$ doesn't act freely on $V^s(G)$. In a more complicated example, Pandharipande (\cite{Pa1}) used equivariant Chow groups to compute the rational Chow ring of the moduli space, $M_{{\bf P}^r}({\bf P}^k,d)$ of maps ${\bf P}^k \rightarrow {\bf P}^r$ of degree $d$. This moduli space is the quotient $U(k,r,d)/GL(k+1)$, where $U(k,r,d) \subset \oplus^r_0 H^0({\bf P}^k, {\cal O}_{{\bf P}^k}(d))$ is the open set parameterizing base-point free $r+1$-tuples of polynomials of degree $d$ on ${\bf P}^k$. His result is that for any $d$, $A^*(M_{{\bf P}^r}({\bf P}^k,d))_{{\bf Q}}$ is canonically isomorphic to the rational Chow ring of the Grassmannian $Gr({\bf P}^k, {\bf P}^r)$. \subsection{Intersecting equivariant cycles, an example} \label{noinprod} Let $X = k^3 - \{ 0 \}$ and let $T$ denote the $1$-dimensional torus acting with weights 1,2,2. We let $A^i[X/T]$ denote the group of invariant cycles on $X$ of codimension $i$, modulo the relation $\mbox{div}(f) = 0$, where $f$ is a $T$-invariant rational function on an invariant subvariety of $X$. We will show that there is no (reasonable) intersection product, with integer coefficients, on $A^*[X/T]$. We will also compare $A^*[X/T]$ to $A^*_T(X)$, which does have an integral intersection product. Clearly $A^0[X/T] = {\bf Z} \cdot [X]$. An invariant codimension $1$ subvariety is the zero set of a weighted homogeneous polynomials $f(x,y,z)$, where $x$ has weight $1$ and $y$ and $z$ have weights $2$. If $f$ has weight $n$ then the cycle defined by $f$ is equivalent to the cycle $n \cdot p$, where $p$ is the class of the plane $x = 0$. Thus $A^1[X/T] = {\bf Z} \cdot p$. The invariant codimension $2$ subvarieties are just the $T$-orbits. If we let $l$ denote the class of the line $x=y=0$, then we see that the orbit $T \cdot (a,b,c)$ is equivalent to $l$ if $a=0$, and to $2l$ otherwise. Thus $A^2[X/T] = {\bf Z} \cdot l$. Finally, $A^i[X/T] = 0$ for $i \geq 3$. If $Z_1$ and $Z_2$ are the cycles defined by $x=0$ and $y=0$, then $Z_1$ and $Z_2$ intersect transversely in the line $x=y=0$. Thus, in a ``reasonable'' intersection product we would want $2 p^2 = [Z_1] \cdot [Z_2] = l$ or $p \cdot p = \frac{1}{2}l$. But $\frac{1}{2}l$ is not an integral class in $A^2[X/T]$, so such an intersection product does not exist. Now consider the equivariant groups $A^*_T(X)$. We model $BT$ by ${\bf P}^N$, where $N$ is arbitrarily large; then the mixed space $X_T$ corresponds to the complement of the $0$-section in the vector bundle ${\cal O}(1) \oplus {\cal O}(2) \oplus {\cal O}(2)$. Thus $A^*_T(X) = {\bf Z}[t] / (4 t^3)$. Each invariant cycle on $X$ defines an element of $A^*_T(X)$, so there is a natural map $A^*[X/T] \rightarrow A^*_T(X)$. This map takes $p$ to $t$ and $l$ to $2 t^2$. The equivariant theory includes the extra cycle, $t^2$, necessary to define an integral intersection product. We can view elements of $A^*_T(X)$ as cycles on $X \times V$, where $V$ is a representation of $T$ with all weights $1$. The class $t^2$ is represented by the cycle $x = 0, \phi = 0$ where $\phi$ is any linear function on $V$. \section{Intersection theory on quotients} \label{qint} One of the most important properties of equivariant Chow groups is that they compute the rational Chow groups of a quotient by a group acting with finite stabilizer. They can also be used to show that the rational Chow groups of a moduli space which is a quotient (by a group) of a smooth algebraic space have an intersection product -- even when there are infinitesimal automorphisms. \subsection{Chow groups of quotients} Let $G$ be a $g$-dimensional group acting on a algebraic space $X$. Following Vistoli, we define a {\em quotient} $X \stackrel{\pi} \rightarrow Y$ to be a map which satisfies the following properties (cf. \cite[Definition 0.6(i - iii)]{GIT}): $\pi$ commutes with the action of $G$, the geometric fibers of $\pi$ are the orbits of the geometric points of $X$, and $\pi$ is submersive, i.e., $U \subset Y$ is open if and only if $\pi^{-1}(U)$ is. (This is called a topological quotient in \cite[Definition 2.7]{Kollar}.) Unlike what Mumford calls a geometric quotient, we do not require that ${\cal O}_Y = \pi_*({\cal O} _X)^G$. The advantage of this definition is that it is preserved under base change. In characteristic 0 there are no inseparable extensions, so our quotient is in fact a geometric quotient (\cite[Prop. 0.2]{GIT}). For proper actions, a geometric quotient is unique (\cite{GIT}, \cite{Kollar}). If $L \subset K$ is an inseparable extension and $G_K$ is a group defined over $K$, then both $\mbox{Spec } L$ and $\mbox{Spec } K$ are quotients of $G_K$ by $G_K$. This example shows that quotients need not be unique in characteristic $p$. \begin{prop} \label{p.quotient} (a) If $G$ acts with trivial stabilizer on an algebraic space $X$ and $X \rightarrow Y$ is the principal bundle quotient then $A_{i+g}^G(X) = A_{i}(Y)$. (b) If in addition $X$ is quasi-projective and $G$ acts linearly, then $A_{i+g}^G(X,m) = A_i^G(X,m)$ for all $m \geq 0$. \end{prop} Proof. If the stabilizers are trivial, then $(V \times X)/G$ is a vector bundle over the quotient $Y$. Thus $X_G$ is an open set in this bundle with arbitrarily high codimension, and the proposition follows from homotopy properties of (higher) Chow groups. $\Box$ \medskip \begin{thm} \label{quotient} (a) Let $X$ be an algebraic space with a (locally) proper $G$-action and let $X \stackrel{\pi} \rightarrow Y$ be a quotient. Then $$A_{i+g}^G(X) \otimes {\bf Q} \simeq A_i(Y) \otimes {\bf Q}.$$ (b) If in addition $X$ is quasi-projective with a linearized $G$-action, and the quotient $Y$ is quasi-projective, then $$A_{i+g}^G(X,m) \otimes {\bf Q} \simeq A_{i}^G(Y,m) \otimes {\bf Q}$$ \end{thm} \begin{thm} \label{quotient.cor} With the same hypotheses as in Theorem \ref{quotient}(a), there is an isomorphism of operational Chow rings $$\pi^*:A^*(Y)_{{\bf Q}} \stackrel{\simeq} \rightarrow A^*_G(X)_{{\bf Q}}.$$ Moreover if $X$ is smooth, then the map $A^*(Y)_{{\bf Q}} \stackrel{\cap [Y]} \rightarrow A_*(Y)_{{\bf Q}}$ is an isomorphism. In particular, if $X$ is smooth, the rational Chow groups of the quotient space $Y=X/G$ have a ring structure, which is independent of the presentation of $Y$ as a quotient of $X$ by $G$. \end{thm} {\bf Remarks.} (1) By \cite{Kollar} or \cite{KM}, if $G$ acts locally properly on a (locally separated) algebraic space, then a (locally separated) geometric quotient $X \stackrel{\pi} \rightarrow Y$ always exists in the category of algebraic spaces. Moreover, the result of \cite{KM} holds under the weaker hypothesis that $G$ acts on $X$ with finite stabilizer. However, the quotient $Y$ need not be locally separated. We expect that Theorems \ref{quotient} and \ref{quotient.cor} still hold in this case, but our proof does not go through. (2) The hypotheses in Theorem \ref{quotient}(b) are purely technical. They are necessary because the localization theorem for higher Chow groups has only been proved for quasi-projective schemes. If the localization theorem were proved for algebraic spaces, Theorem \ref{quotient}(b) would hold in this case. (3) Checking that an action is proper can be difficult. If $G$ is reductive, then \cite[Proposition 0.8 and Converse 1.13]{GIT} give criteria for properness when $G$ is reductive. In particular if $X$ is contained in the set of stable points for some linearized action of $G$ on $X$ then the action is proper. If $X \rightarrow Y$ is affine and $Y$ is quasi-projective the action is also proper. Not surprisingly, checking that an action is locally proper is slightly easier. In particular if $G$ is a reductive and $X \rightarrow Y$ is a geometric quotient such that $Y$ is a scheme, then the action is locally proper if $X$ can be covered by invariant affine open sets. (4) In practice, many interesting varieties arise as quotients of smooth varieties by connected algebraic groups which act with finite stabilizers. Examples include simplicial toric varieties and various moduli spaces such as curves, vector bundles, stable maps, etc. Theorem \ref{quotient.cor} provides a tool to compute Chow groups of such varieties (see Section \ref{examples} for some examples). (5) As noted above, Theorem \ref{quotient.cor} shows that there exists an intersection product on the rational Chow group of quotients of smooth varieties and algebraic spaces. There is a long history of work on this problem. In characteristic 0, Mumford \cite{Mu} proved the existence of an intersection product on the rational Chow groups of $\overline{{\cal M}}_g$, the moduli space of stable curves. Gillet \cite{Gi} and Vistoli \cite {Vi} constructed intersection products on quotients in arbitrary characteristic, provided that the stabilizers of geometric points are reduced. (In characteristic 0 this condition is automatic, but it can fail in positive characteristic.) In characteristic 0, Gillet (\cite[Thm 9.3]{Gi}) showed that his product on $\overline{{\cal M}}_g$ agreed with Mumford's, and in \cite[Lemma 1.1]{Ed} it was shown that Vistoli's product also agreed with Mumford's. If the stabilizers are reduced, we show that our product agrees with Gillet's and Vistoli's (Proposition \ref{triprod}). Hence, Gillet's product and Vistoli's agree for quotient stacks, answering a question in \cite{Vi}. Moreover, Theorem \ref{quotient.cor} does not require that the stabilizers be reduced and is therefore true in arbitrary characteristic, answering \cite[Conjecture 6.6]{Vi} affirmatively for moduli spaces of quotient stacks. (6) Equivariant intersection theory gives a nice way of working with cycles on a singular moduli space ${\cal M}$ which is a quotient $X/G$ of a smooth variety by a group acting with finite stabilizer. Given a subvariety $W \subset {\cal M}$ and a family $Y \stackrel{p}\rightarrow B$ of schemes parameterized by ${\cal M}$, there is a map $B \stackrel{f} \rightarrow {\cal M}$. We wish to define a class $f^*([W]) \in A_*B$ corresponding to how the image of $B$ intersects $W$. This can be done (after tensoring with ${\bf Q}$) using equivariant theory. By Theorem \ref{quotient}, there is an isomorphism $A_*({\cal M})_{{\bf Q}}= A_*^G(X)$ which takes $[W]$ to the equivariant class $w= \frac{e_W}{i_W}[f^{-1}W]_G$. Let $E \rightarrow B$ be the principal $G$-bundle $B \times_{[X/G]} X$ (the fiber product is a scheme, although the product is taken over the quotient stack $[X/G]$). Typically, $E$ is the structure bundle of a projective bundle ${\bf P}(p_*L)$ for a relatively very ample line bundle $L$ on $Y$). Since $X$ is smooth, there is an equivariant pullback $f^*_G: A_*^G(X) \rightarrow A_*^G(E)$ of the induced map $E \stackrel{f_G} \rightarrow X$, so we can define a class $f_G^*(w) \in A_*^G(E)$. Identifying $A_*^G(E)$ with $A_*(B)$ we obtain our class $f^*(W)$. When ${\cal M} = \overline{{\cal M}}_g$ is the moduli space of stable curves of genus $g$, these methods can be used to re-derive formulas of \cite[Section 3]{Ed} for intersections with various nodal loci. \subsection{Preliminaries} This section contains some results about quotients that will be used in proving Theorem \ref{quotient}. The reader may wish to read the proofs after the proof of Theorem \ref{quotient}. Let $G$ act locally properly on $X$ with quotient $X \stackrel{\pi}\rightarrow Y$. The field extension $K(Y) \subset K(X)^G$ is purely inseparable by \cite[p.43]{Borel}, and thus finite because both $K(Y)$ and $K(X)^G$ are intermediate extensions of $k \subset K(X)$ and $K(X)$ is a finitely generated extension of $k$. Set $i_X = [K(X)^G:K(Y)]$. Write $e_X$ for the order of the stabilizer at a general point of $X$. This is the degree of the finite map $S(id_X) \rightarrow X$ where $S(id_X)$ is the stabilizer of the identity morphism as defined in \cite[Definition 0.4]{GIT}. Note that the map $S(id_X) \rightarrow X$ can be totally ramified. This occurs exactly when the stabilizer of a general geometric point is non-reduced. Finally, set $\alpha_X = \frac{e_X}{i_X}$. \begin{lemma} \label{l.factor} Let $K = K(Y)$ be the ground field, and suppose $\pi: X \rightarrow Y = \mbox{Spec }K$ is a quotient of a variety $X$ by a group $G$ over $K$. Then $\pi$ factors as $X \rightarrow \mbox{Spec }K(X)^G \rightarrow Spec\; K(Y)$. \end{lemma} Proof. First, $X$ is normal. To see this, let $Z \subset X$ be the set of non-normal points, a proper $G$-invariant subset of $X$. If $L$ is an algebraically closed field containing $K$, write $X_L = X \times_{Spec \; K} \mbox{Spec }L$, $G_L = G \times_{Spec \; K} \mbox{Spec }L$. Now, $Z_L$ is a proper $G$-invariant subset of $X_L$. Since $X_L$ is a single $G_L$-orbit, $Z_L$ is empty. The map $X_L \rightarrow X$ is surjective, by the going up theorem; hence $Z$ is empty, so $X$ is normal. Now if $\mbox{Spec }A \subset X$ is an open affine subset, then $K(Y) \subset A$, $K(X)^G$ is integral over $K(Y)$, and $A$ is integrally closed in $K(X)$. We conclude that $K(X)^G \subset A$, which implies the result. $\Box$ \medskip {\bf Remark.} The fact that $X_L$ is a single $G_L$-orbit is essential to the result. For example, suppose $K = {\bf F}_p(t)$, $A = K[u,v]/(u^p - t v^p)$, $X = \mbox{Spec }A$, $G = {\bf G}_m$ acting by $g \cdot(u,v) = (gu,gv)$. The geometric points of $X_L$ form two $G_L$-orbits, since $(X_L)_{red} = {\bf A}^1_L$. The conclusion of the lemma fails since $\frac{u}{v}$ is in $K(X)^G$ but not in $A$. \bigskip \begin{prop} \label{p.technical.quotient} Let $$ \begin{array}{ccc} X' & \stackrel{g} \rightarrow &X \\ \small{\pi'} \downarrow & & \small{\pi} \downarrow\\ Y' & \stackrel{f} \rightarrow & Y \end{array} $$ be a commutative diagram of quotients with $f$ and $g$ finite and surjective. Then $$ \frac{[K(X'):K(X)]}{[K(Y'):K(Y)]} = \frac{\alpha_X}{\alpha_{X'}} = \frac{e_X}{e_{X'}} \cdot \frac{i_{X'}}{i_X} $$ \end{prop} Proof. Since we are checking degrees, we may replace $Y'$ and $Y$ by $K(Y')$ and $K(Y)$, and $X'$ and $X$ by their generic fibers over $Y'$ and $Y$ respectively. By the above lemma, we have a commutative diagram of varieties $$\begin{array}{ccc} X' & \rightarrow & X \\ \downarrow & & \downarrow\\ \mbox{spec}(K(X')^G) & \rightarrow & \mbox{spec}(K(X)^G)\\ \downarrow & & \downarrow\\ \mbox{spec}(K(Y')) & \rightarrow & \mbox{spec}(K(Y)). \end{array}$$ Since $i_{X'} := [K(X')^G:K(Y')]$ and $i_X :=[K(X)^G:K(Y)]$, it suffices to prove that $$ \frac{[K(X'):K(X)]}{[K(X')^G:K(X)^G]} = \frac{e_X}{e_{X'}}. $$ By \cite[Prop. 2.4]{Borel} the extensions $K(X')^G \subset K(X')$ and $K(X)^G \subset K(X)$ are separable (transcendental). Thus, after finite separable base extensions, we may assume that there are sections $s':\mbox{spec}(K(X')^G) \rightarrow X'$ and $s: \mbox{spec}(K(X)^G) \rightarrow X$. The section $s$ gives us a finite surjective map $G_K \rightarrow X$, where $G_K = G \times_{Spec \; k} \mbox{Spec }K(X)^G$. The degree of this map is $e_X = [K(G_K): K(X)]$ because $G_K \rightarrow X$ is the pullback of the action morphism $G \times X \rightarrow X$ via the map $X \rightarrow X \times X$ given by $x \mapsto (x,s(\mbox{Spec }K))$. Likewise, $e_{X'} = [K(G_{K'}): K(X)]$, where $G_{K'} = G \times_{Spec \; k} \mbox{Spec }K(X')^G$. Therefore, $$ \frac{e_X}{e_{X'}} = \frac{[K(G_K): K(X)]}{[K(G_{K'}): K(X')]} = \frac{[K(X'):K(X)]}{[K(X')^G:K(X)^G]}, $$ since $[K(G_{K'}):K(G_K)] = [K(X')^G:K(X)^G]$. This completes the proof. $\Box$ \medskip The following proposition is an analogue of \cite[Prop. 2.6]{Vi} and \cite[Thm 6.1]{Seshadri}. Our proof is similar to Vistoli's. \begin{prop} \label{whizzbang} Suppose that $G$ acts locally properly on an algebraic space $X$. Let $X \rightarrow Y$ be a quotient. Then there is a commutative diagram of quotients, with $X'$ a normal algebraic space: $$\begin{array}{ccc} X' & \rightarrow & X\\ \downarrow & & \downarrow\\ Y' & \rightarrow & Y \end{array}$$ where $X' \rightarrow Y'$ is a principal $G$-bundle (in particular $G$ acts with trivial stabilizer on $X'$) and the horizontal maps are finite and surjective. \end{prop} {\bf Remark.} If $X$ and $Y$ are quasi-projective, then so are $X'$ and $Y'$. If the action on $X$ is proper, then the action on $X'$ is free. Proof. By \cite[Lemma, p.14]{GIT}, there is a finite map $Y' \rightarrow Y$, with $Y'$ normal, so that the pullback $X_1 \stackrel{\pi}\rightarrow Y'$ has a section in a neighborhood of every point. Cover $Y'$ by a finite number of open sets $\{U_\alpha\}$ so that $X_1 \rightarrow Y'$ has a section $U_\alpha \stackrel{s_{\alpha}} \rightarrow V_{\alpha}$ where $V_{\alpha} = \pi^{-1}(U_{\alpha})$. Define a $G$-map $$\phi_{\alpha}: G \times U_\alpha \rightarrow V_\alpha$$ by the Cartesian diagram $$\begin{array}{ccc} G \times U_\alpha & \stackrel{\phi_\alpha} \rightarrow & V_\alpha\\ \downarrow & & \small{id \times s_\alpha \circ \pi} \downarrow\\ G \times V_\alpha & \rightarrow & V_{\alpha} \times V_{\alpha}. \end{array}$$ The action is locally proper so we can, by shrinking $V_\alpha$, assume that $\phi_\alpha$ is proper. Since it is also quasi-finite, it is finite. To construct a principal bundle $X' \rightarrow Y'$ we must glue the $G \times U_{\alpha}$'s along their fiber product over $X$. To do this we will find isomorphisms $\phi_{\alpha\beta}: s_\alpha(U_{\alpha\beta}) \rightarrow s_{\beta}(U_{\alpha\beta})$ which satisfy the cocycle condition. For each $\alpha, \beta$, let $I_{\alpha\beta}$ be the space which parameterizes isomorphisms of $s_\alpha$ and $s_\beta$ over $U_{\alpha\beta}$ (i.e. a section $U_{\alpha\beta} \rightarrow I_{\alpha\beta}$ corresponds to a global isomorphism $s_\alpha(U_{\alpha\beta}) \rightarrow s_\beta(U_{\alpha\beta})$). The space $I_{\alpha\beta}$ is finite over $U_{\alpha\beta}$ (but possibly totally ramified in characteristic $p$) since it is defined by the cartesian diagram $$\begin{array}{ccc} I_{\alpha\beta} & \rightarrow & U_{\alpha\beta} \\ \downarrow & & \small{1 \times s_\beta} \downarrow\\ G\times U_{\alpha\beta} & \stackrel{1 \times \phi_\alpha} \rightarrow & U_{\alpha\beta} \times V_{\alpha\beta} \end{array}$$ (Note that $I_{\alpha\alpha}$ is the stabilizer of $s_\alpha(U_\alpha)$.) Over $U_{\alpha\beta\gamma}$ there is a composition $$I_{\alpha\beta} \times_{U_{\alpha\beta\gamma}} I_{\beta\gamma} \rightarrow I_{\alpha\gamma}$$ which gives multiplication morphisms which are surjective when $\gamma = \beta$. After a suitable finite (but possibly inseparable) base change, we may assume that there is a section $U_{\alpha\beta} \rightarrow I_{\alpha\beta}$ for every irreducible component of $I_{\alpha\beta}$. (Note that $I_{\alpha\beta}$ need not be reduced.) Fix an open set $U_{\alpha}$. For $\beta \neq \alpha$ choose a section $\phi_{\alpha\beta}: U_{\alpha\beta} \rightarrow I_{\alpha\beta}$. Since the $I_{\alpha\beta}$'s split completely and $I_{\alpha\alpha}$ is a group over $U_{\alpha\alpha}$ (in the sense of \cite[Definition 0.1]{GIT}), there are sections $\phi_{\beta\alpha}:U_{\alpha\beta} \rightarrow I_{\beta\alpha}$ so that $\phi_{\alpha\beta} \cdot \phi_{\beta\alpha}$ is the identity section of $U_{\alpha\alpha}$. For any $\beta, \gamma$ we can define a section of $I_{\beta\gamma}$ over $U_{\alpha\beta\gamma}$ as the composition $\phi_{\beta\alpha} \cdot \phi_{\alpha\gamma}$. Because $I_{\beta\gamma}$ splits, the $\phi_{\beta\alpha}$'s extend to sections over $U_{\beta\gamma}$. By construction, the $\phi_{\beta\gamma}$'s satisfy the cocycle condition. We can now define $X'$ by gluing the $G \times U_{\beta}$'s along the $\phi_{\beta\gamma}$'s. $\Box$ \subsection{Proof of Theorems 3 and 4} To simplify the notation we give the proofs assuming that $G$ is connected (so the inverse image in $X$ of a subvariety of $Y$ is irreducible). All coefficients, including those of cycle groups, are assumed to be rational. \paragraph{Proof of Theorem \ref{quotient}} Let $\Delta^m$ be the $m$-simplex of \cite{Bl}. If $G$ acts locally properly on $X$, then $G$ acts locally properly on $X \times \Delta^m$ by acting trivially on the second factor. In this case, the boundary map of the higher Chow group complex preserves invariant cycles, so there is a subcomplex of invariant cycles $Z_*(X,\cdot)^G$. Set $$A_*([X/G],m) = H_m(Z_*(X, \cdot)^G,\partial).$$ This construction is well defined even when $X$ is an arbitrary algebraic space. Now if $X \rightarrow Y$ is a quotient, then so is $X \times \Delta^m \stackrel{\pi} \rightarrow Y \times \Delta^m$. Define a map $\pi^*: Z_k(X,m) \otimes {\bf Q} \rightarrow Z_{k+g}(X,m)^G \otimes {\bf Q}$ for all $m$ as follows. If $F \subset Y \times \Delta^m$ is a $k+m$-dimensional subvariety intersecting the faces properly, then $H = (\pi^{-1}F)_{red}$ is a $G$-invariant $(k+m+g)$-dimensional subvariety of $X \times \Delta^m$ which intersects the faces properly. Thus, $[H] \in Z_{k+g}(X,m)^G $. Set $\pi^*[F] = \alpha_H[ H] \in Z_{k+g}^G(X,m)$, where $\alpha_H = \frac{e_H}{i_H}$ is defined as above. Since $G$-invariant subvarieties of $X \times \Delta^m$ correspond exactly to subvarieties of $Y \times \Delta^m$, $\pi^*$ is an isomorphism of cycles for all $m$. This pullback has good functorial properties: \begin{prop} \label{p.functorial} $(a)$ If $$ \begin{array}{ccc} X' & \stackrel{g} \rightarrow &X \\ \small{\pi'} \downarrow & & \small{\pi} \downarrow\\ Y' & \stackrel{f} \rightarrow & Y \end{array} $$ is a commutative diagram of quotients with $f$ and $g$ proper, then $f_* {\pi '}^* = \pi^{*} g_*$ as maps $Z(Y',m) \rightarrow Z_*(X,m)^G$. $(b)$ Suppose $T \subset X$ is a $G$-invariant subvariety. Let $S \subset Y$ be its image under the quotient map. Set $U = X -T$ and $V=U/G$, so there is a diagram of quotients: $$ \begin{array}{ccc} U & \stackrel{j} \rightarrow & X \\ \small{\pi} \downarrow & & \small{\pi} \downarrow \\ V & \stackrel{j} \rightarrow & Y . \end{array} $$ Then $\pi^*j^* = j^* \pi^*$ as maps from $Z_k(Y,m)$ to $Z_{k+g}^G(U,m)$. \end{prop} Proof. Part (a) follows immediately from Proposition \ref{p.technical.quotient}. For (b), if $\alpha = [F]$ and $H= \pi^{-1}(F)_{red}$, then $\pi^*j^*\alpha$ and $j^*\pi^*\alpha$ are both multiples of $[H \cap U]$. Since $e_{[H \cap U]} = e_{[H]}$, and $i_{[H \cap U]} = i_{[H]}$, the multiples are the same, proving (b). $\Box$ \begin{prop} \label{squiggy} $(a)$ The map $\pi^*$ commutes with the boundary operator defining higher Chow groups. In particular, there is an induced isomorphism of Chow groups $$A_k(Y,m) \simeq A_{k+g}([X/G],m)$$ (note that the higher Chow groups $A_k(Y,m)$ are defined as groups even if $Y$ is only an algebraic space). $(b)$ In the setting of Proposition \ref{p.functorial}(b), if $X$ is quasi-projective with a linearized $G$-action, and the quotient $Y$ is quasi-projective, then there is a commutative diagram of higher Chow groups $$ \begin{array}{ccccccc} \ldots \rightarrow & A_*([T/G],m) & \rightarrow & A_*([X/G],m) & \rightarrow & A_*([U/G],m) & \rightarrow \ldots \\ & \uparrow & & \uparrow & & \uparrow & \\ \ldots \rightarrow & A_*(S,m) & \rightarrow & A_*(Y,m) & \rightarrow & A_*(V,m) & \rightarrow \ldots \end{array}$$ where the vertical maps are isomorphisms. Hence the top row of this diagram is exact. \end{prop} Proof. To prove (a), since $\pi^*$ is an isomorphism on the level of cycles, once we show that $\pi^*$ commutes with the boundary operator, it will follow that the induced map on Chow groups is an isomorphism. If $$\begin{array}{ccc} X' & \stackrel{g} \rightarrow & X \\ \small{\pi'} \downarrow & & \small{\pi} \downarrow\\ Y' & \stackrel{f} \rightarrow & Y \end{array}$$ is a commutative diagram of quotients with $f$ and $g$ finite and surjective, then $f_*$ and $g_*$ are surjective as maps of cycles. By Proposition \ref{p.functorial}(a) it suffices to prove ${\pi '}^*:Z_*(Y') \rightarrow Z_*(X')^G$ commutes with $\partial$. By Proposition \ref{whizzbang}, there exists such a commutative diagram of quotients with $\pi':X' \rightarrow Y'$ a principal bundle. Since $\pi'$ is flat, ${\pi '}^*$ commutes with $\partial$. This proves (a). Part (b) follows from (a), Proposition \ref{p.functorial} and the localization theorem for higher Chow groups. $\Box$ \medskip Define a map $\alpha:Z_*([X/G],m) \rightarrow Z_*(X_G,m)$ by $[F] \mapsto [F]_G$. This map commutes with proper pushforward and flat pullback. Arguing as in Proposition \ref{squiggy}(a), we see that $\alpha$ commutes with the boundary operator defining higher Chow groups and hence induces a map on Chow groups (again denoted $\alpha$). The proof of Theorem \ref{quotient} is completed by the following proposition. \begin{prop} \label{warhol} (a) If $X$ is an algebraic space with a (locally) proper $G$-action, then the map $\alpha: A_*([X/G]) \rightarrow A^G_*(X)$ is an isomorphism. (b) If $X$ is quasi-projective with a proper linearized $G$-action and a quasi-projective quotient $X \rightarrow Y$ exists, then $\alpha: A_*([X/G],m) \rightarrow A^G_*(X,m)$ is an isomorphism for $m > 0$. \end{prop} Proof of (a). Let $Y$ be the quotient of $X$ by $G$. We will prove that the composition $\alpha \circ \pi^*:A_*(Y) \rightarrow A_*^G(X)$ is an isomorphism. By Proposition \ref{whizzbang} there is a commutative diagram of quotients with $g$ and $f$ finite and surjective $$ \begin{array}{ccc} X' & \stackrel{g} \rightarrow &X \\ \small{\pi'} \downarrow & & \small{\pi} \downarrow\\ Y' & \stackrel{f} \rightarrow & Y \end{array} $$ and $X' \rightarrow Y'$ is a principal bundle. Since $X' \rightarrow X$ and $Y' \rightarrow Y$ are finite and surjective, \cite[Theorem 1.8]{Kimura} (which extends to the equivariant setting) says that there are exact sequences $$\begin{array}{c} A_*^G(X' \times_X X') \stackrel{g_{1*} - g_{2*}}\rightarrow A_*^G(X') \stackrel{g_*} \rightarrow A_*^G(X) \rightarrow 0\\ A_*^G(Y' \times_Y Y') \stackrel{f_{1*} - f_{2*}}\rightarrow A_*^G(Y') \stackrel{f_*} \rightarrow A_*^G(Y) \rightarrow 0 \end{array}$$ where $g_i$ and $f_i$ are the projections to $X'$ and $Y'$. Set $X'' = X' \times_X X'$, and set $Y'' = X''/G$. The natural map $p:Y'' \rightarrow (Y' \times_Y Y')$ is finite and surjective, so the pushforward $A_*(Y'') \rightarrow A_*(Y' \times_Y Y')$ is a surjection. Hence the second sequence remains exact if we replace $Y' \times_Y Y'$ by $Y''$ (and $f_i$ by $f_i \circ p$). We have a commutative diagram of exact sequences $$\begin{array}{cccccc} A_*^G(X'') & \rightarrow & A_*^G(X') & \rightarrow & A_*^G(X) & \rightarrow 0\\ \uparrow & & \uparrow & & \uparrow & \\ A_*(Y'') & \rightarrow & A_*(Y') & \rightarrow & A_*(Y) & \rightarrow 0 , \end{array}$$ where the vertical maps are $\pi^{''*} \circ \alpha{''}$, $\pi^{'*} \circ \alpha{'}$, and $\pi \circ \alpha$, respectively. By Proposition \ref{p.quotient}, the first two maps are isomorphisms, so by the 5-lemma, the third is as well. \medskip Proof of (b): We are going to use the localization exact sequences for higher equivariant Chow groups, and for the invariant Chow groups $A_*([X/G],m)$ (Proposition \ref{squiggy}(b)). This is why we must assume that $X$ is quasi-projective and that a quasi-projective quotient exists. By Proposition \ref{whizzbang}, we can find a finite surjective map $g: X' \rightarrow X$, where $G$ acts freely on $X'$. The map $\alpha': A_*([X'/G],m) \rightarrow A^G_*(X',m)$ is an isomorphism, as noted in (a). By Noetherian induction and the localization long exact sequences it suffices to prove the result when $X$ is replaced by the open subset over which $g$ is flat, so assume this. Because $g$ is also finite, $g_* g^*$ is multiplication by the degree of $g$. Hence (since we are using rational coefficients) the flat pullback $g^*$ makes $A_*^G(X,m)$ a summand in $A_*^G(X',m)$ and $A_*([X/G],m)$ a summand in $A_*([X'/G],m)$. Since $\alpha' \circ g^* = g^* \circ \alpha$, the summand $A_*^G([X/G],m) \subset A_*^G( [X'/G],m)$ is isomorphic to the summand $A_*^G(X,m) \subset A_*^G([X/G],m)$. This proves the proposition, and with it Theorem \ref{quotient}. $\Box$. \medskip \paragraph{Proof of Theorem \ref{quotient.cor}} The proof is similar to \cite[Proposition 6.1]{Vi}. Let $\pi: X \rightarrow Y$ be the quotient map. We define a pullback $\pi^*:A^*(Y) \rightarrow A^*_G(X)$ as follows: Suppose $c \in A^i(Y)$, $Z \rightarrow X$ is a $G$-equivariant morphism, and $z \in A_*^G(Z)$. For any representation, there are maps $Z_G \rightarrow X_G \rightarrow Y$. The class $z$ is represented by a class $z_V \in A_{*+l-g}(Z_G)$ for some mixed space $Z_G$. Define $$\pi^*c \cap z = c \cap z_V \in A_{*+l-g-i}(Z_G) \simeq A_{*-i}^G(Z) $$ As usual, this definition is independent of the representation, so $\pi^*c \cap \alpha \in A_*^G(Z)$. Let $\hat{\pi}: A_* (Y) \rightarrow A_{*+g}^G(X)$ denote the isomorphism of Theorem \ref{quotient}. \begin{lemma} \label{l.compatibility} If $c \in A^*(Y)$ and $y \in A_*(Y)$, then $$\hat{\pi} (c \cap y) = \pi^*c \cap \hat{\pi} y.$$ \end{lemma} Proof of Lemma \ref{l.compatibility}. We first prove the lemma when $X \rightarrow Y$ is a principal $G$-bundle. In this case the map $\hat{\pi}: A_*(Y) \rightarrow A_{*+g}^G(X) \simeq A_{* + g}(X_G)$ is just the pullback induced by flat map $X_G \rightarrow Y$. Since the operations in $A^*(Y)$ are compatible with flat pullback the lemma follows in this case. By proposition \ref{whizzbang} there is a commutative diagram of quotients with $g$ and $f$ finite and surjective. $$ \begin{array}{ccc} X' & \stackrel{g} \rightarrow &X \\ \small{\pi'} \downarrow & & \small{\pi} \downarrow\\ Y' & \stackrel{f} \rightarrow & Y \end{array} $$ and $X' \rightarrow Y'$ is a principal bundle. Then $y = f_*(y')$ for some $y' \in A_*(Y')$. Since $c \cap y' = f^*c \cap y'$, we have by the first case $$\hat{\pi'}(c \cap y') = \pi'^*f^*c \cap \hat{\pi'}y'$$ Let $x'_V \in A_*^G(X'_GXS)$ be the class corresponding to $\hat{\pi'}y'$. Then $\pi'^*f^*c \cap \hat{\pi'}y' = c \cap x'_V = g^*\pi^*c \cap \hat{\pi'}y$. Thus we obtain the equation $$(*) \mbox{ }\hat{\pi'}(c \cap y') =g^*\pi^*c \cap \hat{\pi'}y.$$ By Proposition \ref{p.functorial} $g_*(\hat{\pi'}(c \cap y')) = \hat{\pi}(c \cap f_*y') = \hat{\pi}(c \cap y)$. The lemma follows by applying $g_*$ to both sides of (*). $\Box$ Given the lemma, we show that $\pi^*$ is an isomorphism as follows. For injectivity, it suffices (by base change) to show that if $\pi^* c = 0$ then $c \cap y = 0$ for all $y \in A_*(Y)$; this follows since $\hat{\pi} (c \cap y) = 0$ by the lemma, and $\hat{\pi}$ is an isomorphism. The proof of surjectivity is more subtle. Given $d \in A^*_G(X)$ define $c \in A^*(Y)$ as follows: If $Y' \rightarrow Y$ and $y' in A_*(Y')$, set $$ c \cap y' = \hat{\pi}'^{-1} (d \cap \hat{\pi}' y). $$ where $\pi': X' = X \times_Y Y' \rightarrow Y'$ is the pullback quotient. We must now show $\pi^*c = d$. We begin with a preliminary construction. For any mixed space $X_G$ define a map $r:A^i_G(X) \rightarrow A^i(X_G)$ as follows: Given $Z \rightarrow X_G$ let $Z_U \rightarrow Z$ be the pullback of the principal $G$-bundle $X \times U \rightarrow X_G$. Since $Z_U \rightarrow Z$ is a principal bundle we can identify $A_*(Z)$ with $A_{* +g}^G(Z_U)$ and view a class $z \in A_*(Z)$ as an equivariant class in $A_{*+g}^G(Z_U)$. Now set $r(c) \cap z = c \cap z \in A_*(Z_U) \simeq A_*(Z)$. {}From the construction it is clear that if $X' \rightarrow X$ is equivariant and $x' \in A_*^G(X')$ corresponds to $x'_V \in A_*(X'_G)$ then $r(c) \cap x'_V$ corresponds to $c \cap x'$ under the identification of $A_*^G(X')$ $A_*^G(X'_G)$. Suppose $Z \rightarrow X$ is equivariant and $z \in A_*^G(Z)$ is defined by the class $z_V \in A_*^G(Z_G)$. Let $\pi':X' \rightarrow Z_G$ be the quotient induced by pulling back $X \rightarrow Y$ along the map $Z_G \rightarrow Y$. By definition we must show $\hat{\pi}'^{-1}(d \cap \hat{\pi}'z_V) \in A_*(Z_G) \simeq A_*^G(Z)$ defines the same class as $d \cap z \in A_*^G(Z)$. By construction, $d \cap z$ is defined by the class $r(d) \cap z_V \in A_*(Z_G)$. Since $\hat{\pi}'$ is an isomorphism it suffices to prove that $d \cap \hat{\pi}'z_V$ equals $\hat{\pi}'(r(d) \cap z_V)$. By Lemma \ref{l.compatibility} $\hat{\pi}'(r(d) \cap z_V) = \pi'^{*}r(d) \cap \hat{\pi}'z_V$. The class $\hat{\pi}'z_V$ is represented by a class $(\hat{\pi}'z_V)_V \in A_*(X'_G)$ and $\pi^{'*}r(d) \cap \hat{\pi}'z_V = r(d) \cap (\hat{\pi}'z_V)_V$. Finally $r(d) \cap (\hat{\pi}'z_V)_V = d \cap \hat{\pi}'z_V$ under the identification of $A_*(X'_G)$ with $A_*^G(X')$. This proves that $\pi^*$ is surjective and with it part (a) of the theorem. To prove (b) recall that $\hat{\pi}(c \cap y) = \pi^*c \cap \hat{\pi}y$. By Proposition \ref{opsmooth} the map $A^*_G(X) \stackrel{\cap [X]_G} \rightarrow A_*(X)$ is an isomorphism (with ${\bf Z}$ coefficients). Since $\hat{\pi}([Y]) = \alpha_X [X]_G$, the map the map $c \mapsto c \cap [Y]$ is an isomorphism (with ${\bf Q}$ coefficients). $\Box$. \section{Intersection theory on quotient stacks and their moduli} \label{itstacks} If $G$ acts on an algebraic space $X$, a quotient $[X/G]$ exists in the category of stacks (\cite[Example 4.8]{D-M}; see below). This section relates equivariant Chow groups to Chow groups of quotient stacks. We show that for proper actions, with rational coefficients, equivariant Chow groups coincide with the Chow groups defined by Gillet in terms of integral substacks. Thus, in this case the intersection products of Gillet and Vistoli are the same. For an arbitrary action, the equivariant Chow groups are an invariant of the quotient stack. As an application, we calculate the Chow rings of the moduli stacks of elliptic curves. \subsection {Definition of quotient stacks} We recall the definition of quotient stack. For an introduction to stacks see \cite{D-M}, \cite{Vi}. Let $G$ be a linear algebraic group acting on a scheme (or algebraic space). The quotient stack $[X/G]$ is the stack associated to the groupoid $G \times X \rightarrow X \times X$. If $B$ is a scheme, sections of $[X/G](B)$ are principal $G$-bundles $E \rightarrow B$ together with an equivariant map $E \rightarrow X$. Morphisms in $[X/G](B)$ are $G$-bundle isomorphisms which preserve the map to $X$. In particular, if $G$ acts on $X$ with trivial stabilizers then all morphisms are trivial, $[X/G]$ is a sheaf in the \'etale topology and the quotient is in fact an algebraic space (Proposition \ref{l.algspacequotient}). If $G$ acts with finite reduced stabilizers then the diagonal $[X/G] \rightarrow [X/G] \times [X/G]$ is unramified and $[X/G]$ is a Deligne-Mumford stack. If $G$ acts (locally) properly then the diagonal $[X/G] \rightarrow [X/G] \times [X/G]$ is (locally) proper and the stack is (locally) separated. In characteristic $0$ any (locally) separated stack is Deligne-Mumford. However, in characteristic $p$ this need not be true since the stabilizers of geometric points can be non-reduced. \subsection{Quotient stacks for proper actions} If $G$ acts locally properly on $X$ with reduced stabilizers then the quotient stack $[X/G]$ is locally separated and Deligne-Mumford. The rational Chow groups $A_*([X/G]) \otimes {\bf Q}$ were first defined by Gillet \cite{Gi} and coincide with the groups $A_*([X/G]) \otimes {\bf Q}$ defined above. More generally, if $G$ acts with finite stabilizers which are not reduced, then Gillet's definition can be extended and we can define the ``naive'' Chow groups $A_k([X/G])_{{\bf Q}}$ as the group generated by $k$-dimensional integral substacks modulo rational equivalences. In this context, Proposition \ref{warhol} can be restated in the language of stacks as \begin{prop} Let $G$ be a $g$-dimensional group which acts locally properly on an algebraic space $X$ (so the quotient $[X/G]$ is a locally separated Artin stack). Then $A_i^G(X) \otimes {\bf Q} = A_{i-g}([X/G]) \otimes {\bf Q}$. $\Box$ \end{prop} {\bf Remark.} Although $A_*^G(X) \otimes {\bf Q} = A_*([X/G]) \otimes {\bf Q}$, the integral Chow groups may have non-zero torsion for all $i < \mbox{dim } X$. \medskip With the identification of $A_*^G(X) \otimes {\bf Q}$ and $A_*([X/G]) \otimes {\bf Q}$ there are three intersection products on the rational Chow groups of a smooth Deligne-Mumford quotient stack -- the equivariant product, Vistoli's product defined using a Gysin pullback for regular embeddings of stacks, and Gillet's product defined using the product in higher $K$-theory. The next proposition shows that they are identical. \begin{prop} \label{triprod} If $X$ is smooth and $[X/G]$ is a locally separated Deligne-Mumford stack (so $G$ acts with finite, reduced stabilizers) then the intersection products on $A_*([X/G])_{{\bf Q}}$ defined by Vistoli and Gillet are the same as the equivariant product on $A_*^G(X)_{{\bf Q}}$. \end{prop} Proof. If $V$ is an $l$-dimensional representation, then all three products agree on the smooth quotient space $(X \times U)/G$ (\cite{Vi}, \cite{Grayson}). Since the flat pullback of stacks $f:A^*([X/G])_{{\bf Q}} \rightarrow A^*((X \times U)/G)_{{\bf Q}}$ commutes with all 3 products, and is an isomorphism up to arbitrarily high codimension, the proposition follows. $\Box$ \subsection{Integral Chow groups of quotient stacks} \label{intstack} Suppose $G$ acts arbitrarily on $X$. Consider the (possibly non-separated Artin) quotient stack $[X/G]$. The next proposition shows that the equivariant Chow groups do not depend on the presentation as a quotient, so they are an invariant of the stack. \begin{prop} \label{qstacks} Suppose that $[X/G] \simeq [Y/H]$ as quotient stacks. Then $A_{i+g}^G(X) \simeq A_{i+h}^H(Y)$, where $\mbox{dim } G = g$ and $\mbox{dim } H = h$. \end{prop} Proof. Let $V_1$ be an $l$-dimensional representation of $G$, and $V_2$ an $M$ dimensional representation of $H$. Let $X_G = X \times^G U_1$ and $Y_H = X \times^H U_2$, where $U_1$ (resp. $U_2$) is an open set on which $G$ (resp. $H$) acts freely. Since the diagonal of a quotient stack is representable, the fiber product $Z=X_G \times_{[X/G]} Y_H$ is an algebraic space. This space is a bundle over $X_G$ and $Y_H$ with fiber $U_2$ and $U_1$ respectively. Thus, $A_{i+l}(X_G) = A_{i+l+m}(Z) = A_{i+m}(Y_H)$ and the proposition follows. $\Box$ \medskip As a consequence of Proposition \ref{qstacks} we can define the integral Chow groups of a quotient stack ${\cal F} = [X/G]$ by $A_i({\cal F}) = A_{i-g}^G(X)$ where $g = \mbox{dim }G$. \begin{prop} If ${\cal F}$ is smooth, then $\oplus A_*({\cal F})$ has an integral ring structure. $\Box$ \end{prop} Following \cite[Definition, p. 64]{Mu} we define the Picard group $Pic_{fun}({\cal F})$ of an algebraic stack ${\cal F}$ as follows. A element ${\cal L} \in Pic_{fun}( {\cal F})$ assigns to any map $S \stackrel{F} \rightarrow {\cal F}$ of a scheme $S$, an isomorphism class of a line bundle $L(F)$ on $S$. Moreover the assignment must satisfy the following compatibility conditions. (i) Let $S_1 \stackrel{F_1} \rightarrow {\cal F}$, $S_2 \stackrel{F_2} \rightarrow {\cal F}$ and $S_2 \stackrel{F_2} \rightarrow {\cal F}$ be maps of schemes to ${\cal F}$. If there is a map $S_1 \stackrel{f} \rightarrow S_2$ such that $F_1 = F_2 \circ f$ then there is an isomorphism $\phi(f): L(F_1) \simeq f^*(L(F_2))$. (ii) If $S_1 \stackrel{f} \rightarrow S_2 \stackrel{g} \rightarrow S_3$ are maps of schemes such that $F_2 = F_3 \circ g$ and $F_1 = F_2 \circ f$ then there is a commutative diagram of isomorphisms $$\begin{array}{ccc} L(F_1) & \stackrel{\phi(F_1)} \rightarrow & f^*(L(F_2)) \\ \small{\phi(g \circ f)} \downarrow & & \small{f^*\phi(g)} \downarrow\\ (g \circ f)^*(L(F_3)) & = & f^*(g^*(L(F_3))). \end{array}$$ The product ${\cal L} \otimes {\cal M}$ assigns to $S \stackrel{f} \rightarrow {\cal F}$ the line bundle $L_f \otimes M_f$. \begin{prop} Let $X$ be a smooth variety with a $G$-action. Then $A^1_G(X) = Pic_{fun}([X/G])$. \end{prop} Proof. By Theorem \ref{piciscool}, $A^1_G(X) = Pic^G(X)$. The latter group is naturally isomorphic to $Pic_{fun}([X/G])$. $\Box$ \medskip More generally, if ${\cal F}$ is any stack, we can define the integral operational Chow ring $A^*({\cal F})$ as follows. An element $c \in A^k({\cal F})$ defines an operation $A_*B \stackrel{c_f} \rightarrow A_{*-k}B$ for any map of a scheme $B \stackrel{f} \rightarrow {\cal F}$. The operations should be compatible with proper pushforward, flat pullback and intersection products for maps of schemes to ${\cal F}$ (cf. \cite[Definition 17.1]{Fulton} and \cite[Definition 5.1]{Vi}). (This definition differs slightly from Vistoli's because we use integer coefficients and only consider compatibility with maps of schemes.) \begin{prop} Let ${\cal F} \simeq [X/G]$ be a smooth quotient stack. Then $A^*({\cal F}) = A^*_G(X)$. \end{prop} Proof. Giving a map $B \stackrel{f} \rightarrow [X/G]$ is equivalent to giving a principal $G$-bundle $E \rightarrow B$ together with an equivariant map $E \rightarrow X$. An element of $A_G^*(X)$ defines an operation on $A_*^G(E) = A_*(B)$, hence an operational class in $A^*({\cal F})$. Conversely an operational class $c \in A^k({\cal F})$ defines an operation on $A_*(X_G)$ corresponding to the map $X_G \rightarrow {\cal F}$ associated to the principal bundle $X \times U \rightarrow X_G$. Set $d = c \cap [X_G] \in A_{dim \; X -k}^G(X)$. Since $X$ is smooth, the latter group is isomorphic to $A^k_G(X)$. $\Box$ \medskip {\bf Remark.} Proposition \ref{qstacks} suggests that there should be a notion of Chow groups of an arbitrary algebraic stack which can be non-zero in arbitrarily high degree. This situation would be analogous to the cohomology of quasi-coherent sheaves on the \'etale (or flat) site (cf. \cite[p. 101]{D-M}). \subsection{The Chow ring of the moduli stack of elliptic curves} In this section we will work over a field of characteristic not equal to 2 or 3, and compute the Chow ring of the moduli stacks ${\cal M}_{1,1}$ and $\overline{{\cal M}}_{1,1}$ of elliptic curves. A. Vistoli has independently obtained the results of this section, also using equivariant intersection theory. He has also calculated the Chow ring of ${\cal M}_2$, the moduli stack of elliptic curves. This calculation will appear as an appendix to this article \cite{Viap}. \paragraph{Construction of the moduli stack} The stacks ${\cal M}_{1,1}$ and $\overline{{\cal M}}_{1,1}$ are defined as follows. A section of ${\cal M}_{1,1}$ (resp. $\overline{{\cal M}}_{1,1}$) over $S$ is a family $(X \stackrel{\pi} \rightarrow S, \sigma)$ where $X \rightarrow S$ is a smooth (resp. possibly nodal) curve of genus $1$ and $\sigma: S \rightarrow X$ is a smooth section. Our construction of ${\cal M}_{1,1}$ and $\overline{{\cal M}}_{1,1}$ is similar to \cite[Section 4]{Mu}. Let ${\bf P}(V) = {\bf P}^9$ be the projective space of homogeneous degree $3$ forms in variables $x, y, z$. Let $X \simeq {\bf A}^3 \subset {\bf P}(V)$ be the affine subspace parameterizing forms proportional to $$y^2 z - (x^3 + e_1 x^2 z + e_2 x z^2 + e_3 z^3),$$ with $e_1, e_2, e_3$ arbitrary elements of the field $k$. Let $G = \{ \left( \begin{array}{ccc} a & 0 & b \\ 0 & c & 0 \\ 0 & 0 & d \end{array} \right) | a^3 = c^2d \neq 0\}$. The image of $G$ in $PGL(3)$ consists of projective transformations which stabilize $X$. Since $a \neq 0$ for all $g \in G$, we can normalize and identify $G$ with the subgroup\\ $\{ \left( \begin{array}{ccc} 1 & 0 & B \\ 0 & A & 0 \\ 0 & 0 & A^{-2} \end{array} \right) | A \neq 0 \}$ of $GL(3)$. An element $g = \left( \begin{array}{ccc} 1& 0 & B \\ 0 & A & 0 \\ 0 & 0 & A^{-2} \end{array} \right)$ acts on $(e_1, e_2, e_3)$ by $$(e_1, e_2, e_3) \mapsto (A^{-2} e_1 + 3B, A^{-4} e_2 + 2 A^{-2} B e_1 + 3B^2, A^{-6} e_3 + A^{-4} B e_2 + A^{-2} B^2 e_1 + B^3).$$ Let $U \subset X$ be the open set where the polynomial $(x^3 + e_1 x^2 + e_2 x + e_3)$ has distinct roots over the algebraic closure of the field of definition. Likewise, let $W \subset X$ be the open set where $x^3 + e_1 x^2 + e_2 x$ has at least 2 distinct roots. \begin{prop} (a) ${\cal M}_{1,1} \simeq [U/G]$. (b) $\overline{{\cal M}}_{1,1} \simeq [W/G]$. \end{prop} Proof of (a). Let $Y^{\cdot}$ be the functor such that a section of $Y^{\cdot}$ over $S'$ is a triple $(X' \stackrel{f'} \rightarrow S', \sigma', \phi')$ where $(X',\sigma')$ is an elliptic curve with section $\sigma'$ and $\phi'$ is an isomorphism of the flag ${\cal O}_{S'} \subset {\cal O}_{S'}^{\oplus 2} \subset {\cal O}_{S'}^{\oplus 3}$ with the flag $f'_*({\cal O}_{X'}(\sigma)) \subset f'_*({\cal O}_{X'}(\sigma)^{\otimes 2}) \subset f'_*({\cal O}_{X'}(\sigma)^{\otimes 3})$. This functor is represented by the scheme $Y$ parameterizing cubic forms in $3$ variables with a flex at $(1:0:0)$, since there is a universal triple $({\cal X} \stackrel{F} \rightarrow Y, \sigma_Y, \Phi)$ on $Y$. There is an obvious action of the group $B \subset GL(3)$ of upper triangular matrices on $Y$. Claim: The quotient stack $[Y/B]$ is ${\cal M}_{1,1}$. Proof: Given a family $X \stackrel{f} \rightarrow S$ of elliptic curves, there is a canonical principal $B$-bundle $S' \rightarrow S$ associated to the flag of vector bundles ${\cal O}_S \subset {\cal O}_S^{\oplus 2} \subset {\cal O}_S^{\oplus 3}$. By construction the pullback of this flag to $S'$ is equipped with an isomorphism with the standard flag in ${\cal O}_{S'}^{\oplus 3}$. Thus from a family of elliptic curves over $S$, we obtain a principal $B$-bundle $S' \rightarrow S$ together with an equivariant map $S' \rightarrow Y$, i.e., a section of $[Y/B]$ over $S$. The converse is similar. Thus, it suffices to prove that $[U/G] \simeq [Y/B]$. Let $Y'$ be the quotient of $Y$ by the natural ${\bf G}_m$-action. Then $[Y/B] = [Y'/B']$, where $B'$ is the group of upper-triangular matrices in $PGL(3)$. Identify $Y'$ with the locally closed subscheme of ${\bf P}^9$ corresponding to cubics with a flex at $(1:0:0)$. Identifying $G \subset B'$, the inclusion $U \subset Y'$ is $G$-equivariant for the corresponding actions of $G$ and $B'$. Thus we obtain a smooth (representable) morphisms of quotient stacks $$[U/G] \rightarrow [Y'/G] \rightarrow [Y'/B'] \simeq {\cal M}_{1,1}.$$ Two curves in $U$ are isomorphic iff they are in the same $G$-orbit, so the map $[U/G] \rightarrow {\cal M}_{1,1}$ is quasi-finite. Moreover, every elliptic curve can be embedded in ${\bf P}^2$ as double cover of ${\bf P}^1$ branched at $\infty$ and 3 finite points. Thus the map is surjective. Finally, a direct check shows that the stabilizer of the $G$-action on a point of $U$ is exactly the automorphism group of the corresponding elliptic curve. Thus the geometric fibers are single points. Hence, the map is \'etale, surjective of degree 1, and thus an isomorphism of stacks. The proof of (b) is essentially identical. $\Box$ \medskip We now compute the Chow ring of this stack. \begin{prop} \label{didwediss} (a) $A^*({\cal M}_{1,1}) = {\bf Z}[t]/12t$. (b) $A^*(\overline{{\cal M}}_{1,1})= {\bf Z}[t]/24t^2$. \end{prop} Proof. (a) By definition, $A^*({\cal M}_{1,1}) = A^*_G(U)$. The equivariant Chow ring is not hard to calculate. Let $T \subset G$ be the maximal torus. Then $G$ is a unipotent extension of $T$, so $A^*_T(U) = A^*_G(U)$. Now $T$ acts diagonally on $X = \{(e_1,e_2,e_3)\}$ with weights $(-2,-4,-6)$. Let $S = X - U$, then $S$ is the discriminant locus in ${\bf A}^3$, which can be identified as the image of the big diagonal (i.e., the image under the $S_3$ quotient map $A^3 \rightarrow X \simeq A^3$ which maps $(a,b,c) \mapsto (a+b+c, ab + bc + ac, abc))$ and has equation $f(e_1, e_2, e_3) = 4e_2^3 + 27 e_3^2 - 18e_1 e_2 e_3 - e_1^2 e_2^2 + 4e_1^3e_3$. The form $f$ is homogeneous of weighted degree $-12$ with respect to the $T$-action on $X$. Since $S \subset X$ is a divisor, $A^*_T(U) = A^*_T(X)/([S]_T)$ where $([S]_T)$ denotes the $T$-equivariant fundamental class of $S$. Since $f$ has weight $12$, $[S]_T = 12t \in A^*_T(X) = Z[t]$. Therefore, $A^*_T(U) = {\bf Z}[t]/12t$. (b) The complement of $W$ in $X$ is the image of the small diagonal $a = b = c$ under the degree $6$ map ${\bf A}^3 \rightarrow {\bf A}^3$. The small diagonal has $T$-equivariant fundamental class $4t^2$, the $T$ fundamental class of $X-W$ is $24t^2$. Hence $A^*(\overline{{\cal M}}_{1,1}) = {\bf Z}[t]/24t^2$ as claimed. $\Box$ \medskip {\bf Remark.} From our computation we see $A^1({\cal M}_{1,1}) = Pic_{fun}({\cal M}_{1,1}) = {\bf Z}/12$, a fact which was originally proved by Mumford \cite{Mu}. With an appropriate sign convention, the class $t \in Z[t]/12t$ is just $c_1(L)$ where $L$ is the generator of $Pic_{fun}({\cal M}_{1,1})$ which assigns to a family of elliptic curves $X \stackrel{\pi} \rightarrow S$ the line bundle $\pi_*(\omega_{X/S})$ (the Hodge bundle). Thus monomial $at^n$ corresponds to the class which assigns to a family ${\cal X} \rightarrow S$ of elliptic curves, the class $c_1(L^{\otimes a})^n \cap [S] \in A_*(S)$. Angelo Vistoli observed that this can be seen directly as follows: The unipotent radical of $G$ acts freely on $U$ (or $W$) and the quotient is the space of forms $y^2z = x^3 + \alpha xz^2 + \beta z^3$ with no double (or triple) roots. The torus action is given by $t\cdot \alpha = t^{-4}\alpha$ and $t\cdot \beta = t^{-6}\beta$, and the space is the total space of the ${\bf G}_m$ bundle over ${\cal M}_{1,1}$ (or $\overline{{\cal M}}_{1,1}$) corresponding to the Hodge bundle. Thus, the Chow ring of ${\cal M}_{1,1}$ (resp. $\overline{{\cal M}}_{1,1}$) is generated by the first Chern class of the Hodge bundle. \section{Some technical facts} \label{appendix} \subsection{Intersection theory on algebraic spaces} \label{algspace} Unfortunately, while most results about schemes generalize to algebraic spaces, most references deal exclusively with schemes. In particular, this is the case for \cite{Fulton}, the basic reference for the intersection theory used in this paper. The purpose of this section is to indicate very briefly how this theory generalizes to algebraic spaces. We recall from \cite{Knutson} the definition of algebraic spaces, and basic facts about them. If $X$ is a scheme, the functor $X^{\cdot} = Hom(\cdot, X)$ from $(\mbox{Schemes})^{opp}$ to (Sets) is a sheaf in either the Zariski or \'etale topologies. With this as motivation, an algebraic space is defined to be a functor $A: (\mbox{Schemes})^{opp} \rightarrow (\mbox{Sets})$ such that: (1) $A$ is a sheaf in the \'etale topology. (2) (Local representability) There is a scheme $U$ and a sheaf map $U^{\cdot} \rightarrow A$ such that for any scheme $V$ with a map $V^{\cdot} \rightarrow A$, the fiber product (of sheaves) $U^{\cdot} \times_{A} V^{\cdot}$ is represented by a scheme, and the map $U^{\cdot} \times_{A} V^{\cdot} \rightarrow V^{\cdot}$ is induced by an \'etale surjective map of schemes. Knutson also imposes a technical hypothesis of quasi-separatedness, which states that the map $U^{\cdot} \times_{A} U^{\cdot} \rightarrow U^{\cdot} \times U^{\cdot}$ is quasi-compact. A morphism of algebraic spaces is a natural transformation of functors. The map $X \mapsto X^{\cdot}$ is a fully faithful embedding of (Schemes) into (Algebraic spaces). We identify $X$ with $X^{\cdot}$ and henceforth use the same notation $X$ for both of these. The scheme $X$ is called a representable \'etale covering of $A$ (or an \'etale atlas for $A$); it can be chosen to be a disjoint union of affine schemes. Thus, just as a scheme has a Zariski covering by affine schemes, an algebraic space has an \'etale covering by affine schemes. There are several ways to think of algebraic spaces in relation to schemes. One way is to think of a (normal) algebraic space as a quotient of a scheme by a finite group (see \cite{Kollar}). Another is to think of an algebraic space as a quotient of a scheme by an \'etale equivalence relation. More precisely, in the setting of the above definition, let $R$ denote the scheme $U \times_{A} U$; then $A$ is a categorical quotient of $R \rightarrow U \times U$ \cite[II.1.3]{Knutson}. Finally, any algebraic space has an open dense subset which is isomorphic to a scheme \cite[II.6.7]{Knutson}. A key fact of algebraic spaces that we use is: \begin{prop} \label{l.algspacequotient} Let $X$ be an algebraic space with a set theoretically free action of $G$. Then a quotient $X \rightarrow Y$ exists in the category of algebraic spaces. Moreover this quotient is a principal bundle. \end{prop} Sketch of proof. Consider the functor $Y=[X/G]$ whose sections over $B$ are principal $G$-bundles $E \rightarrow B$ together with an equivariant map $E \rightarrow X$. Since principal bundles can be constructed locally in the \'etale topology, and $G$ acts without stabilizers, one can check that $Y$ is a sheaf in the \'etale topology (cf. \cite[Example 4.8]{D-M}). To show that $Y$ is an algebraic space, we must construct an \'etale atlas for $Y$. This follows from \cite[Theorem 4.21]{D-M}. We sketch a proof assuming that $X$ is normal. Consider the surjective map of \'etale sheaves $X \rightarrow Y$. It suffices to show that every closed point $x \in X$ is contained in a locally closed subscheme $Z \subset X$ such that $Z \rightarrow Y$ is \'etale. Since $X$ is an algebraic space and we are working locally in the \'etale topology we may assume $X$ is a scheme. Let $Gx \simeq G$ be the the $G$ orbit of $X$. Then $Gx$ is the fiber of $X \rightarrow Y$ containing $x$. Let $Z$ be a locally closed subscheme of $X$ defined by lifts to ${\cal O}_X$ of the local equations for $x \in Gx$ so that $\mbox{dim }Z = \mbox{dim }X - \mbox{dim }G$ and the scheme theoretic intersection $Z \cap Gx$ is $x \in Gx$. Since $G$ is smooth, the point $x \in G$ is cut out by a regular sequence of length $g$ where $g = \mbox{dim }G$. Now consider the equivariant map (where $G$ acts on $G \times Z$ by $g(g_1,z) = (gg_1,z)$) $G \times Z \stackrel{\psi} \rightarrow X$ given by $(g,z) \mapsto gz$. This map is the restriction to $G \times Z$ of the action map $G \times X \stackrel{\Psi} \rightarrow X$. Since $G$ acts without stabilizers, the fiber of $\Psi$ over $x$ is $\{(g^{-1},gx)\} \simeq Gx$. The fiber of $\psi$ over $x$ is then $(G \times Z \cap Gx)$. By construction of $Z$, $Z \cap Gx \simeq x$. Thus, the scheme-theoretic fiber of $\psi$ over $x$ is $(1,x)$, so $\psi$ is unramified $x$. Thus we have a surjection of complete local rings ${\cal O}_{x,X} \rightarrow {\cal O}_{(1,x),G \times Z} \rightarrow 0.$ Since $X$ and $G \times Z$ have the same dimension, and we assume $X$ is normal, the map is also an injection since $\widehat{\cal O}_{x,X}$ is integral domain. Therefore $\psi$ is \'etale at $x$. Since $G$ acts by automorphisms, the open neighborhood of $(1,x)$ where $\psi$ is \'etale is $G$-invariant. The group acts transitively on itself so any invariant neighborhood of $(1,x)$ is of the form $G \times U$ where $x \in U \subset Z$. Since $G \times U \rightarrow X$ is \'etale the map $U \rightarrow Y$ is \'etale. Thus $Y =[X/G]$ has an \'etale cover by schemes and is therefore an algebraic space. Finally, $X \rightarrow Y$ is a principal bundle, since $Y = [X/G]$ and (tautologically) we have $X \times_{[X/G]} X = X \times G$. $\Box$ \medskip Using representable \'etale coverings, one can extend basic definitions about schemes to algebraic spaces. Much of this is done in \cite{Knutson}, where more complete definitions and details can be found. Here are some examples. Any sheaf on the category of schemes (e.g. ${\cal O}$) extends uniquely to a sheaf on the category of algebraic spaces: if $U$ is an affine \'etale covering of the space $A$, and $R$ is as above, then ${\cal O}(A) = \mbox{Ker}( {\cal O}(U) \rightarrow {\cal O}(R))$. Likewise, a property $P$ of schemes is called stable if given an \'etale covering $\{X_i \rightarrow X \}$, $X$ has $P$ if and only if $X_i$ has $P$. Any stable property of schemes extends to a property of algebraic spaces by defining it in terms of representable \'etale coverings. Thus, one can speak of algebraic spaces which are normal, smooth, reduced, $n$-dimensional, etc. Similarly, if $P$ is a stable property of maps of schemes such that $P$ either (a) is local on the domain, or (b) satisfies effective descent, then $P$ extends to a property of maps of algebraic spaces. For example, one can speak of maps of algebraic spaces which are (a) faithfully flat, flat, \'etale, universally open, etc., or (b) open immersions, closed immersions, affine or quasi-affine morphisms, etc. Likewise, again using representable \'etale coverings, one can extend facts and constructions about schemes, e.g. $\mbox{Proj}$, fiber products, divisors, etc., to algebraic spaces; again much of this is done in \cite{Knutson}. The definition of Chow groups of schemes given in \cite{Fulton} generalizes immediately to algebraic spaces. (A similar definition was given for stacks in \cite{Gi}.) If $X$ is an algebraic space, define the group of $k$-cycles $Z_k(X)$ to be the free abelian group generated by integral subspaces of dimension $k$. To define rational equivalence, first note that if $X$ is an integral algebraic space, then the group of rational functions on $X$ is defined. (Indeed, by the above remarks, $X$ has an open dense subspace $X^0$ which is a variety, and the rational functions on $X$ are the same as those on $X^0$.) If $Y$ is an integral subspace of $X$ of codimension $1$ and $f$ is a rational function on $X$, then the order of vanishing of $f$ along $Y$, denoted $\mbox{ord}_Y(f)$, can be defined by taking an \'etale map $\phi: U \rightarrow X$, where $U$ is a variety and $\phi(U)$ has nonempty intersection with $Y$, and setting $\mbox{ord}_Y(f) = \mbox{ord}_{\phi^{-1}(Y)}(\phi^*f)$, where the right hand side is the definition for schemes in \cite{Fulton}. If $W$ is a $k+1$-dimensional integral subspace of $X$ and $f$ is a rational function on $W$, define $\mbox{div}(f) \in Z_k(X)$ to be $\sum \mbox{ord}_Y(f) [Y]$ where the sum is over all codimension $1$ integral subspaces of $Y$. Then, exactly as in \cite{Fulton}, define $\mbox{Rat}_k(X)$ to be the subgroup of $Z_k(X)$ generated by all $\mbox{div}(f)$, for $f$ and $W$ as above, and define the Chow groups $A_k(X) = Z_k(X) / \mbox{Rat}_k(X)$. The arguments of \cite[Chapters 1-6]{Fulton} can be carried over almost unchanged to show that Chow groups of algebraic spaces have the same functorial properties as Chow groups of schemes. Many of the facts about schemes needed to prove this are extended to algebraic spaces in \cite{Knutson}. As a illustration, we will discuss the construction of Gysin homomorphisms for regular embeddings, which is the central construction of the first six chapters of \cite{Fulton}. If $X \rightarrow Y$ is any closed embedding of algebraic spaces then we can define the cone $C_XY$ as for schemes, since the $\mbox{Spec}$ construction for sheaves (in the \'etale topology) of ${\cal O}_Y$ algebras defines an algebraic space over $Y$. If $X \rightarrow Y$ is a regular embedding of codimension $d$, then $C_XY = N_XY$ is a vector bundle of rank $d$. We can then define a specialization homomorphism $Z_k(Y) \rightarrow Z_k(C_XY)$ as in \cite[Section 5.2]{Fulton}. The deformation to the normal bundle construction of \cite[Section 5.1]{Fulton} goes through unchanged, since the blow-up $Y \times {\bf P}^1$ along the subspace $X \times \infty$ is defined in the category of algebraic spaces. (The existence of blow-ups is a consequence of the $\mbox{Proj}$ construction for graded algebras over algebraic spaces.) Thus as in Fulton, the specialization map passes to rational equivalence. In particular if $X \rightarrow Y$ is a regular embedding of codimension $d$, then the construction of \cite[Chapter 6]{Fulton} goes through, and we obtain a (refined) Gysin homomorphism. If $X$ is a (separated) smooth algebraic space, then the diagonal map is a regular embedding. Therefore, the integral Chow groups of $X$ have an intersection product. Note also that algebraic spaces have an operational Chow ring with the same formal properties as that of \cite[Chapter 17]{Fulton}. This follows from the fact that the ordinary Chow groups of algebraic spaces have the same functorial properties as Chow groups of schemes. In particular, if $X$ is a smooth algebraic space of dimension $n$, then $A^i(X) = A_{n-i}(X)$, with the map defined as in \cite{Fulton}. {\bf Remark.} For algebraic stacks which have automorphisms, the diagonal is not a regular embedding in the sense we have defined. If the stack is Deligne-Mumford, then the diagonal is a local embedding (i.e. unramified). For such morphisms, Vistoli constructed a Gysin pullback with ${\bf Q}$ coefficients on the Chow groups of integral substacks. To obtain a good intersection theory with ${\bf Z}$ coefficients on arbitrary algebraic stacks, a different definition of Chow groups is required. For quotient stacks, equivariant Chow groups give a good definition. This point is discussed in Section \ref{itstacks}. \subsection{Actions of group schemes over a Dedekind domain} \label{mixed} Let $R$ be a Dedekind domain, and set $S = Spec(R)$. Fulton's intersection theory remains valid for schemes and thus algebraic spaces defined over $S$ (\cite[Section 20.2]{Fulton}). Thus, the equivariant theory will work for actions of smooth affine group schemes over $S$, provided we can find finite-dimensional representations of $G/S$ where $G$ acts generically freely. The following lemma shows that this can always be done if the fibers of $G/S$ are connected. \begin{lemma} \label{ded} Let $G/S$ be a smooth affine group scheme defined over $S =\mbox{Spec }R$, where $R$ is Dedekind domain. Then there exists a finitely generated projective $S$-module $E$ , such that $G/S$ acts freely on an open set $U \subset E$ whose complement has arbitrarily high codimension. \end{lemma} Proof. By \cite[Lemma 1]{Seshadri1} the coordinate ring $R(G)$ is a projective $R$-module with a $G$ action. The group $G$ embeds into a finitely generated $R(G)$ submodule $F$. Since $R$ is a Dedekind domain $F$ is also projective. By \cite[Proposition 3]{Seshadri} $F$ is contained in an invariant finitely generated submodule $E$ (which is also projective). $G$ acts freely on itself it acts freely on an open of each fiber of $E/S$. Replacing $E$ by $E \times_S \ldots \times_S E$ we obtain a representation on which $G$ acts freely on an open set $U \subset E$, such that $E-U$ has arbitrarily high codimension. $\Box$ \medskip Thus if $X/S$ is an algebraic space over $S$, we can construct a mixed space $X_G = X \times_G U$, where $U$ is as in the lemma. We then define the $i$-th equivariant Chow group as $A_{i+l-g}(X_G)$ where $l = \mbox{dim }(U/S)$ and $g = \mbox{dim }(G/S)$. Since most of the results of intersection theory hold for algebraic spaces over a Dedekind domain, most of the results on equivariant Chow groups also hold, including the following: (1) The functorial properties with respect to proper, flat and l.c.i maps hold. (2) If $X/S$ is smooth, there is an intersection product on $A_*^G(X)$ for $X/S$ smooth. (3) If $G$ acts freely on $X$ with quotient $X \rightarrow Y$ then $A_{*+g}^G(X) = A_*(Y)$. (4) If $G/S$ acts with (locally) properly on $X/S$, then the theorem of \cite{Kollar}, \cite{KM} implies that a quotient $X \rightarrow Y$ exists as an algebraic space over $S$. The results of Section \ref{qint} (for ordinary Chow groups) generalize, and $A_{*+g}^G(X)_{{\bf Q}} = A_*(Y)_{{\bf Q}}$. {\bf Remark.} Facts (3) and (4) imply that any moduli space over $\mbox{Spec {\bf Z}}$ which is the quotient of a smooth algebraic space by a proper action has a rational Chow ring. \subsection{Some facts about group actions and quotients} \label{whenascheme} Here we collect some useful results about actions of algebraic groups. \begin{lemma} \label{l.free} Suppose that $G$ acts properly on an an algebraic space $X$. If the stabilizers are trivial, then the action is free. \end{lemma} Proof. We must show that the action map $G \times X \rightarrow X \times X$ is a closed embedding. The properness of the action implies that this map is proper and quasi-finite, hence finite. Since the stabilizers are trivial, the map is unramified so it is an embedding in a neighborhood of every point of $G \times X$. Finally, the map is set theoretically injective, hence an embedding. $\Box$ \medskip \begin{lemma} \label{q.exist} (\cite{E-G}) Let $G$ be an algebraic group. For any $i > 0$, there is a representation $V$ of $G$ and an open set $U \subset V$ such that $V-U$ has codimension more than $i$ and such that a principal bundle quotient $U \rightarrow U/G$ exists in the category of schemes. \end{lemma} Proof. Embed $G$ into $GL(n)$ for some $n$. Assume that $V$ is a representation of $GL(n)$ and $U \subset V$ is an open set such that a principal bundle quotient $U \rightarrow U/GL(n)$ exists. Since $GL(n)$ is special, this principal bundle is locally trivial in the Zariski topology. Thus $U$ is locally isomorphic to $W \times GL(n)$ for some open $W \subset U/GL(n)$. A quotient $U/G$ can be constructed by patching the quotients $W \times GL(n) \rightarrow W \times (GL(n)/G)$. (It is well-known that a quotient $GL(n)/G$ exists \cite{Borel}.) We have thus reduced to the case $G=GL(n)$. Let $V$ be the vector space of $n \times p$ matrices with $p > i+ n$, and let $U \subset V$ be the open set of matrices of maximal rank. Then $V - U$ has codimension $p - n + 1$ and the quotient $U/G$ is the Grassmannian $Gr(n,p)$. $\Box$ \medskip \medskip The following proposition gives conditions under which the mixed space $X_G$ is a scheme. Recall that a group is special if every principal bundle is locally trivial in the Zariski topology. The groups $GL(n)$, $SL(n)$, $Sp(2n)$, as well as solvable groups, are special; $PGL(n)$ and $SO(n)$, as well as finite groups, are not \cite{Sem-Chev}. \begin{prop} \label{inap} Let $G$ be an algebraic group, let $U$ be a scheme on which $G$ acts freely, and suppose that a principal bundle quotient $U \rightarrow U/G$ exists. Let $X$ be a scheme with a $G$-action. Assume that one of the following conditions holds: (1) $X$ is (quasi)-projective with a linearized $G$-action, or (2) $G$ is connected and $X$ is equivariantly embedded as a closed subscheme of a normal variety, or (3) $G$ is special. \noindent Then a principal bundle quotient $X \times U \rightarrow X \times^G U$ exists in the category of schemes. \end{prop} Proof. If $X$ is quasi-projective with a linearized action, then there is an equivariant line bundle on $X \times U$ which is relatively ample for the projection $X \times U \rightarrow U$. By \cite[Prop. 7.1]{GIT} a principal bundle quotient $X \times^G U$ exists. Now suppose that $X$ is normal and $G$ is connected. By Sumihiro's theorem \cite{Sumihiro}, $X$ can be covered by invariant quasi-projective open sets which have a linearized $G$-action. Thus, by \cite[Prop. 7.1]{GIT} we can construct a quotient $X_G = X \times^{G} U$ by patching the quotients of the quasi-projective open sets in the cover. If $X$ equivariantly embeds in a normal variety $Y$, then by the above paragraph a principal bundle quotient $Y \times U \rightarrow Y \times^G U$ exists. Since $G$ is affine, the quotient map is affine, and $Y \times U$ can be covered by affine invariant open sets. Since $X \times U$ is an invariant closed subscheme of $Y \times U$, $X \times U$ can also be covered by invariant affines. A quotient $X \times^G U$ can then be constructed by patching the quotients of the invariant affines. Finally, if $G$ is special, then $U \rightarrow U/G$ is a locally trivial bundle in the Zariski topology. Thus $U = \bigcup\{U_\alpha\}$ where $\phi_{\alpha}:U_\alpha \simeq G \times W_\alpha$ for some open $W_\alpha \subset U/G$. Then $\psi_{\alpha}: X \times U_{\alpha} \rightarrow X \times W_{\alpha}$ is a quotient, where $\psi_{\alpha}$ is defined by the formula $(x,w,g) \mapsto (g^{-1}x,w)$. (Here we assume that $G$ acts on the left on both factors of $X \times G$.) $\Box$

9609

alg-geom/9609007

en

https://arxiv.org/abs/alg-geom/9609007

alg-geom/9609007

Frank Sottile

Frank Sottile

Enumerative geometry for real varieties

Based upon the Author's talk at 1995 AMS Summer Research Institute in Algebraic geometry. To appear in the Proceedings. 11 pages, extended version with Postscript figures and appendix available at http://www.msri.org/members/bio/sottile.html, or by request from Author ([email protected])

Proc. Sympos. Pure Math., Vol 62.1, 1997 pp. 435-447

We discuss the problem of whether a given problem in enumerative geometry can have all of its solutions be real. In particular, we describe an approach to problems of this type, and show how this can be used to show some enumerative problems involving the Schubert calculus on Grassmannians may have all of their solutions be real. We conclude by describing the work of Fulton and Ronga-Tognoli-Vust, who (independently) showed that there are 5 real plane conics such that each of the 3264 conics tangent to all five are real.

alg-geom

\section{Introduction} Of the geometric figures in a given family satisfying real conditions, some figures are real while the rest occur in complex conjugate pairs, and the distribution of the two types depends subtly upon the configuration of the conditions. Despite this difficulty, applications~(\cite{Byrnes},\cite{Mourrain_MEGA94},\cite{Ronga_Vust}) may demand real solutions. Fulton~\cite{Fulton_introduction_intersection} asked how many solutions of an enumerative problem can be real, and we consider a special case of his question: Given a problem of enumerative geometry, are there real conditions such that every figure satisfying them is real? Such an enumerative problem is {\em fully real}. B\'{e}zout's Theorem, or rather the problem of intersecting hypersurfaces in ${\Bbb P}^n$, is fully real. This is readily seen for ${\Bbb P}^2$, and the argument generalizes to ${\Bbb P}^n$. Suppose $X_0$ consists of $d$ real lines, $Y_0$ of $e$ real lines, and $X_0$ meets $Y_0$ transversally in (necessarily) $d\cdot e$ real points. Let $X$ and $Y$ be defined by suitably small generic real deformations of the forms defining $X_0$ and $Y_0$. Then $X$ and $Y$ are smooth real plane curves of degrees $d$ and $e$ meeting transversally in $d\cdot e$ real points. This argument used a degenerate case free of multiplicities; $X_0$ and $Y_0$ are reduced and meet transversally. While it is typical to introduce multiplicities (for example, in the proof of B{\'e}zout's Theorem in~\cite{Mumford_cpv}) to establish enumerative formulas, multiplicities may lead to complex conjugate pairs of solutions, complicating the search for real solutions. All Schubert-type enumerative problems involving lines in ${\Bbb P}^n$ are fully real~\cite{sottile_real_lines}. This follows from the existence of (multiplicity-free) deformations of generically transverse intersections of Schubert varieties into sums of Schubert varieties. Refining this method of multiplicity-free deformations~\cite{sottile_mega96} yields techniques for showing other enumerative problems are fully real. Fulton, and more recently, Ronga, Tognoli, and Vust~\cite{Ronga_Tognoli_Vust}, have shown the problem of 3264 conics tangent to five general plane conics is fully real. Their analysis utilizes degenerate conditions having multiplicities. Enumerative problems that we know are not fully real share a common flaw: they do not involve intersecting general subvarieties. For example, Klein~\cite{Klein} showed that at most $n(n-2)$ of the $3n(n-2)$ flexes on a real plane curve of degree $n$ can be real. These flexes are the intersection of the curve with its Hessian determinant, {\em not} with a general curve of degree $3(n-2)$. Also, Khovanskii~\cite{Khovanskii_fewnomials} showed that if hypersurfaces in a complex torus are defined by polynomials with few monomials, then the real points of intersection are at most a fraction of all points of intersection. However, these are not generic hypersurfaces with given Newton polytope. Little is known about fully real enumerative problems. For instance, we are unaware of a good theoretical framework for studying fully real enumerative problems. Also, it is not known how common it is for an enumerative problem to be fully real. Here are some examples of enumerative problems worth considering: \begin{enumerate} \item Is the Kouchnirenko-Bernstein Theorem~(\cite{Bernstein},~\cite{Kouchnirenko}) for hypersurfaces in a torus fully real? That is, given lattice polytopes $\Delta_1,\ldots,\Delta_n$ in ${\Bbb Z}^n$ are there real polynomials $f_1,\ldots,f_n$ where $\Delta_i$ is the Newton polytope of $f_i$ and all solutions to the system $f_1=\cdots=f_n=0$ in $({\Bbb C}^\times)^n$ are real? \item Generalize the results of~\cite{sottile_real_lines} and~\cite{sottile_mega96}: Are other (all?) Schubert-type enumerative problems on flag varieties fully real? \item All known examples involve spherical varieties~(\cite{Brion_spherical_introduction},% ~\cite{Knop_spherical_expository},~\cite{Luna_Vust_plongements}). Which enumerative problems on other spherical varieties are fully real? \item In~\cite{sottile_mega96} all problems of enumerating lines incident upon subvarieties of fixed dimension and degree in ${\Bbb P}^n$ are shown to be fully real. What is the situation for rational curves of higher degree? (Degree 0 is B{\'e}zout's Theorem.) For example, for which positive integers $d$ do there exist $3d-1$ real points in ${\Bbb P}^2$ such that the Kontsevich number $N_d$ of degree $d$ rational curves passing through these points~(\cite{Kontsevich_Manin}~\cite{Ruan_Tian}) are all real? For an introduction to these questions of quantum cohomology, see the paper by Fulton and Pandharipande~\cite{Fulton_Rahul} in this volume. \end{enumerate} This technique of multiplicity-free deformations may have applications beyond showing the existence of real solutions. When the deformations are explicitly described (which is the case in most known results), it may be possible to obtain explicit solutions to the enumerative problem using continuation methods of numerical analysis~\cite{Allgower_Georg_1990} to follow real points in the degenerate configuration backwards along the deformation. Algorithms to accomplish this have recently been developed in the case of intersecting hypersurfaces in a {\em complex\/} torus~\cite{CVVerschelde,Huber_Sturmfels}. This note is organized as follows: In \S 2 we discuss some examples of fully real enumerative problems for which multiplicity-free deformations play a central role. This technique is illustrated in \S 3, where we show that there are nine real Veronese surfaces in ${\Bbb P}^5$ such that the $11010048$ planes meeting all nine are real. We conclude with a discussion of the work of Fulton and of Ronga, Tognoli, and Vust~\cite{Ronga_Tognoli_Vust} on the problem of conics tangent to five conics and show that the multiplicities they introduce are unavoidable. \section{Effective Rational Equivalence} A common feature of many fully real enumerative problems is multiplicity-free deformations of intersection cycles. Effective rational equivalence is a precise formulation of this for Grassmannians and flag varieties. \subsection{Real effective rational equivalence} Varieties will be quasi-projective, reduced, complex and defined over the real numbers, ${\Bbb R}$. Let $X$ be a Grassmannian or flag variety, $G$ a linear algebraic group which acts transitively on $X$, and $B$ a Borel subgroup of $G$. The letters $U$ and $V$ denote smooth rational varieties. Let the real points $Y({\Bbb R})$ of a variety $Y$ be equipped with the classical topology. A subvariety $\Xi\subset U\times X$ (or $\Xi\rightarrow U$) with generically reduced equidimensional fibres over $U$ is a {\em family of (multiplicity-free) cycles on $X$ over $U$}. We assume all families are $G$-stable; if $Y$ is a fibre of $\Xi$ over $U$, then so are all translates of $Y$. Associating a point $u$ of $U$ to the fundamental cycle of the fibre $\Xi_u$ determines a morphism $\phi: U\rightarrow \mbox{\it Chow}\, X$. Here, $\mbox{\it Chow}\, X$ is the Chow variety of $X$ parameterizing cycles of the same dimension and degree as $\Xi_u$ (\cite{Samuel}, \S I.9). A priori, $\phi$ is only a function. However, if $C\subset U$ is a smooth curve, then $\Xi|_C$ is flat and the canonical map of the Hilbert scheme to the Chow variety (\cite{Mumford_Fogarty}, \S 5.4) shows $\phi|_C$ is a morphism. By Hartogs' Theorem on separate analyticity, $\phi$ is in fact a morphism. In fact, if $U$ is normal, then $\phi$ is a morphism~(\cite[\S 1]{Kollar_rational} or~\cite[\S 3]{Friedlander_Mazur}). For a discussion of Chow varieties in the analytic category (which suffices for our purposes), see~\cite{Barlet}. Any cycle $Y$ on $X$ is rationally equivalent to an integral linear combination of Schubert classes. As Hirschowitz~\cite{Hirschowitz} observed, this rational equivalence occurs within the closure of $B\cdot Y$ in $\mbox{\it Chow}\, X$ since $B$-stable cycles of $X$ ($B$-fixed points in $B\cdot Y$) are integral linear combinations of Schubert varieties. If any coefficients in this linear combination exceed 1, this stable cycle has multiplicities. A family $\Xi\rightarrow U$ of multiplicity-free cycles on $X$ has {\em effective rational equivalence} with {\em witness} $Z$ if there is a cycle $Z\in \overline{\phi(U)}$ which is a sum of distinct Schubert varieties, and hence multiplicity-free. An effective rational equivalence is {\em real} if $Z\in \overline{\phi(U({\Bbb R}))}$ and each component of $Z$ is a Schubert variety defined by a real flag. Suppose $\Xi_1\rightarrow U_1,\ldots,\Xi_b\rightarrow U_b$ are $G$-stable families of multiplicity-free cycles on $X$. By Kleiman's Transversality Theorem~\cite{Kleiman}, there is a nonempty open set $U\subset \prod_{i=1}^b U_i$ consisting of $b$-tuples $(u_1,\ldots,u_b)$ such that the fibres $(\Xi_1)_{u_1},\ldots,(\Xi_b)_{u_b}$ meet generically transversally. Let $\Xi\subset U\times X$ be the resulting family of intersection cycles and call $\Xi\rightarrow U$ the {\em intersection problem} given by $\Xi_1,\ldots,\Xi_b$. \begin{thm}\label{thm:real_lines}\ Any intersection problem given by families of Schubert varieties in the Grassmannian of lines in projective space has real effective rational equivalence. \end{thm} We present a synopsis of the proof in~\cite{sottile_real_lines}: Let $X$ be the Grassmannian of lines in ${\Bbb P}^n$ and suppose $\Xi\rightarrow U$ is an intersection problem given by families of Schubert varieties. A sequence $\Psi_0\rightarrow V_0,\ldots,\Psi_c\rightarrow V_c$ of families of multiplicity-free cycles on $X$ is constructed with each $V_i$ rational, where $\Psi_0\rightarrow V_0$ is the family $\Xi\rightarrow U$, $V_c$ is a point, and $\Psi_c$ a union of distinct real Schubert varieties. For each $i=0,\ldots, c$, let ${\cal G}_i\subset \mbox{\it Chow}\, X$ be $\phi(V_i({\Bbb R}))$, the set of fibres of the family $\Psi_i\rightarrow V_i$ over $V_i({\Bbb R})$. Then ${\cal G}_i\subset \overline{{\cal G}_{i-1}}$: For any $v\in V_i({\Bbb R})$ a family $\Gamma\rightarrow C$ of cycles is constructed with $C$ a smooth rational curve, the cycle $(\Psi_i)_v$ a fibre over $C({\Bbb R})$, and all other fibres of $\Gamma$ are fibres of $\Psi_{i-1}$. This family induces a morphism $\phi:C\rightarrow \mbox{\it Chow}\, X$, which shows $(\Psi_i)_v\in \overline{{\cal G}_{i-1}}$ since $\phi(C({\Bbb R}))-\{(\Psi_i)_v\} \subset {\cal G}_{i-1}$. It follows that $\Psi_c \in \overline{{\cal G}_0} = \overline{\phi(U({\Bbb R}))}$, showing $\Xi\rightarrow U$ has real effective rational equivalence. \QED It is typically difficult to describe an intersection of several Schubert varieties. While this task is easier when they are in special position, even this may be too hard. It is better yet to consider the limiting position of intersection cycles as the subvarieties being intersected degenerate to the point of attaining excess intersection. This is the aim of effective rational equivalence. For example, the `limit cycle' $\Psi_c$ of the previous proof is generally not an intersection of Schubert varieties, however, it is a deformation of such cycles. An {\em enumerative problem} of degree $d$ is an intersection problem $\Xi\rightarrow U$ with finite fibres of cardinality $d$. It is {\em fully real\,} if there is a fibre $\Xi_u$ with $u\in U({\Bbb R})$ consisting entirely of real points. Here, $u=(u_1,\ldots,u_b)$ with $u_i\in U_i({\Bbb R})$ and $\Xi_u$ is the transverse intersection of the cycles $(\Xi_1)_{u_1},\ldots,(\Xi_b)_{u_b}$. The set ${\cal M}\subset \mbox{\it Sym}^d X$ of degree $d$ zero cycles consisting of $d$ distinct real points of $X$ is an open subset of $(\mbox{\it Sym}^d X)({\Bbb R})$. Thus $\Xi\rightarrow U$ is fully real if and only if it has real effective rational equivalence. Hence Theorem~\ref{thm:real_lines} has the following consequence: \begin{cor}\label{cor:real_lines} Any enumerative problem given by Schubert conditions on lines in projective space is fully real. \end{cor} \subsection{Products in $A^*X$} $X$ is the quotient $G/P$ of $G$ by a parabolic subgroup $P$. A Schubert subvariety $\Omega_w{F\!_{\DOT}}$ of $X$ is given by a complete flag ${F\!_{\DOT}}$ and a coset $w$ of the corresponding parabolic subgroup of the symmetric group (\cite{Bourbaki_Groupes_IV}, Ch.~IV, \S 2.5). Call $w$ the {\em type} of $\Omega_w{F\!_{\DOT}}$. A Schubert class $\sigma_w$ is the cycle class of $\Omega_w{F\!_{\DOT}}$. Let $\Xi_1\rightarrow U_1,\ldots,\Xi_b\rightarrow U_b$ be families of cycles on $X$ giving an intersection problem $\Xi\rightarrow U$. Then fibres of $\Xi\rightarrow U$ have cycle class $\prod_i \beta_i$, where $\beta_i$ is the cycle class of fibres of $\Xi_i\rightarrow U_i$. Suppose $\Xi\rightarrow U$ has effective rational equivalence with witness $Z$. Let $c_w$ count the components of $Z$ of type $w$. Since $Z$ is rationally equivalent to fibres of $\Xi\rightarrow U$, we deduce the formula in $A^*X$. $$ \prod_{i=1}^b \beta_i\quad =\quad \sum_w c_w\cdot \sigma_w. $$ \subsection{Pieri-type formulas} Given a such product formula with each $c_w\leq 1$, the action of a real Borel subgroup $B$ of $G$ shows that the family of intersection cycles $\Xi\rightarrow U$ has real effective rational equivalence: Let $Y$ be a fibre of $\Xi\rightarrow U$ over a real point of $U$. Then the closure of the orbit $B({\Bbb R})\cdot Y$ in $\mbox{\it Chow}\, X({\Bbb R})$ contains a $B({\Bbb R})$-fixed point $Z$, as Borel's fixed point Theorem (\cite{Borel_groups}, III.10.4), holds for $B({\Bbb R})$-stable real analytic sets. Moreover, $Z$ is multiplicity-free as $c_w\leq 1$. In the Grassmannian of $k$-planes in ${\Bbb P}^n$, a {\em special Schubert variety} is the locus of $k$-planes having excess intersection with a fixed linear subspace. A {\em special Schubert variety} of a flag variety is the inverse image of a special Schubert variety in a Grassmannian projection. Pieri's formula for Grassmannians~(\cite{Griffiths_Harris},~\cite{Hodge_Pedoe}) and the Pieri-type formulas for flag varieties~(\cite{Lascoux_Schutzenberger_polynomes_schubert ,~\cite{sottile_pieri_schubert}) show that the coefficients $c_w$ in a product of a Schubert class with a special Schubert class are either 0 or 1. Thus any intersection problem given by a Schubert variety and a special Schubert variety has real effective rational equivalence. We use this to prove the following theorem. \begin{thm}\label{thm:three_special} Any enumerative problem in any flag variety given by five Schubert varieties, three of which are special, is fully real. \end{thm} {\sc Proof}. First pair each non-special Schubert variety with a special Schubert variety. The associated families $\Xi\rightarrow U$ and $\Xi'\rightarrow U'$ of intersection cycles have real effective rational equivalence with witnesses $Z$ and $Z'$, respectively. Since the coefficients $c_w$ in the Pieri-type formulas are either 0 or 1, a zero-dimensional intersection of three real Schubert varieties in general position where one is special is a single real point. Considering components of $Z$ and $Z'$ separately, we see that if $Z$, $Z'$, and the third special Schubert variety $Y$ are in general position with $Y$ real, then they intersect transversally with all points of intersection real. Suitably small deformations of $Z$ and $Z'$ into real fibres of $\Xi$ and $\Xi'$ preserve the number of real points of intersection, completing the proof. \QED \section{The Grassmannian of planes in ${\Bbb P}^5$} The Grassmannian of planes in ${\Bbb P}^5$, ${\Bbb G}\,_{2,5}$, is a 9-dimensional variety. If $K$ is a plane in ${\Bbb P}^5$, then the set $\Omega(K)$ of planes which meet $K$ is a hyperplane section of ${\Bbb G}\,_{2,5}$ in its Pl{\"u}cker embedding. Thus the number of planes which meet 9 general planes is the degree of ${\Bbb G}\,_{2,5}$, which is $\frac{1!2!9!}{3!4!5!}=42$~\cite{Schubert_degree}. This variety is the smallest dimensional flag variety for which an analog of Corollary~\ref{cor:real_lines} is not known. We illustrate the methods of \S 2 to prove the following result: \begin{thm}\label{thm:42_planes} There are 9 real planes in ${\Bbb P}^5$ such that the 42 planes meeting all 9 are real. \end{thm} The Veronese surface in ${\Bbb P}^5$ is the image of ${\Bbb P}^2$ under the embedding induced by the complete linear system $|{\cal O}(2)|$, and so it has degree 4. \begin{cor} There are 9 real Veronese surfaces in ${\Bbb P}^5$ such that the\ 11010048 ($= 4^9\cdot 42$) planes meeting all 9 are real. \end{cor} {\sc Proof}. Let $x_{ij}$ , $1\leq i\leq j\leq 3$, be real coordinates for ${\Bbb P}^5$. For $t\neq 0$ \begin{multline}\label{eq:ideal_family} \langle \underline{x_{11}x_{33}}-t^4\,x_{13}^2,\ \underline{x_{11}x_{22}}-t^2\,x_{12}^2,\ \underline{x_{11}x_{23}}-t\, x_{12}x_{13},\\ \underline{x_{12}x_{33}}-t\, x_{13}x_{23},\ \underline{x_{13}x_{22}}-t\, x_{12}x_{23},\ \underline{x_{22}x_{33}}-t^2\,x_{23}^2 \rangle \end{multline} generates the ideal of a Veronese surface, ${\cal V}(t)$ (cf.~\cite{Sturmfels_grobner_ULS}, p.~142), which is real for $t\in{\Bbb R}$. This family of Veronese surfaces is induced by the (real) ${\Bbb C}^\times$-action on the space of linear forms on ${\Bbb P}^5$: $$ x_{ij}\quad \mapsto\quad t^{j-i}x_{ij}\qquad\mbox{for }t\in {\Bbb C}^\times . $$ The ideal of the special fibre ${\cal V}(0)$ of this family is generated by the underlined terms, so ${\cal V}(0)$ is the union of the four planes given by the ideals: \begin{equation}\label{eq:four_planes} \langle x_{11},x_{22},x_{33}\rangle \qquad \langle x_{ii},x_{jj},x_{ij}\rangle,\quad ij=12, 13, 23. \end{equation} By Theorem~\ref{thm:42_planes}, there exist 9 real planes $K_1,\ldots,K_9$ such that $\bigcap_{i=1}^9 \Omega(K_i)$ is a transverse intersection consisting of 42 real planes. This property of $K_1,\ldots, K_9$ is preserved by small real deformations. So for each $1\leq i\leq 9$, there is a neighborhood $W_i$ of $K_i$ in ${\Bbb G}\,_{2,5}({\Bbb R})$ such that if $K'_i\in V_i$ for $1\leq i\leq 9$, then $\bigcap_{i=1}^9 \Omega(K'_i)$ is transverse and consists of 42 real planes. For each $1\leq i\leq 9$, choose a set of real coordinates for ${\Bbb P}^5$ so that the four planes, $K_{i,j}$, for $j=1,2,3,4$, defined by the ideals of~(\ref{eq:four_planes}) are in $W_i$. In these same coordinates, consider the family ${\cal V}_i(t)$ of real Veronese surfaces given by the ideals~(\ref{eq:ideal_family}) with special member ${\cal V}_i(0) = K_{i,1} + K_{1,2}+ K_{i,3}+ K_{i,4}$. If the sets of coordinates are chosen sufficiently generally, there exists $\epsilon >0$ such that whenever $t\in (0,\epsilon)$, there are exactly $4^9\cdot 42$ real planes meeting each of ${\cal V}_1(t), \ldots,{\cal V}_9(t)$. This is because there are $4^9\cdot 42$ real planes meeting each of ${\cal V}_1(0),\ldots,{\cal V}_9(0)$, as $$ \bigcap_{i=1}^9 \left(\Omega(K_{i,1})+\Omega(K_{i,2})+ \Omega(K_{i,3})+\Omega(K_{i,4}) \right) $$ is a transverse intersection consisting of $4^9\cdot 42$ real planes: Since $K_{i,j}\in W_i$ for $1\leq i\leq 9$ and $1\leq j\leq 4$, this follows if the $4^9$ sets of 42 planes $\bigcap_{i=1}^9\Omega(K_{i,l_i})$ given by all sequences $l_i$, where $1\leq l_i\leq 4$ for $1\leq i\leq 9$, are pairwise disjoint. But this may be arranged when choosing the seta of coordinates. \QED \begin{lemma}\label{lemma:real_eff_rat_equiv} The intersection problem of planes meeting 4 given planes in ${\Bbb P}^5$ has real effective rational equivalence. \end{lemma} {\sc Proof of Theorem~\ref{thm:42_planes} using Lemma~\ref{lemma:real_eff_rat_equiv}}. Partition the 9 planes into two sets of 4, and a singleton. Apply Lemma~\ref{lemma:real_eff_rat_equiv} to the intersection problems $\Xi\rightarrow U$, $\Xi'\rightarrow U'$ given by each set of 4, obtaining witnesses $Z$ and $Z'$. Arguing as for Theorem~\ref{thm:three_special} completes the proof. \QED {\sc Proof of Lemma~\ref{lemma:real_eff_rat_equiv}}. We use an economical notation for Schubert varieties. A partial flag $A_0\subsetneqq A_1\subsetneqq A_2\subset {\Bbb P}^5$ determines a Schubert subvariety of ${\Bbb G}\,_{2,5}$: $$ \Omega(A_0,A_1,A_2) := \{H\in {\Bbb G}\,_{2,5}\,|\, \dim H\bigcap A_i \geq i, \mbox{ for }i=0,1,2\}. $$ If $A_j$ is a hyperplane in $A_{j+1}$ or if $A_j={\Bbb P}^5$, then it is no additional restriction for $\dim H\bigcap A_j\geq j$. We omit such inessential conditions. Thus, if $\mu\subsetneqq M\subsetneqq\Lambda$ is a partial flag, then $\Omega(\mu), \Omega(\dotc,M)$, and $\Omega(\mu,\dotc,\Lambda)$ are, respectively, those planes $H$ which meet $\mu$, those $H$ with $\dim H\cap M\geq 1$, and those $H\subset\Lambda$ which also meet $\mu$. Let $\Xi\subset U\times {\Bbb G}\,_{2,5}$ be the intersection problem of planes meeting four given planes. Then $U\subset ({\Bbb G}\,_{2,5})^4$ is the set of 4-tuples of planes $(K_1,K_2,K_3,K_4)$ such that $\bigcap_{i=1}^4 \Omega(K_i)$ is a generically transverse intersection and the fibre of $\Xi$ over $(K_1,K_2,K_3,K_4)$ is $\bigcap_{i=1}^4 \Omega(K_i)$. We show $\Xi\rightarrow U$ has real effective rational equivalence by exhibiting a family $\Psi\subset V\times {\Bbb G}\,_{2,5}$, satisfying the four conditions: \begin{enumerate} \item[(a)] $V$ is rational. In fact $V$ is a dense subset of Magyar's configuration variety ${\cal F}_D$~\cite{Magyar_Borel-Weil}, where $D$ is the diagram $$ \DIAG $$ \item[(b)] $V$ has a dense open subset $V^\circ$ such that the fibres of $\Psi|_{V^{\circ}}$ are also fibres in the family $\Xi\rightarrow U$. \item[(c)] $V$ has a rational subset $V'$ such that the fibres of $\Psi|_{V'}$ are unions of distinct Schubert varieties, real for real points of $V'$. \item [(d)]$V'({\Bbb R})\subset\overline{V^\circ({\Bbb R})}$. Hence $\phi(V'({\Bbb R}))\subset \overline{\phi(U({\Bbb R}))}$. Together with (c), this shows $\Xi\rightarrow U$ has effective rational equivalence. \end{enumerate} Let $V\subset ({\Bbb G}_{1,5})^3\times({\Bbb G}\,_{3,5})^3$ be the locus of sextuples $(\mu_1,\mu_2,\lambda;M_1,M_2,L)$ such that $\mu_i\subset M_i$, $i=1,2$, $\mu_1,\mu_2\subset L$, $\lambda \subset M_1\bigcap M_2$, and $\mu_i\not\subset M_j, i\neq j$. We illustrate the inclusions: $$ \begin{picture}(100,50) \put(0,0){$\mu_1$} \put(40,0){$\lambda$} \put(85,0){$\mu_2$} \put(0,38){$M_1$} \put(40,38){$L$} \put(85,38){$M_2$} \put(5,10){\line(0,1){25}} \put(90,10){\line(0,1){25}} \put(40,10){\line(-1,1){25}} \put(50,10){\line(1,1){25}} \put(40,35){\line(-1,-1){11}} \put(26,21){\line(-1,-1){11}} \put(50,35){\line(1,-1){11}} \put(64,21){\line(1,-1){11}} \end{picture} $$ Let $V^\circ\subset V$ be the dense locus where $\langle \mu_i,M_j\rangle ={\Bbb P}^5$ for $i\neq j$. Then $\lambda = M_1\bigcap M_2$ and $L=\langle \mu_1,\mu_2\rangle$. Let $V'\subset V$ be the locus where $\mu_1\bigcap \mu_2$ is a point, so that $\langle M_1,M_2\rangle$ is a hyperplane. Then $V'$ is rational and $V'({\Bbb R})\subset \overline{V^\circ({\Bbb R})}$, proving (d). We define the family $\Psi$. For $v=(\mu_1,\mu_2,\lambda;M_1,M_2,L)\in V$, let $\Psi_v$ be the cycle \begin{multline}\label{eq:Psi_v} \Omega(\mu_1)\bigcap\Omega(\dotc,M_2) \ +\ \Omega(\mu_2)\bigcap\Omega(\dotc,M_1) \ +\\ \{H\in \Omega(\mu_1,L)\,|\, H\bigcap \mu_2\neq \emptyset\} \ +\ \{H\in \Omega(\lambda,M_1)\,|\, \dim H\bigcap M_2\geq 1\}. \end{multline} Let $\Psi\subset {\Bbb G}\,_{2,5}\times V$ be the subvariety with fibre $\Psi_v$ over points $v\in V$. Suppose $v\in V^\circ$. Since $L=\langle \mu_1,\mu_2\rangle$ and $\mu_1\bigcap\mu_2=\emptyset$, $$ \Omega(\mu_1)\bigcap\Omega(\mu_2) \ =\ \{H\in \Omega(\mu_1,L)\,|\, H\bigcap \mu_2\neq \emptyset\}, $$ as any plane meeting both $\mu_1$ and $\mu_2$ must intersect their span $L$ in at least a line. Similarly, $\Omega(\dotc,M_1)\bigcap\Omega(\dotc,M_2)$ is the fourth term of the cycle (\ref{eq:Psi_v}): If $l_i$ is a line in $H\bigcap M_i$ for $i=1,2$, then $l_1\bigcap l_2 \subset \lambda = M_1\bigcap M_2$. Thus $\Psi_v=\bigcap_{i=1}^2\left(\Omega(\mu_i)+\Omega(\dotc,M_i)\right)$. Since the pairs of subspaces $(\mu_1,\mu_2)$, $(M_1,M_2)$, and $(\mu_i,M_j)$ for $i\neq j$ are in general position, this intersection is generically transverse. We claim $\Psi_v$ is a fibre of $\Xi\rightarrow U$: Let $K_i,K_i'$ for $i=1,2$ be planes such that $\mu_i=K_i\bigcap K'_i$ and $M_i=\langle K_i,K_i'\rangle$. Then $\Omega(K_i)\bigcap\Omega(K'_i) = \Omega(\mu_i)+\Omega(\dotc,M_i)$: If a plane $H$ meets both $K_i$ and $K_i'$, either it meets their intersection $\mu_i$, or else it intersects their span $M_i$ in at least a line. Moreover, while $K_i,K_i'$ are not in general position, this intersection is generically transverse as a proper intersection of a Schubert variety with a special Schubert variety is necessarily generically transverse (\cite{sottile_explicit_pieri}, \S 2.7). Thus $\Psi_v = \Xi_{(K_1,K'_1,K_2,K'_2)}$, proving (b). To show (c), let $v=(\mu_1,\mu_2,\lambda;M_1,M_2,L)\in V'$. Set $p=\mu_1\bigcap \mu_2$, a point and $\Lambda=\langle M_1,M_2\rangle$, a hyperplane. Then $\langle \mu_1,\mu_2\rangle$ is a plane $\nu$ contained in $L$ and $M_1\bigcap M_2$ is a plane $N$ containing $\lambda$. We illustrate these inclusions: $$ \begin{picture}(108,100)(-3,-5) \put(33,0){$p$} \put(-1,20){$\mu_1$}\put(32,18){$\lambda$}\put(91,20){$\mu_2$} \put(31,41){$N$}\put(60,40){$\nu$}\put(-3,60){$M_1$} \put(60,61){$L$}\put(89,60){$M_2$}\put(60,82){$\Lambda$} \put(30,5){\line(-3,2){18}}\put(30,48){\line(-3,2){18}} \put(88,25){\line(-3,2){20}}\put(88,70){\line(-3,2){18}} \put(45,5){\line(3,1){40}}\put(45,48){\line(3,1){42}} \put(13,25){\line(3,1){21}}\put(37,33){\line(3,1){20}} \put(13,68){\line(3,1){43}} \put(35.5,7){\line(0,1){9}}\put(35.5,29){\line(0,1){9}} \put(64,48){\line(0,1){5}}\put(64,56){\line(0,1){3}} \put(64,71){\line(0,1){8}} \put(5,28){\line(0,1){29}}\put(95,28){\line(0,1){29}} \end{picture} $$ To complete the proof, we show $\Psi_v$ is the sum of Schubert varieties \begin{multline*} \Omega(\mu_1,\dotc,\Lambda) \ +\ \Omega(p,M_2) \ +\ \Omega(\mu_2,\dotc,\Lambda) \ +\ \Omega(p,M_1) \ +\\ \Omega(\dotc,\nu) \ +\ \Omega(p,L) \ +\ \Omega(\lambda,\dotc,\Lambda) \ +\ \Omega(\dotc,N). \end{multline*} First note that $$\Omega(\mu_1)\bigcap \Omega(\dotc,M_2)\ =\ \Omega(\mu_1,\dotc,\Lambda) + \Omega(p,M_2): $$ If $H\in \Omega(\mu_1)\bigcap \Omega(\dotc,M_2)$, then either $H\bigcap \mu_1\not\subset M_2$, so that $H\subset \langle \mu_1,M_2\rangle = \Lambda$, or else $p\in H$ so that $H\in \Omega(p,M_2)$. Similarly, we have $\Omega(\mu_2)\bigcap \Omega(\dotc,M_1)= \Omega(\mu_2,\dotc,\Lambda) + \Omega(p,M_1)$. These intersections are generically transverse, as they are proper. Furthermore, $$ \{H\in \Omega(\mu_1,L)\,|\, H\bigcap \mu_2\neq \emptyset\} \ =\ \Omega(\dotc,\nu) \ +\ \Omega(p,L): $$ Either $H\bigcap \mu_1\bigcap \mu_2= \emptyset$, thus $\dim H\bigcap\langle \mu_1,\mu_2\rangle\geq 1$, and so $H\in\Omega(\dotc,\nu)$, or else $p\in H$, so that $H\in\Omega(p,L)$. Finally, $$ \{H\in \Omega(\lambda,M_1)\,|\, \dim H\bigcap M_2\geq 1\}\ =\ \Omega(\lambda,\dotc,\Lambda) \ +\ \Omega(\dotc,N): $$ Either $H\bigcap M_1\not\subset M_2$ thus $H\subset \langle M_1,M_2\rangle=\Lambda$ and so $H\in \Omega(\lambda,\dotc,\Lambda)$, or else $\dim H\bigcap M_1\bigcap M_2\geq 1$, so that $H\in \Omega(\dotc,N)$. \QED Note that for $v\in V'$, the fibre $\Psi_v$ is {\em not} an intersection of four Schubert varieties of type $\Omega(K)$, for $K$ a plane: The Schubert subvariety $\Omega(\dotc,N)$ is the locus of planes which contain a line $l\subset N$ and hence it consists of all planes of the form $\langle q,l\rangle$, where $l\subset N$ is a line and $q\in {\Bbb P}^5\setminus l$ is a point. Suppose $\Psi_v\subset \Omega(K)$ so that $\Omega(\dotc,N)\subset\Omega(K)$. Then for every line $l\subset N$ and point $q\in {\Bbb P}^5\setminus l$, we have $K\bigcap\langle q,l\rangle\neq\emptyset$. This implies that $K\bigcap l\neq \emptyset$ for every line $l\subset N$, and hence that $\dim K\bigcap N\geq 1$. Similarly, $\dim K\bigcap \nu\geq 1$, and so $K\bigcap N\bigcap\nu\neq\emptyset$, thus $p\in K$. This shows $\Omega(p)\subset \Omega(K)$ and so if $\Psi_v\subset \Omega(K_1)\bigcap\Omega(K_2) \bigcap\Omega(K_3)\bigcap\Omega(K_4)$, then this intersection must contain $\Omega(p)$. Hence $\Psi_v$ is a proper subset of the intersection. If $a_i=\dim A_i$, then $\sigma_{a_1a_2a_3}$ is the rational equivalence class of $\Omega(A_0,A_1,A_2)$. By the observation of \S 2.2, Lemma~\ref{lemma:real_eff_rat_equiv} implies the formula in $A^*{\Bbb G}\,_{2,5}$: $$ (\sigma_{245})^4\ =\ 3\cdot \sigma_{035}\ +\ 2\cdot \sigma_{125}\ +\ 3\cdot \sigma_{134}, $$ which may be determined by other means from the classical Schubert calculus. \section{Real Plane Conics} In 1864 Chasles~\cite{Chasles} showed there are 3264 plane conics tangent to five general conics. Fulton~\cite{Fulton_introduction_intersection} asked how many of the 3264 conics tangent to five general (real) conics can be real. He later determined that all can be real, but did not publish that result. More recently, Ronga, Tognoli, and Vust~\cite{Ronga_Tognoli_Vust} rediscovered this result. We conclude this note with an outline of their work. The author is grateful to Bill Fulton and Felice Ronga for explaining these ideas. Let $X$ be the variety of complete plane conics, a smooth variety of dimension 5. Let the hypersurfaces $H_p$, $H_l$, and $H_C$, be, respectively those conics containing a point $p$, those tangent to a line $l$, and those tangent to a conic, $C$. If $\hat{p},\hat{l}$, and $\widehat{C}$ are, respectively, their cycle classes in $A^1 X$, then $$ \widehat{C}\ =\ 2 \hat{p}\ +\ 2 \hat{l}, $$ which may be seen by degenerating a conic into two lines. Then the number of conics tangent to five general conics is the degree of $$ \widehat{C}^5 \ =\ 32 (\hat{p}\,^5\ +\ 5\hat{p}\,^4\cdot\hat{l}\ +\ 10\hat{p}\,^3\cdot\hat{l}\,^2\ +\ 10\hat{p}\,^2\cdot\hat{l}\,^3\ +\ 5\hat{p}\cdot\hat{l}^4\ +\ \hat{l}\,^5). $$ The monomials $\hat{p}\,^j\cdot\hat{l}^{5-j}$ for $j=0,\ldots,5$, have degrees $1,2,4,4,2,1$, giving Chasles' number of $32(1+10+40+40+10+1)=3264$~\cite[\S 9]{Kleiman_pspum}. \begin{thm}[Fulton, Ronga-Tognoli-Vust] There are five real conics in general position such that all of the 3264 conics tangent to the five conics are real. \end{thm} {\sc Proof.} The strategy is to realize the five conics as a deformation of five degenerate conics giving a maximal number of real conics. The first step is to show that for each $j$, there are $j$ lines and $5-j$ points such that the $2^{\min\{j, 5-j\}}$ conics tangent to the lines and containing the points are real. In~\cite{Ronga_Tognoli_Vust}, this step is done explicitly with a precise determination of which configurations of points and lines are `maximal'; that is, have all solutions real. Remarkably, there are five lines $l_1,\ldots,l_5$ and five real points $p_1,\ldots,p_5$ with $p_i\in l_i$ such that each of the 32 terms in $$ \bigcap_{i=1}^5 \left (H_{p_i}\ +\ H_{l_i} \right) $$ is a transverse intersection with all points of intersection real. Such a configuration is illustrated in Figure~\ref{fig:one}. \setcounter{figure}{0} \begin{figure}[htb] $$ \begin{picture}(170,130)(-5,-5) \put(60,0){$p_1$}\put(63,10){\circle*{2}} \put(5,15){$l_1$} \put(0,10){\line(1,0){160}} \put(12,40){$p_2$}\put(8,86){$l_2$}\put(25,45){\circle*{2}} \put(25,45){\line(-1,2){25}}\put(25,45){\line(1,-2){22}} \put(45,90){$p_3$}\put(85,113){$l_3$}\put(59,86){\circle*{2}} \put(59,86){\line(-3,-2){59}}\put(59,86){\line(3,2){51}} \put(118,80){$p_4$}\put(145,57){$l_4$}\put(114,74){\circle*{2}} \put(114,74){\line(3,-2){46}}\put(114,74){\line(-3,2){69}} \put(118,20){$p_5$}\put(140,105){$l_5$}\put(112,24){\circle*{2}} \put(112,24){\line(1,2){48}}\put(112,24){\line(-1,-2){12}} \end{picture} $$ \caption{A Maximal Configuration}\label{fig:one} \end{figure} The maximality of such a configuration is stable under small real deformations of its points and lines. Thus we may choose real lines $l_1',\ldots,l_5'$ where \begin{enumerate} \item $p_i\in l_i'$ and $l_i'$ is distinct from $l_i$, for $i=1,\ldots,5$, \item Any configuration obtained from a maximal configuration by substituting some primed lines for the corresponding unprimed lines is maximal. \item The lines $l_i$ and $l_i'$ partition the real tangent directions at $p_i$ into two intervals. The configurations described in condition (2) give finitely many real conics passing through $p_i$. We require that all tangent directions to these conics at $p_i$ lie within the interior of {\em one} of these two intervals. \end{enumerate} The relation $\widehat{C} = 2\hat{p} + 2\hat{l}$ may be obtained by considering a conic $C$ near a degenerate conic consisting of two lines $l,l'$ meeting at a point $p$, and a pencil of conics.\footnote{This version of this manuscript does not contain all of the postscript files of the original, in particular, it does not have an illustration of this degenerate conics $C$ and the nearby conics. To visualize this, think of a real hyperbola in ${\Bbb R}^2$ defined by $x^2 -y^2=t$, where $t$ is a small positive real number. Then the two lines are $x=\pm y$ and the point $p$ is the origin. The condition that the real tangent line to $Q$ at $p$ does not intersect $C$ means the absolute value of its slope exceeds 1.} For any conic $q$ in that pencil tangent to one of the lines, there is a nearby conic $q'$ in that pencil tangent to $C$. However, for every conic $Q$ in the pencil containing $p$, there are {\em two} nearby conics $Q', Q''$ in that pencil tangent to $C$. Moreover, if $Q$ is real, then $Q'$ and $Q''$ are real if and only if the real tangent line to $Q$ at $p$ does not intersect $C$. By condition (3), we may choose real conics $C_1,\ldots,C_5$ with $C_i$ near the degenerate conic $l_i + l_i'$ and, if $Q$ is a conic in \begin{equation}\label{eq:last} \bigcap_{i=1}^5 \left( H_{p_i}\ +\ H_{l_i}\ +\ H_{l_i'}\right) \end{equation} containing $p_i$, the the real tangent line to $Q$ at $p_i$ does not intersect $C_i$. If, in addition, the conics $C_i$ are sufficiently close to each degenerate conic, then there will be 3264 real conics tangent to each of $C_1,\ldots,C_5$. Indeed, suppose $H_{C_1}$ replaces $H_{p_1} + H_{l_1} + H_{l_1'}$ in the intersection (\ref{eq:last}). Then for any conic $q$ in (\ref{eq:last}) that is tangent to either $l_1$ or $l_1'$, there is a nearby real conic $q'$ tangent to $C$ which satisfies the other conditions on $q$ (since these other conditions determine a pencil of conics). If $Q$ is a conic in (\ref{eq:last}) containing $p_1$, then there are two nearby real conics $Q'$ and $Q''$ tangent to $C$ which satisfy the other conditions on $Q$. Similarly, if $H_{C_2}$ now replaces $H_{p_2} + H_{l_2}+ H_{l_2'}$ in the new intersection $H_{C_1}\cap\bigcap_{i=2}^5\left( H_{p_i}+H_{l_i}+H_{l_i'}\right)$, then each conic tangent to $l_2$ and $l_2'$ gives a conic tangent to $C_2$, but each conic through $p_2$ gives two conics tangent to $C_2$. Replacing $H_{C_3}, H_{C_4}$, and $H_{C_5}$ in turn completes the argument. \QED This proof used the effective rational equivalence: $$ H_C \ \sim\ 2 H_p\ +\ H_l\ +\ H_{l'}, $$ where $l, l'$ form a degenerate conic with $p= l\bigcap l'$. This deformation to a cycle having multiplicities (the coefficient 2 of $H_p$) is unavoidable: The variety $X$ and thus $\mbox{\it Chow}\, X$ has an action of $G=PGL(3,{\Bbb C}\,)$. The locus of hypersurfaces $H_C$ on $\mbox{\it Chow}\, X$ is a single 5-dimensional $G$-orbit. This family cannot have effective rational equivalence. If $Z$ is a cycle in the closure of this locus, then $Z$ is in a $G$-orbit of dimension at most 4. Thus if $Z= H_p+H_{p'}+H_l+H_{l'}$, then the dimension of the $G$-orbit of $(p,p',l,l')$ in the product of ${\Bbb P}^2$'s and their duals is at most 4. But this is impossible unless either $p=p'$ or $l=l'$.

9609

alg-geom/9609017

en

https://arxiv.org/abs/alg-geom/9609017

alg-geom/9609017

Arnaud Beauville

Arnaud Beauville

The Verlinde formula for PGL(p)

Plain TeX with the macro package xypic

Let G be a complex semi-simple group, X a Riemann surface, M_G the moduli space of principal G-bundles on X. When G is simply-connected, there exists a closed formula expressing the dimension of the space H^0(M_G,L) for any line bundle L on M_G (this is usually called among mathematicians, somewhat incorrectly, the "Verlinde formula").In this paper we find an analogous formula for G = PGL(r) where r is prime.

alg-geom

\global\def\currenvir{subsection{\global\defth}\smallskip{subsection} \global\advance\prmno by1 \par\hskip 1truecm\relax{ (\number\secno.\number\prmno) }} \def\global\def\currenvir{subsection{\global\defth}\smallskip{subsection} \global\advance\prmno by1 { (\number\secno.\number\prmno)\ }} \def\proclaim#1{\global\advance\prmno by 1 {\bf #1 \the\secno.\the\prmno$.-$ }} \long\def\th#1 \enonce#2\endth{% \medbreak\proclaim{#1}{\it #2}\global\defth}\smallskip{th}\smallskip} \def\rem#1{\global\advance\prmno by 1 {\it #1} \the\secno.\the\prmno$.-$ } \def\isinlabellist#1\of#2{\notfoundtrue% {\def\given{#1}% \def\\##1{\def\next{##1}% \lop\next\to\za\lop\next\to\zb% \ifx\za\given{\zb\global\notfoundfalse}\fi}#2}% \ifnotfound{\immediate\write16% {Warning - [Page \the\pageno] {#1} No reference found}}% \fi}% \def\ref#1{\ifx\empty{\immediate\write16 {Warning - No references found at all.}} \else{\isinlabellist{#1}\of}\fi} \def\newlabel#1#2{\rightappenditem{\\{#1}\\{#2}}\to} \def{} \def\label#1{\relax} \def\openout\lbl=\jobname.lbl{\openout\lbl=\jobname.lbl} \newread\testfile \def\lookatfile#1{\openin\testfile=\jobname.#1 \ifeof\testfile{\immediate\openout\nameuse{#1}\jobname.#1 \write\nameuse{#1}{} \immediate\closeout\nameuse{#1}}\fi% \immediate\closein\testfile}% \def\newlabel{theta}{1.2{\newlabel{theta}{1.2} \newlabel{Pic}{1.3} \newlabel{lin}{1.4} \newlabel{Trd}{2.1} \newlabel{Tr0}{2.2} \newlabel{normal}{2.3} \newlabel{D_r}{2.4} \newlabel{Verlinde}{3.1} \newlabel{sum}{3.2} \newlabel{formule}{3.3} \newlabel{total}{3.4} \newlabel{rem}{4.3}} \magnification 1250 \mathcode`A="7041 \mathcode`B="7042 \mathcode`C="7043 \mathcode`D="7044 \mathcode`E="7045 \mathcode`F="7046 \mathcode`G="7047 \mathcode`H="7048 \mathcode`I="7049 \mathcode`J="704A \mathcode`K="704B \mathcode`L="704C \mathcode`M="704D \mathcode`N="704E \mathcode`O="704F \mathcode`P="7050 \mathcode`Q="7051 \mathcode`R="7052 \mathcode`S="7053 \mathcode`T="7054 \mathcode`U="7055 \mathcode`V="7056 \mathcode`W="7057 \mathcode`X="7058 \mathcode`Y="7059 \mathcode`Z="705A \def\spacedmath#1{\def\packedmath##1${\bgroup\mathsurround =0pt##1\egroup$} \mathsurround#1 \everymath={\packedmath}\everydisplay={\mathsurround=0pt}} \def\nospacedmath{\mathsurround=0pt \everymath={}\everydisplay={} } \spacedmath{2pt} \font\eightrm=cmr8 \font\eighti=cmmi8 \font\eightsy=cmsy8 \font\eightbf=cmbx8 \font\eighttt=cmtt8 \font\eightit=cmti8 \font\eightsl=cmsl8 \font\sixrm=cmr6 \font\sixi=cmmi6 \font\sixsy=cmsy6 \font\sixbf=cmbx6 \catcode`\@=11 \def\eightpoint{% \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm \def\rm{\fam\z@\eightrm}% \textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy \textfont\itfam=\eightit \def\it{\fam\itfam\eightit}% \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}% \abovedisplayskip=9pt plus 3pt minus 9pt \belowdisplayskip=\abovedisplayskip \abovedisplayshortskip=0pt plus 3pt \belowdisplayshortskip=3pt plus 3pt \smallskipamount=2pt plus 1pt minus 1pt \medskipamount=4pt plus 2pt minus 1pt \bigskipamount=9pt plus 3pt minus 3pt \normalbaselineskip=9pt \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}% \let\bigf@nt=\eightrm \let\smallf@nt=\sixrm \normalbaselines\rm} \def\note#1#2{\footnote{\parindent 0.4cm$^#1$}{\vtop{\eightpoint\baselineskip12pt\hsize15.5truecm\noindent #2}} \parindent 0cm} \def\sdir_#1^#2{\mathrel{\mathop{\kern0pt\oplus}\limits_{#1}^{#2}}} \def\pc#1{\tenrm#1\sevenrm} \def\up#1{\raise 1ex\hbox{\smallf@nt#1}} \def\kern-1.5pt -{\kern-1.5pt -} \def\cqfd{\kern 2truemm\unskip\penalty 500\vrule height 4pt depth 0pt width 4pt\medbreak} \def\vrule height 4pt depth 0pt width 4pt{\vrule height 4pt depth 0pt width 4pt} \defn\up{o}\kern 2pt{n\up{o}\kern 2pt} \def\par\hskip 1truecm\relax{\par\hskip 1truecm\relax} \def\par\hskip 0.5cm\relax{\par\hskip 0.5cm\relax} \def\mathrel{\hbox{\vrule height 3pt depth -2pt width 6pt}}{\mathrel{\hbox{\vrule height 3pt depth -2pt width 6pt}}} \def\kern 1pt{\scriptstyle\circ}\kern 1pt{\kern 1pt{\scriptstyle\circ}\kern 1pt} \def\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}}{\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}}} \def\mathop{\rm End}\nolimits{\mathop{\rm End}\nolimits} \def\mathop{\rm Hom}\nolimits{\mathop{\rm Hom}\nolimits} \def\mathop{\rm Aut}\nolimits{\mathop{\rm Aut}\nolimits} \def\mathop{\rm Im}\nolimits{\mathop{\rm Im}\nolimits} \def\mathop{\rm Ker}\nolimits{\mathop{\rm Ker}\nolimits} \def\mathop{\rm Coker}{\mathop{\rm Coker}} \def\mathop{\rm det}\nolimits{\mathop{\rm det}\nolimits} \def\mathop{\rm Pic}\nolimits{\mathop{\rm Pic}\nolimits} \def\mathop{\rm Div}\nolimits{\mathop{\rm Div}\nolimits} \def\mathop{\rm dim}\nolimits{\mathop{\rm dim}\nolimits} \def\mathop{\rm Card}\nolimits{\mathop{\rm Card}\nolimits} \def\mathop{\rm Tr}\nolimits{\mathop{\rm Tr}\nolimits} \def\mathop{\rm rk\,}\nolimits{\mathop{\rm rk\,}\nolimits} \def\mathop{\rm div\,}\nolimits{\mathop{\rm div\,}\nolimits} \def\mathop{\rm Ad}\nolimits{\mathop{\rm Ad}\nolimits} \def\mathop{\rm ad}\nolimits{\mathop{\rm ad}\nolimits} \def\mathop{\rm codim}\nolimits{\mathop{\rm codim}\nolimits} \def\mathop{\rm ch}\nolimits{\mathop{\rm ch}\nolimits} \font\gragrec=cmmib10 \def\hbox{\gragrec \char22}{\hbox{\gragrec \char22}} \font\san=cmssdc10 \def\hbox{\san \char3} {\hbox{\san \char3}} \def\hbox{\san \char83}{\hbox{\san \char83}} \input amssym.def \input amssym \vsize = 25truecm \hsize = 16truecm \voffset = -.5truecm \parindent=0cm \baselineskip15pt \overfullrule=0pt \newlabel{theta}{1.2 \centerline{\bf The Verlinde formula for ${\bf PGL}_p$} \smallskip \smallskip \centerline{Arnaud {\pc BEAUVILLE\note{1}{Partially supported by the European HCM project ``Algebraic Geometry in Europe" (AGE).}}} \vskip0.9cm \hfill\vbox{\hsize4cm\eightpoint\baselineskip12pt\line{\hfill\it To the memory of\hfill} \line{\hfill Claude ITZYKSON\hfill}} \vskip0.7cm {\bf Introduction} \smallskip \par\hskip 1truecm\relax The Verlinde formula expresses the number of linearly independent conformal blocks in any rational conformal field theory. I am concerned here with a quite particular case, the Wess-Zumino-Witten model associated to a complex semi-simple group\note{2}{This group is the complexification of the compact semi-simple group considered by physicists.} $G$. In this case the space of conformal blocks can be interpreted as the space of holomorphic sections of a line bundle on a particular projective variety, the moduli space $M_G$ of holomorphic $G$\kern-1.5pt - bundles on the given Riemann surface. The fact that the dimension of this space of sections can be explicitly computed is of great interest for mathematicians, and a number of rigorous proofs of that formula (usually called by mathematicians, somewhat incorrectly, the ``Verlinde formula") have been recently given (see e.g.\ [F], [B-L], [L-S]). \par\hskip 1truecm\relax These proofs deal only with simply-connected groups. In this paper we treat the case of the projective group ${\bf PGL}_r$ when $r$ is prime. \par\hskip 1truecm\relax Our approach is to relate to the case of ${\bf SL}_r$, using standard algebro-geometric methods. The components $M_{{\bf PGL}_r}^d$ $(0\le d<r)$ of the moduli space $M_{{\bf PGL}_r}$ can be identified with the quotients $M_r^d/J_r$ , where $M_r^d$ is the moduli space of vector bundles on $X$ of rank $r$ and fixed determinant of degree $d$, and $J_r$ the finite group of holomorphic line bundles $\alpha $ on $X$ such that $\alpha ^{\otimes r}$ is trivial. The space we are looking for is the space of $J_r$\kern-1.5pt - invariant global sections of a line bundle ${\cal L}$ on $M_r^d$; its dimension can be expressed in terms of the character of the representation of $J_r$ on $H^0(M_r^d,{\cal L})$. This is given by the Lefschetz trace formula, with a subtlety for $d=0$, since $M_r^0$ is not smooth. The key point (already used in [N-R]) which makes the computation quite easy is that the fixed point set of any non-zero element of $J_r$ is an abelian variety -- this is where the assumption on the group is essential. Extending the method to other cases would require a Chern classes computation on the moduli space $M_H$ for some semi-simple subgroups $H$ of $G$; this may be feasible, but goes far beyond the scope of the present paper. Note that the case of $M_{{\bf PGL}_2}^1$ has been previously worked out in [P] (with an unfortunate misprint in the formula). \par\hskip 1truecm\relax In the last section we check that our formulas agree with the predictions of Conformal Field Theory, as they appear for instance in [F-S-S]. Note that our results are slightly more precise (in this particular case): we get a formula for $\mathop{\rm dim}\nolimits H^0(M_{{\bf PGL}_r}^d,{\cal L})$ for every $d$, while CFT only predicts the sum of these dimensions (see Remark \ref{rem}). \vskip1cm \section{The moduli space $M_{{\bf PGL}_r}$} \global\def\currenvir{subsection Throughout the paper we denote by $X$ a compact (connected) Riemann surface, of genus $g\ge 2$; we fix a point $p$ of $X$. Principal ${\bf PGL}_r$\kern-1.5pt - bundles on $X$ correspond in a one-to-one way to projective bundles of rank $r-1$ on $X$, i.e. bundles of the form ${\bf P}(E)$, where $E$ is a rank $r$ vector bundle on $X$; we say that ${\bf P}(E)$ is semi-stable if the vector bundle $E$ is semi-stable. The semi-stable projective bundles of rank $r-1$ on $X$ are parameterized by a projective variety, the moduli space $M_{{\bf PGL}_r}$. \par\hskip 1truecm\relax Two vector bundles $E$, $F$ give rise to isomorphic projective bundles if and only if $F$ is isomorphic to $E\otimes \alpha $ for some line bundle $\alpha $ on $X$. Thus a projective bundle can always be written as ${\bf P}(E)$ with $\mathop{\rm det}\nolimits E={\cal O}_X(dp)$, $0\le d<r$; the vector bundle $E$ is then determined up to tensor product by a line bundle $\alpha $ with $\alpha ^r={\cal O}_X$. In particular, the moduli space $M_{{\bf PGL}_r}$ has $r$ connected components $M_{{\bf PGL}_r}^d$ $(0\le d<r)$. Let us denote by $M_r^d$ the moduli space of semi-stable vector bundles on $X$ of rank $r$ and determinant ${\cal O}_X(dp)$, and by $J_r$ the kernel of the multiplication by $r$ in the Jacobian $JX$ of $X$; it is a finite group, canonically isomorphic to $H^1(X,{\bf Z}/(r))$. The group $J_r$ acts on $M_r^d$, by the rule $(\alpha ,E)\mapsto E\otimes\alpha $; it follows from the above remarks that the component $M_{{\bf PGL}_r}^d$ is isomorphic to the quotient $M_r^d/J_r$. \global\def\currenvir{subsection\label{theta} We will need a precise description of the line bundles on $M_{{\bf PGL}_r}$. Let me first recall how one describes line bundles on $M_r^d$ [D-N]: a simple way is to mimic the classical definition of the theta divisor on the Jacobian of $X$ (i.e.\ in the rank $1$ case). Put $\delta=(r,d)$; let $A$ be a vector bundle on $X$ of rank $r/\delta$ and degree $(r(g-1)-d)/\delta$. These conditions imply $\chi (E\otimes A)=0$ for all $E$ in $M_r^d$; if $A$ is general enough, it follows that the condition $H^0(X,E\otimes A)\not=0$ defines a (Cartier) divisor $\Theta _A$ in $M_r^d$. The corresponding line bundle ${\cal L}_d:={\cal O}(\Theta _A)$ does not depend on the choice of $A$, and generates the Picard group $\mathop{\rm Pic}\nolimits(M_r^d)$. \global\def\currenvir{subsection\label{Pic} The quotient map $q:M_r^d\rightarrow M_{{\bf PGL}_r}^d $ induces a homomorphism\break $q^*:\mathop{\rm Pic}\nolimits(M_{{\bf PGL}_r}^d)\rightarrow \mathop{\rm Pic}\nolimits(M_r^d)$, which is easily seen to be injective. Its image is determined in [B-L-S]: it is generated by ${\cal L}_d^\delta$ if $r$ is odd, by ${\cal L}_d^{2\delta}$ if $r$ is even. \global\def\currenvir{subsection\label{lin} Let ${\cal L}'$ be a line bundle on $M_{{\bf PGL}_r}^d$. The line bundle ${\cal L}:=q^*{\cal L}'$ on $M_r^d$ admits a natural action of $J_r$, compatible with the action of $J_r$ on $M_r^d$ (this is often called a $J_r$\kern-1.5pt - linearization of ${\cal L}$). This action is characterized by the property that every element $\alpha $ of $J_r$ acts trivially on the fibre of ${\cal L}$ at a point of $M_r^d$ fixed by $\alpha $. In the sequel we will always consider line bundles on $M_r^d$ of the form $q^*{\cal L}'$, and endow them with the above $J_r$\kern-1.5pt - linearization. \par\hskip 1truecm\relax This linearization defines a representation of $J_r$ on the space of global sections; essentially by definition, the global sections of ${\cal L}'$ correspond to the $J_r$\kern-1.5pt - invariant sections of ${\cal L}$. Therefore our task will be to compute the dimension of the space of invariant sections; as indicated in the introduction, we will do that by computing, for any $\alpha \in J_r$ of order $r$, the trace of $\alpha $ acting on $H^0(M_r^d,{\cal L})$. \vskip1cm \section {The action of $J_r$ on $H^0(M_r^d,{\cal L}_d^k)$} \par\hskip 1truecm\relax We start with the case when $r$ and $d$ are coprime, which is easier to deal with because the moduli space is smooth. \th Proposition \enonce Assume $r$ and $d$ are coprime. Let $k$ be an integer; if $r$ is even we assume that $k$ is even. Let $\alpha $ be an element of order $r$ in $JX$. Then the trace of $\alpha $ acting on $H^0(M_r^d,{\cal L}_d^{k})\ $ is $(k+1)^{(r-1)(g-1)}$. \endth\label{Trd} {\it Proof}: The Lefschetz trace formula reads [A-S]$$\mathop{\rm Tr}\nolimits(\alpha\, |\,H^0(M_r^d,{\cal L}_d^k))=\int_{P}{\rm Todd}(T_P)\ \lambda (N_{P/M_r^d},\alpha )^{-1}\ \widetilde{\mathop{\rm ch}\nolimits}({\cal L}^k_{d\,|P},\alpha )\ .$$ Here $P$ is the fixed subvariety of $\alpha $; whenever $F$ is a vector bundle on $P$ and $\varphi $ a diagonalizable endomorphism of $F$, so that $F$ is the direct sum of its eigen-sub-bundles $F_\lambda $ for $\lambda \in {\bf C}$, we put $$\widetilde{\mathop{\rm ch}\nolimits}(F,\varphi )=\sum \lambda \,{\mathop{\rm ch}\nolimits}(F_\lambda )\quad ;\quad \lambda (F,\varphi )=\prod_\lambda \sum_{p\ge 0} (-\lambda )^p \mathop{\rm ch}\nolimits(\hbox{\san \char3}^pF_\lambda ^*)\ .$$ \par\hskip 1truecm\relax We have a number of informations on the right hand side thanks to [N-R]: (\ref{Trd} {\it a}) \ Let $\pi :\widetilde{X}\rightarrow X$ be the \'etale $r$\kern-1.5pt - sheeted covering associated to $\alpha $; put $\xi =\alpha ^{r(r-1)/2}\in JX$. The map $L\mapsto \pi _*(L)$ identifies any component of the fibre of the norm map ${\rm Nm}:J^d\widetilde{X}\rightarrow J^dX$ over $\xi(dp)$ with $P$. In particular, $P$ is isomorphic to an abelian variety, hence the term ${\rm Todd}(T_P)$ is trivial. (\ref{Trd} {\it b}) \ Let $\theta \in H^2(P,{\bf Z})$ be the restriction to $P$ of the class of the principal polarization of $J^d\widetilde{X}$. The term $\lambda (N_{P/M_r^d},\alpha )$ is equal to $r^{r(g-1)}e^{-r\theta }$. (\ref{Trd} {\it c}) \ The dimension of $P$ is $N=(r-1)(g-1)$, and one has $\int_P {\theta^N \over N!}=r^{g-1}$. \par\hskip 1truecm\relax With our convention the action of $\alpha $ on ${\cal L}^k_{d\,|P}$ is trivial. The class $c_1({\cal L}_{d\,|P})$ is equal to $r\theta $: the pull back to $P$ of the theta divisor $\Theta _A$ (\ref{theta}) is the divisor of line bundles $L$ in $P$ with $H^0(L\otimes \pi ^*A)\not=0$; to compute its cohomology class we may replace $\pi ^*A$ by any vector bundle with the same rank and degree, in particular by a direct sum of $r$ line bundles of degree $r(g-1)-d$, which gives the required formula. \par\hskip 1truecm\relax Putting things together, we find $$\mathop{\rm Tr}\nolimits(\alpha\, |\,H^0(M_r^d,{\cal L}_d^k))=\int_{P} r^{-r(g-1)}e^{r\theta } e^{kr\theta } = (k+1)^{(r-1)(g-1)}\ .\quad \vrule height 4pt depth 0pt width 4pt$$ \bigskip \par\hskip 1truecm\relax We now consider the degree $0$ case: \th Proposition \enonce Let $k$ be a multiple of $r$, and of $2r$ if $r$ is even; let $\alpha $ be an element of order $r$ in $JX$. Then the trace of $\alpha $ acting on $H^0(M_r^0,{\cal L}_0^{k})\ $ is $ ({k\over r}+1)^{(r-1)(g-1)}$. \endth\label{Tr0} {\it Proof}: We cannot apply directly the Lefschetz trace formula since it is manageable only for smooth projective varieties; instead we use another well-known tool, the Hecke correspondence (this idea appears for instance in [B-S]). For simplicity we write $M_d$ instead of $M_r^d$. There exists a Poincar\'e bundle ${\cal E}$ on $X\times M_1$, i.e.\ a vector bundle whose restriction to $X\times\{E\}$, for each point $E$ of $M_1$, is isomorphic to $E$. Such a bundle is determined up to tensor product by a line bundle coming from $M_1$; we will see later how to normalize it. We denote by ${\cal E}_p$ the restriction of ${\cal E}$ to $\{p\}\times M_1$, and by ${\cal P}$ the projective bundle ${\bf P}({\cal E}_p^*)$ on $M_1$. A point of ${\cal P}$ is a pair $(E,\varphi )$ where $E$ is a vector bundle in $M_1$ and $\varphi :E\rightarrow {\bf C}_p$ a non-zero homomorphism, defined up to a scalar; the kernel of $\varphi $ is then a vector bundle $F\in M_1$, and we can view equivalently a point of ${\cal P}$ as a pair of vector bundles $(F,E)$ with $F\in M_0$, $E\in M_1$ and $F\i E$. The projections $p_d$ on $M_d$ $(d=0,1)$ give rise to the ``Hecke diagram" \input xypic $$\diagram &{\cal P}\dlto_{p_1} \drto^{p_0} &&\\ M_1 &&M_0&\kern-40pt.\\ \enddiagram$$ \smallskip \th Lemma \enonce The Poincar\'e bundle ${\cal E}$ can be normalized (in a unique way) so that $\mathop{\rm det}\nolimits {\cal E}_p={\cal L}_1\ ;$ then ${\cal O}_{\cal P}(1) \cong p_0^*{\cal L}_0$. \endth \label{normal} {\it Proof}: Let $E\in M_1$. The fibre $p_1^{-1}(E)$ is the projective space of non-zero linear forms $\ell :E_p\rightarrow {\bf C}$, up to a scalar. The restriction of $p_0^*{\cal L}_0$ to this projective space is ${\cal O}(1)$ (choose a line bundle $L$ of degree $g-1$ on $X$; if $E$ is general enough, $H^0(X,E\otimes L)$ is spanned by a section $s$ with $s(p)\not=0$, and the condition that the bundle $F$ corresponding to $\ell $ belongs to $\Theta _L$ is the vanishing of $\ell (s(p))$). Therefore $p_0^*{\cal L}_0$ is of the form ${\cal O}_{\cal P}(1)\otimes p_1^*{\cal N}$ for some line bundle ${\cal N}$ on $M_1$. Replacing ${\cal E}$ by ${\cal E}\otimes{\cal N}$ we ensure ${\cal O}_{\cal P}(1) \cong p_0^*{\cal L}_0$. \par\hskip 1truecm\relax An easy computation gives $K_{\cal P}=p_1^*{\cal L}_1^{-1}\otimes p_0^*{\cal L}_0^{-r}$ ([B-L-S], Lemma 10.3). On the other hand, since ${\cal P}={\bf P}({\cal E}_p^*)$, one has $K_{\cal P}= p_1 ^*(K_{M_1}\otimes \mathop{\rm det}\nolimits {\cal E}_p)\otimes {\cal O}_{\cal P}(-r)$; using $K_{ M_1}={\cal L}^{-2}_1$ [D-N], we get $\mathop{\rm det}\nolimits{\cal E}_p={\cal L}_1$. \cqfd \medskip \par\hskip 1truecm\relax We normalize ${\cal E}$ as in the lemma; this gives for each $k\ge 0$ a canonical isomorphism $p_{1*}p_0^*{\cal L}_0^k\cong \hbox{\san \char83}^k{\cal E}_p$. Let $\alpha $ be an element of order $r$ of $JX$. It acts on the various moduli spaces in sight; with a slight abuse of language, I will still denote by $\alpha $ the corresponding automorphism. There exists an isomorphism $\alpha ^*{\cal E}\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}} {\cal E}\otimes\alpha $, unique up to a scalar ([N-R], lemma 4.7); the induced isomorphism $u:\alpha ^*{\cal E}_p\mathrel{\mathop{\kern 0pt\longrightarrow }\limits^{\sim}} {\cal E}_p$ induces the action of $\alpha $ on ${\cal P}$. Imposing $u^r={\rm Id}$ determines $u$ up to a $r$\kern-1.5pt - th root of unity, hence determines completely $\hbox{\san \char83}^ku$ when $k$ is a multiple of $r$. Since the Hecke diagram is equivariant with respect to $\alpha $, it gives rise to a diagram of isomorphisms $$\diagram & H^0({\cal P},p_0^*{\cal L}^k_0) & \\ H^0(M_1,\hbox{\san \char83}^k{\cal E}_p) \urto^{p_1^*} & & H^0( M_0,{\cal L}_0^k)\ulto_{p_0^*} \enddiagram$$ which is compatible with the action of $\alpha $; in particular, the trace we are looking for is equal to the trace of $\alpha $ on $H^0(M_1,\hbox{\san \char83}^k{\cal E}_p)$. \par\hskip 1truecm\relax We are now in the situation of Prop.\ \ref{Trd}, and the Lefschetz trace formula gives: $$\mathop{\rm Tr}\nolimits(\alpha\, |\,H^0(M_1,\hbox{\san \char83}^k{\cal E}_p))=\int_{P}{\rm Todd}(T_P)\ \lambda (N_{P/M_1},\alpha )^{-1}\ \widetilde{\mathop{\rm ch}\nolimits}(\hbox{\san \char83}^k{\cal E}_{p\,|P},\alpha )\ .$$ \par\hskip 1truecm\relax The only term we need to compute is $\widetilde{\mathop{\rm ch}\nolimits}(\hbox{\san \char83}^k{\cal E}_{p\,|P},\alpha )$. Let ${\cal N}$ be the restriction to $\widetilde{X}\times P$ of a Poincar\'e line bundle on $\widetilde{X}\times J^1\widetilde{X}$; let us still denote by $\pi : \widetilde{X}\times P\rightarrow X\times P$ the map $\pi \times {\rm Id}_P$. The vector bundles $\pi _*({\cal N})$ and ${\cal E}_{|X\times P}$ have the same restriction to $ X\times \{\gamma \}$ for all $\gamma \in P$, hence after tensoring ${\cal N}$ by a line bundle on $P$ we may assume they are isomorphic ([R], lemma 2.5). Restricting to $\{p\} \times P$ we get $\displaystyle {\cal E}_{p\,|P}=\sdir_{\pi (q)=p}^{}{\cal N}_{q}$, with ${\cal N}_q={\cal N}_{|\{q\}\times P}$. \smallskip \par\hskip 1truecm\relax We claim that the ${\cal N}_q$'s are the eigen-sub-bundles of ${\cal E}_{p\,|P}$ relative to $\alpha $. By (\ref{Trd} {\it a}), a pair $(E,F)\in{\cal P}$ is fixed by $\alpha $ if and only if $E=\pi_*L$, $F=\pi_*L'$, with ${\rm Nm}(L)=\xi(p) $, ${\rm Nm}(L')=\xi $; because of the inclusion $F\i E$ we may take $L'$ of the form $L(-q)$, for some point $q\in \pi^{-1}(p)$. In other words, the fixed locus of $\alpha $ acting on ${\cal P}$ is the disjoint union of the sections $(\sigma _q)_{q\in \pi^{-1}(p)}$ of the fibration $p_1^{-1}(P)\rightarrow P$ characterized by $\sigma _q(\pi_*L)=(\pi_*L,\pi_*(L(-q)))$. Viewing ${\cal P}$ as ${\bf P}({\cal E}^*_{p\,|P})$, the section $\sigma _q$ corresponds to the exact sequence $$0\rightarrow \pi _*({\cal N}(-q))_{\,|\{p\}\times P}\longrightarrow \pi _*({\cal N})_{\,| \{p\}\times P}\cong {\cal E}_{\,|\{p\}\times P}\longrightarrow {\cal N}_q\rightarrow 0\ .$$ Therefore on each fibre ${\bf P}(E_p)$, for $E\in P$, the automorphism $\alpha $ has exactly $r$ fixed points, corresponding to the $r$ sub-spaces ${\cal N}_{(q,E)}$ for $q\in \pi ^{-1} (p)$; this proves our claim. \par\hskip 1truecm\relax The line bundles ${\cal N}_q$ for $q\in \widetilde{X}$ are algebraically equivalent, and therefore have the same Chern class. We thus have $c_1( {\cal E}_{p\,|P})=r\,c_1({\cal N}_q)$. On the other hand we know that $\mathop{\rm det}\nolimits {\cal E}_{p}={\cal L}_1$ (lemma \ref{normal}), and that $c_1({\cal L}_{1\,|P})=r\theta $ (proof of Prop.\ \ref{Trd}). By comparison we get $c_1({\cal N}_q)=\theta $. Putting things together we obtain $$\widetilde{\mathop{\rm ch}\nolimits}(\hbox{\san \char83}^k{\cal E}_{p\,|P},\alpha )=\int_P \mathop{\rm Tr}\nolimits \hbox{\san \char83}^kD_r\ e^{k\theta }r^{-r(g-1)}e^{r\theta }$$ where $D_r$ is the diagonal $r$\kern-1.5pt - by-$\!r$ matrix with entries the $r$ distinct $r$\kern-1.5pt - th roots of unity. \th Lemma \enonce The trace of $\hbox{\san \char83}^kD_r$ is $1$ if $r$ divides $k$ and $0$ otherwise. \endth\label{D_r} \par\hskip 1truecm\relax Consider the formal series $\displaystyle s(T):=\sum_{i\ge 0}T^i\mathop{\rm Tr}\nolimits\hbox{\san \char83}^iu$ and $\displaystyle \lambda (T):=\sum_{i\ge 0} T^i\mathop{\rm Tr}\nolimits\hbox{\san \char3}^iu$. The formula $s(T)\lambda (-T)=1$ is well-known (see e.g. [Bo], \S 9, formula (11)). But $$\lambda (-T)=\sum_{i=0}^r(-T)^i\mathop{\rm Tr}\nolimits\hbox{\san \char3}^iu=\prod_{\zeta^r=1}(1-\zeta T)=1-T^r\ ,$$ hence the lemma. Using (\ref{Trd} {\it c}) the Proposition follows. \cqfd \vskip1cm \section{Formulas} \par\hskip 1truecm\relax In this section I will apply the above results to compute the dimension of the space of sections of the line bundle ${\cal L}_d^k$ on the moduli space $M_{{\bf PGL}_r}^d$. Let me first recall the corresponding Verlinde formula for the moduli spaces $M_r^d$. Let $\delta=(r,d)$; we write ${\cal L}_d={\cal D}^{r/\delta}$, with the convention that we only consider powers of ${\cal D}$ which are multiple of $r/\delta$ (the line bundle ${\cal D}$ actually makes sense on the {\it moduli stack} ${\cal M}_r^d$, and generates its Picard group). We denote by $\hbox{\gragrec \char22}_r$ the center of ${\bf SL}_r$, i.e.\ the group of scalar matrices $\zeta {\it I}_r$ with $\zeta ^r=1$. \th Proposition \enonce Let $T_k$ be the set of diagonal matrices $t={\rm diag}(t_1,\ldots,t_r)$ in ${\bf SL}_r({\bf C}) $ with $t_i\not=t_j$ for $i\not=j$, and $t^{k+r}\in \hbox{\gragrec \char22}_r$; for $t\in T_k$, let $\displaystyle \delta(t)=\prod_{i<j}(t_i-t_j)$. Then $$\mathop{\rm dim}\nolimits H^0(M_r^d,{\cal D}^k)=r^{g-1}(k+r)^{(r-1)(g-1)}\ \sum_{ t\in T_k/{\goth S}_r} { ((-1)^{r-1} t^{k+r})^{-d} \over |\delta(t)|^{2g-2}}\quad \cdot$$ \endth\label{Verlinde} {\it Proof}: According to [B-L], Thm. 9.1, the space $H^0(M_r^d,{\cal D}^k)$ for $0<d<r$ is canonically isomorphic to the space of conformal blocks in genus $g$ with the representation $V_{k\varpi_{r-d}}$ of ${\bf SL}_r$ with highest weight $k\varpi_{r-d}$ inserted at one point. The Verlinde formula gives therefore (see [B], Cor. 9.8\note{1}{There is a misprint in the first equality of that corollary, where one should read $T_\ell ^{\rm reg}/W$ instead of $T_\ell ^{\rm reg}$; the second equality (and the proof!) are correct.}\kern-4pt): $$\mathop{\rm dim}\nolimits H^0(M_r^d,{\cal D}^k)=r^{g-1}(k+r)^{(r-1)(g-1)}\sum_{t\in T_k/{\goth S}_r } {\mathop{\rm Tr}\nolimits^{}_{V_{k\varpi_{r-d}}}(t)\over |\delta(t)|^{2g-2}}\ ;$$ this is still valid for $d=0$ with the convention $\varpi_r=0$. \par\hskip 1truecm\relax The character of the representation $V_{k\varpi_{r-d}}$ is given by the Schur formula (see e.g. [F-H], Thm. 6.3): $$\mathop{\rm Tr}\nolimits^{}_{V_{k\varpi_{r-d}}}(t)={1\over \delta (t)}\ \left| \matrix{t_1^{k+r-1} & t_2^{k+r-1} & \ldots & t_r^{k+r-1}\cr t_1^{k+r-2} &t_2^{k+r-2}&\ldots & t_r^{k+r-2}\cr \vdots & \vdots & & \vdots \cr t_1^{k+d} &t_2^{k+d}&\ldots & t_r^{k+d}\cr t_1^{d-1} &t_2^{d-1}&\ldots & t_r^{d-1}\cr \vdots & \vdots & & \vdots \cr 1 &1&\ldots & 1\cr } \right|\ .$$ \par\hskip 1truecm\relax Writing $t^{k+r}=\zeta{\it I}_r \in\hbox{\gragrec \char22}_r$, the big determinant reduces to \break $\zeta ^{r-d}(-1)^{d(r-d)}\mathop{\rm det}\nolimits(t_j^{d-i})$, and finally, since $\prod t_i=1$, to $((-1)^{r-1}\zeta) ^{-d}\delta (t)$, which gives the required formula. \cqfd \smallskip \th Corollary \enonce Let $T'_k$ be the set of matrices $t={\rm diag}(t_1,\ldots,t_r)$ in ${\bf SL}_r({\bf C})$ with $t_i\not=t_j$ if $i\not=j$, and $t^{k+r}=(-1)^{r-1}{\it I}_r$. Then $$\sum_{d=0}^{r-1}\mathop{\rm dim}\nolimits H^0(M_r^d,{\cal D}^k)=r^{g}(k+r)^{(r-1)(g-1)}\ \sum_{t\in T'_k/{\goth S}_r} { 1 \over |\delta(t)|^{2g-2}}\ \cdot\quad \vrule height 4pt depth 0pt width 4pt$$ \endth \label{sum}\smallskip \par\hskip 1truecm\relax We now consider the moduli space $M_{{\bf PGL}_r}$. We know that the line bundle ${\cal D}^k$ on $M_r^d$ descends to $M_{{\bf PGL}_r}^d=M_r^d/J_r$ exactly when $k$ is a multiple of $r$ if $r$ is odd, or of $2r$ if $r$ is even (\ref{Pic}). When this is the case we obtain a line bundle on $M_{{\bf PGL}_r}^d$, that we will still denote by ${\cal D}^k$; its global sections correspond to the $J_r$\kern-1.5pt - invariant sections of $H^0(M_r^d,{\cal D}^{k})$. \par\hskip 1truecm\relax We will assume that $r$ is {\it prime}, so that every non-zero element $\alpha $ of $J_r$ has order $r$. Then Prop.\ \ref{Trd} and \ref{Tr0} lead immediately to a formula for the dimension of the $J_r$\kern-1.5pt - invariant subspace of $H^0(M_r^d,{\cal D}^{k})$ as the average of the numbers $\mathop{\rm Tr}\nolimits(\alpha )$ for $\alpha $ in $J_r$. Using Prop.\ \ref{Verlinde} we conclude: \th Proposition \enonce Assume that $r$ is prime. Let $k$ be a multiple of $r$; if $r=2$ assume $4\mid k$. Then $$\nospacedmath\displaylines{\mathop{\rm dim}\nolimits H^0(M_{{\bf PGL}_r}^d,{\cal D}^{k}) = r^{-2g}\,\mathop{\rm dim}\nolimits H^0(M_r^d,{\cal D}^k)\ +\ (1-r^{-2g})({k\over r}+1)^{(r-1)(g-1)}\cr \hfill = r^{-2g}\,({k\over r}+1)^{(r-1)(g-1)}\ \Bigl(r^{r(g-1)} \sum_{ t\in T_k/{\goth S}_r} { ((-1)^{r-1}t^{k+r})^{-d} \over |\delta(t)|^{2g-2}} \, +\, r^{2g}-1 \Bigr)\ .}$$ \endth\label{formule} \smallskip \par\hskip 1truecm\relax Summing over $d$ and plugging in Cor.\ \ref{sum} gives the following rather complicated formula: \th Corollary \enonce $$\mathop{\rm dim}\nolimits H^0(M_{{\bf PGL}_r},{\cal D}^{k})=r^{1-2g}\,({k\over r}+1)^{(r-1)(g-1)}\ \Bigl( r^{ r(g-1)}\sum_{t\in T'_k/{\goth S}_r} { 1 \over |\delta(t)|^{2g-2}}\ +\ r^{2g}-1\Bigr)\ .$$ \endth\label{total} \par\hskip 1truecm\relax As an example, if $k$ is an integer divisible by $4$, we get $$\mathop{\rm dim}\nolimits H^0(M_{{\bf PGL}_2},{\cal D}^{k})=2^{1-2g}\,({k\over 2}+1)^{g-1}\bigl( \sum_{l\ {\rm odd}\atop 0<l<k+2} { 1 \over (\sin{l\pi \over k+2})^{2g-2}}\,+\,2^{2g}-1\bigr)\ .\leqno(3.5)$$ \vskip1cm \section{Relations with Conformal Field Theory} \global\def\currenvir{subsection According to Conformal Field Theory, the space $H^0(M_{{\bf PGL}_r},{\cal D}^{k})$ should be canonically isomorphic to the space of conformal blocks for a certain Conformal Field Theory, the WZW model associated to the projective group. This implies in particular that its dimension should be equal to $\sum_j|S_{0j}|^{2-2g}$, where $(S_{ij})$ is a unitary symmetric matrix. For instance in the case of the WZW model associated to ${\bf SL}_2$, one has $$S_{0j}={\sin {(j+1)\pi \over k+2}\over \sqrt{{k\over 2}+1}}\qquad\hbox{, with}\quad 0\le j\le k\ , $$ where the index $j$ can be thought as running through the set of irreducible representations $\hbox{\san \char83}^1,\ldots,\hbox{\san \char83}^k$ of ${\bf SL}_2$ (or equivalently ${\bf SU}_2$), with $\hbox{\san \char83}^j:=\hbox{\san \char83}^j({\bf C}^2)$. \par\hskip 1truecm\relax We deduce from (3.5) an analogous expression for ${\bf PGL}_2$: we restrict ourselves to even indices and write $$S'_{0j}=2\,S_{0j}\qquad {\rm for}\quad j\hbox { even }<k/2\quad {\rm ;}\qquad S'_{0,{k\over 2}^{(1)}}=S'_{0,{k\over 2}^{(2)}} =S_{0{k\over 2}}\quad \cdot$$ In other words, we consider only those representations of ${\bf SL}_2$ which factor through ${\bf PGL}_2$ and we identify the representation $\hbox{\san \char83}^{2j}$ with $\hbox{\san \char83}^{k-2j}$, doubling the coefficient $S_{0j}$ when these two representations are distinct, and counting twice the representation which is fixed by the involution (this process is well-known, see e.g.\ [M-S]). \global\def\currenvir{subsection The case of ${\bf SL}_r$ is completely analogous; we only need a few more terminology from representation theory (we follow the notation of [B]). The primary fields are indexed by the set $P_k$ of dominant weights $\lambda $ with $\lambda (H_\theta )\le k$, where $H_\theta $ is the matrix $\ {\rm diag}(1,0,\ldots,0,-1)$. For $\lambda \in P_k$, we put $\displaystyle t_\lambda =\exp 2\pi i\,{\lambda+\rho\over k+r}$ (we identify the Cartan algebra of diagonal matrices with its dual using the standard bilinear form); the map $\lambda \mapsto t_\lambda $ induces a bijection of $P_k$ onto $T_k/{\goth S}_r$ ([B], lemma 9.3 {\it c})). In view of Prop.\ \ref{Verlinde}, the coefficient $S_{0\lambda }$ for $\lambda \in P_k$ is given by $$S_{0\lambda } = {\delta(t_\lambda )\over \sqrt{r}(k+r)^{(r-1)/2}}\ \cdot$$ \par\hskip 1truecm\relax Passing to ${\bf PGL}_r$, we first restrict the indices to the subset $P'_k$ of elements $\lambda \in P_k$ such that $t_\lambda $ belongs to $T'_k$; this means that $\lambda $ belongs to the root lattice, i.e.\ that the representation $V_\lambda$ factors through ${\bf PGL}_r$. The center $\hbox{\gragrec \char22}_r$ acts on $T_k$ by multiplication; this action preserves $T'_k$, and commutes with the action of ${\goth S}_r$. The corresponding action on $P_k$ is deduced, via the bijection $\lambda \mapsto {\lambda +\rho \over k+r}$, from the standard action of $\hbox{\gragrec \char22}_r$ on the fundamental alcove $A$ with vertices $\{0,\varpi_1,\ldots,\varpi_{r-1}\}$.\note{1}{The element $\exp \varpi_1$ of the center gives the rotation of $A$ which maps $0$ to $\varpi_1$, $\varpi_1$ to $\varpi_2$, $\ldots,$ and $\varpi_{r-1}$ to $0$.} \par\hskip 1truecm\relax We identify two elements of $P'_k$ if they are in the same orbit with respect to this action. The action has a unique fixed point, the weight ${k\over r}\rho $, which corresponds to the diagonal matrix $D_r$ (\ref{D_r}); we associate to this weight $r$ indices $\nu ^{(1)},\ldots,\nu ^{(r)}$, and put $$S'_{0\lambda }=r\,S_{0\lambda }\quad {\rm for}\ \lambda \in P'_k/\hbox{\gragrec \char22}_r\ ,\ \lambda \not={k\over r}\rho\ {\rm ;} \qquad S'_{0,\nu ^{(i)} }= S_{0,{k\over r}\rho}\quad {\rm for}\ i=1,\ldots,r\ .$$ One deduces easily from Cor.\ \ref{total} the formula $\mathop{\rm dim}\nolimits H^0(M_{{\bf PGL}_r},{\cal D}^{k})=\sum |S'_{0\lambda }|^{2-2g}$, where $\lambda $ runs over $P'_k/\hbox{\gragrec \char22}_r\cup\{\nu ^{(1)},\ldots,\nu ^{(r)}\}$. \bigskip \rem {Remark}\label{rem} It is not clear to me what is the physical meaning of the space $H^0(M_{{\bf PGL}_r}^d,{\cal D}^{k})$, in particular if its dimension can be predicted in terms of the $S$\kern-1.5pt - matrix. It is interesting to observe that the number $N(g)$ given by Prop.\ \ref{formule}, which is equal to $\mathop{\rm dim}\nolimits H^0(M_{{\bf PGL}_r}^d,{\cal D}^{k})$ for $g\ge 2$, {\it is not necessarily an integer} for $g=1$: for $d=0$ one finds $\displaystyle N(1)=1+{(k+1)^{r-1}-1\over r^2}$, which is not an integer unless $r^2\mid k$. \vfill\eject \centerline{ REFERENCES} \vglue15pt\baselineskip12.8pt \def\num#1{\item{\hbox to\parindent{\enskip [#1]\hfill}}} \parindent=1.3cm \num{A-S} M.F.\ {\pc ATIYAH}, I.M.\ {\pc SINGER}: {\sl The index of elliptic operators III}. Ann.\ of Math.\ {\bf 87}, 546-604 (1968). \smallskip \num{B} A.\ {\pc BEAUVILLE}: {\sl Conformal blocks, Fusion rings and the Verlinde formula.} Proc.\ of the Hirzebruch 65 Conf.\ on Algebraic Geometry, Israel Math.\ Conf.\ Proc.\ {\bf 9}, 75-96 (1996). \smallskip \num{B-L} A.\ {\pc BEAUVILLE}, Y.\ {\pc LASZLO}: {\sl Conformal blocks and generalized theta functions.} Comm.\ Math.\ Phys.\ {\bf 164}, 385-419 (1994). \smallskip \num{B-L-S} A.\ {\pc BEAUVILLE}, Y.\ {\pc LASZLO}, Ch.\ {\pc SORGER}: {\sl The Picard group of the moduli of $G$\kern-1.5pt - bundles on a curve}. Preprint alg-geom/9608002. \smallskip \num{B-S} A. {\pc BERTRAM}, A. {\pc SZENES}: {\sl Hilbert polynomials of moduli spaces of rank $2$ vector bundles II.} Topology {\bf 32}, 599-609 (1993). \smallskip \num{Bo} N.\ {\pc BOURBAKI}: {\sl Alg\`ebre}, Chap.\ X (Alg\`ebre homologique). Masson, Paris (1980). \smallskip \num{D-N} J.M.\ {\pc DREZET}, M.S.\ {\pc NARASIMHAN}: {\sl Groupe de Picard des vari\'et\'es de modules de fibr\'es semi-stables sur les courbes alg\'ebriques.} Invent.\ math.\ {\bf 97}, 53-94 (1989). \smallskip \num {F} G.\ {\pc FALTINGS}: {\sl A proof for the Verlinde formula.} J.\ Algebraic Geometry {\bf 3}, 347-374 (1994). \smallskip \num{F-S-S} J.\ {\pc FUCHS}, B.\ {\pc SCHELLEKENS}, Ch.\ {\pc SCHWEIGERT}: {\sl From Dynkin diagram symmetries to fixed point structures}. Preprint hep-th/9506135. \smallskip \num{F-H} W.\ {\pc FULTON}, J.\ {\pc HARRIS}: {\sl Representation theory}. GTM {\bf 129}, Springer-Verlag, New York Berlin Heidelberg (1991). \smallskip \num{L-S} Y.\ {\pc LASZLO}, Ch.\ {\pc SORGER}: {\sl The line bundles on the moduli of parabolic $G$\kern-1.5pt - bundles over curves and their sections}. Annales de l'ENS, to appear; preprint alg-geom/9507002. \smallskip \num{M-S} G. {\pc MOORE}, N. {\pc SEIBERG}: {\sl Taming the conformal zoo}. Phys. Letters B {\bf 220}, 422-430 (1989). \smallskip \num{N-R} M.S.\ {\pc NARASIMHAN}, S.\ {\pc RAMANAN}: {\sl Generalized Prym varieties as fixed points}. J.\ of the Indian Math.\ Soc.\ {\bf 39}, 1-19 (1975). \smallskip \num{P} T.\ {\pc PANTEV}: {\sl Comparison of generalized theta functions}. Duke Math.\ J.\ {\bf 76}, 509-539 (1994). \smallskip \num{R} S.\ {\pc RAMANAN}: {\sl The moduli spaces of vector bundles over an algebraic curve}. Math. Ann. {\bf 200}, 69-84 (1973). \vskip1cm \def\pc#1{\eightrm#1\sixrm} \hfill\vtop{\eightrm\hbox to 5cm{\hfill Arnaud {\pc BEAUVILLE}\hfill} \hbox to 5cm{\hfill DMI -- \'Ecole Normale Sup\'erieure\hfill} \hbox to 5cm{\hfill (URA 759 du CNRS)\hfill} \hbox to 5cm{\hfill 45 rue d'Ulm\hfill} \hbox to 5cm{\hfill F-75230 {\pc PARIS} Cedex 05\hfill}} \end

9609

alg-geom/9609011

en

https://arxiv.org/abs/alg-geom/9609011

alg-geom/9609011

Misha S. Verbitsky

Misha Verbitsky

Algebraic structures on hyperkaehler manifold

reference to Fujiki paper added in revision. LaTeX2e, 6 pages

Math. Res. Lett. 3 763-767 1996

Let $M$ be a compact hyperkaehler manifold. The hyperkaehler structure equips $M$ with a set $R$ of complex structures parametrized by $CP^1$, called "the set of induced complex structures". It was known previously that induced complex structures are non-algebraic, except may be a countable set. We prove that a countable set of induced complex structures is algebraic, and this set is dense in $R$. A more general version of this theorem was proven by Fujiki.

alg-geom

\section{Introduction.} \label{_Intro_Section_} We give the basic definitions and cite the results relevant to this paper. \subsection{Hyperk\"ahler manifolds} \label{_hype_defi_Subsection_} \hfill \definition \label{_hyperkahler_manifold_Definition_} (\cite{_Besse:Einst_Manifo_}) A {\bf hyperk\"ahler manifold} is a Riemannian manifold $M$ endowed with three complex structures $I$, $J$ and $K$, such that the following holds. \begin{description} \item[(i)] the metric on $M$ is K\"ahler with respect to these complex structures and \item[(ii)] $I$, $J$ and $K$, considered as endomorphisms of a real tangent bundle, satisfy the relation $I\circ J=-J\circ I = K$. \end{description} \hfill The notion of a hyperk\"ahler manifold was introduced by E. Calabi (\cite{_Calabi_}). \hfill Clearly, a hyperk\"ahler manifold has the natural action of quaternion algebra ${\Bbb H}$ in its real tangent bundle $TM$. Therefore its complex dimension is even. For each quaternion $L\in \Bbb H$, $L^2=-1$, the corresponding automorphism of $TM$ is an almost complex structure. It is easy to check that this almost complex structure is integrable (\cite{_Besse:Einst_Manifo_}). \hfill \definition \label{_indu_comple_str_Definition_} Let $M$ be a hyperk\"ahler manifold, $L$ a quaternion satisfying $L^2=-1$. The corresponding complex structure on $M$ is called {\bf an induced complex structure}. The $M$ considered as a complex manifold is denoted by $(M, L)$. \hfill Let $M$ be a hyperk\"ahler manifold. We identify the group $SU(2)$ with the group of unitary quaternions. This gives a canonical action of $SU(2)$ on the tangent bundle, and all its tensor powers. In particular, we obtain a natural action of $SU(2)$ on the bundle of differential forms. \hfill \lemma \label{_SU(2)_commu_Laplace_Lemma_} The action of $SU(2)$ on differential forms commutes with the Laplacian. {\bf Proof:} This is Proposition 1.1 of \cite{Verbitsky:Symplectic_II_}. \blacksquare Thus, for compact $M$, we may speak of the natural action of $SU(2)$ in cohomology. \subsection{Appendix: Induced complex structures of general type.} \label{_gene_appe_Subsection_} We cite results useful for understanding of behaviour of induced complex structures. The function of this appendix is illustrative. We do not refer to this subsection in the body of the article. This appendix is perfectly safe to skip. \hfill \definition \label{_generic_manifolds_Definition_} Let $M$ be a hyperk\"ahler manifold, $\c R$ the set of all induced complex structures. With each $I\in \c R$, we associate the Hodge decomposition $H^*(M) = \oplus H^{p,q}_I(M)$ on the cohomology of $M$. We say that $I$ is {\bf of general type} when all elements of the group \[ H^{p,p}_I(M)\cap H^{2p}(M,{\Bbb Z})\] are $SU(2)$-invariant. \hfill As \ref{_gene_type_co_div_by2_Remark_} below implies, the manifolds $(M, I)$ have no Weil divisors when $I$ is of general type. In particular, induced complex structures of general type are never algebraic. \hfill \proposition \label{_generic_are_dense_Proposition_} Let $M$ be a hyperk\"ahler manifold and $\c R$ be the set of induced complex structures over $M$. Let $\c R_{ng}\subset \c R$ be the set of all induced complex structures {\bf not} of general type. Then $\c R_{ng}$ is no more than countable. {\bf Proof:} This is Proposition 2.2 from \cite{Verbitsky:Symplectic_II_} \blacksquare \hfill Let $M$ be a compact hyperk\"ahler manifold, $dim_{\Bbb R} M =2m$. \hfill \definition\label{_trianalytic_Definition_} Let $N\subset M$ be a closed subset of $M$. Then $N$ is called {\bf trianalytic} if $N$ is an analytic subset of $(M,L)$ for any induced complex structure $L$. \hfill Let $I$ be an induced complex structure on $M$, and $N\subset(M,I)$ be a closed analytic subvariety of $(M,I)$, $dim_{\Bbb C} N= n$. Denote by $[N]\in H_{2n}(M)$ the homology class represented by $N$. Let $\inangles N\in H^{2m-2n}(M)$ denote the Poincare dual cohomology class. Recall that the hyperk\"ahler structure induces the action of the group $SU(2)$ on the space $H^{2m-2n}(M)$. \hfill \theorem\label{_G_M_invariant_implies_trianalytic_Theorem_} Assume that $\inangles N\in H^{2m-2n}(M)$ is invariant with respect to the action of $SU(2)$ on $H^{2m-2n}(M)$. Then $N$ is trianalytic. {\bf Proof:} This is Theorem 4.1 of \cite{Verbitsky:Symplectic_II_}. \blacksquare \remark \label{_triana_dim_div_4_Remark_} Trianalytic subvarieties have an action of quaternion algebra in the tangent bundle. In particular, the real dimension of such subvarieties is divisible by 4. \hfill \ref{_G_M_invariant_implies_trianalytic_Theorem_} has the following immediate corollary, also proven by a Fujiki (\cite{_Fujiki_}, Theorem 4.8 (1)): \corollary \label{_hyperkae_embeddings_Corollary_} Let $M$ be a compact hyperk\"ahler manifold, $I$ induced complex structure of general type, and $S\subset (M,I)$ its complex analytic subvariety. Then $S$ is trianalytic. \blacksquare \remark \label{_gene_type_co_div_by2_Remark_} {}From \ref{_hyperkae_embeddings_Corollary_} and \ref{_triana_dim_div_4_Remark_}, it follows that a holomorphically symplectic manifold of general type has no closed complex analytic subvarieties of odd dimension; in particular, such a manifold has no divisors. \section{Induced complex structures which are algebraic} \label{_algebra_Section_} \hfill \definition Let $M$ be a compact hyperk\"ahler manifold. Then $M$ is called {\bf simple} if $M$ is simply connected and cannot be non-trivially represented as a direct product of hyperk\"ahler manifolds. \hfill The general version of the following theorem was proven by A. Fujiki (\cite{_Fujiki_}, Theorem 4.8 (2)). Let $\pi:\; \c M {\:\longrightarrow\:} S$ be a deformation of a simple holomorphically simplectic manifold, with arbitrary base of positive dimension. Assume that $\c M{\:\longrightarrow\:} S$ is not isotrivial (not trivial on periods). Fujiki proves that for a dense subset $S_a\subset S$, the fibers $\pi^{-1}(s_a)$ are algebraic for all $s_a\in S_a$. I am grateful to Daniel Huybrechts, who provided me with this reference. Also, a similar (but weaker) result was proven by F. Campana (\cite{_Campana_}). \hfill \theorem \label{_alge_dense_Theorem_} Let $M$ be a compact simple hyperk\"ahler manifold and $\c R$ be the set of induced complex structures $\c R \cong {\Bbb C} P^1$. Let $\c R_{alg}\subset \c R$ be the set of all $L\in \c R$ such that the complex manifold $(M, L)$ is algebraic. Then $\c R_{alg}$ is countable and dense in $\c R$. \hfill {\bf Proof:} For each $L\in R$, consider the K\"ahler cone of $(M,L)$, denoted by $K(L)\subset H^2(M, {\Bbb R})$. By definition, $K(L)$ is the set of all cohomology classes $\omega\subset H^2(M,{\Bbb R})$ which are K\"ahler classes with respect to some metric on $(M,L)$. Let \[ K:= \bigcup\limits_{L\in \c R} K(L). \] By \cite{_main_}, Lemma 5.6, $K$ is open in $H^2(M, {\Bbb R})$. Therefore, the intersection $H^2(M, {\Bbb Q}) \cap K$ is dense in $K$. By Kodaira, a compact K\"ahler manifold is algebraic if and only if there exist a rational K\"ahler class on $M$ (\cite{_Griffi_Harri_}). By \ref{_K_proje_on_R_Lemma_} below, every cohomology class $\omega \in K$ corresponds to a unique induced complex structure $I(\omega)\in \c R$ such that $\omega$ is K\"ahler with respect to $I(\omega)$. We also prove that thus obtained map $\pi:\; K {\:\longrightarrow\:} \c R$ is continuous. Thus, $\pi\left( H^2(M, {\Bbb Q} \cap K)\right)$ is dense in $\c R$. On the other hand, by Kodaira, $\pi\left( H^2(M, {\Bbb Q} \cap K)\right)$ coinsides with $\c R_{alg}$. This proves \ref{_alge_dense_Theorem_}. \blacksquare \hfill The following lemma is implicit from \cite{_main_}. We decided to spell out its proof, for clarity; for missing details the reader is referred to \cite{_main_}. \hfill \lemma \label{_K_proje_on_R_Lemma_} Let $\omega\in K$ be a cohomology class which is K\"ahler with respect to $I\in \c R$. Then \begin{description} \item [(i)] such $I$ is unique, \item[(ii)] and the obtained map $\pi:\; K {\:\longrightarrow\:} \c R$ is continuous. \end{description} \hfill {\bf Proof:} Consider the positively defined scalar product on the cohomology space $H^2(M)$ induced by the standard scalar product on harmonic forms. This scalar product is clearly $SU(2)$-invariant. For an induced complex structure $L$, denote by $\omega_L\in H^2(M, {\Bbb R})$ the K\"ahler class of the K\"ahler structure associated with $L$ and the hyperk\"ahler structure. The corresponding harmonic form can be expressed as $\omega_L(x, y) = (x, L(y))$, where $(\cdot,\cdot)$ is the Riemannian form on $M$. Let $V\subset H^2(M, {\Bbb R})$ be the 3-dimensional subspace generated by $\omega_L$, for all $L\in \c R$ (see \cite{_main_}, Section 4), and $p:\; H^2(M) {\:\longrightarrow\:} V$ be the orthogonal projection to $V$. For a K\"ahler class $\omega$ on $(M, I)$, the product $(\omega, \omega_I)$ is positive, by Hodge--Riemann relations (\cite{_Griffi_Harri_}). Thus, for all $\omega\in K$, the vector $p(\omega)\in V$ is non-zero. Now, for each $I\in \c R$, the intersection $H^{1,1}_I(M) \cap V$ is one-dimensional, because in notation of \ref{_hyperkahler_manifold_Definition_}, the cohomology class $\omega_J+\sqrt{-1}\: \omega_K\in V$ belongs to $H^{2,0}_I(M)$ and $\omega_J-\sqrt{-1}\: \omega_K\in V$ belongs to $H^{0,2}_I(M)$. Therefore, $\omega$ and $\omega_I$ are collinear. Clearly, a complex structure $L$ is uniquely defined by the form $\omega_L$, and for all $L\in \c R$, the vectors $\omega_L\in V$ all have the same length. Thus, for each line $l\in V$, there exist no more than two induced complex structures $L\in \c R$ satisfying $l\in H^{1,1}_L(M)$. It is easy to check that these induced complex structures are opposite: we have $H^{1,1}_{L}(M) = H^{1,1}_{-L}(M)$. This implies that in $\c R$, only $L=I$ and $L=-I$ satisfy $\omega\in H^{1,1}_L(M)$. On the other hand, $\omega_{L} = - \omega_{-L}$. Thus, the numbers $(\omega, \omega_I)$ and $(\omega, \omega_{-I})$ cannot be positive simultaneously. This implies that $\omega$ cannot be a K\"ahler class for $I$ and $-I$ at the same time. We proved \ref{_K_proje_on_R_Lemma_} (i). \hfill To prove \ref{_K_proje_on_R_Lemma_} (ii), consider the composition $s$ of $p:\; K {\:\longrightarrow\:} V\backslash 0$ and the natural projection map from $V\backslash 0 $ to the sphere $S^2\subset V \cong {\Bbb R}^3$. If we identify $S^2$ with $\c R\cong{\Bbb C} P^1$, we find that $\pi$ is equal to $s$ (see, for instance, \cite{_main_}, the proof of Sublemma 5.6). On the other hand, $s$ is continous by construction. This proves \ref{_K_proje_on_R_Lemma_} (ii). \blacksquare \hfill I am grateful to Daniel Huybrechts, who provided me with reference to \cite{_Campana_} and \cite{_Fujiki_} after the preliminary version of this paper appeared in alg-geom preprint server. \hfill

9302

alg-geom/9302005

en

https://arxiv.org/abs/alg-geom/9302005

alg-geom/9302005

Claude LeBrun

Claude LeBrun

A Finiteness Theorem for Quaternionic-Kaehler Manifolds with Positive Scalar Curvature

17 pages, LaTeX

We study the topology and geometry of those compact Riemannian (4n)-manifolds (M,g), n > 1, with positive scalar curvature and holonomy in Sp(n)Sp(1). Up to homothety, we show that there are only finitely many such manifolds of any dimension 4n.

alg-geom

\section{Preliminaries} \begin{defn} Let $(M, g)$ be a connected Riemannian $4n$-manifold, $n\geq 2$. We will say that $(M,g)$ is a quaternionic-K\"ahler manifold iff the holonomy group ${\cal H}(M,g)$ is conjugate to $H\cdot Sp(1)$ for some Lie subgroup $H\subset Sp(n)\subset SO (4n)$. \end{defn} \begin{example} The quaternionic projective spaces $${\Bbb HP}_n=Sp(n+1)/(Sp(n)\times Sp(1))$$ are quaternionic-K\"ahler manifolds. So are the complex Grassmannians $$Gr_2({\Bbb C}^{n+2})=SU(n+2)/S(U(n)\times U(2))$$ and the oriented real Grassmannians $$\tilde{Gr}_4({\Bbb R}^{n+4}) = SO(n+4)/(SO(n)\times SO(4)).$$ \end{example} In fact, these examples very nearly exhaust the compact homogeneous examples of quaternionic-K\"ahler manifolds. Indeed \cite{a2}, every such homogeneous space is a symmetric space, and \cite{wo} there is exactly one such symmetric space for each compact simple Lie algebra. They can be constructed as follows: let $G$ be a compact simple centerless group, and let $Sp(1)$ be mapped to $G$ so that its root vector is mapped to a root of highest weight. If $H$ is the centralizer of this $Sp(1)$, then the symmetric space $M=G/(H\cdot Sp(1))$ is quaternionic-K\"ahler, and every compact homogeneous quaternionic-K\"ahler manifold arises this way. How typical are these symmetric examples? One geometric feature of any irreducible symmetric space is that it must be Einstein, with non-zero scalar curvature. This, it turns out, also happens for quaternionic-K\"ahler manifolds: \begin{prop}[Berger] Every quaternionic-K\"ahler manifold is Einstein, with non-zero scalar curvature. \end{prop} \noindent For details, see \cite{besse}. In particular, a complete quaternionic-K\"ahler manifold has constant scalar curvature. \begin{defn} We will say that a quaternionic-K\"ahler manifold is {\em positive} if it is complete and has positive scalar curvature. \end{defn} \noindent It is now an an immediate consequence of Myers' theorem that a positive quaternionic-K\"ahler manifold is compact and has finite fundamental group. Unfortunately, however, the only known positive quaternionic-K\"ahler manifolds are the previously mentioned symmetric spaces! The main objective of the present article will be to explain why this situation is hardly surprising. The main tool in our investigation will be the next result: \begin{thm}[Salamon {\rm \cite{S}}; B\'erard-Bergery {\rm \cite{beber}}] Let $(M^{4n},g)$ be a quaternionic-K\"ahler manifold. Then there is a complex manifold $(Z,J)$ of complex dimension $2n+1$, called the {\em twistor space} of $(M,g)$, such that \begin{itemize} \item there is a smooth fibration $\wp : Z\to M$ with fiber $S^2$; \item each fiber of $\wp$ is a complex curve in $(Z,J)$ with normal bundle holomorphically isomorphic to $[{\cal O}(1)]^{\oplus 2n}$, where ${\cal O}(1)$ is the point-divisor line bundle on ${\Bbb CP}_1$; and \item there is a complex-codimension 1 holomorphic sub-bundle $D\subset TZ$ which is maximally non-integrable and transverse to the fibers of $\wp$. \end{itemize} Moreover, if $(M,g)$ is {\em positive}, then $Z$ carries a K\"ahler-Einstein metric of positive scalar curvature such that \begin{itemize} \item $\wp$ is a Riemannian submersion; \item $D$ is the orthogonal complement of the vertical tangent bundle of $\wp$; and \item the induced metric on each fiber of $\wp$ has constant curvature. \end{itemize} If $(M,g)$ is instead negative, there is an {\em indefinite} K\"ahler-Einstein pseudo-metric on $Z$ with all these properties.\label{sal} \end{thm} In particular, the twistor space $Z$ of a positive quaternionic-K\"ahler manifold is {\em Fano}: \begin{defn} A Fano manifold is a compact complex manifold $Z$ such that $c_1(Z)$ can be represented by a positive (1,1)-form. \end{defn} \noindent That is, a Fano manifold is a compact complex manifold which admits K\"ahler metrics of positive Ricci curvature. Every Fano manifold is simply connected, since $c_1>0~\Rightarrow ~\chi ({\cal O}) =h^0 ({\cal O}) =1$ by the Kodaira vanishing theorem, thus forbidding the possibility that the manifold might have a finite cover. Applying the exact homotopy sequence of $Z\to M$, we now conclude the following: \begin{prop} Any positive quaternionic-K\"ahler manifold is compact and simply connected. \end{prop} A completely different and extremely important feature of our twistor spaces is the holomorphic hyperplane distribution $D$, which gives a so-called {\em complex contact structure} to $Z$. Such structures will be discussed systematically in \S \ref{cont}. Our definition of quaternionic has carefully avoided the case of $n=1$; after all, $Sp(1)\cdot Sp(1)$ is all of $SO(4)$, so such a holonomy restriction says nothing at all. Instead, we choose the our definition in order to insure that Theorem \ref{sal} remains valid: \begin{defn} A Riemannian 4-manifold $(M,g)$ is called quaternionic-K\"ahler if it is Einstein, with non-zero scalar curvature, and half-conformally flat. \end{defn} \section{Complex Contact Manifolds} \label{cont} \begin{defn} A complex contact manifold is a pair $(X,D)$, where $X$ is a complex manifold and $D\subset TX=T^{1,0}X$ is a codimension-one holomorphic sub-bundle which is maximally non-integrable in the sense that the O'Neill tensor \begin{eqnarray*} D\times D &\to& TX/D\\ (v,w)&\mapsto & [v,w]\bmod D \end{eqnarray*} is everywhere non-degenerate. \end{defn} \begin{example} Let $Y_{n+1}$ be any complex manifold, and let $X_{2n+1}={\Bbb P}(T^{\ast}Y)$ be its projectived holomorphic cotangent bundle; dually stated, $X$ is the Grassmann bundle of complex $n$-planes in $TY$. Let $\pi :X\to Y$ be the canonical projection, and let $D\subset TX$ be the sub-bundle defined by $D|_P:=\pi^{-1}_{\ast}(P)$ for all complex n-planes $P\subset TY$. Then $D$ is a complex contact structure on $X$. \end{example}\bigskip The condition of non-integrability has a very useful reformulation, which we shall now describe. Given a codimension-one holomorphic sub-bundle $D\subset TX$, let $L:=TX/D$ denote the quotient line bundle. Letting $\theta : TX\to L$ be the tautological projection, we may think of $\theta$ as a line-bundle-valued 1-form $$\theta \in \Gamma (X, \Omega^1(L))~,$$ and so attempt to form its exterior derivative $d\theta$. Unfortunately, this ostensibly depends on a choice of local trivialization; for if $\vartheta$ is any 1-form, $d(f\vartheta)=fd\vartheta +df\wedge\vartheta$. However, it is now clear that $d\theta|_D$ {\em is} well defined as a section of $L\otimes \wedge^2D^{\ast}$, and an elementary computation, which we leave to the reader, shows that $d\theta|_D$, thought of in this way, is exactly the O'Neill tensor mentioned above. Now if the skew form $d\theta|_D$ is to be non-degenerate, $D$ must have positive even rank $2n$, so that $X$ must have odd complex dimension $2n+1\geq 3$. Moreover, the non-degeneracy exactly requires that $$\theta \wedge (d\theta )^{\wedge n}\in \Gamma (X, \Omega^{2n+1}(L^{n+1}))$$ is nowhere zero. But this provides a bundle isomorphism between $L^{\otimes (n+1)}$ and the anti-canonical line bundle $\kappa^{-1}=\wedge^{2n+1}T^{1,0}X$. Conversely, let $X$ be a simply-connected compact complex $(2n+1)$-manifold, and suppose that $c_1(X)$ is divisible by $n+1$. Then there is a unique holomorphic line bundle $L:=\kappa^{-1/(n+1)}$ such that $L^{\otimes (n+1)}\cong \kappa^{-1}$. If we are then given a twisted holomorphic 1-form $$\theta \in \Gamma (X, \Omega^1(\kappa^{-1/(n+1)}))$$ we may then construct $$\theta \wedge (d\theta )^{\wedge n}\in \Gamma (X, \Omega^{2n+1}(\kappa^{-1}))= \Gamma (X,{\cal O})= {\Bbb C}~.$$ If this constant is non-zero, $D=\ker \theta$ is then a complex contact structure. This simple observation has powerful consequences: \begin{prop} Let $X_{2n+1}$ be a simply connected compact complex manifold, and let $\cal G$ denote the identity component of the group of biholomorphisms $X\to X$. Then $\cal G$ acts transitively on the set of complex contact structures on $X$. \label{trans} \end{prop} \begin{proof} We may assume that there is at least one complex contact structure on $X$, since otherwise there is nothing to prove. In this case, the canonical line bundle $\kappa$ has a root $\kappa^{1/(n+1)}$, and there is only one such root because $H^1(X, {\Bbb Z}_{n+1})=0.$ Thus any complex contact structure is determined by a class $[\theta]\in {\Bbb P }\Gamma (X, \Omega^1\otimes (\kappa^{-1/(n+1)}))$ satisfying $\theta\wedge (d\theta)^n\neq 0$. The group $\cal G$ acts on this projective space $\cong {\Bbb P}_m$ in a manner preserving the hypersurface $S$ defined by $\theta\wedge (d\theta)^n= 0$, and so partitions ${\Bbb P}_m-S$ into orbits; since ${\Bbb P}_m-S$ is connected, it therefore suffices to prove that each orbit is open, and for this it would be enough to prove that the holomorphic vector fields generating the action of the the Lie algebra of $\cal G$ on ${\Bbb P}_m-S$ span the tangent space at each point. To prove the last statement, let $\theta\in \Gamma (X, \Omega^1\otimes (\kappa^{-1/(n+1)}))$ be any contact form, and let $\phi \in \Gamma (X, \Omega^1\otimes (\kappa^{-1/(n+1)}))$ be any other section. If $D$ denotes the kernel of $\theta$, $d\theta |_D: D\to D\otimes \kappa^{-1/(n+1)}$ is an isomorphism of holomorphic vector bundles, so we can define a holomorphic vector field $v\in \Gamma (X, {\cal O} (D))$ by $v=(d\theta |_D)^{-1}(\phi)$. We then have $\pounds_{\xi} \theta\equiv {\xi} \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\theta\equiv \phi \bmod\theta$, so that action of the Lie algebra of $\cal G$ spans the tangent space of ${\Bbb P }\Gamma (X, \Omega^1\otimes (\kappa^{-1/(n+1)}))$ at $[\theta ]$, thus proving the proposition. \end{proof} \begin{cor} Two simply-connected compact complex manifolds are complex-contact isomorphic iff the underlying complex manifolds are biholomorphically equivalent. \end{cor} This will now yield a result which is crucial for our purposes. \begin{defn} We will say that two Riemannian manifolds $(M_1,g_1)$ and $(M_2,g_2)$ are {\em homothetic} if there exists a diffeomorphism $\Phi : M_1\to M_2$ such that $\Phi^{\ast}g_2=cg_1$ for some constant $c>0$. Such a map $\Phi$ will be called a {\em homothety}. \end{defn} \begin{prop} Two positive quaternionic-K\"ahler manifolds are homothetic iff their twistor spaces are biholomorphic. \label{perfect} \end{prop} \begin{proof} \label{final} Let $(M, g)$ and $(\tilde{M}, \tilde{g})$ be two given quaternionic-K\"ahler manifolds, $\wp :Z\to M$ and $\tilde{\wp} :\tilde{Z}\to\tilde{M}$ their twistor spaces, $h$ and $\tilde{h}$ the K\"ahler-Einstein metrics of $Z$ and $\tilde{Z}$. We also suppose that a biholomorphism $\Phi: Z\to \tilde{Z}$ is given to us. For some positive constant $c>0$, $h$ and $c\tilde{h}$ have the same scalar curvature; and notice that replacing $\tilde{h}$ with $c\tilde{h}$ just corresponds to replacing $\tilde{g}$ with $c\tilde{g}$. Now $\Phi^{\ast}c\tilde{h}$ is a K\"ahler-Einstein metric on $Z$ with the same scalar curvature as $h$, and the Bando-Mabuchi theorem \cite{BM} on the uniqueness of K\"ahler-Einstein metrics now asserts that there exists a biholomorphism $\Psi: Z\to Z$ such that $\Psi^{\ast}(\Phi^{\ast}c\tilde{h})=h$. Let $N\subset\Gamma (Z, \Omega^1(\kappa^{-1/(n+1)}))$ be defined by $\phi\wedge (d\phi)^{\wedge n}=1$. Proposition \ref{trans} implies that a finite connected cover $\cal G$ of the connected component of the automorphism group of $(Z, J)$ acts transitively on $N$, since, in the notation of the proof of that proposition, $N\to {\Bbb P}_m-S$ is a finite covering. Because $h$ is K\"ahler-Einstein, with positive scalar curvature, the Killing fields are a real form of the algebra of holomorphic vector fields, and a finite cover $G$ of the connected component of the isometry group of $(Z,h)$ is therefore a compact real form of $\cal G$. Morse theory now predicts that one orbit of the action of $G$ on $N$ is precisely the set of critical points of the $G$-invariant strictly plurisubharmonic function $f:N\to {\Bbb R}$ given by $\phi\mapsto \|\phi\|^2_{L^2,h}$. On the other hand, the derivative of $f$ at a contact form $\phi$ in the direction of a real-holomorphic vector field $\xi$ on $Z$ is given by $$df(\xi )|_{\phi}={\textstyle \frac{1}{(2n)!}} \int_X d (\xi \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm \omega) \wedge |\phi|^2_h\left(\beta_{\phi}-{\textstyle \frac{n+2}{n+1}} \omega^{2n}\right)~,$$ where $\omega$ is the K\"ahler form of $(Z,J,h)$ and $\beta_{\phi}$ is the $(2n,2n)$ form obtained by orthogonally extending the restriction of $\omega^{2n}$ from $D=\ker \phi$ to $TZ$. For the canonical contact form $\theta$ associated with the quaternionic-K\"ahler metric $g$ by the twistor construction, $|\theta |_h$ is constant and $\beta_{\theta}=\wp^{\ast}(2n)! d\mbox{\rm vol}_g$ is closed, so that $\theta$ is a critical point of $f$; but the same argument applies equally to the contact from of $\tilde{Z}$, and hence to the pull-back of this contact form via the holomorphic isometry $\Phi\Psi$. Hence there is a holomorphic isometry $\Xi: Z\to Z$ sending the first of these contact structures to the second, and $\Phi\Psi\Xi : Z\to \tilde{Z}$ is a then biholomorphism which sends $h$ to $c\tilde{h}$ and $D$ to $\tilde{D}$. Since the vertical tangent spaces of $\wp$ and $\tilde{\wp}$ are the orthogonal complements of $D$ and $\tilde{D}$ with respect to $h$ and $\tilde{h}$, respectively, it follows that $\Phi\Psi\Xi$ sends fibers of $\wp$ to fibers of $\tilde{\wp}$, and so covers a diffeomorphism $F:M\to \tilde{M}$. Moreover, since the $\wp$ and $\tilde{\wp}$ are Riemannian submersions, one has $F^{\ast}c\tilde{g}=g$, and $F$ is thus a homothety between $(M,g)$ and $(\tilde{M}, \tilde{g})$. \end{proof} For less precise but more broadly applicable theorems on the invertibility of the twistor construction, cf. \cite{L0}\cite{BE}. \begin{defn} Let $(X_{2n+1},D)$ be a complex contact manifold. An $n$-dimensional submanifold $\Sigma_n\subset X_{2n+1}$ is called {\em Legendrian} if $T\Sigma\subset D$. \end{defn} \begin{lem} Let $(X_{2n+1},D)$ be a complex contact manifold, and let \linebreak $\pi: X\to Y_{n+1}$ be a proper holomorphic submersion with Legendrian fibers. Then $X\cong {\Bbb P}(T^{\ast} Y)$ as complex contact manifolds. \label{class} \end{lem} \begin{proof} Define $\Psi : X\to Gr_{n}(TY)$ by $x\to \pi_{\ast}(D_x)$. This map preserves the contact structure; and since the pull-back of the contact form of $Gr_{n}(TY)$ via $\Psi$ is the contact form of $X$, $\Psi^{\ast}$ induces an isomorphism between forms of top degree. In other words, $\Psi$ is a submersion onto its image, and, in particular, induces a submersion from each fiber of $X$ onto its image in the fiber of $Gr_{n}(TY)={\Bbb P}(T^{\ast} Y)$. By the properness assumption, $\Psi$ is fiber-wise therefore a covering map. But the fibers of ${\Bbb P}(T^{\ast} Y)$ are projective spaces, and so simply connected. Hence $\Psi$ is an injective holomorphic submersion, and so biholomorphic. \end{proof} \begin{defn} If $(X,D)$ is a complex contact manifold such that $X$ is Fano, we will say that $(X,D)$ is a Fano contact manifold. \end{defn} \begin{lem} Let $\varpi: {\cal X} \to {\cal B}$ be a holomorphic family of Fano contact manifolds with smooth connected parameter space--- that is, let $\cal B$ be a connected complex manifold, $\varpi$ a proper holomorphic submersion with Fano fibers, and assume that $\cal X$ is equipped with a maximally non-integrable, complex codimension 1 sub-bundle $D\subset \ker \varpi_{\ast}$ of the vertical tangent bundle. Then any two fibers $(X_0, D|_{X_0})$ and $(X_t,D|_{X_t})$ are isomorphic as complex contact manifolds.\label{rig} \end{lem} \begin{proof} Since any two points in $\cal B$ can be joined by a finite chain of holomorphic images of the unit disk $\Delta \subset {\Bbb C}$, it suffices to prove the lemma when the base $\cal B$ is a disk $\Delta$. We now proceed as in \cite{L1}. By Darboux's theorem, any complex contact structure in dimension $2n+1$ is locally complex-contact isomorphic to the one on ${\Bbb C}^{2n+1}$ determined by the 1-form $$\vartheta = dz^{2n+1}+\sum_{j=1}^{n}z^jdz^{n+j}~, $$ so we may cover our family $\varpi :{\cal X}\to \Delta$ by Stein sets $U_j$ on which we have holomorphic charts $\Phi_j: U_j\hookrightarrow {\Bbb C}^{2n+1}\times \Delta$ such that the last coordinate is given by $\varpi$ and the fiber-wise contact structure on $\cal X$ agrees with that induced by $\Phi_j^{\ast} \vartheta$. Letting $t$ denote the standard complex coordinate on $\Delta$, we lift $d/dt$ to each $U_j$ as the vector field $v_j(\Phi_j^{-1})_{\ast}d/dt$, and observe that the $t$-dependent vertical vector field $w_{jk}:=v_j-v_k$ satisfies $\pounds_{w_{jk}}\theta \propto \theta$. Let $f_{jk}:=\theta (w_{jk})\in \Gamma (U_j\cap U_k, {\cal O} (L))$, and notice that the collection $\{ f_{jk}\}$ is a \v{C}ech cocycle representing an element of $H^1({\cal X}, {\cal O} (L))$. On the other hand, since $L$ is a fiber-wise $(n+1)^{st}$-root of the vertical anti-canonical bundle $\kappa^{-1}$, and since each fiber $X_t$ of $\varpi$ is assumed to be a Fano manifold, the bundle $\kappa^{-1}\otimes L$ is fiber-wise positive, and $H^1(X_t, {\cal O}(L))=0$ $\forall t\in \Delta$ by the Kodaira vanishing theorem. Thus the first direct image sheaf $\varpi_{\ast}^1{\cal O}(L)$ is zero. Since $\Delta$ is Stein, the Leray spectral sequence now yields $H^1({\cal X}, {\cal O}(L))=0$. Hence there exist sections $h_j\in \Gamma (U_j\, {\cal O} (L))$ such that $f_{jk}=h_j-h_k$ on $U_j\cap U_k$. On $U_j$ there is now a unique vertical holomorphic vector field $u_j$ such that $\theta (u_j)=h_j$ and $\pounds_{u_j}\theta\propto\theta$. Indeed, taking a local trivialization of $L$ so as to locally represent $\theta$ by a holomorphic 1-form $\vartheta$, a vector field $u$ satisfies $\pounds_u\theta\propto\theta$ iff $$u \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm d\vartheta \equiv -d(u \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm \vartheta ) \bmod \vartheta~,$$ so that such a field is uniquely determined by an arbitrary local function $f=\vartheta (u)= u \, \mbox{\raisebox{-.19ex}{\rule[.1mm]{2.15mm \vartheta$. We therefore conclude that $v_j-v_k=w_{jk}=u_j-u_k$ on $U_j\cap U_k$, and the vector field $v=v_j-u_j$ is therefore globally defined. Since $\pounds_{v_j}\vartheta =0$ and $\pounds_{u_j}\vartheta \equiv 0 \pmod{\vartheta, dt},$ the flow of $v=v_j-u_j$ preseves the fiberwise contact structure on $\cal X$. And since $\varpi$ is a proper map, we can now integrate the flow of our lift $v$ of $d/dt$ to produce a fiber-wise contact biholomorphism between $\cal X$ and $ X_0\times \Delta$. In particular, any fiber $X_t$ is complex-contact equivalent to the central fiber $X_0$. \end{proof} \section{Mori Theory}\label{this} Mori's theory of extremal rays \cite{M} has led to a startling series of advances in the classification of complex algebraic varieties, especially in the Fano case which interests us. One beautiful consequence of this is the so-called {\em contraction theorem}: if $X$ is a Fano manifold, there is always a map $\Upsilon : X\to Y$ to some other variety $Y$ which decreases the second Betti number $b_2$ by one, and where the kernel of $ \Upsilon_{\ast} :H_2(X, {\Bbb R})\to H_2(Y, {\Bbb R})$ is generated by the class of a rational holomorphic curve ${\Bbb CP}_1\subset X$. (The positive half of such a one-dimensional subspace $\ker \Upsilon_{\ast}\subset H_2(X, {\Bbb R})$ is called an ``extremal ray''.) If $b_2(X)=1$, this tells us next to nothing, because we can take $Y$ to be a point; but for $b_2(X)\geq 2$, it is quite a powerful tool. In particular, it gives rise to the following very useful result of Wi\'sniewski \cite{W}: \begin{thm}[Wi\'sniewski] Let $X$ be a Fano manifold of dimension $2r-1$ for which $r|c_1$. Then $b_2(X)=1$ unless $X$ is one of the following: (i) ${\Bbb CP}_{r-1}\times Q_r$\ ; (ii) $ {\Bbb P}(T^{\ast}{\Bbb CP}_r)$\ ; or (iii) ${\Bbb CP}_{2r-1}$ blown up along ${\Bbb CP}_{r-2}$.\label{wis} \end{thm} Here $Q_r\subset {\Bbb CP}_{r+1}$ denotes the r-quadric, while the projectivization of a bundle $E\to Y$ is defined by ${\Bbb P}(E):=(E-0_Y)/({\Bbb C}-0)$. The essence of the proof is that, since the rational curves collapsed by the Mori contraction have, in these circumstances, normal bundles of rather large index, they are so mobile that they sweep out projective spaces of comparatively large dimension, and these must therefore be the fibers of the contraction map. The following is now an easy consequence: \begin{cor} Let $(X_{2n+1},D)$ be a Fano contact manifold. If \linebreak $b_2(X)>1$, then $X= {\Bbb P}(T^{\ast}{\Bbb CP}_{n+1})$. \label{boop} \end{cor} \begin{proof} Setting $r=n+1$, we notice that the existence of a contact structure implies that $(n+1)|c_1$. We may therefore invoke Theorem \ref{wis}. On the other hand, spaces (i) and (iii) aren't complex contact manifolds, since $\Gamma ({\Bbb CP}_{r-1} , \Omega^1 (1))=0$ and therefore the obvious foliations by ${\Bbb CP}_{r-1}$'s would necessarily have Legendrian leaves, implying (by Lemma \ref{class}) that these spaces would then have to be of the form ${\Bbb P}(T^{\ast}Y)$, where $Y$ is the leaf space $Q_r$ or ${\Bbb CP}_{r}$--- a contradiction. So the only candidate left is (ii), and this {\em is} in fact a contact manifold. \end{proof} \begin{thm}{\rm \cite{L3}} Let $(M,g)$ be a compact quaternionic-K\"ahler $4n$-manifold with $s>0$. Then either \begin{description} \item{(a)} $b_2(M)=0$; or else \item{(b)} $M=Gr_2({\Bbb C}^{n+2})$ with its symmetric-space metric.\label{grass} \end{description} \end{thm} \begin{proof} By the Leray-Hirsch theorem on sphere bundles, the second Betti numbers of $M^{4n}$ and its twistor space $Z_{2n+1}$ are related by $b_2(Z)=b_2(M)+1$. Since $Z$ is a Fano contact manifold, $b_2(M)>0\Rightarrow Z={\Bbb P}(T^{\ast}{\Bbb CP}_{n+1})$ by Corollary \ref{boop}. But this is the twistor space of $Gr_2({\Bbb C}^{n+2})$. The result therefore follows by Proposition \ref{perfect}. \end{proof} \setcounter{main}{1} \begin{main}[Strong Rigidity] Let $M$ be a compact quaternionic-K\"ahler manifold of positive scalar curvature. Then $\pi_1(M)=0$ and $$H_2(M, {\Bbb Z} )= \left\{ \begin{array}{cl} 0&M={\Bbb HP}_n\\ {\Bbb Z}&M=Gr_2({\Bbb C}^{n+2})\\ \mbox{finite}\supset {\Bbb Z}_2&\mbox{otherwise.} \end{array} \right. $$\label{tor} \end{main} \begin{proof} If $(M, g)$ is not homothetic to the symmetric space $Gr_2({\Bbb C}^{n+2})=SU(n+2)/S(U(n)\times U(2))$, $b_2(M)=0$ by Theorem \ref{grass}, so that $H^2(M, {\Bbb Z})=0$ and $H_2(M, {\Bbb Z})$ is finite. Since we also know that $H_1(M, {\Bbb Z})=0$, $H^2(M, {\Bbb Z}_2)$ is exactly the the 2-torsion of $H_2(M, {\Bbb Z})$ by the universal coefficients theorem. If, on the other hand, $(M, g)$ is not homothetic to the symmetric space ${\Bbb HP}_n$, the class $\varepsilon\in H^2(M, {\Bbb Z}_2)$ must be non-zero \cite{S}, and the finite group $\pi_2(M)=H_2(M, {\Bbb Z})$ must therefore contain an element of order 2. \end{proof} \section{The Finiteness Theorem} \begin{thm} Up to biholomorphism, there are only finitely many Fano contact manifolds of any given dimension $2n+1$.\label{fanite} \end{thm} \begin{proof} By Wisniewski's theorem, we may restrict our attention to Fano manifolds with $b_2=1$. A theorem of Nadel \cite{nad} then asserts that there are only a finite number of deformation types of any fixed dimension.\footnote{It is now known \cite{kmm} that this is true even {\em without} the restriction $b_2=1$.} For any fixed deformation type, we may embed each Fano manifold in a fixed projective space ${\Bbb CP}_N$ in such a manner that the restriction of the generator $\alpha \in H^2({\Bbb CP}_N, {\Bbb Z})$ is a fixed multiple $\ell c_1(Z)/q$ of the anti-canonical class, and we may freely choose the positive integers $\ell$ and $q$ as long as $q|c_1$ and $\ell$ is sufficiently large. Thus, let $\cal F$ denote the set of all complex submanifolds $Z\subset {\Bbb CP}_N$ of some fixed dimension $m$ and degree $d$, and with the additional property that, for fixed integers $\ell, q$, the restriction of the hyperplane class is $\ell c_1(Z)/q$. Thus $\cal F$ is a Zariski-open subset in a component of the Chow variety, and so, in particular, is quasi-projective. There is now a tautological family \begin{eqnarray*} {\cal Z} & \hookrightarrow & {\cal F}\times {\Bbb CP}_N \\ {\varpi}\downarrow& & ~~~~\downarrow \\ {\cal F} & ~~= & ~~~~{\cal F} \end{eqnarray*} such that the fiber of $Z\in {\cal F}$ is the submanifold $Z\subset {\Bbb CP}_N$. We now assume moreover that the dimension $m$ is an odd number $2n+1$, take $q=n+1$, and choose $\ell \gg 0$ such that $\gcd (n+1, \ell)=1$. Letting $V\to {\cal Z}$ denote the vertical tangent bundle $\ker \varpi_{\ast}$, the vertical anti-canonical line bundle $\kappa^{-1}:= \wedge^{2n+1}V$ has a consistent fiber-wise $(n+1)^{st}$-root $L\to {\cal Z}$; indeed, using the Euclidean algorithm to write $1=a(n+1)+b\ell$, we may define $L$ by $L=\kappa^{-a}\otimes {\cal H}^{b}$, where $\cal H$ is the pull-back of ${\cal O}(1)$ from ${\Bbb CP}_N$ to $\cal Z$. Let ${\cal F}_j\subset {\cal F}$ denote the locus $${\cal F}_j :=\left\{ ~Z\in {\cal F}~~|~~h^0(Z, \Omega^1_Z \otimes \kappa^{-1/(n+1)}) \geq j~\right\} $$ where the space of candidate contact forms has dimension at least $j$. Since $\Omega^1_Z\otimes \kappa^{-1/(n+1)}$ is just the restriction of $V^{\ast}\otimes L$ to the appropriate fiber of $\varpi$, and since $\varpi$ is a flat morphism, it follows from the semi-continuity theorem \cite{hartsh} that each ${\cal F}_j$ is a Zariski-closed subset of the quasi-projective variety $\cal F$, and so, in particular, has only finitely many components. Let $\tilde{\cal F}_j:= {\cal F}_j-{\cal F}_{j-1}$, $j\geq 1$. Since $\tilde{\cal F}_j$ is a quasi-projective variety, it is a finite union of irreducible strata ${\cal F}_{jk}$, each of which is a connected complex manifold. On each stratum ${\cal F}_{jk}$, define a vector bundle ${ E}_{jk}$ as the zero-th direct image ${\cal O}(E_{jk})=\varpi^0_{\ast}{\cal O} (V^{\ast}\otimes L)$ of the fiber-wise 1-forms with values in $L$. Let $\cal L$ denote the line bundle $\varpi^0_{\ast}{\cal O}(L^{n+1}\otimes \wedge^{2n+1}V)$ on $\cal F$. Then $\theta \mapsto \theta\wedge (d\theta)^{n}$ is defines a canonical holomorphic section of the symmetric-product bundle ${\cal L}\otimes \bigodot^{n+1}E_{jk}^{\ast }$; let ${\cal E}_{jk}$ denote the open subset in the total space of $E_{jk}\to {\cal F}_{jk}$ where this homogeneous function is non-zero. Thus each ${\cal E}_{jk}$ is either a connected complex manifold or is empty. Each ${\cal E}_{jk}$ may now be viewed as the smooth parameter space of a connected family of Fano contact manifolds by taking the fiber over $\theta\in \Gamma (Z, \Omega^1(L))$ to be the pair $(Z,\ker \theta)$. On the other hand, every Fano contact manifold $(Z, D)$, where $Z$ is of the fixed deformation type, appears in one of these families--- albeit many times. Applying Lemma \ref{rig}, each of these families is of constant contact type. Since we must construct $\cal F$ only for a finite number of degrees in order to account for all Fano deformation types of the given dimension, and since, for each $\cal F$ we only have a finite number of contact families ${\cal E}_{jk}$, the result now follows. \end{proof} \setcounter{main}{0} \begin{main}[Finiteness Theorem] Up to homothety, there are only finitely many compact quaternionic-K\"ahler manifolds of positive scalar curvature in any given dimension $4n$. \end{main} \begin{proof} By Proposition \ref{perfect}, two positive quaternionic-K\"ahler manifolds are homothetic iff their twistor spaces are biholomorphic. Since the twistor space of any such manifold is a Fano contact manifold, the result now follows immediately from Theorem \ref{fanite}.\end{proof} \section{Other Results} We have seen in \S \ref{this} that the second homology of a positive quaternionic-K\"ahler manifold is far from arbitrary, and may by itself contain enough information to determine the metric up to isometry. Recent calculations of Salamon show that the higher homology groups are similarly constrained, in the following remarkable manner: \begin{thm}[Salamon] Let $(M^{4n},g)$ be a compact quaternionic-K\"ahler manifold with positive scalar curvature. Then the ``odd'' Betti numbers $b_{2k+1}$ of $M$ vanish, and the ``even'' Betti numbers $b_{2k}=b_{2(2n-k)}$ are subject to the linear constraint $$ \sum_{k=0}^n a_k b_{2k} = 0 ~,$$ where $a_k = \left\{\begin{array}{ll}1 + 2k + 2k ^2 - 4n/3 - 2kn + n^2 /3 &k < n\\(n^2 - n)/6 &k=n. \end{array}\right. $ \end{thm} The proof of this result involves an intricate interplay between the Kodaira vanishing theorem and the Penrose transform. Details will appear elsewhere \cite{ls}. \bigskip \noindent {\bf Acknowledgements.} The author would like to thank Shigeru Mukai, Jano\v{s} Koll\'ar, and Alan Nadel for their helpful explanations of Fano theory, and Simon Salamon for many, many helpful conversations.

9302

alg-geom/9302003

en

https://arxiv.org/abs/alg-geom/9302003

alg-geom/9302003

Alexander Sardo-Infirri

Sacha Sardo-Infirri

Lefschetz Fixed Point Theorem and Lattice Points in Convex Polytopes

29 pages, latex 2.09

A simple convex lattice polytope $\Box$ defines a torus-equivariant line bundle $\LB$ over a toric variety $\XB.$ Atiyah and Bott's Lefschetz fixed-point theorem is applied to the torus action on the $d''$-complex of $\LB$ and information is obtained about the lattice points of $\Box$. In particular an explicit formula is derived, computing the number of lattice points and the volume of $\Box$ in terms of geometric data at its extreme points. We show this to be equivalent the results of Brion \cite{brion} and give an elementary convex geometric interpretation by performing Laurent expansions similar to those of Ishida \cite{ishida}.

alg-geom

\section{Introduction} \subsection{The problem} Let $\Box$ be a convex polytope all of whose vertices belong to a lattice $M$. The question of calculating the number of points of $M$ contained in $\Box$ is a well-known one in convex geometry. The oldest formula appears to be Pick's classical result \cite{pick}, valid for arbitrary polygons in 2 dimensions: $$\#(\Box\cap M)={\rm Area\,}(\Box)+{1\over2}\#({\rm boundary}(\Box)\cap M)+1.$$ Following Ehrhart's work on Hilbert polynomials, Macdonald \cite{mac:lattice,mac:poly} subsequently generalised Pick's formula to arbitrary $n$. His formula expresses the volume of $\Box$ in terms of the number of lattice points of its' multiples $k\Box$ for finitely many integers $k$. Although these formulae are valid for arbitrary (non-convex) polygons, they do not give any convenient way of calculating either the volume, or the number of lattice points of $\Box$. A review of this and other problems concerning lattice points can be found in \cite{hammer,erdos}. {}From an elementary point of view, for large polytopes one expects the volume to be a good approximation to the number of lattice points, so that one can imagine a general formula of the form \begin{equation}\label{eq:RR} \mbox{number of points = volume + correction terms} \end{equation} where the corrections terms are negligible in the large limit. The formula we present here is however quite different in nature. \subsection{The results} Given a parameter $\zeta$, to each extreme point $\alpha$ of a simple convex polytope $\Box$, we associate a rational number depending the local geometry of $\Box$ at $\alpha$. Their sum is independent of $\zeta$ and yields the number of lattice points in $\Box$ (Theorem \ref{thm:number}). Our main formula (Theorem \ref{thm:formula-sing}) is more general, since it expresses not just the number, but {\em which\/} points of the lattice belong to the polytope, as a finite Laurent polynomial in $n$ variables (the lattice points corresponding to the monomials via $m\mapsto x^m=x_1^{m_1}x_2^{m_2}\cdots x_n^{m_n}$). I give an initial form of this using the Lefschetz-fixed point theorem for orbifolds. By expanding in Laurent series this is shown to be equivalent to another formulation (Theorem \ref{thm:formula-sing-b}) given by Brion \cite{brion} which doesn't involve cyclotomic sums. I use this form to calculate the number of lattice points. The volume of $\Box$ is obtained by taking the leading order terms for finer and finer subdivisions of the lattice. The Laurent series expansions extend Ishida's \cite{ishida} and provide a convex geometric interpretation of the formula (Theorem \ref{thm:chi-decomposition}). This in turn suggests a proof of the formula involving {\em no toric geometry\/} --- only convex geometry and elementary Laurent expansions. This could be considered as a variation of Ishida's proof \cite{ishida} based on the contractibility of convex sets. This paper is an amplification of my 1990 transfer dissertation at Oxford university \cite{sacha}. This was originally written whilst I was unaware of Michel Brion's 1988 paper \cite{brion}, where a toric approach is used to calculate the number of lattice points. There has also been a paper by Ishida \cite{ishida} where similar Laurent expansions similar to mine are performed. This is the revised version of my original which takes these works into account. Let me briefly mention their relationship to this paper. Brion relies on the Lefschetz-Rieman-Roch theorem for equivariant K-theory \cite{BFQ} and obtains theorem \ref{thm:formula-sing-b}. He calculates the number of lattice points by subdividing the tangent cones into basic cones. The formula that I obtain using the Lefschetz fixed point theorem involves instead cyclotomic sums for the action of the finite quotient group. By extending Ishida's Laurent series expansions \cite{ishida} in section \ref{sec:laurentexpansions} of this article, I prove that the two are equivalent, and provide a combinatorial interpretation of the formula. It is also not necessary for me to subdivide the tangent cones in order to obtain a formula for the number of lattice points. \subsection{The method} Our main tool is the theory of toric varieties. This associates a holomorphic line bundle $L_\Box$ over a complex orbifold $X_\Box$ to any n-dimensional simple polytope $\Box$ on a lattice $M$. The variety comes equipped with the action of an algebraic $n$-torus $T_N$ (the character group of $M$) and $L_\Box$ is equivariant with respect to this action. Its cohomology is trivial in positive dimension, whereas its space of sections is naturally isomorphic to a vector space generated by the lattice points in $\Box$. In \cite{bern,khov}, the Rieman-Roch theorem is used to calculate the number of lattice points in $\Box$. This yields a formula similar to equation (\ref{eq:RR}) above. The problem with this approach, however, is that the correction terms are not readily computable. In this paper I follow an idea of Atiyah and exploit the torus action. I apply Atiyah \& Bott's Lefschetz fixed point theorem \cite{ab:lefI} --- suitably extended to orbifolds \cite{kawasaki} --- to the (geometric endomorphism induced by the) action of $t\in T_N$ on the $d''$-complex of $L_\Box$. The $d''$-complex is elliptic \cite{ab:lefII} and its cohomology groups are (canonically isomorphic to) those of $(X_\Box,L_\Box)$. The fixed points of the torus action on $X_\Box$ correspond to the extreme points of $\Box$. The Lefschetz theorem in this case expresses the equality between the Lefschetz number (an element of ${\Bbb C}[M]$) and the sum of the indexes $\nu_\alpha$ for $\alpha$ in the set of extreme points. The $\nu_\alpha$ define elements of ${\Bbb C}(M)$. The formula I obtain initially involves sum over the characters of the finite abelian groups which charaterise the singularities at the points $P_\alpha\inX_\Box$ corresponding to $\alpha\in{\rm ext}\,\Box$. By studying characteristic series for cones in section \ref{sec:laurentexpansions} I eliminate the summation over group elements. If one restricts $t$ to the one-parameter subgroup of the torus determined by an element $\zeta$ of its' Lie algebra one obtains an equality between a polynomial and a sum of rational functions in one variable. When $t\to 1$ the polynomial tends to the the number of lattice points of $\Box$, and this is given by the sum of the constant terms in the one variable Laurent series for the rational functions: this gives theorem \ref{thm:number}. By identifying the coefficient of the leading order terms in the asymptotic expansions of the formula for submultiples of the lattice --- the `classical limit' in quantum terminology --- I derive a formula for the volume of $\Box$ in Theorem \ref{thm:volume}. I review the toric geometry results I shall need in the first part of this paper. The reader who is familiar with the notation in Oda \cite{oda} can {\tt GOTO PART II}, which contains the application proper. \subsection{Acknowledgments} I would like to thank Michael Atiyah and Peter Kronheimer for their stimulating ideas and encouraging support. Thanks also to Frances Kirwan for her suggestions and to Mark Lenssen, Jorgen Andersen and Jorge Ramirez-Alfonsin for interesting discussions. I was supported by a Rhodes Scholarship while I did this research. \subsection{Notation} \label{subsec:notation} Throughout this paper, let $N\cong {\Bbb Z}^n$ denote an n-dimensional integral lattice, $M\cong\hom_{\Bbb Z}(N,{\Bbb Z})$ it's dual and $N_{\RR}=N\otimes_{{\Bbb Z}}{\Bbb R}$ its' associated real vector space. The complex torus $N\otimes_{{\Bbb Z}} {\Bbb C}^\times\cong\hom_{{\Bbb Z}}(M,{\Bbb C}^\times)$ is denoted $T_N$ and the compact real sub-torus $N\otimes S^1\subset N\otimes {\Bbb C}^\times$ is denoted $CT_N$. If $A$ is any commutative ring with identity and $S$ any additive semi-group, we write $A[S]$ for the group algebra of $S$ with coefficents in $A$; this is generated by elements ${\bf e}(s)$ for $s\in S$ satisfying the relations ${\bf e}(s){\bf e}(s')={\bf e}(s+s')$. We write $A(S)$ for its total quotient ring (i.e., its' field of fractions if $A={\Bbb C}$). Occasionally I choose coordinates $t_i$ for $T_N$. This is equivalent to choosing generators $n_i$ for $N$. I denote the dual generators by $m^j\in M$. Then if $\alpha\in M$ and $z \in T_N$ have coordinates $\vect\alpha1n$ and $\vect z1n$ with respect to the appropriate bases we have $$\alpha(z)=z^\alpha=z_1^{\alpha_1}z_2^{\alpha_2}\cdots z_n^{\alpha_n}.$$ This identifies ${\Bbb C}[M]$ with the Laurent polynomials in the variables $t_i$. \newpage \part{Toric Geometry} The theory of toric varieties establishes correspondances between convex geometry in $n$ real dimensions and the geometry of compactifications of $n$-dimensional complex tori. I refer to \cite{kempf,oda,danilov}. Briefly, there is a functor that associates, to a pair $(N,\Sigma)$ (where $\Sigma$ is a fan in $N$), an irreducible normal Hausdorff complex analytic space $X_{N,\Sigma}$. A convex polytope $\Box$ in $M$ determines a unique fan $\Sigma$ in $N$, and we set \(X_\Box = X_{N,\Sigma} \). The polytope contains more information than simply its cone structure, and this determines a piecewise linear function $h=h_{\Box}$ on the support $|\Sigma|\subset N_{\RR}$ of $\Sigma$. This corresponds under the functorial construction above to an equivariant line bundle $L_h$ on $X_{N,\Sigma}$, which we denote by $L_{\Box}$. \section{Cones and Affine Toric Varieties} \subsection{Cones} Let $V$ denote a vector space and $V^*$ its dual. A {\em cone\/} in a vector space $V$ is a finite intersection of half-spaces in $V$. Cones are always convex and polyhedral. I shall take them to be also strongly convex, namely such that they do not contain any proper subspace of $V$. For $\ntup{v_1}{v_k} \in N_{\RR}$, let $\gen{v_1}{v_k}$ denote the smallest cone containing $\ntup{v_1}{v_k}$. Any cone is generated in this way. A cone is said to be {\em simplicial\/} if it can be generated by linearly independent elements of $N_{\RR}$. If it can be generated by part of a ${\Bbb Z}$-basis of $N$, then the cone is called {\em basic\/}. Finally, a cone is said to be {\em integral\/} with respect to $N$ if it can be generated by elements of $N$. When we speak of a {\em cone in a lattice\/} $N$ we mean a cone in $N_{\RR}$ which is integral with respect to $N$. I only consider such cones henceforth. The {\em dimension\/} of a cone is the dimension of the subspace it generates. By the {\em interior\/} of a cone we usually mean the relative interior in the subspace it generates. \subsection{Duality} Given a subset $A\subset V$ its {\em dual\/} $A{}^{\vee}\subset V{}^{\ast}$ is defined by: \[ A{}^{\vee}=\{\theta\in V{}^{\ast} : \forall v\in V, \ip{\theta}{v} \ge 0\}.\] \begin{prop} \label{prop:duality} The dual of a cone (respectively, a simplicial cone, a basic cone, or an integral cone) is a cone (respectively a simplicial cone, a basic cone, or an integral cone). Moreover, for any cone $\sigma$ we consider, we have $\sigma{}^{\vee}\dual=\sigma$. \end{prop} For a proof of all the results regarding cones, see \cite{rockaf}. A summary of the results I require will be found in \cite{oda}. \subsection{Affine Toric Varieties} Let $\sigma$ be a cone in $N$. Recall \cite[Prop.\ 1.1]{oda} that the subset of $M$ given by \[ S_{\sigma} = M\cap \sigma{}^{\vee} \] is finitely generated as an additive semigroup, generates $M$ as a group, and is saturated. Such semigroups are in one-one correspondance with cones in $N$. Denote by $U_{\sigma}=U_{N,\sigma}$ the set of semigroup homomorphisms from $(S_{\sigma}, +)$ to $({\Bbb C}, \cdot)$, namely \[U_{\sigma}=\{u:S_{\sigma}\to{\Bbb C}: u(0)=1, u(m+m^{\prime})=u(m)u(m^{\prime}),\forall m,m^{\prime} \in S_{\sigma}\}.\] This can be given the structure of an n-dimensional irreducible normal complex analytic space by choosing generators $\ntup{m_1}{m_p}$ for $S_{\sigma}$ and embedding $U_{\sigma}$ in ${\Bbb C}^p$ via the evaluation maps ${\bf ev}(m_i) : u\mapsto u(m_i)$ on the generators $m_i$. The structure is inherited from the usual structure on ${\Bbb C}^p$ and is independent of the generators chosen. In other words, $U_{\sigma}$ is just equal to the (set of points of the) affine scheme $\mbox{Spec}({\Bbb C}[S_{\sigma}])$. Identifying $U_{\sigma}$ with its ${\Bbb C}$-points corresponds to identifying ${\bf ev}(m)$ with ${\bf e}(m)$. I spend little effort making the distinction. The following proposition is easy to show \cite[Th. 1.10]{oda}: \begin{prop} \label{prop:non-sing} The variety $U_{\sigma}$ is non-singular if and only if $\sigma$ is basic. \end{prop} \section{Fans and General Toric Varieties} \subsection{Faces, Fans and Gluing} Let $\sigma$ be a cone in $N$. \begin{dfn} A {\em face\/} of $\sigma$ is a subset of the form $\sigma\cap\{m_0\}{}^{\bot}$, where $m_0\in M=\hom(N,{\Bbb Z})$ is non-negative on $\sigma$. A face of a cone is also a cone. \end{dfn} We immediately have: \begin{lemma} \label{lemma:open} If $\tau$ is a face of $\sigma$ then, for some $m_0\in M$, we have \[ U_{\tau} = \{u\inU_{\sigma} : u(m_0)\ne 0\},\] so that $U_{\tau}$ is naturally an open subset of $U_{\sigma}$. \end{lemma} Given this, one constructs collections of cones (called {\em fans\/}) which have the property that their corresponding varieties fit together in a natural way: \begin{dfn} A {\em fan\/} in $N$ is a collection $\Sigma=\{\sigma: \sigma \mbox{ a cone in }N\}$ satisfying the following conditions: \begin{itemize} \item if $\tau$ is a face of $\sigma$ and $\sigma \in\Sigma$, then $\tau\in \Sigma$. \item $\sigma\cap\sigma^{\prime}$ is a face of both $\sigma$ and $\sigma^{\prime}$, for all $\sigma,\sigma^{\prime}\in\Sigma$. \end{itemize} \end{dfn} The set of cones of $\Sigma$ of dimension $k$ is called the {\em k-skeleton\/} of $\Sigma$ and is denoted $\Sigma^{(k)}$. The union of all the cones of $\Sigma$ is called the {\em support\/} of $\Sigma$ and is denoted $|\Sigma|\subset N_{\RR}$. \begin{thm} \label{thm:general_toric} The {\em toric variety\/} associated to $(N,\Sigma)$ is the space obtained by gluing together the affine varieties $U_{N,\sigma}$ for $\sigma\in\Sigma$, using lemma \ref{lemma:open}. It is an n-dimensional Hausdorff complex analytic space $X_{N,\Sigma}$ which is irreducible and normal \cite[Theorem 1.4]{oda}. It is compact if and only if $\Sigma$ is {\em complete\/}, namely if and only if $|\Sigma|=N_{\RR}$. \end{thm} \subsection{The torus action} The torus $T_N$ acts on $U_{\sigma}$ by \((t\cdot u)(m)=t(m)u(m)\), and this gives an action on $X_{N,\Sigma}$. For $\sigma=\{0\}$, one has $U_{\{0\}}=T_N$, and the action coincides with group multiplication on the torus. The $T_N$-orbits on $X_{N,\Sigma}$ are given by the quotient algebraic tori \begin{equation} \label{eq:orb} \mbox{orb}(\tau)=\hom_{Z}(M\cap\tau{}^{\bot},{\Bbb C}^\times), \end{equation} for each $\tau\in\Sigma$. The orbit corresponding to $\tau$ has dimension equal to the codimension of $\tau$ in $N_{\RR}$. It is also easy to see that $U_{\sigma}$ decomposes as the disjoint union of the orbits corresponding to its faces, and that $\mbox{orb}(\sigma)$ is the only closed orbit in $U_{\sigma}$. I record a special case of this for later use: \begin{lemma} \label{lemma:fixpts} The fixed points of the $T_N$ action on $X_{N,\Sigma}$ are in one- one correspondance with the orbits $\mbox{orb}(\sigma)\inU_{\sigma}$, for the cones $\sigma$ in the $n$-skeleton $\Sigma^{(n)}$. \end{lemma} \subsection{Functoriality} Recall the following characterisation of toric varieties: \begin{quote} \em $X$ is a toric variety if and only if it is an irreducible normal variety, locally of finite type over ${\Bbb C}$, with a densely embedded torus whose action on itself extends to the whole variety. \end{quote} The assignment $(N,\Sigma) \mapsto X_{N,\Sigma}$ is a functor of categories: \begin{dfn} A {\em map of fans\/} \(\phi:(N^{\prime},\Sigma^{\prime})\to(N,\Sigma)\) is a ${\Bbb Z}$-linear homomorphism \(\phi:N^{\prime}\to N\) whose scalar extension \(\phi_{R}:N^{\prime}_{R}\to N_{R}\) satisfies the following property: for each $\sigma^{\prime}\in\Sigma^{\prime}$, there exists $\sigma\in\Sigma$ such that $\phi_{R}(\sigma^{\prime})\subset\sigma$. \end{dfn} \begin{thm} \cite[page 19]{oda} A map of fans \(\phi:(N^{\prime},\Sigma^{\prime})\to(N,\Sigma)\) gives rise to a holomorphic map \[\phi{}_{\ast}:X_{N^{\prime},\Sigma^{\prime}}\toX_{N,\Sigma}\] whose restriction to the open subset $T_{N^{\prime}}$ coincides with the homomorphism of algebraic tori \(\phi_{{\Bbb C}^\times}:N^{\prime}\otimes_{{\Bbb Z}}{\Bbb C}^\times\to N\otimes_{{\Bbb Z}}{\Bbb C}^\times.\) Through this homomorphism, $\phi{}_{\ast}$ is $(T_{N^{\prime}}, T_N)$- equivariant. Conversely any holomorphic map $\psi:X'\to X$ between toric varieties which restricts to a homomorphism $\chi: T'\to T$ on the algebraic tori $T'$ and $T$ in such a way that $\psi$ is $\chi$-equivariant corresponds to a unique ${\Bbb Z}$-linear homomorphism \(f:N^{\prime}\to N\) giving rise to a map of fans \((N^{\prime},\Sigma^{\prime})\to(N,\Sigma)\) such that $f{}_{\ast}=\psi$. \end{thm} \subsection{Finite Quotients} I will be interested in the case when $N^{\prime}$ is a ${\Bbb Z}$-submodule of $N$ of finite index and $\Sigma^{\prime}=\Sigma$. I write $X^{\prime}$ and $X$ for the corresponding varieties: \begin{prop} With the data as above, $X^{\prime}\to X$ coincides \label{prop:quotient} with the projection of $X^{\prime}$ with respect to natural action of the finite group \[K= N/N^{\prime} \cong\hom_{Z}(M^{\prime}/M,{\Bbb C}^\times)= \ker[T_{N^{\prime}}\to T_N].\] \end{prop} \begin{proof} \cite[Cor. 1.16, p.22]{oda} \end{proof} \section{Toric Varieties, Equivariant Line Bundles and Convex Polytopes} \subsection{Polytopes} \label{subsec:polytopes} Recall first some basic notions of convex geometry. A {\em convex polytope\/} $\Box$ in a vector space $V$ is a bounded intersection of a finite number of affine half- spaces of $V$. The set of extreme points of $\Box$ is denoted ${\rm ext}\,\Box$. Since $\Box$ is bounded, it is equal to the convex hull of ${\rm ext}\,\Box$. By a {\em polytope on the lattice\/} $M$ we mean a polytope in $M_{\Bbb R}$ such that ${\rm ext}\,\Box\subset M$. Suppose $\Box$ is such a polytope, and let $\alpha$ be an extreme point. I define the {\em (tangent) cone of $\Box$ at $\alpha$\/} to be the cone $C_\alpha$ in $M$ given by: \begin{equation} \label{eq:calpha} C_\alpha={\Bbb R}_{\ge 0}(\Box-\alpha)=\{r(v- \alpha):r\ge0,v\in\Box\}. \end{equation} Let $\lambda_{\alpha}^{i}, i=1,\dots,k$ be the shortest generators for $C_{\alpha}$ which belong to the lattice $M$. I call these the {\em edges of $\Box$ emanating from}\/ $\alpha$, or simply the {\em edge vectors for $\Box$ at $\alpha$}. If $C_\alpha$ is simplicial (respectively, basic), then $\Box$ is called {\em simple\/} (respectively, {\em basic}) at $\alpha$. Henceforth, all the polytopes we consider are convex, integral and simple at all extreme points. They may be non-basic. \subsection{Toric Varieties Defined by Polytopes} \subsubsection{The Fan Defined by a Polytope} The construction of $C_\alpha$ described in the previous section can be generalised to show that a polytope $\Box$ in $M$ defines a complete fan in $N$. To each face $\Gamma$ of $\Box$ we associate the cone $C_{\Gamma}$ in $M$ defined by \[ C_{\Gamma}={\Bbb R}_{\ge 0}(\Box-m_{\Gamma}),\] where $m_{\Gamma}$ is any element of $M$ strictly in the interior of the face $\Gamma$. If $F=\{\alpha\}$ we set $C_{\{\alpha\}}=C_\alpha$, as defined previously in equation (\ref{eq:calpha}). Taking duals one obtains a collection of cones in $N$ \[\Sigma_\Box=\{\sigma_{\Gamma}=C{}^{\vee}_{\Gamma} : \Gamma \mbox{ a face of }\Box\}.\] One has the following easy lemma: \begin{lemma} \label{lemma:fanBX} $\Sigma_{\Box}$ is equal to the fan consisting of the cones $\sigma_{\alpha}=C_{\alpha}{}^{\vee}$, for $\alpha\in{\rm ext}\,\Box$ and all their faces. It is complete, and its n-skeleton is \(\Sigma^{(n)}=\{\sigma_{\alpha}: \alpha\in{\rm ext}\,\Box\}\) \end{lemma} \subsubsection{The Variety Defined by a Polytope} I define $X_\Box$ to be $X_{\Sigma}$, for $\Sigma=\Sigma_{\Box}$. By \cite[Theorem 2.22]{oda}, $X_\Box$ is an orbifold (i.e., it has at worst quotient singularities) if $\Box$ is simple. \begin{prop} \label{prop:XBaction} The variety $X_\Box$ is compact, and is covered by affine pieces $$U_{\alpha}=U_{\sigma_{\alpha}}=\mbox{Spec}({\Bbb C}[M\cap C_\alpha]),$$ for $\alpha\in{\rm ext}\,\Box,$ each containing a unique $T_N$--fixed point $P_{\alpha}={\rm orb}\,(\sigma_\alpha)$ (see equation (\ref{eq:orb})). Furthermore, when $U_{\alpha}$ is non-singular, the weights of the $T_N$ action on the tangent space $T_{P_{\alpha}}U_{\alpha}$ are given by the edges vectors for $\Box$ at $\alpha$. \end{prop} \begin{proof} The first claim follows directly from theorem \ref{thm:general_toric} and lemmas \ref{lemma:fixpts} and \ref{lemma:fanBX}. For the second part, observe (prop.\ \ref{prop:non-sing} and \ref{prop:duality}) that $U_{\alpha}$ is non-singular if and only if the edge vectors at $\alpha$ generate $M$ as a group. The semigroup $C_\alpha$ is then free on these generators. They correspond to the weights of $T_N$ on $U_\alpha$, and hence, by linearity, to the weights on $T_{P_{\alpha}}U_{\alpha}.$ \end{proof} \subsection{Equivariant Line Bundles} The polytope $\Box$ contains more information than the fan $\Sigma_\Box$. This extra information turns out to be exactly what one needs to specify a $T_N$--equivariant line bundle $L_\Box$ over $X_\Box$. \subsubsection{Line Bundles and Piecewise Linear Functions} \label{subsub:LBPLF} In general (equivalence classes of) equivariant line bundles over $X_{N,\Sigma}$ are in one-one correspondence with the space $PL(N,\Sigma)$ of {\em piecewise linear functions} on $(N,\Sigma)$, namely functions \(h:|\Sigma|\to{\Bbb R}\) that are linear on each $\sigma\in\Sigma$ and which take integer values on the integer points of $|\Sigma|$. Defining an element $h\in PL(N,\Sigma)$ involves, by definition, specifying an element $l_\sigma\in M$ for each $\sigma\in\Sigma$ such that $h(n)=\ip{l_\sigma}{n}$ for all $n\in\sigma$. These elements determine a line bundle $L_h$ equiped with a $T_N$-action and whose projection $L_h\toX_{\Sigma}$ is equivariant with respect to that action. Note that in general, the elements $l_\sigma$ are not uniquely determined by $h$, but different choices give rise to equivariantly equivalent bundles. The bundle $L_h$ is defined to be trivial over the varieties $U_\sigma$, with transition functions given by \begin{equation} g_{\tau\sigma}(x)={\bf e}(l_\sigma-l_\tau)(x). \label{eq:trans} \end{equation} The action of $T_N$ on the piece $U_\sigma\times{\Bbb C}\subset L_h$ is defined by \begin{equation} t(x,c)=(tx,{\bf e}(-l_\sigma)(t)c). \label{eq:actionL} \end{equation} \subsubsection{Cohomology} The cohomology groups for equivariant line bundles decompose under the action of $T_N$ into weight spaces, and can be expressed as a direct sum (see \cite[Th. 2.6]{oda}): \[H^q(X_{\Sigma},{\cal O}_{X_{\Sigma}}(L_h))=\oplus_{m\in M} H^q_{Z(h,m)}(N_R,{\Bbb C}){\bf e}(m),\] where $Z(h,m)=\{n\in N_R:\ip{m}{n} \ge h(n)\},$ and $H^q_{Z(h,m)}(N_R,{\Bbb C})$ denotes the $q$-th cohomology group of $N_R$ with support in $Z(h,m)$ and coefficients in ${\Bbb C}$. \paragraph{The Line Bundle $L_\Box$} The polytope $\Box$ defines a piecewise linear function $h_\Box$ on $\Sigma_\Box$ by putting $l_{\sigma_{\alpha}}=\alpha$ (and $l_\sigma=\alpha$ for the faces $\sigma$ of $\sigma_{\alpha}$). The corresponding bundle is denoted $L_\Box$. Its cohomology is given by \cite[Cor. 2.9]{oda} \begin{equation} \label{eq:coho} H^q(X_\Box,{\cal O}_{X_\Box}(L_\Box))=\left\{ \begin{array}{ll} {\Bbb C}[M]_\Box = \oplus_{m\in M\cap\Box} {\Bbb C}{\bf e}(m) & \mbox{if $q=0$} \\ 0 & \mbox{otherwise} \end{array}\right. \end{equation} \newpage \part{The Polytope Formula} \section{The Lefschetz Fixed-Point Theorem} Recall \cite[Theorem 4.12]{ab:lefII} the following application of the Lefschetz fixed point theorem to the case of holomorphic vector bundles: \begin{thm} \label{thm:lefschetz} Let $X$ be a compact complex manifold $X$, $F$ a holomorphic vector bundle over $X$, $f:X\to X$ a holomorphic map with simple fixed points and $\phi:f{}^{\ast} F\to F$ a holomorphic bundle homomorphism. Let $L(T)$ be the Lefschetz number of the endomorphism $T$ of the $d''$-complex of $F$: \[L(T)=\sum(-1)^q\mbox{trace}H^q T|_{H^q(X;F)}.\] Then \(L(T)=\sum_{P=f(P)}\nu_{P}\), where \[\nu_{P}={{\mbox{trace}_{{\Bbb C}} \phi_{P}}\over{\mbox{det}_{{\Bbb C}}(1-df_{P})}}. \] \end{thm} (Recall that since $P$ is a fixed point, $\phi_{P}$ and $df_{P}$ are endomorphisms of $F_{P}$ and $T_{P}X$ respectively.) \subsection{Application} I apply this to the case where $X=X_\Box$, $L=L_\Box$ and $f: X \to X$ is given by the action of a non-trivial element of $t\in T_N$. The fixed points are simple and are given by $P_{\alpha}=\mbox{orb}(\sigma_{\alpha})\in U_{\sigma_{\alpha}}$, for $\alpha\in{\rm ext}\,\Box$. The bundle homorphism $\phi_{t}: t{}^{\ast}\L\to L$ is given by the action of $-t$ (recall that $T_N$ acts on line bundles). The cohomology groups are all zero, except $H^0(X_\Box,L_\Box)$ which is isomorphic to the subspace ${\Bbb C}[M]_\Box$ of ${\Bbb C}[M]$ determined by $\Box$. In this context, the Lefschetz number is an element of ${\Bbb C}[M]$ and the indexes $\nu_\alpha=\nu_{P_\alpha}$ are elements of ${\Bbb C}(M).$ (As we shall see in section~\ref{sec:laurentexpansions}, they are characteristic functions for the tangent cones to $\Box$.) \begin{lemma} \label{lemma:Laction} We have \[{\rm trace}\,(\phi_{t})_{P_\alpha} = \alpha(t).\] \end{lemma} \begin{proof} Recall (equation (\ref{eq:actionL})), that $t$ acts on the fibres of $L$ over $U_{\sigma}$ by multipication by ${\bf e}(-l_{\sigma})(t)$, where $l_{\sigma}$ are the elements of $M$ corresponding to $L$ as in \ref{subsub:LBPLF}. In the present case, at a fixed point $P_\alpha\in U_{\alpha}$ we have $l_{\sigma_{\alpha}}=\alpha$, so $\phi_t$ acts by ${\bf e}(-\alpha)(-t)=\alpha(t)$. \end{proof} In the case of a basic polytope $\Box$ in $M$, applying Theorem \ref{thm:lefschetz} directly one obtains: \begin{thm} \label{thm:non-sing} For a basic simple convex polytope $\Box$ in $M$, we have \begin{equation}\label{eq:sum-non-sing} \sum_{m\in\Box}m(t)=\sum_{\alpha\in {\rm ext}\,\Box} \nu_\alpha(t) \end{equation} where \begin{equation}\label{eq:nu-non-sing} \nu_\alpha(t)=\sum_{\alpha\in {\rm ext}\,\Box} {\alpha(t) \over (1-{\lambda_{\alpha}^1}(t)) \cdots (1-{\lambda_{\alpha}^n}(t))}, \end{equation} the vectors $\ntup{\lambda_{\alpha}^1}{\lambda_{\alpha}^n}$ are the edge vectors of $\Box$ at $\alpha$. \end{thm} \begin{proof} The decomposition of $H^0(X_\Box;{\cal O}_{X_\Box}(L_\Box))$ given by equation (\ref{eq:coho}) shows that the left-hand side of equation (\ref{eq:sum-non-sing}) is equal to the Lefschetz number of the endomorphism induced by the action of $t$. Lemma \ref{lemma:Laction} and Proposition \ref{prop:XBaction} yield equation (\ref{eq:nu-non-sing}). \end{proof} \subsection{The Lefschetz Fixed-Point Theorem for Orbifolds} In \cite{kawasaki} the Lefschetz formula is generalised to orbifolds (also known as V-manifolds), using zeta-function techniques. As I do not need the full power of this approach, I present an alternative more elementary argument. The Lefschetz fixed-point formula is essentially local in nature, the formula for the multiplicities $\nu_\alpha$ only involving the properties of $f$ and $\phi$ at the point $P_\alpha$. This fact is clearly apparent in Atiyah and Bott's proof in \cite{ab:lefI} (see their remarks at the beginning of section 5, and Proposition 5.3). To extend the formula to orbifolds, it is sufficient therefore to extend it to global quotient spaces, of the form $X=X'/K$. \begin{prop} \label{prop:Lef-Quotient} Suppose that a finite abelian group $K$ acts on a smooth $X'$ and equivariantly on a holomorphic bundle $F'$ over $X'$. Let $f':X'\to X'$ and $\phi':f'{}^{\ast} F'\to F'$ be as in Theorem \ref{thm:lefschetz}, and suppose they are $K$-equivariant. Denote by $L'(T')$ the Lefschetz number of the corresponding endomorphism $T'$ of the $d''$-complex of $F'$. Because of the $K$-equivariance, we can define $X=X'/K$, $f:X\to X$, $F=(F')^K$, $\phi:f{}^{\ast} F\to F$ and the corresponding Lefschetz number \[L(T)=\sum(-1)^q{\rm trace}\, H^qT|_{H^q(X;F)}.\] Then we have \begin{equation}\label{eq:L-Quotient} L(T)={1\over{|K|}}\sum_{k\in K}L'(k\circ T).\end{equation} \end{prop} \begin{proof} Note that since $T$ determines an endomorphism of the primed complex, it makes sense to write $L'(T)$. The claim then follows by applying the following easy lemma of linear algebra, recalling that $H^q(X;F)$ is just the $K$-invariant part of $H^q(X';F')$. \end{proof} \begin{lemma} Suppose we have a linear action of a finite abelian group $K$ on a finite dimensional vectorspace $V$, commuting with an endomorphism $T$ of $V$. Denote by $V^K$ the $K$-invariant subspace of $V$. Then $T$ is an endomorphism of $V^K$ and we have $${\rm trace}\, T|_{V^K}={1\over{|K|}}\sum_{k\in K} {\rm trace}\, (k\circ T)|_V .$$ \end{lemma} \begin{proof} Define $P$ to be the following endomorphism of $V$: $$Pv = {1\over{|K|}}\sum_{k\in K} k\cdot v.$$ Then $P^2=P$, so $P$ is the projection $V\to V^K$. Since $T$ commutes with $P$, it follows that $T$ respects the decomposition $V=V^K\oplus \ker P.$ Furthermore we have $${\rm trace}\, T|_{V^K}={\rm trace}\, TP|_V = {\rm trace}\, PT|_V ,$$ so the result follows. \end{proof} Now, given a general orbifold $X$, at each point $P\in X$, choose a {\em local model\/} $(U'_P,f'_P,K_P,L'_P)$ as follows: Let $U_P$ be an $f$-invariant neighbourhood of $P$ in $X$ and $U'_P$ be a smooth cover with an action of a finite group $K_P$, free away from $P$, such that $U_P=U'_P/K_P$. Thus $X$ has a quotient singularity of type $K_P$ at $P$. Let $f'_P:U'_P\to U'_P$ be a $K_P$-equivariant lifting of $f|_{U_P}$. A line bundle $L$ over $X$ is understood to be an invertible sheaf $L$ over $X$ such that for any $P\in X$ with local model $(U'_P,f'_P,K_P)$, there exists a line bundle $L'_P\to U'_P$ such that $L|_{U_P}=(L'_P)^{K_P}$. With these definitions, our remarks at the beginning of the section and Proposition \ref{prop:Lef-Quotient} imply the following: \begin{thm} \label{thm:lefschetz-orbifold} Let $X$ be a compact complex orbifold $X$, $F$ a holomorphic vector bundle over $X$, $f:X\to X$ a holomorphic map with simple fixed points and $\phi:f{}^{\ast} F\to F$ a holomorphic bundle homomorphism. Let $L(T)$ be the Lefschetz number of the endomorphism $T$ of the $d''$-complex of $F$: \[L(T)=\sum(-1)^q\mbox{trace}H^q T|_{H^q(X;F)}.\] Then \(L(T)=\sum_{P=f(P)}\nu_{P}\), where \[\nu_{P}={1\over |K_P|} \sum_{k\in K_P} {{\mbox{trace}_{{\Bbb C}} (k\circ\phi'_{P})}\over{\mbox{det}_{{\Bbb C}}(1-(k\circ df')_{P})}}, \] and $\phi', f'$ are lifts for $\phi, f$ respectively, in the same spirit as that of the local models above. \end{thm} \subsection{Singular Case} Suppose that $\Box$ is not basic relative to $M$ at $\alpha$. Then $X=X_\Box$ has a singularity at the point $P_\alpha$. Let $C_\alpha$ be the cone of $\Box$ at $\alpha$ and let $\sigma_\alpha$ be the dual cone. \begin{dfn} \label{dfn:dual-edge-vectors} I define the {\em dual edge vectors for $\Box$ at\/} $\alpha$ to be the primitive generators of the cone $\sigma_\alpha$ in $N$. When $\sigma_\alpha$ is not basic, the dual edge vectors do not generate $N$ as a group, but instead a sublattice $N'_\alpha$ of $N$ of finite index, which I call the {\em dual edge lattice for $\Box$ at\/} $\alpha$. \end{dfn} The cone $\sigma_\alpha$ is basic with respect to $N'_\alpha$, and the corresponding variety $X'_\alpha= X_{\sigma_\alpha,N'_\alpha}$ is smooth at $P_\alpha$. By Corollary \ref{prop:quotient}, the map $X'_\alpha \to X_\alpha = X_{\sigma_\alpha,N}$ is the quotient map by the action of the finite abelian group $K_\alpha=N/N'_\alpha\cong\hom_{{\Bbb Z}}(M'_\alpha/M,{\Bbb Q}/{\Bbb Z})$. Here $M'_\alpha$ is the dual of $N'_\alpha$ and is naturally a superlattice of $M$. There is a unique pairing $M'\times N \to {\Bbb Q}/{\Bbb Z}$ which extends the pairings $M\times N \to {\Bbb Z}$ and $M'\times N'\to {\Bbb Z}$. We then use the morphism ${\Bbb Q}/{\Bbb Z} \to {\Bbb C}^\times$ given by the exponential map to identify $K_\alpha$ with $\hom_{{\Bbb Z}}(M'_\alpha/M,{\Bbb C}^\times)$. If we identify $k\in K$ with the morphism $k:M'_\alpha\to{\Bbb Q}/{\Bbb Z}$ such that $k(M)=0$, the action is given by \begin{equation} \label{eq:Kaction} k\cdot u'(m')=\exp(2\pi i\ip{k}{m'})u'(m'), \end{equation} for $u'\in U_{\sigma}'$. Since the invariant part of $M'_\alpha$ under $K_\alpha$ is $M$, the line bundles $L_\alpha$ and $L'_\alpha$ over $X_\alpha$ and $X'_\alpha$ defined by the polytope $\Box$ are related by $L_\alpha=L_{\alpha}'^K$. Equation (\ref{eq:coho}) shows that the cohomology of $L_\alpha$ can be identified with the $K_\alpha$-invariant part of that of $L_\alpha'$. In summary, $(U'_\alpha, t, K_\alpha, L'_\alpha)$ is a local model for $X$ at $P_\alpha$. Applying the Lefschetz formula for orbifolds, one deduces: \begin{thm} \label{thm:formula-sing} For a simple convex polytope $\Box$ in $M$, we have \begin{equation} \label{eq:lef-fns} \sum_{m\in\Box}m(t)=\sum_{\alpha\in {\rm ext}\,\Box} \nu_\alpha(t) \end{equation} where \begin{equation}\label{eq:nu-sing} \nu_\alpha(t)= {1\over{|K_{\alpha}|}}\sum_{k\in K_{\alpha}} {\alpha(t) \over (1-e_k(\lambda_{\alpha}^{\prime 1}) \lambda_{\alpha}^{\prime 1}(t)) \cdots (1-e_k(\lambda_{\alpha}^{\prime n}) \lambda_{\alpha}^{\prime n}(t))}, \end{equation} and we write $e_k(\lambda)$ for $\exp(2\pi i{\ip{k}{\lambda}}).$ Here, the vectors $\ntup{{\lambda_{\alpha}^{\prime 1}}}{\lambda_{\alpha}^{\prime n}}$ are the edge vectors of $\Box$ at $\alpha$ in the dual $M'_\alpha$ of the dual edge lattice $N'_\alpha$ of definition \ref{dfn:dual-edge-vectors}, and $K_\alpha$ is the finite abelian group $N/N'_\alpha$ acting according to equation (\ref{eq:Kaction}). \end{thm} \section{Laurent Expansions} \label{sec:laurentexpansions} In this section I expand the rational functions $\nu_\alpha$ away from their poles, i.e., in the domains where $|\lambda_\alpha^i(t)|$ is not $1$, for $i=1,\dots,n$. This has two benefits. Firstly, it produces another formula which does not involve sums over roots of unity. We shall use this in calculating the number of lattice points and the volume. Secondly it leads us to interpret the formula as a combinatorial statement, decomposing the (characteristic polynomial for the) polytope $\Box$ as an algebraic sum of the (characteristic series for the) cones $C_\alpha$ for each extreme point. Ultimately this could be used to prove the formula using elementary convex geometric reasoning. We don't attempt this here, as Ishida has already reduced the proof to the contractibility of convex sets \cite{ishida}. We begin by some general remarks about characteristic series for convex cones. \subsection{Characteristic functions and series for convex cones} \label{subsec:characteristic} We recall some notation, following \cite{ishida}. Let $A$ be a commutative ring with identity. Recall that $A[M]$ denotes the {\em group algebra of $M$\/} generated by elements ${\bf e}(m)$ for $m\in M$ satisfying relations ${\bf e}(m){\bf e}(m')={\bf e}(m+m')$ and ${\bf e}(0)=1$. We denote by $A(M)$ the total quotient ring of $A[M]$. We define $A[[M]]={\rm Map}(M,A)$. Elements $f\in A[[M]]$ can also be expressed as formal Laurent series $f=\sum_{m\in M} f(m){\bf e}(m)$ and this defines a $A[M]$-module structure on $A[[M]]$ by: $${\bf e}(x)(\sum f(m){\bf e}(m)) = \sum f(m-x) {\bf e}(m).$$ The relationship of $A[[M]]$ to $A(M)$ is as follows. To a given element $\nu\in A(M)$ correspond (possibly) several elements of $A[[M]]$ called the {\em Laurent expansions\/} of $\nu$. As we see below a convex cone $C$ in $M$ gives rise to elements $\nu^M_C\in A(M)$ and $\chi_{C\cap M}\in A[[M]]$ and the latter is a Laurent expansion of the former. \begin{dfn} For $S$ a subset of $M$, we define the {\em characteristic series of $S$\/} to be the element $\chi[S]=\chi_S$ of $A[[M]]$ corresponding to the set-theoretic chacteristic function of $S$ (the function which takes values 1 on $S$ and 0 elsewhere), namely to the series $$\chi_S = \sum_{m \in S} {\bf e}(m).$$ \end{dfn} Let $C$ be a (strongly convex rational simplicial) cone in $M_{\Bbb R}$. We write ${\rm gen}^M_C=\{\lambda_1,\dots,\lambda_n\}$ for the primitive generators in $M$ of $C$. The unit parallelepiped $$Q^M_C=\{\sum a_i\lambda_i: 0\leq a_i < 1\}$$ defined by $C$ in $M$ intersects $M$ in $\{c_1,\dots,c_{k}\}$. Here $k=|K|$, the order of the finite abelian group which is the quotient of the dual lattice $N$ to $M$ by the lattice generated by the primitive generators ${\rm gen}^N_{C{}^{\vee}}=\{\sigma^1,\dots,\sigma_n\}$ of $C{}^{\vee}$ in $N$. \begin{dfn} For $C$ strictly convex, we define the {\em characteristic function for $C$ with respect to $M$\/} is the following element of $A(M)$: \begin{eqnarray*} \nu^M_C & = & \sum_{c\in Q^M_C\cap M} {\bf e}(c) \prod_{\lambda\in{\rm gen}^M_C} (1-{\bf e}(\lambda)){}^{-1}.\\ & = & \sum_{j=1}^{|K|} {\bf e}(c_j) \prod_{i=1}^n (1-{\bf e}(\lambda_i)){}^{-1}. \end{eqnarray*} For the translate of a cone $C$ by $\alpha\in M$, we set $\nu^M_{\alpha+C}={\bf e}(\alpha)\nu^M_C$. \end{dfn} Denote by ${\rm PL}_A(M)$ the $A[M]$-submodule of $A[[M]]$ generated by the set of {\em polyhedral Laurent series\/}: $$\{\chi_{C\cap M} : C \mbox{ a basic cone in }M_{\Bbb R}\}.$$ Ishida proves that the following \cite{ishida} \begin{prop} There exists a unique $A[M]$-homomorphism $$\varphi:{\rm PL}_A(M) \to A(M)$$ such that $\varphi(\chi_{C\cap M})=\nu^M_C$, for all basic cones $C$ in $M_{\Bbb R}$. \end{prop} Actually, we have: \begin{prop} For {\em any\/} cone $C$, $\chi_{C\cap M}\in {\rm PL}_A(M)$ and $\varphi(\chi_{C\cap M})=\nu^M_C$ for $\varphi$ defined above. \end{prop} \begin{proof} This follows from the remark that any element of $m\in M$ can be expressed uniquely as $q+\sum x_i \lambda_i$ with $q\in Q^M_C\cap M$ and $x_i\in {\Bbb N}$. \end{proof} The existence of $\varphi$ says essentially that we loose no information by passing from the characteristic function of a cone to its' Laurent series, even though the latter might not always have a well defined convergence on all of $T_N$ (in the case $A={\Bbb C}$). \paragraph{Remark} Whereas Ishida \cite{ishida} uses open cones, we find it more convenient to use closed ones. The correpondence between the two is of course that $C\cap M =\cup_{F < C} ({\rm int} F)\cap M$, where the union runs over the faces of $C$. \subsubsection{Action of $K$} The group $K$ acts on $M'$ and hence on $A[[M']]$ by $$k\cdot f = \sum_{m \in M} e_k(m)f(m){\bf e}(m),$$ and we have $A[[M]] = A[[M']]^K$. The following elementary remark gives the relationship between the characteristic series for $C$ with respect to the two lattices $M$ and $M'$. \begin{prop} \label{prop:chi} For any cone $C$, we have $$\chi_{C\cap M} = {1\over |K|} \sum_{k\in K} k\cdot \chi_{C\cap M'}.$$ \end{prop} \begin{proof} Note that $k\cdot\chi(m')=e_k(m')\chi(m')$. Since $e_k$, for $k\in K$, are nothing but the characters of the finite abelian group $M'/M$, we have $e_k(M)=|K|$ and $e_k(m'+M)=0$, for all $m'\not\in M$. Hence the formula follows. \end{proof} By the uniqueness of $\varphi$ we deduce that the same equality holds between the characteristic functions of $C$: \begin{cor} For any cone $C$, we have $$\nu^M_C={1\over |K|} \sum_{k\in K} k\cdot \nu^{M'}_C.$$ \end{cor} \subsection{Recovery of Brion's result} We apply the results of the previous section with $A={\Bbb C}$. Then ${\Bbb C}[M]$ is the affine coordinate ring for the algebraic torus $T_N$ and its' field of fractions ${\Bbb C}(M)$ is the ring of rational functions on $T_N$. The Lefschetz formula is expressing the chacteristic series $\chi_\Box$ of $\Box$ as a sum of elements of ${\Bbb C}(M)$. The theorem below says that these are simply the characteristic functions for the tangent cones of $\Box$ at its' extreme points. See \cite[Th\'eor\`eme 2.2]{brion} \begin{thm} \label{thm:formula-sing-b} Let $\Box$ be a simple convex polytope $\Box$ in $M$. Denote by $C_\alpha$ its' tangent cone at $\alpha\in{\rm ext}\,\Box$. Then we have \begin{equation} \label{eq:lef-fns-b} \chi_{\Box\cap M}= \sum_{\alpha\in{\rm ext}\,\Box} \nu^M_{C_\alpha}. \end{equation} \end{thm} \begin{proof} By theorem \ref{thm:formula-sing} we have $\nu_\alpha= {1\over|K|}\sum_{k\in K} k\cdot\nu^{M'}_{C_\alpha}$, which by the corollary of the previous section is nothing but $\nu^M_{C_\alpha}$. \end{proof} \subsection{Laurent expansions of $\nu_C$ and their domains of validity} We take $A={\Bbb C}$ and give all the different possible Laurent expansions of $\nu^M_C$ for a cone $C$. When we attempt to evaluate these on elements of $T_N$ these series only converge on certain open subsets which we specify here. \subsubsection{The expansions} We adopt the same notation as in section \ref{subsec:characteristic}. The primitive generators of $C_\alpha$ are ${\lambda_\alpha}^i$ in $M$ and $\lambda_\alpha^{\prime i}$ in $M'_\alpha$. \begin{prop}[Basic Expansion] For $|\lambda_\alpha^{\prime i}(t)|<1,$ for $i=1,\dots,n,$ we have \begin{equation} \label{eq:nu-basic-expansion} \nu_\alpha(t)=\chi_{\alpha+C_\alpha\cap M}(t). \end{equation} \end{prop} \begin{proof} Applying the elementary expansion (valid for $|z|<1$) $$ (1-z){}^{-1}= 1 + z + z^2 + z^3 + \cdots $$ to the individual factors $(1-e_k(\lambda_{\alpha}^{\prime i})\lambda_{\alpha}^{\prime i}(t)){}^{-1}$ gives: $$\nu_\alpha(t)=\alpha(t) {1\over{|K_{\alpha}|}}\sum_{k\in K_{\alpha}}(\sum_{c_1,\dots,c_n=0}^{\infty} e_k(c\cdot \lambda'_\alpha) (c\cdot \lambda'_\alpha)(t)),$$ where I have written $c\cdot \lambda'_\alpha$ for $\sum_{i=1}^{n}c_i \lambda_{\alpha}^{\prime i}$. Since the series is convergent, one has $$ \nu_\alpha(t) = \sum_{c_1,\dots,c_n=0}^{\infty} (\alpha+c\cdot \lambda'_\alpha)(t) {1\over{|K_{\alpha}|}}\sum_{k\in K_{\alpha}} e_k(c\cdot \lambda'_\alpha),$$ and the result follows from the proof of proposition \ref{prop:chi}. \end{proof} There are in fact $2^n$ different possible expansions for $\nu_\alpha(t)$ depending on whether we expand about $\lambda_\alpha^{\prime i}(t)=0$ or $\infty$, each expansion being valid for $ |\lambda_\alpha^{\prime i}(t)|<1$ or $>1$ respectively. \paragraph{Notation:} Let $s$ be an $n$-tuple $s\in \{\pm1\}^n$. As a shorthand, I will write: \begin{eqnarray*} \lambda'_\alpha & \stackrel{{\rm def}}{=} & (\lambda_\alpha^{\prime 1}, \dots, \lambda_\alpha^{\prime n})\\ s\lambda'_\alpha & \stackrel{{\rm def}}{=} & (s_1 \lambda_\alpha^{\prime 1}, \dots, s_n\lambda_\alpha^{\prime n}). \end{eqnarray*} I also write $\langle\lambda'_\alpha\rangle$ for the cone $\langle\lambda_\alpha^{\prime 1},\dots,\lambda_\alpha^{\prime n}\rangle$. I define the quantity $s_{-\kern -0.2em}\cdot\lambda'_\alpha$ by: $$s_{-\kern -0.2em}\cdot\lambda'_\alpha = \sum_{s_i=-1} s_i\lambda_\alpha^{\prime i}.$$ An element $m\in M'$ defines a region $T_{m}$ of $T_{N'}$ by: $$T_{m}=\{t\in T_{N'}: |m(t)|<1\}.$$ I also write, for a cone $C$ in $M$, $$T_C=\{t\in T_{N'} : |m(t)|<1, \forall m\in C\cap M\}.$$ Thus, for example, $$T_{\langle\lambda'_\alpha\rangle} = T_{\lambda_\alpha^{\prime 1}}\cap\cdots\cap T_{\lambda_\alpha^{\prime n}}.$$ \begin{prop}[General Expansion] \label{prop:general-exp} Given $s\in \{\pm1\}^n$, we have, for $t\in T_{\langle s\lambda'_\alpha\rangle},$ \begin{equation} \label{eq:nu-general-exp} \nu_\alpha(t)=(\prod_{i=1}^n s_i)\chi[{\alpha + s_{-\kern -0.2em}\cdot\lambda'_\alpha + \langle s\lambda'_\alpha\rangle \cap M}](t). \end{equation} \end{prop} \begin{proof} In order to expand $\nu_\alpha$ when, for some $i$, we have $|\lambda_\alpha^{\prime i}(t)|>1,$ I use the other expansion of $(1-z){}^{-1}$, valid for $|z|>1$: $$(1-z){}^{-1}= -z -z^2 -z^3 - z^4 - \cdots. $$ The result follows in the same way as the basic expansion. Note that compared to the basic expansion, the cone whose characteristic series we end up with undergoes a reflection plus a translation: $\langle\lambda'_\alpha\rangle\cap M$ becomes $s_{-\kern -0.2em}\cdot\lambda'_\alpha + \langle s\lambda'_\alpha\rangle \cap M$. This is due to the shift from $1+z+z^2+\cdots$ to $-z^1-z^2-z^3-\cdots$. \end{proof} \subsubsection{Consistency of expansions} It doesn't make sense to expand all the $\nu_\alpha$ according to (\ref{eq:nu-basic-expansion}) because the variable $t$ can't satisfy the condition $ |\lambda_\alpha^{\prime i}(t)|<1$ for all $i$ and $\alpha$. For one thing, if $\alpha$ and $\beta$ are two extreme vertices of $\Box$ connected by an edge, we will have $\lambda_\alpha^{\prime i}=-\lambda_\beta^{\prime j}$ for some $i$ and $j$, so that $ |\lambda_\alpha^{\prime i}(t)|<1 \iff |\lambda_\beta^{\prime j}(t)|>1$. I we can find a domain for $t\in T_{N'}$ such that {\em all\/} the expansions we perform are valid {\em at the same time,} then when we sum up all the $\nu_\alpha(t)$, all but a finite number of terms in the infinite series cancel, and we get the characteristic polynomial $\chi_\Box$ evaluated on $t$. For each $\beta\in{\rm ext}\,\Box$, we choose an element $s^\beta\in \{\pm1\}^n$, and expand according to (\ref{eq:nu-general-exp}). We require that the set \begin{equation} \bigcap_{\beta\in{\rm ext}\,\Box} T_{\langle s^\beta \lambda'_\beta\rangle} = T_{\cup \{\langle s^\beta \lambda'_\beta\rangle : {\beta\in{\rm ext}\,\Box}\}} \end{equation} be non-empty. I turn next to the necessary conditions for this to be so. \subsubsection{Neccessary conditions for a consistent expansion} \label{subsub:necc-cond} The above requirement implies, for instance, that if $\lambda^{\prime i}_\alpha =-\lambda^{\prime j}_\beta$, as it happens for ajdacent vertices, then $s^\alpha_i=-s^\beta_j$. This can be thought of graphically as choosing a direction for each edge of the polytope $\Box$ and sticking to it throughout the expansion. For each vertex $\alpha$ if the $i$-th edge is pointing into $\alpha$ then we set $s^\alpha_i=-1$, if it is pointing out, we set $s^\alpha_i=+1.$ Another necessary condition is that we choose $s^\alpha=(1,1,\dots,1)$ for some $\alpha\in{\rm ext}\,\Box$. This can be seen easily, if one thinks for a moment of decomposing $\chi_\Box$ as a sum of characteristic series for cones: \begin{equation} \label{eq:sum-chi} \chi_\Box = \sum_{\beta\in{\rm ext}\,\Box} \pm \chi_{C'_\beta\cap M} \end{equation} where the cones $C'_\beta$ are obtained from the tangent cones $C_\beta$ eventually by the `reflection + translation' process prescribed in the general expansion in proposition \ref{prop:general-exp} and the sign is determined by the number of reflections specified by $s^\beta$. One of the cones involved must be $C_\alpha$, for some $\alpha\in{\rm ext}\,\Box$. It will have all of its' edges pointing outwards in the above orientation and will correspond to the characteristic series $+\chi_{C_\alpha\cap M}$. I will call this the {\em base vertex\/} for the expansion. The non-emptiness requirement above then implies that the following condition on the orientations be satisfied: \paragraph{Orientation condition} Let $\lambda^{\prime i}_\alpha$ for $i=1,\dots p$ be any set of edges emanating from $\alpha$ that have been oriented so that they are {\em all outgoing with respect to $\alpha$.\/ } Then we require that for all $\beta\neq\alpha,$ \begin{equation} \label{eq:exp-condition} \hbox{if }(\lambda^{\prime}_\beta)^j \in \pm \langle\lambda^{\prime 1}_\alpha,\dots,\lambda^{\prime p}_\alpha\rangle \hbox{ then }(s^\beta)_j=\pm 1. \end{equation} In words, this says that if an edge $\lambda^{\prime j}_\beta$ is a linear combination, all of whose coefficients are of the same sign or zero, of oriented edges $\lambda^{\prime i}_\alpha$ all going outwards from a given vertex $\alpha$, then it should be oriented in the direction which includes it in the cone spanned by these outgoing edges. This is because, if it were oriented oppositely, it would mean that $T_{\langle\lambda^{\prime 1}_\alpha,\dots,\lambda^{\prime p}_\alpha\rangle} \cap T_{s^\beta_j \lambda^{\prime j}_\beta} = \emptyset,$ since one cannot have both $|\lambda^{\prime j}_\beta(t)|<1$ and $|-\lambda^{\prime j}_\beta(t)|<1.$ \subsubsection{Domain of validity of simultaneous expansions} It is always possible to choose at least one orientation of the edges of $\Box$ which satisfies the orientation condition (\ref{eq:exp-condition}). Suppose we have chosen such an orientation. For what values of $t\in T_N$ is it valid ? In order to answer this, let us first make some remarks about the regions $T_C\subset T_N,$ for $C$ a cone in $M$. It is helpful, to describe $T_C$, to decompose $T_N$ as $CT_N\times H$, corresponding to the Lie algebra decomposition ${\frak t}_{\Bbb C}={\frak t}\oplus i{\frak t}$. By identifying the second factor in the Lie algebra decomposition with $N_{\Bbb R}$, we have the exponential map $$N_{\Bbb R} \stackrel{\exp}{\to} H.$$ \begin{lemma} If $C$ is a cone in $M$, then $T_C$ is given by $$T_C=CT_N\times\exp(-{\rm int}(C{}^{\vee}))\subset CT_N\times H.$$ \end{lemma} \begin{proof} The interior of $C{}^{\vee}$ is the set of $n\in N_{\Bbb R}$ such that $\ip{n}{c}>0, \forall c\in C.$ Under the exponential map, the orbit $CT_N\times \{-n\}$ corresponds to an orbit of constant modulus strictly less than $1$. \end{proof} {}From this, we see that $$\bigcap_{\beta\in{\rm ext}\,\Box} T_{\langle s^\beta \lambda'_\beta\rangle} = CT_N\times\exp(-{\rm int}(\sigma)),$$ where \begin{equation} \label{eq:sigma} \sigma=\left(\bigcup_{\beta\in{\rm ext}\,\Box} \langle s^\beta \lambda'_\beta\rangle\right)^\vee. \end{equation} If we respect condition (\ref{eq:exp-condition}), we see that $\bigcup_{\beta\in{\rm ext}\,\Box} \langle s^\beta \lambda'_\beta\rangle$ never contains a whole subspace, so that $\sigma$ is non-zero. The expansion determined by $s^\beta$ for $\beta\in \Box$ is thus valid in the region $T_\sigma\subset T_N$ given by equation (\ref{eq:sigma}). \subsection{Elementary convex geometric interpretation} According to the work we have done in the previous sections, one can prove the extreme point formula as follows: Begin by orienting the edges of $\Box$ such as to respect condition (\ref{eq:exp-condition}). This defines a cone (with a sign) for each extreme vertex, according to proposition \ref{prop:general-exp}, and the algebraic sum of their characteristic series should yield the characteristic polynomial for the polytope $\Box$. If one can prove this for one admissible orientation of the edges of $\Box$, then the formula for the characteristic functions follows by the existence of Ishida's ${\Bbb C}[M]$-homomorphism in the previous section. This gives a proof of the formula involving only elementary convex geometry. We won't bother with this, as Ishida \cite{ishida} already gives a proof which reduces the problem to the contractibility of convex sets. Instead we can deduce the following result in convex geometry: \begin{thm} \label{thm:chi-decomposition} For all orientations $\{s^\alpha\}$ of the edges of $\Box$ satisfying the orientation condition (\ref{eq:exp-condition}) we have $$\chi_{\Box\cap M} = \sum_{\alpha\in{\rm ext}\,\Box} \pm\chi_{C^s_\alpha\cap M}$$ where $\pm=\prod_i (s^\alpha)_i$ and $$ C^s_\alpha = \alpha + s_{-\kern -0.2em}\cdot\lambda'_\alpha + \langle s\lambda'_\alpha\rangle.$$ \end{thm} \section{Number of Lattice Points and Volume} In this section I expand the functions $\nu_\alpha(t)$ around $t=1$ and derive formulae for the number of lattice points and volume of $\Box$. \subsection{The Number of Lattice Points} Equation (\ref{eq:lef-fns-b}) expresses an equality between the finite Laurent polynomial determined by $\Box$ and a sum a rational functions. When evaluated on $t\in T_N$ with $t\to 1$ the left-hand side tends to the number of lattice points of $\Box$ whereas on the right-hand side the rational functions may have poles. I choose a one-parameter subgroup $\{\exp(s\zeta) : s\in{\Bbb R}\}$ determined by some element $\zeta$ of the Lie algebra $\bf t$ of $CT_N$. Substituting $\exp(s\zeta)$ for $t$, the formula reduces to an equality between rational functions of $s$ --- provided I choose a one-parameter subgroup that does not coincide with the singular loci of the $\nu_\alpha$. \begin{dfn} For short, I call $\zeta\in {\frak t}$ {\em generic\/} if $\ip\zeta{{\lambda_\alpha}^i}\ne0$, for all $i$ and $\alpha$. (This is indeed the case generically). \end{dfn} For generic $\zeta$, the functions $\nu_{\alpha,\zeta}^\Box: s \mapsto \nu_\alpha^\Box(e^{s\zeta})$ can expanded in Laurent series: $$ \nu_{\alpha,\zeta}^\Box(s)=\sum_{i=-\infty}^{\infty} \nu_{\alpha,\zeta,i}^\Box s^i, $$ and their sum as $s\to 0$ is obviously given by the sum of the constant terms $\nu_{\alpha,\zeta,0}$ in each expansion. Denote by $C_\alpha$ the tangent cone of $\Box$ at $\alpha\in{\rm ext}\,\Box$, and by $\lambda^i_\alpha$ for $ i=1,\dots,n$, its' primitive generators in $M$. The semi-open unit parallelepiped determied by the generators of $C_\alpha$ in $M$ is denoted \begin{equation} \label{eq:Qalpha} Q_\alpha=Q^M_{C_\alpha}=\{\sum a_i{\lambda_\alpha}^i: 0\leq a_i < 1\}. \end{equation} We have $$\nu^\Box_{\alpha,\zeta}(s)= {\sum_{q\in Q_\alpha\cap M} e^{s\ip{\zeta}{\alpha+q}} \over (1-e^{s\ip\zeta{{\lambda_\alpha}^1}})\cdots(1- e^{s\ip\zeta{{\lambda_\alpha}^n}} )},$$ provided $\ip\zeta{{\lambda_\alpha}^i}\ne0$. The zero-th order term in the expansion of $\nu^\Box_{\alpha,\zeta}(s)$ is a homogeneous function of $\zeta$, which is equal to: $$ {\sum_{q\in Q_\alpha\cap M} e^{s\ip{\zeta}{\alpha+q}}\over s^n\prod_i (-\ip\zeta{{\lambda_\alpha}^i})}{\prod_i (-s\ip\zeta{{\lambda_\alpha}^i})\over\prod_i(1-\exp(s\ip\zeta{{\lambda_\alpha}^i}))} $$ which gives $${1\over\prod_i \ip\zeta{{\lambda_\alpha}^i}} \sum_{j=0}^n {(-1)^j \over j!} \sum_{q\in Q_\alpha\cap M} \ip{\zeta}{\alpha+q}^j {\cal T}_{n-j}(\ip\zeta{\lambda_\alpha}), $$ where ${\cal T}_k $ are the {\em Todd polynomials}, homogeneous polynomials of degree $k$ whose coefficients can be expressed in terms of the Bernoulli numbers \cite{hirz}. They are defined by the formal series $$\sum_{k=0}^\infty s^k {\cal T}_k(x_1,x_2,\dots) = \prod_{i\geq 1} {sx_i\over{1-\exp(-sx_i)}}. $$ By ${\cal T}_k(\ip\zeta{\lambda_\alpha})$ I mean $T_k(\ip\zeta{{\lambda_\alpha}^1},\dots,\ip\zeta{{\lambda_\alpha}^n})$. \begin{thm}\label{thm:number} Let $\Box$ be a simple convex lattice polytope. Denote by $C_\alpha$ the tangent cone of $\Box$ at $\alpha\in{\rm ext}\,\Box$, and by $\lambda^i_\alpha,$ for $ i=1,\dots,n$, the primitive generators of $C_\alpha$ in $M$. The semi-open unit parallelepiped determied by the generators of $C_\alpha$ in $M$ as in equation \ref{eq:Qalpha} is denoted $Q_\alpha$. Then, for generic $\zeta\in {\frak t}$, the number of lattice points in $\Box$ is given by $$\sum_{\alpha\in{\rm ext}\, \Box}{1\over\prod_i \ip\zeta{{\lambda_\alpha}^i}} \sum_{j=0}^n {(-1)^j \over j!} \sum_{q_\alpha\in Q_\alpha\cap M} \ip{\zeta}{\alpha+q_\alpha}^j {\cal T}_{n-j}(\ip\zeta{\lambda_\alpha}).$$ \end{thm} \paragraph{Remark 1} It might be more convenient in some cases to subdivide the tangent cone into non-singular cones. One obtains a similar formula (see \cite[Th\'eor\`eme 3.1]{brion}). \paragraph{Remark 2} Putting $t=\exp(s\zeta)$ corresponds to considering the Lefschetz number for the action of the one-parameter subgroup $G_\zeta$ of $CT_N$ generated by $\zeta\in{\frak t} = {\rm Lie }\,CT_N$. Generically this has a dense orbit, and therefore the same fixed points on $X$ as the whole real torus $CT_N$, and so the Lefschetz formula for $G_\zeta$ is the same as that obtained by substituting $\exp(s\zeta)$ for $t$. This is not true of course when $\ip\zeta{{\lambda_\alpha}^i}=0$, for some $i$ and $\alpha$. Indeed in that case the group $G_\zeta$ has whole circles of fixed points. Restricting to $G_\zeta$ corresponds to projecting the vertices and edges of $\Box$ onto the hyperplane in $M_{\Bbb R}$ defined by the form $\zeta\in N_{\Bbb R}$. \subsection{The Volume} \subsubsection{The ``Classical Limit''} In the introduction I mentioned the fact that for larger and larger polytopes (or finer and finer lattices) the number of points is asymptotically equal to their volume --- I call this ``the classical limit'' by analogy with the limit $\hbar\to 0$ in quantum mechanics. More precisely, for any $n$-dimensional polytope $\Box$, the volume of $\Box$ is given by \begin{equation} \label{eq:vol_lim} {\rm vol}_n(\Box)=\lim_{k\to\infty}{\#(k{}^{-1} M\cap \Box)\over k^n} = \lim_{k\to\infty}{\#(M\cap k\Box)\over k^n}. \end{equation} Indeed \cite{mac:poly}, the function $$H_{\Box}(k)=\#(k{}^{-1} M\cap \Box)=\#(M\cap (k\Box))$$ is a polynomial of degree $n$, for $k\in{\Bbb N}$, with leading coefficient ${\rm vol}_n(\Box),$ and is called the {\em Hilbert polynomial\/} for $\Box$. The polynomial $H_{\Box}$ is in fact equal to the {\em Hilbert polynomial\/} $H_{(X_\Box,L_\Box)}$ for the pair $(X_\Box,L_\Box)$, namely $$H_{(X_\Box,L_\Box)}(k)=\chi(X_\Box,{\cal O}_{X_\Box}(kL_\Box))= \sum(-1)^i\dim H^i(X_\Box,{\cal O}_{X_\Box}(kL_\Box)).$$ This follows from equation (\ref{eq:coho}) and because taking tensor powers $L_\Box^{\otimes k}$ of $L_\Box$ corresponds to taking multiples of $kN$ of $N$, and hence submultiples $k{}^{-1} M$ of $M$. \begin{thm} \label{thm:volume} Let $\Box$ be a simple convex lattice polytope and adopt the same notation as theorem \ref{thm:number}. Let $|K_\alpha|$ denote the order of the singularity of $\Box$ at $\alpha$. Then for generic $\zeta$ the volume of $\Box$ is given by $${\rm vol}_n(\Box)= {(-1)^n\over n!} \sum_{\alpha\in{\rm ext}\,\Box}{\ip\zeta{\alpha}^n |K_\alpha| \over \ip\zeta{{\lambda_\alpha}^1}\cdots\ip\zeta{{\lambda_\alpha}^n}}.$$ \end{thm} \begin{proof} The proposition follows from taking the coefficients of the $k^n$ terms in theorem \ref{thm:number} applied to the polytope $k\Box$. Note that ${\rm ext}\, k\Box=k({\rm ext}\,\Box)$ and that $C^{k\Box}_{k\alpha}=C^\Box_\alpha.$ Note that the order $|K_\alpha|$ of the singularity at $\alpha$ is equal to the cardinality of $Q_\alpha\cap M$. See \cite[Corollaire 2]{brion}. \end{proof} \subsubsection{The Riemann-Roch approach} The volume of $\Box$ appears if one uses the same geometric approach based on the $d''$-complex but directly applies the Riemann-Roch theorem, instead of computing the Lefschetz number for the action of $t\in T$ and then letting $t\to1$. The Riemann-Roch theorem expresses the Euler characteristic of a holomorphic vector bundle $E$ over a complex manifold $X$ in terms of characteristic classes of $E$ and (tangent bundle to the) $X$: \begin{equation} \chi(X,E)=\{\hbox{ch}(E)\cdot{\cal T}(X)\}[X], \end{equation} where ch$(E)$ and ${\cal T}(X)$ are the Chern character of $E$ and the Todd class of $X$, respectively. If $E$ has rank $n$ and $c_1,\dots,c_n$ denote the characteristic classes of $E$ then the {\em Chern character} can be defined by the power series $$\sum_{i=1}^n e^{x_i}= n+\sum x_i+{\sum x_i^2\over 2!}+\cdots,$$ where the $c_i$ are to be thought of formally as the elementary symmetric functions in the $x_i$. Since we are in a one-dimensional situation and $c_1(L_\Box)$ is represented by the K\"ahler form $\omega$, the Chern character is given by $${\rm ch}(L_\Box)=1+\omega+{\omega^2\over2!}+{\omega^3\over3!}+\dots+{\omega^n\over n!}.$$ The {\em Todd class} is a polynomial in the characteristic classes $c'_i$ of the tangent bundle of $X$. If the $c'_i$ are regarded formally as the elementary symmetric functions of the $x'_i$ (as in the case above), the Todd class can be expressed as $${\cal T}(X)=\prod_i {x'_i\over 1-e^{-x'_i}}.$$ (Presumably, there is some relationship between these and the Todd polynomials of theorem \ref{thm:number} which in this case exhibits the Riemann-Roch formula as the ``classical limit'' of the Lefschetz fixed point formula.) By multiplying the two series selecting the terms of order $n$ and evaluating them on $[X]$, we get $$\chi(X,L_\Box) ={\rm vol}_n(X)+ \hbox{\em lower order terms},$$ where the ``lower order terms" are terms involving powers of $\omega$ of order less than $n$. Again, because refining the lattice $M$ corresponds to multipying $\omega$, we see that $\chi(X,tL_\Box)$ is given asymptotically by ${\rm vol}_n(X)t^n$.

9302

alg-geom/9302006

en

https://arxiv.org/abs/alg-geom/9302006

alg-geom/9302006

Claude LeBrun

Claude LeBrun and Michael Singer

Existence and Deformation Theory for Scalar-Flat Kaehler Metrics on Compact Complex Surfaces

60 pages, LaTeX

10.1007/BF01232436

Let M be a compact complex surface which admits a Kaehler metric whose scalar curvature has integral zero; and suppose the fundamental group of M does not contain an Abelian subgroup of finite index. Then if M is blown up at sufficiently many points, the resulting surface M' admits scalar-flat Kaehler metrics.

alg-geom

\section{Introduction} \subsection{Motivation} The classical uniformization theorem provides a complete translation dictionary for the etymologically unrelated languages of complex 1-manifolds and constant curvature Riemannian 2-manifolds. In higher dimensions, there are a number of natural ways in which one might try to generalize this remarkable theorem; unfortunately, these various potential generalizations remain, for the most part, programs rather than established bodies of fact. However, the subject of the present article, namely {\sl the existence problem for zero-scalar-curvature K\"ahler metrics on compact complex 2-manifolds}, occupies the cross-roads of several such avenues of research; and by clearing up a substantial piece of this problem, we thereby hope to facilitate the flow of traffic heading on to a number interesting destinations. Purely in the context of Riemannian geometry, the most optimistic programs to generalize the classical uniformization theorem would try to equip every compact smooth manifold of a given dimension with a (small!) class of ``optimal'' or ``canonical'' metrics. In dimension four, one of the most natural versions would have us seek extrema (or perhaps just critical points) of the squared $L^2$-norm $${\cal R}(g)= \int_M\|R\|^2 \vol$$ of the Riemann curvature tensor $R$ over the space of smooth Riemannian metrics $g$ on a given smooth, compact, oriented 4-manifold $M$. Using the Chern-Gauss-Bonnet formulas for the Euler characteristic $\chi$ and signature $\tau$ of our manifold $M$, one may easily show \cite{L5} that \begin{eqnarray*} {\cal R}(g)&=& -8\pi^2(3\tau +\chi )+\int_M (4\|W_{+}\|^2+\frac{s^2}{12})\vol \\ &\geq& -8\pi^2(3\tau +\chi ) ~,\end{eqnarray*} with equality iff $W_+=s=0$; here $s$ denotes the scalar curvature and $W_+$ the self-dual Weyl curvature (cf. \S \ref{asd}) of $g$. Metrics with $W_+=s=0$, when they exist, are thus absolute minima of ${\cal R}$, and it is therefore natural to try to determine which manifolds $M$ can admit such metrics. However, if the intersection form $$\cup : H^2(M, {\Bbb R})\times H^2(M, {\Bbb R})\to {\Bbb R}$$ is indefinite, an elementary Weitzenb\"ock argument \cite{L0} shows that such a manifold admits an integrable complex structure with respect to which the metric is K\"ahler; conversely \cite{G}, any K\"ahler manifold of complex dimension 2 with $s\equiv 0$ automatically satisfies $W_+=0$ and has indefinite intersection form. Thus the problem of minimizing $\cal R$ on a smooth manifold leads us quite naturally\footnote{It should be pointed out that many manifolds which do not admit scalar-flat K\"ahler metrics nonetheless admit metrics which are absolute minima of $\cal R$. In particular \cite{besse}, any {\em Einstein metric} on a compact 4-manifold provides an absolute minimum of $\cal R$. It is this fact which explains much of the current interest in this Riemannian functional.} to the problem of classifying compact K\"ahler manifolds of complex dimension 2 and scalar curvature zero--- henceforth referred to as scalar-flat K\"ahler surfaces. A related four-dimensional program would instead seek to optimize the {\em conformal} geometry of Riemannian metrics by seeking to minimize the conformally-invariant squared $L^2$-norm $${\cal W}(g)=\int_M\| W\|^2\vol $$ of the conformal curvature over the space of conformal classes of Riemannian metrics on $M$. Since $${\cal W}(g)= - 12\pi^2 \tau + 2\int_M\| W_+\|^2\vol ~,$$ anti-self-dual metrics (i.e. metrics satisfying $W_+=0$) are obviously absolute minima of ${\cal W}$, scalar-flat K\"ahler surfaces again provide examples of absolute minima. Note that while there are strong topological constraints on anti-self-dual metrics with non-negative scalar curvature \cite{DF} \cite{flo} \cite{L0} \cite{L} \cite{poon}, the situation is radically different once the scalar curvature condition is dropped; in fact, it has recently been shown \cite{taubes} that the obstructions to the existence of anti-self-dual metrics on any oriented smooth 4-manifold are so weak that they can always be killed off by ``blowing up points,'' in the differentiable sense of taking connected sums with enough $\overline{\Bbb CP}_2$'s. Our own results in this article will have something of a similar ring to them--- while there are a number of obstructions to the existence of scalar-flat K\"ahler metrics on a compact complex surface, we will see that all but the crudest can be killed off by blowing up points. In search of a natural bridge between complex and differential geometry, Calabi has proposed the problem of representing K\"ahler classes on compact complex manifolds by K\"ahler metrics of constant scalar curvature. Here again there is a natural variational approach to the problem, since such metrics are absolute minima of the functional $${\cal C}(g)=\int_M s^2 \vol$$ among metrics in a fixed K\"ahler class; more generally, critical points of this functional have come to be known \cite{besse} \cite{cal} as {\em extremal K\"ahler metrics}. However, the existence of constant scalar curvature K\"ahler metrics is, in general, obstructed \cite{burnsbart}\cite{besse}\cite{fut}\cite{cal2}, and the known obstructions will necessarily play a central r\^ole in the present article--- although perhaps not quite in the way the reader might expect. It is hoped that our present existence results will provide a useful way station, {\sl en route} to a more general understanding of Calabi's problem. From a quite different perspective, namely that of Hawking's Euclideanization program in gravitational physics, a fundamental problem is that of classifying compact Riemannian solutions of the {\em Einstein-Maxwell equations} $$-{\textstyle \frac{1}{2}} r^{\sharp}=\mbox{{\em trace-free part}~}(F^{\sharp}\circ F^{\sharp}) $$ $$ dF = d\star F=0 $$ governing the interaction of the gravitational field, represented by a Riemannian metric $g$, with the electromagnetic field, represented by a harmonic 2-form $F$; here the metric has been used to identify the Ricci curvature and electromagnetic field with endomorphisms $r^{\sharp}$ and $F^{\sharp}$ of the tangent bundle. Any scalar-flat K\"ahler surface provides a solution of these equations once one sets $$F=\rho +{\textstyle \frac{1}{4}}\omega ~,$$ where $\omega$ and $\rho$ are respectively the K\"ahler and Ricci forms\footnote{If $F$ is to be viewed as a ${\bf U}(1)$-gauge field, as it must be in realistic physical theories, our K\"ahler metric must also be of Hodge type--- that is, cohomologous to the metric induced by some projective embedding.}; conversely \cite{fla}, these are essentially the only solutions with $W_+=0$. Thus the classification problem for scalar-flat K\"ahler surfaces may be seen as part of a quest to classify electro-gravitational instantons. Finally, the Penrose twistor correspondence \cite{P} gives quite a different way of generalizing the conformal surface/complex curve dictionary to dimension four. If $M$ is a smooth oriented 4-manifold and $$[g]=\{ e^u g\}$$ is a conformal class of Riemannian metrics on $M$, the space $Z$ of orthogonal complex structures on $TM$ compatible with the orientation is an almost-complex manifold of real dimension 6; it then turns out \cite{AHS} that $Z$ is a complex manifold iff $(M, [g])$ satisfies $W_+=0$. In this case, $Z$ is called the {\em twistor space} of $(M, [g])$, and it turns out that both $M$ and its anti-self-dual conformal structure can be reconstructed from this complex manifold. In particular, every scalar-flat K\"ahler surface $(M,g, J)$ has associated to it a compact complex 3-fold $Z$; moreover, $(M,J)$ is naturally a complex submanifold of $Z$. While this construction has elsewhere served \cite{LP}\cite{taubes} primarily as an excellent source of complex 3-folds with various ``pathologically'' non-K\"ahlerian properties, the deformation theory of $Z$ will here serve as our guiding light as we trek through the realm of scalar-flat K\"ahler geometry. \subsection{Outline} {\sl We now provide the reader with a statement of the central result of the paper, followed by an indication of the structure of the argument. } \bigskip \noindent {\bf Main Theorem}~~ {\em Let $M$ be a compact complex surface which admits a K\"ahler metric whose scalar curvature has integral zero. Suppose $\pi_1 (M)$ does not contain an Abelian subgroup of finite index. Then if $M$ is blown up at sufficiently many points, the resulting surface $\tilde{M}$ admits scalar-flat K\"ahler metrics.} \begin{description} \item{\S \ref{flag} } We study the behavior of the solution space under small deformations of complex structure of the complex surface in question. Our approach uses the twistor correspondence and a modified version of Kodaira-Spencer theory. This deformation theory is generally obstructed, but we are able to describe the obstructions completely in terms of the Futaki character of the algebra of holomorphic vector fields. \item{\S\ref{foo} } We compute the Futaki character for all relevant complex surfaces, and use this to show that the deformation theory is unobstructed for all non-minimal surfaces. \item{\S\ref{next} } We describe a large class of exact solutions previously found in \cite{L2}, and an improvement on those results which gives a classification of scalar-flat K\"ahler surfaces with non-trivial automorphism algebra. \item{\S\ref{tag} } Using the bimeromorphic classification theory of surfaces, we show that every surface satisfying the hypotheses of the Main Theorem has blow-ups which are arbitrarily small deformations of surfaces on which we have exact solutions. Applying our deformation theory then proves the Main Theorem. \end{description} \noindent It should be emphasized that the Main Theorem's fundamental-group hypothesis reflects the limitations of our techniques rather than a known obstruction to the existence of scalar-flat K\"ahler metrics. In fact, the article concludes with some speculations (Conjecture \ref{better}, \S\ref{tag}) to the effect that this restriction is essentially superfluous. \medskip \noindent Incidental to the main course of the argument, we will also encounter a remarkable empirical relationship, observed (\S\ref{next}) in two quite different classes of explicit examples, which seems to link the existence problem for scalar-flat K\"ahler metrics to the stability of vector bundles with parabolic structure in the sense of Seshadri \cite{sesh}; it is our belief, as expressed in Conjecture 2, \S\ref{tag}, that this relationship will actually turn out to hold in complete generality. \subsection{Notation and Conventions} For the purposes of this section, $(M,g)$ will denote an oriented Riemannian $2m$-manifold, although in the sequel we will specialize to the case $m=2$. We use $\volm$ to denote the volume form of $g$, and $\nabla$ to denote its Levi-Civit\`{a} connection. The $C^{\infty}$ sections of any smooth vector bundle ${\cal V}\to M$ will be denoted by ${\cal E}({\cal V})$. The operation of raising (lowering) an index will be indicated by $\sharp$ ($\flat$). We give the curvature tensor $R$ the usual Riemannian sign: \begin{equation} (\nabla_{c}\nabla_{d}-\nabla_{d}\nabla_{c})\xi^a={R^a}_{bcd}\xi^b {}~~~\forall~\xi\in {\cal E}(TM). \label{n1} \end{equation} The Ricci tensor ${R^c}_{acb}$ is denoted by $r_{ab}$ and the scalar curvature ${R^{ab}}_{ab}$ by $s$. The pointwise inner product induced by $g$ on the tensor bundles will be denoted $(~,~)$, whereas the the global $L^2$ inner product will be denoted by $\langle~,~\rangle$. The formal adjoint of $d$ with respect to this inner product will be denoted $\delta$ and is given by the usual formula \begin{equation} \delta=-\star d\star.\label{n2} \end{equation} The metric $g$ is said to be {\em K\"{a}hler} if its holonomy group is conjugate to a subgroup of ${\bf U}(m)\subset {\bf O }(2m)$. More concretely, this means that there is a compatible parallel almost-complex structure $J$: \begin{equation} J^2=-{\bf 1}, ~~g(J\xi , J\eta )=g(\xi , \eta)~\forall \xi , \eta\in TM, {}~~~\nabla J=0.\label{n3} \end{equation} If the holonomy of $g$ happens to be a proper subgroup of ${\bf U}(m)$, there may be more than one $J$ which satisfies (\ref{n3}); nonetheless, when we speak of a K\"ahler metric we will henceforth always assume that a {\em particular choice} of $J$ has been made. We therefore have a decomposition $ {\Bbb C}\otimes TM=T^{1,0}\oplus T^{0,1} $ into the $\pm i$ eigenspaces of $J$, thereby inducing a decomposition \begin{equation}\wedge^r_{\Bbb C}=\bigoplus_{r=p+q}\wedge^{p,q}\label{decomp} \end{equation} of the bundle of $r$-forms into forms of type $(p,q)$, as defined by $\wedge^{p,q}:= (\wedge^pT^{1,0})^{\ast}\otimes (\wedge^qT^{0,1})^{\ast}$; in particular, $J$ induces a ``standard'' orientation of $M$ by requiring that the $2m$-form $i^m\varphi \wedge\bar{\varphi}$ be positive for any non-zero element $\varphi$ of the {\em canonical line bundle} $\kappa = \wedge^{m,0}$. For brevity, we will use ${\cal E}^r$ and ${\cal E}^{p,q}$ to respectively denote ${\cal E}(\wedge^r)$ and ${\cal E}(\wedge^{p,q})$. Because $\nabla$ is torsion-free, $[ {\cal E} (T^{1,0}),{\cal E} (T^{1,0})] \subset {\cal E} (T^{1,0})$, and the the Newlander-Nirenberg \cite{NN} theorem therefore asserts that $J$ is integrable--- that is, there exists a system of local coordinates for which $J$ becomes the standard almost-complex structure on ${\Bbb C}^m$, making $M$ a complex $m$-manifold. The K\"{a}hler form $\omega$ and Ricci form $\rho$ are then defined by the formulae \begin{equation} \omega(X,Y)=g(J\xi, Y),~~~\rho(\xi,\eta)=r(J\xi, \eta) {}~~~\forall \xi,\eta\in TM~;\label{n4} \end{equation} both are real closed forms of type (1,1). Conversely, a closed real (1,1)-form on complex manifold is a K\"ahler form iff the symmetric form $g$ it defines implicitly via (\ref{n4}) is positive definite. The deRham class $[\omega ]\in H^{2}(M)$ of the K\"ahler form is called the K\"{a}hler class. It is a central fact of K\"ahler geometry that $\rho$ is exactly the curvature of the Chern connection on $\kappa^{-1}=\wedge^mT^{1,0}$; in particular, $\rho$ is completely determined by $J$ and $\volm$ alone, and the deRham class of $\rho /2\pi$ is just the first Chern class $c_{1}(M):=c_1(T^{1,0}M)=c_1(\kappa^{-1})$. Composing $J$ with $d$ yields a new real operator \begin{equation} d^{c}:=i(\overline{\partial}-\partial ).\label{n5} \end{equation} The formal adjoint of $d^{c}$ is denoted by $\delta^{c}$ and is given by \begin{equation} \delta^{c}=-\star d^{c}\star.\label{n6} \end{equation} On a K\"{a}hler manifold $d,d^{c}$ and $\delta,\delta^{c}$ are related by the so-called K\"{a}hler identities \begin{equation} \hphantom{-}\delta=[\Lambda,d^{c}],~~~~-\delta^{c}=[\Lambda,d]\label{n7} \end{equation} \begin{equation} -d=[L,\delta^{c}],~~~~\hphantom{-}d^{c}=[L,\delta]\label{n8} \end{equation} where $L$ is the algebraic operation \begin{equation} L\varphi=\omega\wedge\varphi\label{n9} \end{equation} and $\Lambda$ is its adjoint (contraction with $\omega$). We note \begin{equation} [\Lambda,L]=(m-r){\bf 1}\label{n10} \end{equation} on $r$-forms. Finally, the Laplace-Beltrami operator $\Delta=\delta d$ on functions may be re-expressed in the useful form \begin{equation} \Delta f=-\Lambda dd^{c}f=-(\omega,dd^{c}f)~ .\label{n11} \end{equation} If ${\cal V}\to M$ is a holomorphic vector bundle over a complex manifold $M$, its sheaf of sections will be denoted by ${\cal O} ({\cal V})$. We define the {\em projectivization} of ${\cal V}$ by ${\Bbb P} ({\cal V})=({\cal V}-{\bf 0})/{{\Bbb C}^{\times }}$, where ${\bf 0}$ is the zero section; notice that this differs from a competing convention which replaces ${\cal V}$ with its dual ${\cal V}\*$ on the right-hand side. Finally, depending on the context, we will use ${\cal O} (k)$ to denote either the degree $k$ holomorphic line bundle on ${\Bbb CP}_m$, or its sheaf of holomorphic sections. \subsection{Anti-self-duality}\label{asd} On an oriented Riemannian 4-manifold $(M,g)$, the bundle ${\wedge^{2}}$ of 2-forms breaks up as \begin{equation} {\wedge^{2}}={\wedge^{+}}\oplus{\wedge^{-}}~,\label{n12} \end{equation} where $\wedge^{\pm}$ is the eigenspace of the Hodge operator $\star$ with eigenvalue $\pm 1$. We will call ${\wedge^{+}}$ the bundle of self-dual (SD) 2-forms and ${\wedge^{-}}$ the bundle of anti-self-dual (ASD) 2-forms. This decomposition is conformally invariant, in the sense that it is invariant under conformal rescalings $g\to e^u g$. The decomposition (\ref{n12}) allows us to define differential operators $ d^{\pm}: {\cal E}^1\to {\cal E}(\wedge^{\pm})$ by following the exterior derivative with projection $\wedge^2\to\wedge^{\pm}$. Since a closed anti-self-dual form is automatically harmonic, the following useful vanishing result is an immediate consequence of Hodge theory: \begin{propn} If $M^4$ is compact and $\beta \in {\cal E}^1(M)$, $d^{+}\beta =0 \Leftrightarrow d\beta =0$. {\hfill \rule{.5em}{1em} \\} \label{sd2} \end{propn} Applying (\ref{n12}) to the curvature operator ${R^{ab}}_{cd}: \wedge^2 \to \wedge^2$ results in a block-matrix decomposition \begin{center} \begin{tabular}{cccc} \hphantom{~$\wedge_-$}& ~$\wedge^{+\ast}$~~ & ~~~~~$\wedge^{-\ast}$& \hphantom{$\wedge_-$}\\ \end{tabular} \begin{tabular}{c|c|c|} \cline{2-3}&&\\ $\wedge^+$&$W_++\frac{s}{12}$&$\Phi$\\ &&\\ \cline{2-3}&&\\ $\wedge^-$&$\Phi$ & $W_-+\frac{s}{12}$\\&&\\ \cline{2-3} \end{tabular} \hphantom{$\wedge^-$} \end{center} where $W_{\pm}$ are trace-free and $2\Phi$ is the trace-free part of the Ricci curvature $r$. If $W_+=0$, the metric $g$ is said to be {\em anti-self-dual}, or ASD. Since the Weyl curvature $W=W_++W_-$ is precisely the conformally invariant piece of the curvature tensor $R$, the anti-self-duality condition is invariant under conformal rescalings $g\to e^u g$; thus it makes sense to speak of ASD conformal (classes of) metrics. If the Riemannian manifold $(M,g)$ is actually K\"{a}hler, and is given its canonical orientation, the decompositions (\ref{decomp}) and (\ref{n12}) are compatible in the sense that \begin{equation} {\wedge^{+}_{{\Bbb C}}}={\Bbb C}\omega\oplus \wedge^{0,2}\oplus {\wedge^{2,0}}\label{n13} \end{equation} and \begin{equation} {\wedge^{-}_{{\Bbb C}}}={\wedge^{1,1}_{0}},\label{n14} \end{equation} where ${\wedge^{1,1}_{0}}=\{ \varphi\in {\wedge^{1,1}}~|~\omega\wedge\varphi =0\}$ is the bundle of ``primitive'' (1,1)-forms. But, as a consequence of (\ref{n3}), the curvature operator of a K\"ahler manifold is in $\mbox{End} (\wedge^{1,1})$; thus, the upper left-hand block of the curvature operator must just be a multiple of $\omega\otimes \omega^{\sharp}$. This immediately leads to the following: \begin{propn} {\rm \cite{fla}\cite{G}} In complex dimension 2, a K\"{a}hler metric $g$ is anti-self-dual iff it is scalar-flat $(s \equiv 0)$. {\hfill \rule{.5em}{1em} \\} \label{sd1} \end{propn} We conclude this section with a closely related observation. If $\varphi$ is any form of type (1,1), we can write \begin{equation} \varphi={\textstyle \frac{1}{2}}~(\Lambda\varphi)\omega+\varphi_{0}\label{n15} \end{equation} where \begin{equation} \Lambda\varphi :=(\varphi,\omega)\label{n16} \end{equation} and $\varphi_{0}\in{\wedge^{1,1}_{0}}$. Applied to the Ricci form, this yields \begin{equation} \rho={\textstyle \frac{1}{4}}~s\omega+\rho_{0},\label{n17} \end{equation} so that \begin{equation} \rho~\mbox{is ASD}~~\Longleftrightarrow~~s=0~ .\label{n18} \end{equation} In particular, the Riemannian connections on $\kappa =\wedge^{2,0}$ and $\wedge^+ \cong\kappa \oplus {\Bbb R}$ are ASD iff the K\"ahler manifold $(M,g)$ is scalar-flat. \subsection{Admissible K\"ahler Classes} The fact that the Ricci form $\rho$ of a K\"ahler manifold represents $2\pi c_1$ in deRham cohomology leads to serious constraints on the scalar curvature of K\"ahler metrics. Indeed, on a compact complex surface with K\"ahler form $\omega$, the integral of the scalar curvature $s$, henceforth called the {\em total scalar curvature}, must be given by \begin{equation} \int_Ms\vol= 4\pi c_1\cdot [\omega ] \label{tsc}\end{equation} as an immediate consequence of (\ref{n17}). Since the volume of $M$ is just $[\omega ]^{2}/2$, we conclude that the {\sl average value of the scalar curvature is determined by the K\"ahler class alone}. (In complex dimension $m$, the total scalar curvature is similarly given by $\int s\vol = 4\pi c_1 \cdot [\omega]^{m-1}/{(m-1)!}$, while the volume is $[\omega]^{ m}/m!$; thus the average scalar curvature is still completely determined by the K\"ahler class.) In trying to classify scalar-flat K\"ahler surfaces, the first logical step is therefore to limit ourselves to those K\"ahler classes with total scalar curvature zero. This motivates the following definition: \begin{defn} Let $M$ be a compact complex surface. A K\"ahler class $[\omega ]\in H^{1,1}(M, {\Bbb R})$ will be said to be {\em admissible} iff $c_1\cup [\omega ]=0$, i.e. iff the total scalar curvature $\int s\vol$ vanishes for K\"ahler metrics in $[\omega ]$. The set of admissible K\"ahler classes will be denoted by ${\cal A}_M\subset H^{1,1}(M)$. \end{defn} If $c_1^{{\Bbb R}}=0$, any K\"ahler class is admissible, and Yau's solution \cite{yau2} of the Calabi conjecture asserts that every such class is represented by a unique Ricci-flat metric; moreover, this Ricci-flat metric is the only scalar-flat metric in the class, since (\ref{n18}) tells us that $\rho/2\pi$ is the unique harmonic representative of $c_1^{{\Bbb R}}=0$ when $s=0$. If, on the other hand, $c_1^{{\Bbb R}}\neq 0$, the set ${\cal A}_M$ of admissible K\"ahler classes is evidently an open set in a hyperplane in $H^{1,1}(M, {\Bbb R})$. However, this open set is often {\em empty}, as is shown by the following arguments of Yau \cite{yau1}; {\em cf.} \cite{G2} \cite{h1}. \begin{propn} Let $(M, J, \omega)$ be a compact K\"ahler m-manifold, and suppose that $L\to M$ is a holomorphic line bundle such that $c_1(L) \cup [\omega]^{m-1}=0$. Then either $\Gamma (M, {\cal O}(L^{\ell }))=0$ $\forall {\ell }\neq 0$, or else $c_1^{\Bbb R}(L)=0$.\label{ox} \end{propn} \begin{proof} Suppose that $u\in\Gamma (M, {\cal O}(L^{\ell }))$, $u\not\equiv 0$. Using Poincar\'e duality, the volume of the zero locus $u=0$, counted with multiplicity, must equal $(c_1(L^{\ell }) \cup [\omega]^{m-1})/(m-1)!=0$. Thus $u\neq 0$, and $L^{\ell }$ is trivial. \end{proof} \begin{cor} Let $(M, J, \omega)$ be a compact K\"ahler manifold of total scalar curvature zero. Then either $c_1^{\Bbb R}=0$, or else $\Gamma (M, {\cal O}(\kappa^{\ell }))=0$ for all $\ell \neq 0$. In particular, the Kodaira dimension of $M$ is either $0$ or $-\infty$. \label{yup} \end{cor} \begin{proof} Since the hypothesis may be interpreted as stating that $[\omega ]^{m-1}\cup c_1(\kappa)=0$, we may apply Proposition \ref{ox} with $L=\kappa$. \end{proof} \begin{thm} {\rm \cite{yau1}} Let $(M, J)$ be a compact complex surface which carries an admissible K\"ahler class $[\omega ]$. Suppose, moreover, that $M$ is not covered by a complex torus or a K3 surface. Then $M$ is a ruled surface. In particular, $H^{0,2}(M)=H^{2,0}(M)=0$, and $b^{+}(M)=1$. \label{sd3} \end{thm} \begin{proof} From the conclusions of Corollary \ref{yup}, the Kodaira-Enriques classification \cite{bpv} allows us to conclude, in the first instance, that $M$ is covered by a complex torus or a K3 surface, or that, in the second instance, $M$ is either a ruled surface or ${\Bbb CP}_2$. Since we also have $H^0 (M, {\cal O}(\kappa^{\ell }))=0$ for ${\ell }<0$, it follows that $M\not\cong {\Bbb CP}_2$, so that, in the second instance, $M$ is a ruled surface--- i.e. $M$ is obtained from the projectivization ${\Bbb P}(E)\to \Sigma_{\bf g}$ of a rank 2 holomorphic vector bundle $E$ over a compact complex curve $\Sigma_{\bf g}$ by blowing up $|\tau (M)|$ points. \end{proof} While the biregular classification of ruled surfaces over a curve $\Sigma_{\bf g}$ involves the classification of holomorphic vector bundles on the curve, the {\em bimeromorphic} classification is extremely simple. In fact \cite{bpv}, {\sl every ruled surface over $\Sigma_{\bf g}$ can be obtained from $\Sigma_{\bf g}\times {\Bbb CP}_1$ by blowing up and blowing down.} To see this, first observe that, since $E\otimes {\cal L}$ is generated by its sections when ${\cal L}$ is sufficiently positive, there are sections of ${\Bbb P}(E)\to \Sigma_{\bf g}$ passing through any given point. Choose three distinct such sections, and successively blow up each of their points of intersection while at each step blowing down the proper transform of the fiber through it. In finitely many steps this leads us to a minimal model equipped with a projection to $\Sigma_{\bf g}$ which admits three disjoint sections, and it is then easy to see that this model must be the product surface $\Sigma_{\bf g}\times {\Bbb CP}_1$. Although this knowledge will not prove essential for our purposes, we conclude this section by pointing out that the set of admissible K\"ahler classes can be described in extremely concrete terms: \begin{propn} Let $(M, \omega_0)$ be a compact K\"ahler surface. Then ${\cal A}_M$ is precisely the set of real $(1,1)$-classes $[\omega ]\in H^{1,1}(M, {\Bbb R})$ satisfying the following cohomological conditions: \begin{description} \item{\rm (i)} $[\omega ]\cdot c_1 =0$; \item{\rm (ii)} $[\omega ]\cdot [\omega_0] > 0$; \item{\rm (iii)} $[\omega ]^2 > 0$; and \item{\rm (iv)} $[\omega ]\cdot C > 0$ for every curve $C\subset M$. \end{description}\label{nakai} \end{propn} \begin{proof} If $[\omega ]$ satisfies {(i)} and {(iv)}, the proof of Proposition \ref{ox} shows that either $H^{2,0}=0$ or else $\kappa$ is trivial. If the former happens, $H^{1,1}=H^2$, and hence $H^2(M, {\Bbb Q})$ is dense in $H^{1,1}(M, {\Bbb R})$; in particular, $M$ is algebraic. When $[\omega ]\in H^2(M, {\Bbb Q})$, $k[\omega ]\in H^2(M, {\Bbb Z})$ for some $k\in {\Bbb N}$, and, by Nakai's criterion \cite{bpv}, $k[\omega ]$ is thus a K\"ahler class iff {(iii)} and {(iv)} hold. Thus the cone of K\"ahler classes and the cone determined by conditions {(iii)} and {(iv)} intersect $H^2(M, {\Bbb Q})$ in the same set. But both these cones are open and convex; since they contain the same set of rational points, and since the open boxes with rational corners form a basis for the topology of the Euclidean space $H^{1,1}(M, {\Bbb R})= H^2(M, {\Bbb R} )$, they must therefore coincide. Conditions {(iii)} and {(iv)} are therefore equivalent to the class $[\omega ]$ being K\"ahler, and in particular imply {(ii)}. Condition {(i)} is thus the only additional condition needed to assure a class is admissible. If, on the other hand, $\kappa$ is trivial, $M$ is either a torus or a K3 surface. In the torus case every class is uniquely represented by a form with constant coefficients, so that {(ii)} and {(iii)} are easily seen to be necessary and sufficient for a class to be K\"ahler (and automatically admissible). For the K3 case, the claim follows from Todorov's surjectivity of the refined period map \cite{bpv}. \end{proof} \subsection{Holomorphic Vector Fields and Scalar Curvature} \label{key} The space $\Gamma (M, {\cal O} (T^{1,0}M))$ of holomorphic vector fields on a complex manifold $(M,J)$ is equipped by the Lie bracket with the structure of a complex Lie algebra, denoted by $a(M)$. If $M$ is compact, this is precisely the Lie algebra of the group of biholomorphisms of $(M,J)$, since a vector field $\Xi$ of type (1,0) is holomorphic iff ${\pounds}_{\Re \Xi}J=0$, where ${\pounds}$ denotes the Lie derivative. In particular, if $g$ is a K\"ahler metric on $M$, the Lie algebra $\imath(M,g)$ of real Killing fields is canonically identified with a real sub-algebra of $a(M)$ because the K\"ahler form, being the unique harmonic representative of its deRham class, is automatically invariant under the isometry group of $M$. If, in addition, the scalar curvature of $M$ is constant, the following remarkable result of Lichnerowicz \cite{lich}, which generalizes work of Matsushima \cite{mat}, says that, modulo parallel fields, $a(M)$ is in fact just the complexification of $\imath(M,g)$: \begin{propn} {\rm (Matsushima-Lichnerowicz Theorem)} If $(M, J, g)$ is a compact K\"ahler manifold of constant scalar curvature, $a(M)$ is the direct sum of the space of parallel (1,0)-vector fields and the space of vector fields of the form $(\bar{\partial}f)^{\sharp}$ where $f$ is any (complex) solution of the equation \begin{equation}\Delta^{2}f+2(dd^{c}f, \rho)=0.\label{vf2}\end{equation} Moreover, a solution $f$ of (\ref{vf2}) corresponds to a Killing field iff it is purely imaginary ($\Re f=0$). In particular, the algebra $a(M)$ is reductive (semi-simple plus Abelian), and the identity component of the group of biholomorphisms of $M$ has a compact real form. \label{vf1} {\hfill \rule{.5em}{1em} \\} \end{propn} In the above proposition, $\sharp$ denotes, as always, the inverse of $$\flat :{\Bbb C}\otimes TM\to {\Bbb C}\otimes T^{\ast}M : X\mapsto g(X, \cdot )~,$$ and in particular induces a complex-linear isomorphism $T^{0,1\ast}M\stackrel{\sim }{\to} T^{1,0}M$. \begin{defn} Let $M$ be a compact complex surface. We will say that the {\em Matsushima-Lichnerowicz obstruction vanishes} if the identity component of the group of biholomorphisms of $M$ has a compact real form. \label{matlic} \end{defn} While the Matsushima-Lichnerowicz Theorem gives us an important obstruction to the existence of constant-scalar curvature K\"ahler metrics on a compact complex manifold $M$ in terms of the algebra $a(M)$ of holomorphic vector fields, a more subtle such obstruction was later discovered by Futaki \cite{fut}\cite{foot}. The {\em Futaki character} ${\cal F}(\cdot , [\omega ]): a(M)\rightarrow {\Bbb C}$ is defined by \begin{equation} {\cal F}(\Xi , [\omega ])= \int_{M}\Xi (\phi_{\omega })\vol \label{vf3} \end{equation} where $\Xi $ is any holomorphic vector field and $\phi_{\omega }$ is the Ricci potential: \begin{equation} \rho=\rho_{ \rm H}+dd^{c}\phi_{\omega }\label{vf4} \end{equation} where $\rho_{ \rm H}$ is harmonic and $\phi_{\omega }$ is $C^{\infty}$, real and normalized so that \begin{equation} \int \phi_{\omega }\vol=0.\label{vf5} \end{equation} Notice, by taking the trace of (\ref{vf4}), that \begin{equation} s=\mbox{constant}~\Longleftrightarrow~\phi_{\omega }=0.\label{vf6} \end{equation} The most remarkable property of the Futaki character ${\cal F}(\cdot , [\omega ])$, implicit in our notation but not evident from the definition, is \cite{cal2} \cite{fut} that it depends only upon the K\"{a}hler {\em class}; for this reason it is sometimes referred to as the Futaki invariant. From (\ref{vf6}), it now follows immediately that the vanishing of ${\cal F}(\cdot , [\omega ])$ is a necessary condition for the K\"{a}hler class $[\omega ]$ to contain a representative with constant scalar curvature. We now give a way of rewriting the Futaki invariant that is particularly useful when $\Xi = 2(\bar{\partial}f)^{\sharp}$ for a complex-valued function $f$ on $M$ with $\int f\vol =0$. Our calculations will actually work, however, for an arbitrary holomorphic vector field, provided we define its {\em holomorphy potential} $f$ by $f:= \overline{\partial }\* {\bf G}\Xi^{\flat}= {\bf G} \bar{\partial }\* \Xi^{\flat}$, where ${\bf G}$ is the Green's operator of the Hodge Laplacian. Now notice that the Ricci potential can be written in terms of the Green's operator and the scalar curvature as $\phi_{\omega }= -\frac{1}{2}{\bf G}s$. Hence \begin{eqnarray*} {\cal F}(\Xi , [\omega ])&=&\int_{M}\Xi (\phi_{\omega })\vol = \langle \Xi^{\flat}, d\phi_{\omega }\rangle \\&=& \langle \Xi^{\flat}, \bar{\partial }\phi_{\omega }\rangle = \langle \Delta {\bf G}\Xi^{\flat}, \bar{\partial }\phi_{\omega }\rangle\\&=& \langle 2{\bar{\partial }} { \bar{\partial }\*} {\bf G}\Xi^{\flat}, \bar{\partial }\phi_{\omega }\rangle = \langle 2\bar{\partial }f, \bar{\partial }\phi_{\omega }\rangle\\&=& \langle f, 2\bar{\partial }\* \bar{\partial }\phi_{\omega }\rangle = \langle f, \Delta\phi_{\omega }\rangle\\&=& \langle f, -{\textstyle \frac{1}{2}}(s-s_H)\rangle = -{\textstyle \frac{1}{2}}\int_{M}f(s-s_{ \rm H})\vol \\&=& -{\textstyle \frac{1}{2}}\int_{M}fs\vol \end{eqnarray*} where $s_{ \rm H}$ is the average value of $s$ on $(M,g)$. Notice, incidentally, that the conclusion is insensitive to the normalization $\int f\vol =0$ if the total scalar curvature $\int s\vol =s_{ \rm H}\int \volm$ happens to vanish. As a consequence we deduce an innocuous-looking fact (cf. \cite{besse}, Proposition 2.159) which will later turn out to be surprisingly important: \begin{propn} Let $(M, \omega)$ be a compact K\"ahler manifold of {\em constant} scalar curvature $s=c$. Then, for any closed (1,1)-form $\alpha$ one has $$\left.\frac{d}{dt}{\cal F}(\Xi , [\omega +t\alpha])\right|_{t=0}= \langle f\rho , \alpha_H \rangle~,$$ where $f$ is the holomorphy potential of $\Xi$ as defined above, and $\alpha_H$ is the harmonic part of $\alpha$.\label{besser} \end{propn} \begin{proof} Since the $\cal F$ only depends on the K\"ahler class, we might as well assume that $\alpha$ is harmonic. Since ${\cal F}(\Xi ,[ \omega + t\omega])= (1+t)^m {\cal F}(\Xi , [\omega ])=0$ by the assumption that $\omega$ has constant scalar curvature, whereas the corresponding right-hand side $\langle f\rho , \omega \rangle =\frac{1}{2}\int f s\vol =-{\cal F}(\Xi , [\omega ])$ vanishes for the same reason, we may assume that the harmonic form $\alpha$ is {\em primitive}. This assumption has the effect that the volume form of $\omega (t)$ is $$\vol (t)=\frac{(\omega +t\alpha )^m}{m!}= \vol + \frac{t\omega^{m-1}\wedge \alpha }{(m-1)!} + O(t^2)=\vol + O(t^2)~,$$ so that the Ricci form, being determined by the volume form and $J$, similarly satisfies $$\rho (t) =\rho + O(t^2)~ .$$ The normalization of the holomorphy potential reads $\int f(t)\vol =O(t^2)$ for the same reason. Hence \begin{eqnarray*} \left.\frac{d}{dt}{\cal F}(\Xi , [\omega +t\alpha])\right|_{t=0} &=&{ -\frac{1}{2}} \left.\frac{d}{dt} \left[\int_{M}f(t)s(t)\vol (t)\right]\right|_{t=0} \\&=& {\textstyle -\frac{1}{2}}\left.\int_{M}\frac{df}{dt}\right|_{t=0}c\vol - {\textstyle \frac{1}{2}}\left.\int_{M}f\frac{d}{dt}\left[s(t)\vol (t)) \right]\right|_{t=0} \\&=&{\textstyle -\frac{1}{2}} \left.\int_{M}f\frac{d}{dt}\left[s(t)\vol (t)\right]\right|_{t=0} \\&=&-{\textstyle\frac{1}{(m-1)!}} \int_{M}f\left.\frac{d}{dt}\left[\rho (t) \wedge \omega^{m-1} (t) \right] \right|_{t=0} \\&=&-{\textstyle\frac{1}{(m-1)!}} \int_{M}f\rho\wedge \left. \frac{d}{dt} \omega^{m-1} (t) \right|_{t=0} \\&=&-{\textstyle\frac{1}{(m-2)!}} \int_{M}f\rho \wedge \omega^{m-2} \wedge \alpha \\&=& \int_{M}f\rho \wedge \star \alpha =\langle f\rho , \alpha_H \rangle~ . \end{eqnarray*} \end{proof} \subsection{Twistor Spaces} \label{twistors} The Penrose correspondence \cite{AHS}\cite{besse}\cite{P} is a dictionary between anti-self-dual conformal Riemannian 4-manifolds and a special class of complex 3-folds. We will begin by briefly explaining how the translation works in each direction. If $(M,[g])$ is an anti-self-dual Riemannian 4-manifold, its {\em twistor space} is a complex 3-manifold $Z$ whose underlying smooth 6-manifold is the total space of the sphere bundle of the rank-three real vector-bundle of self-dual 2-forms: \begin{eqnarray*} S^2\to &Z&:= \{ \omega\in{\wedge}_+~|~\|\omega\| =\sqrt 2 \}\\ &\hphantom{\wp}\downarrow\wp& \\ &M&\end{eqnarray*} We now give $Z$ an almost-complex structure $J: TZ \to TZ$, $J^2=-1$, by first observing that, for each $x \in M$, there is a natural one-to-one correspondence between $\wp^{-1}(x)$ and the set of $g$-orthogonal complex structures $\jmath: T_xM\to T_xM$ inducing the given orientation on $T_xM$; namely, in the spirit of (\ref{n4}), any such $\jmath$ corresponds to the 2-form $\omega_{\jmath}$ defined by $$ \omega_{\jmath} (\xi , \eta) = g ({\jmath}\xi, \eta)~ .$$ Since the Levi-Civit\`a connection of $g$ induces a splitting $TZ=H\oplus V$ of the tangent bundle of $Z$ into horizontal and vertical parts, and we have a canonical isomorphism $\wp_{\ast}: H\to \wp^{\ast}TM$, we may define $J_H: H\to H$, $J_H^2=-1$ by $J_H|_{\omega_\jmath}:=\jmath$. Since the fibers of $\wp$ are oriented metric 2-spheres, we may also define $J_V:V\to V$, $J_V^2=-1$ to be rotation by $-90^{\circ}$ in the tangent space of the fiber. Defining $J=J_H\oplus J_V$ then makes $Z$ an almost-complex manifold. Quite remarkably, this almost-complex structure is conformally invariant. Even more remarkably, it is {\em integrable} because its Nijenhuis tensor may be identified with $W_+$, and so vanishes precisely by the assumption that $(M,g)$ is anti-self-dual. The fibers of $\wp$ have become ${\Bbb CP}_1$'s with normal bundle ${\cal O} (1)\oplus {\cal O} (1)$ in the complex manifold $Z$, while the fiber-wise antipodal map $Z\to Z : \omega\mapsto -\omega$ has become a free anti-holomorphic involution $\sigma : Z\stackrel{\bar{{\cal O}}}{\to} Z$. Conversely, let $Z$ be a complex 3-fold with free anti-holomorphic involution $\sigma : Z\stackrel{\bar{{\cal O}}}{\to} Z$, $\sigma^2=\mbox{id}_Z$, and suppose that there is a smooth $\sigma$-invariant rational curve in $Z$ with normal bundle ${\cal O} (1)\oplus {\cal O} (1)$. Let ${\Bbb C}M$ denote the connected component of this curve in the space of all ${\Bbb CP}_1\subset Z$ with normal bundle bundle ${\cal O} (1)\oplus {\cal O} (1)$. Invoking \cite{K}, ${\Bbb C}M$ is a complex 4-manifold. Moreover, $\sigma$ induces an anti-holomorphic involution $\hat{\sigma}: {\Bbb C}M\to {\Bbb C}M$ for which our original curve corresponds to a fixed point; the fixed-point set of $\hat{\sigma}$ is therefore a (non-empty) real-analytic 4-manifold, the obvious connected component of which we denote by $M$. There is then an anti-self-dual conformal class of metrics $[g]$ on $M$ determined by requiring that a complex tangent vector $\xi\in{\Bbb C}\otimes TM=T{\Bbb C}M|_M$ satisfies $g(\xi , \xi)=0$ iff the corresponding section of the normal bundle of ${\Bbb CP}_1\subset Z$ has a zero. If $Z$ actually arises by the construction of the preceding paragraph, and if our initial curve is a fiber of $\wp$, this exactly reconstructs the given manifold $M$ and conformal structure $[g]$. Now recall from \S \ref{asd} that a K\"ahler manifold of complex dimension 2 is anti-self-dual iff its scalar-curvature vanishes. (This might be considered rather remarkable insofar as neither the K\"ahler condition nor the scalar-curvature condition are themselves conformally invariant, and yet their coincidence is reflected by a conformally invariant property.) The Penrose correspondence thus provides an unexpected link between K\"ahler surfaces and complex 3-folds. As will now be explained, the speciality of the metric being scalar-flat K\"ahler is echoed by a specialty of the twistor space $Z$ in a manner simple enough to allow us to study scalar-flat K\"ahler geometry by using Kodaira-Spencer theory. The central result here is a theorem of Pontecorvo: \begin{propn} {\rm \cite{Pt}} Let $\wp : Z\to M$ be the twistor fibration of a (perhaps non-compact) anti-self-dual conformal Riemannian 4-manifold $(M, [g])$. Suppose we are given a complex hypersurface $D_1\subset Z$ for which the restriction $\wp|_{D_1}$ of the twistor projection is a diffeomorphism onto $M$, and let $J$ denote the complex structure on $M$ given by this section of $Z$. (Thus $\wp|_{D_1}$ becomes a biholomorphism of $D_1$ and $(M,J)$.) Let $D_2=\overline{D_1}$ denote the image of $D_1$ under the real structure $\sigma : Z\to Z$, and let $D=D_1\cup D_2$. Then there exists a metric $g$ in the conformal class $[g]$ such that $(M,g,J)$ is K\"ahler iff the divisor line bundle of $D$ is isomorphic to the half-anti-canonical line bundle $\kappa^{-1/2}$ of $Z$. \label{wanna} \end{propn} The implication $\Rightarrow$ is relatively straightforward; indeed, the K\"ahler form $\omega$, being parallel and self-dual, is a solution of the twistor equation, and so defines, via the Penrose transform, a holomorphic section of $\kappa^{-1/2}$ vanishing precisely at $D$. For an analogous proof in the $\Leftarrow$ direction, cf. \cite{L3}. This implies the following key result, originally discovered in a somewhat different guise by Boyer \cite{boyer}: \begin{thm} \label{bythm} Let $\wp : Z\to M$ be the twistor fibration of a compact anti-self-dual 4-manifold $(M, [g])$, and suppose that $b_1(M)$ is even. Let $D_1\subset Z$ be a complex hypersurface which meets every fiber in exactly one point. Then, for any $c\in {\Bbb R}^+$, the conformal class $[g]$ contains a unique scalar-flat K\"ahler metric of volume $c$. Conversely, every scalar-flat K\"ahler surface arises in this way. \end{thm} \begin{proof} Let $[D]$ denote the divisor line bundle of the hypersurface $D=D_1\cup \sigma (D_1)$. Then $c_1([D])=c_1(\kappa^{-1/2})$, so that $[D]\otimes \kappa^{1/2}$ is an element of $\mbox{Pic}_0(Z)$. By the Penrose transform, and using Proposition \ref{sd2}, $H^1(Z, {\cal O})=\{ \beta \in {\cal E}^1(M)~|~d^+\beta =0\}/d{\cal E}^0(M) =H^1(M, {\Bbb C})$, so that one has $\mbox{Pic}_0(Z)=H^1(Z, {\Bbb C}^{\times })=H^1(M, {\Bbb C}^{\times })$; in other words, every topologically trivial holomorphic line bundle on $Z$ admits a compatible flat connection, and these all come from the base. In particular, $[D]\otimes \kappa^{1/2}$ admits a flat ${\Bbb C}^{\times}$-connection, and is the pull-back of a flat ${\Bbb C}^{\times}$-bundle on $M$. In particular, $[D]\otimes \kappa^{1/2}$ is trivial on $\wp^{-1}(U)$ for any sufficiently small open set $U\subset M$. By Proposition \ref{wanna}, $g$ is therefore conformal to a K\"ahler metric on a sufficiently small open set $U$. We have thus shown that there are locally-defined smooth functions $u\in {\cal E}_U$ for which the (1,1)-form $\omega$ associated to $(g,J)$ satisfies $0=d(e^u\omega )= e^u(du\wedge\omega + d\omega )$. Since any two local choices of $u$ differ by an additive constant, the 1-form $\beta=-du$ is {\em globally} defined on $M$, and $$d\omega = \beta \wedge \omega ~ .$$ Because the Fr\"ohlicher spectral sequence of any compact complex surface degenerates \cite{bpv}, the hypothesis that $b_1(M)\equiv 0\bmod 2$ implies that $H^1_{d}(M, {\Bbb C} )= H^0(M, \Omega^1)\oplus \overline{H^0(M, \Omega^1)}$. (A less elementary but deeper explanation of this decomposition stems from the fact \cite{siu} that a compact complex surface admits K\"ahler metrics iff $b_1$ is even.) Thus the closed real 1-form $\beta$ can be written as $$\beta = \Re \alpha + df$$ for some holomorphic 1-form $\alpha$ and some smooth function $f$. Introducing the conformally rescaled metric $\hat{g}:=e^{-f}g$, we now have $d\hat{\omega }= \Re \alpha \wedge \hat{\omega }$. But then $$ 0= \int_M d(\alpha \wedge \hat{\omega})= -{\textstyle \frac{1}{2}} \int_M \alpha \wedge \overline{\alpha} \wedge \hat{\omega}= {\textstyle \frac{i}{2}} \|\alpha\|^2_{L^2,\hat{g}}~,$$ so that $\alpha =0$, and $\hat{g}$ is K\"ahler. Since $\hat{g}$ is also ASD, it is automatically scalar-flat by Proposition \ref{sd1}. \end{proof} \subsection{Deformation Problems} For a compact manifold $M$ which admits a scalar-flat K\"{a}hler metric $g$, a number of moduli problems are now obviously of interest: \begin{description} \item{(a)} the moduli of scalar-flat K\"{a}hler metrics in the given K\"{a}hler class; \item{(b)} the moduli of scalar-flat K\"{a}hler metrics for the given complex structure; \item{(c)} the moduli of scalar-flat K\"{a}hler metrics, with the complex structure allowed to vary; and \item{(d)} the moduli of ASD conformal structures on $M$. \end{description} One might also be tempted to add the following: \begin{description} \item{(b$'$)} the moduli of ASD Hermitian conformal structures for a given complex structure; \item{(c$'$)} the moduli of ASD Hermitian conformal structures for some complex structure. \end{description} However, as we saw in proving Theorem \ref{bythm}, a result of Boyer \cite{boyer} states that, because $b_1(M)$ is even, (b$'$) and (c$'$) are respectively equivalent to (b) and (c), so nothing is to be gained by considering these problems separately. Of these problems, (a) can be tackled quite easily within the standard framework of K\"{a}hler geometry, but for (b)--(d) very valuable information comes from the twistor description. As we saw in the previous section, the twistor space $Z$ of a scalar-flat K\"{a}hler surface $M$ is a complex 3-manifold equipped with a real structure $\sigma$ and a $\sigma$-invariant divisor $D$. The complex structure of $Z$ completely determines the conformal structure of $M$ while the divisor $D$ specifies the given complex structure on $M$. Thus the moduli problems (b)--(d) correspond to the following problems in terms of $(Z,D)$: \begin{description} \item{(b$^{*}$)} moduli of complex structures on $Z$ with $D$ as a fixed $\sigma$-invariant divisor; \item{(c$^{*}$)} moduli of complex structures on $Z$ which admit a $\sigma$-invariant divisor with divisor line-bundle isomorphic to $\kappa^{-1/2}$; \item{(d$^{*}$)} moduli of complex structures on $Z$, which admit a compatible real structure $\sigma$. \end{description} Note that this point of view imposes different equivalence relations on the metrics occurring in the different problems; in problems (a)--(b), two metrics of the same total volume will be considered equivalent iff they are literally {\em equal}, whereas in problems (c)--(d) two metrics will be equivalent if they are in the same orbit of the diffeomeorphism group cross conformal rescalings. Certainly an advantage of the starred formulation over the original one is that one can appeal to the machinery of Kodaira-Spencer deformation theory to get local information about the moduli spaces in terms of certain sheaf cohomology groups of the twistor spaces. But from our point of view the most significant advantage of this description is that the cohomology groups involved in these distinct problems are related by exact sequences; once problem (b) is thoroughly understood, problems (c) and (d) can also be solved with relatively little further effort. \subsection{Deformation Theory} \label{ks} Let $Z$ be a compact complex manifold. For us, a {\em deformation} of $Z$ will consist of the following: a ``parameter'' manifold ${\cal T}$ with basepoint $o$; a smooth manifold ${\cal Z}$; a proper submersion $\varpi:{\cal Z}\rightarrow {\cal T}$; an integrable fiber-wise complex structure on ${\cal Z}$; and an identification of the central fiber $\pi^{-1}(o)$ with $Z$. If ${\cal T}'$ is another manifold, with basepoint $o'$, and $\varphi:{\cal T}'\rightarrow {\cal T}$ is a basepoint-preserving smooth map, there is an induced deformation $\varphi^{*}({\cal Z})\rightarrow {\cal T}'$. The deformation $\varpi:{\cal Z}\rightarrow {\cal T}$ is called {\em complete} if any other deformation can be induced from it by a smooth map $\varphi$, {\em versal} if, in addition, the derivative of $\varphi$ at $o'$ is always uniquely determined, and {\em universal} if, in addition, the inducing $\varphi$ is always unique. When a universal deformation of $Z$ exists, a neighborhood of $o$ in the parameter space $\cal T$ gives a model for the moduli space of complex structures on $Z$ in a neighborhood of the given structure. If $\varpi:{\cal Z}\rightarrow {\cal T}$ is any deformation in the above sense, the {\em Kodaira-Spencer map} at $o\in {\cal T}$ is an ${\Bbb R}$-linear map ${\bf ks}: T_o{\cal T}\to H^1(Z_o, \Theta )$ obtained in \v{C}ech cohomology by differentiating the transition functions of a fiber-wise complex coordinate atlas on ${\cal Z}$. The first basic result of Kodaira-Spencer theory is that a deformation is complete (respectively, versal) if ${\bf ks}$ is surjective (respectively, bijective). Notice that, by virtue of its definition, the Kodaira-Spencer map behaves functorially under pull-backs. The main results of Kodaira-Spencer theory \cite{KS} assert that any versal deformation may be made into a holomorphic map $\varpi :{\cal Z}\to {\cal T}$ between complex manifolds (in an essentially unique manner), and, more importantly, give sufficient conditions \cite{KNS} for the existence of a versal or universal deformation of $Z$ in terms of the sheaf cohomology groups $H^{\jmath}(Z,\Theta)$, where $\Theta ={\cal O}(T^{1,0}Z)$ is the sheaf of holomorphic vector fields on $Z$. These results may be summed up as follows: \noindent \begin{thm} Suppose $H^{2}(Z,\Theta)=0$. Then a (holomorphic) versal deformation exists, with parameter space ${\cal T}$ an open neighborhood of $o=0$ in $H^{1}(Z,\Theta)$. This deformation is universal if $H^{0}(Z,\Theta)=0$. {\hfill \rule{.5em}{1em} \\} \label{ks1} \end{thm} Unfortunately, this will not suffice for our purposes, because we will be primarily interested in deformations of complex manifolds {\em with real structure}. By a real structure on a compact complex manifold $Z$, we always mean an anti-holomorphic involution of $Z$--- i.e. an anti-holomorphic map $\sigma : Z\stackrel{\overline{\cal O}}{\to }Z$ such that $\sigma^2=\mbox{id}_Z$. We will further assume that $\sigma$ {\em acts freely}--- i.e. without fixed points. This in particular means that $Z/\sigma$ is a smooth manifold, and while the complex structure tensor $J$ of $Z$ cannot descend to the quotient, the unordered pair $\{ J,-J\}$ {\em is} globally well-defined downstairs. We therefore introduce the following concept: \begin{defn} A {\em semi-complex manifold} is a smooth manifold $P$, together with a 1-dimensional sub-bundle $L\subset \mbox{End}(TP)$ such that, in a neighborhood of any point $x\in P$, $L$ is spanned by an integrable complex structure $J$. \end{defn} Of course, near any point there are then exactly 2 choices of the complex structure spanning $L$--- if $J$ is one, $-J$ is the other. If $P^{2m}$ is a semi-complex manifold, we can thus equip $P$ with an atlas for which all the transition functions are either holomorphic or anti-holomorphic diffeomorphisms of domains in ${\Bbb C}^m$; and conversely, any manifold equipped with such an atlas has an induced semi-complex structure. \begin{example} Let $P$ be a smooth, unoriented surface, and let $[g]$ be a conformal class of Riemannian metrics on $P$. Then $[g]$ determines a unique semi-complex structure on $P$. \end{example} \begin{example} Let $Z$ be the twistor space of a half-conformally-flat Riemannian 4-manifold $(M, g)$, and let $Z$ be its twistor space. Let $\sigma : Z\stackrel{\overline{\cal O}}{\to }Z$ be its real structure, acting on the fibers of $\wp : Z\to M$ by the antipodal map. Then $P:=Z/\sigma$ is a semi-complex manifold. Notice, incidentally, that $P\to M$ is an ${\Bbb RP}^2$-bundle. \end{example} The following observation will be as crucial as it is trivial: {\em every semi-complex manifold $P$ is double-covered by a complex manifold in a manner that makes the non-trivial deck transformation a free anti-holomorphic involution.} Indeed, one simply takes the cover to consist of the elements $J\in L\subset \mbox{End}(TP)$ such that $J^2=-1$. Thus, our basic example $P=Z/\sigma$ of a semi-complex manifold, where $Z$ is complex and $\sigma : Z\to Z$ is a free anti-holomorphic involution, actually represents the general case. If $Z$ is a complex manifold, the sheaf of holomorphic vector fields is, as mentioned above, denoted by $\Theta :={\cal O} (T^{1,0}Z)$. However, we may identify the underlying real vector bundle of $T^{1,0}Z$ with the real tangent bundle $TZ$ by $2\Re : T^{1,0}Z\to TZ: \Xi\mapsto \Xi + \overline{\Xi }$, and in the process we identify $\Theta$ with the sheaf $$\Re \Theta : =\{ \xi \in {\cal E}(TZ)~|~{\pounds}_{\xi}J=0\}$$ of ``real holomorphic'' vector fields; of course, this only identifies them as sheaves of real Lie algebras. The interesting observation is that $\Re\Theta$ is {\em exactly the same} for the conjugate complex manifolds $(Z, J)$ and $(Z, -J)$, and is thus even well defined on a semi-complex manifold. This, of course, happens precisely because $\Re\Theta$ is the sheaf of infinitesimal automorphisms of the semi-complex structure. If we repeat our previous definitions of deformations and versality for semi-complex manifolds, with the fiber-wise structures only required to be semi-complex instead of complex, we immediately get the following result: \begin{propn} Let $P$ be a compact semi-complex manifold such that $H^2(P, \Re\Theta)=0$. Then there exists a versal deformation of $P$ with a neighborhood of $0\in H^1(P, \Re\Theta)$ as parameter space. This deformation is universal if $H^0(P, \Re\Theta)=0$. \end{propn} \begin{proof} The Forster-Knorr power-series proof \cite{fok} of Theorem \ref{ks} goes through without any essential changes. \end{proof} \begin{lemma} Let $Z$ be a complex manifold with free anti-holomorphic involution $\sigma : Z\to Z$, and let $P=Z/\sigma$ be the associated semi-complex manifold. Then $H^j(Z, \Theta)= H^j(P, \Re\Theta)\otimes_{{\Bbb R} } {\Bbb C}$. \end{lemma} \begin{proof} There are arbitrarily fine covers $\cal V$ of $Z$ which are equivariant under $\sigma$, and any such cover descends to a cover $\cal W$ of $P$. For any such cover $\cal V$, $\sigma$ acts as on $\check{H}^j({\cal V}, \Theta)$ via the anti-liner map $\{ f_{\alpha\cdots \beta}\}\mapsto \{ \sigma\* \overline{f_{\alpha\cdots \beta}}\}$, and the fixed-point set of this action can be identified with $\check{H}^j({\cal W}, \Re\Theta)$. Hence $\check{H}^j({\cal V}, \Theta)= \check{H}^j({\cal W}, \Re\Theta)\otimes_{{\Bbb R} } {\Bbb C}$. The lemma now follows by taking direct limits. \end{proof} \begin{thm} Let $Z$ be a compact complex manifold with $H^2(Z, \Theta)=0$. Suppose that $\sigma : Z\to Z$ is an anti-holomorphic involution without fixed points. Then $\sigma$ can be extended as an anti-holomorphic involution $\sigma_{\cal Z}:{\cal Z}\to {\cal Z}$ of the total space of the versal deformation of $Z$ which covers an anti-holomorphic involution $\sigma_{\cal T}:{\cal T}\to {\cal T}$ of the base. The fixed-point set of $\sigma_{\cal T}$ is a totally real subspace ${\cal T}_{\sigma}$ of real dimension $h^1(Z, \Theta )$, and the restriction of $\varpi : {\cal Z}\to {\cal T}$ to this subspace is a versal deformation of $({\cal Z}, \sigma )$. \end{thm} \begin{remark} When $H^0(Z, \Theta )=0$, this is an immediate consequence \cite{DF} of the universal property of the versal deformation. \end{remark} The importance of real deformations stems from the following observation, the essence of which was discovered by Penrose \cite{P}: \begin{thm} Let $\varpi : {\cal Z}\to {\cal T}$ be a deformation of the twistor space $Z=Z_o$ of a compact ASD conformal Riemannian 4-manifold $(M, [g])$. Suppose, moreover, that ${\cal Z}$ is equipped with a fiber-wise anti-holomorphic involution which restricts to the twistor real structure on $Z_o$. Then there is a neighborhood ${\cal U}$ of $o\in {\cal T}$ and a family of ASD Riemannian metrics $g_t$ on $M$, depending smoothly on $t\in {\cal U}$, such that $Z_t=\varpi^{-1}(t)$ is biholomorphic to the twistor space of $(M, [g_t])$.\label{Pen} \end{thm} \begin{proof} The point is that the normal bundle $\nu$ of a twistor fiber $C=\wp^{-1}(x)\in Z$ is isomorphic to the bundle ${\cal O} (1)\oplus {\cal O} (1)$ on ${\Bbb CP}_1$, and so satisfies $H^1(C, \nu )=0$. By Kodaira's stability theorem \cite{K}, the complete analytic family generated by the twistor fibers is stable under deformations. Because ${\cal O} (1)\oplus {\cal O} (1)$ is a rigid bundle on a rigid manifold, we have a 4-complex parameter family of ${\Bbb CP}_1$'s with normal bundle ${\cal O} (1)\oplus {\cal O} (1)$ in any small deformation $Z_t$ of $Z$; the $\sigma$-invariant curves in these families then foliate a manifold containing $Z_o$ and spread over an open neighborhood of $Z_o\subset {\cal Z}$. There is therefore a neighborhood of $Z_o\subset{\cal Z}$ foliated by these curves and projecting properly to a neighborhood ${\cal U}$ of $o\in {\cal T}$. Invoking the inverse twistor correspondence described in \S\ref{twistors} finishes the proof. \end{proof} In short, the moduli space for problem (d) is locally the same as the moduli space for semi-complex structures on $Z/\sigma$. In order to attack the moduli problems (b) and (c), we shall instead require two `relative versions' of the above deformation theory. Suppose we are given a compact complex manifold $Z$ and a nonsingular complex hypersurface $D$ in $Z$. A deformation of $(Z,D)$ is given by the following: a deformation $\varpi:{\cal Z}\rightarrow {\cal T}$ of $Z$ and a deformation ${\cal D} \rightarrow {\cal T}$ of $D$, together with a commutative diagram \begin{eqnarray} {\cal D} & \longrightarrow & {\cal Z} \nonumber \\ \displaystyle\downarrow& & \displaystyle\downarrow{\varpi} \nonumber \\ {\cal T} & = & {\cal T} \label{ks2} \end{eqnarray} which restricts to the inclusion of $D$ in $Z$ at the central fibers. A deformation of $(Z,D)$ with {\em fixed divisor} $D$ is a deformation of the above type with ${\cal D}\cong D\times {\cal T}$, and $\varpi |_{\cal D}$ corresponding to projection on the second factor. The notions of versal and universal deformations exist also for these relative deformations and there are analogues of Theorem \ref{ks1}. To state them, let $\Theta_{Z,D}$ be the sheaf of holomorphic vector fields on $Z$ that are tangent to $D$ along $D$; and let ${\Theta}_{Z}\otimes {\cal I}_D$ be the subsheaf of vector fields which vanish along $D$. Finally, we can modify all the above definitions so as to replace $Z$ with a semi-complex manifold $P$ and $D$ with a semi-complex submanifold $M\subset P$ of real codimension 2. The same reasoning as before then yields \noindent \begin{thm} Let $(P,M)$ be a semi-complex manifold with nonsingular semi-complex hypersurface, and let $(Z, D, \sigma )$ be the complex manifold with hypersurface and real structure which covers it. (i) Suppose that $H^{2}(Z,\Theta_{Z,D})=0$. Then there are versal deformations of $(P,M)$ and $(Z,D)$. Moreover, the parameter space for the former deformation is a real slice in that of the latter, which may be taken to be a neighborhood of $0\in H^1(Z,\Theta_{Z,D})$. These deformations are both universal if $H^{0}(Z,\Theta_{Z,D})=0$. (ii) Suppose that $H^{2}(Z,{\Theta}_{Z}\otimes {\cal I}_D)=0$. Then there are versal deformations of $P$ with fixed divisor $M$, and of $Z$ with fixed divisor $D$. Moreover, the parameter space for the former deformation is a real slice in that of the latter, which may be taken to be a neighborhood of $0\in H^1(Z,{\Theta}_{Z}\otimes {\cal I}_D)$. These deformations are both universal if $H^{0}(Z,{\Theta}_{Z}\otimes {\cal I}_D)=0$. \label{ks3} \end{thm} Notice that these deformation problems are all related in that there exist exact sequences: \begin{equation} 0\rightarrow\Theta_{Z,D}\rightarrow\Theta_{Z}\rightarrow N_{D}\rightarrow 0 \label{ks4} \end{equation} where $N_{D}$ is the normal bundle of $D$ in $Z$, and \begin{equation} 0\rightarrow{\Theta}_{Z}\otimes {\cal I}_D\rightarrow \Theta_{Z,D}\rightarrow\Theta_{D} \rightarrow 0.\label{ks5} \end{equation} Moreover, the induced long-exact sequences exactly intertwine the Kodaira-Spencer maps of the deformation theories involved. In particular, given a deformation $\varpi : ({\cal Z}, {\cal D})\to {\cal T}$ of $({ Z}, { D})$, there is a Kodaira-Spencer map ${\bf ks}_{Z,D}\in \mbox{Hom} (T_o{\cal T}, H^1(Z, \Theta_{Z,D}))$, gotten by differentiating the transition functions of a fiber-wise complex atlas which sends open sets of $Z_t$ to ${\Bbb C}^m$ and open sets of $D_t$ to ${\Bbb C}^{m-1}\subset {\Bbb C}^m$; this is obviously related to the Kodaira-Spencer maps of $\varpi : {\cal Z}\to {\cal T}$ and $\varpi |_{\cal D}:{\cal D} \to {\cal T}$ by composition with the natural homomorphisms $H^1(Z, \Theta_{Z,D})\to H^1(Z, \Theta_{Z})$ and $H^1(Z, \Theta_{Z,D}) \to H^1(D, \Theta_{D})$. In particular, if $H^2 (Z, {\Theta}_{Z}\otimes {\cal I}_D) =0$, the resulting surjectivity of $H^1(Z, \Theta_{Z,D}) \to H^1(D, \Theta_{D})$ implies that if $\varpi : ({\cal Z}, {\cal D})\to {\cal T}$ is assumed to be a versal deformation of $({ Z}, { D})$, the induced deformation $\varpi |_{\cal D}:{\cal D} \to {\cal T}$ is complete. A similar argument in the semi-complex case will feature prominently in our proof of the Main Theorem. \pagebreak \setcounter{equation}{0} \section{Deformations of Scalar-flat K\"{a}hler Surfaces}\label{flag} The standard treatment of these problems is as follows. Given a scalar-flat K\"{a}hler metric $g$, K\"{a}hler form $\omega$, normalized so that the total volume is 1, we identify the tangent space to the space of volume-1 K\"{a}hler forms as \begin{equation} K=\left\{\varphi\in{\wedge^{1,1}}(M):d\varphi=0~ \mbox{ and }~\int~(\Lambda\varphi) \vol=0\right\}.\label{d1} \end{equation} The derivative in the direction $\varphi$ of the scalar curvature is (cf. \cite{besse}, Lemma 2.158(iii)) \begin{equation} s'(\varphi)=\Delta (\Lambda\varphi)-2(\rho,\varphi)\label{d2} \end{equation} (where $\rho$ is the Ricci form as in (\ref{n4})). If it is required to preserve the K\"{a}hler class then $\varphi\in K$ is taken to have the form \begin{equation} \varphi=-dd^{c}f\label{d3} \end{equation} (for some real $C^{\infty}$ function $f$) and (\ref{d2}) reduces to \begin{equation} s'(f)=\Delta^{2}f+2(dd^{c}f, \rho)\label{d4} \end{equation} This we recognize as Lichnerowicz's differential equation (\ref{vf2}). Invoking Proposition \ref{vf1}, we can thus immediately solve problem (a): \begin{propn} The tangent space to the moduli space of scalar-flat K\"{a}hler metrics in a given K\"{a}hler class is precisely the space $\imath(M)^{\perp}\subset a(M)$ of holomorphic vector fields orthogonal to the space $\imath(M)$ of Killing fields. {\hfill \rule{.5em}{1em} \\} \label{dt1} \end{propn} Let us now turn to the more general problem (b). For this, we shall study the equivalent problem of deforming $Z$ with fixed divisor $D$. Referring to Theorem \ref{ks3} we see that the first task is to identify ${\Theta}_{Z}\otimes {\cal I}_D$ and its cohomology groups. Since the ideal sheaf of $D$ is isomorphic to ${\cal O}(-2):=\kappa_Z^{1/2}$, we have ${\Theta}_{Z}\otimes {\cal I}_D=\Theta(-2)$. To study the cohomology groups we use the Penrose transform to relate them to data on $M$. A straightforward application of the techniques of \cite{be}\cite{bs} or \cite{h2} yields: \noindent \begin{propn} Let $(M,g)$ be any anti-self-dual manifold. For each $j=0,\ldots,3$, the Penrose transform identifies $H^{j}(Z,\Theta(-2))$ with the ${j}$-th cohomology group of the complex \begin{equation} 0\rightarrow{\wedge^{-}}(M)~ \stackrel{S}{\rightarrow}~{\wedge^{+}}(M) \rightarrow 0 \end{equation} where \begin{equation} S(\alpha)=d^{+}\delta\alpha + \Phi\alpha \end{equation} for $\alpha\in{\wedge^{-}}(M)$. Here $\Phi : \wedge^-\to \wedge^+$ denotes one-half the trace-free Ricci curvature, acting by $\alpha_{ab}\mapsto \Phi^c_{b}\alpha_{ac}-\Phi^c_{a}\alpha_{bc}$. \label{dt0} {\hfill \rule{.5em}{1em} \\} \end{propn} \noindent{\bf Remark}. The operator $S$ is conformally invariant, provided that $\alpha$ is conformal weighted as follows: $\alpha \mapsto e^{u/2}\alpha$ when $g\mapsto e^{u}g$. \begin{cor} Let $(M,g)$ be a scalar-flat K\"ahler surface. The Penrose transform then identifies $H^{j}(Z,\Theta(-2))$ with the ${j}$-th cohomology group of the complex \begin{equation} 0\rightarrow{\wedge^{-}}(M)~ \stackrel{S}{\rightarrow}~{\wedge^{+}}(M) \rightarrow 0\label{d5} \end{equation} where \begin{equation} S(\alpha)=d^{+}\delta\alpha-{\textstyle \frac{1}{2}}~(\rho,\alpha)\omega\label{d6} \end{equation} for $\alpha\in{\wedge^{-}}(M)$. {\hfill \rule{.5em}{1em} \\} \label{dt2} \end{cor} While the twistor theory predicts that the operator $S$ completely governs problem (b), it is perhaps not obvious why this is so. Let us therefore digress for a moment in order to observe the kernel of $S$ can indeed be identified with space of $\varphi$ given by (\ref{d1}) and (\ref{d2}). \begin{thm} The map $$ K\rightarrow{\wedge^{-}}(M) $$ given by $\varphi\mapsto\varphi_{0}$ (see (\ref{n15})) induces an isomorphism of $\ker(s')$ with $\ker(S)$. \label{dt3} \end{thm} \noindent{\bf Proof}. We begin by noticing that, as a consequence of (\ref{n8}) and (\ref{n13}), the equation \begin{equation} d^{+}\delta\alpha=\lambda\omega\label{d11} \end{equation} is equivalent to the two equations \begin{equation} {\Lambda}d\delta\alpha={-\Lambda}\delta d\alpha=2\lambda\label{d9} \end{equation} and \begin{equation} d^{+} \Lambda d\alpha=0.\label{d10} \end{equation} Now suppose that $\varphi$ is in $K$ and write $$ \varphi={\textstyle \frac{1}{2}}~(\Lambda\varphi)\omega+\varphi_{0}. $$ We have to show that equation (\ref{d2}) implies that $\varphi_{0}$ is in the kernel of $S$. But $d\varphi =0$ is equivalent to \begin{equation} d\varphi_{0}=-{\textstyle \frac{1}{2}}Ld\,(\Lambda\varphi)\label{d12} \end{equation} and so to \begin{equation} {\Lambda}d\varphi_{0}=-{\textstyle \frac{1}{2}}~[{\Lambda},L]d\Lambda\varphi =-{\textstyle \frac{1}{2}} {}~d(\Lambda\varphi)\label{d14a} \end{equation} by (\ref{n10}) (remember $\Lambda$ of any 1-form is zero). This implies that $\varphi_{0}$ satisfies equation (\ref{d10}). On the other hand, by applying ${\Lambda}\delta$ to (\ref{d12}) we get $$ \begin{array}{lcll} {\Lambda}\delta d\varphi_{0} & = & - & {\textstyle \frac{1}{2}} {\Lambda}\delta Ld(\Lambda\varphi) \\[+12pt] & = & - & {\textstyle \frac{1}{2}}{\Lambda}[\delta,L]d\Lambda\varphi- {\textstyle \frac{1}{2}}{\Lambda}L\delta d(\Lambda\varphi) \\[+12pt] & = & & {\textstyle \frac{1}{2}}~ {\Lambda}d^{c}d(\Lambda\varphi)- \delta d(\Lambda\varphi) \\[+12pt] & = & & {\textstyle \frac{1}{2}}~ \Delta (\Lambda\varphi)-\Delta (\Lambda\varphi)=- {\textstyle \frac{1}{2}}~\Delta (\Lambda\varphi) \end{array} $$ where we have used the K\"{a}hler identities (\ref{n8}), (\ref{n10}) and (\ref{n11}). Hence by equation (\ref{d2}) $$ {\Lambda}\delta d\varphi_{0}=-(\rho,\varphi)=-(\rho,\varphi_{0}) $$ because $\rho$ is ASD (cf. (\ref{n18})), and this is equation (\ref{d9}) with $\lambda={\textstyle \frac{1}{2}}~(\rho,\varphi_{0})$ as required. To go in the other direction we suppose $\alpha\in\ker(S)$ so that it satisfies (\ref{d9}) and (\ref{d10}) with $\lambda= {\textstyle \frac{1}{2}}~(\rho,\alpha)$. Let $u$ be the unique solution of $\Delta u=2(\rho,\alpha)$ with $\int u\vol=0$, and put $\varphi={\textstyle \frac{1}{2}}~u\omega+\alpha$. Then $\varphi$ automatically satisfies (\ref{d2}): all that remains is to check $d\varphi=0$. By Proposition \ref{sd2} and equation (\ref{d10}), ${\Lambda}d\alpha$ is $d$-closed. Accordingly its Hodge decomposition takes the form \begin{equation} {\Lambda}d\alpha=h+dv\label{d13} \end{equation} where $h$ is a harmonic 1-form and $v$ is a $C^{\infty}$ function which is unique if we insist that $\int v\vol=0$. We claim that $h=0$. Indeed $$ ||h||^{2}=\langle h,{\Lambda}d\alpha+dv\rangle=\langle h,-\delta^{c}\alpha \rangle=-\langle d^{c}h,\alpha\rangle=0 $$ where we've used the K\"{a}hler identity (\ref{n7}) and the basic fact that on a K\"{a}hler manifold any $\Delta$-harmonic form is also $\Delta^{c}$-harmonic. Now if we compare (\ref{d13}) with (\ref{d14a}) and the definition of $u$ we see that $d\varphi=0$ iff $v=-{\textstyle \frac{1}{2}}~u$. To see that this is the case we use (\ref{d13}) to compute $$ \Delta(-2v) = -2\delta{\Lambda}d\alpha=2{\Lambda}d\delta\alpha=2(\rho, \alpha) = \Delta u $$ as required. In the above we have used the K\"{a}hler identities and equation (\ref{d9}). {\hfill \rule{.5em}{1em} \\} \begin{remark} Aside from the twistor-theoretic argument, the relevance of the operator $S$ to problem (b) can best be seen by first restating the problem as problem (b$'$). One then observes that $S$ is the linearization of the operator which sends a Hermitian metric $g$, with associated 2-form $\omega_g$, to $W_{+g}(\omega_g )\in {\cal E}(\wedge^+)$. From Boyer's calculations \cite{boyer} one then reads off the fact that the kernel of this non-linear operator is precisely the space of ASD Hermitian conformal classes, and, since $b_1(M)$ is even, these are all represented by unit-volume scalar-flat K\"ahler metrics. \end{remark} \noindent \begin{propn} The operator $S$ of (\ref{d6}) is elliptic with index equal to $-\tau(M)$, where $\tau(M)$ is the signature of $M$. \label{dt2a} \end{propn} \begin{proof} Both statements depend only on the top-order term $d^{+}\delta$ of $S$. Now the kernel of $d^{+}\delta$ is the space $H^-$ of ASD harmonic 2-forms and similarly the kernel of its adjoint $d^-\delta$ is the space $H^+$ of SD harmonic 2-forms. The proof that the symbol is an isomorphism $\wedge^-\rightarrow \wedge^+$ is left to the reader. \end{proof} We now complete our analysis of problem (b) by identifying the cokernel of $S$. \begin{propn} Suppose that $M$ is not Ricci-flat. The cokernel of $S$ can then be identified with the space of $C^{\infty}$ functions $f$ which satisfy the following conditions: \noindent$\!\!\!$\begin{tabular}{lp{5.9in}} (i) & the Lichnerowicz equation (\ref{vf2}): $~\Delta^{2}f=-2(dd^{c}f, \rho)$; \\ (ii) & the orthogonality conditions $\langle f\rho,\alpha\rangle=0$ for all ASD harmonic 2-forms $\alpha$.\label{crux} \end{tabular} \noindent In particular, if $M$ supports no non-parallel holomorphic vector fields, then coker$(S)=0$. \label{dt4} \end{propn} \noindent{\bf Proof}. By the Fredholm alternative for elliptic operators, the cokernel of $S$ can be identified with the kernel of the adjoint $S^{*}$. Now for any $\psi\in{\Lambda^{+}}(M)$ and $\alpha\in{\wedge^{-}}(M)$, $$ \begin{array}{lcl} \langle S^{*}\psi,\alpha\rangle & = & \langle\psi,S\alpha\rangle=\langle\psi, d^{+}\delta\alpha-{\textstyle \frac{1}{2}}~(\rho,\alpha)\omega\rangle \\ & = & \langle d^{-}\delta\psi,\alpha\rangle- {\textstyle \frac{1}{2}}~\langle(\psi,\omega) \rho,\alpha\rangle \end{array} $$ so \begin{equation} S^{*}\psi=d^{-}\delta\psi-{\textstyle \frac{1}{2}}~(\psi,\omega)\rho.\label{d14} \end{equation} To analyze the equation $S^{*}\psi=0$, we shall invoke (\ref{n13}) to write \begin{equation} \psi=f\omega+\chi\label{d15} \end{equation} (where $f$ is a $C^{\infty}$ function and $\chi$ lies in ${\wedge^{2,0}} \oplus{\wedge^{0,2}})$. We shall also need to write the operator $d^{-} \delta$ in terms of $d$ and $d^{c}$. This is an exercise involving the K\"{a}hler identities (\ref{n7}). Indeed, as an operator ${\wedge^{+}} \rightarrow{\wedge^{-}}$, \begin{eqnarray} d^{-}\delta & = & {\textstyle \frac{1}{2}}~(1-\star)d\delta \nonumber \\ & = & {\textstyle \frac{1}{2}}~d\delta- {\textstyle \frac{1}{2}}~\delta d \nonumber \\ & = & {\textstyle \frac{1}{2}}~d[{\Lambda},d^{c}]-{\textstyle \frac{1}{2}}~ [{\Lambda},d^{c}]d \nonumber \\ & = & {\textstyle \frac{1}{2}}~(d{\Lambda}d^{c}+d^{c}{\Lambda}d)+ {\textstyle \frac{1}{2}}~ ({\Lambda}dd^c-dd^{c}{\Lambda}). \label{d16} \end{eqnarray} Consider the second bracketed term in (\ref{d16}). For reasons of bidegree, it annihilates $\chi$ in (\ref{d15}). On the other hand \begin{eqnarray} d^{-}\delta(f\omega) & = & d^{-}\delta Lf \nonumber \\ & = & -d^{-}[L,\delta]f \nonumber \\ & = & -{\textstyle \frac{1}{2}} {}~(1-\star)dd^{c}f \nonumber \\ & = & -dd^{c}f-{\textstyle \frac{1}{2}} {}~\omega\Delta f \label{d17} \end{eqnarray} where we have used the K\"{a}hler identities, the relation (\ref{n14}) to identify the ASD part of $dd^{c}f$ with its projection perpendicular to $\omega$, and (\ref{n11}) to relate this to the Laplacian. Combining (\ref{d15}), (\ref{d16}) and (\ref{d17}), we obtain \begin{equation} \label{d18} S^{*}(f\omega+\chi)= d^-\delta \chi -dd^{c}f-{\textstyle \frac{1}{2}}~\omega\Delta f-\rho f \end{equation} where we can also write \begin{equation} d^{-}\delta\chi={\textstyle \frac{1}{2}}~(d{\Lambda}d^{c}+d^{c}{\Lambda}d)\chi. \label{d19} \end{equation} Suppose (\ref{d18}) vanishes. Applying $dd^{c}$ (and using (\ref{d19})) we find that $f$ satisfies condition (i) of the Theorem: $$ \begin{array}{lcl} 0 & = & -~{\textstyle \frac{1}{2}} {}~\omega{\wedge}dd^{c}\Delta f-\rho{\wedge}dd^cf \\ & = & -~{\textstyle \frac{1}{2}} {}~(\omega,dd^{c}\Delta f)\vol+(dd^{c}f, \rho)\vol \\ & = & \left({\textstyle \frac{1}{2}}~\Delta^{2}f+(dd^{c}f, \rho)\right)\vol \end{array} $$ (the change of sign in the term in $\rho$ arises because $\rho$ is ASD cf. (\ref{n18})). The orthogonality conditions (ii) are just the conditions that the equation (\ref{d18}) \begin{equation} d^{-}\delta\chi=dd^{c}f+{\textstyle \frac{1}{2}}~\omega\Delta f+\rho f \label{d20} \end{equation} be soluble for $\chi$. By the Fredholm alternative this equation is soluble iff the right-hand side is orthogonal to the kernel of the adjoint operator $d^{+}\delta$. But we have already identified this kernel in the Proof of Proposition \ref{dt2a} with the space $H^{-}$ of ASD harmonic 2-forms. Since the inner product of the first two terms on the right-hand side with any such form is zero we get condition (ii) of the Theorem. The proof is completed by noting that if $\chi$ satisfying (\ref{d20}) exists, it is unique. This is because the kernel of $d^{-}\delta$ is $H^{+}={\Bbb C} \omega$ by Theorem \ref{sd3}, and by definition $\chi$ is orthogonal to $\omega$. {\hfill \rule{.5em}{1em} \\} \vspace{12pt} \begin{defn} Let $M$ be a compact complex surface. If $\Xi$ is any holomorphic vector field on $M$, the {\em restricted Futaki invariant} of $(M, \Xi )$ is defined to be the map \begin{eqnarray*} \hat{\cal F}_{\Xi}: {\cal A}_M & \to & {\Bbb C} \\ ~~ [\omega ] & \mapsto & {\cal F}(\Xi , [\omega ]) ~ .\end{eqnarray*}\label{two} \end{defn} Here ${\cal A}_M:=\{ [\omega ]\in H^{1,1}~|~[\omega ]>0, c_1\cup [\omega ]=0\}$ again denotes the set of admissible K\"ahler classes. \begin{thm} Let $(M, \omega )$ be a compact scalar-flat K\"ahler surface, and let $Z$ be its twistor space. Assume that $M$ is not Ricci-flat. Then the cohomology groups $H^{2}(Z,{\Theta}_{Z}\otimes {\cal I}_D)$, $H^{2}(Z, {\Theta}_{Z,D})$, and $H^{2}(Z,{\Theta}_{Z})$ are all equal, and can be identified with the space of holomorphic vector fields $\Xi $ on $M$ such that $d\hat{\cal F}_{\Xi}|_{[\omega ]}=0.$ \label{dt5}\end{thm} \begin{proof} Let us first observe that $M$ cannot carry a non-zero parallel vector field. If it did, $g$ would locally be a Riemannian product of the flat metric and some other K\"ahler metric on ${\Bbb C}$; and since $s=0$, the second factor would also have to be flat. Thus $g$ would itself be flat, contradicting the assumption that $\rho \not\equiv 0$. Since $g$ has constant scalar curvature and there are now no parallel vector fields on $M$, we may therefore, by Proposition \ref{vf1}, write each holomorphic vector field $\Xi$ in the form $2(\bar{\partial} f)^{\sharp}$ for a unique $f$ satisfying the Lichnerowicz equation (\ref{vf2}) and $\int f\vol =0$. Now, in accordance with Definition \ref{two} above, the restricted Futaki invariant $\hat{\cal F}_{\Xi}$ is just the restriction of ${\cal F}(\Xi,\cdot )$, defined by (\ref{vf3}), to the admissible K\"ahler classes ${\cal A}_M\subset H^{1,1}$. The tangent space of ${\cal A}_M$ at $[\omega ]$ is just the $\cup$-orthogonal complement of $\rho$ in the harmonic (1,1)-forms; but since the Futaki invariant vanishes for all multiples of $[\omega ]$, we might as well restrict ourselves to admissible classes of {\em fixed volume}, which corresponds to cutting the tangent space down to the $L^2$-orthogonal complement of $\rho$ in the closed ASD 2-forms $H^{-}$. If $\Xi =2(\bar{\partial} f)^{\sharp}$, $\int f\vol =0$, then, by Proposition \ref{besser} we have $$\frac{d}{dt}{\cal F}(\Xi , [\omega + t\alpha ])= \langle f\rho,\alpha\rangle . $$ However, if $C$ is any constant and $\hat{f}:=f+C$, this becomes $$\frac{d}{dt}{\cal F}(\Xi , [\omega + t\alpha ]) =\langle\hat{f}\rho,\alpha\rangle-C\langle\rho,\alpha \rangle $$ so that the right-hand side is independent of the representative $\hat{f}$ provided $\langle \rho,\alpha\rangle =0$. Thus $$d\hat{\cal F}_{\Xi}(\alpha)=\langle f\rho,\alpha\rangle {}~~\forall \alpha\in H^-~s.t.~\langle \rho,\alpha\rangle =0$$ for any $f$ with $\Xi =2(\bar{\partial} f)^{\sharp}$, independent of any statement concerning $\int f\vol$. On the other hand, if $\Xi =2(\bar{\partial} f)^{\sharp}$ is a holomorphic vector field with $d\hat{\cal F}_{\Xi}|_{[\omega ]}=0$, there is exactly one $C$ for which $\hat{f}:=f+C$ satisfies $\langle\hat{f}\rho,\rho\rangle=0$, since $\langle\rho,\rho\rangle=-4\pi^{2}c_{1}^{2}>0$. This allows us to identify the space of holomorphic vector fields $\Xi$ satisfying $d\hat{\cal F}_{\Xi}|_{[\omega ]}=0$ with the space of solutions $f$ of the Lichnerowicz equation such that $\langle f\rho,\alpha\rangle=0$ for all $\alpha \in H^-$. Using Corollary \ref{dt2} and Theorem \ref{crux}, this in turn identifies $H^{2}(Z,{\Theta}_{Z}\otimes {\cal I}_D)$ with the space of holomorphic vector fields $\Xi$ on $M$ such that $d\hat{\cal F}_{\Xi}|_{[\omega ]}=0$, as promised. Now, using Serre duality, we observe that, since $M$ is ruled, $H^2(M, \Theta_M)\cong H^0 (M, \Omega^1 ({\kappa}_M ))=0$ because $\Omega^1 (\kappa )$ becomes $ {\cal O} (-2 )\oplus {\cal O} (-4 )$ when restricted to a smooth rational curve with trivial normal bundle. Since $[\omega ]$ has total scalar curvature 0, we also have $H^2(M, {\cal O} (\kappa^{-1}_M)) \cong H^0 (M, {\cal O} (\kappa^2_M))=0$ by Corollary \ref{yup}. The isomorphisms $$H^{2}(Z,{\Theta}_{Z}\otimes {\cal I}_D)\cong H^{2}(Z, {\Theta}_{Z,D}) \cong H^{2}(Z,{\Theta}_{Z})$$ now follow immediately from the short exact sequences (\ref{ks4}) and (\ref{ks5}), since $D$ consists of $M$, embedded in $Z$ with normal bundle $\kappa^{-1}_M$, together with the image of this surface via the anti-holomorphic map $\sigma$. \end{proof} \begin{example} Let $M={\Bbb CP}_1\times {\Sigma}_{\bf g}$ be the product of the Riemann sphere with a curve of genus ${\bf g}\geq 2$. Equip the factors with metrics of curvature $\pm 1$, and let $g$ be the product metric, which is a scalar-flat K\"ahler metric on $M$. Because there is only one admissible K\"ahler class on $M$ of a given volume, $\hat{\cal F} _{\Xi}\equiv 0$ for any holomorphic vector field $\Xi$. Since $a(M)={\bf sl}(2, {\Bbb C})$, we therefore have $$h^{2}(Z,{\Theta}_{Z}\otimes {\cal I}_D)= h^{2}(Z, {\Theta}_{Z,D})=h^{2}(Z,{\Theta}_{Z})=3~ .$$ On the other hand, if $(M, g)$ is instead the twisted version of the above example constructed on the ${\Bbb CP}_1$-bundle associated to any flat connection on a principal ${\bf SU}(2)$-bundle over ${\Sigma}_{\bf g}$, then, provided that the given flat connection is generic in the sense that its holonomy acts irreducibly on ${\bf su} (2)$, there are no non-trivial holomorphic vector fields on $M$, and $$h^{2}(Z,{\Theta}_{Z}\otimes {\cal I}_D)= h^{2}(Z, {\Theta}_{Z,D})=h^{2}(Z,{\Theta}_{Z})=0~ .$$ \end{example} \begin{remark} If $(M,g)$ is Ricci-flat, the story is utterly different from that described in Theorem \ref{dt5}. Instead, one may immediately read off from Corollary \ref{dt2} that $$r\equiv 0 \Rightarrow h^2 (Z, {\Theta}_{Z}\otimes {\cal I}_D)=b^+\neq 0.$$ The deformation techniques we are developing here are thus ill-suited to, say, a K3 surface. Instead, in this hyper-K\"ahler case, when there is more than one choice of parallel complex structure available, an unobstructed deformation theory can be obtained by considering deformations of $Z$ relative to a fibration over ${\Bbb CP}_1$. However, in light of the quite definitive theory of Ricci-flat K\"ahler metrics one obtains from Yau's solution of the Calabi conjecture \cite{yau2}, there is little reason to pursue this point of view. \end{remark} As our first application of this result, let $M$ be a compact K\"{a}hler surface with a fixed complex structure $J$ and $c^{2}_{1}<0$. Introduce ${\cal S}$, the moduli space of scalar-flat K\"{a}hler metrics modulo homothety, and ${\cal A}_M/{\Bbb R}^+$, the projectivized cone of K\"ahler classes which are $\cup$-orthogonal to $c_{1}$. There is a natural map $\mu:{\cal S}\rightarrow {\cal A}_M/{\Bbb R}^+$ induced by mapping a metric to its K\"{a}hler class. \noindent \begin{thm} Let $g$ be a scalar-flat K\"{a}hler metric on $M$. Assume that $g$ is not Ricci-flat. (i) If $d\hat{\cal F}_{\Xi}|_{[\omega_g ]}\neq 0$ for every non-zero holomorphic vector field $\Xi$ on $M$, the deformation theories for problems (b), (c), and (d) are all unobstructed. In particular, $g$ is a smooth point of the moduli space ${\cal S}$ of problem (b), and ${\cal S}$ has dimension $|\tau (M)|$ near $g$. (ii) If $M$ carries no non-trivial holomorphic vector fields, the moduli space ${\cal S}$ is smooth, and $\mu$ is a local diffeomorphism between ${\cal S}$ and ${\cal A}_M/{\Bbb R}^+$. The set of K\"ahler classes represented by scalar-flat metrics is therefore open in the space ${\cal A}_M$ of admissible K\"ahler classes. \label{dt6} \end{thm} \noindent{\bf Proof}. The first part is a consequence of the relevant Kodaira-Spencer Theorem \ref{ks3}(ii), Corollary \ref{dt2} and Theorem \ref{dt5}. The second part follows from Proposition \ref{dt1} and a simple count of dimensions: from its definition, ${\cal A}_M/{\Bbb R}^+$ is a manifold of dimension $b_{2}-2$ and this coincides with the dimension $|\tau(M)|$ by Yau's Theorem \ref{sd3}. {\hfill \rule{.5em}{1em} \\} \pagebreak \section{Ruled Surfaces} \setcounter{equation}{0} \subsection{Computing the Futaki Invariant} \label{foo} Let $(M, J)$ be a compact complex surface with $c_1^{\Bbb R}(M)\neq 0$, and suppose that $[\omega ]$ is a K\"ahler class on $M$ such that the total scalar curvature vanishes--- equivalently, such that $c_1\cup [\omega ] =0$. Then, by Theorem \ref{sd3}, $M$ must be a ruled surface, which is to say that $(M, J)$ is obtained from a projectivized rank-2 vector bundle ${\Bbb P}(E)\to \Sigma_{\bf g}$ over a compact complex curve $\Sigma_{\bf g}$ by blowing up $m= |\tau (M)|$ points. As our eventual goal is to study scalar-flat K\"ahler surfaces, we will only wish to consider surfaces $M$ with vanishing Matsushima-Lichnerowicz obstruction in the sense of Definition \ref{matlic}. The search is therefore considerably narrowed by the following result: \begin{propn} Let $M$ be a compact complex surface with an admissible K\"ahler class, vanishing Matsushima-Lichnerowicz obstruction, and non-trivial automorphism algebra $a(M)$. Suppose also that $M$ is not finitely covered by a a complex torus. Then, for some holomorphic line bundle ${\cal L}\to \Sigma_{\bf g}$ over a compact complex curve of genus ${\bf g}\geq 2$, $M$ is obtained from the minimal ruled surface ${\Bbb P}({\cal L}\oplus {\cal O} )\to \Sigma_{\bf g}$ by blowing up $|\tau (M)|$ points along the zero section of ${\cal L}\subset{\Bbb P}({\cal L}\oplus {\cal O} )$. Moreover, unless $M= {\Bbb CP}_1\times\Sigma_{\bf g}$, the space $a(M)$ of holomorphic vector fields is 1-dimensional, and is spanned by the Euler vector field of ${\cal L}$.\label{lem} \end{propn} \begin{proof} Theorem \ref{sd3} tells us immediately that $M$ is either ruled or covered by a K3 surface. The latter possibility, however, is excluded because $\Gamma ( {\bf K3}, \Theta )=0$. Since $c_1\cdot [\omega ]=0$, $c_1^{\Bbb R}$ is a non-zero primitive class in $H^{1,1}$, and $c_1^2 <0$. Thus, with respect to the complex orientation, $2\chi +3\tau < 0$. If the curve $\Sigma_{\bf g}$ has genus $<2$, we therefore have an estimate of the number of (-1)-curves contained in $M$. Specifically, if $M$ is obtained by blowing up a Hirzebruch surface ${\Bbb P}({\cal O}(k)\oplus {\cal O}) \to {\Bbb CP}_1$ at $\ell$ points, we must have $\ell\geq 9$; and if $M$ is instead obtained by blowing up a minimal ruled surface $\check{M}\to {\Bbb E}$ over an elliptic curve ${\Bbb E}={\Bbb C}/\Lambda$ at $\ell$ points, then $\ell>0$. Let $\Xi\neq 0$ denote a holomorphic vector field on $M$, and let $\pi : \check{M}\to \Sigma_{\bf g}$ denote a $ {\Bbb CP}_1$-bundle from which $M$ can be obtained by a blow-up $b: M\to \check{M}$. We then consider the component of $b_{\ast}\Xi$ normal to the fibers of $\pi$. Since the normal bundle of each such fiber is trivial, this normal component is constant up the fibers, so that $(\pi b)_{\ast}\Xi$ is a well-defined holomorphic vector field on $\Sigma_{\bf g}$. If $\Sigma_{\bf g}$ has genus $>1$, this vector field must vanish. If, on the other hand, $\Sigma_{\bf g}$ has genus 1, the fact that $b$ involves blowing up at at least one point forces $b_{\ast}\Xi$, and hence $(\pi b)_{\ast}\Xi$, to have at least one zero, implying that $(\pi b)_{\ast}\Xi\equiv 0$. Finally, if $\Sigma_{\bf g}$ has genus 0, and if $(\pi b)_{\ast}\Xi\not\equiv 0$, the $\ell\geq 9$ blown-up points of $\pi :F_k\to {\Bbb CP}_1$ must be located on at most 2 fibers of $\pi$; but then $F_k={\Bbb P}({\cal O} \oplus{\cal O} (k))$ admits sections of $\kappa^{-\ell}$ which vanishes along these two fibers to order $\ell$, and this section lifts to a non-zero element of $H^0(M, {\cal O} (\kappa^{-\ell}))$, contradicting Corollary \ref{yup}. The vector field $(\pi b)_{\ast}\Xi$ must therefore vanish identically, and $ b_{\ast}\Xi$ is tangent to the fibers of $\pi$. In short $a(M)$ consists strictly of {\em vertical} vector fields. By the Matsushima-Lichnerowicz assumption, the identity component of the automorphism group of $M$ is the complexification of a compact group. We therefore have a non-trivial holomorphic vector field $\Xi$ on $M$ whose imaginary part $\xi$ generates an $S^1$-action, and which itself generates a ${\Bbb C}^{\times}$-action; for brevity's sake, we shall henceforth refer to any such $\Xi$ as a {\em periodic holomorphic vector field}. On the other hand, the minimal model $\pi : \check{M}\to \Sigma_{\bf g}$ may be represented in the form ${\Bbb P}(E)\to \Sigma_{\bf g}$ for a rank 2 holomorphic vector bundle $E\to \Sigma_{\bf g}$ which is completely specified once an arbitrary line bundle $\wedge^2E$ is chosen, subject to the condition $c_1(E)\equiv w_2(\pi)\bmod 2$. The vector field $ b_{\ast}\Xi$ is then uniquely specified by a trace-free holomorphic section $A$ of ${\cal E}nd (E)$. The determinant of $A$ is a holomorphic function on $\Sigma_{\bf g}$, hence a constant. On the other hand, since $\Xi$ is periodic, $A$ is diagonalizable, and $A$ must be a half-integer multiple of $$\left[\begin{array}{cc}1&0\\0&-1 \end{array} \right]~ .$$ The vector bundle $E$ thus globally splits as a direct sum of the eigenspaces of $A$, and, twisting by a line bundle, we may therefore take $E={\cal L}\oplus {\cal O}$, so that $\Xi$ becomes a constant multiple of the Euler vector field on ${\cal L}$. We henceforth normalize this constant to be 1. The blown-up points must all occur at zeroes of $\Xi$, namely either at the zero section of ${\cal L}$ or at the ``infinity section'' corresponding to the ${\cal O}$ factor. The latter possibility may be reduced to the former by noticing that the proper transform of a fiber through exactly one blown-up point is a (-1)-curve, which may therefore be blown down, thereby leading to a different minimal model. In our case, iteration of this procedure allows us to replace blown-up points ``at the infinity section'' by blown-up points ``at the zero section,'' at the small price of twisting our line bundle $\cal L$ by the divisor of the relevant points of $\Sigma_{\bf g}$. The space of vertical vector fields on the minimal model $\check{M}$ is now precisely $\Gamma (\Sigma_{\bf g}, {\cal O} \oplus {\cal L}\oplus {\cal L}\* )$, with the Lie algebra structure induced by identifying $(u,v,w)\in\Gamma (\Sigma_{\bf g}, {\cal O} \oplus {\cal L}\oplus {\cal L}\* )$ with the matrix $$\left[\begin{array}{cc} u&v\\w&-u\end{array}\right]~ .$$ If $M=\check{M}$, this algebra is itself required to be reductive, implying $H^0(\Sigma_{\bf g}, {\cal L})\neq 0 \Leftrightarrow H^0(\Sigma_{\bf g}, {\cal L}\* )\neq 0$; we conclude that either ${\cal L}$ is trivial and $M={\Bbb CP}_1\times \Sigma_{\bf g}$, or else $a(M)$ is 1-dimensional. If, on the other hand, $M$ is obtained by blowing up points on the zero section of $\cal L$, the vector field $(u, v, w)$ lifts to $M$ iff $v=0$, and $a(M)=\{ (u,0,w )\}$; thus, if $\tau (M)\neq 0$, $a(M)$ is reductive iff $\Gamma (\Sigma_{\bf g},{\cal L}\* )=0$. Thus, provided that $M\neq {\Bbb CP}_1\times \Sigma_{\bf g}$, $a(M)$ is 1-dimensional, with the Euler vector fields $\Xi$ (corresponding to $(u,v,w)=(\frac{1}{2},0,0)$) as a basis. Finally, we observe that $M$ must have genus $\geq 2$. Indeed, the Euler field $\Xi$ is a vector field on $\check{M}$ which vanishes at all the points which are to be blown up. Let $\ell$ be the greatest multiplicity with which any point is to be blown up, and, assuming ${\bf g}=0,1$, let $\Upsilon\not\equiv 0$ be any vector field on $\Sigma_{\bf g}$. Then $(\Xi\wedge\Upsilon)^{\otimes \ell }$ lifts to $M$ as a non-trivial section of $\kappa^{-\ell }$, contradicting Proposition \ref{yup}. Hence ${\bf g}\geq 2$. \end{proof} Our goal is now to calculate the Futaki invariant of $(M, [\omega ], \Xi )$, where $M$ is in normal form described in the above Proposition, $[\omega ]$ is an admissible K\"ahler class and $\Xi$ is the Euler vector field. We proceed by a symplectic quotient construction in the spirit of \cite{L}. The invariant we seek to compute is known \cite{cal2} to be independent of the representative $\omega\in [\omega ]$, so we may assume (by averaging) that $\omega$ is invariant under the $S^1$-action generated by $\xi=\Im \Xi$. Since $0={\pounds}_{\xi } \omega = d (\xi\rfloor\omega )$, we see that $ \nu : =\omega ( \xi , \cdot )$ is closed; and, on the other hand, any real harmonic 1-form on our compact K\"ahler manifold is the real part of a holomorphic 1-form, and so must everywhere be orthogonal to $\xi =\Im \Xi $, as may either be seen directly from the our explicit description of $(M,\Xi )$, or deduced as a consequence of the maximum principle for pluriharmonic functions and the fact that $\Xi $ has zeroes. Harmonic theory therefore yields $\nu := (dd^{\ast}+d^{\ast}d)G\nu=dd^{\ast}G\nu$, where $G$ is the Green's operator, and $\Re\Xi := \mbox{grad} f$ for a unique function $f:= d^{\ast}G\nu$, called the holomorphy potential \cite{besse}\cite{lich} of $\Xi $, such that $\int_M f\vol=0$. As we saw in \S \ref{key}, the Futaki invariant ${\cal F}(\Xi , [\omega ])$ of $(M,[ \omega ])$ is then given by $${\cal F}(\Xi , [\omega ])= -{\textstyle \frac{1}{2}} \int_M fs\vol~ . $$ The symplectic vector field $\xi = \Im \Xi $ generating the $S^1$-action is now a globally Hamiltonian vector field, meaning that $$\omega ( \xi , \cdot ) = dt$$ for a smooth (``Hamiltonian'') function $t:M\to {\Bbb R}$; indeed, anything of the form of the form $t=f+c$ will do. We could, of course, choose our constant $c$ to vanish, but we will instead find it convenient to choose $c$ so that $\max t=-\min t=a$, and $t: M \raisebox{3pt [-a,a]$. (As we shall see in a moment, the intrinsic significance of the number $a$ is that $\int_F [\omega ]=4\pi a$, where $F$ is any fiber of $M\to \Sigma_{\bf g}$.) Fortunately, because we have assumed that $c_1\cup [\omega ]=0$, this will not interfere with our calculation of the Futaki invariant because \begin{eqnarray*}\int_Mts\vol&=&\int_M (f+c)s\vol\\&=& \int_M fs\vol+c\int_M s\vol\\&=& \int_M fs\vol\\&=& -2{\cal F}(\Xi , [\omega ])~ .\end{eqnarray*} The only isolated critical points of $t$ occur at those zeroes of $\Xi$ which occur at the intersection of an exceptional curve and the proper transform of a fiber; since such a fixed point is attractive along the exceptional curve and repulsive along the proper transform of a fiber, such a critical point has index 2. On the other hand, the maxima and minima of $t$ occur along a pair of holomorphic curves, $C_0=t^{-1}(-a)$ and $C_{\infty}=t^{-1}(a)$, which are just the proper transforms of the ``zero'' and ``infinity'' sections of $ {\Bbb P}({\cal O}\oplus {\cal L})\to \Sigma_{\bf g}$. We now have a projection $\pi : M\to \Sigma_{\bf g}\times [-a,a]$ given by the product of the ruling $M\to \Sigma_{\bf g}$ and the Hamiltonian $t$. Because the ${\Bbb C}^{\times}$-action preserves the ruling, every fiber of $\pi$ consists of exactly one orbit of the $S^1$-action. (This is really a consequence \cite{atiyah}\cite{mum} of the fact that $\Sigma_{\bf g}$ is both the symplectic and stable quotient of $M$ by the ${\Bbb C}^{\times}$-action.) Let $q_1 , \ldots , q_m\in \Sigma_{\bf g}\times (-a,a)$ be the images of the isolated fixed points of the action, and let $X := [\Sigma_{\bf g}\times (-a,a)]-\{ q_1 , \ldots , q_m\}$ denote the set of regular values of $\pi$. If $Y\subset M$ is the set of regular points, then $\pi : Y\to X$ is a principal $S^1$-bundle, and, by taking the orthogonal complement of the $S^1$ orbits with respect to the K\"ahler metric, we endow $Y\to X$ with a connection form $\theta$. If $z=x+iy$ is any complex local coordinate on $\Sigma_{\bf g}$, we may then express the given K\"ahler metric $g$ on $Y\subset M$ in the form\begin{equation} g=vw(dx^{\otimes 2}+dy^{\otimes 2})+w~dt^{\otimes 2}+w^{-1}\theta^{\otimes 2}~, \label{met}\end{equation} for positive functions $v, w>0$ on $X$, while the complex structure $J$ is given by \begin{eqnarray*} dx&\mapsto &dy\\frac{d}{dt}&\mapsto& w^{-1}\theta \end{eqnarray*} so that the K\"ahler form is given by $$\omega = dt\wedge \theta +vw~ dx\wedge dy~ .$$ Since the complex structure $J$ is integrable, the differential ideal $${\cal J} = \langle dx + i dy , w dt + i\theta\rangle$$ must satisfy $d{\cal J}\subset {\cal J}$; explicitly, this means that \begin{eqnarray*} d (w dt + i\theta)&=& dw\wedge dt + id\theta \\ &=&\varphi\wedge (dx +i dy)\end{eqnarray*} for some complex-valued 1-form $\varphi$ on $X$, and, because $\theta$ and $w$ are real, this is in turn equivalent to \begin{equation} d\theta \equiv w_x dy\wedge dt+ w_y dt\wedge dx \bmod dx\wedge dy~ .\label{ka2} \end{equation} The K\"ahler condition $d\omega=0$ now reads \begin{equation} 0=d(dt\wedge \theta +vw dx\wedge dy)= -dt\wedge d\theta + (vw)_t dt\wedge dx\wedge dy~, \label{ka1}\end{equation} so that the curvature of our $S^1$-connection $\theta$ is now completely determined by $v$ and $w$: $$d\theta = w_x dy\wedge dt+ w_y dt\wedge dx + (vw)_t dx\wedge dy~ .$$ In particular, we conclude that $w_{xx}+w_{yy}+(vw)_{tt}=0$. Notice that equation (\ref{met}) says that the metric on any fiber $F$ of $M\to \Sigma_{\bf g}$ is given by $$g|_F=wdt^2+ w^{-1}d\vartheta^2~,$$ where $\vartheta\in [0,2\pi ]$ is a fiber coordinate in a gauge chosen such that the connection form $\theta$ has no $dt$ component; the area form on $F$ is just\footnote{This generalizes to a simple relationship between volumes and moment maps for torus actions that is sometimes called the ``Archimedes Principle'' \cite{archie}\cite{atiyah}.} $$\omega |_F= dt\wedge d\vartheta ~,$$ and the area of $F$ is therefore $4\pi a$. On the other hand, since this metric is smooth at the ``south pole'' $t=-a$ of the 2-sphere $F$, letting $r$ denote the Riemannian distance from the south pole, we have $$w~dt^2+ w^{-1}~d\vartheta^2=dr^2+(r^2+O(r^4))~d\vartheta^2~,$$ so that $dt=r(1+O(r^2))~dr$, $t+a=\frac{r^2}{2}+O(r^4)$, and $w^{-1}=2(t+a)+O((t+a)^2)$. Similarly, $w^{-1}=-2(t-a)+O((t-a)^2)$ near $t=a$. Thus $\ell = w^{-1}$, which is a smooth function on $M$ because it represents the square of the length of the Killing field $\xi$, descends to a differentiable function on $\Sigma_{\bf g}\times [-a,a]$ which vanishes at the boundary and satisfies $\frac{d\ell }{dt}=\mp 2$ at $t=\pm a$. At the same time, equation (\ref{met}) tells us that $\lim_{t\to -a} vw~ dx\wedge dy= \omega |_{C_0}$, while $\lim_{t\to a} vw~ dx\wedge dy= \omega |_{C_{\infty}}$. Thus $v$ is smooth up to the boundary of $\Sigma_{\bf g}\times [-a,a]$, and moreover \begin{eqnarray*} \left. v\right|_{t=\pm a}&=&0 \\ \left. v_t dx\wedge dy\right|_{t=-a} &=& \left. \hphantom{-}2\omega \right|_{C_0} \\ \left. v_t dx\wedge dy\right|_{t=a\hphantom{-}} &=& \left. -2\omega \right|_{C_{\infty }}~ . \end{eqnarray*} Since $\Xi $ is a holomorphic vector field and the $(2,0)$-form $\mu := dz\wedge (w~dt + i\theta )$ has the property that $\Xi \rfloor \mu = 2 ~dz$ is a holomorphic form, $\mu$ must itself be holomorphic; thus $\frac{1}{4} \mu\wedge\overline{\mu }= w dx\wedge dy\wedge dt \wedge \theta$ is the volume form of a holomorphic frame. On the other hand, the metric volume form is $${\textstyle \frac{1}{2}}\omega\wedge\omega=vw~ dx\wedge dy\wedge dt\wedge \theta~, $$ so that the Ricci form of $g$ must be $$\rho = -i\partial \overline{\partial }\log \left( \frac{vw~ dx\wedge dy\wedge dt\wedge \theta}{w~ dx\wedge dy\wedge dt\wedge \theta} \right)= -i\partial \overline{\partial }\log v~ .$$ The scalar curvature density of $g$ thus is given in terms of $u:=\log v$ by \begin{eqnarray*} s\vol&=& {\textstyle \frac{1}{2}} s~\omega\wedge\omega\\ &=&2 \omega\wedge\rho\\ &=& -2i\omega\wedge \partial \overline{\partial }u \\ &=& -2i\omega\wedge [-{\textstyle\frac{i}{2}} dJd u ]\\ &=& -\omega\wedge dJd u \\ &=& -\omega\wedge d[u_xdy- u_ydx+u_tw^{-1}\theta ]\\ &=& -[dt\wedge\theta + vw~dx\wedge dy ]\wedge d[u_xdy- u_ydx+u_tw^{-1}\theta ]\\ &=& [u_{xx}+ u_{yy}+ vw(w^{-1}u_t)_t]~dx\wedge dy\wedge dt \wedge\theta + (u )_tw^{-1}d\theta \wedge dt \wedge \theta \\ &=& [u_{xx}+ u_{yy}+ vw(w^{-1}u_t)_t +w^{-1}u_t (vw)_t ]~dx\wedge dy\wedge dt \wedge\theta \\ &=& [u_{xx}+ u_{yy}+ (vu_t)_t ]~dx\wedge dy\wedge dt \wedge\theta \\ &=& [( \log v)_{xx}+ ( \log v)_{yy}+ v_{tt}]~dx\wedge dy\wedge dt \wedge\theta {}~ . \end{eqnarray*} In particular, \begin{equation} s= \frac{( \log v)_{xx}+ ( \log v)_{yy}+ v_{tt}}{vw} ~ . \label{scal} \end{equation} Let us now rephrase the above results in more global terms. Because $g$, $w$ and $dt$ are globally defined, it follows from equation (\ref{met}) that, for $-a< t < a$, $$g^v(t):=v~(dx^2+dy^2)$$ is a well-defined $t$-dependent K\"ahler metric on $\Sigma_{\bf g}$, with K\"ahler form $$\omega^v (t):= v~dx\wedge dy~ .$$ Moreover, \begin{eqnarray*} \left.\omega^v\right|_{t=\pm a}&=&0 \\ \left.\frac{d }{d t} \omega^v \right|_{t=-a} &=& \left.\hphantom{-}2\omega \right|_{C_0} \\ \left.\frac{d }{d t} \omega^v \right|_{t=a\hphantom{-}} &=& \left. - 2\omega \right|_{C_{\infty }}~ . \end{eqnarray*} The Ricci form of this metric is \begin{eqnarray*} \rho^v(t)&=& -i(\frac{\partial^2}{\partial z\partial \overline{z}} \log v )dz\wedge d\overline{z}\\ &=& -{\textstyle \frac{1}{2}}[(\log v )_{xx}+ (\log v )_{yy}]~dx\wedge dy ~, \end{eqnarray*} so that our formula for the density of scalar curvature may be written globally on $Y\subset M$ as $$ s\vol = [-2\rho^v + \frac{d^2}{dt^2} \omega^v]\wedge dt\wedge \theta~ .$$ Hence \begin{eqnarray*} \int_M ts\vol &=& \int_Y ts\vol \\&=& \int_Y t[-2\rho^v + \frac{d^2}{dt^2} \omega^v]\wedge dt\wedge \theta \\&=& 2\pi \int_{-a}^at[\int_{\Sigma_{\bf g}} -2\rho^v + \frac{d^2}{dt^2} \omega^v]~dt \\&=& 2\pi \int_{-a}^at\frac{d^2}{dt^2}[\int_{\Sigma_{\bf g}}\omega^v]~dt {}~~+2\pi \int_{-a}^at [\int_{\Sigma_{\bf g}} -2\rho^v ]~dt \\&=& 2\pi \int_{-a}^at\frac{d^2}{dt^2}[\int_{\Sigma_{\bf g}}\omega^v]~dt -8\pi^2 \chi (\Sigma_{\bf g})\int_{-a}^at ~dt \\&=& 2\pi \left[t\frac{d}{dt}\int_{\Sigma_{\bf g}}\omega^v\right]^a_{-a}- 2\pi \int_{-a}^a\frac{d}{dt}[\int_{\Sigma_{\bf g}}\omega^v]~dt \\&=& 2\pi \left[t\frac{d}{dt}\int_{\Sigma_{\bf g}}\omega^v\right]^a_{-a}- 2\pi \left[\int_{\Sigma_{\bf g}}\omega^v\right]^a_{-a} \\&=& 2\pi \left[t\frac{d}{dt}\int_{\Sigma_{\bf g}}\omega^v\right]^a_{-a} \\&=& 2\pi a\left[\int_{\Sigma_{\bf g}} \left.\frac{d}{dt}\omega^v\right|_{t=a} + \int_{\Sigma_{\bf g}} \left.\frac{d}{dt}\omega^v\right|_{t=-a}\right] \\&=& 2\pi a\left[\int_{C_0}2\omega - \int_{C_{\infty}}2\omega \right] = 4\pi a\left[\int_{C_0}\omega - \int_{C_{\infty}}\omega \right] \\&=& \left[\int_{C_0}\omega - \int_{C_{\infty}}\omega \right]\int_F\omega \end{eqnarray*} In conclusion, we have \begin{eqnarray*} {\cal F}(\Xi , [\omega ])&=& -{\textstyle \frac{1}{2}}\int_M fs\vol \\&=& -{\textstyle \frac{1}{2}}\int_M ts\vol\\&=& -{\textstyle \frac{1}{2}}\left[\int_{C_0}\omega - \int_{C_{\infty}} \omega\right]\int_F\omega \\&=& {\textstyle \frac{1}{2}}\left[\int_{C_{\infty}} \omega -\int_{C_0}\omega\right]\int_F\omega \end{eqnarray*} where $F$ is any fiber of $M\to \Sigma_{\bf g}$. We have thus proved the following: \begin{thm} Let $M$ be any compact complex surface equipped with an admissible K\"ahler class $[\omega ]$ and a holomorphic ${\Bbb C}^{\times}$-action which is free on an open dense set. Assume that $c_1^{\Bbb R}(M)\neq 0$, and let $\Xi \in \Gamma (M, {\cal O} (TM))$ denote the holomorphic vector field which generates the action. Thus $M$ is a ruled surface $M\to \Sigma_{\bf g}$ of genus ${\bf g}\geq 2$, the generic fiber $F$ of which is the closure of an orbit, while the ``attractive'' and ``repulsive'' fixed curves $C_0$ and $C_{\infty}$ of the action are sections of the projection $M\to \Sigma_{\bf g}$. The Futaki invariant of $(M, J, [\omega ], \Xi )$ is then given by $${\cal F}(\Xi , [\omega ])= {\textstyle \frac{1}{2}}\left[\int_{C_{\infty}} \omega -\int_{C_0}\omega\right]\int_F\omega ~ .$$ \label{local} \end{thm} \hfill \rule{.5em}{1em} \begin{remark} We have computed the Futaki invariant only for a single vector field $\Xi$. However, if the Matsushima-Lichnerowicz obstruction vanishes, Proposition \ref{lem} tells us that either $a(M)$ is spanned by $\Xi$ or else $M={\Bbb CP}_1\times \Sigma_{\bf g}$. In the former case, the ${\Bbb C}$-linearity of ${\cal F}(\cdot, [\omega ])$ tells us that our computation completely determines the ${\cal F}(\cdot, [\omega ])$. In the exceptional case $M={\Bbb CP}_1\times \Sigma_{\bf g}$, the Futaki character vanishes, since the product of two constant curvature metrics has constant scalar curvature. \end{remark} Use of the fact that $c_1\cdot [\omega ]=0$ allows one to rewrite the Futaki invariant in interesting equivalent ways. As previously indicated, we will always normalize the minimal model of our ruled surface with holomorphic vector field by putting it in the form ${\Bbb P}({\cal L}\oplus {\cal O})\to \Sigma_{\bf g}$, where all the blown-up points {\em are on the zero section of} ${\cal L}\hookrightarrow {\Bbb P}({\cal L}\oplus {\cal O}): \zeta\mapsto [\zeta ,1]$, and so correspond to fibers of ${\cal O}\subset {\cal L}\oplus {\cal O}$. For simplicity, let us assume for the moment that the blown-up points in ${\Bbb P}({\cal L}\oplus {\cal O})$ are all distinct, and so give rise to $m=|\tau (M)|$ exceptional rational curves $E_j$ of self-intersection $-1$. Let us associate $m$ ``weights'' $w_j>0$, $j=1, \ldots , m$, to the K\"ahler class $[\omega ]$ by defining $$w_j:=\frac{\int_{E_j}\omega }{\int_{F}\omega }~ .$$ The homology classes of $C_{\infty}, F, E_1, \ldots , E_m$ form a basis for $H_2 (M)$, and the intersection form of $M$ is $$ \left[\begin {array}{ccccc} -k&1&&&\\1&0&&\\&&-1&&\\&&&\ddots&\\&&&&-1 \end{array}\right]$$ with respect to this basis, where $k:=\mbox{deg}({\cal L})$; it follows that the Poincar\'e dual of $[\omega ]$ can be expressed in this basis as $A (1, B+k, -w_1, \ldots , -w_m)$, where $A=\int_F\omega $ and $AB=\int_{C_{\infty }}\omega$. Since $C_{\infty }$ has genus $\bf g$ and self-intersection $-k$, whereas $F$, $E_1, \ldots , E_m$ have genus 0 and self-intersection $0, -1, \ldots , -1$, the adjunction formula may be used to rewrite the condition $c_1\cdot [\omega ]=0$ as $$0=[2(1-{\bf g})-k, 2, 1, \ldots , 1] \left[ \begin{array}{c}1\{\cal B}+k\\ -w_1\\ \vdots \\-w_m\end{array}\right] =2(1-{\bf g})+k+2B-\sum_{j=1}^mw_j~,$$ so that $B=[-k+2({\bf g}-1)+\sum_{j=1}^mw_j ]/2$, and $$\int_{C_{\infty }}\omega = \frac{A}{2}[-k+2({\bf g}-1)+\sum_{j=1}^mw_j ]~ .$$ On the other hand, the picture is symmetrical between ${C_{\infty }}$ and $C_0$ as long as we remember to replace $k=-C_{\infty }\cdot C_{\infty }$ with $m-k =-C_{0}\cdot C_{0}$ and replace the $E_j$ with new exceptional curves $\hat{E}_j$ such that $[E_j]+[\hat{E}_j]=[F]$, resulting in a replacement of the weights $w_j$ by new weights $1-w_j$; the upshot is that \begin{eqnarray*}\int_{C_{0 }}\omega &=& \frac{A}{2}[k-m+2({\bf g}-1)+\sum_{j=1}^m(1-w_j) ] \\&=&\frac{A}{2}[k+2({\bf g}-1)-\sum_{j=1}^mw_j ]~ .\end{eqnarray*} The Futaki invariant can therefore be rewritten as \begin{eqnarray*} {\cal F}(\Xi , [\omega ])&=& {\textstyle \frac{1}{2}}\left[\int_{C_{\infty}} \omega -\int_{C_0}\omega\right]\int_F\omega \\&=& -\frac{A^2}{4}\left[ (k+2({\bf g}-1)-\sum_{j=1}^mw_j)- (-k+2({\bf g}-1)+\sum_{j=1}^mw_j)\right]\\&=& -\frac{A^2}{2} [k-\sum_{j=1}^mw_j]\\&=& -\frac{A^2}{2} [\mbox{deg}({\cal L})-\sum_{j=1}^mw_j]~ . \end{eqnarray*} We therefore have the following : \begin{propn} Let $M$ be obtained from the minimal model \linebreak ${\Bbb P}({\cal L}\oplus {\cal O})\to \Sigma_{\bf g}$, ${\bf g}\geq 2$, by blowing up $m$ points on the zero section of $\cal L$. Let $[\omega ]$ be a K\"ahler class on $M$ such that $c_1\cup [\omega ]=0$, and let the weights $w_j\in ~]0,1[$ be defined by $\int_{E_j}\omega =w_j\int_F\omega$, where $F$ is a typical fiber of $M\to \Sigma_{\bf g}$ and the $E_j$, $j=1,\ldots ,m$ are the exceptional curves corresponding to the blown-up points. Let $\Xi $ denote the vector field on $M$ corresponding to the Euler field on $\cal L$. Then $${\cal F}(\Xi , [\omega ])=0~~\Longleftrightarrow {}~~\sum_{j=1}^mw_j=\mbox{deg}({\cal L})~ .$$\label{foot} \end{propn} \begin{proof} We assumed for simplicity in the previous discussion that $M$ is obtained from its minimal model by blowing up {\em distinct} points. However, the argument goes through without change provided that, in defining the weights $w_j$, one replaces the integrals of $\omega$ over the exceptional divisors $E_j$ with integrals of $\omega$ over the corresponding homology classes, each of which can be represented by a chain of rational curves with intersection matrix $$\left[ \begin{array}{cccc} -2\hphantom{-}&1&&\\1&\ddots&1&\\ &1&-2\hphantom{-}&1\\&&1&-1\hphantom{-}\end{array}\right]~ .$$ \end{proof} \begin{cor} Let $(M, \Xi)$ be as in Proposition \ref{foot}. Then $M$ carries an admissible K\"ahler class $[\omega ]$ for which ${\cal F}( \Xi , [\omega ])=0$ iff one of the following holds: \begin{description} \item{(a)} $0=\mbox{deg}({\cal L})=m$; or \item{(b)} $0<\mbox{deg}({\cal L})<m$. \end{description}\label{tri} \end{cor} \begin{proof} The necessity of these conditions is an immediate consequence of Proposition \ref{foot}. For sufficiency, one may either invoke Proposition \ref{nakai}, or else wait for the explicit construction in \S \ref{next} below. \end{proof} \begin{remark} Notice that $m=1$ is excluded, since $(b)\Rightarrow m\geq 2$. Also notice that the vanishing of the Futaki invariant implies the vanishing of the Matsushima-Lichnerowicz obstruction. Indeed, if $m=0$ we either have $\cal L$ is trivial, and $M={\Bbb CP}_1\times \Sigma_{\bf g}$, or else $\Gamma (\Sigma_{\bf g}, {\cal L})= \Gamma (\Sigma_{\bf g}, {\cal L}\* )=0$, so that $a(M)$ is generated by the Euler field $\Xi$; if $m>0$, $\Gamma (\Sigma_{\bf g}, {\cal L}\* )=0$ and again $a(M)$ is generated by the Euler field $\Xi$. \end{remark} \begin{cor} Suppose that $(M, [\omega ])$ is a compact complex surface with admissible K\"ahler class and vanishing Matsushima-Lichnerowicz obstruction. Let $\Xi\not\equiv 0$ be any non-trivial holomorphic vector field, and consider the restricted Futaki functional $\hat{\cal F}_{\Xi}$ of Definition \ref{two}. Then $$d\hat{\cal F}_{\Xi}|_{[\omega ]}=0 ~\Longleftrightarrow ~ \tau (M)=0.$$ \label{van} \end{cor} \begin{proof} By Proposition \ref{lem}, the only non-ruled surfaces we need consider are tori and their (hyper-elliptic) quotients, for which both $\tau$ and $\cal F$ vanish. For the ruled surfaces, the result follows immediately from Theorem \ref{local}. \end{proof} Finally, as an aside, let us observe that the condition ${\cal F}(\Xi , [\omega ])=0$ can now be restated in terms of parabolic stability in the sense of Seshadri \cite{sesh}\cite{mehta}. We consider the vector bundle ${\cal V}={\cal L}\oplus {\cal O}$ of our minimal model, equipped with 1-dimensional subspaces $L_j$ in some fibers of ${\cal V}$ which represent the exceptional divisors $E_1, \ldots , E_m\subset M$. (Thus the subspaces $L_j$ are contained the ${\cal O}$ factor of ${\cal L}\oplus {\cal O}$.) Let $w_1, \ldots , w_m$ denote the weights as before, and let $\alpha_j < \beta_j$ be arbitrary numbers in $[0,1]$ such that $w_j=\beta_j- \alpha_j$. The criterion $$\sum_{j=1}^mw_j=\mbox{deg}({\cal L})~,$$ is precisely equivalent to the statement that $({\cal V}, \{( L_j,\alpha_j, \beta_j)\})$ is parabolically quasi-stable\footnote{ With this choice of terminology, stable $\Rightarrow$ quasi-stable $\Rightarrow$ semi-stable.}, in the sense that, for every line sub-bundle $L\subset {\cal V}$, we have $$\mbox{pardeg}(L)\leq \frac{1}{2}\mbox{pardeg}({\cal V}),$$ with equality iff $L$ is a {\em direct summand} of ${\cal V}$; here the {\em parabolic degree} of a line sub-bundle $L\subset {\cal V}$ is defined to be $$\mbox{pardeg}(L):=\mbox{deg}(L)+\sum_{\{j|L_j\not\subset L\}}\alpha_j +\sum_{\{j|L_j\subset L\}}\beta_j~,$$ whereas $$\mbox{pardeg}({\cal V}):=\mbox{deg}({\cal V})+\sum_{j=0}^m\alpha_j +\sum_{j=0}^m\beta_j~ .$$ \begin{cor} The vanishing of the Futaki invariant for a non-trivial ${\Bbb C}^{\times }$-action on a ruled surface with admissible K\"ahler class is equivalent to the quasi-stability of the the corresponding parabolic bundle. \end{cor} \subsection{Scalar-Flat Ruled Surfaces with Vector Fields} \label{next} In the last section, we analyzed the Futaki invariant of compact complex surfaces with periodic holomorphic vector fields which admit admissible K\"ahler classes. (Recall that {\em admissible} means that the total scalar curvature of a metric in the class vanishes.) Since the Matsushima-Lichnerowicz Theorem and the Futaki invariant are obstructions to the existence of constant scalar curvature K\"ahler metrics in the given class, this gives us a rough classification of those surfaces with holomorphic vector fields which might admit scalar flat K\"ahler metrics. In this section we will review a construction \cite{L2} \cite{L5} of compact scalar-flat K\"ahler surfaces, and use it to observe that this ``rough'' classification is in fact perfectly sharp. The idea is to reverse the symplectic quotient construction of the last section. Let $\Sigma_{\bf g}$ be any compact complex curve of genus $\geq 2$, and let $h_{\Sigma}$ be the unique Hermitian metric on $\Sigma_{\bf g}$ of constant curvature $-1$. We can then give the 3-manifold $\Sigma_{\bf g}\times (-1,1)$ a hyperbolic structure by introducing the constant curvature $-1$ metric $$h:=\frac{h_{\Sigma}}{(1-t^2)}+\frac{dt^2}{(1-t^2)^2}~ .$$ Let $q_1, \ldots , q_m\in \Sigma_{\bf g}\times (-1,1)$ be arbitrary points, and associate to each the Green's function $G_j$, defined by $$\Delta G_j=2\pi \delta_{q_j}, ~~~\lim_{t\to \pm 1} G_j = 0 ~,$$ where $\Delta =-\star d\star d$ is the (positive) Laplace-Beltrami operator of $h$. We define $V:=1+\sum_{j=1}^mG_j$, so that $$\Delta V=2\pi \sum_{j=1}^m\delta_{q_j}, ~~~\lim_{t\to \pm 1} V=1 ~ .$$ $V$ extends smoothly to $(\Sigma_{\bf g}\times [-1,1])-\{ q_1, \ldots , q_m\}$ and satisfies $V\geq 1$. On $[\Sigma_{\bf g}\times (-1,1)]-\{ q_1, \ldots , q_m\}$, the 2-form $$\alpha := \frac{1}{2\pi }\star dV$$ is now closed, and its integral on a small sphere around any one of the $q_j$'s is $-1$. On the other hand, if $1-\epsilon > \max_{j} t(q_j)$, then one may check \cite{L2} that \begin{equation}\int_{\Sigma_{\bf g}\times \{ 1-\epsilon\}}\alpha = -\sum_{j=1}^m\left( \frac{1+t(q_j )}{2}\right) ~,\label{int}\end{equation} so that, setting $w_j:=[1+t( q_j)]/2$, we have \begin{equation} [\frac{1}{2\pi }\star dV]\in H^2_d \left( [\Sigma_{\bf g}\times (-1,1)]-\{ q_1, \ldots , q_m\}, {\Bbb Z} \right) ~~\Longleftrightarrow ~~\sum_{j=1}^mw_j\in {\Bbb Z}~, \label{foutre}\end{equation} since the second homology of $[\Sigma_{\bf g}\times (-1,1)]-\{ q_1, \ldots , q_m\}$ is generated by the homology classes of $\Sigma_{\bf g}\times \{ 1-\epsilon\}$ and $m$ small spheres centered at the punctures $q_1, \ldots , q_m$. If we assume this condition is met, the Chern-Weil theorem guarantees that we can then find a principal $S^1$-bundle $$\pi_0: M_0\to [\Sigma_{\bf g}\times (-1,1)]-\{ q_1, \ldots , q_m\}$$ with a connection 1-form $\theta$ for which the curvature is $$d\theta = \star dV~ .$$ Notice that, even modulo gauge equivalence, the pair $(M_0, \theta )$ is by no means unique, since our base is not simply connected; instead, the group $H^1( \Sigma_{\bf g}, S^1)$ of flat circle bundles on $\Sigma_{\bf g}$ acts freely and transitively on the orbits by tensor product. Given a choice of $(M_0, \theta )$ we then equip $M_0$ with the Riemannian metric $$g:= (1-t^2)[Vh+V^{-1}\theta^2]~ . $$ If we identify the universal cover of $\Sigma_{\bf g}$ with the upper half-plane $y=\Im z >0$ in $\Bbb C$, the metric can be more explicitly written in the form \begin{eqnarray*} g &=& (1-t^2)\left[ V \frac{dx^2+dy^2}{y^2(1-t^2)}+V\frac{dt^2}{(1-t^2)^2} +V^{-1}\theta^2\right] \\ &=&vw~(dx^2+dy^2)+w~dt^2+w^{-1}\theta^2, \end{eqnarray*} where $$w=\frac{V}{1-t^2}$$ and $$v=\frac{1-t^2}{y^2}~ .$$ Since the equation $d\theta =\star dV$ can now be rewritten as $$d\theta =w_x~dy\wedge dt+w_y~dt\wedge dx+(vw)_t~dx\wedge dy~,$$ our calculations (\ref{ka2}) and (\ref{ka1}) show that $g$ is K\"ahler with respect to the integrable complex structure \begin{eqnarray*} dx&\mapsto &dy\\frac{d}{dt}&\mapsto& w^{-1}\theta \end{eqnarray*} Moreover, since $$(\log v)_{xx}+(\log v)_{yy}+v_{tt}=0~,$$ we conclude from equation (\ref{scal}) that $g$ is scalar-flat. We can now compactify $M_0$ by adding two copies of $\Sigma_{\bf g}$, corresponding to $t=\pm 1$, and $m$ isolated points, corresponding to $q_1, \ldots , q_m$. This compactification $M$ can then \cite{L2} be made into a smooth manifold in such a way that the metric $g$ and the complex structure $J$ extend to $M$, giving us a compact scalar-flat K\"ahler surface $(M,g)$. The bundle projection $\pi_0$ now extends to a smooth map $$\pi: M \to \Sigma_{\bf g}\times [-1,1]~,$$ and the original $S^1$-action extends to an action on $M$ for which $\pi$ is projection to the orbit space; the points added to $M_0$ in order to obtain $M$ are precisely the fixed points of the action. The tautological projection $\mbox{pr}_1\pi: M\to \Sigma_{\bf g}$ induced by $\pi$ and the first-factor projection $\mbox{pr}_1 :\Sigma_{\bf g}\times [-1,1]\to \Sigma_{\bf g}$ is now holomorphic, with rational curves as fibers. To get a minimal model for $M$, we can proceed by observing that, for any ``puncture point'' $q_j$, the inverse image $\pi^{-1}(\{ \mbox{pr}_1(q_j)\}\times [-1, t(q_j)])$ of the vertical line segment joining the lower boundary of $\Sigma_{\bf g}\times [-1,1]$ to $q_j$ is a rational curve in $M$, and, provided the segment does not pass through any other puncture point, this rational curve is smooth, with self-intersection $-1$. (In the non-generic situation in which several of the puncture points project to the same point of $\Sigma_{\bf g}$, the line segment between any two such consecutive points similarly corresponds to a smooth rational curve in $M$ of self-intersection $-2$.) By blowing down all such $(-1)$-curves (and iteratively blowing down the $(-1)$-curves that then arise from $(-2)$-curves in the non-generic case) we eventually arrive at a minimal model ${\Bbb P}({\cal L}\oplus {\cal O})\to \Sigma_{\bf g}$ with holomorphic vector field, where all the blow-ups occur at the the zero section of ${\cal L}$. The proper transform $C_0$ of the zero section now corresponds to $t=-1$, whereas the infinity section $C_{\infty }$ corresponds to $t=+1$. Meanwhile, the line bundle ${\cal L}^{\ast}$ is exactly the holomorphic line bundle associated to the $U(1)$-connection obtained by restricting $(M_0, \theta)$ to $\Sigma_{\bf g} \times \{ 1-\epsilon\}$, so that equation (\ref{int}) yields \begin{equation} \mbox{deg} ({\cal L})=\sum_{j=1}^mw_j~ .\label{deg}\end{equation} But a different choice of $M_0$ would change this $U(1)$-connection by twisting it by an arbitrary flat $U(1)$-connection; since $\mbox{Pic}_0(\Sigma_{\bf g})=H^1(\Sigma_{\bf g}, {\cal O})/H^1(\Sigma_{\bf g}, {\Bbb Z})$ is canonically identified with $H^1(\Sigma_{\bf g}, S^1)= H^1(\Sigma_{\bf g}, {\Bbb R})/H^1(\Sigma_{\bf g}, {\Bbb Z})$ by the Hodge decomposition, this means that the holomorphic line-bundles which arise for a fixed configuration $q_1, \ldots , q_m$ fill out the entire connected component of $\mbox{Pic}(\Sigma_{\bf g})$ specified by the degree formula (\ref{deg}). And since the fiber-wise K\"ahler form is just\footnote{This is again a manifestation of the ``Archimedes Principle'' \cite{archie}\cite{atiyah} for symplectic torus actions.} $$\left.\omega \right|_{ fiber}= dt\wedge\theta ~,$$ the area of the holomorphic curve $E_j=\pi^{-1}(\{ \mbox{pr}_1 (q_j)\}\times [-1, t(q_j)])$ is just $$\int_{E_j}\omega = 2\pi (t(q_j)-(-1))= 4\pi w_j~ .$$ Since, by the same reasoning, the typical fiber $F$ of $M\to \Sigma_{\bf g}$ has area $4\pi$, the numbers $w_j$ are precisely the weights we associated with the exceptional divisors in \S \ref{foo}, and equation (\ref{deg}) is therefore, by Proposition \ref{foot}, just the assertion that the Futaki invariant vanishes--- as of course it must, since our K\"ahler manifold has constant scalar curvature zero! Since we are free to choose the numbers $w_j$, subject only to the constraint (\ref{foutre}), and since, by multiplying $g$ by an arbitrary constant, we can make the typical fiber $F$ have any area we choose, the above explicit construction produces a scalar-flat K\"ahler metric in any K\"ahler class on $M$ such that both $c_1\cdot [\omega ]$ and the Futaki invariant are zero: \begin{thm} Let $M$ be a compact complex surface with $a(M)\neq 0$. Then a K\"ahler class $[\omega ]\in H^{1,1}(M)$ contains a scalar-flat K\"ahler metric iff the total scalar curvature, the Matsushima-Lichnerowicz obstruction, and the Futaki invariant all vanish. When such a metric exists, it is unique modulo biholomorphisms of $M$. \label{class} \end{thm} \begin{proof} For the existence part, it remains only to observe that the only non-ruled cases are tori and hyperelliptic surfaces, and these admit flat metrics in every K\"ahler class. For the uniqueness result, which we shall never use in this article, we refer the reader to \cite{L5}, Theorem 3. \end{proof} \noindent The following simple application will prove to be particularly useful: \begin{cor} Let the product surface $\Sigma_{\bf g}\times {\Bbb CP}_1$ be blown up at any $k>0$ points along $\Sigma_{\bf g}\times\{ [1:0]\}$ and any $\ell > 0$ points along $\Sigma_{\bf g}\times\{ [0:1]\}$. (Some or all of the given points are allowed to coincide, but in this case the iterated blow-ups are required to occur along proper transforms of the $\Sigma_{\bf g}$ or ${\Bbb CP}_1$ factors.) The resulting surface then admits scalar-flat K\"ahler metrics. \label{triv} \end{cor} \begin{proof} If the set of blown-up points is $$\{ (p_1, [1:0]), \ldots , (p_k, [1:0]), (q_1, [0:1]), \ldots , (q_{\ell}, [0:1])\}~,$$ then, letting ${\cal L}\to \Sigma_{\bf g}$ denote the divisor line bundle of $\{ q_1, \ldots , q_{\ell}\}$, the surface in question can also be described as the blow-up of ${\Bbb P}({\cal O} \oplus {\cal L})$ at the points $\{ p_1, \ldots , p_k,q_1, \ldots , q_{\ell}\}$ on the zero section. Since $0<\mbox{deg}({\cal L})=\ell<m=\ell +k$, the result follows from Corollary \ref{tri} and Theorem \ref{class}. \end{proof} \noindent In light of \cite{burnsbart}, the following restatement of the above theorem seems particularly tantalizing: \begin{cor} An admissible K\"ahler class on a (blown-up) ruled surface with periodic holomorphic vector field contains a scalar-flat K\"ahler metric iff the corresponding parabolic bundle is quasi-stable. \label{corn} \end{cor} \subsection{Scalar-Flat Metrics on Generic Ruled Surfaces }\label{tag} The key technical result of this paper is as follows: \begin{thm} Let $\varpi : {\cal M}\to {\cal U}$ be a family of non-minimal ruled surfaces of genus $\geq 2$. Suppose that, for some $o\in {\cal U}$, the corresponding fiber $M=M_o:= \varpi^{-1}(o)$ admits a scalar-flat K\"ahler metric. Then there is a neighborhood $\tilde{\cal U}\subset {\cal U}$ of $o$ such that $M_t:= \varpi^{-1}(t)$ admits a scalar-flat K\"ahler metric for all $t\in \tilde{\cal U}$. Moreover, relative to local trivializations of the real-analytic fiber-bundle underlying $\varpi$, these metrics can be chosen so as to depend real-analytically on $t$. \label{tech} \end{thm} \begin{proof} Let $Z$ be the twistor space of $M$, $D=M\coprod \bar{M}$ its standard divisor, and $\sigma$ its real structure. Using Theorem \ref{dt5} and Corollary \ref{van}, we have $H^{2}(Z,{\Theta}_{Z}\otimes {\cal I}_D)=H^{2}(Z, {\Theta}_{Z,D})=0$. The first statement can be reinterpreted on $Z/\sigma$ as saying that $H^{2}(Z/\sigma,\Re{\Theta}_{Z}\otimes {\cal I}_D)=0$, so that the long exact sequence induced by the short exact sequence $$0\to \Re [{\Theta}_{P}\otimes {\cal I}_D ] \to \Re{\Theta}_{P}\to \Theta_M\to 0$$ on $P=Z/\sigma$ predicts that the natural restriction map $$H^1(Z/\sigma , \Re{\Theta})\to H^1(M, \Theta_M)$$ is surjective. Applying this morphism to restrict the Kodaira-Spencer map to $M$, we see that the versal family for $(Z/\sigma , M)$ given by Theorem \ref{ks3} induces a complete deformation of $M$--- i.e. a deformation of $M$ which contains a versal deformation as a subspace. Thus any small deformation of $M$ can be extended as a deformation of $(Z, D, \sigma)$. Applying Theorem \ref{Pen} then finishes the proof. \end{proof} \begin{remark} The analogous statement fails \cite{burnsbart} for {\em minimal} ruled surfaces. \end{remark} \noindent We now prove our main result: \begin{thm} Let $M$ be any ruled surface of genus $\geq 2$. If $M$ is blown up at sufficiently many points, the resulting complex surface $\tilde{M}$ admits scalar-flat K\"ahler metrics. \label{mainline} \end{thm} \begin{proof} Since any ruled surface \cite{bpv} is bimeromorphic to a product surface, some blow-up $\tilde{M}$ of the given surface $M$ is biregularly equivalent to an iterated blow-up of $\Sigma_{\bf g}\times {\Bbb CP}_1$, ${\bf g}\geq 2$, at $r_1\ldots r_m\in \Sigma_{\bf g}\times {\Bbb CP}_1$, where repetition of a point indicates that we are also given certain directional information at the multiple point. By blowing up extra points if necessary, we may assume that the projection of $\{r_1\ldots r_m \}$ to ${\Bbb CP}_1$ consists of more than one point. By changing to another homogeneous coordinate system $[\zeta_1 : \zeta_2]$ on ${\Bbb CP}_1$, we may also assume that $[1:0]$ is in the image of $\{r_1\ldots r_m \}$. For $t\in {\Bbb C}$, and if $r_j$ projects to $[1:0]$, set $r_j(t):= r_j$; otherwise let $r_j(t):= \mu_t (r_j)$, where \begin{eqnarray*} \mu_t : \Sigma_{\bf g}\times ({\Bbb CP}_1- \{ [1:0]\} )&\to& \Sigma_{\bf g}\times {\Bbb CP}_1 \\(p , [\zeta_1 : \zeta_2])&\mapsto& (p , [t\zeta_1 : \zeta_2])~ . \end{eqnarray*} Define a family $\varpi :{\cal M}\to {\Bbb C}$ of complex surfaces by blowing up $\Sigma_{\bf g}\times {\Bbb CP}_1 \times {\Bbb C}$ along the graphs of $t\mapsto r_j(t) $. For $t\neq 0$, the manifold $M_t$ is then biholomorphic to $\tilde{M}$. However, for $t=0$, the fiber is a ruled surface with a holomorphic vector field, namely the blow-up of $\Sigma_{\bf g}\times {\Bbb CP}_1$ at non-empty collections of points along $\Sigma_{\bf g}\times \{ [0:1]\}$ and $\Sigma_{\bf g}\times \{ [1:0]\}$. (When several of the blown-up points project to the same point of $\Sigma_{\bf g}$, one should think of the blow-ups as happening successively rather than simultaneously; for $t=0$, we are, at each stage, blowing up the previous surface at a zero of the vector field $\Xi = \zeta_1\partial / \partial \zeta_1$, and $\Xi$ therefore lifts to the blow-up.) Corollary \ref{triv} then asserts that $M_0$ admits scalar-flat K\"ahler metrics. The result therefore follows immediately from Theorem \ref{tech}. \end{proof} This can be repackaged as follows: \begin{cor} {\rm \bf (Main Theorem)} Let $M$ be a compact complex surface which admits a K\"ahler metric whose scalar curvature has integral zero. Suppose $\pi_1 (M)$ does not contain an Abelian subgroup of finite index. Then if $M$ is blown up at sufficiently many points, the resulting surface $\tilde{M}$ admits scalar-flat K\"ahler metrics. \end{cor} \begin{proof} By Theorem \ref{sd3}, a complex surface with admissible K\"ahler class is either ruled or covered by a torus or K3. Thus the fundamental group hypothesis forces $M$ to be a ruled surface of genus $\geq 2$. The statement therefore follows from Theorem \ref{mainline}. \end{proof} We now conclude this article with a pair of conjectures intended to remind the reader that the study of scalar-flat K\"ahler surfaces is still in its infancy. First, one would like to understand what happens for ruled surfaces of genus 0 and 1. It is our hope and expectation that the genus hypothesis in Theorem \ref{mainline} is actually superfluous: \begin{conjecture} The blow-up of {\em any} ruled surface at sufficiently many points admits scalar-flat K\"ahler metrics. \label{better} \end{conjecture} On the other hand, one would really like to understand {\em precisely} when an admissible class contains a scalar-flat K\"ahler metric. In light of Corollary \ref{corn} and known results on relatively minimal ruled surfaces \cite{burnsbart}, the following would seem very natural: \begin{conjecture} An admissible K\"ahler class on a (blown-up) ruled surface of genus $\geq 2$ contains a scalar-flat K\"ahler metric iff the corresponding parabolic bundle is quasi-stable. \end{conjecture} \vfill \noindent {\bf Acknowledgements.} The authors would like to thank the Australian Research Council for funding their visits to the University of Adelaide, where much of the actual writing was done. They would also like to thank N P.\ Buchdahl, M.G.\ Eastwood, M.S.\ Narasimhan, Y.-T.\ Siu, and G.\ Tian for their helpful comments and suggestions. \pagebreak

9701

alg-geom/9701009

en

https://arxiv.org/abs/alg-geom/9701009

alg-geom/9701009

Paul Bressler

P.Bressler, J.-L.Brylinski

On the singularities of the theta divisors on Jacobians

LaTeX Version 2.09

We study the intersection cohomology of the theta divisors on Jacobians of nonhyperelliptic curves.

alg-geom

\section{Introduction}\label{section:intro} The theta divisor $\Theta$ of the Jacobian variety of a complex curve $X$ is best viewed as a divisor inside the component $\operatorname{Pic}^{g-1}(X)$ consisting of (isomorphism classes of) line bundles of degree $g-1$. Then a line bundle $L$ belongs to $\Theta$ if and only if it has non-zero sections. Riemann proved that the multiplicity of $\Theta$ at a point $L$ is equal to $\dim H^0(X;L)-1$. Kempf (\cite{Ke}) obtained a geometric proof of Riemann's theorem and a beautiful description of the tangent cone to $\Theta$ at any point. In this paper we study the intersection cohomology of $\Theta$ when $X$ is not hyperelliptic. Our starting point is a theorem of Martens concerning the geometry of the Abel-Jacobi mapping $\phi:S^{g-1}(X)\to \operatorname{Pic}^{g-1}(X)$ and of its fibers. We interpret this theorem as saying that $\phi$ is small in the sense of Goresky and MacPherson. This means that the intersection cohomology $IH^\bullet(\Theta;\Bbb Q)$ is isomorphic to the cohomology $H^\bullet(S^{g-1}(X);\Bbb Q)$. The cohomology of $S^{g-1}(X)$, including the algebra structure, was completely determined by MacDonald in \cite{Mac} . From the evaluation of the differential of $\phi$ we deduce (Theorem \ref{thm:main}) that the intersection complex has the property that its characteristic variety (inside the cotangent bundle of $\operatorname{Pic}^{g-1}(X)$) is irreducible. This is a rather unusual phenomenon; it is known to be true for Schubert varieties in classical grassmannians (\cite{BFL}) and more generally in hermitian symmetric spaces of simply-laced groups (\cite{BF}). In Section \ref{section:invol} we study the effect on the intersection cohomology of the involution $\iota$ of $\Theta$ given by \[ \iota(L)=\Omega^1_X\otimes L^{\otimes -1}\ . \] We show that, for curves of even genus, the action of $\iota$ on $IH^\bullet(\Theta;\Bbb Q)$ does not preserve the algebra structure of $H^\bullet(S^{g-1}(X);\Bbb Q)$. This gives another interpretation of the well-known ``calcul triste'' of Verdier (\cite{BG}), leading to more examples of singular varieties with two different small resolutions yielding different algebra structures on intersection cohomology. In Section \ref{section:invol} interpret the classical computation of the number of odd $\Theta$-characteristics, which is equal to $2^{2g-1}-2^{g-1}$ according to \cite{W} and \cite{Mu1}, in terms of intersection cohomology. A $\Theta$-characteristic $L$ is a square root of the canonical bundle $\Omega^1_X$. Thus, a $\Theta$-characteristic with $\dim H^0(X;L) > 0$ determines a fixed point of $\iota$ in $\Theta$. A $\Theta$-characteristic $L$ is called odd or even depending on the parity of $\dim H^0(X;L)$. Our idea is to apply the Lefschetz fixed point formula to the action of $\iota$ on $IH^\bullet(\Theta;\Bbb Q)$. We show that the contribution of the theta characteristic $L$ to the fixed point formula is equal to $1$ if $L$ is odd and $0$ if $L$ is even. On the other hand we show that the (super)trace of $\iota$ acting on intersection cohomology is equal to $2^{2g-1}-2^{g-1}$. We would like to thank Alberto Collino for communicating his calculations to us. \section{Algebraic curves and their Jacobians}\label{section:jacs} In what follows $X$ will denote a connected smooth projective algebraic curve of genus $g$ over the field $\Bbb C$ of complex numbers, i.e. a compact Riemann surface. \subsection{Line bundles} Let $\operatorname{Pic}(X)$ denote the set of isomorphism classes of algebraic (equivalently holomorphic) line bundles on $X$. The operation of tensor product of line bundles endows $\operatorname{Pic}(X)$ with the structure of an algebraic group. The identity element is given by the (isomorphism class of) the structure sheaf $\cal O_X$. The map \begin{eqnarray*} \deg : \operatorname{Pic}(X) & \to & \Bbb Z \\ L & \mapsto & \deg(L) = \int_X c_1(L) \end{eqnarray*} induces an isomorphism on the respective groups of connected components \[ \pi_0(\operatorname{Pic}(X))\overset{\sim}{\to}\Bbb Z\ . \] As is usual, we denote by $\operatorname{Pic}^d(X)$ the component of $\operatorname{Pic}(X)$ consisting of (the isomorphism classes of) line bundles of degree $d$. The connected component of the identity $\operatorname{Pic}^0(X)$ is an Abelian variety isomorphic to the quotient $H^1(X;\cal O_X)/H^1(X;\Bbb Z)\overset{\sim}{=} H^1(X;\Bbb C)/\left(H^0(X;\Omega^1_X)+H^1(X;\Bbb Z)\right)$. In particular the Lie algebra of $\operatorname{Pic}^0(X)$ is $H^1(X;\cal O_X)$, and the cotangent space at the identity is $H^0(X;\Omega^1_X)$. The component $\operatorname{Pic}^d(X)$ is a principal homogeneous space under $\operatorname{Pic}^0(X)$. There is an involution \begin{eqnarray*} \iota : \operatorname{Pic}(X) & \to & \operatorname{Pic}(X) \\ L & \mapsto & \Omega^1_X\otimes L^{\otimes -1} \end{eqnarray*} which maps $\operatorname{Pic}^d(X)$ to $\operatorname{Pic}^{2g-2-d}(X)$ and, therefore preserves $\operatorname{Pic}^{g-1}(X)$. \subsection{Divisors} Let $\operatorname{Div}(X)$ denote the free Abelian group generated by the (closed) points of $X$. An element $D = \sum_i m_i\cdot p_i$ (where $m_i\in\Bbb Z$ and $p_i\in X$, and the sum is finite) is called a {\em divisor} on $X$. The divisor $D$ is said to be {\em effective} if $m_i\geq 0$ for all $i$. We will denote this fact by $D\geq 0$. The {\em degree} of the divisor $D$ is the integer defined by $\deg(D) = \sum_i m_i$. To a nonzero meromorphic function $f$ on $X$ one associates the divisor $\operatorname{div}(f) = \sum_{p\in X} \operatorname{ord}_p(f)\cdot p$. Note that $\deg(\operatorname{div}(f)) = 0$. The divisors of the form $\operatorname{div}(f)$ are called {\em principal} and form the subgroup $P(X)$ of $\operatorname{Div}(X)$. The quotient group $\operatorname{Div}(X)/P(X)$ is denoted by $\operatorname{Cl}(X)$ and is called the {\em divisor class group} of $X$. To a nonzero meromorphic section $s$ of a line bundle $L$ on $X$ one associates the divisor $\operatorname{div}(s) = \sum_{p\in X} \operatorname{ord}_p(s)\cdot p$. If, in addition, $f$ is a nonzero meromorphic function on $X$, then $\operatorname{div}(fs) = \operatorname{div}(f) + \operatorname{div}(s)$. Since for any two nonzero meromorphic sections $s_1$ and $s_2$ of $L$ one has a (necessarily nonzero) meromorphic function $f$ such that $s_1 = fs_2$, the divisors $\operatorname{div}(s_1)$ and $\operatorname{div}(s_2)$ are in the same divisor class in $\operatorname{Cl}(X)$. Moreover, if $s_1$ and $s_2$ are nonzero meromorphic sections of line bundles $L_1$ and $L_2$ respectively, then $s_1\otimes s_2$ is a nonzero meromorphic section of $L_1\otimes L_2$ and $\operatorname{div}(s_1\otimes s_2) = \operatorname{div}(s_1) + \operatorname{div}(s_2)$. Thus, the association $L\mapsto \operatorname{div}(s)$, where $s$ is a nonzero meromorphic section of $L$, gives rise to a well defined homomorphism \begin{eqnarray*} \operatorname{Pic}(X) & \to & \operatorname{Cl}(X) \\ L & \mapsto & D(L)\ . \end{eqnarray*} This map is in fact an isomorphism. The inverse is given by the map \begin{eqnarray*} \operatorname{Cl}(X) & \to & \operatorname{Pic}(X) \\ D & \mapsto & \cal O_X(D)\ , \end{eqnarray*} where $\cal O_X(D)$ is the subsheaf of the sheaf of meromorphic functions on $X$ consisting of functions $f$ which are holomorphic in the complement of $D$ and satisfy $\operatorname{div}(f)\geq D$. In particular, if $D$ is effective the line bundle $\cal O_X(D)$ has nonzero holomorphic sections, i.e. $H^0(X;L)\neq 0$. \subsection{Effective divisors} The set of effective divisors of degree $d$ on $X$ is easily identified with the $d$-th symmetric power $S^d(X)$ of the curve $X$. The variety $S^d(X) = X^{\times d}/\Sigma_d$ is smooth of dimension $d$ for all $d\geq 0$. For an effective divisor $D$, the {\em complete linear system} $\vert D\vert$ is defined as the set of all effective divisors in the class of $D$ in $\operatorname{Cl}(X)$. The association $s\mapsto \operatorname{div}(s)$ for a nonzero section $s\in H^0(X;\cal O_X(D))$ gives rise to the natural isomorphism $\Bbb P(H^0(X;L))\overset{\sim}{\to}\vert D\vert$. The map \begin{eqnarray*} \phi : S^d(X) & \to & \operatorname{Pic}^d(X) \\ D & \mapsto & \cal O_X(D) \end{eqnarray*} is a morphism of algebraic varieties. It is surjective for $d\geq g$ and birational onto its image for $d\leq g$. For $L\in\operatorname{Pic}^d(X)$ the fiber $\phi^{-1}(L)$ is naturally identified with the complete linear system $\Bbb P(H^0(X;L))$. \subsection{The $\Theta$-divisor} The $\Theta$-divisor is defined as the image of the map \[ \phi : S^{g-1}(X) \to \operatorname{Pic}^{g-1}(X) \] and will be denoted by $\Theta$. It is an irreducible closed subvariety of $\operatorname{Pic}^{g-1}(X)$ (for $X$ smooth connected) of codimension one. From the discussion above it follows that $\Theta$ is the locus of (isomorphism classes of) line bundles $L$ of degree $g-1$ with $H^0(X;L)\neq 0$. The hypersurface $\Theta$ is singular in general. It is known that $\dim Sing(\Theta)\geq g-4$. The singular locus is determined from the following result of Riemann (see \cite{Ke}). \begin{thm} The multiplicity of $\Theta$ at the point $L$ is equal to $\dim H^0(X;L)-1$. \end{thm} \begin{cor} A line bundle $L$ of degree $g-1$ determines a singular point of $\Theta$ if and only if $\dim H^0(X;L)\geq 2$. \end{cor} For a line bundle $L$ of degree $g-1$ the Riemann-Roch theorem shows that \linebreak $\dim H^0(X;L) = \dim H^0(X;\Omega^1_X\otimes L^{\otimes -1})$. Therefore the $\Theta$-divisor is preserved by the involution $\iota$. The $\Theta$-divisor is naturally stratified by the closed subvarieties $W^r_{g-1}$ defined as the locus of lined bundles $L$ of degree $g-1$ with $\dim H^0(X;L) - 1\geq r$. In particular $\Theta = W^0_{g-1}$ and $W^1_{g-1}$ is the singular locus of $\Theta$. The following theorem of Martens, stated here in the particular case of interest, gives an estimate on the dimension of $W^r_{g-1}$. Note that, by Clifford's theorem, $2\cdot\dim H^0(X;L)\leq g-1$. \begin{thm} Suppose that $g\geq 3$, $X$ not hyperelliptic and $2r\leq g-1$. Then all components of $W^r_{g-1}$ have the same dimension, and $\dim W^r_{g-1}\leq g - 2r - 2$. \end{thm} Recall that a map $f: Y\to Z$ of algebraic varieties is called {\em small} if \[ \operatorname{codim} \lbrace z\in Z\vert\ \dim f^{-1}(z)\geq d\rbrace > 2d\ . \] Thus Martens' theorem has the following corollary. \begin{cor}\label{cor:small} Suppose that $X$ is not hyperelliptic. Then the map \linebreak $\phi : S^{g-1}(X)\to\Theta$ is small. \end{cor} \section{The characteristic variety of the intersection complex} \label{section:char-var} From now on we will assume that $X$ is not hyperelliptic. \subsection{The Riemann-Hilber correspondence} For an algebraic variety $Y$ over $\Bbb C$ let $\operatorname{D}^b_c(Y;k)$ denote the bounded derived category of complexes of sheaves of $k$-vector spaces on $Y(\Bbb C)$ with (algebraically) constructible cohomology. Let $\operatorname{Perv}(Y;k)$ denote the full (Abelian) subcategory of perverse sheaves. For a smooth algebraic variety $Y$ over $\Bbb C$ let $\operatorname{D}^b_{rh}(\cal D_Y)$ denote the bounded derived category of complexes of left $\cal D_Y$-modules with regular holonomic cohomology. Let $\operatorname{RH}(\cal D_Y)$ denote the full (Abelian) subcategory of left holonomic $\cal D_Y$-modules with regular singularities. Recall that the de Rham functor \begin{eqnarray*} \operatorname{DR} : \operatorname{D}^b_{rh}(\cal D_Y) & \to & \operatorname{D}^b_c(Y;\Bbb C) \\ M^\bullet & \mapsto & \omega_Y\otimes^{\Bbb L}_{\cal D_Y}M^\bullet[\dim Y] \end{eqnarray*} is an equivalence of categories called the {\em Riemann-Hilbert correspondence} (\cite{KK}, \cite{Me1}, \cite{Me2}, see also \cite{B})), and restricts to the exact equivalence of Abelian categories \[ \operatorname{DR} : \operatorname{RH}(\cal D_Y)\to\operatorname{Perv}(Y;\Bbb C)\ . \] \subsection{The intersection cohomology of the $\Theta$-divisor} Let $\Theta^{reg} = \Theta\setminus W^1_{g-1}$ denote the nonsingular part of the $\Theta$-divisor and let $i : \Theta^{reg}\hookrightarrow \operatorname{Pic}^{g-1}(X)$ denote the (locally closed) inclusion map. From Corollary \ref{cor:small} and \cite{GM1} we obtain the following proposition. \begin{prop}\label{prop:IC-is-dir-im} There is an isomorphism $\operatorname{\bold R}\phi_*(\Bbb Q_{S^{g-1}(X)})\overset{\sim}{=} i_{!*}(\Bbb Q_{\Theta^{reg}})$ in \linebreak $\operatorname{D}^b_c(\operatorname{Pic}^{g-1}(X);\Bbb Q)$ \end{prop} Here $i_{!*}(\Bbb Q_{\Theta^{reg}})$ is the ``middle'' (\cite{GM1}, \cite{BBD}) extension of the constant sheaf $\Bbb Q_{\Theta^{reg}}$ such that, in particular, there is an isomorphism $H^\bullet(\operatorname{Pic}^{g-1}(X);i_{!*}(\Bbb Q_{\Theta^{reg}})) \overset{\sim}{=} IH^\bullet(\Theta;\Bbb Q)$. It follows from Proposition \ref{prop:IC-is-dir-im} that there is an isomorphism \[ IH^\bullet(\Theta;\Bbb Q)\overset{\sim}{=} H^\bullet(S^{g-1}(X);\Bbb Q)\ . \] The $\cal D_{S^{g-1}(X)}$-module associated (under the Riemann-Hilbert correspondence) to the constant sheaf $\Bbb C_{S^{g-1}(X)}$ is the sructure sheaf $\cal O_{S^{g-1}(X)}$. The Corollary \ref{cor:small} implies that the cohomology of the direct image (in $\cal D$-modules) $\phi_+\cal O_{S^{g-1}(X)}$ is a complex with cohomology concentrated only in degree zero so that $\phi_+\cal O_{S^{g-1}(X)}\overset{\sim}{=} H^0\phi_+\cal O_{S^{g-1}(X)}$ in $\operatorname{D}^b_{rh}(\cal D_{\operatorname{Pic}^{g-1}(X)})$. Let $\cal L$ denote the regular holonomic $\cal D_{\operatorname{Pic}^{g-1}(X)}$-module $H^0\phi_+\cal O_{S^{g-1}(X)}$, so that $\operatorname{DR}(\cal L)\overset{\sim}{=} i_{!*}\Bbb C_{\Theta^{reg}}$. The $\cal D$-module $\cal L$ may be characterized (up to a unique isomorphism) as the smallest nontrivial submodule of the $\cal D$-module $i_+\cal O_{\Theta^{reg}}$ (see \cite{Ka} and also \cite{B}). \subsection{The characteristic variety} Recall that to a holonomic $\cal D_Y$-module $M$ on a complex algebraic variety $Y$ one associates the characteristic cycle $\operatorname{SS}(M)$ which is an effective conic Lagrangian cycle on the cotangent bundle $T^*Y$ by a theorem of Sato-Kashiwara-Kawai, Malgrange and Gabber (\cite{SKK}, \cite{Mal}, \cite{G}). It is known (\cite{Ka}, see also \cite{B}) that it is of the form \[ \operatorname{SS}(M) = \sum_i m_i\cdot \overline{T^*_{Y_i}Y} \] for suitable smooth locally closed subvarieties $Y_i$ of $\operatorname{Supp} M$ and positive integers $m_i$. For example, if $f:Z\hookrightarrow Y$ is the inclusion of a closed smooth subvariety $Z$, then $\operatorname{SS}(f_+\cal O_Z) = T^*_ZY$. The multiplicities of the components of the characteristic cycle are local in the sence that $m_i$ depends only on the restriction of $M$ to any open (in $Y$) neighborhood of any point of $Y_i$. From the discussion above we may conclude that the characteristic cycle of $\cal L$ is of the form \[ \operatorname{SS}(\cal L) = \overline{T^*_{\Theta^{reg}}\operatorname{Pic}^{g-1}(X)} + \sum_i m_i\cdot\overline{T^*_{Y_i}\operatorname{Pic}^{g-1}(X)} \] for suitable smooth locally closed subvarieties $Y_i$ of $\Theta$. The main result of this note is the following. \begin{thm}\label{thm:main} Suppose that $X$ is a smooth connected projective algebraic curve over $\Bbb C$. Let $\cal L$ denote the simple holonomic $\cal D_{\operatorname{Pic}^{g-1}(X)}$-module with regular singularities which restricts to $\cal O_{\Theta^{reg}}$ on the nonsingular part $\Theta^{reg}$ of the $\Theta$-divisor. Then the characteristic cycle of $\cal L$ is irreducible, i.e. $\operatorname{SS}(\cal L) = \overline{T^*_{\Theta^{reg}}\operatorname{Pic}^{g-1}(X)}$. \end{thm} \begin{pf} We have the isomorphism (in the derived category) $\phi_+\cal O_{S^{g-1}(X)}\overset{\sim}{=} \cal L$ and $\operatorname{SS}(\cal O_{S^{g-1}(X)}) = T^*_{S^{g-1}(X)}{S^{g-1}(X)}$. Thus, according to Kashiwara, there is an inclusion \[ \operatorname{Supp}\operatorname{SS}(\cal L)\subseteq\operatorname{pr}((d\phi^t)^{-1}(T^*_{S^{g-1}(X)}{S^{g-1}(X)})) \] where the maps \[ T^*S^{g-1}(X) @<{d\phi^t}<< S^{g-1}(X)\times_{\operatorname{Pic}^{g-1}(X)} T^*\operatorname{Pic}^{g-1}(X) @>{\operatorname{pr}}>> T^*\operatorname{Pic}^{g-1}(X) \] are the (transpose of) the differential of the map $\phi$ and the projection on the second factor. We have the following classical identification. \begin{lemma} $\ker(d\phi^t)\overset{\sim}{=} H^0(X;\Omega^1_X\otimes\cal O_X(-D))\subset H^0(X;\Omega^1_X)$ \end{lemma} Therefore the subvariety \[ (d\phi^t)^{-1}(T^*_{S^{g-1}(X)}{S^{g-1}(X)}) = \lbrace (D,\xi)\vert\ D\in S^{g-1}(X),\ \xi\in T^*_{\phi(D)}\operatorname{Pic}^{g-1}(X), \ d\phi^t(\xi) = 0\rbrace \] is naturally described as \[ (d\phi^t)^{-1}(T^*_{S^{g-1}(X)}{S^{g-1}(X)}) = \lbrace (D,\omega)\vert\ D\in S^{g-1}(X),\ \omega\in H^0(X;\Omega^1_X\otimes\cal O_X(-D))\rbrace\ . \] From this description one sees immediately that $(d\phi^t)^{-1}(T^*_{S^{g-1}(X)}{S^{g-1}(X)})$ is the union of irreducible components $\overline Z^r$, where the locally closed subvariety $Z^r$ of \linebreak $S^{g-1}(X)\times_{\operatorname{Pic}^{g-1}(X)}T^*\operatorname{Pic}^{g-1}(X)$ is given by \[ Z^r = \lbrace (D,\omega)\vert\ \phi(D)\in W^r_{g-1}\setminus W^{r+1}_{g-1}, \ \omega\in H^0(X;\Omega^1_X\otimes\cal O_X(-D))\rbrace\ . \] In particular $\dim Z^r = \dim W^r_{g-1} + \dim\Bbb P(H^0(X;L)) + \dim H^0(X;\Omega^1_X\otimes L^{\otimes -1})$, where $L$ is a general point of $W^r_{g-1}$. From the theorems of Clifford and Martens one deduces immediately that $\dim Z^0 = g$ and $\dim Z^r\leq g-1$ for $r>0$. Therefore $\dim\operatorname{pr}(\overline Z^r)\leq g-1$ for $r>0$. It is easy to see that $\operatorname{pr}(Z^0) = \overline{T^*_{\Theta^{reg}}\operatorname{Pic}^{g-1}(X)}$. Since all components of $\operatorname{Supp}\operatorname{SS}(\cal L)$ are Lagrangian and, therefore, of dimension exactly $g$, one must have the inclusion $\operatorname{Supp}\operatorname{SS}(\cal L)\subseteq\operatorname{pr}(Z^0)$. \end{pf} We also obtain the following description of $\overline{T^*_{\Theta^{reg}}\operatorname{Pic}^{g-1}(X)}$: it is a conical Lagrangian subvariety of $T^*\operatorname{Pic}^{g-1}(X)$ which projects to $\Theta$; the fiber over $L$ is given by \[ \overline{T^*_{\Theta^{reg}}\operatorname{Pic}^{g-1}(X)}\cap T^*_L\operatorname{Pic}^{g-1}(X) = \bigcup_{D\in\Bbb P(H^0(X;L))}H^0(X;\Omega^1_X(-D))\ , \] where we use the natural identification $T^*_L\operatorname{Pic}^{g-1}(X)\overset{\sim}{=} H^0(X;\Omega^1_X)$. \begin{remark} It follows from Theorem \ref{thm:main} and a theorem of Andronikof (\cite{A}) that the wave-front set of the current defined by integrating over $\Theta$ is irreducible. \end{remark} \subsection{The universal $\Theta$-divisor} Let $\cal M_g^{(n)}$ denote the moduli space of curves of genus $g$ with level $n$ structure. Let $\cal X\to\cal M_g^{(n)}$ denote the universal curve of genus $g$. One has the varieties $\operatorname{Pic}^d(\cal X/\cal M_g^{(n)})$ for $d\in\Bbb Z$, the universal $\Theta$-divisor $\Theta_{univ}\hookrightarrow\operatorname{Pic}^{g-1}(\cal X/\cal M_g^{(n)})$ and the map $\phi : S^{g-1}(\cal X/\cal M_g^{(n)})\to\operatorname{Pic}^{g-1}(\cal X/\cal M_g^{(n)})$ of varieties over $\cal M_g^{(n)}$ which is birational onto $\Theta_{univ}$. In this setting we have the analog of Theorem \ref{thm:main} \begin{thm} The map $\phi : S^{g-1}(\cal X/\cal M_g^{(n)})\to\operatorname{Pic}^{g-1}(\cal X/\cal M_g^{(n)})$ is small. Consequently there is a canonical isomorphism (of $H^\bullet(\cal M_g^{(n)};\Bbb Q)$-modules)\linebreak $IH^\bullet(\Theta_{univ};\Bbb Q)\overset{\sim}{=} H^\bullet(S^{g-1}(\cal X/\cal M_g^{(n)});\Bbb Q)$. The characteristic variety (cycle) of the simple $\cal D_{\operatorname{Pic}^{g-1}(\cal X/\cal M_g^{(n)})}$-module which restricts to $\cal O_{\Theta_{univ}^{reg}}$ on the nonsingular part $\Theta_{univ}^{reg}$ of $\Theta_{univ}$ is irreducible. \end{thm} \section{Action of the involution}\label{section:invol} In this section we discuss the induced action of the involution \begin{eqnarray*} \iota: \operatorname{Pic}^{g-1}(X) & @>>> & \operatorname{Pic}^{g-1}(X) \\ L & \mapsto & \Omega^1_X\otimes L^{\otimes -1} \end{eqnarray*} on the intersection cohomology of the $\Theta$-divisor. By Corollary \ref{cor:small} there is an isomorphism \[ H^\bullet(S^{g-1}(X);\Bbb Q)\overset{\sim}{=} IH^\bullet(\Theta;\Bbb Q) \] which induces an algebra structure on $IH^\bullet(\Theta;\Bbb Q)$. We will show that the involution $\iota$ does not preserve this algebra structure. In what follows we will not make notational distictions between homology classes of cycles and their Poincare duals in cohomology. \subsection{The Riemann-Roch correspondence} Let $\rho\in H^{2g-2}(S^{g-1}(X)\times S^{g-1}(X);\Bbb Q)$ denote the class of the cycle \[ \left\lbrace (D_1,D_2)\in S^{g-1}(X)\times S^{g-1}(X)\vert D_1+D_2\in\vert K\vert\right\rbrace\ , \] where $K$ denotes the canonical divisor, which we call {\em the Riemann-Roch correspondence}. The canonical linear system \[ \vert K\vert\overset{\sim}{=} \Bbb P(H^0(X;\Omega^1_X))\hookrightarrow S^{2g-2}(X) \] determines the class $\kappa\in H^{2g-2}(S^{2g-2}(X);\Bbb Q)$. The class $\rho$ is the image of the class $\kappa$ under the pullback map \[ \Sigma^* : H^{2g-2}(S^{2g-2}(X);\Bbb Q) @>>> H^{2g-2}(S^{g-1}(X)\times S^{g-1};\Bbb Q) \] induced by the map \begin{eqnarray*} \Sigma : S^{g-1}(X)\times S^{g-1}(X) & @>>> & S^{2g-2}(X) \\ (D_1,D_2) & \mapsto & D_1+D_2 \ . \end{eqnarray*} The Riemann-Roch correspondence acts on $H^\bullet(S^{g-1}(X);\Bbb Q)$ by \[ \alpha\mapsto (\operatorname{pr}_2)_*(pr_1^*(\alpha)\smile\rho)\ , \] where $\operatorname{pr}_i: S^{g-1}(X)\times S^{g-1}(X)\to S^{g-1}(X)$ denotes the projection on the $i^{\text{th}}$ factor. \begin{prop} Under the isomorphism $IH^\bullet(\Theta;\Bbb Q)\overset{\sim}{=} H^\bullet(S^{g-1}(X); \Bbb Q)$ the action of the involution $\iota$ on $IH^\bullet(\Theta;\Bbb Q)$ corresponds to the action of the Riemann-Roch correspondence on $H^\bullet(S^{g-1}(X);\Bbb Q)$. \end{prop} \begin{remark} It follows, in particular, that the action of the Riemann-Roch correspondence is, in fact, an involution of $H^\bullet(S^{g-1}(X);\Bbb Q)$.\qed \end{remark} \subsection{Cohomology of symmetric powers of a curve} A complete study of the cohomology of a symmetric power of a curve may be found in \cite{Mac}. We will need the following facts. The Abel-Jacobi map \[ \phi_d : S^d(X) @>>> \operatorname{Pic}^d(X) \] induces on $H^\bullet(S^d(X);\Bbb Q)$ the structure of a module over $H^\bullet(\operatorname{Pic}^d(X);\Bbb Q)$. A point $p\in X$ determines the embedding \begin{eqnarray*} j : S^{d-1} & @>>> & S^d(X) \\ D & \mapsto & D+p \end{eqnarray*} and the isomorphism \[ \otimes\cal O(p) : \operatorname{Pic}^{d-1}(X) @>>> Pic^d(X) \] such that $\phi_d\circ j= (\otimes\cal O(p))\circ\phi_{d-1}$. The homotopy classes of the maps $j$ and $\otimes\cal O(p)$ do not depend on the point $p$ for $X$ connected. In particular the induced map $j^* : H^\bullet(S^d(X);\Bbb Q)\to H^\bullet(S^{d-1}(X);\Bbb Q)$ and the Gysin map $j_* : H^\bullet(S^{d-1}(X);\Bbb Q)\to H^{\bullet +2}(S^d(X);\Bbb Q)$ are well defined. The map $(\otimes\cal O(p))^*$ provides the canonical identification of algebras $H^\bullet(\operatorname{Pic}^d(X);\Bbb Q)$ for various $d$. Note that there is a canonical isomorphism $H^\bullet(\operatorname{Pic}^d(X);\Bbb Q)\overset{\sim}{=}{\bigwedge}^\bullet H^1(X;\Bbb Q)$. Moreover, $j^*$ and $j_*$ are maps of modules over ${\bigwedge}^\bullet H^1(X;\Bbb Q)$, $j^*$ is surjective and $j_*$ is injective. Let $\eta_d = j_*(1) = j_*([S^{d-1}(X)])\in H^2(S^d(X);\Bbb Q)$. Then the identity $j^*(\eta_d) = \eta_{d-1}$ holds, hence $j_*(\eta_{d-1})= \eta_d^2$ Consider the map \begin{eqnarray*} \Sigma : S^{d_1}(X)\times S^{d_2}(X) & @>>> & S^{d_1+d_2}(X) \\ (D_1,D_2) & \mapsto & D_1+D_2\ . \end{eqnarray*} Then $\Sigma^*(\eta_{d_1+d_2}) = \operatorname{pr}_1^*(\eta_{d_1})+\operatorname{pr}_2^*(\eta_{d_2})$. Multiplication by $\eta_d$ commutes with the action of ${\bigwedge}^\bullet H^1(X;\Bbb Q)$. Moreover, the natural map \[ {\bigwedge}^\bullet H^1(X;\Bbb Q)\otimes\Bbb Q [\eta] @>>> H^\bullet(S^d(X); \Bbb Q) \] is surjective in all degrees and an isomorphism in degrees up through $d$. The class $\theta = [\Theta]\in H^2(\operatorname{Pic}^{g-1}(X);\Bbb Q)\overset{\sim}{=}{\bigwedge}^2 H^1(X;\Bbb Q)$ corresponds to the (symplectic) intersection pairing on $X$. The Poincar\'e's formula (\cite{ACGH}) says that \[ (\phi_{g-m})_*(1) = (\phi_{g-m+i})_*(\eta_{g-m+i}^i) = \frac{\theta^m}{m!}\ . \] In particular, for $m=g$ we find that $\displaystyle\frac{\theta^g}{g!}$ is the class of a point. The map $\phi_{2g-1}:S^{2g-1}(X)\to\operatorname{Pic}^{2g-1}(X)$ is a projective space bundle. The map $j:S^{2g-2}(X)\hookrightarrow S^{2g-1}(X)$ restricts to an embedding of $\vert K\vert$ as a fiber of $\phi_{2g-1}$. Therefore the identities $j_*(\kappa) = \phi_{2g-1}^*(\displaystyle\frac{\theta^g}{g!})$, and, consequently, $\phi_{2g-2}^*(\displaystyle\frac{\theta^g}{g!}) = j^*\phi_{2g-1}^*(\displaystyle\frac{\theta^g}{g!}) =\eta_{2g-2}\smile\kappa$ hold. \subsection{The action of the Riemann-Roch correspondence} The action of the involution $\iota$ on $H^p(\operatorname{Pic}^{g-1}(X);\Bbb Q)$ is given by multiplication by $(-1)^p$ and may be realized as the action of a correspondence as follows. Let $\Delta^-\in H^{2g}(\operatorname{Pic}^{g-1}(X)\times\operatorname{Pic}^{g-1}(X);\Bbb Q)$ denote the ``antidiagonal'', i.e. the cycle \[ \left\lbrace (L_1,L_2)\in\operatorname{Pic}^{g-1}(X)\times\operatorname{Pic}^{g-1}(X)\vert L_1\otimes L_2\overset{\sim}{=}\Omega^1_X\right\rbrace\ . \] The class $\Delta^-$ is the image of the class of the point $[\Omega^1_X]\in H^{2g}(\operatorname{Pic}^{2g-2}(X);\Bbb Q)$ under the map \[ \otimes^*: H^\bullet(\operatorname{Pic}^{2g-2}(X);\Bbb Q) @>>> H^\bullet(\operatorname{Pic}^{g-1}\times\operatorname{Pic}^{g-1}(X);\Bbb Q) \] induced by the map \begin{eqnarray*} \otimes : \operatorname{Pic}^{g-1}(X)\times\operatorname{Pic}^{g-1}(X) & @>>> & \operatorname{Pic}^{2g-2}(X) \\ (L_1, L_2) & \mapsto & L_1\otimes L_2\ . \end{eqnarray*} Then, clearly, the action of the involution $\iota$ is given by \[ \iota(\alpha) = (\operatorname{pr}_2^\prime)_*((\operatorname{pr}_1^\prime)^*(\alpha)\smile\Delta^-)\ , \] where $\operatorname{pr}_i^\prime : \operatorname{Pic}^{g-1}(X)\times\operatorname{Pic}^{g-1}(X)\to\operatorname{Pic}^{g-1}(X)$ denotes the projection on the $i^{\text{th}}$ factor. The action of the Riemann-Roch correspondence on $H^\bullet(S^{g-1}(X);\Bbb Q)$ is compatible with the action of $\iota$ on $H^\bullet(\operatorname{Pic}^{g-1}(X);\Bbb Q)$ and the action of $H^\bullet(\operatorname{Pic}^{g-1}(X);\Bbb Q)$ on $H^\bullet(S^{g-1}(X);\Bbb Q)$. Thus, the action of the Riemann-Roch correspondence is determined by its values on the powers of the class $\eta_{g-1}$. The following proposition is due to Alberto Collino. \begin{prop} \begin{equation}\label{formula:RR-eta} (\operatorname{pr}_2)_*(\operatorname{pr}_1^*(\eta_{g-1}^m)\smile\rho) = \sum_{i=0}^{m}\frac{(-1)^i}{(m-i)!}\phi_{g-1}^*(\theta^{m-i})\smile \eta_{g-1}^i \end{equation} \end{prop} \begin{pf} Consider the commutative diagram \[ \begin{CD} S^{g-1}(X)\times S^{g-1}(X) @>{\Sigma}>> S^{2g-2}(X) \\ @V{\phi_{g-1}\times\phi_{g-1}}VV @VV{\phi_{2g-2}}V \\ \operatorname{Pic}^{g-1}(X)\otimes\operatorname{Pic}^{g-1}(X) @>{\otimes}>> \operatorname{Pic}^{2g-2}(X)\ . \end{CD} \] We have \begin{eqnarray*} (\phi_{g-1}\times\phi_{g-1})^*\Delta^- & = & (\phi_{g-1}\times\phi_{g-1})^* \otimes^*(\frac{\theta^g}{g!}) \\ & = & \Sigma^*\phi_{2g-2}^*(\frac{\theta^g}{g!}) \\ & = & \Sigma^*(\eta_{2g-2}\smile\kappa) \\ & = & (\operatorname{pr}_1^*(\eta_{g-1})+\operatorname{pr}_2^*(\eta_{g-1}))\smile\Sigma^*\kappa \\ & = & \operatorname{pr}_1^*(\eta_{g-1})\smile\rho + \operatorname{pr}_2^*(\eta_{g-1})\smile\rho\ . \end{eqnarray*} Therefore, for $\alpha\in H^\bullet(S^{g-1}(X);\Bbb Q)$, we have \begin{multline*} (\operatorname{pr}_2)_*(\operatorname{pr}_1^*(\alpha)\smile(\phi_{g-1}\times\phi_{g-1})^*\Delta^-) \\ = (\operatorname{pr}_2)_*(\operatorname{pr}_1^*(\alpha\smile\eta_{g-1})\smile\rho) + (-1)^{\deg\alpha}\cdot\eta_{g-1}\smile(\operatorname{pr}_2)_*(\operatorname{pr}_1^*(\alpha)\smile\rho)\ . \end{multline*} On the other hand, by Lemma \ref{lemma:ind-cor} below (with $X=Y=S^{g-1}(X)$, $A=B=\operatorname{Pic}^{g-1}(X)$, $f=g=\phi_{g-1}$ and $\Gamma=\Delta^-$), we have \[ (\operatorname{pr}_2)_*(\operatorname{pr}_1^*(\alpha)\smile(\phi_{g-1}\times\phi_{g-1})^*\Delta^-) = \phi_{g-1}^*\iota^*(\phi_{g-1})_*(\alpha) = (-1)^{\deg\alpha}\phi_{g-1}^*(\phi_{g-1})_*(\alpha)\ . \] Thus, putting two calculations together we obtain the identity \[ (-1)^{\deg\alpha}\phi_{g-1}^*(\phi_{g-1})_*(\alpha) = (\operatorname{pr}_2)_*(\operatorname{pr}_1^*(\alpha\smile\eta_{g-1}) \smile\rho)+(-1)^{\deg\alpha}\cdot\eta_{g-1}\smile(\operatorname{pr}_2)_*(\operatorname{pr}_1^*(\alpha) \smile\rho)\ . \] In particular, for $\alpha=\eta_{g-1}^{m-1}$ we have \[ \phi_{g-1}^*\left(\frac{\theta^m}{m!}\right) = (\operatorname{pr}_2)_*(\operatorname{pr}_1^*(\eta_{g-1}^m)\smile\rho)+ \eta_{g-1}\smile(\operatorname{pr}_2)_*(\operatorname{pr}_1^*(\eta_{g-1}^{m-1})\smile\rho) \] or, equivalently, \begin{equation}\label{formula:rec-rel} (\operatorname{pr}_2)_*(\operatorname{pr}_1^*(\eta_{g-1}^m)\smile\rho) = -\eta_{g-1}\smile(\operatorname{pr}_2)_*(\operatorname{pr}_1^*(\eta_{g-1}^{m-1})\smile\rho) +\phi_{g-1}^*\left(\frac{\theta^m}{m!}\right)\ . \end{equation} In particular, for $m=1$ we have \[ (\operatorname{pr}_2)_*(\operatorname{pr}_1^*(\eta_{g-1})\smile\rho) = -\eta_{g-1} +\phi^*(\theta) \] which is \eqref{formula:RR-eta} in this case. Proceding by induction on $m$ with the help of \eqref{formula:rec-rel} we obtain \eqref{formula:RR-eta}. \end{pf} \begin{lemma}\label{lemma:ind-cor} Suppose given oriented manifolds $X,\ Y,\ A,\ B$, maps $f:X\to A$ and $g:Y\to B$, and a class $\Gamma\in H^\bullet(A\times B; \Bbb Q)$. Then, for any class $\alpha\in H^\bullet(X;\Bbb Q)$, \[ (\operatorname{pr}_Y)_*(\operatorname{pr}_X^*(\alpha)\smile (f\times g)^*\Gamma) = g^*((\operatorname{pr}_B)_*(\operatorname{pr}_A^*(f_*\alpha)\smile\Gamma))\ . \] \end{lemma} Let \[ STr(\iota, IH^\bullet(\Theta;\Bbb Q)) = \sum_p (-1)^p Tr(\iota : IH^p(\Theta;\Bbb Q)\to IH^p(\Theta;\Bbb Q))\ . \] \begin{prop}\label{prop:STr} \[ STr(\iota, IH^\bullet(\Theta;\Bbb Q)) = 2^{2g-1}-2^{g-1}\ . \] \end{prop} \begin{pf} Using Poincare duality we can write \begin{multline} STr(\iota, IH^\bullet(\Theta;\Bbb Q)) = \sum_{p < g-1} (-1)^p\cdot 2\cdot Tr(\iota : IH^p(\Theta;\Bbb Q)\to IH^p(\Theta;\Bbb Q)) \\ + (-1)^{g-1} Tr(\iota : IH^{g-1}(\Theta;\Bbb Q)\to IH^{g-1}(\Theta;\Bbb Q))\ . \end{multline} For $p\leq g-1$ we have \[ IH^p(\Theta;\Bbb Q) = \bigoplus_{j} H^{p-2j}(\operatorname{Pic}^{g-1}(X);\Bbb Q) \smile\eta_{g-1}^j\ . \] Formula \eqref{formula:RR-eta} shows that $\iota$ preserves the filtration $F_\bullet IH^p(\Theta;\Bbb Q)$ defined by \[ F_qIH^p(\Theta;\Bbb Q) = \bigoplus_{j\leq q} H^{p-2j}(\operatorname{Pic}^{g-1}(X);\Bbb Q)\smile\eta_{g-1}^j \] and acts by $(-1)^{p-q}$ on $Gr^F_q IH^p(\Theta;\Bbb Q)$. Therefore \begin{multline*} Tr(\iota : IH^p(\Theta;\Bbb Q)\to IH^p(\Theta;\Bbb Q)) = \\ = Tr(Gr^F_\bullet\iota : Gr^F_\bullet IH^p(\Theta;\Bbb Q)\to Gr^F_\bullet IH^p(\Theta;\Bbb Q)) = \\ = \sum_{i+2j = p}(-1)^{i+j}{2g\choose i} \end{multline*} and \begin{multline} STr(\iota, IH^\bullet(\Theta;\Bbb Q)) = \\ = \sum_{p < g-1} (-1)^p\cdot 2\cdot \sum_{i+2j = p}(-1)^{i+j}{2g\choose i} + (-1)^{g-1} \sum_{i+2j = g-1}(-1)^{i+j}{2g\choose i} = \\ = \sum_{i = 0}^{g-1}\delta(g-1-i){2g\choose i}\ , \end{multline} where $\delta:\Bbb Z/4\cdot\Bbb Z\to\Bbb Z$ is defined by \[ \delta(0) = 1,\ \delta(1) = 2,\ \delta(2) = 1\ \delta(3) = 0\ . \] The proposition now follows from Lemma \ref{lemma:calc} below. \end{pf} \begin{lemma}\label{lemma:calc} $\sum_{i = 0}^{g-1}\delta(g-1-i){2g\choose i} = 2^{2g-1}-2^{g-1}$ \end{lemma} \begin{pf} First observe that \[ 2^{2g-1} = \frac12 (1+1)^{2g}= \sum_{i=0}^{g-1}{2g\choose i} +\frac12{2g\choose g}\ . \] Now consider separately the four cases corresponding to the residue of $g$ modulo $4$. Suppose that $g$ is divisible by $4$. Then \[ 2^{g-1} = \frac12 (1+\sqrt{-1})^{2g} = \sum_{i=0}^{g-1}\frac12 (\sqrt{-1}^i + \sqrt{-1}^{-i}){2g\choose i} +\frac12{2g\choose g}\ . \] Therefore \[ 2^{2g-1} - 2^{g-1} = \sum_{i=0}^{g-1}\left( 1 - \frac12 (\sqrt{-1}^i + \sqrt{-1}^{-i})\right){2g\choose i}\ . \] Now observe that \[ 1 - \frac12 (\sqrt{-1}^i + \sqrt{-1}^{-i}) = \delta(-1-i) = \delta(g-1-i) \] and the equality follows. Other cases follow from similar calculations and we omit the details. If $g\equiv 2\bmod 4$, then we find that $$2^{2g-1}-2^{g-1}={1\over 2}(1+1)^{2g}+{1\over 2} (1+\sqrt{-1})^{2g}=\sum_{j=1}^{g-1}\delta(1-j){2g\choose j}.$$ If $g\equiv 1\bmod 4$ , then we find that $$2^{2g-1}-2^{g-1}={1\over 2}(1+1)^{2g} +{\sqrt{-1}\over 4}(1+\sqrt{-1})^{2g} -{\sqrt{-1}\over 4}(1-\sqrt{-1})^{2g} =\sum_{j=0}^{g-1}\delta(-j){2g\choose j}.$$ If $g\equiv 3\bmod 4$ , then we find that $$2^{2g-1}-2^{g-1}={1\over 2}(1+1)^{2g} -{\sqrt{-1}\over 4}(1+\sqrt{-1})^{2g} +{\sqrt{-1}\over 4}(1-\sqrt{-1})^{2g} =\sum_{j=0}^{g-1}\delta(2-j){2g\choose j}.$$ \end{pf} \subsection{$\Theta$-characteristics} Recall that the {\em $\Theta$-characteristics} are the fixed points of the involution $\iota$ acting on $\operatorname{Pic}^{g-1}(X)$, i.e. they are the (isomorphism classes of) line bundles $L$ on $X$ such that there is an isomorphism $L^{\otimes 2}\overset{\sim}{=} \Omega^1_X$. A $\Theta$-characteristic is called {\em even} (respectively {\em odd}) if $\dim H^0(X;L)$ is even (respectively odd). The total number of $\Theta$-characteristics on a curve of genus $g$ is equal to $2^{2g}$. The fixed points of the action of $\iota$ on $\Theta$ correspond to the $\Theta$-characteristics $L$ with $\dim H^0(X;L)\geq 1$. Note that the fixed point set $\Theta^\iota$ contains all of the odd $\Theta$-characteristics. From Proposition \ref{prop:STr} we obtain the formula to the number of odd $\Theta$-characteristics. Naturally, this is a classical result, first proved by Wirtinger in \cite{W} using theta functions. An algebro-geometric proof valid in all odd characteristics was given by Mumford in \cite{Mu1}. \begin{prop} The number of odd $\Theta$-characteristics is equal to $2^{2g-1}-2^{g-1}$. \end{prop} \begin{pf} The Lefschetz Fixed Point Formula (\cite{GM2}, \cite{V}) applied to $\iota$ gives \[ STr(\iota, IH^\bullet(\Theta:\Bbb Q)) = \sum_{L\in\Theta^\iota}STr(\iota, H^\bullet((i_{!*}\Bbb Q_{\Theta^{reg}})_L)) \] where $(i_{!*}\Bbb Q_{\Theta^{reg}})_L)$ is the stalk of the the sheaf $i_{!*}\Bbb Q_{\Theta^{reg}}$ at the point $L$. By Proposition \ref{prop:IC-is-dir-im} there is an isomorphism $H^\bullet((i_{!*}\Bbb Q_{\Theta^{reg}})_L)\overset{\sim}{=} H^\bullet(\phi^{-1}(L);\Bbb Q)$, and there is a natural identification $\phi^{-1}(L)\overset{\sim}{=}\Bbb P(H^0(X;L))$. Let $r(L) = \dim H^0(X;L) - 1$. Then we have \[ H^p((i_{!*}\Bbb Q_{\Theta^{reg}})_L)\overset{\sim}{=}\left\lbrace\begin{array}{lll} \Bbb Q & \text{if} & 0\leq p=2j\leq r(L)\\ 0 & \text{otherwise} \end{array}\right. \] By Lemma \ref{lemma:str-fp} below \[ STr(\iota, H^\bullet((i_{!*}\Bbb Q_{\Theta^{reg}})_L)) = \sum_{j=0}^{r(L)} (-1)^j = \left\lbrace\begin{array}{lll} 1 & \text{if} & \text{$L$ is odd} \\ 0 & \text{if} & \text{$L$ is even} \end{array}\right. \] and the proposition follows immediately from Proposition \ref{prop:STr}. \end{pf} \begin{lemma}\label{lemma:str-fp} \[ Tr(\iota : H^{2j}((i_{!*}\Bbb Q_{\Theta^{reg}})_L)\to H^{2j}((i_{!*}\Bbb Q_{\Theta^{reg}})_L)) = (-1)^j\ . \] \end{lemma} \begin{pf} Consider the commutative diagram \[ \begin{CD} H^{2j}(S^{g-1}(X);\Bbb Q) @>>> H^{2j}(\phi^{-1}(L);\Bbb Q) \\ @V{\overset{\sim}{=}}VV @V{\overset{\sim}{=}}VV \\ IH^{2j}(\Theta;\Bbb Q) @>>> H^{2j}((i_{!*}\Bbb Q_{\Theta^{reg}})_L) \end{CD} \] The top horizontal map is nonzero for $j=1$ since the class of an ample divisor on $S^{g-1}(X)$ must have a nontrivial restriction. Therefore it is nontrivial for all $j\leq r(L)$, hence surjective because the target is one-dimensional. Since the vertical maps are isomorphisms so is the bottom horizontal map. Since the composition \[ H^k(\operatorname{Pic}^{g-1}(X);\Bbb Q) @>>> H^k(S^{g-1}(X);\Bbb Q) @>>> H^k(\phi^{-1}(L);\Bbb Q) \] is trivial for all $k$ it follows that the compostion \[ H^1(\operatorname{Pic}^{g-1}(X);\Bbb Q)\otimes IH^{2j-1}(\Theta) @>>> IH^{2j}(\Theta;\Bbb Q) @>>> H^{2j}((i_{!*}\Bbb Q_{\Theta^{reg}})_L) \] must be trivial. Note that $\operatorname{Coker}(H^1(\operatorname{Pic}^{g-1}(X);\Bbb Q) \otimes IH^{2j-1}(\Theta)\to IH^{2j}(\Theta;\Bbb Q))$ is generated by the image of $1\otimes\eta_{g-1}^j$. The induced map \[ \operatorname{Coker}(H^1(\operatorname{Pic}^{g-1}(X);\Bbb Q)\otimes IH^{2j-1}(\Theta) @>>> IH^{2j}(\Theta;\Bbb Q)) @>>> H^{2j}((i_{!*}\Bbb Q_{\Theta^{reg}})_L) \] is an isomorphism which is clearly $\iota$-equivariant and the lemma follows from \eqref{formula:RR-eta}. \end{pf} \subsection{Un calcul encore plus triste} Now we study the relation between the involution $\iota$ acting on $IH^\bullet(\Theta;\Bbb Q)$ and the algebra structure on $IH^\bullet(\Theta;\Bbb Q)$ induced by the isomorphism $IH^\bullet(\Theta;\Bbb Q)\tilde{\to}H^\bullet(S^{g-1}(X);\Bbb Q)$. \begin{prop} For $g$ even, $\iota$ does not preserve the algebra structure. \end{prop} \begin{pf} According to \eqref{formula:RR-eta} we have $\iota(\eta_{g-1}) = -\eta_{g-1} + \phi_{g-1}^*(\theta)$. Therefore the class $2\eta_{g-1} +\phi_{g-1}^*(\theta)$ satisfies \[ \iota(\eta_{g-1}+\phi_{g-1}^*(\theta)) = -(\eta_{g-1} +\phi_{g-1}^*(\theta))\ . \] On the other hand, the class $(2\eta_{g-1} +\phi_{g-1}^*(\theta))^{g-1}$ satisfies \[ \iota((\eta_{g-1}+\phi_{g-1}^*(\theta))^{g-1}) = (\eta_{g-1} +\phi_{g-1}^*(\theta))^{g-1} = (-1)^{g-1}(\iota (\eta_{g-1} +\phi_{g-1}^*(\theta))^{g-1}) \] since $\iota$ acts trivially on the one dimensional space $IH^{2g-2}(\Theta;\Bbb Q)$. It remains to observe that $(\eta_{g-1} +\phi_{g-1}^*(\theta))^{g-1}\neq 0$. \end{pf}

9701

alg-geom/9701012

en

https://arxiv.org/abs/alg-geom/9701012

alg-geom/9701012

Elham Izadi

E. Izadi

A Prym construction for the cohomology of a cubic hypersurface

AMS-Latex, 39 pages

Mumford defined a natural isomorphism between the intermediate jacobian of a conic-bundle over $P^2$ and the Prym variety of a naturally defined \'etale double cover of the discrminant curve of the conic-bundle. Clemens and Griffiths used this isomorphism to give a proof of the irrationality of a smooth cubic threefold and Beauville later generalized the isomorphism to intermediate jacobians of odd-dimensional quadric-bundles over $P^2$. We further generalize the isomorphism to the primitive cohomology of a smooth cubic hypersurface in $P^n$.

alg-geom

\section{The variety $D_l$ of lines incident to $l$} \label{secDl} For a smooth cubic hypersurface $X \subset {\Bbb P}^n$ of equation $G$, we let $\delta : {\Bbb P}^n \longrightarrow ({\Bbb P}^n)^*$ be the dual morphism of $X$. In terms of a system of projective coordinates $\{x_0, ..., x_n \}$ on ${\Bbb P}^n$, the morphism $\delta$ is given by \[ \delta (x_0, ..., x_n) = \left( \partial_0 G(x_0, ..., x_n), ..., \partial_n G(x_0, ..., x_n) \right) \] where $\partial_i = \frac{\partial}{\partial x_i}$. Let $l \subset X$ be a line. Following \cite{CG} (page 307 Definition 6.6, Lemma 6.7 and page 310 Proposition 6.19), we make the definition: \begin{definition} \begin{enumerate} \item The line $l$ is of first type if the normal bundle to $l$ in $X$ is isomorphic to ${\cal O}_l^{\oplus 2} \oplus {\cal O}_l(1)^{\oplus (n-4)}$. Equivalently, the intersection ${\Bbb T}_l$ of the projective tangent spaces to $X$ along $l$ is a linear subspace of ${\Bbb P}^n$ of dimension $n-3$. Equivalently, the dual morphism $\delta$ maps $l$ isomorphically onto a conic in $({\Bbb P}^n)^*$, i.e., the restriction map $\langle \partial_0 G, ..., \partial_n G \rangle \longrightarrow H^0(l, {\cal O}_l(2))$ is onto where $\langle \partial_0 G, ..., \partial_n G \rangle$ is the span of $\partial_0 G, ..., \partial_n G$ in $H^0({\Bbb P}^n, {\cal O}_{{\Bbb P}^n}(2))$. \item The line $l$ is of second type if the normal bundle to $l$ in $X$ is isomorphic to ${\cal O}_l(-1) \oplus {\cal O}_l(1)^{\oplus (n-3)}$. Equivalently, the space ${\Bbb T}_l$ is a linear subspace of ${\Bbb P}^n$ of dimension $n-2$. Equivalently, the dual morphism $\delta$ has degree $2$ on $l$ and maps $l$ onto a line in $({\Bbb P}^n)^*$, i.e., the restriction map $\langle \partial_0 G, ..., \partial_n G \rangle \longrightarrow H^0(l, {\cal O}_l(2))$ has rank $2$. \end{enumerate} \end{definition} By \cite{CG} (Lemma 7.7 page 312), the variety $F$ of lines in $X$ is smooth of dimension $2(n-3)$. An easy dimension count shows that the dimension of $D_l$ is at least $n-3$ for any $l \in F$. Suppose that $l$ is of first type. We have \begin{lemma} Let $l' \in D_l$ be distinct from $l$. If $l'$ is of first type or if $l'$ is of second type and $l$ is {\em not} contained in ${\Bbb T}_{l'}$, then the dimension of $T_{l'}D_l$ is $n-3$ (i.e., $D_l$ is smooth of dimension $n-3$ at $l'$). If $l'$ is of second type and $l$ is contained in ${\Bbb T}_{l'}$, then the dimension of $T_{l'}D_l$ is $n-2$. \label{Dlsmooth1} \end{lemma} {\em Proof :} The variety $D_l$ is the intersection of $F$ with the variety $G_l$ parametrizing all lines in ${\Bbb P}^n$ which are incident to $l$. Therefore $T_{l'}D_l = T_{l'}G_l \cap T_{l'}F \subset T_{l'}G(2,n+1)$. Let $V$ and $V'$ be the vector spaces in ${\Bbb C}^{n+1}$ whose projectivizations are respectively $l$ and $l'$. Then $T_{l'}G_l$ can be identified with the subvector space of $T_{l'}G(2,n+1) = Hom(V', {\Bbb C}^{n+1} / V')$ consisting of those homomorphisms $f$ such that $f(V \cap V') \subset (V+V') / V'$ (see e.g. \cite{harris2}, Ex. 16.4 pages 202-203). It follows that the set of homomorphisms $f$ such that $f(V \cap V') = 0 $ is a codimension one subspace of $T_{l'}G_l$ and therefore its intersection $H$ with $T_{l'}D_l$ has codimension one or less in $T_{l'}D_l$. The space $T_{l'}F$ can be identified with the subvector space of $T_{l'}G(2,n+1) = Hom(V', {\Bbb C}^{n+1} / V')$ consisting of those homomorphisms $f$ such that for any vector $v \in V' \setminus \{ 0 \}$ mapping to a point $p \in l'$, we have $f(v) \in T'_pX / V'$ (see \cite{harris2} Ex. 16.21 and 16.23 pages 209-210). If $f : V' \longrightarrow {\Bbb C}^{n+1} / V'$ verifies $f(V \cap V') =0$, then $f(V') = {\Bbb C} f(v)$ for $v$ a general vector in $V'$. Hence, if $f \in H$, then $f(V') \subset \bigcap_{p \in l'} T_p'X / V'$. If $l'$ is of first type, then $\bigcap_{p \in l'} T_p'X$ has dimension $n-2$, hence $\bigcap_{p \in l'} T_p'X / V'$ has dimension $n-4$. So $H$ has dimension $n-4$ and, since $H$ has codimension one or less in $T_{l'}D_l$, we deduce that $T_{l'}D_l$ has dimension at most $n-3$ hence it has dimension equal to $n-3$ (since $D_l$ has dimension $\geq n-3$). If $l'$ is of second type, then the tangent space $T_{l'}F$ can be identified with $Hom(V', \bigcap_{p \in l'} T_p'X / V')$ (because, for instance, the latter is contained in $T_{l'}F$ and the two spaces have the same dimension). If $V$ is not contained in $\bigcap_{p \in l'} T_p'X$, then $f(V \cap V') \subset (V+V') / V'$ for $f \in Hom(V', \bigcap_{p \in l'} T_p'X / V')$ implies $f(V \cap V') =0$. So $T_{l'}D_l = T_{l'}F \cap T_{l'}G_l$ has dimension equal to the dimension of $\bigcap_{p \in l'} T_p'X / V'$ which is $n-3$. So in this case $D_l$ is smooth at $l'$. If $V \subset \bigcap_{p \in l'} T_p'X / V'$, then the requirement $f(V \cap V') \subset (V+V') / V'$ imposes $n-4$ conditions on $f$ and the dimension of $T_{l'}D_l$ is $n-2$. \hfill $\qed$ \vskip10pt Since ${\Bbb T}_l$ has dimension $n-3$, we see that, as soon as $n \geq 5$, we have $l \in D_l$. We have the following \begin{lemma} If $n \geq 6$, then $D_l$ is singular at $l$. If $n = 5$, then $D_l$ is smooth at $l$ if $X$ does not have contact multiplicity $3$ along $l$ with the plane ${\Bbb T}_l$ and if there is no line $l'$ of second type in ${\Bbb T}_l$. \end{lemma} {\em Proof :} The case $n=5$ is Lemme 1 on page 590 of \cite{voisin}. Suppose $n \geq 6$. For $l$ general, consider a plane section of $X$ of the form $l+l'+l''$ such that $l \cap l'$ and $l \cap l''$ are general points on $l$. The set of lines through $l \cap l'$ is a divisor in $D_l$ and meets the set of lines through $l \cap l''$ only at $l \in D_l$. So we have two divisors in $D_l$ which meet only at a point and $D_l$ has dimension $\geq 3$. Therefore $D_l$ is not smooth at $l$ for $l$ general and hence for all $l$. \hfill $\qed$ \vskip10pt We now prove an existence result, namely, \begin{lemma} The set of lines $l \in F$ such that $l$ is contained in ${\Bbb T}_{l'} $ for some line $l' \in F$ of second type is a proper closed subset of $F$. In other words (by Lemma \ref{Dlsmooth1}), for $l \in F$ general, the variety $D_l \setminus \{ l \}$ is smooth of dimension $n-3$. \label{lemTl'} \end{lemma} {\em Proof :} Since the dimension of $F$ is $2(n-3)$ and the dimension of the variety $F_0 \subset F$ parametrizing lines of second type is $n-3$ (see \cite{CG} page 311 Corollary 7.6), if the lemma fails, then for any line $l' \in F_0$, the dimension of the family of lines in $X \cap {\Bbb T}_{l'}$ which intersect $l'$ is at least $n-3$. The variety ${\Bbb T}_{l'}$ is a linear subspace of codimension $2$ of ${\Bbb P}^n$. Any plane in ${\Bbb T}_{l'}$ which contains $l'$ is tangent to $X$ along $l'$. The intersection of a general such plane $P$ with $X$ is the union of $l'$ and a line $l$, the line $l'$ having multiplicity $2$ (or $3$ if $l=l'$) in the intersection cycle $[P \cap X]$. Conversely, any line $l$ in $X \cap {\Bbb T}_{l'}$ which intersects $l'$ is contained in such a plane. The family of planes in ${\Bbb T}_{l'}$ which contain $l'$ has dimension $n-4$. Therefore, if the family of lines $l$ in $X \cap {\Bbb T}_{l'}$ which intersect $l'$ has dimension $\geq n-3$, then for each such line $l \neq l'$, the plane $\langle l, l' \rangle$ contains a positive-dimensional family of lines in $X \cap {\Bbb T}_{l'}$ and hence $\langle l, l' \rangle$ is contained in $X \cap {\Bbb T}_{l'}$. Therefore $X \cap {\Bbb T}_{l'}$ is a cone over a cubic hypersurface in ${\Bbb T}_{l'} / l'$ and, for each plane $P \subset X \cap {\Bbb T}_{l'}$ which contains $l'$, there is a hyperplane in ${\Bbb T}_{l'}$ tangent to $X \cap {\Bbb T}_{l'}$ along $P$. Therefore ${\Bbb T}_P \stackrel{def}{=} \cap_{p \in P} {\Bbb P} T'_p X$ has codimension $3$ in ${\Bbb P}^n$. Hence the restriction of the dual morphism of $X$ to $P$ is a morphism of degree $4$ from $P$ onto a plane in $({\Bbb P}^n)^*$. It follows from \cite{CG} Lemma 5.15 page 304 that all such planes are contained in a proper closed subset of $X$. Therefore a general line $l \in F$ is not contained in such a plane and hence not in ${\Bbb T}_{l'}$. Contradiction. \hfill $\qed$ \section{Desingularizing $D_l$} \label{secSl} Let $X_l$ and ${\Bbb P}^n_l$ be the blow ups of $X$ and ${\Bbb P}^n$ respectively along $l$. Then the projection from $l$ gives a projective bundle structure on ${\Bbb P}^n_l$ and a conic bundle structure on $X_l$ (i.e., a general fiber of $\pi_X : X_l \longrightarrow {\Bbb P}^{n-2}$ is a conic in the corresponding fiber of $\pi : {\Bbb P}^n_l \longrightarrow {\Bbb P}^{n-2}$): \[ \begin{array}{crl} X_l & \hookrightarrow & {\Bbb P}^n_l \\ & \pi_X \searrow & \downarrow \pi \\ & & {\Bbb P}^{n-2} \end{array} \] Let $E$ be the locally free sheaf ${\cal O}_{{\Bbb P}^{n-2}}(-1) \oplus {\cal O}_{{\Bbb P}^{n-2}}^{\oplus 2}$. Then it is easily seen (as in e.g. \cite{hartshorne} page 374 Example 2.11.4) that $\pi : {\Bbb P}^n_l \longrightarrow {\Bbb P}^{n-2}$ is isomorphic to the projective bundle ${\Bbb P}(E) \longrightarrow {\Bbb P}^{n-2}$. The variety $X_l \subset {\Bbb P}^n_l$ is the divisor of zeros of a section $s$ of ${\cal O}_{{\Bbb P} E}(2) \otimes \pi^* {\cal O}_{{\Bbb P}^{n-2}}(m)$ for some integer $m$ because the general fibers of $\pi_X : X_l \longrightarrow {\Bbb P}^{n-2}$ are smooth conics in the fibers of $\pi$. Since $\pi_* ({\cal O}_{{\Bbb P} E}(2) \otimes \pi^* {\cal O}_{{\Bbb P}^{n-2}}(m)) \cong Sym^2E^* \otimes {\cal O}_{{\Bbb P}^{n-2}}(m)$, the section $s$ defines a (``symmetric'') morphism of vector bundles $\phi : E \longrightarrow E^* \otimes {\cal O}_{{\Bbb P}^{n-2}}(m)$. The degeneracy locus $Q_l \subset {\Bbb P}^{n-2}$ of this morphism is the locus over which the fibers of $\pi_X$ are singular conics (or have dimension $\geq 2$). By, for instance, intersecting $Q_l$ with a general line, we see that $Q_l$ is a quintic hypersurface (see \cite{segre} pages 3-5). Therefore $m=1$. Let $S_l$ be the variety parametrizing lines in the fibers of $\pi_X$. We have a morphism $S_l \longrightarrow D_l$ defined by sending a line in a fiber of $\pi$ to its image in ${\Bbb P}^n$. Let $E_1 \subset X_l$ be the exceptional divisor of $\epsilon_1 : X_l \rightarrow X$ and let $P_1 \subset S_l$ be the variety parametrizing lines which lie in $E_1$. Then the morphism $S_l \longrightarrow D_l$ induces an isomorphism $S_l \setminus P_1 \cong D_l \setminus \{ l \} $. We have \begin{lemma} Suppose that $l$ is of first type and $D_l \setminus \{ l \}$ is smooth. Then $S_l$ is smooth and irreducible and admits a {\em morphism} of generic degree $2$ onto $Q_l$. The variety $S_l$ can also be defined as the closure of the subvariety of $G(2,n+1) \times G(3,n+1)$ parametrizing pairs $(l',L')$ such that $l' \in D_l \setminus \{ l \}$ and $L' = \langle l,l' \rangle$. \label{lemSlsm} \end{lemma} {\em Proof :} The morphism $S_l \longrightarrow Q_l$ is defined by sending a line in a fiber of $\pi$ to its image in ${\Bbb P}^{n-2}$. It is of generic degree $2$ because the rational map $D_l \longrightarrow Q_l$ is of generic degree $2$. The variety $S_l$ is irreducible because $Q_l$ is irreducible and $S_l \longrightarrow Q_l$ is not split (intersect $Q_l$ with a general plane and use \cite{B5} ). For $l' \in S_l \setminus P_1$, the variety $S_l$ is smooth at $l'$ since $S_l \setminus P_1 \cong D_l \setminus \{ l \} $. For $l' \in P_1$ we determine the Zariski tangent space to $S_l$ at $l'$. Since $l'$ maps to a point in ${\Bbb P}^{n-2}$, it corresponds to a plane $L'$ in ${\Bbb P}^n$ which contains $l$. Since $l'$ is also contained in $E_1$, it maps onto $l$ in ${\Bbb P}^n$ under the blow up morphism ${\Bbb P}^n_l \longrightarrow {\Bbb P}^n$ and $L'$ is tangent to $X$ along $l$. So we easily see that we can identify $S_l$ with the closure of the subvariety of the product of the Grassmannians $G(2,n+1) \times G(3,n+1)$ parametrizing pairs $(l',L')$ such that $l' \in D_l \setminus \{ l \}$ and $L' = \langle l,l' \rangle$. Let $W'$ and $V$ be the vector spaces in ${\Bbb C}^{n+1}$ whose projectivizations are $L'$ and $l$ respectively. The tangent space to $G(2,n+1) \times G(3,n+1)$ at $(l,L')$ can be canonically identified with $ Hom(V, {\Bbb C}^{n+1}/V) \oplus Hom(W', {\Bbb C}^{n+1}/W') $. As in \cite{harris2} Ex. 16.3 pages 202-203 and Ex. 16.21, 16.23 pages 209-210, one can see that the tangent space to $S_l$ at $(l,L')$ can be identified with the set of pairs of homomorphisms $(f,g)$ such that for every nonzero vector $v \in V$ mapping to a point $p$ of $l$, we have $f(v) \in T'_pX / V$, $g(V) = 0$, $g |_V = f(\hbox{mod} W')$ and $g(W') \subset \cap_{p \in l} T'_pX /W'$ (this last condition expresses the fact that the deformation of $L'$ contains a deformation of $l$ which is contained in $X$, hence the deformation of $L'$ is tangent to $X$ along $l$, i.e., is contained in ${\Bbb T}_l$). Equivalently, $g(V) =0$, $f(V) \subset W' /V$ and $g(W') \subset \cap_{p \in l} T'_pX /W'$. Assuming $l$ of first type, we see that the space of such pairs of homomorphisms has dimension $n-3$. \hfill $\qed$ \section{The planes in $X$} \label{secplanes} Let ${\cal P}$ be the variety parametrizing planes in $X$. For $P \in {\cal P}$, we say that $\delta$ has rank $r_P$ on $P$ if the span of $\delta(P)$ has dimension $r_P$. Since $\delta$ is defined by quadrics, we have $r_P \leq 5$. Since $X$ is smooth, we have $r_P \geq 2$. Consider the commutative diagram \[ \begin{array}{rcl} & & {\Bbb P}^5 \\ & v \nearrow & \downarrow p \\ P & \stackrel{\delta_P}{\longrightarrow} & {\Bbb P}^{r_P} \subset ({\Bbb P}^n)^* \end{array} \] where $v$ is the Veronese map, $\delta_P$ is the restriction of $\delta$ to $P$ and $p$ is the projection from a linear space $L \subset {\Bbb P}^5$ of dimension $4 - r_P$ (with the convention that the empty set has dimension $-1$). Note that $L$ does not intersect $v(P)$ because $\delta$ is a morphism. Let ${\cal P}_r$ be the subvariety of ${\cal P}$ parametrizing planes $P$ for which $r_P \leq r$. In this section we will prove a few facts about ${\cal P}$ and ${\cal P}_r$ which we will need later. We begin with \begin{lemma} Let $T \stackrel{def}{=} \cup_{l \subset P} \langle v(l) \rangle \subset {\Bbb P}^5$ be the secant variety of $v(P)$. Then there is a bijective morphism from $T \cap L$ to the parameter space of the family of lines of second type in $P$ and $T \cap L$ contains no positive-dimensional linear spaces. In particular, \begin{enumerate} \item if $r_P = 5$, then $P$ contains no lines of second type, \item if $r_P=4$, then $P$ contains at most one line of second type and this happens exactly when $L$ (which is a point in this case) is in $T$, \item if $r_P = 3$, then $P$ contains one, two or three distinct lines of second type, \item if $r_P=2$, then $P$ contains exactly a one-parameter family of lines of second type whose parameter space is the bijective image of an irreducible and reduced plane cubic. \end{enumerate} \label{linesinP} \end{lemma} {\em Proof :} A line $l \subset P$ is of second type if and only if $\delta_P(l) \subset {\Bbb P}^{r_P}$ is a line, i.e., if and only if the span $\langle v(l) \rangle \cong {\Bbb P}^2$ of the smooth conic $v(l)$ intersects $L$. Consider the universal line $f_1: L_P \rightarrow P^*$ and its embedding $L_P \hookrightarrow V_P$ where $f_2:V_P \rightarrow P^*$ is the projectivization of the bundle $f_* {\cal O}_{L_P}(2)^*$. Then $T$ is the image of $V_P$ in ${\Bbb P}^5$ by a morphism, say $g$, which is an isomorphism on the complement of $L_P$ and contracts $L_P$ onto $v(P)$. Since $L \cap v(P) = \emptyset$, the morphism $g |_{g^{-1}(T \cap L)}$ is an isomorphism, say $g'$. The morphism from $T \cap L$ onto the parameter space of the family of lines of second type in $P$ is the composition of ${g'}^{-1}$ with $f_2$. This morphism is bijective because (since $L \cap v(P) = \emptyset$) the space $L$ intersects any $\langle v(l) \rangle$ in at most one point and any two planes $\langle v(l_1) \rangle$ and $\langle v(l_2) \rangle$ intersect in exactly one point which is $v(l_1 \cap l_2) \in v(P)$. To show that $T \cap L$ contains no positive-dimensional linear spaces, recall that $T$ is the image of the Segre embedding of $P \times P$ in ${\Bbb P}^8 = {\Bbb P} \left( H^0(P, {\cal O}_P (1))^{\otimes 2} \right)^*$ by the projection from ${\Bbb P} \left( \Lambda^2 H^0(P, {\cal O}_P (1)) \right)^*$. Let $R_1$ be the ruling of $T$ by planes which are images of the fibers of the two projections of $P \times P$ onto $P$. Let $R_2$ be the ruling of $T$ by planes of the form $\langle v(l) \rangle$ for some line $l \subset P$. Then a simple computation (determining all the pencils of conics which consist entirely of singular conics) shows that every linear subspace contained in $T$ is contained either in an element of $R_1$ or an element of $R_2$. Therefore, if $L \cap T$ contains a linear space $m$, then either $m \subset \langle v(l) \rangle$ for some line $l \subset P$ or $m \subset L'$ for some element $L'$ of $R_1$. In the first case, the space $m$ is a point because otherwise it would intersect $v(P)$. In the second case, the space $m$ is either a point or a line because any element of $R_1$ contains exactly one point of $v(P)$. It is easily seen that there is an element $s_0 \in H^0(P,{\cal O}_P(1))$ such that $L'$ parametrizes the hyperplanes in $|{\cal O}_P(2)|$ containing all the conics of the form $Z(s.s_0)$ for some $s \in H^0(P,{\cal O}_P(1))$. If $m \subset L'$ is a line, then it is easily seen that the codimension, in $\langle \partial_0 G, ..., \partial_n G \rangle |_P$, of the set of elements of the form $s.s_0$ is one. Restricting to $Z(s_0)$, we see that the dimension of $\langle \partial_0 G, ..., \partial_n G \rangle |_{Z(s_0)}$ is $1$ which is impossible since then $X$ would have a singular point on $Z(s_0)$. Therefore $m$ is always a point if non-empty. \hfill $\qed$ \begin{proposition} The space of infinitesimal deformations of $P$ in $X$ has dimension $3n-15$ if $r_P = 2$. In particular, if $n=5$, then $X$ contains at most a finite number of planes. \label{rP2inf} \end{proposition} {\em Proof :} The intersection ${\Bbb T}_P$ of the projective tangent spaces to $X$ along $P$ has dimension $n-3$. It follows that we have an exact sequence \[ 0 \longrightarrow {\cal O}_P(1)^{n-5} \longrightarrow N_{P/X} \longrightarrow V_2 \longrightarrow 0 \] where $V_2$ is a locally free sheaf of rank $2$. We need to show that $h^0(P,V_2) = 0$. Suppose that there is a nonzero section $u \in H^0(P,V_2)$. We will first show that the restriction of $u$ to any line $l$ in $P$ is nonzero. This will follow if we show that the restriction map $H^0(P,V_2) \longrightarrow H^0(l, V_2|_l)$ is injective, i.e., $h^0(P, V_2(-1)) = 0$. Consider therefore the exact sequence of normal sheaves \[ 0 \longrightarrow N_{P/X} \longrightarrow N_{P/{\Bbb P}^n} \longrightarrow N_{X/{\Bbb P}^n} |_P \longrightarrow 0 \] After tensoring by ${\cal O}_P(-1)$ we obtain the exact sequence \[ 0 \longrightarrow N_{P/X}(-1) \longrightarrow {{\cal O}_P}^{\oplus (n-2)} \longrightarrow {\cal O}_P(2) \longrightarrow 0. \] We can choose our system of coordinates (on ${\Bbb P}^n$) in such a way that $x_3 = ... = x_n = 0$ are the equations for $P$ and the map ${{\cal O}_P}^{\oplus (n-2)} \longrightarrow {\cal O}_P(2)$ in the sequence above is given by multiplication by $\partial_3 G |_P, ... \partial_n G |_P$. So we see that, since $r_P = 2$, the map on global sections $H^0({{\cal O}_P}^{\oplus (n-2)}) \longrightarrow H^0({\cal O}_P(2))$ has rank $3$. Therefore $h^0(P,N_{P/X}(-1)) = n-5$ and $h^0(P, V_2(-1)) = 0$. By Lemma \ref{linesinP}, the plane $P$ contains lines of first type. For any line $l \subset P$ which is of first type, it is easily seen that $V_2 |_l \cong {\cal O}_l^{\oplus 2}$. Hence $u$ has no zeros on $l$. It follows that $Z(u)$ is finite. We compute the total Chern class of $V_2$ as \[ c(V_2) = \frac{c(N_{P/X})}{(1+ \zeta)^{n-5}} = 1+3 \zeta^2 \] where $\zeta = c_1({\cal O}_P(1))$. Therefore $Z(u)$ is a finite subscheme of length $3$ of $P$. Let $l_u$ be a line in $P$ such that $l_u \cap Z(u)$ has length $\geq 2$. Then, by what we saw above, $l_u$ is of second type. It is easily seen that $V_2 |_{l_u} \cong {\cal O}_{l_u}(-1) \oplus {\cal O}_{l_u}(1)$. Restricting $u$ to $l_u$, we see that $Z(u|_{l_u})= l_u \cap Z(u)$ has length $1$ which is a contradiction. So $h^0(P,V_2)=0$ and $h^0(P,N_{P/X}) = 3n-15$. \hfill $\qed$ \vskip20pt The next result we will need is \begin{lemma} The dimension of ${\cal P}_2$ is at most $Min(n-4, 5)$. \label{rP2} \end{lemma} {\em Proof :} The proof of the part $dim({\cal P}_2) \leq n-4$ is similar to the proof of Corollary 7.6 on page 311 of \cite{CG}. To prove $dim({\cal P}_2) \leq 5$, we may suppose that $n \geq 10$. Let $P$ be an element of ${\cal P}_2$. We will show that the space of infinitesimal deformations of $P$ for which the rank of $\delta$ does not increase has dimension at most $5$. Let $x_0, x_1, x_2$ be coordinates on $P$, let $x_0, x_1, x_2, x_3, ..., x_{n-3}$ be coordinates on ${\Bbb T}_P$ and $x_0, ..., x_{n-3}, x_{n-2}, x_{n-1}, x_n$ coordinates on ${\Bbb P}^n$. Then the conditions $P \subset X$ and ${\Bbb T}_P$ is tangent to $X$ along $P$ can be written \[ \partial_i \partial_j \partial_k G = 0 \] for all $i,j \in \{ 0,1,2 \}$, $k \in \{ 0, ..., n-3 \}$, where $G$ is, as before, an equation for $X$ and $\partial_i= \frac{\partial}{\partial x_i}$. We need to determine the infinitesimal deformations of $P$ for which there is an infinitesimal deformation of ${\Bbb T}_P$ which is tangent to $X$ along the deformation of $P$. The infinitesimal deformations of $P$ in ${\Bbb P}^n$ are parametrized by \[ Hom_{{\Bbb C}} \left( \langle \partial_0, \partial_1, \partial_2 \rangle, \frac{{\Bbb C}^{n+1}}{\langle \partial_0, \partial_1, \partial_2 \rangle} \right) \cong Hom_{{\Bbb C}} \left( \langle \partial_0, \partial_1, \partial_2 \rangle, \langle \partial_3, ..., \partial_n \rangle \right) \] and those of ${\Bbb T}_P$ in ${\Bbb P}^n$ are parametrized by \[ Hom_{{\Bbb C}} \left( \langle \partial_0, ..., \partial_{n-3} \rangle, \frac{{\Bbb C}^{n+1}}{\langle \partial_0, ..., \partial_{n-3} \rangle} \right) \cong Hom_{{\Bbb C}} \left( \langle \partial_0, ..., \partial_{n-3} \rangle, \langle \partial_{n-2}, \partial_{n-1}, \partial_n \rangle \right) \] where we also denote by $\partial_i$ the vector in ${\Bbb C}^{n+1}$ corresponding to the differential operator $\partial_i$. We need to determine the homomorphisms $\{ \partial_i \mapsto \partial'_i \in \langle \partial_3, ..., \partial_n \rangle : i \in \{ 0, 1, 2 \} \}$ for which there is a homomorphism $\{ \partial_i \mapsto \partial''_i \in \langle \partial_{n-2}, \partial_{n-1}, \partial_n \rangle : i \in \{ 0, ..., n-3 \} \}$ such that the following conditions hold. \begin{enumerate} \item The vector $\partial''_i$ is the projection of $\partial_i'$ to $\langle \partial_{n-2}, \partial_{n-1}, \partial_n \rangle$ for $i \in \{0,1,2 \}$. This expresses the condition that the infinitesimal deformation of ${\Bbb T}_P$ contains the infinitesimal deformation of $P$. \item For all $i,j \in \{0,1,2 \}$ and $k \in \{0, ..., n-3 \}$, \[ \left( \partial_i + \epsilon \partial'_i \right) \left( \partial_j + \epsilon \partial'_j \right) \left( \partial_k + \epsilon \partial''_k \right) G = 0 \] where, as usual, $\epsilon$ has square $0$. Here we are ``differentiating'' the relations $\partial_i \partial_j \partial_k G =0$. Developing, we obtain \[ \left( \partial_i \partial_j \partial_k'' + \partial_i \partial_j' \partial_k + \partial_i' \partial_j \partial_k \right) G = 0 \: \: . \] \end{enumerate} \vskip10pt Writing $\partial_i' = \sum_{j=3}^n a_{ij} \partial_j$ and $\partial_i'' = \sum_{j=n-2}^n b_{ij} \partial_j$, the above conditions can be written as \begin{enumerate} \item For all $i \in \{0,1,2 \}$ and $j \in \{n-2, n-1, n \}$, \[ a_{ij} = b_{ij} \: \: . \] \item For all $i,j \in \{0,1,2 \}$ and $k \in \{0, ..., n-3 \}$, \[ \sum_{l=n-2}^n b_{kl} \partial_i \partial_j \partial_l G + \sum_{l=3}^n a_{jl} \partial_i \partial_l \partial_k G + \sum_{l=3}^n a_{il} \partial_l \partial_j \partial_k G = 0 \: \: . \] \end{enumerate} Incorporating the first set of conditions in the second and using the relations $\partial_i \partial_j \partial_k G = 0$ for $i,j \in \{0,1,2 \}, k \in \{ 0, ... n-3 \}$, we divide our conditions into two different sets of conditions as follows. We are looking for matrices $(a_{il})_{0 \leq i \leq 2, 3 \leq l \leq n}$ for which there is a matrix $(b_{kl})_{3 \leq k \leq n-3, n-2 \leq l \leq n}$ such that, for all $i,j,k \in \{ 0,1,2 \}$, \[ \sum_{l=n-2}^n \left( a_{kl} \partial_i \partial_j \partial_l + a_{jl} \partial_i \partial_l \partial_k + a_{il} \partial_l \partial_j \partial_k \right) G = 0 \] and, for all $i,j \in \{ 0,1,2 \}$, $k \in \{ 3, ..., n-3 \}$, \[ \sum_{l=n-2}^n b_{kl} \partial_i \partial_j \partial_l G + \sum_{l=3}^n \left( a_{jl} \partial_i \partial_l \partial_k + a_{il} \partial_l \partial_j \partial_k \right) G = 0 \:\: . \] Consider the matrix whose columns are indexed by the $a_{lm}, b_{su}$ ($0 \leq l \leq 2$, $3 \leq m \leq n$, $3 \leq s \leq n-3$, $n-2 \leq u \leq n$), whose rows are indexed by {\em unordered} triples $(i,j,k)$ with $i,j \in \{ 0,1,2 \}$, $k \in \{ 0, ..., n-3 \}$ and whose entries are the $\partial_i \partial_j \partial_m G$, $\partial_i \partial_m \partial_k G$, $\partial_m \partial_j \partial_k G$ or $\partial_i \partial_j \partial_u G$. The entry in the column of $a_{lm}$ and the row of $(i,j,k)$ is nonzero only if $l=i,j$ or $k$. We can, and will, suppose that $l=i$. Here is the list of possibly nonzero such entries. \[ 3 \leq m \leq n \:\: , \:\:\: 3 \leq k \leq n-3 \] \[ \begin{array}{ll} l=i \neq j & \partial_m \partial_j \partial_k G \\ l = i = j & 2 \partial_m \partial_l \partial_k G \end{array} \] \[ n-2 \leq m \leq n \:\: , \:\:\: 0 \leq k \leq 2 \] \[ \begin{array}{ll} l=i \neq j,k & \partial_m \partial_j \partial_k G \\ l = i = j \neq k & 2 \partial_m \partial_l \partial_k G \\ l = i = j = k & 3 \partial_m \partial_l^2 G \: \: . \end{array} \] The entry in the column of $b_{su}$ and the row of $\{i,j,k\}$ is nonzero only if $s=k$. These possibly nonzero entries are the following. \[ n-2 \leq u \leq n \:\: , \:\:\: 3 \leq k \leq n-3 \] \[ \begin{array}{ll} s = k & \partial_i \partial_j \partial_u G \: \: . \end{array} \] An easy dimension count shows that we need to prove that there are at most $6$ relations between the rows of the matrix. Suppose that there are $t$ relations with coefficients \[ \{ \{ \lambda_{ijk}^r \}\begin{Sb} 0 \leq i,j \leq 2 \\ 0 \leq k \leq n-3 \end{Sb} \}_{1 \leq r \leq t} \] between the rows of our matrix. Each relation can be written as a collection $3 \leq m \leq n-3$, $0 \leq i \leq 2$ \[ \sum\begin{Sb} 3 \leq k \leq n-3 \\ 0 \leq j \leq 2 \end{Sb} \lambda_{ijk}^r \partial_m \partial_j \partial_k G = 0 \] $n-2 \leq m \leq n$, $0 \leq i \leq 2$ \begin{equation} \sum\begin{Sb} 0 \leq k \leq n-3 \\ 0 \leq j \leq 2 \end{Sb} \lambda_{ijk}^r \partial_m \partial_j \partial_k G = 0 \label{secondset} \end{equation} $n-2 \leq u \leq n$, $3 \leq k \leq n-3$ \[ \sum\begin{Sb} 0 \leq i,j \leq 2 \end{Sb} \lambda_{ijk}^r \partial_i \partial_j \partial_u G = 0 \] Each expression $\sum_{0 \leq i,j \leq 2} \lambda_{ijk}^r \partial_i \partial_j$ defines a hyperplane in $H^0(P, {\cal O}_P(2))$ which contains the polynomials $\partial_u G |_P$. Since we have $3$ independant such polynomials, the vector space of hyperplanes containing them has dimension $3$. Hence, after a linear change of coordinates, we can suppose that, for $r \in \{0, ..., t-3 \}$, we have $\lambda^r_{ijk} = 0$ if $0 \leq i,j \leq 2, 3 \leq k \leq n-3$. The relations (\ref{secondset}) now become $0 \leq r \leq t-3$, $0 \leq i \leq 2$ \[ \sum\begin{Sb} 0 \leq k \leq 2 \\ 0 \leq j \leq 2 \end{Sb} \lambda_{ijk}^r \partial_j \partial_k G = 0 \: . \] If, for a fixed $r \in \{1, ..., t-3 \}$, the three relations $\sum\begin{Sb} 0 \leq k \leq 2 \\ 0 \leq j \leq 2 \end{Sb} \lambda_{ijk}^r \partial_j \partial_k G = 0$, for $0 \leq i \leq 2$, are not independent, then after a linear change of coordinates, we may suppose that, for instance, $\lambda_{2jk}^r=0$ for all $j,k \in \{0,1,2 \}$. Since the coefficients $\lambda_{ijk}^r$ are symmetric in $i,j,k$, we obtain $0 \leq i \leq 1$ \[ \sum\begin{Sb} 0 \leq k \leq 1 \\ 0 \leq j \leq 1 \end{Sb} \lambda_{ijk}^r \partial_j \partial_k G = 0 \: . \] If $l$ is the line in $P$ obtained as the projectivization of $\langle \partial_0, \partial_1 \rangle$, then $\langle \partial_{n-2} G, \partial_{n-1} G, \partial_n G \rangle |_l$ has dimension at least $2$ and there can be at most one hyperplane in $H^0(l, {\cal O}_l(2))$ containing $\langle \partial_{n-2} G, \partial_{n-1} G, \partial_n G \rangle |_l$. In other words, up to multiplication by a scalar, there is at most one nonzero relation $\sum\begin{Sb} 0 \leq k \leq 1 \\ 0 \leq j \leq 1 \end{Sb} \lambda_{ijk}^r \partial_j \partial_k G = 0$. Hence, we can suppose $\lambda_{1jk}^r=0$ for all $j,k \in \{ 0,1 \}$. Again, by symmetry, we are reduced to $\lambda_{000}^r \partial_0^2 G = 0$ which implies $\lambda_{000}^r = 0$ because $X$ is smooth. Hence all the $\lambda_{ijk}^r$ are zero. Therefore, if the $\lambda_{ijk}^r$ are not all zero, the three relations $0 \leq i \leq 2$ \[ \sum\begin{Sb} 0 \leq k \leq 2 \\ 0 \leq j \leq 2 \end{Sb} \lambda_{ijk}^r \partial_j \partial_k G = 0 \] are independent. If $t-3 \geq 4$, then, after a linear change of coordinates, for some $r \in \{1, ..., t-3 \}$, one of the above three relations will be trivial and we are reduced to the previous case. Therefore $t-3 \leq 3$ and $t \leq 6$. \hfill $\qed$ \vskip20pt \begin{proposition} Suppose that $n \geq 6$. Then ${\cal P}$ has pure dimension equal to the expected dimension $3n-16$. If $r_P \geq 3$, then ${\cal P}$ is smooth at $P$. \label{cPsmooth} \end{proposition} {\em Proof :} Since the dimension of ${\cal P}_2$ is at most $Min(n-4,5)$ by Lemma \ref{rP2} and the dimension of every irreducible component of ${\cal P}$ is at least $3n-16$, it is enough to show that for every $P$ such that $r_P \geq 3$, the space $H^0(P, N_P)$ of infinitesimal deformations of $P$ in $X$ has dimension $3n-16$. \vskip10pt Suppose that $r_P = 3$. As in the proof of Proposition \ref{rP2inf}, we have an exact sequence \[0 \longrightarrow {\cal O}_P(1)^{\oplus (n-6)} \longrightarrow N_{P/X} \longrightarrow V_3 \longrightarrow 0 \] where $V_3$ is a locally free sheaf of rank $3$. Since $h^0(P, N_{P/X}) \geq 3n-16$, we have $h^0(P,V_3) \geq 2$. We need to show that $h^0(P,V_3) = 2$. As in the proof of Proposition \ref{rP2inf} we have $h^0(P, V_3(-1))=0$ so that, for any line $l \subset P$, \[ H^0(P, V_3) \hookrightarrow H^0(l, V_3|_l) \:\: . \] Suppose that $h^0(P, V_3) \geq 3$ and let $u_1, u_2, u_3$ be three linearly independent elements of $H^0(P, V_3)$. By Lemma \ref{linesinP}, the plane $P$ contains at least one line $l_0$ of second type. It is easily seen that $V_3|_{l_0} \cong {\cal O}_{l_0}(-1) \oplus {\cal O}_{l_0}(1)^{\oplus 2}$. Therefore $\langle u_1, u_2, u_3 \rangle |_{l_0}$ generates a subsheaf of the ${\cal O}_{l_0}(1)^{\oplus 2}$ summand of $V_3 |_{l_0}$ isomorphic to ${\cal O}_{l_0} \oplus {\cal O}_{l_0}(1)$. The quotient of ${\cal O}_{l_0}(1)^{\oplus 2}$ by ${\cal O}_{l_0} \oplus {\cal O}_{l_0}(1)$ is a skyscraper sheaf supported on a point $p$ of $l_0$ (with fiber at $p$ isomorphic to ${\Bbb C}$). So the images of $u_1, u_2$ and $u_3$ by the evaluation map at $p$ generate a one-dimensional vector subspace of the fiber of $V_3$ at $p$. By Lemma \ref{linesinP}, there is a line $l$ of first type in $P$ which contains $p$. It is easily seen that $V_3|_l \cong {\cal O}_l^{\oplus 2} \oplus {\cal O}_l(1)$. Restricting $u_1, u_2, u_3$ to $l$ we see that their images by the evaluation map at $p$ generate a vector subspace of dimension $\geq 2$ of the fiber of $V_3$ at $p$. Contradiction. \vskip10pt Suppose now that $r_P =4$. Then $n \geq 7$ and we have the exact sequence \[ 0 \longrightarrow {\cal O}_P(1)^{\oplus (n-7)} \longrightarrow N_{P/X} \longrightarrow V_4 \longrightarrow 0 \] where $V_4$ is a locally free sheaf of rank $4$. Since $h^0(P, N_{P/X}) \geq 3n-16$, we have $h^0(P, V_4) \geq 5$. We need to show that $h^0(P, V_4) = 5$. As before, $h^0(P, V_4(-1)) = 0$, hence, for any line $l \subset P$, we have $H^0(P, V_4) \hookrightarrow H^0(l, V_4|_l)$. It is easily seen that when $l$ is of first type $V_4|_l \cong {\cal O}_l^{\oplus 2} \oplus {\cal O}_l(1)^{\oplus 2}$ and when $l$ is of second type $V_4|_l \cong {\cal O}_l(-1) \oplus {\cal O}_l(1)^{\oplus 3}$. Thus $h^0(P, V_4) \leq 6$. Suppose that $h^0(P, V_4) = 6$. Then $H^0(P, V_4) \stackrel{\cong}{\longrightarrow} H^0(l, V_4 |_l)$ for every line $l \subset P$. Suppose first that $P$ contains a line $l_0$ of second type and let $l$ be a line of first type in $P$. We see that $V_4$ is not generated by its global sections anywhere on $l_0$ whereas $V_4 |_l$ is generated by its global sections. This gives a contradiction at the point of intersection of $l$ and $l_0$. So every line $l$ in $P$ is of first type, $V_4|_l \cong {\cal O}_l^{\oplus 2} \oplus {\cal O}_l(1)^{\oplus 2}$ and $V_4$ is generated by its global sections. Let $s$ be a general global section of $V_4$. We claim that $s$ does not vanish at any point of $P$. Indeed, since $V_4$ is generated by its global sections, for every point $p$ of $P$, the vector space of global sections of $V_4$ vanishing at $p$ has dimension $2$. Hence the set of all global sections of $V_4$ vanishing at some point of $P$ has dimension $\leq 2+2 = 4 < 6$. So we have the exact sequence \[ 0 \longrightarrow {\cal O}_P \stackrel{s}{\longrightarrow} V_4 \longrightarrow V \longrightarrow 0 \] where $V$ is a locally free sheaf of rank $3$. Since $V_4$ is generated by its global sections, so is $V$ and we have $h^0(P, V) = 5$. As before a general global section $s'$ of $V$ does not vanish anywhere on $P$ and we have the exact sequence \[ 0 \longrightarrow {\cal O}_P \stackrel{s'}{\longrightarrow} V \longrightarrow V' \longrightarrow 0 \] where $V'$ is a locally free sheaf of rank $2$. We have $h^0(P, V') = 4$ and $h^0(V'(-1))=h^0(V(-1))=h^0(V_4(-1))=0$. Hence for every line $l \subset P$, $H^0(P,V') \hookrightarrow H^0(l,V'|_l)$. Since $V'|_l \cong {\cal O}_l(1)^{\oplus 2}$, for a nonzero section $s$ of $V'$ the scheme $Z(s|_l)=Z(s) \cap l$ has length $\leq 1$. The scheme $Z(s)$ is not a line because $H^0(P,V') \rightarrow H^0(Z(s),V'|_{Z(s)})$ is injective. Hence for a general line $l \subset P$, $Z(s) \cap l$ is empty. Therefore $Z(s)$ is finite. We compute $c(V') = c(V) = c(V_4) = 1 + 2 \zeta + 4 \zeta^2$. Therefore $Z(s)$ has length $4$. Hence there is a line $l$ such that $Z(s_l)$ has length $\geq 2$ and this contradicts $length(Z(s_l)) \leq 1$. \vskip10pt If $r_P = 5$, consider again the exact sequence of normal sheaves \[ 0 \longrightarrow N_{P/X} \longrightarrow N_{P/{\Bbb P}^n} \longrightarrow N_{X/ {\Bbb P}^n} |_P \longrightarrow 0 \] which after tensoring by ${\cal O}_P(-1)$ becomes \[ 0 \longrightarrow N_{P/X}(-1) \longrightarrow {\cal O}_P^{\oplus (n-2)} \longrightarrow {\cal O}_P(2) \longrightarrow 0 \: . \] Then the map on glabal sections \[ H^0(P, {\cal O}_P^{\oplus (n-2)}) \longrightarrow H^0(P, {\cal O}_P(2)) \] is onto (see the proof of Proposition \ref{rP2inf}). A fortiori, the map \[ H^0(P, N_{P/{\Bbb P}^n})=H^0(P, {\cal O}_P(1)^{\oplus (n-2)}) = \] \[ = H^0(P, {\cal O}_P^{\oplus (n-2)}) \otimes H^0(P, {\cal O}_P(1)) \longrightarrow H^0(P, {\cal O}_P(3))= H^0(P, N_{X/ {\Bbb P}^n} |_P) \] is onto and $H^0(P, N_{P/X})$ has dimension $3n-16$. \hfill $\qed$ \begin{corollary} If $n \geq 8$, then ${\cal P}$ is irreducible. \end{corollary} {\em Proof :} As before, let $G$ be an equation for $X$. Choose a linear embedding ${\Bbb P}^n \hookrightarrow {\Bbb P}^{n+1}$. Choose coordinates $\{ x_0, ..., x_n \}$ on ${\Bbb P}^n$ and coordinates $\{ x_0, ..., x_n, x_{n+1} \}$ on ${\Bbb P}^{n+1}$. Let $Y \subset {\Bbb P}^{n+1}$ be the cubic of equation $G+ x_{n+1} Q$ where $Q$ is the equation of a general quadric in ${\Bbb P}^{n+1}$ and let ${\cal P}_Y \supset {\cal P}$ be the variety of planes in $Y$. Then, by Proposition \ref{cPsmooth}, the codimension of ${\cal P}$ in ${\cal P}_Y$ is $3$. The singular locus of ${\cal P}$ is ${\cal P}_2$ (\ref{rP2inf} and \ref{cPsmooth}) which has codimension at least $4$ in ${\cal P}$ by \ref{rP2} and \ref{cPsmooth}. Therefore, since ${\cal P}$ is connected (\cite{borcea1} Theorem 4.1 page 33 or \cite{debarremanivel} Th\'eor\`eme 2.1 pages 2-3), it is sufficient to show that ${\cal P}_Y$ is smooth at a general point of ${\cal P}_2$. Since $Q$ does not contain a general plane $P \in {\cal P}_2$, the rank of the dual morphism of $Y$ on $P$ is at least $3$. Hence ${\cal P}_Y$ is smooth at a general point of ${\cal P}_2$ (\ref{cPsmooth}). \hfill $\qed$ \begin{lemma} The dimension of ${\cal P}_3$ is at most $n-2$. \label{rP3} \end{lemma} {\em Proof :} It is enough to show that at any $P$ with $r_P \leq 3$ the dimension of the tangent space to ${\cal P}_3$ is at most $n-2$. By \ref{rP2} it is enough to prove this for $r_P = 3$. The proof of this is very similar to (and simpler than) the proof of Lemma \ref{rP2}. \hfill $\qed$ \begin{proposition} If $n \geq 7$, then ${\cal P}_4$ has pure dimension $2n-9$. \label{rP4} \end{proposition} {\em Proof :} For $n=7$ there is nothing to prove since ${\cal P}$ has pure dimension $5=3.7-16=2.7-9$ and ${\cal P} = {\cal P}_4$. Suppose $n \geq 8$. By an easy dimension count, the dimension of every irreducible component of ${\cal P}_4$ is at least $2n-9$. Since the dimension of ${\cal P}_3$ is at most $n-2 < 2n-9$ (see \ref{rP3}), for a general element $P$ of any irreducible component of ${\cal P}_4$ we have $r_P = 4$. We first show \begin{lemma} Suppose $n \geq 8$. Then the subscheme ${\cal P}_4'$ of ${\cal P}_4$ parametrizing planes which contain a line of second type has pure dimension $2n-10$. \label{rP42} \end{lemma} {\em Proof :} Again by a dimension count, the dimension of every irreducible component of ${\cal P}'_4$ is at least $2n - 10$. Let $P$ be an element of ${\cal P}_4'$. By \ref{rP3}, the scheme ${\cal P}_3 \subset {\cal P}_4'$ has dimension $\leq n-2 \leq 2n-10$, so we may suppose that $r_P =4$. Let $l$ be the unique line of second type contained in $P$ (see Lemma \ref{linesinP}). Since the family of lines of second type in $X$ has dimension $n-3$ (see \cite{CG} Corollary 7.6), it is enough to show that the space of infinitesimal deformations of $P$ in $X$ which contain $l$ has dimension $n-7$. Consider the exact sequence of sheaves \[ 0 \longrightarrow N_{P/X}(-1) \longrightarrow N_{P/X} \longrightarrow N_{P/X} |_l \longrightarrow 0 \] with associated cohomology sequence \[ 0 \longrightarrow H^0(P, N_{P/X}(-1)) \longrightarrow H^0(P, N_{P/X}) \longrightarrow H^0(P, N_{P/X} |_l) \longrightarrow H^1(P, N_{P/X}(-1)) \longrightarrow ... \] The space of infinitesimal deformations of $P$ in $X$ which contain $l$ can be identified with the kernel of the homomorphism $H^0(P, N_{P/X}) \longrightarrow H^0(P, N_{P/X} |_l) $ which, by the above sequence, can be identified with $H^0(P, N_{P/X}(-1))$. Recall the exact sequence \[ 0 \longrightarrow N_{P/X}(-1) \longrightarrow {\cal O}_P^{\oplus (n-2)} \longrightarrow {\cal O}_P(2) \longrightarrow 0 \] where the map ${\cal O}_P^{\oplus (n-2)} \longrightarrow {\cal O}_P(2)$ is given by multiplication by $\partial_3 G, ..., \partial_n G$ (see the proof of \ref{rP2inf}). It immediately follows that $h^0(P, N_{P/X}(-1)) = n-7$ if and only if $r_P=4$. \hfill $\qed$ \vskip20pt Note that containing a line of second type imposes at most one condition on planes $P$ with $r_P \leq 4$. Therefore Proposition \ref{rP4} follows from Lemma \ref{rP42}. \hfill $\qed$ \section{Resolving the indeterminacies of the rational involution on $S_l$} \label{secSl'} A good generalization of the Prym construction for cubic threefolds to cubic hypersurfaces of higher dimension would be to realize the cohomology of $X$ as the anti-invariant part of the cohomology of $S_l$ for the involution exchanging two lines whenever they are in the same fiber of $\pi$. However, this is only a rational involution and we need to resolve its indeterminacies. This involution is not well-defined exactly at the lines $l'$ such that $\pi^{-1}(\pi(l')) \subset X_l$, i.e., the plane $L' \subset {\Bbb P}^n$ corresponding to $\pi(l')$ is contained in $X$. Let $T_l \subset Q_l \subset {\Bbb P}^{n-2}$ be the variety parametrizing the planes in ${\Bbb P}^n$ which contain $l$ and are contained in $X$ (equivalently, the variety $T_l$ parametrizes the fibers of $\pi$ which are contained in $X_l$). Recall that $X_l \subset {\Bbb P}^n_l$ is the divisor of zeros of $s \in H^0({\Bbb P} E, {\cal O}_{{\Bbb P} E}(2) \otimes \pi^* {\cal O}_{{\Bbb P}^{n-2}}(1)) = H^0({\Bbb P}^{n-2}, \pi_*({\cal O}_{{\Bbb P} E}(2)) \otimes {\cal O}_{{\Bbb P}^{n-2}}(1)) = H^0({\Bbb P}^{n-2}, Sym^2E^* \otimes {\cal O}_{{\Bbb P}^{n-2}}(1))$. Since $E \cong {\cal O}_{{\Bbb P}^{n-2}}(-1) \oplus {\cal O}_{{\Bbb P}^{n-2}}^{\oplus 2}$, we have $Sym^2E^* \otimes {\cal O}_{{\Bbb P}^{n-2}}(1) \cong {\cal O}_{{\Bbb P}^{n-2}}(3) \oplus {\cal O}_{{\Bbb P}^{n-2}}(2)^{\oplus 2} \oplus {\cal O}_{{\Bbb P}^{n-2}}(1)^{\oplus 3}$. The variety $T_l$ is the locus of common zeros of all the components of $s$ in the above direct sum decomposition. Therefore $T_l$ is the scheme-theoretic intersection of three hyperplanes, two quadrics and one cubic in ${\Bbb P}^{n-2}$. We have \begin{lemma} There is a Zariski-dense open subset of $F$ parametrizing lines $l$ such that $l$ is of first type and $r_P=5$ for every plane $P$ in $X$ containing $l$. For $l$ in this Zariski-dense open subset, the variety $T_l$ is the smooth complete intersection of the six hypersurfaces obtained as the zero loci of the components of $s$ in the direct sum decomposition $Sym^2E^* \otimes {\cal O}_{{\Bbb P}^{n-2}}(1) \cong {\cal O}_{{\Bbb P}^{n-2}}(3) \oplus {\cal O}_{{\Bbb P}^{n-2}}(2)^{\oplus 2} \oplus {\cal O}_{{\Bbb P}^{n-2}}(1)^{\oplus 3}$. \label{Tlsmooth} \end{lemma} {\em Proof :} The first part of the lemma follows from Proposition \ref{rP4}. For the second part we need to show that $T_l$ is smooth of the expected dimension $n-8$. In other words, for any plane $P$ containing $l$, the space of infinitesimal deformations of $P$ in $X$ containing $l$ has dimension $n-8$. The proof of this is similar to the proof of Lemma \ref{rP42}. \hfill \qed \vskip20pt \begin{definition} Let $U_0$ be the subvariety of $F$ parametrizing lines $l$ such that $l$ is of first type, is not contained in ${\Bbb T}_{l'}$ for any line $l'$ of second type and every plane containing $l$ is an element of ${\cal P} \setminus {\cal P}_4$. \end{definition} By Lemmas \ref{lemTl'} and \ref{Tlsmooth}, the variety $U_0$ is an open dense subvariety of $F$. Suppose $l \in U_0$. By Lemmas \ref{Dlsmooth1}, \ref{lemSlsm} and \ref{Tlsmooth}, the varieties $S_l$ and $T_l$ are smooth of the expected dimensions $n-3$ and $n-8$ respectively. Let $X'_l \subset {{\Bbb P}^n_l}'$ be the blow ups of $X_l \subset {\Bbb P}^n_l$ along $\pi^{-1}(T_l)$ and let ${{\Bbb P}^{n-2}}'$ be the blow up of ${\Bbb P}^{n-2}$ along $T_l$. Then we have morphisms \[ \begin{array}{crl} X'_l & \subset & {{\Bbb P}^n_l}' \\ & \pi_X' \searrow & \downarrow \pi' \\ & & {{\Bbb P}^{n-2}}' \end{array} \] where $\pi' : {{\Bbb P}^n_l}' \longrightarrow {{\Bbb P}^{n-2}}'$ is again a ${\Bbb P}^2$-bundle. Since $T_l$ is the zero locus of $s \in H^0({\Bbb P}^{n-2}, \pi_* {\cal O}_{{\Bbb P} E}(2) \otimes {\cal O}_{{\Bbb P}^{n-2}}(1))$, we have $N_{T_l/{\Bbb P}^{n-2}} \cong \pi_* {\cal O}_{{\Bbb P} E}(2) \otimes {\cal O}_{{\Bbb P}^{n-2}}(1) |_{T_l}$. Therefore, the exceptional divisor $E'$ of ${{\Bbb P}^{n-2}}' \longrightarrow {\Bbb P}^{n-2}$ is a ${\Bbb P}^5$-bundle over $T_l$ whose fiber at a point $t \in T_l$ corresponding to the plane $P_t \subset X_l$ is $| {\cal O}_{P_t}(2) |$. We have \begin{lemma} Suppose that $l \in U_0$. For all $t \in T_l$, the restriction of $\pi_X' : X_l' \longrightarrow {{\Bbb P}^{n-2}}'$ to $| {\cal O}_{P_t}(2) | \subset {{\Bbb P}^{n-2}}'$ is the universal conic on $| {\cal O}_{P_t}(2) |$. In particular, the fibers of $\pi_X' : X'_l \longrightarrow {{\Bbb P}^{n-2}}'$ are always one-dimensional. \label{fibonedim} \end{lemma} {\em Proof :} The restriction of $\pi'$ to the inverse image of a point $t \in T_l$ is the second projection $P_t \times | {\cal O}_{P_t}(2) | \longrightarrow | {\cal O}_{P_t}(2) |$. Let $N_{X,p}$ be the normal space in $X_l$ to $\pi^{-1}(T_l)$ at $p \in P_t$ and let $\rho_t : P_t \longrightarrow | {\cal O}_{P_t}(2)|^* \cong {\Bbb P}^5$ be the map which to $p \in P_t$ associates ${\Bbb P} N_{X,p} \in | {\cal O}_{P_t}(2) |^*$. For $n \in | {\cal O}_{P_t}(2) |$, the fiber of $\pi'_X$ at $(t,n) \in E'$ is equal to $\rho_t^{-1}(\rho_t(P_t) \cap H_n)$ where $H_n$ is the hyperplane in $| {\cal O}_{P_t}(2) |^*$ corresponding to $n$. It is immediately seen that $\rho_t$ is induced by the dual morphism $\delta$ of $X$. Hence, since $r_{P_t} = 5$, the map $\rho_t$ is the Veronese morphism $P_t \longrightarrow | {\cal O}_{P_t}(2) |^*$. Hence $\rho_t^{-1}(\rho_t(P_t) \cap H_n)$ is the conic in $P_t$ corresponding to $n$. \hfill $\qed$ \vskip20pt It follows from lemma \ref{fibonedim} that if we let $S'_l$ be the variety parametrizing lines in the fibers of $\pi_X' : X'_l \longrightarrow {{\Bbb P}^{n-2}}'$, then there is a well-defined involution $i_l : S'_l \longrightarrow S_l'$ which sends $l'$ to $l''$ when $l'+l''$ is a fiber of $X'_l \longrightarrow {{\Bbb P}^{n-2}}'$. Sending a line in a fiber of $\pi'_X$ to its image in $X_l$ defines a morphism $S_l' \longrightarrow S_l$. Let ${\cal P}_l \rightarrow T_l$ be the family of planes in $X$ containing $l$, then the inverse image of $T_l$ in $S_l$ by the morphism $S_l \rightarrow Q_l$ is the projective bundle ${\cal P}_l^*$ of lines in the fibers of ${\cal P}_l \rightarrow T_l$. We have \begin{proposition} Suppose that $l \in U_0$. The morphism $S_l' \longrightarrow S_l$ is the blow up of $S_l$ along ${\cal P}^*_l$. In particular, the variety $S_l'$ is smooth. The fixed point locus $R_l'$ of $i_l$ in $S_l'$ is a smooth subvariety of codimension $2$ of $S_l'$. The projective bundle ${\Bbb P}(N_{R_l'/S_l'}) \longrightarrow R_l'$ is isomorphic to the family of lines in the fibers of $\pi_X'$ parametrized by $R_l'$. \label{everythingsmooth} \end{proposition} {\em Proof :} In Lemma \ref{lemSlsm}, we saw that $S_l$ can be identified with the closure of the subvariety $G(2,n+1) \times G(3,n+1)$ parametrizing pairs $(l',L')$ of a line and a plane such that $l \neq l'$ and $l \cup l' \subset L'$. In the same way, we see that $S_l'$ can be identified with the closure of the subvariety of $G(2,n+1) \times G(2,n+1) \times G(3,n+1)$ parametrizing triples $(l',l'',L')$ such that $L' \cap X \supset l \cup l' \cup l''$ and $l, l', l''$ are distinct. Furthermore, the morphism $S'_l \rightarrow S_l$ is the restriction of the projection to the second and third factors of $G(2,n+1) \times G(2,n+1) \times G(3,n+1)$. Again as in the proof of Lemma \ref{lemSlsm} we see that $S_l'$ is smooth. Blowing up ${\cal P}_l^*$ and its inverse image in $S_l'$ we obtain the commutative diagram \[ \begin{array}{ccc} \widetilde{S}_l' & \longrightarrow & \widetilde{S}_l \\ \downarrow & & \downarrow \\ S_l' & \longrightarrow & S_l . \end{array} \] Since the inverse image of ${\cal P}_l^*$ is a divisor in $S_l'$, the blow up morphism $\widetilde{S}_l' \rightarrow S_l'$ is an isomorphism. The morphism $S_l' \rightarrow \widetilde{S}_l$ thus obtained is a birational morphism of smooth varieties with constant fiber dimension hence it is an isomorphism. This proves the first part of the Proposition. Now let $\Delta$ be the diagonal of $G(2,n+1) \times G(2,n+1)$. Then the variety $R_l'$ is identified with $S'_l \cap \left( \Delta \times G(3,n+1) \right)$. One now computes the tangent space to $R_l'$ as in the proof of Lemma \ref{lemSlsm} and see that $N_{R_l'/S_l'}$ is isomorphic to $I^* \otimes J/I$ where $I$ is the restriction of the universal bundle on $G(2,n+1)$ and $J$ is the restriction of the universal bundle on $G(3,n+1)$. Therefore ${\Bbb P}(N_{R_l'/S_l'})$ is isomorphic to ${\Bbb P}(I)$ which is the family of lines in the fibers of $\pi_X'$ parametrized by $R_l'$. \hfill \qed \vskip15pt Let $Q_l'$ be the blow up of $Q_l$ along $T_l$. Sending a line $l \in S_l'$ to the fiber of $X'_l \longrightarrow {{\Bbb P}^{n-2}}'$ which contains it defines a finite morphism $S'_l \longrightarrow Q_l'$ of degree $2$ with ramification locus $R_l'$. Blowing up $R_l'$ in $Q_l'$ and $S_l'$ we obtain the morphism $S_l'' \longrightarrow Q_l''$. We have \begin{proposition} \label{propQ'''} The variety $R_l'$ is an ordinary double locus for $Q_l'$. In particular, $Q_l''$ is smooth and (by \ref{everythingsmooth}) the projectivization ${\Bbb P}(C_{R_l'/Q_l''})$ of the normal cone to $R_l'$ in $Q_l'$ is isomorphic to ${\Bbb P}(N_{R_l'/S_l'})$. \end{proposition} {\em Proof :} The fact that $R_l \setminus T_l$ is an ordinary double locus for $Q_l \setminus T_l$ can be proved, for instance, by intersecting $Q_l$ with a general plane through a point $p$ of $R_l \setminus T_l$. The resulting curve has an ordinary double point at $p$ by \cite{B2} Proposition 1.2 page 321. At a point $q$ of the exceptional divisor of $R_l' \rightarrow R_l$, locally trivialize the pull-back of $E={\cal O}_{{\Bbb P}^{n-2}}^{\oplus 2} \oplus {\cal O}_{{\Bbb P}^{n-2}}(-1)$ to obtain a morphism from a neighborhood $U$ of $q$ to $|{\cal O}_{{\Bbb P}^2}(2)|$. It easily follows from \ref{Tlsmooth} and \ref{fibonedim} that this morphism is dominant and the restriction of $X_l' \rightarrow {{\Bbb P}^{n-2}}'$ to $U$ is the inverse image of the universal conic on $|{\cal O}_{{\Bbb P}^2}(2)|$. The assertion of the Proposition now follows from the corresponding fact for the cubic fourfold parametrizing singular conics in ${\cal P}^2$. \hfill \qed \vskip15pt \section{The main Theorem} \label{secmaintheorem} Let $L_l \longrightarrow S'_l$ and $\overline{L}_l \longrightarrow S_l$ be the families of lines in the fibers of $\pi'_X$ and $\pi_X$ respectively. The blow-up morphism $\epsilon_2 : X_l' \longrightarrow X_l$ defines a morphism $L_l \longrightarrow \overline{L}_l$ which fits into the commutative diagram \[ \begin{array}{rclcc} X_l' & \stackrel{\epsilon_2}{\longrightarrow} & X_l & \stackrel{\epsilon_1}{\longrightarrow} & X \\ \rho \uparrow & & \uparrow \overline{\rho} & & \\ L_l & \longrightarrow & \overline{L}_l & & \\ p \downarrow & & \downarrow \overline{p} & & \\ S_l' & \longrightarrow & S_l & & \end{array} \] where the squares are Cartesian. Put $q = \epsilon_1 \epsilon_2 \rho$ and let $\psi' = q_* p^* : H^{n-3}(S'_l, {\Bbb Z}) \longrightarrow H^{n-1}(X, {\Bbb Z})$ and $\psi = (\epsilon_1 \overline{\rho})_* \overline{p}^*: H^{n-3}(S_l, {\Bbb Z}) \longrightarrow H^{n-1}(X, {\Bbb Z})$ be the Abel-Jacobi maps. The map $\psi$ is the composition of $\psi'$ with the inclusion $H^{n-3}(S_l, {\Bbb Z}) \hookrightarrow H^{n-3}(S'_l, {\Bbb Z})$ because the bottom (or top) square above is Cartesian. We have \begin{theorem} The maps $\psi : H^{n-3}(S_l, {\Bbb Z}) \longrightarrow H^{n-1}(X, {\Bbb Z})$ and $\psi' : H^{n-3}(S'_l, {\Bbb Z}) \longrightarrow H^{n-1}(X, {\Bbb Z})$ are onto. \label{thmpsionto} \end{theorem} {\em Proof :} Consider the rational map $Q_l' \longrightarrow X_l'$ which to the singular conic $l'+l''$ associates the point of intersection $l' \cap l''$. An easy local computation shows that the closure of the image of this map is smooth, hence, by a reasoning analogous to the proof of Proposition \ref{everythingsmooth}, it can be identified with $Q_l''$. Let $\epsilon_3 : X_l'' \longrightarrow X_l'$ be the blow up of $X_l'$ along $Q_l''$ and, for each $i$ ($1 \leq i \leq 3$), let $E_i$ be the exceptional divisor of the blow up map $\epsilon_i$. Then we have a factorization \[ \begin{array}{crl} & & X_l'' \\ & \widetilde{q} \nearrow & \downarrow \epsilon_3\\ L_l & \stackrel{\rho}{\longrightarrow} & X_l' \end{array} \] so that $\psi' = q_*p^* = {\epsilon_1}_* {\epsilon_2}_* \rho_* p^* = {\epsilon_1}_* {\epsilon_2}_* {\epsilon_3}_* \widetilde{q}_* p^* $. Note that $\widetilde{q}$ is an embedding so that we can, and will, identify $L_l$ with $\widetilde{q}(L_l)$. Put $U_l = X''_l \setminus (E_3 \cup L_l) = X'_l \setminus \rho(L_l)$. Let $m_l : U_l \longrightarrow X''_l$ be the inclusion. We have the spectral sequence \[ E^{p,q}_2 = H^p(X''_l, R^q{m_l}_* {\Bbb Z}_{U_l}) \implies H^{p+q}(U_l, {\Bbb Z}) \] and by \cite{deligne}, $\S$3.1, we have $R^0{m_l}_* {\Bbb Z}_{U_l} = {\Bbb Z}_{X''_l}, R^1{m_l}_* {\Bbb Z}_{U_l} = {\Bbb Z}_{E_3} \oplus {\Bbb Z}_{L_l}, R^2{m_l}_* {\Bbb Z}_{U_l} = {\Bbb Z}_{E_3 \cap L_l}$ and $R^q {m_l}_* {\Bbb Z}_{U_l} = 0$ for $q > 2$. Note that $E_3 \cap L_l \cong S_l''$. Therefore \[ \begin{array}{c} E^{p,0}_2 = H^p(X''_l, {\Bbb Z}) \: , \\ E^{p,1}_2 = H^p(X''_l, {\Bbb Z}_{E_3} \oplus {\Bbb Z}_{L_l}) = H^p(L_l, {\Bbb Z}) \oplus H^p(E_3, {\Bbb Z}) \: , \\ E^{p,2}_2 = H^p(X''_l, {\Bbb Z}_{S_l''} ) = H^p(S''_l, {\Bbb Z}) \: , \\ E^{p,q}_2 = 0 \hbox{ for } q > 2 \end{array} \] So the $E_2^{.,.}$ complex is \[ 0 \longrightarrow H^{p-2}(S''_l, {\Bbb Z}) \longrightarrow H^p(L_l, {\Bbb Z}) \oplus H^p(E_3, {\Bbb Z}) \longrightarrow H^{p+2}(X''_l, {\Bbb Z}) \longrightarrow 0 \] where the maps are obtained by Poincar\'e Duality from the natural push-forwards on homology induced by the inclusions. We have (see, for instance, \cite{B2}, 0.1.3, page 312) \begin{eqnarray}H^{p+2}(X''_l, {\Bbb Z}) \cong H^{p+2}(X'_l, {\Bbb Z}) \oplus H^p(Q''_l, {\Bbb Z}) \:, \label{cohomXlll} \\ H^{p+2}(X'_l, {\Bbb Z}) \cong H^{p+2}(X_l, {\Bbb Z}) \oplus \left( \oplus\begin{Sb} p-6 \leq i \leq p \\ i \equiv p [2] \end{Sb} H^i(\pi^{-1}(T_l), {\Bbb Z}) \right) \: , \label{cohomXll} \\ H^{p+2}(X_l, {\Bbb Z}) \cong H^{p+2}(X, {\Bbb Z}) \oplus \left( \oplus\begin{Sb} p-2(n-4) \leq i \leq p \\ i \equiv p [2] \end{Sb} H^i(l, {\Bbb Z}) \right) \label{cohomXl} \end{eqnarray} and \begin{eqnarray} H^{p-2}(S''_l, {\Bbb Z}) \cong H^{p-2}(S'_l, {\Bbb Z}) \oplus H^{p-4}(R'_l, {\Bbb Z}). \end{eqnarray} Since $E_3$ and $L_l$ are ${\Bbb P}^1$-bundles over $Q''_l$ and $S'_l$ respectively, \begin{eqnarray} H^p(E_3, {\Bbb Z}) \cong H^p(Q''_l, {\Bbb Z}) \oplus H^{p-2}(Q''_l, {\Bbb Z}) \end{eqnarray} and \begin{eqnarray} H^p(L_l, {\Bbb Z}) \cong H^p(S'_l, {\Bbb Z}) \oplus H^{p-2}(S'_l, {\Bbb Z}) \: . \label{cohomLl} \end{eqnarray} The map $\psi'$ is the composition of the inclusion $H^{n-3}(S'_l, {\Bbb Z}) \hookrightarrow H^{n-3}(L_l, {\Bbb Z})$ obtained from (\ref{cohomLl}) with the differential $E^{n-3,1}_2 \longrightarrow E^{n-1,0}_2$ and the projection $H^{n-1}(X_l'', {\Bbb Z}) \hspace{3pt}\to \hspace{-19pt}{\rightarrow} \:\: H^{n-1}(X, {\Bbb Z})$ obtained from (\ref{cohomXlll}), (\ref{cohomXll}) and (\ref{cohomXl}). We first study the cokernel of the differential $E^{n-3,1}_2 \longrightarrow E^{n-1,0}_2$. By \cite{deligne}, 3.2.13, the differentials $E^{p,q}_3 \longrightarrow E^{p+3, q-2}_3$ are zero. Therefore $E^{.,.}_{\infty} = E^{.,.}_3$ and, in particular, \[ Coker \left( H^{n-3}(L_l, {\Bbb Z}) \oplus H^{n-3}(E_3, {\Bbb Z}) \longrightarrow H^{n-1}(X''_l, {\Bbb Z}) \right) = \] \[ = Coker \left( E_2^{n-3,1} \longrightarrow E_2^{n-1,0} \right) \] \[ = E_3^{n-1,0} = E_{\infty}^{n-1,0} = Gr^{n-1}(H^{n-1}(U_l, {\Bbb Z})) \: . \] This is the image of $H^{n-1}(X''_l, {\Bbb Z})$ in $H^{n-1}(U_l, {\Bbb Z})$ and, by \cite{deligne} 3.2.17, it is the piece $W_{n-1}(H^{n-1}(U_l, {\Bbb Z}))$ of weight $n-1$ of the mixed Hodge structure on $H^{n-1}(U_l, {\Bbb Z})$. Define $V_l := {{\Bbb P}^{n-2}}' \setminus Q_l'$. The fibers of the conic-bundle $U_l \longrightarrow V_l$ are all smooth, hence \[H^{n-1}(U_l, {\Bbb Z}) \cong H^{n-3}(V_l, {\Bbb Z}) \oplus H^{n-1}(V_l, {\Bbb Z}) \] \begin{claim} Under this isomorphism, the space $W_{n-1}(H^{n-1}(U_l, {\Bbb Z}))$ is isomorphic to \linebreak $W_{n-3}(H^{n-3}(V_l, {\Bbb Z})) \oplus W_{n-1}(H^{n-1}(V_l, {\Bbb Z}))$. \end{claim} To prove this it is sufficient to show that the maps $H^{n-1}(V_l, {\Bbb Z}) \longrightarrow H^{n-1}(U_l, {\Bbb Z})$ and $H^{n-3}(V_l, {\Bbb Z}) \longrightarrow H^{n-1}(U_l, {\Bbb Z})$ are morphisms of mixed Hodge structures of type $(0,0)$ and $(1,1)$ respectively. By \cite{deligne} pages 37-38, the pull-backs on cohomology $H^{n-3}(V_l, {\Bbb Z}) \longrightarrow H^{n-3}(U_l, {\Bbb Z})$ and $H^{n-1}(V_l, {\Bbb Z}) \longrightarrow H^{n-1}(U_l, {\Bbb Z})$ are morphisms of mixed Hodge structures of type $(0,0)$. To see that the map $H^{n-3}(V_l, {\Bbb Z}) \longrightarrow H^{n-1}(U_l, {\Bbb Z})$ is a morphism of mixed Hodge structures of type $(1,1)$ choose a bisection $B$ of the conic bundle $U_l \longrightarrow V_l$ and let $\eta$ be a half of the cohomology class of $B$. Then the map \[ H^{n-3}(V_l, {\Bbb Z}) \longrightarrow H^{n-1}(U_l, {\Bbb Z}) \] is the composition of pull-back \[ H^{n-3}(V_l, {\Bbb Z}) \longrightarrow H^{n-3}(U_l, {\Bbb Z}) \] with cup-product with $\eta$ \[ H^{n-3}(U_l, {\Bbb Z}) \longrightarrow H^{n-1}(U_l, {\Bbb Z}) \: .\] The class $2 \eta$ is the restriction to $U_l$ of the cohomology class of the closure of $B$ in $X'_l$. Therefore $2 \eta$ is in the image of \[ H^2(X'_l, {\Bbb Z}) \longrightarrow H^2(U_l, {\Bbb Z}) \] and hence has pure weight $2$ and Hodge type $(1,1)$. Therefore $\eta$ has pure weight $2$ and Hodge type $(1,1)$ in the mixed Hodge structure on $H^2(U_l, {\Bbb Z})$, the map $H^{n-3}(V_l, {\Bbb Z}) \longrightarrow H^{n-1}(U_l, {\Bbb Z})$ is a morphism of mixed Hodge structures of type $(1,1)$ and sends $W_{n-3}(H^{n-3}(V_l, {\Bbb Z}))$ into $W_{n-1}( H^{n-1}(U_l, {\Bbb Z}))$. \vskip20pt We now determine $W_{n-3}(H^{n-3}(V_l, {\Bbb Z})) \oplus W_{n-1}(H^{n-1}(V_l, {\Bbb Z}))$. In the following we let $p$ be equal to $n-3$ or $n-1$. Let ${{\Bbb P}^{n-2}}'' \longrightarrow {{\Bbb P}^{n-2}}'$ be the blow up of ${{\Bbb P}^{n-2}}'$ along $R_l'$ with exceptional divisor $E''$ and identify $Q_l''$ with its image in ${{\Bbb P}^{n-2}}''$. Then $V_l = {{\Bbb P}^{n-2}}'' \setminus \left( E'' \cup Q_l'' \right)$ and the divisors $E''$ and $Q_l''$ are smooth and meet transversally. Therefore $W_p(H^p(V_l, {\Bbb Z}))$ is the image of $H^p({{\Bbb P}^{n-2}}'', {\Bbb Z})$ in $H^p(V_l, {\Bbb Z})$, i.e., it is isomorphic to the cokernel of the map \[ H^{p-2}(Q_l'', {\Bbb Z}) \oplus H^{p-2}(E'', {\Bbb Z}) \longrightarrow H^p({{\Bbb P}^{n-2}}'', {\Bbb Z}) \] obtained by Poincar\'e Duality from push-forward on homology. Since $E''$ is a ${\Bbb P}^2$-bundle over $R_l'$, we have \begin{eqnarray} H^{p-2}(E'', {\Bbb Z}) \cong H^{p-2}(R_l', {\Bbb Z}) \oplus H^{p-4}(R_l', {\Bbb Z}) \oplus H^{p-6}(R_l', {\Bbb Z}) \: , \end{eqnarray} By e.g. \cite{B2} 0.1.3, we have the isomorphism \[ H^p({{\Bbb P}^{n-2}}'', {\Bbb Z}) \cong H^p({{\Bbb P}^{n-2}}', {\Bbb Z}) \oplus H^{p-2}(R_l', {\Bbb Z}) \oplus H^{p-4}(R_l', {\Bbb Z}) \: . \] Under the map $H^{p-2}(E'', {\Bbb Z}) \longrightarrow H^p({{\Bbb P}^{n-2}}'', {\Bbb Z})$ above, the summand $H^{p-2}(R_l', {\Bbb Z}) \oplus H^{p-4}(R_l', {\Bbb Z})$ in $H^{p-2}(E'', {\Bbb Z})$ maps isomorphically onto the same summand in $H^p({{\Bbb P}^{n-2}}'', {\Bbb Z})$. Therefore $W_p(H^p(V_l, {\Bbb Z}))$ is a quotient of $H^p({{\Bbb P}^{n-2}}',{\Bbb Z})$. The summand $H^{p-6}(R_l', {\Bbb Z})$ in $H^{p-2}(E'', {\Bbb Z})$ maps into the summand $H^p({{\Bbb P}^{n-2}}', {\Bbb Z})$ of $H^p({{\Bbb P}^{n-2}}'', {\Bbb Z})$, the map $H^{p-6}(R_l', {\Bbb Z}) \longrightarrow H^{p}({{\Bbb P}^{n-2}}', {\Bbb Z})$ being again obtained by Poincar\'e Duality from push-forward on homology. Since the degree of $R_l$ in ${\Bbb P}^{n-2}$ is $16$, the image of the composition of $H^{p-6}(R_l', {\Bbb Z}) \hookrightarrow H^{p}({{\Bbb P}^{n-2}}', {\Bbb Z})$ with the isomorphism \[ H^p({{\Bbb P}^{n-2}}', {\Bbb Z}) \cong H^p({\Bbb P}^{n-2}, {\Bbb Z}) \oplus \left( \oplus\begin{Sb} p-10 \leq i \leq p-2 \\ i \equiv p [2] \end{Sb} H^i(T_l, {\Bbb Z}) \right) \] contains an element whose component in the summand $H^p({\Bbb P}^{n-2}, {\Bbb Z})$ is $16$ times a generator of $H^p({\Bbb P}^{n-2}, {\Bbb Z})$. Since the degree of $Q_l$ is $5$, the image of the composition of the direct sum embedding \[ H^{p-2}(Q_l'', {\Bbb Z}) \hookrightarrow H^{p-2}(E'', {\Bbb Z}) \oplus H^{p-2}(Q_l'', {\Bbb Z}) \] with the map \[ H^{p-2}(E'', {\Bbb Z}) \oplus H^{p-2}(Q_l'', {\Bbb Z}) \longrightarrow H^p({{\Bbb P}^{n-2}}'', {\Bbb Z}) \] contains an element whose component in the summand $H^p({{\Bbb P}^{n-2}}, {\Bbb Z})$ is $5$ times a generator of $H^{p}({\Bbb P}^{n-2}, {\Bbb Z})$. Since $16$ and $5$ are coprime, we deduce that the image of $H^{p-2}(E'', {\Bbb Z}) \oplus H^{p-2}(Q_l'', {\Bbb Z})$ in $H^{p}({{\Bbb P}^{n-2}}'', {\Bbb Z})$ contains an element whose component in the summand $H^{p}({{\Bbb P}^{n-2}}, {\Bbb Z})$ is a generator of $H^p({{\Bbb P}^{n-2}}, {\Bbb Z})$. \vskip10pt {\em So far we have obtained that $W_{p}(H^p(V_l, {\Bbb Z}))$ is a quotient of} \[ \oplus\begin{Sb} p-10 \leq i \leq p-2 \\ i \equiv p [2] \end{Sb} H^i(T_l, {\Bbb Z}) \subset H^p({{\Bbb P}^{n-2}}'', {\Bbb Z})\: . \] It is now easily seen that $\left( \oplus\begin{Sb} n-11 \leq i \leq n-3 \\ i \equiv n-1 [2] \end{Sb} H^i(T_l, {\Bbb Z}) \right) \oplus \left( \oplus\begin{Sb} n-13 \leq i \leq n-5 \\ i \equiv n-1 [2] \end{Sb} H^i(T_l, {\Bbb Z}) \right)$ maps into the summand \[ \oplus\begin{Sb} n-9 \leq i \leq n-3 \\ i \equiv n-3 [2] \end{Sb} H^i(\pi^{-1}(T_l), {\Bbb Z}) \] of $H^{n-1}(X_l'', {\Bbb Z})$. Therefore $W_{n-1}(H^{n-1}(U_l, {\Bbb Z}))= W_{n-3}(H^{n-3}(V_l, {\Bbb Z})) \oplus W_{n-1}(H^{n-1}(V_l, {\Bbb Z}))$ is a subquotient of \[ \oplus\begin{Sb} n-9 \leq i \leq n-3 \\ i \equiv n-3 [2] \end{Sb} H^i(\pi^{-1}(T_l), {\Bbb Z}) \subset H^{n-1}(X_l'', {\Bbb Z}) = \] \[ H^{n-1}(X_l, {\Bbb Z}) \oplus H^{n-3}(Q_l'', {\Bbb Z}) \oplus \left( \oplus\begin{Sb} n-9 \leq i \leq n-3 \\ i \equiv n-3 [2] \end{Sb} H^i(\pi^{-1}(T_l), {\Bbb Z}) \right) \] and the map \[ H^{n-3}(L_l, {\Bbb Z})\oplus H^{n-3}(E_3, {\Bbb Z}) \longrightarrow H^{n-1}(X_l, {\Bbb Z}) \] is onto. So, in particular, we have proved \begin{claim} The map \[ H^{n-3}(L_l, {\Bbb Z}) \oplus H^{n-3}(E_3, {\Bbb Z}) \longrightarrow H^{n-1}(X, {\Bbb Z}) \] is onto. \end{claim} Since $E_3$ is the exceptional divisor of the blow up $X_l'' \longrightarrow X_l'$, the image of \[ H^{n-3}(E_3, {\Bbb Z}) \longrightarrow H^{n-1}(X, {\Bbb Z}) \] is equal to the image of \[ H^{n-5}(Q_l'', {\Bbb Z}) \longrightarrow H^{n-1}(X, {\Bbb Z}) \: . \] We will prove that the image of this map is algebraic. Since $H^{n-1}(X,{\Bbb Z})$ is torsion-free, it is enough to prove this after tensoring with ${\Bbb Q}$. Since, by Poincar\'e Duality, $H^{n-5}(Q_l'', {\Bbb Q}) \cong H^{n-1}(Q_l'', {\Bbb Q})^*$, we first determine $H^{n-1}(Q_l'', {\Bbb Q})$. For this we use the spectral sequence \[ E^{p,q}_2 = H^p({{\Bbb P}^{n-2}}'', R^q{u}_* {\Bbb Z}) \implies H^{p+q}(W, {\Bbb Z}) \] where $W := {\Bbb P}^{n-2} \setminus Q_l = {{\Bbb P}^{n-2}}'' \setminus \left( \widetilde{E}' \cup E'' \cup Q_l'' \right)$ with $\widetilde{E}'$ the proper transform of $E'$ in ${{\Bbb P}^{n-2}}''$ and $u : W \hookrightarrow {{\Bbb P}^{n-2}}''$ is the inclusion. Recall that such a spectral sequence degenerates at $E_3$ (\cite{deligne}, 3.2.13). By \cite{goreskymcpherson} pages 23-24, we have $H^i(W, {\Bbb Z}) = 0$ for $i > dim(W) = n-2$. Therefore we obtain the following exact sequence from the spectral sequence \begin{eqnarray} H^{n-5}(\widetilde{E}' \cap E'' \cap Q_l'', {\Bbb Z}) \stackrel{d_{n-3}}{\longrightarrow} H^{n-3}(\widetilde{E}' \cap E'', {\Bbb Z}) \oplus H^{n-3}(\widetilde{E}' \cap Q_l'', {\Bbb Z}) \oplus H^{n-3}(E'' \cap Q_l'', {\Bbb Z}) \stackrel{d_{n-1}}{\longrightarrow} \nonumber \end{eqnarray} \begin{eqnarray} \stackrel{d_{n-1}}{\longrightarrow} H^{n-1}(\widetilde{E}', {\Bbb Z}) \oplus H^{n-1}(E'', {\Bbb Z}) \oplus H^{n-1}(Q_l'', {\Bbb Z}) \stackrel{d_{n+1}}{\longrightarrow} H^{n+1}({{\Bbb P}^{n-2}}'', {\Bbb Z}) \longrightarrow 0 \:\: . \label{Qlcoh} \end{eqnarray} We have \begin{lemma} The varieties whose cohomologies appear in sequence (\ref{Qlcoh}) are described as follows. \label{descrinters} \begin{description} \item[$\bold \widetilde{E}' \cap E'' \cap Q_l''$] ${\Bbb P}^1$-bundle over ${\cal V}_l$ where ${\cal V}_l := E' \cap R_l'$. The variety ${\cal V}_l$ is a ${\Bbb P}^2$-bundle over $T_l$ and each of its fibers over $T_l$ embeds into the corresponding fiber of $E'$ as the Veronese surface. Hence \[ H^{n-5}(\widetilde{E}' \cap E'' \cap Q_l'', {\Bbb Z}) \cong H^{n-5}({\cal V}_l, {\Bbb Z}) \oplus H^{n-7}({\cal V}_l, {\Bbb Z}) \] and \[ H^i({\cal V}_l, {\Bbb Z}) \cong H^i(T_l, {\Bbb Z}) \oplus H^{i-2}(T_l, {\Bbb Z}) \oplus H^{i-4}(T_l, {\Bbb Z}) \:\: . \] \item[$\bold T_l'' := \widetilde{E}' \cap Q_l''$] bundle over $T_l$ with fibers isomorphic to the blow up $\widehat{S}^2 {\Bbb P}^2$ of the symmetric square $S^2 {\Bbb P}^2$ of ${\Bbb P}^2$ along the diagonal of $S^2 {\Bbb P}^2$. A fiber of $\widetilde{E}' \cap E'' \cap Q_l''$ embeds into the corresponding fiber of $\widetilde{E}' \cap Q_l''$ as the exceptional divisor of the blow up $\widehat{S}^2 {\Bbb P}^2 \longrightarrow S^2 {\Bbb P}^2$. We have \[ H^{n-3}(T_l'', {\Bbb Z}) \cong H^{n-3}(T_l, {\Bbb Z}) \oplus H^{n-5}(T_l, {\Bbb Z}) \oplus H^{n-7}(T_l, {\Bbb Z})^{\oplus 2} \oplus H^{n-9}(T_l, {\Bbb Z}) \oplus H^{n-11}(T_l, {\Bbb Z}) \oplus H^{n-5}({\cal V}_l, {\Bbb Z}) \] \[ \cong H^{n-3}(T_l, {\Bbb Z}) \oplus H^{n-5}(T_l, {\Bbb Z}) \oplus H^{n-7}(T_l, {\Bbb Z}) \oplus H^{n-7}({\cal V}_l, {\Bbb Z}) \oplus H^{n-5}({\cal V}_l, {\Bbb Z}) \] and, under $d_{n-3}$, the summand $H^{n-7}({\cal V}_l, {\Bbb Z}) \oplus H^{n-5}({\cal V}_l, {\Bbb Z})$ in $H^{n-5}(\widetilde{E}' \cap E'' \cap Q_l'', {\Bbb Z})$ maps into the same summand in $H^{n-3}(T_l'', {\Bbb Z})$. \item[$\bold E'' \cap Q_l''$] ${\Bbb P}^1$-bundle over $R_l'$. Hence \[ H^{n-3}(E'' \cap Q_l'', {\Bbb Z}) \cong H^{n-3}(R_l', {\Bbb Z}) \oplus H^{n-5}(R_l', {\Bbb Z}) \:\: . \] \item[$\bold \widetilde{E}' \cap E''$] ${\Bbb P}^2$-bundle over ${\cal V}_l$ which contains $\widetilde{E}' \cap E'' \cap Q_l''$ as a conic-bundle over ${\cal V}_l$. We have \[ H^{n-3}(\widetilde{E}' \cap E'', {\Bbb Z}) \cong H^{n-3}({\cal V}_l, {\Bbb Z}) \oplus H^{n-5}({\cal V}_l, {\Bbb Z}) \oplus H^{n-7}({\cal V}_l, {\Bbb Z})\:\: . \] \item[$\bold \widetilde{E}'$] the blow up of $E'$ along ${\cal V}_l$, i.e., bundle over $T_l$ with fibers isomorphic to the blow up of ${\Bbb P}^5$ along the Veronese surface. This contains $\widetilde{E}' \cap E''$ as its exceptional divisor. Hence \[ H^{n-1}(\widetilde{E}', {\Bbb Z}) \cong H^{n-3}({\cal V}_l, {\Bbb Z}) \oplus H^{n-5}({\cal V}_l, {\Bbb Z}) \oplus H^{n-1}(T_l, {\Bbb Z}) \oplus \] \[ \oplus H^{n-3}(T_l, {\Bbb Z}) \oplus H^{n-5}(T_l, {\Bbb Z}) \oplus H^{n-7}(T_l, {\Bbb Z}) \oplus H^{n-9}(T_l, {\Bbb Z}) \oplus H^{n-11}(T_l, {\Bbb Z}) \:\: . \] \item[$\bold E''$] ${\Bbb P}^2$-bundle over $R_l'$ which contains $E'' \cap Q_l''$ as a conic-bundle over $R_l'$. Hence \[ H^{n-1}(E'', {\Bbb Z}) \cong H^{n-1}(R_l', {\Bbb Z}) \oplus H^{n-3}(R_l', {\Bbb Z}) \oplus H^{n-5}(R_l', {\Bbb Z}) \:\: . \] \end{description} \end{lemma} {\em Proof :} Easy. \hfill \qed \begin{lemma} There is a natural exact sequence \[ 0 \longrightarrow H^{n-3}(T_l, {\Bbb Q}) \oplus H^{n-5}(T_l, {\Bbb Q}) \oplus H^{n-7}(T_l, {\Bbb Q})^{\oplus 2} \oplus H^{n-9}(T_l, {\Bbb Q}) \oplus H^{n-3}(R_l', {\Bbb Q}) \longrightarrow \] \[ \longrightarrow H^{n-1}(Q_l'', {\Bbb Q}) \longrightarrow H^{n+1}({\Bbb P}^{n-2}, {\Bbb Q}) \longrightarrow 0 \] where the map \[ H^{n-3}(T_l, {\Bbb Q}) \oplus H^{n-5}(T_l, {\Bbb Q}) \oplus H^{n-7}(T_l, {\Bbb Q})^{\oplus 2} \oplus H^{n-9}(T_l, {\Bbb Q}) \longrightarrow H^{n-1}(Q_l'', {\Bbb Q}) \] is obtained from the inclusion $T_l'' \subset Q_l''$. \label{lemHQl} \end{lemma} {\em Proof :} From the description of $\widetilde{E}' \cap Q_l''$ in Lemma \ref{descrinters} it follows that the map $d_{n-3}$ in sequence (\ref{Qlcoh}) is injective and we have the exact sequence \begin{eqnarray} 0 \longrightarrow H^{n-5}(\widetilde{E}' \cap E'' \cap Q_l'', {\Bbb Z}) \stackrel{d_{n-3}}{\longrightarrow} H^{n-3}(\widetilde{E}' \cap E'', {\Bbb Z}) \oplus H^{n-3}(\widetilde{E}' \cap Q_l'', {\Bbb Z}) \oplus H^{n-3}(E'' \cap Q_l'', {\Bbb Z}) \stackrel{d_{n-1}}{\longrightarrow} \nonumber \end{eqnarray} \begin{eqnarray} \stackrel{d_{n-1}}{\longrightarrow} H^{n-1}(\widetilde{E}', {\Bbb Z}) \oplus H^{n-1}(E'', {\Bbb Z}) \oplus H^{n-1}(Q_l'', {\Bbb Z}) \stackrel{d_{n+1}}{\longrightarrow} H^{n+1}({{\Bbb P}^{n-2}}'', {\Bbb Z}) \longrightarrow 0 \:\: . \nonumber \end{eqnarray} Tensoring the exact sequence (\ref{Qlcoh}) with ${\Bbb Q}$ and using Lemma \ref{descrinters} and the isomorphism \[ H^{n+1}({{\Bbb P}^{n-2}}'', {\Bbb Z}) \cong H^{n+1}({\Bbb P}^{n-2}, {\Bbb Z}) \oplus \] \[ \oplus H^{n-1}(T_l, {\Bbb Z}) \oplus H^{n-3}(T_l, {\Bbb Z}) \oplus H^{n-5}(T_l, {\Bbb Z}) \oplus H^{n-7}(T_l, {\Bbb Z}) \oplus H^{n-9}(T_l, {\Bbb Z}) \oplus \] \[ \oplus H^{n-1}(R_l', {\Bbb Z}) \oplus H^{n-3}(R_l', {\Bbb Z}) \:\: , \] we easily deduce Lemma \ref{lemHQl}. \hfill $\qed$ \begin{remark} In fact we have the exact sequence \[ 0 \longrightarrow H^{n-3}(T_l, {\Bbb Z} [\frac{1}{30}]) \oplus H^{n-5}(T_l, {\Bbb Z} [\frac{1}{30}]) \oplus H^{n-7}(T_l, {\Bbb Z} [\frac{1}{30}])^{\oplus 2} \oplus H^{n-9}(T_l, {\Bbb Z} [\frac{1}{30}]) \oplus H^{n-3}(R_l', {\Bbb Z} [\frac{1}{30}]) \] \[ \longrightarrow H^{n-1}(Q_l'', {\Bbb Z} [\frac{1}{30}]) \longrightarrow H^{n+1}({\Bbb P}^{n-2}, {\Bbb Z} [\frac{1}{30}]) \longrightarrow 0 \:\: . \] \end{remark} It follows from the previous lemma (since the cohomology of $X$ has no torsion) that the image of \[ H^{n-5}(Q_l'', {\Bbb Z}) \longrightarrow H^{n-1}(X, {\Bbb Z}) \] is algebraic. Hence the image of the composition $H^{n-5}(Q_l'', {\Bbb Z}) \longrightarrow H^{n-1}(X, {\Bbb Z}) \hspace{3pt}\to \hspace{-19pt}{\rightarrow} \:\: H^{n-1}(X, {\Bbb Z})^0$ is algebraic. For $X$ generic $H^{n-1}(X, {\Bbb Z})^0$ has no nonzero algebraic part. Hence for $X$ generic and therefore, for all $X$, the image of $H^{n-5}(Q_l'', {\Bbb Z}) \longrightarrow H^{n-1}(X, {\Bbb Z})^0$ is zero. Hence the map \[ H^{n-3}(L_l, {\Bbb Z}) \longrightarrow H^{n-1}(X, {\Bbb Z})^0 \] is onto. We have \[ H^{n-3}(L_l, {\Bbb Z}) \cong H^{n-3}(S_l', {\Bbb Z}) \oplus H^{n-5}(S_l', {\Bbb Z}) \: \] and the restriction $H^{n-5}(S_l', {\Bbb Z}) \longrightarrow H^{n-1}(X, {\Bbb Z})^0$ is the composition of pull-back $H^{n-5}(S_l', {\Bbb Z}) \longrightarrow H^{n-5}(S_l'', {\Bbb Z})$, and push-forward $H^{n-5}(S_l'', {\Bbb Z}) \longrightarrow H^{n-5}(Q_l'', {\Bbb Z}) \longrightarrow H^{n-1}(X, {\Bbb Z})^0$. Hence the map $H^{n-5}(S_l', {\Bbb Z}) \longrightarrow H^{n-1}(X, {\Bbb Z})^0$ is zero and the map \[ H^{n-3}(S_l', {\Bbb Z}) \longrightarrow H^{n-1}(X, {\Bbb Z})^0 \] is onto. Now, we have \[ H^{n-3}(S_l', {\Bbb Z}) \cong H^{n-3}(S_l, {\Bbb Z}) \oplus H^{n-5}({\cal P}^*_l, {\Bbb Z}) \oplus H^{n-7}({\cal P}^*_l, {\Bbb Z}) \: . \] recall that ${\cal P}^*_l$ is the variety parametrizing lines in the fibers of $\pi^{-1}(T_l) \longrightarrow T_l$. Therefore ${\cal P}^*_l$ is a ${\Bbb P}^2$-bundle over $T_l$. Using the fact that $T_l$ is a smooth complete intersection of dimension $n-8$ in ${\Bbb P}^{n-2}$, one immediately sees that the image of the summand $H^{n-5}({\cal P}^*_l, {\Bbb Z}) \oplus H^{n-7}({\cal P}^*_l, {\Bbb Z})$ of $H^{n-3}(S_l', {\Bbb Z})$ in $H^{n-1}(X, {\Bbb Z})^0$ is zero. Therefore the map \[ H^{n-3}(S_l, {\Bbb Z}) \longrightarrow H^{n-1}(X, {\Bbb Z})^0 \] is onto. This proves the theorem in the case where $n$ is even, since in that case $H^{n-1}(X, {\Bbb Z})^0 = H^{n-1}(X, {\Bbb Z})$. Let $\sigma_1$ be the inverse image in $S_l$ of the hyperplane class on the Grassmannian $G(2,n+1)$ by the composition $S_l \longrightarrow D_l \hookrightarrow G(2,n+1)$. If $n$ is odd, one easily computes that the image of $\sigma_1^{(n-3)/2}$ in $H^{n-1}(X, {\Bbb Z})$ is $5 \zeta^{(n-1)/2}$ where $\zeta$ is the hyperplane class on $X$. On the other hand, let $x$ be a general point on $l$ and let $L_x$ be the union of the lines in $X$ through $x$. Then $L_x$ is the intersection of $X$ with the hyperplane tangent to $X$ at $x$ and a quadric (it is the second osculating cone to $X$ at $x$). The cohomology class of a linear section (through $x$) of $L_x$ of codimension $\frac{n-1}{2} - 2$ is $2 \zeta^{(n-1)/2}$ in $X$ and it is in the image of $H^{n-3}(S_l, {\Bbb Z})$. Since $2$ and $5$ are coprime, the image of $H^{n-3}(S_l, {\Bbb Z})$ in $H^{n-1}(X, {\Bbb Z})$ contains $\zeta^{(n-1)/2}$ and the map \[ \psi : H^{n-3}(S_l, {\Bbb Z}) \longrightarrow H^{n-1}(X, {\Bbb Z}) \] in onto for $n$ odd as well. It is now immediate that $\psi'$ is also onto for $n$ odd. \hfill $\qed$ \vskip20pt Let $h$ be the first Chern class of the pull-back of ${\cal O}_{{\Bbb P}^{n-2}}(1)$ to $S_l'$, let $\sigma_i$ be the pull-back to $S'_l$ of the $i$-th Chern class of the universal quotient bundle on the grassmannian $G(2,n+1) \supset D_l$ and let $e_2$ be the first Chern class of the exceptional divisor of $S'_l \longrightarrow S_l$. We make the \begin{definition} For a positive integer $k$ the $k-th$ primitive cohomologies of $S_l$ and $S_l'$ are \[ H^k(S_l, {\Bbb Z})^0 := ({\Bbb Z} h \oplus {\Bbb Z} \sigma_1)^{\perp} \subset H^k(S_l, {\Bbb Z}) \] and \[ H^k(S_l', {\Bbb Z})^0 := ({\Bbb Z} h \oplus {\Bbb Z} \sigma_1 \oplus {\Bbb Z} e_2)^{\perp} \subset H^k(S_l', {\Bbb Z}) \] where $\perp$ means orthogonal complement with respect to cup-product. \label{defprim} \end{definition} Composing the map $\psi'$ with restriction to $H^{n-3}(S_l', {\Bbb Z})^0$ on the right and with the projection $H^{n-1}(X, {\Bbb Z}) \hspace{3pt}\to \hspace{-19pt}{\rightarrow} \:\: H^{n-1}(X, {\Bbb Z})^0$ on the left, we get ${\psi'}^0 : H^{n-3}(S_l', {\Bbb Z})^0 \longrightarrow H^{n-1}(X, {\Bbb Z})^0$. Our goal is to prove the following generalization of the results of Clemens and Griffiths. \begin{theorem} The map ${\psi'}^0$ is onto and its kernel is the $i_l$-invariant part $H^{n-3}(S_l', {\Bbb Z})^{0+}$ of $H^{n-3}(S_l', {\Bbb Z})^0$. \label{maintheorem} \end{theorem} The first step for proving the theorem is \begin{theorem} Let $a$ and $b$ be two elements of $H^{n-3}(S_l', {\Bbb Z})^0$. Then \[ \psi'(a). \psi'(b) = a. i_l^* b - a . b \: . \] \label{thmaibab} \end{theorem} {\em Proof :} We have \[ \psi'(a). \psi'(b) = (\epsilon_1 \epsilon_2 \rho)_* p^* a . (\epsilon_1 \epsilon_2 \rho)_* p^* b = (\epsilon_2 \rho)_* p^* a. \epsilon_1^* {\epsilon_1}_* (\epsilon_2 \rho)_* p^* b \: . \] Let $\xi_1$ be the first Chern class of the tautological invertible sheaf for the projective bundle $g_1 : E_1 \longrightarrow l$. Let $\gamma_i^1$ be the Chern classes of the universal quotient bundle on the projective bundle $g_1 : E_1 \longrightarrow l$, i.e., \[ \gamma^1_i = \xi_1^i + \xi_1^{i-1}.g_1^* c_1(N_{l/X}) + ... + g_1^* c_i(N_{l/X}) \: . \] Define $\xi_2, \gamma^2_i$ and $\xi_3, \gamma^3_i$ similarly for the projective bundles $g_2 : E_2 \longrightarrow \pi^{-1}(T_l)$ and $g_3 : E_3 \longrightarrow Q_l''$ respectively. By, e.g., \cite{B2}, 0.1.3, we have \[ \epsilon_1^* {\epsilon_1}_* (\epsilon_2 \rho)_* p^* b = (\epsilon_2 \rho)_* p^* b + {i_1}_* \left( \sum_{r=0}^{n-4} \xi_1^r.g_1^* {g_1}_* \left( \gamma^1_{n-4-r}.i_1^* \left( (\epsilon_2 \rho)_* p^* b \right) \right) \right) \] where $i_1 : E_1 \hookrightarrow X_l$ is the inclusion. We also let $i_2 : E_2 \hookrightarrow X_l'$ and $i_3 : E_3 \hookrightarrow X_l''$ be the inclusions. For any $r$, ($0 \leq r \leq n-4$), we have \[ {g_1}_* \left( \gamma_{n-4-r} . i_1^* (\epsilon_2 \rho)_* p^* b \right) \in H^{n-3-2r}(l, {\Bbb Z}) \: . \] Therefore ${g_1}_* \left( \gamma_{n-4-r} . i_1^* (\epsilon_2 \rho)_* p^* b \right) \neq 0$ only if $n-3-2r = 0$ or $n-3-2r = 2$. This is impossible if $n$ is even so {\em we now suppose that $n$ is odd}. So if we put \[ B := {i_1}_* \left( \xi_1^{(n-3)/2}.g_1^* {g_1}_* \left( \gamma^1_{(n-5)/2}.i_1^* \left( (\epsilon_2 \rho)_* p^* b \right) \right) \hbox{\linebreak} + \xi_1^{(n-5)/2}.g_1^* {g_1}_* \left( \gamma^1_{(n-3)/2}.i_1^* \left( (\epsilon_2 \rho)_* p^* b \right) \right) \right) \: , \] we have \[ \epsilon_1^* {\epsilon_1}_* (\epsilon_2 \rho)_* p^* b = (\epsilon_2 \rho)_* p^* b + B \: . \] If $n \geq 7$, replacing $\gamma^1_{(n-5)/2}$ and $\gamma^1_{(n-3)/2}$ in terms of $\xi_1$, we obtain \[ B = {i_1}_* \left( \xi_1^{(n-3)/2}.g_1^* {g_1}_* \left( \xi_1^{(n-5)/2} . i_1^* \left( (\epsilon_2 \rho)_* p^* b \right) + \xi_1^{(n-7)/2} . g_1^* c_1(N_{l/X}) . i_1^* \left( (\epsilon_2 \rho)_* p^* b \right) \right) \right) + \] \[ + {i_1}_* \left( \xi_1^{(n-5)/2}.g_1^* {g_1}_* \left( \xi_1^{(n-3)/2} . i_1^* \left( (\epsilon_2 \rho)_* p^* b \right) + \xi_1^{(n-5)/2} . g_1^* c_1(N_{l/X}) . i_1^* \left( (\epsilon_2 \rho)_* p^* b \right) \right) \right) \: . \] We have $c_1(N_{l/X}) = (n-4)j_1^* \zeta$ where $\zeta= c_1({\cal O}_{{\Bbb P}^n}(1))$ and $j_1 : l \hookrightarrow X$ is the inclusion. Similarly we define $j_2 : \pi^{-1}(T_l) \hookrightarrow X_l$ and $j_3 : Q_l'' \hookrightarrow X_l'$ to be the inclusions. Therefore we obtain \[ B = {i_1}_* \left( \xi_1^{(n-3)/2}.g_1^* {g_1}_* \left( \xi_1^{(n-5)/2} . i_1^* \left( (\epsilon_2 \rho)_* p^* b \right) + \xi_1^{(n-7)/2} . (n-4) g_1^* j_1^* \zeta . i_1^* \left( (\epsilon_2 \rho)_* p^* b \right) \right) \right) + \] \[ + {i_1}_* \left( \xi_1^{(n-5)/2}.g_1^* {g_1}_* \left( \xi_1^{(n-3)/2} . i_1^* \left( (\epsilon_2 \rho)_* p^* b \right) + \xi_1^{(n-5)/2} . (n-4) g_1^* j_1^* \zeta . i_1^* \left( (\epsilon_2 \rho)_* p^* b \right) \right) \right) \: . \] Or, since $j_1 g_1 = \epsilon_1 i_1$, \[ B = {i_1}_* \left( \xi_1^{(n-3)/2}.g_1^* {g_1}_* \left( \xi_1^{(n-5)/2} . i_1^* \left( (\epsilon_2 \rho)_* p^* b \right) + \xi_1^{(n-7)/2} . (n-4) i_1^* \epsilon_1^* \zeta . i_1^* \left( (\epsilon_2 \rho)_* p^* b \right) \right) \right) + \] \[ + {i_1}_* \left( \xi_1^{(n-5)/2}.g_1^* {g_1}_* \left( \xi_1^{(n-3)/2} . i_1^* \left( (\epsilon_2 \rho)_* p^* b \right) + \xi_1^{(n-5)/2} . (n-4) i_1^* \epsilon_1^* \zeta . i_1^* \left( (\epsilon_2 \rho)_* p^* b \right) \right) \right) \: . \] Let $E_1$ also denote the first Chern class of the invertible sheaf ${\cal O}_{X_l}(E_1)$. Since $\xi_1 = - i_1^* E_1$, we can write \[ B =(-1)^{n} {i_1}_* \left( i_1^* E_1^{(n-3)/2}.g_1^* {g_1}_* i_1^*\left( E_1^{(n-5)/2} . \left( (\epsilon_2 \rho)_* p^* b \right) - E_1^{(n-7)/2} . (n-4) \epsilon_1^* \zeta . \left( (\epsilon_2 \rho)_* p^* b \right) \right) \right) + \] \[ + (-1)^n {i_1}_* \left( i_1^* E_1^{(n-5)/2}.g_1^* {g_1}_* i_1^*\left( E_1^{(n-3)/2} . \left( (\epsilon_2 \rho)_* p^* b \right) - E_1^{(n-5)/2} . (n-4) \epsilon_1^* \zeta . \left( (\epsilon_2 \rho)_* p^* b \right) \right) \right) \: . \] Or, since ${g_1}_*i_1^*={j_1}^*{\epsilon_1}_*$, \[ B =(-1)^{n} {i_1}_* \left( i_1^* E_1^{(n-3)/2}.g_1^* {j_1}^*{\epsilon_1}_* \left( E_1^{(n-5)/2} . \left( (\epsilon_2 \rho)_* p^* b \right) - E_1^{(n-7)/2} . (n-4) \epsilon_1^* \zeta . \left( (\epsilon_2 \rho)_* p^* b \right) \right) \right) + \] \[ + (-1)^n {i_1}_* \left( i_1^* E_1^{(n-5)/2}.g_1^* {j_1}^*{\epsilon_1}_* \left( E_1^{(n-3)/2} . \left( (\epsilon_2 \rho)_* p^* b \right) - E_1^{(n-5)/2} . (n-4) \epsilon_1^* \zeta . \left( (\epsilon_2 \rho)_* p^* b \right) \right) \right) \: . \] Now \[ {\epsilon_1}_* \left( E_1^{(n-5)/2} . \left( (\epsilon_2 \rho)_* p^* b \right) - E_1^{(n-7)/2} . (n-4) \epsilon_1^* \zeta . \left( (\epsilon_2 \rho)_* p^* b \right)\right) \] is an element of $H^{2n-6}(X, {\Bbb Z})$. Hence its image by $j_1^*$ is zero unless $2n-6 \leq 2$, i.e., $n \leq 4$. We supposed $n \geq 7$. Similarly, \[ j_1^*{\epsilon_1}_* \left( E_1^{(n-5)/2} . \left( (\epsilon_2 \rho)_* p^* b \right) - E_1^{(n-7)/2} . (n-4) \epsilon_1^* \zeta . \left( (\epsilon_2 \rho)_* p^* b \right)\right) \] is zero unless $2n-4 \leq 2$ which implies $n \leq 3$. Hence $B$ is zero for $n \geq 7$. Similarly, $B$ is zero for $n=5$. \vskip10pt Therefore \[ \psi'(a) . \psi'(b) = (\epsilon_2 \rho)_* p^* a. (\epsilon_2 \rho)_* p^* b \: . \] \vskip10pt Now write \[ \psi'(a) . \psi'(b) = \rho_* p^* a. \epsilon_2^* {\epsilon_2}_* \rho_* p^* b \] and, as before, \[ \epsilon_2^* {\epsilon_2}_* \rho_* p^* b = \rho_* p^* b + {i_2}_* \left( \sum_{r=0}^3 \xi_2^r . g_2^* {g_2}_* \left( \gamma^2_{3-r} . i_2^* \rho_* p^* b \right) \right) \: . \] So \[ \psi'(a) . \psi'(b) = \rho_* p^* a. \rho_* p^* b + \rho_* p^* a . {i_2}_* \left( \sum_{r=0}^3 \xi_2^r . g_2^* {g_2}_* \left( \gamma^2_{3-r} . i_2^* \rho_* p^* b \right) \right) \] or \[ \psi'(a) . \psi'(b) = \rho_* p^* a. \rho_* p^* b + i_2^* \rho_* p^* a . \left( \sum_{r=0}^3 \xi_2^r . g_2^* {g_2}_* \left( \gamma^2_{3-r} . i_2^* \rho_* p^* b \right) \right) \: . \] We have $a. e_2 = 0$. Hence $p^* a . p^* e_2 = 0$. Let $E_2$ also denote the cohomology class of $E_2$. Then it is easily seen that $\rho^* E_2 = p^* e_2$. Therefore $p^* a . \rho^* E_2 = 0$. In order to use this, we need to modify the above expression a bit. We first need to write the first three Chern classes of $N_{\pi^{-1}(T_l)/X_l}$ as inverse images of cohomology classes by $j_2$. Consider the exact sequence \[ 0 \longrightarrow N_{\pi^{-1}(T_l)/X_l} \longrightarrow N_{\pi^{-1}(T_l)/ {\Bbb P}^n_l} \longrightarrow N_{X_l/ {\Bbb P}^n_l}|_{\pi^{-1}(T_l)} \longrightarrow 0 \: . \] We have \[ N_{X_l/ {\Bbb P}^n_l} \cong {\cal O}_{{\Bbb P} E}(2) \otimes \pi^* {\cal O}_{{\Bbb P}^{n-2}}(1) \] where $E = {\cal O}_{{\Bbb P}^{n-2}}(-1) \oplus {\cal O}_{{\Bbb P}^{n-2}}^{\oplus 2}$, so that ${\Bbb P} E \cong {\Bbb P}^n_l$. Also \[ N_{\pi^{-1}(T_l)/ {\Bbb P}^n_l} \cong \pi^*N_{T_l/ {\Bbb P}^{n-2}} \cong \pi^* \left( {\cal O}_{{\Bbb P}^{n-2}}(3) \oplus {\cal O}_{{\Bbb P}^{n-2}}(2)^{\oplus 2} \oplus {\cal O}_{{\Bbb P}^{n-2}}(1)^{\oplus 3} \right) \: . \] It follows that we can write $c_i(N_{\pi^{-1}(T_l)/X_l}) = j_2^* c_i$ where the $c_i$ are cohomology classes on $X_l$. So \[ \gamma^2_r = \xi_2^r + \xi_2^{r-1} . g_2^* j_2^* c_1 + ... + g_2^* j_2^* c_r \] and, since $\xi_2 = - i_2^* E_2$ and $j_2 g_2 = \epsilon_2 i_2$, we have \[ \gamma^2_r = i_2^* \alpha^2_r \] where \[ \alpha^2_r = (-1)^r E_2^r + (-1)^{r-1} E_2^{r-1} . \epsilon_2^* c_1 + ... + \epsilon_2^* c_r \: . \] Therefore, using ${g_2}_* i_2^* = j_2^* {\epsilon_2}_*$ and $j_2 g_2 = \epsilon_2 i_2$, \[ i_2^* \rho_* p^* a . \left( \sum_{r=0}^3 \xi_2^r . g_2^* {g_2}_* \left( \gamma^2_{3-r} . i_2^* \rho_* p^* b \right) \right) = i_2^* \left( \rho_* p^* a . \left( \sum_{r=0}^3 (-1)^r E_2^r . \epsilon_2^* {\epsilon_2}_* \left( \alpha^2_{3-r} . \rho_* p^* b \right) \right) \right) \] \[ = \rho_* p^* a . E_2 . \left( \sum_{r=0}^3 (-1)^r E_2^r . \epsilon_2^* {\epsilon_2}_* \left( \alpha^2_{3-r} . \rho_* p^* b \right) \right) = p^* a . \rho^* E_2 . \rho^* \left( \sum_{r=0}^3 (-1)^r E_2^r . \epsilon_2^* {\epsilon_2}_* \left( \alpha^2_{3-r} . \rho_* p^* b \right) \right) = 0 \] and we obtain \[ \psi'(a) . \psi'(b) = \rho_* p^* a. \rho_* p^* b \: . \] Writing $\rho = \epsilon_3 \widetilde{q}$, we have \[ \psi'(a) . \psi'(b) = (\epsilon_3 \widetilde{q})_* p^* a. (\epsilon_3 \widetilde{q})_* p^* b = \widetilde{q}_* p^* a . \epsilon_3^* {\epsilon_3}_* \widetilde{q}_* p^* b \] and, as before, \[ \psi'(a) . \psi'(b) = \widetilde{q}_* p^* a . \widetilde{q}_* p^* b + \widetilde{q}_* p^* a . {i_3}_* g_3^* {g_3}_* i_3^* \widetilde{q}_* p^* b = \widetilde{q}_* p^* a . \widetilde{q}_* p^* b + i_3^* \widetilde{q}_* p^* a . g_3^* {g_3}_* i_3^* \widetilde{q}_* p^* b \: . \] Consider the commutative diagram \[ \begin{array}{ccccccc} & & S_l'' & \stackrel{q'}{\longrightarrow} & E_3 & \stackrel{g_3}{\longrightarrow} & Q_l'' \\ & \stackrel{\epsilon_4}{\swarrow} & \downarrow i_3' & & \downarrow i_3 & & \downarrow j_3 \\ S_l' & \stackrel{p}{\longleftarrow} & L_l & \stackrel{\widetilde{q}}{\longrightarrow} & X_l'' & \stackrel{\epsilon_3}{\longrightarrow} & X_l' \end{array} \] where the two squares are fiber squares. Using the diagram, we modify $\psi'(a). \psi'(b)$ as follows \[ \psi'(a) . \psi'(b) = \widetilde{q}_* p^* a . \widetilde{q}_* p^* b + q'_* {i_3'}^* p^* a . g_3^* {g_3}_* q'_* {i_3'}^* p^* b = \] \[ = \widetilde{q}_* p^* a . \widetilde{q}_* p^* b + q'_* {\epsilon_4}^* a . g_3^* {g_3}_* q'_* {\epsilon_4}^* b = \widetilde{q}_* p^* a . \widetilde{q}_* p^* b + {\epsilon_4}^* a . (g_3 q')^* (g_3 q')_* {\epsilon_4}^* b \: . \] The morphism $g_3q' : S_l'' \longrightarrow Q_l''$ is a double cover whose involution $i_l'$ is the lift of $i_l$. Therefore \[ (g_3 q')^* (g_3 q')_* {\epsilon_4}^* b = \epsilon_4^* b + {i_l'}^* \epsilon_4^* b = \epsilon_4^* b + \epsilon_4^* i_l^* b \] and \[ {\epsilon_4}^* a . (g_3 q')^* (g_3 q')_* {\epsilon_4}^* b = \epsilon_4^* a . \left( \epsilon_4^* b + \epsilon_4^* i_l^* b \right) = a . {\epsilon_4}_* \left( \epsilon_4^* b + \epsilon_4^* i_l^* b \right) = a . \left( b + i_l^* b \right) \: . \] On the other hand \[ \widetilde{q}_* p^* a . \widetilde{q}_* p^* b = p^* a . p^* b . \widetilde{q}^* L_l \] where we also denote by $L_l$ the cohomology class of $L_l$ in $X_l''$. We have the following \begin{lemma} The cohomology class of $L_l$ in $X_l''$ is equal to \[ 5 \left(\epsilon_1 \epsilon_2 \epsilon_3 \right)^* \zeta - 5 \left( \epsilon_2 \epsilon_3 \right)^* E_1 - 2 E_3 - k \epsilon_3^* E_2 \] for some nonnegative integer $k$. \end{lemma} {\em Proof :} To compute the coefficient of $\left(\epsilon_1 \epsilon_2 \epsilon_3 \right)^* \zeta$, we push $L_l$ forward to $X$ and compute its degree in ${\Bbb P}^n$. The image of $L_l$ in $X$ is the union of all the lines in $X$ which are incident to $l$. Since any such line maps to a point of $Q_l$ by the projection from $l$, the image of $L_l$ is the intersection with $X$ of the cone of vertex $l$ over $Q_l$. Since $Q_l$ has degree $5$, this proves that the coefficient of $\left(\epsilon_1 \epsilon_2 \epsilon_3 \right)^* \zeta$ is $5$. The coefficient of $\left( \epsilon_2 \epsilon_3 \right)^* E_1$ is the negative of the multiplicity of the image of $L_l$ in $X$ along $l$. Intersecting $X$ with a general linear subspace of dimension $3$ which contains $l$, we see that this linear subspace contains $10$ distinct lines which are distinct from $l$ and are in the image of $L_l$. Therefore, the multiplicity of the image of $L_l$ along $l$ is exactly $5 = 5.3 - 10$. The coefficient of $E_3$ is the negative of the multiplicity of the image of $L_l$ in $X_l'$ along $Q_l''$. This is $2$ since $L_l$ is smooth and $\rho$ is an embedding outside $S_l''$ and has degree $2$ on $S_l''$. \hfill $\qed$ \vskip30pt Now we will use the hypothesis $a.h = 0$. It implies $p^* a . p^*h = 0$. One easily sees that \[ p^* h = \left( \epsilon_2 \rho \right)^* \pi_X^* c_1({\cal O}_{{\Bbb P}^{n-2}}(1)) \: . \] On the other hand $\epsilon_1^* \zeta - E_1 = \pi^* c_1({\cal O}_{{\Bbb P}^{n-2}}(1))$. Therefore \[ p^* a . \left(\epsilon_1 \epsilon_2 \rho \right)^* \zeta = p^* a . \left( \epsilon_2 \rho \right)^* E_1 \: . \] Furthermore, we saw that $p^* a . \rho^* E_2 = 0$, hence, \[ \widetilde{q}_* p^* a . \widetilde{q}_* p^* b = p^* a . p^* b . \widetilde{q}^* L_l = p^* a . p^* b . \left( - 2 \widetilde{q}^* E_3 \right) = - 2 a . b \: \: . \] Finally, \[ \psi'(a) . \psi'(b) = - 2 a . b + a . \left( b + i_l^* b \right) = a . i_l^* b - a . b \: \: . \] \hfill $\qed$ \vskip.5in \begin{corollary} \label{kerpsi0} If ${\psi'}^0$ is onto, the kernel of ${\psi'}^0$ is equal to the set of $i_l$-invariant elements of $H^{n-3}(S_l', {\Bbb Z})$. \end{corollary} {\em Proof :} Let $b$ be an element of $H^{n-3}(S_l', {\Bbb Z})^0$. Then ${\psi'}^0(b)$ is zero if and only if, \[ \hbox{for every element }c \hbox{ of } H^{n-1}(X, {\Bbb Z})^0, \:\: \psi'(b) . c = 0 \: \: . \] If ${\psi'}^0$ is onto, this is equivalent to \[ \hbox{for every element }a \hbox{ of } H^{n-3}(S_l', {\Bbb Z})^0, \: \: \psi'(a) . \psi'(b) = 0 \: \: . \] By theorem \ref{thmaibab}, this is equivalent to \[ \hbox{for every element }a \hbox{ of } H^{n-3}(S_l', {\Bbb Z})^0, \:\: a . \left( i_l^* b - b \right) = 0 \] which is in turn equivalent to \[ b = i_l^* b \: \: . \] \hfill \qed \vskip20pt We are now ready to prove \begin{lemma} \label{NSSl} Suppose $n \geq 6$, then \[ H^2(S_l, {\Bbb Q}) = {\Bbb Q} h \oplus {\Bbb Q} \sigma_1 \] \[ H^2(S_l', {\Bbb Q}) = {\Bbb Q} h \oplus {\Bbb Q} \sigma_1 \oplus {\Bbb Q} e_2 \] and, if $n=5$, we have the exact sequence \[ 0 \longrightarrow H^2(Q_l , {\Bbb Z})^0 \longrightarrow H^2(S_l, {\Bbb Z})^0 \longrightarrow H^4(X, {\Bbb Z})^0 \longrightarrow 0 \] and \[ H^2(S_l, {\Bbb Q}) = H^2(S_l, {\Bbb Q})^0 \oplus {\Bbb Q} h \oplus {\Bbb Q} \sigma_1 \] (note that $T_l = \emptyset$ for $n \leq 7$ so that $Q_l = Q_l'$ and $S_l = S_l'$). \end{lemma} {\em Proof :} First suppose $n=5$. Then the direct sum decomposition above is clear. To prove the exactness of the sequence, note that $H^2(S_l, {\Bbb Z}) \longrightarrow H^4(X, {\Bbb Z})^0$ is onto by Theorem \ref{thmpsionto}. Since ${\Bbb Z} h \oplus {\Bbb Z} \sigma_1$ is algebraic, its image in $H^4(X, {\Bbb Z})^0$ is algebraic. For $X$ generic, the group $H^4(X, {\Bbb Z})^0$ has no nonzero algebraic part. Therefore for $X$ generic and hence for all $X$, the image of ${\Bbb Z} h \oplus {\Bbb Z} \sigma_1$ in $H^4(X, {\Bbb Z})^0$ is zero. It follows that the sequence is exact on the right. The exactness of the rest of the sequence now follows from Corollary \ref{kerpsi0}. Now suppose $n \geq 6$. Since $H^2(S_l', {\Bbb Q}) \cong H^2(S_l, {\Bbb Q}) \oplus {\Bbb Q} e_2$, we only need to compute $H^2(S_l, {\Bbb Q})$. Let $H_1$ be a general hyperplane in ${\Bbb P}^{n-2}$ and let $H_2$ be its inverse image in ${\Bbb P}^n$. The inverse image $S_{l,H}$ of $H_1$ in $S_l$ parametrizes the lines in the fibers of $X_{l,H} \longrightarrow H_1$ where $X_{l,H}$ is the proper transform of $X_H := X \cap H_2$ in $X_l$. By \cite{goreskymcpherson} pages 23-25, we have $H^2(S_l, {\Bbb Z}) \cong H^2(S_{l,H}, {\Bbb Z})$ for $n \geq 7$ and $H^2(S_l, {\Bbb Z}) \hookrightarrow H^2(S_{l,H}, {\Bbb Z})$ for $n=6$. Suppose therefore that $n=6$. If we choose a general pencil of hyperplanes in ${\Bbb P}^{n-2}$ of which $H_1$ is a member, then $H^2(S_l, {\Bbb Z})$ maps into the part of $H^2(S_{l,H}, {\Bbb Z})$ which is invariant under monodromy. Since $H^4(X_H, {\Bbb Z})^0$ has no nonzero elements invariant under monodromy, we see that $H^2(S_l, {\Bbb Z})^0$ lies in $H^2(Q_{l,H}, {\Bbb Z})^0$. Since $H^2(Q_{l,H}, {\Bbb Z})^0$ has no nonzero element invariant under monodromy, we have $H^2(S_l, {\Bbb Z})^0=0$ and $H^2(S_l, {\Bbb Q}) = {\Bbb Q} h \oplus {\Bbb Q} \sigma_1$. \hfill \qed \vskip20pt We will prove Theorem \ref{maintheorem} in conjunction with some results on the cohomology of $S_l$ and by induction as follows. \begin{theorem} \label{thmpsi0onto} \begin{enumerate} \item The maps $\psi^0 : H^{n-3}(S_l, {\Bbb Z})^0 \longrightarrow H^{n-1}(X, {\Bbb Z})^0$ and ${\psi'}^0 : H^{n-3}(S_l', {\Bbb Z})^0 \longrightarrow H^{n-1}(X, {\Bbb Z})^0$ are onto. The kernel of ${\psi'}^0$ is the $i_l$-invariant part $H^{n-3}(S_l', {\Bbb Z})^{0+}$ of $H^{n-3}(S_l', {\Bbb Z})^0$ and therefore the kernel of $\psi^0$ is $H^{n-3}(S_l, {\Bbb Z}) \cap H^{n-3}(S_l', {\Bbb Z})^{0+}$. \item The cohomology of $S_l$ is torsion in odd degree except in degree $n-3$. \item In even degree the rational cohomology of $S_l$ is generated by monomials in $h$ and $\sigma_1$ except in degree $n-3$. \end{enumerate} \end{theorem} {\em Proof :} As mentioned above, we proceed by induction on $n$. We first show that, for any given $n \geq 5$, parts $2$ and $3$ of the theorem imply part $1$. Indeed, assume that parts $2$ and $3$ are true for any smooth cubic hypersurface in ${\Bbb P}^n$ for a fixed $n$. Let $Sym(h,\sigma_1)$ be the subvector space of $H^{n-3}(S_l, {\Bbb Q})$ generated by monomials in $h$ and $\sigma_1$ ($Sym(h, \sigma_1)=0$ if $n$ is even). Then, if $n$ is odd, it follows from numbers $2$ and $3$ that we have the decomposition \[ H^{n-3}(S_l, {\Bbb Q}) \cong H^{n-3}(S_l, {\Bbb Q})^0 \oplus Sym(h,\sigma_1) \: . \] Since $Sym(h,\sigma_1)$ is algebraic, its image in $H^{n-1}(X, {\Bbb Z})$ is also algebraic. For $X$ generic $H^{n-1}(X, {\Bbb Z})^0$ has no algebraic part. Therefore for $X$ generic and hence for all $X$, the image of $Sym(h,\sigma_1)$ is zero in $H^{n-1}(X, {\Bbb Z})^0$. Since the cohomology of $X$ has no torsion and, by Theorem \ref{thmpsionto}, the map $\psi : H^{n-3}(S_l, {\Bbb Z}) \longrightarrow H^{n-1}(X, {\Bbb Z})$ is onto, it follows that \[ \psi^0 : H^{n-3}(S_l, {\Bbb Z})^0 \longrightarrow H^{n-1}(X, {\Bbb Z})^0 \] is onto. Since $\psi^0$ is the composition of ${\psi'}^0$ with the inclusion $H^{n-3}(S_l, {\Bbb Z})^0 \hookrightarrow H^{n-3}(S_l', {\Bbb Z})^0$, we deduce that ${\psi'}^0$ is also onto. The rest of part $1$ is Corollary \ref{kerpsi0}. \vskip10pt Now we prove that parts $1$, $2$ and $3$ for $n-1 \geq 5$ imply parts $2$ and $3$ for $n$. Let $H_1, H_2, X_{l,H}, S_{l, H}$ be as in the proof of Lemma \ref{NSSl}, let $H_1'$ be the proper transform of $H_1$ in ${{\Bbb P}^{n-2}}'$ and let $X_{l,H}'$ and $S_{l,H}'$ be the proper transforms of $X_{l,H}$ and $S_{l,H}$ in $X_l'$ and $S_l'$ respectively. By \cite{goreskymcpherson} pages 23-25, for every $k \leq n-5$, we have \[ H^k(S_l, {\Bbb Z}) \cong H^k(S_{l,H}, {\Bbb Z}) \] and \[ H^{n-4}(S_l, {\Bbb Z}) \hookrightarrow H^{n-4}(S_{l,H}, {\Bbb Z}) \:\: . \] In particular, it follows from this and our induction hypothesis that $H^{n-3}(S_l, {\Bbb Q})$ and $H^{n-4}(S_l, {\Bbb Q})$ are the direct sums of their primitive parts and their subvector spaces generated by the monomials in $h$ and $\sigma_1$. Now it is enough to show that $H^{n-4}(S_l, {\Bbb Q})^0=0$. If we choose a general pencil of hyperplanes in ${\Bbb P}^{n-2}$ of which $H_1$ is a member, then $H^{n-4}(S_l, {\Bbb Z})$ maps into the part of $H^{n-4}(S_{l,H}, {\Bbb Z})$ which is invariant under monodromy. By our induction hypothesis, we have the exact sequence \[ 0 \longrightarrow H^{n-4}(S_{l,H}, {\Bbb Z})^0 \cap H^{n-4}(S_{l,H}', {\Bbb Z})^{0+} \longrightarrow H^{n-4}(S_{l,H}, {\Bbb Z})^0 \longrightarrow H^{n-2}(X_H, {\Bbb Z})^0 \longrightarrow 0 \:\: . \] Since $H^{n-2}(X_H, {\Bbb Z})^0$ has no nonzero elements invariant under monodromy, we see that $H^{n-4}(S_l, {\Bbb Z})^0$ lies in $H^{n-4}(S_{l,H}, {\Bbb Z})^0 \cap H^{n-4}(S_{l,H}', {\Bbb Z})^{0+}$. Therefore all the elements of $H^{n-4}(S_l, {\Bbb Z})^0$ are $i_l$-invariant and hence are contained in $H^{n-4}(Q_l', {\Bbb Z})^0 \subset H^{n-4}(S_l', {\Bbb Z})^0$. Now let \[ \begin{array}{ccc} {\Bbb P}^n & \subset & {\Bbb P}^{n+1} \\ \downarrow & & \downarrow \\ {\Bbb P}^{n-2} & \subset & {\Bbb P}^{n-1} \end{array} \] be a commutative diagram of linear embeddings and projections from $l$. Let $Y$ be a general cubic hypersurface in ${\Bbb P}^{n+1}$ such that $Y \cap {\Bbb P}^n = X$, let $Y_l$ be the blow up of $Y$ along $l$ and let $S_{l,Y}$ be the variety parametrizing lines in the fibers of $Y_l \longrightarrow {\Bbb P}^{n-1}$. Then, again by \cite{goreskymcpherson} pages 23-25, we have \[ H^{n-4}(S_l, {\Bbb Z}) \cong H^{n-4}(S_{l,Y}, {\Bbb Z}) \:\: . \] Let $T_{l,Y}$ be the variety parametrizing the planes in the fibers of $Y_l \longrightarrow {\Bbb P}^{n-1}$ and similarly define $Q_{l,Y}$, $Q_{l,Y}'$, $R_{l,Y}'$ and $Q_{l,Y}''$. By Lemma \ref{lemHQl} we have the exact sequence \[ 0 \longrightarrow H^{n-2}(T_{l,Y}, {\Bbb Q}) \oplus H^{n-4}(T_{l,Y}, {\Bbb Q}) \oplus H^{n-6}(T_{l,Y}, {\Bbb Q})^{\oplus 2} \oplus H^{n-8}(T_{l,Y}, {\Bbb Q}) \oplus H^{n-2}(R_{l,Y}',{\Bbb Q}) \longrightarrow \] \[ \longrightarrow H^n(Q_{l,Y}'', {\Bbb Q}) \longrightarrow H^{n+2}({\Bbb P}^{n-1}, {\Bbb Q}) \longrightarrow 0 \:\: . \] It is easily seen that the intersection of the subspace \[ H^{n-2}(T_{l,Y}, {\Bbb Q}) \oplus H^{n-4}(T_{l,Y}, {\Bbb Q}) \oplus H^{n-6}(T_{l,Y}, {\Bbb Q})^{\oplus 2} \oplus H^{n-8}(T_{l,Y}, {\Bbb Q}) \oplus H^{n-2}(R_{l,Y}',{\Bbb Q}) \] of $H^n(Q_{l,Y}'', {\Bbb Q}) \supset H^n(Q_{l,Y}', {\Bbb Q})$ with $H^n(S_{l,Y}, {\Bbb Q}) \subset H^n(S_{l,Y}'', {\Bbb Q})$ is zero. It immediately follows that $H^{n-4}(S_{l,Y}, {\Bbb Q})^0=H^{n-4}(S_l, {\Bbb Q})^0=0$. \vskip10pt To finish the proof of the theorem all we need to do is to prove the theorem in the case $n=5$. Suppose therefore that $n=5$. Then part $3$ is clear. Part $2$ is proved in \cite{voisin} Lemme 3 page 591. Part $1$ is Lemma \ref{NSSl}. \hfill $\qed$

9701

alg-geom/9701013

en

https://arxiv.org/abs/alg-geom/9701013

alg-geom/9701013

Richard Borcherds

Richard E. Borcherds, Ludmil Katzarkov, Tony Pantev, and N. I. Shepherd-Barron

Families of K3 surfaces

10 pages AMSLaTeX v 1.2b

J. Algebraic Geometry 7 (1998) 183-193

We use automorphic forms to prove that a compact family of Kaehler K3 surfaces with constant Picard number is isotrivial.

alg-geom

\section{Automorphic forms on moduli spaces of K3 surfaces} In this section we construct some automorphic forms with known zeros on certain period spaces of marked K3 surfaces with extra structure, which gives explicit examples of ample divisors on the corresponding moduli spaces since the set of zeros of an automorphic form is an ample divisor. The extra structure consists of a fixed primitive embedding of some lattice $S$ of signature $(1,m)$ in the Picard lattice of the K3 surface. We call such a K3 surface an $S$-K3 surface. We regard $S$ as a fixed sublattice of the lattice $II_{3,19}$, and write $T$ for the lattice $S^\perp$ of signature $(2,19-m)$. The (hermitean) symmetric space of the lattice $T$ is the set of norm 0 points in the complex projective space of $T\otimes{\mathbb C}$ whose real and imaginary parts span a 2 dimensional positive definite subspace of $T\otimes {\mathbb R}$. We recall from \cite{BOV} that the moduli space of $S$-K3 surfaces can be identified with the quotient of the symmetric space of the lattice $T=S^\perp$ by some arithmetic group. The Baily-Borel theorem implies that the zero locus of an automorphic form is an ample divisor on a compactification of the moduli space, and hence the set of points of the quotient where an automorphic form does not vanish is a quasiaffine variety. The main result of this section is a proof of theorem 1.2, which states that there is an automorphic form on the space of marked $S$-K3 surfaces which vanishes only on divisors of vectors $t\in T'$ with $0>(t,t)\ge -2$. We will construct this automorphic form by first embedding the lattice $T$ in the lattice $II_{2,26}$ and then restricting a certain automorphic form of weight 12 for this lattice to the symmetric space of $T$. We can find a geometric interpretation of the K3 surfaces whose period points lie on the divisors in theorem 1.2 as follows. We know that the Picard lattice contains $S$. As $t\in T'$ and $II_{3,19}$ is unimodular we can find a vector $D\in II_{3,19}$ whose projection into $T$ is $t$. Then the lattice $\langle S,D\rangle$ generated by $S$ and $D$ has the properties \begin{itemize} \item $\langle S,D\rangle$ has signature $(1,m+1)$ \item $|\det(\langle S,D\rangle)|\le 2|\det (S)|$. \end{itemize} because the projection of $v$ into the orthogonal complement of $S$ has norm of absolute value at most 2. Hence the Picard lattice of the K3 surface contains a lattice with the properties above. In particular any K3 surface for which there is a norm $-2$ vector in $S^\perp$ satisfies the condition above, as we can take $D$ to be this norm $-2$ vector. We now prove theorem 1.2. {\bf Proof.} We first construct some primitive embeddings of $T$ into $II_{2,26}$. Corollary 1.12.3 of Nikulin \cite{NIK2} implies that we can primitively embed any lattice $T$ into the unimodular lattice $II_{2,26}$ provided that $T\otimes \mathbb{R}$ embeds into $II_{2,26}\otimes \mathbb{R}$ and the minimum number of generators of $T'/T$ is less than $\dim(II_{2,26})-\dim(T)$. We can therefore find a primitive embedding of our lattice $T$ into $II_{2,26}$ because the rank of the group $T'/T$ is at most the dimension of $S$, so this rank plus the dimension of $T$ is less than the dimension of $II_{2,26}$. We will write $U$ for the orthogonal complement $T^\perp$ of $T$ in $II_{2,26}$. Then $T$ and $U$ have the same determinant as $T$ is a primitive sublattice of $II_{2,26}$. We recall some properties of the function $\Phi$ defined in example 2 of section 10 of \cite{RB1}. The properties of $\Phi$ we will use are that $\Phi$ is an automorphic form on the hermitian symmetric space of $II_{2,26}$ and its only zeros lie on the divisors of norm $-2$ vectors of $II_{2,26}$. Some other properties of $\Phi$ which we will not use are that its zeros all have multiplicity 1, it has weight 12, it is the denominator function of the fake monster Lie algebra, its Fourier series is explicitly known, and it can be written explicitly as an infinite product. The restriction of $\Phi$ to the hermitian symmetric space of $T$ is an automorphic form, but will be identically 0 whenever $U$ contains a norm $-2$ vector. We can get around this by first dividing $\Phi$ by a product of linear functions vanishing on the divisors of each of these norm $-2$ vectors before restricting it. This restriction is an automorphic form as in pages 200-201 of \cite{RB1}. So in all cases we get an automorphic form $\Phi_T$ on the hermitian symmetric space of $T$ whose only zeros lie on the hyperplanes of norm $-2$ vectors of $II_{2,26}$. Although we do not need it we can work out the weight of $\Phi_T$ as follows: the weight is increased by 1 each time we divide $\Phi$ by a linear function, so the final weight is the weight (=12) of $\Phi$ plus half the number of norm $-2$ vectors of $U$. We would like to know the zeros of $\Phi_T$ in terms of vectors of $T$ rather than in terms of norm $-2$ vectors $r$ of the larger lattice $II_{2,26}$. These zeros correspond to the hyperplanes of the negative norm projections of the norm $-2$ vectors $r$ of $II_{2,26}$ into $T$. The projection of $r$ into $U$ has norm at most 0 as $U$ is negative definite, so the projection $t$ of $r$ into $T$ is a vector of $T'$ with $0>(t,t)\ge -2$. This proves theorem 1.2. For the period space of marked Enriques surfaces there is an automorphic form vanishing exactly on the points orthogonal to $-2$ vectors \cite{RB2}, so it is natural to ask if there is an automorphic form for polarized K3 surfaces vanishing exactly on the points orthogonal to a norm $-2$ vector in $S^\perp=T$, which would be much stronger than the result above. Theorem 1.3 says there is such a form for K3 surfaces with a polarization of degree 2, but Nikulin \cite{NIK1} has shown that no such form can exist for some large values of the polarization. The zeros of the form in theorem 1.2 do not always have multiplicity one; in fact they often have some zeros of high multiplicity. We can work out the multiplicity of the zeros by counting numbers of vectors in the dual $U'$ of the lattice $U$ with given norm and given image in $U'/U$. (But notice that some hyperplanes can have higher multiplicity than one might expect because they get zeros from more than one vector $t$.) We now give some examples for polarized K3 surfaces, so we take $S$ to be a one dimensional lattice spanned by a primitive vector of norm $2n$ for some positive integer $n$. We can parameterize embeddings of $T$ into $II_{2,26}$ by primitive norm $-2n$ vectors $v$ in $-E_8$. To do this we simply identify $T=(-2n)\oplus(-E_8)\oplus (-E_8)\oplus H\oplus H$ with the sublattice $\mathbb{Z} v\oplus(-E_8)\oplus (-E_8)\oplus H\oplus H$ of $II_{2,26}=(-E_8)\oplus(-E_8)\oplus (-E_8)\oplus H\oplus H$. The lattice $U$ is then the orthogonal complement of $v$ in $-E_8$. Here is a table of the number of norm $-2$ roots of $U$ and the numbers of vectors $a$ in the lattice $U'$ of norm greater than $-2$ for values of $k=(a,v)$ between 0 and $n$. $ \begin{array}{cccccccccc} 2n &roots&k=0&k=1&k=2&k=3&k=4&k=5&k=6&k=7\\ 2&126&1&56\\ 4&84&1&64&14\\ 6&74&1&54&27&2\\ 8&126&1&0&56&0&1\\ 8&56&1&56&28&8&0\\ 10&60&1&44&33&12&1&0\\ 12&46&1&48&30&16&3&48&10\\ 14&44&1&42&35&14&7&0&21&2\\ 14&72&1&28&27&27&1&1&27&0\\ 2n &roots&k=0&k=1&k=2&k=3&k=4&k=5&k=6&k=7 \end{array} $ The numbers of vectors for other values of $k$ can be worked out using the fact that this number does not change if $k$ is replaced by $2n+k$ or by $-k$. For some values of $n$ there is more than one line because the lattice $E_8$ can have several orbits of vectors of the same norm, corresponding to several different automorphic forms. The first line for $2n=8$ corresponds to a non primitive norm $8$ vector of $E_8$ so does not correspond to a primitive embedding of $T$ into $II_{2,26}$. Some of the entries are 0, corresponding to the fact that the automorphic forms do not always vanish on all the divisors of theorem 1.2 (so the divisor in theorem 1.2 is not necessarily a minimal ample divisor). {\bf Example 2.1.} We will work out exactly what the automorphic form for $2n=2$ looks like. Its weight is (weight of $\Phi$)+(number of roots of $E_7)/2=12+126/2=75$. The zeros of this form come by taking $k=0$ or 1 in theorem 1.2. For $k=0$ we get a contribution of 1 to the multiplicity of the divisor of each norm $-2$ vector in $T$. For $k=1$ we get a contribution of 56 for each norm $-1/2$ vector in the dual $T'$ of $T$. This does not mean that the automorphic form has zeros of multiplicity 56, because twice a norm $-1/2$ vector of $T'$ is a norm $-2$ vector of $T$ which has even inner product with all vectors of $T$. In particular the divisor of the norm $-2$ vectors of $T$ is reducible: it has two components $E_1$ and $E_2$ corresponding to norm $-2$ vectors which have odd inner product with some vector of $T$ and to norm $-2$ vectors which have even inner product with all vectors of $T$. The divisor $E_1$ is a zero of the automorphic form of multiplicity 1, and the divisor $E_2$ is a zero of multiplicity $1+56=57$, and these are all the zeros. In particular this example proves theorem 1.3 of the introduction. {\bf Example 2.2.} Nikulin conjectured in \cite{NIK1} that there are only a finite number of lattices $S$ such that there is an automorphic form vanishing only on $S$-K3 surfaces which have a norm $-2$ vector in $S^\perp$. We can find a few examples of lattices $S$ with this property. Firstly if $S$ is a unimodular Lorentzian lattice in $L$ then $T=T'$ so $S$ has this property. The unimodular Lorentzian lattices in $L$ are $II_{1,1}$, $II_{1,9}$, and $II_{1,17}$. Secondly if $S$ has determinant 2 and dimension 1 mod 8 then it has the property above as in example 1, so we also get the lattices $(2)$, $(2)\oplus (-E_8)$, and $(2)\oplus (-E_8)\oplus (-E_8)$. {\bf Example 2.3.} If $2n$ is 4,6,8, or 10 then we can see from the table above that we can assume that either $(D,D)=-2, (D,P)=0$ or $(D,D)=0$. Hence if the period of the K3 surface is on a zero of the automorphic form then either the surface is singular (in the sense that the Picard group contains a $-2$ vector orthogonal to $S$) or its Picard lattice contains a nonzero element with zero self intersection number. \begin{lemma} Any family of K3 surfaces with constant Picard number is, after a finite \'etale base change, a family of $S$-K3 surfaces for some Lorentzian lattice $S$. \end{lemma} Proof. By the assumption about constant rank, and since the Picard group of a K3 surface is always a primitive sublattice of $H^2$, the Picard groups form a sub-local system of the local system of $H^2$'s. Since the monodromy action on the Picard group is finite we can, after an \'etale base change, assume that this subsystem is constant. This means that there is a primitive embedding of a constant local system with fiber $S$ into the local system of $H^2$'s. By definition this is a family of $S$-K3 surfaces. This proves lemma 2.1. We now prove theorem 1.1. By the remarks after theorem 1.1 we can assume that all the surfaces in the family are K3 surfaces. By lemma 2.1 we can assume that we have a family of $S$-K3 surfaces for some lattice $S$. As the K3 surfaces are projective, the lattice $S$ is Lorentzian. By theorem 1.2 and the remarks near the beginning of this section, if the family is not isotrivial there must be surfaces whose Picard lattice is at least 1 more than the dimension of $S$. This contradicts the fact that all surfaces in the family have the same Picard number and proves theorem 1.1. \section{Some examples} In this section we construct an example of a complete non isotrivial family of smooth polarized K3 surface, to show that the hypothesis about the Picard number in theorem 1.1 cannot be left out. Suppose that $A\to C$ is a complete one parameter family of principally polarized family of abelian surfaces. Such a family exists because the boundary of the Satake compactification of the moduli space has codimension $2>1$. Set $f: K=A/\{\pm 1\}\to C$. Since $f$ is isotrivial in a neighborhood of its critical locus, we can simultaneously resolve the singularities via a map $\sigma: \tilde K\to K$ to get $\tilde K\to C$, a family of smooth K3 surfaces. Let $\Theta$ be the relative principal polarization on $A$. It is well known that $2\Theta$ is the pullback of a relatively ample divisor $H$ on $K$. Let $E$ denote the exceptional locus of $\sigma$. Then it is well known that on each geometric fiber $E$ is uniquely even. So on the geometric generic fiber $\tilde K_{\bar\eta}$ there is a unique divisor class $\bar L$ such that $E_{\bar \eta} \sim 2\bar L$. Recall that the \'etale cohomology group $H^1_{et}(\cdot,{\mathbb G}_m) $ is the Picard group $Pic(\cdot)$, and $H^2_{et}(\cdot,{\mathbb G}_m)$ is the Brauer group $Br(\cdot)$. {}From the Hochschild-Serre spectral sequence $$ E^{pq}_2=H^p(Gal({\mathbb C}(\bar \eta)/{\mathbb C}(\eta)), H^q_{et}(\tilde K_{\bar\eta},{\mathbb G}_m)) \Rightarrow H^{p+q}_{et}(\tilde K_\eta,{\mathbb G}_m) $$ we get an exact sequence $$ 0\rightarrow Pic(\tilde K_\eta)\rightarrow Pic(\tilde K_{\bar\eta})^{Gal({\mathbb C}(\bar \eta)/{\mathbb C}(\eta))} \rightarrow Br({\mathbb C}(\eta)). $$ Since the Brauer group $Br({\mathbb C}(\eta))$ of the function field of a complex curve is trivial (Tsen's theorem) it follows that $E$ is even on the generic fiber $\tilde K_\eta$. It follows that $E\sim 2L+V$ for some $L$ and for some $V$ supported on fibers. By taking a ramified double cover of the base if necessary we can assume that $V$ is even so that $E$ is even. Say $E\sim 2M$. Put $B=2\sigma^*(H)-M$, so that $B$ is a polarization of degree 8 on $\tilde K$ provided that $H$ is very ample on $K$, which is equivalent to the abelian surface $A$ being indecomposable as a principally polarized abelian variety. (See \cite{GH}, pages 773-787.) If, however, $A$ is decomposable and therefore a product of elliptic curves then $B$ is merely semi ample. Nevertheless take $S$ to be the lattice generated by $B$. We have constructed a complete non isotrivial family of $S$-K3 surfaces. (The divisor $B$ is not ample on every fiber; if we want this as well we can take the divisor class $D=B+\sigma^*(H)$ which provides a polarization of degree $(3H)^2+M^2 = 3^2\times 4 -8=28$.) {\bf Example 3.1.} Suppose that $0\in C$ is such that $A_0$ is isomorphic to $X\times X'$ for elliptic curves $X$ and $X'$. Then there is a morphism $\alpha:K\to {\mathbb P}^1\times{\mathbb P}^1$ of degree 2, since ${\mathbb P}^1$ is the Kummer variety of an elliptic curve. Let $F_1,F_2$ be fibers of the projections ${\mathbb P}^1\times {\mathbb P}^1\to {\mathbb P}^1$. Let $D_i$ be the pullback of $F_i$ to $\tilde K_0$. We know that $(D_i,D_i)=0$ as $D_i$ is a fiber of a morphism onto a curve. Moreover $(D_i,E_0)=0$, where $E_0$ is the exceptional locus on $\tilde K_0$. Since $B=2(D_1+D_2)-E/2$ it follows that $(B,D_i)=4$, so that the lattice generated by $S=\langle B\rangle$ and $D_1$ has discriminant $-16$. This is an example of an $S$-bad fiber. {\bf Remark.} Each of the K3 surfaces in the family above is a Kummer surface and so by a result of \cite{JHK} has a fixed point free involution (not necessarily unique!) such that the quotient is an Enriques surface. But by theorem 1.1 we cannot find a global fixed point free involution acting on the whole family, otherwise we would get a complete nonisotrivial family of Enriques surfaces.